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Full text of "The principle of relativity; original papers"

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LIBRARY 

OF THE 

MASSACHUSETTS INSTITUTE 

OF TECHNOLOGY 



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THE 




SEP k 



PRINCIPLE OF RELATJ^^{jK[Ji 



ORIGINAL PAPERS 



BY 



A. EINSTEIN ANT) H. MINKOWSKI 



)' 



TRANSLATED INTO ENGLISH 



BY 

M. N. SAHA AND S. N. BOSE 

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



WITH A HISTORICAL INTRODUCTION 



BY 

P. C. MAHALANOBIS 

professor of physics, presidency college, CALCU- 




PUBLISHED BY THE 

UNIVERSITY OF CALCUTTA 

1920 



Sole Agents 
R. CAMBRAY &: CO. 



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



TABLE OF CONTENTS 



1. Historical Introduction 

[By Mr. P. C. Mahalanobis.] 

2. On the Electrodynamics of Moving Bodies... 

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

3. Albreeht Einstein 

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

4. Principle of Relativity 

[H. Minkowski's original paper on the 
restricted Principle of Relativity first 
published in 1909. Translated from 
the original German by Dr. Meghnad 
Saha.] 

5. Appendix to the above by H. Minkowski ... 

[Translated by Dr. Meghnad Saha.] 

6. The Generalised Principle of Relativity 

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

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



PAGE 

i-xxiii 
1-34 



35-39 



1-52 



53-88 



89-l()3 



165-^185 



124281 



-N \^\ 



y 



J 



HISTORICAL INTRODUCTION 



ooj:^0<>- 

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

Ten years later, in 1905, we find Einstein declarinsj 
that " the ether will be proved to be superflous." At 
first sight the revolution in scientific thought brought 
about in the course of a single decade appears to be almost 
too violent. A more careful even though a rapid review 
of the subject will, however, show how the Theory of 
Relativity gradually became a historical necessity. 

Towards the beginning of the nineteenth century, 
the luminiferous ether came into prominence as a result of 
the brilliant successes of the wave theory in the hands 
of Young and Fresnel. In its stationary aspect the 
elastic solid ether was the outcome of the search for a 
medium in which the light waves may "undulate." This 
stationary ether, as shown by Young, also afforded a 
satisfactory explanation of astronomical aberration. But 
its very success gave rise to a host of new questions all 
bearing on the central problem of relative motion of ether 
and matter. 



11 PRINCIPLE OF RELATIVITY 

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

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

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

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



t « • 



HISTORICAL INTRODUCTION 111 

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

Stokes^ viseous ether. — It should be remembered, however, 
that Fresnel 's stationary ether is absolutelv fixed and is not 
at all disturbed bv the motion of matter throusfh it. In this 
respect Fresnelian ether cannot be said to behave in any 
respectable physical fashion, and this led Stokes, in 
1845-46, to construct a more material type of medium. 
Stokes assumed that viscous motion ensues near the surface 
of separation of ether and moving matter, w^hile at 
sufficiently distant regions the ether remains wholly 
undisturbed. He showed how such a viscous ether would 
explain aberration if all motion in it were differentially 
irrotational. But in order to explain the null Arago 
effect, Stokes was compelled to assume the convection 
hypothesis of Fresnel with an identical numerical value 
for kj namely 1 — V/^'- '^hus the prestige of the Fresnelian 
convection-coefficient was enhanced, if anything, by the 
theoretical investigations of Stokes. 



IV PRINCIPLE OF RELATIVITY 

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

If a divided beam of light is re-nnited after passin<)j 
through two adjacent cylinders filled with water, ordinary 
interference fringes will be produced. If the water in one 
of the cylinders is now nriade to fiow^, the " condensed" 
ether within the flowing water wonld be conveeted and 
would produce a shift in the interference fringes. The 
shift actuallv observed agreed verv well with a value of 
k=l— V/Jt^. The Fresnelian eonveetion-eoeffieient now 
became firmly established as a consequence of a direct 
positive effect. On the other hand, the negative evidences 
in favour of the convection-coefficient had also multiplied. 
Mascart, Hoek, Maxwell and others sought for definite 
changes in different optical effects induced by the motion 
of the earth relative to the stationary ether. But all such 
attempts failed to reveal the slightest trace of any optical 
disturbance due to the "absolute" velocity of the earthy, 
thus proving conclusively that all tne different optical 
effects shared in the general compensation arising out of 
the Fresnelian convection of the excess ether. It must be 
carefully noted that the Fresnelian convection -coefficient 
implicitly assumes the existence of a fixed ether (Fresnel) or 
at least a wholly stagnant medium at sufficiently distant 
regions (Stokes), with reference to which alone a convection 
velocity can have any significance. Thus the convection- 
coefficient implying some type of a stationary or viscous, 
yet nevertheless "absolute" ether, succeeded in explaining 
satisfactorily all known optical facts down to 1880. 

Mic/iehov-Morley Eopperiment. — In 1881, Michelson 
and Morley performed their classical experiments which 
undermined the whole structure of the old ether theory 
and thus served to introduce the new theory of relativity. 



HISTOKICAL INTRODUCTION V 

The fiiiidameiital idea underlyia^^'' tliib experiment is quite 
sim[de. In all old expeiiments the velocity of light 
situated in free ether \Vas corn[)ared with the veloeitv 
of waves actually situated m a piece of moving matter 
and presumably carried away by it. The compensatory 
effect of the Fresnelian convection of ether afforded a 
satisfactory explanation of all neo^ative results. 

In the Michelson-Morley experiment the arrangement is 
quite different. If there is a definite gap in a rigid body, 
light waves situated in free ether will take a delinite time 
in crossing the gap. If the rigid platform carrying the 
gap is set in motion with respect to the ether in the direc- 
tion of light propagation;, light waves (which are even now 
situated in free ether) should presumably take a longer 
time to cross the gap. 

We cannot do better than quote Eddiugton's descrip- 
tion of this famous experiment. " The principle of the 
experiment may be illustrated by considering a swimmer in 
a river. It is easily realized that it takes longer to swim 
to a point 50 yards up-stream and back than to a point 50 
vards acioss-stream and back. If the earth is movino- 
through the ether there is a river of ether flowing- throuopli 
the laboratory, and a wave of light may be compared to a 
swnmmer travelling with constant velocity relative to the 
current. If, then, we divide a beam of light into two parts, 
and send one-half swimming up the stream for a certain 
distance and then (by a mirror) back to the starting 
point, and send the other half an equal distance across 
stream and back, the across-stream beam should arrive 
back first. 

Let the ether be flowing relative to 

oi the apparatus with velocity u in the 

^ direction Or, and let OA, OB, be 

B the two arms of the apparatus of equal 



A 



VI PKiNClPLE 0¥ RELATIVITY 

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

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

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

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

t'l = /7-~^^ = —A SO that t^>t^. 

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

x\fter a most careful series of observations, Michelson 
and Morle^^ failed to detect the slightest trace of any 
effect due to earth's motion throus^h ether. 

The Michelson-Morley experiment seems to show that 
there is no relative motion of ether and matter. Fresnel's 
stagnant ether requires a relative velocity of — n. Thus 
Michelson and Morlev themselves thought at first that their 
experiment conhrmed Stokes^ viscous ether, in wliieh no 
relative motion can ensue on account of the absence of 
slip])ing of ether at the surface of separation. But even 
on Stokes' theory this viscous How of ether would fall 
ofP at a very rapid rate as we recede from the surface 
of separation. Michelson and Morley repeated their experi- 
ment at different heights from the surface of the earth, but 
invariably obtained the same negative results, thus failing 
to confirm Stokes' theory of viscous How. 



HTSTORICAL TNTHODUCTTON TU 

Loflgt!^ experimevi, — Further, in 1893, Lodge per- 
formed bis rotating' sphere experiment which showed 
complete absence of any viscous How of ether due to 
moviuo' masses of matter. A divided beam of light, after 
repeated reflections within a ver}^ narrow gap between two 
massive hemispheres, was allowed to re-unite and thus 
produce interference bands. When the two hemispheres 
are set rotating, it is conceivable that the ether in the gap 
would be disturbed due to viscous flow, and any such flow 
would be immediately detected by a distru'bance of the 
interference bands. But actual observation failed to 
detect the slightest disturbance of the ether in the gap, 
due to the motion of the hemispheres. Lodge's experi- 
ment thus seems to show a complete absence of any viscous 
flow of ether. 

Apart from these experimental discrepancies, grave 
theoretical objections were urged against a viscous ether. 
Stokes himself had shown that his ether must be incom- 
pressible and all motion in it differentially irrotational, 
at the same time there should be absolutely no slipping at 
the surface of separation. Now all these conditions cannot 
be simultaneously satisfied for any conceivable material 
medium without certain very special and arbitrary assump- 
tions. Thus Stokes' ether failed to satisfy the very motive 
which had led Stokes to formulate it^ namely, the desirabi- 
lity of constructing a "physical" medium. Planck offered 
modified forms of Stokes' theory which seemed capable of 
being reconciled with the Miehelson-Morley experiment, 
but required very sjiecial assumptions. The very complexity 
and the very arbitrariness of these assumptions prevented 
Planck's ether from attaining any degree of practical 
importance in the further development of the subject. 

The sole criterion of the value of any scientific theory 
must ultimately be its capacity for offering a simple. 



Vlll PRINCIPLE OF RELATIVITY 

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

We must now consider the problem of relativf motion of 
ether and matter from the point of view of electrical theory. 
From 1860 the identitv of lisht as an electromagnetic 
vector became o-radualh' established as a result of the 
brilliant '^ displacement current" hypothesis of Clerk 
Maxwell and his further analytical investigations. The 
elastic solid ether became gradually transformed into the 
electromagnetic one. Maxwell succeeded in giving a fairly 
.satisfactory account of all ordinary optical phenomena 
and little room was left for any serious doubts as regards 
the general validity of Maxwell's theory. Hertz's re- 
searches on dectric waves, first carried out in 1886, 
succeeded in furnishing a strong experimental conlh-mation 



HISTORICAL INTRODUCTION II 

of Maxwell's theory. Electric waves behaved generally 
like light waves of very large wave length. 

The orthodox Maxwellian view located the dielectric 
polarisation in the electromagnetic ether which was merely 
a transformation of Fresnel's stag-nant ether. The mag- 
netic polarisation was looked upon as wholly secondary in 
origin, being due to the relative motion of the dielectric 
tubes of polarisation. On this view the Fresnelian con- 
vection coefficient comes out to be i, as shown by J. J. 
Thomson in 1880, instead of 1 — ^//x- as required by 
optical experiments. This obviously implies a complete 
failure to account for all those optical experiments which 
depend for their satisfactory explanation on the assumption 
of a value for the convection coefficient equal to 1 — V/*^' 

The modifications proposed independently by Hertz and 
Heaviside fare no better."^ They postulated the actual 
medium to be the seat of all electric polarisation and further 
emphasised the reciprocal relation subsisting between 
electricity and magnetism, thus making the field equations 
more symmetrical. On this view the whole of the 
polarised ether is carried away by the moving medium, 
and consequently, the convection co-efficient naturally 
becomes unity in this theory, a value quite as discrepant 
as that obtained on the original Maxwellian assumption. 

Thus neither Maxwell's original theory nor its subse- 
quent modifications as developed by Hertz and Heaviside 
succeeded in obtainiuii; a value for Fresnelian co-efficient 
equal to 1— V/^^j ^^^ consequently stood totall3^ condemned 
from the optical point of view. 

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

* See Note 1. 



X PRINCIPLE OF RELATIVITY 

According to Hertz a moving dielectric would carry away 
the whole of its electric displacement with it. Hence the 
electromagnetic effect near the moving dielectric would 
be proportional to the total electric displacement, that is 
to K, the specific inductive capacity of the dielectric. In 
)901, Blondlot working with a stream of moving gas 
could not detect any such effect. H. A. Wilson repeated 
the experiment in an improved form in 1903 and working 
with ebonite found that the observed effect was pro- 
portional to K — 1 instead of to K. For gases K is nearly 
equal to 1 and hence practically no effect will be observed 
in their case. This gives a satisfactory explanation of 
Blondlot's negative results. 

Rowland had shown in 1876 that the magnetic force 
due to a rotating condenser (the dielectric remaining 
stationary) was proportional to K, the sp. ind. cap. On 
the other hand, Rontgen found in 1888 the magnetic 
effect due to a rotating dielectric (the condenser remain- 
ing stationary) to be proportional to K— 1, and not to 
K. Finally Eichenwald in 1903 found that when both 
condenser and dielectric are rotated together, the effect 
observed was quite independent of K, a result quite 
consistent with the two previous experiments. The Row- 
land effect proportional to K, together with the opposite 
Rontgen effect proportional to 1 — K, makes the Eichenwald 
effect independent of K. 

All these experiments together with those of Blondlot 
and Wilson made it clear that the electromagnetic 
effect due to a moving dielectric was proportional to 
K— 1, and not to K as required by Hertz's theory. Thus 
the .above group of experiments with moving dielectrics 
directly contradicted the Hertz- Heaviside theory. The 
internal discrepancies inherent in the classic ether theory 
had now become too prominent. It was clear that the 



HISTORICAL INTRODUCTION XI 

ether concept had finally outgrown its usefulness. The 
observed fleets had become too contradictory and too 
heterogeneous to be reduced to an organised whole with 
the help of the ether concept alone. Radical departures 
from the classical theory had become absolutely necessary. 

There were several outstandmg difficulties in connec- 
tion with anomalous dispersion, selective reflection and 
selective absorption which could not be satisfactory 
explained in the classic electromagnetic theory. It 
was evident that the assumption of some kind of 
discreteness in the optical meduim had become inevit- 
able. Such an assumption naturally gave rise to an 
atomic theory of electricity, namely, the modern electron 
theory. Lorentz had postulated the existence of electrons 
so early as 1878, but it was not until some years later that 
the electron theory became firmly established on a satisfac- 
tory basis. 

Lorentz assumed that a moving dielectric merely carried 
away its own '' polarivsation doublets," which on his theory 
gave rise to the induced field proportional to K— 1. The 
field near a moving dielectric is naturally proportional to 
K — 1 and not to K. Lorentz's theory thus gave a 
satisfactory explanation of all those experiments with 
moving dielectrics which required effects proportional to 
K — 1. Lorentz further succeeded in obtaining a value for 
the Fresnelian convection coefficient equal to 1 — ^//a^, the 
exact value required by all optical experiments of the 
moving type. 

We must now go back to Michelson and Morley's 
experiment. We have seen that both parts of the beam 
are situated in free ether ; no material meduim is involved 
in any portion of the paths actually traversed by the beam. 
Consequently no compensation due to Fresnelian convection 



Xll ^PRINCIPLE OP RELATIVITY 

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

In Miphelson and Morley^s experiment, the distance 
travelled by the beam along OA (that is, in a direction 
parallel to the motion of the platform) is 2/^^, while the 
distance travelled by the beam along OB, perpendicular to 
the direction of motion of the platform, is ^lip. Yet the 
most careful experiments showed, as Eddington says, " that 
both parts of the beam took the same time as tested by the 
interference bands produced. It would seem that OA and 
OB could not really have been of the same length ; and if 
OB was of length I, OA must have been of length IjP. The 
apparatus was now rotated through 90°, so that OB became 
the up-stream. The time for the two journeys was again 
the same, so that OB must now be the shorter length. The 
plain meaning of the experiment is that both arms have a 
length I when placed along 0^ (perpendicular to the direc- 
tion of motion), and automatically contract to a length 
Ijpf when placed along 0/ (parallel to the direction of 
motion). This explanation was first given by Fitz-Gerald." 

This Fitz-Gerald contraction^, startling enough in 
itself, does not suffice. Assuming this contraction to be a 
real one, the distance travelled with respect to the ether is 
%lp and the time taken for this journey is 2l^/c. But the 
distance travelled with respect to the platform is always 
21. Hence the velocity of light with respect to the plat- 

form is 21/ — ^ —c/^, a variable quantity depending on 

the " absolute " velocity of the platform. But no trace 
of such an effect has ever been found. The velocity of 
light is always found to be quite independent of the velocity 



HISTOBICAL INTRODUCTION XUl 

of the platform. The present difficulty cannot be solved 
by any further alteration in the measure of space. The 
only recourse left open is to alter the measure of time as 
well, that is, to adopt the concept of "local time." If a mov- 
inoj clock goes slower so that one 'real' second becomes 1/^ 
second as measured in the moving system, the velocity of 
light relative to the platform will always remain c. We 
must adopt two very startling hypotheses, namely, the 
Fitz -Gerald contraction and the concept of "local time," 
in order to give a satisfactory explanation of the 
Miehelson-Morley experiment. 

These results were already reached by Lorentz in the 
course of further developments of his electron theory. 
Lorentz used a special set of transformation equations"^ for 
time which implicitly introduced the concept of local time. 
But he himself failed to attach any special significance to 
it, and looked upon it rather as a mere mathematical 
artifice like imaginary quantities in analysis or the circle 
at infinity in projective geometry. The originality of 
Einstein at this stage consists in his successful physical 
interpretation of these results, and viewing them as the 
coherent organised consequences of a single general 
principle. Lorentz established the Relativity Theoremt 
(consisting merely of a set of transformation equations) 
while Einstein generalised it into a Universal Principle. In 
addition Einstein introduced fundamentally new concepts 
of space and time, which served to destroy old fetishes and 
demanded a wholesale revision of scientific concepts and 
thus opened up new possibilities in the synthetic unification 
of natural processes. 

Newton had framed his laws of motion in such a way 
as to make them quite independent of the absolute velocity 

* See Note 2. 
t See Note 4. 



XIV PRINCIPLE or RELATIVITY 

of the earth. Uniform relative motion of ether and matter 
could not be detected with the help of dynamical laws. 
According to Einstein neither could it be detected with the 
help of optical or electromagnetic experiments. Thus the 
Einsteinian Principle of Relativity asserts that all physical 
laws are independent of the ^absolute' velocity of an observer. 

For different systems, the form of all physical laws is 
conserved. If we chose the velocity of light"^ to be the 
fundamental unit of measurement for all observers (that is, 
assume the constancy of the velocity of light in all systems) 
we can establish a metric ^^ one — one ^' correspondence 
between any two observed systems, such correspondence 
depending only the relative velocity of the two systems. 
Einstein's Relativity is thus merely the consistent logical 
application of the well known physical principle that we 
can know nothing but relative motion. In this sense it is 
a further extension of Newtonian Relativity. 

On this interpretation, the Lorentz- Fitzgerald contrac- 
tion and "local time" lose their arbitrary character. Space 
and time as measured by two different observers are natur- 
ally diverse, and the difference depends only on their relative 
motion. Both are equally valid; they are merely different 
descriptions of the same physical reality. This is essentially 
the point of view adopted by Minkowski. He considers time 
itself to be one of the co-ordinate axes, and in his four- 
dimensional world, that is in the space-time reality, relative 
motion is reduced to a rotation of the axes of reference. 
Thus, the diversity in the measurement of lengths and 
temporal rates is merely due to the static difference in the 
" frame- work ^' of the different observers. 

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

* See Notes 9 and 12. 



HISTORICAL INTRODUCTION XV 

Maxwell-Lorentz system of field equations. It combined 
all the advantages of classic Maxwellian theory together 
with an electronic hypothesis. The Lorentz assumption of 
polarisation doublets had furnished a satisfactory explana- 
tion of the Fresnelian convection of ether, but in the new 
theory this is deduced merely as a consequence of the altered 
concept of relative velocity. In addition, the theory of 
Relativity accepted the results of Michelson and Morley's 
experiments as a definite principle, namely, the principle of 
the constancy of the velocity of light, so that there was 
nothing left for explanation in the Michelson-Morle3^ 
experiment. But even more than all this, it established a 
single general principle which served to connect together 
in a simple coherent and fruitful manner the known facts 
of Physics. 

The theory of Relativity received direct experimental 
confiimation in several directions. Repeated attempts were 
made to detect the Lorentz-Fitzgerald contraction. Any 
ordinary physical contraction will usually have observable 
physical results ; for example, the total electrical resistance 
of a conductor will diminish. Trouton and Noble, Trouton 
and Rankine, Rayleigh and Brace, and others employed 
a variety of different methods to detect the Lorentz- 
Fitzgerald contraction, but invariably with the same 
negative results. Whether there is an ether or not, 
uniform velocity ivith respect to it can never he detected. 
This does not prove that there is no such thing as an 
ether but certainly does render the ether entirely super- 
fluous. Universal compensation is due to a change in local 
units of length and time, or rather, being merely different 
descriptions of the same reality, there is no compensation 
at all. 

There was another group of observed phenomena which 
could scarcely be fitted into a Newtonian scheme of 



XVI PRINCIPLE OF RELATIVITY 

dynamics without doing violence to it. The experimental 
work of Kaufmann, in 1901, made it abundantly clear that 
the " mass '^ of an electron dei)ended on its velocity. So 
early as 1881, J. J. Thomson had shown that the inertia of 
a charged })article increased with its velocity. Abraham 
now deduced a formula for the variation of mass with 
velocity, on the hypothesis that an electron always remain- 
ed a rigid sphere. Lorentz proceeded on the assumption 
that the electron shared in the Lorentz-Fitz2:erald eontrae- 
tion and obtained a totally di:fferent formula. A very 
careful series of measurements carried out independently b}^ 
Biicherer, Wolz, Hupka and finally Neumann in 1913, 
decided conclusively in favour of the Lorentz formula. 
This "contractile^"' formula follows immediately as a direct 
consequence of the new Theory of Relativity, without any 
assumption as regards the electrical origin of inertia. Thus 
the complete agreement of experimental facts witli the 
predictions of the new theory must be considered as 
confirming it as a principle which goes even beyond the 
electron itself. The greatest triumph of this new theory 
consists, indeed, in the fact that a large number of results, 
which had formerly required all kinds of special hypotheses 
for their explanation, are now deduced very simply as 
inevitable consequences of one single general principle. 

We have now traced the history of the development of 
the restricted or special theory of Relativity, which is 
mainly concerned with optical and electrical phenomena. 
It was first offered by Einstein in 1905. Ten years later, 
Einstein formulated his second theory, the Generalised 
Principle of Relativity. This new theory is mainly a theory 
of gravitation and has very little connection with optics 
and electricity. In one sense, the second theory is indeed 
a further generalisation of the restricted princijole, but the 
former does not really contain the latter as a special ease. 



HISTORICAL INTRODUCTION Xvii 

Einstein's first theory is restricted in the sense that it 
only refers to uniform reetiliniar motion and has no appli- 
cation to any kind of accelerated movements. Einstein in 
his second theory extends the Relativity Principle to cases 
of accelerated motion. If Relativity is to be universally 
true, then even accelerated motion must be merely relative, 
motion tjetioeen matter and matter. Hence the Generalised 
Principle of Relativity asserts that " absolute " motion 
cannot be detected even with the help of gravitational laws. 

All movements must be referred to definite sets of 
co-ordinate axes. If there is any change of axes, the 
numerical magnitude of the movements will also chano'e. 
But according to Newtonian dynamics, such alteration in 
physical movements can only be due to the effeet of ceitain 
forces in the tield.^ Thus any change of axes will introduce 
new '• geometrical" forces in the field which are quite 
independent of the nature of the body acted on. Gravitation- 
al forces also have this same remarkable property, and 
gravitation itself may be of essentially the same nature as 
these '^ geometrical" forces introduced by a change of axes. 
This leads to Einstein's famous Principle of Equivalence. 
A gravitational field of force is strictl/j equivole^it to one 
introduced tjy a transformation of co-ordinates and no possitjle 
experiment can distinguish fjetween the tioo. 

Thus it may become possible to " transform away '' 
gravitational effects, at least for sufficiently small regions of 
space, by referring all movements to a new set of axes. This 
new "framework" may of course have all kinds of very 
complicated movements when referred to the old Galilean 
or *' rectangular unaccelerated system of co-ordinates." 

But there is no reason why we should look upon the 
Galilean system as more fundamental than any other. If it 

* Note A. 



XVlll PEIXCIPLE OF EELATIYITY 

is found simpler to refer all motion in a gravitational field 
to a special set of co-ordinates, we may certainly look upon 
this special ^'framework" (at least for the particular region 
concerned), to be more fundamental and more natural. We 
may, still more simply, identify this particular framework 
with the special local properties of space in that region. 
That is, we can look upon the effects of a gravitational 
field as simply due to the local properties of space and time 
itself. The very presence of matter implies a modification 
of the characteristics of space and time in its neighbour- 
hood. As Eddington saj^s ^' matter does uot cause the 
curvature of space-time. It is the curvature. Just as 
light does not cause electromagnetic oscillations ; it is the 
oscillations." 

We may look upon this from a slightly different point 
of view. The General Principle of Relativity asserts that 
all motion is merely relative motion between matter and 
matter, and as all movements must be referred to definite 
sets of co-ordinates, the ground of any possible framework 
must ultimately be material in character, it /v convenient 
to take the matter actually present in a field as the 
fundamental ground of our framework. If this is done, 
the special characteristics of our framework would naturally 
depend on the actual distribution of matter in the field. 
But physical space and time is completely defined by the 
•' framework." In other words the '' framework " itself is 
space and time. Hence w^e see how pit i/sical space and time 
is aetuallv defined bv the local distribution of matter. 

There are certain magnitudes which remain constant by 
any change of axes. In ordinary geometry distance 
between two points is one such magnitude ; so that 
hx'^ +^^^ H-5,e'^ is an invariant. In the restricted theory of 
light, the principle of constancy of light velocity demands 
that 8ir2 +8^^ -|.8^2 __^2g^,2 should remain constant. 



HISTORICAL INTUODUCTION XIX 

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

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

We have seen that Einstein's theory requires local 
curvature of space-time in the neighbourhood of matter. 
Such altered characteristics of space and time give a 
satisfactory explanation of an outstanding discrepancy in 
the observed advance of perihelion of Mercury. The large 
discordance is almost completely removed by Einstein's 
theory. 

Again, in an intense gravitational field, a beam of light 
will be affected by the local curvature of space, so that to 
an observer who is referring all phenomena to a Newtonian 
system, the beam of light will appear to deviate from its 
path along an Euclidean straight line. 

This famous prediction of Einstein about the deflection 
of a beam of light by the sun's gravitational field was 
tested during the total solar eclipse of May, 1919. The 
observed deflection is decisively in favour of the Generalised 
Theory of Relativity. 



XX PRINCIPLE OF RELATIVITY 

It should be uotecl however that the veloeitv of li^ht 
itself would decrease in a gravitational field. This may 
appear at first sight to be a violation of the principle of 
constancy of light-velocity. But when we remember that 
the Special Theory is explicitly restricted to the case of 
unaecelerated motion, the difficulty vanishes. In the 
absence of a gravitational field, that is in any unaecelerated 
system, the velocity of light will always remain constant. 
Thus the validity of the Special Theory is completely 
preserved within its own restricted field. 

Einstein has proposed a third crucial test. He has 
predicted a shift of spectral lines towards the red, due to an 
intense gravitational potential. Experimental difficulties 
are very considerable here, as the shift of spectral lines is a 
complex phenomenon. Evidence is conflicting and nothing 
conclusive can yet be asserted. Einstein thought that a 
gravitational displacement of the Fraunhofer lines is a 
necessary and fundamental condition for the acceptance of 
his theorv. But Eddino'ton has pointed out that even if 
this test fails, the logical conclusion would seem to be that 
while Einstein's law of gravitation is true for matter in 
bulk, it is not true for such small material systems as 
atomic oscillator. 

CONCLI SIGN 

From the conceptual stand-point there are several 
important consequences of the Generalised or Gravitational 
Theory of Relativity. Physical space-time is perceived to 
be intimatel}' connected with the actual local distribution 
of matter. Euclid-Newtonian space-time is itot the actual 
space-time of Physics, simply because the former completely 
neglects the actual presence of matter. Euclid-Newtonian 
continuum is merely an abstraction, while physical space- 
time is the actual framework which has some definite 



HISTORICAL INTRODUCTION XXI 

curvature due to the presence of matter. Gravitational 
Theory of Relativity thus brings out clearly the funda- 
mental distinction between actual physical space-time 
(which is non-isotropie and non-Euclid-Newtonian) on one 
hand and the abstract Euclid-Newtonian continuum (which 
is homogeneous, isotropic and a purely intellectual construc- 
tion) on the other. 

The measurements of the rotation of the earth reveals a 
fundamental framework which may be called the ^' inertial 
framework." This constitutes the actual physical universe. 
This universe approaches Galilean space-time at a great 
distance from matter. 

The properties of this physical universe may be referred 
to some world-distribution of matter or the "inertial frame- 
work" may be constructed by a suitable modification of the 
law of gravitation itself. In Einstein's theory the actual 
curvature of the ** inertia! framework " is referred to vast 
quantities of undetected world-matter. It has interesting 
consequences. The dimensions of Einsteinian universe 
would depend on the quantity of matter in it ; it would 
vanish to a point in the total absence of matter. Then 
again curvature depends on the quantity of matter, and 
hence in the presence of a sufficient quantity of matter space- 
time may curve round and close up. Einsteinian universe 
will then reduce to a finite system without boundaries, like 
the surface of a sphere. In this " closed up " system, 
light rays will come to a focus after travelling round the 
universe and we should see an ''anti-sun'"' (corresponding to 
the back surface of the sun) at a point in the sk}^ opposite 
to the real sun. This anti-sun would of course be equally 
large and equally bright if there is no absorption of hght 
in free space. 

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



XXll PRINCIPLE OF RELATIVITY 

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

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

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

P. C. Mahalanobis. 



HISTORICAL INTRODUCTION XXlll 



Note A. 

For example consider a massive particle resting on a 
circular disc. If we set the disc rotating, a centrifugal force 
appears in the field. On the other hand, if we transform 
to a set of rotating axes, we must introduce a centrifugal 
force in order to correct for the change of axes. This 
newly introduced centrifugal force is usually looked upon 
as a mathematical fiction — as '' geometrical" rather than 
physical. The presence of such a geometrical force is usually 
interpreted us being due to the adoption of a fictitious 
framework. On the other hand a gravitational force is 
considered quite real. Thus a fundamental distinction is 
made between geometrical and gravitational forces. 

In the General Theory of Relativity, this fundamental 
distinction is done away with. The very possibility of 
distinguishing between geometrical and gravitational forces 
is denied. All axes of reference may now be regarded as 
equally valid. 

In the Restricted Theory, all '^unaccelerated" axes of 
reference were recognised as equally valid, so that physical 
laws were made independent of uniform absolute velocity. 
In the General Theory, physical laws are made independent 
of "absolute" motion of any kind. 



On 

The Electrodynamics of Moving Bodies 



BY 
A. EjNSTEIJf. 



INTRODUCTION. 

It is well known that if we attempt to apply Maxwell's 
electrodynamics, as conceived at the present time, to 
moving bodies, we are led to assy met ry which does not 
ao^ree with observed phenomena. Let us think of the 
mutual action between a magi-net and a conductor. The 
observed phenomena in this case depend only on the 
relative motion of the conductor and the magnet, while 
according to the usual conception, a distinction must be 
made between the cases where the one or the other of the 
bodies is in motion. If, for example, the magnet moves 
and the conductor is at rest, then an electric field of certain 
energy-value is produced in the neighbourhood of the 
magnet, which excites a current in those parts of the 
field where a conductor exists. But if the magnet be at 
rest and the conductor be set in motion, no electric field 
is produced in the neighbourhood of the magnet, but an 
electromotive force which corresponds to no energy in 
itself is produced in the conductor; this causes an electric" 
current of the same magnitude and the same career as the 
electric force, it being of course assumed that the relative 
motion in both of these cases is the same. 



il PRINCIPLE OF RELATIVITY 

*2. Examples of a similar kind such as the uusueeessful 
attempt to substantiate the motiou of the earth relative 
to the " Light-medium " lead us to the supposition that 
not only in mechanics, but also in electrodynamics, no 
properties of observed facts correspond to a concept of 
absolute rest: but that for all coordinate svstems for which 
the mechanical equations hold, the equivalent electrodyna- 
mieal and optical equations hold also, as has already been 
shown for magnitudes of the first order. In the following 
we make these assumptions (w^hich we shall subsequently 
call the Principle of Relativity) and introduce the further 
assumption, — an assumption which is at the first sight 
quite irreconcilable with the former one — that light is 
propagated in vacant space, with a velocity c which is 
independent of the nature of motion of the emitting 
bod}'. These tw^o assumptions are quite sufficient to give 
us a simple and consistent theor^^ of electrodynamics of 
movino' bodies on the basis of the Maxwellian theory for 

a t,' 

bodies at rest. The introduction of a ^^ Lightather" 
will be proved to be superfluous, for according to the 
conceptions which will 'be developed, we shall introduce 
neith er a space absolutely at rest, and endowed with 
special properties, nor shall we associate a velocity -vector 
with a point in which electro-magnetic processes take 
place. 

3. Like every other theory in electrodynamics, the 
theory is based on the kinematics of rigid bodies; in the 
enunciation of every theory, Ave have to do with relations 
betw^een rigid bodies (co-ordinate system), clocks, and 
electromagnetic processes. An insufficient consideration 
of these circumstances is the cause of difficulties with 
which the electrodynamics of moving bodies have to fight 
at present. 



ON THE ELECTKODYXA MlCS Oh' 3I0VlNa BODIES 3 

I.-KINEMATIOAL PORTION. 

§ 1. Definition of Synchronism. 

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

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

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

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



4 PKINCIPLE OF RELATIVITY 

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

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

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

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

clocks are synchronous^ if 

t - 1 = t' - t . 

B A A B 



02^ THE ELECTRODYNAMICS OF MOVING BODIES 5 

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

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

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

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

In accordance with experience we shall assume that the 
magnitude 

2 AB 

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

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

§ 2. On the Relativity of Length and Time. 

« 

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

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



6 PRINCIPLE or- RELATIVITY 

which have a uniform translatorv motion relative to each 
other. 

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

, .. Path of Li<yht 

velocity = T—r , ^ , . , 

^ Interval or tmie 

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

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

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

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

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

* Vide Note 4>. 



ON THE ELBCTIIODYNAMICS OF MOVING BODIES I 

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

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

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

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

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

from I. 

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

Relativity of Time. 

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

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

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

B^ 

time t' . Taking into consideration the principle of^ 

A 

constancy of the velocity of light, we have 



and 



t - 

B 


■f = 
A 


'^B 


c-v' 


t' ■ 
A 


-t = 
B 


r 

AB 



b PRINCIPLE OF RELATIVITY 

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

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



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

from a stationary system to a system which 

moves relatively to this with 

uniform velocity. 

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

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

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



ON THE ELECTRODYNAMICS OF MOVING BODIES 9 

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

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

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

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

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



10 PRINCIPLE OF REI ATIVITY 

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

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

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

C — V /. 

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

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

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

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



/, III-' \ 



where a is an unknown function of v. 

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



I 



ON THE ELECTRODYNAMICS OF MOVING BODIES 11 

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



^=.c T , i.e. i=:ac i t— — — \, 



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



therefore we have 






C — V 



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



c2 






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



7] = CT = aC I t — J 



where • , =^, i>;'=^j 

c c 

Therefore t?=a ., . y, l=a • ■ z. 

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

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



c2 



of V. 



12 PRINCIPLE OF RELATIVITY 

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

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

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

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

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

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

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

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

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

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

* Yxde Note 9. 



ON THE ELECTRODYNAMICS OF MOVING BODIES 13 

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

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

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

It appears that the systems K and ¥ are identical. 

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

I 

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

we have on grounds of symmetry, 

I I 



cfi{v) <f>{—v) 



l4 PRINCIPLE OF RELATIVITY / 

§ 4. The physical significance of the equations 

obtained concerning moving rigid 

bodies and moving clocks. 

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

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



'13 



( Vi-^J 



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






K V 1--^, R, R. 

• 

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

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

c 

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



ON THE ELECTRODYNAMICS OF MOVING BODIES 15 

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

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

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

1 / ^ \ -, 

T= — zznzr I ^~"~2^ 15 ^^d x=.vty 



...,■,=[._ V.-g 



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



of motion, to the second order of approximation. 

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



v^ 



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



16 PRINCIPLE OF RELATIVITY 

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

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

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

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

§ 5. Addition-Theorem of Velocities. 

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

where w^ and lu are constants. 

■n 

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






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

i+_i 1+ « 



c"" ' c2 



ON THE ELECTRODYNAMICS OF MOVING BODIES 17 

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

w 
and a = tan~^ - . 

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



a -1 

2 



u= 



[(i'2+2i;2+2 vw cos a)— I "■ J I 



-, . viv cos a 



c^ 



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



1+ "^ 

^2 



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



* 



IJ=c — 2c-x-A_ <^ 

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

U= -^±^ =c. 
1+ ''' 



c^ 



* Vide Note 12. 



3 



18 PRINCIPLE OF RELATIVITY 

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



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



We see that such a parallel transformation forms a 
group. 

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

II.-ELECTBOBYNAMICAL FART. 

§ 6. Transformation of Maxwell's equations for 

Pure Vacuum. 

On the nature of the Electromotive Force caused hy motion 

in a magnetic field. 

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



\ |,[^'Y,^]= 



a 


6 


6 


9;c 


92/ 


a^ 


L 


M 


N 



ON THE ELECTRODYNAMICS OF MOVING BODIES 



19 



and 



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



a. 

a.^ 


a 

dy 


a 
a^ 


X 


Y 


z 



(1) 



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

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

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



a 

a^ 



dv 



a^ 
a^ 



c c 



and 



1 a^ 



[L, (3(M+ ^IZ), «N 



-1-Y)] 



a^ 



d_ 



a_ 
a^ 



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



(2) 



where /?: 



vl — i'Vc' 



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



20 



PRINCIPLE Oi^ RELATIVITY 



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



1 9 
c 'Qi 



(X', Y', Z') ^ 



6^ 



6^ 
9^ 



I ■ 



M 



\i 



1 

N' 



C OT 



6 6 6^ 

6^' dr; 94 



X' 



Z' 



... (3) 



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



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

c c 



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



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

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

c c 



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



ON THE ELECTRODYNAMICS OF MOVING BODIES 21 

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

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

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

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



22 PRINCIPLE OF RELATIVITY 

v 

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

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

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

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

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

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

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

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

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

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

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



ON THE ELECTRODYNAMICS OF MOVING BODIES 



23 



where 



l- 



V 



lv\ 



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



m 



vi 



n — 



n 



1 Iv 



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



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



l—cos^ 



V 



V 



V 



1- 



V 



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

1 v\-s. 



V =v 



1+ 



C 



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

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



cos$— 



cos ^'= 



V 



c 



1— -cos <l> 
c 



24 PRINCIPLE OF RELATIVITY 

This equation expresses the law of observation in its 



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

cos $ = — - . 



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



A'2=A' 



j 1 — - cos <i> I 



2 



1- ^' 



c^ 



If $=o, this reduces to the simple form 



1-'-! 

C 

A'*=A« 

1+^ 



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

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

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

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



ON THE ELECTRODYNAMICS OF MOVING BODIES 25 

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

A. 

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

(x — cUy + (y-cmty + (:-~cnfy =11^ 

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

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

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

S 



'JH 



cos $ 

c 



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

4 



26 



PRINCIPLE OP RELATIVITY 



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



A'' 



E 



8 



S' 



TT 



8 



S 



1— - cos $ 

c 



TT 



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



E' 
E 



1- 



V 



1 + 



V 



J 



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

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

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



V 



1 — COR <J> 



A' = A 



a/ 



J. 2 



COS $ — 



c 

v 



COS $' = 



- COS 4> 



1 — - COS 9 

I c 

V =V =.=rr:^ 

,2 



.\/ 1-^; 



ON THE ELECTRODYJSAxMICS 0¥ AJOVlNG BODIES 27 

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

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

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



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

^2 1 ^^ 



■N 



V -s 



C2 C^' 



cos $'" = 



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

C \ (''■'J c 

1+ 1 ■.„ 1 — 2-cos$H 

C COS $" c c^ 



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



/ -S ( -I )' 



1- 



\ 



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

system) is . The amount of energy going 

STr{c cos ^—v) 

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



A 2 

P = 2 — 



(cos ^ - 0' 

Hi)' 



28 



PKINCIPLE OF EEL.VHV1TY 

i. » 



As a first approximatioD^ we obtain 



A2 

P=2 ^ 



bir 



coa^ 4>. 



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

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

§ 9. Transformation of the Maxwell-Hertz Equations. 

Let us start from the equations : — 



u 



PUx + 



6x\ _aN 8M 



6^ 



7 dy 



dz 



1/ _l9^\ 



a^i_aL 

6 .'.' 6 y 



1 6L 6Y 6Z 



c dt 63 


dy 


laM az 


ax 


c dt dx.\ 


a^ 


1 aN_ax 


aY 



c dt dy d -v 



)■ 



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

a.'= a^ a~ 

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



ON THE IfiLECTHODYNAMlCS 01' MOVING BODIES ^9 

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

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






where 



Uc. 


,ax'-i aN' 
ar J a^ 


aM' 
a^ ' 


a L' a Y' 
ar a^ 


az' 
a^? 


u 


,aY'-i aL' 
ar J dc 


aN' 
a^ ' 


a M' a z' 
ar a^ 


ax' 


Wc, 


, az'-] aM' 
ar J a^ 

u^ — V 


aL' 

dv ' 


a N' a X' 
ar a^ 


aY' 
a^ ' 



u 



y 



,(i- ^^) 



' 6X' aY'.dZ' 

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



:^(l-t)'" 



« 



,(l-Fii.^) 



"i, 



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

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



30 PRINCIPLE 0¥ RELATIVITY 

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



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

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

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



df" dt^ df" 



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

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

« 

immediately following this epoch. 

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



ON THE ELECTRODYNAMICS OF MOVING BODIES 31 

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

From the suppositions made above, in combination 

with the principle of relativity, it is clear that regarded 

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



dr^ dT^ ' dT"" 



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






y 



_/ 



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



dt^ m ^3 ■' di'' m ft \ c ) 



(A) 

d\ 



= 1 i(z+rM) 

m B \ c 7 



dt^ m /5 



32 



PRINCIPLE OF RELATIVITY 



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



myS' 



d\c 



dt'' 



■.eX = eX' 



^ 



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



r 



dt^ 



- '^] =^Y' y 



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



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

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



Longitudinal mass = 



m 



( V'- %y 



# 



Transversal mass = 



■m 



V^- % 



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



* Vide Note 21. 



ON THE ELECTRODYNAMICS OF MOVING BODIES 38 

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

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

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

V 

V c^ 

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

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

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

5 



34 PRINCIPLE OF RELATIVITY 

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

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

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

^ =- . 
A, c 

This relation can be tested by means of experiments 

because the velocity of the electron can be directly 

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

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

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






«N 



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



ALBRECHT EINSTEIN 

[^ short hiograpJiical note.~\ 

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

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

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



36 PRINCIPLE OF RELATIVITY 

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

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

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

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



ALBERT EINSTEIN 37 

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

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

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

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

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

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

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



38 PRINCIPLE OF HELATIVITT 

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

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

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

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



ALBERT EINSTEIN 39 

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

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



INTRODUCTION. 

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

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

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

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



3 PRINCIPLE OF RELATIVITY 

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

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

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

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

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

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

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

* Yiie Note 4. f Note 5. 



INTRODUCTION 3 

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

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

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

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

* See uQtes on § S and 10. 



4' PRINCIPLE OF RELATIVITY 

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

NOTATIONS. 

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

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

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



NOTATIONS 

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

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

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



5 PRINCIPLE OP RELATIVITY 

PAET I § 2. 

The Limiting Case. 

The Fundcwiental Equations for Ather. 

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

(i) Curl m— -»- = pu 

(ii) div e= p 

(iii) Curl^ +.||' = 

(iv) div m = (? 

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

(/>nP2; ^3; P4) for 

(pu,, puy, pu,, ip) 

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

Further I shall write « 

for 

m,, m^, m,, — ie„ — ie , — ie,. 

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

* See note 9 



THE FUNDAMENTAL EQUATIONS FOR ATHER 



7 



Therefore 

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

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



8Xc 



3xj 



+ 



+ 



¥. 



32 



8x 



g/4t . ?/ 

Sxj 



+ 



42 



Sx, 



+ 



0/l3 
8X3 


+ 


S/t4 
8X4 


?^2 3 
8X3 


X 


S/24 
8X4 




+ 


0/34 
^^4 


?A3 







8x, 



■Pi 



= P2 



= P; 



= ^4 



(A) 



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



¥,, 






8xj 

?A4 



+ 



^^4 2 , ?/2_3 

8X3 8X4 

^14 , ?Al 

Sx„ ^ 



+ - 



S/4, 



8x, 



+ 



8x4 

?/l2 



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



8/*. 



2 1 



8x, 



= 

= 

= 

= 



(B) 



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

§ 3. 

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



8 PRINCIPLE OF RELATIVITY 

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

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

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

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

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

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

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

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

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

^^4' = — ic, sin ixjf 4- .^4 cos i\^, 

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

(? -f ^ 



THE FUNDAMENTAL EQUATIONS FOR ARTHER 9 

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



^l-q^ x/l-q 



2 



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

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

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

the coefficients being essentially real. 

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

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

» 

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

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

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

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

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



(5) 



■qu^ +1 



p =p — • \ , P^^r -p \ ^ZIZZIIl 



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



10 PRINCIPLE OF RELATIVITY 



(6) ^j = ?i^i^, ,„V = 2^4^, e.'=e 



» ' 



(7) w',' = ■ , e'/ = , m','=m 



z • 



VI — q^ VI — q"" 

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

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

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

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

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

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

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

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

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

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

Special libnENTZ Transformation. 

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



SPECIAL LORENTZ TRANSFORMATION ll 

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

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

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

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

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

V 1 — q^ 

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

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

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



12 PRINCIPLE OF RELATIVITY 



further 

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

Vl — q" 

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

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

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

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

Now we shall make a very important observation 

about the vectors u and u. We can again introduce 

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

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

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

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

is transformed into 

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

(18) p vr^r:i? 

is an invariant in a Lorentz-transformation. 

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

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



VT 



u 



so that Wi' +(u,^ +W3' 4-W4* = — 1. 

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



SPECIAL LOEENTZ TRANSFORMATION 13 

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

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

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

The differentials donoting the displacements of matter 

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

adjacent space-time point. 

After the Lorentz-transfornation is accomplished the 

voeocity of matter in the new system of reference for the 

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

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

Now it is quite apparent that the system of values 



X^—Oi^, ■f'2=<^25 aJ3=W3J '^'4=W 



4 



<• 



is transformed into the values 

a 

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

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

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

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

* See Note. 



14 



PRINCIPLE OF RELATIVITY 



§ 5. Space-time Vectors. 

Of the 1st and 2nd kind. 

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

Let us take a linear homogeneous transformation, 



X. 



■r,. 



X. 



V.^4V 



a 



a 



a 



1 1 



2 1 



3 1 



s ^41 



a 



a 



a 



a 



1 2 



2 2 



33 



42 



a 



a 



a 



a 



13 



23 



33 



43 



a 



a 



a 



a 



1 4 



2 4 



34 



4 4.^ 



■V 



X, 



X, 



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

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

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



i 



SPACE-TIME VECTORS I^ 

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

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

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

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

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

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



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

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

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

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

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



16 PRINCIPLE OF RELATIVITY 

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

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

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

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

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

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

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

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

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

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

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



SPECIAL LORENTZ TRANSFORMATION 17 

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

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

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

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

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

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

{m,, — i e,y -\-{7n,,--i e^y=^o, 

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

tan(</>-fiV)=?iL±t^v^ 



Vide Note. 



18 • PRINCIPLE OF RELATIVITY 

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

§ Concept op Time. 

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

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

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



CONCEPT OF TIME 19 

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

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

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

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

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



^0 PRINCIPLE OF RELATIVITY 

PART II. ELECTRO-MAGNETIC .. 
PHENOMENA. 

§ 7. Fundamental Equations for bodies 

AT REST. 

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

us when proper fundamental data are given — to 

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

The relations in question can be divided into two 
classes. 

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

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

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



FUNDAMENTAL EQUATIONS FOE BODIES AT REST 21 

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

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

(i) Curl m — — r— = C, (u) div e =]p. 

ot 



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

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

(V) e = € E, M = />iw, C = crE, 

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

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

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

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



u 



PRINCIPLE OF EELATIVITY 



Then the fundamental Equations can be written as 



9/12 ^ 9/1 



9. 



9 



3 ^ 9/i.i _ g 



^^) lt"'+ 



+ 



8/ 



2 3 



9.t 



+ -, 



9/ 



2 4 



9. 



V 



9/3_X 

9.<^i 



+ 



9/ 



3 2 



9.< 



+ 



+ 



9/3. ^ 

9r. 



9Ax + 9/ 
9.t^i 



4 2 



+ 



3/ 



4 3 



• C, 



9 <. 



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

9F34. , 9F4,, , 9F23 



^ 



-\ 



9 .t^^ 



+ 



+ 



aF.3 


9*1 


9F,, 


9.C1 


9F35 



+ 



+ 9Z- + 



9F, 



9ic. 



+ 



+ 



9^:4 


8F„ 


a.. 


aF,, 



= 



= 



y 



9ic, 



= 



9 «i 



+ 



9F, 

9aj„ 



+ 



9F^ 

9 ^'3 



= 



§ 8. The Fundamental Equations. 

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

The first Axion shall be, — 

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



* Einzelue stelle der Materie. 



THE FUNDAMENTAL EQUATIONS / 23 

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

;. 

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

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

Shortly I can signify the third axiom as ; — 

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

This axiom T call the Principle of Relativity. 

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

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





". 


J ^2 


_ ^y 


y Wg 


_ ". 




(Oj 


\/i-ir^ 


^/l-u^ 


\/l-u'-^ 
















W4 



\/l-u' 



* Vide Note. 



24 PRINCIPLE OF RELATIVITY 

with the relation 

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

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

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

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

(2) and the equations 

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

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



'rut i'UNDAiiEKTAL EqUATlONS 25 

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

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

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

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

mentil equations sought by us for the moving bodies. 

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

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

I I curl m — - = Ci, II J div ex=p 

III \ curl E + ^97 = « IvVliv M=o 

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

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

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



L 



2d PRINCIPLE OF RELATIVITY 

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

(C) 6'+[2^ wz]=e(E +[/'M]), 

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

(D) M-[^^E]=/.Oyz- [«.']), 

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

hy ^^rr^ 

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

(E+kM])„ 



C,7 =cr 



v^l -«^ 



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

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



FUNDAMENTAL EQUATIONS IN LORENTZ THEORY 27 

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

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

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

lit 

— curl [^^D]. 
(TO div X)=p 

{l\") curl E =. — ^ , Div B-0 (V) 

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

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

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

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

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

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



28 PEfN-CIPLE OF RELATIVITY 

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

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

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

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

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

(,_1) [E + (..B)]„-(l-.rO (D-E), 

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

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

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



FUNDAMENTAL EQUATIONS OF E. COHEN 29 

§10. Fundamental Equations of E. Cohn. 

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

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

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

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

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

transform into 

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

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

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

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

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

(53) E = €E,M=/.M, J = (rE. 



30 



PRINCIPLE OF RELATIYlTr 



§11. Typical Representations of the 
eundamental equations. 

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

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



a 



1 1 



.a 



1 9 



a 



p 1 



,a 



P H 



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

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

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



TYPICAL KEPKESENTATIO]S^S 



31 



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



A = hhi 



.«,; 



1 



a 



1 ? •■ 



a 



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



B = 



1 1 



'^i 



V',.. 



p 1 



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

members are ai, k+hi,k. 



2^ If w^e have two matrices 



A = 



a 



a 



1 1 



p 1 



,a 



.a 



1 1 



p 'I 



B = 



h 



1 1 



•^1 V 



^1 



P r 



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



C 



\c 



1 1 



^ i» r 



1 '• 



• ^' l> V 



* <? ^ U h 



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

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



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



^2 



PRInCIPIE of feELATlVltr 



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



rows. 



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

(3 JO 



1 1 



e. 



'3 1 



1 2 



Gg o 



3 2 



1 3 



1 4 



^^2 



e., 



^34 



10 
10 
10 

41 ^42 "=43 ^44 1 

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

I 

A matrix I 

/ . '^ J 12 /is /l4 



2 4 



3 4 



I/31/32O /j 

! 

'Al /*2 /is O 

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

the dual, alternating matrix 

(35) 



/ 



■jf— 



o 



J Si J 4 2 .7 2 



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

/s 3/13/ 21 ^ ' 



TYPICAL REPRESENTATIONS 



33 



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

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

The determinant of /is therefore the square of the 



-A 

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



4°. A linear transformation 

which is accomplished by the matrix 



A = 



^11' ^125 ^135 *14 



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



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



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



4 4 



will be denoted as the transformation A 

By the transformation A, the expression 

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



for III 






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

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

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



34 



PRINCIPLE OF JIELATIVITY 



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

From the condition (39) we obtain 

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

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

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

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

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



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



(4.2) 



/= 



O 



./12 JiS /l4 



/21 ^ ./2 3 ./: 



24 



./31 /s2 o ./; 



34 



o 



./ 4 1 y42 ./4: 

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

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



* Vide note 13. 
t Vide note 14. 



TYPICAL REPRESENTATIONS 35 

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

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

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

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

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

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

10^ lu^ 10.. lU ^ 

8 1 S c) So 4 

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

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

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

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

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

written as [w, 5].^ 

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



36 PUINCIPLE OF EELATIVITY 

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

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

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

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

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

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

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

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

21 J. 71 y u^ i 



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



u^ . 



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

and — 2,*(o^>0. 

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

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

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



I 

I 



* Vide note 15. 



TYPICAL UEPRESENTATIONS 37 

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



of the space-vector 



\/X^y 2 



u- 



and further ci , = — - J— . — =L 



Because F is an alternating matrix, 

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

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

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

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

{ C } a./=ewF. 

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

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

^^ = — i( wj3^,+ 0)3/^2+^^23) 1 

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

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

I 

^4,I=—i ((0j32+W2/i3-f (03/21 ) J 

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



vr 



■u 



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



^\—u'^' 



* Vide note 16. 



38 



PRINCIPLE OP UELATIVITY 



Among these there is the relation 

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

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

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

Relations analogous to these hold among the quantities 

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

formula 

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

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

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

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

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

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

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

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

Let us now take the vector of the 1st kind 

with the components i2j_ = — L w. 



etc. 



^2 


W3 


<^4 


^. 


^3 


^4 


*2 


^S 


^4 



TYPICAL REPRESENTATIONS .'J 9 

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

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

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

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



[n„ 03,0.3 = 



(J) <I> <b 



^1 ^2*^3 



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

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

_ — I ^H C„+p , 

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

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

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



C„-^ 



^H p' _ ^« — I « I p _ J, 



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

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



40 



PRINCIPLE OF RELATIVITY 



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



m 



S+ (co,?)a)=r — cro)F, 



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

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



2 4 



§12. The Differential Operator Lor. 

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

Qi O q q 

'•-'21 2 2 *^ 2 3 ^-^ 

q q a q 

^31 ^32 ^^33 ^~34 

q q q q 

•^41 *^42 '^43 ^^44 

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

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

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

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

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

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



I'Hfi DIFFERENTIAL OPERATOR LOR 



41 



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



d a a a 



a " a 2/ a^ ^a^' 



or (6;^) 



a a 



a.t'i a.t^2 a«s a** 



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

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

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

matrix 

Kj Kj Kg K^^ 



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



a .' 1 



'2 



'3 



'■ 4 



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



lor'= 



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



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



K'. = ^3^ + 



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



i + ^^p-^ + 



4, k 



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

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

a _ a 



we have the rule 



'<-k 



a « 1 , a a 



" 3 



aa?i a^ryt' a.»'a a.t'A-' 



+ 



a^'s • a 



-r + 



a«4 



a.t'a Q^Vk' . a»4 Q'^'k 



a.ii 



^i/t + 



dx, 



^2 A "^ 



Q X 



Cl^k + 



s 



dx, 



a 



ik' 



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



* Vide note 17. 



42 



PRINCIPLE OF RELATIVITY 



Therefore it follows that 

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

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

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



aL aL aL aL 



a .( 1 a 



.(■< 



Qx, 



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



lor s 



asi , a^ 



2 1 a s s a^ 4 



a<'-'i a i«2 ' a«?3 

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

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

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

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



a ^('' . 



a.<^; 



^•X'^ 



a./.x 
a .*' 1 

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



+ 



a/. 



+ 



a/ 



2^3 I 

a <v , a .'( 



2 4 



'2 



a/a 

9 a;. 



Qf~ ^ a/ 



4 2 



Qx^ 



a* 



+ 



a/. 
a^K, 



THE DIFFERENTIAL OPERATOR, LOR 43 

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

{A} \ovf=-s, 

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

{B} log F-^ = 0. 

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

(68) 9£i + 91. + 9_l3, + 9i'* = 0, 

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

while the relation 

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

I shall now collect the results. 

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






(?^= velocity of matter), 

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



u 



PUINCIPLE OE RELATIVITY 



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

{A} \0Yf=—S 

{B} log ¥^ = 

{C} (o/=€a)F 

{D} <oF^ = /xCo/^ 

{E} S+{o)s), ?(;=— o-toF. 

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

lor (lor F-^) = 0. 

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

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

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



/- 





A. 


/l 3 


A« 


F = 





F 

^ 1 S 


F F 




/.. 





/a 3 


J 2 4 




F 





F F 

^23 ^24 




/s 1 


/>. 





JS4- 




F 

-■- 3 1 


F 

^ S2 


^ F3, 




^^ 


A. 


/* s 







F 


F 

-"■42 


P*, 










* Via 


le note 1 


9. 







THE PRODUCT OF THE FIELD- VECTORS /f 



4t5 



are of much importance. Let us write. 



(70) fF = 



Sj 1 — L Sj 

S21 
S31 



S 



I s 



O T Q 



s 



3 2 



Oo Q — Jj 



'3 3 



^14 
S24 

s 



84 



S41 



S 



4, 2 



s 



4 3 



S. - — L 



'4 4 



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

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



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



1 4 



+/2.F 



2 4 



' /a 4 ■'^3 4 I 



Then we shall have 



'12 ^12 



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



In order to express in a real form, we write 



(74) S: 



Now X , = 



^11 ^12 


Sis 


s.. 




X. 


^2 1 ^2 2 


^2 3 


S24 




Xy 


S31 S32 


S3 3 


S3 4 




X, 


S41 S^2 


S+3 


S44 




— ^ 


1 r 

=2 ^ "^- 


M.- 


TTlyMy 


— 7 


M,Ik1 



z 



z 



z 



-2T. 

-zT, 

-2T. 



■^X, ^lY, -iZ, T, 



— m,M- +<?^E^— CyE,— e,E, 



46 



PRINCIPLE OF RELATIVITY 



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

Xt=ey'M.,—eMy, T^=m^Ey—m,jE, etc, 

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

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

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



(77) Y*f*= 



— Sj ^ — L, — S 

— 84.^ — S 



1 2 



-s, 



'^912 ^1 >^ 



3 2 



2 R 






— ^33 — L, — S 



34 



4. 2 



-S 



4 ?< 



-S..-L 



and hence, we can put 

(78) /F=S-L, 



F^/^=-S-L, 



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

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

Since here SL = LS, we deduce that, 

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

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



* Vide note 18. 



THE PRODUCT OF 'JHK FIELD-VECTORS /f ii 

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

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

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

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

Det ^/ Det ^'f, 

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

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

(82) L = — ie$¥+^/x*^ 

(83) S,, = - I €^"$e,, - i/x*¥e,, 

-f € {<^,, $/, — ^4> (0/, <0j 

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

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

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



48 PRINCIPLE OF RELATIVITY 

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

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

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

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

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

expression as Det * S. 

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

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

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



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

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

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

(87) 0) (S -"S) = (€ /a - 1) n 

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

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



THE PRODUCT OP THE PIELD- VECTORS /f 49 

and from 83), 

T.-1},, T,=0„ T,,.:=fi3 

X^=:e/xOj, Y^^e/xfig, Zf=€jjLCi^ 

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

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

According to 71), we have 

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

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

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

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

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

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

(see also 83) that 

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

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

The space-time vector of the iirst kind 

(89) K=lor S, 

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

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

lor S=lor L + lor/F. 



50 PRINCIPLE OF RELATIVITY 

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

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

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

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

where N is the vector with the components 

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

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

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

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

ay 2 3 a /SI ~~a ^12 ■' c^ J 1 4 

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

d./ S 4 ~^ ■ J S - h 

••■■/< O.'A / 

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

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

(91) lorS = -5F + N. 

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



THE PRODUCT OF THE FIELD- VECTORS /'f 51 

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

dt OXh ^ OX,, 

for h = l, 2, 3,4. 

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



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



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

§ 14. The Ponderomotive Force.* 

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

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



a^ 2 6'^ vi. 



■ 

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



Vide note 40. 



52 tAINCIPLE OF RELATIVITY 

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

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

K + (a)K)aj, 

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

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

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



APPENDIX 
Mechanics and the Pv;Elativity- Postulate. 

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

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

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

,/___,,._ ,.' _ c ()\—qt) ,,_ qi\-hcH 

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

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

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



54 PRINCIPLE OF RELATIVITY 

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

_a;«_2/2 — -s+r^ ... ... (1) 

when • (?=qo. 

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

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

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

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

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

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



APPENDIX 55 

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

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

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

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

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

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

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

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



dT=Vdt^'-d.io''-dy^ —d;^=dtVl—ir" 

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



jdT=f V — idx^^+dr^'^+dr^^-^dx^^) 



56 PRINCIPLE OF RELATIVITY 

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

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

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

-(K)"-(f:)-(©HiT)'>»'if>» 

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

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

a® _o 6® _n 9® -0 9® ^0 

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

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



APPENDIX 57 

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

We shall now formulate the principle of conservation 
of mass. 

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

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

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



vl— ?t2 dr 



an 



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



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



58 PEINCIPLE OF RELATIVITY 

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

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

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

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

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

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

lor ,-= .§.^ + ^ + 4^^ + ^""^ 



dx^ Q.Cg 9^3 6. 



4- 



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

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



APPENDIX 59 

which is the coudition of cortinuitv 

a^j dy ' d: 67~' 

Further let us form the intefrral 

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

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

Then the integral (7) can be denoted by 

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

taken over all the elements of the sichel. 

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

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



60 PIIINCIPLE OF llELATIVITY 

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

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

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

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

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

k o.'a 6 a 

/(• = !, 2, 3, 4 

/^ = 1, 2, 3, 4 

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

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



- \S\S '^ 



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

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

by means of (9) and 

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



APPENDIX 61 

thfirel'ore : — 



dr 0>tA 



^•=1, 2,3,4. 
4. 



[k — 1, 2, 3, 
7i=l, 2, 3, 



We shall now subject the value of the differential 
quotient 

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



o:eneral ->r— ^ =c, for X = o. 



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



A=o 



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

Ms: /9f»„ J. 9^'.,. J.9**,, j.9fA 

cZ,i! cZi/ dz dt 

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



M-( 



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

(^J3 dy dz dt 



6'^ j:>RiNciPLfe oi' helativity 

the expression within the bracket may be written as 






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

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

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

wherebv — is meant the differential quotient in the 
dr 

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

dx dij dz dt. 

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

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

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



APPENDIX 



03 



3 i 



X, 


r. 


z. 




-iT. 


X, 


Y. 


Z, 




-iT, 


X, 


Y, 


z. 




-zT, 


-iX, 


-iY, 


— ' 


iZt 


T, 



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

^11 ^12 ^13 ^1 

(16) S= S21 S22 S23 S21 

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

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

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

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

a >-h 

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

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

X.+Y,+Z„+T,+X. -1^' +X, |?i>...Z,-|^^ 

ox oy 9z 



-X, 



a ;^ d .(^ a ^ 



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

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

The meaning is, that for each virtual displacement^ 



( 



c^(-8y-h8W) 
dX 



) -' 



(18) 



64 PEINCIPLE OF RELATIVITY 

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

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

O.t'i 0-«'2 0.<'3 0'<'4 

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

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

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

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

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

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

vector as 

X, Y, Z, • /T 

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

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



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



©•- (I)'- (!)•> ©■=-■• 

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

dr dr ar dr 

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

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



APlPENDiX 65 

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

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

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

d.e dy dz ■ dt 
dr ' dr ^ dr * dr ' 

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

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

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

dt \ dr / dt dt dt 

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



66 PRINCIPLE OF RELATIVITY 

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



m 



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



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

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

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

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

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

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

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

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



APPENDIX 6? 

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

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

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

Let us conceive that a material point F of mass m 
may, owing to the presence of another material point F"^, 
experience a moving force according to the following law. 
Let us picture to ourselves the space-time filaments of F 
and F"^ along with the principal lines of the filaments. Let 
BC be an infinitely small element of the princi})al line of 
F ; further let B^ be the light point of B, C^ be the 
light })oint of C on the principal line of F^; so that 
OA' is the radius vector of the hyperboloidal fundamental 
figure (23) parallel to B"^C^, finally D^ is the point of 
intersection of line B^C^ with the space normal to itself 
and passing through B. The moving force of the mass- 
point F in the space-time point B is now the space- 
time vector of the first kind which is normal to BC, 
and which is composed of the vectors 

3 

(24) mm^f^^'^] BD"^ in the direction of BD^ and 
another vector of suitable value in direction of B'^C"^. 



68 PRINCIPLE OP RELxiTIVITY 

Now by ( — if—o ) is to be understood the ratio of the two 

vectors in question. It is clear that this proposition at 
once shows the covariant character with respect to a 
Lorentz-group. 

Let us now ask how the space-time tilament of F 
behaves when the material point F"^ has a uniform 
trauslatory motion, /.(?., the principal line of the filament 
of F* is a line. Let us take the space time null-point in 
this, and by means of a Lorentz-transformation, we can 
take this axis as the /-axis. Let x, y, z, /, denote the point 
B, let T"^ denote the proper time of B^, reckoned from O. 
Our proposition leads to the equations 

d'^z _ ^ m^z (oa\^__ jz!^ d{t-r^) 
dr'^ ~ {t—r^Y <^^' "" {t-r'^y dt 

where (27) .c^ -fj/' 4--?2 = (j{-t^)2 



"^<-' (4;)'- (*)'-©■=(!)■- 



1 



\\\ consideration of (27), the three equations (25) are 
of the same form as the equations for the motion of a 
material point subjected to attraction from a fixed centre 
according to the Newtonian Law, only that instead of the 
time t) the proper time t of the material point occurs. The 
fourth equation (26) gives then the connection between 
proper time and the time for the material point. 

Now for different values of t\ the orbit of the space- 
point (,(• y z) is an ellipse with the semi-major axis a and 
the eccentricity e. Let E denote the excentric anomaly, T 



APPENDIX 69 

the increment of the proper time for a complete description 
of the orbit, finally nT =27r, so that from a properly chosen 
initial point t, we have the Kepler-equation 

(29) }iT=zE-e sin E. 

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

Now neglecting c~* with regard to 1, it follows that 
7/ ^ r 1 . i ^'^^ l + ^cosE~| 

from which, by applying (29), 

(31 ) nt 4- const =f 1 -f- \~ \ nr-\- —,^ SinE. 



m^ 



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



ac- 



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

The law of mass attraction which has been just describ- 
ed and which is formulated in accordance with the 
relativity postulate would signify that gravitation is 
propagated with the velocity of light. In view of the fact 
that the periodic terms in (31) are very small, it is not 
possible to decide out of astronomical observations between 
sueh a law (with the modified mechanics proposed above) 
and the Newtonian law of attraction with Newtonian 
mechanics. 



70 l^RiNClPLE OF RELATIVITY 

SPACE AND TIME 

A Lecture delivered before the Naturforsclier Yer- 
sammlung (Congress of Natural Philosophers) at Cologne — 
(21st September, 1908). , 

Gentlemen, 

The eoneeptious about time and space, which I hope 
to develop before you to-day, has grown on experimental 
physical grounds. Herein lies its strength. The tendency 
is radical. Henceforth, the old conception of space for 
itself, and time for itself shall reduce to a mere shadow, 
and some sort of union of the two will be found consistent 
with facts. 

I 

Now I want to show 3 ou how we can arrive at the 
changed concepts about time and space from mechanics, as 
accepted now-a-days, from purely mathematical considera- 
tions. The equations of Newtonian mechanics show a two- 
fold invariance, (?') their form remains unaltered when 
we subject the fundamental space-coordinate system to 
any possible change of position, {ii) when we change the 
system in its nature of motion, /. e., when we impress upon 
it any uniform motion of translation, the null-point of time 
plays no part. We are accustomed to look upon the axioms 
of geometry as settled once for all, while we seldom have the 
same amount of conviction regarding the axioms of mecha- 
nics, and therefore the two invariants are seldom mentioned 
in the same breath. Each one of these denotes a certain 
group of transformations for the differential equations of 
mechanics. We look upon the existence of the first group 
as a fundamental characteristics of space. We always 
prefer to leave off the second group to itself, and w^ith a 
lisht heart conclude that we can never decide from physical 
considerations whether the si)ace, which is supposed to be 



APPENDIX 71 

at rest, may not finally t>e in uniform motion. So these two 
groups lead quite separate existences besides each other. 
Their totally heterogeneous character may scare us away 
from the attempt to compound them. Yet it is the whole 
compouuded group which as a whole gives us occasion for 
thought. 

We wish to picture to ourselves the whole relation 
graphically. Let (,<', y, z) be the rectangular coordinates of 
space, and t denote the time. Subjects of our perception 
are always connected with place and time. No one has 
observed a place e, cept at a pariicnlar iime, or has obserred 
a time exce^A at a particular place. Yet I respect the 
dogma that time and space have independent existences. I 
will call a space-point plus a time-point, i.e., a system of 
values X, y^ r, /, as a world-point. The manifoldness of all 
possible values of x, y, z, t, will be the world. I can draw 
four world-axes with the chalk. Now any axis drawn 
consists of quickly vibrating molecules, and besides, takes 
part in all the journeys of the earth ; and therefore giyes 
us occasion for reflection. The greater abstraction required 
for the four-axes does not cause the mathematician any 
trouble. In order not to allow any yawning gap to 
exist, we shall suppose that at every place and time, 
something perceptible exists. In order not to specify 
either matter or electricity, we shall simply style these as 
substances. We direct our attention to the world -point 
^, y, z, t, and suppose that we are in a position to recognise 
this substantial point at any subsequent time. Let dt be 
the time element corresponding to the changes of space 
coordinates of this point [d.v, dy, dz]. Then we obtain (as 
a picture, so to speak, of the perennial life-career of the 
substantial point), — a curve in the 2Vorld — the ivorld-line, 
the points on which unambiguously correspond to the para- 
meter t from -f 00 to— <^. The whole world appears to be 



72 PRINCIPLE OF RELATIVITY 

resolved in such 70orld4ineSy and I may just deviate from 
my point if I say that according to my opinion the physical 
laws would find their fullest expression as mutual relations 
among these lines. 

By this conception of time and space, the (", y, z) mani- 
foldness t = o and its two sides /<o and t>o falls asunder. 
If for the sake of simplicity, we keep the null-point of time 
and space fixed, then the first named group of mechanics 
signifies that at f — o we can give the ,'•, y, and ^-axes any 
possible rotation about the null-point corresponding to the 
homogeneous linear transformation of the expression 



^2+^2_^ 



^2 



The second group denotes that without changing the 
expression for the mechanical laws, we can substitute 
{x — atyy—ptj z—yt^ for ('■, y, z) where (a, ^, y) are any 
constants. According to this we can give the time-axis 
any possible direction in the upper half of the woild />o. 
Now what have the demands of orthogonality in space to 
do with this perfect freedom of the time-axis towards the 
upper half ? 

To establish this connection, let us take a positive para- 
meter c y and let us consider the figure 

According to the analogy of the hyperboloid of two 
sheets, this consists of two sheets separated by t-=^o. Let us 
consider the sheet, in the region of ^>o, and let us now 
conceive the transformation of ,>•, y, z, i in the new system 
of variables ; (.</, y', z ^ t') by means of which the form of 
the expression will remain unaltered. Clearly the rotation 
of space round the null-point belongs to this group of 
transformations. Now we can have a full idea of the trans- 
formations which we picture to ourselves from a particular 



\ 



APPENDTJC 73 

transformation in which (y, z) remain unaltered. Let 
us draw the cross section of the upper sheets with the 
plane of the .r- and /-axes, i.e., the upper half of 
the hyperbola <?-/2_,2_.]^ with its asymptotes {vide 

fig. I). . 

Then let us draw the radius rector OA', the tansrent 
A' B' at A', and let us complete the parallelogram OA' 
B' C ; also produce W C to meet the f -axis at D'. 
Let us now take Ox', OA' as new axes with the unit mea- 
suring rods 0C' = 1, 0A'= ; then the h^^perbola is again 

expressed in the form c^t'-— ■'^ = ], t'>o and the transi- 
tion from ( r, ;f/j ;, t) to ( ' y'^'t^ is one of the transitions in 
question. Let us add to this characteristic transformation 
any possible displacement of the space and time null-points ; 
then we get a group of transformation depending only on 
c, which we may denote by Gc. 

Now let us increase c to infinity. Thus ~ becomes zero 

c 

and it appears from the figure that the hyperbola is gradu- 
ally shrunk into the /-axis, the asymptotic angle be- 
comes a straight one, and every special transformation in 
the limit changes in such a manner that the /-axis can 
have any possible direction upwards, and ,'' more and 
more approximates to .'''. Remembering this point it is 
clear that the full group belonging to Newtonian Mechanics 
is simply the group G^, with the value of c=oo. In this 
state of affairs, and since Gc is mathematically more in- 
telligible than G oo, a mathematician may, by a free play 
of imagination, hit upon the thought that natural pheno- 
mena possess an invariance not onl}^ for the group G^, 
but in fact also for a group G^, where c is finite, but yet 
10 



74 PRINCIPLE OF KELATIVITY 

exceedingly large compared to the usual measuring units. 
Such a preconception would be an extraordinary triumph 
for pure mathematics. 

At the same time I shall remark for which value of c, 
this invariance can be conclusively held to be true. For c, 
we shall substitute the velocity of light c in free space. 
In order to avoid speaking either of space or of vacuum, 
we may take this quantity as the ratio between the electro- 
static and eleetro-mas:netie units of electricity. 

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

Out of the totality of natural phenomena, we can, by 
successive higher approximations, deduce a coordinate 
system (,r, ^, ^, t) ; by means of this coordinate system, we 
can represent the phenomena according to definite laws. 
This system of reference is by no means uniquely deter- 
mined by the phenomena. JFe can change the system of 
reference in any possifjle manner corresjjonding to the above- 
mentioned group transformation Gc, but the expressions for 
natttral laws ivill not be changed thereby. 

For example, corresponding to the above described 
figure, we can call // the time, but then necessarily the 
space connected with it must be expressed by the mani- 
foldness {/ y z). The physical laws are now expressed by 
means of ■<', y, ^, i' , — and the expressions are just the 
same as in the case of <<■, y^ z, t. According to this, we 
shall have in the world, not one space, but many spaces, — 
quite analogous to the case that the three-dimensional 
space consists of an infinite number of planes. The three- 
dimensional geometry will be a chapter of four-dimensional 
physics. Now you perceive, why I said in the beginning 



AtPE^BlX 76 

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

II 

Now the question may be asked, — what circumstances 
lead us to these changed views about time and space, are 
they not in contradiction with observed phenomena, do 
they finally guarantee us advantages for the description of 
natural phenomena ? 

Before we enter into the discussion, a very important 
point must be noticed. Suppose we have individualised 
time and space in any manner; then a world-line parallel 
to the ^-axis will correspond to a stationar}^ point ; a 
world-line inclined to the /f-axis will correspond to a 
point moving uniformly ; and a world-curve will corres- 
pond to a point moving in any manner. Let us now picture 
to our mind the world-line passing through any world 
point ■''if/,z,tj now if we find the world-line parallel 
to the radius vector OA' of the hyperboloidal sheet, then 
we can introduce OA' as a new time-axis, and then 
according to the new conceptions of time and space the 
substance will appear to be at rest in the world point 
concerned. AVe shall now introduce this fundamental 
axiom : — 

Th<! ■^lihstance eiisllnij at (uuf world j^oiui can always 
be conceived to he at rest, if we esta/ilifih. our time wml 
s^pace xtdtatjlf/. The axiom denotes that in a world-point 
the expression 

ciflfi —dx"^ —fh^ —dz"^ 

shall always be positive or what is eipiivalent to the 
same thing, every velocity V should be snialler than c, 
c shall therefore be the up[)er limit for all substantial 
velocities and herein lies a deep significance for tlie 



76 PJliNClPLE Oi^ KELATlVlTV 

quantity c. At the first impression, the axiom seems to 
be rather unsatisfactory. It is to be remembered that 
only a modified mechanics will occur, in which the square 
root of this differential combination takes the place of 
time, so that cases in which the velocity is greater than c 
will play no part, something like imaginary coordinates 
in ofeometry. 

The im'piihe and real cause of inducement for the 
assumption of the group-traiuf or }iLatio}i Gc is the fact that 
the differential equation for the propagation of light in 
va-^ant spa'je possesses the group-transformation Gc. On 
the oth-n* hand, the idei of rig^id bodies has anv sense 
only in a system mechanics with the group G^,.. Now 
if we have an optics with G,, and on the other hand 
if there are rigid bodies, it is easy to see that a 
/^-direction can be defined by the two hyperboloidal 
shells common to the groups G^^, and G^, which has 
got the further consequence, that by means of suitable 
rigid instruments in the laboratory, we can perceive a 
change in natural phenomena, in case of different orienta- 
tions, with regard to the direction of progressive motion 
of the earth. But all efforts directed towards this 
object, and even the celebrated interference-experiment 
of Michelson have sj'iven nciirative results. In order to 
supply an explanation for this result, H. A. Lorentz 
formed a hypothesis which practically amounts to an 
invariance of optics for the group G,, According to 
Lorentz every substance shall suffer a contraction 

1 \ V ^ r P^i length, in the direction of its motion 

T= "THE ''={'- 3- • 



This hypothesis sounds rather }jhaotastical. For the 
contraction is not to be thought of as a consequence of the 
resistance of ether, but purely as a gift from the skies, as a 
sort of eundition always accompanying a state of motion. 

I shall show in our figure that Lorentz's hypothesis 

is fully equivalent to the new conceptions about time and 

space. Thereby it may appear more intelligible. Let us 

now, for the sake of simplicity, neglect (j/, z) and fix our 

attention on a two dimensional world, in which let upright 

strips parallel to the ^^-axis represent a state of rest and 

another parallel strip inclined to the /.-axis represent a 

state of uniform motion for a body, which has a constant 

spatial extension (see fig. 1). If OA' is parallel to the second 

y strip, we can take f/ as the .-^-axis and x' as the a;-axis, then 

the se<^ond body will appear to be at rest, and the first body 

in uniform motion. We shall now assume that the first 

body supposed to be at rest, has the length /, i.e., the 

cross section PP of the first strip upon the .-axis^/* OC, 

where OC is the unit measuring rod upon the j^-axis — and 

the second body also, when supposed to be at rest, has the 

same length I, this means that, the cross section Q'Q' of 

the second strip has a cross-section I'OC, when measured 

parallel to the ''-axis. In these two bodies, we have 

now images of twD Lorentz-electrons, one of which is at 

rest and the other moves uniformly. Now if we stick 

to our original coordinates, then the extension of the 

second electron is given by the cross section QQ of the 

strip belonging to it measured parallel to the '-axis. 

Now it is clear since a'Q' = ^OC', that QQ = /-OD'. 

If -— = r, an easv calculation li'ives that 
dt " 



\/l ' 



PP 

jj ^- I -» ^-k ft ft ^-fc 4- ^-k «* ^ I 



OD' = 0C '\' " c2, therefore QQ / v^ 

' \/ 1— . 



c 



2 



78 . PRINCIPLE OF IlELxiTIViTY 

This is the sense of Lorentz's hypothesis about the 
contraction of electrons in ease of motion. On the other 
hand, if we conceive the second electron to be at rest, 
and therefore adopt the system (.0', i\) then the cross-section 
PT' of the strip of the electron parallel to OC is to be 
regarded as its length and we shall find the first electron 
shortened with reference to the second in the same propor- 
tion, for it is, 

P'P' _0D _0p'_ QQ 

(ra'~oc'~oc - pp 

Lorentz called the combination /-' of {t and ,* ) as the 
local ti'tie {Ortszeit) of the uniformly moving electron, and 
used a physical construction of this idea for a better compre- 
hension of the contraction-hypothesis. But to perceive 
clearlv that the time of an electron is as ijood as the time 
of any other electron, i,e. t, i' are to be regarded as equi- 
valent, has been the service of A. Einstein [Ann. d. 
Phys. 891, p. 1905, Jahrb. d. Radis... 4-4-1 1—1907] There 
the concept of time was shown to be completely and un- 
arabio'uouslv established bv natural phenomena. But the 
concept of space was not arrived at, either by Einstein 
or Lorentz, probably because in the case of th^ above- 
mentioned spatial transformations, where the ( </, /') plane 
coincides with the ••'-/ plane, the significance is possible 
that the -^-axis of space some-how remains conserved in 
its position. 

We can approach the idea of space in a corresponding 
manner, though some may regard the attempt as rather 
fantastical. 

AccordiniT to these ideas, the word '' Relativitv-Pastu- 
late'' which has been coined for the demands of invariance 
in the group G, seetus to be rather inexpressive for a true 
understanding of the group Gc, and tor further progress. 



APPENDIX 79 

Because the sense of the postulate is that the four- 
dimensional world is given in space and time by pheno- 
mena only, but the projection in time and space can 
be handled with a certain freedom, and therefore I would 
rather hke to ojive to this assertion the name " The 
Post uJ ate of the Ahsohde worliV [World- Postulate]. 



Ill 



By the world-postulate a similar treatment of the four 
determining quantities .r, ?/, 0, t, of a world-point is pos- 
sible. Thereby the forms under which the physical laws 
come forth, gain in intelligibility, as I shall presently show. 
Above all, the idea of acceleration becomes much more 
strikins: and clear. 

I shall agai!i use the geometrical method of expression. 
Let us call any world-point O as a " Spaee-time-null- 
point.'' The cone 

consists of two parts with O as apex, one part having 
/<0', the other having />0. The first, which we may call 
t\\e fore-cone consists of all those points which send light 
towards O, the second, which we ma}' call the aft-cone. 
consists of all those points which receive their light from 
O. The region bounded by the fore-cone may be called 
the fore-side of O, and the region bounded by the aft-cone 
may be called the aft-side of O. [Vide fig. 2). 

On the aft-side of O e have the already considered 
hyperboloidal shell F = c^^ -x^- -y- —z"" = '[, t>0. 



80 PRINCIPT^E OP EELATIVITY 

The region inside the two cones will be occupied by the 
hyperboloid of one sheet 

where k^ can have all possible positive values. The 
hyperbolas which lie upon this fiss'nre with O as centre, 
are important for us. For the sake of clearness the indivi- 
dual branches of this hyperbola will be called the " Inter- 
hi/perbola with centra- 0^ Such a hyperbolic branch, 
when thought of as a world-line, would represent a 
motion which for /=— oo and t = oo^ asymptotically 
approaches the velocit}^ of light c. 

If, by way of analogy to the idea of vectors in space, 
we call any directed length in the manifoldness i',^,z,l a 
vector, then we have to distinguish between a time-vector 
directed from O towards the sheet +F=1, ^>Oand a 
space-vector directed from O towards the sheet —F=l. 
The time-axis can be parallel to any vector of the first 
kind. Any world-point between the fore and aft cones 
of O, may bv means of the system of reference be res^arded 
either as synchronous with O, as well as later or earlier 
than O. Every world-point on the fore-side of O is 
necessarily always earlier, every point on the aft side of 
O, later than O. The limit c = oo corresponds to a com- 
plete folding up of the wedge-shaped cross-section between 
the fore and aft cones in the manifoldness / = 0. In the 
fiojure drawn, this cross-section has been intentionally 
drawn with a different breadth. 

Let us decompose a vector drawn from O towards 
{a',]/,z,t) into its components. If the directions of the two 
vectors are respectively the directions of the radius vector 
OR to one of the surfaces -|-F=1, and of a tangent RS 



APPENDIX ' 81 

at the point R of the surface^ then the vectors shall be 
called normal to each other. Accordinsjlv 

» 

which is the condition that the vectors with the com- 
ponents ((', y, Zy t) and {x ^ y^ z^ t^) are normal to each 
other. 

For the measurement of vectors in different directions^ 
the unit measuring rod is to be fixed in the following 
manner; — a space-like vector from to — F = I is always 
to have the measure unity, and a time-like vector from 

O to +F= 1 , />0 is always to have the measure — . 

Let us now fix our attention upon the world-line of a 
substantive point running through the world-point (t, y, 
z, t) ; then as we follow the -progress of the line, the 
quantity 



c 
corresponds to the time-like vector-element {clc, dy, dz, dt). 

The integral T= fr/r, taken over the world-line from 

any fixed initial point P^ to any variable final point P, 
may be called the " Proper-time " of the substantial point 
at Po upon t,he icorld-line. We may regard (r, y, z, t), i.e., 
the components of the vector OP, as functions of the 

" proper-time " r; let (.r, y^ i, denote the first differential- 
quotients, and {x, y\ z, f) the second differential quotients 

of ( ', 'f, -, t) with regard to r, then these may respectively 
11 



82 • PRINCIPLE OF RELATIVITY 

be called the Velocity-vector, and the Accelercition-vector 
of the substantial point at P. Now we haye 

••• •«• ••• ••• 

c2 t t ^ X X — y y — z ^=0 

i.e., the ' Velocity'Vector ' is the time-like vector of unit 
measure in the direction of the world-line at P, the ' Accele- 
ration-vector^ at P is normal to the velocity-vector at P, 
and is in any case, a space-like vector. 

Now there is, as can be easily seen, a certain hyperbola, 
which has three infinitely contiguous points in common 
with the world-line at P, and of which the asymptotes 
are the generators of a 'fore-cone^ and an 'aft-cone.' 
This hyperbola may be called the " hyperbola of curvature " 
at P (^vide fig. 3). If M be the centre of this hyperbola, 
then we have to deal here .with an ' Inter-hyperbola ' with 
centre M. Let P = measure of the vector MP, then we 
easily perceive that the acceleration-vector at P is a vector 



c^ . 
of magnitude — in the direction of MP. 

P 



If .r, y, z, t are nil, then the hyperbola of curvature 
at P reduces to the straight line touching the world-line 
at P, and p=oc. 

IV 

In order to demonstrate that the assumption of the 
crroup Gc fo^' ^^^® physical laws does not possibly lead to 
any contradiction, it is unnecessary to undertake a revision 
of the whole of physics on the basis of the assumptions 
underlying this group. The revision has already been 
successful!}' made in the case of " Thermodjmamics and 



APPENDIX 80 

Radiation,"^ for "Eleetromagnetie phenomena '^,t and 
finally for "Mechanics with the maintenance of the idea of 
mass." 

For this last mentioned province of physics, the ques- 
tion may be asked : if there is a force with the components 
X, Y, Z (in the direction of the space-axes) at a world- 

• • • • 

point (c?', y, z, f), v^rhere the velocity-vector is (r, y, Zj t), 

then how are we to resrard this force when the svstem of 
reference is changed in any possible manner ? Now it is 
known that there are certain well-tested theorems about 
the ponderomotive force in electromagnetic fields, where 
the group G^ is undoubtedly permissible. These theorems 
lead us to the following simple rule ; if the i^ijdem of 
'reference he changed in an// loay^ then the supposed force is 
to be put as a force in, the new sjMce- coordinates in such a 
manner, that the corresponding vector with the components 

tX ^'Y, tZ, tT, 

• « • 

ivhere T=— f4x + ^Y + ^z"^ = ^ {the rate of 
c^ \ t t t ) c^ 

tohicli work is done at the toorld-point), remains unaltered. 
This vector is always normal to the velocity-vector at P. 
Such a force-vector, representing a force at P, may be 
called a moving force-vector at P. 

Now the world-line passing through P will be described 
by a substantial point with the constant mechanical mass 
m. Let us call m-times the velocity-vector at P as the 

* Planck, Ziir Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 

1908, p. 1. 

f H. Minkowski ; the passage refers to paper (2) of the present 

edition. 



84 PRINCIPLE OF RELATIVITY 

impidse -vector, and m-iimes the acceleration-vector at P as 
the force-vector of motion^ at P. According- to these 
definitions, the following law tells us how the motion of 
a point-mass takes place under any moving force-vector"^ : 

The force-vector of motion is equal to the moving force- 
vector. 

This enunciation comprises four equations for the com- 
ponents in the four directions, of which the fourth can be 
deduced from the first three, because both of the above- 
mentioned vectors are perpendicular to the velocity-vector. 
From the definition of T, we see that the fourth simply 
expresses the " Ener2:y-law.'" Accordingly c'^ -times the 
component of the impulse-vector in the direction of the 
t-axis is to be defined as the hinetic-energ)/ of the point- 
mass. The expression for this is 



dr 



v^-^ 



i.e., if we deduct from this the additive constant w<?-, we 
obtain the expression \ inv^ of Newtonian-mechanics upto 

magnitudes of the order of -^. Hence it appears that the 

energij depends upon the system of reference. But since the 
^-axis can be laid in the direction of any time-like axis, 
therefore the energy-law comprises, for any possible system 
of reference, the whoL.^ system of equations of motion. 
This fact retains its significance even in the limiting: ease 
C=oo, for the axiomatic construction of Newtonian 
mechanics, as has already been pointed out by T. R. 
Sehiitz.t 

* Minkowski — Mechanics, appendix, page 65 of paper (2). 

Planck— Yerh. d. D. P. G. Vol. 4, 1906, p. 136. 
t Schutz, Gott. Nachr. 1897, p. 110. 



APPENDIX 85 

From the very beginning, we can establish the ratio 

between the units of time and space in such a manner, that 

the velocity of light becomes unity. If we now write 

a/HI t = lj in the place of I, then the differential expression 

dr"- = -(c?ic2 +%2 +(/2;2 +^^2)^ 

becomes symmetrical in (.- , 3/. ^, /) ; this symmetry then 
enters into each law, which does not contradict the ?rr)rA/- 
2J0stnla{e. We can clothe the " essential nature of this 
postulate in the mystical, but mathematically significant 
formula 



• The advantages arising from the formulation of the 
world-] )0.>tulate are illustrated bv nothing so strikinglv 
as by the expressions which tell us about the reactions 
exerted by a point-charge moving in any manner accord- 
ing to the Maxwell-Lorentz theory. 

Let us conceive of the world-line of such an electron 
with the charge [e), and let us introduce upon it the 
'^ Propjr-time " r reckoned from any possible initial point. 
In order to obtain the field caused by tlie electron at any 
world-point P^ let us construct the fore-cone belonging 
to Pj {vide fig. 4). Clearly this cuts the unlimited 
world-line of the electron at a single point P, because these 
directions are all time-like vectors. At P, let us draw the 
tangent to the world-line, and let us draw from P^ the 
normal to this tangent. Let f be the measure ofP,Q. 
According to the definition of a fore-cone, rje is to be 
reckoned as the measure of PQ. Now at the world-point Pj, 



86 PHINCIPLE OF RELATIVITY 

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



e 



cr i 



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



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

If the field caused by the electron be described in the 
above-mentioned way, then it will appear that the division 
of the field into electric and magnetic forces is a relative 
one, and depends upon the time-axis assumed ; the two 
forces considered together bears some analogy to the 
force-screw in mechanics ; the analog}^ is, however, im- 
perfect. 

I shall now describe the ponder omoiive force whicJi is 
exerted hij one moving electron upon Q7iother moving electron. 
Let us suppose that the world-line of a second point- 
electron passes through the world-point Pj. Let us 
determine P, Q, r as before, construct the middle-point M 
of the hyperbola of curvature at P, and finally the normal 
MN upon a line through P which is parallel to QPj. 
With P as the initial point, we shall establish a system 
of reference in the following way : the /-axis will be laid 
along PQ, the a -axis in the direction of QP^. The ^'-axis 
in the direction of MN, then the r-axis is automatically 
determined, as it is normal to the .» -, t/-, ^-axes. Let 



;c, 1/, Zy /be the acceleration-vector at P, x^^y^^z^^t^ 

be the velocity-vector at P^. Then the force-vector exerted 
by the first election r^ (moving in any possible manner) 

* Lienard, L'Eolairage electriqne T.16, 1896, p. 53, 
Wiechert, Ann. d. Physik, Vol. 4. 



APPENBIX 87 

upon the second election e, (likewise moving in any 
possible manner) at Pj is represented by 



» 



F, 



For the coiujwnenls F,^ Fy, F:, Ft of the vector F the 
folloiving three relations hold : — 

cF,-F.= i,F,= 4-,F.=0, 

and fourthly this vector F is normal to the velocity -vector 
P^, a]id through this circumstance alone, its dependence on 
this last velocity -vector arises. 

I£ we compare with this expression the previous for- 
mulie"^ giving the elementary law about the pouderomotive 
action of moving electric charges upon each other, then we 
cannot but admit, that the relations which occur here 
reveal the inner essence of full simplicity first in four 
dimensions ; but in three dimensions, they have very com- 
plicated projections. 

In the mechanics reformed according to the world- 
postulate, the disharmonies which have disturbed the 
relations between Newtonian mechanics, and modern 
electrodynamics automatically disappear. I shall now con- 
sider the position of the Newtonian law of attraction to 
this postulate. I will assume that two point-masses 7}i and 
m^ describe their world-lines ; a moving force-vector is 
exercised by m upon m^, and the expression is just the same 
as in the case of the electron, only we have to write 
■\-mm^ instead of— 6'6'i. We shall consider only the special 
case in which the acceleration-vector of m is always zero ; 

* K. Schwarzschild. Gott-Nachr. 1903. 

II. A. Lorentz, Enzyklopadie der Math. Wisscnschaftcn V. Art 14, 
p. 199. 



88 PHINCIPLE OF RELATIVITY 

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

• 1 

given vector by writing . . instead of / (? = 1 up 

to magnitudes of the order —17 ), then it a})pears that 

Ke2:)Ier\s laws hold good for tlie position {^n^i, ^j), of 
m^ at any time, only in place of the time t^, we have to 
write the proper time t^ oi m^. On the basis of this 
simple remark, it can be seen that the proposed law of 
attraction in combination with new mechanics is not less 
suited for the explanation of astronomical phenomena than 
the Newtonian law of attraction in combination with 
Newtonian mechanics. 

Also the fundamental equations for electro-magnetic 
processes in moving bodies are in accordance with the 
world-postulate. I shall also show on a later occasion 
that the deduction of these equations, as taught by 
Lorentz, are by no means to be given up. 

The fact that the world-postulate holds without excep- 
tion is, 1 believe, the true essence of an electromagnetic 
picture of the world ; the idea first occurred to Lorentz, its 
essence was first picked out by Einstein, and is now gradu- 
ally fully manifest. In course of time, the mathematical 
consequences will be gradually deduced, and enough 
suggestions will be forthcoming for the experimental 
verification oi' the postulate ; in this way even those, who 
find it uncongenial, or even painful to give up the old, 
time-honoured concepts^ will be reconciled to the new ideas 
of time and space, — in the prospect that they will lead to 
pre-established harmony between pure mathematics and 
physics. 



The Foundation of the Generalised 
Theory of Relativity 

By a. Einstein. 

From Annalen der Physik 4.49,1916. 

The theory which is sketched in the following pages 
forms the most wide-going generalization conceivable of 
what is at present known as " the theory of Relativity ; " 
this latter theory I differentiate from the former 
"Special Relativity theory," and suppose it to be known. 
The generalization of the Relativity theory has been made 
much easier through the farm given to the special Rela- 
tivity theory by Minkowski, which mathematician was the 
first to recognize clearly the formal equivalence of the space 
like and time-like co-ordinates, and who made use of it in 
the building up of the theory. The mathematical apparatus 
useful for the general relativity theory, lay already com- 
plete in the "Absolute Differential Calculus/' which were 
based on the researches of Gauss, Riemann and Christoffel 
on the tibn-EucHdean manifold, and which have been 
shaped into a system by Rieci and Levi-civita, and already 
applied to the problems of theoretical physics. I have in 
part B of this communication developed in the simplest 
and clearest manner, all the supposed mathematical 
auxiliaries, not known to Physicists, which will be useful 
for our purpose, so that, a study of the mathematical 
literature is not necessary for an understanding of this 
paper. Finally in this place I thank my friend Grossmann, 
by whose help I was not only spared the study of the 
mathematical literature pertinent to this subject, but who 
also aided me in the researches on the field equations of 
gravitation. > ? 



90 PRINCIPLE OF EELATIVITT 



A 



Principal considerations about the Postulate 

OF Relativity. 

§ 1. Remarks on the Special Relativity Theory. 

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

If a co-ordinate system K be so chosen that when re- 
ferred to it^ the physical laws hold in their simplest forms 
these laws would be also valid when referred to another 
system of co-ordinates K' which is subjected to an uniform 
trauslational motion relative to K. We call this postulate 
** The Special Kelativity Principle.'' By the word special, 
it is sij^nilied that the principle is limited to the ease, 
when K' has nniform trandatory motion with reference to 
K., but the equivalence of K and K' does not extend to the 
ease of no n -uniform motion of K' relative to K. 

The Special Relativity Theory does not differ from the 
classical mechanics through the assumption of this postu- 
late, but only through the postulate of the constancy of 
light-velocity in vacuum which, when combined with the 
special relativity postulate, gives in a well-known way, the 
relativity of synchronism as well as the Lore nz- transfor- 
mation, with all the relations between moving rigid bodies 
and clocks. 

The modification which the theory of space and time 
has undergone through the special relativity theory, is 
indeed a profound one, but a weightier point remains 
untouched. According to the special relativity theory, the 
theorems of geometry are to be looked upon as the laws 
about any possible relative positions of solid bodies at rest, 
and more generally the theorems of kinematics, as theorems 
which describe the relation between measurable bodies and 



GBNEEATJSED THEORY OF RELATIVITY 91 . 

clocks. Consider two material points of a solid body at 
rest ; then according' to these conceptions their corres- 
ponds to these points a wholly definite extent of length, 
independent of kind, position, orientation and time of the 
body. 

Similarly let us consider two positions of the pointers of 
a clock which is at rest with reference to a co-ordinate 
system ; then to these positions, there always curresponds, 
a time-interval of a definite length, independent of time 
and place. It would be soon shown that the general rela- 
tivity theory can not hold fast to this simple physical 
significance of space and time. 

§ 2. About the reasons which explain the extension 
of the relativity-postulate. 

To the- classical mechanics (no less than) to the special 
relativity theory, is attached an episteomologioal defect, 
which was perhaps first clea»'ly pointed out by E. Mach. 
We shall illusti*ate it by the following example ; Let 
two fluid bodies of equal kind and magnitude swim freeh^ 
in space at such a great distance from one another (and 
from all other masses) that only that sort of gravitational 
forces are to be taken into account which the parts of any 
of these bodies exert upon each other. The distance of 
the bodies from one another is in\^riable. The relative 
motion of the different parts of each body is not to occur. 
But each mass is seen to rotate by an observer at rest re- 
lative to the other mass round the connecting line of .the 
masses with a constant angular velocity (definite relative 
motion for both the masses). Now let us think that the 
surfaces of both the bodies (S^ and S.J are measured 
with the help of measuring rods (relatively at rest) ; it is 
then found that the surface of S^ is a sphere and the 
surface of the other is an ellipsoid of rotation. We now 



92 PRINCIPLE OF BELATIVITT 



/ 



ask, why is this difference between the two bodies ? An 
answer to this question can only then be regarded As satis- 
factory from the episteomological standpoint when the 
thin 2: adduced as the cause is an observable fact of ex- 
perience. The law of causality has the sense of a definite 
statement about the world of experience only when 
observable facts alone appear as causes and effects. 

The Newtonian mechanics does not give to this question 
any satisfactory answer. For example, it says ! — The laws 
of mechanics hold true for a space R^ relative to which 
the body S^ is at rest, not however for a space relative ta 
which S3 is at rest. , ^ 

The Galiliean space, which is here introduced is how- 
ever only a purely imaginary cause, not an observable thing. 
It is thus clear that the Newtonian mechanics does not, 
in the case treated here, actually fulfil the requirements 
of causality, but produces on the mind a fictitious com- 
placency, in that it makes responsible a wholly imaginaryi 
cause Ri for the different behaviours of the bodies S, and 
Sg which are actually observable. 

A satisfactory explanation to the question put forvvard 
above can only be thus given : — that the physical system- 
composed of S^ and S^ shows for itself alone no con- 
ceivable cause to which the different behaviour of S, and 
Sg can be attributed. The cause must thus lie outside the 
system. We are therefore led to the conception that the 
general laws of motion which determine specially the 
forms of S^ and Sg must be of such a kind, that the 
mechanical behaviour of S^ and S^ must be essentially 
conditioned by the distant masses, which we had not 
brought into the system considered. These distant masses, 
(and their relative motion as* regards the bodies under con- 
sideration) are then to be looked upon as the seat of the 
principal observable causes for the different behaviours- 



GENERALISED THEORY OE RELATIVITY .93 , 

of the bodies under consideration. They take the place 
of the imaginary cause R^. Among all the conceivable 
spaces Ri and Rg moving in any manner relative to one 
another, there is a priori, no one set which can be regarded 
as affording c reater advantages, against which the objection 
which was already raised from the standpoint of the 
theory of knowledge cannot be again revived. The laws 
of physics must be so constituted that they should remain 
valid for any system of co-ordinates moving in any manner. 
We thus arrive at an extension of the relativity postulate. 

Besides this momentous episteomological argument> 
there is also a well-known physical fact which speaks in 
favour of an extension of the relativity theory. Let there 
be a Galiliean co-ordinate system K relative to which (at 
least in the four-dimensional- region considered) a ma^s at 
a sufficient distance from other masses move uniformly in 
a line. Let K' be a second co-ordinate system which has 
a uniformly accelerated motion relative to K. Relative tq 
K' any mass at a sufficiently great distance experiences 
an accelerated motion such that its acceleration and ihq 
direction of acceleration is independent of its material com- 
position and its physical conditions. 

Can any observer, at rest relative to K', then conclude 
that he is in an actually accelerated reference-system ? 
This is to be answered in the negative ; the above-named 
behaviour of the freely moving masses relative to K' esu} 
be explained in as good a manner in the following way. 
The reference-system K' has no acceleration. In the space- 
time region considered there is a gravitation-fiekl which 
generates the accelerated motion relative to K'. 

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



94 PRINCIPLE OF RELATIVITY 

raecbapica] behaviour of the bodies relative to K' is the 
i^me as experience would expect of them with reference 
to systems which we assume from habit as stationary; 
thus it explains why from the physical stand-point it can 
be assumed that the systems K and K' can both with the 
same legitimacy be taken as at rest, that is, they will be 
equivalent as systems of reference for a description of 
physical phenomena. 

From these discussions we see, that the working out 
of the general relativity theory must, at the same time, 
lead to a theory of gravitation ; for we can " create " 
a gravitational field by a simple variation of the co-ordinate 
system. Also we see immediately that the principle 
of the constancy of light- velocity must be modified, 
for we recognise easily that the path of a ray of light 
with reference to K' must be, in general, curved, when 
light travels with a definite and constant velocity in a 
straight line with reference to K. 

§ 3. The time-space continuum. Requirements of the 
general Co-variance for the equations expressing 
the laws of Nature in general. 

In the classical mechanics as well as in the special 
relativity theory", the co-ordinates of time and space have 
an immediate ph3^sical significance ; when we say that 
any arbitrary point has .>\ as its X^ co-ordinate, it signifies 
that the projection of the point-event on the X^-axis 
a»certained by means of a solid rod according to the rules 
of Euclidean Geometry is reached when a definite measur- 
ing rod, the unit rod, can be carried ,e^ times from the 
origin of co-ordinates along the X^ axis. 4 point having 
r^ — t-^ as the X^ co-ordinate signifies that a unit clock 
which is adjusted to be at rest relative to. the system of 
co-ordinates, and coinciding in its spatial position , with the 



GENERALISED THEORY OY RELATIVITY '95 

point-event and set according to some definite standard has 
gone over .v^=i periods before the occurence of the 
point-event. 

This conception of time and space is continually present 
in the mind of the physicist, though often in an unconsci- 
ous way, as is clearly recognised from the role which this 
conception has played in physical measurements. This 
conception must also appear to the reader to be lying at 
the basis of the second consideration of the last para- 
graph and imparting a sense to these conceptions. But 
we wish to show that we are to abandon it and in ireneral 
to replace it by more general conceptions in order to be. 
able to work out thoroughly the postulate of general relati- 
vity,— the case of special relativity appearing as a limiting 
case when there is no gravitation. 

We introduce in a space, which is free from Gravita- 
tion-field, a Galiliean Co-ordinate System K ( < , y, z, t) and 
also, another system K' (y' y' z' t') rotating uniformly rela- 
tive to K. The origin of both the systems as well as their 
2-axes might continue to coincide. We will show that for 
a space-time measurement in the system K', the above 
established rules for the physical significance of time and 
space can not be maintained. On grounds of symmetry 
it is clear that a circle round the origin in the -XY plane 
of K, can also be looked upon as a circle in the plane 
(X', Y') of K'. Let us now think of measuring the circum- 
ference and the diameter of these circles, with a unit 
measuring rod (infinitely small compared with the raidius) 
and take the quotient of both the results of measurement. 
If this experiment be carried out with a measuring rod 
at rest relatively to the Galiliean system K we would get 
TT, as the quotient. The result of measurement with a rod 
relatively at rest as regards K' would be a number which 
is greater than tt. This can be seen easily when we 



96 PRINCIPLE OF RELATIVITY 

regard the whole measurement- process from the system K 
and remember that the rod placed on the periphery 
suffers a Loreuz-contraction, not however when the rod 
is placed along the radius. Euclidean Geometry therefore 
does not hold for the system K' ; the above Hxed concep- 
tions of co-ordinates which assume the validity of 
Euclidean Greometry fail with regard to the system K'. 
We cannot similarly introduce in K' a time corresponding to 
physical requirements, which will be shown by all similarly 
prepared clocks at rest relative to the system K'. In order 
to see this we suppose that two similarly made clocks are 
arranged one at the centre and one at the periphery of 
the circle, and considered from the stationary system 
K. According to the well-known results of the special 
relativity theory it follows — (as viewed from K) — that the 
clock placed at the periphery will go slower than the 
second one which is at rest. The observer at the common 
origin of co-ordinates who is able to see the clock at the 
periphery by means of light will see the clock at the 
periphery going slower than the clock beside him. Since he 
cannot allow the velocity of light to depend explicitly upon 
the time in the way under consideration he will interpret 
his observation by saying that the clock on the periphery 
actully goes slower than the clock at the origin. He 
cannot therefore do otherwise than define time in such 
a way that the rate of going of a clock depends on its 
position. 

We therefore arrive at this result. In the oreneral 
relativity theory time and space magnitudes cannot be so 
defined that the difference in spatial co-ordinates can be 
immediately measured by the unit-measuring rod, and time- 
like co-ordinate difference with the aid of a normal clock. 

The means hitherto at our disposal, for placing our 
co-ordinate system in the time-space continuum, in a 



GENERALISED THEORY OP RELATIVITY 1^7 

definite way, therefore completely fail and it appears that 
there is no other way which will enable us to fit the 
co-ordinate sjstem to the four-dimensional world in such 
a way, that by it we can expect to get a specially simple 
formulation of the laws of Nature. So that nothing remains 
for us but to regard all conceivable co-ordinate systems 
as equally suitable for the description of natural phenomena. 
This amounts to the following law:* — 

That in general^ Laws of I^ature are e:f pressed hy means of 
equations which are valid for all co-ordinate systems^ that is, 
which are covariant for all possible transformations. It is 
clear that a physics which satisfies this postulate will be 
unobjectionable from the standpoint of the general 
relativity postulate. Because among all substitutions 
there are, in every case, contained those, which correspond 
to all relative motions of the co-ordinate system (in 
three dimensions). This condition of general covarianee 
which takes away the last remnants of physical objectivity 
from space and time, is a natural requirement, as seen 
from the following considerations. All our icelUsnhstantiated 
space-time propositions amount to the determination 
of space-time coincidences. If, for example, the event 
consisted in the motion of material points, then, for this 
last case, nothing else are really observable except the 
encounters between tw^o or more of these material points. 
The results of our measurements are nothing else than 
well-proved theorems about such coincidences of material 
points, of our measuring rods with other material points, 
coincidences between the hands of a clock, dial-marks and 
point-events occuring at the same position and at the same 
time. 

The introduction of a system of co-ordinates serves no 
other purpose than an easy description of totality of such 
coincidences. We fit to the world our space- time variables 
13 



98 PEINCIPLE OF ELLATIVITY 

(•^1 '^8 '"s '^4) such that to any and every point-event^ 
corresponds a system of values of (tj r^ ,(3 .c^). Two co- 
incident point-events correspond to the same value of the 
variables {.c^ x^ x^ -i'^) ; i.e., the coincidence is cha- 
racterised by the equality of the co-ordinates. If we now 
introduce any four functions (./i i\ t'g t;'^) as co- 
ordinates, so that there is an unique correspondence between 
them, the equality of all the four co-ordinates in the new 
system will still be the expression of the space-time 
coincidence of two material points. As the purpose of 
all physical laws is to allow us to remember such coinci- 
dences, there is a priori no reason present, to prefer a 
certain co-ordinate system to another ; i.e., we get the 
condition of o^eneral covariance. 

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

Analytical expression for the Gravitaiion»field. 

I am not trying in this communication to deduce the 
general Relativity-theory as the simplest logical system 
possible, with a minimum of axioms. But it is my chief 
aim to develop the theory in such a manner that the 
reader perceives the psychological naturalness of the way 
proposed, and the fundamental assumptions appear to be 
most reasonable according to the light of experience. In 
this sense, we shall now introduce the following supposition; 
that for an infinitely small four-dimensional region, the 
relativity theory is valid in the special sense when the axes 
are suitably chosen. 

The nature of acceleration of an infinitely small (posi- 
tional) co-ordinate system is hereby to be so chosen, that 
the gravitational field does not apipear; this is possible for 
an infinitely small region. Xi, Xg, Xg are the spatial 



- / 

GENERALISED THEORY OF RELATIVITY 99 

co-ordinates ; X^^ is the corresponding time-co-ordinate 
measured by some suitable measuring clock. These co- 
ordinates have, with a given orientation of the S3^stem, an 
immediate physical significance in the sense of the special 
relativity theory (when we take a rigid rod as our unit of 
measure), llie expression 

(1) ds'^ = -dX,^ -dX^ 2 -dX^ ' +^X^ • 

had then, according to the special relativity theory, a value 
which may be obtained by space-time measurement, and 
which is independent of the orientation of the local 
co-ordinate system. Let us take ds as the magnitude of the 
line-element belonging to two infinitely near points in the 
four-dimensional region. If ds"^ belonging to the element 
(^Xj dX^fdX^, ff'^i) he positive we call it with Minkowski, 
time-like, and in the contrary ease space-like. 

To the line-element considered, i.e., to both the infi- 
nitely near point-events belong also definite differentials 
<^Xj, d.c^, dx^, do^, of the four-dimensional co-ordinates of 
any chosen system of reference. If there be also a local 
system of the above kind given for the case under consi- 
deration, dX's would then be represented by definite linear 
homogeneous expressions of the form 



(2) dX =^ a dx 

V / V (T vcr (T 

If we substitute the expression in (1) we get 

(3) ds''='^ g d.v d.v 

where a will be functions of .c, but will no longer depend 



(TT 



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



100 PEIXCIPLE OP HELATIVITY 

measurements with rods and clocks. The g 's are hereto 

ht so chosen, that n =n - the summation is to be 

extended over all values of o- and t, so that the sum is to 
he extended, over 4x4 terms, of which 12 are equal in 
pairs. 

From the method adopted here, the ease of the usual 
relativity theory comes out when owing to the special 
behaviour of ff in 2i> finite region it is possible to choose the 

system of co-ordinates in such a way that g assumes 

eonstanf values — 

--1, 0, 0, 



{*) 



0-100 
0-10 
0+1 



Wfe would afterwards see that the choice of such a system 
of co-ordinates for a finite region is in general not possible. 

From the considerations in § 2 and § H it is clear, 
that from the physical stand-point the quantities g are to 

be looked upon as magnitudes wliich describe the gravita- 
tion-field with reference to the chosen system of axes. 
We assume firstly, that in a certain four-dimensional 
region considered, the special relativity theory is true for 
some particular choice of co-ordinates. Tiie g 's then 

have the values given in (4). A free material point moves 
with reference to such a system uniformly in a straight- 
line. If we now introduce, by any substitution, the space- 
time co-ordinates x^ ...-^'4, then in the new system g ^s are 

no longer constants, but functions of space and time. At 
the same time, the motion of a free point-mass in the new 



GENEEALTSED TITEOBY OF RELATIVITY lOl 

co-ordinates, will appear as curvilinear, and not uniform, in 
which the law of motion, will be independent of the 
nature of the moving mass-points. We can thus signify this 
motion as one under the influence of a gravitation field. 
We see that the app^^arance of a gravitation-field is con- 
nected with space- time variability of g ^s. In the general 

ease, we can not by any suitable choice of axes, make 
special rela^^ivity theory valid throughout any finite region. 
We thus deduce the conception that g *s describe the 

gravitational field. According to the general relativity 
theory, gravitation thus plays an exceptional role as dis- 
tinguished from the others, specially the electromagnetie 
forces, in as much as the 10 functions g representing 



gravitation, define immediately the metrical properties of 



the four-dimensional region. 



B 

Mathematical Auxill\kies for Establishing the 
General Covartant Equations. 

We have seen before that the general relativity-postu- 
late leads to the condition that the system of equations 
for Physics, must be C9- variants for any possible substitu- 
tion of co-ordinates .<,, ... j^ ; we have now to see 
how such general co-variant equations can be obtained. 
We shall now turn our attention to these purely matheniati- 
cal propositions. It will be shown that in the solution, the 
invariant ds, given in equation (3) plays a fundamental 
role, which we, following Gauss's Theory of Surfaces, 
style as the line-element. 

The fundamental idea of the general co-v^ariant theory 
is this : — With reference to any co-ordinate system, let 
certain things (tensors) be defined by a number of func- 
tions of co-ordinates which are called the components of 



102 PRINCIPLE OF RELATIVITY 

the tensor. There are now certain rules according to which 
the components can be calculated in a new system of 
co-ordinates, when these are known for the original 
system, and when the transformation connecting the two 
systems is known. The things herefrom designated as 
" Tensors " have further the property that the transforma- 
tion equation of their components are linear and homogene- 
ous ; so that all the components in the new s^^stem vanish 
if they are all zero in the original system. Thus a law 
of Nature can be formulated by putting all the components 
of a tensor equal to zero so that it is a general co-variant 
equation ; thus while we seek the laws of formation of 
the tensors, we also reach the means of establishing general 
CO- variant laws. 

5. Contra-variant and co-imriant Four-vector. 

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

is expressed by the equation 



(5) dx! =^ -^ d. 



V 



The dx' '« are expressed as linear and homogeneous func- 
tion of dr ^s ; we can look upon the differentials of the 

co-ordinates as the components of a tensor, which we 
designate specially as a eontravariant Four-vector. Every- 
thing which is defined by Four quantities A , with reference 
to a co-ordinate system, and transforms according to 
the same law, 

/ 

(5a) A =^^-^ A 

V 



GENEEALISBD TilEORT OF RELATIVITY 103 

we may call a contra- variant Four-vector. From (5. a), 
it follows at once that the sums (A 4 ^ ) ^^^ ^^so com- 
ponents of a four-vector, when A^ and B*^ are so ; cor- 
responding relations hold also for all systems afterwards 
introduced as " tensors " (Rule of addition and subtraction 

of Tensors). 

Co-variant Four-vector. 

We call four quantities A as the components of a co- 
variant four- vector, when for any choice of the contra- 
variant four vector B (6) > A B = Invariant. 

V V 

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



^ A' B*^ =^ A 

cr cr V V 


B^ 


the expressions 

a ^ 
0- a,,. 


• 



for B following from the inversion of the equation (5a) 
we get 

^ B^ ^ — ^ A =^ B^ A' 

*^(r »' 9-13 , V ^ <^ 



or 



As in the above equation B are independent of one another 

and perfectly arbitrary, it follows that the transformation 

law is : — 

9 



A' =^ ^ A 

- ^ 9v ' 



lU* PRINCIPLE OF RELATIVITY 

: fiemafka on the simplification of the mode of loriting 
the expressions. A glance at the equations of this 
paragraph will show that the indices which appear twice 
within the sign of summation [for example v in (5)] are 
those over which the summation is to be made and that 
gnly .over the indices which appear twice. It is therefore 
possible, without loss of clearness, to leave off the summation 
sign ; so that we introduce the rule : wherever the 
index in any term of an expression appears twice, it is to 
be summed over all of them except when it is not oxpress- 
edly said to the contrary. • 

The difference between the co- variant and the contra- 
variant four- vector lies in the transformation laws [ (7) 
and (5)]. Both the quantities are tensors according to the 
above general remarks ; in it lies its significance. In 
accordance with Rieei and Levi-eivita, the contravariafits 
and co-variants are designated by the over and under 
indices. 

§ 6. Tensors of the second and highei ranks. 

Contra variant tensor : — If we now calculate all the 16 

products A^ of the components A'^ B^ , of two eon- 
travariant four- vectors 

a'**', will according to (8) and (5 a) satisfy the following 
transformation law. 

(9) A^ = -^--^ -^ A^^^ 

We call a thing which, with reference to any reference 
system is defined by 16 quantities and fulfils the transfor- 
mation relation (9), a contra variant tensor of the second 



GENERALISED THEORY OF RELATIVITY l5h 

rank. Not every such tensor can be built from two four- 
vectors, (according to 8). But it is easy to show that any 

16 quantities A'^^, can be represented as the sum of A'^ 

B of properly chosen four pairs of four-vectors. From it, 
we can prove in the simplest way all laws which hold true 
for the tensor of the second rank defined through (9), by 
proving it only for the special tensor of the type (8). 

Contravariant Tensor of anij rank : — If is clear that 
corresponding to (8) and (9j, we can define contravariant 
tensors of the 3rd and higher ranks, with 4^, etc. com- 
ponents. Thus it is clear from (8) and (9) that in this 
sense, we can look upon contravariant four-vectors, as 
eontra variant tensors of the first rank. 

Co'Variant tensor. 

If on the other hand, we take the 16 products A of 

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

V 

(10) A =A B . 

for them holds the transformation law 

(J T 

By means of these transforma;tion laws, the co-variant 
tensor of the second rank is defined. All re-marks which 
we have already made concerning tbe contravariant tensors, 
hold also for co- variant tensors. 

Remark : — 

It is convenient to treat the scalar Invariant either 
as a contravariant or a co-variant tensor of zero rank. 
14 



106 . PKINCIPLE OF RELATIVITY 

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

(12) a' =AB'' 

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

(13) a" = -s- • a- ^ 

Naturally there are mixed tensors with any number of 
co-variant indices, and with any number of contra- variant 
indices. The co-variant and contra-variant tensors can be 
looked upon as special cases of mixed tensors. 

Symmetrical tensors : — 

A contravariant or a co-variant tensor of the second 
or higher rank is called symmetrical when any two com- 
ponents obtained by the mutual interchange of two indices 

IXV 

are equal. The tensor A or A is symmetrical, when 

> . 

we have for any combination of indices 

(U) A''''=A'''' 

or 
(14a) A =A . 

It must be proved that a symmetry so defined is a property 
independent of the system of reference. It follows in fact 
from (9) remembering (14) 

A"^ = — -^ I A'^*'=: - ~ A^^^ A^"^ 

y. V /A. V 



GENEHALISED THEORY OF EELATIVITY 107 



» 



Aniiiymmetriaal tensor. 

A contravariant or co-variant tensor of the 2nd, 3r(l or 
■ith rank is called antuy mmetrical when the two com- 
ponents got by mutually interchanging any two indicjs 

are equal and opposite. The tensor A or A is thus 

an tisy mmetrical when we have 

(15) A''*' = -A'''^ 



or 



(15a) A =~.A 



Of the 16 components A'^ , the four components A'^^ 
vanish, the rest are equal and opposite in pairs ; so that 
there are only 6 numerically different components present 
(Six-vector). 

Thus we also see that the antisymmetrical tensor 

^^^ (3rd rank) has only 4 components numerically 

different, and the antisymmetrical tensor A only one. 

Symmetrical tensors of ranks higher than the fourth, do 
not exist in a continuum of 4 dimensions. 

§ 7. Multiplication of Tensors. 

Outer multiplication of Tensors : — We get from the 
components of a tensor of rank z^ and another of a rank 
-', the components of a tensor of rank {z-^z') for which 
we multiply all the components of the first with all the 
components of the second in pairs. For example, we 



108 ' PRINCIPLE OF RELATIVITi' 

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

T = A B , 

fxvcr /xv (T 

The proof of the tensor character of T, follows imme- 
diately from the expressions (8), (10) or (12), or the 
transformation equations (9), (11), (13); equations (8), 
(10) and (12) are themselves exaftiples of the outer 
multiplication of tensors of the first rank. 

Reduction in rank of a 7mxed Tenmr. 

From every mixed tensor we can tret a tensor which is 
two ranks lower, when we put an index of eo- variant 
character equal to an index of the contravariant character 
and sum according to these indices (Reduction). We get 
for example, out of the mixed tensor of the fourth rank 

A , the mixed tensor of the second rank 

A =A =(SA ) 

/5 a^ V^ a/3/ 

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

A= A = A 

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



GENERALISED THEORY OF RELATIVITY 109 

of tensor according to the generalisation of (12) in combi- 
nation with (6) or out of the generalisation of (13). 

Inner and mixed muUiplicatiori of Tensors. 

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

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



o" or 

D = A B . 



Through reduction according to indices v and o- {I.e., put- 
ting v = a"), the co-variant four vector 



V y 

D = D = A B is generated. 



These we denote as the inner product of the tensor A 

^ fXV 

and B . Similarly we get from the tensors A and B^^ 
through outer multiplication and two-fold reduction the 
inner product A B^*' . Through outer multiplication 

and one-fold reduction we get out of A and B^^ , the 

^ jXV ' 

mixed tensor of the second rank D = A B^*" . We 

can fitly call this operation a mixed one ; for it is outer 
with reference to the indices ju, and t, and inner with 
respect to the indices v and q-. 



110 PRINCIPLE OF RELATIVITY 

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

to the abuve representation, A B is a scalar, when A 
and B are tensors. We also remark that when A B is 



an invariant for every choice of the tensor B , then A 

has a tensorial character. 

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



A , B-^" =A B'^''. 

err fxv 

\ 

From the inversion of (9) we have however 

9 a; / 9^' ^ 

O" T 



Substitution of this for B'^*' in the above equation gives 
9 ^ 9 .{' t 

(^ err a a-^, 9 ^V f"" ) 



This can be true, for any choice of B only when 
the term within the bracket vanishes. From which by 
referring to (11), the thtorem at once follows. This law 
correspondingly holds for tensors of any rank and character. 
The proof is quite similar, The law can also be put in, the 

following from. If B'^ and C are any two vectors, and 



OENEEALISED THEOEY OF RELATIVITY 111 



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

is a scalar, then A is a co-variant tensor. The last 

law holds even when there is the more special formulation, 

that with any arbitrary choice of the four- vector B alone 

the scalav product A B'^ B is a scalar, in which case 

we have the additional condition that A satisfies the 

symmetry condition. According to the method givien 
above, we prove the tensor character of (A 4- A ), from 

which on account of symmetry follows the tensor- character 
of A . This law can easily be generalized in the case of 

CO- variant and contravariant tensors of any rank. 

Finally, from what has been proved, we can deduce the 
following law which can be easily generalized for any kind 

of tensor : If the quanties A B form a tensor of the 

first rank, when B is any arbitrarily chosen four-vector, 
then A is a tensor of the second rank. If for example, 

C'* is any four-vector, then owing to the tensor character 

of A B*' , the inner product A C'^ B is a scalar, 

both the four- vectors C and B being arbitrarily chosen. 
Hence the proposition follows at once. 

A few words about the Fundamental Tensor g . 

The co-variant fundamental tensor — In the invariant 
expression of the square of the linear element 

ds^-=ig dx dx 



112 PRINCIPLE OF RELATIVITY 

(U plays the role of any arbitarily chosen eontravariant 
vector, since further g —q , it follows from the eonsi- 

[XV '^ VfX 

derations of the last paragraph that g is a symmetrical 

co-variant tensor of the second rank. We call it the 
" fundamental tensor/^ Afterwards we shall deduce 
some properties of this tensor, which will also be true for 
any tensor of the second rank. But the special role of the 
fundamental tensor in our Theory, which has its physical 
basis on the particularly exceptional character of gravita- 
tion makes it clear that those relations are to be developed 
which will be required only in the case of the fundamental 
tensor. 

The co-variant fundamental tensor. 

If we form from the determinant scheme I a \ the 

minors of ^ and divide them by the determinat ^= | g j 

we get certain quantities g^^ = g^^ , which as we shall 
prove generates a eontravariant tensov- 

Accordino: to the well-known law of Determinants 



'» 



(16) ,^„r^i' 

where o is 1, or 0, according asV = ^ or not. Instead 

fX 

of the above expression for ds^ y we can also write 

a S d.v dx 

-" IX<T y fX V 

or according to (16) also in the form 

goo dx dx 



GENERALISED THEORY OF RELATIVITY 



118 



Now according to the rules of multiplication, of the 
fore- going paragraph, the magnitudes 

d^ ^q dx 

foims a co-variant four-vector, and in fact (on account 
of the arbitrary choice of dx ) any arbitrary four- vector. 

If we introduce it in our expression, we get 



ds^ ^g^'^d^^ r/|^. 



For any choice of the vectors d^ d^ this is scalar, and 



(TT 



g , according, to its defintion is a symmetrical thing in o- 



(TT 



and T, so it follows from the above results, that g is a 
contravariant tensor. Out of (16) it also follows that S 



V 



is a tensor which we may call the mixed fundamental 
tensor. 

Determinant of the fundamental tensor. 

According to the law of multiplication of determinants, 
we have 



^ 9 



av 



= \ 9^J \ 9 



av 



On the other hand we have 



^/xa*^ 



av 



h 



=1 



So that it follows (17) that 
15 



9 



ixv 



9 



fXV 



= 1. 



114 



PRINCIPLE OF RELATIVITY 



Invariant of volume. 

We see k first the transformation law for the determinant 



i^= \9 



fA.V 



According to (II) 





dx 


a- 




9'^ 




V 

9 .' 


^/XV 




(T 


T 





Frorti this by applyiug the law of mutiplication twice, 
we obtain. 



9' = 


9V 


1 


a.r 

V 




^„ 






a .r / 

T 



or 



V/^ 



a ^^ 

H 

9V' 



vA 



... V'"^/ 



On the other hand the law of transformation of the 
volume element 

dT'=fdx^ dr^ dr^ dx^^ 
is aecordinff to the wellknown law of Jacobi. 



dr'^ 



d ' 
c 

dx 



P- 



di 



... (B) 



by multiplication of the two last equation (A) and (B) we 



get. 



(18) 



= Vg £Zt'= Vg dr. 



Insted of ^g, we shall afterwards introduce \/^g 
which has a real value on account of the hyperbolic character 
of the time-space continuum. The invariant ^'ZTgdr, is 
equal in magnitude to the four-dimensional volume-element 



GENERALISED TtlEORY OF RELATIVITY 115 

measured with solid rods and clocks, in accordance with 
the special relativity theory. 

EemarJcs on the character of the spacC'time conthnmrn — 
Our assumption that in an infinitely small region the 
special relativity theory holds, leads us to conelude that ds^ 
can always, according to (1) be exprersed in real magni- 
tudes r/X,..Y/X . If we call dr o t^Q ^' natural ^* \o\\xme 
eleinent ^Xj r/Xg ^Xg d^^ we have thus (18a) ^t. 

Should \/ —g vanish at any point of the four-dimensional 
continuum it would si^nifv that to a finite co-ordinate 
volume at the place corresponds an iiifiuitely small 
" natural volume." This can nevei' be the ca^e ; so that g 
can never chan^(? i's sijLin; we would, according to tlie special 
relativity thtory assume that ff has a finite negative 
value. It is a hypothesis about the physical nature of the 
continuum consid^iieJ, and also a pre-establislied rule for 
tiie choice of co-ordinates. 

If however {—g) remains po.-itive and finite, it is 
clear that the choice of co-ordinatts can be so made that 
this quantity becomes equal to one. We would afterwards 
see that sueh a limitation of the choice of co-ordinates 
would produce a significant simplification in expressions 
for laws of nature. 

In place of (18) it fellows then simply that 

dr'^d 

from this it follows, remembering the law of Jacobi, 



(19) 



cr 

dx 



= 1 



116 PIIINCIPLE of ilELATltlTY 

With this choice of co-ordinates, only substitutions with 
determinant 1, are allowable. 

It would however be erroneous to think that this step 
signifies a partial renunciation of the general relativity 
postulate. We do not seek those laws of nature which are 
co-variants with regard to the tranformations having 
the determinant 1, but we ask : what are the general 
co-variant laws of nature ? First we get the law, and then 
we simplify its expression by a special choice of the system 
of reference. 

Building up of neio tensors wit/i the help of the fundamental 
tensor. 

Through inner, outer and mixed multiplications of a 
tensor with the fundamental tensor, tensors of other 
kinds and of other ranks can be formed. 

Example : — 



k.= g A 



fXV 



We would point out specially the following combinations: 



A'^' = /" /^ A 



A — g g Q ^ 

jxv ^fxa'^vp 

(complement to the, co-variant or eontravariant tensors) 

and, B •= a q^'^ A ^ 
We can call B the reduced tensor related to A . 



GENERALlSt:i) THiEOUY OP RELATIVITY 



117 



Similarly 






It is to be remarked that g is no other than the " com- 
plement " of ^ , for we have, — 



§ 9. Equation of the geodetic line 
(or of point-motion). 

As the " line element *' ds is a definite magnitude in- 
dependent of the co-ordinate system, we have also between 
two points Pj and P.2 of a four dimensional continuum a 
line for which ItU is an extremum (geodetic line), i.e., one 
which has got a significance independent of the choice of 
co-ordinates. 

Its equation is 



(20) 



u 



p. 



^ 



S 



LP. J 



I 



From this equation, we can in a wellknown way 
deduce 4 total differential equations which define the 
geodetic line ; this deduction is given here for the sake 
of completeness. 

Let A_, be a function of the co-ordinates x^ ; This 

defines a series of surfaces which cut the geodetic line 
sought-for as well as all neighbouring lines from P, to P^. 
We can suppose that all such curves are given when the 
vahie of its co-ordinates x^ are siven in terms of \. The 



ii8 



PRINCIPLE OF RELBTIVriTY 



sign S corresponds to a passage from a point of the 
geodetic curve soiight-for to a point of the contiguous 
curve, both lying on the same surface A,. 

Then (20) can be replaced by 

8w d\-0 



(20a) 






v^ 



dr dx 
ty*=qf — L 



But 






uA c^A 



V dA ^ rfA ^ j 



So we get by the substitution of hw in (^Oa), remem- 



bering that 



^ d\ ^ 



± (Be ) 



after partial integration, 



(20b) 



1 



d\ k Bx =0 
<r or 



L 



^ ( 3 ^^ 

where k =-— < . — - 

,<^ dX I w dX 



dg 



fXV 



2w a 



.r 



y-.^ . 



c^j; 



dX dX 



GENEllALfSED THEORY OF RELATIVITY 1 I9 

From which it follows, since the choice of 8 * is per- 
fectly arbitrary that k \ should vanish ; Then 

(20c) k =0 (cr=l, 2, 3, 4) 



<r 



are the e|uations of geodetic line; since along the 
geodetic line considered we have ^5=^0, we can choose the 
parameter A, as the length of the arc measured along the 
geodetic line. Then w = ], and we would get in place of 
•(20c) 

^ ^v ^> ^ 

^t^v a*" d^c^ ds ds 

1 dg d'^ 6^ 

_i _/^ Ij^ ? -_0. 

2 Q.f 6* 6^ 

Or by merely changing the notation suitably, 

d^x - - dx dr- 

(20d) g -/ + \^'^ -J^ . -r =0 

where we have put, following Christoffel, 

.on M -1 r ®^'"^+ ®'''"^- ®^'''''! 



(TT 



Multiply finally (^Od) with g (outer multiplication with 
reference to t, and inner with respect to <r) we gtt at 
last the final form of the equation of the geodetic line — 

' d^x ( ^ d^' da 

ds^ (t ) ^* ds 

Here we have put, following Christoffel, 



120 PRIXCIPLE OF EELATIVITY 

§ 10. Formation of Tensors through Differentiation. 

Relying on the equation of the ;^eodetie line, we can 
now easily deduce laws according to which new tensors can 
be formed from given tensors by differentiation. For this 
purpose, we would first establish the general co-variant 
differential equations. We achieve this through a repeated 
application of the following simple law. If a certain 
curve be given in our continuum whose points are character- 
ised by the arc-distances s. measured from a fixed point on 
the curve, and if further <f>, be an invariant space function", 

then ~ is also an invariant. The proof follows from 
as 

the fact that d<f> as well as ds, are both invariants 



Since 



d(f> __ 6 <^ At 

ds Qx Q s 



so that i/a= ~— ' ~- is also an invariant for all curves 
OX ds 

^hich go out from a point in the continuum, i.e., for 
any choice of the vector d.c . From which follows imme- 
diately that 

A = -M 

is a co-variant four-vector (gradient of ^). 

According to our law, the differential-quotient x= -S-^ 

OS 

taken along any curve is likewise an invariant. 
Substituting the value of if/, we get 

9;c Qa' ds ds 9»' ds^ 



GENERALISED THEORY OF RELATIVITY 



HI 



Here however we can not at once deduce the existence 
of any tensor. If we however take that the curves along 
which we are differentiating are geodesies, we get from it 



by replacing 



17^ 



according to (22) 



-[ 









dnV d'X 

ds ds 



Prom the interchan^eabilitv of the differentiation with 
regard to /x and v, and also according to (23_) and (21) we see 



that the bracket 



and V. 



l-i 



is sj'mmetrical with respect to ^ 



As we can draw a geodetic line in any direction from any 

point in the continuum, — -^ is thus a four-vector, with an 

ds 

arbitrary ratio of components, so that it follows from the 
results of §7 that 



(25) 



A = 



_ 6'<A 



ft V 






is a co-viiriant tensor of the second rank. We have thus got 
the result that out of the co-variant tensor of the first rank 

A = 5-^ we can get by differentiation a co- variant tensor 



of 2nd rank 



(26) 



A 



ixv 



dA ( 



ixv 



16 



1^-^ PRINCIPLE OF RELATIVITY 

We call the tensor A the *' extension " of the tensor 

A . Then Ave can easily show that this combination also 

leads to a tensor, when the vector A is not representable 

as a gradient. In order to see this we first remark that 

o^ .^ ^ co-variant four-vector when \p- and tjy are 

acalars. This is also the case for a sum of four such 
terms : — 

when j/^^^), <^(i)...«i^(4) ^(4) are scalars. Now it is however 
clear that every co- variant four- vector is representable in 
the form of S 

If for example. A is a four-vector whose components 
are any given functions of i« , we have, (with reference to 
the chosen co-ordinate system) only to put 

i/.W=A3 <3S»(3)=,i53 
in order to arrive at the result that S is equal to A . 

fX fX 

In order to prove then that A in a tensor when on the 

right aide of (26) we substitute any co-variant four-vector 
for A we have only to show that this is true for the 



GENERALISED THJIORY OF RELATIVITY H3 

four-vector S . For this latter case, however, a glance on 

the right hand side of (26) will show that we have only to 
bring forth the proof for the case when 

Now the right hand side of (25) maltiplied by i/^ is 

which has a tensor character. Similarlv, 5^ -S-^ is 

' 6.'- 6a; 
/^ ^ 

also a tensor (outer product of two foui'- vectors). 
Through addition follows the tensor character of 






Thus we get the desired proof for the fonr-vector, 

*A ^ J^^d hence for any four-vectors A as shown above. 
/^ 

With the help of the extension of the four- vector, we 
can easily define ''extension" of a co-variant tensor of any 
rank. This is a generalisation of the extension of the four- 
vector. We confine ourselves to the case of the extension 
of the tensors of the 2nd rank for which the law of for- 
mation can be clearly seen. 

As already remarked every co- variant tensor of the 2nd 
rank can be represented as a sum of the tensors of the type 
A B . 



1*24 PRINCIPLE OF RELAT1\'ITY 

It would therefore be sufficient to deduce the expression 
of extension, for one such special tensor. According to 
(26) we have the expressions 

aA ( ) 



6B 

V \ (XV 



B 



" cr y. "T ) 

are tensors. Through outer multiplication of the first 
with B and the 2nd with A we ffet tensors of the 

V fX ^ 

third rank. Their addition gives the tensor of the third 
rank 

A =:^Z£^-\''''] A -{""Ia ... (27) 



/xi'cr 



^\ It) " (t) '^^ 



where A ^ is put=:A B . The right hand side of (27) 

is linear and homogeneous with reference to A .and its 

fb-st differential co-efficient so that this law of foi-mation leads 
to a tensor not only in the case of a tensor of the type A 

B but also in the case of a summation for all such 

tensors, ^,e.J in the case of any co-variant tensor of the 

second rank. We call A the extension of the tensor A . 

fxva fxv 

It is clear that (26) and (24) are only special cases of 

(27) (extension of the tensors of the first and zero rank). 

In general we can get all special laws of formation of 

tensors from (27) combined with tensor multiplication. 



GENER.ALISED THEORY OF RELATiVlEY 



1 25 



Some special cases of Particular Importance. 

A few auxiliary lemmas concerning the fwlda mental 
tensor. We shall first deduce some of the lemmas much used 
afterwards. Accoi'diiig to the law of differentiation of 
determinants, we have 

(28) dg=:g^'' gdg^^=^g^^ gdg^"" . 

The last form follows from the first when we remember 
that 



a qf^^z=^^ , and therefore a g^^ = -1. 
consequently g dg^^-^g^^ dg =0- 



From (28), it follows that 



(29) 






(T 



■i 9. *^ 



>»' 6- 



Again, since g q =8 . we have, by differentiation, 



r 



a da ^=^--q dq 



(30) i '"" - 

OQ vcr ^ 



jxcr 



L 



a.> 



err 



By mixed multiplication with g and .7 v respectiyely 
we obtain (changing the mode of writing the indices). 



126 



Principle of uelativity 



(31) 



r 



dg^'^=:—gf^"- /^ dg 



ay3 



< 



.Z^" 



6<7 fta v/? J 



# and 



(32) 



"S 



rfa =—(7 (( n dg "^ 



6(7 a tt/^ 






The expression (31) allows a transfonnation which we 
shall often use; according to (21) 



(33) 



8? 



[ 






+ 



" /5 (T 



a -' 



If we substitute this in the second of the formnla (31), 
we get, remembering (23), 



(34) 






flV 



i MT > T Cr f , VT ^ T 



.) 



A^ 



S) 



By substituting the right-hand side of (34) in (29), we 



get 



(29a) 






Generalised 'fHEOny of relativity 1:^7 

Divergence of the contravarimit four -vector. 

Lefc us multiply (26) with the con ti'a variant fnndaniental 

tensor ^'^^^(inner multiplication), then by a transformation 
of the first member, the right-hand side takes the form 

9(/^ 1 Tu / ^'^a 






According to (31) and (29). the last member can take 
the form 

Both the first members of the expression (B), and the 
second member of the expression (A) cancel each other, 
since the naming of the summation-indices is immaterial. 
The last member of (B) can then be united with fii»st of 
(A). If we put 

r A^ = A^ ■ 

where k^ as well as A are vectors which can be arbi- 
trarily chosen, we obtain finally 

1 



^=: 



( V'-^g A^' ) . 



This scalar is the Divergence of the contravariant four- 
Toctor A , 



128 PRINCIPLE OP UELATIVITY 

^ notation of the [covariant) fowr^vector. 

The second membw in (26) ie symmetrical in the indiceR 
/A, and V, Hence A ,— A is an antisymmetrical tensor 

built up in a very simple manner. We obtain 

6A 6A 

^^^^ ^Mr= -^~ -^ S/ 



Anti.si/mmefTical Extension of a Six-reHor. 

If Ave apply the operation (27) on an antisymmetrical 
tensor of the second rank A , and form all the equations 

arising from the cyclic interchange of the indices /a, v, cr. and 
add all them, we obtain a tensor of the thini rank 

6A 
(37) B =:A + A + A = ~~^ 

^ ^ fxya {lycT - vatx (t/xv Q^ 

(T 



aA 6A 

+ "L^^ ^/^ 

6 '<>• 6 •« 

fX V 

from which it in easy to see that the tensor is antisymmetri- 
cal. 

Divergence of the Six-vector. 

If (27) is multiplied by ^'^^ ^*'' (mixed multiplication), 
then a tensor is obtained. The first member of the right 
hand side of (27) can be written in the form 



GKXEIJALISED THEORY OV RELATIVITY 1:29 

If we replace g^^^ 7^^ A by A ,/ q^^'' 1/^' A by 

A ' and replace in the transformed first member 

with the help of (•'^4), then from the right-hand side of (27) 
there arises an expression with seven terms, of which four 
cancel. There remains 

(38) a"^= %^- + i"^ '} A''-/^+ ^^ " j A''^ 

This is the expression for the extension of a contravariant 
tensor of the second rank; extensions can also be formed -for 
corresponding- contravni'iant tensors of higher and lower 
ranks. 

We lemark that in the same way, we can also form the 

a 

extension of a mixed tensor A 



a 



^^ ^^ ^} .^ ^' ^^ .- 

r39) A- = ---/" - ^ A -f ^ A . 



By the reduction of (38) with reference to the indices 
(3 and o- ( inner multiplication with 6 I , we get a con- 
travariant four-vector 

17 



130 PRINCIPLE OF itELATlVITY 

On the account of the symraetrv of -^ • witli 

■ ( " ) 

reference to the indices (3, and k, the third member of the 

right hand side vanishes when A '^ is an antisymmetrical 
tensor, which we assume here ; the second member can be 
transformed according to (29a) ; we therefore get 

(40) ^/^-g dx^ 

This is the expression of the direro'ence of a contra - 
variant six-vector. 

Divergence of the mixed tensor of the second rank. 

Let us form the reduction of (89) with reference to the 
indices a and <r, we obtain remembering (29a) 

Tf we introduce into the last term the contravariant 
tensor A" =17" A , it takes the fori 



'm 



[or ^ 



If further A'^ is symmetrical it is reduced to 



CIENEllALISED THEORY OF KEf-ATIVlTY 13i 



If instead of A' , we iiitrocliice in a similar way the 
symmetrical co-variant tensor A ■=.g g r* A ^ , then 
owing to (31) the last member can take the form 






In the symmetrical case treated, ("11) can be replaced by 
either of the forms T 



6 ( v^-y A-^ ) 



or 



(Ua) s^-g A = ^ 

/A 



a ( v^-^ A^ ) 



(41b) ^/-^ A = -^^ .- ^ 

(T 



.P^ 



+ 1 1^ sf-g A 
which we shall have to make use of afterwards. 

§12. The Riemann-Christoffel Tensor. 

We now seek only those tensors, which can be 
obtained from the fundaiiiental tensor </^ ^by differentiation 
alone. It is found easily. We put in (37) instead of 
any tensor A'^*'' the fundamental tensor g^^ and get from 



132 



PlllNClPLE OF IIELATIVITY 



it a new tensor, namely the extension of the fundamental 
tensor. We ean easily convince ourselves that this 
vanishes identically. We prove it in the following way; we 
substitute in (27) 




i.e. J the extension of a four- vector. 

Thus we get (by slightly changing the indices) the 
tensor of the third rank 

a'^A (iKT^b^ CfJiT^ dA. ((XT') 6A 

Mcrr a.^6.', l^ ^a., I, 56.^ Ip ^ 6.^ 



•f 



6.' 



fid') (flT 

p ) (a 




acr 






We ,use these expressions for the formation of the tensor 



A 



— A 



/i.<7T 



/xTcr 



Therebv the followin"r terms in A 



fJi(TT 



cancel the corresponding terms in A ; the lirst member, 
the fourth member, as well as the member corresponding 
to the last term within the square bracket. These are all 
symmetrical in o-, and r. The same is true for the sum uf 
the second and third members. We thus get 



fxar 



A = B^ A 

jXTCr fJ.CT p 



(^3)^ 



fXCTT 



6_ 



iidf 



_6_ ^/XT 



to- ) tp J (a 




GENERALISED THEOllY OF RELATIVITY 138 

The essential thing in this result is that on the 
right hand side of (42) we have only A , but not its 

differential co-efficients. From the tensor-character of A 

— A , and from the fact that A is an arbitrary four 
vector, it follows, on account of the result of §7, that 
B ^ is a tensor (Iliemann-Christoft'el Tensor). 

fXO-T 

The mathematical signilicance of this tensor is as 
follows; when the continuum is so shaped, that there is a 

co-ordinate system for which o 's are constants, B^ all 

vanish. 

If we choose instead of the oriijiual co-ordinate svstem 

I 
any new one, so would the ^ 's referred to this last system 

be no Ioniser constants. The tensor character of B^ 

^ /X(TT 

shows us, however, that these components vanish collectively 
also in any other chosen system of reference. The 
vanishing of the Riemann Tensor is thus a necessary con- 
dition that for some choice of the axis-system </ 's can be 
taken as constants. In our problem it corresponds to the 
ease when b}^ a suitable choice of the co-ordinate system, 
the special relativity theory holds throughout any finite 
region. By the reduction of (i-i) with reference to indices 
to T and p, we get the eo variant tensor of the second rank 

B =R 4-S 

fXV fiv flV 

S = Q tog- \/^j _ >/^^7 9 lug ^/.Zy ^ 
p. V v.a J a 



134 PlllXCrPLE OF EELATIVITY 

Eemnrks upon the choice of co-ordinates. — It has already 
been remarked in §8, with reference to the equation (18a), 
that the co-ordinates can with advantage be so chosen that 
^ — </ = 1. A glance at the equations got in the last two 
paragraphs shows that, through such a choice, the law of 
formation of the tensors suffers a significant simplifica- 
tion. It is specially true for the tensor B , which plays 
a fundamental role in the theory. By this simplifica- 
tion, S vanishes of itself so that tensor B reduces to 

[XV 

I 

I shall give in the following pages all relations in the 
jriimplified form, with the above-named specialisation of 
the co-ordinates. It is then very easy to go back to the 
general covariant equations, if it appears desirable in 
any special ease. 



C. THE THEORY OF THE GRAVITATION-FIELD 

§13. Equation of motion of a material point in a 
gravitation-field. Expression for the field-components 
of gravitation. 

A freely moving body not acted on by external forces 
moves, according to the special relativity theory, along a 
straight line and uniformly. This also holds for the 
generalised relativity theory for any part of the four-dimen- 
sional region, in which the co-ordinates Ko can be^ and 

are, so chosen that (j /s have special constant values of 

the expression (4). 

Let us discuss this motion from the stand-jmint of any 
arbitrary co-ordinate-system K;; it moves with reference to 
Kj (as explained in ^'l) in a gravitational field. The laws 



• GEXEllALISED TIIEOJJY OF RELATIVITY 135 

of motion with reference to K, follow easily from the 
following consideration. With reference to K^,, the law 
of motion is a four-dimensional straight line and thus a 
geodesic. As a geodetic-line is defined independently 
of the system of co-ordinates, it would also be the law of 
motion for the motion of the material-point* with reference 
to Kj ; If we put 



(45) p^ ^ _ 



• we get the motion of Ihe point with reference to K^ 
given by 



2 

d ," (1 " (J.v 



V 






We now make the very simple assumption that this 
general covariant system of equations defines also the 
motion of the point in the gravitational field, when there 
exists no reference-system K^, with reference to which 
the special relativity theory holds throughout a finite 
region. The assumption seems to us to be all the more 
legitimate, as (46) contains only the first differentials of 

(/ , among which there is no relation in the special ease 
when Kq exists. 



If r ^ 's vanish, the point moves uniformly and in a 

fJLV 

straight line ; these magnitudes therefore determine the 
deviation from uniformity. They are the components of 
the gravitational field. 



1-36 PllIXClPLE OF llELATIVITi 

§14. The Field-equation of Gravitation in the 

absence of matter. 

In the following, we differentiate gravitation-field from 
matter in. the sense that everything besides the gravita- 
tion-field will be signified as matter ; therefore the term 
includes not only matter in the usual sense, but also the 
electro-dynamie field. Our next problem is to seek the 
field-equations of gravitation in the absence of matter. For 
this we apply the same method as employed in the fore- 
going paragraph for the deduction of the equations of 
motion for material points. A special case in w^hich the 
field-equations sought-for are evidently satisfied is that of 

the special relativity theorv in which q 's have certain 

fXV 

constant values. This would be the case in a certain 
finite region with reference to a definite co-ordinate 
system K^,. With reference to this system, all the com- 
ponents B^^ of the Riemann's Tensor [equation i'3] 

vanish. These vanish then also in the region considered, 
with reference to every other co-ordinate svstem. 

The equations of the gravitation-field free from matter 

must thus be in everv case satisfied when all & vanish. 

But this condition is clearly one which goes too far.. For 
it is clear that the o^ravitati on -field srenerated bv a material 
point in its own neighbourhood can never be transformed 
aivai/ by any choice of axes, i.e., it cannot be transformed 

to a case of constant g 's. 

Therefore it is clear that, for a gravitational field free 
from matter, it is desirable that the symmetrical ten- 
sors B deduced from the tensors B„^^ should vanish. 



GEXEK-UiTSED THEORY OF RELATIVITY 137 

We tlius get 10 equations for 10 cinantities g which are 
I'ulhlled in the special ease when B^ 's all vanish. 

^ fXCTT 

Rerae«ibering (44) we see that in a})senee o£ matter 
the field-eqiiations come out as follows ; (when referred 
to the special co-ordinate-system chosen.) 



6r" . 

(47) ^^ + r\ r^ =o; 

a 



/ — — 1 r " — ) f^^l 

^ -^ ' ' /XL' / " \ 



It can also be shown that the choice of these equa- 
tions is connected with a minimum of arbitrariness. For 
besides B , there is no tensor of the second rank, which 



fX 



V 



can be built out of a ^s and their derivatives no his/her 

fjLV 

than the second, and which is also linear in them. 

It will be shown that the equations arising in a purely 
mathematical way out of the conditions of the general 
relativity, together with equations (46), give us the New- 
tonian law of attraction as a first approximation, and lead 
in the second approximation to the explanation of the 
perihelion-motion of mercury discovered by Leverrier 
(the residual effect which could not be accounted for by 
the consideration of all sorts of disturbing factors). My 
view is that these are convincing proofs of the physical 

correctness of my theory. 
18 



138 



PRIXCIPLE OF I^ELATIVITY 



^15. Hamiltonian Function for the Gravitation-field. 
Laws of Impulse and Energy. 

In order to sliow that the field equations correspond to 
the laws of impulse and ener^ry, it is most convenient to 
write it in the following Hamiltonian form : — 



f 



Hr/T = o 



(47a) 



•' ' VOL 

^13 



Here the variations vanish at the limits of the finite 
four-dimensional integration-space considered. 

It is first necessary to show that the form (47a) is 
equivalent to equations (47). For this purpose, let us 

consider H as a function of g^^ and g^^' I :■- ^ 

We have at first 



a 



(T 



8H=r" r^ 8/^+2/Y'' sr^ 

ix(3 va fji^ va 

= -r" r'^s,r+2r"ga( rrr'^) 



va 



But <rrry= -1 3[rr. 



.iSX 






ar/ \ a.^z . 

'' 'xA av \ 

a.'- a.i\ / 



GiENEkALlSED THEORY OE iJELATiVlTY 



];39 



The terms arising out of the two last terms witiiin the 
round bracket are of different signs, and change into one 
another by the interchange of the indices /x and /3. They 
cancel each other in the expression for 3H, when they are 

multiplied by F q, which is symmetrical with respect to 

/x and ft so that only the first member of the bracket 
remains for our consideration. Remembering (31), we 
thus have : — 



Therefore 



(48) 



r an ^ _ f-a p/5 
^. an 



<T 



9.^/ 



fXV f^^ 



<r 



If we now carry out the variations in (47a), we obtain 
the system of equations 



(47b) 



a / ^ H \ an 



a 



a^ 



/jti 

a 



dg' 






which, owing to the relations (48), coincide with (47), 
as was required to be proved. 



If (47b) is multiplied by g^ , 



suice 



Qg 



jXV 



(T 



d-^ 



a 



dg 



a 



140 PRINCIPLE or uelativitY 

aud consequently 



i) 



(T 






a 



9f/ 



a 



a 



99' 



a 



9H 



6'/ 



/Xl' 



a 



9 c/ 
we obtain the equation 

6 / «^ 8H \_ 8H _,^ 



/ui' 9 i<; 



a 



6. 



- { r ^) 



6? 



or 



a 



or 



(49) 



9^ 



a 

(T 



9-' 



:0 



a 



.1 ^a /^''' 



9H 



9ry 



a 



8*" H. 

(T 



Owirrp: to the relations (48), the equations (47) and (34), 



(50) 



,a I o>a ur _ a ^ ^ 



-^ ' /x^ ' vo- 



lt is to be noticed that /^ is not a tensor, so that the 

equation (49) holds only for systems +or which ^/— (^ = 1. 
This equation expresses the laws of conservation of impulse 
and energy in a gravitation-held. In fact, the integra- 
tion of this equation over a three-dimensional volume V 
leads to the four equations 



(49a) 



d.v 



■{ 



1 



^ 



^ dV ['^ 



j( C - 



+ f a, + / 
cr 



.^"'O 



'IS 



(JENEE.VLISEID THEORY OE EEEATIVITY Ijl 

where a^, a^^ a.^ are the direetion-eosines of the inward- 
drawn normal to the sarface-elemeiit ^^S in the Euchdean 
Sense. We recognise in this the usnal expression for the 

laws of conservation. AVe denote the maofnitudes t as the 

energy-components of the gravitation-field. 

I will now put the equation (47) in a third form which 
will be very serviceable for a quick realisation of our object. 

By multiplying the iield-equations (47) with g , these are 

obtained in the mixed forms. If we remember that 



j'o- 9 r 9 / \ 9 9' 

9 r _ 






a a ' a 



which owing to (o4) is e(jual to 



9 



) { vo- _ a \ i'/8 __ (J ^ <i 



9 

— (I ^ 

or slightly altering the notation equal to 



•^ ^ fta ^ ,uP 



9 
9- 



a 



The third member of this expression cancel with the 
second member of the field-equations (47). In place of 
the second term of this expression, we can, on account of 
the relations (50), put 



K i f — — 8 /^j, where t =: f 
\ fj, 2 /^ / 'i 



ii2 



i'klNClPLE OF RLLAl'lViTY 



Tlierei'ore iii the ])]aee of the equations (47), we obtain 



(51) 



6 



a-' 



{'- 



a 



f3 a 






v/-r/=i. 



§16. General formulation of the field-equation 

of Gravitation. 

The field-ec[iiations established in the preceding para- 
graph for spaces free from matter is to be compared with 
the e((uation v^<^=Oof the Newtonian theory. AVe have 
now to find the equations which wall correspond to 
Poisson's Equation \/^(fi = 4TrKp, (p signifies the density of 
matter) . 

The special relativity theory has led to the conception 
that the inertial mass (Trage Masse) is no other than 
energ}'. It can also be fully expressed mathematically by 
a symmetrical tensor of the second rank, the energy-tensor. 
We have therefore to introduce in our generalised theory 

energy-tensor t"' associated with matter, which like the 



energy components t _ of the gravitation-field (equations 

49, and oO"! have a mixed character but which however can 
be connected »with symmetrical covariant tensors. The 
ecpiation (.51) teaches us how to introduce the energy-tensor 
(corresponding to the density of Poisson's equation) in the 
field equations of gravitation. If we consider a complete 
system (for example the Solar-system) its total mass, as 
also its total gravitating action, will depend on the total 
energy of the system, ponderable as well as gravitational. 



GEXEllALTSED TITEOEY OF RELATIVITY 



14:5 



This can be expressed, b}^ pnttino^ in (51), in place of 
energy-components t of li^ravitation-fleld alone the sum 
of tlie eneri^y-components of matter and gravitation, i.e., 
t ^ + T^. 

fX fX 

We thus get instead of (51), the tensor-ecpiation 



r a / cr/S a\ 
(52)^ ^'"aV f^P^ 



/ o- ■ rr \ 1 a- n 

f +T )-:, 8 (f + T) 
\ fx ix / ^ . u. 



I ' V-g=l 



,fX 



where T=:T (Lane's Scalar). These are the general tield- 

ecpiations of gravitation in the mixed form. In place of 
(47), we get by working backwards the system 






/xv 2 -'ixv J 



V^g = l. 



It must be admitted, that this introduction of the 
energy-tensor of matter cannot be justified by means of the 
Relativity-Postulate alone ; for we have in the foregoing 
analvs's deduced it from the condition that the eners^v of 
the gravitalion-field should exert gravitating action in the 
same way as every other kind of (^nergy. The strongest 
ground for the choice of the above equation however lies in 
this, that they lead, as their consequences, to equations 
expressing the conservation ■ of the components of total 
energy (the impulses and the energy) which exactly 
correspond to the equations (49) and (4 9a). This shall be 
shown afterwards. 



144 FKTXCTPLE OF HELATTVTTY 

ii 

^17. The laws of conservation in the general case. 

The equations (52) can be easily so transformed that 
the second member on the right-hand side vanishes. Me 
reduce (52) with reference to the indices /x and o- and 
subtract the equation so obtained after multiplication with 

i B from (52). 
We obtain. 



V a IX. J 



we 



operate on it b}' ^-^ . Now, 



9^- / ./3 



6 .'' a ■ „ 

u a 



( ''' r;, ) 



a 



2 d ,r a -'' 
a tr 



aX / '^/xA 






9.^/?X ^O..R N -| 



.d« a-x 

/x A 



The first and the third member of the round bracket 
i lead to expressions which cancel one another, as can be 
easily seen by interchanging the summation-indices a, and 
(f, on the one hand^ and /? and A, on the other. 



GEXERATJSRD THEOnY OF TJELATTVTTY 



14:^ 



The second term can be transformed according' to (-il). 
So that we got^ 



<T 



fj^jS 



1 



6 



.7 



afS 



2 6.(„. dXn 6.'' 

<r p jx 



The second member of the expression on the left-hand 
side of (j^a) leads first to 



a 



1 



2 9-'^ d 

a fx 



( .M^ 



:, { '■•' r l^ ) 



(tr 



to 



a 



4 9 ,r ^x 

a jji 



a.i'^ 



+ 



a^ 



6^/ 



;(' 



A 



a 






The expression arisini^^ out of the last member within 
the round bracket vanishes a<?cordiug to ('^9) on account 
of the choice of axes. The two others can be taken 
too'ether and give us on account of (-M)^ the expression 

1 6» </"^ " 



i 6- 6..« d. 

a p jx 



So that remembering (54) we have 



(55) 



a '■ a ->' 






/5 






1 jjo- A/5 

— TV 6 <l ' 



^Ip ) =^- 

identically 



19 



1 H) PRINCIPLE OP KELATIA'HY 

From (55) and (52a) it follows that 
(5H) _A ( /^^ 



9 ^ '^^ + T" I ^ o. 



From the field equations of (^gravitation, it also follows 
that the conservation-laws of impulse and energy are 
satisfied. AVe see it most simply following the same 
reasoning which lead to equations (f9a) ; only instead of 
the energy-components of the gravitational-field, we are to 
introduce the total energy-components of matter and gravi- 
tational field. 

§18. The Impulse- energy law for matter as a 
consequence of the field-equations. 

If we multiply (53) with ^^- , we get in a way 
similar to ^15, remembering that 



a/xv 
•^/v ^ — vanishes, 



6 / ?^ /^^ 

the equations __i'" _ i ^0 T' =; o 

a cr 

or remembering (56) ^ 

('^7) ^ + i ^^ T =o 

a (T 

A comparison with (41b) shows that these equations 
for the above choice of co-ordinates i\/—y = 1) asserts 
nothing but the vanishing of the divergence of the tensor 
of the energy-components of matter. 



(;exeijali>sI':d theory of kelativity li? 

Physically the appearance of the second term on the 
lel't-hand side shows that for matter alone the law of con- 
servation of impulse and energy cannot hold ; or can only 

hold when f/^'''s are constants ; i.e., when the field of gravi- 
tation vanishes. The second member is an expression for 
impulse and energy which the gravitation-field exeits per 
time and per volume upon matter. This comes out clearer 
when instead of (57) we write it in the Form of (47). 

8T^ /-? 

a 

The right-hand side expresses the interaction of the energy 
of the gravitational-field on matter. The field-equations of 
irravitation contain thus at the same time 4 conditions 
which are to be satisfied by all material phenomena. We 
get the equations of the material phenomena completely 
when the latter is characterised by four other differential 
equations independent of one another. 

D. THE '' MATEEIAL " PHENOMENA. 

The Mathematical auxiliaries developed under ^ B ' at 
once enables us to generalise, according to the generalised 
theory of relativity, the physical laws of matter (Hydrody- 
namics, Maxwell's Electro-dynamics) as they lie already 
formulated ^ according to the special-relativit^'-theorA'. • 
The ireneralised Relativitv Principle leads us to no further 
limitation of ])ossibilities ; but it enables us to know 
exactly the inHuence of gravitation on all processes with- 
out the introduction of any new h3q3othesis. 

It is owing to this, that as regards the physical nature 
of matter (in a narrow sense) no definite necessary assump- 
tions are to be introduced. The question may lie open 



118 I'lU^CIl'LE OF KELATIVlTV 

whether the theories of the electro-magnetic lield and the 
gravitational-liekl together, will form a sufficient basis fur 
the theory of matter. The general relativity postulate can 
teach us no new principle. But by building up the 
theorv it must be shown whether olectro-mao-netism and 
gravitation together can achieve what the former alone 
did not succeed in doing. 



§19. Euler's equations for fhctionless adiabatic 

liquid. 

Let j^y and p, be two scalars, of which the first denotes 
the pressure and the last the density of the liuid ; between 
them there is a relation. Let the contravariant symmetrical 
tensor 

rnap al3 " a ' ^ ^^-ov 

T "^ = -(1 ' p + p J- -j^ ... (58) 

' as «*■ 

be the contra-variant energy-tensor of the liquid. To it 
also belonijrs the covariant tensor 

rSSa) T =— V I, -f .V "- rf f, -^ p 

as well as the mixed tensor 

(581)) T^^--^"- P + g o -^^ ~ P- 



If we |)ut the right-hand side of (58b) in (57a) we 
get the general hydrodynaniieal ei] nations of Euler accord- 
iuo" to the ireneralised relativity theor\ . This in t)rinciple 
eom])letely solves the problem of motion ; for the four 



GENERALLSEi) THEORY 0¥ RELATIVITY 149 

equations (57a) together with the i^iveii e([uatioii between 
jj and p, and the equation 

(If (li'n 

^ P _ 1 

are sufficient, with the given values of g n, for finding 
out the six unknowns 

dx^ d>\ • (Ix ^ dx^ 
^ ^ ^'*' ds ' dn ' c^.s" ' ds 



If ^ ^s are unknown we have also to take the equ- 
tions (53). There are now 11 e([uations for finding out 
10 functions // , so that the number is more than suffi- 
cient. Now it is be noticed that the equation (57a) is 
ah'eady contained in (53), so that the latter only represents 
(7) independent equations. This indehniteness is due to 
the wide freedom in the choice of co-ordinates, so that 
mathematically the [)roblern is indelinite in the sense that 
three of the S[)ace-functions can be arbitrarily chosen. 

§20. Maxwell's Electro-Magnetic field-equations. 

Let c^ be the components of a covariant four-vector, 
the electro-magnetic potential ; from it let us form accord- 
ing to (36) the Components F of the covariant six-vector 

of the electro-maa;netic Held accordinsr to the svstem of 
equations 



(59) F 



_ iL — ^ 

o- p 



150 t'RlNXiPLE Oi' KELATIVITY 

From (5t))j it follows that the system of ecjuatioiis 



(60) 



6F 



pa- 



6F 



a.^' 



+ 



(TT 



61^ 



6 



+ 



TO 
^ ' =0 



a. 



(T 



is satisfied of which the left-hand side, according to 
(37), is an anti-symmetrical tensor of the third kind. 
This system (HO) contains essentially four equatioDs, which 
can be thus written : — 



(GOa) < 



6F^3 

a.'-," 

aF,, 
"a.'. 

a-f. 



+ 



aF 



3 i 



a.'^ 



a-' 



4- 2 



O 



4- 



aF, , aF, 

a ''3 a. ''4 



3 — 



^ o 



+ 



aF, , aF, , 



=: O 



aF, , _^ aF,3 a_F3i 



This system of equations corresponds to the second 
system of equations of Maxwell. We see it at once if we 

])ut 



f(3l) 



r ^'23 = 

1 


H, 


l'\. = 


K, 


■ -i 1*',, = 


H, 


F,, - 


E, 


v-l',, - 


H, 


l\v. - 


K, 



Instead of (GOa) we can therefore write according to 
the usual notation of three-dimensional veetor-analvsis : — 



/ 



(GOl)j 



aH 

a7 



+ l-ot Erz:(. 



div H=:o. 



CENEEALTSKD TTIEOT^Y OF RELATIVITY 



151 



Tlie first MaxwelHaii system is obtained bv a genera- 
lisation of the form given by Minkowski." 



We introduce the contra- variant six-vector F ^ bv 

a/5 



the equation 



(62) 



^.f^v ^ ^/xa ^^vp ^, 



tt^' 



and also a contra-variant four-vector J _, which is the 
electrical current-densitv in vacuum. Then rememberinir 
(40) we can establish the system of equations, which 
remains invariant for any substitution with determinant 1 
(according to our choice of co-ordinates). 



(63) 



6F 



fXl^ 



9''. 



^^' 



If we put 



(64) 



' ^2 3 _ JJ' 



F^^ = — E' 



■{ F^^ = H'„ F^^ = - E' 



' F12 - H', F^^ 



- E' 



which quantities become equal to H,. ..E, in the rase of 
the special relativity theory, and besides 

J^ = ^^, ... .7^ = p 

we get instead of (63) 



(68a) 



rot H'- 



a/ 



L div E' = p 



152 l>TlTXnVLE OF RELATIVTTY 

The equations (60), (62) and (63) give thus a i^enerali- 
sation of Maxwell's field- equations in Aaeuum, which 
remains true in our chosen system of* co-ordinates. 

TAe eHcv(ju-c(>))ipo}L6nl% of I he el ectro- id a (j netic jichL 

Let us form the inner-product 

(65) K = F .1^. 

According- to (61) its components can be written down 
in the three-dimensional notation. 

I K, ^ pE„-j-[/, H], 

\ 
(65a) I - - 

. [ K, = - (i, E). • 

K is a covariant four-vector whose components are eqnal 

to tlie nes^ative impulse and energy which are transferred 
to th<^ electro-magnetic Held per unit ol time, and per unit 
of volume, by the electrical masses. If the electrical 
masses be free, that is, under the influence of the eleetro- 
maofnetic field only, then the covariant four-vector 

K will vanish. 
(J 



In order to 2:et the energv components T of the elec- 
tro-magnetic field, we recpiire only to give to the equation 
K =0, the form of the equation (57). 

From (63) and (65) we get first, 



K = F 



/xv 



o- ^\^ ^x 



V V 



GENERALISED THEORY OF RELATIVITY 153 

On aeeoimt of (60) the second member on the risjht-hand 
side admits of the transformation — 



6F d^ 

V <T 



n 8F 

1 fxa v/j T-, tiv 

— ■> 9 9 ^ o ^^ ■ 



Owinof to symmotry, this expression can also be written in 
the form 

aF 



* L -^ ^ «^ aT~ 






a« 



or 



which can also be put in the form 






+ * *a;8 %. aT V " ^ )■ 



The first of these terms can be written shortly as 



X a 



- ( Ff'¥ \ 



9 . 



and the second after differentiation can be transformed in 
the form 

J . : ■ 

6.7 



- iF^^^F .. /^ 



(TT 



•20 



^54 PRINCIPLE OF RELATIVITY 

If we take all the three terras together, we t^et the 
relation 



ax'' dg 

V (J 



where 



(66a) ■ t'^-F F-'V \ f F « F''^. 



On aeeount of (30) the equation (66) becomes equivalent 



V 



to (57) and (57a) when K vanishes. Thus T 's are the 
energy-components of the electro-magnetic field. With 
the help of (61) and (64?) we can easily show that the 
energy -components of the electro-magnetic field, in the case 
of the special relativity theory, give rise to the well-known 
Maxwell-Poynting expressions. 

We have now deduced the most general laws which 
the o'ravitation-field and matter satisfv when we use a 
co-ordinate system for which \/ —g = 1. Thereby we 
achieve an important simplification in all our formulas and 
calculations, without renouncing the conditions of general 
covariance, as we have obtained the equations through a 
specialisation of the co-ordinate system from the general 
c'ovariant-equations. Still the question is not without formal 
interest, whether, when the energy-components of the 
gravitation -field and matter is defined in a generalised manner 
without any specialisation of co-ordinates, the laws of con- 
servation have the form of the equation (56), and the fiela- 
equations of gravitation hold in the form (52) or (52a) ; 
such that on the left-hand side, we have a divergence in the 
usual sense, and on the right-hand side, the sum of the 
energy-components of matter and gravitation. I have 



fetENERALlSED THEORY OF RELATIVITY loS 

found out that this is indeed the case. But I am of opinion 
that the communication of my rather comprehensive work 
on this subject will not pay, for nothing essentially new 
comes out of it. 



E. §21. Newton's theory as a first approximation. 

We have already mentioned several times that the 
special relativity theory is to be looked upon as a special 
case of the s^eneral, in which a ^s have constant values (4). 

This signifies, according to what has been said before, a 
total neglect of the influence of gravitation. We get 
one important approximation if we consider the case 

when (I 's differ from (4) onlv bv small masrnitudes (com- 

pared to 1) where we can neglect small quantities of the 
second and higher orders (first aspect of the approxima- 
tion.) 

Further it should be assumed that within the space- 
time reojion considered, a 's at infinite distances (using 

the word infinite in a spatial sense) can, by a suitable choice 
of co-ordinates, tend to the limiting values (4); i.e,, we con- 
sider only those gravitational fields which can be regarded 
as produced by masses distributed over finite regions. 

We can assume that this approximation should lead to 
Newton's theory. For it however, it is necessary to treat 
the fundamental equations from another point of view. 
Let us consider the motion of a ])article according to the 
equation (46). In the case of the special relativity theory, 
the components 

<^.<;^ dx^ dx^ 

ds ds ds 



156 PIUKCIPLE OF llELATlVitY 

can take any values ; This signifies that any velocity 

can appear which is less than the velocity of light in 
vacuum (i^ <1). If we finally limit ourselves to the 
consideration of the case when v is small compared to the 
velocity of liglit, it signifies that the components 



dx^ dx^ d,v. 



ds ds ' ds 



ti > 



can be treated as small (juantities, whereas ^- is equal to 

1, up to the second-order magnitudes (the second point of 
view for approximation). 

Now we see that, according to the first view of approxi- 

mation^ the magnitudes f 's are all small quantities of 

at least the first order. A glance at (46) will also show, 
that in this equation according to the second view of 
approximation, we are only to take into account those 
terms for which /x=v=4. 

By limiting ourselves only to terms of the lowest order 
we get instead of (46)^ first, the equations : — 

d^x 

= r . .. where ds=dx. =df. 

dt^ I 4.x. 

or by limiting ourselves only to those terms which according 
to the first stand-point are approximations of the first 
order, 

d*:( 

di" 



^ =[t'] -(^-1,2,:^) 



dt' 



= -[']■ 



GENERALiiJEb TltEOltY OE RELATIVITY 15t 

If we further assume that the gravitation-iield is 
quasi-static, i.e., it is limited only to the case when the 
matter producing the gravitation-field is moving slowly 
(relative to the velocity of light) we can neglect the 
differentiations of the positional co-ordinates on the right- 
hand side with respect to time, so that we get 

(67) -^ = - 1^ Oy- (r, = 1, 2, 3) 

This is the equation of motion of a material point 
according to Newton's theory, where ff^^/^ plays the part of 
gravitational potential. The remarkable thing in the 
result is that in the first-approximation of motion of the 
material pointy only the component ^^^ of the fundamental 
tensor appears. ; 

Let us now turn to the field-equation (5o). In this 

ease, we have to remember that the energy-tensor of 

matter is exclusively defined in a narrow sense by the 

density p of matter, i.e., by the second member on the 

right-hand side of 58 [(58a, or 5 Sb)]. If we make the 

necessary approximations, then all component vanish 

except 

' T^^ = p = T. 

On the left-hand side of (^o) the second term is an 
infinitesimal of the second order, so that the first leads to 
the following terms in the approximation, which are rather 
interesting for us ; 

^ f /^i^i , ^ r /xi^i , 6_ r /xvi _ 6_ r p^^'] 

^y neglecting all differentiations with regard to time, 
this leads, when /x==v=4, to the expression 






' 9'l4 



12 3 



158 PRINCIPLE OF RELATIVITY 

The last of the equations (53) thus leads to 

(68) V'cj,,^Kp. 

The equations (67) and (68) together, are equivalent to 
Newton's law of gravitation. 

For the gravitation-potential we get from (67) and (68) 
the exp. 



(68a.) 



K I pdr 



whereas the Newtonian theory for the chosen unit of time 



gives 



K 1 pdr 



where K denotes usually the 



gravitation-constant. 6 7 x 10 ^ ; equating them we get 
(69) K = ^^ =1-87 X 10-2 ^ 



§22. Behaviour of measuring rods and clocks in a 
statical gravitation-field. Curvature of light-rays. 
Perihelion-motion of the paths of the Planets. 

In order to obtain Newton's theory as a first approxi- 
mation we had to calculate only g^^^ out of the 10 compo- 
nents (J of the gravitation-potential, for that is the only 

component which conies in the first approximate equations 
of motion of a material point in a gravitational field. 

We see however, that the other components of g 

should also differ from the values given in (4) as required by 
the condition y/ = — 1 . 



GEXERAUSED THEORY OF RELATIVITY 



159 



For a heavy particle at the origin of co-ordinates and 
generating the gravitational field, we get as a first approxi- 
mation the symmetrical solution of the equation : — 



(70) 



.3 



q == — 8 — a '^ (p and <t 1, 2, 3) 

'^pcr pa * ^^ 5 : / 



- ^p^^-'Up 



V 



•^44 



= 1 



= 

a 

r 



(P 1, 2, 3) 



S is 1 or 0, according as p=cr or not and r is the quantity 
pa 



On account of (68a) we have 



(70a) 



a 



47r 



where M denotes the mass generating the field. It is easy 
to verify that this solution satisfies approximately the 
field-equation outside the mass M. 

Let us now investisrate the infiuences which the field 
of mass M will have upon the metrical properties of the 
field. Between the lena:ths and times measured locallv on 

the one hand, and the differences in co-ordinates dx on the 
other, we have the relation 

ds^ ■= a (I >' d,r . 

■' fXV fX V 

For a unit measuring rod, for example, placed parallel to 
the " axis, we have to put 

ds^ = — 1, d.r^=zdd'^=:d.i\=:o 



then 



-1=^11 ^ 



ir»0 PBJXCIPLE OF KEIATIVITV 

m; If the unit measurinio^ rod lies on the < axis, the first of* 
the equations (~0) gives 



1 1 



= -(•-.■)■ 



From both these relations it follows as a first approxi- 
mation that 

(71) ,l" = l- ^ . 

The unit measuring rod appears^ when referred to the 
eo-ordinate-system, shortened by the calculated magnitude 
through the presence of the gravitational field, when we 
place it radially in the field. 

Similarly we can get its co-ordinate-length in a 
tangential position, if we put for example 

we then get 

(71a) — l = f/22 '^K = —<^ 



2 T 2 

2 



The gravitational field has no influence upon the length 
of the rod, when we put it tangeatially in the field. 

Thus Euclidean geometry does not hold in the gravi- 
tational field even in the first approximation, if we conceive 
that one and the same rod independent of its position and 
its orientation can serve as the measure of the same 
extension. But a glance at (70a) and (69) shows that the 
expected difference is much too small to be noticeable 
in the measurement of earth's surface. 

We would further investigate the rate of going of a 
unit-clock Avhich is placed in a statical gravitational field. 
Here we have for a period of the clock 

^9 = 1, d.r^=^di\=^d."^=o -, 



GENERALISED THEORY OF REI.ATIVITY 161 

then we have 



d.,= ,2= = . L ^\-^±^--l 



or lU^ 



= 1+ i' ( P^ 

8. J r 



Therefore the eloek o^oes slowly what it is placed in 
the neighbourhood of ponderable masses. It follows from 
this that the spectral lines in the light coming to us from 
the surfaces of big stars should appear shifted towards the 
red end of the spectrum. 

Let us further investigate the path of light-rays in a 
statical gravitational field. According to the special relati- 
vity theory, the velocity of light is given by the equation 

— d^^^ — d,f — rfi; -f-rf.c =o ; 

1 2 3 4 

thus also according to the generalised relativity theory it 
is given by the equation 

(73) ih^^q d.c d.v =zo. 

^ ' jXV fji V 

■ t 

If the direction, i.e., the ratio d-'^ : d^'.^ • d.i'^ is given, 
the e(|uation (73) gives the magnitudes 

dj'y^ d,v^ dd-^ 

div^ ' dx^ d.i-^ 

i 

and with it the velocity, 



^^( fe h( k y+( ft H 



w> 



I 



lf;>i PRIXriPT.K OF RELATIVITY 

in the sense of the Enelidean Cjeometry. We can easily see 
that, with reference to the co-ordinate system, the rays of 

light must appear curved in ease y 's are not constants. 

If n be the direction perpendicular to the direction 
of propa<jjation, we have, from Huygen's principle, that 
light-rays (taken in the plane (y, >?)] must suffer a 

curvature -^ -I 
9^i ? 



X2 



A I^iglit-ray 



, ), A 

Let us find out the curvature which a light-rav suffers 
when it goes hy a mass M at a distance A from it. If we 
use the co-ordinate system according to the above scheme, 
then- the total bending R of light-rays (reckoned positive 
when it is concave to the origin) is given as a sufficient 
approximation by 



oo 



S 



1) >!* 

OC 



where (7'i) and (70) gives 

y = J -a^ = 1 - "L / 1 + l' ") 



:i - /( 



The oalf'ulntion srives 



■!-<' 



p_ 2a _ Ol 
'"" A ~ 2irA 

A ray of light just grazing the sun would suffer a bend- 
ing of J-7'^ whereas one coming by Jupiter would have 
fi deviation of about '02'^ 



GENERALISED THEOT?y OF RELATTVTTY 168 

If we calculate the oravitation -field to a sjreater order 
of approximation and with it the corresponding path 
of a material particle of a relatively small (infinitesimal) 
mass we set a deviation of the folio wins; kind from the 
Repler-Newtonian Laws of Planetary motion. The Ellipse 
of Planetary motion suffers a slow rotation in the direction 
of motion, of amount 

(75 ) .s'= — per revolution. 

In this Formula ' a ' signifies the semi-major axis, r, 
the velocity of light, measured in the usual way, e, the 
eccentricity, T, the time of revolution in seconds. 

The calculation gives for the planet Mercury, a rotation 
of path of amount 43" per century, corresponding suflii- 
ciently to what has been found by astronomers (Leverrier). 
They found a residual i)erihelion motion of this planet of 
the given magnitude which can not be explained by the 
perturbation of the other planets. 



n1*>(!»?f*! 



N0TE5 

Note 1. The fundamental oleetro-niaiL^'Udtic e<| nations 
of Maxwell for stationary media are : — 



curl 



«-Ua^^-) - ''^ 



eurl £=- 1 oL? ... (^j 

div B=p B=/iH 

div Brrro T)-J:F. 

AeeordioG;' to Hertz and TIeavIside, Ihese recjuire modi- 
fleation in the case of moving* bodies. 

Now it is known that due to motion alone there h a 
change in a vector It given by 

I — — ) ^^"^ ^^ motion — //, div H -(-eurl TT^w] 

where u is the vector velocity of the moving- body and 
[R?/] the vector product of II and //. 

Hence equations (1) and (2) become 
e curl H= ?i^ -I // div D + curl Veet. [D^*] -f pv (M) 

and 

-r- curl E= gy- -1-^^ div B + eurl Veot. [Bn] (-M) 
which gives finally, for p = o and div B = 0, 
^~ +u div D=:^- curl (H- 1 Veet. [D?^l ) (1-i) 

^1 ^ -.curlfE- ^ Veet. [uH] ) (f-o) 



166 PRINriPT.E OF EELATTVITY 

Let us consider a beam travellin«j^ along the .?'-axis, 
with apparent velocity r {i.e., velocity with respect to the 
fixed ether) in mpdi\im moving with velocity n, = i{ in the 
same direction. 

Then if the electric and magnetic vectors are 

i A {x — vt) 
proportional to e , we have 

^-=?A, ^- =— /A?', ^- = ^ —0,v^ — v,—0 
ox. ' dt ' dy dz ' " 

Then -^=-^'^-— ^^^ ••• (I'-l) 

Ot ox Oz ^ ^ 

and -^T = ~'^' -^~ ^'' ^~ ••• (^'Sl) 

Since D = K E and B = /^- H, we have 

i A.V {KEy)=-ci A (H,+^^KE,) ... (1-2.2) 

i Av (y-U. )=-ci ACE, +u/iB,) ... (2-22) 

or viK-7()E,=cli. ... (1-23) 

/x (t'-^O H,=:cE, ... (2-23) 

Multiplying (1*23), by (2-28) 

/x K (l-7^)2=C*^ 

Hence {v — /f)-=c-/fxk=Vn^ 

making Fresnelian convection co-efFicient simply unity. 

Equations (1*21), and (2"21) may be obtained more 
simply from j)hysical considerations. 

According to Heaviside and Hertz, the real seat of 
both electric and magnetic polarisation is the moving 
medium itself. Now at a point which is fixed with respect 
to the ether, the rate of change of electric polarisation is 
BD 



NOTES 167 

Consider a slab of matter moving with velocity n, 
along- the .r-axis, then even in a stationary field of 
electrostatic polarisation, that is, for a field in which 

-^ =0, there will be some change in the polarisation of 

ot 

the body due to its motion, given by u r ^- . Hence we 

o \; 

must add this term to a purely temporal rate of change 

-r^ . Doing this we immediately arrive at equations 

ct 

(1'21) and (2'21) for the special case considered there. 

Thus the Hertz- Heaviside form of field equations gives 
unity as the value for the Fresnelian convection co-efficient. 
It has been shown in the historical introduction how this 
is entirely at variance with the observed optical facts. As 
a matter of fact, liarmor lias shown (Aether and Alatter) 
that I — 1/V^ is not only sufficient but is also necessary, in 
order to explain experiments of the Arago prism type. 

A short summary of the electromagnetic experiments 
bearing on this question, has already been given in the 
introduction. 

According to Hertz and Heaviside the total polarisa- 
tion is situated in the medium itself and is completely 
carried awav bv it. Thus tlie electromagnetic efPect 
outside a moving medium should be proportional to K, the 
specific inductive capacity. 

Rowland showed in 18/ G that when a ciiarged condenser 
is rapidly rotated (the dielectric remaining stationary), 
the magnetic effect outside is proportional to K, the Sp. 
Ind. Cap. 

^^'^////d^/i (Annalen der Physik 1888, 1890) found that 
if the dielectric is rotated while the condenser remains 
stationarv, the effect is proportional to K — 1 . 



168 PRINCIPLK OF RELATIVITY 

Eichemcakl (Aunaleu der Physik 1905, IVtQi) rotated 
together botU condenser and dielectric and found that the 
magnetic effect was proportional to the potential difference 
and to the aniLi^ular velocity, but was completely independent 
of K. This i^ of course quite consistent with Rowland 
and Rontgcu. 

Bloiidlot (Comptes llcndus, lUU]) passed a current 
of air in a steady magnetic field PI ,,, (H =H.. =0). If 
this current of air moves with velocity //, along the 
■r-axis, an electromotive force would be set up along the 
c;-axib, due to the relative mutioji of matter and magnetic 
tubes of induction. A pair of plates at .:=+»'/, will be 
charged up with density p=D,=KE =K. n, Hy/c. 
BuL Blondlot failed to detect any such eft'ect. 

//. ./. )Vihoii (Phil. Trans, lloyal Soc. 1901-) repeated 
the experiment with a cylindrical condenser made of 
ebony, rotating in :«, magnetic held parallel to its own 
axi-^'. Ho observed a change proportional toK— 1 and 
not to K, 

Thus the above set of electro-n)agnetic experiments 
contradict the Mertz-Hcaviside equations, and these must 
be abandoned. 

I P. (;. M.] 

Note 2. Lornniz Tra)i>ifoYiii(i.lU)u, 

Lorentz. Versueh einer theorie der elektrisehen uud 
optitehen Erseheinungon im bewegten Korpern. 

(Leiden— 1895). 

Lorentz. Theory of Electrons (English edition), 
])ages iy7-:iOO, :ioO, also notes 7:j, 86, pages 318, 328. 

Lorentz wanted to explain the Michelson-^NIorley 
null-effect. \\\ order to do so, it was obviously necessary 
to explain the Eitzgerald contraction. Lorentz worked 
on the hypothesis that an electron itself undergoes 



NOTES 169 

contraction when moving. He introduced new variables 
for the raoving system defined by the following set of 
equations. 



x-=.j^{.v--iit),t/^ =y, z^=z, l'=l3{f-y,^^) 



and for velocities, used 

v,''=P''u, + i(, Vy'' =/3v,, v..^=Pt\ andpi=p//5. 

With the help of the above set of equations, which is 
known as the Lorentz transformation, he succeeded in 
showinsc how the P'itzc^erald contraction results as a 
consequence of " fortuitous compensation of opposing 
effects." 

It should be observed that the Lorentz transformation 
is not identical with the Einstein transformation. The 
Einsteinian addition of velocities is quite different as 
also the expression for the ''relative^' density of electricity. 

It is true that the Maxwell-Lorentz field equations 
remain practically/ uncliauged by the Lorentz transforma- 
tion, but they arc changed to some sliglit extent. One 
marked advantage of the Einstein transformation consists 
in the fact that the field equations of a moving system 
preserve exactly the same form as those of a stationary 
system. 

It should also be noted that the Fresneliau convection 
coefficient comes out in the theory of relativity as a direct 
consequence of Einstein's addition of velocities and is 
quite independent of any electrical theory of matter. 

[P. C. M.] 

Note 3. 

See Lorentz, Theory of Electrons (English edition), 
§ 181, page tllS. 



170 I'JlTXCirLE 01' tlELATIYtTY 

H. Poincare, Sur la dynamique 'electron, Rendiconti 
del circolo matematico di Palermo 21 (1906). 

[P. C. M] 

Note 4. Iielativitf/ Theorem and 'Relativity 'Principle. 

Lorentz showed that the Maxwell-Lorentz system 
of electromagnetic tield-equations remained practically 
unchanged by the Lorentz transformation. Thus the 
electromafrneric laws of Maxwell and Lorentz can he 
(lefinitehj jiroved " to be independent of the manner in 
which they are referred to two coordinate systems whicb 
have a uniform translatory motion relative to each other." 
(See '' Electrodynamics of Gloving Bodies/^ P^-ge 5.) Thus 
so far as the electromagnetic laws are concerned, the 
princi])le of relativity cau he proveiJ to he irue. 

But it is not known whether this principle will remain 
true in the case of other ])hysical laws. We can always 
proceed on the assumption that it does remain true. Thus 
it is always possible to construct physical laws in such a 
way that Ihey retain their f(»rm when referred to moving 
coordinates, "^riie ultimate ground for formulating physi- 
cal laws in this way is merely a subjective conviction that 
the principle of relativity is uuiversally true. There is 
no rt7;;wy logical necessity that it should be >^o. Hence 
the Principle of Relativity (so far as it is applied to 
ohenomena other than electromagnetic) must be resrarded 
as ^ pn.^tHlafe, which we have assumed to be true, but for 
which we cannot adduce any definite proof, until after 
the generalisation is made and its consequences tested in 
the light of actual experience. 

[P. C. M.] 

Note 5. 

See '' Electrodynamics of Afoving Bodies," p. 5-S. 



NOTES 



17 



Not© 6. Field EiiuatiouK in Miukon-^l-ih Form. 

Equations (/) and (//) ])eeoine when oxjiandor] into 
Cartesians : — -,_ 



and 



6 '»' r 

9?/ 


9^>ii, 
" 9^' 


9-. 

9t" 


] 


9 rn , 

9^' 


9 m , 


_9^ 
9t = 


=pn, 


9.e 


9 m , 

dy 


9.^. 
9t - 


-pu, j 


9e, 
~9.r + 


dy'^ 


9. P 



(1-1) 



... (2'1) 



Substituting x^^ x^^ •'■'^i ^^ for .c, y, :, and /r; and pj, 
^2* Pai Pa ^0^ P^^j P^^'/5 P^^'j W, where /= y/— 1, 



We get, 



9w» 9-^^^/ -9'^ '^ 

4 



9 



.T, 



9r<'.H 9.1' 



9w^, 9^ 
9''t^i 9.'* 



J v9^v _ 1 



9 my 9i>i., . 9'\ 



9.'?*i 9'T 



*97:='"'-=p= J 



and multiplying (2*1) by / we get 



... (1-2) 



bie^ 9?>y ^ie, 
9a^+ 9.<'«+ ^:e,'=''P='P^ •• 



... (2'2) 



Now substitute 



w.=/2 3=-/s8 and /V,=/,, = -/^^ 












17:Z 



PIJIXCTPLE OP EELATIVITY 



and we o-et finall 



V :■ 






~1 



9.V2 9.> 



9a'^ 



= Pi 



9/ai , 9/33 , 9/24 



a.-. 



6.T. 



dx^ 



y ... (3) 



9 /sj 9/3 8 , 9/34 

9.''i 9. ^'2 dd'i 



= Pi 



9/41 ,9/42 9/. 



* s 



8»'. 



6.T 



= P. 



J 



[P. O. M.] 



Note 9. Oh the Condancy of the Velocity of Light. 

Pao^e vl — refer also to page C, of Einstein's paper. 

One of tlie two fundamental Postulates of the Principle 
of Relatlvitv is that the velocity of lisfht should remain 
oonstant whether the source is moving or stationary. It 
follows that even if a radiant source S move with a velocity 
?/, it should always remain the centre of spherical waves 
expanding outwards with velocity c. 

At first sight, it may not appear clear why the 
velocity should remain constant. Indeed according to the 
theory of Ritz, the velocity should become c + n, when the 
source or light moves towards the observer with the 
velocity n. 

Prof, de Sitter has ojiven an astronomical arsjument for 
decidinoj between these two diverejent views. Let us 
suppose there is a double star of which one is revolving 
about the common centre of gravity in a circular orbit. 



NOTES 173 

Let the observer be in the plane of the orbit^ at a great 
distance A. 




The light emitted by the star when at the position A 

will be received by the observer after a time , while 

c + u ., 

the light emitted by the star when at the position B will 

be received after a time — . Let T be the real half- 

c — u 

period of the star. Then the observed half-2:>eriod from 
B to A is approximately T — '^-— - and from A to B is 

T + — — . Now if ~— — be comparable to T, then it 

is impossible that the observations should satisfy 
Kepler^s Law. In most of the spectroscopic binary stars, 

— ^^— are not only of the same order as T, but are mostly 

much larger. For example, if /i = 100 km /sec, T = 8 days, 
^|6' = 33 years (corresponding to an annual parallax of 'l'^)^ 
then T — '2nAjc^=o. The existence of the Spectroscopic 
binaries, and the fact that they follow Kepler's Law is 
therefore a proof that c is not affected by the motion of 
the source. 

In a later memoir, replying to the criticisms of 
Freundlich and Giinthick that an apparent eccentricity 
occurs in the motion proportional to ^v.Aq, u^-^ being the 



l?t PUINCirLE Of llBLxVnVlrY 

maximum value of /', (lie velocity oL' li'^hl emitted bein^ 

u^ =6' + kiij /(' = Lorentz-Einstein 

/•=! Ritz. 

. / . 
Prof, de Sitteradunts the validity of the eritieisms. But 

he remarks that aii upper value of k may be calculated from 

the observations of the double sar ^-Aurigae. For this star. 

The parallax 7r = '011", 6 = -00o, /^,=:110 kwj&eG T = 3-96, 

A > 65 light-years, 

k is < -OO-^. 

Fur an experimental proof, see a paper by C Majorana. 

Phil. Mag., Vol. 35, p. 163. 

[M. N. S.] 

Note 10. Rest-density of Electricity. 

\i p is the volume density in a moving system then 
p\'^{l — u'-) is the corresj)onding cpiantity in the correspond- 
ing volume in the fixed system, that is, in the system at 
rest, and hence it is termed the rest-density of electricity. 

I'P. C. M.] 
' Note 11 (page 17). 

Space-time vectors of the fir ■'<f and the second kind. 

As we had alreadv occasion to mention, Sommerfeld 
has, in two papers on four dimensional geometry {vide, 
Annalen der Physik, Ed. 32, p. 74-9 ; and Bd. 33, p. 649), 
tj'anslated the ideas of Minkowski into the lanaruaofe of four 
dimensional geometry. Instead of Minkowski's space-time 
vector of the first kind, he uses the more expressive term 
' four-vector,' thereby making ifc quite clear that it 
represents a directed quantity like a straight line, a force 
or a momentum, and has got 4 components, three in the 
direction of space-axes, and one in the direction of the 
time-axis. 



NOTES J 75 

The representation of the plane (defined by two strai^'ht 
lines) is much more difficult. In three dimensions, the 
plane can be represented by the vector ])er})endicu1ar to 
itself. But that artifice is not available in four dimensions. 
For the perpendicular to a plane, we now have not a sini^le 
line, but an infinite number of lines constitutimG^; a plane. 
This diffieultv has been overcome bv Minkowski in a verv 
elegant manner which w^ill become clear later on. 
Meanwhile we oifer the followino^ extract from the 
above mentioned work of Sommerfdd. 

(Pp. 755, Bd. :i:2, Ann. d. Physik.) 

" In order to have a better knowledge about the nature 
of the six- vector (which is the same thing as Minkowski's 
space-time vector of the 2n(l kind) let us take the special 
ease oP a piece of piano, having unit area (contents), and 
the form of a parallelogram, bounded by the four-veetors 
21, V, passing through the origin. Then the projection of 
this piece of plane on the :)'// plane is given hy the 
projections ?/,, ?/,^, r^, r,, of the four veetoi:" in the 
combination 

Let us form in a similar manner all the six components of 
this plane <A. Then six components are not all indepejident 
but are connected bv the folio wins' relation 

Further the contents | <^ | of the piece of a plane is to 
be defined as the square root of the sum of the squares of 
these six cpiantities. In fact, 

Let us now on the olhei iiand take the ease of the tinit 
plane fj>^ normal to </> ; we can call this plane the 



176 PlUxXCIPLE OF RELATIVITY 

Complement of </>. Then we have the followinoj relations 
between the components of the two plane : — 

The proof of these assertions is as follows. Let ?f^^, ?'"^ 
be the four vectors defining^ (f>^. Then we have the 
following relations : — 

2i^ n, + n^; Uy + n*; n , -f ?^t ?0 =0 

?^* v,-\-u'fj ?',+< Vr.+n'^i vi=0 

v't v.+v""^ Vy+v't v^-\-ifl t',=0 

I£ we multiply these equations by Vi, Ui, i\, and 
subtract the second from the first, the fourth from the 
third we obtain 

< ^.i + < <f>yr + n't c^,,=0 

multiplying: these equations by rf . ?^*- , or by v* . ?^* . 
we obtain 

from which we have 

In a eorrespondinoj w^ay w^e have 

when the subscript {il') denotes the component of <^ in 
the plane contained by the lines other than {ik). Therefore 
the theorem is proved. 

We have (<^ <^^)=<^y, c^*, + ... 

= 



NOTES 177 

The general six-veetor / is composed fmm the A^ectors 
ff>f(f)^ in the followinoj wa>y : — 

p and p'^ denotins^ the contents of the pieoos oP mutually 
perpendicular planes composing f. The ^' conjugate 
Vector" _/^ (or it may be called the complement of /) is 
obtained by interchanging p and p^ 

We have, 

/* =/- 4, + ,, 4'* 

We can verify that 

/!.- =,/:,, etc. 
and/2 =p2+p*^(/p) = 2pp* 

I /' I - and (JX^) may be said to be invariants of the six 
vectors, for their values are independent of the choice of 

the svstem of co-ordinates. 

[M. N. S.] 

Note 12. Light -V el ocitij a^ a vinxiwuni. 

Pasre -23, and Electro-dvnamies of Moving Bodies, 
p. 17. 

Putting v — c — .Vy and iv = c — \, we get 

_ 2c — C^' + A) 



2c — (.^' + A) + .rA/c 

Thus 2^<c, so long as | xX | >0. 

Thus the velocity of light is the absolute maximum 
velocity. We sh^ll now see the consequences of admitting 
a velocity W > c. 

Let A and B be separated by distance /, and let 
velocity of a ^^sij^nal " in the system S be W>(". Let the 



178 PRINCIPLE OF RELATIVITY 

(observing) system S' have velocity +?? with respect to 
the system S. 

Then velocity of signal with respect to system S' is 

given bv VV = „, , ^ 

1 - Wv/c^ 

Thus "time " from A to B as measured in S', is given 



Now if v is less thau c, then W being^ Q^reater than c 
(by hypothesis) W is greater than v, i.e., W>v. 

Let W = (? + /x and ?? = <?— X. 

Then Wv = (c-\-fjL)(c-\)=c'~+{fj, + \)c-,jcX. 

Now we can always choose v in such a way that Wv is 
greater than c-, since Wv is >c'- if {ix ■\- X)c — jjlX is >0. 

that is^ if /x + /\> —. which can always be satisfied by 
a suitable choice of \. 

Thus for W>c we can alwaj-s choose X in such a 
way as to make Wv>c^f i.e., l—Wr/c- negative. But 
W— r is always positive. Hence with W>c, we can 
always make t' , the time from A to B in equation (1) 
" negative." That is, the signal starting from A Avill reach 
15 (as observed in system S') in less than no time. Thus the 
effect will be perceived before the cause commences to act, 
i.e., the future will precede the past. Which is absurd. 
Hence we conclude that W>c is an impossibility, there 
can be no velocity greater than that of light. 

It is conceptually possible to imagine velocities greater 

than that of light, but such velocities cannot occur in 

reality. Velocities greater than c, will not produce 

any effect. Causal effect of any physical type can never 

travel with a velocity greater than that of light. 

[P. C. M.] 



NOTE S 



179 



Notes 13 and 14. 

We have denoted the four-vector w by the matrix 

I (o^ oi,2 <^i w^ [ . It is then at onee seen that oo denotes 
the reciprocal matrix 



(Ol 


0)2 


0), 


^i 



It is now evident that while co^ =wA, w^=A ^ 



w 



\jo,s] The vector-product of the four-vector w and -*? 
may be represented by the combination 



[cos] 



OJ'J 6'OJ 



It is now easy to verify the formula ./^=A'^/A. 
Supposing for the sake of simplicity that /' represents the 
vector-product of two four- vectors oi, s, we have 

= [A~' w^A— A~'6'wA] 

= A-^[«5-5w]A = A-yA. 
Now remembering that generally 

Where o, p^ are scalar quantities, (f>, (^"^ are two 
mutually perpendicular unit planes, there is no difficulty 

/«=A-yA. 

Note 15. The vector product (in/). (P. 36). 
This represents the vector product of a four-vector and 
a six-vector. Now as combinations of this type are of 



in seeming that 



180 PIUNCU'LE Oh' RELATIVITY 

frequent oeeurreuce in this paper, it will be better to form 
an idea of their geometrical meaning. The following 
is taken from the above mentioned paper of Sommerfeld. 

^' We can also form a vectorial combination of a four- 
vector and a six-vector, giving us a vector of the third 
type. If the six-vector be of a special type, i.e., a piece 
of plane, then this vector of the third type denotes the 
parallelopiped formed of this four-vector and the comple- 
ment of this piece of plane. In the general case, the 
product will be the geometric sum of two parallelopipeds, 
but it can always be represented by a four-vector of the 
1st type. For two pieces of 3 -space volumes can always 
be added together by the vectorial addition of their com- 
ponents. So by the addition of two 3-space volumes, 
w^e do not obtain a vector of a more general type, but 
one which can alwavs be represented bv a four- vector 
(h)C, cit. p. 759). The state of affairs here is the same as 
in the ordinary vector calculus, where by the vector- 
multiplication of a vector of the first, and a vector of the 
second type (t.e., a polar vectoi), we obtain a vector of the 
first type (axial vector). The formal scheme of this 
multiplication is taken from the three-dimensional case. 

Let A = (A,,, A.,,, A.) denote a vector of the first 
type, B = (B„,, B,^, B^y) denote a vector of the second 
type. From this last, let us form three special vectors of 
the lirst kind^ namely-^- 

B.=(B.,, B,,, B,;n 

B, = (B,.., B,,, B,,)KB,,--B,., B,,=0). 
B.:=(B..., B.,,B....)J 

Since B,, is zero, B, is perpendicular to the ^-axis. 
The /-component of the vector-product of A and B is 
equivalent to the scalar product of A and B,, i.e., 

(ABj^A. B,, + A^B,,+A.- B,,. 



NOTES 



181 



We see easily that this coincides with the usual rule 
for the vector-product; c. g., iov j — ^c. 

Correspondingly let us deline in the four-dimensional 
ease the product (P/) of any four- vector P and the six- 
vector./*. The /-component (/ = ,r, j/, .v\ or /) is j^iven by 

(ly, ) = p,/;, ., + p,,/, , + p,/, , + p./- , 

Each one of these components is obtained as the scalar 
product of P, and the vector /', which is perpendicular to 
j-axis, and is obtained from ,/' by the rule/'.; = [_{f,y, fjyi 

fj^.i f } ././7 =0.] 



■5f 



^ 



a. 



•K- 



^ 



We can also find out here the geometrical significance 
of vectors of the third type, when f=^, i.<?.,y' represents 
only one plane. 

We replace (/> by the parallelogram defined by the two 
four-vectors U, V, and let us pass over to the conjugate 
plane </>'", which is formed by the perpendicular four-vectors 
U"^, V."^^ Ttie components of (P<A) are then equal to the 
4 three-rowed under-determiuants D., D,, D^ Di of the 
matrix 



P. 



P, 



U.^ U/>^ U/^ u,^ 



V/^ 



-x- 



V.,-x- Y.-x- Y,7f 



Leaving aside the first column we obtain 

D,=p,(u,^v,^-u,^v.^)+p.(u,n^-^--u,-v,-^) 

-hP/(U/^-V,*-U.^\%^) 

= P,c^,,^ + P.-c/>^.+P/c^^.. 

=p,>.,+p.-<?!>.. + p,<a;;, 

which coincides with (P«/>.) according to our definition. 



I8:i 



PEINCIPLE OF RELATIVITY 



Examples of this type of vectors will be found on 
page '5i5, ^ = i/;Yy the electrical-rest-foree_, and i/' = 2to/'," 
the magnetic-rest-foree. The rest-ray 12 = t'w [$iJ^] * also 
belong to the same type (page 39). It is easy to show 
that 



n=z -{ 



(0. 



^. 



OJ, 



^, 



w. 



$, 



to. 



* 



y 4 



I »Al ^2 ^3 

When (ojj, ro^, (05)=o, 0^^=:/, 12 reduces to the three- 
dimensional vector 



12,, 12^^ 12, 



^1 



«>, 



$, 



«Al ^2 ^, 

Since in this case, 4>i=oj^ ^14 =<•« (the electric force) 

i/^i ="^'^^4/2 s =''''^r (the magnetic force) 



we have (12J = 



<?,r 



1/« 



m. 



w. 



e, I , /.«., analogous to the 
Poynting-vector. 

[M."K S.] 

Note 16. T/ic eUdric-red force. (Page 37.) 

The four- vector ^ = 0'F which is called by 3Iinko\vski 
the cleetric-rest-force (elehtrische Paih-Kraft) is very 
closelv eonuccicd to Lorentz's Ponderomotive force, or 
the force acting on a moving charge. If p is the density 
of charge, we have, when € = 1, /;. = ], i.e., for free space 



_ Po 



^/i-vvc^ L 



^?.r-f--(^'2^'3— ^'S^^s) 



Now since p^ =p \^l—Y^/c^ 

We have p^(f)^=p\ d,+ - (>\Jh — ^\^^2) 

JS^.B. — We have put the components of e equivalent 
lo ('f.r, (ly, d ,), and the components of vi equivalent to 



X0TE8 18 '3 

^^. -^y ^'-•)) iii accordance with the uofaliou used in 
Lorentz's Theory of Electrons. 

We have therefore 

2.6'. , po (<Ai5 ^2> ^Aa) represents the force acting on tlie 
electron. Compare Lorentz, Theory- of Electrons, V^r^^ l"^- 

The fourth component <^4, when multiplied by p^ 
represents /-times the rate at which w^ork is done bv 
the moving electron, for Po <?^4, =/p ['\,f^.. +'t'^(/y +r-f?,] = 
^'x po</>i -t-?",,/ po^'z + i\ Po9.v — ^^-■^ times the power pos- 
sessed b\' the electron therefore represents the fourth 
component, or the time component of the force-four- 
vector. This component was iirst introduced bv Poincare 
in 1906. 

The four-vector i//=:/ojF* has a similar relation to 

the force acting on a moving magnetic pole. 

[M. X. S.] 

Note 17. Opera/or ''- Lor '' (§ 1:>, p. 11). 

The operation ^ -g^, ^ ;. ^^ | which plays in 

four-dimensional mechanics a rule similar to that of 

the operator ( / 7^,+ / t:--,+ h -—-— v ) in three-dim en- 

sional geometry has been called by Minkow^sld ^ Lorentz- 
Operation ' or shortly Mor ' in honour of H. A. Lorentz, 
the discoverer of the theorem of relativity. Later writers 
have sometimes used the sj^mbol n to denote this 
operation. In the above-mentioned paper (Annalen der 
I'hysik, p. 649, Bd. 38) Sommerfeld has introduced the 
terms, Div (divergence), Rot (Rotation), Grad (gradient) 
as four-dimensional extensions of the corresponding three- 
dimensional operations in place of the general symbol 
lor. The physical significance of these ojierations will 



184 



PKINCIPLK OF RELATIVITY 



beeomo elfar when alono; witli ]\[inko\vh^ki's mfithod of 
treatment we also study the geometrical method of 
Sommerfeld. Minkowski begins here with tlie case of 
lor S, where S is a six- vector (space-time vector of the 
2nd kind). 

This being a complicated case, we take the simpler 
ease of lor .s^, 

w^here s- is a fonr-veetor= | .9^, s^. .9^ s^ | 

and s = 



^ 


'^2 


.«?„ 

.» 


'^'4 



The following geometrical method is taken from Som- 
m erf eld. 

Scalar Divergence — Let A^ denote a small four-dimen- 
sional volume of anj- shape in the neighbourhood of the 
space-time point Q^ ,'/S denote the three-dimensional 
bounding surface of A^> " ^^ ^^^^' outer normal to dS. 
Let S be any four-vector, P„ its norm.al component. 
Then 



D 



ivS = Lim r?ii^ 
AS=0 J AS 



Now if for AS ^ve choose the four-dimensional paral- 
lelepiped wdth sides {df\, <lr,^, div^^ dx^), we have then 



Div S = -^-i^-+^'-^-^'-^-^'-- 



9 '1 9 



+ 



a.r^ 



+ 



a^u 



lor S. 



If /'denotes a space-time vector of the second kind, lor 
/'is equivalent to a space-time vector of the first kind. The 
o'eometrical si<2:nificanee can be thus brouq-htout. We have 
seen t lat the operator ' lor' behaves in every respect like 
a four- vector. The vector-product of a four-vector and a 
six-vector is again a four-vector. Therefore it is easy 



NOTES 185 

to see that. Jor S will be a four-vector. Let. ns tind 
the component of this four-vector in any direction -s-. 
Let S denote the three-space which passes through the 
point Q {a\, .Vo, .^o, x^) and is perpendicular to .^^ AS a 
very small part of it in the region of Q, da- is an element 
of its two-dimensional surface. Let the perpendicular 
to tl:-is surface lying in the space be denoted by j/, and 
let /,.„ denote the component of /in the plane of (<?//) 
which is evidently conjugate to the plane dcr. Then the 
5-eomponent of the vector divergence of /' because the 
operator lor multiplies /' veetorially) 

= Div/^,=:Lim ili^. 
As=0 AS 

AY here the integration in //o- is to be extended over 
the whole surface. 

If now s is selected as the .r-direetion, /\,s' is then 
a three-dimensional parallelepiped with the sides (I//j dz, 
(IJ, then we have 

DiY /,= — i— \dz. dJ. %^ dy + dl dy ^' d: 
ay dz at L oy Os 

+ dy d, ?A_' dl I = ^/- + ^^- H- ?Ai , 

'^ a/ ) dy dz ^ 6/ ' 

and generall}' 

o •■ oy o- oi 

Hence the four-components of the four-vector lor S 
or Div. / is a four-vector with the components given on 
page 42. 

According to the formulae of space geometry, D^ 
denotes a parallelepiped laid in the (;/-^'-0 space, formed 

out of the vectors (P, P, PJ, (u* U* 11^) (v, V^ V* ). 



186 PRINCIPLE OF IlELATIVITY 

D, is therefore the projection on the y-z-l space of 
the perallelopiped formed out of these three four-vectors 
(P, U"^, V"^), and could as well be denoted by Dyzl. 
We see directly that the four-vector of the kind represent- 
ed by (D,, Dy, D., D,) is perpendicular to the parallele- 
piped formed by (P U^ V^")- 

Generally we have 

(P/) = PD + P^D^. 

.-. The vector of the third type represented by (P/*) 
is o-iven bv the ijeometrical sum of the tw^o four-vectors of 
the fir^t type PD and P^D^. 

[M. N. S.] 



VP^ 



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