iOO

T £ -Pri

5 b

ti-

e

f

i ^ *c ;> o

^ uUo £-y^Si'

"A! \ft-Wdu Hi 0/0 bu V,C. Mr -b\5

HISTORICAL INTRODUCTION

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

Ten years later, in 1905, we find Einstein declaring 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 ver\ success; g:ivc rise to a host of new questions all bearing on the central problem of relative motion of ether and matter.

11 I'lilXC II'I.F. OK KKI.m \ m

Aragd'n prism experiment. The refractive index of a glass prism depends on the incident velocity of light outside the prism and its velocity inside the prism after refraction. On Fresnel's fixed ether hypothesis, the incident light waves are situated in the stationary ether outside the prism and move with velocity c with respect to the ether. If the prism moves with a velocity u with respect to this fixed ether, then the incident velocity of light with respect to the prism should be C + H. Thus the refractive index of the glass prism should depend on «, the absolute velocity of the prism, i.e., its velocity with respect to the fixed ether. Arago performed the experiment in 1819, but failed to detect the expected change.

Airy- Boacovitc/i water -telescope experiment. Boscovitch had still earlier in i7b't>, raised the very important question of the dependence of aberration on the refractive index of the medium filling the telescope. Aberration depends on the difference in the velocity of light outside the telescope and its velocity inside the telescope. If the latter velocity changes owing to a change in the medium filling the telescope, aberration itself should change, that is, aberration should depend on the nature of the medium.

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

Fresnel's conrection coefficient /{•=] '//x2. Possibly some form of compensation is taking place. \Vorking on this hypothesis, Fresnel Offered his famous ether convec tion theory. According to Fresnel, the presence of matter implies a definite condensation of ether within the

IIISTOKK \l, I.VII.'OIU TI'ION 111

region occupied by matter. This " condensed " or e\cr<s portion of ether is supposed to he carried awav with its own pice, of moving matter. It should be observed that only the "excess" portion is carried away, while the rest remains a- -taunant ;is 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 £. u. Fresnel showed that if this convection coefficient k is l l/fj.'2 (p. being the refractive index of the prism), then the velocity of light after refraction within the moving prism would be altered to just such extent as would make the refractive index of the moving prism quite indepen dent of its "absolute" velocity u. The non-dependence of aberration on the " absolute " velocity u, is also very easily explained with the help of this Fresnelian convection- coeflicieut k.

Stoke** viscous ether. It should be remembered, however, that Fresnel's stationary ether is absolutely fixed and is not at all disturbed by the motion of matter through it. In this respect Fresnelian ether cannot be said to behave in any respectable physical fashion, and this led Stokes, in 1845-H), to construct a more material type of medium. Stokes assumed that viscous motion ensues near the surface of separation of ether and moving matter, while at sulficifiitlv distant unions the ether remains wholly undisturbed. He >howed how such a viscous ether would explain aberration if all motion in it were differentially irrotational. But in order to explain the null Arago efl'cct. Stoke:- ua.- compelled to assume the convection liNpothesis of Fresnel with an identical numerical value for k, namely I './'-. Thus the prestige of the Fresnelian convection-cocllicient \va> enhanced, il' anything, by the theoretical inveBti&fctiona o!' stokes.

IV I'KINI U'l.K OK RELATIVITY

Fizratf* experiment.. Soon after, in J8ol, it received direct experimental confirmation in a brilliant piece of work by Fizeau.

If a divided beam of light is re-united after passing through two adjacent cylinders filled with water, ordinary interference fringes will be produced. If the water in one of the cylinders is now made to flow, the " condensed " ether within the flowing water would be convected and would produce a shift in the interference fringes. The shift actually observed agreed very well with a value of k=l— ll^1. The Fresnelian convection-coefficient now became firmly established as a consequence of a direct positive effect. On the other hand, the negative evidences in favour of the convection-coefficient had also multiplied. Mascart, Hoek, Maxwell and others sought for definite changes in different optical effects induced by the motion of the earth relative to the stationary ether. But all such attempts failed to reveal the slightest trace of any optical disturbance due to the "absolute" velocity of the earth thus proving conclusively that all tne different optical effects shared in the general compensation arising out of the Fresuelian convection of the excess ether. It must be earefully 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.

Micftelxoti- M or ley Experiment. In 1881, Michelson and Morley performed their classical experiments which undermined the whole structure of the old ether theory and thus served to introduce the new theory of relativity.

III.VTOUU-U. 1NTIIOIH ( Tlo.N V

The fundamental idea underlying thift experiment is quite

simple. In all old experiments the velocity of light situated in free ether was compared with the velocity of waves actually situated in a piece of moving matter and presumably curried away by it. The compensatory effect of the Fresnelian convection of ether afforded a satisfactory explanation of all negative results.

In the Michelson-Morley experiment the arrangement is quite different. If there is a definite gap jn a rigid body, light waves situated in free ether will take a definite time in crossing the gap. If the rigid platform carrying the gap is set in motion with respect to the ether in the 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 Eddington'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 yards across-stream and back. If the earth is moving through the ether there is a river of ether Howing through the laboratory, and a wave of light may be compared to a swimmer travelling with constant velocity relative to the current. If, then, we divide a beam of light into two parts, and send one-half swimming up the stream for a certain distance and then (by a mirror) back to the starting jx)int, and setul the other half an equal distance across stream and back, the across-stivam beam should arrive back first.

Let the ether be llowinir relative to o the apparatus with velocity -n. in the

n' direct ion () . and let OA , OB, be B the two arms of the apparatus of equal

VI I'KINCIL'LK OF KKLATUITY

length /, OA being placed up-stream. Let <• be the velocity of light. The time for the double journey along OA and back is

/ = -- + / = ^ = '2I B* c if r + n rs ^lz c

where 8= (1 n'z /c" )"*, a factor greater than unity.

For the transverse journey the light must have a compo nent velocity n up-stream (relative to the ether) in order to avoid being carried below OB : and since its total velocity is <•, its component across-stream must be \/(c"—»'2), the time for the double journey OB is accordingly

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

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

The Michelson-Morley experiment seems to show that there is no relative motion of ether and matter. Fresnel's stagnant ether requires a relative velocity of 11. Thus Miehelsou and Morley themselves thought at iirst that their e\]H rinient eoniirmed Stokes' viscous ether, in which no relative motion can ensue on account of the abs.MK'r of slipping of ether :i,t the :- -niTrice of separation. Hut even on Stokes' theory this viscous How of ether would fall off 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 sttme negative results, thus failing to confirm Stokes' theory of viscous Mow.

II ISTOIMCAI, INTKOlim TION' Til

o//. Further, in l.V.'-'i, Lodge per formed his rotating sphere experiment wiiich should complete absence of any viscous How of ether due to moving masses of matter. A divided beam of light, after repeated reflections within a very narrow gap between two massive hemispheres. \\-;is 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 How, and any such flow would be immediately detected by a disturbance of the interference bands. But actual observation failed to detect the slightest disturbance of the ether in the gap, due to the motion of tbe 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 >tokes' theory which seemed capable of being reconciled with the Micheleon-Morley experiment, but required very special assumpt ions. The very complexitv 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 OK It K NATIVITY

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

We must now consider the problem of relative motion of ether and matter from the point of view of electrical theory. From 1860 the identity of light as an electromagnetic' vector became gradually established as a result of the brilliant "displacement current" hypothesis of Clerk Maxwell and his further -analytical investigations. The elastic solid ether became gradually transformed into the electromagnetic one. Maxwell succeeded in giving a fairly satisfactory account of all ordinary optical phenomena ;vnd little room was left for any serious doubts as regards the general validity of Maxwell's theory. Hertz's re searches on >lectric waves, first carried out in l^Mi, succeeded in furnishing a Mrong experimental confirmation

MMi U, INTRODUCTION' IX

of Maxwell's thenn. Klef-tric waves behaved like light waves of ven lar^e wave length.

Tlie orthodox Maxwellian view located the dielectric polarisation in fte electromagnetic el her which was merely a transformation of iVe-ncl's stagnant ether. The Mag

netic polarisation was looked upon as wholly Secondary m origin, being due to the relative motion of the dielectric tul.i- of polarisation. On this view the Fresnelian con- vcetion r-octllcient comes out to he i, as shown by J. J. Thomson in 1880, instead of I 1//*- 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 eonvect.ion coefficient equal to 1 l //*2.

The modifications proposed independently by Her!/ 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 consequentty, 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 llertx. and Heaviside succeeded in obtaining a \alne for Fresnelian co-efficient equal to 1 1 1^- , and consequently stood totally condemned from the optical point of view.

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

* See Note 1.

X HKIXC1PLE 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. 1901, Blondlot working with a stream of moving gas could not detect any such effect. H. A. Wilson repeated the experiment in an improved form in 1903 and working with ebonite found that the observed effect was 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 L876 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 wao clear that the

III-TOUK \i. i\ii:oi>rcTifW xi

H her row-opt had finally outgrown its usefulness. The observed !';,<•(> had become too contradictory and too heterogeneous to be reduced to an organised whole with tin- help of the ether concept alone. Radical departures from the classical theory had become absolutely necessary.

There were several outstanding 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 " polarisation doublets," which on his theory gave rise to the induced field proportional to K 1. The field near a moving dielectric is naturally proportional to K 1 and not to K. Lorentz's theory thus gave a satisfactory explanation of all those experiments with moving dielectrics which required effects proportional to K— 1. Lorentz further succeeded in obtaining a value for the Fresnelian convection coefficient equal to 1 *//**, the exact value required by all optical experiments of the moving type.

\VC 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 raeduim is involved in any portion of the paths actually traversed by the beam. Consequently no compensation due to Fresnelian convection

Til I'lMM II'IK OK lU-.LATIVITY

of ether by moving medium is poi-sible. Thus Kre-nelian convection compensation can have no possible application in this case. Yet some marvellous compensation has evidvuily taken place which has completely m;i<ke.d the " absolute " velocity of the earth.

In Michelson and Morley's experiment, the distance travelled by the beam along OA (that is, in a direction parallel to the motion of the platform) is £//82, while the distance travelled by the beam along1 OB, perpendicular to the direction of motion of the platform, is %l(3. 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 /, OA must have been of length 1 1 ft. 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 Oi/ (perpendicular to the direc tion of motion), and automatically contract to a length 1/J3, when placed along O/ (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 he a real one, the distance travelled with respect to the ether is "11(3 and the time taken for this journey is :V/^ r. But the distance travelled with respect to the platform is always 2/. Hence the velocity of light with respect to the plat-

'2,1 8 form is 2// " —fi/P, 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

HI>Toi:l< A I IVil:«M»l ' 'IIM\ XIII

1^1' ilic plat I'm in. The < precent dfffioolty 'cannot \>u solved

h\ .my further alteration in the measure of space. The recount left open i-: to alter the measure ol' time an well, that K, to adopt t lie conecpt of '* loea I t line." If a mov ing clock goes slower so that one 'real' second becomes 1 ft second as measured in the moving system, I lie velocity of light relative to the platform will always remain c. We must ad.-pt two very startling hypotheses, namely, the Fit/ (lerald 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. Lorentx 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 artiliee like imaginary quantities in analysis or the circle at infinity in projective geometry. The originality of Einstein at tins singe consists in his successful physical interpretation of these result?, and viewing them as the coherent organised consequences of a single general principle. Lorentz established the Relativity Theoremt (consisting men-iy . )' a set of transformation equations) while Kinstein 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 \\-.\\ as to make then; quite Independent of the absolute velocity

* See \

t See Note 4.

XIV I'RINTIPLE OF 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 thai \ve 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 praeticalh the whole of the electromagnetic theory based on the

* See Notes 9 and 12.

HISTORICAL INTRODUCTION XV

M;i\\\ell-|,orenU system of field equations. It combined all the advantages (>| classic Maxvvelliaa theory together with an electronic hypothesis. The Lorentx assumption of polarisation doublets had furnished a satisfactory explana tion ..f flu- r're^nelian convection of ether, but in the new theory 1 his is deduced merely as a consequence of the altered concept of relative velocity. In addition, the theory of Relativity accepted the results of Michelson and Morley's experiments as a definite principle, namely, the principle of the constancy of the velocity of light, so that there was nothing left for explanation in the Michelson-Morley experiment. But even more than all this, it established a single general principle which served to connect together in a simple coherent and fruitful manner the known facts of Physics.

The theory of Relativity received direct experimental coiifii -illation in several directions. Repeated attempts were made to detect the Lorentz-Fitzgerald contraction. Any ordinary physical contraction will usual Iv 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 ix an, ether or not, uniform rrforily //•//// r,'x]><>ct to if can never be 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, IxMtig 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 PRIXCIPLE OF

dynamic^ without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that the " mass v of an electron depended on its velocity. "••<> early as 1881, J. J. Thomson had shown that tlie inert in of a charged particle increased with its velocity. Abraham now deduced a formula for the variation of mass with velocity, on the hypothesis that an electron always remain ed a riyid sphere. Lorentz proceeded on the assumption that the electron shared in the Lorentz- Fitzgerald contrac tion and obtained a totally different formula. A very careful series of measurements carried out independently b\ Biicherer, Wolz, Htipka 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 theVlectrical origin of inertia. Thus the complete agreement of experimental facts with the predictions of the new theory must be considered as confirming it as a principle which goes even beyond the electron itself. The greatest triumph of this new theory consists, indeed, in the fact that a large number of results, winch 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 Kiustein in 1905. Ten years later, Einstein formulated his second theorv, the ( Jcncralised 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 principle, but the former does not really contain the latter as a special case.

>i;u AI, iNTHonn riOV

iii's li'-v.t theon is re-tricted in tlu- EWDBC that it only refers to uniform rect iliniar motion and has no appli cation to any kind of accelerated movements. Einstein in hi- second theory extends the Relativity Principle to cases ol' accelerated motion. If Relativity is to be universallv true, ti.en e\vn accelerated motion must be merely rt'lativf nint in, i ln'l n'ct'ii mutter and HI after. Hence the Generalised Principle of Relativity asserts that "absolute" motion cannot be detected even with the help of gravitational IRWH.

All in 'vements must be referred to definite sets of co-ordinate axes. If there is any chaiiLjf of axes, the numerical magnitude of the movements will also change. Bui according to Newtonian dynamics, such alteration in physical movements can only be due to the effect of ceitain 1'orces in the field.* 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 the.-e "'geometrical" forces introduced by a change of axes. 'This loids to Einstein's famous Principle of Equivalence. A i/i'<ifl/nl innal Jield of force in xtrictly equivalent to one introduced /»/ </ tfantforatation <>f co-ordtunfey and nopQwiltfo t'.r/if.'i'/'; fiif ft it ilixtiiKinixh between the two.

Thus it mav hecome possible to "transform away'" gravitational elTect-^. at least For sufficiently small region^ ol' space, bv 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 unarceleraled syslem of co-ordinate-."

But there is no reason why we should look upt.n the Galilean s\>tein ,i> more I'liiHlainenlal than any other. If it

XV1I1 PUIXCIPLE OF BELATIVITY

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 h'eld 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 not cause the curvature of space-time. It is the curvature. Just as light does not cause electromagnetic oscillations; it is the oscillations."

We may look upon this from a slightly different point of view. The General Principle of Relativity asserts that all motion is merely relative motion between matter and matter, and as all movements must be referred to definite sets of co-ordinates, the ground of any possible framework must ultimately be material in character. It /* 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 ?'•< space and time. Hence we see how //////*/'v// space and time is actually defined by the local distribution of matter.

There are certain magnitudes which remain constant by any change of axes. In ordinary geometry distance between two points is one such magnitude ; so that &r2 +Sy- +5?- is an invariant. In the restricted theory of light, the principle of constancy of light velocity demands that Saf*+fy*+Sa*—c*Sl* should remain constant.

IIISIOI;K \i. i vntoiM ( iiox xix

The *i'/m?nti'ni </\ oi' adjacent events is defined by ftx* = -,/./• 2 -////-' _,/:v+,.-V/-. It is :ui extrusion of tin- notion of distance and this is the ne\v invariant. Now if .*•, //, :, I are transformed to any set of new variables j-,, .'•,, .»•.., ./.,, we shall got a quadratic expression for ,/.s- = //! rr,- + fy,8.»v.j + .•• = ;></, i,.<v, where them's are functions of ./•,,./•«,, .r.p #4 depending on the transforma tion.

The special properties of space and time in any region are defined by these //'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 merely 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, 19 IS*. The observed deflection is decisively in favour of the Generalised Theory of Relativity.

XX I'KI Vril'I.K OK KKLATIVITY

It should be noted however that the velocity of light itself would decrease in a gravitational field. This may appear at first sight to be a violation of the principle of constancy of light-velocity. But when we remember that the Special Theory is explicitly restricted to the case of unaccelerated motion, the difficulty vanishes. In the absence of a gravitational field, that is in any unaccelerated system, the velocity of light will always remain constant. Thus the validity of the Special Theory is completely preserved within its own rextrictwl 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 Praunhofer lines is a necessary and fundamental condition for the acceptance of his theory. But Eddington has pointed out that even if this test fails, the logical conclusion would seem to be that while Einstein's law of gravitation is true for matter in bulk, it is not true for such small material systems as atomic oscillator.

CONCLUSION

From the conceptual stand-point theiv are several important consequences of the Generalised or Gravitational Theory of Relativity. Physical space-time is perceived to be intimately connected with the actual local distribution of matter. Euclid-Newtonian space-time is not the actual space-time of Physics, simply because the former completely neglects the actual presence of matter. Euclid-Newtonian continuum is merely an abstraction, while physical spnn- time is the actual framework which has some definite

MlST.HiU-.M. INTKOIirrnnN XXI

curvature due to the presence of matter Gravitational Thrnrv of Relativity tlius brings out clearly the funda mental distinction between actual physical space-time (which is non-isotropk" and non- Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian continuum (which is homogeneous, iaotropio 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.1' 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 " inertial framework " is referred to vast quantities of undetected world-matter. It has interesting consequences. The dimensions of Einsteinian universe would depend on the quantity of matter in it; it would vanish to a roint 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. Eiusteinian 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 <f anti-sun" (corresponding to the back surface of the sun) at a point in the sky opposite to the real sun. This anti-sun would of course be equally large and equally bright if there is no absorption of light in free space.

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

Xxii I'Rl. \CIPLK OF HKL.VTIVITY

distance from an observer, time itself stands still, so that to the observer nothing can ever " happen " there. All these theories are still highly speculative in character, but they have certainly extended the scope of theoretical physics to the central problem of the ultimate nature of the universe itself.

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

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

P. C. MAHALAXOBIS.

II I-TOKIC A I, INTRODUCTION XX111

Note A.

For example consider a massive particle resting on a circular disc. If \ve set the disc rotating, a centrifugal force appears in the Held. 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 velouitv. In the General Theory, physical laws are made independent of "absolute" motion of any kind.

On The Electrodynamics of Moving Bodies

Bl

A. El \STKIN.

INTRODUCTION.

It is well known thai if we attempt to apply Maxwell's electrodynamics, as conceived at the present time, to moving bodies, we are led to assymetry which does not aijrec with observed phenomena. Let u* think of the mutual action between n magnet and a conductor. The observed phenomena, in this case depend only on the relative motion of the conductor and the magnet, while according to the usual conception, a distinction must be made between the cases where the one or the other of the bodies is in motion. If, for example, the magnet moves and the conductor is at rest, then an electric held 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 tield 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 <:ime career as the electric force, it being of course assumed that the iclative motion in both of these eas.^ is the

I'RINCIPLE OF RRLATIV1TY

2. Examples of a similar kind such as the unsuccessful attempt to substantiate the motion of the earth relative to the " Light-medium " lead us to the supposition that not only in mechanics, but also in electrodynamics, no properties of observed facts correspond to a concept of absolute rest; but that for all coordinate systems for which the mechanical equations hold, the equivalent electrodyna- mical and optical equations hold also, as has already been shown for magnitudes of the first order. In the following we make these assumptions (which we shall subsequently call the Principle of Relativity) and introduce the further assumption, an assumption which is at the first sight quite irreconcilable with the former one that light is propagated in vacant space, with a velocity '• which is independent, of the nature of motion of the emitting body. These two assumptions are quite sufficient to gu-e us a simple and consistent theory of electrodynamics of moving bodies on the basis of the Maxwellian theory for bodies at rest. The introduction of a " 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 t he- enunciation of every theory, we have to do with relations between 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 fi^ht at present.

.HI ELECTRODYXAMH^ "F MoVfVO BODIES 4

I.-KINEMATXOAL PORTION.

§ 1. Definition of Synchronism.

l^t us hrivc a co-ordinate system, in which 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 nan be found out by a measuring rod, and can be expressed by the methods of Euclidean Oeometry, or in Cartesian co-ordiuates.

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 such a'

nt/h-ntultcrtt ilcfniKlnn has a physical sense t only when ice lhi\-i' it i.'f.i;i,- ,><>//>, /i ///' n'/i'il. /.v tiifttut l>y tiMf. Jt'f have to dike into consideration l.h<> fact that those of our conceptions, in />•///<>// time /i!<ii/x a part, are al ^nyx conception* of synchronism For example, we say that a train arrives here at 7 o'clock ; this moans 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 ilelinr timt- exclusively for the place at which the clock is stationed, hut the definition is not sufficient when it is required to connect by time events taking place at different station*, or what amounts to the same thing, to estimate by means of time (zeitlich werten) the occurrence of events, take place at stations distant from the clock.

4 JMIINCII'LL <)i Khl.ATIVm

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. AVe can attain to a more practicable result by 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 syrchrouous 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 light 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 A-time t towards B,

arrives and is reflected from B at B-time t and returns to A at A-time t' . According to the definition, both

A

clocks are synchronous, it'

u.Y Till, KLI.t IKOm N'.V.MIO «'(• JKiVIM. IKHMKS 0

\\ •• as-ume that tin's 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.

:>. If the clock at A as well as the clock at B are both synchronous with the clock at C, then the clocks at A and B an- synchronous.

Thus with the help of certain physical experiences, we 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

, where r is a universal constant.

\\ "e have defined time essentially with a clock at rest in a stationary system. Ou 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 Kelatmt\ and on the Principle of Constancy of the velocity of light, both of which we define in the following way :

1. The laws according tu which the uuture of physical >\>tmis alter are independent of the manner in which those flian^s ;,n. referred to two co-ordinate »\stem>

6 PRINCIPLE OF RELATIVITY

which have a uniform translatory motion relative to each other.

2. Every ray of light moves in the " stationary co-ordinate system " with the same velocity c, the velocity being independent of the condition whether this my of light is emitted by a body at rest or in motion.* Therefore

Path of Light

velocity == =— -• ,

Interval or time

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.

(6) The observer finds out, by means of clocks placed in a system at rest (the clocks being synchronous as defined in § 1), the points of this system where the ends of the rod to be measured occm at a particular time f. The distance between these two points, measured .by the previously used measuring rod, this time it being at rest, is a length, which wt> may call the " length of the rod."

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

* Vide Note V.

OX THE EI.FOTRODYXV.Mlrs OF MOVING HOUIKS /

length of the roil in the moving >ystem " is equal to the length I of the rod in the stationary system.

The length which is found out by the second method,

mav he callo<l ' the ffnyth of Ike m<>i'ir></ roil mounted from

I he xf'tlitn/irij xi/xft-m.' This length i> to he estimated on the !>;isis of our principle, and <"V &h<i 1.1 find it /<> If different from (.

In the generally recognised kinematics, we silently

assume that the lengths defined by these two operations

jiial, or in other words, that at an epoch of time (,

a moving rigid body is geometrically replaceable by the

s-imc body, which can replace it in the condition of rest.

Eelativity 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, /.<?., the time of the clocks correspond to the time <>f the stationary system at the points where they happen to arrive ; these clocks are therefore synchronous iti the stationary system.

\Ve 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 clui'k-. At the time t , a ray of light goes out from A, is

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

B

time /' . Taking into consideration the principle of constancy of the velocity oi' light, we have

,, AB

and t —t= .

A B c+v

S PRIXOn'LE OF RELATIVITY

where /• } is the length of the moving rod, meu-mv<i in the stationary system. Therefore the observers stationed with the watches will not find the clocks synchronous, though the observer in the stationary system must declare the docks to be synchronous, \Ve therefore sei« that we can attach no absolute significance to the concept of synchro nism ; but two events which are synchronous when viewed from one system, will not be synchronous when viewed from a system moving relatively to this system.

$ 3. Theory of Co-ordinate and Time-Transformation

from a stationary system to a system which

moves relatively to this with

uniform velocity.

Let there be given, in the stationary system two co-ordinate systems, I.e., two series of three mutually perpendicular lines issuing from a point. Let the X-axes of each coincide with one another, and the V 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 systems (/?•) have a constant velocity in the direction t)f the X-axis of the other which is stationary system K, the motion being also communicated to the rods and clocks in the system (/•). 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 /, we always mean the time in the stationary 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 ILlOTHODYNAMIOa OK MOYl.M; BODIES

system, and \ve thus obtain the co-ordinates (j',y, z) for the stationary s\>tcm, and ($,rj,£) 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 system 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 are clocks) in the manner described in § 1.

To every value of (<•,//, -, /) which fully determines the position and time of an event in the stationary system, there correspond-; a system of values (£,r/,CT) ; now the problem is to find out the system of equations connect ing these magnitudes.

Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linea1.

If we put .?•':= '• rf, then -it is clear that at a point relatively at rest in the system K, we have a system of values (.<•' y z] which are independent of time. Now let us find out T as a function of (',y,z,(}. For this purpose we have to express 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 § I.

Let a ray of light be sent at time TO from the origin of the system k along the X-axis towards .c' and let it be reflected from that place at time rl towards the origin of moving co-ordinates and let it arrive there at time T2 •' then we must have

+ T,)=T1

10 PRINCIPLE OF RELATIVITY

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

| JT (0, 0, 0, 0+T 0, 0, 0, {t+ ^—.+^L- } \ 1

( c— v c + v ) / J

=T(*', o, o, t +— I 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 {^y,z,t,},

A similar conception, being applied to they- and r-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is alnays propogated along those axes with the velocity ^/c* —v*, we have the questions

From these equations it follows that T is a linear func tion of and t. From equations (1) we obtain

where a is an unknown function of v.

With the help of these results it is easy to obtain the magnitudes (£,i?,£)> tf we express by means of equations the 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=O, if the ray is sent in the direction of increasing £, we have

Now the ray of light moves relative to the origin of Jc with a velocity c— r, measured in the stationary system ; therefore we have

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,

where -y ,

c ., c

If for .</, we substitute its value x tv, we obtain

/ v.i- \ V c* /

t=4> (»). /8 (*-»«),

where ff= .• •— t , and (f)= =r-!=r^ = is a function vi— v* Vc*—v* P

^ of v.

12 PRINCIPLE OF RELATIVITY

/

If we make no assumption about the initial position of tlu- moving system and about the null-point of tt 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 principle of relativity is reconcilable with the principle of constant light-velocity.

Atatimer = / = o 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

(•>'} >/> *)> De a point reached by the wave, we have

a.2+2/2 + .»_C2^>

with the aid of our transformation-equations, let us transform this equation, and we obtain by a simple

calculation,

£2+772+£2=czTs.

Therefore the wave is propagated in the moving system with the same velocity c, and as a spherical wave.* Therefore we show that the two principles are mutually reconcilable.

In the transformations we have go: an undetermined function <j> (v), and wo now proceed to find it out.

Let us introduce for this purpose a third co-ordinate system k1 ' , which is set in motion relative to the system X-, the motion being parallel to the £-axis. Let the velocity of the origin be ( r). At the time t = o, all the initial co-ordinate points coincide, and for f = ,< =y = z = n, the time t' of the system k' = o. We shall say that (./ y' ;' /') are the co-ordinates measured in the system k' , then by a * Vide Note 9.

ON THE ELECTRODYNAMICS OF MOVING BODIES 13

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

-v>, etc.

Since the relations between (/, y', z' , t'}, 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 k' are identical.

Let us now turn our attention to the part of the y-axis between = 0, 77 = 0, £ = o), and (£=o, >/ = ], £=0). Let this piece of the y-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 ~T7~y For the system moving with velocity ( ?'), we have on grounds of symmetry, Z Z

14 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., 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 K then the equa tion of the surface of this sphere, which is moving with a velocity v relative to K, is

At time t o, the equation is expressed by means of

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

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 x dimension is shortened in the ratio

1: 'V 1 ; the shortening is the larger, the larger

is v. For 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 OK 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 system gives the time f, and lying at rest relative to the moving system is capable of giving the time r ; suppose it to be placed at the origin of the moving system X-, and to be so arranged that it gives the time T. How much does the clock gain, when viewed from the stationary system K ? We have,

V I-

| t— 7i» \,

7i» \, and x=vt,

Therefore the clock loses by an amount ^"2 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 lag behind the clock

p1 which had been all along at B by an amount $t -$, 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-

01

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 small 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 ;r-axis of the system K) according to the equation

£='Y' 1? = M',T, £ = 0,

'where t/'| and w are constants.

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

w f + v \ / "" ;= —1 t, y—

ON THE ELECTRODYNAMICS OF Mo VI. Mi I'.ODIES 17

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

1C

and u = tan"1

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

u=

r/ o \ / vw sin a \ "1 : [(»• + /<••+ 2 vw cos a)— I j

, /•/(• cos -i

.,

It should be noticed that r- and w outer into the expression for velocity symmetrically. If iv has the direction of the v-axis of the moving system,

TT f+"'

From this equation, \ve see that by combining two velocities, each of which is smaller than c, we ojbtain a velocity which is always smaller than <•. If we put v=c— %, and w=c—\, where x and A. are each smaller than c,

It is also clear that the velocity of light c cannot be altered by adding to it a velocity smaller than c. For (his case,

Fufo Note 12.

18 PRINCIPLE OF RELATIVITY

We have obtained the formula for U for the case when v and w 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 w, 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 r, we shall have to write,

We see that such a parallel transformation forms a group.

Wo 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.— ELECTKODYNAMICAL PART.

§ 6. Transformation of Maxwell's equations for Pure Vacuum.

On the nature of lite Electromotive Force caused by motion in a magnetic field.

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

- -?- [X, Y, /] =

9 JL

9< dy

M N

ON THE KLE( THODVXA.Mli s oi .\[()VlN(i BODIES

and

i a

a a j

3-« 6y 3 X Y Z

.- (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 r, we obtain,

[X,

- - N),

and

a al

X /8(Y- ? N) j8(Z+ -M)

c <•

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 OF RELATIVITY

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

1 J (X', Y', Z') = c o T

a. 6 _a

6f di) L' M' N'

a

9*

y

... (3)

Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these systems 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 $ (v), which depends only on v, and is independent of (£» V> £> T)' Hence the relations,

[X', Y', Z']=V (r) [X, ft (Y- ?X), ft (Z+ ^M)],

[L', M', N']=* (r) [L, /i (M + ?Z;, 0 (X- '' Y)].

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

.-. [X', Y', Z'] = [X, ft (Y- f N), j8 (/+ '; M )j

[ L'. W. N'] = [L,

.^Z), £(N--rY)].

ON THE ELECTRODYNAMICS OF MOVING BODIES 21

For the interpretation of these equations, we make the following remarks. Let us have a point-mass of electricity which is of magnitude unity in the stationary system K, i.e., it exerts a unit force upon a similar quantity placed at a distance of 1 cm. If this quantity of electricity be at rest in the stationary system, then the force acting upon it is equivalent to the vector (X, V, 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), (2), (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 v/c, 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 st-e that in the theory developed the electro magnetic force plays the psirt 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 occurs 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 system 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 :

I

Z = Z0 sin * J N=N0 sin * }

Here (X0, Y0, Z0) and (L0, M0, N0) are the vectors which determine the amplitudes of the train of waves, (I, m, 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 :

L/=Ln sin*'

ON THE ELECTRODYNAMICS OF MOVING BODIES

where

, n =

1-'

From the equation for w' it follows : If an observer moves with the velocity r 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 4> 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

\

This is Doppler's principle for any velocity. then the equation takes the simple form

If 4>=

1 +

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 /' takes the form

1 - cos 4> e

24 PRINCIPLE OK RELATIVITY

This equation expresses the law of observation in its most gen eral form. If <£=-, the equation takes the simple form

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

(1 cos 4> I c )

If 4>=o, this reduces to the simple form

A'8=A'

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 Bays of

Light. Theory of the Radiation-pressure

on a perfect mirror.

Since is equal to the energy of light per unit

07T

volume, we have to regard J as the energy of light in

ON THE ELECTRODYNAMICS OF MOVING BODIES 25

A'*

the moving system. ' _ would therefore denote the A.

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 /, m, u 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

(.1-

which expands with the velocity oi' 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 svstem k, i.e., the energy of the light-complex relative to the svstem A-.

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

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

S' ft

s "

- OOS <t>

If E denotes the quantity of light energy measured in the stationary system, E' the quantity measured in the 4

26 PRINCIPLE OF RELATIVITY

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

E A2 Q Vl-v*/c*

8^S

If 4>=0, we have the simple formula :

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 by the magnitudes A cos $, r (referred to the system K). Regarded from A-, we have the corresponding magnitudes :

i v

1— cos 4>

A' = A

cos 4>— 1 cos <*>' =

v' = v

,•••

~c~«

ON TKE KLECTKOUYNAMUS ()!• MOV1N(; 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 rellected light :

A"' = A" - =A

/ 1 v*

V l~^

COS <t>

Cos4>" + » (1+ -

,„_ _ c_ __ \ c'

"

1+ ^ cos $" 1 2 r cos + '"*

" C '•'

(H)'

•/ =•/ =!/

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

87r(c cos <£— «)

away from unit surface of the mirror per unit of time is A'""/8w (— c cos <I>"+r). The difference of these two expressions is, according to the Energy principle, the amount of work exerted, by the pressure of light ]>er unit of time. If we put this eijiial to P.r, where P= pressure of light, we have

(cos * - | ,,_., Aa V _£/

1-

28 PRINCIPLE OF RELATIVITY

As a first approximation, we obtain

P=-2 ¥- <*>»* 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 :

M 1 1 ?>L r>Y ft 7 1

,

'"

dt

I/ _i.9Y\ -91 -9-? iaM_8Z_ax

c \pu* dt) ~ d= a.<- r c a^ a.'1 a-

l( 4-9j?\ =9^I_9JJ c \pu> dt ) Q.e dy

_ c~dt "a.-

_

Qy 6

where =

oy

, denotes ±TT times the density

of electricity, and (?t,, ?«„, ttr) are the velocity-components of electricity. If we now suppose that the electrical- masses are bound unchangeably to small, rigid bodies

UN THK KLLCTKODVNAMKS OF MUVINC BODIES '29

(Ion*, electrons), then these equations form the electromag 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 /,• with the aid of the transfor mation-equations given in § -3 and § (5, then we obtain the equations :

1 f ,'

C I P

_ ___

67? dr 8* 67;

9Z.'l =

Qr J

* T8r J- 87 dr, '

where

-•„'-

Since the vector (/^ « //x ) is nothing but the £ > T? > 4

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

30 PRINCIPLE OF RELATIVITY

our kineinatical principle as the basis, the electromagnetic basis of Loreutz'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

n d*i = w *• = z rf<- •=

Where (a-, y, .?) are the co-ordinates of the electron, and m is its mass.

Let the electron possess the velocity v 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 THK ELECTRODYNAMICS OF MOVING BODIES 31

the electron is at the origin of co-ordinates, and moves with the velocity v along the X-axis of the system. It is clear 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 r.

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*

in the time immediately following the moment, where the symliols (£, rj, £, T, X', Y', Z') refer to the system A\ If we now fix, that for t .v=y-z=(\ T=g = i) = £=Q, then the equations of transformation ^iven in •'{ (and (5) 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 :

Z8,- _ <•_ 1 y d*y e_ I ( y_ r <//« vn J5» ' (//" ,n f

(A)

32 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

"• =eX=eX'

? M] =

and let us" first remark, that <?X', eY', ?Z' 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 aeceleration-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

Transversal mass =

Naturally, when other definitions are given of the force and the acceleration, other numbers are obtained f^r the

* Vide Note 21.

o.N THK ELECTRODYNAMICS or MOVING BODIES 33

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

\Ve 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 charge which may be as small as possible.

Let us now determine the kinetic energy of the electron. It' 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 /<?X^--. 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 iirst of equations A) holds, we obtain :

For v = c, *\\ is infinite!] great. A> our former result shows, vel'iritir.- >'\<vco!iiig that of light can have no possibility of existent'

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

ponderable masse*.

\Ye can now enumerate the characteristics of the

motion of the electron H\,ulaMe 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. t> 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 magnetic fields.

2. vFrom 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 :

p=( xd<;=™c2 r l -ii .

J L0-? J

8. 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 :

_d*y ^ ^ rK

<«• R ,». c ,

H= ""^

. \

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

ALBRECHT EINSTEIN

[y/ s/turf l>iot/i-a/j//i<'<it t/o/e.]

The name of Prof. Albrecht Einstein has now spread far beyond the narrow pale of scientific investigators owing to the brilliant confirmation of his predicted deflection of light-rays by the gravitational field of the sun during the total solar eclipse of May 29, 1919. But to the serious / student of science, he has been known from the beginning 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 Hashes 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 University of Prague in Bohemia / as ausser-ordentliche (or associate) Professor. In 19! 4, * through the exertions of Prof. M. Planck of the Berlin University, he was appointed a paid member of the Hoyal (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 Yan't ilofY, 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 gained him distinction was an investigation on Brownian Movement. An admirable account will be found in Perrin's book 'The Atoms.' Starting from Boltzmann's

86 PRINCIPLE OP 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 Sorbonne, 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 correspond 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 falling upon an individual atom, liberates electrons according to the energy equation

liv=- -ftnv^ + A,

where (in, r) 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

88 , PRINCIPLE OF RELATIVITY

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 191."), 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 generally 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

tried, in a series of papers beginiiin^ from 1011, 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 Annalen der Physik in 1916 on this subject, and gives, in his own words, the Principles of Generalized Relativity. The triumphs of this theory are now matters 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 further achievements of his genius.

INTRODUCTION.

A* the present time, different opinions are being held about the fundamental equations of Electro-dynamics for moving bodies. The Hertzian1 forms must be given up, for it has appeared that they are contrary to many experi mental results.

In 1895 H. A. Lorentz- published his theory of optical and electrical phenomena in moving bodies; this theory uas 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 however 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 gobd account of the non-exietence of relative motion of the earth and the lumiiiiferous " Ather '' ; it shows that this fact is intimately connected with the covarianee of the original equations when certain simultaneous transformations of the space and time co-ordinates are effected; these transformations have therefore obtained from II. l'»iueare; the name of Lbrentz- transformations. The covarjance of the-e fundamental

equations, when subjected to the Lomit/-traiisfunnation is a purely mathematical fart i.e. not based on any physi- cal considerations; 1 will call this the Theorem of Rela tivity ; this theorem re-t> essentially on the form of the

1 n,i. No* l. •-' Note •_'. 3 Vide Nute ::.

I'KIXCIPT.E OF RELATIVITY

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

Now without recognising 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 anyhow 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. Lorentz1 has found out the" Relativity theorem" and has created the Relativity-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.

A. Einstein2 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-dynamics of moving bodies in the sense

i F«cieNote4, * Note 6.

IVTKODUCTION 3

characterized by me. "In the present essay, while formu lating this principle, I shall obtain the fundamental equa- \\«\\< 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-magnetisation 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 very

* See notes on § 8 aud 10.

4 I'lUN'Oll'LK OK RELATIVITY

.

surprising consequences. By laying down tin1 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 foiras). -,. +

NOTATIONS.

Let a rectangular system (s, //, ~, /,)• of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the nnit of length that the velocity of light in space becomes unity.

Although I would prefer not to change the notations used by Lorent/, 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 c and the magnetic force bv MI, so that (E, M, e, ;#) 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, ?'. £., instead of operating with (/), I shall operate with (/'/), where /denotes \/-*-\. If now instead of (./•, //, 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 suffixss (1, 2, 3, 1). The advantage of this method will be, as I expressly emphasize here, tha^ 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 thr symlbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those

S'OTATIOXS O

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

An individual system of values of (.*•, y, ~, /) /. e., of (•ri >r-.' '3 ^'4) sna^ called a space-time point.

Further let « denote the velocity vector of matter, c the dielectric constant, //. the magnetic permeability, <r the conductivity of matter, while p denotes the density of electricity in space, and .v the vector of "Electric Current" which we shall some across in §7 and §8.

O PUNOIPLS OF RELATIVITY

PART I § 2.

THE LIMITING CASE.

The Fundamental Equations for Atlier.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-dynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limitting case «=-*, /A=-/,(T = O, they should constitute the laws for ponderable bodies. In this ideal limitting case e=l, /*=!, <r=o, E will be equal to e, and M to m. At every space time point (t, y, ~, f) we shall have the equations*

(i) Curl m— gr=/>u (ii) div e= p (iii) Curlr +-.*? = 0 (iv) div m = o I shall now write (.rt xy .r3 .r4) for (.r, y, z, t) and

0>i>P2> Ps> Pi) for

(Pu,,puy,pur, ip)

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

Further 1 shall wriU-

./23<./3 ltfl 2> /I 4>./2 l'./34'

for

m,, mv, m., ie,, ie , ie..

i.e., the components of m and ( i.e.} along the three axes; now *f we take any two indices (h. k) out of the series (1,2,3,4), /„=-/..,

* See note 9.

THK FUNDAMENTAL EQUATIONS FOR ATIIEK 7

Therefore

./ ,1 -.' = ~~./2 s> f\ 3 = ~"./s i > y 2 1 = *vi •-•

/4 1 = ~V I 4> ./ 4 4 = ~/2 4> ./ 4 U = ~~.A 4

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

&i x 8A?

8xj 8x2

*/_4J. 8/4_2 8/i3

8x, 8x2 8xn

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

^4 , 8/4. «/t. .

8x, 8xa ' 8x4

(B)

. .

Sx, " 8x2 8x3 By means of this method of writing \ve 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).

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

8 PKI NT I I'M. ol liKI.ATIVITY

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 .r/, .ra' .r3' .r/ instead of ^( .r2 .r,s .r4, where

;r, = ;r, cos + .r2 sn <>, .r2 = .r, sn<£ + .r2 ,r'3 =.r3.r'4 = «4, and introduce magnitudes p' t, p' 2, p'3 p'4, where p,' = p, cos ^> + pL, sin^>, p2' = p, sin<^> + p2 cos«/> aud/".,o, ...... /'8 4,' where

/'•H=/t4 COS 'A +/M sil1 */»'/' 2 4 = "/I 4 SU1 ^ +

/24 COS ^,./'34=/:Mi

/,, = -fkh (h I k = 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 suffices (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by i'\j/ a purely imaginary magnitude, and consider the substitution

*i' = *i> •''•/= *>>> •' a' = ^3 COR lVr + -"4 H'n T'^ (1)

*/ = x3 sin ity + ,r4 cos ?'i//,

Putting - / tan ^ = T = ^ ^ = lo^ _

THE FUNDAMENTAL EQUATIONS FOR ARTHEtt

\Vi; shall liavo cos i\f/ =

where i < q < i, and v/l~?a *s always to be taken with the positive sign.

Let us now write x , = /, ,7/2 —y'i x' 3 saz') x' i—it' (3) then the substitution 1) takes the form

,,

*=• ••> y=y,z=- —>t=- =, (4)

v/l-^ -x/l-f*

the coofficients 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 by t'^, we at once perceive that simultaneously, new magnitudes p'j, p'2, p'3, p'4, where

(P/i=Pn P'»=P2> P s=Ps cos e't/' + p4 sin t^r, p'4 =

P3 sin ty + p4 cos t^r), and/' 12 .../34, where /'4i=/4i cos ty +/13 sin 7>,/13= -/41 sin »V +/t g

COS f «A, /' 3 4 =/3 4 , / 3 2 =/3 o COS ^ + /4 2 «in t ^, /' 4 2 = ~/32 Sin ^ + /42 COS ^ /I 2 =/18i /** = -/*»t

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 taken, and .i/ y z' t' be takes as a new frame of reference, then we shall have

,KX » r-qrn.+l-]

(5) P=P - , P»« =P

p«.=pu.,

10 PRINCIPLE OF RELATIVITY

e,— qm, , , oe.+m,

(6) e,' =

q* ' VI— '

Then we have for these newly introduced vectors u', e , m' (with components ux', ur', u/ ; ex', <?/, e/ mx', my', «»/), 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, ev+qm, are components of the vector e+ [ym], where v is a vector in the direction of the positive Z-axis, and i v \=q, and [vm~\ is the vector product of v and m ; similarly —qet + m,,mf+ge, are the components of the vector m—[ve].

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

e,' + t'mV = (e. +«"•») cos i\ff + (ey+tmy) sin i\[/,

ey' + im'y' = (e.+t'm,) sin i\j/ + (ey+»my) cos ty,

e,' -\-irn' ,'~e' t -}-imt.

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

t'mY) cos. <t>+(es/" + im']/') sin </>

cos. (^ + ^) + (ey+tmy) sin (<f> + i\j/), ') sin </> + (e'/ + t'm'/) cos. <^>

sin (<t> + i^) + (eif +im, ) cos. (<^ + ).

SPECIAL LORENTZ TRANSFORMATION.

The rl 3 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 11

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

Let v be a vector with the components vx, vy, v,, and let \ v \ =q<l. By » we shall denote any vector which is perpendicular to v, and by rv, r-^ we shall denote components of r in direction of ~y and v.

Instead of (.r, y, z, t), new magnetudes (x y' 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, 2) 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 y, we have

T^r

and for the perpendicular direction v,

(11) r'T=r7

ar.+t

and further (12) t'= / .

v I q*

The notations (/-?, r'r) are to be understood in the sense that with the directions v, and every direction y~ perpendi cular to v in the system (.r, y, z) are always associated the directions with the same direction cosines in the system

(*' y, O-

A transformation which is accomplished by means of (10), (11), (12) with the condition Q<q<\ 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 u , e ' , pt', in the system (x'y'z'} are so defined that,

12 PRINCIPLE OF RELATIVITY

further

(15) (e + «V) , = (e + m) i [u, (e + /m)] ,. .

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

(16) rc

q2 q

Now we shall make a very important observation about the vectors u and u'. We can again introduce the indices 1, 2, 3, 4, so that we write (.*',', #2', vz' , #/) instead of (.c', t/', ;', it') and p/, p2', p3', p^' instead of (p1 u',', p u'y', p' u',', ip.

Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determiuant + 1, so that

(17) .T1a+.r22+.tsa+.«42 i. e. x* + y*+z*-t\ is transformed into

*!" +.'.'," + .''s'' + 'O1'' e- x'*+y'*+z'2-t'*. On the basis of the equations (13), (14), we shall have (pl*+pS+P>*+P<*)=P*(l-ut>,-uv\-u,*,)=p*(l-u*) transformed into ps(l MS) or in other words,

(18) p A/l^T*

is an invariant in a Lorentz-transformation.

If we divide (pl, ps, ps, p4) by this magnitude, we obtain

the four values f(»1, wa, o>,, o>4) = . . (u,, uy, «,, i)

VI _ M*

so that a)12+u)a*+w31+to4f = 1.

It is apparent that these four values, are determined by the vector u and inversely the vector // of magnitude

SPECIAL L011ENTZ TRANSFORMATION 13

<i follows from the -1 values o^, oja, o>3, w.t ; where (<•>„ 01,, wj are real, *w4 real and positive and condition (ID) is fulfilled.

The meaning of (<•>!,<•»,, o»3, <•>,) here is, that they are the ratios of <Arlf ilr,, d „, rfx4 to

(20) V_(d.,:l

The differentials denoting the displaeements of matter occupying the spacetimc point (.clf .••„ ^3, .i-J to the adjacent space-time point.

After the Lorentz-transfornation is accomplished the vococity of matter in the ne\v system of reference for the same space-time point (.</ y -J t') is the vector n' with the

dx' <ly' dz' dl' ratios -^-,, -J,, ^p, ^p, as components.

Now it is quite apparent that the system of values

.('=

is transformed into the values

*i' = "i'i -I'a^Wi'. »3' = W3'5 *4' = o)4'

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

The dashed system has got the same meaning for the velocity //' after the transformation as the first system of values lias got for n before transformation.

If in particular the vector v of the special Lorentz- transformation be equal to the velecity vector u of matter at tin- space-time point (.e,, a?,, r3, x4) then it follows out of (10\ (11), (12) that

<o1'=:o, 0)a' = 0, ti)3' = 0, o>4'=l

Under these circumstances therefore, the corresponding space-time point has the velocity v' = u after the trans formation, it is as if we transform to rest. We may call the invariant f> ^/ \ as the re.-t-dcn-ity of Mlectricity.*

* See Note.

PRINCIPLE OF RELATIVITY

§ 5. SPACE-TIME VECTORS.

Of the hf, and 2nd kind.

If we take the principal result of the Lorentz transfor mation together with the fact that ;he 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 easi.ly comprehensible, it may 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, a n a , 55 a t 3 a ,

a21 «22 «23 a2 a31 «33 a33 a3 _ __ La41 «42 a43 «4

the Determinant of the matrix is +1, all co-efficients with out the index 4 occurring once are real, while a4 ,, a42, a4 3, are purely imaginary, but a4 4 is real and >o, and a^2 + #22 + ''32 +^42 transforms into ^,'2 +#2'2 +'C3'2 + #4'2. The operation shall be called a general Lorentz transformation.

If we put xl' = ,ct ,r 2 '=//', .'-3 ' = z', ,c4' = it', then immediately there occurs a homogeneous linear transfor mation of (*, y, z, f) to (r'} y', z', t') with essentially real co-efficients, whereby the aggregrate .c2 y* ;2 +t2 transforms into v'2 y'~* z'- +t'*, and to every such system of valut^ £, y, :, ./ with a positive I, for which this aggregate o, there always corresponds a positive t' ;

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

SPACE-TIME VECTORS

15

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

The last vertical column of co-efficients has to fulfil, the condition :!2) al 42 +a.J48 + tf34a -f-a442 = l.

If ,i

= fl34 = o, then «44 = i, and the Lorentz

transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If 0l4, as4, «S4 are not all zoro, and if we put «i4 : fl»4 : «34 : «44 = v, : -v,:i

Ou the other hand, with every set of value of ai4> rt»4> a34» fl44 which in this way fulfil the condition I'l) with real values of v,, vv, v,, we can construct the special Lorentz transformation (.16) with (a^ 4, «24, 034,a44) as the last vertical column, and then every Lorentz- transformatiou with the same last vertical column (al4, a24, a34, a44) can be supposed to be composed of the special Lor(?ntz-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 Pi,p8,ps, p4) with the condition that in case of a Lorentz-transformation it is to be replaced by the set p,', p2', p3', p4'), where thes; are tlii! v.ilue; oP .'•/, c./, .-.,', .e/), obtaiael by sab^tituting (p,, p.}, p.,, p ) for (-,, xj} .»-3, .«-4) in the expression (il).

Besides the time-space vector of the 1st kind (xlt o;2, *3> -''4) we shall also make use of another space-time vector of the first kind (y,, //.,,. //3, y4), and let us form the linear combination

-*« y.)

«* ys)

16

PRINCIPLE OP RELATIVITY

with six coefficients /„.,—/, v. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors.

•Ci,as*,#a ; yi,y.,ys ;/»3./si./n ; /i*. A*' A* and

the constants ,«4 and y4, at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject (ajlf .r,, ,«3, ,r4) and (yt, ya, ya, y4) simul taneously to the Lorentz transformation f21), the combina tion (23) is changed to.

(24) /„' (,-,' y,'-.r,' y.) +/sl (a,' <//-../ ys) + /, ,

whei'e the coefficients /83', /3 /, /x 2', /14', /,4', /84', depend solely on (/8S /94) and the coefficients alt ...a+4.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes /2 3,t/s t ...... /34, with the

condition that when subjected to a Lorentz transformation, it is changed to a new system y^./ ...... /"34J... 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 ,--, i/, z, it (space co-ordinates, and time if) is sub jected to a Lorentz transformation, and at the same time (/)/*,, pity, pit',, ip] (convection-current, and charge density pi) is transformed as a space time vector of the I st kind, further (mx, mf, )»..-, '*',, ie,, if.) (magnetic force, and electric induction x ( /) is transformed as a space time vector of the :!nd kind, then the system of equations (1), (II), and the system of c |inti > is (III), (IV) trans forms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system.

SPECIAL LORENTZ TRANSFORMATION 17

These facts can be more concisely expressed iu these words : the system of equations (I, and II) as well as the system of equations (III) (IV) are covariant in all eases of Lorentz-transformation, where (pu, ip) is to be trans formed as a space time vector of the 1st kind, (ni ie) is to be treated as a vector of the 2nd kind, or more significantly,

(pit, ip) is a space time vector of the 1st kind, (m 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 in variants for such a vector when subjected to a group of Lorentz transformation.

A space-time vector of the second kind (m ie), where (m, and e) are real magnitudes, may be called singular, when the scalar square (in ie)*=0) ie ms—et=oi and at the >;une time (m e)=o, ie the vector wand e are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector oi 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 \me\ coincides with the Z-axis, i.e. mjt) = o, e,-=o. Then

Therefore (i-t +i m,, )j(et +i cx) is different from +i, and we can therefore define a complex argument <j> + i$) in such a manner that

=

'„ +t m,

Vide Note.

18 PRINCIPLE OF RELATIVITY

If then, by referring back to equations (9), we carry out the transformation (1) through the angle <A, and a subsequent rotation round the Z-axis through the angle <£, we perform a Lorentz-transformation at the end of which mv = o, ev=o, and therefore m and e shall both coincide with the new Z-axis. Then by means of the invariants m"2 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 OF 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, t) for space time points (events) be somehow known. Now if a space point A (x,,y0, O at the time ta be compared with a space point P (.f, y, z) at the time /, and if the difference of time t—tt, (let t> I.) be less than the length A P i.e. less than the time required for the propogation of light from

* Just as beings which aro confined within a narrow region surrounding a point on a shperical surface, may fall into the error that a sphere is a geometric figure in which ouo diameter is particularly distinguished from the rest.

CONCEPT OP TIME 19

A to P, and if q= - ° < 1, then by a special Lorentz A i

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, t0), and P, t). So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time t0 = 0, 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 tt (t > t0] be less than the time which light requires for propogation from the line A B, or the plane A B C) 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, I.), [B, t0), (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 t,, 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 of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Kinstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.

20 PRINCIPLE OP 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- = 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 m, 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 u, 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 (.<-, y, r, (,], lead us to a knowledge of other magnitude as functions of .,-, 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

*, y, *, 0-

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

FUNDAMENTAL EQUATIONS FOR BODIES AT REST 21

Loreoti lead to the same fundamental equations. They are ;

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

(i) Curl m ~- - C, (iV) div e =>.

(in) Curl E + - = o, (ir) 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 E, M = urn, C = <rE.

where « = dielectric constant, /j. = magnetic permeability, a- = the conductivity of matter, all given as function of •cj > "> ^j * is h«re 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 *», st, 53, *4 for C,, Cy, C, V _ 1 p. further /,,,/,,,/, ,,/,,,/„,,/,. form,,m,,m, t (e., ey, c. ), andFls,F31,F19,F14,F,.,FS4 for M., M,, M,, - i (E,, Ey, E.)

lastly we shall have the relation fkk = —,./**, Fkll = —F)tk, (tin- letter/, F shall denote the field, * the (i.e. current).

22 PRINCIPLE OF RELATIVITY

Then the fundamental Equations can be written as

a/sl

3/t, 3.c4

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

3F,. .

a.., a 3

= 0

8F,t .

3*, ' a,4

+ ^^-8 +

3F,,

= 0

= 0

= o

§ 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 " is zero, for a system * Einzelne stelle der Materie.

THE FUNDAMENTAL EQUATIONS 23

(j-, y, :, /) the neighbourhood may be supposed to be in motion in any possible manner, then for the space- time point x, y, z, tt the samo relations (A) (B) (V) which hold in the case when all matter is at rest, shall 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 propogation of light.*

The fundamental equations are of such a kind that when (f, y, zt 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 magnitudes.

Shortly I can signify the third axiom as :

(/#, 1<?), and (W, iE] are space-time vectors of the second kind, (C, ip) is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.

In fact thes^ 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 | it \ is <1 at any space-time point. In consequence, we can always write, instead of the vector u, the following set of four allied quantities

11 - - u' -

x/n^T'

11 »

* Vido Note.

24 PRINCIPLE OF RELATIVITY

with the relation

(27) Wl2+W22+w,2+W42= |

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

Let us now iix our attention on a certain point (r, 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 (A), (5) (V) of §7. It u t o, then there exists according to 16), in case | n \ <1, a special Lorentz-trans- formation, whose vector v is equal to this vector u (.r, y, z, t), and we pass on to a new system of reference (?' y' z I') in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values 28) 0)^ = 0, o/2 = 0, u/3:=0, u/4=f, therefore the new velocity vector o/ = 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 (.r' / z I) involves the newly introduced magnitude (n p', C', e', m', E' , M'} and their differential quotients with respect to (x', y' , z', t'} in the same manner as the original equations for the point (x, y, z, t). But according to the first axiom, when u' = o, these equations must be exactly equivalent to

(1) the differential equations (^'), (#'), which are obtained from the equations (.-/), (/?) by simply dashing the symbols in (./) and (B).

(2) and the equations

(V') <?' = cA", M' = pm', C' = a/-'

where e, ^, o- are the dielectric constant, magnetic permea bility, and conductivity for the system (./ //' :' t'} i.e. in the space-time point (t y, z t) of matter.

Illl. II MM \II.\T\I. Kor \TIOXS 20

Now let us return, by means of the- reciprocal Lorentz- trausl'onnation to (lie original variables (.»-, y, :, /), and the magnitudes (/', p, C, <•, ///, A', I/) and (he equations, which we then obtain from the last mentioned, will be the funda mental equations sought by us for the moving bodies.

Now from § 4, and § 0, it is to be seen that the equa tions .-/), as well as the equations //) are covariant for a Lorentz-transformation, />. the equations, which we obtain backwards from A'} B'}, must be exactly of the same form am the equations ./) and />'), as we take them for bodies at rest. \Ve have therefore as the first result :

The differential equations expressing the fundamental equations of electrodynamics for moving bodie?, when written in p and the vectors C, >% ///, K, M, are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the rectorial wav of writing, we have

curl ni ' ' = C,, II) div f=

n i I

III VMH-! K -f 90M = o lV)divM=o

/ U /

The velocity of matter occut> only in the anxilliary equations which characterise the influence of matter mi the basis of their characteristic constants e, /<, <r. Let us now

transform these aiixilliary equations \ ') into the original co-ordinates ( , //,:, and /.)

Acconlin^ to formula 15) in ^ I, the (-omponent of >•' in the direction of tin- vector /' i- the same us llial n|' (?+[H tn']), the component of /;/' is th" same as that of «/ [// r], but for the perpendicular direction r>, the com ponents of f' , ni are the same as those of (••+[> ///i)and (///

[UP], multiplied by / . On the other hand !•'/

26 IMUXC II'I.K 01 i; I.I \livm

and M' shall stand to E + ["M,], and M [«Kj in the same relation us <•' aud ///' to <!'+[«>#], and m (//>•). From the relation c>/ = « E', the following- equations i'uliow

(C) r+[W//,]=c(E+[//M]),

and from the relation M'=//. «/, we have

(D) M-[ttE]=/t(/y/-[/^]),

For the components in the directions perpendicular to /', and to each other, the equations are to be multiplied by v'fll^:

Then the following equations follow from the transfer- mation equations (12), 10), (11) in $ 4, when we replace q, rt, r-r-, t, r',., r'r, (' by | n \ , C,,, C,, p, C'. C'v, p

,- C + , CM-

In cunsideration of the manner in which o- enters into these relations, it will be convenient to call the vector C— p /' with the components C,— p | /' | in the direction of /', and C',, in the directions T> [lerpendicular to H the "Convection current.'' This last vanishes for <r=o.

\\'c remark that for «=1, //-=! the ecjuations r' = E', /«' = M' immediately lead to the equations / E, //i = M bv means of a rcciprofid Liu'cnt/.-transformatiun with it as vector; and for o-=o, the equation C' = o leads to C=p i' ; that the fundamental equations of Ather discussed in § :i becomes in fact the limitting case of the equations obtained here with « = 1, /* = !, ^—o.

FUNDAMENTAL EQUATIONS IN LOKKXT2 THKORY 27

§1). THE FUNDAMENTAL EQUATIONS IN LORENTZ'S THEORY.

Let us now see how far the fundamental equations assumed by Lorentz correspond to the Relativity postulate, as defined in sjS. 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 200, Eq" XXX', formula (11) on page 78 of the same (part).

(Ul»'") Curl (H-[//E]) = J+ ) + /'divD

-curl OD]. (1") div !) = /.

(IV") curl E = -^ , Div 13 = 0 (V)

Then for moving non-magnetised bodies, Lorentz pubs (page :2:!3, :i) ^= 1, B = H, and in addition to that takes account of the occurrence of the di-eleetric constant e, and conductivity o- according to equations

Lort'iit/'s 1-], 1), II ;irt- here denoted i>v E, M, r, m while J denotes the :-onduction cuiTeiit.

The three last equations which have been just cited here coincide with eq" (II), (III), (IV), the first equation would be, if J i> identified with C, /'/; (the current being y.ero for (/ = (),

(2U) Curl i II -(.,K; j =('f-> -curl[«l)i,

28 1'itfxcii'T.K or i! i;r. VTFVITY

and thus comes out to he in a different form than (1) here. Therefore for magnetised bodies, Lorentz's equations do not correspond to the Relnti vity Principle.

On the other hand, tlie form corresponding to the relativity principle, for the condition of non -magnetisation is to be taken out of (D) in ^S, with i>.= \, not as B = H, as Lorentz takes, but as (30) B [>!)] = H [»D] (M [«E]=/;/ [«<*] Now by putting 11 = B, the d.flW- ential equation ('-•') is transformed into the same form as e<j" (1) liere when ;// [//,»] =M [«K]. 'Pherefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for moving non-magnetised bodies at last agree with the relat.ivit v postulate.

If \ve make use of (-'HI) for non-magnetic bodies, and pat accordingly Il = IJ-r-[", (D E)], then in conse(|iience of (C) in §8,

(e_]) (K + r,,,B;) = 0_E+r,,. [;/,D— E]], is. for the direction of n

and for a perpendicular direct ion fi,

i.r. it coincides with L^rent/'s assumjition, if we neglect /' - 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 ; eomp. (K) ^ 8]— that the components of J,,; -I ftW eijual to the components of <r(E-f [n B])

multiplied by ^fZ^i or ^j^TT respectively.

i I \UAMKXT\T. i;OI'ATIONS <>K K. COHEN* 29

vjlO. I'YNDAMKNTAL EQUATIONS OF E. COHN.

K. Cohn assumes the following fundamental equations.

(31) Curl (M-f 0 E]) = '^ + u div. E + .T

Curl [E— (//. M)]= _ + n div. M. ill

(32) J = ,r E, = cE-|> M], M = /*(y//+[> E.]) where I1} M are the electric and magnetic field intensities (forces), E, M are the electric and magnetic polarisation (induction). The equations also permit the existence of true magnetism ; if we do not take into account this eoiHiderat ion, div. M. is to be put = ft.

An objection to this svstem of equations is that according to these, for «= 1, /i =1, the vectors force and induction do not coincide. If in the equations, we conceive E and M and not E-(t*. M), and M+[T E] as electric and magnetic forces, and with a glance to this we substitute for E, M, E, M, div. E, the symbols c, M, E -fCU M], >// [/'<"], p, then the differential equations transform to our equations, and the conditions (32)

transform into

J = «r(E+0 M])

,+ [*, (*-[**])] =«(E+[«M])

then in fart the equations of (John become the same as those required by the relativity principle, if errors of the order •' 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 auxiliary conditions are

.'50 I'lil.N'l II'I.K OF KM I,

§11. TYPICAL HKPKKSKXTATIONS OF THK EUND A MENTAL EQUATIONS.

In the statement of the fundamental equations, our leading idea had been that they should retain a eovariance of form, when subjected to a group of Lorentz-trans- formations. Xo\v we have to deal with ponoeromotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that, the settlement of this question is in some way connected with the simplest forms which can be given to (he fundamental equations, satisfying the conditions of covarianee. In order to arrive at such forms, I shall first of all puf the fundamental equations in a typical form which brings out clearly their covariaucein case of a Lorentz-transformation. Here I am using a method of calculation, 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 iikk formed into the matrix

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

If all the quantities </,.,< are multiplied by (\ the resulting matrix will be denoted by CA.

It' the roles of the horizontal rows and vertical columns be interchanged, we obtain a yx;; series matrix, which

AI i;i,i'i;i -i,.\ i > i lOHS

ill

will In- known ;i> the transposed matrix oi A, aiul will be »!fiu>1etl 1'V A.

It' we have a second jj x ij series matrix B,

tl-.en A + B shall denote the j> x y series matrix whose members are a,, k -\-bhk.

If we have two matrices A=|aM a

ki a

where the number of horizontal 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

'••'.' i

,;;f I

these elements bein^r formed l>y eombinat ion of the hori/onlal rows of A with the vertical columns of B. For such a point, the associative law (AB) S = A(HS) holds, where S is a third matrix which has -ot u many hori/ontal rows as B (or A l>) has -^ot vortical columns.

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

.K or RELAIIVITV

3". We shall have principally to deal with matrir-rs with at most four vertical columns and for horizontal

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.

(:H) e,, eI2 e18 c14 = I 1 0 o 0

0100 00 1 01

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 may denote by A'1 so that A-1A = 1

A matrix

./»i/s>° y.,,

/, , /, a /, ,

in which the elements fulfil the relation //, < = /»«,is called an alternating matrix. Those relations say that the transposed matrix f = ./. Then by /* will be the il/'df, alternating matrix (35)

; /,,

TYPICAL i;i:i'iM>i:\Tm<)N.s

33

/>. We shall have a 4x4 series matrix in which all the elements except those on the diagonal from left up to right clown are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).

The determinant of /is therefore the square of the

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

4°. A linear transformation

which is accomplished by the matrix

A=| a,,. a,2, als, a,+ «n, « 2> "a si a*4

will be denoted as the transformation A By the transformation A, the expression

.,•»-}- ,»+ ,i«4- l is changed into the quadrat ie

for /// ^'i/,/ ••/ ."/,

where aAi=alt «,*+«„/. asA+usAa3l +«+*«4lt.

are the members of Jl 1-x !• series matrix which is the

produrt of A A, the transposed matrix of A into A. li \\\ the tranrformation, the rxpn-sion i- cliaiiged to

inu- t h:ue A A = 1.

PRINCIPLE OF RELATIVITY

A has to correspond to the following relition, if trans formation (38) is to be a Lorentz-transformation. For the determinant of A) it follows out of (39) that, (Det A)2 = 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 involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and /*44>0.

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

(41) *= | *, *s *,, *4 |

is to be replaced by ft A in case of a Lorentz transformation A. i.e. -f'= | •?/ •?.>' *../ */ | = | f> i .'2 *s •% | A; A space-time vector of the 2nd kindt with components /*2 3 . . . /'34 shall be represented by the alternating matrix

/,, /42 -A, o and is to be replaced by A"1 /'A in case of a Lorentx. tr;insformation [see the rules in § 5 ('23) (24)]. Therefore referring to the expression (37), we have the identity

Det^ (A/A) = DetA. Det*/ Therefore Det V be comes an invariant in the case of a Lorentz transformation [•eeeq. (:Jfi) Sec. § 5].

* riiJr nut.. 13. t T'l'f/c note 14.

M en \i fcBPJUU I-.N r.vnoNs 35

Looking back to (-M), we have i'or the dual matrix (A./1*A)(A-'/A) = A-1/VA-Det" /. A-1A = Dofi / from which it is to he seen that the dual matrix/* behaves exactly like the primary matrix/, and is therefore a space time vector of the II kind; /* is therefore kiiown as the dual space-time vector of/ with components (/', 4>/34,/34>)»

OWso/ij).

0.* If w and ,v are two space-time rectors of the 1st kind then by w ,1 (as well as by -v //> ) will be understood tin- combination (1-3) w i sv +10,, s.j-Mfg iV3 + #>4 ,v,.

In case of a Loreiitz transformation A, since (wA) (A-v) ' = 10 s, this expression is invariant. If tr x = o, then w and s are perpendicular to each other.

Two space-time rectors of the lirst kind (10, .v) gives us a 2 x 1 .series matrix

W, IV.) 10., 10:

Then it follows immediately that the system of six magnitudes (14) /«._, .v3 to.j *._,, w3 -v, wl -v3, 10 , s.,—io* s,,

behaves in case of a Lorentz-transformation as a space-time vector of the II kind. The vector of the second kind with the components (It) are denoted by [w, .v]. We see easily

that Del" [«?, v]=o. The dual vector of ["Vs'] shall be written as [ /'•, ,v].*

If //; is a space-time vector of the 1st kind, /' of the .-eeond kind, /'•/ si^nilies a 1 x 1 series matrix. In case of a Ijoivnt/.-traiisfonnation A, f is changed into tr' = icA, /'into /" = \~l /'A, therefore /'•'/" become> =(/'-AA~'/' A) = /'•/' \ /.'. "'/'is transformed as a >pace-time vector of

36 PRINCIPLE OF RELATIVITY

the 1st kind.* We can verify, when w is a space-time vector of the 1st kind, /' of the 2nd kind, the important identity (45) O, wf } + O, »/*] * = (m w ) 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 (al=o, o;a=:o. w3=o, ta.t=i,

<«/= I */*!» */*„, *!/4S, ° I ; <«/* = I #3 •' */!»» #21 > ° I

|> eo/J =0, o, o, /., t , /, 2 , /t , ; [w a>/*]* = o, o, o, /, 2,fi;>J.2l.

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 LoreNitz transformation, and is homogeneous in (wj, wa, o)3. wt).

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

Instead cf the space vector u, the velocity of matter, we shall introduce the space-time vector of the first kind to with the components.

(40) where w 1 2 + w2 a + co3 2 + w and— iw.t >0.

By F and / shall be understood (lie space time vectors of tin- second kind M i\']. n> ic.

In 4> = wF. \vu have a space time vector of the first kind with components

3., .ln>

4jl+w.F^

I'idc note 15.

I-YPICM. I;H'I;I;SK\TATIONS 37

The first three quantities (<£,, <£a, <£,) are the components of the space-vector IrL+JJi'JH .

^1-1T»~

and further <£4 = -IfiLjEL . ^1— w"

Because F is an alternating matrix,

(49) O}* = w1^»1 +wa4>2-|-<jJ34>3 +oj44>t r=o.

i.e. & is perpendicular to the vector w ; we can also write 4>t=i [wx4>j +wy4>a +w.4>3].

I shall call the space-time vector <I> of the first kind as the Electric Rest Force*

Relations analogous to those holding between wF, E, M, Uphold amongst w/, v, m, u, and in particular otf is normal to w. The relation (C) can be written as

{ C } o)/=e<oF.

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 ^ = iw/*, whose components are 0

*!=-*'( »J*,* +"3/4, +«*»/„) 1

*a = -''('",/,.,+ «•!/*• *«4/»»)

*.=-»(«l/.*+«Ji, +",/12) I

** = *(*»!/• •+»•/!» +*»|/g, ) J

Of these, thu first three «!',. >!/,. »J/S) .irc the .i\ //. \ i'(iinpo;ients of the space-vector 51) "i ('"')

and further (52; ft = '/'nil/ /

Fide note 16.

38 ri;i\( ii'i.i. 01 KKI.VIIMTY

Among these there is tlie relation

(53) t1>V (J>ltyi+M2ty.2+ 0)^.1+ c>4*4=o which can also be written as 4'4=:i (uxfyl + "y^a ~J""»^'s)'

The vector * is perpendicular to « ; we can call it the \Iay>ietic rest-force.

Relations analogous to Ihese hold amon^ the quantities iwF*, M, E, u and Relation (D) can be replaced by the formula

{ D } -o,F*=/4w/*.

We can use the relations (C) and (D) to calculate F and f from <J> and * we have

WF - _ $. up* = _ i'^i/. ,,,/= _ €<i,, „,;• *--/*.

and applying the relation (45) and (46), we have F= [CD. *J + z>[o». *]* 55)

/= e[<o. *] + *[«.*]* 5G;.

i.e. F12=(Wl$1-wi!*1) + .>[w/l't-on*3]! etc. /la=e(wI*a-wa*l) + A [o)3*4-w4*3]. etc. Let us uo\v consider the space-time vector of the second kind [$ *], with the components

*8*3-*3*2, *,*1-*1*3, ^^j,-*,*, -, 4>ixi/i_4>4^/l, <J>2\J/4_4)^\I>2. cj>3»l/+— <j>t\i/3 J

Then the corresponding space-time vector of the tir-t kind w[*, *] vanishes identically owing to c.(juation> ;>) and 33)

for

Let us now take the vector of the 1st kind

the

. etc.

TYPICAL i:r.i'i:i-si;.\T \TI<>\

Then In' applying 1'11'(1 (^-"O' wo 'l^ (58) [*.*] = i [u>n]»

i>. ^j^,— <P.2>I'1=«(wr,O+— o>+$2.,) etc. Tho vector fi fulfils the relation

(which we can write as n+=t (to^Qj +ojyQ2 +a>2Q'3) and n is also normal (o o>. In case w=o. we have *4=o, *.v=o, n4=o. and

*i *a *

I shall call fi. which is a space-time vector 1st kind the Rest- Ray.

As for the relation E), which introduces the conductivity a- we have u>S=: (wi*i +Was2 "r^s** +W4-S'.«.)

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

Then r»l)=s+(t,w)«

represents a space-time vector of the 1st kind, which since axi>= 1, is normal to «,>, and which I may call the rest- cnrrent. Let us now conceive of the lirst three component uf this vector as the (./• // c) co-ordinates of the -pace- vector, then the component in the direction ui' // is

C - I " I P = ~ I u I P -1"

Vll^ A/1— «» l-«"

and the comiionent in a per])endicular direction is ('., Ju.

This space-\ector i- conneeted with the space-veefur .1 C fiit, which we denoted in ^ as the current.

40

PRINrC[IT,K OK KKI.ATIVITY

Now by comparing with <f>= --o>F, the relation (li!) cm be brought into the form

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.

§12. THE DIFFERENTIAL OPERATOR LOR.

A 4x4 series matrix 62) S= S,, S12 S1S S14 = | S,A | SS1 Sa, S2S S,4

S41 S42 S4;t S44

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 alternating matrix f, which corresponds to the space-time vector of the II kind,

2) the product f F of two such matrices, for by a transfor mation A, it is replaced by (A-1/A-A~IFA)=A~I/F A, M) further when (MI. uis. cua, co4) and (ni; O2, «s, U4) aiv l \vn spare-time veotiu-s of llie 1st kind, the 4x4 matrix with tlie element SAA. =wAnA,

lastly in a multiple L of the unit matrix of 4x4 series in which all the elements in the principal diagonal are equal to L, and the rest are xero.

\Yc shall have to do constantly with functions of the F pace-time point (.r, >/, :, if], and we may with advantage

III! IH I KKUK.VIIAI. OI'MJATOR LOR

employ the Ixl series matrix, formed of differential symbols,

a a a a

or (63)

a a a a

3 dy 3:

For this matrix T 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 1 x 4 series

matrix

I K, K, K< K. |

where K4 = ^L* + *i* +

When by a Lorentz transformation A, a new reference system (.K\ .<:', x' s ,i-4) is introduced, we can use the operator

lor'=

^ a a a

8."/ 8.V",' 8*,' a-,'

Then S is transformed to S'=A S A= | S'»j | , so bj lor 'S' is meant the 1x4 series matrix, whose element are

*ri aS|4.oSjt.as.tj i a s * *

Now foi1 the differentiation of any function of (x y t f)

a a a , , a a ,

we have the rule

ST . ^ -^ , "T j? W i

a .»•* 3', 3 .••* 3 .--, 3 *

t a 3-''3 , a 3««

+ 3.8 87T' 3^ 3A

3 3 L 3 , , 3

3 i 3 •*' $ 3

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

Vide note 17.

42 PRINCIPLK OK 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 matrix with the elements

8L 6L 6L 6L

8-«i 8.I-, 9.t 3 9**

If # is a space-time vector of the 1st kind, then lori =»il+»£.+|il + »i...

8*1 9-c2 80, o «-4

In case of a Lorentz transformation A, we have lor V=lor A. As = lor *.

i.e., lor * 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 f denotes a space-time vector of the first kind with the components

8/i, a/,, , 8/l4

a + a + "a O ••, O , 0.' +

a/ , +._8At..

9-f.-, 9-(i,

a*

3.' + l^' +8Z'

a/.,

THK DIFFERENTIAL OPKRATOR LOR 43

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

{A} lor/=-»,

and the system (B) can be expressed in the form {B} log F* = U.

Referring back to the definition (H7) for log *, we find that the combinations lor (for/), and lor (lor F* vanish identically, when f and F* are alternating matrices. Accordingly it follows out of {A}, that

^ & + -& + 1;:- + -& = °>

while the relation

0>9) lor (lor F*) = 0, signities 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

«' \

l_W2 Vl-?<2 /

(// = 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 (///, />) (/« = magnetic force, f= Electric Induction. s the space-time vector of the first kind (C. ip) (p = electrical space-den>it\', (' —f>/>= com 1 net ivity curn-nt, = dielectric constant, // = manin'tic penne:il)ility, a = conductivity,

44 PEINCIPLK OK RELATIVITY

then the fundamental equations for electromagnetic processes in moving bodies are*

{A} lor/=—v {B}logF* = o {C} w/'=«oF * T-V i TiMf" /'"X*

{B}^^»=-«*P.

ww = 1, and WF, w/, o>F*, w/**, « + (O>*)<D which are space-time vectors of the first 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 arc mvt-- sary for determining the motion of matter as well as the vector 11 as a function of •, //, , 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 u> as a function of (;,y,:,/.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

/=

o A* As

A*

F=

0 Flt P18 F,,

/.i 0 /,„

/.;

Fsl 0 Pfa F14

/si A* °

A*

F31 F32 0 F,4

A, A, A,

0

F+i ^+8 F+s 0

* Vide note 19.

THK PRODUCT OF T1IK KIELU-VKCTORS /F

are of much importance. Let us write.

(70) fF=

8

S.*-L sls s84

S3, S33-L S..

S*a S*., S44-L

Then (71) SI1+S,,+S33+S44=0.

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

=•• /„ F13+/3I Psl+/lf pif +/14 Fi4

)

Then we shall have

- t;;±':;F;;)

In order to express in a real form, we write

(74) S =

S81 S82 S,3 S2 + S»i S,, S31 S34

X, Y, Z, _,T, X, Y, Z, -iT, X.. >" Z -zT; -iX, -iT, -tZ, T,

nrJJC.=? r/^M,-^,,^ -mrMr+<.,E.-«yEv-,. K.l

40 PRINCIPLE OF RELATIVITY

(75) X,=tn,M,+e.E,, Y,=wi,M,+?,E, etc. X,=e,M,-erM,, T,=m,E,-»»,Et etc.

li,=~

These quantities are all real. In the theory for bodies at rest, the combinations (X,, X,, X., Y., Y,, Yr, Z,, Zv, ZJ are known as "Maxwell's Stresses," T,, T,, T, are known as the Poynting's Vector, T, as the electro magnetic energy-density, and L as the Langrangian function.

On the other hand, by multiplying the alternating matrices of,/"* and F*, we obtain

(77) P*/*=-S1I-L, -Slt , -Stl . -S14

SS1 . Sg2 L, S2S , S24

S1 S,2 , S,,— L, S,,.

S41 S4S S4S S44— L and hence, we can put

(78) /F = S-L, F*.r*=-S-L,

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

| L^< . | , (c4A=l, e**=0, A^A- /*, A-=l, 2, 3,4). Since here SL = L^, we deduce that,

F*/*/F = ( -S-L) (S-L) = - SS + L«,

and find, since/*/ = Dot '-/. F* F = Det * F, we arrive at the intonating conclusion

* Vide note 18.

THE PRODUCT OF 'NIK M ELU-VKCTORS 47

(79) SS = - Det * / Det * F

i.r. 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*1 elements in the principal diagonal are all equal and have the value given on the right-hand side of. (7(J). Therefore the general relations

(80) SM Slt + S*. Sat + 6A, S,*+S4l S4»=«,

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

(81) SM SlA+S*, S,*+S*S SS»+S*, S4» =La-

Det * / Det * F, for// = 1,2, 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 tt [(55), (56) and (57)1, we can pass over to the expressions,

(82) L = i c * ¥ + J p. * *7 (83 j S»4

-f- c

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

<|><t> = 4ii»+«t»/+<l>:, »+*,', * * = ^,'J-*!1 +*•*+*« ' eAA = 1, ekk o (h=f=k).

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

48 PRINCII'LK OF RELATIVITY

right side of (83) as well as S*A represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) gener ally, to prove it for the special case o>l= o, w, =o, ?/?s =o, w4=/. 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 c = eE, M = ps// on the other hand.

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

[-5- (m M - <?E)«] + (em) (EM),

^is = o, because (cm c <S> ^, (EM) = /x *; now referring

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

expression as Det * S.

Since 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 - S~= - (c/t - 1) fa>,n],

from which we deduce that [see (57), (58)].

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

(87) w (S -~S) = (e p - 1) O

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

Zy=Y,, X,=Z,, Y,=X,,

THE PRODUCT OF THE FIELD-VECTORS /> 4'J

and from 83),

T,=nif T,=n2, T..=«:|

x^t^n,, Y,=f/,n2. #,=^0,

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

Z,=Y.=o. Xt=Z,=0, X,=X,=0.

. According to 71), we have '88) X,+Y,+Z.. + T,=o.

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

and if T, and one of the three magnitudes X,, Y,. Z. are = ± Det * Sr the two other.s = - Det ^ S. If fi does not vanish let O ^=0, then we have in particular from 80)

Tr X,=0, Ts Y,=0, Z,Ts+TtTf=0. and if f), =0, «1=0, Zr=-T, It follows from (81), (see also 83) that

and -Z,=T, = VDet "S-f-e,*!),'

The space-t-'me vector of the first kind f89) K = lorS,

is of very o^reat imjwrtanpe for which we now want to demons! rnte a very important transformation

Accord in- to 7*\ S = L+/F, and it follows that lor S=lor L + lor/F.

50 PRINCIPLE OK RELATIVITY

The symbol ' lor ' denotes a differential process which in lor /F, operates on the one hand upon the components of f, on the other hand also upon the components of F. Accordingly lor 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 where N is the vector with the components

8F.. f 8FS1 6F,, f _

-~-~ ~^~

<// = !, 2, 3,4)

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

(91) lor S = *F + N.

In the limitting case < = 1. /x = l. /=F. X vanishes identically.

1HI. PRODUCT OF THK FIELD-VECTORS /K 51

Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of \,_

,92) Xt=- * ** l_l **_£.

for A=l, 2. 3, 4.

Now if we make use of (59), and denote the space- vector which has 11,, i!2, 113 as the •-,//, - components by the symbol W, then the third component of 92) can be expressed in the form

,93)

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

§ 1A. THE PONDEROMOTIVE FORCE.*

Let us now write out the relation K=lor S = . in a more practical form ; we have the four equations

Vide note 40.

52 PRINCIPLE OF RELATIVITY

0#v 9Z. QZ,

_ +_.. _a_

/ 07 \ 1 -K- _ 0 T 9 T y 9 T . 9 T , I?

(97) K* " <-t^i:

It is my opinion that when we calculate the pondero- motive force which acts upon a unit volume at the space- time point .', y, :, /, it lias got, .<•, y, - components as the first three components of the space-time vector

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

The establishment of this opinion i^ reserved for a separate tract.

In the limitting case €=1, /*=!, <r=0, the vector N=0, S=p(D, wK^^O, and we obtain the ordinary equations in the theory of electrons.

APPENDIX MECHANICS AND THE RELATIVITY- POSTULATE.

It would be very unsatisfactory, if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Phvsics.

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

In order to decide this let us fix our attention upon a spe cial Lorentz transformation represented by (Hi), (H), (1 -)> with a vector v in any direction and of any magnitude <y<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 /, /', <y, we shall write ct, ct' , and q/c, where c represents a certain positive constant, and -7 is <c. The above mentioned equations arc transformed into

, c1— ' </''

They denote. as we remember, that r is the space-voMor (•>y> -)> '' 's th'1 space-vector (•''' y ')

If in these equations, kcej>ing '• constant we approach the limit c = oo, then we obtain from these

The new e(|Uatiniis would now denote the transforma tion of a -patial co-ordinate system (*, y, :) to another spatial co-ordinate »\>t. in ( ' y' -') with parallel axt-s, the

54 IMUNCIl'LK 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

+* (1)

when c = <x.

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

It is evident that according to Newtonian Mechanics, this covariance holds for c=oo} and not for £== velocity of light.

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

I may here point out that by if instead of the Newtonian Relativity-Postulate with e=oc, \ve 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 ( , //, -, /), it may be convenient to leave (y, ) out «-F account, and to treat s and t as any possible pair of co-ordinates in a plane, refered to oblique axes.

APl'KXUIX 55

\ space time null point 0 (.r, y, -, /=0, 0, 0, 0) will be kept fixed in a Lorentx. transformation.

The figure— * '-ya -z* +£* = !, />0 ... (2)

which represents a hyper holoidal shell, contains the space- time points A (j-, y, :, / = 0, 0, 0, 1), and all points A' which after a Lorentz-transformation enter into the newly introduced system of reference as (.r' , y', -.', /' = 0, 0, 0, 1).

The direction of a radius vector OA' drawn from 0 to the point A' of (2), and the directions of the tangents to (2) at A' are to be called normal to each other.

Let us now follow a definite position of matter in its course through all time t. The totality of the space-time points ('', t/, r, 0 which correspond to the positions at different times /, 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 point the direction of the space-time line passing through it.

To transform a space-time point P (*•, y, ~, f) to rest is equivalent to introducing, by means of a Lorentx transfor mation, a new system of reference (•', y', -', t'}, in which the t' axis has the direction OA', 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 spuce-time line through P.

To the increment /// of the time of P corresponds the increment

of the' newly introduced time parameter /". The value of the intesrral

56 PRINCIPLE OF RELATIVITY

when calculated upon the space-time line from a fixed initial point P0 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 Lorentz for uniform motion.)

If we take a body R* which has got extension in space at time t0, then the region comprising all the space-time line passing- through R* and /„ shall be called a space-time filament.

If we have an anatylical expression 6(.\> y, ?, /) so that Q(x, y z t) = ft is intersected by every space time line of the filament at one point, whereby

*®.\ -( a \

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

At any point. P of such across-section, we can introduce by means of a Lorentz transformation a system of refer ence (/, y, :' t), so that according to this

.

8..' 8y'

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 r/J=/ J / <L>' ,hj ,f:' 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 / axis of the filament at the point /=i", and the volume of the body R' is to be regarded as the contents of the cross- section.

APPENDIX 57

If we allow to converge to a point, we oome to the conception of an infinitely thin space-time filament. In such a case, a space-time line will be thought 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 through 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 t. If R converges to a point (.c, y, :, /), then the quotient of this mass, and the volume of R approaches a limit p.(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 /*//.}, where /x= mass-density at the point (.r, yt :, /) of the fila ment (i.e., the principal line of the filament), //.I =eontents of the cross-section normal to the / axis, and passing through ('',y,:, t), is constant along the whole filament.

Now the contents f/J. of the normal cross-section of the filament which is laid through (•••, t/, :, f) is

(4)

</!—„«

and the function v= ^— =n\/\_.l* =/*-2l. (5) iw4 O'

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

58 PRINCIPLE OF RELATIVITY

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

For an infinitely thin space-time fi/amenl, tJie product of the rest-mass density and the contents of the normal cross-section is constant atony the whole filament.

In any space-time filament, let us 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 i on Q' than on Q.0. The finite range enclosed between ti° and Q' shall be called a space-time sic/tel* Q' is the lower boundary, and Q' is the upper boundary of the nickel.

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 nickel, 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 jW/n (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

//// lor vu dfdydzdt,

the integration being extended over tin' whole range of the Kic/icl, and (comp. (67), § 15)

, -_ 8vo>1 8»'<i)2 , 8^3 , 3va)4

D11"'-- + - + - +--

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

* Sichel a German word meanintr a crescent or a scythe. The original terra is retained as there is no snitnble Kn»li.sli equivalent.

\iTi\m\ 59

which is the- condition of cortinuity

-* , . r , / _0

a* a^ a-- f<8<

Further let us form the integial

N=SISJvd«lyd:dt (7)

extending over the whole range of the space-time .tic/tcl. We shall decompose the sichel into elementary space-time filaments, and every one of these filaments in small elements dr 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 vdJ n=dm and write T" , rl for the 'Proper-time' of the upper and lower boundary of the sichel.

Then the integral (?) cau be denoted by

IJvdJn </T=/(T'-T') dm. taken over all the elements of the sichel.

Now let us conceive of the space-time lines inside a space-time nichd 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 c-urves are to be varied in any possible manner inside the *ic/tc(, while the end points on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they al \va\s move forward normal to the curves. The whole pit, cess may be analytically repre sented by means t,f a parameter \, nnd to the value \ = o, shall correspond the actual curves inside the .v/'7/t7. Such a process may be called a virtual displacement in the sichel.

Let the point (.r, _//, r, /) in the sichel \ = o have the values .r + S--, .// + fy, .- + 8-, f + «/, when the parameter has

60 PRINCIPLE OF RELATIVITY

the value A ; these magnitudes are then functions of (./•, y, z, /, A). Let us now conceive of an infinitely thin space- time filament at the point (j- y : f) with the normal section of contents djn, and if dJn+&dJK 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 + dv being the rest-mass-deusity at the varied position),

(8) (v + Sv) (dJ + Sd J ) = i/rfj = dm.

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 + SN as the mass action of the virtual displacement.

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

(9) rf(a4+8.-4)=cl,-4 + 2 !?£* + ^ d\

k O'r'k OA

*= 1,2,3,4 ^=1,2,3,4

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

(10)

the integration extending over the whole sichel <( where ^(r + Sr) denotes the magnitude, which is deduced from

Jd .»•, + d$xt ) « _(dxs +d8x~)*r^(~d.i> 4 +dS .> by means of (9) and

APPENDIX

therefore :

= 1, 2, 3, 4.

1 = 1, 2, 3, 4.

We shall now subject the value of the differential quotient

(12)

to a transformation. Since each 8- A as a function of (r, y, z, 0 vanishes for the zero-value of the paramater A., so in

, do.l't

general ---- =o, for \ = o.

O.'-*

Let us now put ( - - * ) = £A (/* = !, 2, 3, 4) (13)

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

,1 B (Jy d: dt

for the system (a-, ./-., ./• , r4) on the boundary of the Kic/i'-f, (8.r, 8r2 '&r3 8 }) shall vanish for every value of \aiul therefore £,,£a,f,,£4 are nil. Then by partial integration, the integral is transformed into the form

*w' -U rw*«tf, , 8"w*«,

-i "^v ~a^r

Jj; <hj 1 1 .

6-1 PRINCIPLE OF NEGATIVITY

the expression within the bracket may be written as

The first sum vanishes in consequence of the continuity equation (6). The second may be written as

a«u (l>'l Qwk dcy Qwh di's a<oA dc4

~

(l*>k d (drk

dr dr\dr

whereby is meant the differential quotient in the

t/T

direction of the space-time line at any position. For the differential quotient (12), we obtain the final expression

dx d>/ //•- tit.

For a virtual displacement in the Kiclicl we have postulated the condition that the points Hijipnsi'd to be substantial shall advance normally to the curves i^ivin«jf their actual motion, which is \ o; this condition denotes that the £A is to satisfy the condition

H'I ^i+«'8 ^a+w3 £3+«;* f»=«'. (15)

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

M'PKXDIX

At every space-time point (as in § 13), let a space time matrix of the 'ind kind be known

(16) S=

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

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

slt

BM

S18 SI4 |=

x,

Y,

z, -n\

stl

S2Z

SQ 83 °8 +

X,

Yy

Zv -iTy

S3l

S39

So 33 to3*

x..

Y,

Zr -iT:

S41

S4,

S43 S4*

-;x

/ —ft

r, -«z, Tr

(17)

dzdt

extended over the whole range of the sichel, may be called the tensional work of the virtual displacement.

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

Xi V i f~I i rn * xr O O " . -\y- U O.F , rw O ^* j+lyT^t'rA/'r-A-j -^— + -". y f- ...Li.

o.i ay 3r

X a.— '— -l-T ^8'+ T ^8/

a / a .'• a /

we can now postulate the following minimum principle in mechanics.

If any space-timr Si,-//,-/ /,<• Lovndeil , tlicu for each rirtitiit jfaplacement /// th<- $i<-h>'l , ilte mini of Hie w/r/.v.s1- works, and tension works s/nilf ahcm/x lie mi c>tremum for that process of the spacr-lime line in the 8ie/n occurs.

The meaning is, that for each virtual displacement,.

A=0

(1*5

64. PRINCIPLE OF RELATIVITY

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

(19) v 9™'. =K,+x!t-, (7, = l,a:S4) ,,,onee K, = + *

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 tv^j= 1. By multiplying (19) by wk, and summing the four, we obtain X = Kw, and therefore clearly K + (Kw}w 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,

(21) =X, =T ; ,- =Z,

dr \drj dr

( | =T, and we have also ar \dr/

and X f + Y + ZI:=T<.

</T (Zr </T (?T

On the basis of this condition, the fourth of equations (21) is to be regarded as a direct consequence of the first three.

From (21), we can deduce the law for the motion of a material point, i.e., the law for the career of an infinitely thin space-time filament.

APPENDIX 65

Let *, v, 2, f, 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 styt z, /, 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 R, Ry Rt R, and if m be the constant mass of the filament, we obtain ^

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.f dy dz . dt

,r > ~T ~j » * T ' (IT <ir <IT </T

We may call this vector R t/ie moving force of the

material point.

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 ( s.4V,0, then [See (4)] the equations (2:!) art- obtained, but

are now multiplied by ; in part ioular, the last equa tion comes out in the form,

The right side is to be looked upon «x the amount of u-ork (hue per unit <>f li/ne 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

may be called the kinetic energy of the material point. Since tit is always greater than tlr we may call the

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

over the proper-time of the material point and the law can then be thus expressed ; The kinetic energy 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) a^ain shows the sym metry in (s&tsj), which is demanded by the relativity postulate; to the fourth equation however, a higher phy sical significance is to be attached, as we have already seen in the analogous case in electrodynamics. 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*(.>*, /*, z*t /*) be a solid (fester) space-time point, then the region of all those space-time points B (./•, i/, z, /), for which

VIM'KXDIX 67

may be called a " Hay-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 ( •• y z /) be supposed to be fixed, the point (t* t/* c* (•*) be supposed to be variable, then the relation (:J#) \vould represent the loeus of all the space- time points ft*, 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 principal line of F ; further let B* be the light point of B, C* be the light point of C on the principal line of F*; so that OA' i.s 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

3

(21) ;«;»*( (j\ ) HI)* in the direction of BD*, and another vector of Mutable \;i.lue in direction of B*C*.

68 PRINCIPLE OF RELATIVITY

Now by ( - - # J 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 filament of F behaves when the material point F* has a uniform translate ry motion, i.e., 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 .*', y, z, t, denote the point B, let T* denote the proper time of B*, reckoned from O. Our proposition leads to the equations

/ae\ <* _

where (27) ,c* +y* + ^2 =(t-

In consideration of (27), the three equations (25) are of the same form as the equations for the motion of a material point subjected te attraction from a fixed centre according to the Newtonian Law, only that instead of the time t, the proper time r of the material jx>int occurs. The fourth equation (2(>) 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 (.r y z] is an ellipse with the semi-major axis a and the eccentricity e. Let E denote the excentric anomaly, T

A1TKND1X 69

the increment of the proper time for a complete description of the orbit, finally n~T ='2tr, so that from a properly chosen initial point T, we have the Kepler-equation

(29) nr=E-e sin E.

If we now change the unit of time, and denote the velocity of light by c, then from (28), we obtain

i* 1-fecosE

Now neglecting c~* with regard to 1, it follows that 7, 7 f i i m* l-f£cosE~l

•*="*L1+t«« i^ssffiJ

from which, by applying (29),

(31) nt + const =( 1 + ^-^— \ wr-f '— SinE. \ ac* / ac2

the factor ^ is here the square of the ratio of a certain

average velocity of F in its orbit to the velocity of light. If now w'f denote the mass of the sun, a the semi major axis of the earth's orbit, then this factor amounts to 10~s.

The law of mass attraction which has bn-n just describ ed and which is foimnlated in accordance with the relativity j>ostu!a<e would signifv 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 such a law (with the modified mechanics pro|K>sed above) Hnd the Newtonian law of attraction with Newtonian mechanics.

70 PRINCIPLE OF RELATIVITY

SPACE AND TIME

A Lecture delivered before the Naturforscher Yer- sarnmlung (Congress of Natural Philosophers) at Cologne (21st September, 1908). Gentlemen,

The conceptions 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.

, I

Now I want to show you 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 in variance, (i) their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, (//) when we change the system in its nature of motion, /'. e., when we impress upon itanv 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. Wo look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off tin: second group to itself, and with a li^ht In-art, conclude that we can never decide from physical considerations whether the space, which is supposed to be

APPENDIX 71

at rest, may not finally be in uniform motion. So those two groups lead quite separate existences besides each other. Tlu-ir totally heterogeneous character mav scare us away from the attempt to compound them. Vet it is the whole compounded group which as a whole gives us occasion for thought.

We wish to picture to ourselves the whole relation graphically. Let ( < , i/, z) be the rectangular coordinates of space, and / denote the time. Subjects of our perception are always connected with place and time. No one has observed a place except at a particular time, or Tias obserred a time except 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, ;/, -, /, as a n-or Id-point. The manifoldness of all possible values of r, //, z, t, will be the world. I can draw four world-axes with the chalk. Now any axis drawn (••onsi..ts of quickly vibrating molecules, and besides, takes part in all the journevs of the earth ; and therefore gives 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, wo shall suppose that at every place and time, something perceptible exists. In order not to specify either matter or elect ric.it \ , we shall simplv style these as substances. \Vo direct our attention to the ti-nrlil-pohil , y, :, t, and suppose that wo are in a position to recognise this substantial point at any subsequent time. Let- t/f he the time element corresponding to the changes of space coordinates of this point [t/.r, /It/, if:]. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point), a curve in the world the Karltl-linr, the points on which unambiguously correspond to the para meter / from + oo to— oc. Tho whole world appears to be

72 PRINCIPLE OF RELATIVITY

resolved in sucli world-lives, 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 (•••,#, c) mani- folduess / = o and its two sides /<o and ^>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 r-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression

The second group denotes that without changing the expression for the mechanical laws, we can substitute (.v-at,y-pt, z-yt} for (.•-, y, z) where (a, p, 7) are any constants. According to this we can give the time-axis any possible direction in the upper half of the world />o. Now what have the demands of orthogonality in spaco 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, and let us consider the figure

According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by ( = o. Let us consider the sheet, in the region of l>o, and let us now conceive the transformation of .-•, y, r, / 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

AI'HKMHX 73

tHUuforaialkm in which (y, r) 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 -c-l •— - = ]f witli its asymptotes (/•////» fig. 1).

Then let draw the radius rector OA', the tangent A' B' at A', and let us complete the parallelogram OA' B' C' ; also produce B' C' to meet the -axis at D'. Let us now take Ox', OA' as new axes with the unit mea suring rods OC' = 1, OA'= ; then the hyperbola is again

expressed in the fonii <•''('-— '- = 1, t'>o and the transi tion from (.-, i/y -, /) to ( -'i/z't} is one of the transitions in |iif-iioii. L-it 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 '•, which we may denote by CJr.

N"MW let as, increase c to infinity. Thus - becomes zero

c

:uxl it :i])]icii-> i'rdu the figure that the h\'pei-l)ol:i is j^radu- all\- shrunk into the -;i\i<. the asymptot ic nnglc lic- i-n'iie-- :i -tnii^hi one, and everv <|irf-[;>.! 1 nmsfninrit ion in tlie limit changes in s'ich :i manner that the /-axis (-in Miv possible dir;"-tiun upwiirds, and •' m«>iv -ind

i '")'-oxiin:ites tn . Remembering this point it is "leiv that the full group belonging to Newtonian Mechanics is siiii)>ly the n'roup (ir, with the value of c=oo. In this state iii' aiTairs, :in«l sinre ( ! is in:tt hemat ically more iu- iellin'ible thii'i (r -x>, a mat hem ; . , b\ a i'n-e p!:iv

ol' imagination, hit upon the thought that natural pheno mena possess nn invarianre not only for the group ('<,, but in i'ai-l also for a uroup d , , where c is lintte, but yet

10

74 PRINCIPLE OF RELATIVITY

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 <•, 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 electro-magnetic units of electricity.

We can form an idea of the invariant character of the expression for natural laws for the group-transformation G, in the following manner.

Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (.»•, y, z, t) ; by means of this coordinate system, we can represent the phenomena according 1o definite laws. This system of reference is by no means uniquely deter mined by the phenomena. We can change the system of reference in any possible manner corresponding to the abore- meniioned group transformation Gc, bnt the expressions for natural laws will not be changed thereby.

For example, corresponding to the above described figure, we can call t' the time, but then necessarily the space connected with it must be expressed by the mani- foldness (,/ y :). The physical laws are now expressed by means of ,<•', ?/, :, t1 ', 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

APPENDIX 75

that time and .space shall reduce to mere shadows and we shall have a world complete in itself.

II

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 a,ny manner; then a world-line parallel to the £-axis will correspond to a stationary point ; a world-line inclined to the £-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 r,y,z,i; 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. We shall now introduce this fundamental axiom :

Tin', substance <?</*////// at am/ worfil point can always !,<• fiiiK-fircil to be, af resf, if ice «'.flatifix\\ vnr lime and space Kiiifafj///. The axiom denotes that in a world-point tho expression

,.*,/(.* _,/.,-! _,///' _,/;*

shall always be positive or what is equivalent to the

same thing, every velocity V should be smaller than c.

>hall therefore be the upper limit for all substantial

velocities and herein lies a deep significance for the

76 I'KINTIM.K 01- I!

quantity <•. At the first impression, the axio: 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 geometry.

The impulse and real cause of inducement for tlie assumption, of the yroup-kranisforniatioii Gr is the fact that the differential equation for the propagation of light in vacant spase possesses the group- transformation Gr. On the oth-3r hand, the idea of rigid bodies has any sense only in a system mechanics with the group Gx. Now if we have an optics with G,., and on the other h;'inl 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 Gx, and Gf, which Ins got the further consequence, that by means of suitable rigid instruments in the laborafory, we can perceive a change in natural uhcnomena, in case of different orienta tions, with regard to the direction of progres-ive motion of the earth. But all efforts directed towards this object, and even the celebrate! interference-experiment of Michelson have u'iven negative results. In order to supply an explanation for this result, II. A. L. rent/ formed a hypothesis -vhich practically amounts to an invanance of optics i'or ihe «;roup G,.. Aeeordi Lorent/ every substance shall Mifi'er a contraction

1 •( v '~ ~ ) '" ^'ll'-tn> '" the direction of its motion

\ITI,.\|l|\ 7?

l'in> hypothesis sounds rather phantastical. For the .rtion is nut to be thought of as a consequence of tin; resistance of ether, but purely as a gift from the skies, as a sort oF condition 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 (y, 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 s:nti:il extension (see fipf.'l). IF O.\' is parallel to the second strip, we can take /' as the /-axis and x' as the *-axis, then •ond body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supp is,- 1 to be at rest, has the length /, i.e., the cross seutidn PP of the first strip upon the , -axis = /' OC, \vlu-re OC is the unit measuring rod upon the s-axig and the second bodv also, when supposed to beat rest, has the same l.-u^th /, this means that, the cross section Q'Q' of the -e .-oud strip has :i <Tnss-soct,ion /'OC', when measured p;irall'-l 1 > tin- '-axis. In th ;se two bodies, we have no\v images of \\\- •> Lorentz-ele'ctr6bs, one of which ifl a; rest ;»nd the oilier moves iinifurinlx . Now if we stick t.» our original Coordinates, then the extension of the iveii ov (lie cross section (Id of the

strip lielon^iii'^ to it nie::sui-»-ii p:-. rallel to the -;i\is. •Nov. if i| ,-!,•;,!• -iuc,. (i'U'=r/()C'. that U(4 = /'O1)'.

If if—1'" ;in (>;ls.v falcnhtion u'ives that

01)' = OC x' ,•-' therefore

78 fRlNCiPLK OK RELATIVITY

This is the sense of Lorentz's hypothesis about tlie contraction of electrons in case of motion. On the other hand, if we conceive the second electron to be at rest, and therefore adopt the system (*', (' ',) then the cross-section P'P' 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^L_2H_°P _ QQ CPQ?-OC/-OC ~ PP

Lorentz called the combination I' of (t and ,») as the local Li 'ne (Ortszeil) 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 clearly that the time of an electron is as good as the time of any other electron, i.e. t, t' are to be regarded as equi valent, has been the service of A. Einstein [Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis... 4-1-1 1— 1907] There the concept of time was shown to be completely and un ambiguously established by natural phenomena. But the concept of space was not arrived at, either by Einstein or Lorentz, probably because in the case of the 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.

According to these ideas, the word " Relativity-Postu late" which has been coined for the demands of invariance in the group (r, seems to be rather inexpressive for a true understanding of the group Gf, and tor further progress.

\PPENDIX 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 like to give to this assertion the name " The Populate of the Absolute worM" [World- Postulate].

Ill

By the world-postulate a similar treatment of the four determining quantities x,i/t z, 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 striking and clear.

I shall again use the geometrical method of expression. Let us call any world-point O as a " Space-time-null- point." The cone

consists of two parts with O as apex, one part having /<()', the other having />0. The first, which we may call f\\e fore-cant' consists of all those points which send light towards O, the second, which we may 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. (Fide fig. :>).

On the aft-side of O -p have the already considered hyperboloidal shell F = rV - r2 -y* -z" = 1, t>0.

80 HlMXCIl'I.K Ol' RELATIVITY

Hie region inside the two cones will be ocoupibd by the

hvperboloid «f one sheet

' F= s+j/' + v-2 ^t^—k-,

where £9 can liave all possible positive values. The hyperbolas which lie upon this figure with O as centre, are important for us. For the sake of clearness the indivi dual branches of this hyperbola will be called the " Tnter- liyperbola. with centre 0." Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for / = °o and t = <x>, asymptotically approaches the velocity of light <•.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness -,//,:,/ a vector, then we have to distinguish between a time-vector directed from O towards the sheet + F 1, />0 and a space-vector directed from O towards the sheet F = 1. The time-axis can be parallel to any vector of the first kind. Any world-point between the fore and aft cones of O, mav by means of the system of reference be regard --d either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is nece-ssarilv always earlier, every point on the nft side of O, later thin O. Tin limit f oa corresponds to a com plete folding up of fcbfe wcdgc-shawd cross-section bet \\.v.i the. fore and aft cones in the taanifoldofess / = (). In the 'figure drawn, this cvos-s-scc! ion Ins been intentional!;- drawn with a different breadth.

Let us decompose a vector drawn iV.>in O t "\\-ards (.r,//,/,/) into its components. If the directions of the two ve< tors are respectively the directions of the radius vector OR to one of tin- surfaces +F=l,find of a tangt-nt IIS

APPENDIX 81

at tin- point R of the surface, then the vectors shall be called normal to each other. Accordingly

which is the condition that the vectors with the com ponents (-, y, :, /) and (fl // L z^ /j) 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, />() 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 (r, y, ;, 1} ; then as we follow the progress of the line, the quantity

Vc*dt* dx* dy* dz* , c

corresponds to the time-like vector-element (dr, dy, dz, dt}. The integral T= I dr, 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 P0 upon the irnrlil-H.i*-. We may regard (r, y, :, t], I.e., the components of the vector OP, as functions of the

" proper-time " T; let (.r, //, ?, f) denote the first different ial- ((iiotifMts, and (.r, //, r, /) the s(>cond differential quotients

of ( , '/, :, 0 with regard to T, then tlu'si- may respectively 11

82 I'UIXCIL'Lli OF RELATIVITY

he called the Velocity -vector, and the Acceleration-rector of the substantial point at P. Now we have

2 t 'f— X ,'(' —y'y—'z O ) ,

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 tig. 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

of magnitude - in the direction of MP. P

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 = <x .

IV

In order to demonstrate that the assumption of the group Crr for the physical laws does not possibly lead to anv 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 successfully made in the case of " Thermodynamics and

APPENDIX

Radiation,"* i'or "Electromagnetic phenomena ",t and linally i'or "Mechanics with the maintenance of the idea of

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 (x, y} z, f), where the velocity-vector is (f, y, z, t),

then how are we to regard this force when the system 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 Gr is undoubtedly permissible. These theorems lead us to the following simple rule ; {/ the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such n warmer, that the corresponding rector with the components

*X, (?Y, t'Z, /T,

where T=— ( -4- X -f ~J- Y + T- Z^ = l (the rate of r1 \ / t t c-

irork is done at the world-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

calk-il a inuriiiii /'oro'-recfur <it P.

X'»\v the world-line passing through P will be described 1)\- a substantial point with the constant mechaincut ///</vv ///. Let us call tn-ii>ii<'x the velocity-vector at P as the

I'liiiii k, YAM- Dynnmik bcwrgtor svRtemo, Ann. d. physik, Bd. 'JO, 1908, p. 1.

t II. MinkoNvski ; the ji:issn«,'o refers to paper (2) of the present edition.

84- PRINCIPLE OF RELATIVITY

iwpuhe-r.er.tor, and m-times 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 cnn 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 " Energy-law/' Accordingly cz -times the component of the impulse-vector in the direction of the t-avis is to be defined as the kinetic-energy of the point- mass. The expression for this is

i.e., if we deduct from this the additive constant me2, we obtain the expression 4 mv- of Newtonian-mechanics upto

magnitudes of the order of . Hence it appears that the

energy depends upon the xyxlcm oj' reference. But since the /-axis can be laid in the direction of any time-like MM'S, therefore the energy-law comprises, for any possible system of reference, thr> whole system of equations of motion. This fact retains its significance even in the limitinir rase C = oo, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schiiix.t '

* Minkowski— Mechanics, appendix, page G."> of paper (U).

Planck -\Vrh. d. I). P. (I. Vol. 4, 1906, p. 136 f St-hutz, fiott. Nuclir. 1897, p. 110.

Al'I'KMMX 85

From the very beginning, we can establish the ratio between the units of time and space in such a nr.mner, thai the velocity of light becomes unity. If we now write \/~l / = /, in the place of /, then the differentia] expression

,h •-• = - (dx - + <ly a + ilz * + ill - ),

becomes symmetrical in ( , //, .", /) ; this symmetry then enters into each law, which does not contradict the worfit- jjoxtnlafe. We can clothe the "essential nature of this postulate in the mystical, but mathematically significant formula

rH05frm=i'^l Sec.

The advantages arising from the formulation of the world-postulate are illustrated by 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 Ma \\vell-Fjorentz theory.

Let us conceive of the world-line of such an electron with the charge (f), and let us introduce upon it the " Pr.'p r-(ime " T reckoned from any possible initial point. In order to obtnin the lieM oau.-ed bv the electron at anv world-point I', let us construct the fore-cone belonging to P, (ridi' tig. 4). Clearly this etits tin- nnliinit-d 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 tin's tangent. Let r be the measure of P,Q. According to the definition of a fore-cone, rfc is to be reckoned as th«- measure of PU. Now at the world-point P,,

Ob PRINCIPLE OF RELATIVITY

the vector-potential of the field excited by e is represented by the vector in direction PQ., having the magnitude

in its three space components along the x-, y-, --axes ;

the scalar-potential is represented by the component along the £-axis. This is the elementary law found out by A. Lienard, and K. "Wiechert.*

If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the h'eld 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 analogy is, however, im perfect.

I shall now describe the ponderonwlivc force which is exerted by one moving electron upon another 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 QPj. The t?/-axis in the direction of MN, then the r-axis is automatically determined, as it is normal to the .'•-, t/-, ^-axes. Let

'1,'y, z, 7 be the acceleration-vector at P, .< ,, yl} z ,, /,

be the velocity- vector at P,. Then the force-vector exerted bv the first election e, (moving in any possible manner)

* Lionard, L'Eolnirnpi' clcctriqne T 10, 1896, p. '•:{. Wirchert, Ann. <1. Pliysik, Vol. 4.

Al'I'ENDIX 87

upon the ><•(•< .mil election >-, (likewise moving in any 1 o>>iblc manner) at I', is represented by

For the components FJt 7"7y, F., F, of the rector F the following three relations hold :

and fourthly this vector F is normal to the velocity-vector P,, a ltd through this circumstance alone, its dependence on f/iis laxf relitcity-vector arises.

If we compare with this expression the previous for mula3* giving the elementary law about the ponderomotive 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 tin- 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 m and w, describe their world-lines ; a moving force-vector is exercised by m upon /#,, and the expression is just tin- saun as in the case of the electron, only we have to write + >//;;/, instead of— ee^. We shall consider only the special C;IM- in which tlu- acceleration-vector of in is always zero :

* K. Sdiwar/.si-liild. (iott-N'aclir. 1903.

II. A. Lorcntz, Ens/Uopidie dor Math. \Vi.<.-i iiM-haftm V. Art 14,

PRINCIPLE OF KKI.AT1VITY

then / may be introduced in such a manner that m may be regarded as fixed, the motion of m is now subjected to the moving-force vector of m alone. If we now modify this

given vector by writing f - i

instead of /

to magnitudes of the order -j ), then it appears that

Kepler's laws hold good for the position (r,,^, zv), of MI at any time, only in place of the time ll} we have to write the proper time T, of ml. 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 consequent's will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of 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 ranch easier through the form given to the special Rela tivity theory by Miukowski, 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 (jauss, Riemaun and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci 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 lor 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. It

90 PRINCIPLE OF RELATIVITY

PRINCIPAL CONSIDERATIONS ABOUT THE POSTULATE o* RELATIVITY.

§ 1. Remarks on the Special Relativity Theory.

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

Tf a co-ordinate system K he 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 translational motion relative to K. We call this postulate " The Special Relativity Principle." By the word special, it is signified that the principle is limited to the case, when K' has uniform trandalory motion with reference to K, but the equivalence of K and K' does not extend to the case of non-uniform motion of K' relative to K.

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this j>ostu- 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 Lorenz-transfor- mation, with all the relations between moving rigid bodie« 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 jwssible relative positions of solid bodies at rest, and more generally the theorems of kinematics, as theorems which describe the relation between measurable bodies and

GENRRAI.1.41U TIIMMn »h RKT.ATIVITY 91

clocks. Consider two material points of a solid bodv at rest ; then according to these conceptions their corres- jx>nd8 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 syetem ; then to these positions, there always corresponds, a time-interval of a definite length, independent of time and place. It would he 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 episteomological defect, which was perhai* lirst cleanly pointed out by E. Mach. We shall illustrate it by the following example ; Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another (and from all other masses) that only that sort of gravitational forces art- to be taken into account which the |>art« of any of these bodies exert UJKJII each other. The distance of the bodies from one .another is invariable. 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 S8) 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 •uiface of the other i* an ellipsoid of rotation. We now

92 PRINCIPLE OF RELATIVITY

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 thing 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 Rj relative to which the body S, is at rest, not however for a space relative to which S8 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 imaginary cause B, for the different behaviour? of the bodies S, and Sa which are actually observable.

A satisfactory explanation to the question put forward above can only be thus given : that the physical system composed of St and Ss shows for itself alone no con ceivable cause to which the different behaviour of S, and S, 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 S, 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

KALISED THEORY OF RELATIVITY 9S

of the bodies under consideration. They take the place of the imaginary cause R,. Among all the conceivable spaces Rt and Ra moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater 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 muss at a sufficient distance from other masses move uniformlv in a line. Let K' be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K' any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and the 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' can 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-field which generates the accelerated motion relative to K.'.

This conception is feasible, because to us the experience of the existence of a field of force (namely the gravitation field) has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. The

i>4 PRINCIPLE OP RELATIVITY

mechanical behaviour of the bodies relative to K' is the same 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 physical significance ; when we say that any arbitrary point has .ri as its X, co-ordinate, it signifies that the projection of the point-event on the X,-axis ascertained 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 .c^ times from the origin of co-ordinates along the Xi axis. A. point having st—t, as the X4 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 OF RELATIVITY 95

point-event and set according to some definite standard has gone over .<-4=J periods before the occurence of the point-event.

This conception of time and SIWLCC 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 aud in general 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, s, t,) and also, another system K' (.«' y' :' t') rotating uniformly rela tive to K. The origin of both the systems as well as their ~-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 aud 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 plaur (X', Y') of K'. Let us now think of measuring the circum ference aud the diameter of these circles, with a unit measuring rod (infinitely small compared with the radius) 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 ijet IT, as the quotient. The result of measurement with a rod relatively at rest as regards K' would be a number which is greater than *. 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 Lorenz-contraction, not however when the rod is placed along the radius. Euclidean Geometry therefore does not hold for the system K' ; the above fixed concep tions of co-ordinates which assume the validity of Euclidean Geometry 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 svstem 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 general 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 OF RELATIVITY 97

definite way, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate system 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 repaid 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 .\<if//re are expressed ly mean* of equations which are valid for till co-ordinal e systems, that is, which are covariant for all po*xi/j/<- fraasfortudtioiU. It ig 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 covariance 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 icell-sntjstantiatfd 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 two 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 IS

98 PRINCIPLE OF RKLATIVm

(.«! ./•, ,us .c4) such that to any and every point-event corresponds a system of values of (<., .. , , 4). Two co incident point-events correspond to the same value of the variables (./ , .< s ,r,s .»:4) ; i.e., the coincidence is cha racterised by the equality of the co-ordinates. If we now introduce any four functions (. ' l •' % .'3 .<-'4) as co ordinates, so that there is an unique correspondence between them, the equality of all the four eo-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 general covariance.

$ 4. Relation of four co-ordinates to spatial and time-like measurements.

Analytical expression for the Gravitation-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 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 appear; this is possible for au infinitely small region. X,, X,, X, are the apmtiaJ

GENERALISED THEORY OP RELATIVITY 99

co-ordinates ; Xt is the corresponding time-co-ordinate measured by some suitable measuring clock. These co ordinates have, with a given orientation of the system, an immediate physical significance in the sense of the special relativity theory (when we take a rigid rod as our unit of measure). The expression

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 '/.« as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. If ds* belonging to the element (r/X, dXt, ^X3, ^X4) be positive we call it with Minkowski, time-like, and in the contrary case space-like.

To the line-element considered, i.e., to both the infi nitely near point-events belong also definite differentials (I < , , d< , . dxs, d'\, 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, '/X's would then be represented by definite linear homogeneous expressions of the form

*2) dX =2 a d.r

V (T V<T (T

If we substitute the expression in (1) we get

where g will be functions of .<• ^ but will no longer depend

upon the orientation and motion of the 'local' co-ordinates; for d*' is a definite magnitude belonging to two point- events infinitely near in space and time and can be got by

100 PRINCIPLE OF RELATIVITY

measurements with rods and clocks. The g 's are here to

be so chosen, that n = n ; the summation is to be "or rw

extended over all value? of o- and r, so that the sum is to be extended over 4x4- terms, of which 12 are equal in pairs.

From the method adopted here, the case of the usual relativity theory comes out when owing to the special behaviour of g in a finite region it is possible to choose 'the

system of co-ordinates in such a way that g ^ assume* constant values

f -1, 0, 0, 0 0-100

(*)

00-10

ooo+i

We would afterwards sec 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 § X it is clear, that from the physical stand-point the quantities g are to

.be looked upon as magnitudes which 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. The g '& then

have the values given in (I). A free material point moves with reference to such a system uniformly in a straight- line. If we now intro:lu3J, by u'.iy substitution, the space- time co-ordinates .r, . ...r4, 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 ne.w

GENERALISED THEORY OF RELATIVITY

101

co-ordinates, will appear as curvilinear, and not uniform, in which the law of motion, will be iwiependetd of the ini.fii.rr nl' I In- innrJ,i<i nt<txx-])uinl*. We can thus signify this motion as on*- under the influence of a gravitation field. We see that the appearance of a gravitation-field is con nected with space-time variability of g 's. In the general

case, we can not by any suitable choice of axes, make special relativity theory valid throughout any finite region. We thus deduce the conception that y '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 electromagnetic forces, in as much as the 10 functions g representing

gravitation, define immediately the metrical properties of the four-dimensional region.

MATHEMATICAL AUXILIARIES FOR ESTABLISHING THE GENERAL COVARIANT EQUATIONS.

We have seen before that the general relativity-postu late leads to the condition that the system of equations for Physics, must be co-variants for arnr possible substitu tion of co-ordinates .- , , . , : \ve have now to see how such general co- variant equations can be obtained. \Vr shall now turn our attention to these purely mathemati cal pro positions. Ft will be shown that in the solution, the invariant ifs, given in <'tjiiat.ion (:}) plays a fundamental role, which \vr. following (iau-Vs Theory of Surfaces, style as the line-element.

The fundamental idea of the general co-variant theorv is this : With reference to any co-ordinate system, let certain tiling (tensors) be defined by a number of func tions of co-ordinates which are called the components of

10? PRINCIPLE OF KKI.ATIVITY

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 u 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 system vanish if they are all zero in the original system. Thus a law of Nature can be formulated by puttjng 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-variant Four-vector.

Contra-variant Four-vector. The line-element is denned by the four components d.-'v whose transformation law

is expressed by the equation

V.

<*> "'-=*„ T^ ",. .

The d<' ',< are expressed as linear and homogeneous func tion of <l> '* ; we can look upon the differentials of the

co-ordinates as the components of a tensor, which we designate specially as a contravariant 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,

i.KNKKAUSBO THEORY OK RELATIVITY 103

we may call a contra-variant Four-vector. From (5. a), it follows at once that the sums (A 4 B ) are also com

ponents of a four-vector, when \a and B'T are so ; cor responding relations hold also for all systems afterwards introduced as " tensors " (Rule of addition and subtraction of Tensors).

Co-r<inanf-

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 •= Invm-ianL

From this definition follows the law of transformation of the co-variant four-vectors. If we substitute in the right hand side of the equation

the expressions

e-v

for Bv following from the inversion of the equation (5») we get

a

B^ 5 -^ A =5 B" A

<T _ ~ V , nO* A »

<r

<7

As in the above equation BCT are independent of one another and perfectly arbitrary, it follows that the transformation law is :

104 PEINCIPLE OF RELATIVITY

Remarks on the simplification <>f tl/c mode of wriliny 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 only 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 Ricci and Levi-civita, the contravariants and co-variants are designated by the over and under indices.

§ 6. Tensors of the second and highei ranks. Contravariant tensor : If we now calculate all the Ifi

products A^v of the components A^ Bv , of two con- trava riant four- vectors

f8) A^ = A^B*

A'*", will according to (8) and (ft a) satisfy the following transformation law.

6 *' 8 .- ' (9) A" = 3-? g-I 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 105

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^v, can be represented as the sum of A

Bv 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 any rank : If is clear that corresponding to (8) and (9), we can define contravariant tensors of the 3rd and higher ranks, with 43, etc. com- l>onents. Thus it is clear from (8) and (9) that in this sense, we can look upon contravariant four-vectors, as contravariant tensors of the first rank.

Co-variant tensor,

If ou the other hand, we take the 16 products A^ of

the components of two co. variant four- vectors A and

for them holds the transformation law

Q'u 8'rv <"> ^r =-97/67;, V

By means of these transformation laws, the co-variant tensor of the second rank is defined. All re-marks which we have already made concerning tbe contravariaut 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

]66 HllXCIPLK OF KELATIVITV

Mixed tensor. We can also define a tensor of the second rank of the type

(12) A = A BV

P. P-

which is co-variant with reference to p. and contravariant with reference to v. Its transformation law is

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.

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

are equal. The tensor A ^ or A ^ is symmetrical, when we have for any combination of indices

(14) A^W7*

(14a) A =A

p.V Vfi

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-)

i 9 .<• - 9 .T Q ,i , Q ,»• ,

.or a- T >p.v a T /.vu. .TO-

- e,,, e.v ' a,,, 8,;

CKN'Ktf ALIsKlt TIIKOllI oh II KI,.\TI VITV 107

Anti-til iniii''lri<-nl

A contravariant or co-variant tensor of the 2nd, 3rd or 4th rank is called <iiifi*i/nnit<>trical when the two com ponents got by mutually interchanging any two indices

are equal and opposite. The tensor A.1*" or A ^ is thus antisymmetrical when we have

(15) A^ = -A^

(15a)

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-rector).

Thus we also see that the antisymmetrical tensor A!*"" (3rd rank) lias only 4- components numerically

different, and the antisymmetrioal 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.

On/ft- i)ii(/(iji/n-<iti<>/i of T''iix,n-x : \Vc ^ct from the components of a tensor of rank :, and another of a rank c', the components of a tensor of rank (r-fc') for which we multiply all the components of the first with all the components of the second in pairs. Fur example, we

108 PRINCIPLE OF RELATIVIT*

obtain the tensor T from the tensors A and B of different kinds :

T = A B

UV<T flV (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), (18) ; equations (S), (10) and (12) are themselves examples of the outer multiplication of tensors of the first rank. Reduction in rank of a mixed Tensor.

From every mixed tensor \ve can get a tensor which is two ranks lower, when we put an index of co-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

sv

A , the mixed tensor of the second rank

8 a8 /

A =A =(SA

a \

ft \a aft

and from it again by " reduction " the tensor of the zero rank

.0 .«/* A= A = A

ft aft

The proof that the result of reduction retains a truly tensorial character, follows either from the representation

t, I NKI5AL1SK1) THKOKY OF HKLATIVITY

109

of tensor according to the generalisation of (1~) in combi nation with (H) or out of the generalisation of (13).

Imier a<ul mixed multiplication of Tensors.

This consists in the combination of outer multiplication with reduction. Examples :— From the co-variant tensor of the seconj.1 rank A and the contravariant tensor of

the first rank B we get by outer multiplication the mixed tensor

B"

Through reduction according to indices v and o- (i.e., put ting i' = <r), the co-variant four vector

D = D = A B is generated. /* .v /"

These we denote as the inner product of the tensor A v

and B . Similarly we get from the tenters A and B

ILV

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

mixed tensor of the second rank D = A B . \Ve

can fitly Pall this operation a mixed one ; for it is outer with reference to the indices M and rt and inner with respect to the indices v and o-.

110 I'KI.Ni I I'l.K <>!' I! KI, \TIVm

We now prove a law, which will be often applicable for provingthe tensor-character of certain quantities. According

to the above representation. A B^vis a scalar, when A

P-V /JLV

and B are tensors. We also remark that when A B^' is

an invariant for everv choice of the tensor B^v, then A

(Of

has a tensorial character.

Proof : According to the above assumption, for any substitution we have

A , B" = A

OT fJiV

From the inversion of (9) we have however

o & o */* /

Substitution of this for B^" in the above equation ^i

A 8% 9^ A

A , r - A

°"r 8 .»' f 8-'y /

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

<ih\l,KAI.ISKI. THEORY OF RELATIVITY III

if for every choice of them the inner product A B C

is a scalar, then A is a co-variant tensor. The last py

law holds even when there is the more special formulation, that with any arbitrary choice of the four-vector B" alone the scala: product A B'* Bv is a scalar, in which case we have the additional condition that A satisfies the

symmetry condition. According to the method given above, we prove the tensor character of (A v + ^v }, 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 qualities' A B form a tensor of the

first rank, when B is any arbitrarily chosen four-vector, then A v is a tensor of the second rank. If for example,

C'1 is any four-vector, then owing to the tensor character of A n B1 , the inner product A v C^ B1' 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 //

U.V

The co-variant fundamental tensor In the invariant expression of the square of the linear element

<lx- =7 df 1 1. 1 M* /*

112 PRINCIPLE OF RELATIVITY

A A plays the role of any arbitarily chosen contravariant vector, since further g y=ffv , it follows from the consi 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 teusor.

The co-variant fundamental tensor.

If we form from the determinant scheme | g v | the

minors of g yand divide them by the determinat ^ = j y^ \

we get certain quantities f* gv^ , which as we shall prove generates a contra variant tensor.

According to the well-known law of Determinants

(16) g /a=8

/XO"

where S is 1, or 0, according as /* = v or not. Instead of the above expression for rf.»2, we can also write

V 6V rf V ^'" or according to (16) also in the form

0T . *

a a a (M •/

ytt«7 JVT * U. V

GENERALISED THEORY OF RELATIVITY

118

Now according to the rules of multiplication, of the fore-going paragraph, the magnitudes

p.<T fJL

foims a co-variant four- vector, and in fact (on account of the arbitrary choice of d< ) any arbitrary four-vector.

If we introduce it in our expression, we get

For any choice of the vectors d) this is scalar, and

y*"", according to its defintibn is a symmetrical thing in a and r, so it follows from the above results, that g is »

contravariant tensor. Out of (16) it also follows that 8

^

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

On the other hand we have

So that it follows (17) that \9 v\ I/"

15

114

PRINCIPLE OF RELATIVITY

of ruin me.

We see K first the transformation law for the determinant 0= | 0 v 1 . According to (11)

dx 6-c

U V

9,

9 =

6,v a,v -/-

From this by applying the law of mutiplication twice, we obtain.

9*,

F

H-

a^

rf.r /

V7f =

_j*

v-,

°"

... (A)

On the other hand the law of transformation of the volume element

is according to the wellknown law of Jacobi. dtl

dr' =

... (B)

by multiplication of the two last equation (A) and (B) we get.

(18)

=<Sf dr'^V dr.

lusted of v/^-, we shall afterwards introduce \/~g which has area! value on account of the hyperbolic character of the time-space continuum. The invariant ^/—gdr, is equal in magnitude to the four-dimensional volume-element

GENERALISED THEORY OF RELATIVITY llo

measured with solid rods and clocks, in accordance with the special relativity theory.

Remark* on the character of Hie space-time continuum Our assumption that in an infinitely small region the special relativity theory holds, leads us to conclude that ds- can always, according to (1) be expressed in real magni tudes fJX,...<JX . If we call dra the " natural " volume element rfXt </X2 dX3 dX4 we have thus (18a) tlr*

Should Y/ g vanish at any point of the four-dimensional continuum it would signify that to a finite co-ordinate volume at the place corresponds an infinitely small " natural volume." This can never be the case ; so that g can never change its sign; we would, according to the special relativity theory assume that g has a finite negative value. It is a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice of co-ordinates.

If however (— g) remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one. We would afterwards see that such a limitation of the choice of co-ordinates* would produce a significant simplification in expressions for laws of nature.

In place of (18) it follows then simply that

dr' = d

from this it follows, remembering the law of Jacobi, -— =1.

116 PRINCIPLE OF RELATIVITY

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 lenunciation of the general relativity postulate. We do not seek thoj=e laws of nature which are co- variants with regard to the Iran formations 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 new tensor* with 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 :

We would point out specially the following combinations: A <f g A n

ap

A =g g Aaft

(complement to the, co-variant or contravariant tensors) and, B o Q A a

We can call B the reduced tensor related to A

GENERALISED THEOEY OP HELATIVITY 117

Similarly

,-V/AV

B

a/3

It is to be remarked that f* is no other than the " com plement " of y v , for we have,

ua vB S p-v

Q Q 9 o~ 9 *> 9 y y v J a

§ 9. Equation of the geodetic line (or of point-motion).

At the " line element " ds is a definite magnitude in dependent of the co-ordinate system, we have also between two points Pt and P2 of a four dimensional continuum a line for which />/•? is an extremum (geodetic line), i.e., one which has got a significance independent of the choice of co-ordinates.

Its equation is

(20)

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 X, be a function of the co-ordinates xv ; This defines a series of surfaces which cut the geodetic line sou^ht-for as well as all neighbourin<r lines from P, to P3. We can suppose that all such curves are given when the value of its co-ordinates ^ are given m terms of X. The

118 PRINCIPLE of RELBTIVITY

sign S corresponds to a passage from a point of the geodetic curve sotight-for to a point of the contiguous curve, both lying on the same surface A.. Then (20) can be replaced by

u

(20»)

dx dxy

^ l ~9fJiV ~(j\ ~rfX

But

«LlL

2 a,

So we get by the substitution of 8w in (20a), remem bering that

( ii>'v ^ a

H - /=— (8-iv^

after partial integration, X.,

(20b)

J

d\ k So; =0 o- o-

a.'/A

1. 1 AKIIALISED TIIEOKY OF RELATIVITY 119

From which it follows, since the choice of 8.<- is per fectly arbitrary that k^ '* should vanish ; Then

(20c) ka =0 (o-=l, 2, 3, 4)

are the equations of geodetic line ; since along the geodetic line considered we have ^,<?=/=0, we can choose the parameter A, as the length of the arc measured along the geodetic line. Then w = l, and we would get in place of (20c)

V 8** 8.<- 8* 8*

<r

_i av 8 V ar«- _0

Or by merely changing the notation suitably,

<ZV _ _ (L- (ft

(20d) ga(f t-a -4- K" -^ . ~ =0

where we have put, following Christoffel,

(9i\ F/xvl * I ' M0"^ ' vcr ' P-v~\

' S a* a~~ ~ ~ai '

LO- J L O v fi

Multiply finally (20d) with 0*™ (outer multiplication with reference to T, and inner with respect to rr) we get at last the final form of the equation of the geodetic line

-

ds* Here we have put, following Christoffel,

120 PRINCIPLE OF RELATIVITY

§ 10. Formation of Tensors through Differentiation.

Relying on the equation of the geodetic 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 pertain 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 <£, be an invariant space function,

then is also an invariant. The proof follows from the fact that d<j> as well as ds, are both invariants

Since £ = 9* |> ds Qx 9 *

a , d. so that = 5^— . is also an invariant for all curves

which go out from a point in the continuum, i.e., for any choice o]

diately that.

any choice of the vector d c . From which follows imme /*

A =1*

. Vj

is a co-variant four-vector (gradient of <£).

According to our law, the differential-quotient x= taken along any curve is likewise an invariant Substituting the value of iff, we get

d- dr. d** O <P /* v . o <P A4 v= r 7^ . ^ . +

A Q., 9, d* 9- «•

M * f1

<;F.\K1UUSK1> THKOKY OJ- HKL.Vl'mTY

Here however we cannot at once deduce the existence of any tensor It' we however tiikc that the curves along which we are differentiating are geodesies, we pet from it

dfx

by replacing -- - according to (4J'J

= r aa^_ - ^v) -9*

I 3* 9-'v <- r > 9-«T

From the interrliangeability of the differentiation with regard to ^ and v, and also according to (2o) and (21 ) svc iin-

\ M1' J that tlie bracket - is symmetrical with respect to u.

(T ) and t-.

As we can draw ;t uvoilrt ic line in any direction from any point in the continuum. 'x is thus a four-vector, with an

arbitrary ratio of components. that it follows from Mir results of §7 that

(25)

is a co- variant tensor of the second rank. We have thus got the result that out of the co-variant tensor ot the tirst rank

A = "-^ we can get I>V differentiation a co-\ariant tensor of 2nd rank

V= 6/ 16

\:l'l PRINCIPLE OF RELATIVriY

W»> call rhe tensor A the " extension " «>f the tensor ta>

A . Then we can easily show that this combination also leads to a. tensor, when the vector A is not repvesentable us a gradient. In order to see this we first remark that \^ ~~ is a co-variant four-vector when \L and are

scalars. This i> uiso the case for a sum of four such

terms :

when i/rd), <j>(l)...\f/(*) <£(*) are scalar*. Now it is however clear that every co-variant four-vector is j-epresentable in

the form of S P-

If for example, A is a fonr- vector whose components are any given functions of x , we have, (with reference to the chosen co-ordinate system) only to pat

= A,

in order to arrive at the result that S is equal to A .

V- P

In order to prove then that A in a tensor when on the right side of (26Nt we substitute any co-variant four-vector for A we have only to show that this is true for the

Til ROll Y OF BRLATIVITY 1:

four- vector S . For this latter, case, howevtjr, a glance on

the right hand side of (26) will nhow that we have only to bring forth the proof for the case when

Now the right hand side of (25) multiplied by i/' is

which has a tensor character. Similarly, ~* f * is

also a tensor (outer product of two four- vectors). Through addition follows the tensor character of

Thus we get the desired proof for the four-vector,

•S l

\1/ ££- and hence f«->r 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 i-ank. 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.

Aft already remarked every co-variant tensor of th^ 2nd rank can be represented as a sum of the tensors of the type A B .

fi V

124 PRINCIPLE OF EELATIVITY

It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions

8 A ( )

9 ^ - ( r ) ^

are tensors. Through outer multiplication of the first with B and the 2nd -with A we get tensors of the

third rank. Their addition gives the tensor of the third rank

A - /xv- A - J av A

V" srr

where A y is put=A B^, . The right hand side of (27) is linear and homogeneous with reference to A . and its

first differential co-efficient so that this law of formation 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, i.e.., in the case of any co-variant tensor of the

second rank. We call A the extension of the tensor A . p.va /iv

It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the tirst and zero rank). In general we can get all specinl Ijtws of formation of tensors from (27) combined with tensor multiplication.

(JKXKH.M.ISKD TUKOliY Ol KKLAT1V1KY 125

Some special cases of Particular Importance.

A f'-ic inuiJiai'n It'miiia* con<:rr,iui<i lite fundamental t«n*<',-. W<> shall first deduce some of tin- h-mnms much xised ut'fci -\\ards. According to the law of differentiation uf determinants, we have

28) ,,9 = /V^v=-f//xv ^/V-

Tlu- la.-t foi-ni follows From tho first when we remember

that

,/ ,fv = tiL , and there fun- ,-/ gfa' =

ULV U.V

consequently;/ fly**1 + >j^v <l<i

From (28), it folhnvn that

~t i'^ (-r/) _,,/•• V 6. -^ 6,,

/"• 6-

Again, since ^ / <j —^ , we havi-, Ity differentiation,

vcr

(/

a,-.

(30) - 01

By mixed multiplication with (j nnd ;/ we obtain (Hwn^inir the mode of wntinir the

126 I'KINTIPLE OF RELATIVITY

Id ftv=- Pa <VP d< (31)

Q7~ ~;/ ^ Va/S

I '

V

p.- p.w K^,

(32)

,

Tlio expression (31) allows a transformation which we shall often use ; according to (21)

If we substitute this in the second of the foi-mula (31). we get, remembering (23).

By substituting the right-hand side of (H4) in (29), we get

.lsKM TUKOKV OF KEI.AT1VIIV I"'?

nf ///>' eo^tfavoriani four^vcctor, Let u> multiply i ii'» ) with the conttavariani fandanumta] ten-Mir if (inner mult ijilicittion'). then l»v ;t

of the firnt incjn^ci1. th«' ritrlit-lianrl side tnkes the form

;,a_V r.

8' 8-' '

According to (.'U ) ;nul (-'.*). th»« lust member cun the form

Both the first members of the exj)rcssion (B), and the second member of the expression (A) cancel each other. since the naming of the summation-indices immaterial. The laHt meiuhei- of (B; cun then he united with first of (A). If we put

where A* as well as A are vectors which «-an be »trbi- M

trarily chosen, we obtain finally

9

-" A"

a

This scalar is the Di'vcrym-' ut the contra variant four- vector AV ,

128 I'lUM ll'i.l. OK UKLATTVITY

ttnfdiin,/ a/' t/tc (covarianfy fjouy»vectof,

Tin- seron.l iiHMiiiifi in :!>»'•) iv -\ mmetrical in the indict

u, and v. Hence A A is an antisymmetrical tensor / f p.* vp.

built up iu a vrry simple niainier. \Ve ol)tain

a A 9 A

(36) B = - /x -

t>f ii Six-vector.

Jf we apply the operation ('27) on an nntisynimetrical

tensor of tht- sri-ond I'jink A . ;uid form all the equations pr

arising from the cyclic* interchange of the indices /x, v. <r, and add all them, we obtain a tensor of thr third rank

a A

(.37) B =A 4- A + A = -— ^

/Aj'ir fiver V<TJJ. ir/j.v Q<T

from which it is easy to see that the tensor is antisymmetri- eal.

<-/> nf tfir S

If (27) is multiplied by <^°' f]V ( mixed multiplicjition), then a tensor is obtained. Tho tirst member of the right hand side of (27) can be written in the form

6 A

a,(r

t.t \i I;M.ISI;D TIIKOUY <>i if KMTI\ 1 1 \ J:><)

If we rc,d:iee /" '/'^.\ I'v A"^. /" v'^ A hy

/"•<r ,r /'M J

A and rojthicc in llic ti-insforiiind first inoinliop

^'/i,,,,, 9-""" 9", 9-V

with tlio help of (^M-), then from tlie right-hand side of (27) t here arises ;xn expression with seven terms, of which four t-ancel. '1'here remnifis

This is the expression for the ext elision of n contra variant of the second rank : extensions can also he formed I'm em-responding cont ravariant tensors of higher and lower ranks.

We remark that in the same way. we can also form the extension of a mixed tensor A

9A" i, ,) (, r)

(:>':h A ' ~ '' ~ 'V + / A '

15\ the reduction of (US) with reference to the indices

^tr v /i -tnd tr ( inner multiplication with £ I , we tret a con-

V ft /

t ravariant four- vector

9 17

I'uixrii'i.i: OK in;i, \TI\ITY

v of - with

On tlu1 ;iccouiit of flic svmiiirtrv

( " ) reference to the indices (3. and K. the third member of tht

right hand side vanishes when Au" is an antisymrnetrical tensor, which we assume here ; the second member can be transformed according to (29a) ; we therefore get

A a _ i o I v a j

(4°) : 7^~~8^7

p

This is the expression of the divergence of a oontra- variant six-vector.

of t//f t//ic<:il ti'/i.wr of i/ie second rank. Let us form the reduction of (89) with reference to the indices a and <r, we obtain remembering (29;i)

If we introduce into the last term the contravnriant tensor Ap<r = r/^T A , it takes the form

T

-[">-» A-.

L ^ J

If further A^*7 is symmetrical it is reduced t<>

(,l \ I i; M.IM.D Til K( ' in u| UK I \TI \ IT1 1 •'> i

If instead <>i A: . «»• introduce in ;i >imilar way the symmetrical co-variant tensor A ^ —g a </ « A°" . then owing to (31) the last member can take the form

-,j A

J 8 r f*r

In the symmetrical case treated, (41) can be replaced by -it her of the forms

I In.

( /-j A' )

'

which we snail have to make use of afterwards.

$12. The Riemann-Christoffel Tensor. \Ve now seek only those tensors, which can be obtained from the fundamental tensor ^ \by differentiation alone. It is found ea-il\ . Wo ]>ut in (27) instead of i- A'"' tin- fundamental t«-nsor //'" and i^et from

IW PIMNCIPU. 01 l!i;i.\Tl\ m

it a new tensor, namely the extension of the fundamental tensor. We can easily convince ourselves that this vanishes identically. We prove it in the following way; we substitute in (27)

8 A (^

[IV Q j: / \ P

f r (^p J

i.e., the extension of a four- vector.

Thus we get (by slightly changing the indices) the tensor of the third rank

^ _ /' _ ) r P _ y 'T r p _ \ (TT f _p

/xtrr a .'-a- /• ija,'- " ;0 (a- )n ( a..-

6,

Wre use these expressions for the formation of the ten>ur A A . Thereby the following terms in A

/AOT I>.T(T fJ(TT

cancel the corresponding terms in A ^; the iirst member, the fourth member, as well as the member corresponding to tlie last term within the square bracket. These are all symmetrical in <r, and T. The same is true for the sum of the second and third members. We thus get

A - A = ttp A .

/iOT /ATO" fJUTT p

/HOT

I.I.M.I; \i.i>i.n TIII:OI;N or UKi,ATi\rn

The essential thing in this result is that on the right hand nde of ( 1:2) we have only A , but not its

differential C0r»efficientf. From the tensor-character of A

fi(TT

A , and from the fact that A is an arbitrary four

flTO- ' p

vector, it follows, on account of the result of §7, that B is a tensor (Kiemann-Christoffel Tensor).

fUTT

The mathematical significance of this tensor is as follows; when the continuum is so shaped, that there is a co-ordinate system for which y 's are constants, Bp all

[1.V /XCTT

vanish.

If \ve choose instead of the original co-ordinate system any new one, so would the </ 's referred to this last system

be no longer constants. The tensor character of B''

/XCTT

shows us, however, that these components vanish collectively also in any other chosen system of reference. The vanishing of the Kiemann Tensor is thus a necessary con dition that for -some choice of the axis-system y V can be taken a> constant^ In our problem it corresponds to the case when by a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region. By the reduction of (1-3) with reference to indicc> to r and p, we get the covarinnt tensor of the second rank

1-'J1- ]'iu.\( ii'i.i; 01 KI.I. nivrn

ti/io/t I In- c/io/ce o/' co-ordinoti-cs. It has alrcadv been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so choseu that x/— 0 = !• 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

/it' fJLV

R .

/J.V

I shall give in the following pages all relations in the simplified 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 case.

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 freelv moving body not acted on by external torcr- 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 K,, can be, and are, so chosen that // 's have special constant values of the expression (4).

Let us discuss this motion from the stand-point of any arbitrary co-ordinal O-M -stem K;: it moves with reference to K, (as explained in §2) in a gravitational field. The la\\>

(ii:\KI! M.ISKD THEORY (>K It Kl. \TI \ m

1:55

of motion' With reference to K, follow easily from the following consideration. \Yith reference to K,,, the law

of motion is a four-dimenfcional straight lino and tlm* a geodesic1. As a geodetic-line is defined independently

of the system of oo-ordinates, it would also be the law of motion for the motion of the material-point with reference to K! ; If we put

v= - T ^

we get tho motion of the point with reference to K jjiven by

J 'if. ./ dx

__T =rT /(

,/X» " /'l' »/s- </>

\\'e now make the very simplo assumption that this general oovariant system oi' equations defines also the motion of the point in the gravitational Held, 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 (1(5) contains only the first differentials of // , among which there is no relation in the special case when K,, exists.

If r T 's vanish, the point moves uniformlv and in a

/*"

straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field.

l-'iC) I'HI.M IIM.I. ol Kill. \Tlvm

§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 :is matter : therefore the term includes not only matter in the usual sense, but also the electro-dynamic field. Our next problem is to seek the Reid-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 which the field-equations sought-for are evidently satisfied is that of the special relativity theory in which </ 's have certain

constant values. This would be the case in a certain finite region with reference to a definite co-ordinate >vstem K,,. With reference to this system, all the com ponents B/J of the Riemann's Tensor ("equation \:>

vanish. These vanish then also in the region considered. with reference to every other co-ordinate system.

The equations of the gravitation-field free from matter

must thus be in everv case satisfied when all B^ vanish.

P.<TT

But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed ,111;, t/ by any olioiiv of a\«-s, i.e., it cannot be transformed to a case of constant ft 's.

Therefore it i> clear that, for a gravitational field free from matter, it is desirable that the symmetrical ten-

BOW H deduced from the tensors 1^ should vanish.

.v r-

B ILI8! I) TIIF.oin Hi' i!i:r. \TI\m 137

\\ethu-iget 1 (I equations for 1 0 quantities // which are fulfilled in the special case when IV V all vanish.

Remembering ( I I) we see that in absence of matter the field-equations come out as follows ; (when referred to ihe special co-ordinate-system chosen.)

er;"

It fan also he shown that the ohoiee of these equa tions is connected with a minimum of arbitrariness. For

liesides l> , there is no tensor of the second rank, which

/"''

can lie built out of // 's and their derivatives no higher /"'

than the second, and which is also linear in them.

It will be shown that the equations arising in a, purely mathematical wa\ out of the conditions of the general relativity, together with equations (Hi), givo us the Xe\\- loman law of attraction as a tirst approximation, and lead in the second approximation t<> the explanation of the perihelion-motion of mercury diseovertd by Leverrier (the residual ellect 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 cornctne-s i.f my theory.

l-'is I'Hl N'CIIT.I. Or I!K1. \TIVITV

vH5. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy.

In order to show that the field equations correspond to the laws of impulse ;uid energy, it is most convenient to write it in the following Hamiltonian form :

f f

| 8 I Hdr=o

Here the variations vanish at the limits of the finite four-dimensional integration-space considered.

It is first necessary to show that the form (47 n) is equivalent to equations (47). For this purpose, I.-t us

consider H as a function of f and <fv ( •:•:. ® •> We have at I list

, 9^, A , 9",,A _9'',,

\ 3, 6- 3S

<.i MI; \Usi.u TIIKOIIS DI IM-.I. \II\IM 1 •)•'

The terms arising out of the two last terms within tin- bracket are of different si^ris, ami change into one another bv the interchange of the indices /* and ft. They cancel each other in the expression forSH, when they are multiplied by f „», which is symmetrical with respect to fj and ft, so that only the first member of the bracket remains for our consideration. Remembering ('31), we thus have :

ftft ra i> ft a

Therefore

f 9H rft

uv~ f*P i'«

! 9 f

(48) 8H r,r

[ 9.'/r

If we now cam out thr \ariation> in ( IJa), we obtain

the system of equations

/

a / an v an

(•^h) o , ( , ) - , =<>.

O ,<• v ~ ii r I -> // r

a o ,'/ O f/

which, nwinir to the relation.- (|s\ eoineide with (1-7), as was rr(|iiired to be proved.

If (47b) is multiplied by /''.

14-0 i'JM.NTII'l.l. (M KM, \TIVm

aud consequently

r a / an { a / /•• 9"

'<r a.-' V «.,/«•/ a.-;. V •" a

we obtain the equation

_a_ a.*

8H

37.

L **•

Owinsr to the relations (48), the equations (4?) and (:>4).

fjifh „/"• ' ,,/"V a r-/^

It is to be noticed that f'L is not a tensor, so that (lie <r

''<iuatioii (4'9) holds only for systems t'or which \ —,/ = ]. This equation expresses tin- laws of conservation of i in pulse and energy in » -ravitation-field. In fact, the inte-ra- tiou of this equation over ;i tlncc-dimcnsional vohnne V leads to the four equations

(40a)

r

(I] \l.|i \I.1M-.I» 1 MKollV (H-' IJKI.ATI VITY

where <> , , >i .. , «'., are the direction-cosines <>F the inward- drawn normal to the -urFaee-elemeut i/S in the Kuclidean Sense. \V«- reco^ni-e iii this the usual expression For the laws ol conservation. \\ e denote the ma^nituiles I ^ as the energy-components oF the gravitation-field.

I will now put the equation (1-7) in a third Form which will be ver\ -erviceable For a quick realisation of our object. By multiplxiii"1 the field-equations (17) with <j'r , these are obtained in the mixed Forms. IF we remember that

,.< a r ;„.__ 6 , a/"

9,._ -a.,£V' rf)" a r/-'

which ownm to (:U) is equal to

^a (rri>"] "/r^r^

- v'r/V' r"

or slight ly altering the notation equal to

6

The third member oF tin- expression cancel with the second member of the field-equations (17V In place oF the second term of this r\pre><ion. we ran, on account oF the relations (")H\ put

( " -1 Ar f\ . V /' -j /' '

I'll I XUI'I.K OK I!M.\ |'|\ rn

Therefore in UK- place of the equations (17), wo obtain

a

^16. General formulation of the field-equation of Gravitation.

The lield-e<|uations established in the preceding para- graph for spaces free from matter is to be compared with the equation V '-'<£=" of the Newtonian theorv. We have now to find the equations which will correspond to Poisson's Equation V2<j!> = ^7r^. (/, signifies the density of matter) .

The special relativity theorv has led to the conception that the inertial mass (Triige Masse) is no other than energy. 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 ~Ya associated with matter, which like the energy components 1 ir of (he gravitation-field (equal ion-.

10, and •")()) have a mixeil character but which however can be connected with synimetrieal eovariant tensors. The equation (•")!) teaches us how to introduce the energy-tensor (corresponding to tin density of 1'oisson's equation) in the field e»|iiatioiis of gravitation. If we consider a complete system (for example the Sular->\ stem its total ina^-. a- aiso it> total i;ra\ itatin^ action, v.ill depend on the total .•neiL,r\ of the system, ponderable as well a> ^ravitat ional.

CKVI'.I; VI.KKD TIIKOUY or KKI. \Ti\m l-l-"-

'\'\\\> can !><• expi •(•s>cil, hv putting in (•")!), in place of energy-components / ct' gravitation-field alone the ^um of I hi- energy-components of matter and gravitation, i.e.,

\Ye thus i;et instead of (51), the teiisor-pijnation

where T T1" (Laife's Sealar^. The<«- ;nv tin- general tiehl-

M

e(|tialions i)l' gravitation in the mixed form. In plaei of (47), we Ljet.ltv working l);iekvvards the svstem

It must be admitted, that this introduction of the rLT\ -tensor of matter cannot he justified l>\ means of the Relativity- Postulate alone; for we have in the fore^oin^- analysis deduced it from the condition that the ••ner-\ of the gravitation-field should exert gravitating action in the same \\a\- a- every other kind of energy. The stroii^e-! ground foi' the choice id' the al'ove eijiiilion however lies in this, that they lead, as their <-..iisri|iirnce>, to equations --IIIL; the eon>crvatioM of the comjioneiit > of total energy (the ini|iul>e- and the ciieru\ which »\aetly correspond to the e.|iiati"ii- (I!') and i M'a'. This shall lie shown afterward-

1 t I IMMXrlN.h. i»| i;i.I, \T| VITY

vj 17. The laws of conservation in the general case.

The equations (5:2) can be easily so transformed that the second member on the right-band side vanishes. \\ c reduce (or!) with reference to the indices p- and o- and subtract the equation so obtained after multiplication with .1 ^ from (5-2).

AYe obtain.

we i>|)eratc on il l>y ,., . Now.

9'. 9V

. r * """( °">

•2 8... 6. .. L » P V 8*,

u rr

'I'lic t'nst and the third member of the round bracket lead to t.\]ifc>sions which cancel one another. a> can be eusilv >e«-n by interchaii-in- the -nmniation-indices u, and ,r. on the one hand, and ft and A. on the other.

! i; u.is|.-.|i TiiKui,^ in 1:1:1. \Ti\rn ll.'t

The M'cond term c;in be transformed according to (•>!). Sr, that u,. et

_

- •-• a .„ a^ a.v

The second member of the t-xprcssioii on the Ifi't-liand -idr of ("):2a) leads first to

1 a1 / ,//*

2 a. a.r I f/

,(/

a--, a, a

+ a..

The exprerokm arising ont of the last incmlicr \\-itliin the round brackel vanishes ftcooiding to (%29) on aooonnt

of tlic flioiot- of axes. The two others can l>e taken ton-ether and ni\r u> on account of (•'>!), the rxpression

So that reroetnbering (•">!) we li

a,9^a, (^-

i ^<r A/^ ,_ « \

'V '' rw ) **

identical I v. 19

1 1C, I'lilN'rUM.r. ()! ItKI. \TIYI I Y

I'Yoni (5.)) jind (~>:l-,i} i! follows I hat

From the field equations of gravitation, it also follows that the conservation-laws of impulse and energy are satisfied. \Ve see it most simply following the >atne reasoning which lead to equations (I'.hi); onlv 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.

8 ^' If we multiply (5-5) with . we L;-et ill a w;iv

(T

similar to §15, remembering that

a/1

:/,.„ =H vanishes,

a/" ~ ^

the equations 'I" =_ 0

8' ' 8.. /"'

or remembering (:>(>)

(57) g-/:+: f£V=°

4 comparison with (Ml>) shows that these equations Pof the above choice of co-onlinates (x ,/ = 1) asserts nothiiiLl hut the vanishing of the di\eruence .if the tensor (jf the eiier^'v-coiiiponents ul' matlei.

(. K\ i i; \ i.isi- D IIII.OIM ni KI;I, \n\iTY 117

IMivsieallv the appearance of the second term on the left-hand >ide shows that for matter alone the law of con-

iervation of impulse and energy cannot hold; or can only

hold when g^'a are eon>tants ; i.e., when the tield of gravi tation vanishes. The second memhei i> ;in e\|H'.^-ion for

impulse and euergj whic!ithe gravitation-field exe:ls pel- time and per volume upon matter. This comes out clearer when instead of (57 we write it in the Form of (17).

9'C- a

9; =-r:fi ^

The ri-ht-liand side expresses the interaction of the energy of the gravitatioiiftl-iifld on matter. The field-equations of gravitation contain thus at the same time I conditions

which are to be satisfied by all material phenomena. We

^•{ the ei|iiations of the material phenomena completely when the latter i> characterix-d In four other differential equation;- independent of one another.

D THE ''MATERIAL" PHENOMENA.

The Mathematical auxiliaries developed under ' H ! at once enal)ie> u> to ^cnei-ali-e, according to the generalised theor\ of relativitx, the physical law> of matter (Hydrody- namies, Maxwell's I'llectro-dynainies a> the\ lie already formulated according to the special-retain ity-theory. The generalised Kelativitx Principle leads us to no further limitation of po->il>ilitics ; hut it eiiahles u> to know exactly the influence of gravitation on all processes with out the introduction of any new hypothesis.

It is owinii to tins, that as regard- the physical nature of matter 1,111 a narrou >en>e) :.o delinitt iie:-e--ar\ a»nni|i- tion> are to he introduced. The i|ite-tioii ma\ he open

I Is i'i; IN (i i '!,!•; (>!• i;i<;r,\TiviTY

whether the theories of the clectro-maguetic field and the gravitational-field together, will Jonn a sufficient basis for the theory of matter. The general relativity j»o$tulate can teach us no new principle. But by building up tin- theory it must be shown whether electro-magnetism and gravitation together can achieve what the former alone did not succeed in doin<r.

§19. Euler's equations for frictionless adiabatic liquid.

Let // and p, be two scalars, of which the first denotes the pressure and the last the density of the fluid ; between them there is a relation. Let the eontrawriant symmetrical

tensor

,j ,j il.r '/•''/)

ff

be the contra-variant energy -tensor of the liquid. To it also belongs the covarianl tensor

/ /i + '/ , '/ ,, ,- p

fir ' ' i><> ,/.. ' fl> ,/>• '

> well a> the mixed tensor

It' we put the ri^ht-liand side ol' ^OSb) in i^j/a) we •^t t the ^eiieriil h\ drod\ ii;iinical ei|iialiuns of Kuler accord ing tn the g --ne raided relativity theory. This in \>\ inciplc Completely solves the problem of motion: for the four

i,KNKKAI,IM',l> 'I lltdin 01 IJKI.A'I l\ ITV II1.'

equations (.")?a) together \vitli the ^iveii equation between // and f>, and the equation

./ p - 1

•V ,ls ,/*

are Millieient, with the ^iven values of y ,,, for finding out the six unknowns

' ' ' ' (/X I/S ' l If l/S

If // 's are unknown we have also to take the equ- tions (*)-i). There are now 11 equations for rinding out 10 functions // , so that the number is more than sufii- cient. Now it is be noticed that the equation (o?a) is already contained in (•">•'$), so that the latter only represents (7) independent equations. This indeliniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the problem is indefinite in the sense that of the gpace-fuuetiona can be arbitrarily chosen.

$20. Maxwell's Electro-Magnetic field- equations.

Let <f> be the components of a covariant four-vector. the electro-tnagnetic potential ; from it let iis form accord ing to (.'US) the components I-' ul the eovariant >i \-vector of the electro-magnetic field aeeordin^- \<> the >\>tem of

J50 i-ui \CIPI.K oi I;KI, \TI\ITV

From (•")'.»), it follows that the system ol' equations

9F

9F,

a<

is satisfied of which the left-hand side, according to (o?), is MM anti-s\ mmetrical tensor of the third kind. This system (HO) contains essentially four equations, which can be thus written :

((lOa)

a-.

This system of eqaatioDfl correeponds to the second system of equations of Maxwell. \Ve see it at once if we put

( I'Y, = II, !•',< = K,

Instead of (OOa) we can therefore write according to the usual notation of three-dimensional vector-analysis:

an

a/

.it K=(

div 1I=,

OBNERALTABD TIITOIM or RBT*ATIVIT\ 1 •> I

The litst Maxwellian system is obtained by a genera - lisation of the t'onn n'iven liy Miukowski.

\Ve introduce the (-out ra-variant six-vector 1'' 0 ''\

•P

tin- f«|iu(i()n

and also a contra-variant four-vector .l/', \\liicli is the electrical current-deusitv in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant I g to our choice of co-ordinates).

It' we put

, I'11 = H'., K" = }'/.

(••I )

F" = H KS1 = - M'

which »iuantities become e(|iial to II, . !•] in the ra>e of the special relativity theory, and besides

.1 ' = we Gjet instead of ((>•'$)

r

= P

,-ot H'- ' = *

a/

152 NMxni'i.i: or 1:1:1. \TI VITY

The p(|uations (('•()). ((>:!) and (<>.">) give ihus a generali sation of Maxwell's field-equations in vacuum, which remains true iti our chosen system of co-ordinates.

T/ti' energy-components <>/' I In- electTO-ntttffneftc fi''/<I .

Let us form 1 he inner-product

«>:>) K = K -I'''.

,<r <TJ>

According to (01) its components ean l>e written down in the three-dimensional notation.

/. H].

(. K K is a eovariant four-vector whose components are equal

cr

to the negative impulse and energy which are transferred to ill" electro-magnetic field per unit oi time, and per unit of volume, bv the electrical masses. If the electrical masses be free, that is, under the influence of the electro magnetic field onlv, then the eovariant four-vector K will vanish.

(T

In order to get the energy components T of the elec tro-magnetic field, we require only to give to the equation K o, the form of the equation (">7).

(T

From (»!•'}) and ((>•">) we get first,

6 ,, ,,,W<"9'"'

a, ( v

riE.VElMUSED THKORY OF RELATIVITY 1 53

On account «f (60) the second member on the right-hand side admits of the transformation

ft V

to symmetry, this expression can also be written in the form

which can also be put in the form

1 9 .<• \ llft /*•' /

+ ' F ft F 2 ( ^ 'lVft} ap ^v 6 •'• V

The first of these terms r-:m be written short lv :i-~

:in« I the second after difl'erentiation can be transformed in the form

154 NM\( I I'Li: ul

11' we take all the throe terms together, we get the relation

<«»

where

(<Hk) T"=- F Fv" + \ 81' F

(Td

«

a/?

On account of (30) the equation (66) become? 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 gravitation-field and matter satisfy when we use a co-ordinate system for which \/— .'/ = 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'O variant-equations. Still the question is not without formal interest, whether, when the energy-components of the gravitation-Held 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 (06), and the field- 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. J have

tiBNfcK \USKH TIIMilM (>!• KKLATIVm I -V)

found out thai this is indeed the ease, lint I am of opinion that the eominnnicat ion of my rather comprehensive work- on this subject will not pay, for nothing essentially ne\\ conies 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 general, in which q 's have constant values (4). p.v

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 // \ differ from (4) only by small magnitudes (com pared to I) where we can neglect small quantities of the second and higher orders (first asjK'e.t of the approxima tion.)

Further it should be assumed that within the >pace-

, time region considered, f] 's at infinite distances (JIMIIL:

the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); /.<•., we con sider only those gravitational fields which can be regarded as produced by masses distributed over finite regions.

\Ve can assume that this approximation should lead to Newton's theory. For it however, it is necessary to treat the fundamental equations from another |M>in<t <>f view. Let us consider the motion of a particle according to the equation (4fi). In the case of the special relativity theory, the components

156 I'ltl.NCll'Lh OF KELATivm

can take any values : This signifies that any

-- y + (;;;; >%< g y

can appear which is less than the velocity of light in vacuum (r <1). If we finally limit ourselves to the consideration of the case when •/,- is small compared to the velocity of light, it signifies that the components

'/(1 <k2 '/-''.I

.>) 8oiil«v Juiit*iio-» ivf r^ ui fl«v»<wx

can be treated as small quantities, whereas ' ' is equal In

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 \~ 's are all small quantities of at least the first order. A glance at (4(5) 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 :

= f \ i- where ds=<1st =<//.

or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,

,/'.x, r 44 -I

,//• -i J-

<;I.M.K.\I.I>I.I> niKoio 01 KKLATIVITY l">?

It' we further ;t—ume that the gravitation-field is quasi-static, i.e., it is limited only to the case when the matter producing the gravitation-Hold is moving slowh (relative to the velocity of light) we can 'neglect the differentiations of the positional co-ordinates on the ni;ht- hand side with respect to time, so that \ve get -idj lt

'<"' 9</4 +

11171 ,fi.-T =-i a^ (», = », a, »)

Tins i> the equation [of motion of a material point according to Newton's theory, where fftt/t plays the part of gravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the material point, only the component ytl of the fundamental tensor appears.

Let us now turn to the h'eld-equation (5o). In this case, we have to re mem her that the energy-tensor of matter is exclusively defined in a narrow sense bv the density/* of matter, >.<-., by the second member on the right-hand side of 58 [(">Sa, or 5M>)]. If we make the necessary approximations, then all component vanish except

T , , = i* ~ T

On the left-hand sidr of (r,:)) the second term is an infinitesimal of the second order, so that the first leads to the following term- in the approximation, which are rather interesting fur us ;

6 r^'i 4 a [>"! 4 a r/u/i a r^i

a-l J f e-.L 2J + a.',L J 8^L .J-

By neglecting all different iat ions with regard to time, this leads, when ^=^=4, to the expression

_! / 62.'/, ,6 •-'>,,, , a2:/,, v x

-^ ( 6"" + 6'- + 6 ' <7"-

158 i IM\( IIM.K OK ULLATIMIN

The last of the equations (53) thus leads to (68) VY,,=^-

The equations (67) and (68) together, ai>e equivalent to Newton's law of gravitation.

For the gravitation-potential we get from (67) and (68) the exp.

(68a.)

whereas the Newtonian theory for the chosen unit of time gi ves

% I .

wliere K denotes usually the

gravitation-constant. 67xlO~s : equating them we tret

SirK

(6!») * = = Ks7xlO-".

$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 a theory as a first approxi mation we had to calculate only //,.,, out of the 10 coni|X)- nt'iits // of the gravitation-potential, for that is the only

component which comes in the first approximate equal ion>- of motion of a material point in a gravitational field.

We M-e however, that the other components of </

should also differ from the values- given in (1-) as required by the condition «/ = 1 .

flF.N'ERALISF.i) T1IKOHY OK HKLATIVJTY l">y

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 :

S is 1 or 0. according as p = o- or not and /• is the quantity

... On account of (68a) \VP have

(70a) . . = £'

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 investigate the influences which the field ef mass M will have upon the metrical pro|>erties of the Held. Between the lengths and times measured locally on the one hand, and the differences in co-ordinutes d.r on the

V

other, \ve have the relation

For a unit me:i8iiring rod, for example, placed parallel to the axis, we have to put

./»•'= -1. i/.r,=,f.-s=,/.,4=o then -!=</ '/-••".

100 PKixriPi.i. 01 i;i;i.\Ti\m

If the unit measuring rod lies on the axis, the tirst of the equations (rO) gives

From both these relations it follows as a first approxi mation that

(71) ,/.• = !- .

The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational Held, when we place it radially in the field.

Similarly we can get its co-ordinate-length in a tangential |x>sition, if we put for example

(/x2 = 1. </.",_ =</.( •_, =</••, =o', .i.1!—!'. .!•.,=. ''3=O

we then get

i71a.) -!=.'/», '/••!; = -'' "I

The gravitational field has no influence upon the length of the rod, \vhon we put it tangentially in the field.

Thus Euclidean geometry docs nol hold in the giavi- tational lield O\»MI 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 (fi!)) shows that the ex]>ected difference is much too small to l>e noticeable in the measurement of earth's surface.

AVe would further investigate the rate of going of ;i i mil -clock- which is placed in a statical gravitational field. Mere we have for a period of the cluck

:i \Ki; VI.ISKH TIIKiHiY OF KKI.ATI VITY Kil

then we h:ive

•/••,=! +

I f "•

Therefore the clock goes slowly what it is place*! 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 Held. According to the special relati vity theory, tin* velocity of light is given by the equation

thus also according to the generalised relativity theory it is given by the e<|ii:it ion

(73 *>9V*J' (/'' =0'

If the direction, »>., the ratio d , : <l,<\ : //-, is gi\Mi. ihe e»jii!ition (7:J) gives the magnitudes

and with it the velocity,

a

jl'.-> PR1NTIPLK OK KEJ-A.T1VTTV

in the sense of the Kuclidean Geometry. vVe can easily see that, with reference to the co-ordinate system, the rays of light must appear curved in case // *s are not constants.

If a be the direction perpendicular to the direction of propagation, we have, from II uy gen's principle, that light-rays ] (taken in Ihe plane (y, »)] must suffer a

curvature ^-? .

A Light-ray

Let us Hnd out the curvature which a light-ray suffers when it goes by a mass M at a distance A from it. If we use the co-ordinate system according to the above scheme, then the totsil bending B of light-rays (reckoned positive when it is eonca.ve to the origin) is given as ;i sufficient approximation bv

where (?.'') and (70) gives

, ,

\ ray of light just gra/ing the sun would suffer :i beml- ing of 1'7", whereas one coming by Jupiter would have a deviation of about -0:2".

KD THKOKV OF KKLVTIVITY IM

If we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of a material particle of a relatively small (infinitesimal) mass we get a deviation of the following kind from the Kepler-Newtonian Laws of Planetary motion. The Kllipse of Planetary motion suffers ;i slow rotation in the direction of motion, of amount

( 7.') i .<= 7r,"S per revolution.

T"c*(l— •" )

In this Formula ' ft ' signifies the semi-major axis, c, the velocity of light, measured in the usual way, f, 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 suffi ciently to what has been found by astronomers (Leverrier). They found a residual perihelion motion of this planet of the given magnitude which can not I>e explained b\ the perturbation of the other planet-.

NOTES

Note 1. The fundamental electro-magnetic equation of Maxwell for stationary media an> :

<mrlH=I

D-//K

According to Hertx and Hoaviside, these require modi fication in the case of moving bodies.

Now it is known that due to motion alone there is a change in a vector /? given by

( _ - - ) due to motion //, div \\ +enrl \Uu\ \ o' /

where « is the vector velocity of the moving bodv and [Rw] the vector product, of \\ and //. Hence equations (1) and (2) become

I- curl H= ~ +v div D-fcurl \ ect. [Dv]+pv (1-1)

and

K= -- |- « div H -j- curl \Vrti.

which gives linally, for p o and div H = O,

+ n div l)=:r curl 111- Vwfc, ' D«l )

O / '•

,H ) -

166 PKINCIPT.K OK ItET.ATIVrn

Let us consider a beam travelling along the ./--axis, with apparent velocity /• (i.e., velocity with respect to the Hxed ether) in medium moving with velocity n, = n in the same direction.

Then if the electric and magnetic vectors are

f- A (x-vt) proportional to e , we have

f-=/A, |j = -/Ar, |- = £ =0, ,/, = ;,. =0

5." 6* dy 8-

Since D = K E and B = /A H, we have

. ... (1-22)

/x (»;-«.) H,=cEv ...' (2-23)

Multiplying (1'23), by (2'28)

/A K (f 7')2=C2

Hence (i'-//)2=cs/^ = »'.5f

.'. '• = «„ + w, making Fresnelian convection co-efficient simply unity.

Equations (1'21), and (^'^l) may bo obtained more simply from physical 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 tixed with respect to the ether, the rate of change of electric polarisation is 5D

NOTE* 167

Consider a siab of matter moving with velocity n, along- the ./--axis, then even in a stationary field of electrostatic polarisation, that is, for a Held in which

= o, there will be some- change in the polarisation of

or

the body due to its motion, given by ur . Hence we

O <:

must add this term to a purely temporal rate of change -^- . Doing this we immediately arrive at equations

Of

(T21) and (2'21) for the special case considered there.

Thus the Hert/-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, Larmor lias shown (Aether and Matter) that I— I/ft2 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 away by it. Thus the electromagnetic effect outside a moving medium should be proportional to K, the specific inductive capacity.

Rvic/mtil showed in l^itl that when a charged condenser is rapidly rotated (the dielectric remaining stationary), the magnetic effect outside is proportional to K, the Sp. Ind. Cap.

/,',•;/////,-// (Annaleu .lev IMiysik 1SSS, 1890) found that if the dielectric is rotated while the condenser remains stationary, the effect is proportional to K 1.

168 PRINCIPLE OF RELATIVITY

Kickenwalil (Annaleu der Physik 1903, IVOV) rotated together both condenser and dielectric and fouud that the magnetic effect was proportional to the potential difference and to the angular velocity, but was completely independent of K. This is of course quite consistent with Rowland and Rb'ntgeu.

Blotidlot (Coinptes Reudus, 1'JOl) passed a current of air in a steady magnetic Held II,, (H =H,=0). If this current of air moves with velocity /', along the .F-axis, an electromotive force would be set up along the r-axis, due to the relative motion of matter and magnetic f-ubt's <>f induction. A pair of plates at .*= + a, will be charged up with density /i=D, = KE =K. n. Hy/c. But Blundlot failed to detect any such effect.

//. A. intsnn (Phil. Trans. Kuyal Sot-. 190i) repeated the experiment with a cylindrical condenser made of ebouy, rotating in a magnetic Held parallel to its own axi^ He observed a change proportional to K— I and not to K.

Thus the above set of electro-magnet ic experiments contradict the llcrt/.-lleaviside equations, and these must be abandoned.

[P. C. M.]

Not© 2. Lur«'iii: Ti'ii itxfoi'm/i/ i<nt.

Lorentz. \rrsuch einer theorie der elektri-chen uud optibfhon Ki'-'C'lu'ijinn^fn im bewegteii Korpern.

(Leiden— 1895).

Lorent/.. Theory of .MK-ctrnns (English eilition), pages l:»r---Dii. i30,also notes '<••'>. 86, pages :;is. :;-:s.

Lorentx. wanted to explain the (Vfichelson-Morley null-effect. In order to do 50, it was obviously necessary to explain tin- I'it/.gerald COUtrACtlOD. Lorentx, worked on the hypothesis (hat an electron itself undergoes

NOTES 169

contraction when moving. He introduced new variables for the moving system defined by the following set of equations.

and for velocities, used

v.*=p*v, + i', v,l=(3r,, rrl=(3r, and pl=p/fi.

With the help of the above set of equations, which is. known as the Lorentz transformation, he succeeded in showing how the Fitzgerald 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 Kinsteinian addition of velocities is quite different as also the expression for the ''relative" density of electricity.

It is true that the Max we 11- Lorentz Held equations remain practical/ 1/ unchanged by the Lorentz transforma tion, but they ar>- changed to some slight extent. UUP marked advantage of the Einstein transformation con^i-t> in the fact that the Held equations of a moving system preserve exactly the same form as those of a stationary M -torn.

It should also be noted that the l; resin-Han convection coctKcient comes out in the theory of relativity as a direct consequence of Kin-tern'- addition of velocities- and is quite independent o!' any electrical theory nf mattrr.

[P.C If.]

Note 3.

Lorent/., Theory of Klectrons (English edition), § 181, page 513.

170 I'ttlNrill.F MI RELATIVITY

H. Poiucare, Sur la dynamujue 'electron, Rendiconti del circolo matematico di Palermo 21 (1906).

[P. C. M.]

Note 4. lielaticity Theorem <ai<l Kelativiiy-Principle.

Lorentz showed that the Maxwell-Lorentz system of electromagnetic field-equations remained practically unchanged bv the Lorentz transformation. Thus the electromagnetic laws of Maxwell and Lorentz can be definitely proved " to be independent of the manner in which they arc referred to two coordinate systems which have a uniform translatory motion relative to each other." (See " Electrodynamics of Moving Bodies,"' page 5.) Thus so far as the electromagnetic laws are concerned, the principle of relativity ant he proved lo be fntf.

But it is not known whether this principle will remain true in the case of other physical 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 they retain their form wln-n referred to moving coordinates. The ultimate ground for formulating physi cal laws in this way is merely a subjective conviction that the principle of relativity is universally true. There is no a priori logical neccs^it \ t!:;it it should be so. Hence the Principle of Relativity (so I'ar as it is applied to phenomtMin other than electromagnetic') must be regarded as a jjostiiliil.t.-, which we have u.-Mimed to be true, but for which \\v cannot adduce any definite proof, until after the generalisation is made and its consequences tested in the light of actual experience.

[P. C. M.j Note 5.

See " Kli-ctrodynamic> oi' Moving Hodies," p. 5-8.

VOTES 171

Note 6. /•'/>/,/ fymi/f aim in l/////-0jrj»/r* l-'onii.

Equation! (/) :md (//) become when expanded into Cartesians :—

8*1 8*», a«,

^ a o " />'' ,

o y o - o T

,_.

T a^ av^"" r '

8 MI, 8>", 9«',

o >s~" ~" ^ =/>»-

a.-- a// 8r j

Substituting ;c, , ,,'2, .-i-3> .r , for .<•, ;/, r, and ir ; and p, ,

f.jj pj, p^ for pi( . t />"yi pit:) ipi where /^= \/ lt

We get,

a »' s a^i, a^j "^

"8;r, ~ ?8

a?»: 9m, ,-9j\ ar,~ aa-,~*9^

and multiplyinG: (2'1) by / \v<> get

a/'*', a'-', a/'-1

Now substitute

»»• =/a * = —/« I a' id /V,=/4I=— /it

711 » =/S 1 = ""/I S * '' , =/4 * = /, «

172 HI? I \CTPT.K OP RELATIVITY

and \ve irct finally :

8./. i a/,, 8/S4

d^t-^+iSt88^

... (3)

a/,,

, 8/,4

+*5*

8/41 , s/*. , a/..,

[P. C. M.j

Note 9. On the Constancy of the Velocity of /,//////.

Page i:> refer also to page 6, of Einstein's paper.

One of the two fundamental Postulates of the Principle of Relativity is that the velocity of light should remain constant whether the source is moving or stationary. It follows that even if a radiant source S move with a velocity u, it should always remain the centre of spherical wave- expanding outwards with velocitv 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 + it, when the source o!' light moves towards the observer with the velocity n.

Prof, de Sitter has given an astronomical argument for deciding between these two divergent views. Let u< suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit.

VOTES 173

Let the observer lie 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 + it

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 ha If- period from B to A is approximately T '*-—' and from A to B is

T+ 2-All. 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 onlv of the same <>nler a> T, but are mostly ca

much larger. For example, if ;<=!()() /•/// sec, T = b da\s, A/c = 33 years (corresponding to an annual parallax of -1"), then T i/^A/c.- - =o. The existence of the Spectroscupio binaries, and the fact that they follow Kepler's Law is therefore a proof thai c is not ad'octed by the motion of the source.

In a later memoir, replying t ^ the criticisms of Freun'llich and (liinthiek that an apparent ecoontripity occurs in the motion proportional to /'"A,;*, ",i being the

17 t rillNCll'LE Ot IIELATIVIIY

maximum value of t> , the velocity of light emitted being 7/,,=c + X"«, /t = 0 Lorentz- Einstein /•=! Ritz.

Prof, de Sitter admits the validity of the criticisms. But he remarks that an upper value of k may be calculated from the observations of the double sar /3-Aurigae. For this star.

The parallax 7r = 'OU", e = '005, wn = 110 /f-w/sec T = 3'96, A > 05 light-years, /• is < -OOe.

F«ji an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol. 35, p. 1C,:}.

[M. N. S.] Note 10. Urtl-tlcnuili, <>f Muctricity.

It' /> is the volume density in a moving system then p \/([ n'-} is the corresponding quantity 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.

[P. a li.]

Note 11 (page 17).

SpaM-limc Ir <}/•<> r^ of lk»: //>*/ uml I /»' .svv.W khnl, As we had already occasion to mention, Sommerfeld hag, in two papers on four dimensional geometry ('vWr, Annalen der Physik, Bd. «2, p. Tli) ; and Hd. :i:1,, p. 640), translated the ideas of Minkowski into the language of four dimensional geometry. Instead of M inkowski's space-time vector of the first kind, he uses the more expressive term 'four-vector,' thereby making it 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.

VOTKS I7.r)

The representation of the plane (defined by two straight lines) is much more difficult. In three dimensions, the plane can he represented by the vector perpendicular to itself. Hut that artifice is not available in four dimensions. For the perpendicular to a plane, we now have not a single line, but an infinite number of lines constituting a plane. This difficulty has been overcome by Minkowski in a very elegant manner which will become clear later on. Meanwhile we offer the following extract from the above mentioned work of Sommerfeld.

(Pp. 755, Hd. Z-2, Ann. d. Pliysik.)

" In order to have a better knowledge about the nature of the six-vector (which is the same thing HS Minkowski's space-time vector of the 2n<l kind) let us take the special case of a piece of piano, having unit area (contents), and the form of a parallelogram, bounded by the four-vectors //, »', passing through the origin. Then the projection of this piece of plane on the .r// plane is given hv the projections w,, //„, /•,, r a of the four vectors in the combination

Let us form in a similar manner all the six component of this plane <£. Then six component* are not all independent but are connected 1>\ the following relation

Further the contents ) | of the piece of a plane is to be defined as the square rout of the sum of the squares of these six ipiantitio. In fact,

i 4| =*«, +#,+#,+«««+#. +<*>*.-

Let us now on the other hand take the ea-^e of the unit plane <j>* normal to $ ; we can call this plane the

176 PRINCIPLE OF RELATIVITY

Complement of <f>. Then we have the following relations between the components of the two plane :

The proof of these assertions is as follows. Let ;/*, r* be the four vectors defining <£*. Then we have the following relations :

multiplying these equations by r* . ;/*-,or by r* . ?/* . we obtain

<i>*: .$,1 + **, £,,=0 and <£*y ^,, +^?. ^..,=0 from which we have

In a corresponding way we have

whon the subscript (//•) denotes the component of </> in the plane contained by the lines other than (;'£). Therefore the theorem is proved. We have ^^*=^, <•+..,

r* r,+v* ?•„+?'* r.+rfr,=0

If we multiply these equations by v,, n,, rt, and subtract the second from the first, the fourth from the third we obtain

XOTKS 177

The ^eneral si i\- vector /' is composed from the vectors *" in the following way :

p and p* denoting the contents of the pieces of mutually perpend ic « ihir planes composing /'. The "conjugate Vector"' f* (or it may be called the comi)lement of /*) is obtained by interchanging p and //* We have,

f* = ,,* 4, + ,, $*

We can verify that

/*, = /.i etc.

| / | - and (J'f*} may be said to bo invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates.

[M. N. 8.] Note 12. Ligkt-tcfocity H* ti maximum.

Page -2-J, and Electro-dynamics of Moving Bodies, p. 17.

Flitting r = c .»•, and /r = c A, \ve get

1 + (f-.r) (r-A)/c-' c- -(-r-'-(.r-fA)^-f , I

= , ^-(.r + A) 2<- -(.<• + A)+J-A/c

Thus f<r, so lorig as | .rA | >().

Thus the velocity of light is the absolute maximum velocity. We sh.il! i:n\v see the consequences of admitting a velocity W >r.

Let A and B be separated by distance /, and let velocity of a -Signal " in the system S be W>r. Let the

178 PRINCIPLE OF 11ELATIVITY

(observing) system S' have velocity -H1 with respect to the system S.

Then velocity of signal with respect to system S' is Kivenbv 1T« '!*£*.,

Thus " time " from A to B as measured in S', i3 given

^ nw-'zggD -,• ... •;_; ;•.':.: 0)

Now if r is less than e, then W being greater than c (by hypothesis) W is greater than v, i.e., W>v.

Let W=? + /A and v = c A.

Then Wr = (c {- n)(c A) = c- + (/* + X)c /zA.

Now we can always choose v in such a way that Wv is greater than c-, since Wr is >c- if (/* + A)c yuA is >0.

that is, if fji + \>— .which can always be satisfied by a suitable choice of A.

Thus for W><? we can always choose A in such a way as to make W?->e2, i.e., I— Wr/c2 negative. But \V /• i>; always positive. Hence with W>c, we can always make f' , the time from A lo B in equation (1) " negative." That is, the signal starting from A will reach B (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 uot produce any effect. Causal effect of any physical type can never

travel with a velocity greater than that of light.

[P. C. M.]

NOTES 179

Notes 13 and 14.

We have denoted the four-vector by the matrix I wi <02 W3 I It- is then at once seen that &> denotes the reciprocal matrix

'• !

>4

It is now evident that while <•>' =w\, = A~'w

[<»..»] The vector-product of the four-vector w and -t may be represented by the combination

[w»] = Sta

It is now easy to verify the formula ./'! = A M/A. Supposing for the sake of simplicity that / represents the vector-product of two four-vectors a>, <v, we have

= [ A~ a w,sA A " ' *w

= A-1[(^-,va>JA = A Now remembering that generally

Where p, p* are scalar (|iiantities, <^, <£* are two mutually perpendicular unit planes, there is no difficulty in seeming that

/• = A-VA.

Note 15. The rector product (trf). (L\ 3(>). This represents the vector product of four- vector and a six-vector. Now as combinations of this type are of

180 l'i;l\( ll'I.K OF KEI.ATIVITY

frequent occurrence 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 parallelepiped 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 parallelepipeds, but it can always bo 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, we do not obtain a vector of a more general type, but one which can always be represented by a four-vector (loc, cit. p. 75!)). The state of affairs here is Hie same as in the ordinary vector calculus, where by the veetor- mu Implication of a vector of the first, and a vector of the second type (/.''., a polar vector), we obtain a vector of the first tvpe (axial vector). Tl;c I'unnal scheme of this multiplication is taken from the three-dimensional ease.

Let A=(A,. A,,, A.) denote a vector of the first type, B = (B,, . , B -.,, B., ,) denote a vector of the second type. From this lust, let us form three special vectors of the first kind, namely— -

B,=(Brr, .B,,, B,sn

B, = (B,., B,,, By.V-(Bl4 = -Btl. B,,=0).

B.. =(B,,, 1} .„, B..)J

Since H , , 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 U,, i.e., ^A, B,,+A B,, +A. B,,.

MHKS 181

We see easily that this coincides with the usual rule for the vector-product, c. y., fory = .'•.

= A, B,,-A.. B..,,

Correspondingly let us define in the four-dimciibional case the product (P/) of any four- vector P and the six- vector./". The /-component (j = -r, //, :, or /) is given by

(P/, ) = p,/v , + ?,/, , + p,,/, , + p./;, ,

Eac-h one of these components is obtained as the scalar product of P, and the vector f , which is perpendicular to j-axis, and is obtained from /' bv the ruley', = [(f}1, J ,v>

We can also find out here the geometrical significance of vectors of the third type, when ,/'=<£, /.<?.,./' 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.* Tiie components of (P</>) are then eijual to the 1 three-rowed under-determiuants 1), Dv D; Di of the matrix

P, P. P. P/

U,* U,* U,* I

\ Vv* V:* \ ,*

Leavin aside the tirst column we obtain

which coincides with (P'M according to our di-linition.

182

I'llJNCll'LE OF RELATIVITY

Examples of this type of vectors will be found ou page 36, * = /t-F, the electrical-rest-force, and ^ = i>o/,* ibe magnetic- rest-force. The rest-ray I) = ?o> [<I>i/T] * also belong to the same type (page 89). It is easy to show that

When (wn w2, o),)=o, dimensional vector

=/. il reduces to the thi-ee-

Since in this case, *!=oj4 F14 =«r, (the electric force)

^l = i<1}J\^r=mJ, (the magnetic force)

we have (Q) =

e,

m,

, analogous to the Poynting-vector. [M. N. S.] Note 16. 27*e eleclric-rcd force. (Page 37.)

The four- rector ^> = o-F which is called b\r Minkowaki the clectric-rest-force (elektrische Rub-Kraft) is very closely connected to Lorentz'a Ponderomotive force, or the force acting on a moving charge. If i> i? the density of charge, we have, when e=l, //.= ], /'.<"., for free space

Nn\v since p(,=: \Vu have

l V2 V2

o*4=p[d.+ j («•*•'.-••*•)]

J>. ^ c liavc j)ul the c

X.J>. \\\' have j)ut the component- of e equivalent to ('/,, dy, </:), and the components of in equivalent to

NOTES 18«i

// , // a //.), in accordance with the notation used in Lorent/'s Theory of Electrons, \Ye have therefore

i. /•., /j(, (<£,, </>„, </>., ) represents the force acting on the electron. Compare Lorent/, Theory of Electrons, page 14. The fourth component <£, when multiplied by /j() represents /-times the rate at which work is done by the moving electron, for />,, <£4 .=i'p [>',<!* + ?v7» + r.c?.] = Pn</>i+ry Po^2 + r.- Po^s- -%/_l times the power pos sessed by the electron therefore represents the fourth component, or the time component, of the force-four- vector. This component was first introduced by Poineare in 1900.

The four-vector ^ = /wK* has a similar relation to the force acting on a moving magnetic pole.

Note 17. Operator " Lor" 1:2, p. 11).

£\ £\ p\

The operation ^- -5—, Q ~, g (, j which plays in four-dimensional mechanics a role similar to that of

/ o cs o \

the operator I / fC-, + j o >+ ^' "aT= V ) m three-dimen

sional geometry has been called by ^TinUowski ' Lorent /- Operation ' or shortly ' lor ' in honour of H. A. Lorent/, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol Q to denote this operation. In the above-mentioned paper (Annalen der Physik, p. (ill), Bd. 33) Sommerfeld has introduced the terms, Div (divergence), Hot ( Rotation), (I rad (gradient) as four-dimensional extensions of the corresponding three- dimensional operations in place of the general svmho! lor. The ph\'sical si^ni'ii-iiu-e of thes- operations will

184- I'lUNCIl'I.K ()! UKLATIVITY

become clear when along with Minkowski's met hod of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with the case of lor S, where 8 is a six- vector (^pace-time vector of the 2nd kind).

This being a complicated case, we take the simpler catffe of lor *,

where A- is a four- vector = j -fj, *„, A-O *4 | and .s = i *,

The following geometrical method is taken from Som- m erf eld.

Scalar Divergence Let A 2 denote a small four-dimen sional volume of any shape in the neighbourhood of the space-time point Q, //S denote the three-dimensional bounding surface of A2, /' be the outer normal to ifS. Let S be anv four-vector, PN its normal component. Then

Div S = Lim

Now if for A 2 we choose the four-dimensional paral lelepiped with sides (//>-,, //.>•„, ^.rn, (?.r4}} we have then

O '' i O 2 C/ •' 3 O •'' i

If f denotes a space-time vector of the second kind, lor /'is equivalent to a space-time vector of the first kind. The geometrical significance can be thus brought out. We have seen fiat the operator ' lor' beirives 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

XOTKS 185

tn see that lor S will lie a four-vector. Let us find thr component of tliis f on r- vector in any direction *. l.i't S denote the three-space whieh passes through the point Q (./•,, .r,, .r.,, .r4) and is perpendicular to ,v, AS a very small part of it in the region of Q, tfn- is an element of its two-dimensional surface. Let the perpendicular to this surface lying in the space be denoted by //, and let /'.„ denote the component of f in the plane of (*//) which is evidently conjugate to the plane tlv. Then the v-component of tho vector divergence of /' because the operator lor multiplies /' vectorially)

= Divf.=Lim Ih^?.

A*=0 AS

Where the integration in </<r is to be extended over the whole surface.

If now x is selected as the .r-direction, A* is then a three-dimensional parallelopiped with the sides t1i/} <h, i//, then we have

and generallv

' = ' + +- +

H.-nce the four-components of the four- vector lor S or Div. /' is a four-vector with the components given on pag.- 1-2.

According to the formulae of space geometry, D, denote-; a parallelopiped laid in the (//->0 space, formed

out of the vectors (P, P. ?,), (IT* r* U*) (v* vt V*,).

18H I'lilXCII'LK 01 UKLATIVITY

D, is therefore the projection on the y-:-l space of 1he peralielopiped formed out of these three four- vectors (P, U*, V*), and could as well be denoted by Dyxl. 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 given by the geometrical sum of the two four-vectors of the first type PD and P*D*.

[M. N. S.]

0

PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKE'

UNIVERSITY OF TORONTO LIBRARY

QC 6

Einstein, Albert

The principle of relat

P&A Sci