Conceptio › Archive › Internet Archive

Internet Archiveopen access

The principle of relativity; original papers

Einstein, Albert, 1879-1955

Internet Archive · Books · License: Open Access

Open Source ↗

relativity (physics)

;. i».. •' •Mii'-marvF^i'rv ■; u ■' ' t.^».-ot Mnafco

LIBRARY

OF THE

MASSACHUSETTS INSTITUTE

OF TECHNOLOGY

^0' <

THE

SEP k

PRINCIPLE OF RELATJ^^{jK[Ji

ORIGINAL PAPERS

A. EINSTEIN ANT) H. MINKOWSKI

TRANSLATED INTO ENGLISH

M. N. SAHA AND S. N. BOSE

lecturers on physics and applied jiathematics Univebsity College of Science, Calcutta Univeksity

WITH A HISTORICAL INTRODUCTION

P. C. MAHALANOBIS

professor of physics, presidency college, CALCU-

PUBLISHED BY THE

UNIVERSITY OF CALCUTTA

1920

Sole Agents R. CAMBRAY &: CO.

PRINTED BY ATULCHANDRA BHATTACHAKYYA, AT THE CALCUTTA UNIVERSITY PRE3B. SENATE HOUSE. CALCUTTA

TABLE OF CONTENTS

1. Historical Introduction

[By Mr. P. C. Mahalanobis.]

2. On the Electrodynamics of Moving Bodies...

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

3. Albreeht Einstein

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

4. Principle of Relativity

[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 ...

[Translated by Dr. Meghnad Saha.]

6. The Generalised Principle of Relativity

[A. Einstein's second paper on the Genera- lised Principle first published in 1916. Translated from the orijjina] German by Mr. Satyendranath Bose.]

/, iNotes ,,, ,,, ...

PAGE

i-xxiii 1-34

35-39

1-52

53-88

89-l()3

165-^185

124281

-N \^\

HISTORICAL INTRODUCTION

ooj:^0<>-

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 bnild up the nineteenth century school of plenuDij one etiier 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 cons ^mmation now realised."

Ten years later, in 1905, we find Einstein declarinsj that " the ether will be proved to be superflous." 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.

11 PRINCIPLE OF RELATIVITY

Arago^s prison 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 ethei outside the prism and move with veloeit)' c with respeci to the ether. If the prism moves with a velocity n with respect to this fixed ether, then the incident velocity of light with respect to the prism should be c + n. ThuE the refractive index of the glass prism should depend on m the absolute velocity of the prism, i.e., its velocity witl respect to the fixed ether. Arago performed the experimeni in 1819, but failed to detect the expected change.

Airy- Boscovitch ivaler-telescoije experimeni. — Boscovitcl had still earlier in 1766, raised the very importan question of the dependence of aberration on the refractive index of the medium filling the telescope. Aberratior depends on the difference in the velocity of light outsid» the telescope and its velocity inside the telescope. If thi latter velocity changes owing to a change in the medium filling the telescope, aberration itself should change, thai is, aberration should depend on the nature of the medium.

Airy, in 1871 filled up a telescope with water — but failed to detect any chansje in the aberration. Thus w< get both in the case of Arago prism experiment an( Airy -Boscovitch water-telescope experiment, the ver startling result that optical effects in a moving mediun seem to be quite independent of the volocit}^ of th medium with respect to Fresnel's stationary ether.

FresneVs convection coefficient /(:=1 — ^/^^. — Possibb some form of compensation is taking place. Working oi this hypothesis, Fresnel effered his famous ether convee tion theory. According to Fresnel, the presence of matte: implies a definite condensation of ether within th(

t « •

HISTORICAL INTRODUCTION 111

region occupied by matter. This " condensed " or excess portion of ether is supposed to be carried away with its own piece of movino" 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 p with the full velocity u is optically equivalent to a partial convection of the total ether p, with only a fraction of the velocity k. u. Fresnel showed that if this convection coefficient k is 1 — *//x'-^ (/x being the refractive index of the prism), then the velocitv of lio^ht after retraction within the movin"; prism would be altered to just such extent as would make the refractive index of the moving prism quite indepen- dent of its "absolute" velocity u. The non-depeudence of aberration on the '" absolute " velocity it, is also very easily explained with the help of this Fi-esnelian convection- coefficient k.

Stokes^ viseous ether. — It should be remembered, however, that Fresnel 's stationary ether is absolutelv fixed and is not at all disturbed bv the motion of matter throusfh 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, w^hile 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 kj namely 1 — V/^'- '^hus the prestige of the Fresnelian convection-coefficient was enhanced, if anything, by the theoretical investigations of Stokes.

IV PRINCIPLE OF RELATIVITY

Fizeaic^s experin/cnl, — Soon aftur, in 1851, it received direct experimental eonHrmation in a brilliant piece of work by Fizeau.

If a divided beam of light is re-nnited after passin<)j through two adjacent cylinders filled with water, ordinary interference fringes will be produced. If the water in one of the cylinders is now nriade to fiow^, the " condensed" ether within the flowing water wonld be conveeted and would produce a shift in the interference fringes. The shift actuallv observed agreed verv well with a value of k=l— V/Jt^. The Fresnelian eonveetion-eoeffieient 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 earthy, thus proving conclusively that all tne 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.

Mic/iehov-Morley Eopperiment. — 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.

HISTOKICAL INTRODUCTION V

The fiiiidameiital idea underlyia^^'' tliib experiment is quite sim[de. In all old expeiiments the velocity of light situated in free ether \Vas corn[)ared with the veloeitv of waves actually situated m 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 neo^ative 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 delinite time in crossing the gap. If the rigid platform carrying the gap is set in motion with respect to the ether in the direc- tion 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 Eddiugton's descrip- tion 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 vards acioss-stream and back. If the earth is movino- through the ether there is a river of ether flowing- throuopli the laboratory, and a wave of light may be compared to a swnmmer 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

oi the apparatus with velocity u in the

^ direction Or, and let OA, OB, be

B the two arms of the apparatus of equal

VI PKiNClPLE 0¥ RELATIVITY

length L Oi^. being placed up-stream. Let c be tbe velocity of lig;ht. The time for the double journev alon^' OA and back is

t,= ± + -A = J^= ^/S^ G — If. c~\ru c^ — u^ c

where f3=:(l—u'^/c^)~'^, a factor greater than unity.

For tbe transverse journey the light must have a compo- nent velocity n up-stream (relative to the ether) in order to avoid beins: carried below OB : and since its total velocity is c, its component across-stream must be \/{c'^ —u'^), the time for the double journey OB is accordingly

t'l = /7-~^^ = —A SO that t^>t^.

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."

x\fter a most careful series of observations, Michelson and Morle^^ failed to detect the slightest trace of any effect due to earth's motion throus^h 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 — n. Thus Michelson and Morlev themselves thought at first that their experiment conhrmed Stokes^ viscous ether, in wliieh no relative motion can ensue on account of the absence of slip])ing of ether at the surface of separation. But even on Stokes' theory this viscous How of ether would fall ofP at a very rapid rate as we recede from the surface of separation. Michelson and Morley repeated their experi- ment at different heights from the surface of the earth, but invariably obtained the same negative results, thus failing to confirm Stokes' theory of viscous How.

HTSTORICAL TNTHODUCTTON TU

Loflgt!^ experimevi, — Further, in 1893, Lodge per- formed bis rotating' sphere experiment which showed complete absence of any viscous How of ether due to moviuo' masses of matter. A divided beam of light, after repeated reflections within a ver}^ 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 distru'bance 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 experi- ment 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 incom- pressible 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 assump- tions. Thus Stokes' ether failed to satisfy the very motive which had led Stokes to formulate it^ namely, the desirabi- lity of constructing a "physical" medium. Planck offered modified forms of Stokes' theory which seemed capable of being reconciled with the Miehelson-Morley experiment, but required very sjiecial 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.

Vlll PRINCIPLE OF RELATIVITY

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 aibitrary 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 nine- teenth century showed the impossibility of constructing a simple ether theory, which would be jsmenable to analytic treatment and would at the same time stimulate funher progress. It should be observed that it could scarcely be shown that no looieallv consistent ether theorv was possible ; indeed ill 1910, H. A. Wilson offered a consis- sent ether ilieor\ which was at least quite neutral with respect to all available optical data. But Wilson's ether is almost whollv nesfative — its onlv virtue beinoj that it does not directly contradict observed facts. Neither any direct conhrmation nor a direct refutation is possible and it does not throw any light on the various optical pheno- mena. A theory like this being practicall}' useless stands self-condemned.

We must now consider the problem of relativf motion of ether and matter from the point of view of electrical theory. From 1860 the identitv of lisht as an electromagnetic vector became o-radualh' 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 re- searches on dectric waves, first carried out in 1886, succeeded in furnishing a strong experimental conlh-mation

HISTORICAL INTRODUCTION II

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 stag-nant ether. The mag- netic 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 con- vection coefficient comes out to be i, as shown by J. J. Thomson in 1880, instead of 1 — ^//x- 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 — V/*^'

The modifications proposed independently by Hertz and Heaviside fare no better."^ 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 co-efficient 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 subse- quent modifications as developed by Hertz and Heaviside succeeded in obtainiuii; a value for Fresnelian co-efficient equal to 1— V/^^j ^^^ consequently stood totall3^ condemned from the optical point of view.

Certain direct electromagnetic experiments invohing relative motion of polarised dielectrics were no less conclu- sive against the generalised theory of Hertz and Heaviside.

* See Note 1.

X PRINCIPLE OF RELATIVITY

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 )901, 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 pro- portional 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, Rontgen found in 1888 the magnetic effect due to a rotating dielectric (the condenser remain- ing 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 Row- land effect proportional to K, together with the opposite Rontgen 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

HISTORICAL INTRODUCTION XI

ether concept had finally outgrown its usefulness. The observed fleets 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 outstandmg difficulties in connec- tion 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 meduim had become inevit- able. 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 satisfac- tory basis.

Lorentz assumed that a moving dielectric merely carried away its own '' polarivsation 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 — ^//a^, 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 meduim is involved in any portion of the paths actually traversed by the beam. Consequently no compensation due to Fresnelian convection

Xll ^PRINCIPLE OP RELATIVITY

of ether by moving medium is possible. Thus Presneliao convection compensation can have no possible application in this ease. Yet some marvellous compensation has evidently tai^en place which has completely masked the " absolute '"' velocity of the earth.

In Miphelson 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/^^, while the distance travelled by the beam along OB, perpendicular to the direction of motion of the platform, is ^lip. 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 I, OA must have been of length IjP. 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 OB must now be the shorter length. The plain meaning of the experiment is that both arms have a length I when placed along 0^ (perpendicular to the direc- tion of motion), and automatically contract to a length Ijpf when placed along 0/ (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 %lp and the time taken for this journey is 2l^/c. But the distance travelled with respect to the platform is always 21. Hence the velocity of light with respect to the plat-

form is 21/ — ^ —c/^, 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

HISTOBICAL INTRODUCTION XUl

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 mov- inoj clock goes slower so that one 'real' second becomes 1/^ 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 Miehelson-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"^ 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 organised consequences of a single general principle. Lorentz established the Relativity Theoremt (consisting merely of a set of transformation equations) while Einstein generalised 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

* See Note 2. t See Note 4.

XIV PRINCIPLE or RELATIVITY

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"^ 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 contrac- tion and "local time" lose their arbitrary character. Space and time as measured by two different observers are natur- ally 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 " frame- work ^' of the different observers.

The above theory of Relativity absorbed practically the whole of the electromagnetic theory based on the

* See Notes 9 and 12.

HISTORICAL INTRODUCTION XV

Maxwell-Lorentz system of field equations. It combined all the advantages of classic Maxwellian theory together with an electronic hypothesis. The Lorentz assumption of polarisation doublets had furnished a satisfactory explana- tion 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-Morle3^ 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 confiimation 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 ivith respect to it can never he detected. This does not prove that there is no such thing as an ether but certainly does render the ether entirely super- fluous. 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

XVI PRINCIPLE OF RELATIVITY

dynamics without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that the " mass '^ of an electron dei)ended on its velocity. So early as 1881, J. J. Thomson had shown that the inertia of a charged })article increased with its velocity. Abraham now deduced a formula for the variation of mass with velocity, on the hypothesis that an electron always remain- ed a rigid sphere. Lorentz proceeded on the assumption that the electron shared in the Lorentz-Fitz2:erald eontrae- tion and obtained a totally di:fferent formula. A very careful series of measurements carried out independently b}^ Biicherer, 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 witli 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 Generalised 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 generalisation of the restricted princijole, but the former does not really contain the latter as a special ease.

HISTORICAL INTRODUCTION Xvii

Einstein's first theory is restricted in the sense that it only refers to uniform reetiliniar motion and has no appli- cation 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 tjetioeen matter and matter. Hence the Generalised 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 chano'e. But according to Newtonian dynamics, such alteration in physical movements can only be due to the effeet of ceitain forces in the tield.^ 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. Gravitation- al 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 strictl/j equivole^it to one introduced tjy a transformation of co-ordinates and no possitjle experiment can distinguish fjetween the tioo.

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 unaccelerated 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

* Note A.

XVlll PEIXCIPLE OF EELATIYITY

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 neighbour- hood. As Eddington saj^s ^' matter does uot 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 /v 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 w^e see how pit i/sical space and time is aetuallv defined bv 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 hx'^ +^^^ H-5,e'^ is an invariant. In the restricted theory of light, the principle of constancy of light velocity demands that 8ir2 +8^^ -|.8^2 __^2g^,2 should remain constant.

HISTORICAL INTUODUCTION XIX

The 'Sejjaration ds of adjacent events is defined by ds'^ = —(Lv^ —di/'^ —dz" -\-c^dt^ , It is an extension of the notion of distance and this is the new invariant. Now if Xy ijy Zy t are Iransformed to any set of new variables ji'j, ti'g, i'g, x^, we shall get a quadratic expression for ds^ =y J j.r J 2 H- 2-7j 2=^'i'''2 + • • • = >'J i .i'V i ^Vj where the ^^s are functions of d'^, x^, .^'3, ii\ depending on the transforma- tion.

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

We have seen that Einstein's theory requires local curvature of space-time in the neighbourhood 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 favour of the Generalised Theory of Relativity.

XX PRINCIPLE OF RELATIVITY

It should be uotecl however that the veloeitv of li^ht 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 unaecelerated motion, the difficulty vanishes. In the absence of a gravitational field, that is in any unaecelerated 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 theorv. But Eddino'ton 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.

CONCLI SIGN

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 intimatel}' connected with the actual local distribution of matter. Euclid-Newtonian space-time is itot 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

HISTORICAL INTRODUCTION XXI

curvature due to the presence of matter. Gravitational Theory of Relativity thus brings out clearly the funda- mental distinction between actual physical space-time (which is non-isotropie and non-Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian continuum (which is homogeneous, isotropic and a purely intellectual construc- tion) 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 frame- work" may be constructed by a suitable modification of the law of gravitation itself. In Einstein's theory the actual curvature of the ** inertia! 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 sk}^ opposite to the real sun. This anti-sun would of course be equally large and equally bright if there is no absorption of hght in free space.

In de Sitter's theory, the existence of vast quantities of world-matter is not required. But beyond a definite

XXll PRINCIPLE OF RELATIVITY

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 phj^sics 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 awav " bv 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 separatel}" are more abstractions. Physical reality consists of a synthesis of all three.

P. C. Mahalanobis.

HISTORICAL INTRODUCTION XXlll

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 us 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 recognised 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.

The Electrodynamics of Moving Bodies

BY A. EjNSTEIJf.

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 assy met ry which does not ao^ree with observed phenomena. Let us think of the mutual action between a magi-net 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.

il PRINCIPLE OF RELATIVITY

*2. Examples of a similar kind such as the uusueeessful attempt to substantiate the motiou 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 svstems for which the mechanical equations hold, the equivalent electrodyna- mieal and optical equations hold also, as has already been shown for magnitudes of the first order. In the following we make these assumptions (w^hich 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 bod}'. These tw^o assumptions are quite sufficient to give us a simple and consistent theor^^ of electrodynamics of movino' bodies on the basis of the Maxwellian theory for

a t,'

bodies at rest. The introduction of a ^^ Lightather" will be proved to be superfluous, for according to the conceptions which will 'be developed, we shall introduce neith er a space absolutely at rest, and endowed with special properties, nor shall we associate a velocity -vector with a point in which electro-magnetic 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, Ave have to do with relations betw^een rigid bodies (co-ordinate 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.

ON THE ELECTKODYXA MlCS Oh' 3I0VlNa BODIES 3

I.-KINEMATIOAL PORTION.

§ 1. Definition of Synchronism.

Let us have a eo-ordinate system, in wliieh the New- tonian equations hold. For distinguishing this system from another which will be introduced hereafter, we shall always call it " the stationary system,"

If a material point be at rest in this system, then its position in this system can be found out by a measuring rod, and can be expressed by the methods of Euclidean Geometry, or in Cartesian co-ordinates.

If we wish to describe the motion of a material point, the values of its coordinates must be expressed as functions of time. It is always to be borne in mind that sicc/i a ■ *• atliemaiical (lefinition has a physical senses only lohen loe have a clear )iotio7i of what is meant by time. We have to fake into consideration the fact that those of our conceptions^ in lohich time plays a part, are alioays conceptions of synchronism For example, we say that a train arrives here at 7 o'clock ; this means that the exact pointing of the little hand of my watch to 7, and the arrival of the train are synchronous events.

It may appear that all difficulties connected with the definition of time can be removed when in place of time, we substitute the position of the little hand of my watch. Such a definition is in fact sufficient, when it is required to define time exclusively for the place at which the clock is stationed. But the definition is not sufficient when it is required to connect by time events taking place at different stations,-— -or what amounts to the same thing,- — to estimate by means of time (zeitlich werten) the occurrence of events, which take place at stations distant from the clock.

4 PKINCIPLE OF RELATIVITY

Now with regard to this attempt; — the time-estimation of events^ we can satisfy ourselves in the following manner. Suppose an observer — who is stationed at the origin of coordinates with the clock — associates a ray of light which comes to him through space, and gives testimony to the event of which the time is to be estimated, — with the corresponding position of the hands of the clock. But such an association has this defect^ — it depends on the position of the observer provided with the clock, as we know by experience. We can attain to a more practicable result bv the following- treatment.

If an observer be stationed at A with a clock, he can estimate the time of events occurring in the immediate neighbourhood of A, by looking for the position of the hands of the clock, which are syrchronous with the event. If an observer be stationed at B with a clock, — we should add that the clock is of the same nature as the one at A, — he can estimate the time of events occurring about B. But without further premises, it is not possible to compare, as far as time is concerned, the events at B with the events at A. We have hitherto an A-time, and a B-time, but no time common to A and B. This last time {i.e., common time) can be defined, if we establish by definition that the time which Hght requires in travelling from A to B is equivalent to the time which light requires in travelling from B to A. For example, a ray of light proceeds from A at xl-time t towards B,

arrives and is reflected from B at B-time t and returns

to A at A-time t' . Accordin£c to the definition, both

clocks are synchronous^ if

t - 1 = t' - t .

B A A B

02^ THE ELECTRODYNAMICS OF MOVING BODIES 5

We assume tbal this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold : —

1. If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B.

2. If the clock at A as w^ell as the clock at B are both synchronous with the clock at C, then the clocks at A and B are svnchronous.

Thus with the help of certain physical experiences, w^e have established what we understand when we speak of clocks at rest at different stations, and synchronous with one another ; and thereby we have arrived at a definition of synchronism and time.

In accordance with experience we shall assume that the magnitude

2 AB

77 ~^ =zc, where c is a universal constant. A A "

We have defined time essentially w^ith a clock at rest in a stationary system. On account of its adaptability to the stationary system, we call the time defined in this way as " time of the stationary system.'^

§ 2. On the Relativity of Length and Time.

The following reflections are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following w^ay :—

1. The laws according to which the nature of physical systems alter are independent of the manner in which these changes are referred to two co-ordinate systems

6 PRINCIPLE or- RELATIVITY

which have a uniform translatorv motion relative to each other.

2. Every ray of light moves ^ in the '^^ stationary co-ordinate system " with the same velocity c-j the velocity being independent of the condition whether this ray of light is emitted by a bod}^ at rest or in motion.^' Therefore

, .. Path of Li<yht

velocity = T—r , ^ , . ,

^ Interval or tmie

where, by ^ interval of time,' we mean time as defined in § 1.

Let us have a rigid rod at rest; this has a length /, when measured by a measuring rod at rest ; we suppose that the axis of the rod is laid along the X-axis of the system at rest, and then a uniform velocity /', parallel to the axis of X, is imparted to it. Let us now enquire ^ about the length of the moving rod ; this can be obtained by either of these operations. —

(a) The observer provided with the measuring rod moves along with the rod to be measured, and measures by direct superposition the length of the rod : — just as if the observer, the measuring rod, and the rod to be measured were at rest.

{b) The observer finds out, by means of clocks placed in a system at rest (the clocks being synchronous as defined in § ]), the points of this system where the ends of the rod to be measured oceui at a particular time t. The distance between these two points, measured by the previously used measuring rod, this time it being at rest, is a length, which we may call the ** length of the rod."

According to the Principle of Relativity, the length found out by the operation «), which we may call " the

* Vide Note 4>.

ON THE ELBCTIIODYNAMICS OF MOVING BODIES I

length of the rod in the moving system " i^ equal to the length^/ of the rod in the station aiy system.

The leno-th which is foand out bv the second method,

may be called * f^fe length of the moving rod 'measured from

the sfatiomr^ si/dem/ This leni^th is to be estimated on

the basis of our principle, and we shall find it to he different

from I.

In the generally recognised kinematics, we silently assume that the lengths defined by these two operations are equal, or in other words, that at an epoch of time t, a moving rigid body is geometrically replaceable by the same body, which can replace it in the condition of rest.

Relativity of Time.

Let us suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A, and B of a rod, i.e., the time of the clocks correspond to the time of the stationary system at the points where they happen to arrive ; these clocks are therefore synchronous in the stationary system.

We further imagine that there are two observers at the two watches, and moving with them, and that these observers apply the criterion for synchronism to the two clocks. At the time ^ , a ray of light goes out fi^m A, is.

reflected from B at the time t , and arrives back at A at

time t' . Taking into consideration the principle of^

constancy of the velocity of light, we have

and

t -

■f = A

'^B

c-v'

t' ■ A

-t = B

b PRINCIPLE OF RELATIVITY

where r is the lens^th of the movins^ rod, measured

in the stationary system. Therefore the observers stationed with the watches will not find the clocks Fj-nchrouous, thoiio-h the observer in the stationarv system must declare the clocks to be svnehronous. We therefore see that we can attach no absolute signiticanee to the concept of synchro- nism ; but two events which ara synchronous v»dien viewed from one system, will not be synchronous when viewed from a system movin<^ relatival v to this svstem.

§ 3. Theory of Co-prdinate and Time- Transformation

from a stationary system to a system which

moves relatively to this with

uniform velocity.

Let there be sjiven, in the stationarv svstem two co-ordinate systems, I.e., two series o{" three mutually perpendicular lines issuing from a point. Let the X-axes of each coincide with one another, and the Y and Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be given to each of the systems, and let the rods and clocks in each be exactly alike each other.

Let the initial point of one of the sj^stems (k) have a constant velocity in the direction of the X-axis of the other which is stationary system K, the motion being also communicated to the rods and clocks in the system (k). Any time t of the stationary system K corresponds to a definite position of the axes of the moving system, which are always parallel to the axes of the stationary system. By I, we alwaj^s mean the time in the stationaiy system.

We suppose that the space is measured by the stationary measuring rod placed in the stationary system, as well as by the moving measuring rod placed in the moving

ON THE ELECTRODYNAMICS OF MOVING BODIES 9

system, and we thus obtain the co-ordinates (3c,y^z) for the stationary system, and (^, yy, ^) for the moving system. Let the time t be determined for each point of the stationary system (which are provided with clocks) by means of the •clocks which are placed in the stationary system, with the help of light-signals as described in § 1. Let also the time t of the moving^ svstem be determined for each point of the moving system (in which there are clocks which are at rest relative to the moving system), by means of the method of light signals between these points (in which there ar^^ clocks) in the manner described in § 1.

To every value of (r, y, z, t) which fully determines the position and time of^ an event in the static uary system, there correspond-; a system of values {^,y],'C'T) ; now the problem is to find out the system of equations connect- ing these magnitudes.

Primarily it is clear that on account of the j^roperty of homogeneity which we ascribe to time and space, the . equations must be linear

If we put .r'rrx — ?;^, then it i clear that at a point relatively at rest in the system -J§^,^A^e have a system of values (,(/ y z) which are independent of time. Now let us find out r as a function of (%,y,z,t). For this purpose we have to exp'fess in equations the fact that t is not other than the time given by the clocks which are at rest in the system k which must be made synchron- ous in the manner described in § L

Let a ray of light be sent at time r^ from the origin of the system A,- along the- X-axis towards iv' and let it be reflected from that place at time t^ towards the origin of moving co-ordinates and let it arrive there at time t^ ; then we must have

10 PRINCIPLE OF REI ATIVITY

If we now introduce the condition that t is a function (?f co-orrdinates, and apply the principle of constancy of the velocity of light in the stationary system, we have

i ]t (o, o, 0, t)+T (o, 0, 0, {t+ il— + J!__ [ ) 1 C c—v c-{-v -) / J

=T(a;', 0, 0,t + -^ )

C — V /.

It is to be noticed that instead of the origin of co- ordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of (0/^,2",^,).

A similar conception, being applied to the y- and -s'-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is always ppopogated along those axes with the velocity^c^— i;^, we have the questions

^- =0, ^- =0. . oy oz

Prom these equations it follows that t is a linear func- tion of .c'and t. From equations (1) we obtain

/, III-' \

where a is an unknown function of v.

With the help of these results it is easy to obtain the magnitudes (i,r]X), if we express by means of equations t!ie fact that light, when measured in the moving system is always propagated with the constant velocity c (as the principle of constancy of light velocity in conjunc- tion with the principle of relativity requires). For a

ON THE ELECTRODYNAMICS OF MOVING BODIES 11

time T=Oy if the ray is sent in the direction of increasing ^, we have

^=.c T , i.e. i=:ac i t— — — \,

Now the ray of light moves relative to the origin of k with a velocity c— t;, measured in the stationary system ;

therefore we have

C — V

Substituting these values of t in the equation for $, we obtain

In an analogous manner, we obtain by considering the ray of light which moves along the ^-axis,

7] = CT = aC I t — J

where • , =^, i>;'=^j

c c

Therefore t?=a ., . y, l=a • ■ z.

If for .t;', we substitute its value x—tv, we obtain

r}=4> (v) y where S= . - — , and (f> (v)=z — =r«r is a function

of V.

12 PRINCIPLE OF RELATIVITY

If we make no assumption about the initial position of the moving system and about the null-point of t^ then an additive constant is to be added to the right hand side.

We have now to show, that every ray of light moves in the moving system with a velocity c (when measured in the moving system), in case, as we have actually assumed, c is also the velocity in the stationary system ; for we have not as yet adduced any proof in support of the assump- tion that the j)rincip]e of relativity is reconcilable with the principle of constant light-velocity.

At a time T = ^ = i> let a spherical wave be sent out

' from the common origin of the two systems of co-ordinates,

and let it spread with a velocity c in the system K. If

{,c, y, z)y be a point reached by the wave, we have

with the aid of our transformation-equations, let us transform this equation, and we obtain by a sin^ple calculation,

Therefore the wave is propagated in the moving system with the same velocit}' e, and as a spherical wave.^ Therefore we show that the two principles are mutually reconcilable.

In the transformations we have go; an undetermined function <^ (?;), and wo now proceed to find it out.

Let us" introduce for this purpose a third co-ordinate system k' , which is set in motion relative to the system h, the motion being parallel to the ^-axis. Let the velocity of the origin be { — v). At the time t = Oy all the initial co-ordinate points coincide, and for t=j=y=zz = o, the time t' of the system k' =^o. We shtill say that {x y' z t') are the co-ordinates measured in the system k' ^ then by a

* Yxde Note 9.

ON THE ELECTRODYNAMICS OF MOVING BODIES 13

two-fold application of the transformation-equations, we obtain

x'=<f>\^v)/S(v)'($+vT)=4>(v)<l>(^v)x, etc.

Since the relations between (,(/, ^', z\ f), and (x, y, z, t) do not contain time explicitly, therefore K and k' are relatively at rest.

It appears that the systems K and ¥ are identical.

Let us now turn our attention to the part of the ^-axis between (^^—o,y] = o,t, = o), and (^=0, ry = l, ^=o). Let this piece of the ^-axis be covered with a rod moving with the velocity v relative to the system K and perpendicular to its axis ; — the ends of the rod having therefore the co-ordinates

Therefore the length of the rod measured in the system K is ~r7~Y For the system moving with velocity (—v),

we have on grounds of symmetry,

I I

cfi{v) <f>{—v)

l4 PRINCIPLE OF RELATIVITY /

§ 4. The physical significance of the equations

obtained concerning moving rigid

bodies and moving clocks.

Let us consider a rigid sphere {i.e.y one having a spherical figure when tested in the stationary system) of radius R which is at rest relative to the system (K), and whose centre coincides with the origin of ^ then the equa- tion of the surface of this sphere, which is moving with a velocity v relative to K, is ;

At time t = Oj the equation is expressed by means of (ar, y, Zy t,) as

'13

( Vi-^J

A rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition — when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes

K V 1--^, R, R.

•

Therefore the y and z dimensions of the sphere (there- fore of any figure also) do not appear to be modified by the motion, but the a^ dimension is shortened in the ratio

1 : \'^ 1 ; the shortening is the larger, the larger

is V. ¥oY v = c, all moving bodies, when considered from a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become

ON THE ELECTRODYNAMICS OF MOVING BODIES 15

meaningless ; in our theory c plays the part of infinite velocity.

It is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system.

Let us now consider that a clock which is lying at rest in the stationary sj'stem gives the time t^ and lying at rest relative to the moving system is capable of giving the time t ; suppose it to be placed at the origin of the moving system k, and to be so arranged that it gives the time r. How much does the clock gain, when viewed from the stationary system K ? We have,

1 / ^ \ -,

T= — zznzr I ^~"~2^ 15 ^^d x=.vty

...,■,=[._ V.-g

Therefore the clock loses by an amount ^-^ per second

of motion, to the second order of approximation.

From this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense explained in § 3 when viewed from the stationary system. Suppose the clock at A to be set in motion in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A will las: behind the clock

which had been all along at B by an amount ^t -g, where t is the time required for the journey.

16 PRINCIPLE OF RELATIVITY

We see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide.

If we assume that the result obtained for a polygonal line holds also for a curved line, we obtain the following law. If at A, there be two synchronous clocks, and if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in /^-seconds, then after arrival, the last men-

tioned clock will be behind the stationary one by \t ~

seconds. From this, we conclude that a clock placed at the equator must be slower by a very &mall amount than a similarly constructed clock which is placed at the pole, all other conditions being identical.

§ 5. Addition-Theorem of Velocities.

Let a point move in the system k (which moves with velocity v along the ^-axis of the system K) according to the equation

where w^ and lu are constants.

■n

It is required to find out the motion of the point relative to the system K. If we now introduce the system of equations in § 3 in the equation of motion of the point, we obtain

aj=_J t, y~ ,0=0.

i+_i 1+ «

c"" ' c2

ON THE ELECTRODYNAMICS OF MOVING BODIES 17

The law of parallelogram of velocities hold up to the first order of approximation. We can put

w and a = tan~^ - .

i.e.f a is put equal to the angle between the velocities v, and w. Then we have —

a -1

[(i'2+2i;2+2 vw cos a)— I "■ J I

-, . viv cos a

It should be noticed that v and 2v enter into the expression for velocity symmetrically, li 2v has the direction of the ^-axis of the nioving system,

1+ "^

From this equation, we see that by combining two velocities, each of which is smaller than c, we obtain a velocity which is always smaller than c. If we put v=c—Xj *and w—c~\y where x and A are each smaller than c,

IJ=c — 2c-x-A_ <^

It is also clear that the veloeitv of lis^ht c cannot be altered by adding to it a velocity smaller than c. For this ease,

U= -^±^ =c. 1+ '''

* Vide Note 12.

18 PRINCIPLE OF RELATIVITY

We have obtained the formula for U for the ease when V and tv have the same direction; it can also be obtained by combining two transformations according to section § 3. If in addition to the systems K, and k, we intro- duce the system k', of which the initial point moves parallel to the ^-axis with velocity 2v, then between the magnitudes, x, y^ z, t and the corresponding magnitudes of k', we obtain a system of equations, which differ from the equations in §3, only in the respect that in place of V, we shall have to write,

(.+.)/( 1+ ^'^ )

We see that such a parallel transformation forms a group.

We have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electro- dynamics.

II.-ELECTBOBYNAMICAL FART.

§ 6. Transformation of Maxwell's equations for

Pure Vacuum.

On the nature of the Electromotive Force caused hy motion

in a magnetic field.

The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that

\ |,[^'Y,^]=

9;c

92/

ON THE ELECTRODYNAMICS OF MOVING BODIES

and

-0 a-rf^''''^^=-

a.^

a a^

(1)

where [X, Y, Z] are the components of the electric force, L, M, N are the components of the magnetic force.

If we apply the transformations in §3 to these equa- tions, and if we refer the electromagnetic processes to the co-ordinate system moving with velocity v, we obtain,

i I- [X, AY- - N), 13(Z + "i M)] =

a^ a^

c c

and

1 a^

[L, (3(M+ ^IZ), «N

-1-Y)]

a_ a^

X y8(Y--N) i8(Z4- -M) c c

(2)

where /?:

vl — i'Vc'

The principle of Relativity requires that the Maxwell- Hertzian equations for pure vacuum shall hold also for the system k, if they hold for 'he system K, i.e., for the vectors of the electric and magnetic forces acting upon electric and magnetic masses in the moving system k,

PRINCIPLE Oi^ RELATIVITY

which are defined by their pondermotive reaction, the same equations hold, ... i.e. ...

1 9 c 'Qi

(X', Y', Z') ^

6^ 9^

I ■

C OT

6 6 6^

6^' dr; 94

... (3)

Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these sj^stems are equivalent to the Maxwell- Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor \l/ (^?), which depends only on v^ and is independent of {^y Vy L ''■)• Hence the relations,

[X', y, Z']=4' (v) [X, p (Y- ^'N), 13 (Z+ fM)],

c c

[h', M', X']=:.A W [L, /^ (M-f ^Z;, /3 (N- ^ Y)].

Then by reasoning similar to that followed in §(3), it can be shown that ^/^(^;) = l.

.-. [X\ r, Z'] = [X, p (Y- ^N), 13 (Z+ ^M)]

c c

[V, W, N'] = [L, 13 (M+ - Z), /3 (N- -^' Y)].

ON THE ELECTRODYNAMICS OF MOVING BODIES 21

For the interpretation of these equations, we make the followini^ remarks. Let us have a point-mass of electricity which is of magnitude unity in the stationary system K, i.e.f it exerts a unit force upon a similar quantity placed at a distance of 1 em. If this quantity of electricity be at rest in the stationary system, then the force acting upon it is equivalent to the vector (X, Y, Z) of electric force. But if the quantity of electricity be at rest relative to the moving system (at least for the moment considered), then the force acting upon it, and measured in the moving system is equivalent to the vector (X', Y', Z'). The first three of equations (1), ('Z), (3), can be expressed in the following way : — '

1. If a point-mass of electric unit pole moves in an electro-magnetic field, then besides the electric force, an electromotive force acts upon it, which, neglecting the numbers involving the second and higher powers of !;/(?, is equivalent to the vector-product of the velocity vector, and the magnetic force divided by the velocity of light (Old mode of expression).

2. If a point-mass of electric unit pole moves in an electro-magnetic field, then the force acting upon it is equivalent to the electric force existing at the position of the unit pole, which we obtain by the transformation of the field to a co-ordinate system which is at rest relative to the electric unit pole [New mode of expression].

Similar theorems hold with reference to the magnetic force. We see that in the theory developed the electro- magnetic force plays the part of an auxiliary concept, which owes its introduction in theory to the circumstance that the electric and magnetic forces possess no existence independent of the nature of motion of the co-ordinate system.

22 PRINCIPLE OF RELATIVITY

It is further clear that the assymetry mentioned in the introduction which oc-curs when we treat of the current excited by the relative motion of a magnet and a con- ductor disappears. Also the question about the seat of electromagnetic energy is seen to be without any meaning.

§ 7. Theory of Doppler's Principle and Aberration.

In the sj^stem K, at a great distance from the origin of co-ordinates, let there be a source of electrodynamic waves, which is represented with sufficient approximation in a part of space not containing the origin, by the equations : —

X=Xo sin ^ "] L=Lo sin <l> ^

Y=Yo sin $ y M=MoSin$ ^ ^=o>(^-^£±!!!:2^±!!!'|

Z = Zo sin ^ J N=No sin $ J

Here (X^, Yq, Zq) and (Lq, M^, Nq) are the vectors which determine the amplitudes of the train of waves, {Ij Mj n) are the direction-cosines of the wave-normal.

Let us now ask ourselves about the composition of these waves, when they are investigated by an observer at rest in a moving medium A- : — By applying the equations of transformation obtained in §6 for the electric and magnetic forces, and the equations of transformation obtained in § 3 for the co-ordinates, and time, we obtain immediately : —

X'=Xo sin ^' L' = Lo sin $'

Y' = i3/'Yo-.- No") sin<I>' M'=^ Cm.^+ ^ Z^\ sin ^'

Z' =:^/'Zo+-Mo') sin<3^' N'=/3 /" No-i' Yo") sin«l>',

ON THE ELECTRODYNAMICS OF MOVING BODIES

where

lv\

u)' = a)^(l-^) , l' =

n —

1 Iv

,(i-'H) ,a-%)

From the equation for w' it follows : — If an observer nioves with the velocity v relative to an infinitely distant source of light emitting waves of frequency v, in such a manner that the line joining the source of light and the observer makes an angle of $ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency v which is perceived by the observer is represented by the formula

l—cos^

This is l)op pier's principle for any velocity. If ^—oj then the equation takes the simple form

1 v\-s.

V =v

We see that — contrary to the usual conception — v=oo, for v = —c.

If $'=angle between the wave-normal (direction of the ray) in the moving system, and the line of motion of the observer, the equation for I' takes the form

cos$—

cos ^'=

1— -cos <l> c

24 PRINCIPLE OF RELATIVITY

This equation expresses the law of observation in its

most general form. If $= - , the equation takes the simple form

cos $ = — - .

We have still to investigate the , amplitude of the waves, which occur in these equations. If A and A' be the amplitudes in the stationarj' and the moving systems (either electrical or magnetic), we have

A'2=A'

j 1 — - cos <i> I

1- ^'

If $=o, this reduces to the simple form

1-'-!

A'*=A«

1+^

From these equations, it appears that for an observer, which moves with the velocity c towards the source of light, the source should appear infinitely intense.

§ 8. Transformation of the Energy of the Rays of Light. Theory of the Radiation-pressure on a perfect mirror.

A^ Since ^- is equal to the energy of light per unit

volume, we have to regard ^— - as the energy of light in

ON THE ELECTRODYNAMICS OF MOVING BODIES 25

A'" the moving system. -— would therefore denote the

ratio between the energies of a definite light-complex "measured when moving "" and ^^ measured when stationary/' the volumes of the light-complex measured in K and k being equal. Yet this is not the case. If /, w;,, n are the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface

(x — cUy + (y-cmty + (:-~cnfy =11^

which expands with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses, when regarded from the system ^, i.e., the energy of the light-complex relative to the system A;.

Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time T=0, the equation : —

If S=volume of the sphei-e, S'=volume of this ellipsoid, then a simple calculation shows that :

'JH

cos $

If E denotes the quantity of light energy measured in the stationary system, E' the quantity measured in the

PRINCIPLE OP RELATIVITY

moving system, which are enclosed by the surfaces mentioned above, then

A''

1— - cos $

If <l> = 0, we have the simple formula : —

E' E

1 +

It is to be noticed that the energy and the frequency of a light-complex vary according to the same law with the state of motion of the observer.

Let there be a perfectly reflecting mirror at the co-or- dinate-plane ^=0, from which the plane-wave considered in the last paragraph is reflected. Let us now ask ourselves about the light-pressure exerted on the reflecting surface and the direction, frequency, intensity of the light after reflexion.

Let the incident light be defined b}^ the magnitudes A cos ^, r (referred to the system K). Regarded from A-, we have the corresponding magnitudes :

1 — COR <J>

A' = A

J. 2

COS $ —

COS $' =

- COS 4>

1 — - COS 9

I c

V =V =.=rr:^

.\/ 1-^;

ON THE ELECTRODYJSAxMICS 0¥ AJOVlNG BODIES 27

For' the reflected light we obtain, when the process is referred to the system k : —

A" = A', cos $"= -cos *', v" = v'.

By means of a back-transformation to the stationary system, we obtain K, for the reflected light : —

1+ - cos $" 1-2 - cos ^ + — A'" = A" " =A ^ '-

^2 1 ^^

■N

V -s

C2 C^'

cos $'" =

cos4>" + "^ ("H- '^^ cos 4>-2 !^

C \ (''■'J c

1+ 1 ■.„ 1 — 2-cos$H

C COS $" c c^

1+ -cos<^" 1-2 H COS <^ 4-^

/ -S ( -I )'

The amount or energy falling upon the unit surface of the mirror per unit of time (measured in the stationary

system) is . The amount of energy going

STr{c cos ^—v)

away from unit surface of the mirror per unit of time is A'"V?7r {—c cos ^"+v). The difference of these two expressions is, according to the Energy principle, the amount of work exerted, by the pressure of light per unit of time. If we put this equal to P.?*, where P= pressure of light, we have

A 2

P = 2 —

(cos ^ - 0'

Hi)'

PKINCIPLE OF EEL.VHV1TY

i. »

As a first approximatioD^ we obtain

P=2 ^

bir

coa^ 4>.

which is in accordance with facts, and with other theories.

All problems of optics of moving bodies can be solved after the method used here. The essential point is, that the electric and magnetic forces of light, which are influenced by a moving body, should be transformed to a system of co-ordinates which is stationary relative to the body. In this way, every problem of the optics of moving bodies would be reduced to a series of problems of the optics of stationary bodies.

§ 9. Transformation of the Maxwell-Hertz Equations.

Let us start from the equations : —

PUx +

6x\ _aN 8M

7 dy

1/ _l9^\

a^i_aL

6 .'.' 6 y

1 6L 6Y 6Z

c dt 63

laM az

c dt dx.\

1 aN_ax

c dt dy d -v

)■

where p=%~ +2— + 4^?- , denotes 47r times the density

a.'= a^ a~

of electricity, and {u.,, Uy^ u.) are the velocity-components of electricity. If we now suppose that the electrical- masses are bound unchangeably to small, rigid bodies

ON THE IfiLECTHODYNAMlCS 01' MOVING BODIES ^9

(Ions, electrons), then these equations form the electrom^-j^- netic basis of Lorentz's electrodynamics and optics for moving bodies.

If these equations which hold in the system K, are transformed to the system k with the aid of the transfor- mation-equations given in § 3 and § 6, then we obtain the equations : —

where

Uc.

,ax'-i aN' ar J a^

aM' a^ '

a L' a Y' ar a^

az' a^?

,aY'-i aL' ar J dc

aN' a^ '

a M' a z' ar a^

ax'

Wc,

, az'-] aM' ar J a^

u^ — V

aL'

dv '

a N' a X' ar a^

aY' a^ '

,(i- ^^)

' 6X' aY'.dZ'

= %,"= 6?"*" 9^"*" a?

:^(l-t)'"

,(l-Fii.^)

"i,

Since the vector U. ic Hy ) is nothing but the

velocity of the electrical mass measured in the system A:, as can be easily seen from the addition-theorem of velocities in § 4 — so it is hereby shown, that by taking

30 PRINCIPLE 0¥ RELATIVITY

onr kinematical principle as the basis, the electromagnetic basis of Lorentz^s theory of electrodynamics of moving bodies correspond to the relativity-postulate. It can be briefly remarked here that the following important law follows easily from the equations developed in the present section : — if an electrically charged body moves in any manner in space, and if its charge does not change thereby, when regarded from a system moving along with it, then the charge remains constant even when it is regarded from the stationary system K.

§ 10. Dynamics of the Electron (slowly accelerated).

Let us suppose that a point-shaped particle, having the electrical charge e (to be called henceforth the electron) moves in the electromagnetic field ; we assume the following about its law of motion.

If the electron be at rest at any definite epoch, then in the next "particle of time,^^ the motion takes place according to the equations

df" dt^ df"

Where (.r, ^, z) are the co-ordinates of the electron, and m is its mass. •

Let the electron possess the velocity z; at a certain epoch of time. Let us now investigate the laws according to which the electron will move in the ^particle of time ^

immediately following this epoch.

Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering,

ON THE ELECTRODYNAMICS OF MOVING BODIES 31

the electron is at the origin o£ co-ordinates^ and moves with the velocity v along the X-axis of the system. It is clfear that at this moment (^ = 0) the .electron is at rest relative to the system A-, which moves parallel to the X-axis with the constant velocity v.

From the suppositions made above, in combination

with the principle of relativity, it is clear that regarded

from the system k, the electron moves according to the equations

dr^ dT^ ' dT""

in the time immediately following the moment, where the symbols (^, 77, I, t, X', Y', Z') refer to the system A'. If we now fix, tliat for t—v = y = z=^0, T = ^=:r; = ^=0, then the equations of transformation given in 3 (and 6) hold, and we have :

With the aid of these equations, we can transform the above equations of motion from the system A- to the system K, and obtain : —

dt^ m ^3 ■' di'' m ft \ c )

(A)

= 1 i(z+rM)

m B \ c 7

dt^ m /5

PRINCIPLE OF RELATIVITY

Let US now consider, following the usual method of treatment, the longitudinal and transversal mass of a moving electron. We write the equations (A) in the form

myS'

d\c

dt''

■.eX = eX'

m/S' ^4-^ =e/3

dt^

- '^] =^Y' y

mp' ^; =e/3 rZ+ ^' mJ =eZ'

and let us first remark, that ^X', eY', eZ' are the com- ponents of the ponderomotive force acting upon the electron, and are considered in a moving system which, at this moment, moves with a velocity which is equal to that of the electron. This force can, for example, be measured by means of a spring-balance which is at rest in this last system. If we briefly call this force as ^^the force acting upon the electron," and maintain the equation : —

Mass-number x acceleration-number=force-number, and if we further -fix that the accelerations are measured in the stationary system K, then from the above equations, we obtain : —

Longitudinal mass =

( V'- %y

Transversal mass =

■m

V^- %

Naturally, when other definitions are given of the force and the acceleration, other numlers are obtained for the

* Vide Note 21.

ON THE ELECTRODYNAMICS OF MOVING BODIES 38

mass ; hence we see that we must proceed very carefully in comparing the different theories of the motion of the electron.

We remark that this result about the mass hold also for ponderable material mass ; for in our sense, a ponder- able material point may be made into an electron by the addition of an electrical charo^e which mav be as small as possible.

Let us now determine the kinetic energy of the electron. If the electron moves from the origin of co-or- dinates of the system K with the initial velocity 0 steadily along the X-axis under the action of an electromotive force X, then it is clear that the energy drawn from the electrostatic field has the value SelLd>\ Since the electron is only slowly accelerated, and in consequence, no energy is given out in the form of radiation, therefore the energy drawn from the electro-static field may be put equal to the energy W of motion. Considering the whole process of motion in questions, the first of equations A) holds, we obtain : —

0 V c^

For v=c, W is infinitely great. As our former result shows, velocities exceeding that of light can have no possibility of existence.

In consequence of the arguments mentioned above, this expression for kinetic energy must also hold .for ponderable masses.

We can now enumerate the characteristics of the motion of the electrons available for experimental verifica- tion, which follow from equations A).

34 PRINCIPLE OF RELATIVITY

1. From the second of equations A) ; it follows that an electrical force Y, and a magnetic force N produce equal deflexions of an electron moving with the velocity

V, when Y= — . Therefore we see that according to

our theory, it is possible to obtain the velocity of an electron from the ratio of the magnetic deflexion Am, and the electric deflexion A^, by applying the law : —

^ =- . A, c

This relation can be tested by means of experiments

because the velocity of the electron can be directly

measured by means of rapidly oscillating electric and mag:netic fields.

%. From the value which is deduced for the kinetic energy of the electron, it follows that when the electron falls through a potential difference of P, the velocity v which is acquired is given by the following relation : —

3. We calculate the radius of curvature R of the path, where the only deflecting force is a magnetic force N acting perpendicular to the velocity of projection. From the second of equations A) we obtain :

«N

These three relations are complete expressions for the law of motion of the electron according to the above theory.

ALBRECHT EINSTEIN

[^ short hiograpJiical note.~\

The name of Prof. Albreelit Einstein has now spread far beyond the narrow pale of scientific investigators owing to the brilliant confirmation of his predicted deflection of liojht-ravs bv the ^gravitational field of the sun durins: the total solar eclipse of May 29, 1919. But to the serious student of science, he has been known from the beffinnino* of the current century, and many dark problems in physics has been illuminated with the lustre of his genius, before, owing to the latest sensation just mentioned, he flashes out before public imagination as a scientific star of the first magnitude.

Einstein is a Swiss-German of Jewish extraction, and began his scientific career as a privat-dozent in the Swiss University of ZUrich about the year 1902. Later on, he migrated to the German Universitv of Prague in Bohemia as ausser-ordentliche (or associate) Professor. In 1914, through the exertions of Prof. M. Planck of the Berlin University, he was appointed a paid member of the Koyal (now National) Prussian Academy of Sciences, on a salary of 18^000 marks per year. In this post, he has only to do and guide research work. Another distinguished occupant of the same post was Van't Hoff, the eminent physical chemist.

It is rather difficult to give a detailed, and consistent chronological account of his scientific activities, — they are so variegated, and cover such a wide field. The. first work which sjained him distinction was an investiscation on Brownian Movement. An admirable account will be found in Perrin's book ^The Atoms.' Starting from Boltzmann's

36 PRINCIPLE OF RELATIVITY

theorem connecting the entropy, and the probability of a state, he deduced a formula on the mean displacement of small particles (colloidal) suspended in a liquid. This formula gives us one of the best methods for finding out a very fundamental number in physics — namely — the number of molecules in one gm. molecule of gas (Avogadro's number). The formula was shortly afterwards verified by Perrin, Prof, of Chemical Physics in the Sorboniie, Paris.

To Einstein is also due the resusciation of Planck's quantum theory of energy-emission. This theory has not yet caught the popular imagination to the same extent as the new theory of Time, and Space, but it is none the less iconoclastic in its scope as far as classical concepts are concerned. It was known for a long time that the observed emission of light from a heated black body did not corrospond to the formula which could be deduced from the older classical theories of continuous emission and propagation. In the year 1900, Prof. Planck of the Berlin University worked out a formula which was based on the bold assumption that energy was emitted and absorbed by the molecules in multiples of the quantity hv^ where // is a constant (which is universal like the constant of gravitation), and v is the frequency of the light.

The conception was so radically different from all accepted theories that in spite of the great success of Planck's radiation formula in explaining the observed facts of black-body radiation, it did not meet with much favour from the physicists. In fact, some one remarked jocularly that according to Planck, energy flies out of a radiator like a swarm of gnats.

But Einstein found a support for the new-born concept in another direction. It was known that if green or ultraviolet light was allowed to fall on a plate of some alkali metal, the plate lost electrons. The electrons were emitted with

ALBERT EINSTEIN 37

all velocities, but there is generally a maximum limit. From the investigations of Lenard and Ladenburg, the curious discovery was made that this maximum velocity of emission did not at all depend upon the intensity of light, but upon its wavelength. The more violet was the light, the greater was the velocity of emission.

To account for this fact, Einstein made the bold assumption that the light is propogated in space as a unit pulse (he calls it a Light-cell), and falHng upon an individual atom, liberates electrons according to the energy equation

hv=-;^mv^ -\- A,

where (iu, v) are the mass and velocity of the electron. A is a constant characteristic of the metal plate.

There was little material for the confirmation of this law when it was first proposed (1905), and eleven years elapsed before Prof. Millikan established, by a set of experiments scarcely rivalled for the ingenuity, skill, and care displayed, the absolute truth of the law. As results of this confirmation, and other brilliant triumphs, the quantum law is now regarded as a fundamental law of Energetics. In recent years, X-rays have been added to the domain of light, and in this direction also, Einstein's photo-electric formula has proved to be one of the most fruitful conceptions in Physics.

The quantum law was next extended by Einstein to the problems of decrease of specific heat at low temperature, and here also his theory was confirmed in a brilliant manner.

We pass over his other contributions to the equation of state, to the problems of null-point energy, and photo- chemical reactions. The recent experimental works of

38 PRINCIPLE OF HELATIVITT

Nernst and Warburg seem to indicate that through Einstein's genius, we are probably for the first time having a satisfactory theory of photo-chemical action.

In 1915, Einstein made an excursion into Experimental Physics, and here also, in his characteristic way, he tackled one of the most fundamental concepts of Physics. It is well-known that according to Ampere, the magnetisation of iron and iron-like bodies, when placed within a coil carrying an electric current is due to the excitation in the metal of small electrical circuits. But the conception though a very fruitful one, long remained without a trace of experimental proof, though after the discovery of the electron, it was srenerallv believed that these molecular currents may be due to the rotational motion of free electrons within the metal. It is easily seen that if in the process of magnetisation, a number of electrons be set into rotatory motion, then these will impart to the metal itself a turning couple. The experiment is a rather difficult one, and many physicists tried in vain to observe the effect. But in collaboration with de Haas, Einstein planned and successfully carried out this experiment, and proved the essential correctness of Ampere's views.

Einstein's studies on Relativity were commenced in the year 1905, and has been continued up to the present time. The first paper in the present collection forms Einstein's first great contribution to the Principle of Special Relativity. We have recounted in the introduction how out of the chaos and disorder into which the electrodynamics and optics of moving bodies had fallen previous to 1895, Lorentz, Einstein and Minkowski have succeeded in building up a consistent, and fruitful new theory of Time and Space.

But Einstein was not satisfied with the study of the special problem of Relativity for uniform motion, but

ALBERT EINSTEIN 39

tried, in a series of papers beginning from 1911, to extend it to the case of non-uniform motion. The last paper in the present collection is a translation of a comprehensive article which he contributed to the Anualen der Physik in 1916 on this subject, and gives, in his own words, the Principles of Generalized Kelativity. The triumphs of this theory are now mat<^ers of public knowledge.

Einstein is now only 45, and it is to be hoped that science will continue to be enriched, for a long time to come, with farther achievements of his genius.

INTRODUCTION.

At the present time, different opinions are being held about the fundamental equations of Eleetro-dynamics for moving" bodies. The Hertzian^ forms must be given up, for it has appeared that they are contrary to many experi- mental results.

In 1895 H. A. Lorentzf published his theory of optical and electrical phenomena in moving bodies; this theory was based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been ushered into existence. In his theory, Lorentz proceeds from certain equations, which must hold at every point of ^'Ather'^; then by forming the average values over *^^ Phy- sically infinitely small " regions, which how^ever contain large numbers of electrons, the equations for electro-mag- netic processes in moving bodies can be successfully built up.

In particular, Lorentz's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous " Ather ^' ; it shows that this fact is intimately connected with the covariance of the original equation, when certain simultaneous transformations of the space and time co-ordinates are effected; these transfoi;mations have therefore obtained from H. PoincareJ the name of Lorentz- transformations. The covariance of these fundamental equations, when subjected to tbe Lorentz-transformation is a purely mathematical fact i.e. not based on any physi- cal considerations; I will call this the Theorem of Rela- tivity ; this theorem rests essentially on the form of the

* Vid,e Note 1. f Note 2. % Vide Note 3.

3 PRINCIPLE OF RELATIVITY

differential equations for the propagation of waves with the velocity of light.

Now without recognizing any hypothesis about the con- nection between " Ather " and matter, we can expect these mathematically evident theorems to have their consequences so far extended — 'that thereby even those laws of ponder- able media which are yet unknown may anj^how possess this covariance when subjected to a Lorentz-transformation ; by saying this, we do not indeed express an opinion, but rather a conviction, — and this conviction I may be permit- ted to call the Postulate of Relativity. The position of affairs here is almost the same as when the Principle of Conservation of Energy was poslutated in cases, where the corresponding forms of energy were unknown.

Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite con- nection may be styled ' the Principle of Relativity.'

These differentiations seem to me to be necessary for enabling us to characterise the present day position of the electro-dynamics for moving bodies.

H. A. Lorentz"^ has found out the " Relativity theorem'' and has created the Relativitj^-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.

A. Einstein t has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept which is forced upon us by observation of natural pheno- mena.

The Principle of Relativity has not yet been formu- lated for electro-dvnamics of moviug: bodies in the sense

* Yiie Note 4. f Note 5.

INTRODUCTION 3

characterized by me. "In the present essay, while formu- lating- this principle, I shall obtain the fundamental equa- tions for moving bodies in a sense which is uniquely deter- mined by this principle.

But it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle."^

We would at first expect that the fundamental equa- tions which are assumed by Lorentz for moving bodies would correspond to the Relativity Principle. But it will be shown that this is not the case for the general equations which Lorentz has for any possible, and also for magnetic bodies ; but this is approximately the case (if neglect the square of the velocity of matter in comparison to the velocity of light) for those equations which Lorentz here- after infers for non-magnetic bodies. But this latter accordance with the Relativity Principle is due to the fact that the condition of non-mag^netisation has been formula- ted in a way not corresponding to the Relativity Principle; therefore the accordance is due to the fortuitous compensa- tion of two contradictions to the Relalivity-Postulate. But meanwhile enunciation of the Principle in a rigid manner does not signify any contradiction to the hypotheses of Lorentz's molecular theory, but it shall become clear that the assumption of the contraction of the electron in Lorentz^s theory must be introduced* at an earlier stage than Lorentz has actually dene.

In an appendix, I have gone into discussion of the position of Classical Mechanics with respect to the Relativity Postulate. Any easily perceivable modification of mechanics for satisfying the requirements of the Relativity theory would hardly afford any noticeable difference in observable processes ; but would lead to rery

* See uQtes on § S and 10.

4' PRINCIPLE OF RELATIVITY

surprising consequences. By laying down the Relativity- Postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of Energy alone (the form of the Energy being given in explicit forms).

NOTATIONS.

Let a rectangular system {.r, y, z, t,) of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the unit of length that the velocity of light in space becomes unity.

Although I would prefer not to change the notations used by Lorentz^ it appears important to me to use a different selection of symbols, for thereby certain homo- geneity will appear from the very beginning. I shall denote the vector electric force by E,' the magnetic induction by M_, the electric induction by e and the magnetic force by 7n, so that (E, M, »?, m) are used instead of Lorentz's (E, B, D, H) respectively.

I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, i.e., instead of operating with {t), I shall operate with {it), where i denotes ^ — \. If now instead of {x, y, z, it), I use the method of writing with indices, certain essential circumstances will come into evidence ; on this will be based a general use of the suffixes (1, 2, 3, ^). The advantage of this method will be, as I expresslj' emphasize here, that we shall have to handle symbols which have apparently a purely real appearance ; we can however at any moment pass to real equations if it is understood that of the symlbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those

NOTATIONS 0

which have not at all the suffix 4, or have it twice denote real quantities.

An individual system of values of {x, y, Zy t) i. e.^ of {x^ x^ rg Xj^) shall be called a space-time point.

Further let u denote the velocity vector of matter, e the dielectric constant, /u, the magnetic permeability, a- the conductivity of matter, while p denotes the density of electricity in space, and s the vector of "Electric Current" which we shall some across in §7 and §8.

5 PRINCIPLE OP RELATIVITY

PAET I § 2.

The Limiting Case.

The Fundcwiental Equations for Ather.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-d3mamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limitting case e=i, ix=1,(t = o, they should constitute the laws for ponderable bodies. In this ideal limitting case €=1, fji=l, o-=:o, E will be equal to e, and M to m. At every space time point {j-, y^ z, t) we shall have the equations*

(i) Curl m— -»- = pu

(ii) div e= p

(iii) Curl^ +.||' = 0

(iv) div m = (?

I shall now write {x^ x^ x^ x ^) for {x^y, z, t) and

(/>nP2; ^3; P4) for

(pu,, puy, pu,, ip)

i.e. the components of the convection current pu, and the electric density multiplied by \/— 1.

Further I shall write «

for

m,, m^, m,, — ie„ — ie , — ie,.

i.c.y the components of m and ( — i.e.) along the three axes; now if we take any two indices (h. k) out of the series

* See note 9

THE FUNDAMENTAL EQUATIONS FOR ATHER

Therefore

/s 2 ^^ ~'J 1 3 > ./ 1 3 ~ ~~J Z \i J 2 1^^ ~/ 1 2 ..4 1 — ~Jl 45 ../ 4 4 — ~/2 4J /4 3 " ""/ 3 4

Then the three equations comprised in (i), and the equation (ii) multiplied by / becomes

8Xc

3xj

¥.

g/4t . ?/

Sxj

Sx,

0/l3 8X3

S/t4 8X4

?^2 3 8X3

S/24 8X4

0/34 ^^4

?A3

8x,

■Pi

= P2

= P;

= ^4

(A)

On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by {i) becomes

¥,,

8xj

?A4

^^4 2 , ?/2_3

8X3 8X4

^14 , ?Al

Sx„ ^

+ -

S/4,

8x,

8x4

?/l2

^3 2 , ?/j_3_ , SXi "^ 8x2 "^

8/*.

2 1

8x,

= 0

(B)

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices. (1,2,3,4).

§ 3.

It is well-known that by writing the equations i) tc iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the

8 PRINCIPLE OF RELATIVITY

system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis. through an amount <f>, keeping e, m fixed in space, and introduce new variables x^', cc^ x^ Xi^ instead of X:^ x^ x^ x ^, where

x\ •=^x^ cos <^ H-^2 sin ^, ;r'2 = — ^i sin<^ + x^ cos<^, jr' ^ =Xqx\= x^, and introduce magnitudes p\, p\j p s p\, where p^' = p^ cos i> -i- P2 sin<^, p^' = — p^ sin^ + p2 cos<^ *nd/i2, 7^3 4, where

/% 3 =A 3 cos (^ + /g 1 sin <!>,/. 1 r: -/j 3 sin <^ + /'i4=/i4 COS <^ +/24 sin ct>,/\^ - -/,4 sitt <t> -f

/2 4 COS <f>,,/\^=/s4y

fu. = -/.A (hlk = 1,2,3,4).

then out of the equations (A) would follow a corres- ponding system of dashed equations (A') composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equa- tions (A) and (B) concerning the sitffiies (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by «V^ a purely imaginary magnitude, and consider the substitution

^i—^\i ^s'=*2> ^^^' = xz cos i\if-\-x^ sin iyj/, (1)

^^4' = — ic, sin ixjf 4- .^4 cos i\^,

Putting - i tan i^^ = '\^ "^ _^ = ^' ^^ = 9 ^og jz^r (2)

(? -f ^

THE FUNDAMENTAL EQUATIONS FOR ARTHER 9

We shall have cos i\\/ = — , sin z^ = — ■

^l-q^ x/l-q

where — i < q < \, and \/l— ^^ is always to be taken with the positive sign.

Let us now write x\=-/j ^o 2=^^' , x ^=z'y x\^=it' (3) then the substitution 1) takes the form

^ =.r, y =y,z ^ , t = , (4)

the coefficients being essentially real.

If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1^ 2, and <f> by i^, we at once perceive that simultaneously, new magnitudes p\, p'2, p 3, p' 4, where

{p\=Pi, P2=P2^ P3=P3 cos ii}/ + P4 sin iif/, p\ =

— Pg sin t\l/ + P4 cos iij/), and/ 12 •••/34. where /4i=/4i cos ^^A +/13 sin ixlf,f\^= -/41 sin «V +/13

e0StlA,/3 4=/3 4,/3 2=/3 2 COS /l/^ 4-/42 siu t'l/^, /42 = -/32 sin ^> + /42 COS ?lA, /12 =/i2^ /*A = -fkky

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A'), and (B'), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are t^en, and ^' y' z' t' be takes as a new frame of reference, then we shall have

(5)

■qu^ +1

p =p — • \ , P^^r -p \ ^ZIZZIIl

p'uj=pu^, p'uy'=pUy.

10 PRINCIPLE OF RELATIVITY

(6) ^j = ?i^i^, ,„V = 2^4^, e.'=e

» '

(7) w',' = ■, e'/ = , m','=m

z •

VI — q^ VI — q""

Then we have for these newly introduced vectors tc', e', m' (with components %ij , uj , uj \ ej , ^/, ej ) mj, m/, m/)y and the quantity p a series of equations I'), II'), III'), IV) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that e^—qmy, ey+qm^ are components of the vector e-\- \_vm'\, where v is a vector in the direction of the positive Z-axis, and i v i=^, and [vfu'] is the vector product of y and W2 ; similarly —qe^-\-myym,,+qey are the components of the vector m—\ye].

The equations 6) and 7), as they stand in pairs, can be expressed as. -

eJ-\-i'ni'J=.{e^+im^) cos i\^ + {Cy+imy) sin ix^/,

Sy' + im'y' = — (e^+zw,) sin ii(/ + (gy+imy) cos lij/,

If (^ denotes any other real angle, we can form the following combinations : —

{eJ + im'J) cos. ^+(ey" + zWy') sin <;^

= (e,+/w,) cos. (ct> + i^) + (ey+imy) sin ((j^ + iif/),

= (e,' + zW,') sin ^+(ey' + zWy') cos. ^

= — (e:.^-^mJ sin (cfi + iif/) + (ey-\-zmy) cos, (cf> + {\ff).

Special libnENTZ Transformation.

The role which is played by the Z-axis in the transfor- mation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation

SPECIAL LORENTZ TRANSFORMATION ll

about this last axis. So we came to a more general law : —

Let ?; be a vector with the components v^, Vy, v^, and let \ v \ =q<l. By v we shall denote any vector which is perpendicular to v, and by i\, r^ we shall denote components o£ r in direction of^ and v.

Instead of {x, y^ z, t), new magnetudes {x' ij z t') will be introduced in the following way. If for the sake of shortness, r is written for the vector with the components {x, y, z) in the first system of reference, r' for the same vector with the components (x' y' z) in the second system of reference, then for the direction of Vy we have

and for the perpendicular direction i"), (11) r^ = r^

and further (12) \! = ~f ^ "^/ .

V 1 — q^

The notations (rV, ^\>) are to be understood in the sense that with the directions v, and every direction v perpendi- cular to V in the system {x, y, z) are always associated the directions with the same direction cosines in the system [x' y, z),

A transformation which is accomplished by means of (10), (11), (12) with the condition 0<^<1 will be called a special Lorentz-transformation. We shall call v the vector, the direction of v the axis, and the magnitude of V the moment of this transformation.

If further p and the vectors w', e' , in, in the system {xy'z) are so defined that,

12 PRINCIPLE OF RELATIVITY

further

(14) (/ + m')^ = ^^ + ''"'^-i^^^ + "'^K

Vl — q"

(15) {e' 4- iffi'') » = (^ + ^'^^) — i [u, {e + ini)] ^ .

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

(U\ r -!Ljl±1!i_ r- =/- t= TL^±L^ V \.—q- Vl — q^

Now we shall make a very important observation

about the vectors u and u. We can again introduce

the indices 1, 2, 8, 4, so that we write (^/, ^^2? ^3? *^*'4 0

instead of (,u', ?/'? -') ^'^') a^nd p^', pg'? Ps'? P4' ii^stead of

Like the rotation round the Z-axis, the transformation (4), and more geaeraily the transformations (10), (1 1), (12), are also linear transformations with the determinant -|-1, so that

(17) x^^+x^^+x^^+x^"" i. e. x^ + y''+z^—t'',

is transformed into

On the basis of the equations (13), (14), we shall have (p,'+P,'+P,'+P,'')=pHl-u^\-u,\-ur^,)=p'a-u') transformed into p^(l — u^) or in other words,

(18) p vr^r:i?

is an invariant in a Lorentz-transformation.

If we divide (p^, p^, P3, p^) by this magnitude, we obtain

the four values (w^, co,, w,, w^^) = . _ {u^, u^, u^, i)

so that Wi' +(u,^ +W3' 4-W4* = — 1.

It ■'is apparent that these four values, are determined by the vector 10 and inversely the vector it of magnitude

SPECIAL LOEENTZ TRANSFORMATION 13

<i follows from the 4 values co^, 0)3, 003, w^ ; where (oji, W2J (^3) ai'6 real, — ^'0)4 real and positive and condition (19) is fulfilled.

The meaning of (m^, Wg, 0)3, Wa) here is, that they are the ratios of da\, dx^, d • ^, d,c^ to

(20) V—{^clc^^ + dx^ '-^ + d.c3 2 + dx^ =^ =dt Vl — u\

The differentials donoting the displacements of matter

occupying the spacetime point (.f.^, .i^g^ -^'3; '^u) ^^ ^^le

adjacent space-time point.

After the Lorentz-transfornation is accomplished the

voeocity of matter in the new system of reference for the

same space-time point (u' y -J t') is the vector tt' with the

,. dx' dy' dz' dV ^^^^"^ -dt'^lU'^li'^ d^'^""' components.

Now it is quite apparent that the system of values

X^—Oi^, ■f'2=<^25 aJ3=W3J '^'4=W

<•

is transformed into the values

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity 71^' after the transformation as the first system of values has o:ot for it before transformation.

If in particular the vector v of the special Lorentz- transformation be equal to the velecity vector u of matter at the space-time point {x^, x^, ;«3, x^) then it follows out of (10), (11), (12) that

Under these' circumstances therefore, the corresponding space-time point has the velocity v' = 0 after the trans- formation, it is as if we transform to rest. We may call the invariant p ^/l — u^ as the rest-density of Electricity.^

* See Note.

PRINCIPLE OF RELATIVITY

§ 5. Space-time Vectors.

Of the 1st and 2nd kind.

I£ we take the priucipal result of the Lorentz traDsfor- mation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it ma}^ be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magni- tudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

■r,.

V.^4V

1 1

2 1

3 1

s ^41

1 2

2 2

1 4

2 4

4 4.^

■V

the Determinant of the matrix is +1, all co-efficients with- out the index 4 occurring once are real, while a^^, <^^2i 043, are purely imaginary, but a 4^^ is real and >o, and ^1^ +'^2" + ^"3^ +-^4^ transforms into x^'^ +x^'- -{- ,v.^"^ -\-x^"^. The operation shall be called a general Lorentz transformation.

If we put aj/=:,c', x^' =y\ ,v^' = z\ x^=^it\ then immediately there occurs a homogeneous linear transfor- mation of («, y, z, t) to (r', y' y z y t') with essentially real co-efficients, whereby the aggregrate — c^ — ^2 _~2 _|_^2 transforms into — ^'f ^ — y' ^ — z"^ -\- 1"^ , and to every such systetn of values ■», y, Zy t with a positive t, for which this aggregate >o, there always corresponds a positive t' ;

This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski's notation,

SPACE-TIME VECTORS I^

this last is quite evident from the continuity of the aggregate x, y, z, t.

The last vertical column of co-efficients has to fulfil,

the condition 22) <^i 4^+^24^ +^34^ +'^4 4^ = 1.

If «^^=<3^2^=<X3 4=0^ then (244 = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If «^4, ^2 4? ^^s4 ^^'® ",^^ ^^^ zoro, and if we put

^ X 4t • ^24 • ^3 4 • ^^44""^! • ^y • ^s • ^

q=-\/v.^-\-Vy'^v,' <1. On the other hand, with every set of value of ^14^ ^24J ^34' ^44 w^iich in this way fulfil the condition 22) with real values of ^^, Vy, v,, we can construct the special Lorentz transformation (L6) with (^1 4, ^245 ^3 4> ^^44) as the last vertical column, — and then every Lorentz- transformation with the same last vertical column

(^14^ <^2 4? ^^3 4' '^44) ^^^ ^® supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes p^, p^, p^, p^) with the coiidition that in case of a Lorentz-transformation it is to be replaced by the set p/, 132', ps\ pA:')i where thes3 are tho value? oO ^c/, v.^\ ,c^', -^iO' obtained by substituting (p^, p}, p.^, p ) for (^-j, x-^, .Vq, ,^4) in the expression (21).

Besides the time-space vector of the 1st kind (x^, x^i Xqj v-^) we shall also make use of another space- time vector of the first kind (y^, ^.^,^3, ^4), and let us form the linear combination ^

023) Aa C*^2 2/3— ''3 2/2)+/si (^3 2/1— ^ 2/3)+ /l2 (^1 2/2— '^z 2/1)+ /li (■^■1 2/4— ^'«4. 2/x) + /24 (''a 2/4—^^4 2/2) + /s* (-''s 2/4—^4 2/3)

16 PRINCIPLE OF RELATIVITY

with six coefficients /g 3 — f^^. Let us remark that in the vectorial method o£ writing, this can be constructed out of the four vectors.

the constants x^ and y^^ at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject {x^, .c^, ,83, x^) and (2/1, y^, y^, yj simul- taneously to the Lorentz transformation (^21), the combina- tion (23) is changed to.

(24) f^s' ('''2 ys'-'^s y^) +/31 (^3' 2/i'--^i'!/3)+/i2

(^.' yJ-^^Jy.') +frJ(^.yJ)-H'y.') +/2.' i'^^' yJ - ''4' 2/2') + /s/ ('^3' yJ—-^J 2/3'),

where the coefficients As'^ /a i^ /12'' /i*'? /24'r /s*'. depend solely on (/g 3 /a 4) and the coefficients a^^...a^^.

We shall define a space-time Vector of the 2nd kind

as a system of six-magnitudes /"^ 3 j/si fziJ with the

condition that when subjected to a Lorentz transformation, it is changed to a new system /^ 3' /"g^,... which satis- fies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I) — (iv), — which are the fundamental equations for Ather.

If ,«, y, z, it (space co-ordinates, and time it) is sub- jected to a Lorentz transformation, and at the same time {pu^^ pUy, pu,, ip) (convection-current, and chnrge density pi) is transformed as a space time vector of the 1st kind, further {m^^ 711^, 1^ ,-, — i(i ^^—ie y^ — ie ,) (magnetic force, and electric induction x (— 0 is transformed as a space time vector of the 2nd kind, then the system of equations (1), (II), and the system of equations (III), • (IV) trans- forms into essentially corresponding relations between the corresponding magnitudes newly introduced info the system.

SPECIAL LORENTZ TRANSFORMATION 17

These facts can be more concisely exj^ressed in these words : the system of equations (I, and II) as well as the system of equations (III) (IV) are co variant in all cases of Lorentz-transformation, where (p?^, ip) is to be trans- formed as a space time vector of the 1st kind, {m—ie) is to be treated as a vector of the 2nd kind, or more significantly, —

(pfi, ip) is a space time vector of the 1st kind, {vt—ie)^ is a space-time vector of the 2nd kind.

I shall add a fe ,v more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.

(0 ^^'-e' = f.l + f,\ + f.\ + /xl + /L + /.I

A space-time vector of the second kind (m—ie), where {tn, and e) are real magnitudes, may be called singular, when the scalar square Qni—ieY =o, ie m^ —e"^ =o, and at the same time (?;^ <?)=o, ie the vector ?;iand e are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector ol the 2nd kind in every Lorentz-transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product \jne] coincides with the Z-axis, i.e. m,, = o, e^=o. Then

{m,, — i e,y -\-{7n,,--i e^y=^o,

Therefore {e^+i m^,)/(e,-\-i e^) is different from +i, and we can therefore define a complex argument <^ + tV) in such a manner that

tan(</>-fiV)=?iL±t^v^

Vide Note.

18 • PRINCIPLE OF RELATIVITY

If then, by referring back to equations (9), we carry out the transformation (1) through the angle ^j and a subsequent rotation round the Z-axis through tbe angle <^, we perform a Lorentz-transformation at the end of which ;;^^=o_, ey=o, and therefore m and e shall both coincide with the new Z-axis. Then by means of the invariants m'^—e^, [me) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ Concept op Time.

By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.^ In fact, the following considerations will prove conclusive.

Let a reference system {x,y, z, f^ for space time points (events) be somehow known. Now if a space point A {'^'tiVof ^o) ^^ the time t„ be compared with a space point P ( f, ^, z) at the time fy and if the difference of time t—t^, (let t > to) be less than the length A P i.e. less than the time required for the propogation of light from

* Just as being.s which, are confined within a narrow region surrovinding a point on a shperical surface, may fall into the error that a sphere is a geometric figure in which oue diameter is particularly distinguished from the rest.

CONCEPT OF TIME 19

A to P, and if ^= " < 1, then by a special Lorentz

transformation, in which A P is taken as the axis_, and which has the moment^, we can introduce a time parameter t\ which (see equation 11, 12, § 4) has got the same value t' = o for both space-time points (A, t^), and P, t). So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time t„ =o, two different space-points A, B, or three space-points (A, B, C) which are not in the same space-line, and compare therewith a space point P, which is outside the line A B, or the plane A B C^ at another time t, and let the time difference t — t^ (t > t^) be less than the time which light requires for propogation from the line A B, or the plane A B 0) to P. Let q be the quotient of {t — to) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C is the axis, and q is the moment, then all the three (or four) events (A, to), [B, to), (C, t,) and (P, t) are simultaneous.

If four space-points, which do not lie in one plane are conceived to be at the same time to, then it is no longer per- missible to make a change of the time parameter by a Lorentz — transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures ot the so-called non-Euclidean Geometr^y, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.

^0 PRINCIPLE OF RELATIVITY

PART II. ELECTRO-MAGNETIC .. PHENOMENA.

§ 7. Fundamental Equations for bodies

AT REST.

After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case « = 1, /a = 1, o- = o, let us turn to the electro-magnatic phenomena in matter. We look for those relations which make it possible for

us when proper fundamental data are given — to

obtain the following quantities at every place and time, and therefore at every space- time point as functions of {r, y, z, t) : — the vector of the electric force E, the magnetic induction M, the electrical induction <?, the magnetic force /«, the electrical space-density p, the electric current s (whose relation hereafter to the conduc- tion current is known by the manner in which conduc- tivity occurs in the process), and lastly the vector w, the velocity of matter.

The relations in question can be divided into two classes.

Firstly — those equations, which, — when v, the velocity of matter is given as a function of (r, i/, ~, t), — lead us to a knowledge of other magnitude as functions of x, y, r, t — I shall call this first class of equations the fundamental equations —

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector u as functions of a-, y, ~, t).

For the case of bodies at rest, i.e. when u {x, y, z, t) = 0 the theories of Maxwell (Heaviside, Hertz) and

FUNDAMENTAL EQUATIONS FOE BODIES AT REST 21

Loreutz lead to the same fundamental equations. They are ; —

(1) The Differential Equations : — which contain no constant referring to matter : —

(i) Curl m — — r— = C, (u) div e =]p.

{Hi) Curl E -f ^ = o, (tr) Div M = o.

(2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves i.e. for isotopic bodies ; — they are com- prised in the equations

(V) e = € E, M = />iw, C = crE,

where c = dielectric constant, /x = magnetic permeability, (T = the conductivity of matter, all given as function of '*■> ^j 2^> ^J ^ is here the conduction current. ,

By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,

and write ^j, s^^ s^, s^ for C,, C^, C, V _ 1 p, • further/23,/5i,/i,,/i4,/„4,/54 for m,, Wy, m, — i (e., e^, e,), and F33, E31, Fia, F^^, P,^, F,^ forM.,M,,M., -i (E.,E,,E,)

lastly we shall have the relation /^ a = -~ >/'> k, F^k •, = — i^^ *, (the letter /, F shall denote the field, <? the (i.e. current).

PRINCIPLE OF EELATIVITY

Then the fundamental Equations can be written as

9/12 ^ 9/1

3 ^ 9/i.i _ g

^^) lt"'+

2 3

9.t

+ -,

2 4

9/3_X

9.<^i

3 2

9.<

9/3. ^

9r.

9Ax + 9/ 9.t^i

4 2

4 3

• C,

9 <.

and the equations (3) and (4), are

9F34. , 9F4,, , 9F23

9 .t^^

aF.3

9*1

9F,,

9.C1

9F35

+ 9Z- +

9F,

9ic.

9^:4

8F„

a..

aF,,

= 0

9ic,

= 0

9 «i

9F,

9aj„

9F^

9 ^'3

= 0

§ 8. The Fundamental Equations.

We are now in a position to establish in a unique way the fundamental equations for bodies moving in any man- ner by means of these three axioms exclusively.

The first Axion shall be, —

When a detached region"**" of matter is at rest at any moment, therefore the vector n is zero, for a system

* Einzelue stelle der Materie.

THE FUNDAMENTAL EQUATIONS / 23

(^, y, Zf t) — the neighbourhood may be supposed to be in motion in any possible manner, then for the space- time point X, I/, z, t, the same relations (A) (B) (V) which hoM in the case when all matter is at rest, snail also hold between p, the vectors C, e, m, M, E and their differ- entials with respect to x, y, z, t. The second axiom shall be : —

Every velocity of matter is <1, smaller than the velo- city of propo^ation of light."^

The fundamental equations are of such a kind that when {Xy y, z, it) are subjected to a Lorentz transformation and thereby (m — ie) and {M—iE) are transformed into space-time vectors of the second kind, (C, ip) as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed ma2:nitudes.

Shortly I can signify the third axiom as ; —

{m, — ie), and {}f, — iE) are space-time vectors of the second kind, (C, ip) is a space-time vector oP the first kind.

This axiom T call the Principle of Relativity.

In fact thes j three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.

According to the second axiom, the magnitude of the velocity vector | /^ | is <1 at any space-time point. In consequence, we cm always write, instead of the vector 7i, the following set of four allied quantities

J ^2

_ ^y

y Wg

_ ".

(Oj

\/i-ir^

^/l-u^

\/l-u'-^

\/l-u'

* Vide Note.

24 PRINCIPLE OF RELATIVITY

with the relation

(27) o.i2+o)22+<0 32+a),2=:_ I

From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space -time vector of the 1st kind.

Let us now fix our attention on a certain point (a*, y, z) of matter at a certain time (/). If at this space-time point u — o, then we have at once for this point the equa- tions (^), (S) (F) of § 7. It u X 0, then there exists according to 16), in case \ u \ <1, a special Lorentz-trans- formation, whose vector v is equal to this vector n [x, y, Zy t)f and we pass on to a new system of reference {x\ y' z i') in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values 28) o)\ = 0, i^'^^O, o)'q = 0, (ii\=zi^ therefore the new velocity vector oj' = o, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point {x' y' z i) involves the newly introduced magnitude {u p J C, e , m y E' , M') and their differential quotients with respect to {x , y' , *' , t') in the same manner as the original equations for the point {x, y, z^ t). But according to the first axiom^ when u ^=.0^ these equations must be exactly equivalent to

(1) the differential equations (^'), (^')j which are obtained from the equations {A), (B) by simply dashing the symbols in (A) and (B).

(2) and the equations

(V) e' = ,E\ 3r=/im\ C' = ctF . ^

where «, /x, or are the dielectric constant, magnetic permea- bility, and conductivity for the system (x' y' z t') i.e. in the space-time point [x y, z t) of matter.

'rut i'UNDAiiEKTAL EqUATlONS 25

Now let us return, by means of the reciprocal Loreutz-

trausformation to the original variables (.r, ?/, :, f), and the

magnitudes {n, p, C, e, m, E^ M) and the equations, which

we then obtain from the last mentioned, will be the funda-

mentil equations sought by us for the moving bodies.

Now from § 4, and § 6, it is to be seen that the equa- tions A), as well as the equations B) are covariant for a Lorentz-transformation, i.e. the equations, which we obtain backwards from A') B'), must be exactly of the same form as the equations A) and B), as we take them for bodies at rest. We have therefore as the first result : —

The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in p and the vectors C, ^, in, E, M, are exactl}^ of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have

I I curl m — - = Ci, II J div ex=p

III \ curl E + ^97 = « IvVliv M=o

The velocity of matter occurs only in the auxilliary equations which characterise the influence of matter on the basis of their characteristic constants e, /^, a. Let us now transform these auxilliary equations \') into the original co-ordinates ( '■, f/,z, and t.)

According to formula 15) in § 4, the component of e' in the direction of the vector u is the same us that of ((5-f [w w]), the component of m is the same as that of vi — [Hc']y but for the perpendicular direction «, the com- ponents of e\ m are the same as those of (<? + \ii niY) and (^n

— Sji e], multiplied by — ^ • ^^^ ^^^ other hand E'

2d PRINCIPLE OF RELATIVITY

and M' shall stand to E + [«M,], and M— [/^E] in the same relation us e and ;// to 6^+ [?(w], and m-^i^ae). From the relation <?' — e E', the following" equations follow

and from the relation M'=:/x iii\ we have

(D) M-[^^E]=/.Oyz- [«.']),

For the components in the directions perpendicular to V, and to each other, the eijuations are to be multiplied

hy ^^rr^

Then the following equations follow from the transfer- mation equations (12), 10), (11) in § 4, when we replace q, f., r-, f, r ,, r-, f by \n\ , C\,, Cv, p, C'„, C't, p . .

(E+kM])„

C,7 =cr

v^l -«^

In consideration of the manner in which cr enters into these relations, it will be convenient to call the vector C— p n with the components C, — p | " \ in the direction of //, and C „ in the directions v. perpendicular to it the '^Convection current/' This last vanishes for o-=o.

We remark that for €=1, /x=l the equations 6''=:E', ?m' = M' immediately lead to the equations 6' = E, ;>i = M by means of a reciprocal Lurentz-transformatiun with — ii as vector; and for o-=:o, the equation C' = o leads to C=p u; that the fundamental equations of Ather discussed in § '! becomes in fact the limitting case of the equations obtained here with €=1, />t = l, o-=o.

FUNDAMENTAL EQUATIONS IN LORENTZ THEORY 27

§9. The Eundamental Equations in LoKENTz's Theory.

Let us no\v ■ see how far the f'liiidameutal equations assumed by Loreutz correspond to the Relativity postulate, as defined in §8. In the article on Electron-theory (Ency, Math., Wiss., Bd. V. 2, Art 14) Lorentz has given the fundamental equations for any possible, even magnetised bodies (see there page 209, Eq" XXX', formula (14) on page 78 of the same (part).

(IIL/'O Curl (il-[//E]) = J+ — +;/divD

lit

— curl [^^D]. (TO div X)=p

{l\") curl E =. — ^ , Div B-0 (V)

Then for moving non-magnetised bodies, Lorentz pufes (page 223, o) /x=: 1, B = H, and in addition to that takes account of the occurrence of the di-eleetric constant e, and conductivity <j according to equations

(cryXXXIV^ p. 327) D-E = (€-l){E+[/^B]}

(c^XXXIir, p. 223) J = cr(E4- [?^B])

Lorentz's E, D, H are here denoted by E, M, «?, m while J denotes the conduction current.

The three last equations whioh have been just cited here coincide with eq" (II), (III), (IV), the first equation Would be, if J is identified with C, — 2(p (the current being zero for o- = 0,

(29) Curl [H-(/^E)]=C+-^ -curl [uD],

28 PEfN-CIPLE OF RELATIVITY

and thus comes out to ])e in a difl'erent I'ortn than (1) here. Therefore for ma^^netised bodies, Loreutz's equations do not correspond to the Relativity Principle.

On the other hand, the form corresponding to the relativity principle, for the condition of non-magnetisation is to be taken out of (D) in §8, with />i= ], not as B = H, as Lorentz takes, but as (30) B — [/^D] = H — [?eD] (M — \_uYj']=iii — [ne'\ Now by putting H = B, the differ- ential e(piation (:29) is transformed into the same form as eq" (1) here when /// — [//<?] =M — \^u1l']. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for movinG" non-masi'netised bodies at last ao-ree with the relativity postulate.

If we make use of (oO) for non-magnetic bodies, and put accordingly H = B+[//, (D — E)], then in consequence of (C) in §8,

• (c-1) (E+['^B])=:D^E-f [/^ KT)-E]], i.e. for the direction of u

(.-1) (E+[7^B])„=(I)-E), ■ and for a perpendicular direction u,

(,_1) [E + (..B)]„-(l-.rO (D-E),

i.r. it coincides with Lorentz's assumption, if we neglect v.'^ in comparison to 1.

Also to the same order of approximation, Lorentz^s form for J corresponds to the conditions imposed by the relativity principle [comp. (E) § 8] — that the components of J„, Jirare equal to the components of o-(E+[?^B])

multiplied by /fZ72 ov .^/f^;:^ respectively.

FUNDAMENTAL EQUATIONS OF E. COHEN 29

§10. Fundamental Equations of E. Cohn.

E. Colin assumes the i'ollowing fundamental equations. (.11) Curl (M+[^^ E]) = ~4-udiv. E4-J

-Curl [E-(//. M)]=-^-f n div. M.

(U) J = o-E, =€E-[//. M], M = iJ.{w-{-[?f E.])

where E M are the electric and mao-hetic field intensities (forces), E, M are the electric and magnetic polarisation (induction). The equations also permit the existence of true mas^netism ; if we do not take into account this consideration, div. M. is to be put = o.

An objecti<Mi to this sj'stem of equations, is that according to these, for e=:l, /x=l, the vectors force and induction do not coincide. If in the equations, we conceive E and M and not E-(U. M), and M+ [U E] as electric and magnetie forces, and with a glance to this we substitute for E, M, E, M, div. E, the symbols e, M, E -fFU M], M--lf/(^], p, then the differential equations transform to our equations, and the conditions (3:2)

transform into

J = tr(E-f-[?/ M])

.+ [7^(.Z-[7. .])] = <E+[./M])

M - [h, (1^ -f ^/ M ) J = /.(/// - [u e] )

then in fact the equations of Cohn become the same as those re<:|uired by the relativity principle, if errors of the order n^ are neglected in comparison to 1.

It may be mentioned here that the equations of Hertz become the same as those of Cohn, if the auxilliary conditions are

(53) E = €E,M=/.M, J = (rE.

PRINCIPLE OF RELATIYlTr

§11. Typical Representations of the eundamental equations.

III the statement of the fandamental equations^ our leadins^ idea liad been that tliev should retain a covarianee of form, Avhen subjected to a group of Lorentz-trans- formations. Now we have to deal with ponderomotive reactions and enero^v in the electro-maa;netic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected w^ith the simplest forms which can be given to the fundamental equations, satisfying the conditions of covarianee. In order to arrive at such forms, I sliall first of all ]mt the fundamental ecpiations in a typical form which brings out clearly their covarianee in case of a Lorentz-transformation. Here I am using a method of calei'ilation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2ud kind, and of which the rules, as far as required are given below.

A system of magnitudes a/,/, formed into the matrix

1 1

1 9

p 1

P H

arranged in p horizontal rows, and q vertical columns is called a /; X (/ series-matrix, and will be denoted by the letter A.

If all the quantities a,,^ are multiplied bv C, the resulting matrix will be denoted by CA.

If the roles of the horizontal rows and vertical columns be intercharged, we obtain a qxp series matrix, which

TYPICAL KEPKESENTATIO]S^S

will be kuoAvn as the traDsposed matrix of A, and will be denoted by A.

A = hhi

.«,;

1 ? •■

If we have a second p ^ q series matrix B.

B =

1 1

'^i

V',..

p 1

then A + B shall denote the J^xq series matrix whose

members are ai, k+hi,k.

2^ If w^e have two matrices

A =

1 1

p 1

1 1

p 'I

B =

1 1

•^1 V

P r

where the number of liorizontal rows of B_, is equal to the number of vertical columns of A, then by AB, the product of the matrics A and B, will be denoted the matrix

1 1

^ i» r

1 '•

• ^' l> V

* <? ^ U h

where Ci, „ =a^ ^ ^i /, •+ a,^ .. ^^- a + " k , ^ , ^. + . . .</

these elements beini;- formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law (AB) S=A(BS) holds, where S is a third matrix which has got as many horizontal rows as B (or x-VB) has got vertical columns.

For the transposed matrix of .C = BA, we have C = BA

PRInCIPIE of feELATlVltr

S*^. We shall have principally to deal with matrices with at most four vertical columns and for horizontal

rows.

As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 x 4 series) with the elements.

(3 JO

1 1

'3 1

1 2

Gg o

3 2

1 3

1 4

^^2

e.,

^34

10 0 0 0 10 0 0 0 10

41 ^42 "=43 ^44 0 0 0 1

For a 4x4 series-matrix, Det A shall denote the determinant formed of the 4x4 elements of the matrix. If det A { o, then corresponding* to A there is a reciprocal matrix, which we mav denote bv A"^ so that A~^A = 1

A matrix I

/ . '^ J 12 /is /l4

2 4

3 4

I/31/32O /j

'Al /*2 /is O

in which the elements fulfil the relation f,,k = — /w. , is called an alternating matrix. These relations say that the transposed matrix / = — /• Then by /* will be

the dual, alternating matrix

(35)

■jf—

J Si J 4 2 .7 2

/4 5 ^ /l 4 /s 1 J2i Jil ^ J\Q \

/s 3/13/ 21 ^ '

TYPICAL REPRESENTATIONS

Then (3G) ^ f=j\ , /, , +/; , A , + A , /, ,

i e. We shall have a 4x4 series m^itrix in which all tJie elements except those on the dia^'onal from left up to right clown are zero, and the elements in this Jia':^onal agree with each other, and are each equal to the above mentioned combination in (36).

The determinant of /is therefore the square of the

-A

combination, by Det ./we shall denote]the expression

4°. A linear transformation

which is accomplished by the matrix

A =

^11' ^125 ^135 *14

**15 *'3 23^ ^-^o

^31' ^3 25 '^SS' ^34

^4 1 ' ^^4 M' ^431^

4 4

will be denoted as the transformation A

By the transformation A, the expression

•^1+ .'1+ .'3+ ■"I is changed into the quadratic

for III

where a;, ^,— a, ^. a^k+(^2/> «2A+a3/, «3A +"4/- «4A'

are the members of a 4x4 series matrix which is the product of A A, the transposed matrix of A into A. If by the transformation, the expression is changed to

„' 21,, '2 I ^ '2 ij,' 2 we must have A A = l.

PRINCIPLE OF JIELATIVITY

A has to correspond to the following relation, if trans- formation (38) is to be a Lorentz-transforniation. For the determinant of A) it follows out of (39) that (DetA)- = 1;, or Det A=-}-l.

From the condition (39) we obtain

i.e. the reciprocal matrix of A is equivalent to the trans- posed matrix of A.

For A as Lorentz transformation, we have further Det A= +1, the quantities involvinoj the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and n^^'^0.

5°. A space time vector of the first kind^ which s represented by the 1x4 series matrix,

(41) .9= I .9j ,9, .93 s^ I

is to be replaced by 5 A in ease of a Lorentz transformation

A. i.e. s'= I 5/ .92' 5/ ,94' I = I .?! .92 .93 ,94 I A; A space-time vector of the 2nd kindt with components /"^ 3 . . . /34 shall be represented by the alternating matrix

(4.2)

./12 JiS /l4

/21 ^ ./2 3 ./:

./31 /s2 o ./;

./ 4 1 y42 ./4:

and is to be replaced by A"\/*A in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression (37), we have the identity

DetMA/A) = Det A. Det"/. Therefore Det-/be- comes an invariant in the case of a Lorentz transformation [seeeq. (26) Sec. § 5].

* Vide note 13. t Vide note 14.

TYPICAL REPRESENTATIONS 35

Looking back to (-36), we have for the dual matrix

(A/-A)(A-i/A) = A-i/VA = Det^ /. A-iA = Det-/ from which it is to be seen that the claal matrix/'^ behaves exactly like the primary matrix/*, and is therefore a space time vector of the II kind; y'^^ is therefore known as the dual space-time vector of /with components {/^ \-if\ 4?/'3 4>)j

6."^" If 10 and 6' are two space-time rectors of the 1st kind then by w *• (as well as by sw) will be understood the combination (43) w^ 8^ +^^2 ^-2 +^'-^3 8^-^iOj^ 6-4.

In case of a Lorentz transformation A, since {^wK) (A -s)

= /d; ,s, this expression is invariant. — If 10 s =0, then w and 6' are perpendicular to each other.

Two space-time rectors of the first kind {lo^ s) gives us a 2 X 4 series matrix ^

10^ lu^ 10.. lU ^

8 1 S c) So 4

Then it follows immediately that the system of six magnitudes (14) ?c>2 8.^ —io^ 8 2, w^ '^1 "~^'^i *3> ^'^ i ^2 ~'^^'i -^ u

W^ 8^ — 20^8^, 10.2 *'4— ?^4 82, fOg S_^—tn^ Sq,

behaves in case of a Lorentz-transformation as a spaee-time vector of the II kind. The vector of the second kind with

the components (41) are denoted by \_iOj 5]. We see easily

1 that Det''^ \tOj ^^]=:o. The dual vector of \_w, 8'] shall be

written as [w, 5].^

If 2V is a spaee-time vector of the 1st kind, ,/ of the second kind, 10 f signifies a 1x4 series matrix. In case of a Lorentz-transformation A^ 10 is changed into u'' = 2uA, fmto/" = A~^ fA, — therefore w' /' becomes =(wA A"^ /' A) = w/ A i.e. io f is transformed as a space-time vector of

36 PUINCIPLE OF EELATIVITY

the 1st kind.^ We can verify, when 2^ is a space-time vector o£ the 1st kind,/' of the '^nd kind, the important identity

(45) [^W, W f ] + \_10, 10/"^']"^ — (w 20 ) f.

The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.

For example, for co^ =:o, co^, =o, 0J3 =0, w^ =&,

<^/= I ^hx^ ij\i. ^'As. o I ; 0)/*= I, z'f32, ^/la. ^Six^ » I [w • oj/J =0, o, o, fi^.f^^, f., 3 ; [to • (o/*]* = 0, o, o, /g 2 , /i 3 , /a 1 .

The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in (w^, m^. cog. co^).

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants c /x, cr will be introduced.

Instead cf the space vector ?f, the velocity of matter, we shall introduce the space-time vector of the first kind w with the components.

21 J. 71 y u^ i

VY^; ' ' vrr^^ ' ' vr^^ * vi-

u^ .

(40) where w,2+oj2 2_j_j^^2^^^^2__i

and — 2,*(o^>0.

By F and / shall be understood the space time vectors of the second kind M — i'E, vi — ie.

In $=wF, we have a space time vector of the first kind with components

<I>i=w,F2, +w3F23+w.,F2.,

* Vide note 15.

TYPICAL UEPRESENTATIONS 37

The first three quantities (<^i, (jbg, (^3) are the components

of the space-vector

\/X^y 2

and further ci , = — -J— . — =L

Because F is an alternating matrix,

(49) W$ = W,(^1+C02<^2 +0)3^3 +C04$4=0.

i.e. <E> is perpendicuhar to the vector to ; we can also write ^^=i [w.^i +c0y$3 +o),(I>3].

I shall call the space-time vector $ of the first kind as the Electric Best Force.^

Relations analogous to those holding between — -wF, E, M, U, hold amongst —mf, e, m, u, and in particular — w/ is normal to oj. The relation (C) can be written as

{ C } a./=ewF.

The expression (w/) gives four components, but the fourth can be derived from the first three.

Let us now form the time-space vector 1st kind^ <if=^ici)f*, whose components are

^^ = — i( wj3^,+ 0)3/^2+^^23) 1

Vt'^zz:-/ (coJ^3-f W3/1.4+W4/31) '

"^'3 = —^ (^1/2 4 + ^^2/4 1 + ^^4/1 2 ) I

^4,I=—i ((0j32+W2/i3-f (03/21 ) J

Of these, the first three ^1, ^'2, ^3, are the x, y. z components of the space-vector 51) - — ^ — ^/

■u

and lurtlier (0-) i^_^ r-

^\—u'^'

* Vide note 16.

PRINCIPLE OP UELATIVITY

Among these there is the relation

(53) (0^==COi>I\ +CD2^2 ^-Wg^I^g +00^*4 =0

which can also be written as ^j^=^l {njc^\-\-?Oy^^ + i','^^).

The vector ^ is perpendicular to co ; we can call it the Mayuetlo rest-force.

Relations analogous to these hold among the quantities

twP^", M, E, ic and Relation (D) can be replaced by the

formula

{ D } -a)F* = /xco/*-

We can use the relations (C) and (D) to calculate F and / from ^ and ^' we have

0)¥=—^, wF* = — i/X^, Ojf= — €^, (.of *= — L^,

and applying the relation (4^5) and (4^6), we have F= [w. 4>J + i>[w. ^]* 55)

i.e. Fi2=(wi$i— W2$i) + i>[w3vl/^— w^vif J^ etc.

/,2=e(wi^2-<^2^i) + ^' [«3 ^4-^4^3]- etc. Let us now consider the space-time vector of the second kind [<l> ^], with the components

_ ^,^^-^^^l\, <^2*4-^4^'2: ^3*4-^4^3 -J Then the corresponding space-time vector of the first kind wT*^, ^'] vanishes identically owiug to equations 9) and 53)

for co[$.^] = -(wvp)<^+ (w^)^

Let us now take the vector of the 1st kind

with the components i2j_ = — L w.

etc.

<^4

TYPICAL REPRESENTATIONS .'J 9

Then by applying rule (4^5), we have (58) l^:^] = i [ojfi]*

i,e. <J>i^2— ^2*i=?'(^^3^i.— ^4^3) etc. The vector fi fulfils the relation

(wl})=W^O^ +(020^ +0)3^23 +0J^O4 =0,

(which we can write as 0^=/ (^.O^ +oj,^02 +^^2^3) and O is also normal to w. In case w==o, we have ^^=0, ^4=0, 04^=0, and

[n„ 03,0.3 =

(J) <I> <b

^1 ^2*^3

I shall call 0, which is a spaoe-time vector 1 st kind the Rest- Ray.

As for the relation E), which introduces the conductivity a we have — wS== — (w^s^ +0)253 ~1"^3''^3 +^->4-'^4)

_ — I ^H C„+p ,

This expression j^ives us the rest-density of electricity (see §8 and §4).

Then 61)=5+(oj.?)w

represents a space-time vector of the 1st kind, which since o)w= — 1, is normal to m, and which I may call the rest- current. Let us now conceive of the first three component of this vector as the ('?"— ;y~-) co-ordinates of the space- veetor, then the component in the direction of ?/ is

C„-^

^H p' _ ^« — I « I p _ J,

and the component in a perpendicular direction is C„=J^.

This space-vector is connected with the space-vector J = C — pti, which we denoted in §£ as the conduction- current.

PRINCIPLE OF RELATIVITY

Now by comparing with ^~ --wF, the relation (E) can be brouofht into the form

S+ (co,?)a)=r — cro)F,

This formula contains four equations, of which the fourth follows from the first three, since this is a space- time vector which is perpendicular to w.

Lastly, we shall transform the differential equations (A) and (B) into a typical form.

2 4

§12. The Differential Operator Lor.

A 4x4 series matrix 62) S= S^.S^.S^gS,^ = I S,,

Qi O q q

'•-'21 2 2 *^ 2 3 ^-^

q q a q

^31 ^32 ^^33 ^~34

q q q q

•^41 *^42 '^43 ^^44

with the condition that in case of a Lorentz transformation it is to be replaced by ASA, may be called a space-time matrix of the II kind. We have examples of this in : —

1) the alternatint^ matrix /", which corresponds to the space-time vector of the II kind, —

2) the product /F of two such matrices_, for by a transfor- mation A, it is replaced by (A-^A- A-^FA) = A-y F A,

8) further when (w^. u>^, 0)3, w^) and (O^. Q^, fig, fi^) are two space-time vectors of tlie 1st kind, the 4 x 4 matrix with the element S^ ;i. =w/,fi;.,,

lastly in a multiple L of the unit matrix of 4 x 4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.

We shall have to do constantly with functions of the space-time point (^r, y, c, it), and we may with advantage

I'Hfi DIFFERENTIAL OPERATOR LOR

employ the 1x4 series matrix, formed of differential symbols, —

d a a a

a " a 2/ a^ ^a^'

or (6;^)

a a

a.t'i a.t^2 a«s a**

For this matrix I shall use the shortened from " lor."*

Then if S is, as in (62), a space-time matrix of the

II kind, by lor S' will be understood the 1x4 series

matrix

Kj Kj Kg K^^

where K,= |^ + ^3^ + ^^ + 4^

a .' 1

'■ 4

When by a Lorentz transformation A, a new reference system (,c\ d\ x' ^ x^) is introduced, we can use the operator

lor'=

a.'jj' a.^',' a^3' a.-,'

Then S is transformed to S'=A S A=: | S'^^. | , so by lor 'S' is meant the 1x4 series matrix, whose element are

K'. = ^3^ +

9^'2/t _i_ as'a^r, , as'

i + ^^p-^ +

4, k

aa!i' Qx^' a-f's' ^^J

Now for the differentiation of any function of {x y t t)

a _ a

we have the rule

'<-k

a « 1 , a a

" 3

aa?i a^ryt' a.»'a a.t'A-'

a^'s • a

-r +

a«4

a.t'a Q^Vk' . a»4 Q'^'k

a.ii

^i/t +

dx,

^2 A "^

Q X

Cl^k +

dx,

ik'

so that, we have symbolically lor' = lor A,

* Vide note 17.

PRINCIPLE OF RELATIVITY

Therefore it follows that

lor 'S' = lor (A A'^ SA) = (lor S)A.

i.e., lor S behaves like a space-time vector of the first kind.

If L is a multiple of the unit matrix, then by lor L will be denoted the matri'x with the elements

aL aL aL aL

a .( 1 a

.(■<

Qx,

If s is a space-time vector of the ]st kind, then

lor s

asi , a^

2 1 a s s a^ 4

a<'-'i a i«2 ' a«?3

In case of a Lorentz transformation A, we have ^ lor V=Ior A. As = lor s.

i.e., lor s is an invariant in a Lorentz-transformation.

In all these operations the operator lor plays the part of a space-time vector of the first kind.

If / represents a space-time vector of the second kind, — lor / denotes a space-time vector of the first kind with the components

a ^('' .

a.<^;

^•X'^

a./.x a .*' 1

9/31 _|. a/33 a^jj a^t

a/.

2^3 I

a <v , a .'(

2 4

a/a

9 a;.

Qf~ ^ a/

4 2

Qx^

a/. a^K,

THE DIFFERENTIAL OPERATOR, LOR 43

So the system o£ differential equations (A) can be expressed in the concise form

{A} \ovf=-s,

and the system (B) can be expressed in the form

{B} log F-^ = 0.

Referring back to the definition (67) for log .s', we find that the combinations lor (lor/)^ and lor (lor F* vanish identically, when /and F"^ are alternating matrices. Accordingly it follows out of {A}, that

(68) 9£i + 91. + 9_l3, + 9i'* = 0,

0<^'x 0''^'2 OOJg 0.*'4

while the relation

(69) lor (lor F^) = 0, signifies that of the four equations in {B}, only three represent independent conditions.

I shall now collect the results.

Let w denote the space-time vector of the first kind

(?^= velocity of matter),

F the space-time vector of the second kind (M, — ^E) (M = magnetic induction, E = Electric force, /the space-time vector af the second kind (w/, — ?>) (y^^ = magnetic force, (? = Electric Induction. s the space-time vector of the first kind (C, ip) (p = electrical space-density, C—p?^ = conductivity current, € = dielectric constant, /x. = magnetic permeability, 0- = conductivity.

PUINCIPLE OE RELATIVITY

then the fundamental equations for electromagnetic processes in moving bodies are"^

{A} \0Yf=—S

{B} log ¥^ = 0

{C} (o/=€a)F

{D} <oF^ = /xCo/^

{E} S+{o)s), ?(;=— o-toF.

o,w= — I, and wF, w/, mF^, o)f^,s+ (w5)w which are space-time vectors of the tirst kind are all normal to w, and for the system {B}, we have

lor (lor F-^) = 0.

Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are neces- sary for determining the motion of matter as well as the vector 11 as a function of .c, j/, r, f, when proper funda- mental data are given.

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

Finally let us enquire about the laws which lead to the determination of the vector w as a function of {■'(■,i/,z,f.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

/l 3

A«

F =

^ 1 S

F F

/..

/a 3

J 2 4

F F

^23 ^24

/s 1

/>.

JS4-

-■- 3 1

^ S2

0 ^ F3,

/* s

-"■42

P*, 0

* Via

le note 1

THE PRODUCT OF THE FIELD- VECTORS /f

4t5

are of much importance. Let us write.

(70) fF =

Sj 1 — L Sj

S21 S31

I s

O T Q

3 2

Oo Q — Jj

'3 3

^14 S24

S41

4, 2

4 3

S. - — L

'4 4

Then (71) S, , H-S,^ +S33 +S,,=0.

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

(72) L=| (a., P.,+/3,P3,+/,.F,.+A4F

1 4

+/2.F

2 4

' /a 4 ■'^3 4 I

Then we shall have

'12 ^12

(73) 8^1=- //a 3 F23+/3, F3^+/^2 F42— /i Si2=/i8 Fgg+Zi+F^a etc....

In order to express in a real form, we write

(74) S:

Now X , =

^11 ^12

Sis

s..

^2 1 ^2 2

^2 3

S24

S31 S32

S3 3

S3 4

S41 S^2

S+3

S44

— ^

1 r

=2 ^ "^-

M.-

TTlyMy

— 7

M,Ik1

-2T.

-zT,

-2T.

■^X, ^lY, -iZ, T,

— m,M- +<?^E^— CyE,— e,E,

PRINCIPLE OF RELATIVITY

(75) Xy=m,M^+e,E,, Y, =m,M, +6'^E^ etc. *

Xt=ey'M.,—eMy, T^=m^Ey—m,jE, etc,

L,=:i rm,,M,,+m2,M,+m,M,— e,,E^— 6yEj,— e,E,l

These quantities are all real. In the theor}^ for bodies at rest, the combinations (X^, X^, X., Y,, 1^, Y., Z^, Z,, Z,) are known as '^Maxwell's Stresses/' T„ T,, T, are known as the Poynting's Vector, T/ as the electro- magnetic energ^^-density, and L as the Langrangian function.

On the other hand, by multiplying the alternating matrices of _/^ and P^, we obtain

(77) Y*f*=

— Sj ^ — L, — S

— 84.^ — S

1 2

-s,

'^912 ^1 >^

3 2

2 R

— ^33 — L, — S

4. 2

-S

4 ?<

-S..-L

and hence, we can put

(78) /F=S-L,

F^/^=-S-L,

where by L, we mean L-times the unit matrix, i.e. the matrix with elements

|Le,,|, (e,,=l, e,,=0, h=l=:k /., A-1, 2, 3, 4).

Since here SL = LS, we deduce that,

F*/*/F = (-S-L) (S-L) = - SS + L%

and find, since/*/ = Det "^ f, F* F = Det ^ F, we arrive at the interesting conclusion

* Vide note 18.

THE PRODUCT OF 'JHK FIELD-VECTORS /f ii

(79) SS = L^ - Det ^"/ Det ^ F

i.e. the product of the matrix S into itself can be ex- pressed as the multiple of a unit matrix — a matrix in which all the elements except those in the principal diagonal are zero, th*^ elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations

k, k being unequal indices in the series 1, 2, 3, 4, and

(81) S/ji Si/, + S/,2 Sg/, +S/, 3 S3/,-{-Sa4^, S^/, =:L''^ —

Det ^/ Det ^'f,

for/^ = l, 2, .3, 4.

Now if instead of F, and / in the combinations (72) and (73), we introduce the electrical rest-force ^, the magnetic rest-force "^^ and the rest-ray O [(55), (56) and (57)], we can pass over to the expressions, —

(82) L = — ie$¥+^/x*^

(83) S,, = - I €^"$e,, - i/x*¥e,,

-f € {<^,, $/, — ^4> (0/, <0j

, + fi (*A 4^ — * 4* <o, oj,) - Qk <^k — e/x (Ok Qk

(h. A- = 1, 2, 3, 4). Here we have

The right side of (82) as well as L is an invariant in a Lorentz transformation^ and the 4x4 element on the

48 PRINCIPLE OF RELATIVITY

right side of (83) as well as Ski, represent a space time vector of the second kind. Remembering this faet^ it suffices, for establishing the theorems (82) and (83) gener- ally, to prove it for the special case <t>i=o, w^=o, ta^=Of (ii^=i. But for this case w = o, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and 6^ = eE, M = /xm on the other hand.

The expression on the right-hand side of (81), which equals

[I (m M - eE)2] + (em) (EM),

is = 0, because (evi =: e ^ ^, (EM) = // ^ ^ • now referring

> back to 79), we can denote the positive square root of this

expression as Det * S.

Since f = — f^ and F = — F, we obtain for S, the transposed matrix of S, the following relations from (78),

(84) F/ = S-L,/* F* = -"S-L,

ThenisS-S= | S.^-S,,

an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,

(85) S - 8"= - (c /x - 1) [w, 12], from which we deduce that [see (57), (58)].

(86) o)(S-S)* = o,

(87) 0) (S -"S) = (€ /a - 1) n

When the matter is at rest at a space-time point, w=o, then the equation 86) denotes the existence of the follow- ing equations

Zy=Yj, X^=Z,, Yx=:X.j,,

THE PRODUCT OP THE PIELD- VECTORS /f 49

and from 83),

T.-1},, T,=0„ T,,.:=fi3

X^=:e/xOj, Y^^e/xfig, Zf=€jjLCi^

Now by means of a rotation of the space co-ordinate system round the null-point, we can make,

Zy=Y-=o, Xj,=Z^, =o, X^=:Xj, =o.

According to 71), we have

(88) X,+Y, + Z,-f T.=o,

and according to 83), T<>o. In special eases, where Q vanishes it follows from 81) that

X,^=:Y,«=Z,^ = T,^=:(Det^S)^ and if T^ and one of the three mag-nitudes X^^,, Yy. Z. are

= + Det ^ S, the two others = — Det * S. If 12 does not vanish let O =^0, then we have in particular from 80)

T, X,=0, T, Y,=0, Z,T,+T,T,=0,

and if fii=0, 0^=0, Z,=-T, It follows from (81),

(see also 83) that

X,=:-Y, = +Def^S,

and -Z,=T, = '' Det^ S + e/iOg^" >Det^S.—

The space-time vector of the iirst kind

(89) K=lor S,

is of very great importance for which we now want to demonstrate a very important transformation

According to 78), S=L-|-/F, and it follows that

lor S=lor L + lor/F.

50 PRINCIPLE OF RELATIVITY

The symbol ^ lor ' denotes a differential process which in lor fY, operatt^s on the one hand upon the components of fi on the other hand also upon the components of F. Accordingly lor f¥ can be expressed as the sum of two parts. The first part is the product of the matrices (lor /') F, lor /' being regarded as a 1 x 4 series matrix. The second part is that part of lor f¥, in which the diffentiations operate upon the components of F alone. From 78) we obtain

/F=-F*/*-2L;

hence the second part of lor / F = — (lor F*)/*+ the part of — 2 lor L, in which the differentiations operate upon the components of F alone We thus obtain

lor S = ( lor / ) F - (lor F* )/* + N,

where N is the vector with the components

N, =:JL / Q/23 F JL. 0/31 W I 0/12 p I Of\i F

\ 0"h 0''-h 0'<-h O^-h

4- ^/g -i F 4- Q/34 p •^ "a ^ 3 1 ~ "a ^3*

_ Q^aa f _ QFg^ f _ 9 F 1 2 . _ 9F 14 ^

ay 2 3 a /SI ~~a ^12 ■' c^ J 1 4

't'A O'Ca O^;/, 0.;a

d./ S 4 ~^ ■ J S - h

••■■/< O.'A /

(/. = !, 2, 3, 4)

By using the fundamental relations A) and B), 90) is transformed into the fundamental relation

(91) lorS = -5F + N.

In the limitting case €=1, />t=l, /=F, N vanishes identically.

THE PRODUCT OF THE FIELD- VECTORS /'f 51

Now upon the basis of the equations (55) and (56), and referring back to the expression (8£) for L, and from 57) we obtain the following- expressions as components of N,—

dt OXh ^ OX,,

for h = l, 2, 3,4.

Now if we make use of (59), and denote the space- vector which has O^, O3, O3 as the c, j/, z components bj the symbol W, then the third component of 92) can be expressed in the form

^93) ^^-^ (W^IL)

The round bracket denoting the scalar product of the vectors within it.

§ 14. The Ponderomotive Force.*

Let us now write out the relation K=lor S = — -^F + N in a more practical form ; we have the four equations

_l $$ P^ _1 vi/^ 9jf +^/^-l / w9u^

a^ 2 6'^ vi.

■

2 61/ 2 at/ ^^i-ti^V a?//

Vide note 40.

52 tAINCIPLE OF RELATIVITY

(97) ^K, = 1^-' - 1^" -|-- -1^-^ =s,..E, +,v,E, +S..E..

It is my opiuion that when we calculate the pondero- motive force which acts upon a unit volume at the space- time point ..", y, :, I, it has got, .c, y, :: components as the first three components of the space-time vector

K + (a)K)aj,

This vector is perpendicular to w ; the law of Energy finds its expression in the fourth relation.

The establishment of this opinion is reserved for a separate tract.

In the limitting case €=1, /x=l, cr=:0, the vector N=0, S=pa), a)K=0, and we obtain the ordinary equations in the theory of electrons.

APPENDIX Mechanics and the Pv;Elativity- Postulate.

It would be very imsatisfactoiy^ if the new way o£ looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.

Now many authors say that clas^jieal mechanics stand in opposition to the relativity postulate, which is taken to be the basii of the new Electro-dyiiamics.

In order to decide this let us fix our attention upon a spe- cial Lorentz transformation re])resented by (10), (11), (1"^)? •with a vector v in anv direction and of anv maonitude a<l but different from zero. For a moment we shall not suppose any special relation to hold between the unit of length and the unit of time, so that instead of t, f, q, we shall write ct, cl', and q/c, where c represents a certain positive constant, and q is <c. The above mentioned equations are transformed into

,/___,,._ ,.' _ c ()\—qt) ,,_ qi\-hcH

They denote, as we remember, that r is the space- vector (•^'i V) -):> ^'' is the space-vector (■^' y' z)

If in these equations, keeping v constant we approach the limit c = oo, then we obtain from these

The new equations would now denote the transforma- tion of a spatial co-ordinate system (x, y, :) to another spatial co-ordinate system ( t' y' -') with parallel axes, the

54 PRINCIPLE OF RELATIVITY

null point of the second system moving with constant velocity in a straight line, while the time parameter remains unchanged. We can, therefore, say that classical mechanics postulates a covariance of Physical laws for the group of homogeneous linear transformations of the expression

_a;«_2/2 — -s+r^ ... ... (1)

when • (?=qo.

Now it is rather confusing to find that in one branch of Physics, we shall find a covariance of the laws for the transformation of expression (1) with a finite value of 6', in another part for c = oo.

It is evident that according to Newtonian Mechanics, this covariance holds for c=^oo^ and not for c*=volocity of Hght.

May we not then regard those traditional co variances for c' = oo only as an approximation consistent with experience, the actual covariance of natural laws holding for a certain finite value of e.

I may here point out that by if instead of the Newtonian Relativity-Postulate with c~oc^ we assume a relativity- postulate with a finite c, then the axiomatic construction of Mechanics appears to gain considerably in perfection.

The ratio of the time unit to the length unit is chosen in a manner so as to make the velocity of light equivalent to unity.

While now I want to introduce geometrical figures in the manifold of the variables ( , y, z, t)^ it may be convenient to leave {y, ~) out of account, and to treat .r and t as any possible pair of co-ordinates in a plane, refered to oblique axes.

APPENDIX 55

A space time null point 0 (.r, y, :, r = 0, 0, 0, 0) will be kept fixed in a Lorentz transformation.

The figure-.r^-^'^-z2+?2 = l, i'>0 ■ (£)

which represents a hjper boloidal shell, contains the space- time points A {iv, y, z, / = 0, 0, 0, 1), and all points A' whicli after a Lorentz-transformation enter into the newly introduced system of reference as {.r , y' , J, /'=0, 0, 0, !).

The direction of a radius vector OA' drawn from 0 to the point A' of ("2), and the directions of the tan<?ents to {%) at A' are to be called normal to each other.

Let us now follow a definite position*. of matter in its course thi'ough all time t. The totality of the space-time points (', y, :, f) which correspond to the positions at different times t, shall be called a space-time line.

The task of determining the motion of matter is com- prised in the following problem: — It is required to establish for every space-time jioiut the direction of the space-time line passing through it.

To transform a space-time point P {x^ y, :, i) to rest is equivalent to introducing, by means of a Lorentz transfor- mation, a new system of reference ( ■ ', y' , z' , t'), in which the t' axis has the direction Oc\', OA' indicating the direc- tion of the space-time line passing through P. The space ^' = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P.

To the increment dt of the time of P corresponds the increment

dT=Vdt^'-d.io''-dy^ —d;^=dtVl—ir"

of the newly introduced time parameter /'. The value of the inte.orral

jdT=f V — idx^^+dr^'^+dr^^-^dx^^)

56 PRINCIPLE OF RELATIVITY

when calculated upon the space-time line from a fixed initial point P^ to the variable point P, (both being on the space-time line), is known as the ' Proper-time ' of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional- time which was introduced by Loi'entz for uniform motion.)

If we take a body R* which has got extension in space at time t^, then the region comprising all the space-time line passing through R* and ( „ shall be called a space-time filament.

If wo have an anatylical expression 6{x y^ r, t) so that B{xy y z ^) = 0 is intersected by every space time line of the filament at one pointy — whereby

-(K)"-(f:)-(©HiT)'>»'if>»

then the tolality of the intersecting points will be called a cross section of the filament. •

At any ])oint P of such across-section, we can introduce by means of a Lorentz transformation a system of refer- ence (o', y, :' i)i so that according to this

a® _o 6® _n 9® -0 9® ^0

-^7 -'^' a? ~ ' d^ ~ ' a7" ^

The direction of the uniquely determined ^'— axis in question here is known as the upper normal of the cross- section at the point P and the value of cU—\ f f d.r' dy' dz for the surrounding points of P on the cross-section is known as the elementary contents (Inhalts-element) of the cross-section. In this sense R" is to be regarded as the cross-section normal to the t axis of the filament at the point t=t' y and the volume of the body R" is to be resrarded as the contents of the cross- section.

APPENDIX 57

If we allow R" to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a ease, a spaoe-time line will be thouo^ht of as a principal line and by the term ' Proper- time ' of the filament will be understood the ^ Proper-time ' which is laid along this principal line ; under the term normal cross-section of the filament, we shall understand the cross-section upon the space which is normal to the principal line tbrousfh P. ■

We shall now formulate the principle of conservation of mass.

To every space R at a time t, belongs a positive quantity — the mass at R at the time /. If R converges to a point (c, ^, r, t), then the quotient of this mass, and the volume of R approaches a limit /x(.t, ^, :, t), which is known as the mass-density at the space-time point

The principle of conservation of mass says — that for an infinitely thin space-time filament, the product /xr/J, where /a = mass-density at the point {^, y^ z, t) of the fila- ment {i.e., the principal line of the filament), ^/J=contents of the cross-section normal to the t axis, and passing through (^^, 2r, t), is constant along the whole filament.

Now the contents ^?J„ of the normal cross-section of the filament which is laid through ( r, ?/, r, f) is

vl— ?t2 dr

d the function v= — ^ =/x a^i _ 2 =/x -^ . (5)

may be defined as the rest-mass density at the position 8

58 PEINCIPLE OF RELATIVITY

(xyzt). Then the principle of conservation of mass can be formulated in this manner : —

For an infinite! 1/ thin ^pace-time filament, the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament .

In any space-time filament, let ns consider two cross- sections Q" and Q', which have only the points on the boundary common to each other ; let the space-time lines inside the filament have a larger value of t on Q' than on Q". The finite range enclosed between Q" and Q' shall be called a space-time sichel^ Ql is the lower boundary, and Q' is the upper boundary of the sichel.

If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of the sichel, there corresponds an exit point of the same by the upper boundary, whereby for both, the product vdJ„ taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals /v^/„ (the first being extended over the upper, the second upon the lower boundary) vanishes. According to a well-known theorem of Integral Calculus the difference is equivalent to

//// ^^^' ^^ ^''' ^^y ^~ ^^

the integrration beins: extended over the whole ranofe of the sichel, and (comp. (67), § 1:2)

lor ,-= .§.^ + ^ + 4^^ + ^""^

dx^ Q.Cg 9^3 6.

If the sichel reduces to a jDoint, then the differential equation lor vw=0, (6)

* Sichel — a German word meaning a crescent or a scythe. The original term is retained as there is no snitable English equivalent.

APPENDIX 59

which is the coudition of cortinuitv

a^j dy ' d: 67~'

Further let us form the intefrral

^=S!S!vdulyd:cU (7)

extending over the whole range of the space-time sic/iel. We shall decompose the sic/iel into elementary space-time filaments^ and every one of these filaments in small elements (It of its proper-time, which are however large compared to the linear dimensions of the normal cross-section; let us assume that the mass of such a lilament vdJn=dm and write t", t^ for the ^Proper-time' of the upper and lower boundary of the slc/iel.

Then the integral (7) can be denoted by

// vdJn dT=j (t'-t") dm.

taken over all the elements of the sichel.

Now let us conceive of the space-time lines inside a space-time dcliel as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sichel in the follow- ing manner. The entire curves are to be varied in any possible manner inside the >^icliel, while the end p)oints on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they alwavs move forward normal to the curves. The whole process may be analytically repre- sented by means of a parameter A, and to the value A = o, shall correspond the actual curves inside the sicheL Such a ])rocess may be called a virtual displacement in the sichel.

Let the point (:-^\ i/, z, i) in the sichel X = o have the values i?' -f 8 v, y + 8^^ z + 8-, t + U, when the parameter has

60 PIIINCIPLE OF llELATIVITY

the value X ; these magnitudes are then functions oE {w, j/, Zj I, \). Let us now conceive of an infinitely thin space- time filament at the point (^* f/ z f) with the n'^rmal section of contents r^J„, and if f/J„+8r(?J„ be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass — (v + ^/v being the rest-mass-density at the varied position),

(8) {v-\-hv) {(U „-\-hcUn) — vdi n—fl^it"

In consequence of this condition, the integral (7) taken over the whole range of the sichel, varies on account of the displacement as a definite function N + 8N of X, and we may call this function N + 8N as the mass action of the virtual displacement.

If we now introduce the method of writiuor with indices, we shall have

(9) d{x,A-^:,)=:d,,^-> |^+ ^ d\

k o.'a 6 a

/(• = !, 2, 3, 4

/^ = 1, 2, 3, 4

Now on the basis of the remarks already made, it is clear that the value of N + 8N, when the value of the parameter is A, will be : —

(10) X -I- 8N = \ \ U '^^Kl+Sr ) ^^ ^ j^y ^^^ ^^^^

- \S\S '^

the integration extending over the whole sichel (l{r-\-hT) where ^^(t + St) denotes the magnitude, which is deduced from

^~(tZr,-hfZ8i«J2_^,cZr2+rf8a;2>_((^a;3-{-rf8a;3)2_(cZ,.^+^S,.j2

by means of (9) and

(Ix^-^in^ fhy ^/.«*2=W2 (It, (LVq=w^ (It, (Lv^~oi_^ (It, d\=^0

APPENDIX 61

thfirel'ore : —

dr 0>tA

^•=1, 2,3,4. 4.

[k — 1, 2, 3, 7i=l, 2, 3,

We shall now subject the value of the differential quotient

to a transformation. Since each S'/, as a function of (;r, ^, 0, ^) vanishes for the zero-value of the paramater A., so in

o:eneral ->r— ^ =c, for X = o.

Let us now put ( ^^ ") = ^u (^=1, 2, 3, 4) (13)

A=o

then on the basis of (10) and (11), we have the expression (12) :-

Ms: /9f»„ J. 9^'.,. J.9**,, j.9fA

cZ,i! cZi/ dz dt

for the system {a\ d\, x^ r ^) on the boundary of the sicliel, {hx^ 8i'2 S.rg 3 ^) shall vanish for every value of \ and therefore ^j, |2> ^s? ^4 ^^^ i^^l* Then by partial integration, the intei^ral is transformed into the form

M-(

9'"'i 6 -'2 ^^5 9 '-4 /

(^J3 dy dz dt

6'^ j:>RiNciPLfe oi' helativity

the expression within the bracket may be written as

The first sum vanishes in consequence of the continuity iH{uation (b). The second may be written as

d<Jik ^1 , 9t^A de^ 6oJ/, dj-j , do)k dcj, 9.i'i cIt 9<'2 dr Q'i's dr 9 .c^. dr

_ dijii, _ d^ (drj\ cIt dr \ dr J

wherebv — is meant the differential quotient in the dr

direction of the space-time line at any position. For the differential quotient (1^), we obtain the final expression

dx dij dz dt.

For a virtual displacement in the ^ichel we have postulated the condition that the points supposed to be substantial shall advance normally to the curves jxivins: their actual motion, which is \ = o:, this condition denotes that the ^h is to satisfy the condition

iL\ ^^-\-iL\ ^^-]rio^ ^^-^-w^ ^^=0, (15)

Let us now turn our attention to the Maxwellian tensions in the electrodynamics of stationary bodies, and let us consider the results in § 1'! and 13; then we find that Hamilton's Principle can be reconciled to the relativity postulate for continuously extended elastic media.

APPENDIX

3 i

-iT.

-iT,

-zT,

-iX,

-iY,

— '

iZt

At every space-time point (as in § 18), let a space time matrix of the 2nd kind be known

^11 ^12 ^13 ^1

(16) S= S21 S22 S23 S21

^31 ^32 ^33 ^-^ ^41 S^2 S^3 S4

where X^ Y^ X^,, T^ are real magnitudes.

For a virtual displacement in a space-time siehel (with the previously applied designation) the value of the integral

(17) W+8W:^fffJ(^S,, ^^'^'-^^'"'^ dcdydzdt

a >-h

extended over the whole rans^e of the siehel, mav be called the tensional work of the virtual displacement.

The sum' which comes forth here, written in real magnitudes, is

X.+Y,+Z„+T,+X. -1^' +X, |?i>...Z,-|^^

ox oy 9z

-X,

a ;^ d .(^ a ^

we can now postulate the following 7ainim.um principle in mechanics.

If any sjiace-time Siehel he bounded, then for each virtual displacement in the Siehel^ the snm of the mass- works, and tension ivorks shall always he an eHremnm for that process of the space-time line in the Siehel ivhich actnally occurs.

The meaning is, that for each virtual displacement^

(

c^(-8y-h8W) dX

) -'

(18)

64 PEINCIPLE OF RELATIVITY

By applying the methods of the Caleuhis of Varia- tions, the following four differential equations at onee follow from this minimal principle by means of the trans- formation (11), and the condition (15).

(19) . ^^^- =K, +XW, (h = l, 2, 3 4) \ whence K, =.^-^ + ^ii' + ^^ + ^±^. (20}

O.t'i 0-«'2 0.<'3 0'<'4

are components of the space-time vector 1st kind K=lor S,

and X is a factor, which is to be determined from the

relation wm;=— 1. ^j multiplying (19) by tv^, and

summing the four, we obtain X = K2y, and therefore clearly

K + (Kw)iy will be a space-time vector of the Jst kind which

is normal to w. Let us write out the components of this

vector as

X, Y, Z, • /T

Then we arrive at the following equation for the motion of matter,

(^^> ^J(j:)=^' 'iiry^-' ^1.(1)=^'

v^ (^\ =T, and we have also cLt xdrj

©•- (I)'- (!)•> ©■=-■•

, -^ dx .-yj- dy .r/ dz __ny dt and A — --l-i-^-t-Zi -- = 1 — -.

dr dr ar dr

On the basis of this condition, the fourth of equations {t\) is to be regarded as a direct consequence of the first three.

From (ril), we can deduce the law for the motion of a material point, 2".^., the law for the career of an infinitely thin space-time filament.

APlPENDiX 65

Let X, y, z, tf denote a point on a principal line chosen in any manner within the filament. We shall form the equations (21) for the points of the normal cross section of the filament through .<■, y^ z, t, and integrate them, multiply- ing by the elementary contents of the cross section over the whole space of the normal section. If the integrals of the right side be K^. R^ R, R, and if m be the constant mass of the filament, we obtain

(22) w— — =R,, m- /=Rj,, w— -— =R,, m- ^ =R, uT dr dr dr dr dr dr dr

R is now a space-time vector of the 1st kind with the components (R„ Ry R^ R^) which is normal to the space- time vector of the 1st kind w, — the velocity of the material point with the components

d.e dy dz ■ dt dr ' dr ^ dr * dr '

We may call this vector R f/ie moving force of the material jioinf.

If instead of integrating over the normal section, we integrate the equations over that cross section of the fila- ment which is normal to the / axis, and passes through {■(\y,z,t), then [See (4)] the equations (22) are obtained, but

dr are now multiplied by — ; in particular, the last equa- tion comes out in the form,

dt \ dr / dt dt dt

The right side is to be looked upon as the amount of work done per unit of time at the material point. In this 9

66 PRINCIPLE OF RELATIVITY

equation, we obtain the energy-law for the motion of the material point and the expression

e-')"[7i=. ->]-('j-=+i^**)

may be called the kinetic energy of the material point. Since (It is always greater than cIt we may call the

quotient — - — '^ as the ^^ Gain " (vorgehen) of the time (It

over the proper-time of the material point and the law can then be thus expressed ; — The kinetic energ}- of a ma- terial point is the product of its mass into the gain of the time over its proper-time.

The set of four equations (22) again shows the sym- metry in (^',^,-,0? which is demanded by the relativity postulate; to the fourth equation however, a higher phy- sical si2:nificance is to be attached, as we have alreadv seen in the analoojous case in electrodvnamics. On the ground of this demand for symmetry, the triplet consisting of the first three equations are to be constructed after the model of the fourth ; remembering this circumstance, we are justified in saying, —

" If the relativity-postulate be placed at the head of mechanics, then the whole set of laws of motion follows from the law of energy."

I cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation.

If B^(.('^, ?/"^, e"^, /^) be a solid (fester) space-time point, then the region of all those space-time points B (.r, //, ?, /), for which

(•23) (,.-.,:*)= +(;;_y»)5 +(^_,*)2 =(/-/*)2

APPENDIX 6?

Ill ay be called a ^' Kay- figure " (Strahl-gebilde) of the space lime point B"^.

A space-time line taken in any manner can be cut by this figure only at one particular point ; this easily follows from the convexity of the figure on the one hand, and on the other hand from the fact that all directions of the space- time lines are only directions from B^ towards to the concave side of the figure. Then B^ may be called the light-point of B.

If in (23), the point ( " ^ z I) be su})p«>sed to be fixed, the point (:^•^ j/^ z^ t^) be supposed to be variable, then the relation (:Zo) would represent the locus of all the space- time points B"^, which are light-points of B.

Let us conceive that a material point F of mass m may, owing to the presence of another material point F"^, experience a moving force according to the following law. Let us picture to ourselves the space-time filaments of F and F"^ along with the principal lines of the filaments. Let BC be an infinitely small element of the princi})al line of F ; further let B^ be the light point of B, C^ be the light })oint of C on the principal line of F^; so that OA' is the radius vector of the hyperboloidal fundamental figure (23) parallel to B"^C^, finally D^ is the point of intersection of line B^C^ with the space normal to itself and passing through B. The moving force of the mass- point F in the space-time point B is now the space- time vector of the first kind which is normal to BC, and which is composed of the vectors

(24) mm^f^^'^] BD"^ in the direction of BD^ and another vector of suitable value in direction of B'^C"^.

68 PRINCIPLE OP RELxiTIVITY

Now by ( — if—o ) is to be understood the ratio of the two

vectors in question. It is clear that this proposition at once shows the covariant character with respect to a Lorentz-group.

Let us now ask how the space-time tilament of F behaves when the material point F"^ has a uniform trauslatory motion, /.(?., the principal line of the filament of F* is a line. Let us take the space time null-point in this, and by means of a Lorentz-transformation, we can take this axis as the /-axis. Let x, y, z, /, denote the point B, let T"^ denote the proper time of B^, reckoned from O. Our proposition leads to the equations

d'^z _ ^ m^z (oa\^__ jz!^ d{t-r^) dr'^ ~ {t—r^Y <^^' "" {t-r'^y dt

where (27) .c^ -fj/' 4--?2 = (j{-t^)2

"^<-' (4;)'- (*)'-©■=(!)■-

\\\ consideration of (27), the three equations (25) are of the same form as the equations for the motion of a material point subjected to attraction from a fixed centre according to the Newtonian Law, only that instead of the time t) the proper time t of the material point occurs. The fourth equation (26) gives then the connection between proper time and the time for the material point.

Now for different values of t\ the orbit of the space- point (,(• y z) is an ellipse with the semi-major axis a and the eccentricity e. Let E denote the excentric anomaly, T

APPENDIX 69

the increment of the proper time for a complete description of the orbit, finally nT =27r, so that from a properly chosen initial point t, we have the Kepler-equation

(29) }iT=zE-e sin E.

If we now change the unit of time, and denote the velocity of light by c, then from (28), we obtain

Now neglecting c~* with regard to 1, it follows that 7/ ^ r 1 . i ^'^^ l + ^cosE~|

from which, by applying (29),

(31 ) nt 4- const =f 1 -f- \~ \ nr-\- —,^ SinE.

the factor — ^ is here the square of the ratio of a certain

ac-

average velocity of F in its orbit to the velocity of light. If now m^ denote the mass of the sun, a the semi major axis of the earth's orbit, then this factor amounts to 10~®.

The law of mass attraction which has been just describ- ed and which is formulated in accordance with the relativity postulate would signify that gravitation is propagated with the velocity of light. In view of the fact that the periodic terms in (31) are very small, it is not possible to decide out of astronomical observations between sueh a law (with the modified mechanics proposed above) and the Newtonian law of attraction with Newtonian mechanics.

70 l^RiNClPLE OF RELATIVITY

SPACE AND TIME

A Lecture delivered before the Naturforsclier Yer- sammlung (Congress of Natural Philosophers) at Cologne — (21st September, 1908). ,

Gentlemen,

The eoneeptious about time and space, which I hope to develop before you to-day, has grown on experimental physical grounds. Herein lies its strength. The tendency is radical. Henceforth, the old conception of space for itself, and time for itself shall reduce to a mere shadow, and some sort of union of the two will be found consistent with facts.

Now I want to show 3 ou how we can arrive at the changed concepts about time and space from mechanics, as accepted now-a-days, from purely mathematical considera- tions. The equations of Newtonian mechanics show a two- fold invariance, (?') their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, {ii) when we change the system in its nature of motion, /. e., when we impress upon it any uniform motion of translation, the null-point of time plays no part. We are accustomed to look upon the axioms of geometry as settled once for all, while we seldom have the same amount of conviction regarding the axioms of mecha- nics, and therefore the two invariants are seldom mentioned in the same breath. Each one of these denotes a certain group of transformations for the differential equations of mechanics. We look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off the second group to itself, and w^ith a lisht heart conclude that we can never decide from physical considerations whether the si)ace, which is supposed to be

APPENDIX 71

at rest, may not finally t>e in uniform motion. So these two groups lead quite separate existences besides each other. Their totally heterogeneous character may scare us away from the attempt to compound them. Yet it is the whole compouuded group which as a whole gives us occasion for thought.

We wish to picture to ourselves the whole relation graphically. Let (,<', y, z) be the rectangular coordinates of space, and t denote the time. Subjects of our perception are always connected with place and time. No one has observed a place e, cept at a pariicnlar iime, or has obserred a time exce^A at a particular place. Yet I respect the dogma that time and space have independent existences. I will call a space-point plus a time-point, i.e., a system of values X, y^ r, /, as a world-point. The manifoldness of all possible values of x, y, z, t, will be the world. I can draw four world-axes with the chalk. Now any axis drawn consists of quickly vibrating molecules, and besides, takes part in all the journeys of the earth ; and therefore giyes us occasion for reflection. The greater abstraction required for the four-axes does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to specify either matter or electricity, we shall simply style these as substances. We direct our attention to the world -point ^, y, z, t, and suppose that we are in a position to recognise this substantial point at any subsequent time. Let dt be the time element corresponding to the changes of space coordinates of this point [d.v, dy, dz]. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point), — a curve in the 2Vorld — the ivorld-line, the points on which unambiguously correspond to the para- meter t from -f 00 to— <^. The whole world appears to be

72 PRINCIPLE OF RELATIVITY

resolved in such 70orld4ineSy and I may just deviate from my point if I say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines.

By this conception of time and space, the (", y, z) mani- foldness t = o and its two sides /<o and t>o falls asunder. If for the sake of simplicity, we keep the null-point of time and space fixed, then the first named group of mechanics signifies that at f — o we can give the ,'•, y, and ^-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression

^2+^2_^

The second group denotes that without changing the expression for the mechanical laws, we can substitute {x — atyy—ptj z—yt^ for ('■, y, z) where (a, ^, y) are any constants. According to this we can give the time-axis any possible direction in the upper half of the woild />o. Now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half ?

To establish this connection, let us take a positive para- meter c y and let us consider the figure

According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by t-=^o. Let us consider the sheet, in the region of ^>o, and let us now conceive the transformation of ,>•, y, z, i in the new system of variables ; (.</, y', z ^ t') by means of which the form of the expression will remain unaltered. Clearly the rotation of space round the null-point belongs to this group of transformations. Now we can have a full idea of the trans- formations which we picture to ourselves from a particular

APPENDTJC 73

transformation in which (y, z) remain unaltered. Let us draw the cross section of the upper sheets with the plane of the .r- and /-axes, i.e., the upper half of the hyperbola <?-/2_,2_.]^ with its asymptotes {vide

fig. I). .

Then let us draw the radius rector OA', the tansrent A' B' at A', and let us complete the parallelogram OA' B' C ; also produce W C to meet the f -axis at D'. Let us now take Ox', OA' as new axes with the unit mea- suring rods 0C' = 1, 0A'= ; then the h^^perbola is again

expressed in the form c^t'-— ■'^ = ], t'>o and the transi- tion from ( r, ;f/j ;, t) to ( ' y'^'t^ is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points ; then we get a group of transformation depending only on c, which we may denote by Gc.

Now let us increase c to infinity. Thus ~ becomes zero

and it appears from the figure that the hyperbola is gradu- ally shrunk into the /-axis, the asymptotic angle be- comes a straight one, and every special transformation in the limit changes in such a manner that the /-axis can have any possible direction upwards, and ,'' more and more approximates to .'''. Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group G^, with the value of c=oo. In this state of affairs, and since Gc is mathematically more in- telligible than G oo, a mathematician may, by a free play of imagination, hit upon the thought that natural pheno- mena possess an invariance not onl}^ for the group G^, but in fact also for a group G^, where c is finite, but yet 10

74 PRINCIPLE OF KELATIVITY

exceedingly large compared to the usual measuring units. Such a preconception would be an extraordinary triumph for pure mathematics.

At the same time I shall remark for which value of c, this invariance can be conclusively held to be true. For c, we shall substitute the velocity of light c in free space. In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electro- static and eleetro-mas:netie units of electricity.

We can form an idea of the invariant character of the expression for natural laws for tlie group-transformation G^ in the following manner.

Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (,r, ^, ^, t) ; by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely deter- mined by the phenomena. JFe can change the system of reference in any possifjle manner corresjjonding to the above- mentioned group transformation Gc, but the expressions for natttral laws ivill not be changed thereby.

For example, corresponding to the above described figure, we can call // the time, but then necessarily the space connected with it must be expressed by the mani- foldness {/ y z). The physical laws are now expressed by means of ■<', y, ^, i' , — and the expressions are just the same as in the case of <<■, y^ z, t. According to this, we shall have in the world, not one space, but many spaces, — quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three- dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning

AtPE^BlX 76

that time and space shall reduce to mere shadows and we shall have a world complete in itself.

Now the question may be asked, — what circumstances lead us to these changed views about time and space, are they not in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena ?

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualised time and space in any manner; then a world-line parallel to the ^-axis will correspond to a stationar}^ point ; a world-line inclined to the /f-axis will correspond to a point moving uniformly ; and a world-curve will corres- pond to a point moving in any manner. Let us now picture to our mind the world-line passing through any world point ■''if/,z,tj now if we find the world-line parallel to the radius vector OA' of the hyperboloidal sheet, then we can introduce OA' as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. AVe shall now introduce this fundamental axiom : —

Th<! ■^lihstance eiisllnij at (uuf world j^oiui can always be conceived to he at rest, if we esta/ilifih. our time wml s^pace xtdtatjlf/. The axiom denotes that in a world-point the expression

ciflfi —dx"^ —fh^ —dz"^

shall always be positive or what is eipiivalent to the same thing, every velocity V should be snialler than c, c shall therefore be the up[)er limit for all substantial velocities and herein lies a deep significance for tlie

76 PJliNClPLE Oi^ KELATlVlTV

quantity c. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in ofeometry.

The im'piihe and real cause of inducement for the assumption of the group-traiuf or }iLatio}i Gc is the fact that the differential equation for the propagation of light in va-^ant spa'je possesses the group-transformation Gc. On the oth-n* hand, the idei of rig^id bodies has anv sense only in a system mechanics with the group G^,.. Now if we have an optics with G,, and on the other hand if there are rigid bodies, it is easy to see that a /^-direction can be defined by the two hyperboloidal shells common to the groups G^^, and G^, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orienta- tions, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have sj'iven nciirative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group G,, According to Lorentz every substance shall suffer a contraction

1 \ V ^ r P^i length, in the direction of its motion

T= "THE ''={'- 3- •

This hypothesis sounds rather }jhaotastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of eundition always accompanying a state of motion.

I shall show in our figure that Lorentz's hypothesis

is fully equivalent to the new conceptions about time and

space. Thereby it may appear more intelligible. Let us

now, for the sake of simplicity, neglect (j/, z) and fix our

attention on a two dimensional world, in which let upright

strips parallel to the ^^-axis represent a state of rest and

another parallel strip inclined to the /.-axis represent a

state of uniform motion for a body, which has a constant

spatial extension (see fig. 1). If OA' is parallel to the second

y strip, we can take f/ as the .-^-axis and x' as the a;-axis, then

the se<^ond body will appear to be at rest, and the first body

in uniform motion. We shall now assume that the first

body supposed to be at rest, has the length /, i.e., the

cross section PP of the first strip upon the .-axis^/* OC,

where OC is the unit measuring rod upon the j^-axis — and

the second body also, when supposed to be at rest, has the

same length I, this means that, the cross section Q'Q' of

the second strip has a cross-section I'OC, when measured

parallel to the ''-axis. In these two bodies, we have

now images of twD Lorentz-electrons, one of which is at

rest and the other moves uniformly. Now if we stick

to our original coordinates, then the extension of the

second electron is given by the cross section QQ of the

strip belonging to it measured parallel to the '-axis.

Now it is clear since a'Q' = ^OC', that QQ = /-OD'.

If -— = r, an easv calculation li'ives that dt "

\/l '

jj ^- I -» ^-k ft ft ^-fc 4- ^-k «* ^ I

OD' = 0C '\' " c2, therefore QQ / v^

' \/ 1— .

78 . PRINCIPLE OF IlELxiTIViTY

This is the sense of Lorentz's hypothesis about the contraction of electrons in ease of motion. On the other hand, if we conceive the second electron to be at rest, and therefore adopt the system (.0', i\) then the cross-section PT' of the strip of the electron parallel to OC is to be regarded as its length and we shall find the first electron shortened with reference to the second in the same propor- tion, for it is,

P'P' _0D _0p'_ QQ

(ra'~oc'~oc - pp

Lorentz called the combination /-' of {t and ,* ) as the local ti'tie {Ortszeit) of the uniformly moving electron, and used a physical construction of this idea for a better compre- hension of the contraction-hypothesis. But to perceive clearlv that the time of an electron is as ijood as the time of any other electron, i,e. t, i' are to be regarded as equi- valent, has been the service of A. Einstein [Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis... 4-4-1 1—1907] There the concept of time was shown to be completely and un- arabio'uouslv established bv natural phenomena. But the concept of space was not arrived at, either by Einstein or Lorentz, probably because in the case of th^ above- mentioned spatial transformations, where the ( </, /') plane coincides with the ••'-/ plane, the significance is possible that the -^-axis of space some-how remains conserved in its position.

We can approach the idea of space in a corresponding manner, though some may regard the attempt as rather fantastical.

AccordiniT to these ideas, the word '' Relativitv-Pastu- late'' which has been coined for the demands of invariance in the group G, seetus to be rather inexpressive for a true understanding of the group Gc, and tor further progress.

APPENDIX 79

Because the sense of the postulate is that the four- dimensional world is given in space and time by pheno- mena only, but the projection in time and space can be handled with a certain freedom, and therefore I would rather hke to ojive to this assertion the name " The Post uJ ate of the Ahsohde worliV [World- Postulate].

Ill

By the world-postulate a similar treatment of the four determining quantities .r, ?/, 0, t, of a world-point is pos- sible. Thereby the forms under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more strikins: and clear.

I shall agai!i use the geometrical method of expression. Let us call any world-point O as a " Spaee-time-null- point.'' The cone

consists of two parts with O as apex, one part having /<0', the other having />0. The first, which we may call t\\e fore-cone consists of all those points which send light towards O, the second, which we ma}' call the aft-cone. consists of all those points which receive their light from O. The region bounded by the fore-cone may be called the fore-side of O, and the region bounded by the aft-cone may be called the aft-side of O. [Vide fig. 2).

On the aft-side of O e have the already considered hyperboloidal shell F = c^^ -x^- -y- —z"" = '[, t>0.

80 PRINCIPT^E OP EELATIVITY

The region inside the two cones will be occupied by the hyperboloid of one sheet

where k^ can have all possible positive values. The hyperbolas which lie upon this fiss'nre with O as centre, are important for us. For the sake of clearness the indivi- dual branches of this hyperbola will be called the " Inter- hi/perbola with centra- 0^ Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for /=— oo and t = oo^ asymptotically approaches the velocit}^ of light c.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness i',^,z,l a vector, then we have to distinguish between a time-vector directed from O towards the sheet +F=1, ^>Oand a space-vector directed from O towards the sheet —F=l. The time-axis can be parallel to any vector of the first kind. Any world-point between the fore and aft cones of O, may bv means of the system of reference be res^arded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily always earlier, every point on the aft side of O, later than O. The limit c = oo corresponds to a com- plete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness / = 0. In the fiojure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards {a',]/,z,t) into its components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces -|-F=1, and of a tangent RS

APPENDIX ' 81

at the point R of the surface^ then the vectors shall be called normal to each other. Accordinsjlv

which is the condition that the vectors with the com- ponents ((', y, Zy t) and {x ^ y^ z^ t^) are normal to each other.

For the measurement of vectors in different directions^ the unit measuring rod is to be fixed in the following manner; — a space-like vector from 0 to — F = I is always to have the measure unity, and a time-like vector from

O to +F= 1 , />0 is always to have the measure — .

Let us now fix our attention upon the world-line of a substantive point running through the world-point (t, y, z, t) ; then as we follow the -progress of the line, the quantity

c corresponds to the time-like vector-element {clc, dy, dz, dt).

The integral T= fr/r, taken over the world-line from

any fixed initial point P^ to any variable final point P, may be called the " Proper-time " of the substantial point at Po upon t,he icorld-line. We may regard (r, y, z, t), i.e., the components of the vector OP, as functions of the

" proper-time " r; let (.r, y^ i, 0 denote the first differential- quotients, and {x, y\ z, f) the second differential quotients

of ( ', 'f, -, t) with regard to r, then these may respectively 11

82 • PRINCIPLE OF RELATIVITY

be called the Velocity-vector, and the Accelercition-vector of the substantial point at P. Now we haye

••• •«• ••• •••

c2 t t ^ X X — y y — z ^=0

i.e., the ' Velocity'Vector ' is the time-like vector of unit measure in the direction of the world-line at P, the ' Accele- ration-vector^ at P is normal to the velocity-vector at P, and is in any case, a space-like vector.

Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a 'fore-cone^ and an 'aft-cone.' This hyperbola may be called the " hyperbola of curvature " at P (^vide fig. 3). If M be the centre of this hyperbola, then we have to deal here .with an ' Inter-hyperbola ' with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is a vector

c^ . of magnitude — in the direction of MP.

If .r, y, z, t are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and p=oc.

In order to demonstrate that the assumption of the crroup Gc fo^' ^^^® physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successful!}' made in the case of " Thermodjmamics and

APPENDIX 80

Radiation,"^ for "Eleetromagnetie phenomena '^,t and finally for "Mechanics with the maintenance of the idea of mass."

For this last mentioned province of physics, the ques- tion may be asked : if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-

• • • •

point (c?', y, z, f), v^rhere the velocity-vector is (r, y, Zj t),

then how are we to resrard this force when the svstem of reference is changed in any possible manner ? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group G^ is undoubtedly permissible. These theorems lead us to the following simple rule ; if the i^ijdem of 'reference he changed in an// loay^ then the supposed force is to be put as a force in, the new sjMce- coordinates in such a manner, that the corresponding vector with the components

tX ^'Y, tZ, tT,

• « •

ivhere T=— f4x + ^Y + ^z"^ = ^ {the rate of c^ \ t t t ) c^

tohicli work is done at the toorld-point), remains unaltered. This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a moving force-vector at P.

Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the

* Planck, Ziir Dynamik bewegter systeme, Ann. d. physik, Bd. 26,

1908, p. 1.

f H. Minkowski ; the passage refers to paper (2) of the present

edition.

84 PRINCIPLE OF RELATIVITY

impidse -vector, and m-iimes the acceleration-vector at P as the force-vector of motion^ at P. According- to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector"^ :

The force-vector of motion is equal to the moving force- vector.

This enunciation comprises four equations for the com- ponents in the four directions, of which the fourth can be deduced from the first three, because both of the above- mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the " Ener2:y-law.'" Accordingly c'^ -times the component of the impulse-vector in the direction of the t-axis is to be defined as the hinetic-energ)/ of the point- mass. The expression for this is

v^-^

i.e., if we deduct from this the additive constant w<?-, we obtain the expression \ inv^ of Newtonian-mechanics upto

magnitudes of the order of -^. Hence it appears that the

energij depends upon the system of reference. But since the ^-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whoL.^ system of equations of motion. This fact retains its significance even in the limiting: ease C=oo, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Sehiitz.t

* Minkowski — Mechanics, appendix, page 65 of paper (2).

Planck— Yerh. d. D. P. G. Vol. 4, 1906, p. 136. t Schutz, Gott. Nachr. 1897, p. 110.

APPENDIX 85

From the very beginning, we can establish the ratio

between the units of time and space in such a manner, that

the velocity of light becomes unity. If we now write

a/HI t = lj in the place of I, then the differential expression

dr"- = -(c?ic2 +%2 +(/2;2 +^^2)^

becomes symmetrical in (.- , 3/. ^, /) ; this symmetry then enters into each law, which does not contradict the ?rr)rA/- 2J0stnla{e. We can clothe the " essential nature of this postulate in the mystical, but mathematically significant formula

• The advantages arising from the formulation of the world-] )0.>tulate are illustrated bv nothing so strikinglv as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner accord- ing to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge [e), and let us introduce upon it the '^ Propjr-time " r reckoned from any possible initial point. In order to obtain the field caused by tlie electron at any world-point P^ let us construct the fore-cone belonging to Pj {vide fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P^ the normal to this tangent. Let f be the measure ofP,Q. According to the definition of a fore-cone, rje is to be reckoned as the measure of PQ. Now at the world-point Pj,

86 PHINCIPLE OF RELATIVITY

the vector-potential of the field excited by e is represented by the vector in direction PQ., having the magnitude

cr i

; in its three space components along the x-j y-, c-axes ;

the scalar-potential is represented by the component along the ^-axis. This is the elementary law found out by A. Lienard, and E. Wiechert."^"

If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed ; the two forces considered together bears some analogy to the force-screw in mechanics ; the analog}^ is, however, im- perfect.

I shall now describe the ponder omoiive force whicJi is exerted hij one moving electron upon Q7iother moving electron. Let us suppose that the world-line of a second point- electron passes through the world-point Pj. Let us determine P, Q, r as before, construct the middle-point M of the hyperbola of curvature at P, and finally the normal MN upon a line through P which is parallel to QPj. With P as the initial point, we shall establish a system of reference in the following way : the /-axis will be laid along PQ, the a -axis in the direction of QP^. The ^'-axis in the direction of MN, then the r-axis is automatically determined, as it is normal to the .» -, t/-, ^-axes. Let

;c, 1/, Zy /be the acceleration-vector at P, x^^y^^z^^t^

be the velocity-vector at P^. Then the force-vector exerted by the first election r^ (moving in any possible manner)

* Lienard, L'Eolairage electriqne T.16, 1896, p. 53, Wiechert, Ann. d. Physik, Vol. 4.

APPENBIX 87

upon the second election e, (likewise moving in any possible manner) at Pj is represented by

For the coiujwnenls F,^ Fy, F:, Ft of the vector F the folloiving three relations hold : —

cF,-F.= i,F,= 4-,F.=0,

and fourthly this vector F is normal to the velocity -vector P^, a]id through this circumstance alone, its dependence on this last velocity -vector arises.

I£ we compare with this expression the previous for- mulie"^ giving the elementary law about the pouderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions ; but in three dimensions, they have very com- plicated projections.

In the mechanics reformed according to the world- postulate, the disharmonies which have disturbed the relations between Newtonian mechanics, and modern electrodynamics automatically disappear. I shall now con- sider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses 7}i and m^ describe their world-lines ; a moving force-vector is exercised by m upon m^, and the expression is just the same as in the case of the electron, only we have to write ■\-mm^ instead of— 6'6'i. We shall consider only the special case in which the acceleration-vector of m is always zero ;

* K. Schwarzschild. Gott-Nachr. 1903.

II. A. Lorentz, Enzyklopadie der Math. Wisscnschaftcn V. Art 14, p. 199.

88 PHINCIPLE OF RELATIVITY

then i may be introduced in such a manner that m may be regarded as fixed, the motion of w. is now subjected to the moving.force vector of m alone. If we now modify this

• 1

given vector by writing .. instead of / (? = 1 up

to magnitudes of the order —17 ), then it a})pears that

Ke2:)Ier\s laws hold good for tlie position {^n^i, ^j), of m^ at any time, only in place of the time t^, we have to write the proper time t^ oi m^. On the basis of this simple remark, it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the Newtonian law of attraction in combination with Newtonian mechanics.

Also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate. I shall also show on a later occasion that the deduction of these equations, as taught by Lorentz, are by no means to be given up.

The fact that the world-postulate holds without excep- tion is, 1 believe, the true essence of an electromagnetic picture of the world ; the idea first occurred to Lorentz, its essence was first picked out by Einstein, and is now gradu- ally fully manifest. In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification oi' the postulate ; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts^ will be reconciled to the new ideas of time and space, — in the prospect that they will lead to pre-established harmony between pure mathematics and physics.

The Foundation of the Generalised Theory of Relativity

By a. Einstein.

From Annalen der Physik 4.49,1916.

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as " the theory of Relativity ; " this latter theory I differentiate from the former "Special Relativity theory," and suppose it to be known. The generalization of the Relativity theory has been made much easier through the farm given to the special Rela- tivity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already com- plete in the "Absolute Differential Calculus/' which were based on the researches of Gauss, Riemann and Christoffel on the tibn-EucHdean manifold, and which have been shaped into a system by Rieci and Levi-civita, and already applied to the problems of theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend Grossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation. > ?

90 PRINCIPLE OF EELATIVITT

Principal considerations about the Postulate

OF Relativity.

§ 1. Remarks on the Special Relativity Theory.

The special relativity theory rests on the following poetulate which also holds valid for the Gialileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when re- ferred to it^ the physical laws hold in their simplest forms these laws would be also valid when referred to another system of co-ordinates K' which is subjected to an uniform trauslational motion relative to K. We call this postulate ** The Special Kelativity Principle.'' By the word special, it is sij^nilied that the principle is limited to the ease, when K' has nniform trandatory motion with reference to K., but the equivalence of K and K' does not extend to the ease of no n -uniform motion of K' relative to K.

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postu- late, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lore nz- transfor- mation, with all the relations between moving rigid bodies and clocks.

The modification which the theory of space and time has undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. According to the special relativity theory, the theorems of geometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest, and more generally the theorems of kinematics, as theorems which describe the relation between measurable bodies and

GBNEEATJSED THEORY OF RELATIVITY 91 .

clocks. Consider two material points of a solid body at rest ; then according' to these conceptions their corres- ponds to these points a wholly definite extent of length, independent of kind, position, orientation and time of the body.

Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system ; then to these positions, there always curresponds, a time-interval of a definite length, independent of time and place. It would be soon shown that the general rela- tivity theory can not hold fast to this simple physical significance of space and time.

§ 2. About the reasons which explain the extension of the relativity-postulate.

To the- classical mechanics (no less than) to the special relativity theory, is attached an episteomologioal defect, which was perhaps first clea»'ly pointed out by E. Mach. We shall illusti*ate it by the following example ; Let two fluid bodies of equal kind and magnitude swim freeh^ in space at such a great distance from one another (and from all other masses) that only that sort of gravitational forces are to be taken into account which the parts of any of these bodies exert upon each other. The distance of the bodies from one another is in\^riable. The relative motion of the different parts of each body is not to occur. But each mass is seen to rotate by an observer at rest re- lative to the other mass round the connecting line of .the masses with a constant angular velocity (definite relative motion for both the masses). Now let us think that the surfaces of both the bodies (S^ and S.J are measured with the help of measuring rods (relatively at rest) ; it is then found that the surface of S^ is a sphere and the surface of the other is an ellipsoid of rotation. We now

92 PRINCIPLE OF BELATIVITT

ask, why is this difference between the two bodies ? An answer to this question can only then be regarded As satis- factory from the episteomological standpoint when the thin 2: adduced as the cause is an observable fact of ex- perience. The law of causality has the sense of a definite statement about the world of experience only when observable facts alone appear as causes and effects.

The Newtonian mechanics does not give to this question any satisfactory answer. For example, it says ! — The laws of mechanics hold true for a space R^ relative to which the body S^ is at rest, not however for a space relative ta which S3 is at rest. , ^

The Galiliean space, which is here introduced is how- ever only a purely imaginary cause, not an observable thing. It is thus clear that the Newtonian mechanics does not, in the case treated here, actually fulfil the requirements of causality, but produces on the mind a fictitious com- placency, in that it makes responsible a wholly imaginaryi cause Ri for the different behaviours of the bodies S, and Sg which are actually observable.

A satisfactory explanation to the question put forvvard above can only be thus given : — that the physical system- composed of S^ and S^ shows for itself alone no con- ceivable cause to which the different behaviour of S, and Sg can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specially the forms of S^ and Sg must be of such a kind, that the mechanical behaviour of S^ and S^ must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as* regards the bodies under con- sideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours-

GENERALISED THEORY OE RELATIVITY .93 ,

of the bodies under consideration. They take the place of the imaginary cause R^. Among all the conceivable spaces Ri and Rg moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording c reater advantages, against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate.

Besides this momentous episteomological argument> there is also a well-known physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional- region considered) a ma^s at a sufficient distance from other masses move uniformly in a line. Let K' be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative tq K' any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and ihq direction of acceleration is independent of its material com- position and its physical conditions.

Can any observer, at rest relative to K', then conclude that he is in an actually accelerated reference-system ? This is to be answered in the negative ; the above-named behaviour of the freely moving masses relative to K' esu} be explained in as good a manner in the following way. The reference-system K' has no acceleration. In the space- time region considered there is a gravitation-fiekl which generates the accelerated motion relative to K'.

This conception is feasible, because to us the experience of the existence of a lield of force (namely the gravitation field) has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. The

94 PRINCIPLE OF RELATIVITY

raecbapica] behaviour of the bodies relative to K' is the i^me as experience would expect of them with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can be assumed that the systems K and K' can both with the same legitimacy be taken as at rest, that is, they will be equivalent as systems of reference for a description of physical phenomena.

From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation ; for we can " create " a gravitational field by a simple variation of the co-ordinate system. Also we see immediately that the principle of the constancy of light- velocity must be modified, for we recognise easily that the path of a ray of light with reference to K' must be, in general, curved, when light travels with a definite and constant velocity in a straight line with reference to K.

§ 3. The time-space continuum. Requirements of the general Co-variance for the equations expressing the laws of Nature in general.

In the classical mechanics as well as in the special relativity theory", the co-ordinates of time and space have an immediate ph3^sical significance ; when we say that any arbitrary point has .>\ as its X^ co-ordinate, it signifies that the projection of the point-event on the X^-axis a»certained by means of a solid rod according to the rules of Euclidean Geometry is reached when a definite measur- ing rod, the unit rod, can be carried ,e^ times from the origin of co-ordinates along the X^ axis. 4 point having r^ — t-^ as the X^ co-ordinate signifies that a unit clock which is adjusted to be at rest relative to. the system of co-ordinates, and coinciding in its spatial position , with the

GENERALISED THEORY OY RELATIVITY '95

point-event and set according to some definite standard has gone over .v^=i periods before the occurence of the point-event.

This conception of time and space is continually present in the mind of the physicist, though often in an unconsci- ous way, as is clearly recognised from the role which this conception has played in physical measurements. This conception must also appear to the reader to be lying at the basis of the second consideration of the last para- graph and imparting a sense to these conceptions. But we wish to show that we are to abandon it and in ireneral to replace it by more general conceptions in order to be. able to work out thoroughly the postulate of general relati- vity,—the case of special relativity appearing as a limiting case when there is no gravitation.

We introduce in a space, which is free from Gravita- tion-field, a Galiliean Co-ordinate System K ( < , y, z, t) and also, another system K' (y' y' z' t') rotating uniformly rela- tive to K. The origin of both the systems as well as their 2-axes might continue to coincide. We will show that for a space-time measurement in the system K', the above established rules for the physical significance of time and space can not be maintained. On grounds of symmetry it is clear that a circle round the origin in the -XY plane of K, can also be looked upon as a circle in the plane (X', Y') of K'. Let us now think of measuring the circum- ference and the diameter of these circles, with a unit measuring rod (infinitely small compared with the raidius) and take the quotient of both the results of measurement. If this experiment be carried out with a measuring rod at rest relatively to the Galiliean system K we would get TT, as the quotient. The result of measurement with a rod relatively at rest as regards K' would be a number which is greater than tt. This can be seen easily when we

96 PRINCIPLE OF RELATIVITY

regard the whole measurement- process from the system K and remember that the rod placed on the periphery suffers a Loreuz-contraction, not however when the rod is placed along the radius. Euclidean Geometry therefore does not hold for the system K' ; the above Hxed concep- tions of co-ordinates which assume the validity of Euclidean Greometry fail with regard to the system K'. We cannot similarly introduce in K' a time corresponding to physical requirements, which will be shown by all similarly prepared clocks at rest relative to the system K'. In order to see this we suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle, and considered from the stationary system K. According to the well-known results of the special relativity theory it follows — (as viewed from K) — that the clock placed at the periphery will go slower than the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light will see the clock at the periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he will interpret his observation by saying that the clock on the periphery actully goes slower than the clock at the origin. He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position.

We therefore arrive at this result. In the oreneral relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time- like co-ordinate difference with the aid of a normal clock.

The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a

GENERALISED THEORY OP RELATIVITY 1^7

definite way, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate sjstem to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation of the laws of Nature. So that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena. This amounts to the following law:* —

That in general^ Laws of I^ature are e:f pressed hy means of equations which are valid for all co-ordinate systems^ that is, which are covariant for all possible transformations. It is clear that a physics which satisfies this postulate will be unobjectionable from the standpoint of the general relativity postulate. Because among all substitutions there are, in every case, contained those, which correspond to all relative motions of the co-ordinate system (in three dimensions). This condition of general covarianee which takes away the last remnants of physical objectivity from space and time, is a natural requirement, as seen from the following considerations. All our icelUsnhstantiated space-time propositions amount to the determination of space-time coincidences. If, for example, the event consisted in the motion of material points, then, for this last case, nothing else are really observable except the encounters between tw^o or more of these material points. The results of our measurements are nothing else than well-proved theorems about such coincidences of material points, of our measuring rods with other material points, coincidences between the hands of a clock, dial-marks and point-events occuring at the same position and at the same time.

The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences. We fit to the world our space- time variables 13

98 PEINCIPLE OF ELLATIVITY

(•^1 '^8 '"s '^4) such that to any and every point-event^ corresponds a system of values of (tj r^ ,(3 .c^). Two co- incident point-events correspond to the same value of the variables {.c^ x^ x^ -i'^) ; i.e., the coincidence is cha- racterised by the equality of the co-ordinates. If we now introduce any four functions (./i i\ t'g t;'^) as co- ordinates, so that there is an unique correspondence between them, the equality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence of two material points. As the purpose of all physical laws is to allow us to remember such coinci- dences, there is a priori no reason present, to prefer a certain co-ordinate system to another ; i.e., we get the condition of o^eneral covariance.

§ 4. Relation of four co-ordinates to spatial and ^ time-like measurements.

Analytical expression for the Gravitaiion»field.

I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible, with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonable according to the light of experience. In this sense, we shall now introduce the following supposition; that for an infinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen.

The nature of acceleration of an infinitely small (posi- tional) co-ordinate system is hereby to be so chosen, that the gravitational field does not apipear; this is possible for an infinitely small region. Xi, Xg, Xg are the spatial

- /

GENERALISED THEORY OF RELATIVITY 99

co-ordinates ; X^^ is the corresponding time-co-ordinate measured by some suitable measuring clock. These co- ordinates have, with a given orientation of the S3^stem, an immediate physical significance in the sense of the special relativity theory (when we take a rigid rod as our unit of measure), llie expression

(1) ds'^ = -dX,^ -dX^ 2 -dX^ ' +^X^ •

had then, according to the special relativity theory, a value which may be obtained by space-time measurement, and which is independent of the orientation of the local co-ordinate system. Let us take ds as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. If ds"^ belonging to the element (^Xj dX^fdX^, ff'^i) he positive we call it with Minkowski, time-like, and in the contrary ease space-like.

To the line-element considered, i.e., to both the infi- nitely near point-events belong also definite differentials <^Xj, d.c^, dx^, do^, of the four-dimensional co-ordinates of any chosen system of reference. If there be also a local system of the above kind given for the case under consi- deration, dX's would then be represented by definite linear homogeneous expressions of the form

(2) dX =^ a dx

V / V (T vcr (T

If we substitute the expression in (1) we get

(3) ds''='^ g d.v d.v

where a will be functions of .c, but will no longer depend

(TT

upon the orientation and motion of the 'local' co-ordinates; for ds^ is a definite magnitude belonging to two point- events infinitely near in space and time and can be got by

100 PEIXCIPLE OP HELATIVITY

measurements with rods and clocks. The g 's are hereto

ht so chosen, that n =n - the summation is to be

extended over all values of o- and t, so that the sum is to he extended, over 4x4 terms, of which 12 are equal in pairs.

From the method adopted here, the ease of the usual relativity theory comes out when owing to the special behaviour of ff in 2i> finite region it is possible to choose the

system of co-ordinates in such a way that g assumes

eonstanf values —

--1, 0, 0, 0

{*)

0-100 0 0-10 0 0 0+1

Wfe would afterwards see that the choice of such a system of co-ordinates for a finite region is in general not possible.

From the considerations in § 2 and § H it is clear, that from the physical stand-point the quantities g are to

be looked upon as magnitudes wliich describe the gravita- tion-field with reference to the chosen system of axes. We assume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for some particular choice of co-ordinates. Tiie g 's then

have the values given in (4). A free material point moves with reference to such a system uniformly in a straight- line. If we now introduce, by any substitution, the space- time co-ordinates x^ ...-^'4, then in the new system g ^s are

no longer constants, but functions of space and time. At the same time, the motion of a free point-mass in the new

GENEEALTSED TITEOBY OF RELATIVITY lOl

co-ordinates, will appear as curvilinear, and not uniform, in which the law of motion, will be independent of the nature of the moving mass-points. We can thus signify this motion as one under the influence of a gravitation field. We see that the app^^arance of a gravitation-field is con- nected with space- time variability of g ^s. In the general

ease, we can not by any suitable choice of axes, make special rela^^ivity theory valid throughout any finite region. We thus deduce the conception that g *s describe the

gravitational field. According to the general relativity theory, gravitation thus plays an exceptional role as dis- tinguished from the others, specially the electromagnetie forces, in as much as the 10 functions g representing

gravitation, define immediately the metrical properties of

the four-dimensional region.

Mathematical Auxill\kies for Establishing the General Covartant Equations.

We have seen before that the general relativity-postu- late leads to the condition that the system of equations for Physics, must be C9- variants for any possible substitu- tion of co-ordinates .<,, ... j^ ; we have now to see how such general co-variant equations can be obtained. We shall now turn our attention to these purely matheniati- cal propositions. It will be shown that in the solution, the invariant ds, given in equation (3) plays a fundamental role, which we, following Gauss's Theory of Surfaces, style as the line-element.

The fundamental idea of the general co-v^ariant theory is this : — With reference to any co-ordinate system, let certain things (tensors) be defined by a number of func- tions of co-ordinates which are called the components of

102 PRINCIPLE OF RELATIVITY

the tensor. There are now certain rules according to which the components can be calculated in a new system of co-ordinates, when these are known for the original system, and when the transformation connecting the two systems is known. The things herefrom designated as " Tensors " have further the property that the transforma- tion equation of their components are linear and homogene- ous ; so that all the components in the new s^^stem vanish if they are all zero in the original system. Thus a law of Nature can be formulated by putting all the components of a tensor equal to zero so that it is a general co-variant equation ; thus while we seek the laws of formation of the tensors, we also reach the means of establishing general CO- variant laws.

5. Contra-variant and co-imriant Four-vector.

Contra- variant Four- vector. The line-element is defined by the four components d>' whose transformation law

is expressed by the equation

(5) dx! =^ -^ d.

The dx' '« are expressed as linear and homogeneous func- tion of dr ^s ; we can look upon the differentials of the

co-ordinates as the components of a tensor, which we designate specially as a eontravariant Four-vector. Every- thing which is defined by Four quantities A , with reference to a co-ordinate system, and transforms according to the same law,

(5a) A =^^-^ A

GENEEALISBD TilEORT OF RELATIVITY 103

we may call a contra- variant Four-vector. From (5. a), it follows at once that the sums (A 4 ^ ) ^^^ ^^so com- ponents of a four-vector, when A^ and B*^ are so ; cor- responding relations hold also for all systems afterwards introduced as " tensors " (Rule of addition and subtraction

of Tensors).

Co-variant Four-vector.

We call four quantities A as the components of a co- variant four- vector, when for any choice of the contra- variant four vector B (6) > A B = Invariant.

V V

From this definition follows the law of transformation of the CO- variant four-vectors. If we substitute in the right band side of the equation-

^ A' B*^ =^ A

cr cr V V

the expressions

a ^ 0- a,,.

•

for B following from the inversion of the equation (5a) we get

^ B^ ^ — ^ A =^ B^ A'

*^(r »' 9-13 , V ^ <^

As in the above equation B are independent of one another

and perfectly arbitrary, it follows that the transformation

law is : —

A' =^ ^ A

- ^ 9v '

lU* PRINCIPLE OF RELATIVITY

: fiemafka on the simplification of the mode of loriting the expressions. A glance at the equations of this paragraph will show that the indices which appear twice within the sign of summation [for example v in (5)] are those over which the summation is to be made and that gnly .over the indices which appear twice. It is therefore possible, without loss of clearness, to leave off the summation sign ; so that we introduce the rule : wherever the index in any term of an expression appears twice, it is to be summed over all of them except when it is not oxpress- edly said to the contrary. •

The difference between the co- variant and the contra- variant four- vector lies in the transformation laws [ (7) and (5)]. Both the quantities are tensors according to the above general remarks ; in it lies its significance. In accordance with Rieei and Levi-eivita, the contravariafits and co-variants are designated by the over and under indices.

§ 6. Tensors of the second and highei ranks.

Contra variant tensor : — If we now calculate all the 16

products A^ of the components A'^ B^ , of two eon- travariant four- vectors

a'**', will according to (8) and (5 a) satisfy the following transformation law.

(9) A^ = -^--^ -^ A^^^

We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transfor- mation relation (9), a contra variant tensor of the second

GENERALISED THEORY OF RELATIVITY l5h

rank. Not every such tensor can be built from two four- vectors, (according to 8). But it is easy to show that any

16 quantities A'^^, can be represented as the sum of A'^

B of properly chosen four pairs of four-vectors. From it, we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through (9), by proving it only for the special tensor of the type (8).

Contravariant Tensor of anij rank : — If is clear that corresponding to (8) and (9j, we can define contravariant tensors of the 3rd and higher ranks, with 4^, etc. com- ponents. Thus it is clear from (8) and (9) that in this sense, we can look upon contravariant four-vectors, as eontra variant tensors of the first rank.

Co'Variant tensor.

If on the other hand, we take the 16 products A of

the components of two co. variant four-vectors A and B ,

(10) A =A B .

for them holds the transformation law

(J T

By means of these transforma;tion laws, the co-variant tensor of the second rank is defined. All re-marks which we have already made concerning tbe contravariant tensors, hold also for co- variant tensors.

Remark : —

It is convenient to treat the scalar Invariant either as a contravariant or a co-variant tensor of zero rank. 14

106 . PKINCIPLE OF RELATIVITY

Mixed tensor. We can also define a tensor of the second rank of the type

(12) a' =AB''

which is co-variant with reference to ^ and contravariant with reference to v. Its transformation law is

(13) a" = -s- • a- ^

Naturally there are mixed tensors with any number of co-variant indices, and with any number of contra- variant indices. The co-variant and contra-variant tensors can be looked upon as special cases of mixed tensors.

Symmetrical tensors : —

A contravariant or a co-variant tensor of the second or higher rank is called symmetrical when any two com- ponents obtained by the mutual interchange of two indices

IXV

are equal. The tensor A or A is symmetrical, when

> .

we have for any combination of indices

(U) A''''=A''''

or (14a) A =A .

It must be proved that a symmetry so defined is a property independent of the system of reference. It follows in fact from (9) remembering (14)

A"^ = — -^ I A'^*'=: - ~ A^^^ A^"^

y. V /A. V

GENEHALISED THEORY OF EELATIVITY 107

Aniiiymmetriaal tensor.

A contravariant or co-variant tensor of the 2nd, 3r(l or ■ith rank is called antuy mmetrical when the two com- ponents got by mutually interchanging any two indicjs

are equal and opposite. The tensor A or A is thus

an tisy mmetrical when we have

(15) A''*' = -A'''^

(15a) A =~.A

Of the 16 components A'^ , the four components A'^^ vanish, the rest are equal and opposite in pairs ; so that there are only 6 numerically different components present (Six-vector).

Thus we also see that the antisymmetrical tensor

^^^ (3rd rank) has only 4 components numerically

different, and the antisymmetrical tensor A only one.

Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions.

§ 7. Multiplication of Tensors.

Outer multiplication of Tensors : — We get from the components of a tensor of rank z^ and another of a rank -', the components of a tensor of rank {z-^z') for which we multiply all the components of the first with all the components of the second in pairs. For example, we

108 ' PRINCIPLE OF RELATIVITi'

obtain the tensor T from the tensors A and B of different kinds ; —

T = A B ,

fxvcr /xv (T

The proof of the tensor character of T, follows imme- diately from the expressions (8), (10) or (12), or the transformation equations (9), (11), (13); equations (8), (10) and (12) are themselves exaftiples of the outer multiplication of tensors of the first rank.

Reduction in rank of a 7mxed Tenmr.

From every mixed tensor we can tret a tensor which is two ranks lower, when we put an index of eo- variant character equal to an index of the contravariant character and sum according to these indices (Reduction). We get for example, out of the mixed tensor of the fourth rank

A , the mixed tensor of the second rank

A =A =(SA )

/5 a^ V^ a/3/

and from it again by '* reduction " the tensor of the zero rank

A= A = A

The proof that the result of reduction retains a truly tensorial character, follows either from the representation

GENERALISED THEORY OF RELATIVITY 109

of tensor according to the generalisation of (12) in combi- nation with (6) or out of the generalisation of (13).

Inner and mixed muUiplicatiori of Tensors.

This consists in the combination of outer multiplication with reduction. Examples: — From the co-variant tensor of the second rank A and the contravariant tensor of

the first rank B we get by outer multiplication the mixed tensor

o" or

D = A B .

Through reduction according to indices v and o- {I.e., put- ting v = a"), the co-variant four vector

V y

D = D = A B is generated.

These we denote as the inner product of the tensor A

^ fXV

and B . Similarly we get from the tensors A and B^^ through outer multiplication and two-fold reduction the inner product A B^*' . Through outer multiplication

and one-fold reduction we get out of A and B^^ , the

^ jXV '

mixed tensor of the second rank D = A B^*" . We

can fitly call this operation a mixed one ; for it is outer with reference to the indices ju, and t, and inner with respect to the indices v and q-.

110 PRINCIPLE OF RELATIVITY

We now prove a law, which will be often applicable for proving the tensor-character of certain quantities. According

to the abuve representation, A B is a scalar, when A and B are tensors. We also remark that when A B is

an invariant for every choice of the tensor B , then A

has a tensorial character.

Proof : — According to the above assumption, for any substitution we have

A , B-^" =A B'^''.

err fxv

From the inversion of (9) we have however

9 a; / 9^' ^

O" T

Substitution of this for B'^*' in the above equation gives 9 ^ 9 .{' t

(^ err a a-^, 9 ^V f"" )

This can be true, for any choice of B only when the term within the bracket vanishes. From which by referring to (11), the thtorem at once follows. This law correspondingly holds for tensors of any rank and character. The proof is quite similar, The law can also be put in, the

following from. If B'^ and C are any two vectors, and

OENEEALISED THEOEY OF RELATIVITY 111

if for every choice of them the inner product A ^ B C

is a scalar, then A is a co-variant tensor. The last

law holds even when there is the more special formulation,

that with any arbitrary choice of the four- vector B alone

the scalav product A B'^ B is a scalar, in which case

we have the additional condition that A satisfies the

symmetry condition. According to the method givien above, we prove the tensor character of (A 4- A ), from

which on account of symmetry follows the tensor- character of A . This law can easily be generalized in the case of

CO- variant and contravariant tensors of any rank.

Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind

of tensor : If the quanties A B form a tensor of the

first rank, when B is any arbitrarily chosen four-vector, then A is a tensor of the second rank. If for example,

C'* is any four-vector, then owing to the tensor character

of A B*' , the inner product A C'^ B is a scalar,

both the four- vectors C and B being arbitrarily chosen. Hence the proposition follows at once.

A few words about the Fundamental Tensor g .

The co-variant fundamental tensor — In the invariant expression of the square of the linear element

ds^-=ig dx dx

112 PRINCIPLE OF RELATIVITY

(U plays the role of any arbitarily chosen eontravariant vector, since further g —q , it follows from the eonsi-

[XV '^ VfX

derations of the last paragraph that g is a symmetrical

co-variant tensor of the second rank. We call it the " fundamental tensor/^ Afterwards we shall deduce some properties of this tensor, which will also be true for any tensor of the second rank. But the special role of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravita- tion makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor.

The co-variant fundamental tensor.

If we form from the determinant scheme I a \ the

minors of ^ and divide them by the determinat ^= | g j

we get certain quantities g^^ = g^^ , which as we shall prove generates a eontravariant tensov-

Accordino: to the well-known law of Determinants

'»

(16) ,^„r^i'

where o is 1, or 0, according asV = ^ or not. Instead

of the above expression for ds^ y we can also write

a S d.v dx

-" IX<T y fX V

or according to (16) also in the form

goo dx dx

GENERALISED THEORY OF RELATIVITY

118

Now according to the rules of multiplication, of the fore- going paragraph, the magnitudes

d^ ^q dx

foims a co-variant four-vector, and in fact (on account of the arbitrary choice of dx ) any arbitrary four- vector.

If we introduce it in our expression, we get

ds^ ^g^'^d^^ r/|^.

For any choice of the vectors d^ d^ this is scalar, and

(TT

g , according, to its defintion is a symmetrical thing in o-

(TT

and T, so it follows from the above results, that g is a contravariant tensor. Out of (16) it also follows that S

is a tensor which we may call the mixed fundamental tensor.

Determinant of the fundamental tensor.

According to the law of multiplication of determinants, we have

^ 9

= \ 9^J \ 9

On the other hand we have

^/xa*^

So that it follows (17) that 15

ixv

fXV

= 1.

114

PRINCIPLE OF RELATIVITY

Invariant of volume.

We see k first the transformation law for the determinant

i^= \9

fA.V

According to (II)

9'^

9 .'

^/XV

Frorti this by applyiug the law of mutiplication twice, we obtain.

9' =

a.r

^„

a .r /

V/^

a ^^

9V'

... V'"^/

On the other hand the law of transformation of the volume element

dT'=fdx^ dr^ dr^ dx^^ is aecordinff to the wellknown law of Jacobi.

dr'^

d ' c

P-

... (B)

by multiplication of the two last equation (A) and (B) we

get.

(18)

= Vg £Zt'= Vg dr.

Insted of ^g, we shall afterwards introduce \/^g which has a real value on account of the hyperbolic character of the time-space continuum. The invariant ^'ZTgdr, is equal in magnitude to the four-dimensional volume-element

GENERALISED TtlEORY OF RELATIVITY 115

measured with solid rods and clocks, in accordance with the special relativity theory.

EemarJcs on the character of the spacC'time conthnmrn — Our assumption that in an infinitely small region the special relativity theory holds, leads us to conelude that ds^ can always, according to (1) be exprersed in real magni- tudes r/X,..Y/X . If we call dr o t^Q ^' natural ^* \o\\xme eleinent ^Xj r/Xg ^Xg d^^ we have thus (18a) ^t.

Should \/ —g vanish at any point of the four-dimensional continuum it would si^nifv that to a finite co-ordinate volume at the place corresponds an iiifiuitely small " natural volume." This can nevei' be the ca^e ; so that g can never chan^(? i's sijLin; we would, according to tlie special relativity thtory assume that ff has a finite negative value. It is a hypothesis about the physical nature of the continuum consid^iieJ, and also a pre-establislied rule for tiie choice of co-ordinates.

If however {—g) remains po.-itive and finite, it is clear that the choice of co-ordinatts can be so made that this quantity becomes equal to one. We would afterwards see that sueh a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature.

In place of (18) it fellows then simply that

dr'^d

from this it follows, remembering the law of Jacobi,

(19)

= 1

116 PIIINCIPLE of ilELATltlTY

With this choice of co-ordinates, only substitutions with determinant 1, are allowable.

It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate. We do not seek those laws of nature which are co-variants with regard to the tranformations having the determinant 1, but we ask : what are the general co-variant laws of nature ? First we get the law, and then we simplify its expression by a special choice of the system of reference.

Building up of neio tensors wit/i the help of the fundamental tensor.

Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed.

Example : —

k.= g A

fXV

We would point out specially the following combinations:

A'^' = /" /^ A

A — g g Q ^

jxv ^fxa'^vp

(complement to the, co-variant or eontravariant tensors)

and, B •= a q^'^ A ^ We can call B the reduced tensor related to A .

GENERALlSt:i) THiEOUY OP RELATIVITY

117

Similarly

It is to be remarked that g is no other than the " com- plement " of ^ , for we have, —

§ 9. Equation of the geodetic line (or of point-motion).

As the " line element *' ds is a definite magnitude in- dependent of the co-ordinate system, we have also between two points Pj and P.2 of a four dimensional continuum a line for which ItU is an extremum (geodetic line), i.e., one which has got a significance independent of the choice of co-ordinates.

Its equation is

(20)

LP. J

From this equation, we can in a wellknown way deduce 4 total differential equations which define the geodetic line ; this deduction is given here for the sake of completeness.

Let A_, be a function of the co-ordinates x^ ; This

defines a series of surfaces which cut the geodetic line sought-for as well as all neighbouring lines from P, to P^. We can suppose that all such curves are given when the vahie of its co-ordinates x^ are siven in terms of \. The

ii8

PRINCIPLE OF RELBTIVriTY

sign S corresponds to a passage from a point of the geodetic curve soiight-for to a point of the contiguous curve, both lying on the same surface A,.

Then (20) can be replaced by

8w d\-0

(20a)

dr dx ty*=qf — L

But

uA c^A

V dA ^ rfA ^ j

So we get by the substitution of hw in (^Oa), remem-

bering that

^ d\ ^

± (Be )

after partial integration,

(20b)

d\ k Bx =0 <r or

^ ( 3 ^^

where k =-— < . — -

,<^ dX I w dX

fXV

2w a

y-.^ .

c^j;

dX dX

GENEllALfSED THEORY OF RELATIVITY 1 I9

From which it follows, since the choice of 8 * is per- fectly arbitrary that k \ should vanish ; Then

(20c) k =0 (cr=l, 2, 3, 4)

are the e|uations of geodetic line; since along the geodetic line considered we have ^5=^0, we can choose the parameter A, as the length of the arc measured along the geodetic line. Then w = ], and we would get in place of •(20c)

^ ^v ^> ^

^t^v a*" d^c^ ds ds

1 dg d'^ 6^

_i _/^ Ij^ ? -_0.

2 Q.f 6* 6^

Or by merely changing the notation suitably,

d^x - - dx dr-

(20d) g -/ + \^'^ -J^ . -r =0

where we have put, following Christoffel,

.on M -1 r ®^'"^+ ®'''"^- ®^'''''!

(TT

Multiply finally (^Od) with g (outer multiplication with reference to t, and inner with respect to <r) we gtt at last the final form of the equation of the geodetic line —

' d^x ( ^ d^' da

ds^ (t ) ^* ds

Here we have put, following Christoffel,

120 PRIXCIPLE OF EELATIVITY

§ 10. Formation of Tensors through Differentiation.

Relying on the equation of the ;^eodetie line, we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation. For this purpose, we would first establish the general co-variant differential equations. We achieve this through a repeated application of the following simple law. If a certain curve be given in our continuum whose points are character- ised by the arc-distances s. measured from a fixed point on the curve, and if further <f>, be an invariant space function",

then ~ is also an invariant. The proof follows from as

the fact that d<f> as well as ds, are both invariants

Since

d(f> __ 6 <^ At

ds Qx Q s

so that i/a= ~— ' ~- is also an invariant for all curves OX ds

^hich go out from a point in the continuum, i.e., for any choice of the vector d.c . From which follows imme- diately that

A = -M

is a co-variant four-vector (gradient of ^).

According to our law, the differential-quotient x= -S-^

taken along any curve is likewise an invariant. Substituting the value of if/, we get

9;c Qa' ds ds 9»' ds^

GENERALISED THEORY OF RELATIVITY

Here however we can not at once deduce the existence of any tensor. If we however take that the curves along which we are differentiating are geodesies, we get from it

by replacing

17^

according to (22)

dnV d'X

ds ds

Prom the interchan^eabilitv of the differentiation with regard to /x and v, and also according to (23_) and (21) we see

that the bracket

and V.

l-i

is sj'mmetrical with respect to ^

As we can draw a geodetic line in any direction from any

point in the continuum, — -^ is thus a four-vector, with an

arbitrary ratio of components, so that it follows from the results of §7 that

(25)

A =

_ 6'<A

ft V

is a co-viiriant tensor of the second rank. We have thus got the result that out of the co-variant tensor of the first rank

A = 5-^ we can get by differentiation a co- variant tensor

of 2nd rank

(26)

ixv

dA (

ixv

1^-^ PRINCIPLE OF RELATIVITY

We call the tensor A the *' extension " of the tensor

A . Then Ave can easily show that this combination also

leads to a tensor, when the vector A is not representable

as a gradient. In order to see this we first remark that

o^ .^ ^ co-variant four-vector when \p- and tjy are

acalars. This is also the case for a sum of four such terms : —

when j/^^^), <^(i)...«i^(4) ^(4) are scalars. Now it is however clear that every co- variant four- vector is representable in the form of S

If for example. A is a four-vector whose components are any given functions of i« , we have, (with reference to the chosen co-ordinate system) only to put

i/.W=A3 <3S»(3)=,i53 in order to arrive at the result that S is equal to A .

fX fX

In order to prove then that A in a tensor when on the

right aide of (26) we substitute any co-variant four-vector for A we have only to show that this is true for the

GENERALISED THJIORY OF RELATIVITY H3

four-vector S . For this latter case, however, a glance on

the right hand side of (26) will show that we have only to bring forth the proof for the case when

Now the right hand side of (25) maltiplied by i/^ is

which has a tensor character. Similarlv, 5^ -S-^ is

' 6.'- 6a; /^ ^

also a tensor (outer product of two foui'- vectors). Through addition follows the tensor character of

Thus we get the desired proof for the fonr-vector,

*A ^ J^^d hence for any four-vectors A as shown above. /^

With the help of the extension of the four- vector, we can easily define ''extension" of a co-variant tensor of any rank. This is a generalisation of the extension of the four- vector. We confine ourselves to the case of the extension of the tensors of the 2nd rank for which the law of for- mation can be clearly seen.

As already remarked every co- variant tensor of the 2nd rank can be represented as a sum of the tensors of the type A B .

1*24 PRINCIPLE OF RELAT1\'ITY

It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions

aA ( )

V \ (XV

" cr y. "T )

are tensors. Through outer multiplication of the first with B and the 2nd with A we ffet tensors of the

V fX ^

third rank. Their addition gives the tensor of the third rank

A =:^Z£^-\''''] A -{""Ia ... (27)

/xi'cr

^\ It) " (t) '^^

where A ^ is put=:A B . The right hand side of (27)

is linear and homogeneous with reference to A .and its

fb-st differential co-efficient so that this law of foi-mation leads to a tensor not only in the case of a tensor of the type A

B but also in the case of a summation for all such

tensors, ^,e.J in the case of any co-variant tensor of the

second rank. We call A the extension of the tensor A .

fxva fxv

It is clear that (26) and (24) are only special cases of

(27) (extension of the tensors of the first and zero rank).

In general we can get all special laws of formation of

tensors from (27) combined with tensor multiplication.

GENER.ALISED THEORY OF RELATiVlEY

1 25

Some special cases of Particular Importance.

A few auxiliary lemmas concerning the fwlda mental tensor. We shall first deduce some of the lemmas much used afterwards. Accoi'diiig to the law of differentiation of determinants, we have

(28) dg=:g^'' gdg^^=^g^^ gdg^"" .

The last form follows from the first when we remember that

a qf^^z=^^ , and therefore a g^^ = -1. consequently g dg^^-^g^^ dg =0-

From (28), it follows that

(29)

■i 9. *^

>»' 6-

Again, since g q =8 . we have, by differentiation,

a da ^=^--q dq

(30) i '"" -

OQ vcr ^

jxcr

a.>

err

By mixed multiplication with g and .7 v respectiyely we obtain (changing the mode of writing the indices).

126

Principle of uelativity

(31)

dg^'^=:—gf^"- /^ dg

ay3

.Z^"

6<7 fta v/? J

# and

(32)

rfa =—(7 (( n dg "^

6(7 a tt/^

The expression (31) allows a transfonnation which we shall often use; according to (21)

(33)

[

" /5 (T

a -'

If we substitute this in the second of the formnla (31), we get, remembering (23),

(34)

flV

i MT > T Cr f , VT ^ T

By substituting the right-hand side of (34) in (29), we

get

(29a)

Generalised 'fHEOny of relativity 1:^7

Divergence of the contravarimit four -vector.

Lefc us multiply (26) with the con ti'a variant fnndaniental

tensor ^'^^^(inner multiplication), then by a transformation of the first member, the right-hand side takes the form

9(/^ 1 Tu / ^'^a

According to (31) and (29). the last member can take the form

Both the first members of the expression (B), and the second member of the expression (A) cancel each other, since the naming of the summation-indices is immaterial. The last member of (B) can then be united with fii»st of (A). If we put

r A^ = A^ ■

where k^ as well as A are vectors which can be arbi- trarily chosen, we obtain finally

^=:

( V'-^g A^' ) .

This scalar is the Divergence of the contravariant four- Toctor A ,

128 PRINCIPLE OP UELATIVITY

^ notation of the [covariant) fowr^vector.

The second membw in (26) ie symmetrical in the indiceR /A, and V, Hence A ,— A is an antisymmetrical tensor

built up in a very simple manner. We obtain

6A 6A

^^^^ ^Mr= -^~ -^ S/

Anti.si/mmefTical Extension of a Six-reHor.

If Ave apply the operation (27) on an antisymmetrical tensor of the second rank A , and form all the equations

arising from the cyclic interchange of the indices /a, v, cr. and add all them, we obtain a tensor of the thini rank

6A (37) B =:A + A + A = ~~^

^ ^ fxya {lycT - vatx (t/xv Q^

aA 6A

+ "L^^ ^/^

6 '<>• 6 •«

fX V

from which it in easy to see that the tensor is antisymmetri- cal.

Divergence of the Six-vector.

If (27) is multiplied by ^'^^ ^*'' (mixed multiplication), then a tensor is obtained. The first member of the right hand side of (27) can be written in the form

GKXEIJALISED THEORY OV RELATIVITY 1:29

If we replace g^^^ 7^^ A by A ,/ q^^'' 1/^' A by

A ' and replace in the transformed first member

with the help of (•'^4), then from the right-hand side of (27) there arises an expression with seven terms, of which four cancel. There remains

(38) a"^= %^- + i"^ '} A''-/^+ ^^ " j A''^

This is the expression for the extension of a contravariant tensor of the second rank; extensions can also be formed -for corresponding- contravni'iant tensors of higher and lower ranks.

We lemark that in the same way, we can also form the

extension of a mixed tensor A

^^ ^^ ^} .^ ^' ^^ .-

r39) A- = ---/" - ^ A -f ^ A .

By the reduction of (38) with reference to the indices (3 and o- ( inner multiplication with 6 I , we get a con- travariant four-vector

130 PRINCIPLE OF itELATlVITY

On the account of the symraetrv of -^ • witli

■ ( " )

reference to the indices (3, and k, the third member of the

right hand side vanishes when A '^ is an antisymmetrical tensor, which we assume here ; the second member can be transformed according to (29a) ; we therefore get

(40) ^/^-g dx^

This is the expression of the direro'ence of a contra - variant six-vector.

Divergence of the mixed tensor of the second rank.

Let us form the reduction of (89) with reference to the indices a and <r, we obtain remembering (29a)

Tf we introduce into the last term the contravariant tensor A" =17" A , it takes the fori

[or ^

If further A'^ is symmetrical it is reduced to

CIENEllALISED THEORY OF KEf-ATIVlTY 13i

If instead of A' , we iiitrocliice in a similar way the symmetrical co-variant tensor A ■=.g g r* A ^ , then owing to (31) the last member can take the form

In the symmetrical case treated, ("11) can be replaced by either of the forms T

6 ( v^-y A-^ )

(Ua) s^-g A = ^

a ( v^-^ A^ )

(41b) ^/-^ A = -^^ .- ^

.P^

+ 1 1^ sf-g A which we shall have to make use of afterwards.

§12. The Riemann-Christoffel Tensor.

We now seek only those tensors, which can be obtained from the fundaiiiental tensor </^ ^by differentiation alone. It is found easily. We put in (37) instead of any tensor A'^*'' the fundamental tensor g^^ and get from

132

PlllNClPLE OF IIELATIVITY

it a new tensor, namely the extension of the fundamental tensor. We ean easily convince ourselves that this vanishes identically. We prove it in the following way; we substitute in (27)

i.e. J the extension of a four- vector.

Thus we get (by slightly changing the indices) the tensor of the third rank

a'^A (iKT^b^ CfJiT^ dA. ((XT') 6A

Mcrr a.^6.', l^ ^a., I, 56.^ Ip ^ 6.^

•f

6.'

fid') (flT

p ) (a

acr

We ,use these expressions for the formation of the tensor

— A

/i.<7T

/xTcr

Therebv the followin"r terms in A

fJi(TT

cancel the corresponding terms in A ; the lirst member, the fourth member, as well as the member corresponding to the last term within the square bracket. These are all symmetrical in o-, and r. The same is true for the sum uf the second and third members. We thus get

fxar

A = B^ A

jXTCr fJ.CT p

(^3)^

fXCTT

iidf

_6_ ^/XT

to- ) tp J (a

GENERALISED THEOllY OF RELATIVITY 138

The essential thing in this result is that on the right hand side of (42) we have only A , but not its

differential co-efficients. From the tensor-character of A

— A , and from the fact that A is an arbitrary four vector, it follows, on account of the result of §7, that B ^ is a tensor (Iliemann-Christoft'el Tensor).

fXO-T

The mathematical signilicance of this tensor is as follows; when the continuum is so shaped, that there is a

co-ordinate system for which o 's are constants, B^ all

vanish.

If we choose instead of the oriijiual co-ordinate svstem

I any new one, so would the ^ 's referred to this last system

be no Ioniser constants. The tensor character of B^

^ /X(TT

shows us, however, that these components vanish collectively also in any other chosen system of reference. The vanishing of the Riemann Tensor is thus a necessary con- dition that for some choice of the axis-system </ 's can be taken as constants. In our problem it corresponds to the ease when b}^ a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region. By the reduction of (i-i) with reference to indices to T and p, we get the eo variant tensor of the second rank

B =R 4-S

fXV fiv flV

S = Q tog- \/^j _ >/^^7 9 lug ^/.Zy ^ p. V v.a J a

134 PlllXCrPLE OF EELATIVITY

Eemnrks upon the choice of co-ordinates. — It has already been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so chosen that ^ — </ = 1. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplifica- tion. It is specially true for the tensor B , which plays a fundamental role in the theory. By this simplifica- tion, S vanishes of itself so that tensor B reduces to

[XV

I shall give in the following pages all relations in the jriimplified form, with the above-named specialisation of the co-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any special ease.

C. THE THEORY OF THE GRAVITATION-FIELD

§13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation.

A freely moving body not acted on by external forces moves, according to the special relativity theory, along a straight line and uniformly. This also holds for the generalised relativity theory for any part of the four-dimen- sional region, in which the co-ordinates Ko can be^ and

are, so chosen that (j /s have special constant values of

the expression (4).

Let us discuss this motion from the stand-jmint of any arbitrary co-ordinate-system K;; it moves with reference to Kj (as explained in ^'l) in a gravitational field. The laws

• GEXEllALISED TIIEOJJY OF RELATIVITY 135

of motion with reference to K, follow easily from the following consideration. With reference to K^,, the law of motion is a four-dimensional straight line and thus a geodesic. As a geodetic-line is defined independently of the system of co-ordinates, it would also be the law of motion for the motion of the material-point* with reference to Kj ; If we put

(45) p^ ^ _

• we get the motion of Ihe point with reference to K^ given by

d ," (1 " (J.v

We now make the very simple assumption that this general covariant system of equations defines also the motion of the point in the gravitational field, when there exists no reference-system K^, with reference to which the special relativity theory holds throughout a finite region. The assumption seems to us to be all the more legitimate, as (46) contains only the first differentials of

(/ , among which there is no relation in the special ease when Kq exists.

If r ^ 's vanish, the point moves uniformly and in a

fJLV

straight line ; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field.

1-36 PllIXClPLE OF llELATIVITi

§14. The Field-equation of Gravitation in the

absence of matter.

In the following, we differentiate gravitation-field from matter in. the sense that everything besides the gravita- tion-field will be signified as matter ; therefore the term includes not only matter in the usual sense, but also the electro-dynamie field. Our next problem is to seek the field-equations of gravitation in the absence of matter. For this we apply the same method as employed in the fore- going paragraph for the deduction of the equations of motion for material points. A special case in w^hich the field-equations sought-for are evidently satisfied is that of

the special relativity theorv in which q 's have certain

fXV

constant values. This would be the case in a certain finite region with reference to a definite co-ordinate system K^,. With reference to this system, all the com- ponents B^^ of the Riemann's Tensor [equation i'3]

vanish. These vanish then also in the region considered, with reference to every other co-ordinate svstem.

The equations of the gravitation-field free from matter

must thus be in everv case satisfied when all & vanish.

But this condition is clearly one which goes too far.. For it is clear that the o^ravitati on -field srenerated bv a material point in its own neighbourhood can never be transformed aivai/ by any choice of axes, i.e., it cannot be transformed

to a case of constant g 's.

Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical ten- sors B deduced from the tensors B„^^ should vanish.

GEXEK-UiTSED THEORY OF RELATIVITY 137

We tlius get 10 equations for 10 cinantities g which are I'ulhlled in the special ease when B^ 's all vanish.

^ fXCTT

Rerae«ibering (44) we see that in a})senee o£ matter the field-eqiiations come out as follows ; (when referred to the special co-ordinate-system chosen.)

6r" .

(47) ^^ + r\ r^ =o;

/ — — 1 r " — ) f^^l

^ -^ ' ' /XL' / " \

It can also be shown that the choice of these equa- tions is connected with a minimum of arbitrariness. For besides B , there is no tensor of the second rank, which

can be built out of a ^s and their derivatives no his/her

fjLV

than the second, and which is also linear in them.

It will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity, together with equations (46), give us the New- tonian law of attraction as a first approximation, and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier (the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors). My view is that these are convincing proofs of the physical

correctness of my theory. 18

138

PRIXCIPLE OF I^ELATIVITY

^15. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy.

In order to sliow that the field equations correspond to the laws of impulse and ener^ry, it is most convenient to write it in the following Hamiltonian form : —

Hr/T = o

(47a)

•' ' VOL

^13

Here the variations vanish at the limits of the finite four-dimensional integration-space considered.

It is first necessary to show that the form (47a) is equivalent to equations (47). For this purpose, let us

consider H as a function of g^^ and g^^' I :■- ^

We have at first

8H=r" r^ 8/^+2/Y'' sr^

ix(3 va fji^ va

= -r" r'^s,r+2r"ga( rrr'^)

But <rrry= -1 3[rr.

.iSX

ar/ \ a.^z .

'' 'xA av \

a.'- a.i\ /

GiENEkALlSED THEORY OE iJELATiVlTY

];39

The terms arising out of the two last terms witiiin the round bracket are of different signs, and change into one another by the interchange of the indices /x and /3. They cancel each other in the expression for 3H, when they are

multiplied by F q, which is symmetrical with respect to

/x and ft so that only the first member of the bracket remains for our consideration. Remembering (31), we thus have : —

Therefore

(48)

r an ^ _ f-a p/5 ^. an

9.^/

fXV f^^

If we now carry out the variations in (47a), we obtain the system of equations

(47b)

a / ^ H \ an

/jti

dg'

which, owing to the relations (48), coincide with (47), as was required to be proved.

If (47b) is multiplied by g^ ,

suice

jXV

d-^

140 PRINCIPLE or uelativitY

aud consequently

9f/

99'

6'/

/Xl'

9 c/ we obtain the equation

6 / «^ 8H \_ 8H _,^

/ui' 9 i<;

- { r ^)

(49)

9-'

.1 ^a /^'''

9ry

8*" H.

Owirrp: to the relations (48), the equations (47) and (34),

(50)

,a I o>a ur _ a ^ ^

-^ ' /x^ ' vo-

lt is to be noticed that /^ is not a tensor, so that the

equation (49) holds only for systems +or which ^/— (^ = 1. This equation expresses the laws of conservation of impulse and energy in a gravitation-held. In fact, the integra- tion of this equation over a three-dimensional volume V leads to the four equations

(49a)

d.v

■{

^ dV ['^

j( C -

+ f a, + / cr

.^"'O

'IS

(JENEE.VLISEID THEORY OE EEEATIVITY Ijl

where a^, a^^ a.^ are the direetion-eosines of the inward- drawn normal to the sarface-elemeiit ^^S in the Euchdean Sense. We recognise in this the usnal expression for the

laws of conservation. AVe denote the maofnitudes t as the

energy-components of the gravitation-field.

I will now put the equation (47) in a third form which will be very serviceable for a quick realisation of our object.

By multiplying the iield-equations (47) with g , these are

obtained in the mixed forms. If we remember that

j'o- 9 r 9 / \ 9 9'

9 r _

a a ' a

which owing to (o4) is e(jual to

) { vo- _ a \ i'/8 __ (J ^ <i

— (I ^

or slightly altering the notation equal to

•^ ^ fta ^ ,uP

9 9-

The third member of this expression cancel with the second member of the field-equations (47). In place of the second term of this expression, we can, on account of the relations (50), put

K i f — — 8 /^j, where t =: f \ fj, 2 /^ / 'i

ii2

i'klNClPLE OF RLLAl'lViTY

Tlierei'ore iii the ])]aee of the equations (47), we obtain

(51)

a-'

{'-

f3 a

v/-r/=i.

§16. General formulation of the field-equation

of Gravitation.

The field-ec[iiations established in the preceding para- graph for spaces free from matter is to be compared with the e((uation v^<^=Oof the Newtonian theory. AVe have now to find the equations which wall correspond to Poisson's Equation \/^(fi = 4TrKp, (p signifies the density of matter) .

The special relativity theory has led to the conception that the inertial mass (Trage Masse) is no other than energ}'. It can also be fully expressed mathematically by a symmetrical tensor of the second rank, the energy-tensor. We have therefore to introduce in our generalised theory

energy-tensor t"' associated with matter, which like the

energy components t _ of the gravitation-field (equations

49, and oO"! have a mixed character but which however can be connected »with symmetrical covariant tensors. The ecpiation (.51) teaches us how to introduce the energy-tensor (corresponding to the density of Poisson's equation) in the field equations of gravitation. If we consider a complete system (for example the Solar-system) its total mass, as also its total gravitating action, will depend on the total energy of the system, ponderable as well as gravitational.

GEXEllALTSED TITEOEY OF RELATIVITY

14:5

This can be expressed, b}^ pnttino^ in (51), in place of energy-components t of li^ravitation-fleld alone the sum of tlie eneri^y-components of matter and gravitation, i.e., t ^ + T^.

fX fX

We thus get instead of (51), the tensor-ecpiation

r a / cr/S a\ (52)^ ^'"aV f^P^

/ o- ■ rr \ 1 a- n

f +T )-:, 8 (f + T) \ fx ix / ^ . u.

I ' V-g=l

,fX

where T=:T (Lane's Scalar). These are the general tield-

ecpiations of gravitation in the mixed form. In place of (47), we get by working backwards the system

/xv 2 -'ixv J

V^g = l.

It must be admitted, that this introduction of the energy-tensor of matter cannot be justified by means of the Relativity-Postulate alone ; for we have in the foregoing analvs's deduced it from the condition that the eners^v of the gravitalion-field should exert gravitating action in the same way as every other kind of (^nergy. The strongest ground for the choice of the above equation however lies in this, that they lead, as their consequences, to equations expressing the conservation ■ of the components of total energy (the impulses and the energy) which exactly correspond to the equations (49) and (4 9a). This shall be shown afterwards.

144 FKTXCTPLE OF HELATTVTTY

^17. The laws of conservation in the general case.

The equations (52) can be easily so transformed that the second member on the right-hand side vanishes. Me reduce (52) with reference to the indices /x and o- and subtract the equation so obtained after multiplication with

i B from (52). We obtain.

V a IX. J

operate on it b}' ^-^ . Now,

9^- / ./3

6 .'' a ■ „

u a

( ''' r;, )

2 d ,r a -'' a tr

aX / '^/xA

9.^/?X ^O..R N -|

.d« a-x

/x A

The first and the third member of the round bracket i lead to expressions which cancel one another, as can be easily seen by interchanging the summation-indices a, and (f, on the one hand^ and /? and A, on the other.

GEXERATJSRD THEOnY OF TJELATTVTTY

14:^

The second term can be transformed according' to (-il). So that we got^

fj^jS

afS

2 6.(„. dXn 6.''

<r p jx

The second member of the expression on the left-hand side of (j^a) leads first to

2 9-'^ d

a fx

( .M^

:, { '■•' r l^ )

(tr

4 9 ,r ^x

a jji

a.i'^

6^/

;('

The expression arisini^^ out of the last member within the round bracket vanishes a<?cordiug to ('^9) on account of the choice of axes. The two others can be taken too'ether and give us on account of (-M)^ the expression

1 6» </"^ "

i 6- 6..« d.

a p jx

So that remembering (54) we have

(55)

a '■ a ->'

1 jjo- A/5

— TV 6 <l '

^Ip ) =^-

identically

1 H) PRINCIPLE OP KELATIA'HY

From (55) and (52a) it follows that (5H) _A ( /^^

9 ^ '^^ + T" I ^ o.

From the field equations of (^gravitation, it also follows that the conservation-laws of impulse and energy are satisfied. AVe see it most simply following the same reasoning which lead to equations (f9a) ; only instead of the energy-components of the gravitational-field, we are to introduce the total energy-components of matter and gravi- tational field.

§18. The Impulse- energy law for matter as a consequence of the field-equations.

If we multiply (53) with ^^- , we get in a way similar to ^15, remembering that

a/xv •^/v ^ — vanishes,

6 / ?^ /^^

the equations __i'" _ i ^0 T' =; o

a cr

or remembering (56) ^

('^7) ^ + i ^^ T =o

a (T

A comparison with (41b) shows that these equations for the above choice of co-ordinates i\/—y = 1) asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter.

(;exeijali>sI':d theory of kelativity li?

Physically the appearance of the second term on the lel't-hand side shows that for matter alone the law of con- servation of impulse and energy cannot hold ; or can only

hold when f/^'''s are constants ; i.e., when the field of gravi- tation vanishes. The second member is an expression for impulse and energy which the gravitation-field exeits per time and per volume upon matter. This comes out clearer when instead of (57) we write it in the Form of (47).

8T^ /-?

The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations of irravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another.

D. THE '' MATEEIAL " PHENOMENA.

The Mathematical auxiliaries developed under ^ B ' at once enables us to generalise, according to the generalised theory of relativity, the physical laws of matter (Hydrody- namics, Maxwell's Electro-dynamics) as they lie already formulated ^ according to the special-relativit^'-theorA'. • The ireneralised Relativitv Principle leads us to no further limitation of ])ossibilities ; but it enables us to know exactly the inHuence of gravitation on all processes with- out the introduction of any new h3q3othesis.

It is owing to this, that as regards the physical nature of matter (in a narrow sense) no definite necessary assump- tions are to be introduced. The question may lie open

118 I'lU^CIl'LE OF KELATIVlTV

whether the theories of the electro-magnetic lield and the gravitational-liekl together, will form a sufficient basis fur the theory of matter. The general relativity postulate can teach us no new principle. But by building up the theorv it must be shown whether olectro-mao-netism and gravitation together can achieve what the former alone did not succeed in doing.

§19. Euler's equations for fhctionless adiabatic

liquid.

Let j^y and p, be two scalars, of which the first denotes the pressure and the last the density of the liuid ; between them there is a relation. Let the contravariant symmetrical tensor

rnap al3 " a ' ^ ^^-ov

T "^ = -(1 ' p + p J- -j^ ... (58)

' as «*■

be the contra-variant energy-tensor of the liquid. To it also belonijrs the covariant tensor

rSSa) T =— V I, -f .V "- rf f, -^ p

as well as the mixed tensor

(581)) T^^--^"- P + g o -^^ ~ P-

If we |)ut the right-hand side of (58b) in (57a) we get the general hydrodynaniieal ei] nations of Euler accord- iuo" to the ireneralised relativity theor\ . This in t)rinciple eom])letely solves the problem of motion ; for the four

GENERALLSEi) THEORY 0¥ RELATIVITY 149

equations (57a) together with the i^iveii e([uatioii between jj and p, and the equation

(If (li'n

^ P _ 1

are sufficient, with the given values of g n, for finding out the six unknowns

dx^ d>\ • (Ix ^ dx^ ^ ^ ^'*' ds ' dn ' c^.s" ' ds

If ^ ^s are unknown we have also to take the equ- tions (53). There are now 11 e([uations for finding out 10 functions // , so that the number is more than suffi- cient. Now it is be noticed that the equation (57a) is ah'eady contained in (53), so that the latter only represents (7) independent equations. This indehniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the [)roblern is indelinite in the sense that three of the S[)ace-functions can be arbitrarily chosen.

§20. Maxwell's Electro-Magnetic field-equations.

Let c^ be the components of a covariant four-vector, the electro-magnetic potential ; from it let us form accord- ing to (36) the Components F of the covariant six-vector

of the electro-maa;netic Held accordinsr to the svstem of equations

(59) F

_ iL — ^

o- p

150 t'RlNXiPLE Oi' KELATIVITY

From (5t))j it follows that the system of ecjuatioiis

(60)

pa-

a.^'

(TT

61^

TO ^ ' =0

is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (HO) contains essentially four equatioDs, which can be thus written : —

(GOa) <

6F^3

a.'-,"

aF,, "a.'.

a-f.

3 i

a.'^

a-'

4- 2

aF,, aF,

a ''3 a. ''4

3 —

^ o

aF,, aF,,

=: O

aF,, _^ aF,3 a_F3i

This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we

])ut

f(3l)

r ^'23 =

l'\. =

■ -i 1*',, =

F,, -

v-l',, -

l\v. -

Instead of (GOa) we can therefore write according to the usual notation of three-dimensional veetor-analvsis : —

(GOl)j

+ l-ot Erz:(.

div H=:o.

CENEEALTSKD TTIEOT^Y OF RELATIVITY

151

Tlie first MaxwelHaii system is obtained bv a genera- lisation of the form given by Minkowski."

We introduce the contra- variant six-vector F ^ bv

a/5

the equation

(62)

^.f^v ^ ^/xa ^^vp ^,

tt^'

and also a contra-variant four-vector J _, which is the electrical current-densitv in vacuum. Then rememberinir (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates).

(63)

fXl^

9''.

^^'

If we put

(64)

' ^2 3 _ JJ'

F^^ = — E'

■{ F^^ = H'„ F^^ = - E'

' F12 - H', F^^

- E'

which quantities become equal to H,. ..E, in the rase of the special relativity theory, and besides

J^ = ^^, ... .7^ = p

we get instead of (63)

(68a)

rot H'-

L div E' = p

152 l>TlTXnVLE OF RELATIVTTY

The equations (60), (62) and (63) give thus a i^enerali- sation of Maxwell's field- equations in Aaeuum, which remains true in our chosen system of* co-ordinates.

TAe eHcv(ju-c(>))ipo}L6nl% of I he el ectro- id a (j netic jichL

Let us form the inner-product

(65) K = F .1^.

According- to (61) its components can be written down in the three-dimensional notation.

I K, ^ pE„-j-[/, H],

\ (65a) I - -

. [ K, = - (i, E). •

K is a covariant four-vector whose components are eqnal

to tlie nes^ative impulse and energy which are transferred to th<^ electro-magnetic Held per unit ol time, and per unit of volume, by the electrical masses. If the electrical masses be free, that is, under the influence of the eleetro- maofnetic field only, then the covariant four-vector

K will vanish. (J

In order to 2:et the energv components T of the elec- tro-magnetic field, we recpiire only to give to the equation K =0, the form of the equation (57).

From (63) and (65) we get first,

K = F

/xv

o- ^\^ ^x

V V

GENERALISED THEORY OF RELATIVITY 153

On aeeoimt of (60) the second member on the risjht-hand side admits of the transformation —

6F d^

V <T

n 8F

1 fxa v/j T-, tiv

— ■> 9 9 ^ o ^^ ■

Owinof to symmotry, this expression can also be written in the form

* L -^ ^ «^ aT~

a«

which can also be put in the form

+ * *a;8 %. aT V " ^ )■

The first of these terms can be written shortly as

X a

- ( Ff'¥ \

9 .

and the second after differentiation can be transformed in the form

J . : ■

6.7

- iF^^^F .. /^

(TT

•20

^54 PRINCIPLE OF RELATIVITY

If we take all the three terras together, we t^et the relation

ax'' dg

V (J

where

(66a) ■ t'^-F F-'V \ f F « F''^.

On aeeount of (30) the equation (66) becomes equivalent

to (57) and (57a) when K vanishes. Thus T 's are the energy-components of the electro-magnetic field. With the help of (61) and (64?) we can easily show that the energy -components of the electro-magnetic field, in the case of the special relativity theory, give rise to the well-known Maxwell-Poynting expressions.

We have now deduced the most general laws which the o'ravitation-field and matter satisfv when we use a co-ordinate system for which \/ —g = 1. Thereby we achieve an important simplification in all our formulas and calculations, without renouncing the conditions of general covariance, as we have obtained the equations through a specialisation of the co-ordinate system from the general c'ovariant-equations. Still the question is not without formal interest, whether, when the energy-components of the gravitation -field and matter is defined in a generalised manner without any specialisation of co-ordinates, the laws of con- servation have the form of the equation (56), and the fiela- equations of gravitation hold in the form (52) or (52a) ; such that on the left-hand side, we have a divergence in the usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation. I have

fetENERALlSED THEORY OF RELATIVITY loS

found out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive work on this subject will not pay, for nothing essentially new comes out of it.

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

We have already mentioned several times that the special relativity theory is to be looked upon as a special case of the s^eneral, in which a ^s have constant values (4).

This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get one important approximation if we consider the case

when (I 's differ from (4) onlv bv small masrnitudes (com-

pared to 1) where we can neglect small quantities of the second and higher orders (first aspect of the approxima- tion.)

Further it should be assumed that within the space- time reojion considered, a 's at infinite distances (using

the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); i.e,, we con- sider only those gravitational fields which can be regarded as produced by masses distributed over finite regions.

We can assume that this approximation should lead to Newton's theory. For it however, it is necessary to treat the fundamental equations from another point of view. Let us consider the motion of a ])article according to the equation (46). In the case of the special relativity theory, the components

<^.<;^ dx^ dx^

ds ds ds

156 PIUKCIPLE OF llELATlVitY

can take any values ; This signifies that any velocity

can appear which is less than the velocity of light in vacuum (i^ <1). If we finally limit ourselves to the consideration of the case when v is small compared to the velocity of liglit, it signifies that the components

dx^ dx^ d,v.

ds ds ' ds

ti >

can be treated as small (juantities, whereas ^- is equal to

1, up to the second-order magnitudes (the second point of view for approximation).

Now we see that, according to the first view of approxi-

mation^ the magnitudes f 's are all small quantities of

at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which /x=v=4.

By limiting ourselves only to terms of the lowest order we get instead of (46)^ first, the equations : —

d^x

= r . .. where ds=dx. =df.

dt^ I 4.x.

or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,

d*:(

di"

^ =[t'] -(^-1,2,:^)

dt'

= -[']■

GENERALiiJEb TltEOltY OE RELATIVITY 15t

If we further assume that the gravitation-iield is quasi-static, i.e., it is limited only to the case when the matter producing the gravitation-field is moving slowly (relative to the velocity of light) we can neglect the differentiations of the positional co-ordinates on the right- hand side with respect to time, so that we get

(67) -^ = - 1^ Oy- (r, = 1, 2, 3)

This is the equation of motion of a material point according to Newton's theory, where ff^^/^ plays the part of gravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the material pointy only the component ^^^ of the fundamental tensor appears. ;

Let us now turn to the field-equation (5o). In this

ease, we have to remember that the energy-tensor of

matter is exclusively defined in a narrow sense by the

density p of matter, i.e., by the second member on the

right-hand side of 58 [(58a, or 5 Sb)]. If we make the

necessary approximations, then all component vanish

except

' T^^ = p = T.

On the left-hand side of (^o) the second term is an infinitesimal of the second order, so that the first leads to the following terms in the approximation, which are rather interesting for us ;

^ f /^i^i , ^ r /xi^i , 6_ r /xvi _ 6_ r p^^']

^y neglecting all differentiations with regard to time, this leads, when /x==v=4, to the expression

' 9'l4

12 3

158 PRINCIPLE OF RELATIVITY

The last of the equations (53) thus leads to

(68) V'cj,,^Kp.

The equations (67) and (68) together, are equivalent to Newton's law of gravitation.

For the gravitation-potential we get from (67) and (68) the exp.

(68a.)

K I pdr

whereas the Newtonian theory for the chosen unit of time

gives

K 1 pdr

where K denotes usually the

gravitation-constant. 6 7 x 10 ^ ; equating them we get (69) K = ^^ =1-87 X 10-2 ^

§22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature of light-rays. Perihelion-motion of the paths of the Planets.

In order to obtain Newton's theory as a first approxi- mation we had to calculate only g^^^ out of the 10 compo- nents (J of the gravitation-potential, for that is the only

component which conies in the first approximate equations of motion of a material point in a gravitational field.

We see however, that the other components of g

should also differ from the values given in (4) as required by the condition y/ = — 1 .

GEXERAUSED THEORY OF RELATIVITY

159

For a heavy particle at the origin of co-ordinates and generating the gravitational field, we get as a first approxi- mation the symmetrical solution of the equation : —

(70)

q == — 8 — a '^ (p and <t 1, 2, 3)

'^pcr pa * ^^ 5 : /

- ^p^^-'Up

•^44

= 1

= 0

(P 1, 2, 3)

S is 1 or 0, according as p=cr or not and r is the quantity pa

On account of (68a) we have

(70a)

47r

where M denotes the mass generating the field. It is easy to verify that this solution satisfies approximately the field-equation outside the mass M.

Let us now investisrate the infiuences which the field of mass M will have upon the metrical properties of the field. Between the lena:ths and times measured locallv on

the one hand, and the differences in co-ordinates dx on the other, we have the relation

ds^ ■= a (I >' d,r .

■' fXV fX V

For a unit measuring rod, for example, placed parallel to the " axis, we have to put

ds^ = — 1, d.r^=zdd'^=:d.i\=:o

then

-1=^11 ^

ir»0 PBJXCIPLE OF KEIATIVITV

m; If the unit measurinio^ rod lies on the < axis, the first of* the equations (~0) gives

1 1

= -(•-.■)■

From both these relations it follows as a first approxi- mation that

(71) ,l" = l- ^ .

The unit measuring rod appears^ when referred to the eo-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational field, when we place it radially in the field.

Similarly we can get its co-ordinate-length in a tangential position, if we put for example

we then get

(71a) — l = f/22 '^K = —<^

2 T 2

The gravitational field has no influence upon the length of the rod, when we put it tangeatially in the field.

Thus Euclidean geometry does not hold in the gravi- tational field even in the first approximation, if we conceive that one and the same rod independent of its position and its orientation can serve as the measure of the same extension. But a glance at (70a) and (69) shows that the expected difference is much too small to be noticeable in the measurement of earth's surface.

We would further investigate the rate of going of a unit-clock Avhich is placed in a statical gravitational field. Here we have for a period of the clock

^9 = 1, d.r^=^di\=^d."^=o -,

GENERALISED THEORY OF REI.ATIVITY 161

then we have

d.,= ,2= = . L ^\-^±^--l

or lU^

= 1+ i' ( P^

8. J r

Therefore the eloek o^oes slowly what it is placed in the neighbourhood of ponderable masses. It follows from this that the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the red end of the spectrum.

Let us further investigate the path of light-rays in a statical gravitational field. According to the special relati- vity theory, the velocity of light is given by the equation

— d^^^ — d,f — rfi; -f-rf.c =o ;

1 2 3 4

thus also according to the generalised relativity theory it is given by the equation

(73) ih^^q d.c d.v =zo.

^ ' jXV fji V

■ t

If the direction, i.e., the ratio d-'^ : d^'.^ • d.i'^ is given, the e(|uation (73) gives the magnitudes

dj'y^ d,v^ dd-^

div^ ' dx^ d.i-^

and with it the velocity,

^^( fe h( k y+( ft H

lf;>i PRIXriPT.K OF RELATIVITY

in the sense of the Enelidean Cjeometry. We can easily see that, with reference to the co-ordinate system, the rays of

light must appear curved in ease y 's are not constants.

If n be the direction perpendicular to the direction of propa<jjation, we have, from Huygen's principle, that light-rays (taken in the plane (y, >?)] must suffer a

curvature -^ -I 9^i ?

A I^iglit-ray

, ), A

Let us find out the curvature which a light-rav suffers when it goes hy a mass M at a distance A from it. If we use the co-ordinate system according to the above scheme, then- the total bending R of light-rays (reckoned positive when it is concave to the origin) is given as a sufficient approximation by

1) >!*

where (7'i) and (70) gives

y = J -a^ = 1 - "L / 1 + l' ")

:i - /(

The oalf'ulntion srives

■!-<'

p_ 2a _ Ol '"" A ~ 2irA

A ray of light just grazing the sun would suffer a bend- ing of J-7'^ whereas one coming by Jupiter would have fi deviation of about '02'^

GENERALISED THEOT?y OF RELATTVTTY 168

If we calculate the oravitation -field to a sjreater order of approximation and with it the corresponding path of a material particle of a relatively small (infinitesimal) mass we set a deviation of the folio wins; kind from the Repler-Newtonian Laws of Planetary motion. The Ellipse of Planetary motion suffers a slow rotation in the direction of motion, of amount

(75 ) .s'= — per revolution.

In this Formula ' a ' signifies the semi-major axis, r, the velocity of light, measured in the usual way, e, the eccentricity, T, the time of revolution in seconds.

The calculation gives for the planet Mercury, a rotation of path of amount 43" per century, corresponding suflii- ciently to what has been found by astronomers (Leverrier). They found a residual i)erihelion motion of this planet of the given magnitude which can not be explained by the perturbation of the other planets.

n1*>(!»?f*!

N0TE5

Note 1. The fundamental oleetro-niaiL^'Udtic e<| nations of Maxwell for stationary media are : —

curl

«-Ua^^-) - ''^

eurl £=- 1 oL? ... (^j

div B=p B=/iH

div Brrro T)-J:F.

AeeordioG;' to Hertz and TIeavIside, Ihese recjuire modi- fleation in the case of moving* bodies.

Now it is known that due to motion alone there h a change in a vector It given by

I — — ) ^^"^ ^^ motion — //, div H -(-eurl TT^w]

where u is the vector velocity of the moving- body and [R?/] the vector product of II and //.

Hence equations (1) and (2) become e curl H= ?i^ -I // div D + curl Veet. [D^*] -f pv (M)

and

-r- curl E= gy- -1-^^ div B + eurl Veot. [Bn] (-M) which gives finally, for p = o and div B = 0, ^~ +u div D=:^- curl (H- 1 Veet. [D?^l ) (1-i)

^1 ^ -.curlfE- ^ Veet. [uH] ) (f-o)

166 PRINriPT.E OF EELATTVITY

Let us consider a beam travellin«j^ along the .?'-axis, with apparent velocity r {i.e., velocity with respect to the fixed ether) in mpdi\im moving with velocity n, = i{ in the same direction.

Then if the electric and magnetic vectors are

i A {x — vt) proportional to e , we have

^-=?A, ^- =— /A?', ^- = ^ —0,v^ — v,—0 ox. ' dt ' dy dz ' "

Then -^=-^'^-— ^^^ ••• (I'-l)

Ot ox Oz ^ ^

and -^T = ~'^' -^~ ^'' ^~ ••• (^'Sl)

Since D = K E and B = /^- H, we have

i A.V {KEy)=-ci A (H,+^^KE,) ... (1-2.2)

i Av (y-U. )=-ci ACE, +u/iB,) ... (2-22)

or viK-7()E,=cli. ... (1-23)

/x (t'-^O H,=:cE, ... (2-23)

Multiplying (1*23), by (2-28)

/x K (l-7^)2=C*^

Hence {v — /f)-=c-/fxk=Vn^

making Fresnelian convection co-efFicient simply unity.

Equations (1*21), and (2"21) may be obtained more simply from j)hysical considerations.

According to Heaviside and Hertz, the real seat of both electric and magnetic polarisation is the moving medium itself. Now at a point which is fixed with respect to the ether, the rate of change of electric polarisation is BD

NOTES 167

Consider a slab of matter moving with velocity n, along- the .r-axis, then even in a stationary field of electrostatic polarisation, that is, for a field in which

-^ =0, there will be some change in the polarisation of

the body due to its motion, given by u r ^- . Hence we

o \;

must add this term to a purely temporal rate of change

-r^ . Doing this we immediately arrive at equations

(1'21) and (2'21) for the special case considered there.

Thus the Hertz- Heaviside form of field equations gives unity as the value for the Fresnelian convection co-efficient. It has been shown in the historical introduction how this is entirely at variance with the observed optical facts. As a matter of fact, liarmor lias shown (Aether and Alatter) that I — 1/V^ is not only sufficient but is also necessary, in order to explain experiments of the Arago prism type.

A short summary of the electromagnetic experiments bearing on this question, has already been given in the introduction.

According to Hertz and Heaviside the total polarisa- tion is situated in the medium itself and is completely carried awav bv it. Thus tlie electromagnetic efPect outside a moving medium should be proportional to K, the specific inductive capacity.

Rowland showed in 18/ G that when a ciiarged condenser is rapidly rotated (the dielectric remaining stationary), the magnetic effect outside is proportional to K, the Sp. Ind. Cap.

^^'^////d^/i (Annalen der Physik 1888, 1890) found that if the dielectric is rotated while the condenser remains stationarv, the effect is proportional to K — 1 .

168 PRINCIPLK OF RELATIVITY

Eichemcakl (Aunaleu der Physik 1905, IVtQi) rotated together botU condenser and dielectric and found that the magnetic effect was proportional to the potential difference and to the aniLi^ular velocity, but was completely independent of K. This i^ of course quite consistent with Rowland and Rontgcu.

Bloiidlot (Comptes llcndus, lUU]) passed a current of air in a steady magnetic field PI ,,, (H =H.. =0). If this current of air moves with velocity //, along the ■r-axis, an electromotive force would be set up along the c;-axib, due to the relative mutioji of matter and magnetic tubes of induction. A pair of plates at .:=+»'/, will be charged up with density p=D,=KE =K. n, Hy/c. BuL Blondlot failed to detect any such eft'ect.

//. ./. )Vihoii (Phil. Trans, lloyal Soc. 1901-) repeated the experiment with a cylindrical condenser made of ebony, rotating in :«, magnetic held parallel to its own axi-^'. Ho observed a change proportional toK— 1 and not to K,

Thus the above set of electro-n)agnetic experiments contradict the Mertz-Hcaviside equations, and these must be abandoned.

I P. (;. M.]

Note 2. Lornniz Tra)i>ifoYiii(i.lU)u,

Lorentz. Versueh einer theorie der elektrisehen uud optitehen Erseheinungon im bewegten Korpern.

(Leiden— 1895).

Lorentz. Theory of Electrons (English edition), ])ages iy7-:iOO, :ioO, also notes 7:j, 86, pages 318, 328.

Lorentz wanted to explain the Michelson-^NIorley null-effect. \\\ order to do so, it was obviously necessary to explain the Eitzgerald contraction. Lorentz worked on the hypothesis that an electron itself undergoes

NOTES 169

contraction when moving. He introduced new variables for the raoving system defined by the following set of equations.

x-=.j^{.v--iit),t/^ =y, z^=z, l'=l3{f-y,^^)

and for velocities, used

v,''=P''u, + i(, Vy'' =/3v,, v..^=Pt\ andpi=p//5.

With the help of the above set of equations, which is known as the Lorentz transformation, he succeeded in showinsc how the P'itzc^erald contraction results as a consequence of " fortuitous compensation of opposing effects."

It should be observed that the Lorentz transformation is not identical with the Einstein transformation. The Einsteinian addition of velocities is quite different as also the expression for the ''relative^' density of electricity.

It is true that the Maxwell-Lorentz field equations remain practically/ uncliauged by the Lorentz transforma- tion, but they arc changed to some sliglit extent. One marked advantage of the Einstein transformation consists in the fact that the field equations of a moving system preserve exactly the same form as those of a stationary system.

It should also be noted that the Fresneliau convection coefficient comes out in the theory of relativity as a direct consequence of Einstein's addition of velocities and is quite independent of any electrical theory of matter.

[P. C. M.]

Note 3.

See Lorentz, Theory of Electrons (English edition), § 181, page tllS.

170 I'JlTXCirLE 01' tlELATIYtTY

H. Poincare, Sur la dynamique 'electron, Rendiconti del circolo matematico di Palermo 21 (1906).

[P. C. M]

Note 4. Iielativitf/ Theorem and 'Relativity 'Principle.

Lorentz showed that the Maxwell-Lorentz system of electromagnetic tield-equations remained practically unchanged by the Lorentz transformation. Thus the electromafrneric laws of Maxwell and Lorentz can he (lefinitehj jiroved " to be independent of the manner in which they are referred to two coordinate systems whicb have a uniform translatory motion relative to each other." (See '' Electrodynamics of Gloving Bodies/^ P^-ge 5.) Thus so far as the electromagnetic laws are concerned, the princi])le of relativity cau he proveiJ to he irue.

But it is not known whether this principle will remain true in the case of other ])hysical laws. We can always proceed on the assumption that it does remain true. Thus it is always possible to construct physical laws in such a way that Ihey retain their f(»rm when referred to moving coordinates, "^riie ultimate ground for formulating physi- cal laws in this way is merely a subjective conviction that the principle of relativity is uuiversally true. There is no rt7;;wy logical necessity that it should be >^o. Hence the Principle of Relativity (so far as it is applied to ohenomena other than electromagnetic) must be resrarded as ^ pn.^tHlafe, which we have assumed to be true, but for which we cannot adduce any definite proof, until after the generalisation is made and its consequences tested in the light of actual experience.

[P. C. M.]

Note 5.

See '' Electrodynamics of Afoving Bodies," p. 5-S.

NOTES

Equations (/) and (//) ])eeoine when oxjiandor] into Cartesians : — -,_

and

6 '»' r

9?/

9^>ii, " 9^'

9-.

9t"

]

9 rn ,

9^'

9 m ,

_9^ 9t =

=pn,

9.e

9 m ,

9.^. 9t -

-pu, j

9e, ~9.r +

dy'^

9. P

(1-1)

... (2'1)

Substituting x^^ x^^ •'■'^i ^^ for .c, y, :, and /r; and pj, ^2* Pai Pa ^0^ P^^j P^^'/5 P^^'j W, where /= y/— 1,

We get,

9w» 9-^^^/ -9'^ '^

.T,

9r<'.H 9.1'

9w^, 9^ 9''t^i 9.'*

J v9^v _ 1

9 my 9i>i., . 9'\

9.'?*i 9'T

*97:='"'-=p= J

and multiplying (2*1) by / we get

... (1-2)

bie^ 9?>y ^ie, 9a^+ 9.<'«+ ^:e,'=''P='P^ ••

... (2'2)

Now substitute

w.=/2 3=-/s8 and /V,=/,, = -/^^

17:Z

PIJIXCTPLE OP EELATIVITY

and we o-et finall

V :■

9.V2 9.>

9a'^

= Pi

9/ai , 9/33 , 9/24

a.-.

6.T.

dx^

y ... (3)

9 /sj 9/3 8 , 9/34

9.''i 9. ^'2 dd'i

= Pi

9/41 ,9/42 9/.

* s

8»'.

6.T

= P.

[P. O. M.]

Note 9. Oh the Condancy of the Velocity of Light.

Pao^e vl — refer also to page C, of Einstein's paper.

One of tlie two fundamental Postulates of the Principle of Relatlvitv is that the velocity of lisfht should remain oonstant whether the source is moving or stationary. It follows that even if a radiant source S move with a velocity ?/, it should always remain the centre of spherical waves expanding outwards with velocity c.

At first sight, it may not appear clear why the velocity should remain constant. Indeed according to the theory of Ritz, the velocity should become c + n, when the source or light moves towards the observer with the velocity n.

Prof, de Sitter has ojiven an astronomical arsjument for decidinoj between these two diverejent views. Let us suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit.

NOTES 173

Let the observer be in the plane of the orbit^ at a great distance A.

The light emitted by the star when at the position A

will be received by the observer after a time , while

c + u .,

the light emitted by the star when at the position B will

be received after a time — . Let T be the real half-

c — u

period of the star. Then the observed half-2:>eriod from B to A is approximately T — '^-— - and from A to B is

T + — — . Now if ~— — be comparable to T, then it

is impossible that the observations should satisfy Kepler^s Law. In most of the spectroscopic binary stars,

— ^^— are not only of the same order as T, but are mostly

much larger. For example, if /i = 100 km /sec, T = 8 days, ^|6' = 33 years (corresponding to an annual parallax of 'l'^)^ then T — '2nAjc^=o. The existence of the Spectroscopic binaries, and the fact that they follow Kepler's Law is therefore a proof that c is not affected by the motion of the source.

In a later memoir, replying to the criticisms of Freundlich and Giinthick that an apparent eccentricity occurs in the motion proportional to ^v.Aq, u^-^ being the

l?t PUINCirLE Of llBLxVnVlrY

maximum value of /', (lie velocity oL' li'^hl emitted bein^

u^ =6' + kiij /(' = 0 Lorentz-Einstein

/•=! Ritz.

. / . Prof, de Sitteradunts the validity of the eritieisms. But

he remarks that aii upper value of k may be calculated from

the observations of the double sar ^-Aurigae. For this star.

The parallax 7r = '011", 6 = -00o, /^,=:110 kwj&eG T = 3-96,

A > 65 light-years,

k is < -OO-^.

Fur an experimental proof, see a paper by C Majorana.

Phil. Mag., Vol. 35, p. 163.

[M. N. S.]

Note 10. Rest-density of Electricity.

\i p is the volume density in a moving system then p\'^{l — u'-) is the corresj)onding cpiantity in the correspond- ing volume in the fixed system, that is, in the system at rest, and hence it is termed the rest-density of electricity.

I'P. C. M.] ' Note 11 (page 17).

Space-time vectors of the fir ■'<f and the second kind.

As we had alreadv occasion to mention, Sommerfeld has, in two papers on four dimensional geometry {vide, Annalen der Physik, Ed. 32, p. 74-9 ; and Bd. 33, p. 649), tj'anslated the ideas of Minkowski into the lanaruaofe of four dimensional geometry. Instead of Minkowski's space-time vector of the first kind, he uses the more expressive term ' four-vector,' thereby making ifc quite clear that it represents a directed quantity like a straight line, a force or a momentum, and has got 4 components, three in the direction of space-axes, and one in the direction of the time-axis.

NOTES J 75

The representation of the plane (defined by two strai^'ht lines) is much more difficult. In three dimensions, the plane can be represented by the vector ])er})endicu1ar to itself. But that artifice is not available in four dimensions. For the perpendicular to a plane, we now have not a sini^le line, but an infinite number of lines constitutimG^; a plane. This diffieultv has been overcome bv Minkowski in a verv elegant manner which w^ill become clear later on. Meanwhile we oifer the followino^ extract from the above mentioned work of Sommerfdd.

(Pp. 755, Bd. :i:2, Ann. d. Physik.)

" In order to have a better knowledge about the nature of the six- vector (which is the same thing as Minkowski's space-time vector of the 2n(l kind) let us take the special ease oP a piece of piano, having unit area (contents), and the form of a parallelogram, bounded by the four-veetors 21, V, passing through the origin. Then the projection of this piece of plane on the :)'// plane is given hy the projections ?/,, ?/,^, r^, r,, of the four veetoi:" in the combination

Let us form in a similar manner all the six components of this plane <A. Then six components are not all indepejident but are connected bv the folio wins' relation

Further the contents | <^ | of the piece of a plane is to be defined as the square root of the sum of the squares of these six cpiantities. In fact,

Let us now on the olhei iiand take the ease of the tinit plane fj>^ normal to </> ; we can call this plane the

176 PlUxXCIPLE OF RELATIVITY

Complement of </>. Then we have the followinoj relations between the components of the two plane : —

The proof of these assertions is as follows. Let ?f^^, ?'"^ be the four vectors defining^ (f>^. Then we have the following relations : —

2i^ n, + n^; Uy + n*; n , -f ?^t ?0 =0

?^* v,-\-u'fj ?',+< Vr.+n'^i vi=0

v't v.+v""^ Vy+v't v^-\-ifl t',=0

I£ we multiply these equations by Vi, Ui, i\, and subtract the second from the first, the fourth from the third we obtain

< ^.i + < <f>yr + n't c^,,=0

multiplying: these equations by rf . ?^*- , or by v* . ?^* . we obtain

from which we have

In a eorrespondinoj w^ay w^e have

when the subscript {il') denotes the component of <^ in the plane contained by the lines other than {ik). Therefore the theorem is proved.

We have (<^ <^^)=<^y, c^*, + ...

= 0

NOTES 177

The general six-veetor / is composed fmm the A^ectors ff>f(f)^ in the followinoj wa>y : —

p and p'^ denotins^ the contents of the pieoos oP mutually perpendicular planes composing f. The ^' conjugate Vector" _/^ (or it may be called the complement of /) is obtained by interchanging p and p^

We have,

/* =/- 4, + ,, 4'*

We can verify that

/!.- =,/:,, etc. and/2 =p2+p*^(/p) = 2pp*

I /' I - and (JX^) may be said to be invariants of the six vectors, for their values are independent of the choice of

the svstem of co-ordinates.

[M. N. S.]

Note 12. Light -V el ocitij a^ a vinxiwuni.

Pasre -23, and Electro-dvnamies of Moving Bodies, p. 17.

Putting v — c — .Vy and iv = c — \, we get

_ 2c — C^' + A)

2c — (.^' + A) + .rA/c

Thus 2^<c, so long as | xX | >0.

Thus the velocity of light is the absolute maximum velocity. We sh^ll now see the consequences of admitting a velocity W > c.

Let A and B be separated by distance /, and let velocity of a ^^sij^nal " in the system S be W>(". Let the

178 PRINCIPLE OF RELATIVITY

(observing) system S' have velocity +?? with respect to the system S.

Then velocity of signal with respect to system S' is

given bv VV = „, , ^

1 - Wv/c^

Thus "time " from A to B as measured in S', is given

Now if v is less thau c, then W being^ Q^reater than c (by hypothesis) W is greater than v, i.e., W>v.

Let W = (? + /x and ?? = <?— X.

Then Wv = (c-\-fjL)(c-\)=c'~+{fj, + \)c-,jcX.

Now we can always choose v in such a way that Wv is greater than c-, since Wv is >c'- if {ix ■\- X)c — jjlX is >0.

that is^ if /x + /\> —. which can always be satisfied by a suitable choice of \.

Thus for W>c we can alwaj-s choose X in such a way as to make Wv>c^f i.e., l—Wr/c- negative. But W— r is always positive. Hence with W>c, we can always make t' , the time from A to B in equation (1) " negative." That is, the signal starting from A Avill reach 15 (as observed in system S') in less than no time. Thus the effect will be perceived before the cause commences to act, i.e., the future will precede the past. Which is absurd. Hence we conclude that W>c is an impossibility, there can be no velocity greater than that of light.

It is conceptually possible to imagine velocities greater

than that of light, but such velocities cannot occur in

reality. Velocities greater than c, will not produce

any effect. Causal effect of any physical type can never

travel with a velocity greater than that of light.

[P. C. M.]

NOTE S

179

Notes 13 and 14.

We have denoted the four-vector w by the matrix

I (o^ oi,2 <^i w^ [ . It is then at onee seen that oo denotes the reciprocal matrix

(Ol

0)2

0),

It is now evident that while co^ =wA, w^=A ^

\jo,s] The vector-product of the four-vector w and -*? may be represented by the combination

[cos]

OJ'J 6'OJ

It is now easy to verify the formula ./^=A'^/A. Supposing for the sake of simplicity that /' represents the vector-product of two four- vectors oi, s, we have

= [A~' w^A— A~'6'wA]

= A-^[«5-5w]A = A-yA. Now remembering that generally

Where o, p^ are scalar quantities, (f>, (^"^ are two mutually perpendicular unit planes, there is no difficulty

/«=A-yA.

Note 15. The vector product (in/). (P. 36). This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of

in seeming that

180 PIUNCU'LE Oh' RELATIVITY

frequent oeeurreuce in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.

^' We can also form a vectorial combination of a four- vector and a six-vector, giving us a vector of the third type. If the six-vector be of a special type, i.e., a piece of plane, then this vector of the third type denotes the parallelopiped formed of this four-vector and the comple- ment of this piece of plane. In the general case, the product will be the geometric sum of two parallelopipeds, but it can always be represented by a four-vector of the 1st type. For two pieces of 3 -space volumes can always be added together by the vectorial addition of their com- ponents. So by the addition of two 3-space volumes, w^e do not obtain a vector of a more general type, but one which can alwavs be represented bv a four- vector (h)C, cit. p. 759). The state of affairs here is the same as in the ordinary vector calculus, where by the vector- multiplication of a vector of the first, and a vector of the second type (t.e., a polar vectoi), we obtain a vector of the first type (axial vector). The formal scheme of this multiplication is taken from the three-dimensional case.

Let A = (A,,, A.,,, A.) denote a vector of the first type, B = (B„,, B,^, B^y) denote a vector of the second type. From this last, let us form three special vectors of the lirst kind^ namely-^-

B.=(B.,, B,,, B,;n

B, = (B,.., B,,, B,,)KB,,--B,., B,,=0). B.:=(B..., B.,,B....)J

Since B,, is zero, B, is perpendicular to the ^-axis. The /-component of the vector-product of A and B is equivalent to the scalar product of A and B,, i.e.,

(ABj^A. B,, + A^B,,+A.- B,,.

NOTES

181

We see easily that this coincides with the usual rule for the vector-product; c. g., iov j — ^c.

Correspondingly let us deline in the four-dimensional ease the product (P/) of any four- vector P and the six- vector./*. The /-component (/ = ,r, j/, .v\ or /) is j^iven by

(ly, ) = p,/;, ., + p,,/, , + p,/, , + p./- ,

Each one of these components is obtained as the scalar product of P, and the vector /', which is perpendicular to j-axis, and is obtained from ,/' by the rule/'.; = [_{f,y, fjyi

fj^.i f } 0 ././7 =0.]

■5f

•K-

We can also find out here the geometrical significance of vectors of the third type, when f=^, i.<?.,y' represents only one plane.

We replace (/> by the parallelogram defined by the two four-vectors U, V, and let us pass over to the conjugate plane </>'", which is formed by the perpendicular four-vectors U"^, V."^^ Ttie components of (P<A) are then equal to the 4 three-rowed under-determiuants D., D,, D^ Di of the matrix

U.^ U/>^ U/^ u,^

V/^

-x-

V.,-x- Y.-x- Y,7f

Leaving aside the first column we obtain

D,=p,(u,^v,^-u,^v.^)+p.(u,n^-^--u,-v,-^)

-hP/(U/^-V,*-U.^\%^)

= P,c^,,^ + P.-c/>^.+P/c^^..

=p,>.,+p.-<?!>.. + p,<a;;,

which coincides with (P«/>.) according to our definition.

I8:i

PEINCIPLE OF RELATIVITY

Examples of this type of vectors will be found on page '5i5, ^ = i/;Yy the electrical-rest-foree_, and i/' = 2to/'," the magnetic-rest-foree. The rest-ray 12 = t'w [$iJ^] * also belong to the same type (page 39). It is easy to show that

n=z -{

(0.

OJ,

to.

y 4

I »Al ^2 ^3

When (ojj, ro^, (05)=o, 0^^=:/, 12 reduces to the three- dimensional vector

12,, 12^^ 12,

«>,

«Al ^2 ^,

Since in this case, 4>i=oj^ ^14 =<•« (the electric force)

i/^i ="^'^^4/2 s =''''^r (the magnetic force)

we have (12J =

<?,r

1/«

e, I , /.«., analogous to the Poynting-vector.

[M."K S.]

Note 16. T/ic eUdric-red force. (Page 37.)

The four- vector ^ = 0'F which is called by 3Iinko\vski the cleetric-rest-force (elehtrische Paih-Kraft) is very closelv eonuccicd to Lorentz's Ponderomotive force, or the force acting on a moving charge. If p is the density of charge, we have, when € = 1, /;. = ], i.e., for free space

_ Po

^/i-vvc^ L

^?.r-f--(^'2^'3— ^'S^^s)

Now since p^ =p \^l—Y^/c^

We have p^(f)^=p\ d,+ - (>\Jh — ^\^^2)

JS^.B. — We have put the components of e equivalent lo ('f.r, (ly, d ,), and the components of vi equivalent to

X0TE8 18 '3

^^. -^y ^'-•)) iii accordance with the uofaliou used in Lorentz's Theory of Electrons.

We have therefore

2.6'. , po (<Ai5 ^2> ^Aa) represents the force acting on tlie electron. Compare Lorentz, Theory- of Electrons, V^r^^ l"^-

The fourth component <^4, when multiplied by p^ represents /-times the rate at which w^ork is done bv the moving electron, for Po <?^4, =/p ['\,f^.. +'t'^(/y +r-f?,] = ^'x po</>i -t-?",,/ po^'z + i\ Po9.v — ^^-■^ times the power pos- sessed b\' the electron therefore represents the fourth component, or the time component of the force-four- vector. This component was iirst introduced bv Poincare in 1906.

The four-vector i//=:/ojF* has a similar relation to

the force acting on a moving magnetic pole.

[M. X. S.]

Note 17. Opera/or ''- Lor '' (§ 1:>, p. 11).

The operation ^ -g^, ^ ;. ^^ | which plays in

four-dimensional mechanics a rule similar to that of

the operator ( / 7^,+ / t:--,+ h -—-— v ) in three-dim en-

sional geometry has been called by Minkow^sld ^ Lorentz- Operation ' or shortly Mor ' in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the sj^mbol n to denote this operation. In the above-mentioned paper (Annalen der I'hysik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three- dimensional operations in place of the general symbol lor. The physical significance of these ojierations will

184

PKINCIPLK OF RELATIVITY

beeomo elfar when alono; witli ]\[inko\vh^ki's mfithod of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with tlie case of lor S, where S is a six- vector (space-time vector of the 2nd kind).

This being a complicated case, we take the simpler ease of lor .s^,

w^here s- is a fonr-veetor= | .9^, s^. .9^ s^ |

and s =

'^2

.«?„

.»

'^'4

The following geometrical method is taken from Som- m erf eld.

Scalar Divergence — Let A^ denote a small four-dimen- sional volume of anj- shape in the neighbourhood of the space-time point Q^ ,'/S denote the three-dimensional bounding surface of A^> " ^^ ^^^^' outer normal to dS. Let S be any four-vector, P„ its norm.al component. Then

ivS = Lim r?ii^ AS=0 J AS

Now if for AS ^ve choose the four-dimensional paral- lelepiped wdth sides {df\, <lr,^, div^^ dx^), we have then

Div S = -^-i^-+^'-^-^'-^-^'--

9 '1 9

a.r^

a^u

lor S.

If /'denotes a space-time vector of the second kind, lor /'is equivalent to a space-time vector of the first kind. The o'eometrical si<2:nificanee can be thus brouq-htout. We have seen t lat the operator ' lor' behaves in every respect like a four- vector. The vector-product of a four-vector and a six-vector is again a four-vector. Therefore it is easy

NOTES 185

to see that. Jor S will be a four-vector. Let. ns tind the component of this four-vector in any direction -s-. Let S denote the three-space which passes through the point Q {a\, .Vo, .^o, x^) and is perpendicular to .^^ AS a very small part of it in the region of Q, da- is an element of its two-dimensional surface. Let the perpendicular to tl:-is surface lying in the space be denoted by j/, and let /,.„ denote the component of /in the plane of (<?//) which is evidently conjugate to the plane dcr. Then the 5-eomponent of the vector divergence of /' because the operator lor multiplies /' veetorially)

= Div/^,=:Lim ili^. As=0 AS

AY here the integration in //o- is to be extended over the whole surface.

If now s is selected as the .r-direetion, /\,s' is then a three-dimensional parallelepiped with the sides (I//j dz, (IJ, then we have

DiY /,= — i— \dz. dJ. %^ dy + dl dy ^' d: ay dz at L oy Os

+ dy d, ?A_' dl I = ^/- + ^^- H- ?Ai ,

'^ a/ ) dy dz ^ 6/ '

and generall}'

o •■ oy o- oi

Hence the four-components of the four-vector lor S or Div. / is a four-vector with the components given on page 42.

According to the formulae of space geometry, D^ denotes a parallelepiped laid in the (;/-^'-0 space, formed

out of the vectors (P, P, PJ, (u* U* 11^) (v, V^ V* ).

186 PRINCIPLE OF IlELATIVITY

D, is therefore the projection on the y-z-l space of the perallelopiped formed out of these three four-vectors (P, U"^, V"^), and could as well be denoted by Dyzl. We see directly that the four-vector of the kind represent- ed by (D,, Dy, D., D,) is perpendicular to the parallele- piped formed by (P U^ V^")-

Generally we have

(P/) = PD + P^D^.

.-. The vector of the third type represented by (P/*) is o-iven bv the ijeometrical sum of the tw^o four-vectors of the fir^t type PD and P^D^.

[M. N. S.]

VP^

Einstoin^A, & Minkowski >H> 530 > 11

E35

Prinoiple, of relativity

1920

M. I. T. LIBRARY 124281

This book is due on the last date stamped below.

Subject to fine if kept beyond date due.

9 AOh

^ ^?\ 1^38

^,,.i

'-^ Q

"£6

'340

\9^

&^fe

T 69

flOV2

-5 9.^'>

LSS to. 00^-90 JE'S4

MASSA(

Rule A

MIT LIBRARIES

3 9080 02417 8235

OOLOGY

17,1910

If any book, the property of the Institute, shall be lost or seriously injured, as by any marks or writing made therein, the person to whom it stands charged shall replace it by a new copy, or by a new set if it forms a part of a set.

C 53-3000 24 OC '34

LI STORAGE.

L4-. A^R'i 5-0000

Record · ID 646012
Conceptio Open Knowledge Archive — every document is proof-bundled with source, license, and retrieval metadata.