1915. Gravitational field equations.
Explanation of the motion of Mercury's perihelion.
[INTRODUCTION]
TOWARDS the end of 1915 Albert Einstein brought to its conclusion a theory of gravitation on the basis of a general principle of relativity of all motions. His object was to create not a visual picture of the action of an attractive force between bodies, but rather a mechanics of the motions of the bodies relative to one another under the influence of inertia and gravity. To attain this difficult goal, it is true, many time-honoured views had to be sacrificed, but as a reward a standpoint was reached which had long seemed the highest aim of all who had occupied their minds with theoretical physics. The fact that these sacrifices are demanded by the new theory must, indeed, inspire confidence in it. For the unsuccessful attempts that have been made during the last centuries to fit the doctrine of gravitation satisfactorily into the scheme of natural science necessarily lead to the conclusion that this would not be possible without giving up many deeply-rooted ideas. As a matter of fact, Einstein reverted to the foundation pillars of mechanics as starting-points on which to build his theory, and he did not satisfy himself by merely reforming the Newtonian law in order to establish a link with the more recent views.
To get at an understanding of Einstein's ideas, we must compare the fundamental point of view adopted by Einstein with that of classical mechanics. We then recognize that a logical development leads from "the special" principle of relativity to the general theory, and simultaneously to a theory of gravitation.
THEORY OF GRAVITATION
§ 1
THE "SPECIAL" THEORY OF RELATIVITY AS A STEPPING STONE TO THE "GENERAL" THEORY OF RELATIVITY
THE complete upheaval which we are witnessing in the world of physics at the present time received its impulse from obstacles which were encountered in the progress of electrodynamics. Yet the important point in the later development was that an escape from these difficulties was possible[1] only by founding mechanics on a new basis.
[1]Note.—Most of the objections to the new development have, it is admitted, been raised because a branch of science which was not considered to have a just claim to deal with questions of mechanics, asserted the right of exercising a far-reaching influence upon the latter, extending even to its foundation. If, however, we trace these objections to their source, we discover that they are due to a wish to give mechanics the form of a purely mathematical science, similar to geometry, in spite of the fact that it is founded upon hypotheses which are essentially physical: up to the present, certainly, these hypotheses have not been recognized to be such.
The development of electrodynamics took place essentially without being influenced by the results of mechanics, and without itself exerting any influence upon the latter, so long as its range of investigation remained confined to the electrodynamic phenomena of bodies at rest. Only after Maxwell's equations had furnished a foundation for these did it become possible to take up the study of the electrodynamic phenomena of moving media. All optical occurrences—and according to Maxwell's theory all these also belong to the sphere of electrodynamics—take place either between stellar bodies which are in motion relatively to one another, or upon the earth, which revolves about the sun with a velocity of about 30 kilometres per second, and performs, together with the sun, a translational motion of about the same order of magnitude through the region of the stellar system. Hence questions of great fundamental importance at once asserted themselves. Does the motion of a light-source leave its trace on the velocity of the light emitted by it? And what is the influence of the earth's motion on the optical phenomena which occur on its surface, for example, in optical experiments in a laboratory? An endeavour was therefore to be made to find a theory of these phenomena in which electrodynamic and mechanical effects occurred simultaneously (vide [Note 1]). Mechanics, which had long stood as a structure complete in every detail, had to stand the test as to whether it was capable of supplying the fitting arguments for a description of such phenomena. Thus the problem of electrodynamic events in the case of moving matter became at the same time a decisive problem of mechanics.
The first outstanding attempt to describe these phenomena for moving bodies was made by H. Hertz. He extended Maxwell's equations by additional terms so as also to express the influence of the motion of matter on electrodynamic phenomena, and in his extensions he adopted the view, characteristic for his theory, that the carrier of the electromagnetic field, the ether, everywhere participates in the motion of matter. Consequently, in his equations the state of motion of the ether, as denoting the state of the ether, occurs as well as the electromagnetic field. As is well known, Hertz's extensions cannot be brought into harmony with the results of observation, for example, that of Fizeau's experiment ([Note 2]), so that they excite merely an historic interest as a land-mark on the road to an electrodynamics of moving matter. Lorentz was the first to derive from Maxwell's theory fundamental electrodynamic equations for moving matter which were in essential agreement with observation. He, indeed, succeeded in this only by renouncing a principle of fundamental importance, namely, by disallowing that Newton's and Galilei's principle of relativity of classical mechanics also holds for electrodynamics. The practical success of Lorentz's theory at first almost made us fail to see this sacrifice, but then the disintegration set in at this point which finally made the position of classical mechanics untenable. To understand this development we therefore require a detailed treatment of the principle of relativity in the fundamental equations of physics.