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| committer | nfenwick <nfenwick@pglaf.org> | 2025-01-22 09:46:48 -0800 |
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} - </style> - </head> - <body> - -<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of The Principle of Relativity, by Albert Einstein</div> - -<div style='display:block; margin:1em 0'> -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online -at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: The Principle of Relativity</p> - -<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: Albert Einstein and Hermann Minkowski</div> - -<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Translator: Meghnad N. Saha and Satyendra N. Bose</div> - -<div style='display:block; margin:1em 0'>Release Date: December 14, 2021 [eBook #66944]</div> - -<div style='display:block; margin:1em 0'>Language: English</div> - -<div style='display:block; margin:1em 0'>Character set encoding: UTF-8</div> - -<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Richard Tonsing, David King, and the Online Distributed Proofreading Team at http://www.pgdp.net. (This file was produced from images generously made available by The Internet Archive.)</div> - -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK THE PRINCIPLE OF RELATIVITY ***</div> - -<div class='figcenter id001'> -<img src='images/cover.jpg' alt='' class='ig001' /> -</div> -<div class='pbb'> - <hr class='pb c000' /> -</div> -<div> - <h1 class='c001'>The Principle of Relativity</h1> -</div> - -<div class='nf-center-c1'> -<div class='nf-center c002'> - <div><span class='xxlarge'><b>THE PRINCIPLE OF RELATIVITY</b></span></div> - <div class='c000'><span class='large'><b>ORIGINAL PAPERS BY</b></span></div> - <div class='c000'><span class='xxlarge'><b>A. EINSTEIN AND H. MINKOWSKI</b></span></div> - <div class='c000'><span class='large'><b>TRANSLATED INTO ENGLISH BY</b></span></div> - <div class='c000'><span class='xlarge'><b>M. N. SAHA AND S. N. BOSE</b></span></div> - <div class='c000'><b>LECTURERS ON PHYSICS AND APPLIED MATHEMATICS</b></div> - <div><b>University College of Science, Calcutta University</b></div> - <div class='c000'><span class='large'><b>WITH A HISTORICAL INTRODUCTION BY</b></span></div> - <div class='c000'><span class='xlarge'><b>P. C. MAHALANOBIS</b></span></div> - <div><b>PROFESSOR OF PHYSICS, PRESIDENCY COLLEGE, CALCU.</b></div> - <div class='c000'>PUBLISHED BY THE</div> - <div>UNIVERSITY OF CALCUTTA</div> - <div>1920</div> - <div class='c000'><i>Sole Agents</i></div> - <div>R. CAMBRAY & CO.</div> - </div> -</div> - -<div class='nf-center-c1'> -<div class='nf-center c002'> - <div>PRINTED BY ATULCHANDRA BHATTACHARYYA,</div> - <div class='c000'>AT THE CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA</div> - </div> -</div> - -<div class='chapter'> - <h2 class='c003'>TABLE OF CONTENTS</h2> -</div> -<p class='c004'><a href='#art1'>1. Historical Introduction i-xxiii</a></p> - -<p class='c005'>[By Mr. P. C. Mahalanobis.]</p> - -<p class='c005'><a href='#art2'>2. On the Electrodynamics of Moving Bodies 1-34</a></p> - -<p class='c005'>[Einstein’s first paper on the restricted -Theory of Relativity, originally published -in the Annalen der Physik in -1905. Translated from the original -German by Dr. Meghnad Saha.]</p> - -<p class='c005'><a href='#art3'>3. Albrecht Einstein 35-39</a></p> - -<p class='c005'>[A short biographical note by Dr. -Meghnad Saha.]</p> - -<p class='c005'><a href='#art4'>4. Principle of Relativity 1-52</a></p> - -<p class='c005'>[H. Minkowski’s original paper on the -restricted Principle of Relativity first -published in 1909. Translated from -the original German by Dr. Meghnad -Saha.]</p> - -<p class='c005'><a href='#art5'>5. Appendix to the above by H. Minkowski 53-88</a></p> - -<p class='c005'>[Translated by Dr. Meghnad Saha.]</p> - -<p class='c005'><a href='#art6'>6. The Generalised Principle of Relativity 89-163</a></p> - -<p class='c005'>[A. Einstein’s second paper on the Generalised -Principle first published in 1916. -Translated from the original German -by Mr. Satyendranath Bose.]</p> - -<p class='c005'><a href='#notes'>7. Notes 165-185</a></p> -<div> - -<div class='nf-center-c1'> - <div class='nf-center'> - <div>Transcriber’s Note:</div> - </div> -</div> - -<p class='c005'>This ebook includes complex mathematical -formulas. Some are simple in-line expressions like k = 1 - 1/μ<sup>2</sup>. -They may include special notations such as [=a] for an 'a' with a bar across the top, -[.a] for an 'a' with a dot over it, [..a] for an 'a' with two dots over it. -Others are more complex “ASCII Art” like this:</p> -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>l l 2lc 2l</div> - <div class='line'>t₁ = ------- + ---------- = ----------- = --- β²</div> - <div class='line in5'>c - u c + u c² - u² c</div> - </div> - </div> -</div> - -</div> -<div class='chapter'> - <h2 id='art1' class='c003'>HISTORICAL INTRODUCTION</h2> -</div> -<p class='c004'>Lord Kelvin writing in 1893, in his preface to the -English edition of Hertz’s Researches on Electric Waves, -says “many workers and many thinkers have helped to -build up the nineteenth century school of <i>plenum</i>, one -ether for light, heat, electricity, magnetism; and the -German and English volumes containing Hertz’s electrical -papers, given to the world in the last decade of the -century, will be a permanent monument of the splendid -consummation now realised.”</p> - -<p class='c005'>Ten years later, in 1905, we find Einstein declaring -that “the ether will be proved to be superfluous.” At -first sight the revolution in scientific thought brought -about in the course of a single decade appears to be almost -too violent. A more careful even though a rapid review -of the subject will, however, show how the Theory of -Relativity gradually became a historical necessity.</p> - -<p class='c005'>Towards the beginning of the nineteenth century, -the luminiferous ether came into prominence as a result of -the brilliant successes of the wave theory in the hands -of Young and Fresnel. In its stationary aspect the -elastic solid ether was the outcome of the search for a -medium in which the light waves may “undulate.” This -stationary ether, as shown by Young, also afforded a -satisfactory explanation of astronomical aberration. But -its very success gave rise to a host of new questions all -bearing on the central problem of relative motion of ether -and matter.</p> - -<p class='c005'><i>Arago’s prism experiment.</i>—The refractive index of a -glass prism depends on the incident velocity of light -outside the prism and its velocity inside the prism after -refraction. On Fresnel’s fixed ether hypothesis, the -incident light waves are situated in the stationary ether -outside the prism and move with velocity <i>c</i> with respect -to the ether. If the prism moves with a velocity <i>u</i> -with respect to this fixed ether, then the incident velocity -of light with respect to the prism should be <i>c</i> + <i>u</i>. Thus -the refractive index of the glass prism should depend on <i>u</i> -the absolute velocity of the prism, <i>i.e.</i>, its velocity with -respect to the fixed ether. Arago performed the experiment -in 1819, but failed to detect the expected change.</p> - -<p class='c005'><i>Airy-Boscovitch water-telescope experiment.</i>—Boscovitch -had still earlier in 1766, raised the very important -question of the dependence of aberration on the refractive -index of the medium filling the telescope. Aberration -depends on the difference in the velocity of light outside -the telescope and its velocity inside the telescope. If the -latter velocity changes owing to a change in the medium -filling the telescope, aberration itself should change, that -is, aberration should depend on the nature of the medium.</p> - -<p class='c005'>Airy, in 1871 filled up a telescope with water—but -failed to detect any change in the aberration. Thus we -get both in the case of Arago prism experiment and -Airy-Boscovitch water-telescope experiment, the very -startling result that optical effects in a moving medium -seem to be quite independent of the velocity of the -medium with respect to Fresnel’s stationary ether.</p> - -<p class='c005'><i>Fresnel’s convection coefficient k = 1 - 1/μ<sup>2</sup>.</i>—Possibly -some form of compensation is taking place. Working on -this hypothesis, Fresnel offered his famous ether convection -theory. According to Fresnel, the presence of matter -implies a definite condensation of ether within the -region occupied by matter. This “condensed” or -excess portion of ether is supposed to be carried away -with its own piece of moving matter. It should be -observed that only the “excess” portion is carried away, -while the rest remains as stagnant as ever. A complete -convection of the “excess” ether ρ′ with the full velocity -<i>u</i> is optically equivalent to a partial convection of the -total ether ρ, with only a fraction of the velocity <i>k</i>. <i>u</i>. -Fresnel showed that if this convection coefficient <i>k</i> is -1 - 1/μ<sup>2</sup> (μ being the refractive index of the prism), then -the velocity of light after refraction within the moving -prism would be altered to just such extent as would make -the refractive index of the moving prism quite independent -of its “absolute” velocity <i>u</i>. The non-dependence -of aberration on the “absolute” velocity <i>u</i>, is also very -easily explained with the help of this Fresnelian convection-coefficient -<i>k</i>.</p> - -<p class='c005'><i>Stokes’ viscous ether.</i>—It should be remembered, however, -that Fresnel’s stationary ether is absolutely fixed and is not -at all disturbed by the motion of matter through it. In this -respect Fresnelian ether cannot be said to behave in any -respectable physical fashion, and this led Stokes, in -1845-46, to construct a more material type of medium. -Stokes assumed that viscous motion ensues near the surface -of separation of ether and moving matter, while at -sufficiently distant regions the ether remains wholly -undisturbed. He showed how such a viscous ether would -explain aberration if all motion in it were differentially -irrotational. But in order to explain the null Arago -effect, Stokes was compelled to assume the convection -hypothesis of Fresnel with an identical numerical value -for <i>k</i>, namely 1 - 1/μ<sup>2</sup>. Thus the prestige of the Fresnelian -convection-coefficient was enhanced, if anything, by the -theoretical investigations of Stokes.</p> - -<p class='c005'><i>Fizeau’s experiment.</i>—Soon after, in 1851, it received -direct experimental confirmation in a brilliant piece of -work by Fizeau.</p> - -<p class='c005'>If a divided beam of light is re-united after passing -through two adjacent cylinders filled with water, ordinary -interference fringes will be produced. If the water in one -of the cylinders is now made to flow, the “condensed” -ether within the flowing water would be convected and -would produce a shift in the interference fringes. The -shift actually observed agreed very well with a value of -k = 1 - 1/μ<sup>2</sup>. The Fresnelian convection-coefficient now -became firmly established as a consequence of a direct -positive effect. On the other hand, the negative evidences -in favour of the convection-coefficient had also multiplied. -Mascart, Hoek, Maxwell and others sought for definite -changes in different optical effects induced by the motion -of the earth relative to the stationary ether. But all such -attempts failed to reveal the slightest trace of any optical -disturbance due to the “absolute” velocity of the earth, -thus proving conclusively that all the different optical -effects shared in the general compensation arising out of -the Fresnelian convection of the excess ether. It must be -carefully noted that the Fresnelian convection-coefficient -implicitly assumes the existence of a fixed ether (Fresnel) or -at least a wholly stagnant medium at sufficiently distant -regions (Stokes), with reference to which alone a convection -velocity can have any significance. Thus the convection-coefficient -implying some type of a stationary or viscous, -yet nevertheless “absolute” ether, succeeded in explaining -satisfactorily all known optical facts down to 1880.</p> - -<p class='c005'><i>Michelson-Morley Experiment.</i>—In 1881, Michelson -and Morley performed their classical experiments which -undermined the whole structure of the old ether theory -and thus served to introduce the new theory of relativity. -The fundamental idea underlying this experiment is quite -simple. In all old experiments the velocity of light -situated in free ether was compared with the velocity -of waves actually situated in a piece of moving matter -and presumably carried away by it. The compensatory -effect of the Fresnelian convection of ether afforded a -satisfactory explanation of all negative results.</p> - -<p class='c005'>In the Michelson-Morley experiment the arrangement is -quite different. If there is a definite gap in a rigid body, -light waves situated in free ether will take a definite time -in crossing the gap. If the rigid platform carrying the -gap is set in motion with respect to the ether in the direction -of light propagation, light waves (which are even now -situated in free ether) should presumably take a longer -time to cross the gap.</p> - -<p class='c005'>We cannot do better than quote Eddington’s description -of this famous experiment. “The principle of the -experiment may be illustrated by considering a swimmer in -a river. It is easily realized that it takes longer to swim -to a point 50 yards up-stream and back than to a point 50 -yards across-stream and back. If the earth is moving -through the ether there is a river of ether flowing through -the laboratory, and a wave of light may be compared to a -swimmer travelling with constant velocity relative to the -current. If, then, we divide a beam of light into two parts, -and send one-half swimming up the stream for a certain -distance and then (by a mirror) back to the starting -point, and send the other half an equal distance across -stream and back, the across-stream beam should arrive -back first.</p> - -<div class='figcenter id002'> -<img src='images/fig01.png' alt='Experiment.' class='ig001' /> -</div> - -<p class='c005'>Let the ether be flowing relative to -the apparatus with velocity <i>u</i> in the -direction O<i>x</i>, and let OA, OB, be -the two arms of the apparatus of equal -length <i>l</i>, OA being placed up-stream. Let <i>c</i> be the -velocity of light. The time for the double journey along -OA and back is</p> - -<div class='figcenter id001'> -<img src='images/form-001.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>where</p> - -<div class='figcenter id002'> -<img src='images/form-002.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>a factor greater than unity.</p> - -<p class='c005'>For the transverse journey the light must have a component -velocity <i>n</i> up-stream (relative to the ether) in order to -avoid being carried below OB: and since its total velocity -is <i>c</i>, its component across-stream must be √(<i>c²</i> - <i>u²</i>), the -time for the double journey OB is accordingly</p> - -<div class='figcenter id003'> -<img src='images/form-003.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>so that <i>t₁</i> > <i>t₂</i>.</p> - -<p class='c005'>But when the experiment was tried, it was found that -both parts of the beam took the same time, as tested by -the interference bands produced.”</p> - -<p class='c005'>After a most careful series of observations, Michelson -and Morley failed to detect the slightest trace of any -effect due to earth’s motion through ether.</p> - -<p class='c005'>The Michelson-Morley experiment seems to show that -there is no relative motion of ether and matter. Fresnel’s -stagnant ether requires a relative velocity of—<i>u</i>. Thus -Michelson and Morley themselves thought at first that their -experiment confirmed Stokes’ viscous ether, in which no -relative motion can ensue on account of the absence of -slipping of ether at the surface of separation. But even -on Stokes’ theory this viscous flow of ether would fall -off at a very rapid rate as we recede from the surface -of separation. Michelson and Morley repeated their experiment -at different heights from the surface of the earth, but -invariably obtained the same negative results, thus failing -to confirm Stokes’ theory of viscous flow.</p> - -<p class='c005'><i>Lodge’s experiment.</i>—Further, in 1893, Lodge performed -his rotating sphere experiment which showed -complete absence of any viscous flow of ether due to -moving masses of matter. A divided beam of light, after -repeated reflections within a very narrow gap between two -massive hemispheres, was allowed to re-unite and thus -produce interference bands. When the two hemispheres -are set rotating, it is conceivable that the ether in the gap -would be disturbed due to viscous flow, and any such flow -would be immediately detected by a disturbance of the -interference bands. But actual observation failed to -detect the slightest disturbance of the ether in the gap, -due to the motion of the hemispheres. Lodge’s experiment -thus seems to show a complete absence of any viscous -flow of ether.</p> - -<p class='c005'>Apart from these experimental discrepancies, grave -theoretical objections were urged against a viscous ether. -Stokes himself had shown that his ether must be incompressible -and all motion in it differentially irrotational, -at the same time there should be absolutely no slipping at -the surface of separation. Now all these conditions cannot -be simultaneously satisfied for any conceivable material -medium without certain very special and arbitrary assumptions. -Thus Stokes’ ether failed to satisfy the very motive -which had led Stokes to formulate it, namely, the desirability -of constructing a “physical” medium. Planck offered -modified forms of Stokes’ theory which seemed capable of -being reconciled with the Michelson-Morley experiment, -but required very special assumptions. The very complexity -and the very arbitrariness of these assumptions prevented -Planck’s ether from attaining any degree of practical -importance in the further development of the subject.</p> - -<p class='c005'>The sole criterion of the value of any scientific theory -must ultimately be its capacity for offering a simple, -unified, coherent and fruitful description of observed facts. -In proportion as a theory becomes complex it loses in -usefulness—a theory which is obliged to requisition a -whole array of arbitrary assumptions in order to explain -special facts is practically worse than useless, as it serves -to disjoin, rather than to unite, the several groups of facts. -The optical experiments of the last quarter of the nineteenth -century showed the impossibility of constructing a -simple ether theory, which would be amenable to analytic -treatment and would at the same time stimulate further -progress. It should be observed that it could scarcely be -shown that no logically consistent ether theory was -possible; indeed in 1910, H. A. Wilson offered a consistent -ether theory which was at least quite neutral with -respect to all available optical data. But Wilson’s ether -is almost wholly negative—its only virtue being that it -does not directly contradict observed facts. Neither any -direct confirmation nor a direct refutation is possible and -it does not throw any light on the various optical phenomena. -A theory like this being practically useless stands -self-condemned.</p> - -<p class='c005'>We must now consider the problem of relative motion of -ether and matter from the point of view of electrical theory. -From 1860 the identity of light as an electromagnetic -vector became gradually established as a result of the -brilliant “displacement current” hypothesis of Clerk -Maxwell and his further analytical investigations. The -elastic solid ether became gradually transformed into the -electromagnetic one. Maxwell succeeded in giving a fairly -satisfactory account of all ordinary optical phenomena -and little room was left for any serious doubts as regards -the general validity of Maxwell’s theory. Hertz’s researches -on electric waves, first carried out in 1886, -succeeded in furnishing a strong experimental confirmation -of Maxwell’s theory. Electric waves behaved generally -like light waves of very large wave length.</p> - -<p class='c005'>The orthodox Maxwellian view located the dielectric -polarisation in the electromagnetic ether which was merely -a transformation of Fresnel’s stagnant ether. The magnetic -polarisation was looked upon as wholly secondary in -origin, being due to the relative motion of the dielectric -tubes of polarisation. On this view the Fresnelian convection -coefficient comes out to be ½, as shown by J. J. -Thomson in 1880, instead of 1 - (1/μ²) as required by -optical experiments. This obviously implies a complete -failure to account for all those optical experiments which -depend for their satisfactory explanation on the assumption -of a value for the convection coefficient equal to 1 - (1/μ²).</p> - -<p class='c005'>The modifications proposed independently by Hertz and -Heaviside fare no better.<a id='r1' /><a href='#f1' class='c007'><sup>[1]</sup></a> They postulated the actual -medium to be the seat of all electric polarisation and further -emphasised the reciprocal relation subsisting between -electricity and magnetism, thus making the field equations -more symmetrical. On this view the whole of the -polarised ether is carried away by the moving medium, -and consequently, the convection coefficient naturally -becomes unity in this theory, a value quite as discrepant -as that obtained on the original Maxwellian assumption.</p> - -<p class='c005'>Thus neither Maxwell’s original theory nor its subsequent -modifications as developed by Hertz and Heaviside -succeeded in obtaining a value for Fresnelian coefficient -equal to 1 - (1/μ<sup>2</sup>), and consequently stood totally condemned -from the optical point of view.</p> - -<p class='c005'>Certain direct electromagnetic experiments involving -relative motion of polarised dielectrics were no less conclusive -against the generalised theory of Hertz and Heaviside. -According to Hertz a moving dielectric would carry away -the whole of its electric displacement with it. Hence the -electromagnetic effect near the moving dielectric would -be proportional to the total electric displacement, that is -to K, the specific inductive capacity of the dielectric. In -1901, Blondlot working with a stream of moving gas -could not detect any such effect. H. A. Wilson repeated -the experiment in an improved form in 1903 and working -with ebonite found that the observed effect was proportional -to K - 1 instead of to K. For gases K is nearly -equal to 1 and hence practically no effect will be observed -in their case. This gives a satisfactory explanation of -Blondlot’s negative results.</p> - -<p class='c005'>Rowland had shown in 1876 that the magnetic force -due to a rotating condenser (the dielectric remaining -stationary) was proportional to K, the sp. ind. cap. On -the other hand, Röntgen found in 1888 the magnetic -effect due to a rotating dielectric (the condenser remaining -stationary) to be proportional to K - 1, and not to -K. Finally Eichenwald in 1903 found that when both -condenser and dielectric are rotated together, the effect -observed was quite independent of K, a result quite -consistent with the two previous experiments. The Rowland -effect proportional to K, together with the opposite -Röntgen effect proportional to 1 - K, makes the Eichenwald -effect independent of K.</p> - -<p class='c005'>All these experiments together with those of Blondlot -and Wilson made it clear that the electromagnetic -effect due to a moving dielectric was proportional to -K - 1, and not to K as required by Hertz’s theory. Thus -the above group of experiments with moving dielectrics -directly contradicted the Hertz-Heaviside theory. The -internal discrepancies inherent in the classic ether theory -had now become too prominent. It was clear that the -ether concept had finally outgrown its usefulness. The -observed facts had become too contradictory and too -heterogeneous to be reduced to an organised whole with -the help of the ether concept alone. Radical departures -from the classical theory had become absolutely necessary.</p> - -<p class='c005'>There were several outstanding difficulties in connection -with anomalous dispersion, selective reflection and -selective absorption which could not be satisfactory -explained in the classic electromagnetic theory. It -was evident that the assumption of some kind of -discreteness in the optical medium had become inevitable. -Such an assumption naturally gave rise to an -atomic theory of electricity, namely, the modern electron -theory. Lorentz had postulated the existence of electrons -so early as 1878, but it was not until some years later that -the electron theory became firmly established on a satisfactory -basis.</p> - -<p class='c005'>Lorentz assumed that a moving dielectric merely carried -away its own “polarisation doublets,” which on his theory -gave rise to the induced field proportional to K - 1. The -field near a moving dielectric is naturally proportional to -K - 1 and not to K. Lorentz’s theory thus gave a -satisfactory explanation of all those experiments with -moving dielectrics which required effects proportional to -K - 1. Lorentz further succeeded in obtaining a value for -the Fresnelian convection coefficient equal to 1 - 1/μ<sup>2</sup>, the -exact value required by all optical experiments of the -moving type.</p> - -<p class='c005'>We must now go back to Michelson and Morley’s -experiment. We have seen that both parts of the beam -are situated in free ether; no material medium is involved -in any portion of the paths actually traversed by the beam. -Consequently no compensation due to Fresnelian convection -of ether by moving medium is possible. Thus Fresnelian -convection compensation can have no possible application -in this case. Yet some marvellous compensation has -evidently taken place which has completely masked the -“absolute” velocity of the earth.</p> - -<p class='c005'>In Michelson and Morley’s experiment, the distance -travelled by the beam along OA (that is, in a direction -parallel to the motion of the platform) is 2<i>l</i>β², while the -distance travelled by the beam along OB, perpendicular to -the direction of motion of the platform, is 2<i>l</i>β. Yet the -most careful experiments showed, as Eddington says, “that -both parts of the beam took the same time as tested by the -interference bands produced. It would seem that OA and -OB could not really have been of the same length; and if -OB was of length <i>l</i>, OA must have been of length <i>l</i>/β. The -apparatus was now rotated through 90°, so that OB became -the up-stream. The time for the two journeys was again -the same, so that 0B must now be the shorter length. The -plain meaning of the experiment is that both arms have a -length <i>l</i> when placed along O<i>y</i> (perpendicular to the direction -of motion), and automatically contract to a length -<i>l</i>/β, when placed along O<i>x</i> (parallel to the direction of -motion). This explanation was first given by Fitz-Gerald.”</p> - -<p class='c005'>This Fitz-Gerald contraction, startling enough in -itself, does not suffice. Assuming this contraction to be a -real one, the distance travelled with respect to the ether is -2<i>l</i>β and the time taken for this journey is 2<i>l</i>β/<i>c</i>. But the -distance travelled with respect to the platform is always -2<i>l</i>. Hence the velocity of light with respect to the platform -is</p> - -<div class='figcenter id002'> -<img src='images/form-004.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>a variable quantity depending on -the “absolute” velocity of the platform. But no trace -of such an effect has ever been found. The velocity of -light is always found to be quite independent of the velocity -of the platform. The present difficulty cannot be solved -by any further alteration in the measure of space. The -only recourse left open is to alter the measure of time as -well, that is, to adopt the concept of “local time.” If a moving -clock goes slower so that one ‘real’ second becomes 1/β -second as measured in the moving system, the velocity of -light relative to the platform will always remain <i>c</i>. We -must adopt two very startling hypotheses, namely, the -Fitz-Gerald contraction and the concept of “local time,” -in order to give a satisfactory explanation of the -Michelson-Morley experiment.</p> - -<p class='c005'>These results were already reached by Lorentz in the -course of further developments of his electron theory. -Lorentz used a special set of transformation equations<a id='r2' /><a href='#f2' class='c007'><sup>[2]</sup></a> for -time which implicitly introduced the concept of local time. -But he himself failed to attach any special significance to -it, and looked upon it rather as a mere mathematical -artifice like imaginary quantities in analysis or the circle -at infinity in projective geometry. The originality of -Einstein at this stage consists in his successful physical -interpretation of these results, and viewing them as the -coherent organised consequences of a single general -principle. Lorentz established the Relativity Theorem<a id='r3' /><a href='#f3' class='c007'><sup>[3]</sup></a> -(consisting merely of a set of transformation equations) -while Einstein generalised it into a Universal Principle. In -addition Einstein introduced fundamentally new concepts -of space and time, which served to destroy old fetishes and -demanded a wholesale revision of scientific concepts and -thus opened up new possibilities in the synthetic unification -of natural processes.</p> - -<p class='c005'>Newton had framed his laws of motion in such a way -as to make them quite independent of the absolute velocity -of the earth. Uniform relative motion of ether and matter -could not be detected with the help of dynamical laws. -According to Einstein neither could it be detected with the -help of optical or electromagnetic experiments. Thus the -Einsteinian Principle of Relativity asserts that all physical -laws are independent of the ‘absolute’ velocity of an observer.</p> - -<p class='c005'>For different systems, the <i>form</i> of all physical laws is -conserved. If we chose the velocity of light<a id='r4' /><a href='#f4' class='c007'><sup>[4]</sup></a> to be the -fundamental unit of measurement for all observers (that is, -assume the constancy of the velocity of light in all systems) -we can establish a <i>metric</i> “one-one” correspondence -between any two observed systems, such correspondence -depending only the <i>relative</i> velocity of the two systems. -Einstein’s Relativity is thus merely the consistent logical -application of the well known physical principle that we -can know nothing but <i>relative</i> motion. In this sense it is -a further extension of Newtonian Relativity.</p> - -<p class='c005'>On this interpretation, the Lorentz-Fitzgerald contraction -and “local time” lose their arbitrary character. Space -and time as measured by two different observers are naturally -diverse, and the difference depends only on their relative -motion. Both are equally valid; they are merely different -descriptions of the same physical reality. This is essentially -the point of view adopted by Minkowski. He considers time -itself to be one of the co-ordinate axes, and in his four-dimensional -world, that is in the space-time reality, relative -motion is reduced to a rotation of the axes of reference. -Thus, the diversity in the measurement of lengths and -temporal rates is merely due to the static difference in the -“frame-work” of the different observers.</p> - -<p class='c005'>The above theory of Relativity absorbed practically -the whole of the electromagnetic theory based on the -Maxwell-Lorentz system of field equations. It combined -all the advantages of classic Maxwellian theory together -with an electronic hypothesis. The Lorentz assumption of -polarisation doublets had furnished a satisfactory explanation -of the Fresnelian convection of ether, but in the new -theory this is deduced merely as a consequence of the altered -concept of relative velocity. In addition, the theory of -Relativity accepted the results of Michelson and Morley’s -experiments as a definite principle, namely, the principle of -the constancy of the velocity of light, so that there was -nothing left for explanation in the Michelson-Morley -experiment. But even more than all this, it established a -single general principle which served to connect together -in a simple coherent and fruitful manner the known facts -of Physics.</p> - -<p class='c005'>The theory of Relativity received direct experimental -confirmation in several directions. Repeated attempts were -made to detect the Lorentz-Fitzgerald contraction. Any -ordinary physical contraction will usually have observable -physical results; for example, the total electrical resistance -of a conductor will diminish. Trouton and Noble, Trouton -and Rankine, Rayleigh and Brace, and others employed -a variety of different methods to detect the Lorentz-Fitzgerald -contraction, but invariably with the same -negative results. <i>Whether there is an ether or not, -uniform velocity with respect to it can never be detected.</i> -This does not prove that there is no such thing as an -ether but certainly does render the ether entirely superfluous. -Universal compensation is due to a change in local -units of length and time, or rather, being merely different -descriptions of the same reality, there is no compensation -at all.</p> - -<p class='c005'>There was another group of observed phenomena which -could scarcely be fitted into a Newtonian scheme of -dynamics without doing violence to it. The experimental -work of Kaufmann, in 1901, made it abundantly clear that -the “mass” of an electron depended on its velocity. So -early as 1881, J. J. Thomson had shown that the inertia of -a charged particle increased with its velocity. Abraham -now deduced a formula for the variation of mass with -velocity, on the hypothesis that an electron always remained -a <i>rigid</i> sphere. Lorentz proceeded on the assumption -that the electron shared in the Lorentz-Fitzgerald contraction -and obtained a totally different formula. A very -careful series of measurements carried out independently by -Bücherer, Wolz, Hupka and finally Neumann in 1913, -decided conclusively in favour of the Lorentz formula. -This “contractile” formula follows immediately as a direct -consequence of the new Theory of Relativity, without any -assumption as regards the electrical origin of inertia. Thus -the complete agreement of experimental facts with the -predictions of the new theory must be considered as -confirming it as a principle which goes even beyond the -electron itself. The greatest triumph of this new theory -consists, indeed, in the fact that a large number of results, -which had formerly required all kinds of special hypotheses -for their explanation, are now deduced very simply as -inevitable consequences of one single general principle.</p> - -<p class='c005'>We have now traced the history of the development of -the restricted or special theory of Relativity, which is -mainly concerned with optical and electrical phenomena. -It was first offered by Einstein in 1905. Ten years later, -Einstein formulated his second theory, the Generalised -Principle of Relativity. This new theory is mainly a theory -of gravitation and has very little connection with optics -and electricity. In one sense, the second theory is indeed -a further generalisation of the restricted principle, but the -former does not really contain the latter as a special case.</p> - -<p class='c005'>Einstein’s first theory is restricted in the sense that it -only refers to uniform rectilinear motion and has no application -to any kind of accelerated movements. Einstein in -his second theory extends the Relativity Principle to cases -of accelerated motion. If Relativity is to be universally -true, then even accelerated motion must be merely <i>relative -motion between matter and matter</i>. Hence the Generalised -Principle of Relativity asserts that “absolute” motion -cannot be detected even with the help of gravitational laws.</p> - -<p class='c005'>All movements must be referred to definite sets of -co-ordinate axes. If there is any change of axes, the -numerical magnitude of the movements will also change. -But according to Newtonian dynamics, such alteration in -physical movements can only be due to the effect of certain -forces in the field.<a id='r5' /><a href='#f5' class='c007'><sup>[5]</sup></a> Thus any change of axes will introduce -new “geometrical” forces in the field which are quite -independent of the nature of the body acted on. Gravitational -forces also have this same remarkable property, and -gravitation itself may be of essentially the same nature as -these “geometrical” forces introduced by a change of axes. -This leads to Einstein’s famous Principle of Equivalence. -<i>A gravitational field of force is strictly equivalent to one -introduced by a transformation of co-ordinates and no possible -experiment can distinguish between the two.</i></p> - -<p class='c005'>Thus it may become possible to “transform away” -gravitational effects, at least for sufficiently small regions of -space, by referring all movements to a new set of axes. This -new “framework” may of course have all kinds of very -complicated movements when referred to the old Galilean -or “rectangular unaccelerated system of co-ordinates.”</p> - -<p class='c005'>But there is no reason why we should look upon the -Galilean system as more fundamental than any other. If it -is found simpler to refer all motion in a gravitational field -to a special set of co-ordinates, we may certainly look upon -this special “framework” (at least for the particular region -concerned), to be more fundamental and more natural. We -may, still more simply, identify this particular framework -with the special local properties of space in that region. -That is, we can look upon the effects of a gravitational -field as simply due to the local properties of space and time -itself. The very presence of matter implies a modification -of the characteristics of space and time in its neighbourhood. -As Eddington says “matter does not cause the -curvature of space-time. It is the curvature. Just as -light does not cause electromagnetic oscillations; it is the -oscillations.”</p> - -<p class='c005'>We may look upon this from a slightly different point -of view. The General Principle of Relativity asserts that -all motion is merely relative motion between matter and -matter, and as all movements must be referred to definite -sets of co-ordinates, the ground of any possible framework -must ultimately be material in character. It <i>is</i> convenient -to take the matter actually present in a field as the -fundamental ground of our framework. If this is done, -the special characteristics of our framework would naturally -depend on the actual distribution of matter in the field. -But physical space and time is completely defined by the -“framework.” In other words the “framework” itself <i>is</i> -space and time. Hence we see how <i>physical</i> space and time -is actually defined by the local distribution of matter.</p> - -<p class='c005'>There are certain magnitudes which remain constant by -any change of axes. In ordinary geometry distance -between two points is one such magnitude; so that -δ<i>x²</i> + δ<i>y²</i> + δ<i>z²</i> is an invariant. In the restricted theory of -light, the principle of constancy of light velocity demands -that δ<i>x²</i> + δ<i>y²</i> + δ<i>z²</i> - <i>c²</i>δ<i>t²</i> should remain constant.</p> - -<p class='c005'>The <i>separation ds</i> of adjacent events is defined by -<i>ds²</i> = -<i>dx²</i> - <i>dy²</i> - <i>dz²</i> + <i>c²dt²</i>. It is an extension of the -notion of distance and this is the new invariant. Now if -<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> are transformed to any set of new variables -<i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i>, we shall get a quadratic expression for</p> - -<div class='figcenter id004'> -<img src='images/form-005.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>where the <i>g</i>’s are -functions of <i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i> depending on the transformation.</p> - -<p class='c005'>The special properties of space and time in any region -are defined by these <i>g</i>’s which are themselves determined -by the actual distribution of matter in the locality. Thus -from the Newtonian point of view, these <i>g</i>’s represent the -gravitational effect of matter while from the Relativity -stand-point, these merely define the non-Newtonian (and -incidentally non-Euclidean) space in the neighbourhood of -matter.</p> - -<p class='c005'>We have seen that Einstein’s theory requires local -curvature of space-time in the neighbourhood of matter. -Such altered characteristics of space and time give a -satisfactory explanation of an outstanding discrepancy in -the observed advance of perihelion of Mercury. The large -discordance is almost completely removed by Einstein’s -theory.</p> - -<p class='c005'>Again, in an intense gravitational field, a beam of light -will be affected by the local curvature of space, so that to -an observer who is referring all phenomena to a Newtonian -system, the beam of light will appear to deviate from its -path along an Euclidean straight line.</p> - -<p class='c005'>This famous prediction of Einstein about the deflection -of a beam of light by the sun’s gravitational field was -tested during the total solar eclipse of May, 1919. The -observed deflection is decisively in favour of the Generalised -Theory of Relativity.</p> - -<p class='c005'>It should be noted however that the velocity of light -itself would decrease in a gravitational field. This may -appear at first sight to be a violation of the principle of -constancy of light-velocity. But when we remember that -the Special Theory is explicitly <i>restricted</i> to the case of -unaccelerated motion, the difficulty vanishes. In the -absence of a gravitational field, that is in any unaccelerated -system, the velocity of light will always remain constant. -Thus the validity of the Special Theory is completely -preserved within its own <i>restricted</i> field.</p> - -<p class='c005'>Einstein has proposed a third crucial test. He has -predicted a shift of spectral lines towards the red, due to an -intense gravitational potential. Experimental difficulties -are very considerable here, as the shift of spectral lines is a -complex phenomenon. Evidence is conflicting and nothing -conclusive can yet be asserted. Einstein thought that a -gravitational displacement of the Fraunhofer lines is a -necessary and fundamental condition for the acceptance of -his theory. But Eddington has pointed out that even if -this test fails, the logical conclusion would seem to be that -while Einstein’s law of gravitation is true for matter in -bulk, it is not true for such small material systems as -atomic oscillator.</p> -<h3 class='c008'>Conclusion</h3> -<p class='c004'>From the conceptual stand-point there are several -important consequences of the Generalised or Gravitational -Theory of Relativity. Physical space-time is perceived to -be intimately connected with the actual local distribution -of matter. Euclid-Newtonian space-time is <i>not</i> the actual -space-time of Physics, simply because the former completely -neglects the actual presence of matter. Euclid-Newtonian -continuum is merely an abstraction, while physical space-*time -is the actual framework which has some definite -curvature due to the presence of matter. Gravitational -Theory of Relativity thus brings out clearly the fundamental -distinction between actual physical space-time -(which is non-isotropic and non-Euclid-Newtonian) on one -hand and the abstract Euclid-Newtonian continuum (which -is homogeneous, isotropic and a purely intellectual construction) -on the other.</p> - -<p class='c005'>The measurements of the rotation of the earth reveals a -fundamental framework which may be called the “inertial -framework.” This constitutes the actual physical universe. -This universe approaches Galilean space-time at a great -distance from matter.</p> - -<p class='c005'>The properties of this physical universe may be referred -to some world-distribution of matter or the “inertial framework” -may be constructed by a suitable modification of the -law of gravitation itself. In Einstein’s theory the actual -curvature of the “inertial framework” is referred to vast -quantities of undetected world-matter. It has interesting -consequences. The dimensions of Einsteinian universe -would depend on the quantity of matter in it; it would -vanish to a point in the total absence of matter. Then -again curvature depends on the quantity of matter, and -hence in the presence of a sufficient quantity of matter space-time -may curve round and close up. Einsteinian universe -will then reduce to a finite system without boundaries, like -the surface of a sphere. In this “closed up” system, -light rays will come to a focus after travelling round the -universe and we should see an “anti-sun” (corresponding to -the back surface of the sun) at a point in the sky opposite -to the real sun. This anti-sun would of course be equally -large and equally bright if there is no absorption of light -in free space.</p> - -<p class='c005'>In de Sitter’s theory, the existence of vast quantities of -world-matter is not required. But beyond a definite -distance from an observer, time itself stands still, so that -to the observer nothing can ever “happen” there. All -these theories are still highly speculative in character, but -they have certainly extended the scope of theoretical physics -to the central problem of the ultimate nature of the -universe itself.</p> - -<p class='c005'>One outstanding peculiarity still attaches to the concept -of electric force—it is not amenable to any process of being -“transformed away” by a suitable change of framework. -H. Weyl, it seems, has developed a geometrical theory (in -hyper-space) in which no fundamental distinction is made -between gravitational and electrical forces.</p> - -<p class='c005'>Einstein’s theory connects up the law of gravitation -with the laws of motion, and serves to establish a very -intimate relationship between matter and physical space-*time. -Space, time and matter (or energy) were considered -to be the three ultimate elements in Physics. The restricted -theory fused space-time into one indissoluble whole. The -generalised theory has further synthesised space-time and -matter into one fundamental physical reality. Space, time -and matter taken separately are more abstractions. Physical -reality consists of a synthesis of all three.</p> - -<p class='c005'><span class='sc'>P. C. Mahalanobis.</span></p> -<h3 id='note-a' class='c008'>Note A.</h3> -<p class='c004'>For example consider a massive particle resting on a -circular disc. If we set the disc rotating, a centrifugal force -appears in the field. On the other hand, if we transform -to a set of rotating axes, we must introduce a centrifugal -force in order to correct for the change of axes. This -newly introduced centrifugal force is usually looked upon -as a mathematical fiction—as “geometrical” rather than -physical. The presence of such a geometrical force is usually -interpreted as being due to the adoption of a fictitious -framework. On the other hand a gravitational force is -considered quite real. Thus a fundamental distinction is -made between geometrical and gravitational forces.</p> - -<p class='c005'>In the General Theory of Relativity, this fundamental -distinction is done away with. The very possibility of -distinguishing between geometrical and gravitational forces -is denied. All axes of reference may now be regarded as -equally valid.</p> - -<p class='c005'>In the Restricted Theory, all “unaccelerated” axes of -reference were recognised as equally valid, so that physical -laws were made independent of uniform absolute velocity. -In the General Theory, physical laws are made independent -of “absolute” motion of any kind.</p> - -<div class='chapter'> - <h2 id='art2' class='c003'>On The Electrodynamics of Moving Bodies <br />By<br /> A. Einstein.</h2> -</div> -<h3 class='c008'>INTRODUCTION.</h3> -<p class='c004'>It is well known that if we attempt to apply Maxwell’s -electrodynamics, as conceived at the present time, to -moving bodies, we are led to asymmetry which does not -agree with observed phenomena. Let us think of the -mutual action between a magnet and a conductor. The -observed phenomena in this case depend only on the -relative motion of the conductor and the magnet, while -according to the usual conception, a distinction must be -made between the cases where the one or the other of the -bodies is in motion. If, for example, the magnet moves -and the conductor is at rest, then an electric field of certain -energy-value is produced in the neighbourhood of the -magnet, which excites a current in those parts of the -field where a conductor exists. But if the magnet be at -rest and the conductor be set in motion, no electric field -is produced in the neighbourhood of the magnet, but an -electromotive force which corresponds to no energy in -itself is produced in the conductor; this causes an electric -current of the same magnitude and the same career as the -electric force, it being of course assumed that the relative -motion in both of these cases is the same.</p> - -<p class='c005'>2. Examples of a similar kind such as the unsuccessful -attempt to substantiate the motion of the earth relative -to the “Light-medium” lead us to the supposition that -not only in mechanics, but also in electrodynamics, no -properties of observed facts correspond to a concept of -absolute rest; but that for all coordinate systems for which -the mechanical equations hold, the equivalent electrodynamical -and optical equations hold also, as has already been -shown for magnitudes of the first order. In the following -we make these assumptions (which we shall subsequently -call the Principle of Relativity) and introduce the further -assumption,—an assumption which is at the first sight -quite irreconcilable with the former one—that light is -propagated in vacant space, with a velocity <i>c</i> which is -independent of the nature of motion of the emitting -body. These two assumptions are quite sufficient to give -us a simple and consistent theory of electrodynamics of -moving bodies on the basis of the Maxwellian theory for -bodies at rest. The introduction of a “Lightäther” -will be proved to be superfluous, for according to the -conceptions which will be developed, we shall introduce -neither a space absolutely at rest, and endowed with -special properties, nor shall we associate a velocity-vector -with a point in which electro-magnetic processes take -place.</p> - -<p class='c005'>3. Like every other theory in electrodynamics, the -theory is based on the kinematics of rigid bodies; in the -enunciation of every theory, we have to do with relations -between rigid bodies (co-ordinate system), clocks, and -electromagnetic processes. An insufficient consideration -of these circumstances is the cause of difficulties with -which the electrodynamics of moving bodies have to fight -at present.</p> -<h3 class='c008'>I.—KINEMATICAL PORTION.</h3> -<h4 id='sect2-1' class='c009'>§ 1. Definition of Synchronism.</h4> -<p class='c004'>Let us have a co-ordinate system, in which the Newtonian -equations hold. For distinguishing this system -from another which will be introduced hereafter, we -shall always call it “the stationary system.”</p> - -<p class='c005'>If a material point be at rest in this system, then its -position in this system can be found out by a measuring -rod, and can be expressed by the methods of Euclidean -Geometry, or in Cartesian co-ordinates.</p> - -<p class='c005'>If we wish to describe the motion of a material point, -the values of its coordinates must be expressed as functions -of time. It is always to be borne in mind that <i>such a -mathematical definition has a physical sense, only when we -have a clear notion of what is meant by time. We have to -take into consideration the fact that those of our conceptions, in -which time plays a part, are always conceptions of synchronism.</i> -For example, we say that a train arrives here at 7 o’clock; -this means that the exact pointing of the little hand of my -watch to 7, and the arrival of the train are synchronous -events.</p> - -<p class='c005'>It may appear that all difficulties connected with the -definition of time can be removed when in place of time, -we substitute the position of the little hand of my watch. -Such a definition is in fact sufficient, when it is required to -define time exclusively for the place at which the clock is -stationed. But the definition is not sufficient when it is -required to connect by time events taking place at different -stations,—or what amounts to the same thing,—to estimate -by means of time (zeitlich werten) the occurrence of events, -which take place at stations distant from the clock.</p> - -<p class='c005'>Now with regard to this attempt;—the time-estimation -of events, we can satisfy ourselves in the following -manner. Suppose an observer—who is stationed at the -origin of coordinates with the clock—associates a ray of -light which comes to him through space, and gives testimony -to the event of which the time is to be estimated,—with -the corresponding position of the hands of the clock. But -such an association has this defect,—it depends on the -position of the observer provided with the clock, as we -know by experience. We can attain to a more practicable -result by the following treatment.</p> - -<p class='c005'>If an observer be stationed at A with a clock, he can -estimate the time of events occurring in the immediate -neighbourhood of A, by looking for the position of -the hands of the clock, which are synchronous with -the event. If an observer be stationed at B with a -clock,—we should add that the clock is of the same nature -as the one at A,—he can estimate the time of events -occurring about B. But without further premises, it is -not possible to compare, as far as time is concerned, the -events at B with the events at A. We have hitherto an -A-time, and a B-time, but no time common to A and B. -This last time (<i>i.e.</i>, common time) can be defined, if we -establish by definition that the time which light requires -in travelling from A to B is equivalent to the time which -light requires in travelling from B to A. For example, -a ray of light proceeds from A at A-time t<sub>A</sub> towards B, -arrives and is reflected from B at B-time t<sub>B</sub>, and returns -to A at A-time t′<sub>A</sub>. According to the definition, both -clocks are synchronous, if</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>t<sub>B</sub> - t<sub>A</sub> = t′<sub>A</sub> - t<sub>B</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>We assume that this definition of synchronism is possible -without involving any inconsistency, for any number of -points, therefore the following relations hold:—</p> - -<p class='c005'>1. If the clock at B be synchronous with the clock -at A, then the clock at A is synchronous with the clock -at B.</p> - -<p class='c005'>2. If the clock at A as well as the clock at B are -both synchronous with the clock at C, then the clocks at -A and B are synchronous.</p> - -<p class='c005'>Thus with the help of certain physical experiences, we -have established what we understand when we speak of -clocks at rest at different stations, and synchronous with -one another; and thereby we have arrived at a definition of -synchronism and time.</p> - -<p class='c005'>In accordance with experience we shall assume that the -magnitude</p> - -<div class='figcenter id005'> -<img src='images/form-006.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>where <i>c</i> is a universal constant.</p> - -<p class='c005'>We have defined time essentially with a clock at rest -in a stationary system. On account of its adaptability -to the stationary system, we call the time defined in this -way as “time of the stationary system.”</p> -<h4 class='c009'>§ 2. On the Relativity of Length and Time.</h4> -<p class='c004'>The following reflections are based on the Principle -of Relativity and on the Principle of Constancy of the -velocity of light, both of which we define in the following -way:—</p> - -<p class='c005'>1. The laws according to which the nature of physical -systems alter are independent of the manner in which -these changes are referred to two co-ordinate systems -which have a uniform translators motion relative to each -other.</p> - -<p class='c005'>2. Every ray of light moves in the “stationary -co-ordinate system” with the same velocity <i>c</i>, the velocity -being independent of the condition whether this ray of -light is emitted by a body at rest or in motion.<a id='r6' /><a href='#f6' class='c007'><sup>[6]</sup></a> Therefore</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>velocity = Path of Light/Interval of time,</div> - </div> - </div> -</div> - -<p class='c010'>where, by ‘interval of time’ we mean time as defined -in <a href='#sect2-1'>§1</a>.</p> - -<p class='c005'>Let us have a rigid rod at rest; this has a length <i>l</i>, -when measured by a measuring rod at rest; we suppose -that the axis of the rod is laid along the X-axis of the -system at rest, and then a uniform velocity <i>v</i>, parallel -to the axis of X, is imparted to it. Let us now enquire -about the length of the moving rod; this can be obtained -by either of these operations.—</p> - -<p class='c005'>(<i>a</i>) The observer provided with the measuring rod -moves along with the rod to be measured, and measures -by direct superposition the length of the rod:—just as if -the observer, the measuring rod, and the rod to be measured -were at rest.</p> - -<p class='c005'>(<i>b</i>) The observer finds out, by means of clocks placed -in a system at rest (the clocks being synchronous as defined -in <a href='#sect2-1'>§1</a>), the points of this system where the ends of the -rod to be measured occur at a particular time <i>t</i>. The -distance between these two points, measured by the -previously used measuring rod, this time it being at rest, -is a length, which we may call the “length of the rod.”</p> - -<p class='c005'>According to the Principle of Relativity, the length -found out by the operation <i>a</i>), which we may call “the -length of the rod in the moving system” is equal to the -length <i>l</i> of the rod in the stationary system.</p> - -<p class='c005'>The length which is found out by the second method, -may be called ‘<i>the length of the moving rod measured from -the stationary system</i>.’ This length is to be estimated on -the basis of our principle, and <i>we shall find it to be different -from l</i>.</p> - -<p class='c005'>In the generally recognised kinematics, we silently -assume that the lengths defined by these two operations -are equal, or in other words, that at an epoch of time <i>t</i>, -a moving rigid body is geometrically replaceable by the -same body, which can replace it in the condition of rest.</p> -<h5 class='c009'>Relativity of Time.</h5> -<p class='c004'>Let us suppose that the two clocks synchronous with -the clocks in the system at rest are brought to the ends A, -and B of a rod, <i>i.e.</i>, the time of the clocks correspond to -the time of the stationary system at the points where they -happen to arrive; these clocks are therefore synchronous -in the stationary system.</p> - -<p class='c005'>We further imagine that there are two observers at the -two watches, and moving with them, and that these -observers apply the criterion for synchronism to the two -clocks. At the time <i>t</i><sub>A</sub>, a ray of light goes out from A, is -reflected from B at the time <i>t</i><sub>B</sub>, and arrives back at A at -time <i>t′</i><sub>A</sub>. Taking into consideration the principle of, -constancy of the velocity of light, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in10'><i>t</i><sub>B</sub> - <i>t</i><sub>A</sub> = <i>r</i><sub>AB</sub>/(<i>c</i> - <i>v</i>),</div> - </div> - <div class='group'> - <div class='line'>and <i>t′</i><sub>A</sub> - <i>t</i><sub>B</sub> = <i>r</i><sub>AB</sub>/(<i>c</i> + <i>v</i>),</div> - </div> - </div> -</div> - -<p class='c010'>where <i>r</i><sub>AB</sub> is the length of the moving rod, measured -in the stationary system. Therefore the observers stationed -with the watches will not find the clocks synchronous, -though the observer in the stationary system must declare -the clocks to be synchronous. We therefore see that we can -attach no absolute significance to the concept of synchronism; -but two events which are synchronous when viewed -from one system, will not be synchronous when viewed -from a system moving relatively to this system.</p> -<h4 id='sect2-3' class='c009'>§ 3. Theory of Co-ordinate and Time-Transformation from a stationary system to a system which moves relatively to this with uniform velocity.</h4> -<p class='c004'>Let there be given, in the stationary system two -co-ordinate systems, <i>i.e.</i>, two series of three mutually -perpendicular lines issuing from a point. Let the X-axes -of each coincide with one another, and the Y and Z-axes -be parallel. Let a rigid measuring rod, and a number -of clocks be given to each of the systems, and let the rods -and clocks in each be exactly alike each other.</p> - -<p class='c005'>Let the initial point of one of the systems (<i>k</i>) have -a constant velocity in the direction of the X-axis of -the other which is stationary system K, the motion being -also communicated to the rods and clocks in the system (<i>k</i>). -Any time <i>t</i> of the stationary system K corresponds to a -definite position of the axes of the moving system, which -are always parallel to the axes of the stationary system. By -<i>t</i>, we always mean the time in the stationary system.</p> - -<p class='c005'>We suppose that the space is measured by the stationary -measuring rod placed in the stationary system, as well as -by the moving measuring rod placed in the moving -system, and we thus obtain the co-ordinates (<i>x</i>, <i>y</i>, <i>z</i>) for the -stationary system, and (ξ, η, ζ) for the moving system. Let -the time <i>t</i> be determined for each point of the stationary -system (which are provided with clocks) by means of the -clocks which are placed in the stationary system, with -the help of light-signals as described in <a href='#sect2-1'>§ 1</a>. Let also -the time τ of the moving system be determined for each -point of the moving system (in which there are clocks which -are at rest relative to the moving system), by means of -the method of light signals between these points (in -which there are clocks) in the manner described in <a href='#sect2-1'>§ 1</a>.</p> - -<p class='c005'>To every value of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) which fully determines -the position and time of an event in the stationary system, -there correspond a system of values (ξ, η, ζ, τ); now the -problem is to find out the system of equations connecting -these magnitudes.</p> - -<p class='c005'>Primarily it is clear that on account of the property -of homogeneity which we ascribe to time and space, the -equations must be linear.</p> - -<p class='c005'>If we put <i>x′</i> = <i>x</i> - <i>vt</i>, then it is clear that at a point -relatively at rest in the system <i>k</i>, we have a system of -values (<i>x′</i> <i>y</i> <i>z</i>) which are independent of time. Now -let us find out τ as a function of (<i>x′</i>, <i>y</i>, <i>z</i>, <i>t</i>). For this -purpose we have to express in equations the fact that τ is -not other than the time given by the clocks which are -at rest in the system <i>k</i> which must be made synchronous -in the manner described in <a href='#sect2-1'>§ 1</a>.</p> - -<p class='c005'>Let a ray of light be sent at time τ₀ from the origin -of the system <i>k</i> along the X-axis towards <i>x′</i> and let it be -reflected from that place at time τ₁ towards the origin -of moving co-ordinates and let it arrive there at time τ₂; -then we must have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>½ (τ<sub>0</sub> + τ<sub>2</sub>) = τ<sub>1</sub></div> - </div> - </div> -</div> - -<p class='c010'>If we now introduce the condition that τ is a function -of co-ordinates, and apply the principle of constancy of -the velocity of light in the stationary system, we have</p> - -<div class='figcenter id006'> -<img src='images/form-007.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>It is to be noticed that instead of the origin of co-ordinates, -we could select some other point as the exit -point for rays of light, and therefore the above equation -holds for all values of (<i>x′</i>, <i>y</i>, <i>z</i>, <i>t</i>,).</p> - -<p class='c005'>A similar conception, being applied to the <i>y</i>- and <i>z</i>-axis -gives us, when we take into consideration the fact that -light when viewed from the stationary system, is always -propagated along those axes with the velocity √(<i>c²</i> - <i>v²</i>), -we have the questions</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in1'>∂ ∂τ</div> - <div class='line'>---- = 0, ---- = 0.</div> - <div class='line in1'>∂y ∂z</div> - </div> - </div> -</div> - -<p class='c010'>From these equations it follows that τ is a linear function -of <i>x′</i> and <i>t</i>. From equations (1) we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in10'>vx′</div> - <div class='line'>τ = a (t - --------- )</div> - <div class='line in9'>c<sup>2</sup> - v<sup>2</sup></div> - </div> - </div> -</div> - -<p class='c010'>where <i>a</i> is an unknown function of <i>v</i>.</p> - -<p class='c005'>With the help of these results it is easy to obtain the -magnitudes (ξ, η, ζ) if we express by means of equations -the fact that light, when measured in the moving system -is always propagated with the constant velocity <i>c</i> (as -the principle of constancy of light velocity in conjunction -with the principle of relativity requires). For a -time τ = 0, if the ray is sent in the direction of increasing -ξ, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in23'><i>vx′</i></div> - <div class='line'>ξ = <i>c</i>τ, <i>i.e.</i> ξ = <i>a c</i>(<i>t</i> - ------------ )</div> - <div class='line in23'><i>c<sup>2</sup></i> - <i>v<sup>2</sup></i></div> - </div> - </div> -</div> - -<p class='c010'>Now the ray of light moves relative to the origin of <i>k</i> -with a velocity <i>c</i> - <i>v</i>, measured in the stationary system; -therefore we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in2'><i>x′</i></div> - <div class='line'>-------- = <i>t</i></div> - <div class='line in1'><i>c</i> - <i>v</i></div> - </div> - </div> -</div> - -<p class='c010'>Substituting these values of <i>t</i> in the equation for ξ, -we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in10'><i>c<sup>2</sup></i></div> - <div class='line'>ξ = <i>a</i> ------------ <i>x′</i></div> - <div class='line in7'><i>c<sup>2</sup></i> - <i>v<sup>2</sup></i></div> - </div> - </div> -</div> - -<p class='c010'>In an analogous manner, we obtain by considering the -ray of light which moves along the <i>y</i>-axis,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in17'><i>vx′</i></div> - <div class='line'>η = <i>c</i>τ = <i>a c</i>(<i>t</i> - ----------- )</div> - <div class='line in16'><i>c<sup>2</sup></i> - <i>v<sup>2</sup></i></div> - </div> - </div> -</div> - -<p class='c010'>where</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in4'><i>y</i></div> - <div class='line'>------------- = <i>t</i>, <i>x′</i> = 0,</div> - <div class='line in1'>√ (<i>c<sup>2</sup></i> - <i>v<sup>2</sup></i>)</div> - </div> - </div> -</div> - -<p class='c010'>Therefore</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in10'><i>c</i></div> - <div class='line'>η = <i>a</i> ------------- <i>y</i>,</div> - <div class='line in6'>√ (<i>c<sup>2</sup></i> - <i>v<sup>2</sup></i>)</div> - </div> - <div class='group'> - <div class='line in11'><i>c</i></div> - <div class='line'>ζ = <i>a</i> --------------- <i>z</i> .</div> - <div class='line in7'>√ (<i>c<sup>2</sup></i> - <i>v<sup>2</sup></i>)</div> - </div> - </div> -</div> - -<p class='c010'>If for <i>x′</i>, we substitute its value <i>x</i> - <i>tv</i>, we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in16'><i>v</i>.<i>c</i></div> - <div class='line'>τ = φ (<i>v</i>). β (<i>t</i> - ------- ,</div> - <div class='line in16'>c<sup>2</sup></div> - </div> - <div class='group'> - <div class='line'>ξ = φ (<i>v</i>). β (<i>x</i> - <i>vt</i>) ,</div> - </div> - <div class='group'> - <div class='line'>η = φ (<i>v</i>) <i>y</i></div> - </div> - <div class='group'> - <div class='line'>ζ = φ (<i>v</i>) <i>z</i> ,</div> - </div> - </div> -</div> - -<p class='c010'>where</p> - -<div class='figcenter id005'> -<img src='images/form-008.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>and</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ (<i>v</i>) = <i>ac</i> / √ (<i>c<sup>2</sup></i> - <i>v<sup>2</sup></i>) = <i>a</i> / β</div> - </div> - </div> -</div> - -<p class='c010'>is a function -of <i>v</i>.</p> - -<p class='c005'>If we make no assumption about the initial position -of the moving system and about the null-point of <i>t</i>, -then an additive constant is to be added to the right -hand side.</p> - -<p class='c005'>We have now to show, that every ray of light moves -in the moving system with a velocity <i>c</i> (when measured in -the moving system), in case, as we have actually assumed, -<i>c</i> is also the velocity in the stationary system; 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-velocity.</p> - -<p class='c005'>At a time τ = <i>t</i> = 0 let a spherical wave be sent out -from the common origin of the two systems of co-ordinates, -and let it spread with a velocity <i>c</i> in the system K. If -(<i>x</i>, <i>y</i>, <i>z</i>), be a point reached by the wave, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x<sup>2</sup></i> + <i>y<sup>2</sup></i> + <i>z<sup>2</sup></i> = <i>c<sup>2</sup></i><i>t<sup>2</sup></i></div> - </div> - </div> -</div> - -<p class='c010'>with the aid of our transformation-equations, let us -transform this equation, and we obtain by a simple -calculation,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ξ<sup>2</sup> + η<sup>2</sup> + ζ<sup>2</sup> = <i>c<sup>2</sup></i>τ<sup>2</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>Therefore the wave is propagated in the moving system -with the same velocity <i>c</i>, and as a spherical wave.<a id='r7' /><a href='#f7' class='c007'><sup>[7]</sup></a> Therefore -we show that the two principles are mutually reconcilable.</p> - -<p class='c005'>In the transformations we have got an undetermined -function φ(<i>v</i>), and we now proceed to find it out.</p> - -<p class='c005'>Let us introduce for this purpose a third co-ordinate -system <i>k′</i>, which is set in motion relative to the system <i>k</i>, -the motion being parallel to the ξ-axis. Let the velocity of -the origin be (-<i>v</i>). At the time <i>t</i> = 0, all the initial -co-ordinate points coincide, and for <i>t</i> = <i>x</i> = <i>y</i> = <i>z</i> = 0, the -time <i>t′</i> of the system <i>k′</i> = 0. We shall say that (<i>x′</i> <i>y′</i> <i>z′</i> <i>t′</i>) -are the co-ordinates measured in the system <i>k′</i>, then by a -two-fold application of the transformation-equations, we -obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in19'><i>v</i></div> - <div class='line'>τ′ = φ(-<i>v</i>)β(-<i>v</i>){τ + --- ξ}</div> - <div class='line in19'><i>c<sup>2</sup></i></div> - <div class='line in5'>= φ(<i>v</i>)φ(-<i>v</i>)t,</div> - </div> - <div class='group'> - <div class='line'><i>x′</i> = φ](<i>v</i>)β(<i>v</i>)(ξ + <i>v</i>τ)</div> - <div class='line in5'>= φ(<i>v</i>)φ(-<i>v</i>)<i>x</i>, etc.</div> - </div> - </div> -</div> - -<p class='c010'>Since the relations between (<i>x′</i>, <i>y′</i>, <i>z′</i>, <i>t′</i>), and (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) -do not contain time explicitly, therefore K and <i>k′</i> are -relatively at rest.</p> - -<p class='c005'>It appears that the systems K and <i>k′</i> are identical.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∴ φ(<i>v</i>)φ(-<i>v</i>) = 1.</div> - </div> - </div> -</div> - -<p class='c010'>Let us now turn our attention to the part of the ξ-axis -between (ξ = 0, η = 0, ζ = 0), and (ξ = 0, η = 1, ζ = 0). Let -this piece of the <i>y</i>-axis be covered with a rod moving with -the velocity <i>v</i> relative to the system K and perpendicular -to its axis;—the ends of the rod having therefore the -co-ordinates</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x<sub>1</sub></i> = <i>vt</i>, <i>y<sub>1</sub></i> = <i>l</i> / φ(<i>v</i>), <i>z<sub>1</sub></i> = 0</div> - </div> - <div class='group'> - <div class='line'><i>x<sub>2</sub></i> = <i>vt</i>, <i>y<sub>2</sub></i> = 0, <i>z<sub>2</sub></i> = 0</div> - </div> - </div> -</div> - -<p class='c010'>Therefore the length of the rod measured in the system -K is <i>l</i>/φ(<i>v</i>). For the system moving with velocity (-<i>v</i>), -we have on grounds of symmetry,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in2'><i>l</i> <i>l</i></div> - <div class='line'>------ = ------</div> - <div class='line'> φ(<i>v</i>) φ(-<i>v</i>)</div> - </div> - <div class='group'> - <div class='line'>∴ φ(<i>v</i>) = φ(-<i>v</i>), ∴ φ(<i>v</i>) = 1.</div> - </div> - </div> -</div> - -<h4 id='sect2-4' class='c009'>§ 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks.</h4> -<p class='c004'>Let us consider a rigid sphere (<i>i.e.</i>, one having a -spherical figure when tested in the stationary system) of -radius R which is at rest relative to the system (K), and -whose centre coincides with the origin of K then the equation -of the surface of this sphere, which is moving with a -velocity <i>v</i> relative to K, is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ξ<sup>2</sup> + η<sup>2</sup> + ζ<sup>2</sup> = R<sup>2</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>At time <i>t</i> = 0, the equation is expressed by means of -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>,) as</p> - -<div class='figcenter id002'> -<img src='images/form-009.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>A rigid body which has the figure of a sphere when -measured in the moving system, has therefore in the -moving condition—when considered from the stationary -system, the figure of a rotational ellipsoid with semi-axes</p> - -<div class='figcenter id002'> -<img src='images/form-010.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Therefore the <i>y</i> and <i>z</i> dimensions of the sphere (therefore -of any figure also) do not appear to be modified by the -motion, but the <i>x</i> dimension is shortened in the ratio</p> - -<div class='figcenter id002'> -<img src='images/form-011.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>the shortening is the larger, the larger -is <i>v</i>. For <i>v</i> = <i>c</i>, all moving bodies, when considered from -a stationary system shrink into planes. For a velocity -larger than the velocity of light, our propositions become -meaningless; in our theory <i>c</i> plays the part of infinite -velocity.</p> - -<p class='c005'>It is clear that similar results hold about stationary -bodies in a stationary system when considered from a -uniformly moving system.</p> - -<p class='c005'>Let us now consider that a clock which is lying at rest -in the stationary system gives the time <i>t</i>, and lying -at rest relative to the moving system is capable of giving -the time τ; suppose it to be placed at the origin of the -moving system <i>k</i>, and to be so arranged that it gives the -time τ. How much does the clock gain, when viewed from -the stationary system K? We have,</p> - -<div class='figcenter id004'> -<img src='images/form-012.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Therefore the clock loses by an amount ½(<i>v²</i>/<i>c²</i>) per second -of motion, to the second order of approximation.</p> - -<p class='c005'>From this, the following peculiar consequence follows. -Suppose at two points A and B of the stationary system -two clocks are given which are synchronous in the sense -explained in § 3 when viewed from the stationary system. -Suppose the clock at A to be set in motion in the line -joining it with B, then after the arrival of the clock at B, -they will no longer be found synchronous, but the clock -which was set in motion from A will lag behind the clock -which had been all along at B by an amount ½<i>t</i>(<i>v²</i>/<i>c²</i>), where -<i>t</i> is the time required for the journey.</p> - -<p class='c005'>We see forthwith that the result holds also when the -clock moves from A to B by a polygonal line, and also -when A and B coincide.</p> - -<p class='c005'>If we assume that the result obtained for a polygonal -line holds also for a curved line, we obtain the following -law. If at A, there be two synchronous clocks, and if we -set in motion one of them with a constant velocity along a -closed curve till it comes back to A, the journey being -completed in <i>t</i>-seconds, then after arrival, the last mentioned -clock will be behind the stationary one by ½<i>t</i>(<i>v²</i>/<i>c²</i>) -seconds. From this, we conclude that a clock placed at -the equator must be slower by a very small amount than a -similarly constructed clock which is placed at the pole, all -other conditions being identical.</p> -<h4 class='c009'>§ 5. Addition-Theorem of Velocities.</h4> -<p class='c004'>Let a point move in the system <i>k</i> (which moves with -velocity <i>v</i> along the <i>x</i>-axis of the system K) according to -the equation</p> - -<div class='figcenter id002'> -<img src='images/form-013.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>where <i>w</i><sub>ξ</sub> and <i>w</i><sub>η</sub> are constants.</p> - -<p class='c005'>It is required to find out the motion of the point -relative to the system K. If we now introduce the system -of equations in § 3 in the equation of motion of the point, -we obtain</p> - -<div class='figcenter id003'> -<img src='images/form-014.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>The law of parallelogram of velocities hold up to the -first order of approximation. We can put</p> - -<div class='figcenter id004'> -<img src='images/form-015.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>and</p> - -<div class='figcenter id002'> -<img src='images/form-016.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'><i>i.e.</i>, α is put equal to the angle between the velocities <i>v</i>, -and <i>w</i>. Then we have—</p> - -<div class='figcenter id004'> -<img src='images/form-017.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>It should be noticed that <i>v</i> and <i>w</i> enter into the -expression for velocity symmetrically. If <i>w</i> has the direction -of the ξ-axis of the moving system,</p> - -<div class='figcenter id005'> -<img src='images/form-018.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>From this equation, we see that by combining two -velocities, each of which is smaller than <i>c</i>, we obtain a -velocity which is always smaller than <i>c</i>. If we put <i>v</i> = <i>c</i> - χ, -and <i>w</i> = <i>c</i> - λ, where χ and λ are each smaller than <i>c</i>,<a id='r8' /><a href='#f8' class='c007'><sup>[8]</sup></a></p> - -<div class='figcenter id003'> -<img src='images/form-019.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>It is also clear that the velocity of light <i>c</i> cannot be -altered by adding to it a velocity smaller than <i>c</i>. For this -case,</p> - -<div class='figcenter id002'> -<img src='images/form-020.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>We have obtained the formula for U for the case when -<i>v</i> and <i>w</i> have the same direction; it can also be obtained -by combining two transformations according to section -<a href='#sect2-3'>§ 3</a>. If in addition to the systems K, and k, we introduce -the system k´, of which the initial point moves -parallel to the ξ-axis with velocity <i>w</i>, then between the -magnitudes, <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> and the corresponding magnitudes -of k´, we obtain a system of equations, which differ from -the equations in <a href='#sect2-3'>§ 3</a>, only in the respect that in place of -<i>v</i>, we shall have to write,</p> - -<div class='figcenter id002'> -<img src='images/form-021.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>We see that such a parallel transformation forms a -group.</p> - -<p class='c005'>We have deduced the kinematics corresponding to our -two fundamental principles for the laws necessary for us, -and we shall now pass over to their application in electrodynamics.</p> -<h3 class='c008'>II.—ELECTRODYNAMICAL PART.</h3> -<h4 id='sect2-6' class='c009'>§ 6. Transformation of Maxwell’s equations for Pure Vacuum.</h4> -<h5 class='c009'>On the nature of the Electromotive Force caused by motion in a magnetic field.</h5> -<p class='c004'>The Maxwell-Hertz equations for pure vacuum may -hold for the stationary system K, so that</p> - -<div class='figcenter id003'> -<img src='images/form-022.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>and</p> - -<p class='c005'><a id='eqn-2-1'></a></p> -<div class='figcenter id003'> -<img src='images/form-023.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"Equation 1."</p> -</div> -</div> - -<p class='c005'>where [X, Y, Z] are the components of the electric -force, L, M, N are the components of the magnetic force.</p> - -<p class='c005'>If we apply the transformations in <a href='#sect2-3'>§3</a> to these equations, -and if we refer the electromagnetic processes to the -co-ordinate system moving with velocity <i>v</i>, we obtain,</p> - -<div class='figcenter id008'> -<img src='images/form-024.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>and</p> - -<p class='c005'><a id='eqn-2-2'></a></p> -<div class='figcenter id008'> -<img src='images/form-025.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"Equation 2."</p> -</div> -</div> - -<p class='c005'>where</p> - -<div class='figcenter id002'> -<img src='images/form-026.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>The principle of Relativity requires that the Maxwell-Hertzian -equations for pure vacuum shall hold also for the -system k, if they hold for the system K, <i>i.e.</i>, for the -vectors of the electric and magnetic forces acting upon -electric and magnetic masses in the moving system k, -which are defined by their pondermotive reaction, the same -equations hold, ... <i>i.e.</i> ...</p> - -<p class='c005'><a id='eqn-2-3'></a></p> -<div class='figcenter id008'> -<img src='images/form-027.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>" Equation 3."</p> -</div> -</div> - -<p class='c005'>Clearly both the systems of equations (<a href='#eqn-2-2'>2</a>) and (<a href='#eqn-2-3'>3</a>) -developed for the system k shall express the same things, -for both of these systems are equivalent to the Maxwell-Hertzian -equations for the system K. Since both the -systems of equations (<a href='#eqn-2-2'>2</a>) and (<a href='#eqn-2-3'>3</a>) agree up to the symbols -representing the vectors, it follows that the functions -occurring at corresponding places will agree up to a certain -factor ψ(<i>v</i>), which depends only on <i>v</i>, and is independent of -(ξ, η, ζ, τ). Hence the relations,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in28'><i>v</i> <i>v</i></div> - <div class='line'>[X′, Y′, Z′] = ψ (<i>v</i>) [X, β(Y - ----- N), β(Z + ------ M)],</div> - <div class='line in28'><i>c</i> <i>c</i></div> - </div> - <div class='group'> - <div class='line in28'><i>v</i> <i>v</i></div> - <div class='line'>[L′, M′, N′] = ψ (<i>v</i>) [L, β(M - ----- Z), β(N + ----- Y)],</div> - <div class='line in28'><i>c</i> <i>c</i></div> - </div> - </div> -</div> - -<p class='c010'>Then by reasoning similar to that followed in <a href='#sect2-3'>§(3)</a>, -it can be shown that ψ(<i>v</i>) = 1.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in22'><i>v</i> <i>v</i></div> - <div class='line'>[X′, Y′, Z′] = [X, β(Y - ---- N), β(Z + ----- M)]</div> - <div class='line in22'><i>c</i> <i>c</i></div> - </div> - <div class='group'> - <div class='line in23'><i>v</i> <i>v</i></div> - <div class='line'>[L′, M′, N′] = [L, β(M - ---- Z), β(N + ----- Y)],</div> - <div class='line in23'><i>c</i> <i>c</i></div> - </div> - </div> -</div> - -<p class='c010'>For the interpretation of these equations, we make the -following remarks. Let us have a point-mass of electricity -which is of magnitude unity in the stationary system K, -<i>i.e.</i>, it exerts a unit force upon a similar quantity placed at -a distance of 1 cm. If this quantity of electricity be at -rest in the stationary system, then the force acting upon it -is equivalent to the vector (X, Y, Z) of electric force. But -if the quantity of electricity be at rest relative to the -moving system (at least for the moment considered), then -the force acting upon it, and measured in the moving -system is equivalent to the vector (X′, Y′, Z′). The first -three of equations (<a href='#eqn-2-1'>1</a>), (<a href='#eqn-2-2'>2</a>), (<a href='#eqn-2-3'>3</a>), can be expressed in the -following way:—</p> - -<p class='c005'>1. If a point-mass of electric unit pole moves in an -electro-magnetic field, then besides the electric force, an -electromotive force acts upon it, which, neglecting the -numbers involving the second and higher powers of <i>v</i>/<i>c</i>, -is equivalent to the vector-product of the velocity vector, -and the magnetic force divided by the velocity of light -(Old mode of expression).</p> - -<p class='c005'>2. If a point-mass of electric unit pole moves in -an electro-magnetic field, then the force acting upon it is -equivalent to the electric force existing at the position of -the unit pole, which we obtain by the transformation of -the field to a co-ordinate system which is at rest relative -to the electric unit pole [New mode of expression].</p> - -<p class='c005'>Similar theorems hold with reference to the magnetic -force. We see that in the theory developed the electro-magnetic -force plays the part of an auxiliary concept, -which owes its introduction in theory to the circumstance -that the electric and magnetic forces possess no existence -independent of the nature of motion of the co-ordinate -system.</p> - -<p class='c005'>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 -disappears. Also the question about the seat of -electromagnetic energy is seen to be without any meaning.</p> -<h4 class='c009'>§ 7. Theory of Döppler’s Principle and Aberration.</h4> -<p class='c004'>In the system K, at a great distance from the origin of -co-ordinates, let there be a source of electrodynamic waves, -which is represented with sufficient approximation in a part -of space not containing the origin, by the equations:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>X = X₀ sin Φ</div> - <div class='line'>Y = Y₀ sin Φ</div> - <div class='line'>Z = Z₀ sin Φ</div> - <div class='line'>L = L₀ sin Φ</div> - <div class='line'>M = M₀ sin Φ</div> - <div class='line'>N = N₀ sin Φ</div> - <div class='line in11'>lx + my + nz</div> - <div class='line'>Φ = ω(t - --------------------- )</div> - <div class='line in15'>c</div> - </div> - </div> -</div> - -<p class='c010'>Here (X₀, Y₀, Z₀) and (L₀, M₀, N₀) are the vectors -which determine the amplitudes of the train of waves, -(<i>l</i>, <i>m</i>, <i>n</i>) are the direction-cosines of the wave-normal.</p> - -<p class='c005'>Let us now ask ourselves about the composition of -these waves, when they are investigated by an observer at -rest in a moving medium <i>k</i>:—By applying the equations of -transformation obtained in <a href='#sect2-6'>§6</a> for the electric and magnetic -forces, and the equations of transformation obtained in <a href='#sect2-3'>§ 3</a> -for the co-ordinates, and time, we obtain immediately:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>X′ = X₀ sin Φ′</div> - </div> - <div class='group'> - <div class='line in11'>v</div> - <div class='line'>Y′ = β(Y₀ - --- N₀) sin Φ′</div> - <div class='line in11'>c</div> - </div> - <div class='group'> - <div class='line in11'>v</div> - <div class='line'>Z′ = β(Z₀ - --- M₀) sin Φ′</div> - <div class='line in11'>c</div> - </div> - <div class='group'> - <div class='line'>L′ = L₀ sin Φ′</div> - </div> - <div class='group'> - <div class='line in12'>v</div> - <div class='line'>M′ = β(M₀ - --- Z₀) sin Φ′</div> - <div class='line in12'>c</div> - </div> - <div class='group'> - <div class='line in12'>v</div> - <div class='line'>N′ = β(N₀ - --- Y₀) sin Φ′</div> - <div class='line in12'>c</div> - </div> - <div class='group'> - <div class='line in11'>l′ξ + m′η + n′ζ</div> - <div class='line'>Φ′ = ω′(t - ---------------------- )</div> - <div class='line in16'>c</div> - </div> - </div> -</div> - -<p class='c010'>where</p> - -<div class='figcenter id008'> -<img src='images/form-028.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>From the equation for ω′ it follows:—If an observer moves -with the velocity <i>v</i> relative to an infinitely distant source -of light emitting waves of frequency ν, in such a manner -that the line joining the source of light and the observer -makes an angle of Φ with the velocity of the observer -referred to a system of co-ordinates which is stationary -with regard to the source, then the frequency ν′ which -is perceived by the observer is represented by the formula</p> - -<div class='figcenter id005'> -<img src='images/form-029.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>This is Döppler’s principle for any velocity. If Φ = 0, -then the equation takes the simple form</p> - -<div class='figcenter id005'> -<img src='images/form-030.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>We see that—contrary to the usual conception—ν = ∞, -for <i>v</i> = -<i>c</i>.</p> - -<p class='c005'>If Φ′ = angle between the wave-normal (direction of the -ray) in the moving system, and the line of motion of the -observer, the equation for <i>l´</i> takes the form</p> - -<div class='figcenter id002'> -<img src='images/form-031.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>This equation expresses the law of observation in its -most general form. If Φ = π/2, the equation takes the -simple form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in9'>v</div> - <div class='line'>cos Φ′ = ---</div> - <div class='line in9'>c</div> - </div> - </div> -</div> - -<p class='c010'>We have still to investigate the amplitude of the -waves, which occur in these equations. If A and A′ be -the amplitudes in the stationary and the moving systems -(either electrical or magnetic), we have</p> - -<div class='figcenter id002'> -<img src='images/form-032.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If Φ = 0, this reduces to the simple form</p> - -<div class='figcenter id002'> -<img src='images/form-033.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>From these equations, it appears that for an observer, -which moves with the velocity c towards the source of -light, the source should appear infinitely intense.</p> -<h4 class='c009'>§ 8. Transformation of the Energy of the Rays of Light. Theory of the Radiation-pressure on a perfect mirror.</h4> -<p class='c004'>Since A²/8π is equal to the energy of light per unit -volume, we have to regard A²/8π as the energy of light in -the moving system. A′²/A² would therefore denote the -ratio between the energies of a definite light-complex -“measured when moving” and “measured when stationary,” -the volumes of the light-complex measured in K and <i>k</i> -being equal. Yet this is not the case. If <i>l</i>, <i>m</i>, <i>n</i> are the -direction-cosines of the wave-normal of light in the -stationary system, then no energy passes through the -surface elements of the spherical surface</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(<i>x</i> - <i>clt</i>)² + (<i>y</i> - <i>cmt</i>)² + (<i>z</i> - <i>cnt</i>)² = R²,</div> - </div> - </div> -</div> - -<p class='c010'>which expands with the velocity of light. We can therefore -say, that this surface always encloses the same light-complex. -Let us now consider the quantity of energy, which this -surface encloses, when regarded from the system <i>k</i>, <i>i.e.</i>, -the energy of the light-complex relative to the system -<i>k</i>.</p> - -<p class='c005'>Regarded from the moving system, the spherical -surface becomes an ellipsoidal surface, having, at the time -τ = 0, the equation:—</p> - -<div class='figcenter id008'> -<img src='images/form-034.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If S = volume of the sphere, S′ = volume of this -ellipsoid, then a simple calculation shows that:</p> - -<div class='figcenter id002'> -<img src='images/form-035.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If E denotes the quantity of light energy measured in -the stationary system, E′ the quantity measured in the -moving system, which are enclosed by the surfaces -mentioned above, then</p> - -<div class='figcenter id003'> -<img src='images/form-036.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If Φ = 0, we have the simple formula:—</p> - -<div class='figcenter id005'> -<img src='images/form-037.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>It is to be noticed that the energy and the frequency -of a light-complex vary according to the same law with -the state of motion of the observer.</p> - -<p class='c005'>Let there be a perfectly reflecting mirror at the co-ordinate-plane -ξ = 0, from which the plane-wave considered -in the last paragraph is reflected. Let us now ask ourselves -about the light-pressure exerted on the reflecting surface -and the direction, frequency, intensity of the light after -reflexion.</p> - -<p class='c005'>Let the incident light be defined by the magnitudes -A cos Φ, <i>v</i> (referred to the system K). Regarded from <i>k</i>, -we have the corresponding magnitudes:</p> - -<div class='figcenter id002'> -<img src='images/form-038.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>For the reflected light we obtain, when the process -is referred to the system <i>k</i>:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A″ = A′, cos Φ″ = -cos Φ″, ν″ = ν′</div> - </div> - </div> -</div> - -<p class='c010'>By means of a back-transformation to the stationary -system, we obtain K, for the reflected light:—</p> - -<div class='figcenter id008'> -<img src='images/form-039.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>The amount or energy falling upon the unit surface -of the mirror per unit of time (measured in the stationary -system) is A²/(8π (c cos Φ - <i>v</i>)). The amount of energy going -away from unit surface of the mirror per unit of time is -A‴²/(8π (-c cos Φ″ + <i>v</i>)). The difference of these two -expressions is, according to the Energy principle, the -amount of work exerted, by the pressure of light per unit -of time. If we put this equal to P.<i>v</i>, where P = pressure -of light, we have</p> - -<div class='figcenter id003'> -<img src='images/form-040.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>As a first approximation, we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in7'>A²</div> - <div class='line'>P = 2 ---- cos² Φ</div> - <div class='line in6'>8π</div> - </div> - </div> -</div> - -<p class='c010'>which is in accordance with facts, and with other -theories.</p> - -<p class='c005'>All problems of optics of moving bodies can be solved -after the method used here. The essential point is, that -the electric and magnetic forces of light, which are -influenced by a moving body, should be transformed to a -system of co-ordinates which is stationary relative to the -body. In this way, every problem of the optics of moving -bodies would be reduced to a series of problems of the -optics of stationary bodies.</p> -<h4 class='c009'>§ 9. Transformation of the Maxwell-Hertz Equations.</h4> -<p class='c004'>Let us start from the equations:—</p> - -<div class='figcenter id008'> -<img src='images/form-041.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>where</p> - -<div class='figcenter id002'> -<img src='images/form-042.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>denotes 4π times the density -of electricity, and (<i>u</i><sub><i>x</i></sub>, <i>u</i><sub><i>y</i></sub>, <i>u</i><sub><i>z</i></sub>) are the velocity-components -of electricity. If we now suppose that the electrical-masses -are bound unchangeably to small, rigid bodies -(Ions, electrons), then these equations form the electromagnetic -basis of Lorentz’s electrodynamics and optics for -moving bodies.</p> - -<p class='c005'>If these equations which hold in the system K, are -transformed to the system <i>k</i> with the aid of the transformation-equations -given in <a href='#sect2-3'>§ 3</a> and <a href='#sect2-6'>§ 6</a>, then we obtain -the equations:—</p> - -<div class='figcenter id008'> -<img src='images/form-043.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>where</p> - -<div class='figcenter id008'> -<img src='images/form-044.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Since the vector (<i>u</i><sub>ξ</sub>, <i>u</i><sub>η</sub>, <i>u</i><sub>ζ</sub>) is nothing but the -velocity of the electrical mass measured in the system <i>k</i>, -as can be easily seen from the addition-theorem of -velocities in <a href='#sect2-4'>§ 4</a>—so it is hereby shown, that by taking -our kinematical principle as the basis, the electromagnetic -basis of Lorentz’s theory of electrodynamics of moving -bodies correspond to the relativity-postulate. It can be -briefly remarked here that the following important law -follows easily from the equations developed in the present -section:—if an electrically charged body moves in any -manner in space, and if its charge does not change thereby, -when regarded from a system moving along with it, then -the charge remains constant even when it is regarded from -the stationary system K.</p> -<h4 class='c009'>§ 10. Dynamics of the Electron (slowly accelerated).</h4> -<p class='c004'>Let us suppose that a point-shaped particle, having -the electrical charge <i>e</i> (to be called henceforth the electron) -moves in the electromagnetic field; we assume the -following about its law of motion.</p> - -<p class='c005'>If the electron be at rest at any definite epoch, then -in the next “<i>particle of time</i>,” the motion takes place -according to the equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in3'><i>d²x</i> <i>d²y</i> <i>d²z</i></div> - <div class='line'><i>m</i> ----- = <i>e</i>X, <i>m</i> ----- = <i>e</i>Y, <i>m</i> ----- = <i>e</i>Z</div> - <div class='line in3'><i>dt²</i> <i>dt²</i> <i>dt²</i></div> - </div> - </div> -</div> - -<p class='c010'>Where (<i>x</i>, <i>y</i>, <i>z</i>) are the co-ordinates of the electron, and -<i>m</i> is its mass.</p> - -<p class='c005'>Let the electron possess the velocity <i>v</i> at a certain -epoch of time. Let us now investigate the laws according -to which the electron will move in the ‘particle of time’ -immediately following this epoch.</p> - -<p class='c005'>Without influencing the generality of treatment, we can -and we will assume that, at the moment we are considering, -the electron is at the origin of co-ordinates, and moves -with the velocity <i>v</i> along the X-axis of the system. It is -clear that at this moment (<i>t</i> = 0) the electron is at rest -relative to the system <i>k</i>, which moves parallel to the X-axis -with the constant velocity <i>v</i>.</p> - -<p class='c005'>From the suppositions made above, in combination -with the principle of relativity, it is clear that regarded -from the system <i>k</i>, the electron moves according to the -equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in3'><i>d²</i>ξ <i>d²</i>η <i>d²</i>ζ</div> - <div class='line'><i>m</i> ----- = <i>e</i>X′, <i>m</i> ----- = <i>e</i>Y′, <i>m</i> ----- = <i>e</i>Z′ ,</div> - <div class='line in3'><i>d</i>τ² <i>d</i>τ² <i>d</i>τ²</div> - </div> - </div> -</div> - -<p class='c010'>in the time immediately following the moment, where the -symbols (ξ, η, ζ, τ, X’, Y’, Z’) refer to the system <i>k</i>. If we -now fix, that for <i>t</i> = <i>v</i> = <i>y</i> = <i>z</i> = 0, τ = ξ = η = ζ = 0, then the -equations of transformation given in <a href='#sect2-3'>§ 3</a> (and <a href='#sect2-6'>§ 6</a>) hold, and we -have:</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in9'><i>v</i></div> - <div class='line'>τ = β(<i>t</i> - ---- <i>x</i>), ξ = β(<i>x</i> - <i>vt</i>), η = <i>y</i>, ζ = <i>z</i>,</div> - <div class='line in9'><i>c²</i></div> - </div> - <div class='group'> - <div class='line in18'><i>v</i> <i>v</i></div> - <div class='line'>X′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M)</div> - <div class='line in18'><i>c</i> <i>c</i></div> - </div> - </div> -</div> - -<p class='c010'>With the aid of these equations, we can transform the -above equations of motion from the system <i>k</i> to the system -K, and obtain:—</p> - -<p class='c005'><a id='eqn-2-A'></a></p> -<div class='figcenter id008'> -<img src='images/form-045.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Let us now consider, following the usual method of -treatment, the longitudinal and transversal mass of a -moving electron. We write the equations (<a href='#eqn-2-A'>A</a>) in the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'><i>d²x</i></div> - <div class='line'><i>m</i>β² ----- = <i>e</i>X = <i>e</i>X′</div> - <div class='line in5'><i>dt²</i></div> - </div> - <div class='group'> - <div class='line in5'><i>d²y</i> <i>v</i></div> - <div class='line'><i>m</i>β² ----- = <i>e</i>β (Y - --- N) = <i>e</i>Y′</div> - <div class='line in5'><i>dt²</i> <i>c</i></div> - </div> - <div class='group'> - <div class='line in5'><i>d²z</i> <i>v</i></div> - <div class='line'><i>m</i>β² ----- = <i>e</i>β (Z - --- M) = <i>e</i>Z′</div> - <div class='line in5'><i>dt²</i> <i>c</i></div> - </div> - </div> -</div> - -<p class='c010'>and let us first remark, that <i>e</i>X′, <i>e</i>Y′, <i>e</i>Z′ are the components -of the ponderomotive force acting upon the -electron, and are considered in a moving system which, at -this moment, moves with a velocity which is equal to that -of the electron. This force can, for example, be measured -by means of a spring-balance which is at rest in this last -system. If we briefly call this force as “the force acting -upon the electron,” and maintain the equation:—</p> - -<p class='c005'>Mass-number × acceleration-number = force-number, and -if we further fix that the accelerations are measured in -the stationary system K, then from the above equations, -we obtain:—</p> - -<p class='c005'>Longitudinal mass:</p> - -<div class='figcenter id002'> -<img src='images/form-046.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Transversal mass:</p> - -<div class='figcenter id002'> -<img src='images/form-047.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Naturally, when other definitions are given of the force -and the acceleration, other numbers are obtained for the -mass; hence we see that we must proceed very carefully -in comparing the different theories of the motion of the -electron.</p> - -<p class='c005'>We remark that this result about the mass hold also -for ponderable material mass; for in our sense, a ponderable -material point may be made into an electron by the -addition of an electrical charge which may be as small as -possible.</p> - -<p class='c005'>Let us now determine the kinetic energy of the -electron. If the electron moves from the origin of co-ordinates -of the system K with the initial velocity 0 steadily -along the X-axis under the action of an electromotive -force X, then it is clear that the energy drawn from the -electrostatic field has the value ∫<i>e</i>X<i>dx</i>. Since the electron -is only slowly accelerated, and in consequence, no energy -is given out in the form of radiation, therefore the energy -drawn from the electro-static field may be put equal to -the energy W of motion. Considering the whole process of -motion in questions, the first of equations <a href='#eqn-2-A'>A)</a> holds, we -obtain:—</p> - -<div class='figcenter id008'> -<img src='images/form-048.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>For <i>v</i> = <i>c</i>, W is infinitely great. As our former result -shows, velocities exceeding that of light can have no -possibility of existence.</p> - -<p class='c005'>In consequence of the arguments mentioned above, -this expression for kinetic energy must also hold for -ponderable masses.</p> - -<p class='c005'>We can now enumerate the characteristics of the -motion of the electrons available for experimental verification, -which follow from equations <a href='#eqn-2-A'>A)</a>.</p> - -<p class='c005'>1. From the second of equations <a href='#eqn-2-A'>A)</a>, it follows that -an electrical force Y, and a magnetic force N produce -equal deflexions of an electron moving with the velocity -<i>v</i>, when Y = N<i>v</i>/<i>c</i>. Therefore we see that according to -our theory, it is possible to obtain the velocity of an -electron from the ratio of the magnetic deflexion A<sub><i>m</i></sub>, and -the electric deflexion A<sub><i>e</i></sub>, by applying the law:—</p> - -<div class='figcenter id005'> -<img src='images/form-049.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>This relation can be tested by means of experiments -because the velocity of the electron can be directly -measured by means of rapidly oscillating electric and -magnetic fields.</p> - -<p class='c005'>2. From the value which is deduced for the kinetic -energy of the electron, it follows that when the electron -falls through a potential difference of P, the velocity <i>v</i> -which is acquired is given by the following relation:—</p> - -<div class='figcenter id004'> -<img src='images/form-050.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>3. We calculate the radius of curvature R of the -path, where the only deflecting force is a magnetic force N -acting perpendicular to the velocity of projection. From -the second of equations <a href='#eqn-2-A'>A)</a> we obtain:</p> - -<div class='figcenter id004'> -<img src='images/form-051.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>or</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'><i>mv</i>β<i>c</i></div> - <div class='line'>R = --------</div> - <div class='line in7'><i>e</i>N</div> - </div> - </div> -</div> - -<p class='c010'>These three relations are complete expressions for the -law of motion of the electron according to the above -theory.</p> - -<div class='chapter'> - <h2 id='art3' class='c003'>ALBRECHT EINSTEIN <br /> [<i>A short biographical note.</i>]</h2> -</div> -<p class='c004'>The name of Prof. Albrecht Einstein has now spread far -beyond the narrow pale of scientific investigators owing to -the brilliant confirmation of his predicted deflection of -light-rays by the gravitational field of the sun during the -total solar eclipse of May 29, 1919. But to the serious -student of science, he has been known from the beginning -of the current century, and many dark problems in physics -has been illuminated with the lustre of his genius, before, -owing to the latest sensation just mentioned, he flashes out -before public imagination as a scientific star of the first -magnitude.</p> - -<p class='c005'>Einstein is a Swiss-German of Jewish extraction, and -began his scientific career as a privat-dozent in the Swiss -University of Zürich about the year 1902. Later on, he -migrated to the German University of Prague in Bohemia -as ausser-ordentliche (or associate) Professor. In 1914, -through the exertions of Prof. M. Planck of the Berlin -University, he was appointed a paid member of the Royal -(now National) Prussian Academy of Sciences, on a -salary of 18,000 marks per year. In this post, he has -only to do and guide research work. Another distinguished -occupant of the same post was Van’t Hoff, the eminent -physical chemist.</p> - -<p class='c005'>It is rather difficult to give a detailed, and consistent -chronological account of his scientific activities,—they are -so variegated, and cover such a wide field. The first work -which gained him distinction was an investigation on -Brownian Movement. An admirable account will be found -in Perrin’s book ‘The Atoms.’ Starting from Boltzmann’s -theorem connecting the entropy, and the probability of a -state, he deduced a formula on the mean displacement of -small particles (colloidal) suspended in a liquid. This -formula gives us one of the best methods for finding out a -very fundamental number in physics—namely—the number -of molecules in one gm. molecule of gas (Avogadro’s -number). The formula was shortly afterwards verified by -Perrin, Prof. of Chemical Physics in the Sorbonne, Paris.</p> - -<p class='c005'>To Einstein is also due the resuscitation of Planck’s -quantum theory of energy-emission. This theory has not -yet caught the popular imagination to the same extent as -the new theory of Time, and Space, but it is none the less -iconoclastic in its scope as far as classical concepts are -concerned. It was known for a long time that the -observed emission of light from a heated black body did -not correspond to the formula which could be deduced from -the older classical theories of continuous emission and -propagation. In the year 1900, Prof. Planck of the Berlin -University worked out a formula which was based on the -bold assumption that energy was emitted and absorbed by -the molecules in multiples of the quantity <i>h</i>ν, where <i>h</i> -is a constant (which is universal like the constant of -gravitation), and ν is the frequency of the light.</p> - -<p class='c005'>The conception was so radically different from all -accepted theories that in spite of the great success of -Planck’s radiation formula in explaining the observed facts -of black-body radiation, it did not meet with much favour -from the physicists. In fact, some one remarked jocularly -that according to Planck, energy flies out of a radiator like -a swarm of gnats.</p> - -<p class='c005'>But Einstein found a support for the new-born concept -in another direction. It was known that if green or ultraviolet -light was allowed to fall on a plate of some alkali metal, -the plate lost electrons. The electrons were emitted with -all velocities, but there is generally a maximum limit. -From the investigations of Lenard and Ladenburg, the -curious discovery was made that this maximum velocity of -emission did not at all depend upon the intensity of light, -but upon its wavelength. The more violet was the light, -the greater was the velocity of emission.</p> - -<p class='c005'>To account for this fact, Einstein made the bold -assumption that the light is propagated in space as a unit -pulse (he calls it a Light-cell), and falling upon an -individual atom, liberates electrons according to the energy -equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>1</div> - <div class='line'><i>h</i>ν = --- <i>mv²</i> + A,</div> - <div class='line in6'>2</div> - </div> - </div> -</div> - -<p class='c010'>where (<i>m</i>, <i>v</i>) are the mass and velocity of the electron. -A is a constant characteristic of the metal plate.</p> - -<p class='c005'>There was little material for the confirmation of this -law when it was first proposed (1905), and eleven years -elapsed before Prof. Millikan established, by a set of -experiments scarcely rivalled for the ingenuity, skill, and -care displayed, the absolute truth of the law. As results of -this confirmation, and other brilliant triumphs, the quantum -law is now regarded as a fundamental law of Energetics. -In recent years, X-rays have been added to the domain of -light, and in this direction also, Einstein’s photo-electric -formula has proved to be one of the most fruitful -conceptions in Physics.</p> - -<p class='c005'>The quantum law was next extended by Einstein to the -problems of decrease of specific heat at low temperature, -and here also his theory was confirmed in a brilliant -manner.</p> - -<p class='c005'>We pass over his other contributions to the equation of -state, to the problems of null-point energy, and photo-chemical -reactions. The recent experimental works of -Nernst and Warburg seem to indicate that through -Einstein’s genius, we are probably for the first time having -a satisfactory theory of photo-chemical action.</p> - -<p class='c005'>In 1915, Einstein made an excursion into Experimental -Physics, and here also, in his characteristic way, he tackled -one of the most fundamental concepts of Physics. It is -well-known that according to Ampere, the magnetisation -of iron and iron-like bodies, when placed within a coil -carrying an electric current is due to the excitation in the -metal of small electrical circuits. But the conception -though a very fruitful one, long remained without a trace -of experimental proof, though after the discovery of the -electron, it was generally believed that these molecular -currents may be due to the rotational motion of free -electrons within the metal. It is easily seen that if in the -process of magnetisation, a number of electrons be set into -rotatory motion, then these will impart to the metal itself -a turning couple. The experiment is a rather difficult one, -and many physicists tried in vain to observe the effect. -But in collaboration with de Haas, Einstein planned and -successfully carried out this experiment, and proved the -essential correctness of Ampere’s views.</p> - -<p class='c005'>Einstein’s studies on Relativity were commenced in the -year 1905, and has been continued up to the present time. -The first paper in the present collection forms Einstein’s -first great contribution to the Principle of Special -Relativity. We have recounted in the introduction how out -of the chaos and disorder into which the electrodynamics -and optics of moving bodies had fallen previous to 1895, -Lorentz, Einstein and Minkowski have succeeded in -building up a consistent, and fruitful new theory of Time -and Space.</p> - -<p class='c005'>But Einstein was not satisfied with the study of the -special problem of Relativity for uniform motion, but -tried, in a series of papers beginning from 1911, to extend -it to the case of non-uniform motion. The last paper in -the present collection is a translation of a comprehensive -article which he contributed to the Annalen der Physik in -1916 on this subject, and gives, in his own words, the -Principles of Generalized Relativity. The triumphs of -this theory are now matters of public knowledge.</p> - -<p class='c005'>Einstein is now only 45, and it is to be hoped that -science will continue to be enriched, for a long time to -come, with further achievements of his genius.</p> -<div class='chapter'> - <h2 id='art4' class='c003'>Principle of Relativity</h2> -</div> -<h3 class='c008'>INTRODUCTION.</h3> -<p class='c004'>At the present time, different opinions are being held -about the fundamental equations of Electro-dynamics for -moving bodies. The Hertzian<a id='r9' /><a href='#f9' class='c007'><sup>[9]</sup></a> forms must be given up, -for it has appeared that they are contrary to many experimental -results.</p> - -<p class='c005'>In 1895 H. A. Lorentz<a id='r10' /><a href='#f10' class='c007'><sup>[10]</sup></a> published his theory of optical -and electrical phenomena in moving bodies; this theory -was based upon the atomistic conception (vorstellung) of -electricity, and on account of its great success appears to -have justified the bold hypotheses, by which it has been -ushered into existence. In his theory, Lorentz proceeds -from certain equations, which must hold at every point of -“Äther”; then by forming the average values over “Physically -infinitely small” regions, which however contain -large numbers of electrons, the equations for electro-magnetic -processes in moving bodies can be successfully built -up.</p> - -<p class='c005'>In particular, Lorentz’s theory gives a good account of -the non-existence of relative motion of the earth and the -luminiferous “Äther”; it shows that this fact is intimately -connected with the covariance of the original equation, -when certain simultaneous transformations of the space and -time co-ordinates are effected; these transformations have -therefore obtained from H. Poincare<a id='r11' /><a href='#f11' class='c007'><sup>[11]</sup></a> the name of Lorentz-transformations. -The covariance of these fundamental -equations, when subjected to the Lorentz-transformation -is a purely mathematical fact <i>i.e.</i> not based on any physical -considerations; I will call this the Theorem of Relativity; -this theorem rests essentially on the form of the -differential equations for the propagation of waves with -the velocity of light.</p> - -<p class='c005'>Now without <i>recognizing</i> any hypothesis about the connection -between “Äther” and matter, 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-transformation; -by saying this, we do not indeed express an opinion, but -rather a conviction,—and this conviction I may be permitted -to call the Postulate of Relativity. The position of -affairs here is almost the same as when the Principle of -Conservation of Energy was postulated in cases, where the -corresponding forms of energy were unknown.</p> - -<p class='c005'>Now if hereafter, we succeed in maintaining this -covariance as a definite connection between pure and simple -observable phenomena in moving bodies, the definite connection -may be styled ‘the Principle of Relativity.’</p> - -<p class='c005'>These differentiations seem to me to be necessary for -enabling us to characterise the present day position of the -electro-dynamics for moving bodies.</p> - -<p class='c005'>H. A. Lorentz<a id='r12' /><a href='#f12' class='c007'><sup>[12]</sup></a> has found out the “Relativity theorem” -and has created the Relativity-postulate as a hypothesis -that electrons and matter suffer contractions in consequence -of their motion according to a certain law.</p> - -<p class='c005'>A. Einstein<a id='r13' /><a href='#f13' class='c007'><sup>[13]</sup></a> has brought out the point very clearly, -that this postulate is not an artificial hypothesis but is -rather a new way of comprehending the time-concept -which is forced upon us by observation of natural phenomena.</p> - -<p class='c005'>The Principle of Relativity has not yet been formulated -for electro-dynamics of moving bodies in the sense -characterized by me. In the present essay, while formulating -this principle, I shall obtain the fundamental equations -for moving bodies in a sense which is uniquely determined -by this principle.</p> - -<p class='c005'>But it will be shown that none of the forms hitherto -assumed for these equations can exactly fit in with this -principle.<a id='r14' /><a href='#f14' class='c007'><sup>[14]</sup></a></p> - -<p class='c005'>We would at first expect that the fundamental equations -which are assumed by Lorentz for moving bodies -would correspond to the Relativity Principle. But it will -be shown that this is not the case for the general equations -which Lorentz has for any possible, and also for magnetic -bodies; but this is approximately the case (if neglect the -square of the velocity of matter in comparison to the -velocity of light) for those equations which Lorentz hereafter -infers for non-magnetic bodies. But this latter -accordance with the Relativity Principle is due to the fact -that the condition of non-magnetisation has been formulated -in a way not corresponding to the Relativity Principle; -therefore the accordance is due to the fortuitous compensation -of two contradictions to the Relativity-Postulate. -But meanwhile enunciation of the Principle in a rigid -manner does not signify any contradiction to the hypotheses -of Lorentz’s molecular theory, but it shall become clear that -the assumption of the contraction of the electron in -Lorentz’s theory must be introduced at an earlier stage -than Lorentz has actually done.</p> - -<p class='c005'>In an appendix, I have gone into discussion of the -position of Classical Mechanics with respect to the -Relativity Postulate. Any easily perceivable modification -of mechanics for satisfying the requirements of the -Relativity theory would hardly afford any noticeable -difference in observable processes; but would lead to very -surprising consequences. By laying down the Relativity-Postulate -from the outset, sufficient means have been -created for deducing henceforth the complete series of -Laws of Mechanics from the principle of conservation of -Energy alone (the form of the Energy being given in -explicit forms).</p> -<h4 class='c009'>NOTATIONS.</h4> -<p class='c004'>Let a rectangular system (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>,) of reference be -given in space and time. The unit of time shall be chosen -in such a manner with reference to the unit of length that -the velocity of light in space becomes unity.</p> - -<p class='c005'>Although I would prefer not to change the notations -used by Lorentz, it appears important to me to use a -different selection of symbols, for thereby certain homogeneity -will appear from the very beginning. I shall -denote the vector electric force by E, the magnetic -induction by M, the electric induction by <i>e</i> and the -magnetic force by <i>m</i>, so that (E, M, <i>e</i>, <i>m</i>) are used instead -of Lorentz’s (E, B, D, H) respectively.</p> - -<p class='c005'>I shall further make use of complex magnitudes in a -way which is not yet current in physical investigations, -<i>i.e.</i>, instead of operating with (<i>t</i>), I shall operate with (<i>i t</i>), -where <i>i</i> denotes √(-1). If now instead of (<i>x</i>, <i>y</i>, <i>z</i>, <i>i t</i>), I -use the method of writing with indices, certain essential -circumstances will come into evidence; on this will be -based a general use of the suffixes (1, 2, 3, 4). The -advantage of this method will be, as I expressly emphasize -here, that we shall have to handle symbols which have -apparently a purely real appearance; we can however at -any moment pass to real equations if it is understood that -of the symbols with indices, such ones as have the suffix -4 only once, denote imaginary quantities, while those -which have not at all the suffix 4, or have it twice denote -real quantities.</p> - -<p class='c005'>An individual system of values of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) <i>i. e.</i>, of -(<i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i>) shall be called a space-time point.</p> - -<p class='c005'>Further let <i>u</i> denote the velocity vector of matter, ε the -dielectric constant, μ the magnetic permeability, σ the -conductivity of matter, while ρ denotes the density of -electricity in space, and <i>x</i> the vector of “Electric Current” -which we shall some across in <a href='#sect4-7'>§7</a> and <a href='#sect4-8'>§8</a>.</p> -<h3 id='sect4-2' class='c001'>PART I <br /> § 2. <br /> The Limiting Case. <br /> The Fundamental Equations for Äther.</h3> -<p class='c004'>By using the electron theory, Lorentz in his above -mentioned essay traces the Laws of Electro-dynamics of -Ponderable Bodies to still simpler laws. Let us now adhere -to these simpler laws, whereby we require that for the -limiting case ε = 1, μ = 1, σ = 0, they should constitute the -laws for ponderable bodies. In this ideal limiting case -ε = 1, μ = 1, σ = 0, E will be equal to <i>e</i>, and M to <i>m</i>. At -every space time point (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) we shall have the -equations<a id='r15' /><a href='#f15' class='c007'><sup>[15]</sup></a></p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(i) Curl <i>m</i> - (δ<i>e</i>/δ<i>t</i>) = ρu</div> - </div> - <div class='group'> - <div class='line'>(ii) div <i>e</i> = ρ</div> - </div> - <div class='group'> - <div class='line'>(iii) Curl <i>e</i> + δ<i>m</i>/δ<i>t</i> = 0</div> - </div> - <div class='group'> - <div class='line'>(iv) div m = 0</div> - </div> - </div> -</div> - -<p class='c010'>I shall now write (<i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i>) for (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) and -(ρ₁, ρ₂, ρ₃, ρ₄) for</p> - -<div class='figcenter id003'> -<img src='images/form-052.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'><i>i.e.</i> the components of the convection current ρu, and the -electric density multiplied by √ -1</p> - -<p class='c005'>Further I shall write</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i><sub>2 3</sub>, <i>f</i><sub>3 1</sub>, <i>f</i><sub>1 2</sub>, <i>f</i><sub>1 4</sub>, <i>f</i><sub>2 4</sub>, <i>f</i><sub>3 4</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>for</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>m<sub><i>x</i></sub>, m<sub><i>y</i></sub>, m<sub><i>z</i></sub>, -ie<sub><i>x</i></sub>, -ie<sub><i>y</i></sub>, -ie<sub><i>z</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i>, the components of m and (-<i>i.e.</i>) along the three axes; -now if we take any two indices (h. k) out of the series</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>3, 4), <i>f</i><sub><i>k h</i></sub> = -<i>f</i><sub><i>k h</i></sub>,</div> - </div> - </div> -</div> - -<p class='c010'>Therefore</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f₃₂</i> = -<i>f₂₃</i>, <i>f₁₃</i> = -<i>f₃₁</i>, <i>f₂₁</i> = -<i>f₁₂</i></div> - <div class='line'><i>f₄₁</i> = -<i>f₁₄</i>, <i>f₄₄</i> = -<i>f₂₄</i>, <i>f₄₃</i> = -<i>f₃₄</i></div> - </div> - </div> -</div> - -<p class='c010'>Then the three equations comprised in (i), and the -equation (ii) multiplied by i becomes</p> - -<p class='c005'><a id='eqn-4-A'></a></p> -<div class='figcenter id004'> -<img src='images/form-053.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"Formula A."</p> -</div> -</div> - -<p class='c005'>On the other hand, the three equations comprised in (iii) -and the (iv) equation multiplied by (<i>i</i>) becomes</p> - -<p class='c005'><a id='eqn-4-B'></a></p> -<div class='figcenter id004'> -<img src='images/form-054.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"Formula B."</p> -</div> -</div> - -<p class='c005'>By means of this method of writing we at once notice -the perfect symmetry of the 1st as well as the 2nd system -of equations as regards permutation with the indices, -(1, 2, 3, 4).</p> -<h3 class='c008'>§ 3.</h3> -<p class='c004'>It is well-known that by writing the equations i) to -iv) in the symbol of vector calculus, we at once set in -evidence an invariance (or rather a (covariance) of the -system of equations <a href='#eqn-4-A'>A)</a> as well as of <a href='#eqn-4-B'>B)</a>, when the co-ordinate -system is rotated through a certain amount round the -null-point. For example, if we take a rotation of the -axes round the z-axis, through an amount φ, keeping -e, m fixed in space, and introduce new variables <i>x₁′</i> <i>x₂′</i> <i>x₃′</i> -<i>x₄′</i> instead of <i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i> where -<i>x′₁</i> = <i>x₁</i> cos φ + <i>x₂</i> sin φ, <i>x′₂</i> = -<i>x₁</i> sin φ + <i>x₂</i> cos φ, -<i>x′₃</i> = <i>x₃</i>, <i>x′₄</i> = <i>x₄</i>, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, -where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ -and <i>f′</i><sub>1 2</sub>, ... ... <i>f′</i><sub>3 4</sub>, where</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f′₂₃</i> = <i>f₂₃</i> cos φ + <i>f₃₁</i> sin φ,</div> - <div class='line'><i>f′₃₁</i> = - <i>f₂₃</i> sin φ + <i>f₃₁</i> cos φ,</div> - <div class='line'><i>f′₁₂</i> = <i>f₁₂</i>,</div> - <div class='line'><i>f′₁₄</i> = <i>f₁₄</i> cos φ + <i>f₂₄</i> sin φ,</div> - <div class='line'><i>f′₂₄</i> = - <i>f₁₄</i> sin φ + <i>f₂₄</i> cos φ,</div> - <div class='line'><i>f′₃₄</i> = <i>f₃₄</i><sub>3 4</sub>,</div> - <div class='line'><i>f′</i><sub><i>k h</i></sub> = - <i>f</i><sub><i>k h</i></sub> (h l k = 1, 2, 3, 4).</div> - </div> - </div> -</div> - -<p class='c010'>then out of the equations (A) would follow a corresponding -system of dashed equations (A´) composed of the -newly introduced dashed magnitudes.</p> - -<p class='c005'>So upon the ground of symmetry alone of the equations -(A) and (B) concerning the <i>suffixes</i> (1, 2, 3, 4), the -theorem of Relativity, which was found out by Lorentz, -follows without any calculation at all.</p> - -<p class='c005'>I will denote by <i>i</i>ψ a purely imaginary magnitude, -and consider the substitution</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x₁′</i> = <i>x₁</i>,</div> - <div class='line'><i>x₂′</i> = <i>x₂</i>,</div> - <div class='line'><i>x₃′</i> = <i>x₃</i> cos <i>i</i>ψ + <i>x₄</i> sin <i>i</i>ψ, (1)</div> - <div class='line'><i>x₄′</i>´ = - <i>x₃</i> sin <i>i</i>ψ + <i>x₄</i> cos <i>i</i>ψ,</div> - </div> - </div> -</div> - -<p class='c010'>Putting</p> - -<div class='figcenter id004'> -<img src='images/form-055.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(2)."</p> -</div> -</div> - -<p class='c005'>We shall have cos <i>i</i>ψ = 1/√(1 - <i>q²</i>), sin <i>i</i>ψ = <i>iq</i>/√(1 - <i>q²</i>) -where -1 < <i>q</i> < 1, and √(1 - <i>q²</i>) is always to be taken -with the positive sign.</p> - -<p class='c005'>Let us now write <i>x′₁</i> = <i>x′</i>, <i>x′₂</i> = <i>y′</i>, <i>x′₃</i> = <i>z′</i>, <i>x′₄</i> = <i>it′</i> (3)</p> - -<p class='c005'>then the substitution 1) takes the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x′</i> = <i>x</i>, <i>y′</i> = <i>y</i>, <i>z′</i> = (<i>z</i> - <i>qt</i>)/√(1 - <i>q²</i>), <i>t′</i> = (-<i>qz</i> + <i>t</i>)/√(1 - <i>q²</i>), (4)</div> - </div> - </div> -</div> - -<p class='c010'>the coefficients being essentially real.</p> - -<p class='c005'>If now in the above-mentioned rotation round the -Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and -φ by <i>i</i>ψ, we at once perceive that simultaneously, new -magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos <i>i</i>ψ + ρ₄ sin <i>i</i>ψ,</div> - <div class='line'>ρ′₄ = - ρ₃ sin <i>i</i>ψ + ρ₄ cos <i>i</i>ψ),</div> - </div> - </div> -</div> - -<p class='c010'>and <i>f′</i><sub>1 2</sub> ... <i>f′</i><sub>3 4</sub>, where</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f′</i><sub>4 1</sub> = <i>f</i><sub>4 1</sub> cos <i>i</i>ψ + <i>f</i><sub>1 3</sub> sin <i>i</i>ψ,</div> - <div class='line'><i>f′</i><sub>1 3</sub> = - <i>f</i><sub>4 1</sub> sin <i>i</i>ψ + <i>f</i><sub>1 3</sub> cos <i>i</i>ψ,</div> - <div class='line'><i>f′</i><sub>3 4</sub> = <i>f</i><sub>3 4</sub>,</div> - <div class='line'><i>f′</i><sub>3 2</sub> = <i>f</i><sub>3 2</sub> cos <i>i</i>ψ + <i>f</i><sub>4 2</sub> sin <i>i</i>ψ,</div> - <div class='line'><i>f′</i><sub>4 2</sub> = - <i>f</i><sub>3 2</sub> sin <i>i</i>ψ + <i>f</i><sub>4 2</sub> cos <i>i</i>ψ,</div> - <div class='line'><i>f′</i><sub>1 2</sub> = <i>f</i><sub>1 2</sub>, <i>f</i><sub><i>k h</i></sub> = - <i>f′</i><sub><i>k h</i></sub>,</div> - </div> - </div> -</div> - -<p class='c010'>must be introduced. Then the systems of equations in -(A) and (B) are transformed into equations (A´), and (B´), -the new equations being obtained by simply dashing the -old set.</p> - -<p class='c005'>All these equations can be written in purely real figures, -and we can then formulate the last result as follows.</p> - -<p class='c005'>If the real transformations 4) are taken, and <i>x´</i> <i>y´</i> <i>z´</i> <i>t´</i> -be taken as a new frame of reference, then we shall have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(5) ρ´ = ρ [(-<i>qu</i><sub><i>z</i></sub> + 1)/√(1 - <i>q²</i>)],</div> - <div class='line'>ρ´<i>u</i><sub><i>z</i></sub>´ = ρ[(<i>u</i><sub><i>z</i></sub> - <i>q</i>)/√(1 - <i>q²</i>)],</div> - <div class='line'>ρ´<i>u</i><sub><i>x</i></sub>´ = ρ<i>u</i><sub><i>x</i></sub>,</div> - <div class='line'>ρ´<i>u</i><sub><i>y</i></sub>´ = ρ<i>u</i><sub><i>y</i></sub>.</div> - </div> - <div class='group'> - <div class='line'>(6) <i>e´</i><sub><i>x´</i></sub> = (<i>e</i><sub><i>x</i></sub> - <i>qm</i><sub><i>y</i></sub>)/(√(1 - <i>q²</i>)),</div> - <div class='line'><i>m´</i><sub><i>r´</i></sub> = (<i>qe</i><sub><i>x</i></sub> + <i>m</i><sub><i>y</i></sub>)/(√(1 - <i>q²</i>)),</div> - <div class='line'><i>e´</i><sub><i>z´</i></sub> = <i>e</i><sub><i>z</i></sub>.</div> - </div> - <div class='group'> - <div class='line'>(7) <i>m´</i><sub><i>x´</i></sub> = (<i>m</i><sub><i>x</i></sub> - <i>qe</i><sub><i>y</i></sub>)/(√(1 - <i>q²</i>)),</div> - <div class='line'><i>e´</i><sub><i>y</i>´</sub> = (<i>qm</i><sub><i>x</i></sub> + <i>e</i><sub><i>y</i></sub>)/(√(1 - <i>q²</i>)),</div> - <div class='line'><i>m</i>´<sub><i>z</i>´</sub> = <i>m</i><sub><i>z</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Then we have for these newly introduced vectors <i>u´</i>, <i>e´</i>, -<i>m´</i> (with components <i>u</i><sub><i>x</i></sub>´, <i>u</i><sub><i>y</i></sub>´, <i>u</i><sub><i>z</i></sub>´; <i>e</i><sub><i>x</i></sub>´, <i>e</i><sub><i>y</i></sub>´, <i>e</i><sub><i>z</i></sub>´; <i>m</i><sub><i>x</i></sub>´, <i>m</i><sub><i>y</i></sub>´, -<i>m</i><sub><i>z</i></sub>´), and the quantity ρ´ a series of equations I´), II´), -III´), IV´) which are obtained from I), II), III), IV) by -simply dashing the symbols.</p> - -<p class='c005'>We remark here that <i>e</i><sub><i>x</i></sub> - <i>qm</i><sub><i>y</i></sub>, <i>e</i><sub><i>y</i></sub> + <i>qm</i><sub><i>x</i></sub> are components -of the vector <i>e</i> + [<i>vm</i>], where <i>v</i> is a vector in the direction -of the positive Z-axis, and | <i>v</i> | = <i>q</i>, and [<i>vm</i>] is the vector -product of <i>v</i> and <i>m</i>; similarly -<i>qe</i><sub><i>x</i></sub> + <i>m</i><sub><i>y</i></sub>, <i>m</i><sub><i>x</i></sub> + <i>qe</i><sub><i>y</i></sub> are the -components of the vector <i>m</i> - [<i>ve</i>].</p> - -<p class='c005'>The equations 6) and 7), as they stand in pairs, can be -expressed as.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>e′</i><sub><i>x′</i></sub> + <i>im′</i><sub><i>x′</i></sub> = (<i>e</i><sub><i>x</i></sub> + <i>im</i><sub><i>x</i></sub>) cos <i>i</i>ψ + (<i>e</i><sub><i>y</i></sub> + <i>im</i><sub><i>y</i></sub>) sin <i>i</i>ψ,</div> - </div> - <div class='group'> - <div class='line'><i>e′</i><sub><i>y′</i></sub> + <i>im′</i><sub><i>y′</i></sub> = - (<i>e</i><sub><i>x</i></sub> + <i>im</i><sub><i>x</i></sub>) sin <i>i</i>ψ + (<i>e</i><sub><i>y</i></sub> + <i>im</i><sub><i>y</i></sub>) cos <i>i</i>ψ,</div> - </div> - <div class='group'> - <div class='line'><i>e′</i><sub><i>z′</i></sub> + <i>im′</i><sub><i>z′</i></sub> = <i>e′</i><sub><i>z</i></sub> + <i>im</i><sub><i>z</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'>If φ denotes any other real angle, we can form the -following combinations:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(<i>e′</i><sub><i>x′</i></sub> + <i>im′</i><sub><i>x′</i></sub>) cos. φ + (<i>e′</i><sub><i>y″</i></sub> + <i>im′</i><sub><i>y′</i></sub>) sin φ</div> - </div> - <div class='group'> - <div class='line in2'>= (<i>e</i><sub><i>x</i></sub> + <i>im</i><sub><i>x</i></sub>) cos. (φ + <i>i</i>ψ) + (<i>e</i><sub><i>y</i></sub> + <i>im</i><sub><i>y</i></sub>) sin (φ + <i>i</i>ψ),</div> - </div> - <div class='group'> - <div class='line in2'>= (<i>e′</i><sub><i>x′</i></sub> + <i>im′</i><sub><i>x′</i></sub>) sin φ + (<i>e′</i><sub><i>y′</i></sub> + <i>im′</i><sub><i>y′</i></sub>) cos. φ</div> - </div> - <div class='group'> - <div class='line in2'>= - (<i>e</i><sub><i>x</i></sub> + <i>im</i><sub><i>x</i></sub>) sin (φ + <i>i</i>ψ) + (<i>e</i><sub><i>y</i></sub> + <i>im</i><sub><i>y</i></sub>) cos. (φ + <i>i</i>ψ).</div> - </div> - </div> -</div> - -<h3 id='sect4-4' class='c008'>§ 4. Special Lorentz Transformation.</h3> -<p class='c004'>The rôle which is played by the Z-axis in the transformation -(4) can easily be transferred to any other axis -when the system of axes are subjected to a transformation -about this last axis. So we came to a more general -law:—</p> - -<p class='c005'>Let <i>v</i> be a vector with the components <i>v</i><sub><i>x</i></sub>, <i>v</i><sub><i>y</i></sub>, <i>v</i><sub><i>z</i></sub>, -and let | <i>v</i> | = <i>q</i> < 1. By <i>ṽ</i> we shall denote any vector -which is perpendicular to <i>v</i>, and by <i>r</i><sub><i>v</i></sub>, <i>r</i><sub><i>ṽ</i></sub> we shall denote -components of <i>r</i> in direction of <i>ṽ</i> and <i>v</i>.</p> - -<p class='c005'>Instead of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), new magnetudes (<i>x′</i> <i>y′</i> <i>z′</i> <i>t′</i>) will -be introduced in the following way. If for the sake of -shortness, <i>r</i> is written for the vector with the components -(<i>x</i>, <i>y</i>, <i>z</i>) in the first system of reference, <i>r′</i> for the same -vector with the components (<i>x′</i> <i>y′</i> <i>z′</i>) in the second system -of reference, then for the direction of <i>v</i>, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(10) <i>r′</i><sub><i>v</i></sub> = (<i>r</i><sub><i>v</i></sub> - <i>qt</i>)/√(1 - <i>q²</i>)</div> - </div> - </div> -</div> - -<p class='c010'>and for the perpendicular direction <i>ṽ</i>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(11) <i>r′</i><sub><i>ṽ</i></sub> = <i>r</i><sub><i>ṽ</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>and further (12) <i>t′</i> = (-<i>qr</i><sub><i>v</i></sub> + <i>t</i>)/√(1 - <i>q²</i>).</p> - -<p class='c005'>The notations (<i>r′</i><sub><i>ṽ</i></sub>, <i>r′</i><sub><i>v</i></sub>) are to be understood in the sense -that with the directions <i>v</i>, and every direction <i>ṽ</i> perpendicular -to <i>v</i> in the system (<i>x</i>, <i>y</i>, <i>z</i>) are always associated -the directions with the same direction cosines in the system -(<i>x′</i> <i>y′</i> <i>z′</i>).</p> - -<p class='c005'>A transformation which is accomplished by means of -(10), (11), (12) with the condition 0 < <i>q</i> < 1 will be called -a special Lorentz-transformation. We shall call <i>v</i> the -vector, the direction of <i>v</i> the axis, and the magnitude -of <i>v</i> the moment of this transformation.</p> - -<p class='c005'>If further ρ′ and the vectors <i>u′</i>, <i>e′</i>, <i>m′</i>, in the system -(<i>x′</i> <i>y′</i> <i>z′</i>) are so defined that,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(13) ρ′ = ρ[(-<i>qu</i><sub><i>v</i></sub> + 1)/√(1 - <i>q²</i>)],</div> - <div class='line'>ρ′<i>u</i>′<sub><i>v</i></sub> = ρ(<i>u</i><sub><i>v</i></sub> - <i>q</i>)/√(1 - <i>q²</i>),</div> - <div class='line'>ρ′<i>u</i><sub><i>ṽ</i></sub> = ρ′<i>u</i><sub><i>v</i></sub>,</div> - </div> - </div> -</div> - -<p class='c010'>further</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(14) (<i>e′</i> + <i>im′</i>)<sub><i>ṽ</i></sub> = ((<i>e</i> + <i>im</i>) - <i>i</i>[<i>v</i>, (<i>e</i> + <i>im</i>])']<sub><i>ṽ</i></sub>)/√(1 - <i>q²</i>).</div> - </div> - <div class='group'> - <div class='line'>(15) (<i>e′</i> + <i>im′</i>)<sub><i>v</i></sub> = (<i>e</i> + <i>im</i>) - <i>i</i>[<i>u</i>, (<i>e</i> + <i>im</i>)]<sub><i>v</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Then it follows that the equations I), II), III), IV) are -transformed into the corresponding system with dashes.</p> - -<p class='c005'>The solution of the equations (10), (11), (12) leads to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(16) <i>r</i><sub><i>v</i></sub> = (<i>r′</i><sub><i>v</i></sub> + <i>qt′</i>)/√(1 - <i>q²</i>),</div> - <div class='line'><i>r</i><sub><i>ṽ</i></sub> = <i>r′</i><sub><i>ṽ</i></sub>,</div> - <div class='line'><i>t</i> = (<i>qr′</i><sub><i>v</i></sub> + <i>t′</i>)/√(1 - <i>q²</i>),</div> - </div> - </div> -</div> - -<p class='c010'>Now we shall make a very important observation -about the vectors <i>u</i> and <i>u′</i>. We can again introduce -the indices 1, 2, 3, 4, so that we write (<i>x₁</i>′, <i>x₂</i>′, <i>x₃</i>′, <i>x₄</i>′) -instead of (<i>x′</i>, <i>y′</i>, <i>z′</i>, <i>it′</i>) and ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of -(ρ′<i>u′</i>{<i>x′</i>}, ρ′<i>u′</i>{<i>y′</i>}, ρ′<i>u′</i>{<i>z′</i>}, <i>i</i>ρ′).</p> - -<p class='c005'>Like the rotation round the Z-axis, the transformation -(4), and more generally the transformations (10), (11), -(12), are also linear transformations with the determinant -+ 1, so that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(17) <i>x₁²</i> + <i>x₂²</i> + <i>x₃²</i> + <i>x₄²</i> <i>i. e.</i> <i>x²</i> + <i>y²</i> + <i>z²</i> - <i>t²</i>,</div> - </div> - </div> -</div> - -<p class='c010'>is transformed into</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x₁′²</i> + <i>x₂′²</i> + <i>x₃′²</i> + <i>x₄′²</i> <i>i. e.</i> <i>x′²</i> + <i>y′²</i> + <i>z′²</i> - <i>t′²</i>.</div> - </div> - </div> -</div> - -<p class='c010'>On the basis of the equations (13), (14), we shall have -(ρ₁² + ρ₂² + ρ₃² + ρ₄²) = ρ²(1 - <i>u</i><sub><i>x²</i></sub>, -<i>u</i><sub><i>y²</i></sub>, -<i>u</i><sub><i>z²</i></sub>) = ρ²(1 - <i>u²</i>) -transformed into ρ²(1 - <i>u²</i>) or in other words,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(18) ρ√(1 - <i>u²</i>)</div> - </div> - </div> -</div> - -<p class='c010'>is an invariant in a Lorentz-transformation.</p> - -<p class='c005'>If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain -the four values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - <i>u²</i>))(<i>u</i><sub><i>x</i></sub>, <i>u</i><sub><i>y</i></sub>, <i>u</i><sub><i>z</i></sub>, <i>i</i>) -so that ω₁² + ω₂² + ω₃² + ω₄² = -1.</p> - -<p class='c005'>It is apparent that these four values are determined -by the vector <i>u</i> and inversely the vector <i>u</i> of magnitude -< 1 follows from the 4 values ω₁, ω₂, ω₃, ω₄; where -(ω₁, ω₂, ω₃) are real, -<i>i</i>ω₄ real and positive and condition -(19) is fulfilled.</p> - -<p class='c005'>The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are -the ratios of <i>dx₁</i>, <i>dx₂</i>, <i>dx₃</i>, <i>dx₄</i> to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(20) √(-(<i>dx₁²</i> + <i>dx₂²</i> + <i>dx₃²</i> + <i>dx₄²</i>)) = <i>dt</i>√(1 - <i>u²</i>).</div> - </div> - </div> -</div> - -<p class='c010'>The differentials denoting the displacements of matter -occupying the spacetime point (<i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i>) to the -adjacent space-time point.</p> - -<p class='c005'>After the Lorentz-transformation is accomplished the -velocity of matter in the new system of reference for the -same space-time point (<i>x′</i> <i>y′</i> <i>z′</i> <i>t′</i>) is the vector <i>u′</i> with the -ratios <i>dx′</i>/<i>dt′</i>, <i>dy′</i>/<i>dt′</i>, <i>dz′</i>/<i>dt′</i>, <i>dl′</i>/<i>dt′</i>, as components.</p> - -<p class='c005'>Now it is quite apparent that the system of values</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x₁</i> = ω₁, <i>x₂</i> = ω₂, <i>x₃</i> = ω₃, <i>x₄</i> = ω₄</div> - </div> - </div> -</div> - -<p class='c010'>is transformed into the values</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x₁′</i> = ω₁′, <i>x₂′</i> = ω₂′, <i>x₃′</i> = ω₃′, <i>x₄′</i> = ω₄′</div> - </div> - </div> -</div> - -<p class='c010'>in virtue of the Lorentz-transformation (10), (11), (12).</p> - -<p class='c005'>The dashed system has got the same meaning for the -velocity <i>u′</i> after the transformation as the first system -of values has got for <i>u</i> before transformation.</p> - -<p class='c005'>If in particular the vector <i>v</i> of the special Lorentz-transformation -be equal to the velocity vector <i>u</i> of matter at -the space-time point (<i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i>) then it follows out of -(10), (11), (12) that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ = <i>i</i></div> - </div> - </div> -</div> - -<p class='c010'>Under these circumstances therefore, the corresponding -space-time point has the velocity <i>v′</i> = 0 after the transformation, -it is as if we transform to rest. We may call -the invariant ρ√(1 - <i>u²</i>) the rest-density of Electricity.<a id='r16' /><a href='#f16' class='c007'><sup>[16]</sup></a></p> -<h3 id='sect4-5' class='c008'>§ 5. Space-time Vectors. <br /> Of the 1st and 2nd kind.</h3> -<p class='c004'>If we take the principal result of the Lorentz transformation -together with the fact that the system (A) as well -as the system (B) is covariant with respect to a rotation -of the coordinate-system round the null point, we obtain -the general <i>relativity theorem</i>. In order to make the -facts easily comprehensible, it may be more convenient to -define a series of expressions, for the purpose of expressing -the ideas in a concise form, while on the other hand -I shall adhere to the practice of using complex magnitudes, -in order to render certain symmetries quite evident.</p> - -<p class='c005'>Let us take a linear homogeneous transformation,</p> - -<div class='figcenter id003'> -<img src='images/form-056.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>the Determinant of the matrix is +1, all co-efficients without -the index 4 occurring once are real, while <i>a₄₁</i>, <i>a₄₂</i>, -<i>a₄₃</i>, are purely imaginary, but <i>a₄₄</i> is real and > 0, and -<i>x₁²</i> + <i>x₂²</i> + <i>x₃²</i> + <i>x₄²</i> transforms into <i>x₁′²</i> + <i>x₂′²</i> + <i>x₃′²</i> -+ <i>x₄′²</i>. The operation shall be called a general Lorentz -transformation.</p> - -<p class='c005'>(This notation, which is due to Dr. C. E. Cullis of the Calcutta -University, has been used throughout instead of Minkowski’s notation, -<i>x₁</i> = <i>a₁₁x₁′</i> + <i>a₁₂x₂′</i>+ <i>a₁₃x₃′</i>+ <i>a₁₄x₄′</i>.)</p> - -<p class='c005'>If we put <i>x₁′</i> = <i>x′</i>, <i>x₂′</i> = <i>y′</i>, <i>x₃′</i> = <i>z′</i>, <i>x₄′</i> = <i>it′</i>, then -immediately there occurs a homogeneous linear transformation -of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) to (<i>x′</i>, <i>y′</i>, <i>z′</i>, <i>t′</i>) with essentially real -co-efficients, whereby the aggregate -<i>x²</i> - <i>y²</i> - <i>z²</i> + <i>t²</i> -transforms into -<i>x′²</i> - <i>y′²</i> - <i>z′²</i> + <i>t′²</i>, and to every such -system of values <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> with a positive <i>t</i>, for which -this aggregate > 0, there always corresponds a positive <i>t’</i>; -this last is quite evident from the continuity of the -aggregate <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>.</p> - -<p class='c005'>The last vertical column of co-efficients has to fulfil -the condition 22) <i>a₁₄²</i> + <i>a₂₄²</i> + <i>a₃₄²</i> + <i>a₄₄²</i> = 1.</p> - -<p class='c005'>If <i>a₁₄</i> = <i>a₂₄</i> = <i>a₃₄</i> = 0, then <i>a₄₄</i> = 1, and the Lorentz -transformation reduces to a simple rotation of the spatial -co-ordinate system round the world-point.</p> - -<p class='c005'>If <i>a₁₄</i>, <i>a₂₄</i>, <i>a₃₄</i> are not all zero, and if we put -<i>a₁₄</i> : <i>a₂₄</i> : <i>a₃₄</i> : <i>a₄₄</i> = <i>v</i><sub><i>x</i></sub> : <i>v</i><sub><i>y</i></sub> : <i>v</i><sub><i>z</i></sub> : <i>i</i></p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>q</i> = √(<i>v</i><sub><i>x</i></sub>² + <i>v</i><sub><i>y</i></sub>² +<i>v</i><sub><i>z</i></sub>²) < 1.</div> - </div> - </div> -</div> - -<p class='c010'>On the other hand, with every set of values of -<i>a₁₄</i>, <i>a₂₄</i>, <i>a₃₄</i>, <i>a₄₄</i> which in this way fulfil the condition -22) with real values of <i>v</i><sub><i>x</i></sub>, <i>v</i><sub><i>y</i></sub>, <i>v</i><sub><i>z</i></sub>, we can construct the -special Lorentz transformation (16) with (<i>a₁₄</i>, <i>a₂₄</i>, <i>a₃₄</i>, <i>a₄₄</i>) -as the last vertical column,—and then every Lorentz-transformation -with the same last vertical column -(<i>a₁₄</i>, <i>a₂₄</i>, <i>a₃₄</i>, <i>a₄₄</i>) can be supposed to be composed of -the special Lorentz-transformation, and a rotation of the -spatial co-ordinate system round the null-point.</p> - -<p class='c005'>The totality of all Lorentz-Transformations forms a -group. Under a space-time vector of the 1st kind shall -be understood a system of four magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) -with the condition that in case of a Lorentz-transformation -it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where -these are the values of (<i>x₁′</i>, <i>x₂′</i>, <i>x₃′</i>, <i>x₄′</i>), obtained by -substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (<i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i>) in the -expression (21).</p> - -<p class='c005'>Besides the time-space vector of the 1st kind (<i>x₁</i>, <i>x₂</i>, -<i>x₃</i>, <i>x₄</i>) we shall also make use of another space-time vector -of the first kind (<i>y₁</i>, <i>y₂</i>, <i>y₃</i>, <i>y₄</i>), and let us form the linear -combination</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(23) <i>f₂₃</i>(<i>x₂</i><i>y₃</i> - <i>x₃</i><i>y₂</i>) + <i>f₃₁</i>(<i>x₃</i><i>y₁</i> - <i>x₁</i><i>y₃</i>) + <i>f₁₂</i>(<i>x₁</i><i>y₂</i></div> - <div class='line'>- <i>x₂</i><i>y₁</i>) + <i>f₁₄</i>(<i>x₁</i><i>y₄</i> - <i>x₄</i><i>y₁</i>) + <i>f₂₄</i>(<i>x₂</i><i>y₄</i> - <i>x₄</i><i>y₂</i>) +</div> - <div class='line'><i>f₃₄</i>(<i>x₃</i><i>y₄</i> - <i>x₄</i><i>y₃</i>)</div> - </div> - </div> -</div> - -<p class='c010'>with six coefficients <i>f₂₃</i>--<i>f₃₄</i>. Let us remark that in the -vectorial method of writing, this can be constructed out of -the four vectors.</p> - -<p class='c005'><i>x₁</i>, <i>x₂</i>, <i>x₃</i>; <i>y₁</i>, <i>y₂</i>, <i>y₃</i>; <i>f₂₃</i>, <i>f₃₁</i>, <i>f₁₂</i>; <i>f₁₄</i>, <i>f₂₄</i>, <i>f₃₄</i> and -the constants <i>x₄</i> and <i>y₄</i>, at the same time it is symmetrical -with regard the indices (1, 2, 3, 4).</p> - -<p class='c005'>If we subject (<i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i>) and (<i>y₁</i>, <i>y₂</i>, <i>y₃</i>, <i>y₄</i>) simultaneously -to the Lorentz transformation (21), the combination -(23) is changed to:</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(24) <i>f₂₃′</i>(<i>x₂′</i><i>y₃′</i> - <i>x₃′</i><i>y₂′</i>) + <i>f₃₁</i>(<i>x₃′</i><i>y₁′</i> - <i>x₁′</i><i>y₃′</i>) + <i>f₁₂</i></div> - <div class='line'>(<i>x₁′</i><i>y₂′</i> - <i>x₂′</i><i>y₁′</i>) + <i>f₁₄′</i>(<i>x₁′</i><i>y₄′</i>) - <i>x₄′</i><i>y₁′</i>) + <i>f₂₄′</i>(<i>x₂′</i><i>y₄′</i></div> - <div class='line'>- <i>x₄′</i><i>y₂′</i>) + <i>f₃₄′</i>(<i>x₃′</i><i>y₄′</i> - <i>x₄′</i><i>y₃′</i>),</div> - </div> - </div> -</div> - -<p class='c010'>where the coefficients <i>f₂₃′</i>, <i>f₃₁′</i>, <i>f₁₂′</i>, <i>f₁₄′</i>, <i>f₂₄′</i>, <i>f₃₄′</i>, depend -solely on (<i>f₂₃</i> <i>f₂₄</i>) and the coefficients <i>a₁₁</i> ... <i>a₄₄</i>.</p> - -<p class='c005'>We shall define a space-time Vector of the 2nd kind -as a system of six-magnitudes <i>f₂₃</i>, <i>f₃₁</i> ... <i>f₃₄</i>, with the -condition that when subjected to a Lorentz transformation, -it is changed to a new system <i>f₂₃′</i> ... f₃₄, ... which satisfies -the connection between (23) and (24).</p> - -<p class='c005'>I enunciate in the following manner the general -theorem of relativity corresponding to the equations (I)-(iv),—which -are the fundamental equations for Äther.</p> - -<p class='c005'>If <i>x</i>, <i>y</i>, <i>z</i>, <i>it</i> (space co-ordinates, and time <i>it</i>) is subjected -to a Lorentz transformation, and at the same time -(<i>pu</i><sub><i>x</i></sub>, <i>pu</i><sub><i>y</i></sub>, <i>pu</i><sub><i>z</i></sub>, <i>i</i>ρ) (convection-current, and charge density -ρ<i>i</i>) is transformed as a space time vector of the 1st kind, -further (<i>m</i><sub><i>x</i></sub>, <i>m</i><sub><i>y</i></sub>, <i>m</i><sub><i>z</i></sub>, -<i>ie</i><sub><i>x</i></sub>, -<i>ie</i><sub><i>y</i></sub>, -<i>ie</i><sub><i>z</i></sub>) (magnetic force, -and electric induction × (-<i>i</i>) is transformed as a space -time vector of the 2nd kind, then the system of equations -(I), (II), and the system of equations (III), (IV) transforms -into essentially corresponding relations between the -corresponding magnitudes newly introduced into the -system.</p> - -<p class='c005'>These facts can be more concisely expressed in these -words: the system of equations (I and II) as well as the -system of equations (III) (IV) are covariant in all cases -of Lorentz-transformation, where (ρ<i>u</i>, <i>i</i>ρ) is to be transformed -as a space time vector of the 1st kind, (<i>m</i> - <i>ie</i>) is -to be treated as a vector of the 2nd kind, or more -significantly,—</p> - -<p class='c005'>(ρ<i>u</i>, <i>i</i>ρ) is a space time vector of the 1st kind, (<i>m</i> - <i>ie</i>)<a id='r17' /><a href='#f17' class='c007'><sup>[17]</sup></a> -is a space-time vector of the 2nd kind.</p> - -<p class='c005'>I shall add a few more remarks here in order to elucidate -the conception of space-time vector of the 2nd kind. -Clearly, the following are invariants for such a vector when -subjected to a group of Lorentz transformation.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(<i>i</i>) <i>m²</i> - <i>e²</i> = <i>f₂₃²</i> + <i>f₃₁²</i> + <i>f₁₂²</i> + <i>f₁₄²</i> + <i>f₂₄²</i> + <i>f₂₄²</i></div> - </div> - <div class='group'> - <div class='line'><i>me</i> = <i>i</i>(<i>f₂₃</i><i>f₁₄</i> + <i>f₃₁</i><i>f₂₄</i> + <i>f₁₂</i><i>f₃₄</i>).</div> - </div> - </div> -</div> - -<p class='c010'>A space-time vector of the second kind (<i>m</i> - <i>ie</i>), where -(<i>m</i> and <i>e</i>) are real magnitudes, may be called singular, -when the scalar square (<i>m</i> - <i>ie</i>)² = 0, <i>ie</i> <i>m²</i> - <i>e²</i> = 0, and at -the same time (<i>m e</i>) = 0, <i>ie</i> the vector <i>m</i> and <i>e</i> are equal and -perpendicular to each other; when such is the case, these -two properties remain conserved for the space-time vector -of the 2nd kind in every Lorentz-transformation.</p> - -<p class='c005'>If the space-time vector of the 2nd kind is not -singular, we rotate the spacial co-ordinate system in such -a manner that the vector-product [<i>me</i>] coincides with -the Z-axis, <i>i.e.</i> <i>m</i><sub><i>x</i></sub> = 0, <i>e</i><sub><i>x</i></sub> = 0. Then</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(<i>m</i><sub><i>x</i></sub>, -<i>i e</i><sub><i>x</i></sub>)² + (<i>m</i><sub><i>y</i></sub>, -<i>i e</i><sub><i>y</i></sub>)² ≠ 0.</div> - </div> - </div> -</div> - -<p class='c010'>Therefore (<i>e</i><sub><i>y</i></sub> + <i>i m</i><sub><i>y</i></sub>)/(<i>e</i><sub><i>x</i></sub> + <i>i e</i><sub><i>x</i></sub>) is different from +<i>i</i>, -and we can therefore define a complex argument (φ + <i>i</i>ψ) -in such a manner that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>tan (φ + <i>i</i>ψ)</div> - </div> - <div class='group'> - <div class='line in8'><i>e</i><sub><i>y</i></sub> + <i>i m</i><sub><i>y</i></sub></div> - <div class='line in4'>= ---------------</div> - <div class='line in8'><i>e</i><sub><i>x</i></sub> + <i>i m</i><sub><i>x</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>If then, by referring back to equations (9), we carry out -the transformation (1) through the angle ψ and a subsequent -rotation round the Z-axis through the angle φ, we perform a -Lorentz-transformation at the end of which <i>m</i><sub><i>y</i></sub> = 0, <i>e</i><sub><i>y</i></sub> = 0, -and therefore <i>m</i> and <i>e</i> shall both coincide with the new -Z-axis. Then by means of the invariants <i>m²</i> - <i>e²</i>, (<i>me</i>) -the final values of these vectors, whether they are of the -same or of opposite directions, or whether one of them is -equal to zero, would be at once settled.</p> -<h3 id='sect4-6' class='c008'>§ 6. Concept of Time.</h3> -<p class='c004'>By the Lorentz transformation, we are allowed to effect -certain <i>changes</i> of the time parameter. In consequence -of this fact, it is no longer permissible to speak of the -absolute simultaneity of two events. The ordinary idea -of simultaneity rather presupposes that six independent -parameters, which are evidently required for defining a -system of space and time axes, are somehow reduced to -three. Since we are accustomed to consider that these -limitations represent in a unique way the actual facts -very approximately, we maintain that the simultaneity of -two events exists of themselves.<a id='r18' /><a href='#f18' class='c007'><sup>[18]</sup></a> In fact, the following -considerations will prove conclusive.</p> - -<p class='c005'>Let a reference system (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) for space time points -(events) be somehow known. Now if a space point A -(<i>x₀</i>, <i>y₀</i>, <i>z₀</i>) the time <i>t₀</i> be compared with a space -point P (<i>x</i>, <i>y</i>, <i>z</i>) at the time <i>t</i>, and if the difference of -time <i>t</i> - <i>t₀</i>, (let <i>t</i> > <i>t₀</i>) be less than the length A P <i>i.e.</i> less -than the time required for the propagation of light from -A to P, and if <i>q</i> = (<i>t</i> - <i>t₀</i>)/(A P) < 1, then by a special Lorentz -transformation, in which A P is taken as the axis, and which -has the moment <i>q</i>, we can introduce a time parameter <i>t′</i>, which -(see equation 11, 12, § 4) has got the same value <i>t′</i> = <i>0</i> for -both space-time points (A, <i>t₀</i>), and (P, t). So the two -events can now be comprehended to be simultaneous.</p> - -<p class='c005'>Further, let us take at the same time <i>t₀</i> = 0, two -different space-points A, B, or three space-points (A, B, C) -which are not in the same space-line, and compare -therewith a space point P, which is outside the line A B, -or the plane A B C, at another time <i>t</i>, and let the time -difference <i>t</i> - <i>t₀</i> (t > <i>t₀</i>) be less than the time which light -requires for propagation from the line A B, or the plane -(A B C) to P. Let q be the quotient of (<i>t</i> - <i>t₀</i>) by the -second time. Then if a Lorentz transformation is taken -in which the perpendicular from P on A B, or from P on -the plane A B C is the axis, and q is the moment, then -all the three (or four) events (A, <i>t₀</i>), (B, <i>t₀</i>), (C, <i>t₀</i>) and -(P, t) are simultaneous.</p> - -<p class='c005'>If four space-points, which do not lie in one plane, are -conceived to be at the same time <i>t₀</i>, then it is no longer permissible -to make a change of the time parameter by a Lorentz-transformation, -without at the same time destroying the -character of the simultaneity of these four space points.</p> - -<p class='c005'>To the mathematician, accustomed on the one hand to -the methods of treatment of the poly-dimensional -manifold, and on the other hand to the conceptual figures -of the so-called non-Euclidean Geometry, there can be no -difficulty in adopting this concept of time to the application -of the Lorentz-transformation. The paper of Einstein which -has been cited in the Introduction, has succeeded to some -extent in presenting the nature of the transformation -from the physical standpoint.</p> -<h3 id='sect4-7' class='c001'>PART II. ELECTRO-MAGNETIC PHENOMENA. <br /> § 7. Fundamental Equations for bodies at rest.</h3> -<p class='c004'>After these preparatory works, which have been first -developed on account of the small amount of mathematics -involved in the limiting case ε = 1, μ = 1, σ = 0, let -us turn to the electro-magnetic phenomena in matter. -We look for those relations which make it possible for -us—when proper fundamental data are given—to -obtain the following quantities at every place and time, -and therefore at every space-time point as functions of -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>):—the vector of the electric force E, the -magnetic induction M, the electrical induction <i>e</i>, the -magnetic force <i>m</i>, the electrical space-density ρ, the -electric current s (whose relation hereafter to the conduction -current is known by the manner in which conductivity -occurs in the process), and lastly the vector <i>v</i>, the -velocity of matter.</p> - -<p class='c005'>The relations in question can be divided into two -classes.</p> - -<p class='c005'>Firstly—those equations, which,—when <i>v</i>, the velocity -of matter is given as a function of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>),—lead us to -a knowledge of other magnitude as functions of <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>—I -shall call this first class of equations the fundamental -equations—</p> - -<p class='c005'>Secondly, the expressions for the ponderomotive force, -which, by the application of the Laws of Mechanics, gives -us further information about the vector <i>u</i> as functions of -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>).</p> - -<p class='c005'>For the case of bodies at rest, <i>i.e.</i> when <i>u</i> (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) -= 0 the theories of Maxwell (Heaviside, Hertz) and -Lorentz lead to the same fundamental equations. They -are;—</p> - -<p class='c005'>(1) The Differential Equations:—which contain no -constant referring to matter:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(<i>i</i>) Curl <i>m</i> - δ<i>e</i>/δ<i>t</i> = C,</div> - <div class='line'>(<i>ii</i>) div <i>e</i> = lρ.</div> - <div class='line'>(<i>iii</i>) Curl E + δM/δ<i>t</i> = 0,</div> - <div class='line'>(<i>iv</i>) Div M = 0.</div> - </div> - </div> -</div> - -<p class='c010'>(2) Further relations, which characterise the influence -of existing matter for the most important case to which -we limit ourselves <i>i.e.</i> for isotopic bodies;—they are comprised -in the equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(V) <i>e</i> = ε E, M = μ<i>m</i>, C = σE.</div> - </div> - </div> -</div> - -<p class='c010'>where ε = dielectric constant, μ = magnetic permeability, -σ = the conductivity of matter, all given as function of -<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>; <i>s</i> is here the conduction current.</p> - -<p class='c005'>By employing a modified form of writing, I shall now -cause a latent symmetry in these equations to appear. -I put, as in the previous work,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x₁</i> = <i>x</i>, <i>x₂</i> = <i>y</i>, <i>x₃</i> = <i>z</i>, <i>x₄</i> = <i>it</i>,</div> - </div> - </div> -</div> - -<p class='c010'>and write <i>s₁</i>, <i>s₂</i>, <i>s₃</i>, <i>s₄</i> for C<sub><i>x</i></sub>, C<sub><i>y</i></sub>, C<sub><i>z</i></sub> (√-1)ρ.</p> - -<p class='c005'>Further <i>f₂₃</i>, <i>f₃₁</i>, <i>f₁₂</i>, <i>f₁₄</i>, <i>f₂₄</i>, <i>f₃₄</i></p> - -<p class='c005'>for <i>m</i><sub><i>x</i></sub>, <i>m</i><sub><i>y</i></sub>, <i>m</i><sub><i>z</i></sub>, -<i>i</i>(<i>e</i><sub><i>x</i></sub>, <i>e</i><sub><i>y</i></sub>, <i>e</i><sub><i>z</i></sub>),</p> - -<p class='c005'>and F₂₃, F₃₁, F₁₂, F₁₄, F₂₄, F₃₄</p> - -<p class='c005'>for M<sub><i>x</i></sub>, M<sub><i>y</i></sub>, M<sub><i>z</i></sub>, -<i>i</i>(E<sub><i>x</i></sub>, E<sub><i>y</i></sub>, E<sub><i>z</i></sub>)</p> - -<p class='c005'>lastly we shall have the relation <i>f</i><sub>k h</sub> = - <i>f</i><sub><i>h k</i></sub>, <i>F</i><sub><i>k h</i></sub> = -<i>F</i><sub><i>h k</i></sub>, -(the letter <i>f</i>, F shall denote the field, <i>s</i> the (<i>i.e.</i> current).</p> - -<p class='c005'>Then the fundamental Equations can be written as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(A)</div> - <div class='line'> ∂<i>f₁₂</i>/∂<i>x₂</i> + ∂<i>f₁₃</i>/∂<i>x₃</i> + ∂<i>f₁₄</i>/∂<i>x₄</i> = s₁</div> - </div> - <div class='group'> - <div class='line'>∂<i>f₂₁</i>/∂<i>x₁</i> + + ∂<i>f₂₃</i>/∂<i>x₃</i> + ∂<i>f₂₄</i>/∂<i>x₄</i> = s₂</div> - </div> - <div class='group'> - <div class='line'>∂<i>f₃₁</i>/∂<i>x₁</i> + ∂<i>f₃₂</i>/∂<i>x₂</i> + + ∂<i>f₃₄</i>/∂<i>x₄</i> = s₃</div> - </div> - <div class='group'> - <div class='line'>∂<i>f₄₁</i>/∂<i>x₁</i> + ∂<i>f₄₂</i>/∂<i>x₂</i> + ∂<i>f₄₃</i>/∂<i>x₃</i> = s₄</div> - </div> - </div> -</div> - -<p class='c010'>and the equations (3) and (4), are</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'> ∂F₃₄/∂<i>x₂</i> + ∂F₄₂/∂<i>x₃</i> + ∂F₂₃/∂<i>x₄</i> = 0</div> - </div> - <div class='group'> - <div class='line'>∂F₄₃/∂<i>x₁</i> + + ∂F₁₄/∂<i>x₃</i> + ∂F₃₁∂<i>x₄</i> = 0</div> - </div> - <div class='group'> - <div class='line'>∂F₂₄/∂<i>x₁</i> + ∂F₄₁/∂<i>x₂</i> + + ∂F₁₂/∂<i>x₄</i> = 0</div> - </div> - <div class='group'> - <div class='line'>∂F₃₂/∂<i>x₁</i> + ∂F₁₃/∂<i>x₂</i> + ∂F₂₁/∂<i>x₃</i> = 0</div> - </div> - </div> -</div> - -<h3 id='sect4-8' class='c008'>§ 8. The Fundamental Equations.</h3> -<p class='c004'>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 exclusively.</p> - -<p class='c005'>The first Axion shall be,—</p> - -<p class='c005'>When a detached region<a id='r19' /><a href='#f19' class='c007'><sup>[19]</sup></a> of matter is at rest at any -moment, therefore the vector <i>u</i> is zero, for a system -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>)—the neighbourhood may be supposed to be -in motion in any possible manner, then for the space-time -point <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, the same relations (A) (B) (V) which -hold in the case when all matter is at rest, shall also -hold between ρ, the vectors C, <i>e</i>, <i>m</i>, <i>M</i>, <i>E</i> and their differentials -with respect to <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>. The second axiom shall -be:—</p> - -<p class='c005'>Every velocity of matter is < 1, smaller than the velocity -of propagation of light.<a id='r20' /><a href='#f20' class='c007'><sup>[20]</sup></a></p> - -<p class='c005'>The fundamental equations are of such a kind that -when (<i>x</i>, <i>y</i>, <i>z</i>, <i>it</i>) are subjected to a Lorentz transformation -and thereby (<i>m</i> - <i>ie</i>) and (<i>M</i> - <i>iE</i>) are transformed into -space-time vectors of the second kind, (C, <i>i</i>ρ) as a space-time -vector of the 1st kind, the equations are transformed into -essentially identical forms involving the transformed -magnitudes.</p> - -<p class='c005'>Shortly I can signify the third axiom as:—</p> - -<p class='c005'>(<i>m</i>, -<i>ie</i>), and (<i>M</i>, -<i>iE</i>) are space-time vectors of the -second kind, (C, <i>i</i>p) is a space-time vector of the first kind.</p> - -<p class='c005'>This axiom I call the Principle of Relativity.</p> - -<p class='c005'>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 way.</p> - -<p class='c005'>According to the second axiom, the magnitude of the -velocity vector | <i>u</i> | is < 1 at any space-time point. In -consequence, we can always write, instead of the vector <i>u</i>, -the following set of four allied quantities</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ω₁ = u<sub><i>x</i></sub>/√(1 - <i>u²</i>),</div> - <div class='line'>ω₂ = u<sub><i>y</i></sub>/√(1 - u²),</div> - <div class='line'>ω₃ = u<sub><i>z</i></sub>/√(1 - u²),</div> - <div class='line'>ω₄ = <i>i</i>/√(1 - u²)</div> - </div> - </div> -</div> - -<p class='c010'>with the relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(27) ω₁² + ω₂² + ω₃² + ω₄² = - |</div> - </div> - </div> -</div> - -<p class='c010'>From what has been said at the end of <a href='#sect4-4'>§ 4</a>, it is clear -that in the case of a Lorentz-transformation, this set -behaves like a space-time vector of the 1st kind.</p> - -<p class='c005'>Let us now fix our attention on a certain point (<i>x</i>, <i>y</i>, <i>z</i>) -of matter at a certain time (<i>t</i>). If at this space-time -point <i>u</i> = 0, then we have at once for this point the equations -(<i>A</i>), (<i>B</i>) (<i>V</i>) of <a href='#sect4-7'>§ 7</a>. If <i>u</i> ≠ 0, then there exists -according to 16), in case | <i>u</i> | < 1, a special Lorentz-transformation, -whose vector <i>v</i> is equal to this vector <i>u</i> (<i>x</i>, <i>y</i>, <i>z</i>, -<i>t</i>), and we pass on to a new system of reference (<i>x′</i> <i>y′</i> <i>z′</i> <i>t′</i>) -in accordance with this transformation. Therefore for -the space-time point considered, there arises as in <a href='#sect4-4'>§ 4</a>, -the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = <i>i</i>, -therefore the new velocity vector ω′ = 0, the space-time -point is as if transformed to rest. Now according to the -third axiom the system of equations for the transformed -point (<i>x′</i> <i>y′</i> <i>z′</i> <i>t</i>) involves the newly introduced magnitude -(<i>u′</i> ρ′, C′, <i>e′</i>, <i>m′</i>, <i>E′</i>, <i>M′</i>) and their differential quotients -with respect to (<i>x′</i>, <i>y′</i>, <i>z′</i>, <i>t′</i>) in the same manner as the -original equations for the point (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>). But according -to the first axiom, when <i>u′</i> = 0, these equations must be -exactly equivalent to</p> - -<p class='c005'>(1) the differential equations (<i>A′</i>), (<i>B′</i>), which are -obtained from the equations (<i>A</i>), (<i>B</i>) by simply dashing -the symbols in (<i>A</i>) and (<i>B</i>).</p> - -<p class='c005'>(2) and the equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(V′) <i>e′</i> = ε<i>E′</i>, <i>M’</i> = μ<i>m′</i>, <i>C′</i> = σ<i>E′</i></div> - </div> - </div> -</div> - -<p class='c010'>where ε, μ, σ are the dielectric constant, magnetic permeability, -and conductivity for the system (<i>x′</i> <i>y′</i> <i>z′</i> <i>t′</i>) <i>i.e.</i> in -the space-time point (<i>x</i> <i>y</i>, <i>z</i> <i>t</i>) of matter.</p> - -<p class='c005'>Now let us return, by means of the reciprocal Lorentz-transformation -to the original variables (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), and the -magnitudes (<i>u</i>, ρ, C, <i>e</i>, <i>m</i>, <i>E</i>, <i>M</i>) and the equations, which -we then obtain from the last mentioned, will be the fundamental -equations sought by us for the moving bodies.</p> - -<p class='c005'>Now from <a href='#sect4-4'>§ 4</a>, and <a href='#sect4-6'>§ 6</a>, it is to be seen that the equations -<i>A</i>), as well as the equations <i>B</i>) are covariant for a -Lorentz-transformation, <i>i.e.</i> the equations, which we obtain -backwards from <i>A′</i>) <i>B′</i>), must be exactly of the same form -as the equations <i>A</i>) and <i>B</i>), as we take them for bodies -at rest. We have therefore as the first result:—</p> - -<p class='c005'>The differential equations expressing the fundamental -equations of electrodynamics for moving bodies, when -written in ρ and the vectors C, <i>e</i>, <i>m</i>, E, M, are exactly of -the same form as the equations for moving bodies. The -velocity of matter does not enter in these equations. In -the vectorial way of writing, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>I) curl <i>m</i> - ∂<i>e</i>/∂<i>t</i> = C₁,</div> - </div> - <div class='group'> - <div class='line'>II) div <i>e</i> = ρ</div> - </div> - <div class='group'> - <div class='line'>III) curl E + ∂M/∂<i>t</i> = 0</div> - </div> - <div class='group'> - <div class='line'>IV) div M = 0</div> - </div> - </div> -</div> - -<p class='c010'>The velocity of matter occurs only in the auxiliary -equations which characterise the influence of matter on the -basis of their characteristic constants ε, μ, σ. Let us now -transform these auxiliary equations V′) into the original -co-ordinates (<i>x</i>, <i>y</i>, <i>z</i>, and <i>t</i>.)</p> - -<p class='c005'>According to formula 15) in <a href='#sect4-4'>§ 4</a>, the component of <i>e′</i> -in the direction of the vector <i>u</i> is the same as that of -(<i>e</i> + [<i>u</i> <i>m</i>]), the component of <i>m′</i> is the same as that of -<i>m</i> - [<i>u</i> <i>e</i>], but for the perpendicular direction <i>ū</i>, the components -of <i>e′</i>, <i>m′</i> are the same as those of (<i>e</i> + [<i>u</i> <i>m</i>]) and (<i>m</i> -- [<i>u</i> <i>e</i>], multiplied by 1/√(1 - <i>u²</i>). On the other hand E′ -and M′ shall stand to E + [<i>u</i>M], and M - [<i>u</i>E] in the -same relation as <i>e′</i> and <i>m′</i> to <i>e</i> + [<i>um</i>], and <i>m</i> - (<i>ue</i>). -From the relation <i>e′</i> = εE′, the following equations follow</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(C) <i>e</i> + [<i>um</i>] = ε(E + [<i>u</i>M]),</div> - </div> - </div> -</div> - -<p class='c010'>and from the relation M′ = μ<i>m′</i>, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(D) M - [<i>u</i> E] = μ(<i>m</i> - [<i>ue</i>]),</div> - </div> - </div> -</div> - -<p class='c010'>For the components in the directions perpendicular -to <i>u</i>, and to each other, the equations are to be multiplied -by √(1 - <i>u²</i>).</p> - -<p class='c005'>Then the following equations follow from the transformation? -equations (12), (10), (11) in <a href='#sect4-4'>§ 4</a>, when we replace -q, <i>r</i><sub><i>v</i></sub>, <i>r</i><sub><i>ṽ</i></sub>, <i>t</i>, <i>r′</i><sub><i>v</i></sub>, <i>r′</i><sub><i>ṽ</i></sub>, <i>t’</i> by |<i>u</i>|, C<sub><i>u</i></sub>, C<sub><i>ū</i></sub>, ρ, C′<sub><i>u</i></sub>, C′<sub><i>ū</i></sub>, ρ′</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ρ′ = (-|<i>u</i>| C<sub><i>u</i></sub> + ρ)/√(1 - <i>u²</i>),</div> - <div class='line'>C’<sub><i>u</i></sub> = (C<sub><i>u</i></sub> - |<i>u</i>|ρ)/√(1 - <i>u²</i>),</div> - <div class='line'>C′<sub><i>ū</i></sub> = C<sub><i>ū</i></sub>,</div> - </div> - <div class='group'> - <div class='line'>E) (C<sub><i>u</i></sub> - |<i>u</i>|ρ)/√(1 - <i>u²</i>) = σ(E + [<i>u</i>M])<sub><i>u</i></sub>,</div> - </div> - <div class='group'> - <div class='line'>C<sub><i>ū</i></sub> = σ (E + [<i>u</i>M])<sub><i>u</i></sub>/√(1 - <i>u²</i>).</div> - </div> - </div> -</div> - -<p class='c010'>In consideration of the manner in which σ enters into -these relations, it will be convenient to call the vector -C - ρ<i>u</i> with the components C<sub><i>u</i></sub> - ρ|<i>u</i>| in the direction of -<i>u</i>, and C′<sub><i>ū</i></sub> in the directions <i>ū</i> perpendicular to <i>u</i> the -“Convection current.” This last vanishes for σ = 0.</p> - -<p class='c005'>We remark that for ε = 1, μ = 1 the equations <i>e′</i> = E′, -<i>m′</i> = M′ immediately lead to the equations <i>e</i> = E, <i>m</i> = M -by means of a reciprocal Lorentz-transformation with -<i>u</i> -as vector; and for σ = 0, the equation C′ = 0 leads to C = ρ<i>u</i>; -that the fundamental equations of Äther discussed in <a href='#sect4-2'>§ 2</a> -becomes in fact the limitting case of the equations -obtained here with ε = 1, μ = 1, σ = 0.</p> -<h3 class='c008'>§ 9. The Fundamental Equations in Lorentz’s Theory.</h3> -<p class='c004'>Let us now see how far the fundamental equations -assumed by Lorentz correspond to the Relativity postulate, -as defined in <a href='#sect4-8'>§8</a>. In the article on Electron-theory (Ency., -Math., Wiss., Bd. V. 2, Art 14) Lorentz has given the -fundamental equations for any possible, even magnetised -bodies (see there page 209, Eqn XXX′, formula (14) on -page 78 of the same (part).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(III<i>a″</i>) Curl (H - [<i>u</i>E]) = J + <i>d</i>D/<i>dt</i> + <i>u</i> div D</div> - <div class='line'>- curl [<i>u</i>D].</div> - </div> - <div class='group'> - <div class='line'>(I″) div D = ρ</div> - </div> - <div class='group'> - <div class='line'>(IV″) curl E = - <i>d</i>B/<i>dt</i>, Div B = 0 (V′)</div> - </div> - </div> -</div> - -<p class='c010'>Then for moving non-magnetised bodies, Lorentz puts -(page 223, 3) μ = 1, B = H, and in addition to that takes -account of the occurrence of the di-electric constant ε, and -conductivity σ according to equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(ε<i>q</i>XXXIV″, p. 327) D - E = (ε - 1) {E + [<i>u</i>B]}</div> - </div> - <div class='group'> - <div class='line'>(ε<i>q</i>XXXIII′, p. 223) J = σ(E + [<i>u</i>B])</div> - </div> - </div> -</div> - -<p class='c010'>Lorentz’s E, D, H are here denoted by E, M, <i>e</i>, <i>m</i> -while J denotes the conduction current.</p> - -<p class='c005'>The three last equations which have been just cited -here coincide with eqn (II), (III), (IV), the first equation -would be, if J is identified with C, = <i>u</i>ρ (the current being -zero for σ = 0,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(29) Curl [H - (<i>u</i>, E)] = C + <i>d</i>D/<i>dt</i> - curl [<i>u</i>D],</div> - </div> - </div> -</div> - -<p class='c010'>and thus comes out to be in a different form than (1) here. -Therefore for magnetised bodies, Lorentz’s equations do not -correspond to the Relativity Principle.</p> - -<p class='c005'>On the other hand, the form corresponding to the -relativity principle, for the condition of non-magnetisation -is to be taken out of (D) in <a href='#sect4-8'>§8</a>, with μ = 1, not as B = H, -as Lorentz takes, but as (30) B - [<i>u</i>D] = H - [<i>u</i>D] -(M - [<i>u</i>E] = <i>m</i> - [<i>ue</i>]. Now by putting H = B, the differential -equation (29) is transformed into the same form as -eqn (1) here when <i>m</i> - [<i>ue</i>] = M - [<i>u</i>E]. Therefore it so -happens that by a compensation of two contradictions to -the relativity principle, the differential equations of Lorentz -for moving non-magnetised bodies at last agree with the -relativity postulate.</p> - -<p class='c005'>If we make use of (30) for non-magnetic bodies, and -put accordingly H = B + [<i>u</i>, (D - E)], then in consequence -of (C) in <a href='#sect4-8'>§8</a>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(ε - 1) (E + [<i>u</i>, B]) = D - E + [<i>u</i>. [<i>u</i>, D - E]],</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i> for the direction of <i>u</i>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(ε - 1) (E + [<i>u</i>B])<sub><i>u</i></sub> = (D - E)<sub><i>u</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>and for a perpendicular direction ū,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(ε - 1) [E + (<i>u</i>B)]<sub><i>u</i></sub> = (1 - <i>u²</i>) (D - E)<sub><i>u</i></sub></div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i> it coincides with Lorentz’s assumption, if we neglect -<i>u²</i> in comparison to 1.</p> - -<p class='c005'>Also to the same order of approximation, Lorentz’s -form for J corresponds to the conditions imposed by the -relativity principle [comp. (E) <a href='#sect4-8'>§ 8</a>]—that the components -of J<sub><i>u</i></sub>, J<sub><i>ū</i></sub> are equal to the components of σ (E + [<i>u</i> B]) -multiplied by √(1 - <i>u²</i>) or 1 / √(1 - <i>u²</i>) respectively.</p> -<h3 id='sect4-10' class='c008'>§10. Fundamental Equations of E. Cohn.</h3> -<p class='c004'>E. Cohn assumes the following fundamental equations.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(31) Curl (M + [<i>u</i> E]) = <i>d</i>E/<i>dt</i> + u div. E + J</div> - </div> - <div class='group'> - <div class='line'>- Curl [E - (<i>u</i>. M)] = <i>d</i>M/<i>dt</i> + u div. M.</div> - </div> - <div class='group'> - <div class='line'>(32) J = σ E, = ε E - [<i>u</i> M], M = μ (<i>m</i> + [<i>u</i> E.])</div> - </div> - </div> -</div> - -<p class='c010'>where E M are the electric and magnetic field intensities -(forces), E, M are the electric and magnetic polarisation -(induction). The equations also permit the existence of -true magnetism; if we do not take into account this -consideration, div. M. is to be put = 0.</p> - -<p class='c005'>An objection to this system of equations, is that -according to these, for ε = 1, μ = 1, the vectors force and -induction do not coincide. If in the equations, we conceive -E and M and not E - (U. M), and M + [U E] as electric -and magnetic forces, and with a glance to this we -substitute for E, M, E, M, div. E, the symbols <i>e</i>, M, E -+ [U M], <i>m</i> - [<i>u</i> <i>e</i>], ρ, then the differential equations -transform to our equations, and the conditions (32) -transform into</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>J = σ(E + [<i>u</i> M])</div> - <div class='line'><i>e</i> + [<i>u</i>, (<i>m</i> - [<i>u</i> <i>e</i>])] = ε(E + [<i>u</i> M])</div> - <div class='line'>M - [<i>u</i>, (E + <i>u</i> M)] = μ(<i>m</i> - [<i>u</i> <i>e</i>])</div> - </div> - </div> -</div> - -<p class='c010'>then in fact the equations of Cohn become the same as -those required by the relativity principle, if errors of the -order <i>u²</i> are neglected in comparison to 1.</p> - -<p class='c005'>It may be mentioned here that the equations of Hertz -become the same as those of Cohn, if the auxiliary -conditions are</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(33) E = εE, M = μM, J = σE.</div> - </div> - </div> -</div> - -<h3 class='c008'>§11. Typical Representations of the Fundamental Equations.</h3> -<p class='c004'>In the statement of the fundamental equations, our -leading idea had been that they should retain a covariance -of form, when subjected to a group of Lorentz-transformations. -Now we have to deal with ponderomotive -reactions and energy in the electro-magnetic field. Here -from the very first there can be no doubt that the -settlement of this question is in some way connected with -the simplest forms which can be given to the fundamental -equations, satisfying the conditions of covariance. In -order to arrive at such forms, I shall first of all put the -fundamental equations in a typical form which brings out -clearly their covariance in case of a Lorentz-transformation. -Here I am using a method of calculation, which enables us -to deal in a simple manner with the space-time vectors of -the 1st, and 2nd kind, and of which the rules, as far as -required are given below.</p> - -<p class='c005'>A system of magnitudes <i>a</i><sub><i>h</i> <i>k</i></sub> formed into the matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| <i>a₁₁</i>.......<i>a</i><sub>1 <i>q</i></sub> |</div> - <div class='line'>| |</div> - <div class='line'>| |</div> - <div class='line'>| |</div> - <div class='line'>| <i>a</i><sub><i>p</i> 1</sub>.....<i>a</i><sub><i>p</i> <i>q</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>arranged in <i>p</i> horizontal rows, and <i>q</i> vertical columns is -called a <i>p</i> × <i>q</i> series-matrix, and will be denoted by the -letter A.</p> - -<p class='c005'>If all the quantities <i>a</i><sub><i>h</i> <i>k</i></sub> are multiplied by C, the -resulting matrix will be denoted by CA.</p> - -<p class='c005'>If the roles of the horizontal rows and vertical columns -be intercharged, we obtain a <i>q</i> × <i>p</i> series matrix, which -will be known as the transposed matrix of A, and will be -denoted by Ā.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Ā = | <i>a₁₁</i> ...... <i>a</i><sub><i>p</i> 1</sub> |</div> - <div class='line in4'>| |</div> - <div class='line in4'>| <i>a</i><sub>1 <i>q</i></sub> ...... <i>a</i><sub><i>p</i> <i>q</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>If we have a second <i>p</i> × <i>q</i> series matrix B,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>B = | <i>b₁₁</i> .......... <i>b₁</i><sub><i>q</i></sub> |</div> - <div class='line in4'>| |</div> - <div class='line in4'>| <i>b</i><sub><i>p</i> 1</sub> .......... <i>b</i><sub><i>p</i> <i>q</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>then A + B shall denote the <i>p</i> × <i>q</i> series matrix whose -members are <i>a</i><sub><i>h</i> <i>k</i></sub> + <i>b</i><sub><i>h</i> <i>k</i></sub>.</p> - -<p class='c005'>2⁰ If we have two matrices</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A = | <i>a₁₁</i> .......... <i>a₁</i><sub><i>q</i></sub> |</div> - <div class='line in4'>| |</div> - <div class='line in4'>| <i>a</i><sub><i>p</i> 1</sub> .......... <i>a</i><sub><i>p</i> <i>q</i></sub> |</div> - </div> - <div class='group'> - <div class='line'>B = | <i>b₁₁</i> .......... <i>b₁</i><sub><i>r</i></sub> |</div> - <div class='line in4'>| |</div> - <div class='line in4'>| <i>b</i><sub><i>q</i> 1</sub> .......... <i>b</i><sub><i>p</i> <i>r</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>where the number of horizontal rows of B, is equal to the -number of vertical columns of A, then by AB, the product -of the matrices A and B, will be denoted the matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>C = | <i>c₁₁</i> .......... <i>c₁</i><sub><i>r</i></sub> |</div> - <div class='line in4'>| |</div> - <div class='line in4'>| <i>c</i><sub><i>pr</i></sub> .......... <i>c</i><sub><i>p</i> <i>p</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>where <i>c</i><sub><i>h</i> <i>k</i></sub> = <i>a</i><sub><i>h</i> 1</sub> <i>b₁</i><sub><i>k</i></sub> + <i>a</i><sub><i>h</i> 2</sub> <i>b</i><sub>2 <i>h</i></sub> + ... <i>a</i><sub><i>k</i> <i>s</i></sub> <i>b</i><sub><i>s</i> <i>k</i></sub> + ... + <i>a</i><sub><i>k</i> <i>q</i></sub> <i>b</i><sub><i>q</i> <i>h</i></sub></p> - -<p class='c005'>these elements being formed by combination of the -horizontal rows of A with the vertical columns of B. For -such a point, the associative law (AB)S = A(BS) holds, -where S is a third matrix which has got as many horizontal -rows as B (or AB) has got vertical columns.</p> - -<p class='c005'>For the transposed matrix of C = BA, we have Ċ = ḂĀ</p> - -<p class='c005'>3⁰. We shall have principally to deal with matrices -with at most four vertical columns and for horizontal -rows.</p> - -<p class='c005'>As a unit matrix (in equations they will be known for -the sake of shortness as the matrix 1) will be denoted the -following matrix (4 × 4 series) with the elements.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(34) | e₁₁ e₁₂ e₁₃ e₁₄ | = | 1 0 0 0 |</div> - <div class='line in6'>| e₂₁ e₂₂ e₂₃ e₂₄ | | 0 1 0 0 |</div> - <div class='line in6'>| e₃₁ e₃₂ e₃₃ e₃₄ | | 0 0 1 0 |</div> - <div class='line in6'>| e₄₁ e₄₂ e₄₃ e₄₄ | | 0 0 0 1 |</div> - </div> - </div> -</div> - -<p class='c010'>For a 4 × 4 series-matrix, Det A shall denote the -determinant formed of the 4 × 4 elements of the matrix. -If det A ≠ 0, then corresponding to A there is a reciprocal -matrix, which we may denote by A⁻¹ so that A⁻¹A = 1.</p> - -<p class='c005'>A matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i> = | 0 <i>f₁₂</i> <i>f</i>₁₃ <i>f₁₄</i> |</div> - <div class='line in5'>| <i>f</i>₂₁ 0 <i>f₂₃</i> <i>f₂₄</i> |</div> - <div class='line in5'>| <i>f₃₁</i> <i>f</i>₃₂ 0 <i>f₃₄</i> |</div> - <div class='line in5'>| <i>f</i>₄₁ <i>f</i>₄₂ <i>f</i>₄₃ 0 |</div> - </div> - </div> -</div> - -<p class='c010'>in which the elements fulfil the relation <i>f</i><sub><i>h</i> <i>k</i></sub> = -<i>f</i><sub><i>h</i> <i>k</i></sub>, is -called an alternating matrix. These relations say that -the transposed matrix <i>ḟ</i> = -<i>f</i>. Then by -<i>f</i><sup>*</sup> -will be -the <i>dual</i>, alternating matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(35)</div> - </div> - <div class='group'> - <div class='line'><i>f</i><sup>*</sup> = | 0 <i>f₃₄</i> <i>f</i>₄₂ <i>f₂₃</i> |</div> - <div class='line in5'>| <i>f</i>₄₃ 0 <i>f₁₄</i> <i>f₃₁</i> |</div> - <div class='line in5'>| <i>f₂₄</i> <i>f</i>₄₁ 0 <i>f₁₂</i> |</div> - <div class='line in5'>| <i>f</i>₃₂ <i>f</i>₁₃ <i>f</i>₂₁ 0 |</div> - </div> - </div> -</div> - -<p class='c010'>Then (36) <i>f</i>* <i>f</i> = <i>f₃₄</i> <i>f₂₂</i> + <i>f₄₂</i> <i>f₃₁</i> + <i>f₃₂</i> <i>f₂₄</i></p> - -<p class='c005'><i>i.e.</i> We shall have a 4 × 4 series matrix in which all the -elements except those on the diagonal from left up to -right down are zero, and the elements in this diagonal -agree with each other, and are each equal to the above -mentioned combination in (36).</p> - -<p class='c005'>The determinant of <i>f</i> is therefore the square of the -combination, by -Det<sup>½</sup><i>f</i> -we shall denote the expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Det<sup>½</sup><i>f</i></div> - <div class='line'>= <i>f₃₂</i> <i>f₁₄</i> <i>f₁₃</i> <i>f₂₄</i> + <i>f₂₁</i> <i>f₃₄</i>·</div> - </div> - </div> -</div> - -<p class='c010'>4⁰. A linear transformation</p> - -<p class='c005'><i>x</i><sub><i>h</i></sub> = α<sub><i>h</i>1</sub> <i>x₁′</i> + α<sub><i>h</i>2</sub> <i>x₂</i>′ + α<sub><i>h</i>3</sub> <i>x₃′</i> + α<sub><i>h</i>4</sub> <i>x₄′</i> (<i>h</i> = 1,2,3,</p> - -<p class='c005'>which is accomplished by the matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A = | α₁₁, α₁₂, α₁₃, α₁₄ |</div> - <div class='line in5'>| |</div> - <div class='line in5'>| α₂₁, α₂₂, α₂₃, α₂₄ |</div> - <div class='line in5'>| |</div> - <div class='line in5'>| α₃₁, α₃₂, α₃₃, α₃₄ |</div> - <div class='line in5'>| |</div> - <div class='line in5'>| α₄₁, α₄₂, α₄₃, α₄₄ |</div> - </div> - </div> -</div> - -<p class='c010'>will be denoted as the transformation A.</p> - -<p class='c005'>By the transformation A, the expression</p> - -<p class='c005'><i>x²₁</i> + <i>x²₂</i> + <i>x²₃</i> + <i>x²₄</i> is changed into the quadratic -for <i>m</i> ∑ α<sub><i>hk</i></sub> <i>x</i><sub><i>h</i></sub>′ <i>x</i><sub><i>k</i></sub>′,</p> - -<p class='c005'>where α<sub><i>hk</i></sub> = α<sub>1<i>k</i></sub> α<sub>1<i>k</i></sub> + α<sub>2<i>h</i></sub> α<sub>2<i>k</i></sub> + α<sub>3<i>h</i></sub> α<sub>3<i>k</i></sub> + α<sub>4<i>h</i></sub> α<sub>4<i>k</i></sub> -are the members of a 4 × 4 series matrix which is the -product of Ā A, the transposed matrix of A into A. If by -the transformation, the expression is changed to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x′₁²</i> + <i>x₂′</i><sup>2</sup> + <i>x₃′</i><sup>2</sup> + <i>x′₄²</i>,</div> - </div> - </div> -</div> - -<p class='c010'>we must have Ā A = 1.</p> - -<p class='c005'>A has to correspond to the following relation, if transformation -(38) is to be a Lorentz-transformation. For the -determinant of A) it follows out of (39) that (Det A)² = -1, or Det A = ± 1.</p> - -<p class='c005'>From the condition (39) we obtain</p> -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A⁻¹ = Ā,</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i> the reciprocal matrix of A is equivalent to the transposed -matrix of A.</p> - -<p class='c005'>For A as Lorentz transformation, we have further -Det A = +1, the quantities involving the index 4 once in -the subscript are purely imaginary, the other co-efficients -are real, and <i>a₄₄</i> > 0.</p> -<p class='c004'>5⁰. A space time vector of the first kind<a id='r21' /><a href='#f21' class='c007'><sup>[21]</sup></a> which s -represented by the 1 × 4 series matrix,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(41) <i>s</i> = |<i>s₁</i> <i>s₂</i> <i>s₃</i> <i>s₄</i>|</div> - </div> - </div> -</div> - -<p class='c010'>is to be replaced by <i>s</i>A in case of a Lorentz transformation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A. <i>i.e.</i> <i>s′</i> = | <i>s₁′</i> <i>s₂′</i> <i>s₃′</i> <i>s₄′</i>| = |<i>s₁</i> <i>s₂</i> <i>s₃</i> <i>s₄</i>| A;</div> - </div> - </div> -</div> - -<p class='c010'>A space-time vector of the 2nd kind<a id='r22' /><a href='#f22' class='c007'><sup>[22]</sup></a> with components <i>f₂₃</i> ... -<i>f₃₄</i> shall be represented by the alternating matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(42) <i>f</i> = | 0 <i>f</i>₁₂ <i>f₁₃</i> <i>f₁₄</i> |</div> - </div> - <div class='group'> - <div class='line in12'>|<i>f₂₁</i> 0 <i>f</i>₂₃ <i>f₂₄</i> |</div> - </div> - <div class='group'> - <div class='line in12'>|<i>f</i>₃₁ <i>f₃₂</i> 0 <i>f₃₄</i> |</div> - </div> - <div class='group'> - <div class='line in12'>|<i>f</i>₄₁ <i>f</i>₄₂ <i>f</i>₄₃ 0 |</div> - </div> - </div> -</div> - -<p class='c010'>and is to be replaced by A⁻¹ <i>f</i> A in case of a Lorentz -transformation [see the rules in <a href='#sect4-5'>§ 5</a> (23) (24)]. Therefore -referring to the expression (37), we have the identity -Det<sup>½</sup> -(Ā <i>f</i> A) = Det A. -Det<sup>½</sup> -<i>f</i>. Therefore -Det<sup>½</sup> -<i>f</i> becomes -an invariant in the case of a Lorentz transformation -[see eq. (26) See. <a href='#sect4-5'>§ 5</a>].</p> - -<p class='c005'>Looking back to (36), we have for the dual matrix -(Ā<i>f</i>*A) (A⁻¹<i>f</i>A) = A⁻¹<i>f</i>*<i>f</i>A = -Det<sup>½</sup> -function. A⁻¹A = -Det<sup>½</sup><i>f</i> -from which it is to be seen that the dual matrix <i>f</i>* behaves -exactly like the primary matrix <i>f</i>, and is therefore a space -time vector of the II kind; <i>f</i>* is therefore known as the -dual space-time vector of <i>f</i> with components (<i>f₁₄</i>, <i>f₂₄</i>, <i>f₃₄</i>,), -(<i>f₂₃</i>}, <i>f₃₁</i>, <i>f₁₂</i>).</p> - -<p class='c005'>6. If <i>w</i> and <i>s</i> are two space-time rectors of the 1st kind -then by <i>w</i> <i>ṡ</i> (as well as by <i>s</i> <i>ẇ</i>) will be understood the -combination (43) <i>w₁</i> <i>s₁</i> + <i>w₂</i> <i>s₂</i> + <i>w₃</i> <i>s₃</i> + <i>w₄</i> <i>s₄</i>.</p> - -<p class='c005'>In case of a Lorentz transformation A, since (<i>w</i>A) (Ā<i>ṡ</i>) -= <i>w</i> <i>s</i>, this expression is invariant.—If <i>w</i> <i>ṡ</i> = 0, then <i>w</i> -and <i>s</i> are perpendicular to each other.</p> - -<p class='c005'>Two space-time rectors of the first kind (<i>w</i>, <i>s</i>) gives us -a 2 × 4 series matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| <i>w₁</i> <i>w₂</i> <i>w₃</i> <i>w₄</i> |</div> - <div class='line'>| <i>s₁</i> <i>s₂</i> <i>s₃</i> <i>s₄</i> |</div> - </div> - </div> -</div> - -<p class='c010'>Then it follows immediately that the system of six -magnitudes (44)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>w₂</i> <i>s₃</i> - <i>w₃</i> <i>s₂</i>,</div> - <div class='line'><i>w₃</i> <i>s₁</i> - <i>w₁</i> <i>s₃</i>,</div> - <div class='line'><i>w₁</i> <i>s₂</i> - <i>w₂</i> <i>s₁</i>,</div> - <div class='line'><i>w₁</i> <i>s₄</i> - <i>w₄</i> <i>s₁</i>,</div> - <div class='line'><i>w₂</i> <i>s₄</i> - <i>w₄</i> <i>s₂</i>,</div> - <div class='line'><i>w₃</i> <i>s₄</i> - <i>w₄</i> <i>s₃</i>,</div> - </div> - </div> -</div> - -<p class='c010'>behaves in case of a Lorentz-transformation as a space-time -vector of the II kind. The vector of the second kind with -the components (44) are denoted by [<i>w</i>, <i>s</i>]. We see easily -that -Det<sup>½</sup> -[<i>w</i>, <i>s</i>] = 0. The dual vector of [<i>w</i>, <i>s</i>] shall be -written as [<i>w</i>, <i>s</i>].</p> - -<p class='c005'>If <i>ẇ</i> is a space-time vector of the 1st kind, <i>f</i> of the -second kind, <i>w</i> <i>f</i> signifies a 1 × 4 series matrix. In case -of a Lorentz-transformation A, <i>w</i> is changed into <i>w′</i> = <i>w</i>A, -<i>f</i> into <i>f′</i> = A⁻¹ <i>f</i> A,—therefore <i>w′</i> <i>f′</i> becomes = (<i>w</i>A A⁻¹ <i>f</i> -A) = <i>w</i> <i>f</i> A <i>i.e.</i> <i>w</i> <i>f</i> is transformed as a space-time vector of -the 1st kind.<a id='r23' /><a href='#f23' class='c007'><sup>[23]</sup></a> We can verify, when <i>w</i> is a space-time vector -of the 1st kind, <i>f</i> of the 2nd kind, the important identity</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(45) [<i>w</i>, <i>w</i><i>f</i>] + [<i>w</i>, <i>w</i><i>f</i>*]* = (<i>w</i>] <i>ẇ</i>)<i>f</i>.</div> - </div> - </div> -</div> - -<p class='c010'>The sum of the two space time vectors of the second kind -on the left side is to be understood in the sense of the -addition of two alternating matrices.</p> - -<p class='c005'>For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = <i>i</i>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ω<i>f</i> = | <i>i</i><i>f</i>₄₁, <i>i</i><i>f</i>₄₂, <i>i</i><i>f</i>₄₃, 0 |;</div> - <div class='line'>ω<i>f</i>* = | <i>i</i><i>f</i>₃₂, <i>i</i><i>f</i>₁₃, <i>i</i><i>f</i>₂₁, 0 |</div> - </div> - <div class='group'> - <div class='line'>[ω · ω<i>f</i>] = 0, 0, 0, <i>f</i>₄₁, <i>f</i>₄₂, <i>f</i>₄₃;</div> - <div class='line'>[ω · ω<i>f</i>*]* = 0, 0, 0, <i>f</i>₃₂, <i>f</i>₁₃, <i>f</i>₂₁.</div> - </div> - </div> -</div> - -<p class='c010'>The fact that in this special case, the relation is satisfied, -suffices to establish the theorem (45) generally, for this -relation has a covariant character in case of a Lorentz -transformation, and is homogeneous in (ω₁, ω₂, ω₃, ω₄).</p> - -<p class='c005'>After these preparatory works let us engage ourselves -with the equations (C,) (D,) (E) by means which the constants -ε μ, σ will be introduced.</p> - -<p class='c005'>Instead of the space vector <i>u</i>, the velocity of matter, we -shall introduce the space-time vector of the first kind ω with -the components.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ω₁ = <i>u</i><sub><i>x</i></sub>/√(1 - <i>u²</i>),</div> - <div class='line'>ω₂ = <i>u</i><sub><i>y</i></sub>/√(1 - <i>u²</i>),</div> - <div class='line'>ω₃ = <i>u</i><sub><i>z</i></sub>/√(1 - <i>u²</i>),</div> - <div class='line'>ω₄ = <i>i</i>/√(1 - <i>u²</i>).</div> - </div> - </div> -</div> - -<p class='c010'>(40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 -and -<i>i</i>ω₄ > 0.</p> - -<p class='c005'>By F and <i>f</i> shall be understood the space time vectors -of the second kind M - <i>i</i>E, <i>m</i> - <i>ie</i>.</p> - -<p class='c005'>In Φ = ωF, we have a space time vector of the first kind -with components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄</div> - </div> - <div class='group'> - <div class='line'>Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄</div> - </div> - <div class='group'> - <div class='line'>Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄</div> - </div> - <div class='group'> - <div class='line'>Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃</div> - </div> - </div> -</div> - -<p class='c010'>The first three quantities (φ₁, φ₂, φ₃) are the components -of the space-vector (E + [<i>u</i>, M])/√(1 - <i>u²</i>),</p> - -<p class='c005'>and further (φ₄ = <i>i</i>[<i>u</i> E]/√(1 - <i>u²</i>).</p> - -<p class='c005'>Because F is an alternating matrix,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(49) ωΦ = ω₁ φ₁ + ω₂ Φ₂ + ω₃ Φ₃ + ω₄ Φ₄ = 0.</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i> Φ is perpendicular to the vector ω; we can also -write Φ₄ = <i>i</i>[ω<sub>x</sub> Φ₁ + ω<sub>y</sub> Φ₂ + ω<sub>z</sub> Φ₃].</p> - -<p class='c005'>I shall call the space-time vector Φ of the first kind as -the <i>Electric Rest Force</i>.<a id='r24' /><a href='#f24' class='c007'><sup>[24]</sup></a></p> - -<p class='c005'>Relations analogous to those holding between -ωF, -E, M, U, hold amongst -ω<i>f</i>, <i>e</i>, <i>m</i>, <i>u</i>, and in particular -ω<i>f</i> -is normal to ω. The relation (C) can be written as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>{C} ω<i>f</i> = εωF.</div> - </div> - </div> -</div> - -<p class='c010'>The expression (ω<i>f</i>) gives four components, but the -fourth can be derived from the first three.</p> - -<p class='c005'>Let us now form the time-space vector 1st kind, -ψ - <i>i</i>ω<i>f</i>*, whose components are</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ψ₁ = -<i>i</i>(ω₂ <i>f₃₄</i> + ω₃ <i>f</i>₄₂ + ω₄ <i>f₂₃</i>)</div> - <div class='line'>ψ₂ = -<i>i</i>(ω₁ <i>f</i>₄₃ + ω₃ <i>f</i>₄₄ + ω₄ <i>f₃₁</i>)</div> - <div class='line'>ψ₃ = -<i>i</i>(ω₁ <i>f₂₄</i> + ω₂ <i>f</i>₄₁ + ω₄ <i>f₁₂</i>)</div> - <div class='line'>ψ₄ = -<i>i</i>(ω₁ <i>f</i>₃₂ + ω₂ <i>f</i>₁₃ + ω₃ <i>f</i>₂₁)</div> - </div> - </div> -</div> - -<p class='c010'>Of these, the first three ψ₁, ψ₂, ψ₃, are the <i>x</i>, <i>y</i>, <i>z</i> -components of the space-vector 51) (m - (<i>ue</i>))/√(1 - <i>u²</i>) -and further (52) ψ₄ = <i>i</i>(<i>u</i>m)/√(1 - <i>u²</i>).</p> - -<p class='c005'>Among these there is the relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(53) ωψ = ω₁ ψ₁ + ω₂ ψ₂ + ω₃ ψ₃ + ω₄ ψ₄ = 0</div> - </div> - </div> -</div> - -<p class='c010'>which can also be written as ψ₄ = <i>i</i> (<i>u</i><sub><i>x</i></sub> ψ₁ + <i>u</i><sub><i>y</i></sub> ψ₂ + <i>u</i><sub><i>z</i></sub> ψ₃).</p> - -<p class='c005'>The vector ψ is perpendicular to ω; we can call it the -<i>Magnetic rest-force</i>.</p> - -<p class='c005'>Relations analogous to these hold among the quantities -ωF*, M, E, <i>u</i> and Relation (D) can be replaced by the -formula</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>{ D } -ωF* = μψ<i>f</i>*.</div> - </div> - </div> -</div> - -<p class='c010'>We can use the relations (C) and (D) to calculate -F and <i>f</i> from Φ and ψ we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ωF = -Φ, ωF* = -<i>i</i>μψ, ω<i>f</i> = -εΦ, ω<i>f</i>* = -<i>i</i>ψ.</div> - </div> - </div> -</div> - -<p class='c010'>and applying the relation (45) and (46), we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>F = [ω. Φ] + <i>i</i>μ[ω. ψ]* 55)</div> - <div class='line'><i>f</i> = ε[ω. Φ] + <i>i</i>[ω. ψ]* 56)</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i></p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + <i>i</i>μ [ω₃ Ψ₄ - ω₄ ψ₃], etc.</div> - <div class='line'><i>f₁₂</i> = ε(ω₁ Φ₂ - ω₂ φ₁) + <i>i</i> [ω₃ ψ₄ - ω₄ ψ₃]., etc.</div> - </div> - </div> -</div> - -<p class='c010'>Let us now consider the space-time vector of the -second kind [Φ ψ], with the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>[ Φ₂ ψ₃ - Φ₃ ψ₂, Φ₃ ψ₁ - Φ₁ ψ₃, Φ₁ ψ₂ - Φ₂ ψ₁ ]</div> - <div class='line'>[ Φ₁ ψ₄ - Φ₄ ψ₁, Φ₂ ψ₄ - Φ₄ ψ₂, Φ₃ ψ₄ - Φ₄ ψ₃ ]</div> - </div> - </div> -</div> - -<p class='c010'>Then the corresponding space-time vector of the first -kind ω[Φ, ψ] vanishes identically owing to equations 9) -and 53)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>for ω[Φ.ψ] = -(ωψ)Φ + (ωΦ)ψ</div> - </div> - </div> -</div> - -<p class='c010'>Let us now take the vector of the 1st kind</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(57) Ω = <i>i</i>ω[Φψ]*</div> - </div> - </div> -</div> - -<p class='c010'>with the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Ω₁ = -<i>i</i> | ω₂ ω₃ ω₄ |</div> - <div class='line in10'>| Φ₂ Φ₃ Φ₄ |</div> - <div class='line in10'>| ψ₂ ψ₃ ψ₄ |, etc.</div> - </div> - </div> -</div> - -<p class='c010'>Then by applying rule (45), we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(58) [Φ.ψ] = <i>i</i>[ωΩ]*</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i> Φ₁ψ₂ - Φ₂ψ₁ = <i>i</i>(ω₃Ω₄ - ω₄Ω₃) etc.</p> - -<p class='c005'>The vector Ω fulfils the relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0,</div> - </div> - </div> -</div> - -<p class='c010'>(which we can write as Ω₄ = <i>i</i>(ω<sub>x</sub>Ω₁ + ω<sub>y</sub>Ω₂ + ω<sub>z</sub>Ω₃) -and Ω is also normal to ω. In case ω = 0, -we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>[Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ |</div> - <div class='line in13'>| ψ₁ ψ₂ ψ₃ |.</div> - </div> - </div> -</div> - -<p class='c010'>I shall call Ω, which is a space-time vector 1st kind the -Rest-Ray.</p> - -<p class='c005'>As for the relation E), which introduces the conductivity σ -we have -ωS = -(ω₁<i>s₁</i> + ω₂<i>s₂</i> + ω₃<i>s₃</i> + ω₄<i>s₄</i>) - = (- | <i>u</i> | C<sub><i>u</i></sub> + ρ)/√(1 - <i>u²</i>) = ρ′.</p> - -<p class='c005'>This expression gives us the rest-density of electricity -(see <a href='#sect4-8'>§8</a> and <a href='#sect4-4'>§4</a>).</p> - -<p class='c005'>Then 61) = <i>s</i> + (ω<i>ṡ</i>)ω -represents a space-time vector of the 1st kind, which since -ωω = -1, is normal to ω, and which I may call the rest-current. -Let us now conceive of the first three component -of this vector as the (<i>x</i>-<i>y</i>-<i>z</i>) co-ordinates of the space-vector, -then the component in the direction of <i>u</i> is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>C<sub><i>u</i></sub> - (| <i>u</i> | ρ′)/√(1 - <i>u²</i>)</div> - <div class='line in2'>= (<i>c</i><sub><i>u</i></sub> - | <i>u</i> |ρ)/√(1 - <i>u²</i>)</div> - <div class='line in2'>= J<sub><i>u</i></sub>/(1 - <i>u²</i>)</div> - </div> - </div> -</div> - -<p class='c010'>and the component in a perpendicular direction is C<sub><i>u</i></sub> = J<sub><i>ū</i></sub>.</p> - -<p class='c005'>This space-vector is connected with the space-vector -J = C - ρ<i>u</i>, which we denoted in <a href='#sect4-8'>§8</a> as the conduction-current.</p> - -<p class='c005'>Now by comparing with Φ = -ωF, the relation (E) can -be brought into the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>{E} <i>s</i> + (ω<i>ṡ</i>)ω = - σωF,</div> - </div> - </div> -</div> - -<p class='c010'>This formula contains four equations, of which the -fourth follows from the first three, since this is a space-time -vector which is perpendicular to ω.</p> - -<p class='c005'>Lastly, we shall transform the differential equations -(A) and (B) into a typical form.</p> -<h3 class='c008'>§12. The Differential Operator Lor.</h3> -<div class='lg-container-b c011'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S<sub><i>kh</i></sub> |</div> - <div class='line in30'>| S₂₁ S₂₂ S₂₃ S₂₄ |</div> - <div class='line in30'>| S₃₁ S₃₂ S₃₃ S₃₄ |</div> - <div class='line in30'>| S₄₁ S₄₂ S₄₃ S₄₄ |</div> - </div> - </div> -</div> - -<p class='c010'>with the condition that in case of a Lorentz transformation -it is to be replaced by ĀSA, may be called a space-time -matrix of the II kind. We have examples of this in:—</p> - -<p class='c005'>1) the alternating matrix <i>f</i>, which corresponds to the -space-time vector of the II kind,—</p> - -<p class='c005'>2) the product <i>f</i>F of two such matrices, for by a transformation -A, it is replaced by (A⁻¹<i>f</i>A·A⁻¹FA) = A⁻¹<i>f</i>FA,</p> - -<p class='c005'>3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are -two space-time vectors of the 1st kind, the 4 × 4 matrix with -the element S<sub><i>hk</i></sub> = ω<sub><i>h</i></sub>Ω<sub><i>k</i></sub>,</p> - -<p class='c005'>lastly in a multiple L of the unit matrix of 4 × 4 series -in which all the elements in the principal diagonal are -equal to L, and the rest are zero.</p> - -<p class='c005'>We shall have to do constantly with functions of the -space-time point (<i>x</i>, <i>y</i>, <i>z</i>, <i>it</i>), and we may with advantage</p> - -<p class='c005'>employ the 1 × 4 series matrix, formed of differential -symbols,—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in10'>| ∂/∂<i>x</i>, ∂/∂<i>y</i>, ∂/∂<i>z</i>, ∂/<i>i</i>∂<i>t</i>,|</div> - <div class='line in2'>or (63) | ∂/∂<i>x₁</i> ∂/∂<i>x₂</i> ∂/∂<i>x₃</i> ∂/∂<i>x₄</i> |</div> - </div> - </div> -</div> - -<p class='c010'>For this matrix I shall use the shortened from “lor.”<a id='r25' /><a href='#f25' class='c007'><sup>[25]</sup></a></p> - -<p class='c005'>Then if S is, as in (62), a space-time matrix of the -II kind, by lor S′ will be understood the 1 × 4 series -matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| K₁ K₂ K₃ K₄ |</div> - </div> - </div> -</div> - -<p class='c010'>where K<sub><i>k</i></sub> = ∂S<sub>1<i>k</i></sub>/∂<i>x₁</i> + ∂S<sub>2<i>k</i></sub>/∂<i>x₂</i> + ∂S<sub>3<i>k</i></sub>/∂<i>x₃</i> + ∂S<sub>4<i>h</i></sub>/∂<i>x₄</i>.</p> - -<p class='c005'>When by a Lorentz transformation A, a new reference -system (<i>x′₁</i> <i>x′₂</i> <i>x′₃</i> <i>x₄</i>) is introduced, we can use the operator</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor′ = | ∂/∂<i>x₁′</i> ∂/∂<i>x₂′</i> ∂/∂<i>x₃′</i> ∂/∂<i>x₄′</i> |</div> - </div> - </div> -</div> - -<p class='c010'>Then S is transformed to S′= Ā S A = | S′<sub><i>hk</i></sub> |, so by -lor 'S′ is meant the 1 × 4 series matrix, whose element are</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>K’<sub><i>k</i></sub> = ∂S′<sub>1<i>k</i></sub>/∂<i>x₁′</i> + ∂S′<sub>2<i>k</i></sub>/∂<i>x₂′</i></div> - <div class='line in2'>+ ∂S′<sub>3<i>k</i></sub>/∂<i>x₃′</i> + ∂S′<sub>4<i>k</i></sub>/∂<i>x₄′</i>.</div> - </div> - </div> -</div> - -<p class='c010'>Now for the differentiation of any function of (<i>x</i> <i>y</i> <i>z</i> <i>t</i>) -we have the rule ∂/∂<i>x</i><sub><i>k</i></sub>′ = ∂/∂<i>x₁</i> ∂<i>x₁</i>/∂<i>x</i><sub><i>k</i></sub>′ + ∂/∂<i>x₂</i> ∂<i>x₂</i>/∂<i>x</i><sub><i>k</i></sub>′ -+ ∂/∂<i>x₃</i> ∂<i>x₃</i>/∂<i>x</i><sub><i>k</i></sub>′ + ∂/∂<i>x₄</i> ∂<i>x₄</i>/∂<i>x</i><sub><i>k</i></sub>′ -= ∂/∂<i>x₁</i> <i>a</i><sub>1<i>k</i></sub> + ∂/∂<i>x₂</i> <i>a</i><sub>2<i>k</i></sub> + ∂/∂<i>x₃</i> <i>a</i><sub>3<i>k</i></sub> + ∂/∂<i>x₄</i> <i>a</i><sub>4<i>k</i></sub>.</p> - -<p class='c005'>so that, we have symbolically lor′ = lor A.</p> - -<p class='c005'>Therefore it follows that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor ′S′ = lor (A A⁻¹ SA) = (lor S)A.</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i>, lor S behaves like a space-time vector of the first -kind.</p> - -<p class='c005'>If L is a multiple of the unit matrix, then by lor L will -be denoted the matrix with the elements</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| ∂L/∂<i>x₁</i> ∂L/∂<i>x₂</i> ∂L/∂<i>x₃</i> ∂L/∂<i>x₄</i> |</div> - </div> - </div> -</div> - -<p class='c010'>If <i>s</i> is a space-time vector of the 1st kind, then</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor <i>ṡ</i> = ∂<i>s₁</i>/∂<i>x₁</i> + ∂<i>s₂</i>/∂<i>x₂</i> + ∂<i>s₃</i>/∂<i>x₃</i> + ∂<i>s₄</i>/∂<i>x₄</i>.</div> - </div> - </div> -</div> - -<p class='c010'>In case of a Lorentz transformation A, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor ′<i>ṡ′</i> = lor A. Ā<i>s</i> = lor <i>s</i>.</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i>, lor <i>s</i> is an invariant in a Lorentz-transformation.</p> - -<p class='c005'>In all these operations the operator lor plays the part -of a space-time vector of the first kind.</p> - -<p class='c005'>If <i>f</i> represents a space-time vector of the second kind,—lor -<i>f</i> denotes a space-time vector of the first kind with -the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>f₁₂</i>/∂<i>x₂</i> + ∂<i>f₁₃</i>/∂<i>x₃</i> + ∂<i>f₁₄</i>/∂<i>x₄</i>,</div> - <div class='line'>∂<i>f₂₁</i>/∂<i>x₁</i> + ∂<i>f₂₃</i>/∂<i>x₃</i> + ∂<i>f₂₄</i>/∂<i>x₄</i>,</div> - <div class='line'>∂<i>f₃₁</i>/∂<i>x₁</i> + ∂<i>f₃₂</i>/∂<i>x₂</i> + ∂<i>f₃₄</i>/∂<i>x₄</i>,</div> - <div class='line'>∂<i>f₄₁</i>/∂<i>x₁</i> + ∂<i>f₄₂</i>/∂<i>x₂</i> + ∂<i>f₄₃</i>/∂<i>x₃</i></div> - </div> - </div> -</div> - -<p class='c010'>So the system of differential equations (A) can be -expressed in the concise form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>{A} lor f = -<i>s</i>,</div> - </div> - </div> -</div> - -<p class='c010'>and the system (B) can be expressed in the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>{B} log F* = 0.</div> - </div> - </div> -</div> - -<p class='c010'>Referring back to the definition (67) for log <i>ṡ</i>, we -find that the combinations lor ([=(lor <i>f</i>)=]), and lor ([=(lor F*)]) -vanish identically, when <i>f</i> and F* are alternating matrices. -Accordingly it follows out of {A}, that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(68) (∂<i>s₁</i>/∂<i>x₁</i>) + (∂<i>s₂</i>/∂<i>x₂</i>) + (∂<i>s₃</i>/∂<i>x₃</i>) + (∂<i>s₄</i>/∂<i>x₄</i>) = 0,</div> - </div> - </div> -</div> - -<p class='c010'>while the relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(69) lor (lor F*) = 0,</div> - </div> - </div> -</div> - -<p class='c010'>signifies that of the four -equations in {B}, only three represent independent -conditions.</p> - -<p class='c005'>I shall now collect the results.</p> - -<p class='c005'>Let ω denote the space-time vector of the first kind</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(<i>u</i>/√(1 - <i>u²</i>}), <i>i</i>/√(1 - <i>u²</i>))</div> - </div> - <div class='group'> - <div class='line'>(<i>u</i> = velocity of matter),</div> - </div> - </div> -</div> - -<p class='c010'>F the space-time vector of the second kind (M,-<i>i</i>E)</p> - -<p class='c005'>(M = magnetic induction, E = Electric force,</p> - -<p class='c005'><i>f</i> the space-time vector of the second kind (<i>m</i>,-<i>ie</i>)</p> - -<p class='c005'>(<i>m</i> = magnetic force, <i>e</i> = Electric Induction.</p> - -<p class='c005'><i>s</i> the space-time vector of the first kind (C, <i>i</i>ρ)</p> - -<p class='c005'>(ρ = electrical space-density, C - ρ<i>u</i> = conductivity current,</p> - -<p class='c005'>ε = dielectric constant, μ = magnetic permeability,</p> - -<p class='c005'>σ = conductivity,</p> - -<p class='c005'>then the fundamental equations for electromagnetic -processes in moving bodies are<a id='r26' /><a href='#f26' class='c007'><sup>[26]</sup></a></p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>{A} lor <i>f</i> = -<i>s</i></div> - </div> - <div class='group'> - <div class='line'>{B} log F* = 0</div> - </div> - <div class='group'> - <div class='line'>{C} ω<i>f</i> = εωF</div> - </div> - <div class='group'> - <div class='line'>{D} ωF* = μω<i>f</i>*</div> - </div> - <div class='group'> - <div class='line'>{E} <i>s</i> + (ω<i>ṡ</i>), <i>w</i> = - σωF.</div> - </div> - </div> -</div> - -<p class='c010'>ω ῶ = -1, and ωF, ω<i>f</i>, ωF*, ω<i>f</i>*, <i>s</i> + (ω<i>s</i>)ω which -are space-time vectors of the first kind are all normal to -ω, and for the system {B}, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor (lor F*) = 0.</div> - </div> - </div> -</div> - -<p class='c010'>Bearing in mind this last relation, 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 <i>u</i> as a function of <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, when proper fundamental -data are given.</p> -<h3 class='c008'>§ 13. The Product of the Field-vectors <i>f</i> F.</h3> -<p class='c004'>Finally let us enquire about the laws which lead to the -determination of the vector ω as a function of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>.) -In these investigations, the expressions which are obtained -by the multiplication of two alternating matrices</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i> = | 0 <i>f₁₂</i> <i>f₁₃</i> <i>f₁₄</i> |</div> - <div class='line in6'>| <i>f₂₁</i> 0 <i>f₂₃</i> <i>f₂₄</i> |</div> - <div class='line in6'>| <i>f₃₁</i> <i>f₃₂</i> 0 <i>f₃₄</i> |</div> - <div class='line in6'>| <i>f₄₁</i> <i>f₄₂</i> <i>f₄₃</i> 0 |</div> - </div> - <div class='group'> - <div class='line'>F = | 0 F₁₂ F₁₃ F₁₄ |</div> - <div class='line in4'>| F₂₁ 0 F₂₃ F₂₄ |</div> - <div class='line in4'>| F₃₁ F₃₂ 0 F₃₄ |</div> - <div class='line in4'>| F₄₁ F₄₂ F₄₃ 0 |</div> - </div> - </div> -</div> - -<p class='c010'>are of much importance. Let us write,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(70) <i>f</i>F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |</div> - </div> - <div class='group'> - <div class='line in11'>| S₂₁ S₂₂ - L S₂₃ S₂₄ |</div> - </div> - <div class='group'> - <div class='line in11'>| S₃₁ S₃₂ S₃₃ - L S₃₄ |</div> - </div> - <div class='group'> - <div class='line in11'>| S₄₁ S₄₂ S₄₃ S₄₄ - L |</div> - </div> - </div> -</div> - -<p class='c010'>Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.</p> - -<p class='c005'>Let L now denote the symmetrical combination of the -indices 1, 2, 3, 4, given by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(72) L = ½(<i>f₂₃</i> F₂₃ + <i>f₃₁</i>F₃₁ + <i>f₁₂</i> + F₁₂ + <i>f₁₄</i> F₁₄</div> - <div class='line'>+ <i>f₂₄</i> F₂₄ + <i>f₃₄</i> F₃₄)</div> - </div> - </div> -</div> - -<p class='c010'>Then we shall have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(73) S₁₁ = ½(<i>f₂₃</i> F₂₃ + <i>f₃₄</i> F₃₄ + <i>f₄₂</i> F₄₂ - <i>f₁₂</i> F₁₂</div> - <div class='line'>- <i>f₁₃</i> F₁₃ <i>f₁₄</i> F₁₄)</div> - </div> - <div class='group'> - <div class='line'>S₁₂ = <i>f₁₃</i> F₃₂ + <i>f₁₄</i> F₄₂ etc....</div> - </div> - </div> -</div> - -<p class='c010'>In order to express in a real form, we write</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |</div> - </div> - <div class='group'> - <div class='line in9'>| S₂₁ S₂₂ S₂₃ S₂₄ |</div> - </div> - <div class='group'> - <div class='line in9'>| S₃₁ S₃₂ S₃₃ S₃₄ |</div> - </div> - <div class='group'> - <div class='line in9'>| S₄₁ S₄₂ S₄₃ S₄₄ |</div> - </div> - <div class='group'> - <div class='line in7'>= | X<sub><i>x</i></sub> Y<sub><i>x</i></sub> Z<sub><i>x</i></sub> -<i>i</i>T<sub><i>x</i></sub> |</div> - </div> - <div class='group'> - <div class='line in9'>| X<sub><i>y</i></sub> Y<sub><i>y</i></sub> Z<sub><i>y</i></sub> -<i>i</i>T<sub><i>y</i></sub> |</div> - </div> - <div class='group'> - <div class='line in9'>| X<sub><i>z</i></sub> Y<sub><i>z</i></sub> Z<sub><i>z</i></sub> -<i>i</i>T<sub><i>z</i></sub> |</div> - </div> - <div class='group'> - <div class='line in9'>| -<i>i</i>X<sub><i>t</i></sub> -<i>i</i>Y<sub><i>t</i></sub> -<i>i</i>Z<sub><i>t</i></sub> T<sub><i>t</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>Now X<sub><i>x</i></sub> = ½[<i>m</i><sub><i>x</i></sub>M<sub><i>x</i></sub> - <i>m</i><sub><i>y</i></sub>M<sub><i>y</i></sub> - <i>m</i><sub><i>z</i></sub>M<sub><i>z</i></sub> + <i>e</i><sub><i>x</i></sub>E<sub><i>x</i></sub> - <i>e</i><sub><i>y</i></sub>E<sub><i>y</i></sub> - <i>e</i><sub><i>z</i></sub>E<sub><i>z</i></sub>]</p> - -<p class='c005'>so</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(75) X<sub><i>y</i></sub> = <i>m</i><sub><i>x</i></sub>M<sub><i>y</i></sub> + <i>e</i><sub><i>y</i></sub>E<sub><i>x</i></sub>, Y<sub><i>x</i></sub> = <i>m</i><sub><i>y</i></sub>M<sub><i>x</i></sub> + <i>e</i><sub><i>x</i></sub>E<sub><i>y</i></sub> etc.</div> - </div> - <div class='group'> - <div class='line in5'>X<sub><i>t</i></sub> = <i>e</i><sub><i>y</i></sub>M<sub><i>z</i></sub> - <i>e</i><sub><i>z</i></sub>M<sub><i>y</i></sub>, T<sub><i>x</i></sub> = <i>m</i><sub><i>x</i></sub>E<sub><i>y</i></sub> - <i>m</i><sub><i>y</i></sub>E<sub><i>z</i></sub>, etc.</div> - </div> - <div class='group'> - <div class='line in5'>T<sub><i>t</i></sub> = ½[<i>m</i><sub><i>x</i></sub>M<sub><i>x</i></sub> + <i>m</i><sub><i>y</i></sub>M<sub><i>y</i></sub> + <i>m</i><sub><i>z</i></sub>M<sub><i>z</i></sub> + <i>e</i><sub><i>x</i></sub>E<sub><i>x</i></sub> + <i>e</i><sub><i>y</i></sub>E<sub><i>y</i></sub> + <i>e</i><sub><i>z</i></sub>E<sub><i>z</i></sub>]</div> - </div> - <div class='group'> - <div class='line in5'>L<sub><i>t</i></sub> = ½[<i>m</i><sub><i>x</i></sub>M<sub><i>x</i></sub> + <i>m</i><sub><i>y</i></sub>M<sub><i>y</i></sub> + <i>m</i><sub><i>z</i></sub>M<sub><i>z</i></sub> - <i>e</i><sub><i>x</i></sub>E<sub><i>x</i></sub> - <i>e</i><sub><i>y</i></sub>E<sub><i>y</i></sub> - <i>e</i><sub><i>z</i></sub>E<sub><i>z</i></sub>]</div> - </div> - </div> -</div> - -<p class='c010'>These quantities<a id='r27' /><a href='#f27' class='c007'><sup>[27]</sup></a> are all real. In the theory for bodies -at rest, the combinations (X<sub><i>x</i></sub>, X<sub><i>y</i></sub>, X<sub><i>z</i></sub>, Y<sub><i>z</i></sub>, Y<sub><i>y</i></sub>, Y<sub><i>z</i></sub>, Z<sub><i>x</i></sub>, -Z<sub><i>y</i></sub>, Z<sub><i>z</i></sub>) are known as “Maxwell’s Stresses,” T<sub><i>x</i></sub>, T<sub><i>y</i></sub>, T<sub><i>z</i></sub> -are known as the Poynting’s Vector, T<sub><i>t</i></sub> as the electromagnetic -energy-density, and L as the Langrangian -function.</p> - -<p class='c005'>On the other hand, by multiplying the alternating -matrices of <i>f</i>* and F*, we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |</div> - </div> - <div class='group'> - <div class='line in11'>| -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |</div> - </div> - <div class='group'> - <div class='line in11'>| -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |</div> - </div> - <div class='group'> - <div class='line in11'>| -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |</div> - </div> - </div> -</div> - -<p class='c010'>and hence, we can put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(78) <i>f</i>F = S - L, F*<i>f</i>* = -S - L,</div> - </div> - </div> -</div> - -<p class='c010'>where by L, we mean L-times the unit matrix, <i>i.e.</i> the -matrix with elements</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| L<i>e</i><sub><i>hk</i></sub> |, (<i>e</i><sub><i>hh</i></sub> = 1, <i>e</i><sub><i>hk</i></sub> = 0, <i>h</i> ≠ <i>k</i> <i>h</i>, <i>k</i> = 1, 2, 3, 4).</div> - </div> - </div> -</div> - -<p class='c010'>Since here SL = LS, we deduce that,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>F*<i>f</i>*<i>f</i>F = (-S - L)(S - L) = -SS + L²,</div> - </div> - </div> -</div> - -<p class='c010'>and find, since <i>f</i>*<i>f</i> = -Det<sup>½</sup><i>f</i>, -F*F = -Det<sup>½</sup>F, -we arrive -at the interesting</p> - -<p class='c005'>conclusion</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(79) SS = L² - Det<sup>½</sup><i>f</i> Det<sup>½</sup>F</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i> 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 -zero, the elements in the principal diagonal are all equal -and have the value given on the right-hand side of (79). -Therefore the general relations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(80) S<sub><i>h</i>1</sub> S<sub>1<i>k</i></sub> + S<sub><i>h</i>2</sub> S<sub>2<i>k</i></sub> + S<sub><i>h</i>3</sub> S<sub>3<i>k</i></sub> + S<sub><i>h</i>4</sub> S<sub>4<i>k</i></sub> = 0,</div> - </div> - </div> -</div> - -<p class='c010'><i>h</i>, <i>k</i> being unequal indices in the series 1, 2, 3, 4, and</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(81) S<sub><i>h</i>1</sub> S<sub>1<i>h</i></sub> + S<sub><i>h</i>2</sub> S<sub>2<i>h</i></sub> + S<sub><i>h</i>3</sub> S<sub>3<i>h</i></sub> + S{<i>h</i>4} S<sub>4<i>h</i></sub> = L² -</div> - <div class='line'>Det<sup>½</sup><i>f</i> Det<sup>½</sup>F,</div> - </div> - </div> -</div> - -<p class='c010'>for <i>h</i> = 1, 2, 3, 4.</p> - -<p class='c005'>Now if instead of F, and <i>f</i> in the combinations (72) -and (73), we introduce the electrical rest-force Φ, the -magnetic rest-force ψ, and the rest-ray Ω [(55), (56) and -(57)], we can pass over to the expressions,—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],</div> - </div> - <div class='group'> - <div class='line'>(83) S<sub><i>hk</i></sub> = - ½ ε Φ [=Φ] <i>e</i><sub><i>hk</i></sub> - ½ μ ψ [=ψ] <i>e</i><sub><i>hk</i></sub></div> - <div class='line'>+ ε (Φ<sub><i>h</i></sub> Φ<sub><i>k</i></sub> - Φ ([=Φ]) ω<sub><i>h</i></sub> Ω<sub><i>k</i></sub></div> - <div class='line'>+ μ (ψ<sub><i>h</i></sub> ψ<sub><i>k</i></sub> - Ψ [=ψ] Ω{<i>h</i>} ω<sub><i>k</i></sub>) - ω<sub><i>h</i></sub> ω<sub><i>k</i></sub> - εμ ω<sub><i>h</i></sub> Ω<sub><i>k</i></sub></div> - <div class='line'>(<i>h₁</i> <i>k</i> = 1, 2, 3, 4).</div> - </div> - </div> -</div> - -<p class='c010'>Here we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²</div> - </div> - <div class='group'> - <div class='line'><i>e</i><sub><i>hh</i></sub> = 1, <i>e</i><sub><i>hk</i></sub> = 0 (<i>h</i> ≠ <i>k</i>).</div> - </div> - </div> -</div> - -<p class='c010'>The right side of (82) as well as L is an invariant -in a Lorentz transformation, and the 4 × 4 element on the -right side of (83) as well as S<sub><i>k</i> <i>h</i></sub> represent a space time -vector of the second kind. Remembering this fact, it -suffices, for establishing the theorems (82) and (83) generally, -to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, -ω₄ = <i>i</i>. But for this case ω = 0, we immediately arrive at -the equations (82) and (83) by means (45), (51), (60) -on the one hand, and <i>e</i> = εE, M = μ<i>m</i> on the other hand.</p> - -<p class='c005'>The expression on the right-hand side of (81), which -equals</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>[½ (<i>m</i> M - <i>e</i>E)²] + (<i>em</i>) (EM),</div> - </div> - </div> -</div> - -<p class='c010'>is >= 0, because (<i>em</i> = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring -back to 79), we can denote the positive square root of this -expression as -Det<sup>1/4</sup> -S.</p> - -<p class='c005'>Since <i>ḟ</i> = -<i>f</i>, and Ḟ = -F, we obtain for Ṡ, the -transposed matrix of S, the following relations from (78),</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(84) F<i>f</i> = Ṡ - L, <i>f</i>* F* = -Ṡ - L,</div> - </div> - </div> -</div> - -<p class='c010'>Then is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Ṡ - S = | S<sub><i>h</i> <i>k</i></sub> - S<sub><i>t</i> <i>k</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>an alternating matrix, and denotes a space-time vector -of the second kind. From the expressions (83), we -obtain,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(85) S - Ṡ = - (εμ - 1) [ω, Ω],</div> - </div> - </div> -</div> - -<p class='c010'>from which we deduce that [see (57), (58)].</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(86) ω (S - Ṡ)* = 0,</div> - </div> - <div class='group'> - <div class='line'>(87) ω (S - Ṡ) = (εμ - 1) Ω</div> - </div> - </div> -</div> - -<p class='c010'>When the matter is at rest at a space-time point, ω = 0, -then the equation 86) denotes the existence of the following -equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Z<sub><i>y</i></sub> = Y<sub><i>z</i></sub>, X<sub><i>z</i></sub> = Z<sub><i>x</i></sub>, Y<sub><i>x</i></sub> = X<sub><i>y</i></sub>,</div> - </div> - </div> -</div> - -<p class='c010'>and from 83),</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>T<sub><i>x</i></sub> = Ω₁, T<sub><i>y</i></sub> = Ω₂, T<sub><i>z</i></sub> = Ω₃</div> - </div> - <div class='group'> - <div class='line'>X<sub><i>t</i></sub> = εμΩ₁, Y<sub><i>t</i></sub> = εμΩ₂, Z<sub><i>t</i></sub> = εμΩ₃</div> - </div> - </div> -</div> - -<p class='c010'>Now by means of a rotation of the space co-ordinate -system round the null-point, we can make,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Z<sub><i>y</i></sub> = Y<sub><i>z</i></sub> = 0, X<sub><i>z</i></sub> = Z<sub><i>x</i></sub> = 0, X<sub><i>x</i></sub> = X<sub><i>y</i></sub> = 0,</div> - </div> - </div> -</div> - -<p class='c010'>According to 71), we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(88) X<sub><i>x</i></sub> + Y<sub><i>y</i></sub> + Z<sub><i>z</i></sub> + T<sub><i>t</i></sub> = 0,</div> - </div> - </div> -</div> - -<p class='c010'>and according to 83), T<sub><i>t</i></sub> > 0. In special cases, where ω -vanishes it follows from 81) that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>X<sub><i>x</i></sub>² = Y<sub><i>y</i></sub>² = Z<sub><i>z</i></sub>² = T<sub><i>t</i></sub>², = (Det<sup>1/4</sup> S)²,</div> - </div> - </div> -</div> - -<p class='c010'>and if T, and one of the three magnitudes X<sub><i>x</i></sub>, Y<sub><i>y</i></sub>, Z<sub><i>z</i></sub> are -= -±Det<sup>1/4</sup> -S, the two others = --Det<sup>1/4</sup> -S. If Ω does not -vanish let Ω ≠ 0, then we have in particular from 80)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>T<sub><i>z</i></sub> X<sub><i>t</i></sub> = 0, T<sub><i>z</i></sub> Y<sub><i>t</i></sub> = 0, Z<sub><i>z</i></sub> T<sub><i>z</i></sub> + T<sub><i>z</i></sub> T<sub><i>t</i></sub> = 0,</div> - </div> - </div> -</div> - -<p class='c010'>and if Ω₁ = 0, Ω₂ = 0, Z<sub><i>z</i></sub> = -T<sub><i>t</i></sub> It follows from (81), -(see also 83) that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>X<sub><i>x</i></sub> = -Y<sub><i>y</i></sub> = ±Det<sup>1/4</sup> S,</div> - </div> - </div> -</div> - -<p class='c010'>and -Z<sub><i>z</i></sub> = T<sub><i>t</i></sub> = √(Det<sup>½</sup> S + εμΩ₃²) > Det<sup>1/4</sup>S.</p> - -<p class='c005'>The space-time vector of the first kind</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(89) K = lor S,</div> - </div> - </div> -</div> - -<p class='c010'>is of very great importance for which we now want to -demonstrate a very important transformation</p> - -<p class='c005'>According to 78), S = L + <i>f</i>F, and it follows that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor S = lor L + lor <i>f</i>F.</div> - </div> - </div> -</div> - -<p class='c010'>The symbol ‘lor’ denotes a differential process which -in lor <i>f</i>F, operates on the one hand upon the components -of <i>f</i>, on the other hand also upon the components of F. -Accordingly lor <i>f</i>F can be expressed as the sum of two -parts. The first part is the product of the matrices -(lor <i>f</i>) F, lor <i>f</i> being regarded as a 1 × 4 series matrix. -The second part is that part of lor <i>f</i>F, in which the -diffentiations operate upon the components of F alone. -From 78) we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i>F = -F*<i>f</i>* - 2L;</div> - </div> - </div> -</div> - -<p class='c010'>hence the second part of lor <i>f</i>F = -(lor F*)<i>f</i>* + the part -of -2 lor L, in which the differentiations operate upon the -components of F alone. We thus obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor S = (lor <i>f</i>)F - (lor F*)<i>f</i>* + N,</div> - </div> - </div> -</div> - -<p class='c010'>where N is the vector with the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>N<sub><i>h</i></sub> = ½(∂<i>f₂₃</i>/∂<i>x</i><sub><i>h</i></sub> F₂₃ + ∂<i>f₃₁</i>/∂<i>x</i><sub><i>h</i></sub> F₃₁ + ∂<i>f₁₂</i>/∂<i>x</i><sub><i>h</i></sub> F₁₂ + ∂<i>f₁₄</i>/∂<i>x</i><sub><i>h</i></sub> F₁₄</div> - <div class='line'>+ ∂<i>f₂₄</i>/∂<i>x</i><sub><i>h</i></sub> F₂₄ + ∂<i>f₃₄</i>/∂<i>x</i><sub><i>h</i></sub> F₃₄</div> - <div class='line'>- ∂F₂₃/∂<i>x</i><sub><i>h</i></sub> <i>f₂₃</i> - ∂F₃₁/∂<i>x</i><sub><i>h</i></sub> <i>f</i>₃₁ - ∂F₁₂/∂<i>x</i><sub><i>h</i></sub> <i>f₁₂</i> - ∂F₁₄/∂<i>x</i><sub><i>h</i></sub> <i>f₁₄</i></div> - <div class='line'>- ∂F₂₄/∂<i>x</i><sub><i>h</i></sub> <i>f₂₄</i> - ∂F₃₄/∂<i>x</i><sub><i>h</i></sub> <i>f₃₄</i>),</div> - </div> - <div class='group'> - <div class='line'>(<i>h</i> = 1, 2, 3, 4)</div> - </div> - </div> -</div> - -<p class='c010'>By using the fundamental relations A) and B), 90) -is transformed into the fundamental relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(91) lor S = -<i>s</i>F + N.</div> - </div> - </div> -</div> - -<p class='c010'>In the limitting case ε = 1, μ = 1, <i>f</i> = F, N vanishes -identically.</p> - -<p class='c005'>Now upon the basis of the equations (55) and (56), -and referring back to the expression (82) for L, and from -57) we obtain the following expressions as components -of N,—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(92) N<sub><i>h</i></sub> = - ½ Φ[=Φ]∂ε/∂<i>x</i><sub><i>h</i></sub> - ½ ψ[=ψ]∂μ/∂<i>x</i><sub><i>h</i></sub></div> - <div class='line'>+ (εμ - 1)(Ω₁ ∂ω₁/∂<i>x</i><sub><i>h</i></sub> + Ω₂ ∂ω₂/∂<i>x</i><sub><i>h</i></sub> + Ω₃ ∂ω₃/∂<i>x</i><sub><i>h</i></sub> + Ω₄ ∂ω₄/∂<i>x</i><sub><i>h</i></sub>)</div> - </div> - <div class='group'> - <div class='line'>for <i>h</i> = 1, 2, 3, 4.</div> - </div> - </div> -</div> - -<p class='c010'>Now if we make use of (59), and denote the space-vector -which has Ω₁, Ω₂, Ω₃ as the <i>x</i>, <i>y</i>, <i>z</i> components by -the symbol W, then the third component of 92) can be -expressed in the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(93) (εμ - 1)/√(1 - <i>u²</i>) (W ∂<i>u</i>/∂<i>x</i><sub><i>h</i></sub>),</div> - </div> - </div> -</div> - -<p class='c010'>The round bracket denoting the scalar product of the -vectors within it.</p> -<h3 class='c008'>§ 14. The Ponderomotive Force.<a id='r28' /><a href='#f28' class='c007'><sup>[28]</sup></a></h3> -<p class='c004'>Let us now write out the relation K = lor S = -<i>s</i>F + N -in a more practical form; we have the four equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(94) K₁ = ∂X<sub><i>x</i></sub>/∂<i>x</i> + ∂X<sub><i>y</i></sub>/∂<i>y</i> + ∂X<sub><i>y</i></sub>/∂<i>z</i> - ∂X<sub><i>t</i></sub>/∂<i>t</i> = ρE<sub><i>x</i></sub> + <i>s</i><sub><i>y</i></sub>M<sub><i>z</i></sub> - <i>s</i><sub><i>z</i></sub>M<sub><i>x</i></sub></div> - </div> - <div class='group'> - <div class='line in1'>- ½ Φ[=Φ] ∂ε/∂<i>x</i> - ½ ψ[=ψ]∂μ/∂<i>x</i> + (εμ - 1)/√(1 - <i>u²</i>) (W∂<i>u</i>/∂<i>x</i>),</div> - </div> - <div class='group'> - <div class='line'>(95) K₂ = ∂Y<sub><i>x</i></sub>/∂<i>x</i> + ∂Y<sub><i>y</i></sub>/∂<i>y</i> + ∂Y<sub><i>z</i></sub>/∂<i>z</i> - ∂Y<sub><i>t</i></sub>/∂<i>t</i> = ρE<sub><i>y</i></sub> + <i>s</i><sub><i>z</i></sub>M<sub><i>x</i></sub> - <i>s</i><sub><i>x</i></sub>M<sub><i>y</i></sub></div> - </div> - <div class='group'> - <div class='line in1'>- ½ Φ[=Φ]∂ε/∂<i>y</i> - ½ ψ[=ψ]∂μ/∂<i>y</i> + (εμ - 1)/√(1 - <i>u²</i>) (W∂<i>u</i>/∂<i>y</i>),</div> - </div> - <div class='group'> - <div class='line'>(96) K₃ = ∂Z<sub><i>x</i></sub>/∂<i>x</i> + ∂Z<sub><i>y</i></sub>/∂<i>y</i> + ∂Z<sub><i>z</i></sub>/∂<i>z</i> - ∂Z<sub><i>t</i></sub>/∂<i>t</i> = ρE₂ + <i>s</i><sub><i>x</i></sub>M<sub><i>y</i></sub> - <i>s</i><sub><i>y</i></sub>M₄</div> - </div> - <div class='group'> - <div class='line in1'>- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂<i>z</i> + (εμ - 1)/√(1 - <i>u²</i>) (W∂<i>u</i>/∂<i>z</i>),</div> - </div> - <div class='group'> - <div class='line'>(97) (1/<i>i</i>)K₄ = ∂T<sub><i>y</i></sub>/∂<i>x</i> - ∂T<sub><i>y</i></sub>/∂<i>y</i> - ∂T<sub><i>z</i></sub>/∂<i>z</i> - ∂T<sub><i>t</i></sub>/∂<i>t</i> = <i>s</i><sub><i>x</i></sub>E<sub><i>x</i></sub> + <i>s</i><sub><i>y</i></sub>E<sub><i>y</i></sub> + <i>s</i><sub><i>z</i></sub>E<sub><i>z</i></sub></div> - </div> - <div class='group'> - <div class='line in1'>- ½ Φ[=Φ]∂ε/∂<i>t</i> - ½ ψ[=ψ]∂μ/∂<i>t</i> + (εμ - 1)/√(1 - <i>u²</i>) (W∂<i>u</i>/∂<i>t</i>).</div> - </div> - </div> -</div> - -<p class='c010'>It is my opinion that when we calculate the ponderomotive -force which acts upon a unit volume at the space-time -point <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, it has got, <i>x</i>, <i>y</i>, <i>z</i> components as the -first three components of the space-time vector</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>K + (ωK)ω,</div> - </div> - </div> -</div> - -<p class='c010'>This vector is perpendicular to ω; the law of Energy -finds its expression in the fourth relation.</p> - -<p class='c005'>The establishment of this opinion is reserved for a -separate tract.</p> - -<p class='c005'>In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, -S = ρω, ωK = 0, and we obtain the ordinary equations in the -theory of electrons.</p> - -<div class='chapter'> - <h2 id='art5' class='c003'>APPENDIX <br /> Mechanics and the Relativity-Postulate.</h2> -</div> -<p class='c004'>It would be very unsatisfactory, if the new way of -looking at the time-concept, which permits a Lorentz -transformation, were to be confined to a single part of -Physics.</p> - -<p class='c005'>Now many authors say that classical mechanics stand -in opposition to the relativity postulate, which is taken -to be the basis of the new Electro-dynamics.</p> - -<p class='c005'>In order to decide this let us fix our attention upon a special -Lorentz transformation represented by (10), (11), (12), -with a vector <i>v</i> in any direction and of any magnitude <i>q</i> < 1 -but different from zero. For a moment we shall not suppose -any special relation to hold between the unit of length -and the unit of time, so that instead of <i>t</i>, <i>t′</i>, <i>q</i>, we shall -write <i>ct</i>, <i>ct′</i>, and <i>q</i>/<i>c</i>, where <i>c</i> represents a certain positive -constant, and <i>q</i> -is < -<i>c</i>. The above mentioned equations -are transformed into</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>r′</i><sub><i>ṽ</i></sub> = <i>r</i><sub><i>ṽ</i></sub>,</div> - <div class='line'><i>r′</i><sub><i>v</i></sub> = <i>c</i>(<i>r</i><sub><i>v</i></sub> - <i>qt</i>)/√(<i>c²</i> - <i>q²</i>),</div> - <div class='line'><i>t′</i> = (<i>qr</i><sub><i>v</i></sub> + <i>c²</i><i>t</i>)/<i>c</i>√(<i>c²</i> - <i>q²</i>)</div> - </div> - </div> -</div> - -<p class='c010'>They denote, as we remember, that <i>r</i> is the space-vector -(<i>x</i>, <i>y</i>, <i>z</i>), <i>r′</i> is the space-vector (<i>x′</i> <i>y′</i> <i>z′</i>)</p> - -<p class='c005'>If in these equations, keeping <i>v</i> constant we approach -the limit <i>c</i> = ∞, then we obtain from these</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>r′</i><sub><i>ṽ</i></sub> = <i>r</i><sub><i>ṽ</i></sub>,</div> - <div class='line'><i>r′</i><sub><i>v</i></sub> = <i>r</i><sub><i>v</i></sub> - <i>qt</i>,</div> - <div class='line'><i>t′</i> = <i>t</i>.</div> - </div> - </div> -</div> - -<p class='c010'>The new equations would now denote the transformation -of a spatial co-ordinate system (<i>x</i>, <i>y</i>, <i>z</i>) to another -spatial co-ordinate system (<i>x′</i> <i>y′</i> <i>z′</i>) with parallel axes, the -null point of the second system moving with constant -velocity in a straight line, while the time parameter -remains unchanged. We can, therefore, say that classical -mechanics postulates a covariance of Physical laws for -the group of homogeneous linear transformations of the -expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-<i>x²</i> - <i>y²</i> - <i>z²</i> + <i>c²</i> (1)</div> - </div> - </div> -</div> - -<p class='c010'>when <i>c</i> = ∞.</p> - -<p class='c005'>Now it is rather confusing to find that in one branch -of Physics, we shall find a covariance of the laws for the -transformation of expression (1) with a finite value of <i>c</i>, -in another part for <i>c</i> = ∞.</p> - -<p class='c005'>It is evident that according to Newtonian Mechanics, -this covariance holds for <i>c</i> = ∞ and not for <i>c</i> = velocity of light.</p> - -<p class='c005'>May we not then regard those traditional covariances -for <i>c</i> = ∞ only as an approximation consistent with -experience, the actual covariance of natural laws holding -for a certain finite value of <i>c</i>.</p> - -<p class='c005'>I may here point out that by if instead of the Newtonian -Relativity-Postulate with <i>c</i> = ∞, we assume a relativity-postulate -with a finite <i>c</i>, then the axiomatic construction -of Mechanics appears to gain considerably in perfection.</p> - -<p class='c005'>The ratio of the time unit to the length unit is chosen -in a manner so as to make the velocity of light equivalent -to unity.</p> - -<p class='c005'>While now I want to introduce geometrical figures -in the manifold of the variables (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), it may be -convenient to leave (<i>y</i>, <i>z</i>) out of account, and to treat <i>x</i> -and <i>t</i> as any possible pair of co-ordinates in a plane, -referred to oblique axes.</p> - -<p class='c005'>A space time null point 0 (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> = 0, 0, 0, 0) will be -kept fixed in a Lorentz transformation.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>The figure -<i>x²</i> - <i>y²</i> - <i>z²</i> + <i>t²</i> = 1, <i>t</i> > 0 ... (2)</div> - </div> - </div> -</div> - -<p class='c010'>which represents a hyper boloidal shell, contains the space-time -points A (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> = 0, 0, 0, 1), and all points A′ -which after a Lorentz-transformation enter into the newly -introduced system of reference as (<i>x′</i>, <i>y′</i>, <i>z′</i>, <i>t′</i> = 0, 0, 0, 1).</p> - -<p class='c005'>The direction of a radius vector 0A′ drawn from 0 to -the point A′ of (2), and the directions of the tangents to -(2) at A′ are to be called normal to each other.</p> - -<p class='c005'>Let us now follow a definite position of matter in its -course through all time <i>t</i>. The totality of the space-time -points (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) which correspond to the positions at -different times <i>t</i>, shall be called a space-time line.</p> - -<p class='c005'>The task of determining the motion of matter is comprised -in the following problem:—It is required to establish -for every space-time point the direction of the space-time -line passing through it.</p> - -<p class='c005'>To transform a space-time point P (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) to rest is -equivalent to introducing, by means of a Lorentz transformation, -a new system of reference (<i>x′</i>, <i>y′</i>, <i>z′</i>, <i>t′</i>), in which -the <i>t′</i> axis has the direction 0A′, 0A′ indicating the direction -of the space-time line passing through P. The space -<i>t′</i> = const, which is to be laid through P, is the one which -is perpendicular to the space-time line through P.</p> - -<p class='c005'>To the increment <i>dt</i> of the time of P corresponds the -increment</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d</i>τ = √(<i>dt²</i> - <i>dx²</i> - <i>dy²</i>) - <i>dz²</i> = <i>dt</i>√(1 - <i>u²</i>)</div> - </div> - </div> -</div> - -<p class='c010'>of the newly introduced time parameter <i>t′</i>. The value of -the integral</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∫ <i>dτ</i> = ∫ √(-(<i>dx₁²</i> + <i>dx₂²</i> + <i>dx₃²</i> + <i>dx₄²</i>))</div> - </div> - </div> -</div> - -<p class='c010'>when calculated upon the space-time line from a fixed -initial point P₀ to the variable point P, (both being on the -space-time line), is known as the ‘Proper-time’ of the -position of matter we are concerned with at the space-time -point P. (It is a generalization of the idea of Positional-time -which was introduced by Lorentz for uniform -motion.)</p> - -<p class='c005'>If we take a body R₀ which has got extension in space -at time <i>t₀</i>, then the region comprising all the space-time -line passing through R₀ and <i>t₀</i> shall be called a space-time -filament.</p> - -<p class='c005'>If we have an analytical expression θ(<i>x</i> <i>y</i>, <i>z</i>, <i>t</i>) so that -θ(<i>x</i>, <i>y</i> <i>z</i> <i>t</i>) = 0 is intersected by every space time line of the -filament at one point,—whereby</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-(∂Θ/∂<i>x</i>)², -(∂Θ/∂<i>y</i>)², -(∂Θ/∂<i>z</i>)²,</div> - <div class='line'>-(∂Θ/∂<i>t</i>)² > 0, ∂Θ/∂<i>t</i> > 0.</div> - </div> - </div> -</div> - -<p class='c010'>then the totality of the intersecting points will be called -a cross section of the filament.</p> - -<p class='c005'>At any point P of such across-section, we can introduce -by means of a Lorentz transformation a system of reference -(<i>x′</i>, <i>y</i>, <i>z′</i> <i>t</i>), so that according to this</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂Θ/∂<i>x′</i> = 0, ∂Θ/∂<i>y′</i> = 0, ∂Θ/∂<i>z′</i> = 0, ∂Θ/∂<i>t′</i> > 0.</div> - </div> - </div> -</div> - -<p class='c010'>The direction of the uniquely determined <i>t′</i>—axis in -question here is known as the upper normal of the cross-section -at the point P and the value of <i>d</i>J = ∫∫∫ <i>dx′ dy′ dz′</i> -for the surrounding points of P on the cross-section is -known as the elementary contents (Inhalts-element) of the -cross-section. In this sense R₀ is to be regarded as the -cross-section normal to the <i>t</i> axis of the filament at the -point <i>t</i> = <i>t₀</i>, and the volume of the body R₀ is to be -regarded as the contents of the cross-section.</p> - -<p class='c005'>If we allow R₀ to converge to a point, we come to the -conception of an infinitely thin space-time filament. In -such a case, a space-time line will be thought of as a -principal line and by the term ‘Proper-time’ of the filament -will be understood the ‘Proper-time’ which is laid along -this principal line; under the term normal cross-section -of the filament, we shall understand the cross-section -upon the space which is normal to the principal line -through P.</p> - -<p class='c005'>We shall now formulate the principle of conservation -of mass.</p> - -<p class='c005'>To every space R at a time <i>t</i>, belongs a positive -quantity—the mass at R at the time <i>t</i>. If R converges -to a point (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), then the quotient of this mass, and -the volume of R approaches a limit μ(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), which is -known as the mass-density at the space-time point -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>).</p> - -<p class='c005'>The principle of conservation of mass says—that for -an infinitely thin space-time filament, the product μ<i>d</i>J, -where μ = mass-density at the point (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) of the filament -(<i>i.e.</i>, the principal line of the filament), <i>d</i>J = contents -of the cross-section normal to the <i>t</i> axis, and passing -through (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), is constant along the whole filament.</p> - -<p class='c005'>Now the contents <i>d</i>J<sub>n</sub> of the normal cross-section of -the filament which is laid through (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(4) <i>d</i>J<sub>n</sub> = (1/√(1 - <i>u²</i>))<i>d</i>J = -<i>i</i>ω₄ <i>d</i>J = (<i>dt</i>/<i>d</i>τ)<i>d</i>J.</div> - </div> - </div> -</div> - -<p class='c010'>and the function</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ν = μ/-<i>i</i>ω₄ = μ√(1 - <i>u²</i>)) = μ(∂τ/∂<i>t</i>. (5)</div> - </div> - </div> -</div> - -<p class='c010'>may be defined as the rest-mass density at the position -(<i>x</i> <i>y</i> <i>z</i> <i>t</i>). Then the principle of conservation of mass can -be formulated in this manner:—</p> - -<p class='c005'><i>For an infinitely thin space-time filament, the product -of the rest-mass density and the contents of the normal -cross-section is constant along the whole filament.</i></p> - -<p class='c005'>In any space-time filament, let us consider two cross-sections -Q° and Q′, which have only the points on the -boundary common to each other; let the space-time lines -inside the filament have a larger value of <i>t</i> on Q′ than -on Q°. The finite range enclosed between Q° and Q′ -shall be called a space-time <i>sichel</i>,<a id='r29' /><a href='#f29' class='c007'><sup>[29]</sup></a> Q′ is the lower -boundary, and Q′ is the upper boundary of the <i>sichel</i>.</p> - -<p class='c005'>If we decompose a filament into elementary space-time -filaments, then to an entrance-point of an elementary -filament through the lower boundary of the <i>sichel</i>, there -corresponds an exit point of the same by the upper boundary, -whereby for both, the product νdJ<sub>n</sub> taken in the sense of -(4) and (5), has got the same value. Therefore the difference -of the two integrals ∫ν<i>dJ</i><sub>n</sub> (the first being extended over -the upper, the second upon the lower boundary) vanishes. -According to a well-known theorem of Integral Calculus -the difference is equivalent to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∫∫∫∫ lor ν[=ω] <i>dx dy dz dt</i>,</div> - </div> - </div> -</div> - -<p class='c010'>the integration being extended over the whole range of -the <i>sichel</i>, and (comp. (67), § 12)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor ν[=ω] = (∂νω₁/∂<i>x₁</i>) + (∂νω₂/∂<i>x₂</i>) + (∂νω₃/∂<i>x₃</i>) + (∂νω₄/∂<i>x₄</i>).</div> - </div> - </div> -</div> - -<p class='c010'>If the <i>sichel</i> reduces to a point, then the differential -equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>lor ν[=ω] = 0, (6)</div> - </div> - </div> -</div> - -<p class='c010'>which is the condition of continuity</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(∂μ<i>u</i><sub><i>x</i></sub>/∂<i>x</i>) + (∂μ<i>u</i><sub><i>y</i></sub>/∂<i>y</i>) + (∂μ<i>u</i><sub><i>z</i></sub>/∂<i>z</i>) + (∂μ/∂<i>t</i>) = 0.</div> - </div> - </div> -</div> - -<p class='c010'>Further let us form the integral</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>N = ∫ ∫∫∫ ν <i>dx dy dz dt</i> (7)</div> - </div> - </div> -</div> - -<p class='c010'>extending over the whole range of the space-time <i>sichel</i>. -We shall decompose the <i>sichel</i> into elementary space-time -filaments, and every one of these filaments in small elements -<i>d</i>τ of its proper-time, which are however large compared -to the linear dimensions of the normal cross-section; let -us assume that the mass of such a filament ν<i>d</i>J<sub><i>n</i></sub> = <i>dm</i> and -write τ⁰, τ<sup>l</sup> for the ‘Proper-time’ of the upper and lower -boundary of the <i>sichel</i>.</p> - -<p class='c005'>Then the integral (7) can be denoted by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∫∫ ν<i>d</i>J<sub><i>n</i></sub> <i>d</i>τ = ∫ (τ′-τ⁰) <i>dm</i>.</div> - </div> - </div> -</div> - -<p class='c010'>taken over all the elements of the sichel.</p> - -<p class='c005'>Now let us conceive of the space-time lines inside a -space-time <i>sichel</i> as material curves composed of material -points, and let us suppose that they are subjected to a -continual change of length inside the sichel in the following -manner. The entire curves are to be varied in any -possible manner inside the <i>sichel</i>, while the end points -on the lower and upper boundaries remain fixed, and the -individual substantial points upon it are displaced in such a -manner that they always move forward normal to the -curves. The whole process may be analytically represented -by means of a parameter λ, and to the value λ = 0, -shall correspond the actual curves inside the <i>sichel</i>. Such a -process may be called a virtual displacement in the sichel.</p> - -<p class='c005'>Let the point (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) in the sichel λ = 0 have the -values <i>x</i> + δ<i>x</i>, <i>y</i> + δ<i>y</i>, <i>z</i> + δ<i>z</i>, <i>t</i> + δ<i>t</i>, when the parameter has -the value λ; these magnitudes are then functions of (<i>x</i>, <i>y</i>, -<i>z</i>, <i>t</i>, λ). Let us now conceive of an infinitely thin space-time -filament at the point (<i>x</i> <i>y</i> <i>z</i> <i>t</i>) with the normal section -of contents <i>d</i>J<sub><i>n</i></sub> and if <i>d</i>J<sub><i>n</i></sub> + δ<i>d</i>J<sub><i>n</i></sub> be the contents of the -normal section at the corresponding position of the varied -filament, then according to the principle of conservation -of mass—(ν + <i>d</i>ν being the rest-mass-density at the varied -position),</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(8) (ν + δν) (<i>d</i>J<sub><i>n</i></sub> + δ<i>d</i>J<sub><i>n</i></sub>) = ν<i>d</i>J<sub><i>n</i></sub> = <i>dm</i>.</div> - </div> - </div> -</div> - -<p class='c010'>In consequence of this condition, the integral (7) -taken over the whole range of the <i>sichel</i>, varies on account -of the displacement as a definite function N + δN of λ, -and we may call this function N + δN as the <i>mass action</i> -of the virtual displacement.</p> - -<p class='c005'>If we now introduce the method of writing with -indices, we shall have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(9) <i>d</i>(<i>x</i><sub><i>h</i></sub> + δ<i>x</i><sub><i>h</i></sub>) = <i>dx</i><sub><i>h</i></sub> + ∑<sub><i>k</i></sub> ∂δ<i>x</i><sub><i>h</i></sub>/∂<i>x</i><sub><i>k</i></sub> + ∂δ<i>x</i><sub><i>h</i></sub>/∂λ <i>d</i>λ</div> - </div> - <div class='group'> - <div class='line'><i>k</i> = 1, 2, 3, 4</div> - <div class='line'><i>h</i> = 1, 2, 3, 4</div> - </div> - </div> -</div> - -<p class='c010'>Now on the basis of the remarks already made, it is -clear that the value of N + δN, when the value of the -parameter is λ, will be:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(10) N + δN = ∫∫∫∫ ((ν<i>d</i>(τ + δτ))/<i>d</i>τ)<i>dx</i> <i>dy</i> <i>dz</i> <i>dt</i>,</div> - </div> - </div> -</div> - -<p class='c010'>the integration extending over the whole sichel <i>d</i>(τ + δτ) -where <i>d</i>(τ + δτ) denotes the magnitude, which is deduced from</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>√(-(<i>dx₁</i> + <i>d</i>δ<i>x₁</i>)² - (<i>dx₂</i> + <i>d</i>δ<i>x₂</i>)² - (<i>dx₃</i> + <i>d</i>δ<i>x₃</i>)² - (<i>dx₄</i> + <i>d</i>δ<i>x₄</i>)²)</div> - </div> - </div> -</div> - -<p class='c010'>by means of (9) and</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx₁</i> = ω₁ <i>d</i>τ, <i>dx₂</i> = ω₂ <i>d</i>τ,</div> - <div class='line'><i>dx₃</i> = ω₃ <i>d</i>τ, <i>dx₄</i> = ω₄ <i>d</i>τ, <i>d</i>λ = 0</div> - </div> - </div> -</div> - -<p class='c010'>therefore:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(11) (<i>d</i>(τ + δτ))/<i>d</i>τ = √( -∑(ω<sub><i>h</i></sub> + ∑(∂δ<i>x</i><sub><i>h</i></sub>/∂<i>x</i><sub><i>k</i></sub>)ω<sub><i>k</i></sub>)²)</div> - </div> - <div class='group'> - <div class='line'><i>k</i> = 1, 2, 3, 4.</div> - <div class='line'><i>h</i> = 1, 2, 3, 4.</div> - </div> - </div> -</div> - -<p class='c010'>We shall now subject the value of the differential -quotient</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(12) ((<i>d</i>(N + δN))/<i>d</i>λ) (λ = 0)</div> - </div> - </div> -</div> - -<p class='c010'>to a transformation. Since each δ<i>x</i><sub><i>h</i></sub> as a function of (<i>x</i>, <i>y</i>, -<i>z</i>, <i>t</i>) vanishes for the zero-value of the parameter λ, so in -general <i>d</i>δ<i>x</i><sub><i>k</i></sub>/(∂<i>x</i><sub><i>h</i></sub> = 0, for λ = 0.</p> - -<p class='c005'>Let us now put (∂δ<i>x</i><sub><i>h</i></sub>/∂λ) = ξ<sub><i>h</i></sub> (<i>h</i> = 1, 2, 3, 4) (13)</p> - -<p class='c005'>λ = 0</p> - -<p class='c005'>then on the basis of (10) and (11), we have the expression -(12):—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in2'>= -∫∫∫∫ ∑ ω<sub><i>h</i></sub>((∂ξ<sub><i>h</i></sub>/∂<i>x₁</i>)ω₁ + (∂ξ<sub><i>h</i></sub>/∂<i>x₂</i>)ω₂ +(∂ξ<sub><i>h</i></sub>/∂<i>x₃</i>)ω₃ + (∂ξ<sub><i>h</i></sub>/∂<i>x₄</i>)ω₄)</div> - <div class='line'><i>dx dy dz dt</i></div> - </div> - </div> -</div> - -<p class='c010'>for the system (<i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i>) on the boundary of the -<i>sichel</i>, (δ<i>x₁</i> δ<i>x₂</i> δ<i>x₃</i> δ<i>x₄</i>) shall vanish for every value of -λ and therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial -integration, the integral is transformed into the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∫∫∫∫ ∑ ξ<sub><i>h</i></sub>(∂νω<sub><i>h</i></sub>ω₁/∂<i>x₁</i> + ∂νω<sub><i>h</i></sub>ω₂/∂<i>x₂</i> + ∂νω<sub><i>h</i></sub>ω₃/∂<i>x₃</i> + ∂νω<sub><i>h</i></sub>ω₄/∂<i>x₄</i>)</div> - <div class='line'><i>dx dy dz dt</i></div> - </div> - </div> -</div> - -<p class='c010'>the expression within the bracket may be written as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>= ω<sub><i>h</i></sub> ∑ ∂νω<sub><i>k</i></sub>/∂<i>x</i><sub><i>k</i></sub> + ν∑ω<sub><i>k</i></sub>∂ω<sub><i>h</i></sub>/∂<i>x</i><sub><i>k</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'>The first sum vanishes in consequence of the continuity -equation (<i>b</i>). The second may be written as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(∂ω<sub><i>h</i></sub>/∂<i>x₁</i>)(<i>dx₁</i>/<i>d</i>τ) + (∂ω<sub><i>h</i></sub>/∂<i>x₂</i>)(<i>dx₂</i>/<i>d</i>τ) + (∂ω<sub><i>h</i></sub>/∂<i>x₃</i>)(<i>dx₃</i>/<i>d</i>τ) + (∂ω<sub><i>h</i></sub>/∂<i>x₄</i>)(<i>dx₄</i>/<i>d</i>τ)</div> - </div> - <div class='group'> - <div class='line in3'>= <i>d</i>ω<sub><i>h</i></sub>/<i>d</i>τ = (<i>d</i>/<i>d</i>τ)(<i>dx</i><sub><i>h</i></sub>/<i>d</i>τ)</div> - </div> - </div> -</div> - -<p class='c010'>whereby (<i>d</i>/<i>d</i>τ) is meant the differential quotient in the -direction of the space-time line at any position. For the -differential quotient (12), we obtain the final expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)</div> - </div> - <div class='group'> - <div class='line'><i>dx dy dz dt</i>.</div> - </div> - </div> -</div> - -<p class='c010'>For a virtual displacement in the <i>sichel</i> we have -postulated the condition that the points supposed to be -substantial shall advance normally to the curves giving -their actual motion, which is λ = 0; this condition denotes -that the ξ<sub><i>h</i></sub> is to satisfy the condition</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>w₁</i>ξ₁ + <i>w₂</i>ξ₂ + <i>w₃</i>ξ₃ + <i>w₄</i>ξ₄ = 0. (15)</div> - </div> - </div> -</div> - -<p class='c010'>Let us now turn our attention to the Maxwellian -tensions in the electrodynamics of stationary bodies, and -let us consider the results in § 12 and 13; then we find -that Hamilton’s Principle can be reconciled to the relativity -postulate for continuously extended elastic media.</p> - -<p class='c005'>At every space-time point (as in § 13), let a space time -matrix of the 2nd kind be known</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(16) S =</div> - <div class='line'>| S₁₁ S₁₂ S₁₃ S₁₄ | = | X<sub><i>x</i></sub> Y<sub><i>x</i></sub> Z<sub><i>x</i></sub> -<i>i</i>T<sub><i>x</i></sub> |</div> - </div> - <div class='group'> - <div class='line'>| S₂₁ S₂₂ S₂₃ S₂₄ | = | X<sub><i>y</i></sub> Y<sub><i>y</i></sub> Z<sub><i>y</i></sub> -<i>i</i>T<sub><i>y</i></sub> |</div> - </div> - <div class='group'> - <div class='line'>| S₃₁ S₃₂ S₃₃ S₃₄ | = | X<sub><i>z</i></sub> Y<sub><i>z</i></sub> Z<sub><i>z</i></sub> -<i>i</i>T<sub><i>z</i></sub> |</div> - </div> - <div class='group'> - <div class='line'>| S₄₁ S₄₂ S₄₃ S₄₄ | = | -<i>i</i>X<sub><i>t</i></sub> -<i>i</i>Y<sub><i>t</i></sub> -<i>i</i>Z<sub><i>t</i></sub> T<sub><i>t</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>where X<sub><i>n</i></sub> Y<sub><i>x</i></sub> .....X<sub><i>z</i></sub>, T<sub><i>t</i></sub> are real magnitudes.</p> - -<p class='c005'>For a virtual displacement in a space-time sichel -(with the previously applied designation) the value of -the integral</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(17) W + δW = ∫∫∫∫ (∑S<sub><i>h k</i></sub> (∂(<i>x</i><sub><i>k</i></sub> + δ<i>x</i><sub><i>k</i></sub>))/∂<i>x</i><sub><i>h</i></sub> <i>dx dy dz dt</i></div> - </div> - </div> -</div> - -<p class='c010'>extended over the whole range of the <i>sichel</i>, may be called -the tensional work of the virtual displacement.</p> - -<p class='c005'>The sum which comes forth here, written in real -magnitudes, is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>X<sub><i>x</i></sub> + Y<sub><i>y</i></sub> + Z<sub><i>z</i></sub> + T<sub><i>t</i></sub> + X<sub><i>x</i></sub> (∂δ<i>x</i>)/∂<i>x</i> + X<sub><i>y</i></sub> (∂δ<i>x</i>)/∂<i>y</i> + ... Z<sub><i>z</i></sub> (∂δ<i>z</i>)/∂<i>z</i></div> - </div> - <div class='group'> - <div class='line'>- X<sub><i>t</i></sub> (∂δ<i>x</i>/∂<i>t</i> - ... + T<sub><i>x</i></sub> (∂δ<i>t</i>)/∂<i>x</i> + ... T<sub><i>t</i></sub> (∂δ<i>t</i>)/∂<i>t</i></div> - </div> - </div> -</div> - -<p class='c010'>we can now postulate the following <i>minimum principle in -mechanics</i>.</p> - -<p class='c005'><i>If any space-time Sichel be bounded, then for each -virtual displacement in the Sichel, the sum of the mass-works, -and tension works shall always be an extremum -for that process of the space-time line in the Sichel which -actually occurs.</i></p> - -<p class='c005'>The meaning is, that for each virtual displacement,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>([<i>d</i>(·δN + δW)]/<i>d</i>λ)<sub>λ = 0</sub> = 0 (18)</div> - </div> - </div> -</div> - -<p class='c010'>By applying the methods of the Calculus of Variations, -the following four differential equations at once -follow from this minimal principle by means of the transformation -(14), and the condition (15).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(19) ν ∂<i>w</i><sub><i>h</i></sub>/∂τ = K<sub><i>h</i></sub> + χ<i>w</i><sub><i>h</i></sub> (<i>h</i> = 1, 2, 3, 4)</div> - </div> - <div class='group'> - <div class='line'>whence K<sub><i>h</i></sub> = ∂S<sub>1 <i>h</i></sub>/∂<i>x₁</i> + ∂S<sub>2 <i>h</i></sub>/∂<i>x₂</i> + ∂S<sub>3 <i>h</i></sub>/∂<i>x₃</i> + ∂S<sub>4 <i>h</i></sub>/∂<i>x₄</i>, (20)</div> - </div> - </div> -</div> - -<p class='c010'>are components of the space-time vector 1st kind K = lor S, -and X is a factor, which is to be determined from the -relation <i>w</i><i>ẇ</i> = - 1. By multiplying (19) by <i>w</i><sub><i>h</i></sub>, and -summing the four, we obtain X = K<i>ẇ</i>, and therefore clearly -K + (K<i>ẇ</i>)<i>w</i> will be a space-time vector of the 1st kind which -is normal to <i>w</i>. Let us write out the components of this -vector as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>X, Y, Z, ·<i>i</i>T</div> - </div> - </div> -</div> - -<p class='c010'>Then we arrive at the following equation for the motion -of matter,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(21) ν <i>d</i>/<i>d</i>τ (<i>dx</i>/<i>d</i>τ) = X, ν <i>d</i>/<i>d</i>τ (<i>dy</i>/<i>d</i>τ) = Y, ν <i>d</i>/<i>d</i>τ (<i>dz</i>/<i>d</i>τ) = Z,</div> - </div> - <div class='group'> - <div class='line'>ν <i>d</i>/<i>d</i>τ (<i>dx</i>/<i>d</i>τ) = T, and we have also</div> - </div> - <div class='group'> - <div class='line'>(<i>dx</i>/<i>d</i>τ)² + (<i>dy</i>/<i>d</i>τ)² + (<i>dz</i>/<i>d</i>τ)² > (<i>dt</i>/<i>d</i>τ)² = -1,</div> - </div> - <div class='group'> - <div class='line'>and X <i>dx</i>/<i>d</i>τ + Y <i>dy</i>/<i>d</i>τ + Z <i>dz</i>/<i>d</i>τ = T <i>dt</i>/<i>d</i>τ.</div> - </div> - </div> -</div> - -<p class='c010'>On the basis of this condition, the fourth of equations (21) -is to be regarded as a direct consequence of the first three.</p> - -<p class='c005'>From (21), we can deduce the law for the motion of -a material point, <i>i.e.</i>, the law for the career of an infinitely -thin space-time filament.</p> - -<p class='c005'>Let <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, denote a point on a principal line chosen -in any manner within the filament. We shall form the -equations (21) for the points of the normal cross section of -the filament through <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, and integrate them, multiplying -by the elementary contents of the cross section over the -whole space of the normal section. If the integrals of the -right side be R<sub><i>x</i></sub> R<sub><i>y</i></sub> R<sub><i>z</i></sub> R<sub><i>t</i></sub> and if <i>m</i> be the constant mass -of the filament, we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(22) <i>m</i> <i>d</i>/<i>d</i>τ <i>dx</i>/<i>d</i>τ = R<sub><i>x</i></sub>,</div> - <div class='line in5'><i>m</i> <i>d</i>/<i>d</i>τ <i>dy</i>/<i>d</i>τ = R<sub><i>y</i></sub>,</div> - <div class='line in5'><i>m</i> <i>d</i>/<i>d</i>τ <i>dz</i>/<i>d</i>τ = R<sub><i>z</i></sub>,</div> - <div class='line in5'><i>m</i> <i>d</i>/<i>d</i>τ <i>dt</i>/<i>d</i>τ = R<sub><i>t</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>R is now a space-time vector of the 1st kind with the -components (R<sub><i>x</i></sub> R<sub><i>y</i></sub> R<sub><i>z</i></sub> R<sub><i>t</i></sub>) which is normal to the space-time -vector of the 1st kind <i>w</i>,—the velocity of the material -point with the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx</i>/<i>d</i>τ, <i>dy</i>/<i>d</i>τ, <i>dz</i>/<i>d</i>τ, <i>i</i> <i>dt</i>/<i>d</i>τ.</div> - </div> - </div> -</div> - -<p class='c010'>We may call this vector R <i>the moving force of the -material point</i>.</p> - -<p class='c005'>If instead of integrating over the normal section, we -integrate the equations over that cross section of the filament -which is normal to the <i>t</i> axis, and passes through -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), then [See (4)] the equations (22) are obtained, but</p> - -<p class='c005'>are now multiplied by <i>d</i>τ/<i>dt</i>; in particular, the last equation -comes out in the form,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>m</i> <i>d</i>/<i>dt</i> (<i>dt</i>/<i>d</i>τ) = <i>w</i><sub><i>x</i></sub> R<sub><i>x</i></sub> <i>d</i>τ/<i>dt</i> + <i>w</i><sub><i>y</i></sub> R<sub><i>y</i></sub> <i>d</i>τ/<i>dt</i> + <i>w</i><sub><i>z</i></sub> R<sub><i>z</i></sub> <i>d</i>τ/<i>dt</i>.</div> - </div> - </div> -</div> - -<p class='c010'>The right side is to be looked upon <i>as the amount of work -done per unit of time</i> at the material point. In this -equation, we obtain the energy-law for the motion of -the material point and the expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>m</i> (<i>dt</i>/<i>d</i>τ - 1) = <i>m</i> [1/√(1 - <i>w²</i>) - 1]</div> - <div class='line in2'>= <i>m</i> (½ |<i>w₁²</i> + 3/8 |<i>w₁⁴</i> + )</div> - </div> - </div> -</div> - -<p class='c010'>may be called the kinetic energy of the material point.</p> - -<p class='c005'>Since <i>dt</i> is always greater than <i>d</i>τ we may call the -quotient (<i>dt</i> - <i>d</i>τ)/<i>d</i>τ as the “Gain” (vorgehen) of the time -over the proper-time of the material point and the law can -then be thus expressed;—The kinetic energy of a material -point is the product of its mass into the gain of the -time over its proper-time.</p> - -<p class='c005'>The set of four equations (22) again shows the symmetry -in (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), which is demanded by the relativity -postulate; to the fourth equation however, a higher physical -significance is to be attached, as we have already -seen in the analogous case in electrodynamics. On the -ground of this demand for symmetry, the triplet consisting -of the first three equations are to be constructed after the -model of the fourth; remembering this circumstance, we -are justified in saying,—</p> - -<p class='c005'>“If the relativity-postulate be placed at the head of -mechanics, then the whole set of laws of motion follows -from the law of energy.”</p> - -<p class='c005'>I cannot refrain from showing that no contradiction -to the assumption on the relativity-postulate can be -expected from the phenomena of gravitation.</p> - -<p class='c005'>If B*(<i>x</i>*, <i>y</i>*, <i>z</i>*, <i>t</i>*) be a solid (fester) space-time point, -then the region of all those space-time points B (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), -for which</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(23) (<i>x</i> - <i>x</i>*)² + (<i>y</i> - <i>y</i>*)² + (<i>z</i> - <i>z</i>*)² = (<i>t</i> - <i>t</i>*)²</div> - </div> - <div class='group'> - <div class='line'><i>t</i> - <i>t</i>* >= 0</div> - </div> - </div> -</div> - -<p class='c010'>may be called a “Ray-figure” (Strahl-gebilde) of the space -time point B*.</p> - -<p class='c005'>A space-time line taken in any manner can be cut by this -figure only at one particular point; this easily follows from -the convexity of the figure on the one hand, and on the -other hand from the fact that all directions of the space-time -lines are only directions from B* towards to the -concave side of the figure. Then B* may be called the -light-point of B.</p> - -<p class='c005'>If in (23), the point (<i>x</i> <i>y</i> <i>z</i> <i>t</i>) be supposed to be fixed, -the point (<i>x</i>* <i>y</i>* <i>z</i>* <i>t</i>*) be supposed to be variable, then -the relation (23) would represent the locus of all the space-time -points B*, which are light-points of B.</p> - -<p class='c005'>Let us conceive that a material point F of mass <i>m</i> -may, owing to the presence of another material point F*, -experience a moving force according to the following law. -Let us picture to ourselves the space-time filaments of F -and F* along with the principal lines of the filaments. Let -BC be an infinitely small element of the principal line of -F; further let B* be the light point of B, C* be the -light point of C on the principal line of F*; so that -OA′ is the radius vector of the hyperboloidal fundamental -figure (23) parallel to B*C*, finally D* is the point of -intersection of line B*C* with the space normal to itself -and passing through B. The moving force of the mass-point -F in the space-time point B is now the space-time -vector of the first kind which is normal to BC, -and which is composed of the vectors</p> - -<p class='c005'>(24) <i>mm</i>*(OA′/B*D*)³ BD* in the direction of BD*, and -another vector of suitable value in direction of B*C*.</p> - -<p class='c005'>Now by (OA′/B*D*) is to be understood the ratio of the two -vectors in question. It is clear that this proposition at -once shows the covariant character with respect to a -Lorentz-group.</p> - -<p class='c005'>Let us now ask how the space-time filament of F -behaves when the material point F* has a uniform -translatory motion, <i>i.e.</i>, the principal line of the filament -of F* is a line. Let us take the space time null-point in -this, and by means of a Lorentz-transformation, we can -take this axis as the t-axis. Let <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, denote the point -B, let τ* denote the proper time of B*, reckoned from O. -Our proposition leads to the equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(25) <i>d²</i><i>x</i>/<i>d</i>τ² = - <i>m</i>*<i>x</i>/(<i>t</i> - τ*)², <i>d²</i><i>y</i>/<i>d</i>τ² = - <i>m</i>*<i>y</i>/(<i>t</i> - τ*)³</div> - </div> - <div class='group'> - <div class='line'><i>d²</i><i>z</i>/<i>d</i>τ² = -<i>m</i>*<i>z</i>/(<i>t</i> - τ*)³,</div> - <div class='line'>(26) <i>d²</i><i>t</i>/<i>d</i>τ² = -<i>m</i>*/(<i>t</i> - τ*)² <i>d</i>(<i>t</i> - τ*)/<i>dt</i></div> - </div> - </div> -</div> - -<p class='c010'>where (27) <i>x²</i> + <i>y²</i> + <i>z²</i> = (<i>t</i> - τ*)²</p> - -<p class='c005'>and (28) (<i>dx</i>/<i>d</i>τ)² + (<i>dy</i>/<i>d</i>τ)² + (<i>dz</i>/<i>d</i>τ)² = (<i>dt</i>/<i>d</i>τ)² - 1.</p> - -<p class='c005'>In consideration of (27), the three equations (25) are -of the same form as the equations for the motion of a -material point subjected to attraction from a fixed centre -according to the Newtonian Law, only that instead of the -time <i>t</i>, the proper time τ of the material point occurs. The -fourth equation (26) gives then the connection between -proper time and the time for the material point.</p> - -<p class='c005'>Now for different values of τ′, the orbit of the space-point -(<i>x</i> <i>y</i> <i>z</i>) is an ellipse with the semi-major axis <i>a</i> and -the eccentricity <i>e</i>. Let E denote the eccentric anomaly, Τ -the increment of the proper time for a complete description -of the orbit, finally <i>n</i>Τ = 2π, so that from a properly chosen -initial point τ, we have the Kepler-equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(29) <i>n</i>τ = E - <i>e</i> sin E.</div> - </div> - </div> -</div> - -<p class='c010'>If we now change the unit of time, and denote the -velocity of light by <i>c</i>, then from (28), we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(30) (<i>dt</i>/<i>d</i>τ)² - 1</div> - <div class='line in2'>= (<i>m</i>*/<i>ac²</i>) (1 + <i>e</i> cos E)/(1 - <i>e</i> cos E)</div> - </div> - </div> -</div> - -<p class='c010'>Now neglecting <i>c⁻⁴</i> with regard to 1, it follows that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ndt</i> = <i>nd</i>τ [ 1 + ½ <i>m</i>*/<i>ac²</i> (1 + <i>e</i> cos E)/(1 - <i>e</i> cos E) ]</div> - </div> - </div> -</div> - -<p class='c010'>from which, by applying (29),</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(31) <i>nt</i> + const = (1 + ½ <i>m</i>*/<i>ac²</i>) <i>n</i>τ + <i>m</i>*/<i>ac²</i> Sin E.</div> - </div> - </div> -</div> - -<p class='c010'>the factor <i>m</i>*/<i>ac²</i> is here the square of the ratio of a certain -average velocity of F in its orbit to the velocity of light. -If now <i>m</i>* denote the mass of the sun, <i>a</i> the semi major -axis of the earth’s orbit, then this factor amounts to 10⁻⁸.</p> - -<p class='c005'>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 light. In view of the fact -that the periodic terms in (31) are very small, it is not -possible to decide out of astronomical observations between -such a law (with the modified mechanics proposed above) -and the Newtonian law of attraction with Newtonian -mechanics.</p> - -<div class='chapter'> - <h2 class='c003'>SPACE AND TIME</h2> -</div> -<p class='c004'>A Lecture delivered before the Naturforscher Versammlung -(Congress of Natural Philosophers) at Cologne—(21st -September, 1908).</p> - -<p class='c005'>Gentlemen,</p> - -<p class='c005'>The conceptions about time and space, which I hope -to develop before you to-day, has grown on experimental -physical grounds. Herein lies its strength. The tendency -is radical. Henceforth, the old conception of space for -itself, and time for itself shall reduce to a mere shadow, -and some sort of union of the two will be found consistent -with facts.</p> -<h3 class='c008'>I</h3> -<p class='c004'>Now I want to show you how we can arrive at the -changed concepts about time and space from mechanics, as -accepted now-a-days, from purely mathematical considerations. -The equations of Newtonian mechanics show a twofold -invariance, (<i>i</i>) their form remains unaltered when -we subject the fundamental space-coordinate system to -any possible change of position, (<i>ii</i>) when we change the -system in its nature of motion, <i>i. e.</i>, when we impress upon -it any uniform motion of translation, the null-point of time -plays no part. We are accustomed to look upon the axioms -of geometry as settled once for all, while we seldom have the -same amount of conviction regarding the axioms of mechanics, -and therefore the two invariants are seldom mentioned -in the same breath. Each one of these denotes a certain -group of transformations for the differential equations of -mechanics. We look upon the existence of the first group -as a fundamental characteristics of space. We always -prefer to leave off the second group to itself, and with a -light heart conclude that we can never decide from physical -considerations whether the space, which is supposed to be -at rest, may not finally be in uniform motion. So these two -groups lead quite separate existences besides each other. -Their totally heterogeneous character may scare us away -from the attempt to compound them. Yet it is the whole -compounded group which as a whole gives us occasion for -thought.</p> - -<p class='c005'>We wish to picture to ourselves the whole relation -graphically. Let (<i>x</i>, <i>y</i>, <i>z</i>) be the rectangular coordinates of -space, and <i>t</i> denote the time. Subjects of our perception -are always connected with place and time. <i>No one has -observed a place except at a particular time, or has observed -a time except at a particular place.</i> Yet I respect the -dogma that time and space have independent existences. I -will call a space-point plus a time-point, <i>i.e.</i>, a system of -values <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, as a <i>world-point</i>. The manifoldness of all -possible values of <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, will be the <i>world</i>. I can draw -four world-axes with the chalk. Now any axis drawn -consists of quickly vibrating molecules, and besides, takes -part in all the journeys of the earth ; and therefore gives -us occasion for reflection. The greater abstraction required -for the four-axes does not cause the mathematician any -trouble. In order not to allow any yawning gap to -exist, we shall suppose that at every place and time, -something perceptible exists. In order not to specify -either matter or electricity, we shall simply style these as -substances. We direct our attention to the <i>world-point</i> -<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, and suppose that we are in a position to recognise -this substantial point at any subsequent time. Let <i>dt</i> be -the time element corresponding to the changes of space -coordinates of this point [<i>dx</i>, <i>dy</i>, <i>dz</i>]. Then we obtain (as -a picture, so to speak, of the perennial life-career of the -substantial point),—a curve in the <i>world</i>—the <i>world-line</i>, -the points on which unambiguously correspond to the parameter -<i>t</i> from +∞ to -∞. The whole world appears to be -resolved in such <i>world-lines</i>, and I may just deviate from -my point if I say that according to my opinion the physical -laws would find their fullest expression as mutual relations -among these lines.</p> - -<p class='c005'>By this conception of time and space, the (<i>x</i>, <i>y</i>, <i>z</i>) manifoldness -<i>t</i> = 0 and its two sides <i>t</i> < 0 and <i>t</i> > 0 falls asunder. -If for the sake of simplicity, we keep the null-point of time -and space fixed, then the first named group of mechanics -signifies that at <i>t</i> = 0 we can give the <i>x</i>, <i>y</i>, and <i>z</i>-axes any -possible rotation about the null-point corresponding to the -homogeneous linear transformation of the expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x²</i> + <i>y²</i> + <i>z²</i>.</div> - </div> - </div> -</div> - -<p class='c010'>The second group denotes that without changing the -expression for the mechanical laws, we can substitute -(<i>x</i> - α<i>t</i>, <i>y</i> - β<i>t</i>, <i>z</i> - γ<i>t</i> for (<i>x</i>, <i>y</i>, <i>z</i>) where (α, β, γ) are any -constants. According to this we can give the time-axis -any possible direction in the upper half of the world <i>t</i> > 0. -Now what have the demands of orthogonality in space to -do with this perfect freedom of the time-axis towards the -upper half?</p> - -<p class='c005'>To establish this connection, let us take a positive parameter -c, and let us consider the figure</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c²</i><i>t²</i> - <i>x²</i> - <i>y²</i> - <i>z²</i> = 1</div> - </div> - </div> -</div> - -<p class='c010'>According to the analogy of the hyperboloid of two -sheets, this consists of two sheets separated by <i>t</i> = 0. Let us -consider the sheet, in the region of <i>t</i> > 0, and let us now -conceive the transformation of <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> in the new system -of variables; (<i>x’</i>, <i>y’</i>, <i>z’</i>, <i>t’</i>) by means of which the form of -the expression will remain unaltered. Clearly the rotation -of space round the null-point belongs to this group of -transformations. Now we can have a full idea of the transformations -which we picture to ourselves from a particular -transformation in which (<i>y</i>, <i>z</i>) remain unaltered. Let -us draw the cross section of the upper sheets with the -plane of the <i>x</i>- and <i>t</i>-axes, <i>i.e.</i>, the upper half of -the hyperbola <i>c²</i><i>t²</i> - x² = 1, with its asymptotes (<i>vide</i> -fig. 1).</p> - -<p class='c005'>Then let us draw the radius rector OA′, the tangent -A′ B′ at A′, and let us complete the parallelogram OA′ -B′ C′; also produce B′ C′ to meet the x-axis at D′. -Let us now take Ox′, OA′ as new axes with the unit measuring -rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again -expressed in the form <i>c²</i><i>t′²</i> - x′² = 1, t′ > 0 and the transition -from (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) to (<i>x′</i> <i>y′</i> <i>z′</i> <i>t</i>) is one of the transitions in -question. Let us add to this characteristic transformation -any possible displacement of the space and time null-points; -then we get a group of transformation depending only on -<i>c</i>, which we may denote by G<sub><i>c</i></sub>.</p> - -<p class='c005'>Now let us increase <i>c</i> to infinity. Thus (1/c) becomes zero -and it appears from the figure that the hyperbola is gradually -shrunk into the <i>x</i>-axis, the asymptotic angle becomes -a straight one, and every special transformation in -the limit changes in such a manner that the <i>t</i>-axis can -have any possible direction upwards, and <i>x′</i> more and -more approximates to <i>x</i>. Remembering this point it is -clear that the full group belonging to Newtonian Mechanics -is simply the group G<sub><i>c</i></sub>, with the value of <i>c</i> = ∞. In this -state of affairs, and since G<sub><i>c</i></sub> is mathematically more intelligible -than G<sub>∞</sub>, a mathematician may, by a free play -of imagination, hit upon the thought that natural phenomena -possess an invariance not only for the group G<sub>∞</sub>, -but in fact also for a group G<sub><i>c</i></sub>, where <i>c</i> is finite, but yet -exceedingly large compared to the usual measuring units. -Such a preconception would be an extraordinary triumph -for pure mathematics.</p> - -<p class='c005'>At the same time I shall remark for which value of <i>c</i>, -this invariance can be conclusively held to be true. <i>For c, -we shall substitute the velocity of light c in free space.</i> -In order to avoid speaking either of space or of vacuum, -we may take this quantity as the ratio between the electrostatic -and electro-magnetic units of electricity.</p> - -<p class='c005'>We can form an idea of the invariant character of the -expression for natural laws for the group-transformation -G<sub><i>c</i></sub> in the following manner.</p> - -<p class='c005'>Out of the totality of natural phenomena, we can, by -successive higher approximations, deduce a coordinate -system (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>); by means of this coordinate system, we -can represent the phenomena according to definite laws. -This system of reference is by no means uniquely determined -by the phenomena. <i>We can change the system of -reference in any possible manner corresponding to the above-mentioned -group transformation G<sub>c</sub>, but the expressions for -natural laws will not be changed thereby.</i></p> - -<p class='c005'>For example, corresponding to the above described -figure, we can call <i>t′</i> the time, but then necessarily the -space connected with it must be expressed by the manifoldness -(<i>x′</i> <i>y</i> <i>z</i>). The physical laws are now expressed by -means of <i>x′</i>, <i>y</i>, <i>z</i>, <i>t′</i>,—and the expressions are just the -same as in the case of <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>. According to this, we -shall have in the world, not one space, but many spaces,—quite -analogous to the case that the three-dimensional -space consists of an infinite number of planes. The three-dimensional -geometry will be a chapter of four-dimensional -physics. Now you perceive, why I said in the beginning -that time and space shall reduce to mere shadows and we -shall have a world complete in itself.</p> -<h3 class='c008'>II</h3> -<p class='c004'>Now the question may be asked,—what circumstances -lead us to these changed views about time and space, are -they not in contradiction with observed phenomena, do -they finally guarantee us advantages for the description of -natural phenomena?</p> - -<p class='c005'>Before we enter into the discussion, a very important -point must be noticed. Suppose we have individualised -time and space in any manner; then a world-line parallel -to the <i>t</i>-axis will correspond to a stationary point; a -world-line inclined to the <i>t</i>-axis will correspond to a -point moving uniformly; and a world-curve will correspond -to a point moving in any manner. Let us now picture -to our mind the world-line passing through any world -point <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>; now if we find the world-line parallel -to the radius vector OA′ of the hyperboloidal sheet, then -we can introduce OA′ as a new time-axis, and then -according to the new conceptions of time and space the -substance will appear to be at rest in the world point -concerned. We shall now introduce this fundamental -axiom:—</p> - -<p class='c005'><i>The substance existing at any world point can always -be conceived to be at rest, if we establish our time and -space suitably.</i> The axiom denotes that in a world-point, -the expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c²</i><i>dt²</i> - <i>dx²</i> - <i>dy²</i> - <i>dz²</i></div> - </div> - </div> -</div> - -<p class='c010'>shall always be positive or what is equivalent to the -same thing, every velocity V should be smaller than <i>c</i>. -<i>c</i> shall therefore be the upper limit for all substantial -velocities and herein lies a deep significance for the -quantity <i>c</i>. At the first impression, the axiom seems to -be rather unsatisfactory. It is to be remembered that -only a modified mechanics will occur, in which the square -root of this differential combination takes the place of -time, so that cases in which the velocity is greater than <i>c</i> -will play no part, something like imaginary coordinates -in geometry.</p> - -<p class='c005'>The <i>impulse</i> and real cause of inducement <i>for the -assumption of the group-transformation G<sub>c</sub></i> is the fact that -the differential equation for the propagation of light in -vacant space possesses the group-transformation G<sub><i>c</i></sub>. On -the other hand, the idea of rigid bodies has any sense -only in a system mechanics with the group G<sub>infinity</sub>. Now -if we have an optics with G<sub><i>c</i></sub>, and on the other hand -if there are rigid bodies, it is easy to see that a -<i>t</i>-direction can be defined by the two hyperboloidal -shells common to the groups G<sub>∞</sub>, and G<sub><i>c</i></sub>, which has -got the further consequence, that by means of suitable -rigid instruments in the laboratory, we can perceive a -change in natural phenomena, in case of different orientations, -with regard to the direction of progressive motion -of the earth. But all efforts directed towards this -object, and even the celebrated interference-experiment -of Michelson have given negative results. In order to -supply an explanation for this result, H. A. Lorentz -formed a hypothesis which practically amounts to an -invariance of optics for the group G<sub><i>c</i></sub>. According to -Lorentz every substance shall suffer a contraction</p> - -<p class='c005'>1:(√(1 - v²/<i>c²</i>)) in length, in the direction of its motion</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>l</i>/<i>l′</i> = 1/√(1 - <i>v²</i>/<i>c²</i>) <i>l′</i> = <i>l</i>(1 - <i>v²</i>/<i>c²</i>).</div> - </div> - </div> -</div> - -<p class='c010'>This hypothesis sounds rather phantastical. For the -contraction is not to be thought of as a consequence of the -resistance of ether, but purely as a gift from the skies, as a -sort of condition always accompanying a state of motion.</p> - -<p class='c005'>I shall show in our figure that Lorentz’s hypothesis -is fully equivalent to the new conceptions about time and -space. Thereby it may appear more intelligible. Let us -now, for the sake of simplicity, neglect (<i>y</i>, <i>z</i>) and fix our -attention on a two dimensional world, in which let upright -strips parallel to the <i>t</i>-axis represent a state of rest and -another parallel strip inclined to the <i>t</i>-axis represent a -state of uniform motion for a body, which has a constant -spatial extension (see fig. 1). If OA′ is parallel to the second -strip, we can take <i>t′</i> as the <i>t</i>-axis and <i>x′</i> as the <i>x</i>-axis, then -the second body will appear to be at rest, and the first body -in uniform motion. We shall now assume that the first -body supposed to be at rest, has the length <i>l</i>, <i>i.e.</i>, the -cross section PP of the first strip upon the <i>x</i>-axis = <i>l</i><sup>.</sup> OC, -where OC is the unit measuring rod upon the <i>x</i>-axis—and -the second body also, when supposed to be at rest, has the -same length <i>l</i>, this means that, the cross section Q′Q′ of -the second strip has a cross-section <i>l</i><sup>·</sup> OC′, when measured -parallel to the <i>x′</i>-axis. In these two bodies, we have -now images of two Lorentz-electrons, one of which is at -rest and the other moves uniformly. Now if we stick -to our original coordinates, then the extension of the -second electron is given by the cross section QQ of the -strip belonging to it measured parallel to the <i>x</i>-axis. -Now it is clear since Q′Q′ = <i>l</i><sup>·</sup> OC′, that QQ = <i>l</i><sup>·</sup> OD′.</p> - -<p class='c005'>If (<i>dc</i>/<i>dt</i>) = <i>v</i>, an easy calculation gives that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>OD′ = OC √(1-(<i>v²</i>/<i>c²</i>)), therefore (PP/QQ) = (1/√(1-(<i>v²</i>/<i>c²</i>))</div> - </div> - </div> -</div> - -<p class='c010'>This is the sense of Lorentz’s hypothesis about the -contraction of electrons in case of motion. On the other -hand, if we conceive the second electron to be at rest, -and therefore adopt the system (<i>x′</i>, <i>t′</i>,) 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 proportion, -for it is,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP</div> - </div> - </div> -</div> - -<p class='c010'>Lorentz called the combination <i>t′</i> of (<i>t</i> and <i>x</i>) as the -<i>local time</i> (<i>Ortszeit</i>) of the uniformly moving electron, and -used a physical construction of this idea for a better comprehension -of the contraction-hypothesis. But to perceive -clearly that the time of an electron is as good as the time -of any other electron, <i>i.e.</i> <i>t</i>, <i>t′</i> are to be regarded as equivalent, -has been the service of A. Einstein [Ann. d. -Phys. 891, p. 1905, Jahrb. d. Radis. ... 4-4-11-1907.] There -the concept of time was shown to be completely and unambiguously -established by natural phenomena. But the -concept of space was not arrived at, either by Einstein -or Lorentz, probably because in the case of the above-mentioned -spatial transformations, where the (<i>x′</i>, <i>t′</i>) plane -coincides with the <i>x</i>-<i>t</i> plane, the significance is possible -that the <i>x</i>-axis of space some-how remains conserved in -its position.</p> - -<p class='c005'>We can approach the idea of space in a corresponding -manner, though some may regard the attempt as rather -fantastical.</p> - -<p class='c005'>According to these ideas, the word “Relativity-Postulate” -which has been coined for the demands of invariance -in the group G, seems to be rather inexpressive for a true -understanding of the group G<sub><i>c</i></sub>, and for further progress. -Because the sense of the postulate is that the four-dimensional -world is given in space and time by phenomena -only, but the projection in time and space can -be handled with a certain freedom, and therefore I would -rather like to give to this assertion the name “<i>The -Postulate of the Absolute world</i>” [World-Postulate].</p> -<h3 class='c008'>III</h3> -<p class='c004'>By the world-postulate a similar treatment of the four -determining quantities <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>, of a world-point is possible. -Thereby the forms under which the physical laws -come forth, gain in intelligibility, as I shall presently show. -Above all, the idea of acceleration becomes much more -striking and clear.</p> - -<p class='c005'>I shall again use the geometrical method of expression. -Let us call any world-point O as a “Space-time-null-point.” -The cone</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c²</i><i>t²</i> - <i>x²</i> - <i>y²</i> - <i>z²</i> = O</div> - </div> - </div> -</div> - -<p class='c010'>consists of two parts with O as apex, one part having -<i>t</i> < 0, the other having <i>t</i> > 0. The first, which we may call -<i>the fore-cone</i> consists of all those points which send light -towards O, the second, which we may call <i>the aft-cone</i>, -consists of all those points which receive their light from -O. The region bounded by the fore-cone may be called -the fore-side of O, and the region bounded by the aft-cone -may be called the aft-side of O. (<i>Vide</i> fig. 2).</p> - -<p class='c005'>On the aft-side of O we have the already considered -hyperboloidal shell F = <i>c²</i><i>t²</i> - <i>x²</i> - <i>y²</i> - <i>z²</i> = 1, <i>t</i> > 0.</p> - -<p class='c005'>The region inside the two cones will be occupied by the -hyperboloid of one sheet</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-F = <i>x²</i> + <i>y²</i> + <i>z²</i> - <i>c²</i><i>t²</i> = <i>k²</i>,</div> - </div> - </div> -</div> - -<p class='c010'>where <i>k²</i> can have all possible positive values. The -hyperbolas which lie upon this figure with O as centre, -are important for us. For the sake of clearness the individual -branches of this hyperbola will be called the “<i>Inter-hyperbola -with centre O</i>.” Such a hyperbolic branch, -when thought of as a world-line, would represent a -motion which for <i>t</i> = -∞ and <i>t</i> = ∞, asymptotically -approaches the velocity of light <i>c</i>.</p> - -<p class='c005'>If, by way of analogy to the idea of vectors in space, -we call any directed length in the manifoldness <i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> a -vector, then we have to distinguish between a time-vector -directed from O towards the sheet ±F = 1, <i>t</i> > 0 and a -space-vector directed from O towards the sheet -F = 1. -The time-axis can be parallel to any vector of the first -kind. Any world-point between the <i>fore</i> and <i>aft cones</i> -of O, may by means of the system of reference be regarded -either as synchronous with O, as well as later or earlier -than O. Every world-point on the fore-side of O is -necessarily always earlier, every point on the aft side of -O, later than O. The limit <i>c</i> = ∞ corresponds to a complete -folding up of the wedge-shaped cross-section between -the fore and aft cones in the manifoldness <i>t</i> = 0. In the -figure drawn, this cross-section has been intentionally -drawn with a different breadth.</p> - -<p class='c005'>Let us decompose a vector drawn from O towards -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) into its components. If the directions of the two -vectors are respectively the directions of the radius vector -OR to one of the surfaces ±F = 1, and of a tangent RS -at the point R of the surface, then the vectors shall be -called normal to each other. Accordingly</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c²</i><i>tt₁</i> - <i>xx₁</i> - <i>yy₁</i> - <i>zz₁</i> = 0,</div> - </div> - </div> -</div> - -<p class='c010'>which is the condition that the vectors with the components -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) and (<i>x₁</i> <i>y₁</i> <i>z₁</i> <i>t₁</i>) are normal to each -other.</p> - -<p class='c005'>For the <i>measurement</i> of vectors in different directions, -the unit measuring rod is to be fixed in the following -manner;—a space-like vector from 0 to -F = I is always -to have the measure unity, and a time-like vector from -O to +F = 1, <i>t</i> > 0 is always to have the measure 1/<i>c</i>.</p> - -<p class='c005'>Let us now fix our attention upon the world-line of a -substantive point running through the world-point (<i>x</i>, <i>y</i>, -<i>z</i>, <i>t</i>); then as we follow the <i>progress</i> of the line, the -quantity</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d</i>τ = (1/<i>c</i>) √(<i>c²</i><i>dt²</i> - <i>dx²</i> - <i>dy²</i> - <i>dz²</i>),</div> - </div> - </div> -</div> - -<p class='c010'>corresponds to the time-like vector-element (<i>dx</i>, <i>dy</i>, <i>dz</i>, <i>dt</i>).</p> - -<p class='c005'>The integral τ = ∫<i>d</i>τ, taken over the world-line from -any fixed initial point P₀ to any variable final point P, -may be called the “Proper-time” of the substantial point -at P₀ upon the <i>world-line</i>. We may regard (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), <i>i.e.</i>, -the components of the vector OP, as functions of the -“proper-time” τ; let ([.<i>x</i>], [.<i>y</i>], [.<i>z</i>], [.<i>t</i>]) denote the first differential-quotients, -and ([..<i>x</i>], [..<i>y</i>], [..<i>z</i>], [..<i>t</i>]) the second differential quotients -of (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) with regard to τ, then these may respectively -be called the <i>Velocity-vector</i>, and the <i>Acceleration-vector</i> -of the substantial point at P. Now we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c²</i> [.<i>t²</i>] - [.<i>x²</i>] - [.<i>y²</i>] - [.<i>z²</i>] = <i>c²</i></div> - </div> - <div class='group'> - <div class='line'><i>c²</i> [.<i>t</i>][..<i>t</i>] - [.<i>x</i>][..<i>x</i>] - [.<i>y</i>][..<i>y</i>] - [.<i>z</i>][..<i>z</i>] = 0</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i>, the ‘<i>Velocity-vector</i>’ is the time-like vector of unit -measure in the direction of the world-line at P, the ‘<i>Acceleration-vector</i>’ -at P is normal to the velocity-vector at P, -and is in any case, a space-like vector.</p> - -<p class='c005'>Now there is, as can be easily seen, a certain hyperbola, -which has three infinitely contiguous points in common -with the world-line at P, and of which the asymptotes -are the generators of a ‘fore-cone’ and an ‘aft-cone.’ -This hyperbola may be called the “hyperbola of curvature” -at P (<i>vide</i> fig. 3). If M be the centre of this hyperbola, -then we have to deal here with an ‘Inter-hyperbola’ with -centre M. Let P = measure of the vector MP, then we -easily perceive that the acceleration-vector at P is <i>a vector -of magnitude</i> <i>c²</i>/ρ <i>in the direction of</i> MP.</p> - -<p class='c005'>If [..<i>x</i>], [..<i>y</i>], [..<i>z</i>], [..<i>t</i>] are nil, then the hyperbola of curvature -at P reduces to the straight line touching the world-line -at P, and ρ = ∞.</p> -<h3 class='c008'>IV</h3> -<p class='c004'>In order to demonstrate that the assumption of the -group G<sub><i>c</i></sub> for the physical laws does not possibly lead to -any contradiction, it is unnecessary to undertake a revision -of the whole of physics on the basis of the assumptions -underlying this group. The revision has already been -successfully made in the case of “Thermodynamics and -Radiation,”<a id='r30' /><a href='#f30' class='c007'><sup>[30]</sup></a> for “Electromagnetic phenomena”,<a id='r31' /><a href='#f31' class='c007'><sup>[31]</sup></a> and -finally for “Mechanics with the maintenance of the idea of -mass.”</p> - -<p class='c005'>For this last mentioned province of physics, the question -may be asked: if there is a force with the components -X, Y, Z (in the direction of the space-axes) at a world-point -(<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>), where the velocity-vector is ([.<i>x</i>], [.<i>y</i>], [.<i>z</i>], [.<i>t</i>]), -then how are we to regard this force when the system of -reference is changed in any possible manner? Now it is -known that there are certain well-tested theorems about -the ponderomotive force in electromagnetic fields, where -the group G<sub><i>c</i></sub> is undoubtedly permissible. These theorems -lead us to the following simple rule; <i>if the system of -reference be changed in any way, then the supposed force is -to be put as a force in the new space-coordinates in such a -manner, that the corresponding vector with the components</i></p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>[.<i>t</i>]X, [.<i>t</i>]Y, [.<i>t</i>]Z, [.<i>t</i>]T,</div> - </div> - <div class='group'> - <div class='line'><i>where</i> T = 1/<i>c²</i> ([.<i>x</i>]/[.<i>t</i>] X + [.<i>y</i>]/[.<i>t</i>] Y + [.<i>z</i>]/[.<i>t</i>] Z) = 1/<i>c²</i></div> - <div class='line'>(<i>the rate of</i></div> - <div class='line'><i>which work is done at the world-point</i>), <i>remains unaltered</i>.</div> - </div> - </div> -</div> - -<p class='c010'>This vector is always normal to the velocity-vector at P. -Such a force-vector, representing a force at P, may be -called a <i>moving force-vector at</i> P.</p> - -<p class='c005'>Now the world-line passing through P will be described -by a substantial point with the constant <i>mechanical mass -m</i>. Let us call <i>m-times</i> the velocity-vector at P as the -<i>impulse-vector</i>, and <i>m-times</i> the acceleration-vector at P as -the <i>force-vector of motion</i>, at P. According to these -definitions, the following law tells us how the motion of -a point-mass takes place under any moving force-vector<a id='r32' /><a href='#f32' class='c007'><sup>[32]</sup></a>:</p> - -<p class='c005'><i>The force-vector of motion is equal to the moving force-vector.</i></p> - -<p class='c005'>This enunciation comprises four equations for the components -in the four directions, of which the fourth can be -deduced from the first three, because both of the above-mentioned -vectors are perpendicular to the velocity-vector. -From the definition of T, we see that the fourth simply -expresses the “Energy-law.” Accordingly <i>c²</i>-<i>times the -component of the impulse-vector in the direction of the -t-axis is</i> to be defined as <i>the kinetic-energy</i> of the point-mass. -The expression for this is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>mc²</i> <i>dt</i>/<i>d</i>τ = <i>mc²</i> /√(1 - <i>v²</i>/<i>c²</i>)</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i>, if we deduct from this the additive constant <i>mc²</i>, we -obtain the expression ½ <i>mv²</i> of Newtonian-mechanics up to -magnitudes of <i>the order of</i> 1/<i>c²</i>. Hence it appears that <i>the -energy</i> depends <i>upon the system of reference</i>. But since the -<i>t</i>-axis can be laid in the direction of any time-like axis, -therefore the energy-law comprises, for any possible system -of reference, the whole system of equations of motion. -This fact retains its significance even in the limiting case -c = ∞, for the axiomatic construction of Newtonian -mechanics, as has already been pointed out by T. R. -Schütz.<a id='r33' /><a href='#f33' class='c007'><sup>[33]</sup></a></p> - -<p class='c005'>From the very beginning, we can establish the ratio -between the units of time and space in such a manner, that -the velocity of light becomes unity. If we now write -√-1 <i>t</i> = <i>l</i>, in the place of <i>l</i>, then the differential expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d</i>τ² = -(<i>dx²</i> + <i>dy²</i> + <i>dz²</i> + <i>dl²</i>),</div> - </div> - </div> -</div> - -<p class='c010'>becomes symmetrical in (<i>x</i>, <i>y</i>, <i>r</i>, <i>l</i>); this symmetry then -enters into each law, which does not contradict the <i>world-postulate</i>. -We can clothe the “essential nature of this -postulate in the mystical, but mathematically significant -formula</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>3·10⁵ <i>km</i> = √-1 Sec.</div> - </div> - </div> -</div> - -<h4 class='c009'>V</h4> -<p class='c004'>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 theory.</p> - -<p class='c005'>Let us conceive of the world-line of such an electron -with the charge (<i>e</i>), and let us introduce upon it the -“Proper-time” τ reckoned from any possible initial point. -In order to obtain the field caused by the electron at any -world-point P₁ let us construct the fore-cone belonging -to P₁ (<i>vide</i> fig. 4). Clearly this cuts the unlimited -world-line of the electron at a single point P, because these -directions are all time-like vectors. At P, let us draw the -tangent to the world-line, and let us draw from P₁ the -normal to this tangent. Let <i>r</i> be the measure of P₁Q. -According to the definition of a fore-cone, <i>r</i>/<i>e</i> is to be -reckoned as the measure of PQ. Now at the world-point P₁, -the vector-potential of the field excited by <i>e</i> is represented -by the vector in direction PQ, having the magnitude -<i>e</i>/<i>cr</i>, in its three space components along the <i>x</i>-, <i>y</i>-, <i>z</i>-axes; -the scalar-potential is represented by the component along -the <i>t</i>-axis. This is the elementary law found out by -A. Lienard, and E. Wiechert.<a id='r34' /><a href='#f34' class='c007'><sup>[34]</sup></a></p> - -<p class='c005'>If the field caused by the electron be described in the -above-mentioned way, then it will appear that the division -of the field into electric and magnetic forces is a relative -one, and depends upon the time-axis assumed; the two -forces considered together bears some analogy to the -force-screw in mechanics; the analogy is, however, imperfect.</p> - -<p class='c005'>I shall now describe <i>the ponderomotive force which is -exerted by one moving electron upon another moving electron</i>. -Let us suppose that the world-line of a second point-electron -passes through the world-point P₁. Let us -determine P, Q, <i>r</i> as before, construct the middle-point M -of the hyperbola of curvature at P, and finally the normal -MN upon a line through P which is parallel to QP₁. -With P as the initial point, we shall establish a system -of reference in the following way: the <i>t</i>-axis will be laid -along PQ, the <i>x</i>-axis in the direction of QP₁. The <i>y</i>-axis -in the direction of MN, then the <i>z</i>-axis is automatically -determined, as it is normal to the <i>x</i>-, <i>y</i>-, <i>z</i>-axes. Let -[:<i>x</i>], [:<i>y</i>], [:<i>z</i>], [:<i>t</i>] be the acceleration-vector at P, [.<i>x</i>]₁, [.<i>y</i>]₁ [.<i>z</i>]₁, [.<i>t</i>]₁ -be the velocity-vector at P₁. Then the force-vector exerted -by the first election <i>e</i>, (moving in any possible manner) -upon the second election <i>e</i>, (likewise moving in any -possible manner) at P₁ is represented by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-<i>e e₁</i>([.<i>t₁</i>] - [.<i>x₁</i>]/<i>c</i>)F,</div> - </div> - </div> -</div> - -<p class='c010'><i>For the components F<sub>x</sub>, F<sub>y</sub>, F<sub>z</sub>, F<sub>t</sub> of the vector F the -following three relations hold</i>:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c</i>F<sub><i>t</i></sub> - F<sub><i>x</i></sub> = 1/<i>r²</i>, F<sub><i>y</i></sub> = [:<i>y</i>]/(<i>c²</i><i>r</i>), F<sub><i>z</i></sub> = 0,</div> - </div> - </div> -</div> - -<p class='c010'><i>and fourthly this vector F is normal to the velocity-vector</i> -P₁, <i>and through this circumstance alone, its dependence on -this last velocity-vector arises</i>.</p> - -<p class='c005'>If we compare with this expression the previous formulæ<a id='r35' /><a href='#f35' class='c007'><sup>[35]</sup></a> -giving the elementary law about the ponderomotive -action of moving electric charges upon each other, then we -cannot but admit, that the relations which occur here -reveal the inner essence of full simplicity first in four -dimensions; but in three dimensions, they have very complicated -projections.</p> - -<p class='c005'>In the mechanics reformed according to the world-postulate, -the disharmonies which have disturbed the -relations between Newtonian mechanics and modern -electrodynamics automatically disappear. I shall now consider -the position of the Newtonian law of attraction to -this postulate. I will assume that two point-masses <i>m</i> and -<i>m₁</i> describe their world-lines; a moving force-vector is -exercised by <i>m</i> upon <i>m₁</i>, and the expression is just the same -as in the case of the electron, only we have to write -+<i>mm₁</i> instead -<i>ee₁</i>. We shall consider only the special -case in which the acceleration-vector of <i>m</i> is always zero: -then <i>t</i> may be introduced in such a manner that <i>m</i> may be -regarded as fixed, the motion of <i>m</i> is now subjected to the -moving-force vector of <i>m</i> alone. If we now modify this -given vector by writing -([<sup>.</sup>]1/√(1-(<i>v</i>²/<i>c²</i>)) instead of [.<i>t</i>] ([.<i>t</i>] = 1 up -to magnitudes of the order (1[<sup>.</sup>]/<i>c²</i>)), then it appears that -Kepler’s laws hold good for the position (<i>x₁</i>, <i>y₁</i>, <i>z₁</i>), of -<i>m₁</i> at any time, only in place of the time <i>t₁</i>, we have to -write the proper time τ₁ of <i>m₁</i>. On the basis of this -simple remark, it can be seen that the proposed law of -attraction in combination with new mechanics is not less -suited for the explanation of astronomical phenomena than -the Newtonian law of attraction in combination with -Newtonian mechanics.</p> - -<p class='c005'>Also the fundamental equations for electro-magnetic -processes in moving bodies are in accordance with the -world-postulate. I shall also show on a later occasion -that the deduction of these equations, as taught by -Lorentz, are by no means to be given up.</p> - -<p class='c005'>The fact that the world-postulate holds without exception -is, I believe, the true essence of an electromagnetic -picture of the world; the idea first occurred to Lorentz, its -essence was first picked out by Einstein, and is now gradually -fully manifest. In course of time, the mathematical -consequences will be gradually deduced, and enough -suggestions will be forthcoming for the experimental -verification of the postulate; in this way even those, who -find it uncongenial, or even painful to give up the old, -time-honoured concepts, will be reconciled to the new ideas -of time and space,—in the prospect that they will lead to -pre-established harmony between pure mathematics and -physics.</p> - -<div class='chapter'> - <h2 id='art6' class='c003'>The Foundation of the Generalised Theory of Relativity <br /> By A. Einstein. <br /> From Annalen der Physik 4.49.1916.</h2> -</div> -<p class='c004'>The theory which is sketched in the following pages -forms the most wide-going generalization conceivable of -what is at present known as “the theory of Relativity;” -this latter theory I differentiate from the former -“Special Relativity theory,” and suppose it to be known. -The generalization of the Relativity theory has been made -much easier through the form given to the special Relativity -theory by Minkowski, which mathematician was the -first to recognize clearly the formal equivalence of the space -like and time-like co-ordinates, and who made use of it in -the building up of the theory. The mathematical apparatus -useful for the general relativity theory, lay already complete -in the “Absolute Differential Calculus,” which were -based on the researches of Gauss, Riemann and Christoffel -on the non-Euclidean manifold, and which have been -shaped into a system by Ricci and Levi-civita, and already -applied to the problems of theoretical physics. I have in -part B of this communication developed in the simplest -and clearest manner, all the supposed mathematical -auxiliaries, not known to Physicists, which will be useful -for our purpose, so that, a study of the mathematical -literature is not necessary for an understanding of this -paper. Finally in this place I thank my friend Grossmann, -by whose help I was not only spared the study of the -mathematical literature pertinent to this subject, but who -also aided me in the researches on the field equations of -gravitation.</p> -<h3 class='c008'>A <br /> Principal considerations about the Postulate of Relativity.</h3> -<h4 class='c009'>§ 1. Remarks on the Special Relativity Theory.</h4> -<p class='c004'>The special relativity theory rests on the following -postulate which also holds valid for the Galileo-Newtonian -mechanics.</p> - -<p class='c005'>If a co-ordinate system K be so chosen that when referred -to it, the physical laws hold in their simplest forms -these laws would be also valid when referred to another -system of co-ordinates K′ which is subjected to an uniform -translational motion relative to K. We call this postulate -“The Special Relativity Principle.” By the word special, -it is signified that the principle is limited to the case, -when K′ has <i>uniform translatory</i> motion with reference to -K, but the equivalence of K and K′ does not extend to the -case of non-uniform motion of K′ relative to K.</p> - -<p class='c005'>The Special Relativity Theory does not differ from the -classical mechanics through the assumption of this postulate, -but only through the postulate of the constancy of -light-velocity in vacuum which, when combined with the -special relativity postulate, gives in a well-known way, the -relativity of synchronism as well as the Lorenz-transformation, -with all the relations between moving rigid bodies -and clocks.</p> - -<p class='c005'>The modification which the theory of space and time -has undergone through the special relativity theory, is -indeed a profound one, but a weightier point remains -untouched. According to the special relativity theory, the -theorems of geometry are to be looked upon as the laws -about any possible relative positions of solid bodies at rest, -and more generally the theorems of kinematics, as theorems -which describe the relation between measurable bodies and -clocks. Consider two material points of a solid body at -rest; then according to these conceptions there corresponds -to these points a wholly definite extent of length, -independent of kind, position, orientation and time of the -body.</p> - -<p class='c005'>Similarly let us consider two positions of the pointers of -a clock which is at rest with reference to a co-ordinate -system; then to these positions, there always corresponds, -a time-interval of a definite length, independent of time -and place. It would be soon shown that the general relativity -theory can not hold fast to this simple physical -significance of space and time.</p> -<h4 class='c009'>§ 2. About the reasons which explain the extension of the relativity-postulate.</h4> -<p class='c004'>To the classical mechanics (no less than) to the special -relativity theory, is attached an episteomological defect, -which was perhaps first cleanly pointed out by E. Mach. -We shall illustrate it by the following example; Let -two fluid bodies of equal kind and magnitude swim freely -in space at such a great distance from one another (and -from all other masses) that only that sort of gravitational -forces are to be taken into account which the parts of any -of these bodies exert upon each other. The distance of -the bodies from one another is invariable. The relative -motion of the different parts of each body is not to occur. -But each mass is seen to rotate by an observer at rest relative -to the other mass round the connecting line of the -masses with a constant angular velocity (definite relative -motion for both the masses). Now let us think that the -surfaces of both the bodies (S₁ and S₂) are measured -with the help of measuring rods (relatively at rest); it is -then found that the surface of S₁ is a sphere and the -surface of the other is an ellipsoid of rotation. We now -ask, why is this difference between the two bodies? 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 experience. -The law of causality has the sense of a definite -statement about the world of experience only when -observable facts alone appear as causes and effects.</p> - -<p class='c005'>The Newtonian mechanics does not give to this question -any satisfactory answer. For example, it says:—The laws -of mechanics hold true for a space R₁ relative to which -the body S₁ is at rest, not however for a space relative to -which S₂ is at rest.</p> - -<p class='c005'>The Galiliean space, which is here introduced is however -only a purely imaginary cause, not an observable thing. -It is thus clear that the Newtonian mechanics does not, -in the case treated here, actually fulfil the requirements -of causality, but produces on the mind a fictitious complacency, -in that it makes responsible a <i>wholly imaginary -cause</i> R₁ for the different behaviours of the bodies S₁ and -S₂ which are actually observable.</p> - -<p class='c005'>A satisfactory explanation to the question put forward -above can only be thus given:—that the physical system -composed of S₁ and S₂ shows for itself alone no conceivable -cause to which the different behaviour of S₁ and -S₂ can be attributed. The cause must thus lie outside the -system. We are therefore led to the conception that the -general laws of motion which determine specially the -forms of S₁ and S₂ must be of such a kind, that the -mechanical behaviour of S₁ and S₂ must be essentially -conditioned by the distant masses, which we had not -brought into the system considered. These distant masses, -(and their relative motion as regards the bodies under consideration) -are then to be looked upon as the seat of the -principal observable causes for the different behaviours -of the bodies under consideration. They take the place -of the imaginary cause R₁. Among all the conceivable -spaces R₁ and R₂ moving in any manner relative to one -another, there is a priori, no one set which can be regarded -as affording greater advantages, against which the objection -which was already raised from the standpoint of the -theory of knowledge cannot be again revived. The laws -of physics must be so constituted that they should remain -valid for any system of co-ordinates moving in any manner. -We thus arrive at an extension of the relativity postulate.</p> - -<p class='c005'>Besides this momentous episteomological argument, -there is also a well-known physical fact which speaks in -favour of an extension of the relativity theory. Let there -be a Galiliean co-ordinate system K relative to which (at -least in the four-dimensional region considered) a mass at -a sufficient distance from other masses move uniformly in -a line. Let K′ be a second co-ordinate system which has -a uniformly accelerated motion relative to K. Relative 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 conditions.</p> - -<p class='c005'>Can any observer, at rest relative to K′, then conclude -that he is in an actually accelerated reference-system? -This is to be answered in the negative; the above-named -behaviour of the freely moving masses relative to K′ can -be explained in as good a manner in the following way. -The reference-system K′ has no acceleration. In the space-time -region considered there is a gravitation-field which -generates the accelerated motion relative to K′.</p> - -<p class='c005'>This conception is feasible, because to us the experience -of the existence of a field of force (namely the gravitation -field) has shown that it possesses the remarkable property -of imparting the same acceleration to all bodies. The -mechanical behaviour of the bodies relative to K′ is the -same as experience would expect of them with reference -to systems which we assume from habit as stationary; -thus it explains why from the physical stand-point it can -be assumed that the systems K and K′ can both with the -same legitimacy be taken as at rest, that is, they will be -equivalent as systems of reference for a description of -physical phenomena.</p> - -<p class='c005'>From these discussions we see, that the working out -of the general relativity theory must, at the same time, -lead to a theory of gravitation; for we can “create” -a gravitational field by a simple variation of the co-ordinate -system. Also we see immediately that the principle -of the constancy of light-velocity must be modified, -for we recognise easily that the path of a ray of light -with reference to K′ must be, in general, curved, when -light travels with a definite and constant velocity in a -straight line with reference to K.</p> -<h4 class='c009'>§ 3. The time-space continuum. Requirements of the general Co-variance for the equations expressing the laws of Nature in general.</h4> -<p class='c004'>In the classical mechanics as well as in the special -relativity theory, the co-ordinates of time and space have -an immediate physical significance; when we say that -any arbitrary point has <i>x₁</i> as its X₁ co-ordinate, it signifies -that the projection of the point-event on the X₁-axis -<i>ascertained</i> by means of a solid rod according to the rules -of Euclidean Geometry is reached when a definite measuring -rod, the unit rod, can be carried <i>x₁</i> times from the -origin of co-ordinates along the X₁ axis. A point having -<i>x₄</i> = <i>t₁</i> as the X₄ co-ordinate signifies that a unit clock -which is adjusted to be at rest relative to the system of -co-ordinates, and coinciding in its spatial position with the -point-event and set according to some definite standard has -gone over <i>x₄</i> = <i>t</i> periods before the occurrence of the -point-event.</p> - -<p class='c005'>This conception of time and space is continually present -in the mind of the physicist, though often in an unconscious -way, as is clearly recognised from the role which this -conception has played in physical measurements. This -conception must also appear to the reader to be lying at -the basis of the second consideration of the last paragraph -and imparting a sense to these conceptions. 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 relativity,—the -case of special relativity appearing as a limiting -case when there is no gravitation.</p> - -<p class='c005'>We introduce in a space, which is free from Gravitation-field, -a Galiliean Co-ordinate System K (<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i>) and -also, another system K′ (<i>x′</i> <i>y′</i> <i>z′</i> <i>t′</i>) rotating uniformly relative -to K. The origin of both the systems as well as their -<i>z</i>-axes might continue to coincide. We will show that for -a space-time measurement in the system K′, the above -established rules for the physical significance of time and -space can not be maintained. On grounds of symmetry -it is clear that a circle round the origin in the XY plane -of K, can also be looked upon as a circle in the plane -(X′, Y′) of K′. Let us now think of measuring the circumference -and the diameter of these circles, with a unit -measuring rod (infinitely small compared with the radius) -and take the quotient of both the results of measurement. -If this experiment be carried out with a measuring rod -at rest relatively to the Galiliean system K we would get -π, as the quotient. The result of measurement with a rod -relatively at rest as regards K′ would be a number which -is greater than π. 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-contraction, not however when the rod -is placed along the radius. Euclidean Geometry therefore -does not hold for the system K′; the above fixed conceptions -of co-ordinates which assume the validity of -Euclidean Geometry fail with regard to the system K′. -We cannot similarly introduce in K′ a time corresponding to -physical requirements, which will be shown by all similarly -prepared clocks at rest relative to the system K′. In order -to see this we suppose that two similarly made clocks are -arranged one at the centre and one at the periphery of -the circle, and considered from the stationary system -K. According to the well-known results of the special -relativity theory it follows—(as viewed from K)—that the -clock placed at the periphery will go slower than the -second one which is at rest. The observer at the common -origin of co-ordinates who is able to see the clock at the -periphery by means of light will see the clock at the -periphery going slower than the clock beside him. Since he -cannot allow the velocity of light to depend explicitly upon -the time in the way under consideration he will interpret -his observation by saying that the clock on the periphery -actually goes slower than the clock at the origin. He -cannot therefore do otherwise than define time in such -a way that the rate of going of a clock depends on its -position.</p> - -<p class='c005'>We therefore arrive at this result. In the general -relativity theory time and space magnitudes cannot be so -defined that the difference in spatial co-ordinates can be -immediately measured by the unit-measuring rod, and time-like -co-ordinate difference with the aid of a normal clock.</p> - -<p class='c005'>The means hitherto at our disposal, for placing our -co-ordinate system in the time-space continuum, in a -definite way, therefore completely fail and it appears that -there is no other way which will enable us to fit the -co-ordinate system to the four-dimensional world in such -a way, that by it we can expect to get a specially simple -formulation of the laws of Nature. So that nothing remains -for us but to regard all conceivable co-ordinate systems -as equally suitable for the description of natural phenomena. -This amounts to the following law:—</p> - -<p class='c005'><i>That in general, Laws of Nature are expressed by means of -equations which are valid for all co-ordinate systems, that is, -which are covariant for all possible transformations.</i> It is -clear that a physics which satisfies this postulate will be -unobjectionable from the standpoint of the general -relativity postulate. Because among all substitutions -there are, in every case, contained those, which correspond -to all relative motions of the co-ordinate system (in -three dimensions). This condition of general covariance -which takes away the last remnants of physical objectivity -from space and time, is a natural requirement, as seen -from the following considerations. All our <i>well-substantiated</i> -space-time propositions amount to the determination -of space-time coincidences. If, for example, the event -consisted in the motion of material points, then, for this -last case, nothing else are really observable except the -encounters between two or more of these material points. -The results of our measurements are nothing else than -well-proved theorems about such coincidences of material -points, of our measuring rods with other material points, -coincidences between the hands of a clock, dial-marks and -point-events occurring at the same position and at the same -time.</p> - -<p class='c005'>The introduction of a system of co-ordinates serves no -other purpose than an easy description of totality of such -coincidences. We fit to the world our space-time variables -(<i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i>) such that to any and every point-event -corresponds a system of values of (<i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i>). Two coincident -point-events correspond to the same value of the -variables (<i>x₁</i> <i>x₂</i> <i>x₃</i> <i>x₄</i>); <i>i.e.</i>, the coincidence is characterised -by the equality of the co-ordinates. If we now -introduce any four functions (<i>x′₁</i> <i>x′₂</i> <i>x′₃</i> <i>x′₄</i>) as co-ordinates, -so that there is an unique correspondence between -them, the equality of all the four co-ordinates in the new -system will still be the expression of the space-time -coincidence of two material points. As the purpose of -all physical laws is to allow us to remember such coincidences, -there is a priori no reason present, to prefer a -certain co-ordinate system to another; <i>i.e.</i>, we get the -condition of general covariance.</p> -<h4 class='c009'>§ 4. Relation of four co-ordinates to spatial and time-like measurements.</h4> -<p class='c004'><i>Analytical expression for the Gravitation-field.</i></p> -<p class='c004'>I am not trying in this communication to deduce the -general Relativity-theory as the simplest logical system -possible, with a minimum of axioms. But it is my chief -aim to develop the theory in such a manner that the -reader perceives the psychological naturalness of the way -proposed, and the fundamental assumptions appear to be -most reasonable according to the light of experience. In -this sense, we shall now introduce the following supposition; -that for an infinitely small four-dimensional region, the -relativity theory is valid in the special sense when the axes -are suitably chosen.</p> - -<p class='c005'>The nature of acceleration of an infinitely small (positional) -co-ordinate system is hereby to be so chosen, that -the gravitational field does not appear; this is possible for -an infinitely small region. X₁, X₂, X₃ are the spatial -co-ordinates; X₄ is the corresponding time-co-ordinate -measured by some suitable measuring clock. These co-ordinates -have, with a given orientation of the system, an -immediate physical significance in the sense of the special -relativity theory (when we take a rigid rod as our unit of -measure). The expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(1) <i>ds²</i> = - <i>d</i>X₁² - <i>d</i>X₂² - <i>d</i>X₃² + <i>d</i>X₄²</div> - </div> - </div> -</div> - -<p class='c010'>had then, according to the special relativity theory, a value -which may be obtained by space-time measurement, and -which is independent of the orientation of the local -co-ordinate system. Let us take <i>ds</i> as the magnitude of the -line-element belonging to two infinitely near points in the -four-dimensional region. If <i>ds²</i> belonging to the element -(<i>d</i>X₁, <i>d</i>X₂, <i>d</i>X₃, <i>d</i>X₄) be positive we call it with Minkowski, -time-like, and in the contrary case space-like.</p> - -<p class='c005'>To the line-element considered, <i>i.e.</i>, to both the infinitely -near point-events belong also definite differentials -<i>dx₁</i>, <i>dx₂</i>, <i>dx₃</i>, <i>dx₄</i>, of the four-dimensional co-ordinates of -any chosen system of reference. If there be also a local -system of the above kind given for the case under consideration, -<i>d</i>X’s would then be represented by definite linear -homogeneous expressions of the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(2) <i>d</i>X<sub>ν</sub> = σ<sub>σ</sub><i>a</i><sub>νσ</sub><i>dx</i><sub>σ</sub></div> - </div> - </div> -</div> - -<p class='c010'>If we substitute the expression in (1) we get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(3) <i>ds²</i> = σ<sub>στ</sub><i>g</i><sub>στ</sub><i>dx</i><sub>σ</sub><i>dx</i><sub>τ</sub></div> - </div> - </div> -</div> - -<p class='c010'>where <i>g</i><sub>στ</sub> will be functions of <i>x</i><sub>σ</sub>, but will no longer depend -upon the orientation and motion of the ‘local’ co-ordinates; -for <i>ds²</i> is a definite magnitude belonging to two point-events -infinitely near in space and time and can be got by -measurements with rods and clocks. The <i>g</i><sub>τσ</sub>’s are here to -be so chosen, that <i>g</i><sub>τσ</sub> = <i>g</i><sub>στ</sub>; the summation is to be -extended over all values of σ and τ, so that the sum is to -be extended, over 4 × 4 terms, of which 12 are equal in -pairs.</p> - -<p class='c005'>From the method adopted here, the case of the usual -relativity theory comes out when owing to the special -behaviour of <i>g</i><sub>στ</sub> in a <i>finite</i> region it is possible to choose the -system of co-ordinates in such a way that <i>g</i><sub>στ</sub> assumes -constant values—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ -1, 0, 0, 0</div> - <div class='line'>(4) { 0 -1 0 0</div> - <div class='line in6'>{ 0 0 -1 0</div> - <div class='line in6'>{ 0 0 0 +1</div> - </div> - </div> -</div> - -<p class='c010'>We would afterwards see that the choice of such a system -of co-ordinates for a finite region is in general not possible.</p> - -<p class='c005'>From the considerations in § 2 and § 3 it is clear, -that from the physical stand-point the quantities <i>g</i><sub>στ</sub> are to -be looked upon as magnitudes which describe the gravitation-field -with reference to the chosen system of axes. -We assume firstly, that in a certain four-dimensional -region considered, the special relativity theory is true for -some particular choice of co-ordinates. The <i>g</i><sub>στ</sub>’s then -have the values given in (4). A free material point moves -with reference to such a system uniformly in a straight-line. -If we now introduce, by any substitution, the space-time -co-ordinates <i>x₁</i>...<i>x₄</i> then in the new system <i>g</i><sub>μν</sub>’s are -no longer constants, but functions of space and time. At -the same time, the motion of a free point-mass in the new -co-ordinates, will appear as curvilinear, and not uniform, in -which the law of motion, will be <i>independent of the -nature of the moving mass-points</i>. We can thus signify this -motion as one under the influence of a gravitation field. -We see that the appearance of a gravitation-field is connected -with space-time variability of <i>g</i><sub>στ</sub>’s. In the general -case, we can not by any suitable choice of axes, make -special relativity theory valid throughout any finite region. -We thus deduce the conception that <i>g</i><sub>στ</sub>’s describe the -gravitational field. According to the general relativity -theory, gravitation thus plays an exceptional rôle as distinguished -from the others, specially the electromagnetic -forces, in as much as the 10 functions <i>g</i><sub>στ</sub> representing -gravitation, define immediately the metrical properties of -the four-dimensional region.</p> -<h3 class='c008'>B <br /> Mathematical Auxiliaries for Establishing the General Covariant Equations.</h3> -<p class='c004'>We have seen before that the general relativity-postulate -leads to the condition that the system of equations -for Physics, must be co-variants for any possible substitution -of co-ordinates <i>x₁</i>, ... <i>x₄</i>; we have now to see -how such general co-variant equations can be obtained. -We shall now turn our attention to these purely mathematical -propositions. It will be shown that in the solution, the -invariant <i>ds</i>, given in equation (3) plays a fundamental -rôle, which we, following Gauss’s Theory of Surfaces, -style as the line-element.</p> - -<p class='c005'>The fundamental idea of the general co-variant theory -is this:—With reference to any co-ordinate system, let -certain things (tensors) be defined by a number of functions -of co-ordinates which are called the components of -the tensor. There are now certain rules according to which -the components can be calculated in a new system of -co-ordinates, when these are known for the original -system, and when the transformation connecting the two -systems is known. The things herefrom designated as -“Tensors” have further the property that the transformation -equation of their components are linear and homogeneous; -so that all the components in the new system vanish -if they are all zero in the original system. Thus a law -of Nature can be formulated by putting all the components -of a tensor equal to zero so that it is a general co-variant -equation; thus while we seek the laws of formation of -the tensors, we also reach the means of establishing general -co-variant laws.</p> -<h4 class='c009'>5. Contra-variant and co-variant Four-vector.</h4> -<p class='c004'>Contra-variant Four-vector. The line-element is defined -by the four components <i>dx</i><sub>ν</sub>, whose transformation law -is expressed by the equation</p> - -<div class='figcenter id002'> -<img src='images/form-057.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(5)."</p> -</div> -</div> - -<p class='c005'>The <i>dx′</i><sub>σ</sub>’<i>s</i> are expressed as linear and homogeneous function -of <i>dx</i><sub>ν</sub>’<i>s</i>; we can look upon the differentials of the -co-ordinates as the components of a tensor, which we -designate specially as a contravariant Four-vector. Everything -which is defined by Four quantities -A<sup>σ</sup>, -with reference -to a co-ordinate system, and transforms according to -the same law,</p> - -<div class='figcenter id005'> -<img src='images/form-058.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(5a)."</p> -</div> -</div> - -<p class='c005'>we may call a contra-variant Four-vector. From (5. a), -it follows at once that the sums -(A<sup>σ</sup> ± B<sup>σ</sup>) -are also components -of a four-vector, when -A<sup>σ</sup> and B<sup>σ</sup> -are so; corresponding -relations hold also for all systems afterwards -introduced as “tensors” (Rule of addition and subtraction -of Tensors).</p> -<p class='c004'><i>Co-variant Four-vector.</i></p> -<p class='c004'>We call four quantities A<sub>ν</sub> as the components of a covariant -four-vector, when for any choice of the contra-variant -four vector -B<sup>ν</sup> (6) ∑<sub>ν</sub> A<sub>ν</sub> B<sup>ν</sup> = <i>Invariant</i>. -From this definition follows the law of transformation of -the co-variant four-vectors. If we substitute in the right -hand side of the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∑<sub>σ</sub> A′<sub>σ</sub> B<sup>σ′</sup> = ∑<sub>ν</sub> A<sub>ν</sub> B<sup>ν</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>the expressions</p> - -<div class='figcenter id005'> -<img src='images/form-059.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>for -B<sup>ν</sup> -following from the inversion of the equation (5a) -we get</p> - -<div class='figcenter id002'> -<img src='images/form-060.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>As in the above equation -B<sup>σ′</sup> -are independent of one another -and perfectly arbitrary, it follows that the transformation -law is:—</p> - -<div class='figcenter id002'> -<img src='images/form-061.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'><i>Remarks on the simplification of the mode of writing -the expressions.</i> A glance at the equations of this -paragraph will show that the indices which appear twice -within the sign of summation [for example ν in (5)] are -those over which the summation is to be made and that -only over the indices which appear twice. It is therefore -possible, without loss of clearness, to leave off the summation -sign; so that we introduce the rule: wherever the -index in any term of an expression appears twice, it is to -be summed over all of them except when it is not expressedly -said to the contrary.</p> - -<p class='c005'>The difference between the co-variant and the contra-variant -four-vector lies in the transformation laws [(7) -and (5)]. Both the quantities are tensors according to the -above general remarks; in it lies its significance. In -accordance with Ricci and Levi-civita, the contravariants -and co-variants are designated by the over and under -indices.</p> -<h4 class='c009'>§ 6. Tensors of the second and higher ranks.</h4> -<p class='c004'>Contravariant tensor:—If we now calculate all the 16 -products -A<sup>μν</sup> of the components A<sup>μ</sup> B<sup>ν</sup>, of two contravariant -four-vectors</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(8) A<sup>μν</sup> = A<sup>μ</sup>B<sup>ν</sup></div> - </div> - </div> -</div> - -<p class='c010'>A<sup>μν</sup>, -will according to (8) and (5 a) satisfy the following -transformation law.</p> - -<div class='figcenter id002'> -<img src='images/form-062.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(9)."</p> -</div> -</div> - -<p class='c005'>We call a thing which, with reference to any reference -system is defined by 16 quantities and fulfils the transformation -relation (9), a contravariant tensor of the second -rank. Not every such tensor can be built from two four-vectors, -(according to 8). But it is easy to show that any -16 quantities A<sup>μν</sup>, can be represented as the sum of A<sup>μ</sup>B<sup>ν</sup> -of properly chosen four pairs of four-vectors. From it, -we can prove in the simplest way all laws which hold true -for the tensor of the second rank defined through (9), by -proving it only for the special tensor of the type (8).</p> - -<p class='c005'><i>Contravariant Tensor of any rank</i>:—It is clear that -corresponding to (8) and (9), we can define contravariant -tensors of the 3rd and higher ranks, with 4³, etc. components. -Thus it is clear from (8) and (9) that in this -sense, we can look upon contravariant four-vectors, as -contravariant tensors of the first rank.</p> -<p class='c004'><i>Co-variant tensor.</i></p> -<p class='c004'>If on the other hand, we take the 16 products A<sub>μν</sub> of -the components of two co-variant four-vectors A<sub>μ</sub> and -B<sub>ν</sub>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(10) A<sub>μν</sub> = A<sub>μ</sub> B<sub>ν</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>for them holds the transformation law</p> - -<div class='figcenter id003'> -<img src='images/form-063.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(11)."</p> -</div> -</div> - -<p class='c005'>By means of these transformation laws, the co-variant -tensor of the second rank is defined. All re-marks which -we have already made concerning the contravariant tensors, -hold also for co-variant tensors.</p> -<p class='c004'><i>Remark</i>:—</p> -<p class='c004'>It is convenient to treat the scalar Invariant either -as a contravariant or a co-variant tensor of zero rank.</p> - -<p class='c005'><i>Mixed tensor.</i> We can also define a tensor of the -second rank of the type</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(12) A<sub>μ</sub><sup>ν</sup> = A<sub>μ</sub>B<sup>ν</sup></div> - </div> - </div> -</div> - -<p class='c010'>which is co-variant with reference to μ and contravariant -with reference to ν. Its transformation law is</p> - -<div class='figcenter id003'> -<img src='images/form-064.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(13)."</p> -</div> -</div> - -<p class='c005'>Naturally there are mixed tensors with any number of -co-variant indices, and with any number of contra-variant -indices. The co-variant and contra-variant tensors can be -looked upon as special cases of mixed tensors.</p> -<p class='c004'><i>Symmetrical tensors</i>:—</p> -<p class='c004'>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 equal. The tensor -A<sup>μν</sup> -or A<sub>μν</sub> is symmetrical, when -we have for any combination of indices</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(14) A<sup>μν</sup> = A<sup>νμ</sup></div> - </div> - </div> -</div> - -<p class='c010'>or</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(14a) A<sub>μν</sub> = A<sub>νμ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>It must be proved that a symmetry so defined is a property -independent of the system of reference. It follows in fact -from (9) remembering (14)</p> - -<div class='figcenter id004'> -<img src='images/form-065.png' alt='Formula.' class='ig001' /> -</div> -<p class='c004'><i>Antisymmetrical tensor.</i></p> -<p class='c004'>A contravariant or co-variant tensor of the 2nd, 3rd or -4th rank is called <i>antisymmetrical</i> when the two components -got by mutually interchanging any two indices -are equal and opposite. The tensor or -A<sup>μν</sup> or A<sub>μν</sub> is thus -antisymmetrical when we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(15) A<sup>μν</sup> = -A<sup>νμ</sup></div> - </div> - </div> -</div> - -<p class='c010'>or</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(15a) A<sub>μν</sub> = -A<sub>νμ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Of the 16 components -A<sup>μν</sup>, the four components A<sup>μμ</sup> -vanish, the rest are equal and opposite in pairs; so that -there are only 6 numerically different components present -(Six-vector).</p> - -<p class='c005'>Thus we also see that the antisymmetrical tensor -A<sup>μνσ</sup> -(3rd rank) has only 4 components numerically -different, and the antisymmetrical tensor -A<sup>μνστ</sup> -only one. -Symmetrical tensors of ranks higher than the fourth, do -not exist in a continuum of 4 dimensions.</p> -<h4 class='c009'>§ 7. Multiplication of Tensors.</h4> -<p class='c004'><i>Outer multiplication of Tensors</i>:—We get from the -components of a tensor of rank <i>z</i>, and another of a rank -<i>z′</i>, the components of a tensor of rank (<i>z</i> + <i>z′</i>) for which -we multiply all the components of the first with all the -components of the second in pairs. For example, we -obtain the tensor Τ from the tensors A and B of different -kinds:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Τ<sub>μνσ</sub> = A<sub>μν</sub>B<sub>σ</sub>,</div> - </div> - <div class='group'> - <div class='line'>Τ<sup>αβγδ</sup> = A<sup>αβ</sup>B<sup>γδ</sup>,</div> - </div> - <div class='group'> - <div class='line'>Τ<sub>αβ</sub><sup>γδ</sup> = A<sub>αβ</sub>B<sup>γδ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>The proof of the tensor character of Τ, follows immediately -from the expressions (8), (10) or (12), or the -transformation equations (9), (11), (13); equations (8), -(10) and (12) are themselves examples of the outer -multiplication of tensors of the first rank.</p> -<p class='c004'><i>Reduction in rank of a mixed Tensor.</i></p> -<p class='c004'>From every mixed tensor we can get a tensor which is -two ranks lower, when we put an index of co-variant -character equal to an index of the contravariant character -and sum according to these indices (Reduction). We get -for example, out of the mixed tensor of the fourth rank -A<sub>αβ</sub><sup>γδ</sup>, -the mixed tensor of the second rank</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A<sub>β</sub><sup>δ</sup> = A<sub>αβ</sub><sup>αδ</sup> = (∑<sub>α</sub> A<sub>αβ</sub><sup>αδ</sup>)</div> - </div> - </div> -</div> - -<p class='c010'>and from it again by “reduction” the tensor of the zero -rank</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A = A<sub>β</sub><sup>β</sup> = A<sub>αβ</sub><sup>αβ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>The proof that the result of reduction retains a truly -tensorial character, follows either from the representation -of tensor according to the generalisation of (12) in combination -with (6) or out of the generalisation of (13).</p> -<p class='c004'><i>Inner and mixed multiplication of Tensors.</i></p> -<p class='c004'>This consists in the combination of outer multiplication -with reduction. Examples:—From the co-variant tensor of -the second rank A<sub>μν</sub> and the contravariant tensor of -the first rank -B<sup>σ</sup> -we get by outer multiplication the -mixed tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>D<sup>σ</sup><sub>μν</sub> = A<sub>μν</sub> B<sup>σ</sup> .</div> - </div> - </div> -</div> - -<p class='c010'>Through reduction according to indices ν and σ (<i>i.e.</i>, putting -ν = σ), the co-variant four vector</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>D<sub>μ</sub> = D<sup>ν</sup><sub>μν</sub> = A<sub>μν</sub> B<sup>ν</sup> is generated.</div> - </div> - </div> -</div> - -<p class='c010'>These we denote as the inner product of the tensor -A<sub>μν</sub> and B<sup>σ</sup>. Similarly we get from the tensors A<sub>μν</sub> and B<sup>στ</sup> -through outer multiplication and two-fold reduction the -inner product -A<sub>μν</sub> B<sup>μν</sup>. -Through outer multiplication -and one-fold reduction we get out of -A<sub>μν</sub> and B<sup>στ</sup>, -the -mixed tensor of the second rank -D<sup>τ</sup><sub>μ</sub> = A<sub>μν</sub> B<sup>τν</sup>. -We -can fitly call this operation a mixed one; for it is outer -with reference to the indices μ and τ and inner with -respect to the indices ν and σ.</p> - -<p class='c005'>We now prove a law, which will be often applicable for -proving the tensor-character of certain quantities. According -to the above representation, -A<sub>μν</sub> B<sup>μν</sup> is a scalar, when A<sub>μν</sub> and B<sup>στ</sup> -are tensors. We also remark that when -A<sub>μν</sub> B<sup>μν</sup> is -an invariant for every choice of the tensor -B<sup>μν</sup>, then A<sub>μν</sub> -has a tensorial character.</p> - -<p class='c005'>Proof:—According to the above assumption, for any -substitution we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A<sub>στ′</sub> B<sup>στ′</sup> = A<sub>μν</sub> B<sup>μν</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>From the inversion of (9) we have however</p> - -<div class='figcenter id002'> -<img src='images/form-066.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Substitution of this for -B<sup>μν</sup> -in the above equation gives</p> - -<div class='figcenter id003'> -<img src='images/form-067.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>This can be true, for any choice of -B<sup>στ′</sup> -only when -the term within the bracket vanishes. From which by -referring to (11), the theorem at once follows. This law -correspondingly holds for tensors of any rank and character. -The proof is quite similar. The law can also be put in the -following form. If -B<sup>μ</sup> and C<sup>ν</sup> are any two vectors, and -if for every choice of them the inner product -A<sub>μν</sub> B<sup>μ</sup> C<sup>ν</sup> -is a scalar, then A<sub>μν</sub> is a co-variant tensor. The last -law holds even when there is the more special formulation, -that with any arbitrary choice of the four-vector -B<sup>μ</sup> alone the scalar product A<sub>μν</sub> B<sup>μ</sup> B<sup>ν</sup> -is a scalar, in which case -we have the additional condition that A<sub>μν</sub> satisfies the -symmetry condition. According to the method given -above, we prove the tensor character of (A<sub>μν</sub> + A<sub>νμ</sub>), from -which on account of symmetry follows the tensor-character -of A<sub>μν</sub>. This law can easily be generalized in the case of -co-variant and contravariant tensors of any rank.</p> - -<p class='c005'>Finally, from what has been proved, we can deduce the -following law which can be easily generalized for any kind -of tensor: If the quantities A<sub>μν</sub> B<sup>ν</sup> form a tensor of the first rank, when B<sup>ν</sup> -is any arbitrarily chosen four-vector, -then A<sub>μν</sub> is a tensor of the second rank. If for example, -C<sup>μ</sup> is any four-vector, then owing to the tensor character of A<sub>μν</sub> B<sup>ν</sup>, -the inner product A<sub>μν</sub> C<sup>μ</sup> B<sup>ν</sup> is a scalar, -both the four-vectors C<sup>μ</sup> and B<sup>ν</sup> being arbitrarily chosen. -Hence the proposition follows at once.</p> -<p class='c004'>A few words about the Fundamental Tensor <i>g</i><sub>μν</sub>.</p> -<p class='c004'>The co-variant fundamental tensor—In the invariant -expression of the square of the linear element</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ds²</i> = <i>g</i><sub>μν</sub> <i>dx</i><sub>μ</sub> <i>dx</i><sub>ν</sub></div> - </div> - </div> -</div> - -<p class='c010'><i>dx</i><sub>μ</sub> plays the rôle of any arbitrarily chosen contravariant -vector, since further <i>g</i><sub>μν</sub> = <i>g</i><sub>νμ</sub>, it follows from the considerations -of the last paragraph that <i>g</i><sub>μν</sub> is a symmetrical -co-variant tensor of the second rank. We call it the -“fundamental tensor.” Afterwards we shall deduce -some properties of this tensor, which will also be true for -any tensor of the second rank. But the special rôle of the -fundamental tensor in our Theory, 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 -tensor.</p> -<p class='c004'><i>The co-variant fundamental tensor.</i></p> -<p class='c004'>If we form from the determinant scheme | <i>g</i><sub>μν</sub> | the -minors of <i>g</i><sub>μν</sub> and divide them by the determinant <i>g</i> = | <i>g</i><sub>μν</sub> | -we get certain quantities <i>g</i><sup>μν</sup> = <i>g</i><sup>νμ</sup>, which as we shall -prove generates a contravariant tensor.</p> - -<p class='c005'>According to the well-known law of Determinants</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(16) <i>g</i><sub>μσ</sub> <i>g</i><sup>νσ</sup> = δ<sub>μ</sub><sup>ν</sup></div> - </div> - </div> -</div> - -<p class='c010'>where -δ<sub>μ</sub><sup>ν</sup> -is 1, or 0, according as μ = ν or not. Instead -of the above expression for <i>ds²</i>, we can also write</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>μσ</sub> δ<sub>ν</sub><sup>σ</sup> <i>dx</i><sub>μ</sub> <i>dx</i><sub>ν</sub></div> - </div> - </div> -</div> - -<p class='c010'>or according to (16) also in the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>μσ</sub> <i>g</i><sub>ντ</sub> <i>g</i><sup>στ</sup> <i>dx</i><sub>μ</sub> <i>dx</i><sub>ν</sub></div> - </div> - </div> -</div> - -<p class='c010'>Now according to the rules of multiplication, of the -fore-going paragraph, the magnitudes</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d</i>ξ<sub>σ</sub> = <i>g</i><sub>μσ</sub> <i>dx</i><sub>μ</sub></div> - </div> - </div> -</div> - -<p class='c010'>forms a co-variant four-vector, and in fact (on account -of the arbitrary choice of <i>dx</i><sub>μ</sub>) any arbitrary four-vector.</p> - -<p class='c005'>If we introduce it in our expression, we get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ds²</i> = <i>g</i><sup>στ</sup> <i>d</i>ξ<sub>σ</sub> <i>d</i>ξ<sub>τ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>For any choice of the vectors <i>d</i>ξ<sub>σ</sub> <i>d</i>ξ<sub>τ</sub> this is scalar, and -<i>g</i><sup>στ</sup>, according to its definition is a symmetrical thing in σ -and τ, so it follows from the above results, that <i>g</i><sup>στ</sup> is a -contravariant tensor. Out of (16) it also follows that δ<sup>ν</sup><sub>μ</sub> -is a tensor which we may call the mixed fundamental -tensor.</p> -<p class='c004'><i>Determinant of the fundamental tensor.</i></p> -<p class='c004'>According to the law of multiplication of determinants, -we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| <i>g</i><sub>μα</sub> <i>g</i><sup>αν</sup> | = | <i>g</i><sub>μα</sub> | | <i>g</i><sup>αν</sup> |</div> - </div> - </div> -</div> - -<p class='c010'>On the other hand we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| <i>g</i><sub>μα</sub> <i>g</i><sup>αν</sup> | = | δ<sup>ν</sup><sub>μ</sub> | = 1</div> - </div> - </div> -</div> - -<p class='c010'>So that it follows (17) that | <i>g</i><sub>μν</sub> | | -<i>g</i><sup>μν</sup> -| = 1.</p> -<p class='c004'><i>Invariant of volume.</i></p> -<p class='c004'>We see first the transformation law for the determinant -<i>g</i> = | <i>g</i><sub>μν</sub> |. According to (11)</p> - -<div class='figcenter id003'> -<img src='images/form-068.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>From this by applying the law of multiplication twice, -we obtain</p> - -<div class='figcenter id004'> -<img src='images/form-069.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>or</p> - -<div class='figcenter id002'> -<img src='images/form-070.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(A)."</p> -</div> -</div> - -<p class='c005'>On the other hand the law of transformation of the -volume element</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d</i>τ′ = ∫ <i>dx₁</i> <i>dx₂</i> <i>dx₃</i> <i>dx₄</i></div> - </div> - </div> -</div> - -<p class='c010'>is according to the wellknown law of Jacobi.</p> - -<div class='figcenter id002'> -<img src='images/form-071.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(B)."</p> -</div> -</div> - -<p class='c005'>by multiplication of the two last equations (A) and (B) we -get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(18) = √<i>g</i> <i>d</i>τ′ = √<i>g</i> <i>d</i>τ.</div> - </div> - </div> -</div> - -<p class='c010'>Instead of √<i>g</i>, we shall afterwards introduce √(-<i>g</i>) -which has a real value on account of the hyperbolic character -of the time-space continuum. The invariant √(-<i>g</i>)<i>d</i>τ, is -equal in magnitude to the four-dimensional volume-element -measured with solid rods and clocks, in accordance with -the special relativity theory.</p> - -<p class='c005'><i>Remarks on the character of the space-time continuum</i>—Our -assumption that in an infinitely small region the -special relativity theory holds, leads us to conclude that <i>ds²</i> -can always, according to (1) be expressed in real magnitudes -<i>d</i>X₁ ... <i>d</i>X<sub><i>h</i></sub>. If we call <i>d</i>τ₀ the “<i>natural</i>” volume -element <i>d</i>X₁ <i>d</i>X₂ <i>d</i>X₃ <i>d</i>X₄ we have thus (18a) <i>d</i>τ₀ -= √(<i>g</i>)<i>i</i>τ.</p> - -<p class='c005'>Should √(-<i>g</i>) vanish at any point of the four-dimensional -continuum it would signify that to a finite co-ordinate -volume at the place corresponds an infinitely small -“natural volume.” This can never be the case; so that <i>g</i> -can never change its sign; we would, according to the special -relativity theory assume that <i>g</i> has a finite negative -value. It is a hypothesis about the physical nature of the -continuum considered, and also a pre-established rule for -the choice of co-ordinates.</p> - -<p class='c005'>If however (-<i>g</i>) remains positive and finite, it is -clear that the choice of co-ordinates can be so made that -this quantity becomes equal to one. We would afterwards -see that such a limitation of the choice of co-ordinates -would produce a significant simplification in expressions -for laws of nature.</p> - -<p class='c005'>In place of (18) it follows then simply that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d</i>τ′ = <i>d</i></div> - </div> - </div> -</div> - -<p class='c010'>from this it follows, remembering the law of Jacobi,</p> - -<div class='figcenter id002'> -<img src='images/form-072.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(19)."</p> -</div> -</div> - -<p class='c005'>With this choice of co-ordinates, only substitutions with -determinant 1 are allowable.</p> - -<p class='c005'>It would however be erroneous to think that this step -signifies a partial renunciation of the general relativity -postulate. We do not seek those laws of nature which are -co-variants with regard to the transformations having -the determinant 1, but we ask: what are the general -co-variant laws of nature? First we get the law, and then -we simplify its expression by a special choice of the system -of reference.</p> -<p class='c004'><i>Building up of new tensors with the help of the fundamental tensor.</i></p> -<p class='c004'>Through inner, outer and mixed multiplications of a -tensor with the fundamental tensor, tensors of other -kinds and of other ranks can be formed.</p> - -<p class='c005'>Example:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A<sup>μ</sup> = <i>g</i><sup>μσ</sup> A<sub>σ</sub></div> - </div> - <div class='group'> - <div class='line'>A = <i>g</i><sub>μν</sub> A<sup>μν</sup></div> - </div> - </div> -</div> - -<p class='c010'>We would point out specially the following combinations:</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A<sup>μν</sup> = <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> A<sub>αβ</sub></div> - </div> - <div class='group'> - <div class='line'>A<sub>μν</sub> = <i>g</i><sub>μα</sub> <i>g</i><sub>νβ</sub> A<sup>αβ</sup></div> - </div> - </div> -</div> - -<p class='c010'>(complement to the co-variant or contravariant tensors)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>and B<sub>μν</sub> = <i>g</i><sub>μν</sub> <i>g</i><sup>αβ</sup> A<sub>αβ</sub></div> - </div> - </div> -</div> - -<p class='c010'>We can call B<sub>μν</sub> the reduced tensor related to A<sub>μν</sub>.</p> - -<p class='c005'>Similarly</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>B<sup>μν</sup> = <i>g</i><sup>μν</sup><i>g</i><sub>αβ</sub>A<sup>αβ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>It is to be remarked that <i>g</i><sup>μν</sup> is no other than the “complement” -of <i>g</i><sub>μν</sub> for we have,—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sup>μα</sup><i>g</i><sup>νβ</sup><i>g</i><sub>αβ</sub> = <i>g</i><sub>μα</sub>δ<sup>ν</sup><sub>α</sub> = <i>g</i><sup>μν</sup>.</div> - </div> - </div> -</div> - -<h4 class='c009'>§ 9. Equation of the geodetic line (or of point-motion).</h4> -<p class='c004'>As the “line element” <i>ds</i> is a definite magnitude independent -of the co-ordinate system, we have also between -two points P₁ and P₂ of a four dimensional continuum a -line for which ∫<i>ds</i> is an extremum (geodetic line), <i>i.e.</i>, one -which has got a significance independent of the choice of -co-ordinates.</p> - -<p class='c005'>Its equation is</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(20) δ{ ∫<sup>P₂</sup><sub>P₁</sub> <i>ds</i> } = 0</div> - </div> - </div> -</div> - -<p class='c010'>From this equation, we can in a wellknown way -deduce 4 total differential equations which define the -geodetic line; this deduction is given here for the sake -of completeness.</p> - -<p class='c005'>Let λ, be a function of the co-ordinates <i>x</i><sub>ν</sub>; this -defines a series of surfaces which cut the geodetic line -sought-for as well as all neighbouring lines from P₁ to P₂. -We can suppose that all such curves are given when the -value of its co-ordinates <i>x</i><sub>ν</sub> are given in terms of λ. The -sign δ corresponds to a passage from a point of the -geodetic curve sought-for to a point of the contiguous -curve, both lying on the same surface λ.</p> - -<p class='c005'>Then (20) can be replaced by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in8'>{ λ₃</div> - <div class='line in8'>{ ∫δω <i>d</i>λ = 0</div> - <div class='line'>(20a) { λ₁</div> - <div class='line in8'>{</div> - <div class='line in8'>{ ω² = <i>g</i><sub>μν</sub>(<i>dx</i><sub>μ</sub>/<i>d</i>λ)(<i>dx</i><sub>ν</sub>/<i>d</i>λ)</div> - </div> - </div> -</div> - -<p class='c010'>But</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>δω = (1/ω){½(∂<i>g</i><sub>μν</sub>/∂<i>x</i><sub>σ</sub>) · (<i>dx</i><sub>μ</sub>/<i>d</i>λ) · (<i>dx</i><sub>ν</sub>/<i>d</i>λ) · δ<i>x</i><sub>σ</sub></div> - <div class='line in2'>+ <i>g</i><sub>μν</sub>(<i>dx</i><sub>μ</sub>/<i>d</i>λ)δ(<i>dx</i><sub>ν</sub>/<i>d</i>λ)}</div> - </div> - </div> -</div> - -<p class='c010'>So we get by the substitution of δω in (20a), remembering -that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>δ(<i>dx</i><sub>ν</sub>/<i>d</i>λ) = (<i>d</i>/<i>d</i>λ)(δ<i>x</i><sub>ν</sub>)</div> - </div> - </div> -</div> - -<p class='c010'>after partial integration,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in8'>{ λ₃</div> - <div class='line in8'>{ ∫ <i>d</i>λ <i>k</i><sub>σ</sub> δ<i>x</i><sub>σ</sub> = 0</div> - <div class='line'>(20b) { λ₁</div> - <div class='line in8'>{</div> - <div class='line in8'>{ where <i>k</i><sub>σ</sub> = (<i>d</i>/<i>d</i>λ){(<i>g</i><sub>μν</sub>/ω) · (<i>dx</i><sub>μ</sub>/<i>d</i>λ)} - (1/(2ω))(∂<i>g</i><sub>μν</sub>/∂<i>x</i><sub>σ</sub></div> - </div> - <div class='group'> - <div class='line in12'>× (<i>dx</i><sub>μ</sub>/<i>d</i>λ) · (<i>dx</i><sub>ν</sub>/<i>d</i>λ).</div> - </div> - </div> -</div> - -<p class='c010'>From which it follows, since the choice of δν<sub>σ</sub> is perfectly -arbitrary that <i>k</i><sub>σ</sub><i>’s</i> should vanish. Then</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(20c) <i>k</i><sub>σ</sub> = 0 (σ = 1, 2, 3, 4)</div> - </div> - </div> -</div> - -<p class='c010'>are the equations of geodetic line; since along the -geodetic line considered we have <i>ds</i> ≠ 0, we can choose the -parameter λ, as the length of the arc measured along the -geodetic line. Then <i>w</i> = 1, and we would get in place of -(20c)</p> - -<div class='figcenter id004'> -<img src='images/form-073.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Or by merely changing the notation suitably,</p> - -<div class='figcenter id004'> -<img src='images/form-074.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"20d"</p> -</div> -</div> - -<p class='c005'>where we have put, following Christoffel,</p> - -<div class='figcenter id002'> -<img src='images/form-075.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"21"</p> -</div> -</div> - -<p class='c005'>Multiply finally (20d) with -<i>g</i><sup>στ</sup> -(outer multiplication with -reference to τ, and inner with respect to σ) we get at -last the final form of the equation of the geodetic line—</p> - -<div class='figcenter id002'> -<img src='images/form-076.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Here we have put, following Christoffel,</p> - -<div class='figcenter id003'> -<img src='images/form-077.png' alt='Formula.' class='ig001' /> -</div> -<h4 class='c009'>§ 10. Formation of Tensors through Differentiation.</h4> -<p class='c004'>Relying on the equation of the geodetic line, we can -now easily deduce laws according to which new tensors can -be formed from given tensors by differentiation. For this -purpose, we would first establish the general co-variant -differential equations. We achieve this through a repeated -application of the following simple law. If a certain -curve be given in our continuum whose points are characterised -by the arc-distances <i>s</i>, measured from a fixed point on -the curve, and if further φ, be an invariant space function, -then <i>d</i>φ/<i>ds</i> is also an invariant. The proof follows from -the fact that <i>d</i>φ as well as <i>ds</i>, are both invariants</p> - -<p class='c005'>Since</p> - -<div class='figcenter id003'> -<img src='images/form-078.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>so that</p> - -<div class='figcenter id002'> -<img src='images/form-079.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>is also an invariant for all curves -which go out from a point in the continuum, <i>i.e.</i>, for -any choice of the vector <i>dx</i><sub>μ</sub>. From which follows immediately -that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A<sub>μ</sub> = ∂φ/∂<i>x</i><sub>μ</sub></div> - </div> - </div> -</div> - -<p class='c010'>is a co-variant four-vector (gradient of φ).</p> - -<p class='c005'>According to our law, the differential-quotient χ = ∂ψ/∂<i>s</i> -taken along any curve is likewise an invariant.</p> - -<p class='c005'>Substituting the value of ψ, we get</p> - -<div class='figcenter id008'> -<img src='images/form-080.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Here however we can not at once deduce the existence -of any tensor. If we however take that the curves along -which we are differentiating are geodesics, we get from it -by replacing <i>d²</i><i>x</i><sub>ν</sub>/<i>ds²</i> according to (22)</p> - -<div class='figcenter id008'> -<img src='images/form-081.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>From the interchangeability of the differentiation with -regard to μ and ν, and also according to (23) and (21) we see -that the bracket</p> - -<div class='figcenter id009'> -<img src='images/form-082.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>is symmetrical with respect to μ -and ν.</p> - -<p class='c005'>As we can draw a geodetic line in any direction from any -point in the continuum, ∂<i>x</i><sub>μ</sub>/<i>ds</i> is thus a four-vector, with an -arbitrary ratio of components, so that it follows from the -results of §7 that</p> - -<div class='figcenter id002'> -<img src='images/form-083.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"25"</p> -</div> -</div> - -<p class='c005'>is a co-variant tensor of the second rank. We have thus got -the result that out of the co-variant tensor of the first rank -A<sub>μ</sub> = ∂φ/∂<i>x</i><sub>μ</sub> we can get by differentiation a co-variant tensor -of 2nd rank</p> - -<div class='figcenter id002'> -<img src='images/form-084.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"26"</p> -</div> -</div> - -<p class='c005'>We call the tensor A<sub>μν</sub> the “extension” of the tensor -A<sub>μ</sub>. Then we can easily show that this combination also -leads to a tensor, when the vector A<sub>μ</sub> is not representable -as a gradient. In order to see this we first remark that -ψ (<i>d</i>φ/∂<i>x</i><sub>μ</sub>) is a co-variant four-vector when ψ and φ are -scalars. This is also the case for a sum of four such -terms :—</p> - -<div class='figcenter id004'> -<img src='images/form-085.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>when ψ<sup>(1)</sup>, φ<sup>(1)</sup> ... ψ<sup>(4)</sup>, φ<sup>(4)</sup> -are scalars. Now it is however -clear that every co-variant four-vector is representable in -the form of S<sub>μ</sub>.</p> - -<p class='c005'>If for example, A<sub>μ</sub> is a four-vector whose components -are any given functions of <i>x</i><sub>ν</sub>, we have, (with reference to -the chosen co-ordinate system) only to put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ψ<sup>(1)</sup> = A₁ φ<sup>(1)</sup> = <i>x₁</i></div> - </div> - <div class='group'> - <div class='line'>ψ<sup>(2)</sup> = A₂ φ<sup>(2)</sup> = <i>x₂</i></div> - </div> - <div class='group'> - <div class='line'>ψ<sup>(3)</sup> = A₃ φ<sup>(3)</sup> = <i>x₃</i></div> - </div> - <div class='group'> - <div class='line'>ψ<sup>(4)</sup> = A₄ φ<sup>(4)</sup> = <i>x₄</i>.</div> - </div> - </div> -</div> - -<p class='c010'>in order to arrive at the result that S<sub>μ</sub> is equal to A<sub>μ</sub>.</p> - -<p class='c005'>In order to prove then that A<sub>μν</sub> is a tensor when on the -right side of (26) we substitute any co-variant four-vector -for A<sub>μ</sub> we have only to show that this is true for the -four-vector S<sub>μ</sub>. For this latter case, however, a glance on -the right hand side of (26) will show that we have only to -bring forth the proof for the case when</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>A<sub>μ</sub> = ψ ∂φ/∂<i>x</i><sub>μ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Now the right hand side of (25) multiplied by ψ is</p> - -<div class='figcenter id003'> -<img src='images/form-086.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>which has a tensor character. Similarly, (∂φ/∂<i>x</i><sub>μ</sub>) (∂φ/∂<i>x</i><sub>ν</sub>) is -also a tensor (outer product of two four-vectors).</p> - -<p class='c005'>Through addition follows the tensor character of</p> - -<div class='figcenter id003'> -<img src='images/form-087.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>Thus we get the desired proof for the four-vector, -ψ ∂φ/∂<i>x</i><sub>μ</sub> and hence for any four-vectors A<sub>μ</sub> as shown above.</p> - -<p class='c005'>With the help of the extension of the four-vector, we -can easily define “extension” of a co-variant tensor of any -rank. This is a generalisation of the extension of the four-vector. -We confine ourselves to the case of the extension -of the tensors of the 2nd rank for which the law of formation -can be clearly seen.</p> - -<p class='c005'>As already remarked every co-variant tensor of the 2nd -rank can be represented as a sum of the tensors of the type -A<sub>μ</sub> B<sub>ν</sub>.</p> - -<p class='c005'>It would therefore be sufficient to deduce the expression -of extension, for one such special tensor. According to -(26) we have the expressions</p> - -<div class='figcenter id002'> -<img src='images/form-088.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>are tensors. Through outer multiplication of the first -with B<sub>ν</sub> and the 2nd with A<sub>μ</sub> we get tensors of the -third rank. Their addition gives the tensor of the third -rank</p> - -<div class='figcenter id004'> -<img src='images/form-089.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(27)"</p> -</div> -</div> - -<p class='c005'>where A<sub>μν</sub> is put = A<sub>μ</sub> B<sub>ν</sub>. The right hand side of (27) -is linear and homogeneous with reference to A<sub>μν</sub>, and its -first differential co-efficient, so that this law of formation leads -to a tensor not only in the case of a tensor of the type A<sub>μ</sub> -B<sub>ν</sub> but also in the case of a summation for all such -tensors, <i>i.e.</i>, in the case of any co-variant tensor of the -second rank. We call A<sub>μνσ</sub> the extension of the tensor A<sub>μν</sub>. -It is clear that (26) and (24) are only special cases of -(27) (extension of the tensors of the first and zero rank). -In general we can get all special laws of formation of -tensors from (27) combined with tensor multiplication.</p> -<p class='c004'>Some special cases of Particular Importance.</p> -<p class='c004'><i>A few auxiliary lemmas concerning the fundamental -tensor.</i> We shall first deduce some of the lemmas much used -afterwards. According to the law of differentiation of -determinants, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(28) <i>dg</i> = <i>g</i><sup>μν</sup> <i>g dg</i><sub>μν</sub> = -<i>g</i><sub>μν</sub> <i>gdg</i><sup>μν</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>The last form follows from the first when we remember -that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>μν</sub> <i>g</i><sup>μ′ν</sup> = δ<sup>μ′</sup><sub>μ</sub> , and therefore <i>g</i><sub>μν</sub><i>g</i><sup>μν</sup> = 4,</div> - </div> - <div class='group'> - <div class='line'>consequently <i>g</i><sub>μν</sub><i>dg</i><sup>μν</sup> + <i>g</i><sup>μν</sup> <i>dg</i><sub>μν</sub> = 0.</div> - </div> - </div> -</div> - -<p class='c010'>From (28), it follows that</p> - -<div class='figcenter id004'> -<img src='images/form-090.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(29)"</p> -</div> -</div> - -<p class='c005'>Again, since <i>g</i><sub>μν</sub> <i>g</i><sup>νσ</sup> = δ<sup>ν</sup><sub>μ</sub> , we have, by differentiation,</p> - -<div class='figcenter id003'> -<img src='images/form-091.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>By mixed multiplication with <i>g</i><sup>στ</sup> and <i>g</i><sub>νλ</sub> respectively -we obtain (changing the mode of writing the -indices).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dg</i><sup>μν</sup> = -<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> <i>dg</i><sub>αβ</sub></div> - </div> - <div class='group'> - <div class='line'>∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> = -<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> <i>dg</i><sub>αβ</sub></div> - </div> - </div> -</div> - -<p class='c010'>and</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(32)</div> - <div class='line'><i>dg</i><sub>μν</sub> = -<i>g</i><sub>μα</sub> <i>g</i><sub>νβ</sub> <i>dg</i><sup>αβ</sup></div> - </div> - <div class='group'> - <div class='line'>∂<i>g</i><sub>μν</sub>/∂<i>x</i><sub>σ</sub> = -<i>g</i><sub>μα</sub> <i>g</i><sub>νβ</sub> ∂<i>g</i><sup>αβ</sup>/∂<i>x</i><sub>σ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>The expression (31) allows a transformation which we -shall often use; according to (21)</p> - -<div class='figcenter id003'> -<img src='images/form-092.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(33)"</p> -</div> -</div> - -<p class='c005'>If we substitute this in the second of the formula (31), -we get, remembering (23),</p> - -<div class='figcenter id004'> -<img src='images/form-093.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(34)"</p> -</div> -</div> - -<p class='c005'>By substituting the right-hand side of (34) in (29), we -get</p> - -<div class='figcenter id002'> -<img src='images/form-094.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(29a)"</p> -</div> -</div> -<p class='c004'><i>Divergence of the contravariant four-vector.</i></p> -<p class='c004'>Let us multiply (26) with the contravariant fundamental -tensor -<i>g</i><sup>μν</sup> -(inner multiplication), then by a transformation -of the first member, the right-hand side takes the form</p> - -<div class='figcenter id004'> -<img src='images/form-095.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(A)"</p> -</div> -</div> - -<p class='c005'>According to (31) and (29), the last member can take -the form</p> - -<div class='figcenter id004'> -<img src='images/form-096.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(B)"</p> -</div> -</div> - -<p class='c005'>Both the first members of the expression (B), and the -second member of the expression (A) cancel each other, -since the naming of the summation-indices is immaterial. -The last member of (B) can then be united with first of -(A). If we put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sup>μν</sup> A<sub>μ</sub> = A<sup>ν</sup>,</div> - </div> - </div> -</div> - -<p class='c010'>where -A<sup>ν</sup> -as well as A<sub>μ</sub> are vectors which can be arbitrarily -chosen, we obtain finally</p> - -<div class='figcenter id003'> -<img src='images/form-097.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>This scalar is the <i>Divergence</i> of the contravariant four-vector -A<sup>ν</sup>.</p> -<p class='c004'><i>Rotation of the (covariant) four-vector.</i></p> -<p class='c004'>The second member in (26) is symmetrical in the indices -μ, and ν. Hence A<sub>μν</sub> - A<sub>νμ</sub> is an antisymmetrical tensor -built up in a very simple manner. We obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in11'>∂A<sub>μ</sub> ∂A<sub>ν</sub></div> - <div class='line'>(36) B<sub>μν</sub> = --------- - -------</div> - <div class='line in11'>∂<i>x</i><sub>ν</sub> ∂<sub><i>x</i>μ</sub></div> - </div> - </div> -</div> - -<p class='c004'><i>Antisymmetrical Extension of a Six-vector.</i></p> -<p class='c004'>If we apply the operation (27) on an antisymmetrical -tensor of the second rank A<sub>μ{ν²</sub>} and form all the equations -arising from the cyclic interchange of the indices μ, ν, σ, and -add all them, we obtain a tensor of the third rank</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(37) B<sub>μνσ</sub> = A<sub>μνσ</sub> + A<sub>νσμ</sub> + A<sub>σμν</sub></div> - </div> - <div class='group'> - <div class='line in9'>∂A<sub>μν</sub> ∂A<sub>νσ</sub> ∂A<sub>σμ</sub></div> - <div class='line in7'>= --------- + ---------- + ---------</div> - <div class='line in9'>∂<i>x</i><sub>σ</sub> ∂<i>x</i><sub>μ</sub> ∂<i>x</i><sub>ν</sub></div> - </div> - </div> -</div> - -<p class='c010'>from which it is easy to see that the tensor is antisymmetrical.</p> -<p class='c004'><i>Divergence of the Six-vector.</i></p> -<p class='c004'>If (27) is multiplied by -<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> -(mixed multiplication), -then a tensor is obtained. The first member of the right -hand side of (27) can be written in the form</p> - -<div class='figcenter id004'> -<img src='images/form-098.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If we replace <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> A<sub>μνσ</sub> by A<sub>σ</sub><sup>αβ</sup>, <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> A<sub>μν</sub> by -A<sup>αβ</sup> and replace in the transformed first member</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>g</i><sup>νβ</sup>/∂<i>x</i><sub>σ</sub> and ∂<i>g</i><sup>μα</sup>/∂<i>x</i><sub>σ</sub></div> - </div> - </div> -</div> - -<p class='c010'>with the help of (34), then from the right-hand side of (27) -there arises an expression with seven terms, of which four -cancel. There remains</p> - -<div class='figcenter id008'> -<img src='images/form-099.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(38)"</p> -</div> -</div> - -<p class='c005'>This is the expression for the extension of a contravariant -tensor of the second rank; extensions can also be formed for -corresponding contravariant tensors of higher and lower -ranks.</p> - -<p class='c005'>We remark that in the same way, we can also form the -extension of a mixed tensor -A<sub>μ</sub><sup>α</sup></p> - -<div class='figcenter id004'> -<img src='images/form-100.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(39)"</p> -</div> -</div> - -<p class='c005'>By the reduction of (38) with reference to the indices -β and σ(inner multiplication with -δ<sub>β</sub><sup>σ</sup>), -we get a contravariant -four-vector</p> - -<div class='figcenter id004'> -<img src='images/form-101.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>On the account of the symmetry of</p> - -<div class='figcenter id009'> -<img src='images/form-102.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>with -reference to the indices β and κ, the third member of the -right hand side vanishes when -A<sup>αβ</sup> -is an antisymmetrical -tensor, which we assume here; the second member can be -transformed according to (29a); we therefore get</p> - -<div class='figcenter id003'> -<img src='images/form-103.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(40)"</p> -</div> -</div> - -<p class='c005'>This is the expression of the divergence of a contravariant -six-vector.</p> -<p class='c004'><i>Divergence of the mixed tensor of the second rank.</i></p> -<p class='c004'>Let us form the reduction of (39) with reference to the -indices α and σ, we obtain remembering (29a)</p> - -<div class='figcenter id004'> -<img src='images/form-104.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(41)"</p> -</div> -</div> - -<p class='c005'>If we introduce into the last term the contravariant -tensor A<sup>ρσ</sup> = <i>g</i><sup>ρτ</sup> A<sup>σ</sup><sub>τ</sub>, it takes the form</p> - -<div class='figcenter id002'> -<img src='images/form-105.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If further -A<sup>ρσ</sup> -or is symmetrical it is reduced to</p> - -<div class='figcenter id002'> -<img src='images/form-106.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If instead of -A<sup>ρσ</sup>, -we introduce in a similar way the -symmetrical co-variant tensor A<sub>ρσ</sub> = <i>g</i><sub>ρα</sub> <i>g</i><sub>σβ</sub> -A<sup>αβ</sup>, -then -owing to (31) the last member can take the form</p> - -<div class='figcenter id002'> -<img src='images/form-107.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>In the symmetrical case treated, (41) can be replaced by -either of the forms</p> - -<div class='figcenter id003'> -<img src='images/form-108.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(41a)"</p> -</div> -</div> - -<p class='c005'>or</p> - -<div class='figcenter id001'> -<img src='images/form-109.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(41b)"</p> -</div> -</div> - -<p class='c005'>which we shall have to make use of afterwards.</p> -<h4 class='c009'>§12. The Riemann-Christoffel Tensor.</h4> -<p class='c004'>We now seek only those tensors, which can be -obtained from the fundamental tensor -<i>g</i><sup>μν</sup> -by differentiation -alone. It is found easily. We put in (27) instead of -any tensor A<sup>μν</sup> the fundamental tensor <i>g</i><sup>μν</sup> and get from -it a new tensor, namely the extension of the fundamental -tensor. We can easily convince ourselves that this -vanishes identically. We prove it in the following way; we -substitute in (27)</p> - -<div class='figcenter id002'> -<img src='images/form-110.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'><i>i.e.</i>, the extension of a four-vector.</p> - -<p class='c005'>Thus we get (by slightly changing the indices) the -tensor of the third rank</p> - -<div class='figcenter id008'> -<img src='images/form-111.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>We use these expressions for the formation of the tensor -A<sub>μστ</sub> - A<sub>μτσ</sub>. Thereby the following terms in A<sub>μστ</sub> -cancel the corresponding terms in A<sub>μτσ</sub>; the first member, -the fourth member, as well as the member corresponding -to the last term within the square bracket. These are all -symmetrical in σ, and τ. The same is true for the sum of -the second and third members. We thus get</p> - -<div class='figcenter id008'> -<img src='images/form-112.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(43)"</p> -</div> -</div> - -<p class='c005'>The essential thing in this result is that on the -right hand side of (42) we have only A<sub>ρ</sub>, but not its -differential co-efficients. From the tensor-character of A<sub>μστ</sub> -- A<sub>μτσ</sub>, and from the fact that A<sub>ρ</sub> is an arbitrary four -vector, it follows, on account of the result of §7, that -B<sup>ρ</sup><sub>μστ</sub> -is a tensor (Riemann-Christoffel Tensor).</p> - -<p class='c005'>The mathematical significance of this tensor is as -follows; when the continuum is so shaped, that there is a -co-ordinate system for which <i>g</i><sub>μν</sub><i>’s</i> are constants, -B<sup>ρ</sup><sub>μστ</sub> all -vanish.</p> - -<p class='c005'>If we choose instead of the original co-ordinate system -any new one, so would the <i>g</i><sub>μν</sub>’s referred to this last system -be no longer constants. The tensor character of -B<sup>ρ</sup><sub>μστ</sub> -shows us, however, that these components vanish collectively -also in any other chosen system of reference. The -vanishing of the Riemann Tensor is thus a necessary condition -that for some choice of the axis-system <i>g</i><sub>μν</sub>’s can be -taken as constants. In our problem it corresponds to the -case when by a suitable choice of the co-ordinate system, -the special relativity theory holds throughout any finite -region. By the reduction of (43) with reference to indices -to τ and ρ, we get the covariant tensor of the second rank</p> - -<div class='figcenter id008'> -<img src='images/form-113.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(44)"</p> -</div> -</div> -<p class='c004'><i>Remarks upon the choice of co-ordinates.</i>—It has already -been remarked in §8, with reference to the equation (18a), -that the co-ordinates can with advantage be so chosen that -√(-<i>g</i>) = 1. A glance at the equations got in the last two -paragraphs shows that, through such a choice, the law of -formation of the tensors suffers a significant simplification. -It is specially true for the tensor B<sub>μν</sub>, which plays -a fundamental rôle in the theory. By this simplification, -S<sub>μν</sub> vanishes of itself so that tensor B<sub>μν</sub> reduces to -R<sub>μν</sub>.</p> - -<p class='c005'>I shall give in the following pages all relations in the -simplified form, with the above-named specialisation of -the co-ordinates. It is then very easy to go back to the -general covariant equations, if it appears desirable in -any special case.</p> -<h3 class='c008'>C. THE THEORY OF THE GRAVITATION-FIELD</h3> -<h4 class='c009'>§13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation.</h4> -<p class='c004'>A freely moving body not acted on by external forces -moves, according to the special relativity theory, along a -straight line and uniformly. This also holds for the -generalised relativity theory for any part of the four-dimensional -region, in which the co-ordinates K<sub>0</sub> can be, and -are, so chosen that <i>g</i><sub>μν</sub>’s have special constant values of -the expression (4).</p> - -<p class='c005'>Let us discuss this motion from the stand-point of any -arbitrary co-ordinate-system K₁; it moves with reference to -K₁ (as explained in §2) in a gravitational field. The laws -of motion with reference to K₁ follow easily from the -following consideration. With reference to K₀, the law -of motion is a four-dimensional straight line and thus a -geodesic. As a geodetic-line is defined independently -of the system of co-ordinates, it would also be the law of -motion for the motion of the material-point with reference -to K₁. If we put</p> - -<div class='figcenter id003'> -<img src='images/form-114.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(45)"</p> -</div> -</div> - -<p class='c005'>we get the motion of the point with reference to K₁, -given by</p> - -<div class='figcenter id004'> -<img src='images/form-115.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(46)"</p> -</div> -</div> - -<p class='c005'>We now make the very simple assumption that this -general covariant system of equations defines also the -motion of the point in the gravitational field, when there -exists no reference-system K₀, with reference to which -the special relativity theory holds throughout a finite -region. The assumption seems to us to be all the more -legitimate, as (46) contains only the first differentials of -<i>g</i><sub>μν</sub>, among which there is no relation in the special case -when K₀ exists.</p> - -<p class='c005'>If -γ<sub>μν</sub><sup>τ</sup>’s -vanish, the point moves uniformly and in a -straight line; these magnitudes therefore determine the -deviation from uniformity. They are the components of -the gravitational field.</p> -<h4 class='c009'>§14. The Field-equation of Gravitation in the absence of matter.</h4> -<p class='c004'>In the following, we differentiate gravitation-field from -matter in the sense that everything besides the gravitation-field -will be signified as matter; therefore the term -includes not only matter in the usual sense, but also the -electro-dynamic field. Our next problem is to seek the -field-equations of gravitation in the absence of matter. For -this we apply the same method as employed in the foregoing -paragraph for the deduction of the equations of -motion for material points. A special case in which the -field-equations sought-for are evidently satisfied is that of -the special relativity theory in which <i>g</i><sub>μν</sub>’s have certain -constant values. This would be the case in a certain -finite region with reference to a definite co-ordinate -system K₀. With reference to this system, all the components -B<sup>ρ</sup><sub>μστ</sub> -of the Riemann’s Tensor [equation 43] -vanish. These vanish then also in the region considered, -with reference to every other co-ordinate system.</p> - -<p class='c005'>The equations of the gravitation-field free from matter -must thus be in every case satisfied when all -B<sup>ρ</sup><sub>μστ</sub> vanish. -But this condition is clearly one which goes too far. For -it is clear that the gravitation-field generated by a material -point in its own neighbourhood can never be transformed -<i>away</i> by any choice of axes, <i>i.e.</i>, it cannot be transformed -to a case of constant <i>g</i><sub>μν</sub>’s.</p> - -<p class='c005'>Therefore it is clear that, for a gravitational field free -from matter, it is desirable that the symmetrical tensors -B<sub>μν</sub> deduced from the tensors -B<sup>ρ</sup><sub>μστ</sub> -should vanish. -We thus get 10 equations for 10 quantities <i>g</i><sub>μν</sub> which are -fulfilled in the special case when -B<sup>ρ</sup><sub>μστ</sub>’s -all vanish.</p> - -<p class='c005'>Remembering (44) we see that in absence of matter -the field-equations come out as follows; (when referred -to the special co-ordinate-system chosen.)</p> - -<div class='figcenter id004'> -<img src='images/form-116.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(47)"</p> -</div> -</div> - -<p class='c005'>It can also be shown that the choice of these equations -is connected with a minimum of arbitrariness. For -besides B<sub>μν</sub>, there is no tensor of the second rank, which -can be built out of <i>g</i><sub>μν</sub>’s and their derivatives no higher -than the second, and which is also linear in them.</p> - -<p class='c005'>It will be shown that the equations arising in a purely -mathematical way out of the conditions of the general -relativity, together with equations (46), give us the Newtonian -law of attraction as a first approximation, and lead -in the second approximation to the explanation of the -perihelion-motion of mercury discovered by Leverrier -(the residual effect which could not be accounted for by -the consideration of all sorts of disturbing factors). My -view is that these are convincing proofs of the physical -correctness of my theory.</p> -<h4 class='c009'>§15. Hamiltonian Function for the Gravitation-field. <br /> Laws of Impulse and Energy.</h4> -<p class='c004'>In order to show that the field equations correspond to -the laws of impulse and energy, it is most convenient to -write it in the following Hamiltonian form:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(47a)</div> - </div> - <div class='group'> - <div class='line'>δ∫ H<i>d</i>τ = 0</div> - </div> - <div class='group'> - <div class='line'>H = <i>g</i><sup>μν</sup> γ<sup>α</sup><sub>μβ</sub> γ<sup>β</sup><sub>να</sub></div> - </div> - <div class='group'> - <div class='line'>√(-<i>g</i>) = 1</div> - </div> - </div> -</div> - -<p class='c010'>Here the variations vanish at the limits of the finite -four-dimensional integration-space considered.</p> - -<p class='c005'>It is first necessary to show that the form (47a) is -equivalent to equations (47). For this purpose, let us -consider H as a function of -<i>g</i><sup>μν</sup> and <i>g</i><sup>μν</sup><sub>σ</sub> (= ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub>)</p> - -<p class='c005'>We have at first</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>δH = Γ<sup>α</sup><sub>μβ</sub> Γ<sup>β</sup><sub>να</sub> δ<i>g</i><sup>μν</sup> + 2<i>g</i><sup>μν</sup> Γ<sup>α</sup><sub>μβ</sub> δΓ<sup>β</sup><sub>να</sub></div> - </div> - <div class='group'> - <div class='line'>= - Γ<sup>α</sup><sub>μβ</sub> Γ<sup>β</sup><sub>να</sub> δ<i>g</i><sup>μν</sup> + 2Γ<sup>α</sup><sub>μβ</sub> δ(<i>g</i><sup>μν</sup>Γ<sup>β</sup><sub>να</sub>).</div> - </div> - </div> -</div> - -<p class='c010'>But</p> - -<div class='figcenter id004'> -<img src='images/form-117.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>The terms arising out of the two last terms within the -round bracket are of different signs, and change into one -another by the interchange of the indices μ and β. They -cancel each other in the expression for δH, when they are -multiplied by Γ<sub>μβ</sub><sup>α</sup>, which is symmetrical with respect to -μ and β, so that only the first member of the bracket -remains for our consideration. Remembering (31), we -thus have:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>δH = -Γ<sub>μβ</sub><sup>α</sup> Γ<sub>να</sub><sup>β</sup> δ<i>g</i><sup>μν</sup> + Γ<sub>μβ</sub><sup>α</sup> δ<i>g</i><sub>α</sub><sup>μβ</sup></div> - </div> - </div> -</div> - -<p class='c010'>Therefore</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(48)</div> - <div class='line'>∂H/∂<i>g</i><sup>μν</sup> = -Γ<sub>μβ</sub><sup>α</sup> Γ<sub>να</sub><sup>β</sup></div> - </div> - <div class='group'> - <div class='line'>∂H/∂<i>g</i><sub>σ</sub><sup>μν</sup> = Γ<sub>μν</sub><sup>σ</sup></div> - </div> - </div> -</div> - -<p class='c010'>If we now carry out the variations in (47a), we obtain -the system of equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(47b) ∂/∂<i>x</i><sub>α</sub> ( ∂H/∂<i>g</i><sub>α</sub><sup>μν</sup> ) - ∂H/∂<i>g</i><sup>μν</sup> = 0,</div> - </div> - </div> -</div> - -<p class='c010'>which, owing to the relations (48), coincide with (47), -as was required to be proved.</p> - -<p class='c005'>If (47b) is multiplied by <i>g</i><sub>σ</sub><sup>μν</sup>, since</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>g</i><sub>σ</sub><sup>μν</sup>/∂<i>x</i><sub>α</sub> = ∂<i>g</i><sub>α</sub><sup>μν</sup>/∂<i>x</i><sub>σ</sub></div> - </div> - </div> -</div> - -<p class='c010'>and consequently</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>σ</sub><sup>μν</sup> ∂/∂<i>x</i><sub>α</sub> (∂H/∂<i>g</i><sub>α</sub><sup>μν</sup>) = ∂/∂<i>x</i><sub>α</sub> (<i>g</i><sub>σ</sub><sup>μν</sup> ∂H/∂<i>g</i><sub>α</sub><sup>μν</sup>)</div> - <div class='line'>- ∂H/∂<i>g</i><sub>α</sub><sup>μν</sup> ∂<i>g</i><sub>α</sub><sup>μν</sup>/∂<i>x</i><sub>σ</sub></div> - </div> - </div> -</div> - -<p class='c010'>we obtain the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂/∂<i>x</i><sub>α</sub> (<i>g</i><sub>σ</sub><sup>μν</sup> ∂H/∂<i>g</i><sub>α</sub><sup>μν</sup>) - ∂H/∂<i>x</i><sub>σ</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>or</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'>{ ∂<i>t</i><sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> = 0</div> - </div> - <div class='group'> - <div class='line'>(49) { -2κ<i>t</i><sub>σ</sub><sup>α</sup> = <i>g</i><sub>σ</sub><sup>μν</sup> ∂H/∂<i>g</i><sub>α</sub><sup>μν</sup> - δ<sub>σ</sub><sup>α</sup> H.</div> - </div> - </div> -</div> - -<p class='c010'>Owing to the relations (48), the equations (47) and (34),</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(50) κ<i>t</i><sub>σ</sub><sup>α</sup> = ½ δ<sub>σ</sub><sup>α</sup> <i>g</i><sup>μν</sup> Γ<sub>μβ</sub><sup>α</sup> Γ<sub>να</sub><sup>β</sup></div> - <div class='line'>- <i>g</i><sup>μν</sup> Γ<sub>μβ</sub><sup>α</sup> Γ<sub>νσ</sub><sup>β</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>It is to be noticed that -<i>t</i><sub>σ</sub><sup>α</sup> -is not a tensor, so that the -equation (49) holds only for systems for which √-<i>g</i> = 1. -This equation expresses the laws of conservation of impulse -and energy in a gravitation-field. In fact, the integration -of this equation over a three-dimensional volume V -leads to the four equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(49a) <i>d</i>/<i>dx₄</i> {∫<i>t</i><sub>σ</sub><sup>4</sup> <i>d</i>V} = ∫(<i>t</i><sub>σ</sub><sup>1</sup> α₁</div> - <div class='line'>+ <i>t</i><sub>σ</sub>² α₂ + <i>t</i><sub>σ</sub>³ α₃)<i>d</i>S</div> - </div> - </div> -</div> - -<p class='c010'>where α₁, α₂, α₂ are the direction-cosines of the inward-drawn -normal to the surface-element <i>d</i>S in the Euclidean -Sense. We recognise in this the usual expression for the -laws of conservation. We denote the magnitudes <i>t</i><sup>α</sup><sub>σ</sub> as the -energy-components of the gravitation-field.</p> - -<p class='c005'>I will now put the equation (47) in a third form which -will be very serviceable for a quick realisation of our object. -By multiplying the field-equations (47) with <i>g</i><sup>νσ</sup>, these are -obtained in the mixed forms. If we remember that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sup>νσ</sup> ∂Γ<sup>α</sup><sub>μν</sub>/∂<i>x</i><sub>α</sub> = ∂/∂<i>x</i><sub>α</sub> (<i>g</i><sup>νσ</sup> Γ<sup>α</sup><sub>μν</sub>) - ∂<i>g</i><sup>νσ</sup>/∂<i>x</i><sub>α</sub> Γ<sup>α</sup><sub>μν</sub>,</div> - </div> - </div> -</div> - -<p class='c010'>which owing to (34) is equal to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂/∂<i>x</i><sub>α</sub> (.<i>g</i><sup>νσ</sup> Γ<sup>α</sup><sub>μν</sub>) - <i>g</i><sup>νβ</sup> Γ<sup>σ</sup><sub>αβ</sub> Γγ<sup>α</sup><sub>μν</sub></div> - <div class='line'>- <i>g</i><sup>σβ</sup> Γ<sup>ν</sup><sub>βα</sub> Γ<sup>α</sup><sub>μν</sub>,</div> - </div> - </div> -</div> - -<p class='c010'>or slightly altering the notation, equal to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂/∂<i>x</i><sub>α</sub> (<i>g</i><sup>σβ</sup> Γ<sup>α</sup><sub>μβ</sub>) - <i>g</i><sup>mn</sup> Γ<sup>σ</sup><sub>mβ</sub> Γ<sup>β</sup><sub><i>n</i>μ</sub></div> - <div class='line'>- <i>g</i><sup>νσ</sup> Γ<sup>α</sup><sub>μβ</sub> Γ<sup>β</sup><sub>να</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>The third member of this expression cancels with the -second member of the field-equations (47). In place of -the second term of this expression, we can, on account of -the relations (50), put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>κ (<i>t</i><sup>σ</sup><sub>μ</sub> - ½ δ<sup>σ</sup><sub>μ</sub> <i>t</i>), where <i>t</i> = <i>t</i><sup>α</sup><sub>α</sub></div> - </div> - </div> -</div> - -<p class='c010'>Therefore in the place of the equations (47), we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(51) { ∂/∂<i>x</i><sub>α</sub> (<i>g</i><sup>σβ</sup> Γ<sup>α</sup><sub>μβ</sub>) = -κ(<i>t</i><sup>σ</sup><sub>μ</sub> - ½ δ<sup>σ</sup><sub>μ</sub> <i>t</i>)</div> - </div> - <div class='group'> - <div class='line in5'>{ √(-<i>g</i>) = 1.</div> - </div> - </div> -</div> - -<h4 class='c009'>§16. General formulation of the field-equation of Gravitation.</h4> -<p class='c004'>The field-equations established in the preceding paragraph -for spaces free from matter is to be compared with -the equation ▽²φ = 0 of the Newtonian theory. We have -now to find the equations which will correspond to -Poisson’s Equation ▽²φ = 4πκρ (ρ signifies the density of -matter).</p> - -<p class='c005'>The special relativity theory has led to the conception -that the inertial mass (Träge Masse) is no other than -energy. It can also be fully expressed mathematically by -a symmetrical tensor of the second rank, the energy-tensor. -We have therefore to introduce in our generalised theory -energy-tensor τ<sup>α</sup><sub>σ</sub> associated with matter, which like the -energy components <i>t</i><sup>α</sup><sub>σ</sub> of the gravitation-field (equations -49, and 50) have a mixed character but which however can -be connected with symmetrical covariant tensors. The -equation (51) teaches us how to introduce the energy-tensor -(corresponding to the density of Poisson’s equation) in the -field equations of gravitation. If we consider a complete -system (for example the Solar-system) its total mass, as -also its total gravitating action, will depend on the total -energy of the system, ponderable as well as gravitational. -This can be expressed, by putting in (51), in place of -energy-components <i>t</i><sub>μ</sub><sup>σ</sup> of gravitation-field alone the sum -of the energy-components of matter and gravitation, <i>i.e.</i>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>t</i><sub>μ</sub><sup>σ</sup> + T<sub>μ</sub><sup>σ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>We thus get instead of (51), the tensor-equation</p> - -<div class='figcenter id004'> -<img src='images/form-118.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(52)"</p> -</div> -</div> - -<p class='c005'>where T = T<sub>μ</sub><sup>μ</sup> (Laue’s Scalar). These are the general field-equations -of gravitation in the mixed form. In place of -(47), we get by working backwards the system</p> - -<div class='figcenter id004'> -<img src='images/form-119.png' alt='Formula.' class='ig001' /> -<div class='ic007'> -<p>"(53)"</p> -</div> -</div> - -<p class='c005'>It must be admitted, that this introduction of the -energy-tensor of matter cannot be justified by means of the -Relativity-Postulate alone; for we have in the foregoing -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 energy. The strongest -ground for the choice of the above equation however lies in -this, that they lead, as their consequences, to equations -expressing the conservation of the components of total -energy (the impulses and the energy) which exactly -correspond to the equations (49) and (49a). This shall be -shown afterwards.</p> -<h4 class='c009'>§17. The laws of conservation in the general case.</h4> -<p class='c004'>The equations (52) can be easily so transformed that -the second member on the right-hand side vanishes. We -reduce (52) with reference to the indices μ and σ and -subtract the equation so obtained after multiplication with -½ δ<sub>μ</sub><sup>σ</sup> from (52).</p> - -<p class='c005'>We obtain,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(52a) ∂/∂<i>x</i><sub>α</sub>(<i>g</i><sup>σβ</sup> Γ<sub>μβ</sub><sup>α</sup> - ½ δ<sub>μ</sub><sup>σ</sup> <i>g</i><sup>λβ</sup> Γ<sub>λβ</sub><sup>α</sup>)</div> - <div class='line'>= -κ(<i>t</i><sub>μ</sub><sup>σ</sup> + T<sub>μ</sub><sup>σ</sup>)</div> - </div> - </div> -</div> - -<p class='c010'>we operate on it by ∂/∂<i>x</i><sub>σ</sub>. Now,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> (<i>g</i><sup>σβ</sup>Γ<sub>μβ</sub><sup>α</sup>)</div> - <div class='line'>= -½ ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> [<i>g</i><sup>σβ</sup> <i>g</i><sup>αλ</sup>(∂<i>g</i><sub>μλ</sub>/∂<i>x</i><sub>β</sub></div> - <div class='line in1'>+ ∂<i>g</i><sub>βλ</sub>/∂<i>x</i><sub>μ</sub> - ∂<i>g</i><sub>μβ</sub>/∂<i>x</i><sub>λ</sub>)].</div> - </div> - </div> -</div> - -<p class='c010'>The first and the third member of the round bracket -lead to expressions which cancel one another, as can be -easily seen by interchanging the summation-indices α, and -σ, on the one hand, and β and λ, on the other.</p> - -<p class='c005'>The second term can be transformed according to (31). -So that we get,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(54) ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> (<i>g</i><sup>σβ</sup>γ<sub>μβ</sub><sup>α</sup>)</div> - <div class='line in1'>= ½ ∂³<i>g</i><sup>αβ</sup>/∂<i>x</i><sub>σ</sub>∂<i>x</i><sub>β</sub>∂<i>x</i><sub>μ</sub></div> - </div> - </div> -</div> - -<p class='c010'>The second member of the expression on the left-hand -side of (52a) leads first to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- ½ ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>μ</sub> (<i>g</i><sup>λβ</sup>Γ<sub>λβ</sub><sup>α</sup>) or</div> - </div> - <div class='group'> - <div class='line'>to 1/4 ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>μ</sub> [<i>g</i><sup>λβ</sup><i>g</i><sup>αδ</sup>( ∂<i>g</i><sub>δλ</sub>/∂<i>x</i><sub>β</sub></div> - <div class='line in1'>+ ∂<i>g</i><sub>δβ</sub>/∂<i>x</i><sub>λ</sub> - ∂<i>g</i><sub>λβ</sub>/∂<i>x</i><sub>δ</sub>)].</div> - </div> - </div> -</div> - -<p class='c010'>The expression arising out of the last member within -the round bracket vanishes according to (29) on account -of the choice of axes. The two others can be taken -together and give us on account of (31), the expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-½ ∂³<i>g</i><sup>αβ</sup>/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>β</sub>∂<i>x</i><sub>μ</sub></div> - </div> - </div> -</div> - -<p class='c010'>So that remembering (54) we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(55) ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> (<i>g</i><sup>σβ</sup>Γ<sub>μβ</sub><sup>α</sup></div> - <div class='line in1'>- ½ δ<sub>μ</sub><sup>σ</sup> <i>g</i><sup>λβ</sup> Γ<sub>λβ</sub><sup>α</sup>) = 0.</div> - </div> - </div> -</div> - -<p class='c010'>identically.</p> - -<p class='c005'>From (55) and (52a) it follows that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(56) ∂/∂<i>x</i><sub>σ</sub> (<i>t</i><sub>μ</sub><sup>σ</sup> + T<sub>μ</sub><sup>σ</sup>) = 0</div> - </div> - </div> -</div> - -<p class='c010'>From the field equations of gravitation, it also follows -that the conservation-laws of impulse and energy are -satisfied. We see it most simply following the same -reasoning which lead to equations (49a); only instead of -the energy-components of the gravitational-field, we are to -introduce the total energy-components of matter and gravitational -field.</p> -<h4 class='c009'>§18. The Impulse-energy law for matter as a consequence of the field-equations.</h4> -<p class='c004'>If we multiply (53) with ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub>, we get in a way -similar to §15, remembering that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>μν</sub> ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> vanishes,</div> - </div> - <div class='group'> - <div class='line'>the equations ∂<i>t</i><sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> - ½ ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> T<sub>μν</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>or remembering (56)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(57) ∂T<sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> + ½ ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> T<sub>μν</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>A comparison with (41b) shows that these equations -for the above choice of co-ordinates (√(-<i>g</i>) = 1) asserts -nothing but the vanishing of the divergence of the tensor -of the energy-components of matter.</p> - -<p class='c005'>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 hold; or can only -hold when <i>g</i><sup>μν</sup>’s are constants; <i>i.e.</i>, when the field of gravitation -vanishes. The second member is an expression for -impulse and energy which the gravitation-field exerts per -time and per volume upon matter. This comes out clearer -when instead of (57) we write it in the form of (47).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(57a) ∂T<sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> = -Γ<sub>σβ</sub><sup>α</sup> T<sub>α</sub><sup>β</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>The right-hand side expresses the interaction of the energy -of the gravitational-field on matter. The field-equations of -gravitation contain thus at the same time 4 conditions -which are to be satisfied by all material phenomena. We -get the equations of the material phenomena completely -when the latter is characterised by four other differential -equations independent of one another.</p> -<h3 class='c008'>D. THE “MATERIAL” PHENOMENA.</h3> -<p class='c004'>The Mathematical auxiliaries developed under ‘B’ at -once enables us to generalise, according to the generalised -theory of relativity, the physical laws of matter (Hydrodynamics, -Maxwell’s Electro-dynamics) as they lie already -formulated according to the special-relativity-theory. -The generalised Relativity Principle leads us to no further -limitation of possibilities; but it enables us to know -exactly the influence of gravitation on all processes without -the introduction of any new hypothesis.</p> - -<p class='c005'>It is owing to this, that as regards the physical nature -of matter (in a narrow sense) no definite necessary assumptions -are to be introduced. The question may lie open -whether the theories of the electro-magnetic field and the -gravitational-field together, will form a sufficient basis for -the theory of matter. The general relativity postulate can -teach us no new principle. 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 doing.</p> -<h4 class='c009'>§19. Euler’s equations for frictionless adiabatic liquid.</h4> -<p class='c004'>Let <i>p</i> and ρ, be two scalars, of which the first denotes -the pressure and the last the density of the fluid; between -them there is a relation. Let the contravariant symmetrical -tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>T<sup>αβ</sup> = -<i>g</i><sup>αβ</sup> <i>p</i> + ρ <i>dx</i><sub>α</sub>/<i>ds</i> <i>dx</i><sub>β</sub>/<i>ds</i> (58)</div> - </div> - </div> -</div> - -<p class='c010'>be the contra-variant energy-tensor of the liquid. To it -also belongs the covariant tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(58a) T<sub>μν</sub> = -<i>g</i><sub>μν</sub> <i>p</i> + <i>g</i><sub>μα</sub> <i>dx</i><sub>α</sub>/<i>ds</i> <i>g</i><sub>μβ</sub> <i>dx</i><sub>β</sub>/<i>ds</i> ρ</div> - </div> - </div> -</div> - -<p class='c010'>as well as the mixed tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(58b) T<sup>α</sup><sub>σ</sub> = -δ<sup>α</sup><sub>σ</sub> <i>p</i> + <i>g</i><sub>σβ</sub> <i>dx</i><sub>β</sub>/<i>ds</i> <i>dx</i><sub>α</sub>/<i>ds</i> ρ.</div> - </div> - </div> -</div> - -<p class='c010'>If we put the right-hand side of (58b) in (57a) we -get the general hydrodynamical equations of Euler according -to the generalised relativity theory. This in principle -completely solves the problem of motion; for the four -equations (57a) together with the given equation between -<i>p</i> and ρ, and the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>αβ</sub> <i>dx</i>_α/<i>ds</i> <i>dx</i><sub>β</sub>/<i>ds</i> = 1,</div> - </div> - </div> -</div> - -<p class='c010'>are sufficient, with the given values of <i>g</i><sub>αβ</sub>, for finding -out the six unknowns</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>p</i>, ρ, <i>dx₁</i>/<i>ds</i>, <i>dx₂</i>/<i>ds</i>, <i>dx₃</i>/<i>ds</i> <i>dx₄</i>/<i>ds</i>.</div> - </div> - </div> -</div> - -<p class='c010'>If <i>g</i><sub>μν</sub>’s are unknown we have also to take the equations -(53). There are now 11 equations for finding out -10 functions <i>g</i>, so that the number is more than sufficient. -Now it is be noticed that the equation (57a) is -already contained in (53), so that the latter only represents -(7) independent equations. This indefiniteness is due to -the wide freedom in the choice of co-ordinates, so that -mathematically the problem is indefinite in the sense that -three of the space-functions can be arbitrarily chosen.</p> -<h4 class='c009'>§20. Maxwell’s Electro-Magnetic field-equations.</h4> -<p class='c004'>Let φ<sub>ν</sub> be the components of a covariant four-vector, -the electro-magnetic potential; from it let us form according -to (36) the components F<sub>ρσ</sub> of the covariant six-vector -of the electro-magnetic field according to the system of -equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(59) F<sub>ρσ</sub> = ∂φ<sub>ρ</sub>/∂<i>x</i><sub>σ</sub> - ∂φ<sub>σ</sub>/∂<i>x</i><sub>ρ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>From (59), it follows that the system of equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(60) ∂F<sub>ρσ</sub>/∂<i>x</i><sub>τ</sub> + ∂F<sub>στ</sub>/∂<i>x</i><sub>ρ</sub> + ∂F<sub>τρ</sub>/∂<i>x</i><sub>σ</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>is satisfied of which the left-hand side, according to -(37), is an anti-symmetrical tensor of the third kind. -This system (60) contains essentially four equations, which -can be thus written:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ ∂F₂₃/∂<i>x₄</i> + ∂F₃₄/∂<i>x₂</i> ∂F₄₂/∂<i>x₃</i> = 0</div> - <div class='line in6'>{</div> - <div class='line in6'>{ ∂F₃₄/∂<i>x₁</i> + ∂F₄₁/∂<i>x₃</i> ∂F₁₃/∂<i>x₄</i> = 0</div> - <div class='line'>(60a) {</div> - <div class='line in6'>{ ∂F₄₁/∂<i>x₂</i> + ∂F₁₂/∂<i>x₄</i> ∂F₂₄/∂<i>x₁</i> = 0</div> - <div class='line in6'>{</div> - <div class='line in6'>{ ∂F₁₂/∂<i>x₃</i> + ∂F₂₃/∂<i>x₁</i> ∂F₃₁/∂<i>x₂</i> = 0.</div> - </div> - </div> -</div> - -<p class='c010'>This system of equations corresponds to the second -system of equations of Maxwell. We see it at once if we -put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'>{ F₂₃ = H<sub><i>x</i></sub> F₁₄ = E<sub><i>x</i></sub></div> - <div class='line in5'>{</div> - <div class='line'>(61) { F₃₁ = H<sub><i>y</i></sub> F₂₄ = E<sub><i>y</i></sub></div> - <div class='line in5'>{</div> - <div class='line in5'>{ F₁₂ = H<sub><i>z</i></sub> F₃₄ = E<sub><i>z</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>Instead of (60a) we can therefore write according to -the usual notation of three-dimensional vector-analysis:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ ∂H/∂<i>t</i> + rot E = 0</div> - <div class='line'>(60b) {</div> - <div class='line in6'>{ div H = 0.</div> - </div> - </div> -</div> - -<p class='c010'>The first Maxwellian system is obtained by a generalisation -of the form given by Minkowski.</p> - -<p class='c005'>We introduce the contra-variant six-vector F<sub>αβ</sub> by -the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(62) F<sup>μν</sup> = <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub>,</div> - </div> - </div> -</div> - -<p class='c010'>and also a contra-variant four-vector J<sup>μ</sup>, which is the -electrical current-density in vacuum. Then remembering -(40) we can establish the system of equations, which -remains invariant for any substitution with determinant 1 -(according to our choice of co-ordinates).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(63) ∂F<sup>μν</sup>/∂<i>x</i><sub>ν</sub> = J<sup>μ</sup></div> - </div> - </div> -</div> - -<p class='c010'>If we put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'>{ F²³ = H′<sub><i>x</i></sub> F¹⁴ = -E′<sub><i>x</i></sub></div> - <div class='line in5'>{</div> - <div class='line'>(64) { F³¹ = H′<sub><i>y</i></sub> F²⁴ = -E′<sub><i>y</i></sub></div> - <div class='line in5'>{</div> - <div class='line in5'>{ F¹² = H′<sub><i>z</i></sub> F³⁴ = -E′<sub><i>z</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>which quantities become equal to H<sub><i>x</i></sub> ... E<sub><i>x</i></sub> in the case of -the special relativity theory, and besides</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>J<sup>1</sup> = <i>i</i><sub><i>x</i></sub> ... J<sup>4</sup> = ρ</div> - </div> - </div> -</div> - -<p class='c010'>we get instead of (63)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ rot H′ - ∂E′/∂<i>t</i> = <i>i</i></div> - <div class='line'>(63a) {</div> - <div class='line in6'>{ div E′ = ρ</div> - </div> - </div> -</div> - -<p class='c010'>The equations (60), (62) and (63) give thus a generalisation -of Maxwell’s field-equations in vacuum, which -remains true in our chosen system of co-ordinates.</p> -<p class='c004'><i>The energy-components of the electro-magnetic field.</i></p> -<p class='c004'>Let us form the inner-product</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(65) K<sub>σ</sub> = F<sub>σμ</sub> J<sup>μ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>According to (61) its components can be written down -in the three-dimensional notation.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ K₁ = ρE<sub><i>x</i></sub> + [<i>i</i>, H]<sub>x</sub></div> - </div> - <div class='group'> - <div class='line'>(65a) { — — —</div> - </div> - <div class='group'> - <div class='line in6'>{ K₄ = — (<i>i</i>, E).</div> - </div> - </div> -</div> - -<p class='c010'>K<sub>σ</sub> 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 time, and per unit -of volume, by the electrical masses. If the electrical -masses be free, that is, under the influence of the electro-magnetic -field only, then the covariant four-vector -K<sub>σ</sub> will vanish.</p> - -<p class='c005'>In order to get the energy components T<sub>σ</sub><sup>ν</sup> of the electro-magnetic -field, we require only to give to the equation -K<sub>σ</sub> = 0, the form of the equation (57).</p> - -<p class='c005'>From (63) and (65) we get first,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>K<sub>σ</sub> = F<sub>σμ</sub> ∂F<sub>μν</sub>/∂<i>x</i><sub>ν</sub></div> - </div> - <div class='group'> - <div class='line'>= ∂/∂<i>x</i><sub>ν</sub> (F<sub>σμ</sub> F<sup>μν</sup>) - F<sup>μν</sup> ∂F<sub>σμ</sub>/∂<i>x</i><sub>ν</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>On account of (60) the second member on the right-hand -side admits of the transformation—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>F<sup>μν</sup> ∂F<sub>σμ</sub>/∂<i>x</i><sub>ν</sub> = -½ F<sup>μν</sup> ∂F<sub>μν</sub>/∂<i>x</i><sub>σ</sub></div> - </div> - <div class='group'> - <div class='line'>= -½ <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub> ∂F<sub>μν</sub>/∂<i>x</i><sub>σ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Owing to symmetry, this expression can also be written in -the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>= -1/4 [<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub> ∂F<sub>μν</sub>/∂<i>x</i><sub>σ</sub></div> - </div> - <div class='group'> - <div class='line'>+ <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> ∂F<sub>αβ</sub>/∂<i>x</i><sub>σ</sub> F<sub>μν</sub>],</div> - </div> - </div> -</div> - -<p class='c010'>which can also be put in the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- 1/4 ∂/∂<i>x</i><sub>σ</sub> (<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub> F<sub>μν</sub>)</div> - </div> - <div class='group'> - <div class='line'>+ 1/4 F<sub>αβ</sub> F<sub>μν</sub> ∂/∂<i>x</i><sub>σ</sub> (<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup>).</div> - </div> - </div> -</div> - -<p class='c010'>The first of these terms can be written shortly as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- 1/4 ∂/∂<i>x</i><sub>σ</sub> (F<sup>μν</sup> F<sub>μν</sub>),</div> - </div> - </div> -</div> - -<p class='c010'>and the second after differentiation can be transformed in -the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- ½ F<sup>μτ</sup> F<sub>μν</sub> <i>g</i><sup>νρ</sup> ∂<i>g</i><sub>στ</sub>/∂<i>x</i><sub>σ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>If we take all the three terms together, we get the -relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(66) K<sub>σ</sub> = ∂τ<sub>σ</sub><sup>ν</sup>/∂<i>x</i><sub>ν</sub> - ½ <i>g</i><sup>τμ</sup> ∂<i>g</i><sub>μν</sub>/∂<i>x</i><sub>σ</sub> τ<sub>τ</sub><sup>ν</sup></div> - </div> - </div> -</div> - -<p class='c010'>where</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(66a) τ<sub>σ</sub><sup>ν</sup> = -F<sub>σα</sub> F<sup>να</sup> + 1/4 δ<sub>σ</sub><sup>ν</sup> F<sub>αβ</sub> F<sup>αβ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>On account of (30) the equation (66) becomes equivalent -to (57) and (57a) when K<sub>σ</sub> vanishes. Thus τ<sub>σ</sub><sup>ν</sup>’s are the -energy-components of the electro-magnetic field. With -the help of (61) and (64) we can easily show that the -energy-components of the electro-magnetic field, in the case -of the special relativity theory, give rise to the well-known -Maxwell-Poynting expressions.</p> - -<p class='c005'>We have now deduced the most general laws which -the gravitation-field and matter satisfy when we use a -co-ordinate system for which √(-<i>g</i>) = 1. Thereby we -achieve an important simplification in all our formulas and -calculations, without renouncing the conditions of general -covariance, as we have obtained the equations through a -specialisation of the co-ordinate system from the general -covariant-equations. Still the question is not without formal -interest, whether, when the energy-components of the -gravitation-field and matter is defined in a generalised manner -without any specialisation of co-ordinates, the laws of conservation -have the form of the equation (56), and the field-equations -of gravitation hold in the form (52) or (52a); -such that on the left-hand side, we have a divergence in the -usual sense, and on the right-hand side, the sum of the -energy-components of matter and gravitation. I have -found out that this is indeed the case. But I am of opinion -that the communication of my rather comprehensive work -on this subject will not pay, for nothing essentially new -comes out of it.</p> -<h3 class='c008'>E. §21. Newton’s theory as a first approximation.</h3> -<p class='c004'>We have already mentioned several times that the -special relativity theory is to be looked upon as a special -case of the general, in which <i>g</i><sub>μν</sub>’s have constant values (4). -This signifies, according to what has been said before, a -total neglect of the influence of gravitation. We get -one important approximation if we consider the case -when <i>g</i><sub>μν</sub>’s differ from (4) only by small magnitudes (compared -to 1) where we can neglect small quantities of the -second and higher orders (first aspect of the approximation.)</p> - -<p class='c005'>Further it should be assumed that within the space-time -region considered, <i>g</i><sub>μν</sub>’s at infinite distances (using -the word infinite in a spatial sense) can, by a suitable choice -of co-ordinates, tend to the limiting values (4); <i>i.e.</i>, we consider -only those gravitational fields which can be regarded -as produced by masses distributed over finite regions.</p> - -<p class='c005'>We can assume that this approximation should lead to -Newton’s theory. For it however, it is necessary to treat -the fundamental equations from another point of view. -Let us consider the motion of a particle according to the -equation (46). In the case of the special relativity theory, -the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx₁</i>/<i>ds</i>, <i>dx₂</i>/<i>ds</i>, <i>dx₃</i>/<i>ds</i>,</div> - </div> - </div> -</div> - -<p class='c010'>can take any values. This signifies that any velocity</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>v</i> = √((<i>dx₁</i>/<i>dx₄</i>)² + (<i>dx₂</i>/<i>dx₄</i>)² + (<i>dx₃</i>/<i>dx₄</i>)²)</div> - </div> - </div> -</div> - -<p class='c010'>can appear which is less than the velocity of light in -vacuum (<i>v</i> < 1). If we finally limit ourselves to the -consideration of the case when <i>v</i> is small compared to the -velocity of light, it signifies that the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx₁</i>/<i>ds</i>, <i>dx₂</i>/<i>ds</i>, <i>dx₃</i>/<i>ds</i>,</div> - </div> - </div> -</div> - -<p class='c010'>can be treated as small quantities, whereas <i>dx₄</i>/<i>ds</i> is equal to -1, up to the second-order magnitudes (the second point of -view for approximation).</p> - -<p class='c005'>Now we see that, according to the first view of approximation, -the magnitudes γ<sub>μν</sub><sup>τ</sup>’s are all small quantities of -at least the first order. A glance at (46) will also show, -that in this equation according to the second view of -approximation, we are only to take into account those -terms for which μ = ν = 4.</p> - -<p class='c005'>By limiting ourselves only to terms of the lowest order -we get instead of (46), first, the equations:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d²</i><i>x</i><sub>τ</sub>/<i>dt²</i> = Γ₄₄<sup>τ</sup>, where <i>ds</i> = <i>dx₄</i> = <i>dt</i>,</div> - </div> - </div> -</div> - -<p class='c010'>or by limiting ourselves only to those terms which according -to the first stand-point are approximations of the first -order,</p> - -<p class='c005'>It must be admitted, that this introduction of the -energy-tensor of matter cannot be justified by means of the -Relativity-Postulate alone; for we have in the foregoing -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 energy. The strongest -ground for the choice of the above equation however lies in -this, that they lead, as their consequences, to equations -expressing the conservation of the components of total -energy (the impulses and the energy) which exactly -correspond to the equations (49) and (49a). This shall be -shown afterwards.</p> -<h4 class='c009'>§17. The laws of conservation in the general case.</h4> -<p class='c004'>The equations (52) can be easily so transformed that -the second member on the right-hand side vanishes. We -reduce (52) with reference to the indices μ and σ and -subtract the equation so obtained after multiplication with -½ δ<sub>μ</sub><sup>σ</sup> from (52).</p> - -<p class='c005'>We obtain,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(52a) ∂/∂<i>x</i><sub>α</sub>(<i>g</i><sup>σβ</sup> Γ<sub>μβ</sub><sup>α</sup> - ½ δ<sub>μ</sub><sup>σ</sup> <i>g</i><sup>λβ</sup> Γ<sub>λβ</sub><sup>α</sup>)</div> - <div class='line'>= -κ(<i>t</i><sub>μ</sub><sup>σ</sup> + T<sub>μ</sub><sup>σ</sup>)</div> - </div> - </div> -</div> - -<p class='c010'>we operate on it by ∂/∂<i>x</i><sub>σ</sub>. Now,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> (<i>g</i><sup>σβ</sup>Γ<sub>μβ</sub><sup>α</sup>)</div> - <div class='line'>= -½ ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> [<i>g</i><sup>σβ</sup> <i>g</i><sup>αλ</sup>(∂<i>g</i><sub>μλ</sub>/∂<i>x</i><sub>β</sub></div> - <div class='line in1'>+ ∂<i>g</i><sub>βλ</sub>/∂<i>x</i><sub>μ</sub> - ∂<i>g</i><sub>μβ</sub>/∂<i>x</i><sub>λ</sub>)].</div> - </div> - </div> -</div> - -<p class='c010'>The first and the third member of the round bracket -lead to expressions which cancel one another, as can be -easily seen by interchanging the summation-indices α, and -σ, on the one hand, and β and λ, on the other.</p> - -<p class='c005'>The second term can be transformed according to (31). -So that we get,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(54) ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> (<i>g</i><sup>σβ</sup>γ<sub>μβ</sub><sup>α</sup>)</div> - <div class='line in1'>= ½ ∂³<i>g</i><sup>αβ</sup>/∂<i>x</i><sub>σ</sub>∂<i>x</i><sub>β</sub>∂<i>x</i><sub>μ</sub></div> - </div> - </div> -</div> - -<p class='c010'>The second member of the expression on the left-hand -side of (52a) leads first to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- ½ ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>μ</sub> (<i>g</i><sup>λβ</sup>Γ<sub>λβ</sub><sup>α</sup>) or</div> - </div> - <div class='group'> - <div class='line'>to 1/4 ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>μ</sub> [<i>g</i><sup>λβ</sup><i>g</i><sup>αδ</sup>( ∂<i>g</i><sub>δλ</sub>/∂<i>x</i><sub>β</sub></div> - <div class='line in1'>+ ∂<i>g</i><sub>δβ</sub>/∂<i>x</i><sub>λ</sub> - ∂<i>g</i><sub>λβ</sub>/∂<i>x</i><sub>δ</sub>)].</div> - </div> - </div> -</div> - -<p class='c010'>The expression arising out of the last member within -the round bracket vanishes according to (29) on account -of the choice of axes. The two others can be taken -together and give us on account of (31), the expression</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-½ ∂³<i>g</i><sup>αβ</sup>/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>β</sub>∂<i>x</i><sub>μ</sub></div> - </div> - </div> -</div> - -<p class='c010'>So that remembering (54) we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(55) ∂²/∂<i>x</i><sub>α</sub>∂<i>x</i><sub>σ</sub> (<i>g</i><sup>σβ</sup>Γ<sub>μβ</sub><sup>α</sup></div> - <div class='line in1'>- ½ δ<sub>μ</sub><sup>σ</sup> <i>g</i><sup>λβ</sup> Γ<sub>λβ</sub><sup>α</sup>) = 0.</div> - </div> - </div> -</div> - -<p class='c010'>identically.</p> - -<p class='c005'>From (55) and (52a) it follows that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(56) ∂/∂<i>x</i><sub>σ</sub> (<i>t</i><sub>μ</sub><sup>σ</sup> + T<sub>μ</sub><sup>σ</sup>) = 0</div> - </div> - </div> -</div> - -<p class='c010'>From the field equations of gravitation, it also follows -that the conservation-laws of impulse and energy are -satisfied. We see it most simply following the same -reasoning which lead to equations (49a); only instead of -the energy-components of the gravitational-field, we are to -introduce the total energy-components of matter and gravitational -field.</p> -<h4 class='c009'>§18. The Impulse-energy law for matter as a consequence of the field-equations.</h4> -<p class='c004'>If we multiply (53) with ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub>, we get in a way -similar to §15, remembering that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>μν</sub> ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> vanishes,</div> - </div> - <div class='group'> - <div class='line'>the equations ∂<i>t</i><sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> - ½ ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> T<sub>μν</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>or remembering (56)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(57) ∂T<sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> + ½ ∂<i>g</i><sup>μν</sup>/∂<i>x</i><sub>σ</sub> T<sub>μν</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>A comparison with (41b) shows that these equations -for the above choice of co-ordinates (√(-<i>g</i>) = 1) asserts -nothing but the vanishing of the divergence of the tensor -of the energy-components of matter.</p> - -<p class='c005'>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 hold; or can only -hold when <i>g</i><sup>μν</sup>’s are constants; <i>i.e.</i>, when the field of gravitation -vanishes. The second member is an expression for -impulse and energy which the gravitation-field exerts per -time and per volume upon matter. This comes out clearer -when instead of (57) we write it in the form of (47).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(57a) ∂T<sub>σ</sub><sup>α</sup>/∂<i>x</i><sub>α</sub> = -Γ<sub>σβ</sub><sup>α</sup> T<sub>α</sub><sup>β</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>The right-hand side expresses the interaction of the energy -of the gravitational-field on matter. The field-equations of -gravitation contain thus at the same time 4 conditions -which are to be satisfied by all material phenomena. We -get the equations of the material phenomena completely -when the latter is characterised by four other differential -equations independent of one another.</p> -<h3 class='c008'>D. THE “MATERIAL” PHENOMENA.</h3> -<p class='c004'>The Mathematical auxiliaries developed under ‘B’ at -once enables us to generalise, according to the generalised -theory of relativity, the physical laws of matter (Hydrodynamics, -Maxwell’s Electro-dynamics) as they lie already -formulated according to the special-relativity-theory. -The generalised Relativity Principle leads us to no further -limitation of possibilities; but it enables us to know -exactly the influence of gravitation on all processes without -the introduction of any new hypothesis.</p> - -<p class='c005'>It is owing to this, that as regards the physical nature -of matter (in a narrow sense) no definite necessary assumptions -are to be introduced. The question may lie open -whether the theories of the electro-magnetic field and the -gravitational-field together, will form a sufficient basis for -the theory of matter. The general relativity postulate can -teach us no new principle. 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 doing.</p> -<h4 class='c009'>§19. Euler’s equations for frictionless adiabatic liquid.</h4> -<p class='c004'>Let <i>p</i> and ρ, be two scalars, of which the first denotes -the pressure and the last the density of the fluid; between -them there is a relation. Let the contravariant symmetrical -tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>T<sup>αβ</sup> = -<i>g</i><sup>αβ</sup> <i>p</i> + ρ <i>dx</i><sub>α</sub>/<i>ds</i> <i>dx</i><sub>β</sub>/<i>ds</i> (58)</div> - </div> - </div> -</div> - -<p class='c010'>be the contra-variant energy-tensor of the liquid. To it -also belongs the covariant tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(58a) T<sub>μν</sub> = -<i>g</i><sub>μν</sub> <i>p</i> + <i>g</i><sub>μα</sub> <i>dx</i><sub>α</sub>/<i>ds</i> <i>g</i><sub>μβ</sub> <i>dx</i><sub>β</sub>/<i>ds</i> ρ</div> - </div> - </div> -</div> - -<p class='c010'>as well as the mixed tensor</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(58b) T<sup>α</sup><sub>σ</sub> = -δ<sup>α</sup><sub>σ</sub> <i>p</i> + <i>g</i><sub>σβ</sub> <i>dx</i><sub>β</sub>/<i>ds</i> <i>dx</i><sub>α</sub>/<i>ds</i> ρ.</div> - </div> - </div> -</div> - -<p class='c010'>If we put the right-hand side of (58b) in (57a) we -get the general hydrodynamical equations of Euler according -to the generalised relativity theory. This in principle -completely solves the problem of motion; for the four -equations (57a) together with the given equation between -<i>p</i> and ρ, and the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g</i><sub>αβ</sub> <i>dx</i>_α/<i>ds</i> <i>dx</i><sub>β</sub>/<i>ds</i> = 1,</div> - </div> - </div> -</div> - -<p class='c010'>are sufficient, with the given values of <i>g</i><sub>αβ</sub>, for finding -out the six unknowns</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>p</i>, ρ, <i>dx₁</i>/<i>ds</i>, <i>dx₂</i>/<i>ds</i>, <i>dx₃</i>/<i>ds</i> <i>dx₄</i>/<i>ds</i>.</div> - </div> - </div> -</div> - -<p class='c010'>If <i>g</i><sub>μν</sub>’s are unknown we have also to take the equations -(53). There are now 11 equations for finding out -10 functions <i>g</i>, so that the number is more than sufficient. -Now it is be noticed that the equation (57a) is -already contained in (53), so that the latter only represents -(7) independent equations. This indefiniteness is due to -the wide freedom in the choice of co-ordinates, so that -mathematically the problem is indefinite in the sense that -three of the space-functions can be arbitrarily chosen.</p> -<h4 class='c009'>§20. Maxwell’s Electro-Magnetic field-equations.</h4> -<p class='c004'>Let φ<sub>ν</sub> be the components of a covariant four-vector, -the electro-magnetic potential; from it let us form according -to (36) the components F<sub>ρσ</sub> of the covariant six-vector -of the electro-magnetic field according to the system of -equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(59) F<sub>ρσ</sub> = ∂φ<sub>ρ</sub>/∂<i>x</i><sub>σ</sub> - ∂φ<sub>σ</sub>/∂<i>x</i><sub>ρ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>From (59), it follows that the system of equations</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(60) ∂F<sub>ρσ</sub>/∂<i>x</i><sub>τ</sub> + ∂F<sub>στ</sub>/∂<i>x</i><sub>ρ</sub> + ∂F<sub>τρ</sub>/∂<i>x</i><sub>σ</sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>is satisfied of which the left-hand side, according to -(37), is an anti-symmetrical tensor of the third kind. -This system (60) contains essentially four equations, which -can be thus written:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ ∂F₂₃/∂<i>x₄</i> + ∂F₃₄/∂<i>x₂</i> ∂F₄₂/∂<i>x₃</i> = 0</div> - <div class='line in6'>{</div> - <div class='line in6'>{ ∂F₃₄/∂<i>x₁</i> + ∂F₄₁/∂<i>x₃</i> ∂F₁₃/∂<i>x₄</i> = 0</div> - <div class='line'>(60a) {</div> - <div class='line in6'>{ ∂F₄₁/∂<i>x₂</i> + ∂F₁₂/∂<i>x₄</i> ∂F₂₄/∂<i>x₁</i> = 0</div> - <div class='line in6'>{</div> - <div class='line in6'>{ ∂F₁₂/∂<i>x₃</i> + ∂F₂₃/∂<i>x₁</i> ∂F₃₁/∂<i>x₂</i> = 0.</div> - </div> - </div> -</div> - -<p class='c010'>This system of equations corresponds to the second -system of equations of Maxwell. We see it at once if we -put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'>{ F₂₃ = H<sub><i>x</i></sub> F₁₄ = E<sub><i>x</i></sub></div> - <div class='line in5'>{</div> - <div class='line'>(61) { F₃₁ = H<sub><i>y</i></sub> F₂₄ = E<sub><i>y</i></sub></div> - <div class='line in5'>{</div> - <div class='line in5'>{ F₁₂ = H<sub><i>z</i></sub> F₃₄ = E<sub><i>z</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>Instead of (60a) we can therefore write according to -the usual notation of three-dimensional vector-analysis:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ ∂H/∂<i>t</i> + rot E = 0</div> - <div class='line'>(60b) {</div> - <div class='line in6'>{ div H = 0.</div> - </div> - </div> -</div> - -<p class='c010'>The first Maxwellian system is obtained by a generalisation -of the form given by Minkowski.</p> - -<p class='c005'>We introduce the contra-variant six-vector F<sub>αβ</sub> by -the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(62) F<sup>μν</sup> = <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub>,</div> - </div> - </div> -</div> - -<p class='c010'>and also a contra-variant four-vector J<sup>μ</sup>, which is the -electrical current-density in vacuum. Then remembering -(40) we can establish the system of equations, which -remains invariant for any substitution with determinant 1 -(according to our choice of co-ordinates).</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(63) ∂F<sup>μν</sup>/∂<i>x</i><sub>ν</sub> = J<sup>μ</sup></div> - </div> - </div> -</div> - -<p class='c010'>If we put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'>{ F²³ = H′<sub><i>x</i></sub> F¹⁴ = -E′<sub><i>x</i></sub></div> - <div class='line in5'>{</div> - <div class='line'>(64) { F³¹ = H′<sub><i>y</i></sub> F²⁴ = -E′<sub><i>y</i></sub></div> - <div class='line in5'>{</div> - <div class='line in5'>{ F¹² = H′<sub><i>z</i></sub> F³⁴ = -E′<sub><i>z</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>which quantities become equal to H<sub><i>x</i></sub> ... E<sub><i>x</i></sub> in the case of -the special relativity theory, and besides</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>J<sup>1</sup> = <i>i</i><sub><i>x</i></sub> ... J<sup>4</sup> = ρ</div> - </div> - </div> -</div> - -<p class='c010'>we get instead of (63)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ rot H′ - ∂E′/∂<i>t</i> = <i>i</i></div> - <div class='line'>(63a) {</div> - <div class='line in6'>{ div E′ = ρ</div> - </div> - </div> -</div> - -<p class='c010'>The equations (60), (62) and (63) give thus a generalisation -of Maxwell’s field-equations in vacuum, which -remains true in our chosen system of co-ordinates.</p> -<p class='c004'><i>The energy-components of the electro-magnetic field.</i></p> -<p class='c004'>Let us form the inner-product</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(65) K<sub>σ</sub> = F<sub>σμ</sub> J<sup>μ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>According to (61) its components can be written down -in the three-dimensional notation.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in6'>{ K₁ = ρE<sub><i>x</i></sub> + [<i>i</i>, H]<sub>x</sub></div> - </div> - <div class='group'> - <div class='line'>(65a) { — — —</div> - </div> - <div class='group'> - <div class='line in6'>{ K₄ = — (<i>i</i>, E).</div> - </div> - </div> -</div> - -<p class='c010'>K<sub>σ</sub> 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 time, and per unit -of volume, by the electrical masses. If the electrical -masses be free, that is, under the influence of the electro-magnetic -field only, then the covariant four-vector -K<sub>σ</sub> will vanish.</p> - -<p class='c005'>In order to get the energy components T<sub>σ</sub><sup>ν</sup> of the electro-magnetic -field, we require only to give to the equation -K<sub>σ</sub> = 0, the form of the equation (57).</p> - -<p class='c005'>From (63) and (65) we get first,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>K<sub>σ</sub> = F<sub>σμ</sub> ∂F<sub>μν</sub>/∂<i>x</i><sub>ν</sub></div> - </div> - <div class='group'> - <div class='line'>= ∂/∂<i>x</i><sub>ν</sub> (F<sub>σμ</sub> F<sup>μν</sup>) - F<sup>μν</sup> ∂F<sub>σμ</sub>/∂<i>x</i><sub>ν</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>On account of (60) the second member on the right-hand -side admits of the transformation—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>F<sup>μν</sup> ∂F<sub>σμ</sub>/∂<i>x</i><sub>ν</sub> = -½ F<sup>μν</sup> ∂F<sub>μν</sub>/∂<i>x</i><sub>σ</sub></div> - </div> - <div class='group'> - <div class='line'>= -½ <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub> ∂F<sub>μν</sub>/∂<i>x</i><sub>σ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Owing to symmetry, this expression can also be written in -the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>= -1/4 [<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub> ∂F<sub>μν</sub>/∂<i>x</i><sub>σ</sub></div> - </div> - <div class='group'> - <div class='line'>+ <i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> ∂F<sub>αβ</sub>/∂<i>x</i><sub>σ</sub> F<sub>μν</sub>],</div> - </div> - </div> -</div> - -<p class='c010'>which can also be put in the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- 1/4 ∂/∂<i>x</i><sub>σ</sub> (<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup> F<sub>αβ</sub> F<sub>μν</sub>)</div> - </div> - <div class='group'> - <div class='line'>+ 1/4 F<sub>αβ</sub> F<sub>μν</sub> ∂/∂<i>x</i><sub>σ</sub> (<i>g</i><sup>μα</sup> <i>g</i><sup>νβ</sup>).</div> - </div> - </div> -</div> - -<p class='c010'>The first of these terms can be written shortly as</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- 1/4 ∂/∂<i>x</i><sub>σ</sub> (F<sup>μν</sup> F<sub>μν</sub>),</div> - </div> - </div> -</div> - -<p class='c010'>and the second after differentiation can be transformed in -the form</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>- ½ F<sup>μτ</sup> F<sub>μν</sub> <i>g</i><sup>νρ</sup> ∂<i>g</i><sub>στ</sub>/∂<i>x</i><sub>σ</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>If we take all the three terms together, we get the -relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(66) K<sub>σ</sub> = ∂τ<sub>σ</sub><sup>ν</sup>/∂<i>x</i><sub>ν</sub> - ½ <i>g</i><sup>τμ</sup> ∂<i>g</i><sub>μν</sub>/∂<i>x</i><sub>σ</sub> τ<sub>τ</sub><sup>ν</sup></div> - </div> - </div> -</div> - -<p class='c010'>where</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(66a) τ<sub>σ</sub><sup>ν</sup> = -F<sub>σα</sub> F<sup>να</sup> + 1/4 δ<sub>σ</sub><sup>ν</sup> F<sub>αβ</sub> F<sup>αβ</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>On account of (30) the equation (66) becomes equivalent -to (57) and (57a) when K<sub>σ</sub> vanishes. Thus τ<sub>σ</sub><sup>ν</sup>’s are the -energy-components of the electro-magnetic field. With -the help of (61) and (64) we can easily show that the -energy-components of the electro-magnetic field, in the case -of the special relativity theory, give rise to the well-known -Maxwell-Poynting expressions.</p> - -<p class='c005'>We have now deduced the most general laws which -the gravitation-field and matter satisfy when we use a -co-ordinate system for which √(-<i>g</i>) = 1. Thereby we -achieve an important simplification in all our formulas and -calculations, without renouncing the conditions of general -covariance, as we have obtained the equations through a -specialisation of the co-ordinate system from the general -covariant-equations. Still the question is not without formal -interest, whether, when the energy-components of the -gravitation-field and matter is defined in a generalised manner -without any specialisation of co-ordinates, the laws of conservation -have the form of the equation (56), and the field-equations -of gravitation hold in the form (52) or (52a); -such that on the left-hand side, we have a divergence in the -usual sense, and on the right-hand side, the sum of the -energy-components of matter and gravitation. I have -found out that this is indeed the case. But I am of opinion -that the communication of my rather comprehensive work -on this subject will not pay, for nothing essentially new -comes out of it.</p> -<h3 class='c008'>E. §21. Newton’s theory as a first approximation.</h3> -<p class='c004'>We have already mentioned several times that the -special relativity theory is to be looked upon as a special -case of the general, in which <i>g</i><sub>μν</sub>’s have constant values (4). -This signifies, according to what has been said before, a -total neglect of the influence of gravitation. We get -one important approximation if we consider the case -when <i>g</i><sub>μν</sub>’s differ from (4) only by small magnitudes (compared -to 1) where we can neglect small quantities of the -second and higher orders (first aspect of the approximation.)</p> - -<p class='c005'>Further it should be assumed that within the space-time -region considered, <i>g</i><sub>μν</sub>’s at infinite distances (using -the word infinite in a spatial sense) can, by a suitable choice -of co-ordinates, tend to the limiting values (4); <i>i.e.</i>, we consider -only those gravitational fields which can be regarded -as produced by masses distributed over finite regions.</p> - -<p class='c005'>We can assume that this approximation should lead to -Newton’s theory. For it however, it is necessary to treat -the fundamental equations from another point of view. -Let us consider the motion of a particle according to the -equation (46). In the case of the special relativity theory, -the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx₁</i>/<i>ds</i>, <i>dx₂</i>/<i>ds</i>, <i>dx₃</i>/<i>ds</i>,</div> - </div> - </div> -</div> - -<p class='c010'>can take any values. This signifies that any velocity</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>v</i> = √((<i>dx₁</i>/<i>dx₄</i>)² + (<i>dx₂</i>/<i>dx₄</i>)² + (<i>dx₃</i>/<i>dx₄</i>)²)</div> - </div> - </div> -</div> - -<p class='c010'>can appear which is less than the velocity of light in -vacuum (<i>v</i> < 1). If we finally limit ourselves to the -consideration of the case when <i>v</i> is small compared to the -velocity of light, it signifies that the components</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx₁</i>/<i>ds</i>, <i>dx₂</i>/<i>ds</i>, <i>dx₃</i>/<i>ds</i>,</div> - </div> - </div> -</div> - -<p class='c010'>can be treated as small quantities, whereas <i>dx₄</i>/<i>ds</i> is equal to -1, up to the second-order magnitudes (the second point of -view for approximation).</p> - -<p class='c005'>Now we see that, according to the first view of approximation, -the magnitudes γ<sub>μν</sub><sup>τ</sup>’s are all small quantities of -at least the first order. A glance at (46) will also show, -that in this equation according to the second view of -approximation, we are only to take into account those -terms for which μ = ν = 4.</p> - -<p class='c005'>By limiting ourselves only to terms of the lowest order -we get instead of (46), first, the equations:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>d²</i><i>x</i><sub>τ</sub>/<i>dt²</i> = Γ₄₄<sup>τ</sup>, where <i>ds</i> = <i>dx₄</i> = <i>dt</i>,</div> - </div> - </div> -</div> - -<p class='c010'>or by limiting ourselves only to those terms which according -to the first stand-point are approximations of the first -order,</p> - -<div class='figcenter id004'> -<img src='images/form-120.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>If we further assume that the gravitation-field is -quasi-static, <i>i.e.</i>, it is limited only to the case when the -matter producing the gravitation-field is moving slowly -(relative to the velocity of light) we can neglect the -differentiations of the positional co-ordinates on the right-hand -side with respect to time, so that we get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(67) <i>d²</i><i>x</i><sub>τ</sub>/<i>dt²</i> = -½ ∂<i>g₄₄</i>/∂<i>x</i><sub>τ</sub> (τ, = 1, 2, 3)</div> - </div> - </div> -</div> - -<p class='c010'>This is the equation of motion of a material point -according to Newton’s theory, where <i>g</i>₄₄/₂ plays the part of -gravitational potential. The remarkable thing in the -result is that in the first-approximation of motion of the -material point, only the component <i>g₄₄</i> of the fundamental -tensor appears.</p> - -<p class='c005'>Let us now turn to the field-equation (53). In this -case, we have to remember that the energy-tensor of -matter is exclusively defined in a narrow sense by the -density ρ of matter, <i>i.e.</i>, by the second member on the -right-hand side of 58 [(58a, or 58b)]. If we make the -necessary approximations, then all component vanish -except</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>τ₄₄ = ρ = τ.</div> - </div> - </div> -</div> - -<p class='c010'>On the left-hand side of (53) the second term is an -infinitesimal of the second order, so that the first leads to -the following terms in the approximation, which are rather -interesting for us:</p> - -<div class='figcenter id004'> -<img src='images/form-121.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>By neglecting all differentiations with regard to time, -this leads, when μ = ν =4, to the expression</p> - -<div class='figcenter id004'> -<img src='images/form-122.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>The last of the equations (53) thus leads to</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(68) ▽² <i>g₄₄</i> = κρ.</div> - </div> - </div> -</div> - -<p class='c010'>The equations (67) and (68) together, are equivalent to -Newton’s law of gravitation.</p> - -<p class='c005'>For the gravitation-potential we get from (67) and (68) -the exp.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(68a.) -κ/(8π) ∫ ρ<i>d</i>τ/<i>r</i></div> - </div> - </div> -</div> - -<p class='c010'>whereas the Newtonian theory for the chosen unit of time -gives</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-K/<i>c²</i> ∫ρ<i>d</i>τ/<i>r</i>,</div> - </div> - </div> -</div> - -<p class='c010'>where K denotes usually the -gravitation-constant. 6.7 x 10⁻⁸; equating them we get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(69) κ = 8πK/<i>c²</i> = 1.87 x 10⁻²⁷.</div> - </div> - </div> -</div> - -<h4 class='c009'>§22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature of light-rays. Perihelion-motion of the paths of the Planets.</h4> -<p class='c004'>In order to obtain Newton’s theory as a first approximation -we had to calculate only <i>g₄₄</i>, out of the 10 components -<i>g</i><sub>μν</sub> of the gravitation-potential, for that is the only -component which comes in the first approximate equations -of motion of a material point in a gravitational field.</p> - -<p class='c005'>We see however, that the other components of <i>g</i><sub>μν</sub> -should also differ from the values given in (4) as required by -the condition <i>g</i> = -1.</p> - -<p class='c005'>For a heavy particle at the origin of co-ordinates and -generating the gravitational field, we get as a first approximation -the symmetrical solution of the equation:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line in5'>{ <i>g</i><sub>ρσ</sub> = -δ<sub>ρσ</sub> - α(<i>x</i><sub>ρ</sub> <i>x</i><sub>σ</sub>)/<i>r³</i> (ρ and σ 1, 2, 3)</div> - <div class='line in5'>{</div> - <div class='line'>(70) { <i>g</i><sub>ρ4</sub> = <i>g</i><sub>4ρ</sub> = 0 (ρ 1, 2, 3)</div> - <div class='line in5'>{</div> - <div class='line in5'>{ <i>g₄₄</i> = 1 - α/<i>r</i>.</div> - </div> - </div> -</div> - -<p class='c010'>δ<sub>ρσ</sub> is 1 or 0, according as ρ = σ or not and <i>r</i> is the quantity</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>+√(<i>x₁²</i> + <i>x₂²</i> + <i>x₃²</i>).</div> - </div> - </div> -</div> - -<p class='c010'>On account of (68a) we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(70a) α = κM/4π</div> - </div> - </div> -</div> - -<p class='c010'>where M denotes the mass generating the field. It is easy -to verify that this solution satisfies approximately the -field-equation outside the mass M.</p> - -<p class='c005'>Let us now investigate the influences which the field -of mass M will have upon the metrical properties of the -field. Between the lengths and times measured locally on -the one hand, and the differences in co-ordinates <i>dx</i><sub>ν</sub> on the -other, we have the relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ds²</i> = <i>g</i><sub>μν</sub> <i>dx</i><sub>μ</sub> <i>dx</i><sub>ν</sub>.</div> - </div> - </div> -</div> - -<p class='c010'>For a unit measuring rod, for example, placed parallel to -the <i>x</i> axis, we have to put</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ds²</i> = -1, <i>dx₂</i> = <i>dx₃</i> = <i>dx₄</i> = 0</div> - </div> - <div class='group'> - <div class='line'>then -1 = <i>g</i>₁₁<i>dx₁²</i>.</div> - </div> - </div> -</div> - -<p class='c010'>If the unit measuring rod lies on the <i>x</i> axis, the first of -the equations (70) gives</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>g₁₁</i> = -(1 + α/<i>r</i>).</div> - </div> - </div> -</div> - -<p class='c010'>From both these relations it follows as a first approximation -that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(71) <i>dx</i> = 1 - α/2<i>r</i>.</div> - </div> - </div> -</div> - -<p class='c010'>The unit measuring rod appears, when referred to the -co-ordinate-system, shortened by the calculated magnitude -through the presence of the gravitational field, when we -place it radially in the field.</p> - -<p class='c005'>Similarly we can get its co-ordinate-length in a -tangential position, if we put for example</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ds²</i> = -1, <i>dx₁</i> = <i>dx₃</i> = <i>dx₄</i> = 0, <i>x₁</i> = <i>r</i>, <i>x₂</i> = <i>x₃</i> = 0</div> - </div> - </div> -</div> - -<p class='c010'>we then get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(71a) -1 = <i>g₂₂</i> <i>dx₂²</i> = -<i>dx₂²</i>.</div> - </div> - </div> -</div> - -<p class='c010'>The gravitational field has no influence upon the length -of the rod, when we put it tangentially in the field.</p> - -<p class='c005'>Thus Euclidean geometry does not hold in the gravitational -field even in the first approximation, if we conceive -that one and the same rod independent of its position and -its orientation can serve as the measure of the same -extension. But a glance at (70a) and (69) shows that the -expected difference is much too small to be noticeable -in the measurement of earth’s surface.</p> - -<p class='c005'>We would further investigate the rate of going of a -unit-clock which is placed in a statical gravitational field. -Here we have for a period of the clock</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>ds</i> = 1, <i>dx₁</i> = <i>dx₂</i> <i>dx₃</i> = 0;</div> - </div> - </div> -</div> - -<p class='c010'>then we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>1 = <i>g₄₄</i><i>dx₄²</i></div> - </div> - <div class='group'> - <div class='line'><i>dx₄</i> = 1/√(<i>g</i>₄₄) = 1/√(1 + (<i>g</i>₄₄ - 1)) = 1 - (<i>g</i>₄₄ - 1)/2</div> - </div> - <div class='group'> - <div class='line'>or <i>dx₄</i> = 1 + <i>k</i>/8π ∫ ρ<i>d</i>τ/<i>r</i>.</div> - </div> - </div> -</div> - -<p class='c010'>Therefore the clock goes slowly what it is placed in -the neighbourhood of ponderable masses. It follows from -this that the spectral lines in the light coming to us from -the surfaces of big stars should appear shifted towards the -red end of the spectrum.</p> - -<p class='c005'>Let us further investigate the path of light-rays in a -statical gravitational field. According to the special relativity -theory, the velocity of light is given by the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-<i>dx₁²</i> - <i>dx₂²</i> - <i>dx₃²</i> + <i>dx₄²</i> = 0;</div> - </div> - </div> -</div> - -<p class='c010'>thus also according to the generalised relativity theory it -is given by the equation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(73) <i>ds²</i> = <i>g</i><sub>μν</sub> <i>dx</i><sub>μ</sub> <i>dx</i><sub>ν</sub> = 0.</div> - </div> - </div> -</div> - -<p class='c010'>If the direction, <i>i.e.</i>, the ratio <i>dx₁</i> : <i>dx₂</i> : <i>dx₃</i> is given, -the equation (73) gives the magnitudes</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>dx₁</i>/<i>dx₄</i>, <i>dx₂</i>/<i>dx₄</i>, <i>dx₃</i>/<i>dx₄</i>,</div> - </div> - </div> -</div> - -<p class='c010'>and with it the velocity,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>√((<i>dx₁</i>/<i>dx₄</i>)² + (<i>dx₂</i>/<i>dx₄</i>)² + (<i>dx₃</i>/<i>dx₄</i>)²) = γ,</div> - </div> - </div> -</div> - -<p class='c010'>in the sense of the Euclidean Geometry. We can easily see -that, with reference to the co-ordinate system, the rays of -light must appear curved in case <i>g</i><sub>μν</sub>’s are not constants. -If <i>n</i> be the direction perpendicular to the direction -of propagation, we have, from Huygen’s principle, that -light-rays (taken in the plane (γ, <i>n</i>)] must suffer a -curvature ∂λ/∂<i>n</i>.</p> - -<p class='c005'>Let us find out the curvature which a light-ray suffers -when it goes by a mass M at a distance Δ from it. If we -use the co-ordinate system according to the above scheme, -then the total bending B of light-rays (reckoned positive -when it is concave to the origin) is given as a sufficient -approximation by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>B = ∫<sub>-∞</sub><sup>∞</sup> ∂γ/∂[<i>x</i>]₁ <i>dx₂</i></div> - </div> - </div> -</div> - -<p class='c010'>where (73) and (70) gives</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>γ = √(-<i>g₄₄</i>/<i>g₂₂</i>) = 1 - α/2<i>r</i> (1 + <i>x₂²</i>/<i>r²</i>).</div> - </div> - </div> -</div> - -<p class='c010'>The calculation gives</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>B = 2α/Δ = KM/2πΔ.</div> - </div> - </div> -</div> - -<p class='c010'>A ray of light just grazing the sun would suffer a bending -of 1·7″, whereas one coming by Jupiter would have -a deviation of about ·02″.</p> - -<p class='c005'>If we calculate the gravitation-field to a greater order -of approximation and with it the corresponding path -of a material particle of a relatively small (infinitesimal) -mass we get a deviation of the following kind from the -Kepler-Newtonian Laws of Planetary motion. The Ellipse -of Planetary motion suffers a slow rotation in the direction -of motion, of amount</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(75) <i>s</i> = 24π³<i>a²</i>/τ²<i>c²</i>(1 - <i>e²</i>) per revolution.</div> - </div> - </div> -</div> - -<p class='c010'>In this Formula ‘<i>a</i>’ signifies the semi-major axis, <i>c</i>, -the velocity of light, measured in the usual way, <i>e</i>, the -eccentricity, τ, the time of revolution in seconds.</p> - -<p class='c005'>The calculation gives for the planet Mercury, a rotation -of path of amount 43″ per century, corresponding sufficiently -to what has been found by astronomers (Leverrier). -They found a residual perihelion motion of this planet of -the given magnitude which can not be explained by the -perturbation of the other planets.</p> -<div class='chapter'> - <h2 id='notes' class='c003'>NOTES</h2> -</div> -<h3 id='note-1' class='c008'>Note 1.</h3> -<p class='c004'>The fundamental electro-magnetic equations of Maxwell for stationary media are:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>curl H = 1/<i>c</i> (∂D/∂<i>t</i> + ρν) (1)</div> - </div> - <div class='group'> - <div class='line'>curl E = -1/<i>c</i> ∂B/∂<i>t</i> (2)</div> - </div> - <div class='group'> - <div class='line'>div D = ρ</div> - <div class='line'>B = μH</div> - <div class='line'>div B = 0</div> - <div class='line'>D = kE</div> - </div> - </div> -</div> - -<p class='c010'>According to Hertz and Heaviside, these require modification -in the case of moving bodies.</p> - -<p class='c005'>Now it is known that due to motion alone there is a -change in a vector <i>R</i> given by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(∂<i>R</i>/∂<i>t</i>) due to motion = <i>u</i>. div R + curl [<i>Ru</i>]</div> - </div> - </div> -</div> - -<p class='c010'>where <i>u</i> is the vector velocity of the moving body and -[R<i>u</i>] the vector product of R and <i>u</i>.</p> - -<p class='c005'>Hence equations (1) and (2) become</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>c</i> curl H = ∂D/∂<i>t</i> + <i>u</i> div D + curl Vect. [D<i>u</i>] + ρν (1·1)</div> - </div> - </div> -</div> - -<p class='c010'>and</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>-<i>c</i> curl E = ∂B/∂<i>t</i> + <i>u</i> div B + curl Vect. [B<i>u</i>] (2·1)</div> - </div> - </div> -</div> - -<p class='c010'>which gives finally, for ρ = 0 and div B = 0,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂D/∂<i>t</i> + <i>u</i> div D = <i>c</i> curl (H - 1/<i>c</i> Vect. [D<i>u</i>]) (1·2)</div> - </div> - <div class='group'> - <div class='line'>∂B/∂<i>t</i> = -<i>c</i> curl (E - 1/<i>c</i> Vect. [<i>u</i>B]) (2·2)</div> - </div> - </div> -</div> - -<p class='c010'>Let us consider a beam travelling along the <i>x</i>-axis, -with apparent velocity <i>v</i> (<i>i.e.</i>, velocity with respect to the -fixed ether) in medium moving with velocity <i>u</i><sub><i>x</i></sub> = <i>u</i> in the -same direction.</p> - -<p class='c005'>Then if the electric and magnetic vectors are -proportional to <i>e</i><sup><i>i</i>A(<i>x</i> - <i>vt</i>)</sup>, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂/∂<i>x</i> = <i>i</i>A, ∂/∂<i>t</i> = -<i>i</i>A<i>v</i>, ∂/∂<i>y</i> = ∂/∂<i>z</i> = 0, <i>u</i><sub><i>y</i></sub> = <i>u</i><sub><i>z</i></sub> = 0</div> - </div> - <div class='group'> - <div class='line'>Then ∂D_<i>y</i>/∂<i>t</i> = -<i>c</i>∂H<sub><i>z</i></sub>/∂<i>x</i> - <i>u</i>∂D<sub><i>y</i></sub>/∂<i>z</i> ... (1·21)</div> - </div> - <div class='group'> - <div class='line'>and ∂B<sub><i>z</i></sub>/∂<i>t</i> = -<i>c</i>∂E<sub><i>y</i></sub>/∂<i>x</i> - <i>u</i>∂B<sub><i>z</i></sub>/∂<i>x</i> (2·21)</div> - </div> - </div> -</div> - -<p class='c010'>Since D = KE and B = μH, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>i</i>A<i>v</i>(κE<i>y</i>) = -<i>ci</i>A(H<sub><i>z</i></sub> + <i>u</i>KE<sub><i>y</i></sub>) (1·22)</div> - </div> - <div class='group'> - <div class='line'><i>i</i>A<i>v</i>(μH<sub><i>z</i></sub>) = -<i>ci</i>A(E<sub><i>y</i></sub> + <i>u</i>μH<sub><i>z</i></sub>) (2·22)</div> - </div> - <div class='group'> - <div class='line'>or <i>v</i>(K - <i>u</i>)E<sub><i>y</i></sub> = <i>c</i>H<sub><i>z</i></sub> (1·23)</div> - </div> - <div class='group'> - <div class='line'>μ(<i>v</i> - <i>u</i>)H<sub><i>z</i></sub> = <i>c</i>E<sub><i>y</i></sub> (2·23)</div> - </div> - </div> -</div> - -<p class='c010'>Multiplying (1·23) by (2·23)</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>μK(<i>v</i> - <i>u</i>)² = <i>c²</i></div> - </div> - </div> -</div> - -<p class='c010'>Hence (<i>v</i> - <i>u</i>)² = <i>c²</i>/μ<i>k</i> = <i>v₀</i>²</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∴ <i>v</i> = <i>v₀</i> + <i>u</i>,</div> - </div> - </div> -</div> - -<p class='c010'>making Fresnelian convection co-efficient simply unity.</p> - -<p class='c005'>Equations (1·21) and (2·21) may be obtained more -simply from physical considerations.</p> - -<p class='c005'>According to Heaviside and Hertz, the real seat of -both electric and magnetic polarisation is the moving -medium itself. Now at a point which is fixed with respect -to the ether, the rate of change of electric polarisation is -δD/δ<i>t</i>.</p> - -<p class='c005'>Consider a slab of matter moving with velocity <i>u</i><sub><i>x</i></sub> -along the <i>x</i>-axis, then even in a stationary field of -electrostatic polarisation, that is, for a field in which -δD/δ<i>t</i> = 0, there will be some change in the polarisation of -the body due to its motion, given by <i>u</i><sub><i>x</i></sub>(δD/δ<i>x</i>). Hence we -must add this term to a purely temporal rate of change -δD/δ<i>t</i>. Doing this we immediately arrive at equations -(1·21) and (2·21) for the special case considered there.</p> - -<p class='c005'>Thus the Hertz-Heaviside form of field equations gives -<i>unity</i> as the value for the Fresnelian convection co-efficient. -It has been shown in the historical introduction how this -is entirely at variance with the observed optical facts. As -a matter of fact, Larmor has shown (Aether and Matter) -that 1 - 1/μ² is not only sufficient but is also necessary, in -order to explain experiments of the Arago prism type.</p> - -<p class='c005'>A short summary of the electromagnetic experiments -bearing on this question, has already been given in the -introduction.</p> - -<p class='c005'>According to Hertz and Heaviside the total polarisation -is situated in the medium itself and is completely -carried away by it. Thus the electromagnetic effect -outside a moving medium should be proportional to K, the -specific inductive capacity.</p> - -<p class='c005'><i>Rowland</i> showed in 1876 that when a charged condenser -is rapidly rotated (the dielectric remaining stationary), -the magnetic effect outside is proportional to K, the Sp. -Ind. Cap.</p> - -<p class='c005'><i>Röntgen</i> (Annalen der Physik 1888, 1890) found that -if the dielectric is rotated while the condenser remains -stationary, the effect is proportional to K - 1.</p> - -<p class='c005'><i>Eichenwald</i> (Annalen der Physik 1903, 1904) rotated -together both condenser and dielectric and found that the -magnetic effect was proportional to the potential difference -and to the angular velocity, but was completely independent -of K. This is of course quite consistent with Rowland -and Röntgen.</p> - -<p class='c005'><i>Blondlot</i> (Comptes Rendus, 1901) passed a current -of air in a steady magnetic field H<sub><i>y</i></sub>, (H = H<sub><i>z</i></sub> = 0). If -this current of air moves with velocity <i>u</i><sub><i>x</i></sub> along the -<i>x</i>-axis, an electromotive force would be set up along the -<i>z</i>-axis, due to the relative motion of matter and magnetic -tubes of induction. A pair of plates at <i>z</i> = ±<i>a</i>, will be -charged up with density ρ = D<sub><i>z</i></sub> = KE = K. <i>u</i><sub><i>s</i></sub> H<sub><i>y</i></sub>/c. -But Blondlot failed to detect any such effect.</p> - -<p class='c005'><i>H. A. Wilson</i> (Phil. Trans. Royal Soc. 1904) repeated -the experiment with a cylindrical condenser made of -ebony, rotating in a magnetic field parallel to its own -axis. He observed a change proportional to K — 1 and -not to K.</p> - -<p class='c005'>Thus the above set of electro-magnetic experiments -contradict the Hertz-Heaviside equations, and these must -be abandoned.</p> - -<p class='c005'>[P. C. M.]</p> -<h3 id='note-2' class='c008'>Note 2. <br /> Lorentz Transformation.</h3> -<p class='c004'>Lorentz. Versuch einer theorie der elektrischen und -optischen Erscheinungen im bewegten Körpern.</p> - -<p class='c005'>(Leiden—1895).</p> - -<p class='c005'>Lorentz. Theory of Electrons (English edition), -pages 197-200, 230, also notes 73, 86, pages 318, 328.</p> - -<p class='c005'>Lorentz wanted to explain the Michelson-Morley -null-effect. In order to do so, it was obviously necessary -to explain the Fitzgerald contraction. Lorentz worked -on the hypothesis that an electron itself undergoes -contraction when moving. He introduced new variables -for the moving system defined by the following set of -equations.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>x¹</i> = β(<i>x</i> - <i>ut</i>), <i>y¹</i> = <i>y</i>, <i>z¹</i> = <i>z</i>, <i>t¹</i> = β(<i>t</i> - (<i>u</i>/<i>c²</i>)·<i>x</i>)</div> - </div> - </div> -</div> - -<p class='c010'>and for velocities, used</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>v</i><sub><i>x</i></sub>¹ = β²<i>v</i><sub><i>x</i></sub> + <i>u</i>, <i>v</i><sub><i>y</i></sub>¹ = β<i>v</i><sub><i>y</i></sub>, <i>v</i><sub><i>z</i></sub>¹ = β<i>v</i><sub><i>z</i></sub> and ρ¹ = ρ/β.</div> - </div> - </div> -</div> - -<p class='c010'>With the help of the above set of equations, which is -known as the Lorentz transformation, he succeeded in -showing how the Fitzgerald contraction results as a -consequence of “fortuitous compensation of opposing -effects.”</p> - -<p class='c005'>It should be observed that the Lorentz transformation -is not identical with the Einstein transformation. The -Einsteinian addition of velocities is quite different as -also the expression for the “relative” density of electricity.</p> - -<p class='c005'>It is true that the Maxwell-Lorentz field equations -remain <i>practically</i> unchanged by the Lorentz transformation, -but they <i>are</i> changed to some slight extent. One -marked advantage of the Einstein transformation consists -in the fact that the field equations of a moving system -preserve <i>exactly</i> the same form as those of a stationary -system.</p> - -<p class='c005'>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 matter.</p> - -<p class='c005'>[P. C. M.]</p> -<h3 id='note-3' class='c008'>Note 3.</h3> -<p class='c004'>See Lorentz, Theory of Electrons (English edition), -§ 181, page 213.</p> - -<p class='c005'>H. Poincare, Sur la dynamique ‘electron, Rendiconti -del circolo matematico di Palermo 21 (1906).</p> - -<p class='c005'>[P. C. M.]</p> -<h3 id='note-4' class='c008'>Note 4. <br />Relativity Theorem and Relativity-Principle.</h3> -<p class='c004'>Lorentz showed that the Maxwell-Lorentz system -of electromagnetic field-equations remained practically -unchanged by the Lorentz transformation. Thus the -electromagnetic laws of Maxwell and Lorentz <i>can be -definitely proved</i> “to be independent of the manner in -which they are referred to two coordinate systems which -have a uniform translatory motion relative to each other.” -(See “Electrodynamics of Moving Bodies,” page 5.) Thus -so far as the electromagnetic laws are concerned, the -principle of relativity <i>can be proved to be true</i>.</p> - -<p class='c005'>But it is not known whether this principle will remain -true in the case of other physical laws. We can always -proceed on the assumption that it does remain true. Thus -it is always possible to construct physical laws in such a -way that they retain their form when referred to moving -coordinates. The ultimate ground for formulating physical -laws in this way is merely a subjective conviction that -the principle of relativity is universally true. There is -no <i>a priori</i> logical necessity that it should be so. Hence -the Principle of Relativity (so far as it is applied to -phenomena other than electromagnetic) must be regarded -as a <i>postulate</i>, which we have assumed to be true, but for -which we cannot adduce any definite proof, until after -the generalisation is made and its consequences tested in -the light of actual experience.</p> - -<p class='c005'>[P. C. M.]</p> -<h3 id='note-5' class='c008'>Note 5.</h3> -<p class='c004'>See “Electrodynamics of Moving Bodies,” p. 5-8.</p> -<h3 class='c008'>Note 6. <br />Field Equations in Minkowski’s Form.</h3> -<p class='c004'>Equations (<i>i</i>) and (<i>ii</i>) become when expanded into -Cartesians:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>m</i><sub><i>z</i></sub>/∂<i>y</i> - ∂<i>m</i><sub><i>y</i></sub>/∂<i>z</i> - ∂<i>e</i><sub><i>x</i></sub>/∂τ = ρν<sub><i>x</i></sub> }</div> - <div class='line'>∂<i>m</i><sub><i>x</i></sub>/∂<i>z</i> - ∂<i>m</i><sub><i>z</i></sub>/∂<i>x</i> - ∂<i>e</i><sub><i>y</i></sub>/∂τ = ρν<sub><i>y</i></sub> } ... (1·1)</div> - <div class='line'>∂<i>m</i><sub><i>y</i></sub>/∂<i>x</i> - ∂<i>m</i><sub><i>x</i></sub>/∂<i>y</i> - ∂<i>e</i><sub><i>z</i></sub>/∂τ = ρν<sub><i>z</i></sub> }</div> - </div> - </div> -</div> - -<p class='c010'>and ∂<i>e</i><sub><i>x</i></sub>/∂<i>x</i> + ∂<i>e</i><sub><i>y</i></sub>/∂<i>y</i> + ∂<i>e</i><sub><i>z</i></sub>/∂<i>z</i> = ρ (2·1)</p> - -<p class='c005'>Substituting <i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i> and <i>x</i>, <i>y</i>, <i>z</i>, and <i>i</i>τ; and ρ₁, -ρ₂, ρ₃, ρ₄ for ρν<sub><i>x</i></sub>, ρν<sub><i>y</i></sub>, ρν<sub><i>z</i></sub>, <i>i</i>ρ, where <i>i</i> = √(-1).</p> - -<p class='c005'>We get,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>m</i><sub><i>z</i></sub>/∂<i>x₂</i> - ∂<i>m</i><sub><i>y</i></sub>/∂<i>x₃</i> - <i>i</i>(∂<i>e</i><sub><i>x</i></sub>/∂<i>x₄</i>) = ρν<sub><i>x</i></sub>{ = ρ₁ }</div> - <div class='line'>- ∂<i>m</i><sub><i>z</i></sub>/∂<i>x₁</i> + ∂<i>m</i><sub><i>x</i></sub>/∂<i>x₃</i> - <i>i</i>(∂<i>e</i><sub><i>y</i></sub>/∂<i>x₄</i>) = ρν<sub><i>y</i></sub> = ρ₂ } ... (1·2)</div> - <div class='line'>∂<i>m</i><sub><i>y</i></sub>/∂<i>x₁</i> - ∂<i>m</i><sub><i>x</i></sub>/∂<i>x₂</i> - <i>i</i>(∂<i>e</i><sub><i>z</i></sub>/∂<i>x₄</i>) = ρν<sub><i>z</i></sub>{ = ρ₃ }</div> - </div> - </div> -</div> - -<p class='c010'>and multiplying (2·1) by i we get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>ie</i><sub><i>x</i></sub>/∂<i>x₁</i> + ∂<i>ie</i><sub><i>y</i></sub>/∂<i>x₂</i> + ∂<i>ie</i><sub><i>z</i></sub>/∂<i>x₃</i> = <i>i</i>ρ = ρ₄ ... ... (2·2)</div> - </div> - </div> -</div> - -<p class='c010'>Now substitute</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>m</i><sub><i>x</i></sub> = <i>f₂₃</i> = -<i>f₃₂</i> and <i>ie</i><sub><i>x</i></sub> = <i>f</i>₄₁ = -<i>f₁₄</i></div> - <div class='line'><i>m</i><sub><i>y</i></sub> = <i>f₃₁</i> = -<i>f₁₃</i> <i>ie</i><sub><i>y</i></sub> = <i>f</i>₄₂ = -<i>f₂₄</i></div> - <div class='line'><i>m</i><sub><i>z</i></sub> = <i>f₁₂</i> = -<i>f₂₁</i> <i>ie</i><sub><i>z</i></sub> = <i>f</i>₄₃ = -<i>f₃₄</i></div> - </div> - </div> -</div> - -<p class='c010'>and we get finally:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>∂<i>f₁₂</i>/∂<i>x₂</i> + ∂<i>f₁₃</i>/∂<i>x₃</i> + ∂<i>f₁₄</i>/∂<i>x₄</i> = ρ₁ }</div> - </div> - <div class='group'> - <div class='line'>∂<i>f₂₁</i>/∂<i>x₁</i> + ∂<i>f₂₃</i>/∂<i>x₃</i> + ∂<i>f₂₄</i>/∂<i>x₄</i> = ρ₂ } ... (3)</div> - </div> - <div class='group'> - <div class='line'>∂<i>f₃₁</i>/∂<i>x₁</i> + ∂<i>f₃₂</i>/∂<i>x₂</i> + ∂<i>f₃₄</i>/∂<i>x₄</i> = ρ₃ }</div> - </div> - <div class='group'> - <div class='line'>∂<i>f₄₁</i>/∂<i>x₁</i> + ∂<i>f₄₂</i>/∂<i>x₂</i> + ∂<i>f₄₃</i>/∂<i>x₃</i> = ρ₄ }</div> - </div> - </div> -</div> - -<h3 id='note-9' class='c008'>Note 9. <br />On the Constancy of the Velocity of Light.</h3> -<p class='c004'>Page 12—refer also to page 6, of Einstein’s paper.</p> - -<p class='c005'>One of the two fundamental Postulates of the Principle -of Relativity is that the velocity of light should remain -constant whether the source is moving or stationary. It -follows that even if a radiant source S move with a velocity -<i>u</i>, it should always remain the centre of spherical waves -expanding outwards with velocity <i>c</i>.</p> - -<p class='c005'>At first sight, it may not appear clear why the -velocity should remain constant. Indeed according to the -theory of Ritz, the velocity should become <i>c</i> + <i>u</i>, when the -source of light moves towards the observer with the -velocity <i>u</i>.</p> - -<p class='c005'>Prof. de Sitter has given an astronomical argument for -deciding between these two divergent views. Let us -suppose there is a double star of which one is revolving -about the common centre of gravity in a circular orbit. -Let the observer be in the plane of the orbit, at a great -distance Δ.</p> - -<div class='figcenter id003'> -<img src='images/fig02.png' alt='Experiment.' class='ig001' /> -</div> - -<p class='c005'>The light emitted by the star when at the position A -will be received by the observer after a time, Δ/(<i>c</i> + <i>u</i>) while -the light emitted by the star when at the position B will -be received after a time Δ/(<i>c</i> - <i>u</i>). Let T be the real half-period -of the star. Then the observed half-period from -B to A is approximately T - 2Δ<i>u</i>/<i>c²</i> and from A to B is -T + 2Δ<i>u</i>/<i>c²</i>. Now if 2<i>u</i>Δ/<i>c²</i> be comparable to T, then it -is impossible that the observations should satisfy -Kepler’s Law. In most of the spectroscopic binary stars, -2<i>u</i>Δ/<i>c²</i> are not only of the same order as T, but are mostly -much larger. For example, if <i>u</i> = 100 <i>km</i>/sec, T = 8 days, -Δ/<i>c</i> = 33 years (corresponding to an annual parallax of ·1″), -then T - 2<i>u</i>Δ/<i>c²</i> = 0. The existence of the Spectroscopic -binaries, and the fact that they follow Kepler’s Law is -therefore a proof that <i>c</i> is not affected by the motion of -the source.</p> - -<p class='c005'>In a later memoir, replying to the criticisms of -Freundlich and Günthick that an apparent eccentricity -occurs in the motion proportional to <i>ku</i>Δ₀, <i>u₀</i> being the -maximum value of <i>u</i>, the velocity of light emitted being</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>u₀</i> = <i>c</i> + <i>ku</i>,</div> - <div class='line'><i>k</i> = 0 Lorentz-Einstein</div> - <div class='line'><i>k</i> = 1 Ritz.</div> - </div> - </div> -</div> - -<p class='c010'>Prof. de Sitter admits the validity of the criticisms. But -he remarks that an upper value of <i>k</i> may be calculated from -the observations of the double star β-Aurigae. For this star, -the parallax π = ·014″, <i>e</i> = ·005, <i>u₀</i> = 110 <i>km</i>/sec, T = 3·96,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Δ > 65 light-years,</div> - <div class='line'><i>k</i> is < ·002.</div> - </div> - </div> -</div> - -<p class='c010'>For an experimental proof, see a paper by C. Majorana. -Phil. Mag., Vol. 35, p. 163.</p> - -<p class='c005'>[M. N. S.]</p> -<h3 class='c008'>Note 10. <br />Rest-density of Electricity.</h3> -<p class='c004'>If ρ is the volume density in a moving system then -ρ√(1 - <i>u²</i>) is the corresponding quantity in the corresponding -volume in the fixed system, that is, in the system at -rest, and hence it is termed the rest-density of electricity.</p> - -<p class='c005'>[P. C. M.]</p> -<h3 class='c008'>Note 11 <br /> (page 17) <br /> Space-time vectors of the first and the second kind.</h3> -<p class='c004'>As we had already occasion to mention, Sommerfeld -has, in two papers on four dimensional geometry (<i>vide</i>, -Annalen der Physik, Bd. 32, p. 749; and Bd. 33, p. 649), -translated the ideas of Minkowski into the language of four -dimensional geometry. Instead of Minkowski’s space-time -vector of the first kind, he uses the more expressive term -‘four-vector,’ thereby making it quite clear that it -represents a directed quantity like a straight line, a force -or a momentum, and has got 4 components, three in the -direction of space-axes, and one in the direction of the -time-axis.</p> - -<p class='c005'>The representation of the plane (defined by two straight -lines) is much more difficult. In three dimensions, the -plane can be represented by the vector perpendicular to -itself. But that artifice is not available in four dimensions. -For the perpendicular to a plane, we now have not a single -line, but an infinite number of lines constituting a plane. -This difficulty has been overcome by Minkowski in a very -elegant manner which will become clear later on. -Meanwhile we offer the following extract from the -above mentioned work of Sommerfeld.</p> - -<p class='c005'>(Pp. 755, Bd. 32, Ann. d. Physik.)</p> - -<p class='c005'>“In order to have a better knowledge about the nature -of the six-vector (which is the same thing as Minkowski’s -space-time vector of the <i>2nd</i> kind) let us take the special -case of a piece of plane, having unit area (contents), and -the form of a parallelogram, bounded by the four-vectors -<i>u</i>, <i>v</i>, passing through the origin. Then the projection of -this piece of plane on the <i>xy</i> plane is given by the -projections <i>u</i><sub><i>x</i></sub>, <i>u</i><sub><i>y</i></sub>, <i>v</i><sub><i>x</i></sub>, <i>v</i><sub><i>y</i></sub> of the four vectors in the -combination</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ<sub><i>x</i> <i>y</i></sub> = <i>u</i><sub><i>x</i></sub><i>v</i><sub><i>y</i></sub> - <i>u</i><sub><i>y</i></sub><i>v</i>{<i>x</i>}.</div> - </div> - </div> -</div> - -<p class='c010'>Let us form in a similar manner all the six components of -this plane φ. Then six components are not all independent -but are connected by the following relation</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ<sub><i>y</i> <i>z</i></sub> φ<sub><i>x</i> <i>l</i></sub> + φ<sub><i>z</i> <i>x</i></sub> φ<sub><i>y</i> <i>l</i></sub> + φ<sub><i>x</i> <i>y</i></sub> φ<sub><i>z</i> <i>l</i></sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>Further the contents | φ | of the piece of a plane is to -be defined as the square root of the sum of the squares of -these six quantities. In fact,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| φ |² = φ<sub><i>y</i> <i>z</i></sub>² + φ<sub><i>z</i> <i>x</i></sub>² + φ<sub><i>x</i> <i>y</i></sub>² + φ<sub><i>x</i> <i>l</i></sub>² + φ<sub><i>y</i> <i>l</i></sub>² + φ<sub><i>z</i> <i>l</i></sub>².</div> - </div> - </div> -</div> - -<p class='c010'>Let us now on the other hand take the case of the unit -plane φ<sup>*</sup> normal to φ; we can call this plane the -Complement of φ. Then we have the following relations -between the components of the two plane:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ<sub><i>y</i> <i>z</i></sub><sup>*</sup> = φ<sub><i>x</i> <i>l</i></sub>, φ<sub><i>z</i> <i>x</i></sub><sup>*</sup> = φ<sub><i>y</i> <i>l</i></sub>, φ<sub><i>x</i> <i>y</i></sub><sup>*</sup> = φ<sub><i>z</i> <i>l</i></sub> φ<sub><i>z</i> <i>l</i></sub><sup>*</sup> = φ<sub><i>y</i> <i>x</i></sub> ...</div> - </div> - </div> -</div> - -<p class='c010'>The proof of these assertions is as follows. Let <i>u</i><sup>*</sup>, <i>v</i><sup>*</sup> -be the four vectors defining φ<sup>*</sup>. Then we have the -following relations:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>u</i><sub><i>x</i></sub><sup>*</sup> <i>u</i><sub><i>x</i></sub> + <i>u</i><sub><i>y</i></sub><sup>*</sup> <i>u</i><sub><i>y</i></sub> + <i>u</i><sub><i>z</i></sub><sup>*</sup> <i>u</i><sub><i>z</i></sub> + <i>u</i><sub><i>l</i></sub><sup>*</sup> <i>u</i><sub><i>l</i></sub> = 0</div> - </div> - <div class='group'> - <div class='line'><i>u</i><sub><i>x</i></sub><sup>*</sup> <i>v</i><sub><i>x</i></sub> + <i>u</i><sub><i>y</i></sub><sup>*</sup> <i>v</i><sub><i>y</i></sub> + <i>u</i><sub><i>z</i></sub><sup>*</sup> <i>v</i><sub><i>z</i></sub> + <i>u</i><sub><i>l</i></sub><sup>*</sup> <i>v</i><sub><i>l</i></sub> = 0</div> - </div> - <div class='group'> - <div class='line'><i>v</i><sub><i>x</i></sub><sup>*</sup> <i>u</i><sub><i>x</i></sub> + <i>v</i><sub><i>y</i></sub><sup>*</sup> <i>u</i><sub><i>y</i></sub> + <i>v</i><sub><i>z</i></sub><sup>*</sup> <i>u</i><sub><i>z</i></sub> + <i>v</i><sub><i>l</i></sub><sup>*</sup> <i>u</i><sub><i>l</i></sub> = 0</div> - </div> - <div class='group'> - <div class='line'><i>v</i><sub><i>x</i></sub><sup>*</sup> <i>v</i><sub><i>x</i></sub> + <i>v</i><sub><i>y</i></sub><sup>*</sup> <i>v</i><sub><i>y</i></sub> + <i>v</i><sub><i>z</i></sub><sup>*</sup> <i>v</i><sub><i>z</i></sub> + <i>v</i><sub><i>l</i></sub><sup>*</sup> <i>v</i><sub><i>l</i></sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>If we multiply these equations by <i>v</i><sub><i>l</i></sub>, <i>u</i><sub><i>l</i></sub>, <i>v</i><sub><i>s</i></sub>, and -subtract the second from the first, the fourth from the -third we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>u</i><sub><i>x</i></sub><sup>*</sup> φ<sub><i>x</i> <i>l</i></sub> + <i>u</i><sub><i>y</i></sub><sup>*</sup> φ<sub><i>y</i> <i>l</i></sub> + <i>u</i><sub><i>z</i></sub><sup>*</sup> φ<sub><i>z</i> <i>l</i></sub> = 0</div> - </div> - <div class='group'> - <div class='line'><i>v</i><sub><i>x</i></sub><sup>*</sup> φ<sub><i>z</i> <i>l</i></sub> + <i>v</i><sub><i>y</i></sub><sup>*</sup> φ<sub><i>y</i> <i>l</i></sub> + <i>v</i><sub><i>z</i></sub><sup>*</sup> φ<sub><i>z</i> <i>l</i></sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>multiplying these equations by <i>v</i><sub><i>x</i></sub><sup>*</sup> . <i>u</i><sub><i>x</i></sub><sup>*</sup>, or by <i>v</i><sub><i>y</i></sub><sup>*</sup> . <i>u</i><sub><i>y</i></sub><sup>*</sup>, -we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ<sub><i>x</i> <i>z</i></sub><sup>*</sup> φ<sub><i>x</i> <i>l</i></sub> + φ<sub><i>y</i> <i>z</i></sub><sup>*</sup> φ<sub><i>y</i> <i>l</i></sub> = 0 and φ<sub><i>x</i> <i>y</i></sub><sup>*</sup> φ<sub><i>x</i> <i>l</i></sub> + φ<sub><i>z</i> <i>x</i></sub><sup>*</sup> φ<sub><i>z</i> <i>l</i></sub> = 0</div> - </div> - </div> -</div> - -<p class='c010'>from which we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ<sub><i>y</i> <i>z</i></sub><sup>*</sup> : φ<sub><i>x</i> <i>y</i></sub><sup>*</sup> : φ<sub><i>z</i> <i>x</i></sub><sup>*</sup> = φ<sub><i>x</i> <i>l</i></sub> : φ<sub><i>z</i> <i>l</i></sub> : φ<sub><i>y</i> <i>l</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>In a corresponding way we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>φ<sub><i>y</i> <i>z</i></sub> : φ<sub><i>x</i> <i>y</i></sub> : φ<sub><i>z</i> <i>x</i></sub> = φ<sub><i>x</i> <i>l</i></sub><sup>*</sup> : φ<sub><i>z</i> <i>l</i></sub><sup>*</sup> : φ<sub><i>y</i> <i>l</i></sub><sup>*</sup>.</div> - </div> - </div> -</div> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>i.e.</i> φ<sub><i>i</i> <i>k</i></sub><sup>*</sup> = λφ(<sub><i>i</i> <i>k</i></sub>)</div> - </div> - </div> -</div> - -<p class='c010'>when the subscript (<i>ik</i>) denotes the component of φ in -the plane contained by the lines other than (<i>ik</i>). Therefore -the theorem is proved.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>We have (φ φ*) = φ<sub><i>y</i> <i>z</i></sub> φ<sub><i>y</i> <i>z</i></sub><sup>*</sup> + ...</div> - </div> - <div class='group'> - <div class='line'>= 2 (φ<sub><i>y</i> <i>z</i></sub> φ<sub><i>z</i> <i>l</i></sub> + ...)</div> - </div> - <div class='group'> - <div class='line'>= 0</div> - </div> - </div> -</div> - -<p class='c010'>The general six-vector <i>f</i> is composed from the vectors -φ, φ<sup>*</sup> in the following way:—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i> = ρφ + ρ<sup>*</sup> φ<sup>*</sup>,</div> - </div> - </div> -</div> - -<p class='c010'>ρ and ρ<sup>*</sup> denoting the contents of the pieces of mutually -perpendicular planes composing <i>f</i>. The “conjugate -Vector” <i>f</i><sup>*</sup> (or it may be called the complement of <i>f</i>) is -obtained by interchanging ρ and ρ<sup>*</sup>.</p> - -<p class='c005'>We have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i><sup>*</sup> = ρ<sup>*</sup>φ + ρφ<sup>*</sup></div> - </div> - </div> -</div> - -<p class='c010'>We can verify that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i><sub><i>y z</i></sub><sup>*</sup> = <i>f</i><sub><i>x l</i></sub> etc.</div> - </div> - </div> -</div> - -<p class='c010'>and <i>f²</i> = ρ² + ρ<sup>*</sup>², (<i>f</i><i>f</i><sup>*</sup>) = 2ρρ<sup>*</sup>.</p> - -<p class='c005'>| <i>f</i> |² and (<i>f</i><i>f</i><sup>*</sup>) may be said to be invariants of the six -vectors, for their values are independent of the choice of -the system of co-ordinates.</p> - -<p class='c005'>[M. N. S.]</p> -<h3 id='note-12' class='c008'>Note 12. <br />Light-velocity as a maximum.</h3> -<p class='c004'>Page 23, and Electro-dynamics of Moving Bodies, -p. 17.</p> - -<p class='c005'>Putting <i>v</i> = <i>c</i> - <i>x</i>, and <i>w</i> = <i>c</i> - λ, we get</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>V = (2<i>c</i> - (<i>x</i> + λ))/(1 + (<i>c</i> - <i>x</i>)(<i>c</i> - λ)/<i>c²</i>) = (2<i>c</i> - (<i>x</i> + λ))/(<i>c²</i> + <i>c²</i> - (<i>x</i> + λ)<i>c</i> + <i>x</i>λ/<i>c²</i>)</div> - </div> - <div class='group'> - <div class='line in6'>= <i>c</i> (2<i>c</i> - (<i>x</i> + λ))/(2<i>c</i> - (<i>x</i> + λ) + <i>x</i>λ/<i>c</i>)</div> - </div> - </div> -</div> - -<p class='c010'>Thus <i>v</i> lt; <i>c</i>, so long as | <i>x</i>λ | > 0.</p> - -<p class='c005'>Thus the velocity of light is the absolute maximum -velocity. We shall now see the consequences of admitting -a velocity W > <i>c</i>.</p> - -<p class='c005'>Let A and B be separated by distance <i>l</i>, and let -velocity of a “signal” in the system S be W > <i>c</i>. Let the -(observing) system S′ have velocity +<i>v</i> with respect to -the system S.</p> - -<p class='c005'>Then velocity of signal with respect to system S′ is -given by W′ = (W - <i>v</i>)/(1 - W<i>v</i>/<i>c²</i>)</p> - -<p class='c005'>Thus “time” from A to B as measured in S′, is given -by <i>l</i>/W′ = <i>l</i>(1 - W<i>v</i>/<i>c²</i>)/(W - <i>v</i>) = <i>t′</i> (1)</p> - -<p class='c005'>Now if <i>v</i> is less than <i>c</i>, then W being greater than <i>c</i> -(by hypothesis) W is greater than <i>v</i>, <i>i.e.</i>, W > <i>v</i>.</p> - -<p class='c005'>Let W = <i>c</i> + μ and <i>v</i> = <i>c</i> - λ.</p> - -<p class='c005'>Then W<i>v</i> = (<i>c</i> + μ)(<i>c</i> - λ) = <i>c²</i> + (μ + λ)<i>c</i> - μλ.</p> - -<p class='c005'>Now we can always choose <i>v</i> in such a way that W<i>v</i> is -greater than <i>c²</i>, since W<i>v</i> is > <i>c²</i> if (μ + λ)<i>c</i> - μλ is > 0, -that is, if μ + λ > μλ/<i>c</i>; which can always be satisfied by -a suitable choice of λ.</p> - -<p class='c005'>Thus for W > <i>c</i> we can always choose λ in such a -way as to make W<i>v</i> > <i>c²</i>, <i>i.e.</i>, λ - W<i>v</i>/<i>c²</i> negative. But -W - <i>v</i> is always positive. Hence with W > <i>c</i>, we can -always make <i>t′</i>, the time from A to B in equation (1) -“negative.” That is, the signal starting from A will reach -B (as observed in system S′) in less than no time. Thus the -effect will be perceived before the cause commences to act, -<i>i.e.</i>, the future will precede the past. Which is absurd. -Hence we conclude that W > <i>c</i> is an impossibility, there -can be no velocity greater than that of light.</p> - -<p class='c005'>It is <i>conceptually</i> possible to imagine velocities greater -than that of light, but such velocities cannot occur in -reality. Velocities greater than <i>c</i>, will not produce -any effect. Causal effect of any physical type can never -travel with a velocity greater than that of light.</p> - -<p class='c005'>[P. C. M.]</p> -<p class='c004'><a id='note-14'></a></p> -<h3 id='note-13' class='c008'>Notes 13 and 14.</h3> -<p class='c004'>We have denoted the four-vector ω by the matrix -| ω₁ ω₂ ω₃ ω₄ |. It is then at once seen that [=ω] denotes -the reciprocal matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| ω₁ |</div> - <div class='line'>| ω₂ |</div> - <div class='line'>| ω₃ |</div> - <div class='line'>| ω₄ |</div> - </div> - </div> -</div> - -<p class='c010'>It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]</p> - -<p class='c005'>[ω, <i>s</i>] The vector-product of the four-vector ω and <i>s</i> -may be represented by the combination</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>[ω<i>s</i>] = [=ω]<i>s</i> - <i>ṡ</i>ω</div> - </div> - </div> -</div> - -<p class='c010'>It is now easy to verify the formula <i>f</i>¹ = A⁻¹<i>f</i>A. -Supposing for the sake of simplicity that <i>f</i> represents the -vector-product of two four-vectors ω, <i>s</i>, we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f¹</i> = [ω¹<i>s¹</i>] = [[=ω]¹<i>s¹</i> - [=<i>s</i>]<sup>1</sup>ω<sup>1</sup>]</div> - </div> - <div class='group'> - <div class='line in13'>= [A⁻¹ [=ω]<i>s</i>A - A⁻¹<i>s</i>[=ω]A]</div> - </div> - <div class='group'> - <div class='line in13'>= A⁻¹[[=ω]<i>s</i> - <i>s</i>[=ω]]A = A⁻¹<i>f</i>A.</div> - </div> - </div> -</div> - -<p class='c010'>Now remembering that generally</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i> = ρφ + ρ*φ*.</div> - </div> - </div> -</div> - -<p class='c010'>Where ρ, ρ* are scalar quantities, φ, φ* are two -mutually perpendicular unit planes, there is no difficulty -in seeming that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'><i>f</i><sup>1</sup> = A⁻¹<i>f</i>A.</div> - </div> - </div> -</div> - -<h3 id='note-15' class='c008'>Note 15. <br />The vector product (<i>w</i><i>f</i>). (P. 36).</h3> -<p class='c004'>This represents the vector product of a four-vector and -a six-vector. Now as combinations of this type are of -frequent occurrence in this paper, it will be better to form -an idea of their geometrical meaning. The following -is taken from the above mentioned paper of Sommerfeld.</p> - -<p class='c005'>“We can also form a vectorial combination of a four-vector -and a six-vector, giving us a vector of the third -type. If the six-vector be of a special type, <i>i.e.</i>, a piece -of plane, then this vector of the third type denotes the -parallelopiped formed of this four-vector and the complement -of this piece of plane. In the general case, the -product will be the geometric sum of two parallelopipeds, -but it can always be represented by a four-vector of the -1st type. For two pieces of 3-space volumes can always -be added together by the vectorial addition of their components. -So by the addition of two 3-space volumes, -we do not obtain a vector of a more general type, but -one which can always be represented by a four-vector -(loc. cit. p. 759). The state of affairs here is the same as -in the ordinary vector calculus, where by the vector-multiplication -of a vector of the first, and a vector of the -second type (<i>i.e.</i>, a polar vector), we obtain a vector of the -first type (axial vector). The formal scheme of this -multiplication is taken from the three-dimensional case.</p> - -<p class='c005'>Let A = (A<sub><i>x</i></sub>, A<sub><i>y</i></sub>, A<sub><i>z</i></sub>) denote a vector of the first -type, B = (B<sub><i>y z</i></sub>, B<sub><i>z x</i></sub>, B<sub><i>x y</i></sub>) denote a vector of the second -type. From this last, let us form three special vectors of -the first kind, namely—</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>B<sub><i>x</i></sub> = (B<sub><i>x x</i></sub>, B<sub><i>x y</i></sub>, B<sub><i>x z</i></sub>) }</div> - <div class='line'>B<sub><i>y</i></sub> = (B<sub><i>y x</i></sub>, B<sub><i>y y</i></sub>, B<sub><i>y z</i></sub>) } (B<sub><i>i k</i></sub> = - B<sub><i>k i</i></sub>, B<sub><i>i i</i></sub> = 0).</div> - <div class='line'>B<sub><i>z</i></sub> = (B<sub><i>z x</i></sub>, B<sub><i>z y</i></sub>, B<sub><i>z z</i></sub>) }</div> - </div> - </div> -</div> - -<p class='c010'>Since B<sub><i>j j</i></sub> is zero, B<sub><i>j</i></sub> is perpendicular to the <i>j</i>-axis. -The <i>j</i>-component of the vector-product of A and B is -equivalent to the scalar product of A and B<sub><i>j</i></sub>, <i>i.e.</i>,</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(A B<sub><i>j</i></sub>,) = A<sub><i>x</i></sub> B<sub><i>j x</i></sub> + A<sub><i>y</i></sub> B<sub><i>j y</i></sub> + A<sub><i>z</i></sub> B<sub><i>j z</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'>We see easily that this coincides with the usual rule -for the vector-product; <i>e. g.</i>, for <i>j</i> = <i>x</i>.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(AB<sub><i>x</i></sub>) = A<sub><i>y</i></sub> B<sub><i>x</i> <i>y</i></sub> - A<sub><i>z</i></sub> B<sub><i>z</i> <i>x</i></sub>.</div> - </div> - </div> -</div> - -<p class='c010'>Correspondingly let us define in the four-dimensional -case the product (P<i>f</i>) of any four-vector P and the six-vector -<i>f</i>. The <i>j</i>-component (<i>j</i> = <i>x</i>, <i>y</i>, <i>z</i>, or <i>l</i>) is given by</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(P<i>f</i><sub><i>j</i></sub>) = P<sub><i>x</i></sub><i>f</i><sub><i>j</i> <i>x</i></sub> + P<sub><i>y</i></sub><i>f</i><sub><i>j</i> <i>y</i></sub> + P<sub><i>w</i></sub><i>f</i><sub><i>j</i> <i>z</i></sub> + P<sub><i>z</i></sub><i>f</i><sub><i>j</i> <i>l</i></sub></div> - </div> - </div> -</div> - -<p class='c010'>Each one of these components is obtained as the scalar -product of P, and the vector <i>f</i><sub><i>j</i></sub> which is perpendicular to -j-axis, and is obtained from <i>f</i> by the rule <i>f</i><sub><i>j</i></sub> = [(<i>f</i><sub><i>j</i> <i>x</i></sub>, <i>f</i><sub><i>j</i> <i>y</i></sub>, -<i>f</i><sub><i>j</i> <i>z</i></sub>, <i>f</i><sub><i>j</i> <i>l</i></sub>) <i>f</i><sub><i>j</i> <i>j</i></sub> = 0.]</p> - -<p class='c005'>We can also find out here the geometrical significance -of vectors of the third type, when <i>f</i> = φ, <i>i.e.</i>, <i>f</i> represents -only one plane.</p> - -<p class='c005'>We replace φ by the parallelogram defined by the two -four-vectors U, V, and let us pass over to the conjugate -plane φ<sup>*</sup>, which is formed by the perpendicular four-vectors -U<sup>*</sup>, V<sup>*</sup>. The components of (Pφ) are then equal to the -4 three-rowed under-determinants D<sub><i>x</i></sub> D<sub><i>y</i></sub> D<sub><i>z</i></sub> D<sub><i>l</i></sub> of the -matrix</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| P<sub><i>x</i></sub> P<sub><i>y</i></sub> P<sub><i>z</i></sub> P<sub><i>l</i></sub> |</div> - <div class='line'>| |</div> - <div class='line'>| U<sub><i>x</i></sub><sup>*</sup> U<sub><i>y</i></sub><sup>*</sup> U<sub><i>z</i></sub><sup>*</sup> U<sub><i>l</i></sub><sup>*</sup> |</div> - <div class='line'>| |</div> - <div class='line'>| V<sub><i>x</i></sub><sup>*</sup> V<sub><i>y</i></sub><sup>*</sup> V<sub><i>z</i></sub><sup>*</sup> V<sub><i>l</i></sub><sup>*</sup> |</div> - </div> - </div> -</div> - -<p class='c010'>Leaving aside the first column we obtain</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>D<sub><i>x</i></sub> = P<sub><i>y</i></sub>(U<sub><i>z</i></sub><sup>*</sup> V<sub><i>l</i></sub><sup>*</sup> - U<sub><i>l</i></sub><sup>*</sup> V<sub><i>z</i></sub><sup>*</sup>) + P<sub><i>z</i></sub>(U<sub><i>l</i></sub><sup>*</sup> V<sub><i>y</i></sub><sup>*</sup> - U<sub><i>y</i></sub><sup>*</sup> V<sub><i>l</i></sub><sup>*</sup>)</div> - <div class='line'>+ P<sub><i>l</i></sub>(U<sub><i>y</i></sub><sup>*</sup> V<sub><i>z</i></sub><sup>*</sup> - U<sub><i>z</i></sub><sup>*</sup> V<sub><i>y</i></sub><sup>*</sup>)</div> - <div class='line'>= P<sub><i>y</i></sub> φ<sub><i>z</i> <i>y</i></sub><sup>*</sup> + P<sub><i>z</i></sub><sup>*</sup> φ<sub><i>l</i> <i>y</i></sub> + P<sub><i>l</i></sub> φ<sup>*</sup><sub><i>y</i> <i>z</i></sub>.</div> - <div class='line'>= P<sub><i>y</i></sub> φ<sub><i>x</i> <i>y</i></sub> + P<sub><i>z</i></sub> φ<sub><i>x</i> <i>z</i></sub> + <i>P</i><sub><i>l</i></sub> φ<sub><i>x</i> <i>l</i></sub>,</div> - </div> - </div> -</div> - -<p class='c010'>which coincides with (Pφ<sub><i>x</i></sub>) according to our definition.</p> - -<p class='c005'>Examples of this type of vectors will be found on -page 36, Φ = wF, the electrical-rest-force, and ψ = 2wf<sup>*</sup>, -the magnetic-rest-force. The rest-ray Ω = iw[Φψ]<sup>*</sup> also -belong to the same type (page 39). It is easy to show -that</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Ω = -<i>i</i> | w₁ w₂ w₃ w₄ |</div> - <div class='line in6'>| Φ₁ Φ₂ Φ₃ Φ₄ |</div> - <div class='line in6'>| ψ₁ ψ₂ ψ₃ ψ₄ |</div> - </div> - </div> -</div> - -<p class='c010'>When (Ω₁, Ω₂, Ω₃) = 0, w₄ = <i>i</i>, Ω reduces to the three-dimensional -vector</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |</div> - <div class='line in15'>| |</div> - <div class='line in15'>| ψ₁ ψ₂ ψ₃ |</div> - </div> - </div> -</div> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Since in this case, Φ₁ = w₄ F₁₄ = <i>e</i><sub><i>n</i></sub> (the electric force)</div> - <div class='line in20'>ψ₁ = -<i>i</i>w₄ f₂₃ = <i>m</i><sub><i>x</i></sub> (the magnetic force)</div> - <div class='line'>we have (Ω) = | <i>e</i><sub><i>x</i></sub> <i>e</i><sub><i>y</i></sub> <i>e</i><sub><i>z</i></sub> |</div> - <div class='line in14'>| <i>m</i><sub><i>x</i></sub> <i>m</i><sub><i>y</i></sub> <i>m</i><sub><i>z</i></sub> |</div> - </div> - </div> -</div> - -<p class='c010'>[M. N. S.]</p> -<h3 id='note-16' class='c008'>Note 16. <br />The electric-rest force. (Page 37.)</h3> -<p class='c004'>The four-vector φ = wF which is called by Minkowski -the electric-rest-force (elektrische Ruh-Kraft) is very -closely connected to Lorentz’s Ponderomotive force, or -the force acting on a moving charge. If ρ is the density -of charge, we have, when ε = 1, μ = 1, <i>i.e.</i>, for free space</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]</div> - </div> - <div class='group'> - <div class='line in15'>= ρ₀/(√(1 - V²/<i>c²</i>)) [<i>d</i><sub><i>x</i></sub> + 1/<i>c</i> (<i>v₂</i> <i>h₃</i> - <i>v₃</i> <i>h₂</i>)]</div> - </div> - </div> -</div> - -<p class='c010'>Now since ρ₀ = ρ√(1 - V²/<i>c²</i>)</p> - -<p class='c005'>We have ρ₀φ₁ = ρ[<i>d</i><sub><i>x</i></sub> + 1/<i>c</i> (<i>v₂</i> <i>h₃</i> - <i>v₃</i> <i>h₂</i>)]</p> - -<p class='c005'>N. B.—We have put the components of <i>e</i> equivalent -to (<i>d</i><sub><i>x</i></sub>, <i>d</i><sub><i>y</i></sub>, <i>d</i><sub><i>z</i></sub>), and the components of <i>m</i> equivalent to -<i>h</i><sub><i>x</i></sub> <i>h</i><sub><i>y</i></sub> <i>h</i><sub><i>z</i></sub>), in accordance with the notation used in -Lorentz’s Theory of Electrons.</p> - -<p class='c005'>We have therefore</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>ρ₀ (φ₁, φ₂, φ₃) = ρ (<i>d</i> + 1/<i>c</i> [<i>v</i>·<i>h</i>]),</div> - </div> - </div> -</div> - -<p class='c010'><i>i.e.</i>, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the -electron. Compare Lorentz, Theory of Electrons, page 14.</p> - -<p class='c005'>The fourth component φ₄ when multiplied by ρ₀ -represents <i>i</i>-times the rate at which work is done by -the moving electron, for ρ₀ φ₄ = <i>i</i>ρ [<i>v</i><sub><i>x</i></sub><i>d</i><sub><i>x</i></sub> + <i>v</i><sub><i>y</i></sub><i>d</i><sub><i>y</i></sub> + <i>v</i><sub><i>z</i></sub><i>d</i><sub><i>z</i></sub>] = -<i>v</i><sub><i>x</i></sub> ρ₀φ₁ + <i>v</i><sub><i>y</i></sub> ρ₀φ₂ + <i>v</i><sub><i>z</i></sub> ρ₀φ₃. -√(-1) times the power possessed -by the electron therefore represents the fourth -component, or the time component of the force-four-vector. -This component was first introduced by Poincare -in 1906.</p> - -<p class='c005'>The four-vector ψ = <i>i</i>ωF<sup>*</sup> has a similar relation to -the force acting on a moving magnetic pole.</p> - -<p class='c005'>[M. N. S.]</p> -<h3 id='note-17' class='c008'>Note 17. <br />Operator “Lor” (§ 12, p. 41).</h3> -<p class='c004'>The operation | ∂/∂<i>x₁</i> ∂/∂<i>x₂</i> ∂/∂<i>x₃</i> ∂/∂<i>x₄</i> | which plays in -four-dimensional mechanics a rôle similar to that of -the operator (<i>i</i>∂/∂<i>x</i>, + <i>j</i>∂/∂<i>y</i>, + <i>k</i>∂/∂<i>z</i> = ▽) in three-dimensional -geometry has been called by Minkowski ‘Lorentz-Operation’ -or shortly ‘lor’ in honour of H. A. Lorentz, -the discoverer of the theorem of relativity. Later writers -have sometimes used the symbol □ to denote this -operation. In the above-mentioned paper (Annalen der -Physik, p. 649, Bd. 38) Sommerfeld has introduced the -terms, Div (divergence), Rot (Rotation), Grad (gradient) -as four-dimensional extensions of the corresponding three-dimensional -operations in place of the general symbol -lor. The physical significance of these operations will -become clear when along with Minkowski’s method of -treatment we also study the geometrical method of -Sommerfeld. Minkowski begins here with the case of -lor S, where S is a six-vector (space-time vector of the -2nd kind).</p> - -<p class='c005'>This being a complicated case, we take the simpler -case of lor <i>s</i>,</p> - -<p class='c005'>where <i>s</i> is a four-vector = | <i>s₁</i>, <i>s₂</i>, <i>s₃</i>, <i>s₄</i> |</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>and <i>s</i> = | <i>s₁</i> |</div> - <div class='line in9'>| <i>s₂</i> |</div> - <div class='line in9'>| <i>s₃</i> |</div> - <div class='line in9'>| <i>s₄</i> |</div> - </div> - </div> -</div> - -<p class='c010'>The following geometrical method is taken from Sommerfeld.</p> - -<p class='c005'>Scalar Divergence—Let ΔΣ denote a small four-dimensional -volume of any shape in the neighbourhood of the -space-time point Q, <i>d</i>S denote the three-dimensional -bounding surface of ΔΣ, <i>n</i> be the outer normal to <i>d</i>S. -Let S be any four-vector, P<sub><i>n</i></sub> its normal component. -Then</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Div S = Lim ∫ P<sub><i>n</i></sub><i>d</i>S/ΔΣ.</div> - <div class='line in4'>ΔΣ = 0</div> - </div> - </div> -</div> - -<p class='c010'>Now if for ΔΣ we choose the four-dimensional parallelopiped -with sides (<i>dx₁</i>, <i>dx₂</i>, <i>dx₃</i>, <i>dx₄</i>), we have then</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Div S = ∂<i>s₁</i>/∂<i>x₁</i> + ∂<i>s₂</i>/∂<i>x₂</i> + ∂<i>s₃</i>/∂<i>x₃</i> + ∂<i>s₄</i>/∂<i>x₄</i> = lor S.</div> - </div> - </div> -</div> - -<p class='c010'>If <i>f</i> denotes a space-time vector of the second kind, lor -<i>f</i> is equivalent to a space-time vector of the first kind. The -geometrical significance can be thus brought out. We have -seen that the operator ‘lor’ behaves in every respect like -a four-vector. The vector-product of a four-vector and a -six-vector is again a four-vector. Therefore it is easy -to see that lor S will be a four-vector. Let us find -the component of this four-vector in any direction <i>s</i>. -Let S denote the three-space which passes through the -point Q (<i>x₁</i>, <i>x₂</i>, <i>x₃</i>, <i>x₄</i>) and is perpendicular to <i>s</i>, ΔS a -very small part of it in the region of Q, <i>d</i>σ is an element -of its two-dimensional surface. Let the perpendicular -to this surface lying in the space be denoted by <i>n</i>, and -let <i>f</i><sub><i>s</i> <i>n</i></sub> denote the component of <i>f</i> in the plane of (<i>sn</i>) -which is evidently conjugate to the plane <i>d</i>σ. Then the -<i>s</i>-component of the vector divergence of <i>f</i> because the -operator lor multiplies <i>f</i> vectorially.</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>= Div <i>f</i><sub><i>s</i></sub> = Lim (∫ <i>f</i><sub><i>s</i> <i>n</i></sub><i>d</i>σ)/ΔS.</div> - <div class='line in4'>Δ<i>s</i> = 0</div> - </div> - </div> -</div> - -<p class='c010'>Where the integration in <i>d</i>σ is to be extended over -the whole surface.</p> - -<p class='c005'>If now <i>s</i> is selected as the <i>x</i>-direction, Δ<i>s</i> is then -a three-dimensional parallelopiped with the sides <i>dy</i>, <i>dz</i>, -<i>dl</i>, then we have</p> - -<div class='figcenter id004'> -<img src='images/form-123.png' alt='Formula.' class='ig001' /> -</div> - -<p class='c005'>and generally</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>Div <i>f</i><sub><i>j</i></sub> = ∂<i>f</i><sub><i>j</i> <i>x</i></sub>/∂<i>x</i> + ∂<i>f</i><sub><i>j</i> <i>y</i></sub>/∂<i>y</i> + ∂<i>f</i><sub><i>j</i> <i>z</i></sub>/∂<i>z</i> + ∂<i>f</i><sub><i>j</i> <i>l</i></sub>/∂<i>l</i> (where <i>f</i><sub><i>j</i>, <i>j</i></sub> = 0).</div> - </div> - </div> -</div> - -<p class='c010'>Hence the four-components of the four-vector lor S -or Div. <i>f</i> is a four-vector with the components given on -page 42.</p> - -<p class='c005'>According to the formulae of space geometry, D<sub><i>x</i></sub> -denotes a parallelopiped laid in the (<i>y</i>-<i>z</i>-<i>l</i>) space, formed -out of the vectors (P<sub><i>y</i></sub> P<sub><i>z</i></sub> P<sub><i>l</i></sub>), (U<sub><i>y</i></sub><sup>*</sup> U<sub><i>z</i></sub><sup>*</sup> U<sub><i>l</i></sub><sup>*</sup>) (V<sub><i>y</i></sub><sup>*</sup> V<sub><i>z</i></sub><sup>*</sup> V<sub><i>l</i></sub><sup>*</sup>).</p> - -<p class='c005'>D<sub><i>x</i></sub> is therefore the projection on the <i>y-z-l</i> space of -the parallelopiped formed out of these three four-vectors -(P, U<sup>*</sup>, V<sup>*</sup>), and could as well be denoted by Dyzl. -We see directly that the four-vector of the kind represented -by (D<sub><i>x</i></sub>, D<sub><i>y</i></sub>, D<sub><i>z</i></sub>, D<sub><i>l</i></sub>) is perpendicular to the parallelopiped -formed by (P U<sup>*</sup> V<sup>*</sup>).</p> - -<p class='c005'>Generally we have</p> - -<div class='lg-container-b c006'> - <div class='linegroup'> - <div class='group'> - <div class='line'>(P<i>f</i>) = PD + P<sup>*</sup>D<sup>*</sup>.</div> - </div> - </div> -</div> - -<p class='c010'>∴ The vector of the third type represented by (P<i>f</i>) -is given by the geometrical sum of the two four-vectors of -the first type PD and P<sup>*</sup>D<sup>*</sup>.</p> - -<p class='c005'>[M. N. S.]</p> - -<div class='nf-center-c1'> -<div class='nf-center c002'> - <div><span class='large'>Footnotes</span></div> - </div> -</div> - -<div class='footnote' id='f1'> -<p class='c005'><span class='label'><a href='#r1'>1</a>. </span>See <a href='#note-1'>Note 1</a>.</p> -</div> -<div class='footnote' id='f2'> -<p class='c005'><span class='label'><a href='#r2'>2</a>. </span>See <a href='#note-2'>Note 2</a>.</p> -</div> -<div class='footnote' id='f3'> -<p class='c005'><span class='label'><a href='#r3'>3</a>. </span>See <a href='#note-4'>Note 4</a>.</p> -</div> -<div class='footnote' id='f4'> -<p class='c005'><span class='label'><a href='#r4'>4</a>. </span>See Notes <a href='#note-9'>9</a> and <a href='#note-12'>12</a>.</p> -</div> -<div class='footnote' id='f5'> -<p class='c005'><span class='label'><a href='#r5'>5</a>. </span><a href='#note-a'>Note A</a>.</p> -</div> -<div class='footnote' id='f6'> -<p class='c005'><span class='label'><a href='#r6'>6</a>. </span><i>Vide</i> <a href='#note-9'>Note 9</a>.</p> -</div> -<div class='footnote' id='f7'> -<p class='c005'><span class='label'><a href='#r7'>7</a>. </span><i>Vide</i> <a href='#note-9'>Note 9</a>.</p> -</div> -<div class='footnote' id='f8'> -<p class='c005'><span class='label'><a href='#r8'>8</a>. </span><i>Vide</i> <a href='#note-12'>Note 12</a>.</p> -</div> -<div class='footnote' id='f9'> -<p class='c005'><span class='label'><a href='#r9'>9</a>. </span><i>Vide</i> <a href='#note-1'>Note 1</a>.</p> -</div> -<div class='footnote' id='f10'> -<p class='c005'><span class='label'><a href='#r10'>10</a>. </span><a href='#note-2'>Note 2</a>.</p> -</div> -<div class='footnote' id='f11'> -<p class='c005'><span class='label'><a href='#r11'>11</a>. </span><i>Vide</i> <a href='#note-3'>Note 3</a>.</p> -</div> -<div class='footnote' id='f12'> -<p class='c005'><span class='label'><a href='#r12'>12</a>. </span><i>Vide</i> <a href='#note-4'>Note 4</a>.</p> -</div> -<div class='footnote' id='f13'> -<p class='c005'><span class='label'><a href='#r13'>13</a>. </span><a href='#note-5'>Note 5.</a></p> -</div> -<div class='footnote' id='f14'> -<p class='c005'><span class='label'><a href='#r14'>14</a>. </span>See notes on § <a href='#sect4-8'>8</a> and <a href='#sect4-10'>10</a>.</p> -</div> -<div class='footnote' id='f15'> -<p class='c005'><span class='label'><a href='#r15'>15</a>. </span>See <a href='#note-9'>note 9</a>.</p> -</div> -<div class='footnote' id='f16'> -<p class='c005'><span class='label'><a href='#r16'>16</a>. </span>See Note.</p> -</div> -<div class='footnote' id='f17'> -<p class='c005'><span class='label'><a href='#r17'>17</a>. </span>Vide Note.</p> -</div> -<div class='footnote' id='f18'> -<p class='c005'><span class='label'><a href='#r18'>18</a>. </span>Just as beings which are confined within a narrow region -surrounding a point on a spherical surface, may fall into the error that -a sphere is a geometric figure in which one diameter is particularly -distinguished from the rest.</p> -</div> -<div class='footnote' id='f19'> -<p class='c005'><span class='label'><a href='#r19'>19</a>. </span>Einzelne stelle der Materie.</p> -</div> -<div class='footnote' id='f20'> -<p class='c005'><span class='label'><a href='#r20'>20</a>. </span>Vide Note.</p> -</div> -<div class='footnote' id='f21'> -<p class='c005'><span class='label'><a href='#r21'>21</a>. </span><i>Vide</i> <a href='#note-13'>note 13</a>.</p> -</div> -<div class='footnote' id='f22'> -<p class='c005'><span class='label'><a href='#r22'>22</a>. </span><i>Vide</i> <a href='#note-14'>note 14</a>.</p> -</div> -<div class='footnote' id='f23'> -<p class='c005'><span class='label'><a href='#r23'>23</a>. </span><i>Vide</i> <a href='#note-15'>note 15</a>.</p> -</div> -<div class='footnote' id='f24'> -<p class='c005'><span class='label'><a href='#r24'>24</a>. </span><i>Vide</i> <a href='#note-16'>note 16</a>.</p> -</div> -<div class='footnote' id='f25'> -<p class='c005'><span class='label'><a href='#r25'>25</a>. </span><i>Vide</i> <a href='#note-17'>note 17</a>.</p> -</div> -<div class='footnote' id='f26'> -<p class='c005'><span class='label'><a href='#r26'>26</a>. </span><i>Vide</i> note 19.</p> -</div> -<div class='footnote' id='f27'> -<p class='c005'><span class='label'><a href='#r27'>27</a>. </span><i>Vide</i> note 18.</p> -</div> -<div class='footnote' id='f28'> -<p class='c005'><span class='label'><a href='#r28'>28</a>. </span>Vide note 40.</p> -</div> -<div class='footnote' id='f29'> -<p class='c005'><span class='label'><a href='#r29'>29</a>. </span>Sichel—a German word meaning a crescent or a scythe. The -original term is retained as there is no suitable English equivalent.</p> -</div> -<div class='footnote' id='f30'> -<p class='c005'><span class='label'><a href='#r30'>30</a>. </span>Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, -1908, p. 1.</p> -</div> -<div class='footnote' id='f31'> -<p class='c005'><span class='label'><a href='#r31'>31</a>. </span>H. Minkowski; the passage refers to paper (2) of the present -edition.</p> -</div> -<div class='footnote' id='f32'> -<p class='c005'><span class='label'><a href='#r32'>32</a>. </span>Minkowski—Mechanics, appendix, page 65 of paper (2). -Planck—Verh. d. D. P. G. Vol. 4, 1906, p. 136.</p> -</div> -<div class='footnote' id='f33'> -<p class='c005'><span class='label'><a href='#r33'>33</a>. </span>Schütz, Gött. Nachr. 1897, p. 110.</p> -</div> -<div class='footnote' id='f34'> -<p class='c005'><span class='label'><a href='#r34'>34</a>. </span>Lienard, L’Eclairage électrique T. 16, 1896, p. 53. -Wiechert, Ann. d. Physik, Vol. 4.</p> -</div> -<div class='footnote' id='f35'> -<p class='c005'><span class='label'><a href='#r35'>35</a>. </span>K. Schwarzschild. Gött-Nachr. 1903. -H. A. Lorentz, Enzyklopädie der Math. Wissenschaften V. 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