;. i».. •' •Mii'-marvF^i'rv ■; u ■' ' t.^».-ot Mnafco LIBRARY OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY y ^0' < THE SEP k PRINCIPLE OF RELATJ^^{jK[Ji ORIGINAL PAPERS BY A. EINSTEIN ANT) H. MINKOWSKI )' TRANSLATED INTO ENGLISH BY M. N. SAHA AND S. N. BOSE lecturers on physics and applied jiathematics Univebsity College of Science, Calcutta Univeksity WITH A HISTORICAL INTRODUCTION BY P. C. MAHALANOBIS professor of physics, presidency college, CALCU- PUBLISHED BY THE UNIVERSITY OF CALCUTTA 1920 Sole Agents R. CAMBRAY &: CO. PRINTED BY ATULCHANDRA BHATTACHAKYYA, AT THE CALCUTTA UNIVERSITY PRE3B. SENATE HOUSE. CALCUTTA TABLE OF CONTENTS 1. Historical Introduction [By Mr. P. C. Mahalanobis.] 2. On the Electrodynamics of Moving Bodies... [Einstein's first paper on the restricted Theory of Relativity, originally pub- lished in the Annalen der Physik in 1905, Translated from the original German by Dr. Meghnad Saha.] 3. Albreeht Einstein [A short biographical note by Dr. Meghnad Saha.] 4. Principle of Relativity [H. Minkowski's original paper on the restricted Principle of Relativity first published in 1909. Translated from the original German by Dr. Meghnad Saha.] 5. Appendix to the above by H. Minkowski ... [Translated by Dr. Meghnad Saha.] 6. The Generalised Principle of Relativity [A. Einstein's second paper on the Genera- lised Principle first published in 1916. Translated from the orijjina] German by Mr. Satyendranath Bose.] /, iNotes ,,, ,,, ... PAGE i-xxiii 1-34 35-39 1-52 53-88 89-l()3 165-^185 124281 -N \^\ y J HISTORICAL INTRODUCTION ooj:^0<>- Lord Kelvin writing- in 1893, in his preface to the English edition of Hertz's Researches on Electric Waves, says " many workers and many thinkers have helped to bnild up the nineteenth century school of plenuDij one etiier for light, heat, electricity, magnetism ; and the German and English volumes containing Hertz's electrical papers, given to the world in the last decade of the century, will be a permanent monument of the splendid cons ^mmation now realised." Ten years later, in 1905, we find Einstein declarinsj that " the ether will be proved to be superflous." At first sight the revolution in scientific thought brought about in the course of a single decade appears to be almost too violent. A more careful even though a rapid review of the subject will, however, show how the Theory of Relativity gradually became a historical necessity. Towards the beginning of the nineteenth century, the luminiferous ether came into prominence as a result of the brilliant successes of the wave theory in the hands of Young and Fresnel. In its stationary aspect the elastic solid ether was the outcome of the search for a medium in which the light waves may "undulate." This stationary ether, as shown by Young, also afforded a satisfactory explanation of astronomical aberration. But its very success gave rise to a host of new questions all bearing on the central problem of relative motion of ether and matter. 11 PRINCIPLE OF RELATIVITY Arago^s prison experiment. — The refractive index of a glass prism depends on the incident velocity of light outside the prism and its velocity inside the prism after refraction. On Fresnel's fixed ether hypothesis, the incident light waves are situated in the stationary ethei outside the prism and move with veloeit)' c with respeci to the ether. If the prism moves with a velocity n with respect to this fixed ether, then the incident velocity of light with respect to the prism should be c + n. ThuE the refractive index of the glass prism should depend on m the absolute velocity of the prism, i.e., its velocity witl respect to the fixed ether. Arago performed the experimeni in 1819, but failed to detect the expected change. Airy- Boscovitch ivaler-telescoije experimeni. — Boscovitcl had still earlier in 1766, raised the very importan question of the dependence of aberration on the refractive index of the medium filling the telescope. Aberratior depends on the difference in the velocity of light outsid» the telescope and its velocity inside the telescope. If thi latter velocity changes owing to a change in the medium filling the telescope, aberration itself should change, thai is, aberration should depend on the nature of the medium. Airy, in 1871 filled up a telescope with water — but failed to detect any chansje in the aberration. Thus w< get both in the case of Arago prism experiment an( Airy -Boscovitch water-telescope experiment, the ver startling result that optical effects in a moving mediun seem to be quite independent of the volocit}^ of th medium with respect to Fresnel's stationary ether. FresneVs convection coefficient /(:=1 — ^/^^. — Possibb some form of compensation is taking place. Working oi this hypothesis, Fresnel effered his famous ether convee tion theory. According to Fresnel, the presence of matte: implies a definite condensation of ether within th( t « • HISTORICAL INTRODUCTION 111 region occupied by matter. This " condensed " or excess portion of ether is supposed to be carried away with its own piece of movino" matter. It should be observed that only the " excess " portion is carried away, while the rest remains as stagnant as ever. A complete convection of the ''excess " ether p with the full velocity u is optically equivalent to a partial convection of the total ether p, with only a fraction of the velocity k. u. Fresnel showed that if this convection coefficient k is 1 — *//x'-^ (/x being the refractive index of the prism), then the velocitv of lio^ht after retraction within the movin"; prism would be altered to just such extent as would make the refractive index of the moving prism quite indepen- dent of its "absolute" velocity u. The non-depeudence of aberration on the '" absolute " velocity it, is also very easily explained with the help of this Fi-esnelian convection- coefficient k. Stokes^ viseous ether. — It should be remembered, however, that Fresnel 's stationary ether is absolutelv fixed and is not at all disturbed bv the motion of matter throusfh it. In this respect Fresnelian ether cannot be said to behave in any respectable physical fashion, and this led Stokes, in 1845-46, to construct a more material type of medium. Stokes assumed that viscous motion ensues near the surface of separation of ether and moving matter, w^hile at sufficiently distant regions the ether remains wholly undisturbed. He showed how such a viscous ether would explain aberration if all motion in it were differentially irrotational. But in order to explain the null Arago effect, Stokes was compelled to assume the convection hypothesis of Fresnel with an identical numerical value for kj namely 1 — V/^'- '^hus the prestige of the Fresnelian convection-coefficient was enhanced, if anything, by the theoretical investigations of Stokes. IV PRINCIPLE OF RELATIVITY Fizeaic^s experin/cnl, — Soon aftur, in 1851, it received direct experimental eonHrmation in a brilliant piece of work by Fizeau. If a divided beam of light is re-nnited after passin<)j through two adjacent cylinders filled with water, ordinary interference fringes will be produced. If the water in one of the cylinders is now nriade to fiow^, the " condensed" ether within the flowing water wonld be conveeted and would produce a shift in the interference fringes. The shift actuallv observed agreed verv well with a value of k=l— V/Jt^. The Fresnelian eonveetion-eoeffieient now became firmly established as a consequence of a direct positive effect. On the other hand, the negative evidences in favour of the convection-coefficient had also multiplied. Mascart, Hoek, Maxwell and others sought for definite changes in different optical effects induced by the motion of the earth relative to the stationary ether. But all such attempts failed to reveal the slightest trace of any optical disturbance due to the "absolute" velocity of the earthy, thus proving conclusively that all tne different optical effects shared in the general compensation arising out of the Fresnelian convection of the excess ether. It must be carefully noted that the Fresnelian convection -coefficient implicitly assumes the existence of a fixed ether (Fresnel) or at least a wholly stagnant medium at sufficiently distant regions (Stokes), with reference to which alone a convection velocity can have any significance. Thus the convection- coefficient implying some type of a stationary or viscous, yet nevertheless "absolute" ether, succeeded in explaining satisfactorily all known optical facts down to 1880. Mic/iehov-Morley Eopperiment. — In 1881, Michelson and Morley performed their classical experiments which undermined the whole structure of the old ether theory and thus served to introduce the new theory of relativity. HISTOKICAL INTRODUCTION V The fiiiidameiital idea underlyia^^'' tliib experiment is quite sim[de. In all old expeiiments the velocity of light situated in free ether \Vas corn[)ared with the veloeitv of waves actually situated m a piece of moving matter and presumably carried away by it. The compensatory effect of the Fresnelian convection of ether afforded a satisfactory explanation of all neo^ative results. In the Michelson-Morley experiment the arrangement is quite different. If there is a definite gap in a rigid body, light waves situated in free ether will take a delinite time in crossing the gap. If the rigid platform carrying the gap is set in motion with respect to the ether in the direc- tion of light propagation;, light waves (which are even now situated in free ether) should presumably take a longer time to cross the gap. We cannot do better than quote Eddiugton's descrip- tion of this famous experiment. " The principle of the experiment may be illustrated by considering a swimmer in a river. It is easily realized that it takes longer to swim to a point 50 yards up-stream and back than to a point 50 vards acioss-stream and back. If the earth is movino- through the ether there is a river of ether flowing- throuopli the laboratory, and a wave of light may be compared to a swnmmer travelling with constant velocity relative to the current. If, then, we divide a beam of light into two parts, and send one-half swimming up the stream for a certain distance and then (by a mirror) back to the starting point, and send the other half an equal distance across stream and back, the across-stream beam should arrive back first. Let the ether be flowing relative to oi the apparatus with velocity u in the ^ direction Or, and let OA, OB, be B the two arms of the apparatus of equal A VI PKiNClPLE 0¥ RELATIVITY length L Oi^. being placed up-stream. Let c be tbe velocity of lig;ht. The time for the double journev alon^' OA and back is t,= ± + -A = J^= ^/S^ G — If. c~\ru c^ — u^ c where f3=:(l—u'^/c^)~'^, a factor greater than unity. For tbe transverse journey the light must have a compo- nent velocity n up-stream (relative to the ether) in order to avoid beins: carried below OB : and since its total velocity is c, its component across-stream must be \/{c'^ —u'^), the time for the double journey OB is accordingly t'l = /7-~^^ = —A SO that t^>t^. But when the experiment was tried, it was found that both parts of the beam took the same time, as tested by the interference bands produced." x\fter a most careful series of observations, Michelson and Morle^^ failed to detect the slightest trace of any effect due to earth's motion throus^h ether. The Michelson-Morley experiment seems to show that there is no relative motion of ether and matter. Fresnel's stagnant ether requires a relative velocity of — n. Thus Michelson and Morlev themselves thought at first that their experiment conhrmed Stokes^ viscous ether, in wliieh no relative motion can ensue on account of the absence of slip])ing of ether at the surface of separation. But even on Stokes' theory this viscous How of ether would fall ofP at a very rapid rate as we recede from the surface of separation. Michelson and Morley repeated their experi- ment at different heights from the surface of the earth, but invariably obtained the same negative results, thus failing to confirm Stokes' theory of viscous How. HTSTORICAL TNTHODUCTTON TU Loflgt!^ experimevi, — Further, in 1893, Lodge per- formed bis rotating' sphere experiment which showed complete absence of any viscous How of ether due to moviuo' masses of matter. A divided beam of light, after repeated reflections within a ver}^ narrow gap between two massive hemispheres, was allowed to re-unite and thus produce interference bands. When the two hemispheres are set rotating, it is conceivable that the ether in the gap would be disturbed due to viscous flow, and any such flow would be immediately detected by a distru'bance of the interference bands. But actual observation failed to detect the slightest disturbance of the ether in the gap, due to the motion of the hemispheres. Lodge's experi- ment thus seems to show a complete absence of any viscous flow of ether. Apart from these experimental discrepancies, grave theoretical objections were urged against a viscous ether. Stokes himself had shown that his ether must be incom- pressible and all motion in it differentially irrotational, at the same time there should be absolutely no slipping at the surface of separation. Now all these conditions cannot be simultaneously satisfied for any conceivable material medium without certain very special and arbitrary assump- tions. Thus Stokes' ether failed to satisfy the very motive which had led Stokes to formulate it^ namely, the desirabi- lity of constructing a "physical" medium. Planck offered modified forms of Stokes' theory which seemed capable of being reconciled with the Miehelson-Morley experiment, but required very sjiecial assumptions. The very complexity and the very arbitrariness of these assumptions prevented Planck's ether from attaining any degree of practical importance in the further development of the subject. The sole criterion of the value of any scientific theory must ultimately be its capacity for offering a simple. Vlll PRINCIPLE OF RELATIVITY unified^ coherent and fruitful description of observed facts. In proportion as a theory becomes complex it loses in usefulness — a theory which is obliged to requisition a whole array of aibitrary assumptions in order to explain special facts is practically worse than useless, as it serves to disjoin, rather than to unite, the several groups of facts. The optical experiments of the last quarter of the nine- teenth century showed the impossibility of constructing a simple ether theory, which would be jsmenable to analytic treatment and would at the same time stimulate funher progress. It should be observed that it could scarcely be shown that no looieallv consistent ether theorv was possible ; indeed ill 1910, H. A. Wilson offered a consis- sent ether ilieor\ which was at least quite neutral with respect to all available optical data. But Wilson's ether is almost whollv nesfative — its onlv virtue beinoj that it does not directly contradict observed facts. Neither any direct conhrmation nor a direct refutation is possible and it does not throw any light on the various optical pheno- mena. A theory like this being practicall}' useless stands self-condemned. We must now consider the problem of relativf motion of ether and matter from the point of view of electrical theory. From 1860 the identitv of lisht as an electromagnetic vector became o-radualh' established as a result of the brilliant '^ displacement current" hypothesis of Clerk Maxwell and his further analytical investigations. The elastic solid ether became gradually transformed into the electromagnetic one. Maxwell succeeded in giving a fairly .satisfactory account of all ordinary optical phenomena and little room was left for any serious doubts as regards the general validity of Maxwell's theory. Hertz's re- searches on dectric waves, first carried out in 1886, succeeded in furnishing a strong experimental conlh-mation HISTORICAL INTRODUCTION II of Maxwell's theory. Electric waves behaved generally like light waves of very large wave length. The orthodox Maxwellian view located the dielectric polarisation in the electromagnetic ether which was merely a transformation of Fresnel's stag-nant ether. The mag- netic polarisation was looked upon as wholly secondary in origin, being due to the relative motion of the dielectric tubes of polarisation. On this view the Fresnelian con- vection coefficient comes out to be i, as shown by J. J. Thomson in 1880, instead of 1 — ^//x- as required by optical experiments. This obviously implies a complete failure to account for all those optical experiments which depend for their satisfactory explanation on the assumption of a value for the convection coefficient equal to 1 — V/*^' The modifications proposed independently by Hertz and Heaviside fare no better."^ They postulated the actual medium to be the seat of all electric polarisation and further emphasised the reciprocal relation subsisting between electricity and magnetism, thus making the field equations more symmetrical. On this view the whole of the polarised ether is carried away by the moving medium, and consequently, the convection co-efficient naturally becomes unity in this theory, a value quite as discrepant as that obtained on the original Maxwellian assumption. Thus neither Maxwell's original theory nor its subse- quent modifications as developed by Hertz and Heaviside succeeded in obtainiuii; a value for Fresnelian co-efficient equal to 1— V/^^j ^^^ consequently stood totall3^ condemned from the optical point of view. Certain direct electromagnetic experiments invohing relative motion of polarised dielectrics were no less conclu- sive against the generalised theory of Hertz and Heaviside. * See Note 1. X PRINCIPLE OF RELATIVITY According to Hertz a moving dielectric would carry away the whole of its electric displacement with it. Hence the electromagnetic effect near the moving dielectric would be proportional to the total electric displacement, that is to K, the specific inductive capacity of the dielectric. In )901, Blondlot working with a stream of moving gas could not detect any such effect. H. A. Wilson repeated the experiment in an improved form in 1903 and working with ebonite found that the observed effect was pro- portional to K — 1 instead of to K. For gases K is nearly equal to 1 and hence practically no effect will be observed in their case. This gives a satisfactory explanation of Blondlot's negative results. Rowland had shown in 1876 that the magnetic force due to a rotating condenser (the dielectric remaining stationary) was proportional to K, the sp. ind. cap. On the other hand, Rontgen found in 1888 the magnetic effect due to a rotating dielectric (the condenser remain- ing stationary) to be proportional to K— 1, and not to K. Finally Eichenwald in 1903 found that when both condenser and dielectric are rotated together, the effect observed was quite independent of K, a result quite consistent with the two previous experiments. The Row- land effect proportional to K, together with the opposite Rontgen effect proportional to 1 — K, makes the Eichenwald effect independent of K. All these experiments together with those of Blondlot and Wilson made it clear that the electromagnetic effect due to a moving dielectric was proportional to K— 1, and not to K as required by Hertz's theory. Thus the .above group of experiments with moving dielectrics directly contradicted the Hertz- Heaviside theory. The internal discrepancies inherent in the classic ether theory had now become too prominent. It was clear that the HISTORICAL INTRODUCTION XI ether concept had finally outgrown its usefulness. The observed fleets had become too contradictory and too heterogeneous to be reduced to an organised whole with the help of the ether concept alone. Radical departures from the classical theory had become absolutely necessary. There were several outstandmg difficulties in connec- tion with anomalous dispersion, selective reflection and selective absorption which could not be satisfactory explained in the classic electromagnetic theory. It was evident that the assumption of some kind of discreteness in the optical meduim had become inevit- able. Such an assumption naturally gave rise to an atomic theory of electricity, namely, the modern electron theory. Lorentz had postulated the existence of electrons so early as 1878, but it was not until some years later that the electron theory became firmly established on a satisfac- tory basis. Lorentz assumed that a moving dielectric merely carried away its own '' polarivsation doublets," which on his theory gave rise to the induced field proportional to K— 1. The field near a moving dielectric is naturally proportional to K — 1 and not to K. Lorentz's theory thus gave a satisfactory explanation of all those experiments with moving dielectrics which required effects proportional to K — 1. Lorentz further succeeded in obtaining a value for the Fresnelian convection coefficient equal to 1 — ^//a^, the exact value required by all optical experiments of the moving type. We must now go back to Michelson and Morley's experiment. We have seen that both parts of the beam are situated in free ether ; no material meduim is involved in any portion of the paths actually traversed by the beam. Consequently no compensation due to Fresnelian convection Xll ^PRINCIPLE OP RELATIVITY of ether by moving medium is possible. Thus Presneliao convection compensation can have no possible application in this ease. Yet some marvellous compensation has evidently tai^en place which has completely masked the " absolute '"' velocity of the earth. In Miphelson and Morley^s experiment, the distance travelled by the beam along OA (that is, in a direction parallel to the motion of the platform) is 2/^^, while the distance travelled by the beam along OB, perpendicular to the direction of motion of the platform, is ^lip. Yet the most careful experiments showed, as Eddington says, " that both parts of the beam took the same time as tested by the interference bands produced. It would seem that OA and OB could not really have been of the same length ; and if OB was of length I, OA must have been of length IjP. The apparatus was now rotated through 90°, so that OB became the up-stream. The time for the two journeys was again the same, so that OB must now be the shorter length. The plain meaning of the experiment is that both arms have a length I when placed along 0^ (perpendicular to the direc- tion of motion), and automatically contract to a length Ijpf when placed along 0/ (parallel to the direction of motion). This explanation was first given by Fitz-Gerald." This Fitz-Gerald contraction^, startling enough in itself, does not suffice. Assuming this contraction to be a real one, the distance travelled with respect to the ether is %lp and the time taken for this journey is 2l^/c. But the distance travelled with respect to the platform is always 21. Hence the velocity of light with respect to the plat- form is 21/ — ^ —c/^, a variable quantity depending on the " absolute " velocity of the platform. But no trace of such an effect has ever been found. The velocity of light is always found to be quite independent of the velocity HISTOBICAL INTRODUCTION XUl of the platform. The present difficulty cannot be solved by any further alteration in the measure of space. The only recourse left open is to alter the measure of time as well, that is, to adopt the concept of "local time." If a mov- inoj clock goes slower so that one 'real' second becomes 1/^ second as measured in the moving system, the velocity of light relative to the platform will always remain c. We must adopt two very startling hypotheses, namely, the Fitz -Gerald contraction and the concept of "local time," in order to give a satisfactory explanation of the Miehelson-Morley experiment. These results were already reached by Lorentz in the course of further developments of his electron theory. Lorentz used a special set of transformation equations"^ for time which implicitly introduced the concept of local time. But he himself failed to attach any special significance to it, and looked upon it rather as a mere mathematical artifice like imaginary quantities in analysis or the circle at infinity in projective geometry. The originality of Einstein at this stage consists in his successful physical interpretation of these results, and viewing them as the coherent organised consequences of a single general principle. Lorentz established the Relativity Theoremt (consisting merely of a set of transformation equations) while Einstein generalised it into a Universal Principle. In addition Einstein introduced fundamentally new concepts of space and time, which served to destroy old fetishes and demanded a wholesale revision of scientific concepts and thus opened up new possibilities in the synthetic unification of natural processes. Newton had framed his laws of motion in such a way as to make them quite independent of the absolute velocity * See Note 2. t See Note 4. XIV PRINCIPLE or RELATIVITY of the earth. Uniform relative motion of ether and matter could not be detected with the help of dynamical laws. According to Einstein neither could it be detected with the help of optical or electromagnetic experiments. Thus the Einsteinian Principle of Relativity asserts that all physical laws are independent of the ^absolute' velocity of an observer. For different systems, the form of all physical laws is conserved. If we chose the velocity of light"^ to be the fundamental unit of measurement for all observers (that is, assume the constancy of the velocity of light in all systems) we can establish a metric ^^ one — one ^' correspondence between any two observed systems, such correspondence depending only the relative velocity of the two systems. Einstein's Relativity is thus merely the consistent logical application of the well known physical principle that we can know nothing but relative motion. In this sense it is a further extension of Newtonian Relativity. On this interpretation, the Lorentz- Fitzgerald contrac- tion and "local time" lose their arbitrary character. Space and time as measured by two different observers are natur- ally diverse, and the difference depends only on their relative motion. Both are equally valid; they are merely different descriptions of the same physical reality. This is essentially the point of view adopted by Minkowski. He considers time itself to be one of the co-ordinate axes, and in his four- dimensional world, that is in the space-time reality, relative motion is reduced to a rotation of the axes of reference. Thus, the diversity in the measurement of lengths and temporal rates is merely due to the static difference in the " frame- work ^' of the different observers. The above theory of Relativity absorbed practically the whole of the electromagnetic theory based on the * See Notes 9 and 12. HISTORICAL INTRODUCTION XV Maxwell-Lorentz system of field equations. It combined all the advantages of classic Maxwellian theory together with an electronic hypothesis. The Lorentz assumption of polarisation doublets had furnished a satisfactory explana- tion of the Fresnelian convection of ether, but in the new theory this is deduced merely as a consequence of the altered concept of relative velocity. In addition, the theory of Relativity accepted the results of Michelson and Morley's experiments as a definite principle, namely, the principle of the constancy of the velocity of light, so that there was nothing left for explanation in the Michelson-Morle3^ experiment. But even more than all this, it established a single general principle which served to connect together in a simple coherent and fruitful manner the known facts of Physics. The theory of Relativity received direct experimental confiimation in several directions. Repeated attempts were made to detect the Lorentz-Fitzgerald contraction. Any ordinary physical contraction will usually have observable physical results ; for example, the total electrical resistance of a conductor will diminish. Trouton and Noble, Trouton and Rankine, Rayleigh and Brace, and others employed a variety of different methods to detect the Lorentz- Fitzgerald contraction, but invariably with the same negative results. Whether there is an ether or not, uniform velocity ivith respect to it can never he detected. This does not prove that there is no such thing as an ether but certainly does render the ether entirely super- fluous. Universal compensation is due to a change in local units of length and time, or rather, being merely different descriptions of the same reality, there is no compensation at all. There was another group of observed phenomena which could scarcely be fitted into a Newtonian scheme of XVI PRINCIPLE OF RELATIVITY dynamics without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that the " mass '^ of an electron dei)ended on its velocity. So early as 1881, J. J. Thomson had shown that the inertia of a charged })article increased with its velocity. Abraham now deduced a formula for the variation of mass with velocity, on the hypothesis that an electron always remain- ed a rigid sphere. Lorentz proceeded on the assumption that the electron shared in the Lorentz-Fitz2:erald eontrae- tion and obtained a totally di:fferent formula. A very careful series of measurements carried out independently b}^ Biicherer, Wolz, Hupka and finally Neumann in 1913, decided conclusively in favour of the Lorentz formula. This "contractile^"' formula follows immediately as a direct consequence of the new Theory of Relativity, without any assumption as regards the electrical origin of inertia. Thus the complete agreement of experimental facts witli the predictions of the new theory must be considered as confirming it as a principle which goes even beyond the electron itself. The greatest triumph of this new theory consists, indeed, in the fact that a large number of results, which had formerly required all kinds of special hypotheses for their explanation, are now deduced very simply as inevitable consequences of one single general principle. We have now traced the history of the development of the restricted or special theory of Relativity, which is mainly concerned with optical and electrical phenomena. It was first offered by Einstein in 1905. Ten years later, Einstein formulated his second theory, the Generalised Principle of Relativity. This new theory is mainly a theory of gravitation and has very little connection with optics and electricity. In one sense, the second theory is indeed a further generalisation of the restricted princijole, but the former does not really contain the latter as a special ease. HISTORICAL INTRODUCTION Xvii Einstein's first theory is restricted in the sense that it only refers to uniform reetiliniar motion and has no appli- cation to any kind of accelerated movements. Einstein in his second theory extends the Relativity Principle to cases of accelerated motion. If Relativity is to be universally true, then even accelerated motion must be merely relative, motion tjetioeen matter and matter. Hence the Generalised Principle of Relativity asserts that " absolute " motion cannot be detected even with the help of gravitational laws. All movements must be referred to definite sets of co-ordinate axes. If there is any change of axes, the numerical magnitude of the movements will also chano'e. But according to Newtonian dynamics, such alteration in physical movements can only be due to the effeet of ceitain forces in the tield.^ Thus any change of axes will introduce new '• geometrical" forces in the field which are quite independent of the nature of the body acted on. Gravitation- al forces also have this same remarkable property, and gravitation itself may be of essentially the same nature as these '^ geometrical" forces introduced by a change of axes. This leads to Einstein's famous Principle of Equivalence. A gravitational field of force is strictl/j equivole^it to one introduced tjy a transformation of co-ordinates and no possitjle experiment can distinguish fjetween the tioo. Thus it may become possible to " transform away '' gravitational effects, at least for sufficiently small regions of space, by referring all movements to a new set of axes. This new "framework" may of course have all kinds of very complicated movements when referred to the old Galilean or *' rectangular unaccelerated system of co-ordinates." But there is no reason why we should look upon the Galilean system as more fundamental than any other. If it * Note A. XVlll PEIXCIPLE OF EELATIYITY is found simpler to refer all motion in a gravitational field to a special set of co-ordinates, we may certainly look upon this special ^'framework" (at least for the particular region concerned), to be more fundamental and more natural. We may, still more simply, identify this particular framework with the special local properties of space in that region. That is, we can look upon the effects of a gravitational field as simply due to the local properties of space and time itself. The very presence of matter implies a modification of the characteristics of space and time in its neighbour- hood. As Eddington saj^s ^' matter does uot cause the curvature of space-time. It is the curvature. Just as light does not cause electromagnetic oscillations ; it is the oscillations." We may look upon this from a slightly different point of view. The General Principle of Relativity asserts that all motion is merely relative motion between matter and matter, and as all movements must be referred to definite sets of co-ordinates, the ground of any possible framework must ultimately be material in character, it /v convenient to take the matter actually present in a field as the fundamental ground of our framework. If this is done, the special characteristics of our framework would naturally depend on the actual distribution of matter in the field. But physical space and time is completely defined by the •' framework." In other words the '' framework " itself is space and time. Hence w^e see how pit i/sical space and time is aetuallv defined bv the local distribution of matter. There are certain magnitudes which remain constant by any change of axes. In ordinary geometry distance between two points is one such magnitude ; so that hx'^ +^^^ H-5,e'^ is an invariant. In the restricted theory of light, the principle of constancy of light velocity demands that 8ir2 +8^^ -|.8^2 __^2g^,2 should remain constant. HISTORICAL INTUODUCTION XIX The 'Sejjaration ds of adjacent events is defined by ds'^ = —(Lv^ —di/'^ —dz" -\-c^dt^ , It is an extension of the notion of distance and this is the new invariant. Now if Xy ijy Zy t are Iransformed to any set of new variables ji'j, ti'g, i'g, x^, we shall get a quadratic expression for ds^ =y J j.r J 2 H- 2-7j 2=^'i'''2 + • • • = >'J i .i'V i ^Vj where the ^^s are functions of d'^, x^, .^'3, ii\ depending on the transforma- tion. The special properties of space and time in any region are defined by these r/s which are themselves determined, by the actual distribution of matter in the locality. Thus from the Newtonian point of view, these //'s represent the gravitational effect of matter while from the Relativity stand-point, these mereh' define the non-Newtonian (and incidentally non-Euclidean) spice in the neighbourhood of matter. We have seen that Einstein's theory requires local curvature of space-time in the neighbourhood of matter. Such altered characteristics of space and time give a satisfactory explanation of an outstanding discrepancy in the observed advance of perihelion of Mercury. The large discordance is almost completely removed by Einstein's theory. Again, in an intense gravitational field, a beam of light will be affected by the local curvature of space, so that to an observer who is referring all phenomena to a Newtonian system, the beam of light will appear to deviate from its path along an Euclidean straight line. This famous prediction of Einstein about the deflection of a beam of light by the sun's gravitational field was tested during the total solar eclipse of May, 1919. The observed deflection is decisively in favour of the Generalised Theory of Relativity. XX PRINCIPLE OF RELATIVITY It should be uotecl however that the veloeitv of li^ht itself would decrease in a gravitational field. This may appear at first sight to be a violation of the principle of constancy of light-velocity. But when we remember that the Special Theory is explicitly restricted to the case of unaecelerated motion, the difficulty vanishes. In the absence of a gravitational field, that is in any unaecelerated system, the velocity of light will always remain constant. Thus the validity of the Special Theory is completely preserved within its own restricted field. Einstein has proposed a third crucial test. He has predicted a shift of spectral lines towards the red, due to an intense gravitational potential. Experimental difficulties are very considerable here, as the shift of spectral lines is a complex phenomenon. Evidence is conflicting and nothing conclusive can yet be asserted. Einstein thought that a gravitational displacement of the Fraunhofer lines is a necessary and fundamental condition for the acceptance of his theorv. But Eddino'ton has pointed out that even if this test fails, the logical conclusion would seem to be that while Einstein's law of gravitation is true for matter in bulk, it is not true for such small material systems as atomic oscillator. CONCLI SIGN From the conceptual stand-point there are several important consequences of the Generalised or Gravitational Theory of Relativity. Physical space-time is perceived to be intimatel}' connected with the actual local distribution of matter. Euclid-Newtonian space-time is itot the actual space-time of Physics, simply because the former completely neglects the actual presence of matter. Euclid-Newtonian continuum is merely an abstraction, while physical space- time is the actual framework which has some definite HISTORICAL INTRODUCTION XXI curvature due to the presence of matter. Gravitational Theory of Relativity thus brings out clearly the funda- mental distinction between actual physical space-time (which is non-isotropie and non-Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian continuum (which is homogeneous, isotropic and a purely intellectual construc- tion) on the other. The measurements of the rotation of the earth reveals a fundamental framework which may be called the ^' inertial framework." This constitutes the actual physical universe. This universe approaches Galilean space-time at a great distance from matter. The properties of this physical universe may be referred to some world-distribution of matter or the "inertial frame- work" may be constructed by a suitable modification of the law of gravitation itself. In Einstein's theory the actual curvature of the ** inertia! framework " is referred to vast quantities of undetected world-matter. It has interesting consequences. The dimensions of Einsteinian universe would depend on the quantity of matter in it ; it would vanish to a point in the total absence of matter. Then again curvature depends on the quantity of matter, and hence in the presence of a sufficient quantity of matter space- time may curve round and close up. Einsteinian universe will then reduce to a finite system without boundaries, like the surface of a sphere. In this " closed up " system, light rays will come to a focus after travelling round the universe and we should see an ''anti-sun'"' (corresponding to the back surface of the sun) at a point in the sk}^ opposite to the real sun. This anti-sun would of course be equally large and equally bright if there is no absorption of hght in free space. In de Sitter's theory, the existence of vast quantities of world-matter is not required. But beyond a definite XXll PRINCIPLE OF RELATIVITY distance from an observer^ time itself stands still, so that to the observer nothing can ever " happen " there. All these theories are still highly speculative in character, but they have certainly extended the scope of theoretical phj^sics to the central problem of the ultimate nature of the universe itself. One outstanding peculiarity still attaches to the concept of electric force — it is not amenable to any process of being " transformed awav " bv a suitable change of framework. H. Weyl, it seems, has developed a geometrical theory (in hyper-space) in which no fundamental distinction is made between gravitational and electrical forces. Einstein's theory connects up the law of gravitation with the laws of motion, and serves to establish a very intimate relationship between matter and physical space- time. Space, time and matter (or energy) were considered to be the three ultimate elements in Physics. The restricted theory fused space-time into one indissoluble whole. The generalised theory has further synthesised space-time and matter into one fundamental physical reality. Space, time and matter taken separatel}" are more abstractions. Physical reality consists of a synthesis of all three. P. C. Mahalanobis. HISTORICAL INTRODUCTION XXlll Note A. For example consider a massive particle resting on a circular disc. If we set the disc rotating, a centrifugal force appears in the field. On the other hand, if we transform to a set of rotating axes, we must introduce a centrifugal force in order to correct for the change of axes. This newly introduced centrifugal force is usually looked upon as a mathematical fiction — as '' geometrical" rather than physical. The presence of such a geometrical force is usually interpreted us being due to the adoption of a fictitious framework. On the other hand a gravitational force is considered quite real. Thus a fundamental distinction is made between geometrical and gravitational forces. In the General Theory of Relativity, this fundamental distinction is done away with. The very possibility of distinguishing between geometrical and gravitational forces is denied. All axes of reference may now be regarded as equally valid. In the Restricted Theory, all '^unaccelerated" axes of reference were recognised as equally valid, so that physical laws were made independent of uniform absolute velocity. In the General Theory, physical laws are made independent of "absolute" motion of any kind. On The Electrodynamics of Moving Bodies BY A. EjNSTEIJf. INTRODUCTION. It is well known that if we attempt to apply Maxwell's electrodynamics, as conceived at the present time, to moving bodies, we are led to assy met ry which does not ao^ree with observed phenomena. Let us think of the mutual action between a magi-net and a conductor. The observed phenomena in this case depend only on the relative motion of the conductor and the magnet, while according to the usual conception, a distinction must be made between the cases where the one or the other of the bodies is in motion. If, for example, the magnet moves and the conductor is at rest, then an electric field of certain energy-value is produced in the neighbourhood of the magnet, which excites a current in those parts of the field where a conductor exists. But if the magnet be at rest and the conductor be set in motion, no electric field is produced in the neighbourhood of the magnet, but an electromotive force which corresponds to no energy in itself is produced in the conductor; this causes an electric" current of the same magnitude and the same career as the electric force, it being of course assumed that the relative motion in both of these cases is the same. il PRINCIPLE OF RELATIVITY *2. Examples of a similar kind such as the uusueeessful attempt to substantiate the motiou of the earth relative to the " Light-medium " lead us to the supposition that not only in mechanics, but also in electrodynamics, no properties of observed facts correspond to a concept of absolute rest: but that for all coordinate svstems for which the mechanical equations hold, the equivalent electrodyna- mieal and optical equations hold also, as has already been shown for magnitudes of the first order. In the following we make these assumptions (w^hich we shall subsequently call the Principle of Relativity) and introduce the further assumption, — an assumption which is at the first sight quite irreconcilable with the former one — that light is propagated in vacant space, with a velocity c which is independent of the nature of motion of the emitting bod}'. These tw^o assumptions are quite sufficient to give us a simple and consistent theor^^ of electrodynamics of movino' bodies on the basis of the Maxwellian theory for a t,' bodies at rest. The introduction of a ^^ Lightather" will be proved to be superfluous, for according to the conceptions which will 'be developed, we shall introduce neith er a space absolutely at rest, and endowed with special properties, nor shall we associate a velocity -vector with a point in which electro-magnetic processes take place. 3. Like every other theory in electrodynamics, the theory is based on the kinematics of rigid bodies; in the enunciation of every theory, Ave have to do with relations betw^een rigid bodies (co-ordinate system), clocks, and electromagnetic processes. An insufficient consideration of these circumstances is the cause of difficulties with which the electrodynamics of moving bodies have to fight at present. ON THE ELECTKODYXA MlCS Oh' 3I0VlNa BODIES 3 I.-KINEMATIOAL PORTION. § 1. Definition of Synchronism. Let us have a eo-ordinate system, in wliieh the New- tonian equations hold. For distinguishing this system from another which will be introduced hereafter, we shall always call it " the stationary system," If a material point be at rest in this system, then its position in this system can be found out by a measuring rod, and can be expressed by the methods of Euclidean Geometry, or in Cartesian co-ordinates. If we wish to describe the motion of a material point, the values of its coordinates must be expressed as functions of time. It is always to be borne in mind that sicc/i a ■ *• atliemaiical (lefinition has a physical senses only lohen loe have a clear )iotio7i of what is meant by time. We have to fake into consideration the fact that those of our conceptions^ in lohich time plays a part, are alioays conceptions of synchronism For example, we say that a train arrives here at 7 o'clock ; this means that the exact pointing of the little hand of my watch to 7, and the arrival of the train are synchronous events. It may appear that all difficulties connected with the definition of time can be removed when in place of time, we substitute the position of the little hand of my watch. Such a definition is in fact sufficient, when it is required to define time exclusively for the place at which the clock is stationed. But the definition is not sufficient when it is required to connect by time events taking place at different stations,-— -or what amounts to the same thing,- — to estimate by means of time (zeitlich werten) the occurrence of events, which take place at stations distant from the clock. 4 PKINCIPLE OF RELATIVITY Now with regard to this attempt; — the time-estimation of events^ we can satisfy ourselves in the following manner. Suppose an observer — who is stationed at the origin of coordinates with the clock — associates a ray of light which comes to him through space, and gives testimony to the event of which the time is to be estimated, — with the corresponding position of the hands of the clock. But such an association has this defect^ — it depends on the position of the observer provided with the clock, as we know by experience. We can attain to a more practicable result bv the following- treatment. If an observer be stationed at A with a clock, he can estimate the time of events occurring in the immediate neighbourhood of A, by looking for the position of the hands of the clock, which are syrchronous with the event. If an observer be stationed at B with a clock, — we should add that the clock is of the same nature as the one at A, — he can estimate the time of events occurring about B. But without further premises, it is not possible to compare, as far as time is concerned, the events at B with the events at A. We have hitherto an A-time, and a B-time, but no time common to A and B. This last time {i.e., common time) can be defined, if we establish by definition that the time which Hght requires in travelling from A to B is equivalent to the time which light requires in travelling from B to A. For example, a ray of light proceeds from A at xl-time t towards B, arrives and is reflected from B at B-time t and returns to A at A-time t' . Accordin£c to the definition, both clocks are synchronous^ if t - 1 = t' - t . B A A B 02^ THE ELECTRODYNAMICS OF MOVING BODIES 5 We assume tbal this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold : — 1. If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B. 2. If the clock at A as w^ell as the clock at B are both synchronous with the clock at C, then the clocks at A and B are svnchronous. Thus with the help of certain physical experiences, w^e have established what we understand when we speak of clocks at rest at different stations, and synchronous with one another ; and thereby we have arrived at a definition of synchronism and time. In accordance with experience we shall assume that the magnitude 2 AB 77 ~^ =zc, where c is a universal constant. A A " We have defined time essentially w^ith a clock at rest in a stationary system. On account of its adaptability to the stationary system, we call the time defined in this way as " time of the stationary system.'^ § 2. On the Relativity of Length and Time. « The following reflections are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following w^ay :— 1. The laws according to which the nature of physical systems alter are independent of the manner in which these changes are referred to two co-ordinate systems 6 PRINCIPLE or- RELATIVITY which have a uniform translatorv motion relative to each other. 2. Every ray of light moves ^ in the '^^ stationary co-ordinate system " with the same velocity c-j the velocity being independent of the condition whether this ray of light is emitted by a bod}^ at rest or in motion.^' Therefore , .. Path of Li. ON THE ELBCTIIODYNAMICS OF MOVING BODIES I length of the rod in the moving system " i^ equal to the length^/ of the rod in the station aiy system. The leno-th which is foand out bv the second method, may be called * f^fe length of the moving rod 'measured from the sfatiomr^ si/dem/ This leni^th is to be estimated on the basis of our principle, and we shall find it to he different from I. In the generally recognised kinematics, we silently assume that the lengths defined by these two operations are equal, or in other words, that at an epoch of time t, a moving rigid body is geometrically replaceable by the same body, which can replace it in the condition of rest. Relativity of Time. Let us suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A, and B of a rod, i.e., the time of the clocks correspond to the time of the stationary system at the points where they happen to arrive ; these clocks are therefore synchronous in the stationary system. We further imagine that there are two observers at the two watches, and moving with them, and that these observers apply the criterion for synchronism to the two clocks. At the time ^ , a ray of light goes out fi^m A, is. reflected from B at the time t , and arrives back at A at B^ time t' . Taking into consideration the principle of^ A constancy of the velocity of light, we have and t - B ■f = A '^B c-v' t' ■ A -t = B r AB b PRINCIPLE OF RELATIVITY where r is the lens^th of the movins^ rod, measured in the stationary system. Therefore the observers stationed with the watches will not find the clocks Fj-nchrouous, thoiio-h the observer in the stationarv system must declare the clocks to be svnehronous. We therefore see that we can attach no absolute signiticanee to the concept of synchro- nism ; but two events which ara synchronous v»dien viewed from one system, will not be synchronous when viewed from a system movin<^ relatival v to this svstem. § 3. Theory of Co-prdinate and Time- Transformation from a stationary system to a system which moves relatively to this with uniform velocity. Let there be sjiven, in the stationarv svstem two co-ordinate systems, I.e., two series o{" three mutually perpendicular lines issuing from a point. Let the X-axes of each coincide with one another, and the Y and Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be given to each of the systems, and let the rods and clocks in each be exactly alike each other. Let the initial point of one of the sj^stems (k) have a constant velocity in the direction of the X-axis of the other which is stationary system K, the motion being also communicated to the rods and clocks in the system (k). Any time t of the stationary system K corresponds to a definite position of the axes of the moving system, which are always parallel to the axes of the stationary system. By I, we alwaj^s mean the time in the stationaiy system. We suppose that the space is measured by the stationary measuring rod placed in the stationary system, as well as by the moving measuring rod placed in the moving ON THE ELECTRODYNAMICS OF MOVING BODIES 9 system, and we thus obtain the co-ordinates (3c,y^z) for the stationary system, and (^, yy, ^) for the moving system. Let the time t be determined for each point of the stationary system (which are provided with clocks) by means of the •clocks which are placed in the stationary system, with the help of light-signals as described in § 1. Let also the time t of the moving^ svstem be determined for each point of the moving system (in which there are clocks which are at rest relative to the moving system), by means of the method of light signals between these points (in which there ar^^ clocks) in the manner described in § 1. To every value of (r, y, z, t) which fully determines the position and time of^ an event in the static uary system, there correspond-; a system of values {^,y],'C'T) ; now the problem is to find out the system of equations connect- ing these magnitudes. Primarily it is clear that on account of the j^roperty of homogeneity which we ascribe to time and space, the . equations must be linear If we put .r'rrx — ?;^, then it i clear that at a point relatively at rest in the system -J§^,^A^e have a system of values (,(/ y z) which are independent of time. Now let us find out r as a function of (%,y,z,t). For this purpose we have to exp'fess in equations the fact that t is not other than the time given by the clocks which are at rest in the system k which must be made synchron- ous in the manner described in § L Let a ray of light be sent at time r^ from the origin of the system A,- along the- X-axis towards iv' and let it be reflected from that place at time t^ towards the origin of moving co-ordinates and let it arrive there at time t^ ; then we must have 10 PRINCIPLE OF REI ATIVITY If we now introduce the condition that t is a function (?f co-orrdinates, and apply the principle of constancy of the velocity of light in the stationary system, we have i ]t (o, o, 0, t)+T (o, 0, 0, {t+ il— + J!__ [ ) 1 C c—v c-{-v -) / J =T(a;', 0, 0,t + -^ ) C — V /. It is to be noticed that instead of the origin of co- ordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of (0/^,2",^,). A similar conception, being applied to the y- and -s'-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is always ppopogated along those axes with the velocity^c^— i;^, we have the questions ^- =0, ^- =0. . oy oz Prom these equations it follows that t is a linear func- tion of .c'and t. From equations (1) we obtain /, III-' \ where a is an unknown function of v. With the help of these results it is easy to obtain the magnitudes (i,r]X), if we express by means of equations t!ie fact that light, when measured in the moving system is always propagated with the constant velocity c (as the principle of constancy of light velocity in conjunc- tion with the principle of relativity requires). For a I ON THE ELECTRODYNAMICS OF MOVING BODIES 11 time T=Oy if the ray is sent in the direction of increasing ^, we have ^=.c T , i.e. i=:ac i t— — — \, Now the ray of light moves relative to the origin of k with a velocity c— t;, measured in the stationary system ; therefore we have C — V Substituting these values of t in the equation for $, we obtain c2 In an analogous manner, we obtain by considering the ray of light which moves along the ^-axis, 7] = CT = aC I t — J where • , =^, i>;'=^j c c Therefore t?=a ., . y, l=a • ■ z. If for .t;', we substitute its value x—tv, we obtain r}=4> (v) y where S= . - — , and (f> (v)=z — =r«r is a function c2 of V. 12 PRINCIPLE OF RELATIVITY If we make no assumption about the initial position of the moving system and about the null-point of t^ then an additive constant is to be added to the right hand side. We have now to show, that every ray of light moves in the moving system with a velocity c (when measured in the moving system), in case, as we have actually assumed, c is also the velocity in the stationary system ; for we have not as yet adduced any proof in support of the assump- tion that the j)rincip]e of relativity is reconcilable with the principle of constant light-velocity. At a time T = ^ = i> let a spherical wave be sent out ' from the common origin of the two systems of co-ordinates, and let it spread with a velocity c in the system K. If {,c, y, z)y be a point reached by the wave, we have with the aid of our transformation-equations, let us transform this equation, and we obtain by a sin^ple calculation, Therefore the wave is propagated in the moving system with the same velocit}' e, and as a spherical wave.^ Therefore we show that the two principles are mutually reconcilable. In the transformations we have go; an undetermined function <^ (?;), and wo now proceed to find it out. Let us" introduce for this purpose a third co-ordinate system k' , which is set in motion relative to the system h, the motion being parallel to the ^-axis. Let the velocity of the origin be { — v). At the time t = Oy all the initial co-ordinate points coincide, and for t=j=y=zz = o, the time t' of the system k' =^o. We shtill say that {x y' z t') are the co-ordinates measured in the system k' ^ then by a * Yxde Note 9. ON THE ELECTRODYNAMICS OF MOVING BODIES 13 two-fold application of the transformation-equations, we obtain x'=\^v)/S(v)'($+vT)=4>(v)(^v)x, etc. Since the relations between (,(/, ^', z\ f), and (x, y, z, t) do not contain time explicitly, therefore K and k' are relatively at rest. It appears that the systems K and ¥ are identical. Let us now turn our attention to the part of the ^-axis between (^^—o,y] = o,t, = o), and (^=0, ry = l, ^=o). Let this piece of the ^-axis be covered with a rod moving with the velocity v relative to the system K and perpendicular to its axis ; — the ends of the rod having therefore the co-ordinates I Therefore the length of the rod measured in the system K is ~r7~Y For the system moving with velocity (—v), we have on grounds of symmetry, I I cfi{v) {—v) l4 PRINCIPLE OF RELATIVITY / § 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks. Let us consider a rigid sphere {i.e.y one having a spherical figure when tested in the stationary system) of radius R which is at rest relative to the system (K), and whose centre coincides with the origin of ^ then the equa- tion of the surface of this sphere, which is moving with a velocity v relative to K, is ; At time t = Oj the equation is expressed by means of (ar, y, Zy t,) as '13 ( Vi-^J A rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition — when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes K V 1--^, R, R. • Therefore the y and z dimensions of the sphere (there- fore of any figure also) do not appear to be modified by the motion, but the a^ dimension is shortened in the ratio 1 : \'^ 1 ; the shortening is the larger, the larger c is V. ¥oY v = c, all moving bodies, when considered from a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become ON THE ELECTRODYNAMICS OF MOVING BODIES 15 meaningless ; in our theory c plays the part of infinite velocity. It is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system. Let us now consider that a clock which is lying at rest in the stationary sj'stem gives the time t^ and lying at rest relative to the moving system is capable of giving the time t ; suppose it to be placed at the origin of the moving system k, and to be so arranged that it gives the time r. How much does the clock gain, when viewed from the stationary system K ? We have, 1 / ^ \ -, T= — zznzr I ^~"~2^ 15 ^^d x=.vty ...,■,=[._ V.-g Therefore the clock loses by an amount ^-^ per second of motion, to the second order of approximation. From this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense explained in § 3 when viewed from the stationary system. Suppose the clock at A to be set in motion in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A will las: behind the clock v^ which had been all along at B by an amount ^t -g, where t is the time required for the journey. 16 PRINCIPLE OF RELATIVITY We see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide. If we assume that the result obtained for a polygonal line holds also for a curved line, we obtain the following law. If at A, there be two synchronous clocks, and if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in /^-seconds, then after arrival, the last men- tioned clock will be behind the stationary one by \t ~ seconds. From this, we conclude that a clock placed at the equator must be slower by a very &mall amount than a similarly constructed clock which is placed at the pole, all other conditions being identical. § 5. Addition-Theorem of Velocities. Let a point move in the system k (which moves with velocity v along the ^-axis of the system K) according to the equation where w^ and lu are constants. ■n It is required to find out the motion of the point relative to the system K. If we now introduce the system of equations in § 3 in the equation of motion of the point, we obtain aj=_J t, y~ ,0=0. i+_i 1+ « c"" ' c2 ON THE ELECTRODYNAMICS OF MOVING BODIES 17 The law of parallelogram of velocities hold up to the first order of approximation. We can put w and a = tan~^ - . i.e.f a is put equal to the angle between the velocities v, and w. Then we have — a -1 2 u= [(i'2+2i;2+2 vw cos a)— I "■ J I -, . viv cos a c^ It should be noticed that v and 2v enter into the expression for velocity symmetrically, li 2v has the direction of the ^-axis of the nioving system, 1+ "^ ^2 From this equation, we see that by combining two velocities, each of which is smaller than c, we obtain a velocity which is always smaller than c. If we put v=c—Xj *and w—c~\y where x and A are each smaller than c, * IJ=c — 2c-x-A_ <^ It is also clear that the veloeitv of lis^ht c cannot be altered by adding to it a velocity smaller than c. For this ease, U= -^±^ =c. 1+ ''' c^ * Vide Note 12. 3 18 PRINCIPLE OF RELATIVITY We have obtained the formula for U for the ease when V and tv have the same direction; it can also be obtained by combining two transformations according to section § 3. If in addition to the systems K, and k, we intro- duce the system k', of which the initial point moves parallel to the ^-axis with velocity 2v, then between the magnitudes, x, y^ z, t and the corresponding magnitudes of k', we obtain a system of equations, which differ from the equations in §3, only in the respect that in place of V, we shall have to write, (.+.)/( 1+ ^'^ ) We see that such a parallel transformation forms a group. We have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electro- dynamics. II.-ELECTBOBYNAMICAL FART. § 6. Transformation of Maxwell's equations for Pure Vacuum. On the nature of the Electromotive Force caused hy motion in a magnetic field. The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that \ |,[^'Y,^]= a 6 6 9;c 92/ a^ L M N ON THE ELECTRODYNAMICS OF MOVING BODIES 19 and -0 a-rf^''''^^=- a. a.^ a dy a a^ 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 v, we obtain, i I- [X, AY- - N), 13(Z + "i M)] = a a^ dv a^ a^ c c and 1 a^ [L, (3(M+ ^IZ), «N -1-Y)] a^ d_ a_ a^ X y8(Y--N) i8(Z4- -M) c c (2) where /?: vl — i'Vc' The principle of Relativity requires that the Maxwell- Hertzian equations for pure vacuum shall hold also for the system k, if they hold for 'he system K, i.e., for the vectors of the electric and magnetic forces acting upon electric and magnetic masses in the moving system k, 20 PRINCIPLE Oi^ RELATIVITY which are defined by their pondermotive reaction, the same equations hold, ... i.e. ... 1 9 c 'Qi (X', Y', Z') ^ 6^ 6^ 9^ I ■ M \i 1 N' C OT 6 6 6^ 6^' dr; 94 X' Z' ... (3) Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these sj^stems are equivalent to the Maxwell- Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor \l/ (^?), which depends only on v^ and is independent of {^y Vy L ''■)• Hence the relations, [X', y, Z']=4' (v) [X, p (Y- ^'N), 13 (Z+ fM)], c c [h', M', X']=:.A W [L, /^ (M-f ^Z;, /3 (N- ^ Y)]. Then by reasoning similar to that followed in §(3), it can be shown that ^/^(^;) = l. .-. [X\ r, Z'] = [X, p (Y- ^N), 13 (Z+ ^M)] c c [V, W, N'] = [L, 13 (M+ - Z), /3 (N- -^' Y)]. ON THE ELECTRODYNAMICS OF MOVING BODIES 21 For the interpretation of these equations, we make the followini^ remarks. Let us have a point-mass of electricity which is of magnitude unity in the stationary system K, i.e.f it exerts a unit force upon a similar quantity placed at a distance of 1 em. If this quantity of electricity be at rest in the stationary system, then the force acting upon it is equivalent to the vector (X, Y, Z) of electric force. But if the quantity of electricity be at rest relative to the moving system (at least for the moment considered), then the force acting upon it, and measured in the moving system is equivalent to the vector (X', Y', Z'). The first three of equations (1), ('Z), (3), can be expressed in the following way : — ' 1. If a point-mass of electric unit pole moves in an electro-magnetic field, then besides the electric force, an electromotive force acts upon it, which, neglecting the numbers involving the second and higher powers of !;/(?, is equivalent to the vector-product of the velocity vector, and the magnetic force divided by the velocity of light (Old mode of expression). 2. If a point-mass of electric unit pole moves in an electro-magnetic field, then the force acting upon it is equivalent to the electric force existing at the position of the unit pole, which we obtain by the transformation of the field to a co-ordinate system which is at rest relative to the electric unit pole [New mode of expression]. Similar theorems hold with reference to the magnetic force. We see that in the theory developed the electro- magnetic force plays the part of an auxiliary concept, which owes its introduction in theory to the circumstance that the electric and magnetic forces possess no existence independent of the nature of motion of the co-ordinate system. 22 PRINCIPLE OF RELATIVITY v It is further clear that the assymetry mentioned in the introduction which oc-curs when we treat of the current excited by the relative motion of a magnet and a con- ductor disappears. Also the question about the seat of electromagnetic energy is seen to be without any meaning. § 7. Theory of Doppler's Principle and Aberration. In the sj^stem K, at a great distance from the origin of co-ordinates, let there be a source of electrodynamic waves, which is represented with sufficient approximation in a part of space not containing the origin, by the equations : — X=Xo sin ^ "] L=Lo sin ^ Y=Yo sin $ y M=MoSin$ ^ ^=o>(^-^£±!!!:2^±!!!'| Z = Zo sin ^ J N=No sin $ J Here (X^, Yq, Zq) and (Lq, M^, Nq) are the vectors which determine the amplitudes of the train of waves, {Ij Mj n) are the direction-cosines of the wave-normal. Let us now ask ourselves about the composition of these waves, when they are investigated by an observer at rest in a moving medium A- : — By applying the equations of transformation obtained in §6 for the electric and magnetic forces, and the equations of transformation obtained in § 3 for the co-ordinates, and time, we obtain immediately : — X'=Xo sin ^' L' = Lo sin $' Y' = i3/'Yo-.- No") sin' M'=^ Cm.^+ ^ Z^\ sin ^' Z' =:^/'Zo+-Mo') sin<3^' N'=/3 /" No-i' Yo") sin«l>', ON THE ELECTRODYNAMICS OF MOVING BODIES 23 where l- V lv\ u)' = a)^(l-^) , l' = m vi n — n 1 Iv ,(i-'H) ,a-%) From the equation for w' it follows : — If an observer nioves with the velocity v relative to an infinitely distant source of light emitting waves of frequency v, in such a manner that the line joining the source of light and the observer makes an angle of $ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency v which is perceived by the observer is represented by the formula l—cos^ V V V 1- V This is l)op pier's principle for any velocity. If ^—oj then the equation takes the simple form 1 v\-s. V =v 1+ C We see that — contrary to the usual conception — v=oo, for v = —c. If $'=angle between the wave-normal (direction of the ray) in the moving system, and the line of motion of the observer, the equation for I' takes the form cos$— cos ^'= V c 1— -cos c 24 PRINCIPLE OF RELATIVITY This equation expresses the law of observation in its most general form. If $= - , the equation takes the simple form cos $ = — - . We have still to investigate the , amplitude of the waves, which occur in these equations. If A and A' be the amplitudes in the stationarj' and the moving systems (either electrical or magnetic), we have A'2=A' j 1 — - cos I 2 1- ^' c^ If $=o, this reduces to the simple form 1-'-! C A'*=A« 1+^ From these equations, it appears that for an observer, which moves with the velocity c towards the source of light, the source should appear infinitely intense. § 8. Transformation of the Energy of the Rays of Light. Theory of the Radiation-pressure on a perfect mirror. A^ Since ^- is equal to the energy of light per unit volume, we have to regard ^— - as the energy of light in ON THE ELECTRODYNAMICS OF MOVING BODIES 25 A'" the moving system. -— would therefore denote the 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 /, w;,, n are the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface (x — cUy + (y-cmty + (:-~cnfy =11^ which expands with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses, when regarded from the system ^, i.e., the energy of the light-complex relative to the system A;. Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time T=0, the equation : — If S=volume of the sphei-e, S'=volume of this ellipsoid, then a simple calculation shows that : S 'JH cos $ c If E denotes the quantity of light energy measured in the stationary system, E' the quantity measured in the 4 26 PRINCIPLE OP RELATIVITY moving system, which are enclosed by the surfaces mentioned above, then A'' E 8 S' TT 8 S 1— - cos $ c TT If = 0, we have the simple formula : — E' E 1- V 1 + V J It is to be noticed that the energy and the frequency of a light-complex vary according to the same law with the state of motion of the observer. Let there be a perfectly reflecting mirror at the co-or- dinate-plane ^=0, from which the plane-wave considered in the last paragraph is reflected. Let us now ask ourselves about the light-pressure exerted on the reflecting surface and the direction, frequency, intensity of the light after reflexion. Let the incident light be defined b}^ the magnitudes A cos ^, r (referred to the system K). Regarded from A-, we have the corresponding magnitudes : V 1 — COR A' = A a/ J. 2 COS $ — c v COS $' = - COS 4> 1 — - COS 9 I c V =V =.=rr:^ ,2 .\/ 1-^; ON THE ELECTRODYJSAxMICS 0¥ AJOVlNG BODIES 27 For' the reflected light we obtain, when the process is referred to the system k : — A" = A', cos $"= -cos *', v" = v'. By means of a back-transformation to the stationary system, we obtain K, for the reflected light : — 1+ - cos $" 1-2 - cos ^ + — A'" = A" " =A ^ '- ^2 1 ^^ ■N V -s C2 C^' cos $'" = cos4>" + "^ ("H- '^^ cos 4>-2 !^ C \ (''■'J c 1+ 1 ■.„ 1 — 2-cos$H C COS $" c c^ 1+ -cos<^" 1-2 H COS <^ 4-^ / -S ( -I )' 1- \ The amount or energy falling upon the unit surface of the mirror per unit of time (measured in the stationary system) is . The amount of energy going STr{c cos ^—v) away from unit surface of the mirror per unit of time is A'"V?7r {—c cos ^"+v). The difference of these two expressions is, according to the Energy principle, the amount of work exerted, by the pressure of light per unit of time. If we put this equal to P.?*, where P= pressure of light, we have A 2 P = 2 — (cos ^ - 0' Hi)' 28 PKINCIPLE OF EEL.VHV1TY i. » As a first approximatioD^ we obtain A2 P=2 ^ bir coa^ 4>. which is in accordance with facts, and with other theories. All problems of optics of moving bodies can be solved after the method used here. The essential point is, that the electric and magnetic forces of light, which are influenced by a moving body, should be transformed to a system of co-ordinates which is stationary relative to the body. In this way, every problem of the optics of moving bodies would be reduced to a series of problems of the optics of stationary bodies. § 9. Transformation of the Maxwell-Hertz Equations. Let us start from the equations : — u PUx + 6x\ _aN 8M 6^ 7 dy dz 1/ _l9^\ a^i_aL 6 .'.' 6 y 1 6L 6Y 6Z c dt 63 dy laM az ax c dt dx.\ a^ 1 aN_ax aY c dt dy d -v )■ where p=%~ +2— + 4^?- , denotes 47r times the density a.'= a^ a~ of electricity, and {u.,, Uy^ u.) are the velocity-components of electricity. If we now suppose that the electrical- masses are bound unchangeably to small, rigid bodies ON THE IfiLECTHODYNAMlCS 01' MOVING BODIES ^9 (Ions, electrons), then these equations form the electrom^-j^- netic basis of Lorentz's electrodynamics and optics for moving bodies. If these equations which hold in the system K, are transformed to the system k with the aid of the transfor- mation-equations given in § 3 and § 6, then we obtain the equations : — where Uc. ,ax'-i aN' ar J a^ aM' a^ ' a L' a Y' ar a^ az' a^? u ,aY'-i aL' ar J dc aN' a^ ' a M' a z' ar a^ ax' Wc, , az'-] aM' ar J a^ u^ — V aL' dv ' a N' a X' ar a^ aY' a^ ' u y ,(i- ^^) ' 6X' aY'.dZ' = %,"= 6?"*" 9^"*" a? :^(l-t)'" « ,(l-Fii.^) "i, Since the vector U. ic Hy ) is nothing but the velocity of the electrical mass measured in the system A:, as can be easily seen from the addition-theorem of velocities in § 4 — so it is hereby shown, that by taking 30 PRINCIPLE 0¥ RELATIVITY onr kinematical principle as the basis, the electromagnetic basis of Lorentz^s theory of electrodynamics of moving bodies correspond to the relativity-postulate. It can be briefly remarked here that the following important law follows easily from the equations developed in the present section : — if an electrically charged body moves in any manner in space, and if its charge does not change thereby, when regarded from a system moving along with it, then the charge remains constant even when it is regarded from the stationary system K. § 10. Dynamics of the Electron (slowly accelerated). Let us suppose that a point-shaped particle, having the electrical charge e (to be called henceforth the electron) moves in the electromagnetic field ; we assume the following about its law of motion. If the electron be at rest at any definite epoch, then in the next "particle of time,^^ the motion takes place according to the equations df" dt^ df" Where (.r, ^, z) are the co-ordinates of the electron, and m is its mass. • Let the electron possess the velocity z; at a certain epoch of time. Let us now investigate the laws according to which the electron will move in the ^particle of time ^ « immediately following this epoch. Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering, ON THE ELECTRODYNAMICS OF MOVING BODIES 31 the electron is at the origin o£ co-ordinates^ and moves with the velocity v along the X-axis of the system. It is clfear that at this moment (^ = 0) the .electron is at rest relative to the system A-, which moves parallel to the X-axis with the constant velocity v. From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the system k, the electron moves according to the equations dr^ dT^ ' dT"" in the time immediately following the moment, where the symbols (^, 77, I, t, X', Y', Z') refer to the system A'. If we now fix, tliat for t—v = y = z=^0, T = ^=:r; = ^=0, then the equations of transformation given in 3 (and 6) hold, and we have : y _/ With the aid of these equations, we can transform the above equations of motion from the system A- to the system K, and obtain : — dt^ m ^3 ■' di'' m ft \ c ) (A) d\ = 1 i(z+rM) m B \ c 7 dt^ m /5 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 myS' d\c dt'' ■.eX = eX' ^ m/S' ^4-^ =e/3 r dt^ - '^] =^Y' y mp' ^; =e/3 rZ+ ^' mJ =eZ' and let us first remark, that ^X', eY', eZ' are the com- ponents of the ponderomotive force acting upon the electron, and are considered in a moving system which, at this moment, moves with a velocity which is equal to that of the electron. This force can, for example, be measured by means of a spring-balance which is at rest in this last system. If we briefly call this force as ^^the force acting upon the electron," and maintain the equation : — Mass-number x acceleration-number=force-number, and if we further -fix that the accelerations are measured in the stationary system K, then from the above equations, we obtain : — Longitudinal mass = m ( V'- %y # Transversal mass = ■m V^- % Naturally, when other definitions are given of the force and the acceleration, other numlers are obtained for the * Vide Note 21. ON THE ELECTRODYNAMICS OF MOVING BODIES 38 mass ; hence we see that we must proceed very carefully in comparing the different theories of the motion of the electron. We remark that this result about the mass hold also for ponderable material mass ; for in our sense, a ponder- able material point may be made into an electron by the addition of an electrical charo^e which mav be as small as possible. Let us now determine the kinetic energy of the electron. If the electron moves from the origin of co-or- dinates of the system K with the initial velocity steadily along the X-axis under the action of an electromotive force X, then it is clear that the energy drawn from the electrostatic field has the value SelLd>\ Since the electron is only slowly accelerated, and in consequence, no energy is given out in the form of radiation, therefore the energy drawn from the electro-static field may be put equal to the energy W of motion. Considering the whole process of motion in questions, the first of equations A) holds, we obtain : — V V c^ For v=c, W is infinitely great. As our former result shows, velocities exceeding that of light can have no possibility of existence. In consequence of the arguments mentioned above, this expression for kinetic energy must also hold .for ponderable masses. We can now enumerate the characteristics of the motion of the electrons available for experimental verifica- tion, which follow from equations A). 5 34 PRINCIPLE OF RELATIVITY 1. From the second of equations A) ; it follows that an electrical force Y, and a magnetic force N produce equal deflexions of an electron moving with the velocity V, when Y= — . Therefore we see that according to our theory, it is possible to obtain the velocity of an electron from the ratio of the magnetic deflexion Am, and the electric deflexion A^, by applying the law : — ^ =- . A, c This relation can be tested by means of experiments because the velocity of the electron can be directly measured by means of rapidly oscillating electric and mag:netic fields. %. From the value which is deduced for the kinetic energy of the electron, it follows that when the electron falls through a potential difference of P, the velocity v which is acquired is given by the following relation : — 3. We calculate the radius of curvature R of the path, where the only deflecting force is a magnetic force N acting perpendicular to the velocity of projection. From the second of equations A) we obtain : «N These three relations are complete expressions for the law of motion of the electron according to the above theory. ALBRECHT EINSTEIN [^ short hiograpJiical note.~\ The name of Prof. Albreelit Einstein has now spread far beyond the narrow pale of scientific investigators owing to the brilliant confirmation of his predicted deflection of liojht-ravs bv the ^gravitational field of the sun durins: the total solar eclipse of May 29, 1919. But to the serious student of science, he has been known from the beffinnino* of the current century, and many dark problems in physics has been illuminated with the lustre of his genius, before, owing to the latest sensation just mentioned, he flashes out before public imagination as a scientific star of the first magnitude. Einstein is a Swiss-German of Jewish extraction, and began his scientific career as a privat-dozent in the Swiss University of ZUrich about the year 1902. Later on, he migrated to the German Universitv of Prague in Bohemia as ausser-ordentliche (or associate) Professor. In 1914, through the exertions of Prof. M. Planck of the Berlin University, he was appointed a paid member of the Koyal (now National) Prussian Academy of Sciences, on a salary of 18^000 marks per year. In this post, he has only to do and guide research work. Another distinguished occupant of the same post was Van't Hoff, the eminent physical chemist. It is rather difficult to give a detailed, and consistent chronological account of his scientific activities, — they are so variegated, and cover such a wide field. The. first work which sjained him distinction was an investiscation on Brownian Movement. An admirable account will be found in Perrin's book ^The Atoms.' Starting from Boltzmann's 36 PRINCIPLE OF RELATIVITY theorem connecting the entropy, and the probability of a state, he deduced a formula on the mean displacement of small particles (colloidal) suspended in a liquid. This formula gives us one of the best methods for finding out a very fundamental number in physics — namely — the number of molecules in one gm. molecule of gas (Avogadro's number). The formula was shortly afterwards verified by Perrin, Prof, of Chemical Physics in the Sorboniie, Paris. To Einstein is also due the resusciation of Planck's quantum theory of energy-emission. This theory has not yet caught the popular imagination to the same extent as the new theory of Time, and Space, but it is none the less iconoclastic in its scope as far as classical concepts are concerned. It was known for a long time that the observed emission of light from a heated black body did not corrospond to the formula which could be deduced from the older classical theories of continuous emission and propagation. In the year 1900, Prof. Planck of the Berlin University worked out a formula which was based on the bold assumption that energy was emitted and absorbed by the molecules in multiples of the quantity hv^ where // is a constant (which is universal like the constant of gravitation), and v is the frequency of the light. The conception was so radically different from all accepted theories that in spite of the great success of Planck's radiation formula in explaining the observed facts of black-body radiation, it did not meet with much favour from the physicists. In fact, some one remarked jocularly that according to Planck, energy flies out of a radiator like a swarm of gnats. But Einstein found a support for the new-born concept in another direction. It was known that if green or ultraviolet light was allowed to fall on a plate of some alkali metal, the plate lost electrons. The electrons were emitted with ALBERT EINSTEIN 37 all velocities, but there is generally a maximum limit. From the investigations of Lenard and Ladenburg, the curious discovery was made that this maximum velocity of emission did not at all depend upon the intensity of light, but upon its wavelength. The more violet was the light, the greater was the velocity of emission. To account for this fact, Einstein made the bold assumption that the light is propogated in space as a unit pulse (he calls it a Light-cell), and falHng upon an individual atom, liberates electrons according to the energy equation hv=-;^mv^ -\- A, where (iu, v) are the mass and velocity of the electron. A is a constant characteristic of the metal plate. There was little material for the confirmation of this law when it was first proposed (1905), and eleven years elapsed before Prof. Millikan established, by a set of experiments scarcely rivalled for the ingenuity, skill, and care displayed, the absolute truth of the law. As results of this confirmation, and other brilliant triumphs, the quantum law is now regarded as a fundamental law of Energetics. In recent years, X-rays have been added to the domain of light, and in this direction also, Einstein's photo-electric formula has proved to be one of the most fruitful conceptions in Physics. The quantum law was next extended by Einstein to the problems of decrease of specific heat at low temperature, and here also his theory was confirmed in a brilliant manner. We pass over his other contributions to the equation of state, to the problems of null-point energy, and photo- chemical reactions. The recent experimental works of 38 PRINCIPLE OF HELATIVITT Nernst and Warburg seem to indicate that through Einstein's genius, we are probably for the first time having a satisfactory theory of photo-chemical action. In 1915, Einstein made an excursion into Experimental Physics, and here also, in his characteristic way, he tackled one of the most fundamental concepts of Physics. It is well-known that according to Ampere, the magnetisation of iron and iron-like bodies, when placed within a coil carrying an electric current is due to the excitation in the metal of small electrical circuits. But the conception though a very fruitful one, long remained without a trace of experimental proof, though after the discovery of the electron, it was srenerallv believed that these molecular currents may be due to the rotational motion of free electrons within the metal. It is easily seen that if in the process of magnetisation, a number of electrons be set into rotatory motion, then these will impart to the metal itself a turning couple. The experiment is a rather difficult one, and many physicists tried in vain to observe the effect. But in collaboration with de Haas, Einstein planned and successfully carried out this experiment, and proved the essential correctness of Ampere's views. Einstein's studies on Relativity were commenced in the year 1905, and has been continued up to the present time. The first paper in the present collection forms Einstein's first great contribution to the Principle of Special Relativity. We have recounted in the introduction how out of the chaos and disorder into which the electrodynamics and optics of moving bodies had fallen previous to 1895, Lorentz, Einstein and Minkowski have succeeded in building up a consistent, and fruitful new theory of Time and Space. But Einstein was not satisfied with the study of the special problem of Relativity for uniform motion, but ALBERT EINSTEIN 39 tried, in a series of papers beginning from 1911, to extend it to the case of non-uniform motion. The last paper in the present collection is a translation of a comprehensive article which he contributed to the Anualen der Physik in 1916 on this subject, and gives, in his own words, the Principles of Generalized Kelativity. The triumphs of this theory are now mat<^ers of public knowledge. Einstein is now only 45, and it is to be hoped that science will continue to be enriched, for a long time to come, with farther achievements of his genius. INTRODUCTION. At the present time, different opinions are being held about the fundamental equations of Eleetro-dynamics for moving" bodies. The Hertzian^ forms must be given up, for it has appeared that they are contrary to many experi- mental results. In 1895 H. A. Lorentzf published his theory of optical and electrical phenomena in moving bodies; this theory was based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been ushered into existence. In his theory, Lorentz proceeds from certain equations, which must hold at every point of ^'Ather'^; then by forming the average values over *^^ Phy- sically infinitely small " regions, which how^ever contain large numbers of electrons, the equations for electro-mag- netic processes in moving bodies can be successfully built up. In particular, Lorentz's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous " Ather ^' ; it shows that this fact is intimately connected with the covariance of the original equation, when certain simultaneous transformations of the space and time co-ordinates are effected; these transfoi;mations have therefore obtained from H. PoincareJ the name of Lorentz- transformations. The covariance of these fundamental equations, when subjected to tbe Lorentz-transformation is a purely mathematical fact i.e. not based on any physi- cal considerations; I will call this the Theorem of Rela- tivity ; this theorem rests essentially on the form of the * Vid,e Note 1. f Note 2. % Vide Note 3. 3 PRINCIPLE OF RELATIVITY differential equations for the propagation of waves with the velocity of light. Now without recognizing any hypothesis about the con- nection between " Ather " and matter, we can expect these mathematically evident theorems to have their consequences so far extended — 'that thereby even those laws of ponder- able media which are yet unknown may anj^how possess this covariance when subjected to a Lorentz-transformation ; by saying this, we do not indeed express an opinion, but rather a conviction, — and this conviction I may be permit- ted to call the Postulate of Relativity. The position of affairs here is almost the same as when the Principle of Conservation of Energy was poslutated in cases, where the corresponding forms of energy were unknown. Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite con- nection may be styled ' the Principle of Relativity.' These differentiations seem to me to be necessary for enabling us to characterise the present day position of the electro-dynamics for moving bodies. H. A. Lorentz"^ has found out the " Relativity theorem'' and has created the Relativitj^-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law. A. Einstein t has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept which is forced upon us by observation of natural pheno- mena. The Principle of Relativity has not yet been formu- lated for electro-dvnamics of moviug: bodies in the sense * Yiie Note 4. f Note 5. INTRODUCTION 3 characterized by me. "In the present essay, while formu- lating- this principle, I shall obtain the fundamental equa- tions for moving bodies in a sense which is uniquely deter- mined by this principle. But it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle."^ We would at first expect that the fundamental equa- tions which are assumed by Lorentz for moving bodies would correspond to the Relativity Principle. But it will be shown that this is not the case for the general equations which Lorentz has for any possible, and also for magnetic bodies ; but this is approximately the case (if neglect the square of the velocity of matter in comparison to the velocity of light) for those equations which Lorentz here- after infers for non-magnetic bodies. But this latter accordance with the Relativity Principle is due to the fact that the condition of non-mag^netisation has been formula- ted in a way not corresponding to the Relativity Principle; therefore the accordance is due to the fortuitous compensa- tion of two contradictions to the Relalivity-Postulate. But meanwhile enunciation of the Principle in a rigid manner does not signify any contradiction to the hypotheses of Lorentz's molecular theory, but it shall become clear that the assumption of the contraction of the electron in Lorentz^s theory must be introduced* at an earlier stage than Lorentz has actually dene. In an appendix, I have gone into discussion of the position of Classical Mechanics with respect to the Relativity Postulate. Any easily perceivable modification of mechanics for satisfying the requirements of the Relativity theory would hardly afford any noticeable difference in observable processes ; but would lead to rery * See uQtes on § S and 10. 4' PRINCIPLE OF RELATIVITY surprising consequences. By laying down the Relativity- Postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of Energy alone (the form of the Energy being given in explicit forms). NOTATIONS. Let a rectangular system {.r, y, z, t,) of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the unit of length that the velocity of light in space becomes unity. Although I would prefer not to change the notations used by Lorentz^ it appears important to me to use a different selection of symbols, for thereby certain homo- geneity will appear from the very beginning. I shall denote the vector electric force by E,' the magnetic induction by M_, the electric induction by e and the magnetic force by 7n, so that (E, M, »?, m) are used instead of Lorentz's (E, B, D, H) respectively. I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, i.e., instead of operating with {t), I shall operate with {it), where i denotes ^ — \. If now instead of {x, y, z, it), I use the method of writing with indices, certain essential circumstances will come into evidence ; on this will be based a general use of the suffixes (1, 2, 3, ^). The advantage of this method will be, as I expresslj' emphasize here, that we shall have to handle symbols which have apparently a purely real appearance ; we can however at any moment pass to real equations if it is understood that of the symlbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those NOTATIONS which have not at all the suffix 4, or have it twice denote real quantities. An individual system of values of {x, y, Zy t) i. e.^ of {x^ x^ rg Xj^) shall be called a space-time point. Further let u denote the velocity vector of matter, e the dielectric constant, /u, the magnetic permeability, a- the conductivity of matter, while p denotes the density of electricity in space, and s the vector of "Electric Current" which we shall some across in §7 and §8. 5 PRINCIPLE OP RELATIVITY PAET I § 2. The Limiting Case. The Fundcwiental Equations for Ather. By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-d3mamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limitting case e=i, ix=1,(t = o, they should constitute the laws for ponderable bodies. In this ideal limitting case €=1, fji=l, o-=:o, E will be equal to e, and M to m. At every space time point {j-, y^ z, t) we shall have the equations* (i) Curl m— -»- = pu (ii) div e= p (iii) Curl^ +.||' = (iv) div m = (? I shall now write {x^ x^ x^ x ^) for {x^y, z, t) and (/>nP2; ^3; P4) for (pu,, puy, pu,, ip) i.e. the components of the convection current pu, and the electric density multiplied by \/— 1. Further I shall write « for m,, m^, m,, — ie„ — ie , — ie,. i.c.y the components of m and ( — i.e.) along the three axes; now if we take any two indices (h. k) out of the series * See note 9 THE FUNDAMENTAL EQUATIONS FOR ATHER 7 Therefore /s 2 ^^ ~'J 1 3 > ./ 1 3 ~ ~~J Z \i J 2 1^^ ~/ 1 2 ..4 1 — ~Jl 45 ../ 4 4 — ~/2 4J /4 3 " ""/ 3 4 Then the three equations comprised in (i), and the equation (ii) multiplied by / becomes 8Xc 3xj + + ¥. 32 8x g/4t . ?/ Sxj + 42 Sx, + 0/l3 8X3 + S/t4 8X4 ?^2 3 8X3 X S/24 8X4 + 0/34 ^^4 ?A3 8x, ■Pi = P2 = P; = ^4 (A) On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by {i) becomes ¥,, 8xj ?A4 + ^^4 2 , ?/2_3 8X3 8X4 ^14 , ?Al Sx„ ^ + - S/4, 8x, + 8x4 ?/l2 ^3 2 , ?/j_3_ , SXi "^ 8x2 "^ 8/*. 2 1 8x, = = = = (B) By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices. (1,2,3,4). § 3. It is well-known that by writing the equations i) tc iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the 8 PRINCIPLE OF RELATIVITY system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis. through an amount , keeping e, m fixed in space, and introduce new variables x^', cc^ x^ Xi^ instead of X:^ x^ x^ x ^, where x\ •=^x^ cos <^ H-^2 sin ^, ;r'2 = — ^i sin<^ + x^ cos<^, jr' ^ =Xqx\= x^, and introduce magnitudes p\, p\j p s p\, where p^' = p^ cos i> -i- P2 sin<^, p^' = — p^ sin^ + p2 cos<^ *nd/i2, 7^3 4, where /% 3 =A 3 cos (^ + /g 1 sin ,/. 1 r: -/j 3 sin <^ + /'i4=/i4 COS <^ +/24 sin ct>,/\^ - -/,4 sitt -f /2 4 COS ,,/\^=/s4y fu. = -/.A (hlk = 1,2,3,4). then out of the equations (A) would follow a corres- ponding system of dashed equations (A') composed of the newly introduced dashed magnitudes. So upon the ground of symmetry alone of the equa- tions (A) and (B) concerning the sitffiies (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all. I will denote by «V^ a purely imaginary magnitude, and consider the substitution ^i—^\i ^s'=*2> ^^^' = xz cos i\if-\-x^ sin iyj/, (1) ^^4' = — ic, sin ixjf 4- .^4 cos i\^, Putting - i tan i^^ = '\^ "^ _^ = ^' ^^ = 9 ^og jz^r (2) (? -f ^ THE FUNDAMENTAL EQUATIONS FOR ARTHER 9 We shall have cos i\\/ = — , sin z^ = — ■ ^l-q^ x/l-q 2 where — i < q < \, and \/l— ^^ is always to be taken with the positive sign. Let us now write x\=-/j ^o 2=^^' , x ^=z'y x\^=it' (3) then the substitution 1) takes the form ^ =.r, y =y,z ^ , t = , (4) the coefficients being essentially real. If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1^ 2, and by i^, we at once perceive that simultaneously, new magnitudes p\, p'2, p 3, p' 4, where {p\=Pi, P2=P2^ P3=P3 cos ii}/ + P4 sin iif/, p\ = » — Pg sin t\l/ + P4 cos iij/), and/ 12 •••/34. where /4i=/4i cos ^^A +/13 sin ixlf,f\^= -/41 sin «V +/13 e0StlA,/3 4=/3 4,/3 2=/3 2 COS /l/^ 4-/42 siu t'l/^, /42 = -/32 sin ^> + /42 COS ?lA, /12 =/i2^ /*A = -fkky must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A'), and (B'), the new equations being obtained by simply dashing the old set. All these equations can be written in purely real figures, and we can then formulate the last result as follows. If the real transformations 4) are t^en, and ^' y' z' t' be takes as a new frame of reference, then we shall have (5) ■qu^ +1 p =p — • \ , P^^r -p \ ^ZIZZIIl p'uj=pu^, p'uy'=pUy. 10 PRINCIPLE OF RELATIVITY (6) ^j = ?i^i^, ,„V = 2^4^, e.'=e » ' (7) w',' = ■ , e'/ = , m','=m z • VI — q^ VI — q"" Then we have for these newly introduced vectors tc', e', m' (with components %ij , uj , uj \ ej , ^/, ej ) mj, m/, m/)y and the quantity p a series of equations I'), II'), III'), IV) which are obtained from I), II), III), IV) by simply dashing the symbols. We remark here that e^—qmy, ey+qm^ are components of the vector e-\- \_vm'\, where v is a vector in the direction of the positive Z-axis, and i v i=^, and [vfu'] is the vector product of y and W2 ; similarly —qe^-\-myym,,+qey are the components of the vector m—\ye]. The equations 6) and 7), as they stand in pairs, can be expressed as. - eJ-\-i'ni'J=.{e^+im^) cos i\^ + {Cy+imy) sin ix^/, Sy' + im'y' = — (e^+zw,) sin ii(/ + (gy+imy) cos lij/, If (^ denotes any other real angle, we can form the following combinations : — {eJ + im'J) cos. ^+(ey" + zWy') sin <;^ = (e,+/w,) cos. (ct> + i^) + (ey+imy) sin ((j^ + iif/), = (e,' + zW,') sin ^+(ey' + zWy') cos. ^ = — (e:.^-^mJ sin (cfi + iif/) + (ey-\-zmy) cos, (cf> + {\ff). Special libnENTZ Transformation. The role which is played by the Z-axis in the transfor- mation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation SPECIAL LORENTZ TRANSFORMATION ll about this last axis. So we came to a more general law : — Let ?; be a vector with the components v^, Vy, v^, and let \ v \ =q) are to be understood in the sense that with the directions v, and every direction v perpendi- cular to V in the system {x, y, z) are always associated the directions with the same direction cosines in the system [x' y, z), A transformation which is accomplished by means of (10), (11), (12) with the condition 0<^<1 will be called a special Lorentz-transformation. We shall call v the vector, the direction of v the axis, and the magnitude of V the moment of this transformation. If further p and the vectors w', e' , in, in the system {xy'z) are so defined that, 12 PRINCIPLE OF RELATIVITY further (14) (/ + m')^ = ^^ + ''"'^-i^^^ + "'^K Vl — q" (15) {e' 4- iffi'') » = (^ + ^'^^) — i [u, {e + ini)] ^ . Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes. The solution of the equations (10), (11), (12) leads to (U\ r -!Ljl±1!i_ r- =/- t= TL^±L^ V \.—q- Vl — q^ Now we shall make a very important observation about the vectors u and u. We can again introduce the indices 1, 2, 8, 4, so that we write (^/, ^^2? ^3? *^*'4 instead of (,u', ?/'? -') ^'^') a^nd p^', pg'? Ps'? P4' ii^stead of Like the rotation round the Z-axis, the transformation (4), and more geaeraily the transformations (10), (1 1), (12), are also linear transformations with the determinant -|-1, so that (17) x^^+x^^+x^^+x^"" i. e. x^ + y''+z^—t'', is transformed into On the basis of the equations (13), (14), we shall have (p,'+P,'+P,'+P,'')=pHl-u^\-u,\-ur^,)=p'a-u') transformed into p^(l — u^) or in other words, (18) p vr^r:i? is an invariant in a Lorentz-transformation. If we divide (p^, p^, P3, p^) by this magnitude, we obtain the four values (w^, co,, w,, w^^) = . _ {u^, u^, u^, i) VT u so that Wi' +(u,^ +W3' 4-W4* = — 1. It ■'is apparent that these four values, are determined by the vector 10 and inversely the vector it of magnitude SPECIAL LOEENTZ TRANSFORMATION 13 o, and ^1^ +'^2" + ^"3^ +-^4^ transforms into x^'^ +x^'- -{- ,v.^"^ -\-x^"^. The operation shall be called a general Lorentz transformation. If we put aj/=:,c', x^' =y\ ,v^' = z\ x^=^it\ then immediately there occurs a homogeneous linear transfor- mation of («, y, z, t) to (r', y' y z y t') with essentially real co-efficients, whereby the aggregrate — c^ — ^2 _~2 _|_^2 transforms into — ^'f ^ — y' ^ — z"^ -\- 1"^ , and to every such systetn of values ■», y, Zy t with a positive t, for which this aggregate >o, there always corresponds a positive t' ; This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski's notation, i SPACE-TIME VECTORS I^ this last is quite evident from the continuity of the aggregate x, y, z, t. The last vertical column of co-efficients has to fulfil, the condition 22) <^i 4^+^24^ +^34^ +'^4 4^ = 1. If «^^=<3^2^= ^^44) as the last vertical column, — and then every Lorentz- transformation with the same last vertical column (^14^ <^2 4? ^^3 4' '^44) ^^^ ^® supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point. The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes p^, p^, p^, p^) with the coiidition that in case of a Lorentz-transformation it is to be replaced by the set p/, 132', ps\ pA:')i where thes3 are tho value? oO ^c/, v.^\ ,c^', -^iO' obtained by substituting (p^, p}, p.^, p ) for (^-j, x-^, .Vq, ,^4) in the expression (21). Besides the time-space vector of the 1st kind (x^, x^i Xqj v-^) we shall also make use of another space- time vector of the first kind (y^, ^.^,^3, ^4), and let us form the linear combination ^ 023) Aa C*^2 2/3— ''3 2/2)+/si (^3 2/1— ^ 2/3)+ /l2 (^1 2/2— '^z 2/1)+ /li (■^■1 2/4— ^'«4. 2/x) + /24 (''a 2/4—^^4 2/2) + /s* (-''s 2/4—^4 2/3) 16 PRINCIPLE OF RELATIVITY with six coefficients /g 3 — f^^. Let us remark that in the vectorial method o£ writing, this can be constructed out of the four vectors. the constants x^ and y^^ at the same time it is symmetrical with regard the indices (1, 2, 3, 4). If we subject {x^, .c^, ,83, x^) and (2/1, y^, y^, yj simul- taneously to the Lorentz transformation (^21), the combina- tion (23) is changed to. (24) f^s' ('''2 ys'-'^s y^) +/31 (^3' 2/i'--^i'!/3)+/i2 (^.' yJ-^^Jy.') +frJ(^.yJ)-H'y.') +/2.' i'^^' yJ - ''4' 2/2') + /s/ ('^3' yJ—-^J 2/3'), where the coefficients As'^ /a i^ /12'' /i*'? /24'r /s*'. depend solely on (/g 3 /a 4) and the coefficients a^^...a^^. We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes /"^ 3 j/si fziJ with the condition that when subjected to a Lorentz transformation, it is changed to a new system /^ 3' /"g^,... which satis- fies the connection between (23) and (24). I enunciate in the following manner the general theorem of relativity corresponding to the equations (I) — (iv), — which are the fundamental equations for Ather. If ,«, y, z, it (space co-ordinates, and time it) is sub- jected to a Lorentz transformation, and at the same time {pu^^ pUy, pu,, ip) (convection-current, and chnrge density pi) is transformed as a space time vector of the 1st kind, further {m^^ 711^, 1^ ,-, — i(i ^^—ie y^ — ie ,) (magnetic force, and electric induction x (— is transformed as a space time vector of the 2nd kind, then the system of equations (1), (II), and the system of equations (III), • (IV) trans- forms into essentially corresponding relations between the corresponding magnitudes newly introduced info the system. SPECIAL LORENTZ TRANSFORMATION 17 These facts can be more concisely exj^ressed in these words : the system of equations (I, and II) as well as the system of equations (III) (IV) are co variant in all cases of Lorentz-transformation, where (p?^, ip) is to be trans- formed as a space time vector of the 1st kind, {m—ie) is to be treated as a vector of the 2nd kind, or more significantly, — (pfi, ip) is a space time vector of the 1st kind, {vt—ie)^ is a space-time vector of the 2nd kind. I shall add a fe ,v more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation. (0 ^^'-e' = f.l + f,\ + f.\ + /xl + /L + /.I A space-time vector of the second kind (m—ie), where {tn, and e) are real magnitudes, may be called singular, when the scalar square Qni—ieY =o, ie m^ —e"^ =o, and at the same time (?;^ -fiV)=?iL±t^v^ Vide Note. 18 • PRINCIPLE OF RELATIVITY If then, by referring back to equations (9), we carry out the transformation (1) through the angle ^j and a subsequent rotation round the Z-axis through tbe angle <^, we perform a Lorentz-transformation at the end of which ;;^^=o_, ey=o, and therefore m and e shall both coincide with the new Z-axis. Then by means of the invariants m'^—e^, [me) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled. § Concept op Time. By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.^ In fact, the following considerations will prove conclusive. Let a reference system {x,y, z, f^ for space time points (events) be somehow known. Now if a space point A {'^'tiVof ^o) ^^ the time t„ be compared with a space point P ( f, ^, z) at the time fy and if the difference of time t—t^, (let t > to) be less than the length A P i.e. less than the time required for the propogation of light from * Just as being.s which, are confined within a narrow region surrovinding a point on a shperical surface, may fall into the error that a sphere is a geometric figure in which oue diameter is particularly distinguished from the rest. CONCEPT OF TIME 19 A to P, and if ^= " < 1, then by a special Lorentz transformation, in which A P is taken as the axis_, and which has the moment^, we can introduce a time parameter t\ which (see equation 11, 12, § 4) has got the same value t' = o for both space-time points (A, t^), and P, t). So the two events can now be comprehended to be simultaneous. Further, let us take at the same time t„ =o, two different space-points A, B, or three space-points (A, B, C) which are not in the same space-line, and compare therewith a space point P, which is outside the line A B, or the plane A B C^ at another time t, and let the time difference t — t^ (t > t^) be less than the time which light requires for propogation from the line A B, or the plane A B 0) to P. Let q be the quotient of {t — to) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C is the axis, and q is the moment, then all the three (or four) events (A, to), [B, to), (C, t,) and (P, t) are simultaneous. If four space-points, which do not lie in one plane are conceived to be at the same time to, then it is no longer per- missible to make a change of the time parameter by a Lorentz — transformation, without at the same time destroying the character of the simultaneity of these four space points. To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures ot the so-called non-Euclidean Geometr^y, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint. ^0 PRINCIPLE OF RELATIVITY PART II. ELECTRO-MAGNETIC .. PHENOMENA. § 7. Fundamental Equations for bodies AT REST. After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case « = 1, /a = 1, o- = o, let us turn to the electro-magnatic phenomena in matter. We look for those relations which make it possible for us when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every space- time point as functions of {r, y, z, t) : — the vector of the electric force E, the magnetic induction M, the electrical induction iw, C = crE, where c = dielectric constant, /x = magnetic permeability, (T = the conductivity of matter, all given as function of '*■> ^j 2^> ^J ^ is here the conduction current. , By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work, and write ^j, s^^ s^, s^ for C,, C^, C, V _ 1 p, • further/23,/5i,/i,,/i4,/„4,/54 for m,, Wy, m, — i (e., e^, e,), and F33, E31, Fia, F^^, P,^, F,^ forM.,M,,M., -i (E.,E,,E,) lastly we shall have the relation /^ a = -~ >/'> k, F^k •, = — i^^ *, (the letter /, F shall denote the field,