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.
[14]"Kosmologische Betrachtungen zur allgemeinen Relativitäts-theorie" Sitz. Ber. d. Preuss. Akad. der Wiss., 1917, p. 142.
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.
§ 6
THE VERIFICATION OF THE NEW THEORY BY ACTUAL EXPERIENCE
AS far as can be seen at present, there are three possible experiments for verifying Einstein's theory of gravitation; all three can be performed only by the agency of astronomy. One of them—arising from the deviation of the motion of a material point in the gravitational field according to Einstein's theory, as compared with that required by Newton's theory—has already decided in favour of the new theory: not less so one of the other two that arise through a combination of electromagnetic and gravitational phenomena.