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THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY
By A. Einstein
The theory which is presented in the following pages conceivably constitutes the farthest-reaching generalization of a theory which, today, is generally called the “theory of relativity”; I will call the latter one—in order to distinguish it from the first named—the “special theory of relativity,” which I assume to be known. The generalization of the theory of relativity has been facilitated considerably by Minkowski, a mathematician who was the first one to recognize the formal equivalence of space coordinates and the time coordinate, and utilized this in the construction of the theory. The mathematical tools that are necessary for general relativity were readily available in the “absolute differential calculus,” which is based upon the research on non-Euclidean manifolds by Gauss, Riemann, and Christoffel, and which has been systematized by Ricci and Levi-Civita and has already been applied to problems of theoretical physics. In section B of the present paper I developed all the necessary mathematical tools—which cannot be assumed to be known to every physicist—and I tried to do it in as simple and transparent a manner as possible, so that a special study of the mathematical literature is not required for the understanding of the present paper. Finally, I want to acknowledge gratefully my friend, the mathematician Grossmann, whose help not only saved me the effort of studying the pertinent mathematical literature, but who also helped me in my search for the field equations of gravitation.
The following two translations are reproduced from the English edition of the Collected Papers of Albert Einstein (Doc. 30 and Doc. 41, vol. 6). The first paragraph of the first document and the entire second document have been translated by Alfred Engel; the remaining text of the first document is reprinted from H. A. Lorentz et al., The Principle of Relativity, trans. W. Perrett and G. B. Jeffery (Methuen, 1923; Dover rpt., 1952).
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A. FUNDAMENTAL CONSIDERATIONS ON THE POSTULATE OF RELATIVITY

§ I. Observations on the Special Theory of Relativity
The special theory of relativity is based on the following postulate, which is also satisfied by the mechanics of Galileo and Newton.
If a system of co-ordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws also hold good in relation to any other system of co-ordinates K′ moving in uniform translation relatively to K. This postulate we call the “special principle of relativity.” The word “special” is meant to intimate that the principle is restricted to the case when K′ has a motion of uniform translation relatively to K, but that the equivalence of K′ and K does not extend to the case of non-uniform motion of K′ relatively to K.
Thus the special theory of relativity does not depart from classical mechanics through the postulate of relativity, but through the postulate of the constancy of the velocity of light in vacuo, from which, in combination with the special principle of relativity, there follow, in the well-known way, the relativity of simultaneity, the Lorentzian transformation, and the related laws for the behavior of moving bodies and clocks.
The modification to which the special theory of relativity has subjected the theory of space and time is indeed far-reaching, but one important point has remained unaffected. For the laws of geometry, even according to the special theory of relativity, are to be interpreted directly as laws relating to the possible relative positions of solid bodies at rest; and, in a more general way, the laws of kinematics are to be interpreted as laws which describe the relations of measuring bodies and clocks. To two selected material points of a stationary rigid body there always corresponds a distance of quite definite length, which is independent of the locality and orientation of the body, and is also independent of the time. To two selected positions of the hands of a clock at rest relatively to the privileged system of reference there always corresponds an interval of time of a definite length, which is independent of place and time. We shall soon see that the general theory of relativity cannot adhere to this simple physical interpretation of space and time.

§ 2. The Need for an Extension of the Postulate of Relativity
In classical mechanics, and no less in the special theory of relativity, there is an inherent epistemological defect which was, perhaps for the first time, clearly pointed out by Ernst Mach. We will elucidate it by the following example:—Two fluid bodies of the same size and nature hover freely in space at so great a distance from each other and from all other masses that only those gravitational forces need be taken into account which arise from the interaction of different parts of the same body. Let the distance between the two bodies be invariable, and in neither of the bodies let there be any relative movements of the parts with respect to one another. But let either mass, as judged by an observer at rest relatively to the other mass, rotate with constant angular velocity about the line joining the masses. This is a verifiable relative motion of the two bodies. Now let us imagine that each of the bodies has been surveyed by means of measuring instruments at rest relatively to itself, and let the surface of S1 prove to be a sphere, and that of S2 an ellipsoid of revolution.

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Thereupon we put the question—What is the reason for this difference in the two bodies? No answer can be admitted as epistemologically satisfactory,1 unless the reason given is an observable fact of experience. The law of causality has not the significance of a statement as to the world of experience, except when observable facts ultimately appear as causes and effects.
Newtonian mechanics does not give a satisfactory answer to this question. It pronounces as follows:—The laws of mechanics apply to the space R1, in respect to which the body S1 is at rest, but not to the space R2, in respect to which the body S2 is at rest. But the privileged space R1 of Galileo, thus introduced, is a merely factitious cause, and not a thing that can be observed. It is therefore clear that Newton’s mechanics does not really satisfy the requirement of causality in the case under consideration, but only apparently does so, since it makes the factitious cause R1 responsible for the observable difference in the bodies S1 and S2.
The only satisfactory answer must be that the physical system consisting of S1 and S2 reveals within itself no imaginable cause to which the differing behavior of S1 and S2 can be referred. The cause must therefore lie outside this system. We have to take it that the general laws of motion, which in particular determine the shapes of S1 and S2, must be such that the mechanical behavior of S1 and S2 is partly conditioned, in quite essential respects, by distant masses which we have not included in the system under consideration. These distant masses and their motions relative to S1 and S2 must then be regarded as the seat of the causes (which must be susceptible to observation) of the different behavior of our two bodies S1 and S2. They take over the rô1e of the factitious cause R1. Of all imaginable spaces R1, R2, etc., in any kind of motion relatively to one another, there is none which we may look upon as privileged a priori without reviving the above-mentioned epistemological objection. The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity.
In addition to this weighty argument from the theory of knowledge, there is a wellknown physical fact which favors an extension of the theory of relativity. Let K be a Galilean system of reference, i.e. a system relatively to which (at least in the four-dimensional region under consideration) a mass, sufficiently distant from other masses, is moving with uniform motion in a straight line. Let K′ be a second system of reference which is moving relatively to K in uniformly accelerated translation. Then, relatively to K′, a mass sufficiently distant from other masses would have an accelerated motion such that its acceleration and direction of acceleration are independent of the material composition and physical state of the mass.
Does this permit an observer at rest relatively to K′ to infer that he is on a “really” accelerated system of reference? The answer is in the negative; for the above-mentioned relation of freely movable masses to K′ may be interpreted equally well in the following way. The system of reference K′ is unaccelerated, but the space-time territory in question

1 Of course an answer may be satisfactory from the point of view of epistemology, and yet be unsound physically, if it is in conflict with other experiences.

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is under the sway of a gravitational field, which generates the accelerated motion of the bodies relatively to K′.
This view is made possible for us by the teaching of experience as to the existence of a field of force, namely, the gravitational field, which possesses the remarkable property of imparting the same acceleration to all bodies.2 The mechanical behavior of bodies relatively to K′ is the same as presents itself to experience in the case of systems which we are wont to regard as “stationary” or as “privileged.” Therefore, from the physical standpoint, the assumption readily suggests itself that the systems K and K′ may both with equal right be looked upon as “stationary,” that is to say, they have an equal title as systems of reference for the physical description of phenomena.
It will be seen from these reflections that in pursuing the general theory of relativity we shall be led to a theory of gravitation, since we are able to “produce” a gravitational field merely by changing the system of co-ordinates. It will also be obvious that the principle of the constancy of the velocity of light in vacuo must be modified, since we easily recognize that the path of a ray of light with respect to K′ must in general be curvilinear, if with respect to K light is propagated in a straight line with a definite constant velocity.

§ 3. The Space-Time Continuum. Requirement of General Co-Variance for the Equations Expressing General Laws of Nature
In classical mechanics, as well as in the special theory of relativity, the co-ordinates of space and time have a direct physical meaning. To say that a point-event has the X1 coordinate x1 means that the projection of the point-event on the axis of X1, determined by rigid rods and in accordance with the. rules of Euclidean geometry, is obtained by measuring off a given rod (the unit of length) x1 times from the origin of coordinates along the axis of X1. To say that a point-event has the X4 co-ordinate x4 =  t, means that a standard clock, made to measure time in a definite unit period, and which is stationary relatively to the system of co-ordinates and practically coincident in space with the point-event,3 will have measured off x4 =  t periods at the occurrence of the event.
This view of space and time has always been in the minds of physicists, even if, as a rule, they have been unconscious of it. This is clear from the part which these concepts play in physical measurements; it must also have underlain the reader’s reflections on the preceding paragraph (§ 2) for him to connect any meaning with what he there read. But we shall now show that we must put it aside and replace it by a more general view, in order to be able to carry through the postulate of general relativity, if the special theory of relativity applies to the special case of the absence of a gravitational field.
In a space which is free of gravitational fields we introduce a Galilean system of reference K (x, y, z, t), and also a system of co-ordinates K′ (x′, y′, z′, t′) in uniform rotation relatively to K. Let the origins of both systems, as well as their axes of Z, permanently

2 Eötvös has proved experimentally that the gravitational field has this property in great accuracy. 3 We assume the possibility of verifying “simultaneity” for events immediately proximate in space, or—to speak more precisely—for immediate proximity or coincidence in space-time, without giving a definition of this fundamental concept.

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coincide. We shall show that for a space-time measurement in the system K′ the above definition of the physical meaning of lengths and times cannot be maintained. For reasons of symmetry it is clear that a circle around the origin in the X, Y plane of K may at the same time be regarded as a circle in the X′, Y′ plane of K′. We suppose that the circumference and diameter of this circle have been measured with a unit measure infinitely small compared with the radius, and that we have the quotient of the two results. If this experiment were performed with a measuring-rod at rest relatively to the Galilean system K, the quotient would be r. With a measuring-rod at rest relatively to K′, the quotient would be greater than r. This is readily understood if we envisage the whole process of measuring from the “stationary” system K, and take into consideration that the measuring-rod applied to the periphery undergoes a Lorentzian contraction, while the one applied along the radius does not. Hence Euclidean geometry does not apply to K′. The notion of co-ordinates defined above, which presupposes the validity of Euclidean geometry, therefore breaks down in relation to the system K′. So, too, we are unable to introduce a time corresponding to physical requirements in K′, indicated by clocks at rest relatively to K′. To convince ourselves of this impossibility, let us imagine two clocks of identical constitution placed, one at the origin of co-ordinates, and the other at the circumference of the circle, and both envisaged from the “stationary” system K. By a familiar result of the special theory of relativity, the clock at the circumference—judged from K—goes more slowly than the other, because the former is in motion and the latter at rest. An observer at the common origin of co-ordinates, capable of observing the clock at the circumference by means of light, would therefore see it lagging behind the clock beside him. As he will not make up his mind to let the velocity of light along the path in question depend explicitly on the time, he will interpret his observations as showing that the clock at the circumference “really” goes more slowly than the clock at the origin. So he will be obliged to define time in such a way that the rate of a clock depends upon where the clock may be.
We therefore reach this result:—In the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial co-ordinates can be directly measured by the unit measuring-rod, or differences in the time co-ordinate by a standard clock.
The method hitherto employed for laying co-ordinates into the space-time continuum in a definite manner thus breaks down, and there seems to be no other way which would allow us to adapt systems of co-ordinates to the four-dimensional universe so that we might expect from their application a particularly simple formulation of the laws of nature. So there is nothing for it but to regard all imaginable systems of co-ordinates, on principle, as equally suitable for the description of nature. This comes to requiring that:—
The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant).
It is clear that a physical theory which satisfies this postulate will also be suitable for the general postulate of relativity. For the sum of all substitutions in any case includes those which correspond to all relative motions of three-dimensional systems of co-ordinates. That this requirement of general co-variance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflection. All our space-time verifications invariably amount to a determination of

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space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock dial, and observed point-events happening at the same place at the same time.
The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. We allot to the universe four space-time variables x1, x2, x3, x4 in such a way that for every point-event there is a corresponding system of values of the variables x1 . . . x4. To two coincident point-events there corresponds one system of values of the variables x1 . . . x4, i.e. coincidence is characterized by the identity of the co-ordinates. If, in place of the variables x1 . . . x4, we introduce functions of them, x′1, x′2, x′3, x′4, as a new system of co-ordinates, so that the systems of values are made to correspond to one another without ambiguity, the equality of all four co-ordinates in the new system will also serve as an expression for the space-time coincidence of the two point-events. As all our physical experience can be ultimately reduced to such coincidences, there is no immediate reason for preferring certain systems of co-ordinates to others, that is to say, we arrive at the requirement of general co-variance.

§ 4. The Relation of the Four Co-ordinates to Measurement in Space and Time

It is not my purpose in this discussion to represent the general theory of relativity as a system that is as simple and logical as possible, and with the minimum number of axioms; but my main object is to develop this theory in such a way that the reader will feel that the path we have entered upon is psychologically the natural one, and that the underlying assumptions will seem to have the highest possible degree of security. With this aim in view let it now be granted that:—
For infinitely small four-dimensional regions the theory of relativity in the restricted sense is appropriate, if the coordinates are suitably chosen.
For this purpose we must choose the acceleration of the infinitely small (“local”) system of co-ordinates so that no gravitational field occurs; this is possible for an infinitely small region. Let X1, X2, X3, be the co-ordinates of space, and X4 the appertaining coordinate of time measured in the appropriate unit.4 If a rigid rod is imagined to be given as the unit measure, the co-ordinates, with a given orientation of the system of co-ordinates, have a direct physical meaning in the sense of the special theory of relativity. By the special theory of relativity the expression

ds2

=

−d

X

2 1

−

d

X

2 2

−

d

X

2 3

+

d

X

2 4

(1)

then has a value which is independent of the orientation of the local system of co-ordinates, and is ascertainable by measurements of space and time. The magnitude of the

4 The unit of time is to be chosen so that the velocity of light in vacuo as measured in the “local” system of co-ordinates is to be equal to unity.

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linear element pertaining to points of the four-dimensional continuum in infinite proximity, we call ds. If the ds belonging to the element dX1 . . . dX4 is positive, we follow Minkowski in calling it time-like; if it is negative, we call it space-like.
To the “linear element” in question, or to the two infinitely .proximate point-events, there will also correspond definite differentials dx1 . . . dx4 of the four-dimensional co-ordinates of any chosen system of reference. If this system, as well as the “local” system, is given for the region under consideration, the dXo will allow themselves to be represented here by definite linear homogeneous expressions of the dxv:—

dXo = / aovdxv

(2)

v

Inserting these expressions in (1), we obtain

ds2 = / gvrdxvdxx

(3)

xv

where the gvx will be functions of the xv. These can no longer be dependent on the orientation and the state of motion of the “local” system of co-ordinates, for ds2 is a quantity ascertainable by rod-clock measurement of point-events infinitely proximate in spacetime, and defined independently of any particular choice of co-ordinates. The gvx are to be chosen here so that gvx =  gxv; the summation is to extend over all values of v and x, so that the sum consists of 4 × 4 terms, of which twelve are equal in pairs.
The case of the ordinary theory of relativity arises out of the case here considered, if it is possible, by reason of the particular relations of the gvx in a finite region, to choose the system of reference in the finite region in such a way that the gvx assume the constant values

−1 0 0 0

4 0 −1 0 0
0 0 −1 0

(4)

0 0 0 +1

We shall find hereafter that the choice of such co-ordinates is, in general, not possible for a finite region.
From the considerations of § 2 and § 3 it follows that the quantities gxv are to be regarded from the physical standpoint as the quantities which describe the gravitational field in relation to the chosen system of reference. For, if we now assume the special theory of relativity to apply to a certain four-dimensional region with the co-ordinates properly chosen, then the gvx have the values given in (4). A free material point then moves, relatively to this system, with uniform motion in a straight line. Then if we introduce new space-time co-ordinates xl, x2, x3, x4, by means of any substitution we choose, the gvx in this new system will no longer be constants, but functions of space and time. At the same time the motion of the free material point will present itself in the new co-ordinates as a curvilinear non-uniform motion, and the law of this motion will be independent of the nature of the moving particle. We shall therefore interpret this motion as a motion under the influence of a gravitational field. We thus find the occurrence of a gravitational field connected with a space-time variability of the gv. So, too, in the general case, when we are

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no longer able by a suitable choice of co-ordinates to apply the special theory of relativity to a finite region, we shall hold fast to the view that the gvx describe the gravitational field.
Thus, according to the general theory of relativity, gravitation occupies an exceptional position with regard to other forces, particularly the electromagnetic forces, since the ten functions representing the gravitational field at the same time define the metrical properties of the space measured.

B. MATHEMATICAL AIDS TO THE FORMULATION OF GENERALLY COVARIANT EQUATIONS

Having seen in the foregoing that the general postulate of relativity leads to the requirement that the equations of physics shall be covariant in the face of any substitution of the co-ordinates x1 . . . x4, we have to consider how such generally covariant equations can be found. We now turn to this purely mathematical task, and we shall find that in its solution a fundamental rôle is played by the invariant ds given in equation (3), which, borrowing from Gauss’s theory of surfaces, we have called the “linear element.”
The fundamental idea of this general theory of covariants is the following:—Let certain things (“tensors”) be defined with respect to any system of co-ordinates by a number of functions of the co-ordinates, called the “components” of the tensor. There are then certain rules by which these components can be calculated for a new system of co-ordinates, if they are known for the original system of co-ordinates, and if the transformation connecting the two systems is known. The things hereafter called tensors are further characterized by the fact that the equations of transformation for their components are linear and homogeneous. Accordingly, all the components in the new system vanish, if they all vanish in the original system. If, therefore, a law of nature is expressed by equating all the components of a tensor to zero, it is generally covariant. By examining the laws of the formation of tensors, we acquire the means of formulating generally covariant laws.

§ 5. Contravariant and Covariant Four-vectors

Contravariant Four-vectors.—The linear element is defined by the four “components” dxo, for which the law of transformation is expressed by the equation

dxlv

=

/
o

2xlv 2xo

dxo

(5)

The dx′v are expressed as linear and homogeneous functions of the dxo. Hence we may look upon these co-ordinate differentials as the components of a “tensor” of the particular kind which we call a contravariant four-vector. Any thing which is defined relatively to the system of co-ordinates by four quantities Ao, and which is transformed by the same law

Alv

=

/
o

2xlv 2xo

Ao,

(5a)

we also call a contravariant four-vector. From (5a) it follows at once that the sums Av ± Bv are also components of a four-vector, if Av and Bv are such. Corresponding relations hold for all “tensors” subsequently to be introduced. (Rule for the addition and subtraction of tensors.)

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Covariant Four-vectors.—We call four quantities Ao the components of a covariant four-vector, if for any arbitrary choice of the contravariant four-vector Bo

/ AoBo = Invariant

(6)

o

The law of transformation of a covariant four-vector follows from this definition. For if we replace Bo on the right-hand side of the equation

/ AlvBlv = / AoBo

v

o

by the expression resulting from the inversion of (5a),

/
v

2xo 2xlv

Blv,

we obtain

/
v

Blv

/
o

2xo 2xlv

Ao

=

/
v

BlvAlv.

Since this equation is true for arbitrary values of the B′v, it follows that the law of transformation is

Alv

=

/
o

2xo 2xlv

Ao

(7)

Note on a Simplified Way of Writing the Expressions.—A glance at the equations of this paragraph shows that there is always a summation with respect to the indices which occur twice under a sign of summation (e.g. the index o in (5)), and only with respect to indices which occur twice. It is therefore possible, without loss of clearness, to omit the sign of summation. In its place we introduce the convention:— If an index occurs twice in one term of an expression, it is always to be summed unless the contrary is expressly stated.
The difference between covariant and contravariant four-vectors lies in the law of transformation ((7) or (5) respectively). Both forms are tensors in the sense of the general remark above. Therein lies their importance. Following Ricci and Levi-Civita, we denote the contravariant character by placing the index above, the covariant by placing it below.

§ 6. Tensors of the Second and Higher Ranks

Contravariant Tensors.—If we form all the sixteen products Ano of the components An and Bo of two contravariant four-vectors

Ano = AnBo

(8)

then by (8) and (5a) Ano satisfies the law of transformation

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Alvx

=

2xlv 2xn

2xlx 2xo

Ano

(9)

We call a thing which is described relatively to any system of reference by sixteen quantities, satisfying the law of transformation (9), a contravariant tensor of the second rank. Not every such tensor allows itself to be formed in accordance with (8) from two four-vectors, but it is easily shown that any given sixteen Ano can be represented as the sums of the Ano of four appropriately selected pairs of four-vectors. Hence we can prove nearly all the laws which apply to the tensor of the second rank defined by (9) in the simplest manner by demonstrating them for the special tensors of the type (8).
Contravariant Tensors of Any Rank.—It is clear that, on the lines of (8) and (9), contravariant tensors of the third and higher ranks may also be defined with 43 components, and so on. In the same way it follows from (8) and (9) that the contravariant four-vector may be taken in this sense as a contravariant tensor of the first rank.
Covariant Tensors.—On the other hand, if we take the sixteen products Ano of two covariant four-vectors An and Bo

Ano = AnBo,

(10)

the law of transformation for these is

Alvx

=

2xn 2xlv

2xo 2xlx

Ano

(11)

This law of transformation defines the covariant tensor of the second rank. All our previous remarks on contravariant tensors apply equally to covariant tensors.
Note.—It is convenient to treat the scalar (or invariant) both as a contravariant and a covariant tensor of zero rank.
Mixed Tensors.—We may also define a tensor of the second rank of the type

Ano = AnBo

(12)

which is covariant with respect to the index n, and contravariant with respect to the index o. Its law of transformation is

Alvx

=

2xlx 2xo

2xn 2xlv

A

o n

(13)

Naturally there are mixed tensors with any number of indices of covariant character, and any number of indices of contravariant character. Covariant and contravariant tensors may be looked upon as special cases of mixed tensors.
Symmetrical Tensors.—A contravariant, or a covariant tensor, of the second or higher rank is said to be symmetrical if two components, which are obtained the one from the other by the interchange of two indices, are equal. The tensor Ano, or the tensor Ano, is thus symmetrical if for any combination of the indices n, o

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Ano = Aon,

(14)

or respectively,

Ano = Aon

(14a)

It has to be proved that the symmetry thus defined is a property which is independent of the system of reference. It follows in fact from (9), when (14) is taken into consideration, that

= Alvx 22xx= lvn 22xxlxo Ano 22xx= lvn 22xxlxo Aon 22xx= lvo 22xxnlx Ano Alxv.

The last equation but one depends upon the interchange of the summation indices n and o, i.e. merely on a change of notation.
Antisymmetrical Tensors.—A contravariant or a covariant tensor of the second, third, or fourth rank is said to be anti-symmetrical if two components, which are obtained the one from the other by the interchange of two indices, are equal and of opposite sign The tensor Ano, or the tensor Ano, is therefore antisymmetrical, if always

Ano = −Aon,

(15)

or respectively,

Ano = −Aon

(15a)

Of the sixteen components Ano, the four components Ann vanish; the rest are equal and of opposite sign in pairs, so that there are only six components numerically different (a six-vector). Similarly we see that the antisymmetrical tensor of the third rank Anov has only four numerically different components, while the antisymmetrical tensor Anovx has only one. There are no antisymmetrical tensors of higher rank than the fourth in a continuum of four dimensions.

§ 7. Multiplication of Tensors
Outer Multiplication of Tensors.—We obtain from the components of a tensor of rank n and of a tensor of rank m the components of a tensor of rank n + m by multiplying each component of the one tensor by each component of the other. Thus, for example, the tensors T arise out of the tensors A and B of different kinds,

Tnov = AnoBv, Tnovx = AnoBvx,

Tvx no

=

AnoBvo.

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The proof of the tensor character of T is given directly by the representations (8), (10), (12), or by the laws of transformation (9), (11), (13). The equations (8), (10), (12) are themselves examples of outer multiplication of tensors of the first rank.
“Contraction” of a Mixed Tensor.—From any mixed tensor we may form a tensor whose rank is less by two, by equating an index of covariant with one of contravariant character, and summing with respect to this index (“contraction”). Thus, for example, from the mixed tensor of the fourth rank Anvox, we obtain the mixed tensor of the second rank,

/ = Aox A= nnox b

Annoxl,

n

and from this, by a second contraction, the tensor of zero rank,

A=

A=oo

A

,nx
no

The proof that the result of contraction really possesses the tensor character is given either by the representation of a tensor according to the generalization of (12) in combination with (6), or by the generalization of (13).
Inner and Mixed Multiplication of Tensors.—These consist in a combination of outer multiplication with contraction.
Examples.—From the covariant tensor of the second rank Ano and the contravariant tensor of the first rank Bv we form by outer multiplication the mixed tensor

D

v no

=

AnoBv.

On contraction with respect to the indices o and v, we obtain the covariant four-vector

D=n D=noo AnoBo.
This we call the inner product of the tensors Ano and Bv. Analogously we form from the tensors Ano and Bvx, by outer multiplication and double contraction, the inner product AnoBno. By outer multiplication and one contraction, we obtain from Ano and Bvx the mixed tensor of the second rank Dnx = AnoBox. This operation may be aptly characterized as a mixed one, being “outer” with respect to the indices n and x, and “inner” with respect to the indices o and v.
We now prove a proposition which is often useful as evidence of tensor character. From what has just been explained, AnoBno is a scalar if Ano and Bvx are tensors. But we may also make the following assertion: If AnoBno is a scalar for any choice of the tensor Bno, then Ano has tensor character. For, by hypothesis, for any substitution,
AlvxBlvx = AnoBno.

But by an inversion of (9)

Bno

=

2xn 2xlv

2xo 2xlx

Blvx.

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195

This, inserted in the above equation, gives

fAlvx

−

2xn 2xlv

2xo 2xlx

AnopBlvx

=

0.

This can only be satisfied for arbitrary values of B′vx if the bracket vanishes. The result then follows by equation (11). This rule applies correspondingly to tensors of any rank and character, and the proof is analogous in all cases.
The rule may also be demonstrated in this form: If Bn and Co are any vectors, and if, for all values of these, the inner product AnoBnCo is a scalar, then Ano is a covariant tensor. This latter proposition also holds good even if only the more special assertion is correct, that with any choice of the four-vector Bn the inner product AnoBnBo is a scalar, if in addition it is known that Ano satisfies the condition of symmetry Ano = Aon. For by the method given above we prove the tensor character of (Ano + Aon), and from this the tensor character of Ano follows on account of symmetry. This also can be easily generalized to the case of covariant and contravariant tensors of any rank.
Finally, there follows from what has been proved, this law, which may also be generalized for any tensors: If for any choice of the four-vector Bo the quantities AnoBo form a tensor of the first rank, then Ano is a tensor of the second rank. For, if Cn is any four-vector, then on account of the tensor character of AnoBo, the inner product AnoBoCn is a scalar for any choice of the two four-vectors Bo and Cn. From which the proposition follows.

§ 8. Some Aspects of the Fundamental Tensor

The Covariant Fundamental Tensor.—In the invariant expression for the square of the linear element,

ds2 = gnodxndxo,

the part played by the dxn is that of a contravariant vector which may be chosen at will. Since further, gno =  gon, it follows from the considerations of the preceding paragraph that gno is a covariant tensor of the second rank. We call it the “fundamental tensor.” In what follows we deduce some properties of this tensor which, it is true, apply to any tensor of the second rank. But as the fundamental tensor plays a special part in our theory, which has its physical basis in the peculiar effects of gravitation, it so happens that the relations to be developed are of importance to us only in the case of the fundamental tensor.
The Contravariant Fundamental Tensor.—If in the determinant formed by the elements gno, we take the co-factor of each of the gno and divide it by the determinant g =  | gno |, we obtain certain quantities gno( =  gon) which, as we shall demonstrate, form a contravariant tensor.
By a known property of determinants

gnvgov = dno

(16)

where the symbol dno denotes 1 or 0, according as n =  o or n ≠ o.

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Instead of the above expression for ds2 we may thus write

gnv dovdxndxo

or, by (16)

gnvgoxgvxdxndxo.

But, by the multiplication rules of the preceding paragraphs, the quantities

dpv = gnvdxn

form a covariant four-vector, and in fact an arbitrary vector, since the dxn are arbitrary. By introducing this into our expression we obtain

ds2 = gvxdpvdpx.
Since this, with the arbitrary choice of the vector dξv, is a scalar, and gvx by its definition is symmetrical in the indices v and x, it follows from the results of the preceding paragraph that gvx is a contravariant tensor.
It further follows from (16) that dn is also a tensor, which we may call the mixed fundamental tensor.
The Determinant of the Fundamental Tensor.—By the rule for the multiplication of determinants

| gnagao | = | gna | # | gao |.

On the other hand

| gna ga=o | | d=no | 1.

It therefore follows that

| gno | # | gno | = 1

(17)

The Volume Scalar.—We seek first the law of transformation of the determinant g =  |gno|. In accordance with (11)

gl =

2xn 2xlv

2x 2xlx

gno

.

Hence, by a double application of the rule for the multiplication of determinants, it follows that

= g l

22= xxlvn . 22xxlxo . | gno |

2

2xn 2xlv

g,

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197

or

gl =

2xn 2xlv

g.

On the other hand, the law of transformation of the element of volume
dx = # dx1dx2dx3dx4

is, in accordance with the theorem of Jacobi,

dxl =

2xlv 2xn

dx.

By multiplication of the last two equations, we obtain

gldxl = g dx

(18).

Instead of g, we introduce in what follows the quantity -g, which is always real on account of the hyperbolic character of the space-time continuum. The invariant -g dx is
equal to the magnitude of the four-dimensional element of volume in the “local” system
of reference, as measured with rigid rods and clocks in the sense of the special theory of
relativity.
Note on the Character of the Space-time Continuum.—Our assumption that the special theory of relativity can always be applied to an infinitely small region, implies that ds2 can
always be expressed in accordance with (1) by means of real quantities dX1 . . . dX4. If we denote by dτ0 the “natural” element of volume dX1, dX2, dX3, dX4, then

dx0 = −g dx

(18a)

If -g were to vanish at a point of the four-dimensional continuum, it would mean that at this point an infinitely small “natural” volume would correspond to a finite volume in the co-ordinates. Let us assume that this is never the case. Then g cannot change sign. We will assume that, in the sense of the special theory of relativity, g always has a finite negative value. This is a hypothesis as to the physical nature of the continuum under consideration, and at the same time a convention as to the choice of co-ordinates.
But if –g is always finite and positive, it is natural to settle the choice of co-ordinates a posteriori in such a way that this quantity is always equal to unity. We shall see later that by such a restriction of the choice of co-ordinates it is possible to achieve an important simplification of the laws of nature.
In place of (18), we then have simply dτ′ =  dτ, from which, in view of Jacobi’s theorem, it follows that

2xlv 2xn

=1

(19)

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Thus, with this choice of co-ordinates, only substitutions for which the determinant is unity are permissible.
But it would be erroneous to believe that this step indicates a partial abandonment of the general postulate of relativity. We do not ask “What are the laws of nature which are co-variant in face of all substitutions for which the determinant is unity?” but our question is “What are the generally co-variant laws of nature?” It is not until we have formulated these that we simplify their expression by a particular choice of the system of reference.
The Formation of New Tensors by Means of the Fundamental Tensor.—Inner, outer, and mixed multiplication of a tensor by the fundamental tensor give tensors of different character and rank. For example,
An = guvAv, A = gnoAno.
The following forms may be specially noted:—
Ano = gnagobAab, Ano = gna gobAab
(the “complements” of covariant and contravariant tensors respectively), and
Bno = gno gabAab.
We call Bno the reduced tensor associated with Ano. Similarly,
Bno = gnogabAab.
It may be noted that gno is nothing more than the complement of gno, since
gnago= bgab gn= a dao gno.

§ 9. The Equation of the Geodetic Line. The Motion of a Particle

As the linear element ds is defined independently of the system of co-ordinates, the line drawn between two points P and P′ of the four-dimensional continuum in such a way that ∫ds is stationary—a geodetic line—has a meaning which also is independent of the choice of co-ordinates. Its equation is

d # Pl ds = 0

(20)

P

Carrying out the variation in the usual way, we obtain from this equation four differential equations which define the geodetic line; this operation will be inserted here for the sake of completeness. Let m be a function of the co-ordinates xo, and let this define a family of surfaces which intersect the required geodetic line as well as all the lines in immediate

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199

proximity to it which are drawn through the points P and P′. Any such line may then be supposed to be given by expressing its co-ordinates xo as functions of m. Let the symbol d indicate the transition from a point of the required geodetic to the point corresponding to the same m on a neighboring line. Then for (20) we may substitute

4 # m2 dwdm = 0
m1

w2

=

gno

dxn dm

dxo dm

(20a)

But since and

dw

=

1 w

*1 2

2gno 2xv

dxn dm

dxo dm

dxv

+

gno

dxn dm

d

edxo dm

o4,

deddxmoo

=

d dm

(dxo),

we obtain from (20a), after a partial integration,

# m2
m1

kv dxvdm

=

0,

where

kv

=

d dm

)gno w

dxn3 dm

−

1 2w

2gno 2xv

dxn dm

dxo dm

(20b)

Since the values of δxv are arbitrary, it follows from this that

kv = 0

(20c)

are the equations of the geodetic line. If ds does not vanish along the geodetic line we may choose the “length of the arc” s,
measured along the geodetic line, for the parameter m. Then w =  1, and in place of (20c) we obtain

gno

d2xn ds2

+

2gno 2xv

dxv ds

dxn ds

−

1 2

2gno 2xv

dxn ds

dxo ds

=

0

or, by a mere change of notation,

gav

d2xa ds2

+

[no,

v]

dxn dxo ds ds

=

0

(20d)

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where, following Christoffel, we have written

[no,

v]

=

1 2

f2gnv 2xo

+

2gov 2xn

−

2gnop 2xv

(21)

Finally, if we multiply (20d) by gvx (outer multiplication with respect to x, inner with respect to v), we obtain the equations of the geodetic line in the form

d2xx ds2

+

{no,

x}

dxn ds

dxo ds

=

0

(22)

where, following Christoffel, we have set

{no, x} = gxa [no, a]

(23)

§ 10. The Formation of Tensors by Differentiation

With the help of the equation of the geodetic line we can now easily deduce the laws by which new tensors can be formed from old by differentiation. By this means we are able for the first time to formulate generally covariant differential equations. We reach this goal by repeated application of the following simple law:—
If in our continuum a curve is given, the points of which are specified by the arcual distance s measured from a fixed point on the curve, and if, further, z is an invariant function of space, then dz/ds is also an invariant. The proof lies in this, that ds is an invariant as well as dz.
As

dz ds

=

2z 2xn

dxn ds

therefore

}

=

2z dxn

dxn ds

is also an invariant, and an invariant for all curves starting from a point of the continuum, that is, for any choice of the vector dxn. Hence it immediately follows that

An

=

2z 2xn

(24)

is a covariant four-vector—the “gradient” of z. According to our rule, the differential quotient

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201

|

=

d} ds

taken on a curve, is similarly an invariant. Inserting the value of ψ, we obtain in the first place

|

=

22z 2xn 2xo

dxn ds

dxo ds

+

2z 2xn

d2xn ds2

.

The existence of a tensor cannot be deduced from this forthwith. But if we may take the
curve along which we have differentiated to be a geodetic, we obtain on substitution for d 2xo/ds2 from (22),

|

=

f 22z 2xn 2xo

−

{no,

x}

2z p 2xx

dxn ds

dxv ds

.

Since we may interchange the order of the differentiations, and since by (23) and (21) {no, x} is symmetrical in n and o, it follows that the expression in brackets is symmetrical in n and o. Since a geodetic line can be drawn in any direction from a point of the continuum, and therefore dxn/ds is a four-vector with the ratio of its components arbitrary, it follows from the results of § 7 that

Ano

=

22z 2xn 2xo

−

{no,

x}

2z 2xx

(25)

is a covariant tensor of the second rank. We have therefore come to this result: from the covariant tensor of the first rank

An

=

2z 2xn

we can, by differentiation, form a covariant tensor of the second rank

Ano

=

2An 2xo

−

{no,

x}

Ax

(26)

We call the tensor Ano the “extension” (covariant derivative) of the tensor An. In the first place we can readily show that the operation leads to a tensor, even if the vector An cannot be represented as a gradient. To see this, we first observe that

2z } 2xn

is a covariant vector, if } and z are scalars. The sum of four such terms

Sn

=

} (1)

z2(1) 2xn

+

.

+

.

+

} (4)

2z(4) 2xn ,

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is also a covariant vector, if }(1), z(1) . . . }(4), z(4) are scalars. But it is clear that any covariant vector can be represented in the form Sn. For, if An is a vector whose components are any given functions of the xo, we have only to put (in terms of the selected system of co-ordinates)

= }(1) A= 1, z(1) x1, = }(2) A= 2, z(2) x2, = }(3) A= 3, z(3) x3, = }(4) A= 4, z(4) x4,

in order to ensure that Sn shall be equal to An. Therefore, in order to demonstrate that Ano is a tensor if any covariant vector is
inserted on the right-hand side for An, we only need show that this is so for the vector Sn. But for this latter purpose it is sufficient, as a glance at the right-hand side of (26) teaches us, to furnish the proof for the case

2z An = } 2xn .

Now the right-hand side of (25) multiplied by },

22z

2z

} 2xn 2xo - {no, x} } 2xx

is a tensor. Similarly
2} 2z 2xn 2xo
being the outer product of two vectors, is a tensor. By addition, there follows the tensor character of

2 2x

o

f

}

22xznp

-

{no,

x}

e

}

2z 2xx

o

.

As a glance at (26) will show, this completes the demonstration for the vector

2z } 2xn
and consequently, from what has already been proved, for any vector An. By means of the extension of the vector, we may easily define the “extension” of a
covariant tensor of any rank. This operation is a generalization of the extension of a vector. We restrict ourselves to the case of a tensor of the second rank, since this suffices to give a clear idea of the law of formation.

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203

As has already been observed, any covariant tensor of the second rank can be represented 5 as the sum of tensors of the type AnBo. It will therefore be sufficient to deduce the
expression for the extension of a tensor of this special type. By (26) the expressions

2An 2xv

-

{vn,

x}

Ax

,

2Bo 2xv

-

{vo,

x}

Bx

,

are tensors. On outer multiplication of the first by Bo and of the second by An, we obtain in each case a tensor of the third rank. By adding these, we have the tensor of the third rank

Anov

=

2Bno 2xv

−

{vn, x} Axo

−

{vo, x} Anx

(27)

where we have put Ano =  AnBo As the right-hand side of (27) is linear and homogeneous in the Ano and their first derivatives, this law of formation leads to a tensor, not only in the case of a tensor of the type AnBo but also in the case of a sum of such tensors, i.e. in the case of any covariant tensor of the second rank. We call Anov the extension of the tensor Ano.
It is clear that (26) and (24) concern only special cases of extension (the extension of the tensors of rank one and zero respectively).
In general, all special laws of formation of tensors are included in (27) in combination with the multiplication of tensors.

§ 11. Some Cases of Special Importance

The Fundamental Tensor.—We will first prove some lemmas which will be useful hereafter. By the rule for the differentiation of determinants

dg = gnogdgno = −gnogdgno

(28)

The last member is obtained from the last but one, if we bear in mind that gnognlo = dnnl, so that gnogno =  4, and consequently

gnodgno + gnodgno = 0.

5 By outer multiplication of the vector with arbitrary components A11, A12, A13, A14 by the vector with components 1, 0, 0, 0, we produce a tensor with components
A11 A12 A13 A14 0000 0000 0 0 0 0.
By the addition of four tensors of this type, we obtain the tensor Ano with any assigned components.

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From (28), it follows that

1 −g

2 −g 2xv

=

1 2

2 log (−g) 2xv

=

1 2

gno

2gno 2xv

=

1 2

gno

2gno 2xv

.

(29)

Further, from gnvgov = dno it follows on differentiation that

4 gnvdgov = −govdgnv

gnv

2gov 2xm

= −gov

2gnv 2xm

(30)

From these, by mixed multiplication by gvx and gom respectively, and a change of notation for the indices, we have

4 dgno = − gnagobdgab

2gno 2xv

= − gnagob

2gab xv

(31)

and

4 dgno = −gna gobdgab

2gno 2xv

= −gna

g ob

2gab 2xv

(32)

The relation (31) admits of a transformation, of which we also have frequently to make use: From (21)

2gab 2xv

=

[av,

b]

+

[bv, a]

(33)

Inserting this in the second formula of (31), we obtain, in view of (23)

2gno 2xv

=

−gnx

{xv,

o}

−

g ox

{xv,

n}

(34)

Substituting the right-hand side of (34) in (29), we have

1 −g

2 −g 2xv

=

{nv, n}

(29a)

The “Divergence” of a Contravariant Vector.—If we take the inner product of (26) by the contravariant fundamental tensor gno, the right-hand side, after a transformation of the first term, assumes the form

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205

2 2xo

`gnoAnj

−

An

2gno 2xo

−

1 2

g

xa

f

2gna 2xo

+

2goa 2xn

−

2gnop 2xa

gnoAx.

In accordance with (31) and (29), the last term of this expression may be written

1 2

2g xo 2xo

Ax

+

1 2

2g xn 2xn

Ax

+

1 −g

2 −g 2xa

gnoAx.

As the symbols of the indices of summation are immaterial, the first two terms of this expression cancel the second of the one above. If we then write gno An =  Ao, so that Ao like An is an arbitrary vector, we finally obtain

U=

1 −g

2` 2xo

−g Aoj

(35)

This scalar is the divergence of the contravariant, vector Ao. The “Curl” of a Covariant Vector.—The second term in (26) is symmetrical in the indi-
ces n and o. Therefore Ano - Aon is a particularly simply constructed antisymmetrical tensor. We obtain

Bno

=

2An 2xo

−

2Ao 2xn

(36)

Antisymmetrical Extension of a Six-vector.—Applying (27) to an antisymmetrical tensor of the second rank Ano forming in addition the two equations which arise through cyclic permutations of the indices, and adding these three equations, we obtain the tensor of the third rank

Bnov

=

Anov

+

Aovn

+

Avno

+

2Ano 2xv

+

2Aov 2xn

+

2Avn 2xo

(37)

which it is easy to prove is antisymmetrical. The Divergence of a Six-vector.—Taking the mixed product of (27) by gnagob, we also
obtain a tensor. The first term on the right-hand side of (27) may be written in the form

2 2xv

b g na g obAnol

-

gna

2g ob 2xv

Ano

-

g ob

2gna 2xv

Ano.

If we write Avab for gnagobAnov and Aab for gnagobAno, and in the transformed first term replace

2g ob 2xv

and

2gna 2xv

by their values as given by (34), there results from the right-hand side of (27) an expression consisting of seven terms, of which four cancel, and there remains

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Avab

=

2Aab 2xv

+

{vc, a} Acb

+

{vc, b} Aac

(38)

This is the expression for the extension of a contravariant tensor of the second rank, and corresponding expressions for the extension of contravariant tensors of higher and lower rank may also be formed.
We note that in an analogous way we may also form the extension of a mixed tensor:—

A nav

=

2A

a n

2xv

−

{vn,

x} Axa

+

{vx, a} Anx

(39)

On contracting (38) with respect to the indices b and v (inner multiplication by dbv), we obtain the vector

Aa

=

2Aab 2xb

+

{bc, b} Aac

+

{bc,

a}

Acb.

On account of the symmetry of {bc, a} with respect to the indices b and c, the third term on the right-hand side vanishes, if Aab is, as we will assume, an antisymmetrical tensor. The second term allows itself to be transformed in accordance with (29a). Thus we obtain

Aa =

1 2` −g Aabj −g 2xb

(40)

This is the expression for the divergence of a contravariant six-vector. The Divergence of a Mixed Tensor of the Second Rank.—Contracting (39) with respect
to the indices a and v, and taking (29a) into consideration, we obtain

2b −g An =

−g Anvl 2xv − {vn, x}

−g Axv

(41)

If we introduce the contravariant tensor Atv = gtxAxv in the last term, it assumes the form

- [vn, t] -g Atv.

If, further, the tensor Atv is symmetrical, this reduces to

-

1 2

-g

2gtv 2xn

Atv.

Had we introduced, instead of Atv, the covariant tensor Aρv =  gta gvbAab, which is also symmetrical, the last term, by virtue of (31), would assume the form

1 2

-g

2gtv 2xn

Atv

.

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207

In the case of symmetry in question, (41) may therefore be replaced by the two forms

2b −g An =

−g Anvl 2xv

−

1 2

2gtv 2xn

−g Atv

(41a)

2b −g An =

−g A 2xv

nvl

−

1 2

2gtv 2xn

−g Atv

(41b)

which we have to employ later on.

§ 12. The Riemann-Christoffel Tensor

We now seek the tensor which can be obtained from the fundamental tensor alone, by differentiation. At first sight the solution seems obvious. We place the fundamental tensor of the gno in (27) instead of any given tensor Ano, and thus have a new tensor, namely, the extension of the fundamental tensor. But we easily convince ourselves that this extension vanishes identically. We reach our goal, however, in the following way. In (27) place

Ano

=

2An 2xo

−

{no,

t}

At,

i.e. the extension of the four-vector An. Then (with a somewhat different naming of the indices) we get the tensor of the third rank

Anvx

=

22An 2xv 2xx

−

{nv,

t}

2At 2xx

−

{nx,

t}

2At 2xv

−

{vx,

t}

2An 2xt

+

=−

2 2xx

{nv,

t}

+

{nx,

a}{av,

t}

+

{vx,

a}{an,

t}GAt

.

This expression suggests forming the tensor Anvx - Anxv. For, if we do so, the following terms of the expression for Anvx cancel those of Anxv, the first, the fourth, and the member corresponding to the last term in square brackets; because all these are symmetrical in v and x. The same holds good for the sum of the second and third terms. Thus we obtain

Anvx

−

Anxv

=

Bt nvx

At

(42)

where

Bt nvx

=

2 2xx

{nv,

t}

+

2 2xv

{nx,

t}

−

{nv, a}{ax,

t}

+

{nx,

a}{av,

t}

(43)

The essential feature of the result is that on the right side of (42) the Aρ occur alone, without their derivatives. From the tensor character of Anvx - Anxv in conjunction with

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the

fact

that

At

is

an

arbitrary

vector,

it

follows,

by

reason

of

§

7,

that

Bt nvx

is

a

tensor

(the

Riemann-Christoffel tensor).

The mathematical importance of this tensor is as follows: If the continuum is of such

a nature that there is a co-ordinate system with reference to which the gno constants, then

all

the

Bt nvx

vanish.

If

we

choose

any

new

system

of

co-ordinates

in

place

of

the

origi-

nal ones, the gno referred thereto will not be constants, but in consequence of its tensor

nature,

the

transformed

components

of

Bt nvx

will

still

vanish

in

the

new

system.

Thus

the

vanishing of the Riemann tensor is a necessary condition that, by an appropriate choice

of the system of reference, the gno may be constants. In our problem this corresponds to the case in which,6 with a suitable choice of the system of reference, the special theory of

relativity holds good for a finite region of the continuum.

Contracting (43) with respect to the indices x and t we obtain the covariant tensor of

second rank

where

Gno R no Sno

= = =

−B222nt2loxo2xtnga=2{xn−Roogn, ao−+} {+nSno{on, aa},2bl}o{2ogxba,−ag}_`abbbbbbbbbbbbbbbbbbbb

(44)

Note on the Choice of Co-ordinates.—It has already been observed in § 8, in connection with equation (18a), that the choice of co-ordinates may with advantage be made so that -g =  1. A glance at the equations obtained in the last two sections shows that by such a choice the laws of formation of tensors undergo an important simplification. This applies particularly to Gno, the tensor just developed, which plays a fundamental part in the theory to be set forth. For this specialization of the choice of co-ordinates brings about the vanishing of Sno, so that the tensor Gno reduces to Rno.
On this account I shall hereafter give all relations in the simplified form which this specialization of the choice of coordinates brings with it. It will then be an easy matter to revert to the generally covariant equations, if this seems desirable in a special case.

C. THEORY OF THE GRAVITATIONAL FIELD

§ 13. Equations of Motion of a Material Point in the Gravitational Field. Expression for the Field-components of Gravitation
A freely movable body not subjected to external forces moves, according to the special theory of relativity, in a straight line and uniformly. This is also the case, according to the general theory of relativity, for a part of four-dimensional space in which the system of co-ordinates K0, may be, and is, so chosen that they have the special constant values given in (4).

6 The mathematicians have proved that this is also a sufficient condition.

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209

If we consider precisely this movement from any chosen system of co-ordinates K1, the body, observed from K1, moves, according to the considerations in § 2, in a gravitational field. The law of motion with respect to K1 results without difficulty from the following consideration. With respect to K0 the law of motion corresponds to a four-dimensional straight line, i.e. to a geodetic line. Now since the geodetic line is defined independently
of the system of reference, its equations will also be the equation of motion of the material
point with respect to K1. If we set

C

x no

=−

{no,

x}

(45)

the equation of the motion of the point with respect to K1 becomes

d2xx ds2

=

C nx o

dxn ds

dxo ds

(46)

We now make the assumption, which readily suggests itself, that this covariant system of

equations also defines the motion of the point in the gravitational field in the case when

there is no system of reference K0, with respect to which the special theory of relativity holds good in a finite region. We have all the more justification for this assumption as (46)

contains only first derivatives of the gno, between which even in the special case of the

existence of K0, no relations subsist.7

If

the

C

x no

vanish,

then

the

point

moves

uniformly

in

a

straight

line.

These

quantities

therefore condition the deviation of the motion from uniformity. They are the compo-

nents of the gravitational field.

§ 14. The Field Equations of Gravitation in the Absence of Matter

We make a distinction hereafter between “gravitational field” and “matter” in this way, that

we denote everything but the gravitational field as “matter.” Our use of the word therefore

includes not only matter in the ordinary sense, but the electromagnetic field as well.

Our next task is to find the field equations of gravitation in the absence of matter.

Here we again apply the method employed in the preceding paragraph in formulating the

equations of motion of the material point. A special case in which the required equations

must in any case be satisfied is that of the special theory of relativity, in which the gno have certain constant values. Let this be the case in a certain finite space in relation to a definite

system of co-ordinates K0. Relatively to this system all the components of the Riemann tensor Bntvx, defined in (43), vanish. For the space under consideration they then vanish, also in any other system of co-ordinates.

Thus the required equations of the matter-free gravitational field must in any case be

satisfied

if

all

Bt nvx

vanish.

But

this

condition

goes

too

far.

For

it

is

clear

that,

e.g.,

the

grav-

itational field generated by a material point in its environment certainly cannot be “trans-

formed away” by any choice of the system of co-ordinates, i.e. it cannot be transformed to

the case of constant gno.

7 It is only between the second (and first) derivatives that, by § 12, the relations Bntvx =  0 subsist.

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This prompts us to require for the matter-free gravitational field that the symmetrical tensor Gno, derived from the tensor Bntox, shall vanish. Thus we obtain ten equations for the ten quantities gno, which are satisfied in the special case of the vanishing of all Bntox. With the choice which we have made of a system of co-ordinates, and taking (44) into consideration, the equations for the matter-free field are

4 2C

a no

2xa

+

C

a nb

C

b oa

=

0

(47)

−g = 1

It must be pointed out that there is only a minimum of arbitrariness in the choice of these equations. For besides Gno there is no tensor of second rank which is formed from the gno and its derivatives, contains no derivations higher than second, and is linear in these derivatives.8
These equations, which proceed, by the method of pure mathematics, from the requirement of the general theory of relativity, give us, in combination with the equations of motion (46), to a first approximation Newton’s law of attraction, and to a second approximation the explanation of the motion of the perihelion of the planet Mercury discovered by Leverrier (as it remains after corrections for perturbation have been made). These facts must, in my opinion, be taken as a convincing proof of the correctness of the theory.

§ 15. The Hamiltonian Function for the Gravitational Field. Laws of Momentum and Energy

To show that the field equations correspond to the laws of momentum and energy, it is most convenient to write them in the following Hamiltonian form:—

4 d # Hdx = 0

H

=

g

no

C nab

C

b oa

−g = 1

(47a)

where, on the boundary of the finite four-dimensional region of integration which we

have in view, the variations vanish.

We first have to show that the form (47a) is equivalent to the equations (47). For this

purpose

we

regard

H

as

a

function

of

the

gno

and

the

g

no v

(=

2gno/2xv).

Then in the first place

8 Properly speaking, this can be affirmed only of the tensor Gno + mgnogabGab ,
where m is a constant. If, however, we set this tensor =  0, we come back again to the equations Gno =  0.

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211

dH

=

C

a nb

C oba

dg

no

+

2g

no

C

a nb

dC

b oa

=

−

C

a nb

C

b oa

dg

no

+

2C

a nb

d

`gno

C

obaj.

But

> H d

`gno

C

obaj

=

−

1 2

d

gno

g

bm

f22gxoam

+

2gam 2xo

−

22gxamop

.

The terms arising from the last two terms in round brackets are of different sign, and result from each other (since the denomination of the summation indices is immaterial) through interchange of the indices n and b. They cancel each other in the expression for dH, because they are multiplied by the quantity Cnab, which is symmetrical with respect to the indices n and b. Thus there remains only the first term in round brackets to be considered, so that, taking (31) into account, we obtain

dH

= −Cnab

C

b oa

dgno

+

C

a nb

dganb.

Thus

4 2H
2gno

=−

C

a nb

C

b oa

2H

2g

no v

=

C nvo

(48)

Carrying out the variation in (47a), we get in the first place

2 2xa

e

2H 2gano

o

−

2H 2gno

=

0,

which, on account of (48), agrees with (47), as was to be proved. If we multiply (47b) by gvno, then because

2g

no v

2xa

=

2g

no a

2xv

and, consequently,

g

no v

2 2xa

e22gHanoo

=

2 2xa

e

g

no v

2H 2gano

o

−

2H

2g

no a

2gnao 2xv

,

we obtain the equation

(47b)

2 2xa

e

g

no v

2H 2gano

o

−

2H 2xv

=

0

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or9

2t

a v

2xa

−2lt

a v

= =

0

g

no 2H v 2gano

−

d va H_`abbbbbbbbbbbb

(49)

where, on account of (48), the second equation of (47), and (34)

ltva

=

g 1
2

d

a v

no

C

m nb

C obm

−

g

no

C

a nb

C

b ov

(50)

It

is

to

be

noticed

that

t

a v

is

not

a

tensor;

on

the

other

hand

(49)

applies

to

all

systems

of co-ordinates for which -g =  1. This equation expresses the law of conservation of

momentum and of energy for the gravitational field. Actually the integration of this equa-

tion over a three-dimensional volume V yields the four equations

d dx4

#

t4vdV

=

#

`lt

1 v

+

mt2v

+

nt

3vjds

(49a)

where l, m, n denote the direction-cosines of direction of the inward drawn normal at the

element dS of the bounding surface (in the sense of Euclidean geometry). We recognize

in

this

the

expression

of

the

laws

of

conservation

in

their

usual

form.

The

quantities

t

a v

we

call the “energy components” of the gravitational field.

I will now give equations (47) in a third form, which is particularly useful for a vivid grasp of our subject. By multiplication of the field equations (47) by gov these are obtained

in the “mixed” form. Note that

gov

2C

a no

2xa

=

2 2xa

`gov

Cnaoj

−

2gov 2xa

C nao

,

which quantity, by reason of (34), is equal to

2 2xa

`

gov

C

naoj

-

g

ob

C

v ab

C

a no

-

g

vb

C

o ba

C

a no

,

or (with different symbols for the summation indices)

2 2xa

b g vb

Cnabl

-

gcd

C

v cb

C

b dn

-

gov

C

a nb

Coba.

The third term of this expression cancels with the one arising from the second term of the field equations (47); using relation (50), the second term may be written

9 The reason for the introduction of the factor −2l will be apparent later.

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213

l

`t

v n

-

1 2

d

v n

tj,

where t =taa. Thus instead of equations (47) we obtain

4 2
2x

b
a

gab

C

nabl

=

−l`t

v n

−

1 2

d

v n

tj

(51)

−g = 1

§ 16. The General Form of the Field Equations of Gravitation

The field equations for matter-free space formulated in § 15 are to be compared with the field equation

d2z = 0

of Newton’s theory. We require the equation corresponding to Poisson’s equation

d2 z = 4rlt,

where t denotes the density of matter.

The special theory of relativity has led to the conclusion that inert mass is nothing

more or less than energy, which finds its complete mathematical expression in a symmet-

rical tensor of second rank, the energy-tensor. Thus in the general theory of relativity we must introduce a corresponding energy-tensor of matter Tva, which, like the energy-com-

ponents tv [equations (49) and (50)] of the gravitational field, will have mixed character, but will pertain to a symmetrical covariant tensor.10

The system of equation (51) shows how this energy-tensor (corresponding to the

density t in Poisson’s equation) is to be introduced into the field equations of gravi-

tation. For if we consider a complete system (e.g. the solar system), the total mass of

the system, and therefore its total gravitating action as well, will depend on the total

energy of the system, and therefore on the ponderable energy together with the gravi-

tational energy. This will allow itself to be expressed by introducing into (51), in place

of

the

energ

y-components

of

the

gravitational

field

alone,

the

sums

t

v n

+

T

v n

of

the

ener-

gy-components of matter and of gravitational field. Thus instead of (51) we obtain the

tensor equation

4 2
2xa

b

g

vb

T

nabl

=

−

l:`t

v n

+

T

nvj

−

1 2

d

v n

(t

+

T)D,

(52)

−g = 1

10

g

ax

T

a v

= Tvx

and

g

vb

T

a v

= Tab

are

to

be

symmetrical

tensors.

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ENGLISH TRANSLATION OF EINSTEIN’S PAPER

where

we

have

set

T

=

T

n n

(Laue’s

scalar).

These

are

the

required

general

field

equations

of

gravitation in mixed form. Working back from these, we have in place of (47)

4 2
2xa

C

a no

+

C

a nb

C oba

= −l`Tno

−

1 2

gnoTj,

(53)

−g = 1

It must be admitted that this introduction of the energy-tensor of matter is not justified by the relativity postulate alone. For this reason we have here deduced it from the requirement that the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy. But the strongest reason for the choice of these equations lies in their consequence, that the equations of conservation of momentum and energy, corresponding exactly to equations (49) and (49a), hold good for the components of the total energy. This will be shown in § 17.

§ 17. The Laws of Conservation in the General Case

Equation (52) may readily be transformed so that the second term on the right-hand

side vanishes. Contract (52) with respect to the indices n and v, and after multiplying the

resulting

equation

by

1 2

d

nv,

subtract

it

from

equation

(52).

This

gives

2 2xa

b g vb

C

a nb

−

1 2

d

v n

gmb

Cmabl

=

−l

`t

v n

+

Tnvj.

(52a)

On this equation we perform the operation ∂/∂xv. We have

> H 22
2xa 2xv

`

g

v

C

banj

=

−

1 2

22 2xa 2xv

gvbg

am

f2gnm 2xb

+

2gbm 2xn

−

2gnbp 2xm

.

The first and third terms of the round brackets yield contributions which cancel one another, as may be seen by interchanging, in the contribution of the third term, the summation indices a and v on the one hand, and b and m on the other. The second term may be re-modelled by (31), so that we have

22 2xa 2xv

bgvb Cnabl

=

1 2

23gab 2xa 2xb 2xn

(54)

The second term on the left-hand side of (52a) yields in the first place

-

1 2

22 2xa 2xn

b

gmb

C

mabl

or

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215

> H 1 22
4 2xa 2xn

g

mb

g

ad

f

2gdm 2xb

+

2gdb 2xm

−

2gmbp 2xd

.

With the choice of co-ordinates which we have made, the term deriving from the last term in round brackets disappears by reason of (29). The other two may be combined, and together, by (31), they give

-

1 2

2x

23gab a 2xb 2xn

,

so that in consideration of (54), we have the identity

22 2xa 2xv

bgtb Cnb

-

1 2

d

v n

gmb

Cmabl

/

0

(55)

From (55) and (52a), it follows that

2`t

v n

+

T

nvj

2xv

=

0

(56)

Thus it results from our field equations of gravitation that the laws of conservation of momentum and energy are satisfied. This may be seen most easily from the consideration which leads to equation (49a); except that here, instead of the energy components tv of the gravitational field, we have to introduce the totality of the energy components of matter and gravitational field.

§ 18. The Laws of Momentum and Energy for Matter, as a Consequence of the Field Equations

Multiplying (53) by ∂gno/∂xv, we obtain, by the method adopted in § 15, in view of the vanishing of

gno

2gno 2xv

,

the equation

2tva 2xa

+

1 2

2gno 2xv

Tno

=

0,

or, in view of (56),

2T

a v

2xa

+

1 2

2gno 2xv

Tno

=

0

(57)

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Comparison with (41b) shows that with the choice of system of co-ordinates which we have made, this equation predicates nothing more or less than the vanishing of divergence of the material energy-tensor. Physically, the occurrence of the second term on the left-hand side shows that laws of conservation of momentum and energy do not apply in the strict sense for matter alone, or else that they apply only when the gno are constant, i.e. when the field intensities of gravitation vanish. This second term is an expression for momentum, and for energy, as transferred per unit of volume and time from the gravitational field to matter. This is brought out still more clearly by re-writing (57) in the sense of (41) as

2T

a v

2xa

=

−C

T b a
av b

(57a)

The right side expresses the energetic effect of the gravitational field on matter.
Thus the field equations of gravitation contain four conditions which govern the
course of material phenomena. They give the equations of material phenomena com-
pletely, if the latter is capable of being characterized by four differential equations independent of one another.11

D. MATERIAL PHENOMENA

The mathematical aids developed in part B enable us forthwith to generalize the physical laws of matter (hydrodynamics, Maxwell’s electrodynamics), as they are formulated in the special theory of relativity, so that they will fit in with the general theory of relativity. When this is done, the general principle of relativity does not indeed afford us a further limitation of possibilities; but it makes us acquainted with the influence of the gravitational field on all processes, without our having to introduce any new hypothesis whatever.
Hence it comes about that it is not necessary to introduce definite assumptions as to the physical nature of matter (in the narrower sense). In particular it may remain an open question whether the theory of the electromagnetic field in conjunction with that of the gravitational field furnishes a sufficient basis for the theory of matter or not. The general postulate of relativity is unable on principle to tell us anything about this. It must remain to be seen, during the working out of the theory, whether electromagnetics and the doctrine of gravitation are able in collaboration to perform what the former by itself is unable to do.

§ 19. Euler’s Equations for a Frictionless Adiabatic Fluid

Let p and t be two scalars, the former of which we call the “pressure,” the latter the “density” of a fluid; and let an equation subsist between them. Let the contravariant symmetrical tensor

11 On this question cf. H. Hilbert, Nachr. d. K. Gesellsch. d. Wiss. zu Göttingen, Math.-phys. Klasse, 1915, p. 3.

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217

Tab

=

−gabp

+

t

dxa ds

dxb ds

be the contravariant energy-tensor of the fluid. To it belongs the covariant tensor

Tno

= −gnop

+

gna gnb

dxa ds

dxb ds

t,

(58a)

as well as the mixed tensor 12

T

a v

= −dvap

+

gvb

dxb ds

dxa ds

t

(58b)

Inserting the right-hand side of (58b) in (57a), we obtain the Eulerian hydrodynamical equations of the general theory of relativity. They give, in theory, a complete solution of the problem of motion, since the four equations (57a), together with the given equation between p and t, and the equation

gab

dxa ds

dxb ds

=

1,

are sufficient, gab being given, to define the six unknowns

p,

t,

dx1 ds

,

dx2 ds

,

dx3 ds

,

dx4 ds

.

If the gno are also unknown, the equations (53) are brought in. These are eleven equations for defining the ten functions gno, so that these functions appear over-defined. We must remember, however, that the equations (57a) are already contained in the equations (53),
so that the latter represent only seven independent equations. There is good reason for
this lack of definition, in that the wide freedom of the choice of co-ordinates causes the
problem to remain mathematically undefined to such a degree that three of the functions of space may be chosen at will.13

§ 20. Maxwell’s Electromagnetic Field Equations for Free Space
Let zo be the components of a covariant vector—the electromagnetic potential vector. From them we form, in accordance with (36), the components Ftv of the covariant six-vector of the electromagnetic field, in accordance with the system of equations

12 For an observer using a system of reference in the sense of the special theory of relativity for an infinitely small region, and moving with it, the density of energy T44 equals t - p. This gives the definition of t. Thus t is not constant for an incompressible fluid. 13 On the abandonment of the choice of co-ordinates with g =  –1, there remain four functions of space with
liberty of choice, corresponding to the four arbitrary functions at our disposal in the choice of co-ordinates.

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Ftv

=

2zt 2xv

−

2zv 2xt

(59)

It follows from (59) that the system of equations

2Ftv 2xx

+

2Fvx 2xt

+

2Fxt 2xv

=

0

(60)

is satisfied, its left side being, by (37), an antisymmetrical tensor of the third rank. System (60) thus contains essentially four equations which are written out as follows:—

2F23 2x4 2F34 2x1 2F41 2x2 2F12 2x3

+ + + +

2F34 2x2 2F41 2x3 2F12 2x4 2F23 2x1

+ + + +

2F42 2x3 2F13 2x4 2F24 2x1 2F31 2x2

= = = =

0000_`abbbbbbbbbbbbbbbbbbbbbbbbbb

(60a)

This system corresponds to the second of Maxwell’s systems of equations. We recognize this at once by setting

= F23 H= x , F14 Ex

4 = F31 H= y , F24 E y

(61)

= F12 H= z , F34 Ez

Then in place of (60a) we may set, in the usual notation of three-dimensional vector analysis,

−

2H 2t

=

curl

E4

div H = 0

(60b)

We obtain Maxwell’s first system by generalizing the form given by Minkowski. We introduce the contravariant six-vector associated with Fab

Fno

=

g

nag

F ob ab

(62)

and also the contravariant vector Jn of the density of the electric current. Then, taking (40) into consideration, the following equations will be invariant for any substitution whose invariant is unity (in agreement with the chosen coordinates) :—

2 2xo

Fno

=

Jn

(63)

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219

Let

4 F23 = Hlx, F14 = −Elx
F31 = Hly, F24 = −Ely

(64)

F12 = Hlz, F34 = −Elz

which quantities are equal to the quantities Hx. . . Ez in the special case of the restricted theory of relativity ; and in addition

=J1 jx,=J2 jy,=J3 jz,=J4 t,

we obtain in place of (63)

2El 2t

+

j

=

curl

Hl4

div El = t

(63a)

The equations (60), (62), and (63) thus form the generalization of Maxwell’s field equations for free space, with the convention which we have established with respect to the choice of co-ordinates.
The Energy-components of the Electromagnetic Field.—We form the inner product

lv = FvnJn

(65)

By (61) its components, written in the three-dimensional manner, are

l1 . . l4

= =

tEx + .. .. − ( jE)

[

j

.

H] . .

x_`abbbbbbbbbbbbbb

(65a)

kv is a covariant vector the components of which are equal to the negative momentum, or, respectively, the energy, which is transferred from the electric masses to the

electromagnetic field per unit of time and volume. If the electric masses are free, that

is, under the sole influence of the electromagnetic field, the covariant vector kv will

vanish.

To

obtain

the

energy-components

T

o v

of

the

electromagnetic

field,

we

need

only

give

to equation kv =  0 the form of equation (57). From (63) and (65) we have in the first

place

lv

=

Fvn

2Fno 2xo

=

2 2xo

(FvnFno)

−

Fnt

2Fvn 2xo

.

The second term of the right-hand side, by reason of (60), permits the transformation

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Fno

2Fvn 2xo

=

−

1 2

Fno

2Fno 2xv

=−

1 2

gnagobFab

2Fno 2xv

,

which latter expression may, for reasons of symmetry, also be written in the form

−

1 4

>gna

g

F ob ab

2Fno 2xv

+

gnagob

2Fab 2xv

FnoH.

But for this we may set

−

1 4

2 2xv

bg g na obFabFnol

+

1 4

Fab

Fno

2 2xv

`gnagobj.

The first of these terms is written more briefly

-

1 4

2 2xv

`F

no

Fnoj;

the second, after the differentiation is carried out, and after some reduction, results in

-

1 2

F

nx

Fno

got

2gvx 2xv

.

Taking all three terms together we obtain the relation

lv

=

2T

o v

2xo

−

1 2

g xn

2gno 2xv

T

o x

(66)

where

T

o v

= −FvaFoa

+

1 4

d

o v

Fab

F

ab.

Equation (66), if kv vanishes, is, on account of (30), equivalent to (57) or (57a) respec-

tively.

Therefore

the

T

o v

are

the

energy-components

of

the

electromagnetic

field.

With

the

help of (61) and (64), it is easy to show that these energy-components of the electromag-

netic field in the case of the special theory of relativity give the well-known Maxwell-Poy-

nting expressions.

We have now deduced the general laws which are satisfied by the gravitational field

and matter, by consistently using a system of co-ordinates for which −g =1. We have

thereby achieved a considerable simplification of formulae and calculations, without fail-

ing to comply with the requirement of general covariance; for we have drawn our equa-

tions from generally covariant equations by specializing the system of co-ordinates.

Still the question is not without a formal interest, whether with a correspondingly

generalized definition of the energy-components of gravitational field and matter, even

without specializing the system of co-ordinates, it is possible to formulate laws of conser-

vation in the form of equation (56), and field equations of gravitation of the same nature

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221

as (52) or (52a), in such a manner that on the left we have a divergence (in the ordinary sense), and on the right the sum of the energy-components of matter and gravitation. I have found that in both cases this is actually so. But I do not think that the communication of my somewhat extensive reflections on this subject would be worth while, because after all they do not give us anything that is materially new.

E

§ 21. Newton’s Theory as a First Approximation
As has already been mentioned more than once, the special theory of relativity as a special case of the general theory is characterized by the gno having the constant values (4). From what has already been said, this means complete neglect of the effects of gravitation. We arrive at a closer approximation to reality by considering the case where the gno differ from the values of (4) by quantities which are small compared with 1, and neglecting small quantities of second and higher order. (First point of view of approximation.)
It is further to be assumed that in the space-time territory under consideration the gno at spatial infinity, with a suitable choice of co-ordinates, tend toward the values (4) ; i.e. we are considering gravitational fields which may be regarded as generated exclusively by matter in the finite region.
It might be thought that these approximations must lead us to Newton’s theory. But to that end we still need to approximate the fundamental equations from a second point of view. We give our attention to the motion of a material point in accordance with the equations (16). In the case of the special theory of relativity the components

dx1 ds

,

dx2 ds

,

dx3 ds

may take on any values. This signifies that any velocity

v=

f

dx dx

2
1p
4

+

fddxx

2 4

2
p

+

f

dx dx

2
3p
4

may occur, which is less than the velocity of light in vacuo. If we restrict ourselves to the case which almost exclusively offers itself to our experience, of v being small as compared with the velocity of light, this denotes that the components

dx1 ds

,

dx2 ds

,

dx3 ds

are to be treated as small quantities, while dx4/ds, to the second order of small quantities, is equal to one. (Second point of view of approximation.)
Now we remark that from the first point of view of approximation the magnitudes Cnxo are all small magnitudes of at least the first order. A glance at (46) thus shows that in this equation, from the second point of view of approximation, we have to consider only

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terms for which n =  o =  4. Restricting ourselves to terms of lowest order we first obtain in place of (46) the equations

d2xx dt2

=

C

x 44

where we have set ds =  dx4 =  dt; or with restriction to terms which from the first point of view of approximation are of first order:—

d2xx dt2

=

[44,

x]

(x = 1, 2, 3)

d2x4 dt2

=

−[44,

4]

.

If in addition we suppose the gravitational field to be a quasi-static field, by confining ourselves to the case where the motion of the matter generating the gravitational field is but slow (in comparison with the velocity of the propagation of light), we may neglect on the right-hand side differentiations with respect to the time in comparison with those with respect to the space co-ordinates, so that we have

d2xx dt2

=−

1 2

2g44 2xx

(x = 1, 2, 3)

(67)

This is the equation of motion of the material point according to Newton’s theory, in

which

1 2

g

44

plays

the

part

of

the

gravitational

potential.

What

is

remarkable

in

this

result

is

that the component g44 of the fundamental tensor alone defines, to a first approximation,

the motion of the material point.

We now turn to the field equations (53). Here we have to take into consideration that

the energy-tensor of “matter “is almost exclusively defined by the density of matter in the

narrower sense, i.e. by the second term of the right-hand side of (58) [or, respectively,

(58a) or (58b)]. If we form the approximation in question, all the components vanish with

the one exception of T44 =  t =  T. On the left-hand side of (53) the second term is a small quantity of second order; the first yields, to the approximation in question,

2 2x

1

[no,

1]

+

2 2x2

[no,

2]

+

2 2x

3

[no,

3]

−

2 2x

4

[no,

4]

.

For n =  o =  4, this gives, with the omission of terms differentiated with respect to time,

−

1 2

f222xg1244

+

2 2 g 44

2x

2 2

+

222xg3244p

=

−

1 2

d 2 g 44 .

The last of equations (53) thus yields

d2g44 = lt

(68)

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223

The equations (67) and (68) together are equivalent to Newton’s law of gravitation. By (67) and (68) the expression for the gravitational potential becomes

-

l 8r

#

tdx r

(68a)

while Newton’s theory, with the unit of time which we have chosen, gives

-

K c2

#

tdx r

in which K denotes the constant 6.7 × 10–8, usually called the constant of gravitation. By comparison we obtain

l

=

8rK c2

=

1.87

#

10−27

(69)

§ 22. Behaviour of Rods and Clocks in the Static Gravitational Field. Bending of Light-rays. Motion of the Perihelion of a Planetary Orbit

To arrive at Newton’s theory as a first approximation we had to calculate only one component, g44, of the ten gno of the gravitational field, since this component alone enters into the first approximation, (67), of the equation for the motion of the material point in the gravitational field. From this, however, it is already apparent that other components of the gno must differ from the values given in (4) by small quantities of the first order. This is required by the condition g =  –1.
For a field-producing point mass at the origin of co-ordinates, we obtain, to the first approximation, the radially symmetrical solution

gtv

= − dtv

−

a

x

tx r3

v

gt4 = g4t = 0

g44

=

1

−

a r

(t, v (t =

= 1,

1, 2, 2, 3)

3)_`abbbbbbbbbbbbbb

(70)

where dtv is 1 or 0, respectively, accordingly as t =  v or t  v, and r is the quantity 

+

x

2 1

+

x22

+

x32.

On

account

of

(68a)

a

=

lM 4r

,

(70a)

if M denotes the field-producing mass. It is easy to verify that the field equations (outside the mass) are satisfied to the first order of small quantities.
We now examine the influence exerted by the field of the mass M upon the metrical properties of space. The relation

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ds2 = gnodxndxo.

always holds between the “locally” (§ 4) measured lengths and times ds on the one hand,
and the differences of co-ordinates dxo on the other hand.
For a unit-measure of length laid “parallel” to the axis of x, for example, we should have to set ds2 =  –1; dx2 =  dx3 =  dx4 =  0. Therefore –1 =  g11dx12. If, in addition, the unitmeasure lies on the axis of x, the first of equations (70) gives

g11 = −b1 + ar l.

From these two relations it follows that, correct to a first order of small quantities,

dx

=

1

−

a 2r

(71)

The unit measuring-rod thus appears a little shortened in relation to the system of coordinates by the presence of the gravitational field, if the rod is laid along a radius.
In an analogous manner we obtain the length of co-ordinates in tangential direction if, for example, we set

ds2 = −1; dx1 = dx3 = dx4 = 0; x1 = r, x2 = x3 = 0.

The result is

−1

=

g 22 dx 22

=

−dx

2 2

(71a)

With the tangential position, therefore, the gravitational field of the point of mass has no influence on the length of a rod.
Thus Euclidean geometry does not hold even to a first approximation in the gravitational field, if we wish to take one and the same rod, independently of its place and orientation, as a realization of the same interval; although, to be sure, a glance at (70a) and (69) shows that the deviations to be expected are much too slight to be noticeable in measurements of the earth’s surface.
Further, let us examine the rate of a unit clock, which is arranged to be at rest in a static gravitational field. Here we have for a clock period ds =  1; dxl =  dx2 =  dx3 =  0 Therefore

dx4 =

1

=

g

44

dx

2 4

;

1= g 44

(1+(

1 g 44

−1))

=

1

−

1 2

(

g 44

−

1)

or

dx4

=

1

+

l 8r

#

t

dx r

(72)

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Thus the clock goes more slowly if set up in the neighborhood of ponderable masses.
From this it follows that the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum.14
We now examine the course of light-rays in the static gravitational field. By the special
theory of relativity the velocity of light is given by the equation

−

dx

2 1

−

dx2

−

dx

2 3

+

dx

2 4

=

0

and therefore by the general theory of relativity by the equation

= ds2 gn= odxndxo 0

(73)

If the direction, i.e. the ratio dx1: dx2: dx3 is given, equation (73) gives the quantities

dx1 dx4

,

dx2 dx4

,

dx3 dx4

and accordingly the velocity

fddxx41p2

+

fddxx

2 4

2
p

+

fddxx43p2

=

c

defined in the sense of Euclidean geometry. We easily recognize that the course of the light-rays must be bent with regard to the system of co-ordinates, if the gno are not constant. If n is a direction perpendicular to the propagation of light, the Huyghens principle shows that the light-ray, envisaged in the plane (c, n), has the curvature -∂c/∂n.
We examine the curvature undergone by a ray of light passing by a mass M at the distance ∆. If we choose the system of co-ordinates in agreement with the accompanying diagram, the total bending of the ray (calculated positively if concave towards the origin) is given in sufficient approximation by

B

=

# +3
−3

2c 2x1

dx2

,

while (73) and (70) give

c

=

e−

g g

44 22

o

=

1

−

a 2r

f1

+

x

2 2

r2

p

.

Carrying out the calculation, this gives

14 According to E. Freundlich, spectroscopical observations on fixed stars of certain types indicate the existence of an effect of this kind, but a crucial test of this consequence has not yet been made.

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B=

2Da=

lM 2rD

(74)

Χ2

Χ1 Δ
Fig. 8.

According to this, a ray of light going past the sun undergoes a deflection of 1.7″; and a ray going past the planet Jupiter a deflection of about .02″.
If we calculate the gravitational field to a higher degree of approximation, and likewise with corresponding accuracy the orbital motion of a material point of relatively infinitely small mass, we find a deviation of the following kind from the Kepler-Newton laws of planetary motion. The orbital ellipse of a planet undergoes a slow rotation, in the direction of motion, of amount

f

=

24r3

T2c2

a2 (1− e 2)

(75)

per revolution. In this formula a denotes the major semi-axis, c the velocity of light in the usual measurement, e the eccentricity, T the time of revolution in seconds.15
Calculation gives for the planet Mercury a rotation of the orbit of 43″ per century, corresponding exactly to astronomical observation (Leverrier); for the astronomers have discovered in the motion of the perihelion of this planet, after allowing for disturbances by other planets, an inexplicable remainder of this magnitude.

15 For the calculation I refer to the original papers: A. Einstein, Sitzungsber. d. Preuss. Akad. d. Wiss., 1915, p. 831; K. Schwarzschild, ibid., 1916, p. 189.

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