;. i».. •' •Mii'-marvF^i'rv ■; u ■' ' t.^».-ot Mnafco
LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
y
^0' <
THE
SEP k
PRINCIPLE OF RELATJ^^{jK[Ji
ORIGINAL PAPERS
BY
A. EINSTEIN ANT) H. MINKOWSKI
)'
TRANSLATED INTO ENGLISH
BY
M. N. SAHA AND S. N. BOSE
lecturers on physics and applied jiathematics
Univebsity College of Science, Calcutta Univeksity
WITH A HISTORICAL INTRODUCTION
BY
P. C. MAHALANOBIS
professor of physics, presidency college, CALCU-
PUBLISHED BY THE
UNIVERSITY OF CALCUTTA
1920
Sole Agents
R. CAMBRAY &: CO.
PRINTED BY ATULCHANDRA BHATTACHAKYYA,
AT THE CALCUTTA UNIVERSITY PRE3B. SENATE HOUSE. CALCUTTA
TABLE OF CONTENTS
1. Historical Introduction
[By Mr. P. C. Mahalanobis.]
2. On the Electrodynamics of Moving Bodies...
[Einstein's first paper on the restricted
Theory of Relativity, originally pub-
lished in the Annalen der Physik in
1905, Translated from the original
German by Dr. Meghnad Saha.]
3. Albreeht Einstein
[A short biographical note by Dr.
Meghnad Saha.]
4. Principle of Relativity
[H. Minkowski's original paper on the
restricted Principle of Relativity first
published in 1909. Translated from
the original German by Dr. Meghnad
Saha.]
5. Appendix to the above by H. Minkowski ...
[Translated by Dr. Meghnad Saha.]
6. The Generalised Principle of Relativity
[A. Einstein's second paper on the Genera-
lised Principle first published in 1916.
Translated from the orijjina] German
by Mr. Satyendranath Bose.]
/, iNotes ,,, ,,, ...
PAGE
i-xxiii
1-34
35-39
1-52
53-88
89-l()3
165-^185
124281
-N \^\
y
J
HISTORICAL INTRODUCTION
ooj:^0<>-
Lord Kelvin writing- in 1893, in his preface to the
English edition of Hertz's Researches on Electric Waves,
says " many workers and many thinkers have helped to
bnild up the nineteenth century school of plenuDij one
etiier for light, heat, electricity, magnetism ; and the
German and English volumes containing Hertz's electrical
papers, given to the world in the last decade of the
century, will be a permanent monument of the splendid
cons ^mmation now realised."
Ten years later, in 1905, we find Einstein declarinsj
that " the ether will be proved to be superflous." At
first sight the revolution in scientific thought brought
about in the course of a single decade appears to be almost
too violent. A more careful even though a rapid review
of the subject will, however, show how the Theory of
Relativity gradually became a historical necessity.
Towards the beginning of the nineteenth century,
the luminiferous ether came into prominence as a result of
the brilliant successes of the wave theory in the hands
of Young and Fresnel. In its stationary aspect the
elastic solid ether was the outcome of the search for a
medium in which the light waves may "undulate." This
stationary ether, as shown by Young, also afforded a
satisfactory explanation of astronomical aberration. But
its very success gave rise to a host of new questions all
bearing on the central problem of relative motion of ether
and matter.
11 PRINCIPLE OF RELATIVITY
Arago^s prison experiment. — The refractive index of a
glass prism depends on the incident velocity of light
outside the prism and its velocity inside the prism after
refraction. On Fresnel's fixed ether hypothesis, the
incident light waves are situated in the stationary ethei
outside the prism and move with veloeit)' c with respeci
to the ether. If the prism moves with a velocity n
with respect to this fixed ether, then the incident velocity
of light with respect to the prism should be c + n. ThuE
the refractive index of the glass prism should depend on m
the absolute velocity of the prism, i.e., its velocity witl
respect to the fixed ether. Arago performed the experimeni
in 1819, but failed to detect the expected change.
Airy- Boscovitch ivaler-telescoije experimeni. — Boscovitcl
had still earlier in 1766, raised the very importan
question of the dependence of aberration on the refractive
index of the medium filling the telescope. Aberratior
depends on the difference in the velocity of light outsid»
the telescope and its velocity inside the telescope. If thi
latter velocity changes owing to a change in the medium
filling the telescope, aberration itself should change, thai
is, aberration should depend on the nature of the medium.
Airy, in 1871 filled up a telescope with water — but
failed to detect any chansje in the aberration. Thus w<
get both in the case of Arago prism experiment an(
Airy -Boscovitch water-telescope experiment, the ver
startling result that optical effects in a moving mediun
seem to be quite independent of the volocit}^ of th
medium with respect to Fresnel's stationary ether.
FresneVs convection coefficient /(:=1 — ^/^^. — Possibb
some form of compensation is taking place. Working oi
this hypothesis, Fresnel effered his famous ether convee
tion theory. According to Fresnel, the presence of matte:
implies a definite condensation of ether within th(
t « •
HISTORICAL INTRODUCTION 111
region occupied by matter. This " condensed " or
excess portion of ether is supposed to be carried away
with its own piece of movino" matter. It should be
observed that only the " excess " portion is carried away,
while the rest remains as stagnant as ever. A complete
convection of the ''excess " ether p with the full velocity
u is optically equivalent to a partial convection of the
total ether p, with only a fraction of the velocity k. u.
Fresnel showed that if this convection coefficient k is
1 — *//x'-^ (/x being the refractive index of the prism), then
the velocitv of lio^ht after retraction within the movin";
prism would be altered to just such extent as would make
the refractive index of the moving prism quite indepen-
dent of its "absolute" velocity u. The non-depeudence
of aberration on the '" absolute " velocity it, is also very
easily explained with the help of this Fi-esnelian convection-
coefficient k.
Stokes^ viseous ether. — It should be remembered, however,
that Fresnel 's stationary ether is absolutelv fixed and is not
at all disturbed bv the motion of matter throusfh it. In this
respect Fresnelian ether cannot be said to behave in any
respectable physical fashion, and this led Stokes, in
1845-46, to construct a more material type of medium.
Stokes assumed that viscous motion ensues near the surface
of separation of ether and moving matter, w^hile at
sufficiently distant regions the ether remains wholly
undisturbed. He showed how such a viscous ether would
explain aberration if all motion in it were differentially
irrotational. But in order to explain the null Arago
effect, Stokes was compelled to assume the convection
hypothesis of Fresnel with an identical numerical value
for kj namely 1 — V/^'- '^hus the prestige of the Fresnelian
convection-coefficient was enhanced, if anything, by the
theoretical investigations of Stokes.
IV PRINCIPLE OF RELATIVITY
Fizeaic^s experin/cnl, — Soon aftur, in 1851, it received
direct experimental eonHrmation in a brilliant piece of
work by Fizeau.
If a divided beam of light is re-nnited after passin<)j
through two adjacent cylinders filled with water, ordinary
interference fringes will be produced. If the water in one
of the cylinders is now nriade to fiow^, the " condensed"
ether within the flowing water wonld be conveeted and
would produce a shift in the interference fringes. The
shift actuallv observed agreed verv well with a value of
k=l— V/Jt^. The Fresnelian eonveetion-eoeffieient now
became firmly established as a consequence of a direct
positive effect. On the other hand, the negative evidences
in favour of the convection-coefficient had also multiplied.
Mascart, Hoek, Maxwell and others sought for definite
changes in different optical effects induced by the motion
of the earth relative to the stationary ether. But all such
attempts failed to reveal the slightest trace of any optical
disturbance due to the "absolute" velocity of the earthy,
thus proving conclusively that all tne different optical
effects shared in the general compensation arising out of
the Fresnelian convection of the excess ether. It must be
carefully noted that the Fresnelian convection -coefficient
implicitly assumes the existence of a fixed ether (Fresnel) or
at least a wholly stagnant medium at sufficiently distant
regions (Stokes), with reference to which alone a convection
velocity can have any significance. Thus the convection-
coefficient implying some type of a stationary or viscous,
yet nevertheless "absolute" ether, succeeded in explaining
satisfactorily all known optical facts down to 1880.
Mic/iehov-Morley Eopperiment. — In 1881, Michelson
and Morley performed their classical experiments which
undermined the whole structure of the old ether theory
and thus served to introduce the new theory of relativity.
HISTOKICAL INTRODUCTION V
The fiiiidameiital idea underlyia^^'' tliib experiment is quite
sim[de. In all old expeiiments the velocity of light
situated in free ether \Vas corn[)ared with the veloeitv
of waves actually situated m a piece of moving matter
and presumably carried away by it. The compensatory
effect of the Fresnelian convection of ether afforded a
satisfactory explanation of all neo^ative results.
In the Michelson-Morley experiment the arrangement is
quite different. If there is a definite gap in a rigid body,
light waves situated in free ether will take a delinite time
in crossing the gap. If the rigid platform carrying the
gap is set in motion with respect to the ether in the direc-
tion of light propagation;, light waves (which are even now
situated in free ether) should presumably take a longer
time to cross the gap.
We cannot do better than quote Eddiugton's descrip-
tion of this famous experiment. " The principle of the
experiment may be illustrated by considering a swimmer in
a river. It is easily realized that it takes longer to swim
to a point 50 yards up-stream and back than to a point 50
vards acioss-stream and back. If the earth is movino-
through the ether there is a river of ether flowing- throuopli
the laboratory, and a wave of light may be compared to a
swnmmer travelling with constant velocity relative to the
current. If, then, we divide a beam of light into two parts,
and send one-half swimming up the stream for a certain
distance and then (by a mirror) back to the starting
point, and send the other half an equal distance across
stream and back, the across-stream beam should arrive
back first.
Let the ether be flowing relative to
oi the apparatus with velocity u in the
^ direction Or, and let OA, OB, be
B the two arms of the apparatus of equal
A
VI PKiNClPLE 0¥ RELATIVITY
length L Oi^. being placed up-stream. Let c be tbe
velocity of lig;ht. The time for the double journev alon^'
OA and back is
t,= ± + -A = J^= ^/S^
G — If. c~\ru c^ — u^ c
where f3=:(l—u'^/c^)~'^, a factor greater than unity.
For tbe transverse journey the light must have a compo-
nent velocity n up-stream (relative to the ether) in order to
avoid beins: carried below OB : and since its total velocity
is c, its component across-stream must be \/{c'^ —u'^), the
time for the double journey OB is accordingly
t'l = /7-~^^ = —A SO that t^>t^.
But when the experiment was tried, it was found that
both parts of the beam took the same time, as tested by
the interference bands produced."
x\fter a most careful series of observations, Michelson
and Morle^^ failed to detect the slightest trace of any
effect due to earth's motion throus^h ether.
The Michelson-Morley experiment seems to show that
there is no relative motion of ether and matter. Fresnel's
stagnant ether requires a relative velocity of — n. Thus
Michelson and Morlev themselves thought at first that their
experiment conhrmed Stokes^ viscous ether, in wliieh no
relative motion can ensue on account of the absence of
slip])ing of ether at the surface of separation. But even
on Stokes' theory this viscous How of ether would fall
ofP at a very rapid rate as we recede from the surface
of separation. Michelson and Morley repeated their experi-
ment at different heights from the surface of the earth, but
invariably obtained the same negative results, thus failing
to confirm Stokes' theory of viscous How.
HTSTORICAL TNTHODUCTTON TU
Loflgt!^ experimevi, — Further, in 1893, Lodge per-
formed bis rotating' sphere experiment which showed
complete absence of any viscous How of ether due to
moviuo' masses of matter. A divided beam of light, after
repeated reflections within a ver}^ narrow gap between two
massive hemispheres, was allowed to re-unite and thus
produce interference bands. When the two hemispheres
are set rotating, it is conceivable that the ether in the gap
would be disturbed due to viscous flow, and any such flow
would be immediately detected by a distru'bance of the
interference bands. But actual observation failed to
detect the slightest disturbance of the ether in the gap,
due to the motion of the hemispheres. Lodge's experi-
ment thus seems to show a complete absence of any viscous
flow of ether.
Apart from these experimental discrepancies, grave
theoretical objections were urged against a viscous ether.
Stokes himself had shown that his ether must be incom-
pressible and all motion in it differentially irrotational,
at the same time there should be absolutely no slipping at
the surface of separation. Now all these conditions cannot
be simultaneously satisfied for any conceivable material
medium without certain very special and arbitrary assump-
tions. Thus Stokes' ether failed to satisfy the very motive
which had led Stokes to formulate it^ namely, the desirabi-
lity of constructing a "physical" medium. Planck offered
modified forms of Stokes' theory which seemed capable of
being reconciled with the Miehelson-Morley experiment,
but required very sjiecial assumptions. The very complexity
and the very arbitrariness of these assumptions prevented
Planck's ether from attaining any degree of practical
importance in the further development of the subject.
The sole criterion of the value of any scientific theory
must ultimately be its capacity for offering a simple.
Vlll PRINCIPLE OF RELATIVITY
unified^ coherent and fruitful description of observed facts.
In proportion as a theory becomes complex it loses in
usefulness — a theory which is obliged to requisition a
whole array of aibitrary assumptions in order to explain
special facts is practically worse than useless, as it serves
to disjoin, rather than to unite, the several groups of facts.
The optical experiments of the last quarter of the nine-
teenth century showed the impossibility of constructing a
simple ether theory, which would be jsmenable to analytic
treatment and would at the same time stimulate funher
progress. It should be observed that it could scarcely be
shown that no looieallv consistent ether theorv was
possible ; indeed ill 1910, H. A. Wilson offered a consis-
sent ether ilieor\ which was at least quite neutral with
respect to all available optical data. But Wilson's ether
is almost whollv nesfative — its onlv virtue beinoj that it
does not directly contradict observed facts. Neither any
direct conhrmation nor a direct refutation is possible and
it does not throw any light on the various optical pheno-
mena. A theory like this being practicall}' useless stands
self-condemned.
We must now consider the problem of relativf motion of
ether and matter from the point of view of electrical theory.
From 1860 the identitv of lisht as an electromagnetic
vector became o-radualh' established as a result of the
brilliant '^ displacement current" hypothesis of Clerk
Maxwell and his further analytical investigations. The
elastic solid ether became gradually transformed into the
electromagnetic one. Maxwell succeeded in giving a fairly
.satisfactory account of all ordinary optical phenomena
and little room was left for any serious doubts as regards
the general validity of Maxwell's theory. Hertz's re-
searches on dectric waves, first carried out in 1886,
succeeded in furnishing a strong experimental conlh-mation
HISTORICAL INTRODUCTION II
of Maxwell's theory. Electric waves behaved generally
like light waves of very large wave length.
The orthodox Maxwellian view located the dielectric
polarisation in the electromagnetic ether which was merely
a transformation of Fresnel's stag-nant ether. The mag-
netic polarisation was looked upon as wholly secondary in
origin, being due to the relative motion of the dielectric
tubes of polarisation. On this view the Fresnelian con-
vection coefficient comes out to be i, as shown by J. J.
Thomson in 1880, instead of 1 — ^//x- as required by
optical experiments. This obviously implies a complete
failure to account for all those optical experiments which
depend for their satisfactory explanation on the assumption
of a value for the convection coefficient equal to 1 — V/*^'
The modifications proposed independently by Hertz and
Heaviside fare no better."^ They postulated the actual
medium to be the seat of all electric polarisation and further
emphasised the reciprocal relation subsisting between
electricity and magnetism, thus making the field equations
more symmetrical. On this view the whole of the
polarised ether is carried away by the moving medium,
and consequently, the convection co-efficient naturally
becomes unity in this theory, a value quite as discrepant
as that obtained on the original Maxwellian assumption.
Thus neither Maxwell's original theory nor its subse-
quent modifications as developed by Hertz and Heaviside
succeeded in obtainiuii; a value for Fresnelian co-efficient
equal to 1— V/^^j ^^^ consequently stood totall3^ condemned
from the optical point of view.
Certain direct electromagnetic experiments invohing
relative motion of polarised dielectrics were no less conclu-
sive against the generalised theory of Hertz and Heaviside.
* See Note 1.
X PRINCIPLE OF RELATIVITY
According to Hertz a moving dielectric would carry away
the whole of its electric displacement with it. Hence the
electromagnetic effect near the moving dielectric would
be proportional to the total electric displacement, that is
to K, the specific inductive capacity of the dielectric. In
)901, Blondlot working with a stream of moving gas
could not detect any such effect. H. A. Wilson repeated
the experiment in an improved form in 1903 and working
with ebonite found that the observed effect was pro-
portional to K — 1 instead of to K. For gases K is nearly
equal to 1 and hence practically no effect will be observed
in their case. This gives a satisfactory explanation of
Blondlot's negative results.
Rowland had shown in 1876 that the magnetic force
due to a rotating condenser (the dielectric remaining
stationary) was proportional to K, the sp. ind. cap. On
the other hand, Rontgen found in 1888 the magnetic
effect due to a rotating dielectric (the condenser remain-
ing stationary) to be proportional to K— 1, and not to
K. Finally Eichenwald in 1903 found that when both
condenser and dielectric are rotated together, the effect
observed was quite independent of K, a result quite
consistent with the two previous experiments. The Row-
land effect proportional to K, together with the opposite
Rontgen effect proportional to 1 — K, makes the Eichenwald
effect independent of K.
All these experiments together with those of Blondlot
and Wilson made it clear that the electromagnetic
effect due to a moving dielectric was proportional to
K— 1, and not to K as required by Hertz's theory. Thus
the .above group of experiments with moving dielectrics
directly contradicted the Hertz- Heaviside theory. The
internal discrepancies inherent in the classic ether theory
had now become too prominent. It was clear that the
HISTORICAL INTRODUCTION XI
ether concept had finally outgrown its usefulness. The
observed fleets had become too contradictory and too
heterogeneous to be reduced to an organised whole with
the help of the ether concept alone. Radical departures
from the classical theory had become absolutely necessary.
There were several outstandmg difficulties in connec-
tion with anomalous dispersion, selective reflection and
selective absorption which could not be satisfactory
explained in the classic electromagnetic theory. It
was evident that the assumption of some kind of
discreteness in the optical meduim had become inevit-
able. Such an assumption naturally gave rise to an
atomic theory of electricity, namely, the modern electron
theory. Lorentz had postulated the existence of electrons
so early as 1878, but it was not until some years later that
the electron theory became firmly established on a satisfac-
tory basis.
Lorentz assumed that a moving dielectric merely carried
away its own '' polarivsation doublets," which on his theory
gave rise to the induced field proportional to K— 1. The
field near a moving dielectric is naturally proportional to
K — 1 and not to K. Lorentz's theory thus gave a
satisfactory explanation of all those experiments with
moving dielectrics which required effects proportional to
K — 1. Lorentz further succeeded in obtaining a value for
the Fresnelian convection coefficient equal to 1 — ^//a^, the
exact value required by all optical experiments of the
moving type.
We must now go back to Michelson and Morley's
experiment. We have seen that both parts of the beam
are situated in free ether ; no material meduim is involved
in any portion of the paths actually traversed by the beam.
Consequently no compensation due to Fresnelian convection
Xll ^PRINCIPLE OP RELATIVITY
of ether by moving medium is possible. Thus Presneliao
convection compensation can have no possible application
in this ease. Yet some marvellous compensation has
evidently tai^en place which has completely masked the
" absolute '"' velocity of the earth.
In Miphelson and Morley^s experiment, the distance
travelled by the beam along OA (that is, in a direction
parallel to the motion of the platform) is 2/^^, while the
distance travelled by the beam along OB, perpendicular to
the direction of motion of the platform, is ^lip. Yet the
most careful experiments showed, as Eddington says, " that
both parts of the beam took the same time as tested by the
interference bands produced. It would seem that OA and
OB could not really have been of the same length ; and if
OB was of length I, OA must have been of length IjP. The
apparatus was now rotated through 90°, so that OB became
the up-stream. The time for the two journeys was again
the same, so that OB must now be the shorter length. The
plain meaning of the experiment is that both arms have a
length I when placed along 0^ (perpendicular to the direc-
tion of motion), and automatically contract to a length
Ijpf when placed along 0/ (parallel to the direction of
motion). This explanation was first given by Fitz-Gerald."
This Fitz-Gerald contraction^, startling enough in
itself, does not suffice. Assuming this contraction to be a
real one, the distance travelled with respect to the ether is
%lp and the time taken for this journey is 2l^/c. But the
distance travelled with respect to the platform is always
21. Hence the velocity of light with respect to the plat-
form is 21/ — ^ —c/^, a variable quantity depending on
the " absolute " velocity of the platform. But no trace
of such an effect has ever been found. The velocity of
light is always found to be quite independent of the velocity
HISTOBICAL INTRODUCTION XUl
of the platform. The present difficulty cannot be solved
by any further alteration in the measure of space. The
only recourse left open is to alter the measure of time as
well, that is, to adopt the concept of "local time." If a mov-
inoj clock goes slower so that one 'real' second becomes 1/^
second as measured in the moving system, the velocity of
light relative to the platform will always remain c. We
must adopt two very startling hypotheses, namely, the
Fitz -Gerald contraction and the concept of "local time,"
in order to give a satisfactory explanation of the
Miehelson-Morley experiment.
These results were already reached by Lorentz in the
course of further developments of his electron theory.
Lorentz used a special set of transformation equations"^ for
time which implicitly introduced the concept of local time.
But he himself failed to attach any special significance to
it, and looked upon it rather as a mere mathematical
artifice like imaginary quantities in analysis or the circle
at infinity in projective geometry. The originality of
Einstein at this stage consists in his successful physical
interpretation of these results, and viewing them as the
coherent organised consequences of a single general
principle. Lorentz established the Relativity Theoremt
(consisting merely of a set of transformation equations)
while Einstein generalised it into a Universal Principle. In
addition Einstein introduced fundamentally new concepts
of space and time, which served to destroy old fetishes and
demanded a wholesale revision of scientific concepts and
thus opened up new possibilities in the synthetic unification
of natural processes.
Newton had framed his laws of motion in such a way
as to make them quite independent of the absolute velocity
* See Note 2.
t See Note 4.
XIV PRINCIPLE or RELATIVITY
of the earth. Uniform relative motion of ether and matter
could not be detected with the help of dynamical laws.
According to Einstein neither could it be detected with the
help of optical or electromagnetic experiments. Thus the
Einsteinian Principle of Relativity asserts that all physical
laws are independent of the ^absolute' velocity of an observer.
For different systems, the form of all physical laws is
conserved. If we chose the velocity of light"^ to be the
fundamental unit of measurement for all observers (that is,
assume the constancy of the velocity of light in all systems)
we can establish a metric ^^ one — one ^' correspondence
between any two observed systems, such correspondence
depending only the relative velocity of the two systems.
Einstein's Relativity is thus merely the consistent logical
application of the well known physical principle that we
can know nothing but relative motion. In this sense it is
a further extension of Newtonian Relativity.
On this interpretation, the Lorentz- Fitzgerald contrac-
tion and "local time" lose their arbitrary character. Space
and time as measured by two different observers are natur-
ally diverse, and the difference depends only on their relative
motion. Both are equally valid; they are merely different
descriptions of the same physical reality. This is essentially
the point of view adopted by Minkowski. He considers time
itself to be one of the co-ordinate axes, and in his four-
dimensional world, that is in the space-time reality, relative
motion is reduced to a rotation of the axes of reference.
Thus, the diversity in the measurement of lengths and
temporal rates is merely due to the static difference in the
" frame- work ^' of the different observers.
The above theory of Relativity absorbed practically
the whole of the electromagnetic theory based on the
* See Notes 9 and 12.
HISTORICAL INTRODUCTION XV
Maxwell-Lorentz system of field equations. It combined
all the advantages of classic Maxwellian theory together
with an electronic hypothesis. The Lorentz assumption of
polarisation doublets had furnished a satisfactory explana-
tion of the Fresnelian convection of ether, but in the new
theory this is deduced merely as a consequence of the altered
concept of relative velocity. In addition, the theory of
Relativity accepted the results of Michelson and Morley's
experiments as a definite principle, namely, the principle of
the constancy of the velocity of light, so that there was
nothing left for explanation in the Michelson-Morle3^
experiment. But even more than all this, it established a
single general principle which served to connect together
in a simple coherent and fruitful manner the known facts
of Physics.
The theory of Relativity received direct experimental
confiimation in several directions. Repeated attempts were
made to detect the Lorentz-Fitzgerald contraction. Any
ordinary physical contraction will usually have observable
physical results ; for example, the total electrical resistance
of a conductor will diminish. Trouton and Noble, Trouton
and Rankine, Rayleigh and Brace, and others employed
a variety of different methods to detect the Lorentz-
Fitzgerald contraction, but invariably with the same
negative results. Whether there is an ether or not,
uniform velocity ivith respect to it can never he detected.
This does not prove that there is no such thing as an
ether but certainly does render the ether entirely super-
fluous. Universal compensation is due to a change in local
units of length and time, or rather, being merely different
descriptions of the same reality, there is no compensation
at all.
There was another group of observed phenomena which
could scarcely be fitted into a Newtonian scheme of
XVI PRINCIPLE OF RELATIVITY
dynamics without doing violence to it. The experimental
work of Kaufmann, in 1901, made it abundantly clear that
the " mass '^ of an electron dei)ended on its velocity. So
early as 1881, J. J. Thomson had shown that the inertia of
a charged })article increased with its velocity. Abraham
now deduced a formula for the variation of mass with
velocity, on the hypothesis that an electron always remain-
ed a rigid sphere. Lorentz proceeded on the assumption
that the electron shared in the Lorentz-Fitz2:erald eontrae-
tion and obtained a totally di:fferent formula. A very
careful series of measurements carried out independently b}^
Biicherer, Wolz, Hupka and finally Neumann in 1913,
decided conclusively in favour of the Lorentz formula.
This "contractile^"' formula follows immediately as a direct
consequence of the new Theory of Relativity, without any
assumption as regards the electrical origin of inertia. Thus
the complete agreement of experimental facts witli the
predictions of the new theory must be considered as
confirming it as a principle which goes even beyond the
electron itself. The greatest triumph of this new theory
consists, indeed, in the fact that a large number of results,
which had formerly required all kinds of special hypotheses
for their explanation, are now deduced very simply as
inevitable consequences of one single general principle.
We have now traced the history of the development of
the restricted or special theory of Relativity, which is
mainly concerned with optical and electrical phenomena.
It was first offered by Einstein in 1905. Ten years later,
Einstein formulated his second theory, the Generalised
Principle of Relativity. This new theory is mainly a theory
of gravitation and has very little connection with optics
and electricity. In one sense, the second theory is indeed
a further generalisation of the restricted princijole, but the
former does not really contain the latter as a special ease.
HISTORICAL INTRODUCTION Xvii
Einstein's first theory is restricted in the sense that it
only refers to uniform reetiliniar motion and has no appli-
cation to any kind of accelerated movements. Einstein in
his second theory extends the Relativity Principle to cases
of accelerated motion. If Relativity is to be universally
true, then even accelerated motion must be merely relative,
motion tjetioeen matter and matter. Hence the Generalised
Principle of Relativity asserts that " absolute " motion
cannot be detected even with the help of gravitational laws.
All movements must be referred to definite sets of
co-ordinate axes. If there is any change of axes, the
numerical magnitude of the movements will also chano'e.
But according to Newtonian dynamics, such alteration in
physical movements can only be due to the effeet of ceitain
forces in the tield.^ Thus any change of axes will introduce
new '• geometrical" forces in the field which are quite
independent of the nature of the body acted on. Gravitation-
al forces also have this same remarkable property, and
gravitation itself may be of essentially the same nature as
these '^ geometrical" forces introduced by a change of axes.
This leads to Einstein's famous Principle of Equivalence.
A gravitational field of force is strictl/j equivole^it to one
introduced tjy a transformation of co-ordinates and no possitjle
experiment can distinguish fjetween the tioo.
Thus it may become possible to " transform away ''
gravitational effects, at least for sufficiently small regions of
space, by referring all movements to a new set of axes. This
new "framework" may of course have all kinds of very
complicated movements when referred to the old Galilean
or *' rectangular unaccelerated system of co-ordinates."
But there is no reason why we should look upon the
Galilean system as more fundamental than any other. If it
* Note A.
XVlll PEIXCIPLE OF EELATIYITY
is found simpler to refer all motion in a gravitational field
to a special set of co-ordinates, we may certainly look upon
this special ^'framework" (at least for the particular region
concerned), to be more fundamental and more natural. We
may, still more simply, identify this particular framework
with the special local properties of space in that region.
That is, we can look upon the effects of a gravitational
field as simply due to the local properties of space and time
itself. The very presence of matter implies a modification
of the characteristics of space and time in its neighbour-
hood. As Eddington saj^s ^' matter does uot cause the
curvature of space-time. It is the curvature. Just as
light does not cause electromagnetic oscillations ; it is the
oscillations."
We may look upon this from a slightly different point
of view. The General Principle of Relativity asserts that
all motion is merely relative motion between matter and
matter, and as all movements must be referred to definite
sets of co-ordinates, the ground of any possible framework
must ultimately be material in character, it /v convenient
to take the matter actually present in a field as the
fundamental ground of our framework. If this is done,
the special characteristics of our framework would naturally
depend on the actual distribution of matter in the field.
But physical space and time is completely defined by the
•' framework." In other words the '' framework " itself is
space and time. Hence w^e see how pit i/sical space and time
is aetuallv defined bv the local distribution of matter.
There are certain magnitudes which remain constant by
any change of axes. In ordinary geometry distance
between two points is one such magnitude ; so that
hx'^ +^^^ H-5,e'^ is an invariant. In the restricted theory of
light, the principle of constancy of light velocity demands
that 8ir2 +8^^ -|.8^2 __^2g^,2 should remain constant.
HISTORICAL INTUODUCTION XIX
The 'Sejjaration ds of adjacent events is defined by
ds'^ = —(Lv^ —di/'^ —dz" -\-c^dt^ , It is an extension of the
notion of distance and this is the new invariant. Now if
Xy ijy Zy t are Iransformed to any set of new variables
ji'j, ti'g, i'g, x^, we shall get a quadratic expression for
ds^ =y J j.r J 2 H- 2-7j 2=^'i'''2 + • • • = >'J i .i'V i ^Vj where the ^^s are
functions of d'^, x^, .^'3, ii\ depending on the transforma-
tion.
The special properties of space and time in any region
are defined by these r/s which are themselves determined,
by the actual distribution of matter in the locality. Thus
from the Newtonian point of view, these //'s represent the
gravitational effect of matter while from the Relativity
stand-point, these mereh' define the non-Newtonian (and
incidentally non-Euclidean) spice in the neighbourhood of
matter.
We have seen that Einstein's theory requires local
curvature of space-time in the neighbourhood of matter.
Such altered characteristics of space and time give a
satisfactory explanation of an outstanding discrepancy in
the observed advance of perihelion of Mercury. The large
discordance is almost completely removed by Einstein's
theory.
Again, in an intense gravitational field, a beam of light
will be affected by the local curvature of space, so that to
an observer who is referring all phenomena to a Newtonian
system, the beam of light will appear to deviate from its
path along an Euclidean straight line.
This famous prediction of Einstein about the deflection
of a beam of light by the sun's gravitational field was
tested during the total solar eclipse of May, 1919. The
observed deflection is decisively in favour of the Generalised
Theory of Relativity.
XX PRINCIPLE OF RELATIVITY
It should be uotecl however that the veloeitv of li^ht
itself would decrease in a gravitational field. This may
appear at first sight to be a violation of the principle of
constancy of light-velocity. But when we remember that
the Special Theory is explicitly restricted to the case of
unaecelerated motion, the difficulty vanishes. In the
absence of a gravitational field, that is in any unaecelerated
system, the velocity of light will always remain constant.
Thus the validity of the Special Theory is completely
preserved within its own restricted field.
Einstein has proposed a third crucial test. He has
predicted a shift of spectral lines towards the red, due to an
intense gravitational potential. Experimental difficulties
are very considerable here, as the shift of spectral lines is a
complex phenomenon. Evidence is conflicting and nothing
conclusive can yet be asserted. Einstein thought that a
gravitational displacement of the Fraunhofer lines is a
necessary and fundamental condition for the acceptance of
his theorv. But Eddino'ton has pointed out that even if
this test fails, the logical conclusion would seem to be that
while Einstein's law of gravitation is true for matter in
bulk, it is not true for such small material systems as
atomic oscillator.
CONCLI SIGN
From the conceptual stand-point there are several
important consequences of the Generalised or Gravitational
Theory of Relativity. Physical space-time is perceived to
be intimatel}' connected with the actual local distribution
of matter. Euclid-Newtonian space-time is itot the actual
space-time of Physics, simply because the former completely
neglects the actual presence of matter. Euclid-Newtonian
continuum is merely an abstraction, while physical space-
time is the actual framework which has some definite
HISTORICAL INTRODUCTION XXI
curvature due to the presence of matter. Gravitational
Theory of Relativity thus brings out clearly the funda-
mental distinction between actual physical space-time
(which is non-isotropie and non-Euclid-Newtonian) on one
hand and the abstract Euclid-Newtonian continuum (which
is homogeneous, isotropic and a purely intellectual construc-
tion) on the other.
The measurements of the rotation of the earth reveals a
fundamental framework which may be called the ^' inertial
framework." This constitutes the actual physical universe.
This universe approaches Galilean space-time at a great
distance from matter.
The properties of this physical universe may be referred
to some world-distribution of matter or the "inertial frame-
work" may be constructed by a suitable modification of the
law of gravitation itself. In Einstein's theory the actual
curvature of the ** inertia! framework " is referred to vast
quantities of undetected world-matter. It has interesting
consequences. The dimensions of Einsteinian universe
would depend on the quantity of matter in it ; it would
vanish to a point in the total absence of matter. Then
again curvature depends on the quantity of matter, and
hence in the presence of a sufficient quantity of matter space-
time may curve round and close up. Einsteinian universe
will then reduce to a finite system without boundaries, like
the surface of a sphere. In this " closed up " system,
light rays will come to a focus after travelling round the
universe and we should see an ''anti-sun'"' (corresponding to
the back surface of the sun) at a point in the sk}^ opposite
to the real sun. This anti-sun would of course be equally
large and equally bright if there is no absorption of hght
in free space.
In de Sitter's theory, the existence of vast quantities of
world-matter is not required. But beyond a definite
XXll PRINCIPLE OF RELATIVITY
distance from an observer^ time itself stands still, so that
to the observer nothing can ever " happen " there. All
these theories are still highly speculative in character, but
they have certainly extended the scope of theoretical phj^sics
to the central problem of the ultimate nature of the
universe itself.
One outstanding peculiarity still attaches to the concept
of electric force — it is not amenable to any process of being
" transformed awav " bv a suitable change of framework.
H. Weyl, it seems, has developed a geometrical theory (in
hyper-space) in which no fundamental distinction is made
between gravitational and electrical forces.
Einstein's theory connects up the law of gravitation
with the laws of motion, and serves to establish a very
intimate relationship between matter and physical space-
time. Space, time and matter (or energy) were considered
to be the three ultimate elements in Physics. The restricted
theory fused space-time into one indissoluble whole. The
generalised theory has further synthesised space-time and
matter into one fundamental physical reality. Space, time
and matter taken separatel}" are more abstractions. Physical
reality consists of a synthesis of all three.
P. C. Mahalanobis.
HISTORICAL INTRODUCTION XXlll
Note A.
For example consider a massive particle resting on a
circular disc. If we set the disc rotating, a centrifugal force
appears in the field. On the other hand, if we transform
to a set of rotating axes, we must introduce a centrifugal
force in order to correct for the change of axes. This
newly introduced centrifugal force is usually looked upon
as a mathematical fiction — as '' geometrical" rather than
physical. The presence of such a geometrical force is usually
interpreted us being due to the adoption of a fictitious
framework. On the other hand a gravitational force is
considered quite real. Thus a fundamental distinction is
made between geometrical and gravitational forces.
In the General Theory of Relativity, this fundamental
distinction is done away with. The very possibility of
distinguishing between geometrical and gravitational forces
is denied. All axes of reference may now be regarded as
equally valid.
In the Restricted Theory, all '^unaccelerated" axes of
reference were recognised as equally valid, so that physical
laws were made independent of uniform absolute velocity.
In the General Theory, physical laws are made independent
of "absolute" motion of any kind.
On
The Electrodynamics of Moving Bodies
BY
A. EjNSTEIJf.
INTRODUCTION.
It is well known that if we attempt to apply Maxwell's
electrodynamics, as conceived at the present time, to
moving bodies, we are led to assy met ry which does not
ao^ree with observed phenomena. Let us think of the
mutual action between a magi-net and a conductor. The
observed phenomena in this case depend only on the
relative motion of the conductor and the magnet, while
according to the usual conception, a distinction must be
made between the cases where the one or the other of the
bodies is in motion. If, for example, the magnet moves
and the conductor is at rest, then an electric field of certain
energy-value is produced in the neighbourhood of the
magnet, which excites a current in those parts of the
field where a conductor exists. But if the magnet be at
rest and the conductor be set in motion, no electric field
is produced in the neighbourhood of the magnet, but an
electromotive force which corresponds to no energy in
itself is produced in the conductor; this causes an electric"
current of the same magnitude and the same career as the
electric force, it being of course assumed that the relative
motion in both of these cases is the same.
il PRINCIPLE OF RELATIVITY
*2. Examples of a similar kind such as the uusueeessful
attempt to substantiate the motiou of the earth relative
to the " Light-medium " lead us to the supposition that
not only in mechanics, but also in electrodynamics, no
properties of observed facts correspond to a concept of
absolute rest: but that for all coordinate svstems for which
the mechanical equations hold, the equivalent electrodyna-
mieal and optical equations hold also, as has already been
shown for magnitudes of the first order. In the following
we make these assumptions (w^hich we shall subsequently
call the Principle of Relativity) and introduce the further
assumption, — an assumption which is at the first sight
quite irreconcilable with the former one — that light is
propagated in vacant space, with a velocity c which is
independent of the nature of motion of the emitting
bod}'. These tw^o assumptions are quite sufficient to give
us a simple and consistent theor^^ of electrodynamics of
movino' bodies on the basis of the Maxwellian theory for
a t,'
bodies at rest. The introduction of a ^^ Lightather"
will be proved to be superfluous, for according to the
conceptions which will 'be developed, we shall introduce
neith er a space absolutely at rest, and endowed with
special properties, nor shall we associate a velocity -vector
with a point in which electro-magnetic processes take
place.
3. Like every other theory in electrodynamics, the
theory is based on the kinematics of rigid bodies; in the
enunciation of every theory, Ave have to do with relations
betw^een rigid bodies (co-ordinate system), clocks, and
electromagnetic processes. An insufficient consideration
of these circumstances is the cause of difficulties with
which the electrodynamics of moving bodies have to fight
at present.
ON THE ELECTKODYXA MlCS Oh' 3I0VlNa BODIES 3
I.-KINEMATIOAL PORTION.
§ 1. Definition of Synchronism.
Let us have a eo-ordinate system, in wliieh the New-
tonian equations hold. For distinguishing this system
from another which will be introduced hereafter, we
shall always call it " the stationary system,"
If a material point be at rest in this system, then its
position in this system can be found out by a measuring
rod, and can be expressed by the methods of Euclidean
Geometry, or in Cartesian co-ordinates.
If we wish to describe the motion of a material point,
the values of its coordinates must be expressed as functions
of time. It is always to be borne in mind that sicc/i a
■ *• atliemaiical (lefinition has a physical senses only lohen loe
have a clear )iotio7i of what is meant by time. We have to
fake into consideration the fact that those of our conceptions^ in
lohich time plays a part, are alioays conceptions of synchronism
For example, we say that a train arrives here at 7 o'clock ;
this means that the exact pointing of the little hand of my
watch to 7, and the arrival of the train are synchronous
events.
It may appear that all difficulties connected with the
definition of time can be removed when in place of time,
we substitute the position of the little hand of my watch.
Such a definition is in fact sufficient, when it is required to
define time exclusively for the place at which the clock is
stationed. But the definition is not sufficient when it is
required to connect by time events taking place at different
stations,-— -or what amounts to the same thing,- — to estimate
by means of time (zeitlich werten) the occurrence of events,
which take place at stations distant from the clock.
4 PKINCIPLE OF RELATIVITY
Now with regard to this attempt; — the time-estimation
of events^ we can satisfy ourselves in the following
manner. Suppose an observer — who is stationed at the
origin of coordinates with the clock — associates a ray of
light which comes to him through space, and gives testimony
to the event of which the time is to be estimated, — with
the corresponding position of the hands of the clock. But
such an association has this defect^ — it depends on the
position of the observer provided with the clock, as we
know by experience. We can attain to a more practicable
result bv the following- treatment.
If an observer be stationed at A with a clock, he can
estimate the time of events occurring in the immediate
neighbourhood of A, by looking for the position of
the hands of the clock, which are syrchronous with
the event. If an observer be stationed at B with a
clock, — we should add that the clock is of the same nature
as the one at A, — he can estimate the time of events
occurring about B. But without further premises, it is
not possible to compare, as far as time is concerned, the
events at B with the events at A. We have hitherto an
A-time, and a B-time, but no time common to A and B.
This last time {i.e., common time) can be defined, if we
establish by definition that the time which Hght requires
in travelling from A to B is equivalent to the time which
light requires in travelling from B to A. For example,
a ray of light proceeds from A at xl-time t towards B,
arrives and is reflected from B at B-time t and returns
to A at A-time t' . Accordin£c to the definition, both
clocks are synchronous^ if
t - 1 = t' - t .
B A A B
02^ THE ELECTRODYNAMICS OF MOVING BODIES 5
We assume tbal this definition of synchronism is possible
without involving any inconsistency, for any number of
points, therefore the following relations hold : —
1. If the clock at B be synchronous with the clock
at A, then the clock at A is synchronous with the clock
at B.
2. If the clock at A as w^ell as the clock at B are
both synchronous with the clock at C, then the clocks at
A and B are svnchronous.
Thus with the help of certain physical experiences, w^e
have established what we understand when we speak of
clocks at rest at different stations, and synchronous with
one another ; and thereby we have arrived at a definition of
synchronism and time.
In accordance with experience we shall assume that the
magnitude
2 AB
77 ~^ =zc, where c is a universal constant.
A A "
We have defined time essentially w^ith a clock at rest
in a stationary system. On account of its adaptability
to the stationary system, we call the time defined in this
way as " time of the stationary system.'^
§ 2. On the Relativity of Length and Time.
«
The following reflections are based on the Principle
of Relativity and on the Principle of Constancy of the
velocity of light, both of which we define in the following
w^ay :—
1. The laws according to which the nature of physical
systems alter are independent of the manner in which
these changes are referred to two co-ordinate systems
6 PRINCIPLE or- RELATIVITY
which have a uniform translatorv motion relative to each
other.
2. Every ray of light moves ^ in the '^^ stationary
co-ordinate system " with the same velocity c-j the velocity
being independent of the condition whether this ray of
light is emitted by a bod}^ at rest or in motion.^' Therefore
, .. Path of Li<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^
Einstoin^A, & Minkowski >H> 530 > 11
E35
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