Einstein’s General Principle of Relativity.

V. Frederiks.

The first fundamental work of Einstein on the principle of relativity should be considered the paper that appeared in 1914 in the proceedings of the sessions of the Berlin Academy of Sciences, under the title “Die formale Grundlagen der allgemeinen Relativitätstheorie”¹) (Formal Foundations of the General Theory of Relativity). This work, somewhat corrected and supplemented, was later, in 1916, printed in Annalen d. Physik; separate offprints of it were then put on sale, thanks to which this particular work of Einstein’s enjoys special renown²). Lorentz, who in 1915–1916 delivered lectures on the theory of relativity in Leiden, called them: “Einstein’s theory of gravitation”³); the mathematician Hilbert called his articles, which appeared in 1915–1916, “Die Grundlagen der Physik”⁴) (Foundations of Physics); finally, the mathematician Weyl published in 1918 a book devoted to these theories, under the title “Raum, Zeit, Materie” (Space, Time, Matter). These titles alone show with sufficient clarity that the theory created by Einstein embraces all of physics, and a theory of this kind cannot fail to possess a profound, compelling interest; that this is so is shown also by the circumstance that, from the moment of its appearance, such outstanding physicists and mathematicians as Lorentz, Hilbert, and Weyl took it up. But this theory, for its exposition, more or less complete and thorough, requires a very complex mathematical apparatus, accessible to almost none of the physicists. Popular expositions of this theory, however well they may have been written, can give nothing but vague, imprecise, and hazy images to anyone who would like to have something more than a “glimpse and something” about Einstein’s theory. The present article, by its brevity, cannot lay claim to even a somewhat exhaustive explanation of Einstein’s theory—

¹) Berlin. Sitzungsberichte der Preussischen Akademie der Wissenschaften. 1914. Vol. XLI.

²) In 1920 it was also reproduced in a book published by Teubner under the title: H. A. Lorentz, A. Einstein, H. Minkowski: Das Relativitätsprinzip.

³) H. A. Lorentz. On Einstein Theorie of Gravitation, Amsterdam, 1916.

⁴) Göttingen. Nachrichten der Königl. Gesellschaft der Wissenschaften. 1915, 1916.

stein. Its aim is to clarify the principal fundamental propositions of Einstein and to apply them to the solution of two or three comparatively simple questions, such as, for example, the question that has recently been much in people’s minds concerning the motion of Mercury’s perihelion, or concerning the deflection of a ray in the gravitational field of the Sun. It goes without saying that Einstein’s fundamental propositions should not be regarded as theorems that can be derived, by a purely deductive path, from other, no longer doubtful, first principles. The elucidation of the foundations of the theory amounts to explaining—or, better, to enumerating—the reasons why precisely they should be considered as such. Proofs of the correctness of the theory must be sought not a priori, but a posteriori. Yet neither experimental confirmation of its conclusions nor the prediction of new phenomena hitherto unknown constitutes the most important thing in Einstein’s theory. The foundations of Einstein’s theory have an enormous principial significance; it is in this significance that one must seek the chief value of the theory, and not in the few experiments confirming it, however brilliant these experiments may be.

Geometry and physics. Before Einstein, geometry and physics were regarded as two sciences essentially quite different. Geometry in physics was regarded as something external in relation to physics; the actual content of physics was supplied by experiment and by experiment alone. The Euclidean geometry of three-dimensional space was only a framework, true, a necessary one—since every physical phenomenon takes place in this space—but in any case one not connected in any way with the phenomenon itself. It is true that in the so-called “special” principle of relativity (1905) Minkowski made use of the geometry of a 4-dimensional space, which did not possess all the features of Euclidean geometry and which was linked with physics through the constant speed of light that entered into it. In this geometry the element of length is determined by \(ds^2 = dx^2 + dy^2 + dz^2 - c^2dt\), where \(x\), \(y\), \(z\) denote the spatial coordinates, respectively, \(t\) denotes time, and \(c\) is the speed of light. This geometry is not Euclidean, since in Euclidean geometry it would be \(ds^2 = dx^2 + dy^2 + dz^2 + dc^2t^2\); moreover, it is connected with physics, since the constant \(c\), the speed of light, appears in it; but Minkowski’s geometry was regarded as a thing of a purely formal character, just as in physics one regards \(\sqrt{-1}\), and a closer connection between physics and geometry still did not exist.

Geometry was for physics a framework, something external, extraneous to the content of physics. Conversely, to some geometers physics sometimes seemed a science whose experimental data were necessary for establishing the very foundations of geometry. To examine the foundations of geometry is, of course, beyond the scope of the present article. In Poincaré’s book La science et l’hypothèse¹ one may find an excellent, generally accessible analysis of what, properly speaking, constitute

¹ H. Poincaré, Science and Hypothesis. Translated by A. Bachinsky.

the foundations of geometry—how one must regard the axioms of Euclid’s geometry, as well as the geometries of Lobachevsky, Riemann, and all the other non-Euclidean geometries, infinite in number. Here we shall confine ourselves only to considering the question of the connection between experience and the axioms, or the theorems of geometry derived with their aid. Clarifying this point is important for understanding Einstein’s point of view on geometry.

So long as there existed only one geometry, Euclid’s, there were no doubts about the “physical” truth of its axioms, although even Gauss considered it necessary to carry out a direct experimental verification of the proposition that the sum of the angles of a triangle is equal to two right angles. From the time of the appearance of the geometries of Lobachevsky, Riemann, and others, the question of the experimental verification of geometry acquired a special significance. Lobachevsky’s geometry, as is known, denies Euclid’s postulate according to which through a given point only one straight line can be drawn parallel to a given straight line, and opposes to it a directly opposite postulate: there may be as many such parallel straight lines as one wishes. The so-called spherical geometry of Riemann deviates from Euclid’s geometry in the other direction and completely denies the possibility of the existence of parallel straight lines. Both Lobachevsky and Riemann (in his spherical geometry) accept all the other axioms of Euclid. As is very well and simply shown in the above-mentioned booklet by Poincaré, both geometries are logically quite possible and contain no internal contradictions. For the sum of the angles of a triangle, neither Lobachevsky nor Riemann any longer obtains a magnitude equal to two right angles. In Lobachevsky the sum of the angles is less than two right angles; in Riemann it is greater. Gauss found in his experiment that, within the limits of observational errors, the sum of the angles of a triangle is equal to two right angles. Since measurements of angles can be made with great precision, Gauss’s experiment, as it seems at first glance, shows that the actual “physical” space (as one may call the space in which all physical phenomena occur, in contrast to those which we can imagine to ourselves or construct logically) is ordinary Euclidean space, so familiar to us. But, first, deviations from Euclidean geometry may be so small that, despite their existence and despite the comparative precision of Gauss’s observations, they still could not be detected by precisely this experiment; and, secondly, even if the experiment had given with absolute precision the equality of the sum of the angles of a triangle to two right angles, even then it would not have been possible to assert that physical space is Euclidean without stipulating one fundamentally extremely important circumstance. Indeed, let us suppose that experiment also gives a sum less than two right angles. Would a physicist draw from this the conclusion that Euclid’s geometry is not correct? First of all he would ask how the measurement of the angles had been carried out. He would be answered: by readings on a divided circle and with the aid of a telescope. The application of the telescope

of the tube means that a light ray is used as a straight line joining two vertices of a triangle whose sum of angles is being measured; and the deviation from two right angles for this sum a physicist, of course, could, if he wished, explain not by the falsity of Euclidean geometry, but simply by the “curvature” of the light ray (on the contrary, if it were equal to two right angles, a scholar wishing at all costs to stand on the point of view of Lobachevsky’s geometry could explain the deviation from the corresponding theorem of Lobachevsky also by the “curvature” of the ray). But a physicist speaking of the curvature of a ray assumes that he can in some way detect this curvature; in order for him to be able to do this, he must have some other “physical” apparatus which, in his opinion, would give a “true” straight line; by comparing the light ray with this straight line, he could then show that the ray is indeed curved and that his new apparatus gives a sum of two right angles. But his triumph would be very superficial and short-lived; a scholar standing on the point of view of Lobachevsky would ask him to prove that his new apparatus represents a straight line, and this our physicist, without inventing another new apparatus, could no longer do. And since such apparatuses cannot be invented ad infinitum, it is clear that experiment can give an answer to our question only insofar as we ascribe to our basic apparatus—say, the light ray—the properties of a straight line. But to ascribe the properties of a straight line specifically to the light ray, and not to something else, depends exclusively on our arbitrariness. We draw attention to this circumstance because the geometry used by Einstein is not Euclidean geometry, and it may appear that the truth or falsity of Einstein’s theory serves as proof of the incorrectness or correctness of Euclid’s geometry. Yet this is not so: anyone who wishes to regard Euclid’s geometry as something exceptional may continue to do so without being troubled by Einsteinian reasoning and theories, but then he must renounce regarding as straight those lines which are given to us by our basic measuring instruments: the light ray, the edge of a ruler, and the like. We shall see that, if the light ray is regarded as straight, if one accepts that the edge of a ruler is straight, then observations—the more precise ones, such as those made by Gauss—will yield deviations from Euclid’s geometry.

But, regardless of what actually performed observations give or may give, it is important from the principled point of view to establish that, once a straight line is defined physically, say, by means of a light ray, then only experiment can indicate to us what kind of geometry is valid for physical space. But there exist infinitely many geometries; in what way is one to carry out their experimental verification, and which conclusions and propositions of the geometries would be most convenient and best of all to verify? Geometries in which the motion of invariantly...

figures (the transfer of a figure from one place in space to another, the displacement of figures or motion, the existence of a rigid body) is very small; chief among them are the geometries of Lobachevsky, Riemann, and Euclid. An analysis of the basic propositions of these geometries may be found in Helmholtz, Sophus Lie, B. Russell, and others. A considerably larger class of geometries is comprised by the so-called Riemannian geometries. For each of his geometries Riemann takes as a basis the definition of the elements of length.

Let us have a space of \(n\) dimensions, and let \(x_1, x_2, \ldots, x_n\) be \(n\) coordinates determining the position of a point in this space, and let \(ds\) be an element of arc length. The expression

\[ ds^2 = \sum_{ik} a_{ik}\, dx_i\, dx_k \]

where \(a_{ik}\) are functions of \(x_1, x_2, \ldots, x_n\), is characteristic of the geometry under consideration. For each given geometry the functions \(a_{ik}\) have a certain quite definite form. For example, for Euclidean geometry, and in three-dimensional space,

\[ ds^2 = dx_1^2 + dx_2^2 + dx_3^2; \]

for Lobachevsky geometry

\[ ds^2 = \frac{dx_1^2 + dx_2^2 + dx_3^2} {1 - \dfrac{a^2}{4}\left(x^2 + y^2 + z^2\right)} \]

where \(a\) is some constant quantity.

By assuming in general

\[ ds^2 = \sum_{ik} a_{ik}\, dx_i\, dx_k, \]

we thereby discard not only Euclid’s axiom of parallels, but also certain other axioms; thus, in the most general case, the axioms admitting the possibility of transfer or superposition of figures will not be true (in Euclid these axioms are plainly not formulated; details on these questions may be found in Hilbert’s book Die Grundlagen der Geometrie).

According to Einstein, this relation, fundamental for every geometry, must also be tested by experiment. Of course, only an ideal thought experiment is possible here; in reality the test is made not on the expression for \(ds^2\) itself, but on those conclusions which are drawn from it. If experiment gives, in three-dimensional space, \(a_{ik}=0\) for \(i \ne k\), and \(a_{ik}=1\) for \(i=k\), then we have Euclidean geometry; if the \(a_{ik}\) turn out to be functions of \(x_i\), then we have a geometry whose character depends on the form of these functions.

In Einstein’s “special” principle of relativity one has to consider time as a quantity closely connected with spatial measurements, inseparable from them. Naturally, he takes as the basis of his new, more general theories not a space of three dimensions, but a space of four dimensions, in which one of the coordi-

will be temporal. Every physical phenomenon is determined by the place of its occurrence (three spatial coordinates) and by the moment at which it takes place (the time coordinate); from the increments of these four coordinates the expression for the element of length is composed

\[ ds^2=\sum_{ik}a_{ik}\,dx_i\,dx_k \qquad i,k=1,2,3,4; \]

in this expression all four coordinates play an entirely identical role; in reality, time (say \(x_4\)) is not identical with the spatial coordinates \(x_1, x_2, x_3\). Hilbert1 indicated those conditions which the \(a_{ik}\) must satisfy in order that the fourth coordinate, time, should not lose certain characteristics peculiar to it, which it must preserve in any theory.

Einstein’s first proposition. Thus, the first fundamental proposition of Einstein’s theory will be the following:

The element of length is determined by the formula

\[ ds^2=\sum_{ik}a_{ik}\,dx_i\,dx_k \qquad i,k=1,2,3,4, \]

and experiment decides what the functions \(a_{ik}\) are equal to.

Geometry and mechanics. Newtonian mechanics ascribes an entirely exceptional significance to rectilinear and uniform motion. If, instead of the given coordinate system, we take another coordinate system moving with respect to the first uniformly and rectilinearly, then this cannot be detected by direct physical experiment. Any other motion can be detected experimentally, since in that case new forces appear that did not previously exist. Let us consider a very simple case: motion in a circle. This case makes it possible to point to two remarkable facts, one of an extremely important fundamental character, the other of an experimental character.

Let us take a physical body having the form of a sphere, and suppose that in all space it is the only one we have. Can we decide whether this sphere is rotating or not? Outside the sphere there is not a single physical body, not a single physical point, which could help us in this respect. We must turn to what is happening on the very surface of the sphere or inside it. Suppose that on the surface we have detected the presence of a certain centrifugal force, have noticed that the sphere is somewhat compressed at the poles, that the plane of oscillation of a Foucault pendulum rotates. This will compel us to suppose that our sphere is rotating; we shall even calculate the speed of its rotation. But then the question arises: with respect to

toward something, for there are no external physical bodies with respect to which it might rotate. Evidently, there must exist some kind of space, not marked in any way in itself, in which there is not a single physical body and which therefore is inaccessible in itself to observation; and it is with respect to this “absolute,” non-physical space that the rotation takes place. But everything that has no physical reality and is therefore inaccessible to physical observation belongs, if you like, to metaphysics, but in no way to physics. One may believe or not believe in such absolute space, but one cannot use it, as a really physically existing object, to explain physical phenomena. But then one has to say that Newtonian mechanics can give answers to questions to which, in essence, one cannot answer “physically.” This is a paradox to which Mach first drew attention; Einstein brought this paradox out of the oblivion in which it had lain, and gave the following answer to it: Newtonian mechanics, generally speaking, is not correct. True mechanics gives for such a sphere neither a centrifugal force nor a displacement of the plane of oscillation of Foucault’s pendulum, and so forth—all these forces and phenomena will appear only when the rotation is performed with respect to some other “physical” space, which can be discovered from the physical bodies found in it. Rectilinear and uniform motion plays no exceptional role, just as motion in a circle or any other motion does not; all coordinate systems and all possible displacements of them are equivalent to one another. If there is only one sphere and nothing else, then we may assert that it is spinning or is at rest, is jumping or moving about as it pleases. No physical phenomena will reveal this, because all these jumps, rotations, and the like occur not with respect to “physical” space, but are only conceivable with respect to other spaces that do not really exist. To show that mechanics of this kind is possible constitutes Einstein’s greatest merit.

Another remarkable fact following from observations of rotation is the circumstance that the centrifugal force is always proportional to the mass of the rotating body. The force of gravitation, according to Newton’s law, is likewise proportional to mass, but in the expression of Newton’s law mass has the significance of the cause producing gravitation; whereas in the expression for the centrifugal force, caused by rotation, it plays a completely passive role; the mass that actively creates the force and the inert, or passive, mass—a simple numerical coefficient—turn out, when tested by experiment, to be equal to one another with an enormous degree of accuracy1. This fact cannot be of an accidental character—

ter, and Newtonian mechanics gives no explanation of it. In his second law Newton simply sets forth, as a postulate, the requirement that inertial mass, multiplied by acceleration, should be equal to force—in particular, to the force caused by the same mass, the same in quantity, but now acting actively.

The identity of the masses—active, gravitating, and passive, inertial—Einstein elevates to a principle and calls it the principle of equivalence.

Let us imagine one coordinate system \(K\), at rest, and another \(K'\), in a state of uniformly accelerated and rectilinear motion with respect to the first. A material point moving along a straight line in \(K\) will move along a parabola in \(K'\). If the direction of motion of \(K'\), or the axis of the parabola, is taken as the \(x\)-axis, then in the system \(K'\)

\[ \frac{d^2 x'}{dt^2}=g=\text{constant}. \]

If \(m\) is the mass of the point, then every equation written in the form
\(m\dfrac{d^2x'}{dt^2}=mg\) may be regarded as an expression of equality between the product of the inertial mass by the acceleration and the force \(mg\). The force \(mg\) may be regarded, on the basis of the principle of equivalence, as a force of gravity, where \(m\) then already has the meaning not of inertial mass, but of active mass, exciting the force \(mg\), just as the force \(mg\) is excited by the active mass \(m\) of some heavy body on the surface of the earth (\(mg\) means in the latter case \(\dfrac{mM}{r^2}\), where \(m\) is the mass of the heavy body, \(M\) the mass of the earth, \(r\) the radius of the earth; it is clear that \(m\) is here the exciter of the force).

Thus, the accelerated and rectilinear motion of a system may even go unnoticed by observers within it, if one admits that such accelerated motion is equivalent to the presence of a gravitational field and that the observer explains all the phenomena occurring around him precisely by this gravitation. Centrifugal force, on the basis of the principle of equivalence, may likewise be regarded as a force which by its nature coincides with the force of gravity and is in no way essentially different from it; finally, the same may be said of all forces that arise kinematically in a coordinate system connected with a moving body.

In nature there exist masses creating around themselves a so-called gravitational field; if we take any coordinate system \(K''\), then the character of the gravitational field will depend on just which coordinate system we have chosen; in another coordinate system \(K''\), moving with respect to the first, there will exist another gravitational field. Moving together with \(K''\), we may assign

everything that occurs in \(K^{*'}\), not the motion of \(K^{*'}\) with respect to \(K^*\), but that gravitational field which exists in \(K^{*'}\) and which differs from the field in \(K^*\).

But the transition from one coordinate system to another, arbitrarily chosen coordinate system entails a change in the form of those functions \(a_{ik}\) which determine the properties of the geometry of physical space; if in one coordinate system we have

\[ ds^2=\sum_{ik} a_{ik}\,dx_i\,dx_k \qquad i,k=1,2,3,4, \]

then in \(K^{*'}\) we obtain

\[ ds^2=\sum_{ik} a'_{ik}\,dx'_i\,dx'_k \]

and, obviously, \(a'_{ik}\ne a_{ik}\), since the dependence between \(x_i\) and \(x'_i\) is arbitrary.

It must be admitted that the transition from one coordinate system to another changes not only the gravitational field, but also the geometry of physical space, and this shows that between the gravitational field and \(a_{ik}\), i.e. geometry, there must be a connection.

On this basis Einstein calls the quantities \(a_{ik}\) the potentials of gravitation and denotes them, by analogy with terrestrial acceleration, by \(g_{ik}\), but this name contains nothing beyond the just-indicated parallelism between geometry and gravitation.

Einstein’s second fundamental proposition. Thus, in considering Mach’s paradox, Einstein arrives at the conclusion as to the admissibility not only of a transition from one uniformly and rectilinearly moving coordinate system to another such coordinate system, but of all coordinate transformations in general (since motion is included here as well, this means that the new coordinates \(x'_i\), \(i=1,2,3,4\), may be arbitrary functions of the four coordinates \(x_i\), \(i=1,2,3,4\)).

Einstein’s third fundamental proposition. Considering the principle of equivalence, Einstein likewise arrives at the conclusion that the line element determining the properties of physical space, i.e.

\[ ds^2=\sum_{ik} g_{ik}\,dx_i\,dx_k \qquad i,k=1,2,3,4 \]

includes the 10 functions \(g_{ik}\), on which depend not only the form of the geometry, but also the gravitational field in the given coordinate system.

Einstein’s fourth fundamental proposition. In order to construct mechanics and physics on these foundations, one must make one further essential observation. If the choice of coordinate system is arbitrary, then how is nature to be described with its aid? In what way are results to be obtained that do not depend on the admitted arbitrariness? For it is quite clear that the laws of nature do not depend on it. The answer

suggests itself: since the laws of nature do not depend on our arbitrariness, they must be independent of the coordinate system chosen by us. Expressed mathematically, the laws of nature must be invariant with respect to any coordinate transformations. Einstein’s genius succeeded in finding and formulating the laws of mechanics and physics precisely in such an invariant form, independent of the chosen coordinate system. We shall now proceed to the presentation of the fundamental equations of mechanics and physics. What has been said so far explains only the path by which Einstein proceeded, but cannot yet serve as proof of the correctness of his propositions, although from the principled point of view his assertions have an obvious advantage over the corresponding assertions of Newtonian mechanics.

Einstein’s fundamental equations. We shall not continue along the path followed by Einstein, but shall follow Hilbert, who expounded his theory in a more accessible and simple form. In his original works Einstein starts from Poisson’s equation \(\Delta\varphi=4\pi\rho\), where \(\varphi\) is the ordinary gravitational potential, and \(\rho\) is the density of matter. The generalization of this equation, by introducing instead of \(\varphi\) 10 potentials \(g_{ik}\), and instead of \(\rho\) 10 other quantities determining the state of matter, gives Einstein the possibility of obtaining the desired equations and showing their correctness. But the process of generalizing the equation is not so simple and not so unambiguous that one could easily assess the full significance of the results obtained in this way.

Let us assume, together with Hilbert \(^{1}\), that all events taking place in nature depend on a certain “world” function \(H\); this function \(H\) depends on four coordinates \(x_1, x_2, x_3, x_4\); the first three coordinates are purely spatial, while the fourth \(x_4\), in the given coordinate system, denotes time. The function \(H\) does not depend on the coordinate system chosen by us; therefore, as can be proved, it will not explicitly depend on \(x_1, x_2, x_3, x_4\); it will depend on them through the following quantities:

1) the 10 functions \(g_{ik}\) and their derivatives with respect to \(x_i\); one might allow the dependence of \(H\) on derivatives of any order; but, by analogy with Poisson’s equation, we assume that \(H\) depends only on \(g_{ik}\) and their first and second derivatives. We also assume that these \(g_{ik}\), as well as their derivatives, are everywhere single-valued and continuous.

2) Those parameters which determine the state of matter. Such parameters will be, for example, the density of matter, the density of electricity, the electric potentials (the vector potential and the scalar potential); if the theory of matter cannot make do with these parameters alone, then other necessary parameters must also be included here; if one takes the point of view of the electromagnetic

\(^{1}\) Einstein, Lorentz, and others also use this same method of exposition. We cite Hilbert as the first to have applied this method.

theory of matter, it would have sufficed to include among these parameters only a single electrical density and the vector and scalar potentials. If, finally, one takes Mie’s point of view, then for the creation of a theory of matter it is enough to know the vector potential and the scalar potential; and since the former has 3 terms, in all this means knowledge of four parameters \(q_1, q_2, q_3, q_4\), as functions of \(x_1, x_2, x_3, x_4\). On the basis of Mie’s theory Hilbert assumes that \(H\) depends on \(q_1, q_2, q_3, q_4\) and on their first derivatives with respect to \(x_i\). But this assumption is by no means essential for very many questions solved by Einstein’s theory.

Thus let us suppose that there is a “world function”:

\[ H = H\left(g_{ik},\ \frac{\partial g_{ik}}{\partial x_l},\ \frac{\partial q_{ik}}{\partial x_l\,\partial x_m},\ q_i,\ \frac{\partial q_i}{\partial x_k}\right), \]

where

\[ i,\ k,\ l,\ m, = 1,\ 2,\ 3,\ 4. \]

Let us consider the integral

\[ J=\int H\sqrt{g}\,dx_1dx_2dx_3dx_4, \]

in which \(dx_1dx_2dx_3dx_4\) is an element of volume, \(g\) is the determinant formed from all \(g_{ik}\), and \(H\) by definition is an invariant. It can be shown that \(\sqrt{g}\,dx_1dx_2dx_3dx_4\) will also be an invariant, i.e. it does not depend on the coordinate system. It is obvious that \(J\) is also an invariant, as is every variation of this integral.

All events in nature occur in such a way that the variation of this integral \(\delta J\) is equal to zero

\[ \delta J=0 \ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \tag{2} \]

this is the fundamental law of Einsteinian physics. It must contain within itself all the other laws of physics: the law of universal gravitation, Maxwell’s equations, the laws of interaction between masses, and all the like.

In order for this law to have practical significance, one must, of course, know the expression of the function \(H\). Let us suppose that it is known to us. The expression for \(H\) contains 10 unknown functions \(g_{ik}\) and four unknown functions \(q_i\), but from the condition \(\delta J=0\) there follow 14 differential equations; the first 10 of them are obtained by varying the functions \(g_{ik}\); we shall denote them briefly by

\[ G_{ik}=0\ ^1);\quad i,k=1,\ 2,\ 3,\ 4 .\ .\ . \tag{3} \]

the last four equations are obtained by varying the functions \(q_i\); we shall denote them by

\[ Q_i=0 .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \tag{4} \]

\(^1\) We have 10 equations, and not 16, for the reason that \(g_{ik}\), as well as \(G_{ik}\), are symmetric with respect to the indices \(i\) and \(k\).

The system of equations (3) and (4) enables us to determine \(g_{ik}\) and \(q_i\) in a given coordinate system.

Equations (3) and (4), derived from the invariant \(\delta J=0\), will themselves be invariants (we shall not explain the more exact meaning of this assertion here, for lack of space) and do not depend on the coordinate system chosen by us. The arbitrariness of the chosen coordinate system is expressed here in the fact that these 14 equations are not independent of one another, but are connected by 4 identities. This means that 4 of the 14 functions \(g_{ik}\) and \(q_i\) may be chosen arbitrarily and are not determined from equations (3) and (4). By arbitrary values of 4 of the 14 functions the chosen coordinate system is fixed.

At first glance it seems that insurmountable difficulties should arise in finding the form of the “world” function \(H\). Meanwhile the choice of this function \(H\) is almost unambiguous for a very large class of phenomena. In fact, let us consider the case—we assume that such a case is possible, and the results given by the theory show that it is indeed observed—when the parameters \(q_1, q_2, q_3, q_4\) are small quantities; instead of them let us then introduce the parameters \(\varepsilon q_1, \varepsilon q_2, \varepsilon q_3, \varepsilon q_4\), where \(\varepsilon\) will be some small number, while \(q_1, q_2, q_3, q_4\) already have finite values. Then expand \(H\) in increasing powers of \(\varepsilon\); then

\[ H=K' + \varepsilon L + \varepsilon^2 M+\cdots \]

Let us consider only the first terms of this expansion, \(K'\) and \(\varepsilon L\). \(K'\) depends only on \(g_{ik}\) and on the first and second derivatives of these functions with respect to \(x_i\); \(L\) depends on \(g_{ik}\), their derivatives, \(q_i\), and their derivatives. \(H\) is an invariant; \(K, L, M\ldots\) must also be invariants. It turns out that there exists, in all, only one invariant \(K'\) depending on \(g_{ik}\), their first and second derivatives, and containing second derivatives only linearly; this fact is remarkable. This single invariant will be the so-called Riemannian curvature of four-dimensional space. We shall denote it by \(K\). Obviously \(K'\) may be equal to \(K\) or to \(K+\lambda\), where \(\lambda\) is some constant, a number independent of \(x_i\). We shall put \(\lambda=0\); in later works Einstein and Weyl clarified what enormous significance this constant \(\lambda\) has; for lack of space in the present article we must leave this question aside and set

\[ K'=K \]

Let \(D_{\mu\nu}\) be the minor of the determinant \(g\), formed from \(g_{\mu\nu}\), corresponding to the term of the determinant \(g_{\mu\nu}\); denote \(\dfrac{D_{\mu\nu}}{g}\) by \(g^{\mu\nu}\); let us introduce also the following notations.

Let

\[ \left[\begin{matrix} ik \\ m \end{matrix}\right] = \frac12\left(g_{imk}+g_{mki}-g_{ikm}\right) \]

and let

\[ \left\{\begin{matrix} ik \\ m \end{matrix}\right\} = \sum_{m} g^{nm} \left[\begin{matrix} ik \\ m \end{matrix}\right]; \quad i,k,m,n=1,2,3,4 \]

It can be shown that

\[ K=-\frac14\sum_{ik} g^{ik}K_{ik}, \]

where

\[ K_{ik} = \sum_l \frac{\partial}{\partial x_i} \left\{\begin{matrix} kl \\ l \end{matrix}\right\} - \frac{\partial}{\partial x_l} \left\{\begin{matrix} ik \\ l \end{matrix}\right\} + \sum_{lm} \left\{\begin{matrix} kl \\ m \end{matrix}\right\} \left\{\begin{matrix} mi \\ l \end{matrix}\right\} - \left\{\begin{matrix} ik \\ m \end{matrix}\right\} \left\{\begin{matrix} ml \\ l \end{matrix}\right\}. \]

The \(K_{ik}\) are called the Riemann curvature tensor; the derivation of this formula may be found in Bianchi’s differential geometry. We see that the expression for \(K\) in the general case will be extremely complicated, but in solving certain special problems it becomes greatly simplified.

The expression for \(L\) requires special consideration. However, if one follows Mie’s theory, it is not difficult to find it. Schwarzschild had already shown that Maxwell’s equations can be derived from a kind of Hamiltonian principle. We shall denote the components of the vector potential \(A_1,A_2,A_3\) by \(q_1,q_2,q_3\), and the scalar potential \(\varphi\) by \(-q_4\); let the vector \(r_1,r_2,r_3\) denote the electric convection current \(\rho v_1,\rho v_2,\rho v_3\), where \(\rho\) is the electric density and \(v\) the ordinary velocity; \(r_4\) is equal to \(-\rho\); finally, let

\[ M_{ik}=\frac{\partial q_k}{\partial x_i} - \frac{\partial q_i}{\partial x_k}. \qquad \cdots \qquad a) \]

Consider the integral

\[ L'=\int\left(\sum_{ik}M_{ik}^{2}-\sum_i r_i q_i\right)\,dx_1\,dx_2\,dx_3\,dx_4 \]

and set \(\delta L'=0\).

The variation of this integral gives us Maxwell’s equations

\[ \sum_i \frac{\partial M_{ik}}{\partial x_i}=-r_k \qquad \cdots \qquad b) \]

\[ \frac{\partial M_{ik}}{\partial x_l} + \frac{\partial M_{kl}}{\partial x_i} + \frac{\partial M_{li}}{\partial x_k} =0, \qquad \cdots \qquad c) \]

(In ordinary notation, instead of (a) one writes:

\[ \mathbf{E}_1=-\frac{\partial \varphi}{\partial x_1}-\frac{\partial A_1}{\partial t} =\frac{\partial q_4}{\partial x_1}-\frac{\partial q_1}{\partial x_4}=M_{14}; \quad \mathbf{H}_1=\frac{\partial A_3}{\partial x_2}-\frac{\partial A_2}{\partial x_3}=M_{32}, \quad \text{etc.;} \]

instead of (b) and (c) one writes:

\[ \operatorname{curl}\mathbf{H}=\frac{\partial \mathbf{E}}{\partial t}+\rho \mathbf{v} \]

\[ \operatorname{curl}\mathbf{E}=-\frac{\partial \mathbf{H}}{\partial t} \]

\[ \operatorname{div}\mathbf{E}=\rho \]

\[ \operatorname{div}\mathbf{H}=0). \]

In a space containing no electricity, the second term in \(L_1\) drops out, and we obtain Maxwell’s equation for the vacuum.

Mie, in his theory, also considers such a function \(L\), but replaces the second term \(\sum_i r_i q_i\) by some function \(f\) of \(q_i\); thus for him the electric density turns out to be a function of the potential \(q_i\). But Mie did not write his theory for the general principle of relativity; in his works he was guided by the first, “special,” principle of relativity, and therefore his \(L\) cannot be transferred directly into the expression \(H\) for Hilbert’s “world” function. In order to make use of Mie’s function in the general principle of relativity, it must be generalized in the corresponding manner, and in the theory of invariants it is easily proved that such a generalized and, hence, invariant expression with respect to arbitrary transformations will be

\[ L=\int\left[\left(\sum_{iklm} M_{ik}M_{lm}g^{il}g^{km} -f\left(\sum_{ik} g^{ik}q_iq_k\right)\right)\sqrt{g}\,dx_1dx_2dx_3dx_4\right]\ldots \tag{6} \]

Hilbert inserts this expression as the second term in the expression for the function \(H\). This expression does not have the dimensions of the expression \(K\); in order for the dimensions to be the same, \(L\) must still be multiplied by a certain numerical coefficient \(\varepsilon\). It turns out that this \(\varepsilon=\dfrac{8\pi k}{c^2}\), where \(k\) is the constant of gravitation and \(c\) the velocity of light, i.e. \(\varepsilon=1.87\cdot 10^{-27}\), an extremely small quantity, which is precisely what corresponds to our expansion of the world function \(H\) into an infinite series.1

It is a very remarkable fact that there is also a very limited number of such invariants \(L\), obtained with the aid of \(q_i\) and their first derivatives. Mie counts only four of them in all, but chooses from them the one that immediately gives him Maxwell’s equations.

If one does not take the standpoint of Mie’s electrical theory of matter, then the expression for \(L\) can also be given another form. For certain problems, for example astronomical problems, de Sitter, Einstein, and others do indeed proceed thus; but, as we shall soon show, for the solution of the simplest and most interesting astronomical problems the form of the function \(L\) will play no role.

Thus we shall assume that

\[ H=K+\varepsilon L, \]

where \(K\) is the curvature of four-dimensional space and \(L\) is expression (5).

Examples. We can now proceed to the solution of certain separate problems, which should show what Einstein’s theory is capable of providing and how it leads to the solution of mechanical and physical problems.

First example. Let us suppose that space is devoid of matter; then \(L=0\), and we are left with

\[ J=\int K\sqrt{g}\,dx_1dx_2dx_3dx_4. \]

From \(\delta J=0\) there now follow 10 equations

\[ G_{ik}=0. \]

If we admit what in essence also follows from the theory, namely that the \(g_{ik}\) are continuous and single-valued functions, then the solution of these differential equations will be

\[ \begin{cases} g_{ik}=0 & \text{if } i\ne k,\\ g_{ii}=1 & \text{if } i=1,2,3,\\ g_{44}=-1.& \end{cases} \]

(the value \(g_{44}=-1\), and not \(+1\), follows from those requirements which the \(g_{ik}\) must satisfy in order that \(x_4\) denote time). We thus obtain

\[ ds^2=dx_1^2+dx_2^2+dx_3^2-dx_4^2, \]

i.e. the very expression for \(ds^2\) which we have in Einstein’s “special” principle of relativity. The speed of light does not enter here, since we have set it equal to unity, which obviously affects only the choice of the unit for \(x_4\), i.e. for time.

In the absence of matter, we have, therefore, the usual expression for \(ds^2\), i.e. Euclidean geometry in a three-dimensionally extended space1.

2nd example. Suppose that we are considering a space lying inside some very small sphere described from some point \(x_1, x_2, x_3, x_4\); if its radius is sufficiently small, then inside this sphere the quantities \(g_{ik}\) may be regarded as constant. In this case, as is easy to show, the expression \(ds^2\) can always be reduced to the form

\[ ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - dx_4^2, \]

it being sufficient for this to perform certain coordinate transformations. Hence we conclude that in the infinitely small the “small” principle of relativity is always valid. In this case the “world” function will become

\[ H=\varepsilon\int L\,dx_1dx_2dx_3dx_4 \]

and, varying it, we shall obtain the ordinary Maxwell equations, since all \(g_{ik}\) are either equal to unity \((i=k)\) or to zero \((i\ne k)\); if the velocity of light is taken not equal to unity, then

\[ ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - g_{44}dx_4^2; \]

obviously \(g_{44}=c^2\); this same \(g_{44}\), as is seen from expressions (4) for \(L\), will also occur in Maxwell’s equations; we now understand why, in the special principle of relativity, the “physical” quantity enters into the geometrical expression for \(ds^2\); we also understand the assertion that \(c\), the velocity of light, is a constant quantity. It will be constant insofar as we have the right to regard \(g_{ik}\) as independent of the coordinates \(x_i\).

3rd example. Let us consider the so-called one-body problem, i.e. the gravitational field which is produced by a single gravitating mass. Let us place this mass at the origin of the coordinate system and suppose that it has the form of a sphere. The gravitational field created by it must then have the symmetry of a sphere, but, of course, only under the assumption that the sphere is at rest and that everything is in a stationary state, i.e. that all \(g_{ik}\) do not depend on \(t\). Let us introduce the conditions of spherical symmetry into the expression for \(ds^2\). According to Schwarzschild, the most general expression for \(ds^2\) satisfying this condition, if polar coordinates are introduced,

\[ \begin{aligned} x_1 &= r\operatorname{Cos}\vartheta\\ x_2 &= r\operatorname{Sin}\vartheta\operatorname{Cos}\varphi\\ x_3 &= r\operatorname{Sin}\vartheta\operatorname{Sin}\varphi \end{aligned} \]

and one sets

\[ x_4=t, \]

will be the expression \(^{1}\)

\[ ds^2 = F(r)dr^2 + G(r)\left(d\vartheta^2+\operatorname{Sin}^2\vartheta\,d\varphi^2\right)+H(r)dt^2, \]

\(^{1}\) See also Hilbert: loc. cit.

but instead of \(r\) we may take

\[ r'=\sqrt{G(r)}, \]

then

\[ ds^2=M(r)dr^2+r^2(d\vartheta^2+\sin^2\vartheta\,d\varphi^2)-W(r)dt^2 \]

(the prime over \(r\) we omit).

The two arbitrary functions of \(r\), i.e. \(M(r)\) and \(W(r)\), must be determined from the variation of the integral \(J\). To find this variation we must know not only \(K\), but also the function \(L\); but here we may proceed in the same way as is done in potential theory when solving Poisson’s equation \(\Delta\Psi=4\pi\rho\): instead of seeking continuous and single-valued functions \(\Psi\) satisfying this equation, one considers the equation \(\Delta\Psi=0\), finds its solutions, and assumes that the singular points constitute the places of concentration of matter \(\rho\); in exactly the same way we proceed here. We shall discard the function \(L\), but, in solving the remaining equations, we shall admit solutions with singular points and shall suppose that the mass is concentrated at these points.

Thus we must solve the problem

\[ \delta\int K\sqrt{g}\,dr\,d\vartheta\,d\varphi\,dt=0. \]

For this we must compute the curvature \(K\), proceeding from those values of \(g_{ik}\) which occur in the expression for \(ds^2\). These calculations are rather long and ultimately lead to the following expression for \(K\sqrt{g}\):

\[ K\sqrt{g}= \left\{ \left(\frac{r^2W'}{\sqrt{MW}}\right)' -2\,\frac{rM'\sqrt{W}}{M^{3/2}} -2\sqrt{MW} +2\sqrt{\frac{W}{M}} \right\}\sin\vartheta . \]

Instead of the functions \(M\) and \(W\) we introduce the functions \(m(r)\) and \(w(r)\), so that

\[ M=\frac{r}{r-m}\quad \text{and}\quad W=w^2\frac{r-m}{r}, \]

this gives

\[ K\sqrt{g}= \left\{ \left(\frac{rW'}{\sqrt{MW}}\right)' -2m'w \right\}\sin\vartheta . \]

The prime here everywhere denotes differentiation with respect to \(r\). Performing all possible integrations, we finally have left

\[ \delta\int K\sqrt{g}\,dr\,d\vartheta\,d\varphi\,dt = -\delta\int 2m'w\,dr=0, \]

and this gives two differential equations

\[ m'=0\quad \text{and}\quad w'=0, \]

i.e. \(m=\) constant and \(w=\) constant.

Put \(m=a\) and \(w=1\); the latter will not be a restriction of our problem, since the value of \(w\) is obviously connected only with the choice of the unit of time.

This gives us the following expression for \(ds^2\):

\[ ds^2=\frac{r}{r-a}\,dr^2+r^2(d\vartheta^2+\sin^2\vartheta\,d\varphi^2)-\frac{r-a}{r}\,dt^2\ldots\ldots(6). \]

We see that the solution of our problem leads us to functions \(g_{ik}\) having a special surface—the sphere of radius \(a\); on this sphere are situated the masses that produce a gravitational field with spherical symmetry. If one puts \(a=0\), i.e. assumes that the singular surface does not exist, then the functions \(g_{ik}\) become continuous and single-valued functions having no singular points, but at the same time they take precisely the value that they have in Euclidean geometry and which, as we see, corresponds to the absence of matter.

Thus, the gravitational field has been found, and we must now consider the laws of motion of material particles situated in such a field and not disturbing it. In order to find these laws, we shall assume that their motion occurs just as it does for Newton when no force acts on them, i.e. we shall suppose that they move along lines of shortest distance, or geodesic lines: this means that

\[ \delta \int ds=0 \]

and we must solve a new variational problem.

Let us consider \(r,\varphi,\vartheta,t\) as functions of some parameter \(p\); our task will be to solve the system of differential equations obtained from the condition

\[ \delta \int \sqrt{\frac{r}{r-a}\left(\frac{dr}{dp}\right)^2 +r^2\left[\left(\frac{d\vartheta}{dp}\right)^2+\sin^2\vartheta\left(\frac{d\varphi}{dp}\right)^2\right] -\frac{r-a}{r}\left(\frac{dt}{dp}\right)^2}\,dp=0. \]

It is easy to prove that the geodesic curves thus obtained will be plane curves. But then one may restrict oneself to those curves of shortest distance which lie in the plane of the equator, and put \(\vartheta=\frac{\pi}{2}\).

The preceding expression is then rewritten in the form

\[ \delta \int \sqrt{\frac{r}{r-a}\left(\frac{dr}{dp}\right)^2 +r^2\left(\frac{d\varphi}{dp}\right)^2 -\frac{r-a}{r}\left(\frac{dt}{dp}\right)^2}\,dp=0. \]

Whence follow three second-order differential equations

\[ \frac{d}{dp}\left(\frac{2r}{r-\alpha}\frac{dr}{dp}\right) +\frac{2}{(r-\alpha)^2}\left(\frac{dr}{dp}\right)^2 -2r\left(\frac{d\varphi}{dp}\right)^2 +\frac{\alpha}{r^2}\left(\frac{dt}{dp}\right)^2=0. \]

\[ \frac{d}{dp}\,r^2\frac{d\varphi}{dp}=0 \qquad \frac{d}{dp}\,\frac{r-\alpha}{r}\frac{dt}{dp}=0 \tag{7} \]

Their first three integrals will be:

\[ \frac{r}{r-\alpha}\left(\frac{dr}{dp}\right)^2 +r^2\left(\frac{d\varphi}{dp}\right)^2 -\frac{r-\alpha}{r}\left(\frac{dt}{dp}\right)^2 =A \]

\[ r^2\frac{d\varphi}{dp}=B \qquad \frac{r-\alpha}{r}\frac{dt}{dp}=C, \tag{7} \]

where \(A, B, C\) are constants of integration. The value of the constant \(C\) determines only the choice of units of the parameter; therefore we put \(C=1\).

If we eliminate \(p\) and \(t\) from these equations, we obtain an equation for the trajectory of the motion; having obtained it, we make one more substitution, \(\frac{1}{r}=\rho\), and then, finally, obtain:

\[ \left(\frac{d\rho}{d\varphi}\right)^2 =\frac{1+A}{B^2} -\frac{A\alpha}{B^2}\rho -\rho^2+\alpha\rho^3 \tag{8} \]

This expression is very similar to Kepler’s equation for the motion of the planets. The latter is derived and written in the following way:

The law of conservation of energy gives

\[ \frac{m}{2}\left[\left(\frac{dr}{dt}\right)^2 +r^2\left(\frac{d\varphi}{dt}\right)^2\right] -k\frac{Mm}{r}=a \]

where \(a\) is the energy, \(m\) is the mass of the planet, which henceforth we take equal to \(1\), \(M\) is the mass of the sun, \(k\) is the gravitational constant.

The law of conservation of areas gives

\[ r^2\frac{d\varphi}{dt}=b \]

eliminating \(t\) and introducing \(\rho=\frac{1}{r}\), we obtain

\[ \left(\frac{d\rho}{d\varphi}\right)^2 =\frac{a}{b^2}-\frac{kM}{b^2}\rho-\rho^2. \]

Into expression (8), instead of the constants \(A\) and \(B\), we introduce new constants \(a\) and \(b\) so that

\[ -\frac{A\alpha}{B^2}=-\frac{kM}{b^2} \qquad 1+A=\frac{aA}{kM} \]

then it will pass into the expression

\[ \left(\frac{d\rho}{d\varphi}\right)^2 = \frac{a}{b^2} + \frac{kM}{b^2}\rho - \rho^2 + \alpha\rho^3 . \]

It is obvious that the last expression, in the limit as \(\lim \alpha=0\), passes into the Kepler equation for the motion of planets. If \(\alpha\) is very small, then the last term of the equation may be discarded only when \(\rho\) cannot become very large, i.e. in the case when the planet does not pass too close to the sun.

Let us find the physical meaning of the quantity \(\alpha\). For this purpose let us consider motion in a circle. It can be shown that \(r=\) constant will be an integral of the differential equations (7) and that, consequently, motion in a circle is possible; but then equation (7) gives

\[ r^2\left(\frac{d\varphi}{dt}\right)^2 = \frac{\alpha}{2r} \]

an expression in which the unit of time has been chosen so that \(c=1\); if \(c\ne 1\), then

\[ r^2\left(\frac{d\varphi}{dt}\right)^2 = \frac{\alpha}{2r}c^2 . \]

But from Kepler’s equation it follows that for circular motion

\[ r^2\left(\frac{d\varphi}{dt}\right)^2 = k\frac{M}{r^2}, \]

and, comparing the last two formulas, we obtain

\[ \frac{\alpha}{2} = \frac{kM}{c^2} \;(=1.5\cdot 10^5\ \text{cm. for the sun}). \]

The constant \(\alpha\) therefore plays for us the role of the mass of the sun, and is expressed in centimeters. For the sun it is equal to 1.5 kilometers.

For all the planets the quantity \(\alpha\) is indeed very small in comparison with the radius vectors of their orbits, and Kepler’s equation must be correct to a very, very high degree of accuracy. Nevertheless, the additional term \(\alpha\rho^3\), which does not enter into the classical equation, may in certain special cases exert its influence. Since the quantity \(\alpha\) for all the planets known to us is indeed very small in comparison with the radii vectors of their orbits, in order to take account of the influence of \(\alpha\), when solving the differential equations (8) we may use an expansion in powers of \(\alpha\). Let \(e_1, e_2, e_3\) be the roots of the expression

\[ f(\rho) = \frac{a}{b^2} + \frac{kM}{b^2}\rho - \rho^2 + \alpha\rho^3 = 0 . \]

Obviously,

\[ e_1+e_2+e_3=+\frac{1}{a} \]

and

\[ f(\rho)=(\rho-e_1)(e_2-\rho)\{1-\alpha(\rho+e_1+e_2)\}. \]

The equation of motion will be

\[ d\varphi=\frac{d\rho}{\sqrt{(\rho-e_1)(e_2-\rho)[1-\alpha(\rho+e_1+e_2)]}}\ .\ .\ .\ .\ .\ (9). \]

The motion takes place, obviously, between \(\rho=e_1\) and \(\rho=e_2\). Expanding in a series in powers of \(\alpha\), we obtain1

\[ d\varphi=\frac{d\rho}{\sqrt{(\rho-e_1)(e_2-\rho)}}\left[1+\frac{\alpha}{2}(e_1+e_2)+\frac{\alpha}{2}\rho\right] \]

and, integrating,

\[ \varphi-\varphi_0=-\frac{\alpha}{2}\sqrt{(\rho-e_1)(e_2-\rho)} +\left[1+\frac{3}{4}\alpha(e_1+e_2)\right]\arcsin \frac{\dfrac{e_1+e_2}{2}-\rho}{\dfrac{e_1-e_2}{2}}. \]

This formula makes it possible to compute the angle \(\Phi\) between the radius vectors of the points of greatest and least distance from the sun, i.e. between \(\rho=e_1\) and \(\rho=e_2\); obviously

\[ \Phi=\pi\left[1+\frac{3}{4}\alpha(e_1+e_2)\right]. \]

When the planet returns to the place of greatest distance (aphelion), it will therefore have turned through the angle

\[ 2\Phi=2\pi\left[1+\frac{3}{4}\alpha(e_1+e_2)\right]. \]

For Keplerian motion we have the corresponding angle \(2\Phi_k=2\pi\); we thus see that, according to Einstein’s theory, the perihelion of the orbit, in one revolution of the planet around the sun, is displaced by the angle

\[ \omega=\frac{3}{2}\alpha(e_1+e_2)\pi \]

Let \(T\) be the planet’s period of revolution, \(a\) the major semiaxis of the orbit, \(\varepsilon\) the eccentricity of the orbit. We have

\[ \alpha=\frac{kM}{c^2}=\frac{(2\pi)^2a^3}{T^2c^2} \]

\[ e_1+e_2=\frac{2}{a(1-\varepsilon^2)}. \]

Substituting, we find:

\[ \omega = 24\pi^3 \frac{a^2}{T^2 c^2(1-\varepsilon^2)} ; \]

this quantity is very small; for the planet Mercury and for one hundred years, i.e. for the quantity \(\Omega = \dfrac{100}{T'}\omega\), where \(T'\) is the period of revolution of Mercury, expressed in terrestrial years, we obtain

\[ \Omega = 43'' \]

a quantity that agrees excellently with experiment and that no other theory, without introducing new hypotheses specially constructed for this purpose, can explain!^[Indeed, the actually observed motion of Mercury’s perihelion is somewhat larger, but the difference between the observed value and this angle of \(43''\) is explained by the perturbing influence of the other planets; precisely this residual \(43''\) does not admit a satisfactory explanation in the classical theory.]

For the other planets the quantity \(\Omega\) is considerably smaller, and verification by observation cannot have such decisive significance as for Mercury.

Let us consider the rectilinear motion of a material point falling directly toward the sun; it can be shown that the geodesic lines corresponding to such motion are possible. Then \(\varphi =\) constant, and the dependence of \(r\) on \(t\) is determined from the equation

\[ \frac{d^2 r}{dt^2} = \frac{3a}{2r(r-a)} \left(\frac{dr}{dt}\right)^2 - \frac{a(r-a)}{2r^3} \]

In this expression the ordinary velocity of light has been taken as unity; we see that if

\[ \left|\frac{dr}{dt}\right| < \frac{1}{\sqrt{3}}\frac{r-a}{r} = \frac{c_r}{\sqrt{3}} \]

then the acceleration will be positive; but if

\[ \frac{dr}{dt} > \frac{c_r}{\sqrt{3}} \]

then the acceleration will be negative; the quantity \(c_r\), as we shall see below, has the value of the velocity of light at the point \(r\).

In our calculations we have taken the mass of the moving planet to be equal to unity. Therefore, from the point of view of ordinary mechanics, we may regard the expression for \(\dfrac{d^2 r}{dt^2}\) as the expression for the force acting on unit mass. We see that Einstein’s theory does without the concept of force, but the question may arise whether a corresponding change in Newton’s law could give the same results as Einstein’s theory. The answer to this is negative. Indeed, for rectilinear motion, the expression for the Newtonian force would have to have the form

\[ F_a = -\frac{a}{2r^2} + \frac{a^2}{2r^3} + \frac{3a}{2r(r-a)} \left(\frac{dr}{dt}\right)^2 ; \]

this expression, in the limit, for very small \(\alpha\), passes into the classical expression

\[ F=-\frac{kM}{r^2}. \]

But we have seen that for motion in a circle, according to Einstein’s theory, the force ought to be equal to

\[ F_c=-\frac{\alpha}{2r^2}. \]

It is obvious that \(F_c\) and \(F_d\) cannot be special cases of one and the same general law. If in the expression for \(F_d\) one discards the terms containing the radial velocity, i.e. \(\frac{dr}{dt}\), and equal to zero in motion in a circle, then the entire remaining expression will be \(F_c\). The expression for the force according to Einstein’s theory (if one insists on introducing the term “force” and assigning to it the meaning of a quantity equal to the mass multiplied by the acceleration) will depend on the trajectory of the material point, i.e. it does not have the significance of a universal law in the sense in which Newton’s law of universal gravitation has it. In the limit, i.e. for very small \(\alpha\), \(F_c\) and \(F_d\), of course, coincide and give \(F\).

4th example. Let us now turn to the consideration of the motion of light. Light, like a material point, moves along geodesic lines, but unlike it, and exactly as in the special principle of relativity, the length of these geodesic lines is equal to zero, and for them we have

\[ ds^2=0. \]

In accordance with this, in the integrals of equations (7) we must put \(A=0\), and the trajectories of light rays will be curves determined by integrating the expression

\[ \left(\frac{d\rho}{d\varphi}\right)^2=\frac{1}{B^2}-\rho^2-\alpha\rho^3 \]

in the limit for \(\lim \alpha=0\) the expression is integrated extremely simply; we obtain

\[ B\rho=\operatorname{Sin}(\varphi-\varphi_0), \]

where \(\varphi_0\) is a constant of integration, i.e. simply the straight line

\[ r=\frac{B}{\operatorname{Sin}(\varphi-\varphi_0)}. \]

The quantity \(B\) here has the meaning of the shortest distance of the ray from the sun.

Let us consider not the limiting case \(a=0\), but suppose only that \(a\) is sufficiently small in comparison with the point of the trajectory closest to the sun. Let \(e_1, e_2, e_3\) be the roots of the equation

\[ \frac{1}{B^2}-\rho^2+a\rho^3=0 \]

and let \(e_1\) and \(e_2\), in the limit as \(a \to 0\), pass into the roots of the limiting equation

\[ \frac{1}{B^2}-\rho^2=0, \]

i.e., let

\[ \lim e_1=\frac{1}{B} \quad \text{and} \quad \lim e_2=-\frac{1}{B}. \]

The expression

\[ \frac{d\rho}{\sqrt{\frac{1}{B^2}-\rho^2+a\rho^3}}=d\varphi \qquad \ldots \ldots \ldots \ldots \ldots (10) \]

is integrated approximately in exactly the same way as expression (9) on p. [[unclear: page reference]]. Obviously, we obtain

\[ \varphi-\varphi_0 = -\frac{a}{2}\sqrt{(\rho-e_1)(e_2-\rho)} + \left[ 1+\frac{3}{4}a(e_1+e_2) \right] \arcsin \frac{\frac{e_1+e_2}{2}-\rho}{\frac{e_1-e_2}{2}} \qquad \ldots \ldots \ldots (9) \]

the approximate values for \(e_1\) and \(e_2\) are easy to calculate; we obtain

\[ e_1=\frac{1}{B}-\frac{1}{2}\frac{a}{B^2}; \qquad e_2=-\frac{1}{B}-\frac{1}{2}\frac{a}{B^2}. \]

If expression (9) is written in the form

\[ r= \frac{\dfrac{2}{e_1+e_2}} { 1-\dfrac{e_2-e_1}{e_2+e_1} \sin\left[ \varphi-\varphi_0+\frac{a}{2}\sqrt{(\rho-e_1)(e_2-\rho)} \right] } \]

then it is immediately clear that we are dealing with a curve very similar to a hyperbola, whose eccentricity is

\[ \varepsilon=\frac{e_{2}-e_{1}}{e_{2}+e_{1}}=\frac{2B}{a}; \]

\(B\) approximately denotes the shortest distance of the trajectory from the sun, i.e., by assumption, is very large in comparison with \(a\); this hyperbola therefore has a very large eccentricity and differs little from a straight line.

For the asymptote of this hyperbola \(r=\infty\) and \(\rho=0\); hence \(\varphi\) is determined from the condition:

\[ 1-\frac{e_{2}-e_{1}}{e_{2}+e_{1}} \operatorname{Sin}\left(\varphi-\varphi_{0}+\frac{a}{2}\sqrt{-e_{1}e_{2}}\right)=0. \]

Substitute here the values of \(e_{1}\) and \(e_{2}\) and take an arbitrary constant

\[ \varphi_{0}=\frac{a}{2}\sqrt{-e_{1}e_{2}}; \]

the angle \(\varphi\), measured from this arbitrary direction, will be determined from the condition

\[ \operatorname{Sin}\psi=\operatorname{Sin}(\pi-\varphi)=\frac{e_{2}+e_{1}}{e_{2}-e_{1}}=\frac{a}{2B}. \]

The angles formed by the asymptotes with the direction \(\varphi_{0}\) will therefore be very small and equal to

\[ \varphi=\pm\frac{a}{2B}, \]

and the angle between them will be

\[ \psi=\frac{a}{B}. \]

If a ray of light travels along such a hyperbola, then the sun will be situated at its focus. For parts sufficiently distant from the sun, the motion along the hyperbola may be identified with motion along the asymptote. Thus we arrive at the conclusion that a ray of light, passing around the sun, is deflected by it through the angle:

\[ \psi=\frac{a}{B}=\frac{kM}{c^{2}B} \]

where \(B\) is the shortest distance of the ray from the sun. Einstein calculated this angle for a ray tangent to the solar surface and found \(\psi=1''\); the experiments carried out in 1919 by the English expedition in Brazil brilliantly confirmed this result, predicted by Einstein in advance.

The study of equation (10), which gives the motion of light, presents much that is curious; for lack of space we cannot include this study here; we shall indicate only a few interesting cases. If a ray of light approaches sufficiently close to the surface \(r=\frac{3}{2}a\), then it begins to wind around this surface and can no longer move away from it. Not a single ray can penetrate through the surface \(r=a\). If a ray goes in a straight line in the direction toward the center of the sun, then its velocity is determined from the equation

\[ \frac{dr}{dt}=c_r=1-\frac{a}{r}\,{}^{1}), \]

its acceleration is always negative, and it approaches the surface \(r=a\) over an infinitely long time with velocity \(c_r=0\).

Planets revolving around the sun in circles have the greater velocity the closer they are to the sun; a planet revolving in a circle with radius \(r=\frac{3}{2}a\) has the velocity of light, but this velocity of light is not \(c\), but \(\frac{1}{\sqrt{3}}c\); inside the circle \(r=\frac{3}{2}a\), circular motion is impossible.

We see that for the sun \(a=1.5\cdot 10^5\) cm. In comparison with the radius of the sun, this quantity is very small, and therefore the special properties of the surfaces \(r=a\) and \(r=\frac{3}{2}a\) have no practical significance. For the hydrogen molecule, approximately \(a=10^{-49}\).

5th example. The special principle of relativity taught us that the time measured by a moving observer and by an observer at rest do not coincide with one another. Let \(x_1, y_1, z_1\) be three functions of time \(t\) that give us the motion of some point. The element of time measured by an observer at rest will be \(dt\), whereas the element of time \(d\tau\), measured by an observer moving together with the point, is determined from the expression

\[ d\tau^2=dt^2-dx^2-dy^2-dz^2 \]

\(\tau\) is the so-called “proper” time of the point. In the general principle of relativity, we likewise must distinguish the increment of the fourth coordinate \(dt\) from the “proper” time \(d\tau\) of some point. The difference between the “general” and the “special” principle of relativity is only that, in the special one, for a point at rest \(dt=d\tau\), i.e. the “proper” time coincides with the increment of the fourth coordinate—time, whereas in the general principle of relativity this will no longer be so.

\({}^{1})\) In expression \((7)'\) one must set \(A=0\).

Let us take, for example, the gravitational field considered in example 3. If a point is at rest, then \(dx, dy, dz\) are equal to zero (or, in polar coordinates, \(dr, d\varphi, d\vartheta\) are equal to zero), and the “proper” time is determined for each resting point by

\[ d\tau^2=\left(1-\frac{a}{r}\right)dt^2, \]

i.e., in other words, the increment of the fourth coordinate—time—is not equal to the increment of “proper” time, but depends on \(r\) and \(a\), i.e. on the distance from the sun and on the mass.

Let us consider some periodic molecular process, say the process of radiation. It is natural to suppose that, for a molecule or for a particle oscillating in a molecule, the period of radiation characteristic of the given molecule and following from its internal properties will be the period determined from its “proper” time and not dependent on the arbitrarily imposed coordinate system \(x, y, z, t\), and hence on the arbitrarily imposed coordinate—time \(t\); thus, for the molecule itself the period will be the same everywhere. Suppose that this period is so small that it can be denoted by \(d\tau\), and suppose that the velocity of the oscillations is so small that in the expression for \(d\tau\) we may put \(dr=d\varphi=d\vartheta=0\). But we make our observations in the coordinate system chosen by us; the period that we measure will not be \(d\tau\), but \(dt\). Let us now write the condition that \(d\tau\) is the same everywhere, taking \(d\tau\) once on the surface of the sun, i.e. putting \(r=d\) (the radius of the sun), and a second time \(r=D\) (the semi-diameter of the earth’s orbit). Obviously

\[ d\tau^2=\left(1-\frac{a}{d}\right)dt_d^2=\left(1-\frac{a}{D}\right)dt_D^2 \]

where \(dt_d\) and \(dt_D\) are the periods measured respectively on the sun and on the earth. But \(\frac{a}{D}\) is a quantity very small in comparison with \(\frac{a}{d}\); it may be neglected. On the other hand, if \(dt_d\) and \(dt_D\) are the periods, then the reciprocal quantities will be the frequencies \(\nu_d\) and \(\nu_D\), and our condition may be written in the form

\[ \nu_d=\nu_D\left(1-\frac{a}{d}\right)^{\frac{1}{2}}=\nu_D\left(1-\frac{a}{2d}\right). \]

Let us denote \(\nu_D\) simply by \(\nu\) and \(\nu_d-\nu_D\) by \(d\nu\); obviously

\[ d\nu=-\frac{a}{2d}\nu \qquad \text{or, if } \nu=\frac{1}{\lambda} \]

\[ d\lambda=+\frac{a}{2d}\lambda. \]

Light emitted by some luminous gas has precisely the character of the periodic motion considered. We see that the gravitational potential of the sun \(\frac{a}{2d}\) must cause a displacement of the lines emitted by the gas toward the red side \((d\lambda>0)\). Einstein calculated this displacement, and experiments apparently confirmed this result predicted by the theory.

1. We discard the expression containing \(\alpha^2\). ↩↩↩↩↩