CABHB6IB INSTITUTE
OF TBNOL045Y
THS LIBRARY
THE
PHYSICAL SOCIETY
OF
LONDON.
ON THE
RELATIVITY THEORY OF
GRAVITATION.
A. S. EDDINGTON, M.A., M.Sc., F.K.S.
Plumian Professoflf Astronomy and Experimental Philosophy, Cambridge*
Price to Non-Fdlows, 6s. net, post free 6s. 3d g
Bound in doth, Ss. 6d., post free 8s. 9d.
PHYSICAL SOCIETY
LONDON.
EEPOKT
OK THE
RELATIVITY THEOEY OF
GRAVITATION.
A. S. EDDINGTON, M.A., M.Sc., F.R.S.
Plumian Professor o/ Astronomy and Expe,rime.ntal Philosophy, Cambridge.
Price to Non-Fellows, 68. net, post free 6s. 3d,
Sound in cloth, 8s, 6d., post free 8s> 9d.
LONDON:
FLEETWAY PRESS, LTD,
1, 2 AND 3, SALISBURY COURT, FLEET STREET.
1920.
CONTENTS.
CHAPTER I.
PAOB
THE RESTRICTED PRINCIPLE or RELATIVITY 1
1-3. The Miohelson-Morley experiment and its significance.
4. The transformation of co-ordinates for a moving observer.
5. Reciprocity of the transformation. 6. Standpoint oi the
Principle of Relativity. 7. Transformation of velocity, of den-
sity and of mass. 8. Scope of the Principle.
CHAPTER II.
THE RELATIONS OF SPACE, TIME, AND FORCE 14
9-10. Minkowski's transformation. 11. Invarianoe of Se. 12.
Irrelevance of co-ordinate systems to the phenomena. 13-14.
The Principle of Equivalence. 15-16. Definition of a field of
force by gr MV . 17. Purpose of the theory of tensors. 18. Nature
of space and time in the gravitational field.
CHAPTER III.
THE THEORY OP TENSORS 30
19. Notation ; definition and elementary properties of tensors.
20. The fundamental tensors : associated tensors. 21. Auxili-
ary formula for the second derivatives of the co-ordinates. 22.
Covariant differentiation. 23-24. The Riemann-Chrietoffel ten-
sor. 26. Summary.
CHAPTER IV.
EINSTEIN'S LAW OF GRAVITATION 41
26. The contracted Riemann-Christoffel tensor. 27. Limitation
of the Principle of Equivalence. 28. The gravitational field of a
particle.
IV. CONTENTS.
CHAPTEE V.
PAGB
THE CRUCIAL PHENOMENA 48
29-30. The Equations of Motion. 31. Motion of the Perihelion
of Mercury. 32-33. Deflection of a ray of light 34. Displace-
ment of spectral lines.
CHAPTER VI.
THE GRAVITATION OF A CONTINUOUS DISTRIBUTION OF
MATTER 59
35-36 Equations for a continuous medium. 37 The energy,
tensor T^, and the equations of hydrodynamics. 38. The Law
of Conservation 39. Reaction of the gravitational field on
matter. 40. Propagation of gravitation
CHAPTER VII.
THE PRINCIPLE OF LEAST ACTION 71
41. Expression of the law of gravitation in the form of Lagrange's
Equations. 42. Principle of Least Action. 43. Energy of the
gravitational field. 44. Method of Hilbert and Lorentz. 45-46.
Electromagnetic equations. 1 The JEther. 48. Summary of
the last two chapters.
CHAPTER VIII.
THE CURVATURE OF SPACE AND TIME 82
49. Absolute rotation and the limits oi generalised relativity.
50. Einstein's cur.-od space. 51. De Sitter's curved space-time.
52. Boundary Conditions. 53. Conclusion
PREFACE TO FIRST EDITION.
THE relativity theory of gravitation in its complete
form was published by Einstein in November 1915.
Whether the theory ultimately proves to be correct
or not, it claims attention as one of the most
beautiful examples of the power of general mathematical
reasoning. The nearest parallel to it is found in the applications
of the second law of thermo-dynamics, in which remarkable
conclusions are deduced from a single principle without any
inquiry into the mechanism of the phenomena ; similarly, if
the principle of equivalence is accepted, it is possible to stride
over the difficulties due to ignorance of the nature of gravita-
tion and arrive directly at physical results. Einstein's theory
has been successful in explaining the celebrated astronomical
discordance of the motion of the perihelion of Mercury,
without introducing any arbitrary constant ; there is no trace
of forced agreement about this prediction. It further leads to
interesting conclusions with regard to the deflection of light
by a gravitational field, and the displacement of spectral lines
on the sun, which may be tested by experiment.
The arrangement of this Report is guided by the object oi
reaching the theory of these crucial phenomena as directly as
possible. To make the treatment rather more elementarv,
use of the principle of least action and Hamiltonian methods
has been avoided ; and the brief account of these in Chapter
VII. is merely added for completeness. Similarly, the equa-
tions of electro-dynamics are not used in the main part of
the Report. Owing to the historical tradition, there is an
undue tendency to connect the principle of relativity with
the electrical theory of light and matter, and it seems well to
emphasize its independence. The main difficulty of this
subject is that it requires a special mathematical calculus,
which, though not difficult to understand, needs time and
practice to use with facility. In the older theory of relativity
the somewhat forbidding vector products and vector operators
VL PREFACE TO FIRST EDITION.
constantly appear. Happily this can now be avoided alto-
gether ; bit in its place we use the absolute differential
calculus of Ricci and Levi-Civita. This is developed ab initio
so far as required in Chapter III. Attention must be called
to the remark on notation in 19, which concerns almost all
the subsequent formulae.
Extensive use has been made of the following Papers, which
in some places have been followed rather closely :
A. EINSTEIN. Die Grundlage der allgemeinen Relitivitats
theorie. " Annalen der Physik," XLIX., p. 769 (1916).
W. DE SITTER. On Einstein's Theory of Gravitation and
its Astronomical Consequences. " Monthly Notices of the
Eoyal Astr. Soc.," LXXVL, p. 699 (1916) ; LXXV1L, p. 155
(1916); LXXVIIL, p. 3 (1917).
I am especially indebted to Prof, de Sitter, who has kindly
read the proof-sheets of this Report.
The principal deviations in the present treatment of the
subject will be found in Chapter VI. I have ventured to
modify the enunciation of the principle of equivalence in 27
in order to give a precise criterion for the cases in which it is
assumed to apply.
Other important Papers on the subject, most of which have
been drawn on to some extent, are :
H. HILBERT. Die Grundlagen der Physik, " Gottingen
Nachrichten," 1915, Nov. 20.
H. A. LORENTZ. On Einstein's Theory of Gravitation,
' Proc. Amsterdam Acad.," XIX., p. 1341 (1917).
J. DROSTE. The Field of n moving centres on Einstein's
Theory, " Proc. Amsterdam Acad.," XIX., p. 447 (1916).
A. EINSTEIN. Kosmologische Betrachtungen zur allge-
meinen Relitivitatstheorie, " Berlin Sitzungsber.," 1917, Feb. 8.
Ueber Gravitationswellen, ibid, 1918, Feb. 14.
K. SOHWARZSCHILD. Ueber das Gravitationsfeld eines
Massenpunktes nach der Einstein'schen Theorie, " Berlin
Sitzungsber., " 1916, Feb. 3.
T. LEVi-CrvrcA. Statica Einsteiniana, " Rendiconti dei
Lincei," 1917, p. 458.
A. PALATINI. Lo Spostamento del Perielio di Mercuric,
" Nuovo Cimento," 1917, July.
The last two Papers avoid much of the heavy algebra, but
claim a rather extensive knowledge of differential fireometrv.
Silberstein (Macmillan & Co.) and E. Cunningham (Camb.
Univ. Press). A useful review of the mathematical theory of
Chapter III., giving a fuller account from the standpoint of
the pure mathematician, will be found in " Cambridge Mathe-
matical Tracts," No. 9, by J. E. Wright. Finally, for those
who wish to learn more of the outstanding discrepancies
between astronomical observation and gravitational theory,
the following references may be given :
W. DE SITTER. The Secular Variations of the Elements of
the Four Inner Planets, " Observatory," XXXVI., p. 296.
E. W. BROWN. The Problems of the Moon's Motion, " Ob-
servatory," XXXVII., p. 206.
H. GLAUERT. The Rotation of the Earth, " Monthly Notices
of the Royal Astr. Soc.," LXXV., p. 489.
PREFACE TO SECOND EDITION.
THE advances made in the eighteen months since
this Report was written do not seem to call for
any modification in the general treatment. Perhaps
the most notable event is the verification of Ein-
stein's prediction as to the deflection of a ray of light by
the sun's gravitational field. This was tested at the total
eclipse of May 29, 1919, at two stations independently, by
expeditions sent out by the Royal and Royal Astrono-
mical Societies "jointly, under the superintendence of the
Astronomer Royal. The deflection, reduced to the sun's
limb should be l"-75 on the relativity theory, and 0"-87 (or
possibly zero) according to previous theories. At Principe,
where the observations were very much interfered with by
cloud, the value I"- 61 was obtained, with a probable error
of 0"-3 ; the accuracy appears to be sufficient to indicate
fairly decisively Einstein's value. At Sobral, where a clear
sky prevailed, the observed value was l"-98 ; the accordance
of results derived from right ascensions and declinations,
respectively, and the agreement of the displacements of
individual stars with the theoretical law demonstrate in a
particularly satisfactory manner the trustworthiness of the
observations at this station. The full results will be pub-
lished in a Paper by Sir F. W. Dyson, A. S. Eddington, and
C. Davidson in the Philosophical Transactions of the Royal
Society.
The test of the displacement of the Fraunhofer lines to the
red stands where it did, and we still think that judgment must
be reserved. In view of the possibility of a failure in this
test, it is of interest to consider exactly what part of the
theory can now be considered to rest on a definitely experi-
mental basis. I think it may now be stated that Einstein's
law of gravitation is definitely established by observation in
the following form :
Every particle and light-pulse moves so that the integral
of ds between two points on its track is stationary, where
(equation (28-8))
&= (1 2m/r) -W* rW r 2 sin 2 Od<f + (l2m/r)dt*
in appropriate polar co-ordinates, the co-efficient of dr* being
verified to the order m/r, and the co-efficient of dfi to the
order m 2 /r 2 . This is checked for high speeds by the deflection
of light, and for comparatively low speeds by the motion of
perihelion of Mercury, so that unless the true law is of a kind
much more complicated than we have allowed for, our ex-
pression cannot well be in error.
Accepting Einstein's law in this form, the properties of
invariance for transformations of co-ordinates follow, and we
reach the conclusion that the intermediary quantity ds (to
which as yet we have assigned no physical interpretation) is
an invariant, that is to say it has some absolute significance
in external nature.
Einstein's theory (as distinct from his law of gravitation)
gives a physical interpretation to ds, as a quantity that can
be measured with material scales and clocks. It is this
interpretation which the observation of the Fraunhofer lines
should test. The quantity ds is an ideal measure of space and
time ; and it is possible that we have not yet reached finality
as to the right way of realising the ideal practically. It is a
fair prediction that an atomic vibration will register ds like
an ideal clock ; and it is difficult to see now this can be avoided
unless the equations of vibration of an atom involve the
Biemann-ChristoSel tensor. But, if the test fails, the logical
conclusion would seem to be that we know less about the
conditions of atomic vibration than we thought we did.
A very notable extension of the theory to include electro-
magnetic forces and gravitational forces in one geometrical
scheme has been given by Prof. H. Weyl in two Papers
Berlin, Sitzungsberichte, 1918, May 30.
Annalen der Physik, Bd. 59, p. 101.
In Einstein's theory it is assumed that the interval ds has
an absolute value, so that two intervals at different points
of the world can be immediately compared. In practice
the comparison must take place by steps along an intermediate
path ; for example, by moving a material measuring rod
from one point to the other continuously along some path.
It is possible that the result of the comparison may not be
independent of the path followed, and Weyl considers the
electromagnetic field to be the manifestation of this incon-
sistency. This leads to a very beautiful generalized geometry
of the world, in which the electromagnetic field appears as
the sign of non-integrability of gauge, and the gravitational
field as the sign of non-integrability of direction. The theory
has important consequences though it has not suggested
any experimental test. It may be added that it appears
to favour Einstein's view of the curvature of space, which
has been treated, perhaps too unsympathetically, in Chapter
VIII.
The writer holds the view that the fundamental equations
of gravitation (35-8), which on this theory are the sole basis
of mechanics, should be regarded as a definition of matter
rather than as a law of nature. We need not suppose that
the gravitational field has in vacuo some innate tendency to
arrange itself according to the law G^0 ; we should rather
say that in regions of the world where this state happens to
exist we perceive emptiness ; and where the equations fail,
the failure of the equations is itself the cause of our perception
of matter. Matter does not cause the curvature (G) of space-
time ; it is the curvature. Just as light does not cause electro-
magnetic oscillations ; it is the oscillations. This point of
view is developed in a Paper which will appear shortly in
"Mind."
Finally, a word may be added for those who find a difficulty
in the combination of space and time into a static four-
dimensional world, in which events do not " happen " they
are just there, and we come across them successively in our
exploration. " Surely there is a difference between the
irrevocable past and the open future, different in quality
from the arbitrary distinction of right and left." We agree
entirely ; but this difference, whatever it is, does not enter
into the determinate equations of physics. For physics, the
future is -M and the past t, just as right is -f- % and left x.
If we change the place of one particle in our problem we alter
the past as well as the future, in contrast to what appears to
be the ordinary experience of life, that our interference will
alter the future but not the past. The static four-dimensional
representation may thus be not completely adequate, but it
suffices for all that comes within the purview of physics.
December, 1919.
CHAPTEB L
THE RESTRICTED PRINCIPLE OF RELATIVITY.
1. In 1887 the famous Michelson-Morley experiment was
performed with the object of detecting the earth's motion
through the aether. 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 aether there is
a river of sether 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.
Let the aether be flowing relative to the apparatus with
velocity u in the direction Ox (Fig. 1) ; and let OA, OB be
the two arms of the apparatus of equal length a, OA being
placed up-stream. Let v be the velocity of light. The time
for the double journey along OA and back is
__ a a _ 2av _2a,j 2 1X
1 v u v-\-u v 2 u 2 v
where (3= (I u 2 /v 2 )~%, a factor greater than unity.
For the transverse journey the light must have a com-
ponent velocity u up-stream (relative to the aether) in order
to avoid being carried below OB ; and, since its total velocity
is v, its component across-stream must be <\/(v z u z ). The
time for the double journey OB is accordingly
2 RELATIVITY THEORY OF GRAVITATION.
>
But when the experiment was tried, it was found that both
parts of the beam took the same time, as tested by the inter-
ference bands produced. It would seem that OA and OB
could not really have been of the same length ; and if OB
was of length a ls OA must have been of length ajp. The
apparatus was now rotated through 90, so that OB became
the up-stream arm. The time for the two journeys was again
the same, so that OB must now be the shorter arm. The plain
meaning of the experiment is that both arms have a length a l
when placed along Oy, and automatically contract to a length
a x //J when placed along Ox. This explanation was first given
by FitzGerald.
It is not known how much the earth's motion through the
aether amounts to ; but at some time during the year it must
o
-u
-3C
\B
FIG. 1.
be at least 30 km. per sec,, since the earth's velocity changes
by 60 km. per sec. between opposite seasons. The experiment
would have detected a velocity much smaller than this (about
3 km. per sec.), if it were not for the compensating contraction
of the arms of the apparatus. By experimenting at different
times of the year with different orientations the existence of
the contraction has been fully demonstrated. It has been
shown that it is independent of the material used for the arms,
and the contraction is in all cases measured by the ratio
It is now known that this contraction fits in well with the
electrical theorv of matter, and mav be attributed to
as it at first seemed ; and we shall not here discuss the un-
successful attempts at alternative explanations of the
Michelson-Morley experiment, e.g., by assuming a convection
of the sether by the earth.
2. The Michelson-Morley experiment has thus unexpectedly
failed to measure our motion through the aether, and many
other ingenious experiments have failed in like manner. So
far as we can test, the earth's motion makes absolutely no
difference in the observed phenomena ; and we shall not be
led into any contradiction with observation if we assign to the
earth any velocity through the sether that we please. It is
interesting to trace in a general way how this can happen.
Let us assign to the earth a velocity of 161,000 miles a second,
say, in a vertical direction. With this speed /? 2, and the
contraction is one-half. A rod 6 feet long when horizontal
contracts to 3 feet when placed vertically. Yet we never
notice the change. If the standard yard-measure is brought
to measure it, the rod will still be found to measure two yards ;
but then the yard-measure experiences the same contraction
when placed alongside, and represents only half-a-yard in that
position. It might be thought that we ought to see the change
of length when the rod is rotated. But what we perceive is
an image of the rod on the retina of the eye ; we think that
the image occupies the same space of retina in both positions ;
but our retina has contracted in the vertical direction without
our knowing it, and our estimates of length in that direction
are double what they should be. Similarly with other tests.
We might introduce electrical and optical tests, in which the
cause of the compensation is more difficult to trace ; but, in
fact, they all fail. The universal nature of the change makes
it impossible to perceive any change at all.
3. This discussion leads us to consider more carefully what
is meant by the length of an object, and the space which we
consider it to occupy. To the physicist, space means simply
a scaffolding of reference, in which the mind instinctively
locates the phenomena of nature. Our present point of view
assumes that there is a " real " or " absolute " scaffolding,
in which a material body moving with the earth changes its
length according as it is oriented in one direction or another.
On the other hand, the human race (and its predecessors) have
conceived and used a different scaffolding the space of
appearance in which a material body moving with the earth
does not change length as its orientation alters. It often
happens that a primitive conception is ambiguous, and has
to be re-defined when adopted for scientific purposes ; but
there is little justification for doing this in the case of space.
Firstly, the space of appearance is perfectly suitable for
scientific purposes, since we have just seen that it is impossible
to detect experimentally that it is not the absolute space.
Secondly, so long as we cannot detect our motion through
the aether, we do not know how to convert our observations
so as to express them in terms of absolute space. Thirdly,
for all we know, our velocity through the aether may be so
great that the absolute space and the space of appearance do
not even approximately correspond ; thus we might be re-
volutionising rather than re-defining the common conception
of space.
It will therefore be considered legitimate to use the words
" space " and " length " with their current significance. A
rigid body on the earth is generally considered not to change
length when its direction is altered, and by this property we
block out a scaffolding of reference for our measures and
locate objects in our space the space of appearance. Bui
we have learnt one important thing. Our space is not abso-
lute ; it is determined by our motion. If we transfer our-
selves to the star Arcturus, which is moving relatively to us
with a speed of more than 300 km. per sec., our space wil
not suit it, since it was designed to eliminate our own con-
traction effects. The contraction ratio /3 must be diiferenl
for Arcturus ; and the space surveyed with a material yard
measure carried on Arcturus will differ slightly from the spac<
surveyed with the same yard-measure on the earth. It maj
also be noted that there is a slight difference in our own spac<
in summer and winter (owing to the change of the earth 5 !
motion), and this may have to be taken into account in som<
pplications.
Accordingly by " space " we shall mean the space of appear
ance for the observer considered. It becomes definite when w<
specify the motion of the observer. In particular, if th<
observer is at rest in the sether, the corresponding space is
what we have hitherto called the " absolute space."
The possibility of different observers using different spaces
may be illustrated by considering the question, What is t
circle ? Suppose a circle is drawn on paper in the usual waj
with a pair of compasses. An observer S, who believes th<
paper to be moving through the sether with a great velocity
mast, in accordance with, the Michelson-Morley experiment,
suppose that the distance between the points of the compasses
changed as the curve was described ; he will therefore deem
the curve to be an ellipse. Another observer S', who believes
the paper to be at rest in the aether, will deem it to be a circle.
There is no experimental means of finding out which is right
in his hypothesis. We have, therefore, to admit that the
same curve may be arbitrarily regarded as an ellipse or as a
circle. That illustrates our meaning when we say that S and
S' use different spaces, the curve being an ellipse in one space
and a circle in the other.
4. The failure of all experimental tests to decide whether
the space of S or of S' is the more fundamental is summed up
in the restricted Principle of Relativity. This asserts that
it is impossible by any conceivable experiment to detect uniform
motion through the cether. This generalisation is based on a
great amount of experimental evidence, which is fully dis-
cussed in text-books on the older theory of relativity. Here
it is perhaps sufficient to state that experimental confirmation
appears to be sufficient,* except in regard to the question
whether gravitation falls within the scope of the principle.
We shall assume that the principle is true universally.
Let %', y', z', be the co-ordinates of a point in the space
of an observer S' ; and let x, y, z be the co-ordinates of the
same point in the space of an observer 8 at rest in the sether.
Let S' move relatively to S with velocity u in the direction Ox.
S', using his own space, has no knowledge of his motion through
the aether, and he makes all his theoretical calculations as
though he were at rest ; from what has been already said,
he will not discover any contradiction with observation.
According to ordinary kinematics the relation between the
co-ordinates and the times (t f , t) in the two systems would be
x'=x ut, y'=y, z'=z, t't . . . (4-1)
But the first of these must be modified, because in the x-
direction $"s standard of length is contracted in the ratio 1//5.
The equation becomes " A
x'=p(x-ut} (4-15)
in order to satisfy the principle of relativity, it appears that
the time t' used by 8' must differ from the time t used by S.
* /.., sufficient bo assert the uniesraalily, not necessarily fcbe perfect
accuracy, ot cbo principle.
We shall suppose that both observers use the same value for
the velocity of light ; this is merely a matter of co-ordinating
their units, the significance of which will be considered in the
next paragraph. Let S' observe the time t' taken for the
double journey 05=2a l3 in Fig. 1. It must agree with his
calculated time, which is, of course, 2a 1 /v. Thus
But in (1-2), when we were using /S's co-ordinates, we found
the time to be
=2a!jS/t>.
Hence
This also fits the double journey OA. S f , unaware of his
motion, does not allow for any contraction, and calculates the
time for the double journey as
But *S recognises the contraction, and considers the distance
travelled to be 2<z 1 //3. Hence calculating as in (1-1), he makes
the time to be
'=
so that aain
Accordingly S' must use a unit of time longer than that of
S in the ratio /? ; otherwise he would find a discrepancy
between observation and calculation.
There is another difference in time-measurement involved.
According to S, the light completes the half -journey OA in a
time ^- in /S's units, or in /S"s units of time. But
v u vu
_a l (v j ru)__a l djU
But the difference in the time of leaving and reaching A
must be deemed by S' to be ajv ; he must therefore set his
clock at A a^u/v^ slow compared with the clock at 0. He
has no idea that it is slow ; he has attempted to adjust the
two clocks together. But his determination of simultaneity
of events at and A differs from that of S, because he allows
a different correction for the time of transit of the light.
THE RESTRICTED PKINCIPLE OF RELATIVITY. 7
Including both these differences, we see that the relation
between the times adopted by S and S' is
Substituting this value of t in (4-15) we obtain after an easy
reduction
Collecting together our results, we have the formulae of
transformation
/
iC ij\tJO p" 166 ) y 'y == 11 y % "^^ % y v p 1 t "~p" """'"o'^ ) ' 4: * <t )
By the principle of relativity nothing is altered if S is in
motion relative to the aether ; so the relations (4-2) must hold
between the spaces and times of any two observers having
relative velocity u.
By solving (4-2) for x', if, z', ', we obtain the reciprocal
relations
x'=/3(x-ut), y'^y, z'=z, t'=0(t-?g) . . (4-3)
These might have been written down immediately, because
interchanging S and S' is equivalent to reversing the sign of u ;
but it will be seen later that the verification by direct solution
of (4-2) is important.
5. We have supposed that S and S f adopt the same measure
for the velocity of light ; this was in order to secure that the
units of velocity used by S and S' correspond. It is no use
for S to describe his experiences to S' in terms of units which
are outside the knowledge of the latter ; but if S states that a
velocity occurring in his experiment is a certain fraction of the
velocity of light, S' will be able to compare that with his own
experimental results. By the principle of relativity any other
velocities occurring in their experiments under similar con-
ditions will correspond ; and, for example, we see from (4-2)
and (4-3) that they will agree in calling their relative velocity
4-u and u respectively.
Whilst this settles the consistency of the units of velocity
used in (4-2) we have not yet secured that the units of length
correspond. A description of Brobdingnag by a Brobding-
nagian would not have mentioned the most striking feature
ask whether a natural standard of length, say a hydrogen
atom, at rest in $'s system will be of the same size in terms
of x, y, z, as a hydrogen atom at rest in S"s system in terms
of x', y', %' . Clearly it will be misleading if we do not correlate
the co-ordinates so as to satisfy this.
To allow for a possible non-correspondence oi the units of
length in (4-2) we can write the transformation more generally
kx=p(x'+ut'), ki/=if, kz=z f , kt=p(t'+ux'/v z ) . (5-2)
where k depends on the magnitude, but clearly not on the
direction, of u.
But now applying (5-2) the reverse way, i.e., regarding
x, y, z, t as a system moving with velocity u relative to
x', y', z', t', we shall have
kx r =p(x-ut), ky'=y, kz'=z, kt'=p(t-ux/v*) . (5-3)
which is clearly inconsistent with (5-2) unless k=l. Hence (4-2)
gives the only possible correspondence of the units of length.
We thus use the remarkable property of reciprocity possessed
by (4-2) and (4-3), but not by (5-2) and (5-3), to fix the necessary
correspondence of the units. The dimensions of a motionless
hydrogen atom will now be the same in' both systems ; for, if
not, we could find a system in which the dimensions were
either a maximum or a minimum ; and that system would
give us an absolute standard from which we could measure
absolute motion.
It is thus clear that S' will actually measure his space and
time by the variables x', y', z', t' given by (4-3), if he sets
about choosing his units in the same way that S did.
6. We have established the connection between the co-
ordinates used by S and S' by reference to simple criteria.
It is interesting to work out in detail the correspondence of
the two systems for other and more complex phenomena,
showing that the transformation always works consistently.
But the standpoint of the 'principle of relativity rather dis-
courages this procedure. Its view is that the indifference of
all natural phenomena to an absolute translation is something
immediately understandable, whilst the contractions and other
complications entering into our description arise from our own
perversity in not looking at Nature in a broad enough way.
When a rod is started from rest into uniform motion, nothing
whatever happens to the rod. We say that it contracts ; but
length is not a property of the rod ; it is a relation between
the rod and the observer. Until the observer is specified the
length of the rod is quite indeterminate. We ought always to
remember that our experiments reveal only relations, and not
properties inherent in individual objects ; and then the corre-
spondence of two systems, differing only in uniform motion,
becomes axiomatic, so that laborious mathematical verifica-
tions are redundant. Human minds being what they are, that
is a counsel of perfection, and we shall not follow it too strictly.
The only verification that is needed is to show that our
fundamental laws of mechanics and electrodynamics are con-
sistent with the principle of relativity. This will be done in
connection with a much more general principle of relativity
for mechanics in 37, and for electrodynamics in 45. -
7. (a) As "an illustration of the modification of ordinary
views required by this theory, we may notice the law of
composition of velocities. Consider a particle moving relative
to S with velocity w along Ox, so that
dx_
The velocity relative to S' will be
, dx' P(dx-udt) Qx
= - .... (7-2)
luw/v 2 J ' '
The velocity relative to S' is thus not w u, as we should
have assumed in ordinary mechanics.
It has been pointed out by Robb that the addition-law for
motion in one dimension can be restored if we measure motion
by the rapidity, tanh~ 1 (w/v), instead of by the velocity w.
Equation (7-2) gives
(w/t?) . . (7-3)
Since tanh~ 1 l=oo , the velocity of light corresponds to
infinite rapidity, and we may compound any number of
relative velocities less than that of light without obtaining a
resultant greater than the velocity of light.
(6) To find the relation of the densities (o-=number of par-
ticles per unit volume) in the two systems, we can easily verily
that the Jacobian d(x f , y', z', t')/d(x, y, z, t)~l, so that
dx'dy'dz'dt'-^dxdydzdt ..... (7'4)
But the number of particles in a particular element of volume
cannot depend on the co-ordinates used to describe the element,
hence
(7-5)
in r\
(7 ' 6)
since dx/dt=w.
In particular, if w0, so that o- is the density referred to
axes moving with the matter,
rr'=/30- ........ (7-65)
&ince the mass of a particle may depend on its motion, we
cannot assume that the ratio p'/p of the mass-density is the
same as that of the distribution-density o-'/o-.
When the transformation (4-2) was first introduced in
electro-dynamics by Larmor and Lorentz, t' was regarded as
a fictitious time introduced for mathematical purposes, and it
was scarcely realised that it was the actual measured time of
the moving observer. Einstein in 1905 first showed that
velocity and density would be estimated by the moving
observer in the way given above, and thus removed the last
discrepancy between the electrodynamical equations for the
two systems.
(c) In order to find the change (if any) of mass with velocity,
consider a body of mass m^, m-/ (in the two systems of reference)
moving with velocity w l5 w t '. Lot
01 = (1 -*!/*)"*, 0l' = (l-<W*J
Working out /Y by using (7-2), we easily find
PiX'^ito-") ..... (7-71)
Let a number of bodies be moving in a straight line subject
to the conservation of mass and momentum, i.e.,
Sm l and 2m i w l arc conserved.
Then, since u and (3 are constants,
^2m 1 (w l u) will be conserved.
Therefore by (7-71)
fi-^Wi is conserved. . . . (7-72)
Pi
But since momentum must be conserved for the observer /S"
/w/ is conserved .... (7-73)
The results (7-72) and (7-73) will agree if
m 1 _m 1 '__
7T "fl-"/ W'OJ Sa y>
Pi' Pi
and it is easy to show that there is no other solution. Hence
m 1 =m^ l =m (l-w 1 2 /v z )~^ .... (7-8)
where m Q is constant and equal to the mass at rest. This is
the law of dependence of mass on velocity.
Neglecting w^/v*, we have
W 1 =w () +(|-m w 1 2 )/u a .... (7-85)
so that we may regard the mass as made up of a constant
mass m belonging to the particle, together with a mass pro-
portional to, and presumably belonging to, the kinetic energy.
If we choose units so that the velocity of light is unity, the
mass of the energy is the same as the energy, and the dis-
tinction between energy and mass is obliterated. Accordingly
m is also regarded as a form of energy. (It is usually identified
mainly with the electrostatic energy of the electrons forming
the body.)
Since the conservation of mass now implies the conservation
of energy we have to restrict the reactions between the bodies
in the foregoing discussion to perfectly elastic impacts. Other
interactions would require a more general treatment ; in fact,
if the energy is not conserved, the momentum is not perfectly
conserved, because the disappearing energy has mass and
therefore carries off momentum.
In this discussion we are justified in pressing the laws of
conservation of mass and momentum to the utmost limit as
holding with absolute accuracy, since the definition and
measurement of mass (inertia) rests on these laws,* and
unless we have an accurate definition it is meaningless to
investigate change of mass. In astronomy, however, the
masses of heavenly bodies are measured by their gravitational
effects ; naturally we cannot legitimately apply (7-8) to
gravitational mass without a full discussion of the law of
gravitation.
It should be noticed that this change of mass with velocity
is in no way dependent on the electrical theory of matter.
* The mass here discussed is sometimes called the " transverse mass."
The so-called longitudinal mass is of no theoretical importance ; it is tiot
conserved, it does not enter into the expression for the momentum or energy,
and it has no connection with gravitation.
12 RELATIVITY THEORY OP GRAVITATION.
(d) To find the transformation of mass-density, p, we have
which becomes by (7-71)
P _/'Pi \ 2 (7.QT
V~5~j ^' y
p \p l J
In particular, if p is the density in natural measure, i.e.
referred to axes moving with the matter, p the density referred
to axes with respect to which the matter has a velocity u,
p=j8 a Po (7-92
8. Of late years the domain of the electromagnetic theon
has been extended, so that most natural phenomena are nov
attributed to electrical actions. The relativity theory doe
not presuppose an electromagnetic theory either of matter o
of light ; but, if we accept the latter theories, it become
possible to state exactly the points on which experiments
evidence is required in order to establish our hypothesis
The experimental laws of electromagnetism are summed u'
in Maxwell's equations ; and in so far as these cover th
phenomena, the complete equivalence of the sequence of event
in a fixed system described in terms of x, y, z, t, and a movin,
system described in terms of x', y', z', t', has been establishei
analytically. So far as is known, only three kinds of force ar
outside the scope of Maxwell's equations.
(1) The forces which constrain the size and shape of a
electron are not recognised electromagnetic forces. Fo:
tunately the properties of an electron at rest and in e:
tremely rapid motion can be studied experimentally, ar
it is believed that they change in the way required I
relativity.
(2) The phenomena of Quanta appear to obey laws outsic
the scope of Maxwell's equations. Theoretically these lax
fit in admirably with relativity, since Planck's fundament
unit of action is found to be unaltered by the choice of ax*
But on the experimental side, evidence of the relativity
phenomena involving quantum relations has not yet bei
produced. This is particularly unfortunate, because t]
vibration of an atom depends on quantum relations ; and it
practically essential to the relativity theory that an ato
(acting as a natural clock) should keep the time appropria
(3) Gravitation is outside the electromagnetic scheme. The
Michelson-Morley experiment is necessarily confined to solids
of laboratory dimensions, in which internal gravitation has
no appreciable influence. There is, therefore, no experimental
proof that a body such as the earth, whose figure is determined
mainly by gravitation, will undergo the theoretical contraction
owing to motion. The most direct evidence that gravitation,
conforms to relativity comes from a discussion by Lodge* of
the effect of the sun's motion through the sether on the peri-
helia and eccentricities of the inner planets. If gravitation is
outside the relativity theory (the Newtonian law holding
unmodified) a solar motion of 10 km. per sec. would produce
perturbations in the eccentricities and perihelia of the earth
and Venus, which could probably be detected by observation.
The absence of these perturbations seems to show that gravita-
tion must conform to relativity, unless, indeed, the sun happens
to be nearly at rest in the sether. If we confine attention to
our local stellar system the average stellar velocities are not
so much greater than 10 km. per sec. as to render the latter
alternative too improbable ; but the very high velocities f our,d
for the spiral nebulae (which are thought to be distant stellar
systems) makes it improbable that our local system should be
so nearly at rest in the sether.
* " Phil. Mag.," February, 1918.
CHAPTER II
THE KELATIONS OF SPACE, TIME AND FOKCE.
9. An interesting aspect of the transformation of the
variables x, y, z, t to x', y', z', t f has been brought out by
Minkowsld. We consider them as co-ordinates in a four-
diniensional continuum of space and time. Choose the units
of space and time so that the velocity of light is unity, and set
t=it, where i /I.
The equations of transformation (4-2) become
y=y', z=z', T=p(r'-iux') . (9-1)
Let =ntan0, so that 6 is an imaginary angle. Then
/?=cos 6, and (9-1) becomes
x 2/cos T'sin 6, y=y', z=z', t=T'cos 0~f-a/sin B . (9-2)
Thus the transformation is simply a rotation of the axes of
co-ordinates through an imaginary angle B in the plane of XT.
We know that the orientation chosen for the space-axes,
x, y, z, makes no difference in Newtonian mechanics. The
principle of relativity extends this so as to include the axis T.
The continuum formed of space and imaginary time is perfectly
isotropic ; the resolution into space and time separately, which
depends on the motion of the observer, corresponds to the
arbitrary orientation in it of a set of rectangular axes.
10. From this point of view the strange conspiracy of the
forces of Nature to prevent the detection of our absolute
motion disappears There is no conspiracy of concealment,
because there is nothing to conceal. The continuum being
isotropic, there is no orientation more fundamental than any
other ; we cannot pick out any direction as the absolute time
any more than we can pick out a direction in space as the
absolute vertical, Up-and-down, right-and-left, backwards-
and-forwards, sooner-and-later,* equally express relations to
some particular observer, and have no absolute significance.
In Minkowski's famous words, " Henceforth Space and Time
in themselves vanish to shadows, and only a kind of union of
the two preserves an independent existence."
The scientific basis of the idea that some fundamental
division into space and time exists was the conception of the
aether as a material fluid, filling uniformly and isotropically a
particular space. It now seems clear that the aether cannot
have those material properties which would enable it to serve
as a frame of reference. Its functions seem to be limited to
those summed up in the old description " the nominative of
the verb ' to undulate.' "
Unfortunately the simplicity of this conception of the four-
dimensional continuum is only formal ; and natural pheno-
mena make a discrimination between r and the other variables
by relating themselves to an imaginary x, which we call the
time. In natural variables, x, y, z, t, this view of the trans-
formation as a rotation of axes becomes concealed, f
11. In the four-dimensional continuum the interval ds
between two point-events is given by
-ds*~dx z +dy z +dz*+dr* . . . (11-1)
which is unaffected by any rotation of the axes, and is therefore
invariant for all observers. The minus sign given to <5s 2 is an
arbitrary convention, and the formula is simply the generalisa-
tion of the ordinary equation
The fact that ds is measured consistently by all observers
who would obtain discordant results for dx, dy, 6z, dr separately,
is so important in our subsequent work that we shall consider
the nature of the clock-scale needed for its measurement.
"We have a scale AB divided into kilometres, say, and at
each division is placed a clock also registering kilometres.
* This applies to imaginary time. With real time, events which (aa
usually happens) are separated by a greater interval in time than in space
preserve the same order for all observers. But an event on the sun which
we should describe as occurring 2 minutes later than an event on the earth
might be described by another observer as 2 minutes earlier. (Both ob-
servers have corrected their observations for the light-time.)
f For a logical study of the properties of the continuum of space and real
time reference may be made to A. A. Robb, " A Theory of Time and Space "
(Camb. Univ. Press).
C 2
(The velocity of light being unity, a kilometre is also a unit of
time = Tooo"o"o sec -) When the clocks are correctly set and
viewed from A, the sum of the readings of any clock and the
division beside it is the same for all, since the scale-reading
gives the correction for the time taken by light in travelling
to A. This is shown in Fig. 2, where the clock-readings are
given as though they were being viewed from A.
Now lay the scale in line with the two events ; note the
clock and scale-reading, t l} o^, of the first event, and the
corresponding readings -t z , o- 2J of the second event ; then from
(11-1)
&'=(* a -* 1 ) a -(-2-< r i)* .... (H-2)
If the scale had been set in motion in the direction AB,
<r z~ (r i would have been diminished, owing to the divisions
having advanced to meet the second event. But the clocks
would have been adjusted differently, because A is now
FIG. 2.
advancing to meet the light coming from any clock, and the
clock would appear too fast (by the above rule) if it were not
set back. There are other second-order corrections arising
from the contraction of the scale and change of rate of the
clocks owing to motion ; but the net result is a perfect com-
pensation, and 6s 2 determined from (11-2) must be invariant,
as already proved.
It is clear that the whole (restricted) principle of relativity
is summed up in this invariance of ds, and it is possible to
deduce the equation of transformation (4-2) and our other
previous results by taking this as postulate.
When ds refers to the interval between two events in the
history of a particular particle it has a special interpretation
which deserves notice. If we choose axes moving with the
particle, dx, dy, <5z=0, so that ds=dt. Accordingly the variable
s is called the "proper-time," i.e., the time measured by a
clock attached to the particle.
12. Up to the present we have discussed a particular type
of transformation of co-ordinates, viz., that corresponding to
a uniform motion of translation. We now enter on the theory
of more general changes of co-ordinates.
The co-ordinates x, y, z, t of a particle trace a curve in four
dimensions which is called the world-line of the particle. If
we draw the world-lines of all the particles, light- waves and
other entities,we obtain a complete history of the configurations
of the Universe for all time. But such a history contains a
great deal that is necessarily outside experience. All exact
observations are records of coincidences of two entities in
space and time, that is to say,records of intersections of world -
lines.
It is easy to see that this is the case in laboratory experi-
ments or astronomical observations. Electrical measurements,
determinations of temperature, weight, pressure, &c., rest
finally on the coincidence of some indicator with a division
on a scale. Many of our rough observations depend on co-
incidences of light waves with elements of the retina, or the
simultaneous impact of sound-waves on the ear. It is true
that some of our external knowledge is not obviously of this
character. We estimate the weight of a letter, balancing it
in the hand ; this is based on a muscular sensation having no
immediate relation to time and space, but we fit this crude
knowledge into the exact scheme of physics by comparing it
with more accurate measures based on coincidences.
The observation that the world-lines of two particles intersect
is a genuine addition to knowledge, since in general lines in
space of three or four dimensions miss one another. We have
to build up our conception of the location of objects in space
and time from a large number of records of coincidences. It
is clear that we have a great deal of liberty in drawing the
world-lines, whilst satisfying all the intersections. Let us draw
the world-lines in some admissible way, and imagine them
embedded in a jelly. If the jelly is distorted in any way, the
world-lines in their new courses will still agree with observation,
because no intersection is created or destroyed.
Mathematically this can be expressed by saying that we may
make any mathematical transformation of the co-ordinates. If
we choose new co-ordinates a?', y', z', t', which are any four
independent functions of x, y, z, t, a coincidence in x, y, z, t
will also be a coincidence in a;', y', z', t', and vice versa. By
locating objects in the space-time given by x', y', z', t f , we do
not alter the course of events. The events themselves do not
presuppose any particular system of co-ordinates, and the
ourselves.
It is almost a truism to say that we may adopt any system
of co-ordinates we please. We are accustomed to introduce
curvilinear co-ordinates or moving axes without apology,
whenever they simplify the problem. But there is one point
not so generally recognised. Ordinarily when we use curvi-
linear co-ordinates we never allow ourselves to forget that
they are curvilinear ; it is a mathematical device, not a new
space, that we adopt. Perhaps the only case in which we
really take the new co-ordinates seriously is in the trans-
formation to rotating axes ; we then take account of the
rotation by adding a fictitious centrifugal force to the equations,
and thenceforth the rotation is quite put out of mind. From
the standpoint of relativity, when we adopt new co-ordinates
2', y', z', t', we shall adopt a corresponding new space, and think
no more of the old space. For instance, a " straight line " in
the new space will be given by a linear relation between
z', y', s', *'. _
The behaviour of natural objects will no doubt appear very
odd when referred to a space other than that customarily used.
So-called rigid bodies will change dimensions as they move ;
but we are prepared for that by our study of the Michelson-
Morley contraction. Paths of moving particles will for no
apparent reason deviate from the " straight line," but, accept-
ing the definition of a force as that which changes a body's
state of rest or motion, this must be attributed to a field of
force inherent in the new space (cf. the centrifugal force).
Light-rays will also be deflected, so that the field of force acts
on light as well as on material particles ; this is not altogether
a novel idea, because a little reflection shows that the centri-
fugal force deflects light as well as matter- although optical
problems are not usually treated in that way.
13. The laws of mechanics and electrodynamics are usually
enunciated with respect to " unaccelerated rectangular axes,"
or, as they are often called, " Galilean axes." We cannot
regard such axes as recognisable intuitively, and the only
definition of them that can be given is that they are the axes
with respect to which that particular form of the laws holds.
It is part of the method of the present theory to restate the
laws of Nature in a form not confined to Galilean co-ordinates,
so that all systems of co-ordinates are regarded as on the same
footing.
THE RELATIONS OF SPACE, TIME AND FORCE. lU
In unaccelerated rectangular co-ordinates the path, of a
particle is a straight line (apart from the influence of other
matter, or the electromagnetic field). When we transform to
other co-ordinates the path is no longer straight, i.e., it is no
longer given by a linear relation between the co-ordinates ;
and the bending of the path is attributable to a field of force
which comes into existence in the new space. This field of
force has the property that the deflection produced is inde-
pendent of the nature of the body acted on, being a purely
geometrical deformation. Now the same property is shared
by the force of gravitation the acceleration produced by a
given gravitational field is independent of the nature or mass
of the body acted on. This has led to the hypothesis
that gravitation may be of essentially the same nature
as the geometrical forces introduced by the choice of
co-ordinates.
This hypothesis, which was put forward by Einstein, is
called the Principle of Equivalence. It asserts that a gravita-
tional field of force is exactly equivalent to a field of force introduced
by a transformation of the co-ordinates of reference, so'that ~by no
possible experiment can we distinguish between them.
In Jules Verne's story, " Round the Moon," three men are
shot up in a projectile into space. The author describes their
strange experiences when gravity vanishes at the neutral point
between the earth and moon. Pedantic criticism of so de-
lightful a book is detestable ; yet perhaps we may point out
that, for the inhabitants of the projectile, weight would vanish
the moment they left the cannon's mouth. They and their
projectile are falling freely all the time at the same rate, and
they can feel no sensation of weight. They automatically
adopt a new space, referred to the walls and fixtures of their
projectile instead of to the earth. Their axes of reference are
accelerated falling towards the earth ; and this transforma-
tion of axes introduces a field of force which just neutralises
the gravitational field. But, whilst they could ^detect no
gravitational field by ordinary tests, it is not obviously im-
possible for them to detect some effect by optical or electrical
experiments. According to the principle of equivalence, how-
ever, no effect of any kind could be detected inside the pro-
jectile ; the gravitational field cannot be differentiated from a
transformation of co-ordinates, and therefore the same trans-
formation which neutralises mechanical effects neutralises all
other effects.
It will be seen, that this principle of' equivalence is a natural
generalisation of the principle of relativity. An occupant of
the projectile can only observe the relations of the bodies inside
to himself and to each other. The supposed'absolute accelera-
tion of the projectile is just as irrelevant to the phenomena as
a uniform translation is. The mathematical space-scaffolding
of Galilean axes, from which we measure it, has no real
significance. If the projectile were not allowed to fall, gravity
would be detected or rather the force of constraint which
prevents the fall would be detected. I think it is literally true
to say that we never feel the force of the earth's attraction on
our bodies ; what we do feel is the earth shoving against our
feet.
14. A limitation of the Principle of Equivalence must be
noticed. It is clear that we cannot transform away a natural
gravitational field altogether. If we could, we should un-
consciously make the transformation and adopt the new co-
ordinates just as the inhabitants of the projectile did. They
were concerned with a practically infinitesimal region, and
for an infinitesimal region the gravitational force and the
force due to a transformation correspond ; but we cannot find
any transformation which will remove the gravitational field
throughout a finite region. It is like trying to paste a flat
sheet of paper on a sphere, the paper can be applied at any
point, but as you go away from the point you soon come to a
misfit. For this reason it will be desirable to define the exact
scope of the principle of equivalence. Up to what point are
the properties of a gravitational field and a transformation
field identical ? And what properties does a gravitational
field possess which cannot be imitated by a transformation ?
The impossibility of transforming away a gravitational field is,
of course, an experimental property ; so that, in spite of the
principle of equivalence, there is at least one means of making
an experimental distinction-
Space-time in which there is no gravitational field which
cannot be transformed away is called homaloidal. In homa-
loidal space-time then, we can choose axes so that there is no
field of force anywhere. Remembering that we have no means
of defining axes except from the form of the laws of Nature
referred to them, we should naturally take these axes as
fundamental and name them " rectangular and unaccelerated. "
.The dynamics of homaloidal space would not recognise the
existence of gravitation. Our space is not like that, though
we believe that at great distances from all gravitating matter
it tends towards this condition as a limit. The necessary
limitation of the principle of equivalence turns on the number
of consecutive points for which gravitational space-time agrees
with homaloidal space-time ; in other words, the equivalence
will hold only up to a certain order of differential coefficients.
Properties involving differential coefficients up to this order
will be the same in the gravitational field as in a homaloidal
field ; whilst properties of the transformed field involving
differential coefficients of higher order will not necessarily hold
in the gravitational field
The determination of the order of the differential coefficients
for which agreement is possible must be deferred to 27.
Meanwhile it may be noted that we can always choose axes
for which the field at a given point vanishes viz., take rect-
angular axes moving with the acceleration at that point. In
that case we are said to use " natural measure."
15. At a point of space where there is no field of force the
observer's clock-scale, if unconstrained, will be either at rest
or in uniform motion. We have seen that the measured
interval, ds, between two events is independent of uniform
motion, and hence a unique value of ds is determined by the
measures.
Using rectangular co-ordinates, the relation between an
infinitesimal measured interval ds and the inferred co-ordinates
of the event is (11-1).
ds z =--dx z -dy z -dz z +dt z . . . . (15-1)
Introduce new co-ordinates x 1} x 2 , x s , x 4 , which are any
functions of x, y, z, t given by
s=/iK, a> 2 > x*> x *)> y=fz( x i> 2> KZ> x ^ &c -
Then dxJ^dx^dx z + d ^dx s +lldxv &c. . (15-2)
dx-i ox?. 3^3 94
Substituting (15-2) on the right-hand side of (15-1), we
obtain a general quadratic function of the infinitesimals, which
may be written,
g^dxl+g^dxl+g^dxl^g^dx^x^g^dx^
l dxi+2g z3 dx z dx2+2g 2ll dx 2 dXi i - ! r 2gudx 3 dxi . (15-3)
where the </'s are functions of the co-ordinates, depending on
the transformation.
x -, cos (0X4 x z sin
x sin a)x 4 +ic. cos
Whence
efcr cos a)X i .dx l sin ciXE 4 .^av~ eo^ sin
% sin cox^dx-L-}- cos coa; 4 .^a; 2 4-co(a; 1 cos <w# 4 a; 2 sn
Substituting in (15-1)
x^dx t ...... (15-5)
By comparing this with (15-3) we obtain the values of the y's
for this system of co-ordinates.
16. These values of the </'s express the metrical properties of
the space that is being used. But the observer has no im-
mediate perception of them as properties of space. He does
not realise that there is anything geometrically unnatural
about axes rotating with the earth, but he perceives a field of
centrifugal force. Experiments, such as Foucault's pendulum
and the gyro-compass, designed to exhibit the absolute
rotation of the earth, are more naturally interpreted as de-
tecting this field of force.
Thus the coefficients g^, &c., can be taken as specifying a
field of force. That they are sufficient to define it completely
may be seen from the following consideration. The world-line
of a particle under no forces is a straight line in the system
x, y, z, t, and its equation may be written in the form
ds is stationary ; (16-1)
but in this form the equation is independent oi the choice
of co-ordinates, and applies to all systems. If we choose
new co-ordinates, the world-line given by (16-1) becomes
curved and the curvature is attributed to the field of force
introduced ; but clearly the curvature of the path can only
depend on the expression for ds in the new co-ordinates, i.e.,
on the </'s. Thus the force is completely defined by the y's.
It will be noticed that in (15-5)
044=1 -2 Q, (16-2)
where Q=|a) t (^-f ^)--the potential of the centrifugal force.
Thus <7 44 can be regarded as a potential ; and by analogy
all the coefficients are regarded as components of a generalised
potential of the field of force.
According to the principle of equivalence it must also be
possible to specify a gravitational field by a set of values of
the j/'s. It will be our object to find the differential equations
satisfied by the </'s representing a gravitational field. These
differential equations for the generalised potential will express
the law of gravitation, just as the Newtonian law is expressed
by V 2 9-0.
The double aspect of these coefficients, g llt &c., should be
noted. (1) They express the metrical properties of the co-
ordinates. This is the official standpoint of the principle of
relativity, which scarcely recognises the term " force."
(2) They express the potentials of a field of force.
This is the unofficial interpretation which we use when
we want to translate our results in terms of more familiar
conceptions.
Although we deny absolute space, in the sense that we regard
all space-time frameworks in which we can locate natural
phenomena as on the same footing, yet we admit that space
the whole group of possible spaces may have some absolute
properties. It may, for instance, be homaloidal or non-
homaloidal. Whatever the co-ordinates, space near attracting
matter is non-homaloidal, space at an infinite distance from
matter is homaloidal. You cannot use the same co-ordinates
for describing both kinds of space, any more than you can use
rectangular co-ordinates on the surface of a sphere ; that is,
in fact, the geometrical interpretation of the difference.
Homaloidal space-time may be regarded as a four- dimensional
plane drawn in a continuum of five dimensions ; whereas
non-homaloidal space-time must be regarded as a curved
surface in five dimensions.* These considerations apply, of
course, to measured space ; we can always throw the blame
on our measuring rods, and apply theoretical corrections to
* We shall see (44) that in a region, not containing matter, but traversed
by a gravitational field due to matter, the Gaussian or total curvature is
zero ; but such a space-time does not correspond to a plane in five dimen-
sions, or to any surface which can be developed into a plane. The space-
time in a gravitational field lias an essential curvature in the ordinary
sense, although it happens that the particular invariant technically called
" the curvature " vanishes. In three-dimensional space a surface with zero
Gaussian curvature can always be developed into a plane ; but this is not
true for space of higher dimensions, so that the three-dimensional analogy
is b'able to lead to misunderstanding.
our measures so as to make them agree with any kind of space
we please.
It is not necessary, and indeed it is not possible, to draw a
sharp distinction between the portions of the g's arising from
the choice of co-ordinates and the portions arising from the
gravitation of matter. We have seen that, when there is no
field of force, ds z has the form (15-1), so that the #'s have the
values > -1000
0-100
0-1
0001
(16-3)
These values then express that there is no field of force,
and in the absence of a gravitational field produced by matter
it is possible to take our co-ordinates (Galilean co-ordinates)
so that the values (16-3) hold everywhere. We naturally
regard such co-ordinates as. fundamental ; and, if we choose
any other co-ordinates, the deviations of the g's from this
peculiarly simple set of values are regarded as due to the
distortion of the space-time chosen. But by 14, when
gravitating matter is in the neighbourhood, there is no possi-
bility of choosing co-ordinates, so that the values (16-3) hold
everywhere, and there is no criterion for selecting any one of
the possible systems of co-ordinates as more fundamental than
the others.*
Accordingly we shall henceforth apply the term " gravita-
tional field ?) to the whole field of force given by the g's, what-
ever its origin. In the particular case when no part of it is
due to the gravitation of matter, we shall say there is no
permanent gravitational field.
Just as Galilean co-ordinates are defined by the values (16-3)
of the g's, so any other co-ordinates must be defined analytically
by specifying the g's as functions of x 1} x z , x 3 , x it or what
comes to the same thing by giving the expression for ds 2 .
For example, if in two dimensions ds z =dx^-\-x^dx^, the
co-ordinates are recognised as plane polar co-ordinates with
* Thus if we say " take rectangular axes with the sun as origin," the
statement is ambiguous. Unaccelerated rectangular axes imply that do* is
of the form (15-1) -no other means of defining them having yet been given.
Owing to the sun's gravitation there is no system of co-ordinates for -which
this is true, and several different systems present rival claims to be regarded
as the best approximation possible. The difficulty does not arise if we only
have to consider an infinitesimal region of space ; in that case the co-ordinates
{giving " natural measure ") are defined without ambiguity.
x^r, x z =0 ; if ds 2 =dx^-}-cos z x l dx z 2 , the co-ordinates are
latitude (x^) and longitude (x z ) on a sphere. We might take
for the 10 gr's perfectly arbitrary functions of x ls x z , x 3 , a; 4 ,
and so obtain a ten-fold infinity of mathematically conceivable
systems of co-ordinates. But this would include many systems
of co-ordinates which describe kinds of space-time not occurring
in Nature. In any particular problem our choice is restricted
to a four-fold infinity, viz., if x v X 2 , x s , x is a possible system,
then four arbitrary functions of x l} x 2 , x s , cc 4 will form a possible
system. In some other problem there will be an entirely
different group of possible systems ; the space-times in the
two problems have thus certain absolute properties which are
irreconcilable, and we interpret this physically by saying that
the permanent gravitational field is different in the two cases.
Further, taking all possible distributions of permanent gravi-
tational field which can occur in space (in the neighbourhood
of, but not containing, matter), we do not exhaust the con-
ceivable variety of functions expressing the </'s. There is a
general limitation on the g's imposed, not by mathematics,
but by Nature which is expressed by the differential equa-
tions of the law of gravitation which we are about to seek.
The law of gravitation, in fact, expresses certain absolute
properties common to all the measured space-times that can
under any conditions occur in Nature.
The law of gravitation, or general relation connecting the gr's,
must hold for all observed values of the 0's. Since the f/'s
define the system of co-ordinates used, this means that the
relation must hold for all possible systems of co-ordinates.
If new co-ordinates are chosen, we find new values of the g's
as in (15-5) ; and the differential equations between the new
<7's and new co-ordinates must be the same as between the old
g's and old co-ordinates. In mathematical language the equa-
tions must be covariant.
There is a resemblance between this statement and the
statement of 12 which is somewhat deceptive. We there
found that observable events have no reference to any parti-
cular system of co-ordina,tes, and therefore all laws of nature
can be expressed in a form independent of the co-ordinates.
But this alone does not allow us to deduce the covariance of
the equations satisfied by the gravitation-potentials. Without
the principle of equivalence we could no doubt define the field
by certain potentials 9^ cp 2 > <Ps> which satisfy differential
equations independent of the choice of co-ordinates. But that
conveys no information of value, unless we are told how to
find 9(, 9 a ', .... in the co-ordinates x(, x' 2 , x' 3 , x.' from the
values cp l5 <p 2 , .... in the co-ordinates ajj, , 3 , 4 . The
statement in 12 tells us nothing about that. It is the prin-
ciple of equivalence which, by identifying the potentials with
the </'s for which the method of transiormation is known,
supplies the missing link-
17. The Newtonian law of gravitation, \f 2 g M =Q, does not
fulfil the condition of covariance nor does any modification of
it, which immediately suggests itself. We have, therefore, to
seek a new law guided by the condition that it must be ex-
pressed by a covariant set of equations between the g's. It
will be found in Chapter IV. that the choice is so restricted
as to leave little doubt as to what the new law must be.
If we write the required equations in the form
r t =o, r 2 =o, T 3 =o, & c .,
the left-hand sides, 2\, T 2 , jP 3 , may be regarded as components
of a kind of generalised vector, only the number of components
is not, as in a vector, restricted to 4.
The covariance of the equations means that, if all the com-
ponents vanish in one system of co-ordinates, they must vanish
in all systems. To secure this, T, T z , .... must obey a
linear law of transformation ; thus
r i / =A 1 2 1 1 +A 2 T 2 +A 3 T 3 -h ...... (17-1)
where the coefficients are functions of the co-ordinates de-
pending on the transformation. Generalised vectors of this
kind are called tensors ; and it will be necessary for us to
study their properties in the next chapter, in order to select
the one which can represent the new law of gravitation.
We see that if an equation is known to be a tensor-equation,
it is sufficient to prove it for one particular system of co-
ordinates ; it will then automatically hold in any other system
obtainable by a mathematical transiormation.
The more general purpose of the tensor theory is this :
If we are given a set of equations expressing some physical
law in the usual co-ordinates, we may be able to recognise
these as the degenerate form for Galilean co-ordinates of some
tensor equation. Expressed in tensor form, these equations
will then hold for all systems of co-ordinates that can be
derived by a mathematical transformation. Subject to the
. *
of no great interest, since the mere transformation of co-
ordinates leads to nothing new. But by this mode of approach
we obtain the corresponding equations as modified by the action
of a gravitational field. This is a very powerful method of
investigation.
18. I anticipate that some readers will find the next two
chapters difficult, and I therefore place here, out of order, a
brief account of the field of a particle according to the new
law of gravitation ; but I doubt if there is any royal road to
relativity, and it is scarcely possible to make serious progress
except by analytical methods.
We shall find that when a heavy particle is at rest at the
origin the expression for the line-element in plane polar co-
ordinates is
.... (18-1)
where y=l2m/r, and m is the mass of the particle, the
constant of gravitation and the velocity of light being unity.
For the sun, m= 147 kilometres*, so that y generally differs
from unity by a very small quantity.
If l=m/r= Newtonian potential at the point considered,
we have , on , 1Q o\
44 =y=l-2Q ..... (18-2)
just as in the case of the centrifugal force (16-2). The New-
tonian attraction is therefore a consequence of the coefficient
of dt z .
The general meaning of (18-1) is that our measures will not
fit together in Euclidean space. Measuring in the direction r
we have
that is to say we must correct the measured length ids in the
radial direction, multiplying it by j>i in order to obtain a length
dr which will fit into Euclidean space. Or we may say that our
measuring rod contracts when placed radially ; transverse
measures require no correction. Similarly the measured time
must be multiplied by y~\, i.e., our clocks run slow.
* This can be verified roughly as follows : For a circular orbit m/r 2 =v*/r,
the constant of gravitation being unity. Applying this to the earth,
v=earth's orbital velocity=30 km. per sec.= 10~ 4 x velocity of light. Hence
in our units m=10~ 8 r ; and r, the radius of the earth's orbit, is 1-5 x 10 8 km.
But there is more than one way of correcting the measures
to fit Euclidean space, so that we are not really justified in
making precise statements as to the behaviour of our clocks
and measuring rods. It is better not to discuss their defects,
but to accept the measures and examine the properties of the
corresponding non-Euclidean space and time.
If we draw a circle with a heavy particle near the centre,
the ratio of the measured circumference to the measured
diameter will be a little less than TT, owing to the factor y~\
affecting radial measures. It is thus like a circle-drawn on a
sphere, for which the circumference is less than n times the
diameter if we measure along the surface of the sphere. We
may imagine space pervaded by a gravitational field to have
a curvature in some purely mathematical fifth dimension.
If we draw the elliptic orbit of a planet, slit it along a radius
and try to fold it round our curved space there will evidently
be some overlap. For example, take a cone with the sun as
apex as roughly representing the curved space. Starting with
the radius vector SP, the Euclidean space will fold completely
round the cone and overlap to the extent PSP'. Thus the
corresponding radius advances through an angle PSP' each
revolution (Eig. 3). This shows one reason for the advance of
perihelion of a planet, which is one of the most important
effects predicted by the new theory ; but it is not the whole
explanation.
The reader may not unnaturally suspect that there is an
admixture of metaphysics in a theory which thus reduces the
gravitational field to a modification of the metrical properties
of space and time. This suspicion, however, is a complete
misapprehension, due to the confusion of space, as we have
defined it, with some transcendental and philosophical space.
THE RELATIONS OF SPACE, TIME AND FOECE. 29
There is nothing metaphysical in the statement that under
certain circumstances the measured circumference of a circle
is less than n times the measured diameter ; it is purely a
matter for experiment. We have simply been studying the
way in which physical measures of length and time fit together
just as Maxwell's equations describe how electrical and
magnetic forces fit together. The trouble is that we have
inherited a preconceived idea of the way in which measures,
if " true," ought to fit. But the relativity standpoint is that
we do not know, and do not care, whether the measures under
discussion are " true " or not ; and we certainly ought not to
be accused of metaphysical speculation, since we confine our-
selves to the geometry of measures which are strictly practical,
if not strictly practicable. It is desirable to insist that we do
not attribute any causative properties to these distortions of
measured space and time. To hold that a property of our
measuring-rods is the cause of gravitation would be as absurd
as to hold that the fall of the barometer is the cause of the
storm.
CHAPTER HI.
THE THEORY OF TENSORS.
19. We consider transformations from one system of co-
ordinates x l} x z , # 3 , 4 to another system x[, x'. t , x^, x\.
(a) Notation.
The formula (15-3) for ds* may be written
ds*=Z S g^dx^dx, (^=sr w ) . . . (19-11)
jli = l V = l
In the following work we shall omit the signs of summation,
adopting the convention that, whenever a literal suffix appears
twice in a term, the term is to be summed for values of the
suffix 1, 2, 3, 4:. If a suffix appears once only, no summation
is indicated. Thus we shall write (19-11)
ds 2 =g^dx^dx v ...... (19-12)
In rare cases it may be necessary to write a term containing
a suffix twice which is not to be summed ; these cases will
always be specially indicated. In general, however, this coa-
vention anticipates our desires, and actually gives a kind of
momentum in the right direction to the analysis.
As a rule of manipulation it may be noticed that any suffix
appearing twice is a dummy, and can be changed freely to any
other suffix not occurring in the same term.
(b) Covariant and Contravariant Vectors.
The vector (dx v dx z , dx 3 , dx^} is transformed according to
the equations
or, with our convention as to notation
^
contravariant vector ; its character is denoted by the notation
A* (//=!, 2, 3, 4). The law may be written
"dx'
A!*=z-A* ...... (19-21)
0*0-
where, as already explained, summation is indicated by the
double appearance of the dummy o-.
If 9 is a scalar (i.e., invariant) function of position the vector
/3<p 39 89 89 \ . , j. j ,. , ^ .
( ~, - L - > L , - 1 - ) is transformed according to the law
\O3/i c/^g 0X3 9 4 /
89 __ 3a/i^9_ 3 2 89 9ag 3 39 9x 4 99
a<~3< 3^1 3< a^T^^?^ 9<3i
A vector transformed according to this law is called a
covariant vector, denoted by A IJL . The law may be written
<= ...... (19-22)
c-v
A covariant vector is not necessarily the gradient of a scalar.
The customary geometrical conception of a vector does not
reveal the distinction between the two classes of contravariant
and covariant vectors. We usually represent any directed
quantity by a straight line, which should strictly correspond
only to a contravariant vector. The other class of directed
quantities is more properly represented by the reciprocal of a
straight line ; but in elementary applications, when we are
thinking in terms of rectangular co-ordinates, there is no need
to make this distinction. Consider, however, a fluid with a
velocity potential. With rectangular co-ordinates the velocity
is equal to the gradient of the velocity potential. Both these
are directed quantities, i.e., vectors, and the vector relation
extends to their rectangular components ; thus
dx _ 3o dy _ 99 dz _ 39
dt 'dx dt dy 3 dt dz'
But if we use oblique axes or curvilinear co-ordinates, the
relation no longer holds. E.g., it is not true that in polar co-
ordinates ddfdl^=d^/dO ; the actual relation is rdQldtdyfrdQ.
This is because the two vectors are of opposite natures, the
first being contravariant and the second covariant. If they
had been of the same nature the relation must have held for
all systems of co-ordinates. Clearly, since in our work we
consider all systems of co-ordinates as on the same footing,
re have to distinguish carefully between the two types. We
D2
realise at once that the equation dx^/dtd^/dx^, being an
equation between vectors of opposite kinds, is impossible as a
general equation for all systems of co-ordinates, i.e., it is not
a covariant equation.
(c) Tensors of Higher Rank.
We can denote by A^ v a quantity having 16 components,
obtained by giving difierent numerical values to p and ".
Similarly, A^ has 64 components. By a generalisation of
(19-21) and (19-22) we classify quantities of this kind according
to their transformation laws, viz.,
n ... dx a dx T /iQoiv
Covariant tensors A'- -.A . . . . (19-31)
dx^ dx v
Contravariant tensors A'^=~-^- Jf A (rT . . . . (19-32)
fAry j-)/y x '
t/(y \J"^f
Mixed tensors 4^=r-7 "^ .... (19-33)
and similarly for tensors of the third and higher rank. These
equations of transformation are linear, so that the conditions
of 17 are satisfied. Also it is not difficult to see that there
can be no other linear types of transformation-laws having the
necessary transitive property. For example, consider a vector
A,. Introducing a third set of co-ordinates x, we have
i> ^M , , / 3#<r
\J lC\ (j iA/ j
showing that the result is the same whether the transformation
is performed in two steps or directly. Other suggested tvpes
of transformation law have not this necessary property. Thus
all possible types of tensors are included.
Evidently the sum of two tensors of the same character is a
tensor.
The product of two tensors is a tensor, and its character is
the sum of the characters of the component tensors. For
example, consider the product A^B^, we have by (19-31) and
(19-33) ,. ,, ,
Hence (^^)-(^^) - - (19-84)
showing that the law of transformation is that of a tensor of
the fourth rank having the character denoted by C^.
The product of two vectors is a tensor of the second rank,
but a tensor of the second rank is not necessarily the product
of two vectors.
A familiar example of a tensor of the second rank is afforded
by the stresses in a solid or viscous fluid. The component of
stress denoted by p^ represents the traction in the ^-direction
exerted across an interface perpendicular to the x-direction.
Each component involves a specification of two directions.
(d) Inner Multiplication.
If we multiply A^ by B*, the repetition of the suffix involves
summation of the resulting products. The result is called the
inner product in contrast to the ordinary or outer product A^B".
The notation at once shows whether the product is inner or
outer in any formula.
From a mixed tensor such as A^ vff we can form a " contracted "
tensor A^ v<r , which is of the second rank with suffixes fj, and v
(since <r is now a dummy suffix). To show that it is a tensor
we have as in (19-34)
_ ^ fix-fix^ dx y ... ... .
But r- J >TT 2 = 5-^=0 or 1, according as y -/= o or y=o.
Substituting in (1941) we see that A^ follows the law of
transformation (19-31) and is therefore a covariant tensor.
An expression such as A^ ffa . is not a tensor, and no interest
attaches to it.
By a similar argument we see that A, A%, are invariant,
and consequently A^B* is an invariant. An invariant, or
scalar, corresponds to a tensor of zero rank.
(e) Criterion for the Tensor Character.
To prove that a given quantity is a tensor, we either find
directly its equations of transformation, or we express it as
the sum or product of other tensors, or, under certain re-
strictions, as the quotient of two tensors according to the
following theorem : A quantity, which on inner multiplication
by any covariant (alternatively, by any contravariant) vector
always gives a tensor, is itself a tensor.
To prove this, suppose that A llr B v is a covariant vector for
any choice of the contravariant vector B". Then by (19-22)
But by (19-21) applied to the inverse transformation from
accented to unaccented letters,
Hence
Since B' v is arbitrary, the quantity in the bracket must
vanish, showing that A^ is a covariant tensor (19-31). The
proof can evidently be extended to tensors of any character.
20. (a) The Fundamental Tensor.
Since g llv dx l> dx v ds' i , which is an invariant or tensor of zero
order, and dx v is an arbitrary contravariant vector, it follows
from the last theorem that g^dx^ is a covariant tensor of
the first rank. Repeating the argument, since dx^ is an
arbitrary contravariant vector, g^ v must be a covariant tensor
of the second rank.
The determinant formed with the elements g^ is called the
fundamental determinant and is denoted by g.
We define g 1 " 1 to be the minor of g^ v divided by g.
From this definition g^g^ reproduces the fundamental
determinant divided by itself, when or = v, and gives a deter-
minant with two rows identical, when <r=W. We write
S= W^l when o- = v 1
=0 when <r=/=v ' /( 2(M )
Hence if A" is an arbitrary contravariant vector
fcA'^A'+O+Q+O^A* .... (20-15)
This shows by the theorem of 19 (e) that g v v is a tensor,
and it evidently is a mixed tensor as the notation has antici-
pated.*
* In applying the theorem of 19(e), the appropriate notation for the
tensor (expressing its covariant or contravariant character) is found by
inspection. An equation such as (20-15) must have the suffixes on both
sides in corresponding positions ; the upper and lower a- on the left cancel
one another; O/. equations (20-21), (20-22), (20-23). It must be noted,
however, that in an expression such as g^x^ dx^ is contravarianb, so that
the second n is really an upper suffix.
Similarly, since g^A" is a covariant vector, arbitrary on
account of the free choice of A", and cfq^A'A^ <f must
be a contravariant tensor.
We have thus the three fundamental tensors
y^, c/:, and M ",
of covariant, mixed and contravariant characters.
It will be seen from (20-15) that g v , acts as a substitution
operator substituting v for o- in the operand.
(6) Associated Tensors.
With any covariant tensor A M1 , we can associate
a mixed tensor A v a =g v *A IUi (20-21)
a contra variant tensor A+ v =g' ai g v *A oA =tf M A'i . . . (20-22)
a scalar A=g fJ " > 'A /J , v =A^, .... (20-23)
(c) The Jacobian.-
Denoting the determinant formed with elements a^ by
IGU|, the Jacobian of the transformation is
T
Now
X
x
by (19-31)
dx \
^ I X \g n | (not summed),
since in our notation the ordinary rule for multiplying deter-
minants is \A a? | X | B ay I = | A^B^ | (left side not summed).
Hence y'JzV-
If di is an element of four-dimensional volume, we have
dr'Jdr,
so that V^.di^V-g'.di' .... (20-3)
We shall always assume that the Jacobian is finite, i.e.,
that the transformation has no singularity in the region con-
sidered. The determinant g is always negative for real trans-
formations.
21. Auxiliary Formula for the Second Derivatives.
We introduce certain quantities known as Christoffe
3-index symbols, viz.,
1
-1
We have {/*v, # =^ a |>, a] . . . . (21-1
and the reciprocal relation follows by (20-1)
[/tv, X\=g^ \[AV, a] .... (21-1
Since g^ is a covariant tensor
, 9a?g 9^
^""""a-r'ar^"' 3 '
TT O'*'u, C/*-!/
Hence M
^a^^^ /Oil
In the second term in the bracket we have interchang
a and /?, which is legitimate since they are dummies ; in t
last term we have used
_9 __ dXy d
o- -i i 9a?x ~dx( fix'
Similarly, 7
where in the last term we have made some interchanges
the dummy suffixes a, /5, y.
Adding these two equations and subtracting (21-15) we ha
by (21-11) ~ z ~ o a
r n/ w ''a oaJ/3 PX a 02;^ oa/ yr _ n /oii
^fl--Sx"S + ^^^ w> '' 1 (
dx
Multiply through by g f * p 7T L ,, we have
usbg (20-1) and (21-13).
it we diiierentiate a scalar quantity we obtain a tensor (a
covariant vector) ; but if we differentiate a tensor of the first
or higher rank the result is not a tensor. We can, however,
obtain a tensor which plays the part of a derivative by. a more
general process. The process is particularly useful in gene-
ralising results which have been obtained in Galilean co-
ordinates, since the simple derivative is the degenerate form
for Galilean co-ordinates of the covariant derivative here
considered.
If A,,, is a covariant vector, then by (19'22)
A'-^A
A ~d*r-
Whence, differentiating,
dA^Jdx^d^r ddy d 2 x<r
dxt ~a< dxidxr + a< dxt *
Substitute for d z x ff jdx^ dx u ' by (21-2) ; we have
dA t , , 9z ff d' d x r 3 A * d^a 9 /3 , , A /00 n N
M-fav, 9} A *W~wM^~WW v {a ^ }A (224)
(\np
But Ay^-f^A'p by (19-22); and in the last term the dummies
a, fi, a may be replaced by T, r, p. Hence if we write
r\A
A^^-{^ P }A P .... (22-2)
r i t O^<T O^r A
we have A V =^~,A^
showing that A^ is a tensor. This is called the covariant de-
rivative of Ap.
If A\, B^ are covariant vectors, A Kv , B^ their covariant de-
rivatives, then .
is the sum of two tensors, and is therefore a tensor. Sub-
stituting from (22-2) this tensor becomes
-{Av, 8 MA-{/iv,6}4 x JJ. . .(22-3)
nence izz-o] can oe generalised, giving iur
derivative of A
(22-4
In a somewhat similar manner formulae for the covarian
derivatives of contravariant and mixed tensors can be ob
tained, viz., .
(22-5
. . (22-6
. . . (22-7
The unsymmetrical behaviour of covariant and contra
variant indices in these formulae should be noticed. In al
cases differentiation adds one unit of covariant character.
When the </'s have Galilean values (or, more generally, ar<
constants) the Christoffel symbols vanish, and these derivative
reduce in all cases to the ordinary differential coefficients.
23. The Riemann-Christoffel Tensor.
Let us form the second covariant derivative of the vector A f
that is to say in formula (224) we give the tensor A^ th
value (22-2).
p} ^[ p + { ( oa-, e} {ev, p} 4 p -4 P ^- {/l*v, p
OS/o-
The first five terms are unaltered by interchanging v and i
i.e., by changing the order of differentiation. (We can writ
e for p in the second term. ) Hence
A A
fi iai<r ci ij.<rv
P }-{juv,e} {ecr, pj+l-^, p} - s ~{/*v,
The left side is a tensor, and A f is an arbitrary covariant
vector ; therefore, by 19 (e] the quantity in the bracket is a
tensor. This is called the Riemann-Christoffel tensor, and is
denoted by
^={f^r, }{", p} - \fJlv f s] {go-, p}-f JL \fffr f p } -L fa V) p | (23)
OX V OXg-
24. Conditions for Vanishing of the Riemann-Christoffel
Tensor.
From the foregoing definition the primary meaning of the
vanishing of this tensor is that the order of differentiation is
indifferent (as in the ordinary differentiation). But the tensor
has an even more important property. It will be seen on
inspection that it vanishes when the gr's have their constant
Galilean values.* But, since it is a tensor, it must also vanish
in any other system of co-ordinates derivable by a mathe-
matical transformation. Thus the equation
3^=0, ....... (24-1)
is a necessary condition that with suitable choice of co-ordinates
ds* can be reduced to the form
ds*=-dx*-dy*-dz*+dt*. . . . (24-2)
In other words it is a necessary condition for the absence of
a permanent gravitational field.
It can be shown that the condition is also sufficient.
Equation (24-1) contains 96 apparently different equations,
since, owing to the antisymmetry in <r and v, there are only 6
combinations of <r and v to be combined with 16 combinations
of jji and p. But these are not all independent, and the
number can be reduced to 20, which can be shown, to be the
number of conditions required for the transformation to the
form (24-2) to be possible.
The reduction is effected by writing
so that B^ ff =g^(f4^rv) by (20-1)
Equation (24-1) is thus equivalent to
(/iT<ri/)=0,
and vice versa.
* The Christoffel symbols vanish when the CJ'B are constants.
On working out the value of (ptw} it is seen by inspectioj
that the following additional relations exist : r
(/ui(rv)~\- {/MT
which reduce the number of independent conditions to 20.
25. To sum up what has been accomplished in this chaptei
we have discussed the theory of tensors expressions whicj
have the property that a linear relation between tensors o
the same character will hold in all systems of co-ordinates if i
holds in one system. We have shown that the tensor-propert;
can be established either by determining the law of transf orma
tion, or exhibiting the quantity as a sum or product of othe
tensors, or, under certain restrictions, as the quotient of tensors
We have found formulas for tensors which play the part o
derivatives. Finally, we have found the necessary an<
sufficient relation between the g^, which must be satisfied ii
all systems of co-ordinates, when there is no permanen
gravitational field. 4,.-
This last result is an important step towards obtaining thi
law of gravitation. Any set of values of the gr's which satisfy
(24-1) will correspond to a possible set of co-ordinates whicl
can be used for describing space not containing a permaneni
gravitational field. Hence if (24-1) is satisfied the 0's are sucl
as can occur in Nature, and are accordingly not inconsistenl
with the law of gravitation. The required equations of th<
law of gravitation must, therefore, include the vanishing o:
the Riemann-Christoffel tensor as a special case.
CHAPTEK IV.
EINSTEIN'S LAW OF GRAVITATION.
26. We have seen in 16 that the law of gravitation must
be expressed as a set of differential equations satisfied by the
0's. We have further found the equations (24-1) which are
satisfied in the absence of (i.e., at an infinite distance from)
attracting matter. Clearly the general equations between the
</'s must be covariant equations automatically satisfied when
(24-1) is satisfied ; but they must be less stringent, so as to
admit of permanent gravitational fields, which, we know, 3o
not satisfy (24-1).
The simplest set of equations that suggests itself is
GU=BM) ...... (26-1)
GUV being the contracted Biemann-Christoffel tensor, formed
by setting cr=p and summing. It is evidently satisfied when
all components of the Riemann-ChristofEel tensor vanish ; and
it is a less stringent condition.
The equations 6r Ml ,=0 are taken by Einstein for the Law
of Gravitation. Written in full they are, by (23)
o
{/**, P> ", p 9 j ?, ~ ; >
(26-2)
The last two terms can be simplified. We have- 1
the other terms cancelling on summation.
Hence, since g ft g is the minor of the element g pt in the
determinant g,
*=-*^ (26 ' 25)
Equation (26-2) thus becomes
-{^,e}-logV=0 . (26-3)
tfO/e
The equation is symmetrical in p and y, and therefore
represents 10 different equations. Actually there exist four
identical relations between these, so that the number of in-
dependent equations is reduced to six (see 39).
The selection of this law of gravitation is not so arbitrary
as it might appear. There is no other set of equations corre-
sponding to a tensor of the second rank containing only first
and second derivatives of the g^ and linear in the second
derivatives. Moreover, there is no other way of building up a
tensor of lower rank out of the components of B* v(r *
Having regard to the summations involved in (26-3) it will
be seen that the application of the new law of gravitation
must involve a considerable amount of calculation. There are
first to be calculated 40 different Christoffel symbols, each of
which is the sum of 12 terms. Then each of the 10 equations
contains 25 terms chiefly products or derivatives of the
Christoffel symbols. Finally the partial differential equations
have to be solved. It will probably be admitted that it is
worth while to find out whether this suggested law of gravita-
tion will agree with observation before resorting to something
more complicated.
27. We are now in a position to define the Principle of
Equivalence more precisely. The difference between a per-
manent gravitational field and an artificial one arising from a
transformation of Galilean co-ordinates is that in the latter
case (24-1) is satisfied, whereas in the former the less stringent
condition (26-1) is satisfied. These equations determine the
second differential coefficients of the g^, so that we can make
the natural and artificial fields correspond as far as first
differential coefficients, but not in the second differential co-
efficients. We shall therefore state the Principle of Equiva-
lence as follows :
* The tensor E% v<r vanishes identically. Other suggestions such as
g^B^a- merely give a sefc of equations equivalent to (26-1). The single
equation g^G/tvft would obviously be insufficient to determine the gravita-
tional field.
It must be remembered that we give no proof of this ; it
is merely an explicit statement of our assumptions. It would
be quite consistent with the general idea of relativity if the
true expression of such laws involved the Riemann-Christoffel
tensor, which vanishes in the artificial field, and would have
to be replaced before the equations were applied to the
gravitational field. But if we were to admit that, the principle
of equivalence would become absolutely useless.
THE GRAVITATIONAL FIELD OF A PAETICLE.
28. We have seen that the gravitational-potentials satisfy
the equations (26'3)
~
i=to. . (28-1)
We shall now find a solution of these equations corresponding
to the field of a particle at rest at the origin of space-co-
ordinates. We choose polar co-ordinates, viz.,
x l =r, x z =0, 053=9, a?4=k
In making this statement we are departing somewhat from
the standpoint of general relativity. Strictly speaking, we can
only define a system of co-ordinates by the form of the
corresponding expression for ds z , that is by the gravitatioL-
potentiais. So that to vspecify the co-ordinates that are used
involves solving the problem. Further, we have at present no
knowledge of a particle of matter, except that it must be a
point where the equations (28-1), which hold at points outside
matter, break down ; we can only distinguish a particle from
other mathematically possible singularities, such as doublets,
by the symmetry of the resulting field. Thus the logical
course is to find a solution, and afterwards discuss what
distribution of matter and what system of co-ordinates it
represents. We shall, however, find it more profitable to
accept the guidance of our current approximate ideas in ordej
to arrive at the required solution inductively.
44
RELATIVITY THEORY OF GRAVITATION.
The line-element ds can be assumed to be of the form
e I '<ft 2 . (28-21)
where A, [JL, v are functions of r only.
The omission of the product terms, drdO, drdy, dOdy, is
justified by the symmetry of polar co-ordinates ; the omission
of drdt, dOdt, dqdt involves the symmetry of a static field with
respect to past and future time. If the latter products were
present we should interpret the co-ordinates as changing with
the time.
A further simplification can be made by writing rV=r' 2
and adopting / as our new co-ordinate (dropping the accent).
The resulting change in dr 2 is absorbed by taking a new L
Thus the coefficient <f is made to disappear and we have
e v dt 2 . . (28-22)
Comparing (28-22) with (15-3), we have
9ii=~e\ 9**=-*** 233=-r 2 sin 2 0, g u =*. (28-31)
and gar=Q, when <r =/=> x.
The determinant g reduces to its leading diagonal, so that
-gr=^+"r*sin 2 0, ..... (28-32)
and 9^=1/0 ....... (28-33)
We can now calculate the three-index symbols (21-12)
Since the 0's vanish except when the two suffixes agree.
the summation disappears and we have
, , 1 (fyw , 9&T tyrr] . ,
{n ' a] V^ + :-a^! not OTmmei
If <r, r, p are unequal we get the following possible cases :
. (284;
in-T.. 0^=0.
Whence by (28-31), denoting differentiation with respect to
r by accents, we obtain
{11, 1} =-p'
{I2,2l=l/r
113,3} =l/r
{14, 4} =X
^22, l}=-re- x ) . (28-5)
i23, 3} = cot 6
{33, 1} = -r sin 2 0<r x
{33, 2} = - sin cos 6
{44, l}=i*-V
The remaining 31 Christoffel symbols are zero. It should
be noted that {21, 2} is the same as {12, 2} , etc.
It is now not difficult to obtain the equations of the field.
To assist the reader in carrying through the substitutions, we
shall write out in full the equations (28-1) omitting the terms
(223 in number), which obviously vanish. The following come
respectively from 6r n , G 22 , G S3> G M =Q :
-| {11, 1} + {11, 1} {11, 1} + {12, 2} {12, 2} + {13, 3} {13, 3}
+ {14, 4} {14, 4} + log V^g- {11, 1} -log V^=0
{22, 1} +2 {22, 1} {12, 2} + {23, 3} {23, 3} +^- 2 log V^
| {33, 1} -~ {33, 2} +2 {33, 1} {13, 3} +2 {33, 2} {23, 3}
r\
-- r {44, l}+2{44, 1} {14, 4} - {44, 1} ^-log V~g=0.
Of the remaining equations, G lz =Q gives
{13, 3} {23, 3} - {12, 2} ~logV^=0 ?
oo
which disappears when the values of the symbols are substi-
tuted and in the others there are no surviving terms.
E
Substituting from (28-5) and (28-32) the four equations give
immediately
1" I VP I I I '2 I / 1" \ " I
+2" ~rV
cos 2
These reduce to
-r)- sin 2 0=
(28-6)
From the first and last equations A / = V, and since both
A and v must tend to zero at infinity 1 v. The second and
third equations (which are identical) then give
Set e"=y, then
Whence y=l~> ...... (28'7)
where 2m is a constant of integration, m will later be identified
with the mass of the particle in gravitational units. This
solution satisfies the first and fourth equations, and, therefore,
substituting in (28-22), we have as a possible expression for
the line-element
. (28-8)
with y\~ 2m fr.
It will be seen that the measured space around a particle is
not Euclidean. Any actual measurement with our clock-scale
gives the invariant quantity ds. If we lay our measuring-rod
transversely, dsrdQ, so that our transverse measures are
correct in this system of co-ordinates ; but if we lay it radially,
ds=y-*dr, and the measures need to be multiplied by y* to
give dr. Thus, referring our results to Euclidean space, we
may say that a standard measuring rod contracts when turned
from the transverse to the radial direction.
We could, of course, decide to treat the radial measures as
correct, and apply corrections to the transverse measures.
This amounts to substituting dr' for y'^dr in (28-8), and using
/ as the radial co-ordinate. It is impossible to say which
form of (28-8) corresponds to our ordinary polar co-ordinates,
since we have never hitherto had to pay attention to the
ambiguity.
The possibility of using 'any function of r, instead of r, for
the distance is connected with the fact that Einstein's equations
amount to only 6 independent relations between the 10 </'s.
Consequently, quite apart from boundary conditions, there is
a large amount of arbitrariness in choice of gr's, i.e., of co-
ordinates. The reader may meet elsewhere with different
expressions for the line-element due to a particle. The one
adopted here was first given by Schwarzschild.
For some purposes the following analogy is helpful. Instead
of considering continuous space-time, consider that funda-
mentally we are dealing with an aggregate of points. With
Galilean co-ordinates x, y, z, t <\/ 1 the points are uniformly
packed. Any measure that we make is really a counting of
points, and a particle always moves so as to pass through the
fewest possible points between any two positions on its path.
Any mathematical transformation of these co-ordinates dis-
turbs, without disordering, the distribution of the points in
space ; but it is meaningless so long as we consider only the
points and not the arbitrary continuous space we place them
in. In a gravitational field the points are disordered according
to some definite law. We can evidently re-arrange them so
that the number of points in the circumference of a circle is
less than n times the number in the diameter (a circle being a
geodesic on a hypersphere, which is a locus such that the
minimum number of points between any point on it and a
fixed point called the centre is constant).
This representation, however, gives only imaginary time and
therefore imaginary motions. When extended to real motions
it becomes too complex to be of much help.
E 2
CHAPTER
THE CRUCIAL PHENOMENA.
29. The Equations of Motion of a Particle in the Gravitational
Field.
9jc
Denote tlie contravariant vector ^- by A*. Then by (22-5 )
its covariant derivative is
A a 3 /9
Multiply this by dxjds, we have
A'A'-^'+inR <rl dx ^
AA *-W +{a P> ^ T s >
showing that the right-hand side is a contravariant vector.
Consider the equations
+{a/U} 3 =0, (=!, 2, 3, 4), . . (29)
since the left-side is a vector, the equations will be satisfied
(or not) independently of the choice of co-ordinates. In
Galilean co-ordinates, the second term vanishes, and the
equations reduce to 3 2 av/9s 2 =0 5 which are the equations of a
straight line. Equation (29) is thus the general equation of
the locus which in Galilean co-ordinates becomes a straight
line.
The path, of a particle in Galilean co-ordinates (i.e., under no
forces) is a straight line. The equations (29) are accordingly
the equations of motion of a particle referred to any axes,
provided there is no permanent gravitational field. Further,
since they contain only first derivatives of the g's, in accord-
ance with 27, these equations of motion will hold also when
there is a permanent gravitational field.
The equations must evidently correspond to the condition,
/ds is stationary,
variations. The path of a particle is a geodesic in all cases.
It should be noticed that /ds is not generally a minimum.
30. Using the values (28-5) of ChristofEel's symbols, the
equation of motion (29) for o~=2 becomes
\ 2 , 2 dr dO
+=0. . (30-12)
Choose co-ordinates so that the particle moves initially in
the plane Q=n/Z ; then d6/ds=Q initially, and cos 0=0,
so that d 2 0/ds z =Q. The particle therefore continues to move
in this plane. The equations for o-=l, 3, 4 are then
<2 2 9.2 dr dtp
ds 2 r ds ds
d z t ( ,dr dt
=0 (30-13)
(30-14)
ds 2 ds ds
[rating (30-13) and (30-14), we have
. . . (30-21)
ds
dt c
. . . (30-22)
ds y
where h and c are constants of integration.
Instead of troubling to integrate (30-11), we can use (28-8),
which plays the part of an integral of energy, viz.,
--- 1 (30 ' 23)
From these three integrals,
or substituting for y its value (28-7)
/dr\ z , <,/d<p\ 2 ,
i i -j-rM -j- 5 ) =(c
\ds/ \ds /
i. 2 ^9 i
with r*--=ri
Compare these with the ordinary Newtonian equations f of
elliptic motion,
fdr\ z . , fd<p\ 2 m . 2m
1 -4-?* 1 == ---- --
(dt) U + ''
To make them correspond we must take c z =lm/a,
where o is the major semiaxis of the orbit. The term 2m& 2 /r 3
represents a small additional effect not predicted by the
Newtonian theory. Further, the quantity m, introduced as a
constant of integration, is now identified as the mass of the
attracting particle measured in gravitational units. With
regard to the use of ds instead of dt in (30-3), it must be
remembered that ds is the " proper time " for the moving
particle, so it is permissible to take ds as corresponding to the
time in making a comparison with Newtonian dynamics.
Mass, time and distance are all ambiguously denned in
Newtonian dynamics, and in defining them for the present
theory we have some freedom of choice, provided that our
definition agrees with the Newtonian definition in the limiting
case of a vanishing field of force.
31. The Perihelion of Mercury,
The ratio m/a or mfr is very small in all practical applica-
tions. If we take 1 kilometre as the unit of length and time
( ~.-,nn /-inn sec - )j * nen $ O1 *^ e earth's orbit #=149 . 10* and
\ oUU,UUU /
the angular velocity co=6-64: . 10~ 13 . Hence the mass of the
sun,
m=co 2 cs 3 = 147 kilometres. . . . (31-1)
Thus for applications in the solar system m/r is of order 10~ 8
and it is easily seen that A 2 /V 2 is of the same order. Also the
difference between dt and ds is of order I0~ 8 ds.
From (30-3) we have
h fa* h*, 2m tonh*
V
or writing tt=l/r.
duV- c 2 -! 2mu
Differentiating with respect to 9,
n ?/ vy)
B . . . . (31-2)
Since A'-'w 2 is of order 10* 8 we obtain an approximate solution
by neglecting 3mw 2 . This is
w=|;(l+ecos(9-C7)), . . . (31-3)
as in Newtonian dynamics.
For a second approximation, we substitute this value of u
in the small term 3mu 2 , and (31-2) becomes
d"u . m , 3m 3 , 6m 3 . > , 3wV'., , n . ..
(314)
Of the small additional terms the only one which can give
appreciable effects is the term in cos (957), which is of the
proper period to produce a continually increasing effect by
resonance. It is well known that the particular integral of
s
u^Aq> sin 9.
Hence this term gives a part of u,
Adding this to (31-3) we have
*?w
=- (1-f-ecos (9 CT (5nr)),
ft 2
where to=- -9, and (<5s7) 2 is neglected.
/2-
Thus whilst the planet moves through one revolution, the
perihelion advances a fraction of a revolution equal to
3m
where T is the period of the planet, and the velocity of light a
has been re-instated.
Mercury 442"-9 +8"-82
Venus 8-6 0-05
Earth 3-8 0-07
Mars 1-35 : J3
The value of edft is given because this corresponds to the
perturbation which can be measured. Clearly when e is
vanishingly small it is not possible to detect observationally
any change in the position of perihelion. The orbits of Venus
and the Earth are nearly circular so that the predicted effect
is too small to detect.
The following table gives the outstanding discrepancies
between the present theory and observation for ecter and de
(per century) with their probable errors. The seculai changes
de are analogous to ed& ; and the two perturbations may be
regarded as the two rectangular components of a vector. In
the last column we give the outstanding discrepancies of edtt
on the Newtonian theory ; those of de are, of course, unaltered.
Einstein's Theory. Newtonian.
e<5tu de edtt
Mercury -0"-58 0"-29 ~0"-83 0-"33 4-8"-24
Venus -0-11 0-17 4-0-21 0-21 -0-06
Earth 0-00 0-09 4-0-02 0-07 4-0-07
Mars 4-0-51 0-23 4-0-29 0-18 4-0-64
It will be seen that the famous large discordance of the
perihelion of Mercury is removed by Einstein's theory. No
other charge of importance is made except a slight improve-
ment, for the perihelion of Mars. Of the eight residuals, four
exceed the probable error, and none exceed three times the
probable error, so that the agreement is very satisfactory.
It may be noticed that according to (314) the orbit is not
exactly an ellipse, even apart from this progression of the apse.
But this (unlike the motion of perihelion) has no observational
significance, and merely arises from our particular choice of
measurement of r. In any case the curve in non- Euclidean
space, which is to be described as an ellipse, must be a matter
of convention.
THE CRUCIAL PHENOMENA. 53
It will be found (putting dr/ds=Q in (30-11)) that for a
circular orbit Kepler's third law is exactly fulfilled. This
again is not an observable fact. To compare it with obser-
vation we should have to consider the nature of the astro-
nomical observations from which the direct value of the axis
of the orbit is measured.
32. Deflection of a Ray of Light.
In the absence of a gravitational field the velocity of light
is unity, so that
Tt) + (dt) + (dt) =l '
Accordingly ds 2 =--dx 2 -dy z -dz 2 -\-dt z ^O. . . (32-1)
Hence for the motion of light ds=G, and by the principle of
equivalence this invariant equation must hold also in the
gravitational field.
It may be of interest to notice that for an observer travelling
with the light, dx=dy=dz=Q, so that dt=ds=Q. Hence, if
man wishes to achieve immortality and eternal youth, all he
has to do is to cruise about space with the velocity of light.
He will return to the earth after what seems to him an instant
to find many centuries passed away.
, Setting c?s=0 in (28-8) we have (for motion in a plane)
Hence if v is the velocity of light in a direction making an
angle V with the radius vector,
^(y^cos 2 F-fsin* V)=y,
whence v=y(l -(l-y) sin 2 F)- .... (32-3)
The velocity thus depends on the direction ; but it must
be remembered that this co-ordinate velocity is not the velocity
found directly from measures at the point considered. When
we determine the velocity by measures made in a small region,
and use natural measure (i.e., g^ having the values (16-3) at
that point), the measured velocity is necessarily unity.
Since it is inconvenient to have the velocity of light varying
with direction, we shall slightly alter our co-ordinates. Set
(324)
KlfiLiATJ.VJ.Tj:
Substituting in (32-2)
2
so that in these co-ordinates,
t,= y =l -2ro/rAl ~2m/r 1 .... (32-5)
for all directions. We can now drop the suffix of r lt
By Huygens' principle the direction of the ray is determined
by the condition that the time between two points is stationary
for small variations of the path. The course of the ray will
therefore depend only on the variation of velocity, and will be
the same as in a Euclidean space filled with material of suitable
refractive index. The necessary refractive index ju, is given by
(32-6)
^ '
We thus see that the gravitational field round a particle will
act like a converging lens.
The path of a ray through a medium stratified in concentric
spheres is given by ^ COIlst ....... (32 . 71)
where p is the perpendicular from the centre on the tangent.
By (32-6) we have to this order of approximation,
^=l-\-~ ..... . . (32-72)
But (32-71) and (32-72) are the integrals of angular momen-
tum and energy for the Newtonian motion of a particle with
velocity ^ under the attraction of a mass 2m, the orbit being
a hyperbola of semi-axis 2m. This hyperbola, therefore, gives
the path of the light. If the distance from the focus to the
apse is R, we have __
* d l7t }
a(e-l)=R,
. . R R
so that e=L-\-~ =2= ~,
2m 2m
and the very small angle between the asymptotes
Thus a ray of light travelling from oo to + <x> , and passing
at a distance R from a particle of mass m experiences a total
deflection. 4
=* ........ (32-8)
For a star seen close to the limb of the sun, by (31-1)
m=147 km., and #=sun's radius =697,000 km. Hence
a=l*-74.
It is curious to notice the occurrence of the factor 2 (mass =
2m) in the dynamical analogy. The deflection is twice what
we should obtain on the Newtonian theory for a particle
moving through the gravitational field with the velocity of
light. The path of a light ray is not a geodesic (or rather
the notion of a geodesic fails for mot on with the speed of
ight) ; it s determined by stationary values of fdt instead
of fds.
It may also be noted that the velocity of light decreases as
the light f a 's to the attracting body.
'E+dJE
o
Fio. 4.
33. It is hoped to test this prediction by observations of
stars near the limb of the sun during a total eclipse. If the
answer should be in the affirmative, the question will arise
whether this must be considered to confirm Einstein's law of
gravitation, or whether the deflection is sufficiently accounted
for by the simple hypothesis that the mass of the electro-
magnetic energy of light is subject to gravitation. The
unexpected factor 2 suggests that the deflection on Einstein's
theory will be double that which would result from the ordinary
electromagnetic theory. It is worth while to examine this
more closely.
Consider a tube of light of unit cross-section and length ds
(Fig. 4). Let the inclination of the ray to the axis of x be y.
Let g be the acceleration of the gravitational field directed
along Oy. Let E be the energy per unit volume ; and c be
the velocity of light, which on the electromagnetic theory is
absolutely constant.
54 RELATIVITY THEORY OF GRAVITATION.
Substituting in (32-2)
(dr^ ( dc?\ z
Ur) +('>*) =*
so that in these co-ordinates,
v=y=l-2mlt'^l-2m/r l .... (32-5)
for all directions. We can now drop the suffix of r.
By Huygens' principle the direction of the ray is determined
by the condition that the time between two points is stationary
for small variations of the path. The course of the ray will
therefore depend only on the variation of velocity, and will be
the same as in a Euclidean space filled with material of suitable
refractive index. The necessary refractive index ft is given by
/i= ! = l+ 2 M ..... (32-6)
r v r
We thus see that the gravitational field round a particle will
act like a converging lens.
The path of a ray through a medium stratified in concentric
spheres is given by , , QO ,,.
^ & J /^p^const (32-71)
where p is the perpendicular from the centre on the tangent.
By (32-6) we have to this order of approximation,
jw 2 =l+ 4 ^ (32-72)
But (32-71) and (32-72) are the integrals of angular momen-
tum and energy for the Newtonian motion of a particle with
velocity ^ under the attraction of a mass 2m, the orbit being
a hyperbola of semi-axis 2m. This hyperbola, therefore, gives
the path of the light. If the distance from the focus to the
apse is R, we have a= 2m,
a(e-l)=K, 5
., , ~ R R
so that e=l+A =2= f ,
2m 2m
and the very small angle between the asymptotes
2 2 4m
THE CRUCIAL PHENOMENA.
55
For a star seen close to the limb of the sun, by (31-1)
m=147 km., and -R=sun's radius =697,000 km. Hence
a=l"-74.
It is curious to notice the occurrence of the factor 2 (mass
2m) in the dynamical analogy. The deflection is twice what
we should obtain on the Newtonian theory for a particle
moving through the gravitational field with the velocity of
light. The path of a light ray is not a geodesic (or rather
the notion of a geodesic fails for mot on with the speed of
ight) ; it s determined by stationary values of fdt instead
of fds.
It may also be noted that the velocity of light decreases as
the light la 's to the attracting body.
'E+dE
\Tf>+d,T[>
FIG. 4.
33. It is hoped to test this prediction by observations of
stars near the limb of the sun during a total eclipse. If the
answer should be in the affirmative, the question will arise
whether this must be considered to confirm Einstein's law of
gravitation, or whether the deflection is sufficiently accounted
for by the simple hypothesis that the mass of the electro-
magnetic energy of light is subject to gravitation. The
unexpected factor 2 suggests that the deflection on Einstein's
theory will be double that which would result from the ordinary
electromagnetic theory. It is worth while to examine this
more closely.
Consider a tube of light of unit cross-section and length ds
(Fig. 4). Let the inclination of the ray to the axis of x be ip.
Let g be the acceleration of the gravitational field directed
along Oy. Let E be the energy per unit volume ; and c be
the velocity of light, which on the electromagnetic theory is
absolutely constant.
56 RELATIVITY THEORY OF GRAVITATION.
Then the mass of electromagnetic energy E, according to
electromagnetic theory (or by (7-85)), is E/o z , so that if this
is subject to gravity the momentum generated in the tube in
unit time will be
El
- 2 ds.g along Oy.
If the light is stopped by an absorbing screen placed
perpendicular to the ray the radiation-pressure is numerically
equal to E, showing that momentum E in the direction of
the ray passes across a section of the tube in unit time. Thus,
resolving in the x and y directions, the conservation of
momentum gives
-j-(E cos y;) . ds=0,
(33-1)
/n v -7 qEj
^-(E sin w) . ds= y ~ds,
As ^ c z
,, dE . dw
Whence cos y Z?sm y; ~=0,
ds ds
dE . . dw gE
-3-sm w-\-E cos w -- =*- .
as as c 2
Eliminating dE/ds ,
The radius of curvature ds/dy is thus c 2 /<7 cos y>, which is
exactly the same as for a material particle moving with
velocity c in ordinary dynamics. This, as shown in the last
paragraph, is only half the deflection indicated by Einstein's
theory ; and the experimental amount of the deflection should
thus provide a crucial test.
34. Displacement of Spectral Lines.
Consider an atom vibrating at any point of the gravitational
field. It is a natural clock which ought to give an invariant
measure of an interval ds ; that is to say, the interval ds
corresponding to one vibration of the atom is always the same.
Let t/hfi a,t,nm hft mnmfint.Rrilv at, rftst in nnr svsf.p.rn irf p.r-
THE CRUCIAL PHENOMENA. 57
If then dt and dt' are the periods of two simi ar atoms
v brating at different parts of the field where the potentials
are g^ and g\ t) respectively,
Vfl f 44-*=Vfl^44.*' (34-1)
If t refers to an atom vibrating in the photosphere of the sun,
2m
#44 ~R'
and if t f refers to an atom in a terrestrial laboratory, where
#'44 is practically unity,
^-1+5=1.00000212 . . . (34-2)
at K '
The solar atom thus vibrates more slowly, and its spectral
lines will be displaced towards the red. The amount is
equivalent to the Doppler displacement due to a velocity of
0-00000212, or in ordinary units 0-634 km. per sec. In the
part of the spectrum usually investigated the displacement is
about 0-008 tenth-metres.
The effect is of particular importance, because it has been
claimed that the existence of this displacement is disproved by
observations of the solar spectrum.* The difficulties of the
test are so great that we may perhaps suspend judgment ;
but it would be idle to deny the seriousness of this apparent
break-down of Einstein's theory. We shall therefore consider
the phenomenon from a more elementary point of view.
The phenomenon does not depend on the greater intensity
of the field on the sun, but on the potential ; and it can
evidently occur in a uniform gravitational field. Consider an
observer in a uniform field of intensity g and two similar
atoms A! and A z , A^ being close to the observer and A z at a
distance a measured parallel to the field. The observer and
his atoms will, of course, be falling with the acceleration g.
Consider them all enclosed in a room which is also falling ;
then by the principle of equivalence cannot detect any
effect of the field, and he will therefore observe the same
period of vibration T for both atoms. Now refer the pheno-
mena to unaccelerated axes which coincide with the accelerated
axes at the instant t=Q. The vibration emitted by A z at
the time =0 will reach at the time t~a (the velocity of
* C. E. St. John, " Astrophysical Journal," Vol. 46, p. 249.
CABHB6IB INSTITUTE
OF TBNOL045Y
THS LIBRARY
CHAPTER VI.
THE GRAVITATION OF A CONTINUOUS
DISTRIBUTION OF MATTER.
35. In the problems occurring in Nature our data give, not
the distribution of the individual atoms, but the large-scale
average distribution of density. This transition from discrete
particles to the equivalent continuous medium occurs in the
Newtonian theory of attractions, and involves the replacement
of Laplace's equation v 2 ? by Poisson's equation y 2 <p=
47rp. We shall now find the corresponding modification of
Einstein's equations 0^=0.
The equations G fT =Q are not linear in the 0's, and conse-
quently the fields of two or more particles are not strictly
additive. But the deviations produced in the g's by any
natural gravitational field are extremely small, so we shall
neglect the product terms and treat the fields as superposable.
It will be shown below that ultimately this approximation
does not produce any inaccuracy in the application we have in
view.
As in (324) we shall write r==r 1 -j-m in (28-8) and neglect
(m/r) z . Then the line element in the field surrounding the
particle is
. (35-1)
We consider r t to be the actual radius vector, since the
mode of measurement is arbitrary to this extent. Converting
into rectangular co-ordinates,
. (35-2)
The origin is now arbitrary, and r denotes the distance or
the attracting particle from the element ds. The effects of
a number of particles being additive to our order of approxi-
60 RELATIVITY THEOBY OF GEAVITATION.
mation, we shall have for any number of particles at rest
relative to the axes,
ds*=-~(l-i-2&)(dx z +df+dz*)+(l~2&)dt* . (35-3)
where ii=2(m/r)=the Newtonian potential.
Consider a point in the medium where the density is p,
and with as centre describe an infinitely small sphere. If
we neglect the material inside the sphere, the equations of the
gravitational field in free space will be satisfied at 0, i.e.,
G^0, Hence in calculating the values of 6v T at we need
only take account of the material inside the sphere. Accord-
ingly in (35-3) 1 refers to the potential inside an infinitely
small sphere of uniform density p.
Since dQ/d%, &c., vanish at 0, we have only to take account
of terms in (28-1) containing second derivatives of the g's ;
and the calculation of G aT at is quite simple. We have
3) +a
ogV-, . (354)
omitting 33 terms which vanish or cancel.
AtO, ^=^11= -1, g4A =g"=l, . . (35-5)
and by (35-3)
9 _!^J^__Jl ln ,/- 2 9 2 &
2 2 g ff '
Hence substituting in (354)
r J~
11
= 47tp, by Poisson's equation.
Working out the other components similarly (with slight
variations in the ease of (r 44 ) we find
. . (35-6)
The scalar GcQ
(35-7)
Now form the covariant tensor
A ^u^i iiJN uu US JJ1STKJIJ3 UT1OJN OF MATTER. 01
We have by (35-6) and (35-7)
2*44= P,
and all other components vanish.
Having thus found the value of T rr in this special system of
co-ordinates we could find its general value by (19-31). It is,
however, simpler to proceed as follows. If x^ is a co-ordinate
of a point in the material, consider the quantity,
cfx u Jx.,
'5'"5 ....... (35 ' 91 >
Since with respect to our special axes the material is at rest,
^=0 (,1=1,2,3), and ^=1 (^4).
Hence all the components of (35-91) vanish except for
ya = i>=4, for which the component is p just like T or , This,
however, is a contravariant tensor* and (35-8) requires a
covariant tensor.
We therefore form the associated covariant tensor ( 20&)
_, ?*.. dx v
T m =V-g^- , - - - (35-92)
which agrees with (35'91) in our special co-ordinates.
The equations (35-8) and (35-92) are in covariant form, and
are true in one system, hence they are true in all possible
systems of co-ordinates. They are the general equations of
the gravitational field in a continuous medium.
An alternative form of (35-8) is readily obtained, viz.,
-^g n T) t . . . (35-93.
where T is the associated scalar g" T T VT . (This follows since on
inner multiplication of (35-8) by g we obtain G=8nT.)
36. We thus find that in a continuous medium, G 9rt instead
of vanishing, is equal to a tensor expressing the content and
state of motion of the medium at the point considered. On
the equations here found we have two observations to make.
( 1 ) A little consideration will show that notwithstanding the
approximations made at various stages of the proof, the results
are quite rigorous. It is clear that so far as the calculations
for the infinitely small sphere surrounding are concerned,
* p is to be treated as an invariant. Whatever the axes chosen, p is to
Ve the density in natural measure as estimated by an observer moving with
the matter.
lure we iiecu. vjuiy t/Ajjauu imo y o m. J^UWCJLO uj. / 010 xcn 0,0 / ,
but in our units p is of dimensions r~ z , and since the </'s in
rectangular co-ordinates are of zero dimensions, any terms
involving p 2 would be of the form p 2 r 4 , and therefore need
not be retained. The effect of the gravitation of the matter
outside the sphere is eliminated completely by our choice of
co-ordinates. We chose them so that at the 0's have the
values (16-3), i.e., we use "natural measure." Since our axes
move with the matter at 0, the first derivatives of the <?'s
(expressing the force) will not vanish unless the matter at
is moving with the acceleration of the field, which is not the
case if there is any internal stress. These first derivatives
are omitted from our equations after (35-3), because as already
explained the external matter alone contributes nothing to
Gvr ; further, the cross-terms are zero, because the first de-
rivatives of the <7's arising from the matter inside the sphere
vanish. The result is thus rigorous, provided that in measuring
the invariant density p we use natural measure, i.e., the mass
and unit volume must be taken according to the direct
measures made by an observer at moving with the material
there.
The argument may be summarised thus : G^ consists of
terms of types
/a-^-f/^-Ki^i+^iH- terms in 7 +terms in E&
where I and E refer to the matter internal and external to
the small sphere, and the suffixes refer to the order of the
derivatives. Terms in I 1 vanish by the symmetry of the
sphere ; terms in / vanish as the sphere is made infinitely
small ; terms in E n vanish because we use natural measure ;
the terms J5 2 -f-' 1 2 vanish by Einstein's equations for free
space. All that is left is 7 2 > and as the sphere is made infinitely
small our determination of its value becomes rigorous.
(2) In replacing a molecular medium by a continuous
medium, it is not sufficient to average the distribution of mass
and mass-motion only ; we must also represent somehow the
internal motions. This is done by adding another property to
the continuous medium the pressure, or stress-system. The
tensor T ffT will contain terms corresponding to the pressure ;
these are negligible in practical calculations of the gravitational
paragraph.
37. In the dynamics of a continuous medium the most
fundamental part is taken by the associated mixed tensor,
TifT^^r.^, . . . (37-1)
where we have inserted the 2 in order to take account of the
variety of internal motions, and have written p for p in order
to call attention to the fact that it represents the density in
natural measure and not the density referred to the arbitrary
axes chosen.
Tp may be called the energy-tensor, though it is actually an
omnium gatherum of energy, mass, stress and momentum.
First consider the meaning of this tensor in the absence of a
gravitational field, and accordingly choose Galilean axes. If
u, v, w are the component velocities of the particles,
dx dy dz /ds\* , 2220-2 /orr m
=M, ~=v, -TW, -=- ) =l-w 2 v 2 ~w*=p 2 . (37-2)
dt dt dt \dt/
But by (7-92) the density referred to the axes chosen is
TI /Tli. T"t WifJff wWt/ /OFT f> \
Hence T^g^Zp-j- -=~ (37-3)
Putting in the Galilean values of g^, we have
2* = JSpw 2 , Zpvw, Dpwti, Zpw
| ^"^ Zpuv, Zpv 2 , Zpwv, Zpv (374)
I
This tensor may be separated into two parts, the first referring
to the motion, t* , o , w; , of the centre of mass of the particles
in an element, and the second to their internal motions,
WD Ol5 w v relative to the centre of mass. With regard to the
last part, Zpw^ represents the rate of transfer of
w-momentum across unit area parallel to the /-plane, and is
F2
therefore equal to the stress usually denoted by p x , f . Hence
(374) becomes
-P~P u o w o 3 -Pw pwo 2 , P^o
ptt , pt> , PW , P
* (37-5)
where p is now the whole density referred to the axes chosen.
Consider the equations
^T"=0 (37-6)
93, *
Taking ^=4, and using (37-5), we get the well-known
equation of continuity
^o) , alP^o) , alp^o) , ap^o m t . (37 . 7)
dx % 9z at
Taking p=l,
= P~ ? ............ (37-8)
Now (37-7) and (37-8) are the fundamental equations of
hydrodynamics. By assuming Galilean axes we have neglected
any extraneous body-forces, and so the term pZ, which
occurs on the right side of (37-8) in the more general form of
the equation, does not appear in this case.
The equation (37-6) is thus equivalent to the general equa-
tions of a fluid under no forces.
3 \ The equat n 32^/9 ?=0 represents a law of con-
servation. hoose one of the co-o dinates, x 4 , as independent
variab'e, and integrate the equation through a three-dimen-
sional volume marked out in the other co-ordinates. This gives
=the surface integral of the normal con -
pone t of (21 7% 2*).
If the volume is such as to include the whole of the material,
" vanishes on the surface ; the surface-integral therefore
( TjL, T^ Tjt) . It will be seen from (37-5 ) that for the axes there
used T* represents the negative momentum and the mass (or
energy), and that T\, &c., represent the flux of these quantities,
Equation (37-6) therefore gives the law of conservation of
momentum and mass, as may be verified from the correspond-
in i hydromechanical equations.
39. Equation (37-6) is the degenerate form for Galilean co-
ordinates of the covariant equation
r;,=0 ....... (39-11)
where T^ v is the (contracted) covariant derivative of T^ (see
(22-7)). Equation (39-11) thus holds for Galilean co-ordinates,
and it does not contain derivatives of the gr's higher than the
first. Hence by the principle of equivalence it holds generally,
inc uding the case of a permanent gravitational field.
Taking equation (35-8)
multiply by <7 T ". We obtain
<^-W?=-8^ .... (39-12)
Take the covariant derivative of both sides, and contract it,
whence by (20-1)
Clearly this equation will have to be an identity, and it may
be verified analytically, using the values (26-3) of G^ For
o-=l, 2, 3, 4, this identity gives the four relations between
Einstein's ten equations, which have already been mentioned
as reducing the number of independent conditions to six.
Conversely, from the identity (39-14) we can deduce (39-11),
and hence obtain the equations of hydromechanics and the
law of conservation directly from Einstein's law of gravitation.
Further, by applying the hydromechanical equations to an
isolated particle, we obtain the equations of motion (29).
The mass of a particle has been introduced first as a constant
of integration, and afterwards identified with the gravitation-
mass by determining the motion of a particle in its field ; it
now appears that it is also the inertia-mass, because it satisfies
66 RELATIVITY THEORY OF GRAVITATION.
the law of conservation of mass and momentum, which gives-
the recognised definition of inertia.
it is startling to find that the whole of the dynamics of
material systems is contained in the law of gravitation ; at
first sight gravitation seems scarcely relevant in much of our
dynamics. But there is a natural explanation. A particle of
matter is a singularity in the gravitational field, and its mass
is the pole-strength of the singularity ; consequently the laws
of motion of the singularities must be contained in the field-
equations, just as those of electromagnetic singularities (elec-
trons) are contained in the electromagnetic field-equations.
The fact- that Einstein's law predicts these well-known pro-
perties of matter seems to be a valuable confirmation of this
theory.
The general equation (39-11) enables us to pass from the
equations of a fluid under no body forces to the equations of a
fluid in a field of force. It can be simplified considerably. By
(22-7) y *
fr'jr;. - < 39 ' 21 >
By (26-25) the last term becomes
The second term is equal to
<w 9*^u
since the other two terms cancel on summation,
This last result follows, since
9 va 9ve = or 1,
so that 9" a dg vl! -\-g lle dg va =Q.
Multiply by g# and use (20-15), we obtain
gtfg va dg ye =d,g^ (39-4)
Hence inserting (39-22) and (39-3) in (39-21), we have
LJ^Xv. \J I
This equation has its simplest interpretation when we choose
co-ordinates, so that V g1, that is to say, the volume of a
four-dimensional element is to be the same in co-ordinate
measure as in natural measure. Owing to the considerable
freedom of choice of co-ordinates, allowed by Einstein's equa-
tions, it is always possible to do this. In that case (39-45)
becomes
(39 - 5 >
Comparing this with (37 6), which holds when there is no
field of force, we see that the term on the right represents
the momentum and energy transferred from the gravitational
field to the material system. As a first approximation (re-
taining only T^= p, and <7 44 =1 2O) we see that it gives,
for ju1, 2, 3, the terms pX pY, pZ of the usual hydro-
dynamical equations, which were omitted in (37-8).
40. Propagation of Gravitation.
The velocity of light being a fundamental relation between
the measures of time and space, we may expect the strains
representing a varying gravitational field to be propagated
with this velocity. We shall show how to derive the equations
exhibiting the propagation.
In the theory of sound, the general equation of disturbances
propagated with unit velocity is
where (D is zero except at the source of the disturbance Th
general solution is
*r < 40 ' 12 >
the integral being taken through the volume occupied by the
source of disturbance, and the value of ' taken for a time
t /, where / is the distance of the volume dV from the point
considered. Thus 9 is a retarded potential, and (40-12) ex-
hibits the effect as delayed by propagation.
In the case of sound the velocity depends to a slight extent
on the amplitude, and (40-11) is only strictly true if the square
of cp is negligible. Similarly the velocity of light depends to a
slight extent on the gravitational field ( 32) ; consequently we
become linear.
The origin of gravitational waves must be attributed to
moving matter ; and, since G^ vanishes except in a region
occupied by matter, we may take G> as the analogue of <D.
We shall examine whether the disturbance can be represented
by a quantity k^ satisfying
D/V=2GU ..... (40-21)
where the exact significance of h^ is yet to be found. We
shall regard h^ as a small quantity of the first order ; the
deviations of the g^ from their Galilean values will also be of
the first order. Small quantities of the second order will be
neglected.
If, as usual,
A;=rt, w
and lg^h,,.?.
Then, multiplying (40'21) successively by g"* and g*", we have
to this approximation,*
DfcJ=26? ..... (40-22)
and Dfc=2 ..... (40-23)
Hence D(^-WO=2(^-|. 9 ^)
l=-l$7ill by (39-12).
To the present approximation (37-6) holds, so that
Having regard to boundary conditions, the solution is clearly
* The gp behave as conatatifcs until we reach equation (40-5), because
their derivatives, which are small quantities of the first order, only appear
in combination ivith the small quantities ft w or Gp V . The gw accordingly
pass freely under the differential operators.
Consider the expression
.^{^+-
which to our approximation
By (40-3) the first two terms cancel with the last, and for
Galilean values of g afi the third term is simply
Thus by (40-21) the expression (40-4) reduces to G^.
Neglecting squares of small quantities, 0^ (26-3) reduces to
Comparing (40-4) and (40-5) we see that the 7i's must be
equal to the j/'s or rather since the fe's have been treated as
small quantities, they must be the deviations of the (fa from
their constant Galilean values. Writing <5 MJ , for the Galilean
values of g^ (16-3), then
9V=<MA,> ..... (40-6)
and h^ satisfies the equation of wave-propagation (40-21).
By (40-12) the solution of the propagation equation is
r. (40-7)
This can be used for the practical calculation of g^,, due to
an arbitrary distribution of moving matter. It is necessary,
as in the corresponding calculation of retarded electromagnetic
potentials, to allow for the variation of t r' from point to
point of the body ; the boundary of dV does not coincide with
the limits of the body at any one instant. Thus for a particle
of mass m, we have *
* See, for example, Lorentz, "The Theory of Electrons," p. 254; of
Cunningham, " The Principle of Relativity," p 108.
/U RELATIVITY THEORY OF GRAVITATION.
where v r is the velocity in the direction of r, and the square
bracket indicates retarded values. As is well known [r(l ,)]
is to the first order equal to the unretarded distance r, so that
notwithstanding the finite velocity of propagation the force is
directed approximately towards the contemporaneous position
of the attracting body. It was lack of knowledge of this
compensation which led Laplace and many following him to
state that the velocity of gravitation must far exceed the
velocity of light.
The practical application of these formulae is, however, very
limited. In a natural system (e.g., the solar system) the
relative velocities (u) are due to the gravitational field and u 2
is a small quantity of the first order. Consequently our
approximation is not good enough to take account of T ll3 T 12 ,
&c., in natural systems ; it can only include components with
suffix 4.* The fact is that the whole idea of propagation from
a point-source is an abstraction ; actually the motion of the
source, or singularity, is but the symbol of the changes occurring
in all parts of the field ; we cannot say whether the motion is
the cause or effect of the gravitational waves.
The present solution is a particular solution. It gives
unique values of the g^ , but these may, of course, be subjected
to arbitrary transformations.
* For the higher approximations needed in the problems of the solar
system, see De Sitter, " Monthly Notices," Dec. 1916.
CHAPTER YII.
THE PRINCIPLE OF LEAST ACTION.
41. Lagrange's Equations.
We shall again restrict the choice of co-ordinates so that
V g= 1. Einstein's equations (26-3) for the field in free space
then becomes simplified to
^-JrWa}+M }{*, 0=0-' (!)
d^a
We shall regard g*" as a generalised co-ordinate (q), and
x 1} x z , x 3 , x 4 as independent variables a four-dimensional time.
Writing g v for dg^/dx*, which will then be a generalised
velocity (q), we shall show that equations (41-1) can be ex-
pressed in the Laguangian form.
3 / dL\ dL - ,,, 9 ,
( '
where L=gr{tf, a} {va, |8} ..... (41-3)
it being understood that the g^ are expressed as functions of
the g* v .
We have from (41-3)
dL= {rf, a} {va, ft dgv+Zgr {M } {"a, ^} ,
since in the last term ^ and v are dummies.
= - {/*$, a} {va, $ ^+2 {^, a}
But
The last two terms in the bracket will cancel in the summa-
tion after inner multiplication by {^j8, a} , because /u and /?,
t. fl.nrl 5 fl.rp. infflrp.lifl.norflfl.'hlft-s.TTrmltfl.Tifinnslv. Also hv (39-4)
72 RELATIVITY THEORY OP GKAVirATION.
Therefore
(414)
showing that (41-1) and (41-2) axe equivalent.
As in ordinary dynamics, Lagrange's equations are equiva-
lent to
ILdr is stationary .... (41-5)
for variations of g 1 " 1 , di being the four-dimensional element of
volume, here representing the independent variable. It must
be remembered that the variations are limited by the con-
straint V <7=1,
42. Principle of Least Action.*
Following out the dynamical analogy dL/dgZ" or 9L/%
is to be regarded as a momentum (p). The system is dynamic-
ally of the simplest kind, since L does not contain the " time,"
XP, explicitly, and :t is a homogeneous quadratic function of
the " velocities." By the properties of homogeneous functions
Since (pq-\-qp) is a perfect differential,
will be equal to a surface integral ; and it will, therefore be
stationary for variations of g" v (the variations as usua being
supposed to vanish at the boundary).
Thus dJ2qpdr=-dlSqpdr=-2dfLdi; . . (42-1)
Hence, if we write
H=*L+2qp ...... (42-2)
by (41-5) and (42-1)
\Hdr is stationary ..... (42-3)
THE PRINCIPLE OP LEAST ACTION. 73
By (414)
Hence (42-2), (41-3) and (41-1) give
#=<r<3>=0.
We can therefore write the result (42-3) thus
IG.Vg.di is stationary . . (42-4)
since V 0=1. _
But Gr and V g . fa are invariants (20-3) ; so that (42-4) has
110 reference to any particular choice of co-ordinates, and the
restriction V 0=1 can now be removed. It IB thus a more
general result than (41-5).
43. Energy of the Gravitational Field.
Reverting to the restriction V gl, multiply (41-2) by <$'
But <=- + j . . . . (43-2)
3* ar" 9fl 9^" da?
Remembering that
we have, adding (43-1) and (4 -2),
gwQ v== SL(g^ v ^\^- . . . . (43-3)
= _16j C JL ,...... (43-4)
where -16* t^g^^-g^L (43-5)
We have used the property of g? as a substitution operator.
The quantity <| defined by (43-5) is the analogue of the
Hamiltonian integi'al of energy, 2q.-L. In free space
& ,=0, and (43-4) becomes
showing that ^ is conserved ( 38).
When matter is present (434) gives
-le^-se^
a*, 3* "'
do 1 "'
~ 9r~ v ~ 2 ^ A "' ''
since, when 0= 1, g fa ^g f "=^.
Hence by (35-8)
-rj by (39-5).
O^a
Therefore
^+$=0 ...... (43-8)
This is the law of conservation in the general case when
there is interaction between matter and the gravitational
field. We see that the changes of energy and momentum of
the matter can be regarded as due to a transfer from or to the
gravitational field, the total amount being conserved. We
have, in fact, traced the disappearing portion of the material
tensor T"^ and shown that it reappears as the quantity
.$ belonging to the gravitational field.
In order to represent the phenomena in this way we have
had to restrict the choice of co-ordinates by keeping the volume
of a region of space-time invariant (V 0=1). Otherwise
the equation takes the more general form (39-11) which cannot
immediately be interpreted as a law of conservation. It should
be noted that, unlike T%, the quantity $ is not strictly a tensor.
44. The Method of Hilbert and Lorentz.
An alternative method of deriving the fundamental equations
of this theory is based on the postulate that all the laws of
mechanics can be summed up in a generalised principle of
stationary action, viz.,
H t +....)V=g.dT=Q.. . (44-1)
Here H 1 ,H 2 ,H 3 B,i& invariants* involving, respectively, the
parameters describing the gravitational field, the electro-
magnetic field, and the material system. If we consider
* Invariant because the equation must hold mall systems of co-ordinates,
and we already know that the factor V g . fa is invariant.
matter and radiation in bulk we may add a fourth term
involving the entropy, so as to bring in thermodynamical
phenomena, and so on. The variations are taken with respect
to these parameters, their values at the boundary of integration
being kept constant.
It is Avell known that the laws of mechanics of matter and
of electrodynamics can be expressed in this form, so that we
are here chiefly concerned with H v We already know from
(424) that Einstein's theory is given by H l G. Now G is,
in fact, the principle invariant of the quadratic form g^dx^dx,,,
viz., the Gaussian invariant of curvature. This aspect of the
theory seems to eliminate any element of arbitrariness which
may have been felt when we fixed on the contracted Biemann-
Christoffel tensor for the law of gravitation.
To interpret G as a curvature, consider a surface drawn in
space of five dimensions, whose equation referred to the lines
of curvature and the normal (z) at a point on it may be written
2z=& 1 a?+7c 2 #|4-& 3 a;^+& 4 a55+higher powers . (44-2)
where 7c 1? & 2 , & 3 , & 4 are the reciprocals of the principle radii of
curvature.
Then ds*=dz*+2dx\.
Eliminating z by (44-2)
ds 2 =(I+klxl)dx\+ ____ +2fc 1 fc 8 1 3 a <foj 1 dfe t +. . (44'3)
Hence at the origin,
9W=1, &=<) (M aV )> 9&v/dav=0.
The only surviving terms in G=g> LV G /J , v are
-^-r W*>
O x p
We easily find that
In three dimensions we have only two curvatures, and Jc-Jc 2
is known as Gauss's measure of curvature, i.e., the ratio of the
solid angle contained by the normals round the perimeter of
an element to the area of the element. The expression (444)
is a generalisation of this invariant to five dimensions.
The curvature G in ordinary matter is quite considerable.
In water the curvature is the same as that of a spherical space
of radius 570,000,000 km. Presumably, if a globe of water of
this radius existed, there would not be room in space for
anything else.
where F, G, H is tlie vector potential, and <D the scalar
potential of the ordinary theory.
If M^ is the covariant derivative of M , we have by (22-2)
3 M dx v . , ,
=.,.,= a covariant tensor,
dx v d^ *
=*U say (45-2)
The electric and magnetic forces are given in the electro-
magnetic theory by
?0> SF 3H 36
* - a= -- (45 ' 3)
Hence by (45-2) the value of F^ in Galilean co-ordinates is
F^ = -y ft -X . . . (45-41)
y -a -y
-ft a -Z
v X Y Z
and the associated contravariant tensor, F^ v =g fl(l g v ^F^, is
FI = -y ft X . . . (45-42)
y -a y
-ft ' a Z
-x -y -z o
We can now express Maxwell's equations in covariant form.
In the ordinary theory they are
8Z _?! = _?? 9 .? j^ = __? <U _??=__?
9y a/? 8-X aa 3;/ ay a/5 aa az
3 ^ ~ pv j rv A /-,j"T~ M/ J (4:0*0^ I
i/y (jtj QII (jz fix a' 0^* y 9*
az ay az
LI i p .... (45-531
dx oy dz '
da 9/3 dy
_.-_i =0, (45-54)
C-^ Ci7V '^'y V '
C/i*/ (7W c/4
where the velocity of light is unity, and the Heaviside-Lorentz
unit of charge is chosen so that the factor 4n disappears. The
electric current u, v, w and the density of electric charge p
form a contravariant vector, since
fdx dy dz dt \
(u, v, w, p) = 2,e( -J- , -J- , j- , -j-J per unit volume,*
=/*, say (45-6)
Equations (45-51) and (45-54) may be written,
---+ -""M ^=0, .... (45-71)
and the remaining equations (45-52) and (45-53) give
d -j~r= J " (45-72)
Now (45-71) is satisfied identically on substituting the values
of F^ from (45-2), so that (45-2) and (45-72) represent the
fundamental electromagnetic equations. The former is already
covariant, and the latter is made covariant by writing the
covariant derivative for the ordinary derivative. Thus
(45-81)
are the required equations. These hold in the gravitational
field because the conditions for the application of the principle
of equivalence ( 27) are satisfied.
The expression F 1 ? may be simplified as in 39 ; but owing to
the antisymmetry of F"" the term corresponding to (39-3) dis-
appears, and the equation reduces to
_-(yC^.^ = jM t . . (45-9)
V-gfa,, J ' v
The fact that Maxwell's equations can be reduced to a co-
variant form shows that all electromagnetic phenomena
described by them will be in agreement with the principle of
relativity.
* The occurrence of ds instead of di in the denominator is due to the
Michelson-Morley contraction, $dt/ds, which makes the estimate of unit
volume by a fixed observer differ from that made by an observer moving
with the electrons. (Of. equation '7 '65).)
n
iO Kl3iixA.JLl.VJ.TX TtliiUKX US! WWA VilAXIUJN.
46. The Electromagnetic Energy-Tensor.
According to the electromagnetic theory, the components of
mechanical force on unit volume containing electric charges
are
aw-yu,
av,
and the negative rate of doing work is
ki=Xu Yv~Zw,
since the magnetic force does no work.
By (4541) and (45-6), these give
-k v =F ltt J-
=V/> ...... (46-1)
so that k v is a vector.
But k v represents the rate at which the momentum and
negative energy of the material system are being increased,
i.e., in Galilean co-ordinates,
(46-2)
If there exists a corresponding tensor E* for the electromagnetic
field, this must change by an equivalent amount in the opposite
direction in order to satisfy the law of conservation. Thus
Bl=k ........ (46-3)
<#a
It is not difficult to show from (46-1) and (46-3) that
El^-F^'P + tgWF^ . . . (46-4)
We omit the proof as the precise value is not of great interest
to us. It is sufficient to know that the expression is of the
necessary tensor-local, so that an energy-tensor for the electro-
magnetic field exists.
In general co-ordinates (462) and (46-3) are replaced by
the covariant equations,
~Tl a =k v =:E* a ...... (4G-5)
in accordance with the principle of equivalence.
When no matter is present this gives $ a =0, and we can
derive the reaction of the gravitational field just as in (39-5).
It follows that electromagnetic energy in the gravitational
field experiences a force just as material energy does. Further
THE PRINCIPLE OF LEAST ACTION. 79
electromagnetic energy exerts gravitation, because (39-13) and
(46-5) give
the lower a denoting covariant differentiation.
Hence on integrating, (39-12) must be replaced by
In fact the electromagnetic energy-tensor must simply be
added on tojthe material energy-tensor throughout our work.
When V <7=1, we have the most general law of conserva-
tion for triangular interchanges between matter, electro-
magnetism and gravitation.
)=0 ..... (46-6)
Oo
47. The Aether.
The application of the Calculus of Variations to (44-1) gives
a number of differential equations equal to the number of
parameters varied ; but, according to a general theorem due
to Hilbert, there are always four identical relations between
these equations (the number 4 corresponding to the dimensions
of dz). The number of independent equations is thus four
less than the number of unknowns,- so that in addition to
arbitrary boundary conditions we can impose four arbitrary
relations on the parameters. It is this freedom of choice of
co-ordinates that is so fundamental a characteristic of the
generalised principle of relativity.
If we vary H l only we find the ten equations (?^=0. The
identical relations in this case have been given in 39. If we
vary the electromagnetic variable K^ as well, we get 14
equations, of which 10 are independent, to determine 14 un-
knowns Within certain limits we can give arbitrary values to
four of the unknowns, and the other ten will then bedetermined
definitely by the equations and the boundary conditions. If
we elect to fix the values of the four co-ordinates in this
way (so that they are, as it were, disposed of) the g^ will
become fixed, that is to say, there will be only one possible
space-time. The phenomena, electromagnetic as well as
n-iio xri-f-o-f trvn o I TTtnll oil r\a r\ acinm harl r\Tr fcrna rt TXTnir>li TQT--nQCOirii".
This is only a crude indication of the relation of the aether-
theory to our relativity theory. As is well known, the modern
aether-theory involves rotational strains. Moreover, we can-
not get rid of the electromagnetic variables by putting them
equal to zero, because they form a vector, which cannot vanish
in one system of co-ordinates without vanishing in all.
48. Summary of the Last Two Chapters. It may be useful
to review the results which have been obtained from the point
at which we introduced the energy-tensor T of the material
system. Initially it was brought in for the practical purpose
of calculating the gravitational field of a material body ; but
this has led on to a discussion of the general laws of dynamics.
As mentioned in 6, it is important, if we wish to adopt the
principle of relativity, to show that the laws of nature which
we generally accept are consistent with the principle ; or, if
not, to modify them so that they may become consistent. We
have had to modify one law the law of gravitation. The
laws of mechanics (Newton's laws of motion) are equivalent
to the conservation of momentum and the conservation of
mass. We have in 7(c) found it necessary to generalise the
latter by admitting that energy has mass ; and the conservation
of mass is absorbed in the conservation of energy. The most
general -statement of these two principles of conservation for
material systems is found in the general equations of hydro-
dynamics (or of the theory of gases), viz., (37-7) and (37-8),
and it is therefore sufficient to verify these. We have done
that by showing that they may be expressed in tensor-form.
We have even gone further ; we have shown that these laws
can actually be deduced from the law of gravitation. They
correspond to the four identical relations between Einstein's
ten equations of gravitation (39).
It has similarly been verified that our electromagnetic
equations are of tensor-form and are therefore consistent
with relativity. But in this case we have not deduced the
electromagnetic equations from anything else ; we have
merely shown their admissibility. The energy-tensor E^ of
the electromagnetic field is found from the consideration that
in interchanges between the material and electromagnetic
systems the total momentum and energy must remain constant.
When the co-ordinates are not Galilean, gravitational forces
will be acting and the total energy and momentum of the
material and electromagnetic systems will be altering. We
have shown how to find this flux of energy and momentum
(39-5), and in 43 we have traced it into the gravitational
field, showing that it reappears there as the quantity t%, which,
moreover, is conserved when no transfer of this kind is going
on. There is, however, one reservation necessary; unlikely
and E, ^ is not a tensor, and in order that this complete
conservation of energy and momentum may be apparent we
have to choose co-ordinates so that V g=l. This does not
imply any exception to the physical law of conservation,
because we can always choose co-ordinates satisfying this
condition. It is merely that the energy-tensor is slightly
more general than the physical idea of energy and momentum ;
the former may be reckoned with respect to any co-ordinates,
the latter must be reckoned with respect to co-ordinates
satisfying V g=l.
From the existence of an energy-tensor for the electro-
magnetic field, it is deduced that electromagnetic energy must
experience and exert gravitational force.
The remainder of our work has been principally concerned
with showing that our equations are equivalent to a principle
of least action. From a theoretical standpoint there is a great
deal to be said in favour of reversing the whole procedure,
starting from the principle of least action as a postulate ; but
I have preferred the present course as more elementary.
Some difficulty may be found in the fact that the time-
component of a four-dimensional vector is usually called by a
different name from the space-components. The following
table may be useful for reference :
Vector. Space-Components. Time-Component.
Tp negative momentum energy (mass).
T 1 flux of negative momentum flux of energy (mass).
ftp force negative rate of doing work.
%p negative vector potential... electric iscalar potential.
Jp- electric current-density ... electric charge. density.
CHAPTER VIII.
THE CURVATURE OF SPACE AND TIME.
49. We have now presented the laws of gravitation, of
hydromechanics, and of electronaagnetism, in a form which
regards all systems of co-ordinates as on an equal footing.
And yet it is scarcely true to say that all systems are equally
fundamental ; at least we can discriminate between them in a
way which the restricted principle of relativity would not
tolerate.
Imagine the earth to be covered with impervious cloud. By
the gyro-compass we can find two spots on it called the Poles,
and by Foucault's pendulum-experiment we can determine an
angular velocity about the axis through the Poles, which is
usually called the earth's absolute rotation. The name " abso-
lute rotation " may be criticised ; but, at any rate, it is a name
given to something which can be accurately measured. On
the other hand, we fail completely in any attempt to determine
a corresponding " absolute translation " of the earth. It is
not a question of applying the right name there is no measured
quantity to name. It is clear that the equivalence of systems
of axes in relative rotation is in some way less complete than
the equivalence of axes having different translations ; and
this may perhaps be regarded as a failure to reach the ideals
of a philosophical principle of relativity.
This limitation has its practical aspect. We might suppose
fchat from the expression (28-8) for the field of a particle at
rest it would be possible by a transformation of co-ordinates
to deduce the field of a particle, say, in uniform circular motion.
But this is not the case. We may, of course, reduce the
particle to rest by using rotating axes ; but we find it necessary
to take an entirely different solution of the partial differential
equations, satisfying different boundary conditions.
We have not hitherto paid any attention to the invariance
of the boundary conditions ; and it is here that the break-
down occurs. The axes ordinarily used in dynamics are such
THE CUKVATURE OF SPACE AND TIME. 83
that as we recede towards infinity in space the g^ approach
the special set of values (16-3). On transforming to other co-
ordinates the differential equations are unaltered ; but usually
the boundary values of the & and consequently the appro-
priate solutions of the equations, are altered. We can, there-
fore, discriminate between different systems of co-ordinates
according to the boundary values of the 0's ; and those which
at infinity pass into Galilean co-ordinates may properly be
considered the most fundamental, since the boundary values
are most simple. The complete relativity for uniform trans-
lation is due to the boundary values as well as the differential
equations remaining unaltered.*
We have based our theory on two axioms the restricted
principle of relativity and the principle of equivalence. These
taken together may be called the physical principle of relativity.
We have justified, or explained, them by reference to a philo-
sophical principle of relativity, which asserts that experience
is concerned only with the relations of objects to one another
and to the observer and not to the fictitious space-time frame-
work in which we instinctively locate them. We are now led
into a dilemma ; we can save this philosophical principle only
by undermining its practical application. The measurement
of the rotation of the earth detects something of the nature
of a fundamental frame of reference at least in the part of
space accessible to observation. We shall call this the
" inertial frame." Its existence does not necessarily contradict
the philosophical principle, because it may, for instance, be
determined by the general distribution of matter in the
universe ; that is to say, we may be detecting by our experi-
ments relations to matter not generally recognised. But
having recognised the existence of the inertial frame, the
philosophical principle of relativity becomes arbitrary in its
application. It cannot foretell that the Michelson-Morley ex-
periment will fail to detect uniform motion relative to this
frame ; nor does it explain why the acceleration of the earth
relative to this frame is irrelevant, but the rotation of the
earth is important.
The inertial-frame may be attributed (1) to unobserved
world-matter, (2) to the aether, (3) to some absolute character
84 RELATIVITY THEORY OP GRAVITATION.
of space-time. It is doubtful whether the discrimination
between these alternatives is more than word-splitting, but
they lead to rather different points oi view. The last alterna-
tive seems to contradict the philosophical principle of relativity,
but in the light of what has been said the physicist has no
particular interest in preserving the philosophical principle.
In this chapter we shall consider two suggestions towards a
theory of the inertial frame made by Einstein and de Sitter
respectively. These should be regarded as independent specu-
lations, arising out of, but not required by, the theory hitherto
described.
The inertial frame is distinguished by the property that the
g^ referred to it approach the limiting Galilean values (16-3)
as we recede to a great distance from all attracting matter.
This is verified experimentally with considerable accuracy ;
but it does not follow that we can extrapolate to distances
as yet unplumbed, or to infinity. If it is assumed that the
Galilean values still hold at infinite distances, the inertial
frame is virtually ascribed to conditions at infinity, and its
explanation is removed beyond the scope of physical theory.
We may, however, suppose that observational results relate to
only a minute part of the whole world, and that at vaster
distances the g^ v tend to zero values which would be invariant
for all finite transformations. In that case all frames of
reference are alike at infinity, and the property of the inertial
frame arises from conditions within a finite distance. In that
case physical theories of the inertial frame may be developed.
The ascription of the inertial frame to boundary conditions
at infinity may also be avoided by abolishing the boundary.
This is really only another aspect of the vanishing of the g^
at infinity. Our four-dimensional f-pace-time may be regarded
as a closed surface in a five-dimensional continuum ; it will
then be unbounded but finite, just as the surface of a sphere
is unbounded.
We have seen (44) that wherever matter exists space-time
has a curvature. It might seem that if there were sufficient
matter the continuum would curve round until it closed up ;
but it has not been found possible to eliminate the boundary
so simply. I think the difficulty arises because time is not
symmetrical with respect to the other co-ordinates ; in general
matter moves with small velocity, so that the different com-
ponents of the energy-tensor r l\ are not of the same order of
THE CURVATURE OF SPACE AND TIME. 83
50. Einstein suggests that in measurements on a vast scale
the line-element has the form
This expression includes the effects of the general distribution
of matter through space ; but there will be superposed the
local irregularities due to its condensation into stellar systems,
etc.
The expression (50-1) can be interpreted* as belonging to a
three-dimensional space which forms the surface of a hyper-
sphere of radius R in four dimensions, the time being recti-
linear. Let be the origin of co-ordinates (Fig. 5), A the
FIG; 5.
centre of the hypersphere, and % the angle OAG. If is the
azimuthal angle of the plane OAG, the line-element at C for
an ordinary sphere would be
- ds* =#% 2 + -R 2 sin 2 %(Z0 2 .
The expression (50-1) is the extension of this for an extra
dimension measured by 9.
In the figure the circumference of the circle CC' is 2jsfisiny,
but its radius measured along the sphei e is %. Similarly in
our curved space the surface of a sphere of radius R% will be
n 2 % ; successively more distant spheres will increase in
86 RELATIVITY THEORY OF GRAVITATION.
area up to a radius ^nR, and afterwards decrease to a point
for the limiting distance nR. The whole volume of space is
finite and equal to %7i~R 3 in natural measure.*
From (50-1) the values of G^ can be calculated just as in
28. We find, in fact,
2
G w r - v 7^ 9W> except 44 =0
6
so that ~m
Hence by (35-8)
1
-j^, except -W 44 = ,. . (50-3)
Unless we are willing to suppose that the matter in the universe
is moving with speeds approaching that of light, T^ is much
greater than the other components, and it is clearly impossible
to satisfy (50-3). The only possible course is to make a slight
modification of the law of gravitation. Neglecting the motion
of matter we shall have T u =p, and the other components
vanish. The modified law that satisfies (50-2) must then be
. . . (504)
where 1=1 /R 2 and ?=
Equation (504) replaces (35-8). The radius R may be as
great as we please, so that we may satisfy our scruples without
introducing any modification perceptible to observation.
In Hamilton's principle G becomes replaced by 6? 4A,
and space-time has a natural curvature 4A when no matter is
present ; this curvature is increased to 6/1 where there is
matter having the average density. (Cf. (444) with & 4 =0.)
Since the whole volume of space in natural measure is
27i 2 ^ 3 , the total mass of matter is SrfRZp^nR. The mass
of the sun is 1 47 kilometres ; the mass of the stellar system may
be estimated at 10 9 Xsun; let us suppose further that the
spiral nebulae represent 1,000,000 stellar systems having this
mass. Even this total mass will only give us a universe of
radius 10 15 kilometres, or about 30 parsecs much less than
the average distance of the naked-eye stars. Einstein's
hypothesis therefore demands the existence of vast quantities
of undetected matter which we may call world-matter.
Some curious results are obtained by fol >wing out the
properties of this spherical space. The parallax of a star
diminishes to zero as the distance (in natural measure) increases
up to knR ; it then becomes negative and reaches 90 at a
distance ?iR. Apart from absorption of light in space we
should see an anti-sun, at the point of the sky opposite to the
sun equally large and equally bright,* the surface-markings
corresponding to the back of the sun. After travelling " round
the world " the sun's rays come back to a focus. Since p and
R are related, it has been suggested that we can use the
invisibility of this anti-sun to give a lower limit to R, assuming
that no light is lost in space except by the scattering action
of the world-matter. But it appears to have been overlooked
that Einstein's new hypothesis is inconsistent with relativity
in its ordinary sense ; the anti-sun will not be a virtual image
of the sun as it is now, but of the sun as it was when it emitted
the light perhaps millions of years ago, when it was in another
part of the stellar system. Einstein has restored the difta -
entiation between space and time by assuming the space-tin e
world to be cylindrical, so that the linear direction gives an
absolute time. It is only locally that we can still mal.e
Minkowski's transformation ; rigorously the physical piinciple
of relativity is violated since space-time is no longer isotropic.
We regret being unable to recommend this rather picturesque
theory of anti-suns and anti-stars. It suggests that only a
certain proportion of the visible stars are material bodies ;
the remainder are ghosts of stars, haunting the places where
stars used to be in a far-off past.
Owing to this violation of the restricted principle of relativity
we have a feeling that Einstein's new hypothesis throws away
the substance for the shadow. It is also open to the serious
criticism that the law of gravitation is made to involve a
constant A, which depends on the total amount of matter in
the universe (A=ft 2 /4M 3 ). This seems scarce y conceivable ;
and it looks as though the solution involves a very artificial
adjustment.
51. An alternative proposal has been made by de Sitter
which seeing much less open to objection. He takes for the
line element
ds z = -R 2 {d x z -\-sm z x(d6 2 +sm 2 6d^)}+cos 2 x dtZ ' ( 51>1 )
For constant time the three-dimensional space is spherical a&
in (50-1) ; but there is also a curvature in the time-variable.
-"" A- -"" b ~ ~- i (51-2)
tan (it/R)=co3 ttan CD)
we find
ds 2 = -&(da*+6w*ca(d&+&m*&de*+sin 2 8dv*))). (51-3)
which corresponds to spherical polar co-ordinates (R, CD, , 6, cp)
in space of five dimensions. By measuring from different
azimuths we perform an operation corresponding to Min-
kowski's rotation of the time-axis, so that there is here BO
absolute time, and the original principle of relativity is fully
satisfied.
The properties of de Sitter's space-time are best recognised
from (51-1 ). Near the origin we have ordinary Galilean space-
time. As we recede, space has the spherical properties already
mentioned, and in addition measured time (ds) begins DO ran
slow relative to co-ordinate time (dt}. Finally at y=\n, i.e., at
a natural distance f TzR, time stands still. At any fixed point
ds is zero however large dt may be, so that nothing whatever
can happen however long we wait.
Of course, this is merely the point of view of the observer
at the origin of co-ordinates. All parts of this spherical con-
tinuum are interchangeable ; and if our observer could transport
himself to this peaceful abode, he would find Nature there as
active as ever. Moreover, adopting the co-ordinates natural
to his new position, he would judge his old home to be in this
passive state. There is a complete lack of correspondence
between the times at the two places. They are, as it were,
at right angles, so that the progress of time at one point has
no relation to the perception of time at the other point.
The line-element (51-1) leads to
r -?.
" R^^' v>
and accordingly the law of gravitation is taken to be (504),
with
The aggregate curvature due to matter is here neglected in
comparison with the natural curvature due to the modification
of the law of gravitation, and there is no assumption of the
existence of vast quantities of matter not yet recognised.
i UK tsfAUK AND TIME.
^ There is no anti-sun on de Sitter's hypothesis, because light,
like everything else, is reduced to rest at the zone where time
stands still, and it can never get round the world. The region
beyond the distance %nR is altogether shut off from us by
this barrier of time. The parallax of a star at this distance
will be such as corresponds to a distance R in Euclidean space,
and this is the minimum value possible.
The most interesting application of this hypothesis is in
connection with the very large observed velocities of spiral
nebulas, which are believed to be distant sidereal systems.
Since \/S f 44 ==cos /6 the vibrations of the atoms become slower
(in the observer's time) as cos % diminishes, in accordance
with 34. We should thus expect the spectral lines to be
displaced towards the red in very distant objects, an effect
which would in practice be attributed to a great velocity of
recession. It is not possible to say as yet whether the spiral
nebulas show a systematic recession, but so far as determined
up to the present receding nebulas seem to preponderate.
Superposed on the (spurious) systematic radial velocity will
be the individual velocities of the nebulae. It is scarcely
possible to say what these are likely to be without making
some assumption. There is no meaning in absolute motion,
and if two systems are entirely independent, so that their
relative motion has no physical cause, it must be quite arbitrary,
and there is no reason to expect it to be small compared with
the velocity of light. If, however, the systems have separated
from one another, it can be shown by rather laborious calcu-
lations* that their velocities will tend to become more diverse
as they recede, up to the limit ^nR for which the velocities
are comparable with that of light. We should thus have tin
explanation of the large velocities of the spirals, averaging
300-400 km. per sec., and we could perhaps form an estimate
of the value of R.
It must be remembered that in natural measure the internal
motions of stars in a spiral system will be of the same magni-
tude as in our own system, owing to the homogeneous character
of de Sitter's space-time. In co-ordinate measure these in-
ternal motions will be smaller owing to the transformation of
the time. The possibility of large divergent motions of the
systems as a whole depends on the large separation between
them.
* De Sitter, " Monthly Notices," November, 1917.
,i\.i RELATIVITY THEUKY (Jjlf UKAVITATIUJN.
52. So far we have used spherical co-ordinates, but we can
map the spherical space of Einstein or of de Sitter on a flat
space by performing the central projection r=R tan %
r will be represented by OP in Fig. 5, and the variables r, 6, <r.
will satisfy Euclidean geometry. This does not mean that
measured space is Euclidean ; but that we multiply our
measures by suitable factors in order to obtain results which
will fit together in Euclidean space, just as we did for a local
gravitational field in 28. With r as variable (50-1) and
(51-1) become, respectively,
i+P (52 ' 2)
where e=l/R z .
These show that at " infinity " (i.e., r=oo ) the values of g^
in rectangular co-ordinates approach the respective limits.
EINSTEIN. DE SITTER. GALILEO.
0000 0000 -1000
0000 0000 0-100
0000 0000 00-10
0001 0000 0001
the Galilean values being added for comparison.
De Sitter's limiting values are invariant for all transforma-
tions ; Einstein's only for transformations not involving the
time ; the Galilean values for the transformation of uniform
motion and a limited group of other transformations.
De Sitter's hypothesis thus appears to present the greatest
advantages ; but it will not satisfy the followers of Mach's
philosophy. He derives his inertial-frame from the spherical
property of space-time which in turn is derived from the slightly
modified law of gravitation ; it is not determined by anything
material. The followers of Mach maintain that if there were
no matter there could be no inertial frame, and it appears
that this is Einstein's reason for preferring his own suggestion.
In his theory if all matter were abolished, R would become
zero and the world would vanish to a point. There is some-
thing rather fascinating in a theory of space by which, the
more matter there is, the more room is provided. It is
satisfactory, too, from Einstein's standpoint, because he is
unwilling to admit that a thinkable space without matter
could exist. For our part, we feel equally unwilling to assent
to the introduction of vast quantities of world-matter, which
(to quote de Sitter) " fulfils no other purpose than to enable
us to suppose it not to exist."
53. In this discussion of the law of gravitation., we have
not sought, and we have not reached, any ultimate explanation
of its cause. A certain connection between the gravitational
field and the measurement of space lias been postulated, but
this throws light rather on the nature of our measurements
than on gravitation itself. The relativity theory is indifferent
to hypotheses as to the nature of gravitation, just as it is
indifferent to hypotheses as to matter and light. We do not
in these days seek to explain the behaviour of natural forces in
terms of a mechanical model having the familiar characteristics
of matter in bulk ; we have to accept some mathematical
expression as an axiomatic property which cannot be further
analysed. But I do not think we have reached this stage in
the case of gravitation.
There are three fundamental constants of nature which stand
out pre-eminently
The velocity of light, 3-00. 10 10 o.G.S. units ; dimensions XT" 1 .
The quantum, 6-55. lO" 27 ; ML 2 T'\
The constant of gravitation, 6-66. 10- 8 ,, ; M^L*!' 2 .
From these we can construct a fundamental unit of length
whose value is
4xlO- 33 cms.
There are other natural units of length the radii of the
positive and negative unit electric charges but these are of
an altogether higher order of magnitude.
With the possible exception of Osbornelleynolds ; s theory of
matter, no theory has attempted to reach such fine-grainedness.
But it is evident that this length must be the key to some
essential structure. It may not be an unattainable hope that
some day a clearer knowledge of the processes of gravitation
may be reached ; and the extreme generality and detachment
of the relativity theory may be illuminated by the particular
study of a precise mechanism.
