Negative Results of Second and Third Order Tests of the Aether Drift

by Dewitt Bristol Brace

PROFESSOR LARMOR^[2] in his analysis for a system moving through the aether has shown how first and second order effects may be annulled in certain optical and electrical tests of the aether drift. He has not shown, however, how to annul third and higher order effects: but he states, "if indeed it could be proved that the optical effect is null up to the third order, that circumstance would not demolish the theory, but would rather point to some finer adjustment than it provides for: needless to say the attempt would indefinitely transcend existing experimental possibilities".^[3]

Attention should therefore be called to the results in my experiment on the double refraction of water moving through the aether. The sensibility attained was such that the greatest difference in velocity or in index between the two components which could exist, referred to that of water for green light, $\lambda =.00005$ cm, was less than $7.8\times 10^{-13}$ of the whole. If ($10^{-4}$)^[4] be taken as the third order magnitude, this result is then easily within the limit. It was not possible to attain this sensibility with a solid—glass; but, since the physical state of the substance does not enter into the theory under consideration, the results for a liquid should be equally valid.

The conclusion as to the amount of double refraction as deduced from what might be expected in comparison with accidental double refraction in a solid—glass, should not be considered as defining the mode and the amount of any double refraction arising from molecular reactions due to a system moving through the aether.

Granting the FitzGerald-Lorentz "contraction-hypothesis," we should have for this experiment a complete correspondence as regards molecular activities between the moving system thus shrunk and the same at rest, up to and including second order quantities, as the analysis of Larmor shows. But the negative results of such a third order test, showing as it does the absence of any difference between the moving and the fixed system, up to and including third order quantities, may indicate a complete correspondence, to all orders, of the molecular phases in the moving and in the fixed systems.

On the other hand, Lorentz^[5] has shown in his analysis for "electromagnetic phenomena in a system moving with any velocity smaller than that of light that, with the aid of the contraction-hypothesis, many electrical and optical effects will be independent of the motion of the system for all orders. This assumption of a shrinkage, although bold and thus far entirely hypothetical, is not impossible, and is the only suggestion yet made which is capable of reconciling the negative results of second and third order experiments with a quiescent æther. Poincaré^[6] has raised objection to the electromagnetic theory for moving bodies, that each time new facts are brought to light a new hypothesis has to be introduced. This criticism seems to have been fairly met by Lorentz in his latest treatment of the subject. The deductions, however, from his theory make it untenable without further development. The physical consequences, at least, seem at present to be beyond experimental examination. So far no valid reasons have been brought forward which necessitate the shrinkage hypothesis in the electromagnetic theory. In this connexion, reference should be made to the proof which Hasenöhrl,^[7] reasoning from a cyclic process in a moving radiating system, has given, that the second law of thermodynamics is contradicted unless either a second order contraction takes place in the direction of drift or the emission varies with the velocity, which latter he considers impossible.

On the other hand, Abraham^[8] finds, neglecting fourth and higher order quantities, the ratio of the transverse to the longitudinal mass of the moving electrons to be

$1+{\frac {2}{5}}\left({\frac {v}{V}}\right)^{2}:1+{\frac {6}{5}}\left({\frac {v}{V}}\right)^{2}=1-{\frac {4}{5}}\left({\frac {v}{V}}\right)^{2}$,

while Lorentz requires the ratio to be

$\left(1+\left({\frac {v}{V}}\right)^{2}\right)^{-{\frac {1}{2}}}:\left(1-\left({\frac {v}{V}}\right)^{2}\right)^{-{\frac {3}{2}}}=1-\left({\frac {v}{V}}\right)^{2}$,

for perfect compensation: thus leaving a double refraction of the order ${\frac {1}{5}}\left({\frac {v}{V}}\right)^{2}$ to be accounted for, which would have been detected several thousand times over in my experiment. The analysis of both Lorentz and Abraham seems to be equally consistent with Kaufmann's results on the deflexion of Becquerel rays, Lorentz conforms Abraham's theory to his own, by shrinking the undeformable spherical electrons of the latter into flattened ellipsoids in the line of drift; while Abraham himself shows that these will be unstable unless the greatest axis is in this direction. The latter writer shows^[9] that work must be done against the electrical forces to produce this deformation, so that the total energy in any acceleration is greater than that furnished by the outside forces.

Hence there must be inner forces as well which determine the form of the electron. Thus the hypothesis of Lorentz is incomplete without defining the law of forces further. Hence we must either abandon the contraction hypothesis or modify it. The assumption that the quasi-elastic forces, which maintain the electrons in their positions of equilibrium, experience the same changes as the electrical forces, may possibly be varied, and, together with a modification of the previous hypothesis, be adapted so as to agree with all observations.

While the negative results of the first order experiments involving a study of phase relations between periodic disturbances from the same radiant, optical or electrical, moving with the system, are quite as consistent with a mobile—if we neglect second and higher orders—as with a fixed aether, the explanations of the negative results of second and third order tests are still not in full harmony or free from criticisms, notwithstanding the bold assumption in the premises.

It becomes then a serious question, whether to seek still for decisive results with experiments involving the higher order tests, on the one hand, or direct entrainment tests on the other, in order to settle the question.

The recent repetition by Morley and Miller^[10] of the original Michelson-Morley interference experiment, with a sensibility one hundred times the calculated effect, leaves perhaps no question as to the absence of any such second order optical effect. Lorentz's analysis requires a negative result, likewise, if the rays pass through a transparent substance instead of a vacuum: and it seems desirable therefore that this point should be tested for water, say.

The interferometer method might also be used to test for double refraction if the light were polarized so as to make the electric displacements perpendicular to the plane of the interfering rays, i. e. respectively parallel and perpendicular to the greatest axes of the ellipsoidal electrons of the equivalent system at rest. Otherwise the effect would be masked in using natural light, since, with this type of interferometer, the above components may be as low as 25 per cent, of the total interfering light, as Mills has shown.^[11]

The remaining second order tests first proposed by FitzGerald^[12] and developed and carried out by Trouton,^[13] by examining the couple on a suspended condenser, gave negative results. This is completely explained on the assumption of a shrinkage.

Direct experiments on the entrainment of the aether have also given negative results. Thus Lodge^[14] sent two interfering rays in opposite directions several times around a rectangle between two rotating steel disks, without being able to detect any displacement of the interference-bands. He estimates that if the disks had communicated one eight-hundredth part of their velocity to the aether, he would have been able to detect it.

Zehnder,^[15] using a different method, attempted to detect any dragging of the aether by a metal plug moving within a cylinder whose ends were circuited together by parallel branching tubes through which two interfering rays could be sent in opposite directions. I£ the aether had moved entirely with the plug, the effect would have been a thousand times larger than this sensibility. Fizeau's well-known experiment on the entrainment of the aether, repeated by Michelson and Morley,^[16] showed no effect after allowing for the reaction of the moving water itself upon the interfering waves. Had the aether been carried along completely, the displacement would have been nearly two and a half bands, instead of approximately the single band actually observed, due to the reaction of the water alone. We may conclude from these experiments that the aether was not entrained in any way in the experiments of Morley and Miller, and that their results are therefore valid, although performed within an enclosure.

Nordmeyer,^[17] carrying out the experiment first proposed by Fizeau^[18] on the change in intensity of a radiant due to the earth's motion, found that this variation could not have been as great as 1/300000.^[19] This agrees with the analytical theory of a quiescent æther, which shows there should be no effect if second order quantities be neglected.

Mascart^[20] and Rayleigh's^[21] negative results on the difference in rotation in quartz with and against the drift are not decisive, since the calculated effect, although not of the second order, is the difference between two first-order effects. A reexamination of the problem with greater experimental refinements should give important results.

With the present uncertainty on both the analytical and the experimental sides, decisive results, which will be free from any hypothetical explanation, seem only possible in the direct comparison of the velocities of light with and against the æther-drift. (Of course if a negative result were obtained, it might be open to such a hypothetical explanation by saying that the group velocity, relative to the medium itself, was a function of the absolute motion of this medium.) Thus Wien^[22] proposes to use two synchronized Foucault mirrors or two Fizeau toothed wheels. This plan is of course of long standing, but has been recently revived. The mechanical difficulties in the way do not give much encouragement to hope for success; but with present refinements the test is not beyond possibility. The objection which Newcomb and Michelson^[23] have raised to this mode of comparison, that the phases of the synchronizing systems would be affected by the earth's motion in the same way as the propagation of the light, does not seem to be well taken. For granting a certain phase difference in the rotating-mirror or wheel systems, this difference in phase of the two systems can still be so changed as to give an eclipse, say, along the drift. If now we observe simultaneously the light propagated over the identical path in the opposite direction, there should not be a complete eclipse if the æther were at rest. Any method, therefore, which allows a comparison of two rays, propagated over the same path in opposite directions, is a valid test of the problem. It remains then to devise a method which will certainly show a difference between these two intervals of time equal to one part in ten thousand. The method proposed by Michelson,^[24] of determining the difference of time required by two interfering rays to traverse circuits in opposite directions, would require a path one kilometre square in a horizontal plane to give a displacement equal to .7 of a band for latitude 45°. This would necessitate a very high degree of refinement indeed over any previous attempts at interference. The method of determining the velocity of light which I devised us far back as 1889, and tried a number of years ago,^[25] and which consisted of an eclipsing system made up of a rotating double mirror and a grating, thus combining the principles of the toothed wheel and rotating mirror methods of Fizeau and Foucault, would be delicate enough to show a variation of one part in ten thousand, providing synchronism could be maintained during a short interval of time. With a suitable system of mirrors and observing telescopes, a single observer would be able to bring into his field of view both beams of light after their passage through the eclipsing systems. If now one of the rotating mirrors were either gaining or losing on the other, the observer would see, alternately, eclipses of one ray and the other, if their times of transit were different. If their times of transit were the same, then the two fields would maintain a constant relative intensity, each going through its maximum and minimum simultaneously as the relative phases of the eclipsing systems varied. The latter would correspond to the condition of a moving, the former to that of a quiescent æther. Thus the experiment would be possible, even if perfect synchronism were not attainable, but only sufficiently so to make the frequency of the successive maxima in the field of view less than a few times a second, or slow enough for the eye to resolve the fluctuations of intensity. If, by means of mirrors, a common source of light were employed, the half-shade principle in the field of view could be used which would be very sensitive in showing any difference in intensity (even if rapidly fluctuating) between the two portions of the field due to any slight difference in the time of propagation of the two rays with and against the æther drift. If we take the conditions in the experiment referred to,^[26] namely, an aperture of 2.5 cm. and a distance of .02 cm. between the lines of the reflecting grating, with a radius of 1 m. and 250 revolutions per second, 10,000,000 eclipses per second could be obtained; and, if we carried this to the limit in speed and resolving power, four times as many would be possible. With the former conditions, and allowing a frequency in fluctuation of 10 per second for good resolution to the eye, a difference in speed not greater than one part in a million of the two eclipsing systems would be requisite. While this seems extremely small, experience shows that such an approximation is entirely practicable. Thus Newcomb^[27] records a "run" in his measurement of the velocity of light, in which his micrometer showed "beautiful bisection" during the greater portion of the duration (2 minutes) of such a "run." Allowing this setting to one part in ten of his unit, which was 2".4, out of a total deviation of 7500" of arc for a period of 90 seconds, we have a fluctuation in the speed of only one part in 27,000,000. If the two mirrors could be regulated to this degree, we should still have less than one-tenth the fluctuation of our limit, from the two mirrors combined. Hence, so far as speed regulation is concerned, a much higher eclipse frequency, say the forty-million limit, is possible. The other disturbances would be of the same order as encountered by Newcomb over his total "go" and "return" distance of 5000 in. Supposing the eye could detect a difference in intensity of two per cent, between the two fields (under very favourable conditions this sensibility is one-half of one percent.), we should need an interval corresponding to one hundred eclipses to detect a change in velocity equal to the aberration constant, since we have to add the effects from each ray. This would mean a distance between the mirrors of 3000 m. for the lower and 750 m. for the higher eclipse frequency referred to, which is much within the distance given above in Newcomb's experiment.

Of the other methods proposed,^[28] several do not require a return of the ray in the determination of the velocity constant. Thus the rotation of a polarizing system, such as a half-shade nicol or tourmaline system, could be carried up to 3000 revolutions per second. Polishing machines are now run up to 2000 revolutions per second. Allowing a sensibility of 0°.01, a distance of 15 kilometres would give a variation in velocity of one part in ten thousand.

The objection to the above methods and other similar ones, is the great distance required. If we could increase the eclipse frequency, the distance could be reduced accordingly. This can be attained by the use of electric oscillations in conjunction with suitable optical systems. Two methods proposed in the article referred to for measuring the velocity of light can be readily adapted to the problem before us. The first depends on the Faraday "effect," the second on the Kerr electrostatic "effect".^[29] The second method, in conjunction with a half-shade elliptical polarizer^[30] which I devised several years ago, has given preliminary results indicating a superiority over those described above, quite beyond my expectations, in the arrangement as originally planned for determining the velocity of light a number of years ago.

In fig. 1, k, k' are two condensers containing, say, the dielectric, carbon disulphide or nitrobenzol, giving the Kerr "effect," and placed with their azimuths so as to give a "crossed" system. p, c, a and p', c', a' are the polarizer, elliptical half-shade compensator, and analyser of the two respective optical systems placed in juxtaposition. g is the spark-gap and in a half-silvered mirror system for sending identical beams in opposite directions through the optical systems p' k k' c' a' and p k' k c a.

A and A' represent the azimuths of the various elements as seen from the one side or the other. Thus at $A,p_{1}$ and $a_{1}$ are the azimuths of the polarizer and analyser, k and k' the traces of the planes of the condenser, and c the principal axis of the sensitive strip. Similarly at $A',k_{1},p_{1},a_{1},c_{1}$ are new positions of these elements due to any constant rotation of all of the elements of the second system except k' . If now there be any damping of the electric oscillations, the effect of the one condenser on the polarized ray for, say, the first oscillation, may be made equal to that of the other condenser for the second oscillation by such a rotation, and so on. Thus these two condensers become an equivalent "crossed" system, the one compensating completely the effect of the other, even if the electric stresses in each are not the same. This would hold for electric oscillations which produce successive stresses that are in a constant ratio to each other. If now the interval of passage of the beam of light between k and k' is any multiple of a half period of the vibration, we may obtain compensation, and hence retain the settings for a match in the two half-shade systems c and c' . If there should be any difference in the interval of passage in the two opposite directions, we should not obtain a match in the one if we set for the same in the other, after the frequency had been varied so as to give this exact multiple of the half period in this latter system. This would be determined by noting when the intensity of the field approximated a minimum intensity.

For low frequencies sunlight has been used; but for the higher frequencies this source has not given satisfactory results on account of the very brief duration of the spark in the exciter, and, consequently, the integral time of the electric stresses due to such a discharge. In this case the spark itself may be used to advantage, since here we have a very much greater intensity during the period of the electric stresses. The greater uniformity and intensity of a vapour spark-gap, e. g. of mercury, recommend its use in connexion with the half-shade system. To maintain greater uniformity in the amplitude of the oscillations, and hence in the Kerr "effect," a secondary or resonance circuit itself, with its condenser system, may be used, as this will give sufficient double refraction in the dielectric to make accurate settings. Such a half-shade system, where the photometric sensibility is as low, say, as 2 per cent., will show a change of phase of $0.3\times 10^{-4}\lambda$.^[31] Since now the maxima of the Kerr "effect" occur every half oscillation, the distance between the condenser k and k' needs only to be equal to the space traversed by the ray in half an oscillation, or $d={\tfrac {1}{2}}L$, where L is the wavelength of an electric oscillation. This would give us a change of phase in the one direction of $0.3\times 10^{-4}\lambda$ and an opposite amount for the reverse direction; or, if we set for a match in the one system, the other system would show a change in phase of $0.3\times 10^{-4}\lambda \div 2$, which is seven times the required effect $10^{-4}$, the aberration constant. With condensers larger than needed for the optical condition, 50,000,000 oscillations have been obtained. Thus

$d={\frac {1}{2}}L={\frac {3\times 10^{8}M}{2\times 0.5\times 10^{8}}}=3M$,

a distance small enough to allow a rotating mount for the system. Higher frequencies are undoubtedly possible; so that, if the above optical sensibility is not attainable, we can still use a rotating support, with a reduction in this factor of, say, ten times. This method thus contains the requisite conditions, if all the experimental difficulties in connexion with the uniformity of the oscillation and the source of light can be overcome. The present stage of the work seems to warrant this conclusion.

Physical Laboratory,

University of Nebraska, Lincoln.

April 26, 1905.

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1. Communicated by the Author. Read in part before Sect. B. Amer. Ass.oc. for Adv. Sci., Philadelphia Meeting, 1904-5.
2. 'Aether and Matter,' Chap. xi.
3. Phil. Mag. June, 1904, p. 624.
4. Phil. Mag. April, 1904, p. 318.
5. Amsterdam Proc. April 1904, p. 809.
6. Rapports du Congrès de Physique de 1900, Paris, i. pp. 22, 23.
7. Annalen, Band xiii. p. 367.
8. Annalen, Band xiv. p. 236.
9. Phys. Zeit. Band v. p. 570.
10. Phil. Mag. Dec. 1904, p. 753.
11. Annalen. Band xiii. p. 854.
12. 'Scientific Writings,' p. 557.
13. Trouton & Noble, Roy. Soc. Trans. A. 202. p. 165 (1903).
14. Lodge, Roy. Soc. Trans. A. 184. p. 727 (1893).
15. Wied. Ann. Band lv. p. 65.
16. Amer. Journ. Sci. (3) vol. xxxi. p. 377.
17. Annalen, Band xi. p. 284.
18. Pogg. Ann. Band xcii. p. 652.
19. As this experiment was performed in the latter part of November, when the motion of the solar system has to be subtracted from the earth's motion, this limit is far too high, and the experiment ought therefore to repeated at some other time of the year.
20. Annales de l'Ecole Normale, tom. i. p. 157.
21. Phil. Mag. Aug. 1902.
22. Physikal-Zeit. Band v. p. 585
23. Michelson, Phil. Mag. Dec. 1904, p. 716.
24. L. c. p. 717.
25. Vice-Pres. Address, Sect. B, Amer. Assoc. Adv. Sci., Pittsburg Meeting, 1902; Science, July 18, 1902.
26. This apparatus was the phototachometer, modified for the purpose, which Newcomb used in determining the velocity of light.
27. Astronomical Papers of American Ephemeris, vol. ii. pts. iii., iv. p. 172.
28. All the methods referred to in the paper were devised as far back as 1889-90, and have been developed experimentally to a greater or less extent since.
29. Abraham and Lemoine, J. de Phys. 1899, p. 366, used an analogous arrangement to determine the retardation of the Kerr and also of the Faraday "effect."
30. Phys. Rev. vol. xviii. p. 70, also vol. xix. p. 218.
31. Phys. Rev. vol. xviii. p. 85.