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 GPS, Relativity, and pop-Science Mythology
 Popular science frequently promotes the idea that the Global Positioning System serves not only as a means of position determination but also as a convenient test for Einstein’s Theory of Relativity. The claim made is that if certain adjustments were not made to counter the predictions of this theory, calculated positions would quickly become very inaccurate. Hence anyone with a functioning GPS receiver has already confirmed the theory to be correct and the pizza delivery guy would not have arrived at your home were it not for the adjustments.
 The claim however is false and the science gurus who promote it should know better than to do so. Fact is, it would not matter whether such adjustments were made or not. Why? The reason has to do with basic facts about how GPS receivers work.
 To understand this we first need examine the claims more specifically. The following points are made in regards to Relativity and GPS operation [1]:
 1. Relativity theory predicts the clocks aboard the GPS satellites will run faster than clocks on Earth by about 38 microseconds a day. 2. The satellites’ clocks are slowed down by this amount so as to match the speed of clocks on Earth 3. If these adjustments were not made, position determination would be inaccurate by up to 11 km a day 4. This inaccuracy would grow steadily larger each day, e.g. after 10 days the inaccuracy would have grown to 110 km.
 We’ll begin by assuming the first two points are absolutely correct since this essay is not about questioning the truth about relativity theory, only its application.
 Odd as it may sound, the third and fourth points do not follow from the first two.
 Determining your position
 The basics of how GPS receivers work is fairly straightforward. An array of orbiting satellites surrounds Earth. Each satellite holds one or more highly-accurate atomic clocks. As they orbit, a satellite transmits the time signal (or ‘timestamp’) of its on-board clock. Since these signals travel at a finite speed – the speed of light – we can determine how far away each satellite is from the receiver and thus figure out where we are located.
 Calculating the distance from a satellite requires subtracting the time signal of the satellite from the time at the receiver. By subtracting these numbers and multiplying by the speed of light we calculate distance, i.e.
 D = (tr - ts) c
 Where tr is time at the receiver, ts is the (delayed) time signal from the satellite, and D is distance. By
 GPS, Relativity, and pop-Science Mythology http://www.alternativephysics.org/book/GPSmythology.htm
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 determining the distance from at least 4 satellites we can pinpoint our location. This calculation is generally drawn in terms of a set of overlapping spheres:
 In three dimensions, our position will be given by the intersection of four such spheres.
 Now here’s where time dilation comes in. If the clocks on Earth ran at a different rate to those on satellites then the above calculation would be completely out because |tr-ts| would become increasingly larger each day. We can calculate the daily distance error δD by multiplying the daily time error δt by the speed of light:
 δD = δt c ~= 38x10-6 * 3x108 = 11400 m
 Or about 11 km per day.
 This argument appears quite solid. So what are we overlooking?
 What we are overlooking is the phrase ‘time at the receiver’. Problem is, GPS receivers contain no atomic clock because there’s no room to fit one in. Plus it would be very expensive even if possible. That ‘time at the receiver’ must instead be determined from the satellites’ clocks. Therefore if the satellites’ clocks were running faster (or slower) than those on Earth, this ‘time at the receiver’ would likewise run faster or slower by exactly the same degree. Hence there would be no position error due to time dilation, and certainly no cumulative position error.
 Essentially, what the GPS receiver is trying to determine is not just its position but also its timestamp. To state this mathematically, it is trying to determine variables x,y,z, and t. That’s 4 variables which require 4 pieces of data to solve for, and the data for solving it comes from the timestamps of 4 satellites. The receiver computes all 4 of these variables x,y,z,t but then can discard the time variable (t) as unimportant.
 A simpler approach
 Putting aside whatever method it is that GPS receivers actually use, there is a much simpler approach to calculating position that sidesteps the whole ‘time at the receiver’ question and makes it clear why the clocks’ speed relative to Earth is unimportant. It involves comparing differences between received signals.
 Let’s say you are at an unknown location on Earth (or in space) and you receive two identical signals from different satellites. That is, the timestamps from each satellite are exactly the same when they reach you. What would you conclude from this? Quite obviously: that you are at a position equally distant from both satellites. Therefore you must be somewhere on a plane that slices midway between the two satellites and runs at right-angles to a line joining them. Using this information you have
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 narrowed your location down to a two-dimensional plane. And you did so without knowing your local time. Now quite obviously also, it wouldn’t matter if those satellites’ clocks were both running faster or both running slower than yours because your equidistance makes their timestamps identical when they reach you.
 That much is straightforward. So what about a situation where there is a measured difference between time signals, as will normally be the case? In this case it is slightly more complex but again the answer will depend only on the difference between the signals rather than any local clock’s time.
 To simplify this, suppose we have a two-dimensional situation containing three satellites: A, B and C. All satellites are at known locations with their clocks perfectly synchronised and running at identical speeds.
 We are somewhere in this 2D region shown and wish to determine our position. We receive signals from A and B, and find that B is slightly ahead of A. We subtract the time difference between these two signals and call it Δt. What this tells us is that we are closer to B than A, and at a location where the difference between our distance to A and our distance to B is c Δt. Putting aside the math, this places us somewhere along a line looking like this:
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 The line looks a bit like a parabola, although it isn’t [2]. Next we look at the signals from B and C and find they are identical. As discussed above, this places us on a straight line running midway between B and C like so:
 Where these two lines intersect, that’s our position!
 The above is a two dimensional situation. Extending it to three dimensions will require four satellites and the intersection of three overlapping ‘parabolic’ cones. A little more complex but the method is the same.
 The important point here is that we were able to determine our location without making any reference to ‘local time’. Doing so only required knowing of differences between satellite timestamps rather than the timestamps themselves. So long as the satellites’ clocks all run at the same rate, these timestamp differences will be independent of the actual clock speeds.
 That last point might be confusing. After all, if I double the speed of two clocks won’t that also double the magnitude of their difference?
 Short answer: no. To easily demonstrate we’ll take two simple bench clocks, labelled A and B, and set one 10 minutes ahead of the other. We’ll start our experiment when clock A reads 1:00 and clock B reads 1:10. We’ll then observe them every 5 minutes and compare them to ‘real time’, which in this case will also be clock A’s time.
 real time 1:00 1:05 1:10 1:15
 clock A 1:00 1:05 1:10 1:15
 clock B 1:10 1:15 1:20 1:25
 OK, no surprises there: B stays always 10 minutes ahead of A.
 Now we redo the experiment but this time make an adjustment to both clocks so as they run at double their normal speed. That is, for each one minute of real time, A and B move ahead 2 minutes. We again start the experiment when A = 1:00, B = 1:10 and observe them every 5 minutes.
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 real time 1:00 1:05 1:10 1:15
 clock A 1:00 1:10 1:20 1:30
 clock B 1:10 1:20 1:30 1:40
 So the clocks move at double the rate of real time but keep the same 10 minute difference as before.
 The above experiment could instead be done by having A and B set to the same time and then sending B 10 light-minutes away – about double the Earth-Mars distance. In this case a time signal from B would take 10 minutes to reach here, meaning that its signal would read 10 minutes behind A. Using the same reasoning as above it should be clear that doubling the speed of both clocks would not change the observed time difference (of 10 minutes) between the two. The same goes for GPS satellites. Whether they run faster or slower than our clocks, it will not alter their measured time differences. And hence our GPS-calculated position will be the same.
 Determining a Satellite’s position
 Now admittedly there is an aspect in which relativity can become relevant. And this has to do with figuring out where a satellite is.
 The above exercises assume the satellites to be at fixed locations, which is not the case. The satellites are of course in orbit, changing position all the time, and it’s up to the receivers to determine their location. This calculation will be based on a satellite’s orbital characteristics (which are also transmitted) and its timestamp. Essentially the timestamp allows us to determine how far around in an orbit a satellite is. Therefore any inaccuracy in a timestamp – such as a relativistic slow-down – will cause an inaccurate position to be calculated.
 Fortunately such inaccuracies will be mild because the satellites move much slower than light. The satellites move at 3870m/s and a 38.6 microseconds-per-day inaccuracy would yield a position error of only 15cm. Naturally this will accumulate as the days go by, but the orbital times are calculated relative to a ‘reference epoch’ which is reset weekly and the regular timestamp extends plus or minus 3.5 days from this reference time [3]. Hence the satellite’s calculated position could be out by up to 52cm, which is still within the normal GPS inaccuracy of 2-3 metres. And since satellites are moving in different directions even this error should largely cancel; especially when averaged over several readings.
 Latitudes and Altitudes
 Ignoring all the above arguments for the moment, there is another important consideration that warrants attention. And that is the fact that such relativistic effects would render the entire navigation system unworkable, at least in its present configuration. According to the US-DOD [4] the satellites lose 7200 ns a day due to special relativity and gain 45900 ns a day due to general relativity. These effects combine to yield a net speeding up of 38600 ns a day (45900-7200=38600), relative to clocks on Earth. The clocks are then slowed by a suitable degree to negate this daily amount, and everything is fine for GPS receivers.
 However since the Earth is rotating, the speed of a satellite relative to a GPS receiver will be different at different latitudes. Therefore the amount of dilation should also be different. That 7200 ns was calculated relative to a pole, where the Earth is not moving and only the satellite’s speed of 3870 m/s need be considered. If the calculation was done at the equator where the surface moves at 465 m/s
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 and GPS satellites cross at 55 degrees, the SR time dilation works out to 6300 ns. That’s a 900 ns difference between pole and equator which presumably would yield an increasingly inaccurate position determination of 270 meters per day.
 Next we have a problem with altitude because the 45900 ns figure was calculated for sea level. This figure will decrease as we move higher and experience less gravity. For example, at 1060 metres above sea level there will be a decrease of 10 ns per day. This presumably would lead to an inaccuracy of 3 metres per day, accumulating up to 1 km per year. Aeroplanes flying at 10km would be worse off; accumulating a 10 km error per year.
 If it were necessary to accommodate for relativity at different latitudes and altitudes, the calculations would be very complex and would need to be done at the receiver. But they aren’t done there: only a fixed time adjustment is made at the satellites for all points on Earth. This fact alone demonstrates that Relativity is irrelevant to the GPS operation.
 Conclusion
 The presence of Special and General Relativity effects has no bearing on the accuracy of GPS operation. In summary, it wouldn’t matter whether clocks aboard GPS satellites ran faster or slower than Earth’s clocks or even changed their speed each day. Just so long as the satellites’ clocks remained synchronised with each other and the time-difference relative Earth’s clocks didn’t become too large, GPS receivers would continue to calculate their correct position.
 The GPS is certainly an excellent navigational aid. But from an operational viewpoint at least, it doesn’t serve as a test for Relativity. Scientists should stop calling it that.
 The next chapter discusses ‘secondary relativistic effects’:
 GPS Mythology - part 2 (<-- click to read)
 [1] http://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System [2] The shape looks like the function y=sqrt[1+x2] [3] http://www.navcen.uscg.gov/pubs/gps/sigspec/gpssps1.pdf PDF-page 43 [4] http://www.aticourses.com/global_positioning_system.htm
 Previous Contents Next chapter
 Copyright © 2014 Bernard Burchell, all rights reserved.
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 Phase and frequency shift in a Michelson interferometer
 Wolfgang Engelhardta)
 Fasaneriestrasse 8, D-80636 M ̈unchen, Germany
 (Received 26 June 2013; accepted 7 October 2014; published online 21 November 2014)
 Abstract: Traditionally, the outcome of Michelson’s interference experiment has been interpreted as evidence against the existence of a luminiferous medium called “ether.” Einstein, however, emphasized in 1920 that an ether must exist in spite of Michelson’s null result. In this paper, it is shown that a medium theory—be it for light or for sound—actually predicts the observed null result. Michelson expected a gradual fringe shift when his apparatus was turned in the “ether wind.” Such a phase change would, however, require a temporary frequency change in one of the interferometer arms. Since wind does not alter the frequency in the interferometer, a phase shift cannot occur either. CV 2014 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-27.4.586]
 Re ́sume ́: Habituellement, l’issue de l’expe ́rience sur l’interfe ́rence de Michelson a e ́te ́ interpre ́te ́e comme e ́vidence contre l’existence d’un medium luminife`re appele ́ “e ́ther.” Einstein, cependant, a insiste ́, en 1920, sur le fait qu’un e ́ther doit exister en de ́pit du re ́sultat nul de Michelson. Dans cet article il est de ́montre ́ qu’une the ́orie de medium—que ce soit pour la lumie`re ou pour le sonpre ́dit, en fait, le re ́sultat nul observe ́. Michelson s’attendait a` un changement graduel des franges d’interfe ́rence quand son appareil e ́tait tourne ́ dans le “vent de l’e ́ther.” Un tel changement de phase exigerait, cependant, un changement temporaire de fre ́quence dans un des bras de l’interfe ́rome`tre. Puisque le vent ne change pas la fre ́quence dans l’interfe ́rome`tre, un changement de phase ne peut se produire non plus.
 Key words: Interferometry; Phase Measurement; Fundamental Tests; Ether.
 I. INTRODUCTION
 In his Electromagnetic Theory of Light,1 Maxwell derived a wave equation for the vector potential which was supposed to describe the propagation of electromagnetic perturbations. Like air is necessary to carry sound, an analogous medium—called “ether”—was required to transmit electromagnetic waves with a characteristic velocity c. Based on this conception, one could expect an “ether wind” to blow on earth due to the planet’s motion through space. Maxwell, who strongly believed in the ether hypothesis,b) proposed to measure the strength of this wind by determining the round trip velocity of light.c) It was Michelson who took up the idea and invented his famous interferometer which appeared suitable to measure the ether wind by comparison of the light velocity in the two arms of his device. To his amazement, he obtained a null result in 1881,4 which was later confirmed by himself and Morley,5 as well as by many other experimenters. A survey of the various attempts to measure an
 ether drift is given by Galaev.6 Some experimenters claimed having obtained a positive result which, however, could not be confirmed by others. Galaev himself reported a positive result using a Rozhdestvensky (Mach–Zehnder) interferometer, not a Michelson interferometer. The interpretation relies on a theory of some ether viscosity which is beyond the scope of this paper. The special theory of relativity (STR) has a simple ad hoc “explanation” for Michelson’s null result: It postulates the constancy of the light velocity in all reference frames regardless of their motional state. As a consequence, time had to be transformed according to a rule which was proposed by Voigt in 1887.7 Whereas Voigt considered his transformation as valid for waves in all elastic media including sound, Lorentz applied it only to electromagnetic waves.8 This enabled him to explain Michelson’s experiment in a more general way than by his previous contraction hypothesis.9 Since then, Michelson’s experiment attained more and more the status of an experimentum crucis, a cornerstone proving the validity of STR. In 1905, however, Einstein based his own paper on relativity10 on the Voigt–Lorentz transformation, not so much on the outcome of Michelson’s experiment. According to Holton,11 he did either not know of this experiment or he did not regard it as pertinent. He may have read Voigt’s article of 188712 or Cohn’s paper of 190113 which both offered alternative explanations of the null result in terms of classical electrodynamics and the pertaining ether theory.
 a)wolfgangw.engelhardt@t-online.de b)“... Hence all these theories lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.”2 c)“[I]n the terrestrial methods of determining the velocity of light, the light comes back along the same path again, so that the velocity of the earth with respect to the ether would alter the time [interval] of the double passage...”3
 0836-1398/2013/27(4)/586/5/$25.00 586 CV 2014 Physics Essays Publication
 PHYSICS ESSAYS 27, 4 (2014)
 In this paper, we show that Einstein’s primary intuitiveness was correct: It is not possible to measure any ether wind with Michelson’s device. The reason for this is that a phase change can only be brought about by a temporary frequency change, as will be shown in due course. Since the latter does not occur due to wind, as will also be shown, a null result must be expected. This is true not only for light but also for sound, as will be shown explicitly in Sec. III. Before then, in Sec. II, the connection between phase shift and frequency change in a laser interferometer will be studied. Such a device is used as a length measurement apparatus and constructed on the basis of a Michelson interferometer. Finally, concluding remarks will be presented in Sec. IV.
 II. PHASE SHIFT AND THE PERTAINING FREQUENCY CHANGE DUE TO A MIRROR DISPLACEMENT IN A LENGTH COMPARATOR BASED ON MICHELSON’S INTERFEROMETER
 In 1892–1893, Michelson and Benoıˆt14 succeeded in determining the length of the standard meter in units of the red cadmium line. They used a Michelson interferometer and counted the number of fringes recorded by a detector when the reflecting mirror in one of the arms was moved. This method is still used in modern laser interferometers where coherence lengths of the laser light with more than 50 m are exploited to achieve an extremely high accuracy (Fig. 1). The displacement d of the mirror in the horizontal arm is compared to the wavelength of the laser light by counting the number of fringes which are produced by interference of the horizontal with the vertical light beam. There is a simple relationship of the number of fringes m as a function of the wavelength k and the distance d
 m 1⁄4 2d=k: (1)
 In order to derive this, we describe the light beams in terms of plane waves starting at the beam splitter. When they reunite (interfere) after reflection at the solid mirrors, we may write for the vertical beam
 A cosðk2L  xtÞ; (2)
 and for the horizontal beam
 A cosðk2ðL þ dÞ  xtÞ; (3)
 where the wave vector may be expressed by the wavelength: k 1⁄4 2 p=k. Superposition of the two expressions yields
 2A cosðkdÞ cosðkð2L þ dÞ  xtÞ; (4)
 which is a plane wave with a new amplitude and phase oscillating with the same frequency x as the waves of the original beams. The amplitude is a periodic function of the displacement d and has an extremum when (1) is satisfied. Thus, by counting fringes, while the mirror is moved one can express the total distance d in units of the wavelength k. Note that the frequency of the wave in the horizontal arm cannot stay constant during the motion of the mirror due to the Doppler effect of the light reflected at the receding mirror. This dependence of x on the velocity of the moving mirror may be included in Eq. (3) by substituting d 1⁄4 vt, which yields
 A cosðk2L  ðx  2kvÞtÞ 1⁄4 A cosðk2L  ð1  2v=cÞxtÞ;
 (5)
 where we have also inserted the phase velocity c 1⁄4 x=k of the wave. This relationship reflects the frequency change ð1  2v=cÞx (in first order of v=c) caused by the Doppler effect at the receding mirror. It is important to realize that a change in phase is inevitably brought about by a temporary change in frequency. In modern laser interferometers, this frequency shift is in fact measured, in order to determine the direction of the mirror displacement which would remain ambiguous by counting just fringes.14 Hence, it is obvious that a variation of the phase in one of the beams—leading to a fringe shift at the detector—can never occur unless the frequency in one of the beams differs during the time when the phase is shifted. At constant frequency in both arms, the phases would be strictly locked, thus preventing a fringe shift. One may visualize this situation by considering two wheels rotating in phase [Fig. 2(a)]. In order to achieve a phase shift between the two wheels, it is necessary to break the rigid coupling and rotate one of the wheels somewhat faster [Fig. 2(b)]. When a certain phase shift is established, the two wheels are coupled again and rotate at the same frequency as before [Fig. 2(c)].
 FIG. 1. Michelson interferometer as a length comparator.
 FIG. 2. (a) Two rigidly coupled wheels rotating in phase at frequency x1. (b) Wheel 2 decoupled from 1 and rotating at a higher frequency x2. (c) Both wheels coupled again after a phase shift of Du 1⁄4 p=2.
 Physics Essays 27, 4 (2014) 587
 Indeed, precisely this situation occurs when a car travels along a road. The phase relation between the two wheels on one axle remains constant on the straights, as the frequency of the wheels is also equal. As a bend is navigated, the frequency of the outer wheel increases relative to that of the inner wheel, which leads to a phase shift. The phases will be locked again, once the car continues its journey along the next straight. This consideration should be kept in mind when the effect of wind blowing through an interferometer is discussed in Sec. III.
 III. DOPPLER EFFECT IN A MICHELSON INTERFEROMETER
 Now that the principle of intimate connection between continuous phase change and temporary frequency change has been explained, we turn to the Michelson–Morley interferometer where the distances between the mirrors are kept constant, but an ether wind was expected to blow through the interferometer. First, an acoustic interferometer working with sound waves instead of light is considered, since the physics of sound is well known and entirely understood. A wind blowing along the horizontal arm of this interferometer will be equivalent of both the beam splitter and the mirror moving through a still medium. Therefore, the Doppler effect due to the velocity of beam splitter and mirror through the medium may be considered in order to investigate the effect of such a wind on the stationary interferometer. The important point to note here is that the Doppler effect differs depending on whether the detector (Fig. 3) or the source (Fig. 4) is moving. Let us now assume a wave traveling in still air along the horizontal arm from the beam splitter to the mirror. Using plane wave approximation as before, we have
 A cosðkx  x0tÞ: (6)
 Initially, the wind effects on the mirror—which is taken as a detector—and the beam splitter—which acts as a sourcewill be considered separately. A wind blowing away from the mirror toward the beam splitter is analogous to the detector (mirror) traveling in the wave field on the x-axis with constant velocity v (Fig. 3).
 The frequency the detector experiences may then be obtained by including the velocity term in Eq. (6) and resolving accordingly
 A cosðkðx0 þ vtÞ  x0tÞ 1⁄4 A cosðkx0  ðx0  kvÞtÞ: (7)
 Comparing Eq. (7) with Eq. (6) yields the new frequency
 x1 1⁄4 x0 1  v=cS
 ð Þ; (8)
 where cS 1⁄4 x0=k is the phase velocity of sound. Thus, the moving detector (receding mirror) experiences a wave with a reduced frequency x1. Likewise, the effect of wind blowing from the detector toward the source can be described by considering the case of the source (the beam splitter) moving with velocity v toward the detector (mirror). In this case, the wave fronts are compressed in front of the source as sketched in Fig. 4. As the source moves a distance 2pv=x0 between the emissions of two consecutive wave crests, the wavelength is shortened by the amount
 k 1⁄4 k0 1  v=cS
 ð Þ: (9)
 In this case, the detector measures the frequency
 x2 1⁄4 2pcS
 k0 1  v=cS
 ð Þ 1⁄4 x0
 1  v=cS
 : (10)
 Since a wind blowing along the interferometer’s horizontal arm is in effect the case of both source and detector moving with the same velocity v, the Doppler effects of “moving detector” Eq. (8) and “moving source” Eq. (10) cancel out so that the detector measures simply
 x 1⁄4 x0: (11)
 Naturally, the same consideration applies to the wave reflected back from the receding mirror, which is now a source, to the forward moving beam splitter which is now taken as a detector. This means that wind has no influence whatsoever on the frequency in the acoustic Michelson interferometer.
 FIG. 3. Doppler effect with moving detector.
 FIG. 4. Doppler effect with moving source.
 588 Physics Essays 27, 4 (2014)
 Consequently, this result proves that the frequency of the two interfering waves is not influenced when the direction of the wind in the two arms is interchanged by turning the interferometer by 90. Since we already saw in Sec. II that a phase shift cannot occur under constant frequency [Fig. 2(a)], a change in the interference pattern is also prevented by the physics of the Doppler effect during the turning of the interferometer. In the optical Michelson interferometer, similar Doppler formulae apply. In Ref. 15, we have derived the relativistically correct formulae under conservation of momentum and energy, when photons interact with matter. The relevant “collision physics” is the same as one employs in the treatment of the Compton effect. We obtained for a moving detector
 x1 1⁄4 x0
 1  ðv=cÞ cos a
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1  v2=c2
 p ; (12)
 and for a moving source
 x2 1⁄4 x0
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1  v2=c2
 p
 1  ðv=cÞ cos a ; (13)
 where a denotes the angle between the velocity and the momentum vector of the photons (i.e., wave vector in the wave picture). These formulae have been verified in a number of experiments which were quoted in Ref. 15. They were also derived from the Lorentz transformation both in Ref. 15 and by Einstein himself in Ref. 16. In principle, they have the same property as the acoustic formulae (8) and (10): When both source and detector move in the same direction resulting in an ether wind blowing between them, the velocity dependent factors cancel exactly which means that a Doppler effect between source and detector does not arise as there is no relative velocity. Without a temporary frequency change, however, in at least one of the interferometer arms, a phase shift cannot be observed either. This was precisely what Michelson found.
 IV. CONCLUDING REMARKS
 The usual interpretation of Michelson’s experimentwhich is reiterated in all textbooks dealing with the subjectfollows Michelson’s original idea based on a “swimmer” analogy, namely, that the phase shift between the interfering waves can be calculated from the average time a light beam (a swimmer) spends traveling between the beam splitter and the mirror. If there is an ether wind in the horizontal arm, for example, the average travel time between beam splitter and back reflecting mirror was calculated by the relationship
 s1⁄41
 2
 L
 cvþ L
 cþv
 1⁄41
 2
 Lðc þ vÞ þ Lðc  vÞ c2  v2
 1⁄4 Lc
 c2  v2 : (14)
 In first order the velocity cancels, but in second order there is still an influence of the velocity on the average travel time which Michelson attempted to measure. Rather than drawing an analogy with solid particles (swimmers), it is more appropriate in the field of interferometry to consider the average
 wavelength of the beams traveling forth and back with different velocities due to the ether wind. For the purpose of calculating the interference pattern, it is in fact the average wavelength which is relevant and may be determined from Eq. (9) (which holds also for light)
 k1⁄41
 2 1⁄2k0ð1  v=cÞ þ k0ð1 þ v=cÞ 1⁄4 k0: (15)
 The cancellation of the velocity is complete so that wind does not lead to a phase difference between the waves in the horizontal and the vertical arm. Our analysis shows that the outcome of Michelson’s interference experiment has no relevance for the existence of an ether. Einstein himself had reintroduced the ether concept in 1920d) being well aware both of his own theory of 1905 and of the precise MM-results5 of 1887. The measurement of the solar system’s velocity with respect to the CMB by the COBE satellite18 also lends strong support for the existence of an absolute system in which light travels with constant velocity in all directions in agreement with Einstein’s second postulate of 1905. Einstein’s first postulate, however, is at stake when considering that Michelson’s experiment cannot disprove the ether in combination with the experimentally proven existence of a distinguished system that may be identified with Maxwell’s or Einstein’s ether.
 ACKNOWLEDGMENTS
 The author appreciates Dr. Peter Enders’ efforts in scrutinizing the paper and making interesting comments. He is particularly grateful for his son’s, Dr. Markus Engelhardt, careful reading of the manuscript and suggesting many improvements for a clearer presentation.
 1J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 2 (Dover Publications, Inc., New York, 1954), Chap. XX. 2J. C. Maxwell, ibid., Article 866. 3J. C. Maxwell, Nature, XXI, 315 (1880). 4A. A. Michelson, Am. J. Sci. 22, 120 (1881). 5A. A. Michelson and E. W. Morley, Am. J. Sci. 34, 333 (1887). 6Y. M. Galaev, Spacetime Subst. 3, 207 (2002).
 7W. Voigt, Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–Universita ̈t zu G ̈ottingen, Vol. 1887, Nr. 2, p. 41; See http://www.digizeitschriften.de/dms/toc/?PPN=PPN 252457072_1887. 8H. A. Lorentz, Proc. Acad. Sci. Amsterdam 6, 809 (1904).
 9H. A. Lorentz, Zittingsverslagen der Akad. v. Wet. te Amsterdam (Royal Netherlands Academy of Arts and Sciences, Amsterdam, Netherlands, 1892–1893), p. 74. 10A. Einstein, Ann. Phys. 17, 891 (1905). 11G. Holton, Am. J. Phys. 37, 968 (1969).
 12W. Voigt, Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–Universita ̈t zu G ̈ottingen, Vol. 1887,
 d)“Gema ̈ß der allgemeinen Relativit ̈atstheorie ist ein Raum ohne  ̈Ather undenkbar; denn in einem solchen ga ̈be es nicht nur keine Lichtfortpflanzung, sondern auch keine Existenzmo ̈glichkeit von Maßsta ̈ben und Uhren, also auch keine r ̈aumlich-zeitlichen Entfernungen im Sinne der Physik.”17. English translation republished in: Sidelights on Relativity (Methuen, London, 1922): “According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense.”
 Physics Essays 27, 4 (2014) 589
 Nr. 8, p. 177; See http://www.digizeitschriften.de/dms/toc/?PPN=PPN 252457072_1887.
 13E. Cohn, Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften zu G ̈ottingen, Vol. 1901, 74; See http://www.digizeitschriften.de/dms/ toc/?PPN=PPN252457811_1901. 14A. A. Michelson and J.-R. Benoıˆt, Trav. Mem. Bur. Int. Poids Mes. 11, 85 (1895); J. N. Dukes and G. B. Gordon, Hewlett-Packard J., 21(12), 2–8 (1970).
 15W. Engelhardt, Apeiron 10, 29 (2003). 16A. Einstein, Jahrb. Radioaktivita ̈t Elektron. 4, 411 (1907). 17A. Einstein, A ̈ther und Relativit ̈atstheorie (Reichs-Universita ̈t Leiden, Springer, Berlin, 1920). 18G. F. Smoot, M. V. Gorenstein, and R. A. Muller, Phys. Rev. Lett. 39, 898 (1977); B. Levi, Phys. Today 45(5), 17 (1992); See http://aether.lbl.gov/ www/projects/u2/ for COBE DMR-data.
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 Twin Paradoxes
 T. E. Phipps, Jr. 908 S. Busey Avenue Urbana, IL 61801 tephipps@sbcglobal.net
 Several paradoxical aspects of the twin problem of special relativity theory are reviewed, with the conclusion that none has been permanently resolved.
 1. Introduction
 There are several forms and numerous “resolutions” of the famous twin paradox. In the earliest form, due to Einstein, a space traveler traverses at uniform speed v a closed polygonal path, on each leg of which his motion meets the requirement of rigorously inertial motion characteristic of special relativity theory (SRT). At each vertex of the polygon the traveler’s “jump” to the new inertial system is marked by no change in his clock setting—the initial clock setting on each new leg of the journey being stipulated to be identical to the final reading on the previous leg. Under these conditions the theory predicts that if the total duration of the journey is measured by an (inertial) “stay-athome” observer fixed at the initial vertex to be T, then the total elapsed proper time of the traveler will be T /γ, where
 22
 γ = 1/ 1− v / c . Thus the traveler ages less or “stays younger” than
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 the stay-at-home for travel trajectories of any shape (as approximated by a polygon). Later forms have simplified the polygon to a single out-and-back trajectory at constant speed v, either employing a single non-inertial traveler who is accelerated at the turnaround point of the journey, or else employing three inertial observers—one who stays at home, one who goes out, and one who comes back. In either case the clocksetting convention stipulated above for vertices of the polygon is applied at the turnaround point. As before, SRT predicts that the traveler’s clock, on return, will show less elapsed time for the completed journey than the stay-at-home’s clock by a γ-factor.
 2. Primary Paradox: Symmetry-Asymmetry
 Such an asymmetry of aging is not inherently paradoxical. The traveler has manifestly experienced energy state changes not experienced by the stay-at-home. These physical differences may well affect differently the running rates of their co-moving clocks—just as gravity potential energy state changes (are known from other theory and observation to) affect such clocks. Nevertheless, there is a paradox here, in that the theory, SRT, that predicts the observed asymmetry of elapsed time intervals is itself founded upon a relativity postulate generally considered to assert a symmetry among inertial observers and frames regarding their measurements. This postulated symmetry is in fact explicitly expressed by the Lorentz transformation, which shows clock rates between two inertial systems in relative motion to be measured symmetrically. That is to say, timekeeping symmetry is a feature of the Lorentz group. How is it then, that symmetrical clock-rate measurements can give rise to asymmetrical elapsed time-interval measurements? Let us call this the “primary paradox.”
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 Most twin paradox resolvers restrict their attention to this primary paradox. Typically, they show that rigorous applications of the Lorentz transformation to the out-and-back motions, with clock settings matched at the turnaround point (vertex) as stipulated above, lead to the elapsed time asymmetry; whereas clock rates are symmetrical between traveler and stay-at-home during each moment of the journey (i.e., each “measures” the other’s clock as running always slower than his own). This seeming contradiction between clock-rate symmetry and elapsed-time asymmetry is usually explained in terms of the “relativity of simultaneity.” Observers at rest in the outgoing and incoming inertial frames have different ideas about distant simultaneity. At the instant of turnaround the outgoing observer, according to SRT, measures the distant stay-at-home’s
 clock reading as ( ) ( )
 22
 T1 out = (D / v) 1− v / c , given that the trip
 started at time zero, where D is the length of the outgoing or incoming journey as measured in the stay-at-home’s rest frame. [Derivations are omitted here.] At that same instant and place of turnaround the incoming observer measures the distant stay-at-home’s clock reading
 as ( )
 22
 T1(in) = (D / v) 1+ v / c . In passing we note that the average of
 these two distantly-measured time values,
 11
 [T (out) + T (in)] / 2 = D / v , agrees with the stay-at-home’s measured
 time of the distant turnaround event; so in this instance the relativity of simultaneity is removable by averaging. At the turnaround event the incomer’s clock, by stipulation, is set to equal the reading (D / v) /γ of the outgoer’s clock at that event.
 (The γ-factor comes either from the Lorentz contraction of D
 measured by the outgoer or from the time dilation of the outgoer’s clock measured by the stay-at-home.) By symmetry, the incomer ages the same amount as the outgoer in covering the same travel distance
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 D at the same speed v, so the traveler’s total aging is ( )
 2D / v /γ.
 Such is the stay-at-home’s deduction about the traveler’s aging. The stay-at-home’s measurement of his own aging is of course 2D / v , in agreement with the traveler’s “staying young” by a γ-factor.
 The aging attributed by the traveler to the stay-at-home is much trickier. The usual approach is to take the incomer’s word for the stayat-home’s age at the turnaround event, T1(in) , and to add to it the
 incomer’s inference about the stay-at-home’s aging during the return
 leg of the journey, which is seen to be 2
 (D / v) /γ (one gamma factor
 from the Lorentz contraction of the distance D and the other from clock-slowing attributed by the traveler to the clock of the relativelymoving stay-at-home). The total aging attributed by the traveler to the stay-at-home is then
 ( )( )
 2 22 22
 T1(in) + (D / v) /γ = (D / v) 1+ v / c + (D / v) 1− v / c = 2D / v ,
 in agreement with the stay-at-home’s own measurement and with the accepted result (observationally confirmed by CERN and GPS data)
 that the traveler’s total aging noted above, ( )
 2D / v /γ , is less than
 that of the stay-at-home by a γ-factor.
 It will be observed that, although the theory thus comes up with the “right answer,” it does so by a method widely viewed as illegitimate. For the fact that the incoming and outgoing inertial observers disagree on the stay-at-home’s clock reading at the event of turnaround, 1 1
 T (in) ≠ T (out) , is normally construed by relativists to
 mean that such distant clock-readings are “meaningless.” (See McCrea’s[1] refutation” of Dingle. Another term often used is “irrelevant.”) Their meaning is merely a conventional one, born of the “Einstein clock-setting convention.” That is, since different inertial observers disagree about them, they have no objective validity and are
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 not useable for making legitimate deductions about physical facts. Yet that is exactly what we have done above in determining the traveler’s “measurement” (or inference) of the stay-at-home’s aging. We used the supposedly meaningless quantity T1(in) in deducing the right
 answer, 2D / v . Had we used the equally meaningless “measurement” of the outgoer, T1(out) , we would have got a wrong
 answer,
 ()
 2 22
 T1(out) + (D / v) /γ = (2D / v) 1− v / c .
 So, how is it possible that outgoing and incoming inertial observers are “equivalent” (per the symmetry of the Lorentz group), yet only the “measurement” by one of them leads to the right answer? Suppose we require SRT to be strictly consistent with itself. (Here we adopt a purist viewpoint.) The relativity principle, as embodied in the Lorentz group, declares all inertial observers to be in some sense equivalent. When inertial observers disagree on their “measurements,” such measurements—as a reflection of this principle, so embodied—must seemingly be either irrelevant to physics or meaningless. The incomer cannot be “right” and the outgoer “wrong,” for that confers a preference incompatible with the symmetry properties of the Lorentz group. The very fact of observer disagreement is to be taken as proof (or definition) of “meaninglessness” of the associated measurements. The measurements of T1(in) and T1(out) , if either of them is irrelevant to
 physics, may thus be judged equivalently meaningless, hence both irrelevant to physics. Consequently we can rule out the use of either T1(in) or T1(out) in calculations of other observable quantities. But,
 deprived of both these quantities (and of all similar inferences of distant simultaneity)—the theory cannot be used by either the traveler or the stay-at-home to make any legitimate inferences whatever about
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 total agings during the journey. Viewed from this purist perspective, SRT defaults on the topic: It lacks any rigorous way of dealing with it, consistently with its own logical structure as expressed in the measurement symmetry of the Lorentz group. The traveler must thus, strictly speaking, consider the stay-athome’s “total aging” as itself a meaningless concept. The same must be true for the stay-at-home’s inferences of distant simultaneity, once we have ruled out such inferences by inertial observers in general. Since the concept of total aging during the journey is obviously physically meaningful for both traveler and stay-at-home—is in fact “absolute,” in view of co-location of the associated events—paradox is conserved. The claim of SRT’s “self consistency” in explaining how asymmetrical elapsed times result from symmetrical clock rates is, from the purist’s standpoint, void. The attempt logically to deduce asymmetry from symmetry has failed. Asymmetry comes into established analyses[2] only through the asymmetrical introduction of the meaningless number T1(in) . The primary paradox is not resolved,
 despite the fact that a Minkowski space diagram clearly shows the elapsed time asymmetry ... and experimental evidence leaves no room for doubt about the factuality of that asymmetry in nature. Nature is unambiguous; she has simply not been meaningfully described. The reason is presumably SRT’s limitation to description of inertial motions by the symmetrical Lorentz transformationwhich seems to exclude the possibility that physical work done asymmetrically on clocks in altering their states of motion, or any other physical cause, is to be recognized as affecting clock rates asymmetrically in an objective way.
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 3. Second Paradox: Relative Clock Rates
 Paradox resolvers are usually content to show (perhaps spuriously, as argued above) that all paradox vanishes from the twin problem when the symmetry-asymmetry aspect is dealt with. But their resolutions always entail the assumption (based on the Lorentz transformation) that each of two inertial observers measures the other’s clock as running slow. This in itself seems paradoxical enough to merit analysis as a second paradox. The basic scenario pictures a space traveler on some arbitrary closed polygonal path measuring the stayat-home’s clock at every moment throughout the journey as running slower than his (the traveler’s) own clock. Yet upon his return the traveler finds that the stay-at-home has aged more, not less, than himself. He is not supposed to be surprised by this outcome, because SRT has purportedly told him all about the actual (opposite) elapsed time asymmetry to be expected. But I argued above that a rigorous application of the special theory—which bars resort to the “meaningless”—blocks such a deduction. So, the traveler has every right to be surprised. If the traveler’s co-moving (proper-time) clock runs at some rate that we may designate R, the Lorentz transformation tells him that the stay-at-home’s clock runs continually at rate R /γ, whereas
 observation at the end of the journey tells him that the stay-at-home’s clock must instead have run continually at the rate Rγ. [A simple
 time transformation formally reflecting this factual state of affairs is t ' =γt , with inverse t = t '/γ. This shows a reciprocal symmetry, as
 supported, e.g., by GPS evidence[3]. It corresponds to a physically objective asymmetry of clock proper-time running rates in different motional (or gravitational) states—in apparent disagreement with the Lorentz transformation.] How does SRT deal with this apparent rate discrepancy? Many stories are told.
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 One told by Taylor and Wheeler,[2] although not on the cutting edge of current SRT expository fashion, may serve as typical. Their claim is that at the turnaround event of the out-and-back motion a distant clock resetting or clock phase-jump occurs, such that the outgoer’s notion of distant simultaneity is replaced by the incomer’s notion—which subsequently prevails. That is, T1(out) is replaced by
 T1(in) , and the latter is relied upon thereafter by the traveler as his
 basis for inferences about the stay-at-home’s aging. We have seen that this leads to the right answer, but at the cost of ascribing meaning to the “meaningless” quantity T1(in) . To make any operational sense
 of this approach, we must suppose the traveler to recognize that prior to the turnaround event a false story about distant simultaneity told him by SRT has misled him about the stay-at-home’s clock reading; that a clock phase resetting is necessary to correct this misleading impression; and that, after the resetting, the theory has given up its tricks and is subsequently telling him the truth. But the resetting in question is a change from T1(out) to T1(in) attributed by the traveler
 to the reading of the distant stay-at-home’s clock. Thus it corresponds to no operation performed by any observer. Nobody can actually do this resetting. It has to take place not physically but by analytic consensus, under the aegis of a Minkowski diagram or the equivalent. Its locus is the head of the traveler, where it acts to modify his inferences. Being that as it may, there remains the difficulty faced by the traveler to reconcile (in his head, the locus of his inferences) with his personal experience what SRT is telling him. By that theory he has been told, prior to the turnaround event, that the stay-at-home has aged comparatively little, some small amount S = T1(out) (equal to
 the outgoing traveler’s own self-measured aging, (D / v) /γ , divided
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 by γ). He has been told this as a physical (“measured”) fact. At the
 event he is told to forget all that and to switch ideas about distant simultaneity—consequently to infer a sudden great advance of the
 stay-at-home’s age, a jump J in that age. [ 2
 J = 2Dv / c .] He is further told that after the event the stay-at-home ages another small amount S during the return journey (by symmetry of the outgoing and incoming legs of the total journey). Thus the stay-at-home’s total aging inferred by the traveler is not 2S but 2S + J = 2D / v , which is γ times
 greater than the traveler’s own total aging during the journey, as it should be. Suppose, based on an assumption of the steady running of idealized clocks, or upon our experience of actual clocks left undisturbed, we impose a condition of uniform running rates of all clocks, regardless of their motions. If time is measured in seconds, and clocks tick once each second, then the outgoing observer (according to SRT) measures the stay-at-home’s clock as ticking S times during the outbound leg of the journey, and the incoming observer symmetrically measures the stay-at-home’s clock as ticking S times during the inbound leg; so the two of them measure a total of exactly 2S ticks of the stay-at-home’s clock during the whole journey. Nevertheless, we have seen that the traveler attributes to the stay-athome’s clock a total of 2S + J ticks because of a time jump J or clock resetting resulting from the relativity of simultaneity. Yet our condition of uniform clock running, dictated by experience, tells us that if 2S + J ticks of the stay-at-home’s clock occurred during the whole journey then during the outbound half of the journey
 ()
 2S + J / 2 = S + J / 2 ticks of that clock must have occurred. By
 symmetry, since inbound and outbound legs cover the same distance D at the same speed v, S + J / 2 ticks must also have occurred during the inbound half of the journey. But if only S of those ticks were
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 “measured” during each of the inbound and outbound legs, then the outbound and inbound observers must each have lost track of J/2 ticks. A total of J ticks seem to have disappeared. That is of course ridiculous, as shown by any of the numerous published analyses of electromagnetic signals exchanged between traveler and stay-athome, emitted at regular rates throughout the journey. Since the traveler personally measures (receives signals betokening) all 2S + J ticks of the stay-at-home’s clock, it appears to be false that he “measures” only 2S of those ticks or signals. But that is what SRT (the Lorentz transformation) insists upon. Thus relativists accept that 2S ticks are “measured” and 2S + J ticks received. That is indeed an apparent paradox, but it is readily explained within the ambit of SRT. The key to this aspect of the problem is to be found in the details of the measurement procedure. The 2S “measurement” is performed, not by a single observer, but by a spatially extended set of co-moving observers at rest in a moving inertial frame (each of whom is personally present locally at the emission of one of the successive ticks), whereas the 2S + J ticks are received by a single inertial observer (the traveler, a particular member of that set). The fact that space affects time, according to the Lorentz transformation, implies that the spatial displacements of the co-moving set members affect their collectively measured time flow rate, thus accounting for the failure of the moving traveler to “measure” J of the ticks. Such an explanation will satisfy most relativists. However, since all measurement procedures are identical between the traveler and the stay-at-home, whatever may be said in this regard about the one can be said about the other, and no explanation of the observed asymmetry can be given without introduction of the moral equivalent of the one-sided J-jump (an inference of the traveler only).
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 But what is the traveler to make of this quantity J? Let us suppose him to be not a mathematician but an ordinary astronaut. As such, he possesses certain experiences of real life to draw upon, which do not include miracles or magic. He knows that in all his personal experience he has never encountered a biological process that undergoes anything analogous to a clock resetting, time skip, or phase jump. He therefore must be skeptical of any theory incorporating measurement methods that imply an aging discontinuity J. The traveler could believe a genuine aging rate asymmetry—that he stays steadily younger than his earthly twin. That would contradict none of his experiences, because he has no direct experience of travel at “relativistic” speeds. But the claim that time skips, or that biological clock phases can jump, challenges not only his experience but his credulity—not to mention the testimony of his earth-bound twin, the witness most directly concerned. Clock rates thus furnish the material for another form of paradox that the theory has produced but not fully resolved. Defenders of the theory will point out that the clock-phase jump of the stay-at-home is not real because it is only an attribution or inference by the traveler, not something directly observed by anyone. This of course is true, but it is no mitigation of the paradox. What kind of theory is it that obtains its right answers by concatenating manifestly false stories? In this case the right answer given by SRT is the (differing) elapsed times of traveler and stay-at-home between the events of departure and return ... but this truth is attained only through the necessary telling of what appear to be lies about the stayat-home’s clock behavior. In order to fit with the right answer of great aging of the stay-at-home, it must be a lie that the stay-at-home’s clock runs slower than the traveler’s. The theory asks the traveler to celebrate a truth attained through mutual compensation of apparent falsehoods: Falsehood #1, that the stay-at-home’s clock appears to run
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 slower than the traveler’s; falsehood #2, that the stay-at-home’s clock appears to jump. A better face can be put on these apparent falsehoods by recognizing them as artifacts of the traveler’s measurement method (employing a spatially extended set of clocks co-moving with the traveler, as mentioned above). But the simple truth, thoroughly obscured by these artifacts, is that the stay-at-home’s clock runs steadily faster than the traveler’s. In principle, every (idealized proper-time) clock runs uniformly at some rate, without jumps. This is a basic premise of proper time. The observed outcome in the twin problem fits only with the stay-at-home’s clock having run steadily faster (by a γ-factor) than the traveler’s. This is the objective fact,
 embodying experience with clocks as idealized quantifiers of “time.” It is hidden by SRT’s Lorentz transformation, which asserts an apparent clock-rate symmetry, rather than the observed clock-rate reciprocity. This obscuration (call it, to be generous, a paradox) is in no way mitigated by the acknowledged capability of the theory to account correctly for the asymmetry of elapsed times; for the theory itself (by hypothesizing an unrealistic dependence of timekeeping on distance) has unrealistically severed the logical connection between clock rates and elapsed times—by depicting the former as symmetrical and the latter as asymmetrical. No longer, says SRT, do these two have to exhibit mutual consistency—because we have adopted a method of “measurement” that artificially precludes such mutual consistency (a) by employing a spatially extended set of measurement devices instead of a single detector, (b) by attributing to space an influence on time for which there has never been direct observational evidence.
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 4. Another Paradox: The Meaning of Meaninglessness
 The concept of “meaninglessness” alluded to above merits some attention to its paradoxical aspects. It is not a concept that arises spontaneously within any physical theory, but is one introduced from outside to hide (by a master stroke of verbal obfuscation) discrepancies between theory and experience. Besides SRT, is there any other physical theory in which meaninglessness plays a role? Yes, meaninglessness is a well-known last resort for saving cherished theory of any kind. In Newtonian mechanics the concept emerges wherever imaginary velocities occur, and often where unrealistic negative values of positive-definite parameters arise. In Maxwell’s field theory meaninglessness is traditionally applied to the “advanced solutions.” These are in conflict with experience, hence unwanted, hence swept under the rug as physically meaningless. Mathematically, they are just as good (i.e., just as meaningful) as the accepted retarded solutions; but physically they lack plausibility or relevance. They are indispensable to the mathematical structure of electromagnetic theory, but bear no readily discernable relationship to physical experience. Common sense says to junk them and for once common sense is listened to. That any physical theory should incorporate manifestly non-physical elements is surely in some degree paradoxical—although perhaps not more so than the basic premise of theoretical physics, that mathematics can describe physical experience (meaninglessness being always handy to gloss-over any incident shortcomings). Our invention and application of the concept of meaninglessness to the results of a particular theory is actually a vote of confidence in that theory—since without it we should be forced to acknowledge that the theory erred.
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 So it is with the quantity J discussed above. It may best be considered meaningless, not only operationally but conceptually; since it corresponds to nothing either physically existing or operationally measurable, but is purely the product of an arbitrary choice to honor the SRT conception of “distant simultaneity” of the incoming traveler rather than the outgoing one ... although both are inertial observers supposedly “equivalent.” As Orwell might put it, the incoming inertial observer is more equivalent than the outgoing one. Such an arbitrary choice seems directly analogous to that involved in choosing Maxwell’s retarded solutions over his advanced ones. In both cases, no stronger argument can be found than plausibility. One choice gives the “right answer”—the answer we are determined to get—the other the “wrong answer”—the one we are determined not to get. Yet in this case even plausibility suffers, inasmuch as only the twin-problem assertions of a third party, the stay-at-home inertial observer, are truly plausible—for only they show the timekeeping symmetry (reflecting the spatial and speed symmetry) of outgoing and incoming legs of the journey that fits with common sense. The elapsed half-journey times on the two legs of the full journey ought to be equal ... and only the stay-at-home (according to SRT) “measures” them to be so. (Recall that the traveler who attributes symmetrical aging 2S to the stay-at-home thereby gets the wrong answer ... the right answer, 2S + J , being apparently asymmetrical between the two legs of the journey.) It is significant that the dubious “jump” quantity J is entirely absent from the stay-at-home’s account. For this reason we can say that only his account is plausible. And in fact if we listen only to his account we are assured of getting the “right answer.” (As a point of history, I seem to recall that one of the earliest twin paradox “resolutions” labeled the inertial observer a “good observer,” and the
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 accelerated one a “bad observer.” The more it changes, the more it is the same.) We choose to listen to another account involving the fictitious J (if we do listen) only because we wish to make a genuflection toward an ideological principle—that of motional relativity, interpreted as implying equivalence of inertial observers. It is that genuflection which enmeshes us in paradox and forces us to wrestle with implausibility. Indeed, if the genuflectors were to get down on both knees instead of one, they would have to give as much weight to T1(out) as to T1(in) —hence as much to the wrong answer
 as the right one. If we would admit up front that the customary understanding of the relativity principle, as implying a symmetry of proper timekeeping among inertial systems (fostered by a corresponding symmetry of the Lorentz transformation), is just plain wrong, and that the CERN and GPS evidence points directly to an objectively asymmetrical (but reversible) rate-slowing of any clock that has had work done on it, we would be spared need for the above-mentioned genuflection and its attendant paradoxes. (And we would learn some physics.) But we would thereby also be spared SRT and its attendant truths, such as that “now” is an illusion, that we live “in” Minkowski space, or that, when clock A appears to run slower than clock B, clock B appears to run slower than clock A. And no card-carrying, pain-seeking physicist wants to be spared those particular barbed truths. The upshot is that the various twin-related paradoxes force us to re-examine our interpretation of the relativity principle. Trouble arises whenever, impelled by the Lorentz transformation, we insist that time measurements be “equivalent” in different inertial systems. Is there any simple way of avoiding this trouble? Yes, of course there is. As long as we confine our statement of the relativity principle to its original form—that the laws of nature are the same in all inertial systems—trouble cannot arise. For this allows clock running rates to
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 be affected (objectively altered) by the physical energy-imparting actions needed to alter the inertial states of motion of matter. An objective alteration of a clock’s running rate in a new inertial system merely changes the meaning of the “second” in that system—hence it can be corrected by a compensatory alteration (à la GPS[3]) of time units. (So says Newton’s Principle of Similitude.) The laws of motion or of nature remain invariant under changes of inertial system, in conformity with the just-stated, strictly limited, form of the relativity principle. The Lorentz transformation fails to exploit this Similitude loophole in the relativity principle—a loophole whereby an objective asymmetry of proper timekeeping among inertial systems might be introduced into physical theory to honor directly the facts of observation. Instead, the Lorentz transformation goes beyond minimum requirements of the relativity principle to specify a rigorous symmetry of “measured” clock rates incompatible with the steady (non-jumping) running of clocks, measurements being made in an unrealistically contrived way by means of extended sets of detectors, compensated by an equally contrived dependence of time on space. (The latter derives from the assumption that Maxwell’s equations must hold in all inertial systems—an assumption challenged in Ref. 3.) In short the Lorentz group incorporates a symmetry of “measurement” results not demanded by the original form of the relativity principle. The fact that this artificial symmetry can be made compatible with observation through an equally artificial conception of “measurement” via extended clock sets does not alter the possibility of shortcutting past these model artificialities by means of alternative theory descriptive of a factual (objective) timekeeping asymmetry resulting directly from asymmetrical physical causes. In his SRT Einstein postulated the presence on demand of Gedanken reference frames in various inertial states of motion. He
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 devoted no attention to the energy expenditures necessary to produce such states of motion through state alterations in the real world—or to the possible related effects on timekeeping of those energy expenditures. (The real world is a place wherein no matter arrives at a state of motion through postulation, only through being put into it by physical action.) The price exacted for this insouciance is the loss of any possibility of finding a physical cause for the observed phenomenon of time dilation. So enthusiastically has this price been paid by physicists that they no longer seek physical causation and would oppose any suggestion of the possibility, as incompatible with what they “know” about kinematics. If Nature thrust a physical cause upon them, they would not know what to do with it. Thus the door is closed permanently to exploration of alternative descriptive approaches. This closing has been done deliberately by physicists to their own beloved physics. If you doubt it, consult the announced editorial policies of established physics journals. The twin paradoxes provide tantalizing glimpses of possibilities for more physically-based theories directly incorporating observed physical asymmetries of timekeeping. These may, but need not, be built upon ether hypotheses, since we have noted their possible compatibility with the original relativity principle (referring to “laws of nature”). In light of human nature (mob psychology) it is axiomatic that no alternative to SRT, no “test theory,” will receive a fair hearing until arrival of the millennial era (also known as the “Second Coming”) in which physicists become motivated to re-examine the foundations of their discipline. In that imaginary era they will begin to practice some of the humility to which their current arrogance lays claim.
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 5. Incidental Paradoxes: Spacetime
 Among ancillary paradoxes implicated in the twin affair, that of SRT’s central concept of spacetime stands out as worthy of attention. The idea that time is in some deep sense the same thing as space was historically the brainchild of Minkowski, but the geometrical language he employed to embody it proved useful to Einstein for his more general theorizings, and was soon incorporated integrally into the special theory. This idea is from the start paradoxical in the context of SRT conceived as physics, since a spatial dimension is plainly a single-valued quantity, a measure of linear extension marked by a single number. Time, in contrast, is subtler, not only in the sophistication of the devices needed for its measurement but in its basic conceptualization. Unlike space, it refuses to hold still. In the real world it has an inexorable tendency to “flow” or slip away. There is nothing we can do about this, but we can accept it and measure it by means of repetitive processes, called “clocks.” These enable us, with the help of “calendars,” to seize and label (but not to retrieve) a “moment of time” as epoch or date. But they also enable us, entirely apart from moments and labels, to speak of “clock rates.” These we can imagine as correlated with the flow of time—although of course only relative significance can be attached to such a locution. Thus the parameter “t” we use mathematically to represent “time” can be conceived as serving a dual or bi-valued purpose: to designate a particular point in the flow or to represent the flow itself. From the ambiguity created by use of a single word and parameter for these two communicative purposes stems much confusion. Further ambiguity arises from whether “time” is endowed with a local or nonlocal connotation. No such complexity attaches to the orthogonal space parameters x,y,z. Regardless of what may be assumed about the “state of motion”
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 of hypothesized physical ethers, there is no logical compulsion to introduce a concept of “flow” in connection with space. Relatively to our laboratory, a point of space can be revisited, a point of time cannot. When a space dimension is asserted to be equivalent to a time “dimension,” an essentially single-valued descriptor is being equated to an essentially bi-valued one. The subtleties of “time” and its measurement are swept away and discarded as if they did not exist. The only way this can be sold to a gullible public (or profession) is as “insight.” That it may be, or that it may be NOT. It would be wise, or at least prudent, to consider the jury (after a century of SRT’s spacetime symmetry hype) still out. Our era seems to be unique in its ability to manufacture physical “symmetries” out of the unsuspected infelicities of our physical models. We then have the fun of “breaking” those symmetries. That fun has yet to be exploited in regard to spacetime symmetry. The distinction pointed out in Section 2 between clock rates and elapsed times is directly related to the bi-valuedness of time noted here. In the case of sinusoidal signals this bi-valuedness is marked explicitly by phase and frequency parameters. SRT in the twin problem emphasizes the split between these two concepts by making clock rates (when “measured” as symmetrical between two inertial observers) incompatible with elapsed time intervals (asymmetrical). [Cf. the “primary paradox,” above.] This is like finding a method of “measurement” capable of making phase measurements of a sinusoid incompatible with frequency measurements. Thus SRT by its nature forces attention to a distinction between two aspects of “time,” while asserting no such duality of space. That is, the “flow” aspect of time is distinguished from the “date” aspect (which undergoes the jump J), as a typical feature of the twin paradox resolution. Is it not paradoxical, then, that a theory that builds upon the radical concept of “spacetime,” without a hyphen, should invoke the bi-valuedness of the linear time
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 concept, in contrast to the single-valuedness of the linear space concept, merely to treat the first problem of non-uniform motion it encounters? But enough ... Of the making of SRT paradoxes there is no end.
 Acknowledgment
 This paper grew out of extensive discussions among C. Roy Keys, Stephane Baune, and myself. The analysis on which it is based was done by Dr. Baune, who, however, does not subscribe to my lucubrations.
 References
 [1] W. H. McCrea, Nature 216, 122 (1967).
 [2] E. F. Taylor and J. A. Wheeler, Spacetime Physics (Freeman, San Francisco, 1966). [3] T. E. Phipps, Old Physics for New (Apeiron, Montreal, 2006).
@@ -0,0 +1,403 @@
 University of Dayton
 eCommons
 Electrical and Computer Engineering Faculty Publications
 Department of Electrical and Computer Engineering
 2010
 Characterization of Atmospheric Turbulence
 Effects Over 149 km Propagation Path Using
 Multi-Wavelength Laser Beacons
 Mikhail Vorontsov
 University of Dayton, mvorontsov1@udayton.edu
 Gary W. Carhart
 Army Research Laboratory
 Venkata S. Rao Gudimetla
 Air Force Research Laboratory
 Thomas Weyrauch
 University of Dayton, tweyrauch1@udayton.edu
 Eric Stevenson
 Boeing LTS
 See next page for additional authors
 Follow this and additional works at: http://ecommons.udayton.edu/ece_fac_pub
 Part of the Computer Engineering Commons, Electrical and Electronics Commons, Electromagnetics and Photonics Commons, Optics Commons, Other Electrical and Computer Engineering Commons, and the Systems and Communications Commons
 This Conference Paper is brought to you for free and open access by the Department of Electrical and Computer Engineering at eCommons. It has been accepted for inclusion in Electrical and Computer Engineering Faculty Publications by an authorized administrator of eCommons. For more information, please contact frice1@udayton.edu, mschlangen1@udayton.edu.
 eCommons Citation
 Vorontsov, Mikhail; Carhart, Gary W.; Rao Gudimetla, Venkata S.; Weyrauch, Thomas; Stevenson, Eric; Lachinova, Svetlana; Beresnev, Leonid A.; Liu, Jony Jiang; Rehder, Karl; and Riker, Jim F., "Characterization of Atmospheric Turbulence Effects Over 149 km Propagation Path Using Multi-Wavelength Laser Beacons" (2010). Electrical and Computer Engineering Faculty Publications. Paper 96. http://ecommons.udayton.edu/ece_fac_pub/96
 Author(s)
 Mikhail Vorontsov, Gary W. Carhart, Venkata S. Rao Gudimetla, Thomas Weyrauch, Eric Stevenson, Svetlana Lachinova, Leonid A. Beresnev, Jony Jiang Liu, Karl Rehder, and Jim F. Riker
 This conference paper is available at eCommons: http://ecommons.udayton.edu/ece_fac_pub/96
 Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons
 Mikhail A. Vorontsov
 School of Engineering, University of Dayton, Dayton, Ohio 45469, USA Gary W. Carhart
 Computational and Information Sciences Directorate, U.S. Army Research Laboratory, Adelphi, Maryland 20783, USA V. S. Rao Gudimetla
 Directed Energy Directorate, Air Force Research Laboratory, Kihei, Maui, Hawaii 96753, USA Thomas Weyrauch
 School of Engineering, University of Dayton, Dayton, Ohio 45469, USA Eric Stevenson
 Boeing LTS, Kihei, Maui, Hawaii 96753, USA Svetlana L. Lachinova
 Institute for Systems Research, University of Maryland, College Park, Maryland 20742, USA Leonid A. Beresnev
 Computational and Information Sciences Directorate, U.S. Army Research Laboratory, Adelphi, Maryland 20783, USA Jiang Liu
 Computational and Information Sciences Directorate, U.S. Army Research Laboratory, Adelphi, Maryland 20783, USA Karl Rehder
 Boeing LTS, Kihei, Maui, Hawaii 96753, USA and
 Jim F. Riker
 Space Vehicles Directorate, Air Force Research Laboratory, Kirtland AFB, New Mexico 87117, USA
 ABSTRACT
 We describe preliminary results of a set of laser beam propagation experiments performed over a long (149 km) near-horizontal propagation path between Mauna Loa (Hawaii Island) and Haleakala (Island of Maui) mountains in February 2010. The distinctive feature of the experimental campaign referred to here as the Coherent Multi-Beam Atmospheric Transceiver (COMBAT) experiments is that the measurements of the atmospheric-turbulence induced laser beam intensity scintillations at the receiver telescope aperture were obtained simultaneously using three laser sources (laser beacons) with different wavelengths (λ1 = 0.53 μm, λ2 = 1.06 μm, and λ3 = 1.55 μm). The presented experimental results on intensity scintillation characteristics reveal complexity of the observed phenomena that cannot be fully explained based on the existing atmospheric turbulence models.
 1. INTRODUCTION
 There is a growing interest in development of optical systems capable of operation over long atmospheric distances (path lengths up to and over 100 km) in various atmospheric conditions and engagement scenarios. Among these systems are long-range laser communications, remote sensing, active and passive imaging, target tracking and designation, and laser beam projection (directed energy) systems. Propagation of optical waves over a long distance through volume atmospheric turbulence can result in significant changes in characteristics of received or transmitted waves and thus dramatically impact performance of the optical systems.
 The turbulence effects over atmospheric propagation paths are commonly associated with fluctuations of intensity (scintillations), formation of wavefront phase topological singularities (branch points), wander and widening of laser beam, spatial non-uniformity of the turbulence-induced image degradation, etc. [1–4].
 Currently, analysis of atmospheric turbulence-induced effects is based on theoretical and numerical models that are derived from the classical Kolmogorov-Obukhov turbulence theory [5–8], developed in 1940's–1960's. This theory was extensively validated through a number of various atmospheric experiments. The majority of these experiments were performed over relatively short distances (commonly with path lengths of less than several kilometers [9–12]) and only a few experiments over distances exceeding 100 km [13–15]. Nevertheless the models derived from the classical Kolmogorov-Obukhov turbulence theory are commonly used for performance assessment of long-range optical systems. The existing gap in experimental validation of turbulence models over long-range propagation distances may result in significant errors, leading to miscalculations of optical system capabilities and conceptual mistakes in their design.
 The risk related with unconscious extension of the classic Kolmogorov-Obukhov turbulence model for analysis of atmospheric turbulence effect on optical wave propagation over long-range distances has been recognized and experimental evidence of noticeable deviations from this model prediction have already been observed in a few long-range experiments [13–16]. Several physical factors can contribute to such unanticipated (non-classical) behavior of optical wave characteristics that are not accounted for in the existing theories, including dynamics of large-scale air flows leading to strongly non-stationary refraction effects, impact of ocean-land interface, nonuniformity of air pressure, temperature, and wind profiles along the propagation path. Unfortunately, due to complexity (both logistical and technical) and cost which are commonly associated with long-range optical wave propagation experiments, the collected experimental data are not sufficiently representative to determine if some important adjustments can be made to the existing theory to account for extended turbulence effects, or if fundamentally new approaches are required.
 In this paper we intend to partially fill out the existing gap in experimental assessment of extended turbulence effects on laser beam propagation over long-range paths by providing a large amount of intensity scintillation data obtained over a 149 km long propagation path using light sources with different wavelength.
 The intensity scintillations are commonly characterized by the normalized variance of optical wave intensity fluctuations measured using a point photo-detector located at a point r = {x,y} of receiver aperture,
 () () ()
 ()
 ()
 2
 2 2
 I
 II
 I
 σ
 − =
 rr r r
 , (1)
 where I(r) is the short-exposure intensity at the photo-detector point and angular brackets denote averaging over the ensemble of statistically independent realizations of refractive index perturbations of the propagation medium (atmospheric turbulence). It is assumed that random fluctuations of intensity are homogeneous and isotropic and hence the normalized variance ( )
 2
 σI r in Eq. (1) is independent of coordinate vector r, that is, 2 2
 ()
 II
 σ r =σ .
 In accordance with the classical and more recent atmospheric turbulence theories, the normalized intensity scintillation variance 2
 σI can be represented as
 ()
 2 7/6 2 00
 , , ,,
 In
 σ λ− F C l L z h
 = , (2)
 where λ is the wavelength of the propagating quasi-monochromatic wave and 2
 00
 ( , , ,,)
 n
 F C l L z h is a factor
 represented here as a function of the propagation path characteristics (distance z and elevation h) and the major turbulence parameters: structure constant Cn
 2, inner scale l0, and outer scale L0. The particular form of function
 2
 00
 ( , , ,,)
 n
 F C l L z h depends on theoretical models and approximations used to obtain expression (2) [17, 18].
 In this respect, in principle, validity of one or another theoretical model can be tested through measurements of the intensity scintillation variance ( )
 2
 σI r and independent calculation of the factor 2
 00
 ( , , ,,)
 n
 F C l L z h in Eq. (2).
 Nevertheless, such an approach requires knowledge of the propagation path and turbulence characteristics the function 2
 00
 ( , , ,,)
 n
 F C l L z h depends on. In practice, these parameters cannot be measured independently and
 directly.
 This problem however can be overcome using simultaneous measurements of intensity scintillations corresponding to optical waves at different wavelengths that are originating from a single laser beacon and hence propagating along identical atmospheric path. As follows from Eq. (2), the ratio of the normalized variances 2
 ( 1)
 σI λ and 2
 ( 2)
 σI λ
 corresponding to the laser beacon wavelengths λ1 and λ2 represents a value independent of the factor
 2
 00
 ( , , ,,)
 n
 F C l L z h : 2 2 76
 1 2 21
 () ( ) ( )
 II
 σ λ σ λ =λλ .
 The idea of atmospheric turbulence characterization using multi-wavelength laser beacons with simultaneous measurements of intensity scintillations at different wavelengths has been implemented in the COMBAT experiments. In these experiments intensity scintillations were simultaneously measured using three closely located identical laser beacons that generated truncated Gaussian beams at the following wavelengths: λ1 = 0.53 μm, λ2 = 1.06 μm, and λ3 = 1.55 μm. To simplify notations, the corresponding beacons are referred to as green (G), infrared (IR), and laser communications (COM) beacons. In accordance with Eq. (2), the ratios of the corresponding intensity scintillation variances expected from the atmospheric turbulence theory, G IR
 RT , G COM
 RT , and IR COM
 RT , are
 given by
 22 2
 G IR G COM IR COM
 22 2
 (0.53μm) (0.53μm) (1.06μm)
 2.24; 3.50; 1.56
 (1.06μm) (1.55μm) (1.55μm)
 TT T
 II I
 II I
 RR R
 σσ σ σσ σ
 = = = = = = . (3)
 In the COMBAT experiments, the experimental values of the ratios G IR
 R E , G COM
 RE , and IR COM
 RE corresponding to
 Eq. (3) were directly obtained using simultaneous measurements of statistically representative ensembles of shortexposure intensity patterns 1
 I (r,λ) , 2
 I (r,λ ) , and 3
 I (r,λ ) inside the receiver telescope aperture of diameter 90 cm
 located a distance 149 km from the corresponding laser beacons. With the understanding that over long propagation paths the assumptions of stationary and isotropic turbulence with homogeneous statistics – the basis for the Kolmogorov-Obukhov turbulence theory – are not always satisfied, we expected some deviation in values between the ratios G IR
 RT , G COM
 RT , and IR COM
 RT in Eq. (3), as predicted by the theory, and the values G IR
 R E , G COM
 RE , and
 IR COM
 RE obtained experimentally. The real surprise was the degree of disconnect between predictions based on
 homogeneous turbulence theory and measurements as illustrated by the data obtained and discussed below.
 2. COMBAT EXPERIMENTAL SETTING
 Laser beam propagation experiments were performed over the propagation path of length L ≅ 149.2 km between the laser beacon platform located on the Mauna Loa mountain (Hawaii island) at elevation hML = 3397 m (11,140 ft), and the 3.67-m receiver telescope on the top of the Haleakala mountain (Maui island) at elevation hH = 3058 m (see Fig. 1). Due to the Earth's curvature, the minimum height along the propagation path hmin ≅ 2775 m corresponds to the distance 60 km from the receiver site.
 The laser beacon assembly at the Mauna Loa Observatory comprised three single-mode fiber collimators (each with a clear aperture of 26 mm) and the corresponding lasers: a fiber-coupled laser diode at the wavelength of λ1 = 0.532 μm and two fiber lasers with wavelengths of λ2 = 1.064 μm and λ3 = 1.55 μm. The fiber collimators were
 Fig. 1. Elevation profile along the propagation path from the Mauna Loa NOAA observatory to the AEOS telescope on Haleakala.
 mounted together with an aiming telescope into a gimbal system (see Fig. 2), which allowed for pointing with a smallest angular step size of 1.75 μrad, corresponding to a lateral beam deflection of 26 cm at the receiver telescope over 149 km distance. Each fiber collimator had means for manual horizontal, vertical, and axial alignment of the fiber tip with respect to the collimating lens to adjust beam pointing and focus. For co-alignment of the fiber collimators' optical axes with the aiming telescope, a corner-cube retro-reflector was mounted in front of the beacon assembly in such way that some part of the light from two collimators at a time was directed into the aiming telescope and focused onto the camera surface located in its focal plane. Using the 1064 nm collimator as reference, the focal spots of all three beams were brought to coincide in the telescope focus, which in turn co-aligned the collimated beams. After the alignment procedure, the retro-reflector was removed and the beacon assembly was pointed toward the receiver telescope [3.67 m Air Force Advanced Electro-Optical System (AEOS telescope)] using the beams' former spot location in the camera's field-of-view as aimpoint. A searchlight installed in front of the AEOS telescope helped to achieve and maintain correct aiming after sunset, when the telescope was no longer visible with the aiming telescope.
 Another search light installed near the beacon assembly at the Mauna Loa Observatory allowed for alignment of the AEOS telescope toward the beacons. After propagation to Maui site, the optical waves from the beacons were received by the AEOS telescope and sent through the coudé path to the deformable mirror, which is part of the AEOS adaptive optics system operating in the visible wavelength band. The adaptive optics system was off and the deformable mirror was flattened during the experiments. Collimated light from the adaptive optics bench (beam diameter 10 cm) was sent to one of the AEOS coudé rooms where the beam was subdivided into six separate subapertures, each having a usable diameter of 25 mm (Fig. 2). Each subaperture corresponds to an area with a diameter of 91.7 cm at the telescope pupil considering the de-magnification of the receiver area from 3.67 m diameter at the telescope's primary mirror to 10 cm in the coudé room. Three of the subapertures were used to record pupil-plane and focal-plane irradiance distributions of the received beams (Fig. 2). Narrow bandpass filters were used to dedicate each subaperture to a specific wave from a single beacon. In this paper we discuss only measurements of the pupil-plane intensity distributions, which were recorded using three Sensors Unlimited (SU640SDWHvis) 14-bit cameras. The cameras were operating in a windowed mode with 256×256 pixels' window, providing an actual sensor area of 6.4×6.4 mm2 with a pixel size of 25×25 μm2 and a 100% fill factor. A four-to-one beam reducer was installed in the optical train of each subaperture to match the beam (pupil) size with the active window size.
 Fig. 2. Conceptual schematic of the COMBAT experimental system.
 3. MEASUREMENTS AND RESULTS
 3.1 Data Recording and Processing
 The COMBAT measurements were performed from early evening to middle night from Feb. 12 to Feb. 21, 2010. Data collection consisted of a set of experimental trials (typically from 10 to 30) between 7:00 p.m. and 11:30 p.m. during each night of measurements. During each trial a set of nearly 10000 frames of short-exposure intensity patterns in the form of 14-bit digital arrays {In(rm,l,λ1)}, {In(rm,l,λ2)}, and {In(rm,l,λ3)} were recorded simultaneously using three identical cameras located in the image planes of the telescope subapertures that are dedicated to each beacon. Here n is the frame number in the experimental trial (n = 1, ..., 9980), rm,l is the transversal coordinate vector in the plane of photo-array (m,l = 1, ..., 256), and λq is the received light wavelength (q = 1, 2, 3). The time of the first frame capture by all cameras within each trial was triggered from a PC computer. The capture rate (between 200 and 350 frames per second dependent on the trial) and the integration time (between 1 ms and 2 ms) were identical for all cameras. The averaged deviation between capturing of intensity patterns In(rm,l,λ1), In(rm,l,λ2), and In(rm,l,λ3) by different cameras did not exceed 1.0 ms. The physical size of the camera's pixel 25×25 μm2 corresponds to the telescope receiver aperture size of approximately 3.6×3.6 mm2. Thus the measurements obtained at each pixel of the photo-array correspond to the received light intensity averaged over 3.6-mm square aperture in the AEOS telescope pupil plane. According to the camera specification, variation in sensitivity from pixel to pixel did not exceed 1% and therefore was neglected. To avoid aperture boundary effects related with small amplitude slow displacement of the received beam centroid during the time of measurements, the intensity scintillation data were analyzed inside the circle area corresponding to the sub-aperture boundary reduced by 10 pixels of the photoarray.
 To exclude the data recorded by photo-array pixels that are not properly functioning (broken or blocked by presence of dust on optics or sensor), the following procedure was applied. For each measurement trial we calculated variance of the intensity fluctuations at each pixel
 () () 2
 2
 pix m,l n m,l n
 N
 σ ⎡I I ⎤
 =−
 ⎣⎦
 r r , (4)
 where 1
 ,
 1( )
 N
 n n ml
 n
 IN I
 − =
 = ∑ r is the averaged value of pixel intensity corresponding of the set of N short-exposure
 images belonging to a single measurements trial. The pixels were excluded in the further analysis if the
 corresponding values of the intensity fluctuation variance ( )
 2
 pix m,l
 σ r for these pixels were less than 1% from the
 aperture-average value. In addition, we performed the camera offset correction by capturing a "dark map" for each of the cameras at the beginning of each data collection trial. The values of the "dark map" images were then subtracted from all subsequent data frame captures. The dark maps were recorded by blocking all external light from entering the optical system at the experiment room's coudé port. However, this dark map capture procedure does not take into account any external background light such as the light from the sky, which should also be removed. The impact of this external background light was removed by calculating a map of minimum intensities detected during each trial for each pixel. This "min map" data was subtracted from all the frames of the trial. Note that the difference in the intensity variance (4) brought about by using this additional min map correction or not only produces a change of a few percent in its value and the true value must lie in the range between the two possibilities. It is believed that most of the min map detected intensity is in fact sky background illumination and therefore the data presented makes use of the min map correction.
 3.2 Received Power Fluctuations
 Intensity scintillations observed during most of experiments were surprisingly strong. The short-exposure intensity patterns show the presence of speckles of very different spatial scales: from few centimeters or less to large speckles of approximately the size of the receiver aperture (~0.9 m). These large-size speckles were seen as spikes and deep fadings in the received optical power signal. In most experiments the intensity scintillation patterns did not show coherent (directional) motion but rather random appearance and disappearance. As rare exceptions in windy conditions at the receiver site, we observed occasional coherent motion of small-scale speckles.
 Consider first the received light power fluctuations data simultaneously recorded for all three beacons. For each short-exposure intensity pattern (image frame), the optical power averaged over the aperture is defined as
 ()
 M
 n nm m
 P = ∑ I r , (5)
 where M = 256×256. To simplify notation, in Eq. (5) and below we use a single index m to denote the camera pixel
 coordinate vector. The dependencies of the normalized power Pˆ(n) on the consequent frame number n or physical
 time t are shown in Fig. 3 for two representative measurements trials. In this figure, we can clearly see the presence of large spikes on the received power which indicate the occasional appearance of large-size bright and dark speckles at the receiver aperture for all three wavelengths. Bright speckles can result in up to ten-fold increase in received power. Note that for the "3beacon19" trial with λ2 = 1.06 μm (red), the amplitude of spikes appeared to be drastically higher. The appearance of large-amplitude bright or dark spikes in received power can be explained by the existence of slowly changing/moving large-scale coherent structures of refractive index that play a role of large aperture lenses with several kilometer-long focal lengths. As can be seen from Fig. 3, there seems to be no correlation between the appearance of bright and dark speckles for laser beacons of different wavelengths.
 The characteristic snapshots of the received light pupil-plane intensity distributions for beacons with different wavelengths are illustrated in Fig. 4. The snapshots are chosen from the vicinities of bright spikes (1)
 1
 BS , (1)
 2
 BS , and
 (1) 3
 BS indicated in Fig. 3. The images in Fig. 4 show that the spatial scale of intensity scintillations is strongly
 dependent on the wavelength, i.e., the larger the wavelength the larger the speckles. This observation is consistent
 Fig. 3. Temporal dynamics of received optical power fluctuations ˆ max
 n nn
 P = P P for laser beacons with
 wavelengths λ1 = 0.53 μm (top, green), λ2 = 1.06 μm (middle, red), and λ3 = 1.55 μm (bottom, blue) corresponding to the experimental trials labeled as "3beacon19" (left column) and "3 beacon_C" (right column). The date and recording time are shown on the top of each column. During each trial, three sequences of 10000 short-exposure frames with 2 ms integration time were captured simultaneously. Several selected bright and dark spikes in the received power for the trial "3beacon19" are shown by BS and DS marks. The amplitudes of the biggest spikes in the data set "3beacon19" are: 1 1
 Pmax 14.8P
 λλ
 = (green beacon), 2 2
 Pmax 40.0P
 λλ
 = (IR beacon)
 and 3 3
 Pmax 8.7P
 λλ
 = (COM beacon). Here Pλ1 , Pλ2 , and Pλ3 are the averaged received power values for each
 wavelength.
 with the theoretical findings and was also confirmed by our numerical simulations (see the most right column in Fig. 4).
 3.3 Intensity Scintillations
 Spatial non-uniformity of intensity in each short-exposure intensity pattern (frame) of a trial can be characterized by the set of values (one per a single frame) referred to here as spatial variances
 () ( ) 2
 22 S nm n n S
 σ n ⎡I I ⎤ I
 =−
 ⎣⎦
 r , (6)
 where 1
 1()
 M
 n nm
 m
 IM I
 − =
 = ∑ r is the aperture-average intensity for the nth frame. The index S in Eq. (6) denotes
 averaging over spatial coordinate (over pixels). The sequence of 2 ( )
 σS n values characterizes temporal changes of
 spatial variance occurring during the time of measurements trial. Correspondingly, by averaging 2 ( )
 σS n over the set
 of N samples, which corresponds to time averaging, we obtain statistical characteristic 2 2 ( )
 SN S N
 σ = σ n that can be
 used for evaluation of spatial non-uniformity of intensity for the entire trial. Dependencies 2 ( )
 σS n for three different
 wavelengths and two measurement trials are shown in Fig. 5. The averaged values 2
 SN
 σ are given inside the
 corresponding plots.
 Another and more commonly used approach for intensity scintillations analysis is based on computation of the normalized variance of intensity fluctuations at a single photo-array pixel,
 () () () ()
 2
 22
 N m nm m m N
 σ ⎡I I ⎤ I
 =−
 ⎣⎦
 r r r r , (7)
 where ( ) ( )
 1
 1N
 m nm
 n
 I N =I
 =∑
 r r . The averaging over the set of N ≈ 10000 frames in Eq. (7) is equivalent to time
 averaging over nearly 50 seconds. The normalized variance (7) is referred to as the scintillation index. By averaging ( )
 2
 Nm
 σ r over all pixels inside the receiver aperture (with exception of "bad" pixels described above) we
 obtain the aperture-average scintillation index 2 2 ( )
 NS N m S
 σ =σ r .
 Fig. 4. Snapshots of pupil-plane intensity distributions of received light, ( )
 nm
 I r , from beacons with three
 different wavelengths, λ1 = 0.53 μm, λ2 = 1.06 μm, and λ3 = 1.55 μm, for "3beacon19" measurement trial. First three columns represent experimental data; frame numbers n are denoted above the intensity images. Last column contains typical snapshots of intensity distributions within a single subaperture obtained numerically.
 In order to evaluate whether the observed intensity scintillations can be considered as a stationary random process, consider analogues of the scintillation index obtained using either increasing number n of subsequent frames 2 ( )
 σnS n
 or "moving" (window) averaging of n sequential frames 2 ( , )
 WS W
 σ n n , where nW is a number of frames in the selected
 averaging window. Both 2 ( )
 σnS n and 2 ( )
 σWS n dependencies are shown in Fig. 6; the values 2
 NS
 σ are given inside the
 corresponding plots.
 The gray-scale images shown at the bottom of Fig. 6 illustrate spatial non-uniformity of the scintillation index 2( )
 Nm
 σ r obtained by averaging different number of frames in the trial for different beacons. The indication of
 spatial homogeneity of intensity scintillation process is spatial uniformity of the images mapping scintillation index 2( )
 Nm
 σ r . As it can be seen from 2 ( )
 Nm
 σ r -images in Fig. 6, averaging over 10000 frames results in quite uniform
 2( )
 Nm
 σ r -images for the "3 beacon_C" measurement trial, while the corresponding 2 ( )
 Nm
 σ r -images for the
 "3beacon19" trials are still highly non-uniform. This indicates statistical non-stationary character of the intensity scintillation process during this measurement trial. The same conclusion regarding non-stationary character of the scintillation process can be also derived from the analysis of dependencies of 2 ( )
 σnS n and 2 ( , )
 WS W
 σ n n in Fig. 6. This
 highly non-stationary behavior of the scintillations can be associated with the presence of the large-scale intensity scintillations components (large-scale bright and dark speckles).
 Assuming that the ergodic hypothesis can be applied, the obtained values 2
 SN
 σ and 2
 NS
 σ should be approximately
 equal since they are obtained by changing the order of spatial and temporal averaging [15]. However, when comparing the corresponding values 2
 SN
 σ and 2
 NS
 σ calculated for the two selected data sets (trials) in Figs. 5 and 6,
 we see that these values are quite different and so application of the ergodic hypothesis is questionable.
 Bar diagrams summarizing the results for 2
 SN
 σ and 2
 NS
 σ obtained for a number of measurement trials during the
 COMBAT experimental campaign along with the wave-optics simulation results are shown in Fig. 7. Bars' heights
 Fig. 5. Spatial variances of short-exposure intensity non-uniformity inside receiver subaperture, 2
 σS , as functions
 of frame number n for "3beacon19" (left) and "3 beacon_C" (right) measurement trials and three wavelengths, λ1 = 0.53 μm (top, green), λ2 = 1.06 μm (middle, red), and λ3 = 1.55 μm (bottom, blue).
 indicate spatial variance 2
 SN
 σ (top) and scintillation index 2
 NS
 σ (bottom) values, while color of the bars denote the
 wavelength of the laser beacons used in experiments, i.e., λ1 = 0.53 μm (green), λ2 = 1.06 μm (red), and λ3 = 1.55 μm (blue) beacons, respectively. Each set of adjacent green, red, and blue bars corresponds to a different measurement trial whose name is indicated below the set; numbers at the tops of each set denote local time of the experiment. Groups of several sets of bars correspond to different dates of the experiments, specified below the diagram. Numerical simulation results obtained for the HV-5/7 altitude profile model of refractive index structure parameter (see, e.g., [19]) are shown by the first group of bars.
 Several conclusions can be drawn from the comparison of the experimental with the numerical simulation data. First, the experiments showed significantly (2–4 times) larger scintillation index values. Second, contrary to the predictions indicated by the wave optics simulation results, the scintillation index for 1.06 μm laser beam (red bars) was noticeably larger than the one for the shorter wavelength beacon (0.53 μm, green bars) in almost all experiments.
 In addition, using the numerical simulation and experimental data presented in Fig. 7, we computed the ratios of intensity scintillation variances (indices) G IR
 RE , G COM
 RE , and IR COM
 RE as defined by Eq (3). In Table 1 the obtained
 from two measurement trials ratios G IR
 RE , G COM
 RE , and IR COM
 RE are compared with the corresponding theoretical
 values G IR
 RT , G COM
 RT , and IR COM
 RT obtained based on the expression (2) that describes the wavelength dependence
 Fig. 6. Evolution of the aperture-average scintillation indices 2 ( )
 σnS n (solid lines) and 2 ( , )
 WS W
 σ n n for nW = 2000
 (dashed lines) for three different wavelengths, λ1 = 0.53 μm (top, green), λ2 = 1.06 μm (middle, red), and λ3 = 1.55 μm (bottom, blue), and for "3beacon19" (left), and "3 beacon_C" (right). Corresponding intensity variance images, 2 ( )
 Nm
 σ r , are shown at the bottom.
 of the normalized intensity scintillation variance in accordance with the classical and more recent atmospheric turbulence theories. As can be seen from Table 1, the ratios of the intensity scintillation variances for different wavelengths [cf. Eq. (3)] computed using both numerical simulation and experimental data do not match those expected from theoretical calculations.
 Table 1. Ratios of scintillation indices for different wavelengths.
 G IR
 R G COM
 R IR COM
 R
 Theory [Eq. (3)] 2.24 3.50 1.56
 "numerics" 1.23 1.60 1.29
 "3beacon19" 0.64 2.11 3.31
 "3 beacon_C" 0.57 0.34 0.59
 Fig. 7. Bar diagram summarizing experimental results for a large set of experimental trials obtained during three days (nights) of measurements during COMBAT experimental campaign for intensity scintillation spatial variances 2
 SN
 σ (top) and intensity scintillation indices 2
 NS
 σ (bottom) for three different wavelengths,
 λ1 = 0.53 μm (green), λ2 = 1.06 μm (red), and λ3 = 1.55 μm (blue).
 4. SUMMARY
 Results of the COMBAT experiments make questionable the assumption that stationary and isotropic turbulence with homogeneous statistics is valid for long-range propagation paths. Statistical characteristics of the observed intensity scintillation patterns (spatial and temporal) are highly diverse and dependent on local weather conditions and global coherent structures of temperature gradient dynamics along the propagation path. In most experiments the intensity scintillation and spatial variances measured exceeded by 2–4 times the corresponding values obtained using both analytics and numerical simulations based on the classical Kolmogorov-Obukhov turbulence model.
 The observed intensity scintillation patterns for 149-km near-horizontal propagation are characterized by the increased appearance of both large (~1 m) and small (< 10 mm–5 mm) speckles. Their appearance can be associated with the presence of strong turbulent layers near the transmitter (large-scale speckles) and receiver (small-scale speckles). The frequent (0.3 s–1.0 s) appearance of large-amplitude spikes in received power indicates the existence of slowly changing/moving large-scale coherent structures with sharp refractive index changes at their boundaries.
 In all experiments the intensity scintillation patterns do not show coherent (directional) motion but rather random appearance and disappearance. As rare exceptions in windy conditions at the receiver site, we observed occasional coherent motion of small-scale speckles. These results challenge the assumption of "frozen" turbulence (Taylor hypothesis) commonly used in analyses. In addition, the existence of numerous well-developed small-scale speckles makes the possibility of active/adaptive compensation questionable.
 For the three-wavelength beacons (0.53 μm, 1.06 μm and 1.55 μm), the wavelength dependence of the intensity scintillation variance does not follow existing assumptions that the longer the wavelength the less the turbulence effect on the laser beam. Among other "anomalies" observed in the experiments are the mismatches between the predicted and the measured aperture-averaging factor and spatial autocorrelation functions (not discussed in this paper).
 5. ACKNOWLEDGEMENTS
 This project was supported through the HEL Joint Technology Office contract and the Cooperative Agreements between the US Army Research Laboratory and both the University of Dayton and University of Maryland. The authors also want to thank Dr. John Barnes (Mauna Loa Observatory/NOAA), Ms. Laura Ulibarri (AFRL Maui Branch Chief), Dr. Skip Williams (AFRL Maui Deputy Branch Chief), Bob Lytle, Kevin Moore, Ed Walker, Sarah Loney, Eddie Kanai (all Boeing, Maui) for support for the experimental campaigns.
 6. REFERENCES
 [1] V. I. Tatarskii, The Effects of the Turbulence Atmosphere on Wave Propagation. Jerusalem: Israel Program for Scientific Translations, 1971.
 [2] V. E. Zuev, Propagation of visible and infrared radiation in the atmosphere. New York: John Wiley & Sons, 1974. [3] A. Ishimaru, Wave Propagation and Scattering in Random Media. New York: Academic Press, 1978. [4] D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Optics, vol. 31, no. 15, pp. 28652882, 1992. [5] A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR, vol. 30, no. 4, pp. 299–303, 1941, [English translation in Turbulence: Classic Papers on Statistical Theory (ed. S. K. Friedlander and L. Topper), pp. 151–155. Interscience, New York, 1961]. [6] A. M. Obukhov, “On the distribution of energy in the spectrum of turbulent flow,” Dokl. Akad. Nauk SSSR, vol. 32, no. 1, pp. 22–24, 1941. [7] V. I. Tatarskii, Wave Propagation in a Turbulent Medium, ser. McGraw-Hill Series in Electrical Engineering. New York: McGraw-Hill, 1961.
 [8] S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation Through Random Media. Berlin: Springer-Verlag, 1989.
 [9] M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the Earth,” Izv. Vyssh. Uchebn. Zaved. Radiofiz., vol. 8, no. 4, pp. 711–724, 1965, [Radiophys. Quantum Electron., vol. 8, no. 4, 511–515 (1965)]. [10] W. A. Coles and R. G. Frehlich, “Simultaneous measurements of angular scattering and intensity scintillation in the atmosphere,” J. Opt. Soc. Am., vol. 72, no. 8, pp. 1042–1048, 1982. [11] R. L. Phillips and L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am., vol. 71, no. 12, pp. 1440–1445, 1981. [12] A. Consortini, F. Cochetti, J. H. Churnside, and R. J. Hill, “Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation,” J. Opt. Soc. Am. A, vol. 10, no. 11, pp. 2354–2362, 1993. [13] W. A. Bernard, B. M. Welsh, M. C. Roggemann, and R. J. Feldmann, “Atmospheric turbulence characterization of a low-altitude long horizontal path,” in Image Propagation through the Atmosphere, Proc. SPIE vol. 2828, 198–209, C. Dainty and L. R. Bissonnette, Eds. Bellingham, Wash.: SPIE, 1996. [14] N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beamwave fluctuations over a 142 km atmospheric path,” in Free-Space Laser Communications VI, Proc. SPIE vol. 6304, 63041O, A. K. Majumdar and C. C. Davis, Eds. Bellingham, Wash.: SPIE, 2006. [15] W. M. Hughes and R. B. Holmes, “Pupil-plane imager for scintillometry over long horizontal paths,” Appl. Optics, vol. 46, no. 29, pp. 7099–7109, 2007. [16] M. S. Belen'kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Preliminary experimental evidence of anisotropy of turbulence at Maui Space Surveillance Site,” in Proceedings of the 2006 AMOS Conference, S. Ryan, Ed., 2006, pp. 538–547. [17] R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE, vol. 58, no. 10, pp. 1523–1545, 1970. [18] R. J. Hill and R. G. Frehlich, “Onset of strong scintillation with application to remote sensing of turbulence inner scale,” Appl. Optics, vol. 35, no. 6, pp. 986–997, 1996. [19] J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber. Commun. Rep., vol. 3, no. 2, pp. 111–158, 2006.
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 Sentient World Simulation (SWS):
 A Continuously Running Model of the Real World
 A Concept Paper for Comments
 Government POC
 Tony Cerri
 Anthony.Cerri@je.jfcom.mil
 JFCOM J9,
 Experimentation Engineering Lead,
 757-203-3184
 FAX 757-203-3198
 Technical POC
 Dr. Alok Chaturvedi
 alok@purdue.edu
 Purdue University
 West Lafayette, IN 47907
 765-494-9048
 Version 2.0
 August 22, 2006
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 Introduction
 Modeling and simulation quickly becomes out of sync with new events, the emergence of new forces, and newly proposed theories. The goal of the Sentient World Simulation (SWS) is to build a synthetic mirror of the real world with automated continuous calibration with respect to current real-world information, such as major events, opinion polls, demographic statistics, economic reports, and shifts in trends. The ability of a synthetic model of the real world to sense, adapt, and react to real events distinguishes SWS from the traditional approach of constructing a simulation to illustrate a phenomena. Behaviors emerge in the SWS mirror world and are observed much as they are observed in the real world. Basing the synthetic world in theory in a manner that is unbiased to specific outcomes offers a unique environment in which to develop, test, and prove new perspectives.
 SWS consists of components capable of capturing new events as they occur anywhere in the world, focus on any local area of the synthetic world offers sufficient detail. In other words, the set of models that make up the synthetic environment encompass the behavior of individuals, organizations, institutions, infrastructures and geographies while simultaneously capturing the trends emerging from the interaction among entities as well as between entities and the environment. The multi-granularity detail provides a means for inserting new models of any temporal and spatial scales, or for incorporating user-supplied data at any level of granularity. Therefore, SWS can be continuously enriched and refined as new information becomes available.
 SWS consists of the following components:
 • A synthetic environment that supports Effects Based Approach and a comprehensive representation of the real world at all levels of granularity in terms of a Political, Military, Economic, Social, Informational, and Infrastructure (PMESII) framework.
 • A scalable means of integrating heterogeneous components across time and space granularities.
 • Mechanisms that discover, gather, and incorporate new knowledge into the continuously running synthetic environment.
 • A single façade of user interfaces enabling information from all sources (simulation generated data, parameters for models, and data gathered from the real world) to be searched, viewed and modified in an ontology-aware manner.
 • Integrated Development Environments (IDE)s for constructing and configuring new models or modifying existing models and then incorporating these changes into the continuously running synthetic world.
 • A means to take excursions from any point in time in the synthetic world to focus on select regions of the world, leverage private user data, or to research specific theories by simplifying the types of models to employ in the excursion.
 SWS Components
 The Core Synthetic Environment That Supports Pluralism of Thought
 The core component of SWS is an agent-based environment named the Synthetic Environment for Analysis and Simulation (SEAS), designed to be agnostic to the type of simulations and choice of models in order to allow experimentation in the context of multiple and potentially conflicting theories. No single theory can adequately explain complex behavior, such as the rise or splitting of terrorist organizations. Each theory brings another perspective to the same phenomena. Only by combining these theories within the same environment can we gain a comprehensive perspective.
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 Live Input & Knowledge Discovery
 Excursion Manager
 Analysis Visualizations
 SWS Reference World
 Analyst Interfaces & Tools Excursions
 On Ramp
 Off Ramp
 Real World Web Spiders Live Sensors
 Clustering Semantic Extraction Ontology -Based Data Integration Analytics Model Parameterization Model Development Toolkits
 Model Development Toolkits
 Synthetic World Ontological Representation PMESII Models Continuous Refinement
 Models & Data No Longer Needed
 Cross -Excursion Analysis
 Configurable Excursions Collects Initialization Data Selects Subset of Models Persists Results
 Copies of Ref . World Use Private Data Branch from Calendar Date
 Figure 1: A conceptual view of the Sentient World Simulation
 By basing the designing of the synthetic environment on a fractal representation, SWS avoids the most common pitfall faced by simulation environments: designing and optimizing an infrastructure for a parochial purpose and thereby limiting its application to a small range of problems. The fractal approach provides a consistent representation of the environment’s building blocks to models with all granularities of access. The basic building blocks consist of individuals, organizations, institutions, information, and geographies. From these building blocks, communities are built, constructed into societies, nations, and ultimately providing a synthetic world. Coarse-grained models use a small sample of the population and localized fine-grained models use a full sample of the population in an area. The same individual who experiences events from a fine-grained model can be influenced by a coarse-grained model. This approach to enabling heterogeneous models to interact is a Society of Models approach, described below.
 SEAS provides all aspects of behavior within the PMESII categorization and enables the application of Effects Based Thinking to the reference world of SWS.
 A Society Approach to Integration
 The cross-disciplinary nature of SWS fosters collaboration among diverse modeling and simulation developers. This diversity necessitates an approach to integration that does not limit developers of SWS components to adopt a uniform design, especially in terms of the temporal and spatial granularities with which the components interact with the synthetic environment. The Society Approach implemented in the SimBridge technology addresses the need to integrate components in a scalable manner while preserving heterogeneity.
 The Society Approach is applied to three areas in SWS to integrate models, simulations, and components. A Society of Models enables models to operate at diverse temporal and spatial granularities within the same synthetic environment. A Society of Models is employed by the Synthetic Environment for Analysis and Simulation (SEAS) to integrate models ranging from global, regional, and national PMESII behaviors to near real-time emotional arousal.
 A Society of Simulations captures the necessary data exchange among interacting simulations without requiring unscalable and difficult to maintain, centralized management. A Society of Simulations has been used to integrate SEAS with a tactical, kinetics federation of simulations, Joint Semi-Automated Forces (JSAF). This integration is described more fully in Figure 1.
 A Society of Systems extends the Society of Simulations to include non-simulation components, such as third party visualizations, enterprise system databases, analysis tools, and live sensor feeds.
 A Society Approach fulfills the need of integrating with the goals of:
 • independent and concurrent development of components,
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 • heterogeneous and reusable components,
 • and high fidelity of detail in a scalable system.
 Sensitivity to Current Information Through Real-Time Knowledge Discovery
 Continuously incorporating current information in SWS provides three key benefits that distinguish SWS from other initiatives at constructing a simulated environment:
 • SWS remains up to date with respect to events and emerging trends.
 • SWS leverages the prodigious amounts of data from all publicly available data sources, something that is infeasible for a small number of analysts to gather in a timely manner.
 • Models used by SWS are continuously refined, parameterized, and validated, keeping the underlying model base of SWS relevant across time.
 Burgeoning technology in the area of knowledge discovery has matured so that Web crawlers and spiders are now used in research and industry. Applying this technology to news portals, blogs, and other internet sources enables large amounts of data to be gathered and processed in a short amount of time.
 By considering all available data, automated data mining provides an unbiased means of incorporating data originating from multiple sources, and therefore, data from multiple perspectives. Additionally, interesting outliers are discovered through text, video, and transaction analytics.
 Believability and reliability metrics are applied to weight the influence of data from different sources depending on the type of source, experience with data from the source, and the type of data. The believability and reliability are then taken into account when incorporating the data into the SWS synthetic world.
 The discovery technology is coupled with a semantic engine that extracts semantics from the data. The semantics are used to prepare the gathered data for use by the simulations and to relate the data to knowledge already in the synthetic world.
 Other sources of data besides the internet are incorporated using the same knowledge discover and semantic extraction, such as proprietary, enterprise system, and classified data sources (classified and proprietary data would only be incorporated in select excursions.)
 The Ontological Representation of Knowledge
 Using an ontological repository of knowledge, SWS augments analyst knowledge with simulation semantics. The SWS ontological representation differs from traditional database approaches in that it stores ontology as well as data. Ontology is a methodology for categorizing and annotating data based on logical, human conceptualizations of what the data means. Interactions with the ontological annotations of data in the repository are fundamentally different from the interactions allowed by the prevailing RDMS and OODBMS approaches to database management. Traditional approaches coordinate tools with data by identifying the physical data in queries whereas SWS enables coordination based on the semantic meaning of data. A piece of data can be annotated with different keywords from diverse disciplines. Analysts as well as automated processes can describe data needs using keywords from their own domains, enabling crossdisciplinary integration and a unified view of both real and synthetic information. A non-ambiguous data categorization and annotation process is also needed to support the distributed and incremental process.
 Data is also tagged with its informational sources and temporal lifetime. Consequently, multiple experiments can be performed utilizing the same models combined with different points of views by sourcing data from different informational sources or conducting an experiment in the context of a specific calendar date.
 A seminal integration of a knowledge repository with an active simulation environment was successfully deployed in Urban Resolve 2015 (UR 2015). The implementation of an ontological representation of knowledge, called eXtensible Net Assessment (xNA), was used to augment analyst knowledge with
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 simulation semantics. Including xNA as a component of the prototype SWS deployed in UR2015 extends the Service Orientated Architecture (SOA) approach to provide a semantics-based integration of analyst knowledge, stored in a repository named Operational Net Assessment (ONA), with emergent simulation data. xNA supports multi-granularity and multi-disciplinary data to facilitate integration in the context of a diversity of models and data sources. xNA can also receive incremental data updates from analysts and automated data collection tools. More details on the prototype SWS are given in “JFCOM J9 Deployment of SWS 0.5” below.
 Model Development IDEs
 SWS provides a set of tools based on the ontological repository for use by experts from various domains to develop and experiment with models in the synthetic environment. A researcher begins with a set of theories to test. Using algorithms of how the theories interact with the rest of the synthetic world, the researcher creates models. The researcher designs a complete experiment environment using the new models and leveraging existing models by creating a profile consisting of a selection of parameters from the ontological data repository that fulfill the input requirements and support the experiment’s scenario.
 SEAS Integrated Development Environment (SIDE) provides the tools to create and retrofit the artificial agents in SEAS. Researchers will first design new types of agents along with the DNA and memory these types of agents will have using the design interfaces of SIDE. The design interfaces also allow the DNA and memory of existing agent types to be modified. After creating new agent types, SIDE is used to configure the population makeup of the new agents in the synthetic environment.
 After the creation of new agents, researchers use Just-in-time Modeling Environment (JIME) to create and modify the behaviors of agents. Agent behaviors can be mathematical models based on variables that the agents sense from the environment or can be described procedurally using a workflow engine, a customizable system composed of states, transitions, and messages. The behaviors describe how agents interact with their environment and other fellow agents.
 Before incorporating the new agents into the reference world of SWS, agents are prudently tested in the Bullpen tool. Using Bullpen, the behavior of agents is observed in a limited and controlled environment. Researchers create profiles of environment variables for Bullpen experiments to observe the range of decisions an agent makes and agent’s resulting actions.
 Additionally, the modeling environment of SWS is extensible using ModelNet. ModelNet is a collection of web-services that provides the functionalities of the aforementioned tools. Using the web-services, external modeling tools can be integrated into the SWS to enable researchers to use modeling tools that are best for their domain.
 Excursion Management
 SWS provides a configurable interface for configuring an excursion from the continuously running SWS reference world to meet the individual needs of a user. User needs addressed by the Excursion Manager include:
 • Exploring multiple courses of action by taking different sets of actions in identical copies of the reference world.
 • Using proprietary or classified data in a controlled experiment without interfering with the publicly accessible SWS.
 • Constructing a synthetic environment for only a portion of the world or including only certain models, simulations, tools, visualizations, or data sources.
 • Conducting simultaneous excursions in different areas of the world and merging the nonproprietary and unclassified results together.
 Once an excursion is configured, the Excursion Manager fulfills the following:
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 • Designs a Society of Simulations, consisting of the user-configured choice of simulations and components and any other components that the simulations depend on (referred to as members.)
 • Sets up shared reality, the space shared among all members that are active in the excursion.
 • For each excursion, pulls appropriate information for a specified calendar date from the ontological repository to meet the members’ input requirements.
 • Persists significant simulation results in the ontological repository, tagged with this excursion’s identifier, for later analysis and cross-excursion analysis.
 JFCOM J9 Deployment of SWS 0.5 for UR 2015
 JSAF Federation
 FAARS
 Kinetics
 CultureSim
 CBSim
 TacSim
 MARCI
 ONA External Data
 Sources Multiplayer HITL
 Ontological Structure Multi -COA Persistence Dynamic Sim . Updates
 Pull -Based Queries Time Ranges Past & Real -Time Supports Multi -COA
 SDS
 xNA
 Data Selection COA Initialization COA Persistence
 COA Manager
 Action Planner IED Planner
 Input Tools
 Shared Reality
 SimBridge
 Insurgency , IED Networks Dynamic Infra . Tree Emotional Arousal W ill - toFight Multi -Granularity
 SEAS
 Glyph Graph Viz Nexus Leader Viz
 Analysis Tools
 Figure 2: Components of SWS v0.5
 JFCOM showcased the first prototype of SWS, SWS v0.5, in the Urban Resolve 2015 (UR 2015) exercise. UR 2015 was set in a synthetic urban setting and consisted of both simulation and real-time HITL inputs. SWS 0.5 consisted of SEAS environment, a Society of Simulations and Systems implemented through SimBridge, a deployment of xNA with knowledge from a classified repository and data from research on public data, and an Excursion Manager.
 Synthetic Environment for Analysis and Simulation (SEAS)
 JFCOM has used SEAS to conduct a number of HITL experiments, providing the ability to evaluate the Effects Based Approach, in Sea Viking 05, Multinational Experiment 4, and UR 2015. Each deployment built on the previous, serving as stepping stones in the development of SWS.
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 Virtual International System (SEAS-VIS)
 The SEAS-VIS model is a representation of the Institutions, Organizations, Leaders, Individuals, and Infrastructure that make up a society. The geography of a society is modeled at various levels including City, Province, Country, Region, and World in terms of Political, Military, Economic, Social, Information, and Infrastructure (PMESII) nodes. The virtual environment integrates multiple theories from various disciplines to program behaviorally accurate agents with dynamic rules that govern and guide their actions and interactions. SEAS-VIS facilitates a seamless and interchangeable integration of human and software agents and allows for testing the efficacy of theories, decisions, strategies, and tools.
 SEAS-VIS offers the ability to do course of action (COA) analysis, test hypotheses, learn through experimentation, conduct effects-based operations and planning, and develop and refine Operational Net Assessment (ONA). The product allows the user to evaluate and manage the challenges of international crises.
 Near Real Time (SEAS-NRT)
 SEAS-NRT consists of human behavior models that are best suited to a short period of time (within hours), such as crowding and rioting, escalation of emotional arousal, and terrorist personnel collaborating in improvised explosive device attacks. Following the fractal approach to representing an environment, the SEAS-NRT models operate on the same entities that SEAS-VIS models operate on. In this way, SEASNRT and SEAS-VIS are coupled together in a Society of Models. Individuals respond to the media and world public opinion models in SEAS-VIS, placing the individual in a dynamic global context, while also enabling the individuals to sense and react to local events in real time.
 SimBridge
 Model integration within SEAS and application of SEAS to the tactical, kinetic realm, have been facilitated by SimBridge, an implementation of the Society Approach. Through SimBridge, SEAS has been integrated with Joint Semi-Automated Forces (JSAF) federations that implement the High Level Architecture (HLA) approach to simulation integration. SEAS connected to JSAF through a liaison that joined the JSAF federation as a single federate. In this way, SEAS was enabled to interact with CultureSim, CBSim, realtime HITL players, and many other components. SEAS interacts with CultureSim to represent real-time population movement and activity. SEAS with input from a Chemical Biological Simulation (CBSim) capture the influence of chemical and biological dispersions on population behavior.
 Supporting the Society of Systems approach, the SimBridge Data Service (SDS) provides a query interface through which visualizations, analysis tools, and external data base systems can leverage data residing in shared reality. For UR 2015, SDS provided a selection of statistics representing the state of SEAS models at various granularities to an external data repository, Future After Action Review System (FAARS), and to Tactical Simulation (TACSIM), modeling human intelligence gathering. TACSIM used statistics such as public opinion of Blue (one side in the multiplayer war game) to determine the willingness of the population in a city sector to provide intelligence reports.
 eXtensible Net Assessment (xNA)
 Underlying SEAS is an ontological representation of SEAS knowledge within the eXtensible Net Assessment (xNA) component. A repository populated by analysts’ understanding of Operational Net Assessment (ONA) is matched with corresponding entities in xNA, enabling SEAS to draw from classified ONA data to conduct certain classified excursions in UR2015. Also, selected types of changes that occur within the simulation run time are persisted back into xNA, creating a dynamic ONA.
 xNA represents knowledge tagged with the perspective, source, and excursion where the knowledge originates from. This enables analysis tools to perform cross-COA analysis, such as tracking the emergent growth of an insurgent organization in multiple scenarios where different actions were taken against terror cells.
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 Course of Action Manager (COA Manager)
 Users interacted with SWS 0.5 in both real-time interactions as well as for faster-than-real-time analysis. Eight SEAS environments were used simultaneously to investigate the use of different sets of action plans and to provide an unclassified interface with a select subset of events. One of the environments was placed in a Society of Simulations with JSAF to capture events in real-time.
 A COA Manager was implemented to enable each of the environments to be initialized and to enable simulation data to be persisted, tagged with a COA identifier. A specific environment could be restarted by accessing saved data for the desired COA.
 The implementation of COA Manager is a step towards an Excursion Manager needed to facilitate SWS excursions.
 Use Cases of SWS
 Training
 • SWS provides the context within which training excursions are done, presenting the training community with the unprecedented ability to train in a live and comprehensive synthetic environment that is validated by theory and up to date with the real world.
 • For skill refinement, a trainee’s skills can be directly compared with an experienced individual’s skills by immersing the trainee into the synthetic environment in the real-world context where the event actually occurred and analyzing the ensuing reactions of the simulated environment.
 • Skill acquisition is supported by the ability to illustrate the emergent effects achieved by certain doctrinal theories, and the proper contexts in which to apply them.
 • The experimentation platform provides trainers with the ability to store and recall training experiments, start another training experiment where an earlier one left off, and compare results from multiple experiments.
 Analysis
 • Analysts’ theories are continuously weighed against emergent phenomena in SWS. As SWS progresses with new real world events, the divergence between proposed theories or models indicates a need for a change in thinking. Likewise, the synthetic world may confirm the relevance of theories.
 • Cross-excursion analysis reveals the influence of “what if” excursions from a comprehensive perspective.
 • The synthetic reference world of SWS provides immediate access to a mix of details from both the real and synthetic worlds which remain up to date with respect to real world data and include details which are not easily obtainable in the real world, such as measurements of emotional arousal for a group of people.
 • The analysis capabilities of SWS also reveal relationships between entities that emerge over time. Analysts can perform full strategic, operational, and tactical net assessment with access to a dynamic description of the links between nodes and across nodes and resources.
 Planning
 • SWS provides tools to develop, reuse, and compose action plans into playbooks. A playbook database enables planners to recall playbooks.
 • Planning tools enable plans to be developed for temporally and spatially fine-grained actions as well as long-term actions and to place combined plans into a single playbook.
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 • Additionally, SWS facilitates the integration of Interagency participation in computer assisted events.
 Operations
 • The execution environment enables plans to be utilized in a scenario and decision making to be injected.
 • Augmenting real-time information with near real-time and faster than real-time simulations allows RCC to develop and test multiple courses of action to anticipate and shape behaviors of adversaries, neutrals, and partners.
 • SWS will provide the ability to develop and assess DIME campaigns.
 Experimentation
 • Users of SWS can take excursions that are configurable for the user’s unique needs, such as selectively including or excluding models, interfaces, visualizations, and data sources.
 • Users concerned with a localized region of the world can construct an experiment that uses only a subset of SWS.
 • The results from multiple excursions can be merged into a single picture using SWS’ Excursion Manager, providing cross-excursion analysis.
 • A subsequent excursion can be instantiated using the results from a previous experiment.
 • Researchers can construct experiments to evaluate new models or employ user-supplied data without interfering with the SWS reference world. The new models can be tested and refined in an experimentation environment.
 Testing
 • SWS provides an environment for testing Psychological Operations (PSYOP) and Civil Affairs activities, capable of illustrating the impact of these activities on populations.
 • Commercial users can construct experiments to use proprietary data in a controlled environment.
@@ -0,0 +1,105 @@
 arXiv:1404.4075v1 [physics.gen-ph] 15 Apr 2014
 Classical and Relativistic Derivation of the Sagnac Effect
 W. Engelhardt1, retired from:
 Max-Planck-Institut f ̈ur Plasmaphysik, D-85741 Garching, Germany
 Abstract
 Both the classical and the relativistic composition law for velocities are applied to re-calculate the Sagnac Effect. The ensuing formulae for the fringe shift are found to differ already in first order of v/c. Whilst the classical formula is validated by interferometric measurements and verified by the GPS-system, this is not the case for the relativistic result.
 Keywords: Rotating Interferometer, Sagnac Effect, Ether Theory, Special Relativity Theory (SRT), Global Positioning System (GPS)
 P.A.C.S.: 07.60.Ly, 42.81.Pa, 04.20.-q, 03.30.+p
 1 Introduction
 In 1925 Michelson and Gale built a huge earth-fixed Sagnac Interferometer in Illinois [1] demonstrating that the light velocity is anisotropic on the rotating earth. For Sagnac this result did not come as a surprise having explained the underlying effect on the basis of the ether theory in 1913 [2]. The Special Relativity Theory (SRT), however, had predicted on the basis of the Lorentz Transformation (LT) that the velocity of light is isotropic in all inertial systems [3]. Of course, the earth is not an inertial system, but due to the earth’s rotation the motion of Michelson’s device was only about 2 mm during the passage of a light beam round the interferometer. The deviation of the mirrors from a straight orbit was less than one hundredth of an atomic diameter so that non-inertial effects should be entirely negligible. Nevertheless, the linear term x v/c2 in the LT – which is responsible for c = const – was simply not there. The experimental result was consistent with the Galilei Transformation (GT), i.e. light seemed to travel at the velocity c - v from west to east and c + v on the return path from east to west, where v is the local rotational velocity of the earth’s surface. Strictly speaking, the rotating earth must be described by the General Relativity Theory, but it is hard to see how a very slow rotation should invalidate the LT and re-establish the GT as found by Michelson and Gale. Indeed, Post admits in his great review article [4]: “The search for a physically meaningful transformation for rotation is not aided in any way whatever by the principle of general space-time covariance, nor is it true that the space-time theory of gravitation plays any role in establishing physically correct transformations.” Consequently, he sticks to the classical formula valid in the ether theory making only a slight modification which is too small to be verified by experiment.
 1Home address: Fasaneriestrasse 8, D-80636 Mu ̈nchen, Germany Electronic address: wolfgangw.engelhardt@t-online.de
 In this paper we re-derive the Sagnac Effect on the basis both of the classical and of the relativistic composition law of velocities that can also be formulated for a curved light path according to [5]. Calculating the ensuing travel times of light round the interferometer we find that the LT – due to its linear term x v/c2 – does not predict any Sagnac Effect, but results in c = const also in a rotating system as it does in an inertial system. This explains then why Ashby [6], e.g., uses the Newtonian or Galilean time transformation t’ = t rather than t’ = γ (t - x v/c2) when he calculates the Sagnac Effect in the GPS-System. This was also observed by Carroll Alley in a comment at the end of an engineering presentation on GPS and Relativity [7]. In Sect. 2 we derive the classical Sagnac Effect for a circular light beam, and in Sect. 3 we do the same making use of the LT. Sect. 4 discusses the implications of our results.
 2 Sagnac Effect explained in the framework of the ether theory
 The classical transformation law between inertial systems is the GT which reads:
 ~x ′ = ~x − ~v t , t ′ = t (1)
 Although it is formulated only for a linear constant velocity between the systems, it can also be adapted for a constant rotation by deriving a composition law for circular velocities. Considering infinitesimal line elements on a circular orbit one can write:
 ds ′ = ds − v0 dt , dt ′ = dt (2)
 where v0 = Ω R is now the rotational velocity with angular frequency Ω on the circular orbit s at radius R. Dividing by the time increment on both sides of equation (2) one obtains with ds/dt = vφ the composition law for constant circular velocities:
 v′
 φ = vφ − v0 (3)
 In fact, facilitating the analysis most treatments of the Sagnac Effect consider a circular interferometer where the light beam is guided on a circle either by a large amount of tangential mirrors or by optical fibres [4], [5]. The travel time of light to complete the orbit L = 2π R around the circumference of the interferometer is then
 t± = L
 v
 φ ± v0
 =L
 c ± v0
 (4)
 depending on whether the light propagates parallel or anti-parallel to the rotational velocity. Two coherent light beams starting at the beam splitter in opposite directions return at different times leading to a fringe shift due to the time difference as calculated with (4):
 ∆t = L
 c − v0
 −L
 c + v0
 = 2 L v0
 c2 − v02
 = 2 L v0
 c2
 1
 1 − v02/c2 (5)
 The linear dependence on the rotational velocity has been confirmed by many experiments, whereas the quadratic term is too small to be measured in practice. In most textbooks result (5) is expressed by the fringe shift in units of the wavelength λ0
 ∆Z = 4 A~ · ~Ω
 c λ0
 (6)
 where, in general, the scalar product of the oriented area A~ enclosed by the light
 path with the vector angular velocity ~Ω enters. One can show that the fringe shift is
 independent of the shape of A~ and of the position of the rotational axis, but depends
 on the cosine of the angle between A~ and ~Ω. Whereas Sagnac could rotate his device in the laboratory at variable speed in order to verify formula (6), Michelson and Gale had to vary the area enclosed by the light path in order to prove the validity of (6). This experiment was important as it demonstrated that the ether is not co-rotating with the earth like air. Apparently, an “ether wind” must exist due to the rotation of the earth which leads to the observed time difference (5). Any translational motion, which surely exists, cannot be detected by the circular interferometer, as the influence of a constant translational velocity cancels on the roundtrip of the light. In this respect the Sagnac Interferometer is as insensitive as the Michelson-Morley Interferometer [8] where the enclosed area A vanishes. Although the experimental results obtained by Sagnac Interferometers are in full agreement with the ether theory, as shown above, they are generally not considered as disproving the SRT which postulates c = const and is thus at variance with the ether composition law (3). In the next Section we will show why it is commonly held that there is no discrepancy between the classical composition law (3) and the relativistic philosophy concerning the behaviour of light. The reason is a defective application of SRT.
 3 Sagnac Effect interpreted in the framework of the Special Relativity Theory
 Most textbooks assure the reader that the Sagnac Effect is a “relativistic” effect which incidentally may also be derived by the ether theory leading practically to the same result. Looking closer one finds, however, that a truly relativistic derivation is not really attempted. A good example is Post’s procedure in Sect. III [4]. He also uses the circular geometry (see Fig. 8) which our analysis in the previous Section was referring to. For the stationary observer he comes up with the time difference (7 P) which is identical with our formula (5) derived from the classical composition law for velocities. This must also hold for the co-moving observer, if Ashby’s [6] Galilean time transformation t’ = t into the rotating system is valid. Post insists, however, that the time increment between the rotating and the stationary system must transform according to his equation (11 P):
 dt = γ dt ′ , γ = 1
 /√
 1 − v2/
 c2
 which can also be found in Malykin’s paper [5]. Consequently, these authors claim that the correct relativistic time difference as measured on the rotating beam splitter must be Post’s equation (23 P) or Malykin’s (5 M) which is our classical formula (5) divided by the γ-factor. One notices that the LT was only applied in a mutilated form when (23 P) was derived, since (11 P) is at variance with the complete LT formula. A different conclusion is reached by employing the correct expression, i.e. by applying the truly relativistic composition law for velocities that cannot be obtained from (11 P).
 For the case where the light beam propagates parallel to the x-axis the LT reads:
 x′ = γ (x − v t) , t′ = γ
 (
 t− xv
 /
 c2)
 (7)
 In order to adapt these formulae to a curved light path, we consider infinitesimal time increments in analogy to equation (2) assuming a constant rotational velocity v0:
 dt′ = γ
 (
 dt − v0 ds
 /
 c2)
 (8)
 where s denotes the curved light path in the rotating interferometer. This formula can also be found in Post’s article (29 P) where he quotes papers by Langevin and Trocheris. Of course, (8) is only justified as long as the light beam propagates parallel to the rotational velocity, but this is ensured by the use of an optical fibre, for example. The spatial increment
 ds′ = γ (ds − v0 dt) (9)
 is not given by Post, nor does he calculate the composition rule resulting from division of (9) by (8):
 v′
 φ = vφ − v0
 1 − vφ v0/c2 (10)
 Although not explicitly derived, this formula can be found in Malykin’s paper [5] as the second equation (2 M). He does, however, not draw the obvious conclusion: If the phase velocity of the light in the laboratory is
 v
 φ = c (11)
 it follows from (10) that the phase velocity in the rotating system is also
 v′
 φ = c (12)
 This result does not really come as a surprise, since it only reflects Voigt’s postulate c = const from which the LT (7) was derived [3]. If this transformation holds between inertial systems, it must also hold between an inertial system and a system rotating with constant velocity as just demonstrated. The obvious consequence of Malykin’s formula (2 M), or (10) above is that coherent beams leaving the beam splitter at the same time in opposite directions will return at the same time as they both travel at the same speed c according to (12). The relativistically correct result is, therefore, not Post’s formula (7 P), or Malykins formula (5 M), but simply ∆t′ = t′ + − t′ − = L/c − L/c = 0 (13)
 In other words, the SRT correctly applied to a rotating Sagnac Interferometer does not predict the Sagnac Effect.
 4 Implications of the demonstrated results
 The Michelson-Gale experiment of 1925 had already shown that Voigt’s postulate c = const is untenable. When the GPS-system delivered more and more precise measurements based on the propagation of microwaves, it became clear that the Sagnac Effect had to be taken into account. This implied the use of the classical formulae derived in Sect. 2 on the basis of the ether theory. Ashby drew the correct conclusion and
 re-introduced the Galilean time transformation t’ = t between the inertial ECI and the rotating ECEF system [6]. From Alley’s remark at the end of [7] we conclude that the GPS-software – developed by engineers for providing precise position measurements ignores the Lorentz time transformation, following Ashby’s line instead. There are, of course, other indications voiced by Hatch [9] which suggest that the SRT should not be applied to GPS-measurements. In the light of the analysis in Sect. 3 and the irrefutable experimental results it is obvious that Voigt’s postulate of 1887 [10] does not reflect an observed physical law. It was always clear that it was invalid in the case of acoustics, and we have no reason to adopt it in optics either. As the SRT is a consequence of this postulate, its basic tenet is at stake.
 References
 [1] Albert Abraham Michelson, Henry G. Gale: The Effect of the Earth’s Rotation on the Velocity of Light II. In: The Astrophysical Journal 61 (1925) 140-145.
 [2] Georges Sagnac: Sur la preuve de la r ́ealit ́e de l’ ́ether lumineux par l’exp ́erience de l’interf ́erographe tournant. In: Comptes Rendus 157 (1913) 1410-1413.
 [3] Wolfgang Engelhardt, On the Origin of the Lorentz Transformation. In: http://arxiv.org/abs/1303.5309.
 [4] Evert Jan Post, Sagnac Effect. In: Review of Modern Physics 39 (1967) 475-493. (http://www.orgonelab.org/EtherDrift/Post1967.pdf)
 [5] G. B. Malykin, The Sagnac effect: correct and incorrect explanations. In: Physics-Uspekhi 43 (2000) 1229-1252. See also equation (1.1) in: Annemarie Schied: Helium-Neon-Ringlaser-Gyroskop. Staatsexamensarbeit, University of Ulm (2006) http://www.uni-ulm.de/fileadmin/website uni ulm/nawi.inst.220/publikationen/ LehramtAnnemarieSchied2006.pdf.
 [6] Neil Ashby: Relativity in the Global Positioning System. In: Living Rev.Relativity 6 (2003) 1. Sect. 2, eq. (3) (http://www.livingreviews.org/lrr-2003-1).
 [7] Henry F. Fliegel, Raymond S. DiEsposti: GPS and Relativity: An Engineering Overview. In: 28th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, VA, 3-5 Dec 1996, pp. 189-199 (http://tycho.usno.navy.mil/ptti/1996/Vol%2028 16.pdf).
 [8] Wolfgang Engelhardt, Phase and frequency shift in a Michelson Interferometer. Submitted for publication in: Physics Essays (2013).
 [9] Ronald R. Hatch: Clock Behavior and the Search for an Underlying Mechanism for Relativistic Phenomena. In: Proceedings of the 58th Annual Meeting of The Institute of Navigation and CIGTF 21st Guidance Test Symposium, Albuquerque, NM, June 2002, pp. 70-81 (https://www.ion.org/publications/abstract.cfm?articleID=937).
 [10] Woldemar Voigt: Ueber das Doppler’sche Princip. In: Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–Universit ̈at zu G ̈ottingen, No. 2, 10. M ̈arz 1887.
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 20th IMEKO TC4 International Symposium and 18th International Workshop on ADC Modelling and Testing Research on Electric and Electronic Measurement for the Economic Upturn Benevento, Italy, September 15-17, 2014
 The Sagnac effect in GPS, absolute simultaneity
 and the new meaning of time
 Luigi Maxmilian Caligiuri1, Amrit Sorli2
 1 Department of Chemistry and Chemical Technology, University of Calabria, Arcavacata di Rende (CS), Italy, max.caligiuri@gmail.com FoPRC – Foundation of Physics Research Center, Celico (CS), Italy, caligiuri@foprc.org 2 FoPRC – Foundation of Physics Research Center, Celico (CS), Italy, sorli@foprc.org
 Abstract – The implementation of GPS systems has allowed the understanding of important facts about the true meaning of time and simultaneity. In fact GPS has became a fundamental source for the understanding of clock behaviour and synchronization, through the considerations of the relativistic effects affecting it. Among these, a very important role is played by the Sagnac effect, influencing the clocks readings. The correct consideration of such effect is fundamental in every synchronization process where the receiver is moving with respect to the source. In this paper we show how GPS system demonstrates that the explanation of Sagnac effect given by the commonly accepted version of Special Theory of Relativity is not correct and the use of an alternative formulation based on Inertial Transformations must be used. This implies the adoption of a new synchronization procedure, the renounce to the relativity of simultaneity and a novel meaning of physical time.
 I. INTRODUCTION
 The realization of GPS systems has made it possible to develop many different technological facilities as, for example, satellite navigation system, cellular phone networks and so on [1]. Nevertheless, the use of such systems, beyond its technological usefulness, represents one of the most selective and hard tests of the Einstein’s Theory of Relativity able to give valuable insights about the true meaning of physical time and validity of a given synchronization method. On the other hand relativistic corrections themselves must be considered in order to make GPS system correctly working. As known, the GPS (Global Positioning System) is composed by 24 satellites orbiting around Earth on 6 different planes (inclined of 55° respect to equatorial plane) with a revolution period of about 12 hours. Every satellite describes a nearly circular orbit or radius r = 4R (where R is the Earth radius) at
 about 3.9 km s . Each satellite is equipped with four
 atomic clocks (two Cesium clocks and two Rubidium
 clocks) capable of a precision of about 10 ns . In addition
 to this spatial layer, there is the so – called “control layer” composed by a network of control stations at different points on the Earth surface (with “mother” station placed in Colorado). Every satellite emits omnidirectional
 signals at the two frequencies f1 = 1.5GHz (for usual
 purposes) and f2 = 1.2 GHz (for military uses). In few
 words the system operates comparing the time of emission of every signals with the time of reception, this difference, multiplied by the velocity of signal propagation in the direction satellite – receiver, would give their mutual distance and then, by means of triangulation, the receiver position on Earth surface. Obviously this method works only after the synchronization between the GPS clock and the clocks at Earth. This procedure is far from being obvious [2]. Another way by which GPS can give time is by sending a signal containing information about the time it has been sent as measured by the satellite clock; the receiver then compares the time given by its own clock with that received together with the signal of transmitter. These synchronization procedures imply the introduction of different correction factors due to the propagation delays during propagation. One of the most important one is the relativistic correction, arising from the relative motion of the source and receiver and from the gravitational effects due to Earth and Sun. There are substantially three type of relativistic effects to be considered in GPS systems: a) the effect of source velocity (motion of the GPS satellites in their orbits) and receiver velocity upon clocks; b) the effect of gravitational potential upon satellite and receiver clocks; 3) the effect due to the receiver motion upon the signal reception time (Sagnac effect). The first two effects can be explained within the Einstein’s Special Theory of Relativity (STR), although the most part of these “official” explanations are flawed and must be seriously reconsidered [2]. But the Sagnac effect has been proved to be in conflict with the STR [2]. A lot of explanation has been proposed in order to “trace back” this effect to the commonly accepted framework of
 ISBN-14: 978-92-990073-2-7 518
 relativistic physics without success [2]. It has been shown that, in order to correctly explain this effect introducing the necessary correction to GPS, the postulate of isotropy of light velocity with respect the observer in every inertial frame must be abandoned and with it the assumption of absolute synchronization. We have already shown in some previous papers [3] that a perfectly valid alternative to STR can be realized without using the postulate of isotropy of light velocity (that remains valid in a “preferred” inertial frame) considering the time and space Inertial Transformations (IT), deduce by Selleri [3]. This theory is able to simply explain the Sagnac effect (as well as all the other experimental evidences of STR) but it implies the concept of absolute simultaneity with respect a preferred isotropic inertial frame (and, consequently, the inapplicability of Einstein’s synchronization procedure). We have also demonstrated [3] this implies a new understanding of time as numerical order of change of a system. In this paper we’ll show the GPS correction for Sagnac effect can be correctly interpreted only within this novel theoretical framework and that its inevitable presence further confirms the new vision of physical time previously proposed by us.
 II. THE SAGNAC EFFECT IN GPS SYSTEM
 In his original experiment [4] performed in 1913 Sagnac split a light beam into two parts each travelling along a circumference in opposite directions. The interference fringes effect has been then measured when the two light beams were bought back together (see fig. 1).
 Fig.1. Schematic view of Sagnac experiment. Light from S is split in two parts: the first follows the path ABCDAO in the same direction of rotation, the second ADCBAO in the opposite direction, they interfere in O.
 Sagnac observed the interference fringes at different positions and measured their displacement Δs overlapping the two figures. The value of Δs was a function of rotation velocity around the circumference or, equivalently, the light velocity relative to the rotating sensor depended of whether light beam travelled towards or opposite to the rotational velocity. In particular, the displacement of fringes respectively
 occurred on the right or on the left when the rotational velocity was clockwise or anticlockwise and the entity of Δs was directly proportional to the value of rotational velocity and depended on light wavelength. In the GPS system all the time calculations are performed assuming a non rotating Earth – centred frame or reference in which light speed is considered isotropic. In this case the motion of receiver, due to the Earth rotation around its axis during the time the signal travels from the transmitter (satellite) to receiver (control stations or terminals), determines a shifting of the location of the GPS receiver during this time interval and must be carefully accounted for to ensure accurate navigation results. In fact the importance of the corresponding correction can reach even the value of hundreds of ns . The interpretation of Sagnac effect is one of the most important facts in physics because of its deep consequences on its foundations. The key question is if it can be considered as a purely relativistic effect (so perfectly explainable within the Theory of Relativity) or if it indicates the presence of a new law of Nature. So far all the attempts to explain the origin of this effect in the context of commonly accepted formulation of Theory of Relativity has failed for many reasons [2]. The problem is related to one of the most important postulate of STR namely the isotropy of light velocity (respect to the observer) in all the possible inertial frames and the consequent use of the Einstein’s synchronization procedure. In fact, it has been shown [2] that all the attempts to explain the effect by means of STR (as well as General Relativity) inevitably lead to the negation of the constant light velocity postulate and then are incompatible with STR itself. One of the most common mistakes of the above attempts is to think that the anisotropy of light velocity in the Sagnac effect is actually due to rotation and to the presence of non inertial frame in which the STR is not applicable for definition. Nevertheless, as showed by Hatch [2] studying the GPS systems, rotation is only casually involved in the process and the motion of receiver during the signal transit time must be account for independently of the cause and trajectory of motion and the path followed by the receiver during the arrival time is irrelevant. The solution to this dilemma, as we’ll see can be obtained only by adopting a radically different point of view namely the reformulation of STR without using the postulate of isotropy of light velocity and assuming the existence of a preferred inertial frame in which absolute simultaneity holds. The origin of Sagnac effect is in fact due to non-isotropic propagation of light occurring every time an observer (or a measurement instrument) is moving with respect to the unique inertial frame in which light propagation is truly isotropic i.e the preferred one, defined, as we see in the following, by the IT.
 519
 On the other hand, in the GPS system this conception is actually already applied since the rate of the satellite and receivers clocks is adjusted not as a function of their relative velocity but considering their velocity with respect the quasi – inertial non – rotating frame centred to Earth that thus de facto acts as a preferred inertial frame.
 III. THE INERTIAL TRANSORMATIONS AND THE REFORMULATION OF SPECIAL THEORY OF RELATIVITY
 It has been shown that it is possible, considering only homogeneity of empty space and time and the Relativity Principle (RP), to build “alternative” versions of Special Theory of Relativity (STR), without assuming the invariance of light velocity in vacuum [3]. This result is also related to the non – measurability of the velocity of light, independently of conventions concerning the synchronization of distant clocks [3]. This synchronization is then substantially conventional and is not necessarily related to true properties of physical reality [3] as known to Einstein itself [3]. The Lorentz Transformations (LT) realize a complete
 equivalence between an inertial frame S0 , initially
 supposed to be stationary and in which the one - way velocity of light is assumed to be invariant, and all the other infinite possible inertial systems S in relative motion, in which LT leave the one way velocity of light equal to c in all directions (“Einstein synchronization”). The system S0 then loses its “particularity” and becomes
 equivalent to any other inertial frame in the Universe. Nevertheless, the alternatives synchronization methods give rise to different space and time transformations, of which LT only represent a special case. Among these, the IT, founded by Selleri represent the best choice, since they are able to reproduce all the experimental evidences of STR [3]. If we consider two inertial frames S0 and S ,
 respectively specified by the space and time coordinates
 ()
 0 0 00
 x , y , z ,t and ( )
 x, y, z,t , in motion with relative
 velocity v along the x0 = x axis, the IT are defined by
 the following equations:
 x = x0 − βct0
 ( ) 1− β2
 y = y0
 z = z0
 t = 1 − β2t0
 (1)
 where β = v c and c is the invariant one – way velocity
 of light measured in S0 [3].
 The IT (1) implies [3]: the definition of absolute synchronization (namely two events taking place in different points of the system S but at the same time t
 are considered to be simultaneous also in the system S ' and vice versa) and the existence of a “preferred” inertial
 frame S0 . The latter is called “preferred” because it is
 that in which the first synchronization of clocks is made according to the Einstein method, and where the speed of light is isotropic and equal to c , while, in the moving frame S it is given by
 c '(θ ) = c 1 + β cos θ
 ( ) (2)
 where c ' and θ respectively are the speed of light and the polar angle in the system S .
 IV. THE EXPLANATION OF SAGNAC EFFECT AND THE NECESSITY OF NEW PICTURE OF TIME
 In 1980 the CCDS (Comité Consultatif pour la Définition de la Seconde) has stated the rules, internationally accepted, to perform clocks synchronization between different Earth locations. There are two options: the first one is to initially adjust time when the clocks are at the same point and then transport one clock very slowly to the destination point; the second option is to use an electromagnetic signal to be transmitted between the two clock locations. In any case, before the comparison between the two time readings, the codified procedure prescribes, as above discussed, the application of three correction factors: a) the relativistic
 one proportional to v2 2c2 (where v is the speed of
 clock travel) as indicated by STR; b) the “gravitational” one generated by the Earth gravitational field as indicated by General Theory of Relativity (GTR) and proportional
 to g α,h
 ( ) c2 (where g α,h
 ( ) is the total acceleration at
 latitude α and altitude h ); the last one is a correction
 proportional to 2Sω c2 , where S is the equatorial
 projection of the surface delimited by the clock / signal path and the segments connecting the two position considered (initial and final position of clocks) with the Earth centre. The third correction, usually and incorrectly ascribed to earth rotation, cannot be derived, as discussed above, within the commonly accepted framework of Theory of Relativity and represent the consequence of Sagnac effect that, as we’ll see, can be correctly interpreted through the IT, as the anisotropy of light velocity in the rotating Earth frame. In the experiment of Saburi et al. [5] an electromagnetic signal is sent from a city A (east) to a city B (west) placed on the same parallel (see fig. 2). If the two clocks are synchronized by assuming an isotropic light velocity equal to c , we should write
 tA − tB = ΔtAB = DAB c (3)
 where tA is the starting time, tB the time of arrival and
 DAB = rθ the geodetic distance between A and B.
 520
 Nevertheless, the experimental result obtained by Saburi didn’t agree with (3), so he was forced to add a correction term “at hand” in order to give the correct result and whose form is the same of that proposed by CCDS. This expression can be simply explained in terms of IT. According to IT, the one way velocity of light in the east – west direction is different from that in west – east direction and it is equal, by eq. (2) with θ = 0 , to
 cEW = c 1 + β
 ( ) (4)
 with β = ωr c .
 The correction Δt'AB to be applied to (3) is, by means
 of (4)
 Δta' b = DAB cEW (5)
 and the corrected interval time of arriving at B is
 ΔtB = Δt 'AB − ΔtAB =
 = DAB cEW − DAN c = DABβ c (6)
 that is positive, being c > cEW . We can now write (6)
 as
 ΔtB = r 2ωθ c2 = 2ωA c2 (7)
 where A = r 2θ 2 is the area of the triangle AOB in
 fig. 2. The origin and the “mysterious” form of the correction required in GPS system to account for the Sagnac effect is then very simply explained considering the alternative version of STR based on IT.
 Fig.2. Electromagnetic signal from A to B
 V. CONCLUSIONS
 We have seen the occurrence of Sagnac effect, as confirmed by GPS, implies the light velocity respect to Earth is different as a function of the propagation direction, while it results isotropy only when referred to a privileged system as, for example, with a good approximations, a geocentric non – rotating frame. If an observer placed at Earth at a given point could contemporary send two light signals around the Earth the first to East and the second to West they would came back at different times. In this case the effect cannot be determined by a invoking relative motion between source and receiver but only to a property of light that propagates with different velocity in the different directions. This phenomenon is
 obviously crucial in the synchronization procedures of the GPS control stations and it must be accounted for. Actually it is considered but only as a correction to a wrong calculation based on the isotropic light propagation. But it is conceptually and practically meaningless because it would be sufficient to consider, from the beginning, the difference in light velocity in order to automatically have a correct results, and this is just what the IT do. On the other hand a further confirmation of this fact is given by the GPS system itself because, after a first synchronization, they automatically “include” this correction giving the right time interval, since it light itself that transports this information. It is only when we try to calculate time through “standard” STR based on Lorentz transformations that the calculation result wrong, making it necessary to introduce “at hand” the Sagnac correction. A first very important consequence of this reasoning is that a correct interpretation of light behaviour can be achieved only within a theory based on absolute simultaneity (i.e. that defined with respect a preferred inertial frame) in which, as occurs in the GPS, the clocks rates are adjusted not as a function of their relative velocity but of the velocity of each of them with respect the preferred (nearly) inertial Earth – centred non rotating inertial frame. The experimental results derived from GPS then give further experimental evidences of the correctness of IT and of the need for a new formulation of Theory of Relativity based on them. It has, as already shown [3], very deep consequences not only upon the definition of simultaneity but also on the meaning of time itself that must be considered as duration, namely as a mathematical parameter giving the numerical order of physical changes. Within this picture, the correct synchronization of two clocks can be achieved only when they are synchronized in a resting “special” inertial frame and are subsequently used for the synchronization of all the other clocks located in the moving inertial frames in the instant when they meet the rest clocks. This method obviously leads to the inequality of different inertial frames among which the “preferred” one is that where the first synchronization of clocks takes place. This conceptual framework could also be invoked to successfully explain another experimental fault of Einstein’s Theory of Relativity related to GPS operation, that is the question known as “noon – midnight” problem, related to the absence in the difference between the rate of a clock, placed at a given position on Earth surface, at midday and midnight. The solution of this question, within the discussed model, will be considered in a forthcoming paper.
 REFERENCES
 [1] F.Lamonaca, A.Gasparri, E.Garone, D.Grimaldi, "Clock Synchronization in Wireless Sensor Network
 521
 with Selective Convergence Rate for Event Driven Measurement Applications" , IEEE Transaction on Instrumentation and Measurement, 2014. DOI:10.1109/TIM.2014.2304867 [2] R. Hatch, “Those scandalous clocks”, GPS Solutions, vol.8, No.2, July 2004, pp.67-73. [3] L. M. Caligiuri, A. Sorli, “Special Relativity postulated on homogeneity of space and time and on relativity principle”, Am. J. of Mod. Phys., Vol.2 No.6, November 2013, pp. 375 – 382. DOI: 10.11648/j.ajmp.20130206.25 [4] G. Sagnac, “L’éther lumineux démontre par l’effet du vent relative d’éther dans un interféromètre en rotation uniforme”, Comptes Rendus, 157,1913, pp.708-710. [5] Y.Saburi et al., IEEE Trans., IM25, 473 (1976).
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 Hafele & Keating Tests; Did They Prove Anything?
 A. G. Kelly PhD*
 * HDS Energy Ltd, Celbridge, Co. Kildare, Ireland.
 Abstract. The original test results were not published by Hafele & Keating, in their famous 1972 paper; they
 published figures that were radically different from the actual test results which are here published for the first time. An analysis of the real data shows that no credence can be given to the conclusions of Hafele & Keating.
 Key Words: cesium clocks, relativity, accuracy, drift-rate,
 1. Introduction
 Hafele and Keating (1972) [1] (hereafter referred to as H & K) carried out experiments that purported to confirm the Theory of Special Relativity. The evidence provided was derived from the differences in time recorded by cesium clocks transported in aeroplanes, Eastward and Westward, around the Earth.
 Recent university and other texts on Physics quote the tests e.g. Arfken et al (1989) [2], Beiser (1991) [3], Blatt (1992) [4], Cutnell and Johnson (1995) [5], Davies (1995) [6], Giancoli (1998) [7], Halliday et al (1997) [8] Ohanian (1989) [9] and Rindler (1991) [10]. A leader in Nature in 1972 [11] said that "the agreement between theory and experiment was most satisfactory". In the Science Citation Index their work has accumulated over 1000 references.
 H & K avoided giving the actual test results in their paper; they gave figures that were radically altered from those results. These altered results gave the impression that they were consistent with the theory. The original test results are reproduced for the first time in this paper; these do not confirm the theory. The corrections made by H & K to the raw data, are shown to be totally unjustified.
 It is also shown that the clocks used were not of sufficient stability to prove anything. The magnitude of the random alterations in performance, during the air transportation, were such as to make any result useless.
 2. Test Details
 H & K transported atomic clocks around the Earth in aeroplanes; they were sent firstly Eastward and then Westward. To minimise the effect of the variations in the Earth’s magnetic field, the clocks were triple shielded. Four clocks were employed and the average of their times was used, to lessen the effect of changes in individual drift patterns relative to the standard clock-station at Washington, D. C.. The clocks used had serial numbers 120, 361, 408 and 447.
 Figure 1 is a smoothed sketch of the graphical test results, which were in the original H & K 1972 paper. To appreciate the meandering nature of the results requires consulting that paper. The total test period was 26.5 days; the Eastward test began after ten days; the Westward test began nine days after the start of the Eastward test. The times of circumnavigation were 65.4 hours Eastward and 80.3 hours Westward. The vertical axis in
 Hafele & Keating Tests; Did They Prove Anything http://www.cartesio-episteme.net/h&kpaper.htm
 1 of 10 9/10/24, 03:25
 Figure 1 shows the difference in time, as read at hourly intervals, between each of the four clocks and that recorded by the standard station. The difference, between the average of the times recorded by the four clocks and that of the standard station, was used as the final result, and is shown as the heavy dotted line. As will be seen, H & K did not quote this average, but a radically altered version.
 The forecast was that the clocks would lose (-40ns) on the Eastward journey and gain (+275ns) on the Westward journey, relative to the standard station.
 3. Clock Stability
 The times recorded by individual portable cesium clocks diverge or converge when measured against each other, or against the more accurate average of several larger clocks at a standard ground clock-station; this in turn has drift relative to other standard clock-stations. H & K referred to the fact that cesium clocks exhibited small unpredictable changes in rate. These rate changes were typically separated by at least 2 or 3 days for "good" clocks; some clocks had no rate change in the laboratory for several months.
 Winkler et al. (1970) [12] reported that the time scale at Washington, D C (used by H & K as their standard) was obtained by averaging about 16 selected large cesium beam clocks. Clocks were replaced if their performance deteriorated. In a sample of 45 such clocks used at several stations, one failure per six clocks was experienced over two years; improvements were implemented as type faults were identified. During January 1970, three clocks had changed by +16ns, +18ns and -68ns per day. Two others were removed due to poor timekeeping, while nine had shown no change in the month.
 One of the aims of the tests was the determination of the behaviour of portable clocks on scheduled flights around the globe. The Eastward and Westward flights had 13 and 15 landings/takeoffs respectively; there were several changes of aircraft.
 For example, the Westward test brought the clocks from Washington to Dulles airport, then via Los Angeles, Honolulu, Guam, Okinawa, Hong Kong, Bangkok, Bombay, Tel Aviv, Athens, Rome, Shannon (an unscheduled fuel stop) Boston and Dulles, then by road back to the starting point.
 Prior to 1968, a procedure had been adopted whereby the more stable clocks were given more weighting in the calculation of the standard time; thereafter, the simple average of the clocks was adopted. Winkler et al. gave the standard deviation of the mean of the assembly as 2ns to 4ns when tested every 3 hours over several 5 day periods. In that station, the clocks were housed in six vaults, free from vibration, with control of temperature and humidity, elaborate power supplies, vacuum systems and signal sources and a fixed magnetic field. Beehler et al. (1965) [13] record that the accuracy of smaller portable clocks is worse, by a factor of two, than large stationary clocks; they include variations in the magnetic field among the influences that contribute to inaccuracy of cesium clocks.
 H & K claimed that they chose the four clocks because they showed a steady drift rate for at least 24 hrs before the tests. It was hoped that they would continue as a steady rate during the tests.
 4. Changes in Drift-rate (nsec/hr)
 The individual portable clocks used by H & K should have displayed a steady drift-rate, relative to the ground
 Hafele & Keating Tests; Did They Prove Anything http://www.cartesio-episteme.net/h&kpaper.htm
 2 of 10 9/10/24, 03:25
 clock-station. Three of the four clocks were so poor in this regard as to render them useless.
 The original 1971 test report, prepared by Hafele, has been obtained by this author direct from the United States Naval Observatory (USNO), [14]. It is to be wondered why H & K avoided giving the actual test results in their 1972 paper. The drift-rates given in Table 1 are those from this report written a month after the tests were completed, and four months before the H & K papers were submitted for publication. From these driftrates it is possible to analyse in detail the performance of the four clocks.
 The drift-rates before and after a test can be compared to determine the change during a test. The Hafele 1971 report said "Most people (myself included) would be reluctant to agree that the time gained by any one of these clocks is indicative of anything" and "the difference between theory and measurement is disturbing". Also, he said that, for a useful test, the drift rate of any clock should be constant over the whole period of the test. These reservations are not mentioned in the H & K 1972 paper. The relativistic effect should result in an incremental step change in the times recorded by each clock, before and after a flight test, but should not affect the drift rate.
 Table 1. Drift-rates of the clocks (ns per hour)
 Clock No 120 361 408 447
 Before the Eastward test -4.50 +2.66 -1.78 - 7.16
 After the Eastward Test -8.89 +4.38 +3.22 -8.41
 Before the Westward test -8.88 +6.89 +4.84 -7.17
 After the Westward test -4.56 +3.97 +2.16 -9.42
 Three of the clocks did not keep the same drift rate when on the ground, between the Eastward and Westward tests. There can be little confidence that the same clocks would perform far better while they were transported in passenger seats upon commercial planes.
 Clock 120 was a disaster; it had a change (Table 1) from losing 4.50 ns per hour to losing 8.89 ns per hour on the Eastward trip; on the Westward trip it altered from losing 8.88 to losing 4.56 ns per hour.
 An examination of Table 1 shows that, with the single exception of clock 447, the drift rates were so far from being steady as to render the results totally useless. The changes in drift-rates that occurred during a circumnavigation are derived from Table 1 and shown in Table 2. These alterations in drift-rate should be close to zero, to give confidence in any conclusions. An examination of the drift rates before and after the
 Eastward and Westward (Table 1) tests shows that huge differences emerged during the tests.
 Table 2. Alteration in drift-rates during tests
 Clock No 120 361 408 447
 Eastward Test -4.39 +1.72 +5.00 -1.25
 Hafele & Keating Tests; Did They Prove Anything http://www.cartesio-episteme.net/h&kpaper.htm
 3 of 10 9/10/24, 03:25
 Westward Test +4.31 -2.93 -2.68 -2.25
 The wild swings in drift rate should have resulted in the whole test being declared a failure. The authors had recognised that a steady drift rate was a prerequisite of a meaningful test. The 1971 report says "Particularly in the case of 361 after the eastbound flight, it is quite uncertain what the rate is after the flight" , and "Portable cesium clocks cannot be expected to perform as well under travelling conditions as they do in the laboratory. Our results show that changes as large as 120 nsec/day may occur during trips with clocks that have shown considerably better performance in the laboratory".
 An alteration in drift-rate during a flight could be the result of gradual alteration, or of a series of sudden changes.
 The Eastward circumnavigation time of 65.4 hours would accumulate the forecast theoretical alteration of -40ns at a change in drift-rate of -0.6ns per hour. The figures in Table 2 should, as stated above, all read zero, but any change that was not smaller than 0.6ns per hour by at least, say, a factor of at least 5 would obfuscate any result. This means that no clock was of any value in reaching any conclusion on the Eastward test; the accuracy would need to be better by two orders of magnitude. On the Westward test, the forecast change of +275ns would be accumulated in the trip time of 80.3 hours at a change in drift-rate of +3.4ns per hour. The change in drift rate would have to be less than 0.7 ns per hour to be of any use; again, no result can be used with any confidence.
 Figure 2 depicts a sketch of H & K’s enlarged view of the average of the four clocks (as given in the H & K paper) for the period immediately before and after the Westward test. A similar graph was given for the Eastward. As required by the theory, the trend lines that they drew for the periods before and after the tests ware parallel, but with an incremental step change downwards and upwards for the Eastward and Westward tests respectively. The change due to the forecast relativistic effect would, of course, occur gradually during a flight, but can only be determined as the resulting step change that emerges following that test, because there was no way to compare with the ground station during a flight.
 The starting time for the flights was determined by the departure of a commercial aeroplane. Had the Westward test begun some 12 hours earlier (see Figure 2), the trend before that test would be very different. Using the average of the four clocks, for the complete period shown by H & K on Figure 2, a time shift of approximately zero could have been reasonably deduced for the Eastward test; indeed the 1971 report described the Eastward result as "consistently negative or near zero".
 The rate changes are random and could have occurred in either a + or - direction. Clock 120 altered in driftrate by +4.39ns/h on the Eastward test and by -4.31ns/h on the Westward test; we should not say that this clock had an average drift-rate change of 0.04ns/h; indeed this was the clock with the most erratic performance. This is like saying that a watch, which gained ten hours in the first week and lost ten in the second, is a perfect timekeeper! From Figure 1, Clock 447 can be interpreted as having a small alteration in drift from 100 hours into the test period to the end of the Westward test. Had this clock, with the most steady performance, been chosen,the overall result would have been zero.
 The trend shown in Figure 2 was derived from the average of the four clocks. The results from the
 individual clocks was not disclosed; they are published here for the first time in Columns 2 and 5 of
 Hafele & Keating Tests; Did They Prove Anything http://www.cartesio-episteme.net/h&kpaper.htm
 4 of 10 9/10/24, 03:25
 Table 3. Taking the mathematical average of Columns 2 or 5 is meaningless; on the Eastward trip, clock 408
 gained 166ns, while the theory forecast a loss of 40ns; on the Westward trip clock 361 lost 44ns, while the theory forecast a gain of 275ns!
 5. Test Results
 Table 3.
 Original Test Results and H & K alterations (ns)
 Eastward Westward
 Clock Test First Second Test First Second
 No Results Change Change Results Change Change
 120 -196 -52 -57 +413 +240 +277
 361 -54 -110 -74 -44 +74 +284
 408 +166 +3 -55 +101 +209 +266
 447 -97 -56 -51 +26 +116 +266
 Average --- -54 -59 --- +160 +273
 Notes : (1) The -59ns and +273ns averages derived by H & K are to be compared with the -40ns and +275ns predicted by the theory. (2) Standard Deviations were shown for the mean of the second method as 10 and 7 respectively; the 7 should have read 9.
 The data shown as the "Test Results" in Table 3 are the ns alterations evinced by each clock, from what should be the result, at the drift-rate that was present at the start of the flight. For example, the drift rate of clock 361 before the Eastward flight was +2.66ns/hr, compared to the standard station at Washington.. The flight was 65.42 hours and, at that rate, this clock should have gone ahead of Washington by an accumulated +174ns; it had only gone +120ns, thus proving that it had lost 54ns on the flight.
 As seen in Table 3, there are two instances when a clock altered in the opposite direction to the theory; viz. 408 on the Eastward test (shows a gain, marked G in Figure 1 above) and 361 on the Westward test (shows a loss, marked L). The H & K so-called ‘result’ of the Eastward test (-59ns) was less, by a factor of 2.8, than, and opposite in sign to, the time shift of one clock (408).
 The first attempt by H & K to bring the results closer to the theoretical forecasts was to take the average of the drift rates before and after a flight, and assume that this average was the drift rate that applied throughout the flight. This is equivalent to assuming that one single sudden change in drift-rate occurred mid-way. Such an assumption would have some credence had the alteration in drift-rate been very small e.g. a change from +3.34 to +3.35ns/h, which would not significantly affect the end result. The actual drift-rates (Table 1) doubled (one clock from -4.5 to -8.9ns/h) or halved (+4.8 to +2.2ns/h; -8.9 to -4.6ns/h) or reversed (-1.8 to +3.2ns/h). The alterations made to the test results, using this first method, are seen in Columns 3 and 6 of
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 Table 3. Having made those changes, and from that produced Figure 2, H & K correctly dismissed that approach on the basis that it "depended on the unlikely chance that only one rate change occurred during each trip and that this change occurred the midpoint of the trip"; they added that there was no obvious method of estimating the experimental error of such an assumption. They had, as will be discussed later, actually identified seven alterations in drift-rate on one clock, four on another and two on a third.
 Having dismissed this method, they still published graphs (corresponding to Figure 2 above), based upon that method, which they described as producing "convincing qualitative results". It was publ;ished because it looked convincing and not because it gave a legitimate picture of the test results. To the unsuspecting reader, these graphs looked like proof of the success of the tests.
 H & K next used another method of altering the test data. It was not possible, during a flight test, to check the behaviour of clocks relative to the standard station; when a test ended, the measurement of drift, relative to the standard clock-station, resumed. However, comparisons that were made between the four clocks during flights were used by H & K, to decide whether one clock had undergone what was deemed to have been a sudden change in pattern; in such a case it was assumed that the behaviour of the other three was correct. The rationale was that "the chance that two or more clocks will change rate by the same amount in the same direction at the same time is extremely remote". Corrections were made for fourteen changes: clock 120 three changes Eastward and one Westward; clock 361 three Eastward and four Westward; clock 408 two Eastward; clock 447 one Eastward. These corrections were made after the 1971 report was produced. It might have been justifiable to ignore a single isolated sudden change on one clock during the complete 26.5 day period, but to have made corrections for fourteen such alterations in six days of flights and by amounts that exceed the forecast results by up to 5.5 times is breathtaking.
 The USNO standard station had some years previously adopted a practice of replacing at intervals whichever clock was giving the worst performance. On a similar basis, the results of Clock 120 should have been disregarded. That erratic clock had contributed all of the alteration in time on the Eastward test and 83% on the Westward test, as given in the 1971 report. Discounting this one totally unreliable clock, the results would
 have been within 5ns and 28ns of zero on the Eastward and Westward tests respectively. This is a result that could not be interpreted as proving any difference whatever between the two directions of flight.
 The actual test results (Cols. 2 & 5 of Table 3) are very different from the altered figures produced by the second method (Cols. 4 & 7). The figures in Cols. 4 and 7 were the only ones published in 1972; these give the very misleading impression that the results were compatible, as if they were all of the forecast sign and within a narrow band.
 Examples of how unreasonable were the corrections from the actual test results to the amended version are:
 clock 408 (Eastward) ‘corrected’ from +166ns to -55ns;
 clock 361 (Westward) ‘corrected’ from -44ns to +284ns.
 Clock 447 was amended from +26ns to +266ns on the Westward test; this was by a factor of 10. Yet, the H & K paper said that no significant changes in rate were found for clocks 408 and 447 during the westward trip". This barefaced manipulation of the data was outrageous. Clock 447 was the single clock that had a pretty
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 steady drift-rate throughout the tests. The 1971 report stated that "rate changes that are noticeably larger than those typical in the laboratory occurred for each clock during at least one of the trips, except for clock 447". Why then did they not use the results of this one stable clock and abandon the other three?
 On the Eastward test, corrections of +3.5 and -5.5 times the forecast theoretical -40ns result were applied to two of the clocks; on the Westward test, where the forecast was +273ns, corrections of 0.5 to 1.2 times that amount were applied to three of the clocks.
 H & K stated that the number of measured values was too small for a proper statistical analysis; nonetheless, they gave a standard deviation of the four results on either test, as quoted under Table 3. This gives the misleading impression that the results are grouped reliably closely.
 Was the aim of these tests to fabricate a confirmation of the theory, or to give objective reliable experimental results?
 6. Discussion
 Bodily and Hyatt (1967) [15] stated that 2.5ns/hr as less than changes in drift-rate and random errors in mobile clocks. The average change for the four clocks in these tests (Table 2) was 3.07ns/h. Alterations less than 200ns Eastward or 250ns Westward are therefore of no significance whatever.
 The 1971 report showed a graphical proof that any result below about 125ns could not be used on the Eastward test. Yet, H & K used the Eastward results, which were all below this threshold. That report stated that it was "amusing that values for the Eastward trip were in excellent agreement with the theory, despite expectations that they would not be able to detect any definite effect". But, this is not true; the Eastward results varied from +166 to -196ns; to average such wildly divergent results is meaningless.
 H & K recorded that previous tests, reported in 1970, had showed results that were normally distributed zerocentred and with a spread of about 60ns per day of travel. The 1971 report advocated the future use of better clocks and a circumnavigation with less ground time, which would probably reduce the standard deviation of the results by a factor of ten.
 H & K concluded that there seemed to be little basis for further arguments about whether clocks would indicate the same time after a round trip. The clocks certainly altered during the circumnavigations, but the alterations that occurred were random and have no significance.
 Earlier attempts to deduce the changes in drift rates from the graphs in the 1972 H & K paper were made by this author [16], and later found to have been done by Essen in 1977 {17]. Both concluded that the alterations in drift-rates of the clocks made the results useless. These attempts could reasonably have been discounted, on the basis that the original raw test data was not available to these authors. That excuse in now no longer valid.
 7. Conclusions
 The H & K tests prove nothing. The accuracy of the clocks would need to be two orders of magnitude better to give confidence in the results. The actual test results, which were not published, were changed by H & K
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 give the impression that they confirm the theory. Only one clock (447) had a failry steady performance over the whole test period; taking its results gives no difference for the Eastward and the Westward tests.
 References
 [1] J.C. Hafele and R.E. Keating, Science 177, 166-168 and 168-170 (1972)
 [2] G.B. Arfken, D.F. Griffing, D.C. Kelly & J. Priest, University Physics 2 ed.
 (San Diego: Harcourt Brace Jovanov) p 842, 1989
 [3] A. Beiser, Physics 5 ed (Reading: Addison-Wesley) p 747 (1991)
 [4] F.J. Blatt, Modern Physics (New York: McGraw Hill) p 25 (1992)
 [5] J.D. Cutnell and K.W. Johnson, Physics 3 ed. (N. Y.Wiley) p 909 (1995)
 [6] P.C.W. Davies About Time (London: Viking) p 57 (1995)
 [7] D.C. Giancoli, Physics 5 ed. (London: Prentice-Hall) p 755 (1998)
 [8] D. Halliday, R. Resenick and J. Merrill, Fundamentals of Physics
 (New York:Wiley) p 960 (1997)
 [9] H.C. Ohanian, Physics 2nd ed. expanded.(W. W. Norton, N.Y., 1989)
 [10] W. Rindler An Introduction to Special Relativity 2 ed. (Oxford: Clarendon
 Press) p 29 (1991)
 [11] Leader, Nature 238, 2425 (1972)
 [12] G.M.R. Winkler, R.G. Hall and D.B. Percival, Meterologia 6 No 4, 126-134 (1970)
 [13] R.E. Beehler, R.C. Mockler and J.M. Richardson, Meterologia 1 No 3, 114-131 (1965)
 [14] Proc. 3rd Dept. Def. PTTI Meeting 261-288 (1971)
 [15] L.N. Bodily and R.C. Hyatt, Hewlett Packard J. 19, No 4 12-20 (1967)
 [16] A.G. Kelly, Monograph No 3, Inst. Engrs. Irel. (1996)*
 [17] L. Essen, Creation Res. Society Quarterly Vol 14, 46, (1977)
 * Post free, Institution of Engineers of Ireland,, 22 Clyde Rd, Dublin 4, Ireland.
 List of Figure Captions
 Figure 1, Sketch of Results Given by H & K
 Figure 2. Enlarged View
 List of Tables
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 Table 1. Drift-rate of Clocks (ns per hour)
 Table 2. Alteration in Drift-Rates During Tests
 Table 3. Original Test Results and H & K alterations (ns)
 L= Loss; G = Gain.
 Figure 1
 Sketch of Results Given by H & K
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 Figure 2
 Enlarged View
 A. G. Kelly.
 HDS Energy Ltd.,
 Celbridge,
 Co. Kildare,
 IRELAND.
 French Translation of Summary
 Résumé: Les résultats de l'essai original n'ont pas été publiés par Hafele et Keating, dans leur célèbre article de 1972. Ces résultats sont maintenant disponibles et sont publiés dans cet article. On y montre que des changements radicaux apportés par les auteurs aux données d'essai n'étaient pas justifiés. Une analyse des données confirme que les conclusions d'Hafele et Keating sont fausses. On ne peut donner aucune croyance aux déductions fondées sur ces essais.
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 Nature | Vol 591 | 11 March 2021 | 225
 Article
 Measurement of gravitational coupling between millimetre-sized masses
 Tobias Westphal1 ✉, Hans Hepach1,4, Jeremias Pfaff2,4 & Markus Aspelmeyer1,2,3 ✉
 Gravity is the weakest of all known fundamental forces and poses some of the most important open questions to modern physics: it remains resistant to unification within the standard model of physics and its underlying concepts appear to be fundamentally disconnected from quantum theory1–4. Testing gravity at all scales is therefore an important experimental endeavour5–7. So far, these tests have mainly involved macroscopic masses at the kilogram scale and beyond8. Here we show gravitational coupling between two gold spheres of 1 millimetre radius, thereby entering the regime of sub-100-milligram sources of gravity. Periodic modulation of the position of the source mass allows us to perform a spatial mapping of the gravitational force. Both linear and quadratic coupling are observed as a consequence of the nonlinearity of the gravitational potential. Our results extend the parameter space of gravity measurements to small, single source masses and low gravitational field strengths. Further improvements to our methodology will enable the isolation of gravity as a coupling force for objects below the Planck mass. This work opens the way to the unexplored frontier of microscopic source masses, which will enable studies of fundamental interactions9–11 and provide a path towards exploring the quantum nature of gravity12–15.
 The last decades have seen numerous experimental confirmations of Einstein’s theory of relativity, our best working theory of gravity, by observing massive astronomical objects and their dynamics5,16. This culminated in the recent direct detection of gravitational waves from the merger of two black holes17 and the direct imaging of a supermassive black hole18. Meanwhile, Earth-bound experiments have been continuously increasing their sensitivity to gravity phenomena at laboratory scales, including general relativistic effects19,20, tests of the equivalence principle6,21, precision measurements of Newton’s constant22–24 and tests of the validity of Newton’s law at micrometre-scale distances25–27. Although test masses in such experiments span the whole range from macroscopic objects to individual quantum systems19–21,24,28, the gravitational source is typically either Earth or masses at the kilogram scale and beyond8. This is contrasted by an increasing interest to study gravitational phenomena originating from quantum states of source masses, for example, in the form of ‘quantum Cavendish experiments’1,12–14,29. Because quantum coherence is easily lost for increasing system size, it is important to isolate gravity as a coupling force for as small objects as possible. Experiments with smaller source masses are much more difficult—the gravitational force generated at a given distance by a spherical mass of radius R shrinks with R−3—and hence only few experiments have so far observed gravitational signatures of gram-scale mass configurations27,30,31. In one case, a hole pattern in a rotating, 5-cm-diameter attractor disk made from platinum generated a periodic mass modulation of a few hundred milligrams, which was resolved in a torsional balance measurement27. In another case, a single 700-mg tungsten sphere was used to resonantly excite a torsion pendulum31. Isolating gravitational interactions generated by even smaller, single source-mass objects is a challenging task because
 it requires increasing efforts to shield residual contributions from other sources of acceleration, in particular of seismic and electromagnetic nature32. In addition, resonant detection schemes, which are typically employed to amplify the signal above the readout noise, amplify displacement noise as well, and hence do not yield any gain in terms of separating the signal from other force noise sources. Here we overcome this limitation by combining time-dependent gravitational accelerations with an off-resonant detection scheme of a well balanced differential mechanical mode and independent noise estimation. In this way, we can measure the gravitational field of single source masses smaller than 100 mg.
 Experiment
 In our experiment, the gravitational source is a nearly spherical gold mass of radius R=1.07±0.04mm and mass m s=92.1±0.1mg. A similarly sized gold sphere acts as a test mass of mt = 90.7 ± 0.1 mg. The idea is that a periodic modulation of the position of the source mass generates a time-dependent gravitational potential at the location of the test mass, the acceleration of which is measured in a miniature torsion pendulum configuration (Fig. 1). The experiment is conducted in high vacuum (6×10−7mbar), which minimizes residual noise from acoustic coupling and momentum transfer of gas molecules32 (see Methods). To prevent nonlinear coupling of high-frequency vibrations into the relevant low-frequency measurement band around the modulation frequency fmod=12.7mHz, the support structure of the pendulum rests on soft, vacuum-compatible rubber feet33. We optically monitor the angular deflection of the pendulum, which provides a calibrated readout of the motion of the test mass with a
 https://doi.org/10.1038/s41586-021-03250-7
 Received: 17 September 2020
 Accepted: 18 January 2021
 Published online: 10 March 2021
 Check for updates
 1Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Vienna, Austria. 2Faculty of Physics, University of Vienna, Vienna, Austria. 3Research Platform TURIS, University of Vienna, Vienna, Austria. 4These authors contributed equally: Hans Hepach, Jeremias Pfaff. ✉e-mail: tobias.westphal@oeaw.ac.at; markus.aspelmeyer@univie.ac.at
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 detector-noise-limited sensitivity of about 2 nm Hz−1/2. Figure 2 shows an amplitude spectrum of the displacement of the test mass. The torsional mode resembles a damped harmonic oscillator with resonance frequency f0=3.59mHzandmechanicalqualityfactor Q=4.9.Itiswelldecoupledfrom otheroscillationmodesofthependulum,whichdonotappearbelow0.5Hz. Diurnal variations in the low-frequency noise floor limit the times of highest sensitivity to those hours during the night when local public transport and pedestrian and car traffic are minimized (typically between midnight and 5 a.m.). Then, the oscillation of the test mass is mainly governed by thermal noise (see Methods). The corresponding force spectrum is obtained by deconvolution of the test-mass displacement time series with the inverse of the deduced mechanical susceptibility. The spectrum exhibits a flat noise floor that allows off-resonant detection of test-mass accelerations atfrequenciesofupto0.1Hzatasensitivitybetterthan2×10−11 m s−2within half a day. We use this to obtain a live estimate of the pendulum noise conditions, which is combined with inverse-variance weighting of the data to optimize the information obtained per experimental run. The source mass is mounted on a 300-μm-diameter titanium rod that is connected to a bending piezo, which provides a travel range of more than 5 mm in the vicinity of the test mass. The geometric separation of the masses is determined with an accuracy of 20 μm by linking the drive piezo signal to position information from video tracking through a template-matching algorithm (seeMethods). During the experiment, the centre distance between source and test mass is varied between 2.5 mm and 5.8 mm, with a minimal surface distance of about 0.4 mm. To isolate gravity as a coupling force, we need to minimize other influences on the test mass. In addition to seismic and acoustic effects, these
 are predominantly electromagnetic interactions. We ground the source mass by directly connecting it to the vacuum chamber. The test mass is discharged to less than 8 × 104 elementary charges using ionized nitrogen34, a method that was developed for charge mitigation in interferometric gravitational-wave detectors17. Further shielding is provided by a 150-μm-thick conductive Faraday shield of gold-plated aluminium that is mounted between the source and test masses. In that way, electrostatic forces are suppressed to well below 3% of the expected gravitational coupling strength. Other shielding measures include housing the source-mass drive piezo inside a Faraday shield to suppress coupling via the applied electric fields, as well as gold-coating and grounding most surfaces inside the vacuum chamber to exclude excitation from time-varying charge distributions. We also rule out the presence of relevant magnetic forces by independently measuring the magnetic permeabilities of the masses. As expected, permanent magnetic dipoles are negligible in both (diamagnetic) gold and (paramagnetic) titanium. Coupling via induced magnetic moments, either from Earth’s magnetic field or from low-frequency magnetic noise originating in nearby urban tram traffic, is also found to be orders of magnitude below the expected gravitational coupling strength. In the current experimental geometry, Casimir forces are negligible, although they probably become a dominant factor for considerably smaller masses35. Another relevant noise source is Newtonian noise, which is caused by non-stationary environmental gravitational sources and cannot be shielded from. For comparison, at a centre separation of 2.5 mm the static gravitational force between our masses is expected to be 9×10−14N. The same gravitational force is exerted on the test mass by an experimenter standing at a distance of 2.5 m, or by a typical Vienna tram at 50 m distance from the laboratory building. Consequently, our experiment experiences a complex low-frequency gravitational noise of urban origin. It is obvious that such gravitational noise sources will pose an increasing challenge for future experiments. At present, our torsion pendulum is sufficiently small compared to the distance of typical environmental sources to be insensitive owing to common mode rejection.
 Results
 We observe gravitational coupling between the two masses by harmonicallymodulatingthesource-masspositionxs =d0 +dmcos(2πfmod +φ)
 3 mm
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 a
 bc
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 Fig. 1 | Experimental setup. a, Schematic of the experiment. The torsion pendulum, which acts as a transducer for gravitational acceleration, consists of two gold spheres of radius r ≈ 1 mm held at 40 mm centre distance by a glass capillary. One mass serves as a test mass (mt = 90.7 mg) and the other (ma = 91.5 mg) as counterbalance that provides vibrational-noise common-mode rejection. A 4-μm-diameter silica fibre provides a soft ( f0 ≈ 3.6 mHz) torsional resonance that is well separated from other degrees of freedom. The torsion angle is read out via an optical lever directed to a quadrant photodiode. The gravitational interaction is modulated by harmonically moving the source mass (ms = 92.1 mg) by about 3 mm at a frequency of fmod = 12.7 mHz, well above the torsion resonance, using a shear piezo. Direct electrostatic coupling is suppressed by discharging and by a 150-μm-thick Faraday shield. b, The source mass on a 1 euro cent coin. c, Photograph of the torsion pendulum and the mounted source mass.
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 Fig. 2 | Spectrum of a single measurement run. The rotation of the torsional oscillator, measured over a period of 13.5 h, is calibrated via the test-mass displacement (blue solid line) and the corresponding exerted force (red solid line). The measurement is limited by both thermal noise (dashed line) and diurnally varying white force noise below 100 mHz, whereas white displacement readout noise (dotted line) dominates above 100 mHz. The torsional motion is well decoupled from other oscillator modes, starting at 0.5 Hz. Force residuals (yellow solid line) represent the difference between measured and expected gravitational force. The spectrum shows gravitational coupling at a modulation frequency of fmod = 12.7 mHz and nonlinearity of the gravitational potential at 2fmod = 25.4 mHz, well above the resonance frequency f0 = 3.59 mHz of the oscillator. The flatness of the force residuals indicates that both signals are in agreement with the expected gravitational signal. Other sources of nonlinearity are found to be negligible (see Methods).
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 at a frequency of fmod=12.7mHz, well above the fundamental torsional pendulum resonance (mean centre-of-mass distance, d0; modulation amplitude, dm). In this frequency regime, the test-mass response becomes independent of the pendulum properties and behaves essentially as a free mass. According to Newton’s law, the source mass generates a gravitational acceleration of the test mass of a = Gm /x (t)
 G ss
 2
 (G, Newton’s constant). The 1/r dependence of the gravitational potential results in higher-order contributions to the coupling at multiples of the modulation frequency fmod. Figure 2 shows the measured force spectrum of the test mass for a distance modulation of dm ≈ 1.6 mm at a centre separation of d0≈4.2mm, where we resolve the linear and quad ratic acceleration modulations at fmod and 2fmod, respectively. The flat noise residuals clearly indicate that the observed peak heights agree well with the expected force modulation due to Newtonian gravity. This confirms the gravitational origin of the interaction. To more accurately quantify the strength of the coupling, we correlate the measured separation xs between source and test mass with the independently inferred force on the test mass. This provides us with a position-dependent mapping of the gravitational force (Fig. 3). All data evaluation is carried out in post-processing and using non-causal zero-phase filtering (see Methods). Each dataset, consisting of up to 13.7-h-long measurements, is fitted using the expression for Newtonian gravity to obtain a value for the measured coupling strength. Figure 4 summarizes the results of a series of measurements that were taken during the seismically quiet Christmas season. The weighted combination of measurement runs yields a coupling constant of Gcomb = (6.04 ± 0.06) × 10−11 m3 kg−1 s−2, with a statistical uncertainty of 1% (all errors reported are 1 s.d.). The observed coupling deviates from the recommended CODATA value for Newton’s constant (GCODATA = 6.67430(15) × 10−11 m3 kg−1 s−2) by around 9%. This offset is fully covered by the known systematic uncertainties in our experiment, which include unwanted electrostatic, magnetic and gravitational
 influences from the masses and supports, as well as geometric uncertainties in the centre-of-mass distance due to the actual shape of the masses (we provide a detailed list in Methods). Our results show that we are able to isolate the gravity of a single, small source mass, with other influences being below the 10% level. The small statistical error underlines the precision character of our measurement and our ability to measure even smaller source masses with our approach in the future.
 Discussion and outlook
 We have demonstrated gravitational coupling between a test mass and a 90-mg spherical source mass. To our knowledge, this is the smallest single object whose gravitational field has been measured. Our measurements resolve an acceleration modulation of about 3 × 10−10 m s−2 at the drive frequency fmod with high accuracy, quantified by a systematic uncertainty of around 3 × 10−11 m s−2, and high precision, quantified by a statistical uncertainty of about 3 × 10−12 m s−2 over an integration time of 350 h. Contributions of non-gravitational forces were kept below 10% of the observed signal. Our results extend gravity experiments to the regime of small gravitational source masses, with the potential of going considerably smaller. Given that our current sensitivity is limited by the thermal noise floor of the torsion pendulum when operated under optimal ambient conditions, it can be readily improved by increasing the mechanical quality factor Q. Probing the gravity of objects even smaller than the Planck mass (mP ≈ 22 μg = 2.4 × 10−4ms) should become possible for Q≳20,000 (seeMethods). The thermal-noise-limited force sensitivity of such miniature torsion pendula can be improved even further by dissipation dilution, as recently demonstrated36. Another interesting route is to use levitated test masses, which provide almost perfect charge mitigation and similar acceleration sensitivities37–41. To make full use of improvements in thermal noise, it is important to understand and to mitigate other, predominantly anthropogenic, low-frequency noise sources that currently dominate our measurement performance most of the time. Possible countermeasures include the appropriate choice of a remote laboratory location or pushing the experiment to much higher pendulum frequencies32,42. Going to smaller masses will involve additional challenges, because other noise contributions will play an increasing role. For example, even for the ideal case of electrically neutral source and test
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 Measured modulation force (fN)
 Fig. 3 | Spatial mapping of the gravitational force. a, The probability density distribution of the exerted force versus source–test mass separation shows the spatial nonlinearity of the source–mass potential. This method makes full use of our measurement data by taking into account weighting with the current noise conditions. The apparent width of the distribution is mainly given by noise at frequencies other than that of the modulation (a.c. force). b, The actual measurement precision becomes apparent by extracting the mean of the force measured at given distances (red points) along with their standard deviation obtained by bootstrapping (pink band). We note that the small statistical error collapses the error band to a narrow pink line. A full-weighted fit of the 13.5-h-long data segment with Newtonian gravity (black line) yields Gfit = (5.89 ± 0.20) × 10−11 m3 kg−1 s−2, which is in agreement with our long-term combined value (see Fig. 4). Comparison with the literature (CODATA) value GCODATA = 6.67 × 10−11 m3 kg−1 s−2 (dashed line) and our known systematic errors shows that influences other than gravity are below 10%.
 24 December 31 December 7 January 14 January
 6
 8
 Coupling strength
 (×10–11 m3 kg−1 s−2)
 Run 1 Run 2 Run 3 Ruunn 4 Run 5 Run 6
 CODATA valluuue
 Weighted mean Weighted s.d. Systematic uncertainty
 Fig. 4 | Collection of measurement runs. Over a stretch of three weeks during the seismically quiet Christmas season, a total of 29 measurement runs were conducted, with electrostatic coupling being characterized during periods shaded in grey. The dataset presented in Figs. 2, 3 is highlighted in black (run 1). The statistical error (1σ error bar) of the best-fitting Newton-like coupling of each data segment (dark lines) varies because of highly non-stationary noise in the measurement band. Combining individual measurements weighted by their respective data quality yields Gcomb = (6.04 ± 0.06) × 10−11 m3 kg−1 s−2. The plotted systematic uncertainty (red dashed line) includes the identified systematic influences from the experiment as summarized in Extended Data Table 2. We explicitly do not claim a significant systematic deviation of the combined coupling value from the CODATA recommended value (blue dashed). The experiment and its data evaluation was mainly designed with the reduction of statistical errors in mind, and the deviation is fully covered by the known experimental systematics.
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 masses, Casimir forces will become relevant at distances below 100μm. We note, however, that electromagnetic shielding combined with signal modulation can overcome this particular challenge. We also note that our method allowed us to directly measure the position dependence of the gravitational force, which constitutes a one-dimensional mapping of the gravitational field strength. A more complex test-mass geometry may enable full mapping of the metric tensor of such small source masses43. Our experiment provides a viable path to enter and explore a regime of gravitational physics that involves precision tests of gravity with isolated microscopic source masses at or below the Planck mass. This opens up possibilities such as a different approach to determine Newton’s constant32, which so far remains the least well determined of the fundamental constants44. In general, miniaturized precision experiments may allow tests of the inverse-square law of gravity at considerably smaller scales than possible today (the best reported value is of the order of 50 μm)27, thereby probing, for example, fundamental features of string theory such as extra dimensions and massless scalar particles45. Small source masses also allow us to test the consequences of speculative scalar fields that have been discussed in the context of dark energy9–11. More stringent parameter constraints for so-called chameleon forces may already be possible for source masses at the level investigated here9. Another example is experimental access to gravitational accelerations at or even below the level of galaxy rotations. Modification of Newtonian dynamics in this regime has been suggested as an alternative to dark-matter scenarios46, and some of these models can therefore be directly tested47. Finally, the ongoing discussion on the quantum nature of gravity has revived a gedankenexperiment by Feynman to probe gravitational coupling between quantum systems12–14. The underlying question is whether it is possible to probe gravitational phenomena that cannot be explained by a purely classical source-mass configuration1. Such experimental tests have so far been impossible, because they require the ability to prepare quantum states of motion of the source mass. Because decoherence phenomena scale markedly with system size, isolating the gravity of microscopic masses is a prerequisite for such future experiments. Our result provides a first step in this direction—although we stress that, with current quantum experiments using sub-micrometre-sized objects of 10−17kg (refs. 48,49), adding quantum coherence to experiments in the relevant mass regime remains a completely different experimental challenge.
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 Methods
 Torsion pendulum
 Geometry. The torsion pendulum consists of the test mass (mt=90.7mg) and a counterbalance mass ( ma=91.5mg). The masses are rigidly connected by a 37.9-mm-long, square-shaped glass capillary with an outer dimension of 0.4mm and wall thickness of 0.1mm (9.6mg). It is supported by a 35-mm-long quartz fibre pulled down to 3.6μm diameter over a 20-mm section at the centre for maximal torsional compliance. The 5–10-mm-long, up to 50-μm-wide taper region below this torsion section increases the pitch and roll stiffness for better mode separation and provides a bigger cross-section for attachment to the capillary. The attachment point is tuned to minimize static roll to <10 mrad. A 5.85-mm-wide, 0.3-mm-thick, hand-ground and gold-coated mirror (27.0 mg) is attached underneath the capillary for deflection readout of the torsion (yaw) angle. This turns to point downwards by 50 mrad in pitch, potentially coupling the non-torsion motion—in particular roll—into the yaw readout as well. All connections are established by an ultraviolet-curing adhesive. The fundamental mode frequency f0 is determined by the stiffness of the suspension fibre and the effective mass meff = 183.8 mg ≈ mt + ma = 182.2 mg. The effective mass is almost minimal, that is, hugely dominated by the two gold masses for the balanced, differential mode used here. It can only be reduced further by using a single mass in a non-differential setup—which would then, however, be subject to the full environmental translational acceleration.
 Pendulum characterization. Because we aim to characterize the gravitational force exerted by our source mass, we quantify the test-mass motion by its spatial displacement rather than by the pendulum deflection angle. The amplitude spectrum of the test-mass displacement (shown in Fig. 2) follows the slope of the mechanical susceptibility of a damped harmonic oscillator, which is solely defined by its eigenfrequency f0, mechanical quality factor Q and mass meff. (Owing to the low frequency of the observation and the extremely good frequency separation of the mechanical modes of a torsion pendulum in general, all transfer functions except the mechanical susceptibility of the yaw mode can be neglected.) The eigenfrequency and mechanical quality factor were obtained by a fit to the spectrum as f0 = 3.59 mHz and Q = 4.9. Both parameters were verified during measurement runs and independently by a ring-down measurement with relatively small amplitude. For larger amplitudes, we observe a sizable decrease in mechanical losses. Values in excess of Q = 2 × 104 were observed for larger amplitudes (~π rad) and different rest positions. The effective mass was determined by weighing and modelling all components except for the small amounts of glue that were used. The drive frequency fmod was chosen to lie well above the torsion resonance (~3.6 mHz), but far away from other, highly excited modes (for example, around 500 mHz and 700 mHz) and sufficiently below the readout noise (~100 mHz), such that the 2fmod (and eventually the 3fmod) signal could be resolved. The reason for choosing 12.7mHz is to prevent frequency mixing in nonlinear systems, such as discretization and clipping of narrow-band signals, to accidentally match with the gravitational signal. By choosing the signal frequency in such a way, the test mass effectively behaves like a free mass (meff) (more precisely, if the signal frequency is well above the mechanical linewidth, γ0/(2π) = f0/Q, that is, fmod ≫ γ/(2π) ≈ 0.7 mHz). Hence, in this regime the response is mostly independent of the resonant enhancement factor Q and the resonance frequency f0.
 Prospects towards smaller masses. The experiment presented here verifies our approach to isolating gravitational coupling of 100-mg-sized objects. The achieved signal-to-noise ratio (SNR) of 6.04/0.06 ≈ 100 already allows measuring the gravity exerted by 1-mg-sized objects with SNR≈1 for a measurement duration of ~150h.
 We note that our current measurement runs undergo diurnal variations in the noise floor, to which the thermal noise of our Q = 4.9 oscillator assembly poses a hard limit. To make better use of our measurement time (SNR∝ Tmeas), we must improve our understanding and reduce the coupling of daytime environmental noise. Furthermore, mechanical quality values in excess of Q = 2 × 104 have been observed even for this particular pendulum. Because the off-resonant thermal noise amplitude decreases with 1/ Q , this provides a factor of 20,000/5 ≈ 65 improvement over the current sensitivity. Combined, these aspects show the potential of our approach for sensing the gravitational field of a Planck-mass-sized object (mP = 22 μg).
 Calibration
 Optical-lever readout. The angular position of the torsion balance is read out by means of an optical lever using 1.8 mW of laser light (Innolight Mephisto) with an angle of incidence of ~50 mrad. The signal is obtained as the horizontal differential photo-voltage Vx of an off-the-shelf quadrant photodetector (QPD; Thorlabs QPD80A; sum voltage Vsum) with a high dynamic range (0.2–0.5Vx/Vsum) and low noise (~2 × 10−9Vx/Vsum Hz−1/2) in its approximately linear region of 0.5 mrad. The data are sampled by a Picoscope 4824 oscilloscope at 250 and 286 samples per second (anti-aliasing filter at 132 Hz) and digitally downsampled by a factor of 10. The analogue-to-digical converter (ADC) provides 12-bit resolution (±10 V full scale, 4.9 mV resolution). Although we oversample the signal, the high-frequency quantization noise is improved when using even higher data rates. Reducing the full range would also help, but we require it to cover large low-frequency amplitudes, in particular during noisy times of data acquisition. For future experiments, the actual (not quantization-noise-limited) readout sensitivity can be improved by using interferometric methods, although at the cost of limited absolute range. Systematic calibration errors can be substantially reduced by exchanging the current readout with a perfectly linear, self-calibrating auto-collimator at the expense of increased complexity and limited oversampling capability50.
 Position calibration. Position calibration of the test-mass motion was carried out by measuring the pendulum yaw deflection both via the optical lever and, independently, via video tracking. For the latter measurement, the torsion pendulum was recorded from the top with a resolution of 24 μm per pixel, with both gold spheres being pattern-tracked to obtain the deflection (compare Methods section ‘Distance measurement’). Each method contributes its own error: for small amplitudes, the video-tracked deflection angle is relatively noisy, whereas at large deflections the optical transfer function of the QPD readout becomes nonlinear. For our calibration, the pendulum was excited to an amplitude of ~125 μm, largely exceeding the linear detection range. In the ideal case, the rotation angle and the normalized QPD signal are related via the Gauss error function. In our case, clipping and other effects degrade this relation at large amplitudes. Because our experiment is confined to small deflections, we require a good calibration only in the central QPD range. Therefore, a 3rd-order polynomial was fitted to the data surrounding Vx/Vsum≈0.09 to serve as yaw angle calibration for all our QPD data.
 Force calibration and force noise estimate. A major feature of off-resonant detection is in our case the flatness of the force noise floor around the signal. This allows us to obtain a live estimate of the current noise conditions without additional sensors. Known narrow-band signals, such as fmod and harmonics, are removed from the inferred force signal, and the instantaneous amplitude in the 5–75mHz band is obtained by the absolute value of the signal plus its Hilbert transform. Specifically, evaluating amplitude (real part of complex signal) and change of amplitude (imaginary part) gives a continuous estimate for the noise coupling into the system. The goal of the algorithm is to
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 identify excess excitation segments of the data that occur over a long time, in particular to minimize the data degradation from the sometimes up-to-tenfold increased noise conditions during daytime. We therefore low-pass-filter (moving average, 1/fmod≈79s) the instantane ous amplitude and use the result for inverse-variance weighting of the measurement data. Compared to completely vetoing data during noisy conditions, this allows to make use of the full measurement time under these extremely non-stationary conditions. By employing this method, we could optimally combine a total of 350h of non-stationary measurement time, which is worth ~150 h of higher-quality data, as presented in Figs. 2, 3, into the single evaluation shown in Fig. 4.
 Non-stationary environmental conditions. Each data chunk contains up to 13h53min of data (5×10 4s). To display the time evolution of our noise, we perform a spectral density estimate on 1-h segments using Welch’s method with 20-min-long segments and 50% overlap (Extended Data Fig. 2). (We note that the spectral resolution is inversely proportional to the integration time and therefore worse for the segmented spectra than in Fig. 2. In addition, the gravitational signal is smaller for these short segments because less of the signal is integrated. If we choose the amplitude spectrum representation instead, our signal will remain at the same height but the noise floor will rise, because less noise is averaged out.) The low-frequency noise (below ~100mHz) is dominated by the actual motion of our torsion pendulum. Although we reach the theoretical thermal noise limit of the damped harmonic oscillator during quiet times, we observe up to 10 times higher noise levels during other times of the day. The origin of this noise has not been fully identified yet (see also Methods sections ‘Seismic noise’ and ‘Magnetic interaction’ for more details). However, it always exhibits the same pattern: within 1-h segments, the noise appears to be white in terms of the force exerted onto the test mass, but its level varies slowly. The noise power is well correlated with the diurnal variability of anthropogenic noise, be it in higher-frequency seismic activity (evidenced by the excitation of the 0.7-Hz mode) or in 10-mHz magnetic fields. Nights are better than daytimes, even with nobody being present in the laboratory. Weekdays present worse conditions compared to weekends and public holidays. Therefore, the Christmas 2019 time was exceptionally well suited for the low-noise measurements presented here. At higher frequencies, above ~100mHz, the noise flattens out (except for higher-order modes) at a readout noise level of 2 × 10−9 m Hz−1/2. Because this noise further improves with increased sampling frequencies, it is suspected to be quantization noise.
 Data evaluation
 Data handling. All data are sampled by a Picoscope 4824 at 250 or 286 samples per second using the internal clock. The ADC provides 12-bit resolution (±10 V full scale, 4.9 mV resolution). A typical data trace spans 5×104s, just short of 14h. An analogue anti-aliasing filter at 132Hz is applied to the horizontal difference of the photodiode quandrants to remove high 50-Hz harmonics (room light) and other narrow-band large-amplitude signals. The QPD signals are digitally de-rotated by ~20 rad, which maximizes the sensitivity to the torsion motion while minimizing the impact of pitch motion (vertical spot motion). The calibration procedure described earlier provides a signal corresponding to test-mass displacement. From this, the force signal is deduced as described below. At this point, the momentary noise estimation signal is deduced, processed with the other channels. Only after that, the signals are digitally down-sampled by a factor of 10 (a 10th-order Chebyshev type 1 low-pass filter with a cut-off frequency 0.8 times the new, reduced, sampling frequency fs,new, with the response at fmod normalized to unity, is applied before the decimation) to reduce the amount of data. The initial oversampling reduces the digitization noise (optimally by a factor of 10). Finally, the signals are cropped—that is, usually the first 5% are removed. Besides eliminating general edge
 effects of any filters, this mainly reduces the ring-in effect of the force estimation.
 Force inference. In general, a damped harmonic oscillator can be represented by a 2nd-order low-pass filter. The mechanical susceptibility χ(ω) is defined as the transfer function from the externally applied force F (ω)
 ∼ext to the induced displacement x∼(ω),
 ∼ ∼
 χω xω
 ( ) = F ( ω) m ω γω ω
 ( ) = 1 (− − i + ) . (1)
 ext eff
 2 0
 2 −1
 At a given frequency ω, χ is defined by the (viscous) damping rate γ = ω0/Q, the (un-damped) resonance frequency ω0 and the effective mass meff. We map this to a zero-pole gain (zpk) representation
 χ = k(s − p )(s − p* ) (2)
 !
 00
 (where the notation != stands for ‘must be mapped to’) in the Laplace domain (s =a + ib, where a and b are real numbers and we choose a = 0) by means of a pair of complex conjugate poles (p , p*
 0 0) and a d.c.
 response k (which contains the unit conversion). We find
 k m p γ ω γω
 = 1 , = 2 ± i − 2 . (3)
 eff 0 0
 20
 As long as the zpk representation of a system is known, then the time-domain response x(t) can be inferred from the driving force Fext(t) without solving the equation of motion, by the following recipe; for example, using the MATLAB built-in functions zp2tf(∙), bilinear(∙) and filtfilt(∙):
 1. convert the zpk model with zeros z and poles p to its transfer function form defined by the numerator num and the denominator den by [den, num] = zp2tf(z, p, k) (appropriate k must be chosen to take into account the re-normalization by placing poles/zeros), 2. convert a continuous time model to filter the coefficients bd, ad of a discrete time model sampled at frequency fs by [bd,ad] = bilinear(den, num, fs), 3. apply the filter by x = filtfilt(bd, ad, F). Whereas steps 1 and 2 are more or less technical re-definitions, step 3 convolves the force time series with the impulse response in a very resource-friendly manner. We use this method in reverse by inverting the mechanical susceptibility, which is equivalent to exchanging poles and zeros (p → z′
 0 0, z → p′
 00
 and k → 1/k′) to infer the force Fext required to produce the measured displacement x(t). To avoid an unphysical (infinite) response at or above the Nyquist frequency fN = fs/2 from the two zeros, we must limit the response by a pair of compensation poles. By allowing them to be complex as well, their phase loss around the detection band can be limited. In step 2, we apply pre-warping at 100mHz to reduce malicious effects onto the transfer function. The compensation poles are chosen to lie at ωcomp=2π×1Hz with Qcomp=3. This choice is verified not to influence the measured gravitational signal considerably (compensation pole in Extended Data Table 2).
 Data visualization. The spectrum estimate shown in Fig. 2 was produced by Welch’s method (pwelch(∙)). Each time the series is divided into five blocks with 50% overlap and tapered by a Hanning window. The spectrum estimates of the blocks are then averaged. Hence, the noise floor in the figure is elevated but smoothed out compared to that achieved by performing a single spectral estimate on the whole trace at once. The residuals shown in the plot are a spectral estimate of the difference of the measured force signal (red trace) and the expected/ nominal gravity signal. By performing the subtraction before the spectral (power) estimate, these residuals become sensitive to the relative phase of the two signals.
 Because our goal is to quantify a (nonlinear, but stationary) relation between the source–test mass separation and the exerted force, plotting one against the other comes with the implicit assumption of equal significance for each data point. However, because our momentary noise estimation channel enables us to assess the significance (in terms of SNR) of each point, we chose to weight the data points in Fig. 3 with this significance to obtain the most general visual representation of the force versus separation relation. The hidden assumption here is that the momentary noise level estimation procedure does not introduce systematics into the quantitative data evaluation (see next step). We tried to verify this by checking that there is no quantitative dependence of the results on noise-estimation filter parameters. Only after this most general data representation, the relation is subjected to assumptions using Newton’s law to quantify the relation. Hence, the width of the signal band is determined by all noise within the bandwidth of evaluation (5–75 mHz bandpass; see above) and not only at the signal frequency or its harmonics. Although subjecting data observed in dedicated high-precision experiments to the rigid corset of assumptions (fitting parameters) is a well established method, we believe the least-assumptions data representation approach of Fig. 3 to be extremely valuable in the yet-unexplored regime of gravitational sensing.
 Data evaluation. To quantify the measured gravitational force, we assume the validity of Newton’s law and treat the masses as point masses concentrated at their respective centres of mass. Strictly speaking, this assumption is valid only for perfectly rotationally symmetric masses. Possible deviations due to bumpiness are considered as systematic errors. We use the gravitational constant as a fitting parameter resembling the strength of 1/xs
 2 coupling. We further allow for a distance-independent force resembling the measured force averaged over one signal period, given that the experiment was not designed to measure the d.c. attraction, that is, the force with relation to the source mass being infinitely far away. The statistical uncertainty of the data can be inferred by resampling. While often bootstrapping is used, it turns out that without further pre-conditioning, high-frequency temporal correlations such as high-frequency vibrational modes spoil the results as the data points cannot be regarded as sufficiently independent. Subsampling, by contrast, holds under weaker assumptions and appears to work for our data because short-term correlations remain within the same segment. Each time trace is cut into N equally long segments. For each segment, the coupling strength Gi is fitted with the inverse-variance weight factors described in the previous section in non-stationary environmental conditions. Furthermore, a common force offset F0 is fitted to all segments (allowing each segment to have its own d.c. force F0,i leads to similar results, but is less physical, because this force should not drift). The weighted mean G = ∑ G /σ
 ii i
 2 , where 1/σi
 2 are the normalized inverse-variance weights for the segment, gives the same result as a weighted fit to the whole trace. From the (inverse-variance-weighted) standard deviation σw(Gi) we infer the statistical uncertainty for each Gi. The uncertainty for the whole trace is then inferred by means of σ(G) = σ (G )/ N
 w i eff, where Neff is the effective number of segments with independent noise contributing to the weighted standard deviation. The number N of segments per trace is varied between 11 and 30. For small N, drifts within the segments and non-stationarity of noise within the segments may become sizable, whereas for large N the exact resonance frequency and quality factor of the torsion resonance gain importance as the segment length approaches a torsion period. To exclude such effects, for each trace the results are verified to not depend on N, in order to avoid introducing systematic errors by its exact choice, and the fit parameters are averaged over the full range of segmentations. This evaluation gives the independent estimate of the coupling constant and an estimate of the statistical uncertainty for each trace presented in Fig. 4. All deduced coupling constants agree well within
 their statistical error budgets. A coupling constant with even lower statistical uncertainty is obtained by averaging the coupling constants of all traces while using the respective deduced statistical uncertainties as inverse-variance weights. The value of the time-resolved representation of Fig. 4 is that it shows that there are no major (monotonous) time-dependent systematics such as adsorption and continuous electrical charging.
 Vacuum
 The experiment was conducted in high vacuum (10−6–10−7 hPa) to reduce excitation from residual gas molecules and direct momentum transfer. The vacuum chamber is pumped by a turbo molecular pump (Pfeiffer TMU 071 P) running at 25Hz during measurement runs. These narrow-band vibrations were not observed to produce sensitivity degradation of the gravitational signal. The pressure could not be monitored permanently during measurement runs, because the Pirani pressure gauge (Pfeiffer PKR 251) needed to be detached from the chamber; it was observed to constantly charge the torsion balance when in use, and its relatively strong magnetic field was suspected to cause magnetic interaction between the gold masses. The effects of residual gas molecules cause a Brownian force noise that is derived as an additional damping rate51
 γP
 mv r r
 d
 =π
 2 ln + 1 (4)
 r d
 wall air
 2
 3/2 2
 22
 2 
  
  
 (P, pressure; vair = kBT/mair, average velocity of ‘air molecules’ of mass mair; d, separation between the surface of the cylinder and the electromagnetic shield; r, cylinder radius; m, cylinder mass; kB, Boltzmann constant; T, temperature). The additional force noise damping due to the proximity to the electromagnetic shield is taken into account. Moreover, we assume the worst-case scenario of a cylindrical test mass, which would trap considerably more gas molecules compared to a sphere of similar size close to the wall. Using our experimental parameters (P = 6 × 10 −7 mbar, mair = 4.8 × 10 −26 kg, r = 1 mm, m = 91 mg, d = 150 μm,T=300K), we obtain an additional damping of γ wall ≈ 5.1 × 10−8, which is much smaller than the natural damping rate, γ = ω /Q ≈ 5.0 × 10
 m0
 −3
 t , of our test-mass oscillator. Looking ahead, even
 for a projected experiment in the Planck-mass regime, which would require a mechanical quality factor of Q > 20,000, the effect of the force noise from gas collisions would be at the 5% level in this pressure regime—and can be further reduced by pushing vacuum levels into the ultrahigh-vacuum range.
 Vibration isolation
 The vacuum chamber is placed on an optical table with deflated air springs. Although inflating the springs did attenuate high-frequency noise, as expected52, the pendulum yaw mode, by contrast, was observed to get heavily excited. We suspect a relation to the table tilt, which we independently observed to couple strongly into the experiment (3rad of yaw motion per 1rad of tilt motion). In the inflated state, air density fluctuations act differently onto the air springs and can thereby cause excessive table tilt at low frequencies, which could explain the excessive yaw excitation. Operating our experiment with inflated air springs will require an active tilt stabilization system, as described in ref. 53. High-frequency vibrations are known to couple nonlinearly into lower-frequency bands, where the measurement takes place33. To decouple the pendulum from such vibrations, for example, such as those generated by the turbo pump, the 420-g pendulum support structure rests on soft vacuum-compatible rubber feet (Viton).
 Source-mass position drive
 The position of the source-mass gold sphere is modulated by a long-range bending piezo (PI PL140.10, PICMA Bender). Applying a
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 voltage of 0–60 V bends the piezo and provides a travel range of the order of 1mm at the piezo tip, which is connected to a 0.31-mm-diameter non-magnetic titanium rod carrying the source mass (Extended Data Fig. 3). The rod is inserted only half way into the source mass so that its contact potential that faces the test mass is effectively shielded by the surrounding gold. Upon actuation, the piezo bending is translated to a mostly linear source-mass motion of up to 5 mm and a rotation around its centre, which can be neglected. The position of the source mass is modulated harmonically to avoid d.c. noise of the torsion pendulum such as slow thermal drifts.
 Distance measurement
 A major source of uncertainty for the measurement of the gravitational coupling G is the absolute separation between source mass and test mass. We measure the distance with a digital single-lens reflex (DSLR) camera (Canon EOS 60D) that is mounted on top of the vacuum chamber, looking down vertically onto the experiment. This means that only a two-dimensional (2D) projection of the assembly can be monitored. Videos are taken with a 60-mm macro lens and extension tubes. The positions of the masses are tracked using a template-matching algorithm based on the open-source image and video processing library OpenCV. At the beginning of each distance measurement, templates of the masses are defined in the first frame of a video. Their positions are tracked in subsequent frames by integrating the squared pixel value distance over the dimensions of the template
 ∑∑
 R(x, y) = [T(x′, y′) − I (x + x′, y + y′)] . (5)
 i rgb x y
 ii ∈{ , , } ′, ′
 2
 
 
   
   
 Here, Ti(x, y) denotes the 8-bit pixel value of the colour channel i of the template image, and Ii(x, y) the same for the video frame to be evalu
 ated. The minimum ∼ ∼
 min(R) = R(x , y )then corresponds to the position of the best match (x , y )
 ∼ ∼ for the template. This method is limited to an accuracy of 1 pixel, which corresponds to about 24 μm. Therefore, for each dimension we perform a Gaussian fit over the five neighbouring pixels centred at the best-match position to improve the tracking resolution. To limit the effect of changes in laboratory lighting, as well as the influence of reflexes on the source-mass surface, only the outer perimeter of the masses is tracked. We verify the tracking quality by visual inspection of the residuals—that is, by looking at a video with each frame realigned to the tracked position—similar to a digitally stabilized video. The resolution and accuracy of the tracking is limited by the (manual) determination of the edges of the masses in the template, the low contrast inside the vacuum chamber due to its gold-plated interior, and the resolution of the DSLR, and is taken to be 20 μm. We measure the separation of source and test masses via video at the beginning and the end of each measurement run. Furthermore, the source–test mass separation is defined not only by the momentary source-mass position, but also by the motion of the residual test mass. In particular, the centre-of-mass motion of the pendulum varies the distance by more than 100μm under bad conditions. Therefore, the gravitational force is averaged over these distances. The separation modulation changing from this test-mass motion cannot be extracted from the optical-lever signal to be calibrated out. In the future, continuous video tracking will provide this information.
 Piezo calibration. The piezo is operated without feedback. Because the position cannot be monitored continuously, the piezo’s position response to the applied voltage needs to be characterized and modelled. The response of the piezo can be modelled as an ideal mechanical transducer combined with an additional low-pass filter54. Fundamentally, this reflects electron mobility/diffusion into the piezo material.
 As a consequence, the step response is delayed and the drive shows a slow creep towards its final position, as well as hysteresis, when driven cyclically. The creep (knowledge of the exact position) is of particular interest for the charge measurements, whereas the hysteresis (knowledge of the exact amplitude and phase) is important for the gravity measurements, because it results in the source-mass position lagging behind the applied voltage signal and reduces the amplitude. The stationary creep (the step response of the piezo) depends on the size of the applied voltage step and the starting position (the voltage levels applied to the piezo before and after) and can reach up to 30 μm over a period of 15min. As the creep depends on parameters over which we have little control, the creep is fitted for every measurement separately rather than using global piezo parameters. The observed time lag depends on the drive frequency, but does not show any dependence on the modulation amplitude. For a modulation frequency of 12.7mHz we find a time lag of 509 ± 3 ms by correlating the drive voltage with the source-mass position as measured by video tracking. We also observe a drift over time in both the mean position and the modulation amplitude of the source-mass drive. A possible reason is continuous charging of the piezo material, similar to the observed creep for a step response in quasi-static operation, even in dynamic operation. In contrast to the stationary case, in which creep-induced piezo deflection reaches a final state after about 1 h, under cyclic operation the mean deflection and amplitude continue to drift over a measurement period of 100 h. For a peak-to-peak modulation of about 2 mm, the amplitude is reduced by 13 μm and the mean position drifts by 55μm. Although this long-term drift resembles the stationary creep as described in ref. 54 with a much longer time constant, we cannot fit the corresponding function with only the start and end videos. The drift of the mean position, as well as the modulation depth, therefore, are taken to be linear over the period of a measurement run. Additionally, the source-mass position modulation exhibits higher-harmonics contributions. We measure the relative amplitudes and phases of these higher harmonics with respect to the fundamental mode in the start and end videos. We infer the actual position modulation for the gravitational measurement runs from the drive signal by taking the time lag, the amplitude change and mean position drift, and the higher harmonics into account. For example, the amplitude of the 2nd harmonic of the drive position is fitted and found to be below 0.3% of the fundamental amplitude. Remaining higher-order contributions are estimated to be below <0.1% and are therefore irrelevant for our experiment.
 Charge measurements
 We discharge the test mass to less than 8 × 104 elementary charges using a nitrogen gas discharge and ion diffusion as developed in ref. 55. The nitrogen gas is channelled over a set of high-voltage a.c. tips. After passing through an aperture, the nitrogen ions neutralize the surfaces of the vacuum chamber by diffusion. After discharging, the unshielded electrostatic interaction is still approximately three times stronger than gravity (see Extended Data Fig.6). We note that the interaction is enhanced over a pure Coulomb interaction by induced surface charges of the source and test masses. Because of that, even if the source mass is perfectly grounded, a charged test mass will induce charge separation resulting in parasitic attraction. Whereas most torsion experiments aiming to measure small forces conductively coat their suspension filament and thereby sacrifice mechanical quality (statistical error) for charge control (systematic error), we explicitly chose not to do so. One reason is that when down-scaling the torsion balance further, the proportion of lossy coating (surface) grows with respect to the fibre material (volume). The other reason is the long-term prospect to contactlessly levitate the test mass, which would prohibit the use of any electrical contacting. It can be shown by finite-element simulations (Extended Data Fig.5), however, that electrostatic coupling between source-mass position
 and test-mass force can be suppressed well below 3% of the expected gravitational signal by means of a 150-μm-thick conductive Faraday shield inserted between the source and test masses (see Extended Data Fig. 6). The shield comes at the added benefit of reducing other electrodynamic interactions, such as short-range Casimir forces, as well. The drive piezo is housed in a Faraday shield in order to suppress coupling of the applied electric fields onto the pendulum, for example, via residual charges or polarization effects. During an 8.5-h measurement without a source mass attached but with the piezo modulation active, we did not observe any test-mass actuation. Although this verifies the effectiveness of the shield, a full analysis of this drive piezo contribution to the systematic error of our experiment would require the same amount of effective integration time as in the gravity run, that is, a measurement run would have to last for weeks under quiet conditions as well. Because long-term drifts from charge diffusion on non-conductive materials are observed to degrade our low-frequency sensitivity, all surfaces inside the vacuum chamber (except for optical viewports) are made from conductive materials and are gold-plated wherever possible to equalize surface potentials and mitigate thermo-electric potentials.
 Charge characterization. Static charges on our test mass can be characterized by means of induced mirror charges. They can be quantified by a force versus separation measurement without an electrostatic shield and with the source mass being grounded as shown in Extended Data Fig.4. This is effectively a macroscopic realization of a Kelvin probe force microscope, which is typically used to contact-free measure (surface) potentials. The theory for the electrostatic interaction, including polarization effects of extended conductive spheres, is presented in ref. 56. In addition to the expected signals (electrostatic, gravity), we observe an unidentified coupling that is proportional to the sphere separation. To overcome this and other experimental challenges, such as unknown contact potentials, we isolate the effect induced by residual test-mass charges. We therefore quantify them by biasing the source-mass potential and observing the test-mass charge versus source-mass potential interaction at a set of discrete separations. Extended Data Table1 summarizes the results of the intermittent charge measurements over the whole duration of the measurements. A more detailed description of the charge determination will be presented in an upcoming publication. In conjunction with the electrostatic shield between the masses and our finite-element simulation, we thereby verify that electrostatic coupling is suppressed to below 3% of gravity over the whole measurement period.
 Seismic noise
 Even during the environmentally quiet Christmas season, our fundamentally thermal-noise-limited sensitivity is dominated by a yet-unidentified contribution with a strong diurnal variation (see Extended Data Fig.2). Reduction of this noise is important to advance towards smaller detectable source masses. Long-period seismic ground motion in an urban environment is a major concern for extremely sensitive low-frequency experiments57–59. For ideal harmonic motion, the rotational mode of an oscillator will not be directly excited by translational ground motion. Unfortunately, unavoidable imperfections in the construction of the pendulum cause asymmetries that couple horizontal ground motion into rotational motion60,61. Furthermore, down-conversion of noise at higher frequencies, such as cross-coupling from other modes, cannot be neglected33. To monitor horizontal and vertical ground displacement down to 100 mHz, a Raspberry Shake 3D seismometer was installed in close proximity to the experiment. Low-frequency seismic displacements at the experiment site were monitored with a RefTek Observer 60 s broadband seismometer. In addition, we have access to the broadband data of the ZAMG seismic station (Streckeisen STS2) at Hohe Warte, Vienna, roughly 3 km away from the laboratory.
 Correlating strong seismic events, such as the 6.7-mag earthquake in Turkey on 24 January 2020 in the data of the local seismometer as well as the STS2, with the response of our oscillator provides a rough estimate of seismic coupling, which suggests that the diurnal low-frequency noise observable in our data is not due to linear coupling of horizontal or vertical ground motion. This claim is supported by directly comparing the non-stationary low-frequency noise in the STS2 data in Vienna (Extended Data Fig.7a) and the torsion balance (Extended Data Fig.2). There is a strong variability of the seismic noise floor in the 30-mHz band but a relatively steady primary microseismic peak around 70 mHz. The daytime-dependent change in the 30-mHz noise floor is consistent in amplitude and time with the varying white force noise that we see in our test mass below 100mHz. Yet, the frequency distribution is very different. In particular, the microseismic peak at 70 mHz is a very strong, stationary seismic feature within the frequency band of the experiment. Because we do not see it in the torsion pendulum data, we conclude that our rotational mode is not dominantly driven by linear coupling of the horizontal seismic noise.
 Mechanical coupling
 Recoil of the source-mass drive is of major importance for higher modulation frequencies such as 50 Hz (ref. 32). Here we argue that, owing to its strong frequency dependence, this effect can be neglected for our experiment. A 100-kg freely suspended optical table, for example, would move by 1 nm per millimetre of recoil motion of a 100-mg source mass, which is of the order of our observed signal. The coupling into pendulum yaw is attenuated further by the pendulum’s common mode rejection, that is, how well balanced the assembly is. Yet, already a table suspension resonance of 1 Hz, which can roughly be assumed for the air springs, attenuates this by another factor of (f /f ) ≈ 1/6,000
 mod 0
 table 2 . A deflated table, by contrast, typically exhibits internal resonances well beyond the 100-Hz region, yielding even higher resistance to recoil. Another possible coupling at low frequencies is the cyclic deformation of the environment under the changing load distribution of the drive setup. Again, we consider the 1-m-sized, 100-kg optical table suspended by air springs (~4 × 1 kN m −1 stiffness). A 1-mm horizontally shifted 100-mg mass causes 1μN differential load and thereby induces a tilt of approximately 1nrad. In conjunction with the abovementioned tilt-to-yaw coupling, this can result to a test mass motion of the order of 1nm per millimetre of a 100-mg source-mass motion, which is of the order of the observed signal and dominating the recoil effect. In practice, the air springs employ active tilt feedback at low frequencies, stiffening the table tilt motion. When deflated, the table leg stiffness may increase to 1 MN m−1, providing suppression of source-mass-induced table tilt by a factor of 1,000. Internal table flexing modes even in the low 100-Hz region would not allow increased tilt coupling at the modulation frequency. In the future, a more in-depth investigation of tilt-to-yaw coupling and cancellation of both effects by means of a counteracting mass will be investigated. Therefore, both effects are indeed of importance for massive source masses and high accuracy measurements aimed to determine G very accurately. However, to our current understanding, they can be neglected in this experiment and in similar future low-frequency designs aiming at further reduced source masses.
 Magnetic interaction
 To minimize influence from magnetic coupling, the experiment was designed without ferromagnetic materials. All masses are made of diamagnetic gold, and the source mass is mounted on a paramagnetic titanium rod. As a consequence, Earth’s magnetic field (~50 μT at our location), as well as anthropogenic magnetic noise in an urban environment, induce dipoles in the gold spheres, leading to a repulsive force between them.
 Article
 Using literature values62 for the permeability of commercial-grade gold and titanium, we estimate dipole forces of
 F μ mm
 d
 =6
 4π (6)
 mag
 0 12
 c
 4
 between two magnetic dipoles mi at centre distance dc, such as between our two diamagnetic spheres, as well as between the titanium rod and one gold sphere (μ0, vacuum permeability). The strength of the dipoles is determined by the strength of the externally applied field, the susceptibility, and the volume and geometry of the material. Iron impurities in the gold are suggested to dominate the magnetic susceptibility63. Yet gold–iron alloys act diamagnetically up to 1,000 ppm iron, paramagnetically up to 10% iron, and only then start acting as a ferromagnetic material. The gold used in the experiment is specified to have impurities below 1,000ppm but, according to ref. 64, commercial-grade gold has iron impurities between 10 and 100 ppm. As a rough experimental verification for the permeability, a 1-cm cubic NeFeB magnet with a remanent magnetization of around 1.4 T was brought within 7 mm of the test and feedback masses. The literature value of the magnetic susceptibility XAu is confirmed within a factor of ~4. Projecting the dipoles induced by Earth’s magnetic field to the exerted forces by means of equation (6) shows that they are about four orders of magnitudes weaker than gravity in this setup. To estimate magnetic signals induced by anthropogenic fields, we measure the magnetic field close to the experiment with a three-axis magnetic sensor (Stefan Mayer FLC3-70) and two different Samsung phones containing fluxgate magnetometers (see Extended Data Fig. 7b). The 10-mHz frequency comb is correlated with a close-by traffic light period. We verify that this is a magnetic signal by measuring it with different magnetometers using different sampling ADCs. Moreover, the signal strength does not depend on the exact location in the laboratory. Yet, it is not visible in the force signal of the experiment. We therefore conclude that the masses do not couple magnetically via dipoles induced by anthropogenic magnetism.
 Casimir force
 The steady-state Casimir force for a plane–sphere geometry can be calculated65 using
 F R k Tζ
 d
 = (3)
 8 , (7)
 Casimir
 tB
 s
 2
 where ζ is the Riemann zeta function. For our setup, the parameters are T = 300 K, test-mass radius Rt = 1 mm and plane–sphere distance ds ≈ 250 μm. Thus, the Casimir force between the test mass and the Faraday shield is more than four orders of magnitude smaller than Newtonian gravity66. For extending the experiment towards smaller source masses, however, the Casimir force will dominate the gravitational attraction of a Planck-mass-sized source object for shield–sphere separations below ~100μm. Because in our approach the gravitational potential is modulated, even in that source-mass regime we will be able to distinguish Casimir forces exerted onto the test mass (which are static in the presence of an electrostatic shield) from gravity. Furthermore, for reduced shield separations, the Casimir force might become relevant as a static nonlinearity or instability of the torsion balance potential. For a 183-mg oscillator with a natural frequency of 1 mHz, the Casimir force will render the oscillator unstable by surpassing the restoring force only for surface distances below ~100 nm. For any setup without a Faraday shield, Casimir forces set a lower bound for test–source mass separation, which in turn sets an ultimate lower limit on the detectable Newtonian force. For our scheme, these limits are approximately 2 μm for the symmetric setup with 90-mg masses and 20 μm for a source object with m = mP.
 Systematic deviations
 Here we list identified sources of systematic over- or underestimation of the measured gravitational force. Neither of these has been corrected for in the evaluation, given that most of these can only be guessed. In Extended Data Table 2 we assess the most relevant contributions considered. In general, we find that most effects scale with the source mass. Hence, they limit the accuracy of our G estimation but will not pose a limit to measurements of smaller source masses in the future. All involved masses except for the tiny amounts of glue used were measured at an accuracy of 0.1 mg (Kern ABS120-4). Although in the evaluation we treated our source and test mass as point sources of gravity, they also act gravitationally onto the other components, such as the capillary and glue. Gravity between the source-mass support rod (titanium gravity) and the test mass is numerically simulated by integrating over (10 μm)3 elements. Because the mass distribution of the complete source mass assembly deviates from spherical symmetry, it cannot be summarized as an effective point mass co-moving with the source mass. Instead, the modulation caused over a centre-of-mass separation of 2.4–6 mm is put into relation to the gravity modulation caused by the source- or test-mass modulation over the same range. Similarly, the influence of the spatial extent of the hole used for mounting the titanium rod inside the source mass is estimated. The difference between a spherically symmetric source mass with a 300-μm-diameter hole and a spherically symmetric source mass without a hole but with the same mass is evaluated and put into relation with the nominal source- or test-mass gravity. The modulation of gravity between the counterbalance mass of the torsion pendulum and the source mass is quantified over the same 2.4–6 mm range. The calibration of the QPD signal was carried out once on best-effort basis and the results are shown in Extended Data Fig.1. The quality was checked by varying assumptions, but cannot be tested independently or continuously. The source drive position is calibrated to an accuracy of 1 pixel (~20μm) at the beginning and end of each measurement run. The accuracy of the interpolation in between cannot be quantified, because the camera could not be operated continuously. Instead of only a 2D projection, the three-dimensional (3D) mass separation will be monitored continuously in the future, improving the data for close approaches, such as in Extended Data Fig. 4, by giving height offset and high-frequency pendulum and roll mode excitation information. The 3D mass distribution of our gold ‘spheres’ is not known, given that we can only observe 2D projections. From microscopic imaging under different angles, the masses used are known to be bumpy (source-mass radii vary between 1.03 and 1.11mm), whereas in high-accuracy experiments for the determination of G, even the slightest shape and density deviations are known to be relevant. The masses used in the experiment deviate from perfect spheres. To quantify the influence of these shape deviations, we took photographs of the test sphere under a microscope and at different angles, as shown in Extended Data Fig.8. We extract the angle-resolved radius from the geometric centre of mass of each projection. We assume that each combination of angle and radius defines a spherical wedge and we calculate the 2D deviation of the combined centre of masses of the wedges from the geometric centre of mass of the projection. We repeat the evaluation for every projection and take the mean deviation from the geometric centre of mass as a correction for the centre-of-mass distance used in the evaluation. Comparing the fits to the uncorrected and corrected centre-of-mass distance results in an estimated systematic uncertainty of 4.2% for both test- and source-mass shapes. An additional damping rate caused by the impacts of residual gas molecules is estimated according to ref. 51. Using our experimental parameters and taking into account the close proximity of the electromagnetic shield to the test mass, we obtain a damping rate five orders of magnitude smaller than the natural damping rate of our test-mass oscillator.
 A worst-case estimate for sorption effects on uncleaned gold surfaces67,68 results in a relative mass change of the order of 10−7 and can therefore be neglected. We also note that we do not observe time-dependent systematic effects, in agreement with the absence of substantial sorption effects.
 Data availability
 The data that support the findings of this study are available from the corresponding author upon reasonable request.
 50. Turner, M. D., Hagedorn, C. A., Schlamminger, S. & Gundlach, J. H. Picoradian deflection measurement with an interferometric quasi-autocollimator using weak value amplification. Opt. Lett. 36, 1479–1481 (2011).
 51. Schmoele, J. Development of a Micromechanical Proof-Of-Principle Experiment for Measuring the Gravitational Force of Milligram Masses. PhD thesis, Univ. of Vienna (2017). 52. Newport pneumatic optical table performance. Newport https://www.newport.com/n/ compliance-and-transmissibility-curves 53. Lewandowski, C. W., Knowles, T. D., Etienne, Z. B. & D’Urso, B. Active optical table tilt stabilization. Rev. Sci. Instrum. 91, 076102 (2020). 54. Displacement of open loop piezo actuators. PI https://www.pi-usa.us/en/products/ piezo-motors-stages-actuators/piezo-motion-control-tutorial/tutorial-4-20/
 55. Weiss, R. Charging of the Test Masses Past, Present and Future. LIGO Document T1100332 (2011); https://dcc.ligo.org/public/0115/G1401153/002/charging.pdf 56. Lekner, J. Electrostatics of two charged conducting spheres. Proc. R. Soc. A 468, 2829–2848 (2012). 57. Díaz, J., Ruiz, M., Sánchez-Pastor, P. S. & Romero, P. Urban seismology: on the origin of earth vibrations within a city. Sci. Rep. 7, 1 (2017). 58. Groos, J. C. & Ritter, J. R. R. Time domain classification and quantification of seismic noise in an urban environment. Geophys. J. Int. 179, 1213–1231 (2009). 59. Bourdillon, A., Ropars, G., Gaffet, S. & Le Floch, A. Opposite sense ground rotations of a pair of Cavendish balances in earthquakes. Proc. R. Soc. A 471, 20140997 (2015). 60. Shimoda, T., Aritomi, N., Shoda, A., Michimura, Y. & Ando, M. Seismic cross-coupling noise in torsion pendulums. Phys. Rev. D 97, 104003 (2018).
 61. Gettings, C. & Speake, C. An air suspension to demonstrate the properties of torsion balances with fibers of zero length. Rev. Sci. Instrum. 91, 025108 (2020). 62. Lide, D. R. CRC Handbook of Chemistry and Physics Vol. 85 (CRC Press, 2004). 63. Shih, J. W. Magnetic properties of gold-iron alloys. Phys. Rev. 38, 2051–2055 (1931). 64. Henry, W. & Rogers, J. XXI. The magnetic susceptibilities of copper, silver and gold and errors in the Gouy method. Philos. Mag. 1, 223–236 (1956). 65. Sushkov, A., Kim, W., Dalvit, D. & Lamoreaux, S. Observation of the thermal Casimir force. Nat. Phys. 7, 230–233 (2011). 66. Emig, T., Graham, N., Jaffe, R. L. & Kardar, M. Casimir forces between arbitrary compact objects. Phys. Rev. Lett. 99, 170403 (2007). 67. Beer, W. et al. The METAS 1 kg vacuum mass comparator-adsorption layer measurements on gold-coated copper buoyancy artefacts. Metrologia 39, 263 (2002). 68. Gläser, M. & Borys, M. Precision mass measurements. Rep. Prog. Phys. 72, 126101 (2009).
 Acknowledgements We thank E. Adelberger, A. Buikema, P. Graham, N. Kiesel, N. Klein, D. Racco and J. Schmöle for discussions. We are grateful for the suspension fibre provided by A. Rauschenbeutel and T. Hoinkes and for the mechanical design assistance by M. Dragosits. This project was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 649008, ERC CoG QLev4G), by the Austrian Academy of Sciences through Innovationsfonds Forschung, Wissenschaft und Gesellschaft, by the Alexander von Humboldt Foundation through a Feodor Lynen Fellowship (T.W.) and by the Austrian Federal Ministry of Education, Science and Research (project VCQ HRSM).
 Author contributions T.W., J.P. and M.A. conceived the experiment, T.W., H.H. and J.P. constructed and conducted the experiment and analysed the results. All authors wrote and reviewed the manuscript.
 Competing interests The authors declare no competing interests.
 Additional information
 Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41586-021-03250-7.
 Correspondence and requests for materials should be addressed to T.W. or M.A.
 Peer review information Nature thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Reprints and permissions information is available at http://www.nature.com/reprints.
 Article
 Extended Data Fig. 1 | Quadrant photodiode calibration. The video-tracked yaw angle of the torsion pendulum is compared to the QPD readout (horizontal difference signal normalized by the total sum signal to reduce laser intensity noise coupling). A third-order polynomial fit to these data provides our QPD signal calibration. The occurrence probability accumulated over all our data runs shows the region in which a calibration is required, that is, Vx/Vsum ∈ [−0.4, 0.2].
 Extended Data Fig. 2 | One-hour displacement spectra. a, The displacement amplitude spectral density (ASD) estimate of 1-h data chunks of representative data taken between 26 and 27 December shows stationary readout noise at high frequencies and non-stationary white force noise at low frequencies. The level of the latter varies up to tenfold, consistently being the lowest at 0–5 a.m., particularly in the night after public holidays or Sundays, that is, before normal workdays. During such quiet periods, the thermal noise of the pendulum is reached. b, Time-series data spanning half an hour during the 3–4 a.m. segment. The data trace is bandpass-filtered in the frequency range 5–75 mHz with a 6th-order Butterworth filter and magnified by a factor of 20.
 Article
 Extended Data Fig. 3 | Schematic of drive mechanism. The source-mass drive, consisting of a bending piezo and a titanium rod, can modulate the source-mass position by more than 5 mm peak to peak at frequencies as high as 1 Hz. The geometry of the titanium rod amplifies and translates the piezo deformation into an approximately linear motion.
 Extended Data Fig. 4 | Probability distribution of F(d) without an
 electromagnetic shield. Without electrostatic shielding, the probability density distribution of inferred force versus source–test mass separation requires three constituents to describe: Newtonian gravity (see Fig. 3),
 electrostatic interaction between charged (~8 × 104e+; e, elementary charge) test mass and grounded source mass56, and an unexplained force proportional to the separation.
 Article
 Extended Data Fig. 5 | Finite-element method simulation of charge
 distribution. The charge distribution induced by a 3 × 104e+ charged source mass on the grounded test mass was determined by a finite-element simulation (COMSOL; shown for 1 mm surface separation as an example). With the grounded, conductive electrostatic shield in place, many mirror charges are
 induced in the unmodulated shield, resulting in a d.c. force. The induced surface charge density (colour-coded) on the position-modulated source mass is suppressed by a factor of approximately 100. These charges are screened by the shield, resulting in further suppression of the exerted force. The actual suppression could not be quantified owing to numerical inaccuracies.
 Extended Data Fig. 6 | Simulation of electrostatic coupling. Our
 finite-element simulation of the electrostatic interaction of a charged test mass (105e+) and a grounded source mass was validated using the analytical method described in ref. 56. When inserting a 150-μm-thick, conductive electrostatic shield between them, the electrostatic force exerted onto the test mass is dominated by the test mass–shield interaction, that is, it becomes independent of the source-mass position. Residual fluctuations of the force are on the 3–10% level of gravity, without a clear source position dependence, and are expected to stem from numerical errors.
 Article
 Extended Data Fig. 7 | Magnetic and seismic coupling. a, The time-resolved amplitude spectral density of the horizontal ground motion in Vienna, recorded by an STS2 broadband seismometer, shows a strong frequency dependence of the variability. A rise of the noise floor is observed throughout the frequency range, starting from midnight (dark blue) to noon (light blue), but is most prominent at 30–40 mHz. By contrast, the noise floor at the microseismic peak, at around 70 mHz, varies only slightly. The modulation frequency of the drive mass is marked by a vertical dashed line. Each spectrum corresponds to a single 2-h measurement. b, The amplitude spectral density of magnetic field variations at the site of the experiment were measured with a three-axis magnetometer (Stefan Mayer FLC3-70; z = vertical,y = along sourcetest mass axis). The signal at 10 mHz is explained by the period of the traffic light at a nearby crossing, regulating car traffic as well as trams. The inferred force acting on the pendulum (red line), shows no sign of this signal. Each spectrum corresponds to a single 8,000-s measurement.
 Extended Data Fig. 8 | Shape of test-mass sphere. a, 2D projection of the test mass. The centre of mass determined by the geometric centre of mass (mean over all image dimensions), as well as the centre of mass calculated by our shape evaluation, are indicated (‘Volume COM’). The centre of mass used in the determination of G is indicated by a blue cross. b, The radius measured from the geometric COM over all angles is shown.
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 Extended Data Table 1 | Surplus charges over time
 The surplus charges on the test mass were measured between our measurement runs during the time periods marked by grey bands in Fig. 4. All measured charges stay well below 8 × 104e+, which we take as an upper limit in our estimation of electrostatic coupling between source and test mass.
 Extended Data Table 2 | Identified systematics
 Identified systematic deviations in units of expected Newtonian gravity between source and test mass.
@@ -1,803 +0,0 @@
 1
 Invariant Electromagnetism: Necessity and Sufficiency
 Thomas E. Phipps, Jr. 908 South Busey Avenue Urbana, Illinois 61801 and Harold Willis Milnes
 3101 20th Street Lubbock, Texas 79410
 A first-order Galilean-invariant covering theory of Maxwell’s equations of vacuum electromagnetism first proposed by Heinrich Hertz is reappraised in modern context. Physically, when properly formulated and interpreted, it is found to offer improved sufficiency for electromagnetic description. Mathematically, its use of the total time derivative instead of the Maxwellian partial time derivative is shown to be logically necessary under the condition that the (inertial) coordinate transformation precedes the application of the field equations.
 1. Historical Introduction
 The theory of electromagnetism (EM) developed by Maxwell(1) was noninvariant at first order. That is, when subjected to a Galilean (first-order inertial) transformation of coordinates with velocity parameter v, Maxwell’s field equations yielded surplus terms, proportional to v, that were not originally present. False predictions resulted, as of nonexistent fringe shifts in first-order optical experiments. The noninvariant aspects of his formalism apparently did not trouble Maxwell who, as a confirmed etherist, thought naturally in terms of a preferred reference system. But in hindsight it seems that it should have caused some concern, as possible evidence of a first-order flaw in his theory, in
 view of a growing body of observations (for example those by Mascart(2) in 1872-4, several years before Maxwell’s death) that supported universality of the Newtonian relativity principle as a physical fact at first order in v/c.
 The present analysis is founded on the premise that any theory that makes false predictions at first order is broken at first order and needs to be fixed at first order. His
 torically, this is not an original thesis. Heinrich Hertz(3) had the same idea, which led him to modify Maxwell’s formalism so as to make it rigorously invariant under Galilean (inertial) transformations. This turns out mathematically to be quite easy to do. It mainly requires replacement of Maxwell’s partial time derivative operator ∂ /∂t , wherever it appears, with a total time derivative operator. The latter is customarily written as d/dt by
 physicists and engineers(4,5), but will here be denoted D/Dt to avoid confusion with the usual notation for the derivative of a function of a single variable. Thus
 2
 D
 Dt t
 = + ⋅∇
 ∂
 ∂ w , (1.1)
 where, in physics, w is a vector of velocity (of a localized object – such as a test particle not of a coordinate frame) and ∇ = ∂ ∂ ∂ ∂ ∂ ∂
 / ,/ ,/
 xyz
 b g . It is also necessary to make a
 trivial adjustment of the current source term to agree with Galilean velocity addition. The new “convective” velocity parameter w appearing here is assumed for simplicity to be a constant independent of space variables (otherwise, one deals with the more general
 “Helmholtz derivative”(6)).
 The role of w will be a major focus of attention in what follows. As a “new” parameter not previously appearing in EM description, it requires a new physical interpretation; but the latter is of no significance for the mathematical invariance proof (§2) or for considerations of mathematical necessity (§3). Hertz’s was a first-order invariant covering theory of Maxwell’s equations for the vacuum case – inasmuch as Galilean invariance
 (asserted(3) but not explicitly shown by Hertz) has been repeatedly proven(7-10) and will be demonstrated again here for completeness. The “covering theory” aspect derives from the fact that the special case w = 0 reduces D/Dt to ∂ /∂t , thereby recovering Maxwell’s theory of vacuum EM identically. This means that the “physics of one laboratory” (in which field-responsive instruments are at rest) is identical in the two theories.
 Unfortunately, Hertz’s claim of a first-order invariant EM formulation was ob
 scured by the complicated component notation used in his exposition (3) and by a failure to simplify with the help of vector identities. Consequently his formalism was never used by later workers nor was the significance of its first-order invariance recognized. Further, he handicapped its chances of acceptance by proposing a Stokesian ether interpretation of the new parameter w [denoted by components ( , , )
 α β γ in Hertz’s book(3)], according to which it signified the velocity of an ether 100% entrained (or convected) by tangible matter – so that w measured the velocity of just any material “body” in the laboratory. This led to false predictions (e.g., creation of a magnetic field by a charged rotating di
 electric) that were considered observationally disproven(11) soon after his death. His formalism of true invariance was hence discarded, just in time to make way for EinsteinMinkowski’s “universal covariance” – a mathematical substitute readily accepted in lieu of invariance, under the false impression that true first-order invariance was unattainable.
 An even-handed application of the principle of discarding all mathematics proven to be in first-order disagreement with observation would have led, well before the advent of Hertz’s theory, to the discarding of Maxwell’s theory. In fact the ether theories of both Maxwell and Hertz were soon forgotten. Still, it seems regrettable that Hertz’s first-order invariant covering theory was rejected without a hearing of its mathematical merits, while Maxwell’s noninvariant special case was retained, solely on its mathematical merits, and was made the basis for all subsequent field physics. To put it another way, Maxwell’s spurious ether “physics” was rejected for the sake of keeping his noninvariant mathematics, while Hertz’s invariant mathematics was rejected for the sake of discarding his spurious ether “physics.” We suggest that it is never too late to consider rectifying this odd turn of the political history of science.
 3
 Meanwhile the relativity principle, under Einstein’s tutelage (12), received mathematical expression through formal covariance rather than invariance. This further historical oddity involved skipping over the first order entirely and going directly to secondorder expressions (in v c
 22
 / ) affecting kinematics. The underlying “physical” assumption behind this evolution was spacetime symmetry, a concept derived directly and solely (in terms of objective supporting evidence) from the formal symmetry in Maxwell’s equations of the partial derivative operators ∂ ∂ ∂ ∂ ∂ ∂
 / ,/ ,/
 x y z and ∂/∂t (or ∂ /∂ct 
 we here omit factors of c). In Hertzian theory that alleged symmetry is spoiled at first order by the requirement of Galilean invariance, which mandates replacement of ∂/∂t by the D/Dt of Eq. (1.1). No mathematical symmetry is exhibited between partial space derivatives and such a total time derivative, in view of the augmented “fourth-component” parametrization by the 3-vector w, with no such augmentation of the spatial components.
 Thus, if physics is developed systematically, proceeding from lower to higher orders of approximation and insisting on getting the physics right at lower orders before proceeding to higher ones, then EM theory offers no logical justification for hypothesizing covariance at any stage. It is strict invariance that logically expresses relativity at first order, and at that order invariance “breaks” spacetime symmetry. By the nature of orders of approximation, a symmetry broken at first order cannot be restored at higher orders. Therefore it is evident that in a rationally sequenced evolution of EM physics motional relativity would initially have been expressed by (first-order) invariance, not by covariance (which introduces second-order considerations) ... and spacetime symmetry would never have suggested itself as an attribute of the physical world. Most of this was im
 plicit in Hertz’s work(3) a decade before Einstein’s(12). More will be said in §8 about the implied alternative approach to higher-order EM theory.
 In §2 of this paper we address in mathematical terms the sufficiency of the Hertzian D/Dt-based invariant formalism for physical description via field theory. Sections 3-5 then provide our proof of the necessity of total time derivatives from the purely mathematical standpoint. The remainder of the paper treats physical implications and interpretation, which differ nontrivially from those of the Maxwellian theory. In §6 Hertzian and Maxwellian “fields” are contrasted as to their operational definitions, and in §7 the d’Alembertian solution of the Hertzian wave equation is developed. Throughout, attention is restricted to the case of vacuum electromagnetism, this being generally viewed as the most fundamental form of EM theory. For the most part, attention is also restricted to considerations of first order, such that Newtonian kinematics applies.
 However, in §8 we venture to speculate about higher-order EM description. There, our treatment of phase velocity refers to a proposed higher-order (“neo-Hertzian”) form of invariance, with application in §9 to the special topic of stellar aberration, which suggests a crucial experiment. Returning to first-order considerations in §10, we address the invariant form of the field definitions in terms of EM potentials, as well as the implications of an invariance requirement for the EM force law. We deduce that the universally-accepted Lorentz force law may tell only part of the first-order EM force story; and
 cite the so-called Marinov motor(13) as possible evidence of first-order observable depar
 4
 tures(14,15) from the Lorentz law. With reference to physics one can never prove the sufficiency of a mathematical formalism, because the last experiment is never done. But one can establish insufficiency ... and this, we suggest, experiment may already have done in regard to the Maxwell-Einstein (covariant) EM formalism.
 2. First-Order Formal Invariance of Hertz’s Field Equations
 Confining attention, as stated, to the case of vacuum EM, and arbitrarily choosing the symbols E and B to designate the field vectors, we may express the familiar electromagnetic field equations (omitting factors of c) as
 ∇× − − =
 BE j
 δ
 δπ
 tm
 4 0 (2.1a)
 ∇× + =
 E δB
 δt 0 (2.1b)
 ∇⋅ =
 B 0 (2.1c) ∇⋅ − =
 E4 0
 πρ . (2.1d)
 Here all symbols and their physical referents are as in Maxwell’s theory, except that δ /δt is employed as a generic notation to denote the partial derivative ∂ /∂t in the case of Maxwell’s equations and the total time derivative D/Dt, defined by Eq. (1.1), in the case of Hertz’s equations. Similarly, jm denotes the Hertzian current density, which is a generalization of the Maxwellian j to be defined presently. In this section we address Hertz’s equations, for which D replaces δ D → δ
 b g . It should therefore be emphasized
 that the E, B appearing here are Hertzian field quantities, not Maxwellian, since D/Dt has replaced ∂/∂t . In this connection it is instructive to note that in a typical textbook ac
 count(5) of “Faraday’s law of induction” the author first introduces d/dt (our present D/Dt) in order to describe what Faraday observed in regard to arbitrarily-moving circuits and then, by arbitrarily forbidding circuits to move with respect to the observer, eliminates d/dt in favor of ∂/∂t in order to recover the usual Maxwell-Heaviside theory.
 We now verify the form invariance of Eq. (2.1) [with D → δ
 b g ] under the Galilean transformation,
 r rv
 '= − t , t t
 ' = , (2.2)
 which is generally viewed as the correct first-order mathematical description of physical inertial motions. [It therefore suffices for present purposes to consider such irrotational motions, although the Galilean group is less comprehensive than the most general one the orthonormal group – under which Eq. (2.1) is invariant.] Note that the v parameter in (2.2) is conceptually unrelated to the convective parameter w appearing in Eq. (1.1). Confusion on this almost self-evident point (which is in fact crucial to our invariance
 proof) has extended even to a recent two-volume treatise(6) on Hertzian EM. From (2.2) written out in component form we see that
 5
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ x∂
 x
 xx
 y
 xy
 z
 xz x
 = ++=
 ' '
 ' '
 '
 ''
 and similarly
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ y y z ∂z
 ==
 ', ';
 which is equivalent in vector notation to
 ∇=∇
 ' . (2.3)
 Similarly, when (2.2) is written as x x vxt
 ' = − or x x vxt
 = '+ , so that ∂x ∂t vx
 / = , etc., we verify that
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ t∂
 t
 tt
 t t
 x
 t x etc
 '' ' .
 =+ +
 implies
 ∂ ∂
 ∂ t ∂t
 ' = + ⋅∇
 v . (2.4)
 Again we call attention to the different velocity parametrization of Eqs. (1.1) and (2.4), as well as to the distinction between partial and total differentiation. Eqs. (2.3) and (2.4) are
 well-recognized and accepted in the literature(16). Eq. (2.4) makes explicit the first-order noninvariance of the partial time derivative operator.
 The convective velocity parameter w is postulated to obey the Galilean velocity addition law,
 w wv
 ' = − , (2.5)
 (an input essential to the invariance proof, which obviously rests upon the abovementioned conceptual distinction between w and v) and the field quantities are postulated to transform invariantly,
 EE BB
 ', '
 = = . (2.6)
 For the moment we set aside the physics of such relations (to be treated in §6) and address only the matter of form invariance. Similarly, we assume for the source terms invariance of charge density and the first-order Galilean velocity transformation property of Maxwellian current density; i.e.,
 6
 ρρ
 '( ', ') ( , )
 rr
 t = t and j r j r r v
 '( ', ') ( , ) ( , )
 t = t − ρ t . (2.7)
 These are also accepted relations(16). A “convective” current density may be defined by
 jw
 c = −ρ , (2.8)
 which by (2.5), (2.7) transforms according to
 j w wv w v j v
 cc
 =− =− + =− − = −
 ρ ρ ρρ ρ
 (' ) ' '
 bg ,
 or
 jj v
 cc
 ' = + ρ . (2.9)
 Let a “measured” current density jm , appearing in our field equation (2.1a), be defined as
 j jj
 mc
 = + . (2.10)
 We see from (2.7) and (2.9) that jm transforms invariantly,
 j j j jj j v j v j j j
 ' ' ''
 m c c c cm
 = + =+ = − + + =+ =
 b g b gb g
 ρ ρ , (2.11)
 thus justifying our interpretation of jm as a “measured” (objectively physical) quantity. It
 is apparent that our current density measuring instrument (“detector”) – conceptually, a small box with walls permeable to relatively-moving charges – measures the Maxwellian current density j when it is at rest ( w = 0 ) in the unprimed inertial system, and measures the current density jm when it moves relatively to that system with velocity w ≠ 0 . By this relative motion it generates an additional purely “motional” current density jc , given
 by Eq. (2.8), which has a minus sign because a detector moving in one direction is equivalent to a current flowing in the opposite direction. Note that (2.8), (2.10) show that, when w → 0 , jm approaches j, the Maxwellian current density. In the same limit, w → 0 , δ /δt or D Dt
 / approaches ∂ /∂t . The Hertz equations, (2.1), thus reduce identically to Maxwell’s equations in the special case w ≡ 0 , and therefore constitute a covering theory. This observation is adequate to establish the sufficiency of Hertz’s equations for physical description, for all physicists who accept the sufficiency of Maxwell’s equations.
 A result basic to our proof of first-order invariance of the Hertzian field equations is invariance of the total time derivative. This follows from Eqs. (1.1) and (2.3)-(2.5),
 D
 Dt t t t t
 D Dt
 FHG IKJ = + ⋅∇
 FHG IKJ = + ⋅∇ = + ⋅∇
 FHG IKJ + − ⋅∇ = + ⋅∇ =
 ''
 ' ''
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 w w v wv w
 b g . (2.12)
 7
 Although explicit appearance of c has been suppressed here, it is to be understood that this quantity, which may be construed as a units ratio, is Galilean invariant,
 cc
 ' = . (2.13)
 With these preparations we are ready to prove the invariance of the field equations (2.1). Eq. (2.1c) is obvious from (2.3) and (2.6):
 ∇⋅ = ∇⋅ = ∇⋅ =
 B BB
 b g' ' ' 0 . (2.14)
 Invariance of (2.1d) follows from inclusion of (2.7),
 ∇⋅ − = ∇⋅ − =
 '' '
 EE
 4 40
 πρ πρ . (2.15)
 By recalling that for Hertzian theory δ /δ /
 t D Dt
 ≡ , and by use of (2.12), we similarly verify the invariance of (2.1b),
 ∇ × +FHG IKJ = ∇ × + =
 ' ' ''
 E B EB
 D Dt
 D
 Dt 0 . (2.16)
 Finally, the invariance of (2.1a) follows from inclusion of (2.11),
 ∇ × −FHG IKJ − = ∇ × − − =
 ' ' '' '
 B Ej B E j
 D Dt
 D Dt
 mm
 4 40
 π π . (2.17)
 It will be seen that proof of first-order invariance is no more than a matter of inspection. Since each symbol simply transforms in place between primed and unprimed notations, we may elect to term such a transformation property, involving both formal and numerical invariance, “manifest invariance.” This concludes our demonstration that at first order Hertzian EM constitutes an invariant covering theory of Maxwellian vacuum EM. The logical sufficiency of the Hertzian formalism for purposes of EM physics at the “classical” (non-quantum) level of field theory is thus established. We next turn to proof of the necessity (of D → δ ) in a purely mathematical sense, given first-order form invariance as a physically-mandated precondition. Discussion of physical meaning of the Hertzian formalism will be deferred to §6.
 3. Proof of Necessity of the Total Time Derivative
 That the operator δ /δt appearing in the EM equations (2.1) is necessarily equivalent to a total time derivative D/Dt is a strictly mathematical matter to be addressed in this section. The demonstration will assume invariance and from that deduce D → δ .
 A function F(x,y,z;t) is said to be form invariant(17) under a transformation T: ( , , ,; )
 xyz t → ( ', ', '; ')
 x y z t in case its image in the primed space is term-by-term identical with the original function up to a factor dependent solely on the coefficients of the transform.
 8
 This is some power, p, of the ‘modulus’ M of the transformation, which is the same as the Jacobian:
 M x yzt
 xyz t
 ≡∂
 ∂
 ', ', '; ' ,,;
 bg
 b g . (3.1)
 Thus F(x,y,z;t) is an invariant under T in case
 F F x y z t F x y z t M F x yzt M F
 pp
 ' '( ', ', '; ') ( ', ', '; ') ( , , ; )
 = = = = . (3.2)
 In the particular case of a Galilean transformation (2.2) it is seen that M = 1 and the factor M p need not appear.
 The functions of present interest are the dependent variables E and B appearing in the equations (2.1) of vacuum EM theory. Let us presume those equations solved with respect to given mathematical boundary conditions: first in the image space as
 Ee
 ' '( ', ', '; ')
 = x yzt B b
 ' '( ', ', '; ')
 = x y z t , (3.3)
 and secondly in the direct space as
 E = e( , , ; )
 x y z t B = b( , , ; )
 x y z t , (3.4)
 where the notations e b e b
 ', ', , designate the functions of the stated independent vari
 ables that resolve the original partial differential equations. The arguments are the same logically for the electric vector as for the magnetic, so we need consider only one or the other; e.g., the electric. The vectors e' and e are to be considered as invariantly related in form if and only if
 ee
 '( , , ; ) ( , , ; )
 αβγ τ αβγ τ
 ≡ , (3.5)
 where α β γ τ
 , , , are merely symbols introduced to emphasize the mathematical relation
 ships without reference to physical content. Equation (2.2) shows that the time variables transform between themselves independently of the spatial variables and that dt dt
 '/ = 1.
 If, as we assume, e', e are form invariant then by (3.3)-(3.5)
 Ee e e
 ' '( ', ', '; ' ) ( ', ', '; ' ) ( '( , , ; ), '( , , ; ), '( , , ; ); '( ))
 = ==
 x y z t x y z t x x yzt y x yzt z x yzt t t ==
 eE
 (, ,;)
 x y z t , (3.6)
 where the implication of the next to last equality is that after the transformation has been manipulated out in terms of x,y,z and t it reduces to e( , , ; )
 x y z t , with these arguments appearing in it instead of the original x y z
 ', ', ' and t' .
 9
 By elementary calculus, ∂
 ∂
 ∂ t∂
 dt dt t
 ''
 = , so that from (3.6)
 ∂ ∂
 ∂ ∂
 Ee
 '
 ' ( '( , , ; ), '( , , ; ), '( , , ; ); '( )) '
 t t x x y z t y x y z t z x y z t t t dt
 dt
 =⋅
 = +++
 LNM OQP⋅
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 e e ee x
 x
 ty
 y
 tz
 z
 tt
 dt dt
 dt dt
 '
 ' '
 ' '
 ' '
 ' '
 = ⋅= ⋅
 ==
 D
 Dt x y z t dt
 dt t
 dt dt
 x yzt t t t
 eE
 xx
 ( ', ', '; ') ' '
 ' '( , , ; ); ' '( )
 o t∂
 ∂ , (3.7)
 where the notation D Dt
 / represents the total time derivative(18). Since dt dt
 / ' = 1 we have
 ∂ ∂
 ∂ ∂
 Ee E
 xx
 '
 ' ( ', ', '; ' ) ' '( , , ; ); ' '( )
 t
 D
 Dt x y z t t
 x yzt t t t
 ==
 ==
 o t . (3.8)
 This is just an application of the familiar chain rule.
 The total time derivative is therefore explicitly required. When the problem is thus analyzed in detail, it poses no difficulty whatever. Note that the partial differential operators as well as the total time derivative are interpreted as applied after the transformation of coordinates has been made, not before. This is the normal procedure, and it may conceivably be what Maxwell intended by his partial derivative notation – since notations in his day were less standardized than they are now. It is seen that (by virtue of the first-order relationship dt dt
 / ' = 1 ) the total time derivative can be used in place of the partial time derivative, in accord with the Hertzian interpretation of the operator we have written as δ /δt . The meanings of the partial and total notations in (3.8) in this important special case are equivalent. The question is not a serious one of any real significance apart from the confusion that has resulted from a rather pervasive misunderstanding.
 4. Conclusion Concerning Necessity
 It is hard to understand why this question has so fretted electricians, but we suppose there is some excuse for so much befuddled thinking in that the notation ∂ /∂t had a
 dual meaning in the 19th century, which persisted until the mid-20th century. It represented the simple partial derivative as what it is almost universally accepted as meaning today, but it also represented the total time derivative D/Dt, as used above, when a transformation is involved. The reader, formerly, was just supposed to know this and simply to use the meaning appropriate to the context. It was a bad ambiguity of symbolism, which was actually caused by all available printing fonts of the hand typesetting matrix being used up. There still persist similar confusions of meaning in the symbolism used to designate higher partial and total derivatives that continue to be a mine field over which the wary need to tread lightly. The only reliable course to take through the confusions of
 10
 partial differentiation is: to be fully and surely aware of what is a function of what, and, if transformations are involved, when they are to be applied, before or after differentions.
 Let us clarify the cause of the confusion by a simple, purely mathematical, example summing up the conclusions. Suppose e = x + y +t
 ' ' ; then ∂e ∂t
 / = 1 . That is, unless a transformation, such as x x t y y t
 ' ,'
 = + =+
 2 3 5 intervenes. But, if it does, then after this transformation we obtain the entirely different result,
 ∂ ∂
 ∂ ∂
 ∂ ∂
 e
 t te t x t y t t
 x xt
 yy t
 = = + + + +=
 =+
 =+
 '' 2 53 2 3 5 9
 { } b gb g .
 Note that the latter result is the same as
 De Dt
 e x
 x t
 e y
 y t
 e t
 = ⋅ + ⋅ + =⋅+⋅ + =
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 ∂ ∂
 '
 ' '
 ' 13 15 1 9.
 By recognizing the dimensionality of ∂
 ∂
 ∂ ∂
 x t
 y t
 ', '
 FHG IKJ = w as that of velocity, we convert this
 last expression to the familiar, physically useful form (1.1) of the total time derivative.
 Mathematics knows of this bugbear, but it is not well known. It is not regularly taken up in textbooks, especially those of the pot-boiler class such as have been written as yearly editions by editors, rather than by fully competent senior mathematicians who have in mind another generation that might be aspiring to their level one day. If handled at all, the matter is relegated to a few problems appearing obscurely in some exercise. Cou
 rant(19), however, treats of the matter adequately; but, after all, how many students of the calculus, even those intending to become mathematicians, have had the privilege of learning their fundamentals from a bible of the discipline such as his is? Unfortunately, it is now cherished more on the library shelf than in the classroom.
 5. The Inertial Transformation Group; Preserving the Metric
 The demonstrations of the preceding sections may be easily generalized from the Galilean transforms (2.2) to the (first-order) inertial transformation group defined as r rr v
 ' ()
 =+ − −
 00
 t t . The latter is itself a subgroup of all linear transformations of the general form
 T: x a x a y a z v t t k
 '= + + − − +
 11 12 13 1 0 1
 bg
 y ax a y az v t t k
 '= + + − − +
 21 22 23 2 0 2
 bg
 z ax a y az vt t k
 '= + + − − +
 31 32 33 3 0 3
 bg
 t' = f t − t0
 b g (5.1)
 11
 where the a’s, v’s, k’s, and t0 are constants and f is an arbitrary differentiable function. Eq. (5.1) may be compactly written in matrix form as
 x x vk
 '= − − +
 A t t0
 bg
 t ft t
 '( )
 = − 0 , (5.2)
 where the obvious meanings are given the vectors and the matrix A. It is seen on taking A to be the identity matrix that the inertial transformation group is included as a subgroup of (5.2); but additional transforms that are not inertial are contained in the larger group. These other transformations do not preserve the metric and therefore lead to a form of
 related, but non-Newtonian, mechanics in which the laws of Buridan(20) are changed along with the concept of what is “force.”
 Let x0 be any fixed point; then
 xx x x
 ''
 −= −
 00
 b gb g
 A and x x x x
 ' '* * *
 − =−
 00
 b g b g A , (5.3)
 where * denotes transposition. The matrix product of these two equations yields
 xx y y zz AA
 ' ' ' ' ' ' ' '* ' ' * * ( )
 − + − +− = − •− = − −
 0
 2 0
 2 0
 2
 000 0
 b g b g b g b gb gb g
 x x x x x x x x , (5.4)
 where • denotes scalar product of two vectors. The right-hand side is a quadratic form which for all x and x0 equals
 xx yy zz
 − + − +− = − •−
 0
 2 0
 2 0
 2
 00
 b gb gb g b gb g
 xx xx
 * , (5.5)
 if and only if A A I
 * = , where I is the identity matrix. In such case
 xx yy zz x x y y z z
 '' ' ' ''
 − + − + − == − + − +−
 0
 2 0
 2 0
 2 0
 2 0
 2 0
 2
 b gb gb g
 { } b gb gb g
 {}
 ρ . (5.6)
 Thus the metric is invariant if and only if A A I
 * = ; i.e., the transformation is orthonormal. This was stated but not demonstrated already in §2. The transformation group is therefore inertial if and only if A is orthonormal. The metric distance between any two vectors is then preserved timewise as well; and it does not matter if the transformation be of the alias or alibi type.
 Attention may now be given briefly to time and how it affects inertial transforms. It is to be noted that in the Galilean case, by (2.2), t t
 ' = , so that dt dt
 '/ = 1. This is also true if t t t
 '= − 0
 b g . Otherwise there is a change in the rate at which clocks measure time,
 which would be variable unless df/dt is constant. Non-constancy affects the concepts of both velocity and acceleration, changing the relation between force and acceleration that exists in Newtonian mechanics. If df/dt is constant but different from unity, a change of
 12
 time (or clock) rate is implied, which could be compensated by a change in units of the time scale. Unless this is done, it is required that f t t t t
 ()
 − ≡−
 00
 b g , so as to keep the same force units in the image as in the direct space.
 6. Physical Interpretation
 We turn now to the interpretive aspects in which all the “physics” of EM theory resides. An essential ingredient in the invariance demonstration of §2 was the assumed invariance of the Hertzian field quantities, Eq. (2.6); viz., E E B B
 ' ,'
 = = . What can it
 mean that there is no covariant “scrambling” of electric and magnetic field components, as prescribed by Maxwell’s theory? Is not this scrambling an observable physical fact? The answer lies in the different physical definitions of Hertzian and Maxwellian field quantities. (Perhaps it would be wiser to show the distinction by explicit notational differences ... but, if the reader will keep in mind the considerations of this section, that complication will be unnecessary.) Maxwell’s theory, interpreted in the Einstein way, is physically as well as mathematically specialized: Mathematically, it lacks parameters descriptive of field sink (absorber, detector, etc.) motions in the observer’s frame of reference; so such relative motions are forbidden to occur. Thus the field sink is always tacitly at rest at the observer’s field point. Field sources, by contrast, can move, via the jparametrization. Physically, field detection processes describable by a given MaxwellEinstein inertial observer are restricted to those occurring in instruments, or at “test charges,” at rest in that observer’s system (at his field point). It is as if each field sink had rigidly attached to it its own “preferred observer,” or vice versa. The proof of this somewhat startling proposition is in the manifest lack of parameters: If test charges could move relative to Maxwellian observers, the field equations would contain explicit velocity parameters to describe such motions.
 Needless to say, such under-parametrized EM theory provides no foundation for a motional relativity theory – nor did Maxwell himself (we may assume) intend it to do so; for he seems to have thought primarily in terms of an ether “at rest.” The same “preferred observer” limitation implicit in Maxwell’s theory characterizes Einstein’s special relativity theory, which took over the Maxwellian EM formalism intact (including the modern “partial” meaning of ∂/∂t ). As a result, when different inertial laboratories are in different states of motion, each Maxwell-Einstein “inertial observer” must be equipped with his own comoving set of instruments for EM field or radiation detection. Each instrument is in effect associated with its own preferred observer. As it happens, the symmetrizing of preference among inertial observers (covariance) is logically quite different from the elimination of preference (invariance).
 To elaborate, the Einsteinian observer K has his own comoving E,B meter, stationary at his field point, and observer K' has his own E', B' meter, stationary at his field point – these primed and unprimed field detectors being entirely different macroscopic instruments in different states of motion. Since different measuring instruments are involved, it may seem surprising that there exists any relationship at all between their
 13
 readings. It is not surprising that the relationship is somewhat complicated: If the field points comoving with K and K' instantaneously coincide, then at the place and moment of their coincidence the electric and magnetic field vector (or tensor) components in the two systems are related by a mathematical rule of linear combination termed covariance. It is solely because measurements are made by different instruments in different states of motion that a purely electric field in one system “becomes” a mixed electric and magnetic field in the other. And that in turn traces directly to the parametrization of Maxwell’s equations (i.e., to the failure to parametrize sink motions). Covariant scrambling of field components is thus the hallmark of “fields” construed in the Maxwell way: By definition Maxwell-Einstein fields are what is detected by instruments comoving with inertial observers (permanently fixed at their field points).
 A tacit assumption implicit in the Maxwellian world view is the exact reproducibility of experiments or observations. The simultaneous presence of many macroinstruments at or near a momentarily-shared field point demands such reproducibility, if the readings of these different instruments are to be mutually consistent and are to yield numbers that accord with the linear-combination prescription of covariance. For the readings of separate instruments must represent physically independent “experiments” of measurement. As long as the “field” arises from the presence at or near the field point of a superabundance of field quanta, there is no reason to question the validity of such a traditional formulation of EM theory. But there is good reason to do so (as we shall see) in the case of few quanta.
 The definition of Hertzian fields is quite different, as to both physics and mathematics. Mathematically, we deal with a covering theory that is more richly parametrized. The total time derivative introduces an extra velocity-dimensioned parameter w, for which a physical interpretation is required. We do not wish to repeat Hertz’s mistake of interpreting w as a descriptor of ether velocity, nor Maxwell’s mistake of rigidly attaching sink to observer ( w ≡ 0 ). The reader has probably already guessed, then, that the natural employment for this parameter is as a descriptor of field-detector velocity. Thus it seems natural to correct Maxwell’s failure to parametrize sink motions by the following:
 Interpretation. The parameter w describes the field detector, sink, or radiation absorber velocity with respect to the inertial frame of the observer or his field point.
 About the physics of Hertzian fields we can say this: Now that relative motion of sinks or field-detection instruments is permitted through adequate mathematical parametrization, it is no longer necessary for each inertial observer to possess his own “private” field detector. A single “public” instrument will serve, since now each observer has his own independent means of describing the state of motion and predicting the readings of that instrument. Thus a given detector, being in a determinate state of uniform relative motion, will possess its own (different) numerical values of the components of w with respect to each differently-moving inertial observer.
 14
 This clarifies “invariance” and trivializes interpretation of the Hertzian field transformation equation (2.6), E E, B B
 ''
 = = . It means that all differently-moving observers,
 at any event of simultaneous coincidence of their field points with the given detector, must all read simultaneously from that detector’s digital read-out identically the same numbers. (Of course ... no other numbers are there to be read.) This is why form invariance and numerical invariance in this situation amount to the same thing, and we have not chosen to analyze the distinction. Incidentally, at the level of single-quantum detection, the numbers displayed on the readout of the single instrument need not be predictable, but they must be objectively factual, so that all observers agree on them after the fact. That requirement is satisfied by the Hertzian formulation, which therefore fits perfectly with quantum ideology.
 We see that the Hertzian and Maxwellian models differ in their physical pictures of the “field” measurement process as profoundly as in their formal mathematical properties of invariance and covariance. In the Hertzian case there is a single “public” field detector or radiation absorber – a unique instrument interrogated by a multitude of variously-moving inertial observers, each of whom assigns a different numerical value to the relative “detector velocity” parameter w. (Were we to remove our initial simplifying as
 sumption that w is a constant, which could easily be done(6), it is apparent that general relative motions of observers would be allowed – implying EM general invariance. General invariance, throughout physics, follows automatically from a policy of expressing all laws of nature solely in terms of kinematic invariants.) A single event of detection – and of simultaneous coincidence of field points – occurs within this unique instrument. After the fact, it is described invariantly by all observers – who at a given instant all read the same numbers from the instrument’s display.
 By contrast, in the Maxwellian case (as we noted) each inertial observer has his “private” field detector, a macroscopic instrument permanently at rest at the field point in his frame, with which he makes his own independent measurements. In this manyinstrument case there arises the practical problem that when field points coincide these numerous differently-moving detectors must apparently collide. But of course they can be made small and allowed near misses. As long as there are sufficient field quanta present at the momentarily shared field point to be shared among all these observers, leaving enough quanta detectable by each to allow the law of large numbers to smooth-out statistical discrepancies, the covariant relationships among measured field component values predicted by Maxwell-Einstein must obtain. Until about 1925 there was thus no information available to physicists that could have enabled them to make an uncontested choice between the Hertzian and Maxwellian formulations. At that time, however, knowledge of the microscopic non-reproducibility of experiments became common, as well as recognition of the uniqueness of individual quantum events. At the level of classical field theory the two formulations are equivalent – the Maxwellian and its Hertzian covering theory. But, insofar as quantum realities underlie classical EM theory and are to be recognized as ultimately dominant, or as a nonignorable limiting case, there can be no question that the invariant Hertzian singleinstrument approach offers the only possibility of a consistent EM theory extensible to the
 15
 quantum level. For if field theory is to be applied to the description of the individual quantum event, such as detection of a single photon, it is obvious that the Maxwellian many-instrument approach is bankrupt: a single-quantum detection event is physically unique and can therefore occur in at most a single macroscopic instrument. And only Hertz’s theory is parametrically empowered to describe the motion of that single instrument with respect to all inertial observers.
 7. Solution of the Hertzian Wave Equation
 By taking the curl of Eq. (2.1b), and applying (2.1a) and the condition that the field point is in vacuum (free space), we obtain at first order a Hertzian wave equation,
 −∇ + =
 2 2
 2
 2
 10
 EE
 c
 D
 Dt , (7.1)
 as well as a similar equation for B. Let us seek a solution of the form E = E( p) , where p t xk yk zk t
 x yz
 = ⋅− = + + −
 k r ω ω . (7.2) We find that
 ∇ = ++
 FHG IKJ = + + =
 22
 2
 2
 2
 2
 2
 222 2
 E E EE
 ( ) ( ) "( ) "
 p x y z p kkk pk
 x yz
 ∂ ∂
 ∂ ∂
 ∂
 ∂ d i , (7.3)
 where double-prime indicates two differentiations with respect to p. Similarly, using Eq. (1.1) we have D
 Dt p t p t t p
 2
 2
 22
 2
 2
 2
 E wE w w E
 () () ()
 = + ⋅∇
 FHG IKJ = + ⋅∇ + ⋅∇
 LNM OQP
 ∂ ∂
 ∂ ∂
 ∂
 ∂b g b g
 = − ⋅+ ⋅ = − ⋅
 ωω ω
 2 22
 2 wk wk E wk E
 b gb g b g
 " " . (7.4)
 From (7.1) and (7.4) it follows that
 −+ − ⋅
 RST UVW =
 kc
 2 2
 2
 10
 ω wk E
 b g " . (7.5)
 The vanishing of the coefficient of E" implies that
 ck = − ⋅
 ω w k or ω
 k ck
 = ± +k ⋅ w . (7.6)
 We may thus introduce a phase velocity u, measured with respect to the same laboratory system as is w, by means of
 uk c k
 = =± + ⋅
 ω k w . (7.7)
 The most general d’Alembert solution of the wave equation (for constant w) can be written, with the help of (7.2) and (7.7), as
 E Ekr kw E kr kw
 = ⋅+ − ⋅ + ⋅− + ⋅
 12
 kc t kc t
 c h c h , (7.8)
 where E1 E2
 , are arbitrary vector functions. A similar solution applies to the B-field. It
 might appear that these solutions predict a first-order observable effect of detector veloc
 ity on wave propagation speed. But in fact Potier’s principle(8,21), derived from Fermat’s principle, states that when the expression for phase velocity includes an additive scalar
 16
 product such as k ⋅ w , regardless of how the “velocity” vector w is interpreted physically, it can have no effect on observable aspects of EM propagation, such as spatial light paths, interference patterns, etc. This result is well known in connection with an ether wind in
 terpretation(21) of w, but is a mathematical theorem completely independent of the physical meaning assigned to that parameter. First-order laboratory-observable consequences of wave propagation are thus the same in Hertz’s and Maxwell’s theories. Higher-order considerations will be addressed in the next section.
 8. Higher-order (Neo-Hertzian) EM Theory
 Hertz’s invariant covering theory of Maxwell’s EM is physically valid only at first order. In this section (alone) we shall explore the subject of higher-order approximation. Classical mechanics expresses itself solely in terms of kinematic invariants, and it would seem that in this, as in much else, it could furnish a useful paradigm for all physical science. In seeking an EM theory valid at higher-orders of approximation than the first, then, the investigator’s first task must be to identify the higher-order invariants of kinematics. In Einstein’s special theory these are the proper space interval dσ and the proper
 time interval dτ . Here we shall review some speculations one of us has offered(8) about a possible approach to higher-order description. First, we notice an oddity: Einstein does not formulate his special theory explicitly in terms of the alleged kinematic invariants σ,τ . Rather, he employs noninvariant quantities in covariant combinations. This made sense in an era when physicists were convinced that the equations of EM could not be expressed in terms of manifest invariants. But we have seen that Hertz succeeded at first order in doing just that. So, in combination with Newtonian mechanics, Hertzian theory provides a first-order physics, complete in respect to both EM and mechanics, wherein the basic laws are all expressible solely in terms of kinematic invariants.
 The fact that σ,τ are not self-sufficient in expressing the laws of physics at any level of approximation suggests that at least one of these alleged “invariants” may be falsely identified. Because of their expression of spacetime symmetry, herein discredited (“broken”) already at first order, it would be folly in higher-order analysis to accept these symmetrical (alleged) invariants without questioning their empirical antecedents. In the absence of positive empirical evidence for the Lorentz contraction of extended material structures, one is in fact free to try the simplest alternative, which is that the spacelike invariant of kinematics is Euclidean distance (length). The timelike invariant we may take to be Einstein’s proper time interval dτ between two events, provided these events lie on the trajectory of a given particle; for in this case there is much empirical evidence, such
 as that of CERN(22), to support the hypothesis. Moreover, proper time τ possesses a simple and convincing operational definition (as “pocket watch time of the comoving observer”), whereas σ not only lacks an operational definition but lacks even a conceptual rationalization apart from spacetime symmetry. In this section we shall hence drop σ and assume as higher-order invariants:
 Timelike: Proper-time interval along a trajectory
 17
 Spacelike: Euclidean length interval.
 We wish to re-express the EM field equations (2.1) in terms of these hypothesized higher-order kinematic invariants. The first question about proper time is proper time of what? The answer must reflect our identification in §6 of the field detector (indubitably the central physical actor in any field theory) as the object that moves with velocity w with respect to the observer’s frame. We may picture a group of instruments, E and B detectors, source charge and current detectors, etc., as all moving together (idealized as a single instrument) through the field point at the moment of measurement or “detection.” So, how shall that moment be measured? Evidently by a clock comoving with, and part of, that same master instrument (conceived as a “multi-purpose detector”). We may consider the time shown by this comoving clock to be the proper time of the field detector, to be denoted as τd . In this way time measurement is operationally unified with field measurement.
 This τd identification might not be obvious, except for the fact that we have al
 ready specified our guiding principle to be the use of only kinematic invariants in reformulating Eqs. (2.1), and there is no other proper time than instrument time intrinsic to the problem. The only competitor is proper time of the observer, and this, as can readily be seen, is not intrinsic to the problem. The “field” is “created” not by the human observer, who plays no active role (except in certain highly dubious interpretations of quantum mechanics), but by and at the detection instrument or test charge ... this being the physical agent that “makes the measurement” of the photon or responds observably to the field.
 Our tentative higher-order invariant form of the EM equations, which we shall term neo-Hertzian, is identical to Eq. (2.1) but with formal replacement of “t” wherever it appears by “τd .” Thus D/Dt in neo-Hertzian theory is to be understood as D D d
 / τ, which we shall (in this section alone) write as d d d
 / τ , because of the greater familiarity
 to physicists of that notation for the total derivative. It may be computationally convenient to employ the familiar defining relationship, d dt dx dy dz c
 τd2 2 2 2 2 2
 =− + + =
 c h / invariant , (8.1)
 expressing the invariance of dτd (in which it must be remembered that the differentials
 refer to two neighboring events on the trajectory of a single material “particle,” here the detector). This permits expression of the total time derivative as d d
 dt d
 d
 dt w c
 d dt
 d dt
 dd
 d
 ττ γ
 = =− =
 1
 1 22
 / , (8.2)
 which allows the observer’s frame time t to be used – a great convenience for field or for many-body calculations. When this is done, and the wave equation is solved in precise analogy with the procedure of the preceding section, the resulting wave solution is found to have a phase velocity
 uk c w k
 = =± − + ⋅
 ω 2 2 k w , (8.3)
 18
 which generalizes to higher orders our previous Hertzian first-order phase velocity expression, Eq. (7.7). Further discussion and explicit derivation can be found in refer
 ences(8,10).
 The invariance of Eq. (2.6), E E
 ( ) '( ')
 p = p , implies that
 p= p → ⋅ − t = ⋅ − t
 ' ' ' ''
 k r kr
 ω ω , (8.4)
 which describes a constant phase-value of the propagating field in primed and unprimed frames. Suppose the detection instruments and clock are all at rest in the primed inertial system, so that v = w and Eq. (2.2) (spatial part) yields r r w
 ' = − t . Because of this assumed co-motion, detector proper time becomes identical with t' frame time, so that τd = t' = t / γ . Then, using these relations to eliminate r', t' , together with the fact that r, t are arbitrary, so that their coefficients vanish, we find two relations useful for applications,
 (I) k k
 ' = (8.5a) (II) ω γ ω
 '= −k⋅w
 b g . (8.5b)
 The first of these can be used to describe stellar aberration, the second, the Doppler effect. The latter being fairly straightforward, we shall confine our attention here to the problem of aberration, since relation (I) looks at first glance as if it could not possibly predict aberration. Also, it happens that our first-order Hertzian theory [in which (I) is also true] cannot predict stellar aberration, so this exercise will demonstrate the superiority of neo-Hertzian over Hertzian theory.
 9. Stellar Aberration
 Eq. (8.5a) states that the direction of light propagation is not altered by inertial transformations – it remains invariant. That is in strong contrast to special relativity theory, which explains stellar aberration by an effect of inertial transformation on the angle of light propagation. How can the present alternative be justified? Let us consider the simplest case of starlight coming straight down from the zenith, on a vertical which we shall suppose to be normal to the plane of the ecliptic. Referring to Eq. (8.3), and ignoring complications such as earth’s spin, we see that “detector” (our telescope) velocity w is parallel to the earth’s orbital velocity and normal to the invariant light propagation direction k; so k ⋅ w = 0 and (8.3) (representing vertical phase velocity) reduces to
 u cw
 = 2 − 2 , (9.1) where we have chosen the plus sign in (8.3) arbitrarily. We know that in the inertial system comoving with the detector the latter is at rest and Maxwellian physics ( u = c corre
 sponding to the special case w = 0) applies. Thus relative to the telescope tube or to the “inertial” observer at rest on the earth’s surface the speed of light must be precisely c. For the light to move at speed c with respect to the telescope tube (i.e., parallel to its axis) and with speed u<c with respect to the vertical, as implied by (9.1), it is necessary that the telescope tube be tilted from the vertical by some small angle α . In this case we can as
 19
 sign to the detector velocity w a magnitude equal to vorb , the earth’s orbital velocity. By similar Pythagorean triangles it is easily seen that α = −
 sin /
 1v c
 orb
 b g , which agrees with
 the observed aberration constant of about 20.5 arc-second. [This argument holds in neoHertzian but not in Hertzian theory – since the latter describes phase velocity by (7.7), not by (8.3) or (9.1).]
 Actually, it is slightly more complicated than this, since w is detector velocity at time t1 relative to detector velocity at an earlier time t0 , and α is telescope tilt angle
 relative to tilt at that earlier time – the observability of stellar aberration being wholly dependent on non-inertiality of the earth’s motion (a fact that makes it somewhat paradoxical that special relativity, a theory of inertial motions, can predict it at all). Thus all observable quantities are detector-relative. A consequence is that the figure of aberration is traced on the celestial sphere in phase quadrature to the earth’s orbital motion. The velocity of the light source or of any intervening medium does not come into the analysis.
 In summary, in this section we have dealt with modified field equations constituting a possible candidate for higher-order formal EM invariance termed “neo-Hertzian.” In this context we have seen that the new convective parameter w introduced into EM theory to secure first-order invariance has an influence upon the phase velocity of radiation that itself has no direct first-order effects (Potier’s principle), but that does have collateral second-order consequences affecting, for instance, the description of stellar aber
 ration. A more thorough analysis(23) of the latter shows that neo-Hertzian theory predicts the figure of aberration at all angles of light propagation to coincide exactly with a projection of the earth’s orbit onto the celestial sphere, to within errors of second order; whereas Einstein’s theory predicts first-order departures from such a projected figure. Unfortunately, the first-order angular discrepancy term predicted by special relativity, namely, v c
 orb / 2 sin cos
 b g θ θ (where θ is the polar angle measured from the normal to the
 ecliptic plane), is too small to be observed astronomically unless a resolution about 50 times better than the original design value of the Hubble (satellite) telescope can be
 achieved. The present theory predicts(23) zero for this first-order term. It is to be hoped that in the future crucial observations will become technically feasible. If so, they should prove decisive for or against neo-Hertzian EM.
 10. EM Potentials and the Force Law
 We return now to considerations of first-order physics. It is a moot point whether the fields or the potentials are to be regarded as more fundamental in EM theory. Max
 well, as reworked by Heaviside (though not in the original(1)), favored the fields. Observations such as the Aharonov-Bohm effect and the Marinov motor (discussed below) seem to favor the potentials. For present purposes this issue need not be resolved. As with the fields, we assume invariance under inertial transformations, φφ
 ', '
 ==
 A A , (10.1) and define the fields in terms of the potentials by means of the invariant prescriptions
 20
 E A BA
 = −∇ − = ∇ ×
 φD
 Dt , . (10.2)
 [We return here to the use of D for the total time derivative, as given by Eq. (1.1). The use of the total time derivative, written as dA/dt, in the context of E-definition – on the
 evidence of Faraday’s observations – can be found in some older books; e.g., Smythe(24).
 It has also received more recent endorsement (25).] Of particular significance is the above invariant definition of E. It will be useful to clarify its import by means of the standard
 vector identity(26)
 ∇ ⋅ = ⋅∇ + ⋅∇ + × ∇× + × ∇×
 ab a b b a a b b a
 b g b g b g b g b g.
 In this we make the replacements a = v , where v is “test particle” velocity, assumed constant (independent of space variables), and b = A . We then obtain the simpler identity
 v A vA v A
 × ∇× = ∇ ⋅ − ⋅∇
 b g b g b g . (10.3)
 Identifying our field detector with the “test particle,” so that w and v become the same, and applying Eqs. (1.1) and (10.3) to the E-definition in (10.2), with ∇ × =
 A B , we find
 E A Aw A
 = −∇ − = −∇ − − ⋅ ∇
 φ φ∂
 ∂
 D
 Dt t b g , or equivalently (10.4a)
 E A w A wA A w B wA
 = −∇ − + × ∇ × − ∇ ⋅ = −∇ − + × − ∇ ⋅
 φ∂
 ∂ φ∂
 t ∂t
 b g b g b g. (10.4b)
 The form (10.4a) is directly useful in computations, as Wesley13 has shown. The form (10.4b) is more revealing conceptually, as it demonstrates that our invariance requirement does not violate the Lorentz force law based on w × B (usually written v × B , which is the same thing, since here the test particle and the detector co-move), but includes the Lorentz law along with an extra term, −∇ ⋅
 wA
 b g.
 What can be said about this extra term? First, that as the gradient of a scalar function it will in general not be observable in experiments involving currents flowing in closed circuits – since such a gradient will normally integrate to zero around a closed
 loop(27). But not all “currents” flow in closed loops. The random charge motions occurring in a plasma, for instance, do not meet either (1) the closed-loop condition or (2) the w ⋅ A = 0 perpendicularity condition (obeyed by homopolar generators and motors) needed to eliminate the last (gradient) term in the E-field expression, Eq. (10.4b). The extra term should produce some anomaly in the rate of plasma diffusion, but this has never, to our knowledge, been checked empirically.
 Since the Lorentz force law is already incorporated in the invariant electric field, expressed in form (10.4b), it is apparent that our insistence on manifest invariance has
 21
 borne valuable fruit: We have only to multiply the E in (10.4b) by the test charge q to obtain a force law:
 F E A w B wA
 = = −∇ − + × − ∇ ⋅
 LNM OQP
 qq t
 φ∂
 ∂ b g . (10.5)
 According to (10.4a), this can alternatively be written in the simpler form
 FE A
 = = −∇ −
 FHG IKJ
 qq D
 Dt
 φ . (10.6)
 (In case the reader prefers factors of c in such equations, they can be supplied by everywhere replacing t with ct and w with w/c.)
 We trust this improvement in logical economy is not lost on the reader. In Maxwell’s noninvariant theory, as well as in Einstein’s covariant apotheosis of it, the field equations do not suffice to describe the EM force acting on a test charge. A separate force
 law, variously attributed to Grassmann, Lorentz, and Laplace(27), must be postulated. Here we see that by simply requiring manifest first-order formal invariance of the E-field definition in terms of the potentials (implying use of the total time derivative) we automatically obtain a force law, F E
 = q , directly expressible in terms of a solution of the
 field equations. Thus one of the undocumented features of the Hertzian invariant covering theory of Maxwell’s equations (an extra dividend, so to speak) is automatic provision of a force law self-contained within the field equations, without need for extra postulation. Hertz’s covering theory of Maxwell’s equations thus attains the physical sufficiency attributed to Maxwell’s theory, but with improved logical economy.
 However, the new force law is not quite the same as the old one. There is that extra term – an enhancement. What evidence suggests that that may have physical content? For one thing, it could be speculated that the extra gradient force term in (10.5) might to some degree account for the anomalies in plasma diffusion rates that used to be regularly reported in an era of outmoded honesty, before such reporting became politically inexpedient. Another bit of evidence is furnished by the so-called “Marinov motor.” This has been repeatedly observed to operate in a manner very difficult to reconcile with
 the Lorentz force law(15), and possibly compatible with observable action of the extra
 force term(13,14) in Eq. (10.5). Unfortunately, it is as yet impossible to give a final judgment on this hypothesis. The empirical observations have shown problems of reproducibility, there is a sign problem, the assumption of w constancy may be challenged, and the necessity to choose a preferred gauge for the vector potential may occasion aesthetic discomfort. For such reasons, plus the still not fully resolved physics of the distinction between ponderomotive and electromotive forces, it may prove necessary ultimately to resort to really radical explanations of this type of motor, such as might emerge from a
 modernization of Weber-type electrodynamics(28). Such problems take us far beyond the purview of the present paper, so will not be confronted here.
 22
 11. Summation
 The following points merit re-emphasis:
 • A first-order (Galilean) invariant EM formulation due to Hertz, by virtue of being a covering theory of Maxwell’s equations, suffices to describe generally-known (first-order) EM physics.
 • It may also contribute to the description of certain less-known empiricisms, such as the Marinov motor, for the treatment of which Maxwell’s theory, supplemented by the Lorentz force law, appears to be inadequate. (This remains to be shown.)
 • It achieves such possibly enhanced physical sufficiency through improved logical economy, in that the force law, of the simplified form F E
 = q , is given by Esolution of the (Hertzian) field equations, in contrast to being separately postulated.
 • The formal device by which invariance is achieved – the use of a total time derivative – has been shown to be mathematically necessary, under the proviso that the coordinate transformation precedes the time differentiations prescribed by field theory.
 • The Hertzian form of field theory relates (invariantly) the field component readings of a single instrument – the field detector, sink, or radiation absorber – as seen by a plurality of observers whose field points momentarily coincide with that instrument.
 • The Maxwell-Einstein form of field theory relates (covariantly) the field component readings of a plurality of differently-moving field-detection instruments, permanently fixed at momentarily coincident field points. These readings are definite numbers that demand experimental reproducibility in order to acquire physical meaning.
 • Such reproducibility is attainable only in the many-quantum limit. On pushing field physics in the opposite direction, to the single-quantum limit, it becomes obvious that at most a single macro instrument can be involved in detecting the single quantum. Thus there exists an important physical limit in which the Hertzian one-instrument approach (invariance) is plainly superior to the Maxwell-Einstein many-instrument approach (covariance). This is true independently of the manifest ideological superiority of genuine invariance over mathematical simulacra such as covariance.
 • This could explain some of the difficulties that have been experienced in forcing a marriage between quantum mechanics and relativity theory. The Maxwellian paradigm incorporated in relativity theory and hitherto universally endorsed by physicists entails a “spacetime symmetry” (essential to covariance and deriving mathematically from explicit use of the first-order noninvariant operator ∂ /∂t in the field equations, where it appears symmetrically with the first-order invariant operators ∂ ∂ ∂ ∂ ∂ ∂
 /,/,/
 x y z ) that
 exists at first order neither in nature (operationally) nor in the Hertzian world of invariant total time derivatives.
 23
 • If, as we suggest, spacetime symmetry does not physically exist, it becomes possible to formulate all laws of physics explicitly and solely in terms of kinematic invariants. The first step toward progress along this path is identification of the higherorder (spacelike and timelike) invariants of kinematics. These must differ from those of Einstein’s special theory in that they cannot be presumed spacetime symmetrical.
 • An attempt to carry the Hertzian invariant formulation beyond first order, termed neo-Hertzian EM, has been outlined, based on Euclidean length and field detector particle proper time as the assumed higher-order invariants. By this approach all laws of physics are to be expressed solely and explicitly in terms of kinematic invariants.
 • Observations crucial to the decision between neo-Hertzian and MaxwellEinstein EM theory, involving stellar aberration, will become possible when satelliteborne telescope resolution is improved by about two orders of magnitude over that of the present Hubble telescope.
 References
 1. J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954). 2. E. E. N. Mascart, “Sur les modifications qu’éprove la lumière par suite du mouvement de la source lumineuse et du mouvement de l’observateur,” Ann. Sci. École Norm. 1, 157-214 (1872); also E. Mascart and J. Jamin, Ann. Sci. École Norm. 3, 336-347 (1874). 3. H. R. Hertz, Electric Waves (Dover, New York, 1962), Chap. 14. 4. F. B. Hildebrand, Advanced Calculus for Engineers (Prentice-Hall, New York, 1948), p. 348. 5. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1965), p. 173.
 6. C. I. Mocanu, Hertzian Relativistic Electrodynamics and its Associated Mechanics (Hadronic Press, Palm Harbor, FL, 1991). 7. T. E. Phipps, Jr., “Neo-Hertzian Electromagnetism I & II,” Ann. de la Fond. L. de Broglie 8, 325-344 (1983) and 9, 41-64 (1984).
 8. T. E. Phipps, Jr., Heretical Verities: Mathematical Themes in Physical Description (Classic Non-fiction Library, Urbana, IL, 1986). 9. T. E. Phipps, Jr., “On Hertz’s Invariant Form of Maxwell’s Equations,” Phys. Ess. 6, 249 (1993). 10. T. E. Phipps, Jr., “Hertzian Invariant Forms of Electromagnetism,” in Advanced
 Electromagnetism Foundations, Theory and Applications ed. by T. W. Barrett and D. M. Grimes (World Scientific, Singapore, 1995). 11. A. Eichenwald, “Ueber die Magnetische Felde,” Ann. Physik 11, 1-30 and 421-441 (1903). 12. A. Einstein, “Zur Elektrodynamik bewegter Koerper,” Ann. Phys., Lpz. 17, 891-921 (1905). 13. J. P. Wesley, “The Marinov Motor, Motional Induction without a Magnetic B Field,” Apeiron 5, 219-225 (1998).
 24
 14. T. E. Phipps, Jr., “Observations of the Marinov Motor,” Apeiron 5, 193-208 (1998). 15. J. D. Kooistra, “The Marinov Motor – A Brief History of Mine,” Inf. Energy 3, #17, 40-42 (1997). 16. M. Jammer and J. Stachel, “If Maxwell had worked between Ampère and Faraday: an historical fable with a pedagogical moral,” Am. J. Phys. 48, 5 (1980). 17. The term ‘invariant’ is here used in the sense of physics, which is less restrictive than that of mathematics. Mathematically, an invariant is a function solely of the coefficients of F, while the term ‘covariant’ applies if the independent variables are also involved. The latter term has also acquired a different sense in physics. For a detailed statement of the mathematical meanings, cf. Algebra of Quantics by Edwin Bailey Elliott (Oxford,
 1913), 2nd ed., p. 3 et seq. 18. H. W. Milnes, “The Total Time Derivative, the Heat Equation, the Hertz-Phipps Modification of Maxwell’s Equations and the Transmission Theory of Light Propagation," The Toth-Maatian Review 13, #3, 6205-19 (1996). 19. R. Courant, Differential and Integral Calculus (Nordemann, 1938), vol. 2, pp. 76-77, translated by E. J. McShane. 20. The monk Jean Buridan was the first to recognize what are now called the two first laws of motion. Newton’s reputation is in no way tarnished by according Buridan his due, naming the laws after him. The interested reader will find a review of Buridan’s
 work in Le Système du Monde: Histoire des Doctrines Cosmologiques de Platon à Copernic, by Pierre Duhem. A secondary reference is Benevolent Living (Hope Publishing House, Pasadena, 1990), p. 36, by R. Hazelett and D. Turner, where the pertinent passage is quoted from Buridan’s writings. 21. R. Newburgh, “Fresnel Drag and the Principle of Relativity,” Isis 65, 379-386 (1974); also R. G. Newburgh and O. Costa de Beauregard, “Experimental Search for Anisotropy in the Speed of Light,” Am. J. Phys. 43, 528-530 (1975). 22. J. Bailey et al., “Measurements of relativistic time dilatation for positive and negative muons in a circular orbit,” Nature 268, 301-304 (1977). 23. T. E. Phipps, Jr., “Stellar Aberration from the Standpoint of the Radiation Convection Hypothesis,” Phys. Ess. 4, 368-371 (1991). 24. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, NY, 1950), p. 308. 25. R. Angulo, O. Rodriguez and G. Spavieri, “Does the Expression for the Lorentz Force Need to be Modified?,” Hadronic J. 20, 621-630 (1997). 26. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), App. II. 27. O. Costa de Beauregard, Précis of Special Relativity (Academic Press, New York, 1966), p. 61. 28. A. K. T. Assis, Weber’s Electrodynamics (Kluwer, Dordrecht, 1994).
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 Phase and frequency shift in a Michelson interferometer
 Wolfgang Engelhardta)
 Fasaneriestrasse 8, D-80636 M ̈unchen, Germany
 (Received 26 June 2013; accepted 7 October 2014; published online 21 November 2014)
 Abstract: Traditionally, the outcome of Michelson’s interference experiment has been interpreted as evidence against the existence of a luminiferous medium called “ether.” Einstein, however, emphasized in 1920 that an ether must exist in spite of Michelson’s null result. In this paper, it is shown that a medium theory—be it for light or for sound—actually predicts the observed null result. Michelson expected a gradual fringe shift when his apparatus was turned in the “ether wind.” Such a phase change would, however, require a temporary frequency change in one of the interferometer arms. Since wind does not alter the frequency in the interferometer, a phase shift cannot occur either. CV 2014 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-27.4.586]
 Re ́sume ́: Habituellement, l’issue de l’expe ́rience sur l’interfe ́rence de Michelson a e ́te ́ interpre ́te ́e comme e ́vidence contre l’existence d’un medium luminife`re appele ́ “e ́ther.” Einstein, cependant, a insiste ́, en 1920, sur le fait qu’un e ́ther doit exister en de ́pit du re ́sultat nul de Michelson. Dans cet article il est de ́montre ́ qu’une the ́orie de medium—que ce soit pour la lumie`re ou pour le sonpre ́dit, en fait, le re ́sultat nul observe ́. Michelson s’attendait a` un changement graduel des franges d’interfe ́rence quand son appareil e ́tait tourne ́ dans le “vent de l’e ́ther.” Un tel changement de phase exigerait, cependant, un changement temporaire de fre ́quence dans un des bras de l’interfe ́rome`tre. Puisque le vent ne change pas la fre ́quence dans l’interfe ́rome`tre, un changement de phase ne peut se produire non plus.
 Key words: Interferometry; Phase Measurement; Fundamental Tests; Ether.
 I. INTRODUCTION
 In his Electromagnetic Theory of Light,1 Maxwell derived a wave equation for the vector potential which was supposed to describe the propagation of electromagnetic perturbations. Like air is necessary to carry sound, an analogous medium—called “ether”—was required to transmit electromagnetic waves with a characteristic velocity c. Based on this conception, one could expect an “ether wind” to blow on earth due to the planet’s motion through space. Maxwell, who strongly believed in the ether hypothesis,b) proposed to measure the strength of this wind by determining the round trip velocity of light.c) It was Michelson who took up the idea and invented his famous interferometer which appeared suitable to measure the ether wind by comparison of the light velocity in the two arms of his device. To his amazement, he obtained a null result in 1881,4 which was later confirmed by himself and Morley,5 as well as by many other experimenters. A survey of the various attempts to measure an
 ether drift is given by Galaev.6 Some experimenters claimed having obtained a positive result which, however, could not be confirmed by others. Galaev himself reported a positive result using a Rozhdestvensky (Mach–Zehnder) interferometer, not a Michelson interferometer. The interpretation relies on a theory of some ether viscosity which is beyond the scope of this paper. The special theory of relativity (STR) has a simple ad hoc “explanation” for Michelson’s null result: It postulates the constancy of the light velocity in all reference frames regardless of their motional state. As a consequence, time had to be transformed according to a rule which was proposed by Voigt in 1887.7 Whereas Voigt considered his transformation as valid for waves in all elastic media including sound, Lorentz applied it only to electromagnetic waves.8 This enabled him to explain Michelson’s experiment in a more general way than by his previous contraction hypothesis.9 Since then, Michelson’s experiment attained more and more the status of an experimentum crucis, a cornerstone proving the validity of STR. In 1905, however, Einstein based his own paper on relativity10 on the Voigt–Lorentz transformation, not so much on the outcome of Michelson’s experiment. According to Holton,11 he did either not know of this experiment or he did not regard it as pertinent. He may have read Voigt’s article of 188712 or Cohn’s paper of 190113 which both offered alternative explanations of the null result in terms of classical electrodynamics and the pertaining ether theory.
 a)wolfgangw.engelhardt@t-online.de b)“... Hence all these theories lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.”2 c)“[I]n the terrestrial methods of determining the velocity of light, the light comes back along the same path again, so that the velocity of the earth with respect to the ether would alter the time [interval] of the double passage...”3
 0836-1398/2013/27(4)/586/5/$25.00 586 CV 2014 Physics Essays Publication
 PHYSICS ESSAYS 27, 4 (2014)
 In this paper, we show that Einstein’s primary intuitiveness was correct: It is not possible to measure any ether wind with Michelson’s device. The reason for this is that a phase change can only be brought about by a temporary frequency change, as will be shown in due course. Since the latter does not occur due to wind, as will also be shown, a null result must be expected. This is true not only for light but also for sound, as will be shown explicitly in Sec. III. Before then, in Sec. II, the connection between phase shift and frequency change in a laser interferometer will be studied. Such a device is used as a length measurement apparatus and constructed on the basis of a Michelson interferometer. Finally, concluding remarks will be presented in Sec. IV.
 II. PHASE SHIFT AND THE PERTAINING FREQUENCY CHANGE DUE TO A MIRROR DISPLACEMENT IN A LENGTH COMPARATOR BASED ON MICHELSON’S INTERFEROMETER
 In 1892–1893, Michelson and Benoıˆt14 succeeded in determining the length of the standard meter in units of the red cadmium line. They used a Michelson interferometer and counted the number of fringes recorded by a detector when the reflecting mirror in one of the arms was moved. This method is still used in modern laser interferometers where coherence lengths of the laser light with more than 50 m are exploited to achieve an extremely high accuracy (Fig. 1). The displacement d of the mirror in the horizontal arm is compared to the wavelength of the laser light by counting the number of fringes which are produced by interference of the horizontal with the vertical light beam. There is a simple relationship of the number of fringes m as a function of the wavelength k and the distance d
 m 1⁄4 2d=k: (1)
 In order to derive this, we describe the light beams in terms of plane waves starting at the beam splitter. When they reunite (interfere) after reflection at the solid mirrors, we may write for the vertical beam
 A cosðk2L  xtÞ; (2)
 and for the horizontal beam
 A cosðk2ðL þ dÞ  xtÞ; (3)
 where the wave vector may be expressed by the wavelength: k 1⁄4 2 p=k. Superposition of the two expressions yields
 2A cosðkdÞ cosðkð2L þ dÞ  xtÞ; (4)
 which is a plane wave with a new amplitude and phase oscillating with the same frequency x as the waves of the original beams. The amplitude is a periodic function of the displacement d and has an extremum when (1) is satisfied. Thus, by counting fringes, while the mirror is moved one can express the total distance d in units of the wavelength k. Note that the frequency of the wave in the horizontal arm cannot stay constant during the motion of the mirror due to the Doppler effect of the light reflected at the receding mirror. This dependence of x on the velocity of the moving mirror may be included in Eq. (3) by substituting d 1⁄4 vt, which yields
 A cosðk2L  ðx  2kvÞtÞ 1⁄4 A cosðk2L  ð1  2v=cÞxtÞ;
 (5)
 where we have also inserted the phase velocity c 1⁄4 x=k of the wave. This relationship reflects the frequency change ð1  2v=cÞx (in first order of v=c) caused by the Doppler effect at the receding mirror. It is important to realize that a change in phase is inevitably brought about by a temporary change in frequency. In modern laser interferometers, this frequency shift is in fact measured, in order to determine the direction of the mirror displacement which would remain ambiguous by counting just fringes.14 Hence, it is obvious that a variation of the phase in one of the beams—leading to a fringe shift at the detector—can never occur unless the frequency in one of the beams differs during the time when the phase is shifted. At constant frequency in both arms, the phases would be strictly locked, thus preventing a fringe shift. One may visualize this situation by considering two wheels rotating in phase [Fig. 2(a)]. In order to achieve a phase shift between the two wheels, it is necessary to break the rigid coupling and rotate one of the wheels somewhat faster [Fig. 2(b)]. When a certain phase shift is established, the two wheels are coupled again and rotate at the same frequency as before [Fig. 2(c)].
 FIG. 1. Michelson interferometer as a length comparator.
 FIG. 2. (a) Two rigidly coupled wheels rotating in phase at frequency x1. (b) Wheel 2 decoupled from 1 and rotating at a higher frequency x2. (c) Both wheels coupled again after a phase shift of Du 1⁄4 p=2.
 Physics Essays 27, 4 (2014) 587
 Indeed, precisely this situation occurs when a car travels along a road. The phase relation between the two wheels on one axle remains constant on the straights, as the frequency of the wheels is also equal. As a bend is navigated, the frequency of the outer wheel increases relative to that of the inner wheel, which leads to a phase shift. The phases will be locked again, once the car continues its journey along the next straight. This consideration should be kept in mind when the effect of wind blowing through an interferometer is discussed in Sec. III.
 III. DOPPLER EFFECT IN A MICHELSON INTERFEROMETER
 Now that the principle of intimate connection between continuous phase change and temporary frequency change has been explained, we turn to the Michelson–Morley interferometer where the distances between the mirrors are kept constant, but an ether wind was expected to blow through the interferometer. First, an acoustic interferometer working with sound waves instead of light is considered, since the physics of sound is well known and entirely understood. A wind blowing along the horizontal arm of this interferometer will be equivalent of both the beam splitter and the mirror moving through a still medium. Therefore, the Doppler effect due to the velocity of beam splitter and mirror through the medium may be considered in order to investigate the effect of such a wind on the stationary interferometer. The important point to note here is that the Doppler effect differs depending on whether the detector (Fig. 3) or the source (Fig. 4) is moving. Let us now assume a wave traveling in still air along the horizontal arm from the beam splitter to the mirror. Using plane wave approximation as before, we have
 A cosðkx  x0tÞ: (6)
 Initially, the wind effects on the mirror—which is taken as a detector—and the beam splitter—which acts as a sourcewill be considered separately. A wind blowing away from the mirror toward the beam splitter is analogous to the detector (mirror) traveling in the wave field on the x-axis with constant velocity v (Fig. 3).
 The frequency the detector experiences may then be obtained by including the velocity term in Eq. (6) and resolving accordingly
 A cosðkðx0 þ vtÞ  x0tÞ 1⁄4 A cosðkx0  ðx0  kvÞtÞ: (7)
 Comparing Eq. (7) with Eq. (6) yields the new frequency
 x1 1⁄4 x0 1  v=cS
 ð Þ; (8)
 where cS 1⁄4 x0=k is the phase velocity of sound. Thus, the moving detector (receding mirror) experiences a wave with a reduced frequency x1. Likewise, the effect of wind blowing from the detector toward the source can be described by considering the case of the source (the beam splitter) moving with velocity v toward the detector (mirror). In this case, the wave fronts are compressed in front of the source as sketched in Fig. 4. As the source moves a distance 2pv=x0 between the emissions of two consecutive wave crests, the wavelength is shortened by the amount
 k 1⁄4 k0 1  v=cS
 ð Þ: (9)
 In this case, the detector measures the frequency
 x2 1⁄4 2pcS
 k0 1  v=cS
 ð Þ 1⁄4 x0
 1  v=cS
 : (10)
 Since a wind blowing along the interferometer’s horizontal arm is in effect the case of both source and detector moving with the same velocity v, the Doppler effects of “moving detector” Eq. (8) and “moving source” Eq. (10) cancel out so that the detector measures simply
 x 1⁄4 x0: (11)
 Naturally, the same consideration applies to the wave reflected back from the receding mirror, which is now a source, to the forward moving beam splitter which is now taken as a detector. This means that wind has no influence whatsoever on the frequency in the acoustic Michelson interferometer.
 FIG. 3. Doppler effect with moving detector.
 FIG. 4. Doppler effect with moving source.
 588 Physics Essays 27, 4 (2014)
 Consequently, this result proves that the frequency of the two interfering waves is not influenced when the direction of the wind in the two arms is interchanged by turning the interferometer by 90. Since we already saw in Sec. II that a phase shift cannot occur under constant frequency [Fig. 2(a)], a change in the interference pattern is also prevented by the physics of the Doppler effect during the turning of the interferometer. In the optical Michelson interferometer, similar Doppler formulae apply. In Ref. 15, we have derived the relativistically correct formulae under conservation of momentum and energy, when photons interact with matter. The relevant “collision physics” is the same as one employs in the treatment of the Compton effect. We obtained for a moving detector
 x1 1⁄4 x0
 1  ðv=cÞ cos a
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1  v2=c2
 p ; (12)
 and for a moving source
 x2 1⁄4 x0
 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1  v2=c2
 p
 1  ðv=cÞ cos a ; (13)
 where a denotes the angle between the velocity and the momentum vector of the photons (i.e., wave vector in the wave picture). These formulae have been verified in a number of experiments which were quoted in Ref. 15. They were also derived from the Lorentz transformation both in Ref. 15 and by Einstein himself in Ref. 16. In principle, they have the same property as the acoustic formulae (8) and (10): When both source and detector move in the same direction resulting in an ether wind blowing between them, the velocity dependent factors cancel exactly which means that a Doppler effect between source and detector does not arise as there is no relative velocity. Without a temporary frequency change, however, in at least one of the interferometer arms, a phase shift cannot be observed either. This was precisely what Michelson found.
 IV. CONCLUDING REMARKS
 The usual interpretation of Michelson’s experimentwhich is reiterated in all textbooks dealing with the subjectfollows Michelson’s original idea based on a “swimmer” analogy, namely, that the phase shift between the interfering waves can be calculated from the average time a light beam (a swimmer) spends traveling between the beam splitter and the mirror. If there is an ether wind in the horizontal arm, for example, the average travel time between beam splitter and back reflecting mirror was calculated by the relationship
 s1⁄41
 2
 L
 cvþ L
 cþv
 1⁄41
 2
 Lðc þ vÞ þ Lðc  vÞ c2  v2
 1⁄4 Lc
 c2  v2 : (14)
 In first order the velocity cancels, but in second order there is still an influence of the velocity on the average travel time which Michelson attempted to measure. Rather than drawing an analogy with solid particles (swimmers), it is more appropriate in the field of interferometry to consider the average
 wavelength of the beams traveling forth and back with different velocities due to the ether wind. For the purpose of calculating the interference pattern, it is in fact the average wavelength which is relevant and may be determined from Eq. (9) (which holds also for light)
 k1⁄41
 2 1⁄2k0ð1  v=cÞ þ k0ð1 þ v=cÞ 1⁄4 k0: (15)
 The cancellation of the velocity is complete so that wind does not lead to a phase difference between the waves in the horizontal and the vertical arm. Our analysis shows that the outcome of Michelson’s interference experiment has no relevance for the existence of an ether. Einstein himself had reintroduced the ether concept in 1920d) being well aware both of his own theory of 1905 and of the precise MM-results5 of 1887. The measurement of the solar system’s velocity with respect to the CMB by the COBE satellite18 also lends strong support for the existence of an absolute system in which light travels with constant velocity in all directions in agreement with Einstein’s second postulate of 1905. Einstein’s first postulate, however, is at stake when considering that Michelson’s experiment cannot disprove the ether in combination with the experimentally proven existence of a distinguished system that may be identified with Maxwell’s or Einstein’s ether.
 ACKNOWLEDGMENTS
 The author appreciates Dr. Peter Enders’ efforts in scrutinizing the paper and making interesting comments. He is particularly grateful for his son’s, Dr. Markus Engelhardt, careful reading of the manuscript and suggesting many improvements for a clearer presentation.
 1J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 2 (Dover Publications, Inc., New York, 1954), Chap. XX. 2J. C. Maxwell, ibid., Article 866. 3J. C. Maxwell, Nature, XXI, 315 (1880). 4A. A. Michelson, Am. J. Sci. 22, 120 (1881). 5A. A. Michelson and E. W. Morley, Am. J. Sci. 34, 333 (1887). 6Y. M. Galaev, Spacetime Subst. 3, 207 (2002).
 7W. Voigt, Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–Universita ̈t zu G ̈ottingen, Vol. 1887, Nr. 2, p. 41; See http://www.digizeitschriften.de/dms/toc/?PPN=PPN 252457072_1887. 8H. A. Lorentz, Proc. Acad. Sci. Amsterdam 6, 809 (1904).
 9H. A. Lorentz, Zittingsverslagen der Akad. v. Wet. te Amsterdam (Royal Netherlands Academy of Arts and Sciences, Amsterdam, Netherlands, 1892–1893), p. 74. 10A. Einstein, Ann. Phys. 17, 891 (1905). 11G. Holton, Am. J. Phys. 37, 968 (1969).
 12W. Voigt, Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–Universita ̈t zu G ̈ottingen, Vol. 1887,
 d)“Gema ̈ß der allgemeinen Relativit ̈atstheorie ist ein Raum ohne  ̈Ather undenkbar; denn in einem solchen ga ̈be es nicht nur keine Lichtfortpflanzung, sondern auch keine Existenzmo ̈glichkeit von Maßsta ̈ben und Uhren, also auch keine r ̈aumlich-zeitlichen Entfernungen im Sinne der Physik.”17. English translation republished in: Sidelights on Relativity (Methuen, London, 1922): “According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense.”
 Physics Essays 27, 4 (2014) 589
 Nr. 8, p. 177; See http://www.digizeitschriften.de/dms/toc/?PPN=PPN 252457072_1887.
 13E. Cohn, Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften zu G ̈ottingen, Vol. 1901, 74; See http://www.digizeitschriften.de/dms/ toc/?PPN=PPN252457811_1901. 14A. A. Michelson and J.-R. Benoıˆt, Trav. Mem. Bur. Int. Poids Mes. 11, 85 (1895); J. N. Dukes and G. B. Gordon, Hewlett-Packard J., 21(12), 2–8 (1970).
 15W. Engelhardt, Apeiron 10, 29 (2003). 16A. Einstein, Jahrb. Radioaktivita ̈t Elektron. 4, 411 (1907). 17A. Einstein, A ̈ther und Relativit ̈atstheorie (Reichs-Universita ̈t Leiden, Springer, Berlin, 1920). 18G. F. Smoot, M. V. Gorenstein, and R. A. Muller, Phys. Rev. Lett. 39, 898 (1977); B. Levi, Phys. Today 45(5), 17 (1992); See http://aether.lbl.gov/ www/projects/u2/ for COBE DMR-data.
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 CONTENT
 Written 1916
 Translated: Robert W. Lawson (Authorised translation)
 Preface
 Part I: The Special Theory of Relativity 01. Physical Meaning of Geometrical Propositions
 02. The System of Co-ordinates
 03. Space and Time in Classical Mechanics
 04. The Galileian System of Co-ordinates
 05. The Principle of Relativity (in the Restricted Sense)
 06. The Theorem of the Addition of Velocities employed in Classical Mechanics
 07. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity
 08. On the Idea of Time in Physics
 09. The Relativity of Simultaneity
 10. On the Relativity of the Conception of Distance
 11. The Lorentz Transformation
 12. The Behaviour of Measuring-Rods and Clocks in Motion
 13. Theorem of the Addition of Velocities. The Experiment of Fizeau
 14. The Hueristic Value of the Theory of Relativity
 15. General Results of the Theory
 16. Experience and the Special Theory of Relativity
 17. Minkowski's Four-dimensial Space
 Part II: The General Theory of Relativity 18. Special and General Principle of Relativity
 19. The Gravitational Field
 20. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity
 21. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?
 22. A Few Inferences from the General Principle of Relativity
 23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
 24. Euclidean and non-Euclidean Continuum
 25. Gaussian Co-ordinates
 26. The Space-Time Continuum of the Speical Theory of Relativity Considered as a Euclidean Continuum
 27. The Space-Time Continuum of the General Theory of Realtivity is Not a Euclidean Continuum
 28. Exact Formulation of the General Principle of Relativity
 29. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity
 Part III: Considerations on the Universe as a Whole
 30. Cosmological Difficulties of Newton's Theory
 31. The Possibility of a "Finite" and yet "Unbounded" Universe
 32. The Structure of Space According to the General Theory of Relativity
 Appendices: 01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
 02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
 03. The Experimental Confirmation of the General Theory of Relativity
 04. The Structure of Space According to the General Theory of Relativity (sup. ch 32)
 05. Relativity and the Problem of Space Note: The fifth appendix was added by Einstein at the time of the fifteenth re-printing of this book; and as a result is still under copyright restrictions so cannot be added without the permission of the publisher.
 Preface
 (December, 1916)
 The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a "stepmotherly" fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for the trees. May the book bring some one a few happy hours of suggestive thought!
 December, 1916
 A. EINSTEIN
 Introduction by Nigel Calder
 INTRODUCTION “What really interests me is whether God had any choice in the creation of the world.” At the centenary of Albert Einstein’s birth, in 1979, I scripted that comment of his into a TV documentary about relativity. Some broadcasters expunged it from the soundtrack because they thought it was blasphemous. In truth it was a reverential remark, and the question remains a shrewd one. The relativity theories explained by Einstein in this book supply some of the basic rules that any properly functioning universe must obey. They make sure that atoms and natural forces will behave in the same way anywhere, across the great oceans of space and time. A supplementary rule helps to explain how stars and life can go on running for billions of years. When all the rules are known, will it turn out that only one possible kind of universe can be both self-consistent and con-genial for life? For any young would-be Einstein of today, the question is still on the table. And as the maestro commented on another occasion, to persevere with such difficult trains of thought requires feelings like those of “a religious person or a lover.” For you, the inquisitive reader, a sense of trying to read the mind of Godor to chat up Mother Nature, if you prefer—is an encouragement to accompany Einstein through the forest of tricky ideas contained in this slim volume. You will bear with him even when, from time to time, he uses a little high school mathematics to consolidate the reasoning. The payoff is worth the mental effort. Addressing you person-to-person, Einstein certainly wants viii introduction you to join him in his intellectual adventure. Gratuitous mystification and hero worship, which sometimes contaminate other people’s accounts of relativity, are absent here. As a writer, Einstein takes great pains to find examples and analogies to explain his points, but he pursues no prizes for his prose. He shares the opinion of the physicist Ludwig Boltzmann who said that, “Matters of elegance ought to be left to the tailor and to the cobbler.” When he wrote this book in German in 1916, Einstein’s name was scarcely known outside the physics institutes. He had just completed his masterpiece, the general theory of relativity. It provided a brand-new theory of gravity and it promised a new per-spective on the cosmos as a whole. He set out at once to share his excitement with as wide a public as possible. But
 World War I was raging at the time and English-speaking countries scorned all things German. After the war, two British expeditions to the South Atlantic observed the total eclipse of the sun in May 1919. The astronomers photographed stars shifting in the sky, in a way that was said to support the Berlin professor’s outlandish ideas. Newton’s law of gravity was apparently out of date. Announced in London on November 6, 1919, the news made Einstein a celebrity overnight. “The typhoon of publicity crossed the Atlantic,” Ernest Rutherford noted. As the discoverer of the atomic nucleus, he was a rival for fame. Robert W. Lawson, a British physicist who had polished his knowledge of German while a prisoner of war in Austria, translated this book into English. He secured Einstein’s blessing for the book’s publication in 1920, and the eclipse results were included in an appendix. So why not engage in a little time travel? Imagine that it’s the aftermath of World War I. The usual method of longrange travel is still by railroad train. The U.S. government has prohibited alcohol, and bootleggers are admiring the newly developed Thompson submachine gun. Al Jolson and his song “Swanee” are all the rage. And that German chap’s account of how he upstaged the great Isaac Newton is here in your hands.
 EINSTEIN’S FIRST THIRTY-SEVEN YEARS The transition from relative obscurity to a prominence unmatched by movie stars, signaled by the translation of this book, is also a moment to reflect on Einstein’s life till then. Born in Ulm in Germany in 1879, Albert was the son of an entrepreneurial electrician (seldom successful) and a musicallyminded mother. In the following year the family moved to Munich. Albert’s budding brain was enthused by a magnet given to him at the age of five, and by a book on Euclid’s “holy geometry” as he called it, when he was twelve. From sixteen onward, Einstein studied in Switzerland. He was brilliant and negligent by turns and often preoccupied with his own thoughts. In 1900, aged twentyone, he obtained a diploma as a science teacher from the Federal Institute of Technology in Zurich, but then had great difficulty securing a job. His private life was in a mess. Fellow student Mileva Maric
 was pregnant with his child, Einstein’s parents opposed a marriage, and anyway he was broke. Not until the summer of 1902 did Einstein secure a permanent job, as a technical officer at the Swiss Patent Office. He married Mileva, and they settled in Bern. Their first child disappeared mysteriously—she was presumably adopted and/or died—but in 1904 the first of two sons was born. After a solid day’s work at the office, evaluating all sorts of inventions, Einstein would spend his spare time on fundamental physics, working at the table in the family apartment. The resulting papers went to the Annalen der Physik, which tolerantly published his ideas. He also wrote many review articles for that journal, on other people’s physics. Einstein was neither a competent experimenter nor a high-powered mathematician. His intuition about scientific concepts was unequaled, and when logic was on his side he would stick his neck far out, even when his conclusions ran counter to the received wisdom. A reinterpretation of the photoelectric effect was his first spectacular contribution, proving that light can behave as if it consists of particles, not waves. That was the subject of a paper written in March 1905, shortly before his twenty-sixth birthday. During the next few weeks Einstein completed two groundbreaking papers on molecular physics. In May he solved a puzzle about the speed of light that had taunted him for years, and before the end of June he had sent his first paper on relativity to Annalen der Physik. Three months later he followed it with a related paper on the equivalence of energy and mass. No wonder 1905 is called, in retrospect, Einstein’s miracle year. Yet the academic world was extremely slow to react. Not until 1909 was he able to give up his job at the Patent Office, on becoming an assistant professor at Zurich University. Thereafter his career took off, with a quick succession of posts in other universities culminating in a very prestigious appointment in Berlin, where he settled in 1914. All the while he was struggling to generalize his ideas about relativity. The special theory of 1905 dealt with conflicting views of the world that result from relative movements at a steady speed. If he could extend it to accelerated motions, a new theory of gravity would be in his grasp. He had the right ideas from 1907 onward, but the mathematics was so tricky that the general theory of relativity was not perfected until 1915.
 Just as he was beginning to apply it to the overall nature of the universe, he spared the time needed to write the present book. He was then age thirtyseven.
 SPACE, TIME, AND LIGHT ALONG A RAILROAD Einstein starts his book by asking whether Euclid’s geometry is true. The Greek compendium about how lines and shapes relate to one another, on a flat surface, has been the bedrock of practical mathematics for two thousand years. Yet it is definitely correct only in abstract logic. In the real world, so Einstein warns us, its truth may turn out to be limited. This is a distant, ranging shot, and Section 1 may puzzle some readers, because Einstein does not return to Euclid’s vulnerability until halfway through the book. Its early appearance means that the author wants to shock us into thinking about things in his way. Euclid not necessarily correct? That was fighting talk in 1920. Nowadays scientists can directly test geometry across the solar system, because radar echoes from planets, and signals to and from spacecraft, can measure relative distances. The old rules about triangles, for example, don’t work exactly. Rather than complain that Euclid was wrong, the scientists now prefer to say that the empty space in the solar system is not perfectly “flat” in Euclid’s sense. The way gravity deforms space and time is what general relativity is all about. But first Einstein leads us into special relativity and the effects of motionspecial because it excludes accelerated move-ment. In Sections 2 to 4 he erects the traditional scaffolding of the physical world, which his theories are going to shake, namely the system of coordinates used for pinpointing events and tracing movements. There are three dimensions of space x, y, and z (meaning left-right, forward-back, and up-down) and one dimension of time t. Events can look very different to onlookers in different situa-tions, especially if they are moving relative to one another. In Section 3, Einstein introduces the railroad (the “railway” to the British translator) that sets the scene for his reasoning for many pages that follow. The first example of different viewpoints comes when I drop a stone from the window of a moving train. I see the stone go straight down from my hand to the ground. If you watch from the side of the track, you’ll see it following a curved (parabolic) path, because the stone inherits some sideways motion from the train.
 In Section 4, Einstein equips the observers in the train and by the track with clocks, so that each has a complete coordinate system—a personal frame of reference in space and time. So far so obvious, but in Section 5 Einstein gives a preliminary hint that the world will get out of joint when light comes into the story. That is the meaning of his reference to developments in “electrodynamics and optics.” Approaching the tricky bit gradually, Einstein reasons that the laws of nature ought not to depend on who is watching. A raven flying in a straight line will appear to be going straight to an observer in a moving train, even though he’ll reckon the bird’s course and speed differently from a trackside observer. Similarly the laws of nature observed on the earth don’t change between winter and summer, even though our planet reverses its direction of travel around the sun as it hurries along in its orbit at 30 kilometers per second. A century after Einstein formulated these ideas, the absence of any seasonal variation in physical laws has been checked to the highest precision for which you could ever wish. Next, Einstein turns to the way speeds can be combined. In Section 6 a man walks forward along a moving train. An onlooker beside the track may reckon how fast the man is advancing by simply adding his speed of walking to the train’s velocity, but that will turn out to be an oversimplification. The first clue comes in Section 7, when Einstein imagines a beam of light being sent along a railroad embankment in the same direction as a train is traveling. He asks how fast the light goes in relation to the train. “Classical” ideas tempt you to think that the light must be going more slowly as judged from the train, because you should subtract the train’s speed. Not so, says Einstein. If a traveler on the train could measure the speed of the beam of light for himself, the result would be exactly the same as the speed of the same beam measured by someone stationary on the ground.
 THE CONSTANT SPEED OF LIGHT How does Einstein know that the movement of a source of light, or of the detector that registers its arrival, has no effect on the speed of light as measured by any observer? When he developed the special theory of relativity in 1905, his convic-tion about this crucial point depended on his
 intuition and on the theories of a Dutch physicist, Hendrik Lorentz. But in 1913, before he wrote this book, Einstein was rewarded with strong supporting evidence when a Dutch astronomer, Willem de Sitter, considered pairs of stars that orbit around each other. Sometimes a star swings towards the earth, as it circles its partner, and sometimes it’s receding on the other side of its partner. These phases of its orbit are distinguishable by shifts in frequency of the starlight, and they follow at regular intervals. If the light traveled faster when emitted by the star approaching the earth, it would overtake the light from the previous retreating phase and smear out the alternations. As astronomers can easily distinguish the comings and goings, de Sitter reasoned that the constancy of the speed of light was confirmed. Astronomy at invisible frequencies, which travel at the same speed as visible light, has refined de Sitter’s test. An X-ray star in another galaxy, the Small Magellanic Cloud, is orbiting around an unseen companion. It lies so far away that the slightest discrepancy in the light speed, due to the star’s own speed, would be detectable. Gamma ray bursts come from stupendous explosions that occur almost out to the limits of the observable universe. Even after taking billions of years to reach us, some bursts last for only a split second. That means there can be no difference at all in the speeds of emission from rapidly moving parts of the violently erupting source. The brevity of gamma-ray bursts now makes the constancy of light speed in empty space one of the surest facts in the whole of science. With this hindsight, Einstein’s conviction is correct. But the puzzle illustrated by the railroad train and the relative speed of the light beam “has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties.” The special theory of relativity is promised as the solution.
 TIME BECOMES SLIPPERY Two lightning flashes are said to strike the railroad embankment at the same moment at different places, and Einstein spends several pages fretting about the meaning of “simultaneous” (Sections 8 and 9). After a mock dialogue with the reader, who supposedly defends old-fashioned thinking about the idea of time in physics, Einstein offers an exact definition. The lightning strokes are simultaneous if their light rays meet at the midpoint on the embankment between the places where they strike. But from the point
 of view of a rider on a moving train, who happens to be midway between the points of impact when the lightning strikes, the flashes cannot be simultaneous. This observer will see the flash up ahead slightly before the one behind, because the train has moved him forward to meet its approaching rays. Once simultaneity becomes only a relative concept, time itself goes haywire. What the observer on the embankment considers to be one second is not one second for the person on the train. As a result (in Section 10) the speed of a person walking forward along the train, as judged from the embankment, turns out to be different from the speed of walking judged on the train itself. And the length of the train itself will appear different too. To make these crazyseeming propositions precise, Einstein brings in a mathematical device called the Lorentz transformation, named after the Dutch physicist mentioned earlier. Don’t worry if you can’t follow it all. Section 11 (reinforced by Ap-pendix 1) explains why the same beam of light has the same speed whether judged by a stationary or a moving observer. Time runs more slowly for the moving observer, to exactly the extent needed to secure the constancy of light speed. Similar mathematics in Section 12 tells you why no ordinary object could ever travel faster than light. It also shows that a measuring rod moving past you will appear shorter than when it is at rest; hence, the remark about the length of the train. Einstein’s way of putting it suggests a squeezing. In a more modern gloss, the rod, or the train, appears to be slightly rotated away from you as it passes, and so you see it foreshortened to the extent predicted by the formula. Einstein then returns to the question of how velocities are to be added together (Section 13). When this situation cropped up earlier, with a person walking forward along a moving train, the simpleminded answer was to add the man’s speed to the train’s speed. But in relativity the combined speed as gauged by a stationary watcher has to be reduced a little. With some satisfaction, Einstein revisits an experiment first devised by a French physicist, Armand Fizeau, in 1851. By measuring the speed of light in water flowing down a tube, it unwittingly tested Einstein’s formula for adding velocities. Repeated by Pieter Zeeman (yet another Dutchman, and a colleague of Lorentz), the experiment confirmed the formula’s accuracy to within 1 percent.
 MASS AND ENERGY When this book was written, practicable speeds of motion were too slow, and clocks and rulers too imprecise, to test many predictions of special relativity directly. Undismayed, Einstein went on to make some sweeping inferences from his theory. His assertion, in Section 14, that any general law of nature must be consistent with special relativity, reconfirms his early requirement that the laws of nature cannot depend on who is looking. Einstein’s own most famous law is that mass and energy are equivalent. In Section 15 he introduces the idea by way of another remarkable prediction of special relativity, namely that the mass of a body increases when it travels at high speed. In classical physics it gains in energy of motion. In relativity that kinetic energy makes itself felt as additional mass. Radiant energy absorbed by a body also increases its mass. Indeed the total mass becomes a measure of its total energy. But in this reckoning the body starts with inherent energy even when at rest. It is a huge amount, given by the body’s rest mass rest mass multiplied by the square of the speed of light, or E=mc2. Concentrations of energy available in 1916 were too small in relation to experimental masses for this equivalence of mass and energy to be tested. And in Section 16, when Einstein trawls for evidence in support of special relativity as a whole, the haul is meager—just small deviations from classical expectations in experiments with electrons, and a favorable gloss that he can put on the failure to detect any difference in the speed of light in two directions at right angles, in a nineteenth-century experiment. Modern particle accelerators confirm Einstein’s predictions more directly. They prolong the life of unstable subatomic particles by achieving speeds that stretch time for them. The masses of accelerated particles increase to the point where new matter can be created from them. E= mc2 accounts for the long-lasting power of the sun and the stars, and appears as the nuclear energy that power engineers and bombmakers have learned how to tap. To trace all the consequences of special relativity now verified by scientists would be to recapitulate much of the physics and astronomy of the past one hundred years. Perhaps the crowning glory is antimatter, predicted by Paul Dirac in England when he applied special relativity to the theory of subatomic particles.
 Antimatter is now known to shower down from the sky above us, and when a particle meets its antiparticle, both of them disappear in a burst of radiant energy, exactly in accordance with E=mc2.
 ONWARD TO GENERAL RELATIVITY “The non-mathematician is seized by a mysterious shuddering when he hears of ‘four-dimensional’ things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more commonplace statement than that the world in which we live is a four-dimensional space-time continuum.” Einstein makes this remark at the start of Section 17, which is a preamble about geometry for the transition from special relativity to general relativity. Figuring prominently is the mathematician Hermann Minkowski, who had called Einstein “a lazy dog” when teaching him in Zurich. Minkowski invented a mathematical trick that treats time as if it were just an extra dimension of space. Appendix 2 has a little more on this subject. Without this method, the juggling with space-time whereby Einstein revolutionized the theory of gravity would have been much more difficult. The different meanings of special and general relativity appear in Section 18. The democratic principle that Einstein is pursuing requires the laws of nature to remain the same regardless of how the observer is moving. While the special theory compares different views of the world due to uniform motion, as in the railroad train moving steadily along the track, the general theory removes that restriction and allows for all kinds of movement, including acceleration and rotation. Some puzzles about gravity are set out in Section 19. Whatever the “intermediary medium” is, which pulls a stone down when we drop it, it acts equally on any other object. Everything falls with the same acceleration, if there’s no air resistance. And isn’t it odd that the force of gravity acting on a body is propor-tional to its mass—exactly the same quality that crops up when you gauge the body’s resistance to acceleration, its inertia? These features find an explanation in Section 20, when gravity is seen to be very like any other accelerating system. Einstein invites us to visualize a man living in a big chest that’s drifting in empty space. He must tie himself to the floor if he is not to float about. The modern reader has seen videos of astronauts drifting weightlessly in their spaceships, but Einstein has to picture it for himself.
 Unable to invoke a space rocket to propel the box, he imagines a “being” pulling on a rope attached to the lid of the chest and imparting a steady acceleration. The man in the chest can then think himself at home on the earth. He no longer tends to float, and any object he releases will fall to the floor. The steady acceleration through empty space will feel to him just like gravity. What’s more, the simulation fully accounts for the equal effect of gravity on all objects, and for the equality of inertial and gravitational mass. A similar situation prevails for a person on a train when the brakes are applied hard. He can say, if he wishes, that he is jerked forward by a shortlived gravitational field. It also slows down the embankment (and the planet in general) that were rushing past him while he sat stationary in his own frame of reference. By this time the reader may think that the reasoning is quaint, but in Section 21 it’s clear that Einstein is in earnest. He is simply stressing that, in general relativity, no point of view can take preference over any other.
 GRAVITY BENDS LIGHT The similarity between gravity and any other acceleration means that light must be affected by gravity like any other substance. This crucial ingredient of Einstein’s theory first appears in Section 22. There he predicts that the light from a star grazing the sun will be deflected, so that it will change its apparent position in the sky by 1.7 seconds of arc (roughly one twothousandths of a degree). At the time of an eclipse, he says, stars seen beyond the sun ought to appear shifted outwards from the sun to that sort of extent, compared with their normal positions in the sky. This was the prediction that made Einstein famous. He was doubly lucky. When wrestling with his early ideas about general relativity, in 1911, he published a wrong answer for the deflection of starlight—half the correct result. The outbreak of World War I prevented astronomers from testing Einstein’s prediction at the total eclipse of 1914, before he came up with the right number in 1915. The second stroke of luck was that the British astronomer Arthur Eddington, who led the effort to test it at the eclipse of 1919, was predisposed to believe Einstein’s theory. Looking for starshifts of less than a millimeter on the photographic plates, Eddington’s team put aside several plates that gave “wrong” results, and picked ’n’ mixed the rest until the
 average was about right (see Appendix 3b). It was pretty sloppy science, yet Eddington let the message ring out around the world: “Newton’s theory of gravity is dead—long live Einstein’s!” Fortunately, light bending to the extent required by general relativity has been amply verified since then. Radio waves are invisible light, and astronomers used widely spaced radio telescopes in accurate observations of Quasar 3C279, which regularly passes behind the sun. A European starmapping satellite called Hipparcos (1989–93) detected the deflection of starlight even from stars lying far from the sun’s direction in the sky. Hipparcos scientists were able to verify Einstein’s theory to an accuracy of 1/10 of 1 percent. Gravitational lenses lying far away in the universe give an up-to-date demonstration of light bending in the Einsteinian fash-ion. The gravity of a cluster of galaxies, and of invisible dark matter associated with it, acts as an untidy magnifier. It en-hances the view of even more distant galaxies by magnifying them, albeit with distortion and multiple images.
 COMING CLEAN ABOUT THE SPEED OF LIGHT An ordinary lens works by delaying the light passing through it, because light travels more slowly through glass than through air. The same is true of gravitational lenses. Einstein comments in Section 22 that, against all expectations from special relativity, the deflection of light by gravity implies a change in the speed of light in the sun’s vicinity. “A curvature of rays of light can only take place when the velocity of propagation of light varies with position.” What a pity that remark was not printed in italics in Einstein’s book, or painted on balloons for all to see! Researchers and teachers ignored it for half a century, until radar echoes from Venus and Mercury in the late 1960s turned changes in the speed of light into an observed fact. Radar pulses sent out from the Haystack observatory in Massachusetts were clearly delayed whenever the planets were on the far side of the sun, as seen from the earth. The radio waves (a variety of light) slowed down as they passed the sun on their outward and return journey. Even in the 1970s it was hard to get more than the most grudging admission from experts on relativity that gravity slows down light, although Einstein himself was unabashed about it sixty years earlier. Undue emphasis on the constancy of light speed made general relativity unnecessarily opaque to students and the
 general public for several decades. Just come clean, and admit that light dawdles a little near a massive object like the sun, and Einstein’s theory of gravity is far easier to understand. The experts were not being entirely perverse. One of the glories of general relativity is that the speed of light does indeed remain the same, provided you measure it on the spot. If you could station a heat-resistant spacecraft beside the sun to gauge Haystack’s radar pulses whizzing past, they would seem to be traveling at just the usual speed of light. The reason is that time, too, runs more slowly in the spacecraft, under the influence of the sun’s strong gravity. It’s only the distant observer, with a faster clock, who notices the slowdown. In special relativity, you’ll remember, different rates of time on the train and on the embankment enabled the observers to get the same answer for the speed of light. In general relativity, too, changes in clockrates always keep the speed the same, as measured locally. That’s how Einstein ensures that natural laws hold good everywhere. Despite the effects of stronger gravity, atoms, particles, and radiant energy on the sun interact according to exactly the same laws as on the earth. To say so is to run ahead of the chain of explanation in this book. These retrospective hints may nevertheless give you a sense of destination, as Einstein approaches “a serious difficulty” that “lies at the heart of things” and “lays no small claims on the patience and on the power of abstraction of the reader.” Also helpful, perhaps, is to note that the next step in the argument matches an idea illustrated in the science fiction movie 2001: A Space Odyssey, where a large space station simulates normal gravity by centrifugal force. The station rotates at an appropriate speed and the astronauts walk around a floor at the rim, with their feet pointing outwards, away from the center.
 SLITHERING IN SPACE-TIME The usual picture of gravity is turned inside out in Section 23, by putting an observer on a disc that is rotating. He feels a force pushing him outward, and like the man in the accelerated box he is authorized by general relativity to call it gravity—a peculiar kind of gravity that becomes stronger the farther you are from the center. The man is also moving relative to the center of the disc. Recalling the effects of relative motion between a train and the track, Einstein notes that the man’s clock will run more slowly than a clock at the center of the disc.
 What’s more, if he puts a measuring rod along the edge of the disc in the direction of movement, to start measuring the circumference, the rod will be shorter than it would be at the center. On the other hand, the rod is not shortened when pointing toward the center of the disc, to measure the diameter. As a result, the circumference of the disc will seem to be greater than the diameter multiplied by (pi, 3.14 . . . ) which would be the case if the disc were at rest. When the effective length of a measuring rod can change, Euclid’s geometry for flat surfaces no longer works. General relativity needs a suppler frame of reference, which Einstein sets up in Sections 24 to 28. First he imagines a rectangular grid of rods laid out on a marble slab, which goes askew if you heat part of the slab and some of the rods expand in length. The squares of the grid are no longer square. Not to worry. Carl Friedrich Gauss, a German mathemati-cian and physicist who flourished in the early nineteenth century, devised a system of coordinates in which the grid can be crooked and the lines curved. And just as Minkowski added time to the three dimensions of space in a rectangular system, Einstein adds supple time to Gauss’s supple system for describing space. Then, like a child in a floppy climbing-frame, he has a framework of space-time in which to play with his general theory of relativity. Einstein imagines his four-dimensional world to be inhabited by slithery creatures—he calls them molluscs—that can move about and change shape ad lib. General relativity requires that all molluscs should have “equal right and equal success” in formulating the laws of nature. This invertebrate democracy might seem like a recipe for total confusion. Instead it imposes such strict legislation on the universe that the distortions of space-time due to the influences of massive bodies provide a precise and novel theory of gravity.
 IMPROVING ON NEWTON The mathematics that Einstein used to tame his supple space-time is too abstruse for the wide readership he aims for in this book. He contents himself with summarizing some key results, in Section 29. First, Newton’s law of gravity, in which the gravitational force between two bodies is inversely proportional to the square of the distance between them, springs ready-made from the molluscs’ weird world. Unlike Newton, Einstein can explain what is happening. Masses deform space, with the result that
 other masses follow curved tracks—as when the sun forces the planets to orbit around it. Moreover, Newton’s law of gravity is only approximately correct. Deviations become evident where gravity is strong, and they show how Einstein’s theory improves on Newton’s. One, already mentioned, is the extent of the bending of starlight when it passes near the sun. Another improvement concerns the misbehavior of the planet Mercury, first noticed by the French astronomer Urbain Leverrier in 1865, which finds a ready explanation in Einstein’s theory of gravity. The planet’s elliptical orbit around the sun gradually swivels because of interactions with other planets, but this “precession” is greater than predicted by Newton’s theory. Searches for an unknown planet that might explain the discrepancy were unavailing. The explanation is slightly stronger gravity near the sun, provided by Einstein’s theory. All planetary orbits are affected but Mercury’s the most because it is closest to the sun (see also Appendix 3a). Radar observations of Mercury later confirmed that the swiveling matches Einstein’s theory to a high degree of accuracy. More spectacular in this regard is a pulsating radio star, or pulsar, discovered in 1974. It goes very closely around and around a silent companion, on an orbit that swivels far more rapidly than Mercury’s. In a double pulsar reported in 2004, the effect is even greater. These systems are also seen to be shedding energy, supposedly by radiating gravitational waves that Einstein predicted in 1916. The third innovation from general relativity described in this book is nowadays known to scientists as the gravitational red-shift. Einstein mentions it only briefly in Section 29 but gives more detail in Appendix 3c, where he writes: “An atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated.” As a symptom of gravity’s amazing power to slow down time, the characteristic light emitted by atoms and molecules—their spectral lines —will appear to distant observers to have lower frequencies in strong gravity than on the earth or in empty space. They will be shifted towards the red end of the spectrum. Verification of the gravitational red-shift came in 1924. Walter Adams in California discovered that Sirius B is a very dense star, the first white dwarf ever identified. He reported that some emissions from hydrogen atoms showed marked reductions in
 frequency as required by Einstein’s theory. Much as with the 1919 eclipse story, historians of science question the reliability of Adams’s result. The light from Sirius B was contaminated by light from the much brighter Sirius A. Never mind. The gravitational red-shift is now observed routinely in many astronomical objects including the sun. Even more convincingly for non astronomers, the effect of gravity in slowing time is demonstrated directly with atomic clocks. They run faster in high-flying aircraft than they do on the ground.
 A COSMOLOGICAL SKETCH Part III of the book is entitled “Considerations on the Universe as a Whole.” It is very brief, and lest it should disappoint any-one familiar with Einstein’s contributions to cosmology, be aware that when he wrote this book for the general public in 1916, his ideas were still maturing. A key scientific paper, “Cosmological Considerations on the General Theory of Relativity,” did not appear until the following year. Astronomical knowledge of the time, primitive by contemporary standards, misleads Einstein badly. Like Newton, he imagines stars scattered through the immensity of space and moving about only slowly. He is troubled, as Newton was, by the problem that gravity will tend to drag all the stars together. This outcome might be avoided if the stars were very evenly scattered, which was Newton’s own suggestion. By Einstein’s young days, astronomers knew very well that the stars of the Milky Way are not at all uniform in their distribution. To avoid having all the stars fall together in a heap, the German astronomer Hugo von Seeliger suggested that gravity must weaken at long ranges, more rapidly than prescribed by Newton’s inverse square law. This is the main theme of Section 30. Einstein indicates that he might welcome such an idea if only there were a logical reason for it. Another approach to the problem of the collapsing starfield appears in Section 31. Einstein proposes that cosmic space may be folded back upon itself. He invites us to share the worldview of flat beings living on what they perceive as a flat, two-dimensional surface, but which is in fact a sphere of large but finite size. There is no boundary to the flat creatures’ universe. If they traveled far enough they would come back to their starting point, on a great circle. Without having to make a world tour, the flat beings can figure out what
 kind of universe they are living in, and even measure its diameter, by discovering subtle discrepancies between the predictions of Euclid for truly flat surfaces, and what they find in practice. Similarly, you can imagine a super-geometry in which our own three-dimensional space is so folded that it is “finite” yet “unbounded.” Such a universe is congenial for general relativity and promises a way of preventing the stars falling together. This is Einstein’s assertion in the very brief Section 32 that, except for the appendices, brings his book to a close. The story ends abruptly with a cursory description of one possible form for the universe, which Einstein happens to like. The reader is left with a strong sense of unfinished business. To pursue in any detail here the dramatic cosmology that unfolded in the twentieth century would be too lengthy a departure from the main subjects of this book. Yet not to mention akin to Seeliger’s idea of gravity weakening at long ranges. To stop the stars falling together, adjusted the strength of gravity as required. Without , a universe is unstable and it must be either imploding or expanding, which was contrary to the myopic impressions of early twentieth-century astronomy. The discovery in the 1920s of the great cosmic expansion, in which the Milky Way is just one of many galaxies and the spaces between clusters of galaxies grow rapidly, made Einstein think which it apparently began. Yet at the end of the twentieth century the expansion of the universe turned out to be accelerating and Einstein’s has come back in triumph in the driving seat of the cosmos. And what about Euclid, with whom this book starts and finishes? Einstein wanted astronomers to emulate the flat beings of his folded twodimensional universe and discover the overall geometry of the real world. If the universe were not “flat” in Euclid’s ideal sense, it should act as a lens, and very distant objects should appear magnified or shrunk. The most distant observable objects—clumps of hot gas that existed soon after the Big Bang—are now mapped by radio microwaves and they look neither bigger nor smaller than expected. In the geometers’ heaven, it’s a draw. Einstein trumps Euclid in the distorted space-time surrounding planets, stars, and galaxies, but the geometry of the universe at large still conforms very well to what that old Greek taught his students in Alexandria 2,300 years ago.
 GENERAL RELATIVITY STILL THRIVES To say much about the rest of Einstein’s life story in this introduction might break the mental link that I have tried to fashion with the relatively young Einstein who wrote the book. He would divorce Mileva and marry his cousin Elsa. The 1921 No-bel Prize would come his way—not for relativity but for his interpretation of the photoelectric effect. Later he would fall out with his fellow physicists over the interpretation of the quantum theory. In 1932, as a Jewish refugee from the Nazis, he would find sanctuary in Princeton, New Jersey, and live there until his death in 1955. Much more relevant to the reader is the subsequent career of Einstein’s cleverest brainchild, general relativity. Despite many decades of efforts by experimenters and theorists to prove it imperfect, it still holds sway. Oftrepeated promises of a superior quantum theory of gravity remain only a speculation. A great theory should make surprising predictions that can be verified by observation, and go on to take unexpected discoveries in its stride. General relativity has performed supremely well on both counts. In addition to several successful tests already described, a huge effort is now going into the direct detection of gravitational waves, which should squeeze and stretch space as they pass by. Failure to find them would be surprising because, as mentioned, the behavior of orbiting pulsars makes sense only if they are radiating gravitational waves. Another prediction currently under test with a satellite is that the earth should drag space-time around it as it rotates. The finest example of explaining the unexpected came with the discovery of quasars in 1963. These compact sources of radiation in the hearts of some galaxies were far too powerful to rely on the nuclear energy that lights the stars. Ready to hand was an awful possibility implicit in general relativity. A massive object might collapse into a black hole, which would then be capable of squeezing huge amounts of energy out of any stars or gas falling into it. The idea of black holes won acceptance only gradually. Observations established the compactness of the quasars and the presence of material feeding their hearty appetites. Not until 1994 did direct confirmation of the reality of black holes come in results from the Japanese satellite ASCA. Variations in the wavelength of X-ray emissions from iron atoms in a stormy galaxy made a pattern predicted for material orbiting closely around
 a black hole. And a loss of energy by individual X-ray particles showed time slowing down in the intense gravity near the black hole, just as general relativity requires.
 Suggestions for Further Reading
 The bibliography concerning Einstein is huge. Here are a few well-regarded books, written or edited from a modern perspective.
 Calder, Nigel. Einstein’s Universe: The Layperson’s Guide. New York: Penguin, 2005. This is an updated edition of a book first published in 1979. It covers much of the same ground as the present classic by Einstein himself, but with all the bonuses of scientific hindsight.
 Overbye, Dennis. Einstein in Love. New York: Viking Penguin, 2000. Drawing on many unpublished letters, this book tells of Einstein’s romances with his first wife, Mileva, and with his second, Elsa, which took place during his most creative years.
 Schilpp, Paul Arthur (editor and translator). Albert Einstein: PhilosopherScientist. La Salle, Ill.: Open Court, 1982. Of special interest here are Einstein’s own “Autobiographical Notes” written in 1946, where he told how his ideas developed but avoided saying much about his “merely personal” life.
 White, Michael, and Gribbin, John. Einstein: A Life in Science. New York: Plume Books, 2005. First published in 1993, this book skillfully interweaves the personal, public, and scientific strands of Einstein’s whole life, including his persistent misgivings about the quantum theory.
 Part I. The Special Theory of
 Relativity
 01- Physical Meaning of
 Geometrical Propositions
 K = co-ordinate system
 x, y = two-dimensional co-ordinates
 x, y, z = three-dimensional co-ordinates
 x, y, z, t = four-dimensional co-ordinates
 t = time
 I = distance
 v = velocity F = force
 G = gravitational field
 Part I: The Special Theory of Relativity In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What, then, do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little consideration. Geometry sets out form certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of
 these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines," to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves. It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry "true." Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a "distance" two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation. If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically
 rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the
 "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses. Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.
 Footnotes 01
 1) It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when the points A and C being given, B is chosen such that the sum of the distances AB and BC is as short as possible. This incomplete suggestion will suffice for the present purpose.
 02-The System of Co-ordinates
 On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a " distance " (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry ; then, starting from A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement
 of length. 1) Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place
 specification " Times Square, New York," [A] I arrive at the following result. The earth is the rigid body to which the specification of place refers; " Times Square, New York," is a well-defined point, to which a name has
 been assigned, and with which the event coincides in space.2) This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other. But we can free ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering over Times Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed. (a) We imagine the rigid body, to which the place specification is referred, supplemented in such a manner that the object whose position we require is reached by. the completed rigid body. (b) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.
 (c) We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud. From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates. This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry. In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available ; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be
 sought in accordance with the above considerations. 3) We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for "distances;" the "distance" being represented physically by means of the convention of two marks on a rigid body.
 Footnotes 02
 1) Here we have assumed that there is nothing left over i.e. that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.
 [A] Einstein used "Potsdamer Platz, Berlin" in the original text. In the authorised translation this was supplemented with "Tranfalgar Square, London". We have changed this to "Times Square, New York", as this is the most well known/identifiable location to English speakers in the present day. [Note by the janitor.]
 2) It is not necessary here to investigate further the significance of the expression "coincidence in space." This conception is sufficiently obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice.
 3) A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book.
 03-Space and Time in Classical
 Mechanics
 The purpose of mechanics is to describe how bodies change their position in space with "time." I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins. It is not clear what is to be understood here by "position" and "space." I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola? Moreover, what is meant here by motion "in space" ? From the considerations of the previous section the answer is self-evident. In the first place we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference." The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of " body of reference " we insert " system of co-ordinates," which is a useful idea for mathematical description, we are in a position to say : The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an independently existing
 trajectory (lit. "path-curve" 1)), but only a trajectory relative to a particular body of reference. In order to have a complete description of the motion, we must specify how the body alters its position with time ; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction ; the man at the railway-carriage window is holding one of them, and the man on
 the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.
 Footnotes 03
 1) That is, a curve along which the body moves.
 04-The Galileian System of Co
 ordinates
 As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia, can be stated thus: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line. This law not only says something about the motion of the bodies, but it also indicates the reference-bodies or systems of coordinates, permissible in mechanics, which can be used in mechanical description. The visible fixed stars are bodies for which the law of inertia certainly holds to a high degree of approximation. Now if we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed star describes a circle of immense radius in the course of an astronomical day, a result which is opposed to the statement of the law of inertia. So that if we adhere to this law we must refer these motions only to systems of coordinates relative to which the fixed stars do not move in a circle. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a " Galileian system of coordinates." The laws of the mechanics of Galflei-Newton can be regarded as valid only for a Galileian system of co-ordinates.
 05-The Principle of Relativity (in
 the restricted sense)
 In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation ("uniform" because it is of constant velocity and direction, " translation " because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the embankment, is uniform and in a straight line. If we were to observe the flying raven from the moving railway carriage. we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line. Expressed in an abstract manner we may say : If a mass m is moving uniformly in a straight line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co
 ordinate system K1 provided that the latter is executing a uniform translatory motion with respect to K. In accordance with the discussion contained in the preceding section, it follows that:
 If K is a Galileian co-ordinate system. then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform
 motion of translation. Relative to K1 the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.
 We advance a step farther in our generalisation when we express the tenet
 thus: If, relative to K, K1 is a uniformly moving co-ordinate system devoid
 of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense). As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more
 recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. At this juncture the question of the validity of the principle of relativity became ripe for discussion, and it did not appear impossible that the answer to this question might be in the negative. Nevertheless, there are two general facts which at the outset speak very much in favour of the validity of the principle of relativity. Even though classical mechanics does not supply us with a sufficiently broad basis for the theoretical presentation of all physical phenomena, still we must grant it a considerable measure of " truth," since it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful. The principle of relativity must therefore apply with great accuracy in the domain of mechanics. But that a principle of such broad generality should hold with such exactness in one domain of phenomena, and yet should be invalid for another, is a priori not very probable. We now proceed to the second argument, to which, moreover, we shall return later. If the principle of relativity (in the restricted sense) does not
 hold, then the Galileian co-ordinate systems K, K1, K2, etc., which are moving uniformly relative to each other, will not be equivalent for the description of natural phenomena. In this case we should be constrained to believe that natural laws are capable of being formulated in a particularly simple manner, and of course only on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one (K0) of
 a particular state of motion as our body of reference. We should then be justified (because of its merits for the description of natural phenomena) in calling this system " absolutely at rest," and all other Galileian systems K " in motion." If, for instance, our embankment were the system K0 then our
 railway carriage would be a system K, relative to which less simple laws would hold than with respect to K0. This diminished simplicity would be
 due to the fact that the carriage K would be in motion (i.e."really")with respect to K0. In the general laws of nature which have been formulated
 with reference to K, the magnitude and direction of the velocity of the carriage would necessarily play a part. We should expect, for instance, that the note emitted by an organpipe placed with its axis parallel to the
 direction of travel would be different from that emitted if the axis of the pipe were placed perpendicular to this direction. Now in virtue of its motion in an orbit round the sun, our earth is comparable with a railway carriage travelling with a velocity of about 30 kilometres per second. If the principle of relativity were not valid we should therefore expect that the direction of motion of the earth at any moment would enter into the laws of nature, and also that physical systems in their behaviour would be dependent on the orientation in space with respect to the earth. For owing to the alteration in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year. However,
 the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions. This is very powerful argument in favour of the principle of relativity.
 06-The Theorem of the Addition of
 Velocities Employed in Classical
 Mechanics
 Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly or, in other words, with what velocity W does the man advance relative to the embankment during the process ? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total be covers the distance W=v+w relative to the embankment in the second considered. We shall see later that this result, which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained ; in other words, the law that we have just written down does not hold in reality. For the time being, however, we shall assume its correctness.
 07-The Apparent Incompatibility of
 the Law of Propagation of Light
 with the Principle of Relativity
 There is hardly a simpler law in physics than that according to which light is propagated in empty space. Every child at school knows, or believes he knows, that this propagation takes place in straight lines with a velocity c= 300,000 km./sec. At all events we know with great exactness that this velocity is the same for all colours, because if this were not the case, the minimum of emission would not be observed simultaneously for different colours during the eclipse of a fixed star by its dark neighbour. By means of similar considerations based on observa- tions of double stars, the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of motion of the body emitting the light. The assumption that this velocity of propagation is dependent on the direction "in space" is in itself improbable. In short, let us assume that the simple law of the constancy of the velocity of light c (in vacuum) is justifiably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties? Let us consider how these difficulties arise. Of course we must refer the process of the propagation of light (and indeed every other process) to a rigid reference-body (co-ordinate system). As such a system let us again choose our embankment. We shall imagine the air above it to have been removed. If a ray of light be sent along the embankment, we see from the above that the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v, and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that
 we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity w of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. w is the required velocity of light with respect to the carriage, and we have w = c-v. The velocity of propagation ot a ray of light relative to the carriage thus comes cut smaller than c. But this result comes into conflict with the principle of relativity set forth in Section V. For, like every other general law of nature, the law of the transmission of light in vacuo [in vacuum] must, according to the principle of relativity, be the same for the railway carriage as reference-body as when the rails are the body of reference. But, from our above consideration, this would appear to be impossible. If every ray of light is propagated relative to the embankment with the velocity c, then for this reason it would appear that another law of propagation of light must necessarily hold with respect to the carriage — a result contradictory to the principle of relativity. In view of this dilemma there appears to be nothing else for it than to abandon either the principle of relativity or the simple law of the propagation of light in vacuo. Those of you who have carefully followed the preceding discussion are almost sure to expect that we should retain the principle of relativity, which appeals so convincingly to the intellect because it is so natural and simple. The law of the propagation of light in vacuo would then have to be replaced by a more complicated law conformable to the principle of relativity. The development of theoretical physics shows, however, that we cannot pursue this course. The epochmaking theoretical investigations of H. A. Lorentz on the electrodynamical and optical phenomena connected with moving bodies show that experience in this domain leads conclusively to a theory of electromagnetic phenomena, of which the law of the constancy of the velocity of light in vacuo is a necessary consequence. Prominent theoretical physicists were theref ore more inclined to reject the principle of relativity, in spite of the fact that no empirical data had been found which were contradictory to this principle. At this juncture the theory of relativity entered the arena. As a result of an analysis of the physical conceptions of time and space, it became evident that in realily there is not the least incompatibilitiy between the principle of
 relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically rigid theory could be arrived at. This theory has been called the special theory of relativity to distinguish it from the extended theory, with which we shall deal later. In the following pages we shall present the fundamental ideas of the special theory of relativity.
 08-On the Idea of Time in Physics
 Lightning has struck the rails on our railway embankment at two places A and B far distant from each other. I make the additional assertion that these two lightning flashes occurred simultaneously. If I ask you whether there is sense in this statement, you will answer my question with a decided "Yes." But if I now approach you with the request to explain to me the sense of the statement more precisely, you find after some consideration that the answer to this question is not so easy as it appears at first sight. After some time perhaps the following answer would occur to you: "The significance of the statement is clear in itself and needs no further explanation; of course it would require some consideration if I were to be commissioned to determine by observations whether in the actual case the two events took place simultaneously or not." I cannot be satisfied with this answer for the following reason. Supposing that as a result of ingenious considerations an able meteorologist were to discover that the lightning must always strike the places A and B simultaneously, then we should be faced with the task of testing whether or not this theoretical result is in accordance with the reality. We encounter the same difficulty with all physical statements in which the conception " simultaneous " plays a part. The concept does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case. We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.) After thinking the matter over for some time you then offer the following suggestion with which to test simultaneity. By measuring along the rails, the connecting line AB should be measured up and an observer placed at the mid-point M of the distance AB. This observer should be supplied with an
 arrangement (e.g. two mirrors inclined at 900) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.
 I am very pleased with this suggestion, but for all that I cannot regard the matter as quite settled, because I feel constrained to raise the following objection: "Your definition would certainly be right, if only I knew that the light by means of which the observer at M perceives the lightning flashes travels along the length A M with the same velocity as along the length B M. But an examination of this supposition would only be possible if we already had at our disposal the means of measuring time. It would thus appear as though we were moving here in a logical circle." After further consideration you cast a somewhat disdainful glance at me and rightly so — and you declare: "I maintain my previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light requires the same time to traverse the path A M as for the path B M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity." It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the
 body of reference 1) (here the railway embankment). We are thus led also to a definition of " time " in physics. For this purpose we suppose that clocks of identical construction are placed at the points A, B and C of the railway line (co-ordinate system) and that they are set in such a manner that the positions of their pointers are simultaneously (in the above sense) the same. Under these conditions we understand by the " time " of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time-value is associated with every event which is essentially capable of observation. This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly: When two clocks arranged at rest in different places of a reference-body are set in such a manner that a
 particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position, of the pointers of the other clock, then identical " settings " are always simultaneous (in the sense of the above definition).
 Footnotes
 1) We suppose further, that, when three events A, B and C occur in different places in such a manner that A is simultaneous with B and B is simultaneous with C (simultaneous in the sense of the above definition), then the criterion for the simultaneity of the pair of events A, C is also satisfied. This assumption is a physical hypothesis about the the of propagation of light: it must certainly be fulfilled if we are to maintain the law of the constancy of the velocity of light in vacuo.
 09-The Relativity of Simulatneity
 Up to now our considerations have been referred to a particular body of reference, which we have styled a " railway embankment." We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig 1. People travelling in this train will with a vantage view the train as a rigid reference-body (co-ordinate system); they regard all events in
 reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises : Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A B of the embankment. But the events A and B also
 correspond to positions A and B on the train. Let M1 be the mid-point of the distance A B on the travelling train. Just when the flashes (as judged
 from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves towards the right in the diagram with the
 velocity v of the train. If an observer sitting in the position M1 in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him
 simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result: Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time ; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event. Now before the advent of the theory of relativity it had always tacitly been assumed in physics that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference. But we have just seen that this assumption is incompatible with the most natural definition of simultaneity; if we discard this assumption, then the conflict between the law of the propagation of light in vacuo and the principle of relativity (developed in Section 7) disappears. We were led to that conflict by the considerations of Section 6, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment in each second of time. But, according to the foregoing considerations, the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body). Hence it cannot be contended that the man in walking travels the distance w relative to the railway line in a time which is equal to one second as judged from the embankment. Moreover, the considerations of Section 6 are based on yet a second assumption, which, in the light of a strict consideration, appears to be arbitrary, although it was always tacitly made even before the introduction of the theory of relativity.
 10-On the Relativity of the
 Conception of Distance
 Let us consider two particular points on the train 1) travelling along the embankment with the velocity v, and inquire as to their distance apart. We already know that it is necessary to have a body of reference for the measurement of a distance, with respect to which body the distance can be measured up. It is the simplest plan to use the train itself as reference-body (co-ordinate system). An observer in the train measures the interval by marking off his measuring-rod in a straight line (e.g. along the floor of the carriage) as many times as is necessary to take him from the one marked point to the other. Then the number which tells us how often the rod has to be laid down is the required distance. It is a different matter when the distance has to be judged from the railway
 line. Here the following method suggests itself. If we call A1 and B1 the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the embankment. In the first place we require to determine the points A and B of the embankment which
 are just being passed by the two points A1 and B1 at a particular time t judged from the embankment. These points A and B of the embankment can be determined by applying the definition of time given in Section 8. The distance between these points A and B is then measured by repeated application of thee measuring-rod along the embankment. A priori it is by no means certain that this last measurement will supply us with the same result as the first. Thus the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself. This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section 6. Namely, if the man in the carriage covers the distance w in a unit of time measured from the train, — then this distance — as measured from the embankment — is not necessarily also equal to w.
 Footnotes
 1)e.g. the middle of the first and of the hundredth carriage.
 11-The Lorentz Transformation
 The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section 7) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: (1) The time-interval (time) between two events is independent of the condition of motion of the body of reference. (2) The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference. If we drop these hypotheses, then the dilemma of Section 7 disappears, because the theorem of the addition of velocities derived in Section 6 becomes invalid. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises: How have we to modify the considerations of Section 6 in order to remove the apparent disagreement between these two fundamental results of experience? This question leads to a general one. In the discussion of Section 6 we have to do with places and times relative both to the train and to the embankment. How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment ? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity ? In other words : Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to the embankment and relative to the train ? This question leads to a quite definite positive answer, and to a perfectly definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another. Before we deal with this, we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in Section 2 we can imagine this reference-body supplemented laterally and in a vertical direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework.Fig.2
 Similarly, we can imagine the train travelling with the velocity v to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular to each other marked out, and designated as " co-ordinate planes " (" co-ordinate system "). A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K' to the train. An event, wherever it may have taken place, would be fixed in space with respect to K by the three perpendiculars x, y, z on the co-ordinate planes,
 and with regard to time by a time value t. Relative to K1, the same event
 would be fixed in respect of space and time by corresponding values x1, y1,
 z1, t1, which of course are not identical with x, y, z, t. It has already been set forth in detail how these magnitudes are to be regarded as results of physical measurements.
 Obviously our problem can be exactly formulated in the following manner.
 What are the values x1, y1, z1, t1, of an event with respect to K1, when the magnitudes x, y, z, t, of the same event with respect to K are given ? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray of light (and of course for every
 ray) with respect to K and K1. For the relative orientation in space of the co
 ordinate systems indicated in the diagram (Fig. 2), this problem is solved by means of the equations :
 y1 = y y z1 = z
 This system of equations is known as the " Lorentz transformation." 1) If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:
 x1 = x - vt ; y1 = y ; z1 = z ; t1 = t
 This system of equations is often termed the " Galilei transformation." The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light c in the latter transformation. Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body K and for the reference-body
 K1. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation
 x = ct,
 i.e. with the velocity c. According to the equations of the Lorentz transformation, this simple relation between x and t involves a relation
 between x1 and t1. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain:
 from which, by division, the expression
 x1 = ct1
 immediately follows. If referred to the system K1, the propagation of light takes place according to this equation. We thus see that the velocity of
 transmission relative to the reference-body K1 is also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever. Of cause this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.
 Footnotes
 1) A simple derivation of the Lorentz transformation is given in Appendix I.
 12-The Behaviour of Measuring
 Rods and Clocks in Motion
 Place a metre-rod in the x1-axis of K1 in such a manner that one end (the
 beginning) coincides with the point x1=0 whilst the other end (the end of
 the rod) coincides with the point x1=I. What is the length of the metre-rod relatively to the system K? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of the system K. By means of the first equation of the Lorentz transformation the values of these two points at the time t = 0 can be shown to be
 the distance between the points being . But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its
 length with a velocity v is of a metre. The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity v=c we should
 have , and for stiII greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body. Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these became meaningless if we choose values of v greater than c. If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to K, then we should have found that the length of the rod as judged
 from K1 would have been ; this is quite in accordance with the principle of relativity which forms the basis of our considerations.
 A Priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes z, y, x, t, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. If we had based our considerations on the Galileian transformation we should not have obtained a contraction of the rod as a consequence of its motion. Let us now consider a seconds-clock which is permanently situated at the
 origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks : t=0 and
 As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but
 seconds, i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.
 13-Theorem of the Addition of
 Velocities. The Experiment of
 Fizeau
 Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment. In Section 6 we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics- This theorem can also be deduced readily from the Galilei transformation (Section 11). In place of the man walking inside the carriage, we introduce a point moving relatively to the co-ordinate system
 K1 in accordance with the equation
 x1 = wt1 By means of the first and fourth equations of the Galilei transformation we
 can express x1 and t1 in terms of x and t, and we then obtain x = (v + w)t This equation expresses nothing else than the law of motion of the point with reference to the system K (of the man with reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section 6, W=v+w A) But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation
 x1 = wt1 B)
 we must then express x1and t1 in terms of x and t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation
 which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we
 are enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Fig. 3) when the liquid above mentioned is flowing through the tube with a velocity v ? In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube. It is clear that we have the problem of Section 6 again before us. The tube plays the part of the railway embankment or of the co-ordinate system K,
 the liquid plays the part of the carriage or of the co-ordinate system K1, and finally, the light plays the part of the
 Fig. 3
 man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by W, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts.
 Experiment1) decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of
 flow v on the propagation of light is represented by formula (B) to within one per cent. Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of MaxwellLorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed from electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly independent of each other, on which electrodynamics was built.
 Footnotes
 1) Fizeau found , where is the index of refraction of the liquid. On the other hand, owing to the
 smallness of as compared with I,
 we can replace (B) in the first place by , or to the same order of approximation by
 , which agrees with Fizeau's result.
 14-The Heuristic Value of the
 Theory of Relativity
 Our train of thought in the foregoing pages can be epitomised in the following manner. Experience has led to the conviction that, on the one hand, the principle of relativity holds true and that on the other hand the velocity of transmission of light in vacuo has to be considered equal to a constant c. By uniting these two postulates we obtained the law of transformation for the rectangular co-ordinates x, y, z and the time t of the events which constitute the processes of nature. In this connection we did not obtain the Galilei transformation, but, differing from classical mechanics, the Lorentz transformation.
 The law of transmission of light, the acceptance of which is justified by our actual knowledge, played an important part in this process of thought. Once in possession of the Lorentz transformation, however, we can combine this with the principle of relativity, and sum up the theory thus: Every general law of nature must be so constituted that it is transformed into a law of exactly the same form when, instead of the space-time variables x, y, z, t of the original coordinate system K, we introduce new
 space-time variables x1, y1, z1, t1 of a co-ordinate system K1. In this connection the relation between the ordinary and the accented magnitudes is given by the Lorentz transformation. Or in brief : General laws of nature are co-variant with respect to Lorentz transformations. This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found which did not satisfy this condition, then at least one of the two fundamental assumptions of the theory would have been disproved. Let us now examine what general results the latter theory has hitherto evinced.
 15-General Results of the Theory
 It is clear from our previous considerations that the (special) theory of relativity has grown out of electrodynamics and optics. In these fields it has not appreciably altered the predictions of theory, but it has considerably simplified the theoretical structure, i.e. the derivation of laws, and — what is incomparably more important — it has considerably reduced the number of independent hypothese forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter would have been generally accepted by physicists even if experiment had decided less unequivocally in its favour. Classical mechanics required to be modified before it could come into line with the demands of the special theory of relativity. For the main part, however, this modification affects only the laws for rapid motions, in which the velocities of matter v are not very small as compared with the velocity of light. We have experience of such rapid motions only in the case of electrons and ions; for other motions the variations from the laws of classical mechanics are too small to make themselves evident in practice. We shall not consider the motion of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativity the kinetic energy of a material point of mass m is no longer given by the wellknown expression
 but by the expression
 This expression approaches infinity as the velocity v approaches the velocity of light c. The velocity must therefore always remain less than c, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain
 When is small compared with unity, the third of these terms is always small in comparison with the second,
 which last is alone considered in classical mechanics. The first term mc2 does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-mass; depends on the velocity. We shall speak of its essential significance later. The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the canservation of energy and the law of the conservation of mass these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about, and what meaning is to be attached to it. The principle of relativity requires that the law of the concervation of energy should hold not only with reference to a co-ordinate system K, but
 also with respect to every co-ordinate system K1 which is in a state of uniform motion of translation relative to K, or, briefly, relative to every " Galileian " system of co-ordinates. In contrast to classical mechanics; the Lorentz transformation is the deciding factor in the transition from one such system to another. By means of comparatively simple considerations we are led to draw the following conclusion from these premises, in conjunction with the fundamental equations of the electrodynamics of Maxwell: A body moving
 with the velocity v, which absorbs 1) an amount of energy E0 in the form of
 radiation without suffering an alteration in velocity in the process, has, as a consequence, its energy increased by an amount
 In consideration of the expression given above for the kinetic energy of the body, the required energy of the body comes out to be
 Thus the body has the same energy as a body of mass
 moving with the velocity v. Hence we can say: If a body takes up an amount of energy E0, then its inertial mass increases by an amount
 the inertial mass of a body is not a constant but varies according to the change in the energy of the body. The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy. Writing the expression for the energy in the form
 we see that the term mc2, which has hitherto attracted our attention, is
 nothing else than the energy possessed by the body 2) before it absorbed the energy E0.
 A direct comparison of this relation with experiment is not possible at the present time (1920; see Note, p. 48), owing to the fact that the changes in energy E0 to which we can Subject a system are not large enough to make
 themselves perceptible as a change in the inertial mass of the system.
 is too small in comparison with the mass m, which was present before the alteration of the energy. It is owing to this circumstance that classical mechanics was able to establish successfully the conservation of mass as a law of independent validity. Let me add a final remark of a fundamental nature. The success of the Faraday-Maxwell interpretation of electromagnetic action at a distance resulted in physicists becoming convinced that there are no such things as instantaneous actions at a distance (not involving an intermediary medium)
 of the type of Newton's law of gravitation. According to the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission. This is connected with the fact that the velocity c plays a fundamental role in this theory. In Part II we shall see in what way this result becomes modified in the general theory of relativity.
 Footnotes
 1)E0 is the energy taken up, as judged from a co-ordinate system moving
 with the body.
 2) As judged from a co-ordinate system moving with the body.
 [Note] The equation E = mc2 has been thoroughly proved time and again since this time.
 16-Experience and the Special
 Theory of Relativity
 To what extent is the special theory of relativity supported by experience ? This question is not easily answered for the reason already mentioned in connection with the fundamental experiment of Fizeau. The special theory of relativity has crystallised out from the Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of experience which support the electromagnetic theory also support the theory of relativity. As being of particular importance, I mention here the fact that the theory of relativity enables us to predict the effects produced on the light reaching us from the fixed stars. These results are obtained in an exceedingly simple manner, and the effects indicated, which are due to the relative motion of the earth with reference to those fixed stars are found to be in accord with experience. We refer to the yearly movement of the apparent position of the fixed stars resulting from the motion of the earth round the sun (aberration), and to the influence of the radial components of the relative motions of the fixed stars with respect to the earth on the colour of the light reaching us from them. The latter effect manifests itself in a slight displacement of the spectral lines of the light transmitted to us from a fixed star, as compared with the position of the same spectral lines when they are produced by a terrestrial source of light (Doppler principle). The experimental arguments in favour of the Maxwell-Lorentz theory, which are at the same time arguments in favour of the theory of relativity, are too numerous to be set forth here. In reality they limit the theoretical possibilities to such an extent, that no other theory than that of Maxwell and Lorentz has been able to hold its own when tested by experience. But there are two classes of experimental facts hitherto obtained which can be represented in the Maxwell-Lorentz theory only by the introduction of an auxiliary hypothesis, which in itself — i.e. without making use of the theory of relativity — appears extraneous. It is known that cathode rays and the so-called ß-rays emitted by radioactive substances consist of negatively electrified particles (electrons) of very small inertia and large velocity. By examining the deflection of these rays under the influence of electric and magnetic fields, we can study the law of motion of these particles very exactly. In the theoretical treatment of these electrons, we are faced with the difficulty that electrodynamic theory of itself is unable to give an account of their nature. For since electrical masses of one sign repel each other, the
 negative electrical masses constituting the electron would necessarily be scattered under the influence of their mutual repulsions, unless there are forces of another kind operating between them, the nature of which has
 hitherto remained obscure to us.1) If we now assume that the relative distances between the electrical masses constituting the electron remain unchanged during the motion of the electron (rigid connection in the sense of classical mechanics), we arrive at a law of motion of the electron which does not agree with experience. Guided by purely formal points of view, H. A. Lorentz was the first to introduce the hypothesis that the form of the electron experiences a contraction in the direction of motion in consequence of that motion. the contracted length being proportional to the expression
 This, hypothesis, which is not justifiable by any electrodynamical facts, supplies us then with that particular law of motion which has been confirmed with great precision in recent years. The theory of relativity leads to the same law of motion, without requiring any special hypothesis whatsoever as to the structure and the behaviour of the electron. We arrived at a similar conclusion in Section 13 in connection with the experiment of Fizeau, the result of which is foretold by the theory of relativity without the necessity of drawing on hypotheses as to the physical nature of the liquid. The second class of facts to which we have alluded has reference to the question whether or not the motion of the earth in space can be made perceptible in terrestrial experiments. We have already remarked in Section 5 that all attempts of this nature led to a negative result. Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and space did not allow any doubt to arise as to the prime importance of the Galileian transformation for changing over from one body of reference to another. Now assuming that the Maxwell-Lorentz equations hold for a reference-body K, we then find that they do not hold
 for a reference-body K1 moving uniformly with respect to K, if we assume that the relations of the Galileian transformation exist between the co
 ordinates of K and K1. It thus appears that, of all Galileian co-ordinate systems, one (K) corresponding to a particular state of motion is physically unique. This result was interpreted physically by regarding K as at rest with respect to a hypothetical æther of space. On the other hand, all coordinate
 systems K1 moving relatively to K were to be regarded as in motion with
 respect to the æther. To this motion of K1 against the æther ("æther-drift "
 relative to K1) were attributed the more complicated laws which were
 supposed to hold relative to K1. Strictly speaking, such an æther-drift ought also to be assumed relative to the earth, and for a long time the efforts of physicists were devoted to attempts to detect the existence of an æther-drift at the earth's surface. In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reflecting surfaces face each other. A ray of light requires a perfectly definite time T to pass from one mirror to the other and back again, if the whole system be at rest with respect to the
 æther. It is found by calculation, however, that a slightly different time T1 is required for this process, if the body, together with the mirrors, be moving relatively to the æther. And yet another point: it is shown by calculation that
 for a given velocity v with reference to the æther, this time T1 is different when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result — a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the differeace in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a " specially favoured " (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can
 be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses ; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun.
 Footnotes
 1) The general theory of relativity renders it likely that the electrical masses of an electron are held together by gravitational forces.
 17-Minkowski's Four-Dimensional
 Space
 The non-mathematician is seized by a mysterious shuddering when he hears of "four-dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum. Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (coordinales) x, y, z, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by coordinates such as x1, y1, z1, which may be as near as we choose to the
 respective values of the co-ordinates x, y, z, of the first point. In virtue of the latter property we speak of a " continuum," and owing to the fact that there are three co-ordinates we speak of it as being " three-dimensional." Similarly, the world of physical phenomena which was briefly called " world " by Minkowski is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z, and a time coordinate, the time value t. The" world" is in this sense also a continuum; for to every event there are as many "neighbouring" events (realised or at least thinkable) as we care to choose, the co-ordinates x1, y1, z1, t1 of which
 differ by an indefinitely small amount from those of the event x, y, z, t originally considered. That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent role, as compared with the space coordinates. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the
 Galileian transformation (t1 = t) The four-dimensional mode of consideration of the "world" is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transformation:
 Moreover, according to this equation the time difference Δt1 of two events
 with respect to K1 does not in general vanish, even when the time
 difference Δt1 of the same events with reference to K vanishes. Pure " space-distance " of two events with respect to K results in " time-distance " of the same events with respect to K. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical
 space.1) In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a consequence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure. These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes. Minkowski's work is doubtless difficult of access to anyone inexperienced in mathematics, but since it is not necessary to have a very exact grasp of this work in order to understand the fundamental ideas of either the special or the general theory of relativity, I shall leave it here at present, and revert to it only towards the end of Part 2.
 Footnotes
 1) Cf. the somewhat more detailed discussion in Appendix II.
 Part II. The general theory of
 relativity
 18-Special and General Principle of
 Relativity
 The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. Let as once more analyse its meaning carefully. It was at all times clear that, from the point of view of the idea it conveys to us, every motion must be considered only as a relative motion. Returning to the illustration we have frequently used of the embankment and the railway carriage, we can express the fact of the motion here taking place in the following two forms, both of which are equally justifiable : (a) The carriage is in motion relative to the embankment,
 (b) The embankment is in motion relative to the carriage. In (a) the embankment, in (b) the carriage, serves as the body of reference in our statement of the motion taking place. If it is simply a question of detecting or of describing the motion involved, it is in principle immaterial to what reference-body we refer the motion. As already mentioned, this is self-evident, but it must not be confused with the much more comprehensive statement called "the principle of relativity," which we have taken as the basis of our investigations. The principle we have made use of not only maintains that we may equally well choose the carriage or the embankment as our reference-body for the description of any event (for this, too, is self-evident). Our principle rather asserts what follows : If we formulate the general laws of nature as they are obtained from experience, by making use of (a) the embankment as reference-body,
 (b) the railway carriage as reference-body, then these general laws of nature (e.g. the laws of mechanics or the law of the propagation of light in vacuo) have exactly the same form in both cases. This can also be expressed as follows : For the physical description of
 natural processes, neither of the reference bodies K, K1 is unique (lit. " specially marked out ") as compared with the other. Unlike the first, this latter statement need not of necessity hold a priori; it is not contained in the conceptions of " motion" and " reference-body " and derivable from them; only experience can decide as to its correctness or incorrectness.
 Up to the present, however, we have by no means maintained the equivalence of all bodies of reference K in connection with the formulation of natural laws. Our course was more on the following Iines. In the first place, we started out from the assumption that there exists a reference-body K, whose condition of motion is such that the Galileian law holds with respect to it : A particle left to itself and sufficiently far removed from all other particles moves uniformly in a straight line. With reference to K (Galileian reference-body) the laws of nature were to be as simple as
 possible. But in addition to K, all bodies of reference K1 should be given preference in this sense, and they should be exactly equivalent to K for the formulation of natural laws, provided that they are in a state of uniform rectilinear and non-rotary motion with respect to K ; all these bodies of reference are to be regarded as Galileian reference-bodies. The validity of the principle of relativity was assumed only for these reference-bodies, but not for others (e.g. those possessing motion of a different kind). In this sense we speak of the special principle of relativity, or special theory of relativity. In contrast to this we wish to understand by the "general principle of
 relativity" the following statement : All bodies of reference K, K1, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion. But before proceeding farther, it ought to be pointed out that this formulation must be replaced later by a more abstract one, for reasons which will become evident at a later stage. Since the introduction of the special principle of relativity has been justified, every intellect which strives after generalisation must feel the temptation to venture the step towards the general principle of relativity. But a simple and apparently quite reliable consideration seems to suggest that, for the present at any rate, there is little hope of success in such an attempt; Let us imagine ourselves transferred to our old friend the railway carriage, which is travelling at a uniform rate. As long as it is moving uniformly, the occupant of the carriage is not sensible of its motion, and it is for this reason that he can without reluctance interpret the facts of the case as indicating that the carriage is at rest, but the embankment in motion. Moreover, according to the special principle of relativity, this interpretation is quite justified also from a physical point of view.
 If the motion of the carriage is now changed into a non-uniform motion, as for instance by a powerful application of the brakes, then the occupant of the carriage experiences a correspondingly powerful jerk forwards. The retarded motion is manifested in the mechanical behaviour of bodies relative to the person in the railway carriage. The mechanical behaviour is different from that of the case previously considered, and for this reason it would appear to be impossible that the same mechanical laws hold relatively to the non-uniformly moving carriage, as hold with reference to the carriage when at rest or in uniform motion. At all events it is clear that the Galileian law does not hold with respect to the non-uniformly moving carriage. Because of this, we feel compelled at the present juncture to grant a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relatvity. But in what follows we shall soon see that this conclusion cannot be maintained.
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 arXiv:1404.4075v1 [physics.gen-ph] 15 Apr 2014
 Classical and Relativistic Derivation of the Sagnac Effect
 W. Engelhardt1, retired from:
 Max-Planck-Institut f ̈ur Plasmaphysik, D-85741 Garching, Germany
 Abstract
 Both the classical and the relativistic composition law for velocities are applied to re-calculate the Sagnac Effect. The ensuing formulae for the fringe shift are found to differ already in first order of v/c. Whilst the classical formula is validated by interferometric measurements and verified by the GPS-system, this is not the case for the relativistic result.
 Keywords: Rotating Interferometer, Sagnac Effect, Ether Theory, Special Relativity Theory (SRT), Global Positioning System (GPS)
 P.A.C.S.: 07.60.Ly, 42.81.Pa, 04.20.-q, 03.30.+p
 1 Introduction
 In 1925 Michelson and Gale built a huge earth-fixed Sagnac Interferometer in Illinois [1] demonstrating that the light velocity is anisotropic on the rotating earth. For Sagnac this result did not come as a surprise having explained the underlying effect on the basis of the ether theory in 1913 [2]. The Special Relativity Theory (SRT), however, had predicted on the basis of the Lorentz Transformation (LT) that the velocity of light is isotropic in all inertial systems [3]. Of course, the earth is not an inertial system, but due to the earth’s rotation the motion of Michelson’s device was only about 2 mm during the passage of a light beam round the interferometer. The deviation of the mirrors from a straight orbit was less than one hundredth of an atomic diameter so that non-inertial effects should be entirely negligible. Nevertheless, the linear term x v/c2 in the LT – which is responsible for c = const – was simply not there. The experimental result was consistent with the Galilei Transformation (GT), i.e. light seemed to travel at the velocity c - v from west to east and c + v on the return path from east to west, where v is the local rotational velocity of the earth’s surface. Strictly speaking, the rotating earth must be described by the General Relativity Theory, but it is hard to see how a very slow rotation should invalidate the LT and re-establish the GT as found by Michelson and Gale. Indeed, Post admits in his great review article [4]: “The search for a physically meaningful transformation for rotation is not aided in any way whatever by the principle of general space-time covariance, nor is it true that the space-time theory of gravitation plays any role in establishing physically correct transformations.” Consequently, he sticks to the classical formula valid in the ether theory making only a slight modification which is too small to be verified by experiment.
 1Home address: Fasaneriestrasse 8, D-80636 Mu ̈nchen, Germany Electronic address: wolfgangw.engelhardt@t-online.de
 In this paper we re-derive the Sagnac Effect on the basis both of the classical and of the relativistic composition law of velocities that can also be formulated for a curved light path according to [5]. Calculating the ensuing travel times of light round the interferometer we find that the LT – due to its linear term x v/c2 – does not predict any Sagnac Effect, but results in c = const also in a rotating system as it does in an inertial system. This explains then why Ashby [6], e.g., uses the Newtonian or Galilean time transformation t’ = t rather than t’ = γ (t - x v/c2) when he calculates the Sagnac Effect in the GPS-System. This was also observed by Carroll Alley in a comment at the end of an engineering presentation on GPS and Relativity [7]. In Sect. 2 we derive the classical Sagnac Effect for a circular light beam, and in Sect. 3 we do the same making use of the LT. Sect. 4 discusses the implications of our results.
 2 Sagnac Effect explained in the framework of the ether theory
 The classical transformation law between inertial systems is the GT which reads:
 ~x ′ = ~x − ~v t , t ′ = t (1)
 Although it is formulated only for a linear constant velocity between the systems, it can also be adapted for a constant rotation by deriving a composition law for circular velocities. Considering infinitesimal line elements on a circular orbit one can write:
 ds ′ = ds − v0 dt , dt ′ = dt (2)
 where v0 = Ω R is now the rotational velocity with angular frequency Ω on the circular orbit s at radius R. Dividing by the time increment on both sides of equation (2) one obtains with ds/dt = vφ the composition law for constant circular velocities:
 v′
 φ = vφ − v0 (3)
 In fact, facilitating the analysis most treatments of the Sagnac Effect consider a circular interferometer where the light beam is guided on a circle either by a large amount of tangential mirrors or by optical fibres [4], [5]. The travel time of light to complete the orbit L = 2π R around the circumference of the interferometer is then
 t± = L
 v
 φ ± v0
 =L
 c ± v0
 (4)
 depending on whether the light propagates parallel or anti-parallel to the rotational velocity. Two coherent light beams starting at the beam splitter in opposite directions return at different times leading to a fringe shift due to the time difference as calculated with (4):
 ∆t = L
 c − v0
 −L
 c + v0
 = 2 L v0
 c2 − v02
 = 2 L v0
 c2
 1
 1 − v02/c2 (5)
 The linear dependence on the rotational velocity has been confirmed by many experiments, whereas the quadratic term is too small to be measured in practice. In most textbooks result (5) is expressed by the fringe shift in units of the wavelength λ0
 ∆Z = 4 A~ · ~Ω
 c λ0
 (6)
 where, in general, the scalar product of the oriented area A~ enclosed by the light
 path with the vector angular velocity ~Ω enters. One can show that the fringe shift is
 independent of the shape of A~ and of the position of the rotational axis, but depends
 on the cosine of the angle between A~ and ~Ω. Whereas Sagnac could rotate his device in the laboratory at variable speed in order to verify formula (6), Michelson and Gale had to vary the area enclosed by the light path in order to prove the validity of (6). This experiment was important as it demonstrated that the ether is not co-rotating with the earth like air. Apparently, an “ether wind” must exist due to the rotation of the earth which leads to the observed time difference (5). Any translational motion, which surely exists, cannot be detected by the circular interferometer, as the influence of a constant translational velocity cancels on the roundtrip of the light. In this respect the Sagnac Interferometer is as insensitive as the Michelson-Morley Interferometer [8] where the enclosed area A vanishes. Although the experimental results obtained by Sagnac Interferometers are in full agreement with the ether theory, as shown above, they are generally not considered as disproving the SRT which postulates c = const and is thus at variance with the ether composition law (3). In the next Section we will show why it is commonly held that there is no discrepancy between the classical composition law (3) and the relativistic philosophy concerning the behaviour of light. The reason is a defective application of SRT.
 3 Sagnac Effect interpreted in the framework of the Special Relativity Theory
 Most textbooks assure the reader that the Sagnac Effect is a “relativistic” effect which incidentally may also be derived by the ether theory leading practically to the same result. Looking closer one finds, however, that a truly relativistic derivation is not really attempted. A good example is Post’s procedure in Sect. III [4]. He also uses the circular geometry (see Fig. 8) which our analysis in the previous Section was referring to. For the stationary observer he comes up with the time difference (7 P) which is identical with our formula (5) derived from the classical composition law for velocities. This must also hold for the co-moving observer, if Ashby’s [6] Galilean time transformation t’ = t into the rotating system is valid. Post insists, however, that the time increment between the rotating and the stationary system must transform according to his equation (11 P):
 dt = γ dt ′ , γ = 1
 /√
 1 − v2/
 c2
 which can also be found in Malykin’s paper [5]. Consequently, these authors claim that the correct relativistic time difference as measured on the rotating beam splitter must be Post’s equation (23 P) or Malykin’s (5 M) which is our classical formula (5) divided by the γ-factor. One notices that the LT was only applied in a mutilated form when (23 P) was derived, since (11 P) is at variance with the complete LT formula. A different conclusion is reached by employing the correct expression, i.e. by applying the truly relativistic composition law for velocities that cannot be obtained from (11 P).
 For the case where the light beam propagates parallel to the x-axis the LT reads:
 x′ = γ (x − v t) , t′ = γ
 (
 t− xv
 /
 c2)
 (7)
 In order to adapt these formulae to a curved light path, we consider infinitesimal time increments in analogy to equation (2) assuming a constant rotational velocity v0:
 dt′ = γ
 (
 dt − v0 ds
 /
 c2)
 (8)
 where s denotes the curved light path in the rotating interferometer. This formula can also be found in Post’s article (29 P) where he quotes papers by Langevin and Trocheris. Of course, (8) is only justified as long as the light beam propagates parallel to the rotational velocity, but this is ensured by the use of an optical fibre, for example. The spatial increment
 ds′ = γ (ds − v0 dt) (9)
 is not given by Post, nor does he calculate the composition rule resulting from division of (9) by (8):
 v′
 φ = vφ − v0
 1 − vφ v0/c2 (10)
 Although not explicitly derived, this formula can be found in Malykin’s paper [5] as the second equation (2 M). He does, however, not draw the obvious conclusion: If the phase velocity of the light in the laboratory is
 v
 φ = c (11)
 it follows from (10) that the phase velocity in the rotating system is also
 v′
 φ = c (12)
 This result does not really come as a surprise, since it only reflects Voigt’s postulate c = const from which the LT (7) was derived [3]. If this transformation holds between inertial systems, it must also hold between an inertial system and a system rotating with constant velocity as just demonstrated. The obvious consequence of Malykin’s formula (2 M), or (10) above is that coherent beams leaving the beam splitter at the same time in opposite directions will return at the same time as they both travel at the same speed c according to (12). The relativistically correct result is, therefore, not Post’s formula (7 P), or Malykins formula (5 M), but simply ∆t′ = t′ + − t′ − = L/c − L/c = 0 (13)
 In other words, the SRT correctly applied to a rotating Sagnac Interferometer does not predict the Sagnac Effect.
 4 Implications of the demonstrated results
 The Michelson-Gale experiment of 1925 had already shown that Voigt’s postulate c = const is untenable. When the GPS-system delivered more and more precise measurements based on the propagation of microwaves, it became clear that the Sagnac Effect had to be taken into account. This implied the use of the classical formulae derived in Sect. 2 on the basis of the ether theory. Ashby drew the correct conclusion and
 re-introduced the Galilean time transformation t’ = t between the inertial ECI and the rotating ECEF system [6]. From Alley’s remark at the end of [7] we conclude that the GPS-software – developed by engineers for providing precise position measurements ignores the Lorentz time transformation, following Ashby’s line instead. There are, of course, other indications voiced by Hatch [9] which suggest that the SRT should not be applied to GPS-measurements. In the light of the analysis in Sect. 3 and the irrefutable experimental results it is obvious that Voigt’s postulate of 1887 [10] does not reflect an observed physical law. It was always clear that it was invalid in the case of acoustics, and we have no reason to adopt it in optics either. As the SRT is a consequence of this postulate, its basic tenet is at stake.
 References
 [1] Albert Abraham Michelson, Henry G. Gale: The Effect of the Earth’s Rotation on the Velocity of Light II. In: The Astrophysical Journal 61 (1925) 140-145.
 [2] Georges Sagnac: Sur la preuve de la r ́ealit ́e de l’ ́ether lumineux par l’exp ́erience de l’interf ́erographe tournant. In: Comptes Rendus 157 (1913) 1410-1413.
 [3] Wolfgang Engelhardt, On the Origin of the Lorentz Transformation. In: http://arxiv.org/abs/1303.5309.
 [4] Evert Jan Post, Sagnac Effect. In: Review of Modern Physics 39 (1967) 475-493. (http://www.orgonelab.org/EtherDrift/Post1967.pdf)
 [5] G. B. Malykin, The Sagnac effect: correct and incorrect explanations. In: Physics-Uspekhi 43 (2000) 1229-1252. See also equation (1.1) in: Annemarie Schied: Helium-Neon-Ringlaser-Gyroskop. Staatsexamensarbeit, University of Ulm (2006) http://www.uni-ulm.de/fileadmin/website uni ulm/nawi.inst.220/publikationen/ LehramtAnnemarieSchied2006.pdf.
 [6] Neil Ashby: Relativity in the Global Positioning System. In: Living Rev.Relativity 6 (2003) 1. Sect. 2, eq. (3) (http://www.livingreviews.org/lrr-2003-1).
 [7] Henry F. Fliegel, Raymond S. DiEsposti: GPS and Relativity: An Engineering Overview. In: 28th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, VA, 3-5 Dec 1996, pp. 189-199 (http://tycho.usno.navy.mil/ptti/1996/Vol%2028 16.pdf).
 [8] Wolfgang Engelhardt, Phase and frequency shift in a Michelson Interferometer. Submitted for publication in: Physics Essays (2013).
 [9] Ronald R. Hatch: Clock Behavior and the Search for an Underlying Mechanism for Relativistic Phenomena. In: Proceedings of the 58th Annual Meeting of The Institute of Navigation and CIGTF 21st Guidance Test Symposium, Albuquerque, NM, June 2002, pp. 70-81 (https://www.ion.org/publications/abstract.cfm?articleID=937).
 [10] Woldemar Voigt: Ueber das Doppler’sche Princip. In: Nachrichten von der K ̈oniglichen Gesellschaft der Wissenschaften und der Georg–Augusts–Universit ̈at zu G ̈ottingen, No. 2, 10. M ̈arz 1887.
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 The Sagnac-Wang interferometers and
 absolute vs. relative simultaneity
 Facultad de Ciencias Exactas y Naturales,Universidad Nacional de
 Asunci ́on,Ruta Mcal. J. F. Estigarribia, Km 11 Campus de la UNA, San
 Lorenzo-Paraguay
 Justo Pastor Lambare∗
 Abstract
 Gianfranco Spavieri, George T. Gillies & Espen Gaarder Haug (The Sagnac effect and the role of simultaneity in relativity theory, Journal of Modern Optics (2021)), claim that the Sagnac effect reveals the theory of relativity is incorrect and inconsistent. We prove that when standard relativity is appropriately interpreted, it is observationally correct and logically sound.
 Keywords: Sagnac effect, relativity theory, synchronization, absolute simultaneity, relative simultaneity.
 1 Introduction
 The so-called absolute Lorentz transformations (LTA) have been hailed in some quarters as a valid alternative to the usual Lorentz transformations (LT) [1–5]. Considering only one spatial dimension the LT is,
 t′ = γ(t − vx/c2) x′ = γ(x − vt)
 (LT ) (1)
 ∗email: jupalam@gmail.com
 1
 https://doi.org/10.32388/6C4WDR
 while the LTA,
 t′ = t/γ x′ = γ(x − vt)
 (LT A) (2)
 where γ = 1/p1 − β2 and β = v/c. The alleged virtue of (2) would be that it allows us to keep the absolute character of simultaneity. Without discussing their empirical adequacy, the most evident problem with the LTA is that it lacks theoretical and conceptual support. While the LT can be derived from clear physical principles, as was done for the first time by A. Einstein in 1905, the LTA is an ad hoc alteration of the LT with the only purpose of making simultaneity absolute. We shall refrain from referring to the LTA from here. We aim to prove that relativity theory is free from inconsistencies the Sagnac effect purportedly reveals. In Ref. [4], the authors analyze two versions of the Sagnac experiment. The ring interferometer (Fig.1 a) and the Wang, Zheng, and Yao [6] linear version (Fig.1 b). In both instances, we have a light source and a detector in the same position inside a moving system. The experiments purportedly reveal two problems: 1) for the comoving observer, the speed of light is different from its universal value c, and 2) time is “discontinuous” and a clock cannot be synchronized with itself. The first issue would constitute empirical evidence against relativity, while the second would reveal its inconsistency. In section 2, we examine the Wang’s modified Sagnac experiment, and in section 3, we analyze the ring interferometer that purportedly suffers from similar difficulties.
 Figure 1: Sagnac-Wang Interferometers
 2
 2 The Wang linear interferometer
 Spavieri et al. analyze two examples of the Wang modified Sagnac experiment. Since the problems that those examples presumably reveal are very similar, we only reexamine here the first example given in Ref. [4]. Following a similar approch, all calculations are simplified to first order. This means that we take γ = 1 in all relativistic formulas, and we can neglect time dilation and Lorentz contraction effects. Also, the time elapsed at the turning points A and B is assumed to be null, i.e., at those points, both the clock and the photon undergo infinite acceleration. Spavieri et al. note that even with these simplifications within this order of approximation, we cannot neglect the term vx/c2 in the time transformation (1) responsible for the lack of absolute simultaneity. The objective is to evaluate the elapsed time T between the emission and posterior reception of the photon φ, as registered by the clock C (Fig. 1 b). Since the total distance covered by the photon is 2L and the relative velocity is c + v,
 T = 2L
 c + v = 2L
 c(1 + β) (3)
 We shall prove that an observer comoving with the clock C finds the same value (3) applying relativity theory. Indeed, since (3) is the Newtonian result and we disregard second-order effects, both results are expected to coincide. This is most obvious by calculating
 τ=
 Z dt
 γ (4)
 over the clock trajectory since to first order γ = 1. However, Spavieri et al. explicitly analyze the role of relative simultaneity to highlight the alleged inconsistent nature of relativity theory. Thus, we shall use a similar approach to Spavieri et al. We start with the observer in the inertial frame S+ where the clock C is at rest. The observer in S+, sees the device moving at speed v from left to right (Fig 2 b). We assume that point A reaches the clock position at the same time photon φ reaches point B. If the speed of light in S+ is c, the time registered by the clock at this point is,
 Tout = L
 c (5)
 3
 As seen from the inertial frame S0 where the interferometer is stationary (points A and B stationary, Fig 2 a), the clock C undergoes a sudden acceleration and then continues with its inertial motion in the opposite direction in the frame S−. From the point of view of an observer in S+, the clock passed to the the frame S− with relative speed w = 2v (Fig. 2 c). We want to evaluate the time measured by the clock, now in the frame S−,
 Figure 2: Frames S0, S+, S−
 when the photon reaches it. We reset the clock reading to zero and use (1) to first order. The event corresponding to the photon at B when the clock is at A has spacetime coordinates (0, L) in S+. The same event in S− has spacetime coordinates (−2vL/c2, L). Thus, according to the time in S−, the photon was at B when (t−)1 = −2vL/c2. It is irrelevant that the clock was not in S− at t− = (t−)1. Once it joints S−, its time is assumed synchronized with the time of that frame and the photon has already left point B when t− = 0. Let t2 = Tret be the clock reading when the photon arrives at C. If the distance is covered with speed c, we have that (t−)2 − (t−)1 = Tret − (t−)1 =
 4
 L/c,
 Tret = L
 c + (t−)1 (6)
 Tret = L
 c − 2vL
 c2 (7)
 The total elapsed time is,
 T = Tout + Tret (8)
 T=L
 c +L
 c − 2vL
 c2 (9)
 T = 2L
 c (1 − β) (10)
 Since to first order 1 − β = 1/(1 + β), from (10) we have,
 T = 2L
 c(1 + β) (11)
 Thus, the Newtonian result (3) coincides with the relativist one (11). Now, the question is, where do Spavieri et al. see the inconsistency? According to them, since in frame S−, the photon was at B in the past of the event “E ≡ Clock turning around point A in frame S− at t− = 0” we have a missing part of the section AB = L in that frame. The missing part would correspond to the distance the photon traveled during the past of the event E. The idea seems to be that it is impossible or absurd that the photon traveled that distance in the past of E. Therefore, it had to jump from B to its position corresponding to t− = 0. However, that is true only if we assume that the photon was not indeed at B in the past of E (in frame S−), which merely means that we are rejecting the relative character of simultaneity between frames in relative motion. In other words, Spavieri et al.’s reasoning reduces to proving that relative simultaneity leads to inconsistencies by assuming that it is inconsistent. Their reasoning can be considered a more elaborate case of Einstein’s train-embankment thought experiment [7]. Einstein conceived his example to show that if we accept the existence of an invariant universal speed c, we must jettison the absolute character of the simultaneity of distant events. Similarly, the authors of [4] proved that an observer moving with the clock cannot measure the speed c unless the event E is not simultaneous with
 5
 the event “photon at B” in S−. Thus, they present another proof that an invariant universal speed and absolute simultaneity of distant events cannot stand together. From a purely logical stance, it is possible to deny relativity. But we have to reject its postulates from the start. However, it is pointless trying to prove its internal inconsistency, i.e., once we accept its premises, they do not lead to contradictions or, at least, no one so far was able to find one.
 3 The ring interferometer
 We shall prove that the correct application of relativistic formulas gives an invariant speed of light equal to the universal constant c for observers within the rotating platform and that the puzzling time gap is a natural consequence of the relativity of simultaneity and has nothing to do with the alleged impossibility of synchronizing a clock with itself or with time being discontinuous. Doing this correctly requires the use of the formalism introduced in general relativity. This does not mean that we need gravitation theory to describe noninertial systems. The last point is a widespread misconception, as Pepino and Mabile [8] recently pointed out. To avoid possible misunderstandings, we clarify that the following formalism is based on the validity of the LT and the application of special relativity through the principle of locality [9]. The need of the general metric formalism is necessary because when we pass to a noninertial system we need general nonlinear transformations yμ = yμ(x0, x1, x2, x3). The Minkowski metric ημν = diag{1, −1, −1, −1} becomes,
 gμν = ∂xα
 ∂yμ
 ∂xβ
 ∂yν ηαβ (12)
 The metric tensor components gμν become functions of the spacetime coordinates. We assume the Einstein summation convention. Greek letters indicate spacetime indices varying from 0 to 3, while Latin indices are space indices ranging from 1 to 3. The interval ds in a noninertial frame can be expressed
 6
 as,
 ds2 = gμνdxμdxν (13)
 = g00(dx0)2 + 2g0idx0dxi + gikdxidxk (14)
 = g00
 (dx0)2 + 2 g0idx0dxi
 g00
 + gikdxidxk (15)
 = g00
 (dx0)2 + 2 g0idx0dxi
 g00
 ± g0i
 g00
 g0k
 g00
 dxidxk
 + gikdxidxk (16)
 = g00( dx0 − gidxi
 | {z }
 δx0
 )2 +
 gik − g0ig0k
 g00
 | {z }
 −γik
 dxidxk (17)
 where gi = −g0i/g00. The quantities δx0 and γik has relevant physical meanings. The 3-tensor γik determines the geometry of space with respect to the noninertial frame, while the quantity δx0 is the “synchronized” time interval (see, Refs. [10–12]). As is well known by relativists, in a noninertial frame the geometry of space can be non-Euclidean notwithstanding that we are dealing with flat spacetime without gravitation [11]. Furthermore, synchronization of clocks as in inertial systems where all clocks can be set to show the same reading simultaneously is in general impossible inside a noninertial system. The proper elapsed time indicated by a physical clock in the same point of space within the noninertial frame is,
 dτ =
 √g00
 c dx0 (18)
 However, owed to the lack of synchronization of clocks in different points of space, the elapsed time as mark by two different neighboring clocks is
 δτ =
 √g00
 c δx0 ̸= dτ (19)
 Velocities should be calculated with (19), not with (18) (see, Ref. [13]). In particular, the speed of a given particle is,
 v = dl
 δτ (20)
 not the usual formula dl/dτ used in inertial systems with globally synchronized clocks.
 7
 3.1 The speed of light in a rotating platform
 Using cylindrical coordinates in an inertial frame with origin at the center of the platform (Fig. 1 a),
 ds2 = c2dt′2 − dr′2 − r′2dφ′2 − dz′2 (21)
 We perform a transformation to a frame rotating with the platform,
 t = t′, r = r′, φ = φ′ − ωt, z = z′ (22)
 In the rotating frame the interval is,
 ds2 = 1 − β2 c2dt2 − dr2 − r2dφ2 − dz2 − 2βrc dt dφ (23)
 where β = ωr/c. The values of gi and γik are,
 (gi) = (0, βr
 1 − β2 , 0) (24)
 dl2 = γikdxidxk (25)
 = dr2 + r2
 1 − β2 dφ2 + dz2 (26)
 The null geodesics giving the light paths on the rim of the platform is obtained by setting ds = 0 in (23),
 R dφ+
 dt+ = c(1 − β) (27)
 R dφ−
 dt− = −c(1 + β) (28)
 Applying the correct formula for the speed, according to (17), (19), (20), (24), (26), and (27),
 dl+
 δτ + = γc(1 − β)
 p1 − β2(1 − βR
 c(1−β2)
 dφ+ dt+
 ) (29)
 = γ2c(1 − β)
 1− β
 c(1−β2) R dφ+
 dt+
 (30)
 = γ2c(1 − β)
 1− β
 1+β
 (31)
 = γ2c(1 − β2) (32)
 = c (33)
 8
 Similarly, we can prove that dl−/δτ − = c. Thus when we correctly apply the relativistic formulas with the appropriate formalism, we find that observers in the rotating platform measure with their physical instruments, clocks and rulers, the correct invariant value of the speed of light. As a further consistency test, we can calculate the total times T + and T − that take for both light rays to complete a closed path around the platform according to observers inside it. That time must be calculated by summing the actual (corrected) time increments along their paths,
 T+ =
 I
 δτ + (34)
 T+ =
 Z 2π
 0
 p1 − β2
 dt+ − βR
 c(1 − β2) dφ+
 (35)
 T+ =
 Z 2π
 0
 p1 − β2
 R
 c(1 − β) dφ+ − βR
 c(1 − β2) dφ+)
 (36)
 T + = γ 2πR
 c (37)
 According to (26), total length L+ of the closed path is
 L+ =
 I
 dl (38)
 = γ2πR (39)
 The mean speed along the closed path is again L+/T + = c. Similarly,
 T− =
 I
 δτ − (40)
 T− =
 Z −2π
 0
 p1 − β2
 dt− − βR
 c(1 − β2) dφ−
 (41)
 T− =
 Z −2π
 0
 p1 − β2
 −R
 c(1 + β) dφ− − βR
 c(1 − β2) dφ−)
 (42)
 T − = γ 2πR
 c (43)
 Since L− = L+ we also have L−/T − = c for the clockwise beam. Thus the mean speeds, when correctly evaluated, also coincide with their local value c. One could still think that we have a paradox here because as
 9
 is well known the clockwise and counterclockwise beams do not arrive at C simultaneously. But that is also easily explained because the desynchronizing effect is different in both directions.
 3.2 Clock synchronization in a rotating platform
 This is perhaps the only point on which we partially agree with Spavieri et al. They claim that relativity implies an inconsistency arising from the synchronization of distant clocks. Next, we explain that the alleged inconsistency is only a consequence of the inappropriate definition of simultaneity that sometimes is given. The simultaneity of distant events and synchronization of distant clocks cannot be consistently defined in arbitrary reference frames. Specifically, the case that concerns us here is that of stationary fields. A gravitational field is called constant if the components gμν are independent of the time coordinate x0. Besides, when the cross terms g0i = g0i = 0, the field is called static; otherwise they are called stationary. The same classification applies to the metric field gμν when we are in flat spacetime without gravitation. In a rotating platform we have, according to (24), g0φ ̸= 0, so we must be careful when defining simultaneity of distant events. We can always define simultaneity for infinitesimally near points. The difference of coordinate time ∆x0 that correspond to simultaneous infinitesimally near points in space is given by δx0 in (17). The fact that ∆x0 ̸= 0 for simultaneous events simply means that clocks cannot be assumed to be synchronized. What about simultaneity of distant events? Here is where the confusion arises because some authors define simultaneity of distant events by integration of the infinitesimal concept,
 (x0)B = (x0)A +
 Z
 CAB
 gidxi (44)
 According to (44) we have that (x0)A and (x0)B would be time labels for simultaneous events at A and B. Since in general we have that
 I
 gidxi ̸= 0 (45)
 10
 Spavieri and other authors [4, 5, 14] have correctly pointed out such definition leads to the obvious inconsistency that a given event would not be simultaneous with itself! Therefore relativity is inconsistent. As we mentioned at the beginning of this section, the problem does not resides in the theory but in the incorrect definition of simultaneity according to (44). Indeed any consistent definition of simultaneity requires a property that (44) lacks, namely, transitivity. Really, we have that in general (Fig. 3),
 Z
 CAC
 gidxi ̸=
 Z
 CAB
 gidxi +
 Z
 CBC
 gidxi (46)
 (46) means that if A is simultaneous with B, and B simultaneous with C,
 Figure 3: Paths of simultaneity
 A is no longer simultaneous with C. There is one case when (44) can lead to a consistent definition of distant simultaneity. It is when the integration is path independent. In those cases, we have that,
 g⃗ = −∇f (47)
 However, in such cases it can be proved that a change in the time coordinate x′ 0 = x0 + f (x1, x2, x3), leaving invariant the space coordinates, leads to g⃗ = 0, so we are indeed in a static field [13]. In the case that concerns us, i.e., the rotating platform, (47) is not verified because,
 ∂gφ
 ∂r ̸= ∂gr
 ∂φ (48)
 (48) simply means the concept of distant simultaneity does not make sense for observers within the rotating platform.
 11
 4 Conclusions
 We have proved that the alleged inconsistencies and empirical evidence that the Sagnac effect presumably reveals against relativity theory are based on incorrect applications of the theory and the use of erroneous definitions. Regarding the modified Sagnac experiment of Wang et al., the explication is no more puzzling than relative simultaneity itself. So, it cannot be considered inconsistency proof unless we declare relative simultaneity inconsistent. This type of circular reasoning is typical of arguments trying to find internal inconsistencies in relativity theory. As long as the relative nature of distant simultaneity does not lead to observable or logical contradictions, its absolute character shall remain a forsaken relic of our past metaphysical prejudices.
 References
 [1] Reza Mansouri and Roman U. Sexl. A Test Theory of Special Relativity: I. Simultaneity and Clock Synchronization. General Relativity and Gravitation, 8(7):497–513, 1977.
 [2] F. Selleri. Noninvariant one-way speed of light and locally equivalent reference frames. Foundations of Physics Letters, 10(1):73–83, 1997.
 [3] Chunghyoung Lee. Simultaneity in cylindrical spacetime. American Journal of Physics, 88(2):131–136, 02 2020.
 [4] Gianfranco Spavieri, George T. Gillies, and Espen Gaarder Haug. The Sagnac effect and the role of simultaneity in relativity theory. Journal of Modern Optics, 68(4):202–216, 2021.
 [5] Edward T. Kipreos and Riju S. Balachandran. Assessment of the relativistic rotational transformations. Modern Physics Letters A, 36(16):2150113, 2021.
 [6] Ruyong Wang, Yi Zheng, and Aiping Yao. Generalized Sagnac Effect. Phys. Rev. Lett., 93:143901, Sep 2004.
 [7] A. Einstein. Relativity: The Special and General Theory - Scholar’s Choice Edition. Creative Media Partners, LLC, 2015.
 12
 [8] Ron A. Pepino and Risley W. Mabile. A Misconception Regarding the Einstein Equivalence Principle and a Possible Cure Using the Twin Paradox. The Physics Teacher, 61(2):118–121, 02 2023.
 [9] Bahram Mashhoon. The hypothesis of locality in relativistic physics. Physics Letters A, 145(4):147–153, 1990.
 [10] L.D. Landau and E.M. Lifshitz. Course of Theoretical Physics, Volume 1, Mechanics. ELSEVIER, Oxford, UK, third edition, 1976.
 [11] Anatolii A Logunov and Yu V Chugreev. Special theory of relativity and the Sagnac effect. Soviet Physics Uspekhi, 31(9):861, sep 1988.
 [12] Justo Pastor Lambare. On the Sagnac Effect and the Consistency of Relativity Theory. Qeios, 2024.
 [13] L.D. Landau and E.M. Lifshitz. Course of Theoretical Physics, Volume 2, The Classical Theory of Fields. ELSEVIER, Oxford, UK, fourth edition, 1975.
 [14] Robert D. Klauber. Comments regarding recent articles on relativistically rotating frames. American Journal of Physics, 67(2):158–159, 02 1999.
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 The Proof Of (C+V) , (C-V)
 VELOCITY OF LIGHT IS RELATIVE
 23/OCTOBER/2000 By Han Erim
 INTRODUCTION
 The data obtained from GPS (GLOBAL POSITIONING SYSTEM), a very comprehensive satellite communication project developed by United States of America, has clearly demonstrated that light may travel with (c+v) and (c-v) velocities. It would be incorrect to consider that this experimental result obtained from GPS project makes the two physics postulates of Albert Einstein discussible. These postulates are the basis of modern physics and are extremely strong and binding with respect to physics laws.
 Alice Law accommodates these two postulates, expresses easily that the light travels with (c+v) and (c-v) velocity and strongly protects these two postulates on which its own proof of existence is based. Alice Law is a new physics law. Proof of existence for this new physics law is presented on this page.
 EXPERIMENT 1
 TWO PHYSICS POSTULATES OF ALBERT EINSTEIN
 Proof of existence for Alice Law is completely based on the two postulates of Albert Einstein. We disclose the development of proof depending on these postulates.
 EXPERIMENT 1 : There is a box covered from all sides and a man inside of it. The box moves while the man is sleeping. It moves on a straight direction with constant velocity. The box was made in such a way that it does not produce any noise and shake that reveal its movement. The man does not understand that the box is moving when he wakes up. The objective of the experiment is just this. Man should not know whether the box is moving or not moving.
 Man starts his experiment. He marks the centre point (point O) of the box. He emits two lights from points A and B which represent the front and back sides of the box at the same time and measures the time they arrive in point “O”. Both lights arrive in point “O” at the same time.
 Reference Figure
 Man writes down the result of his experiment:
 MEASUREMENT 1 : For the experiment I made in the box, I measured the velocity of light as “c”. The lights I emitted from points “A” and “B” simultaneously, arrived in point “O” simultaneously. The result I obtained within the box is identical with the results of measurement I made outside.
 The event illustrated above, is the first postulate of Albert Einstein. The first postulate is as follows:
 Principle of Relativity: The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
 We will develop this event in all we will explain below. The result of measurement obtained by the man inside the box, is referred as “MEASUREMENT 1” since it constitutes a reference point for proof of Alice Law.
 Man notifies the results he obtained within the box to his friend outside via phone and says; “With the experiments I made in the inactive environment, I found the velocity of light as “c”. This value is identical with the one I measured outside”. His friend replies that “The box where you are, is not inactive. It travels with a constant velocity and on a straight direction.”
 Man in the box got surprised when he learned that the box was moving. He is sure that he made the experiments accurately. However there is a strange state that he can not explain. If the box is moving, the lights emitted from the points A and B simultaneously, should arrive in the centre of the box at different times. But this was not the case. Man does not believe his friend due to his trust in the results of the experiment but he has suspicions. He opens a big window on the wall of the box and sees that the box is actually moving. Then he decides to repeat his experiment. But this time, before he starts the experiment he opens a number of windows on the walls of the box. Then he repeats his experiment perfectly while the box is moving. He obtains the same results with his first experiment.
 Man writes down the result he obtains:
 “The velocity of propagation of light in the box, is independent of the velocity of box. Additionally, inside or outside of box is irrelevant for our case since that opening a number of windows on the walls of box had no effect on the result. Consequently, I
 Velocity of light is relative https://www.aliceinphysics.com/papers/introduction.html
 1 of 6 9/11/24, 04:40
 can say that the velocity of light is independent of the velocity of objects."
 Postulate 2, presented by Einstein:
 Universal Velocity of Light: The light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.
 EXPERIMENT 2
 THE PROVING OF ALICE LAW (c+v) , (c-v)
 It is impossible to change the result as shown by this theoretical experiment. Theoretical experiment is made assuming the centre points of both boxes as reference points. Therefore, putting forward the length contraction of objects does not change the result arisen due to equal velocities of boxes.
 Problem has arisen in this way: There are two identical boxes in all respects, with a man inside of each. There are also small holes in the front and back parts of the boxes through which light may enter.
 1) Both boxes approximate to each other with EQUAL VELOCITIES from opposite directions and a light beam approximates each box from its back. (Figure 1)
 Figure 1
 2) Let's assume that, the arrival time of lights beams to the back side of boxes are when the centre points (O and O’) of boxes intersect while boxes pass side by side. That is, when the light beam B enters into box from behind, the other light beam A approximating from the opposite direction, enters the same box from the front. This is relevant for the other box. (Figure 2).
 Figure 2
 (In reality, the condition in Figure 2, can never be met. But the way the problem is presented, is sufficient to lead us to a conclusion. The reason why the condition in Figure 2 can never be met, is again included in the result of this theoretical experiment.)
 RESULTS OF EXPERIMENT
 We may logically think about the picture of the lights in the boxes as follows:
 When the light entering from the front part of boxes reaches behind boxes, the lights entering from the back parts of boxes, have not reached their front parts yet. (Figure 3)
 Figure 3
 But physics laws tell us, this opinion does not reflect the truth of nature, because due to MEASUREMENT 1, if the men in the boxes determine that two lights enter from back and front holes at the same time, the lights must arrive other part of the boxes at the same time. (Evidence: Reference Figure)
 Let us illustrate the events occurring upon arrival of lights in boxes, with figures.
 Velocity of light is relative https://www.aliceinphysics.com/papers/introduction.html
 2 of 6 9/11/24, 04:40
 Lights arrive in boxes. Both men determine the lights entering from back and front part of their boxes simultaneously (Figure 4 1)
 While lights travel within the boxes, the boxes move on their travel directions.
 Due to MEASUREMENT 1, the man in box 1, determines that the lights entering into the box simultaneously, arrive in the opposite side of the box simultaneously. (Figure 4 – 2).
 Due to MEASUREMENT 1, the man in box 2, determines that the lights entering into the box simultaneously, arrive in the opposite side of box simultaneously (Figure 4 –3).
 RESULT : As it is seen, when we consider the positions of both boxes in space, the same light beam is in two different places in space (Figure 4 – 4)
 Due to Measurement 1 since observations by both men are correct, there should be a valid reason correcting both cases. The hint giving us solution for (c+v) (c-v) which we seek and want to see, is here.
 COMPLETING THE PROOF
 In this theoretical experiment, it is not possible to claim that the result of experiment is wrong based on the Special Relativity.
 1) For a single box, the condition of figure 2 is always satisfied.
 2) Let us define front and back parts of both boxes as A’, A” and B’, B”. Even if we assume that there exists length contraction in both boxes, it must be equal for both. When O an O' intersect, A’ and A” , B’ and B” will intersect, too. (Figure 5)
 Figure 5
 3) If for the condition of “Figure 5”, any man in the boxes, has perceived the lights at the space positions of A and B, the other man can not perceive these lights due to Special Relativity (Let's remember the famous train example of Albert Einstein). Relative to the latter man, the lights are not at space positions of A and B.
 4) If both men in boxes have perceived the lights, the boxes can not be at the same space position in movement direction due to Special Relativity. We reach the same conclusion here. The lights are at different space positions for each man.
 5) If it is claimed that length contraction will not be same for both boxes (in other words, although both are moving, one of the boxes is assumed as nonmoving reference system), we have a situation as expressed in Figure 6 since the condition of figure 2 was met for one box. The question in here is: Which man has perceived the lights? In this case, lights are at different space positions for two men if one or even both men perceived them. (Figure 6)
 Figure 6
 As it is seen, it is impossible to change the result brought by Experiment 2, based on Special Relativity. Thus we have
 proven that a light beam can exist at different space positions at the same time. This proof has brought a basis for explanation of (c+v), (c-v).
 EXPERIMENT 2 (CONTINUE)
 ACTUAL CASE IN NATURE (c+v) , (c-v)
 Velocity of light is relative https://www.aliceinphysics.com/papers/introduction.html
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 Procedure of Experiment: There are two boxes which are identical in all respects. There is one hole on the front and back parts of each box through which light may pass and an observer in each boxes. They move toward each other with constant and equal velocity. When centre points of the boxes come to the same line (point O), two light sources A and B with equal distance from point O are emitted (Figure 7).
 Figure 7
 For both boxes, the event develops as shown in the figure below: Graph is a position-time graph and represents the only case which satisfies all conditions of both postulates. (Figure 8)
 Figure 8
 Development of events:
 t1 – When centre points of boxes come to same line, light sources A, B are emitted at the same time.
 t2 – Lights arrive the front and back parts of boxes at the same time. At this time, positions of boxes in space are different. Observers inside of boxes, determine the lights entering into boxes from front and back parts at the same time.
 t3 – The lights arrive in the centre points of the boxes at the same time.
 t4 – Lights arrive in opposite parts of boxes at the same time.
 The proof of existence of Alice Law is the confirmation of Postulates 1 and 2. Both men measure the velocity of light as "c" in their boxes. In addition, if both men in the boxes will see both lights at the same time, there is only one alternative for emitting time of lights. When the centre points of boxes come to the same line (point O), two light sources A and B with equal distance from point O must be emitted. The outcome is (c+v),(c-v).
 Experiment 2 is the proof of existence of Alice Law.
 There is (c+v) (c-v) in the nature.
 Velocity of light is RELATIVE.
 Original text of this page is registered under Notary Public 37 of Beyoglu – Istanbul – Republic of Turkey on 23/October/2000 with no. 31001, by Han Erim.
 =============================================================================
 Velocity of light is relative https://www.aliceinphysics.com/papers/introduction.html
 4 of 6 9/11/24, 04:40
 You can find many new articles about (c+v)(c-v) mathematics at aliceinhysics.com which are not placed in Alice Law version 5. In addition, Alice Law version 6 is launched since October 2009 (book format. pdf file).
 =============================================================================
 SOME IMPORTANT LINKS
 Lunar Laser Ranging Test Of The Invariance Of C Daniel Y. Gezari NASA/Goddard Space Flight Center, Laboratory for ExoPlanets and Stellar Astrophysics,
 The GPS and the Constant Velocity of Light
 Paul Marmet, Professor, Physics, Laval University, Québec, Canada 1962-83, Senior Research Officer, National Research Council of Canada 1983-90
 One-Way Light Speed Determination Using the Range Measurement Equation of the GPS Stephan J. G. Gift Department of Electrical and Computer Engineering Faculty of Engineering The University of the West Indies St. Augustine, Trinidad, West Indies
 Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia
 Clock Behavior and the Search for an Underlying Mechanism for Relativistic Phenomena Ronald R. Hatch, NavCom Technology, Inc
 What the Global Positioning System Tells Us about Relativity Tom Van Flandern, Univ. of Maryland & Meta Research
 Successful GPS Operations Contradict the Two Principles of Special Relativity and Imply a New Way for Inertial Navigation – Measuring Speed Directly Ruyong Wang, St. Cloud State University, St. Cloud, Minnesota, United States
 Amplified Doppler shift observed in diffraction images as function of the COBE “ether drift” direction Carlos. E. Navia and Carlos. R. A. Augusto Instituto de F ́ısica Universidade Federal Fluminense, 24210-130, Niter ́oi, RJ, Brazil
 Non-time-orthogonality, gravitational orbits, and Thomas precession Robert D. Klauber 1100 University Manor, 38B, Fairfield, Iowa 52556
 Precision test of the isotropy of light propagation
 H. Muller [1,2], S. Herrmann [1,2], C. Braxmaier [3], S. Schiller [4], A. Peters [1] 1 Institut für Physik, Humboldt-Universität zu Berlin, Hausvogteiplatz 5–7, 10117 Berlin, Germany 2 Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany 3 Astrium GmbH, An der B31, 88039 Friedrichshafen, Germany 4 Institut für Experimentalphysik, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
 ============================================================================
 ANNOUNCE
 Alice Law Version 7 has been published (May 7, 2012)
 Download Page
 ============================================================================
 This page was updated on May 7 , 2012
 www.aliceinphysics.com Establish: December 2001 Copyright © 2000-2011. Han Erim. All Rights Reserved.
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 Physics > General Physics
 arXiv:1404.4075 (physics)
 [Submitted on 15 Apr 2014]
 Classical and Relativistic Derivation of the Sagnac Effect
 Wolfgang Engelhardt
 View PDF
 Both the classical and the relativistic composition law for velocities are applied to re-calculate the Sagnac Effect. The ensuing formulae for the fringe shift are found to differ already in first order of v/c. Whilst the classical formula is validated by interferometric measurements and verified by the GPS-system, this is not the case for the relativistic result.
 Comments:	6 pages
 Subjects:	General Physics (physics.gen-ph); Optics (physics.optics)
 Cite as:	arXiv:1404.4075 [physics.gen-ph]
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 https://doi.org/10.48550/arXiv.1404.4075
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 Journal reference:	Annales de la fondation louis de broglie, Volume 40, 2015, 149
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 Download by: [Johann Christian Senckenberg] Date: 23 December 2017, At: 04:34
 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
 ISSN: 1941-5982 (Print) 1941-5990 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm17
 XIII. On unipolar induction: Another experiment and its significance ans evidence for the existence of the æther
 E.H. Kennard Ph.D.
 To cite this article: E.H. Kennard Ph.D. (1917) XIII. On unipolar induction: Another experiment and its significance ans evidence for the existence of the æther , The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 33:194, 179-190, DOI: 10.1080/14786440208635625
 To link to this article: https://doi.org/10.1080/14786440208635625
 Published online: 08 Apr 2009.
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 [ 179 ]
 X l I I . On Unlpolar Induction : AJwther .Experiment and its Significance as Evidence for the Existence of the .,'Ether.
 By E. H. KES~Xm), Ph.D.*
 w 1. Introduction.
 "][TNIPOLAR induction in the general sense may be L J defined as induction due to motion alone., This condition requires that the magnetic induction shall remain constant at all points fixed relative to any part of the material system involved, so as to exclude any possible effects due to a changing magnetic field ; and the only form of motion satisfying this requirement is rotation about an axis of magnetic symmetry. The fundamental problem of unipolar induction is therefore this, whether the induced E.M.F. is determined by the absolute rotation of the system or by the rotation of its parts relative to each other. Theory has answered the question in three principal ways. The oldest view, put forward by Faraday and adopted by Lorentz, refers the effects to an induced electromotive intensity given by
 1-[vxB 7,
 C
 where B~-magnetic induction and v--velocity relative to the ~ether (assumed stationary); the effect will therefore depend in part upon the rotation of the system as a whole. The " moving line" theory adopts the same expression, but interprets v as velocity relative to axes fixed in the material magnetic system ; this view is virtuallyineluded in Neumann's theory of electromagnetism, and is based upon complete relativity, so that the effects depend only upon relative rotation between the parts.
 Experimentally, the question can be answered only by observations upon opeu circuits. The first investigation of this kind seems to ba one made by the author t, in which an iron bar magnetized by a stationary solenoid was set in rotation inside an insulated metal cylinder connected to earth, and a charge was looked for on the cylinder due to a possible E.M.F. in the earthing wire. The result w:~s definitely negative. Barnett's objection :~ that the negative result migl~t conceivably be due to the non-rotation of the solenoid is valid, but seems decidedly weak, for it assumes a
 * Communicated by the Author.
 ~[ E. H. Kennard, Phil. Maff.June 1912~p. 937.
 :~ S. J. Barnett, _Phys.Zeits. Sept. 1, 1912,p. 803.
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 180 Dr. E. H. Kennard on
 radical differe~lce to exist between induced and permanent magnetism, for which there is no other evidence ; it even requires that the remanent magnetism shall not acquire the inducing properties of permanent magnetism until the magnetizing force is removed, for the remanence in the bar employed amounted to over 25 .per cent, This experiment serves, therefore, to throw grave doubt upon the moving-line theory; but beyond that its theoretical significance is limited by the fact that the rotating magnet was necessarily enclosed in a second metallic screen of small diameter, and this might conceivably act so as to cut off the effect.
 I
 ]~ig. 1.
 A more instructive but more difficult arrangement consists of a cylindrical condenser inside a co-~xial ~olenoid (fig. 1), both capable of rotalion about their longitudinal axis; the condenser is connected to an electrometer as shown. Let the condenser be short-circuited and set in rotation with the solenoid energized. Then according to either Lorentz's or the moving-line theory (but not according to Hertz's) the two cylinders of the condenser should come to different potentials, and the inner one at least should be (.harged. According to Lorentz's theory rotation of the solenoid should have no effect upon this charge. According to the movingline theory rotation of the solenoid alone should charge the condenser in the same manner, while the condenser should remain uncharged when condenser and solenoid rotate together with no relative motion between them. Barnett constructed an apparatus of this sort ~, and showed that rotation of tile solenoid alone developed no charge upon the condenser. Later, Fehrle ~ reported confirmatory observations with a rather different arrangement, but his work is marred by several results which must certainly be wrong : one of his results even contradicts Faradays law for closed circuits ! Under these circumstances, while there appeared to be little doubt of the correctness of Lo,'entz's theory, yet the matter
 * S. J. B~rnett,Phys. Rev. Nov. 1912,p. 323.
 t FehEe,.Ann. xlii. p. 1109(t913).
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 Unipolar Induction. 181
 seemed of sufficient interest to justify the attempt to obtain both positive and negative results with an apparatus like that described above, and to prove experimentally that the inner cylinder really would charge up in spite of its being enclosed by a metallic screen rotating with it. Another apparatus has accordingly been constructed in which either the condenser el" the solenoid could be set in rotation. The present paper contains an account of the work and a discussion of the theoretical significance of the result.
 w 2. Apparatus.
 The condenser and solenoid are shown to scale in fig. 2, which is largely self-explanatory. All metal parts, including the base, were o[ brass. Guard-rings in line with the inner cylinder were introduced in order to simplify the calculation. Rotation was et~ected by means of a belt connected to a motor placed about 2 feet away.
 Fig. 2.
 . . . . . . . . . . . . . . . . . . . . _if_ .... /
 ~,c ',~ :.-__-'_--=-_---_-- -_-- = =.-~:J-__='=-_----C~'-~----':-2~ ,'
 J
 I
 C, brush ; D, slip-ring ; ~E,solenoid, 22"5 cm. long, 1150 tur~)sof wire ;
 F, condenser : outer cylinder 20'5 cm. long, 6"64cm. in diam. ; inner
 cylinder 15"1 cm. long, mean diam. 1"54 cm, on amber rings G.
 Diam. of shaft, 1"27cm.
 The principal apparatus was earthed by a brush of brass wires sliding on the knob B, and was otherwise insulated (except for a resistance of 10,000 ohms accidentally left in place between frame and earth); the outer cylinder of the condenser is thus practically earthed at a point on the axis. In the final arrangement the needle tt communicating with the inner cylinder consisted of a copper wire about 1 ram. in diameter, and was connected to the wire I leading to the electrometer by means of an insulated brush of brass wires bearing partly on one side and partly on the other. This
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 182 Dr. E. H. Kennard on
 arrangement was satisfactory, except that sudden large disturbances were rather frequent. An earlier arrangement consisting of a steel needle running in a drop of mercury seemed rather better; the substitution was made in trying to locate a very troublesome disturbance, which was finally traced to a minute looseness of the needle. The electrometer was placed on a separate pier about 6 feet from the apparatus and in line with it, the connecting wire lying approximately along the axis. The sensitiveness was raised by means of a concave lens tu 3000 to about 18,000 mm./volt at 4 m. distance ; the spot could be read when stationary to 0"2 ram. Electrometer and connecting wire were carefully screened and the windings of the solenoid were connected to earth through 10,000 ohms in order to render definite any residual electrostatic effects. No means being at hand to record the instantaneous speed of rotation, a device was adopted which gave a reading proportional to the product of field-strength and speed. A Faraday disk A was mounted on the end of the shaft, and a galvanometer giving ubout 3 • 10l~ mm./amp, was connected through a megohm to a brush of four brass wires bearing on the bottom of the groove in the disk, and to a second brush sliding opposite the earthing brush on the knob B.
 w 3. Method of Observation.
 Since setting the condenser in rotation caused a deflexion of several centimetres, readings were taken only by varying the magnetic field at steady speed. Let Q1 be the charge held on the inner cylinder of the condenser by electromagnetic induction, and let Q=total charge on inner cylinder, connecting wire, and electrometer, C=capacity of this system, and V=potential of electrometer and connecting wire. Then
 Q--QI+CV.
 If, now, the magnetic field is altered~ Q1is altered ; and if the system is insuluted a deflexion of the electrometer will result corresponding to a change in its potential of
 1 AQ1.
 V=-~
 The usual method was to set the apparatus in rotation and insulate the electrometer ; then read the spot and throw on
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 Unipolar Induction. 183
 the current ; after 20 seconds, read again and reverse the current ; after another 20 seconds, read again and throw off the current; after another 20 seconds, take a final reading. This gives, by subtraction, three deflexions. The electrometer was not quite dead-beat, but reached the end of its swing in just 20 seconds; intervals of this length were marked out by a torsion-pendulmn actuating a sounder. The galvanometer also was not quite dead-beat, but reached the end of its swing in 15 seconds, so its extreme position was read in each instance, giving three galvanometer deflexions likewise. Sets of readings were taken in groups of four so as to eliminate all effects that did not change sign both with the current and with the direction of rotation. The apparent deflexion due to throwing the magnetic field on was calculated by adding twice the middle deflexion, reversed in sign, to the other two and dividing the sum by 6; drift is thus eliminated. The result is a little too large because of the underdamping ; but the necessity for making.a correction was avoided by taking comparison readings m exact imitation of the principal ones, using a known source of potential. A current of 25 amp. was usually employed, and full speed was about 2000 R.Y.M.
 w 4. Calculation of Charge on Inner Cylinder.
 i t is easily shown that any point of the apparatus metallic~,llyconnected to earth comes to a potential of Np chug. units, where p~--rev, per sec. and N-----flux encircled by the point in each revolution. Accordingly, the magnetic flux through various sections of each cylinder and the shaft was compared with that through the central section of the outer cylinder by means of a ballistic galvanometer; a null method was employed and great care was taken to eliminate stray effects. It was found that the flux through any section was proportional to the area of the section and to the sum of the angles subtended by tile mean ends (radius 4"93 cm.) of the solenoid, with a maximum correction of 2 per cent., and this fact was utilized in calculating additional values. The calculation now reduces to the solution of the following electrostatic problem : given the potentials over the bounding surfaces of two spaces each having the form of a right cylindrical shell, to find the distribution of electrification.
 The work was done in steps so as to avoid the solution of
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 184 Dr. E. H. Kennard en
 more than two simultaneous equations at a time. The result for the outer space was
 V = V 0 { - - ' 2 5 9 + 1"067 log r
 --cosh ('3272) g'O205Jo('327r) +'0352 K,('327r) ] --cosh ('292x) ['O027 Jo('292r)--'O l11Ke('29'2r) ] },
 whore x=distanee along axis from centre, V0=potential at centre of outer cylinder, and
 2 2 X4
 K0 = J0 log x + ~ (1) -- ~ (1 + 89 . . . . ;
 with a similar result for the inner space (between shaft and cylinder and guard-rings). The charge on the insulated inner cylinder is tile difference between tile electric fluxes just outside and just inside the cylinder ; this turns out to be (--7"62 + 1"40) V0-- - 6"22V0. From the degree of accuracy or' the work and the following analysis of possible sources of error, it appears that the solution result should be accurate to one-half per cent. The residual error in the boundary values of the potential .exceeded 0"003Vo only over the end plates and near the end ~of the outer cylinder, where it rose quickly to O'04V0. The effect of th. rather large error over the end plates may be estimated by assimilating the space to a rectangular slab and employing as an approximate harmonic the expression
 Ae-'~-nsin ~r~_ri. It is easily shown that even if A were
 ~qual to V the effect of such a term upon the insulated cylinder would be negligible. Evidently the same must be true of the small error near the end of the outer cylinder. The connecting wire will also be charged, but the charge on the par~ of it inside the shaft is easily shown to be negligible by treating it as one coating of a small condenser. The longitudinal part attached to the cylinder will be at a lower potential than would exist at that point in the absence of the wire, because the potential due to induction is proportional to r 2, while that due to the charges will contain chiefly log r; hence the wire will be negatively charged. Calculation shows that the difference in potentials would be about
 O'03V0; estimating the capacity of the wire at 2/3 elst. unit, we have a charge on it of -- 0'02V o. Turning now to the amber insula~ors,--if the entire space were filled with amber there would be no effect upon the distribution o[ electrification*; but here we have to do with
 ~*E. H. Kennard, _Phys.Zeits. Dee. 1, 1912, p. 1155; March 15, 1913,
 p. 256.
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 Unipolar Induction. 185
 short rings, and they were also flattened on opposite sides to a depth of some 3 ram. Now a rather long calculation by the mathod of the first article cited shows that an amber shell extending the whole length of the space and from the shaft to within 3 mm. of the inner cylinder would increase the electric flux by something like 40 per cent. In the actual case the amber extended only an eighth of the length of the insulated cylinder, and was'flatten~d over less than half of the circumference, so that the effect could hardly exceed 2 per cent. of the flux ; and this would result in an extra charge of only 0"02 • 0"140V0=0"03V0. This correction just about cancels the one due to the connecting wire.
 w 5. Determination of the Theoretical _Eff~ect.
 A rough estimate o[ the deflexions to be expected can now be made as tbltows. A current of 25 amp. would produce an induction of 1600 gausses inside the solenoid, and a flux through the central section of the inner surface of the outer cylinder of about 55,400. At a speed of 33 rev. per sec. this gives a difference of V0~'0183 volt between outer cylinder and axis ; the charge on the inner cylinder will therefore be 6"22• "0183 x 1/300=0"114x 11300 elst. unit. The total capacity was found to be 138 elst. units; hence with a sensitiveness of 18,000 mm.]volt the establishment of Che m~gnetic field will cause a deflexion of 0"117 • 18,000 .~-138~-14"9 ram. If in tig. 1 the direction of the magnetic field is from left to right and the top of the condenser is moving out of the paper, then the outer cylinder will be at a higher potentialthan the axis, and the inner cylinder will be negatively charged; establishment of the field will cause a deflexion of the electrometer in the direction of positive potential. For un accurate test of theory the ratio between electrometer and galvanometer deflexions was employed. For the comparison readings a potential of V = 0"02 volt (about) was applied to the electrmneter case, which was used as a standard inductor, and simultaneously exactly one-tenth of this was applied through the megohm to the galvanometer; by applying, reversing, and removing the potential, readings were taken in imitation of the principal ones. The electrometer readings agreed within 2 per cent., and averaged 20"5 ram., while the galvanometer deflexion was 62"2 mm. The coefficient of influence of the "electrometer case had been compared previously by a null method with a good guardring condenser, whose capacity was computed to be 40"1 elst. units, and was found to be 7'42. On the other hand, the
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 186 Dr. E./El. K e n n a r d on
 flux through the Faraday disk was found with a ballistic galvanometer to be 0'1197 of that through the central section of the outer cylinder, so that in the main experiment the E.M.F. in the galvanometer circuit was 0"1197V0. Hence, by proportion, the ratio of the deflexions due to induction should be
 20"___~5 "IV .6"22V0 =0"231. 62"2 "1197Vo 7"42V
 w 6. 7"he JLTectromagnetic Induction Effect.
 The final results on the rotation of the solenoid alone and of solenoid and condenser locked together are shown in the annexed table. Colmnns S and I give the direction of rotation and the initial direction of current respectively, E and G are the electrometer and galvanometer deflexions, and R is their ratio. "Rev. av." denotes the average effect reversing with current and direction of rotation. Four sets of readings were rejected for v e r y obvious good reason, and at once, replaced ; and perhaps as many more sets were not completed because of the occurrence of an obvious large disturbance. To illustrate the character of the separate readings: the three electrometer deflexions for the first line in the table were +8"5, +4"8, +3"7; for the second, +22"4, +13"8, +1"8; for the fifth, +5"1,--39"1, +3"3; for the sixth, --12"6, +29"9, --21"5.
 S. I. E. G.
 Sol. . . . . -- + + "4ram. + '5mm.
 -- -- + "6 - "3
 + - -- 1.1 - .6 + + + 3.5 + -2
 R,
 Rev. av. + 1'2 0
 Both... + + +14"4 +60"7 "237 -- + - 15"6 --65"7 '237 -- -- +16"4 +69"4 "236 + -- - 18"0 - 69'8 "258
 Final Observations. Av. .242
 The first group of results confirms Barnett's conclusion that the electromagnetic effect of rotating the solenoid alone is nil. A similar group taken previously ranged from "8 to 2"7 ram., and gave a reversing effect of only + ' 2 ram.
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 Unipola~" Induction. 187
 For the case where solenoid and condenser are set in rotation at the same speed and in the same direction, the mean value of the ratio 0"242 is 5 per cent. larger than the theoretical value of 0"231. But immediately after these observations a similar group of four sets taken at half speed .gave a mean ratio of 0"228 (range "211 to "263), which is 1"4per cent. too small. Then a group was taken at full speed but by a different method: in quick succession the observer threw on the current, insulated the electrometer, read the spot, and reversed the current; after 20 seconds another reading was taken and the current was tarned off,. and after another 20 seconds a final reading was taken. This gives two deflexions in opposite directions, a third of whose numerical stun should be the electromagnetic effect. The values thus obtained ranged from "195 to "271~ and averaged 0"238; including three readings rejected because the first reading was not the greater~ the average becomes 0'241~ Proportionality of the effect to current strength had been tested previously; a group of four sets using 13 amp. gave a mean ratio (not comparable with the previous values) of "25, while a group taken under the same conditions, but using 25 a m p , gave "27. With the condenser alone rotating final observations were not obtained because the solenoid became short-circuited, and it was not thought worth while to rewind it. The case is, however, deducible from the other two by simple superposition; and partial confirmation is furnished by a group of observations which were taken under about the same conditions, and gave a mean electrometer deflexion of 14 ram. The results thus remain somewhat rough, but they seem sufiicient to warrant the conclusion with fair certainty that the condenser becomes charged by its own rotation in the manner required by Lorentz's theory, independently of the rotation of the solenoid.
 w 7. Fate of the Moving-Line Theory.
 The view has been put forward by Poineare *, Abraham ~', and Barnett ~ that even observations on open circuits cannot disprove the moving-line theory; and in reply to criticism by the author, Barnett maintained this view with some heatw
 " H. Poincar~,3~cl.ffSlect,xxiii, p. 41 (1900).
 t Abraham & F5ppl, Theo. der Elektr. p. 420.
 :[ S. J. Barnett, Phys. Rev. Nov. 1912, p. 323.
 w S. J. Barnett, Phys. Rev. Oct. 1913, p. 323.
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 188 Dr. E. H. Kennard on
 The author yields to none in appreciation of Barnett's work on unipolar induction, and in respect for his long experience with electromagnetic theory ; but on this point it seems clear that all of the writers mentioned have, through an oversight, arrived at a wrong conclusion. Tile matter has been fully discussed elsewhere*, but perhaps a smnmary of the argument will not be amiss here. Barnett urges that an adherent of the moving-line theory must suppose tile lines (i. e. tile rotating magnetic system) to set up in the tether the same electromotive intensity that is caused to act upon a material dielectric, and slmws from this ~ssumpr that tile charges developed upon conductors would be the same whether the lines moved or not. The objection which seems to the author conclusive is that this 1
 electromotive intensity, of magnitude c IV x B~, does not in
 general satisfy Laplace's equation. (The proof is simple.) In spite of this fact Barnett employs the two additional assumptions :
 1 IV • B] + e+E, (1)
 /=~ . . . . .
 where f=total electromotive intensity, e = electric force due to changes in the magnetic fis]d and vanishes in the present instance, E =electric force due to eiectrostatic cause3 alone ; and the equation
 div (Kf)=4~rp, . . . . . . (2)
 where O= electric density.
 Now in free tether we know from electrostatics that
 d i v E = 0 . Hence in free ~ether d i v f = - l d i v [ V x B ] , and
 C
 may not vanish, which contradicts (2). We are thus forced to the conclusion that if the moving-line theory is correct, then the lines do not act on the tether, but only on material bodies, and instead of (2) we must write (with Lorentz) div (KE) =47rp. The real root of the matter seems to be that the tether Cannot be ~reated as only a particular species of the genus "dielectric," but must be regarded as complementary to all material dielectrics. The displacement or polarization in matter does not need to be solenoidal, because its sources can be cancelled by sources of opposite sign in the displacement in the m~her, and the total displacement may then be
 * E. H. Kennard,Phys. Rev. May 1913~p. 355.
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 IJ3zipolar Induction. 189
 solenoidal. But free ~ether has no such means of selfde[once, and consequently all forces acting on it must themselves be solenoidal. The moving-line theory seems therefore to be definitely disposed of by the results obtained by Barnett, Fehrle, and the author.
 w 8. Conclusion.
 The practical bearing of these experiments is small, yet they do necessiLato a correction of cert~tin statements that are common in the textbooks. For instance, it is not correct to say that the effect of rotating the armature of a dynamo is the same :~s that of rotatiug the field-magnets in the opposite
 direction. The total E.M.F. is the same, but in the first case it is developed ahnost entirely in tile longitudinal parts of tile winding, while in the second case a large fraction of it is developed in the radial parts, and the distribution of electrification on the armature will be different. The most interesting case theoretically is that where the solenoid and condenser rotate together at the same speed. The charging up of the condenser cannot be conditioned by rotation relative to the connecting wires and electrometer. For suppose that the sliding contacts had been exactly on the axis, which was nearly true, and that they were connected by an axial wire extending through the condenser so as to form a closed circuit. Then it is certain, by Faraday's law, that rotation of the connecting wires and electrometer would cause no deflexion of the latter ; and this negative result can hardly be due to an E.~I.F. in the axial wire which was added. Rotation relative to the e~rth might in some unknown way be responsible for the observed effect, but this seems improbable. If we reject this assumption, then the effect is due to an absolute rotation in the same sense in which the operation of a gyrocompass is due to an absolute rotation of the earth. But rotation of the whole is essentially only translation of the parts. We must suppose each electron in the condenser to experience a radial force proportional to its distance from the axis ; and it is interesting to inquire what can cause such a force. It cannot be due either to the radial acceleration, for that is proportional to the square of the distance, nor to a possible rotation of the electron about its axis, ibr that should be the same for all electrons, nor to their motion relative to other essential parts of the apparatus~ for there is no such motion. Apparently the only renlaining alternative is to ascribe the force to motion of the electrons relative to
 Downloaded by [Johann Christian Senckenberg] at 04:34 23 December 2017
 190 Prof. W. M. Thornton on tile Limits oj
 smnething which is not matter, and which does not share in the rotation of the condenser, and this something must be the stationary ~ether of Lorentz. This phenomenon seems therefore to lend definite support to the existence of an electromagnetic ~ether. It is~ perhaps, the only low-frequency phenomenon which cannot easily he described in terms of " action at a distance " between electrons and atoms. It seems, for this reason, to have some importance as a stumbling-block in the way of those nltrarelativitists who would abandon the conception of an rather
 altogether.
 SuT~ ~q~ale~].
 An experiment is described showing that a cylindrical condenser r~tatlng inside a magnetized coaxial solenoid becomes charged as required by the theory of Lorentz. Rotation of the solenoid has no effect (Barnett). The disproof of the moving-line theory is thus completed ; electromagnetic induction depends in part upon absolute rotation in the mechanical sense. Analysis in terms of electrons seems to make necessary the existence of a stationary ~ether in order to explain the observed effect; so that the phenomenon seems to present difficulties for those
 relativitists who reject the rather.
 It is a pleasure to acknowledge obligations to Mr. (Jhristian Dane, mechanician, whose share in the construction of the apparatus left nothing to be desired; and to Professor Henry A. Erikson for his kindly interest and
 advice during the progress of the work.
 University of Minnesota,
 August, 1916.
 XIV. The Limits of Inflammabillty of GaseousMixtures. By
 W. M. THOI~NTON,J/).Sc., 2). Eng., Professor of Electrical
 Engineering in Armstrong College, 2Vewcastle-u'pon-Tyne ~.
 1. I r ~ H E ignition of an inflammable gas mixed with air ] _ d~pends in a variety of ways upon the proportion
 of oxygen present. With impulsive sparks or condenserdischarge the ignition passes through critical stages when the ratios of the number of oxygen atoms to one molecule of
 Communicated by the Author.
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@@ -0,0 +1,180 @@
 Home
 GPS Mythology – part 2
 Secondary relativistic effects
 The preceding chapter examined claims regarding the necessary application of Relativity to the Global Positioning System. It argued that, providing the satellites orbited at the same altitude and no adjustments were made to take Relativity into account, the largest error that could accumulate over a 24 hour period would be an unnoticeable 15cm.
 This is nowhere near the enormous 11km that is frequently claimed. Take this quote from Relativity promoter Clifford Will for example:
 “But at 38 microseconds per day, the relativistic offset in the rates of the satellite clocks is so large that, if left uncompensated, it would cause navigational errors that accumulate faster than 10 km per day! GPS accounts for relativity by electronically adjusting the rates of the satellite clocks, and by building mathematical corrections into the computer chips which solve for the user's location. Without the proper application of relativity, GPS would fail in its navigational functions within about 2 minutes.” [1]
 Is he kidding? Nothing much is going to accumulate over a two minute period other than a miniscule error of 0.2mm. The only thing this paragraph demonstrates is that the majority of Relativity followers can’t think straight and function by blindly repeating the words of other followers.
 Another problem we have is that the stated 38μs-per-day adjustments are made in the satellites rather than the receivers. The satellites are military equipment and contain special encryption used for accurate missile targeting. Hence a close examination of them would be off-limits to almost everyone including the majority of scientists. This makes the system somewhat inadequate as a test for Relativity. Even those who work at the GPS ground-monitoring stations with super-accurate clocks would be unable to determine what adjustments are made: Perhaps the satellites experience the stated amount of time dilation and adjustments. Or perhaps they experience different amounts of dilation and adjustments. Or perhaps they experience no dilation and no adjustments are made.
 All that aside, some might argue that GPS can still be a good test of Relativity because there are a number of adjustments for ‘secondary relativistic effects’ built into the receivers that we can verify directly. The first has to do with imperfect orbits and the second with the Sagnac Effect. The purpose of this chapter will be to examine them, starting with the former.
 Adjustments for eccentric orbits
 In an ideal world, GPS satellites would be launched so as to sit in perfectly circular orbits at predetermined altitudes. In this way the amount of relativistic effect would be the same for every satellite and have a constant value throughout its orbit. Hence an identical timing adjustment could be programmed into each satellite to negate the effects. Such an adjustment would allow a GPS receiver not have to take relativity into account.
 In the real world however, things don’t always proceed as planned. The satellites instead end up with slightly elliptical orbits. Because of this, the amount of time dilation will vary throughout an orbit as the satellites experience variations in their speeds and altitudes. And because of this, the fixed amount of
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 relativistic adjustment that is built into satellites pre-launch is insufficient to negate the full effects of relativity. Instead, the receiver needs to make some of its own adjustments to get proper accuracy.
 This May 2002 Physics Today article by Neil Ashby describes the situation:
 “If a clock's orbit is not perfectly circular, gravitational and motional frequency shifts combine to give rise to the so-called eccentricity effect, a periodic shift of the clock’s rate with a period of almost 12 hours and an amplitude proportional to the orbit’s eccentricity. For an eccentricity as small as 1%, these effects integrate to produce a periodic variation of amplitude 28 ns in the elapsed time recorded by the satellite clock. If it is not taken properly into account, the eccentricity effect could cause an unacceptable periodic navigational error of more than 8 m.
 During the early development of the GPS, onboard computers had limited capability. It was decided that correcting for the eccentricity effect would be left to the receivers. Inexpensive receivers accurate to no better than 100 m may not need to correct for the eccentricity effect. But, for the best positional accuracy, receiver software must apply a relativistic eccentricity correction to the time signals broadcast by each satellite. The navigation message includes the current eccentricity, orbital elements, and other information that the receiver needs for computing and applying the eccentricity correction.” [2]
 The below diagram shows where these relativistic effects come into play.
 On the right side of the diagram is the perigee where the orbit is closest, and the left side is the apogee where it is farthest. (For visual ease the eccentricity is shown greatly exaggerated.)
 The perigee has the highest speed and the lowest altitude. So it should have high amounts of both motion-based and gravitational-based time dilation.
 The apogee has the lowest speed and the highest altitude. So it should have low amounts of motionbased and gravitational-based time dilation.
 As a result, a clock in such an orbit would appear to run more slowly at the perigee and more quickly at the apogee. Because of this, receivers would need to make adjustments to the received clock signals in order to calculate position correctly.
 Another explanation?
 The fact that adjustments need to be made within the receivers would seem good evidence for relativity. Because, unlike satellite adjustments, what goes on inside the receiver can be easily confirmed – perhaps not by users, but easily enough by receiver manufacturers, of which there are many.
 So does this mean the eccentric adjustments have confirmed relativity after all, or could there be another explanation?
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 In an earlier chapter on light it was proposed that light moves through a vacuum with a speed that is added to the speed of its source, i.e. at a speed c+v where v is the speed of the source toward the target. This is known as the “ballistic” or “emission model” of light.
 Look again at the orbital diagram:
 Note that as the satellite moves from the apogee to the perigee (farthest to nearest point), it is moving toward Earth. So a signal emitted during this phase would have the satellites’ speed added to it and move at speed c+v rather than just c. It would arrive sooner than “expected”.
 Likewise as the satellite moves from the perigee to the apogee (nearest to farthest point), it is moving away from Earth. So a signal emitted during this phase would have the satellites’ speed subtracted from it and move at speed c-v rather than c. It would arrive later than “expected”.
 Now if someone on Earth was under the assumption that the speed of the signal must always be c, they would likely be confused as they see the signals arriving earlier or later than their fixed-speed-of-light model predicted. They might conclude that the satellites’ clocks were somehow running at different speeds at different points in their orbit. So this person would be working under an erroneous assumption. And even though they would be able to use this assumption to accurately predict the arrival times of signals, their assumption would still be wrong.
 Could it be then that this apparent periodic variation in clock rates is in fact just a change in the transmitted signal speed?
 Ballistic model analogy
 To answer this we need to look at a simpler analogy. Consider this classical mechanics situation:
 Here a gun aims at a target at distance R. The gun is motionless with respect to the target and fires a bullet at velocity b (relative to the gun). The bullet will take time T=R/b to reach the target.
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 Now we change things a bit. In the above diagram, the gun moves toward the target at velocity v (relative to the target). When it is at distance R, it fires the bullet. How long will the bullet take to reach the target?
 The bullet now moves at velocity b+v (relative to the target) because the gun’s velocity v is added to the bullet’s. Therefore it takes time T=R/(b+v).
 The difference between these times is:
 ΔT = R/b – R/(b+v) = R v/(b (b+v))
 For values of v much less than b, i.e. when the bullet moves much faster than the gun, this closely approximates to:
 ΔT ≈ R v/b2
 Now translate the above to a satellite situation: replace the gun with a satellite and the bullet with a radio signal moving at speed c (relative to the satellite). Replacing b with c gives:
 (1)
 This tells us that the signal will arrive earlier by a time difference proportional to distance and approaching velocity.
 The calculated position error that would result from this time difference (if one were assuming light’s speed must be c) is determined by multiplying this by c. I.e.:
 (2)
 So how would this translate to a real situation? Let’s say a GPS satellite had an eccentricity of 1%. The below graph shows its relative velocity toward Earth.
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 Here the horizontal scale represents time in hours over a 12 hour orbit. Hour zero is the perigee (closest) and hour 6 is the apogee (farthest). The vertical axis represents velocity toward the Earth. As can be seen, the velocity reaches a maximum of 38.76m/s on its approach and a minimum of negative the same amount on its departure.
 This next chart shows the change in radial distance over the orbit:
 There is very little change, fluctuating by ±265km or 1% at most.
 So what happens when we multiply these two charts in line with equation (2)? We get this:
 This shows the position error varies by plus and minus 3.43m. This is almost half the 8m described in the referenced article. In fact it turns out to be exactly half the relativity-calculated amount.
 The GPS Navigation User Interface guide gives the exact equation that is to be used by receivers [3]. It is:
 (3)
 This corresponds to the amount of cumulative time difference due to special and general relativity. e is the eccentricity, Ek is the “eccentric anomaly”, which is the angle of orbit measured from the centre of the ellipse, and G, M, and A are the gravitational constant, Earth’s mass, and satellite’s semi-major axis respectively.
 So what would be the equivalent equation due to changes in signal speed? It works out to this:
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 (4)
 Comparing this equation with (3), we see it has the same form but it is exactly half the amount. Here is what a chart would look like comparing them for a 2% eccentricity (e=0.02) and multiplying by c to convert to position error.
 As can be seen, the prediction of relativity gives a maximum of 13.73m, while the prediction due to additive velocity gives a maximum of 6.86m. So it would seem that additive velocities can predict half the measured error. But since the receivers are programmed to accommodate for double that amount, does that mean the additive velocity idea is wrong and relativity is correct? Or is the other half adjusted for elsewhere? We’ll get back to that.
 The Sagnac Effect
 The other relativistic effect we apparently need to adjust for is the Sagnac effect. The literature explaining this is somewhat confusing so let’s ease into it with a couple of quotes:
 The Physics Today article mentions:
 “In the GPS, the Sagnac effect can produce discrepancies amounting to hundreds of nanoseconds. Observers in the nonrotating ECI inertial frame would not see a Sagnac effect. Instead, they would see that receivers are moving while a signal is propagating. Receivers at rest on Earth are moving quite rapidly (465 m/s at the equator) through the ECI frame. Correcting for the Sagnac effect in the Earth-fixed frame is equivalent to correcting for such receiver motion in the ECI frame.”
 While Wikipedia says:
 “GPS observation processing must also compensate for the Sagnac effect. The GPS time scale is defined in an inertial system but observations are processed in an Earth-centered, Earth-fixed (co-rotating) system, a system in which simultaneity is not uniquely defined. A coordinate transformation is thus applied to convert from the inertial system to the ECEF system. The resulting signal run time correction has opposite algebraic signs for satellites in the Eastern and Western celestial hemispheres. Ignoring this effect will produce an east-west error on the order of hundreds of nanoseconds, or tens of meters in position.” [4]
 Based on this we can conclude that the Sagnac Effect:
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 • has to do with the rotation of the Earth. • is most significant at the equator. • is very important to compensate for because it can result in “tens of meters” of position error.
 We will now investigate a more detailed source by the same author here [5].
 On page 8 is a calculation discussing sending signals around the equator. It predicts a timing error of 207.4ns (equation 8). The calculation is rather complicated so let’s break it into easier steps:
 1. If a signal was transmitted east along the equator it would take 0.1337 seconds to get back to its starting point (this could be done using mirrors or optic fibres for visible light, or re-transmitters for radio signals). Calculation: divide the circumference of the equator by the speed of light: 2*pi*6378137/(2.998x108) = 0.13367s 2. But the equator is moving, and during that time the original transmission point would have moved east by 62m. Calculation: multiply that time by equatorial velocity: 0.13367*465.1 = 62.17m 3. It would take an additional 207.4ns to cover that distance. Calculation: divide that distance by the speed of light: 62.17/(2.998x108) = 2.074x10-7s = 207.4ns
 At this point any rational person would have to be wondering: “What the heck does this have to do with the GPS?” GPS satellites don’t send signals around the equator!
 When we read the words “Sagnac effect” it normally refers to light travelling around a circle via rotating mirrors, such as this typical laboratory-setup diagram shows:
 Does this diagram resemble any portion of the GPS? Satellites transmit their signal to Earth and that’s that. There’s no transmission back to the satellites, or between satellites, and none of their signals are made to loop the equator. So in what way does the Sagnac effect have any relevance?
 One place where the Sagnac effect would have relevance would be in the synching of groundmonitoring stations. There are six monitoring stations located roughly along the equator and these would need to be kept in-synch with each other. Such a process would need to adjust for the Sagnac effect, as the signal from the designated master station was transmitted to the others.
 But the documentation discussing the effect insists that it is also relevant to GPS observers/users and that the receivers need to make adjustments for it or they will calculate “tens of meters” of position error. How to make sense of this, and what is the 207ns delay about?
 Have a closer look at the motion of a satellite across the sky, this time including the shape of Earth:
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 Here a satellite rises above the right horizon, passes directly overhead a user, and goes down over the left horizon. As it does so, notice that it spends time moving toward and away from the user. This graph shows the velocity in a straight-line direction, as viewed by the user:
 The velocity is zero when the satellite is directly overhead. But as it approaches, the toward-velocity goes up to 927.9m/s, when level with the horizon. And as it recedes, the velocity goes negative by the same amount.
 Returning to our earlier discussion on additive velocities, this could mean the signal on the approach path arrives earlier than expected because net velocity is higher, i.e. c+v. Likewise it could take longer to arrive on the departing path because signal velocity is lower, at c-v.
 Could this be the explanation for the timing differences? Satellites sit at an altitude of 20,184km. Plugging this into equation (1) gives:
 ΔT = R v/c2 = 20184000*927.9/(2.998x108)2 = 2.0837x10-7s = 208.4ns (5)
 This is quite amazing as we have arrived at an almost identical number as what the Sagnac-around-theequator would predict! More importantly it was arrived at using very different inputs. The 207ns number used the equatorial circumference and velocity. Whereas this 208ns number used the satellite’s distance and velocity.
 Could these additive velocities be the real cause of what is labelled “Sagnac effect”? It does seem more logical, especially as there is no reason why the Sagnac effect should have any relevance to the communication between satellites and receivers.
 Near the end of page 9 is a discussion about an experiment done to measure the effect:
 “In 1984 GPS satellites 3, 4, 6, and 8 were used in simultaneous common view between three pairs of earth timing centers, to accomplish closure in performing an around-the-world Sagnac
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 experiment. ... The size of the Sagnac correction varied from 240 to 350 ns.”
 “240 to 350ns”? Shouldn’t that be “0 to 207ns”? The effect is supposed to be due to Earth’s rotation, meaning it should be zero at the poles. And how did it get up to 350ns when 207.4ns represents the maximum possible (at the equator)?
 Refer to equation (5). Here the distance used for R was the satellite altitude. This is only correct when the satellite is directly overhead. In reality the satellite-receiver distance will be larger when near the horizon. This graph shows the actual distance as the satellite moves overhead.
 When level with the horizon, the distance is 25,766km. Using that value in equation (5) gives a timing value of 266ns (207.4ns will occur when the satellite is 29 degrees above the horizon).
 This is above 240ns but still a fair way from 350ns. Perhaps that 350ns reading was done on an aeroplane and the satellite in question had an eccentric orbit. E.g. if the orbit eccentricity was 1%, this would increase the maximum R value to 26,800km. And if the aeroplane moved at 245m/s (a typical jet cruising speed [6]), we could add this to the 928m/s. Plugging this into equation (1) gives a value of 350ns. Now it is true that the reference didn’t say that one of the “timing centers” was on an aeroplane, or that they were all fixed on the ground for that matter. But how can they justify getting 350ns using the equatorial method?
 We’ll look now at another reference here [7]. It appears to be a more detailed version of the previous reference [5]. On page 10 is a diagram that is reproduced below:
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 Notice there is a formula at bottom with the words “Sagnac correction” pointing to one of the terms. This term looks identical in form to equation (1), where R also refers to the distance between the satellite and receiver. This is quite a change from the previous reference where Sagnac corrections were calculated based on the Earth’s circumference.
 The only difference between this and (1) is that v refers to the velocity of the Earth’s surface relative to its centre, rather than the velocity of the satellite relative to the surface. But should that be correct?
 Taking the diagram at its word, the largest value of v will be the equatorial velocity of 465.1m/s. The largest R would occur when the satellite is on the horizon, which will be R=25769km. Plugging this into the suggested “Sagnac correction” gives:
 v R/c2 = 465.1*25769000/(2.998x108)2 = 133.3ns (6)
 133.3ns? That is well short of the 207.4ns calculated using the “equatorial circumference” method. Something fishy is going on here!
 On page 9 it gives the same equation (1.17) and equation 1.14 makes it clear that R refers to the distance between the satellite (rT) and the receiver (rR).
 So what is v? In the paragraph above that it says:
 “... and let the receiver have velocity v in the ECI frame.”
 Where “ECI frame” means velocity relative to Earth’s centre, which corresponds to the velocity described in the above diagram.
 Logically however, v should refer to the relative velocity between the satellite and receiver. After all, these are the components that make up the GPS. The velocity of the receiver relative to Earth’s centre should not be relevant, any more than the velocity of the receiver relative to Jupiter could be relevant. Earth just happens to be where the components are located. If we were to relocate the satellites and receivers to Mars, there is no reason why v should refer to Earth’s centre.
 To understand this better, it would be helpful to refer to some specific calculations. To do this we can
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 refer to the User Interface guide mentioned earlier [3]. Scanning this PDF for the word “sagnac”, we find no mention of the Sagnac effect!
 Quite amazing. On one hand we are told that the Sagnac effect is important to adjust for otherwise it will result in “tens of metres of error”. But the interface guide gives no specifics on how to adjust for it, or even say that it needs to be adjusted for. It should be also pointed out that this manual is what GPS receiver manufactures are meant to rely on, so any necessary adjustments would have to be described there.
 Does this mean the “Sagnac effect” (or additive velocity equivalent) adjustments have no relevance? I would say they do have relevance, but have been left out of the main guide and sent separately to manufacturers.
 The reason for leaving out specific calculations might be to prevent researchers from seeing that the velocity v refers to the relative velocity between the satellite and receivers, rather than equatorial velocity. And the reason for not wanting people seeing this is that they might discover that the GPS shows that light’s speed is not a constant speed c, as seen by all observers, but varies with the speed of the source. Such a discovery would demolish the fundamental postulate of Special Relativity.
 It appears there are other things left out of the manual. For example on page 113 a diagram says you need to compensate for “Tropospheric Adjustments”. However there is no mention in the guide about what adjustments to make (search the PDF for “tropospher”). But we can find plenty of information on the tropospheric adjustments elsewhere on the web e.g. by searching “gps tropospheric delay”. Searching the gps.gov website brings up 50 results over three pages [8].
 By comparison, the only information available on the web for Sagnac adjustments for GPS appears to be copies of Ashby’s papers. Searching the gps.gov site for information on the Sagnac effect turns up only three ‘powerpoint’-type visuals [9]. These say the maximum is 133ns [10], which corresponds to equation (6). The absence of detail on the Sagnac effect on the gps.gov website is surprising to say the least!
 Referring back to the Physics Today article [2], on page 46 is a column “Confusion and consternation”. It talks about the problem of adjusting for fast-moving receivers and says:
 “One can estimate the discrepancies from the approximate synchronization correction: v x/c2, where v is the receiver’s speed through the ECI frame and x is its distance from the GPS satellite in question.”
 This looks similar to the “Sagnac” effect in equation (6). It then says:
 “Within the framework of general relativity, however, one coordinate system should be as legitimate as another.”
 That’s true, so why not make the receiver the reference frame and describe v as the relative velocity between satellite and receiver? It continues:
 “Measurements made by an observer traveling with a moving receiver can just as well be described in another reference frame, by using transformations that relate the two frames. In the special case of two inertial frames in uniform relative motion, these are the familiar Lorentz transformations.”
 This may be a clue. Transforming time using Lorentz transforms uses the formula:
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 (7)
 For values of v much less than c, the expression inside the square root becomes very close to unity. So this is approximately:
 (8)
 The expression on the right-hand side now looks like the additive-speed-of-light adjustment in (1). It’s possible that receivers manufacturers are told to make a Lorentz transform as shown by (7) and this effectively gives them an additive-speed-of-light adjustment.
 It’s also possible that manufacturers are not given the specifics but need to figure it out themselves. The column begins with:
 “Historically, there has been much confusion about properly accounting for relativistic effects. And it is almost impossible to discover how different manufacturers go about it!”
 If the interface guide had given specifics for handling the Sagnac effect there should not be “much confusion” as manufacturers would go about it the same way. Perhaps manufacturers do their own experiments and end up settling with the Lorentz time transformation as shown in (7).
 The TOPEX experiment
 This brings us back to the earlier discussion about relativistic adjustments for orbital eccentricity. In order to resolve the question of reference frames, the Physics Today article goes on to discuss an experiment that tested for the eccentricity effects. It was done in space on a satellite called TOPEX. This diagram is provided:
 It shows the relativistic error for a GPS satellite with 1.5% eccentricity. For an eccentricity of 0.015, this calculates to a relativistic error of 10.3 metres, and this closely matches what is shown in the diagram.
 An obvious question arises however – why do this experiment in space rather than on the ground? We
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 have extremely accurate clocks in many laboratories. If it were possible to do it on the ground, then it could be verified independently. But if it were not possible, then we can’t know whether the eccentricity effect is due to relativity predictions or additive-speed of light predictions. Instead we must take the word of those promoting the relativity theory. This means the magnitude of the eccentricity effect is questionable, just like the existence of the 38.6μs-per-day adjustments.
 Conclusions
 1. There is no reason why GPS receivers should have to adjust for a Sagnac effect. Various ‘official’ calculations for the effect are inconsistent with each other. Whereas actual measurements don’t match the official calculations but instead correspond to an additive speed of light. Information on how to adjust for the Sagnac effect in receivers is strangely absent in the official user guide and gps.gov website. 2. The adjustment for orbital eccentricity is quite likely not a relativistic adjustment but an adjustment for a change in signal speed as the satellite moves toward and away from the receiver. The difference between the relativistic and the change-in-signal-speed predictions should be measurable on Earth but for some reason is done in space where it can’t be verified.
 [1] http://physicscentral.com/explore/writers/will.cfm [2] http://www.fing.edu.uy/if/cursos/fismod/GPS.pdf page 45 (pdf page 5), or here http://www.ipgp.fr/ ~tarantola/Files/Professional/GPS/Neil_Ashby_Relativity_GPS.pdf [3] The user manuals are here: http://www.gps.gov/technical/icwg/ This is the most recent version: http://www.gps.gov/technical/icwg/IS-GPS-200H.pdf See page 96, or 109 on the PDF [4] http://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System [5] http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf pages 7-9 [6] http://hypertextbook.com/facts/2002/JobyJosekutty.shtml [7] http://areeweb.polito.it/ricerca/relgrav/solciclos/ashby_d.pdf [8] A search for "troposphere" on gps.gov [9] A search for "sagnac" on gps.gov [10] http://www.gps.gov/governance/advisory/meetings/2011-11/nelson.pdf page 14
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 Copyright © 2015 Bernard Burchell, all rights reserved.
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 58 E. H. KENNARD AND E. 0. DIETERICH. [SECOND
 LSERIES.
 AN EFFECT OF LIGHT UPON THE CONTACT POTENTIAL OF SELENIUM AND CUPROUS OXIDE.
 BY E. H. KENNARD AND E. O. DIETERICH.
 r 1 ^ H E increased electrical conductivity of selenium under illumination -*- has been explained in terms of t h e electron theory b y assuming t h a t t h e light produced an increase in t h e number of free electrons, a n d this explanation is strongly supported b y observations on t h e rate of decay of t h e conductivity, which suggest t h e recombination of ions according to familiar laws. It occurred to the authors that the increase in t h e n u m b e r of electrons ought t o result in a change of t h e contact potential under illumination, because the electrons would tend to diffuse away from the illuminated region and would therefore leave the selenium more positive than before. A search for this effect was accordingly m a d e ; t h e work was interrupted b y t h e departure of one of t h e authors, b u t it seems worth reporting as a quick survey of a new field.
 METHODS OF OBSERVATION.
 T w o methods of observation were used. T h e first w a s a condenser method for t h e m e a s u r e m e n t of quick changes in contact potential. T h e plate of selenium, which was a b o u t 5 cm. square a n d 0.2 to 0.4 m m . thick, w a s mounted between a sheet of gauze above a n d a copper plate below (cf. Fig. 1), t h e plates being insulated from each other a n d when
 Fig. 1. Fig. 2.
 desired from earth b y means of bits of amber or b y supports tipped with sealing-wax. A n y one of t h e three plates could be connected to an electrometer giving 3,000 mm./volt or to an improvised potentiometer, contact with t h e selenium being made b y means of a wire clip; t h u s a n y deflection of t h e electrometer d u e to a change in t h e potential of t h e selenium surface could be nullified by adjusting the potentiometer and the change was thus determined. Readings were usually taken in pairs,
 VOL. IX.l
 No. i. J EFFECT OF LIGHT ON SELENIUM OXIDE. 59
 the light being thrown on and then off; the two readings generally differed by 10-20 per cent, but the means of successive pairs agreed mostly within a few per cent. The second method was an adaptation of the ionization method of measuring contact potentials. An " ion contact" capsule (Fig. 2) was made by coating a copper strip 5 on top with polonium and mounting it in a fiber ring r so that it would clear when the ring lay upon a plane surface. The ring was closed on top by a sheet of mica, and just below this was mounted a sheet of brass gauze, one strand of which protruded through the ring and was connected to the electrometer. The mica was of sufficient thickness to stop a-particles and all cracks were closed with soft wax, while the arrangement of the parts prevented a-particles from striking the selenium. The ring was about 12 mm. in diameter and 3 mm. high. A null method was employed as suggested by Fig. 2. The equalization of potential between gauze and specimen was only about half effected in a minute, corresponding to a resistance of some io13 ohms, but when the sensitiveness of the electrometer was increased to 18,000 mm./volt by the use of a concave lens in front of the scale the adjustment of the potentiometer within a millivolt could be made in less than a minute. The specimen was illuminated by a 400-watt tungsten lamp placed some 30 cm. away and focused roughly upon it by means of a large lens, a glass dish containing 2 cm. of water being usually interposed to minimize heating effects. Electrostatic screening was carefully attended to, the light being admitted through a sheet of coarse gauze.
 PHOTO-ELECTROMOTIVE EFFECT IN SELENIUM. Various connections of the plates in the condenser arrangement were employed, the selenium being connected either to the electrometer or to the potentiometer or left insulated; while the ion contact capsule was used chiefly on a 12-mm. cube of selenium. All the results consistently indicated that illumination suddenly made the surface of the selenium more negative, instead of more positive as anticipated. The effect on the plate was usually not far from — 0.1 volt, the deflections in the absence of compensation being of the order of 20 mm. Shading the contact on the selenium had no appreciable effect. On the cube mean values of — 0.152, — 0.176 volt were obtained for a certain face on different days, while later, after washing thoroughly with chloroform, the effect was — 0.129 on the same face and — 0.125 volt on the opposite face. The latter observations indicate a variation with.the state of the surface, which we should rather expect if the phenomenon is a change in the contact potential.
 6o E. H. KENNARD AND E. 0. DIETERICH. [SECOND
 LSERIES.
 An E.M.F. in a closed circuit containing selenium was looked for repeatedly but without appreciable success. In one case two brass clips were placed on the selenium plate, each touching the plate on one (but not the same) side and insulated from the other surface by a bit of paper. When one clip was now connected to the electrometer and the other to earth, illumination produced a minute deflection that was consistent as to sign but amounted only to .3 mm. (0.1 mv.); the sign was reversed by shading the upper contact. These minute residual deflections may very well be due to a heating effect, for the thermo-electric power of selenium is quite large. It is to be noted however that the actual point of contact between the selenium and the metal was not here appreciably illuminated. A natural suspicion is that the effect might be due to photo-electric emission from the gauze above the selenium. But aside from the question of quantitative sufriciency, this explanation seems to be ruled out completely by these two facts: first, the effect is sudden and yet no> steady current persisted under continuous illumination even when all potentials were kept the same by adjusting the potentiometer; secondly, the effect at once reverses when the light is removed. The conclusion seems therefore a safe one that the observed effect is due to a sudden change produced by illumination in the contact potential of the selenium surface. In other words, between more and less strongly illuminated selenium there exists what might be called a " photo-electromotive force " tending to drive positive electricity away from the illuminated region.
 VARIATION WITH INTENSITY OF THE LIGHT.
 The variation of the effect with the intensity of illumination was investigated briefly by varying the voltage on the lamp, using the polonium capsule and the selenium cube. An approximate calibration of the lamp was made in terms of an arbitrary standard. A high degree of sensitiveness to faint light was at once apparent, for a change in potential of about 3 mv. was produced with only 13 volts on the lamp (normal, n o ) ; the illumination on the selenium must have been only about p.01 candle-foot. Yet at 25 volts, or 2-3 candle-feet, the effect had increased only to 35 mv. From this point on a series of readings was taken, the light being thrown on and off at 2 min. intervals while its intensity was varied to full value and back. The photoelectromotive potentials are plotted against lamp voltages in Fig. 4, curve / , while in Fig. 3 the corresponding mean values are plotted on logarithmic scale against the intensity of the light. Something like an
 N o ^ X ' ] EFFECT OF LIGHT ON SELENIUM OXIDE. 6 1
 approach to saturation is evident, for the last light intensity is 2,000 times the first while the photo-electromotive effect is only increased by a factor of 3.6; b u t t h e logarithmic curve shows a persistent slow increase proportional to a b o u t the fifth or sixth root of t h e intensity. Closely similar results were obtained when the light was kept on continuously (except for a short break while changing connections). In this case five minutes was allowed between readings, as a previous test had indicated that a fairly steady equilibrium was thus attained. T h e results are shown in Fig. 4, curve 77, the potential of the selenium relative to the gauze in the capsule being plotted against lamp voltage.
 FATIGUE, DRIFT AND TIME LAG.
 Curve I in Fig. 4 shows clear evidence of fatigue with respect to repetition, for the later readings taken after t h e lapse of 4-28 minutes are all 2-5 mv. smaller. A similar difference between readings taken in immediate succession was frequently noticed. Fatigue with respect to
 2X>
 . c1oL2-n
 1,5
 U ^ I 0 A T 1 U 25 50 75 100 125 150 Fig. 3. Fig. 4.
 continuation, also, is indicated b y curve II, Fig. 4. T h e difference in the initial and final d a r k readings points to a drift of the d a r k potential toward the negative, yet during the strong illumination the potential was evidently drifting toward the positive: the most natural explanation is that the negative displacement due to illumination tended to decrease with time. Such a drift of the d a r k potential was frequently noticed; it probably accounted for an a p p a r e n t hysteresis of 10-20 per cent, t h a t was often observed between the on- and off-readings. If any time lag occurred when the light was thrown on, it was a matter of seconds only and escaped detection. On t h e other hand, t h e change in potential seemed usually to,overshoot and then to settle back in the course of several minutes to an equilibrium value.
 T H E CONTACT POTENTIAL OF SELENIUM. Several comparisons of the contact potential of selenium with t h a t of freshly sandpapered copper were made.
 -120 Mv. -80
 -40
 ff
 V
 1^5
 y
 "I^J
 Volts.
 6 2 E. H. KENNARD AND E. 0. DIETERICH. ["SECOND
 LSERIES.
 Measurements on two different plates by the usual variable condenser method gave, in the dark, - 0.34, - 0.45 volt, resp. T w o faces of the cube were found to be at - .27 and - .36, resp., relative to the gauze in the polonium capsule, while on the same day the latter was observed to be — .099 relative to copper, giving for the selenium — 0.37 and — 0.46, resp. T h u s the contact potential of selenium crystallized in mass is not far from — 0.4 volt in the dark relative to freshly sandpapered copper, changing under moderate to intense illumination by about —0.1 volt.
 PHOTO-ELECTROMOTIVE EFFECT IN CUPROUS OXIDE.
 Interesting results were obtained on a plate of cuprous oxide, which is also light-sensitive as regards resistance. The plate was about 2.5 cm. square and 1 mm. thick, and was prepared by heating a copper plate a t a bright red heat for 48 hours and then dissolving off the surface layer of cupric oxide with a q u a regia. T h e time allowed proved insufficient, however, so t h a t the plate contained a layer of metallic copper down t h e middle.
 T h e polonium capsule was placed on top of the plate and two clips were attached, one insulated with paper above and the other below. By connecting one clip to the electrometer and the other to the potentiometer it was found t h a t an E . M . F . (voltaic?) of some 20 m v . existed between these two points; under illumination this E.M.F. increased by
 \/V
 ! 1L '
 \1
 .-41
 1
 '\
 1L
 r•
 cu,o.
 l\-^<r
 vi
 •\
 •' J 'i
 HHH
 TIME IN MINUTES.
 Fig. 5.
 7 mv. when the top contact was unshaded but decreased by the same amount when the contact was shaded. When the capsule was connected to the electrometer and one of the clips to the potentiometer, illumination made the surface under the capsule become more positive b y 20 to 28 m v . As in selenium, the initial effect seemed to overshoot and to be followed by a moderate reaction. There was also a decided drift of the potential
 VOL. IX.1
 No. i. J EFFECT OF LIGHT ON SELENIUM OXIDE. 63
 in the dark, sometimes in one direction, sometimes in the other, but a true hysteresis was not apparent. A series of readings of the potential at intervals of a minute is shown in Fig. 5, the dotted line marked L showing the periods of illumination. Apparently illumination produces, besides the sudden effect, a slow drift of potential followed by recovery in the dark but lagging by 2-3 minutes; this may be a temperature effect. The curve shows decided fatigue with respect to continuation of illumination, the return being practically complete if made at once but amounting to only some 60 per cent, after ten minutes of illumination. Fatigue with respect to repetition does not appear here, but in some of the other readings it was evident to the extent of some 10 per cent.
 CONCLUSION.
 The negative sign of the effect in selenium disposes at once of the diffusion of electrons as a possible cause. A tempting hypothesis is that the light modifies the molecules in such a way as to diminish the potential energy of the electrons in the intermolecular spaces, and that this results in an increase of the number of free electrons, partly at the expense of the surrounding atoms of selenium and partly at the expense of other bodies with which it is in contact. This would explain both the increased conductivity and the change toward the negative in the contact potential. But the lack of proportionality between the two effects and especially the apparent occurrence of saturation in the latter seem to be fatal objections to this hypothesis. Apparently the two effects are due to independent causes. The experimental results may be summarized as follows: 1. The contact potential of selenium changes under illumination by an amount exceeding a millivolt for 0.01 candle-foot and tending to show saturation at a value of about — 0.1 volt under strong illumination. The phenomenon is complicated by fatigue effects and a slow drift of the potential with time. 2. No E.M.F. is produced however in a closed circuit. 3. The contact potential of selenium relative to freshly sandpapered copper is about — 0.4 volt in the dark. 4. The surface potential of a plate of cuprous oxide changed under illumination by about 25 mv.; but the plate also contained a small voltaic (?) E.M.F. which varied under illumination. The authors take pleasure in acknowledging a debt to Professor A. F. Kovarik for his kindness in preparing the polonium deposit for the ion contact capsule.
 UNIVERSITY OF MINNESOTA, August, 1916.
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