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Full text of "Introduction to the Theory of Relativity: Part 3"

PART II 
Trie General Theory of Relativity 






CHAPTER X 

! he Principle of Equivalence 



Introduction. The special theory of relativity had its origin in the 
development of electrodynamics. The general theory of relativity is 
the relativistic theoiy of gravitation. 

Once before* the problem of gravitatien gave rise to a new era in 
physics, the era of Newton's classical mechanics. The three fathers of 
scientific physics, Galileo, Kepler, and Newton, studied gravitation: 
Galileo the quasi-homogeneous gravitational field on the surface of the 
earth, Kepler and Newton the action of one mass point of great mass on 
another of much smaller mass. Galileo formulated the law of inertia 
and established the idea that the force acting on. a body was measured 
by its acceleration (and not by its velocity, as had been assumed before). 
Newton determined the amount of the gravitational action of one mass 
point on another. This gravitational force is always an attraction, 
and its magnitude is 



-o 1 



f=>-*—r» Cio.i) 

where m and M are the masses of the two mass points, r is the distance 

between them, and a is a universal constant, which has the value 

S = 6,66 X 10" E dyn. cm" g~'\ (10.2) 

The force which acts on the body with the mass m is the negative 
gradient of the gravitational potential G M , multiplied by its mass, m, 

where 

G.u = -k— (10.3) 

at the distance r from the mass point with the mass M. The potential 
energy of the two-body system m, M is 

= m(,_v = — K — , (10.4) 

151 



152 



THE PRINCIPLE OF EQUIVALENCE t Chz P . x 



Thfe theory Of gravitation is a typical example of a mechanical theory, 
in fact, the most important example. The force which acts on a mass 
point K at the time t is completely determined by the distances of all 
other mass points from P M at the time t, their masses, and the mass m 
of P,„ itself. It is, therefore, essential that the simultaneity oi distant 
events and the distance of the two mass points have an invariant 
significance. Newton's theory of gravitation is covariant with respect 
to Galilean transformations, but, of course, not with respect to Lorentz 

transformations. 

The work of Faraday, Maxwell, and Hertz m the held of electro- 
dynamics brought about new concepts which differed sharply from those 
of classical mechanics. The action at a distance of one mass point on 
another which is typical for mechanics, was replaced m electrodynamics 
by the action of the held on a mass point and the dependence of the 
field on the positions and the velocities of the mass points. In other 
words int-eraction. does not take place directly between distant mass 
points, but between points of the field which are separated by infini- 
tesimal distances. . . 

In prerelativistic physics, the mechanical theory of gravitation and 
the field theory of electrodynamics were based on the same concepts 
of space and time; and, therefore, though the two theories were funda- 
mentally different, they did not contradict each other. They were no 
longer compatible, however, when the analysis of the transformation 
properties of Maxwell's equations led to the development of the special 
theory of relativitv. While Maxwell's theory merely eliminated action 
at a distance from the realm of electrodynamics, the Lorentz trans- 
formation equations ruled out action at a distance from the whole of 
physics by depriving time and space of their absolute character 11 
the theory of gravitation was to be at all consistent with the other fields 
of physics, it had to be changed into a relativistic held theory. How- 
ever "an examination of the fundamental assumptions of Newton s 
theorv from the point of view of held physics revealed that the "rela,- 
tivization" of the theory of gravitation necessitated an expansion of the 
special theory of relativity into what is known today as the generai 
theory of relativity. We shall now retrace this analysis. 

T> e principle of equivalence. The gravitational force differs from all 
other forces in one respect: It is proportional to the mass of the body on 
which it acts. In the ponderomotive law of classical mechanics, eqs^ 
(2 13) the components of the force acting on a body are proportional 



Chap. X ] 



THE PRINCIPLE OF EQUIVALENCE 



153 



to the mass of that body. The constant factor, jtu , cancels on both 
sides of eqs. (2.13), and thus, the acceleration of a body in a gravitational 
field is independent of its mass. 

NeWtOh's theory Of gravitation accepts this fact, but docs not explain 
it. Within the framework of classical physics, an "explanation" was 
hardly called for. Other force laws, Coulomb's law of electrostatic 
forces, the nature of Van der Waals' forces, had not been "explained," 
either. Nevertheless, Nev r ton's law is in. a class by itself. The mass 
of a body is a constant which is characteristic for its behavior under 
the influence of any force, it is the ratio of force to acceleration. In 
this connection, we may call the mass of a body its "inertial mass," 
because it is a measure of its "inert.ial resistance to acceleration." 
The electrostatic force acting on a particle is the product of the electric. 
field strength, which is independent of the particle, and the charge of the 
particle, which is characteristic of the particle. Likewise, the gravita- 
tional force is the product of the "gravitational field strength," the. 
negative gradient of the gravitational potential, (10.3), and the mass of 
the particle. In its role as a "gravitational charge," We shall call the 
mass the ''gravitational mass" of the particle According to Newton's 
theory of gia vita lion, the inortial mass and the gravitational mass of 
the same body are always equal. This proposition is called the principle 
of equivalence for reasons which will become apparent later. 

Now, it might be that the "inertia! mass" and the "gravitational 
mass" are approximately equal, for most bodies, but that this approxi- 
mate equality is accidental, and that an accurate determination would 
reveal that the two kinds of mass of a body are really different. For- 
tunately, it is possible to subject the asserted equality of inertial mass 
and gravitational mass to veiy accurate tests. What has to be done is 
to find out whether the acceleration of all bodies is the same in the 
same gravitational field. 

Since 1 it is impossible to measure time intervals accurately enough, 
we cannot measure directly the accelerations of freely falling bodies, but 
must employ an indirect method. There is a type of acceleration 
which is certainly independent of the mass of the accelerated body, 
"inertial acceleration." When we refer the motion of bodies to a frame 
of reference which is not an inertial system, we encounter accelerations 
which do not correspond to real forces acting on a body, but which are 
merely reflections of the accelerations of tin; frame of reference relative 
to some inertial system. In Chapter II, we discussed these "inertial 
forces" in a special case, in which the chosen frame of reference was 
rotating with a constant angular velocity relative to an inertial system. 



154 



THE PRINCIPLE OF EQUIVALENCE 



[ Chap. X 



The "inertial force" of a body is proportional to its "inertial mass." 
If we can observe bodies under tin: combined influence of "inertial 
forces" and gravitational forces, the direction of the resultant for a 
particular body will depend on the ratio of its "inertial mass" to its 
"gravitational mass." Tf we observe several different bodies, we have 
an extremely sensitive test which will tell, us whether tills ratio is the 
same for all the bodies tester!. 

The experimental set-up is already provided by nature: The earth 
is not an inertial system, but rotates around its axis with a constant 
angular velocity. A body which is at rest relative to the earth is, 
therefore, subject both to the gravitational attraction of the earth 
and to "centrifugal force." Its total acceleration relative to the earth 
will be the vector sum of the gravitational acceleration and the "centri- 
fugal acceleration." Except for points on the equator, the two constit- 
uent accelerations are not parallel, and the direction of the resultant 
is a measure of the ratio between inertial and gravitational mass. 

Eotvos 1 suspended two weights of different materials, but with 
equal gravitational masses, from the two arms of a torsion balance. 
If the two inertial masses had been unequal, that is, if the resultants 
of the two weights had not been parallel, the balance would have been 
subject to a torque. The absence of such a torque showed, with a 
relative accuracy of about 10" s , that the ratio of inertial and gravita- 
tional mass is the same for various materials. 

The development of the special theory of relativity showed that at 
least part of the inertial mass of a body had to be attributed to internal 
energy. In radioactive materials, the contribution to the total mass 
from this source was bound to be considerable. Did this part of the 
"inertial mass" also show up as "gravitational mass"? The question 
was answered by Southerns, s who repeated Eotvos' experiments with 
radioactive materials. The result was the same as before: The "gravi- 
tational mass" turned out to be equal to the "inertial mass," even though 
the latter was in part caused by great quantities of bound energy. 
The principle of equivalence was ostensibly a fundamental property ol 
the gravitational forces. 

Preparations for a relativistic theory of gravitation. Before we can 
hope to create a relativistic theory of gravitation, we must first attempt 
to reformulate Newton's theory so that action at a distance is eliminated. 
This can be done fairly easily. 



1 Math, unci Naturw. Ber. aus Ungarn, 8, G5 (1800). 
= Proa. Roy. Soe., 84A, 325 (1910). 



Chap. X ] 



THE PRINCIPLE OF EQUIVALENCE 



155 



The gravitational attraction of one body with the mass m by several 
other ones can be represented by the gum of the "gravitational poten- 
tials/' (10.3), of these other bodies; this sum represents the potential 
energy U n of the first body divided by its mass m. The force experienced 
by that body is the negative gradient of its potential energy, 

f = -in grad G. (10.5) 

The gravitational potential depends on the positions of the other 
bodies. The contribution of every mass point is given by eq. (10.3). 
If we introduce a "gravitational field strength," 

g = -gradC, (I0.fi) 

we find, just as in electrostatics, that the gravitational lines of force 
neither originate nor terminate outside of masses, and that, in a mass M, 
4ttkM lines of force terminate. We conclude that the divergence of g is 

div g — —-limp, 

where p is the mass density. The potential G itself satisfies the equation 

div grad G = V'G = 4msp. (10.7) 

This equation, which was first formulated by Toisson, is, then, the 
classical equation of the gravitational field. Eqs. (10.5) and (10.7) 
together are completely equivalent to the equations of Newton's theory, 
which is based on action at a distance. 

Poissoir's equation, (10.7), is not Lorentzdn variant, Wherever p 
vanishes, it seems reasonable to assume that the three dimensional 
Laplacian operator V J has to be replaced by its four dimensional ana- 
logue, the operator 



dx" dx" 



r 

dp 



eV. 



In the presence of matter, we must remember that the mass density 
P is not a scalar, but one component of the tensor P 1 ™. We face the 
alternative of either replacing p by the L ore ntz- invariant scalar' ■q lir P'"', 
or replacing the nonrelativistic scalar G by a world tensor G"". 

On inertial systems. Suppose we were confronted by the task of 
finding a frame of reference which is an inertial system. An inertial 
system, according to the definition of Chapter 11, is a coordinate system, 
With respect to which all bodies not subjected to forces are unaeceler- 
ated. This definition by itself is not very helpful, as we have first 
to determine whether a, given body is subjected to forces or not. Ac- 
cording to classical mechanics, all (real) forces represent the interaction 



150 



THE PRINCIPLE OF EQUIVALENCE 



Chap. X 



of bodies with each other. A body is, therefore;, not subjected to 
forces if it is sufficiently far removed from all other bodies. 

This criterion is satisfactory from the point of view of classical 
mechanics. But in the theory of relativity we must try to eliminate 
all concepts which involve finite spatial distances. A concept such as 
"sufficiently far" has no Lorentz-jnva.riant significance. The defini- 
tion of an inertia! system should be based on the properties of the im- 
mediate neighborhood of the observer. 

We can determine an inertia! system if we can predict the accelera- 
tions of test bodies, that is, if we know the gravitational and electro- 
magnetic fields in the neighborhood. But there is only one method of 
measuring the field, and that is to measure the accelerations of test 
bodies. This is a vicious circle. 

However, there is a profound difference between the electromagnetic 
and the gravitational field. Nothing prevents us from choosing as test 
bodies uncharged and unpola.rized bodies and, thereby, from reducing 
the electromagnetic forces acting on them io zero. The effects of a 
gravitational held on a test body, however, cannot be eliminated, for 
the acceleration of a body in a gravitational field is independent of its 

mass. 

The action of a gravitational field on a body is indistinguishable 
from "inertia! accelerations." Both gravitational and inertial accelera- 
tions are independent of the characteristics of the test body. There- 
fore, we are unable to separate the gravitational from the inertial ac- 
celerations and to find an inertial system. 

The equivalence of gravitational and inertial fields in this respect is 
a consequence of the equality of gravitational and inertial masses. In 
fact, the equivalence of gravitational and inertial fields gave the; prin- 
ciple of equivalence its name. 

From this point of view, inertial systems are not a particular class erf 
coordinate systems; there is no real difference bed-ween a supposed in- 
ertial frame; of reference With- a gravitational held and a nem-inertial 
frame of reference. 

Einstein's "elevator." To illustrate: the equivalence of inertial and 
non-inertial frames of reference, Einstein gives the: example erf' a man 
enclosed in an elevator car. As long as the: elevator is at rest, the man 
can determine, by one of the: usual methods, the field strength of the 
gravitational field on the surface of the earth, which is about 981 cm 
sec" 3 . He can, for instance, determine: the time interval which a body 

> Strictly speaking, "sufficiently far" me&ris at an infinite distance. Our condi- 
tion can be only approximately satisfied. 



Ch 



ep. 



THE PRINCIPLE OF EQUIVALENCE 



157 



takes to drop to the ground from a point 100 cm above the ground. 
The gravitational field strength, in this case, is 



<J = 



100 XJ2 



(10.8) 






Suppose the man had no possibility of obtaining information freim 
outside his car. Instead of concluding that he: and his car are at rest 
and in a gravitational field, he might also argue as follows: "All objects 
in my car underge.) an apparent acceleration of 9S1 cm sec" 2 as soon 
as their motion is not stopped by collision with other bodies or with 
the floor of my car. As this acceleration deies not depend on the in- 
dividual characteristics of my test beidies, it is not likely that the ac- 
celerations correspond to real forces acting on the: test beidies. Prob- 
ably, my frame of reference (which is connected with the car) is not 
an inertia! system, but feir some reason, unknown to nic, is accelerated 
Upward relative to an inertial system at the rate of OS J cm sec"'. 
Those bodies inside my car which, at least temporarily, are not forced 
to participate in this accelerated motion, obey the law of inertia and 
remain behind until the floor eif the e:ar has caught up with them." 

imagine now that the: cable: erf' the: elevator breaks and that the car, 
not equipped with an automatic safety device, is allowed to fall freely 
in the gravitational field of the earth. During this fall, the bodies in- 
side the car undergo the same acceleration as the car itself, and, there- 
fore:, are unaccelerated relatively to the car. The observer inside the 
car might interpret this to indicate that the acceleration of the car has 
ceased and that Ids frame of reference is how an inertial system. 

Conversely, we may consider an even more: fantastic "conceptual 
experiment": The car is now placed in a region of space where the 
gravitational field vanishes. If the car is left alone and if it does not 
happen to rotate around an axis through its center of gravity, it will 
constitute an inertial system. A playful spirit decides to have some 
fun with the car; he begins to pull at the cable which is attached to the 
top of it, with a constant force. The: car is no longe.-r an inertial system. 
If a body inside the car is released from contact with other bodies, it 
will obey the law of inertia and remain behind the accelerated car, that 
is to say, it will "fall" to the: floor. The man inside may mistake the 
apparent acceleration of his test bodies for the effects of a gravitational 
field. 

The principle of general covariance. If we wish to develop a theory 
of gravitation which incorporates the principle of equivalence as an 

integral part, we must discard the concept of inertial frames of reference. 




THE PRINCIPLE OF EQUIVALENCE 



[ Chap. X 



158 

A11 fa ,ne s of «*«»» » «P* ■*-* "" » f " B ™ lati0n °' the kWS 
of nature, .„,..„ rf „u femes of reference in a 

ma thematrcal tarf Sr^e - £J inertial systems in particular by 
reference by B ~^ |J 2£f ' * conclude that we must no longer 
Loreut.au ^^J^^^ ^formations. To go over 
restrict oursehes to J^ ^ fc arbitrary transformation co- 

to linear coordm at £"*^* £* f r0]n one frame f Terence 
efficients is not sufficient, tor ti e represented by a 

to another which is ^f^^^J^^Z to the time 
coordinate transformation which is bpott I 

coordinate. re MiyIty has shown that we have 

called the 9 ™-I fe»» •/ ««• ^ „ bitrary 

In the second pari of Chapter v, i Msa iblo to in- 

But if we introduce curvffinear ^^^^^elatio^ also 
tions, then we must, for the ^^^rfXh « **>#"* 

introduce the metric tensor g^ , the W™ 1 " be characte r- 

-««*Wf f SS^to^ESff^-to- (5.99). 

ized in any simpler fashion than by the a re l at ionship in 

^fean^rrrr of^int ^dmates, rt - usual, 

■«^£^^ Mass 

coordinate transformer, ^^LV the gen- 
tensor calculus which contams fewe! basr ^ , 

era l formalism. As the metric tensor *®*^^*^. In a 
U can eliminate it as an b^.-J^S^ 

In the theory of gravitation, we en ^»unte r a «m ^ ^ 

can formulate the special theory of ^W«Si«M in a four 
connate systems -^^^^^04^ connate 
dimensional world. However, H l 






I 



Chdp. X 1 



THE PRINCIPLE OF EQUIVALENCE 



159 



systems in which the components of the metric tensor take particular, 
constant values, tfa, , and where the components of the affine connection 
vanish. A formalism which is eovariant only with respect to trans- 
formations leading from one coordinate system of this kind to another 
system of the same kind does not require the introduction of a number 
of geometrical concepts which are an integral part of a formalism which 
is eovariant with respect to general coordinate transformations. The 
particular- coordinate systems in which the components of the metric 
tensor take the constant values %, are the inertial systems, and fire 
coordinate transformations which lead from oae inertial system to 
another inertial system are the Lorentz transformations. 

The equivalence of all frames of reference must be represented by the 
equivalence of all coordinate systems. It must be Impossible to intro- 
duce, in the presence of a gravitational field, the privileged Lorentz 
coordinate systems. Extending the terminology of Chapter V, we shall 
call a four dimensional Minkowski space Kiemannian if it is impossible 
to introduce Lorcntzian coordinate systems. 

In a Itiemannian space, the components of the metric tensor, the 
a„ , are non-constant functions of the coordinates in all coordinate 
systems. A restriction to Lorentz transformations would not bring 
about a simplification of the formalism. The hypothesis that the 
geometry of physical space is represented best by a formalism which is 
eovariant with respect to general coordinate transformations, and that 
a restriction to a less general group of transformations would not 
simplify that formalism, is called the principle of general covariance. It 
is the mathematical representation of the principle of equivalence. The 
development of a theory of gravitation which satisfies the principle of 
general covariance has furnished theoretical physics with the most 
satisfactory field theory which has so far been proposed. 

The nature of the gravitational field. From the principle of equiva- 
lence, it might appear that gravitational fields are not real, that they 
are basically nothing more than ''Inertial forces." Everybody feels 
instinctively that that cannot be true. 

If the man in the elevator car were to measure the direction of the 
accelerating force of the earth with great accuracy, he would find that 
the lines of force converge. This discovery would not enable him to 
separate gravitational field and inertial field, but it would tell him that 
the field was not wholly inertial. Because of the convergence of the 
lines of force, there is no frame of reference in which the gravitational 
held of the earth vanishes everywhere. The impossibility of introduc- 
ing a frame of reference which has everywhere the properties of an in- 



160 THE PRINCIPLE OF EQUIVALENCE [ ch aP . X 

ertiai frame of reference is represented by the impttSMg of intro- 
ducing a Lorentzian coordinate system, that is, by the Riemannian 

character of space. , 

If it is impossible to introduce coordinate systems in which the op- 
ponents of the metric tensor assume constant, preassrgned values, then 
the metric tensor itself beeom.es part of the hold, and te-te 
field equations which restrict and determine, to some extent, the func- 
tional dependence of the ^ on the four world coordinates 

Then what is the physical significance of tins tensor field g,„ ! Let 
us consider a region of space in which the gravitational field W^. 
Tf we introduce a non-inertial coordinate system, tree bodies mil be 
accelerated with respect to the chosen coordinate system, although 
they move along straight world lines. If we express the law of ineiiia 
in terms of an arbitrary, curvilinear coordinate system, the equatrons 
of motion, according to eq. (5.99), are 

(10.9) 



dlT 
dr 



= - r Urv 



where the { "} are linear in the first derivatives of the g. 



= hf^Uvm + ?*&* - 9**.$)- 



(10.10) 



The fe, appear, in a manner of speaking, as the potentials of the in- 
ertial field » It is, therefore, reasonable to assume that, m the presence 
f a gravitationai field, the g,„ are again the potentials which determine 
the accelerations of free bodies;: in other words, that the fc are the 
potentials of the gravitational field. These gravitational potentials 
must satisfy differential equations which resemble the Lap aeian or 
Foisson's equation in four dimensions. We shall find later that there 
is only one particular set of equations of this type which is eovamul 
with respect to general coordinate transformations. 

M any rate;, we find that die theory of gravitation will have to deal 
with spaces which are not "quasi-Euclidean," that is, in winch no in- 
ertia! coordinate systems can be introduced. Before we can continue 
our discussion of the gravitational field, we must develop the geomct > 
of Riemannian spaces somewhat further than we did m Chapter n 

particular, we shall have to find a mathematical criterion which tell, 
us whether a space is Euclidean or not. 



\ 



CHAPTER XI 
The Riemann-Criristofrel Curvature Tensor 



The characterization of Riemannian spaces. According to the defini- 
tion which we gave in Chapter V, a Euclidean space is one in which 
it is possible to introduce Cartesian coordinates; all other spaces are 
non- Euclidean. 

Even if we knew, in a specific case, the components of the metric 
tensor as functions of a particular coordinate system, we could not, 
obviously, try out all the conceivable coordinate transformations to 
find whether some of them lead to Cartesian coordinates. We need a 
criterion which can be applied in a systematic fashion to determine 
whether or not a space is Euclidean. 

The non-Euclidean spaces which we encounter in our daily experience 
are curved two dimensional surfaces imbedded in our three dimensional 
space. It mi glit appear that their geometric properties cannot be char- 
acterized without taking into account their relationship to the imbedding 
space: Actually, at least the Euclidean or non-Euclidean character of 
such a two dimensional space is independent of its relationship to the 
three dimensional space. Let us consider, for instance, a plane, which 
we shall represent by a sheet of graph paper. The ruling on the paper 
represents a Cartesian coordinate system, so there is no doubt that the 
plane is Euclidean. Let us now change the relationship of the two 
dimensional space to the imbedding space by rolling up the paper- the 
ruling will still retain all the characteristics of a Cartesian coordinate 
system. The distance between two infinitesimally near points on the 
paper is given by the equation 

ch~ = d-j? + dy~, 

both before and after we have, rolled up the paper. In other words, 
the metric tensor has the components 

CJil: = <>ik - (11.1 ) 

Furthermore, any line on the paper which was straight before we rolled 
the paper remains the shortest line which connects two points on the 
paper and which lies wholly in the two dimensional space. 

161 



162 



THE CURVATURE TENSOR 



[ Chap. XI 



T 



The Euclidean character of a space depends only on Ihe metric. And 
we must develop a method by which we can distinguish a Euclidean 
from a non-Euclidean metric. 

The integrability of the affine connection. To find such a, method, 
we shall return to the concept of the parallel dis.pl acement of vectors, 
which was introduced in Chapter Y. An affine connection with the 
components r« enables us to displace; a vector along a curve uniquely 
according to the differential laws 



ft 1 = _l4 & ^M 

A metric g it . determines a particular affine connection, 
components 



(11.2) 



t 1 . with the 



i i t 

[iki 



iff *(Sf«,s H" 8f3»-.« ~ ?**.«)■ 



(11,3) 



If the affine connection has as its components the Ch rist of fel symbols, 
the result of the parallol displacement of a vector is independent of 
whether the law (11.2) is applied to its covariaut or its contra variant 
representation. 

Let us displace a vector parallel along a closed curve (Fig. 9), until 
it returns to the starting point. Then we shall find either that the 
vector obtained is identical with the vector with which we started, or 
that it is a different vector. Jf it is the same vector, regardless of the 
choice of the initial vector and regardless of the shape of the closed 
curve, the affine connection is said to be integrable. In such a ease, 
we can speak of "distant parallelism," which means that, when wc dis- 
place a vector at a point Pi parallel to itself along some curve to another 
point P 2 , the components of the vector at P, do not depend on the 
choice of' the path of displacement between Pi and I\ . When the 
affine connection is integrable, a vector at one point generates a whole 
field of parallel vectors throughout the space. 

Euclidicity and integrability. If the components of the affine connec- 
tion are connected with the metric tensor by eqs. (11.3), we shall find 
that the Euclidicity of a space is directly related to the integrability 
of the affine connection. 

When a space is Euclidean, we can introduce a Cartesian coordinate 
system, in which the components or the metric tensor are constants, 

(11.4) 



</;■'■• 



Oih 



Chap. XI ] 



THE CURVATURE TENSOR 



163 



According to eqs. (11.3), the {«} vanish in such a coordinate system, 
and the da 1 , Sbi of eqs. (11.2) vanish, too. The parallel vectors have 
the same components at all points; this affine connection is certainly 





Fig, 9. Tin; integrability of an affine connection. In (a), the affine connection. 
is integrable; in (b), it is not. 

integrable. Integrability is, by its definition, an invariant property 
of the affine connection, independent of the choice of the coordinate 

system. We conclude, therefore, that ihe affme connection of a FnuMdean 
space is always integrable. 

Conversely, we shall show, by actual construction, that we can always 



1 



164 



THE CURVATURE TENSOR 



[ Chap. XI 



find a Cartesian coordinate system when the affine connection (11 3) 
is integrable. This statement, to be feme, requires a slight generaliza- 
tion of our definition of a Euclidean space. So far, we have defined as 
Euclidean a space' in which we can, by means of a real coordinate 
transformation, introduce a coordinate system in which the metric 
tensor g ik takes the constant values S lk . According to this definition 
the four dimensional Minkowski space is not Euclidean. The essential 
difference, between a Euclidean space and the Minkowski space is that 
in a Euclidean space the quadratic form of the coordinate differentials 
is positive definite, 



ds 2 = dxidx, > 0. 



(11.5) 



for arbitrary real values of &% ; in the Minkowski space, however, 
with the quadratic forte 



dr 2 = (dx 4 f - \d^dx\ 



(11.0) 



we found that dr~ can take negative as well as positive values, and 
the interval can be "space-like" or "time-like" (see Chapter IV, pUl). 
It is, therefore, impossible to carry out a real coordinate transformation 
leading from eq. (11.5) to eq. (1.1 .G). 

But in their analytical properties, the forms (11.5) and (11. 0) are 
very similar. We pointed out, at the end of Chapter V, that the \h\ 
belonging to the metric form (11.0) vanish ; the components of a parallel 
displaced vector are, therefore, constants, and the parallel displacement 
is integrable. This will be true, in. general, whenever it is possible to 
introduce coordinates in space so that the components of the metric 
tensor become constants. In such a case, we shah call the space flat. 
Flatness is the generalization of Euchdicity for spaces in which 'the 
metric is not necessarily positive definite. 

With this correction in mind, we assert now that whenever the parallel 
displacement defined by eqs. (1.1.2), (U.3) is integrable, the space is flat; 
that is, there exists a coordinate system in which the metric form is 



M % = Xffi^i', 



e; = ±1. 



(11-7) 



The proof will be carried out in two steps. If the components of 
the affine connection are symmetric in their subscripts, the iutegrability 
of the affine connection enables us to construct a coordinate system in 
which the components of the affine connection vanish. This fact is 
independent of the existence of a metric and will- be proved without 



Chap. XI 



THE CURVATURE TENSOR 



1G5 



resorting to cq. (11.3). Then, if a metric is defined, the vanishing of 
the < r is equivalent to the constancy of the components of the metric 
tensor. 

Let us consider at a point P a set of n eo variant vectors (n being the 

number of dimensions), ?>,- , which are linearly independent of each 
other; that is, which satisfy the inequality 

where $**" * is the contravariant Levi-Civita tensor density. Let us 

now displace all the n vectors h along the same path. The change in 
A is 

. ■••• h< 4- ■■• 4- r? .& ■ . ■ ;„1 sf ! 

(11.9) 






This expression may be considerably simplified. First of all, the bracket 

on the right-hand side of eq. (11.9) is skewsy mine trie in ah indices 
| ■ ■ ■ «« . For, if we exchange, for instance, the two indices ii and u , 
the bracket goes over into 

ah i 



-1 kt 



(11.10) 



+ 5- 

= -l^'" in rt +^--'»ri; + ■ - , 

Second, k can assume only the same value as the; displaced i, , because, 
for all other values, the Levi-Civita tensor density component vanishes. 
We can, therefore, replace the square bracket in eq. (11. 9) by the 
expression 

^- i »rU+ ■■• +5 H-*r^ = ^--rL, (ilii) 

and eq. (11.9) goes over into 

6A = A-rL-or'. (11.9a) 

Along any path, A satisfies a linear, homogeneous differential equation 
of the first order. Therefore, it cannot vanish anywhere on that path 
if it does not vanish everywhere. 

We conclude that, if n vectors hi are linearly independent, they preserve 
that property under parallel displacements. 

We shall assume now that the affine connection is symmetric in its 
subscripts and integrable; then each one of the vectors & generates a 
field of parallel vectors. Each of these fields satisfies differential equa- 
tions of the form 



166 



THE CURVATURE TENSOR 



I Chap, X! 

(11.12) 



The right-hand side is symmetric in the subscripts i and k. There- 
fore, the antisymmetric derivatives of the frj vanish, 

bi.it — bk,i = 0. 

s 

From this equation, we conclude that each of the n fields 6; is a gradient 
field, and that there art; n scalars b, so that 

k-h. (11 " 14) 

These u scalars 6 may be considered as the n coordinates of a new 
coordinate system. Because of eq. (11.8), the Jaeobian of the co- 
ordinate transformation does not vanish. Nqw we can show that, in 
the new coordinate system, the rfj, vanish. 

Let us transform the components of the affine connection according 
to eq, (5.81), 

fit* 1 

According to eqs. (11.14), the derivatives — — are the vector components 









be , so that the parenthesis of eq. (11.15) is 

W^-we--** 1 *-**- (1L1G) 

On account of eq. (11.12), this expression vanishes, and, therefore, the 
T.i£ of eq. (11.15) vanish, Loo. 

Returning now to a consideration of the metric tensor, we can solve 
the eqs. (11.3) with respect to the derivatives of g„. n , We add the two 
equations 



\(9ik,l + ffitjs — 9ki,i) = )r.l( G'i 



and 



!(?*>,! + c Jki.i — gu.k) — 



and obtain: 



\il\ 9sk ' 



;-■■■-■ ~ : w-.s* + ($/** 



A;; 



(11.17) 
(11.18) 



If the \-T 7 7 vanish, the ga are constant. 



Chap. XI 1 



THE CURVATURE TENSOR 



167 



To reduce the constant ga to the form (1.1.7) is a purely algebraic 
problem. It is solved by a standard oi'thogonaliaation and normaliza- 
tion process and is of no particular interest to us. Any space in which 
the components of the metric tensor are constants is ipso facto flat. 

The criterion of integrability. When the affine connection of a space 
is symmetric and integrable, the equations of parallel displacement, 
(11.2), may be considered not only as ordinary differential equations 
along a given path, but as partial differential equations for a whole 
vector field. We may write them in the form 



(1L1S) 

These equations 



a'\i = -riv, 

and analogous equations for covariant vector fields. 
are overdetermined : they are n equations for the n components of a 
vector. To have solutions, they most satisfy differential identities. 
The form of these identities is well known. If we differentiate eqs. 
(11,19) with respect to a coordinate £, we obtain 

a'U = -I&itf* ~ tti<t\ H = C— fwi* + r^I^y. (11.20) 

By subtracting the same expression, with the indices i and k exchanged, 
we obtain the conditions which must be satisfied so that the sequence 
of the differentiations is without effect: 







<tU - rr* it -- rr,n» + riv^W. 



(1.1.21) 

As the values of the a at one point may be chosen arbitrarily ^e ob- 
tain the conditions of integrability, 



Rih£ m n\, k - T n lkti - r&rfc + nwh 



0. 



(11.22) 

These conditions are not only necessary, but also sufficient. The proof 
runs along the same lines as the proof of the the*.) rem that a covariant 
vector field is a gradient field when and only when its skewsy mine trie- 
derivatives vanish (see Problem 14, Chapter V). 

The commutation law for covariant differentiation, the tensor char- 
acter of Em< - The vanishing of the expression R ikt '.' of eq, (11.22) is 
equivalent to the integrability of the afline connection, and, therefore, 
must be an invariant property. There are, of course, spaces which are 
not fiat and in which the li,,,/.' are different from zero (for example, the 
spherical surface). How do the quantities R m ? transform? 

To answer this question, we shah derive a tensor equation in which 
the Baa' appear: the commutation law for covariant differentiation. 
We compute the expression 



16S THE CURVATURE TENSOR I Chap. x\ 

According to the definition of covariant differentiation, we have 

A%i = A"„-+rr,-.4 ! ; (11.23) 

and a second covariant differentiation yields the expression 



XV = U" ;i ). t + r; ft /i' ; i- rvr ;s 



1 

i_v + T^,a l + T? i A\ k + v: b A s :l > 

+ TitFuA t ~Ti i A n , i -V%T-7U i 



(11.24) 



Let us assume that the components of the affine eonnection are 
symmetric in their subscripts. When we exchange the indicts i and k 
and subtract the resulting equations from (.1.1.2-1), the underscored 
terms cancel, because of their symmetry in i and k, and wc are left 
with the relation 



A -ik — A .]M — Riw A 



(11.25) 



This equation is the commutation law for covariant differentiation. In 
a flat space, covariant differentiations commute like ordinary partial 
differentiations; we could have predicted this, for in a flat space there 
are coordinate systems with respect to which covariant and ordinary 
differentiation are the same. 

When the space is not fiat, the commutator depends only on the 
undifferentiated vector. 

The commutation law for covariant differentiation of covariant 
vectors is 



A ti tt - A m = -R ikl ?A n 



(11.20) 



The left-hand sides of eqs. (11.25) and (11.2(3) transform as tensors. 
The right-hand sides are, therefore, tensors too; and, as the factors A 1 
and A " are arbitrary.; it follows that the R^ themselves arc the components 
of a tensor. The tensor 



Riki- — V'n.k — V[L,i — T„iTik -T T f a,Tii 
is called the (Riemann-Chrixloffcl) cur nature tensor} 



(11.27) 



1 In this book, the mctex notation of Levi-Civita lias been adopted. Un- 
fortunately, there is no standard for the writing of indices of the curvature 
tensor. Many authors write our last index first, our third index second, and our 
first and second indices ns third and fourth indices, respectively. The notation 
in this book will consistently follow the definition given by eq. (11,27). 









Chap. XI ] 



THE CURVATURE TENSOR 



109 



Properties of the curvature tensor. The curvature tensor is defined 
toge ther with any afnno e on ue ction . 1 1 o we v e r , i t h as c ertai n sy m metry 
properties only on the condition that the components of the affine con- 
nection, are Christoffel symbols (11.3), which are associated "with a 
metric. Wc; shall first consider those; properties of the eurvature tensor 
which are in deps indent of the relationship of the V i; - to a metric. 

(1) Rua- is slcewsymmetric in the indices i and k, 



Rm? + Rut" = 0. 



(11.28) 



This relationship is satisfied by the expression (11.27), regardless of 
any symmetry properties of the \'\h . 

When the components of the affine connection are symmetric in their 
subscripts, the curvature tensor satisfies another symmetry law and, 
furthermore, a set of differential identities. 

(2) When we rotate the first three indices cyclically, the mm of the 
com'pon mi s v an ishes , 



R, S J + gut? + R m ? 



0. 



(11.20) 



The proof is carried out by straightforward computation of the ex- 
pression (1 1.29). 

(3) We obtain the differential identities as follows: We differentiate 
eqs. (11.26) covariantly with respect to a new coordinate, 

A K iu ~ A s - M i = -R i; -J-.A«. - R^JA^, (.11,30) 

rotate the three indices i, k, I cyclically, and add. The result is 
(A,;ijti — A ti iik) + (As-Mi — 4»{*»i) + (A. x; ui ; — A s -t, :i )\ 

= -A n (R if J; i + R kis :- i + R [is -^\ :!:.ol 
-(Ri^A nU + R m ?A ni4 + Rti£A r £. 

The parentheses on the left-hand side are all commutators of the co- 
variant differentiation. As can be readily shown, the commutation 
law for the covariant differentiation of a covariant tensor of rank 2 is 



J 5 lm;ih 



Bi 



— Ri/a- B, ir , 



R 



' l B la 



(11.32) 



Applying this law to each of the parentheses on the left-hand side, 

we obtain, for instance, for the first parenthesis, 

A-intf - A, - iU: = -Ru^A a ;i — Rm?A m . (11.33) 

When we substitute these expressions in eef. (11.31), the first term on 



170 



THE CURVATURE TENSOR 



[ Chap. XI 



the right-hand snide of eq. (11.33) cancels with a term in the last paren- 
thesis on the right-hand side of eq. (11.31). The second term or± the 
right-hand side of eq. (11.33) cancels together with its cyclically rotated 
analogues, because of eq. (11.29); and we are left with the equation 



AJ.Rn ts . ;j + Rkls- \i + Rlis- ;k) = 0. 



(11.34) 



The vector A n is arbitrary; therefore, the curvature tensor must satisfy 
the identities 



R-iks- \l ~T~ R-kls. ;i ~T~ Rli«- 

They are called the Bianchi identities. 



0. 



(II. .35) 



The co variant form of the curvature tensor. So far, we have not 
made use of a metric. When a metric is defined, and when the r'i 
are connected with the metric through eqs. (11-3), the curvature tensor 
satisfies additional algebraic identities. The purely co variant curva- 
ture tensor is obtained by lowering the index n of eq. (11.27), 



Sa 



ttikl- fflnn 



(11.36) 



This co variant curvature tensor can be expressed in terms of the 
"Christoffel symbols of the first kind," which are the components of 



the affine connection, ( ., >, 
W 



multiplied by g>„ , 

A 
E#, A = $&{$} = ¥ikiA + ga.k ~ gik.t). 

The first terms of Tina™ can be written in the form 
'Jr. 



(11.37) 



FwL-i = [U, m\,k 



w 



\li, m], k — L,} ([nk, m] -.- [mk, n}) 



(11.38) 



Substituting these expressions in eq. (11.36), we obtain: 

Em* = [U, m].k - [Ik, mh + g'' s (\nd, r][lk, s) - [mk, s][li, r]). (11.39) 

Once we have obtained the co variant curvature tensor in this form, we 
can verify the following two algebraic identities, in addition to identi- 
ties (.1.) and (2): 

(4) The cQvariant curvature tensor is skewsymmetric in its last index 
pair, 

£*&, + Rik„u - 0. (11.40) 



The parenthesis of eq. (11.39) is obviously skewsymmetric in m and I. 



Chap. XI ! 



THE CURVATURE TENSOR 



171 



The first two terms contain only second derivatives of the components 
of the metric tensor, in the combination 

[li, m],k — [Ik, mj, j = i[(g«ajk — Jn.mi) + (tfn,™; — {jta,«)J, 

This expression is also skewsymmetric in m and I. 

(5) The covariant curvature tensor is symmetric in its two index pairs, 

(11.41) 



Tt-ikl 



himik 



This relation can be verified exactly like eq. (11.40). 

In the remainder of this chapter, we shall consider only metric spaces, 
w r here the components of the affine connection are given by eq. (11.3). 



Contracted forms of the curvature tensor. From the curvature tensor 
we can obtain tensors of lower rank by contraction. We can form the 
tensors Riu- , R<kt , Mna?tf[, and Rik£g k '~] all other contracted tensors 

vanish because of the skewsymmetry of Rmm in (i, k) and in (I, m). 

The four tensors of rank 2 listed above are all identical (except for 
the sign), because they can be obtained from each other by changing 
the sequence of indices in one pair, or by changing the sequence of the 
two pairs. It is customary to designate the contracted tensor S«y' by 
Ru ■ This tensor Ru is symmetric in its two indices, because of the 
symmetry properties of R ih C ■ 

By contracting Rki once more, we obtain the curvature scalar, R, 



E = /% 



Rki — Rikl- 



(11.42) 



Written in terms of the Christoffel symbols, R k i takes the form 



E & 



Isl.k 



Ik 



■SI' 



+ 



rkJXlsj- 



(11.43) 



Except for the first term, the symmetry of each term with respect to 



the indices k and t is obvious. As for the first term, i ' > can be ex- 
pressed in the form 



W'($rl, S + ffrs.I ~ gis.r). 



The first and the third term in the parenthesis, taken together, art; 
skewsymmetric in r and s and vanish when multiplied by §". There 
remains only the second term, 



3 ri _ i g,l 

f y o™,i — s — 



OogVff),!, i7 = !^|. (H-44) 



172 



THE CURVATURE TENSOR 



The first term of eq. (11.43) is, therefore, 

{isj.r {U,gVg)M > 

and is symmetric in I and k. 



1 Chap, XI 



(11.45) 



The contracted Bianchi identities. By contracting the Bia.nchi iden- 
tities (11.35) twice, we obtain identities which contain only the con- 
tracted curvature tensor. Contracting eq. (11.35) first with respect, 
to i a±id n, we get 

By changing in the last term the sequence of I and r, we obtain, because 
of eq. (11.28), 



or 



Rks:l + -ffills>;r — Kts:k = 



R k s .- l + re«:: :r - Bfa = 0. 



As th.e next step, we change the sequence of the contra variant indices 
s and r in the second term, 

tiki'- = ~Rkl" i 

and contract with respect to the indices k and s. We obtain 

R\i — Sfijj.'if = 

or 

(R lx - iff"/?),, a 0. (11.46) 

Tlie expression in. the parenthesis is often denoted by G u , 

G ls = R u - y u R, (11.47) 

The number of algebraically independent components of the curva- 
ture tensor. The components of the covariant curvature tensor, Bmm , 
satisfy tin; algebraic relations (11.28), (11.29) (both with the index n 
lowered), (11.40), and (11.41). The number of the algebraically inde- 
pendent components is thereby reduced, and we shall show in this 
section that their number in an n dimensional space is 



ff= A«V- !)■ 



(11.48) 



In a two dimensional space, the curvature tensor has only one significant 
component; the scalar R is already sufficient to characterize the curva- 



Chop. XI ] 



THE CURVATURE TENSOR 



173 









ture completely." in a three dimensional space, there are six alge- 
braically independent components. This is also the number of Inde- 
pendent components of the contracted tensor, R ki ; the contracted 
tensor characterizes the uncon traeted tensor completely. 3 In a four 
dimensional space, jV equals 20, while the contracted curvature tensor 
has only 10 independent components. Unless a space has at least jour 
components, its curvature is completely characterised by the contracted 
forms R kl . 

We shall now derive eq. (1.1.48). We shall divide the components 
of Rm- into three groups: those components where each index of the 
first pair has the same value as an index of the second pair, such as 
Rim ; those where only one index value is represented twice, such as 
Urns ; and those where all four indices are different, R vai , , and so forth. 
Obviously, not all four indices can be equal. 

In the first type, with only two indices different, the first and the 
second index pairs must be identical, as the two indices of a pair must 
be different (because of eqs. (11.28) and (11.40)). These components 
are of the typo R ;kik (do not sum!). R ikki (lifters from R, kik only with 
respect to the sign. There are as many components R ikik as there are 
different index pairs (i, k), with i ^ fa 

The index i can take n different values, k is different from % there- 
fore, for any given i, k can take only (■« - 1) different values. Since 
the sequence of i and k is of no consequence, we must divide the product 
n{n - 1.) by 2. The number of different index pairs (i, k), i ^ k is, 
therefore, 



Nr = i«(?l - 1), 



(11.49) 



and the number of algebraically independent components with two 
different indices is also 



A 1 "/ = \n(n — 1). 



(1 1 .50) 



The cyclic identities (11.29) do not further decrease this number, be- 
cause they are independent of the other algebraic id entities only when 



s It can be shown that In a two dimensional space the curvature tensor depends 
l R, as follows: 

3 hi a three dimensional space, the »,■«» depend on the R h i , as follows: 
R M * = 6'lR kl - SlR ;l + guK" - g ;l R k : - tf$$#M - ^ fi )B, 



174 



THE CURVATURE TENSOR 



[ Chap, U] 



all four indices are different. If two of the four indices i, k, I, m are 
equal, eqs. (11.29) are either of the form 

Rikti + (Rklii) + Rliki = 0, 

or 

Rikim + Rkiim + \Riihm) — 0. 

Either equation is Satisfied because of eqs. (11.28), (1.1.40), and (11.41). 
Let us now consider the second group of the components, those with 
three different indices. All these components can be brought into the 
form Rikim by applying eqs. (11.28) and (11.40). There arc n different 
choices for the value of i. Of the remaining (n — 1) numbers, we must 
pick two different ones for h and rti. According to eq. (11.49), there 
are J.fttt — l)(n — 2) different choices for (A:, in), and the number of 
algebraically independent components of the second type is 

A'„ = fof« - 1)0 - 2). (11.51) 

Again the cyclic identities do not further decrease this number. 

In the third group, all four indicts are different. We may first pick 
the first index pair in \n(n — 1) different ways. Out of the remaining 
{n — 2) values, we must choose' the second index pair, which can be 
done in l(n - 2)(n — 3) different ways, Because of eq. (11.41), the 
sequence of the two pairs does not matter; we must, therefore, divide 
once more by 2. There are, then, 

|-M« - l)-i(" - 2)(«, - 3) 

different ways of picking two completely different index pairs. 

In this case, the number of algebraically independent components it. 
further decreased by the existence of tin; identities (11.29). Of the 
three components $«& , B&h , and Bim , for instance, each has a dif- 
ferent combination of index pairs, but any one can be expressed in 
terms of the other two. The number of algebraically independent 
components of R iklm with four different indices is, therefore, 

Nu, = %-hin(n - l)-$(« - 2)(w - 3) = - L \n{n - l)(n - 2)(n - 3). 

(11.52) 

The total number of algebraically independent components of R ik h n 
is the sum of the three numbers Ni , N IT , and N nz . This is the expres- 
sion (11.48). 



CHAPTER XII 

The Field Equations of the General 
or Relativity 



Th 



eory 






The ponderomotive equations of the gravitational field. In this chap- 
ter, we shall formulate the field equations and the ponderomotive equa- 
tions of the gravitational field. 

Unfortunately, we cannot treat the ponderomotive equations fully 
at this point. We must, for the present, restrict ourselves to the mo- 
tions of small particles which. contribute only negligible amounts to the 
field. 

The principle of equivalence determines the law of motion of such 
particles. Their motion under the influence of the gravitational field 
must be indistinguishable from inertial motion, that is, their paths are 
geodesic world lines, 

^ (p,/^^< dS = g^ d f, (12,1) 

This law of motion is more involved than, for instance, the law of 
motion of electrically charged particles in the special theoiy of relativity. 
While eq. (7.49) is linear in the field intensities, eq. (12.1) is not linear 
in the g^ and their derivatives. This nonlinearity is characteristic of 
equations which are covariant with respect to general coordinate trans- 
formations; it is, thus, a consequence of the equivalence principle. 

The representation of matter in the field equations. Before we- set up 
the differential equations for the gravitational held, we shall briefly 
consider the representation of gravitating matter in the equations and 
their solutions. 

Just as the gravitational field is generated by gravitating matter, so 
is the electromagnetic field generated by electric charges. These 
charges can be represented in two entirely different ways. When 
Maxwell set up his field equations, the atomic character "of electric 
charges was not yet known. Maxwell assumed that the charge was 
d 1S tnbuted continuously throughout a charged insulating body, or on 

175 



170 



THE FIELD EQUATIONS 



[ Chap. XII 



the surface of a conductor, and so forth. Correspondingly, he intro- 
duced the concepts of charge density and current density. These four 
densities are represented by our world vector I", which enters into the 
system of Maxwell's field equations. 

In a similar fashion, we can set up field equations in which gravitating 
matter is represented by the world tensor P" v , the stress-energy tensor. 
Ten differential expressions of the second order, which are formed from 
the components of the metric tensor, must equal the ten quantities 
P !> ". These ten expressions must, of course, transform like the P"\ 
that is, as the components of a symmetric tensor of rank 2. Only then 
will the field equations be co variant. 

When physi cists discovered that (dec trie charges were necessarily con- 
nected with small, individual particles, electrons and ions (and today 
we can add, mesons), Lorentii described the electromagnetic properties 
.of matter by means of a new model. According to his point of view, 
the greatest part of the space is free of electric charges. The electric 
charges are point-like and constitute singularities of the field. Outside 
the point charges, there is the electromagnetic field, which satisfies 
Maxwell's equations for charge-free space. At the location of each 
point charge, the equations are not satisfied — each point charge con- 
stitutes a singularity of the field. Although the field equations are not 
satisfied at certain points, the charge contained in each of these singular 
regions is conserved, because the field equations are satisfied everywhere 
around the singularity. If we enclose a singularity by a closed surface, 
then the charge in the interior is given by the integral 



*r* 



(E-dS), 



and the change of « is determined by the integral 



dt 

dt 



4-tt J, \ m j 



the expression c curl H, according to eq. (7.4), the right-hand side 



As long as the field equations are satisfied everywhere on. S (that is, 

as long as no electric current flows through S), we may substitute for 

5E 

dt 

of which is assumed to vanish. But the integral of a curl over a closed 

surface vanishes, according to Stokes' law; and we find that e does not 

change, even though no assumptions have been made regarding tin;' 

behavior of the field in the interior. 

Despite the assumption of singular regions, the field outside these 



Chap. XII ] 



THE FIELD EQUATIONS 



177 



regions remains determined to a high degree. That is why Lorentz 
was able to ex-plain the older theory of Maxwell, which assumed a con- 
tinuous distribution of charge and current, as an approximation of his 
own theory, in which point charges were singularities of the field. 

We can apply the point of view of Lorentz' electron theory to the 
representation of matter in the theory of gravitation. Instead of 
representing matter by means of the stress-energy tensor P"", we can 
assume that the gravitating matter is concentrated in small regions of 
space, and that elsewhere space is free of gravitating matter.. The dif- 
ferential equations of the gravitational field will hold only outside the 
mass concentrations: they will be field equations of empty space. The 
mass concentrations themselves, the "mass points," will be singularities 
of the field. 

We may consider the representation of matter by the tensor field P*" 
as a method of averaging over a great number of mass points and their 
states of motion, just as the concept of charge density is to be considered 
as the average number of elementary charges per unit volume. But, 
on the other hand, the description of matter by means of mass points 
may also be used as a convenient approximation when the components 
of the tensor P"' are different from zero only in small, isolated regions 
of space. This condition is realized in the solar system, where the 
bulk of the matter is concentrated in the interior of celestial bodies, 
while outside of these regions all components of P' 1V vanish. Each of 
these regions can be replaced by one mass point, and the treatment of 
the system is thereby greatly simplified. 

Both representations of matter — by mass points and by a continuous 
medium — break down in the face of a, sufficiently detailed treatment, 
for neither does justice to the quantum effects of atomic physics. But 
the usual fields of application of a theory of gravitation— astronomical 
problems — furnish us with both lands of examples. If we wish to de- 
termine the balance of stresses in the interior of a star, or if we wish to 
get an overall picture of the behavior of a nebula which consists of mil- 
lions of individual stars, we may treat matter as a continuous medium. 
If, on the other hand, the problem is one of computing the motions of 
a small number of celestial bodies, for instance, the bodies composing 
the solar system, matter must be represented by mass points. 

Regardless of whether we describe matter as a continuous medium 
or by means of mass points, we shall assume that the number of field 
equations equals the number of field variables, ten. Furthermore, the 
equations must be of the second differential order in the f/ w „ , for they 
must involve the inhom o gen cities of the gravitational field strength; 






178 



THE FIELD EQUATIONS 



[ Chap. XII 



and they must be covariant with respect to general coordinate trans- 
formations. 

If we treat matter as a continuous medium, the tensor field P'"' must 
equal everywhere a certain other tensor field (which we have vet to 
find) which consists of differential expressions of the second order in 
the fifj,v . On the other hand, if we choose the mass point representation 
of matter, then the same differentia] expressions must vanish every- 
where, except in certain isolated regions,' the locations of the mass 
points. In these regions, the solutions of the field equations become 
singular. 

The differential identities. A physical law, such as the equations ol 

the gravitational field, cannot be derived by purely logical processes. 
However, the range of possible field equations has already been limited 
by our assumptions that the field equations be ten differential equations 
of the second order in the $ m and that they be covanant with respect 
to general coordinate transformations. In this section, we shah formu- 
late a further condition for the field equations, which will exclude ail 
possibilities but one. 

The ten differential equations for the g„ v cannot be fully independent of 
each other, but must satisfy four identities. This condition is intimately 
connected with the condition of general covariance. Let us assume 
that we have obtained a set of ten covariant equations for the g^ , and 
that wo know one solution of these equations. Then wo can obtain 
apparently new solutions of the same equations by merely carrying out 
arbitrary coordinate transformations. The transformed components of 
the metric tensor, p M , will be other functions of £** than the original 
g # , are of f--. These formally different solutions are actually equivalent 
representations of the same physical ease, for their diversity reflects 
merely the variety of possible frames of reference with respect to which 
the same gravitational field can be described. The actual diversity of 
gravitational fields is much smaller than the number of formally dif- 
ferent solutions of the field equations. 

To restrict the variety erf formal solutions, one may subject the co- 
ordinate system to auxiliary conditions. As the coordinate transforma- 
tions contain four arbitrary functions (in a four dimensional continuum), 
it is possible to set up four equations for the g m , which must not be 
covariant and which must be chosen so that,, if we start with any set 
of ftp, , we ean satisfy them by merely carrying out a coordinate trans- 
formation. Such equations are called coordinate conditions. 

By adjoining to the; ten covariant field equations four coordinate 



Chap. XII ] 



THE FIELD EQUATIONS 



179 



conditions, we obtain a set of fourteen equations, which have the sa.me 
variety of inequivalent solutions as the ten field equations alone, though 
the number of formally different solutions is smaller. 

Fourteen fully independent equations for ten variables would have 
vciy few solutions, which represent either only a flat metric or at least 
a lesser variety of actually different cases than is required by the variety 
of conceivable distributions of matter in space . The fourteen equa- 
tions must, therefore, satisfy four identities. 

The four coordinate conditions are, to a high degree, arbitrary. 
They ean be any equations, involving the g^ , which are not covariant 
and which can be satisfied by any metric if only a suitable coordinate 
system is chosen. Since the choice of particular coordinate conditions 
has no effect on the nature of the solutions, it is necessary that the 
identities involve only the covariant field equations and that they be 
independent of the coordinate conditions. 

The preceding argument shows that the ten field equations, because 
of their covariance, must satisfy four identities. Bat we have not as 
yet any clue to the form of the equations and the nature of their- identi- 
ties. We ean obtain such a, clue from the properties of the tensor 
i J "". If matter is treated as a, continuous medium, the P* f form the 
right-hand sides of Hits field equations of the gravitational field, just- 
as the components of the current world vector form the right-hand 
sides of Max weir s equations. Just as the conservation law of electric 
charges is expressed in the equation 

I". = 

so the conservation laws of energy and momentum are expressed in 
the equations 



P™., = 0. 



(12.2) 



We shall, therefore, expect the ten left-hand sides of the field equa- 
tions to be the components of a symmetric tensor of rank two, and the 
four identities to have the form of divergences. 

The field equations. In Chapter XI, we have encountered a tensor 
expression with, just these properties. It is the tensor G'"', defined by 
eq. (11.47). It is possible to show that there is no other tensor with 
ten components which depends only on the g^ and the divergences of 
which vanishes identically. We shall, therefore, choose as the field 
equations of the gravitational field the equations 



(T v + aP"" = 0, 



li u " 



kru, 



(1.2.3) 



180 



THE FIELD EQUATIONS 



[ Chap. XII 



if matter is to be represented by the tensor P"'; in the mass point repre- 
sentation of matter, the field equations of the gravitational field will be 



&" = 



(12.4) 



outside the mass points, but will not be satisfied at the locations of the 
mass points tli emsel ves. The eonstant a of eq . ( 1 2 .3) wil 1 be determined 
later. 

The field equations (12.4) satisfy the identities 



(12,5) 



G* p ;„ = 0, 
while the equations (12.3) yield eqs. (12.2), 

= (ff* + a p-) ; , ^ apv. p . (12 _ G) 

The linear approximation and the standard coordinate conditions. 
The proposed field equations and the ponderomotive law of gravitation 
are nonlinear with respect to the field variables g„, . But we know 
that a linear theory— Newton's theory— accounts, with a considerable 
degree of accuracy, for the motions of bodies under the influence of 
forces. We must, therefore, assume that the gravitational fields (that 
is, flic deviations of the actual metric from a flat metric) encountered 
in celestial mechanics and elsewhere are so weak that the nonlinear 
character of the field equations leads only to secondary effects. 

The metric units, on which we usually base our measurements, are 
chosen so that the gravitational accelerations encountered in nature 
are of the order of magnitude of unity, while the speed of light, c, is a 
large quantity. For the purpose of the theory of gravitation, it is 
preferable to employ different units, in which the speed of light in flat 
space equals unity rather than 3 X 10 10 . We shall keep the centimeter 
as the unit of length, but shall measure both time and proper time in 
units which are one 3 X 10 m th part of a second. In these units, the 
fiat metric has the components 



= 



r -i, , , 

0,-1, o , 

, 0,-1, o 

I o , , , +1 



Y! 



(12.7) 



1 For the remain dor of this book, the notation t^ will always be used for tho 
flat metric when the new units of time are employed, while i)^ will denote as 
beforo, the flat metric in terms of metric units. 



Chap. Xlf 



THE FIELD EQUATIONS 



181 



The fact that the velocities of most material bodies are small compared 
with the speed of light is expressed in the new units by the condition 
that the IP, the spatial components of U", are small compared with 
unity. 

Using our new units of time, we shall assume that it is possible to 
introduce coordinate systems so that the components of the metric 
tensor can be expanded into a series, 



ffptr — £ p , + A/[[>„ 4- X hpf ~r ■ ■ ■ , 

1 2 

where X is the parameter of expansion and a small constant. 
The contravariant metric tensor will have the components 

( f° = r + \)r + kV 4- • • ' 

1 2 



(12.8) 



W = 



act (r(J , 



(12.9) 



The determinant of the metric tensor has the value 

s - Iff* I = -(i + X6 p v + ■■■)■ 

i 
Let us now consider the pondero motive law. It is 

dU f 



dr 



= - (f'W, ?] V w 



(12.10) 



(12.11) 



The Christoffel symbols [«c, u) are small quantities of the first order in 
X. If we neglect quantities of higher order, we may replace g"" by «"'. 
Furthermore, as long as the velocities are small compared with c, we 
may neglect terms which contain components U" as factors, while U* 
is approximately equal to unity. We shall, therefore, replace eq. 
(12.11) by the approximate equation 



car 



[44, a] 



i^X(2/t 4 „,4 - h u „). 
i i 



(12.12) 



Finally, if the field does not change quickly with time — if it is created 
by mass points which themselves move only at moderate velocities — 
the derivatives with respect to £ 4 are small, compared with derivatives 
with respect to the spatial coordinates f \ and may be neglected. We 
find, as the first three equations (12.12), 



dJf 

dr 



i 



(12.13) 



182 



THE FIELD EQUATIONS 



[ Chap. XII 



Upon comparing this equation with the ponderomotive law of classical 

mechanics, eq. (10.5), we find that +|XA« takes the place of the New- 

i 

tonian gravitational potential. This remark will help us later to inter- 
pret solutions of the field equations. 

Let us now proceed to the linear approximation of the field equations. 
As we shall limit ourselves to linear expressions, we shall be able to 
simplify the form of these equations considerably, In the tensor 



p 

"PJ.H 



fflf»3 



jtn} :if [pa-j \ptj \^p\ [wrj \t«ftb 

Jp) J-U4+MMN 



) (12.14) 



we may neglect all the terms which are not linear in the h^ . This 

i 

refers to all terms which are not linear in the Christoffel symbols; in 
the remaining terms, we; may replace all undifferentiated fa and g"" 
bv £„, and t". We obtain the "linearized" expressions 



\JtkY ^^ ['^,1'fi "T" € \f&-pPrpt f^iifi t ver ''vp.ptr 

Z 1 1 1 1 



1 



1 



ft - 6 P % 
1 1 



(12.15) 



Eq. (12.15) can be somewhat simplified by the introduction of the 

quantities 



i i i 



%> 



(12.16) 



7* 
i 



1 



If we express the linearized G„, in terms of the y m , we obtain 

i 



G w 



ii i 



l 



i Dl7 S 



(12.17) 



Still it is difficult to obtain solutions of the field equations (12.3) or 
(12,-1), for each component of the linearized 0„„ , eq. (12.17), contains 
several components i> , and all tea field equations must be solved 



Chap. XII ] 



THE FIELD EQUATIONS 



183 



simultaneously. This situation, however, can be greatly improved if 
we make use of the possibility of introducing coordinate conditions . 
We shall show that we can always carry out a coordinate transformation 
$o that the expressions a-^ vanish. 
Let us consider coordinate transformations of the type 



f* a = r + te^tf), 



(12.18) 



which change the coordinate values only by amounts which are propor- 
tional to the parameter; A. The inverse transformations are 



£« = £*» _ lg*ffi _ g*<* _ Xl"(£* P ), 



(12.1.9) 



up to quantities of the first order in X. 

The components of the metric tensor (12.8) transform according to 
the law 



$ = |£ j£ & ~ <& - &#)$, - ^)c«« + « 1 



(12.20) 



«» «„„ + \(A„„ — € ai ,i) a . M . — ^tf tt rf «) 3 | 

up to first order quantities. The transformation law for h^ is, therefore, 

i 

~ m»*.,*-. (12.20a) 



e aP v 



The transformation law of the quantities y Ml , is 

i 



7^v = V»? — taeV 

1 1 



a I p 



(12.21) 



and the expressions a? transform according to the law 






*V = ^ a w 



(12.22) 



again up to quantities of the first order. 

We find that we obtain a coordinate system in which the ^ vanish 
if we carry out a rioordinate transformation (12.18) in which the v a 
satisfy the differential equations 



i Va, 



(12.23) 



These differential equations, Poisson's equations in four dimensions 
always have solutions. 



IS I 



THE FIELD EQUATIONS 



[ Chap. XII 



Tn the linear approximation, the field equations may be replaced by 

the equations 



1 



0, 

e 7w>.* = 0, 



(12.3a) 



and 



i 



(12.4a) 



In the equations of the second differential order, the variables are now 
completely separated; the discussion of their solutions is thereby greatly 

facilitated. 

Solutions of the linearized field equations. Let us first consider static 

solutions of the field equations, that is, solutions which arc independent 
of £\ If we assume that the held variables depend only on the three 
coordinates £ 3 , the linearized equations reduce to the equations 



«wi = o 



(12.21) 



and 



v%„ 



7-0 
i 



0, 



(12.25) 



Ordinarily, the component P& = T jii = p is large, compared with 
the other components of i\ r . We shall, therefore, treat the case 



(12.26} 



These equations can be solved by the assumption that of all the quanti- 
ties 7^ only 744 does not vanish, y^ itself satisfies Poisson's equation 



-XT 


7m 
i 


+ 2ap 


= 









1 


= 









7p3.> 

i 


= 






Chap. XII ] 



THE FIELD EQUATIONS 



185 






hi three dimensions; the solution is given by the integral 

p(r') dV 



X7w(r) 



a_ f p(r') dV- 
2ff Jv> \t - r'l 



(12.27) 



i 2ff Jy< |] 

We found that -JX/144 must be considered as the quantity which as- 
1 

sumes the role of the classical gravitational potential 6 of eqs. (10.5) 

and (10,7). Because only 744 of the y„ f does not vanish, hu has the 

1 1 1 

value 



t, 1 4i 

ft-K = 744 — jtye 744 



57m 
1 



We find, therefore, for ha the differential equation 
1 

-2v 2 A t4 + ap = 0. 



(12.28) 



(12.29) 



By comparing this equation with eq. (10.7), we find that the constant 
a has the value 



= 8-jtk, 



(12.30) 






A theory of gravitation in which matter is represented by continuous 
media remains incomplete unless we know the equations of state of 
the media. If matter is rarefied to such an extent that there is no 
interaction between neighboring volume elements, then we may assume 
that P ft " may be replaced by 

P" r = P U»U", 

where W is subject to the p on dero motive law (12.1), and the change 
of p is determined by the conservation laws: 

(pvin-, = 0, . 
tomfirV, - (pin-, = o. 

In all other eases, we must make assumptions regarding the internal 
forces of matter. Whether these assumptions are compatible with the 
theory of relativity may not be easy to decide. It is impossible, for 
instance, to conceive of rigid solid bodies or of an incompressible liquid. 
Either type of material would transmit elastic waves with an infinite 
speed of propagation, contrary to the fundamental assumption of the 
theory of relativity — that signals cannot be transmitted with a velocity 
greater than c. If we have a relativistic theory describing the inter- 



186 



THE FIELD EQUATIONS 



I Chap. XII 



action of the individual particles which make up the material, wo can 
compute the equation of state, which will then not contradict the prin- 
ciples of relativity. Actually, such a program has so far been carried 

out for very few types of molecular interaction. 

The field of a mass point. Let us now consider the representation of 

matter by mass points; that is, let us consider the linearized field equa 
trans (12.4a), p. 184. 

First, we shall set up the field which is produced by a mass point at 
rest. This field will be static and spherically symmetric. We shall 
choose: the mass point as the point of origin. Solutions of the Laplacian 
equation which vanish at infinity and which have no singularities outside 

the point of origin are all derivatives of the function - , or linear com- 

r 
binations of such derivatives. To solve the first set of equations (12.25) , 
we shall make the assumption 



X744 
1 



Mis 

1 



XTjj 



a 

r 






+ /•-•«* 



(12.3i: 



where a, b, c, and/ are constants to be determined. Let us now satisfy 
the other set of equations (12.25), the coordinate conditions. We find 



"hffi = — b 



Xcr e = 






(12.32) 



The constant /must vanish, while the other constants remain arbitrary. 
However, upon closer examination, we find that the terms which contain 
the constants b and c are dependent on the choice of the coordinate 
system. By Carrying out a coordinate transformation (12.18), we can 
remove these terms if we choose the functions v a as follows: 



v — — . 
r ' 



-(:),. 



(12.33) 






Chap. XII ] 



THE FIELD EQUATIONS 



1.87 



In accordance with the transformation law (12.21), we find that we are 
left with the solution 



AT44 = - , 

1 r 

74! = 0, 



7= 
1 



0. 



(12.31a) 



The remaining arbitrary constant, a, must be related to the mass which 
produces the field. The Newtonian potential winch is produced by a 
mass M is 



G = - 



kM 



Because of eq. (12.28), and because ^Xlu A corresponds to 0, we find 



that a determines the mass M by the equation 

Af = 



a 
'4k 



(12.34) 



Gravitational waves. So far, we have treated only those solutions of 
the field equations which have counterparts in the classical theory of 
gravitation.. However, there are solutions which are typical for a field 

theory. The most important of these are the "gravitational waves," 
rapidly variable fields, which must originate whenever mass points 
u nd ergo ac eel erati on s , 

Let us consider plane wave fields which depend only on f* and J 1 , 
There are waves progressing in the positive H'-direction and waves which 
propagate in the opposite direction. The most general wave which 
propagates in the positive E -direction has the components 



744 
1 



1 



= Y*0? 

1 

T« = Yrs(£ 






(12.35) 



The field equations are automatically satisfied. The coordinate condi- 
tions are 



fpA ,4 

1 



7&A 



-(7^1 + 7^0 



o, 



(12.36) 



where the prime denotes differentiation with respect to the argument 

(£ r — £ 4 )- We obtain the conditions 



188 



THE FIELD EQUATIONS 



7j4 = Tit ' 
1 i 

712 = — y'i 



-'T.U] 



7l3 

i 



-713, 

1 



[ Chap, XII 



(12.36a) 



while the remaining components, ya , 7a 3 , and 733 remain arbitrary func- 

11 .1 

tions of the argument (J 1 — £?). 

Again, it turns out that several of these components do not corre- 
spond to a physical wave held, but can bo eliminated by a coordinate 
transformation. If We carry out a coordinate transformation (12.18), 
and let the v" depend only on the argument (|* — £ 4 ), the transforma- 
tion law (12,21) takes the form 



* 

711 
1 



712 



* 

yu 



722 



723 
1 



7n + *■" + */', 



7wt v , 7i 3 = 713 -j- v , 



7n 
1 



722 
1 



723, 
I 



1* I 4' 



* 

724 
I 



724 ~ 1' , 

1 jt 1 y 

74-i = 7-ii T s + v 



734 

1 



733 ~ 783 
1 1 



3' 
734 — V , 
3 



,y _i_ „ i/ 



(12.37) 



By a suitable choice of the four f mictions v", we can obtain a coordinate 

system in which all components with at least one index 1 or 4 vanish, 

and in' which the expression (722 + 733) is also equal to zero. The only 

1 1 

waves which cannot be eliminated by coordinate transformations are 

those in which 



722 
1 



733 ^ 0, 

1 



and those in which 



733 9* 0. 



(12.3S) 



(12.39) 



These two types of wave can be transformed into each other if the 

T 7T 

spatial coordinates are rotated around the S -axis by an angle of - 

4 

radians (45°). 












CSap. XII ] 



THE FIELD EQUATIONS 



IS!) 



The gravitational waves have no counterpart in classical theory. Un- 
fortunately, the intensity of those waves which are presumably produced 
by oscillating systems, double stars, planets, and so forth, is not strong 
enough to be observed by any method known to date. 

Einstein and Rosen investigated the wave solutions of the rigorous, 
nonlinear field equations. They found that there are no plane waves, 
but that there are cylindrical waves. Though they obtained this result 
by strictly formal methods, a physical explanation can be given. The 
gravitational waves, just like electro dynamic waves, cany energy. 3 
This energy density in turn creates a stationary gravitational field which 
deforms the metric, and the gravitational waves must be superimposed 
on this deformed metric. A plane wave would be connected with a 
constant finite energy density everywhere, and the deviation of the 
metric from flatness, therefore, would increase toward infinity in all 
directions. Cylindrical waves, on the other hand, have a singularity in 
the axis of symmetry, and there are solutions in which the amplitude of 
the waves approaches zero and the amplitude of the stationary field 
becomes infinite for infinite values of the coordinate p (which, in a 
Euclidean space, denotes the distance from the axis) . 

Our discussion of the "linearized" field equations of the relativistic 
theory of gravitation indicates that these equations possess solutions 
which correspond to Newtonian fields; in addition, there are solutions 
which have no counterpart in the classical theory, gravitational waves 
which propagate with the velocity of light. Now that we have found 
that the relativistic equations have solutions which are approximated by 
the classical theory of gravitation, we shall consider some of the formal 
properties of the relativistic field equations. 

The variational principle. The classical field equations, (10.7), for 
empty space can be represented as the Euler-Lagrange equation of a 
variational problem (or IlamiUonian "principle), 



R I (grad GfdY = 0, 



(12.40) 



where the integral is to be extended over a three dimensional volume, F ; 
the variation of G will vanish on the boundary of the domain V of inte- 
gration, but is arbitrary in its inferior. The variation of the integral 
(12.40) can be represented as follows: 



2 "On Gravitational Waves," Journal Franklin Inst,, 223, 43 (1937). 

s Tlie concept of energy in the general theory of relativity will be; treated later 

this chapter 



190 



THE FIELD EQUATIONS [ Ch ap . XII 

5 j (grad Gf dV = 2 J (grad G-d grad G 1 ) dF 

= 2 / (grad grad (&G)) dV 

= .2. £ div (grad G ■ 5G) dV-2j V 3 G & d7 

= 2 A 5G(grad G-dS) - 2 f v'G&G dV. 

The first integral of the last expression vanishes, because 50 vanishes 
on the boundary. We have, therefore, 

5 j (grad 0)*dV = -%j V 2 G-SG-dV; (12.41) 

said, as m is arbitrary in the interior of V, it follows that the integral 
j^ (grad GfdV is stationary only if (} satisfies She equation 

V 2 G = 0. (12.42) 

Likewise, the rolativistic field equations (.124) can be represented as 
the Euler-Lagrange equations of a Hamiltonian principle. The integral 
in this case is a four dimensional integral 



<< D 



5T = 0. 



(12.43) 



The variations of the $ m (and their first derivatives) must again vanish 
on the boundary of the four dimensional domain D, but are arbitrary 
in its interior. 

The integral I is an invariant. The integrand, 

& = V^gR, (12.44) 

is a density of the weight 1, and transforms according to the law 

i &*> i Jt ' 
and / transforms, therefore, as follows; 

I* = f^d^df'dt'df^ fdldot 



Sf 



t* 1 ,7-*2 Jj-*' IH** 



;: ,,„ | *"«"d{*V£' 









Chap. XII ] 



THE FIELD EQUATIONS 



191 



It is shown, in the theory of multiple integrals, that the integrand of a 
multiple integral is multiplied by the Jacobian of the transformation 
whenever new parameters of integration are introduced; in other words, 
I* is the same integral as /, 

I* = I. 

The Euler-Lagrange equations express the conditions which must be 
satisfied if a certain integral is to be stationary with respect to variations 
of the variables which make up the integrand. If the integral itself is 
invariant with respect to coordinate transformations, its Euler-Lagrange 
equations express conditions which cannot depend on the choice of 
coordinates; in other words, the Eider- Lagrange; equations of an in- 
variant Ilamiltonian principle are themselves eovariant differential 
equations. 

Let us now express the variation of the integral 



-!(tl 



{«p) I^j-j [up) \m)J 



> (12.45) 



d£ = d£ df df d?, 

in terms of the variations of the if : . We shall divide the variation of 
the integral into two parts, in this manner: 



f R^5(V- 



+ 



■g 



<r) di 



(12.46) 



First we shall express the variation of li^ in terms of the variations of the 
Christoffel symbols, 






+ 



m) [tip) [up) kwj 



(12.47) 



We know that the Christoffel symbols are not tensors, as they trans- 
form according to the transformation law (5.81). However, if two 



* It is advantageous to introduce the contra variant components, g^", as the 
independent, variables. As the g^" and the g^„ determine each other uniquely, 
the final result of the computation is not affected by this choice. 



THE FIELD EQUATIONS 



' ' ' '"■''' ' r rL v 

different affine connections are defined on the same spare, their differ 
enee, J ,. - r„ , transforms as a mixed tensor of rank 3, since the last 
term m eq. (5.81) cancels. The variation of the Christoffel symbol, 
3U), is the difference between two affine connections, the varied and 
the unvaried Christoffel symbols, and is, therefore a tensor 

As the left-hand side of eq. (12.47) is a tensor, the right-hand side 
can contain only covariant derivatives of the tensor §{£}, and in fact 
straightforward computation shows that the right-hand side of eq'' 



*•-(•&).-(•$), 



(12.48) 



This simplification of eq. (12.47) was first pointed out bv Palatini, hs 
the covariant derivatives of the metric tensor vanish, we may multiply 
by (f under the differentiation, 



r>R "ht)l-(«''{:,K 



I 



9 °-\ f — 9 s i > 

ml \mj 



(12.49) 



This expression is the covariant divergence of a vector. In Chapter V 
Problem 10(b), we stated that a covariant divergence, V.„ can be written 
in the form 



r*-^Wim, 



(12.50) 



Applying this formula, we obtain for the integrand of the first integral 
on the right-hand side of eq. (12.46) 



L V ml w»ij. 



(12.51) 



The integrand is an ordinary divergence. According to Gauss' theo- 
rem (which holds in n dimensional space just as well as in three dimen- 
sional space), the integral 



/v- 



gg^ffiv&i 



can be transformed Into a surface integral over the boundary this 
integral vanishes because the variations S{^} vanish everywhere on the 
boundary 






Chap. XII ] 



THE FIELD EQUATIONS 



193 



There remains the second integral of eq. (12.46). S(\/ — g g"") is 
■simply 

s(V~g<r) * V~g(sf r ~ fefrfV); (12.52) 

and this, multiplied by Bpt , gives 

&,^{\/~g g"") = V~j(H» r ~ ig^RW = V~g «,*" (12.53) 
We find, therefore, that the variation of I is given by the expression 

5 J RV~g d? - / $U Sff'V^ it (12.54) 

The equations (12.4) are the Euler-Lagrango equations of the Hamil- 
tonian principle (12.43). 

The combination of the gravitational and electromagnetic fields. So 

far, we have treated only the gravitational field in the absence of the 
electromagnetic field. When an electromagnetic field is present, we can 
obtain the field equations of the combined fields by replacing P"" in the 
equations (12.3) by the expressions (8.31). 
We have, then, 



CL 



■ (2<p pp f& — f^, ip pa ■/") = 0, 



V :? 



= 0. 



(12.55) 



These field equations are the Euler-Lagrango equations which are ob- 
tained when the integral 



1 = L ( R ~ ? ^ /0 ) ^' g dk (x2 - m: 



is varied with respect to the 14 variables (f and ip^ . 
The pondero motive law of a charged mass point is 

jV 



dr ' \pv) 



mc 



<p* ? U, = 0. 



(12.57) 



The conservation laws in the general theory of relativity. 5 The 
energy-momentum tensor P 1 ™ in the field equations (12.3) represents the 



s In this section, the transfer of the energy concept to the field of general 
relativity is discussed. -\s the concepts of energy and momentum are not of 
great importance in the general theory, the student, may omit this section without 
losing the connection with the following chapters. 



194 



THE HELD EQUATIONS 



[Chap. XI] 

energy and momentum densities and stresses of a continuum, apart 
from the energy, momentum, and stresses which may he associated with 
the gravitational field. It satisfies the divergence relations 



pfp 



0. 



(12.58) 

These equations are eo variant; they transform as the components of a 
vector. But it is precisely for this reason that they are not what can 
properly be called conservation laws. In ft proper conservation law, 
the change of a certain three, dimensional volume integral with time is 
determined by the surface integral of certain other expressions, repre- 
senting a flux, taken over the spatial boundary of the volume. In other 
words, a proper conservation law has the form 



|(/^\«)=-^(F.dS), 



or, if we apply Gauss' law to the surface integral, 

~dP 



L 



at 



+ div F 



dfdfdf - 0, P A 4- r „ = o. (12.59) 



The conservation law of electrical charges in the general theory of rela- 
tivity has this form, 



(v'-ff /').«, = 0, 



(12,00) 



and the expressions £ ( x /-g /") if d?, £ ( V- g ^) , t df df, and so 

forth, represent the charge: contained in the volume V, the current- 
through a face parallel to the X", X'-surface, and so on. 

While the covariant divergence of a vector is equivalent to the ordi- 
nary divergence of a vector density, the covariant divergence of a sym- 
metric tensor of rank 2 is not equivalent to an ordinary divergence of 
some density. However, if; has been possible to find a set of expressions 
which satisfy four ordinary conservation laws without being the com- 
ponents of a tensor. 

The equations 

o = V-lj i\i 



p;...= V-g(iV-,-\-\'\j> 



*)\ 



(12.61) 



have the form of a set of four conservation laws, except for the last- 
term. We shall now show how this last term, — ■y r ~^{i' ii \p I carl [ xi 









Chap. XII ] 



THE FIELD EQUATIONS 



195 



brought into the form of an ordinary divergence. First of all, because 
of the field equations (12.3), the P/. can be replaced by — -O/. . We 



must now consider the expression 



V- 



-glJ r }G p ". , which contains only 



the g„, and their derivatives. Replacing the Christoffel symbol by the 

derivatives of the metric tensor, we have 



^o<°)g. 



W 



V— g U"j p\Q p 



V- 



■8 9?. 



, (r 



(12.62) 



In the proof that this last expression is the ordinary divergence of a set 
of 16 quantities, we shall make use of the fact that the expressions 
V — 9 <j> aro the Euler- Lagrange equations of a variational principle. 

First, we shall show that there is a variational principle which con- 
tains only first derivatives of the metric tensor and which has the same 
Euler- Lagrange equations as the integral (12.43). 

An (ordinary) divergence, added to the integrand R\/ ' —g, contri- 
butes Lo the integral (12.43) an expression which can be written hi the 
form of a surface integral because of Gauss' theorem. When we carry 
out a variation of the integrand so that the variations of the variables 
and their derivatives vanish at the boundary, the contribution of the 
added divergence will remain unchanged. The Eulor-Lagra.nge equa- 
tions will, therefore, remain the same if we add a divergence to (y/ — gTi) 
ineq. (12.43). 

Let us now consider the integral in the form (12,45). The first two 
terms can be changed as follows : 



m 



-(v-»--)..{; p }h-(v- 9 r),fc 



(12.63) 



The first two terms on the right-hand side are divergences. The Eulcr- 
Lagrange equations will, therefore, remain unchanged if we subtract 
them from {-\/ — gE), In the remaining terms, we replace the deriva- 
tives of the metric tensor everywhere by Christoffel symbols, according 
to the equation 



gW> = W, v\ + [vp, p.}. 



(12.61) 



196 



THE FIELD EQUATIONS 



[ Chap, XII 



When they are combined with the other terms of the integrand of 
(12.45), we obtain the equation 



*-^=» «"({;}{;} -{:}{:})•, 



(12.65) 



The function 3< is, of course, not a scalar density, and W is not invariant 
with respect to coordinate transformations. But the Euler-Lagrange 
equations belonging to the integral W are covariant equations. If we 
consider H as a function of the variables g"" and g^ iP , then Q& must 
have the form 



'33A 



Let us multiply this equation by g«\ a . We obtain the equation 

The first and the last terms on the right-hand side are together the 
derivative of M with respect to t (3f depends on the coordinates only 
indirectly, by way of the §p* and g»\ a ). We have, therefore, 



(12.67) 



V-g fiW- = (& * - J^ r.«) ^ - - ( V~g t 



fa. (12.68) 



Herewith, the proof is essentially completed. For the expression on the 
left-hand side is 

V~g G& g"", a = - V~g W*Sm* , (12.69) 

and we find that cqs. (12.61) can be brought into the form 



The expressions 



tl 



[y^+iM 



P 4- +' 

lOJTK 



(12.70) 



(12.71) 



are not the components of a tensor. But because they satisfy the four 
conservation laws (12.70), they are called the components of the stress- 
energy -pseudo-tensor of general relativity. 









Chap. XII ] 



THE FIELD EQUATIONS 



197 



The expressions t* , the stress-energy components of the gravitational 
field, contain only first derivatives of the g f , ; one might say, they are 
algebraic functions of the gravitational "field intensities." It is charac- 
teristic of the general theory of relativity that expressions of this type 
cannot have tensor character. It is always possible to find frames of 
reference relative to which the "gravitational field strength''' vanishes 
locally, and then the stress-energy components t£ vanish locally, too. 

Conversely, in a perfectly fiat space it is possible to choose a frame of 
reference relative to which we observe "inertial forces." According to 
the principle of equivalence, we cannot distinguish these "inertial forces" 
locally from a gravitational field, and the components t* will not vanish 
in a noninertial coordinate system. 



CHAPTER XIII 

Rigorous Solutions of the Field Equations of the 
General Theory of Relativity 



The field equations of the general theory of relativity are nonlinear 
equations. So far, we have solved only their linear approximations. 
In this chapter, we shall consider cases in which it has been possible to 
solve the rigorous equations in a closed form. 

There is no general method of finding rigorous solutions of the field 
equations. However, the equations have been solved in a few cases hi 
which the n Limber of variables is reduced by symmetry conditions. 

The solution of Schwarzs child. Let us first consider the solution 
which represents a mass point at rest. We shall assume that this solu- 
tion has spherical symmetry and that none of the variables depend on g. 
If we introduce the variable 



r = Vf + f + f, (13.1) 

the most general line element with these properties takes the form 

dr 2 = A{r)df + 2B(r) Xs dt; 4 d¥ ~ C(;r}Kd£ dg 

+ lXr) X rXsd?d?, 



r 



(13.2) 



where A, B, C, and D are functions of r. This hne element does not 
change its form if we carry out a spatial rotation of the coordinates £ ! 
around any axis which goes through the point of origin. 

Without destroying either the static character or the spherical sym- 
metry of the line element (13.2), we can eliminate two of the four 'un- 
known functions A, B, C, and D by suitable coordinate transformations. 
First we can eliminate the terms which contain products of spatial 
coordinate differentials, d£ s and df, by carrying out a coordinate trans- 
formation 



f 



*■! 



t+m, 

198 



4* E 



= r 



(13.3) 









Chap. Xiii ] RIGOROUS SOLUTIONS 

The components g is transform according to the equation 



199 



f/.i.< 



or 



B* 



;/H g|& + 0*> 



B-f 4 . 



dr 



By choosing / so that it satisfies the equation 



dr 



B 
A' 



(13.4) 



(13.5) 



we can eliminate the terra whieh contains B. 
Let us now consider a metric with the components 

gu = A , g is = 0, | 

{}„. = -<X + I>XrX*-\ 

By carrying out a trans formation of the spatial coordinates, 
r = g(r)?, j 

r = ^*)r, r = j,-r*, x : = Xs ,] 

we shall be able to obtain a. new coordinate system in which the metric 
has also the form (13.6), but in which the function C is constant and 
equal to unity. The components g ra transform according to the law 



(13.6) 



(13.7) 



d£* r Q%* 



Qik 



rlr 



,x r )(W + r*^ XftX .)(-Cfi tt + D XiXk ) 



-rCS T , + U + r* p ] Jj 



— r" 



dr* 



& + "%y. 



XrXs- 



(13.8) 



We find that we obtain the coordinate system in which C equals unity, 
if we choose as the function f in eq. (13.7) 

* = C~ m . (.13.9) 

There remain only two unknown functions, A and D. Instead of 
these two functions, we shall introduce two other functions of r, because 



200 



RIGOROUS SOLUTIONS 



[ Chap, xilf 



it has been found that the field equations are easier to handle if we set 
them up for these functions, j» and v. The new functions are defined by 
the equations 



(744 = ^= e", 

gi, = o, 

( Jrs = — S rs + DXtXi 

= -&r« + (1 - el***,, » ~ log (1 - Efjf. 
The contra variant metric tensor has the components 



(13.10) 



-&?i + (1 — C ")xrX>- 



(13.11) 



We shall now compute the Ghristoffel symbols and the components 

of G> , The Ghristoffel symbols of the first kind are 



[44, s ] = -^V*., 
[4 5 ,4] = +^'e" Xt , 
1 - e" 



[rs, t] = X t 



(13.12) 



L r 



%* — XrXx) ~ Wv'XrX* 



j'j 



where the primes denote differentiation with respect to r. The com- 
ponents with an odd number of indices 4 vanish. The Ghristoffel sym- 
bols of the second kind have the values 



{44} = ^" 



Xs ) 



lp' Xs 



L\ - xt [- 



(As ~ XrX*) + &&'%* 



(13.13) 



Again the components with an odd number of indices 4 vanish. 









Chap. Xlll ] RIGOROUS SOLUTIONS 201 

The components of the contracted curvature tensor, j^, , are 

b« = -<""' {* a" + £ / + I vita! - /) ! 

(I .. 1 , . 1 . .1 



Ra„ =■ 

Rrz = 



YS-p" "~ -" ^ "t" '2 **'0»' ~ *^3 J 1 X* ?c> 



H^- fi 



i + i &»' - "0 



2r 



(.XrXs ~ 8 rs ), 



(13.14) 



and the components of G^ are 






IV . 1 - /\ , /1 



,^" + ^V-.') 



? (13.15) 



+ 1 W 



/)> e "ixrX« - O- 



Let us first consider the case of a purely gravitational field, where 
eqs. (1.2.4) are satisfied. We must solve the following three equations 
for two variables: 



y> - 1(1 - g) « 0, 



P* + i (1 - e) = 0, 






(13.16) 



These three equations are not completely independent of each other, for 
they fulfill the contracted Bianchi identities, eqs. (11.40). 

The first and the second of these equations show that the sum of ju' 
and v' vanishes, 



U r\- V =0. 



(13.17) 



The first equation, for v, can be solved. Let us introduce the quan- 
tity x, 



x = e , 



-log x. 



(13.18) 



202 



RIGOROUS SOLUTIONS 



[ Chap. XIII 



We obtain, instead of the first equation (13.16), the equation for fc 



^ + ^1 = 0, 

dr r 



and aa the solution 



x - 1 - 



tic 



--*( i -f)<! 



r — t-t 



(.13.19) 



where « is a constant of integration. Because of eq. (13.17) the func- 
tion fi must have the form 



= log ( 1 



+ 8, 



(13.20) 



where is another constant. The solution (13.19). (13.20) also satisfies 
the third equation (13. 16) . The metric tensor then has the components 




1 - 



| XrXa - 5 r3 — -— %rX , , 



J 



(13.21) 



For large values of r, the metric tensor must approach the values ^ , 
cq. (12.7). This is the case only if we choose for fi the value zero The 
remaining constant, a, must characterize the mass of the particle which 
creates the field (13.21). 

In accordance with eq. (1.2.13), the Newtonian potential of the ra^s 
point which creates the field (13.21) is 



(?= 



la 
~2r 



(13.22) 



On the other hand, G depends on the mass according to the equation 

.-, km 

tj — > (13.23) 

we find that the constant a is determined by the mass m, according to 
the equation 

a = 2*771, Q3 24) 












Chap. XIII ] 



RIGOROUS SOLUTIONS 



203 



The gravitational field of a mass point is, therefore, represented by the 
expressions 



2k?» 
Qu : ■=» 1 — — , 

r 



94, = 0, 



%an 



XrXs' 



(13.25) 



r — 2inm 

This solution was found by Schwarzschild. 

Schwarzschild's solution is significant because it is the only solution 
of the field equations in empty space which is static, which has spherical 
symmetry, and which goes over into the flat metric at infinity. Other 
solutions of the field equations for empty space with these properties 
can be carried over into Schwarzschild's solution merely by a coordinate 
transformation. TSirkhoff has even shown that all spherically symmetric 
solutions of the field equations for empty space which satisfy the bound- 
ary conditions at infinity arc equivalent to Schwarzschild's field, that 
is, their time dependence can be eliminated by a suitable coordinate 
tr ansf o rmati on . s 

Therefore, if we consider a concentration of matter of finite di- 
mensions which is spherically symmetric, we know that the gravita- 
tional field outside the region filled with matter must be Schwarzschild's 
field. Inside this region, the matter might even be pulsating (in a 
spherically symmetric manner) without modifying the gravitational 
field outside. It is, of course, assumed that there is no flux of matter 
or electromagnetic radiation in the outside space. 

The "Schwarzschild singularity." The expression (13.23), the solu- 
tion of the classical field equation, (10.7), has a singularity at the point 
r — 0. The Schwarzschild field has a similar singularity at the same 
point. In addition, it has a singular spherical surface at r = 2(cm. On 
this surface;, the component gu vanishes, while some of the spatial com- 
ponents become infinite. 

Robertson has shown that, if a Schwarzschild field could be realized, 
a test body which falls freely toward the center would take only a finite 
pi'oper time to cross the "Schwarzschild singularity," even though the 
coordinate time is infinite; and he has concluded that at least part of 
the singular character of the surface r = l&m, must be attributed to the 
choice of the coordinate system. 



1 Bed. Tier., 1916, p. 1S9. 

'BirKhoff, Relativity and Modern Physics, Harvard "University Press, 1923, 
p. 253. 



204 



RIGOROUS SOLUTIONS 



[Ch 



«p. XIII 



In nature, mass is never sufficiently concentrated to permit a Schwarz- 
schild singularity to occur in empty space. Einstein investigated the 
field of a system of many mass points, each of which is moving along a 
circular path, r = const., under the influence of the held created by the 
ensemble. 3 If the axes of the circular paths are assumed to be oriented 
at random, the whole system or cluster is spherically symmetric. The 
purpose of the investigation was to find out whether the constituent 
particles can be concentrated toward the center so strongly that the 
total field exhibits a Schwarzsehild singularity. The investigation 
showed that even before the critical concentration of particles is reached 
some of the particles (those on the outside) begin to move with the 
velocity of light, that is, along zero world lines. It is, therefore, im- 
possible to concentrate the particles of the cluster to such a degree that 
the field has a singularity. (The singularities connected with each indi- 
vidual mass point are, of course, not considered.) 

Einstein chose this example so that he would not have to consider 
thermodynamical questions, or to introduce a pressure, for the particles 
of his cluster do not undergo collisions, and their individual paths are 
explicitly known. In this respect, Einstein's cluster has properties 
which are nowhere encountered in nature. Nevertheless, it appears 
reasonable to believe that Einstein's result can be extended to con- 
glomerations of particles where the motions of the individual particles 
are not artificially restricted as in Einstein's example. 

The field of an electrically charged mass point. We shall now treat 
a mass point which carries an electric charge. The electrostatic field 
will be characterized by a scalar potential <& , which is a function of r. 
The covariant components of the electromagnetic field are 

¥>4» = ^4,* = £>£& , <p„, = 0. (13.26) 

The components of the electromagnetic stress-energy tensor are 



M, 



4jt 



Li Qm^'p"® 



¥■„„ <?t 



Mu - ~ k- <?is f-i, <]" ~ g- feife ", 
Sir Sir 

M is = 

Mrs = -r- [Wrt<Pit<P — tp T i<Pf\ 

= ~ (^)V ( ^[- e> Xr y.„ + ( Aa - XrXr )]. 



(13.27) 



Annals of Mathematics, 40, 022 (1930). 






Ch ap . xiii j RIGOROUS SOLUTIONS 205 

By combining eqs. (12.3), (12.30), and (13.5), we obtain the equations 






J-e-'^O, 



i iU"+ i w - <o + yy - ^V + ^)V^ 5 



.2'" ' 2r 
In addition, we have the electromagnetic equation, 

A, = 0. 
We shall compute this expression in the form 



(13.28) 



4« , 4m j 



i J p \ _ 



[m p < 



0. 



For <p , we obtain 



4> 44 ts -{p+r) I 

<p = g g <? it = -e ><p iXr , 



and we have 



r — (jt+ij I -, .-] —(ii-i-t) I , i | p. f, 

W <p4xA,s + P <pdp + » ) = 0, 



or 



2 , 



<fi 



+ -ip., — -|(/i' + v')<?i = 0. 



This last equation has a first integral, 



2 -MM-iO ' 
re- ifli = — e, 



(13.29) 



(13.30) 



where e is a constant of integration, the charge. 

With the help of this integral, we can eliminate ^ from the field 
equations (13.28). We have 

-Y 1 A- 1 u^l-r/ 

\r- r / r 2 r 1 
— e \-[i + j- yt — r) + - p, yt — v 



+ % = 0. 



(13.31) 



200 



RIGOROUS SOLUTIONS 



[ Chap. XIII 



Again, the combination of the first two equations yields eq. (13.17). 
The introduction of the variable x, eq. (13.18), into the first equation 
of the set (13.31) leads to the differential equation 



with the solution 



(/.!■ , X — J , lit 



Y T* 1 



(1.3.32) 



(13.33) 



The metric tensor has the components 
£m = 
%*» = 0, 

m = -6, E + (l - 



, _ Bum kc' 
~ + — f , 

r r- 



1 



2-K.m Kk 

I — — — H ; 

r r J 



— , IXrXs 



(13.34) 



% (13.30) takes the form 



r 
pi 



with the solution 



*y 






r 2 — 2&r + k^ 



arcs cot 



t ! m 2 



Kffll 



_V^ a — K 2 m 2 . 



(13.30a) 



(13.35) 



The solutions with rotational symmetry. AVeyl and Lcvi-Civita suc- 
ceeded in finding those static solutions which have only rotational, but 
not spherical, symmetry.' If it is understood at the outset that "static" 
indicates both the independence of the g„, of £ 4 and the vanishing of the 
components g h , it can be shown that any metric tensor with rotational 
symmetry can be brought into the form 

r,M - 1,2 (but not 3),' 



§u = e", g. u = 0, g, s = 0, 
0w = -e"-», y 3s = 0, 



r 



f + f , 



(13.36) 



'Weyl ; A WM fe B d Physih, 54, 117 (1917); B9, 1S5 (1919). Bteh and Wovl, 
Matkernahscke Zntschrift , 13, 142 (1921). Levi-Civita, £«& 4«, <fei £ta«t 
S«waJ Ay(es (1918-1919). 












Ch ap . Kill ] RIGOROUS SOLUTIONS 

{j. and j' are functions of the two variables 

P = VV -4- f 2 and 

The components of $„, take the form 

{ \dp- p dp <lz-J 

T 2 W T W T 4 |_W ^ W 

&)"-(t): 



G33 = 

G r s — 



1 ^ _ 1 

2p3p I 

1 Sv 



1 ^ dp) 
2p 3s 2 3p <fe J Xa ' 



_j_ dv 1 

2p dp 4 

+ 



~/dpY AvYll 

f 1 /a> s%\ 1 r/aA ! 
{ 2W T a*y iLW/ 



+ 



(£)]} 



' ( 5 ™ - XrX«); 



tfls — 0, (?43 — 0. 

In a purely gravitational field, we have the equations 



Kl 



A? 



® p, 1 5(i f)' fj, 
„ -j- — ■ — -p — 

up- p 3p 3s 2 



0, 



«8 = 



K4 



a* 


p 


rn 


dp 


2 


LW 




dp, 3p; 
dp 33 


3% 
dp 2 


+ | 


Z i 2 



= 0, 



m-m-A 



ai- 



ds. 37) 



(13.38) 



These four equations have two identities, the contracted Bianehi iden- 
tities with the indices 3 and 5 (,s = 1,2). It turns out that the last 
equation (13.38) is identical in the remaining three equations, 



&iti . 3k 3 dp, 

K-i = -r 1" — + p — Kl 

op dz da 



(13.311) 



208 RIGOROUS SOLUTIONS [ Ch ap . Xlli 

while the second and the third equations have this identity with kj : 



~dz 



dun 

dp 



Pit *i- 
dz 



(13,40) 



The two functions p, and v must vanish at infinity. Furthermore 
the form of the components g„ , eq. (13,3.0), indicates that the g r [ 
become singular (that is, indeterminate) along the E 3 -axis unless 
(1 - e~") vanishes there, that is, unless v vanishes at p = 0. 

Of the equations (13. 38), the first one, fa , is a linear, homogeneous 
equation for p. only; and, moreover, it is the Laplacian equation in 
cylindrical coordinates for functions with rotational symmetry. We 
know that the solutions of the Laplacian equation, apart from the solu- 
tion n = 0, satisfy the boundary conditions at infinity only if they have 
singularities somewhere for finite coordinate values. Singularities off 
the Hf-axis are necessarily circular, while singularities on the H 3 -axis may 

be pointlike and of the form £ [p 2 + (z ~ a;)T U ~, or nth derivatives 

i 

of such "poles" with respect to z. 

However, not all of these solutions are compatible with the differ- 
ential equations for v, m and k 3 , Because of eq. (13.40), we know that 
if the equation fa is satisfied in any simply connected domain of the 
p, z-spaee (p £ 0), the equations k 2 , k s have solutions. But in the pres- 
ence of singularities, the p, z-space is no longer simply connected. 

Let us first consider singularities off the E 3 -axis. If we take a solu- 
tion p. of the equation ki with an arbitrary circular singularity, the closed 
line integral around this singularity in the p, z-plane, 



0-> + s <fe ) 



/w-g 



&y 



dp dp X 

+ P — —dz> 

dp dz j 



} (13.41) 



will, in general, not vanish. But the function v will not be single,- valued 
outside the singularity, unless the integral (13.41.) vanishes; in other 
words, the vanishing of the integral (13.41) is a necessary condition for 
the existence of a solution. 

Let us now turn to singularities on the S 3 -axis. Outside of this singu- 
larity , — vanishes on the 2 3 -axis, and if the function v is assumed to 
dz 

vanish at one point on the £ s -a-xis, it will vanish everywhere on the 






Chap, XIII ] 



RIGOROUS SOLUTIONS 



209 



S 3 -axis, up to the singularity. The singularity itself must satisfy the 
condition that the line integral of the differential 



2 LW W . 



d P + p'^^dz (13.42) 

op dz 

over a small half-circle, around the singularity, from the E 3 -axis and back 
to it, must vanish. 

Let us consider a typical singularity on the E 3 -axis, 



m 



U = (p + i) 



i\-\n 



(13,43) 



(i) 



The derivatives of p. are 



m 




dp 
dp 




d1 




dp. 


z 


dz ~ 


~?' 



r = Vp 2 + zK 

The differential (13.42) has the form 



*-(g'' ;-'*+£*) 



(13.44) 



Let us carry the integration out along a small half-circle. For that 
purpose, we shall introduce the angle <p, 



r cos <p, 



dp 



-z -dtp, 



r — const. 



z = r sin <p, dz = p-d<p, 
Substituting these expressions in eq. (13.44), we obtain 
1 



dv 



2r 



cos ip sin ip dip, 



const. 



(13.45) 



Let us integrate this expression over the half- circle, that is, from 



— - to I; ■ We have 



M = A 

f =- r /2 2?' 2 J-i/2 



+ WS 



cos & sin <p dip = — [sin 2 <p] = 0. (13.46) 
4r 2 _ x /2 



(i> 



The solution p., eq. (13.43), is compatible with the regularity conditions 
for v. 



210 



RIGOROUS SOLUTIONS 



[ Chap, XIII 



Let us now consider the case of two singularities. At the point of one 
the other can he expanded into a power series in p and z, and we shall 
assume that around the point of origin ft has the form 






m 



(13.47) 



Before we compute the integral again, let us remark that there are 
only certain of the expansion coefficients which can enter into the in- 
tegral along the half-circle. The value of the integral is, of course 
independent of the size of the half-circle, that is, of the value of r as 
long as the circle does not enclose any other singularities hut the one at 
the origin (r = 0). Therefore, all the expansion coefficients a„ ti „ which 
would make the value of the integral depend on r need not be con- 
sidered. Furthermore, the regular part of p, by itself cannot give rise 
to a nonvanishing integral, and we need to consider only tho cross 
products of the singular and the regular part of ( ^'. The derivatives of 
the singular part of p. decrease as r (for a given value of <p). They are 
multiplied by p (increasing as r' 1 ), by the coordinate differentials (in- 
creasing as r" rl ), and by derivatives of the regular part of %l We are, 
therefore, interested only in those powers of tho expansion the deriva- 
tives of which depend only on %, but not on r. The only power with 
this property is pV' 1 . We shall, therefore, replace eq. (13.47) by 



<S) a _L_ i 
fi = - + m 

r 



(13.47a) 



We shall compute the expression 



*" -' b{?)b<W* + 4 ©5 «*+-. 03.48) 

The terms written out are the only ones which contribute to the in- 
tegral. They are 

i» = — p{zdp — pdz) = —M cos <f- dtp = ~abd (sin p). (13.49) 



7T 7T 

V°+2' 



This integral does 



This expression is to be integrated from 

not vanish. 

We find that at the location of one singularity, the derivative with 

respect to z of the regular part of p. must vanish. This excludes the 
simultaneous existence of sever at point-like singularities on ike £?-axis. It 
looks as if the field equations themselves exclude motions (or the lack 
of motion) of mass points which are incompatible with the equations of 
motion. In Chapter XV, we shall find that this is really the case. 









CHAPTER XIV 

The Experimental Tests of the General Theory 
or Relativity 



As we found in the first part of this book, there are many experimental 
confirmations of the Lorentz- covariant physical laws. The m.ost con- 
vincing arguments in favor of the general theory of relativity, however, 
remain, so far, theoretical. Before we delve into the experimental evi- 
dence in favor of the general theory of relativity, it might be well to 
summarize these theoretical arguments. 

Cfaly a theory of gravitation which is covariant with respect to general 
coordinate transformations can explain the principle of equivalence and 
make, it an integral part of its structure. A theory of gravitation which 
accounts for this principle must be considered more satisfactory than 
other theories -which, though compatible with the principle of equiva- 
lence, do not require it and could be maintained with slight modifica- 
tions if the "gravitating mass" and "inertial mass'" were to be con- 
sidered different and independent quantities. 

Moreover, the general theory of relativity presents us with the most 
nearly perfect example of a field theory which is yet known. In the 
next chapter we shall find that the laws of motion in the general theory 
of relativity are not independent of the field equations, but completely 
determined by them. 

Let us now turn to the experimental tests of the general theory of 
relativity. There are three instances in which the general theory of 
relativity leads to observable effects. Each of these effects has been ob- 
served; however 1 , two of them are just outside the limits of experimental 
error, so that the quantitative agreement between observations and theo- 
retical predictions is still doubtful. 

The general theory of relativity accounts for the advance in the peri- 
helion of Mercury, which, was known before the new theory was formu- 
lated. Furthermore, the theory predicted correctly the deflection of 
light rays which pass near the surface of the sun, and the red shift of 
Spectral lines of light originating in dense stars. 

211 



212 



EXPERIMENTAL TESTS 



[ Chap. XIV 



The advance of the perihelion of Mercury. Let us consider the 
motion oi a small body in a Schwarzs child field which is produced by a 

much larger body. It is advantageous to introduce polar coordinates 
by the coordinate transformation 



f 1 = r = V^' + f + ^ ^ 



8 = arc tan 






|* : = „ = arc tan (iVf 1 ), 



(1.4.1) 



In terms of these coordinates, the metric tensor (13.25) goes over into 
the form 



2nm 
gu = 1 - ■ — ■ ., 



Sii = - 



i 



1 2k?» y 



(fa = —V 



Qii — — r COS" 



(14.2) 



all other components being zero. 
The equations of motion of a small particle in this Schwarzschild field 



€?+H*a:_ 



are 



pa) dr dr 

Let us now compute the Chris toff el symbols, {%, 
for the sake of brevity, the expression 

e" = 1 - 2m/t, 
the Christoffel symbols which do not vanish are 



U4.3) 
If we introduce, 

(14.4) 



Chap. XiV ] 



EXPERIMENTAL TESTS 



213 



4 ^-v 

14j " «*■ 



2 e A 1 



1 1 



= -y, 



1221 



33 



= — e r cos-i 



123 



cos sin 6, 



— tanU. 



(14.5) 



If we simplify our mechanical problem by assuming that the whole 
motion takes place in the plane d = 0, we obtain as the equations of 
motion the differential equations 



d" t , , dt dr 
rfr J dr dr 



£+*v@> -*($-"&)'-* 



d <s 2 dr dtp __ . 

dr 1 r dr dr 



(14.0) 



One integral of these equations is furnished by the definition of the 
proper time differential, 



The first equation (14.6) has an integral, 

dr 
The last equation (14.6) has also an integral, 

T dr ' 



(14.8) 



(14.9) 



the integral of angular momentum. The integral (14.8) corresponds to 

the energy integral. The three equations (14.7), (14.8), and (14.9) re- 
place the equations of the second order, (14.6). Finally, we may elimi- 



214 



EXPERIMENTAL TESTS 



l 1 Chap. XIV 



date the coordinate time t witli the help of eq. (14.8), and obtain the 
two equations 



GDM£) , -^-*-> +-©'.] 



7 Tr = *• 



(14.10) 



These equations differ 



Q 

where e" has been replaced by its value, 1 — uK 

i 

from the classical equations of motioTi of a body in a Newtonian field 

only in that the last term in the first equation (14.10) does not occur 

in the nonrelativistic equations, and that all derivatives are taken with 

respect to proper time rather than with respect to coordinate time. The 

classical Integrals of energy and angular momentum are 



.a , 2* 2icm 2E 
r m 



r f = — , 
m 



(14.11] 



where m'is the mass of the moving body. 

The equations ( ! 4.10) cannot be solved in a closed form. But we may 
solve them approximately, so that the first approximation corresponds 
to the classical path of a body; the second approximation will then reveal 
the deviation of the solutions of the relativistic equations (14.10) from 
the classical equations (14.11). 

Wo multiply the first equation (14.10) by ( ^ j and substitute this 

factor itself from the second equation. We obtain the one differential 
equation 



(f)' 



-fl-J&n + 



2wm 



f + 2ianr, 



(14.12) 



Into this equation we introduce the function u = - , and obtain the 

r 
equation 

fduY 1 - Ar 2«m, , . „ , 

By differentiating this equation with respect to £, wc obtain an equation 
of the second order, 

d'u 



dip- 



+ u 



H 



'(1 4- 3AV). 



(14.14) 



Chap. XIV ] 



EXPERIMENTAL TESTS 



215 



The second term in the round bracket, Zh:u , is the one which distin- 
guishes the relativistic equation from the corresponding classical equa- 
tion. According to eq. (14.il), this term has the significance 



3ft V = >i(>-' l f) ; 



(14.15) 



in other words, it is approximately proportional to the square of the 
velocity component which is perpendicular to the radius vector. As we 
are using the "relativistic units" for time, in which the velocity of light 
equals unity, the velocity of a star, for instance, is small, compared with 
unity. The relativistic term in eq. (14.14) has, therefore, the character 
of a higher order correction, 
The solution of the equation 



d uq 



■h- u 



urn 



is 



too 



Ki»- 



■'[1 4- e cos (jp - w)], 



(1446) 



(14.17) 






where e and w are the constants of integration, e is the eccentricity of 

the ellipse, while the value of cu determines the position of the perihelion. 

Those solutions of eq, (14.14) which are approximated by ellipses are 

also periodic solutions. Eq. (14.13) associates with every value of u two 

values of — - , which differ only with respect to the sign. The solutions 
dtp 

will, be periodic if the light-hand side of eq. (14.13) has two zeros for 
positive values of u and is positive between these two zeros. The solu- 
tion will then oscillate between these two zeros. The period of the 
approximate solution (14.17) is equal to 2tt; that is, the paths are closed. 
The period of the rigorous solutions of eq. (14.14), however, will differ 
from 2tt by a small amount. 

Let us expand the periodic solutions of the equation 



+ u = a{\ + \u~) 



into a Fourier series, 



U = «o 4" OLi COS pip + tt 3 COS 2p<p -f- 



(14.18) 



(14.19) 



Tf X is a small constant, the solution will be approximated by 

Wo = a.(\ — ecosic). (14.20) 



216 



EXPERIMENTAL TESTS 



[ Chap. X!V 



We shall, therefore, assume that « is approximately equal to a, and that 
a 2 and the following coefficients are at least of the order of X. In other 
words, we shall replace eq. (14.19) by the series 



u = a + X.So + at cos pv + A 2 !« cos w . 



(14.21) 



Let us substitute this assumption into eq. (14.18) and neglect all terms 
which are multiplied by the second and higher powers of X. Lor u" 

we obtain 



'[■ 



p" «e COS flip 4- % ^F, v p v cos vp<p 



and for Au 2 , we have 



Am 2 <~ \a"\l -|- 2<e cos p P -f e 2 cos 2 p^] 



= \cf 



1 + n + 2 « cos P¥> + ~ cos 2/>p 



Eq. (14.18) becomes 

a + X f % 4- Sell - /) cos pi ? - \J^(/ - 1)& cos w 



1. -\-\a ( 1 + 9 - + 2-€ cos p P + - cos 2 ptp 



(14.. 18a) 



By comparing the terms which are constant, those which are multiplied 
by cos pv, and those multiplied by cos 2 pip , we obtain the equations 



-°'K> 



i - 



2\a' 



-3.3, = -f-ft 



it 



(14.22) 



The only equation of interest to us is the second one, which determines p. 
We find that p is nearly equal to unity, 

P = vT- 2Xa 2 ~ 1 - \a. (14.23) 

Substituting for X and a the values given in eq. (14.14), we find for p 



2 2 
K Til 



(14.24) 



Chap. XiV ] 



EXPERIMENTAL TESTS 



217 



The angle between two succeeding perihelions is, therefore, 



'1= ~ 2jt 1 +3 



k m 

1? 



2t + 6*- 



2 2 



h" 



(14.25) 



The precession of the perihelion of planets by Gx —pr radians per revo- 
lution can be observed in the case of Mercury, where it amounts to 
about 43" per century. The observed and the predicted values of the 
precession agree well within the experimental error of the astronomical 
obsei-vations. 

The special theory of relativity also leads to a precession effect when 

a bodv moves in a field with a potential - . But this precession is 

r 

numerically different from the one predicted by the general theory of 
relativity. 

Let us return, for a moment, to Sommerf eld's treatment of the hydro- 
gen atom, Chapter IX. The equations (9.27) correspond to the eqs. 
(14.7), (14.8), and (14.9) of this chapter. We can write them in the 
form : 

idt e 

mc ^- = E + — 
dr r 



,dd 
dr 



h\ 



%'-m+'(&. 



(14.26) 



In the last equation, we shall replace ■ by the expression which, is 
furnished by the first equation. Thus, we obtain an equation free of t, 

I 1\ 2 



(0 + Of - ' 

We multiply this equation by 



E + 



me' 



- 1 



(14.27) 



\doJ h' 2 ' 



and obtain the equation 



\de) ~ ° h'* 



E + 



mc 



— r 



(14.28) 



21.8 



EXPERIMENTAL TESTS 



[ Chap. XIV 



If we again introduce u — 1/r as a new variable, the differential equa- 
tion for if becomes 



/duY _ i Y/E + ivY _ 

\mj ~ F~ 2 |_\ vie' ) 



(14.29) 



Differentiation with respect to 6 yields the equation of the second 
order: 



d'u 



+ 



i 



e 
mrc-h' 



'E 



m i ~ti % 



(14.30) 



This equation has the following solutions: 



m- n 2 c 1 



y> 



eE 



L + e cos 



m 1 ^ h ! * 



2th 2 c 2 h' i 



t,(v-u) 



\ \ 



is - d 



m 2 h l2 c- 
The precession of the perihelion, therefore, amounts to 

4 



(14.31) 



vi' c a h' 2 



(M.32) 



per revolution. To compare this precession with the one obtained in 
eq, (14.24), we must replace e\ the coefficient of Coulomb's law, by 
Kimn', the coefficient of Newton's law of gravitation. Furthermore, the 

constant h' equals - [h is the constant appearing in eq. (14.24)], because 

the t of eq, (14.25) is measured in metric units. We obtain, instead of 
eq. (14.32), 



2 2 

k m 



(14.32a 



one-sixth of the precession predicted by fee general theory of relativity. 

The deflection of light in a Schwarzschild field. Light rays travel 
along geodesic zero lines. These lines are no longer the solutions of a 
variational principle, for in the case of zero lines, tin; variation of the 
integrand Vffi *£''£"' 0I eq. (5.93) is not a linear function of the varia- 
tions S£ and 5|". However, there are zero lines with a tangential vector 
the co variant derivative of winch in the direction of the tangential 
vector vanishes. This property characterizes non-zero geodesies and 
can be used as the defining property of both zero and non-aero geodesic 
lines. In a iiat metric and in a Lorentzian coordinate system, these 






Chap. XIV ] 



EXPERIMENTAL TESTS 



219 



zero geodesies are "straight" zero lines, that is, £', £ 3 , and £ are linear 

functions of £ . 

In the case of zero lines, the tangential vector' is a zero vector, 
and its magnitude cannot be normalized. We must, therefore, replace 
the parameter r, which we have used so far, by a parameter s, which 
remains to a certain degree undetermined. The differential equations 
of the geodesic aero lines then take the form 



as 1 [pa-) as as 



as as 



(14.3a) 



If the metric is that of a Schwarzschild field, those equations assume 
the form (14.6), except that r must be replaced everywhere by s. Of 
the first integrals, (14.7), (ITS), and. (14.9), eq. (14.7) has to be modified 
insofar as the right-hand side is now 0, not 1. The three integrals are 

'©' '-(f)' '-'(SH 



dt 

ds 
dtp 



fc, 



(14.33) 



If we combine these three equations into one, by means of the same 
method employed before, we obtain again a relationship between r and <p, 



m- 






i - 



2nth 



and if we introduce again the variable u, 
(dttX 1 fc 2 j 



Kinu 



(14.34) 



(14.35) 



The last term on the right-hand side represents the influence of the 
gravitational field on the path of the light rays. The solutions of Hit; 
equation 

'■ 



are 



Up 






n COS (<p 



A 2 



<<*(.' 



), 



R « 



%' 



(14.36) 



220 



EXPERIMENTAL TESTS 



[ Ch,ap, XIV 



where R is the distance of the light path from the point of origin of the 

coordinate system, and ^o is the constant of integration. 

The angular distance between two zeros of Me (that is, between the; 
two directions in which the light path goes toward infinity) is v. ff 
u(<p) is a solution of cq. (14.35), we are interested in the deviation of the 
angular distance between two zeros of u{tp) from sr. This deviation is 
twice the deviation of the angular distance between the maximum of u, 

u and the nearest zero from - . 



The determining equation for u is 

= — — u + 2ninu. 
If. we subtract this equation from eq. (14.35), we obtain 

(fT = ^ ~ ui) ~ 2Km{,f ~ - /} 

and 



^ = [(;7 2 - a*} - 2kw.(v? - u z )\ 
du 



,3\i-y2 



(14.37) 



(14.38) 



(14.39) 



The angular distance between the maximum of u and the nearest zero 
equals the integral of the right-hand side of eq. (14.39), taken from 
u = to u = u. The integral cannot be solved in a closed form. How- 
ever, we know that the same integral, extended over the right-hand side 

of the equation 

dtp 
dm 



U - «!)-** 



(14.40) 



equals -. The deviation of tin: first integral from - is, therefore, 

given by the integral 

W = f l\(u - u<) - 2«(u J - ^)T m - (u 2 - « 5 )- 1/2 } du. (14.41) 

Since we assume that the relativistic term (the one depending on m) is 
small in comparison with the classical terra, we shall replace the ex- 
pression [f(x + e) - f(x)} by e-f'{x). By doing this, we obtain the 
integral 



Ju=0 (u — u ) 



(14,42) 






Chap. XIV] EXPERIMENTAL TESTS 

which can be solved in a closed form, 



221 



r K m(fi - 



(if - u 3 ) 



du — icmu 



5 (w s - v?) m J I= o (1. - x 1 ) 3 ' 2 



f 



1 -x 



d% 



= tenm 



r^f^Une) « ,rnu F i^l de[ (14.43) 

■I o=o CO&9 J B=0 COS" 9 



■= tanu [tan 9 — cos 9 — cos L 9] = 2k 

9-0 — 



The total deviation of the angular distance between two successive seres 
of u from tt equals, therefore, 



&tp <**> iamu ■ 



4 Km 



(14.44) 



This deflection of light rays which pass near great masses can be ob- 
served during eclipses of the sun. when fixed stars in the apparent 
neighborhood become visible. The predicted deflection amounts to not 
more than about 1.75" and is just outside the limits of experimental 
error. A quantitative agreement between the predicted and the ob- 
served effects cannot be regarded as significant. 

The gravitational shift of spectral lines. The interior forces of a free 
atom are not sensibly affected by the inhomogeneities of the surrounding 
gravitational field. If such a free atom goes over from one quantum 
state to another, the frequency of the emitted photon, -measured in 
proper time units of the alom, will also be independent of the surrounding 
gravitational field. 

Let us now consider the atoms which form the (gaseous) outer layer 
of a hot fixed star. Those atoms which emit any of the normal spectral 
lines must be falling freely at the time of emission, and their velocities 
relative to the star will be distributed at random. The mean frequency 
emitted will correspond to the emission by an atom which is momen- 
tarily at rest relative to the star. 

The gravitational field of the star is described by a coordinate system 
in which the units of coordinate time and proper time are not identical. 
The proper frequency of an oscillatory process of an atom, is the number 
of beats per unit proper time, 



Vg = 



dN 



(14.45) 



222 



EXPERIMENTAL TESTS 



[ Chap. XIV 



and the coordinate frequency is the number of beats per unit coordinates 
time. 



dN 



(14.46) 



The two are related by the equation 

d_N = d_N rh 
d? " &t ' d? 



r _m / ^r d? fU4r . 



If we introduce; a coordinate system in which the Star is at rest, and in 
which the; mean velocity of the atoms of the tarter layer is zero, eq. 
(14.47) reduces, for this mean frequency, to the relationship 

v = V^ n ; (14.48) 

for the three differential quotients, — vanish. 

The coordinate frequency v is the frequency which will.be observed 
by an observer who is at rest relative to a star and stationed at a great 
distance, so that gu at his location equals unity. For the coordinate 
time required to transmit light signals from the surface of the star to 
such an observer is constant (because of the static character of the 
Schwarzschild field), and he will receive periodic signals at the same 
coordinate frequency at which they are being emitted at, the surface of 
the star. 

If the radius of the star is R a.nd its mass m, the value of gu on the 



surface of the; star i 



and eq. (.14.48) becomes 






The "gravitational shift" of the spectral lines is, therefore, 



Kin 
5S — ' — -p -jaji , 



(14.49) 



(14.50) 



In the ease of the sun, the shift is barely observable, but appears to be 
in agreement with eq. (14.47). However, in the case of the companion 
to Sirius, winch is an extremely dense star, the red shift is about 30 
times as great as in the case of the sum In this case, the agreement 
between theory and observation is satisfactory. 






CHAPTER XV 

The Equations of Motion in the General Theory 

of Relativity 



Force laws in classical physics and in electrodynamics. Newtonian 
physics is based on the motions of mass points. The force acting on a 
mass point is the resultant of the actions of all other mass points in the 
world on the one considered. This force is uniquely determined by the 
positions of the other mass points, and it is finite as long as none of the 
other mass points coincide with the one considered. 

The development of electrodynamics shows that tin; force; acting on a, 
body is not so simply determined as Kewton thought. The action of 
one charge on another depends not only on the distance between them, 
but on their relative states of motion as well. A change in the state of 
motion of one charge brings about a change in its action on the other 
charge. But this secondary change does not take place; momentarily; 
rather, the disturbance of the elect ro magnetic field spreads with a finite 
velocity, which is equal to the speed of light, c. Therefore, the force 
on a charged mass point is not determined by the position of all other 
point charges, or even by their positions and velocities, but by the 
electromagnetic field in the; immediate neighborhood of the particle 
<: on si tiered. 

We cannot split up this electromagnetic field into partial fields, each 
representing the action of one particle. For the field itself is not 
uniquely determined by the motions of the charges. It is true that the 
field is determined by the; distribution of the charges and their velocities 
if we impose on the; field those boundary and initial conditions which 
exclude waves which travel toward a singular point rather than emanate 
from it (the;se; are formal solutions of Maxwell's field equations). But 
it is doubtful whether these conditions are really satisfied in nature, or 
whether we impose them on the electromagnetic field just because they 
are suggested by our mechanical superstition that eiisturbances must 
always originate; in mass points. 

At any rate, the field can be treated adequately only as a unit, not 
as the sum total of the contributions of individual point charges. 

m