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