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Title: The Principle of Relativity
Author: Albert Einstein
Hermann Minkowski
Translator: Meghnad N. Saha
Satyendra N. Bose
Release Date: December 14, 2021 [eBook #66944]
Language: English
Produced by: Richard Tonsing, David King, and the Online Distributed
Proofreading Team at http://www.pgdp.net. (This file was
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*** START OF THE PROJECT GUTENBERG EBOOK THE PRINCIPLE OF RELATIVITY ***
The Principle of Relativity
THE PRINCIPLE OF RELATIVITY
ORIGINAL PAPERS BY
A. EINSTEIN AND H. MINKOWSKI
TRANSLATED INTO ENGLISH BY
M. N. SAHA AND S. N. BOSE
LECTURERS ON PHYSICS AND APPLIED MATHEMATICS
University College of Science, Calcutta University
WITH A HISTORICAL INTRODUCTION BY
P. C. MAHALANOBIS
PROFESSOR OF PHYSICS, PRESIDENCY COLLEGE, CALCU.
PUBLISHED BY THE
UNIVERSITY OF CALCUTTA
1920
_Sole Agents_
R. CAMBRAY & CO.
PRINTED BY ATULCHANDRA BHATTACHARYYA,
AT THE CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA
TABLE OF CONTENTS
1. Historical Introduction i-xxiii
[By Mr. P. C. Mahalanobis.]
2. On the Electrodynamics of Moving Bodies 1-34
[Einstein’s first paper on the restricted Theory of Relativity,
originally published in the Annalen der Physik in 1905. Translated from
the original German by Dr. Meghnad Saha.]
3. Albrecht Einstein 35-39
[A short biographical note by Dr. Meghnad Saha.]
4. Principle of Relativity 1-52
[H. Minkowski’s original paper on the restricted Principle of Relativity
first published in 1909. Translated from the original German by Dr.
Meghnad Saha.]
5. Appendix to the above by H. Minkowski 53-88
[Translated by Dr. Meghnad Saha.]
6. The Generalised Principle of Relativity 89-163
[A. Einstein’s second paper on the Generalised Principle first published
in 1916. Translated from the original German by Mr. Satyendranath Bose.]
7. Notes 165-185
Transcriber’s Note:
The plain text version of this ebook includes complex mathematical
formulas. Some are simple in-line expressions like k = 1 - 1/μ^2. They
may include special notations such as x^y for x to the power of y, x_{y}
for x with a subscript of y, [=a] for an 'a' with a bar across the top,
[.a] for an 'a' with a dot over it, [..a] for an 'a' with two dots over
it. Others are more complex “ASCII Art” like this:
l l 2lc 2l
t₁ = ------ + ------ = -------- = --- β²
c - u c + u c² - u² c
Some are so complex that they must be rendered in the TeX mathematical
notation, enclosed between double dollar signs, like this:
$$ \beta = (1 - \frac {u^2}{c^2})^{-\frac{1}{2}} $$
HISTORICAL INTRODUCTION
Lord Kelvin writing in 1893, in his preface to the English edition of
Hertz’s Researches on Electric Waves, says “many workers and many
thinkers have helped to build up the nineteenth century school of
_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 consummation now realised.”
Ten years later, in 1905, we find Einstein declaring that “the ether
will be proved to be superfluous.” At first sight the revolution in
scientific thought brought about in the course of a single decade
appears to be almost too violent. A more careful even though a rapid
review of the subject will, however, show how the Theory of Relativity
gradually became a historical necessity.
Towards the beginning of the nineteenth century, the luminiferous ether
came into prominence as a result of the brilliant successes of the wave
theory in the hands of Young and Fresnel. In its stationary aspect the
elastic solid ether was the outcome of the search for a medium in which
the light waves may “undulate.” This stationary ether, as shown by
Young, also afforded a satisfactory explanation of astronomical
aberration. But its very success gave rise to a host of new questions
all bearing on the central problem of relative motion of ether and
matter.
_Arago’s 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_ + _u_. Thus the refractive index of the glass prism should
depend on _u_ 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-Boscovitch water-telescope experiment._—Boscovitch had still
earlier in 1766, raised the very important question of the dependence of
aberration on the refractive index of the medium filling the telescope.
Aberration depends on the difference in the velocity of light outside
the telescope and its velocity inside the telescope. If the latter
velocity changes owing to a change in the medium filling the telescope,
aberration itself should change, that is, aberration should depend on
the nature of the medium.
Airy, in 1871 filled up a telescope with water—but failed to detect any
change in the aberration. Thus we get both in the case of Arago prism
experiment and Airy-Boscovitch water-telescope experiment, the very
startling result that optical effects in a moving medium seem to be
quite independent of the velocity of the medium with respect to
Fresnel’s stationary ether.
_Fresnel’s convection coefficient k = 1 - 1/μ^2._—Possibly some form
of compensation is taking place. Working on this hypothesis, Fresnel
offered his famous ether convection theory. According to Fresnel, the
presence of matter implies a definite condensation of ether within the
region occupied by matter. This “condensed” or excess portion of ether
is supposed to be carried away with its own piece of moving matter. It
should be observed that only the “excess” portion is carried away,
while the rest remains as stagnant as ever. A complete convection of
the “excess” ether ρ′ with the full velocity _u_ is optically
equivalent to a partial convection of the total ether ρ, with only a
fraction of the velocity _k_. _u_. Fresnel showed that if this
convection coefficient _k_ is 1 - 1/μ^2 (μ being the refractive index
of the prism), then the velocity of light after refraction within the
moving prism would be altered to just such extent as would make the
refractive index of the moving prism quite independent of its
“absolute” velocity _u_. The non-dependence of aberration on the
“absolute” velocity _u_, is also very easily explained with the help
of this Fresnelian convection-coefficient _k_.
_Stokes’ 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 1845-46, to construct a more material type of
medium. Stokes assumed that viscous motion ensues near the surface of
separation of ether and moving matter, while at sufficiently distant
regions the ether remains wholly undisturbed. He showed how such a
viscous ether would explain aberration if all motion in it were
differentially irrotational. But in order to explain the null Arago
effect, Stokes was compelled to assume the convection hypothesis of
Fresnel with an identical numerical value for _k_, namely 1 - 1/μ^2.
Thus the prestige of the Fresnelian convection-coefficient was enhanced,
if anything, by the theoretical investigations of Stokes.
_Fizeau’s experiment._—Soon after, in 1851, 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 = 1 - 1/μ^2. The Fresnelian
convection-coefficient now became firmly established as a consequence of
a direct positive effect. On the other hand, the negative evidences in
favour of the convection-coefficient had also multiplied. Mascart, Hoek,
Maxwell and others sought for definite changes in different optical
effects induced by the motion of the earth relative to the stationary
ether. But all such attempts failed to reveal the slightest trace of any
optical disturbance due to the “absolute” velocity of the earth, thus
proving conclusively that all the different optical effects shared in
the general compensation arising out of the Fresnelian convection of the
excess ether. It must be carefully noted that the Fresnelian
convection-coefficient implicitly assumes the existence of a fixed ether
(Fresnel) or at least a wholly stagnant medium at sufficiently distant
regions (Stokes), with reference to which alone a convection velocity
can have any significance. Thus the convection-coefficient implying some
type of a stationary or viscous, yet nevertheless “absolute” ether,
succeeded in explaining satisfactorily all known optical facts down to
1880.
_Michelson-Morley 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. The fundamental idea underlying this experiment is quite
simple. In all old experiments the velocity of light situated in free
ether was compared with the velocity of waves actually situated in a
piece of moving matter and presumably carried away by it. The
compensatory effect of the Fresnelian convection of ether afforded a
satisfactory explanation of all negative results.
In the Michelson-Morley experiment the arrangement is quite different.
If there is a definite gap in a rigid body, light waves situated in free
ether will take a definite time in crossing the gap. If the rigid
platform carrying the gap is set in motion with respect to the ether in
the direction of light propagation, light waves (which are even now
situated in free ether) should presumably take a longer time to cross
the gap.
We cannot do better than quote Eddington’s description of this famous
experiment. “The principle of the experiment may be illustrated by
considering a swimmer in a river. It is easily realized that it takes
longer to swim to a point 50 yards up-stream and back than to a point 50
yards across-stream and back. If the earth is moving through the ether
there is a river of ether flowing through the laboratory, and a wave of
light may be compared to a swimmer travelling with constant velocity
relative to the current. If, then, we divide a beam of light into two
parts, and send one-half swimming up the stream for a certain distance
and then (by a mirror) back to the starting point, and send the other
half an equal distance across stream and back, the across-stream beam
should arrive back first.
——>_u_
O
A—————........
| _x_
|
|B
Let the ether be flowing relative to the apparatus with velocity _u_ in
the direction O_x_, and let OA, OB, be the two arms of the apparatus of
equal length _l_, OA being placed up-stream. Let _c_ be the velocity of
light. The time for the double journey along OA and back is
l l 2lc 2l
t₁ = ------ + ------ = -------- = --- β²
c - u c + u c² - u² c
where
$$ \beta = (1 - \frac {u^2}{c^2})^{-\frac {1}{2}} $$
a factor greater than unity.
For the transverse journey the light must have a component velocity _n_
up-stream (relative to the ether) in order to avoid being carried below
OB: and since its total velocity is _c_, its component across-stream
must be √(_c²_ - _u²_), the time for the double journey OB is
accordingly
$$ t_2 = \frac {2a}{\sqrt {c^2 - u^2}} = \frac {2a}{c} \beta $$
so that _t₁_ > _t₂_.
But when the experiment was tried, it was found that both parts of the
beam took the same time, as tested by the interference bands produced.”
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—_u_. Thus Michelson and Morley themselves thought at first
that their experiment confirmed Stokes’ viscous ether, in which no
relative motion can ensue on account of the absence of slipping of ether
at the surface of separation. But even on Stokes’ theory this viscous
flow of ether would fall off at a very rapid rate as we recede from the
surface of separation. Michelson and Morley repeated their experiment at
different heights from the surface of the earth, but invariably obtained
the same negative results, thus failing to confirm Stokes’ theory of
viscous flow.
_Lodge’s experiment._—Further, in 1893, Lodge performed his rotating
sphere experiment which showed complete absence of any viscous flow of
ether due to moving masses of matter. A divided beam of light, after
repeated reflections within a very narrow gap between two massive
hemispheres, was allowed to re-unite and thus produce interference
bands. When the two hemispheres are set rotating, it is conceivable that
the ether in the gap would be disturbed due to viscous flow, and any
such flow would be immediately detected by a disturbance of the
interference bands. But actual observation failed to detect the
slightest disturbance of the ether in the gap, due to the motion of the
hemispheres. Lodge’s experiment thus seems to show a complete absence of
any viscous flow of ether.
Apart from these experimental discrepancies, grave theoretical
objections were urged against a viscous ether. Stokes himself had shown
that his ether must be incompressible and all motion in it
differentially irrotational, at the same time there should be absolutely
no slipping at the surface of separation. Now all these conditions
cannot be simultaneously satisfied for any conceivable material medium
without certain very special and arbitrary assumptions. Thus Stokes’
ether failed to satisfy the very motive which had led Stokes to
formulate it, namely, the desirability of constructing a “physical”
medium. Planck offered modified forms of Stokes’ theory which seemed
capable of being reconciled with the Michelson-Morley experiment, but
required very special assumptions. The very complexity and the very
arbitrariness of these assumptions prevented Planck’s ether from
attaining any degree of practical importance in the further development
of the subject.
The sole criterion of the value of any scientific theory must ultimately
be its capacity for offering a simple, unified, coherent and fruitful
description of observed facts. In proportion as a theory becomes complex
it loses in usefulness—a theory which is obliged to requisition a whole
array of arbitrary assumptions in order to explain special facts is
practically worse than useless, as it serves to disjoin, rather than to
unite, the several groups of facts. The optical experiments of the last
quarter of the nineteenth century showed the impossibility of
constructing a simple ether theory, which would be amenable to analytic
treatment and would at the same time stimulate further progress. It
should be observed that it could scarcely be shown that no logically
consistent ether theory was possible; indeed in 1910, H. A. Wilson
offered a consistent ether theory which was at least quite neutral with
respect to all available optical data. But Wilson’s ether is almost
wholly negative—its only virtue being that it does not directly
contradict observed facts. Neither any direct confirmation nor a direct
refutation is possible and it does not throw any light on the various
optical phenomena. A theory like this being practically useless stands
self-condemned.
We must now consider the problem of relative motion of ether and matter
from the point of view of electrical theory. From 1860 the identity of
light as an electromagnetic vector became gradually established as a
result of the brilliant “displacement current” hypothesis of Clerk
Maxwell and his further analytical investigations. The elastic solid
ether became gradually transformed into the electromagnetic one. Maxwell
succeeded in giving a fairly satisfactory account of all ordinary
optical phenomena and little room was left for any serious doubts as
regards the general validity of Maxwell’s theory. Hertz’s researches on
electric waves, first carried out in 1886, succeeded in furnishing a
strong experimental confirmation of Maxwell’s theory. Electric waves
behaved generally like light waves of very large wave length.
The orthodox Maxwellian view located the dielectric polarisation in the
electromagnetic ether which was merely a transformation of Fresnel’s
stagnant ether. The magnetic polarisation was looked upon as wholly
secondary in origin, being due to the relative motion of the dielectric
tubes of polarisation. On this view the Fresnelian convection
coefficient comes out to be ½, as shown by J. J. Thomson in 1880,
instead of 1 - (1/μ²) as required by optical experiments. This obviously
implies a complete failure to account for all those optical experiments
which depend for their satisfactory explanation on the assumption of a
value for the convection coefficient equal to 1 - (1/μ²).
The modifications proposed independently by Hertz and Heaviside fare no
better.[1] They postulated the actual medium to be the seat of all
electric polarisation and further emphasised the reciprocal relation
subsisting between electricity and magnetism, thus making the field
equations more symmetrical. On this view the whole of the polarised
ether is carried away by the moving medium, and consequently, the
convection coefficient naturally becomes unity in this theory, a value
quite as discrepant as that obtained on the original Maxwellian
assumption.
Thus neither Maxwell’s original theory nor its subsequent modifications
as developed by Hertz and Heaviside succeeded in obtaining a value for
Fresnelian coefficient equal to 1 - (1/μ^2), and consequently stood
totally condemned from the optical point of view.
Certain direct electromagnetic experiments involving relative motion of
polarised dielectrics were no less conclusive against the generalised
theory of Hertz and Heaviside. According to Hertz a moving dielectric
would carry away the whole of its electric displacement with it. Hence
the electromagnetic effect near the moving dielectric would be
proportional to the total electric displacement, that is to K, the
specific inductive capacity of the dielectric. In 1901, Blondlot working
with a stream of moving gas could not detect any such effect. H. A.
Wilson repeated the experiment in an improved form in 1903 and working
with ebonite found that the observed effect was proportional to K - 1
instead of to K. For gases K is nearly equal to 1 and hence practically
no effect will be observed in their case. This gives a satisfactory
explanation of Blondlot’s negative results.
Rowland had shown in 1876 that the magnetic force due to a rotating
condenser (the dielectric remaining stationary) was proportional to K,
the sp. ind. cap. On the other hand, Röntgen found in 1888 the magnetic
effect due to a rotating dielectric (the condenser remaining stationary)
to be proportional to K - 1, and not to K. Finally Eichenwald in 1903
found that when both condenser and dielectric are rotated together, the
effect observed was quite independent of K, a result quite consistent
with the two previous experiments. The Rowland effect proportional to K,
together with the opposite Röntgen effect proportional to 1 - K, makes
the Eichenwald effect independent of K.
All these experiments together with those of Blondlot and Wilson made it
clear that the electromagnetic effect due to a moving dielectric was
proportional to K - 1, and not to K as required by Hertz’s theory. Thus
the above group of experiments with moving dielectrics directly
contradicted the Hertz-Heaviside theory. The internal discrepancies
inherent in the classic ether theory had now become too prominent. It
was clear that the ether concept had finally outgrown its usefulness.
The observed facts had become too contradictory and too heterogeneous to
be reduced to an organised whole with the help of the ether concept
alone. Radical departures from the classical theory had become
absolutely necessary.
There were several outstanding difficulties in connection with anomalous
dispersion, selective reflection and selective absorption which could
not be satisfactory explained in the classic electromagnetic theory. It
was evident that the assumption of some kind of discreteness in the
optical medium had become inevitable. Such an assumption naturally gave
rise to an atomic theory of electricity, namely, the modern electron
theory. Lorentz had postulated the existence of electrons so early as
1878, but it was not until some years later that the electron theory
became firmly established on a satisfactory basis.
Lorentz assumed that a moving dielectric merely carried away its own
“polarisation doublets,” which on his theory gave rise to the induced
field proportional to K - 1. The field near a moving dielectric is
naturally proportional to K - 1 and not to K. Lorentz’s theory thus gave
a satisfactory explanation of all those experiments with moving
dielectrics which required effects proportional to K - 1. Lorentz
further succeeded in obtaining a value for the Fresnelian convection
coefficient equal to 1 - 1/μ^2, the exact value required by all optical
experiments of the moving type.
We must now go back to Michelson and Morley’s experiment. We have seen
that both parts of the beam are situated in free ether; no material
medium is involved in any portion of the paths actually traversed by the
beam. Consequently no compensation due to Fresnelian convection of ether
by moving medium is possible. Thus Fresnelian convection compensation
can have no possible application in this case. Yet some marvellous
compensation has evidently taken place which has completely masked the
“absolute” velocity of the earth.
In Michelson and Morley’s experiment, the distance travelled by the beam
along OA (that is, in a direction parallel to the motion of the
platform) is 2_l_β², while the distance travelled by the beam along OB,
perpendicular to the direction of motion of the platform, is 2_l_β. Yet
the most careful experiments showed, as Eddington says, “that both parts
of the beam took the same time as tested by the interference bands
produced. It would seem that OA and OB could not really have been of the
same length; and if OB was of length _l_, OA must have been of length
_l_/β. The apparatus was now rotated through 90°, so that OB became the
up-stream. The time for the two journeys was again the same, so that 0B
must now be the shorter length. The plain meaning of the experiment is
that both arms have a length _l_ when placed along O_y_ (perpendicular
to the direction of motion), and automatically contract to a length
_l_/β, when placed along O_x_ (parallel to the direction of motion).
This explanation was first given by Fitz-Gerald.”
This Fitz-Gerald contraction, startling enough in itself, does not
suffice. Assuming this contraction to be a real one, the distance
travelled with respect to the ether is 2_l_β and the time taken for this
journey is 2_l_β/_c_. But the distance travelled with respect to the
platform is always 2_l_. Hence the velocity of light with respect to the
platform is
$$ \frac {2l}{\frac {2l\beta}{c}} = \frac {c}{\beta} $$
a variable quantity depending on the “absolute” velocity of the
platform. But no trace of such an effect has ever been found. The
velocity of light is always found to be quite independent of the
velocity of the platform. The present difficulty cannot be solved by any
further alteration in the measure of space. The only recourse left open
is to alter the measure of time as well, that is, to adopt the concept
of “local time.” If a moving clock goes slower so that one ‘real’ second
becomes 1/β second as measured in the moving system, the velocity of
light relative to the platform will always remain _c_. We must adopt two
very startling hypotheses, namely, the Fitz-Gerald contraction and the
concept of “local time,” in order to give a satisfactory explanation of
the Michelson-Morley experiment.
These results were already reached by Lorentz in the course of further
developments of his electron theory. Lorentz used a special set of
transformation equations[2] for time which implicitly introduced the
concept of local time. But he himself failed to attach any special
significance to it, and looked upon it rather as a mere mathematical
artifice like imaginary quantities in analysis or the circle at infinity
in projective geometry. The originality of Einstein at this stage
consists in his successful physical interpretation of these results, and
viewing them as the coherent organised consequences of a single general
principle. Lorentz established the Relativity Theorem[3] (consisting
merely of a set of transformation equations) while Einstein generalised
it into a Universal Principle. In addition Einstein introduced
fundamentally new concepts of space and time, which served to destroy
old fetishes and demanded a wholesale revision of scientific concepts
and thus opened up new possibilities in the synthetic unification of
natural processes.
Newton had framed his laws of motion in such a way as to make them quite
independent of the absolute velocity 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[4] to be the fundamental unit of
measurement for all observers (that is, assume the constancy of the
velocity of light in all systems) we can establish a _metric_ “one-one”
correspondence between any two observed systems, such correspondence
depending only the _relative_ velocity of the two systems. Einstein’s
Relativity is thus merely the consistent logical application of the well
known physical principle that we can know nothing but _relative_ motion.
In this sense it is a further extension of Newtonian Relativity.
On this interpretation, the Lorentz-Fitzgerald contraction and “local
time” lose their arbitrary character. Space and time as measured by two
different observers are naturally diverse, and the difference depends
only on their relative motion. Both are equally valid; they are merely
different descriptions of the same physical reality. This is essentially
the point of view adopted by Minkowski. He considers time itself to be
one of the co-ordinate axes, and in his four-dimensional world, that is
in the space-time reality, relative motion is reduced to a rotation of
the axes of reference. Thus, the diversity in the measurement of lengths
and temporal rates is merely due to the static difference in the
“frame-work” of the different observers.
The above theory of Relativity absorbed practically the whole of the
electromagnetic theory based on the Maxwell-Lorentz system of field
equations. It combined all the advantages of classic Maxwellian theory
together with an electronic hypothesis. The Lorentz assumption of
polarisation doublets had furnished a satisfactory explanation of the
Fresnelian convection of ether, but in the new theory this is deduced
merely as a consequence of the altered concept of relative velocity. In
addition, the theory of Relativity accepted the results of Michelson and
Morley’s experiments as a definite principle, namely, the principle of
the constancy of the velocity of light, so that there was nothing left
for explanation in the Michelson-Morley experiment. But even more than
all this, it established a single general principle which served to
connect together in a simple coherent and fruitful manner the known
facts of Physics.
The theory of Relativity received direct experimental confirmation in
several directions. Repeated attempts were made to detect the
Lorentz-Fitzgerald contraction. Any ordinary physical contraction will
usually have observable physical results; for example, the total
electrical resistance of a conductor will diminish. Trouton and Noble,
Trouton and Rankine, Rayleigh and Brace, and others employed a variety
of different methods to detect the Lorentz-Fitzgerald contraction, but
invariably with the same negative results. _Whether there is an ether or
not, uniform velocity with respect to it can never be detected._ This
does not prove that there is no such thing as an ether but certainly
does render the ether entirely superfluous. Universal compensation is
due to a change in local units of length and time, or rather, being
merely different descriptions of the same reality, there is no
compensation at all.
There was another group of observed phenomena which could scarcely be
fitted into a Newtonian scheme of dynamics without doing violence to it.
The experimental work of Kaufmann, in 1901, made it abundantly clear
that the “mass” of an electron depended on its velocity. So early as
1881, J. J. Thomson had shown that the inertia of a charged particle
increased with its velocity. Abraham now deduced a formula for the
variation of mass with velocity, on the hypothesis that an electron
always remained a _rigid_ sphere. Lorentz proceeded on the assumption
that the electron shared in the Lorentz-Fitzgerald contraction and
obtained a totally different formula. A very careful series of
measurements carried out independently by Bücherer, Wolz, Hupka and
finally Neumann in 1913, decided conclusively in favour of the Lorentz
formula. This “contractile” formula follows immediately as a direct
consequence of the new Theory of Relativity, without any assumption as
regards the electrical origin of inertia. Thus the complete agreement of
experimental facts with the predictions of the new theory must be
considered as confirming it as a principle which goes even beyond the
electron itself. The greatest triumph of this new theory consists,
indeed, in the fact that a large number of results, which had formerly
required all kinds of special hypotheses for their explanation, are now
deduced very simply as inevitable consequences of one single general
principle.
We have now traced the history of the development of the restricted or
special theory of Relativity, which is mainly concerned with optical and
electrical phenomena. It was first offered by Einstein in 1905. Ten
years later, Einstein formulated his second theory, the Generalised
Principle of Relativity. This new theory is mainly a theory of
gravitation and has very little connection with optics and electricity.
In one sense, the second theory is indeed a further generalisation of
the restricted principle, but the former does not really contain the
latter as a special case.
Einstein’s first theory is restricted in the sense that it only refers
to uniform rectilinear motion and has no application to any kind of
accelerated movements. Einstein in his second theory extends the
Relativity Principle to cases of accelerated motion. If Relativity is to
be universally true, then even accelerated motion must be merely
_relative motion between matter and matter_. Hence the Generalised
Principle of Relativity asserts that “absolute” motion cannot be
detected even with the help of gravitational laws.
All movements must be referred to definite sets of co-ordinate axes. If
there is any change of axes, the numerical magnitude of the movements
will also change. But according to Newtonian dynamics, such alteration
in physical movements can only be due to the effect of certain forces in
the field.[5] Thus any change of axes will introduce new “geometrical”
forces in the field which are quite independent of the nature of the
body acted on. Gravitational forces also have this same remarkable
property, and gravitation itself may be of essentially the same nature
as these “geometrical” forces introduced by a change of axes. This leads
to Einstein’s famous Principle of Equivalence. _A gravitational field of
force is strictly equivalent to one introduced by a transformation of
co-ordinates and no possible experiment can distinguish between the
two._
Thus it may become possible to “transform away” gravitational effects,
at least for sufficiently small regions of space, by referring all
movements to a new set of axes. This new “framework” may of course have
all kinds of very complicated movements when referred to the old
Galilean or “rectangular unaccelerated system of co-ordinates.”
But there is no reason why we should look upon the Galilean system as
more fundamental than any other. If it is found simpler to refer all
motion in a gravitational field to a special set of co-ordinates, we may
certainly look upon this special “framework” (at least for the
particular region concerned), to be more fundamental and more natural.
We may, still more simply, identify this particular framework with the
special local properties of space in that region. That is, we can look
upon the effects of a gravitational field as simply due to the local
properties of space and time itself. The very presence of matter implies
a modification of the characteristics of space and time in its
neighbourhood. As Eddington says “matter does not cause the curvature of
space-time. It is the curvature. Just as light does not cause
electromagnetic oscillations; it is the oscillations.”
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 _is_ convenient
to take the matter actually present in a field as the fundamental ground
of our framework. If this is done, the special characteristics of our
framework would naturally depend on the actual distribution of matter in
the field. But physical space and time is completely defined by the
“framework.” In other words the “framework” itself _is_ space and time.
Hence we see how _physical_ 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 δ_x²_ + δ_y²_ + δ_z²_ is an invariant. In the
restricted theory of light, the principle of constancy of light velocity
demands that δ_x²_ + δ_y²_ + δ_z²_ - _c²_δ_t²_ should remain constant.
The _separation ds_ of adjacent events is defined by _ds²_ = -_dx²_ -
_dy²_ - _dz²_ + _c²dt²_. It is an extension of the notion of distance
and this is the new invariant. Now if _x_, _y_, _z_, _t_ are transformed
to any set of new variables _x₁_, _x₂_, _x₃_, _x₄_, we shall get a
quadratic expression for
$$ ds^2 = g_{1\;1}x_{1}^2 + 2g_{1\;2}x_{1}x_{2} + ... = \sum
g_{i\;j}x_{i}x_{j} $$
where the _g_’s are functions of _x₁_, _x₂_, _x₃_, _x₄_ depending on the
transformation.
The special properties of space and time in any region are defined by
these _g_’s which are themselves determined by the actual distribution
of matter in the locality. Thus from the Newtonian point of view, these
_g_’s represent the gravitational effect of matter while from the
Relativity stand-point, these merely define the non-Newtonian (and
incidentally non-Euclidean) space in the neighbourhood of matter.
We have seen that Einstein’s theory requires local curvature of
space-time in the neighbourhood of matter. Such altered characteristics
of space and time give a satisfactory explanation of an outstanding
discrepancy in the observed advance of perihelion of Mercury. The large
discordance is almost completely removed by Einstein’s theory.
Again, in an intense gravitational field, a beam of light will be
affected by the local curvature of space, so that to an observer who is
referring all phenomena to a Newtonian system, the beam of light will
appear to deviate from its path along an Euclidean straight line.
This famous prediction of Einstein about the deflection of a beam of
light by the sun’s gravitational field was tested during the total solar
eclipse of May, 1919. The observed deflection is decisively in favour of
the Generalised Theory of Relativity.
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 _restricted_ field.
Einstein has proposed a third crucial test. He has predicted a shift of
spectral lines towards the red, due to an intense gravitational
potential. Experimental difficulties are very considerable here, as the
shift of spectral lines is a complex phenomenon. Evidence is conflicting
and nothing conclusive can yet be asserted. Einstein thought that a
gravitational displacement of the Fraunhofer lines is a necessary and
fundamental condition for the acceptance of his 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 there are several important consequences
of the Generalised or Gravitational Theory of Relativity. Physical
space-time is perceived to be intimately connected with the actual local
distribution of matter. Euclid-Newtonian space-time is _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 space-*time is the actual framework which
has some definite curvature due to the presence of matter. Gravitational
Theory of Relativity thus brings out clearly the fundamental distinction
between actual physical space-time (which is non-isotropic and
non-Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian
continuum (which is homogeneous, isotropic and a purely intellectual
construction) on the other.
The measurements of the rotation of the earth reveals a fundamental
framework which may be called the “inertial framework.” This constitutes
the actual physical universe. This universe approaches Galilean
space-time at a great distance from matter.
The properties of this physical universe may be referred to some
world-distribution of matter or the “inertial framework” may be
constructed by a suitable modification of the law of gravitation itself.
In Einstein’s theory the actual curvature of the “inertial framework” is
referred to vast quantities of undetected world-matter. It has
interesting consequences. The dimensions of Einsteinian universe would
depend on the quantity of matter in it; it would vanish to a point in
the total absence of matter. Then again curvature depends on the
quantity of matter, and hence in the presence of a sufficient quantity
of matter space-time may curve round and close up. Einsteinian universe
will then reduce to a finite system without boundaries, like the surface
of a sphere. In this “closed up” system, light rays will come to a focus
after travelling round the universe and we should see an “anti-sun”
(corresponding to the back surface of the sun) at a point in the sky
opposite to the real sun. This anti-sun would of course be equally large
and equally bright if there is no absorption of light in free space.
In de Sitter’s theory, the existence of vast quantities of world-matter
is not required. But beyond a definite distance from an observer, time
itself stands still, so that to the observer nothing can ever “happen”
there. All these theories are still highly speculative in character, but
they have certainly extended the scope of theoretical physics to the
central problem of the ultimate nature of the universe itself.
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 separately
are more abstractions. Physical reality consists of a synthesis of all
three.
P. C. MAHALANOBIS.
Note A.
For example consider a massive particle resting on a circular disc. If
we set the disc rotating, a centrifugal force appears in the field. On
the other hand, if we transform to a set of rotating axes, we must
introduce a centrifugal force in order to correct for the change of
axes. This newly introduced centrifugal force is usually looked upon as
a mathematical fiction—as “geometrical” rather than physical. The
presence of such a geometrical force is usually interpreted as being due
to the adoption of a fictitious framework. On the other hand a
gravitational force is considered quite real. Thus a fundamental
distinction is made between geometrical and gravitational forces.
In the General Theory of Relativity, this fundamental distinction is
done away with. The very possibility of distinguishing between
geometrical and gravitational forces is denied. All axes of reference
may now be regarded as equally valid.
In the Restricted Theory, all “unaccelerated” axes of reference were
recognised as equally valid, so that physical laws were made independent
of uniform absolute velocity. In the General Theory, physical laws are
made independent of “absolute” motion of any kind.
Footnote 1:
See Note 1.
Footnote 2:
See Note 2.
Footnote 3:
See Note 4.
Footnote 4:
See Notes 9 and 12.
Footnote 5:
Note A.
On The Electrodynamics of Moving Bodies
By
A. Einstein.
INTRODUCTION.
It is well known that if we attempt to apply Maxwell’s electrodynamics,
as conceived at the present time, to moving bodies, we are led to
asymmetry which does not agree with observed phenomena. Let us think of
the mutual action between a magnet and a conductor. The observed
phenomena in this case depend only on the relative motion of the
conductor and the magnet, while according to the usual conception, a
distinction must be made between the cases where the one or the other of
the bodies is in motion. If, for example, the magnet moves and the
conductor is at rest, then an electric field of certain energy-value is
produced in the neighbourhood of the magnet, which excites a current in
those parts of the field where a conductor exists. But if the magnet be
at rest and the conductor be set in motion, no electric field is
produced in the neighbourhood of the magnet, but an electromotive force
which corresponds to no energy in itself is produced in the conductor;
this causes an electric current of the same magnitude and the same
career as the electric force, it being of course assumed that the
relative motion in both of these cases is the same.
2. Examples of a similar kind such as the unsuccessful attempt to
substantiate the motion of the earth relative to the “Light-medium” lead
us to the supposition that not only in mechanics, but also in
electrodynamics, no properties of observed facts correspond to a concept
of absolute rest; but that for all coordinate systems for which the
mechanical equations hold, the equivalent electrodynamical and optical
equations hold also, as has already been shown for magnitudes of the
first order. In the following we make these assumptions (which we shall
subsequently call the Principle of Relativity) and introduce the further
assumption,—an assumption which is at the first sight quite
irreconcilable with the former one—that light is propagated in vacant
space, with a velocity _c_ which is independent of the nature of motion
of the emitting body. These two assumptions are quite sufficient to give
us a simple and consistent theory of electrodynamics of moving bodies on
the basis of the Maxwellian theory for bodies at rest. The introduction
of a “Lightäther” will be proved to be superfluous, for according to the
conceptions which will be developed, we shall introduce neither a space
absolutely at rest, and endowed with special properties, nor shall we
associate a velocity-vector with a point in which electro-magnetic
processes take place.
3. Like every other theory in electrodynamics, the theory is based on
the kinematics of rigid bodies; in the enunciation of every theory, we
have to do with relations between rigid bodies (co-ordinate system),
clocks, and electromagnetic processes. An insufficient consideration of
these circumstances is the cause of difficulties with which the
electrodynamics of moving bodies have to fight at present.
I.—KINEMATICAL PORTION.
§ 1. Definition of Synchronism.
Let us have a co-ordinate system, in which the Newtonian equations hold.
For distinguishing this system from another which will be introduced
hereafter, we shall always call it “the stationary system.”
If a material point be at rest in this system, then its position in this
system can be found out by a measuring rod, and can be expressed by the
methods of Euclidean Geometry, or in Cartesian co-ordinates.
If we wish to describe the motion of a material point, the values of its
coordinates must be expressed as functions of time. It is always to be
borne in mind that _such a mathematical definition has a physical sense,
only when we have a clear notion of what is meant by time. We have to
take into consideration the fact that those of our conceptions, in which
time plays a part, are always conceptions of synchronism._ For example,
we say that a train arrives here at 7 o’clock; this means that the exact
pointing of the little hand of my watch to 7, and the arrival of the
train are synchronous events.
It may appear that all difficulties connected with the definition of
time can be removed when in place of time, we substitute the position of
the little hand of my watch. Such a definition is in fact sufficient,
when it is required to define time exclusively for the place at which
the clock is stationed. But the definition is not sufficient when it is
required to connect by time events taking place at different
stations,—or what amounts to the same thing,—to estimate by means of
time (zeitlich werten) the occurrence of events, which take place at
stations distant from the clock.
Now with regard to this attempt;—the time-estimation of events, we can
satisfy ourselves in the following manner. Suppose an observer—who is
stationed at the origin of coordinates with the clock—associates a ray
of light which comes to him through space, and gives testimony to the
event of which the time is to be estimated,—with the corresponding
position of the hands of the clock. But such an association has this
defect,—it depends on the position of the observer provided with the
clock, as we know by experience. We can attain to a more practicable
result by the following treatment.
If an observer be stationed at A with a clock, he can estimate the time
of events occurring in the immediate neighbourhood of A, by looking for
the position of the hands of the clock, which are synchronous with the
event. If an observer be stationed at B with a clock,—we should add that
the clock is of the same nature as the one at A,—he can estimate the
time of events occurring about B. But without further premises, it is
not possible to compare, as far as time is concerned, the events at B
with the events at A. We have hitherto an A-time, and a B-time, but no
time common to A and B. This last time (_i.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_{A} towards B, arrives and is reflected from B at
B-time t_{B}, and returns to A at A-time t′_{A}. According to the
definition, both clocks are synchronous, if
t_{B} - t_{A} = t′_{A} - t_{B}.
We assume that this definition of synchronism is possible without
involving any inconsistency, for any number of points, therefore the
following relations hold:—
1. If the clock at B be synchronous with the clock at A, then the clock
at A is synchronous with the clock at B.
2. If the clock at A as well as the clock at B are both synchronous with
the clock at C, then the clocks at A and B are 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
$$ \frac {2 \overline{AB}}{t'_{A} - t_{A}} = c $$
where _c_ is a universal constant.
We have defined time essentially with a clock at rest in a stationary
system. On account of its adaptability to the stationary system, we call
the time defined in this way as “time of the stationary system.”
§ 2. On the Relativity of Length and Time.
The following reflections are based on the Principle of Relativity and
on the Principle of Constancy of the velocity of light, both of which we
define in the following way:—
1. The laws according to which the nature of physical systems alter are
independent of the manner in which these changes are referred to two
co-ordinate systems which have a uniform translators motion relative to
each other.
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 ray of light is emitted by a body at rest or in motion.[6]
Therefore
velocity = Path of Light/Interval of 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 _l_, 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 _v_,
parallel to the axis of X, is imparted to it. Let us now enquire about
the length of the moving rod; this can be obtained by either of these
operations.—
(_a_) The observer provided with the measuring rod moves along with the
rod to be measured, and measures by direct superposition the length of
the rod:—just as if the observer, the measuring rod, and the rod to be
measured were at rest.
(_b_) The observer finds out, by means of clocks placed in a system at
rest (the clocks being synchronous as defined in §1), the points of this
system where the ends of the rod to be measured occur at a particular
time _t_. The distance between these two points, measured by the
previously used measuring rod, this time it being at rest, is a length,
which we may call the “length of the rod.”
According to the Principle of Relativity, the length found out by the
operation _a_), which we may call “the length of the rod in the moving
system” is equal to the length _l_ of the rod in the stationary system.
The length which is found out by the second method, may be called ‘_the
length of the moving rod measured from the stationary system_.’ This
length is to be estimated on the basis of our principle, and _we shall
find it to be different from l_.
In the generally recognised kinematics, we silently assume that the
lengths defined by these two operations are equal, or in other words,
that at an epoch of time _t_, a moving rigid body is geometrically
replaceable by the same body, which can replace it in the condition of
rest.
Relativity of Time.
Let us suppose that the two clocks synchronous with the clocks in the
system at rest are brought to the ends A, and B of a rod, _i.e._, the
time of the clocks correspond to the time of the stationary system at
the points where they happen to arrive; these clocks are therefore
synchronous in the stationary system.
We further imagine that there are two observers at the two watches, and
moving with them, and that these observers apply the criterion for
synchronism to the two clocks. At the time _t__{A}, a ray of light goes
out from A, is reflected from B at the time _t__{B}, and arrives back at
A at time _t′__{A}. Taking into consideration the principle of,
constancy of the velocity of light, we have
_t__{B} - _t__{A} = _r__{AB}/(_c_ - _v_),
and _t′__{A} - _t__{B} = _r__{AB}/(_c_ + _v_),
where _r__{AB} is the length of the moving rod, measured in the
stationary system. Therefore the observers stationed with the watches
will not find the clocks synchronous, though the observer in the
stationary system must declare the clocks to be synchronous. We
therefore see that we can attach no absolute significance to the concept
of synchronism; but two events which are synchronous when viewed from
one system, will not be synchronous when viewed from a system moving
relatively to this system.
§ 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 Y and
Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be
given to each of the systems, and let the rods and clocks in each be
exactly alike each other.
Let the initial point of one of the systems (_k_) have a constant
velocity in the direction of the X-axis of the other which is stationary
system K, the motion being also communicated to the rods and clocks in
the system (_k_). Any time _t_ of the stationary system K corresponds to
a definite position of the axes of the moving system, which are always
parallel to the axes of the stationary system. By _t_, 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 system, and we thus obtain the co-ordinates (_x_,
_y_, _z_) for the stationary system, and (ξ, η, ζ) 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 τ of the moving system be determined
for each point of the moving system (in which there are clocks which are
at rest relative to the moving system), by means of the method of light
signals between these points (in which there are clocks) in the manner
described in § 1.
To every value of (_x_, _y_, _z_, _t_) which fully determines the
position and time of an event in the stationary system, there correspond
a system of values (ξ, η, ζ, τ); now the problem is to find out the
system of equations connecting these magnitudes.
Primarily it is clear that on account of the property of homogeneity
which we ascribe to time and space, the equations must be linear.
If we put _x′_ = _x_ - _vt_, then it is clear that at a point relatively
at rest in the system _k_, we have a system of values (_x′_ _y_ _z_)
which are independent of time. Now let us find out τ as a function of
(_x′_, _y_, _z_, _t_). For this purpose we have to express in equations
the fact that τ is not other than the time given by the clocks which are
at rest in the system _k_ which must be made synchronous in the manner
described in § 1.
Let a ray of light be sent at time τ₀ from the origin of the system _k_
along the X-axis towards _x′_ and let it be reflected from that place at
time τ₁ towards the origin of moving co-ordinates and let it arrive
there at time τ₂; then we must have
½ (τ₀ + τ₂) = τ₁
If we now introduce the condition that τ is a function of co-ordinates,
and apply the principle of constancy of the velocity of light in the
stationary system, we have
$$ \frac {1}{2} (\tau (0,0,0,t) + \tau (0,0,0,(t + \frac {x'}{c-v} +
\frac {x'}{c+v}))) $$
$$ = \tau (x',0,0, t + \frac {x'}{c-v}) $$
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 (_x′_, _y_, _z_,
_t_,).
A similar conception, being applied to the _y_- and _z_-axis gives us,
when we take into consideration the fact that light when viewed from the
stationary system, is always propagated along those axes with the
velocity √(_c²_ - _v²_), we have the questions
∂τ ∂τ
---- = 0, ---- = 0.
∂y ∂z
From these equations it follows that τ is a linear function of _x′_ and
_t_. From equations (1) we obtain
vx′
τ = a (t - -------- )
c² - v²
where _a_ is an unknown function of _v_.
With the help of these results it is easy to obtain the magnitudes (ξ,
η, ζ) if we express by means of equations the fact that light, when
measured in the moving system is always propagated with the constant
velocity _c_ (as the principle of constancy of light velocity in
conjunction with the principle of relativity requires). For a time τ =
0, if the ray is sent in the direction of increasing ξ, we have
_vx′_
ξ = _c_τ, _i.e._ ξ = _a c_(_t_ - ------------ )
_c²_ - _v²_
Now the ray of light moves relative to the origin of _k_ with a velocity
_c_ - _v_, measured in the stationary system; therefore we have
_x′_
---------- = _t_
_c_ - _v_
Substituting these values of _t_ in the equation for ξ, we obtain
_c²_
ξ = _a_ ------------- _x′_
_c²_ - _v²_
In an analogous manner, we obtain by considering the ray of light which
moves along the _y_-axis,
_vx′_
η = _c_τ = _a c_(_t_ - ------------- )
_c²_ - _v²_
where
_y_
------------------ = _t_, _x′_ = 0,
√ (_c²_ - _v²_)
Therefore
_c_
η = _a_ ------------------ _y_,
√ (_c²_ - _v²_)
_c_
ζ = _a_ ----------------- _z_ .
√ (_c²_ - _v²_)
If for _x′_, we substitute its value _x_ - _tv_, we obtain
_v_._c_
τ = φ (_v_). β (_t_ - ----------- ,
c²
ξ = φ (_v_). β (_x_ - _vt_) ,
η = φ (_v_) _y_
ζ = φ (_v_) _z_ ,
where
$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
and
φ (_v_) = _ac_ / √ (_c²_ - _v²_) = _a_ / β
is a function of _v_.
If we make no assumption about the initial position of the moving system
and about the null-point of _t_, then an additive constant is to be
added to the right hand side.
We have now to show, that every ray of light moves in the moving system
with a velocity _c_ (when measured in the moving system), in case, as we
have actually assumed, _c_ is also the velocity in the stationary
system; for we have not as yet adduced any proof in support of the
assumption that the principle of relativity is reconcilable with the
principle of constant light-velocity.
At a time τ = _t_ = 0 let a spherical wave be sent out from the common
origin of the two systems of co-ordinates, and let it spread with a
velocity _c_ in the system K. If (_x_, _y_, _z_), be a point reached by
the wave, we have
_x²_ + _y²_ + _z²_ = _c²__t²_
with the aid of our transformation-equations, let us transform this
equation, and we obtain by a simple calculation,
ξ² + η² + ζ² = _c²_τ².
Therefore the wave is propagated in the moving system with the same
velocity _c_, and as a spherical wave.[7] Therefore we show that the two
principles are mutually reconcilable.
In the transformations we have got an undetermined function φ(_v_), and
we now proceed to find it out.
Let us introduce for this purpose a third co-ordinate system _k′_, which
is set in motion relative to the system _k_, the motion being parallel
to the ξ-axis. Let the velocity of the origin be (-_v_). At the time _t_
= 0, all the initial co-ordinate points coincide, and for _t_ = _x_ =
_y_ = _z_ = 0, the time _t′_ of the system _k′_ = 0. We shall say that
(_x′_ _y′_ _z′_ _t′_) are the co-ordinates measured in the system _k′_,
then by a two-fold application of the transformation-equations, we
obtain
_v_
τ′ = φ(-_v_)β(-_v_){τ + ----- ξ}
_c²_
= φ(_v_)φ(-_v_)t,
_x′_ = φ](_v_)β(_v_)(ξ + _v_τ)
= φ(_v_)φ(-_v_)_x_, etc.
Since the relations between (_x′_, _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.
∴ φ(_v_)φ(-_v_) = 1.
Let us now turn our attention to the part of the ξ-axis between (ξ = 0,
η = 0, ζ = 0), and (ξ = 0, η = 1, ζ = 0). Let this piece of the _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
_x₁_ = _vt_, _y₁_ = _l_ / φ(_v_), _z₁_ = 0
_x₂_ = _vt_, _y₂_ = 0, _z₂_ = 0
Therefore the length of the rod measured in the system K is _l_/φ(_v_).
For the system moving with velocity (-_v_), we have on grounds of
symmetry,
_l_ _l_
-------- = ---------
φ(_v_) φ(-_v_)
∴ φ(_v_) = φ(-_v_), ∴ φ(_v_) = 1.
§ 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 equation of the surface of this sphere, which is moving
with a velocity _v_ relative to K, is
ξ² + η² + ζ² = R².
At time _t_ = 0, the equation is expressed by means of (_x_, _y_, _z_,
_t_,) as
$$ \frac {x^2}{(\sqrt {1 - \frac {v_2}{c_2}})^2} + y^2 + z^2 = R^2. $$
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
$$ R \sqrt {1 - \frac {v^2}{c^2}}, R, R. $$
Therefore the _y_ and _z_ dimensions of the sphere (therefore of any
figure also) do not appear to be modified by the motion, but the _x_
dimension is shortened in the ratio
$$ 1 : \sqrt {1 - \frac {v^2}{c^2}}; $$
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 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 _t_, and lying at rest relative to the
moving system is capable of giving the time τ; suppose it to be placed
at the origin of the moving system _k_, and to be so arranged that it
gives the time τ. How much does the clock gain, when viewed from the
stationary system K? We have,
$$ \tau = \frac {1}{\sqrt {1-\frac {v^2}{c^2}}} (t - \frac {v}{c^2}x),
$$
and _x_ = _vt_,
$$ \therefore \tau - t = (1 - \sqrt {1 - \frac {v^2}{c^2}}) t. $$
Therefore the clock loses by an amount ½(_v²_/_c²_) per second of
motion, to the second order of approximation.
From this, the following peculiar consequence follows. Suppose at two
points A and B of the stationary system two clocks are given which are
synchronous in the sense explained in § 3 when viewed from the
stationary system. Suppose the clock at A to be set in motion in the
line joining it with B, then after the arrival of the clock at B, they
will no longer be found synchronous, but the clock which was set in
motion from A will lag behind the clock which had been all along at B by
an amount ½_t_(_v²_/_c²_), where _t_ is the time required for the
journey.
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 _t_-seconds, then after arrival, the last mentioned clock
will be behind the stationary one by ½_t_(_v²_/_c²_) 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 _x_-axis of the system K) according to the equation
$$ \xi = w_{\xi} \tau, \eta = w_{\eta} \tau, \zeta = 0, $$
where _w__{ξ} and _w__{η} 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
$$ x = (\frac {w_{\xi} + v}{1+\frac {vw_{\xi}}{c^2}}) t $$,
$$ y = \frac {(1-\frac {v^2}{c^2})^{\frac {1}{2}} w_{\eta}t} {1+\frac
{vw_{\xi}}{c^2}} $$ ,
z = 0 .
The law of parallelogram of velocities hold up to the first order of
approximation. We can put
$$ U^2 = (\frac {\partial x}{\partial t})^2 + (\frac {\partial
y}{\partial t})^2 $$ ,
$$ w^2 = w_{\xi}^2 + w_{\eta}^2 $$ ,
and
$$ \alpha = tan^{-1} \frac {w}{w_{\xi}} $$
_i.e._, α is put equal to the angle between the velocities _v_, and _w_.
Then we have—
$$ U = \frac {[(v^2 + w^2 + 2 vw \cos \alpha) - (\frac {vw \sin
\alpha}{c})^2]^{\frac {1}{2}}} {1 + \frac {vw \cos \alpha}{c^2}} $$
It should be noticed that _v_ and _w_ enter into the expression for
velocity symmetrically. If _w_ has the direction of the ξ-axis of the
moving system,
$$ U = \frac {v + w}{1 + \frac {vw}{c^2}} $$
From this equation, we see that by combining two velocities, each of
which is smaller than _c_, we obtain a velocity which is always smaller
than _c_. If we put _v_ = _c_ - χ, and _w_ = _c_ - λ, where χ and λ are
each smaller than _c_,[8]
$$ U = c \frac {2c - \chi - \lambda}{2c - \chi - \lambda + \frac {\chi
\lambda}{c^2}} < c $$
It is also clear that the velocity of light _c_ cannot be altered by
adding to it a velocity smaller than _c_. For this case,
$$ U = \frac {c + v}{1 + \frac {cv}{c^2}} = c $$
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 introduce the system k´, of which the initial point moves
parallel to the ξ-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 _v_, we shall have to write,
$$ \frac {v + w}{1 + \frac {vw}{c^2}} $$
We see that such a parallel transformation forms a group.
We have deduced the kinematics corresponding to our two fundamental
principles for the laws necessary for us, and we shall now pass over to
their application in electrodynamics.
II.—ELECTRODYNAMICAL PART.
§ 6. Transformation of Maxwell’s equations for Pure Vacuum.
On the nature of the 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
$$ \frac {1}{c} \frac {\partial}{\partial t} [X, Y, Z] = \begin{vmatrix}
\frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac
{\partial}{\partial z} L & M & N \end{vmatrix} $$
and
$$ \frac {1}{c} \frac {\partial}{\partial t} [L, M, N] = \begin{vmatrix}
\frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac
{\partial}{\partial z} X & Y & Z \end{vmatrix} $$ (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 equations, and if we
refer the electromagnetic processes to the co-ordinate system moving
with velocity _v_, we obtain,
$$ \frac {1}{c} \frac {\partial}{\partial \tau} [X, \beta (Y - \frac
{v}{c} N), \beta (Z + \frac {v}{c} M)] = \begin{vmatrix} \frac
{\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac
{\partial}{\partial \zeta} L & \beta(M + \frac {v}{c} Z) & \beta(N -
\frac {v}{c} Y) \end{vmatrix}
and
$$ \frac {1}{c} \frac {\partial}{\partial \tau} [L, \beta(M + \frac
{v}{c} Z), \beta(N - \frac {v}{c} Y)] = - \begin{vmatrix} \frac
{\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac
{\partial}{\partial \zeta} X & \beta(Y - \frac {v}{c} N) & \beta(Z +
\frac {v}{c} M) \end{vmatrix} $$ ... (2)
where
$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
The principle of Relativity requires that the Maxwell-Hertzian equations
for pure vacuum shall hold also for the system k, if they hold for the
system K, _i.e._, for the vectors of the electric and magnetic forces
acting upon electric and magnetic masses in the moving system k, which
are defined by their pondermotive reaction, the same equations hold, ...
_i.e._ ...
$$ \frac {1}{c} \frac {\partial}{\partial \tau} (X', Y', Z')
= \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac
{\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L' & M' &
N' \end{vmatrix} $$ ,
$$ \frac {1}{c} \frac {\partial}{\partial \tau} (L', M', N') =
- \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac
{\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X' & Y' &
Z' \end{vmatrix} $$ ... (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 ψ(_v_), which
depends only on _v_, and is independent of (ξ, η, ζ, τ). Hence the
relations,
_v_ _v_
[X′, Y′, Z′] = ψ (_v_) [X, β(Y - ----- N), β(Z + ------ M)],
_c_ _c_
_v_ _v_
[L′, M′, N′] = ψ (_v_) [L, β(M - ----- Z), β(N + ----- Y)],
_c_ _c_
Then by reasoning similar to that followed in §(3), it can be shown that
ψ(_v_) = 1.
_v_ _v_
[X′, Y′, Z′] = [X, β(Y - ----- N), β(Z + ------ M)]
_c_ _c_
_v_ _v_
[L′, M′, N′] = [L, β(M - ------ Z), β(N + ----- Y)],
_c_ _c_
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, Y, Z) of electric force. But if
the quantity of electricity be at rest relative to the moving system (at
least for the moment considered), then the force acting upon it, and
measured in the moving system is equivalent to the vector (X′, Y′, Z′).
The first three of equations (1), (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 see that
in the theory developed the electro-magnetic force plays the part of an
auxiliary concept, which owes its introduction in theory to the
circumstance that the electric and magnetic forces possess no existence
independent of the nature of motion of the co-ordinate system.
It is further clear that the asymmetry mentioned in the introduction
which occurs when we treat of the current excited by the relative motion
of a magnet and a conductor disappears. Also the question about the seat
of electromagnetic energy is seen to be without any meaning.
§ 7. Theory of Döppler’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:—
X = X₀ sin Φ
Y = Y₀ sin Φ
Z = Z₀ sin Φ
L = L₀ sin Φ
M = M₀ sin Φ
N = N₀ sin Φ
lx + my + nz
Φ = ω(t - ------------ )
c
Here (X₀, Y₀, Z₀) and (L₀, M₀, N₀) are the vectors which determine the
amplitudes of the train of waves, (_l_, _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 _k_:—By
applying the equations of transformation obtained in §6 for the electric
and magnetic forces, and the equations of transformation obtained in § 3
for the co-ordinates, and time, we obtain immediately:—
X′ = X₀ sin Φ′
v
Y′ = β(Y₀ - --- N₀) sin Φ′
c
v
Z′ = β(Z₀ - --- M₀) sin Φ′
c
L′ = L₀ sin Φ′
v
M′ = β(M₀ - --- Z₀) sin Φ′
c
v
N′ = β(N₀ - --- Y₀) sin Φ′
c
l′ξ + m′η + n′ζ
Φ′ = ω′(t - --------------- )
c
where
$$ \omega' = \omega \beta (1 - \frac {lv}{c}) $$ ,
$$ l' = \frac {l - \frac {v}{c}}{1 - \frac {lv}{c}} $$ ,
$$ m' = \frac {m}{\beta (1 - \frac {lv}{c})} $$ ,
$$ n' = \frac {n}{\beta (1 - \frac {lv}{c})} $$
From the equation for ω′ it follows:—If an observer moves with the
velocity _v_ relative to an infinitely distant source of light emitting
waves of frequency ν, in such a manner that the line joining the source
of light and the observer makes an angle of Φ with the velocity of the
observer referred to a system of co-ordinates which is stationary with
regard to the source, then the frequency ν′ which is perceived by the
observer is represented by the formula
$$ \nu' = \nu \frac {1 - cos \Phi \frac {v}{c}} {\sqrt {1 - \frac
{v^2}{c^2}}} $$
This is Döppler’s principle for any velocity. If Φ = 0, then the
equation takes the simple form
$$ \nu' = \nu (\frac {1 - \frac {v}{c}}{1 + \frac {v}{c}})^{\frac
{1}{2}} $$
We see that—contrary to the usual conception—ν = ∞, for _v_ = -_c_.
If Φ′ = angle between the wave-normal (direction of the ray) in the
moving system, and the line of motion of the observer, the equation for
_l´_ takes the form
$$ \cos \Phi' = \frac {\cos \Phi - \frac {v}{c}} {1 - \frac {v}{c} \cos
\Phi} $$
This equation expresses the law of observation in its most general form.
If Φ = π/2, the equation takes the simple form
v
cos Φ′ = ---
c
We have still to investigate the amplitude of the waves, which occur in
these equations. If A and A′ be the amplitudes in the stationary and the
moving systems (either electrical or magnetic), we have
$$ A'^2 = A^2 \frac {(1 - \frac {v}{c} \cos \Phi)^2} {1 - \frac
{v^2}{c^2}} $$
If Φ = 0, this reduces to the simple form
$$ A'^2 = A^2 \frac {1 - \frac {v}{c}} {1 + \frac {v}{c}} $$
From these equations, it appears that for an observer, which moves with
the velocity c towards the source of light, the source should appear
infinitely intense.
§ 8. Transformation of the Energy of the Rays of Light. Theory of the
Radiation-pressure on a perfect mirror.
Since A²/8π is equal to the energy of light per unit volume, we have to
regard A²/8π as the energy of light in the moving system. A′²/A² would
therefore denote the ratio between the energies of a definite
light-complex “measured when moving” and “measured when stationary,” the
volumes of the light-complex measured in K and _k_ being equal. Yet this
is not the case. If _l_, _m_, _n_ are the direction-cosines of the
wave-normal of light in the stationary system, then no energy passes
through the surface elements of the spherical surface
(_x_ - _clt_)² + (_y_ - _cmt_)² + (_z_ - _cnt_)² = R²,
which expands with the velocity of light. We can therefore say, that
this surface always encloses the same light-complex. Let us now consider
the quantity of energy, which this surface encloses, when regarded from
the system _k_, _i.e._, the energy of the light-complex relative to the
system _k_.
Regarded from the moving system, the spherical surface becomes an
ellipsoidal surface, having, at the time τ = 0, the equation:—
$$ (\beta \xi - l \beta \frac {v}{c} \xi)^2 + (\eta - m \beta \frac
{v}{c} \xi)^2 + (\zeta - n \beta \frac {v}{c} \xi)^2 = R^2 $$
If S = volume of the sphere, S′ = volume of this ellipsoid, then a
simple calculation shows that:
$$ \frac {S'}{S} = \frac {\beta}{\sqrt{1 - \frac {v}{c} \cos \Phi}} $$
If E denotes the quantity of light energy measured in the stationary
system, E′ the quantity measured in the moving system, which are
enclosed by the surfaces mentioned above, then
$$ \frac {E'}{E} = \frac {\frac {A'^2}{8\pi} S'}{\frac {A^2}{8\pi}S} =
\frac {1 - \frac {v}{c} \cos \Phi}{\sqrt{1 - \frac {v^2}{c^2}}} $$
If Φ = 0, we have the simple formula:—
$$ \frac {E'}{E} = (\frac{1 - \frac{v}{c}}{1 +
\frac{v}{c}})^{\frac{1}{2}} $$
It is to be noticed that the energy and the frequency of a light-complex
vary according to the same law with the state of motion of the observer.
Let there be a perfectly reflecting mirror at the co-ordinate-plane ξ =
0, from which the plane-wave considered in the last paragraph is
reflected. Let us now ask ourselves about the light-pressure exerted on
the reflecting surface and the direction, frequency, intensity of the
light after reflexion.
Let the incident light be defined by the magnitudes A cos Φ, _v_
(referred to the system K). Regarded from _k_, we have the corresponding
magnitudes:
$$ A' = A \frac{1 - \frac{v}{c} \cos \Phi}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
$$ \cos \Phi' = \frac{\cos \Phi - \frac{v}{c}}{1 - \frac{v}{c} \cos
\Phi} $$
$$ \nu' = \nu \frac{1 - \frac{v}{c} \cos \Phi}{\sqrt{1 -
\frac{v^2}{c^2}}} $$
For the reflected light we obtain, when the process is referred to the
system _k_:—
A″ = A′, cos Φ″ = -cos Φ″, ν″ = ν′
By means of a back-transformation to the stationary system, we obtain K,
for the reflected light:—
$$ A''' = A'' \frac{1 + \frac{v}{c}\cos \Phi''}{\sqrt{1 -
\frac{v^2}{c^2}}} = A \frac{1 - 2\frac{v}{c} \cos \Phi +
\frac{v^2}{c^2}}{1 - \frac{v^2}{c^2}} $$ ,
$$ \cos \Phi''' = \frac{\cos \Phi'' + \frac{v}{c}}{1 + \frac{v}{c}\cos
\Phi''} = - \frac{(1 + \frac{v^2}{c^2}) \cos \Phi - 2 \frac{v}{c}} {1 -
2 \frac{v}{c} \cos \Phi + \frac {v^2}{c^2}} $$ ,
$$ \nu''' = \nu'' \frac{1 + \frac{v}{c} \cos \Phi''}{\sqrt{1 -
\frac{v^2}{c^2}}} = \nu \frac{1 - 2 \frac{v}{c} \cos \Phi +
\frac{v^2}{c^2}} {(1 - \frac{v}{c})^2} $$
The amount or energy falling upon the unit surface of the mirror per
unit of time (measured in the stationary system) is A²/(8π (c cos Φ -
_v_)). The amount of energy going away from unit surface of the mirror
per unit of time is A‴²/(8π (-c cos Φ″ + _v_)). The difference of these
two expressions is, according to the Energy principle, the amount of
work exerted, by the pressure of light per unit of time. If we put this
equal to P._v_, where P = pressure of light, we have
$$ P = 2 \frac{A^2}{8\pi} \frac{(\cos \Phi - \frac{v}{c})^2} {1 -
(\frac{v}{c})^2} $$
As a first approximation, we obtain
A²
P = 2 -- cos² Φ
8π
which is in accordance with facts, and with other theories.
All problems of optics of moving bodies can be solved after the method
used here. The essential point is, that the electric and magnetic forces
of light, which are influenced by a moving body, should be transformed
to a system of co-ordinates which is stationary relative to the body. In
this way, every problem of the optics of moving bodies would be reduced
to a series of problems of the optics of stationary bodies.
§ 9. Transformation of the Maxwell-Hertz Equations.
Let us start from the equations:—
$$ \frac{1}{c}(\rho u_{x} + \frac{\partial X}{\partial t}) =
\frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} $$
$$ \frac{1}{c}(\rho u_{y} + \frac{\partial Y}{\partial t}) =
\frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} $$
$$ \frac{1}{c}(\rho u_{z} + \frac{\partial Z}{\partial t}) =
\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} $$
$$ \frac{1}{c} \frac{\partial L}{\partial t} = \frac{\partial
Y}{\partial z} - \frac{\partial Z}{\partial y} $$
$$ \frac{1}{c} \frac{\partial M}{\partial t} = \frac{\partial
Z}{\partial x} - \frac{\partial X}{\partial z} $$
$$ \frac{1}{c} \frac{\partial N}{\partial t} = \frac{\partial
X}{\partial y} - \frac{\partial Y}{\partial x} $$
where
$$ \rho = \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y}
+ \frac{\partial Z}{\partial z} $$
denotes 4π times the density of electricity, and (_u__{_x_}, _u__{_y_},
_u__{_z_}) are the velocity-components of electricity. If we now suppose
that the electrical-masses are bound unchangeably to small, rigid bodies
(Ions, electrons), then these equations form the electromagnetic basis
of Lorentz’s electrodynamics and optics for moving bodies.
If these equations which hold in the system K, are transformed to the
system _k_ with the aid of the transformation-equations given in § 3 and
§ 6, then we obtain the equations:—
$$ \frac{1}{c} (\rho' u_{\xi} + \frac{\partial X'}{\partial \tau}) =
\frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta}
$$ ,
$$ \frac{\partial L'}{\partial \tau} = \frac{\partial Y'}{\partial
\zeta} - \frac{\partial Z'}{\partial \eta} $$ ,
$$ \frac{1}{c} (\rho' u_{\eta} + \frac{\partial Y'}{\partial \tau}) =
\frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} $$
,
$$ \frac{\partial M'}{\partial \tau} = \frac{\partial Z'}{\partial \xi}
- \frac{\partial X'}{\partial \zeta} $$ ,
$$ \frac{1}{c} (\rho' u_{\zeta} + \frac{\partial Z'}{\partial \tau}) =
\frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} $$
,
$$ \frac{\partial N'}{\partial \tau} = \frac{\partial X'}{\partial \eta}
- \frac{\partial Y'}{\partial \xi} $$ ,
where
$$ \frac{u_{x} - v}{1 - \frac{u_{x}v}{c}} = u_{\xi} $$ ,
$$ \frac{u_{y}}{\beta(1 - \frac{vu_{x}}{c^2})} = u_{\eta} $$ ,
$$ \rho' = \frac{\partial X'}{\partial \xi} + \frac{\partial
Y'}{\partial \eta} + \frac{\partial Z'}{\partial \xi} = \beta(1 -
\frac{vu_{x}}{c^2}) \rho $$ ,
$$ \frac{u_{x}}{\beta(1 - \frac{vu_{x}}{c^2})} = u_{\zeta} $$ ,
Since the vector (_u__{ξ}, _u__{η}, _u__{ζ}) is nothing but the velocity
of the electrical mass measured in the system _k_, as can be easily seen
from the addition-theorem of velocities in § 4—so it is hereby shown,
that by taking our kinematical principle as the basis, the
electromagnetic basis of Lorentz’s theory of electrodynamics of moving
bodies correspond to the relativity-postulate. It can be briefly
remarked here that the following important law follows easily from the
equations developed in the present section:—if an electrically charged
body moves in any manner in space, and if its charge does not change
thereby, when regarded from a system moving along with it, then the
charge remains constant even when it is regarded from the stationary
system K.
§ 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
_d²x_ _d²y_ _d²z_
_m_ ----- = _e_X, _m_ ----- = _e_Y, _m_ ----- = _e_Z
_dt²_ _dt²_ _dt²_
Where (_x_, _y_, _z_) 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, 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 (_t_ = 0) the electron is
at rest relative to the system _k_, which moves parallel to the X-axis
with the constant velocity _v_.
From the suppositions made above, in combination with the principle of
relativity, it is clear that regarded from the system _k_, the electron
moves according to the equations
_d²_ξ _d²_η _d²_ζ
_m_ ----- = _e_X′, _m_ ----- = _e_Y′, _m_ ----- = _e_Z′ ,
_d_τ² _d_τ² _d_τ²
in the time immediately following the moment, where the symbols (ξ, η,
ζ, τ, X’, Y’, Z’) refer to the system _k_. If we now fix, that for _t_ =
_v_ = _y_ = _z_ = 0, τ = ξ = η = ζ = 0, then the equations of
transformation given in § 3 (and § 6) hold, and we have:
_v_
τ = β(_t_ - ---- _x_), ξ = β(_x_ - _vt_), η = _y_, ζ = _z_,
_c²_
_v_ _v_
X′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M)
_c_ _c_
With the aid of these equations, we can transform the above equations of
motion from the system _k_ to the system K, and obtain:—
(A)
$$ \frac{d^2 x}{dt^2} = \frac{e}{m} \frac{1}{\beta} X $$ ,
$$ \frac{d^2 y}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Y - \frac{v}{c} N)
$$ ,
$$ \frac{d^2 z}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Z + \frac{v}{c} M)
$$
Let us now consider, following the usual method of treatment, the
longitudinal and transversal mass of a moving electron. We write the
equations (A) in the form
_d²x_
_m_β² ----- = _e_X = _e_X′
_dt²_
_d²y_ _v_
_m_β² ----- = _e_β (Y - --- N) = _e_Y′
_dt²_ _c_
_d²z_ _v_
_m_β² ----- = _e_β (Z - --- M) = _e_Z′
_dt²_ _c_
and let us first remark, that _e_X′, _e_Y′, _e_Z′ are the components of
the ponderomotive force acting upon the electron, and are considered in
a moving system which, at this moment, moves with a velocity which is
equal to that of the electron. This force can, for example, be measured
by means of a spring-balance which is at rest in this last system. If we
briefly call this force as “the force acting upon the electron,” and
maintain the equation:—
Mass-number × acceleration-number = force-number, and if we further fix
that the accelerations are measured in the stationary system K, then
from the above equations, we obtain:—
Longitudinal mass:
$$ \frac{m}{(\sqrt{1 - \frac{v^2}{c^2}})^{\frac{3}{2}}} $$
Transversal mass:
$$ \frac{m}{\sqrt{1 - \frac{v^2}{c^2}}} $$
Naturally, when other definitions are given of the force and the
acceleration, other numbers are obtained for the mass; hence we see that
we must proceed very carefully in comparing the different theories of
the motion of the electron.
We remark that this result about the mass hold also for ponderable
material mass; for in our sense, a ponderable material point may be made
into an electron by the addition of an electrical charge which may be as
small as possible.
Let us now determine the kinetic energy of the electron. If the electron
moves from the origin of co-ordinates of the system K with the initial
velocity 0 steadily along the X-axis under the action of an
electromotive force X, then it is clear that the energy drawn from the
electrostatic field has the value ∫_e_X_dx_. Since the electron is only
slowly accelerated, and in consequence, no energy is given out in the
form of radiation, therefore the energy drawn from the electro-static
field may be put equal to the energy W of motion. Considering the whole
process of motion in questions, the first of equations A) holds, we
obtain:—
$$ W = \int eXdx = \int_0^v m\beta^3 vdv = mc^2 (\frac{1}{\sqrt{1 -
\frac{v^2}{c^2}}} - 1) $$
For _v_ = _c_, W is infinitely great. As our former result shows,
velocities exceeding that of light can have no possibility of existence.
In consequence of the arguments mentioned above, this expression for
kinetic energy must also hold for ponderable masses.
We can now enumerate the characteristics of the motion of the electrons
available for experimental verification, which follow from equations A).
1. From the second of equations A), it follows that an electrical force
Y, and a magnetic force N produce equal deflexions of an electron moving
with the velocity _v_, when Y = N_v_/_c_. Therefore we see that
according to our theory, it is possible to obtain the velocity of an
electron from the ratio of the magnetic deflexion A_{_m_}, and the
electric deflexion A_{_e_}, by applying the law:—
$$ \frac{A_{m}}{A_{e}} = \frac{v}{c} $$
This relation can be tested by means of experiments because the velocity
of the electron can be directly measured by means of rapidly oscillating
electric and magnetic fields.
2. From the value which is deduced for the kinetic energy of the
electron, it follows that when the electron falls through a potential
difference of P, the velocity _v_ which is acquired is given by the
following relation:—
$$ P = \int Xdx = \frac{m}{e}c^2 (\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} -
1) $$
3. We calculate the radius of curvature R of the path, where the only
deflecting force is a magnetic force N acting perpendicular to the
velocity of projection. From the second of equations A) we obtain:
$$ - \frac{d^2y}{dt^2} = \frac{v^2}{R} = \frac{e}{m} \frac{v}{c} N
\sqrt{1 - \frac{v^2}{c^2}} $$
or
_mv_β_c_
R = ----------
_e_N
These three relations are complete expressions for the law of motion of
the electron according to the above theory.
Footnote 6:
_Vide_ Note 9.
Footnote 7:
_Vide_ Note 9.
Footnote 8:
_Vide_ Note 12.
ALBRECHT EINSTEIN
[_A short biographical note._]
The name of Prof. Albrecht Einstein has now spread far beyond the narrow
pale of scientific investigators owing to the brilliant confirmation of
his predicted deflection of light-rays by the gravitational field of the
sun during the total solar eclipse of May 29, 1919. But to the serious
student of science, he has been known from the beginning of the current
century, and many dark problems in physics has been illuminated with the
lustre of his genius, before, owing to the latest sensation just
mentioned, he flashes out before public imagination as a scientific star
of the first magnitude.
Einstein is a Swiss-German of Jewish extraction, and began his
scientific career as a privat-dozent in the Swiss University of Zürich
about the year 1902. Later on, he migrated to the German University of
Prague in Bohemia as ausser-ordentliche (or associate) Professor. In
1914, through the exertions of Prof. M. Planck of the Berlin University,
he was appointed a paid member of the Royal (now National) Prussian
Academy of Sciences, on a salary of 18,000 marks per year. In this post,
he has only to do and guide research work. Another distinguished
occupant of the same post was Van’t Hoff, the eminent physical chemist.
It is rather difficult to give a detailed, and consistent chronological
account of his scientific activities,—they are so variegated, and cover
such a wide field. The first work which gained him distinction was an
investigation on Brownian Movement. An admirable account will be found
in Perrin’s book ‘The Atoms.’ Starting from Boltzmann’s theorem
connecting the entropy, and the probability of a state, he deduced a
formula on the mean displacement of small particles (colloidal)
suspended in a liquid. This formula gives us one of the best methods for
finding out a very fundamental number in physics—namely—the number of
molecules in one gm. molecule of gas (Avogadro’s number). The formula
was shortly afterwards verified by Perrin, Prof. of Chemical Physics in
the Sorbonne, Paris.
To Einstein is also due the resuscitation of Planck’s quantum theory of
energy-emission. This theory has not yet caught the popular imagination
to the same extent as the new theory of Time, and Space, but it is none
the less iconoclastic in its scope as far as classical concepts are
concerned. It was known for a long time that the observed emission of
light from a heated black body did not correspond to the formula which
could be deduced from the older classical theories of continuous
emission and propagation. In the year 1900, Prof. Planck of the Berlin
University worked out a formula which was based on the bold assumption
that energy was emitted and absorbed by the molecules in multiples of
the quantity _h_ν, where _h_ is a constant (which is universal like the
constant of gravitation), and ν 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 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 propagated in space as a unit pulse (he calls it a Light-cell),
and falling upon an individual atom, liberates electrons according to
the energy equation
1
_h_ν = --- _mv²_ + A,
2
where (_m_, _v_) are the mass and velocity of the electron. A is a
constant characteristic of the metal plate.
There was little material for the confirmation of this law when it was
first proposed (1905), and eleven years elapsed before Prof. Millikan
established, by a set of experiments scarcely rivalled for the
ingenuity, skill, and care displayed, the absolute truth of the law. As
results of this confirmation, and other brilliant triumphs, the quantum
law is now regarded as a fundamental law of Energetics. In recent years,
X-rays have been added to the domain of light, and in this direction
also, Einstein’s photo-electric formula has proved to be one of the most
fruitful conceptions in Physics.
The quantum law was next extended by Einstein to the problems of
decrease of specific heat at low temperature, and here also his theory
was confirmed in a brilliant manner.
We pass over his other contributions to the equation of state, to the
problems of null-point energy, and photo-chemical reactions. The recent
experimental works of Nernst and Warburg seem to indicate that through
Einstein’s genius, we are probably for the first time having a
satisfactory theory of photo-chemical action.
In 1915, Einstein made an excursion into Experimental Physics, and here
also, in his characteristic way, he tackled one of the most fundamental
concepts of Physics. It is well-known that according to Ampere, the
magnetisation of iron and iron-like bodies, when placed within a coil
carrying an electric current is due to the excitation in the metal of
small electrical circuits. But the conception though a very fruitful
one, long remained without a trace of experimental proof, though after
the discovery of the electron, it was 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 tried, in a series of papers
beginning from 1911, to extend it to the case of non-uniform motion. The
last paper in the present collection is a translation of a comprehensive
article which he contributed to the Annalen der Physik in 1916 on this
subject, and gives, in his own words, the Principles of Generalized
Relativity. The triumphs of this theory are now matters of public
knowledge.
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.
Principle of Relativity
INTRODUCTION.
At the present time, different opinions are being held about the
fundamental equations of Electro-dynamics for moving bodies. The
Hertzian[9] forms must be given up, for it has appeared that they are
contrary to many experimental results.
In 1895 H. A. Lorentz[10] published his theory of optical and electrical
phenomena in moving bodies; this theory was based upon the atomistic
conception (vorstellung) of electricity, and on account of its great
success appears to have justified the bold hypotheses, by which it has
been ushered into existence. In his theory, Lorentz proceeds from
certain equations, which must hold at every point of “Äther”; then by
forming the average values over “Physically infinitely small” regions,
which however contain large numbers of electrons, the equations for
electro-magnetic processes in moving bodies can be successfully built
up.
In particular, Lorentz’s theory gives a good account of the
non-existence of relative motion of the earth and the luminiferous
“Äther”; it shows that this fact is intimately connected with the
covariance of the original equation, when certain simultaneous
transformations of the space and time co-ordinates are effected; these
transformations have therefore obtained from H. Poincare[11] the name of
Lorentz-transformations. The covariance of these fundamental equations,
when subjected to the Lorentz-transformation is a purely mathematical
fact _i.e._ not based on any physical considerations; I will call this
the Theorem of Relativity; this theorem rests essentially on the form of
the differential equations for the propagation of waves with the
velocity of light.
Now without _recognizing_ any hypothesis about the connection between
“Äther” and matter, we can expect these mathematically evident theorems
to have their consequences so far extended—that thereby even those laws
of ponderable media which are yet unknown may anyhow possess this
covariance when subjected to a Lorentz-transformation; by saying this,
we do not indeed express an opinion, but rather a conviction,—and this
conviction I may be permitted to call the Postulate of Relativity. The
position of affairs here is almost the same as when the Principle of
Conservation of Energy was postulated in cases, where the corresponding
forms of energy were unknown.
Now if hereafter, we succeed in maintaining this covariance as a
definite connection between pure and simple observable phenomena in
moving bodies, the definite connection may be styled ‘the Principle of
Relativity.’
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[12] 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[13] has brought out the point very clearly, that this
postulate is not an artificial hypothesis but is rather a new way of
comprehending the time-concept which is forced upon us by observation of
natural phenomena.
The Principle of Relativity has not yet been formulated for
electro-dynamics of moving bodies in the sense characterized by me. In
the present essay, while formulating this principle, I shall obtain the
fundamental equations for moving bodies in a sense which is uniquely
determined by this principle.
But it will be shown that none of the forms hitherto assumed for these
equations can exactly fit in with this principle.[14]
We would at first expect that the fundamental equations which are
assumed by Lorentz for moving bodies would correspond to the Relativity
Principle. But it will be shown that this is not the case for the
general equations which Lorentz has for any possible, and also for
magnetic bodies; but this is approximately the case (if neglect the
square of the velocity of matter in comparison to the velocity of light)
for those equations which Lorentz hereafter infers for non-magnetic
bodies. But this latter accordance with the Relativity Principle is due
to the fact that the condition of non-magnetisation has been formulated
in a way not corresponding to the Relativity Principle; therefore the
accordance is due to the fortuitous compensation of two contradictions
to the Relativity-Postulate. But meanwhile enunciation of the Principle
in a rigid manner does not signify any contradiction to the hypotheses
of Lorentz’s molecular theory, but it shall become clear that the
assumption of the contraction of the electron in Lorentz’s theory must
be introduced at an earlier stage than Lorentz has actually done.
In an appendix, I have gone into discussion of the position of Classical
Mechanics with respect to the Relativity Postulate. Any easily
perceivable modification of mechanics for satisfying the requirements of
the Relativity theory would hardly afford any noticeable difference in
observable processes; but would lead to very surprising consequences. By
laying down the Relativity-Postulate from the outset, sufficient means
have been created for deducing henceforth the complete series of Laws of
Mechanics from the principle of conservation of Energy alone (the form
of the Energy being given in explicit forms).
NOTATIONS.
Let a rectangular system (_x_, _y_, _z_, _t_,) of reference be given in
space and time. The unit of time shall be chosen in such a manner with
reference to the unit of length that the velocity of light in space
becomes unity.
Although I would prefer not to change the notations used by Lorentz, it
appears important to me to use a different selection of symbols, for
thereby certain homogeneity will appear from the very beginning. I shall
denote the vector electric force by E, the magnetic induction by M, the
electric induction by _e_ and the magnetic force by _m_, so that (E, M,
_e_, _m_) are used instead of Lorentz’s (E, B, D, H) respectively.
I shall further make use of complex magnitudes in a way which is not yet
current in physical investigations, _i.e._, instead of operating with
(_t_), I shall operate with (_i t_), where _i_ denotes √(-1). If now
instead of (_x_, _y_, _z_, _i t_), I use the method of writing with
indices, certain essential circumstances will come into evidence; on
this will be based a general use of the suffixes (1, 2, 3, 4). The
advantage of this method will be, as I expressly emphasize here, that we
shall have to handle symbols which have apparently a purely real
appearance; we can however at any moment pass to real equations if it is
understood that of the symbols with indices, such ones as have the
suffix 4 only once, denote imaginary quantities, while those which have
not at all the suffix 4, or have it twice denote real quantities.
An individual system of values of (_x_, _y_, _z_, _t_) _i. e._, of (_x₁_
_x₂_ _x₃_ _x₄_) shall be called a space-time point.
Further let _u_ denote the velocity vector of matter, ε the dielectric
constant, μ the magnetic permeability, σ the conductivity of matter,
while ρ denotes the density of electricity in space, and _x_ the vector
of “Electric Current” which we shall some across in §7 and §8.
PART I
§ 2.
The Limiting Case.
The Fundamental Equations for Äther.
By using the electron theory, Lorentz in his above mentioned essay
traces the Laws of Electro-dynamics of Ponderable Bodies to still
simpler laws. Let us now adhere to these simpler laws, whereby we
require that for the limiting case ε = 1, μ = 1, σ = 0, they should
constitute the laws for ponderable bodies. In this ideal limiting case ε
= 1, μ = 1, σ = 0, E will be equal to _e_, and M to _m_. At every space
time point (_x_, _y_, _z_, _t_) we shall have the equations[15]
(i) Curl _m_ - (δ_e_/δ_t_) = ρu
(ii) div _e_ = ρ
(iii) Curl _e_ + δ_m_/δ_t_ = 0
(iv) div m = 0
I shall now write (_x₁_ _x₂_ _x₃_ _x₄_) for (_x_, _y_, _z_, _t_) and
(ρ₁, ρ₂, ρ₃, ρ₄) for
$$ (\rho u_{x}, \rho u_{y}, \rho u_{z}, i\rho) $$
_i.e._ the components of the convection current ρu, and the electric
density multiplied by √ -1
Further I shall write
_f__{2 3}, _f__{3 1}, _f__{1 2}, _f__{1 4}, _f__{2 4}, _f__{3 4}.
for
m_{_x_}, m_{_y_}, m_{_z_}, -ie_{_x_}, -ie_{_y_}, -ie_{_z_}.
_i.e._, the components of m and (-_i.e._) along the three axes; now if
we take any two indices (h. k) out of the series
3, 4), _f__{_k h_} = -_f__{_k h_},
Therefore
_f₃₂_ = -_f₂₃_, _f₁₃_ = -_f₃₁_, _f₂₁_ = -_f₁₂_
_f₄₁_ = -_f₁₄_, _f₄₄_ = -_f₂₄_, _f₄₃_ = -_f₃₄_
Then the three equations comprised in (i), and the equation (ii)
multiplied by i becomes
$$ \begin{vmatrix} & \frac{\delta f_{1 2}}{\delta x_{2}} & +
\frac{\delta f_{1 3}}{\delta x_{3}} & + \frac{\delta f_{1 4}}{\delta
x_{4}} & = \rho_{1} \frac{\delta f_{2 1}}{\delta x_{1}} & & +
\frac{\delta f_{2 3}}{\delta x_{3}} & \times \frac{\delta f_{2
4}}{\delta x_{4}} & = \rho_{2} \frac{\delta f_{3 1}}{\delta x_{1}} &
\times \frac{\delta f_{3 2}}{\delta x_{2}} & & + \frac{\delta f_{3
4}}{\delta x_{4}} & = \rho_{3} \frac{\delta f_{4 1}}{\delta x_{1}} & +
\frac{\delta f_{4 2}}{\delta x_{2}} & + \frac{\delta f_{4 3}}{\delta
x_{3}} & & = \rho_{4} \end{vmatrix} × $$
On the other hand, the three equations comprised in (iii) and the (iv)
equation multiplied by (_i_) becomes
$$ \begin{vmatrix} & \frac{\delta f_{3 4}}{\delta x_{2}} & +
\frac{\delta f_{4 2}}{\delta x_{3}} & + \frac{\delta f_{2 3}}{\delta
x_{4}} & = = \frac{\delta f_{4 3}}{\delta x_{1}} & & + \frac{\delta f_{1
4}}{\delta x_{3}} & + \frac{\delta f_{3 1}}{\delta x_{4}} & =
0 \frac{\delta f_{2 4}}{\delta x_{1}} & + \frac{\delta f_{4 1}}{\delta
x_{2}} & & + \frac{\delta f_{1 2}}{\delta x_{4}} & = 0 \frac{\delta f_{3
2}}{\delta x_{1}} & + \frac{\delta f_{1 3}}{\delta x_{2}} & +
\frac{\delta f_{2 1}}{\delta x_{3}} & & = - \end{vmatrix} × $$
By means of this method of writing we at once notice the perfect
symmetry of the 1st as well as the 2nd system of equations as regards
permutation with the indices, (1, 2, 3, 4).
§ 3.
It is well-known that by writing the equations i) to iv) in the symbol
of vector calculus, we at once set in evidence an invariance (or rather
a (covariance) of the system of equations A) as well as of B), when the
co-ordinate system is rotated through a certain amount round the
null-point. For example, if we take a rotation of the axes round the
z-axis, through an amount φ, keeping e, m fixed in space, and introduce
new variables _x₁′_ _x₂′_ _x₃′_ _x₄′_ instead of _x₁_ _x₂_ _x₃_ _x₄_
where _x′₁_ = _x₁_ cos φ + _x₂_ sin φ, _x′₂_ = -_x₁_ sin φ + _x₂_ cos φ,
_x′₃_ = _x₃_, _x′₄_ = _x₄_, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄,
where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ and _f′__{1
2}, ... ... _f′__{3 4}, where
_f′₂₃_ = _f₂₃_ cos φ + _f₃₁_ sin φ,
_f′₃₁_ = - _f₂₃_ sin φ + _f₃₁_ cos φ,
_f′₁₂_ = _f₁₂_,
_f′₁₄_ = _f₁₄_ cos φ + _f₂₄_ sin φ,
_f′₂₄_ = - _f₁₄_ sin φ + _f₂₄_ cos φ,
_f′₃₄_ = _f₃₄__{3 4},
_f′__{_k h_} = - _f__{_k h_} (h l k = 1, 2, 3, 4).
then out of the equations (A) would follow a corresponding system of
dashed equations (A´) composed of the newly introduced dashed
magnitudes.
So upon the ground of symmetry alone of the equations (A) and (B)
concerning the _suffixes_ (1, 2, 3, 4), the theorem of Relativity, which
was found out by Lorentz, follows without any calculation at all.
I will denote by _i_ψ a purely imaginary magnitude, and consider the
substitution
_x₁′_ = _x₁_,
_x₂′_ = _x₂_,
_x₃′_ = _x₃_ cos _i_ψ + _x₄_ sin _i_ψ, (1)
_x₄′_´ = - _x₃_ sin _i_ψ + _x₄_ cos _i_ψ,
Putting
$$ - i \tan i\psi = \frac{e^{\psi} - e^{-\psi}}{e^{\psi}+e^{-\psi}} = q
$$ ,
$$ \psi = \frac{1}{2} \log \frac{1 + q}{1 - q′} $$ (2)
We shall have cos _i_ψ = 1/√(1 - _q²_), sin _i_ψ = _iq_/√(1 - _q²_)
where -1 < _q_ < 1, and √(1 - _q²_) is always to be taken with the
positive sign.
Let us now write _x′₁_ = _x′_, _x′₂_ = _y′_, _x′₃_ = _z′_, _x′₄_ = _it′_
(3)
then the substitution 1) takes the form
_x′_ = _x_, _y′_ = _y_, _z′_ = (_z_ - _qt_)/√(1 - _q²_), _t′_ =
(-_qz_ + _t_)/√(1 - _q²_), (4)
the coefficients being essentially real.
If now in the above-mentioned rotation round the Z-axis, we replace 1,
2, 3, 4 throughout by 3, 4, 1, 2, and φ by _i_ψ, we at once perceive
that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where
ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos _i_ψ + ρ₄ sin _i_ψ,
ρ′₄ = - ρ₃ sin _i_ψ + ρ₄ cos _i_ψ),
and _f′__{1 2} ... _f′__{3 4}, where
_f′__{4 1} = _f__{4 1} cos _i_ψ + _f__{1 3} sin _i_ψ,
_f′__{1 3} = - _f__{4 1} sin _i_ψ + _f__{1 3} cos _i_ψ,
_f′__{3 4} = _f__{3 4},
_f′__{3 2} = _f__{3 2} cos _i_ψ + _f__{4 2} sin _i_ψ,
_f′__{4 2} = - _f__{3 2} sin _i_ψ + _f__{4 2} cos _i_ψ,
_f′__{1 2} = _f__{1 2}, _f__{_k h_} = - _f′__{_k h_},
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 _x´_ _y´_ _z´_ _t´_ be
taken as a new frame of reference, then we shall have
(5) ρ´ = ρ [(-_qu__{_z_} + 1)/√(1 - _q²_)],
ρ´_u__{_z_}´ = ρ[(_u__{_z_} - _q_)/√(1 - _q²_)],
ρ´_u__{_x_}´ = ρ_u__{_x_},
ρ´_u__{_y_}´ = ρ_u__{_y_}.
(6) _e´__{_x´_} = (_e__{_x_} - _qm__{_y_})/(√(1 - _q²_)),
_m´__{_r´_} = (_qe__{_x_} + _m__{_y_})/(√(1 - _q²_)),
_e´__{_z´_} = _e__{_z_}.
(7) _m´__{_x´_} = (_m__{_x_} - _qe__{_y_})/(√(1 - _q²_)),
_e´__{_y_´} = (_qm__{_x_} + _e__{_y_})/(√(1 - _q²_)),
_m_´_{_z_´} = _m__{_z_}.
Then we have for these newly introduced vectors _u´_, _e´_, _m´_ (with
components _u__{_x_}´, _u__{_y_}´, _u__{_z_}´; _e__{_x_}´, _e__{_y_}´,
_e__{_z_}´; _m__{_x_}´, _m__{_y_}´, _m__{_z_}´), and the quantity ρ´ a
series of equations I´), II´), III´), IV´) which are obtained from I),
II), III), IV) by simply dashing the symbols.
We remark here that _e__{_x_} - _qm__{_y_}, _e__{_y_} + _qm__{_x_} are
components of the vector _e_ + [_vm_], where _v_ is a vector in the
direction of the positive Z-axis, and | _v_ | = _q_, and [_vm_] is the
vector product of _v_ and _m_; similarly -_qe__{_x_} + _m__{_y_},
_m__{_x_} + _qe__{_y_} are the components of the vector _m_ - [_ve_].
The equations 6) and 7), as they stand in pairs, can be expressed as.
_e′__{_x′_} + _im′__{_x′_} = (_e__{_x_} + _im__{_x_}) cos _i_ψ +
(_e__{_y_} + _im__{_y_}) sin _i_ψ,
_e′__{_y′_} + _im′__{_y′_} = - (_e__{_x_} + _im__{_x_}) sin _i_ψ +
(_e__{_y_} + _im__{_y_}) cos _i_ψ,
_e′__{_z′_} + _im′__{_z′_} = _e′__{_z_} + _im__{_z_}.
If φ denotes any other real angle, we can form the following
combinations:—
(_e′__{_x′_} + _im′__{_x′_}) cos. φ + (_e′__{_y″_} + _im′__{_y′_})
sin φ
= (_e__{_x_} + _im__{_x_}) cos. (φ + _i_ψ) + (_e__{_y_} +
_im__{_y_}) sin (φ + _i_ψ),
= (_e′__{_x′_} + _im′__{_x′_}) sin φ + (_e′__{_y′_} +
_im′__{_y′_}) cos. φ
= - (_e__{_x_} + _im__{_x_}) sin (φ + _i_ψ) + (_e__{_y_} +
_im__{_y_}) cos. (φ + _i_ψ).
§ 4. Special Lorentz Transformation.
The rôle which is played by the Z-axis in the transformation (4) can
easily be transferred to any other axis when the system of axes are
subjected to a transformation about this last axis. So we came to a more
general law:—
Let _v_ be a vector with the components _v__{_x_}, _v__{_y_}, _v__{_z_},
and let | _v_ | = _q_ < 1. By _ṽ_ we shall denote any vector which is
perpendicular to _v_, and by _r__{_v_}, _r__{_ṽ_} we shall denote
components of _r_ in direction of _ṽ_ and _v_.
Instead of (_x_, _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_, _z_) in the
first system of reference, _r′_ for the same vector with the components
(_x′_ _y′_ _z′_) in the second system of reference, then for the
direction of _v_, we have
(10) _r′__{_v_} = (_r__{_v_} - _qt_)/√(1 - _q²_)
and for the perpendicular direction _ṽ_,
(11) _r′__{_ṽ_} = _r__{_ṽ_}
and further (12) _t′_ = (-_qr__{_v_} + _t_)/√(1 - _q²_).
The notations (_r′__{_ṽ_}, _r′__{_v_}) are to be understood in the sense
that with the directions _v_, and every direction _ṽ_ perpendicular to
_v_ in the system (_x_, _y_, _z_) are always associated the directions
with the same direction cosines in the system (_x′_ _y′_ _z′_).
A transformation which is accomplished by means of (10), (11), (12)
with the condition 0 < _q_ < 1 will be called a special
Lorentz-transformation. We shall call _v_ the vector, the direction of
_v_ the axis, and the magnitude of _v_ the moment of this
transformation.
If further ρ′ and the vectors _u′_, _e′_, _m′_, in the system (_x′_ _y′_
_z′_) are so defined that,
(13) ρ′ = ρ[(-_qu__{_v_} + 1)/√(1 - _q²_)],
ρ′_u_′_{_v_} = ρ(_u__{_v_} - _q_)/√(1 - _q²_),
ρ′_u__{_ṽ_} = ρ′_u__{_v_},
further
(14) (_e′_ + _im′_)_{_ṽ_} = ((_e_ + _im_) - _i_[_v_, (_e_ +
_im_])']_{_ṽ_})/√(1 - _q²_).
(15) (_e′_ + _im′_)_{_v_} = (_e_ + _im_) - _i_[_u_, (_e_ +
_im_)]_{_v_}.
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) _r__{_v_} = (_r′__{_v_} + _qt′_)/√(1 - _q²_),
_r__{_ṽ_} = _r′__{_ṽ_},
_t_ = (_qr′__{_v_} + _t′_)/√(1 - _q²_),
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
(_x₁_′, _x₂_′, _x₃_′, _x₄_′) instead of (_x′_, _y′_, _z′_, _it′_) and
ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′_u′_{_x′_}, ρ′_u′_{_y′_}, ρ′_u′_{_z′_},
_i_ρ′).
Like the rotation round the Z-axis, the transformation (4), and more
generally the transformations (10), (11), (12), are also linear
transformations with the determinant + 1, so that
(17) _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ _i. e._ _x²_ + _y²_ + _z²_ -
_t²_,
is transformed into
_x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_ _i. e._ _x′²_ + _y′²_ + _z′²_ -
_t′²_.
On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃²
+ ρ₄²) = ρ²(1 - _u__{_x²_}, -_u__{_y²_}, -_u__{_z²_}) = ρ²(1 - _u²_)
transformed into ρ²(1 - _u²_) or in other words,
(18) ρ√(1 - _u²_)
is an invariant in a Lorentz-transformation.
If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four
values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - _u²_))(_u__{_x_}, _u__{_y_},
_u__{_z_}, _i_) so that ω₁² + ω₂² + ω₃² + ω₄² = -1.
It is apparent that these four values are determined by the vector _u_
and inversely the vector _u_ of magnitude < 1 follows from the 4 values
ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -_i_ω₄ real and positive
and condition (19) is fulfilled.
The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios of
_dx₁_, _dx₂_, _dx₃_, _dx₄_ to
(20) √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_)) = _dt_√(1 - _u²_).
The differentials denoting the displacements of matter occupying the
spacetime point (_x₁_, _x₂_, _x₃_, _x₄_) to the adjacent space-time
point.
After the Lorentz-transformation is accomplished the velocity of matter
in the new system of reference for the same space-time point (_x′_ _y′_
_z′_ _t′_) is the vector _u′_ with the ratios _dx′_/_dt′_, _dy′_/_dt′_,
_dz′_/_dt′_, _dl′_/_dt′_, as components.
Now it is quite apparent that the system of values
_x₁_ = ω₁, _x₂_ = ω₂, _x₃_ = ω₃, _x₄_ = ω₄
is transformed into the values
_x₁′_ = ω₁′, _x₂′_ = ω₂′, _x₃′_ = ω₃′, _x₄′_ = ω₄′
in virtue of the Lorentz-transformation (10), (11), (12).
The dashed system has got the same meaning for the velocity _u′_ after
the transformation as the first system of values has got for _u_ before
transformation.
If in particular the vector _v_ of the special Lorentz-transformation be
equal to the velocity vector _u_ of matter at the space-time point
(_x₁_, _x₂_, _x₃_, _x₄_) then it follows out of (10), (11), (12) that
ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ = _i_
Under these circumstances therefore, the corresponding space-time point
has the velocity _v′_ = 0 after the transformation, it is as if we
transform to rest. We may call the invariant ρ√(1 - _u²_) the
rest-density of Electricity.[16]
§ 5. Space-time Vectors.
Of the 1st and 2nd kind.
If we take the principal result of the Lorentz transformation together
with the fact that the system (A) as well as the system (B) is covariant
with respect to a rotation of the coordinate-system round the null
point, we obtain the general _relativity theorem_. In order to make the
facts easily comprehensible, it may be more convenient to define a
series of expressions, for the purpose of expressing the ideas in a
concise form, while on the other hand I shall adhere to the practice of
using complex magnitudes, in order to render certain symmetries quite
evident.
Let us take a linear homogeneous transformation,
$$ \begin{vmatrix} x_{1} x_{2} x_{3} x_{4} \end{vmatrix} =
\begin{vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & a_{1 4} a_{2 1} & a_{2
2} & a_{2 3} & a_{2 4} a_{3 1} & a_{3 2} & a_{3 3} & a_{3 4} a_{4 1} &
a_{4 2} & a_{4 3} & a_{4 4} \end{vmatrix} \begin{vmatrix}
x_{1}' x_{2}' x_{3}' x_{4}' \end{vmatrix} $$
the Determinant of the matrix is +1, all co-efficients without the index
4 occurring once are real, while _a₄₁_, _a₄₂_, _a₄₃_, are purely
imaginary, but _a₄₄_ is real and > 0, and _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_
transforms into _x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_. The operation shall
be called a general Lorentz transformation.
(This notation, which is due to Dr. C. E. Cullis of the Calcutta
University, has been used throughout instead of Minkowski’s notation,
_x₁_ = _a₁₁x₁′_ + _a₁₂x₂′_+ _a₁₃x₃′_+ _a₁₄x₄′_.)
If we put _x₁′_ = _x′_, _x₂′_ = _y′_, _x₃′_ = _z′_, _x₄′_ = _it′_, then
immediately there occurs a homogeneous linear transformation of (_x_,
_y_, _z_, _t_) to (_x′_, _y′_, _z′_, _t′_) with essentially real
co-efficients, whereby the aggregate -_x²_ - _y²_ - _z²_ + _t²_
transforms into -_x′²_ - _y′²_ - _z′²_ + _t′²_, and to every such system
of values _x_, _y_, _z_, _t_ with a positive _t_, for which this
aggregate > 0, there always corresponds a positive _t’_; this last is
quite evident from the continuity of the aggregate _x_, _y_, _z_, _t_.
The last vertical column of co-efficients has to fulfil the condition
22) _a₁₄²_ + _a₂₄²_ + _a₃₄²_ + _a₄₄²_ = 1.
If _a₁₄_ = _a₂₄_ = _a₃₄_ = 0, then _a₄₄_ = 1, and the Lorentz
transformation reduces to a simple rotation of the spatial co-ordinate
system round the world-point.
If _a₁₄_, _a₂₄_, _a₃₄_ are not all zero, and if we put _a₁₄_ : _a₂₄_ :
_a₃₄_ : _a₄₄_ = _v__{_x_} : _v__{_y_} : _v__{_z_} : _i_
_q_ = √(_v__{_x_}² + _v__{_y_}² +_v__{_z_}²) < 1.
On the other hand, with every set of values of _a₁₄_, _a₂₄_, _a₃₄_,
_a₄₄_ which in this way fulfil the condition 22) with real values of
_v__{_x_}, _v__{_y_}, _v__{_z_}, we can construct the special Lorentz
transformation (16) with (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) as the last
vertical column,—and then every Lorentz-transformation with the same
last vertical column (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) can be supposed to be
composed of the special Lorentz-transformation, and a rotation of the
spatial co-ordinate system round the null-point.
The totality of all Lorentz-Transformations forms a group. Under a
space-time vector of the 1st kind shall be understood a system of four
magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a
Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′,
ρ₄′), where these are the values of (_x₁′_, _x₂′_, _x₃′_, _x₄′_),
obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (_x₁_, _x₂_, _x₃_, _x₄_)
in the expression (21).
Besides the time-space vector of the 1st kind (_x₁_, _x₂_, _x₃_, _x₄_)
we shall also make use of another space-time vector of the first kind
(_y₁_, _y₂_, _y₃_, _y₄_), and let us form the linear combination
(23) _f₂₃_(_x₂__y₃_ - _x₃__y₂_) + _f₃₁_(_x₃__y₁_ - _x₁__y₃_) +
_f₁₂_(_x₁__y₂_
- _x₂__y₁_) + _f₁₄_(_x₁__y₄_ - _x₄__y₁_) + _f₂₄_(_x₂__y₄_ -
_x₄__y₂_) +
_f₃₄_(_x₃__y₄_ - _x₄__y₃_)
with six coefficients _f₂₃_--_f₃₄_. Let us remark that in the vectorial
method of writing, this can be constructed out of the four vectors.
_x₁_, _x₂_, _x₃_; _y₁_, _y₂_, _y₃_; _f₂₃_, _f₃₁_, _f₁₂_; _f₁₄_, _f₂₄_,
_f₃₄_ and the constants _x₄_ and _y₄_, at the same time it is
symmetrical with regard the indices (1, 2, 3, 4).
If we subject (_x₁_, _x₂_, _x₃_, _x₄_) and (_y₁_, _y₂_, _y₃_, _y₄_)
simultaneously to the Lorentz transformation (21), the combination (23)
is changed to:
(24) _f₂₃′_(_x₂′__y₃′_ - _x₃′__y₂′_) + _f₃₁_(_x₃′__y₁′_ -
_x₁′__y₃′_) + _f₁₂_
(_x₁′__y₂′_ - _x₂′__y₁′_) + _f₁₄′_(_x₁′__y₄′_) - _x₄′__y₁′_) +
_f₂₄′_(_x₂′__y₄′_
- _x₄′__y₂′_) + _f₃₄′_(_x₃′__y₄′_ - _x₄′__y₃′_),
where the coefficients _f₂₃′_, _f₃₁′_, _f₁₂′_, _f₁₄′_, _f₂₄′_, _f₃₄′_,
depend solely on (_f₂₃_ _f₂₄_) and the coefficients _a₁₁_ ... _a₄₄_.
We shall define a space-time Vector of the 2nd kind as a system of
six-magnitudes _f₂₃_, _f₃₁_ ... _f₃₄_, with the condition that when
subjected to a Lorentz transformation, it is changed to a new system
_f₂₃′_ ... f₃₄, ... which satisfies 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 Äther.
If _x_, _y_, _z_, _it_ (space co-ordinates, and time _it_) is subjected
to a Lorentz transformation, and at the same time (_pu__{_x_},
_pu__{_y_}, _pu__{_z_}, _i_ρ) (convection-current, and charge density
ρ_i_) is transformed as a space time vector of the 1st kind, further
(_m__{_x_}, _m__{_y_}, _m__{_z_}, -_ie__{_x_}, -_ie__{_y_}, -_ie__{_z_})
(magnetic force, and electric induction × (-_i_) is transformed as a
space time vector of the 2nd kind, then the system of equations (I),
(II), and the system of equations (III), (IV) transforms into
essentially corresponding relations between the corresponding magnitudes
newly introduced into the system.
These facts can be more concisely expressed in these words: the system
of equations (I and II) as well as the system of equations (III) (IV)
are covariant in all cases of Lorentz-transformation, where (ρ_u_, _i_ρ)
is to be transformed as a space time vector of the 1st kind, (_m_ -
_ie_) is to be treated as a vector of the 2nd kind, or more
significantly,—
(ρ_u_, _i_ρ) is a space time vector of the 1st kind, (_m_ - _ie_)[17] is
a space-time vector of the 2nd kind.
I shall add a few more remarks here in order to elucidate the conception
of space-time vector of the 2nd kind. Clearly, the following are
invariants for such a vector when subjected to a group of Lorentz
transformation.
(_i_) _m²_ - _e²_ = _f₂₃²_ + _f₃₁²_ + _f₁₂²_ + _f₁₄²_ + _f₂₄²_ +
_f₂₄²_
_me_ = _i_(_f₂₃__f₁₄_ + _f₃₁__f₂₄_ + _f₁₂__f₃₄_).
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 (_m_
- _ie_)² = 0, _ie_ _m²_ - _e²_ = 0, and at the same time (_m e_) = 0,
_ie_ the vector _m_ and _e_ are equal and perpendicular to each other;
when such is the case, these two properties remain conserved for the
space-time vector of the 2nd kind in every Lorentz-transformation.
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__{_x_} = 0, _e__{_x_} = 0.
Then
(_m__{_x_}, -_i e__{_x_})² + (_m__{_y_}, -_i e__{_y_})² ≠ 0.
Therefore (_e__{_y_} + _i m__{_y_})/(_e__{_x_} + _i e__{_x_}) is
different from +_i_, and we can therefore define a complex argument (φ +
_i_ψ) in such a manner that
tan (φ + _i_ψ)
_e__{_y_} + _i m__{_y_}
= -------------------------
_e__{_x_} + _i m__{_x_}
If then, by referring back to equations (9), we carry out the
transformation (1) through the angle ψ and a subsequent rotation round
the Z-axis through the angle φ, we perform a Lorentz-transformation at
the end of which _m__{_y_} = 0, _e__{_y_} = 0, and therefore _m_ and _e_
shall both coincide with the new Z-axis. Then by means of the invariants
_m²_ - _e²_, (_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.
§ 6. 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.[18] 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 (_x₀_, _y₀_, _z₀_) the
time _t₀_ be compared with a space point P (_x_, _y_, _z_) at the time
_t_, and if the difference of time _t_ - _t₀_, (let _t_ > _t₀_) be less
than the length A P _i.e._ less than the time required for the
propagation of light from A to P, and if _q_ = (_t_ - _t₀_)/(A P) < 1,
then by a special Lorentz transformation, in which A P is taken as the
axis, and which has the moment _q_, we can introduce a time parameter
_t′_, which (see equation 11, 12, § 4) has got the same value _t′_ = _0_
for both space-time points (A, _t₀_), and (P, t). So the two events can
now be comprehended to be simultaneous.
Further, let us take at the same time _t₀_ = 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_ - _t₀_ (t > _t₀_) be less than the time which light
requires for propagation from the line A B, or the plane (A B C) to P.
Let q be the quotient of (_t_ - _t₀_) 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, _t₀_), (B, _t₀_), (C, _t₀_) and (P,
t) are simultaneous.
If four space-points, which do not lie in one plane, are conceived to be
at the same time _t₀_, then it is no longer permissible to make a change
of the time parameter by a Lorentz-transformation, without at the same
time destroying the character of the simultaneity of these four space
points.
To the mathematician, accustomed on the one hand to the methods of
treatment of the poly-dimensional manifold, and on the other hand to the
conceptual figures of the so-called non-Euclidean Geometry, there can be
no difficulty in adopting this concept of time to the application of the
Lorentz-transformation. The paper of Einstein which has been cited in
the Introduction, has succeeded to some extent in presenting the nature
of the transformation from the physical standpoint.
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 limiting case
ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in
matter. We look for those relations which make it possible for us—when
proper fundamental data are given—to obtain the following quantities at
every place and time, and therefore at every space-time point as
functions of (_x_, _y_, _z_, _t_):—the vector of the electric force E,
the magnetic induction M, the electrical induction _e_, the magnetic
force _m_, the electrical space-density ρ, the electric current s (whose
relation hereafter to the conduction current is known by the manner in
which conductivity occurs in the process), and lastly the vector _v_,
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 (_x_, _y_, _z_, _t_),—lead us to a knowledge of
other magnitude as functions of _x_, _y_, _z_, _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 (_x_, _y_, _z_, _t_).
For the case of bodies at rest, _i.e._ when _u_ (_x_, _y_, _z_, _t_) = 0
the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same
fundamental equations. They are;—
(1) The Differential Equations:—which contain no constant referring to
matter:—
(_i_) Curl _m_ - δ_e_/δ_t_ = C,
(_ii_) div _e_ = lρ.
(_iii_) Curl E + δM/δ_t_ = 0,
(_iv_) Div M = 0.
(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 comprised in the equations
(V) _e_ = ε E, M = μ_m_, C = σE.
where ε = dielectric constant, μ = magnetic permeability, σ = the
conductivity of matter, all given as function of _x_, _y_, _z_, _t_; _s_
is here the conduction current.
By employing a modified form of writing, I shall now cause a latent
symmetry in these equations to appear. I put, as in the previous work,
_x₁_ = _x_, _x₂_ = _y_, _x₃_ = _z_, _x₄_ = _it_,
and write _s₁_, _s₂_, _s₃_, _s₄_ for C_{_x_}, C_{_y_}, C_{_z_} (√-1)ρ.
Further _f₂₃_, _f₃₁_, _f₁₂_, _f₁₄_, _f₂₄_, _f₃₄_
for _m__{_x_}, _m__{_y_}, _m__{_z_}, -_i_(_e__{_x_}, _e__{_y_},
_e__{_z_}),
and F₂₃, F₃₁, F₁₂, F₁₄, F₂₄, F₃₄
for M_{_x_}, M_{_y_}, M_{_z_}, -_i_(E_{_x_}, E_{_y_}, E_{_z_})
lastly we shall have the relation _f__{k h} = - _f__{_h k_}, _F__{_k h_}
= -_F__{_h k_}, (the letter _f_, F shall denote the field, _s_ the
(_i.e._ current).
Then the fundamental Equations can be written as
(A)
∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = s₁
∂_f₂₁_/∂_x₁_ + + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = s₂
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + + ∂_f₃₄_/∂_x₄_ = s₃
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = s₄
and the equations (3) and (4), are
∂F₃₄/∂_x₂_ + ∂F₄₂/∂_x₃_ + ∂F₂₃/∂_x₄_ = 0
∂F₄₃/∂_x₁_ + + ∂F₁₄/∂_x₃_ + ∂F₃₁∂_x₄_ = 0
∂F₂₄/∂_x₁_ + ∂F₄₁/∂_x₂_ + + ∂F₁₂/∂_x₄_ = 0
∂F₃₂/∂_x₁_ + ∂F₁₃/∂_x₂_ + ∂F₂₁/∂_x₃_ = 0
§ 8. The Fundamental Equations.
We are now in a position to establish in a unique way the fundamental
equations for bodies moving in any manner by means of these three axioms
exclusively.
The first Axion shall be,—
When a detached region[19] of matter is at rest at any moment, therefore
the vector _u_ is zero, for a system (_x_, _y_, _z_, _t_)—the
neighbourhood may be supposed to be in motion in any possible manner,
then for the space-time point _x_, _y_, _z_, _t_, the same relations (A)
(B) (V) which hold in the case when all matter is at rest, shall also
hold between ρ, the vectors C, _e_, _m_, _M_, _E_ and their
differentials with respect to _x_, _y_, _z_, _t_. The second axiom shall
be:—
Every velocity of matter is < 1, smaller than the velocity of
propagation of light.[20]
The fundamental equations are of such a kind that when (_x_, _y_, _z_,
_it_) are subjected to a Lorentz transformation and thereby (_m_ - _ie_)
and (_M_ - _iE_) are transformed into space-time vectors of the second
kind, (C, _i_ρ) 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:—
(_m_, -_ie_), and (_M_, -_iE_) are space-time vectors of the second
kind, (C, _i_p) is a space-time vector of the first kind.
This axiom I call the Principle of Relativity.
In fact these three axioms lead us from the previously mentioned
fundamental equations for bodies at rest to the equations for moving
bodies in an unambiguous way.
According to the second axiom, the magnitude of the velocity vector |
_u_ | 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
ω₁ = u_{_x_}/√(1 - _u²_),
ω₂ = u_{_y_}/√(1 - u²),
ω₃ = u_{_z_}/√(1 - u²),
ω₄ = _i_/√(1 - u²)
with the relation
(27) ω₁² + ω₂² + ω₃² + ω₄² = - |
From what has been said at the end of § 4, it is clear that in the case
of a Lorentz-transformation, this set behaves like a space-time vector
of the 1st kind.
Let us now fix our attention on a certain point (_x_, _y_, _z_) of
matter at a certain time (_t_). If at this space-time point _u_ = 0,
then we have at once for this point the equations (_A_), (_B_) (_V_) of
§ 7. If _u_ ≠ 0, then there exists according to 16), in case | _u_ | <
1, a special Lorentz-transformation, whose vector _v_ is equal to this
vector _u_ (_x_, _y_, _z_, _t_), and we pass on to a new system of
reference (_x′_ _y′_ _z′_ _t′_) in accordance with this transformation.
Therefore for the space-time point considered, there arises as in § 4,
the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = _i_, therefore the
new velocity vector ω′ = 0, the space-time point is as if transformed to
rest. Now according to the third axiom the system of equations for the
transformed point (_x′_ _y′_ _z′_ _t_) involves the newly introduced
magnitude (_u′_ ρ′, 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′_ = 0, these equations must be exactly
equivalent to
(1) the differential equations (_A′_), (_B′_), which are obtained from
the equations (_A_), (_B_) by simply dashing the symbols in (_A_) and
(_B_).
(2) and the equations
(V′) _e′_ = ε_E′_, _M’_ = μ_m′_, _C′_ = σ_E′_
where ε, μ, σ are the dielectric constant, magnetic permeability, and
conductivity for the system (_x′_ _y′_ _z′_ _t′_) _i.e._ in the
space-time point (_x_ _y_, _z_ _t_) of matter.
Now let us return, by means of the reciprocal Lorentz-transformation to
the original variables (_x_, _y_, _z_, _t_), and the magnitudes (_u_, ρ,
C, _e_, _m_, _E_, _M_) and the equations, which we then obtain from the
last mentioned, will be the fundamental equations sought by us for the
moving bodies.
Now from § 4, and § 6, it is to be seen that the equations _A_), as well
as the equations _B_) are covariant for a Lorentz-transformation, _i.e._
the equations, which we obtain backwards from _A′_) _B′_), must be
exactly of the same form as the equations _A_) and _B_), as we take them
for bodies at rest. We have therefore as the first result:—
The differential equations expressing the fundamental equations of
electrodynamics for moving bodies, when written in ρ and the vectors C,
_e_, _m_, E, M, are exactly of the same form as the equations for moving
bodies. The velocity of matter does not enter in these equations. In the
vectorial way of writing, we have
I) curl _m_ - ∂_e_/∂_t_ = C₁,
II) div _e_ = ρ
III) curl E + ∂M/∂_t_ = 0
IV) div M = 0
The velocity of matter occurs only in the auxiliary equations which
characterise the influence of matter on the basis of their
characteristic constants ε, μ, σ. Let us now transform these auxiliary
equations V′) into the original co-ordinates (_x_, _y_, _z_, and _t_.)
According to formula 15) in § 4, the component of _e′_ in the direction
of the vector _u_ is the same as that of (_e_ + [_u_ _m_]), the
component of _m′_ is the same as that of _m_ - [_u_ _e_], but for the
perpendicular direction _ū_, the components of _e′_, _m′_ are the same
as those of (_e_ + [_u_ _m_]) and (_m_ - [_u_ _e_], multiplied by 1/√(1
- _u²_). On the other hand E′ and M′ shall stand to E + [_u_M], and M -
[_u_E] in the same relation as _e′_ and _m′_ to _e_ + [_um_], and _m_ -
(_ue_). From the relation _e′_ = εE′, the following equations follow
(C) _e_ + [_um_] = ε(E + [_u_M]),
and from the relation M′ = μ_m′_, we have
(D) M - [_u_ E] = μ(_m_ - [_ue_]),
For the components in the directions perpendicular to _u_, and to each
other, the equations are to be multiplied by √(1 - _u²_).
Then the following equations follow from the transformation? equations
(12), (10), (11) in § 4, when we replace q, _r__{_v_}, _r__{_ṽ_}, _t_,
_r′__{_v_}, _r′__{_ṽ_}, _t’_ by |_u_|, C_{_u_}, C_{_ū_}, ρ, C′_{_u_},
C′_{_ū_}, ρ′
ρ′ = (-|_u_| C_{_u_} + ρ)/√(1 - _u²_),
C’_{_u_} = (C_{_u_} - |_u_|ρ)/√(1 - _u²_),
C′_{_ū_} = C_{_ū_},
E) (C_{_u_} - |_u_|ρ)/√(1 - _u²_) = σ(E + [_u_M])_{_u_},
C_{_ū_} = σ (E + [_u_M])_{_u_}/√(1 - _u²_).
In consideration of the manner in which σ enters into these relations,
it will be convenient to call the vector C - ρ_u_ with the components
C_{_u_} - ρ|_u_| in the direction of _u_, and C′_{_ū_} in the directions
_ū_ perpendicular to _u_ the “Convection current.” This last vanishes
for σ = 0.
We remark that for ε = 1, μ = 1 the equations _e′_ = E′, _m′_ = M′
immediately lead to the equations _e_ = E, _m_ = M by means of a
reciprocal Lorentz-transformation with -_u_ as vector; and for σ = 0,
the equation C′ = 0 leads to C = ρ_u_; that the fundamental equations of
Äther discussed in § 2 becomes in fact the limitting case of the
equations obtained here with ε = 1, μ = 1, σ = 0.
§ 9. 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 §8. In the article
on Electron-theory (Ency., Math., Wiss., Bd. V. 2, Art 14) Lorentz has
given the fundamental equations for any possible, even magnetised bodies
(see there page 209, Eqn XXX′, formula (14) on page 78 of the same
(part).
(III_a″_) Curl (H - [_u_E]) = J + _d_D/_dt_ + _u_ div D
- curl [_u_D].
(I″) div D = ρ
(IV″) curl E = - _d_B/_dt_, Div B = 0 (V′)
Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) μ = 1,
B = H, and in addition to that takes account of the occurrence of the
di-electric constant ε, and conductivity σ according to equations
(ε_q_XXXIV″, p. 327) D - E = (ε - 1) {E + [_u_B]}
(ε_q_XXXIII′, p. 223) J = σ(E + [_u_B])
Lorentz’s E, D, H are here denoted by E, M, _e_, _m_ while J denotes the
conduction current.
The three last equations which have been just cited here coincide with
eqn (II), (III), (IV), the first equation would be, if J is identified
with C, = _u_ρ (the current being zero for σ = 0,
(29) Curl [H - (_u_, E)] = C + _d_D/_dt_ - curl [_u_D],
and thus comes out to be in a different form than (1) here. Therefore
for magnetised bodies, Lorentz’s equations do not correspond to the
Relativity Principle.
On the other hand, the form corresponding to the relativity principle,
for the condition of non-magnetisation is to be taken out of (D) in §8,
with μ = 1, not as B = H, as Lorentz takes, but as (30) B - [_u_D] = H -
[_u_D] (M - [_u_E] = _m_ - [_ue_]. Now by putting H = B, the
differential equation (29) is transformed into the same form as eqn (1)
here when _m_ - [_ue_] = M - [_u_E]. Therefore it so happens that by a
compensation of two contradictions to the relativity principle, the
differential equations of Lorentz for moving non-magnetised bodies at
last agree with the relativity postulate.
If we make use of (30) for non-magnetic bodies, and put accordingly H =
B + [_u_, (D - E)], then in consequence of (C) in §8,
(ε - 1) (E + [_u_, B]) = D - E + [_u_. [_u_, D - E]],
_i.e._ for the direction of _u_,
(ε - 1) (E + [_u_B])_{_u_} = (D - E)_{_u_}
and for a perpendicular direction ū,
(ε - 1) [E + (_u_B)]_{_u_} = (1 - _u²_) (D - E)_{_u_}
_i.e._ it coincides with Lorentz’s assumption, if we neglect _u²_ in
comparison to 1.
Also to the same order of approximation, Lorentz’s form for J
corresponds to the conditions imposed by the relativity principle [comp.
(E) § 8]—that the components of J_{_u_}, J_{_ū_} are equal to the
components of σ (E + [_u_ B]) multiplied by √(1 - _u²_) or 1 / √(1 -
_u²_) respectively.
§10. Fundamental Equations of E. Cohn.
E. Cohn assumes the following fundamental equations.
(31) Curl (M + [_u_ E]) = _d_E/_dt_ + u div. E + J
- Curl [E - (_u_. M)] = _d_M/_dt_ + u div. M.
(32) J = σ E, = ε E - [_u_ M], M = μ (_m_ + [_u_ E.])
where E M are the electric and magnetic field intensities (forces), E, M
are the electric and magnetic polarisation (induction). The equations
also permit the existence of true magnetism; if we do not take into
account this consideration, div. M. is to be put = 0.
An objection to this system of equations, is that according to these,
for ε = 1, μ = 1, the vectors force and induction do not coincide. If in
the equations, we conceive E and M and not E - (U. M), and M + [U E] as
electric and magnetic forces, and with a glance to this we substitute
for E, M, E, M, div. E, the symbols _e_, M, E + [U M], _m_ - [_u_ _e_],
ρ, then the differential equations transform to our equations, and the
conditions (32) transform into
J = σ(E + [_u_ M])
_e_ + [_u_, (_m_ - [_u_ _e_])] = ε(E + [_u_ M])
M - [_u_, (E + _u_ M)] = μ(_m_ - [_u_ _e_])
then in fact the equations of Cohn become the same as those required by
the relativity principle, if errors of the order _u²_ 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
(33) E = εE, M = μM, J = σE.
§11. Typical Representations of the Fundamental Equations.
In the statement of the fundamental equations, our leading idea had been
that they should retain a covariance of form, when subjected to a group
of Lorentz-transformations. Now we have to deal with ponderomotive
reactions and energy in the electro-magnetic field. Here from the very
first there can be no doubt that the settlement of this question is in
some way connected with the simplest forms which can be given to the
fundamental equations, satisfying the conditions of covariance. In order
to arrive at such forms, I shall first of all put the fundamental
equations in a typical form which brings out clearly their covariance in
case of a Lorentz-transformation. Here I am using a method of
calculation, which enables us to deal in a simple manner with the
space-time vectors of the 1st, and 2nd kind, and of which the rules, as
far as required are given below.
A system of magnitudes _a__{_h_ _k_} formed into the matrix
| _a₁₁_...................._a__{1 _q_} |
| |
| |
| |
| _a__{_p_ 1}..........._a__{_p_ _q_} |
arranged in _p_ horizontal rows, and _q_ vertical columns is called a
_p_ × _q_ series-matrix, and will be denoted by the letter A.
If all the quantities _a__{_h_ _k_} are multiplied by C, the resulting
matrix will be denoted by CA.
If the roles of the horizontal rows and vertical columns be
intercharged, we obtain a _q_ × _p_ series matrix, which will be known
as the transposed matrix of A, and will be denoted by Ā.
Ā = | _a₁₁_ ...................... _a__{_p_ 1} |
| |
| _a__{1 _q_} ............ _a__{_p_ _q_} |
If we have a second _p_ × _q_ series matrix B,
B = | _b₁₁_ ......................... _b₁__{_q_} |
| |
| _b__{_p_ 1} ............. b_{_p_ _q_} |
then A + B shall denote the _p_ × _q_ series matrix whose members are
_a__{_h_ _k_} + _b__{_h_ _k_}.
2⁰ If we have two matrices
A = | _a₁₁_ ..................... _a__{1 _q_} |
| |
| _a__{_p_ 1} ........... _a__{_p_ _q_} |
B = | _b__{1 1} .............. _b__{1 _r_} |
| |
| _b__{_q_ 1} .......... _b__{_p_ _r_} |
where the number of horizontal rows of B, is equal to the number of
vertical columns of A, then by AB, the product of the matrices A and B,
will be denoted the matrix
C = | _c₁₁_ ...................... _c__{1 _r_} |
| |
| _c__{_p_ _r_} ........... _c__{_p_ _p_} |
where _c__{_h_ _k_} = _a__{_h_ 1} _b₁__{_k_} + _a__{_h_ 2} _b__{2 _h_} +
... _a__{_k_ _s_} _b__{_s_ _k_} + ... + _a__{_k_ _q_} _b__{_q_ _h_}
these elements being formed by combination of the horizontal rows of A
with the vertical columns of B. For such a point, the associative law
(AB)S = A(BS) holds, where S is a third matrix which has got as many
horizontal rows as B (or AB) has got vertical columns.
For the transposed matrix of C = BA, we have Ċ = ḂĀ
3⁰. We shall have principally to deal with matrices with at most four
vertical columns and for horizontal rows.
As a unit matrix (in equations they will be known for the sake of
shortness as the matrix 1) will be denoted the following matrix (4 × 4
series) with the elements.
(34) | e₁₁ e₁₂ e₁₃ e₁₄ | = | 1 0 0 0 |
| e₂₁ e₂₂ e₂₃ e₂₄ | | 0 1 0 0 |
| e₃₁ e₃₂ e₃₃ e₃₄ | | 0 0 1 0 |
| e₄₁ e₄₂ e₄₃ e₄₄ | | 0 0 0 1 |
For a 4 × 4 series-matrix, Det A shall denote the determinant formed of
the 4 × 4 elements of the matrix. If det A ≠ 0, then corresponding to A
there is a reciprocal matrix, which we may denote by A⁻¹ so that A⁻¹A =
1.
A matrix
_f_ = | 0 _f₁₂_ _f_₁₃ _f₁₄_ |
| _f_₂₁ 0 _f₂₃_ _f₂₄_ |
| _f₃₁_ _f_₃₂ 0 _f₃₄_ |
| _f_₄₁ _f_₄₂ _f_₄₃ 0 |
in which the elements fulfil the relation _f__{_h_ _k_} = -_f__{_h_
_k_}, is called an alternating matrix. These relations say that the
transposed matrix _ḟ_ = -_f_. Then by _f_^{*} will be the _dual_,
alternating matrix
(35)
_f_^{*} = | 0 _f₃₄_ _f_₄₂ _f₂₃_ |
| _f_₄₃ 0 _f₁₄_ _f₃₁_ |
| _f₂₄_ _f_₄₁ 0 _f₁₂_ |
| _f_₃₂ _f_₁₃ _f_₂₁ 0 |
Then (36) _f_* _f_ = _f₃₄_ _f₂₂_ + _f₄₂_ _f₃₁_ + _f₃₂_ _f₂₄_
_i.e._ We shall have a 4 × 4 series matrix in which all the elements
except those on the diagonal from left up to right down are zero, and
the elements in this diagonal agree with each other, and are each equal
to the above mentioned combination in (36).
The determinant of _f_ is therefore the square of the combination, by
Det^{½}_f_ we shall denote the expression
Det^{½}_f_
= _f₃₂_ _f₁₄_ _f₁₃_ _f₂₄_ + _f₂₁_ _f₃₄_·
4⁰. A linear transformation
_x__{_h_} = α_{_h_1} _x₁′_ + α_{_h_2} _x₂_′ + α_{_h_3} _x₃′_ + α_{_h_4}
_x₄′_ (_h_ = 1,2,3,
which is accomplished by the matrix
A = | α₁₁, α₁₂, α₁₃, α₁₄ |
| |
| α₂₁, α₂₂, α₂₃, α₂₄ |
| |
| α₃₁, α₃₂, α₃₃, α₃₄ |
| |
| α₄₁, α₄₂, α₄₃, α₄₄ |
will be denoted as the transformation A.
By the transformation A, the expression
_x²₁_ + _x²₂_ + _x²₃_ + _x²₄_ is changed into the quadratic for _m_ ∑
α_{_hk_} _x__{_h_}′ _x__{_k_}′,
where α_{_hk_} = α_{1_k_} α_{1_k_} + α_{2_h_} α_{2_k_} + α_{3_h_}
α_{3_k_} + α_{4_h_} α_{4_k_} are the members of a 4 × 4 series matrix
which is the product of Ā A, the transposed matrix of A into A. If by
the transformation, the expression is changed to
_x′₁²_ + _x₂′_^2 + _x₃′_^2 + _x′₄²_,
we must have Ā A = 1.
A has to correspond to the following relation, if transformation (38) is
to be a Lorentz-transformation. For the determinant of A) it follows out
of (39) that (Det A)² = 1, or Det A = ± 1.
From the condition (39) we obtain
A⁻¹ = Ā,
_i.e._ the reciprocal matrix of A is equivalent to the transposed 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 _a₄₄_ > 0.
5⁰. A space time vector of the first kind[21] which s represented by the
1 × 4 series matrix,
(41) _s_ = |_s₁_ _s₂_ _s₃_ _s₄_|
is to be replaced by _s_A in case of a Lorentz transformation
A. _i.e._ _s′_ = | _s₁′_ _s₂′_ _s₃′_ _s₄′_| = |_s₁_ _s₂_ _s₃_ _s₄_|
A;
A space-time vector of the 2nd kind[22] with components _f₂₃_ ... _f₃₄_
shall be represented by the alternating matrix
(42) _f_ = | 0 _f_₁₂ _f₁₃_ _f₁₄_ |
|_f₂₁_ 0 _f_₂₃ _f₂₄_ |
|_f_₃₁ _f₃₂_ 0 _f₃₄_ |
|_f_₄₁ _f_₄₂ _f_₄₃ 0 |
and is to be replaced by A⁻¹ _f_ A in case of a Lorentz transformation
[see the rules in § 5 (23) (24)]. Therefore referring to the expression
(37), we have the identity Det^{½} (Ā _f_ A) = Det A. Det^{½} _f_.
Therefore Det^{½} _f_ becomes an invariant in the case of a Lorentz
transformation [see eq. (26) See. § 5].
Looking back to (36), we have for the dual matrix (Ā_f_*A) (A⁻¹_f_A) =
A⁻¹_f_*_f_A = Det^{½} function. A⁻¹A = Det^{½}_f_ from which it is to be
seen that the dual matrix _f_* behaves exactly like the primary matrix
_f_, and is therefore a space time vector of the II kind; _f_* is
therefore known as the dual space-time vector of _f_ with components
(_f₁₄_, _f₂₄_, _f₃₄_,), (_f₂₃_}, _f₃₁_, _f₁₂_).
6. If _w_ and _s_ are two space-time rectors of the 1st kind then by _w_
_ṡ_ (as well as by _s_ _ẇ_) will be understood the combination (43) _w₁_
_s₁_ + _w₂_ _s₂_ + _w₃_ _s₃_ + _w₄_ _s₄_.
In case of a Lorentz transformation A, since (_w_A) (Ā_ṡ_) = _w_ _s_,
this expression is invariant.—If _w_ _ṡ_ = 0, then _w_ and _s_ are
perpendicular to each other.
Two space-time rectors of the first kind (_w_, _s_) gives us a 2 × 4
series matrix
| _w₁_ _w₂_ _w₃_ _w₄_ |
| _s₁_ _s₂_ _s₃_ _s₄_ |
Then it follows immediately that the system of six magnitudes (44)
_w₂_ _s₃_ - _w₃_ _s₂_,
_w₃_ _s₁_ - _w₁_ _s₃_,
_w₁_ _s₂_ - _w₂_ _s₁_,
_w₁_ _s₄_ - _w₄_ _s₁_,
_w₂_ _s₄_ - _w₄_ _s₂_,
_w₃_ _s₄_ - _w₄_ _s₃_,
behaves in case of a Lorentz-transformation as a space-time vector of
the II kind. The vector of the second kind with the components (44) are
denoted by [_w_, _s_]. We see easily that Det^{½} [_w_, _s_] = 0. The
dual vector of [_w_, _s_] shall be written as [_w_, _s_].
If _ẇ_ is a space-time vector of the 1st kind, _f_ of the second
kind, _w_ _f_ signifies a 1 × 4 series matrix. In case of a
Lorentz-transformation A, _w_ is changed into _w′_ = _w_A, _f_ into
_f′_ = A⁻¹ _f_ A,—therefore _w′_ _f′_ becomes = (_w_A A⁻¹ _f_ A) =
_w_ _f_ A _i.e._ _w_ _f_ is transformed as a space-time vector of
the 1st kind.[23] We can verify, when _w_ is a space-time vector of
the 1st kind, _f_ of the 2nd kind, the important identity
(45) [_w_, _w__f_] + [_w_, _w__f_*]* = (_w_] _ẇ_)_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 ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_,
ω_f_ = | _i__f_₄₁, _i__f_₄₂, _i__f_₄₃, 0 |;
ω_f_* = | _i__f_₃₂, _i__f_₁₃, _i__f_₂₁, 0 |
[ω · ω_f_] = 0, 0, 0, _f_₄₁, _f_₄₂, _f_₄₃;
[ω · ω_f_*]* = 0, 0, 0, _f_₃₂, _f_₁₃, _f_₂₁.
The fact that in this special case, the relation is satisfied, suffices
to establish the theorem (45) generally, for this relation has a
covariant character in case of a Lorentz transformation, and is
homogeneous in (ω₁, ω₂, ω₃, ω₄).
After these preparatory works let us engage ourselves with the equations
(C,) (D,) (E) by means which the constants ε μ, σ will be introduced.
Instead of the space vector _u_, the velocity of matter, we shall
introduce the space-time vector of the first kind ω with the components.
ω₁ = _u__{_x_}/√(1 - _u²_),
ω₂ = _u__{_y_}/√(1 - _u²_),
ω₃ = _u__{_z_}/√(1 - _u²_),
ω₄ = _i_/√(1 - _u²_).
(40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 and -_i_ω₄ > 0.
By F and _f_ shall be understood the space time vectors of the second
kind M - _i_E, _m_ - _ie_.
In Φ = ωF, we have a space time vector of the first kind with components
Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄
Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄
Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄
Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃
The first three quantities (φ₁, φ₂, φ₃) are the components of the
space-vector (E + [_u_, M])/√(1 - _u²_),
and further (φ₄ = _i_[_u_ E]/√(1 - _u²_).
Because F is an alternating matrix,
(49) ωΦ = ω₁ φ₁ + ω₂ Φ₂ + ω₃ Φ₃ + ω₄ Φ₄ = 0.
_i.e._ Φ is perpendicular to the vector ω; we can also write Φ₄ =
_i_[ω_{x} Φ₁ + ω_{y} Φ₂ + ω_{z} Φ₃].
I shall call the space-time vector Φ of the first kind as the _Electric
Rest Force_.[24]
Relations analogous to those holding between -ωF, E, M, U, hold amongst
-ω_f_, _e_, _m_, _u_, and in particular -ω_f_ is normal to ω. The
relation (C) can be written as
{C} ω_f_ = εωF.
The expression (ω_f_) gives four components, but the fourth can be
derived from the first three.
Let us now form the time-space vector 1st kind, ψ - _i_ω_f_*, whose
components are
ψ₁ = -_i_(ω₂ _f₃₄_ + ω₃ _f_₄₂ + ω₄ _f₂₃_)
ψ₂ = -_i_(ω₁ _f_₄₃ + ω₃ _f_₄₄ + ω₄ _f₃₁_)
ψ₃ = -_i_(ω₁ _f₂₄_ + ω₂ _f_₄₁ + ω₄ _f₁₂_)
ψ₄ = -_i_(ω₁ _f_₃₂ + ω₂ _f_₁₃ + ω₃ _f_₂₁)
Of these, the first three ψ₁, ψ₂, ψ₃, are the _x_, _y_, _z_ components
of the space-vector 51) (m - (_ue_))/√(1 - _u²_) and further (52) ψ₄ =
_i_(_u_m)/√(1 - _u²_).
Among these there is the relation
(53) ωψ = ω₁ ψ₁ + ω₂ ψ₂ + ω₃ ψ₃ + ω₄ ψ₄ = 0
which can also be written as ψ₄ = _i_ (_u__{_x_} ψ₁ + _u__{_y_} ψ₂ +
_u__{_z_} ψ₃).
The vector ψ is perpendicular to ω; we can call it the _Magnetic
rest-force_.
Relations analogous to these hold among the quantities ωF*, M, E, _u_
and Relation (D) can be replaced by the formula
{ D } -ωF* = μψ_f_*.
We can use the relations (C) and (D) to calculate F and _f_ from Φ and ψ
we have
ωF = -Φ, ωF* = -_i_μψ, ω_f_ = -εΦ, ω_f_* = -_i_ψ.
and applying the relation (45) and (46), we have
F = [ω. Φ] + _i_μ[ω. ψ]* 55)
_f_ = ε[ω. Φ] + _i_[ω. ψ]* 56)
_i.e._
F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + _i_μ [ω₃ Ψ₄ - ω₄ ψ₃], etc.
_f₁₂_ = ε(ω₁ Φ₂ - ω₂ φ₁) + _i_ [ω₃ ψ₄ - ω₄ ψ₃]., etc.
Let us now consider the space-time vector of the second kind [Φ ψ], with
the components
[ Φ₂ ψ₃ - Φ₃ ψ₂, Φ₃ ψ₁ - Φ₁ ψ₃, Φ₁ ψ₂ - Φ₂ ψ₁ ]
[ Φ₁ ψ₄ - Φ₄ ψ₁, Φ₂ ψ₄ - Φ₄ ψ₂, Φ₃ ψ₄ - Φ₄ ψ₃ ]
Then the corresponding space-time vector of the first kind ω[Φ, ψ]
vanishes identically owing to equations 9) and 53)
for ω[Φ.ψ] = -(ωψ)Φ + (ωΦ)ψ
Let us now take the vector of the 1st kind
(57) Ω = _i_ω[Φψ]*
with the components
Ω₁ = -_i_ | ω₂ ω₃ ω₄ |
| Φ₂ Φ₃ Φ₄ |
| ψ₂ ψ₃ ψ₄ |, etc.
Then by applying rule (45), we have
(58) [Φ.ψ] = _i_[ωΩ]*
_i.e._ Φ₁ψ₂ - Φ₂ψ₁ = _i_(ω₃Ω₄ - ω₄Ω₃) etc.
The vector Ω fulfils the relation
(ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0,
(which we can write as Ω₄ = _i_(ω_{x}Ω₁ + ω_{y}Ω₂ + ω_{z}Ω₃) and Ω is
also normal to ω. In case ω = 0, we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and
[Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ |
|ψ₁ ψ₂ ψ₃ |.
I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.
As for the relation E), which introduces the conductivity σ we have -ωS
= -(ω₁_s₁_ + ω₂_s₂_ + ω₃_s₃_ + ω₄_s₄_) = (- | _u_ | C_{_u_} + ρ)/√(1 -
_u²_) = ρ′.
This expression gives us the rest-density of electricity (see §8 and
§4).
Then 61) = _s_ + (ω_ṡ_)ω represents a space-time vector of the 1st kind,
which since ωω = -1, is normal to ω, and which I may call the
rest-current. Let us now conceive of the first three component of this
vector as the (_x_-_y_-_z_) co-ordinates of the space-vector, then the
component in the direction of _u_ is
C_{_u_} - (| _u_ | ρ′)/√(1 - _u²_)
= (_c__{_u_} - | _u_ |ρ)/√(1 - _u²_)
= J_{_u_}/(1 - _u²_)
and the component in a perpendicular direction is C_{_u_} = J_{_ū_}.
This space-vector is connected with the space-vector J = C - ρ_u_, which
we denoted in §8 as the conduction-current.
Now by comparing with Φ = -ωF, the relation (E) can be brought into the
form
{E} _s_ + (ω_ṡ_)ω = - σωF,
This formula contains four equations, of which the fourth follows from
the first three, since this is a space-time vector which is
perpendicular to ω.
Lastly, we shall transform the differential equations (A) and (B) into a
typical form.
§12. The Differential Operator Lor.
A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S_{_kh_} |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
with the condition that in case of a Lorentz transformation it is to be
replaced by ĀSA, may be called a space-time matrix of the II kind. We
have examples of this in:—
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 transformation A, it
is replaced by (A⁻¹_f_A·A⁻¹FA) = A⁻¹_f_FA,
3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time
vectors of the 1st kind, the 4 × 4 matrix with the element S_{_hk_} =
ω_{_h_}Ω_{_k_},
lastly in a multiple L of the unit matrix of 4 × 4 series in which all
the elements in the principal diagonal are equal to L, and the rest are
zero.
We shall have to do constantly with functions of the space-time point
(_x_, _y_, _z_, _it_), and we may with advantage
employ the 1 × 4 series matrix, formed of differential symbols,—
| ∂/∂_x_, ∂/∂_y_, ∂/∂_z_, ∂/_i_∂_t_,|
or (63) | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ |
For this matrix I shall use the shortened from “lor.”[25]
Then if S is, as in (62), a space-time matrix of the II kind, by lor S′
will be understood the 1 × 4 series matrix
| K₁ K₂ K₃ K₄ |
where K_{_k_} = ∂S_{1_k_}/∂_x₁_ + ∂S_{2_k_}/∂_x₂_ + ∂S_{3_k_}/∂_x₃_ +
∂S_{4_h_}/∂_x₄_.
When by a Lorentz transformation A, a new reference system (_x′₁_ _x′₂_
_x′₃_ _x₄_) is introduced, we can use the operator
lor′ = | ∂/∂_x₁′_ ∂/∂_x₂′_ ∂/∂_x₃′_ ∂/∂_x₄′_ |
Then S is transformed to S′= Ā S A = | S′_{_hk_} |, so by lor 'S′ is
meant the 1 × 4 series matrix, whose element are
K’_{_k_} = ∂S′_{1_k_}/∂_x₁′_ + ∂S′_{2_k_}/∂_x₂′_
+ ∂S′_{3_k_}/∂_x₃′_ + ∂S′_{4_k_}/∂_x₄′_.
Now for the differentiation of any function of (_x_ _y_ _z_ _t_) we have
the rule ∂/∂_x__{_k_}′ = ∂/∂_x₁_ ∂_x₁_/∂_x__{_k_}′ + ∂/∂_x₂_
∂_x₂_/∂_x__{_k_}′ + ∂/∂_x₃_ ∂_x₃_/∂_x__{_k_}′ + ∂/∂_x₄_
∂_x₄_/∂_x__{_k_}′ = ∂/∂_x₁_ _a__{1_k_} + ∂/∂_x₂_ _a__{2_k_} + ∂/∂_x₃_
_a__{3_k_} + ∂/∂_x₄_ _a__{4_k_}.
so that, we have symbolically lor′ = lor A.
Therefore it follows that
lor ′S′ = lor (A A⁻¹ 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
| ∂L/∂_x₁_ ∂L/∂_x₂_ ∂L/∂_x₃_ ∂L/∂_x₄_ |
If _s_ is a space-time vector of the 1st kind, then
lor _ṡ_ = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_.
In case of a Lorentz transformation A, we have
lor ′_ṡ′_ = lor A. Ā_s_ = lor _s_.
_i.e._, lor _s_ is an invariant in a Lorentz-transformation.
In all these operations the operator lor plays the part of a space-time
vector of the first kind.
If _f_ represents a space-time vector of the second kind,—lor _f_
denotes a space-time vector of the first kind with the components
∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_,
∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_,
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_,
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_
So the system of differential equations (A) can be expressed in the
concise form
{A} lor f = -_s_,
and the system (B) can be expressed in the form
{B} log F* = 0.
Referring back to the definition (67) for log _ṡ_, we find that the
combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish
identically, when _f_ and F* are alternating matrices. Accordingly it
follows out of {A}, that
(68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) =
0,
while the relation
(69) lor (lor F*) = 0,
signifies that of the four equations in {B}, only three represent
independent conditions.
I shall now collect the results.
Let ω denote the space-time vector of the first kind
(_u_/√(1 - _u²_}), _i_/√(1 - _u²_))
(_u_ = velocity of matter),
F the space-time vector of the second kind (M,-_i_E)
(M = magnetic induction, E = Electric force,
_f_ the space-time vector of the second kind (_m_,-_ie_)
(_m_ = magnetic force, _e_ = Electric Induction.
_s_ the space-time vector of the first kind (C, _i_ρ)
(ρ = electrical space-density, C - ρ_u_ = conductivity current,
ε = dielectric constant, μ = magnetic permeability,
σ = conductivity,
then the fundamental equations for electromagnetic processes in moving
bodies are[26]
{A} lor _f_ = -_s_
{B} log F* = 0
{C} ω_f_ = εωF
{D} ωF* = μω_f_*
{E} _s_ + (ω_ṡ_), _w_ = - σωF.
ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time
vectors of the first kind are all normal to ω, and for the system {B},
we have
lor (lor F*) = 0.
Bearing in mind this last relation, we see that we have as many
independent equations at our disposal as are necessary for determining
the motion of matter as well as the vector _u_ as a function of _x_,
_y_, _z_, _t_, when proper fundamental data are given.
§ 13. The Product of the Field-vectors _f_ F.
Finally let us enquire about the laws which lead to the determination of
the vector ω as a function of (_x_, _y_, _z_, _t_.) In these
investigations, the expressions which are obtained by the multiplication
of two alternating matrices
_f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ |
| _f₂₁_ 0 _f₂₃_ _f₂₄_ |
| _f₃₁_ _f₃₂_ 0 _f₃₄_ |
| _f₄₁_ _f₄₂_ _f₄₃_ 0 |
F = | 0 F₁₂ F₁₃ F₁₄ |
| F₂₁ 0 F₂₃ F₂₄ |
| F₃₁ F₃₂ 0 F₃₄ |
| F₄₁ F₄₂ F₄₃ 0 |
are of much importance. Let us write,
(70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ - L S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ - L S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ - L |
Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.
Let L now denote the symmetrical combination of the indices 1, 2, 3, 4,
given by
(72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄
+ _f₂₄_ F₂₄ + _f₃₄_ F₃₄)
Then we shall have
(73) S₁₁ = ½(_f₂₃_ F₂₃ + _f₃₄_ F₃₄ + _f₄₂_ F₄₂ - _f₁₂_ F₁₂
- _f₁₃_ F₁₃ _f₁₄_ F₁₄)
S₁₂ = _f₁₃_ F₃₂ + _f₁₄_ F₄₂ etc....
In order to express in a real form, we write
(74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
= | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
| X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
| X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
| -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} |
Now X_{_x_} = ½[_m__{_x_}M_{_x_} - _m__{_y_}M_{_y_} - _m__{_z_}M_{_z_} +
_e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]
so
(75) X_{_y_} = _m__{_x_}M_{_y_} + _e__{_y_}E_{_x_}, Y_{_x_} =
_m__{_y_}M_{_x_} + _e__{_x_}E_{_y_} etc.
X_{_t_} = _e__{_y_}M_{_z_} - _e__{_z_}M_{_y_}, T_{_x_} =
_m__{_x_}E_{_y_} - _m__{_y_}E_{_z_}, etc.
T_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} +
_m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} + _e__{_y_}E_{_y_} +
_e__{_z_}E_{_z_}]
L_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} +
_m__{_z_}M_{_z_} - _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} -
_e__{_z_}E_{_z_}]
These quantities[27] are all real. In the theory for bodies at rest, the
combinations (X_{_x_}, X_{_y_}, X_{_z_}, Y_{_z_}, Y_{_y_}, Y_{_z_},
Z_{_x_}, Z_{_y_}, Z_{_z_}) are known as “Maxwell’s Stresses,” T_{_x_},
T_{_y_}, T_{_z_} are known as the Poynting’s Vector, T_{_t_} as the
electromagnetic energy-density, and L as the Langrangian function.
On the other hand, by multiplying the alternating matrices of _f_* and
F*, we obtain
(77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |
| -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |
| -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |
| -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |
and hence, we can put
(78) _f_F = S - L, F*_f_* = -S - L,
where by L, we mean L-times the unit matrix, _i.e._ the matrix with
elements
| L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_
= 1, 2, 3, 4).
Since here SL = LS, we deduce that,
F*_f_*_f_F = (-S - L)(S - L) = -SS + L²,
and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the
interesting
conclusion
(79) SS = L² - Det^{½}_f_ Det^{½}F
_i.e._ the product of the matrix S into itself can be expressed as the
multiple of a unit matrix—a matrix in which all the elements except
those in the principal diagonal are zero, the elements in the principal
diagonal are all equal and have the value given on the right-hand side
of (79). Therefore the general relations
(80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} +
S_{_h_4} S_{4_k_} = 0,
_h_, _k_ being unequal indices in the series 1, 2, 3, 4, and
(81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} +
S{_h_4} S_{4_h_} = L² -
Det^{½}_f_ Det^{½}F,
for _h_ = 1, 2, 3, 4.
Now if instead of F, and _f_ in the combinations (72) and (73), we
introduce the electrical rest-force Φ, the magnetic rest-force ψ, and
the rest-ray Ω [(55), (56) and (57)], we can pass over to the
expressions,—
(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
(83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_}
+ ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_}
+ μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ
ω_{_h_} Ω_{_k_}
(_h₁_ _k_ = 1, 2, 3, 4).
Here we have
Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²
_e__{_hh_} = 1, _e__{_hk_} = 0 (_h_ ≠ _k_).
The right side of (82) as well as L is an invariant in a Lorentz
transformation, and the 4 × 4 element on the right side of (83) as well
as S_{_k_ _h_} represent a space time vector of the second kind.
Remembering this fact, it suffices, for establishing the theorems (82)
and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃
= 0, ω₄ = _i_. But for this case ω = 0, we immediately arrive at the
equations (82) and (83) by means (45), (51), (60) on the one hand, and
_e_ = εE, M = μ_m_ on the other hand.
The expression on the right-hand side of (81), which equals
[½ (_m_ M - _e_E)²] + (_em_) (EM),
is >= 0, because (_em_ = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back
to 79), we can denote the positive square root of this expression as
Det^{1/4} S.
Since _ḟ_ = -_f_, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of
S, the following relations from (78),
(84) F_f_ = Ṡ - L, _f_* F* = -Ṡ - L,
Then is
Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} |
an alternating matrix, and denotes a space-time vector of the second
kind. From the expressions (83), we obtain,
(85) S - Ṡ = - (εμ - 1) [ω, Ω],
from which we deduce that [see (57), (58)].
(86) ω (S - Ṡ)* = 0,
(87) ω (S - Ṡ) = (εμ - 1) Ω
When the matter is at rest at a space-time point, ω = 0, then the
equation 86) denotes the existence of the following equations
Z_{_y_} = Y_{_z_}, X_{_z_} = Z_{_x_}, Y_{_x_} = X_{_y_},
and from 83),
T_{_x_} = Ω₁, T_{_y_} = Ω₂, T_{_z_} = Ω₃
X_{_t_} = εμΩ₁, Y_{_t_} = εμΩ₂, Z_{_t_} = εμΩ₃
Now by means of a rotation of the space co-ordinate system round the
null-point, we can make,
Z_{_y_} = Y_{_z_} = 0, X_{_z_} = Z_{_x_} = 0, X_{_x_} = X_{_y_} = 0,
According to 71), we have
(88) X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} = 0,
and according to 83), T_{_t_} > 0. In special cases, where ω vanishes it
follows from 81) that
X_{_x_}² = Y_{_y_}² = Z_{_z_}² = T_{_t_}², = (Det^{1/4} S)²,
and if T, and one of the three magnitudes X_{_x_}, Y_{_y_}, Z_{_z_} are
= ±Det^{1/4} S, the two others = -Det^{1/4} S. If Ω does not vanish let
Ω ≠ 0, then we have in particular from 80)
T_{_z_} X_{_t_} = 0, T_{_z_} Y_{_t_} = 0, Z_{_z_} T_{_z_} + T_{_z_}
T_{_t_} = 0,
and if Ω₁ = 0, Ω₂ = 0, Z_{_z_} = -T_{_t_} It follows from (81), (see
also 83) that
X_{_x_} = -Y_{_y_} = ±Det^{1/4} S,
and -Z_{_z_} = T_{_t_} = √(Det^{½} S + εμΩ₃²) > Det^{1/4}S.
The space-time vector of the first kind
(89) K = lor S,
is of very great importance for which we now want to demonstrate a very
important transformation
According to 78), S = L + _f_F, and it follows that
lor S = lor L + lor _f_F.
The symbol ‘lor’ denotes a differential process which in lor _f_F,
operates on the one hand upon the components of _f_, on the other hand
also upon the components of F. Accordingly lor _f_F can be expressed as
the sum of two parts. The first part is the product of the matrices (lor
_f_) F, lor _f_ being regarded as a 1 × 4 series matrix. The second part
is that part of lor _f_F, in which the diffentiations operate upon the
components of F alone. From 78) we obtain
_f_F = -F*_f_* - 2L;
hence the second part of lor _f_F = -(lor F*)_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_)F - (lor F*)_f_* + N,
where N is the vector with the components
N_{_h_} = ½(∂_f₂₃_/∂_x__{_h_} F₂₃ + ∂_f₃₁_/∂_x__{_h_} F₃₁ +
∂_f₁₂_/∂_x__{_h_} F₁₂ + ∂_f₁₄_/∂_x__{_h_} F₁₄
+ ∂_f₂₄_/∂_x__{_h_} F₂₄ + ∂_f₃₄_/∂_x__{_h_} F₃₄
- ∂F₂₃/∂_x__{_h_} _f₂₃_ - ∂F₃₁/∂_x__{_h_} _f_₃₁ - ∂F₁₂/∂_x__{_h_}
_f₁₂_ - ∂F₁₄/∂_x__{_h_} _f₁₄_
- ∂F₂₄/∂_x__{_h_} _f₂₄_ - ∂F₃₄/∂_x__{_h_} _f₃₄_),
(_h_ = 1, 2, 3, 4)
By using the fundamental relations A) and B), 90) is transformed into
the fundamental relation
(91) lor S = -_s_F + N.
In the limitting case ε = 1, μ = 1, _f_ = F, N vanishes identically.
Now upon the basis of the equations (55) and (56), and referring back to
the expression (82) for L, and from 57) we obtain the following
expressions as components of N,—
(92) N_{_h_} = - ½ Φ[=Φ]∂ε/∂_x__{_h_} - ½ ψ[=ψ]∂μ/∂_x__{_h_}
+ (εμ - 1)(Ω₁ ∂ω₁/∂_x__{_h_} + Ω₂ ∂ω₂/∂_x__{_h_} + Ω₃ ∂ω₃/∂_x__{_h_}
+ Ω₄ ∂ω₄/∂_x__{_h_})
for _h_ = 1, 2, 3, 4.
Now if we make use of (59), and denote the space-vector which has Ω₁,
Ω₂, Ω₃ as the _x_, _y_, _z_ components by the symbol W, then the third
component of 92) can be expressed in the form
(93) (εμ - 1)/√(1 - _u²_) (W ∂_u_/∂_x__{_h_}),
The round bracket denoting the scalar product of the vectors within it.
§ 14. The Ponderomotive Force.[28]
Let us now write out the relation K = lor S = -_s_F + N in a more
practical form; we have the four equations
(94) K₁ = ∂X_{_x_}/∂_x_ + ∂X_{_y_}/∂_y_ + ∂X_{_y_}/∂_z_ -
∂X_{_t_}/∂_t_ = ρE_{_x_} + _s__{_y_}M_{_z_} - _s__{_z_}M_{_x_}
- ½ Φ[=Φ] ∂ε/∂_x_ - ½ ψ[=ψ]∂μ/∂_x_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_x_),
(95) K₂ = ∂Y_{_x_}/∂_x_ + ∂Y_{_y_}/∂_y_ + ∂Y_{_z_}/∂_z_ -
∂Y_{_t_}/∂_t_ = ρE_{_y_} + _s__{_z_}M_{_x_} - _s__{_x_}M_{_y_}
- ½ Φ[=Φ]∂ε/∂_y_ - ½ ψ[=ψ]∂μ/∂_y_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_y_),
(96) K₃ = ∂Z_{_x_}/∂_x_ + ∂Z_{_y_}/∂_y_ + ∂Z_{_z_}/∂_z_ -
∂Z_{_t_}/∂_t_ = ρE₂ + _s__{_x_}M_{_y_} - _s__{_y_}M₄
- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂_z_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_z_),
(97) (1/_i_)K₄ = ∂T_{_y_}/∂_x_ - ∂T_{_y_}/∂_y_ - ∂T_{_z_}/∂_z_ -
∂T_{_t_}/∂_t_ = _s__{_x_}E_{_x_} + _s__{_y_}E_{_y_} +
_s__{_z_}E_{_z_}
- ½ Φ[=Φ]∂ε/∂_t_ - ½ ψ[=ψ]∂μ/∂_t_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_t_).
It is my opinion that when we calculate the ponderomotive force which
acts upon a unit volume at the space-time point _x_, _y_, _z_, _t_, it
has got, _x_, _y_, _z_ components as the first three components of the
space-time vector
K + (ωK)ω,
This vector is perpendicular to ω; the law of Energy finds its
expression in the fourth relation.
The establishment of this opinion is reserved for a separate tract.
In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK =
0, and we obtain the ordinary equations in the theory of electrons.
Footnote 9:
_Vide_ Note 1.
Footnote 10:
Note 2.
Footnote 11:
_Vide_ Note 3.
Footnote 12:
_Vide_ Note 4.
Footnote 13:
Note 5.
Footnote 14:
See notes on § 8 and 10.
Footnote 15:
See note 9.
Footnote 16:
See Note.
Footnote 17:
Vide Note.
Footnote 18:
Just as beings which are confined within a narrow region surrounding a
point on a spherical surface, may fall into the error that a sphere is
a geometric figure in which one diameter is particularly distinguished
from the rest.
Footnote 19:
Einzelne stelle der Materie.
Footnote 20:
Vide Note.
Footnote 21:
_Vide_ note 13.
Footnote 22:
_Vide_ note 14.
Footnote 23:
_Vide_ note 15.
Footnote 24:
_Vide_ note 16.
Footnote 25:
_Vide_ note 17.
Footnote 26:
_Vide_ note 19.
Footnote 27:
_Vide_ note 18.
Footnote 28:
Vide note 40.
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 Physics.
Now many authors say that classical mechanics stand in opposition to the
relativity postulate, which is taken to be the basis of the new
Electro-dynamics.
In order to decide this let us fix our attention upon a special Lorentz
transformation represented by (10), (11), (12), with a vector _v_ in any
direction and of any magnitude _q_ < 1 but different from zero. For a
moment we shall not suppose any special relation to hold between the
unit of length and the unit of time, so that instead of _t_, _t′_, _q_,
we shall write _ct_, _ct′_, and _q_/_c_, where _c_ represents a certain
positive constant, and _q_ is < _c_. The above mentioned equations are
transformed into
_r′__{_ṽ_} = _r__{_ṽ_},
_r′__{_v_} = _c_(_r__{_v_} - _qt_)/√(_c²_ - _q²_),
_t′_ = (_qr__{_v_} + _c²__t_)/_c_√(_c²_ - _q²_)
They denote, as we remember, that _r_ is the space-vector (_x_, _y_,
_z_), _r′_ is the space-vector (_x′_ _y′_ _z′_)
If in these equations, keeping _v_ constant we approach the limit _c_ =
∞, then we obtain from these
_r′__{_ṽ_} = _r__{_ṽ_},
_r′__{_v_} = _r__{_v_} - _qt_,
_t′_ = _t_.
The new equations would now denote the transformation of a spatial
co-ordinate system (_x_, _y_, _z_) to another spatial co-ordinate system
(_x′_ _y′_ _z′_) with parallel axes, the null point of the second system
moving with constant velocity in a straight line, while the time
parameter remains unchanged. We can, therefore, say that classical
mechanics postulates a covariance of Physical laws for the group of
homogeneous linear transformations of the expression
-_x²_ - _y²_ - _z²_ + _c²_ (1)
when _c_ = ∞.
Now it is rather confusing to find that in one branch of Physics, we
shall find a covariance of the laws for the transformation of expression
(1) with a finite value of _c_, in another part for _c_ = ∞.
It is evident that according to Newtonian Mechanics, this covariance
holds for _c_ = ∞ and not for _c_ = velocity of light.
May we not then regard those traditional covariances for _c_ = ∞ 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 _c_ = ∞, we assume a relativity-postulate with
a finite _c_, then the axiomatic construction of Mechanics appears to
gain considerably in perfection.
The ratio of the time unit to the length unit is chosen in a manner so
as to make the velocity of light equivalent to unity.
While now I want to introduce geometrical figures in the manifold of the
variables (_x_, _y_, _z_, _t_), it may be convenient to leave (_y_, _z_)
out of account, and to treat _x_ and _t_ as any possible pair of
co-ordinates in a plane, referred to oblique axes.
A space time null point 0 (_x_, _y_, _z_, _t_ = 0, 0, 0, 0) will be kept
fixed in a Lorentz transformation.
The figure -_x²_ - _y²_ - _z²_ + _t²_ = 1, _t_ > 0 ... (2)
which represents a hyper boloidal shell, contains the space-time points
A (_x_, _y_, _z_, _t_ = 0, 0, 0, 1), and all points A′ which after a
Lorentz-transformation enter into the newly introduced system of
reference as (_x′_, _y′_, _z′_, _t′_ = 0, 0, 0, 1).
The direction of a radius vector 0A′ drawn from 0 to the point A′ of
(2), and the directions of the tangents to (2) at A′ are to be called
normal to each other.
Let us now follow a definite position of matter in its course through
all time _t_. The totality of the space-time points (_x_, _y_, _z_, _t_)
which correspond to the positions at different times _t_, shall be
called a space-time line.
The task of determining the motion of matter is comprised in the
following problem:—It is required to establish for every space-time
point the direction of the space-time line passing through it.
To transform a space-time point P (_x_, _y_, _z_, _t_) to rest is
equivalent to introducing, by means of a Lorentz transformation, a new
system of reference (_x′_, _y′_, _z′_, _t′_), in which the _t′_ axis has
the direction 0A′, 0A′ indicating the direction of the space-time line
passing through P. The space _t′_ = const, which is to be laid through
P, is the one which is perpendicular to the space-time line through P.
To the increment _dt_ of the time of P corresponds the increment
_d_τ = √(_dt²_ - _dx²_ - _dy²_) - _dz²_ = _dt_√(1 - _u²_)
of the newly introduced time parameter _t′_. The value of the integral
∫ _dτ_ = ∫ √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_))
when calculated upon the space-time line from a fixed initial point P₀
to the variable point P, (both being on the space-time line), is known
as the ‘Proper-time’ of the position of matter we are concerned with at
the space-time point P. (It is a generalization of the idea of
Positional-time which was introduced by Lorentz for uniform motion.)
If we take a body R₀ which has got extension in space at time _t₀_, then
the region comprising all the space-time line passing through R₀ and
_t₀_ shall be called a space-time filament.
If we have an analytical expression θ(_x_ _y_, _z_, _t_) so that θ(_x_,
_y_ _z_ _t_) = 0 is intersected by every space time line of the filament
at one point,—whereby
-(∂Θ/∂_x_)², -(∂Θ/∂_y_)², -(∂Θ/∂_z_)²,
-(∂Θ/∂_t_)² > 0, ∂Θ/∂_t_ > 0.
then the totality 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 reference (_x′_, _y_, _z′_ _t_), so
that according to this
∂Θ/∂_x′_ = 0, ∂Θ/∂_y′_ = 0, ∂Θ/∂_z′_ = 0, ∂Θ/∂_t′_ > 0.
The direction of the uniquely determined _t′_—axis in question here is
known as the upper normal of the cross-section at the point P and the
value of _d_J = ∫∫∫ _dx′ dy′ dz′_ for the surrounding points of P on the
cross-section is known as the elementary contents (Inhalts-element) of
the cross-section. In this sense R₀ is to be regarded as the
cross-section normal to the _t_ axis of the filament at the point _t_ =
_t₀_, and the volume of the body R₀ is to be regarded as the contents of
the cross-section.
If we allow R₀ to converge to a point, we come to the conception of an
infinitely thin space-time filament. In such a case, a space-time line
will be thought of as a principal line and by the term ‘Proper-time’ of
the filament will be understood the ‘Proper-time’ which is laid along
this principal line; under the term normal cross-section of the
filament, we shall understand the cross-section upon the space which is
normal to the principal line through P.
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 (_x_, _y_, _z_, _t_), then
the quotient of this mass, and the volume of R approaches a limit μ(_x_,
_y_, _z_, _t_), which is known as the mass-density at the space-time
point (_x_, _y_, _z_, _t_).
The principle of conservation of mass says—that for an infinitely thin
space-time filament, the product μ_d_J, where μ = mass-density at the
point (_x_, _y_, _z_, _t_) of the filament (_i.e._, the principal line
of the filament), _d_J = contents of the cross-section normal to the _t_
axis, and passing through (_x_, _y_, _z_, _t_), is constant along the
whole filament.
Now the contents _d_J_{n} of the normal cross-section of the filament
which is laid through (_x_, _y_, _z_, _t_) is
(4) _d_J_{n} = (1/√(1 - _u²_))_d_J = -_i_ω₄ _d_J = (_dt_/_d_τ)_d_J.
and the function
ν = μ/-_i_ω₄ = μ√(1 - _u²_)) = μ(∂τ/∂_t_. (5)
may be defined as the rest-mass density at the position (_x_ _y_ _z_
_t_). Then the principle of conservation of mass can be formulated in
this manner:—
_For an infinitely thin space-time filament, the product of the
rest-mass density and the contents of the normal cross-section is
constant along the whole filament._
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 _t_ on
Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be
called a space-time _sichel_,[29] Q′ is the lower boundary, and Q′ is
the upper boundary of the _sichel_.
If we decompose a filament into elementary space-time filaments, then to
an entrance-point of an elementary filament through the lower boundary
of the _sichel_, there corresponds an exit point of the same by the
upper boundary, whereby for both, the product νdJ_{n} taken in the sense
of (4) and (5), has got the same value. Therefore the difference of the
two integrals ∫ν_dJ__{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 ν[=ω] _dx dy dz dt_,
the integration being extended over the whole range of the _sichel_, and
(comp. (67), § 12)
lor ν[=ω] = (∂νω₁/∂_x₁_) + (∂νω₂/∂_x₂_) + (∂νω₃/∂_x₃_) +
(∂νω₄/∂_x₄_).
If the _sichel_ reduces to a point, then the differential equation
lor ν[=ω] = 0, (6)
which is the condition of continuity
(∂μ_u__{_x_}/∂_x_) + (∂μ_u__{_y_}/∂_y_) + (∂μ_u__{_z_}/∂_z_) +
(∂μ/∂_t_) = 0.
Further let us form the integral
N = ∫ ∫∫∫ ν _dx dy dz dt_ (7)
extending over the whole range of the space-time _sichel_. We shall
decompose the _sichel_ into elementary space-time filaments, and every
one of these filaments in small elements _d_τ of its proper-time, which
are however large compared to the linear dimensions of the normal
cross-section; let us assume that the mass of such a filament
ν_d_J_{_n_} = _dm_ and write τ⁰, τ^l for the ‘Proper-time’ of the upper
and lower boundary of the _sichel_.
Then the integral (7) can be denoted by
∫∫ ν_d_J_{_n_} _d_τ = ∫ (τ′-τ⁰) _dm_.
taken over all the elements of the sichel.
Now let us conceive of the space-time lines inside a space-time _sichel_
as material curves composed of material points, and let us suppose that
they are subjected to a continual change of length inside the sichel in
the following manner. The entire curves are to be varied in any possible
manner inside the _sichel_, while the end points on the lower and upper
boundaries remain fixed, and the individual substantial points upon it
are displaced in such a manner that they always move forward normal to
the curves. The whole process may be analytically represented by means
of a parameter λ, and to the value λ = 0, shall correspond the actual
curves inside the _sichel_. Such a process may be called a virtual
displacement in the sichel.
Let the point (_x_, _y_, _z_, _t_) in the sichel λ = 0 have the values
_x_ + δ_x_, _y_ + δ_y_, _z_ + δ_z_, _t_ + δ_t_, when the parameter has
the value λ; these magnitudes are then functions of (_x_, _y_, _z_, _t_,
λ). Let us now conceive of an infinitely thin space-time filament at the
point (_x_ _y_ _z_ _t_) with the normal section of contents _d_J_{_n_}
and if _d_J_{_n_} + δ_d_J_{_n_} be the contents of the normal section at
the corresponding position of the varied filament, then according to the
principle of conservation of mass—(ν + _d_ν being the rest-mass-density
at the varied position),
(8) (ν + δν) (_d_J_{_n_} + δ_d_J_{_n_}) = ν_d_J_{_n_} = _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 + δN of λ, and we may call this function N + δN as
the _mass action_ of the virtual displacement.
If we now introduce the method of writing with indices, we shall have
(9) _d_(_x__{_h_} + δ_x__{_h_}) = _dx__{_h_} + ∑_{_k_}
∂δ_x__{_h_}/∂_x__{_k_} + ∂δ_x__{_h_}/∂λ _d_λ
_k_ = 1, 2, 3, 4
_h_ = 1, 2, 3, 4
Now on the basis of the remarks already made, it is clear that the value
of N + δN, when the value of the parameter is λ, will be:—
(10) N + δN = ∫∫∫∫ ((ν_d_(τ + δτ))/_d_τ)_dx_ _dy_ _dz_ _dt_,
the integration extending over the whole sichel _d_(τ + δτ) where _d_(τ
+ δτ) denotes the magnitude, which is deduced from
√(-(_dx₁_ + _d_δ_x₁_)² - (_dx₂_ + _d_δ_x₂_)² - (_dx₃_ + _d_δ_x₃_)² -
(_dx₄_ + _d_δ_x₄_)²)
by means of (9) and
_dx₁_ = ω₁ _d_τ, _dx₂_ = ω₂ _d_τ,
_dx₃_ = ω₃ _d_τ, _dx₄_ = ω₄ _d_τ, _d_λ = 0
therefore:—
(11) (_d_(τ + δτ))/_d_τ = √( -∑(ω_{_h_} +
∑(∂δ_x__{_h_}/∂_x__{_k_})ω_{_k_})²)
_k_ = 1, 2, 3, 4.
_h_ = 1, 2, 3, 4.
We shall now subject the value of the differential quotient
(12) ((_d_(N + δN))/_d_λ) (λ = 0)
to a transformation. Since each δ_x__{_h_} as a function of (_x_, _y_,
_z_, _t_) vanishes for the zero-value of the parameter λ, so in general
_d_δ_x__{_k_}/(∂_x__{_h_} = 0, for λ = 0.
Let us now put (∂δ_x__{_h_}/∂λ) = ξ_{_h_} (_h_ = 1, 2, 3, 4) (13)
λ = 0
then on the basis of (10) and (11), we have the expression (12):—
= -∫∫∫∫ ∑ ω_{_h_}((∂ξ_{_h_}/∂_x₁_)ω₁ + (∂ξ_{_h_}/∂_x₂_)ω₂
+(∂ξ_{_h_}/∂_x₃_)ω₃ + (∂ξ_{_h_}/∂_x₄_)ω₄)
_dx dy dz dt_
for the system (_x₁_ _x₂_ _x₃_ _x₄_) on the boundary of the _sichel_,
(δ_x₁_ δ_x₂_ δ_x₃_ δ_x₄_) shall vanish for every value of λ and
therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial integration, the
integral is transformed into the form
∫∫∫∫ ∑ ξ_{_h_}(∂νω_{_h_}ω₁/∂_x₁_ + ∂νω_{_h_}ω₂/∂_x₂_ +
∂νω_{_h_}ω₃/∂_x₃_ + ∂νω_{_h_}ω₄/∂_x₄_)
_dx dy dz dt_
the expression within the bracket may be written as
= ω_{_h_} ∑ ∂νω_{_k_}/∂_x__{_k_} + ν∑ω_{_k_}∂ω_{_h_}/∂_x__{_k_}.
The first sum vanishes in consequence of the continuity equation (_b_).
The second may be written as
(∂ω_{_h_}/∂_x₁_)(_dx₁_/_d_τ) + (∂ω_{_h_}/∂_x₂_)(_dx₂_/_d_τ) +
(∂ω_{_h_}/∂_x₃_)(_dx₃_/_d_τ) + (∂ω_{_h_}/∂_x₄_)(_dx₄_/_d_τ)
= _d_ω_{_h_}/_d_τ = (_d_/_d_τ)(_dx__{_h_}/_d_τ)
whereby (_d_/_d_τ) is meant the differential quotient in the direction
of the space-time line at any position. For the differential quotient
(12), we obtain the final expression
(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)
_dx dy dz dt_.
For a virtual displacement in the _sichel_ we have postulated the
condition that the points supposed to be substantial shall advance
normally to the curves giving their actual motion, which is λ = 0; this
condition denotes that the ξ_{_h_} is to satisfy the condition
_w₁_ξ₁ + _w₂_ξ₂ + _w₃_ξ₃ + _w₄_ξ₄ = 0. (15)
Let us now turn our attention to the Maxwellian tensions in the
electrodynamics of stationary bodies, and let us consider the results in
§ 12 and 13; then we find that Hamilton’s Principle can be reconciled to
the relativity postulate for continuously extended elastic media.
At every space-time point (as in § 13), let a space time matrix of the
2nd kind be known
(16) S =
| S₁₁ S₁₂ S₁₃ S₁₄ | = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |
| S₂₁ S₂₂ S₂₃ S₂₄ | = | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |
| S₃₁ S₃₂ S₃₃ S₃₄ | = | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |
| S₄₁ S₄₂ S₄₃ S₄₄ | = | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_}
|
where X_{_n_} Y_{_x_} .....X_{_z_}, T_{_t_} are real magnitudes.
For a virtual displacement in a space-time sichel (with the previously
applied designation) the value of the integral
(17) W + δW = ∫∫∫∫ (∑S_{_h k_} (∂(_x__{_k_} +
δ_x__{_k_}))/∂_x__{_h_} _dx dy dz dt_
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
X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} + X_{_x_} (∂δ_x_)/∂_x_ +
X_{_y_} (∂δ_x_)/∂_y_ + ... Z_{_z_} (∂δ_z_)/∂_z_
- X_{_t_} (∂δ_x_/∂_t_ - ... + T_{_x_} (∂δ_t_)/∂_x_ + ... T_{_t_}
(∂δ_t_)/∂_t_
we can now postulate the following _minimum principle in mechanics_.
_If any space-time Sichel be bounded, then for each virtual displacement
in the Sichel, the sum of the mass-works, and tension works shall always
be an extremum for that process of the space-time line in the Sichel
which actually occurs._
The meaning is, that for each virtual displacement,
([_d_(·δN + δW)]/_d_λ)_{λ = 0} = 0 (18)
By applying the methods of the Calculus of Variations, the following
four differential equations at once follow from this minimal principle
by means of the transformation (14), and the condition (15).
(19) ν ∂_w__{_h_}/∂τ = K_{_h_} + χ_w__{_h_} (_h_ = 1, 2, 3, 4)
whence K_{_h_} = ∂S_{1 _h_}/∂_x₁_ + ∂S_{2 _h_}/∂_x₂_ + ∂S_{3
_h_}/∂_x₃_ + ∂S_{4 _h_}/∂_x₄_, (20)
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 _w__ẇ_ = - 1. By
multiplying (19) by _w__{_h_}, and summing the four, we obtain X = K_ẇ_,
and therefore clearly K + (K_ẇ_)_w_ will be a space-time vector of the
1st kind which is normal to _w_. Let us write out the components of this
vector as
X, Y, Z, ·_i_T
Then we arrive at the following equation for the motion of matter,
(21) ν _d_/_d_τ (_dx_/_d_τ) = X, ν _d_/_d_τ (_dy_/_d_τ) = Y, ν
_d_/_d_τ (_dz_/_d_τ) = Z,
ν _d_/_d_τ (_dx_/_d_τ) = T, and we have also
(_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² > (_dt_/_d_τ)² = -1,
and X _dx_/_d_τ + Y _dy_/_d_τ + Z _dz_/_d_τ = T _dt_/_d_τ.
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.
Let _x_, _y_, _z_, _t_, 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 _x_, _y_,
_z_, _t_, and integrate them, multiplying by the elementary contents of
the cross section over the whole space of the normal section. If the
integrals of the right side be R_{_x_} R_{_y_} R_{_z_} R_{_t_} and if
_m_ be the constant mass of the filament, we obtain
(22) _m_ _d_/_d_τ _dx_/_d_τ = R_{_x_},
_m_ _d_/_d_τ _dy_/_d_τ = R_{_y_},
_m_ _d_/_d_τ _dz_/_d_τ = R_{_z_},
_m_ _d_/_d_τ _dt_/_d_τ = R_{_t_}
R is now a space-time vector of the 1st kind with the components
(R_{_x_} R_{_y_} R_{_z_} R_{_t_}) which is normal to the space-time
vector of the 1st kind _w_,—the velocity of the material point with the
components
_dx_/_d_τ, _dy_/_d_τ, _dz_/_d_τ, _i_ _dt_/_d_τ.
We may call this vector R _the moving force of the material point_.
If instead of integrating over the normal section, we integrate the
equations over that cross section of the filament which is normal to the
_t_ axis, and passes through (_x_, _y_, _z_, _t_), then [See (4)] the
equations (22) are obtained, but
are now multiplied by _d_τ/_dt_; in particular, the last equation comes
out in the form,
_m_ _d_/_dt_ (_dt_/_d_τ) = _w__{_x_} R_{_x_} _d_τ/_dt_ + _w__{_y_}
R_{_y_} _d_τ/_dt_ + _w__{_z_} R_{_z_} _d_τ/_dt_.
The right side is to be looked upon _as the amount of work done per unit
of time_ at the material point. In this equation, we obtain the
energy-law for the motion of the material point and the expression
_m_ (_dt_/_d_τ - 1) = _m_ [1/√(1 - _w²_) - 1]
= _m_ (½ |_w₁²_ + 3/8 |_w₁⁴_ + )
may be called the kinetic energy of the material point.
Since _dt_ is always greater than _d_τ we may call the quotient (_dt_ -
_d_τ)/_d_τ as the “Gain” (vorgehen) of the time over the proper-time of
the material point and the law can then be thus expressed;—The kinetic
energy of a material point is the product of its mass into the gain of
the time over its proper-time.
The set of four equations (22) again shows the symmetry in (_x_, _y_,
_z_, _t_), which is demanded by the relativity postulate; to the fourth
equation however, a higher physical significance is to be attached, as
we have already seen in the analogous case in electrodynamics. On the
ground of this demand for symmetry, the triplet consisting of the first
three equations are to be constructed after the model of the fourth;
remembering this circumstance, we are justified in saying,—
“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*(_x_*, _y_*, _z_*, _t_*) be a solid (fester) space-time point, then
the region of all those space-time points B (_x_, _y_, _z_, _t_), for
which
(23) (_x_ - _x_*)² + (_y_ - _y_*)² + (_z_ - _z_*)² = (_t_ - _t_*)²
_t_ - _t_* >= 0
may be called a “Ray-figure” (Strahl-gebilde) of the space time 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 (_x_ _y_ _z_ _t_) be supposed to be fixed, the
point (_x_* _y_* _z_* _t_*) be supposed to be variable, then the
relation (23) would represent the locus of all the space-time points B*,
which are light-points of B.
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′ is the radius vector of the
hyperboloidal fundamental figure (23) parallel to B*C*, finally D* is
the point of intersection of line B*C* with the space normal to itself
and passing through B. The moving force of the mass-point F in the
space-time point B is now the space-time vector of the first kind which
is normal to BC, and which is composed of the vectors
(24) _mm_*(OA′/B*D*)³ BD* in the direction of BD*, and another vector of
suitable value in direction of B*C*.
Now by (OA′/B*D*) is to be understood the ratio of the two vectors in
question. It is clear that this proposition at once shows the covariant
character with respect to a Lorentz-group.
Let us now ask how the space-time filament of F behaves when the
material point F* has a uniform translatory 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 t-axis. Let _x_, _y_, _z_, _t_, denote the
point B, let τ* denote the proper time of B*, reckoned from O. Our
proposition leads to the equations
(25) _d²__x_/_d_τ² = - _m_*_x_/(_t_ - τ*)², _d²__y_/_d_τ² = -
_m_*_y_/(_t_ - τ*)³
_d²__z_/_d_τ² = -_m_*_z_/(_t_ - τ*)³,
(26) _d²__t_/_d_τ² = -_m_*/(_t_ - τ*)² _d_(_t_ - τ*)/_dt_
where (27) _x²_ + _y²_ + _z²_ = (_t_ - τ*)²
and (28) (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² = (_dt_/_d_τ)² - 1.
In consideration of (27), the three equations (25) are of the same form
as the equations for the motion of a material point subjected to
attraction from a fixed centre according to the Newtonian Law, only that
instead of the time _t_, the proper time τ of the material point occurs.
The fourth equation (26) gives then the connection between proper time
and the time for the material point.
Now for different values of τ′, the orbit of the space-point (_x_ _y_
_z_) is an ellipse with the semi-major axis _a_ and the eccentricity
_e_. Let E denote the eccentric anomaly, Τ the increment of the proper
time for a complete description of the orbit, finally _n_Τ = 2π, so that
from a properly chosen initial point τ, we have the Kepler-equation
(29) _n_τ = 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
(30) (_dt_/_d_τ)² - 1
= (_m_*/_ac²_) (1 + _e_ cos E)/(1 - _e_ cos E)
Now neglecting _c⁻⁴_ with regard to 1, it follows that
_ndt_ = _nd_τ [ 1 + ½ _m_*/_ac²_ (1 + _e_ cos E)/(1 - _e_ cos E) ]
from which, by applying (29),
(31) _nt_ + const = (1 + ½ _m_*/_ac²_) _n_τ + _m_*/_ac²_ Sin E.
the factor _m_*/_ac²_ is here the square of the ratio of a certain
average velocity of F in its orbit to the velocity of light. If now _m_*
denote the mass of the sun, _a_ the semi major axis of the earth’s
orbit, then this factor amounts to 10⁻⁸.
The law of mass attraction which has been just described and which is
formulated in accordance with the relativity postulate would signify
that gravitation is propagated with the velocity of light. In view of
the fact that the periodic terms in (31) are very small, it is not
possible to decide out of astronomical observations between such a law
(with the modified mechanics proposed above) and the Newtonian law of
attraction with Newtonian mechanics.
Footnote 29:
Sichel—a German word meaning a crescent or a scythe. The original term
is retained as there is no suitable English equivalent.
SPACE AND TIME
A Lecture delivered before the Naturforscher Versammlung (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 considerations. The equations of Newtonian mechanics show a
twofold invariance, (_i_) their form remains unaltered when we subject
the fundamental space-coordinate system to any possible change of
position, (_ii_) when we change the system in its nature of motion, _i.
e._, when we impress upon it any uniform motion of translation, the
null-point of time plays no part. We are accustomed to look upon the
axioms of geometry as settled once for all, while we seldom have the
same amount of conviction regarding the axioms of mechanics, and
therefore the two invariants are seldom mentioned in the same breath.
Each one of these denotes a certain group of transformations for the
differential equations of mechanics. We look upon the existence of the
first group as a fundamental characteristics of space. We always prefer
to leave off the second group to itself, and with a light heart conclude
that we can never decide from physical considerations whether the space,
which is supposed to be at rest, may not finally be in uniform motion.
So these two groups lead quite separate existences besides each other.
Their totally heterogeneous character may scare us away from the attempt
to compound them. Yet it is the whole compounded group which as a whole
gives us occasion for thought.
We wish to picture to ourselves the whole relation graphically. Let
(_x_, _y_, _z_) be the rectangular coordinates of space, and _t_ denote
the time. Subjects of our perception are always connected with place and
time. _No one has observed a place except at a particular time, or has
observed 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_, _y_, _z_,
_t_, as a _world-point_. The manifoldness of all possible values of _x_,
_y_, _z_, _t_, will be the _world_. I can draw four world-axes with the
chalk. Now any axis drawn consists of quickly vibrating molecules, and
besides, takes part in all the journeys of the earth ; and therefore
gives us occasion for reflection. The greater abstraction required for
the four-axes does not cause the mathematician any trouble. In order not
to allow any yawning gap to exist, we shall suppose that at every place
and time, something perceptible exists. In order not to specify either
matter or electricity, we shall simply style these as substances. We
direct our attention to the _world-point_ _x_, _y_, _z_, _t_, and
suppose that we are in a position to recognise this substantial point at
any subsequent time. Let _dt_ be the time element corresponding to the
changes of space coordinates of this point [_dx_, _dy_, _dz_]. Then we
obtain (as a picture, so to speak, of the perennial life-career of the
substantial point),—a curve in the _world_—the _world-line_, the points
on which unambiguously correspond to the parameter _t_ from +∞ to -∞.
The whole world appears to be resolved in such _world-lines_, 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 (_x_, _y_, _z_) manifoldness
_t_ = 0 and its two sides _t_ < 0 and _t_ > 0 falls asunder. If for the
sake of simplicity, we keep the null-point of time and space fixed, then
the first named group of mechanics signifies that at _t_ = 0 we can give
the _x_, _y_, and _z_-axes any possible rotation about the null-point
corresponding to the homogeneous linear transformation of the expression
_x²_ + _y²_ + _z²_.
The second group denotes that without changing the expression for the
mechanical laws, we can substitute (_x_ - α_t_, _y_ - β_t_, _z_ - γ_t_
for (_x_, _y_, _z_) where (α, β, γ) are any constants. According to this
we can give the time-axis any possible direction in the upper half of
the world _t_ > 0. Now what have the demands of orthogonality in space
to do with this perfect freedom of the time-axis towards the upper half?
To establish this connection, let us take a positive parameter c, and
let us consider the figure
_c²__t²_ - _x²_ - _y²_ - _z²_ = 1
According to the analogy of the hyperboloid of two sheets, this consists
of two sheets separated by _t_ = 0. Let us consider the sheet, in the
region of _t_ > 0, and let us now conceive the transformation of _x_,
_y_, _z_, _t_ in the new system of variables; (_x’_, _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 transformations
which we picture to ourselves from a particular transformation in which
(_y_, _z_) remain unaltered. Let us draw the cross section of the upper
sheets with the plane of the _x_- and _t_-axes, _i.e._, the upper half
of the hyperbola _c²__t²_ - x² = 1, with its asymptotes (_vide_ fig. 1).
Then let us draw the radius rector OA′, the tangent A′ B′ at A′, and let
us complete the parallelogram OA′ B′ C′; also produce B′ C′ to meet the
x-axis at D′. Let us now take Ox′, OA′ as new axes with the unit
measuring rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again
expressed in the form _c²__t′²_ - x′² = 1, t′ > 0 and the transition
from (_x_, _y_, _z_, _t_) to (_x′_ _y′_ _z′_ _t_) is one of the
transitions in question. Let us add to this characteristic
transformation any possible displacement of the space and time
null-points; then we get a group of transformation depending only on
_c_, which we may denote by G_{_c_}.
Now let us increase _c_ to infinity. Thus (1/c) becomes zero and it
appears from the figure that the hyperbola is gradually shrunk into the
_x_-axis, the asymptotic angle becomes a straight one, and every special
transformation in the limit changes in such a manner that the _t_-axis
can have any possible direction upwards, and _x′_ more and more
approximates to _x_. Remembering this point it is clear that the full
group belonging to Newtonian Mechanics is simply the group G_{_c_}, with
the value of _c_ = ∞. In this state of affairs, and since G_{_c_} is
mathematically more intelligible than G_{∞}, a mathematician may, by a
free play of imagination, hit upon the thought that natural phenomena
possess an invariance not only for the group G_{∞}, but in fact also for
a group G_{_c_}, where _c_ is finite, but yet exceedingly large compared
to the usual measuring units. Such a preconception would be an
extraordinary triumph for pure mathematics.
At the same time I shall remark for which value of _c_, this invariance
can be conclusively held to be true. _For c, we shall substitute the
velocity of light c in free space._ In order to avoid speaking either of
space or of vacuum, we may take this quantity as the ratio between the
electrostatic 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_{_c_} in the following
manner.
Out of the totality of natural phenomena, we can, by successive higher
approximations, deduce a coordinate system (_x_, _y_, _z_, _t_); by
means of this coordinate system, we can represent the phenomena
according to definite laws. This system of reference is by no means
uniquely determined by the phenomena. _We can change the system of
reference in any possible manner corresponding to the above-mentioned
group transformation G_{c}, but 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 manifoldness (_x′_ _y_ _z_). The physical laws are now
expressed by means of _x′_, _y_, _z_, _t′_,—and the expressions are just
the same as in the case of _x_, _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
that time and space shall reduce to mere shadows and we shall have a
world complete in itself.
II
Now the question may be asked,—what circumstances lead us to these
changed views about time and space, are they not in contradiction with
observed phenomena, do they finally guarantee us advantages for the
description of natural phenomena?
Before we enter into the discussion, a very important point must be
noticed. Suppose we have individualised time and space in any manner;
then a world-line parallel to the _t_-axis will correspond to a
stationary point; a world-line inclined to the _t_-axis will correspond
to a point moving uniformly; and a world-curve will correspond to a
point moving in any manner. Let us now picture to our mind the
world-line passing through any world point _x_, _y_, _z_, _t_; 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:—
_The substance existing at any world point can always be conceived to be
at rest, if we establish our time and space suitably._ The axiom denotes
that in a world-point, the expression
_c²__dt²_ - _dx²_ - _dy²_ - _dz²_
shall always be positive or what is equivalent to the same thing, every
velocity V should be smaller than _c_. _c_ shall therefore be the upper
limit for all substantial velocities and herein lies a deep significance
for the quantity _c_. At the first impression, the axiom seems to be
rather unsatisfactory. It is to be remembered that only a modified
mechanics will occur, in which the square root of this differential
combination takes the place of time, so that cases in which the velocity
is greater than _c_ will play no part, something like imaginary
coordinates in geometry.
The _impulse_ and real cause of inducement _for the assumption of the
group-transformation G_{c}_ is the fact that the differential equation
for the propagation of light in vacant space possesses the
group-transformation G_{_c_}. On the other hand, the idea of rigid
bodies has any sense only in a system mechanics with the group
G_{infinity}. Now if we have an optics with G_{_c_}, and on the other
hand if there are rigid bodies, it is easy to see that a _t_-direction
can be defined by the two hyperboloidal shells common to the groups
G_{∞}, and G_{_c_}, which has got the further consequence, that by means
of suitable rigid instruments in the laboratory, we can perceive a
change in natural phenomena, in case of different orientations, with
regard to the direction of progressive motion of the earth. But all
efforts directed towards this object, and even the celebrated
interference-experiment of Michelson have given negative results. In
order to supply an explanation for this result, H. A. Lorentz formed a
hypothesis which practically amounts to an invariance of optics for the
group G_{_c_}. According to Lorentz every substance shall suffer a
contraction
1:(√(1 - v²/_c²_)) in length, in the direction of its motion
_l_/_l′_ = 1/√(1 - _v²_/_c²_) _l′_ = _l_(1 - _v²_/_c²_).
This hypothesis sounds rather phantastical. For the contraction is not
to be thought of as a consequence of the resistance of ether, but purely
as a gift from the skies, as a sort of condition always accompanying a
state of motion.
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 _t_-axis represent a state of rest and another
parallel strip inclined to the _t_-axis represent a state of uniform
motion for a body, which has a constant spatial extension (see fig. 1).
If OA′ is parallel to the second strip, we can take _t′_ as the _t_-axis
and _x′_ as the _x_-axis, then the second body will appear to be at
rest, and the first body in uniform motion. We shall now assume that the
first body supposed to be at rest, has the length _l_, _i.e._, the cross
section PP of the first strip upon the _x_-axis = _l_^. OC, where OC is
the unit measuring rod upon the _x_-axis—and the second body also, when
supposed to be at rest, has the same length _l_, this means that, the
cross section Q′Q′ of the second strip has a cross-section _l_^· OC′,
when measured parallel to the _x′_-axis. In these two bodies, we have
now images of two Lorentz-electrons, one of which is at rest and the
other moves uniformly. Now if we stick to our original coordinates, then
the extension of the second electron is given by the cross section QQ of
the strip belonging to it measured parallel to the _x_-axis. Now it is
clear since Q′Q′ = _l_^· OC′, that QQ = _l_^· OD′.
If (_dc_/_dt_) = _v_, an easy calculation gives that
OD′ = OC √(1-(_v²_/_c²_)), therefore (PP/QQ) = (1/√(1-(_v²_/_c²_))
This is the sense of Lorentz’s hypothesis about the contraction of
electrons in case of motion. On the other hand, if we conceive the
second electron to be at rest, and therefore adopt the system (_x′_,
_t′_,) then the cross-section P′P′ of the strip of the electron parallel
to OC′ is to be regarded as its length and we shall find the first
electron shortened with reference to the second in the same proportion,
for it is,
P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP
Lorentz called the combination _t′_ of (_t_ and _x_) as the _local time_
(_Ortszeit_) of the uniformly moving electron, and used a physical
construction of this idea for a better comprehension of the
contraction-hypothesis. But to perceive clearly that the time of an
electron is as good as the time of any other electron, _i.e._ _t_, _t′_
are to be regarded as equivalent, has been the service of A. Einstein
[Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis. ... 4-4-11-1907.] There
the concept of time was shown to be completely and unambiguously
established by natural phenomena. But the concept of space was not
arrived at, either by Einstein or Lorentz, probably because in the case
of the above-mentioned spatial transformations, where the (_x′_, _t′_)
plane coincides with the _x_-_t_ plane, the significance is possible
that the _x_-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-Postulate” which has been
coined for the demands of invariance in the group G, seems to be rather
inexpressive for a true understanding of the group G_{_c_}, and for
further progress. Because the sense of the postulate is that the
four-dimensional world is given in space and time by phenomena only, but
the projection in time and space can be handled with a certain freedom,
and therefore I would rather like to give to this assertion the name
“_The Postulate of the Absolute world_” [World-Postulate].
III
By the world-postulate a similar treatment of the four determining
quantities _x_, _y_, _z_, _t_, of a world-point is possible. Thereby the
forms under which the physical laws come forth, gain in intelligibility,
as I shall presently show. Above all, the idea of acceleration becomes
much more striking and clear.
I shall again use the geometrical method of expression. Let us call any
world-point O as a “Space-time-null-point.” The cone
_c²__t²_ - _x²_ - _y²_ - _z²_ = O
consists of two parts with O as apex, one part having _t_ < 0, the other
having _t_ > 0. The first, which we may call _the fore-cone_ 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. (_Vide_ fig. 2).
On the aft-side of O we have the already considered hyperboloidal shell
F = _c²__t²_ - _x²_ - _y²_ - _z²_ = 1, _t_ > 0.
The region inside the two cones will be occupied by the hyperboloid of
one sheet
-F = _x²_ + _y²_ + _z²_ - _c²__t²_ = _k²_,
where _k²_ can have all possible positive values. The hyperbolas which
lie upon this figure with O as centre, are important for us. For the
sake of clearness the individual branches of this hyperbola will be
called the “_Inter-hyperbola with centre O_.” Such a hyperbolic branch,
when thought of as a world-line, would represent a motion which for _t_
= -∞ and _t_ = ∞, asymptotically approaches the velocity of light _c_.
If, by way of analogy to the idea of vectors in space, we call any
directed length in the manifoldness _x_, _y_, _z_, _t_ a vector, then we
have to distinguish between a time-vector directed from O towards the
sheet ±F = 1, _t_ > 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, may by
means of the system of reference be regarded either as synchronous with
O, as well as later or earlier than O. Every world-point on the
fore-side of O is necessarily always earlier, every point on the aft
side of O, later than O. The limit _c_ = ∞ corresponds to a complete
folding up of the wedge-shaped cross-section between the fore and aft
cones in the manifoldness _t_ = 0. In the figure drawn, this
cross-section has been intentionally drawn with a different breadth.
Let us decompose a vector drawn from O towards (_x_, _y_, _z_, _t_) into
its components. If the directions of the two vectors are respectively
the directions of the radius vector OR to one of the surfaces ±F = 1,
and of a tangent RS at the point R of the surface, then the vectors
shall be called normal to each other. Accordingly
_c²__tt₁_ - _xx₁_ - _yy₁_ - _zz₁_ = 0,
which is the condition that the vectors with the components (_x_, _y_,
_z_, _t_) and (_x₁_ _y₁_ _z₁_ _t₁_) 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 0 to -F = I is always to have the measure unity, and a
time-like vector from O to +F = 1, _t_ > 0 is always to have the measure
1/_c_.
Let us now fix our attention upon the world-line of a substantive point
running through the world-point (_x_, _y_, _z_, _t_); then as we follow
the _progress_ of the line, the quantity
_d_τ = (1/_c_) √(_c²__dt²_ - _dx²_ - _dy²_ - _dz²_),
corresponds to the time-like vector-element (_dx_, _dy_, _dz_, _dt_).
The integral τ = ∫_d_τ, taken over the world-line from any fixed
initial point P₀ to any variable final point P, may be called the
“Proper-time” of the substantial point at P₀ upon the _world-line_. We
may regard (_x_, _y_, _z_, _t_), _i.e._, the components of the vector
OP, as functions of the “proper-time” τ; let ([._x_], [._y_], [._z_],
[._t_]) denote the first differential-quotients, and ([.._x_],
[.._y_], [.._z_], [.._t_]) the second differential quotients of (_x_,
_y_, _z_, _t_) with regard to τ, then these may respectively be called
the _Velocity-vector_, and the _Acceleration-vector_ of the
substantial point at P. Now we have
_c²_ [._t²_] - [._x²_] - [._y²_] - [._z²_] = _c²_
_c²_ [._t_][.._t_] - [._x_][.._x_] - [._y_][.._y_] - [._z_][.._z_] =
0
_i.e._, the ‘_Velocity-vector_’ is the time-like vector of unit measure
in the direction of the world-line at P, the ‘_Acceleration-vector_’ at
P is normal to the velocity-vector at P, and is in any case, a
space-like vector.
Now there is, as can be easily seen, a certain hyperbola, which has
three infinitely contiguous points in common with the world-line at P,
and of which the asymptotes are the generators of a ‘fore-cone’ and an
‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at
P (_vide_ fig. 3). If M be the centre of this hyperbola, then we have to
deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of
the vector MP, then we easily perceive that the acceleration-vector at P
is _a vector of magnitude_ _c²_/ρ _in the direction of_ MP.
If [.._x_], [.._y_], [.._z_], [.._t_] are nil, then the hyperbola of
curvature at P reduces to the straight line touching the world-line at
P, and ρ = ∞.
IV
In order to demonstrate that the assumption of the group G_{_c_} for the
physical laws does not possibly lead to any contradiction, it is
unnecessary to undertake a revision of the whole of physics on the basis
of the assumptions underlying this group. The revision has already been
successfully made in the case of “Thermodynamics and Radiation,”[30] for
“Electromagnetic phenomena”,[31] and finally for “Mechanics with the
maintenance of the idea of mass.”
For this last mentioned province of physics, the question may be asked:
if there is a force with the components X, Y, Z (in the direction of the
space-axes) at a world-point (_x_, _y_, _z_, _t_), where the
velocity-vector is ([._x_], [._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_{_c_} is undoubtedly permissible. These theorems lead us to
the following simple rule; _if the system of reference be changed in any
way, then the supposed force is to be put as a force in the new
space-coordinates in such a manner, that the corresponding vector with
the components_
[._t_]X, [._t_]Y, [._t_]Z, [._t_]T,
_where_ T = 1/_c²_ ([._x_]/[._t_] X + [._y_]/[._t_] Y +
[._z_]/[._t_] Z) = 1/_c²_
(_the rate of
which work 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 called a _moving
force-vector at_ P.
Now the world-line passing through P will be described by a substantial
point with the constant _mechanical mass m_. Let us call _m-times_ the
velocity-vector at P as the _impulse-vector_, 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[32]:
_The force-vector of motion is equal to the moving force-vector._
This enunciation comprises four equations for the components in the four
directions, of which the fourth can be deduced from the first three,
because both of the above-mentioned vectors are perpendicular to the
velocity-vector. From the definition of T, we see that the fourth simply
expresses the “Energy-law.” Accordingly _c²_-_times the component of the
impulse-vector in the direction of the t-axis is_ to be defined as _the
kinetic-energy_ of the point-mass. The expression for this is
_mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)
_i.e._, if we deduct from this the additive constant _mc²_, we obtain
the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the
order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the
system of reference_. But since the _t_-axis can be laid in the
direction of any time-like axis, therefore the energy-law comprises, for
any possible system of reference, the whole system of equations of
motion. This fact retains its significance even in the limiting case c =
∞, for the axiomatic construction of Newtonian mechanics, as has already
been pointed out by T. R. Schütz.[33]
From the very beginning, we can establish the ratio between the units of
time and space in such a manner, that the velocity of light becomes
unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the
differential expression
_d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),
becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters
into each law, which does not contradict the _world-postulate_. We can
clothe the “essential nature of this postulate in the mystical, but
mathematically significant formula
3·10⁵ _km_ = √-1 Sec.
V
The advantages arising from the formulation of the world-postulate are
illustrated by nothing so strikingly as by the expressions which tell us
about the reactions exerted by a point-charge moving in any manner
according to the Maxwell-Lorentz theory.
Let us conceive of the world-line of such an electron with the charge
(_e_), and let us introduce upon it the “Proper-time” τ reckoned from
any possible initial point. In order to obtain the field caused by the
electron at any world-point P₁ let us construct the fore-cone belonging
to P₁ (_vide_ fig. 4). Clearly this cuts the unlimited world-line of the
electron at a single point P, because these directions are all time-like
vectors. At P, let us draw the tangent to the world-line, and let us
draw from P₁ the normal to this tangent. Let _r_ be the measure of P₁Q.
According to the definition of a fore-cone, _r_/_e_ is to be reckoned as
the measure of PQ. Now at the world-point P₁, the vector-potential of
the field excited by _e_ is represented by the vector in direction PQ,
having the magnitude _e_/_cr_, in its three space components along the
_x_-, _y_-, _z_-axes; the scalar-potential is represented by the
component along the _t_-axis. This is the elementary law found out by A.
Lienard, and E. Wiechert.[34]
If the field caused by the electron be described in the above-mentioned
way, then it will appear that the division of the field into electric
and magnetic forces is a relative one, and depends upon the time-axis
assumed; the two forces considered together bears some analogy to the
force-screw in mechanics; the analogy is, however, imperfect.
I shall now describe _the ponderomotive 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 P₁.
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 QP₁. With P as the initial point, we
shall establish a system of reference in the following way: the _t_-axis
will be laid along PQ, the _x_-axis in the direction of QP₁. The
_y_-axis in the direction of MN, then the _z_-axis is automatically
determined, as it is normal to the _x_-, _y_-, _z_-axes. Let [:_x_],
[:_y_], [:_z_], [:_t_] be the acceleration-vector at P, [._x_]₁, [._y_]₁
[._z_]₁, [._t_]₁ be the velocity-vector at P₁. Then the force-vector
exerted by the first election _e_, (moving in any possible manner) upon
the second election _e_, (likewise moving in any possible manner) at P₁
is represented by
-_e e₁_([._t₁_] - [._x₁_]/_c_)F,
_For the components F_{x}, F_{y}, F_{z}, F_{t} of the vector F the
following three relations hold_:—
_c_F_{_t_} - F_{_x_} = 1/_r²_, F_{_y_} = [:_y_]/(_c²__r_), F_{_z_} =
0,
_and fourthly this vector F is normal to the velocity-vector_ P₁, _and
through this circumstance alone, its dependence on this last
velocity-vector arises_.
If we compare with this expression the previous formulæ[35] giving the
elementary law about the ponderomotive action of moving electric charges
upon each other, then we cannot but admit, that the relations which
occur here reveal the inner essence of full simplicity first in four
dimensions; but in three dimensions, they have very complicated
projections.
In the mechanics reformed according to the world-postulate, the
disharmonies which have disturbed the relations between Newtonian
mechanics and modern electrodynamics automatically disappear. I shall
now consider the position of the Newtonian law of attraction to this
postulate. I will assume that two point-masses _m_ and _m₁_ describe
their world-lines; a moving force-vector is exercised by _m_ upon _m₁_,
and the expression is just the same as in the case of the electron, only
we have to write +_mm₁_ instead -_ee₁_. We shall consider only the
special case in which the acceleration-vector of _m_ is always zero:
then _t_ 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
-([^.]1/√(1-(_v_²/_c²_)) instead of [._t_] ([._t_] = 1 up to magnitudes
of the order (1[^.]/_c²_)), then it appears that Kepler’s laws hold good
for the position (_x₁_, _y₁_, _z₁_), of _m₁_ at any time, only in place
of the time _t₁_, we have to write the proper time τ₁ of _m₁_. 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 exception is, I believe,
the true essence of an electromagnetic picture of the world; the idea
first occurred to Lorentz, its essence was first picked out by Einstein,
and is now gradually fully manifest. In course of time, the mathematical
consequences will be gradually deduced, and enough suggestions will be
forthcoming for the experimental verification of the postulate; in this
way even those, who find it uncongenial, or even painful to give up the
old, time-honoured concepts, will be reconciled to the new ideas of time
and space,—in the prospect that they will lead to pre-established
harmony between pure mathematics and physics.
Footnote 30:
Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p.
1.
Footnote 31:
H. Minkowski; the passage refers to paper (2) of the present edition.
Footnote 32:
Minkowski—Mechanics, appendix, page 65 of paper (2). Planck—Verh. d.
D. P. G. Vol. 4, 1906, p. 136.
Footnote 33:
Schütz, Gött. Nachr. 1897, p. 110.
Footnote 34:
Lienard, L’Eclairage électrique T. 16, 1896, p. 53. Wiechert, Ann. d.
Physik, Vol. 4.
Footnote 35:
K. Schwarzschild. Gött-Nachr. 1903. H. A. Lorentz, Enzyklopädie der
Math. Wissenschaften V. Art 14, p. 199.
The Foundation of the Generalised Theory of Relativity
By A. Einstein.
From Annalen der Physik 4.49.1916.
The theory which is sketched in the following pages forms the most
wide-going generalization conceivable of what is at present known as
“the theory of Relativity;” this latter theory I differentiate from the
former “Special Relativity theory,” and suppose it to be known. The
generalization of the Relativity theory has been made much easier
through the form given to the special Relativity theory by Minkowski,
which mathematician was the first to recognize clearly the formal
equivalence of the space like and time-like co-ordinates, and who made
use of it in the building up of the theory. The mathematical apparatus
useful for the general relativity theory, lay already complete in the
“Absolute Differential Calculus,” which were based on the researches of
Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which
have been shaped into a system by Ricci and Levi-civita, and already
applied to the problems of theoretical physics. I have in part B of this
communication developed in the simplest and clearest manner, all the
supposed mathematical auxiliaries, not known to Physicists, which will
be useful for our purpose, so that, a study of the mathematical
literature is not necessary for an understanding of this paper. Finally
in this place I thank my friend Grossmann, by whose help I was not only
spared the study of the mathematical literature pertinent to this
subject, but who also aided me in the researches on the field equations
of gravitation.
A
Principal considerations about the Postulate of 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.
If a co-ordinate system K be so chosen that when referred to it, the
physical laws hold in their simplest forms these laws would be also
valid when referred to another system of co-ordinates K′ which is
subjected to an uniform translational motion relative to K. We call this
postulate “The Special Relativity Principle.” By the word special, it is
signified that the principle is limited to the case, when K′ has
_uniform translatory_ 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 postulate, but only through the
postulate of the constancy of light-velocity in vacuum which, when
combined with the special relativity postulate, gives in a well-known
way, the relativity of synchronism as well as the Lorenz-transformation,
with all the relations between moving rigid bodies and clocks.
The modification which the theory of space and time has undergone
through the special relativity theory, is indeed a profound one, but a
weightier point remains untouched. According to the special relativity
theory, the theorems of geometry are to be looked upon as the laws about
any possible relative positions of solid bodies at rest, and more
generally the theorems of kinematics, as theorems which describe the
relation between measurable bodies and clocks. Consider two material
points of a solid body at rest; then according to these conceptions
there corresponds to these points a wholly definite extent of length,
independent of kind, position, orientation and time of the body.
Similarly let us consider two positions of the pointers of a clock which
is at rest with reference to a co-ordinate system; then to these
positions, there always corresponds, a time-interval of a definite
length, independent of time and place. It would be soon shown that the
general relativity theory can not hold fast to this simple physical
significance of space and time.
§ 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 perhaps first
cleanly pointed out by E. Mach. We shall illustrate it by the following
example; Let two fluid bodies of equal kind and magnitude swim freely in
space at such a great distance from one another (and from all other
masses) that only that sort of gravitational forces are to be taken into
account which the parts of any of these bodies exert upon each other.
The distance of the bodies from one another is invariable. The relative
motion of the different parts of each body is not to occur. But each
mass is seen to rotate by an observer at rest relative to the other mass
round the connecting line of the masses with a constant angular velocity
(definite relative motion for both the masses). Now let us think that
the surfaces of both the bodies (S₁ and S₂) are measured with the help
of measuring rods (relatively at rest); it is then found that the
surface of S₁ is a sphere and the surface of the other is an ellipsoid
of rotation. We now ask, why is this difference between the two bodies?
An answer to this question can only then be regarded as satisfactory
from the episteomological standpoint when the thing adduced as the cause
is an observable fact of experience. The law of causality has the sense
of a definite statement about the world of experience only when
observable facts alone appear as causes and effects.
The Newtonian mechanics does not give to this question any satisfactory
answer. For example, it says:—The laws of mechanics hold true for a
space R₁ relative to which the body S₁ is at rest, not however for a
space relative to which S₂ is at rest.
The Galiliean space, which is here introduced is however only a purely
imaginary cause, not an observable thing. It is thus clear that the
Newtonian mechanics does not, in the case treated here, actually fulfil
the requirements of causality, but produces on the mind a fictitious
complacency, in that it makes responsible a _wholly imaginary cause_ R₁
for the different behaviours of the bodies S₁ and S₂ which are actually
observable.
A satisfactory explanation to the question put forward above can only be
thus given:—that the physical system composed of S₁ and S₂ shows for
itself alone no conceivable cause to which the different behaviour of S₁
and S₂ can be attributed. The cause must thus lie outside the system. We
are therefore led to the conception that the general laws of motion
which determine specially the forms of S₁ and S₂ must be of such a kind,
that the mechanical behaviour of S₁ and S₂ must be essentially
conditioned by the distant masses, which we had not brought into the
system considered. These distant masses, (and their relative motion as
regards the bodies under consideration) are then to be looked upon as
the seat of the principal observable causes for the different behaviours
of the bodies under consideration. They take the place of the imaginary
cause R₁. Among all the conceivable spaces R₁ and R₂ moving in any
manner relative to one another, there is a priori, no one set which can
be regarded as affording greater advantages, against which the objection
which was already raised from the standpoint of the theory of knowledge
cannot be again revived. The laws of physics must be so constituted that
they should remain valid for any system of co-ordinates moving in any
manner. We thus arrive at an extension of the relativity postulate.
Besides this momentous episteomological argument, there is also a
well-known physical fact which speaks in favour of an extension of the
relativity theory. Let there be a Galiliean co-ordinate system K
relative to which (at least in the four-dimensional region considered) a
mass at a sufficient distance from other masses move uniformly in a
line. Let K′ be a second co-ordinate system which has a uniformly
accelerated motion relative to K. Relative to K′ any mass at a
sufficiently great distance experiences an accelerated motion such that
its acceleration and the direction of acceleration is independent of its
material composition and its physical conditions.
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 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 _x₁_ as its X₁
co-ordinate, it signifies that the projection of the point-event on the
X₁-axis _ascertained_ by means of a solid rod according to the rules of
Euclidean Geometry is reached when a definite measuring rod, the unit
rod, can be carried _x₁_ times from the origin of co-ordinates along the
X₁ axis. A point having _x₄_ = _t₁_ as the X₄ co-ordinate signifies that
a unit clock which is adjusted to be at rest relative to the system of
co-ordinates, and coinciding in its spatial position with the
point-event and set according to some definite standard has gone over
_x₄_ = _t_ periods before the occurrence of the point-event.
This conception of time and space is continually present in the mind of
the physicist, though often in an unconscious way, as is clearly
recognised from the role which this conception has played in physical
measurements. This conception must also appear to the reader to be lying
at the basis of the second consideration of the last paragraph and
imparting a sense to these conceptions. But we wish to show that we are
to abandon it and in general to replace it by more general conceptions
in order to be able to work out thoroughly the postulate of general
relativity,—the case of special relativity appearing as a limiting case
when there is no gravitation.
We introduce in a space, which is free from Gravitation-field, a
Galiliean Co-ordinate System K (_x_, _y_, _z_, _t_) and also, another
system K′ (_x′_ _y′_ _z′_ _t′_) rotating uniformly relative to K. The
origin of both the systems as well as their _z_-axes might continue to
coincide. We will show that for a space-time measurement in the system
K′, the above established rules for the physical significance of time
and space can not be maintained. On grounds of symmetry it is clear that
a circle round the origin in the XY plane of K, can also be looked upon
as a circle in the plane (X′, Y′) of K′. Let us now think of measuring
the circumference and the diameter of these circles, with a unit
measuring rod (infinitely small compared with the radius) and take the
quotient of both the results of measurement. If this experiment be
carried out with a measuring rod at rest relatively to the Galiliean
system K we would get π, as the quotient. The result of measurement with
a rod relatively at rest as regards K′ would be a number which is
greater than π. This can be seen easily when we regard the whole
measurement-process from the system K and remember that the rod placed
on the periphery suffers a Lorenz-contraction, not however when the rod
is placed along the radius. Euclidean Geometry therefore does not hold
for the system K′; the above fixed conceptions of co-ordinates which
assume the validity of Euclidean Geometry fail with regard to the system
K′. We cannot similarly introduce in K′ a time corresponding to physical
requirements, which will be shown by all similarly prepared clocks at
rest relative to the system K′. In order to see this we suppose that two
similarly made clocks are arranged one at the centre and one at the
periphery of the circle, and considered from the stationary system K.
According to the well-known results of the special relativity theory it
follows—(as viewed from K)—that the clock placed at the periphery will
go slower than the second one which is at rest. The observer at the
common origin of co-ordinates who is able to see the clock at the
periphery by means of light will see the clock at the periphery going
slower than the clock beside him. Since he cannot allow the velocity of
light to depend explicitly upon the time in the way under consideration
he will interpret his observation by saying that the clock on the
periphery actually goes slower than the clock at the origin. He cannot
therefore do otherwise than define time in such a way that the rate of
going of a clock depends on its position.
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 definite way, therefore completely
fail and it appears that there is no other way which will enable us to
fit the co-ordinate system to the four-dimensional world in such a way,
that by it we can expect to get a specially simple formulation of the
laws of Nature. So that nothing remains for us but to regard all
conceivable co-ordinate systems as equally suitable for the description
of natural phenomena. This amounts to the following law:—
_That in general, Laws of Nature are expressed by means of equations
which are valid for all co-ordinate systems, that is, which are
covariant for all possible transformations._ It is clear that a physics
which satisfies this postulate will be unobjectionable from the
standpoint of the general relativity postulate. Because among all
substitutions there are, in every case, contained those, which
correspond to all relative motions of the co-ordinate system (in three
dimensions). This condition of general covariance which takes away the
last remnants of physical objectivity from space and time, is a natural
requirement, as seen from the following considerations. All our
_well-substantiated_ space-time propositions amount to the determination
of space-time coincidences. If, for example, the event consisted in the
motion of material points, then, for this last case, nothing else are
really observable except the encounters between two or more of these
material points. The results of our measurements are nothing else than
well-proved theorems about such coincidences of material points, of our
measuring rods with other material points, coincidences between the
hands of a clock, dial-marks and point-events occurring at the same
position and at the same time.
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 (_x₁_ _x₂_ _x₃_ _x₄_) such that to any
and every point-event corresponds a system of values of (_x₁_ _x₂_ _x₃_
_x₄_). Two coincident point-events correspond to the same value of the
variables (_x₁_ _x₂_ _x₃_ _x₄_); _i.e._, the coincidence is
characterised by the equality of the co-ordinates. If we now introduce
any four functions (_x′₁_ _x′₂_ _x′₃_ _x′₄_) as co-ordinates, so that
there is an unique correspondence between them, the equality of all the
four co-ordinates in the new system will still be the expression of the
space-time coincidence of two material points. As the purpose of all
physical laws is to allow us to remember such coincidences, there is a
priori no reason present, to prefer a certain co-ordinate system to
another; _i.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 be most
reasonable according to the light of experience. In this sense, we shall
now introduce the following supposition; that for an infinitely small
four-dimensional region, the relativity theory is valid in the special
sense when the axes are suitably chosen.
The nature of acceleration of an infinitely small (positional)
co-ordinate system is hereby to be so chosen, that the gravitational
field does not appear; this is possible for an infinitely small region.
X₁, X₂, X₃ are the spatial co-ordinates; X₄ is the corresponding
time-co-ordinate measured by some suitable measuring clock. These
co-ordinates have, with a given orientation of the system, an immediate
physical significance in the sense of the special relativity theory
(when we take a rigid rod as our unit of measure). The expression
(1) _ds²_ = - _d_X₁² - _d_X₂² - _d_X₃² + _d_X₄²
had then, according to the special relativity theory, a value which may
be obtained by space-time measurement, and which is independent of the
orientation of the local co-ordinate system. Let us take _ds_ as the
magnitude of the line-element belonging to two infinitely near points in
the four-dimensional region. If _ds²_ belonging to the element (_d_X₁,
_d_X₂, _d_X₃, _d_X₄) 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 infinitely near
point-events belong also definite differentials _dx₁_, _dx₂_, _dx₃_,
_dx₄_, of the four-dimensional co-ordinates of any chosen system of
reference. If there be also a local system of the above kind given for
the case under consideration, _d_X’s would then be represented by
definite linear homogeneous expressions of the form
(2) _d_X_{ν} = σ_{σ}_a__{νσ}_dx__{σ}
If we substitute the expression in (1) we get
(3) _ds²_ = σ_{στ}_g__{στ}_dx__{σ}_dx__{τ}
where _g__{στ} will be functions of _x__{σ}, but will no longer depend
upon the orientation and motion of the ‘local’ co-ordinates; for _ds²_
is a definite magnitude belonging to two point-events infinitely near in
space and time and can be got by measurements with rods and clocks. The
_g__{τσ}’s are here to be so chosen, that _g__{τσ} = _g__{στ}; the
summation is to be extended over all values of σ and τ, so that the sum
is to be extended, over 4 × 4 terms, of which 12 are equal in pairs.
From the method adopted here, the case of the usual relativity theory
comes out when owing to the special behaviour of _g__{στ} in a _finite_
region it is possible to choose the system of co-ordinates in such a way
that _g__{στ} assumes constant values—
{ -1, 0, 0, 0
(4) { 0 -1 0 0
{ 0 0 -1 0
{ 0 0 0 +1
We would afterwards see that the choice of such a system of co-ordinates
for a finite region is in general not possible.
From the considerations in § 2 and § 3 it is clear, that from the
physical stand-point the quantities _g__{στ} are to be looked upon as
magnitudes which describe the gravitation-field with reference to the
chosen system of axes. We assume firstly, that in a certain
four-dimensional region considered, the special relativity theory is
true for some particular choice of co-ordinates. The _g__{στ}’s then
have the values given in (4). A free material point moves with reference
to such a system uniformly in a straight-line. If we now introduce, by
any substitution, the space-time co-ordinates _x₁_..._x₄_ then in the
new system _g__{μν}’s are no longer constants, but functions of space
and time. At the same time, the motion of a free point-mass in the new
co-ordinates, will appear as curvilinear, and not uniform, in which the
law of motion, will be _independent of the nature of the moving
mass-points_. We can thus signify this motion as one under the influence
of a gravitation field. We see that the appearance of a
gravitation-field is connected with space-time variability of
_g__{στ}’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 _g__{στ}’s describe the gravitational
field. According to the general relativity theory, gravitation thus
plays an exceptional rôle as distinguished from the others, specially
the electromagnetic forces, in as much as the 10 functions _g__{στ}
representing gravitation, define immediately the metrical properties of
the four-dimensional region.
B
Mathematical Auxiliaries for Establishing the General Covariant
Equations.
We have seen before that the general relativity-postulate leads to the
condition that the system of equations for Physics, must be co-variants
for any possible substitution of co-ordinates _x₁_, ... _x₄_; we have
now to see how such general co-variant equations can be obtained. We
shall now turn our attention to these purely mathematical propositions.
It will be shown that in the solution, the invariant _ds_, given in
equation (3) plays a fundamental rôle, which we, following Gauss’s
Theory of Surfaces, style as the line-element.
The fundamental idea of the general co-variant theory is this:—With
reference to any co-ordinate system, let certain things (tensors) be
defined by a number of functions of co-ordinates which are called the
components of the tensor. There are now certain rules according to which
the components can be calculated in a new system of co-ordinates, when
these are known for the original system, and when the transformation
connecting the two systems is known. The things herefrom designated as
“Tensors” have further the property that the transformation equation of
their components are linear and homogeneous; so that all the components
in the new system vanish if they are all zero in the original system.
Thus a law of Nature can be formulated by putting all the components of
a tensor equal to zero so that it is a general co-variant equation; thus
while we seek the laws of formation of the tensors, we also reach the
means of establishing general co-variant laws.
5. Contra-variant and co-variant Four-vector.
Contra-variant Four-vector. The line-element is defined by the four
components _dx__{ν}, whose transformation law is expressed by the
equation
(5) $$ dx'_{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial
x_{\nu}} dx_{\nu} $$
The _dx′__{σ}’_s_ are expressed as linear and homogeneous function of
_dx__{ν}’_s_; we can look upon the differentials of the co-ordinates as
the components of a tensor, which we designate specially as a
contravariant Four-vector. Everything which is defined by Four
quantities A^{σ}, with reference to a co-ordinate system, and transforms
according to the same law,
(5a)
$$ A^{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}}
A^{\nu} $$
we may call a contra-variant Four-vector. From (5. a), it follows at
once that the sums (A^{σ} ± B^{σ}) are also components of a four-vector,
when A^{σ} and B^{σ} are so; corresponding relations hold also for all
systems afterwards introduced as “tensors” (Rule of addition and
subtraction of Tensors).
_Co-variant Four-vector._
We call four quantities A_{ν} as the components of a covariant
four-vector, when for any choice of the contra-variant four vector B^{ν}
(6) ∑_{ν} A_{ν} B^{ν} = _Invariant_. From this definition follows the
law of transformation of the co-variant four-vectors. If we substitute
in the right hand side of the equation
∑_{σ} A′_{σ} B^{σ′} = ∑_{ν} A_{ν} B^{ν}.
the expressions
$$ \sum_{\sigma} \frac{\partial x_{\nu}}{\partial x_{\sigma'}}
B^{\sigma'} $$
for B^{ν} following from the inversion of the equation (5a) we get
$$ \sum_{\sigma} B^{\sigma'} \sum_{\nu} \frac{\partial x_{\nu}}{\partial
x_{\sigma'}} A_{\nu} = \sum_{\sigma} B^{\sigma'} A'_{\sigma} $$
As in the above equation B^{σ′} are independent of one another and
perfectly arbitrary, it follows that the transformation law is:—
$$ A'_{\sigma} = \sum \frac{\partial x_{\nu}}{\partial x_{\sigma'}}
A_{\nu} $$
_Remarks on the simplification of the mode of writing the expressions._
A glance at the equations of this paragraph will show that the indices
which appear twice within the sign of summation [for example ν in (5)]
are those over which the summation is to be made and that only over the
indices which appear twice. It is therefore possible, without loss of
clearness, to leave off the summation sign; so that we introduce the
rule: wherever the index in any term of an expression appears twice, it
is to be summed over all of them except when it is not expressedly said
to the contrary.
The difference between the co-variant and the contra-variant four-vector
lies in the transformation laws [(7) and (5)]. Both the quantities are
tensors according to the above general remarks; in it lies its
significance. In accordance with Ricci and Levi-civita, the
contravariants and co-variants are designated by the over and under
indices.
§ 6. Tensors of the second and higher ranks.
Contravariant tensor:—If we now calculate all the 16 products A^{μν} of
the components A^{μ} B^{ν}, of two contravariant four-vectors
(8) A^{μν} = A^{μ}B^{ν}
A^{μν}, will according to (8) and (5 a) satisfy the following
transformation law.
(9)
$$ A^{\sigma \tau'} = \frac{\partial x'_{\sigma}}{\partial x_{\mu}}
\frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} $$
We call a thing which, with reference to any reference system is defined
by 16 quantities and fulfils the transformation relation (9), a
contravariant tensor of the second rank. Not every such tensor can be
built from two four-vectors, (according to 8). But it is easy to show
that any 16 quantities A^{μν}, can be represented as the sum of
A^{μ}B^{ν} of properly chosen four pairs of four-vectors. From it, we
can prove in the simplest way all laws which hold true for the tensor of
the second rank defined through (9), by proving it only for the special
tensor of the type (8).
_Contravariant Tensor of any rank_:—It is clear that corresponding to
(8) and (9), we can define contravariant tensors of the 3rd and higher
ranks, with 4³, etc. components. Thus it is clear from (8) and (9) that
in this sense, we can look upon contravariant four-vectors, as
contravariant tensors of the first rank.
_Co-variant tensor._
If on the other hand, we take the 16 products A_{μν} of the components
of two co-variant four-vectors A_{μ} and B_{ν},
(10) A_{μν} = A_{μ} B_{ν}.
for them holds the transformation law
(11)
$$ A^{\sigma \tau'} = \frac{\partial x'_{\mu}}{\partial x_{\sigma'}}
\frac{\partial x'_{\nu}}{\partial x_{\tau'}} A^{\mu \nu} $$
By means of these transformation laws, the co-variant tensor of the
second rank is defined. All re-marks which we have already made
concerning the contravariant tensors, hold also for co-variant tensors.
_Remark_:—
It is convenient to treat the scalar Invariant either as a contravariant
or a co-variant tensor of zero rank.
_Mixed tensor._ We can also define a tensor of the second rank of the
type
(12) A_{μ}^{ν} = A_{μ}B^{ν}
which is co-variant with reference to μ and contravariant with reference
to ν. Its transformation law is
(13)
$$ A^{\tau'}_{\sigma} = \frac{\partial x_{\tau'}}{\partial x_{\beta}}
\frac{\partial \alpha}{\partial x_{\sigma'}} A^{\beta}_{\alpha} $$
Naturally there are mixed tensors with any number of co-variant indices,
and with any number of contra-variant indices. The co-variant and
contra-variant tensors can be looked upon as special cases of mixed
tensors.
_Symmetrical tensors_:—
A contravariant or a co-variant tensor of the second or higher rank is
called symmetrical when any two components obtained by the mutual
interchange of two indices are equal. The tensor A^{μν} or A_{μν} is
symmetrical, when we have for any combination of indices
(14) A^{μν} = A^{νμ}
or
(14a) A_{μν} = A_{νμ}.
It must be proved that a symmetry so defined is a property independent
of the system of reference. It follows in fact from (9) remembering (14)
$$ A^{\sigma \tau'} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}}
\frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} = \frac{\partial
x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x_{\tau'}}{\partial
x_{\nu}} A^{\nu \mu} = A^{\tau \sigma'} $$
_Antisymmetrical tensor._
A contravariant or co-variant tensor of the 2nd, 3rd or 4th rank is
called _antisymmetrical_ when the two components got by mutually
interchanging any two indices are equal and opposite. The tensor or
A^{μν} or A_{μν} is thus antisymmetrical when we have
(15) A^{μν} = -A^{νμ}
or
(15a) A_{μν} = -A_{νμ}.
Of the 16 components A^{μν}, the four components A^{μμ} vanish, the rest
are equal and opposite in pairs; so that there are only 6 numerically
different components present (Six-vector).
Thus we also see that the antisymmetrical tensor A^{μνσ} (3rd rank) has
only 4 components numerically different, and the antisymmetrical tensor
A^{μνστ} only one. Symmetrical tensors of ranks higher than the fourth,
do not exist in a continuum of 4 dimensions.
§ 7. Multiplication of Tensors.
_Outer multiplication of Tensors_:—We get from the components of a
tensor of rank _z_, and another of a rank _z′_, the components of a
tensor of rank (_z_ + _z′_) for which we multiply all the components of
the first with all the components of the second in pairs. For example,
we obtain the tensor Τ from the tensors A and B of different kinds:—
Τ_{μνσ} = A_{μν}B_{σ},
Τ^{αβγδ} = A^{αβ}B^{γδ},
Τ_{αβ}^{γδ} = A_{αβ}B^{γδ}.
The proof of the tensor character of Τ, follows immediately from the
expressions (8), (10) or (12), or the transformation equations (9),
(11), (13); equations (8), (10) and (12) are themselves examples of the
outer multiplication of tensors of the first rank.
_Reduction in rank of a mixed Tensor._
From every mixed tensor we can get a tensor which is two ranks lower,
when we put an index of co-variant character equal to an index of the
contravariant character and sum according to these indices (Reduction).
We get for example, out of the mixed tensor of the fourth rank
A_{αβ}^{γδ}, the mixed tensor of the second rank
A_{β}^{δ} = A_{αβ}^{αδ} = (∑_{α} A_{αβ}^{αδ})
and from it again by “reduction” the tensor of the zero rank
A = A_{β}^{β} = A_{αβ}^{αβ}.
The proof that the result of reduction retains a truly tensorial
character, follows either from the representation of tensor according to
the generalisation of (12) in combination with (6) or out of the
generalisation of (13).
_Inner and mixed multiplication of Tensors._
This consists in the combination of outer multiplication with reduction.
Examples:—From the co-variant tensor of the second rank A_{μν} and the
contravariant tensor of the first rank B^{σ} we get by outer
multiplication the mixed tensor
D^{σ}_{μν} = A_{μν} B^{σ} .
Through reduction according to indices ν and σ (_i.e._, putting ν = σ),
the co-variant four vector
D_{μ} = D^{ν}_{μν} = A_{μν} B^{ν} is generated.
These we denote as the inner product of the tensor A_{μν} and B^{σ}.
Similarly we get from the tensors A_{μν} and B^{στ} through outer
multiplication and two-fold reduction the inner product A_{μν} B^{μν}.
Through outer multiplication and one-fold reduction we get out of A_{μν}
and B^{στ}, the mixed tensor of the second rank D^{τ}_{μ} = A_{μν}
B^{τν}. We can fitly call this operation a mixed one; for it is outer
with reference to the indices μ and τ and inner with respect to the
indices ν and σ.
We now prove a law, which will be often applicable for proving the
tensor-character of certain quantities. According to the above
representation, A_{μν} B^{μν} is a scalar, when A_{μν} and B^{στ} are
tensors. We also remark that when A_{μν} B^{μν} is an invariant for
every choice of the tensor B^{μν}, then A_{μν} has a tensorial
character.
Proof:—According to the above assumption, for any substitution we have
A_{στ′} B^{στ′} = A_{μν} B^{μν}.
From the inversion of (9) we have however
$$ B_{\mu \nu} = \frac{\partial x_{\mu}}{\partial x_{\sigma'}}
\frac{\partial x_{\nu}}{\partial \tau'} B^{\sigma \tau'} $$
Substitution of this for B^{μν} in the above equation gives
$$ (A_{\sigma \tau'} - \frac{\partial x_{\mu}}{\partial x_{\sigma'}}
\frac{\partial x_{\nu}}{\partial x_{\tau'}}) B^{\sigma \tau'} = 0 $$
This can be true, for any choice of B^{στ′} only when the term within
the bracket vanishes. From which by referring to (11), the theorem at
once follows. This law correspondingly holds for tensors of any rank and
character. The proof is quite similar. The law can also be put in the
following form. If B^{μ} and C^{ν} are any two vectors, and if for every
choice of them the inner product A_{μν} B^{μ} C^{ν} is a scalar, then
A_{μν} is a co-variant tensor. The last law holds even when there is the
more special formulation, that with any arbitrary choice of the
four-vector B^{μ} alone the scalar product A_{μν} B^{μ} B^{ν} is a
scalar, in which case we have the additional condition that A_{μν}
satisfies the symmetry condition. According to the method given above,
we prove the tensor character of (A_{μν} + A_{νμ}), from which on
account of symmetry follows the tensor-character of A_{μν}. 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
quantities A_{μν} B^{ν} form a tensor of the first rank, when B^{ν} is
any arbitrarily chosen four-vector, then A_{μν} is a tensor of the
second rank. If for example, C^{μ} is any four-vector, then owing to the
tensor character of A_{μν} B^{ν}, the inner product A_{μν} C^{μ} B^{ν}
is a scalar, both the four-vectors C^{μ} and B^{ν} being arbitrarily
chosen. Hence the proposition follows at once.
A few words about the Fundamental Tensor _g__{μν}.
The co-variant fundamental tensor—In the invariant expression of the
square of the linear element
_ds²_ = _g__{μν} _dx__{μ} _dx__{ν}
_dx__{μ} plays the rôle of any arbitrarily chosen contravariant vector,
since further _g__{μν} = _g__{νμ}, it follows from the considerations of
the last paragraph that _g__{μν} is a symmetrical co-variant tensor of
the second rank. We call it the “fundamental tensor.” Afterwards we
shall deduce some properties of this tensor, which will also be true for
any tensor of the second rank. But the special rôle of the fundamental
tensor in our Theory, which has its physical basis on the particularly
exceptional character of gravitation makes it clear that those relations
are to be developed which will be required only in the case of the
fundamental tensor.
_The co-variant fundamental tensor._
If we form from the determinant scheme | _g__{μν} | the minors of
_g__{μν} and divide them by the determinant _g_ = | _g__{μν} | we get
certain quantities _g_^{μν} = _g_^{νμ}, which as we shall prove
generates a contravariant tensor.
According to the well-known law of Determinants
(16) _g__{μσ} _g_^{νσ} = δ_{μ}^{ν}
where δ_{μ}^{ν} is 1, or 0, according as μ = ν or not. Instead of the
above expression for _ds²_, we can also write
_g__{μσ} δ_{ν}^{σ} _dx__{μ} _dx__{ν}
or according to (16) also in the form
_g__{μσ} _g__{ντ} _g_^{στ} _dx__{μ} _dx__{ν}
Now according to the rules of multiplication, of the fore-going
paragraph, the magnitudes
_d_ξ_{σ} = _g__{μσ} _dx__{μ}
forms a co-variant four-vector, and in fact (on account of the arbitrary
choice of _dx__{μ}) any arbitrary four-vector.
If we introduce it in our expression, we get
_ds²_ = _g_^{στ} _d_ξ_{σ} _d_ξ_{τ}.
For any choice of the vectors _d_ξ_{σ} _d_ξ_{τ} this is scalar, and
_g_^{στ}, according to its definition is a symmetrical thing in σ and τ,
so it follows from the above results, that _g_^{στ} is a contravariant
tensor. Out of (16) it also follows that δ^{ν}_{μ} 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
| _g__{μα} _g_^{αν} | = | _g__{μα} | | _g_^{αν} |
On the other hand we have
| _g__{μα} _g_^{αν} | = | δ^{ν}_{μ} | = 1
So that it follows (17) that | _g__{μν} | | _g_^{μν} | = 1.
_Invariant of volume._
We see first the transformation law for the determinant _g_ = | _g__{μν}
|. According to (11)
$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial
x_{\nu}}{\partial x_{\tau'}} g_{\mu u} | $$
From this by applying the law of multiplication twice, we obtain
$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} | |
\frac{\partial x_{\nu}}{\partial x_{\tau'}} | | g_{\mu \nu} | = |
\frac{\partial x_{\mu}}{\alpha_{\sigma'}} | g $$
or
(A)
$$ \sqrt{g'} = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} |
\sqrt{g} $$
On the other hand the law of transformation of the volume element
_d_τ′ = ∫ _dx₁_ _dx₂_ _dx₃_ _dx₄_
is according to the wellknown law of Jacobi.
(B) $$ d\tau' = | \frac{dx'_{\sigma}}{dx_{\mu}} | d\tau $$
by multiplication of the two last equations (A) and (B) we get
(18) = √_g_ _d_τ′ = √_g_ _d_τ.
Instead of √_g_, we shall afterwards introduce √(-_g_) which has a real
value on account of the hyperbolic character of the time-space
continuum. The invariant √(-_g_)_d_τ, is equal in magnitude to the
four-dimensional volume-element measured with solid rods and clocks, in
accordance with the special relativity theory.
_Remarks on the character of the space-time continuum_—Our assumption
that in an infinitely small region the special relativity theory holds,
leads us to conclude that _ds²_ can always, according to (1) be
expressed in real magnitudes _d_X₁ ... _d_X_{_h_}. If we call _d_τ₀ the
“_natural_” volume element _d_X₁ _d_X₂ _d_X₃ _d_X₄ we have thus (18a)
_d_τ₀ = √(_g_)_i_τ.
Should √(-_g_) vanish at any point of the four-dimensional continuum it
would signify that to a finite co-ordinate volume at the place
corresponds an infinitely small “natural volume.” This can never be the
case; so that _g_ can never change its sign; we would, according to the
special relativity theory assume that _g_ has a finite negative value.
It is a hypothesis about the physical nature of the continuum
considered, and also a pre-established rule for the choice of
co-ordinates.
If however (-_g_) remains positive and finite, it is clear that the
choice of co-ordinates can be so made that this quantity becomes equal
to one. We would afterwards see that such a limitation of the choice of
co-ordinates would produce a significant simplification in expressions
for laws of nature.
In place of (18) it follows then simply that
_d_τ′ = _d_
from this it follows, remembering the law of Jacobi,
(19)
$$ | \frac{\partial x'_{\sigma}}{dx_{\mu}} | = 1 $$
With this choice of co-ordinates, only substitutions with determinant 1
are allowable.
It would however be erroneous to think that this step signifies a
partial renunciation of the general relativity postulate. We do not seek
those laws of nature which are co-variants with regard to the
transformations having the determinant 1, but we ask: what are the
general co-variant laws of nature? First we get the law, and then we
simplify its expression by a special choice of the system of reference.
_Building up of new tensors 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:—
A^{μ} = _g_^{μσ} A_{σ}
A = _g__{μν} A^{μν}
We would point out specially the following combinations:
A^{μν} = _g_^{μα} _g_^{νβ} A_{αβ}
A_{μν} = _g__{μα} _g__{νβ} A^{αβ}
(complement to the co-variant or contravariant tensors)
and B_{μν} = _g__{μν} _g_^{αβ} A_{αβ}
We can call B_{μν} the reduced tensor related to A_{μν}.
Similarly
B^{μν} = _g_^{μν}_g__{αβ}A^{αβ}.
It is to be remarked that _g_^{μν} is no other than the “complement” of
_g__{μν} for we have,—
_g_^{μα}_g_^{νβ}_g__{αβ} = _g__{μα}δ^{ν}_{α} = _g_^{μν}.
§ 9. Equation of the geodetic line (or of point-motion).
As the “line element” _ds_ is a definite magnitude independent of the
co-ordinate system, we have also between two points P₁ and P₂ of a four
dimensional continuum a line for which ∫_ds_ is an extremum (geodetic
line), _i.e._, one which has got a significance independent of the
choice of co-ordinates.
Its equation is
(20) δ{ ∫^{P₂}_{P₁} _ds_ } = 0
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 λ, be a function of the co-ordinates _x__{ν}; this defines a series
of surfaces which cut the geodetic line sought-for as well as all
neighbouring lines from P₁ to P₂. We can suppose that all such curves
are given when the value of its co-ordinates _x__{ν} are given in terms
of λ. The sign δ corresponds to a passage from a point of the geodetic
curve sought-for to a point of the contiguous curve, both lying on the
same surface λ.
Then (20) can be replaced by
{ λ₃
{ ∫δω _d_λ = 0
(20a) { λ₁
{
{ ω² = _g__{μν}(_dx__{μ}/_d_λ)(_dx__{ν}/_d_λ)
But
δω = (1/ω){½(∂_g__{μν}/∂_x__{σ}) · (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ)
· δ_x__{σ}
+ _g__{μν}(_dx__{μ}/_d_λ)δ(_dx__{ν}/_d_λ)}
So we get by the substitution of δω in (20a), remembering that
δ(_dx__{ν}/_d_λ) = (_d_/_d_λ)(δ_x__{ν})
after partial integration,
{ λ₃
{ ∫ _d_λ _k__{σ} δ_x__{σ} = 0
(20b) { λ₁
{
{ where _k__{σ} = (_d_/_d_λ){(_g__{μν}/ω) · (_dx__{μ}/_d_λ)}
- (1/(2ω))(∂_g__{μν}/∂_x__{σ}
× (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ).
From which it follows, since the choice of δν_{σ} is perfectly arbitrary
that _k__{σ}_’s_ should vanish. Then
(20c) _k__{σ} = 0 (σ = 1, 2, 3, 4)
are the equations of geodetic line; since along the geodetic line
considered we have _ds_ ≠ 0, we can choose the parameter λ, as the
length of the arc measured along the geodetic line. Then _w_ = 1, and we
would get in place of (20c)
$$ g_{\mu\nu} \frac{\partial^2 x_{\mu}}{\partial s^2} + \frac{\partial
g_{\mu\nu}}{\partial x_{\sigma}} \frac{\partial x_{\sigma}}{\partial s}
\frac{\partial x_{\mu}}{\partial s} - \frac{1}{2} \frac{\partial
g_{\mu\sigma}}{\partial x_{\nu}} \frac{\partial x_{\mu}}{\partial s}
\frac{\partial x_{\sigma}}{\partial s} = 0 $$
Or by merely changing the notation suitably,
(20d) $$ g_{\alpha\sigma} \frac{d^2 x_{\alpha}}{ds^2} +
\begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} \frac{dx_{\mu}}{ds}
\frac{dx_{\nu}}{ds} = 0 $$
where we have put, following Christoffel,
(21)
$$ \begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} = \frac{1}{2}
\begin{bmatrix}\frac{\partial g_{\mu\sigma}}{\partial x_{\nu}} +
\frac{\partial g_{\nu\sigma}}{\partial x_{\mu}} - \frac{\partial
g_{\mu\nu}}{\partial \sigma}\end{bmatrix} $$
Multiply finally (20d) with _g_^{στ} (outer multiplication with
reference to τ, and inner with respect to σ) we get at last the final
form of the equation of the geodetic line—
$$ \frac{d^2 x_{\tau}}{ds^2} + \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix}
\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} = 0 $$
Here we have put, following Christoffel,
$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} = g^{\tau\alpha}
\begin{bmatrix}\mu\nu\\\alpha\end{bmatrix} $$
§ 10. Formation of Tensors through Differentiation.
Relying on the equation of the geodetic line, we can now easily deduce
laws according to which new tensors can be formed from given tensors by
differentiation. For this purpose, we would first establish the general
co-variant differential equations. We achieve this through a repeated
application of the following simple law. If a certain curve be given in
our continuum whose points are characterised by the arc-distances _s_,
measured from a fixed point on the curve, and if further φ, be an
invariant space function, then _d_φ/_ds_ is also an invariant. The proof
follows from the fact that _d_φ as well as _ds_, are both invariants
Since
$$ \frac{d \phi}{ds} = \frac{\partial \phi}{\partial x_{\mu}}
\frac{\partial x_{\mu}}{\partial s} $$
so that
$$ \psi = \frac{\partial \phi}{\partial x_{\mu}} \frac{dx_{\mu}}{ds} $$
is also an invariant for all curves which go out from a point in the
continuum, _i.e._, for any choice of the vector _dx__{μ}. From which
follows immediately that
A_{μ} = ∂φ/∂_x__{μ}
is a co-variant four-vector (gradient of φ).
According to our law, the differential-quotient χ = ∂ψ/∂_s_ taken along
any curve is likewise an invariant.
Substituting the value of ψ, we get
$$ \chi = \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}}
\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} + \frac{\partial \phi}{\partial
x_{\mu}} \frac{d^2 x_{\mu}}{ds^2} $$
Here however we can not at once deduce the existence of any tensor. If
we however take that the curves along which we are differentiating are
geodesics, we get from it by replacing _d²__x__{ν}/_ds²_ according to
(22)
$$ \chi = \begin{bmatrix}\frac{\partial^2 \phi}{\partial x_{\mu}\partial
x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{\partial
\phi}{\partial x_{\tau}} \end{bmatrix} \frac{dx_{\mu}}{ds}
\frac{dx_{\nu}}{ds} $$
From the interchangeability of the differentiation with regard to μ and
ν, and also according to (23) and (21) we see that the bracket
$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} $$
is symmetrical with respect to μ and ν.
As we can draw a geodetic line in any direction from any point in the
continuum, ∂_x__{μ}/_ds_ is thus a four-vector, with an arbitrary ratio
of components, so that it follows from the results of §7 that
(25)
$$ A_{\mu\nu} = \frac{\partial^2 \phi}{\partial x_{\mu} \partial
x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{\partial
\phi}{\partial x_{\tau}} $$
is a co-variant tensor of the second rank. We have thus got the result
that out of the co-variant tensor of the first rank A_{μ} = ∂φ/∂_x__{μ}
we can get by differentiation a co-variant tensor of 2nd rank
(26)
$$ A_{\mu\nu} = \frac{\partial A_{\mu}}{\partial x_{\nu}} -
\begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} A_{\tau} $$
We call the tensor A_{μν} the “extension” of the tensor A_{μ}. Then we
can easily show that this combination also leads to a tensor, when the
vector A_{μ} is not representable as a gradient. In order to see this we
first remark that ψ (_d_φ/∂_x__{μ}) is a co-variant four-vector when ψ
and φ are scalars. This is also the case for a sum of four such terms :—
$$ S_{\mu} = \psi^{(1)} \frac{\partial \phi^{(1)}}{\partial x_{\mu}} +
... + \psi^{(4)} \frac{\partial \phi^{(4)}}{\partial x_{\mu}} $$
when ψ^{(1)}, φ^{(1)} ... ψ^{(4)}, φ^{(4)} are scalars. Now it is
however clear that every co-variant four-vector is representable in the
form of S_{μ}.
If for example, A_{μ} is a four-vector whose components are any given
functions of _x__{ν}, we have, (with reference to the chosen co-ordinate
system) only to put
ψ^{(1)} = A₁ φ^{(1)} = _x₁_
ψ^{(2)} = A₂ φ^{(2)} = _x₂_
ψ^{(3)} = A₃ φ^{(3)} = _x₃_
ψ^{(4)} = A₄ φ^{(4)} = _x₄_.
in order to arrive at the result that S_{μ} is equal to A_{μ}.
In order to prove then that A_{μν} is a tensor when on the right side of
(26) we substitute any co-variant four-vector for A_{μ} we have only to
show that this is true for the four-vector S_{μ}. For this latter case,
however, a glance on the right hand side of (26) will show that we have
only to bring forth the proof for the case when
A_{μ} = ψ ∂φ/∂_x__{μ}.
Now the right hand side of (25) multiplied by ψ is
$$ \psi \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}} -
\begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \psi \frac{\partial
\phi}{\partial x_{\tau}} $$
which has a tensor character. Similarly, (∂φ/∂_x__{μ}) (∂φ/∂_x__{ν}) is
also a tensor (outer product of two four-vectors).
Through addition follows the tensor character of
$$ \frac{\partial}{\partial x_{\nu}} (\psi \frac{\partial \phi}{\partial
x_{\mu}}) - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} (\psi
\frac{\partial \phi}{\partial x_{\tau}}) $$
Thus we get the desired proof for the four-vector, ψ ∂φ/∂_x__{μ} and
hence for any four-vectors A_{μ} as shown above.
With the help of the extension of the four-vector, we can easily define
“extension” of a co-variant tensor of any rank. This is a generalisation
of the extension of the four-vector. We confine ourselves to the case of
the extension of the tensors of the 2nd rank for which the law of
formation can be clearly seen.
As already remarked every co-variant tensor of the 2nd rank can be
represented as a sum of the tensors of the type A_{μ} B_{ν}.
It would therefore be sufficient to deduce the expression of extension,
for one such special tensor. According to (26) we have the expressions
$$ \frac{\partial A_{\mu}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau} $$
$$ \frac{\partial B_{\nu}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} B_{\tau} $$
are tensors. Through outer multiplication of the first with B_{ν} and
the 2nd with A_{μ} we get tensors of the third rank. Their addition
gives the tensor of the third rank
(27)
$$ A_{\mu\nu\sigma} = \frac{\partial A_{\mu\nu}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau\nu} -
\begin{Bmatrix}\sigma\nu\\\tau\end{Bmatrix} A_{\mu\tau} $$
where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear
and homogeneous with reference to A_{μν}, and its first differential
co-efficient, so that this law of formation leads to a tensor not only
in the case of a tensor of the type A_{μ} B_{ν} but also in the case of
a summation for all such tensors, _i.e._, in the case of any co-variant
tensor of the second rank. We call A_{μνσ} the extension of the tensor
A_{μν}. It is clear that (26) and (24) are only special cases of (27)
(extension of the tensors of the first and zero rank). In general we can
get all special laws of formation of tensors from (27) combined with
tensor multiplication.
Some special cases of Particular Importance.
_A few auxiliary lemmas concerning the fundamental tensor._ We shall
first deduce some of the lemmas much used afterwards. According to the
law of differentiation of determinants, we have
(28) _dg_ = _g_^{μν} _g dg__{μν} = -_g__{μν} _gdg_^{μν}.
The last form follows from the first when we remember that
_g__{μν} _g_^{μ′ν} = δ^{μ′}_{μ} , and therefore _g__{μν}_g_^{μν} =
4,
consequently _g__{μν}_dg_^{μν} + _g_^{μν} _dg__{μν} = 0.
From (28), it follows that
(29)
$$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} =
\frac{1}{2} \frac{\log (-g)}{\partial x_{\sigma}} = \frac{1}{2}
g^{\mu\nu} \frac{\partial g_{\mu\nu}}{\partial x_{\sigma}} = -
\frac{1}{2} g_{\mu\nu} \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}}
$$
Again, since _g__{μν} _g_^{νσ} = δ^{ν}_{μ} , we have, by
differentiation,
$$ g_{\mu\sigma} dg^{\nu\sigma} = -g^{\nu\sigma} dg_{\mu\sigma} $$
or
$$ g_{\mu\sigma} \frac{\partial g^{\nu\sigma}}{\partial x_{\lambda}} = -
g^{\nu\sigma} \frac{\partial g_{\mu\sigma}}{\partial x_{\lambda}} $$
By mixed multiplication with _g_^{στ} and _g__{νλ} respectively we
obtain (changing the mode of writing the indices).
(31)
_dg_^{μν} = -_g_^{μα} _g_^{νβ} _dg__{αβ}
∂_g_^{μν}/∂_x__{σ} = -_g_^{μα} _g_^{νβ} _dg__{αβ}
and
(32)
_dg__{μν} = -_g__{μα} _g__{νβ} _dg_^{αβ}
∂_g__{μν}/∂_x__{σ} = -_g__{μα} _g__{νβ} ∂_g_^{αβ}/∂_x__{σ}.
The expression (31) allows a transformation which we shall often use;
according to (21)
(33)
$$ \frac{\partial g_{\alpha\beta}}{\partial x_{\sigma}} =
\begin{bmatrix}\alpha & & \sigma\ & \beta &\end{bmatrix} +
\begin{bmatrix}\beta & & \sigma\ \alpha&\end{bmatrix} $$
If we substitute this in the second of the formula (31), we get,
remembering (23),
(34)
$$ \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}} = - ( g^{\mu\tau}
\begin{Bmatrix}\tau & & \sigma\ \nu&\end{Bmatrix} + g^{\nu\tau}
\begin{Bmatrix}\tau & & \sigma\ \mu&\end{Bmatrix} ) $$
By substituting the right-hand side of (34) in (29), we get
(29a)
$$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} =
\begin{Bmatrix}\mu \sigma\\\mu\end{Bmatrix} $$
_Divergence of the contravariant four-vector._
Let us multiply (26) with the contravariant fundamental tensor _g_^{μν}
(inner multiplication), then by a transformation of the first member,
the right-hand side takes the form
(A)
$$ \frac{\partial}{\partial x_{\nu}} (g^{\mu\nu} A_{\mu}) - A_{\mu}
\frac{\partial g^{\mu\nu}}{\partial x_{\nu}} - \frac{1}{2}
g^{\tau\alpha} (\frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} +
\frac{\partial g_{ u\alpha}}{\partial x_{\mu}} - \frac{\partial
g_{\mu\nu}}{\partial x_{\alpha}}) g^{\mu\nu} A_{\tau} $$
According to (31) and (29), the last member can take the form
(B)
$$ \frac{1}{2} \frac{\partial g^{\tau\nu}}{\partial x_{\nu}} A_{\tau} +
\frac{1}{2} \frac{\partial g^{\mu\tau}}{\partial x_{\mu}} A_{\tau} +
\frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\alpha}}
g^{\mu\alpha} A_{\tau} $$
Both the first members of the expression (B), and the second member of
the expression (A) cancel each other, since the naming of the
summation-indices is immaterial. The last member of (B) can then be
united with first of (A). If we put
_g_^{μν} A_{μ} = A^{ν},
where A^{ν} as well as A_{μ} are vectors which can be arbitrarily
chosen, we obtain finally
$$ \Phi = \frac{1}{\sqrt{-g}} \frac{\partial}{\partial x_{\nu}}
(\sqrt{-g} A^{\nu}) $$
This scalar is the _Divergence_ of the contravariant four-vector A^{ν}.
_Rotation of the (covariant) four-vector._
The second member in (26) is symmetrical in the indices μ, and ν. Hence
A_{μν} - A_{νμ} is an antisymmetrical tensor built up in a very simple
manner. We obtain
∂A_{μ} ∂A_{ν}
(36) B_{μν} = -------------- - ------------
∂_x__{ν} ∂_{_x_μ}
_Antisymmetrical Extension of a Six-vector._
If we apply the operation (27) on an antisymmetrical tensor of the
second rank A_{μ{ν²}} and form all the equations arising from the cyclic
interchange of the indices μ, ν, σ, and add all them, we obtain a tensor
of the third rank
(37) B_{μνσ} = A_{μνσ} + A_{νσμ} + A_{σμν}
∂A_{μν} ∂A_{νσ} ∂A_{σμ}
= ------------ + ------------- + ------------
∂_x__{σ} ∂_x__{μ} ∂_x__{ν}
from which it is easy to see that the tensor is antisymmetrical.
_Divergence of the Six-vector._
If (27) is multiplied by _g_^{μα} _g_^{νβ} (mixed multiplication), then
a tensor is obtained. The first member of the right hand side of (27)
can be written in the form
$$ \frac{\partial}{\partial x_{\sigma}} (g^{\mu\alpha}
g^{\nu\beta} A_{\mu\nu}) - g^{\mu\alpha} \frac{\partial
g^{\nu\beta}}{\partial x_{\sigma}} A_{\mu\nu} - g^{\nu\beta}
\frac{\partial g^{\mu\alpha}}{\partial x_{\sigma}} A_{\mu\nu} $$
If we replace _g_^{μα} _g_^{νβ} A_{μνσ} by A_{σ}^{αβ}, _g_^{μα} _g_^{νβ}
A_{μν} by A^{αβ} and replace in the transformed first member
∂_g_^{νβ}/∂_x__{σ} and ∂_g_^{μα}/∂_x__{σ}
with the help of (34), then from the right-hand side of (27) there
arises an expression with seven terms, of which four cancel. There
remains
(38) $$ A^{\alpha\beta}_{\sigma} = \frac{\partial
A^{\alpha\beta}}{\partial x_{\sigma}} + \begin{Bmatrix}\sigma & &
\kappa\ \alpha end{Bmatrix} A^{\kappa\beta} + \begin{Bmatrix}\sigma & &
\kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} $$
This is the expression for the extension of a contravariant tensor of
the second rank; extensions can also be formed for corresponding
contravariant tensors of higher and lower ranks.
We remark that in the same way, we can also form the extension of a
mixed tensor A_{μ}^{α}
(39) $$ A^{\alpha}_{\mu\sigma} = \frac{\partial
A^{\alpha}_{\mu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & &
\mu\ \tau&\end{Bmatrix} A^{\alpha}_{\tau} + \begin{Bmatrix}\sigma & &
\tau\ \alpha&\end{Bmatrix} A^{\tau}_{\mu} $$
By the reduction of (38) with reference to the indices β and σ(inner
multiplication with δ_{β}^{σ}), we get a contravariant four-vector
$$ A^{\alpha} = \frac{\partial A^{\alpha\beta}}{\partial x_{\beta}} +
\begin{Bmatrix}\beta & & \kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} +
\begin{Bmatrix}\beta & & \kappa\ \alpha&\end{Bmatrix} A^{\kappa\beta} $$
On the account of the symmetry of
$$ \begin{Bmatrix}\beta & &\kappa\ \alpha&\end{Bmatrix} $$
with reference to the indices β and κ, the third member of the right
hand side vanishes when A^{αβ} is an antisymmetrical tensor, which we
assume here; the second member can be transformed according to (29a); we
therefore get
(40) $$ A^{\alpha} = \frac{1}{\sqrt{-g}} \frac{\partial(\sqrt{-g}
A^{\alpha\beta})}{\partial x_{\beta}} $$
This is the expression of the divergence of a contravariant six-vector.
_Divergence of the mixed tensor of the second rank._
Let us form the reduction of (39) with reference to the indices α and σ,
we obtain remembering (29a)
(41) $$ \sqrt{-g} A_{\mu} = \frac{\partial(\sqrt{-g}
A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & &
\mu\ \tau&\end{Bmatrix} \sqrt{-g} A^{\sigma}_{\tau} $$
If we introduce into the last term the contravariant tensor A^{ρσ} =
_g_^{ρτ} A^{σ}_{τ}, it takes the form
$$ - \begin{bmatrix}\sigma & & \mu\ \rho&\end{bmatrix} \sqrt{-g}
A^{\rho\sigma} $$
If further A^{ρσ} or is symmetrical it is reduced to
$$ - \frac{1}{2} \sqrt{-g} \frac{\partial g_{\rho\sigma}}{\partial
x_{\mu}} A^{\rho\sigma} $$
If instead of A^{ρσ}, we introduce in a similar way the symmetrical
co-variant tensor A_{ρσ} = _g__{ρα} _g__{σβ} A^{αβ}, then owing to (31)
the last member can take the form
$$ \frac{1}{2} \sqrt{-g} \frac{\partial g_{\rho\sigma}}{\partial
x_{\mu}} A_{\rho\sigma} $$
In the symmetrical case treated, (41) can be replaced by either of the
forms
(41a)
$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g}
A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \frac{1}{2} \frac{\partial
g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A^{\rho\sigma} $$
or
(41b)
$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g}
A^{\sigma}_{\mu})}{\partial x_{\sigma}} + \frac{1}{2} \frac{\partial
g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A_{\rho\sigma} $$
which we shall have to make use of afterwards.
§12. The Riemann-Christoffel Tensor.
We now seek only those tensors, which can be obtained from the
fundamental tensor _g_^{μν} by differentiation alone. It is found
easily. We put in (27) instead of any tensor A^{μν} the fundamental
tensor _g_^{μν} and get from it a new tensor, namely the extension of
the fundamental tensor. We can easily convince ourselves that this
vanishes identically. We prove it in the following way; we substitute in
(27)
$$ A_{\mu\nu} = \frac{\partial A_{\mu}}{\partial x_{\nu}} -
\begin{Bmatrix}\mu & & \nu\ \rho&\end{Bmatrix} A_{\rho} $$
_i.e._, the extension of a four-vector.
Thus we get (by slightly changing the indices) the tensor of the third
rank
$$ A_{\mu\sigma\tau} = \frac{\partial^2 A_{\mu}}{\partial x_{\sigma}
\partial x_{\tau}} - \begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix}
\frac{\partial A_{\rho}}{\partial x_{\tau}} - \begin{Bmatrix}\mu & &
\tau\ \rho&\end{Bmatrix} \frac{\partial A_{\rho}}{\partial x_{\sigma}} -
\begin{Bmatrix}\sigma & & \tau\ \rho&\end{Bmatrix} \frac{\partial
A_{\mu}}{\partial x_{\rho}} + \begin{bmatrix} - \frac{\partial}{\partial
x_{\tau}} \begin{Bmatrix}\mu&&\sigma\ \rho&\end{Bmatrix} +
\begin{Bmatrix}\mu&&\tau\ \alpha\end{Bmatrix}
\begin{Bmatrix}\alpha&&\sigma\ \rho&\end{Bmatrix} +
\begin{Bmatrix}\sigma&&\tau\ \alpha\end{Bmatrix}
\begin{Bmatrix}\alpha&&\mu\ \rho&\end{Bmatrix} \end{bmatrix} A_{\rho} $$
We use these expressions for the formation of the tensor A_{μστ} -
A_{μτσ}. Thereby the following terms in A_{μστ} cancel the corresponding
terms in A_{μτσ}; the first member, the fourth member, as well as the
member corresponding to the last term within the square bracket. These
are all symmetrical in σ, and τ. The same is true for the sum of the
second and third members. We thus get
(43)
$$ A_{\mu\sigma\tau} - A_{\mu\tau\sigma} = B^{\rho}_{\mu\sigma\tau}
A_{\rho} $$
$$ B^{\rho}_{\mu\sigma\tau} = - \frac{\partial}{\partial x_{\tau}}
\begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix} +
\frac{\partial}{\partial x_{\sigma}} \begin{Bmatrix}\mu & &
\tau\ \rho&\end{Bmatrix} - \begin{Bmatrix}\mu & &
\sigma\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & &
\tau\ \rho&\end{Bmatrix} + \begin{Bmatrix}\mu & &
\tau\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & &
\sigma\ \rho&\end{Bmatrix} $$
The essential thing in this result is that on the right hand side of
(42) we have only A_{ρ}, but not its differential co-efficients. From
the tensor-character of A_{μστ} - A_{μτσ}, and from the fact that A_{ρ}
is an arbitrary four vector, it follows, on account of the result of §7,
that B^{ρ}_{μστ} is a tensor (Riemann-Christoffel Tensor).
The mathematical significance of this tensor is as follows; when the
continuum is so shaped, that there is a co-ordinate system for which
_g__{μν}_’s_ are constants, B^{ρ}_{μστ} all vanish.
If we choose instead of the original co-ordinate system any new one, so
would the _g__{μν}’s referred to this last system be no longer
constants. The tensor character of B^{ρ}_{μστ} shows us, however, that
these components vanish collectively also in any other chosen system of
reference. The vanishing of the Riemann Tensor is thus a necessary
condition that for some choice of the axis-system _g__{μν}’s can be
taken as constants. In our problem it corresponds to the case when by a
suitable choice of the co-ordinate system, the special relativity theory
holds throughout any finite region. By the reduction of (43) with
reference to indices to τ and ρ, we get the covariant tensor of the
second rank
(44)
$$ B_{\mu\nu} = R_{\mu\nu} + S_{\mu\nu} $$
$$ R_{\mu\nu} = - \frac{\partial}{\partial x_{\alpha}}
\begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} + \begin{Bmatrix}\mu
& & \alpha\ \beta&\end{Bmatrix} \begin{Bmatrix}\nu & &
\beta\ \alpha&\end{Bmatrix} $$
$$ S_{\mu\nu} = \frac{\partial \log \sqrt{-g}}{\partial x_{\mu} \partial
x_{\nu}} - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix}
\frac{\partial \log \sqrt{-g}}{\partial x_{\alpha}} $$
_Remarks upon the choice of co-ordinates._—It has already been remarked
in §8, with reference to the equation (18a), that the co-ordinates can
with advantage be so chosen that √(-_g_) = 1. A glance at the equations
got in the last two paragraphs shows that, through such a choice, the
law of formation of the tensors suffers a significant simplification. It
is specially true for the tensor B_{μν}, which plays a fundamental rôle
in the theory. By this simplification, S_{μν} vanishes of itself so that
tensor B_{μν} reduces to R_{μν}.
I shall give in the following pages all relations in the simplified
form, with the above-named specialisation of the co-ordinates. It is
then very easy to go back to the general covariant equations, if it
appears desirable in any special case.
C. THE THEORY OF THE GRAVITATION-FIELD
§13. Equation of motion of a material point in a gravitation-field.
Expression for the field-components of gravitation.
A freely moving body not acted on by external forces moves, according to
the special relativity theory, along a straight line and uniformly. This
also holds for the generalised relativity theory for any part of the
four-dimensional region, in which the co-ordinates K_{0} can be, and
are, so chosen that _g__{μν}’s have special constant values of the
expression (4).
Let us discuss this motion from the stand-point of any arbitrary
co-ordinate-system K₁; it moves with reference to K₁ (as explained in
§2) in a gravitational field. The laws of motion with reference to K₁
follow easily from the following consideration. With reference to K₀,
the law of motion is a four-dimensional straight line and thus a
geodesic. As a geodetic-line is defined independently of the system of
co-ordinates, it would also be the law of motion for the motion of the
material-point with reference to K₁. If we put
(45) $$ \Gamma^{\tau}_{\mu\nu} = - \begin{Bmatrix}\mu & &
\nu\ \tau&\end{Bmatrix} $$
we get the motion of the point with reference to K₁, given by
(46) $$ \frac{d^2 x_{\tau}}{ds^2} = \Gamma^{\tau}_{\mu\nu}
\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} $$
We now make the very simple assumption that this general covariant
system of equations defines also the motion of the point in the
gravitational field, when there exists no reference-system K₀, with
reference to which the special relativity theory holds throughout a
finite region. The assumption seems to us to be all the more legitimate,
as (46) contains only the first differentials of _g__{μν}, among which
there is no relation in the special case when K₀ exists.
If γ_{μν}^{τ}’s vanish, the point moves uniformly and in a straight
line; these magnitudes therefore determine the deviation from
uniformity. They are the components of the gravitational field.
§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 gravitation-field will be signified as
matter; therefore the term includes not only matter in the usual sense,
but also the electro-dynamic field. Our next problem is to seek the
field-equations of gravitation in the absence of matter. For this we
apply the same method as employed in the foregoing paragraph for the
deduction of the equations of motion for material points. A special case
in which the field-equations sought-for are evidently satisfied is that
of the special relativity theory in which _g__{μν}’s have certain
constant values. This would be the case in a certain finite region with
reference to a definite co-ordinate system K₀. With reference to this
system, all the components B^{ρ}_{μστ} of the Riemann’s Tensor [equation
43] 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
every case satisfied when all B^{ρ}_{μστ} vanish. But this condition is
clearly one which goes too far. For it is clear that the
gravitation-field generated by a material point in its own neighbourhood
can never be transformed _away_ by any choice of axes, _i.e._, it cannot
be transformed to a case of constant _g__{μν}’s.
Therefore it is clear that, for a gravitational field free from matter,
it is desirable that the symmetrical tensors B_{μν} deduced from the
tensors B^{ρ}_{μστ} should vanish. We thus get 10 equations for 10
quantities _g__{μν} which are fulfilled in the special case when
B^{ρ}_{μστ}’s all vanish.
Remembering (44) we see that in absence of matter the field-equations
come out as follows; (when referred to the special co-ordinate-system
chosen.)
(47) $$ \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x_{\alpha}} +
\Gamma^{\alpha}_{\mu\beta} \Gamma^{\beta}_{\mu\alpha} = 0 $$
$$ \sqrt{-g} = 1 $$
$$ \Gamma^{\alpha}_{\mu\nu} = - \begin{Bmatrix}\mu & &
\nu\ \alpha&\end{Bmatrix} $$
It can also be shown that the choice of these equations is connected
with a minimum of arbitrariness. For besides B_{μν}, there is no tensor
of the second rank, which can be built out of _g__{μν}’s and their
derivatives no higher than the second, and which is also linear in them.
It will be shown that the equations arising in a purely mathematical way
out of the conditions of the general relativity, together with equations
(46), give us the Newtonian law of attraction as a first approximation,
and lead in the second approximation to the explanation of the
perihelion-motion of mercury discovered by Leverrier (the residual
effect which could not be accounted for by the consideration of all
sorts of disturbing factors). My view is that these are convincing
proofs of the physical correctness of my theory.
§15. Hamiltonian Function for the Gravitation-field.
Laws of Impulse and Energy.
In order to show that the field equations correspond to the laws of
impulse and energy, it is most convenient to write it in the following
Hamiltonian form:—
(47a)
δ∫ H_d_τ = 0
H = _g_^{μν} γ^{α}_{μβ} γ^{β}_{να}
√(-_g_) = 1
Here the variations vanish at the limits of the finite four-dimensional
integration-space considered.
It is first necessary to show that the form (47a) is equivalent to
equations (47). For this purpose, let us consider H as a function of
_g_^{μν} and _g_^{μν}_{σ} (= ∂_g_^{μν}/∂_x__{σ})
We have at first
δH = Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2_g_^{μν} Γ^{α}_{μβ}
δΓ^{β}_{να}
= - Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2Γ^{α}_{μβ}
δ(_g_^{μν}Γ^{β}_{να}).
But
$$ \delta(g^{\mu\nu} \Gamma^{\beta}_{\nu\alpha}) = - \frac{1}{2}
\delta \begin{bmatrix}g^{\mu\nu} & g^{\beta\lambda}\end{bmatrix}
(\frac{\partial g_{\nu\lambda}}{\partial x_{\alpha}} +
\frac{\partial g_{\alpha\lambda}}{\partial x_{\nu}} - \frac{\partial
g_{\alpha\nu}}{\partial x_{\lambda}}) $$
The terms arising out of the two last terms within the round bracket are
of different signs, and change into one another by the interchange of
the indices μ and β. They cancel each other in the expression for δH,
when they are multiplied by Γ_{μβ}^{α}, which is symmetrical with
respect to μ and β, so that only the first member of the bracket remains
for our consideration. Remembering (31), we thus have:—
δH = -Γ_{μβ}^{α} Γ_{να}^{β} δ_g_^{μν} + Γ_{μβ}^{α} δ_g__{α}^{μβ}
Therefore
(48)
∂H/∂_g_^{μν} = -Γ_{μβ}^{α} Γ_{να}^{β}
∂H/∂_g__{σ}^{μν} = Γ_{μν}^{σ}
If we now carry out the variations in (47a), we obtain the system of
equations
(47b) ∂/∂_x__{α} ( ∂H/∂_g__{α}^{μν} ) - ∂H/∂_g_^{μν} = 0,
which, owing to the relations (48), coincide with (47), as was required
to be proved.
If (47b) is multiplied by _g__{σ}^{μν}, since
∂_g__{σ}^{μν}/∂_x__{α} = ∂_g__{α}^{μν}/∂_x__{σ}
and consequently
_g__{σ}^{μν} ∂/∂_x__{α} (∂H/∂_g__{α}^{μν}) = ∂/∂_x__{α}
(_g__{σ}^{μν} ∂H/∂_g__{α}^{μν})
- ∂H/∂_g__{α}^{μν} ∂_g__{α}^{μν}/∂_x__{σ}
we obtain the equation
∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_x__{σ} = 0
or
{ ∂_t__{σ}^α/∂_x__{α} = 0
(49) { -2κ_t__{σ}^{α} = _g__{σ}^{μν} ∂H/∂_g__{α}^{μν} - δ_{σ}^{α} H.
Owing to the relations (48), the equations (47) and (34),
(50) κ_t__{σ}^{α} = ½ δ_{σ}^{α} _g_^{μν} Γ_{μβ}^{α} Γ_{να}^{β}
- _g_^{μν} Γ_{μβ}^{α} Γ_{νσ}^{β}.
It is to be noticed that _t__{σ}^{α} is not a tensor, so that the
equation (49) holds only for systems for which √-_g_ = 1. This equation
expresses the laws of conservation of impulse and energy in a
gravitation-field. In fact, the integration of this equation over a
three-dimensional volume V leads to the four equations
(49a) _d_/_dx₄_ {∫_t__{σ}^4 _d_V} = ∫(_t__{σ}^1 α₁
+ _t__{σ}² α₂ + _t__{σ}³ α₃)_d_S
where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to
the surface-element _d_S in the Euclidean Sense. We recognise in this
the usual expression for the laws of conservation. We denote the
magnitudes _t_^α_{σ} as the energy-components of the gravitation-field.
I will now put the equation (47) in a third form which will be very
serviceable for a quick realisation of our object. By multiplying the
field-equations (47) with _g_^{νσ}, these are obtained in the mixed
forms. If we remember that
_g_^{νσ} ∂Γ^α_{μν}/∂_x__{α} = ∂/∂_x__{α} (_g_^{νσ} Γ^α_{μν}) -
∂_g_^{νσ}/∂_x__{α} Γ^α_{μν},
which owing to (34) is equal to
∂/∂_x__{α} (._g_^{νσ} Γ^α_{μν}) - _g_^{νβ} Γ^σ_{αβ} Γγ^α_{μν}
- _g_^{σβ} Γ^ν_{βα} Γ^α_{μν},
or slightly altering the notation, equal to
∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) - _g_^{mn} Γ^σ_{mβ} Γ^β_{_n_μ}
- _g_^{νσ} Γ^α_{μβ} Γ^β_{να}.
The third member of this expression cancels with the second member of
the field-equations (47). In place of the second term of this
expression, we can, on account of the relations (50), put
κ (_t_^σ_{μ} - ½ δ^σ_{μ} _t_), where _t_ = _t_^α_{α}
Therefore in the place of the equations (47), we obtain
(51) { ∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) = -κ(_t_^σ_{μ} - ½ δ^σ_{μ}
_t_)
{ √(-_g_) = 1.
§16. General formulation of the field-equation of Gravitation.
The field-equations established in the preceding paragraph for spaces
free from matter is to be compared with the equation ▽²φ = 0 of the
Newtonian theory. We have now to find the equations which will
correspond to Poisson’s Equation ▽²φ = 4πκρ (ρ signifies the density of
matter).
The special relativity theory has led to the conception that the
inertial mass (Träge Masse) is no other than energy. It can also be
fully expressed mathematically by a symmetrical tensor of the second
rank, the energy-tensor. We have therefore to introduce in our
generalised theory energy-tensor τ^α_{σ} associated with matter, which
like the energy components _t_^α_{σ} of the gravitation-field (equations
49, and 50) have a mixed character but which however can be connected
with symmetrical covariant tensors. The equation (51) teaches us how to
introduce the energy-tensor (corresponding to the density of Poisson’s
equation) in the field equations of gravitation. If we consider a
complete system (for example the Solar-system) its total mass, as also
its total gravitating action, will depend on the total energy of the
system, ponderable as well as gravitational. This can be expressed, by
putting in (51), in place of energy-components _t__{μ}^σ of
gravitation-field alone the sum of the energy-components of matter and
gravitation, _i.e._,
_t__{μ}^σ + T_{μ}^σ.
We thus get instead of (51), the tensor-equation
(52) $$ \frac{\partial}{\partial x_{\alpha}} (g^{\sigmaeta}
\Gamma^{lpha}_{\mu\beta}) = - \kappa [(t^{\sigma}_{\mu} +
T^{\sigma}_{\mu}) - \frac{1}{2} \delta^{\sigma}_{\mu} (t + T)] $$ $$
\sqrt{-g} = 1 $$
where T = T_{μ}^μ (Laue’s Scalar). These are the general field-equations
of gravitation in the mixed form. In place of (47), we get by working
backwards the system
(53) $$ \frac{\partial \Gamma^{lpha}_{\mu u}}{\partial x_{\alpha}} +
\Gamma^{lpha}_{\mu\beta} \Gamma^{eta}_{\nu\alpha} = - \kappa
(T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T) $$
$$ \sqrt{-g} = 1 $$
It must be admitted, that this introduction of the energy-tensor of
matter cannot be justified by means of the Relativity-Postulate alone;
for we have in the foregoing analysis deduced it from the condition that
the energy of the gravitation-field should exert gravitating action in
the same way as every other kind of energy. The strongest ground for the
choice of the above equation however lies in this, that they lead, as
their consequences, to equations expressing the conservation of the
components of total energy (the impulses and the energy) which exactly
correspond to the equations (49) and (49a). This shall be shown
afterwards.
§17. The laws of conservation in the general case.
The equations (52) can be easily so transformed that the second member
on the right-hand side vanishes. We reduce (52) with reference to the
indices μ and σ and subtract the equation so obtained after
multiplication with ½ δ_{μ}^σ from (52).
We obtain,
(52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α)
= -κ(_t__{μ}^σ + T_{μ}^σ)
we operate on it by ∂/∂_x__{σ}. Now,
∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α)
= -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β}
+ ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})].
The first and the third member of the round bracket lead to expressions
which cancel one another, as can be easily seen by interchanging the
summation-indices α, and σ, on the one hand, and β and λ, on the other.
The second term can be transformed according to (31). So that we get,
(54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α)
= ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
The second member of the expression on the left-hand side of (52a) leads
first to
- ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β}
+ ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].
The expression arising out of the last member within the round bracket
vanishes according to (29) on account of the choice of axes. The two
others can be taken together and give us on account of (31), the
expression
-½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}
So that remembering (54) we have
(55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α
- ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0.
identically.
From (55) and (52a) it follows that
(56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0
From the field equations of gravitation, it also follows that the
conservation-laws of impulse and energy are satisfied. We see it most
simply following the same reasoning which lead to equations (49a); only
instead of the energy-components of the gravitational-field, we are to
introduce the total energy-components of matter and gravitational field.
§18. The Impulse-energy law for matter as a consequence of the
field-equations.
If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to
§15, remembering that
_g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,
the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
or remembering (56)
(57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
A comparison with (41b) shows that these equations for the above choice
of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the
divergence of the tensor of the energy-components of matter.
Physically the appearance of the second term on the left-hand side shows
that for matter alone the law of conservation of impulse and energy
cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._,
when the field of gravitation vanishes. The second member is an
expression for impulse and energy which the gravitation-field exerts per
time and per volume upon matter. This comes out clearer when instead of
(57) we write it in the form of (47).
(57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β.
The right-hand side expresses the interaction of the energy of the
gravitational-field on matter. The field-equations of gravitation
contain thus at the same time 4 conditions which are to be satisfied by
all material phenomena. We get the equations of the material phenomena
completely when the latter is characterised by four other differential
equations independent of one another.
D. THE “MATERIAL” PHENOMENA.
The Mathematical auxiliaries developed under ‘B’ at once enables us to
generalise, according to the generalised theory of relativity, the
physical laws of matter (Hydrodynamics, Maxwell’s Electro-dynamics) as
they lie already formulated according to the special-relativity-theory.
The generalised Relativity Principle leads us to no further limitation
of possibilities; but it enables us to know exactly the influence of
gravitation on all processes without the introduction of any new
hypothesis.
It is owing to this, that as regards the physical nature of matter (in a
narrow sense) no definite necessary assumptions are to be introduced.
The question may lie open whether the theories of the electro-magnetic
field and the gravitational-field together, will form a sufficient basis
for the theory of matter. The general relativity postulate can teach us
no new principle. But by building up the theory it must be shown whether
electro-magnetism and gravitation together can achieve what the former
alone did not succeed in doing.
§19. Euler’s equations for frictionless adiabatic liquid.
Let _p_ and ρ, be two scalars, of which the first denotes the pressure
and the last the density of the fluid; between them there is a relation.
Let the contravariant symmetrical tensor
T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)
be the contra-variant energy-tensor of the liquid. To it also belongs
the covariant tensor
(58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
_dx__{β}/_ds_ ρ
as well as the mixed tensor
(58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_
ρ.
If we put the right-hand side of (58b) in (57a) we get the general
hydrodynamical equations of Euler according to the generalised
relativity theory. This in principle completely solves the problem of
motion; for the four equations (57a) together with the given equation
between _p_ and ρ, and the equation
_g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,
are sufficient, with the given values of _g__{αβ}, for finding out the
six unknowns
_p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.
If _g__{μν}’s are unknown we have also to take the equations (53). There
are now 11 equations for finding out 10 functions _g_, so that the
number is more than sufficient. Now it is be noticed that the equation
(57a) is already contained in (53), so that the latter only represents
(7) independent equations. This indefiniteness is due to the wide
freedom in the choice of co-ordinates, so that mathematically the
problem is indefinite in the sense that three of the space-functions can
be arbitrarily chosen.
§20. Maxwell’s Electro-Magnetic field-equations.
Let φ_{ν} be the components of a covariant four-vector, the
electro-magnetic potential; from it let us form according to (36) the
components F_{ρσ} of the covariant six-vector of the electro-magnetic
field according to the system of equations
(59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.
From (59), it follows that the system of equations
(60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0
is satisfied of which the left-hand side, according to (37), is an
anti-symmetrical tensor of the third kind. This system (60) contains
essentially four equations, which can be thus written:—
{ ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0
{
{ ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0
(60a) {
{ ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0
{
{ ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0.
This system of equations corresponds to the second system of equations
of Maxwell. We see it at once if we put
{ F₂₃ = H_{_x_} F₁₄ = E_{_x_}
{
(61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_}
{
{ F₁₂ = H_{_z_} F₃₄ = E_{_z_}
Instead of (60a) we can therefore write according to the usual notation
of three-dimensional vector-analysis:—
{ ∂H/∂_t_ + rot E = 0
(60b) {
{ div H = 0.
The first Maxwellian system is obtained by a generalisation of the form
given by Minkowski.
We introduce the contra-variant six-vector F_{αβ} by the equation
(62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ},
and also a contra-variant four-vector J^μ, which is the electrical
current-density in vacuum. Then remembering (40) we can establish the
system of equations, which remains invariant for any substitution with
determinant 1 (according to our choice of co-ordinates).
(63) ∂F^{μν}/∂_x__{ν} = J^μ
If we put
{ F²³ = H′_{_x_} F¹⁴ = -E′_{_x_}
{
(64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_}
{
{ F¹² = H′_{_z_} F³⁴ = -E′_{_z_}
which quantities become equal to H_{_x_} ... E_{_x_} in the case of the
special relativity theory, and besides
J^1 = _i__{_x_} ... J^4 = ρ
we get instead of (63)
{ rot H′ - ∂E′/∂_t_ = _i_
(63a) {
{ div E′ = ρ
The equations (60), (62) and (63) give thus a generalisation of
Maxwell’s field-equations in vacuum, which remains true in our chosen
system of co-ordinates.
_The energy-components of the electro-magnetic field._
Let us form the inner-product
(65) K_{σ} = F_{σμ} J^μ.
According to (61) its components can be written down in the
three-dimensional notation.
{ K₁ = ρE_{_x_} + [_i_, H]_{x}
(65a) { — — —
{ K₄ = — (_i_, E).
K_{σ} is a covariant four-vector whose components are equal to the
negative impulse and energy which are transferred to the
electro-magnetic field per unit of time, and per unit of volume, by the
electrical masses. If the electrical masses be free, that is, under the
influence of the electro-magnetic field only, then the covariant
four-vector K_{σ} will vanish.
In order to get the energy components T_{σ}^ν of the electro-magnetic
field, we require only to give to the equation K_{σ} = 0, the form of
the equation (57).
From (63) and (65) we get first,
K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}
= ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.
On account of (60) the second member on the right-hand side admits of
the transformation—
F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
= -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.
Owing to symmetry, this expression can also be written in the form
= -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}
+ _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],
which can also be put in the form
- 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})
+ 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).
The first of these terms can be written shortly as
- 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),
and the second after differentiation can be transformed in the form
- ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.
If we take all the three terms together, we get the relation
(66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
τ_{τ}^ν
where
(66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.
On account of (30) the equation (66) becomes equivalent to (57) and
(57a) when K_{σ} vanishes. Thus τ_{σ}^ν’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 √(-_g_) =
1. Thereby we achieve an important simplification in all our formulas
and calculations, without renouncing the conditions of general
covariance, as we have obtained the equations through a specialisation
of the co-ordinate system from the general covariant-equations. Still
the question is not without formal interest, whether, when the
energy-components of the gravitation-field and matter is defined in a
generalised manner without any specialisation of co-ordinates, the laws
of conservation have the form of the equation (56), and the
field-equations of gravitation hold in the form (52) or (52a); such that
on the left-hand side, we have a divergence in the usual sense, and on
the right-hand side, the sum of the energy-components of matter and
gravitation. I have found out that this is indeed the case. But I am of
opinion that the communication of my rather comprehensive work on this
subject will not pay, for nothing essentially new comes out of it.
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
_g__{μν}’s have constant values (4). This signifies, according to what
has been said before, a total neglect of the influence of gravitation.
We get one important approximation if we consider the case when
_g__{μν}’s differ from (4) only by small magnitudes (compared to 1)
where we can neglect small quantities of the second and higher orders
(first aspect of the approximation.)
Further it should be assumed that within the space-time region
considered, _g__{μν}’s at infinite distances (using the word infinite in
a spatial sense) can, by a suitable choice of co-ordinates, tend to the
limiting values (4); _i.e._, we consider only those gravitational fields
which can be regarded as produced by masses distributed over finite
regions.
We can assume that this approximation should lead to Newton’s theory.
For it however, it is necessary to treat the fundamental equations from
another point of view. Let us consider the motion of a particle
according to the equation (46). In the case of the special relativity
theory, the components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can take any values. This signifies that any velocity
_v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)
can appear which is less than the velocity of light in vacuum (_v_ < 1).
If we finally limit ourselves to the consideration of the case when _v_
is small compared to the velocity of light, it signifies that the
components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up
to the second-order magnitudes (the second point of view for
approximation).
Now we see that, according to the first view of approximation, the
magnitudes γ_{μν}^τ’s are all small quantities of at least the first
order. A glance at (46) will also show, that in this equation according
to the second view of approximation, we are only to take into account
those terms for which μ = ν = 4.
By limiting ourselves only to terms of the lowest order we get instead
of (46), first, the equations:—
_d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,
or by limiting ourselves only to those terms which according to the
first stand-point are approximations of the first order,
It must be admitted, that this introduction of the energy-tensor of
matter cannot be justified by means of the Relativity-Postulate alone;
for we have in the foregoing analysis deduced it from the condition that
the energy of the gravitation-field should exert gravitating action in
the same way as every other kind of energy. The strongest ground for the
choice of the above equation however lies in this, that they lead, as
their consequences, to equations expressing the conservation of the
components of total energy (the impulses and the energy) which exactly
correspond to the equations (49) and (49a). This shall be shown
afterwards.
§17. The laws of conservation in the general case.
The equations (52) can be easily so transformed that the second member
on the right-hand side vanishes. We reduce (52) with reference to the
indices μ and σ and subtract the equation so obtained after
multiplication with ½ δ_{μ}^σ from (52).
We obtain,
(52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α)
= -κ(_t__{μ}^σ + T_{μ}^σ)
we operate on it by ∂/∂_x__{σ}. Now,
∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α)
= -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β}
+ ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})].
The first and the third member of the round bracket lead to expressions
which cancel one another, as can be easily seen by interchanging the
summation-indices α, and σ, on the one hand, and β and λ, on the other.
The second term can be transformed according to (31). So that we get,
(54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α)
= ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
The second member of the expression on the left-hand side of (52a) leads
first to
- ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β}
+ ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].
The expression arising out of the last member within the round bracket
vanishes according to (29) on account of the choice of axes. The two
others can be taken together and give us on account of (31), the
expression
-½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}
So that remembering (54) we have
(55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α
- ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0.
identically.
From (55) and (52a) it follows that
(56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0
From the field equations of gravitation, it also follows that the
conservation-laws of impulse and energy are satisfied. We see it most
simply following the same reasoning which lead to equations (49a); only
instead of the energy-components of the gravitational-field, we are to
introduce the total energy-components of matter and gravitational field.
§18. The Impulse-energy law for matter as a consequence of the
field-equations.
If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to
§15, remembering that
_g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,
the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
or remembering (56)
(57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
A comparison with (41b) shows that these equations for the above choice
of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the
divergence of the tensor of the energy-components of matter.
Physically the appearance of the second term on the left-hand side shows
that for matter alone the law of conservation of impulse and energy
cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._,
when the field of gravitation vanishes. The second member is an
expression for impulse and energy which the gravitation-field exerts per
time and per volume upon matter. This comes out clearer when instead of
(57) we write it in the form of (47).
(57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β.
The right-hand side expresses the interaction of the energy of the
gravitational-field on matter. The field-equations of gravitation
contain thus at the same time 4 conditions which are to be satisfied by
all material phenomena. We get the equations of the material phenomena
completely when the latter is characterised by four other differential
equations independent of one another.
D. THE “MATERIAL” PHENOMENA.
The Mathematical auxiliaries developed under ‘B’ at once enables us to
generalise, according to the generalised theory of relativity, the
physical laws of matter (Hydrodynamics, Maxwell’s Electro-dynamics) as
they lie already formulated according to the special-relativity-theory.
The generalised Relativity Principle leads us to no further limitation
of possibilities; but it enables us to know exactly the influence of
gravitation on all processes without the introduction of any new
hypothesis.
It is owing to this, that as regards the physical nature of matter (in a
narrow sense) no definite necessary assumptions are to be introduced.
The question may lie open whether the theories of the electro-magnetic
field and the gravitational-field together, will form a sufficient basis
for the theory of matter. The general relativity postulate can teach us
no new principle. But by building up the theory it must be shown whether
electro-magnetism and gravitation together can achieve what the former
alone did not succeed in doing.
§19. Euler’s equations for frictionless adiabatic liquid.
Let _p_ and ρ, be two scalars, of which the first denotes the pressure
and the last the density of the fluid; between them there is a relation.
Let the contravariant symmetrical tensor
T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)
be the contra-variant energy-tensor of the liquid. To it also belongs
the covariant tensor
(58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
_dx__{β}/_ds_ ρ
as well as the mixed tensor
(58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_
ρ.
If we put the right-hand side of (58b) in (57a) we get the general
hydrodynamical equations of Euler according to the generalised
relativity theory. This in principle completely solves the problem of
motion; for the four equations (57a) together with the given equation
between _p_ and ρ, and the equation
_g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,
are sufficient, with the given values of _g__{αβ}, for finding out the
six unknowns
_p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.
If _g__{μν}’s are unknown we have also to take the equations (53). There
are now 11 equations for finding out 10 functions _g_, so that the
number is more than sufficient. Now it is be noticed that the equation
(57a) is already contained in (53), so that the latter only represents
(7) independent equations. This indefiniteness is due to the wide
freedom in the choice of co-ordinates, so that mathematically the
problem is indefinite in the sense that three of the space-functions can
be arbitrarily chosen.
§20. Maxwell’s Electro-Magnetic field-equations.
Let φ_{ν} be the components of a covariant four-vector, the
electro-magnetic potential; from it let us form according to (36) the
components F_{ρσ} of the covariant six-vector of the electro-magnetic
field according to the system of equations
(59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.
From (59), it follows that the system of equations
(60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0
is satisfied of which the left-hand side, according to (37), is an
anti-symmetrical tensor of the third kind. This system (60) contains
essentially four equations, which can be thus written:—
{ ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0
{
{ ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0
(60a) {
{ ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0
{
{ ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0.
This system of equations corresponds to the second system of equations
of Maxwell. We see it at once if we put
{ F₂₃ = H_{_x_} F₁₄ = E_{_x_}
{
(61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_}
{
{ F₁₂ = H_{_z_} F₃₄ = E_{_z_}
Instead of (60a) we can therefore write according to the usual notation
of three-dimensional vector-analysis:—
{ ∂H/∂_t_ + rot E = 0
(60b) {
{ div H = 0.
The first Maxwellian system is obtained by a generalisation of the form
given by Minkowski.
We introduce the contra-variant six-vector F_{αβ} by the equation
(62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ},
and also a contra-variant four-vector J^μ, which is the electrical
current-density in vacuum. Then remembering (40) we can establish the
system of equations, which remains invariant for any substitution with
determinant 1 (according to our choice of co-ordinates).
(63) ∂F^{μν}/∂_x__{ν} = J^μ
If we put
{ F²³ = H′_{_x_} F¹⁴ = -E′_{_x_}
{
(64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_}
{
{ F¹² = H′_{_z_} F³⁴ = -E′_{_z_}
which quantities become equal to H_{_x_} ... E_{_x_} in the case of the
special relativity theory, and besides
J^1 = _i__{_x_} ... J^4 = ρ
we get instead of (63)
{ rot H′ - ∂E′/∂_t_ = _i_
(63a) {
{ div E′ = ρ
The equations (60), (62) and (63) give thus a generalisation of
Maxwell’s field-equations in vacuum, which remains true in our chosen
system of co-ordinates.
_The energy-components of the electro-magnetic field._
Let us form the inner-product
(65) K_{σ} = F_{σμ} J^μ.
According to (61) its components can be written down in the
three-dimensional notation.
{ K₁ = ρE_{_x_} + [_i_, H]_{x}
(65a) { — — —
{ K₄ = — (_i_, E).
K_{σ} is a covariant four-vector whose components are equal to the
negative impulse and energy which are transferred to the
electro-magnetic field per unit of time, and per unit of volume, by the
electrical masses. If the electrical masses be free, that is, under the
influence of the electro-magnetic field only, then the covariant
four-vector K_{σ} will vanish.
In order to get the energy components T_{σ}^ν of the electro-magnetic
field, we require only to give to the equation K_{σ} = 0, the form of
the equation (57).
From (63) and (65) we get first,
K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}
= ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.
On account of (60) the second member on the right-hand side admits of
the transformation—
F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
= -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.
Owing to symmetry, this expression can also be written in the form
= -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}
+ _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],
which can also be put in the form
- 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})
+ 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).
The first of these terms can be written shortly as
- 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),
and the second after differentiation can be transformed in the form
- ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.
If we take all the three terms together, we get the relation
(66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
τ_{τ}^ν
where
(66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.
On account of (30) the equation (66) becomes equivalent to (57) and
(57a) when K_{σ} vanishes. Thus τ_{σ}^ν’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 √(-_g_) =
1. Thereby we achieve an important simplification in all our formulas
and calculations, without renouncing the conditions of general
covariance, as we have obtained the equations through a specialisation
of the co-ordinate system from the general covariant-equations. Still
the question is not without formal interest, whether, when the
energy-components of the gravitation-field and matter is defined in a
generalised manner without any specialisation of co-ordinates, the laws
of conservation have the form of the equation (56), and the
field-equations of gravitation hold in the form (52) or (52a); such that
on the left-hand side, we have a divergence in the usual sense, and on
the right-hand side, the sum of the energy-components of matter and
gravitation. I have found out that this is indeed the case. But I am of
opinion that the communication of my rather comprehensive work on this
subject will not pay, for nothing essentially new comes out of it.
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
_g__{μν}’s have constant values (4). This signifies, according to what
has been said before, a total neglect of the influence of gravitation.
We get one important approximation if we consider the case when
_g__{μν}’s differ from (4) only by small magnitudes (compared to 1)
where we can neglect small quantities of the second and higher orders
(first aspect of the approximation.)
Further it should be assumed that within the space-time region
considered, _g__{μν}’s at infinite distances (using the word infinite in
a spatial sense) can, by a suitable choice of co-ordinates, tend to the
limiting values (4); _i.e._, we consider only those gravitational fields
which can be regarded as produced by masses distributed over finite
regions.
We can assume that this approximation should lead to Newton’s theory.
For it however, it is necessary to treat the fundamental equations from
another point of view. Let us consider the motion of a particle
according to the equation (46). In the case of the special relativity
theory, the components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can take any values. This signifies that any velocity
_v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)
can appear which is less than the velocity of light in vacuum (_v_ < 1).
If we finally limit ourselves to the consideration of the case when _v_
is small compared to the velocity of light, it signifies that the
components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up
to the second-order magnitudes (the second point of view for
approximation).
Now we see that, according to the first view of approximation, the
magnitudes γ_{μν}^τ’s are all small quantities of at least the first
order. A glance at (46) will also show, that in this equation according
to the second view of approximation, we are only to take into account
those terms for which μ = ν = 4.
By limiting ourselves only to terms of the lowest order we get instead
of (46), first, the equations:—
_d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,
or by limiting ourselves only to those terms which according to the
first stand-point are approximations of the first order,
$$ \frac{d^2 x_{\tau}}{dt^2} = \begin{bmatrix}44\\\tau\end{bmatrix} $$
(\tau = 1, 2, 3)
$$ \frac{d^2 x_{4}}{dt^2} = - \begin{bmatrix}4^4\\4\end{bmatrix] $$
If we further assume that the gravitation-field is quasi-static, _i.e._,
it is limited only to the case when the matter producing the
gravitation-field is moving slowly (relative to the velocity of light)
we can neglect the differentiations of the positional co-ordinates on
the right-hand side with respect to time, so that we get
(67) _d²__x__{τ}/_dt²_ = -½ ∂_g₄₄_/∂_x__{τ} (τ, = 1, 2, 3)
This is the equation of motion of a material point according to Newton’s
theory, where _g_₄₄/₂ plays the part of gravitational potential. The
remarkable thing in the result is that in the first-approximation of
motion of the material point, only the component _g₄₄_ of the
fundamental tensor appears.
Let us now turn to the field-equation (53). In this case, we have to
remember that the energy-tensor of matter is exclusively defined in a
narrow sense by the density ρ of matter, _i.e._, by the second member on
the right-hand side of 58 [(58a, or 58b)]. If we make the necessary
approximations, then all component vanish except
τ₄₄ = ρ = τ.
On the left-hand side of (53) the second term is an infinitesimal of the
second order, so that the first leads to the following terms in the
approximation, which are rather interesting for us:
$$ \frac{\partial}{\partial x_{1}} \begin{bmatrix}\mu\nu\\1\end{bmatrix}
+ \frac{\partial}{\partial x_{2}} \begin{bmatrix}\mu\nu\\2\end{bmatrix}
+ \frac{\partial}{\partial x_{3}} \begin{bmatrix}\mu\nu\\3\end{bmatrix}
+ \frac{\partial}{\partial x_{4}} \begin{bmatrix}\mu\nu\\4\end{bmatrix}
$$
By neglecting all differentiations with regard to time, this leads, when
μ = ν =4, to the expression
$$ - \frac{1}{2} ( \frac{\partial^2 g_{44}}{\partial x^2_{1}} +
\frac{\partial^2 g_{44}}{\partial x^2_{2}} + \frac{\partial^2
g_{44}}{\partial x^2_{3}} ) = - \frac{1}{2} V^2 g_{44} $$
The last of the equations (53) thus leads to
(68) ▽² _g₄₄_ = κρ.
The equations (67) and (68) together, are equivalent to Newton’s law of
gravitation.
For the gravitation-potential we get from (67) and (68) the exp.
(68a.) -κ/(8π) ∫ ρ_d_τ/_r_
whereas the Newtonian theory for the chosen unit of time gives
-K/_c²_ ∫ρ_d_τ/_r_,
where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating
them we get
(69) κ = 8πK/_c²_ = 1.87 x 10⁻²⁷.
§22. Behaviour of measuring rods and clocks in a statical
gravitation-field. Curvature of light-rays. Perihelion-motion of the
paths of the Planets.
In order to obtain Newton’s theory as a first approximation we had to
calculate only _g₄₄_, out of the 10 components _g__{μν} of the
gravitation-potential, for that is the only component which comes in the
first approximate equations of motion of a material point in a
gravitational field.
We see however, that the other components of _g__{μν} should also differ
from the values given in (4) as required by the condition _g_ = -1.
For a heavy particle at the origin of co-ordinates and generating the
gravitational field, we get as a first approximation the symmetrical
solution of the equation:—
{ _g__{ρσ} = -δ_{ρσ} - α(_x__{ρ} _x__{σ})/_r³_ (ρ and σ 1, 2,
3)
{
(70) { _g__{ρ4} = _g__{4ρ} = 0 (ρ 1, 2, 3)
{
{ _g₄₄_ = 1 - α/_r_.
δ_{ρσ} is 1 or 0, according as ρ = σ or not and _r_ is the quantity
+√(_x₁²_ + _x₂²_ + _x₃²_).
On account of (68a) we have
(70a) α = κM/4π
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 of mass M will
have upon the metrical properties of the field. Between the lengths and
times measured locally on the one hand, and the differences in
co-ordinates _dx__{ν} on the other, we have the relation
_ds²_ = _g__{μν} _dx__{μ} _dx__{ν}.
For a unit measuring rod, for example, placed parallel to the _x_ axis,
we have to put
_ds²_ = -1, _dx₂_ = _dx₃_ = _dx₄_ = 0
then -1 = _g_₁₁_dx₁²_.
If the unit measuring rod lies on the _x_ axis, the first of the
equations (70) gives
_g₁₁_ = -(1 + α/_r_).
From both these relations it follows as a first approximation that
(71) _dx_ = 1 - α/2_r_.
The unit measuring rod appears, when referred to the co-ordinate-system,
shortened by the calculated magnitude through the presence of the
gravitational field, when we place it radially in the field.
Similarly we can get its co-ordinate-length in a tangential position, if
we put for example
_ds²_ = -1, _dx₁_ = _dx₃_ = _dx₄_ = 0, _x₁_ = _r_, _x₂_ = _x₃_ = 0
we then get
(71a) -1 = _g₂₂_ _dx₂²_ = -_dx₂²_.
The gravitational field has no influence upon the length of the rod,
when we put it tangentially in the field.
Thus Euclidean geometry does not hold in the gravitational field even in
the first approximation, if we conceive that one and the same rod
independent of its position and its orientation can serve as the measure
of the same extension. But a glance at (70a) and (69) shows that the
expected difference is much too small to be noticeable in the
measurement of earth’s surface.
We would further investigate the rate of going of a unit-clock which is
placed in a statical gravitational field. Here we have for a period of
the clock
_ds_ = 1, _dx₁_ = _dx₂_ _dx₃_ = 0;
then we have
1 = _g₄₄__dx₄²_
_dx₄_ = 1/√(_g_₄₄) = 1/√(1 + (_g_₄₄ - 1)) = 1 - (_g_₄₄ - 1)/2
or _dx₄_ = 1 + _k_/8π ∫ ρ_d_τ/_r_.
Therefore the clock goes slowly what it is placed in the neighbourhood
of ponderable masses. It follows from this that the spectral lines in
the light coming to us from the surfaces of big stars should appear
shifted towards the red end of the spectrum.
Let us further investigate the path of light-rays in a statical
gravitational field. According to the special relativity theory, the
velocity of light is given by the equation
-_dx₁²_ - _dx₂²_ - _dx₃²_ + _dx₄²_ = 0;
thus also according to the generalised relativity theory it is given by
the equation
(73) _ds²_ = _g__{μν} _dx__{μ} _dx__{ν} = 0.
If the direction, _i.e._, the ratio _dx₁_ : _dx₂_ : _dx₃_ is given, the
equation (73) gives the magnitudes
_dx₁_/_dx₄_, _dx₂_/_dx₄_, _dx₃_/_dx₄_,
and with it the velocity,
√((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²) = γ,
in the sense of the Euclidean Geometry. We can easily see that, with
reference to the co-ordinate system, the rays of light must appear
curved in case _g__{μν}’s are not constants. If _n_ be the direction
perpendicular to the direction of propagation, we have, from Huygen’s
principle, that light-rays (taken in the plane (γ, _n_)] must suffer a
curvature ∂λ/∂_n_.
Let us find out the curvature which a light-ray suffers when it goes by
a mass M at a distance Δ from it. If we use the co-ordinate system
according to the above scheme, then the total bending B of light-rays
(reckoned positive when it is concave to the origin) is given as a
sufficient approximation by
B = ∫_{-∞}^∞ ∂γ/∂[_x_]₁ _dx₂_
where (73) and (70) gives
γ = √(-_g₄₄_/_g₂₂_) = 1 - α/2_r_ (1 + _x₂²_/_r²_).
The calculation gives
B = 2α/Δ = KM/2πΔ.
A ray of light just grazing the sun would suffer a bending of 1·7″,
whereas one coming by Jupiter would have a deviation of about ·02″.
If we calculate the gravitation-field to a greater order of
approximation and with it the corresponding path of a material particle
of a relatively small (infinitesimal) mass we get a deviation of the
following kind from the Kepler-Newtonian Laws of Planetary motion. The
Ellipse of Planetary motion suffers a slow rotation in the direction of
motion, of amount
(75) _s_ = 24π³_a²_/τ²_c²_(1 - _e²_) per revolution.
In this Formula ‘_a_’ signifies the semi-major axis, _c_, the velocity
of light, measured in the usual way, _e_, the eccentricity, τ, the time
of revolution in seconds.
The calculation gives for the planet Mercury, a rotation of path of
amount 43″ per century, corresponding sufficiently to what has been
found by astronomers (Leverrier). They found a residual perihelion
motion of this planet of the given magnitude which can not be explained
by the perturbation of the other planets.
NOTES
Note 1.
The fundamental electro-magnetic equations of Maxwell for stationary
media are:—
curl H = 1/_c_ (∂D/∂_t_ + ρν) (1)
curl E = -1/_c_ ∂B/∂_t_ (2)
div D = ρ
B = μH
div B = 0
D = kE
According to Hertz and Heaviside, these require modification in the case
of moving bodies.
Now it is known that due to motion alone there is a change in a vector
_R_ given by
(∂_R_/∂_t_) due to motion = _u_. div R + curl [_Ru_]
where _u_ is the vector velocity of the moving body and [R_u_] the
vector product of R and _u_.
Hence equations (1) and (2) become
_c_ curl H = ∂D/∂_t_ + _u_ div D + curl Vect. [D_u_] + ρν (1·1)
and
-_c_ curl E = ∂B/∂_t_ + _u_ div B + curl Vect. [B_u_] (2·1)
which gives finally, for ρ = 0 and div B = 0,
∂D/∂_t_ + _u_ div D = _c_ curl (H - 1/_c_ Vect. [D_u_]) (1·2)
∂B/∂_t_ = -_c_ curl (E - 1/_c_ Vect. [_u_B]) (2·2)
Let us consider a beam travelling along the _x_-axis, with apparent
velocity _v_ (_i.e._, velocity with respect to the fixed ether) in
medium moving with velocity _u__{_x_} = _u_ in the same direction.
Then if the electric and magnetic vectors are proportional to
_e_^{_i_A(_x_ - _vt_)}, we have
∂/∂_x_ = _i_A, ∂/∂_t_ = -_i_A_v_, ∂/∂_y_ = ∂/∂_z_ = 0, _u__{_y_} =
_u__{_z_} = 0
Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)
and ∂B_{_z_}/∂_t_ = -_c_∂E_{_y_}/∂_x_ - _u_∂B_{_z_}/∂_x_ (2·21)
Since D = KE and B = μH, we have
_i_A_v_(κE_y_) = -_ci_A(H_{_z_} + _u_KE_{_y_}) (1·22)
_i_A_v_(μH_{_z_}) = -_ci_A(E_{_y_} + _u_μH_{_z_}) (2·22)
or _v_(K - _u_)E_{_y_} = _c_H_{_z_} (1·23)
μ(_v_ - _u_)H_{_z_} = _c_E_{_y_} (2·23)
Multiplying (1·23) by (2·23)
μK(_v_ - _u_)² = _c²_
Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²
∴ _v_ = _v₀_ + _u_,
making Fresnelian convection co-efficient simply unity.
Equations (1·21) and (2·21) may be 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 fixed with respect to the ether, the rate of change of electric
polarisation is δD/δ_t_.
Consider a slab of matter moving with velocity _u__{_x_} along the
_x_-axis, then even in a stationary field of electrostatic polarisation,
that is, for a field in which δD/δ_t_ = 0, there will be some change in
the polarisation of the body due to its motion, given by
_u__{_x_}(δD/δ_x_). Hence we must add this term to a purely temporal
rate of change δD/δ_t_. Doing this we immediately arrive at equations
(1·21) and (2·21) for the special case considered there.
Thus the Hertz-Heaviside form of field equations gives _unity_ as the
value for the Fresnelian convection co-efficient. It has been shown in
the historical introduction how this is entirely at variance with the
observed optical facts. As a matter of fact, Larmor has shown (Aether
and Matter) that 1 - 1/μ² is not only sufficient but is also necessary,
in order to explain experiments of the Arago prism type.
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 polarisation is situated in
the medium itself and is completely carried away by it. Thus the
electromagnetic effect outside a moving medium should be proportional to
K, the specific inductive capacity.
_Rowland_ showed in 1876 that when a charged condenser is rapidly
rotated (the dielectric remaining stationary), the magnetic effect
outside is proportional to K, the Sp. Ind. Cap.
_Röntgen_ (Annalen der Physik 1888, 1890) found that if the dielectric
is rotated while the condenser remains stationary, the effect is
proportional to K - 1.
_Eichenwald_ (Annalen der Physik 1903, 1904) rotated together both
condenser and dielectric and found that the magnetic effect was
proportional to the potential difference and to the angular velocity,
but was completely independent of K. This is of course quite consistent
with Rowland and Röntgen.
_Blondlot_ (Comptes Rendus, 1901) passed a current of air in a steady
magnetic field H_{_y_}, (H = H_{_z_} = 0). If this current of air moves
with velocity _u__{_x_} along the _x_-axis, an electromotive force would
be set up along the _z_-axis, due to the relative motion of matter and
magnetic tubes of induction. A pair of plates at _z_ = ±_a_, will be
charged up with density ρ = D_{_z_} = KE = K. _u__{_s_} H_{_y_}/c. But
Blondlot failed to detect any such effect.
_H. A. Wilson_ (Phil. Trans. Royal Soc. 1904) repeated the experiment
with a cylindrical condenser made of ebony, rotating in a magnetic field
parallel to its own axis. He observed a change proportional to K — 1 and
not to K.
Thus the above set of electro-magnetic experiments contradict the
Hertz-Heaviside equations, and these must be abandoned.
[P. C. M.]
Note 2.
Lorentz Transformation.
Lorentz. Versuch einer theorie der elektrischen und optischen
Erscheinungen im bewegten Körpern.
(Leiden—1895).
Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also
notes 73, 86, pages 318, 328.
Lorentz wanted to explain the Michelson-Morley null-effect. In order to
do so, it was obviously necessary to explain the Fitzgerald contraction.
Lorentz worked on the hypothesis that an electron itself undergoes
contraction when moving. He introduced new variables for the moving
system defined by the following set of equations.
_x¹_ = β(_x_ - _ut_), _y¹_ = _y_, _z¹_ = _z_, _t¹_ = β(_t_ -
(_u_/_c²_)·_x_)
and for velocities, used
_v__{_x_}¹ = β²_v__{_x_} + _u_, _v__{_y_}¹ = β_v__{_y_}, _v__{_z_}¹
= β_v__{_z_} and ρ¹ = ρ/β.
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 Einsteinian addition of velocities
is quite different as also the expression for the “relative” density of
electricity.
It is true that the Maxwell-Lorentz field equations remain _practically_
unchanged by the Lorentz transformation, but they _are_ changed to some
slight extent. One marked advantage of the Einstein transformation
consists in the fact that the field equations of a moving system
preserve _exactly_ the same form as those of a stationary system.
It should also be noted that the Fresnelian convection coefficient comes
out in the theory of relativity as a direct consequence of Einstein’s
addition of velocities and is quite independent of any electrical theory
of matter.
[P. C. M.]
Note 3.
See Lorentz, Theory of Electrons (English edition), § 181, page 213.
H. Poincare, Sur la dynamique ‘electron, Rendiconti del circolo
matematico di Palermo 21 (1906).
[P. C. M.]
Note 4.
Relativity Theorem and Relativity-Principle.
Lorentz showed that the Maxwell-Lorentz system of electromagnetic
field-equations remained practically unchanged by the Lorentz
transformation. Thus the electromagnetic laws of Maxwell and Lorentz
_can be definitely proved_ “to be independent of the manner in which
they are referred to two coordinate systems which have a uniform
translatory motion relative to each other.” (See “Electrodynamics of
Moving Bodies,” page 5.) Thus so far as the electromagnetic laws are
concerned, the principle of relativity _can be proved to be true_.
But it is not known whether this principle will remain true in the case
of other physical laws. We can always proceed on the assumption that it
does remain true. Thus it is always possible to construct physical laws
in such a way that they retain their form when referred to moving
coordinates. The ultimate ground for formulating physical laws in this
way is merely a subjective conviction that the principle of relativity
is universally true. There is no _a priori_ logical necessity that it
should be so. Hence the Principle of Relativity (so far as it is applied
to phenomena other than electromagnetic) must be regarded as a
_postulate_, which we have assumed to be true, but for which we cannot
adduce any definite proof, until after the generalisation is made and
its consequences tested in the light of actual experience.
[P. C. M.]
Note 5.
See “Electrodynamics of Moving Bodies,” p. 5-8.
Note 6.
Field Equations in Minkowski’s Form.
Equations (_i_) and (_ii_) become when expanded into Cartesians:—
∂_m__{_z_}/∂_y_ - ∂_m__{_y_}/∂_z_ - ∂_e__{_x_}/∂τ = ρν_{_x_} }
∂_m__{_x_}/∂_z_ - ∂_m__{_z_}/∂_x_ - ∂_e__{_y_}/∂τ = ρν_{_y_} } ...
(1·1)
∂_m__{_y_}/∂_x_ - ∂_m__{_x_}/∂_y_ - ∂_e__{_z_}/∂τ = ρν_{_z_} }
and ∂_e__{_x_}/∂_x_ + ∂_e__{_y_}/∂_y_ + ∂_e__{_z_}/∂_z_ = ρ (2·1)
Substituting _x₁_, _x₂_, _x₃_, _x₄_ and _x_, _y_, _z_, and _i_τ; and ρ₁,
ρ₂, ρ₃, ρ₄ for ρν_{_x_}, ρν_{_y_}, ρν_{_z_}, _i_ρ, where _i_ = √(-1).
We get,
∂_m__{_z_}/∂_x₂_ - ∂_m__{_y_}/∂_x₃_ - _i_(∂_e__{_x_}/∂_x₄_) =
ρν_{_x_}{ = ρ₁ }
- ∂_m__{_z_}/∂_x₁_ + ∂_m__{_x_}/∂_x₃_ - _i_(∂_e__{_y_}/∂_x₄_) =
ρν_{_y_} = ρ₂ } ... (1·2)
∂_m__{_y_}/∂_x₁_ - ∂_m__{_x_}/∂_x₂_ - _i_(∂_e__{_z_}/∂_x₄_) =
ρν_{_z_}{ = ρ₃ }
and multiplying (2·1) by i we get
∂_ie__{_x_}/∂_x₁_ + ∂_ie__{_y_}/∂_x₂_ + ∂_ie__{_z_}/∂_x₃_ = _i_ρ =
ρ₄ ... ... (2·2)
Now substitute
_m__{_x_} = _f₂₃_ = -_f₃₂_ and _ie__{_x_} = _f_₄₁ = -_f₁₄_
_m__{_y_} = _f₃₁_ = -_f₁₃_ _ie__{_y_} = _f_₄₂ = -_f₂₄_
_m__{_z_} = _f₁₂_ = -_f₂₁_ _ie__{_z_} = _f_₄₃ = -_f₃₄_
and we get finally:—
∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = ρ₁ }
∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = ρ₂ } ... (3)
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_ = ρ₃ }
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = ρ₄ }
Note 9.
On the Constancy of the Velocity of Light.
Page 12—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
waves expanding outwards with velocity _c_.
At first sight, it may not appear clear why the velocity should remain
constant. Indeed according to the theory of Ritz, the velocity should
become _c_ + _u_, when the source of light moves towards the observer
with the velocity _u_.
Prof. de Sitter has given an astronomical argument for deciding between
these two divergent views. Let us suppose there is a double star of
which one is revolving about the common centre of gravity in a circular
orbit. Let the observer be in the plane of the orbit, at a great
distance Δ.
[Illustration.]
The light emitted by the star when at the position A will be received by
the observer after a time, Δ/(_c_ + _u_) while the light emitted by the
star when at the position B will be received after a time Δ/(_c_ - _u_).
Let T be the real half-period of the star. Then the observed half-period
from B to A is approximately T - 2Δ_u_/_c²_ and from A to B is T +
2Δ_u_/_c²_. Now if 2_u_Δ/_c²_ be comparable to T, then it is impossible
that the observations should satisfy Kepler’s Law. In most of the
spectroscopic binary stars, 2_u_Δ/_c²_ are not only of the same order as
T, but are mostly much larger. For example, if _u_ = 100 _km_/sec, T = 8
days, Δ/_c_ = 33 years (corresponding to an annual parallax of ·1″),
then T - 2_u_Δ/_c²_ = 0. The existence of the Spectroscopic binaries,
and the fact that they follow Kepler’s Law is therefore a proof that _c_
is not affected by the motion of the source.
In a later memoir, replying to the criticisms of Freundlich and Günthick
that an apparent eccentricity occurs in the motion proportional to
_ku_Δ₀, _u₀_ being the maximum value of _u_, the velocity of light
emitted being
_u₀_ = _c_ + _ku_,
_k_ = 0 Lorentz-Einstein
_k_ = 1 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 star β-Aurigae. For this star, the parallax π = ·014″, _e_ =
·005, _u₀_ = 110 _km_/sec, T = 3·96,
Δ > 65 light-years,
_k_ is < ·002.
For an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol.
35, p. 163.
[M. N. S.]
Note 10.
Rest-density of Electricity.
If ρ is the volume density in a moving system then ρ√(1 - _u²_) is the
corresponding quantity in the corresponding volume in the fixed system,
that is, in the system at rest, and hence it is termed the rest-density
of electricity.
[P. C. M.]
Note 11
(page 17)
Space-time vectors of the first and the second kind.
As we had already occasion to mention, Sommerfeld has, in two papers on
four dimensional geometry (_vide_, Annalen der Physik, Bd. 32, p. 749;
and Bd. 33, p. 649), translated the ideas of Minkowski into the language
of four dimensional geometry. Instead of Minkowski’s space-time vector
of the first kind, he uses the more expressive term ‘four-vector,’
thereby making it quite clear that it represents a directed quantity
like a straight line, a force or a momentum, and has got 4 components,
three in the direction of space-axes, and one in the direction of the
time-axis.
The representation of the plane (defined by two straight lines) is much
more difficult. In three dimensions, the plane can be represented by the
vector perpendicular to itself. But that artifice is not available in
four dimensions. For the perpendicular to a plane, we now have not a
single line, but an infinite number of lines constituting a plane. This
difficulty has been overcome by Minkowski in a very elegant manner which
will become clear later on. Meanwhile we offer the following extract
from the above mentioned work of Sommerfeld.
(Pp. 755, Bd. 32, Ann. d. Physik.)
“In order to have a better knowledge about the nature of the six-vector
(which is the same thing as Minkowski’s space-time vector of the _2nd_
kind) let us take the special case of a piece of plane, having unit area
(contents), and the form of a parallelogram, bounded by the four-vectors
_u_, _v_, passing through the origin. Then the projection of this piece
of plane on the _xy_ plane is given by the projections _u__{_x_},
_u__{_y_}, _v__{_x_}, _v__{_y_} of the four vectors in the combination
φ_{_x_ _y_} = _u__{_x_}_v__{_y_} - _u__{_y_}_v_{_x_}.
Let us form in a similar manner all the six components of this plane φ.
Then six components are not all independent but are connected by the
following relation
φ_{_y_ _z_} φ_{_x_ _l_} + φ_{_z_ _x_} φ_{_y_ _l_} + φ_{_x_ _y_}
φ_{_z_ _l_} = 0
Further the contents | φ | of the piece of a plane is to be defined as
the square root of the sum of the squares of these six quantities. In
fact,
| φ |² = φ_{_y_ _z_}² + φ_{_z_ _x_}² + φ_{_x_ _y_}² + φ_{_x_ _l_}² +
φ_{_y_ _l_}² + φ_{_z_ _l_}².
Let us now on the other hand take the case of the unit plane φ^* normal
to φ; we can call this plane the Complement of φ. Then we have the
following relations between the components of the two plane:—
φ_{_y_ _z_}^* = φ_{_x_ _l_}, φ_{_z_ _x_}^* = φ_{_y_ _l_}, φ_{_x_
_y_}^* = φ_{_z_ _l_} φ_{_z_ _l_}^* = φ_{_y_ _x_} ...
The proof of these assertions is as follows. Let _u_^*, _v_^* be the
four vectors defining φ^*. Then we have the following relations:—
_u__{_x_}^* _u__{_x_} + _u__{_y_}^* _u__{_y_} + _u__{_z_}^*
_u__{_z_} + _u__{_l_}^* _u__{_l_} = 0
_u__{_x_}^* _v__{_x_} + _u__{_y_}^* _v__{_y_} + _u__{_z_}^*
_v__{_z_} + _u__{_l_}^* _v__{_l_} = 0
_v__{_x_}^* _u__{_x_} + _v__{_y_}^* _u__{_y_} + _v__{_z_}^*
_u__{_z_} + _v__{_l_}^* _u__{_l_} = 0
_v__{_x_}^* _v__{_x_} + _v__{_y_}^* _v__{_y_} + _v__{_z_}^*
_v__{_z_} + _v__{_l_}^* _v__{_l_} = 0
If we multiply these equations by _v__{_l_}, _u__{_l_}, _v__{_s_}, and
subtract the second from the first, the fourth from the third we obtain
_u__{_x_}^* φ_{_x_ _l_} + _u__{_y_}^* φ_{_y_ _l_} + _u__{_z_}^*
φ_{_z_ _l_} = 0
_v__{_x_}^* φ_{_z_ _l_} + _v__{_y_}^* φ_{_y_ _l_} + _v__{_z_}^*
φ_{_z_ _l_} = 0
multiplying these equations by _v__{_x_}^* . _u__{_x_}^*, or by
_v__{_y_}^* . _u__{_y_}^*, we obtain
φ_{_x_ _z_}^* φ_{_x_ _l_} + φ_{_y_ _z_}^* φ_{_y_ _l_} = 0 and φ_{_x_
_y_}^* φ_{_x_ _l_} + φ_{_z_ _x_}^* φ_{_z_ _l_} = 0
from which we have
φ_{_y_ _z_}^* : φ_{_x_ _y_}^* : φ_{_z_ _x_}^* = φ_{_x_ _l_} : φ_{_z_
_l_} : φ_{_y_ _l_}
In a corresponding way we have
φ_{_y_ _z_} : φ_{_x_ _y_} : φ_{_z_ _x_} = φ_{_x_ _l_}^* : φ_{_z_
_l_}^* : φ_{_y_ _l_}^*.
_i.e._ φ_{_i_ _k_}^* = λφ(_{_i_ _k_})
when the subscript (_ik_) denotes the component of φ in the plane
contained by the lines other than (_ik_). Therefore the theorem is
proved.
We have (φ φ*) = φ_{_y_ _z_} φ_{_y_ _z_}^* + ...
= 2 (φ_{_y_ _z_} φ_{_z_ _l_} + ...)
= 0
The general six-vector _f_ is composed from the vectors φ, φ^* in the
following way:—
_f_ = ρφ + ρ^* φ^*,
ρ and ρ^* denoting the contents of the pieces of mutually perpendicular
planes composing _f_. The “conjugate Vector” _f_^* (or it may be called
the complement of _f_) is obtained by interchanging ρ and ρ^*.
We have
_f_^* = ρ^*φ + ρφ^*
We can verify that
_f__{_y z_}^* = _f__{_x l_} etc.
and _f²_ = ρ² + ρ^*², (_f__f_^*) = 2ρρ^*.
| _f_ |² and (_f__f_^*) may be said to be invariants of the six vectors,
for their values are independent of the choice of the system of
co-ordinates.
[M. N. S.]
Note 12.
Light-velocity as a maximum.
Page 23, and Electro-dynamics of Moving Bodies, p. 17.
Putting _v_ = _c_ - _x_, and _w_ = _c_ - λ, we get
V = (2_c_ - (_x_ + λ))/(1 + (_c_ - _x_)(_c_ - λ)/_c²_) = (2_c_ -
(_x_ + λ))/(_c²_ + _c²_ - (_x_ + λ)_c_ + _x_λ/_c²_)
= _c_ (2_c_ - (_x_ + λ))/(2_c_ - (_x_ + λ) + _x_λ/_c_)
Thus _v_ lt; _c_, so long as | _x_λ | > 0.
Thus the velocity of light is the absolute maximum velocity. We shall
now see the consequences of admitting a velocity W > _c_.
Let A and B be separated by distance _l_, and let velocity of a “signal”
in the system S be W > _c_. Let the (observing) system S′ have velocity
+_v_ with respect to the system S.
Then velocity of signal with respect to system S′ is given by W′ = (W -
_v_)/(1 - W_v_/_c²_)
Thus “time” from A to B as measured in S′, is given by _l_/W′ = _l_(1 -
W_v_/_c²_)/(W - _v_) = _t′_ (1)
Now if _v_ is less than _c_, then W being greater than _c_ (by
hypothesis) W is greater than _v_, _i.e._, W > _v_.
Let W = _c_ + μ and _v_ = _c_ - λ.
Then W_v_ = (_c_ + μ)(_c_ - λ) = _c²_ + (μ + λ)_c_ - μλ.
Now we can always choose _v_ in such a way that W_v_ is greater than
_c²_, since W_v_ is > _c²_ if (μ + λ)_c_ - μλ is > 0, that is, if μ + λ
> μλ/_c_; which can always be satisfied by a suitable choice of λ.
Thus for W > _c_ we can always choose λ in such a way as to make W_v_ >
_c²_, _i.e._, λ - W_v_/_c²_ negative. But W - _v_ is always positive.
Hence with W > _c_, we can always make _t′_, the time from A to B in
equation (1) “negative.” That is, the signal starting from A 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 not produce any effect. Causal effect of any physical
type can never travel with a velocity greater than that of light.
[P. C. M.]
Notes 13 and 14.
We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is
then at once seen that [=ω] denotes the reciprocal matrix
| ω₁ |
| ω₂ |
| ω₃ |
| ω₄ |
It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]
[ω, _s_] The vector-product of the four-vector ω and _s_ may be
represented by the combination
[ω_s_] = [=ω]_s_ - _ṡ_ω
It is now easy to verify the formula _f_¹ = A⁻¹_f_A. Supposing for the
sake of simplicity that _f_ represents the vector-product of two
four-vectors ω, _s_, we have
_f¹_ = [ω¹_s¹_] = [[=ω]¹_s¹_ - [=_s_]^1ω^1]
= [A⁻¹ [=ω]_s_A - A⁻¹_s_[=ω]A]
= A⁻¹[[=ω]_s_ - _s_[=ω]]A = A⁻¹_f_A.
Now remembering that generally
_f_ = ρφ + ρ*φ*.
Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular
unit planes, there is no difficulty in seeming that
_f_^1 = A⁻¹_f_A.
Note 15.
The vector product (_w__f_). (P. 36).
This represents the vector product of a four-vector and a six-vector.
Now as combinations of this type are of frequent occurrence in this
paper, it will be better to form an idea of their geometrical meaning.
The following is taken from the above mentioned paper of Sommerfeld.
“We can also form a vectorial combination of a four-vector and a
six-vector, giving us a vector of the third type. If the six-vector be
of a special type, _i.e._, a piece of plane, then this vector of the
third type denotes the parallelopiped formed of this four-vector and
the complement of this piece of plane. In the general case, the
product will be the geometric sum of two parallelopipeds, but it can
always be represented by a four-vector of the 1st type. For two pieces
of 3-space volumes can always be added together by the vectorial
addition of their components. So by the addition of two 3-space
volumes, we do not obtain a vector of a more general type, but one
which can always be represented by a four-vector (loc. cit. p. 759).
The state of affairs here is the same as in the ordinary vector
calculus, where by the vector-multiplication of a vector of the first,
and a vector of the second type (_i.e._, a polar vector), we obtain a
vector of the first type (axial vector). The formal scheme of this
multiplication is taken from the three-dimensional case.
Let A = (A_{_x_}, A_{_y_}, A_{_z_}) denote a vector of the first type, B
= (B_{_y z_}, B_{_z x_}, B_{_x y_}) denote a vector of the second type.
From this last, let us form three special vectors of the first kind,
namely—
B_{_x_} = (B_{_x x_}, B_{_x y_}, B_{_x z_}) }
B_{_y_} = (B_{_y x_}, B_{_y y_}, B_{_y z_}) } (B_{_i k_} = - B_{_k
i_}, B_{_i i_} = 0).
B_{_z_} = (B_{_z x_}, B_{_z y_}, B_{_z z_}) }
Since B_{_j j_} is zero, B_{_j_} is perpendicular to the _j_-axis. The
_j_-component of the vector-product of A and B is equivalent to the
scalar product of A and B_{_j_}, _i.e._,
(A B_{_j_},) = A_{_x_} B_{_j x_} + A_{_y_} B_{_j y_} + A_{_z_} B_{_j
z_}.
We see easily that this coincides with the usual rule for the
vector-product; _e. g._, for _j_ = _x_.
(AB_{_x_}) = A_{_y_} B_{_x_ _y_} - A_{_z_} B_{_z_ _x_}.
Correspondingly let us define in the four-dimensional case the product
(P_f_) of any four-vector P and the six-vector _f_. The _j_-component
(_j_ = _x_, _y_, _z_, or _l_) is given by
(P_f__{_j_}) = P_{_x_}_f__{_j_ _x_} + P_{_y_}_f__{_j_ _y_} +
P_{_w_}_f__{_j_ _z_} + P_{_z_}_f__{_j_ _l_}
Each one of these components is obtained as the scalar product of P, and
the vector _f__{_j_} which is perpendicular to j-axis, and is obtained
from _f_ by the rule _f__{_j_} = [(_f__{_j_ _x_}, _f__{_j_ _y_},
_f__{_j_ _z_}, _f__{_j_ _l_}) _f__{_j_ _j_} = 0.]
We can also find out here the geometrical significance of vectors of the
third type, when _f_ = φ, _i.e._, _f_ represents only one plane.
We replace φ by the parallelogram defined by the two four-vectors U, V,
and let us pass over to the conjugate plane φ^*, which is formed by the
perpendicular four-vectors U^*, V^*. The components of (Pφ) are then
equal to the 4 three-rowed under-determinants D_{_x_} D_{_y_} D_{_z_}
D_{_l_} of the matrix
| P_{_x_} P_{_y_} P_{_z_} P_{_l_} |
| |
| U_{_x_}^* U_{_y_}^* U_{_z_}^* U_{_l_}^* |
| |
| V_{_x_}^* V_{_y_}^* V_{_z_}^* V_{_l_}^* |
Leaving aside the first column we obtain
D_{_x_} = P_{_y_}(U_{_z_}^* V_{_l_}^* - U_{_l_}^* V_{_z_}^*) +
P_{_z_}(U_{_l_}^* V_{_y_}^* - U_{_y_}^* V_{_l_}^*)
+ P_{_l_}(U_{_y_}^* V_{_z_}^* - U_{_z_}^* V_{_y_}^*)
= P_{_y_} φ_{_z_ _y_}^* + P_{_z_}^* φ_{_l_ _y_} + P_{_l_} φ^*_{_y_
_z_}.
= P_{_y_} φ_{_x_ _y_} + P_{_z_} φ_{_x_ _z_} + _P__{_l_} φ_{_x_ _l_},
which coincides with (Pφ_{_x_}) according to our definition.
Examples of this type of vectors will be found on page 36, Φ = wF, the
electrical-rest-force, and ψ = 2wf^*, the magnetic-rest-force. The
rest-ray Ω = iw[Φψ]^* also belong to the same type (page 39). It is easy
to show that
Ω = -_i_ | w₁ w₂ w₃ w₄ |
| Φ₁ Φ₂ Φ₃ Φ₄ |
| ψ₁ ψ₂ ψ₃ ψ₄ |
When (Ω₁, Ω₂, Ω₃) = 0, w₄ = _i_, Ω reduces to the three-dimensional
vector
| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |
| |
| ψ₁ ψ₂ ψ₃ |
Since in this case, Φ₁ = w₄ F₁₄ = _e__{_n_} (the electric force)
ψ₁ = -_i_w₄ f₂₃ = _m__{_x_} (the magnetic force)
we have (Ω) = | _e__{_x_} _e__{_y_} _e__{_z_} |
| _m__{_x_} _m__{_y_} _m__{_z_} |
[M. N. S.]
Note 16.
The electric-rest force. (Page 37.)
The four-vector φ = wF which is called by Minkowski the
electric-rest-force (elektrische Ruh-Kraft) is very closely connected to
Lorentz’s Ponderomotive force, or the force acting on a moving charge.
If ρ is the density of charge, we have, when ε = 1, μ = 1, _i.e._, for
free space
ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]
= ρ₀/(√(1 - V²/_c²_)) [_d__{_x_} + 1/_c_ (_v₂_ _h₃_ -
_v₃_ _h₂_)]
Now since ρ₀ = ρ√(1 - V²/_c²_)
We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)]
N. B.—We have put the components of _e_ equivalent to (_d__{_x_},
_d__{_y_}, _d__{_z_}), and the components of _m_ equivalent to _h__{_x_}
_h__{_y_} _h__{_z_}), in accordance with the notation used in Lorentz’s
Theory of Electrons.
We have therefore
ρ₀ (φ₁, φ₂, φ₃) = ρ (_d_ + 1/_c_ [_v_·_h_]),
_i.e._, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron.
Compare Lorentz, Theory of Electrons, page 14.
The fourth component φ₄ when multiplied by ρ₀ represents _i_-times the
rate at which work is done by the moving electron, for ρ₀ φ₄ = _i_ρ
[_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] =
_v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} ρ₀φ₃. -√(-1) times the power
possessed by the electron therefore represents the fourth component, or
the time component of the force-four-vector. This component was first
introduced by Poincare in 1906.
The four-vector ψ = _i_ωF^* has a similar relation to the force acting
on a moving magnetic pole.
[M. N. S.]
Note 17.
Operator “Lor” (§ 12, p. 41).
The operation | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | which plays in
four-dimensional mechanics a rôle similar to that of the operator
(_i_∂/∂_x_, + _j_∂/∂_y_, + _k_∂/∂_z_ = ▽) in three-dimensional geometry
has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in
honour of H. A. Lorentz, the discoverer of the theorem of relativity.
Later writers have sometimes used the symbol □ to denote this operation.
In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38)
Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation),
Grad (gradient) as four-dimensional extensions of the corresponding
three-dimensional operations in place of the general symbol lor. The
physical significance of these operations will become clear when along
with Minkowski’s method of treatment we also study the geometrical
method of Sommerfeld. Minkowski begins here with the case of lor S,
where S is a six-vector (space-time vector of the 2nd kind).
This being a complicated case, we take the simpler case of lor _s_,
where _s_ is a four-vector = | _s₁_, _s₂_, _s₃_, _s₄_ |
and _s_ = | _s₁_ |
| _s₂_ |
| _s₃_ |
| _s₄_ |
The following geometrical method is taken from Sommerfeld.
Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any
shape in the neighbourhood of the space-time point Q, _d_S denote the
three-dimensional bounding surface of ΔΣ, _n_ be the outer normal to
_d_S. Let S be any four-vector, P_{_n_} its normal component. Then
Div S = Lim ∫ P_{_n_}_d_S/ΔΣ.
ΔΣ = 0
Now if for ΔΣ we choose the four-dimensional parallelopiped with sides
(_dx₁_, _dx₂_, _dx₃_, _dx₄_), we have then
Div S = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_ = lor
S.
If _f_ denotes a space-time vector of the second kind, lor _f_ is
equivalent to a space-time vector of the first kind. The geometrical
significance can be thus brought out. We have seen that the operator
‘lor’ behaves in every respect like a four-vector. The vector-product of
a four-vector and a six-vector is again a four-vector. Therefore it is
easy to see that lor S will be a four-vector. Let us find the component
of this four-vector in any direction _s_. Let S denote the three-space
which passes through the point Q (_x₁_, _x₂_, _x₃_, _x₄_) and is
perpendicular to _s_, ΔS a very small part of it in the region of Q,
_d_σ is an element of its two-dimensional surface. Let the perpendicular
to this surface lying in the space be denoted by _n_, and let _f__{_s_
_n_} denote the component of _f_ in the plane of (_sn_) which is
evidently conjugate to the plane _d_σ. Then the _s_-component of the
vector divergence of _f_ because the operator lor multiplies _f_
vectorially.
= Div _f__{_s_} = Lim (∫ _f__{_s_ _n_}_d_σ)/ΔS.
Δ_s_ = 0
Where the integration in _d_σ is to be extended over the whole surface.
If now _s_ is selected as the _x_-direction, Δ_s_ is then a
three-dimensional parallelopiped with the sides _dy_, _dz_, _dl_, then
we have
$$ Div f_{x} = \frac{1}{dy dz dl} {dz. dl. \frac{\partial
f_{xy}}{\partial y} dy + dl dy \frac{\partial f_{xy}}{\partial z} dz +
dy dz \frac{\partial f_{xy}}{\partial l} dl} = \frac{\partial
f_{xy}}{\partial y} + \frac{\partial f_{xy}}{\partial z} +
\frac{\partial f_{xy}}{\partial l} $$
and generally
Div _f__{_j_} = ∂_f__{_j_ _x_}/∂_x_ + ∂_f__{_j_ _y_}/∂_y_ +
∂_f__{_j_ _z_}/∂_z_ + ∂_f__{_j_ _l_}/∂_l_ (where _f__{_j_, _j_} =
0).
Hence the four-components of the four-vector lor S or Div. _f_ is a
four-vector with the components given on page 42.
According to the formulae of space geometry, D_{_x_} denotes a
parallelopiped laid in the (_y_-_z_-_l_) space, formed out of the
vectors (P_{_y_} P_{_z_} P_{_l_}), (U_{_y_}^* U_{_z_}^* U_{_l_}^*)
(V_{_y_}^* V_{_z_}^* V_{_l_}^*).
D_{_x_} is therefore the projection on the _y-z-l_ space of the
parallelopiped formed out of these three four-vectors (P, U^*, V^*), and
could as well be denoted by Dyzl. We see directly that the four-vector
of the kind represented by (D_{_x_}, D_{_y_}, D_{_z_}, D_{_l_}) is
perpendicular to the parallelopiped formed by (P U^* V^*).
Generally we have
(P_f_) = PD + P^*D^*.
∴ The vector of the third type represented by (P_f_) is given by the
geometrical sum of the two four-vectors of the first type PD and P^*D^*.
[M. N. S.]
● Transcriber’s Notes:
○ The book's idiosyncratic spelling, emphasis, punctuation, and
symbology especially in mathematical formulas, have been retained.
○ Text that was in italics is enclosed by underscores (_italics_).
Text that was in bold face is enclosed by equals signs (=bold=).
○ Footnotes have been moved to follow the chapters in which they are
referenced.
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