The Principle of Relativity ( Original Papers) by Albert Einstein and Hermann Minkowski E-Book

Einstein Albert

• Herausgeber: Balungi Francis
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2024

Beschreibung

This book is by no means a comprehensive study of Minkowski's and Einstein's space-time formalism of special relativity. The mathematician, Hermann Minkowski was Einstein's former mathematics professor at the Zürich Polytechnic. During his studies at the Polytechnic Einstein skipped Minkowski's classes. In 1904 Max Born arrived in the first time to Göttingen. Many years later Born wrote his recollections. In the summer of 1905, Minkowski and Hilbert led an advanced seminar on mathematical physics, on electrodynamical theory. Minkowski told Born later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently. He never made a priority claim and always gave Einstein his full share in the great discovery. In his famous talk, "Space and Time" Minkowski wrote that the credit of first recognizing sharply that t and t' are to be treated the same, is of A. Einstein.
TABLE OF CONTENTS

HISTORICAL INTRODUCTION 8
Conclusion 20
Note A. 22
On The Electrodynamics of Moving Bodies By A. Einstein. 23
INTRODUCTION. 24
I.—KINEMATICAL PORTION. 25
§ 1. Definition of Synchronism. 25
§ 2. On the Relativity of Length and Time. 26
Relativity of Time. 27
§ 3. Theory of Coordinate and Time-Transformation from a stationary system to a system which moves relatively to this with uniform velocity. 28
§ 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks. 34
§ 5. Addition-Theorem of Velocities. 36
II.—ELECTRODYNAMICAL PART. 39
SPACE AND TIME 128
The Foundation of the Generalised Theory of Relativity By A. Einstein. From Annalen der Physik 4.49.1916. 142
A Principal considerations about the Postulate of Relativity. 142
§ 1. Remarks on the Special Relativity Theory. 142
§ 2. About the reasons which explain the extension of the relativity-postulate. 143
§ 3. The time-space continuum. Requirements of the general Covariance for the equations expressing the laws of Nature in general. 145
§ 4. Relation of four co-ordinates to spatial and time-like measurements. 147
B Mathematical Auxiliaries for Establishing the General Covariant Equations. 149
5. Contravariant and covariant Four-vector. 149
§ 6. Tensors of the second and higher ranks. 151
§ 7. Multiplication of Tensors. 154
§ 9. Equation of the geodetic line (or of point-motion). 162
§ 10. Formation of Tensors through Differentiation. 165
§12. The Riemann-Christoffel Tensor. 176
C. THE THEORY OF THE GRAVITATION-FIELD 179
§13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation. 179
§14. The Field-equation of Gravitation in the absence of matter. 180
§15. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy. 181
§16. General formulation of the field-equation of Gravitation. 185
§17. The laws of conservation in the general case. 186
§18. The Impulse-energy law for matter as a consequence of the field-equations. 188
D. THE “MATERIAL” PHENOMENA. 190
D. THE “MATERIAL” PHENOMENA. 202

Leseprobe

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

HISTORICAL INTRODUCTION

Conclusion

Note A.

On The Electrodynamics of Moving Bodies  By  A. Einstein.

INTRODUCTION.

I.—KINEMATICAL PORTION.

II.—ELECTRODYNAMICAL PART.

Principle of Relativity

INTRODUCTION.

§ 3.

§ 4. Special Lorentz Transformation.

§ 5. Spacetime Vectors.  Of the 1st and 2nd kind.

§ 6. Concept of Time.

§ 8. The Fundamental Equations.

§ 9. The Fundamental Equations in Lorentz’s Theory.

§10. Fundamental Equations of E. Cohn.

§11. Typical Representations of the Fundamental Equations.

§12. The Differential Operator Lor.

§ 13. The Product of the Field-vectors f F.

§ 14. The Ponderomotive Force.[28]

APPENDIX  Mechanics and the Relativity-Postulate.

SPACE AND TIME

I

II

III

IV

The Foundation of the Generalised Theory of Relativity  By A. Einstein.  From Annalen der Physik 4.49.1916.

A  Principal considerations about the Postulate of Relativity.

B  Mathematical Auxiliaries for Establishing the General Covariant Equations.

C. THE THEORY OF THE GRAVITATION-FIELD

D. THE “MATERIAL” PHENOMENA.

E. §21. Newton’s theory as a first approximation.

D. THE “MATERIAL” PHENOMENA.

E. §21. Newton’s theory as a first approximation.

NOTES

Note 1.

Note 2.  Lorentz Transformation.

Note 3.

Note 4.  Relativity Theorem and Relativity-Principle.

Note 5.

Note 6.  Field Equations in Minkowski’s Form.

Note 9.  On the Constancy of the Velocity of Light.

Note 10.  Rest-density of Electricity.

Note 11  (page 17)  Space-time vectors of the first and the second kind.

Note 12.  Light-velocity as a maximum.

Notes 13 and 14.

Note 15.  The vector product (wf). (P. 36).

Note 16.  The electric-rest force. (Page 37.)

Note 17.  Operator “Lor” (§ 12, p. 41).

Thank You and Welcome

l l 2lc 2l

t₁ =——-—+—————=————-—= -—β²

c - u c + u c² - u² c

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 realized.”

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.

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/μ2. 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.

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.

LET THE ETHER BE FLOWING relative to the apparatus with velocity u in the direction Ox, 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

WHERE

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

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 irrational, 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 polarization in the electromagnetic ether which was merely a transformation of Fresnel’s stagnant ether. The magnetic polarization was looked upon as wholly secondary in origin, being due to the relative motion of the dielectric tubes of polarization. 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 polarization and further emphasized the reciprocal relation subsisting between electricity and magnetism, thus making the field equations more symmetrical. On this view the whole of the polarized 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/μ2), and consequently stood totally condemned from the optical point of view.

Certain direct electromagnetic experiments involving relative motion of polarized dielectrics were no less conclusive against the generalized 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 organized 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 “polarization 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/μ2, 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 marvelous 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 2lβ², while the distance travelled by the beam along OB, perpendicular to the direction of motion of the platform, is 2lβ. 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 Oy (perpendicular to the direction of motion), and automatically contract to a length l/β, when placed along Ox (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 2lβ and the time taken for this journey is 2lβ/c. But the distance travelled with respect to the platform is always 2l. Hence the velocity of light with respect to the platform is

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.

These results were already reached by Lorentz in the course of further developments of his electron theory. Lorentz used a special set of transformation equations[2] for time which implicitly introduced the concept of local time. But he himself failed to attach any special significance to it, and looked upon it rather as a mere mathematical artifice like imaginary quantities in analysis or the circle at infinity in projective geometry. The originality of Einstein at this stage consists in his successful physical interpretation of these results, and viewing them as the coherent organized consequences of a single general principle. Lorentz established the Relativity Theorem [3] (consisting merely of a set of transformation equations) while Einstein generalized it into a Universal Principle. In addition Einstein introduced fundamentally new concepts of space and time, which served to destroy old fetishes and demanded a wholesale revision of scientific concepts and thus opened up new possibilities in the synthetic unification of natural processes.

Newton had framed his laws of motion in such a way as to make them quite independent of the absolute velocity of the earth. Uniform relative motion of ether and matter could not be detected with the help of dynamical laws. According to Einstein neither could it be detected with the help of optical or electromagnetic experiments. Thus the Einsteinian Principle of Relativity asserts that all physical laws are independent of the ‘absolute’ velocity of an observer.

For different systems, the form of all physical laws is conserved. If we chose the velocity of light[4] to be the fundamental unit of measurement for all observers (that is, assume the constancy of the velocity of light in all systems) we can establish a metric “one-one” correspondence between any two observed systems, such correspondence depending only the relative velocity of the two systems. Einstein’s Relativity is thus merely the consistent logical application of the well known physical principle that we can know nothing but relative motion. In this sense it is a further extension of Newtonian Relativity.

On this interpretation, the Lorentz-Fitzgerald contraction and “local time” lose their arbitrary character. Space and time as measured by two different observers are naturally diverse, and the difference depends only on their relative motion. Both are equally valid; they are merely different descriptions of the same physical reality. This is essentially the point of view adopted by Minkowski. He considers time itself to be one of the co-ordinate axes, and in his four-dimensional world, that is in the space-time reality, relative motion is reduced to a rotation of the axes of reference. Thus, the diversity in the measurement of lengths and temporal rates is merely due to the static difference in the “framework” of the different observers.

The above theory of Relativity absorbed practically the whole of the electromagnetic theory based on the Maxwell-Lorentz system of field equations. It combined all the advantages of classic Maxwellian theory together with an electronic hypothesis. The Lorentz assumption of polarization doublets had furnished a satisfactory explanation of the Fresnelian convection of ether, but in the new theory this is deduced merely as a consequence of the altered concept of relative velocity. In addition, the theory of Relativity accepted the results of Michelson and Morley’s experiments as a definite principle, namely, the principle of the constancy of the velocity of light, so that there was nothing left for explanation in the Michelson-Morley experiment. But even more than all this, it established a single general principle which served to connect together in a simple coherent and fruitful manner the known facts of Physics.

The theory of Relativity received direct experimental confirmation in several directions. Repeated attempts were made to detect the Lorentz-Fitzgerald contraction. Any ordinary physical contraction will usually have observable physical results; for example, the total electrical resistance of a conductor will diminish. Trouton and Noble, Trouton and Rankine, Rayleigh and Brace, and others employed a variety of different methods to detect the Lorentz-Fitzgerald contraction, but invariably with the same negative results. Whether there is an ether or not, uniform velocity with respect to it can never be detected. This does not prove that there is no such thing as an ether but certainly does render the ether entirely superfluous. Universal compensation is due to a change in local units of length and time, or rather, being merely different descriptions of the same reality, there is no compensation at all.

There was another group of observed phenomena which could scarcely be fitted into a Newtonian scheme of dynamics without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that the “mass” of an electron depended on its velocity. So early as 1881, J. J. Thomson had shown that the inertia of a charged particle increased with its velocity. Abraham now deduced a formula for the variation of mass with velocity, on the hypothesis that an electron always remained a rigid sphere. Lorentz proceeded on the assumption that the electron shared in the Lorentz-Fitzgerald contraction and obtained a totally different formula. A very careful series of measurements carried out independently by Bücherer, Wolz, Hupka and finally Neumann in 1913, decided conclusively in favour of the Lorentz formula. This “contractile” formula follows immediately as a direct consequence of the new Theory of Relativity, without any assumption as regards the electrical origin of inertia. Thus the complete agreement of experimental facts with the predictions of the new theory must be considered as confirming it as a principle which goes even beyond the electron itself. The greatest triumph of this new theory consists, indeed, in the fact that a large number of results, which had formerly required all kinds of special hypotheses for their explanation, are now deduced very simply as inevitable consequences of one single general principle.

We have now traced the history of the development of the restricted or special theory of Relativity, which is mainly concerned with optical and electrical phenomena. It was first offered by Einstein in 1905. Ten years later, Einstein formulated his second theory, the Generalized Principle of Relativity. This new theory is mainly a theory of gravitation and has very little connection with optics and electricity. In one sense, the second theory is indeed a further generalization of the restricted principle, but the former does not really contain the latter as a special case.

Einstein’s first theory is restricted in the sense that it only refers to uniform rectilinear motion and has no application to any kind of accelerated movements. Einstein in his second theory extends the Relativity Principle to cases of accelerated motion. If Relativity is to be universally true, then even accelerated motion must be merely relative motion between matter and matter. Hence the Generalized Principle of Relativity asserts that “absolute” motion cannot be detected even with the help of gravitational laws.

All movements must be referred to definite sets of co-ordinate axes. If there is any change of axes, the numerical magnitude of the movements will also change. But according to Newtonian dynamics, such alteration in physical movements can only be due to the effect of certain forces in the field.[5] Thus any change of axes will introduce new “geometrical” forces in the field which are quite independent of the nature of the body acted on. Gravitational forces also have this same remarkable property, and gravitation itself may be of essentially the same nature as these “geometrical” forces introduced by a change of axes. This leads to Einstein’s famous Principle of Equivalence. A gravitational field of force is strictly equivalent to one introduced by a transformation of co-ordinates and no possible experiment can distinguish between the two.

Thus it may become possible to “transform away” gravitational effects, at least for sufficiently small regions of space, by referring all movements to a new set of axes. This new “framework” may of course have all kinds of very complicated movements when referred to the old Galilean or “rectangular uncelebrated system of co-ordinates.”

But there is no reason why we should look upon the Galilean system as more fundamental than any other. If it is found simpler to refer all motion in a gravitational field to a special set of co-ordinates, we may certainly look upon this special “framework” (at least for the particular region concerned), to be more fundamental and more natural. We may, still more simply, identify this particular framework with the special local properties of space in that region. That is, we can look upon the effects of a gravitational field as simply due to the local properties of space and time itself. The very presence of matter implies a modification of the characteristics of space and time in its neighborhood. As Eddington says “matter does not cause the curvature of space-time. It is the curvature. Just as light does not cause electromagnetic oscillations; it is the oscillations.”

We may look upon this from a slightly different point of view. The General Principle of Relativity asserts that all motion is merely relative motion between matter and matter, and as all movements must be referred to definite sets of co-ordinates, the ground of any possible framework must ultimately be material in character. It is convenient to take the matter actually present in a field as the fundamental ground of our framework. If this is done, the special characteristics of our framework would naturally depend on the actual distribution of matter in the field. But physical space and time is completely defined by the “framework.” In other words the “framework” itself is space and time. Hence we see how physical space and time is actually defined by the local distribution of matter.

There are certain magnitudes which remain constant by any change of axes. In ordinary geometry distance between two points is one such magnitude; so that δx² + δy² + δz² is an invariant. In the restricted theory of light, the principle of constancy of light velocity demands that δx² + δy² + δz² - c²δt² should remain constant.

WHERE THE g’s are functions of x₁, x₂, x₃, x₄ depending on the transformation.

The special properties of space and time in any region are defined by these g’s which are themselves determined by the actual distribution of matter in the locality. Thus from the Newtonian point of view, these g’s represent the gravitational effect of matter while from the Relativity stand-point, these merely define the non-Newtonian (and incidentally non-Euclidean) space in the neighborhood of matter.

We have seen that Einstein’s theory requires local curvature of space-time in the neighborhood of matter. Such altered characteristics of space and time give a satisfactory explanation of an outstanding discrepancy in the observed advance of perihelion of Mercury. The large discordance is almost completely removed by Einstein’s theory.

Again, in an intense gravitational field, a beam of light will be affected by the local curvature of space, so that to an observer who is referring all phenomena to a Newtonian system, the beam of light will appear to deviate from its path along an Euclidean straight line.

This famous prediction of Einstein about the deflection of a beam of light by the sun’s gravitational field was tested during the total solar eclipse of May, 1919. The observed deflection is decisively in favor of the Generalized Theory of Relativity.

It should be noted however that the velocity of light itself would decrease in a gravitational field. This may appear at first sight to be a violation of the principle of constancy of light-velocity. But when we remember that the Special Theory is explicitly restricted to the case of unaccelerated motion, the difficulty vanishes. In the absence of a gravitational field, that is in any unaccelerated system, the velocity of light will always remain constant. Thus the validity of the Special Theory is completely preserved within its own restricted field.

Einstein has proposed a third crucial test. He has predicted a shift of spectral lines towards the red, due to an intense gravitational potential. Experimental difficulties are very considerable here, as the shift of spectral lines is a complex phenomenon. Evidence is conflicting and nothing conclusive can yet be asserted. Einstein thought that a gravitational displacement of the Fraunhofer lines is a necessary and fundamental condition for the acceptance of his theory. But Eddington has pointed out that even if this test fails, the logical conclusion would seem to be that while Einstein’s law of gravitation is true for matter in bulk, it is not true for such small material systems as atomic oscillator.

Conclusion

From the conceptual stand-point there are several important consequences of the Generalised or Gravitational Theory of Relativity. Physical space-time is perceived to be intimately connected with the actual local distribution of matter. Euclid-Newtonian space-time is not the actual space-time of Physics, simply because the former completely neglects the actual presence of matter. Euclid-Newtonian continuum is merely an abstraction, while physical space-*time is the actual framework which has some definite curvature due to the presence of matter. Gravitational Theory of Relativity thus brings out clearly the fundamental distinction between actual physical space-time (which is non-isotropic and non-Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian continuum (which is homogeneous, isotropic and a purely intellectual construction) on the other.

The measurements of the rotation of the earth reveals a fundamental framework which may be called the “inertial framework.” This constitutes the actual physical universe. This universe approaches Galilean space-time at a great distance from matter.

The properties of this physical universe may be referred to some world-distribution of matter or the “inertial framework” may be constructed by a suitable modification of the law of gravitation itself. In Einstein’s theory the actual curvature of the “inertial framework” is referred to vast quantities of undetected world-matter. It has interesting consequences. The dimensions of Einsteinian universe would depend on the quantity of matter in it; it would vanish to a point in the total absence of matter. Then again curvature depends on the quantity of matter, and hence in the presence of a sufficient quantity of matter space-time may curve round and close up. Einsteinian universe will then reduce to a finite system without boundaries, like the surface of a sphere. In this “closed up” system, light rays will come to a focus after travelling round the universe and we should see an “anti-sun” (corresponding to the back surface of the sun) at a point in the sky opposite to the real sun. This anti-sun would of course be equally large and equally bright if there is no absorption of light in free space.

In de Sitter’s theory, the existence of vast quantities of world-matter is not required. But beyond a definite distance from an observer, time itself stands still, so that to the observer nothing can ever “happen” there. All these theories are still highly speculative in character, but they have certainly extended the scope of theoretical physics to the central problem of the ultimate nature of the universe itself.

One outstanding peculiarity still attaches to the concept of electric force—it is not amenable to any process of being “transformed away” by a suitable change of framework. H. Weyl, it seems, has developed a geometrical theory (in hyper-space) in which no fundamental distinction is made between gravitational and electrical forces.

Einstein’s theory connects up the law of gravitation with the laws of motion, and serves to establish a very intimate relationship between matter and physical space-*time. Space, time and matter (or energy) were considered to be the three ultimate elements in Physics. The restricted theory fused space-time into one indissoluble whole. The generalised theory has further synthesised space-time and matter into one fundamental physical reality. Space, time and matter taken separately are more abstractions. Physical reality consists of a synthesis of all three.

P. C. Mahalanobis.

Note A.

For example consider a massive particle resting on a circular disc. If we set the disc rotating, a centrifugal force appears in the field. On the other hand, if we transform to a set of rotating axes, we must introduce a centrifugal force in order to correct for the change of axes. This newly introduced centrifugal force is usually looked upon as a mathematical fiction—as “geometrical” rather than physical. The presence of such a geometrical force is usually interpreted as being due to the adoption of a fictitious framework. On the other hand a gravitational force is considered quite real. Thus a fundamental distinction is made between geometrical and gravitational forces.

In the General Theory of Relativity, this fundamental distinction is done away with. The very possibility of distinguishing between geometrical and gravitational forces is denied. All axes of reference may now be regarded as equally valid.

In the Restricted Theory, all “unaccelerated” axes of reference were recognized as equally valid, so that physical laws were made independent of uniform absolute velocity. In the General Theory, physical laws are made independent of “absolute” motion of any kind.

On The Electrodynamics of Moving Bodies By A. Einstein.

INTRODUCTION.

It is well known that if we attempt to apply Maxwell’s electrodynamics, as conceived at the present time, to moving bodies, we are led to asymmetry which does not agree with observed phenomena. Let us think of the mutual action between a magnet and a conductor. The observed phenomena in this case depend only on the relative motion of the conductor and the magnet, while according to the usual conception, a distinction must be made between the cases where the one or the other of the bodies is in motion. If, for example, the magnet moves and the conductor is at rest, then an electric field of certain energy-value is produced in the neighbourhood of the magnet, which excites a current in those parts of the field where a conductor exists. But if the magnet be at rest and the conductor be set in motion, no electric field is produced in the neighbourhood of the magnet, but an electromotive force which corresponds to no energy in itself is produced in the conductor; this causes an electric current of the same magnitude and the same career as the electric force, it being of course assumed that the relative motion in both of these cases is the same.

2. Examples of a similar kind such as the unsuccessful attempt to substantiate the motion of the earth relative to the “Light-medium” lead us to the supposition that not only in mechanics, but also in electrodynamics, no properties of observed facts correspond to a concept of absolute rest; but that for all coordinate systems for which the mechanical equations hold, the equivalent electrodynamical and optical equations hold also, as has already been shown for magnitudes of the first order. In the following we make these assumptions (which we shall subsequently call the Principle of Relativity) and introduce the further assumption,—an assumption which is at the first sight quite irreconcilable with the former one—that light is propagated in vacant space, with a velocity c which is independent of the nature of motion of the emitting body. These two assumptions are quite sufficient to give us a simple and consistent theory of electrodynamics of moving bodies on the basis of the Maxwellian theory for bodies at rest. The introduction of a “Lightäther” will be proved to be superfluous, for according to the conceptions which will be developed, we shall introduce neither a space absolutely at rest, and endowed with special properties, nor shall we associate a velocity-vector with a point in which electromagnetic processes take place.

3. Like every other theory in electrodynamics, the theory is based on the kinematics of rigid bodies; in the enunciation of every theory, we have to do with relations between rigid bodies (coordinate system), clocks, and electromagnetic processes. An insufficient consideration of these circumstances is the cause of difficulties with which the electrodynamics of moving bodies have to fight at present.