Part 1

# The Principle of Relativity ### By Einstein, Albert

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The Principle of Relativity

THE PRINCIPLE OF RELATIVITY

ORIGINAL PAPERS BY

A. EINSTEIN AND H. MINKOWSKI

TRANSLATED INTO ENGLISH BY

M. N. SAHA AND S. N. BOSE

LECTURERS ON PHYSICS AND APPLIED MATHEMATICS University College of Science, Calcutta University

WITH A HISTORICAL INTRODUCTION BY

P. C. MAHALANOBIS PROFESSOR OF PHYSICS, PRESIDENCY COLLEGE, CALCU.

PUBLISHED BY THE UNIVERSITY OF CALCUTTA 1920

_Sole Agents_ R. CAMBRAY & CO.

PRINTED BY ATULCHANDRA BHATTACHARYYA,

AT THE CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA

TABLE OF CONTENTS

1. Historical Introduction i-xxiii

[By Mr. P. C. Mahalanobis.]

2. On the Electrodynamics of Moving Bodies 1-34

[Einstein’s first paper on the restricted Theory of Relativity, originally published in the Annalen der Physik in 1905. Translated from the original German by Dr. Meghnad Saha.]

3. Albrecht Einstein 35-39

[A short biographical note by Dr. Meghnad Saha.]

4. Principle of Relativity 1-52

[H. Minkowski’s original paper on the restricted Principle of Relativity first published in 1909. Translated from the original German by Dr. Meghnad Saha.]

5. Appendix to the above by H. Minkowski 53-88

[Translated by Dr. Meghnad Saha.]

6. The Generalised Principle of Relativity 89-163

[A. Einstein’s second paper on the Generalised Principle first published in 1916. Translated from the original German by Mr. Satyendranath Bose.]

7. Notes 165-185

Transcriber’s Note:

The plain text version of this ebook includes complex mathematical formulas. Some are simple in-line expressions like k = 1 - 1/μ^2. They may include special notations such as x^y for x to the power of y, x_{y} for x with a subscript of y, [=a] for an 'a' with a bar across the top, [.a] for an 'a' with a dot over it, [..a] for an 'a' with two dots over it. Others are more complex “ASCII Art” like this:

l l 2lc 2l t₁ = ------ + ------ = -------- = --- β² c - u c + u c² - u² c

Some are so complex that they must be rendered in the TeX mathematical notation, enclosed between double dollar signs, like this:

$$ \beta = (1 - \frac {u^2}{c^2})^{-\frac{1}{2}} $$

HISTORICAL INTRODUCTION

Lord Kelvin writing in 1893, in his preface to the English edition of Hertz’s Researches on Electric Waves, says “many workers and many thinkers have helped to build up the nineteenth century school of _plenum_, one ether for light, heat, electricity, magnetism; and the German and English volumes containing Hertz’s electrical papers, given to the world in the last decade of the century, will be a permanent monument of the splendid consummation now realised.”

Ten years later, in 1905, we find Einstein declaring that “the ether will be proved to be superfluous.” At first sight the revolution in scientific thought brought about in the course of a single decade appears to be almost too violent. A more careful even though a rapid review of the subject will, however, show how the Theory of Relativity gradually became a historical necessity.

Towards the beginning of the nineteenth century, the luminiferous ether came into prominence as a result of the brilliant successes of the wave theory in the hands of Young and Fresnel. In its stationary aspect the elastic solid ether was the outcome of the search for a medium in which the light waves may “undulate.” This stationary ether, as shown by Young, also afforded a satisfactory explanation of astronomical aberration. But its very success gave rise to a host of new questions all bearing on the central problem of relative motion of ether and matter.

_Arago’s prism experiment._—The refractive index of a glass prism depends on the incident velocity of light outside the prism and its velocity inside the prism after refraction. On Fresnel’s fixed ether hypothesis, the incident light waves are situated in the stationary ether outside the prism and move with velocity _c_ with respect to the ether. If the prism moves with a velocity _u_ with respect to this fixed ether, then the incident velocity of light with respect to the prism should be _c_ + _u_. Thus the refractive index of the glass prism should depend on _u_ the absolute velocity of the prism, _i.e._, its velocity with respect to the fixed ether. Arago performed the experiment in 1819, but failed to detect the expected change.

_Airy-Boscovitch water-telescope experiment._—Boscovitch had still earlier in 1766, raised the very important question of the dependence of aberration on the refractive index of the medium filling the telescope. Aberration depends on the difference in the velocity of light outside the telescope and its velocity inside the telescope. If the latter velocity changes owing to a change in the medium filling the telescope, aberration itself should change, that is, aberration should depend on the nature of the medium.

Airy, in 1871 filled up a telescope with water—but failed to detect any change in the aberration. Thus we get both in the case of Arago prism experiment and Airy-Boscovitch water-telescope experiment, the very startling result that optical effects in a moving medium seem to be quite independent of the velocity of the medium with respect to Fresnel’s stationary ether.

_Fresnel’s convection coefficient k = 1 - 1/μ^2._—Possibly some form of compensation is taking place. Working on this hypothesis, Fresnel offered his famous ether convection theory. According to Fresnel, the presence of matter implies a definite condensation of ether within the region occupied by matter. This “condensed” or excess portion of ether is supposed to be carried away with its own piece of moving matter. It should be observed that only the “excess” portion is carried away, while the rest remains as stagnant as ever. A complete convection of the “excess” ether ρ′ with the full velocity _u_ is optically equivalent to a partial convection of the total ether ρ, with only a fraction of the velocity _k_. _u_. Fresnel showed that if this convection coefficient _k_ is 1 - 1/μ^2 (μ being the refractive index of the prism), then the velocity of light after refraction within the moving prism would be altered to just such extent as would make the refractive index of the moving prism quite independent of its “absolute” velocity _u_. The non-dependence of aberration on the “absolute” velocity _u_, is also very easily explained with the help of this Fresnelian convection-coefficient _k_.

_Stokes’ viscous ether._—It should be remembered, however, that Fresnel’s stationary ether is absolutely fixed and is not at all disturbed by the motion of matter through it. In this respect Fresnelian ether cannot be said to behave in any respectable physical fashion, and this led Stokes, in 1845-46, to construct a more material type of medium. Stokes assumed that viscous motion ensues near the surface of separation of ether and moving matter, while at sufficiently distant regions the ether remains wholly undisturbed. He showed how such a viscous ether would explain aberration if all motion in it were differentially irrotational. But in order to explain the null Arago effect, Stokes was compelled to assume the convection hypothesis of Fresnel with an identical numerical value for _k_, namely 1 - 1/μ^2. Thus the prestige of the Fresnelian convection-coefficient was enhanced, if anything, by the theoretical investigations of Stokes.

_Fizeau’s experiment._—Soon after, in 1851, it received direct experimental confirmation in a brilliant piece of work by Fizeau.

If a divided beam of light is re-united after passing through two adjacent cylinders filled with water, ordinary interference fringes will be produced. If the water in one of the cylinders is now made to flow, the “condensed” ether within the flowing water would be convected and would produce a shift in the interference fringes. The shift actually observed agreed very well with a value of k = 1 - 1/μ^2. The Fresnelian convection-coefficient now became firmly established as a consequence of a direct positive effect. On the other hand, the negative evidences in favour of the convection-coefficient had also multiplied. Mascart, Hoek, Maxwell and others sought for definite changes in different optical effects induced by the motion of the earth relative to the stationary ether. But all such attempts failed to reveal the slightest trace of any optical disturbance due to the “absolute” velocity of the earth, thus proving conclusively that all the different optical effects shared in the general compensation arising out of the Fresnelian convection of the excess ether. It must be carefully noted that the Fresnelian convection-coefficient implicitly assumes the existence of a fixed ether (Fresnel) or at least a wholly stagnant medium at sufficiently distant regions (Stokes), with reference to which alone a convection velocity can have any significance. Thus the convection-coefficient implying some type of a stationary or viscous, yet nevertheless “absolute” ether, succeeded in explaining satisfactorily all known optical facts down to 1880.

_Michelson-Morley Experiment._—In 1881, Michelson and Morley performed their classical experiments which undermined the whole structure of the old ether theory and thus served to introduce the new theory of relativity. The fundamental idea underlying this experiment is quite simple. In all old experiments the velocity of light situated in free ether was compared with the velocity of waves actually situated in a piece of moving matter and presumably carried away by it. The compensatory effect of the Fresnelian convection of ether afforded a satisfactory explanation of all negative results.

In the Michelson-Morley experiment the arrangement is quite different. If there is a definite gap in a rigid body, light waves situated in free ether will take a definite time in crossing the gap. If the rigid platform carrying the gap is set in motion with respect to the ether in the direction of light propagation, light waves (which are even now situated in free ether) should presumably take a longer time to cross the gap.

We cannot do better than quote Eddington’s description of this famous experiment. “The principle of the experiment may be illustrated by considering a swimmer in a river. It is easily realized that it takes longer to swim to a point 50 yards up-stream and back than to a point 50 yards across-stream and back. If the earth is moving through the ether there is a river of ether flowing through the laboratory, and a wave of light may be compared to a swimmer travelling with constant velocity relative to the current. If, then, we divide a beam of light into two parts, and send one-half swimming up the stream for a certain distance and then (by a mirror) back to the starting point, and send the other half an equal distance across stream and back, the across-stream beam should arrive back first.

——>_u_ O A—————........ | _x_ | |B

Let the ether be flowing relative to the apparatus with velocity _u_ in the direction O_x_, and let OA, OB, be the two arms of the apparatus of equal length _l_, OA being placed up-stream. Let _c_ be the velocity of light. The time for the double journey along OA and back is

l l 2lc 2l t₁ = ------ + ------ = -------- = --- β² c - u c + u c² - u² c

where

$$ \beta = (1 - \frac {u^2}{c^2})^{-\frac {1}{2}} $$

a factor greater than unity.

For the transverse journey the light must have a component velocity _n_ up-stream (relative to the ether) in order to avoid being carried below OB: and since its total velocity is _c_, its component across-stream must be √(_c²_ - _u²_), the time for the double journey OB is accordingly

$$ t_2 = \frac {2a}{\sqrt {c^2 - u^2}} = \frac {2a}{c} \beta $$

so that _t₁_ > _t₂_.

But when the experiment was tried, it was found that both parts of the beam took the same time, as tested by the interference bands produced.”

After a most careful series of observations, Michelson and Morley failed to detect the slightest trace of any effect due to earth’s motion through ether.

The Michelson-Morley experiment seems to show that there is no relative motion of ether and matter. Fresnel’s stagnant ether requires a relative velocity of—_u_. Thus Michelson and Morley themselves thought at first that their experiment confirmed Stokes’ viscous ether, in which no relative motion can ensue on account of the absence of slipping of ether at the surface of separation. But even on Stokes’ theory this viscous flow of ether would fall off at a very rapid rate as we recede from the surface of separation. Michelson and Morley repeated their experiment at different heights from the surface of the earth, but invariably obtained the same negative results, thus failing to confirm Stokes’ theory of viscous flow.

_Lodge’s experiment._—Further, in 1893, Lodge performed his rotating sphere experiment which showed complete absence of any viscous flow of ether due to moving masses of matter. A divided beam of light, after repeated reflections within a very narrow gap between two massive hemispheres, was allowed to re-unite and thus produce interference bands. When the two hemispheres are set rotating, it is conceivable that the ether in the gap would be disturbed due to viscous flow, and any such flow would be immediately detected by a disturbance of the interference bands. But actual observation failed to detect the slightest disturbance of the ether in the gap, due to the motion of the hemispheres. Lodge’s experiment thus seems to show a complete absence of any viscous flow of ether.

Apart from these experimental discrepancies, grave theoretical objections were urged against a viscous ether. Stokes himself had shown that his ether must be incompressible and all motion in it differentially irrotational, at the same time there should be absolutely no slipping at the surface of separation. Now all these conditions cannot be simultaneously satisfied for any conceivable material medium without certain very special and arbitrary assumptions. Thus Stokes’ ether failed to satisfy the very motive which had led Stokes to formulate it, namely, the desirability of constructing a “physical” medium. Planck offered modified forms of Stokes’ theory which seemed capable of being reconciled with the Michelson-Morley experiment, but required very special assumptions. The very complexity and the very arbitrariness of these assumptions prevented Planck’s ether from attaining any degree of practical importance in the further development of the subject.

The sole criterion of the value of any scientific theory must ultimately be its capacity for offering a simple, unified, coherent and fruitful description of observed facts. In proportion as a theory becomes complex it loses in usefulness—a theory which is obliged to requisition a whole array of arbitrary assumptions in order to explain special facts is practically worse than useless, as it serves to disjoin, rather than to unite, the several groups of facts. The optical experiments of the last quarter of the nineteenth century showed the impossibility of constructing a simple ether theory, which would be amenable to analytic treatment and would at the same time stimulate further progress. It should be observed that it could scarcely be shown that no logically consistent ether theory was possible; indeed in 1910, H. A. Wilson offered a consistent ether theory which was at least quite neutral with respect to all available optical data. But Wilson’s ether is almost wholly negative—its only virtue being that it does not directly contradict observed facts. Neither any direct confirmation nor a direct refutation is possible and it does not throw any light on the various optical phenomena. A theory like this being practically useless stands self-condemned.

We must now consider the problem of relative motion of ether and matter from the point of view of electrical theory. From 1860 the identity of light as an electromagnetic vector became gradually established as a result of the brilliant “displacement current” hypothesis of Clerk Maxwell and his further analytical investigations. The elastic solid ether became gradually transformed into the electromagnetic one. Maxwell succeeded in giving a fairly satisfactory account of all ordinary optical phenomena and little room was left for any serious doubts as regards the general validity of Maxwell’s theory. Hertz’s researches on electric waves, first carried out in 1886, succeeded in furnishing a strong experimental confirmation of Maxwell’s theory. Electric waves behaved generally like light waves of very large wave length.

The orthodox Maxwellian view located the dielectric polarisation in the electromagnetic ether which was merely a transformation of Fresnel’s stagnant ether. The magnetic polarisation was looked upon as wholly secondary in origin, being due to the relative motion of the dielectric tubes of polarisation. On this view the Fresnelian convection coefficient comes out to be ½, as shown by J. J. Thomson in 1880, instead of 1 - (1/μ²) as required by optical experiments. This obviously implies a complete failure to account for all those optical experiments which depend for their satisfactory explanation on the assumption of a value for the convection coefficient equal to 1 - (1/μ²).

The modifications proposed independently by Hertz and Heaviside fare no better.[1] They postulated the actual medium to be the seat of all electric polarisation and further emphasised the reciprocal relation subsisting between electricity and magnetism, thus making the field equations more symmetrical. On this view the whole of the polarised ether is carried away by the moving medium, and consequently, the convection coefficient naturally becomes unity in this theory, a value quite as discrepant as that obtained on the original Maxwellian assumption.

Thus neither Maxwell’s original theory nor its subsequent modifications as developed by Hertz and Heaviside succeeded in obtaining a value for Fresnelian coefficient equal to 1 - (1/μ^2), and consequently stood totally condemned from the optical point of view.

Certain direct electromagnetic experiments involving relative motion of polarised dielectrics were no less conclusive against the generalised theory of Hertz and Heaviside. According to Hertz a moving dielectric would carry away the whole of its electric displacement with it. Hence the electromagnetic effect near the moving dielectric would be proportional to the total electric displacement, that is to K, the specific inductive capacity of the dielectric. In 1901, Blondlot working with a stream of moving gas could not detect any such effect. H. A. Wilson repeated the experiment in an improved form in 1903 and working with ebonite found that the observed effect was proportional to K - 1 instead of to K. For gases K is nearly equal to 1 and hence practically no effect will be observed in their case. This gives a satisfactory explanation of Blondlot’s negative results.

Rowland had shown in 1876 that the magnetic force due to a rotating condenser (the dielectric remaining stationary) was proportional to K, the sp. ind. cap. On the other hand, Röntgen found in 1888 the magnetic effect due to a rotating dielectric (the condenser remaining stationary) to be proportional to K - 1, and not to K. Finally Eichenwald in 1903 found that when both condenser and dielectric are rotated together, the effect observed was quite independent of K, a result quite consistent with the two previous experiments. The Rowland effect proportional to K, together with the opposite Röntgen effect proportional to 1 - K, makes the Eichenwald effect independent of K.

All these experiments together with those of Blondlot and Wilson made it clear that the electromagnetic effect due to a moving dielectric was proportional to K - 1, and not to K as required by Hertz’s theory. Thus the above group of experiments with moving dielectrics directly contradicted the Hertz-Heaviside theory. The internal discrepancies inherent in the classic ether theory had now become too prominent. It was clear that the ether concept had finally outgrown its usefulness. The observed facts had become too contradictory and too heterogeneous to be reduced to an organised whole with the help of the ether concept alone. Radical departures from the classical theory had become absolutely necessary.

There were several outstanding difficulties in connection with anomalous dispersion, selective reflection and selective absorption which could not be satisfactory explained in the classic electromagnetic theory. It was evident that the assumption of some kind of discreteness in the optical medium had become inevitable. Such an assumption naturally gave rise to an atomic theory of electricity, namely, the modern electron theory. Lorentz had postulated the existence of electrons so early as 1878, but it was not until some years later that the electron theory became firmly established on a satisfactory basis.

Lorentz assumed that a moving dielectric merely carried away its own “polarisation doublets,” which on his theory gave rise to the induced field proportional to K - 1. The field near a moving dielectric is naturally proportional to K - 1 and not to K. Lorentz’s theory thus gave a satisfactory explanation of all those experiments with moving dielectrics which required effects proportional to K - 1. Lorentz further succeeded in obtaining a value for the Fresnelian convection coefficient equal to 1 - 1/μ^2, the exact value required by all optical experiments of the moving type.

We must now go back to Michelson and Morley’s experiment. We have seen that both parts of the beam are situated in free ether; no material medium is involved in any portion of the paths actually traversed by the beam. Consequently no compensation due to Fresnelian convection of ether by moving medium is possible. Thus Fresnelian convection compensation can have no possible application in this case. Yet some marvellous compensation has evidently taken place which has completely masked the “absolute” velocity of the earth.

In Michelson and Morley’s experiment, the distance travelled by the beam along OA (that is, in a direction parallel to the motion of the platform) is 2_l_β², while the distance travelled by the beam along OB, perpendicular to the direction of motion of the platform, is 2_l_β. Yet the most careful experiments showed, as Eddington says, “that both parts of the beam took the same time as tested by the interference bands produced. It would seem that OA and OB could not really have been of the same length; and if OB was of length _l_, OA must have been of length _l_/β. The apparatus was now rotated through 90°, so that OB became the up-stream. The time for the two journeys was again the same, so that 0B must now be the shorter length. The plain meaning of the experiment is that both arms have a length _l_ when placed along O_y_ (perpendicular to the direction of motion), and automatically contract to a length _l_/β, when placed along O_x_ (parallel to the direction of motion). This explanation was first given by Fitz-Gerald.”

This Fitz-Gerald contraction, startling enough in itself, does not suffice. Assuming this contraction to be a real one, the distance travelled with respect to the ether is 2_l_β and the time taken for this journey is 2_l_β/_c_. But the distance travelled with respect to the platform is always 2_l_. Hence the velocity of light with respect to the platform is

$$ \frac {2l}{\frac {2l\beta}{c}} = \frac {c}{\beta} $$

a variable quantity depending on the “absolute” velocity of the platform. But no trace of such an effect has ever been found. The velocity of light is always found to be quite independent of the velocity of the platform. The present difficulty cannot be solved by any further alteration in the measure of space. The only recourse left open is to alter the measure of time as well, that is, to adopt the concept of “local time.” If a moving clock goes slower so that one ‘real’ second becomes 1/β second as measured in the moving system, the velocity of light relative to the platform will always remain _c_. We must adopt two very startling hypotheses, namely, the Fitz-Gerald contraction and the concept of “local time,” in order to give a satisfactory explanation of the Michelson-Morley experiment.