§ 5

EINSTEIN'S THEORY OF GRAVITATION

(a) THE FUNDAMENTAL LAW OF MOTION AND THE PRINCIPLE OF EQUIVALENCE OF THE NEW THEORY

AFTER the foregoing remarks we shall be able to proceed to a short account of Einstein's theory of gravitation. Within the limits of the mathematics assumed in this book we shall, of course, only be able to sketch the outlines so far that the assumptions and hypotheses characteristic of the theory come into clear view and that their relation to the two fundamental postulates of the second section becomes manifest. We start out from the fundamental law of motion in classical mechanics, the law of inertia. Since even in the law of inertia all the weaknesses of the old theory come to light, a new fundamental law of motion becomes an absolute necessity for the new mechanics. It is thus natural that we should start building up the new theory from this point. The new law of motion must be a differential law, which, in the first place, describes the motion of a point-mass under the influence of both inertia and gravity, and which, secondly, always preserves the same form, irrespective of the system of co-ordinates to which it be referred, so that no system of co-ordinates enjoys a preference to any other. The first condition arises from the necessity of ascribing the same importance to gravitational phenomena as to inertial phenomena in the new process of founding mechanics—the law must, therefore, also contain terms which denote the gravitational state of the field from point to point; the second condition is derived from the postulate of the relativity of all motion.

A law of this kind exists in the special theory of relativity in the equation of motion of a single point, not subject to any external influence. According to this equation, the path of a point is the "shortest" or "straightest" line (vide Note 23)—i.e. the "straight line," if the line-element is Euclidean. Written as an equation of variation this law is: If the principle of the shortest path, which is to be followed in actual motions, be elevated in this form to a general differential law for the motion in a gravitational field too, with due regard to the principle of the relativity of all motions, the new fundamental law must run as follows: For only this form of the line-element remains unaltered (invariant) for arbitrary transformations of the , , , . The factors ... , which for the present we leave unexplained, occur in it as something essentially new. Now, the extraordinarily fruitful idea that occurred to Einstein was this: Since the new law is to hold for any arbitrary motions whatsoever, thus also for accelerations, such as we perceive in gravitational fields, we must make the gravitation field, in which the observed motion takes place, responsible for the occurrence of these ten factors . These ten coefficients which will, in general, be functions of the variables ... , must, if the new fundamental law is to be of use, be able to be brought into such relationship to the gravitational field, in which the motion takes place, that they are determined by the field, and that the motion described by equation (1) coincides with the observed motion. This is actually possible. (The 's are the gravitational potentials of the new theory, i.e. they take over the part played by the one gravitational potential in Newton's theory, without, however, having the special properties, which according to our knowledge a potential has, in addition.)

Corresponding to the measure-relations of a space-time manifold based upon the line-element: which is now placed at the foundation of mechanics by virtue of the relativity of all motions, the remaining physical laws must also be so formulated that they remain independent of the accidental choice of the variables. Before we enter into this more closely, the distinguishing features of the theory of gravitation characterized by equation (1) will be considered in greater detail.

The postulate of the new theory, that the laws of mechanics are only to contain statements about the relative motions of bodies, and that, in particular, the motion of a body under the action of the attraction of the remaining bodies is to be symbolically described by the formula: is fulfilled in Einstein's theory by a physical hypothesis concerning the nature of gravitational phenomena, which he calls the hypothesis or principle (respectively) of equivalence (vide Note 24). This asserts the following:

Any change, which an observer perceives in the passing of any event to be due to a gravitational field, would be perceived by him in exactly the same way, if the gravitational field were not present, provided that he—the observer—makes his system of reference move with the acceleration which was characteristic of the gravitation at his point of observation.

For, if the variables , , , in the equation of motion of a point-mass moving uniformly and rectilinearly (i.e. uninfluenced by gravity) be subjected to any transformation corresponding to the change of the , , , into the co-ordinates , , , of a system of reference which has any accelerated motion whatsoever with regard to the initial system , , , ; then, in general, coefficients , will occur in the transformed expression for , and will be functions of the new variables ... , so that the transformed equation will be: Taking into account the extended region of validity of this equation, one will be able to regard the which arise from the accelerational transformation (vide Note 25) just as well, as due to the action of a gravitational field, which asserts its existence in effecting just these accelerations. Gravitational problems thus resolve into the general science of motion of a relativity-theory of all motions.

By thus accentuating the equivalence of gravitational and accelerational events, we raise the fundamental fact, that all bodies in the gravitational field of the earth fall with equal acceleration, to a fundamental assumption for a new theory of gravitational phenomena. This fact, in spite of its being reckoned amongst the most certain of those gathered from experience, has hitherto not been allotted any position whatsoever in the foundations of mechanics. On the contrary, the Galilean law of inertia makes an event which had never been actually observed (the uniform rectilinear motion of a body, which is not subject to external forces) function as the main-pillar amongst the fundamental laws of mechanics. This brought about the strange view that inertial and gravitational phenomena, which are probably not less intimately connected with one another than electric and magnetic phenomena, have nothing to do with one another. The phenomenon of inertia is placed at the base of classical mechanics as the fundamental property of matter, whereas gravitation is only, as it were, introduced by Newton's law as one of the many possible forces of nature. The remarkable fact of the equality of the inertial and gravitational mass of bodies only appears as an accidental coincidence.

Einstein's principle of equivalence assigns to this fact the rank to which it is entitled in the theory of motional phenomena. The new equation of motion (1) is intended to describe the relative motions of bodies with respect to one another under the influence of both inertia and gravity. The gravitational and inertial phenomena are amalgamated in the one principle that the motion take place in the geodetic line . Since the element of arc preserves its form after any arbitrary transformation of the variables, all systems of reference are equally justified as such, i.e. there is none which is more important than any other.

The most important part of the problem, with which Einstein saw himself confronted, was the setting-up of differential equations for the gravitational potentials of the new theory. With the help of these differential equations, the 's were to be unambiguously calculated (i.e. as single-valued functions) from the distribution of the quantities exciting the gravitational field; and the motion (e.g. of the planets) which was described, according to equation (1) by inserting these values for the 's, had to agree with the observed motion, if the theory was to hold true. In setting up these differential equations for the gravitational potentials Einstein made use of hints gathered from Newton's theory, in which the factor which excites the field in Poisson's equation for the Newtonian gravitational potential (viz. the factor represented by , the density of mass in this equation) is put proportional to a differential expression of the second order. This circumstance prescribes, as it were, the method of building up these equations, taking for granted that they are to assume a form similar to that of Poisson's equation.

In conformity with the deepened meaning we have assigned to the mutual relation between inertia and gravity, as well as to the connection between the inertia and latent energy of a body, we find that ten components of the quantity which determines the "energetic" state at any point of the field, and which was already introduced by the special theory of relativity as "stress-energy-tensor," duly make their appearance in place of the density of mass , in Poisson's equation.

Concerning the differential expressions of the second order in the 's which are to correspond to the of Poisson's equation, Riemann has shown the following: the measure-relations of a manifold based on the line-element are in the first place determined by a differential expression of the fourth degree (the Riemann-Christoffel Tensor), which is independent of the arbitrary choice of the variables ... and from which all other differential expressions which are likewise independent of the arbitrary choice of the variables ... and only contain the 's and their derivatives, can be developed (by means of algebraical and differential operations). This differential expression leads unambiguously, i.e. in only one possible way, to ten differential expressions in the 's. And now, in order to arrive at the required differential equations, Einstein puts these ten differential expressions proportional to the ten components of the stress-energy-tensor, regarding the latter ten as the quantities exciting the field. He inserts the gravitational constant as the constant of gravitation. These differential equations for the 's, together with the principle of motion given above, represent the fundamental laws of the new theory. To the first order they, in point of fact, lead to those forms of motion, with which Newton's theory has familiarized us (vide Note 26). More than this, without requiring the addition of any further hypothesis, they mathematically account for the only phenomenon in the theory of planetary motion which could not be explained on the Newtonian theory, viz. the occurrence of the remainder-term in the expression for the motion of Mercury's perihelion. Yet we must bear in mind that there is a certain arbitrariness in these hypotheses just as in that made for the fundamental law of motion. Only the careful elaboration of the new theory in all its consequences, and the experimental testing of it will decide whether the new laws have received their final forms.

Since the formulæ of the new theory are based upon a space-time-manifold, the line-element of which has the general form all other physical laws, in order to bring the general theory of relativity to its logical conclusion, must receive (see p. 46) a form which, in agreement with the new measure-conditions, must be independent of the arbitrary choice of the four variables , , , .

Mathematics has already performed the preliminary work for the solution of this problem in the calculus of absolute differentials; Einstein has elaborated them for his particular purposes (in his essay "Concerning the formal foundations of the general theory of relativity"[12]); Gauss invented the calculus of absolute differentials in order to study those properties of a surface (in the theory of surfaces) which are not affected by the position of the surface in space nor by inelastic continuous deformations of the surface (deformations without tearing), so that the value of the line-element does not alter at any point of the surface.

As such properties depend upon the inner measure-relations of the surface only, one avoids referring, in the theory of surfaces, to the usual system of co-ordinates, i.e. one avoids reference to points which do not themselves lie on the surface. Instead of this, every point in the surface is fixed, by covering the surface with a net-work, consisting of two intersecting arbitrary systems of curves, in which each curve is characterized by a parameter; every point of the surface is then unambiguously, i.e. singly, defined by the two parameters of the two curves (one from each system) which pass through it. According to this view of surfaces, a cylindrical envelope and a plane, for instance, are not to be regarded as different configurations: for each can be unfolded upon the other without stretching, and accordingly the same planimetry holds for both—a criterion that the inner measure-relations of these two manifolds are the same (vide Note 27). The general theory of relativity is based upon the same view; but now not as applied to the two-dimensional manifold of surfaces, but with respect to the four-dimensional space-time manifold. As the four space-time variables are devoid of all physical meaning, and are only to be regarded as four parameters, it will be natural to choose a representation of the physical laws, which provides us with differential laws which are independent of the chance choice of the , , , ; this what is done by the calculus of absolute differentials. The results of the preceding paragraphs, the far-reaching consequences of which can be fully recognized only by a detailed study of the mathematical developments involved, may be summarized as follows:

A mechanics of the relative motions of bodies, which is in harmony with the two fundamental postulates of continuity and relativity, can be built up only on a fundamental law of motion that preserves its form independently of the kind of motion the system is undergoing. An available law of this kind is given if we raise the law of motion along a geodetic line, which, in the special theory of relativity, holds only for a body moving under no forces, to the rank of a general differential law of the motion in the gravitational field, too. In this general law, we must, it is true, give the line-element of the orbit of the moving body the general form: at which we arrive in the second section, using as our basis the two fundamental postulates. The new functions that now occur may be interpreted as the potentials of the gravitational field, if we take our stand on the hypothesis of equivalence. To calculate the quantities from the factors determining the gravitational field, namely, matter and energy, it immediately suggests itself to us to assume a system of differential equations of the second order, that are built up analogously to Poisson's differential equation for the Newtonian gravitational potential. These differential equations, together with the fundamental law of motion, represent the fundamental equations of the new mechanics and the theory of gravitation.

Since the new theory uses the generalized curvilinear co-ordinates , , , , and not the Cartesian co-ordinates of Euclidean geometry, all the other physical laws must also receive a general form that is independent of the special choice of co-ordinates. The mathematical instrument for remoulding these formulæ is given by the general calculus of differentials.

This theory, which is built up from the most general assumptions, leads, for a first approximation, to Newton's laws of motion. Wherever deviations from the theory hitherto accepted reveal themselves, we have possibilities of testing the new theory experimentally. Before we turn to this question, let us look back, and become clear as to the attitude which the general theory of relativity compels us to adopt towards the various questions of principle we have touched upon in the course of this essay.

(b) RETROSPECT

1. The conceptions "inertial" and "gravitational" (heavy) mass no longer have the absolute meaning which was assigned to them in Newton's mechanics. The "mass" of a body depends, on the contrary, exclusively upon the presence and relative position of the remaining bodies in the universe. The equality of inertial and gravitational mass is put at the head of the theory as a rigorously valid principle. The hypothesis of equivalence herein supplements the deduction of the special theory of relativity, that all energy possesses inertia, by investing all energy with a corresponding gravitation. It becomes possible—on the basis (be it said) of certain special assumptions into which we cannot enter here—to regard rotations unrestrictedly as relative motions too, so that the centrifugal field around a rotating body can be interpreted as a gravitational field, produced by the revolution of all the masses in the universe about the non-rotating body in question. In this manner mechanics becomes a perfectly general theory of relative motions. As our statements are concerned only with observations of relative motions, the new mechanics fulfils the postulate that in physical laws observable things only are to be brought into causal connection with one another. It also fulfils the postulate of continuity; since the new fundamental laws of mechanics are differential laws, which contain only the line-element and no finite distances between bodies.

2. The principle of the constancy of the velocity of light in vacuo, which was of particular importance in the special theory of relativity, is no longer valid in the general theory of relativity. It preserves its validity only in regions in which the gravitational potentials are constant, finite portions of which we can never meet with in reality. The gravitational field upon the earth's surface is certainly so far constant that the velocity of light, within the limits of accuracy of our measurements, had to appear to be a universal constant in the results of Michelson's experiments. In a gravitational field, however, in which the gravitational potentials vary from place to place, the velocity of light is not constant; the geodetic lines, along which light propagates itself, will thus in general be curved. The proof of the curvature of a ray of light, which passes by in close proximity to the sun, offers us one of the most important possibilities of confirming the new theory.

3. The greatest change has been brought about by the general theory of relativity in our conceptions of space and time.[13]

According to Riemann the expression for the line-element, viz. determines, in our case, the measure-relations of the continuous space-time manifold; and according to Einstein the coefficients of the line-element have, in the general theory of relativity, the significance of gravitational potentials. Quantities, which hitherto had only a purely geometrical import, for the first time became animated with physical meaning. It seems quite natural that gravitation should herein play the fundamental part, viz. that of predominating over the measure-laws of space and time. For there is no physical event in which it does not co-operate, inasmuch as it rules wherever matter and energy come into play. Moreover, it is the only force, according to our present knowledge, which expresses itself quite independently of the physical and chemical constitution of bodies. It therefore without doubt occupies a unique position, in its outstanding importance for the construction of a physical picture of the world.

According to Einstein's theory, then, gravitation is the "inner ground of the metric relations of space and time" in Riemann's sense (vide the final paragraph of Riemann's essay "On the hypotheses which lie at the bases of geometry" quoted on p. 29). If we uphold the view that the space-time manifold is continuously connected, its measure-relations are not then already contained in its definition as being a continuous manifold of the dimensions "four." These have, on the contrary, yet to be gathered from experience. And it is, according to Riemann, the task of the physicist finally to seek the inner ground of these measure-relations in "binding forces which act upon it." Einstein has discovered in his theory of gravitation a solution to this problem, which was presumably first put forward in such clear terms by Riemann. At the same time he gives an answer to the question of the true geometry of physical space, a question which has exercised physicists for the last century,—but an answer, it is true, of a sort quite different from that which had been expected.

The alternative, Euclidean or non-Euclidean geometry, is not decided in favour of either one or the other; but rather space, as a physical thing with given geometrical properties, is banished out of physical laws altogether: just as ether was eliminated out of the laws of electrodynamics by the Lorentz-Einstein special theory of relativity. This, too, is a further step in the sense of the postulate that only observable things are to have a place in physical laws. The inner ground of metric relations of the space-time manifold, in which all physical events take place, lies, according to Einstein's view, in the gravitational conditions. Owing to the continual motion of bodies relatively to one another, these gravitational conditions are continually altering; and, therefore, one cannot speak of an invariable given geometry of measure or distance—whether Euclidean or non-Euclidean. As the laws of physics preserve their form in the general theory of relativity, independent of how the four variables ... may chance to be chosen, the latter have no absolute physical meaning. Accordingly , , , for instance, will not in general denote three distances in space which can be measured with a metre rule, nor will denote a moment of time which can be ascertained by means of a clock. The four variables have only the character of numbers, parameters, and do not immediately allow of an objective interpretation. Time and space have, therefore, not the meaning of real physical things in the description of the events of physical nature.

And yet it seems as if the new theory may even be able to give a definite answer in favour of one or other of the above alternatives, if, namely, we postulate their validity for the world as a whole. The application of the formulæ of the new theory to the world as a whole at first led to the same difficulties as those revealed in classical mechanics. Boundary conditions for what is infinitely distant could not be set up entirely satisfactorily and at the same time satisfy the condition of general relativity. Yet Einstein[14] succeeded in extending the differential equations for the gravitational potentials in such a way that it became possible to apply his theory of gravitation to the universe. The difficulties that arose for the boundary conditions at infinity here vanished, for an extraordinarily interesting reason. For it was shown that in these new formulæ a space that is filled uniformly with matter which is at rest would, to a first approximation, be built up like an, indeed, unbounded, but finitely closed space, so that boundary conditions would not appear at all for infinity. Even if the assumptions that would lead to this result are not fulfilled in the world, yet it must be remembered that the velocities of matter as ascertained in the case of the stars are extraordinarily small compared with the velocity of light which we now take as our unit. Nor does the distribution of the matter so far show, in the main, irregularities sufficient to place Einstein's view of a stationary, uniformly-filled world quite out of the realm of possible truth.

Thus this deduction of the theory would answer our above alternative in this sense: the geometry that we must use as our basis of spatial happening is, indeed, neither Euclidean nor non-Euclidean, but, as stated above, conditioned by the gravitational states from place to place. But a world built up according to the simplest scheme would in the new theory behave on the whole like a finite closed manifold, that is, as if it were non-Euclidean. Even if this result is only of theoretical importance for the present, since the stellar system that we see around us does not fulfil Einstein's assumptions—in particular, the scarcely-to-be-doubted flattening of the Milky Way is not compatible with these simple assumptions—and since we have at present no knowledge of the stellar systems outside the Milky Way, yet this aspect of the theory opens up undreamed-of perspectives for our view of the world as a whole.

4. The gravitational theory, which emerges out of the general theory of relativity, is, in contradistinction to the Newtonian theory, built up, not upon an elementary law of the gravitational forces, but upon an elementary law of the motion of a body in the gravitational field. Consequently, the expressions which would be interpreted as gravitational forces in the new theory play only a minor part in the building-up of the theory (as indeed the conception of force in mechanics altogether is to be regarded as only an auxiliary or derived conception, if we regard it as the object of mechanics to give a flawless description of the motions occurring in physical events).

Nor does Einstein's theory endeavour to explain the nature of gravitation; it does not seek to give a mechanical model, which would symbolize the gravitational effect of two masses upon one another. This is what the various theories involving ether-impulses attempted to do, by freely using hypothetical quantities which had never been actually observed, such as ether-atoms. It is very doubtful whether such endeavours will ever lead to a satisfactory theory of gravitation. For, the difficulties of Newton's mechanics are not contained only in the fact that it formulates the law of gravitation as a law of forces acting at a distance. Two much more serious points are: first, that the close relationship existing between inertial and gravitational phenomena receives no recognition whatsoever, although Newton was already aware of the fact that inertial and gravitational mass are equal; and second, that Newton's mechanics does not present us with a theory of the relative motions of bodies, although we only observe relative motions of bodies with respect to one another. Re-moulding Newton's law of gravitational force, in order to make the attraction of matter more feasible, would therefore not have helped us finally to a satisfactory theory of the phenomena of motion (vide Note 28).

What distinguishes the Newtonian theory, above all, is the extraordinary simplicity of its mathematical form. Classical mechanics, which is built up on Newton's initial construction, will, for this reason, never lose its importance as an excellent mathematical theory for arithmetically following the observed phenomena of motion.

Einstein's theory, on the other hand, as far as the uniformity of its conceptual foundations is concerned, satisfies all the conditions for a physical theory. The fact that (by abandoning the Euclidean measure of distance) it cuts its connection with the familiar representation by means of Cartesian co-ordinates, will not be felt to be a disturbing factor, as soon as the analytical appliances, which have been called into use as a help, have been more generally adopted. This mathematical elaboration of the theory at the same time gives to the astronomer the task of testing the theory experimentally in those phenomena in which measurable deviations from the results of the classical theory arise.