,the ,principle of ,relativity ,,the ,,principle ,,of ,,relativity ,,original ,,papers ,,by ,a4 ,,einstein ,,and ,h4 ,,minkowski ,,translated ,,into ,,english ,,by ,m4 ,n4 ,,saha ,,and ,s4 ,n4 ,,bose ,,lecturers ,,on ,,physics ,,and ,,applied ,,mathematics ,university ,college of ,science1 ,calcutta ,university ,,with ,a ,,historical ,,introduction ,,by ,p4 ,c4 ,,mahalanobis ,,professor ,,of ,,physics1 ,,presidency ,,college1 ,,calcu4 ,,published ,,by ,,the ,,university ,,of ,,calcutta #aibj ,sole ,agents ,r4 ,,cambray ,,co4 ,,printed ,,by ,,atulchandra ,,bhattacharyya1 ,,at ,,the ,,calcutta ,,university ,,press1 ,,senate ,,house1 ,,calcutta ,,table ,,of ,,contents #a4 ,historical ,introduction i-xxiii ,by ,mr4 ,p4 ,c4 ,mahalanobis4 #b4 ,on the ,electrodynamics of ,moving ,bodies #a-cd ,einstein's first paper on the restricted ,theory of ,relativity1 originally published in the ,annalen der ,physik in #aije4 ,translated from the original ,german by ,dr4 ,meghnad ,saha4 #c4 ,albrecht ,einstein #ce-ci ,a short biographical note by ,dr4 ,meghnad ,saha4 #d4 ,principle of ,relativity #a-eb ,h4 ,minkowski's original paper on the restricted ,principle of ,relativity first published in #aiji4 ,translated from the original ,german by ,dr4 ,meghnad ,saha4 #e4 ,appendix to the above by ,h4 ,minkowski #ec-hh ,translated by ,dr4 ,meghnad ,saha4 #f4 ,the ,generalised ,principle of ,relativity #hi-afc ,a4 ,einstein's second paper on the ,generalised ,principle first published in #aiaf4 ,translated from the original ,german by ,mr4 ,satyendranath ,bose4 #g4 ,notes #afe-ahe ,transcriber's ,note3 ,the plain text version of this ebook includes complex mathematical formulas4 ,some are simple in-line expressions like k #a - #a_/b4 ,they may include special notations such as xy for x to the power of y1 xy for x with a subscript of y1 a for an 'a' with a bar across the top1 4a for an 'a' with a dot over it1 44a for an 'a' with two dots over it4 ,others are more complex 8,,ascii ,art0 like this3 l l #blc #bl t ------ ------ -------- --- c - u c u c - u c ,some are so complex that they must be rendered in the ,te,x mathematical notation1 enclosed between double dollar signs1 like this3 beta 7#a - frac u#b;c#b7-;frac#ab ,,historical ,,introduction ,lord ,kelvin writing in #ahic1 in his preface to the ,english edition of ,hertz's ,researches on ,electric ,waves1 says 8many workers and many thinkers have helped to build up the nineteenth century school of plenum1 one ether for light1 heat1 electricity1 magnetism2 and the ,german and ,english volumes containing ,hertz's electrical papers1 given to the world in the last decade of the century1 will be a permanent monument of the splendid consummation now realised40 ,ten years later1 in #aije1 we find ,einstein declaring that 8the ether will be proved to be superfluous40 ,at first sight the revolution in scientific thought brought about in the course of a single decade appears to be almost too violent4 ,a more careful even though a rapid review of the subject will1 however1 show how the ,theory of ,relativity gradually became a historical necessity4 ,towards the beginning of the nineteenth century1 the luminiferous ether came into prominence as a result of the brilliant successes of the wave theory in the hands of ,young and ,fresnel4 ,in its stationary aspect the elastic solid ether was the outcome of the search for a medium in which the light waves may 8undulate40 ,this stationary ether1 as shown by ,young1 also afforded a satisfactory explanation of astronomical aberration4 ,but its very success gave rise to a host of new questions all bearing on the central problem of relative motion of ether and matter4 ,arago's prism experiment4--,the refractive index of a glass prism depends on the incident velocity of light outside the prism and its velocity inside the prism after refraction4 ,on ,fresnel's fixed ether hypothesis1 the incident light waves are situated in the stationary ether outside the prism and move with velocity c with respect to the ether4 ,if the prism moves with a velocity u with respect to this fixed ether1 then the incident velocity of light with respect to the prism should be c u4 ,thus the refractive index of the glass prism should depend on u the absolute velocity of the prism1 i4e41 its velocity with respect to the fixed ether4 ,arago performed the experiment in #ahai1 but failed to detect the expected change4 ,airy-,boscovitch water-telescope experiment4--,boscovitch had still earlier in #agff1 raised the very important question of the dependence of aberration on the refractive index of the medium filling the telescope4 ,aberration depends on the difference in the velocity of light outside the telescope and its velocity inside the telescope4 ,if the latter velocity changes owing to a change in the medium filling the telescope1 aberration itself should change1 that is1 aberration should depend on the nature of the medium4 ,airy1 in #ahga filled up a telescope with water--but failed to detect any change in the aberration4 ,thus we get both in the case of ,arago prism experiment and ,airy-,boscovitch water-telescope experiment1 the very startling result that optical effects in a moving medium seem to be quite independent of the velocity of the medium with respect to ,fresnel's stationary ether4 ,fresnel's convection coefficient k #a - #a_/b4--,possibly some form of compensation is taking place4 ,working on this hypothesis1 ,fresnel offered his famous ether convection theory4 ,according to ,fresnel1 the presence of matter implies a definite condensation of ether within the region occupied by matter4 ,this 8condensed0 or excess portion of ether is supposed to be carried away with its own piece of moving matter4 ,it should be observed that only the 8excess0 portion is carried away1 while the rest remains as stagnant as ever4 ,a complete convection of the 8excess0 ether with the full velocity u is optically equivalent to a partial convection of the total ether 1 with only a fraction of the velocity k4 u4 ,fresnel showed that if this convection coefficient k is #a - #a_/b 7 being the refractive index of the prism71 then the velocity of light after refraction within the moving prism would be altered to just such extent as would make the refractive index of the moving prism quite independent of its 8absolute0 velocity u4 ,the non-dependence of aberration on the 8absolute0 velocity u1 is also very easily explained with the help of this ,fresnelian convection-coefficient k4 ,stokes' viscous ether4--,it should be remembered1 however1 that ,fresnel's stationary ether is absolutely fixed and is not at all disturbed by the motion of matter through it4 ,in this respect ,fresnelian ether cannot be said to behave in any respectable physical fashion1 and this led ,stokes1 in #ahde-df1 to construct a more material type of medium4 ,stokes assumed that viscous motion ensues near the surface of separation of ether and moving matter1 while at sufficiently distant regions the ether remains wholly undisturbed4 ,he showed how such a viscous ether would explain aberration if all motion in it were differentially irrotational4 ,but in order to explain the null ,arago effect1 ,stokes was compelled to assume the convection hypothesis of ,fresnel with an identical numerical value for k1 namely #a - #a_/b4 ,thus the prestige of the ,fresnelian convection-coefficient was enhanced1 if anything1 by the theoretical investigations of ,stokes4 ,fizeau's experiment4--,soon after1 in #ahea1 it received direct experimental confirmation in a brilliant piece of work by ,fizeau4 ,if a divided beam of light is re-united after passing through two adjacent cylinders filled with water1 ordinary interference fringes will be produced4 ,if the water in one of the cylinders is now made to flow1 the 8condensed0 ether within the flowing water would be convected and would produce a shift in the interference fringes4 ,the shift actually observed agreed very well with a value of k #a - #a_/b4 ,the ,fresnelian convection-coefficient now became firmly established as a consequence of a direct positive effect4 ,on the other hand1 the negative evidences in favour of the convection-coefficient had also multiplied4 ,mascart1 ,hoek1 ,maxwell and others sought for definite changes in different optical effects induced by the motion of the earth relative to the stationary ether4 ,but all such attempts failed to reveal the slightest trace of any optical disturbance due to the 8absolute0 velocity of the earth1 thus proving conclusively that all the different optical effects shared in the general compensation arising out of the ,fresnelian convection of the excess ether4 ,it must be carefully noted that the ,fresnelian convection-coefficient implicitly assumes the existence of a fixed ether 7,fresnel7 or at least a wholly stagnant medium at sufficiently distant regions 7,stokes71 with reference to which alone a convection velocity can have any significance4 ,thus the convection-coefficient implying some type of a stationary or viscous1 yet nevertheless 8absolute0 ether1 succeeded in explaining satisfactorily all known optical facts down to #ahhj4 ,michelson-,morley ,experiment4--,in #ahha1 ,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 relativity4 ,the fundamental idea underlying this experiment is quite simple4 ,in all old experiments the velocity of light situated in free ether was compared with the velocity of waves actually situated in a piece of moving matter and presumably carried away by it4 ,the compensatory effect of the ,fresnelian convection of ether afforded a satisfactory explanation of all negative results4 ,in the ,michelson-,morley experiment the arrangement is quite different4 ,if there is a definite gap in a rigid body1 light waves situated in free ether will take a definite time in crossing the gap4 ,if the rigid platform carrying the gap is set in motion with respect to the ether in the direction of light propagation1 light waves 7which are even now situated in free ether7 should presumably take a longer time to cross the gap4 ,we cannot do better than quote ,eddington's description of this famous experiment4 8,the principle of the experiment may be illustrated by considering a swimmer in a river4 ,it is easily realized that it takes longer to swim to a point #ej yards up-stream and back than to a point #ej yards across-stream and back4 ,if the earth is moving through the ether there is a river of ether flowing through the laboratory1 and a wave of light may be compared to a swimmer travelling with constant velocity relative to the current4 ,if1 then1 we divide a beam of light into two parts1 and send one-half swimming up the stream for a certain distance and then 7by a mirror7 back to the starting point1 and send the other half an equal distance across stream and back1 the across-stream beam should arrive back first4 ----u ,o ,a----------44444444 x ,b ,let the ether be flowing relative to the apparatus with velocity u in the direction ,ox1 and let ,,oa1 ,,ob1 be the two arms of the apparatus of equal length l1 ,,oa being placed up-stream4 ,let c be the velocity of light4 ,the time for the double journey along ,,oa and back is l l #blc #bl t ------ ------ -------- --- c - u c u c - u c where beta 7#a - frac u#b;c#b7-;frac #ab a factor greater than unity4 ,for the transverse journey the light must have a component velocity n up-stream 7relative to the ether7 in order to avoid being carried below ,,ob3 and since its total velocity is c1 its component across-stream must be 7c - u71 the time for the double journey ,,ob is accordingly t#b frac #b;asqrt c#b - u#b frac #b;ac beta so that t t4 ,but when the experiment was tried1 it was found that both parts of the beam took the same time1 as tested by the interference bands produced40 ,after a most careful series of observations1 ,michelson and ,morley failed to detect the slightest trace of any effect due to earth's motion through ether4 ,the ,michelson-,morley experiment seems to show that there is no relative motion of ether and matter4 ,fresnel's stagnant ether requires a relative velocity of--u4 ,thus ,michelson and ,morley themselves thought at first that their experiment confirmed ,stokes' viscous ether1 in which no relative motion can ensue on account of the absence of slipping of ether at the surface of separation4 ,but even on ,stokes' theory this viscous flow of ether would fall off at a very rapid rate as we recede from the surface of separation4 ,michelson and ,morley repeated their experiment at different heights from the surface of the earth1 but invariably obtained the same negative results1 thus failing to confirm ,stokes' theory of viscous flow4 ,lodge's experiment4--,further1 in #ahic1 ,lodge performed his rotating sphere experiment which showed complete absence of any viscous flow of ether due to moving masses of matter4 ,a divided beam of light1 after repeated reflections within a very narrow gap between two massive hemispheres1 was allowed to re-unite and thus produce interference bands4 ,when the two hemispheres are set rotating1 it is conceivable that the ether in the gap would be disturbed due to viscous flow1 and any such flow would be immediately detected by a disturbance of the interference bands4 ,but actual observation failed to detect the slightest disturbance of the ether in the gap1 due to the motion of the hemispheres4 ,lodge's experiment thus seems to show a complete absence of any viscous flow of ether4 ,apart from these experimental discrepancies1 grave theoretical objections were urged against a viscous ether4 ,stokes himself had shown that his ether must be incompressible and all motion in it differentially irrotational1 at the same time there should be absolutely no slipping at the surface of separation4 ,now all these conditions cannot be simultaneously satisfied for any conceivable material medium without certain very special and arbitrary assumptions4 ,thus ,stokes' ether failed to satisfy the very motive which had led ,stokes to formulate it1 namely1 the desirability of constructing a 8physical0 medium4 ,planck offered modified forms of ,stokes' theory which seemed capable of being reconciled with the ,michelson-,morley experiment1 but required very special assumptions4 ,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 subject4 ,the sole criterion of the value of any scientific theory must ultimately be its capacity for offering a simple1 unified1 coherent and fruitful description of observed facts4 ,in proportion as a theory becomes complex it loses in usefulness--a theory which is obliged to requisition a whole array of arbitrary assumptions in order to explain special facts is practically worse than useless1 as it serves to disjoin1 rather than to unite1 the several groups of facts4 ,the optical experiments of the last quarter of the nineteenth century showed the impossibility of constructing a simple ether theory1 which would be amenable to analytic treatment and would at the same time stimulate further progress4 ,it should be observed that it could scarcely be shown that no logically consistent ether theory was possible2 indeed in #aiaj1 ,h4 ,a4 ,wilson offered a consistent ether theory which was at least quite neutral with respect to all available optical data4 ,but ,wilson's ether is almost wholly negative--its only virtue being that it does not directly contradict observed facts4 ,neither any direct confirmation nor a direct refutation is possible and it does not throw any light on the various optical phenomena4 ,a theory like this being practically useless stands self-condemned4 ,we must now consider the problem of relative motion of ether and matter from the point of view of electrical theory4 ,from #ahfj the identity of light as an electromagnetic vector became gradually established as a result of the brilliant 8displacement current0 hypothesis of ,clerk ,maxwell and his further analytical investigations4 ,the elastic solid ether became gradually transformed into the electromagnetic one4 ,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 theory4 ,hertz's researches on electric waves1 first carried out in #ahhf1 succeeded in furnishing a strong experimental confirmation of ,maxwell's theory4 ,electric waves behaved generally like light waves of very large wave length4 ,the orthodox ,maxwellian view located the dielectric polarisation in the electromagnetic ether which was merely a transformation of ,fresnel's stagnant ether4 ,the magnetic polarisation was looked upon as wholly secondary in origin1 being due to the relative motion of the dielectric tubes of polarisation4 ,on this view the ,fresnelian convection coefficient comes out to be 1 as shown by ,j4 ,j4 ,thomson in #ahhj1 instead of #a - 7#a_/7 as required by optical experiments4 ,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 #a - 7#a_/74 ,the modifications proposed independently by ,hertz and ,heaviside fare no better4#a ,they postulated the actual medium to be the seat of all electric polarisation and further emphasised the reciprocal relation subsisting between electricity and magnetism1 thus making the field equations more symmetrical4 ,on this view the whole of the polarised ether is carried away by the moving medium1 and consequently1 the convection coefficient naturally becomes unity in this theory1 a value quite as discrepant as that obtained on the original ,maxwellian assumption4 ,thus neither ,maxwell's original theory nor its subsequent modifications as developed by ,hertz and ,heaviside succeeded in obtaining a value for ,fresnelian coefficient equal to #a - 7#a_/b71 and consequently stood totally condemned from the optical point of view4 ,certain direct electromagnetic experiments involving relative motion of polarised dielectrics were no less conclusive against the generalised theory of ,hertz and ,heaviside4 ,according to ,hertz a moving dielectric would carry away the whole of its electric displacement with it4 ,hence the electromagnetic effect near the moving dielectric would be proportional to the total electric displacement1 that is to ,k1 the specific inductive capacity of the dielectric4 ,in #aija1 ,blondlot working with a stream of moving gas could not detect any such effect4 ,h4 ,a4 ,wilson repeated the experiment in an improved form in #aijc and working with ebonite found that the observed effect was proportional to ,k - #a instead of to ,k4 ,for gases ,k is nearly equal to #a and hence practically no effect will be observed in their case4 ,this gives a satisfactory explanation of ,blondlot's negative results4 ,rowland had shown in #ahgf that the magnetic force due to a rotating condenser 7the dielectric remaining stationary7 was proportional to ,k1 the sp4 ind4 cap4 ,on the other hand1 ,rntgen found in #ahhh the magnetic effect due to a rotating dielectric 7the condenser remaining stationary7 to be proportional to ,k - #a1 and not to ,k4 ,finally ,eichenwald in #aijc found that when both condenser and dielectric are rotated together1 the effect observed was quite independent of ,k1 a result quite consistent with the two previous experiments4 ,the ,rowland effect proportional to ,k1 together with the opposite ,rntgen effect proportional to #a - ,k1 makes the ,eichenwald effect independent of ,k4 ,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 - #a1 and not to ,k as required by ,hertz's theory4 ,thus the above group of experiments with moving dielectrics directly contradicted the ,hertz-,heaviside theory4 ,the internal discrepancies inherent in the classic ether theory had now become too prominent4 ,it was clear that the ether concept had finally outgrown its usefulness4 ,the observed facts had become too contradictory and too heterogeneous to be reduced to an organised whole with the help of the ether concept alone4 ,radical departures from the classical theory had become absolutely necessary4 ,there were several outstanding difficulties in connection with anomalous dispersion1 selective reflection and selective absorption which could not be satisfactory explained in the classic electromagnetic theory4 ,it was evident that the assumption of some kind of discreteness in the optical medium had become inevitable4 ,such an assumption naturally gave rise to an atomic theory of electricity1 namely1 the modern electron theory4 ,lorentz had postulated the existence of electrons so early as #ahgh1 but it was not until some years later that the electron theory became firmly established on a satisfactory basis4 ,lorentz assumed that a moving dielectric merely carried away its own 8polarisation doublets10 which on his theory gave rise to the induced field proportional to ,k - #a4 ,the field near a moving dielectric is naturally proportional to ,k - #a and not to ,k4 ,lorentz's theory thus gave a satisfactory explanation of all those experiments with moving dielectrics which required effects proportional to ,k - #a4 ,lorentz further succeeded in obtaining a value for the ,fresnelian convection coefficient equal to #a - #a_/b1 the exact value required by all optical experiments of the moving type4 ,we must now go back to ,michelson and ,morley's experiment4 ,we have seen that both parts of the beam are situated in free ether2 no material medium is involved in any portion of the paths actually traversed by the beam4 ,consequently no compensation due to ,fresnelian convection of ether by moving medium is possible4 ,thus ,fresnelian convection compensation can have no possible application in this case4 ,yet some marvellous compensation has evidently taken place which has completely masked the 8absolute0 velocity of the earth4 ,in ,michelson and ,morley's experiment1 the distance travelled by the beam along ,,oa 7that is1 in a direction parallel to the motion of the platform7 is #bl1 while the distance travelled by the beam along ,,ob1 perpendicular to the direction of motion of the platform1 is #bl4 ,yet the most careful experiments showed1 as ,eddington says1 8that both parts of the beam took the same time as tested by the interference bands produced4 ,it would seem that ,,oa and ,,ob could not really have been of the same length2 and if ,,ob was of length l1 ,,oa must have been of length l_/4 ,the apparatus was now rotated through #ij1 so that ,,ob became the up-stream4 ,the time for the two journeys was again the same1 so that #j;,b must now be the shorter length4 ,the plain meaning of the experiment is that both arms have a length l when placed along ,oy 7perpendicular to the direction of motion71 and automatically contract to a length l_/1 when placed along ,ox 7parallel to the direction of motion74 ,this explanation was first given by ,fitz-,gerald40 ,this ,fitz-,gerald contraction1 startling enough in itself1 does not suffice4 ,assuming this contraction to be a real one1 the distance travelled with respect to the ether is #bl and the time taken for this journey is #bl_/c4 ,but the distance travelled with respect to the platform is always #bl4 ,hence the velocity of light with respect to the platform is frac #blfrac #blbetac frac cbeta a variable quantity depending on the 8absolute0 velocity of the platform4 ,but no trace of such an effect has ever been found4 ,the velocity of light is always found to be quite independent of the velocity of the platform4 ,the present difficulty cannot be solved by any further alteration in the measure of space4 ,the only recourse left open is to alter the measure of time as well1 that is1 to adopt the concept of 8local time40 ,if a moving clock goes slower so that one #8real' second becomes #a_/ second as measured in the moving system1 the velocity of light relative to the platform will always remain c4 ,we must adopt two very startling hypotheses1 namely1 the ,fitz-,gerald contraction and the concept of 8local time10 in order to give a satisfactory explanation of the ,michelson-,morley experiment4 ,these results were already reached by ,lorentz in the course of further developments of his electron theory4 ,lorentz used a special set of transformation equations#b for time which implicitly introduced the concept of local time4 ,but he himself failed to attach any special significance to it1 and looked upon it rather as a mere mathematical artifice like imaginary quantities in analysis or the circle at infinity in projective geometry4 ,the originality of ,einstein at this stage consists in his successful physical interpretation of these results1 and viewing them as the coherent organised consequences of a single general principle4 ,lorentz established the ,relativity ,theorem#c 7consisting merely of a set of transformation equations7 while ,einstein generalised it into a ,universal ,principle4 ,in addition ,einstein introduced fundamentally new concepts of space and time1 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 processes4 ,newton had framed his laws of motion in such a way as to make them quite independent of the absolute velocity of the earth4 ,uniform relative motion of ether and matter could not be detected with the help of dynamical laws4 ,according to ,einstein neither could it be detected with the help of optical or electromagnetic experiments4 ,thus the ,einsteinian ,principle of ,relativity asserts that all physical laws are independent of the #8absolute' velocity of an observer4 ,for different systems1 the form of all physical laws is conserved4 ,if we chose the velocity of light#d to be the fundamental unit of measurement for all observers 7that is1 assume the constancy of the velocity of light in all systems7 we can establish a metric 8one-one0 correspondence between any two observed systems1 such correspondence depending only the relative velocity of the two systems4 ,einstein's ,relativity is thus merely the consistent logical application of the well known physical principle that we can know nothing but relative motion4 ,in this sense it is a further extension of ,newtonian ,relativity4 ,on this interpretation1 the ,lorentz-,fitzgerald contraction and 8local time0 lose their arbitrary character4 ,space and time as measured by two different observers are naturally diverse1 and the difference depends only on their relative motion4 ,both are equally valid2 they are merely different descriptions of the same physical reality4 ,this is essentially the point of view adopted by ,minkowski4 ,he considers time itself to be one of the co-ordinate axes1 and in his four-dimensional world1 that is in the space-time reality1 relative motion is reduced to a rotation of the axes of reference4 ,thus1 the diversity in the measurement of lengths and temporal rates is merely due to the static difference in the 8frame-work0 of the different observers4 ,the above theory of ,relativity absorbed practically the whole of the electromagnetic theory based on the ,maxwell-,lorentz system of field equations4 ,it combined all the advantages of classic ,maxwellian theory together with an electronic hypothesis4 ,the ,lorentz assumption of polarisation doublets had furnished a satisfactory explanation of the ,fresnelian convection of ether1 but in the new theory this is deduced merely as a consequence of the altered concept of relative velocity4 ,in addition1 the theory of ,relativity accepted the results of ,michelson and ,morley's experiments as a definite principle1 namely1 the principle of the constancy of the velocity of light1 so that there was nothing left for explanation in the ,michelson-,morley experiment4 ,but even more than all this1 it established a single general principle which served to connect together in a simple coherent and fruitful manner the known facts of ,physics4 ,the theory of ,relativity received direct experimental confirmation in several directions4 ,repeated attempts were made to detect the ,lorentz-,fitzgerald contraction4 ,any ordinary physical contraction will usually have observable physical results2 for example1 the total electrical resistance of a conductor will diminish4 ,trouton and ,noble1 ,trouton and ,rankine1 ,rayleigh and ,brace1 and others employed a variety of different methods to detect the ,lorentz-,fitzgerald contraction1 but invariably with the same negative results4 ,whether there is an ether or not1 uniform velocity with respect to it can never be detected4 ,this does not prove that there is no such thing as an ether but certainly does render the ether entirely superfluous4 ,universal compensation is due to a change in local units of length and time1 or rather1 being merely different descriptions of the same reality1 there is no compensation at all4 ,there was another group of observed phenomena which could scarcely be fitted into a ,newtonian scheme of dynamics without doing violence to it4 ,the experimental work of ,kaufmann1 in #aija1 made it abundantly clear that the 8mass0 of an electron depended on its velocity4 ,so early as #ahha1 ,j4 ,j4 ,thomson had shown that the inertia of a charged particle increased with its velocity4 ,abraham now deduced a formula for the variation of mass with velocity1 on the hypothesis that an electron always remained a rigid sphere4 ,lorentz proceeded on the assumption that the electron shared in the ,lorentz-,fitzgerald contraction and obtained a totally different formula4 ,a very careful series of measurements carried out independently by ,bcherer1 ,wolz1 ,hupka and finally ,neumann in #aiac1 decided conclusively in favour of the ,lorentz formula4 ,this 8contractile0 formula follows immediately as a direct consequence of the new ,theory of ,relativity1 without any assumption as regards the electrical origin of inertia4 ,thus the complete agreement of experimental facts with the predictions of the new theory must be considered as confirming it as a principle which goes even beyond the electron itself4 ,the greatest triumph of this new theory consists1 indeed1 in the fact that a large number of results1 which had formerly required all kinds of special hypotheses for their explanation1 are now deduced very simply as inevitable consequences of one single general principle4 ,we have now traced the history of the development of the restricted or special theory of ,relativity1 which is mainly concerned with optical and electrical phenomena4 ,it was first offered by ,einstein in #aije4 ,ten years later1 ,einstein formulated his second theory1 the ,generalised ,principle of ,relativity4 ,this new theory is mainly a theory of gravitation and has very little connection with optics and electricity4 ,in one sense1 the second theory is indeed a further generalisation of the restricted principle1 but the former does not really contain the latter as a special case4 ,einstein's first theory is restricted in the sense that it only refers to uniform rectilinear motion and has no application to any kind of accelerated movements4 ,einstein in his second theory extends the ,relativity ,principle to cases of accelerated motion4 ,if ,relativity is to be universally true1 then even accelerated motion must be merely relative motion between matter and matter4 ,hence the ,generalised ,principle of ,relativity asserts that 8absolute0 motion cannot be detected even with the help of gravitational laws4 ,all movements must be referred to definite sets of co-ordinate axes4 ,if there is any change of axes1 the numerical magnitude of the movements will also change4 ,but according to ,newtonian dynamics1 such alteration in physical movements can only be due to the effect of certain forces in the field4#e ,thus any change of axes will introduce new 8geometrical0 forces in the field which are quite independent of the nature of the body acted on4 ,gravitational forces also have this same remarkable property1 and gravitation itself may be of essentially the same nature as these 8geometrical0 forces introduced by a change of axes4 ,this leads to ,einstein's famous ,principle of ,equivalence4 ,a gravitational field of force is strictly equivalent to one introduced by a transformation of co-ordinates and no possible experiment can distinguish between the two4 ,thus it may become possible to 8transform away0 gravitational effects1 at least for sufficiently small regions of space1 by referring all movements to a new set of axes4 ,this new 8framework0 may of course have all kinds of very complicated movements when referred to the old ,galilean or 8rectangular unaccelerated system of co-ordinates40 ,but there is no reason why we should look upon the ,galilean system as more fundamental than any other4 ,if it is found simpler to refer all motion in a gravitational field to a special set of co-ordinates1 we may certainly look upon this special 8framework0 7at least for the particular region concerned71 to be more fundamental and more natural4 ,we may1 still more simply1 identify this particular framework with the special local properties of space in that region4 ,that is1 we can look upon the effects of a gravitational field as simply due to the local properties of space and time itself4 ,the very presence of matter implies a modification of the characteristics of space and time in its neighbourhood4 ,as ,eddington says 8matter does not cause the curvature of space-time4 ,it is the curvature4 ,just as light does not cause electromagnetic oscillations2 it is the oscillations40 ,we may look upon this from a slightly different point of view4 ,the ,general ,principle of ,relativity asserts that all motion is merely relative motion between matter and matter1 and as all movements must be referred to definite sets of co-ordinates1 the ground of any possible framework must ultimately be material in character4 ,it is convenient to take the matter actually present in a field as the fundamental ground of our framework4 ,if this is done1 the special characteristics of our framework would naturally depend on the actual distribution of matter in the field4 ,but physical space and time is completely defined by the 8framework40 ,in other words the 8framework0 itself is space and time4 ,hence we see how physical space and time is actually defined by the local distribution of matter4 ,there are certain magnitudes which remain constant by any change of axes4 ,in ordinary geometry distance between two points is one such magnitude2 so that x y z is an invariant4 ,in the restricted theory of light1 the principle of constancy of light velocity demands that x y z - ct should remain constant4 ,the separation ds of adjacent events is defined by ds -dx - dy - dz cdt4 ,it is an extension of the notion of distance and this is the new invariant4 ,now if x1 y1 z1 t are transformed to any set of new variables x1 x1 x1 x1 we shall get a quadratic expression for ds#b g#a2ax#ab #b;g#a2bx#ax#b 444 sum gi2jxixj where the g's are functions of x1 x1 x1 x depending on the transformation4 ,the special properties of space and time in any region are defined by these g's which are themselves determined by the actual distribution of matter in the locality4 ,thus from the ,newtonian point of view1 these g's represent the gravitational effect of matter while from the ,relativity stand-point1 these merely define the non-,newtonian 7and incidentally non-,euclidean7 space in the neighbourhood of matter4 ,we have seen that ,einstein's theory requires local curvature of space-time in the neighbourhood of matter4 ,such altered characteristics of space and time give a satisfactory explanation of an outstanding discrepancy in the observed advance of perihelion of ,mercury4 ,the large discordance is almost completely removed by ,einstein's theory4 ,again1 in an intense gravitational field1 a beam of light will be affected by the local curvature of space1 so that to an observer who is referring all phenomena to a ,newtonian system1 the beam of light will appear to deviate from its path along an ,euclidean straight line4 ,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 ,may1 #aiai4 ,the observed deflection is decisively in favour of the ,generalised ,theory of ,relativity4 ,it should be noted however that the velocity of light itself would decrease in a gravitational field4 ,this may appear at first sight to be a violation of the principle of constancy of light-velocity4 ,but when we remember that the ,special ,theory is explicitly restricted to the case of unaccelerated motion1 the difficulty vanishes4 ,in the absence of a gravitational field1 that is in any unaccelerated system1 the velocity of light will always remain constant4 ,thus the validity of the ,special ,theory is completely preserved within its own restricted field4 ,einstein has proposed a third crucial test4 ,he has predicted a shift of spectral lines towards the red1 due to an intense gravitational potential4 ,experimental difficulties are very considerable here1 as the shift of spectral lines is a complex phenomenon4 ,evidence is conflicting and nothing conclusive can yet be asserted4 ,einstein thought that a gravitational displacement of the ,fraunhofer lines is a necessary and fundamental condition for the acceptance of his theory4 ,but ,eddington has pointed out that even if this test fails1 the logical conclusion would seem to be that while ,einstein's law of gravitation is true for matter in bulk1 it is not true for such small material systems as atomic oscillator4 ,conclusion ,from the conceptual stand-point there are several important consequences of the ,generalised or ,gravitational ,theory of ,relativity4 ,physical space-time is perceived to be intimately connected with the actual local distribution of matter4 ,euclid-,newtonian space-time is not the actual space-time of ,physics1 simply because the former completely neglects the actual presence of matter4 ,euclid-,newtonian continuum is merely an abstraction1 while physical space-99time is the actual framework which has some definite curvature due to the presence of matter4 ,gravitational ,theory of ,relativity thus brings out clearly the fundamental distinction between actual physical space-time 7which is non-isotropic and non-,euclid-,newtonian7 on one hand and the abstract ,euclid-,newtonian continuum 7which is homogeneous1 isotropic and a purely intellectual construction7 on the other4 ,the measurements of the rotation of the earth reveals a fundamental framework which may be called the 8inertial framework40 ,this constitutes the actual physical universe4 ,this universe approaches ,galilean space-time at a great distance from matter4 ,the properties of this physical universe may be referred to some world-distribution of matter or the 8inertial framework0 may be constructed by a suitable modification of the law of gravitation itself4 ,in ,einstein's theory the actual curvature of the 8inertial framework0 is referred to vast quantities of undetected world-matter4 ,it has interesting consequences4 ,the dimensions of ,einsteinian universe would depend on the quantity of matter in it2 it would vanish to a point in the total absence of matter4 ,then again curvature depends on the quantity of matter1 and hence in the presence of a sufficient quantity of matter space-time may curve round and close up4 ,einsteinian universe will then reduce to a finite system without boundaries1 like the surface of a sphere4 ,in this 8closed up0 system1 light rays will come to a focus after travelling round the universe and we should see an 8anti-sun0 7corresponding to the back surface of the sun7 at a point in the sky opposite to the real sun4 ,this anti-sun would of course be equally large and equally bright if there is no absorption of light in free space4 ,in de ,sitter's theory1 the existence of vast quantities of world-matter is not required4 ,but beyond a definite distance from an observer1 time itself stands still1 so that to the observer nothing can ever 8happen0 there4 ,all these theories are still highly speculative in character1 but they have certainly extended the scope of theoretical physics to the central problem of the ultimate nature of the universe itself4 ,one outstanding peculiarity still attaches to the concept of electric force--it is not amenable to any process of being 8transformed away0 by a suitable change of framework4 ,h4 ,weyl1 it seems1 has developed a geometrical theory 7in hyper-space7 in which no fundamental distinction is made between gravitational and electrical forces4 ,einstein's theory connects up the law of gravitation with the laws of motion1 and serves to establish a very intimate relationship between matter and physical space-99time4 ,space1 time and matter 7or energy7 were considered to be the three ultimate elements in ,physics4 ,the restricted theory fused space-time into one indissoluble whole4 ,the generalised theory has further synthesised space-time and matter into one fundamental physical reality4 ,space1 time and matter taken separately are more abstractions4 ,physical reality consists of a synthesis of all three4 ,p4 ,c4 ,,mahalanobis4 ,note ,a4 ,for example consider a massive particle resting on a circular disc4 ,if we set the disc rotating1 a centrifugal force appears in the field4 ,on the other hand1 if we transform to a set of rotating axes1 we must introduce a centrifugal force in order to correct for the change of axes4 ,this newly introduced centrifugal force is usually looked upon as a mathematical fiction--as 8geometrical0 rather than physical4 ,the presence of such a geometrical force is usually interpreted as being due to the adoption of a fictitious framework4 ,on the other hand a gravitational force is considered quite real4 ,thus a fundamental distinction is made between geometrical and gravitational forces4 ,in the ,general ,theory of ,relativity1 this fundamental distinction is done away with4 ,the very possibility of distinguishing between geometrical and gravitational forces is denied4 ,all axes of reference may now be regarded as equally valid4 ,in the ,restricted ,theory1 all 8unaccelerated0 axes of reference were recognised as equally valid1 so that physical laws were made independent of uniform absolute velocity4 ,in the ,general ,theory1 physical laws are made independent of 8absolute0 motion of any kind4 ,footnote #a3 ,see ,note #a4 ,footnote #b3 ,see ,note #b4 ,footnote #c3 ,see ,note #d4 ,footnote #d3 ,see ,notes #i and #ab4 ,footnote #e3 ,note ,a4 ,on ,the ,electrodynamics of ,moving ,bodies ,by ,a4 ,einstein4 ,,introduction4 ,it is well known that if we attempt to apply ,maxwell's electrodynamics1 as conceived at the present time1 to moving bodies1 we are led to asymmetry which does not agree with observed phenomena4 ,let us think of the mutual action between a magnet and a conductor4 ,the observed phenomena in this case depend only on the relative motion of the conductor and the magnet1 while according to the usual conception1 a distinction must be made between the cases where the one or the other of the bodies is in motion4 ,if1 for example1 the magnet moves and the conductor is at rest1 then an electric field of certain energy-value is produced in the neighbourhood of the magnet1 which excites a current in those parts of the field where a conductor exists4 ,but if the magnet be at rest and the conductor be set in motion1 no electric field is produced in the neighbourhood of the magnet1 but an electromotive force which corresponds to no energy in itself is produced in the conductor2 this causes an electric current of the same magnitude and the same career as the electric force1 it being of course assumed that the relative motion in both of these cases is the same4 #b4 ,examples of a similar kind such as the unsuccessful attempt to substantiate the motion of the earth relative to the 8,light-medium0 lead us to the supposition that not only in mechanics1 but also in electrodynamics1 no properties of observed facts correspond to a concept of absolute rest2 but that for all coordinate systems for which the mechanical equations hold1 the equivalent electrodynamical and optical equations hold also1 as has already been shown for magnitudes of the first order4 ,in the following we make these assumptions 7which we shall subsequently call the ,principle of ,relativity7 and introduce the further assumption1--an assumption which is at the first sight quite irreconcilable with the former one--that light is propagated in vacant space1 with a velocity c which is independent of the nature of motion of the emitting body4 ,these two assumptions are quite sufficient to give us a simple and consistent theory of electrodynamics of moving bodies on the basis of the ,maxwellian theory for bodies at rest4 ,the introduction of a 8,lightther0 will be proved to be superfluous1 for according to the conceptions which will be developed1 we shall introduce neither a space absolutely at rest1 and endowed with special properties1 nor shall we associate a velocity-vector with a point in which electro-magnetic processes take place4 #c4 ,like every other theory in electrodynamics1 the theory is based on the kinematics of rigid bodies2 in the enunciation of every theory1 we have to do with relations between rigid bodies 7co-ordinate system71 clocks1 and electromagnetic processes4 ,an insufficient consideration of these circumstances is the cause of difficulties with which the electrodynamics of moving bodies have to fight at present4 ,i4--,,kinematical ,,portion4 #a4 ,definition of ,synchronism4 ,let us have a co-ordinate system1 in which the ,newtonian equations hold4 ,for distinguishing this system from another which will be introduced hereafter1 we shall always call it 8the stationary system40 ,if a material point be at rest in this system1 then its position in this system can be found out by a measuring rod1 and can be expressed by the methods of ,euclidean ,geometry1 or in ,cartesian co-ordinates4 ,if we wish to describe the motion of a material point1 the values of its coordinates must be expressed as functions of time4 ,it is always to be borne in mind that such a mathematical definition has a physical sense1 only when we have a clear notion of what is meant by time4 ,we have to take into consideration the fact that those of our conceptions1 in which time plays a part1 are always conceptions of synchronism4 ,for example1 we say that a train arrives here at #g o'clock2 this means that the exact pointing of the little hand of my watch to #g1 and the arrival of the train are synchronous events4 ,it may appear that all difficulties connected with the definition of time can be removed when in place of time1 we substitute the position of the little hand of my watch4 ,such a definition is in fact sufficient1 when it is required to define time exclusively for the place at which the clock is stationed4 ,but the definition is not sufficient when it is required to connect by time events taking place at different stations1--or what amounts to the same thing1--to estimate by means of time 7zeitlich werten7 the occurrence of events1 which take place at stations distant from the clock4 ,now with regard to this attempt2--the time-estimation of events1 we can satisfy ourselves in the following manner4 ,suppose an observer--who is stationed at the origin of coordinates with the clock--associates a ray of light which comes to him through space1 and gives testimony to the event of which the time is to be estimated1--with the corresponding position of the hands of the clock4 ,but such an association has this defect1--it depends on the position of the observer provided with the clock1 as we know by experience4 ,we can attain to a more practicable result by the following treatment4 ,if an observer be stationed at ,a with a clock1 he can estimate the time of events occurring in the immediate neighbourhood of ,a1 by looking for the position of the hands of the clock1 which are synchronous with the event4 ,if an observer be stationed at ,b with a clock1--we should add that the clock is of the same nature as the one at ,a1--he can estimate the time of events occurring about ,b4 ,but without further premises1 it is not possible to compare1 as far as time is concerned1 the events at ,b with the events at ,a4 ,we have hitherto an ,a-time1 and a ,b-time1 but no time common to ,a and ,b4 ,this last time 7i4e41 common time7 can be defined1 if we establish by definition that the time which light requires in travelling from ,a to ,b is equivalent to the time which light requires in travelling from ,b to ,a4 ,for example1 a ray of light proceeds from ,a at ,a-time t,a towards ,b1 arrives and is reflected from ,b at ,b-time t,b1 and returns to ,a at ,a-time t,a4 ,according to the definition1 both clocks are synchronous1 if t,b - t,a t,a - t,b4 ,we assume that this definition of synchronism is possible without involving any inconsistency1 for any number of points1 therefore the following relations hold3-- #a4 ,if the clock at ,b be synchronous with the clock at ,a1 then the clock at ,a is synchronous with the clock at ,b4 #b4 ,if the clock at ,a as well as the clock at ,b are both synchronous with the clock at ,c1 then the clocks at ,a and ,b are synchronous4 ,thus with the help of certain physical experiences1 we have established what we understand when we speak of clocks at rest at different stations1 and synchronous with one another2 and thereby we have arrived at a definition of synchronism and time4 ,in accordance with experience we shall assume that the magnitude frac #b overline,,abt',a - t,a c where c is a universal constant4 ,we have defined time essentially with a clock at rest in a stationary system4 ,on account of its adaptability to the stationary system1 we call the time defined in this way as 8time of the stationary system40 #b4 ,on the ,relativity of ,length and ,time4 ,the following reflections are based on the ,principle of ,relativity and on the ,principle of ,constancy of the velocity of light1 both of which we define in the following way3-- #a4 ,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 which have a uniform translators motion relative to each other4 #b4 ,every ray of light moves in the 8stationary co-ordinate system0 with the same velocity c1 the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion4#f ,therefore velocity ,path of ,light_/,interval of time1 where1 by #8interval of time' we mean time as defined in #a4 ,let us have a rigid rod at rest2 this has a length l1 when measured by a measuring rod at rest2 we suppose that the axis of the rod is laid along the ,x-axis of the system at rest1 and then a uniform velocity v1 parallel to the axis of ,x1 is imparted to it4 ,let us now enquire about the length of the moving rod2 this can be obtained by either of these operations4-- 7a7 ,the observer provided with the measuring rod moves along with the rod to be measured1 and measures by direct superposition the length of the rod3--just as if the observer1 the measuring rod1 and the rod to be measured were at rest4 7b7 ,the observer finds out1 by means of clocks placed in a system at rest 7the clocks being synchronous as defined in #a71 the points of this system where the ends of the rod to be measured occur at a particular time t4 ,the distance between these two points1 measured by the previously used measuring rod1 this time it being at rest1 is a length1 which we may call the 8length of the rod40 ,according to the ,principle of ,relativity1 the length found out by the operation a71 which we may call 8the length of the rod in the moving system0 is equal to the length l of the rod in the stationary system4 ,the length which is found out by the second method1 may be called #8the length of the moving rod measured from the stationary system4' ,this length is to be estimated on the basis of our principle1 and we shall find it to be different from l4 ,in the generally recognised kinematics1 we silently assume that the lengths defined by these two operations are equal1 or in other words1 that at an epoch of time t1 a moving rigid body is geometrically replaceable by the same body1 which can replace it in the condition of rest4 ,relativity of ,time4 ,let us suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends ,a1 and ,b of a rod1 i4e41 the time of the clocks correspond to the time of the stationary system at the points where they happen to arrive2 these clocks are therefore synchronous in the stationary system4 ,we further imagine that there are two observers at the two watches1 and moving with them1 and that these observers apply the criterion for synchronism to the two clocks4 ,at the time t,a1 a ray of light goes out from ,a1 is reflected from ,b at the time t,b1 and arrives back at ,a at time t,a4 ,taking into consideration the principle of1 constancy of the velocity of light1 we have t,b - t,a r,,ab_/7c - v71 and t,a - t,b r,,ab_/7c v71 where r,,ab is the length of the moving rod1 measured in the stationary system4 ,therefore the observers stationed with the watches will not find the clocks synchronous1 though the observer in the stationary system must declare the clocks to be synchronous4 ,we therefore see that we can attach no absolute significance to the concept of synchronism2 but two events which are synchronous when viewed from one system1 will not be synchronous when viewed from a system moving relatively to this system4 #c4 ,theory of ,co-ordinate and ,time-,transformation from a stationary system to a system which moves relatively to this with uniform velocity4 ,let there be given1 in the stationary system two co-ordinate systems1 i4e41 two series of three mutually perpendicular lines issuing from a point4 ,let the ,x-axes of each coincide with one another1 and the ,y and ,z-axes be parallel4 ,let a rigid measuring rod1 and a number of clocks be given to each of the systems1 and let the rods and clocks in each be exactly alike each other4 ,let the initial point of one of the systems 7k7 have a constant velocity in the direction of the ,x-axis of the other which is stationary system ,k1 the motion being also communicated to the rods and clocks in the system 7k74 ,any time t of the stationary system ,k corresponds to a definite position of the axes of the moving system1 which are always parallel to the axes of the stationary system4 ,by t1 we always mean the time in the stationary system4 ,we suppose that the space is measured by the stationary measuring rod placed in the stationary system1 as well as by the moving measuring rod placed in the moving system1 and we thus obtain the co-ordinates 7x1 y1 z7 for the stationary system1 and 71 1 7 for the moving system4 ,let the time t be determined for each point of the stationary system 7which are provided with clocks7 by means of the clocks which are placed in the stationary system1 with the help of light-signals as described in #a4 ,let also the time of the moving system be determined for each point of the moving system 7in which there are clocks which are at rest relative to the moving system71 by means of the method of light signals between these points 7in which there are clocks7 in the manner described in #a4 ,to every value of 7x1 y1 z1 t7 which fully determines the position and time of an event in the stationary system1 there correspond a system of values 71 1 1 72 now the problem is to find out the system of equations connecting these magnitudes4 ,primarily it is clear that on account of the property of homogeneity which we ascribe to time and space1 the equations must be linear4 ,if we put x x - vt1 then it is clear that at a point relatively at rest in the system k1 we have a system of values 7x y z7 which are independent of time4 ,now let us find out as a function of 7x1 y1 z1 t74 ,for this purpose we have to express in equations the fact that is not other than the time given by the clocks which are at rest in the system k which must be made synchronous in the manner described in #a4 ,let a ray of light be sent at time from the origin of the system k along the ,x-axis towards x and let it be reflected from that place at time towards the origin of moving co-ordinates and let it arrive there at time 2 then we must have 7 7 ,if we now introduce the condition that is a function of co-ordinates1 and apply the principle of constancy of the velocity of light in the stationary system1 we have frac #ab 7tau 7#j1j1j1t7 tau 7#j1j1j17t frac x'c-v frac x'cv777 tau 7x'1#j1j1 t frac x'c-v7 ,it is to be noticed that instead of the origin of co-ordinates1 we could select some other point as the exit point for rays of light1 and therefore the above equation holds for all values of 7x1 y1 z1 t174 ,a similar conception1 being applied to the y- and z-axis gives us1 when we take into consideration the fact that light when viewed from the stationary system1 is always propagated along those axes with the velocity 7c - v71 we have the questions ---- #j1 ---- #j4 y z ,from these equations it follows that is a linear function of x and t4 ,from equations 7#a7 we obtain vx a 7t - -------- 7 c - v where a is an unknown function of v4 ,with the help of these results it is easy to obtain the magnitudes 71 1 7 if we express by means of equations the fact that light1 when measured in the moving system is always propagated with the constant velocity c 7as the principle of constancy of light velocity in conjunction with the principle of relativity requires74 ,for a time #j1 if the ray is sent in the direction of increasing 1 we have vx c1 i4e4 a c7t - ------------ 7 c - v ,now the ray of light moves relative to the origin of k with a velocity c - v1 measured in the stationary system2 therefore we have x ---------- t c - v ,substituting these values of t in the equation for 1 we obtain c a ------------- x c - v ,in an analogous manner1 we obtain by considering the ray of light which moves along the y-axis1 vx c a c7t - ------------- 7 c - v where y ------------------ t1 x #j1 7c - v7 ,therefore c a ------------------ y1 7c - v7 c a ----------------- z 4 7c - v7 ,if for x1 we substitute its value x - tv1 we obtain v4c 7v74 7t - ----------- 1 c 7v74 7x - vt7 1 7v7 y 7v7 z 1 where beta frac #asqrt #a - frac v#b;c#b and 7v7 ac _/ 7c - v7 a _/ is a function of v4 ,if we make no assumption about the initial position of the moving system and about the null-point of t1 then an additive constant is to be added to the right hand side4 ,we have now to show1 that every ray of light moves in the moving system with a velocity c 7when measured in the moving system71 in case1 as we have actually assumed1 c is also the velocity in the stationary system2 for we have not as yet adduced any proof in support of the assumption that the principle of relativity is reconcilable with the principle of constant light-velocity4 ,at a time t #j let a spherical wave be sent out from the common origin of the two systems of co-ordinates1 and let it spread with a velocity c in the system ,k4 ,if 7x1 y1 z71 be a point reached by the wave1 we have x y z ct with the aid of our transformation-equations1 let us transform this equation1 and we obtain by a simple calculation1 c4 ,therefore the wave is propagated in the moving system with the same velocity c1 and as a spherical wave4#g ,therefore we show that the two principles are mutually reconcilable4 ,in the transformations we have got an undetermined function 7v71 and we now proceed to find it out4 ,let us introduce for this purpose a third co-ordinate system k1 which is set in motion relative to the system k1 the motion being parallel to the -axis4 ,let the velocity of the origin be 7-v74 ,at the time t #j1 all the initial co-ordinate points coincide1 and for t x y z #j1 the time t of the system k #j4 ,we shall say that 7x y z t7 are the co-ordinates measured in the system k1 then by a two-fold application of the transformation-equations1 we obtain v 7-v77-v7 ----- c 7v77-v7t1 x 7v77v77 v7 7v77-v7x1 etc4 ,since the relations between 7x1 y1 z1 t71 and 7x1 y1 z1 t7 do not contain time explicitly1 therefore ,k and k are relatively at rest4 ,it appears that the systems ,k and k are identical4 7v77-v7 #a4 ,let us now turn our attention to the part of the -axis between 7 #j1 #j1 #j71 and 7 #j1 #a1 #j74 ,let this piece of the y-axis be covered with a rod moving with the velocity v relative to the system ,k and perpendicular to its axis2--the ends of the rod having therefore the co-ordinates x vt1 y l _/ 7v71 z #j x vt1 y #j1 z #j ,therefore the length of the rod measured in the system ,k is l_/7v74 ,for the system moving with velocity 7-v71 we have on grounds of symmetry1 l l -------- --------- 7v7 7-v7 7v7 7-v71 7v7 #a4 #d4 ,the physical significance of the equations obtained concerning moving rigid bodies and moving clocks4 ,let us consider a rigid sphere 7i4e41 one having a spherical figure when tested in the stationary system7 of radius ,r which is at rest relative to the system 7,k71 and whose centre coincides with the origin of ,k then the equation of the surface of this sphere1 which is moving with a velocity v relative to ,k1 is ,r4 ,at time t #j1 the equation is expressed by means of 7x1 y1 z1 t17 as frac x#b7sqrt #a - frac v#b;c#b7b y#b z#b ,r#b4 ,a rigid body which has the figure of a sphere when measured in the moving system1 has therefore in the moving condition--when considered from the stationary system1 the figure of a rotational ellipsoid with semi-axes ,r sqrt #a - frac v#b;c#b1 ,r1 ,r4 ,therefore the y and z dimensions of the sphere 7therefore of any figure also7 do not appear to be modified by the motion1 but the x dimension is shortened in the ratio #a 3 sqrt #a - frac v#b;c#b2 the shortening is the larger1 the larger is v4 ,for v c1 all moving bodies1 when considered from a stationary system shrink into planes4 ,for a velocity larger than the velocity of light1 our propositions become meaningless2 in our theory c plays the part of infinite velocity4 ,it is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system4 ,let us now consider that a clock which is lying at rest in the stationary system gives the time t1 and lying at rest relative to the moving system is capable of giving the time 2 suppose it to be placed at the origin of the moving system k1 and to be so arranged that it gives the time 4 ,how much does the clock gain1 when viewed from the stationary system ,k8 ,we have1 tau frac #asqrt #a-;frac v#b;c#b 7t - frac vc#bx71 and x vt1 therefore tau - t 7#a - sqrt #a - frac v#b;c#b7 t4 ,therefore the clock loses by an amount 7v_/c7 per second of motion1 to the second order of approximation4 ,from this1 the following peculiar consequence follows4 ,suppose at two points ,a and ,b of the stationary system two clocks are given which are synchronous in the sense explained in #c when viewed from the stationary system4 ,suppose the clock at ,a to be set in motion in the line joining it with ,b1 then after the arrival of the clock at ,b1 they will no longer be found synchronous1 but the clock which was set in motion from ,a will lag behind the clock which had been all along at ,b by an amount t7v_/c71 where t is the time required for the journey4 ,we see forthwith that the result holds also when the clock moves from ,a to ,b by a polygonal line1 and also when ,a and ,b coincide4 ,if we assume that the result obtained for a polygonal line holds also for a curved line1 we obtain the following law4 ,if at ,a1 there be two synchronous clocks1 and if we set in motion one of them with a constant velocity along a closed curve till it comes back to ,a1 the journey being completed in t-seconds1 then after arrival1 the last mentioned clock will be behind the stationary one by t7v_/c7 seconds4 ,from this1 we conclude that a clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole1 all other conditions being identical4 #e4 ,addition-,theorem of ,velocities4 ,let a point move in the system k 7which moves with velocity v along the x-axis of the system ,k7 according to the equation xi wxi tau1 eta weta tau1 zeta #j1 where w and w are constants4 ,it is required to find out the motion of the point relative to the system ,k4 ,if we now introduce the system of equations in #c in the equation of motion of the point1 we obtain x 7frac wxi v#a;frac vwxic#b7 t 1 y frac 7#a-;frac v#b;c#b7;frac #ab wetat #a;frac vwxic#b 1 z #j 4 ,the law of parallelogram of velocities hold up to the first order of approximation4 ,we can put ,u#b 7frac partial xpartial t7#b 7frac partial ypartial t7#b 1 w#b wxi#b weta#b 1 and alpha tan-#a frac wwxi i4e41 is put equal to the angle between the velocities v1 and w4 ,then we have-- ,u frac 7v#b w#b #b vw cos alpha7 - 7frac vw sin alphac7#b;frac #ab #a frac vw cos alphac#b ,it should be noticed that v and w enter into the expression for velocity symmetrically4 ,if w has the direction of the -axis of the moving system1 ,u frac v w#a frac vwc#b ,from this equation1 we see that by combining two velocities1 each of which is smaller than c1 we obtain a velocity which is always smaller than c4 ,if we put v c - 1 and w c - 1 where and are each smaller than c1#h ,u c frac #b;c - chi - lambda#b;c - chi - lambda frac chi lambdac#b c ,it is also clear that the velocity of light c cannot be altered by adding to it a velocity smaller than c4 ,for this case1 ,u frac c v#a frac cvc#b c ,we have obtained the formula for ,u for the case when v and w have the same direction2 it can also be obtained by combining two transformations according to section #c4 ,if in addition to the systems ,k1 and k1 we introduce the system k1 of which the initial point moves parallel to the -axis with velocity w1 then between the magnitudes1 x1 y1 z1 t and the corresponding magnitudes of k1 we obtain a system of equations1 which differ from the equations in #c1 only in the respect that in place of v1 we shall have to write1 frac v w#a frac vwc#b ,we see that such a parallel transformation forms a group4 ,we have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us1 and we shall now pass over to their application in electrodynamics4 ,,ii4--,,electrodynamical ,,part4 #f4 ,transformation of ,maxwell's equations for ,pure ,vacuum4 ,on the nature of the ,electromotive ,force caused by motion in a magnetic field4 ,the ,maxwell-,hertz equations for pure vacuum may hold for the stationary system ,k1 so that frac #a;c frac partialpartial t ,x1 ,y1 ,z beginvmatrix frac partialpartial x frac partialpartial y frac partialpartial z ,l ,m ,n endvmatrix and frac #a;c frac partialpartial t ,l1 ,m1 ,n beginvmatrix frac partialpartial x frac partialpartial y frac partialpartial z ,x ,y ,z endvmatrix 7#a7 where ,x1 ,y1 ,z are the components of the electric force1 ,l1 ,m1 ,n are the components of the magnetic force4 ,if we apply the transformations in #c to these equations1 and if we refer the electromagnetic processes to the co-ordinate system moving with velocity v1 we obtain1 frac #a;c frac partialpartial tau ,x1 beta 7,y - frac vc ,n71 beta 7,z frac vc ,m7 beginvmatrix frac partialpartial xi frac partialpartial eta frac partialpartial zeta ,l beta7,m frac vc ,z7 beta7,n - frac vc ,y7 endvmatrix and frac #a;c frac partialpartial tau ,l1 beta7,m frac vc ,z71 beta7,n - frac vc ,y7 - beginvmatrix frac partialpartial xi frac partialpartial eta frac partialpartial zeta ,x beta7,y - frac vc ,n7 beta7,z frac vc ,m7 endvmatrix 444 7#b7 where beta frac #asqrt #a - frac v#b;c#b ,the principle of ,relativity requires that the ,maxwell-,hertzian equations for pure vacuum shall hold also for the system k1 if they hold for the system ,k1 i4e41 for the vectors of the electric and magnetic forces acting upon electric and magnetic masses in the moving system k1 which are defined by their pondermotive reaction1 the same equations hold1 444 i4e4 444 frac #a;c frac partialpartial tau 7,x'1 ,y'1 ,z'7 beginvmatrix frac partialpartial xi frac partialpartial eta frac partialpartial zeta ,l' ,m' ,n' endvmatrix 1 frac #a;c frac partialpartial tau 7,l'1 ,m'1 ,n'7 - beginvmatrix frac partialpartial xi frac partialpartial eta frac partialpartial zeta ,x' ,y' ,z' endvmatrix 444 7#c7 ,clearly both the systems of equations 7#b7 and 7#c7 developed for the system k shall express the same things1 for both of these systems are equivalent to the ,maxwell-,hertzian equations for the system ,k4 ,since both the systems of equations 7#b7 and 7#c7 agree up to the symbols representing the vectors1 it follows that the functions occurring at corresponding places will agree up to a certain factor 7v71 which depends only on v1 and is independent of 71 1 1 74 ,hence the relations1 v v ,x1 ,y1 ,z 7v7 ,x1 7,y - ----- ,n71 7,z ------ ,m71 c c v v ,l1 ,m1 ,n 7v7 ,l1 7,m - ----- ,z71 7,n ----- ,y71 c c ,then by reasoning similar to that followed in 7#c71 it can be shown that 7v7 #a4 v v ,x1 ,y1 ,z ,x1 7,y - ----- ,n71 7,z ------ ,m7 c c v v ,l1 ,m1 ,n ,l1 7,m - ------ ,z71 7,n ----- ,y71 c c ,for the interpretation of these equations1 we make the following remarks4 ,let us have a point-mass of electricity which is of magnitude unity in the stationary system ,k1 i4e41 it exerts a unit force upon a similar quantity placed at a distance of #a cm4 ,if this quantity of electricity be at rest in the stationary system1 then the force acting upon it is equivalent to the vector 7,x1 ,y1 ,z7 of electric force4 ,but if the quantity of electricity be at rest relative to the moving system 7at least for the moment considered71 then the force acting upon it1 and measured in the moving system is equivalent to the vector 7,x1 ,y1 ,z74 ,the first three of equations 7#a71 7#b71 7#c71 can be expressed in the following way3-- #a4 ,if a point-mass of electric unit pole moves in an electro-magnetic field1 then besides the electric force1 an electromotive force acts upon it1 which1 neglecting the numbers involving the second and higher powers of v_/c1 is equivalent to the vector-product of the velocity vector1 and the magnetic force divided by the velocity of light 7,old mode of expression74 #b4 ,if a point-mass of electric unit pole moves in an electro-magnetic field1 then the force acting upon it is equivalent to the electric force existing at the position of the unit pole1 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 expression4 ,similar theorems hold with reference to the magnetic force4 ,we see that in the theory developed the electro-magnetic force plays the part of an auxiliary concept1 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 system4 ,it is further clear that the asymmetry mentioned in the introduction which occurs when we treat of the current excited by the relative motion of a magnet and a conductor disappears4 ,also the question about the seat of electromagnetic energy is seen to be without any meaning4 #g4 ,theory of ,dppler's ,principle and ,aberration4 ,in the system ,k1 at a great distance from the origin of co-ordinates1 let there be a source of electrodynamic waves1 which is represented with sufficient approximation in a part of space not containing the origin1 by the equations3-- ,x ,x sin ,y ,y sin ,z ,z sin ,l ,l sin ,m ,m sin ,n ,n sin lx my nz 7t - ------------ 7 c ,here 7,x1 ,y1 ,z7 and 7,l1 ,m1 ,n7 are the vectors which determine the amplitudes of the train of waves1 7l1 m1 n7 are the direction-cosines of the wave-normal4 ,let us now ask ourselves about the composition of these waves1 when they are investigated by an observer at rest in a moving medium k3--,by applying the equations of transformation obtained in #f for the electric and magnetic forces1 and the equations of transformation obtained in #c for the co-ordinates1 and time1 we obtain immediately3-- ,x ,x sin v ,y 7,y - --- ,n7 sin c v ,z 7,z - --- ,m7 sin c ,l ,l sin v ,m 7,m - --- ,z7 sin c v ,n 7,n - --- ,y7 sin c l m n 7t - --------------- 7 c where omega' omega beta 7#a - frac lvc7 1 l' frac l - frac vc#a - frac lvc 1 m' frac mbeta 7#a - frac lvc7 1 n' frac nbeta 7#a - frac lvc7 ,from the equation for it follows3--,if an observer moves with the velocity v relative to an infinitely distant source of light emitting waves of frequency 1 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 source1 then the frequency which is perceived by the observer is represented by the formula nu' nu frac #a - cos ,phi frac vc sqrt #a - frac v#b;c#b ,this is ,dppler's principle for any velocity4 ,if #j1 then the equation takes the simple form nu' nu 7frac #a - frac vc#a frac vc7frac #ab ,we see that--contrary to the usual conception-- 1 for v -c4 ,if angle between the wave-normal 7direction of the ray7 in the moving system1 and the line of motion of the observer1 the equation for l takes the form cos ,phi' frac cos ,phi - frac vc #a - frac vc cos ,phi ,this equation expresses the law of observation in its most general form4 ,if _/#b1 the equation takes the simple form v cos --- c ,we have still to investigate the amplitude of the waves1 which occur in these equations4 ,if ,a and ,a be the amplitudes in the stationary and the moving systems 7either electrical or magnetic71 we have ,a'#b ,a#b frac 7#a - frac vc cos ,phi7#b #a - frac v#b;c#b ,if #j1 this reduces to the simple form ,a'#b ,a#b frac #a - frac vc #a frac vc ,from these equations1 it appears that for an observer1 which moves with the velocity c towards the source of light1 the source should appear infinitely intense4 #h4 ,transformation of the ,energy of the ,rays of ,light4 ,theory of the ,radiation-pressure on a perfect mirror4 ,since ,a_/#h is equal to the energy of light per unit volume1 we have to regard ,a_/#h as the energy of light in the moving system4 ,a_/,a would therefore denote the ratio between the energies of a definite light-complex 8measured when moving0 and 8measured when stationary10 the volumes of the light-complex measured in ,k and k being equal4 ,yet this is not the case4 ,if l1 m1 n are the direction-cosines of the wave-normal of light in the stationary system1 then no energy passes through the surface elements of the spherical surface 7x - clt7 7y - cmt7 7z - cnt7 ,r1 which expands with the velocity of light4 ,we can therefore say1 that this surface always encloses the same light-complex4 ,let us now consider the quantity of energy1 which this surface encloses1 when regarded from the system k1 i4e41 the energy of the light-complex relative to the system k4 ,regarded from the moving system1 the spherical surface becomes an ellipsoidal surface1 having1 at the time #j1 the equation3-- 7beta xi - l beta frac vc xi7#b 7eta - m beta frac vc xi7#b 7zeta - n beta frac vc xi7#b ,r#b ,if ,s volume of the sphere1 ,s volume of this ellipsoid1 then a simple calculation shows that3 frac ,s',s frac betasqrt#a - frac vc cos ,phi ,if ,e denotes the quantity of light energy measured in the stationary system1 ,e the quantity measured in the moving system1 which are enclosed by the surfaces mentioned above1 then frac ,e',e frac frac ,a'#bhpi ,s'frac ,a#bhpi,s frac #a - frac vc cos ,phisqrt#a - frac v#b;c#b ,if #j1 we have the simple formula3-- frac ,e',e 7frac#a - fracvc#a fracvc7frac#ab ,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 observer4 ,let there be a perfectly reflecting mirror at the co-ordinate-plane #j1 from which the plane-wave considered in the last paragraph is reflected4 ,let us now ask ourselves about the light-pressure exerted on the reflecting surface and the direction1 frequency1 intensity of the light after reflexion4 ,let the incident light be defined by the magnitudes ,a cos 1 v 7referred to the system ,k74 ,regarded from k1 we have the corresponding magnitudes3 ,a' ,a frac#a - fracvc cos ,phisqrt#a - fracv#b;c#b cos ,phi' fraccos ,phi - fracvc#a - fracvc cos ,phi nu' nu frac#a - fracvc cos ,phisqrt#a - fracv#b;c#b ,for the reflected light we obtain1 when the process is referred to the system k3-- ,a ,a1 cos -cos 1 ,by means of a back-transformation to the stationary system1 we obtain ,k1 for the reflected light3-- ,a''' ,a'' frac#a fracvccos ,phi''sqrt#a - fracv#b;c#b ,a frac#a - #b;fracvc cos ,phi fracv#b;c#ba - fracv#b;c#b 1 cos ,phi''' fraccos ,phi'' fracvc#a fracvccos ,phi'' - frac7#a fracv#b;c#b7 cos ,phi - #b fracvc #a - #b fracvc cos ,phi frac v#b;c#b 1 nu''' nu'' frac#a fracvc cos ,phi''sqrt#a - fracv#b;c#b nu frac#a - #b fracvc cos ,phi fracv#b;c#b 7#a - fracvc7#b ,the amount or energy falling upon the unit surface of the mirror per unit of time 7measured in the stationary system7 is ,a_/7#h 7c cos - v774 ,the amount of energy going away from unit surface of the mirror per unit of time is ,a_/7#h 7-c cos v774 ,the difference of these two expressions is1 according to the ,energy principle1 the amount of work exerted1 by the pressure of light per unit of time4 ,if we put this equal to ,p4v1 where ,p pressure of light1 we have ,p #b frac,a#bhpi frac7cos ,phi - fracvc7#b #a - 7fracvc7#b ,as a first approximation1 we obtain ,a ,p #b -- cos #h which is in accordance with facts1 and with other theories4 ,all problems of optics of moving bodies can be solved after the method used here4 ,the essential point is1 that the electric and magnetic forces of light1 which are influenced by a moving body1 should be transformed to a system of co-ordinates which is stationary relative to the body4 ,in this way1 every problem of the optics of moving bodies would be reduced to a series of problems of the optics of stationary bodies4 #i4 ,transformation of the ,maxwell-,hertz ,equations4 ,let us start from the equations3-- frac#a;c7rho ux fracpartial ,xpartial t7 fracpartial ,npartial y - fracpartial ,mpartial z frac#a;c7rho uy fracpartial ,ypartial t7 fracpartial ,lpartial z - fracpartial ,npartial x frac#a;c7rho uz fracpartial ,zpartial t7 fracpartial ,mpartial x - fracpartial ,lpartial y frac#a;c fracpartial ,lpartial t fracpartial ,ypartial z - fracpartial ,zpartial y frac#a;c fracpartial ,mpartial t fracpartial ,zpartial x - fracpartial ,xpartial z frac#a;c fracpartial ,npartial t fracpartial ,xpartial y - fracpartial ,ypartial x where rho fracpartial ,xpartial x fracpartial ,ypartial y fracpartial ,zpartial z denotes #d times the density of electricity1 and 7ux1 uy1 uz7 are the velocity-components of electricity4 ,if we now suppose that the electrical-masses are bound unchangeably to small1 rigid bodies 7,ions1 electrons71 then these equations form the electromagnetic basis of ,lorentz's electrodynamics and optics for moving bodies4 ,if these equations which hold in the system ,k1 are transformed to the system k with the aid of the transformation-equations given in #c and #f1 then we obtain the equations3-- frac#a;c 7rho' uxi fracpartial ,x'partial tau7 fracpartial ,n'partial eta - fracpartial ,m'partial zeta 1 fracpartial ,l'partial tau fracpartial ,y'partial zeta - fracpartial ,z'partial eta 1 frac#a;c 7rho' ueta fracpartial ,y'partial tau7 fracpartial ,l'partial zeta - fracpartial ,n'partial xi 1 fracpartial ,m'partial tau fracpartial ,z'partial xi - fracpartial ,x'partial zeta 1 frac#a;c 7rho' uzeta fracpartial ,z'partial tau7 fracpartial ,m'partial xi - fracpartial ,l'partial eta 1 fracpartial ,n'partial tau fracpartial ,x'partial eta - fracpartial ,y'partial xi 1 where fracux - v#a - fracuxvc uxi 1 fracuybeta7#a - fracvuxc#b7 ueta 1 rho' fracpartial ,x'partial xi fracpartial ,y'partial eta fracpartial ,z'partial xi beta7#a - fracvuxc#b7 rho 1 fracuxbeta7#a - fracvuxc#b7 uzeta 1 ,since the vector 7u1 u1 u7 is nothing but the velocity of the electrical mass measured in the system k1 as can be easily seen from the addition-theorem of velocities in #d--so it is hereby shown1 that by taking our kinematical principle as the basis1 the electromagnetic basis of ,lorentz's theory of electrodynamics of moving bodies correspond to the relativity-postulate4 ,it can be briefly remarked here that the following important law follows easily from the equations developed in the present section3--if an electrically charged body moves in any manner in space1 and if its charge does not change thereby1 when regarded from a system moving along with it1 then the charge remains constant even when it is regarded from the stationary system ,k4 #aj4 ,dynamics of the ,electron 7slowly accelerated74 ,let us suppose that a point-shaped particle1 having the electrical charge e 7to be called henceforth the electron7 moves in the electromagnetic field2 we assume the following about its law of motion4 ,if the electron be at rest at any definite epoch1 then in the next 8particle of time10 the motion takes place according to the equations dx dy dz m ----- e,x1 m ----- e,y1 m ----- e,z dt dt dt ,where 7x1 y1 z7 are the co-ordinates of the electron1 and m is its mass4 ,let the electron possess the velocity v at a certain epoch of time4 ,let us now investigate the laws according to which the electron will move in the #8particle of time' immediately following this epoch4 ,without influencing the generality of treatment1 we can and we will assume that1 at the moment we are considering1 the electron is at the origin of co-ordinates1 and moves with the velocity v along the ,x-axis of the system4 ,it is clear that at this moment 7t #j7 the electron is at rest relative to the system k1 which moves parallel to the ,x-axis with the constant velocity v4 ,from the suppositions made above1 in combination with the principle of relativity1 it is clear that regarded from the system k1 the electron moves according to the equations d d d m ----- e,x1 m ----- e,y1 m ----- e,z 1 d d d in the time immediately following the moment1 where the symbols 71 1 1 1 ,x'1 ,y'1 ,z'7 refer to the system k4 ,if we now fix1 that for t v y z #j1 #j1 then the equations of transformation given in #c 7and #f7 hold1 and we have3 v 7t - ---- x71 7x - vt71 y1 z1 c v v ,x ,x1 ,y 7,y - --- ,n71 ,z 7,z --- ,m7 c c ,with the aid of these equations1 we can transform the above equations of motion from the system k to the system ,k1 and obtain3-- 7,a7 fracd#b xdt#b fracem frac#a;beta ,x 1 fracd#b ydt#b fracem frac#a;beta 7,y - fracvc ,n7 1 fracd#b zdt#b fracem frac#a;beta 7,z fracvc ,m7 ,let us now consider1 following the usual method of treatment1 the longitudinal and transversal mass of a moving electron4 ,we write the equations 7,a7 in the form dx m ----- e,x e,x dt dy v m ----- e 7,y - --- ,n7 e,y dt c dz v m ----- e 7,z - --- ,m7 e,z dt c and let us first remark1 that e,x1 e,y1 e,z are the components of the ponderomotive force acting upon the electron1 and are considered in a moving system which1 at this moment1 moves with a velocity which is equal to that of the electron4 ,this force can1 for example1 be measured by means of a spring-balance which is at rest in this last system4 ,if we briefly call this force as 8the force acting upon the electron10 and maintain the equation3-- ,mass-number acceleration-number force-number1 and if we further fix that the accelerations are measured in the stationary system ,k1 then from the above equations1 we obtain3-- ,longitudinal mass3 fracm7sqrt#a - fracv#b;c#b7;frac#cb ,transversal mass3 fracmsqrt#a - fracv#b;c#b ,naturally1 when other definitions are given of the force and the acceleration1 other numbers are obtained for the mass2 hence we see that we must proceed very carefully in comparing the different theories of the motion of the electron4 ,we remark that this result about the mass hold also for ponderable material mass2 for in our sense1 a ponderable material point may be made into an electron by the addition of an electrical charge which may be as small as possible4 ,let us now determine the kinetic energy of the electron4 ,if the electron moves from the origin of co-ordinates of the system ,k with the initial velocity #j steadily along the ,x-axis under the action of an electromotive force ,x1 then it is clear that the energy drawn from the electrostatic field has the value e,xdx4 ,since the electron is only slowly accelerated1 and in consequence1 no energy is given out in the form of radiation1 therefore the energy drawn from the electro-static field may be put equal to the energy ,w of motion4 ,considering the whole process of motion in questions1 the first of equations ,a7 holds1 we obtain3-- ,w int e,xdx int#jv mbeta#c vdv mc#b 7frac#asqrt#a - fracv#b;c#b - #a7 ,for v c1 ,w is infinitely great4 ,as our former result shows1 velocities exceeding that of light can have no possibility of existence4 ,in consequence of the arguments mentioned above1 this expression for kinetic energy must also hold for ponderable masses4 ,we can now enumerate the characteristics of the motion of the electrons available for experimental verification1 which follow from equations ,a74 #a4 ,from the second of equations ,a71 it follows that an electrical force ,y1 and a magnetic force ,n produce equal deflexions of an electron moving with the velocity v1 when ,y ,nv_/c4 ,therefore we see that according to our theory1 it is possible to obtain the velocity of an electron from the ratio of the magnetic deflexion ,am1 and the electric deflexion ,ae1 by applying the law3-- frac,am,ae fracvc ,this relation can be tested by means of experiments because the velocity of the electron can be directly measured by means of rapidly oscillating electric and magnetic fields4 #b4 ,from the value which is deduced for the kinetic energy of the electron1 it follows that when the electron falls through a potential difference of ,p1 the velocity v which is acquired is given by the following relation3-- ,p int ,xdx fracmec#b 7frac#asqrt#a - fracv#b;c#b - #a7 #c4 ,we calculate the radius of curvature ,r of the path1 where the only deflecting force is a magnetic force ,n acting perpendicular to the velocity of projection4 ,from the second of equations ,a7 we obtain3 - fracd#bydt#b fracv#b,r fracem fracvc ,n sqrt#a - fracv#b;c#b or mvc ,r ---------- e,n ,these three relations are complete expressions for the law of motion of the electron according to the above theory4 ,footnote #f3 ,vide ,note #i4 ,footnote #g3 ,vide ,note #i4 ,footnote #h3 ,vide ,note #ab4 ,,albrecht ,,einstein ,a short biographical note4 ,the name of ,prof4 ,albrecht ,einstein has now spread far beyond the narrow pale of scientific investigators owing to the brilliant confirmation of his predicted deflection of light-rays by the gravitational field of the sun during the total solar eclipse of ,may #bi1 #aiai4 ,but to the serious student of science1 he has been known from the beginning of the current century1 and many dark problems in physics has been illuminated with the lustre of his genius1 before1 owing to the latest sensation just mentioned1 he flashes out before public imagination as a scientific star of the first magnitude4 ,einstein is a ,swiss-,german of ,jewish extraction1 and began his scientific career as a privat-dozent in the ,swiss ,university of ,zrich about the year #aijb4 ,later on1 he migrated to the ,german ,university of ,prague in ,bohemia as ausser-ordentliche 7or associate7 ,professor4 ,in #aiad1 through the exertions of ,prof4 ,m4 ,planck of the ,berlin ,university1 he was appointed a paid member of the ,royal 7now ,national7 ,prussian ,academy of ,sciences1 on a salary of #ah1jjj marks per year4 ,in this post1 he has only to do and guide research work4 ,another distinguished occupant of the same post was ,van't ,hoff1 the eminent physical chemist4 ,it is rather difficult to give a detailed1 and consistent chronological account of his scientific activities1--they are so variegated1 and cover such a wide field4 ,the first work which gained him distinction was an investigation on ,brownian ,movement4 ,an admirable account will be found in ,perrin's book #8,the ,atoms4' ,starting from ,boltzmann's theorem connecting the entropy1 and the probability of a state1 he deduced a formula on the mean displacement of small particles 7colloidal7 suspended in a liquid4 ,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 gm4 molecule of gas 7,avogadro's number74 ,the formula was shortly afterwards verified by ,perrin1 ,prof4 of ,chemical ,physics in the ,sorbonne1 ,paris4 ,to ,einstein is also due the resuscitation of ,planck's quantum theory of energy-emission4 ,this theory has not yet caught the popular imagination to the same extent as the new theory of ,time1 and ,space1 but it is none the less iconoclastic in its scope as far as classical concepts are concerned4 ,it was known for a long time that the observed emission of light from a heated black body did not correspond to the formula which could be deduced from the older classical theories of continuous emission and propagation4 ,in the year #aijj1 ,prof4 ,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 h1 where h is a constant 7which is universal like the constant of gravitation71 and is the frequency of the light4 ,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 radiation1 it did not meet with much favour from the physicists4 ,in fact1 some one remarked jocularly that according to ,planck1 energy flies out of a radiator like a swarm of gnats4 ,but ,einstein found a support for the new-born concept in another direction4 ,it was known that if green or ultraviolet light was allowed to fall on a plate of some alkali metal1 the plate lost electrons4 ,the electrons were emitted with all velocities1 but there is generally a maximum limit4 ,from the investigations of ,lenard and ,ladenburg1 the curious discovery was made that this maximum velocity of emission did not at all depend upon the intensity of light1 but upon its wavelength4 ,the more violet was the light1 the greater was the velocity of emission4 ,to account for this fact1 ,einstein made the bold assumption that the light is propagated in space as a unit pulse 7he calls it a ,light-cell71 and falling upon an individual atom1 liberates electrons according to the energy equation #a h --- mv ,a1 #b where 7m1 v7 are the mass and velocity of the electron4 ,a is a constant characteristic of the metal plate4 ,there was little material for the confirmation of this law when it was first proposed 7#aije71 and eleven years elapsed before ,prof4 ,millikan established1 by a set of experiments scarcely rivalled for the ingenuity1 skill1 and care displayed1 the absolute truth of the law4 ,as results of this confirmation1 and other brilliant triumphs1 the quantum law is now regarded as a fundamental law of ,energetics4 ,in recent years1 ,x-rays have been added to the domain of light1 and in this direction also1 ,einstein's photo-electric formula has proved to be one of the most fruitful conceptions in ,physics4 ,the quantum law was next extended by ,einstein to the problems of decrease of specific heat at low temperature1 and here also his theory was confirmed in a brilliant manner4 ,we pass over his other contributions to the equation of state1 to the problems of null-point energy1 and photo-chemical reactions4 ,the recent experimental works of ,nernst and ,warburg seem to indicate that through ,einstein's genius1 we are probably for the first time having a satisfactory theory of photo-chemical action4 ,in #aiae1 ,einstein made an excursion into ,experimental ,physics1 and here also1 in his characteristic way1 he tackled one of the most fundamental concepts of ,physics4 ,it is well-known that according to ,ampere1 the magnetisation of iron and iron-like bodies1 when placed within a coil carrying an electric current is due to the excitation in the metal of small electrical circuits4 ,but the conception though a very fruitful one1 long remained without a trace of experimental proof1 though after the discovery of the electron1 it was generally believed that these molecular currents may be due to the rotational motion of free electrons within the metal4 ,it is easily seen that if in the process of magnetisation1 a number of electrons be set into rotatory motion1 then these will impart to the metal itself a turning couple4 ,the experiment is a rather difficult one1 and many physicists tried in vain to observe the effect4 ,but in collaboration with de ,haas1 ,einstein planned and successfully carried out this experiment1 and proved the essential correctness of ,ampere's views4 ,einstein's studies on ,relativity were commenced in the year #aije1 and has been continued up to the present time4 ,the first paper in the present collection forms ,einstein's first great contribution to the ,principle of ,special ,relativity4 ,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 #ahie1 ,lorentz1 ,einstein and ,minkowski have succeeded in building up a consistent1 and fruitful new theory of ,time and ,space4 ,but ,einstein was not satisfied with the study of the special problem of ,relativity for uniform motion1 but tried1 in a series of papers beginning from #aiaa1 to extend it to the case of non-uniform motion4 ,the last paper in the present collection is a translation of a comprehensive article which he contributed to the ,annalen der ,physik in #aiaf on this subject1 and gives1 in his own words1 the ,principles of ,generalized ,relativity4 ,the triumphs of this theory are now matters of public knowledge4 ,einstein is now only #de1 and it is to be hoped that science will continue to be enriched1 for a long time to come1 with further achievements of his genius4 ,principle of ,relativity ,,introduction4 ,at the present time1 different opinions are being held about the fundamental equations of ,electro-dynamics for moving bodies4 ,the ,hertzian#i forms must be given up1 for it has appeared that they are contrary to many experimental results4 ,in #ahie ,h4 ,a4 ,lorentz#aj published his theory of optical and electrical phenomena in moving bodies2 this theory was based upon the atomistic conception 7vorstellung7 of electricity1 and on account of its great success appears to have justified the bold hypotheses1 by which it has been ushered into existence4 ,in his theory1 ,lorentz proceeds from certain equations1 which must hold at every point of 8ther02 then by forming the average values over 8,physically infinitely small0 regions1 which however contain large numbers of electrons1 the equations for electro-magnetic processes in moving bodies can be successfully built up4 ,in particular1 ,lorentz's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous 8ther02 it shows that this fact is intimately connected with the covariance of the original equation1 when certain simultaneous transformations of the space and time co-ordinates are effected2 these transformations have therefore obtained from ,h4 ,poincare#aa the name of ,lorentz-transformations4 ,the covariance of these fundamental equations1 when subjected to the ,lorentz-transformation is a purely mathematical fact i4e4 not based on any physical considerations2 ,i will call this the ,theorem of ,relativity2 this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light4 ,now without recognizing any hypothesis about the connection between 8ther0 and matter1 we can expect these mathematically evident theorems to have their consequences so far extended--that thereby even those laws of ponderable media which are yet unknown may anyhow possess this covariance when subjected to a ,lorentz-transformation2 by saying this1 we do not indeed express an opinion1 but rather a conviction1--and this conviction ,i may be permitted to call the ,postulate of ,relativity4 ,the position of affairs here is almost the same as when the ,principle of ,conservation of ,energy was postulated in cases1 where the corresponding forms of energy were unknown4 ,now if hereafter1 we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies1 the definite connection may be styled #8the ,principle of ,relativity4' ,these differentiations seem to me to be necessary for enabling us to characterise the present day position of the electro-dynamics for moving bodies4 ,h4 ,a4 ,lorentz#ab has found out the 8,relativity theorem0 and has created the ,relativity-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law4 ,a4 ,einstein#ac has brought out the point very clearly1 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 phenomena4 ,the ,principle of ,relativity has not yet been formulated for electro-dynamics of moving bodies in the sense characterized by me4 ,in the present essay1 while formulating this principle1 ,i shall obtain the fundamental equations for moving bodies in a sense which is uniquely determined by this principle4 ,but it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle4#ad ,we would at first expect that the fundamental equations which are assumed by ,lorentz for moving bodies would correspond to the ,relativity ,principle4 ,but it will be shown that this is not the case for the general equations which ,lorentz has for any possible1 and also for magnetic bodies2 but this is approximately the case 7if neglect the square of the velocity of matter in comparison to the velocity of light7 for those equations which ,lorentz hereafter infers for non-magnetic bodies4 ,but this latter accordance with the ,relativity ,principle is due to the fact that the condition of non-magnetisation has been formulated in a way not corresponding to the ,relativity ,principle2 therefore the accordance is due to the fortuitous compensation of two contradictions to the ,relativity-,postulate4 ,but meanwhile enunciation of the ,principle in a rigid manner does not signify any contradiction to the hypotheses of ,lorentz's molecular theory1 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 done4 ,in an appendix1 ,i have gone into discussion of the position of ,classical ,mechanics with respect to the ,relativity ,postulate4 ,any easily perceivable modification of mechanics for satisfying the requirements of the ,relativity theory would hardly afford any noticeable difference in observable processes2 but would lead to very surprising consequences4 ,by laying down the ,relativity-,postulate from the outset1 sufficient means have been created for deducing henceforth the complete series of ,laws of ,mechanics from the principle of conservation of ,energy alone 7the form of the ,energy being given in explicit forms74 ,,notations4 ,let a rectangular system 7x1 y1 z1 t17 of reference be given in space and time4 ,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 unity4 ,although ,i would prefer not to change the notations used by ,lorentz1 it appears important to me to use a different selection of symbols1 for thereby certain homogeneity will appear from the very beginning4 ,i shall denote the vector electric force by ,e1 the magnetic induction by ,m1 the electric induction by e and the magnetic force by m1 so that 7,e1 ,m1 e1 m7 are used instead of ,lorentz's 7,e1 ,b1 ,d1 ,h7 respectively4 ,i shall further make use of complex magnitudes in a way which is not yet current in physical investigations1 i4e41 instead of operating with 7t71 ,i shall operate with 7i t71 where i denotes 7-#a74 ,if now instead of 7x1 y1 z1 i t71 ,i use the method of writing with indices1 certain essential circumstances will come into evidence2 on this will be based a general use of the suffixes 7#a1 #b1 #c1 #d74 ,the advantage of this method will be1 as ,i expressly emphasize here1 that we shall have to handle symbols which have apparently a purely real appearance2 we can however at any moment pass to real equations if it is understood that of the symbols with indices1 such ones as have the suffix #d only once1 denote imaginary quantities1 while those which have not at all the suffix #d1 or have it twice denote real quantities4 ,an individual system of values of 7x1 y1 z1 t7 i4 e41 of 7x x x x7 shall be called a space-time point4 ,further let u denote the velocity vector of matter1 the dielectric constant1 the magnetic permeability1 the conductivity of matter1 while denotes the density of electricity in space1 and x the vector of 8,electric ,current0 which we shall some across in #g and #h4 ,,part ,i #b4 ,the ,limiting ,case4 ,the ,fundamental ,equations for ther4 ,by using the electron theory1 ,lorentz in his above mentioned essay traces the ,laws of ,electro-dynamics of ,ponderable ,bodies to still simpler laws4 ,let us now adhere to these simpler laws1 whereby we require that for the limiting case #a1 #a1 #j1 they should constitute the laws for ponderable bodies4 ,in this ideal limiting case #a1 #a1 #j1 ,e will be equal to e1 and ,m to m4 ,at every space time point 7x1 y1 z1 t7 we shall have the equations#ae 7i7 ,curl m - 7e_/t7 u 7ii7 div e 7iii7 ,curl e m_/t #j 7iv7 div m #j ,i shall now write 7x x x x7 for 7x1 y1 z1 t7 and 71 1 1 7 for 7rho ux1 rho uy1 rho uz1 irho7 i4e4 the components of the convection current u1 and the electric density multiplied by -#a ,further ,i shall write f#b #c1 f#c #a1 f#a #b1 f#a #d1 f#b #d1 f#c #d4 for mx1 my1 mz1 -iex1 -iey1 -iez4 i4e41 the components of m and 7-i4e47 along the three axes2 now if we take any two indices 7h4 k7 out of the series #c1 #d71 fk h -fk h1 ,therefore f -f1 f -f1 f -f f -f1 f -f1 f -f ,then the three equations comprised in 7i71 and the equation 7ii7 multiplied by i becomes beginvmatrix fracdelta f#a #b;delta x#b fracdelta f#a #c;delta x#c fracdelta f#a #d;delta x#d rho#a fracdelta f#b #a;delta x#a fracdelta f#b #c;delta x#c times fracdelta f#b #d;delta x#d rho#b fracdelta f#c #a;delta x#a times fracdelta f#c #b;delta x#b fracdelta f#c #d;delta x#d rho#c fracdelta f#d #a;delta x#a fracdelta f#d #b;delta x#b fracdelta f#d #c;delta x#c rho#d endvmatrix ,on the other hand1 the three equations comprised in 7iii7 and the 7iv7 equation multiplied by 7i7 becomes beginvmatrix fracdelta f#c #d;delta x#b fracdelta f#d #b;delta x#c fracdelta f#b #c;delta x#d fracdelta f#d #c;delta x#a fracdelta f#a #d;delta x#c fracdelta f#c #a;delta x#d #j fracdelta f#b #d;delta x#a fracdelta f#d #a;delta x#b fracdelta f#a #b;delta x#d #j fracdelta f#c #b;delta x#a fracdelta f#a #c;delta x#b fracdelta f#b #a;delta x#c - endvmatrix ,by means of this method of writing we at once notice the perfect symmetry of the #ast as well as the #bnd system of equations as regards permutation with the indices1 7#a1 #b1 #c1 #d74 #c4 ,it is well-known that by writing the equations i7 to iv7 in the symbol of vector calculus1 we at once set in evidence an invariance 7or rather a 7covariance7 of the system of equations ,a7 as well as of ,b71 when the co-ordinate system is rotated through a certain amount round the null-point4 ,for example1 if we take a rotation of the axes round the z-axis1 through an amount 1 keeping e1 m fixed in space1 and introduce new variables x x x x instead of x x x x where x x cos x sin 1 x -x sin x cos 1 x x1 x x1 and introduce magnitudes 1 1 1 1 where cos sin 1 - sin cos and f#a #b1 444 444 f#c #d1 where f f cos f sin 1 f - f sin f cos 1 f f1 f f cos f sin 1 f - f sin f cos 1 f f#c #d1 fk h - fk h 7h l k #a1 #b1 #c1 #d74 then out of the equations 7,a7 would follow a corresponding system of dashed equations 7,a7 composed of the newly introduced dashed magnitudes4 ,so upon the ground of symmetry alone of the equations 7,a7 and 7,b7 concerning the suffixes 7#a1 #b1 #c1 #d71 the theorem of ,relativity1 which was found out by ,lorentz1 follows without any calculation at all4 ,i will denote by i a purely imaginary magnitude1 and consider the substitution x x1 x x1 x x cos i x sin i1 7#a7 x - x sin i x cos i1 ,putting - i tan ipsi fracepsi - e-psiepsie-psi q 1 psi frac#ab log frac#a q#a - q 7#b7 ,we shall have cos i #a_/7a - q71 sin i iq_/7#a - q7 where -#a q #a1 and 7#a - q7 is always to be taken with the positive sign4 ,let us now write x x1 x y1 x z1 x it 7#c7 then the substitution #a7 takes the form x x1 y y1 z 7z - qt7_/7#a - q71 t 7-qz t7_/7#a - q71 7#d7 the coefficients being essentially real4 ,if now in the above-mentioned rotation round the ,z-axis1 we replace #a1 #b1 #c1 #d throughout by #c1 #d1 #a1 #b1 and by i1 we at once perceive that simultaneously1 new magnitudes 1 1 1 1 where 1 1 cos i sin i1 - sin i cos i71 and f#a #b 444 f#c #d1 where f#d #a f#d #a cos i f#a #c sin i1 f#a #c - f#d #a sin i f#a #c cos i1 f#c #d f#c #d1 f#c #b f#c #b cos i f#d #b sin i1 f#d #b - f#c #b sin i f#d #b cos i1 f#a #b f#a #b1 fk h - fk h1 must be introduced4 ,then the systems of equations in 7,a7 and 7,b7 are transformed into equations 7,a71 and 7,b71 the new equations being obtained by simply dashing the old set4 ,all these equations can be written in purely real figures1 and we can then formulate the last result as follows4 ,if the real transformations #d7 are taken1 and x y z t be taken as a new frame of reference1 then we shall have 7#e7 7-quz #a7_/7a - q71 uz 7uz - q7_/7#a - q71 ux ux1 uy uy4 7#f7 ex 7ex - qmy7_/77#a - q771 mr 7qex my7_/77#a - q771 ez ez4 7#g7 mx 7mx - qey7_/77#a - q771 ey 7qmx ey7_/77#a - q771 mz mz4 ,then we have for these newly introduced vectors u1 e1 m 7with components ux1 uy1 uz2 ex1 ey1 ez2 mx1 my1 mz71 and the quantity a series of equations ,i71 ,,ii71 ,,iii71 ,,iv7 which are obtained from ,i71 ,,ii71 ,,iii71 ,,iv7 by simply dashing the symbols4 ,we remark here that ex - qmy1 ey qmx are components of the vector e vm1 where v is a vector in the direction of the positive ,z-axis1 and v q1 and vm is the vector product of v and m2 similarly -qex my1 mx qey are the components of the vector m - ve4 ,the equations #f7 and #g71 as they stand in pairs1 can be expressed as4 ex imx 7ex imx7 cos i 7ey imy7 sin i1 ey imy - 7ex imx7 sin i 7ey imy7 cos i1 ez imz ez imz4 ,if denotes any other real angle1 we can form the following combinations3-- 7ex imx7 cos4 7ey imy7 sin 7ex imx7 cos4 7 i7 7ey imy7 sin 7 i71 7ex imx7 sin 7ey imy7 cos4 - 7ex imx7 sin 7 i7 7ey imy7 cos4 7 i74 #d4 ,special ,lorentz ,transformation4 ,the rle which is played by the ,z-axis in the transformation 7#d7 can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis4 ,so we came to a more general law3-- ,let v be a vector with the components vx1 vy1 vz1 and let v q #a4 ,by we shall denote any vector which is perpendicular to v1 and by rv1 r we shall denote components of r in direction of and v4 ,instead of 7x1 y1 z1 t71 new magnetudes 7x y z t7 will be introduced in the following way4 ,if for the sake of shortness1 r is written for the vector with the components 7x1 y1 z7 in the first system of reference1 r for the same vector with the components 7x y z7 in the second system of reference1 then for the direction of v1 we have 7#aj7 rv 7rv - qt7_/7#a - q7 and for the perpendicular direction 1 7#aa7 r r and further 7#ab7 t 7-qrv t7_/7#a - q74 ,the notations 7r1 rv7 are to be understood in the sense that with the directions v1 and every direction perpendicular to v in the system 7x1 y1 z7 are always associated the directions with the same direction cosines in the system 7x y z74 ,a transformation which is accomplished by means of 7#aj71 7#aa71 7#ab7 with the condition #j q #a will be called a special ,lorentz-transformation4 ,we shall call v the vector1 the direction of v the axis1 and the magnitude of v the moment of this transformation4 ,if further and the vectors u1 e1 m1 in the system 7x y z7 are so defined that1 7#ac7 7-quv #a7_/7a - q71 uv 7uv - q7_/7#a - q71 u uv1 further 7#ad7 7e im7 77e im7 - iv1 7e im7'7_/7#a - q74 7#ae7 7e im7v 7e im7 - iu1 7e im7v4 ,then it follows that the equations ,i71 ,,ii71 ,,iii71 ,,iv7 are transformed into the corresponding system with dashes4 ,the solution of the equations 7#aj71 7#aa71 7#ab7 leads to 7#af7 rv 7rv qt7_/7#a - q71 r r1 t 7qrv t7_/7#a - q71 ,now we shall make a very important observation about the vectors u and u4 ,we can again introduce the indices #a1 #b1 #c1 #d1 so that we write 7x1 x1 x1 x7 instead of 7x1 y1 z1 it7 and 1 1 1 instead of 7ux1 uy1 uz1 i74 ,like the rotation round the ,z-axis1 the transformation 7#d71 and more generally the transformations 7#aj71 7#aa71 7#ab71 are also linear transformations with the determinant #a1 so that 7#ag7 x x x x i4 e4 x y z - t1 is transformed into x x x x i4 e4 x y z - t4 ,on the basis of the equations 7#ac71 7#ad71 we shall have 7 7 7#a - ux1 -uy1 -uz7 7#a - u7 transformed into 7#a - u7 or in other words1 7#ah7 7#a - u7 is an invariant in a ,lorentz-transformation4 ,if we divide 71 1 1 7 by this magnitude1 we obtain the four values 71 1 1 7 7#a_/7a - u777ux1 uy1 uz1 i7 so that -#a4 ,it is apparent that these four values are determined by the vector u and inversely the vector u of magnitude #a follows from the #d values 1 1 1 2 where 71 1 7 are real1 -i real and positive and condition 7#ai7 is fulfilled4 ,the meaning of 71 1 1 7 here is1 that they are the ratios of dx1 dx1 dx1 dx to 7#bj7 7-7dx dx dx dx77 dt7#a - u74 ,the differentials denoting the displacements of matter occupying the spacetime point 7x1 x1 x1 x7 to the adjacent space-time point4 ,after the ,lorentz-transformation is accomplished the velocity of matter in the new system of reference for the same space-time point 7x y z t7 is the vector u with the ratios dx_/dt1 dy_/dt1 dz_/dt1 dl_/dt1 as components4 ,now it is quite apparent that the system of values x 1 x 1 x 1 x is transformed into the values x 1 x 1 x 1 x in virtue of the ,lorentz-transformation 7#aj71 7#aa71 7#ab74 ,the dashed system has got the same meaning for the velocity u after the transformation as the first system of values has got for u before transformation4 ,if in particular the vector v of the special ,lorentz-transformation be equal to the velocity vector u of matter at the space-time point 7x1 x1 x1 x7 then it follows out of 7#aj71 7#aa71 7#ab7 that #j1 #j1 #j1 i ,under these circumstances therefore1 the corresponding space-time point has the velocity v #j after the transformation1 it is as if we transform to rest4 ,we may call the invariant 7#a - u7 the rest-density of ,electricity4#af #e4 ,space-time ,vectors4 ,of the #ast and #bnd kind4 ,if we take the principal result of the ,lorentz transformation together with the fact that the system 7,a7 as well as the system 7,b7 is covariant with respect to a rotation of the coordinate-system round the null point1 we obtain the general relativity theorem4 ,in order to make the facts easily comprehensible1 it may be more convenient to define a series of expressions1 for the purpose of expressing the ideas in a concise form1 while on the other hand ,i shall adhere to the practice of using complex magnitudes1 in order to render certain symmetries quite evident4 ,let us take a linear homogeneous transformation1 beginvmatrix x#a x#b x#c x#d endvmatrix beginvmatrix a#a #a a#a #b a#a #c a#a #d a#b #a a#b #b a#b #c a#b #d a#c #a a#c #b a#c #c a#c #d a#d #a a#d #b a#d #c a#d #d endvmatrix beginvmatrix x#a' x#b' x#c' x#d' endvmatrix the ,determinant of the matrix is #a1 all co-efficients without the index #d occurring once are real1 while a1 a1 a1 are purely imaginary1 but a is real and #j1 and x x x x transforms into x x x x4 ,the operation shall be called a general ,lorentz transformation4 7,this notation1 which is due to ,dr4 ,c4 ,e4 ,cullis of the ,calcutta ,university1 has been used throughout instead of ,minkowski's notation1 x ax ax ax ax47 ,if we put x x1 x y1 x z1 x it1 then immediately there occurs a homogeneous linear transformation of 7x1 y1 z1 t7 to 7x1 y1 z1 t7 with essentially real co-efficients1 whereby the aggregate -x - y - z t transforms into -x - y - z t1 and to every such system of values x1 y1 z1 t with a positive t1 for which this aggregate #j1 there always corresponds a positive t'2 this last is quite evident from the continuity of the aggregate x1 y1 z1 t4 ,the last vertical column of co-efficients has to fulfil the condition #bb7 a a a a #a4 ,if a a a #j1 then a #a1 and the ,lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point4 ,if a1 a1 a are not all zero1 and if we put a 3 a 3 a 3 a vx 3 vy 3 vz 3 i q 7vx vy vz7 #a4 ,on the other hand1 with every set of values of a1 a1 a1 a which in this way fulfil the condition #bb7 with real values of vx1 vy1 vz1 we can construct the special ,lorentz transformation 7#af7 with 7a1 a1 a1 a7 as the last vertical column1--and then every ,lorentz-transformation with the same last vertical column 7a1 a1 a1 a7 can be supposed to be composed of the special ,lorentz-transformation1 and a rotation of the spatial co-ordinate system round the null-point4 ,the totality of all ,lorentz-,transformations forms a group4 ,under a space-time vector of the #ast kind shall be understood a system of four magnitudes 71 1 1 7 with the condition that in case of a ,lorentz-transformation it is to be replaced by the set 71 1 1 71 where these are the values of 7x1 x1 x1 x71 obtained by substituting 71 1 1 7 for 7x1 x1 x1 x7 in the expression 7#ba74 ,besides the time-space vector of the #ast kind 7x1 x1 x1 x7 we shall also make use of another space-time vector of the first kind 7y1 y1 y1 y71 and let us form the linear combination 7#bc7 f7xy - xy7 f7xy - xy7 f7xy - xy7 f7xy - xy7 f7xy - xy7 f7xy - xy7 with six coefficients f--f4 ,let us remark that in the vectorial method of writing1 this can be constructed out of the four vectors4 x1 x1 x2 y1 y1 y2 f1 f1 f2 f1 f1 f and the constants x and y1 at the same time it is symmetrical with regard the indices 7#a1 #b1 #c1 #d74 ,if we subject 7x1 x1 x1 x7 and 7y1 y1 y1 y7 simultaneously to the ,lorentz transformation 7#ba71 the combination 7#bc7 is changed to3 7#bd7 f7xy - xy7 f7xy - xy7 f 7xy - xy7 f7xy7 - xy7 f7xy - xy7 f7xy - xy71 where the coefficients f1 f1 f1 f1 f1 f1 depend solely on 7f f7 and the coefficients a 444 a4 ,we shall define a space-time ,vector of the #bnd kind as a system of six-magnitudes f1 f 444 f1 with the condition that when subjected to a ,lorentz transformation1 it is changed to a new system f 444 f1 444 which satisfies the connection between 7#bc7 and 7#bd74 ,i enunciate in the following manner the general theorem of relativity corresponding to the equations 7,i7-7iv71--which are the fundamental equations for ther4 ,if x1 y1 z1 it 7space co-ordinates1 and time it7 is subjected to a ,lorentz transformation1 and at the same time 7pux1 puy1 puz1 i7 7convection-current1 and charge density i7 is transformed as a space time vector of the #ast kind1 further 7mx1 my1 mz1 -iex1 -iey1 -iez7 7magnetic force1 and electric induction 7-i7 is transformed as a space time vector of the #bnd kind1 then the system of equations 7,i71 7,,ii71 and the system of equations 7,,iii71 7,,iv7 transforms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system4 ,these facts can be more concisely expressed in these words3 the system of equations 7,i and ,,ii7 as well as the system of equations 7,,iii7 7,,iv7 are covariant in all cases of ,lorentz-transformation1 where 7u1 i7 is to be transformed as a space time vector of the #ast kind1 7m - ie7 is to be treated as a vector of the #bnd kind1 or more significantly1-- 7u1 i7 is a space time vector of the #ast kind1 7m - ie7#ag is a space-time vector of the #bnd kind4 ,i shall add a few more remarks here in order to elucidate the conception of space-time vector of the #bnd kind4 ,clearly1 the following are invariants for such a vector when subjected to a group of ,lorentz transformation4 7i7 m - e f f f f f f me i7ff ff ff74 ,a space-time vector of the second kind 7m - ie71 where 7m and e7 are real magnitudes1 may be called singular1 when the scalar square 7m - ie7 #j1 ie m - e #j1 and at the same time 7m e7 #j1 ie the vector m and e are equal and perpendicular to each other2 when such is the case1 these two properties remain conserved for the space-time vector of the #bnd kind in every ,lorentz-transformation4 ,if the space-time vector of the #bnd kind is not singular1 we rotate the spacial co-ordinate system in such a manner that the vector-product me coincides with the ,z-axis1 i4e4 mx #j1 ex #j4 ,then 7mx1 -i ex7 7my1 -i ey7 #j4 ,therefore 7ey i my7_/7ex i ex7 is different from i1 and we can therefore define a complex argument 7 i7 in such a manner that tan 7 i7 ey i my ------------------------- ex i mx ,if then1 by referring back to equations 7#i71 we carry out the transformation 7#a7 through the angle and a subsequent rotation round the ,z-axis through the angle 1 we perform a ,lorentz-transformation at the end of which my #j1 ey #j1 and therefore m and e shall both coincide with the new ,z-axis4 ,then by means of the invariants m - e1 7me7 the final values of these vectors1 whether they are of the same or of opposite directions1 or whether one of them is equal to zero1 would be at once settled4 #f4 ,concept of ,time4 ,by the ,lorentz transformation1 we are allowed to effect certain changes of the time parameter4 ,in consequence of this fact1 it is no longer permissible to speak of the absolute simultaneity of two events4 ,the ordinary idea of simultaneity rather presupposes that six independent parameters1 which are evidently required for defining a system of space and time axes1 are somehow reduced to three4 ,since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately1 we maintain that the simultaneity of two events exists of themselves4#ah ,in fact1 the following considerations will prove conclusive4 ,let a reference system 7x1 y1 z1 t7 for space time points 7events7 be somehow known4 ,now if a space point ,a 7x1 y1 z7 the time t be compared with a space point ,p 7x1 y1 z7 at the time t1 and if the difference of time t - t1 7let t t7 be less than the length ,a ,p i4e4 less than the time required for the propagation of light from ,a to ,p1 and if q 7t - t7_/7,a ,p7 #a1 then by a special ,lorentz transformation1 in which ,a ,p is taken as the axis1 and which has the moment q1 we can introduce a time parameter t1 which 7see equation #aa1 #ab1 #d7 has got the same value t #j for both space-time points 7,a1 t71 and 7,p1 t74 ,so the two events can now be comprehended to be simultaneous4 ,further1 let us take at the same time t #j1 two different space-points ,a1 ,b1 or three space-points 7,a1 ,b1 ,c7 which are not in the same space-line1 and compare therewith a space point ,p1 which is outside the line ,a ,b1 or the plane ,a ,b ,c1 at another time t1 and let the time difference t - t 7t t7 be less than the time which light requires for propagation from the line ,a ,b1 or the plane 7,a ,b ,c7 to ,p4 ,let q be the quotient of 7t - t7 by the second time4 ,then if a ,lorentz transformation is taken in which the perpendicular from ,p on ,a ,b1 or from ,p on the plane ,a ,b ,c is the axis1 and q is the moment1 then all the three 7or four7 events 7,a1 t71 7,b1 t71 7,c1 t7 and 7,p1 t7 are simultaneous4 ,if four space-points1 which do not lie in one plane1 are conceived to be at the same time t1 then it is no longer permissible to make a change of the time parameter by a ,lorentz-transformation1 without at the same time destroying the character of the simultaneity of these four space points4 ,to the mathematician1 accustomed on the one hand to the methods of treatment of the poly-dimensional manifold1 and on the other hand to the conceptual figures of the so-called non-,euclidean ,geometry1 there can be no difficulty in adopting this concept of time to the application of the ,lorentz-transformation4 ,the paper of ,einstein which has been cited in the ,introduction1 has succeeded to some extent in presenting the nature of the transformation from the physical standpoint4 ,,part ,,ii4 ,,electro-,,magnetic ,,phenomena4 #g4 ,fundamental ,equations for bodies at rest4 ,after these preparatory works1 which have been first developed on account of the small amount of mathematics involved in the limiting case #a1 #a1 #j1 let us turn to the electro-magnetic phenomena in matter4 ,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 time1 and therefore at every space-time point as functions of 7x1 y1 z1 t73--the vector of the electric force ,e1 the magnetic induction ,m1 the electrical induction e1 the magnetic force m1 the electrical space-density 1 the electric current s 7whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process71 and lastly the vector v1 the velocity of matter4 ,the relations in question can be divided into two classes4 ,firstly--those equations1 which1--when v1 the velocity of matter is given as a function of 7x1 y1 z1 t71--lead us to a knowledge of other magnitude as functions of x1 y1 z1 t--,i shall call this first class of equations the fundamental equations-- ,secondly1 the expressions for the ponderomotive force1 which1 by the application of the ,laws of ,mechanics1 gives us further information about the vector u as functions of 7x1 y1 z1 t74 ,for the case of bodies at rest1 i4e4 when u 7x1 y1 z1 t7 #j the theories of ,maxwell 7,heaviside1 ,hertz7 and ,lorentz lead to the same fundamental equations4 ,they are2-- 7#a7 ,the ,differential ,equations3--which contain no constant referring to matter3-- 7i7 ,curl m - e_/t ,c1 7ii7 div e l4 7iii7 ,curl ,e ,m_/t #j1 7iv7 ,div ,m #j4 7#b7 ,further relations1 which characterise the influence of existing matter for the most important case to which we limit ourselves i4e4 for isotopic bodies2--they are comprised in the equations 7,v7 e ,e1 ,m m1 ,c ,e4 where dielectric constant1 magnetic permeability1 the conductivity of matter1 all given as function of x1 y1 z1 t2 s is here the conduction current4 ,by employing a modified form of writing1 ,i shall now cause a latent symmetry in these equations to appear4 ,i put1 as in the previous work1 x x1 x y1 x z1 x it1 and write s1 s1 s1 s for ,cx1 ,cy1 ,cz 7-#a74 ,further f1 f1 f1 f1 f1 f for mx1 my1 mz1 -i7ex1 ey1 ez71 and ,f1 ,f1 ,f1 ,f1 ,f1 ,f for ,mx1 ,my1 ,mz1 -i7,ex1 ,ey1 ,ez7 lastly we shall have the relation fk h - fh k1 ,fk h -,fh k1 7the letter f1 ,f shall denote the field1 s the 7i4e4 current74 ,then the fundamental ,equations can be written as 7,a7 f_/x f_/x f_/x s f_/x f_/x f_/x s f_/x f_/x f_/x s f_/x f_/x f_/x s and the equations 7#c7 and 7#d71 are ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,fx #j ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j #h4 ,the ,fundamental ,equations4 ,we are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively4 ,the first ,axion shall be1-- ,when a detached region#ai of matter is at rest at any moment1 therefore the vector u is zero1 for a system 7x1 y1 z1 t7--the neighbourhood may be supposed to be in motion in any possible manner1 then for the space-time point x1 y1 z1 t1 the same relations 7,a7 7,b7 7,v7 which hold in the case when all matter is at rest1 shall also hold between 1 the vectors ,c1 e1 m1 ,m1 ,e and their differentials with respect to x1 y1 z1 t4 ,the second axiom shall be3-- ,every velocity of matter is #a1 smaller than the velocity of propagation of light4#bj ,the fundamental equations are of such a kind that when 7x1 y1 z1 it7 are subjected to a ,lorentz transformation and thereby 7m - ie7 and 7,m - i,e7 are transformed into space-time vectors of the second kind1 7,c1 i7 as a space-time vector of the #ast kind1 the equations are transformed into essentially identical forms involving the transformed magnitudes4 ,shortly ,i can signify the third axiom as3-- 7m1 -ie71 and 7,m1 -i,e7 are space-time vectors of the second kind1 7,c1 ip7 is a space-time vector of the first kind4 ,this axiom ,i call the ,principle of ,relativity4 ,in fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way4 ,according to the second axiom1 the magnitude of the velocity vector u is #a at any space-time point4 ,in consequence1 we can always write1 instead of the vector u1 the following set of four allied quantities ux_/7#a - u71 uy_/7#a - u71 uz_/7#a - u71 i_/7#a - u7 with the relation 7#bg7 - ,from what has been said at the end of #d1 it is clear that in the case of a ,lorentz-transformation1 this set behaves like a space-time vector of the #ast kind4 ,let us now fix our attention on a certain point 7x1 y1 z7 of matter at a certain time 7t74 ,if at this space-time point u #j1 then we have at once for this point the equations 7,a71 7,b7 7,v7 of #g4 ,if u #j1 then there exists according to #af71 in case u #a1 a special ,lorentz-transformation1 whose vector v is equal to this vector u 7x1 y1 z1 t71 and we pass on to a new system of reference 7x y z t7 in accordance with this transformation4 ,therefore for the space-time point considered1 there arises as in #d1 the new values #bh7 #j1 #j1 #j1 i1 therefore the new velocity vector #j1 the space-time point is as if transformed to rest4 ,now according to the third axiom the system of equations for the transformed point 7x y z t7 involves the newly introduced magnitude 7u 1 ,c1 e1 m1 ,e1 ,m7 and their differential quotients with respect to 7x1 y1 z1 t7 in the same manner as the original equations for the point 7x1 y1 z1 t74 ,but according to the first axiom1 when u #j1 these equations must be exactly equivalent to 7#a7 the differential equations 7,a71 7,b71 which are obtained from the equations 7,a71 7,b7 by simply dashing the symbols in 7,a7 and 7,b74 7#b7 and the equations 7,v7 e ,e1 ,m' m1 ,c ,e where 1 1 are the dielectric constant1 magnetic permeability1 and conductivity for the system 7x y z t7 i4e4 in the space-time point 7x y1 z t7 of matter4 ,now let us return1 by means of the reciprocal ,lorentz-transformation to the original variables 7x1 y1 z1 t71 and the magnitudes 7u1 1 ,c1 e1 m1 ,e1 ,m7 and the equations1 which we then obtain from the last mentioned1 will be the fundamental equations sought by us for the moving bodies4 ,now from #d1 and #f1 it is to be seen that the equations ,a71 as well as the equations ,b7 are covariant for a ,lorentz-transformation1 i4e4 the equations1 which we obtain backwards from ,a7 ,b71 must be exactly of the same form as the equations ,a7 and ,b71 as we take them for bodies at rest4 ,we have therefore as the first result3-- ,the differential equations expressing the fundamental equations of electrodynamics for moving bodies1 when written in and the vectors ,c1 e1 m1 ,e1 ,m1 are exactly of the same form as the equations for moving bodies4 ,the velocity of matter does not enter in these equations4 ,in the vectorial way of writing1 we have ,i7 curl m - e_/t ,c1 ,,ii7 div e ,,iii7 curl ,e ,m_/t #j ,,iv7 div ,m #j ,the velocity of matter occurs only in the auxiliary equations which characterise the influence of matter on the basis of their characteristic constants 1 1 4 ,let us now transform these auxiliary equations ,v7 into the original co-ordinates 7x1 y1 z1 and t47 ,according to formula #ae7 in #d1 the component of e in the direction of the vector u is the same as that of 7e u m71 the component of m is the same as that of m - u e1 but for the perpendicular direction 1 the components of e1 m are the same as those of 7e u m7 and 7m - u e1 multiplied by #a_/7a - u74 ,on the other hand ,e and ,m shall stand to ,e u,m1 and ,m - u,e in the same relation as e and m to e um1 and m - 7ue74 ,from the relation e ,e1 the following equations follow 7,c7 e um 7,e u,m71 and from the relation ,m m1 we have 7,d7 ,m - u ,e 7m - ue71 ,for the components in the directions perpendicular to u1 and to each other1 the equations are to be multiplied by 7#a - u74 ,then the following equations follow from the transformation8 equations 7#ab71 7#aj71 7#aa7 in #d1 when we replace q1 rv1 r1 t1 rv1 r1 t' by u1 ,cu1 ,c1 1 ,cu1 ,c1 7-u ,cu 7_/7#a - u71 ,c'u 7,cu - u7_/7#a - u71 ,c ,c1 ,e7 7,cu - u7_/7#a - u7 7,e u,m7u1 ,c 7,e u,m7u_/7#a - u74 ,in consideration of the manner in which enters into these relations1 it will be convenient to call the vector ,c - u with the components ,cu - u in the direction of u1 and ,c in the directions perpendicular to u the 8,convection current40 ,this last vanishes for #j4 ,we remark that for #a1 #a the equations e ,e1 m ,m immediately lead to the equations e ,e1 m ,m by means of a reciprocal ,lorentz-transformation with -u as vector2 and for #j1 the equation ,c #j leads to ,c u2 that the fundamental equations of ther discussed in #b becomes in fact the limitting case of the equations obtained here with #a1 #a1 #j4 #i4 ,the ,fundamental ,equations in ,lorentz's ,theory4 ,let us now see how far the fundamental equations assumed by ,lorentz correspond to the ,relativity postulate1 as defined in #h4 ,in the article on ,electron-theory 7,ency41 ,math41 ,wiss41 ,bd4 ,v4 #b1 ,art #ad7 ,lorentz has given the fundamental equations for any possible1 even magnetised bodies 7see there page #bji1 ,eqn ,,xxx1 formula 7#ad7 on page #gh of the same 7part74 7,,iiia7 ,curl 7,h - u,e7 ,j d,d_/dt u div ,d - curl u,d4 7,i7 div ,d 7,,iv7 curl ,e - d,b_/dt1 ,div ,b #j 7,v7 ,then for moving non-magnetised bodies1 ,lorentz puts 7page #bbc1 #c7 #a1 ,b ,h1 and in addition to that takes account of the occurrence of the di-electric constant 1 and conductivity according to equations 7q,,xxxiv1 p4 #cbg7 ,d - ,e 7 - #a7 ,e u,b 7q,,xxxiii1 p4 #bbc7 ,j 7,e u,b7 ,lorentz's ,e1 ,d1 ,h are here denoted by ,e1 ,m1 e1 m while ,j denotes the conduction current4 ,the three last equations which have been just cited here coincide with eqn 7,,ii71 7,,iii71 7,,iv71 the first equation would be1 if ,j is identified with ,c1 u 7the current being zero for #j1 7#bi7 ,curl ,h - 7u1 ,e7 ,c d,d_/dt - curl u,d1 and thus comes out to be in a different form than 7#a7 here4 ,therefore for magnetised bodies1 ,lorentz's equations do not correspond to the ,relativity ,principle4 ,on the other hand1 the form corresponding to the relativity principle1 for the condition of non-magnetisation is to be taken out of 7,d7 in #h1 with #a1 not as ,b ,h1 as ,lorentz takes1 but as 7#cj7 ,b - u,d ,h - u,d 7,m - u,e m - ue4 ,now by putting ,h ,b1 the differential equation 7#bi7 is transformed into the same form as eqn 7#a7 here when m - ue ,m - u,e4 ,therefore it so happens that by a compensation of two contradictions to the relativity principle1 the differential equations of ,lorentz for moving non-magnetised bodies at last agree with the relativity postulate4 ,if we make use of 7#cj7 for non-magnetic bodies1 and put accordingly ,h ,b u1 7,d - ,e71 then in consequence of 7,c7 in #h1 7 - #a7 7,e u1 ,b7 ,d - ,e u4 u1 ,d - ,e1 i4e4 for the direction of u1 7 - #a7 7,e u,b7u 7,d - ,e7u and for a perpendicular direction 1 7 - #a7 ,e 7u,b7u 7#a - u7 7,d - ,e7u i4e4 it coincides with ,lorentz's assumption1 if we neglect u in comparison to #a4 ,also to the same order of approximation1 ,lorentz's form for ,j corresponds to the conditions imposed by the relativity principle comp4 7,e7 #h--that the components of ,ju1 ,j are equal to the components of 7,e u ,b7 multiplied by 7#a - u7 or #a _/ 7#a - u7 respectively4 #aj4 ,fundamental ,equations of ,e4 ,cohn4 ,e4 ,cohn assumes the following fundamental equations4 7#ca7 ,curl 7,m u ,e7 d,e_/dt u div4 ,e ,j - ,curl ,e - 7u4 ,m7 d,m_/dt u div4 ,m4 7#cb7 ,j ,e1 ,e - u ,m1 ,m 7m u ,e47 where ,e ,m are the electric and magnetic field intensities 7forces71 ,e1 ,m are the electric and magnetic polarisation 7induction74 ,the equations also permit the existence of true magnetism2 if we do not take into account this consideration1 div4 ,m4 is to be put #j4 ,an objection to this system of equations1 is that according to these1 for #a1 #a1 the vectors force and induction do not coincide4 ,if in the equations1 we conceive ,e and ,m and not ,e - 7,u4 ,m71 and ,m ,u ,e as electric and magnetic forces1 and with a glance to this we substitute for ,e1 ,m1 ,e1 ,m1 div4 ,e1 the symbols e1 ,m1 ,e ,u ,m1 m - u e1 1 then the differential equations transform to our equations1 and the conditions 7#cb7 transform into ,j 7,e u ,m7 e u1 7m - u e7 7,e u ,m7 ,m - u1 7,e u ,m7 7m - u e7 then in fact the equations of ,cohn become the same as those required by the relativity principle1 if errors of the order u are neglected in comparison to #a4 ,it may be mentioned here that the equations of ,hertz become the same as those of ,cohn1 if the auxiliary conditions are 7#cc7 ,e ,e1 ,m ,m1 ,j ,e4 #aa4 ,typical ,representations of the ,fundamental ,equations4 ,in the statement of the fundamental equations1 our leading idea had been that they should retain a covariance of form1 when subjected to a group of ,lorentz-transformations4 ,now we have to deal with ponderomotive reactions and energy in the electro-magnetic field4 ,here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations1 satisfying the conditions of covariance4 ,in order to arrive at such forms1 ,i shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a ,lorentz-transformation4 ,here ,i am using a method of calculation1 which enables us to deal in a simple manner with the space-time vectors of the #ast1 and #bnd kind1 and of which the rules1 as far as required are given below4 ,a system of magnitudes ah k formed into the matrix a44444444444444444444a#a q ap #a44444444444;ap q arranged in p horizontal rows1 and q vertical columns is called a p q series-matrix1 and will be denoted by the letter ,a4 ,if all the quantities ah k are multiplied by ,c1 the resulting matrix will be denoted by ,,ca4 ,if the roles of the horizontal rows and vertical columns be intercharged1 we obtain a q p series matrix1 which will be known as the transposed matrix of ,a1 and will be denoted by 4 a 4444444444444444444444 ap #a a#a q 444444444444 ap q ,if we have a second p q series matrix ,b1 ,b b 4444444444444444444444444 bq bp #a 4444444444444 bp q then ,a ,b shall denote the p q series matrix whose members are ah k bh k4 #b ,if we have two matrices ,a a 444444444444444444444 a#a q ap #a 44444444444 ap q ,b b#a #a 44444444444444 b#a r bq #a 4444444444 bp r where the number of horizontal rows of ,b1 is equal to the number of vertical columns of ,a1 then by ,,ab1 the product of the matrices ,a and ,b1 will be denoted the matrix ,c c 4444444444444444444444 c#a r cp r 44444444444 cp p where ch k ah #a bk ah #b b#b h 444 ak s bs k 444 ak q bq h these elements being formed by combination of the horizontal rows of ,a with the vertical columns of ,b4 ,for such a point1 the associative law 7,,ab7,s ,a7,,bs7 holds1 where ,s is a third matrix which has got as many horizontal rows as ,b 7or ,,ab7 has got vertical columns4 ,for the transposed matrix of ,c ,,ba1 we have #c4 ,we shall have principally to deal with matrices with at most four vertical columns and for horizontal rows4 ,as a unit matrix 7in equations they will be known for the sake of shortness as the matrix #a7 will be denoted the following matrix 7#d #d series7 with the elements4 7#cd7 e e e e #a #j #j #j e e e e #j #a #j #j e e e e #j #j #a #j e e e e #j #j #j #a ,for a #d #d series-matrix1 ,det ,a shall denote the determinant formed of the #d #d elements of the matrix4 ,if det ,a #j1 then corresponding to ,a there is a reciprocal matrix1 which we may denote by ,a so that ,a,a #a4 ,a matrix f #j f f f f #j f f f f #j f f f f #j in which the elements fulfil the relation fh k -fh k1 is called an alternating matrix4 ,these relations say that the transposed matrix -f4 ,then by f99 will be the dual1 alternating matrix 7#ce7 f99 #j f f f f #j f f f f #j f f f f #j ,then 7#cf7 f99 f f f f f f f i4e4 ,we shall have a #d #d series matrix in which all the elements except those on the diagonal from left up to right down are zero1 and the elements in this diagonal agree with each other1 and are each equal to the above mentioned combination in 7#cf74 ,the determinant of f is therefore the square of the combination1 by ,detf we shall denote the expression ,detf f f f f f f #d4 ,a linear transformation xh h#a x h#b x h#c x h#d x 7h #a1b1c1 which is accomplished by the matrix ,a 1 1 1 1 1 1 1 1 1 1 1 1 will be denoted as the transformation ,a4 ,by the transformation ,a1 the expression x x x x is changed into the quadratic for m hk xh xk1 where hk #ak #ak #b;h #bk #c;h #ck #d;h #dk are the members of a #d #d series matrix which is the product of ,a1 the transposed matrix of ,a into ,a4 ,if by the transformation1 the expression is changed to x x#b x#b x1 we must have ,a #a4 ,a has to correspond to the following relation1 if transformation 7#ch7 is to be a ,lorentz-transformation4 ,for the determinant of ,a7 it follows out of 7#ci7 that 7,det ,a7 #a1 or ,det ,a #a4 ,from the condition 7#ci7 we obtain ,a 1 i4e4 the reciprocal matrix of ,a is equivalent to the transposed matrix of ,a4 ,for ,a as ,lorentz transformation1 we have further ,det ,a #a1 the quantities involving the index #d once in the subscript are purely imaginary1 the other co-efficients are real1 and a #j4 #e4 ,a space time vector of the first kind#ba which s represented by the #a #d series matrix1 7#da7 s s s s s is to be replaced by s,a in case of a ,lorentz transformation ,a4 i4e4 s s s s s s s s s ,a2 ,a space-time vector of the #bnd kind#bb with components f 444 f shall be represented by the alternating matrix 7#db7 f #j f f f f #j f f f f #j f f f f #j and is to be replaced by ,a f ,a in case of a ,lorentz transformation see the rules in #e 7#bc7 7#bd74 ,therefore referring to the expression 7#cg71 we have the identity ,det 7 f ,a7 ,det ,a4 ,det f4 ,therefore ,det f becomes an invariant in the case of a ,lorentz transformation see eq4 7#bf7 ,see4 #e4 ,looking back to 7#cf71 we have for the dual matrix 7f99,a7 7,af,a7 ,af99f,a ,det function4 ,a,a ,detf from which it is to be seen that the dual matrix f99 behaves exactly like the primary matrix f1 and is therefore a space time vector of the ,,ii kind2 f99 is therefore known as the dual space-time vector of f with components 7f1 f1 f171 7f1 f1 f74 #f4 ,if w and s are two space-time rectors of the #ast kind then by w 7as well as by s 7 will be understood the combination 7#dc7 w s w s w s w s4 ,in case of a ,lorentz transformation ,a1 since 7w,a7 77 w s1 this expression is invariant4--,if w #j1 then w and s are perpendicular to each other4 ,two space-time rectors of the first kind 7w1 s7 gives us a #b #d series matrix w w w w s s s s ,then it follows immediately that the system of six magnitudes 7#dd7 w s - w s1 w s - w s1 w s - w s1 w s - w s1 w s - w s1 w s - w s1 behaves in case of a ,lorentz-transformation as a space-time vector of the ,,ii kind4 ,the vector of the second kind with the components 7#dd7 are denoted by w1 s4 ,we see easily that ,det w1 s #j4 ,the dual vector of w1 s shall be written as w1 s4 ,if is a space-time vector of the #ast kind1 f of the second kind1 w f signifies a #a #d series matrix4 ,in case of a ,lorentz-transformation ,a1 w is changed into w w,a1 f into f ,a f ,a1--therefore w f becomes 7w,a ,a f ,a7 w f ,a i4e4 w f is transformed as a space-time vector of the #ast kind4#bc ,we can verify1 when w is a space-time vector of the #ast kind1 f of the #bnd kind1 the important identity 7#de7 w1 wf w1 wf9999 7w 7f4 ,the sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices4 ,for example1 for #j1 #j1 #j1 i1 f if1 if1 if1 #j 2 f99 if1 if1 if1 #j f #j1 #j1 #j1 f1 f1 f2 f9999 #j1 #j1 #j1 f1 f1 f4 ,the fact that in this special case1 the relation is satisfied1 suffices to establish the theorem 7#de7 generally1 for this relation has a covariant character in case of a ,lorentz transformation1 and is homogeneous in 71 1 1 74 ,after these preparatory works let us engage ourselves with the equations 7,c17 7,d17 7,e7 by means which the constants 1 will be introduced4 ,instead of the space vector u1 the velocity of matter1 we shall introduce the space-time vector of the first kind with the components4 ux_/7#a - u71 uy_/7#a - u71 uz_/7#a - u71 i_/7#a - u74 7#dj7 where -#a and -i #j4 ,by ,f and f shall be understood the space time vectors of the second kind ,m - i,e1 m - ie4 ,in ,f1 we have a space time vector of the first kind with components ,f ,f ,f ,f ,f ,f ,f ,f ,f ,f ,f ,f ,the first three quantities 71 1 7 are the components of the space-vector 7,e u1 ,m7_/7#a - u71 and further 7 iu ,e_/7#a - u74 ,because ,f is an alternating matrix1 7#di7 #j4 i4e4 is perpendicular to the vector 2 we can also write ix y z 4 ,i shall call the space-time vector of the first kind as the ,electric ,rest ,force4#bd ,relations analogous to those holding between -,f1 ,e1 ,m1 ,u1 hold amongst -f1 e1 m1 u1 and in particular -f is normal to 4 ,the relation 7,c7 can be written as ,c f ,f4 ,the expression 7f7 gives four components1 but the fourth can be derived from the first three4 ,let us now form the time-space vector #ast kind1 - if991 whose components are -i7 f f f7 -i7 f f f7 -i7 f f f7 -i7 f f f7 ,of these1 the first three 1 1 1 are the x1 y1 z components of the space-vector #ea7 7m - 7ue77_/7#a - u7 and further 7#eb7 i7um7_/7#a - u74 ,among these there is the relation 7#ec7 #j which can also be written as i 7ux uy uz 74 ,the vector is perpendicular to 2 we can call it the ,magnetic rest-force4 ,relations analogous to these hold among the quantities ,f991 ,m1 ,e1 u and ,relation 7,d7 can be replaced by the formula ,d -,f99 f994 ,we can use the relations 7,c7 and 7,d7 to calculate ,f and f from and we have ,f -1 ,f99 -i1 f -1 f99 -i4 and applying the relation 7#de7 and 7#df71 we have ,f 4 i4 99 #ee7 f 4 i4 99 #ef7 i4e4 ,f 7 - 7 i - 1 etc4 f 7 - 7 i - 41 etc4 ,let us now consider the space-time vector of the second kind 1 with the components - 1 - 1 - - 1 - 1 - ,then the corresponding space-time vector of the first kind 1 vanishes identically owing to equations #i7 and #ec7 for 4 -77 77 ,let us now take the vector of the #ast kind 7#eg7 i99 with the components -i 1 etc4 ,then by applying rule 7#de71 we have 7#eh7 4 i99 i4e4 - i7 - 7 etc4 ,the vector fulfils the relation 77 #j1 7which we can write as i7x y z7 and is also normal to 4 ,in case #j1 we have #j1 #j1 #j1 and 1 1 4 ,i shall call 1 which is a space-time vector #ast kind the ,rest-,ray4 ,as for the relation ,e71 which introduces the conductivity we have -,s -7s s s s7 7- u ,cu 7_/7#a - u7 4 ,this expression gives us the rest-density of electricity 7see #h and #d74 ,then #fa7 s 77 represents a space-time vector of the #ast kind1 which since -#a1 is normal to 1 and which ,i may call the rest-current4 ,let us now conceive of the first three component of this vector as the 7x-y-z7 co-ordinates of the space-vector1 then the component in the direction of u is ,cu - 7 u 7_/7#a - u7 7cu - u 7_/7#a - u7 ,ju_/7#a - u7 and the component in a perpendicular direction is ,cu ,j4 ,this space-vector is connected with the space-vector ,j ,c - u1 which we denoted in #h as the conduction-current4 ,now by comparing with -,f1 the relation 7,e7 can be brought into the form ,e s 77 - ,f1 ,this formula contains four equations1 of which the fourth follows from the first three1 since this is a space-time vector which is perpendicular to 4 ,lastly1 we shall transform the differential equations 7,a7 and 7,b7 into a typical form4 #ab4 ,the ,differential ,operator ,lor4 ,a #d #d series matrix #fb7 ,s ,s ,s ,s ,s ,skh ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s with the condition that in case of a ,lorentz transformation it is to be replaced by ,,sa1 may be called a space-time matrix of the ,,ii kind4 ,we have examples of this in3-- #a7 the alternating matrix f1 which corresponds to the space-time vector of the ,,ii kind1-- #b7 the product f,f of two such matrices1 for by a transformation ,a1 it is replaced by 7,af,a,a,,fa7 ,af,,fa1 #c7 further when 71 1 1 7 and 71 1 1 7 are two space-time vectors of the #ast kind1 the #d #d matrix with the element ,shk hk1 lastly in a multiple ,l of the unit matrix of #d #d series in which all the elements in the principal diagonal are equal to ,l1 and the rest are zero4 ,we shall have to do constantly with functions of the space-time point 7x1 y1 z1 it71 and we may with advantage employ the #a #d series matrix1 formed of differential symbols1-- _/x1 _/y1 _/z1 _/it1 or 7#fc7 _/x _/x _/x _/x ,for this matrix ,i shall use the shortened from 8lor40#be ,then if ,s is1 as in 7#fb71 a space-time matrix of the ,,ii kind1 by lor ,s will be understood the #a #d series matrix ,k ,k ,k ,k where ,kk ,s#ak_/x ,s#bk_/x ,s#ck_/x ,s#d;h_/x4 ,when by a ,lorentz transformation ,a1 a new reference system 7x x x x7 is introduced1 we can use the operator lor _/x _/x _/x _/x ,then ,s is transformed to ,s ,s ,a ,shk 1 so by lor ',s is meant the #a #d series matrix1 whose element are ,k'k ,s#ak_/x ,s#bk_/x ,s#ck_/x ,s#dk_/x4 ,now for the differentiation of any function of 7x y z t7 we have the rule _/xk _/x x_/xk _/x x_/xk _/x x_/xk _/x x_/xk _/x a#ak _/x a#bk _/x a#ck _/x a#dk4 so that1 we have symbolically lor lor ,a4 ,therefore it follows that lor ,s lor 7,a ,a ,,sa7 7lor ,s7,a4 i4e41 lor ,s behaves like a space-time vector of the first kind4 ,if ,l is a multiple of the unit matrix1 then by lor ,l will be denoted the matrix with the elements ,l_/x ,l_/x ,l_/x ,l_/x ,if s is a space-time vector of the #ast kind1 then lor s_/x s_/x s_/x s_/x4 ,in case of a ,lorentz transformation ,a1 we have lor lor ,a4 s lor s4 i4e41 lor s is an invariant in a ,lorentz-transformation4 ,in all these operations the operator lor plays the part of a space-time vector of the first kind4 ,if f represents a space-time vector of the second kind1--lor f denotes a space-time vector of the first kind with the components f_/x f_/x f_/x1 f_/x f_/x f_/x1 f_/x f_/x f_/x1 f_/x f_/x f_/x ,so the system of differential equations 7,a7 can be expressed in the concise form ,a lor f -s1 and the system 7,b7 can be expressed in the form ,b log ,f99 #j4 ,referring back to the definition 7#fg7 for log 1 we find that the combinations lor 77lor f771 and lor 77lor ,f9977 vanish identically1 when f and ,f99 are alternating matrices4 ,accordingly it follows out of ,a1 that 7#fh7 7s_/x7 7s_/x7 7s_/x7 7s_/x7 #j1 while the relation 7#fi7 lor 7lor ,f997 #j1 signifies that of the four equations in ,b1 only three represent independent conditions4 ,i shall now collect the results4 ,let denote the space-time vector of the first kind 7u_/7#a - u71 i_/7#a - u77 7u velocity of matter71 ,f the space-time vector of the second kind 7,m1-i,e7 7,m magnetic induction1 ,e ,electric force1 f the space-time vector of the second kind 7m1-ie7 7m magnetic force1 e ,electric ,induction4 s the space-time vector of the first kind 7,c1 i7 7 electrical space-density1 ,c - u conductivity current1 dielectric constant1 magnetic permeability1 conductivity1 then the fundamental equations for electromagnetic processes in moving bodies are#bf ,a lor f -s ,b log ,f99 #j ,c f ,f ,d ,f99 f99 ,e s 771 w - ,f4 -#a1 and ,f1 f1 ,f991 f991 s 7s7 which are space-time vectors of the first kind are all normal to 1 and for the system ,b1 we have lor 7lor ,f997 #j4 ,bearing in mind this last relation1 we see that we have as many independent equations at our disposal as are necessary for determining the motion of matter as well as the vector u as a function of x1 y1 z1 t1 when proper fundamental data are given4 #ac4 ,the ,product of the ,field-vectors f ,f4 ,finally let us enquire about the laws which lead to the determination of the vector as a function of 7x1 y1 z1 t47 ,in these investigations1 the expressions which are obtained by the multiplication of two alternating matrices f #j f f f f #j f f f f #j f f f f #j ,f #j ,f ,f ,f ,f #j ,f ,f ,f ,f #j ,f ,f ,f ,f #j are of much importance4 ,let us write1 7#gj7 f,f ,s - ,l ,s ,s ,s ,s ,s - ,l ,s ,s ,s ,s ,s - ,l ,s ,s ,s ,s ,s - ,l ,then 7#ga7 ,s ,s ,s ,s #j4 ,let ,l now denote the symmetrical combination of the indices #a1 #b1 #c1 #d1 given by 7#gb7 ,l 7f ,f f,f f ,f f ,f f ,f f ,f7 ,then we shall have 7#gc7 ,s 7f ,f f ,f f ,f - f ,f - f ,f f ,f7 ,s f ,f f ,f etc4444 ,in order to express in a real form1 we write 7#gd7 ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,xx ,yx ,zx -i,tx ,xy ,yy ,zy -i,ty ,xz ,yz ,zz -i,tz -i,xt -i,yt -i,zt ,tt ,now ,xx mx,mx - my,my - mz,mz ex,ex - ey,ey - ez,ez so 7#ge7 ,xy mx,my ey,ex1 ,yx my,mx ex,ey etc4 ,xt ey,mz - ez,my1 ,tx mx,ey - my,ez1 etc4 ,tt mx,mx my,my mz,mz ex,ex ey,ey ez,ez ,lt mx,mx my,my mz,mz - ex,ex - ey,ey - ez,ez ,these quantities#bg are all real4 ,in the theory for bodies at rest1 the combinations 7,xx1 ,xy1 ,xz1 ,yz1 ,yy1 ,yz1 ,zx1 ,zy1 ,zz7 are known as 8,maxwell's ,stresses10 ,tx1 ,ty1 ,tz are known as the ,poynting's ,vector1 ,tt as the electromagnetic energy-density1 and ,l as the ,langrangian function4 ,on the other hand1 by multiplying the alternating matrices of f99 and ,f991 we obtain 7#gg7 ,f99f99 -,s - ,l1 -,s1 -,s4 -,s -,s1 -,s - ,l1 -,s1 -,s -,s -,s1 -,s - ,l1 -,s -,s -,s -,s -,s - ,l and hence1 we can put 7#gh7 f,f ,s - ,l1 ,f99f99 -,s - ,l1 where by ,l1 we mean ,l-times the unit matrix1 i4e4 the matrix with elements ,lehk 1 7ehh #a1 ehk #j1 h k h1 k #a1 #b1 #c1 #d74 ,since here ,,sl ,,ls1 we deduce that1 ,f99f99f,f 7-,s - ,l77,s - ,l7 -,,ss ,l1 and find1 since f99f ,detf1 ,f99,f ,det,f1 we arrive at the interesting conclusion 7#gi7 ,,ss ,l - ,detf ,det,f i4e4 the product of the matrix ,s into itself can be expressed as the multiple of a unit matrix--a matrix in which all the elements except those in the principal diagonal are zero1 the elements in the principal diagonal are all equal and have the value given on the right-hand side of 7#gi74 ,therefore the general relations 7#hj7 ,sh#a ,s#ak ,sh#b ,s#bk ,sh#c ,s#ck ,sh#d ,s#dk #j1 h1 k being unequal indices in the series #a1 #b1 #c1 #d1 and 7#ha7 ,sh#a ,s#a;h ,sh#b ,s#b;h ,sh#c ,s#c;h ,sh#d ,s#d;h ,l - ,detf ,det,f1 for h #a1 #b1 #c1 #d4 ,now if instead of ,f1 and f in the combinations 7#gb7 and 7#gc71 we introduce the electrical rest-force 1 the magnetic rest-force 1 and the rest-ray 7#ee71 7#ef7 and 7#eg71 we can pass over to the expressions1-- 7#hb7 ,l - 1 7#hc7 ,shk - ehk - ehk 7h k - 77 h k 7h k - h k7 - h k - h k 7h k #a1 #b1 #c1 #d74 ,here we have 1 ehh #a1 ehk #j 7h k74 ,the right side of 7#hb7 as well as ,l is an invariant in a ,lorentz transformation1 and the #d #d element on the right side of 7#hc7 as well as ,sk h represent a space time vector of the second kind4 ,remembering this fact1 it suffices1 for establishing the theorems 7#hb7 and 7#hc7 generally1 to prove it for the special case #j1 #j1 #j1 i4 ,but for this case #j1 we immediately arrive at the equations 7#hb7 and 7#hc7 by means 7#de71 7#ea71 7#fj7 on the one hand1 and e ,e1 ,m m on the other hand4 ,the expression on the right-hand side of 7#ha71 which equals 7m ,m - e,e7 7em7 7,,em71 is #j1 because 7em 1 7,,em7 2 now referring back to #gi71 we can denote the positive square root of this expression as ,det#a_/d ,s4 ,since -f1 and -,f1 we obtain for 1 the transposed matrix of ,s1 the following relations from 7#gh71 7#hd7 ,ff - ,l1 f99 ,f99 - - ,l1 ,then is - ,s ,sh k - ,st k an alternating matrix1 and denotes a space-time vector of the second kind4 ,from the expressions 7#hc71 we obtain1 7#he7 ,s - - 7 - #a7 1 1 from which we deduce that see 7#eg71 7#eh74 7#hf7 7,s - 799 #j1 7#hg7 7,s - 7 7 - #a7 ,when the matter is at rest at a space-time point1 #j1 then the equation #hf7 denotes the existence of the following equations ,zy ,yz1 ,xz ,zx1 ,yx ,xy1 and from #hc71 ,tx 1 ,ty 1 ,tz ,xt 1 ,yt 1 ,zt ,now by means of a rotation of the space co-ordinate system round the null-point1 we can make1 ,zy ,yz #j1 ,xz ,zx #j1 ,xx ,xy #j1 ,according to #ga71 we have 7#hh7 ,xx ,yy ,zz ,tt #j1 and according to #hc71 ,tt #j4 ,in special cases1 where vanishes it follows from #ha7 that ,xx ,yy ,zz ,tt1 7,det#a_/d ,s71 and if ,t1 and one of the three magnitudes ,xx1 ,yy1 ,zz are ,det#a_/d ,s1 the two others -,det#a_/d ,s4 ,if does not vanish let #j1 then we have in particular from #hj7 ,tz ,xt #j1 ,tz ,yt #j1 ,zz ,tz ,tz ,tt #j1 and if #j1 #j1 ,zz -,tt ,it follows from 7#ha71 7see also #hc7 that ,xx -,yy ,det#a_/d ,s1 and -,zz ,tt 7,det ,s 7 ,det#a_/d,s4 ,the space-time vector of the first kind 7#hi7 ,k lor ,s1 is of very great importance for which we now want to demonstrate a very important transformation ,according to #gh71 ,s ,l f,f1 and it follows that lor ,s lor ,l lor f,f4 ,the symbol #8lor' denotes a differential process which in lor f,f1 operates on the one hand upon the components of f1 on the other hand also upon the components of ,f4 ,accordingly lor f,f can be expressed as the sum of two parts4 ,the first part is the product of the matrices 7lor f7 ,f1 lor f being regarded as a #a #d series matrix4 ,the second part is that part of lor f,f1 in which the diffentiations operate upon the components of ,f alone4 ,from #gh7 we obtain f,f -,f99f99 - #b,l2 hence the second part of lor f,f -7lor ,f997f99 the part of -#b lor ,l1 in which the differentiations operate upon the components of ,f alone4 ,we thus obtain lor ,s 7lor f7,f - 7lor ,f997f99 ,n1 where ,n is the vector with the components ,nh 7f_/xh ,f f_/xh ,f f_/xh ,f f_/xh ,f f_/xh ,f f_/xh ,f - ,f_/xh f - ,f_/xh f - ,f_/xh f - ,f_/xh f - ,f_/xh f - ,f_/xh f71 7h #a1 #b1 #c1 #d7 ,by using the fundamental relations ,a7 and ,b71 #ij7 is transformed into the fundamental relation 7#ia7 lor ,s -s,f ,n4 ,in the limitting case #a1 #a1 f ,f1 ,n vanishes identically4 ,now upon the basis of the equations 7#ee7 and 7#ef71 and referring back to the expression 7#hb7 for ,l1 and from #eg7 we obtain the following expressions as components of ,n1-- 7#ib7 ,nh - _/xh - _/xh 7 - #a77 _/xh _/xh _/xh _/xh7 for h #a1 #b1 #c1 #d4 ,now if we make use of 7#ei71 and denote the space-vector which has 1 1 as the x1 y1 z components by the symbol ,w1 then the third component of #ib7 can be expressed in the form 7#ic7 7 - #a7_/7a - u7 7,w u_/xh71 ,the round bracket denoting the scalar product of the vectors within it4 #ad4 ,the ,ponderomotive ,force4#bh ,let us now write out the relation ,k lor ,s -s,f ,n in a more practical form2 we have the four equations 7#id7 ,k ,xx_/x ,xy_/y ,xy_/z - ,xt_/t ,ex sy,mz - sz,mx - _/x - _/x 7 - #a7_/7a - u7 7,wu_/x71 7#ie7 ,k ,yx_/x ,yy_/y ,yz_/z - ,yt_/t ,ey sz,mx - sx,my - _/y - _/y 7 - #a7_/7a - u7 7,wu_/y71 7#if7 ,k ,zx_/x ,zy_/y ,zz_/z - ,zt_/t ,e sx,my - sy,m - _/z - _/z 7 - #a7_/7a - u7 7,wu_/z71 7#ig7 7#a_/;i7,k ,ty_/x - ,ty_/y - ,tz_/z - ,tt_/t sx,ex sy,ey sz,ez - _/t - _/t 7 - #a7_/7a - u7 7,wu_/t74 ,it is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point x1 y1 z1 t1 it has got1 x1 y1 z components as the first three components of the space-time vector ,k 7,k71 ,this vector is perpendicular to 2 the law of ,energy finds its expression in the fourth relation4 ,the establishment of this opinion is reserved for a separate tract4 ,in the limiting case #a1 #a1 #j1 the vector ,n #j1 ,s 1 ,k #j1 and we obtain the ordinary equations in the theory of electrons4 ,footnote #i3 ,vide ,note #a4 ,footnote #aj3 ,note #b4 ,footnote #aa3 ,vide ,note #c4 ,footnote #ab3 ,vide ,note #d4 ,footnote #ac3 ,note #e4 ,footnote #ad3 ,see notes on #h and #aj4 ,footnote #ae3 ,see note #i4 ,footnote #af3 ,see ,note4 ,footnote #ag3 ,vide ,note4 ,footnote #ah3 ,just as beings which are confined within a narrow region surrounding a point on a spherical surface1 may fall into the error that a sphere is a geometric figure in which one diameter is particularly distinguished from the rest4 ,footnote #ai3 ,einzelne stelle der ,materie4 ,footnote #bj3 ,vide ,note4 ,footnote #ba3 ,vide note #ac4 ,footnote #bb3 ,vide note #ad4 ,footnote #bc3 ,vide note #ae4 ,footnote #bd3 ,vide note #af4 ,footnote #be3 ,vide note #ag4 ,footnote #bf3 ,vide note #ai4 ,footnote #bg3 ,vide note #ah4 ,footnote #bh3 ,vide note #dj4 ,,appendix ,mechanics and the ,relativity-,postulate4 ,it would be very unsatisfactory1 if the new way of looking at the time-concept1 which permits a ,lorentz transformation1 were to be confined to a single part of ,physics4 ,now many authors say that classical mechanics stand in opposition to the relativity postulate1 which is taken to be the basis of the new ,electro-dynamics4 ,in order to decide this let us fix our attention upon a special ,lorentz transformation represented by 7#aj71 7#aa71 7#ab71 with a vector v in any direction and of any magnitude q #a but different from zero4 ,for a moment we shall not suppose any special relation to hold between the unit of length and the unit of time1 so that instead of t1 t1 q1 we shall write ct1 ct1 and q_/c1 where c represents a certain positive constant1 and q is c4 ,the above mentioned equations are transformed into r r1 rv c7rv - qt7_/7c - q71 t 7qrv ct7_/c7c - q7 ,they denote1 as we remember1 that r is the space-vector 7x1 y1 z71 r is the space-vector 7x y z7 ,if in these equations1 keeping v constant we approach the limit c 1 then we obtain from these r r1 rv rv - qt1 t t4 ,the new equations would now denote the transformation of a spatial co-ordinate system 7x1 y1 z7 to another spatial co-ordinate system 7x y z7 with parallel axes1 the null point of the second system moving with constant velocity in a straight line1 while the time parameter remains unchanged4 ,we can1 therefore1 say that classical mechanics postulates a covariance of ,physical laws for the group of homogeneous linear transformations of the expression -x - y - z c 7#a7 when c 4 ,now it is rather confusing to find that in one branch of ,physics1 we shall find a covariance of the laws for the transformation of expression 7#a7 with a finite value of c1 in another part for c 4 ,it is evident that according to ,newtonian ,mechanics1 this covariance holds for c and not for c velocity of light4 ,may we not then regard those traditional covariances for c only as an approximation consistent with experience1 the actual covariance of natural laws holding for a certain finite value of c4 ,i may here point out that by if instead of the ,newtonian ,relativity-,postulate with c 1 we assume a relativity-postulate with a finite c1 then the axiomatic construction of ,mechanics appears to gain considerably in perfection4 ,the ratio of the time unit to the length unit is chosen in a manner so as to make the velocity of light equivalent to unity4 ,while now ,i want to introduce geometrical figures in the manifold of the variables 7x1 y1 z1 t71 it may be convenient to leave 7y1 z7 out of account1 and to treat x and t as any possible pair of co-ordinates in a plane1 referred to oblique axes4 ,a space time null point #j 7x1 y1 z1 t #j1 #j1 #j1 #j7 will be kept fixed in a ,lorentz transformation4 ,the figure -x - y - z t #a1 t #j 444 7#b7 which represents a hyper boloidal shell1 contains the space-time points ,a 7x1 y1 z1 t #j1 #j1 #j1 #a71 and all points ,a which after a ,lorentz-transformation enter into the newly introduced system of reference as 7x1 y1 z1 t #j1 #j1 #j1 #a74 ,the direction of a radius vector #j;,a drawn from #j to the point ,a of 7#b71 and the directions of the tangents to 7#b7 at ,a are to be called normal to each other4 ,let us now follow a definite position of matter in its course through all time t4 ,the totality of the space-time points 7x1 y1 z1 t7 which correspond to the positions at different times t1 shall be called a space-time line4 ,the task of determining the motion of matter is comprised in the following problem3--,it is required to establish for every space-time point the direction of the space-time line passing through it4 ,to transform a space-time point ,p 7x1 y1 z1 t7 to rest is equivalent to introducing1 by means of a ,lorentz transformation1 a new system of reference 7x1 y1 z1 t71 in which the t axis has the direction #j;,a1 #j;,a indicating the direction of the space-time line passing through ,p4 ,the space t const1 which is to be laid through ,p1 is the one which is perpendicular to the space-time line through ,p4 ,to the increment dt of the time of ,p corresponds the increment d 7dt - dx - dy7 - dz dt7#a - u7 of the newly introduced time parameter t4 ,the value of the integral d 7-7dx dx dx dx77 when calculated upon the space-time line from a fixed initial point ,p to the variable point ,p1 7both being on the space-time line71 is known as the #8,proper-time' of the position of matter we are concerned with at the space-time point ,p4 7,it is a generalization of the idea of ,positional-time which was introduced by ,lorentz for uniform motion47 ,if we take a body ,r which has got extension in space at time t1 then the region comprising all the space-time line passing through ,r and t shall be called a space-time filament4 ,if we have an analytical expression 7x y1 z1 t7 so that 7x1 y z t7 #j is intersected by every space time line of the filament at one point1--whereby -7_/x71 -7_/y71 -7_/z71 -7_/t7 #j1 _/t #j4 then the totality of the intersecting points will be called a cross section of the filament4 ,at any point ,p of such across-section1 we can introduce by means of a ,lorentz transformation a system of reference 7x1 y1 z t71 so that according to this _/x #j1 _/y #j1 _/z #j1 _/t #j4 ,the direction of the uniquely determined t--axis in question here is known as the upper normal of the cross-section at the point ,p and the value of d,j dx dy dz for the surrounding points of ,p on the cross-section is known as the elementary contents 7,inhalts-element7 of the cross-section4 ,in this sense ,r is to be regarded as the cross-section normal to the t axis of the filament at the point t t1 and the volume of the body ,r is to be regarded as the contents of the cross-section4 ,if we allow ,r to converge to a point1 we come to the conception of an infinitely thin space-time filament4 ,in such a case1 a space-time line will be thought of as a principal line and by the term #8,proper-time' of the filament will be understood the #8,proper-time' which is laid along this principal line2 under the term normal cross-section of the filament1 we shall understand the cross-section upon the space which is normal to the principal line through ,p4 ,we shall now formulate the principle of conservation of mass4 ,to every space ,r at a time t1 belongs a positive quantity--the mass at ,r at the time t4 ,if ,r converges to a point 7x1 y1 z1 t71 then the quotient of this mass1 and the volume of ,r approaches a limit 7x1 y1 z1 t71 which is known as the mass-density at the space-time point 7x1 y1 z1 t74 ,the principle of conservation of mass says--that for an infinitely thin space-time filament1 the product d,j1 where mass-density at the point 7x1 y1 z1 t7 of the filament 7i4e41 the principal line of the filament71 d,j contents of the cross-section normal to the t axis1 and passing through 7x1 y1 z1 t71 is constant along the whole filament4 ,now the contents d,jn of the normal cross-section of the filament which is laid through 7x1 y1 z1 t7 is 7#d7 d,jn 7#a_/7a - u77d,j -i d,j 7dt_/d7d,j4 and the function _/-i 7#a - u77 7_/t4 7#e7 may be defined as the rest-mass density at the position 7x y z t74 ,then the principle of conservation of mass can be formulated in this manner3-- ,for an infinitely thin space-time filament1 the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament4 ,in any space-time filament1 let us consider two cross-sections ,q and ,q1 which have only the points on the boundary common to each other2 let the space-time lines inside the filament have a larger value of t on ,q than on ,q4 ,the finite range enclosed between ,q and ,q shall be called a space-time sichel1#bi ,q is the lower boundary1 and ,q is the upper boundary of the sichel4 ,if we decompose a filament into elementary space-time filaments1 then to an entrance-point of an elementary filament through the lower boundary of the sichel1 there corresponds an exit point of the same by the upper boundary1 whereby for both1 the product d,jn taken in the sense of 7#d7 and 7#e71 has got the same value4 ,therefore the difference of the two integrals d,jn 7the first being extended over the upper1 the second upon the lower boundary7 vanishes4 ,according to a well-known theorem of ,integral ,calculus the difference is equivalent to lor dx dy dz dt1 the integration being extended over the whole range of the sichel1 and 7comp4 7#fg71 #ab7 lor 7_/x7 7_/x7 7_/x7 7_/x74 ,if the sichel reduces to a point1 then the differential equation lor #j1 7#f7 which is the condition of continuity 7ux_/x7 7uy_/y7 7uz_/z7 7_/t7 #j4 ,further let us form the integral ,n dx dy dz dt 7#g7 extending over the whole range of the space-time sichel4 ,we shall decompose the sichel into elementary space-time filaments1 and every one of these filaments in small elements d of its proper-time1 which are however large compared to the linear dimensions of the normal cross-section2 let us assume that the mass of such a filament d,jn dm and write 1 l for the #8,proper-time' of the upper and lower boundary of the sichel4 ,then the integral 7#g7 can be denoted by d,jn d 7-7 dm4 taken over all the elements of the sichel4 ,now let us conceive of the space-time lines inside a space-time sichel as material curves composed of material points1 and let us suppose that they are subjected to a continual change of length inside the sichel in the following manner4 ,the entire curves are to be varied in any possible manner inside the sichel1 while the end points on the lower and upper boundaries remain fixed1 and the individual substantial points upon it are displaced in such a manner that they always move forward normal to the curves4 ,the whole process may be analytically represented by means of a parameter 1 and to the value #j1 shall correspond the actual curves inside the sichel4 ,such a process may be called a virtual displacement in the sichel4 ,let the point 7x1 y1 z1 t7 in the sichel #j have the values x x1 y y1 z z1 t t1 when the parameter has the value 2 these magnitudes are then functions of 7x1 y1 z1 t1 74 ,let us now conceive of an infinitely thin space-time filament at the point 7x y z t7 with the normal section of contents d,jn and if d,jn d,jn be the contents of the normal section at the corresponding position of the varied filament1 then according to the principle of conservation of mass--7 d being the rest-mass-density at the varied position71 7#h7 7 7 7d,jn d,jn7 d,jn dm4 ,in consequence of this condition1 the integral 7#g7 taken over the whole range of the sichel1 varies on account of the displacement as a definite function ,n ,n of 1 and we may call this function ,n ,n as the mass action of the virtual displacement4 ,if we now introduce the method of writing with indices1 we shall have 7#i7 d7xh xh7 dxh k xh_/xk xh_/ d k #a1 #b1 #c1 #d h #a1 #b1 #c1 #d ,now on the basis of the remarks already made1 it is clear that the value of ,n ,n1 when the value of the parameter is 1 will be3-- 7#aj7 ,n ,n 77d7 77_/d7dx dy dz dt1 the integration extending over the whole sichel d7 7 where d7 7 denotes the magnitude1 which is deduced from 7-7dx dx7 - 7dx dx7 - 7dx dx7 - 7dx dx77 by means of 7#i7 and dx d1 dx d1 dx d1 dx d1 d #j therefore3-- 7#aa7 7d7 77_/d 7 -7h 7xh_/xk7k77 k #a1 #b1 #c1 #d4 h #a1 #b1 #c1 #d4 ,we shall now subject the value of the differential quotient 7#ab7 77d7,n ,n77_/d7 7 #j7 to a transformation4 ,since each xh as a function of 7x1 y1 z1 t7 vanishes for the zero-value of the parameter 1 so in general dxk_/7xh #j1 for #j4 ,let us now put 7xh_/7 h 7h #a1 #b1 #c1 #d7 7#ac7 #j then on the basis of 7#aj7 and 7#aa71 we have the expression 7#ab73-- - h77h_/x7 7h_/x7 7h_/x7 7h_/x77 dx dy dz dt for the system 7x x x x7 on the boundary of the sichel1 7x x x x7 shall vanish for every value of and therefore 1 1 1 are nil4 ,then by partial integration1 the integral is transformed into the form h7h_/x h_/x h_/x h_/x7 dx dy dz dt the expression within the bracket may be written as h k_/xk kh_/xk4 ,the first sum vanishes in consequence of the continuity equation 7b74 ,the second may be written as 7h_/x77dx_/d7 7h_/x77dx_/d7 7h_/x77dx_/d7 7h_/x77dx_/d7 dh_/d 7d_/d77dxh_/d7 whereby 7d_/d7 is meant the differential quotient in the direction of the space-time line at any position4 ,for the differential quotient 7#ab71 we obtain the final expression 7#ad7 77_/7 7_/7 7_/7 7_/77 dx dy dz dt4 ,for a virtual displacement in the sichel we have postulated the condition that the points supposed to be substantial shall advance normally to the curves giving their actual motion1 which is #j2 this condition denotes that the h is to satisfy the condition w w w w #j4 7#ae7 ,let us now turn our attention to the ,maxwellian tensions in the electrodynamics of stationary bodies1 and let us consider the results in #ab and #ac2 then we find that ,hamilton's ,principle can be reconciled to the relativity postulate for continuously extended elastic media4 ,at every space-time point 7as in #ac71 let a space time matrix of the #bnd kind be known 7#af7 ,s ,s ,s ,s ,s ,xx ,yx ,zx -i,tx ,s ,s ,s ,s ,xy ,yy ,zy -i,ty ,s ,s ,s ,s ,xz ,yz ,zz -i,tz ,s ,s ,s ,s -i,xt -i,yt -i,zt ,tt where ,xn ,yx 44444,xz1 ,tt are real magnitudes4 ,for a virtual displacement in a space-time sichel 7with the previously applied designation7 the value of the integral 7#ag7 ,w ,w 7,sh k 77xk xk77_/xh dx dy dz dt extended over the whole range of the sichel1 may be called the tensional work of the virtual displacement4 ,the sum which comes forth here1 written in real magnitudes1 is ,xx ,yy ,zz ,tt ,xx 7x7_/x ,xy 7x7_/y 444 ,zz 7z7_/z - ,xt 7x_/t - 444 ,tx 7t7_/x 444 ,tt 7t7_/t we can now postulate the following minimum principle in mechanics4 ,if any space-time ,sichel be bounded1 then for each virtual displacement in the ,sichel1 the sum of the mass-works1 and tension works shall always be an extremum for that process of the space-time line in the ,sichel which actually occurs4 ,the meaning is1 that for each virtual displacement1 7d7,n ,w7_/d7 #j #j 7#ah7 ,by applying the methods of the ,calculus of ,variations1 the following four differential equations at once follow from this minimal principle by means of the transformation 7#ad71 and the condition 7#ae74 7#ai7 wh_/ ,kh wh 7h #a1 #b1 #c1 #d7 whence ,kh ,s#a h_/x ,s#b h_/x ,s#c h_/x ,s#d h_/x1 7#bj7 are components of the space-time vector #ast kind ,k lor ,s1 and ,x is a factor1 which is to be determined from the relation w - #a4 ,by multiplying 7#ai7 by wh1 and summing the four1 we obtain ,x ,k1 and therefore clearly ,k 7,k7w will be a space-time vector of the #ast kind which is normal to w4 ,let us write out the components of this vector as ,x1 ,y1 ,z1 i,t ,then we arrive at the following equation for the motion of matter1 7#ba7 d_/d 7dx_/d7 ,x1 d_/d 7dy_/d7 ,y1 d_/d 7dz_/d7 ,z1 d_/d 7dx_/d7 ,t1 and we have also 7dx_/d7 7dy_/d7 7dz_/d7 7dt_/d7 -#a1 and ,x dx_/d ,y dy_/d ,z dz_/d ,t dt_/d4 ,on the basis of this condition1 the fourth of equations 7#ba7 is to be regarded as a direct consequence of the first three4 ,from 7#ba71 we can deduce the law for the motion of a material point1 i4e41 the law for the career of an infinitely thin space-time filament4 ,let x1 y1 z1 t1 denote a point on a principal line chosen in any manner within the filament4 ,we shall form the equations 7#ba7 for the points of the normal cross section of the filament through x1 y1 z1 t1 and integrate them1 multiplying by the elementary contents of the cross section over the whole space of the normal section4 ,if the integrals of the right side be ,rx ,ry ,rz ,rt and if m be the constant mass of the filament1 we obtain 7#bb7 m d_/d dx_/d ,rx1 m d_/d dy_/d ,ry1 m d_/d dz_/d ,rz1 m d_/d dt_/d ,rt ,r is now a space-time vector of the #ast kind with the components 7,rx ,ry ,rz ,rt7 which is normal to the space-time vector of the #ast kind w1--the velocity of the material point with the components dx_/d1 dy_/d1 dz_/d1 i dt_/d4 ,we may call this vector ,r the moving force of the material point4 ,if instead of integrating over the normal section1 we integrate the equations over that cross section of the filament which is normal to the t axis1 and passes through 7x1 y1 z1 t71 then ,see 7#d7 the equations 7#bb7 are obtained1 but are now multiplied by d_/dt2 in particular1 the last equation comes out in the form1 m d_/dt 7dt_/d7 wx ,rx d_/dt wy ,ry d_/dt wz ,rz d_/dt4 ,the right side is to be looked upon as the amount of work done per unit of time at the material point4 ,in this equation1 we obtain the energy-law for the motion of the material point and the expression m 7dt_/d - #a7 m #a_/7a - w7 - #a m 7 w #c_/h w 7 may be called the kinetic energy of the material point4 ,since dt is always greater than d we may call the quotient 7dt - d7_/d as the 8,gain0 7vorgehen7 of the time over the proper-time of the material point and the law can then be thus expressed2--,the kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time4 ,the set of four equations 7#bb7 again shows the symmetry in 7x1 y1 z1 t71 which is demanded by the relativity postulate2 to the fourth equation however1 a higher physical significance is to be attached1 as we have already seen in the analogous case in electrodynamics4 ,on the ground of this demand for symmetry1 the triplet consisting of the first three equations are to be constructed after the model of the fourth2 remembering this circumstance1 we are justified in saying1-- 8,if the relativity-postulate be placed at the head of mechanics1 then the whole set of laws of motion follows from the law of energy40 ,i cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation4 ,if ,b997x991 y991 z991 t997 be a solid 7fester7 space-time point1 then the region of all those space-time points ,b 7x1 y1 z1 t71 for which 7#bc7 7x - x997 7y - y997 7z - z997 7t - t997 t - t99 #j may be called a 8,ray-figure0 7,strahl-gebilde7 of the space time point ,b994 ,a space-time line taken in any manner can be cut by this figure only at one particular point2 this easily follows from the convexity of the figure on the one hand1 and on the other hand from the fact that all directions of the space-time lines are only directions from ,b99 towards to the concave side of the figure4 ,then ,b99 may be called the light-point of ,b4 ,if in 7#bc71 the point 7x y z t7 be supposed to be fixed1 the point 7x99 y99 z99 t997 be supposed to be variable1 then the relation 7#bc7 would represent the locus of all the space-time points ,b991 which are light-points of ,b4 ,let us conceive that a material point ,f of mass m may1 owing to the presence of another material point ,f991 experience a moving force according to the following law4 ,let us picture to ourselves the space-time filaments of ,f and ,f99 along with the principal lines of the filaments4 ,let ,,bc be an infinitely small element of the principal line of ,f2 further let ,b99 be the light point of ,b1 ,c99 be the light point of ,c on the principal line of ,f992 so that ,,oa is the radius vector of the hyperboloidal fundamental figure 7#bc7 parallel to ,b99,c991 finally ,d99 is the point of intersection of line ,b99,c99 with the space normal to itself and passing through ,b4 ,the moving force of the mass-point ,f in the space-time point ,b is now the space-time vector of the first kind which is normal to ,,bc1 and which is composed of the vectors 7#bd7 mm997,,oa_/,b99,d997 ,,bd99 in the direction of ,,bd991 and another vector of suitable value in direction of ,b99,c994 ,now by 7,,oa_/,b99,d997 is to be understood the ratio of the two vectors in question4 ,it is clear that this proposition at once shows the covariant character with respect to a ,lorentz-group4 ,let us now ask how the space-time filament of ,f behaves when the material point ,f99 has a uniform translatory motion1 i4e41 the principal line of the filament of ,f99 is a line4 ,let us take the space time null-point in this1 and by means of a ,lorentz-transformation1 we can take this axis as the t-axis4 ,let x1 y1 z1 t1 denote the point ,b1 let 99 denote the proper time of ,b991 reckoned from ,o4 ,our proposition leads to the equations 7#be7 dx_/d - m99x_/7t - 9971 dy_/d - m99y_/7t - 997 dz_/d -m99z_/7t - 9971 7#bf7 dt_/d -m99_/7t - 997 d7t - 997_/dt where 7#bg7 x y z 7t - 997 and 7#bh7 7dx_/d7 7dy_/d7 7dz_/d7 7dt_/d7 - #a4 ,in consideration of 7#bg71 the three equations 7#be7 are of the same form as the equations for the motion of a material point subjected to attraction from a fixed centre according to the ,newtonian ,law1 only that instead of the time t1 the proper time of the material point occurs4 ,the fourth equation 7#bf7 gives then the connection between proper time and the time for the material point4 ,now for different values of 1 the orbit of the space-point 7x y z7 is an ellipse with the semi-major axis a and the eccentricity e4 ,let ,e denote the eccentric anomaly1 the increment of the proper time for a complete description of the orbit1 finally n #b1 so that from a properly chosen initial point 1 we have the ,kepler-equation 7#bi7 n ,e - e sin ,e4 ,if we now change the unit of time1 and denote the velocity of light by c1 then from 7#bh71 we obtain 7#cj7 7dt_/d7 - #a 7m99_/ac7 7#a e cos ,e7_/7#a - e cos ,e7 ,now neglecting c with regard to #a1 it follows that ndt nd #a m99_/ac 7#a e cos ,e7_/7#a - e cos ,e7 from which1 by applying 7#bi71 7#ca7 nt const 7#a m99_/ac7 n m99_/ac ,sin ,e4 the factor m99_/ac is here the square of the ratio of a certain average velocity of ,f in its orbit to the velocity of light4 ,if now m99 denote the mass of the sun1 a the semi major axis of the earth's orbit1 then this factor amounts to #aj4 ,the law of mass attraction which has been just described and which is formulated in accordance with the relativity postulate would signify that gravitation is propagated with the velocity of light4 ,in view of the fact that the periodic terms in 7#ca7 are very small1 it is not possible to decide out of astronomical observations between such a law 7with the modified mechanics proposed above7 and the ,newtonian law of attraction with ,newtonian mechanics4 ,footnote #bi3 ,sichel--a ,german word meaning a crescent or a scythe4 ,the original term is retained as there is no suitable ,english equivalent4 ,,space ,,and ,,time ,a ,lecture delivered before the ,naturforscher ,versammlung 7,congress of ,natural ,philosophers7 at ,cologne--7#bast ,september1 #aijh74 ,gentlemen1 ,the conceptions about time and space1 which ,i hope to develop before you to-day1 has grown on experimental physical grounds4 ,herein lies its strength4 ,the tendency is radical4 ,henceforth1 the old conception of space for itself1 and time for itself shall reduce to a mere shadow1 and some sort of union of the two will be found consistent with facts4 ,i ,now ,i want to show you how we can arrive at the changed concepts about time and space from mechanics1 as accepted now-a-days1 from purely mathematical considerations4 ,the equations of ,newtonian mechanics show a twofold invariance1 7i7 their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position1 7ii7 when we change the system in its nature of motion1 i4 e41 when we impress upon it any uniform motion of translation1 the null-point of time plays no part4 ,we are accustomed to look upon the axioms of geometry as settled once for all1 while we seldom have the same amount of conviction regarding the axioms of mechanics1 and therefore the two invariants are seldom mentioned in the same breath4 ,each one of these denotes a certain group of transformations for the differential equations of mechanics4 ,we look upon the existence of the first group as a fundamental characteristics of space4 ,we always prefer to leave off the second group to itself1 and with a light heart conclude that we can never decide from physical considerations whether the space1 which is supposed to be at rest1 may not finally be in uniform motion4 ,so these two groups lead quite separate existences besides each other4 ,their totally heterogeneous character may scare us away from the attempt to compound them4 ,yet it is the whole compounded group which as a whole gives us occasion for thought4 ,we wish to picture to ourselves the whole relation graphically4 ,let 7x1 y1 z7 be the rectangular coordinates of space1 and t denote the time4 ,subjects of our perception are always connected with place and time4 ,no one has observed a place except at a particular time1 or has observed a time except at a particular place4 ,yet ,i respect the dogma that time and space have independent existences4 ,i will call a space-point plus a time-point1 i4e41 a system of values x1 y1 z1 t1 as a world-point4 ,the manifoldness of all possible values of x1 y1 z1 t1 will be the world4 ,i can draw four world-axes with the chalk4 ,now any axis drawn consists of quickly vibrating molecules1 and besides1 takes part in all the journeys of the earth 2 and therefore gives us occasion for reflection4 ,the greater abstraction required for the four-axes does not cause the mathematician any trouble4 ,in order not to allow any yawning gap to exist1 we shall suppose that at every place and time1 something perceptible exists4 ,in order not to specify either matter or electricity1 we shall simply style these as substances4 ,we direct our attention to the world-point x1 y1 z1 t1 and suppose that we are in a position to recognise this substantial point at any subsequent time4 ,let dt be the time element corresponding to the changes of space coordinates of this point dx1 dy1 dz4 ,then we obtain 7as a picture1 so to speak1 of the perennial life-career of the substantial point71--a curve in the world--the world-line1 the points on which unambiguously correspond to the parameter t from to -4 ,the whole world appears to be resolved in such world-lines1 and ,i may just deviate from my point if ,i say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines4 ,by this conception of time and space1 the 7x1 y1 z7 manifoldness t #j and its two sides t #j and t #j falls asunder4 ,if for the sake of simplicity1 we keep the null-point of time and space fixed1 then the first named group of mechanics signifies that at t #j we can give the x1 y1 and z-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression x y z4 ,the second group denotes that without changing the expression for the mechanical laws1 we can substitute 7x - t1 y - t1 z - t for 7x1 y1 z7 where 71 1 7 are any constants4 ,according to this we can give the time-axis any possible direction in the upper half of the world t #j4 ,now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half8 ,to establish this connection1 let us take a positive parameter c1 and let us consider the figure ct - x - y - z #a ,according to the analogy of the hyperboloid of two sheets1 this consists of two sheets separated by t #j4 ,let us consider the sheet1 in the region of t #j1 and let us now conceive the transformation of x1 y1 z1 t in the new system of variables2 7x'1 y'1 z'1 t'7 by means of which the form of the expression will remain unaltered4 ,clearly the rotation of space round the null-point belongs to this group of transformations4 ,now we can have a full idea of the transformations which we picture to ourselves from a particular transformation in which 7y1 z7 remain unaltered4 ,let us draw the cross section of the upper sheets with the plane of the x- and t-axes1 i4e41 the upper half of the hyperbola ct - x #a1 with its asymptotes 7vide fig4 #a74 ,then let us draw the radius rector ,,oa1 the tangent ,a ,b at ,a1 and let us complete the parallelogram ,,oa ,b ,c2 also produce ,b ,c to meet the x-axis at ,d4 ,let us now take ,ox1 ,,oa as new axes with the unit measuring rods ,,oc #a1 ,,oa 7#a_/;c7 2 then the hyperbola is again expressed in the form ct - x #a1 t #j and the transition from 7x1 y1 z1 t7 to 7x y z t7 is one of the transitions in question4 ,let us add to this characteristic transformation any possible displacement of the space and time null-points2 then we get a group of transformation depending only on c1 which we may denote by ,gc4 ,now let us increase c to infinity4 ,thus 7#a_/;c7 becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the x-axis1 the asymptotic angle becomes a straight one1 and every special transformation in the limit changes in such a manner that the t-axis can have any possible direction upwards1 and x more and more approximates to x4 ,remembering this point it is clear that the full group belonging to ,newtonian ,mechanics is simply the group ,gc1 with the value of c 4 ,in this state of affairs1 and since ,gc is mathematically more intelligible than ,g1 a mathematician may1 by a free play of imagination1 hit upon the thought that natural phenomena possess an invariance not only for the group ,g1 but in fact also for a group ,gc1 where c is finite1 but yet exceedingly large compared to the usual measuring units4 ,such a preconception would be an extraordinary triumph for pure mathematics4 ,at the same time ,i shall remark for which value of c1 this invariance can be conclusively held to be true4 ,for c1 we shall substitute the velocity of light c in free space4 ,in order to avoid speaking either of space or of vacuum1 we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity4 ,we can form an idea of the invariant character of the expression for natural laws for the group-transformation ,gc in the following manner4 ,out of the totality of natural phenomena1 we can1 by successive higher approximations1 deduce a coordinate system 7x1 y1 z1 t72 by means of this coordinate system1 we can represent the phenomena according to definite laws4 ,this system of reference is by no means uniquely determined by the phenomena4 ,we can change the system of reference in any possible manner corresponding to the above-mentioned group transformation ,gc1 but the expressions for natural laws will not be changed thereby4 ,for example1 corresponding to the above described figure1 we can call t the time1 but then necessarily the space connected with it must be expressed by the manifoldness 7x y z74 ,the physical laws are now expressed by means of x1 y1 z1 t1--and the expressions are just the same as in the case of x1 y1 z1 t4 ,according to this1 we shall have in the world1 not one space1 but many spaces1--quite analogous to the case that the three-dimensional space consists of an infinite number of planes4 ,the three-dimensional geometry will be a chapter of four-dimensional physics4 ,now you perceive1 why ,i said in the beginning that time and space shall reduce to mere shadows and we shall have a world complete in itself4 ,,ii ,now the question may be asked1--what circumstances lead us to these changed views about time and space1 are they not in contradiction with observed phenomena1 do they finally guarantee us advantages for the description of natural phenomena8 ,before we enter into the discussion1 a very important point must be noticed4 ,suppose we have individualised time and space in any manner2 then a world-line parallel to the t-axis will correspond to a stationary point2 a world-line inclined to the t-axis will correspond to a point moving uniformly2 and a world-curve will correspond to a point moving in any manner4 ,let us now picture to our mind the world-line passing through any world point x1 y1 z1 t2 now if we find the world-line parallel to the radius vector ,,oa of the hyperboloidal sheet1 then we can introduce ,,oa as a new time-axis1 and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned4 ,we shall now introduce this fundamental axiom3-- ,the substance existing at any world point can always be conceived to be at rest1 if we establish our time and space suitably4 ,the axiom denotes that in a world-point1 the expression cdt - dx - dy - dz shall always be positive or what is equivalent to the same thing1 every velocity ,v should be smaller than c4 c shall therefore be the upper limit for all substantial velocities and herein lies a deep significance for the quantity c4 ,at the first impression1 the axiom seems to be rather unsatisfactory4 ,it is to be remembered that only a modified mechanics will occur1 in which the square root of this differential combination takes the place of time1 so that cases in which the velocity is greater than c will play no part1 something like imaginary coordinates in geometry4 ,the impulse and real cause of inducement for the assumption of the group-transformation ,gc is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation ,gc4 ,on the other hand1 the idea of rigid bodies has any sense only in a system mechanics with the group ,ginfinity4 ,now if we have an optics with ,gc1 and on the other hand if there are rigid bodies1 it is easy to see that a t-direction can be defined by the two hyperboloidal shells common to the groups ,g1 and ,gc1 which has got the further consequence1 that by means of suitable rigid instruments in the laboratory1 we can perceive a change in natural phenomena1 in case of different orientations1 with regard to the direction of progressive motion of the earth4 ,but all efforts directed towards this object1 and even the celebrated interference-experiment of ,michelson have given negative results4 ,in order to supply an explanation for this result1 ,h4 ,a4 ,lorentz formed a hypothesis which practically amounts to an invariance of optics for the group ,gc4 ,according to ,lorentz every substance shall suffer a contraction #a377a - v_/c77 in length1 in the direction of its motion l_/l #a_/7a - v_/c7 l l7#a - v_/c74 ,this hypothesis sounds rather phantastical4 ,for the contraction is not to be thought of as a consequence of the resistance of ether1 but purely as a gift from the skies1 as a sort of condition always accompanying a state of motion4 ,i shall show in our figure that ,lorentz's hypothesis is fully equivalent to the new conceptions about time and space4 ,thereby it may appear more intelligible4 ,let us now1 for the sake of simplicity1 neglect 7y1 z7 and fix our attention on a two dimensional world1 in which let upright strips parallel to the t-axis represent a state of rest and another parallel strip inclined to the t-axis represent a state of uniform motion for a body1 which has a constant spatial extension 7see fig4 #a74 ,if ,,oa is parallel to the second strip1 we can take t as the t-axis and x as the x-axis1 then the second body will appear to be at rest1 and the first body in uniform motion4 ,we shall now assume that the first body supposed to be at rest1 has the length l1 i4e41 the cross section ,,pp of the first strip upon the x-axis l4 ,,oc1 where ,,oc is the unit measuring rod upon the x-axis--and the second body also1 when supposed to be at rest1 has the same length l1 this means that1 the cross section ,q,q of the second strip has a cross-section l ,,oc1 when measured parallel to the x-axis4 ,in these two bodies1 we have now images of two ,lorentz-electrons1 one of which is at rest and the other moves uniformly4 ,now if we stick to our original coordinates1 then the extension of the second electron is given by the cross section ,,qq of the strip belonging to it measured parallel to the x-axis4 ,now it is clear since ,q,q l ,,oc1 that ,,qq l ,,od4 ,if 7dc_/dt7 v1 an easy calculation gives that ,,od ,,oc 7#a-7v_/c771 therefore 7,,pp_/,,qq7 7#a_/7a-7v_/c77 ,this is the sense of ,lorentz's hypothesis about the contraction of electrons in case of motion4 ,on the other hand1 if we conceive the second electron to be at rest1 and therefore adopt the system 7x1 t17 then the cross-section ,p,p of the strip of the electron parallel to ,,oc is to be regarded as its length and we shall find the first electron shortened with reference to the second in the same proportion1 for it is1 ,p,p_/,q,q ,,od_/,,oc ,,od_/,,oc ,,qq_/,,pp ,lorentz called the combination t of 7t and x7 as the local time 7,ortszeit7 of the uniformly moving electron1 and used a physical construction of this idea for a better comprehension of the contraction-hypothesis4 ,but to perceive clearly that the time of an electron is as good as the time of any other electron1 i4e4 t1 t are to be regarded as equivalent1 has been the service of ,a4 ,einstein ,ann4 d4 ,phys4 #hia1 p4 #aije1 ,jahrb4 d4 ,radis4 444 #d-d-aa-aijg4 ,there the concept of time was shown to be completely and unambiguously established by natural phenomena4 ,but the concept of space was not arrived at1 either by ,einstein or ,lorentz1 probably because in the case of the above-mentioned spatial transformations1 where the 7x1 t7 plane coincides with the x-t plane1 the significance is possible that the x-axis of space some-how remains conserved in its position4 ,we can approach the idea of space in a corresponding manner1 though some may regard the attempt as rather fantastical4 ,according to these ideas1 the word 8,relativity-,postulate0 which has been coined for the demands of invariance in the group ,g1 seems to be rather inexpressive for a true understanding of the group ,gc1 and for further progress4 ,because the sense of the postulate is that the four-dimensional world is given in space and time by phenomena only1 but the projection in time and space can be handled with a certain freedom1 and therefore ,i would rather like to give to this assertion the name 8,the ,postulate of the ,absolute world0 ,world-,postulate4 ,,iii ,by the world-postulate a similar treatment of the four determining quantities x1 y1 z1 t1 of a world-point is possible4 ,thereby the forms under which the physical laws come forth1 gain in intelligibility1 as ,i shall presently show4 ,above all1 the idea of acceleration becomes much more striking and clear4 ,i shall again use the geometrical method of expression4 ,let us call any world-point ,o as a 8,space-time-null-point40 ,the cone ct - x - y - z ,o consists of two parts with ,o as apex1 one part having t #j1 the other having t #j4 ,the first1 which we may call the fore-cone consists of all those points which send light towards ,o1 the second1 which we may call the aft-cone1 consists of all those points which receive their light from ,o4 ,the region bounded by the fore-cone may be called the fore-side of ,o1 and the region bounded by the aft-cone may be called the aft-side of ,o4 7,vide fig4 #b74 ,on the aft-side of ,o we have the already considered hyperboloidal shell ,f ct - x - y - z #a1 t #j4 ,the region inside the two cones will be occupied by the hyperboloid of one sheet -,f x y z - ct k1 where k can have all possible positive values4 ,the hyperbolas which lie upon this figure with ,o as centre1 are important for us4 ,for the sake of clearness the individual branches of this hyperbola will be called the 8,inter-hyperbola with centre ,o40 ,such a hyperbolic branch1 when thought of as a world-line1 would represent a motion which for t - and t 1 asymptotically approaches the velocity of light c4 ,if1 by way of analogy to the idea of vectors in space1 we call any directed length in the manifoldness x1 y1 z1 t a vector1 then we have to distinguish between a time-vector directed from ,o towards the sheet ,f #a1 t #j and a space-vector directed from ,o towards the sheet -,f #a4 ,the time-axis can be parallel to any vector of the first kind4 ,any world-point between the fore and aft cones of ,o1 may by means of the system of reference be regarded either as synchronous with ,o1 as well as later or earlier than ,o4 ,every world-point on the fore-side of ,o is necessarily always earlier1 every point on the aft side of ,o1 later than ,o4 ,the limit c corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness t #j4 ,in the figure drawn1 this cross-section has been intentionally drawn with a different breadth4 ,let us decompose a vector drawn from ,o towards 7x1 y1 z1 t7 into its components4 ,if the directions of the two vectors are respectively the directions of the radius vector ,,or to one of the surfaces ,f #a1 and of a tangent ,,rs at the point ,r of the surface1 then the vectors shall be called normal to each other4 ,accordingly ctt - xx - yy - zz #j1 which is the condition that the vectors with the components 7x1 y1 z1 t7 and 7x y z t7 are normal to each other4 ,for the measurement of vectors in different directions1 the unit measuring rod is to be fixed in the following manner2--a space-like vector from #j to -,f ,i is always to have the measure unity1 and a time-like vector from ,o to ,f #a1 t #j is always to have the measure #a_/;c4 ,let us now fix our attention upon the world-line of a substantive point running through the world-point 7x1 y1 z1 t72 then as we follow the progress of the line1 the quantity d 7#a_/;c7 7cdt - dx - dy - dz71 corresponds to the time-like vector-element 7dx1 dy1 dz1 dt74 ,the integral d1 taken over the world-line from any fixed initial point ,p to any variable final point ,p1 may be called the 8,proper-time0 of the substantial point at ,p upon the world-line4 ,we may regard 7x1 y1 z1 t71 i4e41 the components of the vector ,,op1 as functions of the 8proper-time0 2 let 74x1 4y1 4z1 4t7 denote the first differential-quotients1 and 744x1 44y1 44z1 44t7 the second differential quotients of 7x1 y1 z1 t7 with regard to 1 then these may respectively be called the ,velocity-vector1 and the ,acceleration-vector of the substantial point at ,p4 ,now we have c 4t - 4x - 4y - 4z c c 4t44t - 4x44x - 4y44y - 4z44z #j i4e41 the #8,velocity-vector' is the time-like vector of unit measure in the direction of the world-line at ,p1 the #8,acceleration-vector' at ,p is normal to the velocity-vector at ,p1 and is in any case1 a space-like vector4 ,now there is1 as can be easily seen1 a certain hyperbola1 which has three infinitely contiguous points in common with the world-line at ,p1 and of which the asymptotes are the generators of a #8fore-cone' and an #8aft-cone4' ,this hyperbola may be called the 8hyperbola of curvature0 at ,p 7vide fig4 #c74 ,if ,m be the centre of this hyperbola1 then we have to deal here with an #8,inter-hyperbola' with centre ,m4 ,let ,p measure of the vector ,,mp1 then we easily perceive that the acceleration-vector at ,p is a vector of magnitude c_/ in the direction of ,,mp4 ,if 44x1 44y1 44z1 44t are nil1 then the hyperbola of curvature at ,p reduces to the straight line touching the world-line at ,p1 and 4 ,,iv ,in order to demonstrate that the assumption of the group ,gc for the physical laws does not possibly lead to any contradiction1 it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group4 ,the revision has already been successfully made in the case of 8,thermodynamics and ,radiation10#cj for 8,electromagnetic phenomena01#ca and finally for 8,mechanics with the maintenance of the idea of mass40 ,for this last mentioned province of physics1 the question may be asked3 if there is a force with the components ,x1 ,y1 ,z 7in the direction of the space-axes7 at a world-point 7x1 y1 z1 t71 where the velocity-vector is 74x1 4y1 4z1 4t71 then how are we to regard this force when the system of reference is changed in any possible manner8 ,now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields1 where the group ,gc is undoubtedly permissible4 ,these theorems lead us to the following simple rule2 if the system of reference be changed in any way1 then the supposed force is to be put as a force in the new space-coordinates in such a manner1 that the corresponding vector with the components 4t,x1 4t,y1 4t,z1 4t,t1 where ,t #a_/;c 74x_/4t ,x 4y_/4t ,y 4z_/4t ,z7 #a_/;c 7the rate of which work is done at the world-point71 remains unaltered4 ,this vector is always normal to the velocity-vector at ,p4 ,such a force-vector1 representing a force at ,p1 may be called a moving force-vector at ,p4 ,now the world-line passing through ,p will be described by a substantial point with the constant mechanical mass m4 ,let us call m-times the velocity-vector at ,p as the impulse-vector1 and m-times the acceleration-vector at ,p as the force-vector of motion1 at ,p4 ,according to these definitions1 the following law tells us how the motion of a point-mass takes place under any moving force-vector#cb3 ,the force-vector of motion is equal to the moving force-vector4 ,this enunciation comprises four equations for the components in the four directions1 of which the fourth can be deduced from the first three1 because both of the above-mentioned vectors are perpendicular to the velocity-vector4 ,from the definition of ,t1 we see that the fourth simply expresses the 8,energy-law40 ,accordingly c-times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass4 ,the expression for this is mc dt_/d mc _/7#a - v_/c7 i4e41 if we deduct from this the additive constant mc1 we obtain the expression mv of ,newtonian-mechanics up to magnitudes of the order of #a_/;c4 ,hence it appears that the energy depends upon the system of reference4 ,but since the t-axis can be laid in the direction of any time-like axis1 therefore the energy-law comprises1 for any possible system of reference1 the whole system of equations of motion4 ,this fact retains its significance even in the limiting case c 1 for the axiomatic construction of ,newtonian mechanics1 as has already been pointed out by ,t4 ,r4 ,schtz4#cc ,from the very beginning1 we can establish the ratio between the units of time and space in such a manner1 that the velocity of light becomes unity4 ,if we now write -#a t l1 in the place of l1 then the differential expression d -7dx dy dz dl71 becomes symmetrical in 7x1 y1 r1 l72 this symmetry then enters into each law1 which does not contradict the world-postulate4 ,we can clothe the 8essential nature of this postulate in the mystical1 but mathematically significant formula #caj km -#a ,sec4 ,v ,the advantages arising from the formulation of the world-postulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the ,maxwell-,lorentz theory4 ,let us conceive of the world-line of such an electron with the charge 7e71 and let us introduce upon it the 8,proper-time0 reckoned from any possible initial point4 ,in order to obtain the field caused by the electron at any world-point ,p let us construct the fore-cone belonging to ,p 7vide fig4 #d74 ,clearly this cuts the unlimited world-line of the electron at a single point ,p1 because these directions are all time-like vectors4 ,at ,p1 let us draw the tangent to the world-line1 and let us draw from ,p the normal to this tangent4 ,let r be the measure of ,p,q4 ,according to the definition of a fore-cone1 r_/e is to be reckoned as the measure of ,,pq4 ,now at the world-point ,p1 the vector-potential of the field excited by e is represented by the vector in direction ,,pq1 having the magnitude e_/cr1 in its three space components along the x-1 y-1 z-axes2 the scalar-potential is represented by the component along the t-axis4 ,this is the elementary law found out by ,a4 ,lienard1 and ,e4 ,wiechert4#cd ,if the field caused by the electron be described in the above-mentioned way1 then it will appear that the division of the field into electric and magnetic forces is a relative one1 and depends upon the time-axis assumed2 the two forces considered together bears some analogy to the force-screw in mechanics2 the analogy is1 however1 imperfect4 ,i shall now describe the ponderomotive force which is exerted by one moving electron upon another moving electron4 ,let us suppose that the world-line of a second point-electron passes through the world-point ,p4 ,let us determine ,p1 ,q1 r as before1 construct the middle-point ,m of the hyperbola of curvature at ,p1 and finally the normal ,,mn upon a line through ,p which is parallel to ,,qp4 ,with ,p as the initial point1 we shall establish a system of reference in the following way3 the t-axis will be laid along ,,pq1 the x-axis in the direction of ,,qp4 ,the y-axis in the direction of ,,mn1 then the z-axis is automatically determined1 as it is normal to the x-1 y-1 z-axes4 ,let 3x1 3y1 3z1 3t be the acceleration-vector at ,p1 4x1 4y 4z1 4t be the velocity-vector at ,p4 ,then the force-vector exerted by the first election e1 7moving in any possible manner7 upon the second election e1 7likewise moving in any possible manner7 at ,p is represented by -e e74t - 4x_/c7,f1 ,for the components ,fx1 ,fy1 ,fz1 ,ft of the vector ,f the following three relations hold3-- c,ft - ,fx #a_/r1 ,fy 3y_/7cr71 ,fz #j1 and fourthly this vector ,f is normal to the velocity-vector ,p1 and through this circumstance alone1 its dependence on this last velocity-vector arises4 ,if we compare with this expression the previous formul#ce giving the elementary law about the ponderomotive action of moving electric charges upon each other1 then we cannot but admit1 that the relations which occur here reveal the inner essence of full simplicity first in four dimensions2 but in three dimensions1 they have very complicated projections4 ,in the mechanics reformed according to the world-postulate1 the disharmonies which have disturbed the relations between ,newtonian mechanics and modern electrodynamics automatically disappear4 ,i shall now consider the position of the ,newtonian law of attraction to this postulate4 ,i will assume that two point-masses m and m describe their world-lines2 a moving force-vector is exercised by m upon m1 and the expression is just the same as in the case of the electron1 only we have to write mm instead -ee4 ,we shall consider only the special case in which the acceleration-vector of m is always zero3 then t may be introduced in such a manner that m may be regarded as fixed1 the motion of m is now subjected to the moving-force vector of m alone4 ,if we now modify this given vector by writing -74#a_/7a-7v_/c77 instead of 4t 74t #a up to magnitudes of the order 7#a4_/;c771 then it appears that ,kepler's laws hold good for the position 7x1 y1 z71 of m at any time1 only in place of the time t1 we have to write the proper time of m4 ,on the basis of this simple remark1 it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the ,newtonian law of attraction in combination with ,newtonian mechanics4 ,also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate4 ,i shall also show on a later occasion that the deduction of these equations1 as taught by ,lorentz1 are by no means to be given up4 ,the fact that the world-postulate holds without exception is1 ,i believe1 the true essence of an electromagnetic picture of the world2 the idea first occurred to ,lorentz1 its essence was first picked out by ,einstein1 and is now gradually fully manifest4 ,in course of time1 the mathematical consequences will be gradually deduced1 and enough suggestions will be forthcoming for the experimental verification of the postulate2 in this way even those1 who find it uncongenial1 or even painful to give up the old1 time-honoured concepts1 will be reconciled to the new ideas of time and space1--in the prospect that they will lead to pre-established harmony between pure mathematics and physics4 ,footnote #cj3 ,planck1 ,zur ,dynamik bewegter systeme1 ,ann4 d4 physik1 ,bd4 #bf1 #aijh1 p4 #a4 ,footnote #ca3 ,h4 ,minkowski2 the passage refers to paper 7#b7 of the present edition4 ,footnote #cb3 ,minkowski--,mechanics1 appendix1 page #fe of paper 7#b74 ,planck--,verh4 d4 ,d4 ,p4 ,g4 ,vol4 #d1 #aijf1 p4 #acf4 ,footnote #cc3 ,schtz1 ,gtt4 ,nachr4 #ahig1 p4 #aaj4 ,footnote #cd3 ,lienard1 ,l',eclairage lectrique ,t4 #af1 #ahif1 p4 #ec4 ,wiechert1 ,ann4 d4 ,physik1 ,vol4 #d4 ,footnote #ce3 ,k4 ,schwarzschild4 ,gtt-,nachr4 #aijc4 ,h4 ,a4 ,lorentz1 ,enzyklopdie der ,math4 ,wissenschaften ,v4 ,art #ad1 p4 #aii4 ,the ,foundation of the ,generalised ,theory of ,relativity ,by ,a4 ,einstein4 ,from ,annalen der ,physik #d4di4aiaf4 ,the theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as 8the theory of ,relativity20 this latter theory ,i differentiate from the former 8,special ,relativity theory10 and suppose it to be known4 ,the generalization of the ,relativity theory has been made much easier through the form given to the special ,relativity theory by ,minkowski1 which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates1 and who made use of it in the building up of the theory4 ,the mathematical apparatus useful for the general relativity theory1 lay already complete in the 8,absolute ,differential ,calculus10 which were based on the researches of ,gauss1 ,riemann and ,christoffel on the non-,euclidean manifold1 and which have been shaped into a system by ,ricci and ,levi-civita1 and already applied to the problems of theoretical physics4 ,i have in part ,b of this communication developed in the simplest and clearest manner1 all the supposed mathematical auxiliaries1 not known to ,physicists1 which will be useful for our purpose1 so that1 a study of the mathematical literature is not necessary for an understanding of this paper4 ,finally in this place ,i thank my friend ,grossmann1 by whose help ,i was not only spared the study of the mathematical literature pertinent to this subject1 but who also aided me in the researches on the field equations of gravitation4 ,a ,principal considerations about the ,postulate of ,relativity4 #a4 ,remarks on the ,special ,relativity ,theory4 ,the special relativity theory rests on the following postulate which also holds valid for the ,galileo-,newtonian mechanics4 ,if a co-ordinate system ,k be so chosen that when referred to it1 the physical laws hold in their simplest forms these laws would be also valid when referred to another system of co-ordinates ,k which is subjected to an uniform translational motion relative to ,k4 ,we call this postulate 8,the ,special ,relativity ,principle40 ,by the word special1 it is signified that the principle is limited to the case1 when ,k has uniform translatory motion with reference to ,k1 but the equivalence of ,k and ,k does not extend to the case of non-uniform motion of ,k relative to ,k4 ,the ,special ,relativity ,theory does not differ from the classical mechanics through the assumption of this postulate1 but only through the postulate of the constancy of light-velocity in vacuum which1 when combined with the special relativity postulate1 gives in a well-known way1 the relativity of synchronism as well as the ,lorenz-transformation1 with all the relations between moving rigid bodies and clocks4 ,the modification which the theory of space and time has undergone through the special relativity theory1 is indeed a profound one1 but a weightier point remains untouched4 ,according to the special relativity theory1 the theorems of geometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest1 and more generally the theorems of kinematics1 as theorems which describe the relation between measurable bodies and clocks4 ,consider two material points of a solid body at rest2 then according to these conceptions there corresponds to these points a wholly definite extent of length1 independent of kind1 position1 orientation and time of the body4 ,similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system2 then to these positions1 there always corresponds1 a time-interval of a definite length1 independent of time and place4 ,it would be soon shown that the general relativity theory can not hold fast to this simple physical significance of space and time4 #b4 ,about the reasons which explain the extension of the relativity-postulate4 ,to the classical mechanics 7no less than7 to the special relativity theory1 is attached an episteomological defect1 which was perhaps first cleanly pointed out by ,e4 ,mach4 ,we shall illustrate it by the following example2 ,let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another 7and from all other masses7 that only that sort of gravitational forces are to be taken into account which the parts of any of these bodies exert upon each other4 ,the distance of the bodies from one another is invariable4 ,the relative motion of the different parts of each body is not to occur4 ,but each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity 7definite relative motion for both the masses74 ,now let us think that the surfaces of both the bodies 7,s and ,s7 are measured with the help of measuring rods 7relatively at rest72 it is then found that the surface of ,s is a sphere and the surface of the other is an ellipsoid of rotation4 ,we now ask1 why is this difference between the two bodies8 ,an answer to this question can only then be regarded as satisfactory from the episteomological standpoint when the thing adduced as the cause is an observable fact of experience4 ,the law of causality has the sense of a definite statement about the world of experience only when observable facts alone appear as causes and effects4 ,the ,newtonian mechanics does not give to this question any satisfactory answer4 ,for example1 it says3--,the laws of mechanics hold true for a space ,r relative to which the body ,s is at rest1 not however for a space relative to which ,s is at rest4 ,the ,galiliean space1 which is here introduced is however only a purely imaginary cause1 not an observable thing4 ,it is thus clear that the ,newtonian mechanics does not1 in the case treated here1 actually fulfil the requirements of causality1 but produces on the mind a fictitious complacency1 in that it makes responsible a wholly imaginary cause ,r for the different behaviours of the bodies ,s and ,s which are actually observable4 ,a satisfactory explanation to the question put forward above can only be thus given3--that the physical system composed of ,s and ,s shows for itself alone no conceivable cause to which the different behaviour of ,s and ,s can be attributed4 ,the cause must thus lie outside the system4 ,we are therefore led to the conception that the general laws of motion which determine specially the forms of ,s and ,s must be of such a kind1 that the mechanical behaviour of ,s and ,s must be essentially conditioned by the distant masses1 which we had not brought into the system considered4 ,these distant masses1 7and their relative motion as regards the bodies under consideration7 are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration4 ,they take the place of the imaginary cause ,r4 ,among all the conceivable spaces ,r and ,r moving in any manner relative to one another1 there is a priori1 no one set which can be regarded as affording greater advantages1 against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived4 ,the laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner4 ,we thus arrive at an extension of the relativity postulate4 ,besides this momentous episteomological argument1 there is also a well-known physical fact which speaks in favour of an extension of the relativity theory4 ,let there be a ,galiliean co-ordinate system ,k relative to which 7at least in the four-dimensional region considered7 a mass at a sufficient distance from other masses move uniformly in a line4 ,let ,k be a second co-ordinate system which has a uniformly accelerated motion relative to ,k4 ,relative to ,k any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and the direction of acceleration is independent of its material composition and its physical conditions4 ,can any observer1 at rest relative to ,k1 then conclude that he is in an actually accelerated reference-system8 ,this is to be answered in the negative2 the above-named behaviour of the freely moving masses relative to ,k can be explained in as good a manner in the following way4 ,the reference-system ,k has no acceleration4 ,in the space-time region considered there is a gravitation-field which generates the accelerated motion relative to ,k4 ,this conception is feasible1 because to us the experience of the existence of a field of force 7namely the gravitation field7 has shown that it possesses the remarkable property of imparting the same acceleration to all bodies4 ,the mechanical behaviour of the bodies relative to ,k is the same as experience would expect of them with reference to systems which we assume from habit as stationary2 thus it explains why from the physical stand-point it can be assumed that the systems ,k and ,k can both with the same legitimacy be taken as at rest1 that is1 they will be equivalent as systems of reference for a description of physical phenomena4 ,from these discussions we see1 that the working out of the general relativity theory must1 at the same time1 lead to a theory of gravitation2 for we can 8create0 a gravitational field by a simple variation of the co-ordinate system4 ,also we see immediately that the principle of the constancy of light-velocity must be modified1 for we recognise easily that the path of a ray of light with reference to ,k must be1 in general1 curved1 when light travels with a definite and constant velocity in a straight line with reference to ,k4 #c4 ,the time-space continuum4 ,requirements of the general ,co-variance for the equations expressing the laws of ,nature in general4 ,in the classical mechanics as well as in the special relativity theory1 the co-ordinates of time and space have an immediate physical significance2 when we say that any arbitrary point has x as its ,x co-ordinate1 it signifies that the projection of the point-event on the ,x-axis ascertained by means of a solid rod according to the rules of ,euclidean ,geometry is reached when a definite measuring rod1 the unit rod1 can be carried x times from the origin of co-ordinates along the ,x axis4 ,a point having x t as the ,x co-ordinate signifies that a unit clock which is adjusted to be at rest relative to the system of co-ordinates1 and coinciding in its spatial position with the point-event and set according to some definite standard has gone over x t periods before the occurrence of the point-event4 ,this conception of time and space is continually present in the mind of the physicist1 though often in an unconscious way1 as is clearly recognised from the role which this conception has played in physical measurements4 ,this conception must also appear to the reader to be lying at the basis of the second consideration of the last paragraph and imparting a sense to these conceptions4 ,but we wish to show that we are to abandon it and in general to replace it by more general conceptions in order to be able to work out thoroughly the postulate of general relativity1--the case of special relativity appearing as a limiting case when there is no gravitation4 ,we introduce in a space1 which is free from ,gravitation-field1 a ,galiliean ,co-ordinate ,system ,k 7x1 y1 z1 t7 and also1 another system ,k 7x y z t7 rotating uniformly relative to ,k4 ,the origin of both the systems as well as their z-axes might continue to coincide4 ,we will show that for a space-time measurement in the system ,k1 the above established rules for the physical significance of time and space can not be maintained4 ,on grounds of symmetry it is clear that a circle round the origin in the ,,xy plane of ,k1 can also be looked upon as a circle in the plane 7,x1 ,y7 of ,k4 ,let us now think of measuring the circumference and the diameter of these circles1 with a unit measuring rod 7infinitely small compared with the radius7 and take the quotient of both the results of measurement4 ,if this experiment be carried out with a measuring rod at rest relatively to the ,galiliean system ,k we would get 1 as the quotient4 ,the result of measurement with a rod relatively at rest as regards ,k would be a number which is greater than 4 ,this can be seen easily when we regard the whole measurement-process from the system ,k and remember that the rod placed on the periphery suffers a ,lorenz-contraction1 not however when the rod is placed along the radius4 ,euclidean ,geometry therefore does not hold for the system ,k2 the above fixed conceptions of co-ordinates which assume the validity of ,euclidean ,geometry fail with regard to the system ,k4 ,we cannot similarly introduce in ,k a time corresponding to physical requirements1 which will be shown by all similarly prepared clocks at rest relative to the system ,k4 ,in order to see this we suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle1 and considered from the stationary system ,k4 ,according to the well-known results of the special relativity theory it follows--7as viewed from ,k7--that the clock placed at the periphery will go slower than the second one which is at rest4 ,the observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light will see the clock at the periphery going slower than the clock beside him4 ,since he cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he will interpret his observation by saying that the clock on the periphery actually goes slower than the clock at the origin4 ,he cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position4 ,we therefore arrive at this result4 ,in the general relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod1 and time-like co-ordinate difference with the aid of a normal clock4 ,the means hitherto at our disposal1 for placing our co-ordinate system in the time-space continuum1 in a definite way1 therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate system to the four-dimensional world in such a way1 that by it we can expect to get a specially simple formulation of the laws of ,nature4 ,so that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena4 ,this amounts to the following law3-- ,that in general1 ,laws of ,nature are expressed by means of equations which are valid for all co-ordinate systems1 that is1 which are covariant for all possible transformations4 ,it is clear that a physics which satisfies this postulate will be unobjectionable from the standpoint of the general relativity postulate4 ,because among all substitutions there are1 in every case1 contained those1 which correspond to all relative motions of the co-ordinate system 7in three dimensions74 ,this condition of general covariance which takes away the last remnants of physical objectivity from space and time1 is a natural requirement1 as seen from the following considerations4 ,all our well-substantiated space-time propositions amount to the determination of space-time coincidences4 ,if1 for example1 the event consisted in the motion of material points1 then1 for this last case1 nothing else are really observable except the encounters between two or more of these material points4 ,the results of our measurements are nothing else than well-proved theorems about such coincidences of material points1 of our measuring rods with other material points1 coincidences between the hands of a clock1 dial-marks and point-events occurring at the same position and at the same time4 ,the introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences4 ,we fit to the world our space-time variables 7x x x x7 such that to any and every point-event corresponds a system of values of 7x x x x74 ,two coincident point-events correspond to the same value of the variables 7x x x x72 i4e41 the coincidence is characterised by the equality of the co-ordinates4 ,if we now introduce any four functions 7x x x x7 as co-ordinates1 so that there is an unique correspondence between them1 the equality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence of two material points4 ,as the purpose of all physical laws is to allow us to remember such coincidences1 there is a priori no reason present1 to prefer a certain co-ordinate system to another2 i4e41 we get the condition of general covariance4 #d4 ,relation of four co-ordinates to spatial and time-like measurements4 ,analytical expression for the ,gravitation-field4 ,i am not trying in this communication to deduce the general ,relativity-theory as the simplest logical system possible1 with a minimum of axioms4 ,but it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed1 and the fundamental assumptions appear to be most reasonable according to the light of experience4 ,in this sense1 we shall now introduce the following supposition2 that for an infinitely small four-dimensional region1 the relativity theory is valid in the special sense when the axes are suitably chosen4 ,the nature of acceleration of an infinitely small 7positional7 co-ordinate system is hereby to be so chosen1 that the gravitational field does not appear2 this is possible for an infinitely small region4 ,x1 ,x1 ,x are the spatial co-ordinates2 ,x is the corresponding time-co-ordinate measured by some suitable measuring clock4 ,these co-ordinates have1 with a given orientation of the system1 an immediate physical significance in the sense of the special relativity theory 7when we take a rigid rod as our unit of measure74 ,the expression 7#a7 ds - d,x - d,x - d,x d,x had then1 according to the special relativity theory1 a value which may be obtained by space-time measurement1 and which is independent of the orientation of the local co-ordinate system4 ,let us take ds as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region4 ,if ds belonging to the element 7d,x1 d,x1 d,x1 d,x7 be positive we call it with ,minkowski1 time-like1 and in the contrary case space-like4 ,to the line-element considered1 i4e41 to both the infinitely near point-events belong also definite differentials dx1 dx1 dx1 dx1 of the four-dimensional co-ordinates of any chosen system of reference4 ,if there be also a local system of the above kind given for the case under consideration1 d,x's would then be represented by definite linear homogeneous expressions of the form 7#b7 d,x adx ,if we substitute the expression in 7#a7 we get 7#c7 ds gdxdx where g will be functions of x1 but will no longer depend upon the orientation and motion of the #8local' co-ordinates2 for ds is a definite magnitude belonging to two point-events infinitely near in space and time and can be got by measurements with rods and clocks4 ,the g's are here to be so chosen1 that g g2 the summation is to be extended over all values of and 1 so that the sum is to be extended1 over #d #d terms1 of which #ab are equal in pairs4 ,from the method adopted here1 the case of the usual relativity theory comes out when owing to the special behaviour of g in a finite region it is possible to choose the system of co-ordinates in such a way that g assumes constant values-- -#a1 #j1 #j1 #j 7#d7 #j -#a #j #j #j #j -#a #j #j #j #j #a ,we would afterwards see that the choice of such a system of co-ordinates for a finite region is in general not possible4 ,from the considerations in #b and #c it is clear1 that from the physical stand-point the quantities g are to be looked upon as magnitudes which describe the gravitation-field with reference to the chosen system of axes4 ,we assume firstly1 that in a certain four-dimensional region considered1 the special relativity theory is true for some particular choice of co-ordinates4 ,the g's then have the values given in 7#d74 ,a free material point moves with reference to such a system uniformly in a straight-line4 ,if we now introduce1 by any substitution1 the space-time co-ordinates x444x then in the new system g's are no longer constants1 but functions of space and time4 ,at the same time1 the motion of a free point-mass in the new co-ordinates1 will appear as curvilinear1 and not uniform1 in which the law of motion1 will be independent of the nature of the moving mass-points4 ,we can thus signify this motion as one under the influence of a gravitation field4 ,we see that the appearance of a gravitation-field is connected with space-time variability of g's4 ,in the general case1 we can not by any suitable choice of axes1 make special relativity theory valid throughout any finite region4 ,we thus deduce the conception that g's describe the gravitational field4 ,according to the general relativity theory1 gravitation thus plays an exceptional rle as distinguished from the others1 specially the electromagnetic forces1 in as much as the #aj functions g representing gravitation1 define immediately the metrical properties of the four-dimensional region4 ,b ,mathematical ,auxiliaries for ,establishing the ,general ,covariant ,equations4 ,we have seen before that the general relativity-postulate leads to the condition that the system of equations for ,physics1 must be co-variants for any possible substitution of co-ordinates x1 444 x2 we have now to see how such general co-variant equations can be obtained4 ,we shall now turn our attention to these purely mathematical propositions4 ,it will be shown that in the solution1 the invariant ds1 given in equation 7#c7 plays a fundamental rle1 which we1 following ,gauss's ,theory of ,surfaces1 style as the line-element4 ,the fundamental idea of the general co-variant theory is this3--,with reference to any co-ordinate system1 let certain things 7tensors7 be defined by a number of functions of co-ordinates which are called the components of the tensor4 ,there are now certain rules according to which the components can be calculated in a new system of co-ordinates1 when these are known for the original system1 and when the transformation connecting the two systems is known4 ,the things herefrom designated as 8,tensors0 have further the property that the transformation equation of their components are linear and homogeneous2 so that all the components in the new system vanish if they are all zero in the original system4 ,thus a law of ,nature can be formulated by putting all the components of a tensor equal to zero so that it is a general co-variant equation2 thus while we seek the laws of formation of the tensors1 we also reach the means of establishing general co-variant laws4 #e4 ,contra-variant and co-variant ,four-vector4 ,contra-variant ,four-vector4 ,the line-element is defined by the four components dx1 whose transformation law is expressed by the equation 7#e7 dx'sigma sumnu fracpartial x'sigmapartial xnu dxnu ,the dx's are expressed as linear and homogeneous function of dx's2 we can look upon the differentials of the co-ordinates as the components of a tensor1 which we designate specially as a contravariant ,four-vector4 ,everything which is defined by ,four quantities ,a1 with reference to a co-ordinate system1 and transforms according to the same law1 7#e;a7 ,asigma sumnu fracpartial x'sigmapartial xnu ,anu we may call a contra-variant ,four-vector4 ,from 7#e4 a71 it follows at once that the sums 7,a ,b7 are also components of a four-vector1 when ,a and ,b are so2 corresponding relations hold also for all systems afterwards introduced as 8tensors0 7,rule of addition and subtraction of ,tensors74 ,co-variant ,four-vector4 ,we call four quantities ,a as the components of a covariant four-vector1 when for any choice of the contra-variant four vector ,b 7#f7 ,a ,b ,invariant4 ,from this definition follows the law of transformation of the co-variant four-vectors4 ,if we substitute in the right hand side of the equation ,a ,b ,a ,b4 the expressions sumsigma fracpartial xnupartial xsigma' ,bsigma' for ,b following from the inversion of the equation 7#e;a7 we get sumsigma ,bsigma' sumnu fracpartial xnupartial xsigma' ,anu sumsigma ,bsigma' ,a'sigma ,as in the above equation ,b are independent of one another and perfectly arbitrary1 it follows that the transformation law is3-- ,a'sigma sum fracpartial xnupartial xsigma' ,anu ,remarks on the simplification of the mode of writing the expressions4 ,a glance at the equations of this paragraph will show that the indices which appear twice within the sign of summation for example in 7#e7 are those over which the summation is to be made and that only over the indices which appear twice4 ,it is therefore possible1 without loss of clearness1 to leave off the summation sign2 so that we introduce the rule3 wherever the index in any term of an expression appears twice1 it is to be summed over all of them except when it is not expressedly said to the contrary4 ,the difference between the co-variant and the contra-variant four-vector lies in the transformation laws 7#g7 and 7#e74 ,both the quantities are tensors according to the above general remarks2 in it lies its significance4 ,in accordance with ,ricci and ,levi-civita1 the contravariants and co-variants are designated by the over and under indices4 #f4 ,tensors of the second and higher ranks4 ,contravariant tensor3--,if we now calculate all the #af products ,a of the components ,a ,b1 of two contravariant four-vectors 7#h7 ,a ,a,b ,a1 will according to 7#h7 and 7#e a7 satisfy the following transformation law4 7#i7 ,asigma tau' fracpartial x'sigmapartial xmu fracpartial x'taupartial xnu ,amu nu ,we call a thing which1 with reference to any reference system is defined by #af quantities and fulfils the transformation relation 7#i71 a contravariant tensor of the second rank4 ,not every such tensor can be built from two four-vectors1 7according to #h74 ,but it is easy to show that any #af quantities ,a1 can be represented as the sum of ,a,b of properly chosen four pairs of four-vectors4 ,from it1 we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through 7#i71 by proving it only for the special tensor of the type 7#h74 ,contravariant ,tensor of any rank3--,it is clear that corresponding to 7#h7 and 7#i71 we can define contravariant tensors of the #crd and higher ranks1 with #d1 etc4 components4 ,thus it is clear from 7#h7 and 7#i7 that in this sense1 we can look upon contravariant four-vectors1 as contravariant tensors of the first rank4 ,co-variant tensor4 ,if on the other hand1 we take the #af products ,a of the components of two co-variant four-vectors ,a and ,b1 7#aj7 ,a ,a ,b4 for them holds the transformation law 7#aa7 ,asigma tau' fracpartial x'mupartial xsigma' fracpartial x'nupartial xtau' ,amu nu ,by means of these transformation laws1 the co-variant tensor of the second rank is defined4 ,all re-marks which we have already made concerning the contravariant tensors1 hold also for co-variant tensors4 ,remark3-- ,it is convenient to treat the scalar ,invariant either as a contravariant or a co-variant tensor of zero rank4 ,mixed tensor4 ,we can also define a tensor of the second rank of the type 7#ab7 ,a ,a,b which is co-variant with reference to and contravariant with reference to 4 ,its transformation law is 7#ac7 ,atau'sigma fracpartial xtau'partial xbeta fracpartial alphapartial xsigma' ,abetaalpha ,naturally there are mixed tensors with any number of co-variant indices1 and with any number of contra-variant indices4 ,the co-variant and contra-variant tensors can be looked upon as special cases of mixed tensors4 ,symmetrical tensors3-- ,a contravariant or a co-variant tensor of the second or higher rank is called symmetrical when any two components obtained by the mutual interchange of two indices are equal4 ,the tensor ,a or ,a is symmetrical1 when we have for any combination of indices 7#ad7 ,a ,a or 7#ad;a7 ,a ,a4 ,it must be proved that a symmetry so defined is a property independent of the system of reference4 ,it follows in fact from 7#i7 remembering 7#ad7 ,asigma tau' fracpartial xsigma'partial xmu fracpartial x'taupartial xnu ,amu nu fracpartial xsigma'partial xmu fracpartial xtau'partial xnu ,anu mu ,atau sigma' ,antisymmetrical tensor4 ,a contravariant or co-variant tensor of the #bnd1 #crd or #dth rank is called antisymmetrical when the two components got by mutually interchanging any two indices are equal and opposite4 ,the tensor or ,a or ,a is thus antisymmetrical when we have 7#ae7 ,a -,a or 7#ae;a7 ,a -,a4 ,of the #af components ,a1 the four components ,a vanish1 the rest are equal and opposite in pairs2 so that there are only #f numerically different components present 7,six-vector74 ,thus we also see that the antisymmetrical tensor ,a 7#crd rank7 has only #d components numerically different1 and the antisymmetrical tensor ,a only one4 ,symmetrical tensors of ranks higher than the fourth1 do not exist in a continuum of #d dimensions4 #g4 ,multiplication of ,tensors4 ,outer multiplication of ,tensors3--,we get from the components of a tensor of rank z1 and another of a rank z1 the components of a tensor of rank 7z z7 for which we multiply all the components of the first with all the components of the second in pairs4 ,for example1 we obtain the tensor from the tensors ,a and ,b of different kinds3-- ,a,b1 ,a,b1 ,a,b4 ,the proof of the tensor character of 1 follows immediately from the expressions 7#h71 7#aj7 or 7#ab71 or the transformation equations 7#i71 7#aa71 7#ac72 equations 7#h71 7#aj7 and 7#ab7 are themselves examples of the outer multiplication of tensors of the first rank4 ,reduction in rank of a mixed ,tensor4 ,from every mixed tensor we can get a tensor which is two ranks lower1 when we put an index of co-variant character equal to an index of the contravariant character and sum according to these indices 7,reduction74 ,we get for example1 out of the mixed tensor of the fourth rank ,a1 the mixed tensor of the second rank ,a ,a 7 ,a7 and from it again by 8reduction0 the tensor of the zero rank ,a ,a ,a4 ,the proof that the result of reduction retains a truly tensorial character1 follows either from the representation of tensor according to the generalisation of 7#ab7 in combination with 7#f7 or out of the generalisation of 7#ac74 ,inner and mixed multiplication of ,tensors4 ,this consists in the combination of outer multiplication with reduction4 ,examples3--,from the co-variant tensor of the second rank ,a and the contravariant tensor of the first rank ,b we get by outer multiplication the mixed tensor ,d ,a ,b 4 ,through reduction according to indices and 7i4e41 putting 71 the co-variant four vector ,d ,d ,a ,b is generated4 ,these we denote as the inner product of the tensor ,a and ,b4 ,similarly we get from the tensors ,a and ,b through outer multiplication and two-fold reduction the inner product ,a ,b4 ,through outer multiplication and one-fold reduction we get out of ,a and ,b1 the mixed tensor of the second rank ,d ,a ,b4 ,we can fitly call this operation a mixed one2 for it is outer with reference to the indices and and inner with respect to the indices and 4 ,we now prove a law1 which will be often applicable for proving the tensor-character of certain quantities4 ,according to the above representation1 ,a ,b is a scalar1 when ,a and ,b are tensors4 ,we also remark that when ,a ,b is an invariant for every choice of the tensor ,b1 then ,a has a tensorial character4 ,proof3--,according to the above assumption1 for any substitution we have ,a ,b ,a ,b4 ,from the inversion of 7#i7 we have however ,bmu nu fracpartial xmupartial xsigma' fracpartial xnupartial tau' ,bsigma tau' ,substitution of this for ,b in the above equation gives 7,asigma tau' - fracpartial xmupartial xsigma' fracpartial xnupartial xtau'7 ,bsigma tau' #j ,this can be true1 for any choice of ,b only when the term within the bracket vanishes4 ,from which by referring to 7#aa71 the theorem at once follows4 ,this law correspondingly holds for tensors of any rank and character4 ,the proof is quite similar4 ,the law can also be put in the following form4 ,if ,b and ,c are any two vectors1 and if for every choice of them the inner product ,a ,b ,c is a scalar1 then ,a is a co-variant tensor4 ,the last law holds even when there is the more special formulation1 that with any arbitrary choice of the four-vector ,b alone the scalar product ,a ,b ,b is a scalar1 in which case we have the additional condition that ,a satisfies the symmetry condition4 ,according to the method given above1 we prove the tensor character of 7,a ,a71 from which on account of symmetry follows the tensor-character of ,a4 ,this law can easily be generalized in the case of co-variant and contravariant tensors of any rank4 ,finally1 from what has been proved1 we can deduce the following law which can be easily generalized for any kind of tensor3 ,if the quantities ,a ,b form a tensor of the first rank1 when ,b is any arbitrarily chosen four-vector1 then ,a is a tensor of the second rank4 ,if for example1 ,c is any four-vector1 then owing to the tensor character of ,a ,b1 the inner product ,a ,c ,b is a scalar1 both the four-vectors ,c and ,b being arbitrarily chosen4 ,hence the proposition follows at once4 ,a few words about the ,fundamental ,tensor g4 ,the co-variant fundamental tensor--,in the invariant expression of the square of the linear element ds g dx dx dx plays the rle of any arbitrarily chosen contravariant vector1 since further g g1 it follows from the considerations of the last paragraph that g is a symmetrical co-variant tensor of the second rank4 ,we call it the 8fundamental tensor40 ,afterwards we shall deduce some properties of this tensor1 which will also be true for any tensor of the second rank4 ,but the special rle of the fundamental tensor in our ,theory1 which has its physical basis on the particularly exceptional character of gravitation makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor4 ,the co-variant fundamental tensor4 ,if we form from the determinant scheme g the minors of g and divide them by the determinant g g we get certain quantities g g1 which as we shall prove generates a contravariant tensor4 ,according to the well-known law of ,determinants 7#af7 g g where is #a1 or #j1 according as or not4 ,instead of the above expression for ds1 we can also write g dx dx or according to 7#af7 also in the form g g g dx dx ,now according to the rules of multiplication1 of the fore-going paragraph1 the magnitudes d g dx forms a co-variant four-vector1 and in fact 7on account of the arbitrary choice of dx7 any arbitrary four-vector4 ,if we introduce it in our expression1 we get ds g d d4 ,for any choice of the vectors d d this is scalar1 and g1 according to its definition is a symmetrical thing in and 1 so it follows from the above results1 that g is a contravariant tensor4 ,out of 7#af7 it also follows that is a tensor which we may call the mixed fundamental tensor4 ,determinant of the fundamental tensor4 ,according to the law of multiplication of determinants1 we have g g g g ,on the other hand we have g g #a ,so that it follows 7#ag7 that g g #a4 ,invariant of volume4 ,we see first the transformation law for the determinant g g 4 ,according to 7#aa7 g' fracpartial xmupartial xsigma' fracpartial xnupartial xtau' gmu u ,from this by applying the law of multiplication twice1 we obtain g' fracpartial xmupartial xsigma' fracpartial xnupartial xtau' gmu nu fracpartial xmualphasigma' g or 7,a7 sqrtg' fracpartial xmupartial xsigma' sqrtg ,on the other hand the law of transformation of the volume element d dx dx dx dx is according to the wellknown law of ,jacobi4 7,b7 dtau' fracdx'sigmadxmu dtau by multiplication of the two last equations 7,a7 and 7,b7 we get 7#ah7 g d g d4 ,instead of g1 we shall afterwards introduce 7-g7 which has a real value on account of the hyperbolic character of the time-space continuum4 ,the invariant 7-g7d1 is equal in magnitude to the four-dimensional volume-element measured with solid rods and clocks1 in accordance with the special relativity theory4 ,remarks on the character of the space-time continuum--,our assumption that in an infinitely small region the special relativity theory holds1 leads us to conclude that ds can always1 according to 7#a7 be expressed in real magnitudes d,x 444 d,xh4 ,if we call d the 8natural0 volume element d,x d,x d,x d,x we have thus 7#ah;a7 d 7g7i4 ,should 7-g7 vanish at any point of the four-dimensional continuum it would signify that to a finite co-ordinate volume at the place corresponds an infinitely small 8natural volume40 ,this can never be the case2 so that g can never change its sign2 we would1 according to the special relativity theory assume that g has a finite negative value4 ,it is a hypothesis about the physical nature of the continuum considered1 and also a pre-established rule for the choice of co-ordinates4 ,if however 7-g7 remains positive and finite1 it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one4 ,we would afterwards see that such a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature4 ,in place of 7#ah7 it follows then simply that d d from this it follows1 remembering the law of ,jacobi1 7#ai7 fracpartial x'sigmadxmu #a ,with this choice of co-ordinates1 only substitutions with determinant #a are allowable4 ,it would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate4 ,we do not seek those laws of nature which are co-variants with regard to the transformations having the determinant #a1 but we ask3 what are the general co-variant laws of nature8 ,first we get the law1 and then we simplify its expression by a special choice of the system of reference4 ,building up of new tensors with the help of the fundamental tensor4 ,through inner1 outer and mixed multiplications of a tensor with the fundamental tensor1 tensors of other kinds and of other ranks can be formed4 ,example3-- ,a g ,a ,a g ,a ,we would point out specially the following combinations3 ,a g g ,a ,a g g ,a 7complement to the co-variant or contravariant tensors7 and ,b g g ,a ,we can call ,b the reduced tensor related to ,a4 ,similarly ,b gg,a4 ,it is to be remarked that g is no other than the 8complement0 of g for we have1-- ggg g g4 #i4 ,equation of the geodetic line 7or of point-motion74 ,as the 8line element0 ds is a definite magnitude independent of the co-ordinate system1 we have also between two points ,p and ,p of a four dimensional continuum a line for which ds is an extremum 7geodetic line71 i4e41 one which has got a significance independent of the choice of co-ordinates4 ,its equation is 7#bj7 ,p,p ds #j ,from this equation1 we can in a wellknown way deduce #d total differential equations which define the geodetic line2 this deduction is given here for the sake of completeness4 ,let 1 be a function of the co-ordinates x2 this defines a series of surfaces which cut the geodetic line sought-for as well as all neighbouring lines from ,p to ,p4 ,we can suppose that all such curves are given when the value of its co-ordinates x are given in terms of 4 ,the sign corresponds to a passage from a point of the geodetic curve sought-for to a point of the contiguous curve1 both lying on the same surface 4 ,then 7#bj7 can be replaced by d #j 7#bj;a7 g7dx_/d77dx_/d7 ,but 7#a_/77;g_/x7 7dx_/d7 7dx_/d7 x g7dx_/d77dx_/d7 ,so we get by the substitution of in 7#bj;a71 remembering that 7dx_/d7 7d_/d77x7 after partial integration1 d k x #j 7#bj;b7 where k 7d_/d77g_/7 7dx_/d7 - 7#a_/7b777;g_/x 7dx_/d7 7dx_/d74 ,from which it follows1 since the choice of is perfectly arbitrary that k's should vanish4 ,then 7#bj;c7 k #j 7 #a1 #b1 #c1 #d7 are the equations of geodetic line2 since along the geodetic line considered we have ds #j1 we can choose the parameter 1 as the length of the arc measured along the geodetic line4 ,then w #a1 and we would get in place of 7#bj;c7 gmunu fracpartial#b xmupartial s#b fracpartial gmunupartial xsigma fracpartial xsigmapartial s fracpartial xmupartial s - frac#ab fracpartial gmusigmapartial xnu fracpartial xmupartial s fracpartial xsigmapartial s #j ,or by merely changing the notation suitably1 7#bj;d7 galphasigma fracd#b xalphads#b beginbmatrixmunusigmaendbmatrix fracdxmuds fracdxnuds #j where we have put1 following ,christoffel1 7#ba7 beginbmatrixmunusigmaendbmatrix frac#ab beginbmatrixfracpartial gmusigmapartial xnu fracpartial gnusigmapartial xmu - fracpartial gmunupartial sigmaendbmatrix ,multiply finally 7#bj;d7 with g 7outer multiplication with reference to 1 and inner with respect to 7 we get at last the final form of the equation of the geodetic line-- fracd#b xtauds#b begin,bmatrixmunutauend,bmatrix fracdxmuds fracdxnuds #j ,here we have put1 following ,christoffel1 begin,bmatrixmunutauend,bmatrix gtaualpha beginbmatrixmunualphaendbmatrix #aj4 ,formation of ,tensors through ,differentiation4 ,relying on the equation of the geodetic line1 we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation4 ,for this purpose1 we would first establish the general co-variant differential equations4 ,we achieve this through a repeated application of the following simple law4 ,if a certain curve be given in our continuum whose points are characterised by the arc-distances s1 measured from a fixed point on the curve1 and if further 1 be an invariant space function1 then d_/ds is also an invariant4 ,the proof follows from the fact that d as well as ds1 are both invariants ,since fracd phids fracpartial phipartial xmu fracpartial xmupartial s so that psi fracpartial phipartial xmu fracdxmuds is also an invariant for all curves which go out from a point in the continuum1 i4e41 for any choice of the vector dx4 ,from which follows immediately that ,a _/x is a co-variant four-vector 7gradient of 74 ,according to our law1 the differential-quotient _/s taken along any curve is likewise an invariant4 ,substituting the value of 1 we get chi fracpartial#b phipartial xmu partial xnu fracdxmuds fracdxnuds fracpartial phipartial xmu fracd#b xmuds#b ,here however we can not at once deduce the existence of any tensor4 ,if we however take that the curves along which we are differentiating are geodesics1 we get from it by replacing dx_/ds according to 7#bb7 chi beginbmatrixfracpartial#b phipartial xmupartial xnu - begin,bmatrixmunutauend,bmatrix fracpartial phipartial xtau endbmatrix fracdxmuds fracdxnuds ,from the interchangeability of the differentiation with regard to and 1 and also according to 7#bc7 and 7#ba7 we see that the bracket begin,bmatrixmunutauend,bmatrix is symmetrical with respect to and 4 ,as we can draw a geodetic line in any direction from any point in the continuum1 x_/ds is thus a four-vector1 with an arbitrary ratio of components1 so that it follows from the results of #g that 7#be7 ,amunu fracpartial#b phipartial xmu partial xnu - begin,bmatrixmunutauend,bmatrix fracpartial phipartial xtau is a co-variant tensor of the second rank4 ,we have thus got the result that out of the co-variant tensor of the first rank ,a _/x we can get by differentiation a co-variant tensor of #bnd rank 7#bf7 ,amunu fracpartial ,amupartial xnu - begin,bmatrixmunutauend,bmatrix ,atau ,we call the tensor ,a the 8extension0 of the tensor ,a4 ,then we can easily show that this combination also leads to a tensor1 when the vector ,a is not representable as a gradient4 ,in order to see this we first remark that 7d_/x7 is a co-variant four-vector when and are scalars4 ,this is also the case for a sum of four such terms 3-- ,smu psi7#a7 fracpartial phi7#a7partial xmu 444 psi7#d7 fracpartial phi7#d7partial xmu when 7#a71 7#a7 444 7#d71 7#d7 are scalars4 ,now it is however clear that every co-variant four-vector is representable in the form of ,s4 ,if for example1 ,a is a four-vector whose components are any given functions of x1 we have1 7with reference to the chosen co-ordinate system7 only to put 7#a7 ,a 7#a7 x 7#b7 ,a 7#b7 x 7#c7 ,a 7#c7 x 7#d7 ,a 7#d7 x4 in order to arrive at the result that ,s is equal to ,a4 ,in order to prove then that ,a is a tensor when on the right side of 7#bf7 we substitute any co-variant four-vector for ,a we have only to show that this is true for the four-vector ,s4 ,for this latter case1 however1 a glance on the right hand side of 7#bf7 will show that we have only to bring forth the proof for the case when ,a _/x4 ,now the right hand side of 7#be7 multiplied by is psi fracpartial#b phipartial xmu partial xnu - begin,bmatrixmunutauend,bmatrix psi fracpartial phipartial xtau which has a tensor character4 ,similarly1 7_/x7 7_/x7 is also a tensor 7outer product of two four-vectors74 ,through addition follows the tensor character of fracpartialpartial xnu 7psi fracpartial phipartial xmu7 - begin,bmatrixmunutauend,bmatrix 7psi fracpartial phipartial xtau7 ,thus we get the desired proof for the four-vector1 _/x and hence for any four-vectors ,a as shown above4 ,with the help of the extension of the four-vector1 we can easily define 8extension0 of a co-variant tensor of any rank4 ,this is a generalisation of the extension of the four-vector4 ,we confine ourselves to the case of the extension of the tensors of the #bnd rank for which the law of formation can be clearly seen4 ,as already remarked every co-variant tensor of the #bnd rank can be represented as a sum of the tensors of the type ,a ,b4 ,it would therefore be sufficient to deduce the expression of extension1 for one such special tensor4 ,according to 7#bf7 we have the expressions fracpartial ,amupartial xsigma - begin,bmatrixsigmamutauend,bmatrix ,atau fracpartial ,bnupartial xsigma - begin,bmatrixsigmamutauend,bmatrix ,btau are tensors4 ,through outer multiplication of the first with ,b and the #bnd with ,a we get tensors of the third rank4 ,their addition gives the tensor of the third rank 7#bg7 ,amunusigma fracpartial ,amunupartial xsigma - begin,bmatrixsigmamutauend,bmatrix ,ataunu - begin,bmatrixsigmanutauend,bmatrix ,amutau where ,a is put ,a ,b4 ,the right hand side of 7#bg7 is linear and homogeneous with reference to ,a1 and its first differential co-efficient1 so that this law of formation leads to a tensor not only in the case of a tensor of the type ,a ,b but also in the case of a summation for all such tensors1 i4e41 in the case of any co-variant tensor of the second rank4 ,we call ,a the extension of the tensor ,a4 ,it is clear that 7#bf7 and 7#bd7 are only special cases of 7#bg7 7extension of the tensors of the first and zero rank74 ,in general we can get all special laws of formation of tensors from 7#bg7 combined with tensor multiplication4 ,some special cases of ,particular ,importance4 ,a few auxiliary lemmas concerning the fundamental tensor4 ,we shall first deduce some of the lemmas much used afterwards4 ,according to the law of differentiation of determinants1 we have 7#bh7 dg g g dg -g gdg4 ,the last form follows from the first when we remember that g g 1 and therefore gg #d1 consequently gdg g dg #j4 ,from 7#bh71 it follows that 7#bi7 frac#asqrt-g fracpartial sqrt-gpartial xsigma frac#ab fraclog 7-g7partial xsigma frac#ab gmunu fracpartial gmunupartial xsigma - frac#ab gmunu fracpartial gmunupartial xsigma ,again1 since g g 1 we have1 by differentiation1 gmusigma dgnusigma -gnusigma dgmusigma or gmusigma fracpartial gnusigmapartial xlambda - gnusigma fracpartial gmusigmapartial xlambda ,by mixed multiplication with g and g respectively we obtain 7changing the mode of writing the indices74 7#ca7 dg -g g dg g_/x -g g dg and 7#cb7 dg -g g dg g_/x -g g g_/x4 ,the expression 7#ca7 allows a transformation which we shall often use2 according to 7#ba7 7#cc7 fracpartial galphabetapartial xsigma beginbmatrixalpha sigma beta endbmatrix beginbmatrixbeta sigma alphaendbmatrix ,if we substitute this in the second of the formula 7#ca71 we get1 remembering 7#bc71 7#cd7 fracpartial gmunupartial xsigma - 7 gmutau begin,bmatrixtau sigma nuend,bmatrix gnutau begin,bmatrixtau sigma muend,bmatrix 7 ,by substituting the right-hand side of 7#cd7 in 7#bi71 we get 7#bi;a7 frac#asqrt-g fracpartial sqrt-gpartial xsigma begin,bmatrixmu sigmamuend,bmatrix ,divergence of the contravariant four-vector4 ,let us multiply 7#bf7 with the contravariant fundamental tensor g 7inner multiplication71 then by a transformation of the first member1 the right-hand side takes the form 7,a7 fracpartialpartial xnu 7gmunu ,amu7 - ,amu fracpartial gmunupartial xnu - frac#ab gtaualpha 7fracpartial gmualphapartial xnu fracpartial g ualphapartial xmu - fracpartial gmunupartial xalpha7 gmunu ,atau ,according to 7#ca7 and 7#bi71 the last member can take the form 7,b7 frac#ab fracpartial gtaunupartial xnu ,atau frac#ab fracpartial gmutaupartial xmu ,atau frac#asqrt-g fracpartial sqrt-gpartial xalpha gmualpha ,atau ,both the first members of the expression 7,b71 and the second member of the expression 7,a7 cancel each other1 since the naming of the summation-indices is immaterial4 ,the last member of 7,b7 can then be united with first of 7,a74 ,if we put g ,a ,a1 where ,a as well as ,a are vectors which can be arbitrarily chosen1 we obtain finally ,phi frac#asqrt-g fracpartialpartial xnu 7sqrt-g ,anu7 ,this scalar is the ,divergence of the contravariant four-vector ,a4 ,rotation of the 7covariant7 four-vector4 ,the second member in 7#bf7 is symmetrical in the indices 1 and 4 ,hence ,a - ,a is an antisymmetrical tensor built up in a very simple manner4 ,we obtain ,a ,a 7#cf7 ,b -------------- - ------------ x x ,antisymmetrical ,extension of a ,six-vector4 ,if we apply the operation 7#bg7 on an antisymmetrical tensor of the second rank ,a and form all the equations arising from the cyclic interchange of the indices 1 1 1 and add all them1 we obtain a tensor of the third rank 7#cg7 ,b ,a ,a ,a ,a ,a ,a ------------ ------------- ------------ x x x from which it is easy to see that the tensor is antisymmetrical4 ,divergence of the ,six-vector4 ,if 7#bg7 is multiplied by g g 7mixed multiplication71 then a tensor is obtained4 ,the first member of the right hand side of 7#bg7 can be written in the form fracpartialpartial xsigma 7gmualpha gnubeta ,amunu7 - gmualpha fracpartial gnubetapartial xsigma ,amunu - gnubeta fracpartial gmualphapartial xsigma ,amunu ,if we replace g g ,a by ,a1 g g ,a by ,a and replace in the transformed first member g_/x and g_/x with the help of 7#cd71 then from the right-hand side of 7#bg7 there arises an expression with seven terms1 of which four cancel4 ,there remains 7#ch7 ,aalphabetasigma fracpartial ,aalphabetapartial xsigma begin,bmatrixsigma kappa alpha end,bmatrix ,akappabeta begin,bmatrixsigma kappa betaend,bmatrix ,aalphakappa ,this is the expression for the extension of a contravariant tensor of the second rank2 extensions can also be formed for corresponding contravariant tensors of higher and lower ranks4 ,we remark that in the same way1 we can also form the extension of a mixed tensor ,a 7#ci7 ,aalphamusigma fracpartial ,aalphamupartial xsigma - begin,bmatrixsigma mu tauend,bmatrix ,aalphatau begin,bmatrixsigma tau alphaend,bmatrix ,ataumu ,by the reduction of 7#ch7 with reference to the indices and 7inner multiplication with 71 we get a contravariant four-vector ,aalpha fracpartial ,aalphabetapartial xbeta begin,bmatrixbeta kappa betaend,bmatrix ,aalphakappa begin,bmatrixbeta kappa alphaend,bmatrix ,akappabeta ,on the account of the symmetry of begin,bmatrixbeta kappa alphaend,bmatrix with reference to the indices and 1 the third member of the right hand side vanishes when ,a is an antisymmetrical tensor1 which we assume here2 the second member can be transformed according to 7#bi;a72 we therefore get 7#dj7 ,aalpha frac#asqrt-g fracpartial7sqrt-g ,aalphabeta7partial xbeta ,this is the expression of the divergence of a contravariant six-vector4 ,divergence of the mixed tensor of the second rank4 ,let us form the reduction of 7#ci7 with reference to the indices and 1 we obtain remembering 7#bi;a7 7#da7 sqrt-g ,amu fracpartial7sqrt-g ,asigmamu7partial xsigma - begin,bmatrixsigma mu tauend,bmatrix sqrt-g ,asigmatau ,if we introduce into the last term the contravariant tensor ,a g ,a1 it takes the form - beginbmatrixsigma mu rhoendbmatrix sqrt-g ,arhosigma ,if further ,a or is symmetrical it is reduced to - frac#ab sqrt-g fracpartial grhosigmapartial xmu ,arhosigma ,if instead of ,a1 we introduce in a similar way the symmetrical co-variant tensor ,a g g ,a1 then owing to 7#ca7 the last member can take the form frac#ab sqrt-g fracpartial grhosigmapartial xmu ,arhosigma ,in the symmetrical case treated1 7#da7 can be replaced by either of the forms 7#da;a7 sqrt-g ,amu fracpartial 7sqrt-g ,asigmamu7partial xsigma - frac#ab fracpartial grhosigmapartial xmu sqrt-g ,arhosigma or 7#da;b7 sqrt-g ,amu fracpartial 7sqrt-g ,asigmamu7partial xsigma frac#ab fracpartial grhosigmapartial xmu sqrt-g ,arhosigma which we shall have to make use of afterwards4 #ab4 ,the ,riemann-,christoffel ,tensor4 ,we now seek only those tensors1 which can be obtained from the fundamental tensor g by differentiation alone4 ,it is found easily4 ,we put in 7#bg7 instead of any tensor ,a the fundamental tensor g and get from it a new tensor1 namely the extension of the fundamental tensor4 ,we can easily convince ourselves that this vanishes identically4 ,we prove it in the following way2 we substitute in 7#bg7 ,amunu fracpartial ,amupartial xnu - begin,bmatrixmu nu rhoend,bmatrix ,arho i4e41 the extension of a four-vector4 ,thus we get 7by slightly changing the indices7 the tensor of the third rank ,amusigmatau fracpartial#b ,amupartial xsigma partial xtau - begin,bmatrixmu sigma rhoend,bmatrix fracpartial ,arhopartial xtau - begin,bmatrixmu tau rhoend,bmatrix fracpartial ,arhopartial xsigma - begin,bmatrixsigma tau rhoend,bmatrix fracpartial ,amupartial xrho beginbmatrix - fracpartialpartial xtau begin,bmatrixmusigma rhoend,bmatrix begin,bmatrixmutau alphaend,bmatrix begin,bmatrixalphasigma rhoend,bmatrix begin,bmatrixsigmatau alphaend,bmatrix begin,bmatrixalphamu rhoend,bmatrix endbmatrix ,arho ,we use these expressions for the formation of the tensor ,a - ,a4 ,thereby the following terms in ,a cancel the corresponding terms in ,a2 the first member1 the fourth member1 as well as the member corresponding to the last term within the square bracket4 ,these are all symmetrical in 1 and 4 ,the same is true for the sum of the second and third members4 ,we thus get 7#dc7 ,amusigmatau - ,amutausigma ,brhomusigmatau ,arho ,brhomusigmatau - fracpartialpartial xtau begin,bmatrixmu sigma rhoend,bmatrix fracpartialpartial xsigma begin,bmatrixmu tau rhoend,bmatrix - begin,bmatrixmu sigma alphaend,bmatrix begin,bmatrixalpha tau rhoend,bmatrix begin,bmatrixmu tau alphaend,bmatrix begin,bmatrixalpha sigma rhoend,bmatrix ,the essential thing in this result is that on the right hand side of 7#db7 we have only ,a1 but not its differential co-efficients4 ,from the tensor-character of ,a - ,a1 and from the fact that ,a is an arbitrary four vector1 it follows1 on account of the result of #g1 that ,b is a tensor 7,riemann-,christoffel ,tensor74 ,the mathematical significance of this tensor is as follows2 when the continuum is so shaped1 that there is a co-ordinate system for which g's are constants1 ,b all vanish4 ,if we choose instead of the original co-ordinate system any new one1 so would the g's referred to this last system be no longer constants4 ,the tensor character of ,b shows us1 however1 that these components vanish collectively also in any other chosen system of reference4 ,the vanishing of the ,riemann ,tensor is thus a necessary condition that for some choice of the axis-system g's can be taken as constants4 ,in our problem it corresponds to the case when by a suitable choice of the co-ordinate system1 the special relativity theory holds throughout any finite region4 ,by the reduction of 7#dc7 with reference to indices to and 1 we get the covariant tensor of the second rank 7#dd7 ,bmunu ,rmunu ,smunu ,rmunu - fracpartialpartial xalpha begin,bmatrixmu nu alphaend,bmatrix begin,bmatrixmu alpha betaend,bmatrix begin,bmatrixnu beta alphaend,bmatrix ,smunu fracpartial log sqrt-gpartial xmu partial xnu - begin,bmatrixmu nu alphaend,bmatrix fracpartial log sqrt-gpartial xalpha ,remarks upon the choice of co-ordinates4--,it has already been remarked in #h1 with reference to the equation 7#ah;a71 that the co-ordinates can with advantage be so chosen that 7-g7 #a4 ,a glance at the equations got in the last two paragraphs shows that1 through such a choice1 the law of formation of the tensors suffers a significant simplification4 ,it is specially true for the tensor ,b1 which plays a fundamental rle in the theory4 ,by this simplification1 ,s vanishes of itself so that tensor ,b reduces to ,r4 ,i shall give in the following pages all relations in the simplified form1 with the above-named specialisation of the co-ordinates4 ,it is then very easy to go back to the general covariant equations1 if it appears desirable in any special case4 ,c4 ,,the ,,theory ,,of ,,the ,,gravitation-,,field #ac4 ,equation of motion of a material point in a gravitation-field4 ,expression for the field-components of gravitation4 ,a freely moving body not acted on by external forces moves1 according to the special relativity theory1 along a straight line and uniformly4 ,this also holds for the generalised relativity theory for any part of the four-dimensional region1 in which the co-ordinates ,k#j can be1 and are1 so chosen that g's have special constant values of the expression 7#d74 ,let us discuss this motion from the stand-point of any arbitrary co-ordinate-system ,k2 it moves with reference to ,k 7as explained in #b7 in a gravitational field4 ,the laws of motion with reference to ,k follow easily from the following consideration4 ,with reference to ,k1 the law of motion is a four-dimensional straight line and thus a geodesic4 ,as a geodetic-line is defined independently of the system of co-ordinates1 it would also be the law of motion for the motion of the material-point with reference to ,k4 ,if we put 7#de7 ,gammataumunu - begin,bmatrixmu nu tauend,bmatrix we get the motion of the point with reference to ,k1 given by 7#df7 fracd#b xtauds#b ,gammataumunu fracdxmuds fracdxnuds ,we now make the very simple assumption that this general covariant system of equations defines also the motion of the point in the gravitational field1 when there exists no reference-system ,k1 with reference to which the special relativity theory holds throughout a finite region4 ,the assumption seems to us to be all the more legitimate1 as 7#df7 contains only the first differentials of g1 among which there is no relation in the special case when ,k exists4 ,if 's vanish1 the point moves uniformly and in a straight line2 these magnitudes therefore determine the deviation from uniformity4 ,they are the components of the gravitational field4 #ad4 ,the ,field-equation of ,gravitation in the absence of matter4 ,in the following1 we differentiate gravitation-field from matter in the sense that everything besides the gravitation-field will be signified as matter2 therefore the term includes not only matter in the usual sense1 but also the electro-dynamic field4 ,our next problem is to seek the field-equations of gravitation in the absence of matter4 ,for this we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motion for material points4 ,a special case in which the field-equations sought-for are evidently satisfied is that of the special relativity theory in which g's have certain constant values4 ,this would be the case in a certain finite region with reference to a definite co-ordinate system ,k4 ,with reference to this system1 all the components ,b of the ,riemann's ,tensor equation #dc vanish4 ,these vanish then also in the region considered1 with reference to every other co-ordinate system4 ,the equations of the gravitation-field free from matter must thus be in every case satisfied when all ,b vanish4 ,but this condition is clearly one which goes too far4 ,for it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed away by any choice of axes1 i4e41 it cannot be transformed to a case of constant g's4 ,therefore it is clear that1 for a gravitational field free from matter1 it is desirable that the symmetrical tensors ,b deduced from the tensors ,b should vanish4 ,we thus get #aj equations for #aj quantities g which are fulfilled in the special case when ,b's all vanish4 ,remembering 7#dd7 we see that in absence of matter the field-equations come out as follows2 7when referred to the special co-ordinate-system chosen47 7#dg7 fracpartial ,gammaalphamunupartial xalpha ,gammaalphamubeta ,gammabetamualpha #j sqrt-g #a ,gammaalphamunu - begin,bmatrixmu nu alphaend,bmatrix ,it can also be shown that the choice of these equations is connected with a minimum of arbitrariness4 ,for besides ,b1 there is no tensor of the second rank1 which can be built out of g's and their derivatives no higher than the second1 and which is also linear in them4 ,it will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity1 together with equations 7#df71 give us the ,newtonian law of attraction as a first approximation1 and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by ,leverrier 7the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors74 ,my view is that these are convincing proofs of the physical correctness of my theory4 #ae4 ,hamiltonian ,function for the ,gravitation-field4 ,laws of ,impulse and ,energy4 ,in order to show that the field equations correspond to the laws of impulse and energy1 it is most convenient to write it in the following ,hamiltonian form3-- 7#dg;a7 ,hd #j ,h g 7-g7 #a ,here the variations vanish at the limits of the finite four-dimensional integration-space considered4 ,it is first necessary to show that the form 7#dg;a7 is equivalent to equations 7#dg74 ,for this purpose1 let us consider ,h as a function of g and g 7 g_/x7 ,we have at first ,h g #b;g - g #b 7g74 ,but delta7gmunu ,gammabetanualpha7 - frac#ab delta beginbmatrixgmunu gbetalambdaendbmatrix 7fracpartial gnulambdapartial xalpha fracpartial galphalambdapartial xnu - fracpartial galphanupartial xlambda7 ,the terms arising out of the two last terms within the round bracket are of different signs1 and change into one another by the interchange of the indices and 4 ,they cancel each other in the expression for ,h1 when they are multiplied by 1 which is symmetrical with respect to and 1 so that only the first member of the bracket remains for our consideration4 ,remembering 7#ca71 we thus have3-- ,h - g g ,therefore 7#dh7 ,h_/g - ,h_/g ,if we now carry out the variations in 7#dg;a71 we obtain the system of equations 7#dg;b7 _/x 7 ,h_/g 7 - ,h_/g #j1 which1 owing to the relations 7#dh71 coincide with 7#dg71 as was required to be proved4 ,if 7#dg;b7 is multiplied by g1 since g_/x g_/x and consequently g _/x 7,h_/g7 _/x 7g ,h_/g7 - ,h_/g g_/x we obtain the equation _/x 7g ,h_/g7 - ,h_/x #j or t_/x #j 7#di7 -#bt g ,h_/g - ,h4 ,owing to the relations 7#dh71 the equations 7#dg7 and 7#cd71 7#ej7 t g - g 4 ,it is to be noticed that t is not a tensor1 so that the equation 7#di7 holds only for systems for which -g #a4 ,this equation expresses the laws of conservation of impulse and energy in a gravitation-field4 ,in fact1 the integration of this equation over a three-dimensional volume ,v leads to the four equations 7#di;a7 d_/dx t#d d,v 7t#a t t 7d,s where 1 1 are the direction-cosines of the inward-drawn normal to the surface-element d,s in the ,euclidean ,sense4 ,we recognise in this the usual expression for the laws of conservation4 ,we denote the magnitudes t as the energy-components of the gravitation-field4 ,i will now put the equation 7#dg7 in a third form which will be very serviceable for a quick realisation of our object4 ,by multiplying the field-equations 7#dg7 with g1 these are obtained in the mixed forms4 ,if we remember that g _/x _/x 7g 7 - g_/x 1 which owing to 7#cd7 is equal to _/x 74g 7 - g - g 1 or slightly altering the notation1 equal to _/x 7g 7 - gmn m n - g 4 ,the third member of this expression cancels with the second member of the field-equations 7#dg74 ,in place of the second term of this expression1 we can1 on account of the relations 7#ej71 put 7t - t71 where t t ,therefore in the place of the equations 7#dg71 we obtain 7#ea7 _/x 7g 7 -7t - t7 7-g7 #a4 #af4 ,general formulation of the field-equation of ,gravitation4 ,the field-equations established in the preceding paragraph for spaces free from matter is to be compared with the equation #j of the ,newtonian theory4 ,we have now to find the equations which will correspond to ,poisson's ,equation #d 7 signifies the density of matter74 ,the special relativity theory has led to the conception that the inertial mass 7,trge ,masse7 is no other than energy4 ,it can also be fully expressed mathematically by a symmetrical tensor of the second rank1 the energy-tensor4 ,we have therefore to introduce in our generalised theory energy-tensor associated with matter1 which like the energy components t of the gravitation-field 7equations #di1 and #ej7 have a mixed character but which however can be connected with symmetrical covariant tensors4 ,the equation 7#ea7 teaches us how to introduce the energy-tensor 7corresponding to the density of ,poisson's equation7 in the field equations of gravitation4 ,if we consider a complete system 7for example the ,solar-system7 its total mass1 as also its total gravitating action1 will depend on the total energy of the system1 ponderable as well as gravitational4 ,this can be expressed1 by putting in 7#ea71 in place of energy-components t of gravitation-field alone the sum of the energy-components of matter and gravitation1 i4e41 t ,t4 ,we thus get instead of 7#ea71 the tensor-equation 7#eb7 fracpartialpartial xalpha 7gsigmaeta ,gammalphamubeta7 - kappa 7tsigmamu ,tsigmamu7 - frac#ab deltasigmamu 7t ,t7 sqrt-g #a where ,t ,t 7,laue's ,scalar74 ,these are the general field-equations of gravitation in the mixed form4 ,in place of 7#dg71 we get by working backwards the system 7#ec7 fracpartial ,gammalphamu upartial xalpha ,gammalphamubeta ,gammaetanualpha - kappa 7,tmunu - frac#ab gmunu ,t7 sqrt-g #a ,it must be admitted1 that this introduction of the energy-tensor of matter cannot be justified by means of the ,relativity-,postulate alone2 for we have in the foregoing analysis deduced it from the condition that the energy of the gravitation-field should exert gravitating action in the same way as every other kind of energy4 ,the strongest ground for the choice of the above equation however lies in this1 that they lead1 as their consequences1 to equations expressing the conservation of the components of total energy 7the impulses and the energy7 which exactly correspond to the equations 7#di7 and 7#di;a74 ,this shall be shown afterwards4 #ag4 ,the laws of conservation in the general case4 ,the equations 7#eb7 can be easily so transformed that the second member on the right-hand side vanishes4 ,we reduce 7#eb7 with reference to the indices and and subtract the equation so obtained after multiplication with from 7#eb74 ,we obtain1 7#eb;a7 _/x7g - g 7 -7t ,t7 we operate on it by _/x4 ,now1 _/xx 7g7 - _/xx g g7g_/x g_/x - g_/x74 ,the first and the third member of the round bracket lead to expressions which cancel one another1 as can be easily seen by interchanging the summation-indices 1 and 1 on the one hand1 and and 1 on the other4 ,the second term can be transformed according to 7#ca74 ,so that we get1 7#ed7 _/xx 7g7 g_/xxx ,the second member of the expression on the left-hand side of 7#eb;a7 leads first to - _/xx 7g7 or to #a_/d _/xx gg7 g_/x g_/x - g_/x74 ,the expression arising out of the last member within the round bracket vanishes according to 7#bi7 on account of the choice of axes4 ,the two others can be taken together and give us on account of 7#ca71 the expression - g_/xxx ,so that remembering 7#ed7 we have 7#ee7 _/xx 7g - g 7 #j4 identically4 ,from 7#ee7 and 7#eb;a7 it follows that 7#ef7 _/x 7t ,t7 #j ,from the field equations of gravitation1 it also follows that the conservation-laws of impulse and energy are satisfied4 ,we see it most simply following the same reasoning which lead to equations 7#di;a72 only instead of the energy-components of the gravitational-field1 we are to introduce the total energy-components of matter and gravitational field4 #ah4 ,the ,impulse-energy law for matter as a consequence of the field-equations4 ,if we multiply 7#ec7 with g_/x1 we get in a way similar to #ae1 remembering that g g_/x vanishes1 the equations t_/x - g_/x ,t #j or remembering 7#ef7 7#eg7 ,t_/x g_/x ,t #j ,a comparison with 7#da;b7 shows that these equations for the above choice of co-ordinates 77-g7 #a7 asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter4 ,physically the appearance of the second term on the left-hand side shows that for matter alone the law of conservation of impulse and energy cannot hold2 or can only hold when g's are constants2 i4e41 when the field of gravitation vanishes4 ,the second member is an expression for impulse and energy which the gravitation-field exerts per time and per volume upon matter4 ,this comes out clearer when instead of 7#eg7 we write it in the form of 7#dg74 7#eg;a7 ,t_/x - ,t4 ,the right-hand side expresses the interaction of the energy of the gravitational-field on matter4 ,the field-equations of gravitation contain thus at the same time #d conditions which are to be satisfied by all material phenomena4 ,we get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another4 ,d4 ,,the 8,,material0 ,,phenomena4 ,the ,mathematical auxiliaries developed under #8,b' at once enables us to generalise1 according to the generalised theory of relativity1 the physical laws of matter 7,hydrodynamics1 ,maxwell's ,electro-dynamics7 as they lie already formulated according to the special-relativity-theory4 ,the generalised ,relativity ,principle leads us to no further limitation of possibilities2 but it enables us to know exactly the influence of gravitation on all processes without the introduction of any new hypothesis4 ,it is owing to this1 that as regards the physical nature of matter 7in a narrow sense7 no definite necessary assumptions are to be introduced4 ,the question may lie open whether the theories of the electro-magnetic field and the gravitational-field together1 will form a sufficient basis for the theory of matter4 ,the general relativity postulate can teach us no new principle4 ,but by building up the theory it must be shown whether electro-magnetism and gravitation together can achieve what the former alone did not succeed in doing4 #ai4 ,euler's equations for frictionless adiabatic liquid4 ,let p and 1 be two scalars1 of which the first denotes the pressure and the last the density of the fluid2 between them there is a relation4 ,let the contravariant symmetrical tensor ,t -g p dx_/ds dx_/ds 7#eh7 be the contra-variant energy-tensor of the liquid4 ,to it also belongs the covariant tensor 7#eh;a7 ,t -g p g dx_/ds g dx_/ds as well as the mixed tensor 7#eh;b7 ,t - p g dx_/ds dx_/ds 4 ,if we put the right-hand side of 7#eh;b7 in 7#eg;a7 we get the general hydrodynamical equations of ,euler according to the generalised relativity theory4 ,this in principle completely solves the problem of motion2 for the four equations 7#eg;a7 together with the given equation between p and 1 and the equation g dx_/ds dx_/ds #a1 are sufficient1 with the given values of g1 for finding out the six unknowns p1 1 dx_/ds1 dx_/ds1 dx_/ds dx_/ds4 ,if g's are unknown we have also to take the equations 7#ec74 ,there are now #aa equations for finding out #aj functions g1 so that the number is more than sufficient4 ,now it is be noticed that the equation 7#eg;a7 is already contained in 7#ec71 so that the latter only represents 7#g7 independent equations4 ,this indefiniteness is due to the wide freedom in the choice of co-ordinates1 so that mathematically the problem is indefinite in the sense that three of the space-functions can be arbitrarily chosen4 #bj4 ,maxwell's ,electro-,magnetic field-equations4 ,let be the components of a covariant four-vector1 the electro-magnetic potential2 from it let us form according to 7#cf7 the components ,f of the covariant six-vector of the electro-magnetic field according to the system of equations 7#ei7 ,f _/x - _/x4 ,from 7#ei71 it follows that the system of equations 7#fj7 ,f_/x ,f_/x ,f_/x #j is satisfied of which the left-hand side1 according to 7#cg71 is an anti-symmetrical tensor of the third kind4 ,this system 7#fj7 contains essentially four equations1 which can be thus written3-- ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j 7#fj;a7 ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j4 ,this system of equations corresponds to the second system of equations of ,maxwell4 ,we see it at once if we put ,f ,hx ,f ,ex 7#fa7 ,f ,hy ,f ,ey ,f ,hz ,f ,ez ,instead of 7#fj;a7 we can therefore write according to the usual notation of three-dimensional vector-analysis3-- ,h_/t rot ,e #j 7#fj;b7 div ,h #j4 ,the first ,maxwellian system is obtained by a generalisation of the form given by ,minkowski4 ,we introduce the contra-variant six-vector ,f by the equation 7#fb7 ,f g g ,f1 and also a contra-variant four-vector ,j1 which is the electrical current-density in vacuum4 ,then remembering 7#dj7 we can establish the system of equations1 which remains invariant for any substitution with determinant #a 7according to our choice of co-ordinates74 7#fc7 ,f_/x ,j ,if we put ,f ,hx ,f -,ex 7#fd7 ,f ,hy ,f -,ey ,f ,hz ,f -,ez which quantities become equal to ,hx 444 ,ex in the case of the special relativity theory1 and besides ,j#a ix 444 ,j#d we get instead of 7#fc7 rot ,h - ,e_/t i 7#fc;a7 div ,e ,the equations 7#fj71 7#fb7 and 7#fc7 give thus a generalisation of ,maxwell's field-equations in vacuum1 which remains true in our chosen system of co-ordinates4 ,the energy-components of the electro-magnetic field4 ,let us form the inner-product 7#fe7 ,k ,f ,j4 ,according to 7#fa7 its components can be written down in the three-dimensional notation4 ,k ,ex i1 ,hx 7#fe;a7 -- -- -- ,k -- 7i1 ,e74 ,k is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time1 and per unit of volume1 by the electrical masses4 ,if the electrical masses be free1 that is1 under the influence of the electro-magnetic field only1 then the covariant four-vector ,k will vanish4 ,in order to get the energy components ,t of the electro-magnetic field1 we require only to give to the equation ,k #j1 the form of the equation 7#eg74 ,from 7#fc7 and 7#fe7 we get first1 ,k ,f ,f_/x _/x 7,f ,f7 - ,f ,f_/x4 ,on account of 7#fj7 the second member on the right-hand side admits of the transformation-- ,f ,f_/x - ,f ,f_/x - g g ,f ,f_/x4 ,owing to symmetry1 this expression can also be written in the form -#a_/d g g ,f ,f_/x g g ,f_/x ,f1 which can also be put in the form - #a_/d _/x 7g g ,f ,f7 #a_/d ,f ,f _/x 7g g74 ,the first of these terms can be written shortly as - #a_/d _/x 7,f ,f71 and the second after differentiation can be transformed in the form - ,f ,f g g_/x4 ,if we take all the three terms together1 we get the relation 7#ff7 ,k _/x - g g_/x where 7#ff;a7 -,f ,f #a_/d ,f ,f4 ,on account of 7#cj7 the equation 7#ff7 becomes equivalent to 7#eg7 and 7#eg;a7 when ,k vanishes4 ,thus 's are the energy-components of the electro-magnetic field4 ,with the help of 7#fa7 and 7#fd7 we can easily show that the energy-components of the electro-magnetic field1 in the case of the special relativity theory1 give rise to the well-known ,maxwell-,poynting expressions4 ,we have now deduced the most general laws which the gravitation-field and matter satisfy when we use a co-ordinate system for which 7-g7 #a4 ,thereby we achieve an important simplification in all our formulas and calculations1 without renouncing the conditions of general covariance1 as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations4 ,still the question is not without formal interest1 whether1 when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates1 the laws of conservation have the form of the equation 7#ef71 and the field-equations of gravitation hold in the form 7#eb7 or 7#eb;a72 such that on the left-hand side1 we have a divergence in the usual sense1 and on the right-hand side1 the sum of the energy-components of matter and gravitation4 ,i have found out that this is indeed the case4 ,but ,i am of opinion that the communication of my rather comprehensive work on this subject will not pay1 for nothing essentially new comes out of it4 ,e4 #ba4 ,newton's theory as a first approximation4 ,we have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general1 in which g's have constant values 7#d74 ,this signifies1 according to what has been said before1 a total neglect of the influence of gravitation4 ,we get one important approximation if we consider the case when g's differ from 7#d7 only by small magnitudes 7compared to #a7 where we can neglect small quantities of the second and higher orders 7first aspect of the approximation47 ,further it should be assumed that within the space-time region considered1 g's at infinite distances 7using the word infinite in a spatial sense7 can1 by a suitable choice of co-ordinates1 tend to the limiting values 7#d72 i4e41 we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions4 ,we can assume that this approximation should lead to ,newton's theory4 ,for it however1 it is necessary to treat the fundamental equations from another point of view4 ,let us consider the motion of a particle according to the equation 7#df74 ,in the case of the special relativity theory1 the components dx_/ds1 dx_/ds1 dx_/ds1 can take any values4 ,this signifies that any velocity v 77dx_/dx7 7dx_/dx7 7dx_/dx77 can appear which is less than the velocity of light in vacuum 7v #a74 ,if we finally limit ourselves to the consideration of the case when v is small compared to the velocity of light1 it signifies that the components dx_/ds1 dx_/ds1 dx_/ds1 can be treated as small quantities1 whereas dx_/ds is equal to #a1 up to the second-order magnitudes 7the second point of view for approximation74 ,now we see that1 according to the first view of approximation1 the magnitudes 's are all small quantities of at least the first order4 ,a glance at 7#df7 will also show1 that in this equation according to the second view of approximation1 we are only to take into account those terms for which #d4 ,by limiting ourselves only to terms of the lowest order we get instead of 7#df71 first1 the equations3-- dx_/dt 1 where ds dx dt1 or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order1 ,it must be admitted1 that this introduction of the energy-tensor of matter cannot be justified by means of the ,relativity-,postulate alone2 for we have in the foregoing analysis deduced it from the condition that the energy of the gravitation-field should exert gravitating action in the same way as every other kind of energy4 ,the strongest ground for the choice of the above equation however lies in this1 that they lead1 as their consequences1 to equations expressing the conservation of the components of total energy 7the impulses and the energy7 which exactly correspond to the equations 7#di7 and 7#di;a74 ,this shall be shown afterwards4 #ag4 ,the laws of conservation in the general case4 ,the equations 7#eb7 can be easily so transformed that the second member on the right-hand side vanishes4 ,we reduce 7#eb7 with reference to the indices and and subtract the equation so obtained after multiplication with from 7#eb74 ,we obtain1 7#eb;a7 _/x7g - g 7 -7t ,t7 we operate on it by _/x4 ,now1 _/xx 7g7 - _/xx g g7g_/x g_/x - g_/x74 ,the first and the third member of the round bracket lead to expressions which cancel one another1 as can be easily seen by interchanging the summation-indices 1 and 1 on the one hand1 and and 1 on the other4 ,the second term can be transformed according to 7#ca74 ,so that we get1 7#ed7 _/xx 7g7 g_/xxx ,the second member of the expression on the left-hand side of 7#eb;a7 leads first to - _/xx 7g7 or to #a_/d _/xx gg7 g_/x g_/x - g_/x74 ,the expression arising out of the last member within the round bracket vanishes according to 7#bi7 on account of the choice of axes4 ,the two others can be taken together and give us on account of 7#ca71 the expression - g_/xxx ,so that remembering 7#ed7 we have 7#ee7 _/xx 7g - g 7 #j4 identically4 ,from 7#ee7 and 7#eb;a7 it follows that 7#ef7 _/x 7t ,t7 #j ,from the field equations of gravitation1 it also follows that the conservation-laws of impulse and energy are satisfied4 ,we see it most simply following the same reasoning which lead to equations 7#di;a72 only instead of the energy-components of the gravitational-field1 we are to introduce the total energy-components of matter and gravitational field4 #ah4 ,the ,impulse-energy law for matter as a consequence of the field-equations4 ,if we multiply 7#ec7 with g_/x1 we get in a way similar to #ae1 remembering that g g_/x vanishes1 the equations t_/x - g_/x ,t #j or remembering 7#ef7 7#eg7 ,t_/x g_/x ,t #j ,a comparison with 7#da;b7 shows that these equations for the above choice of co-ordinates 77-g7 #a7 asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter4 ,physically the appearance of the second term on the left-hand side shows that for matter alone the law of conservation of impulse and energy cannot hold2 or can only hold when g's are constants2 i4e41 when the field of gravitation vanishes4 ,the second member is an expression for impulse and energy which the gravitation-field exerts per time and per volume upon matter4 ,this comes out clearer when instead of 7#eg7 we write it in the form of 7#dg74 7#eg;a7 ,t_/x - ,t4 ,the right-hand side expresses the interaction of the energy of the gravitational-field on matter4 ,the field-equations of gravitation contain thus at the same time #d conditions which are to be satisfied by all material phenomena4 ,we get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another4 ,d4 ,,the 8,,material0 ,,phenomena4 ,the ,mathematical auxiliaries developed under #8,b' at once enables us to generalise1 according to the generalised theory of relativity1 the physical laws of matter 7,hydrodynamics1 ,maxwell's ,electro-dynamics7 as they lie already formulated according to the special-relativity-theory4 ,the generalised ,relativity ,principle leads us to no further limitation of possibilities2 but it enables us to know exactly the influence of gravitation on all processes without the introduction of any new hypothesis4 ,it is owing to this1 that as regards the physical nature of matter 7in a narrow sense7 no definite necessary assumptions are to be introduced4 ,the question may lie open whether the theories of the electro-magnetic field and the gravitational-field together1 will form a sufficient basis for the theory of matter4 ,the general relativity postulate can teach us no new principle4 ,but by building up the theory it must be shown whether electro-magnetism and gravitation together can achieve what the former alone did not succeed in doing4 #ai4 ,euler's equations for frictionless adiabatic liquid4 ,let p and 1 be two scalars1 of which the first denotes the pressure and the last the density of the fluid2 between them there is a relation4 ,let the contravariant symmetrical tensor ,t -g p dx_/ds dx_/ds 7#eh7 be the contra-variant energy-tensor of the liquid4 ,to it also belongs the covariant tensor 7#eh;a7 ,t -g p g dx_/ds g dx_/ds as well as the mixed tensor 7#eh;b7 ,t - p g dx_/ds dx_/ds 4 ,if we put the right-hand side of 7#eh;b7 in 7#eg;a7 we get the general hydrodynamical equations of ,euler according to the generalised relativity theory4 ,this in principle completely solves the problem of motion2 for the four equations 7#eg;a7 together with the given equation between p and 1 and the equation g dx_/ds dx_/ds #a1 are sufficient1 with the given values of g1 for finding out the six unknowns p1 1 dx_/ds1 dx_/ds1 dx_/ds dx_/ds4 ,if g's are unknown we have also to take the equations 7#ec74 ,there are now #aa equations for finding out #aj functions g1 so that the number is more than sufficient4 ,now it is be noticed that the equation 7#eg;a7 is already contained in 7#ec71 so that the latter only represents 7#g7 independent equations4 ,this indefiniteness is due to the wide freedom in the choice of co-ordinates1 so that mathematically the problem is indefinite in the sense that three of the space-functions can be arbitrarily chosen4 #bj4 ,maxwell's ,electro-,magnetic field-equations4 ,let be the components of a covariant four-vector1 the electro-magnetic potential2 from it let us form according to 7#cf7 the components ,f of the covariant six-vector of the electro-magnetic field according to the system of equations 7#ei7 ,f _/x - _/x4 ,from 7#ei71 it follows that the system of equations 7#fj7 ,f_/x ,f_/x ,f_/x #j is satisfied of which the left-hand side1 according to 7#cg71 is an anti-symmetrical tensor of the third kind4 ,this system 7#fj7 contains essentially four equations1 which can be thus written3-- ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j 7#fj;a7 ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j4 ,this system of equations corresponds to the second system of equations of ,maxwell4 ,we see it at once if we put ,f ,hx ,f ,ex 7#fa7 ,f ,hy ,f ,ey ,f ,hz ,f ,ez ,instead of 7#fj;a7 we can therefore write according to the usual notation of three-dimensional vector-analysis3-- ,h_/t rot ,e #j 7#fj;b7 div ,h #j4 ,the first ,maxwellian system is obtained by a generalisation of the form given by ,minkowski4 ,we introduce the contra-variant six-vector ,f by the equation 7#fb7 ,f g g ,f1 and also a contra-variant four-vector ,j1 which is the electrical current-density in vacuum4 ,then remembering 7#dj7 we can establish the system of equations1 which remains invariant for any substitution with determinant #a 7according to our choice of co-ordinates74 7#fc7 ,f_/x ,j ,if we put ,f ,hx ,f -,ex 7#fd7 ,f ,hy ,f -,ey ,f ,hz ,f -,ez which quantities become equal to ,hx 444 ,ex in the case of the special relativity theory1 and besides ,j#a ix 444 ,j#d we get instead of 7#fc7 rot ,h - ,e_/t i 7#fc;a7 div ,e ,the equations 7#fj71 7#fb7 and 7#fc7 give thus a generalisation of ,maxwell's field-equations in vacuum1 which remains true in our chosen system of co-ordinates4 ,the energy-components of the electro-magnetic field4 ,let us form the inner-product 7#fe7 ,k ,f ,j4 ,according to 7#fa7 its components can be written down in the three-dimensional notation4 ,k ,ex i1 ,hx 7#fe;a7 -- -- -- ,k -- 7i1 ,e74 ,k is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time1 and per unit of volume1 by the electrical masses4 ,if the electrical masses be free1 that is1 under the influence of the electro-magnetic field only1 then the covariant four-vector ,k will vanish4 ,in order to get the energy components ,t of the electro-magnetic field1 we require only to give to the equation ,k #j1 the form of the equation 7#eg74 ,from 7#fc7 and 7#fe7 we get first1 ,k ,f ,f_/x _/x 7,f ,f7 - ,f ,f_/x4 ,on account of 7#fj7 the second member on the right-hand side admits of the transformation-- ,f ,f_/x - ,f ,f_/x - g g ,f ,f_/x4 ,owing to symmetry1 this expression can also be written in the form -#a_/d g g ,f ,f_/x g g ,f_/x ,f1 which can also be put in the form - #a_/d _/x 7g g ,f ,f7 #a_/d ,f ,f _/x 7g g74 ,the first of these terms can be written shortly as - #a_/d _/x 7,f ,f71 and the second after differentiation can be transformed in the form - ,f ,f g g_/x4 ,if we take all the three terms together1 we get the relation 7#ff7 ,k _/x - g g_/x where 7#ff;a7 -,f ,f #a_/d ,f ,f4 ,on account of 7#cj7 the equation 7#ff7 becomes equivalent to 7#eg7 and 7#eg;a7 when ,k vanishes4 ,thus 's are the energy-components of the electro-magnetic field4 ,with the help of 7#fa7 and 7#fd7 we can easily show that the energy-components of the electro-magnetic field1 in the case of the special relativity theory1 give rise to the well-known ,maxwell-,poynting expressions4 ,we have now deduced the most general laws which the gravitation-field and matter satisfy when we use a co-ordinate system for which 7-g7 #a4 ,thereby we achieve an important simplification in all our formulas and calculations1 without renouncing the conditions of general covariance1 as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations4 ,still the question is not without formal interest1 whether1 when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates1 the laws of conservation have the form of the equation 7#ef71 and the field-equations of gravitation hold in the form 7#eb7 or 7#eb;a72 such that on the left-hand side1 we have a divergence in the usual sense1 and on the right-hand side1 the sum of the energy-components of matter and gravitation4 ,i have found out that this is indeed the case4 ,but ,i am of opinion that the communication of my rather comprehensive work on this subject will not pay1 for nothing essentially new comes out of it4 ,e4 #ba4 ,newton's theory as a first approximation4 ,we have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general1 in which g's have constant values 7#d74 ,this signifies1 according to what has been said before1 a total neglect of the influence of gravitation4 ,we get one important approximation if we consider the case when g's differ from 7#d7 only by small magnitudes 7compared to #a7 where we can neglect small quantities of the second and higher orders 7first aspect of the approximation47 ,further it should be assumed that within the space-time region considered1 g's at infinite distances 7using the word infinite in a spatial sense7 can1 by a suitable choice of co-ordinates1 tend to the limiting values 7#d72 i4e41 we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions4 ,we can assume that this approximation should lead to ,newton's theory4 ,for it however1 it is necessary to treat the fundamental equations from another point of view4 ,let us consider the motion of a particle according to the equation 7#df74 ,in the case of the special relativity theory1 the components dx_/ds1 dx_/ds1 dx_/ds1 can take any values4 ,this signifies that any velocity v 77dx_/dx7 7dx_/dx7 7dx_/dx77 can appear which is less than the velocity of light in vacuum 7v #a74 ,if we finally limit ourselves to the consideration of the case when v is small compared to the velocity of light1 it signifies that the components dx_/ds1 dx_/ds1 dx_/ds1 can be treated as small quantities1 whereas dx_/ds is equal to #a1 up to the second-order magnitudes 7the second point of view for approximation74 ,now we see that1 according to the first view of approximation1 the magnitudes 's are all small quantities of at least the first order4 ,a glance at 7#df7 will also show1 that in this equation according to the second view of approximation1 we are only to take into account those terms for which #d4 ,by limiting ourselves only to terms of the lowest order we get instead of 7#df71 first1 the equations3-- dx_/dt 1 where ds dx dt1 or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order1 fracd#b xtaudt#b beginbmatrix#ddtauendbmatrix 7tau #a1 #b1 #c7 fracd#b x#d;dt#b - beginbmatrix#ddd;endbmatrix ,if we further assume that the gravitation-field is quasi-static1 i4e41 it is limited only to the case when the matter producing the gravitation-field is moving slowly 7relative to the velocity of light7 we can neglect the differentiations of the positional co-ordinates on the right-hand side with respect to time1 so that we get 7#fg7 dx_/dt - g_/x 71 #a1 #b1 #c7 ,this is the equation of motion of a material point according to ,newton's theory1 where g_/ plays the part of gravitational potential4 ,the remarkable thing in the result is that in the first-approximation of motion of the material point1 only the component g of the fundamental tensor appears4 ,let us now turn to the field-equation 7#ec74 ,in this case1 we have to remember that the energy-tensor of matter is exclusively defined in a narrow sense by the density of matter1 i4e41 by the second member on the right-hand side of #eh 7#eh;a1 or #eh;b74 ,if we make the necessary approximations1 then all component vanish except 4 ,on the left-hand side of 7#ec7 the second term is an infinitesimal of the second order1 so that the first leads to the following terms in the approximation1 which are rather interesting for us3 fracpartialpartial x#a beginbmatrixmunu#a;endbmatrix fracpartialpartial x#b beginbmatrixmunu#b;endbmatrix fracpartialpartial x#c beginbmatrixmunu#c;endbmatrix fracpartialpartial x#d beginbmatrixmunu#d;endbmatrix ,by neglecting all differentiations with regard to time1 this leads1 when #d1 to the expression - frac#ab 7 fracpartial#b g#ddpartial x#ba fracpartial#b g#ddpartial x#bb fracpartial#b g#ddpartial x#bc 7 - frac#ab ,v#b g#dd ,the last of the equations 7#ec7 thus leads to 7#fh7 g 4 ,the equations 7#fg7 and 7#fh7 together1 are equivalent to ,newton's law of gravitation4 ,for the gravitation-potential we get from 7#fg7 and 7#fh7 the exp4 7#fh;a47 -_/7#h7 d_/r whereas the ,newtonian theory for the chosen unit of time gives -,k_/c d_/r1 where ,k denotes usually the gravitation-constant4 #f4g x #aj2 equating them we get 7#fi7 #h,k_/c #a4hg x #aj4 #bb4 ,behaviour of measuring rods and clocks in a statical gravitation-field4 ,curvature of light-rays4 ,perihelion-motion of the paths of the ,planets4 ,in order to obtain ,newton's theory as a first approximation we had to calculate only g1 out of the #aj components g of the gravitation-potential1 for that is the only component which comes in the first approximate equations of motion of a material point in a gravitational field4 ,we see however1 that the other components of g should also differ from the values given in 7#d7 as required by the condition g -#a4 ,for a heavy particle at the origin of co-ordinates and generating the gravitational field1 we get as a first approximation the symmetrical solution of the equation3-- g - - 7x x7_/r 7 and #a1 #b1 #c7 7#gj7 g#d g#d #j 7 #a1 #b1 #c7 g #a - _/r4 is #a or #j1 according as or not and r is the quantity 7x x x74 ,on account of 7#fh;a7 we have 7#gj;a7 ,m_/#d where ,m denotes the mass generating the field4 ,it is easy to verify that this solution satisfies approximately the field-equation outside the mass ,m4 ,let us now investigate the influences which the field of mass ,m will have upon the metrical properties of the field4 ,between the lengths and times measured locally on the one hand1 and the differences in co-ordinates dx on the other1 we have the relation ds g dx dx4 ,for a unit measuring rod1 for example1 placed parallel to the x axis1 we have to put ds -#a1 dx dx dx #j then -#a gdx4 ,if the unit measuring rod lies on the x axis1 the first of the equations 7#gj7 gives g -7#a _/r74 ,from both these relations it follows as a first approximation that 7#ga7 dx #a - _/#br4 ,the unit measuring rod appears1 when referred to the co-ordinate-system1 shortened by the calculated magnitude through the presence of the gravitational field1 when we place it radially in the field4 ,similarly we can get its co-ordinate-length in a tangential position1 if we put for example ds -#a1 dx dx dx #j1 x r1 x x #j we then get 7#ga;a7 -#a g dx -dx4 ,the gravitational field has no influence upon the length of the rod1 when we put it tangentially in the field4 ,thus ,euclidean geometry does not hold in the gravitational field even in the first approximation1 if we conceive that one and the same rod independent of its position and its orientation can serve as the measure of the same extension4 ,but a glance at 7#gj;a7 and 7#fi7 shows that the expected difference is much too small to be noticeable in the measurement of earth's surface4 ,we would further investigate the rate of going of a unit-clock which is placed in a statical gravitational field4 ,here we have for a period of the clock ds #a1 dx dx dx #j2 then we have #a gdx dx #a_/7;g7 #a_/7a 7g - #a77 #a - 7g - #a7_/b or dx #a k_/#h d_/r4 ,therefore the clock goes slowly what it is placed in the neighbourhood of ponderable masses4 ,it follows from this that the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the red end of the spectrum4 ,let us further investigate the path of light-rays in a statical gravitational field4 ,according to the special relativity theory1 the velocity of light is given by the equation -dx - dx - dx dx #j2 thus also according to the generalised relativity theory it is given by the equation 7#gc7 ds g dx dx #j4 ,if the direction1 i4e41 the ratio dx 3 dx 3 dx is given1 the equation 7#gc7 gives the magnitudes dx_/dx1 dx_/dx1 dx_/dx1 and with it the velocity1 77dx_/dx7 7dx_/dx7 7dx_/dx77 1 in the sense of the ,euclidean ,geometry4 ,we can easily see that1 with reference to the co-ordinate system1 the rays of light must appear curved in case g's are not constants4 ,if n be the direction perpendicular to the direction of propagation1 we have1 from ,huygen's principle1 that light-rays 7taken in the plane 71 n7 must suffer a curvature _/n4 ,let us find out the curvature which a light-ray suffers when it goes by a mass ,m at a distance from it4 ,if we use the co-ordinate system according to the above scheme1 then the total bending ,b of light-rays 7reckoned positive when it is concave to the origin7 is given as a sufficient approximation by ,b - _/x dx where 7#gc7 and 7#gj7 gives 7-g_/g7 #a - _/#br 7#a x_/r74 ,the calculation gives ,b #b_/ ,,km_/#b4 ,a ray of light just grazing the sun would suffer a bending of #ag1 whereas one coming by ,jupiter would have a deviation of about #jb4 ,if we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of a material particle of a relatively small 7infinitesimal7 mass we get a deviation of the following kind from the ,kepler-,newtonian ,laws of ,planetary motion4 ,the ,ellipse of ,planetary motion suffers a slow rotation in the direction of motion1 of amount 7#ge7 s #bd;a_/c7#a - e7 per revolution4 ,in this ,formula #8a' signifies the semi-major axis1 c1 the velocity of light1 measured in the usual way1 e1 the eccentricity1 1 the time of revolution in seconds4 ,the calculation gives for the planet ,mercury1 a rotation of path of amount #dc per century1 corresponding sufficiently to what has been found by astronomers 7,leverrier74 ,they found a residual perihelion motion of this planet of the given magnitude which can not be explained by the perturbation of the other planets4 ,,notes ,note #a4 ,the fundamental electro-magnetic equations of ,maxwell for stationary media are3-- curl ,h #a_/;c 7,d_/t 7 7#a7 curl ,e -#a_/;c ,b_/t 7#b7 div ,d ,b ,h div ,b #j ,d k,e ,according to ,hertz and ,heaviside1 these require modification in the case of moving bodies4 ,now it is known that due to motion alone there is a change in a vector ,r given by 7,r_/t7 due to motion u4 div ,r curl ,ru where u is the vector velocity of the moving body and ,ru the vector product of ,r and u4 ,hence equations 7#a7 and 7#b7 become c curl ,h ,d_/t u div ,d curl ,vect4 ,du 7#aa7 and -c curl ,e ,b_/t u div ,b curl ,vect4 ,bu 7#ba7 which gives finally1 for #j and div ,b #j1 ,d_/t u div ,d c curl 7,h - #a_/;c ,vect4 ,du7 7#ab7 ,b_/t -c curl 7,e - #a_/;c ,vect4 u,b7 7#bb7 ,let us consider a beam travelling along the x-axis1 with apparent velocity v 7i4e41 velocity with respect to the fixed ether7 in medium moving with velocity ux u in the same direction4 ,then if the electric and magnetic vectors are proportional to ei,a7x - vt71 we have _/x i,a1 _/t -i,av1 _/y _/z #j1 uy uz #j ,then ,dy_/t -c,hz_/x - u,dy_/z 444 7#aba7 and ,bz_/t -c,ey_/x - u,bz_/x 7#bba7 ,since ,d ,,ke and ,b ,h1 we have i,av7,ey7 -ci,a7,hz u,,key7 7#abb7 i,av7,hz7 -ci,a7,ey u,hz7 7#bbb7 or v7,k - u7,ey c,hz 7#abc7 7v - u7,hz c,ey 7#bbc7 ,multiplying 7#abc7 by 7#bbc7 ,k7v - u7 c ,hence 7v - u7 c_/k v v v u1 making ,fresnelian convection co-efficient simply unity4 ,equations 7#aba7 and 7#bba7 may be obtained more simply from physical considerations4 ,according to ,heaviside and ,hertz1 the real seat of both electric and magnetic polarisation is the moving medium itself4 ,now at a point which is fixed with respect to the ether1 the rate of change of electric polarisation is ,d_/t4 ,consider a slab of matter moving with velocity ux along the x-axis1 then even in a stationary field of electrostatic polarisation1 that is1 for a field in which ,d_/t #j1 there will be some change in the polarisation of the body due to its motion1 given by ux7,d_/x74 ,hence we must add this term to a purely temporal rate of change ,d_/t4 ,doing this we immediately arrive at equations 7#aba7 and 7#bba7 for the special case considered there4 ,thus the ,hertz-,heaviside form of field equations gives unity as the value for the ,fresnelian convection co-efficient4 ,it has been shown in the historical introduction how this is entirely at variance with the observed optical facts4 ,as a matter of fact1 ,larmor has shown 7,aether and ,matter7 that #a - #a_/ is not only sufficient but is also necessary1 in order to explain experiments of the ,arago prism type4 ,a short summary of the electromagnetic experiments bearing on this question1 has already been given in the introduction4 ,according to ,hertz and ,heaviside the total polarisation is situated in the medium itself and is completely carried away by it4 ,thus the electromagnetic effect outside a moving medium should be proportional to ,k1 the specific inductive capacity4 ,rowland showed in #ahgf that when a charged condenser is rapidly rotated 7the dielectric remaining stationary71 the magnetic effect outside is proportional to ,k1 the ,sp4 ,ind4 ,cap4 ,rntgen 7,annalen der ,physik #ahhh1 #ahij7 found that if the dielectric is rotated while the condenser remains stationary1 the effect is proportional to ,k - #a4 ,eichenwald 7,annalen der ,physik #aijc1 #aijd7 rotated together both condenser and dielectric and found that the magnetic effect was proportional to the potential difference and to the angular velocity1 but was completely independent of ,k4 ,this is of course quite consistent with ,rowland and ,rntgen4 ,blondlot 7,comptes ,rendus1 #aija7 passed a current of air in a steady magnetic field ,hy1 7,h ,hz #j74 ,if this current of air moves with velocity ux along the x-axis1 an electromotive force would be set up along the z-axis1 due to the relative motion of matter and magnetic tubes of induction4 ,a pair of plates at z a1 will be charged up with density ,dz ,,ke ,k4 us ,hy_/c4 ,but ,blondlot failed to detect any such effect4 ,h4 ,a4 ,wilson 7,phil4 ,trans4 ,royal ,soc4 #aijd7 repeated the experiment with a cylindrical condenser made of ebony1 rotating in a magnetic field parallel to its own axis4 ,he observed a change proportional to ,k -- #a and not to ,k4 ,thus the above set of electro-magnetic experiments contradict the ,hertz-,heaviside equations1 and these must be abandoned4 ,p4 ,c4 ,m4 ,note #b4 ,lorentz ,transformation4 ,lorentz4 ,versuch einer theorie der elektrischen und optischen ,erscheinungen im bewegten ,krpern4 7,leiden--#ahie74 ,lorentz4 ,theory of ,electrons 7,english edition71 pages #aig-bjj1 #bcj1 also notes #gc1 #hf1 pages #cah1 #cbh4 ,lorentz wanted to explain the ,michelson-,morley null-effect4 ,in order to do so1 it was obviously necessary to explain the ,fitzgerald contraction4 ,lorentz worked on the hypothesis that an electron itself undergoes contraction when moving4 ,he introduced new variables for the moving system defined by the following set of equations4 x 7x - ut71 y y1 z z1 t 7t - 7u_/c7x7 and for velocities1 used vx vx u1 vy vy1 vz vz and _/4 ,with the help of the above set of equations1 which is known as the ,lorentz transformation1 he succeeded in showing how the ,fitzgerald contraction results as a consequence of 8fortuitous compensation of opposing effects40 ,it should be observed that the ,lorentz transformation is not identical with the ,einstein transformation4 ,the ,einsteinian addition of velocities is quite different as also the expression for the 8relative0 density of electricity4 ,it is true that the ,maxwell-,lorentz field equations remain practically unchanged by the ,lorentz transformation1 but they are changed to some slight extent4 ,one marked advantage of the ,einstein transformation consists in the fact that the field equations of a moving system preserve exactly the same form as those of a stationary system4 ,it should also be noted that the ,fresnelian convection coefficient comes out in the theory of relativity as a direct consequence of ,einstein's addition of velocities and is quite independent of any electrical theory of matter4 ,p4 ,c4 ,m4 ,note #c4 ,see ,lorentz1 ,theory of ,electrons 7,english edition71 #aha1 page #bac4 ,h4 ,poincare1 ,sur la dynamique #8electron1 ,rendiconti del circolo matematico di ,palermo #ba 7#aijf74 ,p4 ,c4 ,m4 ,note #d4 ,relativity ,theorem and ,relativity-,principle4 ,lorentz showed that the ,maxwell-,lorentz system of electromagnetic field-equations remained practically unchanged by the ,lorentz transformation4 ,thus the electromagnetic laws of ,maxwell and ,lorentz can be definitely proved 8to be independent of the manner in which they are referred to two coordinate systems which have a uniform translatory motion relative to each other40 7,see 8,electrodynamics of ,moving ,bodies10 page #e47 ,thus so far as the electromagnetic laws are concerned1 the principle of relativity can be proved to be true4 ,but it is not known whether this principle will remain true in the case of other physical laws4 ,we can always proceed on the assumption that it does remain true4 ,thus it is always possible to construct physical laws in such a way that they retain their form when referred to moving coordinates4 ,the ultimate ground for formulating physical laws in this way is merely a subjective conviction that the principle of relativity is universally true4 ,there is no a priori logical necessity that it should be so4 ,hence the ,principle of ,relativity 7so far as it is applied to phenomena other than electromagnetic7 must be regarded as a postulate1 which we have assumed to be true1 but for which we cannot adduce any definite proof1 until after the generalisation is made and its consequences tested in the light of actual experience4 ,p4 ,c4 ,m4 ,note #e4 ,see 8,electrodynamics of ,moving ,bodies10 p4 #e-h4 ,note #f4 ,field ,equations in ,minkowski's ,form4 ,equations 7i7 and 7ii7 become when expanded into ,cartesians3-- mz_/y - my_/z - ex_/ x mx_/z - mz_/x - ey_/ y 444 7#aa7 my_/x - mx_/y - ez_/ z and ex_/x ey_/y ez_/z 7#ba7 ,substituting x1 x1 x1 x and x1 y1 z1 and i2 and 1 1 1 for x1 y1 z1 i1 where i 7-#a74 ,we get1 mz_/x - my_/x - i7ex_/x7 x - mz_/x mx_/x - i7ey_/x7 y 444 7#ab7 my_/x - mx_/x - i7ez_/x7 z and multiplying 7#ba7 by i we get iex_/x iey_/x iez_/x i 444 444 7#bb7 ,now substitute mx f -f and iex f -f my f -f iey f -f mz f -f iez f -f and we get finally3-- f_/x f_/x f_/x f_/x f_/x f_/x 444 7#c7 f_/x f_/x f_/x f_/x f_/x f_/x ,note #i4 ,on the ,constancy of the ,velocity of ,light4 ,page #ab--refer also to page #f1 of ,einstein's paper4 ,one of the two fundamental ,postulates of the ,principle of ,relativity is that the velocity of light should remain constant whether the source is moving or stationary4 ,it follows that even if a radiant source ,s move with a velocity u1 it should always remain the centre of spherical waves expanding outwards with velocity c4 ,at first sight1 it may not appear clear why the velocity should remain constant4 ,indeed according to the theory of ,ritz1 the velocity should become c u1 when the source of light moves towards the observer with the velocity u4 ,prof4 de ,sitter has given an astronomical argument for deciding between these two divergent views4 ,let us suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit4 ,let the observer be in the plane of the orbit1 at a great distance 4 ,illustration4 ,the light emitted by the star when at the position ,a will be received by the observer after a time1 _/7c u7 while the light emitted by the star when at the position ,b will be received after a time _/7c - u74 ,let ,t be the real half-period of the star4 ,then the observed half-period from ,b to ,a is approximately ,t - #bu_/c and from ,a to ,b is ,t #bu_/c4 ,now if #bu_/c be comparable to ,t1 then it is impossible that the observations should satisfy ,kepler's ,law4 ,in most of the spectroscopic binary stars1 #bu_/c are not only of the same order as ,t1 but are mostly much larger4 ,for example1 if u #ajj km_/sec1 ,t #h days1 _/c #cc years 7corresponding to an annual parallax of #a71 then ,t - #bu_/c #j4 ,the existence of the ,spectroscopic binaries1 and the fact that they follow ,kepler's ,law is therefore a proof that c is not affected by the motion of the source4 ,in a later memoir1 replying to the criticisms of ,freundlich and ,gnthick that an apparent eccentricity occurs in the motion proportional to ku1 u being the maximum value of u1 the velocity of light emitted being u c ku1 k #j ,lorentz-,einstein k #a ,ritz4 ,prof4 de ,sitter admits the validity of the criticisms4 ,but he remarks that an upper value of k may be calculated from the observations of the double star -,aurigae4 ,for this star1 the parallax #jad1 e #jje1 u #aaj km_/sec1 ,t #cif1 #fe light-years1 k is #jjb4 ,for an experimental proof1 see a paper by ,c4 ,majorana4 ,phil4 ,mag41 ,vol4 #ce1 p4 #afc4 ,m4 ,n4 ,s4 ,note #aj4 ,rest-density of ,electricity4 ,if is the volume density in a moving system then 7#a - u7 is the corresponding quantity in the corresponding volume in the fixed system1 that is1 in the system at rest1 and hence it is termed the rest-density of electricity4 ,p4 ,c4 ,m4 ,note #aa 7page #ag7 ,space-time vectors of the first and the second kind4 ,as we had already occasion to mention1 ,sommerfeld has1 in two papers on four dimensional geometry 7vide1 ,annalen der ,physik1 ,bd4 #cb1 p4 #gdi2 and ,bd4 #cc1 p4 #fdi71 translated the ideas of ,minkowski into the language of four dimensional geometry4 ,instead of ,minkowski's space-time vector of the first kind1 he uses the more expressive term #8four-vector1' thereby making it quite clear that it represents a directed quantity like a straight line1 a force or a momentum1 and has got #d components1 three in the direction of space-axes1 and one in the direction of the time-axis4 ,the representation of the plane 7defined by two straight lines7 is much more difficult4 ,in three dimensions1 the plane can be represented by the vector perpendicular to itself4 ,but that artifice is not available in four dimensions4 ,for the perpendicular to a plane1 we now have not a single line1 but an infinite number of lines constituting a plane4 ,this difficulty has been overcome by ,minkowski in a very elegant manner which will become clear later on4 ,meanwhile we offer the following extract from the above mentioned work of ,sommerfeld4 7,pp4 #gee1 ,bd4 #cb1 ,ann4 d4 ,physik47 8,in order to have a better knowledge about the nature of the six-vector 7which is the same thing as ,minkowski's space-time vector of the #bnd kind7 let us take the special case of a piece of plane1 having unit area 7contents71 and the form of a parallelogram1 bounded by the four-vectors u1 v1 passing through the origin4 ,then the projection of this piece of plane on the xy plane is given by the projections ux1 uy1 vx1 vy of the four vectors in the combination x y uxvy - uyvx4 ,let us form in a similar manner all the six components of this plane 4 ,then six components are not all independent but are connected by the following relation y z x l z x y l x y z l #j ,further the contents of the piece of a plane is to be defined as the square root of the sum of the squares of these six quantities4 ,in fact1 y z z x x y x l y l z l4 ,let us now on the other hand take the case of the unit plane 99 normal to 2 we can call this plane the ,complement of 4 ,then we have the following relations between the components of the two plane3-- y z99 x l1 z x99 y l1 x y99 z l z l99 y x 444 ,the proof of these assertions is as follows4 ,let u991 v99 be the four vectors defining 994 ,then we have the following relations3-- ux99 ux uy99 uy uz99 uz ul99 ul #j ux99 vx uy99 vy uz99 vz ul99 vl #j vx99 ux vy99 uy vz99 uz vl99 ul #j vx99 vx vy99 vy vz99 vz vl99 vl #j ,if we multiply these equations by vl1 ul1 vs1 and subtract the second from the first1 the fourth from the third we obtain ux99 x l uy99 y l uz99 z l #j vx99 z l vy99 y l vz99 z l #j multiplying these equations by vx99 4 ux991 or by vy99 4 uy991 we obtain x z99 x l y z99 y l #j and x y99 x l z x99 z l #j from which we have y z99 3 x y99 3 z x99 x l 3 z l 3 y l ,in a corresponding way we have y z 3 x y 3 z x x l99 3 z l99 3 y l994 i4e4 i k99 7i k7 when the subscript 7ik7 denotes the component of in the plane contained by the lines other than 7ik74 ,therefore the theorem is proved4 ,we have 7 997 y z y z99 444 #b 7y z z l 4447 #j ,the general six-vector f is composed from the vectors 1 99 in the following way3-- f 99 991 and 99 denoting the contents of the pieces of mutually perpendicular planes composing f4 ,the 8conjugate ,vector0 f99 7or it may be called the complement of f7 is obtained by interchanging and 994 ,we have f99 99 99 ,we can verify that fy z99 fx l etc4 and f 991 7ff997 #b994 f and 7ff997 may be said to be invariants of the six vectors1 for their values are independent of the choice of the system of co-ordinates4 ,m4 ,n4 ,s4 ,note #ab4 ,light-velocity as a maximum4 ,page #bc1 and ,electro-dynamics of ,moving ,bodies1 p4 #ag4 ,putting v c - x1 and w c - 1 we get ,v 7#b;c - 7x 77_/7#a 7c - x77c - 7_/c7 7#b;c - 7x 77_/7c c - 7x 7c x_/c7 c 7#b;c - 7x 77_/7#b;c - 7x 7 x_/c7 ,thus v lt2 c1 so long as x #j4 ,thus the velocity of light is the absolute maximum velocity4 ,we shall now see the consequences of admitting a velocity ,w c4 ,let ,a and ,b be separated by distance l1 and let velocity of a 8signal0 in the system ,s be ,w c4 ,let the 7observing7 system ,s have velocity v with respect to the system ,s4 ,then velocity of signal with respect to system ,s is given by ,w 7,w - v7_/7#a - ,wv_/c7 ,thus 8time0 from ,a to ,b as measured in ,s1 is given by l_/,w l7#a - ,wv_/c7_/7,w - v7 t 7#a7 ,now if v is less than c1 then ,w being greater than c 7by hypothesis7 ,w is greater than v1 i4e41 ,w v4 ,let ,w c and v c - 4 ,then ,wv 7c 77c - 7 c 7 7c - 4 ,now we can always choose v in such a way that ,wv is greater than c1 since ,wv is c if 7 7c - is #j1 that is1 if _/c2 which can always be satisfied by a suitable choice of 4 ,thus for ,w c we can always choose in such a way as to make ,wv c1 i4e41 - ,wv_/c negative4 ,but ,w - v is always positive4 ,hence with ,w c1 we can always make t1 the time from ,a to ,b in equation 7#a7 8negative40 ,that is1 the signal starting from ,a will reach ,b 7as observed in system ,s7 in less than no time4 ,thus the effect will be perceived before the cause commences to act1 i4e41 the future will precede the past4 ,which is absurd4 ,hence we conclude that ,w c is an impossibility1 there can be no velocity greater than that of light4 ,it is conceptually possible to imagine velocities greater than that of light1 but such velocities cannot occur in reality4 ,velocities greater than c1 will not produce any effect4 ,causal effect of any physical type can never travel with a velocity greater than that of light4 ,p4 ,c4 ,m4 ,notes #ac and #ad4 ,we have denoted the four-vector by the matrix 4 ,it is then at once seen that denotes the reciprocal matrix ,it is now evident that while ,a1 ,a 1 s ,the vector-product of the four-vector and s may be represented by the combination s s - ,it is now easy to verify the formula f ,af,a4 ,supposing for the sake of simplicity that f represents the vector-product of two four-vectors 1 s1 we have f s s - s#aa ,a s,a - ,as,a ,as - s,a ,af,a4 ,now remembering that generally f 99994 ,where 1 99 are scalar quantities1 1 99 are two mutually perpendicular unit planes1 there is no difficulty in seeming that f#a ,af,a4 ,note #ae4 ,the vector product 7wf74 7,p4 #cf74 ,this represents the vector product of a four-vector and a six-vector4 ,now as combinations of this type are of frequent occurrence in this paper1 it will be better to form an idea of their geometrical meaning4 ,the following is taken from the above mentioned paper of ,sommerfeld4 8,we can also form a vectorial combination of a four-vector and a six-vector1 giving us a vector of the third type4 ,if the six-vector be of a special type1 i4e41 a piece of plane1 then this vector of the third type denotes the parallelopiped formed of this four-vector and the complement of this piece of plane4 ,in the general case1 the product will be the geometric sum of two parallelopipeds1 but it can always be represented by a four-vector of the #ast type4 ,for two pieces of #c-space volumes can always be added together by the vectorial addition of their components4 ,so by the addition of two #c-space volumes1 we do not obtain a vector of a more general type1 but one which can always be represented by a four-vector 7loc4 cit4 p4 #gei74 ,the state of affairs here is the same as in the ordinary vector calculus1 where by the vector-multiplication of a vector of the first1 and a vector of the second type 7i4e41 a polar vector71 we obtain a vector of the first type 7axial vector74 ,the formal scheme of this multiplication is taken from the three-dimensional case4 ,let ,a 7,ax1 ,ay1 ,az7 denote a vector of the first type1 ,b 7,by z1 ,bz x1 ,bx y7 denote a vector of the second type4 ,from this last1 let us form three special vectors of the first kind1 namely-- ,bx 7,bx x1 ,bx y1 ,bx z7 ,by 7,by x1 ,by y1 ,by z7 7,bi k - ,bk i1 ,bi i #j74 ,bz 7,bz x1 ,bz y1 ,bz z7 ,since ,bj j is zero1 ,bj is perpendicular to the j-axis4 ,the j-component of the vector-product of ,a and ,b is equivalent to the scalar product of ,a and ,bj1 i4e41 7,a ,bj17 ,ax ,bj x ,ay ,bj y ,az ,bj z4 ,we see easily that this coincides with the usual rule for the vector-product2 e4 g41 for j x4 7,,abx7 ,ay ,bx y - ,az ,bz x4 ,correspondingly let us define in the four-dimensional case the product 7,pf7 of any four-vector ,p and the six-vector f4 ,the j-component 7j x1 y1 z1 or l7 is given by 7,pfj7 ,pxfj x ,pyfj y ,pwfj z ,pzfj l ,each one of these components is obtained as the scalar product of ,p1 and the vector fj which is perpendicular to j-axis1 and is obtained from f by the rule fj 7fj x1 fj y1 fj z1 fj l7 fj j #j4 ,we can also find out here the geometrical significance of vectors of the third type1 when f 1 i4e41 f represents only one plane4 ,we replace by the parallelogram defined by the two four-vectors ,u1 ,v1 and let us pass over to the conjugate plane 991 which is formed by the perpendicular four-vectors ,u991 ,v994 ,the components of 7,p7 are then equal to the #d three-rowed under-determinants ,dx ,dy ,dz ,dl of the matrix ,px ,py ,pz ,pl ,ux99 ,uy99 ,uz99 ,ul99 ,vx99 ,vy99 ,vz99 ,vl99 ,leaving aside the first column we obtain ,dx ,py7,uz99 ,vl99 - ,ul99 ,vz997 ,pz7,ul99 ,vy99 - ,uy99 ,vl997 ,pl7,uy99 ,vz99 - ,uz99 ,vy997 ,py z y99 ,pz99 l y ,pl 99y z4 ,py x y ,pz x z ,pl x l1 which coincides with 7,px7 according to our definition4 ,examples of this type of vectors will be found on page #cf1 w,f1 the electrical-rest-force1 and #bwf991 the magnetic-rest-force4 ,the rest-ray iw99 also belong to the same type 7page #ci74 ,it is easy to show that -i w w w w ,when 71 1 7 #j1 w i1 reduces to the three-dimensional vector 1 1 ,since in this case1 w ,f en 7the electric force7 -iw f mx 7the magnetic force7 we have 77 ex ey ez mx my mz ,m4 ,n4 ,s4 ,note #af4 ,the electric-rest force4 7,page #cg47 ,the four-vector w,f which is called by ,minkowski the electric-rest-force 7elektrische ,ruh-,kraft7 is very closely connected to ,lorentz's ,ponderomotive force1 or the force acting on a moving charge4 ,if is the density of charge1 we have1 when #a1 #a1 i4e41 for free space w ,f w ,f w ,f w ,f _/77#a - ,v_/c77 dx #a_/;c 7v h - v h7 ,now since 7#a - ,v_/c7 ,we have dx #a_/;c 7v h - v h7 ,n4 ,b4--,we have put the components of e equivalent to 7dx1 dy1 dz71 and the components of m equivalent to hx hy hz71 in accordance with the notation used in ,lorentz's ,theory of ,electrons4 ,we have therefore 71 1 7 7d #a_/;c vh71 i4e41 71 1 7 represents the force acting on the electron4 ,compare ,lorentz1 ,theory of ,electrons1 page #ad4 ,the fourth component when multiplied by represents i-times the rate at which work is done by the moving electron1 for i vxdx vydy vzdz vx vy vz 4 -7-#a7 times the power possessed by the electron therefore represents the fourth component1 or the time component of the force-four-vector4 ,this component was first introduced by ,poincare in #aijf4 ,the four-vector i,f99 has a similar relation to the force acting on a moving magnetic pole4 ,m4 ,n4 ,s4 ,note #ag4 ,operator 8,lor0 7 #ab1 p4 #da74 ,the operation _/x _/x _/x _/x which plays in four-dimensional mechanics a rle similar to that of the operator 7i_/x1 j_/y1 k_/z 7 in three-dimensional geometry has been called by ,minkowski #8,lorentz-,operation' or shortly #8lor' in honour of ,h4 ,a4 ,lorentz1 the discoverer of the theorem of relativity4 ,later writers have sometimes used the symbol to denote this operation4 ,in the above-mentioned paper 7,annalen der ,physik1 p4 #fdi1 ,bd4 #ch7 ,sommerfeld has introduced the terms1 ,div 7divergence71 ,rot 7,rotation71 ,grad 7gradient7 as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor4 ,the physical significance of these operations will become clear when along with ,minkowski's method of treatment we also study the geometrical method of ,sommerfeld4 ,minkowski begins here with the case of lor ,s1 where ,s is a six-vector 7space-time vector of the #bnd kind74 ,this being a complicated case1 we take the simpler case of lor s1 where s is a four-vector s1 s1 s1 s and s s s s s ,the following geometrical method is taken from ,sommerfeld4 ,scalar ,divergence--,let denote a small four-dimensional volume of any shape in the neighbourhood of the space-time point ,q1 d,s denote the three-dimensional bounding surface of 1 n be the outer normal to d,s4 ,let ,s be any four-vector1 ,pn its normal component4 ,then ,div ,s ,lim ,pnd,s_/4 #j ,now if for we choose the four-dimensional parallelopiped with sides 7dx1 dx1 dx1 dx71 we have then ,div ,s s_/x s_/x s_/x s_/x lor ,s4 ,if f denotes a space-time vector of the second kind1 lor f is equivalent to a space-time vector of the first kind4 ,the geometrical significance can be thus brought out4 ,we have seen that the operator #8lor' behaves in every respect like a four-vector4 ,the vector-product of a four-vector and a six-vector is again a four-vector4 ,therefore it is easy to see that lor ,s will be a four-vector4 ,let us find the component of this four-vector in any direction s4 ,let ,s denote the three-space which passes through the point ,q 7x1 x1 x1 x7 and is perpendicular to s1 ,s a very small part of it in the region of ,q1 d is an element of its two-dimensional surface4 ,let the perpendicular to this surface lying in the space be denoted by n1 and let fs n denote the component of f in the plane of 7sn7 which is evidently conjugate to the plane d4 ,then the s-component of the vector divergence of f because the operator lor multiplies f vectorially4 ,div fs ,lim 7 fs nd7_/,s4 s #j ,where the integration in d is to be extended over the whole surface4 ,if now s is selected as the x-direction1 s is then a three-dimensional parallelopiped with the sides dy1 dz1 dl1 then we have ,div fx frac#a;dy dz dl dz4 dl4 fracpartial fxypartial y dy dl dy fracpartial fxypartial z dz dy dz fracpartial fxypartial l dl fracpartial fxypartial y fracpartial fxypartial z fracpartial fxypartial l and generally ,div fj fj x_/x fj y_/y fj z_/z fj l_/l 7where fj1 j #j74 ,hence the four-components of the four-vector lor ,s or ,div4 f is a four-vector with the components given on page #db4 ,according to the formulae of space geometry1 ,dx denotes a parallelopiped laid in the 7y-z-l7 space1 formed out of the vectors 7,py ,pz ,pl71 7,uy99 ,uz99 ,ul997 7,vy99 ,vz99 ,vl9974 ,dx is therefore the projection on the y-z-l space of the parallelopiped formed out of these three four-vectors 7,p1 ,u991 ,v9971 and could as well be denoted by ,dyzl4 ,we see directly that the four-vector of the kind represented by 7,dx1 ,dy1 ,dz1 ,dl7 is perpendicular to the parallelopiped formed by 7,p ,u99 ,v9974 ,generally we have 7,pf7 ,,pd ,p99,d994 ,the vector of the third type represented by 7,pf7 is given by the geometrical sum of the two four-vectors of the first type ,,pd and ,p99,d994 ,m4 ,n4 ,s4 ,transcriber's ,notes3 ,the book's idiosyncratic spelling1 emphasis1 punctuation1 and symbology especially in mathematical formulas1 have been retained4 ,text that was in italics is enclosed by underscores 7italics74 ,text that was in bold face is enclosed by equals signs 7bold74 ,footnotes have been moved to follow the chapters in which they are referenced4