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HISTORICAL INTRODUCTION
Lord Kelvin writing in 1893, in his preface to the
English edition of Hert/ s Researches on Electric Waves,
says " many workers and many thinkers have helped to
build up the nineteenth century school of plenum, one
ether for light, heat, electricity, magnetism ; and the
German and English volumes containing Hertz s electrical
papers, given to the world in the last decade of the
century, will be a permanent monument of the splendid
cons jmmation now realised.
Ten years later, in 1905, we find Einstein declaring
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 ver\ success; g:ivc rise to a host of new questions all
bearing on the central problem of relative motion of ether
and matter.
11 I lilXC II I.F. OK KKI.m \ m
Aragd n prism 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 ether
outside the prism and move with velocity c with respect
to the ether. If the prism moves with a velocity u
with respect to this fixed ether, then the incident velocity
of light with respect to the prism should be C + H. Thus
the refractive index of the glass prism should depend on ,
the absolute velocity of the prism, i.e., its velocity with
respect to the fixed ether. Arago performed the experiment
in 1819, but failed to detect the expected change.
Airy- Boacovitc/i water -telescope experiment. Boscovitch
had still earlier in i7b t>, raised the very important
question of the dependence of aberration on the refractive
index of the medium filling the telescope. Aberration
depends on the difference in the velocity of light outside
the telescope and its velocity inside the telescope. If the
latter velocity changes owing to a change in the medium
filling the telescope, aberration itself should change, that
is, aberration should depend on the nature of the medium.
Airy, in 1871 filled up a telescope with water but
failed to detect any change in the aberration. Thus we
get both in the case of Arago pri^m experiment and
Airy- Boscovitch water-telescope experiment, the very
startling result that optical effects in a moving medium
seem to be quite independent of the velocity of the
medium with respect to Fresnel s stationary ether.
Fresnel s conrection coefficient /{=] //x 2 . Possibly
some form of compensation is taking place. \Vorking on
this hypothesis, Fresnel Offered his famous ether convec
tion theory. According to Fresnel, the presence of matter
implies a definite condensation of ether within the
IIISTOKK \l, I.VII. OIU TI ION 111
region occupied by matter. This " condensed " or
e\cr<s portion of ether is supposed to he carried awav
with its own pice, of moving matter. It should be
observed that only the "excess" portion is carried away,
while the rest remains a- -taunant ;is 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 . u.
Fresnel showed that if this convection coefficient k is
l l /fj. 2 (p. being the refractive index of the prism), then
the velocity of light after refraction within the moving
prism would be altered to just such extent as would make
the refractive index of the moving prism quite indepen
dent of its "absolute" velocity u. The non-dependence
of aberration on the " absolute " velocity u, is also very
easily explained with the help of this Fresnelian convection-
coeflicieut k.
Stoke** viscous ether. It should be remembered, however,
that Fresnel s stationary ether is absolutely fixed and is not
at all disturbed by the motion of matter through it. In this
respect Fresnelian ether cannot be said to behave in any
respectable physical fashion, and this led Stokes, in
1845-H), to construct a more material type of medium.
Stokes assumed that viscous motion ensues near the surface
of separation of ether and moving matter, while at
sulficifiitlv distant unions the ether remains wholly
undisturbed. He >howed how such a viscous ether would
explain aberration if all motion in it were differentially
irrotational. But in order to explain the null Arago
efl cct. Stoke:- ua.- compelled to assume the convection
liNpothesis of Fresnel with an identical numerical value
for k, namely I ./ -. Thus the prestige of the Fresnelian
convection-cocllicient \va> enhanced, il anything, by the
theoretical inveBti&fctiona o! stokes.
IV I KINI U l.K OK RELATIVITY
Fizratf* experiment.. Soon after, in J8ol, it received
direct experimental confirmation in a brilliant piece of
work by Fizeau.
If a divided beam of light is re-united after passing
through two adjacent cylinders filled with water, ordinary
interference fringes will be produced. If the water in one
of the cylinders is now made to flow, the " condensed "
ether within the flowing water would be convected and
would produce a shift in the interference fringes. The
shift actually observed agreed very well with a value of
k=l l l^ 1 . The Fresnelian convection-coefficient now
became firmly established as a consequence of a direct
positive effect. On the other hand, the negative evidences
in favour of the convection-coefficient had also multiplied.
Mascart, Hoek, Maxwell and others sought for definite
changes in different optical effects induced by the motion
of the earth relative to the stationary ether. But all such
attempts failed to reveal the slightest trace of any optical
disturbance due to the "absolute" velocity of the earth
thus proving conclusively that all tne different optical
effects shared in the general compensation arising out of
the Fresuelian convection of the excess ether. It must be
earefully 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.
Micftelxoti- M or ley Experiment. 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.
III.VTOUU-U. 1NTIIOIH ( Tlo.N V
The fundamental idea underlying thift experiment is quite
simple. In all old experiments the velocity of light
situated in free ether was compared with the velocity
of waves actually situated in a piece of moving matter
and presumably curried away by it. The compensatory
effect of the Fresnelian convection of ether afforded a
satisfactory explanation of all negative results.
In the Michelson-Morley experiment the arrangement is
quite different. If there is a definite gap j n a rigid body,
light waves situated in free ether will take a definite time
in crossing the gap. If the rigid platform carrying the
gap is set in motion with respect to the ether in the 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 Eddington 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
yards across-stream and back. If the earth is moving
through the ether there is a river of ether Howing through
the laboratory, and a wave of light may be compared to a
swimmer travelling with constant velocity relative to the
current. If, then, we divide a beam of light into two parts,
and send one-half swimming up the stream for a certain
distance and then (by a mirror) back to the starting
jx)int, and setul the other half an equal distance across
stream and back, the across-stivam beam should arrive
back first.
Let the ether be llowinir relative to
o the apparatus with velocity -n. in the
n direct ion () . and let OA , OB, be
B the two arms of the apparatus of equal
VI I KINCIL LK OF KKLATUITY
length /, OA being placed up-stream. Let < be the
velocity of light. The time for the double journey along
OA and back is
/ = -- + / = ^ = 2I B*
c if r + n r s ^l z c
where 8= (1 n z /c" )"*, a factor greater than unity.
For the transverse journey the light must have a compo
nent velocity n up-stream (relative to the ether) in order to
avoid being carried below OB : and since its total velocity
is <, its component across-stream must be \/(c" 2 ), the
time for the double journey OB is accordingly
But when the experiment was tried, it was found that
both parts of the beam took the same time, as tested by
the interference bauds produced."
After a most careful series of observations; Michelson
and Morley failed to detect the slightest trace of any
effect due to earth s motion through ether.
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 11. Thus
Miehelsou and Morley themselves thought at iirst that their
e\]H rinient eoniirmed Stokes viscous ether, in which no
relative motion can ensue on account of the abs.MK r of
slipping of ether :i,t the :- -niTrice of separation. Hut even
on Stokes theory this viscous How of ether would fall
off 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 sttme negative results, thus failing
to confirm Stokes theory of viscous Mow.
II ISTOIMCAI, INTKOlim TION Til
o//. Further, in l.V. - i, Lodge per
formed his rotating sphere experiment wiiich should
complete absence of any viscous How of ether due to
moving masses of matter. A divided beam of light, after
repeated reflections within a very narrow gap between two
massive hemispheres. \\-;is 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 How, and any such flow
would be immediately detected by a disturbance of the
interference bands. But actual observation failed to
detect the slightest disturbance of the ether in the gap,
due to the motion of tbe 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 >tokes theory which seemed capable of
being reconciled with the Micheleon-Morley experiment,
but required very special assumpt ions. The very complexitv
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 OK It K NATIVITY
unified, coherent and fruitful description of observed facts.
In proportion as a theory becomes complex it loses in
usefulness a theory which is obliged to requisition a
whole array of arbitrary assumptions in order to explain
special facts is practically worse than useless, as it serves
to disjoin, rather than to unite, the several groups of facts.
The optical experiments of the last quarter of the nine
teenth century showed the impossibility of constructing a
simple ether theory, which would be simenable to analytic
treatment and would at the same time stimulate further
progress. It should be observed that, it could scarcely be
shown that no logically consistent ether theory was
possible; indeed in 1910, H- A. Wilson offered a consis-
sent ether theory which was at least quite neutral with
respect to all available optical data. But Wilson s ether
is almost wholly negative, its only virtue being that it
does not directly contradict observed facts. Neither any
direct confirmation nor a direct refutation is possible and
it does not throw any light on the various optical pheno
mena. A theory like this being practically useless stands
self-condemned-
We must now consider the problem of relative motion of
ether and matter from the point of view of electrical theory.
From 1860 the identity of light as an electromagnetic
vector became gradually established as a result of the
brilliant "displacement current" hypothesis of Clerk
Maxwell and his further -analytical investigations. The
elastic solid ether became gradually transformed into the
electromagnetic one. Maxwell succeeded in giving a fairly
satisfactory account of all ordinary optical phenomena
;vnd little room was left for any serious doubts as regards
the general validity of Maxwell s theory. Hertz s re
searches on >lectric waves, first carried out in l^Mi,
succeeded in furnishing a Mrong experimental confirmation
MMi U, INTRODUCTION IX
of Maxwell s thenn. Klef-tric waves behaved
like light waves of ven lar^e wave length.
Tlie orthodox Maxwellian view located the dielectric
polarisation in fte electromagnetic el her which was merely
a transformation of iVe-ncl s stagnant ether. The Mag
netic polarisation was looked upon as wholly Secondary m
origin, being due to the relative motion of the dielectric
tul.i- of polarisation. On this view the Fresnelian con-
vcetion r-octllcient comes out to he i, as shown by J. J.
Thomson in 1880, instead of I 1 //*- 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 eonvect.ion coefficient equal to 1 l //* 2 .
The modifications proposed independently by Her!/ 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 consequentty, 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 llertx. and Heaviside
succeeded in obtaining a \alne for Fresnelian co-efficient
equal to 1 1 1^- , and consequently stood totally condemned
from the optical point of view.
Certain direct electromagnetic experiments involving-
relative motion of polarised dielectrics were no less conclu
sive against the generalised theory of Hertx and Heaviside.
* See Note 1.
X HKIXC1PLE 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.
1901, Blondlot working with a stream of moving gas
could not detect any such effect. H. A. Wilson repeated
the experiment in an improved form in 1903 and working
with ebonite found that the observed effect was 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 L876 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 wao clear that the
III-TOUK \i. i\ii:oi>rcTifW xi
H her row-opt had finally outgrown its usefulness. The
observed ! ;,<(> had become too contradictory and too
heterogeneous to be reduced to an organised whole with
tin- help of the ether concept alone. Radical departures
from the classical theory had become absolutely necessary.
There were several outstanding 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 " polarisation doublets," which on his theory
gave rise to the induced field proportional to K 1. The
field near a moving dielectric is naturally proportional to
K 1 and not to K. Lorentz s theory thus gave a
satisfactory explanation of all those experiments with
moving dielectrics which required effects proportional to
K 1. Lorentz further succeeded in obtaining a value for
the Fresnelian convection coefficient equal to 1 *//**, the
exact value required by all optical experiments of the
moving type.
\VC 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 raeduim is involved
in any portion of the paths actually traversed by the beam.
Consequently no compensation due to Fresnelian convection
Til I lMM II IK OK lU-.LATIVITY
of ether by moving medium is poi-sible. Thus Kre-nelian
convection compensation can have no possible application
in this case. Yet some marvellous compensation has
evidvuily taken place which has completely m;i<ke.d the
" absolute " velocity of the earth.
In Michelson and Morley s experiment, the distance
travelled by the beam along OA (that is, in a direction
parallel to the motion of the platform) is //8 2 , while the
distance travelled by the beam along 1 OB, perpendicular to
the direction of motion of the platform, is %l(3. 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 /, OA must have been of length 1 1 ft. 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 Oi/ (perpendicular to the direc
tion of motion), and automatically contract to a length
1/J3, when placed along O/ (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 he a
real one, the distance travelled with respect to the ether is
"11(3 and the time taken for this journey is :V/^ r. But the
distance travelled with respect to the platform is always
2/. Hence the velocity of light with respect to the plat-
2,1 8
form is 2// " fi/P, 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
HI>Toi:l< A I IVil:Ml IIM\ XIII
1^1 ilic plat I m in. The < precent dfffioolty cannot \>u solved
h\ .my further alteration in the measure of space. The
recount left open i-: to alter the measure ol time an
well, that K, to adopt t lie conecpt of * loea I t line." If a mov
ing clock goes slower so that one real second becomes 1 ft
second as measured in the moving system, I lie velocity of
light relative to the platform will always remain c. We
must ad.-pt two very startling hypotheses, namely, the
Fit/ (lerald 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.
Lorentx 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
artiliee like imaginary quantities in analysis or the circle
at infinity in projective geometry. The originality of
Einstein at tins singe consists in his successful physical
interpretation of these result?, and viewing them as the
coherent organised consequences of a single general
principle. Lorentz established the Relativity Theoremt
(consisting men-iy . ) a set of transformation equations)
while Kinstein 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 \\-.\\
as to make then; quite Independent of the absolute velocity
* See \
t See Note 4.
XIV I RINTIPLE OF 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 thai \ve
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 praeticalh
the whole of the electromagnetic theory based on the
* See Notes 9 and 12.
HISTORICAL INTRODUCTION XV
M;i\\\ell-|,orenU system of field equations. It combined
all the advantages ( >| classic Maxvvelliaa theory together
with an electronic hypothesis. The Lorentx assumption of
polarisation doublets had furnished a satisfactory explana
tion ..f flu- r re^nelian convection of ether, but in the new
theory 1 his is deduced merely as a consequence of the altered
concept of relative velocity. In addition, the theory of
Relativity accepted the results of Michelson and Morley s
experiments as a definite principle, namely, the principle of
the constancy of the velocity of light, so that there was
nothing left for explanation in the Michelson-Morley
experiment. But even more than all this, it established a
single general principle which served to connect together
in a simple coherent and fruitful manner the known facts
of Physics.
The theory of Relativity received direct experimental
coiifii -illation in several directions. Repeated attempts were
made to detect the Lorentz-Fitzgerald contraction. Any
ordinary physical contraction will usual Iv 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 ix an, ether or not,
uniform rrforily ////// r, x]><>ct to if can never be 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, IxMtig 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 PRIXCIPLE OF
dynamic^ without doing violence to it. The experimental
work of Kaufmann, in 1901, made it abundantly clear that
the " mass v of an electron depended on its velocity. "<>
early as 1881, J. J. Thomson had shown that tlie inert in of
a charged particle increased with its velocity. Abraham
now deduced a formula for the variation of mass with
velocity, on the hypothesis that an electron always remain
ed a riyid sphere. Lorentz proceeded on the assumption
that the electron shared in the Lorentz- Fitzgerald contrac
tion and obtained a totally different formula. A very
careful series of measurements carried out independently b\
Biicherer, Wolz, Htipka 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 theVlectrical origin of inertia. Thus
the complete agreement of experimental facts with the
predictions of the new theory must be considered as
confirming it as a principle which goes even beyond the
electron itself. The greatest triumph of this new theory
consists, indeed, in the fact that a large number of results,
winch 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 Kiustein in 1905. Ten years later,
Einstein formulated his second theorv, the ( Jcncralised
Principle of Relativity. This new theory is mainly a theory
of gravitation and has very little connection with optics
and electricity. In one sense, the second theory is indeed
a further generalisation of the restricted principle, but the
former does not really contain the latter as a special case.
>i;u AI, iNTHonn riOV
iii s li -v.t theon is re-tricted in tlu- EWDBC that it
only refers to uniform rect iliniar motion and has no appli
cation to any kind of accelerated movements. Einstein in
hi- second theory extends the Relativity Principle to cases
ol accelerated motion. If Relativity is to be universallv
true, ti.en e\vn accelerated motion must be merely rt lativf
nint in, i ln l n ct ii mutter and HI after. Hence the Generalised
Principle of Relativity asserts that "absolute" motion
cannot be detected even with the help of gravitational IRWH.
All in vements must be referred to definite sets of
co-ordinate axes. If there is any chaiiLjf of axes, the
numerical magnitude of the movements will also change.
Bui according to Newtonian dynamics, such alteration in
physical movements can only be due to the effect of ceitain
1 orces in the field.* 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
the.-e " geometrical" forces introduced by a change of axes.
This loids to Einstein s famous Principle of Equivalence.
A i/i <ifl/nl innal Jield of force in xtrictly equivalent to one
introduced // </ tfantforatation <>f co-ordtunfey and nopQwiltfo
t .r/if. i / ; fiif ft it ilixtiiKinixh between the two.
Thus it mav hecome possible to "transform away "
gravitational elTect-^. at least For sufficiently small region^ ol
space, bv 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 unarceleraled syslem of co-ordinate-."
But there is no reason why we should look upt.n the
Galilean s\>tein ,i> more I liiHlainenlal than any other. If it
XV1I1 PUIXCIPLE OF BELATIVITY
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
h eld 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 not 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 /* 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 ? <
space and time. Hence we see how //////*/ v// space and time
is actually defined by 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
&r 2 +Sy- +5?- is an invariant. In the restricted theory of
light, the principle of constancy of light velocity demands
that Saf*+fy*+Sa*c*Sl* should remain constant.
IIISIOI;K \i. i vntoiM ( iiox xix
The *i /m?nti ni </\ oi adjacent events is defined by
ftx* = -,/./ 2 -////- _,/ : v+,.-V/-. It is :ui extrusion of tin-
notion of distance and this is the ne\v invariant. Now if
.*, //, :, I are transformed to any set of new variables
j-,, . ,, ..., ./.,, we shall got a quadratic expression for
,/.s- = //! r r,- + fy, 8 .v.j + . = ;></, i,.<v, where them s are
functions of ./,,./,, .r.p # 4 depending on the transforma
tion.
The special properties of space and time in any region
are defined by these // 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 merely 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, 19 IS*. The
observed deflection is decisively in favour of the Generalised
Theory of Relativity.
XX I KI Vril I.K OK KKLATIVITY
It should be noted however that the velocity of light
itself would decrease in a gravitational field. This may
appear at first sight to be a violation of the principle of
constancy of light-velocity. But when we remember that
the Special Theory is explicitly restricted to the case of
unaccelerated motion, the difficulty vanishes. In the
absence of a gravitational field, that is in any unaccelerated
system, the velocity of light will always remain constant.
Thus the validity of the Special Theory is completely
preserved within its own rextrictwl 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 Praunhofer lines is a
necessary and fundamental condition for the acceptance of
his theory. But Eddington has pointed out that even if
this test fails, the logical conclusion would seem to be that
while Einstein s law of gravitation is true for matter in
bulk, it is not true for such small material systems as
atomic oscillator.
CONCLUSION
From the conceptual stand-point theiv are several
important consequences of the Generalised or Gravitational
Theory of Relativity. Physical space-time is perceived to
be intimately connected with the actual local distribution
of matter. Euclid-Newtonian space-time is not 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 spnn-
time is the actual framework which has some definite
MlST.HiU-.M. INTKOIirrnnN XXI
curvature due to the presence of matter Gravitational
Thrnrv of Relativity tlius brings out clearly the funda
mental distinction between actual physical space-time
(which is non-isotropk" and non- Euclid-Newtonian) on one
hand and the abstract Euclid-Newtonian continuum (which
is homogeneous, iaotropio 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. 1 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 " inertial framework " is referred to vast
quantities of undetected world-matter. It has interesting
consequences. The dimensions of Einsteinian universe
would depend on the quantity of matter in it; it would
vanish to a roint 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. Eiusteinian 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 <f anti-sun" (corresponding to
the back surface of the sun) at a point in the sky opposite
to the real sun. This anti-sun would of course be equally
large and equally bright if there is no absorption of light
in free space.
In de Sitter s theory, the existence of vast quantities of
world-matter is not required. But beyond a definite
Xxii I Rl. \CIPLK OF HKL.VTIVITY
distance from an observer, time itself stands still, so that
to the observer nothing can ever " happen " there. All
these theories are still highly speculative in character, but
they have certainly extended the scope of theoretical physics
to the central problem of the ultimate nature of the
universe itself.
One outstanding peculiarity still attaches to the concept
of electric force it is not amenable to any process of being
"transformed away" by a suitable change of framework.
H. Weyl, it seems, has developed a geometrical theory (in
hyper-space) in which no fundamental distinction is made
between gravitational and electrical forces.
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 sejarately are more abstractions. Physical
reality consist** of a synthesis of all three.
P. C. MAHALAXOBIS.
II I-TOKIC A I, INTRODUCTION XX111
Note A.
For example consider a massive particle resting on a
circular disc. If \ve set the disc rotating, a centrifugal force
appears in the Held. 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 velouitv.
In the General Theory, physical laws are made independent
of "absolute" motion of any kind.
On
The Electrodynamics of Moving Bodies
Bl
A. El \STKIN.
INTRODUCTION.
It is well known thai if we attempt to apply Maxwell s
electrodynamics, as conceived at the present time, to
moving bodies, we are led to assymetry which does not
aijrec with observed phenomena. Let u* think of the
mutual action between n magnet and a conductor. The
observed phenomena, in this case depend only on the
relative motion of the conductor and the magnet, while
according to the usual conception, a distinction must be
made between the cases where the one or the other of the
bodies is in motion. If, for example, the magnet moves
and the conductor is at rest, then an electric held 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 tield
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 <:ime career as the
electric force, it being of course assumed that the iclative
motion in both of these eas.^ is the
I RINCIPLE OF RRLATIV1TY
2. Examples of a similar kind such as the unsuccessful
attempt to substantiate the motion of the earth relative
to the " Light-medium " lead us to the supposition that
not only in mechanics, but also in electrodynamics, no
properties of observed facts correspond to a concept of
absolute rest; but that for all coordinate systems for which
the mechanical equations hold, the equivalent electrodyna-
mical and optical equations hold also, as has already been
shown for magnitudes of the first order. In the following
we make these assumptions (which we shall subsequently
call the Principle of Relativity) and introduce the further
assumption, an assumption which is at the first sight
quite irreconcilable with the former one that light is
propagated in vacant space, with a velocity which is
independent, of the nature of motion of the emitting
body. These two assumptions are quite sufficient to gu-e
us a simple and consistent theory of electrodynamics of
moving bodies on the basis of the Maxwellian theory for
bodies at rest. The introduction of a " 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 t he-
enunciation of every theory, we have to do with relations
between rigid bodies (co-ordinate system), clocks, and
electromagnetic processes. An insufficient consideration
of these circumstances is the cause of difficulties with
which the electrodynamics of moving bodies have to fi^ht
at present.
.HI ELECTRODYXAMH^ "F MoVfVO BODIES 4
I.-KINEMATXOAL PORTION.
1. Definition of Synchronism.
l^t us hrivc a co-ordinate system, in which 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 nan be found out by a measuring
rod, and can be expressed by the methods of Euclidean
Oeometry, or in Cartesian co-ordiuates.
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 such a
nt/h-ntultcrtt ilcfniKlnn has a physical sense t only when ice
lhi\-i it i. f.i;i,- ,><>//>, /i /// n /i il. /.v tiifttut l>y tiMf. Jt f have to
dike into consideration l.h<> fact that those of our conceptions, in
/>///<>// time /i!<ii/x a part, are al ^nyx conception* of synchronism
For example, we say that a train arrives here at 7 o clock ;
this moans 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
ilelinr timt- exclusively for the place at which the clock is
stationed, hut the definition is not sufficient when it is
required to connect by time events taking place at different
station*, or what amounts to the same thing, to estimate
by means of time (zeitlich werten) the occurrence of events,
take place at stations distant from the clock.
4 JMIINCII LL <)i Khl.ATIVm
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. AVe can attain to a more practicable
result by 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 syrchrouous 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 light requires
in travelling from A to B is equivalent to the time which
light requires in travelling from B to A. For example,
a ray of light proceeds from A at A-time t towards B,
arrives and is reflected from B at B-time t and returns
to A at A-time t . According to the definition, both
A
clocks are synchronous, it
u.Y Till, KLI.t IKOm N .V.MIO ( JKiVIM. IKHMKS
\\ as-ume that tin s 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.
:>. If the clock at A as well as the clock at B are
both synchronous with the clock at C, then the clocks at
A and B an- synchronous.
Thus with the help of certain physical experiences, we
have established what we understand when we speak of
clocks at rest at different stations, and synchronous with
one another ; and thereby we have arrived at a definition of
synchronism and time.
In accordance with experience we shall assume that the
magnitude
2 AB
, where r is a universal constant.
\\ "e have defined time essentially with a clock at rest
in a stationary system. Ou 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 Kelatmt\ and on the Principle of Constancy of the
velocity of light, both of which we define in the following
way :
1. The laws according tu which the uuture of physical
>\>tmis alter are independent of the manner in which
those flian^s ;, n . referred to two co-ordinate \stem>
6 PRINCIPLE OF RELATIVITY
which have a uniform translatory motion relative to each
other.
2. Every ray of light moves in the " stationary
co-ordinate system " with the same velocity c, the velocity
being independent of the condition whether this my of
light is emitted by a body at rest or in motion.* Therefore
Path of Light
velocity == = - ,
Interval or time
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.
(6) The observer finds out, by means of clocks placed
in a system at rest (the clocks being synchronous as defined
in 1), the points of this system where the ends of the
rod to be measured occm at a particular time f. The
distance between these two points, measured .by the
previously used measuring rod, this time it being at rest,
is a length, which wt> may call the " length of the rod."
According to the Principle of Relativity, the length
found out by the operation ), which we mav call " the
* Vide Note V.
OX THE EI.FOTRODYXV.Mlrs OF MOVING HOUIKS /
length of the roil in the moving >ystem " is equal to the
length I of the rod in the stationary system.
The length which is found out by the second method,
mav he callo<l the ffnyth of Ike m<>i ir></ roil mounted from
I he xf tlitn/irij xi/xft-m. This length i> to he estimated on
the !>;isis of our principle, and <"V &h<i 1.1 find it /<> If different
from (.
In the generally recognised kinematics, we silently
assume that the lengths defined by these two operations
jiial, or in other words, that at an epoch of time (,
a moving rigid body is geometrically replaceable by the
s-imc body, which can replace it in the condition of rest.
Eelativity 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, /.<?., the time of the clocks correspond to
the time <>f the stationary system at the points where they
happen to arrive ; these clocks are therefore synchronous
iti the stationary system.
\Ve 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
clui k-. At the time t , a ray of light goes out from A, is
rollected from B at the time t , and arrives back at A at
B
time / . Taking into consideration the principle of
constancy of the velocity oi light, we have
,, AB
and t t= .
A B c+v
S PRIXOn LE OF RELATIVITY
where / } is the length of the moving rod, meu-mv<i
in the stationary system. Therefore the observers stationed
with the watches will not find the clocks synchronous,
though the observer in the stationary system must declare
the docks to be synchronous, \Ve therefore sei that we can
attach no absolute significance to the concept of synchro
nism ; but two events which are synchronous when viewed
from one system, will not be synchronous when viewed
from a system moving relatively to this system.
$ 3. Theory of Co-ordinate and Time-Transformation
from a stationary system to a system which
moves relatively to this with
uniform velocity.
Let there be given, in the stationary system two
co-ordinate systems, I.e., two series of three mutually
perpendicular lines issuing from a point. Let the X-axes
of each coincide with one another, and the V 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 systems (/?) have
a constant velocity in the direction t)f the X-axis of
the other which is stationary system K, the motion being
also communicated to the rods and clocks in the system (/).
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
/, we always mean the time in the stationary 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 ILlOTHODYNAMIOa OK MOYl.M; BODIES
system, and \ve thus obtain the co-ordinates (j ,y, z) for the
stationary s\>tcm, and ($,rj,) 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 system be determined for each
point of the moving system (in which there are clocks which
are at rest relative to the moving system), by means of
the method of light signals between these points (in
which there are clocks) in the manner described in 1.
To every value of (<,//, -, /) which fully determines
the position and time of an event in the stationary system,
there correspond-; a system of values (,r/,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 property
of homogeneity which we ascribe to time and space, the
equations must be linea 1 .
If we put .? := rf, then -it is clear that at a point
relatively at rest in the system K, we have a system of
values (.< y z] which are independent of time. Now
let us find out T as a function of ( ,y,z,(}. For this
purpose we have to express 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 I.
Let a ray of light be sent at time T O from the origin
of the system k along the X-axis towards .c and let it be
reflected from that place at time r l towards the origin
of moving co-ordinates and let it arrive there at time T 2
then we must have
+ T,)=T 1
10 PRINCIPLE OF RELATIVITY
If we now introduce the condition that T is a function
of co-orrdinates, and apply the principle of constancy of
the velocity of light in the stationary system, we have
| JT (0, 0, 0, 0+T 0, 0, 0, {t+ ^.+^L- } \ 1
( c v c + v ) / J
=T(* , o, o, t + I
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 {^y,z,t,},
A similar conception, being applied to they- and r-axis
gives us, when we take into consideration the fact that
light when viewed from the stationary system, is alnays
propogated along those axes with the velocity ^/c* v*,
we have the questions
From these equations it follows that T is a linear func
tion of . and t. From equations (1) we obtain
where a is an unknown function of v.
With the help of these results it is easy to obtain the
magnitudes (,i?,)> tf we express by means of equations
the 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
ON THE ELECTRODYNAMICS OF MOVING BODIES 11
time T=O, if the ray is sent in the direction of increasing
, we have
Now the ray of light moves relative to the origin of Jc
with a velocity c r, measured in the stationary system ;
therefore we have
Substituting these values of t in the equation for ,
we obtain
In an analogous manner, we obtain by considering the
ray of light which moves along the ^-axis,
where - y ,
c ., c
If for .</, we substitute its value x tv, we obtain
/ v.i- \
V c* /
t=4> (). /8 (*-),
where ff= . t , and < (f)= =r-!=r^ = is a function
vi v* Vc*v* P
^
of v.
12 PRINCIPLE OF RELATIVITY
/
If we make no assumption about the initial position
of tlu- moving system and about the null-point of t 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 principle of relativity is reconcilable with the
principle of constant light-velocity.
Atatimer = / = o 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
(> } >/> *)> De a point reached by the wave, we have
a .2 +2/ 2 + ._ C 2^ >
with the aid of our transformation-equations, let us
transform this equation, and we obtain by a simple
calculation,
2 +77 2 + 2 =c z T s .
Therefore the wave is propagated in the moving system
with the same velocity c, and as a spherical wave.* Therefore
we show that the two principles are mutually reconcilable.
In the transformations we have go: an undetermined
function <j> (v), and wo now proceed to find it out.
Let us introduce for this purpose a third co-ordinate
system k 1 , which is set in motion relative to the system X-,
the motion being parallel to the -axis. Let the velocity of
the origin be ( r). At the time t = o, all the initial
co-ordinate points coincide, and for f = ,< =y = z = n, the
time t of the system k = o. We shall say that (./ y ; / )
are the co-ordinates measured in the system k , then by a
* Vide Note 9.
ON THE ELECTRODYNAMICS OF MOVING BODIES 13
two-fold application of the transformation-equations, we
obtain
-v>, etc.
Since the relations between (/, y , z , t }, 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 k are identical.
Let us now turn our attention to the part of the y-axis
between ( = 0, 77 = 0, = o), and (=o, >/ = ], =0). Let
this piece of the y-axis be covered with a rod moving with
the velocity v relative to the system K and perpendicular
to its axis ; the ends of the rod having therefore the
co-ordinates
Therefore the length of the rod measured in the system
K is ~T7~y For the system moving with velocity ( ? ),
we have on grounds of symmetry,
Z Z
14 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., one having a
spherical figure when tested in the stationary system) of
radius R which is at rest relative to the system (K), and
whose centre coincides with the origin of K then the equa
tion of the surface of this sphere, which is moving with a
velocity v relative to K, is
At time t o, the equation is expressed by means of
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
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 x dimension is shortened in the ratio
1: V 1 ; the shortening is the larger, the larger
is v. For 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 OK 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 system gives the time f, and lying
at rest relative to the moving system is capable of giving
the time r ; suppose it to be placed at the origin of the
moving system X-, and to be so arranged that it gives the
time T. How much does the clock gain, when viewed from
the stationary system K ? We have,
V I-
| t 7i \,
7i \, and x=vt,
Therefore the clock loses by an amount ^"2 P er 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 lag behind the clock
p 1
which had been all along at B by an amount $t -$, 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-
01
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 small 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 ;r-axis of the system K) according to
the equation
= Y 1? = M ,T, = 0,
where t/ | and w are constants.
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
w f + v \ c / ""
;= 1 t, y
ON THE ELECTRODYNAMICS OF Mo VI. Mi I .ODIES 17
The law of parallelogram of velocities hold up to the
first order of approximation. We can put
1C
and u = tan" 1
i.e., a is put equal to the angle between the velocities v,
and w. Then we have
u=
r/ o \ / vw sin a \ "1 :
[( + /<+ 2 vw cos a) I j
, //( cos -i
.,
It should be noticed that r- and w outer into the
expression for velocity symmetrically. If iv has the direction
of the v-axis of the moving system,
TT f+"
From this equation, \ve see that by combining two
velocities, each of which is smaller than c, we ojbtain a
velocity which is always smaller than <. If we put v=c %,
and w=c\, where x and A. are each smaller than c,
It is also clear that the velocity of light c cannot be
altered by adding to it a velocity smaller than c. For (his
case,
Fufo Note 12.
18 PRINCIPLE OF RELATIVITY
We have obtained the formula for U for the case when
v and w 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 w, 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
r, we shall have to write,
We see that such a parallel transformation forms a
group.
Wo 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. ELECTKODYNAMICAL PART.
6. Transformation of Maxwell s equations for
Pure Vacuum.
On the nature of lite Electromotive Force caused by motion
in a magnetic field.
The Maxwell-Hertz equations for pure vacuum may
hold for the stationary system K, so that
- -?- [X, Y, /] =
9 JL
9< dy
M N
ON THE KLE( THODVXA.Mli s oi .\[()VlN(i BODIES
and
i a
a a j
3- 6y 3
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 r, we obtain,
[X,
- - N),
and
a
al
X /8(Y- ? N) j8(Z+ -M)
c <
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,
PRINCIPLE OF RELATIVITY
which are defined by their pondermotive reaction, the same
equations hold, . . . i.e. . . .
1 J (X , Y , Z ) =
c o T
a. 6 _a
6f di) 3
L M N
a
9*
y
... (3)
Clearly both the systems of equations (2) and (3)
developed for the system k shall express the same things,
for both of these systems are equivalent to the Maxwell-
Hertzian equations for the system K. Since both the
systems of equations (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 $ (v), which depends only on v, and is independent of
( V> > T ) Hence the relations,
[X , Y , Z ]=V (r) [X, ft (Y- ?X), ft (Z+ ^M)],
[L , M , N ]=* (r) [L, /i (M + ?Z;, (X- Y)].
Then by reasoning similar to that followed in (3),
it can be shown that ^(r) = l.
.-. [X , Y , Z ] = [X, ft (Y- f N), j8 (/+ ; M )j
[ L . W. N ] = [L,
.^Z), (N-- r Y)].
ON THE ELECTRODYNAMICS OF MOVING BODIES 21
For the interpretation of these equations, we make the
following remarks. Let us have a point-mass of electricity
which is of magnitude unity in the stationary system K,
i.e., it exerts a unit force upon a similar quantity placed at
a distance of 1 cm. If this quantity of electricity be at
rest in the stationary system, then the force acting upon it
is equivalent to the vector (X, V, 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), (2), (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 v/c,
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 st-e that in the theory developed the electro
magnetic force plays the psirt 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
It is further clear that the assymetry mentioned in the
introduction which occurs 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 system K, at a great distance from the origin of
co-ordinates, let there be a source of electrodynamic waves,
which is represented with sufficient approximation in a part
of space not containing the origin, by the equations :
I
Z = Z sin * J N=N sin * }
Here (X , Y , Z ) and (L , M , N ) are the vectors
which determine the amplitudes of the train of waves,
(I, m, 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 :
L/=L n sin*
ON THE ELECTRODYNAMICS OF MOVING BODIES
where
, n =
1-
From the equation for w it follows : If an observer moves
with the velocity r 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 4> 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
\
This is Doppler s principle for any velocity.
then the equation takes the simple form
If 4>=
1 +
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 / takes the form
1 - cos 4>
e
24 PRINCIPLE OK RELATIVITY
This equation expresses the law of observation in its
most gen eral form. If <=-, the equation takes the
simple form
We have still to investigate the amplitude of the
waves, which occur in these equations. If A and A be
the amplitudes in the stationary and the moving systems
(either electrical or magnetic), we have
(1 cos 4> I
c )
If 4>=o, this reduces to the simple form
A 8 =A
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 Bays of
Light. Theory of the Radiation-pressure
on a perfect mirror.
Since is equal to the energy of light per unit
07T
volume, we have to regard J 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 /, m, u 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
(.1-
which expands with the velocity oi 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 svstem k, i.e.,
the energy of the light-complex relative to the svstem
A-.
Regarded from the moving system, the spherical
surface becomes an ellipsoidal surface, having, at the time
r = 0, the equation :
If S= volume of the sphere, S = volume of this
ellipsoid, then a simple calculation shows that :
S ft
s "
- OOS <t>
If E denotes the quantity of light energy measured in
the stationary system, E the quantity measured in the
4
26 PRINCIPLE OF RELATIVITY
moving system, which are enclosed by the surfaces
mentioned above, then
E A 2 Q Vl-v*/c*
8^ S
If 4>=0, we have the simple formula :
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 by the magnitudes
A cos $, r (referred to the system K). Regarded from A-,
we have the corresponding magnitudes :
i v
1 cos 4>
A = A
cos 4> 1
cos <*> =
v = v
,
~c~
ON TKE KLECTKOUYNAMUS ()! MOV1N(; BODIES 27
For the reflected light we obtain, when the process
is referred to the system k :
A =A , cos <$" = cos < , v" = v .
By means of a back-transformation to the stationary
system, we obtain K, for the rellected light :
A" = A" - =A
/ 1 v *
V l ~^
COS <t>
Cos4 >" + (1+ -
,_ _ c_ __ \ c
"
1+ ^ cos $" 1 2 r cos < + "*
" C
(H)
/ =/ =!/
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
87r(c cos < )
away from unit surface of the mirror per unit of time is
A ""/8w ( c cos <I>"+r). The difference of these two
expressions is, according to the Energy principle, the
amount of work exerted, by the pressure of light ]>er unit
of time. If we put this eijiial to P.r, where P= pressure
of light, we have
(cos * - |
,,_., Aa V _/
1-
28 PRINCIPLE OF RELATIVITY
As a first approximation, we obtain
P=-2 - <*>* 4>.
8
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 :
M 1 1 ?>L r>Y ft 7 1
,
"
dt
I/ _i.9 Y \ -91 -9-? iaM_8Z_ax
c \ pu * dt) ~ d= a.<- r c a^ a. 1 a-
l( 4-9j?\ = 9^ I _9J J
c \ pu> dt ) Q.e dy
_
c ~dt "a.-
_
Qy 6
where =
oy
, denotes TT times the density
of electricity, and (?t,, ?, tt r ) are the velocity-components
of electricity. If we now suppose that the electrical-
masses are bound unchangeably to small, rigid bodies
UN THK KLLCTKODVNAMKS OF MUVINC BODIES 29
(Ion*, electrons), then these equations form the electromag
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 /, with the aid of the transfor
mation-equations given in -3 and (5, then we obtain
the equations :
1 f ,
C I P
_ ___
67? 9 dr 8* 67;
9Z. l =
Qr J
* T 8r J- 87 dr,
where
- -
Since the vector (/^ // x ) is nothing but the
> T? > 4
velocity of the electrical mass measured in the system A-,
as can be easily seen from the addition-theorem of
velocities in \ so it is hereby shown, that by taking
30 PRINCIPLE OF RELATIVITY
our kineinatical principle as the basis, the electromagnetic
basis of Loreutz 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
n d*i = w * = z rf<- =
Where (a-, y, .?) are the co-ordinates of the electron, and
m is its mass.
Let the electron possess the velocity v 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 THK ELECTRODYNAMICS OF MOVING BODIES 31
the electron is at the origin of co-ordinates, and moves
with the velocity v along the X-axis of the system. It is
clear 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 r.
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*
in the time immediately following the moment, where the
symliols (, rj, , T, X , Y , Z ) refer to the system A\ If we
now fix, that for t .v=y-z=(\ T=g = i) = =Q, then the
equations of transformation ^iven in { (and (5) hold, and we
have :
With the aid of these equations, we can transform the
above equations of motion from the system A- to the system
K, and obtain :
Z 8 ,- _ <_ 1 y d*y e_ I ( y _ r
<// vn J5 (//" ,n f
(A)
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
" =eX=eX
? M] =
and let us" first remark, that <?X , eY , ?Z 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 aeceleration-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
Transversal mass =
Naturally, when other definitions are given of the force
and the acceleration, other numbers are obtained f^r the
* Vide Note 21.
o.N THK ELECTRODYNAMICS or MOVING BODIES 33
mass ; hence we see that we must proceed very carefully
in comparing the different theories of the motion of the
electron.
\Ve 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 charge which may be as small as
possible.
Let us now determine the kinetic energy of the
electron. It 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 /<?X^--. 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 iirst of equations A) holds, we
obtain :
For v = c, *\\ is infinite!] great. A> our former result
shows, vel iritir.- > \<vco!iiig that of light can have no
possibility of existent
In consequence of the arguments mentioned above,
this expression for kinetic energy must also hold for
ponderable masse*.
\Ye can now enumerate the characteristics of the
motion of the electron H\,ulaMe for experimental verifica
tion, which follow from equations A).
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 A m , and
the electric deflexion A,, by applying the law :
A. t>
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
magnetic fields.
2. v 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 :
p=( xd<;=c 2 r l -ii .
J L0-? J
8. 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 :
_d*y ^ ^ r K
< R ,. c ,
H= ""^
. \
These three relations are complete expressions for the
law of motion of the electron according to the above
theory.
ALBRECHT EINSTEIN
[y/ s/turf l>iot/i-a/j//i< <it t/o/e.]
The name of Prof. Albrecht Einstein has now spread far
beyond the narrow pale of scientific investigators owing to
the brilliant confirmation of his predicted deflection of
light-rays by the gravitational field of the sun during the
total solar eclipse of May 29, 1919. But to the serious /
student of science, he has been known from the beginning
of the current century, and many dark problems in physics
has been illuminated with the lustre of his genius, before,
owing to the latest sensation just mentioned, he Hashes 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 University of Prague in Bohemia /
as ausser-ordentliche (or associate) Professor. In 19! 4, *
through the exertions of Prof. M. Planck of the Berlin
University, he was appointed a paid member of the Hoyal
(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 Yan t ilofY, 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 gained him distinction was an investigation on
Brownian Movement. An admirable account will be found
in Perrin s book The Atoms. Starting from Boltzmann s
86 PRINCIPLE OP 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 Sorbonne, 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 correspond to the formula which could be deduced from
the older classical theories of continuous emission and
propagation. In the year 1900, Prof\ Planck of the Berlin
University worked out a formula which was based on the
bold assumption that energy was emitted and absorbed by
the molecules in multiples of the quantity 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 falling upon an
individual atom, liberates electrons according to the energy
equation
liv=- -ftnv^ + A,
where (in, r) 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
88 , PRINCIPLE OF RELATIVITY
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 191."), Einstein made an excursion into Experimental
Physics, and here also, in his characteristic way, he tackled
one of the most fundamental concepts of Physics. It is
well-known that according- to Ampere, the magnetisation
of iron and iron-like bodies, when placed within a coil
carrying an electric current is due to the excitation in the
metal of small electrical circuits. But the conception
though a very fruitful one, long remained without a trace
of experimental proof, though after the discovery of the
electron, it was generally believed that these molecular
currents may be due to the rotational motion of free
electrons within the metal. It is easily seen that if in the
process of magnetisation, a number of electrons be set into
rotatory motion, then these will impart to the metal itself
a turning couple. The experiment is a rather difficult one,
and many physicists tried in vain to observe the effect.
But in collaboration with de Haas, Einstein planned and
successfully carried out this experiment, and proved the
essential correctness of Ampere s views.
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
tried, in a series of papers beginiiin^ from 1011, to extend
it to the case of non-uniform motion. The last paper in
the present collection is a translation of a comprehensive
article which he contributed to the Annalen der Physik in
1916 on this subject, and gives, in his own words, the
Principles of Generalized Relativity. The triumphs of
this theory are now matters of public knowledge.
Einstein is now only -45, and it is to be hoped that
science will continue to be enriched, for a long time to
come, with further achievements of his genius.
INTRODUCTION.
A* the present time, different opinions are being held
about the fundamental equations of Electro-dynamics for
moving bodies. The Hertzian 1 forms must be given up,
for it has appeared that they are contrary to many experi
mental results.
In 1895 H. A. Lorentz- published his theory of optical
and electrical phenomena in moving bodies; this theory
uas 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 however 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 gobd account of
the non-exietence of relative motion of the earth and the
lumiiiiferous " Ather ; it shows that this fact is intimately
connected with the covarianee of the original equations
when certain simultaneous transformations of the space and
time co-ordinates are effected; these transformations have
therefore obtained from II. l iueare ; the name of Lbrentz-
transformations. The covarjance of the-e fundamental
equations, when subjected to the Lomit/-traiisfunnation
is a purely mathematical fart i.e. not based on any physi-
cal considerations; 1 will call this the Theorem of Rela
tivity ; this theorem re-t> essentially on the form of the
1 n,i. No* l. - Note _ . 3 Vide Nute ::.
I KIXCIPT.E OF RELATIVITY
differential equations for the propagation of waves with
the velocity of light.
Now without recognising 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 anyhow possess
this covariance when subjected to a Lorentz-transformation ;
by saying this, we do not indeed express an opinion, but
rather a conviction, and this conviction I may be 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 1 has found out the" Relativity theorem"
and has created the Relativity-postulate as a hypothesis
that electrons and matter suffer contractions in consequence
of their motion according to a certain law.
A. Einstein 2 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-dynamics of moving bodies in the sense
i FcieNote4, * Note 6.
IVTKODUCTION 3
characterized by me. "In the present essay, while formu
lating this principle, I shall obtain the fundamental equa-
\\\\< 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-magnetisation 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 very
* See notes on 8 aud 10.
4 I lUN Oll LK OK RELATIVITY
.
surprising consequences. By laying down tin 1 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 foiras). -,. +
NOTATIONS.
Let a rectangular system (s, //, ~, /,) of reference be
given in space and time. The unit of time shall be chosen
in such a manner with reference to the nnit of length that
the velocity of light in space becomes unity.
Although I would prefer not to change the notations
used by Lorent/, 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 c and the
magnetic force bv MI, so that (E, M, e, ;#) 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,
? . ., instead of operating with (/), I shall operate with (/ /),
where /denotes \/-*-\. If now instead of (./, //, 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 suffixss (1, 2, 3, 1). The
advantage of this method will be, as I expressly emphasize
here, tha^ 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 thr symlbols with indices, such ones as have the suffix
4 only once, denote imaginary quantities, while those
S OTATIOXS O
which have not at all the suffix 4, or have it twice denote
real quantities.
An individual system of values of (.*, y, ~, /) /. e., of
( r i >r -. 3 ^ 4) sna ^ b called a space-time point.
Further let denote the velocity vector of matter, c the
dielectric constant, //. the magnetic permeability, <r the
conductivity of matter, while p denotes the density of
electricity in space, and .v the vector of "Electric Current"
which we shall some across in 7 and 8.
O PUNOIPLS OF RELATIVITY
PART I 2.
THE LIMITING CASE.
The Fundamental Equations for Atlier.
By using the electron theory, Lorentz in his above
mentioned essay traces the Laws of Electro-dynamics of
Ponderable Bodies to still simpler laws. Let us now adhere
to these simpler laws, whereby we require that for the
limitting case =-*, /A=-/,(T = O, they should constitute the
laws for ponderable bodies. In this ideal limitting case
e=l, /*=!, <r=o, E will be equal to e, and M to m. At
every space time point (t, y, ~, f) we shall have the
equations*
(i) Curl m gr=/ >u
(ii) div e= p
(iii) Curlr +-.*? =
(iv) div m = o
I shall now write (.r t x y .r 3 .r 4 ) for (.r, y, z, t) and
0>i>P 2 > Ps> Pi) for
( P u,,pu y ,pu r , ip)
i.e. the components of the convection current pu, and the
electric density multiplied by \/l.
Further 1 shall wriU-
./23<./3 ltfl 2> /I 4>./2 l ./34
for
m,, m v , m., ie,, ie , ie..
i.e., the components of m and ( i.e.} along the three axes;
now *f we take any two indices (h. k) out of the series
(1,2,3,4), /=-/..,
* See note 9.
THK FUNDAMENTAL EQUATIONS FOR ATIIEK 7
Therefore
./ ,1 -. = ~~./2 s> f\ 3 = ~"./s i > y 2 1 = *vi -
/4 1 = ~V I 4> ./ 4 4 = ~/2 4> ./ 4 U = ~~.A 4
Then the three equations comprised in (i), and the
equation (ii) multiplied by i becomes
&i x 8 A ?
8xj 8x 2
*/_4J. 8 /4_2 8 /i3
8x, 8x 2 8x n
On the other hand, the three equations comprised in (iii)
and the (iv) equation multiplied by (/) becomes
^4 , 8/4. /t. .
8x, 8x a 8x 4
(B)
. .
Sx, " 8x 2 8x 3
By means of this method of writing \ve 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).
It is well-known that by writing the equations f) to
iv) in the symbol of vector calculus, we at once set in
evidence an invariance (or rather a (covariance) of the
8 PKI NT I I M. ol liKI.ATIVITY
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 .r/, .r a .r 3
.r/ instead of ^ ( .r 2 .r, s .r 4 , where
; r, = ;r, cos < + .r 2 sn <>, .r 2 = .r, sn< + .r 2
,r 3 =.r 3 .r 4 = 4 , and introduce magnitudes p t , p 2 , p 3 p 4 ,
where p, = p, cos ^> + p L , sin^>, p 2 = p, sin<^> + p 2 cos/>
aud/".,o, ...... / 8 4, where
/ H=/t4 COS A +/M sil1 */ / 2 4 = "/I 4 SU1 ^ +
/ 24 COS ^,./ 3 4=/ : Mi
/,, = -f kh (h I k = 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 suffices (1, 2, 3, 4), the
theorem of Relativity, which was found out by Lorentz,
follows without any calculation at all.
I will denote by i \j/ a purely imaginary magnitude,
and consider the substitution
*i = *i> /= *>>> a = ^3 COR l V r + -"4 H n T ^ (1)
*/ = x 3 sin ity + ,r 4 cos ? i//,
Putting - / tan ^ = T = ^ ^ = lo ^ _
THE FUNDAMENTAL EQUATIONS FOR ARTHEtt
\Vi; shall liavo cos i\f/ =
where i < q < i, and v/l~? a * s always to be taken
with the positive sign.
Let us now write x , = /, ,7/ 2 y i x 3 saz ) x iit (3)
then the substitution 1) takes the form
,,
*= > y=y,z=- >t=- =, (4)
v/l-^ -x/l-f*
the coofficients being essentially real.
If now in the above-mentioned rotation round the
Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and
< by t ^, we at once perceive that simultaneously, new
magnitudes p j, p 2 , p 3 , p 4 , where
(P / i=Pn P =P2> P s=Ps cos e t/ + p 4 sin t^r, p 4 =
P 3 sin ty + p 4 cos t^r),
and/ 12 .../ 34 , where
/ 4 i=/4i cos ty +/ 13 sin 7>,/ 13 = -/ 41 sin V +/ t g
COS f A, / 3 4 =/ 3 4 , / 3 2 =/ 3 o COS ^ + / 4 2 in t ^, / 4 2 =
~/32 Sin ^ + /42 COS ^ /I 2 =/18i /** = -/*t
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 taken, and .i/ y z t
be takes as a new frame of reference, then we shall have
,KX r-qrn.+l-]
(5) P=P - , P =P
p.=pu.,
10 PRINCIPLE OF RELATIVITY
e, qm, , , oe.+m,
(6) e, =
q* VI
Then we have for these newly introduced vectors u , e ,
m (with components u x , u r , u/ ; e x , <?/, e/ m x , 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,qm y , e v +qm, are components
of the vector e+ [ym], where v is a vector in the direction
of the positive Z-axis, and i v \=q, and [vm~\ is the vector
product of v and m ; similarly qe t + m,,m f +ge, are the
components of the vector m[ve].
The equations 6) and 7), as they stand in pairs, can be
expressed as.
e, + t mV = ( e . +") cos i\ff + (e y +tm y ) sin i\[/,
ey + im y = (e.+t m,) sin i\j/ + (e y +m y ) cos ty,
e, -\-irn , ~e t -}-im t .
If < denotes any other real angle, we can form the
following combinations :
t mY) cos. <t>+(e s/ " + im ]/ ) sin </>
cos. (^ + ^) + (e y +tm y ) sin (<f> + i\j/),
) sin </> + (e / + t m /) cos. <^>
sin (<t> + i^) + (e if +im, ) cos. (<^ + ).
SPECIAL LORENTZ TRANSFORMATION.
The rl 3 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 11
about this last axis. So we came to a more general
law :
Let v be a vector with the components v x , v y , v,,
and let \ v \ =q<l. By we shall denote any vector
which is perpendicular to v, and by r v , r-^ we shall denote
components of r in direction of ~y and v.
Instead of (.r, y, z, t), new magnetudes (x y 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, 2) 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 y, we have
T^r
and for the perpendicular direction v,
(11) r T =r7
ar.+t
and further (12) t = / .
v I q*
The notations (/-?, r r ) are to be understood in the sense
that with the directions v, and every direction y~ perpendi
cular to v in the system (.r, y, z) are always associated
the directions with the same direction cosines in the system
(* y, O-
A transformation which is accomplished by means of
(10), (11), (12) with the condition Q<q<\ 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 u , e , pt , in the system
(x y z } are so defined that,
12 PRINCIPLE OF RELATIVITY
further
(15) (e + V) , = (e + m) i [u, (e + /m)] ,. .
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
(16) rc
q 2 q
Now we shall make a very important observation
about the vectors u and u . We can again introduce
the indices 1, 2, 3, 4, so that we write (.* , , # 2 , v z , #/)
instead of (.c , t/ , ; , it ) and p/, p 2 , p 3 , p^ instead of
(p 1 u , , p u y , p u , , ip.
Like the rotation round the Z-axis, the transformation
(4), and more generally the transformations (10), (11),
(12), are also linear transformations with the determiuant
+ 1, so that
(17) .T 1 a +.r 2 2 +.t s a +. 4 2 i. e. x * + y*+ z *-t\
is transformed into
*!" +. . ," + . s + O 1 e - x *+y *+z 2 -t *.
On the basis of the equations (13), (14), we shall have
(p l *+pS+P>*+ P <*)=P*(l-u t >,-u v \-u,*,)=p*(l-u*)
transformed into p s (l M S ) or in other words,
(18) p A/l^T*
is an invariant in a Lorentz-transformation.
If we divide (p l , p s , p s , p 4 ) by this magnitude, we obtain
the four values f( 1 , w a , o>,, o> 4 ) = . . (u,, u y , ,, i)
VI _ M *
so that a) 1 2 +u) a *+w 3 1 +to 4 f = 1.
It is apparent that these four values, are determined
by the vector u and inversely the vector // of magnitude
SPECIAL L011ENTZ TRANSFORMATION 13
<i follows from the -1 values o^, oj a , o> 3 , w. t ; where
(<> 01,, wj are real, *w 4 real and positive and condition
(ID) is fulfilled.
The meaning of (<>!,<,, o 3 , <>,) here is, that they are
the ratios of <Ar lf ilr,, d , rfx 4 to
(20) V_(d.,: l
The differentials denoting the displaeements of matter
occupying the spacetimc point (.c lf . ^ 3 , .i-J to the
adjacent space-time point.
After the Lorentz-transfornation is accomplished the
vococity of matter in the ne\v system of reference for the
same space-time point (.</ y -J t ) is the vector n with the
dx <ly dz dl
ratios -^-,, -J,, ^p, ^p, as components.
Now it is quite apparent that the system of values
.( =
is transformed into the values
*i = "i i -I a^Wi . 3 = W 3 5 * 4 = o) 4
in virtue of the Lorentz-transformation (10), (11), (12).
The dashed system has got the same meaning for the
velocity // after the transformation as the first system
of values lias got for n before transformation.
If in particular the vector v of the special Lorentz-
transformation be equal to the velecity vector u of matter at
tin- space-time point (.e,, a?,, r 3 , x 4 ) then it follows out of
(10\ (11), (12) that
<o 1 =:o, 0) a = 0, ti) 3 = 0, o> 4 =l
Under these circumstances therefore, the corresponding
space-time point has the velocity v = u after the trans
formation, it is as if we transform to rest. We may call
the invariant f > ^/ \ M as the re.-t-dcn-ity of Mlectricity.*
* See Note.
PRINCIPLE OF RELATIVITY
5. SPACE-TIME VECTORS.
Of the hf, and 2nd kind.
If we take the principal result of the Lorentz transfor
mation together with the fact that ;he 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 easi.ly comprehensible, it may be more convenient to
define a series of expressions, for the purpose of expressing
the ideas in a concise form, while on the other hand
I shall adhere to the practice of using complex magni
tudes, in order to render certain symmetries quite evident.
Let us take a linear homogeneous transformation,
a n a , 55 a t 3 a ,
a 21 22 23 a 2
a 31 33 a 33 a 3
_ __ L a 41 42 a 43 4
the Determinant of the matrix is +1, all co-efficients with
out the index 4 occurring once are real, while a 4 ,, a 42 ,
a 4 3 , are purely imaginary, but a 4 4 is real and >o, and
a^ 2 + # 2 2 + 3 2 +^ 4 2 transforms into ^, 2 +# 2 2 + C 3 2
+ # 4 2 . The operation shall be called a general Lorentz
transformation.
If we put x l = ,c t ,r 2 =// , . - 3 = z , ,c 4 = it , then
immediately there occurs a homogeneous linear transfor
mation of (*, y, z, f) to (r } y , z , t ) with essentially real
co-efficients, whereby the aggregrate .c 2 y* ; 2 +t 2
transforms into v 2 y ~* z - +t *, and to every such
system of valut^ , y, :, ./ with a positive I, for which
this aggregate o, there always corresponds a positive t ;
This notation, whirli is due to Dr. C. E. Cullis of the Calcutta
University, has been used throughout instead of Minkowski s notation,
SPACE-TIME VECTORS
15
this last is quite evident from the continuity of the
aggregate .r, y, z, t.
The last vertical column of co-efficients has to fulfil,
the condition :!2) a l 4 2 +a. J4 8 + tf 34 a -f-a 44 2 = l.
If ,i
= fl 34 = o, then 44 = i, and the Lorentz
transformation reduces to a simple rotation of the spatial
co-ordinate system round the world-point.
If l4 , a s4 , S4 are not all zoro, and if we put
i4 : fl 4 : 34 : 44 = v , : v -v,:i
Ou the other hand, with every set of value of
a i4> rt 4> a 34 fl 44 which in this way fulfil the condition
I l) with real values of v,, v v , v,, we can construct the
special Lorentz transformation (.16) with (a^ 4 , 24 , 34 ,a 44 )
as the last vertical column, and then every Lorentz-
transformatiou with the same last vertical column
(a l4 , a 24 , a 34 , a 44 ) can be supposed to be composed of
the special Lor(?ntz-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 Pi,p 8 ,p s , p 4 )
with the condition that in case of a Lorentz-transformation
it is to be replaced by the set p, , p 2 , p 3 , p 4 ), where
thes; are tlii! v.ilue; oP . /, c./, .-., , .e/), obtaiael by
sab^tituting (p,, p.}, p.,, p ) for (-,, x j} .- 3 , .- 4 ) in the
expression (il).
Besides the time-space vector of the 1st kind (x lt o; 2 ,
*3> - 4) we shall also make use of another space-time vector
of the first kind (y,, //.,,. // 3 , y 4 ), and let us form the linear
combination
-* y.)
* y s )
16
PRINCIPLE OP RELATIVITY
with six coefficients /.,/, v . Let us remark that in the
vectorial method of writing, this can be constructed out of
the four vectors.
Ci,as*,#a ; yi,y.,y s ;/ 3 ./si./n ; /i*. A* A* and
the constants , 4 and y 4 , at the same time it is symmetrical
with regard the indices (1, 2, 3, 4).
If we subject (aj lf .r,, , 3 , ,r 4 ) and (y t , y a , y a , y 4 ) simul
taneously to the Lorentz transformation f21), the combina
tion (23) is changed to.
(24) / (,-, y, -.r, y.) +/ sl (a, <//-../ y s ) + /, ,
whei e the coefficients / 83 , / 3 /, / x 2 , / 14 , /, 4 , / 84 , depend
solely on (/ 8S / 94 ) and the coefficients a lt ...a +4 .
We shall define a space-time Vector of the 2nd kind
as a system of six-magnitudes / 2 3 , t / s t ...... / 34 , with the
condition that when subjected to a Lorentz transformation,
it is changed to a new system y^./ ...... /" 34J ... 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 ,--, i/, z, it (space co-ordinates, and time if) is sub
jected to a Lorentz transformation, and at the same time
(/)/*,, pity, pit ,, ip] (convection-current, and charge density
pi) is transformed as a space time vector of the I st kind,
further (m x , m f , )..-, * ,, ie,, if.) (magnetic force,
and electric induction x ( /) is transformed as a space
time vector of the :!nd kind, then the system of equations
(1), (II), and the system of c |inti > is (III), (IV) trans
forms into essentially corresponding relations between the
corresponding magnitudes newly introduced into the
system.
SPECIAL LORENTZ TRANSFORMATION 17
These facts can be more concisely expressed iu these
words : the system of equations (I, and II) as well as the
system of equations (III) (IV) are covariant in all eases
of Lorentz-transformation, where (pu, ip) is to be trans
formed as a space time vector of the 1st kind, (ni ie) is
to be treated as a vector of the 2nd kind, or more
significantly,
(pit, ip) is a space time vector of the 1st kind, (m 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 in variants for such a vector when
subjected to a group of Lorentz transformation.
A space-time vector of the second kind (m ie), where
(m, and e) are real magnitudes, may be called singular,
when the scalar square (in ie)*= 0) ie m s e t =o i and at
the >;une time (m e)=o, ie the vector wand e are equal and
perpendicular to each other; when such is the case, these
two properties remain conserved for the space-time vector
oi 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 \me\ coincides with
the Z-axis, i.e. m jt) = o, e,-=o. Then
Therefore (i- t +i m,, )j(e t +i c x ) is different from +i,
and we can therefore define a complex argument <j> + i$)
in such a manner that
=
+t m,
Vide Note.
18 PRINCIPLE OF RELATIVITY
If then, by referring back to equations (9), we carry out
the transformation (1) through the angle <A, and a subsequent
rotation round the Z-axis through the angle <, we perform a
Lorentz-transformation at the end of which m v = o, e v =o,
and therefore m and e shall both coincide with the new
Z-axis. Then by means of the invariants m" 2 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 OF 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, t) for space time points
(events) be somehow known. Now if a space point A
(x,,y , O at the time t a be compared with a space
point P (.f, y, z) at the time /, and if the difference of
time tt t , (let t> I.) be less than the length A P i.e. less
than the time required for the propogation of light from
* Just as beings which aro confined within a narrow region
surrounding a point on a shperical surface, may fall into the error that
a sphere is a geometric figure in which ouo diameter is particularly
distinguished from the rest.
CONCEPT OP TIME 19
A to P, and if q= - < 1, then by a special Lorentz
A i
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 = 0, two
different space-points A, B, or three space-points (A, B, C)
which are not in the same space-line, and compare
therewith a space point P, which is outside the line A B,
or the plane A B C, at another time t, and let the time
difference t t t (t > t ] be less than the time which light
requires for propogation from the line A B, or the plane
A B C) 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, I.), [B, t ), (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 t,, 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
of the so-called non-Euclidean Geometry, there can be no
difficulty in adopting this concept of time to the application
of the Lorentz-transformation. The paper of Kinstein which
has been cited in the Introduction, has succeeded to some
extent in presenting the nature of the transformation
from the physical standpoint.
20 PRINCIPLE OP 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- = 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 m, 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 u, 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 (.<-, y, r, (,], lead us to
a knowledge of other magnitude as functions of .,-, 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
*, y, *, 0-
For the case of bodies at rest, i.e. when u (x, y, :, t)
= o the theories of Maxwell (Heaviside, Hertz) and
FUNDAMENTAL EQUATIONS FOR BODIES AT REST 21
Loreoti lead to the same fundamental equations. They
are ;
(1) The Differential Equations: which contain no
constant referring to matter :
(i) Curl m ~- - C, (iV) div e =>.
(in) Curl E + - = o, (ir) 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 E, M = urn, C = <rE.
where = dielectric constant, /j. = magnetic permeability,
a- = the conductivity of matter, all given as function of
c j > "> ^j * is hre 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 *, s t , 5 3 , * 4 for C,, C y , C, V _ 1 p.
further /,,,/,,,/, ,,/,,,/,,/,.
form,,m,,m, t (e., e y , c. ),
andF ls ,F 31 ,F 19 ,F 14 ,F,.,F S4
for M., M,, M,, - i (E,, E y , E.)
lastly we shall have the relation f kk = ,./**, F kll = F )t k,
(tin- letter/, F shall denote the field, * the (i.e. current).
22 PRINCIPLE OF RELATIVITY
Then the fundamental Equations can be written as
a/ sl
3/t,
3.c 4
and the equations (3) and (4), are
3F,. .
a.., a 3
=
8F, t .
3*, a, 4
+ ^^- 8 +
3F,,
=
=
= o
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 " is zero, for a system
* Einzelne stelle der Materie.
THE FUNDAMENTAL EQUATIONS 23
(j-, y, :, /) the neighbourhood may be supposed to be
in motion in any possible manner, then for the space-
time point x, y, z, t t the samo relations (A) (B) (V) which
hold in the case when all matter is at rest, shall 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 propogation of light.*
The fundamental equations are of such a kind that
when (f, y, z t it) are subjected to a Lorentz transformation
and thereby (m ie) and (MiE) 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
magnitudes.
Shortly I can signify the third axiom as :
(/#, 1<?), and (W, iE] are space-time vectors of the
second kind, (C, ip) is a space-time vector of the first kind.
This axiom I call the Principle of Relativity.
In fact thes^ 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 | it \ is <1 at any space-time point. In
consequence, we can always write, instead of the vector u,
the following set of four allied quantities
11 - - u - u
x/n^T
11
* Vido Note.
24 PRINCIPLE OF RELATIVITY
with the relation
(27) Wl 2+ W2 2+ w ,2+ W4 2= |
From what has been said at the end of 4, it is clear
that in the case of a Lorent/.-transformation, this set
behaves like a space-time vector of the 1st kind.
Let us now iix our attention on a certain point (r, 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 (A), (5) (V) of 7. It u t o, then there exists
according to 16), in case | n \ <1, a special Lorentz-trans-
formation, whose vector v is equal to this vector u (.r, y, z,
t), and we pass on to a new system of reference (? y z I )
in accordance with this transformation. Therefore for
the space-time point considered, there arises as in 4,
the new values 28) 0)^ = 0, o/ 2 = 0, u/ 3 :=0, u/ 4 =f,
therefore the new velocity vector o/ = 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 (.r / z I) involves the newly introduced magnitude
(n p , C , e , m , E , M } and their differential quotients
with respect to (x , y , z , t } in the same manner as the
original equations for the point (x, y, z, t). But according
to the first axiom, when u = o, these equations must be
exactly equivalent to
(1) the differential equations (^ ), (# ), which are
obtained from the equations (.-/), (/?) by simply dashing
the symbols in (./) and (B).
(2) and the equations
(V ) <? = cA", M = pm , C = a/-
where e, ^, o- are the dielectric constant, magnetic permea
bility, and conductivity for the system (./ // : t } i.e. in
the space-time point (t y, z t) of matter.
Illl. II MM \II.\T\I. Kor \TIOXS 20
Now let us return, by means of the- reciprocal Lorentz-
trausl onnation to (lie original variables (.-, y, :, /), and the
magnitudes (/ , p, C, <, ///, A , I/) and (he equations, which
we then obtain from the last mentioned, will be the funda
mental equations sought by us for the moving bodies.
Now from 4, and 0, it is to be seen that the equa
tions .-/), as well as the equations //) are covariant for a
Lorentz-transformation, />. the equations, which we obtain
backwards from A } B }, must be exactly of the same form
am the equations ./) and /> ), as we take them for bodies
at rest. \Ve have therefore as the first result :
The differential equations expressing the fundamental
equations of electrodynamics for moving bodie?, when
written in p and the vectors C, >% ///, K, M, are exactly of
the same form as the equations for moving bodies. The
velocity of matter does not enter in these equations. In
the rectorial wav of writing, we have
curl ni = C,, II) div f=
n i I
III VMH-! K -f 9 M = o lV)divM=o
/ U /
The velocity of matter occut> only in the anxilliary
equations which characterise the influence of matter mi the
basis of their characteristic constants e, /<, <r. Let us now
transform these aiixilliary equations \ ) into the original
co-ordinates ( , //,:, and /.)
Acconlin^ to formula 15) in ^ I, the (-omponent of >
in the direction of tin- vector / i- the same us llial n|
(?+[H tn ]), the component of /;/ is th" same as that of
/ [// r], but for the perpendicular direction r>, the com
ponents of f , ni are the same as those of (+[> ///i)and (///
[UP], multiplied by / . On the other hand ! /
26 IMUXC II I.K 01 i; I.I \livm
and M shall stand to E + ["M,], and M [Kj in the
same relation us < aud /// to <! +[>#], and m (//>).
From the relation c >/ = E , the following- equations i uliow
(C) r+[ W //,]=c(E+[//M]),
and from the relation M =//. /, we have
(D) M-[ttE]=/t(/y/-[/^]),
For the components in the directions perpendicular
to / , and to each other, the equations are to be multiplied
by v fll^:
Then the following equations follow from the transfer-
mation equations (12), 10), (11) in $ 4, when we replace
q, r t , r- r -, t, r ,., r r , ( by | n \ , C,,, C,, p, C . C v, p
,- C + , C M -
In cunsideration of the manner in which o- enters into
these relations, it will be convenient to call the vector
C p / with the components C, p | / | in the direction of
/ , and C ,, in the directions T> [lerpendicular to H the
"Convection current. This last vanishes for <r=o.
\\ c remark that for =1, //-=! the ecjuations r = E ,
/ = M immediately lead to the equations / E, //i = M
bv means of a rcciprofid Liu cnt/.-transformatiun with it
as vector; and for o-=o, the equation C = o leads to C=p i ;
that the fundamental equations of Ather discussed in
:i becomes in fact the limitting case of the equations
obtained here with = 1, /* = !, ^o.
FUNDAMENTAL EQUATIONS IN LOKKXT2 THKORY 27
1). THE FUNDAMENTAL EQUATIONS IN
LORENTZ S THEORY.
Let us now see how far the fundamental equations
assumed by Lorentz correspond to the Relativity postulate,
as defined in sjS. 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 200, Eq" XXX , formula (11) on
page 78 of the same (part).
(Ul ") Curl (H-[//E]) = J+ ) + / divD
-curl OD].
(1") div !) = /.
(IV") curl E = -^ , Div 13 = (V)
Then for moving non-magnetised bodies, Lorentz pubs
(page :2:!3, :i) ^= 1, B = H, and in addition to that takes
account of the occurrence of the di-eleetric constant e, and
conductivity o- according to equations
Lort iit/ s 1-], 1), II ;irt- here denoted i>v E, M, r, m
while J denotes the :-onduction cuiTeiit.
The three last equations which have been just cited
here coincide with eq" (II), (III), (IV), the first equation
would be, if J i> identified with C, / /; (the current being
y.ero for (/ = (),
(2U) Curl i II -(.,K; j =( f- > -curl[l)i,
28 1 itfxcii T.K or i! i;r. VTFVITY
and thus comes out to he in a different form than (1) here.
Therefore for magnetised bodies, Lorentz s equations do not
correspond to the Relnti vity Principle.
On the other hand, tlie form corresponding to the
relativity principle, for the condition of non -magnetisation
is to be taken out of (D) in ^S, with i>.= \, not as B = H,
as Lorentz takes, but as (30) B [>!)] = H [D]
(M [E]=/;/ [<*] Now by putting 11 = B, the d.flW-
ential equation ( - ) is transformed into the same form as
e<j" (1) liere when ;// [//,] =M [K]. Pherefore it so
happens that by a compensation of two contradictions to
the relativity principle, the differential equations of Lorentz
for moving non-magnetised bodies at last agree with the
relat.ivit v postulate.
If \ve make use of (- HI) for non-magnetic bodies, and
pat accordingly Il = IJ-r-[", (D E)], then in conse(|iience
of (C) in 8,
(e _]) ( K + r,,,B;) = 0_E+r,,. [;/,D E]],
is. for the direction of n
and for a perpendicular direct ion fi,
i.r. it coincides with L^rent/ s assumjition, if we neglect
/ - 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 ; eomp. (K) ^ 8] that the components
of J,, ; -I ftW eijual to the components of <r(E-f [n B])
multiplied by ^fZ^i or ^j^TT respectively.
i I \UAMKXT\T. i;OI ATIONS <>K K. COHEN* 29
vjlO. I YNDAMKNTAL EQUATIONS OF E. COHN.
K. Cohn assumes the following fundamental equations.
(31) Curl (M-f E]) = ^ + u div. E + .T
Curl [E (//. M)]= _ + n div. M.
ill
(32) J = , r E, = cE-|> M], M = /*(y//+[> E.])
where I 1 } M are the electric and magnetic field intensities
(forces), E, M are the electric and magnetic polarisation
(induction). The equations also permit the existence of
true magnetism ; if we do not take into account this
eoiHiderat ion, div. M. is to be put = ft.
An objection to this svstem of equations is that
according to these, for = 1, /i =1, the vectors force and
induction do not coincide. If in the equations, we conceive
E and M and not E-(t*. M), and M+[T E] as electric
and magnetic forces, and with a glance to this we
substitute for E, M, E, M, div. E, the symbols c, M, E
-fCU M], >// [/ <"], p, then the differential equations
transform to our equations, and the conditions (32)
transform into
J = r(E+0 M])
,+ [*, (*-[**])] =(E+[M])
then in fart the equations of (John become the same as
those required by the relativity principle, if errors of the
order 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 auxiliary
conditions are
. 50 I lil.N l II I.K OF KM I,
11. TYPICAL HKPKKSKXTATIONS OF THK
EUND A MENTAL EQUATIONS.
In the statement of the fundamental equations, our
leading idea had been that they should retain a eovariance
of form, when subjected to a group of Lorentz-trans-
formations. Xo\v we have to deal with ponoeromotive
reactions and energy in the electro-magnetic field. Here
from the very first there can be no doubt that, the
settlement of this question is in some way connected with
the simplest forms which can be given to (he fundamental
equations, satisfying the conditions of covarianee. In
order to arrive at such forms, I shall first of all puf the
fundamental equations in a typical form which brings out
clearly their covariaucein case of a Lorentz-transformation.
Here I am using a method of calculation, which enables us
to deal in a simple manner with the space-time vectors of
the 1st, and 2ud kind, and of which the rules, as far as
required are given below.
A system of magnitudes ii kk formed into the matrix
arranged in /; horizontal rows, and // vertical columns is
called a ft xq series-matrix, and will be denoted by the
letter A.
If all the quantities </,.,< are multiplied by (\ the
resulting matrix will be denoted by CA.
It the roles of the horizontal rows and vertical columns
be interchanged, we obtain a yx;; series matrix, which
AI i;i,i i;i -i,.\ i > i lOHS
ill
will In- known ;i> the transposed matrix oi A, aiul will be
!fiu>1etl 1 V A.
It we have a second jj x ij series matrix B,
tl-.en A + B shall denote the j> x y series matrix whose
members are a,, k -\-bhk.
2 If we have two matrices
A=|a M a
ki a
where the number of horizontal rows of B, is equal to the
number of vertical columns of A, then by AB, the product
of the matrics A and B, will be denoted the matrix
. i
,;;f I
these elements bein^r formed l>y eombinat ion of the
hori/onlal rows of A with the vertical columns of B. For
such a point, the associative law (AB) S = A(HS) holds,
where S is a third matrix which has -ot u many hori/ontal
rows as B (or A l>) has -^ot vortical columns.
For the transposed matrix of C=BA, we have C = BA
.K or RELAIIVITV
3". We shall have principally to deal with matrir-rs
with at most four vertical columns and for horizontal
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.
(:H) e,, e I2 e 18 c 14 = I 1 o
0100
00 1 01
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 may denote by A 1 so that A -1 A = 1
A matrix
./i/s> y.,,
/, , /, a /, ,
in which the elements fulfil the relation //, < = /,is
called an alternating matrix. Those relations say that
the transposed matrix f = ./. Then by /* will be
the il/ df, alternating matrix
(35)
; /,,
TYPICAL i;i:i iM>i:\Tm<)N.s
33
/>. We shall have a 4x4 series matrix in which all the
elements except those on the diagonal from left up to
right clown are zero, and the elements in this diagonal
agree with each other, and are each equal to the above
mentioned combination in (36).
The determinant of /is therefore the square of the
|
combination, by Det /we shall denote]the expression
4. A linear transformation
which is accomplished by the matrix
A=| a,,. a, 2 , a ls , a, +
n, 2> "a si a *4
will be denoted as the transformation A
By the transformation A, the expression
.,-}- ,+ ,i4- l is changed into the quadrat ie
for /// ^ i/,/ / ."/,
where a Ai =a lt ,*+/. a sA +u sA a 3l ++* 4lt .
are the members of Jl 1-x ! series matrix which is the
produrt of A A, the transposed matrix of A into A. li \\\
the tranrformation, the rxpn-sion i- cliaiiged to
inu- t h:ue A A = 1.
PRINCIPLE OF RELATIVITY
A has to correspond to the following relition, if trans
formation (38) is to be a Lorentz-transformation. For the
determinant of A) it follows out of (39) that, (Det A) 2 =
1, or Det A= + 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 involving the index 4 once in
the subscript are purely imaginary, the other co-efficients
are real, and /* 44 >0.
5. A space time vector of the first kind* which s
represented by the 1x4 series matrix,
(41) *= | *, * s *,, * 4 |
is to be replaced by ft A in case of a Lorentz transformation
A. i.e. -f = | ?/ ?.> *../ */ | = | f> i . 2 *s % | A;
A space-time vector of the 2nd kindt with components /* 2 3 . . .
/ 34 shall be represented by the alternating matrix
/,, / 42 -A, o
and is to be replaced by A" 1 / A in case of a Lorentx.
tr;insformation [see the rules in 5 ( 23) (24)]. Therefore
referring to the expression (37), we have the identity
Det^ (A/A) = DetA. Det*/ Therefore Det V be
comes an invariant in the case of a Lorentz transformation
[eeeq. (:Jfi) Sec. 5].
* riiJr nut.. 13.
t T l f/c note 14.
M en \i fcBPJUU I-.N r.vnoNs 35
Looking back to (-M), we have i or the dual matrix
(A./ 1 *A)(A- /A) = A- 1 /VA-Det" /. A- 1 A = Dof i /
from which it is to he seen that the dual matrix/* behaves
exactly like the primary matrix/, and is therefore a space
time vector of the II kind; /* is therefore kiiown as the
dual space-time vector of/ with components (/ , 4>/ 3 4,/34>)
OWso/ij).
0.* If w and ,v are two space-time rectors of the 1st kind
then by w ,1 (as well as by -v //> ) will be understood tin-
combination (1-3) w i s v +10,, s.j-Mfg iV 3 + #> 4 ,v,.
In case of a Loreiitz transformation A, since (wA) (A-v)
= 10 s, this expression is invariant. If tr x = o, then w
and s are perpendicular to each other.
Two space-time rectors of the lirst kind (10, .v) gives us
a 2 x 1 .series matrix
W, IV.) 10., 10:
Then it follows immediately that the system of six
magnitudes (14) /._, .v 3 to.j *._,, w 3 -v, w l -v 3 , 10 , s.,io* s,,
behaves in case of a Lorentz-transformation as a space-time
vector of the II kind. The vector of the second kind with
the components (It) are denoted by [w, .v]. We see easily
that Del" [?, v]=o. The dual vector of ["V s ] shall be
written as [ / , ,v].*
If //; is a space-time vector of the 1st kind, / of the
.-eeond kind, / / si^nilies a 1 x 1 series matrix. In case
of a Ijoivnt/.-traiisfonnation A, f is changed into tr = icA,
/ into /" = \~ l / A, therefore / /" become> =(/ -AA~ /
A) = / / \ /. . " / is transformed as a >pace-time vector of
36 PRINCIPLE OF RELATIVITY
the 1st kind.* We can verify, when w is a space-time vector
of the 1st kind, / of the 2nd kind, the important identity
(45) O, wf } + O, /*] * = (m w ) 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 (a l =o, o; a =:o. w 3 =o, ta. t =i,
</= I */*! */*, *!/4S, I ; </* = I #3 */! #21 > I
|> eo/J =0, o, o, /., t , /, 2 , / t , ; [ w a>/*]* = o, o, o, /, 2 ,f i; >J. 2l .
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 LoreNitz
transformation, and is homogeneous in (wj, w a , o) 3 . w t ).
After these preparatory works let us engage ourselves
with the equations (C,) (D,) (E) by means which the constants
e /x, a will be introduced.
Instead cf the space vector u, the velocity of matter, we
shall introduce the space-time vector of the first kind to with
the components.
(40) where w 1 2 + w 2 a + co 3 2 + w
and iw. t >0.
By F and / shall be understood (lie space time vectors
of tin- second kind M i\ ]. n> ic.
In 4> = wF. \vu have a space time vector of the first kind
with components
3 ., . ln>
4jl + w .F^
I idc note 15.
I-YPICM. I;H I;I;SK\TATIONS 37
The first three quantities (<,, < a , <,) are the components
of the space-vector IrL+JJi JH .
^1-1T~
and further < 4 = -IfiLjEL .
^1 w"
Because F is an alternating matrix,
(49) O}* = w 1 ^ 1 +w a 4> 2 -|-<jJ 3 4> 3 +oj 4 4> t r=o.
i.e. & is perpendicular to the vector w ; we can also
write 4> t =i [w x 4>j +w y 4> a +w.4> 3 ].
I shall call the space-time vector <I> of the first kind as
the Electric Rest Force*
Relations analogous to those holding between wF,
E, M, Uphold amongst w/, v, m, u, and in particular otf
is normal to w. The relation (C) can be written as
{ C } o)/=e<oF.
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
^ = iw/*, whose components are
*!=-* ( J*,* +"3/4, +*/) 1
* a = - ( ",/,.,+ !/* *4/)
*.=-(l/.*+Ji, +",/ 12 ) I
** = *(*!/ +/! +*|/g, ) J
Of these, thu first three ! ,. >!/,. J/ S) . irc the .i\ //. \
i (iinpo;ients of the space-vector 51) "i ( " )
and further (52; f t = / nil/
/
Fide note 16.
38 ri;i\( ii i.i. 01 KKI.VIIMTY
Among these there is tlie relation
(53) t 1 >V (J > l ty i + M2 ty. 2 + 0)^.1+ c> 4 * 4 =o
which can also be written as 4 4 =:i (u x fy l + " y ^a ~J""^ s)
The vector * is perpendicular to ; we can call it the
\Iay>ietic rest-force.
Relations analogous to Ihese hold amon^ the quantities
iwF*, M, E, u and Relation (D) can be replaced by the
formula
{ D } -o,F*= /4w /*.
We can use the relations (C) and (D) to calculate
F and f from <J> and * we have
W F - _ $. up* = _ i ^i/. ,,,/ = _ <i,, ,; *--/*.
and applying the relation (45) and (46), we have
F= [CD. *J + z>[o. *]* 55)
/= e[<o. *] + *[.*]* 5G;.
i.e. F 12 =( Wl $ 1 -w i! * 1 ) + .>[ w /l t -o n * 3 ] ! etc.
/ la =e(w I * a -w a * l ) + A [o) 3 * 4 -w 4 * 3 ]. etc.
Let us uo\v consider the space-time vector of the
second kind [$ *], with the components
* 8 * 3 -* 3 * 2 , *,* 1 -* 1 * 3 , ^^j,-*,*, -,
4> i xi/ i _4> 4 ^/ l , <J> 2 \J/ 4 _4)^\I> 2 . cj> 3 l/ + <j> t \i/ 3 J
Then the corresponding space-time vector of the tir-t
kind w[*, *] vanishes identically owing to c.(juation> ;>)
and 33)
for
Let us now take the vector of the 1st kind
the
. etc.
TYPICAL i:r.i i:i-si;.\T \TI<>\
Then In applying 1 11 (1 (^-"O wo l ^
(58) [*.*] = i [u>n]
i>. ^j^, <P. 2 >I 1 =(w r ,O + o> + $2.,) etc.
Tho vector fi fulfils the relation
(which we can write as n + =t (to^Qj +oj y Q 2 +a> 2 Q 3 )
and n is also normal (o o>. In case w=o.
we have * 4 =o, *. v =o, n 4 =o. and
*i * a *
I shall call fi. which is a space-time vector 1st kind the
Rest- Ray.
As for the relation E), which introduces the conductivity a-
we have u>S=: ( w i*i + W a s 2 "r^s** + W 4- S ..)
This expression reives us the rest-density of electricity
(see 8 and 4).
Then rl)=s+(t,w)
represents a space-time vector of the 1st kind, which since
axi>= 1, is normal to ,>, and which I may call the rest-
cnrrent. Let us now conceive of the lirst three component
uf this vector as the (./ // c) co-ordinates of the -pace-
vector, then the component in the direction ui // is
C - I " I P = c ~ I u I P - 1 "
Vll^ A/1 l-"
and the comiionent in a per])endicular direction is ( ., J u .
This space-\ector i- conneeted with the space-veefur
.1 C fiit, which we denoted in ^ as the
current.
40
PRIN r C[IT,K OK KKI.ATIVITY
Now by comparing with <f>= --o>F, the relation (li!) cm
be brought into the form
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.
12. THE DIFFERENTIAL OPERATOR LOR.
A 4x4 series matrix 62) S= S,, S 12 S 1S S 14 = | S, A |
S S1 S a , S 2S S, 4
S 41 S 42 S 4;t S 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 alternating matrix f, which corresponds to the
space-time vector of the II kind,
2) the product f F of two such matrices, for by a transfor
mation A, it is replaced by (A- 1 /A-A~ I FA)=A~ I /F A,
M) further when ( MI . ui s . cu a , co 4 ) and (n i; O 2 , s , U 4 ) aiv
l \vn spare-time veotiu-s of llie 1st kind, the 4x4 matrix with
tlie element S AA . =w A n A ,
lastly in a multiple L of the unit matrix of 4x4 series
in which all the elements in the principal diagonal are
equal to L, and the rest are xero.
\Yc shall have to do constantly with functions of the
F pace-time point (.r, >/, :, if], and we may with advantage
III! IH I KKUK.VIIAI. OI MJATOR LOR
employ the Ixl series matrix, formed of differential
symbols,
a a a a
or (63)
a a a a
3 dy 3:
For this matrix T 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 1 x 4 series
matrix
I K, K, K< K. |
where K 4 = ^L* + *i* +
When by a Lorentz transformation A, a new reference
system (.K\ .<: , x s ,i- 4 ) is introduced, we can use the operator
lor =
^ a a a
8."/ 8.V", 8*, a-,
Then S is transformed to S =A S A= | S j | , so bj
lor S is meant the 1x4 series matrix, whose element are
*ri aS|4.oSjt.as. t j i a s * *
Now foi 1 the differentiation of any function of (x y t f)
a a a , , a a ,
we have the rule
ST . ^ -^ , "T j? W i
a .* 3 , 3 .* 3 .--, 3 *
t a 3- 3 , a 3
+ 3. 8 87T 3^ 3A
3 3 L 3 , , 3
3 i 3 * $ 3
so that, we have symbolically lor = lor A.
Vide note 17.
42 PRINCIPLK OK 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 matrix with the elements
8L 6L 6L 6L
8-i 8.I-, 9.t 3 9**
If # is a space-time vector of the 1st kind, then
lori =il + . + |il + i...
8*1 9-c 2 80, o - 4
In case of a Lorentz transformation A, we have
lor V=lor A. As = lor *.
i.e., lor * 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 f denotes a space-time vector of the first kind with
the components
8/i, a/,, , 8/ l4
a + a + "a
O , O , 0. +
a/ , + ._8At..
9-f.-, 9- (i ,
a*
3. + l^ + 8Z
a/.,
THK DIFFERENTIAL OPKRATOR LOR 43
So the system of differential equations (A) can be
expressed in the concise form
{A} lor/=-,
and the system (B) can be expressed in the form
{B} log F* = U.
Referring back to the definition (H7) for log *, we
find that the combinations lor (for/), and lor (lor F*
vanish identically, when f and F* are alternating matrices.
Accordingly it follows out of {A}, that
^ & + -& + 1;:- + -& = >
while the relation
0>9) lor (lor F*) = 0, signities 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
\
l_ W 2 Vl-?< 2 /
(// = 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 (///, />)
(/ = magnetic force, f= Electric Induction.
s the space-time vector of the first kind (C. ip)
(p = electrical space-den>it\ , ( f >/>= com 1 net ivity curn-nt,
= dielectric constant, // = manin tic penne:il)ility,
a = conductivity,
44 PEINCIPLK OK RELATIVITY
then the fundamental equations for electromagnetic
processes in moving bodies are*
{A} lor/=v
{B}logF* = o
{C} w/ =oF
* T-V i TiMf" / "X*
{B}^^=-*P.
ww = 1, and W F, w/, o>F*, w/**, + (O>*)<D which
are space-time vectors of the first 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 arc mvt--
sary for determining the motion of matter as well as the
vector 11 as a function of , //, , 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 u> as a function of (;,y,:,/.)
In these investigations, the expressions which are obtained
by the multiplication of two alternating matrices
/=
o A* As
A*
F=
F lt P 18 F,,
/.i /,
/.;
F sl P fa F 14
/si A*
A*
F 31 F 32 F, 4
A, A, A,
F+i ^+8 F +s
* Vide note 19.
THK PRODUCT OF T1IK KIELU-VKCTORS /F
are of much importance. Let us write.
(70) fF=
8
S .*-L s ls s 84
S 3, S 33 -L S..
S * a S*., S 44 -L
Then (71) S I1 +S,,+S 33 +S 44 =0.
Let L now denote the symmetrical combination of the
indices I, 2, 3, 4, given by
= / F 13 +/ 3I P sl+ / lf p if +/14 Fi4
)
Then we shall have
- t;; :; F ;;)
In order to express in a real form, we write
(74) S =
S 81 S 82 S, 3 S 2 +
Si S,, S 31 S 34
X, Y, Z, _,T,
X, Y, Z, -iT,
X.. >" Z -zT ;
-iX, -iT, -tZ, T,
nrJJC.=? r/^M,-^,,^ -m r M r +<.,E.- y E v -,. K.l
40 PRINCIPLE OF RELATIVITY
(75) X,=tn,M,+e.E,, Y,=wi,M,+?,E, etc.
X,=e,M,-e r M,, T,=m,E,-,E t etc.
li,=~
These quantities are all real. In the theory for bodies
at rest, the combinations (X,, X,, X., Y., Y,, Y r , Z,,
Z v , ZJ are known as "Maxwell s Stresses," T,, T,, T,
are known as the Poynting s Vector, T, as the electro
magnetic energy-density, and L as the Langrangian
function.
On the other hand, by multiplying the alternating
matrices of,/"* and F*, we obtain
(77) P*/*=-S 1I -L, -S lt , -S tl . -S 14
S S1 . S g2 L, S 2S , S 24
S1 S, 2 , S,, L, S,,.
S 41 S 4S S 4S S 44 L
and hence, we can put
(78) /F = S-L, F*.r*=-S-L,
where by L, we mean L-times the unit matrix, i.e. the
matrix with elements
| L^< . | , (c 4A =l, e**=0, A^A- /*, A-=l, 2, 3,4).
Since here SL = L^, we deduce that,
F*/*/F = ( -S-L) (S-L) = - SS + L,
and find, since/*/ = Dot -/. F* F = Det * F, we arrive
at the intonating conclusion
* Vide note 18.
THE PRODUCT OF NIK M ELU-VKCTORS / 47
(79) SS = L - Det * / Det * F
i.r. 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* 1 elements in the principal diagonal are all equal
and have the value given on the right-hand side of. (7 ( J).
Therefore the general relations
(80) S M S lt + S*. S at + 6 A , S,*+S 4l S 4 =,
h, k being unequal indices in the series 1, 2, 3, 4, and
(81) S M S lA +S*, S,*+S* S S S +S*, S 4 =L a -
Det * / Det * F,
for// = 1,2, 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 tt [(55), (56) and
(57)1, we can pass over to the expressions,
(82) L = i c * + J p. * *7
(83 j S 4
-f- c
(h, k = 1, 2, 3, 4).
Here we have
<|><t> = 4i i +t/+<l> : , +*, , * * = ^, J-*! 1 +**+*
e AA = 1, e kk o (h=f=k).
The right side of (82) as well as L is an invariant
in a Lorentz transformation, and the 4x4 element on the
48 PRINCII LK OF RELATIVITY
right side of (83) as well as S* A represent a space time
vector of the second kind. Remembering this fact, it
suffices, for establishing the theorems (82) and (83) gener
ally, to prove it for the special case o> l = o, w, =o, ?/? s =o,
w 4 =/. 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 c = eE, M = ps// on the other hand.
The expression on the right-hand side of (81), which
equals
[-5- (m M - <?E)] + (em) (EM),
^is = o, because (cm c <S> ^, (EM) = /x 4 *; now referring
>
back to 79), we can denote the positive square root of this
expression as Det * S.
Since 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 - S~= - (c/t - 1) fa>,n],
from which we deduce that [see (57), (58)].
(86) <o (S - S)* = o,
(87) w (S -~S) = (e p - 1) O
When the matter is at rest at a space-time point, u=o,
then the equation 86) denotes the existence of the follow
ing equations
Z y =Y,, X,=Z,, Y,=X,,
THE PRODUCT OF THE FIELD-VECTORS /> 4 J
and from 83),
T,=n if T,=n 2 , T..= :|
x^t^n,, Y,= f/ ,n 2 . #,=^0,
Now by means of a rotation of the space co-ordinate
system round the null-point, we can make,
Z,=Y.=o. X t =Z,= , X,=X,= .
. According to 71), we have
88) X,+Y,+Z.. + T,=o.
and according to 83), T,>o. In special cases, where Q
vanishes it follows from 81) that
and if T, and one of the three magnitudes X,, Y,. Z. are
= Det * S r the two other.s = - Det ^ S. If fi does not
vanish let O ^=0, then we have in particular from 80)
T r X,=0, T s Y,=0, Z,T s+ T t T f =0.
and if f), =0, 1= 0, Z r =-T, It follows from (81),
(see also 83) that
and -Z,=T, = V Det "S-f-e,*!),
The space-t- me vector of the first kind
f89) K = lorS,
is of very o^reat imjwrtanpe for which we now want to
demons! rnte a very important transformation
Accord in- to 7*\ S = L+/F, and it follows that
lor S=lor L + lor/F.
50 PRINCIPLE OK RELATIVITY
The symbol lor denotes a differential process which
in lor /F, operates on the one hand upon the components
of f, 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
where N is the vector with the components
8F.. f 8F S1 6F,, f _
-~-~ ~^~
<// = !, 2, 3,4)
By using the fundamental relations A) and B), 90)
is transformed into the fundamental relation
(91) lor S = *F + N.
In the limitting case < = 1. /x = l. /=F. X vanishes
identically.
1HI. PRODUCT OF THK FIELD-VECTORS /K 51
Now upon the basis of the equations (55) and (56),
and referring back to the expression (82) for L, and from
57) we obtain the following expressions as components
of \,_
,92) X t =- * ** l_l **_.
for A=l, 2. 3, 4.
Now if we make use of (59), and denote the space-
vector which has 11,, i! 2 , 11 3 as the -,//, - components by
the symbol W, then the third component of 92) can be
expressed in the form
,93)
The round bracket denoting the scalar product of the
vectors within it.
1A. THE PONDEROMOTIVE FORCE.*
Let us now write out the relation K=lor S = .
in a more practical form ; we have the four equations
Vide note 40.
52 PRINCIPLE OF RELATIVITY
0# v 9Z. QZ,
_ +_.. _ a _
/ 07 \ 1 -K- _ T 9 T y 9 T . 9 T , I?
(97) K * " <-t^i:
It is my opinion that when we calculate the pondero-
motive force which acts upon a unit volume at the space-
time point . , y, :, /, it lias got, .<, y, - components as the
first three components of the space-time vector
This vector is perpendicular to w ; the law of Energy
Jiiids iti expression in the fourth relation.
The establishment of this opinion i^ reserved for a
separate tract.
In the limitting case =1, /*=!, <r=0, the vector N=0,
S=p(D, wK^^O, and we obtain the ordinary equations in the
theory of electrons.
APPENDIX
MECHANICS AND THE RELATIVITY- POSTULATE.
It would be very unsatisfactory, if the new way of
looking at the time-concept, which permits a Lorentz
transformation, were to be confined to a single part of
Phvsics.
Now many authors say that classical mechanics stand
in opposition to the relativity postulate, which is taken
to be the basii of the new Electro-dynamics.
In order to decide this let us fix our attention upon a spe
cial Lorentz transformation represented by (Hi), (H), (1 -)>
with a vector v in any direction and of any magnitude <y<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 /, / , <y, we shall
write ct, ct , and q/c, where c represents a certain positive
constant, and -7 is <c. The above mentioned equations
arc transformed into
,
c 1 q </
They denote. as we remember, that r is the space-voMor
(>y> -)> s th 1 space-vector ( y )
If in these equations, kcej>ing constant we approach
the limit c = oo, then we obtain from these
The new e(|Uatiniis would now denote the transforma
tion of a -patial co-ordinate system (*, y, :) to another
spatial co-ordinate \>t. in ( y - ) with parallel axt-s, the
54 IMUNCIl LK 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
+* (1)
when c = <x.
Now it is rather confusing to find that in one branch
of Physics, we shall find a eovariance of the laws for the
transformation of expression (1) with a finite value of c,
iu another part for c=oc.
It is evident that according to Newtonian Mechanics,
this covariance holds for c=oo } and not for == velocity of
light.
May we not then regard those traditional covariances
for f = oo only as an approximation consistent with
experience, the actual covariance of natural laws holding
for a certain finite value of c.
I may here point out that by if instead of the Newtonian
Relativity-Postulate with e=oc, \ve 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 ( , //, -, /), it may be
convenient to leave (y, ) out -F account, and to treat s
and t as any possible pair of co-ordinates in a plane,
refered to oblique axes.
APl KXUIX 55
\ space time null point (.r, y, -, /=0, 0, 0, 0) will be
kept fixed in a Lorentx. transformation.
The figure * -y a -z* +* = !, />0 ... (2)
which represents a hyper holoidal shell, contains the space-
time points A (j-, y, :, / = 0, 0, 0, 1), and all points A
which after a Lorentz-transformation enter into the newly
introduced system of reference as (.r , y , -. , / = 0, 0, 0, 1).
The direction of a radius vector OA drawn from to
the point A of (2), and the directions of the tangents to
(2) at A are to be called normal to each other.
Let us now follow a definite position of matter in its
course through all time t. The totality of the space-time
points ( , t/, r, which correspond to the positions at
different times /, 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 point the direction of the space-time
line passing through it.
To transform a space-time point P (*, y, ~, f) to rest is
equivalent to introducing, by means of a Lorentx transfor
mation, a new system of reference ( , y , - , t }, in which
the t axis has the direction OA , 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 spuce-time line through P.
To the increment /// of the time of P corresponds the
increment
of the newly introduced time parameter /". The value of
the intesrral
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 Lorentz 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 we have an anatylical expression 6(.\> y, ?, /) so that
Q(x, y z t) = ft is intersected by every space time line of the
filament at one point, whereby
*.\ -(
a \
then the tolality of the intersecting points will be called
a cross section of the filament.
At any point. P of such across-section, we can introduce
by means of a Lorentz transformation a system of refer
ence (/, y, : t), so that according to this
.
8.. 8y
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 r/J=/ J / <L> ,hj ,f :
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 / axis of the filament at the
point /=i", and the volume of the body R is to be
regarded as the contents of the cross- section.
APPENDIX 57
If we allow R to converge to a point, we oome to the
conception of an infinitely thin space-time filament. In
such a case, a space-time line will be thought of as a
principal line and by the term Proper-time of the filament
will be understood the Proper-time which is laid along
this principal line ; under the term normal cross-section
of the filament, we shall understand the cross-section
upon the space which is normal to the principal line
through P.
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 t. If R converges
to a point (.c, y, :, /), then the quotient of this mass, and
the volume of R approaches a limit p.(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 /*//.},
where /x= mass-density at the point (.r, y t : , /) of the fila
ment (i.e., the principal line of the filament), //.I =eontents
of the cross-section normal to the / axis, and passing
through ( ,y,:, t), is constant along the whole filament.
Now the contents f/J. of the normal cross-section of
the filament which is laid through (, t/, :, f) is
(4)
</!
and the function v= ^ =n\ / \_. l * =/*-2l. (5)
iw 4 O
may be defined as the rest-mass density at the position
8
58 PRINCIPLE OF RELATIVITY
(xyzf). Then the principle of conservation of mass can
be formulated in this manner :
For an infinitely thin space-time fi/amenl, tJie product
of the rest-mass density and the contents of the normal
cross-section is constant atony the whole filament.
In any space-time filament, let us consider two cross-
sections Q." and Q , which have only the points on the
boundary common to each other ; let the space-time lines
inside the filament have a larger value of i on Q than
on Q. . The finite range enclosed between ti and Q
shall be called a space-time sic/tel* Q is the lower
boundary, and Q is the upper boundary of the nickel.
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 nickel, 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 jW/ n (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
//// lor vu dfdydzdt,
the integration being extended over tin whole range of
the Kic/icl, and (comp. (67), 15)
, -_ 8vo> 1 8 <i) 2 , 8^3 , 3va) 4
D11 " -- + - + - + --
If the nickel reduces to a point, then the differential
equation lor vw = 0, (6)
* Sichel a German word meanintr a crescent or a scythe. The
original terra is retained as there is no snitnble Knli.sli equivalent.
\iTi\m\ 59
which is the- condition of cortinuity
-* , . r , / _
a* a^ a-- f<8<
Further let us form the integial
N=SISJvdlyd:dt (7)
extending over the whole range of the space-time .tic/tcl.
We shall decompose the sichel into elementary space-time
filaments, and every one of these filaments in small elements
dr 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 vdJ n =dm and
write T" , r l for the Proper-time of the upper and lower
boundary of the sichel.
Then the integral (?) cau be denoted by
IJvdJn </T=/(T -T ) dm.
taken over all the elements of the sichel.
Now let us conceive of the space-time lines inside a
space-time nichd 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 c-urves are to be varied in any
possible manner inside the *ic/tc(, while the end points
on the lower and upper boundaries remain fixed, and the
individual substantial points upon it are displaced in such a
manner that they al \va\s move forward normal to the
curves. The whole pit, cess may be analytically repre
sented by means t ,f a parameter \, nnd to the value \ = o,
shall correspond the actual curves inside the .v/ 7/t7. Such a
process may be called a virtual displacement in the sichel.
Let the point (.r, _//, r, /) in the sichel \ = o have the
values .r + S--, .// + fy, .- + 8-, f + /, when the parameter has
60 PRINCIPLE OF RELATIVITY
the value A ; these magnitudes are then functions of (./, y,
z, /, A). Let us now conceive of an infinitely thin space-
time filament at the point (j- y : f) with the normal section
of contents dj n , and if dJ n +&dJ K 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 + dv being the rest-mass-deusity at the varied
position),
(8) (v + Sv) (dJ + Sd J ) = i/rfj = dm.
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 + SN as the mass action
of the virtual displacement.
If we now introduce the method of writing with
indices, we shall have
(9) rf(a 4 +8.- 4 )=cl,- 4 + 2 !?* + ^ d\
k O r k OA
*= 1,2,3,4
^=1,2,3,4
Now on the basis of the remarks already made, it is
clear that the value of N + SN, when the value of the
parameter is A, will be :
(10)
the integration extending over the whole sichel <(
where ^(r + Sr) denotes the magnitude, which is deduced from
Jd ., + d$x t ) _(dx s +d8x~)* r ^(~d.i> 4 +dS .>
by means of (9) and
APPENDIX
therefore :
= 1, 2, 3, 4.
1 = 1, 2, 3, 4.
We shall now subject the value of the differential
quotient
(12)
to a transformation. Since each 8- A as a function of (r, y,
z, vanishes for the zero-value of the paramater A., so in
, do.l t
general ---- =o, for \ = o.
O. - *
Let us now put ( - - * ) = A (/* = !, 2, 3, 4) (13)
then on the basis of (10) and (11), we have the expression
(12):-
,1 B (Jy d: dt
for the system (a-, ./-., ./ , r 4 ) on the boundary of the
Kic/i -f, (8.r, 8r 2 &r 3 8 } ) shall vanish for every value of
\aiul therefore ,, a ,f,, 4 are nil. Then by partial
integration, the integral is transformed into the form
* w -U rw*tf, , 8"w*,
-i "^v ~a^r
Jj; <hj 1 1 .
6-1 PRINCIPLE OF NEGATIVITY
the expression within the bracket may be written as
The first sum vanishes in consequence of the continuity
equation (6). The second may be written as
au (l> l Qw k dcy Qw h di s a<o A dc 4
~
( l*> k d (dr k
dr dr\dr
whereby is meant the differential quotient in the
t/T
direction of the space-time line at any position. For the
differential quotient (12), we obtain the final expression
dx d>/ //- tit.
For a virtual displacement in the Kiclicl we have
postulated the condition that the points Hijipnsi d to be
substantial shall advance normally to the curves i^ivinjf
their actual motion, which is \ o; this condition denotes
that the A is to satisfy the condition
H I ^i+ 8 ^ a +w 3 3 +;* f= . (15)
Let us now turn our attention to the Mnxwellian
tensions in the electrodynamics of stationary bodies, and
let us consider the results in 1:1 and 13; then we find
that Hamilton s Principle can be reconciled to the relativity
postulate for continuously extended elastic media.
M PKXDIX
At every space-time point (as in 13), let a space time
matrix of the ind kind be known
(16) S=
where X, Y, ...... X x , T, are real magnitudes.
For a virtual displacement in a space-time siohel
(with the previously applied designation) the value of
the integral
s lt
B M
S 18 S I4 |=
x,
Y,
z, -n\
s tl
S 2Z
SQ
83 8 +
X,
Y y
Z v -iT y
S 3l
S 39
So
33 to 3*
x..
Y,
Z r -iT :
S 41
S 4 ,
S 43 S 4*
-;x
/ ft
r, -z, T r
(17)
dzdt
extended over the whole range of the sichel, may be called
the tensional work of the virtual displacement.
The sum which comes forth here, written in real
magnitudes, is
Xi V i f~I i rn * xr O O " . -\y- U O.F , rw O ^*
j+lyT^t rA/ r-A-j -^ + -". y f- ...Li.
o.i ay 3r
X a . -l-T ^ 8 + T ^ 8/
a / a . a /
we can now postulate the following minimum principle in
mechanics.
If any space-timr Si,-//,-/ /,< Lovndeil , tlicu for each
rirtitiit jfaplacement /// th<- $i<-h> l , ilte mini of Hie w/r/.v.s 1 -
works, and tension works s/nilf ahcm/x lie mi c>tremum
for that process of the spacr-lime line in the 8ie/n
occurs.
The meaning is, that for each virtual displacement,.
A=0
(1*5
64. PRINCIPLE OF RELATIVITY
By applying the methods of the Calculus of Varia
tions, the following four differential equations at once
follow from this minimal principle by means of the trans
formation (H), and the condition (15).
(19) v 9 . =K,+ x!t -, (7, = l,a : S4)
,,,onee K, = + *
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 tv^j= 1. By multiplying (19) by w k , and
summing the four, we obtain X = Kw, and therefore clearly
K + (Kw}w 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,
(21) =X , =T ; ,- =Z,
dr \drj dr
( | =T, and we have also
ar \dr/
and X f + Y + Z I : =T < .
</T (Zr </T (?T
On the basis of this condition, the fourth of equations (21)
is to be regarded as a direct consequence of the first three.
From (21), we can deduce the law for the motion of
a material point, i.e., the law for the career of an infinitely
thin space-time filament.
APPENDIX 65
Let *, v, 2, f, 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 s t y t z, /, 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 R, R y R t R, and if m be the constant mass
of the filament, we obtain ^
R is now a space-time vector of the 1st kind with the
components (R, R y 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.f dy dz . dt
,r > ~T ~j * T
(IT <ir <IT </T
We may call this vector R t/ie moving force of the
material point.
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
( s.4V,0, then [See (4)] the equations (2:!) art- obtained, but
are now multiplied by ; in part ioular, the last equa
tion comes out in the form,
The right side is to be looked upon x the amount of u-ork
(hue per unit <>f li/ne 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
may be called the kinetic energy of the material point.
Since tit is always greater than tlr we may call the
quotient - r as the " Gain " (vorgehen) of the time
over the proper-time of the material point and the law can
then be thus expressed ; The kinetic energy of a 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) a^ain shows the sym
metry in (s&tsj), which is demanded by the relativity
postulate; to the fourth equation however, a higher phy
sical significance is to be attached, as we have already
seen in the analogous case in electrodynamics. On the
ground of this demand for symmetry, the triplet consisting
of the first three equations are to be constructed after the
model of the fourth ; remembering this circumstance, we
are justified in saying,
" 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*(.>*, /*, z* t /*) be a solid (fester) space-time point,
then the region of all those space-time points B (./, i/, z, /),
for which
VIM KXDIX 67
may be called a " Hay-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 ( y z /) be supposed to be fixed,
the point (t* t/* c* (*) be supposed to be variable, then
the relation (:J#) \vould represent the loeus of all the space-
time points ft*, 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 principal line of
F ; further let B* be the light point of B, C* be the
light point of C on the principal line of F*; so that
OA i.s 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
(21) ;;*( ( j\ ) HI)* in the direction of BD*, and
another vector of Mutable \;i.lue in direction of B*C*.
68 PRINCIPLE OF RELATIVITY
Now by ( - - # J 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 filament of F
behaves when the material point F* has a uniform
translate ry motion, i.e., 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 .* , y, z, t, denote the point
B, let T* denote the proper time of B*, reckoned from O.
Our proposition leads to the equations
/ae\ <* _
where (27) , c * +y* + ^ 2 =(t-
In consideration of (27), the three equations (25) are
of the same form as the equations for the motion of a
material point subjected te attraction from a fixed centre
according to the Newtonian Law, only that instead of the
time t, the proper time r of the material jx>int occurs. The
fourth equation (2(>) 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 (.r y z] is an ellipse with the semi-major axis a and
the eccentricity e. Let E denote the excentric anomaly, T
A1TKND1X 69
the increment of the proper time for a complete description
of the orbit, finally n~T = 2tr, so that from a properly chosen
initial point T, we have the Kepler-equation
(29) nr=E-e sin E.
If we now change the unit of time, and denote the
velocity of light by c, then from (28), we obtain
i* 1-fecosE
Now neglecting c~* with regard to 1, it follows that
7, 7 f i i m * l-fcosE~l
*="*L 1+t i^ssffiJ
from which, by applying (29),
(31) nt + const =( 1 + ^-^ \ wr-f SinE.
\ ac* / ac 2
the factor ^ is here the square of the ratio of a certain
average velocity of F in its orbit to the velocity of light.
If now w f denote the mass of the sun, a the semi major
axis of the earth s orbit, then this factor amounts to 10~ s .
The law of mass attraction which has bn-n just describ
ed and which is foimnlated in accordance with the
relativity j>ostu!a<e would signifv that gravitation is
propagated with the velocity of light. In view of the fact
that the periodic terms in (31) are very small, it. is not
possible to decide out of astronomical observations between
such a law (with the modified mechanics pro|K>sed above)
Hnd the Newtonian law of attraction with Newtonian
mechanics.
70 PRINCIPLE OF RELATIVITY
SPACE AND TIME
A Lecture delivered before the Naturforscher Yer-
sarnmlung (Congress of Natural Philosophers) at Cologne
(21st September, 1908).
Gentlemen,
The conceptions about time and space, which I hope
to develop before you to-day, has grown on experimental
physical grounds. Herein lies its strength. The tendency
is radical. Henceforth, the old conception of space for
itself, and time for itself shall reduce to a mere shadow,
and some sort of union of the two will be found consistent
with facts.
, I
Now I want to show you how we can arrive at the
changed concepts about time and space from mechanics, as
accepted now-a-days, from purely mathematical considera
tions. The equations of Newtonian mechanics show a two
fold in variance, (i) their form remains unaltered when
we subject the fundamental space-coordinate system to
any possible change of position, (//) when we change the
system in its nature of motion, / . e., when we impress upon
itanv 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. Wo look upon the existence of the first group
as a fundamental characteristics of space. We always
prefer to leave off tin: second group to itself, and with a
li^ht In-art, conclude that we can never decide from physical
considerations whether the space, which is supposed to be
APPENDIX 71
at rest, may not finally be in uniform motion. So those two
groups lead quite separate existences besides each other.
Tlu-ir totally heterogeneous character mav scare us away
from the attempt to compound them. Vet it is the whole
compounded group which as a whole gives us occasion for
thought.
We wish to picture to ourselves the whole relation
graphically. Let ( < , i/, z) be the rectangular coordinates of
space, and / denote the time. Subjects of our perception
are always connected with place and time. No one has
observed a place except at a particular time, or Tias obserred
a time except 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, ;/, -, /, as a n-or Id-point. The manifoldness of all
possible values of r, //, z, t, will be the world. I can draw
four world-axes with the chalk. Now any axis drawn
(onsi..ts of quickly vibrating molecules, and besides, takes
part in all the journevs of the earth ; and therefore gives
us occasion for reflection. The greater abstraction required
for the four-axes does not cause the mathematician any
trouble. In order not to allow any yawning gap to
exist, wo shall suppose that at every place and time,
something perceptible exists. In order not to specify
either matter or elect ric.it \ , we shall simplv style these as
substances. \Vo direct our attention to the ti-nrlil-pohil
, y, :, t, and suppose that wo are in a position to recognise
this substantial point at any subsequent time. Let- t/f he
the time element corresponding to the changes of space
coordinates of this point [t/.r, /It/, if:]. Then we obtain (as
a picture, so to speak, of the perennial life-career of the
substantial point), a curve in the world the Karltl-linr,
the points on which unambiguously correspond to the para
meter / from + oo to oc. Tho whole world appears to be
72 PRINCIPLE OF RELATIVITY
resolved in sucli world-lives, 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 (,#, c) mani-
folduess / = o and its two sides /<o and ^>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 r-axes any
possible rotation about the null-point corresponding to the
homogeneous linear transformation of the expression
The second group denotes that without changing the
expression for the mechanical laws, we can substitute
(.v-at,y-pt, z-yt} for (.-, y, z) where (a, p, 7 ) are any
constants. According to this we can give the time-axis
any possible direction in the upper half of the world />o.
Now what have the demands of orthogonality in spaco 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, and let us consider the figure
According to the analogy of the hyperboloid of two
sheets, this consists of two sheets separated by ( = o. Let us
consider the sheet, in the region of l>o, and let us now
conceive the transformation of .-, y, r, / 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
AI HKMHX 73
tHUuforaialkm in which (y, r) 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 -c-l - = ] f witli its asymptotes (/////
fig. 1).
Then let n draw the radius rector OA , the tangent
A B at A , and let us complete the parallelogram OA
B C ; also produce B C to meet the -axis at D .
Let us now take Ox , OA as new axes with the unit mea
suring rods OC = 1, OA = ; then the hyperbola is again
expressed in the fonii < ( - - = 1, t >o and the transi
tion from (.-, i/ y -, /) to ( - i/z t} is one of the transitions in
|iif-iioii. L-it 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
, which we may denote by CJ r .
N"MW let as, increase c to infinity. Thus - becomes zero
c
:uxl it :i])]icii - > i rdu the figure that the h\ pei - l)ol:i is j^radu-
all\- shrunk into the -;i\i<. the asymptot ic nnglc lic-
i-n iie-- :i -tnii^hi one, and everv <|irf-[;>.! 1 nmsfninrit ion in
tlie limit changes in s ich :i manner that the /-axis (-in
Miv possible dir;"-tiun upwiirds, and m>iv -ind
i ") -oxiin:ites tn . Remembering this point it is
"leiv that the full group belonging to Newtonian Mechanics
is siiii)>ly the n roup (i r , with the value of c=oo. In this
state iii aiTairs, :inl sinre ( ! is in:tt hemat ically more iu-
iellin ible thii i (r -x>, a mat hem ; . , b\ a i n-e p!:iv
ol imagination, hit upon the thought that natural pheno
mena possess nn invarianre not only for the group ( <,,
but in i ai-l also for a uroup d , , where c is lintte, but yet
10
74 PRINCIPLE OF RELATIVITY
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 <,
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 electro-magnetic units of electricity.
We can form an idea of the invariant character of the
expression for natural laws for the group-transformation
G, in the following manner.
Out of the totality of natural phenomena, we can, by
successive higher approximations, deduce a coordinate
system (., y, z, t) ; by means of this coordinate system, we
can represent the phenomena according 1o definite laws.
This system of reference is by no means uniquely deter
mined by the phenomena. We can change the system of
reference in any possible manner corresponding to the abore-
meniioned group transformation G c , bnt the expressions for
natural laws will not be changed thereby.
For example, corresponding to the above described
figure, we can call t the time, but then necessarily the
space connected with it must be expressed by the mani-
foldness (,/ y :). The physical laws are now expressed by
means of ,< , ?/, :, t 1 , 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
APPENDIX 75
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 a,ny manner; then a world-line parallel
to the -axis will correspond to a stationary point ; a
world-line inclined to the -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 r,y,z,i; now if we find the world-line parallel
to the radius vector OA of the hyperboloidal sheet, then
we can introduce OA as a new time-axis, and then
according to the new conceptions of time and space the
substance will appear to be at rest in the world point
concerned. We shall now introduce this fundamental
axiom :
Tin , substance <?</*////// at am/ worfil point can always
!,< fiiiK-fircil to be, af resf, if ice .flatifix\\ vnr lime and
space Kiiifafj///. The axiom denotes that in a world-point
tho expression
,.*,/(.* _,/.,-! _,/// _,/;*
shall always be positive or what is equivalent to the
same thing, every velocity V should be smaller than c.
>hall therefore be the upper limit for all substantial
velocities and herein lies a deep significance for the
76 I KINTIM.K 01- I!
quantity <. At the first impression, the axio:
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 geometry.
The impulse and real cause of inducement for tlie
assumption, of the yroup-kranisforniatioii G r is the fact that
the differential equation for the propagation of light in
vacant spase possesses the group- transformation G r . On
the oth-3r hand, the idea of rigid bodies has any sense
only in a system mechanics with the group G x . Now
if we have an optics with G,., and on the other h; inl
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 x , and G f , which Ins
got the further consequence, that by means of suitable
rigid instruments in the laborafory, we can perceive a
change in natural uhcnomena, in case of different orienta
tions, with regard to the direction of progres-ive motion
of the earth. But all efforts directed towards this
object, and even the celebrate! interference-experiment
of Michelson have u iven negative results. In order to
supply an explanation for this result, II. A. L. rent/
formed a hypothesis -vhich practically amounts to an
invanance of optics i or ihe ;roup G,.. Aeeordi
Lorent/ every substance shall Mifi er a contraction
1 ( v ~ ~ ) " ^ ll - tn > " the direction of its motion
\ITI,.\|l|\ 7?
l in> hypothesis sounds rather phantastical. For the
.rtion is nut to be thought of as a consequence of tin;
resistance of ether, but purely as a gift from the skies, as a
sort oF condition 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 (y, 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
s:nti:il extension (see fipf. l). IF O.\ is parallel to the second
strip, we can take / as the /-axis and x as the *-axis, then
ond body will appear to be at rest, and the first body
in uniform motion. We shall now assume that the first
body supp is,- 1 to be at rest, has the length /, i.e., the
cross seutidn PP of the first strip upon the , -axis = / OC,
\vlu-re OC is the unit measuring rod upon the s-axig and
the second bodv also, when supposed to beat rest, has the
same l.-u^th /, this means that, the cross section Q Q of
the -e .-oud strip has :i <Tnss-soct,ion / OC , when measured
p;irall -l 1 > tin- -axis. In th ;se two bodies, we have
no\v images of \\\- > Lorentz-ele ctr6bs, one of which ifl a;
rest ;nd the oilier moves iinifurinlx . Now if we stick
t. our original Coordinates, then the extension of the
iveii ov (lie cross section (Id of the
strip lielon^iii ^ to it nie::sui--ii p:-. rallel to the -;i\is.
Nov. if i| ,-!,;,! -iuc,. (i U =r/()C . that U(4 = / O1) .
If if 1 " ;in (>;ls . v falcnhtion u ives that
01) = OC x ,- therefore
78 fRlNCiPLK OK RELATIVITY
This is the sense of Lorentz s hypothesis about tlie
contraction of electrons in case of motion. On the other
hand, if we conceive the second electron to be at rest,
and therefore adopt the system (* , ( ,) then the cross-section
P P of the strip of the electron parallel to OC is to be
regarded as its length and we shall find the first electron
shortened with reference to the second in the same propor
tion, for it is,
P ^L_2H_P _ QQ
CPQ ? -OC / -OC ~ PP
Lorentz called the combination I of (t and ,) as the
local Li ne (Ortszeil) 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
clearly that the time of an electron is as good as the time
of any other electron, i.e. t, t are to be regarded as equi
valent, has been the service of A. Einstein [Ann. d.
Phys. 891, p. 1905, Jahrb. d. Radis... 4-1-1 1 1907] There
the concept of time was shown to be completely and un
ambiguously established by natural phenomena. But the
concept of space was not arrived at, either by Einstein
or Lorentz, probably because in the case of the above-
mentioned spatial transformations, where the (./, / ) 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.
According to these ideas, the word " Relativity-Postu
late" which has been coined for the demands of invariance
in the group (r, seems to be rather inexpressive for a true
understanding of the group G f , and tor further progress.
\PPENDIX 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 like to give to this assertion the name " The
Populate of the Absolute worM" [World- Postulate].
Ill
By the world-postulate a similar treatment of the four
determining quantities x,i/ t z, 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
striking and clear.
I shall again use the geometrical method of expression.
Let us call any world-point O as a " Space-time-null-
point." The cone
consists of two parts with O as apex, one part having
/<() , the other having />0. The first, which we may call
f\\e fore-cant consists of all those points which send light
towards O, the second, which we may 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. (Fide fig. :>).
On the aft-side of O -p have the already considered
hyperboloidal shell F = rV - r 2 -y* -z" = 1, t>0.
80 HlMXCIl I.K Ol RELATIVITY
Hie region inside the two cones will be ocoupibd by the
hvperboloid f one sheet
F= s+j/ + v- 2 ^t^k-,
where 9 can liave all possible positive values. The
hyperbolas which lie upon this figure with O as centre,
are important for us. For the sake of clearness the indivi
dual branches of this hyperbola will be called the " Tnter-
liyperbola. with centre 0." Such a hyperbolic branch,
when thought of as a world-line, would represent a
motion which for / = o and t = <x>, asymptotically
approaches the velocity of light <.
If, by way of analogy to the idea of vectors in space,
we call any directed length in the manifoldness -,//,:,/ a
vector, then we have to distinguish between a time-vector
directed from O towards the sheet + F 1, />0 and a
space-vector directed from O towards the sheet F = 1.
The time-axis can be parallel to any vector of the first
kind. Any world-point between the fore and aft cones
of O, mav by means of the system of reference be regard --d
either as synchronous with O, as well as later or earlier
than O. Every world-point on the fore-side of O is
nece-ssarilv always earlier, every point on the nft side of
O, later thin O. Tin limit f oa corresponds to a com
plete folding up of fcbfe wcdgc-shawd cross-section bet \\.v.i
the. fore and aft cones in the taanifoldofess / = (). In the
figure drawn, this cvos-s-scc! ion Ins been intentional!;-
drawn with a different breadth.
Let us decompose a vector drawn iV.>in O t "\\-ards
(.r,//,/,/) into its components. If the directions of the two
ve< tors are respectively the directions of the radius vector
OR to one of tin- surfaces +F=l,find of a tangt-nt IIS
APPENDIX 81
at tin- point R of the surface, then the vectors shall be
called normal to each other. Accordingly
which is the condition that the vectors with the com
ponents (-, y, :, /) and (f l // L z^ /j) 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, />() 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 (r, y,
;, 1} ; then as we follow the progress of the line, the
quantity
Vc*dt* dx* dy* dz* ,
c
corresponds to the time-like vector-element (dr, dy, dz, dt}.
The integral T= I dr, 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 P upon the irnrlil-H.i*-. We may regard (r, y, :, t], I.e.,
the components of the vector OP, as functions of the
" proper-time " T; let (.r, //, ?, f) denote the first different ial-
((iiotifMts, and (.r, //, r, /) the s(>cond differential quotients
of ( , /, :, with regard to T, then tlu si- may respectively
11
82 I UIXCIL Lli OF RELATIVITY
he called the Velocity -vector, and the Acceleration-rector
of the substantial point at P. Now we have
2 t f X , ( y y z O ) ,
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 tig. 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
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 = <x .
IV
In order to demonstrate that the assumption of the
group Cr r for the physical laws does not possibly lead to
anv contradiction, it is unnecessary to undertake a revision
of the whole of physics on the basis of the assumptions
underlying this group. The revision has already been
successfully made in the case of " Thermodynamics and
APPENDIX
Radiation,"* i or "Electromagnetic phenomena ",t and
linally i or "Mechanics with the maintenance of the idea of
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 (x, y } z, f), where the velocity-vector is (f, y, z, t),
then how are we to regard this force when the system of
reference is changed in any possible manner ? Now it is
known that there are certain well-tested theorems about
the ponderomotive force in electromagnetic fields, where
the group G r is undoubtedly permissible. These theorems
lead us to the following simple rule ; {/ the system of
reference be changed in any way, then the supposed force is
to be put as a force in the new space-coordinates in such n
warmer, that the corresponding rector with the components
*X, (?Y, t Z, /T,
where T= ( -4- X -f ~ J - Y + T- Z^ = l (the rate of
r 1 \ / t t c-
irork is done at the world-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
calk-il a inuriiiii / oro -recfur <it P.
X \v the world-line passing through P will be described
1)\- a substantial point with the constant mechaincut ///</vv
///. Let us call tn-ii>ii< x the velocity-vector at P as the
I liiiii k, YAM- Dynnmik bcwrgtor svRtemo, Ann. d. physik, Bd. JO,
1908, p. 1.
t II. MinkoNvski ; the ji:issn, o refers to paper (2) of the present
edition.
84- PRINCIPLE OF RELATIVITY
iwpuhe-r.er.tor, and m-times 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 cnn be
deduced from the first three, because both of the above-
mentioned vectors are perpendicular to the velocity-vector.
From the definition of T, we see that the fourth simply
expresses the " Energy-law/ Accordingly c z -times the
component of the impulse-vector in the direction of the
t-avis is to be defined as the kinetic-energy of the point-
mass. The expression for this is
i.e., if we deduct from this the additive constant me 2 , we
obtain the expression 4 mv- of Newtonian-mechanics upto
magnitudes of the order of . Hence it appears that the
energy depends upon the xyxlcm oj reference. But since the
/-axis can be laid in the direction of any time-like MM S,
therefore the energy-law comprises, for any possible system
of reference, thr> whole system of equations of motion.
This fact retains its significance even in the limitinir rase
C = oo, for the axiomatic construction of Newtonian
mechanics, as has already been pointed out by T. R.
Schiiix.t
* Minkowski Mechanics, appendix, page G."> of paper (U).
Planck -\Vrh. d. I). P. (I. Vol. 4, 1906, p. 136
f St-hutz, fiott. Nuclir. 1897, p. 110.
Al I KMMX 85
From the very beginning, we can establish the ratio
between the units of time and space in such a nr.mner, thai
the velocity of light becomes unity. If we now write
\/~l / = /, in the place of /, then the differentia] expression
,h - = - (dx - + <ly a + ilz * + ill - ),
becomes symmetrical in ( , //, .", /) ; this symmetry then
enters into each law, which does not contradict the worfit-
jjoxtnlafe. We can clothe the "essential nature of this
postulate in the mystical, but mathematically significant
formula
rH0 5 frm=i ^l Sec.
The advantages arising from the formulation of the
world-postulate are illustrated by 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 Ma \\vell-Fjorentz theory.
Let us conceive of the world-line of such an electron
with the charge (f), and let us introduce upon it the
" Pr. p r-(ime " T reckoned from any possible initial point.
In order to obtnin the lieM oau.-ed bv the electron at anv
world-point I , let us construct the fore-cone belonging
to P, (ridi tig. 4). Clearly this etits tin- nnliinit-d
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 tin s tangent. Let r be the measure of P,Q.
According to the definition of a fore-cone, rfc is to be
reckoned as th- measure of PU. Now at the world-point P,,
Ob PRINCIPLE OF RELATIVITY
the vector-potential of the field excited by e is represented
by the vector in direction PQ., having the magnitude
in its three space components along the x-, y-, --axes ;
the scalar-potential is represented by the component along
the -axis. This is the elementary law found out by
A. Lienard, and K. "Wiechert.*
If the field caused by the electron be described in the
above-mentioned way, then it will appear that the division
of the h eld into electric and magnetic forces is a relative
one, and depends upon the time-axis assumed ; the two
forces considered together bears some analogy to the
force-screw in mechanics ; the analogy is, however, im
perfect.
I shall now describe the ponderonwlivc force which is
exerted by one moving electron upon another 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 QPj. The t ?/-axis
in the direction of MN, then the r-axis is automatically
determined, as it is normal to the . -, t/-, ^-axes. Let
1, y, z, 7 be the acceleration-vector at P, .< ,, y l} z ,, /,
be the velocity- vector at P,. Then the force-vector exerted
bv the first election e, (moving in any possible manner)
* Lionard, L Eolnirnpi clcctriqne T 10, 1896, p. :{.
Wirchert, Ann. <1. Pliysik, Vol. 4.
Al I ENDIX 87
upon the ><(< .mil election >-, (likewise moving in any
1 o>>iblc manner) at I , is represented by
For the components F Jt 7" 7 y , F., F, of the rector F the
following three relations hold :
and fourthly this vector F is normal to the velocity-vector
P,, a ltd through this circumstance alone, its dependence on
f/iis laxf relitcity-vector arises.
If we compare with this expression the previous for
mula 3 * giving the elementary law about the ponderomotive
action of moving electric charges upon each other, then we
cannot but admit, that the relations which occur here
reveal the inner essence of full simplicity first in four
dimensions ; but in three dimensions, they have very com
plicated projections.
In the mechanics reformed according to the world-
postulate, the disharmonies which have disturbed tin-
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 m and
w, describe their world-lines ; a moving force-vector is
exercised by m upon /#,, and the expression is just tin- saun
as in the case of the electron, only we have to write
+ >//;;/, instead of ee^. We shall consider only the special
C;IM- in which tlu- acceleration-vector of in is always zero :
* K. Sdiwar/.si-liild. (iott-N aclir. 1903.
II. A. Lorcntz, Ens/Uopidie dor Math. \Vi.<.-i iiM-haftm V. Art 14,
PRINCIPLE OF KKI.AT1VITY
then / may be introduced in such a manner that m may be
regarded as fixed, the motion of m is now subjected to the
moving-force vector of m alone. If we now modify this
given vector by writing f - i
instead of /
to magnitudes of the order -j ), then it appears that
Kepler s laws hold good for the position (r,,^, z v ), of
MI at any time, only in place of the time l l} we have to
write the proper time T, of m l . 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
consequent s will be gradually deduced, and enough
suggestions will be forthcoming for the experimental
verification of the postulate ; in this way even those, who
find it uncongenial, or even painful to give up the old,
time-honoured concepts, will be reconciled to the new ideas
of time and space, in the prospect that they will lead to
pre-established harmony between pure mathematics and
physics.
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
ranch easier through the form given to the special Rela
tivity theory by Miukowski, 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 (jauss, Riemaun and Christoffel
on the non-Euclidean manifold, and which have been
shaped into a system by Ricci and Levi-civita, and already
applied to the problems of theoretical physics. I have in
part B of this communication developed in the simplest
and clearest manner, all the supposed mathematical
auxiliaries, not known to Physicists, which will be useful
for our purpose, so that, a study of the mathematical
literature is not necessary lor 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.
It
90 PRINCIPLE OF RELATIVITY
PRINCIPAL CONSIDERATIONS ABOUT THE POSTULATE
o* RELATIVITY.
1. Remarks on the Special Relativity Theory.
The special relativity theory rests on the following
postulate which also holds valid for the Galileo-Newtonian
mechanics.
Tf a co-ordinate system K he 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
translational motion relative to K. We call this postulate
" The Special Relativity Principle." By the word special,
it is signified that the principle is limited to the case,
when K has uniform trandalory motion with reference to
K, but the equivalence of K and K does not extend to the
case of non-uniform motion of K relative to K.
The Special Relativity Theory does not differ from the
classical mechanics through the assumption of this j>ostu-
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 Lorenz-transfor-
mation, with all the relations between moving rigid bodie
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 jwssible relative positions of solid bodies at rest,
and more generally the theorems of kinematics, as theorems
which describe the relation between measurable bodies and
GENRRAI.1.41U TIIMMn h RKT.ATIVITY 91
clocks. Consider two material points of a solid bodv at
rest ; then according to these conceptions their corres-
jx>nd8 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
syetem ; then to these positions, there always corresponds,
a time-interval of a definite length, independent of time
and place. It would he 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 episteomological defect,
which was perhai* lirst cleanly pointed out by E. Mach.
We shall illustrate it by the following example ; Let
two fluid bodies of equal kind and magnitude swim freely
in space at such a great distance from one another (and
from all other masses) that only that sort of gravitational
forces art- to be taken into account which the |>art of any
of these bodies exert UJKJII each other. The distance of
the bodies from one .another is invariable. The relative
motion of the different parts of each body is not to occur.
But each mass is seen to rotate, by an observer at rest 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 8 ) 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
uiface of the other i* an ellipsoid of rotation. We now
92 PRINCIPLE OF RELATIVITY
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
thing 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 Rj relative to which
the body S, is at rest, not however for a space relative to
which S 8 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 imaginary
cause B, for the different behaviour? of the bodies S, and
S a which are actually observable.
A satisfactory explanation to the question put forward
above can only be thus given : that the physical system
composed of S t and S s shows for itself alone no con
ceivable cause to which the different behaviour of S, and
S, 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 S, 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
KALISED THEORY OF RELATIVITY 9S
of the bodies under consideration. They take the place
of the imaginary cause R,. Among all the conceivable
spaces R t and R a moving in any manner relative to one
another, there is a priori, no one set which can be regarded
as affording greater advantages, against which the objection
which was already raised from the standpoint of the
theory of knowledge cannot be again revived. The laws
of physics must be so constituted that they should remain
valid for any system of co-ordinates moving in any manner.
We thus arrive at an extension of the relativity postulate.
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 muss at
a sufficient distance from other masses move uniformlv in
a line. Let K be a second co-ordinate system which has
a uniformly accelerated motion relative to K. Relative to
K any mass at a sufficiently great distance experiences
an accelerated motion such that its acceleration and the
direction of acceleration is independent of its material 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 can
be explained in as good a manner in the following way.
The reference-system K has no acceleration. In the space-
time region considered there is a gravitation-field which
generates the accelerated motion relative to K. .
This conception is feasible, because to us the experience
of the existence of a field of force (namely the gravitation
field) has shown that it possesses the remarkable property
of imparting the same acceleration to all bodies. The
i>4 PRINCIPLE OP RELATIVITY
mechanical behaviour of the bodies relative to K is the
same as experience would expect of them with reference
to systems which we assume from habit as stationary;
thus it explains why from the physical stand-point it can
be assumed that the systems K and K can both with the
same legitimacy be taken as at rest, that is, they will be
equivalent as systems of reference for a description of
physical phenomena.
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 physical significance ; when we say that
any arbitrary point has .ri as its X, co-ordinate, it signifies
that the projection of the point-event on the X,-axis
ascertained by means of a solid rod according to the rules
of Euclidean Geometry is reached when a definite measur
ing rod, the unit rod, can be carried .c^ times from the
origin of co-ordinates along the Xi axis. A. point having
s t t, as the X 4 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 OF RELATIVITY 95
point-event and set according to some definite standard has
gone over .<- 4 =J periods before the occurence of the
point-event.
This conception of time and SIWLCC 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 aud in general
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, s, t,) and
also, another system K (. y : t ) rotating uniformly rela
tive to K. The origin of both the systems as well as their
~-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 aud
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 plaur
(X , Y ) of K . Let us now think of measuring the circum
ference aud the diameter of these circles, with a unit
measuring rod (infinitely small compared with the radius)
and take the quotient of both the results of measurement.
If this experiment be carried out with a measuring rod
at rest relatively to the Galiliean system K we would ijet
IT, as the quotient. The result of measurement with a rod
relatively at rest as regards K would be a number which
is greater than *. 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 Lorenz-contraction, not however when the rod
is placed along the radius. Euclidean Geometry therefore
does not hold for the system K ; the above fixed concep
tions of co-ordinates which assume the validity of
Euclidean Geometry fail with regard to the system K .
We cannot similarly introduce in K a time corresponding to
physical requirements, which will be shown by all similarly
prepared clocks at rest relative to the system K . In order
to see this we suppose that two similarly made clocks are
arranged one at the centre and one at the periphery of
the circle, and considered from the stationary svstem
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 general
relativity theory time and space magnitudes cannot be so
defined that the difference in spatial co-ordinates can be
immediately measured by the unit-measuring rod, and time-
like co-ordinate difference with the aid of a normal clock.
The means hitherto at our disposal, for placing our
co-ordinate system in the time-space continuum, in a
GENERALISED THEORY OF RELATIVITY 97
definite way, therefore completely fail and it appears that
there is no other way which will enable us to fit the
co-ordinate system to the four-dimensional world in such
a way, that by it we can expect to get a specially simple
formulation of the laws of Nature. So that nothing remains
for us but to repaid 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 .\<if//re are expressed ly mean* of
equations which are valid for till co-ordinal e systems, that is,
which are covariant for all po*xi/j/<- fraasfortudtioiU. It ig
clear that a physics which satisfies this postulate will be
unobjectionable from the standpoint of the general
relativity postulate. Because among all substitutions
there are, in every case, contained those, which correspond
to all relative motions of the co-ordinate system (in
three dimensions). This condition of general covariance
which takes away the last remnants of physical objectivity
from space and time, is a natural requirement, as seen
from the following considerations. All our icell-sntjstantiatfd
space-time propositions amount to the determination
of space-time coincidences. If, for example, the event
consisted in the motion of material points, then, for this
last case, nothing else are really observable except the
encounters between two or more of these material points.
The results of our measurements are nothing else than
well-proved theorems about such coincidences of material
points, of our measuring rods with other material points,
coincidences between the hands of a clock, dial-marks and
point-events 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
IS
98 PRINCIPLE OF RKLATIVm
(.! ./, ,u s .c 4 ) such that to any and every point-event
corresponds a system of values of (<., .. , , 4 ). Two co
incident point-events correspond to the same value of the
variables (./ , .< s , r , s .: 4 ) ; i.e., the coincidence is cha
racterised by the equality of the co-ordinates. If we now
introduce any four functions (. l % . 3 .<- 4 ) as co
ordinates, so that there is an unique correspondence between
them, the equality of all the four eo-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 general covariance.
$ 4. Relation of four co-ordinates to spatial and
time-like measurements.
Analytical expression for the Gravitation-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 b
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 appear; this is possible for
au infinitely small region. X,, X,, X, are the apmtiaJ
GENERALISED THEORY OP RELATIVITY 99
co-ordinates ; X t is the corresponding time-co-ordinate
measured by some suitable measuring clock. These co
ordinates have, with a given orientation of the system, an
immediate physical significance in the sense of the special
relativity theory (when we take a rigid rod as our unit of
measure). The expression
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 /. as the magnitude of the
line-element belonging to two infinitely near points in the
four-dimensional region. If ds* belonging to the element
(r/X, dX t , ^X 3 , ^X 4 ) be positive we call it with Minkowski,
time-like, and in the contrary case space-like.
To the line-element considered, i.e., to both the infi
nitely near point-events belong also definite differentials
(I < , , d< , . dx s , d \, 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, /X s would then be represented by definite linear
homogeneous expressions of the form
*2) dX =2 a d.r
V (T V<T (T
If we substitute the expression in (1) we get
where g will be functions of .< ^ but will no longer depend
upon the orientation and motion of the local co-ordinates;
for d* is a definite magnitude belonging to two point-
events infinitely near in space and time and can be got by
100 PRINCIPLE OF RELATIVITY
measurements with rods and clocks. The g s are here to
be so chosen, that n = n ; the summation is to be
"or r w
extended over all value? of o- and r, so that the sum is to
be extended over 4x4- terms, of which 12 are equal in
pairs.
From the method adopted here, the case of the usual
relativity theory comes out when owing to the special
behaviour of g in a finite region it is possible to choose the
system of co-ordinates in such a way that g ^ assume*
constant values
f -1, 0, 0,
0-100
(*)
00-10
ooo+i
We would afterwards sec 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 X it is clear,
that from the physical stand-point the quantities g are to
.be looked upon as magnitudes which 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. The g & then
have the values given in (I). A free material point moves
with reference to such a system uniformly in a straight-
line. If we now intro:lu3J, by u .iy substitution, the space-
time co-ordinates .r, . ...r 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 ne.w
GENERALISED THEORY OF RELATIVITY
101
co-ordinates, will appear as curvilinear, and not uniform, in
which the law of motion, will be iwiependetd of the
ini.fii.rr nl I In- innrJ,i<i nt<txx-])uinl*. We can thus signify this
motion as on*- under the influence of a gravitation field.
We see that the appearance of a gravitation-field is con
nected with space-time variability of g s. In the general
case, we can not by any suitable choice of axes, make
special relativity theory valid throughout any finite region.
We thus deduce the conception that y 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 electromagnetic
forces, in as much as the 10 functions g representing
gravitation, define immediately the metrical properties of
the four-dimensional region.
MATHEMATICAL AUXILIARIES FOR ESTABLISHING THE
GENERAL COVARIANT EQUATIONS.
We have seen before that the general relativity-postu
late leads to the condition that the system of equations
for Physics, must be co-variants for arn r possible substitu
tion of co-ordinates .- , , . , : \ve have now to see
how such general co- variant equations can be obtained.
\Vr shall now turn our attention to these purely mathemati
cal pro positions. Ft will be shown that in the solution, the
invariant ifs, given in < tjiiat.ion (:}) plays a fundamental
role, which \vr. following (iau-Vs Theory of Surfaces,
style as the line-element.
The fundamental idea of the general co-variant theorv
is this : With reference to any co-ordinate system, let
certain tiling (tensors) be defined by a number of func
tions of co-ordinates which are called the components of
10? PRINCIPLE OF KKI.ATIVITY
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
u 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 system vanish
if they are all zero in the original system. Thus a law
of Nature can be formulated by puttjng 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-variant Four-vector.
Contra-variant Four-vector. The line-element is denned
by the four components d.- v whose transformation law
is expressed by the equation
V.
<*> " -=* T^ ",. .
The d< ,< are expressed as linear and homogeneous func
tion of <l> * ; we can look upon the differentials of the
co-ordinates as the components of a tensor, which we
designate specially as a contravariant Four-vector. Every
thing which is defined by Four quantities A , with reference
to a co-ordinate system, and transforms according to
the same law,
i.KNKKAUSBO THEORY OK RELATIVITY 103
we may call a contra-variant Four-vector. From (5. a),
it follows at once that the sums (A 4 B ) are also com
ponents of a four-vector, when \ a and B T are so ; cor
responding relations hold also for all systems afterwards
introduced as " tensors " (Rule of addition and subtraction
of Tensors).
Co-r<inanf-
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 = Invm-ianL
From this definition follows the law of transformation of
the co-variant four-vectors. If we substitute in the right
hand side of the equation
the expressions
e -v
for B v following from the inversion of the equation (5)
we get
a
B^ 5 -^ A =5 B" A
<T _ ~ V , nO* A
<r
<7
As in the above equation B CT are independent of one another
and perfectly arbitrary, it follows that the transformation
law is :
104 PEINCIPLE OF RELATIVITY
Remarks on the simplification <>f tl/c mode of wriliny
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
only over the indices which appear twice. It is therefore
possible, without loss of clearness, to leave off the summation,
sign ; so that we introduce the rule : wherever the
index in any term of an expression appears twice, it is to
be summed over all of them except when it is not 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 Ricci and Levi-civita, the contravariants
and co-variants are designated by the over and under
indices.
6. Tensors of the second and highei ranks.
Contravariant tensor : If we now calculate all the Ifi
products A^ v of the components A^ B v , of two con-
trava riant four- vectors
f8) A^ = A^B*
A *", will according to (8) and (ft a) satisfy the following
transformation law.
6 * 8 .-
(9) A" = 3-? g-I 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 105
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^ v , can be represented as the sum of A
B v 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 any rank : If is clear that
corresponding to (8) and (9), we can define contravariant
tensors of the 3rd and higher ranks, with 4 3 , etc. com-
l>onents. Thus it is clear from (8) and (9) that in this
sense, we can look upon contravariant four-vectors, as
contravariant tensors of the first rank.
Co-variant tensor,
If ou the other hand, we take the 16 products A^ of
the components of two co. variant four- vectors A and
for them holds the transformation law
Q u 8 r v
<"> ^r =-97/67;, V
By means of these transformation laws, the co-variant
tensor of the second rank is defined. All re-marks which
we have already made concerning tbe contravariaut 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
]66 HllXCIPLK OF KELATIVITV
Mixed tensor. We can also define a tensor of the
second rank of the type
(12) A = A B V
P. P-
which is co-variant with reference to p. and contravariant
with reference to v. Its transformation law is
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.
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
are equal. The tensor A ^ or A ^ is symmetrical, when
we have for any combination of indices
(14) A^W 7 *
(14a) A =A
p.V Vfi
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-)
i 9 .< - 9 .T Q ,i , Q , ,
.or a- T >p.v a T /.vu. .TO-
- e,,, e. v a,,, 8,;
CKN Ktf ALIsKlt TIIKOllI oh II KI,.\TI VITV 107
Anti-til iniii lri<-nl
A contravariant or co-variant tensor of the 2nd, 3rd or
4th rank is called <iiifi*i/nnit<>trical when the two com
ponents got by mutually interchanging any two indices
are equal and opposite. The tensor A. 1 *" or A ^ is thus
antisymmetrical when we have
(15) A^ = -A^
(15a)
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-rector).
Thus we also see that the antisymmetrical tensor
A!*"" (3rd rank) lias only 4- components numerically
different, and the antisymmetrioal 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.
On/ft- i)ii(/(iji/n-<iti<>/i of T iix,n-x : \Vc ^ct from the
components of a tensor of rank :, and another of a rank
c , the components of a tensor of rank (r-fc ) for which
we multiply all the components of the first with all the
components of the second in pairs. Fur example, we
108 PRINCIPLE OF RELATIVIT*
obtain the tensor T from the tensors A and B of different
kinds :
T = A B
UV<T flV (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), (18) ; equations (S),
(10) and (12) are themselves examples of the outer
multiplication of tensors of the first rank.
Reduction in rank of a mixed Tensor.
From every mixed tensor \ve can get a tensor which is
two ranks lower, when we put an index of co-variant
character equal to an index of the contravariant character
and sum according to these indices (Reduction). We get
for example, out of the mixed tensor of the fourth rank
sv
A , the mixed tensor of the second rank
8 a8 /
A =A =(SA
a \
ft \ a aft
and from it again by " reduction " the tensor of the zero
rank
.0 ./*
A= A = A
ft aft
The proof that the result of reduction retains a truly
tensorial character, follows either from the representation
t, I NKI5AL1SK1) THKOKY OF HKLATIVITY
109
of tensor according to the generalisation of (1~) in combi
nation with (H) or out of the generalisation of (13).
Imier a<ul mixed multiplication of Tensors.
This consists in the combination of outer multiplication
with reduction. Examples : From the co-variant tensor of
the seconj.1 rank A and the contravariant tensor of
the first rank B we get by outer multiplication the
mixed tensor
B"
Through reduction according to indices v and o- (i.e., put
ting i = <r), the co-variant four vector
D = D = A B is generated.
/* .v /"
These we denote as the inner product of the tensor A v
and B . Similarly we get from the tenters A and B
ILV
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
mixed tensor of the second rank D = A B . \Ve
can fitly Pall this operation a mixed one ; for it is outer
with reference to the indices M and r t and inner with
respect to the indices v and o-.
110 I KI.Ni I I l.K <>! I! KI, \TIVm
We now prove a law, which will be often applicable for
provingthe tensor-character of certain quantities. According
to the above representation. A B^ v is a scalar, when A
P-V /JLV
and B are tensors. We also remark that when A B^ is
an invariant for everv choice of the tensor B^ v , then A
(Of
has a tensorial character.
Proof : According to the above assumption, for any
substitution we have
A , B" = A
OT fJiV
From the inversion of (9) we have however
o & o * / * /
Substitution of this for B^" in the above equation ^i
A 8 % 9 ^ A
A , r - A
" r 8 . f 8- y /
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
<ih\l,KAI.ISKI. THEORY OF RELATIVITY III
if for every choice of them the inner product A B C
is a scalar, then A is a co-variant tensor. The last
py
law holds even when there is the more special formulation,
that with any arbitrary choice of the four-vector B" alone
the scala: product A B * B v is a scalar, in which case
we have the additional condition that A satisfies the
symmetry condition. According to the method given
above, we prove the tensor character of (A v + ^ v }, 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 qualities A B form a tensor of the
first rank, when B is any arbitrarily chosen four-vector,
then A v is a tensor of the second rank. If for example,
C 1 is any four-vector, then owing to the tensor character
of A n B 1 , the inner product A v C^ B 1 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 //
U.V
The co-variant fundamental tensor In the invariant
expression of the square of the linear element
<lx- =7 df 1 1. 1
M* /*
112 PRINCIPLE OF RELATIVITY
A A plays the role of any arbitarily chosen contravariant
vector, since further g y =ff v , it follows from the consi
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
teusor.
The co-variant fundamental tensor.
If we form from the determinant scheme | g v | the
minors of g y and divide them by the determinat ^ = j y^ \
we get certain quantities f* g v ^ , which as we shall
prove generates a contra variant tensor.
According to the well-known law of Determinants
(16) g / a =8
/XO"
where S is 1, or 0, according as /* = v or not. Instead
of the above expression for rf. 2 , we can also write
V 6 V rf V ^ "
or according to (16) also in the form
0T . *
a a a (M /
y tt7 J VT * U. V
GENERALISED THEORY OF RELATIVITY
118
Now according to the rules of multiplication, of the
fore-going paragraph, the magnitudes
p.<T fJL
foims a co-variant four- vector, and in fact (on account
of the arbitrary choice of d< ) any arbitrary four-vector.
If we introduce it in our expression, we get
For any choice of the vectors d d) this is scalar, and
y*"", according to its defintibn is a symmetrical thing in a
and r, so it follows from the above results, that g is
contravariant tensor. Out of (16) it also follows that 8
^
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
On the other hand we have
So that it follows (17) that \9 v \ I/"
15
114
PRINCIPLE OF RELATIVITY
of ruin me.
We see K first the transformation law for the determinant
0= | v 1 . According to (11)
dx 6-c
U V
9,
9 =
6,v a, v -/-
From this by applying the law of mutiplication twice,
we obtain.
9*,
F
H-
a ^
rf.r /
V7f =
_j*
v-,
"
... (A)
On the other hand the law of transformation of the
volume element
is according to the wellknown law of Jacobi.
dtl
dr =
... (B)
by multiplication of the two last equation (A) and (B) we
get.
(18)
=<Sf dr ^V dr.
lusted of v/^-, we shall afterwards introduce \/~g
which has area! value on account of the hyperbolic character
of the time-space continuum. The invariant ^/gdr, is
equal in magnitude to the four-dimensional volume-element
GENERALISED THEORY OF RELATIVITY llo
measured with solid rods and clocks, in accordance with
the special relativity theory.
Remark* on the character of Hie space-time continuum
Our assumption that in an infinitely small region the
special relativity theory holds, leads us to conclude that ds-
can always, according to (1) be expressed in real magni
tudes fJX,...<JX . If we call dr a the " natural " volume
element rfX t </X 2 dX 3 dX 4 we have thus (18a) tlr*
Should Y/ g vanish at any point of the four-dimensional
continuum it would signify that to a finite co-ordinate
volume at the place corresponds an infinitely small
" natural volume." This can never be the case ; so that g
can never change its sign; we would, according to the special
relativity theory assume that g has a finite negative
value. It is a hypothesis about the physical nature of the
continuum considered, and also a pre-established rule for
the choice of co-ordinates.
If however ( g) remains positive and finite, it is
clear that the choice of co-ordinates can be so made that
this quantity becomes equal to one. We would afterwards
see that such a limitation of the choice of co-ordinates*
would produce a significant simplification in expressions
for laws of nature.
In place of (18) it follows then simply that
dr = d
from this it follows, remembering the law of Jacobi,
- =1.
116 PRINCIPLE OF RELATIVITY
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 lenunciation of the general relativity
postulate. We do not seek thoj=e laws of nature which are
co- variants with regard to the Iran formations 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 new tensor* with 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 :
We would point out specially the following combinations:
A <f g A n
ap
A =g g A aft
(complement to the, co-variant or contravariant tensors)
and, B o Q A a
We can call B the reduced tensor related to A
GENERALISED THEOEY OP HELATIVITY 117
Similarly
,-V/AV
B
a/3
It is to be remarked that f* is no other than the " com
plement " of y v , for we have,
ua vB S p-v
Q Q 9 o~ 9 *> 9
y y v J a
9. Equation of the geodetic line
(or of point-motion).
At the " line element " ds is a definite magnitude in
dependent of the co-ordinate system, we have also between
two points P t and P 2 of a four dimensional continuum a
line for which />/? is an extremum (geodetic line), i.e., one
which has got a significance independent of the choice of
co-ordinates.
Its equation is
(20)
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 X, be a function of the co-ordinates x v ; This
defines a series of surfaces which cut the geodetic line
sou^ht-for as well as all neighbourin<r lines from P, to P 3 .
We can suppose that all such curves are given when the
value of its co-ordinates ^ are given m terms of X. The
118 PRINCIPLE of RELBTIVITY
sign S corresponds to a passage from a point of the
geodetic curve sotight-for to a point of the contiguous
curve, both lying on the same surface A..
Then (20) can be replaced by
u
(20)
dx dx y
^ l ~ 9 fJiV ~(j\ ~rfX
But
LlL
2 a,
So we get by the substitution of 8w in (20a), remem
bering that
( ii> v ^ a
H - /= ( 8 - i v^
after partial integration,
X.,
(20b)
J
d\ k So; =0
o- o-
a. / A
1. 1 AKIIALISED TIIEOKY OF RELATIVITY 119
From which it follows, since the choice of 8.<- is per
fectly arbitrary that k^ * should vanish ; Then
(20c) k a =0 (o-=l, 2, 3, 4)
are the equations of geodetic line ; since along the
geodetic line considered we have ^,<?=/=0, we can choose the
parameter A, as the length of the arc measured along the
geodetic line. Then w = l, and we would get in place of
(20c)
V 8** 8.<- 8* 8*
<r
_i a v 8 V ar - _
Or by merely changing the notation suitably,
<ZV _ _ (L- (ft
(20d) g a(f t - a -4- K" -^ . ~ =0
where we have put, following Christoffel,
(9i\ F/ xv l * I M "^ vcr P- v ~\
S a* a~~ ~ ~ai
LO- J L O v fi
Multiply finally (20d) with 0* (outer multiplication with
reference to T, and inner with respect to rr) we get at
last the final form of the equation of the geodetic line
-
ds*
Here we have put, following Christoffel,
120 PRINCIPLE OF RELATIVITY
10. Formation of Tensors through Differentiation.
Relying on the equation of the geodetic line, we can
now easily deduce laws according to which new tensors can
be formed from given tensors by differentiation. For this
purpose, we would first establish the general co-variant
differential equations. We achieve this through a repeated
application of the following simple law. If a pertain
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 <, be an invariant space function,
then is also an invariant. The proof follows from
the fact that d<j> as well as ds, are both invariants
Since = 9* |>
ds Qx 9 *
a , d.
so that i = 5^ . is also an invariant for all curves
which go out from a point in the continuum, i.e., for
any choice o]
diately that.
any choice of the vector d c . From which follows imme
/*
A =1*
. Vj
is a co-variant four-vector (gradient of <).
According to our law, the differential-quotient x=
taken along any curve is likewise an invariant
Substituting the value of iff, we get
d- dr. d**
O <P /* v . o <P A 4
v= r 7^ . ^ . +
A Q., 9, d* d 9-
M * f 1
<;F.\K1UUSK1> THKOKY OJ- HKL.Vl mTY
Here however we cannot at once deduce the existence
of any tensor It we however tiikc that the curves along
which we are differentiating are geodesies, we pet from it
dfx
by replacing -- - according to ( 4 J J
= r aa ^_ - ^ v) -9*
I 3* 9- v <- r > 9- T
From the interrliangeability of the differentiation with
regard to ^ and v, and also according to (2o) and (21 ) svc iin-
\ M 1 J
that tlie bracket - is symmetrical with respect to u.
( T )
and t-.
As we can draw ;t uvoilrt ic line in any direction from any
point in the continuum. x is thus a four-vector, with an
arbitrary ratio of components. M that it follows from Mir
results of 7 that
(25)
is a co- variant tensor of the second rank. We have thus got
the result that out of the co-variant tensor ot the tirst rank
A = "-^ we can get I>V differentiation a co-\ariant tensor
of 2nd rank
V= 6/
16
\:l l PRINCIPLE OF RELATIVriY
W> call rhe tensor A the " extension " >f the tensor
ta>
A . Then we can easily show that this combination also
leads to a. tensor, when the vector A is not repvesentable
us a gradient. In order to see this we first remark that
\^ ~~ is a co-variant four-vector when \L and < are
scalars. This i> uiso the case for a sum of four such
terms :
when i/rd), <j>( l )...\f/(*) <(*) are scalar*. Now it is however
clear that every co-variant four-vector is j-epresentable in
the form of S
P-
If for example, A is a fonr- vector whose components
are any given functions of x , we have, (with reference to
the chosen co-ordinate system) only to pat
= A,
in order to arrive at the result that S is equal to A .
V- P
In order to prove then that A in a tensor when on the
right side of (26 N t we substitute any co-variant four-vector
for A we have only to show that this is true for the
Til ROll Y OF BRLATIVITY 1:
four- vector S . For this latter, case, howevtjr, a glance on
the right hand side of (26) will nhow that we have only to
bring forth the proof for the case when
Now the right hand side of (25) multiplied by i/ is
which has a tensor character. Similarly, ~* f * is
also a tensor (outer product of two four- vectors).
Through addition follows the tensor character of
Thus we get the desired proof for the four-vector,
S l
\1/ - and hence f->r 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
i-ank. 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.
Aft already remarked every co-variant tensor of th^ 2nd
rank can be represented as a sum of the tensors of the type
A B .
fi V
124 PRINCIPLE OF EELATIVITY
It would therefore be sufficient to deduce the expression
of extension, for one such special tensor. According to
(26) we have the expressions
8 A ( )
9 ^ - ( r ) ^
are tensors. Through outer multiplication of the first
with B and the 2nd -with A we get tensors of the
third rank. Their addition gives the tensor of the third
rank
A - /xv - A - J av A
V" srr
where A y is put=A B^, . The right hand side of (27)
is linear and homogeneous with reference to A . and its
first differential co-efficient so that this law of formation leads
to a tensor not only in the case of a tensor of the type A
B but also in the case of a summation for all such
tensors, i.e.., in the case of any co-variant tensor of the
second rank. We call A the extension of the tensor A .
p.va /iv
It is clear that (26) and (24) are only special cases of
(27) (extension of the tensors of the tirst and zero rank).
In general we can get all specinl Ijtws of formation of
tensors from (27) combined with tensor multiplication.
(JKXKH.M.ISKD TUKOliY Ol KKLAT1V1KY 125
Some special cases of Particular Importance.
A f -ic inuiJiai n It miiia* con<:rr,iui<i lite fundamental
tn*< ,-. W<> shall first deduce some of tin- h-mnms much xised
ut fci -\\ards. According to the law of differentiation uf
determinants, we have
28) ,, 9 = /V^ v =-f/ /xv ^/ V -
Tlu- la.-t foi-ni follows From tho first when we remember
that
,/ ,f v = ti L , and there fun- ,-/ g fa =
ULV U.V
consequently;/ fly** 1 + >j^ v <l<i
From (28), it folhnvn that
~t i ^ (-r/) _,,/ V
6. -^ 6,,
/" 6-
Again, since ^ / <j ^ , we havi-, Ity differentiation,
vcr
(/
a,-.
(30) - 01
By mixed multiplication with (j nnd ;/
we obtain (Hwn^inir the mode of wntinir the
126 I KINTIPLE OF RELATIVITY
Id ftv =- P a < V P d<
(31)
Q7~ ~ ;/ ^ V a/S
I
V
p.- p.w K^,
(32)
,
Tlio expression (31) allows a transformation which we
shall often use ; according to (21)
If we substitute this in the second of the foi-mula (31).
we get, remembering (23).
By substituting the right-hand side of (H4) in (29), we
get
.lsKM TUKOKV OF KEI.AT1VIIV I" ?
nf ///> eo^tfavoriani four^vcctor,
Let u> multiply i ii ) with the conttavariani fandanumta]
ten-Mir if (inner mult ijilicittion ). then lv ;t
of the firnt incjn^ci 1 . th ritrlit-lianrl side tnkes the form
; , a _ V r .
8 8 -
According to (. U ) ;nul (- .*). th lust member cun
the form
Both the first members of the exj)rcssion (B), and the
second member of the expression (A) cancel each other.
since the naming of the summation-indices i immaterial.
The laHt meiuhei- of (B; cun then he united with first of
(A). If we put
where A* as well as A are vectors which -an be trbi-
M
trarily chosen, we obtain finally
9
-" A "
a
This scalar is the Di vcrym- ut the contra variant four-
vector A V ,
128 I lUM ll i.l. OK UKLATTVITY
ttnfdiin,/ a/ t/tc (covarianfy fjouyvectof,
Tin- seron.l iiHMiiiifi in :!> ) iv -\ mmetrical in the indict
u, and v. Hence A A is an antisymmetrical tensor
/ f p.* vp.
built up iu a vrry simple niainier. \Ve ol)tain
a A 9 A
(36) B = - /x -
t>f ii Six-vector.
Jf we apply the operation ( 27) on an nntisynimetrical
tensor of tht- sri-ond I jink A . ;uid form all the equations
pr
arising from the cyclic* interchange of the indices /x, v. <r, and
add all them, we obtain a tensor of thr third rank
a A
(.37) B =A 4- A + A = - ^
/Aj ir fiver V<TJJ. ir/j.v Q <T
from which it is easy to see that the tensor is antisymmetri-
eal.
<-/> nf tfir S
If (27) is multiplied by <^ f] V ( mixed multiplicjition),
then a tensor is obtained. Tho tirst member of the right
hand side of (27) can be written in the form
6 A
a, (r
t.t \i I;M.ISI;D TIIKOUY <>i if KMTI\ 1 1 \ J:><)
If we rc,d:iee /" / ^.\ I v A"^. /" v ^ A hy
/"<r , r / M J
A and rojthicc in llic ti-insforiiind first inoinliop
^ /i ,,,,, 9-"""
9", 9-V
with tlio help of (^M-), then from tlie right-hand side of (27)
t here arises ;xn expression with seven terms, of which four
t-ancel. 1 here remnifis
This is the expression for the ext elision of n contra variant
of the second rank : extensions can also he formed I m
em-responding cont ravariant tensors of higher and lower
ranks.
We remark that in the same way. we can also form the
extension of a mixed tensor A
9A " i, ,) (, r)
(:> :h A ~ ~ V + / A
15\ the reduction of (US) with reference to the indices
^tr v
/i -tnd tr ( inner multiplication with I , we tret a con-
V ft /
t ravariant four- vector
9
17
I uixrii i.i: OK in;i, \TI\ITY
v of - with
On tlu 1 ;iccouiit of flic svmiiirtrv
( " )
reference to the indices (3. and K. the third member of tht
right hand side vanishes when A u " is an antisymrnetrical
tensor, which we assume here ; the second member can be
transformed according to (29a) ; we therefore get
A a _ i o I v a j
(4) : 7^~~8^7
p
This is the expression of the divergence of a oontra-
variant six-vector.
of t//f t//ic<:il ti /i.wr of i/ie second rank.
Let us form the reduction of (89) with reference to the
indices a and <r, we obtain remembering (29;i)
If we introduce into the last term the contravnriant
tensor A p<r = r/^ T A , it takes the form
T
-[">- A-.
L ^ J
If further A^* 7 is symmetrical it is reduced t<>
(,l \ I i; M.IM.D Til K( in u| UK I \TI \ IT1 1 > i
If instead <>i A: . introduce in ;i >imilar way the
symmetrical co-variant tensor A ^ g a </ A" . then
owing to (31) the last member can take the form
-,j A
J 8 r f*r
In the symmetrical case treated, (41) can be replaced by
-it her of the forms
I In.
( /-j A )
which we snail have to make use of afterwards.
$12. The Riemann-Christoffel Tensor.
\Ve now seek only those tensors, which can be
obtained from the fundamental tensor ^ \by differentiation
alone. It is found ea-il\ . Wo ]>ut in (27) instead of
i- A " tin- fundamental t-nsor // " and i^et from
IW PIMNCIPU. 01 l!i;i.\Tl\ m
it a new tensor, namely the extension of the fundamental
tensor. We can easily convince ourselves that this
vanishes identically. We prove it in the following way; we
substitute in (27)
8 A (^
[IV Q j: / \ P
f r (^p J
i.e., the extension of a four- vector.
Thus we get (by slightly changing the indices) the
tensor of the third rank
^ _ / _ ) r P _ y T r p _ \ (TT f _p
/xtrr a . -a- / ija, - " ; (a- ) n ( a..-
6,
W r e use these expressions for the formation of the ten>ur
A A . Thereby the following terms in A
/AOT I>.T(T fJ(TT
cancel the corresponding terms in A ^; the iirst member,
the fourth member, as well as the member corresponding
to tlie last term within the square bracket. These are all
symmetrical in <r, and T. The same is true for the sum of
the second and third members. We thus get
A - A = tt p A .
/iOT /ATO" fJUTT p
/HOT
I.I.M.I; \i.i>i.n TIII:OI;N or UKi,ATi\rn
The essential thing in this result is that on the
right hand nde of ( 1:2) we have only A , but not its
differential C0refficientf. From the tensor-character of A
fi(TT
A , and from the fact that A is an arbitrary four
flTO- p
vector, it follows, on account of the result of 7, that
B is a tensor (Kiemann-Christoffel Tensor).
fUTT
The mathematical significance of this tensor is as
follows; when the continuum is so shaped, that there is a
co-ordinate system for which y s are constants, B p all
[1.V /XCTT
vanish.
If \ve choose instead of the original co-ordinate system
any new one, so would the </ s referred to this last system
be no longer constants. The tensor character of B
/XCTT
shows us, however, that these components vanish collectively
also in any other chosen system of reference. The
vanishing of the Kiemann Tensor is thus a necessary con
dition that for -some choice of the axis-system y V can be
taken a> constant^ In our problem it corresponds to the
case when by a suitable choice of the co-ordinate system,
the special relativity theory holds throughout any finite
region. By the reduction of (1-3) with reference to indicc>
to r and p, we get the covarinnt tensor of the second rank
1- J1- ] iu.\( ii i.i; 01 KI.I. nivrn
ti/io/t I In- c/io/ce o/ co-ordinoti-cs. It has alrcadv
been remarked in 8, with reference to the equation (18a),
that the co-ordinates can with advantage be so choseu that
x/ = ! 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
/it fJLV
R .
/J.V
I shall give in the following pages all relations in the
simplified form, with the above-named specialisation of
the co-ordinates. It is then very easy to go back to the
general covariant equations, if it appears desirable in
any special case.
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 freelv moving body not acted on by external torcr-
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 K,, can be, and
are, so chosen that // s have special constant values of
the expression (4).
Let us discuss this motion from the stand-point of any
arbitrary co-ordinal O-M -stem K ; : it moves with reference to
K, (as explained in 2) in a gravitational field. The la\\>
(ii:\KI! M.ISKD THEORY (>K It Kl. \TI \ m
1:55
of motion With reference to K, follow easily from the
following consideration. \Yith reference to K,,, the law
of motion is a four-dimenfcional straight lino and tlm* a
geodesic 1 . As a geodetic-line is defined independently
of the system of oo-ordinates, it would also be the law of
motion for the motion of the material-point with reference
to K! ; If we put
v = - T ^
we get tho motion of the point with reference to K
jjiven by
J if. ./ dx
__ T =r T /(
,/X " / l /s- </>
\\ e now make the very simplo assumption that this
general oovariant system oi equations defines also the
motion of the point in the gravitational Held, 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 (1(5) contains only the first differentials of
// , among which there is no relation in the special case
when K,, exists.
If r T s vanish, the point moves uniformlv and in a
/*"
straight line; these magnitudes therefore determine the
deviation from uniformity. They are the components of
the gravitational field.
l- iC) I HI.M IIM.I. ol Kill. \Tlvm
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 :is matter : therefore the term
includes not only matter in the usual sense, but also the
electro-dynamic field. Our next problem is to seek the
Reid-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 which the
field-equations sought-for are evidently satisfied is that of
the special relativity theory in which </ s have certain
constant values. This would be the case in a certain
finite region with reference to a definite co-ordinate
>vstem K,,. With reference to this system, all the com
ponents B /J of the Riemann s Tensor ("equation \:>
vanish. These vanish then also in the region considered.
with reference to every other co-ordinate system.
The equations of the gravitation-field free from matter
must thus be in everv case satisfied when all B^ vanish.
P.<TT
But this condition is clearly one which goes too far. For
it is clear that the gravitation-field generated by a material
point in its own neighbourhood can never be transformed
,111;, t/ by any olioiiv of a\-s, i.e., it cannot be transformed
to a case of constant ft s.
Therefore it i> clear that, for a gravitational field free
from matter, it is desirable that the symmetrical ten-
BOW H deduced from the tensors 1^ should vanish.
.v r-
B ILI8! I) TIIF.oin Hi i!i:r. \TI\m 137
\\ethu-iget 1 (I equations for 1 quantities // which are
fulfilled in the special case when IV V all vanish.
Remembering ( I I) we see that in absence of matter
the field-equations come out as follows ; (when referred
to ihe special co-ordinate-system chosen.)
er; "
It fan also he shown that the ohoiee of these equa
tions is connected with a minimum of arbitrariness. For
liesides l> , there is no tensor of the second rank, which
/"
can lie built out of // s and their derivatives no higher
/"
than the second, and which is also linear in them.
It will be shown that the equations arising in a, purely
mathematical wa\ out of the conditions of the general
relativity, together with equations (Hi), givo us the Xe\\-
loman law of attraction as a tirst approximation, and lead
in the second approximation t<> the explanation of the
perihelion-motion of mercury diseovertd by Leverrier
(the residual ellect 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
cornctne-s i.f my theory.
l- i s I Hl N CIIT.I. Or I!K1. \TIVITV
vH5. Hamiltonian Function for the Gravitation-field.
Laws of Impulse and Energy.
In order to show that the field equations correspond to
the laws of impulse ;uid energy, it is most convenient to
write it in the following Hamiltonian form :
f f
| 8 I Hdr=o
Here the variations vanish at the limits of the finite
four-dimensional integration-space considered.
It is first necessary to show that the form (47 n) is
equivalent to equations (47). For this purpose, I.-t us
consider H as a function of f and <f v ( ::. >
We have at I list
, 9^, A , 9 ",,A _ 9 ,,
\ 3, 6- 3 S
<.i MI; \Usi.u TIIKOIIS DI IM-.I. \II\IM 1 )
The terms arising out of the two last terms within tin-
bracket are of different si^ris, ami change into one
another bv the interchange of the indices /* and ft. They
cancel each other in the expression forSH, when they are
multiplied by f , which is symmetrical with respect to
fj and ft, so that only the first member of the bracket
remains for our consideration. Remembering ( 31), we
thus have :
ftft ra i> ft a
Therefore
f 9H r r ft
uv~ f*P i
! 9 f
(48) 8H r ,r
[ 9. /r
If we now cam out thr \ariation> in ( IJa), we obtain
the system of equations
/
a / an v an
(^ h ) o , ( , ) - , =<>.
O ,< v ~ ii r I -> // r
a o , / O f/
which, nwinir to the relation.- (|s\ eoineide with (1-7),
as was rr(|iiired to be proved.
If (47b) is multiplied by / .
14-0 i JM.NTII l.l. (M KM, \TIVm
aud consequently
r a / an { a / / 9"
< r a.- V .,// a.-;. V " a
we obtain the equation
_a_
a.*
8H
37.
L **
Owinsr to the relations (48), the equations (4?) and (:>4).
fjifh /" A ,,/"V a r-/^
It is to be noticed that f L is not a tensor, so that (lie
<r
<iuatioii (4 9) holds only for systems t or which \ ,/ = ].
This equation expresses tin- laws of conservation of i in pulse
and energy in -ravitation-field. In fact, the inte-ra-
tiou of this equation over ;i tlncc-dimcnsional vohnne V
leads to the four equations
(40a)
r
(I] \l.|i \I.1M-.I 1 MKollV (H- IJKI.ATI VITY
where <> , , >i .. , ., are the direction-cosines <>F the inward-
drawn normal to the -urFaee-elemeut i/S in the Kuclidean
Sense. \V- reco^ni-e iii this the usual expression For the
laws ol conservation. \\ e denote the ma^nituiles I ^ as the
energy-components oF the gravitation-field.
I will now put the equation (1-7) in a third Form which
will be ver\ -erviceable For a quick realisation of our object.
By multiplxiii" 1 the field-equations (17) with <j r , these are
obtained in the mixed Forms. IF we remember that
,.< a r ;.__ 6 , a/"
9 ,._ -a., V r f)" a r /-
which ownm to (:U) is equal to
^a ( rr i>" ] " /r ^ r ^
- v r/ V r"
or slight ly altering the notation equal to
6
The third member oF tin- expression cancel with the
second member of the field-equations (17V In place oF
the second term of this r\pre><ion. we ran, on account oF
the relations (")H\ put
( " - 1 A r f \ .
V / -j /
I ll I XUI I.K OK I!M.\ | |\ rn
Therefore in UK- place of the equations (17), wo obtain
a
^16. General formulation of the field-equation
of Gravitation.
The lield-e<|uations established in the preceding para-
graph for spaces free from matter is to be compared with
the equation V - <=" of the Newtonian theorv. We have
now to find the equations which will correspond to
Poisson s Equation V 2 <j!> = ^7r^. (/, signifies the density of
matter) .
The special relativity theorv has led to the conception
that the inertial mass (Triige Masse) is no other than
energy. It can also be fully expressed mathematically by
a symmetrical tensor of the. second rank, the energy-tensor.
We have therefore to introduce in our generalised theory
energy-tensor ~Y a associated with matter, which like the
energy components 1 ir of (he gravitation-field (equal ion-.
10, and ")()) have a mixeil character but which however can
be connected with synimetrieal eovariant tensors. The
equation (")!) teaches us how to introduce the energy-tensor
(corresponding to tin density of 1 oisson s equation) in the
field e|iiatioiis of gravitation. If we consider a complete
system (for example the Sular->\ stem its total ina^-. a-
aiso it> total i;ra\ itatin^ action, v.ill depend on the total
.neiL, r \ of the system, ponderable as well a> ^ravitat ional.
CKVI .I; VI.KKD TIIKOUY or KKI. \Ti\m l-l-"-
\ \\\> can !>< expi (s>cil, hv putting in (")!), in place of
energy-components / ct gravitation-field alone the ^um
of I hi- energy-components of matter and gravitation, i.e.,
\Ye thus i;et instead of (51), the teiisor-pijnation
where T T 1 " (Laife s Sealar^. The<- ;nv tin- general tiehl-
M
e(|tialions i)l gravitation in the mixed form. In plaei of
(47), we Ljet.ltv working l);iekvvards the svstem
It must be admitted, that this introduction of the
rLT\ -tensor of matter cannot he justified l>\ means of the
Relativity- Postulate alone; for we have in the fore^oin^-
analysis deduced it from the condition that the ner-\ of
the gravitation-field should exert gravitating action in the
same \\a\- a- every other kind of energy. The stroii^e-!
ground foi the choice id the al ove eijiiilion however lies in
this, that they lead, as their <-..iisri|iirnce>, to equations
--IIIL; the eon>crvatioM of the comjioneiit > of total
energy (the ini|iul>e- and the ciieru\ which \aetly
correspond to the e.|iiati"ii- (I! ) and i M a . This shall lie
shown afterward-
1 t I IMMXrlN.h. i| i;i.I, \T| VITY
vj 17. The laws of conservation in the general case.
The equations (5:2) can be easily so transformed that
the second member on the right-band side vanishes. \\ c
reduce (or!) with reference to the indices p- and o- and
subtract the equation so obtained after multiplication with
.1 ^ from (5-2).
AYe obtain.
we i>|)eratc on il l>y ,., . Now.
9 . 9 V
. r * """( ">
2 8... 6. .. L P V 8*,
u rr
I lic t nst and the third member of the round bracket
lead to t .\]ifc>sions which cancel one another. a> can be
eusilv >e-n by interchaii-in- the -nmniation-indices u, and
,r. on the one hand, and ft and A. on the other.
! i; u.is|.-.|i TiiKui,^ in 1:1:1. \Ti\rn ll. t
The M cond term c;in be transformed according to (>!).
Sr, that u,. et
_
- - a . a^ a. v
The second member of the t-xprcssioii on the Ifi t-liand
-idr of ("):2a) leads first to
1 a 1 / ,//*
2 a. a.r I f/
, (/
a--, a, a
+ a..
The exprerokm arising ont of the last incmlicr \\-itliin
the round brackel vanishes ftcooiding to ( % 29) on aooonnt
of tlic flioiot- of axes. The two others can l>e taken
ton-ether and ni\r u> on account of ( >!), the rxpression
So that reroetnbering (">!) we li
a, 9 ^a, (^-
i ^<r A/^ ,_ \
V r w ) **
identical I v.
19
1 1C, I lilN rUM.r. ()! ItKI. \TIYI I Y
I Yoni (5.)) jind (~>:l-,i} i! follows I hat
From the field equations of gravitation, it also follows
that the conservation-laws of impulse and energy are
satisfied. \Ve see it most simply following the >atne
reasoning which lead to equations (I .hi); onlv 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.
8 ^
If we multiply (5-5) with . we L;-et ill a w;iv
(T
similar to 15, remembering that
a/ 1
:/,. =H vanishes,
a/" ~ ^
the equations I" =_
8 8.. /"
or remembering (:>(>)
(57) g-/ : + : fV=
4 comparison with (Ml>) shows that these equations
Pof the above choice of co-onlinates ( x ,/ = 1) asserts
nothiiiLl hut the vanishing of the di\eruence .if the tensor
(jf the eiier^ v-coiiiponents ul matlei.
(. K\ i i; \ i.isi- D IIII.OIM ni KI;I, \n\iTY 117
IMivsieallv the appearance of the second term on the
left-hand >ide shows that for matter alone the law of con-
iervation of impulse and energy cannot hold; or can only
hold when g^ a are eon>tants ; i.e., when the tield of gravi
tation vanishes. The second memhei i> ;in e\|H .^-ion for
impulse and euergj whic!ithe gravitation-field exe:ls pel-
time and per volume upon matter. This comes out clearer
when instead of (57 we write it in the Form of (17).
9 C- a
9 ; =-r: f i ^
The ri-ht-liand side expresses the interaction of the energy
of the gravitatioiiftl-iifld on matter. The field-equations of
gravitation contain thus at the same time I conditions
which are to be satisfied by all material phenomena. We
^{ the ei|iiations of the material phenomena completely
when the latter i> characterix-d In four other differential
equation;- independent of one another.
D THE MATERIAL" PHENOMENA.
The Mathematical auxiliaries developed under H ! at
once enal)ie> u> to ^cnei-ali-e, according to the generalised
theor\ of relativitx, the physical law> of matter (Hydrody-
namies, Maxwell s I llectro-dynainies a> the\ lie already
formulated according to the special-retain ity-theory.
The generalised Kelativitx Principle leads us to no further
limitation of po->il>ilitics ; hut it eiiahles u> to know
exactly the influence of gravitation on all processes with
out the introduction of any new hypothesis.
It is owinii to tins, that as regard- the physical nature
of matter 1,111 a narrou >en>e) :.o delinitt iie:-e--ar\ anni|i-
tion> are to he introduced. The i|ite-tioii ma\ he open
I I s i i; IN (i i !,!; (>! i;i<;r,\TiviTY
whether the theories of the clectro-maguetic field and the
gravitational-field together, will Jonn a sufficient basis for
the theory of matter. The general relativity jo$tulate can
teach us no new principle. But by building up tin-
theory it must be shown whether electro-magnetism and
gravitation together can achieve what the former alone
did not succeed in doin<r.
19. Euler s equations for frictionless adiabatic
liquid.
Let // and p, be two scalars, of which the first denotes
the pressure and the last the density of the fluid ; between
them there is a relation. Let the eontrawriant symmetrical
tensor
,j ,j il.r / /)
ff
be the contra-variant energy -tensor of the liquid. To it
also belongs the covarianl tensor
/ /i + / , / ,, ,- p
fir i><> ,/.. fl> ,/>
> well a> the mixed tensor
It we put the ri^ht-liand side ol ^OSb) in i^j/a) we
^t t the ^eiieriil h\ drod\ ii;iinical ei|iialiuns of Kuler accord
ing tn the g --ne raided relativity theory. This in \>\ inciplc
Completely solves the problem of motion: for the four
i,KNKKAI,IM ,l> I lltdin 01 IJKI.A I l\ ITV II 1 .
equations (.")?a) together \vitli the ^iveii equation between
// and f >, and the equation
./ p - 1
V ,ls ,/*
are Millieient, with the ^iven values of y ,,, for finding
out the six unknowns
(/X I/S l If l/S
If // s are unknown we have also to take the equ-
tions (*)-i). There are now 11 equations for rinding out
10 functions // , so that the number is more than sufii-
cient. Now it is be noticed that the equation (o?a) is
already contained in ("> $), so that the latter only represents
(7) independent equations. This indeliniteness is due to
the wide freedom in the choice of co-ordinates, so that
mathematically the problem is indefinite in the sense that
of the gpace-fuuetiona can be arbitrarily chosen.
$20. Maxwell s Electro-Magnetic field- equations.
Let <f> be the components of a covariant four-vector.
the electro-tnagnetic potential ; from it let iis form accord
ing to (. US) the components I- ul the eovariant >i \-vector
of the electro-magnetic field aeeordin^- \<> the >\>tem of
J50 i-ui \CIPI.K oi I;KI, \TI\ITV
From (") .), it follows that the system ol equations
9F
9 F,
a<
is satisfied of which the left-hand side, according to
(o?), is MM anti-s\ mmetrical tensor of the third kind.
This system (HO) contains essentially four equations, which
can be thus written :
((lOa)
a-.
This system of eqaatioDfl correeponds to the second
system of equations of Maxwell. \Ve see it at once if we
put
( I Y, = II, ! ,< = K,
Instead of (OOa) we can therefore write according to
the usual notation of three-dimensional vector-analysis:
an
a/
.it K=(
div 1I=,
OBNERALTABD TIITOIM or RBT*ATIVIT\ 1 > I
The litst Maxwellian system is obtained by a genera -
lisation of the t onn n iven liy Miukowski.
\Ve introduce the (-out ra-variant six-vector 1 \
P
tin- f|iu(i()n
and also a contra-variant four-vector .l / , \\liicli is the
electrical current-deusitv in vacuum. Then remembering
(40) we can establish the system of equations, which
remains invariant for any substitution with determinant I
g to our choice of co-ordinates).
It we put
, I 11 = H ., K" = } /.
(I )
F" = H K S1 = - M
which iuantities become e(|iial to II, . !] in the ra>e of
the special relativity theory, and besides
.1 =
we Gjet instead of ((> $)
r
= P
,-ot H - = *
a/
152 NMxni i.i: or 1:1:1. \TI VITY
The p(|uations (( ()). ((>:!) and (<>.">) give ihus a generali
sation of Maxwell s field-equations in vacuum, which
remains true iti our chosen system of co-ordinates.
T/ti energy-components <>/ I In- electTO-ntttffneftc fi /<I .
Let us form 1 he inner-product
>:>) K = K -I .
,<r <TJ>
According to (01) its components ean l>e written down
in the three-dimensional notation.
/. H].
(. K
K is a eovariant four-vector whose components are equal
cr
to the negative impulse and energy which are transferred
to ill" electro-magnetic field per unit oi time, and per unit
of volume, bv the electrical masses. If the electrical
masses be free, that is, under the influence of the electro
magnetic field onlv, then the eovariant four-vector
K will vanish.
(T
In order to get the energy components T of the elec
tro-magnetic field, we require only to give to the equation
K o, the form of the equation (">7).
(T
From (! }) and ((>">) we get first,
6 ,, ,,, W <" 9 "
a, ( v e
riE.VElMUSED THKORY OF RELATIVITY 1 53
On account f (60) the second member on the right-hand
side admits of the transformation
ft V
to symmetry, this expression can also be written in
the form
which can also be put in the form
1 9 .< \ ll ft /* /
+ F ft F 2 ( ^ l Vft }
a p ^ v 6 V
The first of these terms r-:m be written short lv :i-~
:in I the second after difl erentiation can be transformed in
the form
154 NM\( I I Li: ul
11 we take all the throe terms together, we get the
relation
<
where
(<Hk) T"=- F F v " + \ 8 1 F
(Td
a/?
On account of (30) the equation (66) become? equivalent
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 gravitation-field and matter satisfy when we use a
co-ordinate system for which \/ . / = 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 O variant-equations. Still the question is not without formal
interest, whether, when the energy-components of the
gravitation-Held 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 (06), and the field-
equations of gravitation hold in the form (52) or (52a) ;
such that on the left-hand side, we have a divergence in the
usual sense, and on the right-hand side, the sum of the
energy-components of matter and gravitation. J have
tiBNfcK \USKH TIIMilM (>! KKLATIVm I -V)
found out thai this is indeed the ease, lint I am of opinion
that the eominnnicat ion of my rather comprehensive work-
on this subject will not pay, for nothing essentially ne\\
conies 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 general, in which q s have constant values (4).
p.v
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 // \ differ from (4) only by small magnitudes (com
pared to I) where we can neglect small quantities of the
second and higher orders (first asjK e.t of the approxima
tion.)
Further it should be assumed that within the >pace-
,
time region considered, f] s at infinite distances (JIMIIL:
the word infinite in a spatial sense) can, by a suitable choice
of co-ordinates, tend to the limiting values (4); /.<., we con
sider only those gravitational fields which can be regarded
as produced by masses distributed over finite regions.
\Ve can assume that this approximation should lead to
Newton s theory. For it however, it is necessary to treat
the fundamental equations from another |M>in<t <>f view.
Let us consider the motion of a particle according to the
equation (4fi). In the case of the special relativity theory,
the components
156 I ltl.NCll Lh OF KELATivm
can take any values : This signifies that any
-- y + (;;;; >%< g y
can appear which is less than the velocity of light in
vacuum (r <1). If we finally limit ourselves to the
consideration of the case when /,- is small compared to the
velocity of light, it signifies that the components
/( 1 <k 2 /- .I
.>) 8oiilv Juiit*iio- ivf r^ ui f lv<wx
can be treated as small quantities, whereas is equal In
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 \~ s are all small quantities of
at least the first order. A glance at (4(5) 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 :
= f \ i- where ds=<1s t =<//.
or by limiting ourselves only to those terms which according
to the first stand-point are approximations of the first
order,
,/ .x, r 4 4 -I
,// -i J-
<;I.M.K.\I.I>I.I> niKoio 01 KKLATIVITY l">?
It we further ;tume that the gravitation-field is
quasi-static, i.e., it is limited only to the case when the
matter producing the gravitation-Hold is moving slowh
(relative to the velocity of light) we can neglect the
differentiations of the positional co-ordinates on the ni;ht-
hand side with respect to time, so that \ve get -idj lt
<" 9</4 +
11171 ,fi.- T =-i a^ (, = , a, )
Tins i> the equation [of motion of a material point
according to Newton s theory, where ff tt / t plays the part of
gravitational potential. The remarkable thing in the
result is that in the first-approximation of motion of the
material point, only the component y tl of the fundamental
tensor appears.
Let us now turn to the h eld-equation (5o). In this
case, we have to re mem her that the energy-tensor of
matter is exclusively defined in a narrow sense bv the
density/* of matter, >.<-., by the second member on the
right-hand side of 58 [(">Sa, or 5M>)]. If we make the
necessary approximations, then all component vanish
except
T , , = i* ~ T
On the left-hand sidr of (r,:)) the second term is an
infinitesimal of the second order, so that the first leads to
the following term- in the approximation, which are rather
interesting fur us ;
6 r^ i 4 a [>"! 4 a r /u/ i a r^i
a-l J f e-.L 2 J + a. ,L J 8^L .J-
By neglecting all different iat ions with regard to time,
this leads, when ^=^=4, to the expression
_! / 6 2 . /, ,6 - >,,, , a 2 :/,, v x
-^ ( 6"" + 6 - + 6 <7 "-
158 i IM\( IIM.K OK ULLATIMIN
The last of the equations (53) thus leads to
(68) VY,,=^-
The equations (67) and (68) together, ai>e equivalent to
Newton s law of gravitation.
For the gravitation-potential we get from (67) and (68)
the exp.
(68a.)
whereas the Newtonian theory for the chosen unit of time
gi ves
% I .
wliere K denotes usually the
gravitation-constant. 67xlO~ s : equating them we tret
SirK
(6!) * = = Ks7xlO-".
$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 a theory as a first approxi
mation we had to calculate only //,.,, out of the 10 coni|X)-
nt iits // of the gravitation-potential, for that is the only
component which comes in the first approximate equal ion>-
of motion of a material point in a gravitational field.
We M-e however, that the other components of </
should also differ from the values- given in (1-) as required by
the condition / = 1 .
flF.N ERALISF.i) T1IKOHY OK HKLATIVJTY l">y
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 :
S is 1 or 0. according as p = o- or not and / is the quantity
...
On account of (68a) \VP have
(70a) . . =
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 investigate the influences which the field
ef mass M will have upon the metrical pro|>erties of the
Held. Between the lengths and times measured locally on
the one hand, and the differences in co-ordinutes d.r on the
V
other, \ve have the relation
For a unit me:i8iiring rod, for example, placed parallel to
the axis, we have to put
./ = -1. i/.r,=,f.- s =,/., 4= o
then -!=</ /-".
100 PKixriPi.i. 01 i;i;i.\Ti\m
If the unit measuring rod lies on the axis, the tirst of
the equations ( r O) gives
From both these relations it follows as a first approxi
mation that
(71) ,/. = !- 4 .
The unit measuring rod appears, when referred to the
co-ordinate-system, shortened by the calculated magnitude
through the presence of the gravitational Held, when we
place it radially in the field.
Similarly we can get its co-ordinate-length in a
tangential |x>sition, if we put for example
(/x 2 = 1. </.",_ =</.( _, =</, =o , .i. 1 !! . .!.,=. 3 =O
we then get
i71a.) -!=. /, /!; = - "I
The gravitational field has no influence upon the length
of the rod, \vhon we put it tangentially in the field.
Thus Euclidean geometry docs nol hold in the giavi-
tational lield O\MI 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 (fi!)) shows that the
ex]>ected difference is much too small to l>e noticeable
in the measurement of earth s surface.
AVe would further investigate the rate of going of ;i
i mil -clock- which is placed in a statical gravitational field.
Mere we have for a period of the cluck
:i \Ki; VI.ISKH TIIKiHiY OF KKI.ATI VITY Kil
then we h:ive
/,=! +
I f "
Therefore the clock goes slowly what it is place*! 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 Held. According to the special relati
vity theory, tin* velocity of light is given by the equation
thus also according to the generalised relativity theory it
is given by the e<|ii:it ion
(73 * >9 V*J (/ =0
If the direction, >., the ratio d , : <l,<\ : //-, is gi\Mi.
ihe ejii!ition (7:J) gives the magnitudes
and with it the velocity,
a
jl .-> PR1NTIPLK OK KEJ-A.T1VTTV
in the sense of the Kuclidean Geometry. vVe can easily see
that, with reference to the co-ordinate system, the rays of
light must appear curved in case // *s are not constants.
If a be the direction perpendicular to the direction
of propagation, we have, from II uy gen s principle, that
light-rays ] (taken in Ihe plane (y, )] must suffer a
curvature ^-? .
A Light-ray
Let us Hnd out the curvature which a light-ray suffers
when it goes by a mass M at a distance A from it. If we
use the co-ordinate system according to the above scheme,
then the totsil bending B of light-rays (reckoned positive
when it is eonca.ve to the origin) is given as ;i sufficient
approximation bv
where (?. ) and (70) gives
, ,
\ ray of light just gra/ing the sun would suffer :i beml-
ing of 1 7", whereas one coming by Jupiter would have
a deviation of about -0:2".
KD THKOKV OF KKLVTIVITY IM
If we calculate the gravitation-field to a greater order
of approximation and with it the corresponding path
of a material particle of a relatively small (infinitesimal)
mass we get a deviation of the following kind from the
Kepler-Newtonian Laws of Planetary motion. The Kllipse
of Planetary motion suffers ;i slow rotation in the direction
of motion, of amount
( 7. ) i .<= 7r ," S per revolution.
T"c*(l " )
In this Formula ft signifies the semi-major axis, c,
the velocity of light, measured in the usual way, f, 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 suffi
ciently to what has been found by astronomers (Leverrier).
They found a residual perihelion motion of this planet of
the given magnitude which can not I>e explained b\ the
perturbation of the other planet-.
NOTES
Note 1. The fundamental electro-magnetic equation
of Maxwell for stationary media an> :
<mrlH=I
D-//K
According to Hertx and Hoaviside, these require modi
fication in the case of moving bodies.
Now it is known that due to motion alone there is a
change in a vector /? given by
( _ - - ) due to motion //, div \\ +enrl \Uu\
\ o /
where is the vector velocity of the moving bodv and
[Rw] the vector product, of \\ and //.
Hence equations (1) and (2) become
I- curl H= ~ +v div D-fcurl \ ect. [Dv]+pv (1-1)
and
K= -- |- div H -j- curl \Vrti.
which gives linally, for p o and div H = O,
+ n div l)=:r curl 111- Vwfc, Dl )
O /
,H ) -
166 PKINCIPT.K OK ItET.ATIVrn
Let us consider a beam travelling along the ./--axis,
with apparent velocity / (i.e., velocity with respect to the
Hxed ether) in medium moving with velocity n, = n in the
same direction.
Then if the electric and magnetic vectors are
f - A (x-vt)
proportional to e , we have
f-=/A, |j = -/Ar, |- = =0, ,/, = ;,. =0
5." 6* dy 8-
Since D = K E and B = /A H, we have
. ... (1-22)
/x (;-.) H,=cE v ... (2-23)
Multiplying (1 23), by (2 28)
/A K (f 7 ) 2 =C 2
Hence (i -//) 2 =c s /^ = . 5f
. . = + w,
making Fresnelian convection co-efficient simply unity.
Equations (1 21), and (^ ^l) may bo obtained more
simply from physical 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 tixed with respect
to the ether, the rate of change of electric polarisation is
5D
NOTE* 167
Consider a siab of matter moving with velocity n,
along- the ./--axis, then even in a stationary field of
electrostatic polarisation, that is, for a Held in which
= o, there will be some- change in the polarisation of
or
the body due to its motion, given by u r . Hence we
O <:
must add this term to a purely temporal rate of change
-^- . Doing this we immediately arrive at equations
Of
(T21) and (2 21) for the special case considered there.
Thus the Hert/-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, Larmor lias shown (Aether and Matter)
that I I/ft 2 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 away by it. Thus the electromagnetic effect
outside a moving medium should be proportional to K, the
specific inductive capacity.
Rvic/mtil showed in l^itl that when a charged condenser
is rapidly rotated (the dielectric remaining stationary),
the magnetic effect outside is proportional to K, the Sp.
Ind. Cap.
/, ,;/////,-// (Annaleu .lev IMiysik 1SSS, 1890) found that
if the dielectric is rotated while the condenser remains
stationary, the effect is proportional to K 1.
168 PRINCIPLE OF RELATIVITY
Kickenwalil (Annaleu der Physik 1903, IVOV) rotated
together both condenser and dielectric and fouud that the
magnetic effect was proportional to the potential difference
and to the angular velocity, but was completely independent
of K. This is of course quite consistent with Rowland
and Rb ntgeu.
Blotidlot (Coinptes Reudus, 1 JOl) passed a current
of air in a steady magnetic Held II,, (H =H,=0). If
this current of air moves with velocity / , along the
.F-axis, an electromotive force would be set up along the
r-axis, due to the relative motion of matter and magnetic
f-ubt s <>f induction. A pair of plates at .*= + a, will be
charged up with density /i=D, = KE =K. n. H y /c.
But Blundlot failed to detect any such effect.
//. A. intsnn (Phil. Trans. Kuyal Sot-. 190i) repeated
the experiment with a cylindrical condenser made of
ebouy, rotating in a magnetic Held parallel to its own
axi^ He observed a change proportional to K I and
not to K.
Thus the above set of electro-magnet ic experiments
contradict the llcrt/.-lleaviside equations, and these must
be abandoned.
[P. C. M.]
Not 2. Lur iii: Ti ii itxfoi m/i/ i<nt.
Lorentz. \rrsuch einer theorie der elektri-chen uud
optibfhon Ki - C lu ijinn^fn im bewegteii Korpern.
(Leiden 1895).
Lorent/.. Theory of .MK-ctrnns (English eilition),
pages l:r---Dii. i30,also notes < >. 86, pages :;is. :;-:s.
Lorentx. wanted to explain the (Vfichelson-Morley
null-effect. In order to do 50, it was obviously necessary
to explain tin- I it/.gerald COUtrACtlOD. Lorentx, worked
on the hypothesis (hat an electron itself undergoes
NOTES 169
contraction when moving. He introduced new variables
for the moving system defined by the following set of
equations.
and for velocities, used
v.*=p*v, + i , v, l =(3r,, r r l =(3r, and p l =p/fi.
With the help of the above set of equations, which is.
known as the Lorentz transformation, he succeeded in
showing how the Fitzgerald contraction results-! as a
consequence of fortuitous compensation of opposing
effects."
It should be observed that the Lorentz transformation
is not identical with the Einstein transformation. The
Kinsteinian addition of velocities is quite different as
also the expression for the relative" density of electricity.
It is true that the Max we 11- Lorentz Held equations
remain practical/ 1/ unchanged by the Lorentz transforma
tion, but they ar>- changed to some slight extent. UUP
marked advantage of the Einstein transformation con^i-t>
in the fact that the Held equations of a moving system
preserve exactly the same form as those of a stationary
M -torn.
It should also be noted that the l ; resin-Han convection
coctKcient comes out in the theory of relativity as a direct
consequence of Kin-tern - addition of velocities- and is
quite independent o! any electrical theory nf mattrr.
[P.C If.]
Note 3.
Lorent/., Theory of Klectrons (English edition),
181, page 513.
170 I ttlNrill.F MI RELATIVITY
H. Poiucare, Sur la dynamujue electron, Rendiconti
del circolo matematico di Palermo 21 (1906).
[P. C. M.]
Note 4. lielaticity Theorem <ai<l Kelativiiy-Principle.
Lorentz showed that the Maxwell-Lorentz system
of electromagnetic field-equations remained practically
unchanged bv the Lorentz transformation. Thus the
electromagnetic laws of Maxwell and Lorentz can be
definitely proved " to be independent of the manner in
which they arc referred to two coordinate systems which
have a uniform translatory motion relative to each other."
(See " Electrodynamics of Moving Bodies," page 5.) Thus
so far as the electromagnetic laws are concerned, the
principle of relativity ant he proved lo be fntf.
But it is not known whether this principle will remain
true in the case of other physical laws. We can always
proceed on the assumption that it does remain true. Thus
it is always possible to construct physical laws in such a
way that they retain their form wln-n referred to moving
coordinates. The ultimate ground for formulating physi
cal laws in this way is merely a subjective conviction that
the principle of relativity is universally true. There is
no a priori logical neccs^it \ t!:;it it should be so. Hence
the Principle of Relativity (so I ar as it is applied to
phenomtMin other than electromagnetic ) must be regarded
as a jjostiiliil.t.-, which we have u.-Mimed to be true, but for
which \\v cannot adduce any definite proof, until after
the generalisation is made and its consequences tested in
the light of actual experience.
[P. C. M.j
Note 5.
See " Kli-ctrodynamic> oi Moving Hodies," p. 5-8.
VOTES 171
Note 6. / />/,/ fymi/f aim in l/////-0jrj/r* l- onii.
Equation! (/) :md (//) become when expanded into
Cartesians :
8*1 8*, a,
^ a o " /> ,
o y o - o T
,_.
T a^ av^"" r
8 MI, 8>", 9 ,
o >s~" ~" ^ =/>-
a.-- a// 8r j
Substituting ;c, , ,, 2 , .-i- 3> .r , for .<, ;/, r, and ir ; and p, ,
f.jj pj, p^ for pi( . t />"yi pit : ) ipi where /^= \/ l t
We get,
a s a^i, a^j "^
"8;r, ~ ? 8
a? : 9m, ,-9j\
ar,~ aa-,~*9^
and multiplyinG: (2 1) by / \v<> get
a/ * , a - , a/ - 1
Now substitute
=/a * = / I a id /V,=/ 4I= / it
711 =/S 1 = ""/I S * , =/ 4 * = /,
172 HI? I \CTPT.K OP RELATIVITY
and \ve irct finally :
8./. i a/,, 8/ S4
d^t-^+iSt 88 ^
... (3)
a/,,
, 8/, 4
+* 5 *
8/41 , s/*. , a/..,
[P. C. M.j
Note 9. On the Constancy of the Velocity of /,//////.
Page i:> refer also to page 6, of Einstein s paper.
One of the two fundamental Postulates of the Principle
of Relativity is that the velocity of light should remain
constant whether the source is moving or stationary. It
follows that even if a radiant source S move with a velocity
u, it should always remain the centre of spherical wave-
expanding outwards with velocitv 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 + it, when the
source o! light moves towards the observer with the
velocity n.
Prof, de Sitter has given an astronomical argument for
deciding between these two divergent views. Let u<
suppose there is a double star of which one is revolving
about the common centre of gravity in a circular orbit.
VOTES 173
Let the observer lie 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 + it
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 ha If- period from
B to A is approximately T *- and from A to B is
T+ 2 -Al l . 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 onlv of the same <>nler a> T, but are mostly
c a
much larger. For example, if ;<=!()() //// sec, T = b da\s,
A/c = 33 years (corresponding to an annual parallax of -1"),
then T i/^A/c.- - =o. The existence of the Spectroscupio
binaries, and the fact that they follow Kepler s Law is
therefore a proof thai c is not ad octed by the motion of
the source.
In a later memoir, replying t ^ the criticisms of
Freun llich and (liinthiek that an apparent ecoontripity
occurs in the motion proportional to / "A,;*, ",i being the
17 t rillNCll LE Ot IIELATIVIIY
maximum value of t> , the velocity of light emitted being
7/,,=c + X", /t = Lorentz- Einstein
/=! Ritz.
Prof, de Sitter admits the validity of the criticisms. But
he remarks that an upper value of k may be calculated from
the observations of the double sar /3-Aurigae. For this star.
The parallax 7r = OU", e = 005, w n = 110 /f-w/sec T = 3 96,
A > 05 light-years,
/ is < -OOe.
Fji an experimental proof, see a paper by C. Majorana.
Phil. Mag., Vol. 35, p. 1C,:}.
[M. N. S.]
Note 10. Urtl-tlcnuili, <>f Muctricity.
It /> is the volume density in a moving system then
p \ / ([ n -} is the corresponding quantity 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.
[P. a li.]
Note 11 (page 17).
SpaM-limc Ir <}/<> r^ of lk: //>*/ uml I / .svv.W khnl,
As we had already occasion to mention, Sommerfeld
hag, in two papers on four dimensional geometry ( vWr,
Annalen der Physik, Bd. 2, p. Tli) ; and Hd. :i: 1 ,, p. 640),
translated the ideas of Minkowski into the language of four
dimensional geometry. Instead of M inkowski s space-time
vector of the first kind, he uses the more expressive term
four-vector, thereby making it quite clear that it
represents a directed quantity like a straight line, a force
or a momentum, and has got 4 components, three in the
direction of space-axes, and one in the direction of the
time-axis.
VOTKS I7. r )
The representation of the plane (defined by two straight
lines) is much more difficult. In three dimensions, the
plane can he represented by the vector perpendicular to
itself. Hut that artifice is not available in four dimensions.
For the perpendicular to a plane, we now have not a single
line, but an infinite number of lines constituting a plane.
This difficulty has been overcome by Minkowski in a very
elegant manner which will become clear later on.
Meanwhile we offer the following extract from the
above mentioned work of Sommerfeld.
(Pp. 755, Hd. Z-2, Ann. d. Pliysik.)
" In order to have a better knowledge about the nature
of the six-vector (which is the same thing HS Minkowski s
space-time vector of the 2n<l kind) let us take the special
case of a piece of piano, having unit area (contents), and
the form of a parallelogram, bounded by the four-vectors
//, , passing through the origin. Then the projection of
this piece of plane on the .r// plane is given hv the
projections w,, //, /,, r a of the four vectors in the
combination
Let us form in a similar manner all the six component of
this plane <. Then six component* are not all independent
but are connected 1>\ the following relation
Further the contents ) < | of the piece of a plane is to
be defined as the square rout of the sum of the squares of
these six ipiantitio. In fact,
i 4| =*, +#,+#,++#. +<*>*.-
Let us now on the other hand take the ea-^e of the unit
plane <j>* normal to $ ; we can call this plane the
176 PRINCIPLE OF RELATIVITY
Complement of <f>. Then we have the following relations
between the components of the two plane :
The proof of these assertions is as follows. Let ;/*, r*
be the four vectors defining <*. Then we have the
following relations :
multiplying these equations by r* . ;/*-,or by r* . ?/* .
we obtain
<i>*: .$,1 + **, ,,=0 and <* y ^,, +^?. ^..,=0
from which we have
In a corresponding way we have
whon the subscript (//) denotes the component of </> in
the plane contained by the lines other than (; ). Therefore
the theorem is proved.
We have ^^*=^, <+..,
r* r,+v* ?+? * r.+rfr,=0
If we multiply these equations by v,, n,, r t , and
subtract the second from the first, the fourth from the
third we obtain
XOTKS 177
The ^eneral si i\- vector / is composed from the vectors
*" in the following way :
p and p* denoting the contents of the pieces of mutually
perpend ic ihir planes composing / . The "conjugate
Vector" f* (or it may be called the comi)lement of /*) is
obtained by interchanging p and //*
We have,
f* = ,,* 4, + ,, $*
We can verify that
/*, = /.i etc.
| / | - and (J f*} may be said to bo invariants of the six
vectors, for their values are independent of the choice of
the system of co-ordinates.
[M. N. 8.]
Note 12. Ligkt-tcfocity H* ti maximum.
Page -2-J, and Electro-dynamics of Moving Bodies,
p. 17.
Flitting r = c ., and /r = c A, \ve get
1 + (f-.r) (r-A)/c- c- -(-r- -(.r-fA)^-f , I
= , ^-(.r + A)
2<- -(.< + A) +J -A/c
Thus f<r, so lorig as | .rA | >().
Thus the velocity of light is the absolute maximum
velocity. We sh.il! i:n\v see the consequences of admitting
a velocity W >r.
Let A and B be separated by distance /, and let
velocity of a -Signal " in the system S be W>r. Let the
178 PRINCIPLE OF 11ELATIVITY
(observing) system S have velocity -H 1 with respect to
the system S.
Then velocity of signal with respect to system S is
Kivenbv 1T !**.,
Thus " time " from A to B as measured in S , i3 given
^ nw- zggD -, ... ;_; ;. :.: 0)
Now if r is less than e, then W being greater than c
(by hypothesis) W is greater than v, i.e., W>v.
Let W=? + /A and v = c A.
Then Wr = (c {- n)(c A) = c- + (/* + X)c /zA.
Now we can always choose v in such a way that Wv is
greater than c-, since Wr is >c- if (/* + A)c yuA is >0.
that is, if fji + \> .which can always be satisfied by
a suitable choice of A.
Thus for W><? we can always choose A in such a
way as to make W?->e 2 , i.e., I Wr/c 2 negative. But
\V / i>; always positive. Hence with W>c, we can
always make f , the time from A lo B in equation (1)
" negative." That is, the signal starting from A will reach
B (as observed in system S ) in less than no time. Thus the
effect will be perceived before the cause commences to act,
i.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 uot produce
any effect. Causal effect of any physical type can never
travel with a velocity greater than that of light.
[P. C. M.]
NOTES 179
Notes 13 and 14.
We have denoted the four-vector t by the matrix
I w i <0 2 W 3 w I It- is then at once seen that &&gt; denotes
the reciprocal matrix
!
>4
It is now evident that while <> =w\, w = A~ w
[<..] The vector-product of the four-vector w and -t
may be represented by the combination
[w] = w Sta
It is now easy to verify the formula ./ ! = A M /A.
Supposing for the sake of simplicity that / represents the
vector-product of two four-vectors a>, <v, we have
= [ A~ a w,sA A " *w
= A- 1 [(^-,va>JA = A
Now remembering that generally
Where p, p* are scalar (|iiantities, <^, <* are two
mutually perpendicular unit planes, there is no difficulty
in seeming that
/ = A-VA.
Note 15. The rector product (trf). (L\ 3(>).
This represents the vector product of ; four- vector and
a six-vector. Now as combinations of this type are of
180 l i;l\( ll I.K OF KEI.ATIVITY
frequent occurrence in this paper, it will be better to form
an idea of their geometrical meaning. The following
is taken from the above mentioned paper of Sommerfeld.
" 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
parallelepiped 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 parallelepipeds,
but it can always bo 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,
we do not obtain a vector of a more general type, but
one which can always be represented by a four-vector
(loc, cit. p. 75!)). The state of affairs here is Hie same as
in the ordinary vector calculus, where by the veetor-
mu Implication of a vector of the first, and a vector of the
second type (/. ., a polar vector), we obtain a vector of the
first tvpe (axial vector). Tl;c I unnal scheme of this
multiplication is taken from the three-dimensional ease.
Let A=(A,. A,,, A.) denote a vector of the first
type, B = (B,, . , B -.,, B., ,) denote a vector of the second
type. From this lust, let us form three special vectors of
the first kind, namely -
B,=(B rr , .B,,, B, s n
B, = (B,., B,,, B y .V-(B l4 = -B tl . B,,=0).
B.. =(B,,, 1} ., B..)J
Since H , , 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 U,, i.e.,
^A, B,,+A B,, +A. B,,.
MHKS 181
We see easily that this coincides with the usual rule
for the vector-product, c. y., fory = . .
= A, B,,-A.. B..,,
Correspondingly let us define in the four-dimciibional
case the product (P/) of any four- vector P and the six-
vector./". The /-component (j = -r, //, :, or /) is given by
(P/, ) = p,/v , + ?,/, , + p,,/, , + p./;, ,
Eac-h one of these components is obtained as the scalar
product of P, and the vector f , which is perpendicular to
j-axis, and is obtained from / bv the ruley , = [(f }1 , J , v >
We can also find out here the geometrical significance
of vectors of the third type, when ,/ =<, /.<?.,./ 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.* Tiie components of (P</>) are then eijual to the
1 three-rowed under-determiuants 1), D v D ; Di of the
matrix
P, P. P. P/
U,* U,* U,* I
\ V v * V : * \ ,*
Leavin aside the tirst column we obtain
which coincides with (P M according to our di-linition.
182
I llJNCll LE OF RELATIVITY
Examples of this type of vectors will be found ou
page 36, * = /t-F, the electrical-rest-force, and ^ = i>o/,*
ibe magnetic- rest-force. The rest-ray I) = ?o> [<I>i/T] * also
belong to the same type (page 89). It is easy to show
that
When (w n w 2 , o),)=o,
dimensional vector
=/. il reduces to the thi-ee-
Since in this case, *!=oj 4 F 14 =r, (the electric force)
^ l = i< 1 }J\^r=m J , (the magnetic force)
we have (Q) =
e,
m,
, analogous to the
Poynting-vector.
[M. N. S.]
Note 16. 27*e eleclric-rcd force. (Page 37.)
The four- rector ^> = o-F which is called b\ r Minkowaki
the clectric-rest-force (elektrische Rub-Kraft) is very
closely connected to Lorentz a Ponderomotive force, or
the force acting on a moving charge. If i> i? the density
of charge, we have, when e=l, //.= ], / .<"., for free space
Nn\v since p ( ,=:
\Vu have
l V 2 V 2
o* 4 =p[d.+ j (* .-*)]
J>. ^ c liavc j)ul the c
X.J>. \\\ have j)ut the component- of e equivalent
to ( /,, dy, </ : ), and the components of in equivalent to
NOTES 18i
// , // a //.), in accordance with the notation used in
Lorent/ s Theory of Electrons,
\Ye have therefore
i. /., /j ( , (<,, </>, </>., ) represents the force acting on the
electron. Compare Lorent/, Theory of Electrons, page 14.
The fourth component <, when multiplied by /j ()
represents /-times the rate at which work is done by
the moving electron, for />,, < 4 .=i p [> ,<!* + ?v7 + r.c?.] =
r Pn</ > i+ r y Po^2 + r .- Po^s- -%/ _ l times the power pos
sessed by the electron therefore represents the fourth
component, or the time component, of the force-four-
vector. This component was first introduced by Poineare
in 1900.
The four-vector ^ = /wK* has a similar relation to
the force acting on a moving magnetic pole.
Note 17. Operator " Lor" ( 1:2, p. 11).
\ \ p\
The operation ^- -5, Q ~, g ( , j which plays in
four-dimensional mechanics a role similar to that of
/ o cs o \
the operator I / fC-, + j o >+ ^ "aT= V ) m three-dimen
sional geometry has been called by ^TinUowski Lorent /-
Operation or shortly lor in honour of H. A. Lorent/,
the discoverer of the theorem of relativity. Later writers
have sometimes used the symbol Q to denote this
operation. In the above-mentioned paper (Annalen der
Physik, p. (ill), Bd. 33) Sommerfeld has introduced the
terms, Div (divergence), Hot ( Rotation), (I rad (gradient)
as four-dimensional extensions of the corresponding three-
dimensional operations in place of the general svmho!
lor. The ph\ sical si^ni ii-iiu-e of thes- operations will
184- I lUNCIl I.K ()! UKLATIVITY
become clear when along with Minkowski s met hod of
treatment we also study the geometrical method of
Sommerfeld. Minkowski begins here with the case of
lor S, where 8 is a six- vector (^pace-time vector of the
2nd kind).
This being a complicated case, we take the simpler
catffe of lor *,
where A- is a four- vector = j -fj, *, A-O * 4 |
and .s = i *,
The following geometrical method is taken from Som-
m erf eld.
Scalar Divergence Let A 2 denote a small four-dimen
sional volume of any shape in the neighbourhood of the
space-time point Q, //S denote the three-dimensional
bounding surface of A2, / be the outer normal to ifS.
Let S be anv four-vector, P N its normal component.
Then
Div S = Lim
Now if for A 2 we choose the four-dimensional paral
lelepiped with sides (//>-,, //.>, ^.r n , (?.r 4 } } we have then
O i O 2 C/ 3 O i
If f denotes a space-time vector of the second kind, lor
/ is equivalent to a space-time vector of the first kind. The
geometrical significance can be thus brought out. We have
seen fiat the operator lor beirives 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
XOTKS 185
tn see that lor S will lie a four-vector. Let us find
thr component of tliis f on r- vector in any direction *.
l.i t S denote the three-space whieh passes through the
point Q (./,, .r,, .r.,, .r 4 ) and is perpendicular to ,v, AS a
very small part of it in the region of Q, tfn- is an element
of its two-dimensional surface. Let the perpendicular
to this surface lying in the space be denoted by //, and
let / . denote the component of f in the plane of (*//)
which is evidently conjugate to the plane tlv. Then the
v-component of tho vector divergence of / because the
operator lor multiplies / vectorially)
= Divf.=Lim Ih^?.
A *=0 AS
Where the integration in </<r is to be extended over
the whole surface.
If now x is selected as the .r-direction, A* is then
a three-dimensional parallelopiped with the sides t1i/ } <h,
i//, then we have
and generallv
= + + - +
H.-nce the four-components of the four- vector lor S
or Div. / is a four-vector with the components given on
pag.- 1-2.
According to the formulae of space geometry, D,
denote-; a parallelopiped laid in the (//->0 space, formed
out of the vectors (P, P. ?,), ( IT * r* U*) (v* vt V*,).
18H I lilXCII LK 01 UKLATIVITY
D, is therefore the projection on the y-:-l space of
1he peralielopiped formed out of these three four- vectors
(P, U*, V*), and could as well be denoted by Dyxl.
We see directly that the four-vector of the kind represent
ed by (D,, D y , 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 given by the geometrical sum of the two four-vectors of
the first type PD and P*D*.
[M. N. S.]
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The principle of relat
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