Here, however, we must recall that Einstein’s theory is not one of loose, unsupported guesses. Had Einstein merely been interested in making the universe this or that, without taking into account the totality of facts at the scientist’s disposal, his cosmological speculations would be no more worthy of consideration than those of the Greeks, or of Dante, or of the philosophers of the seventeenth and eighteenth centuries. We may be quite certain, however, that with a conscientious scientist like Einstein the methodological procedure will be very different. And so, when we find that the theory he advocates entails a finite universe, we may anticipate that there will be very strong reasons for his choice. Indeed, as we shall see, a number of separate considerations lead to this solution. First of all, there is the apparent stability of configuration of the star distribution. Then again, we have such philosophical considerations as the probable physical relativity of all motion and of inertia in accordance with Mach’s ideas; the intrinsically amorphous nature of all empty continuous extensions as demanded by Riemann; the desire to discover unity in nature by proving that matter can be reduced to electricity and gravitation. All these facts, together with many others, can be satisfied only provided we assume that the universe is finite. Unfortunately, a proper understanding of the problems involved requires an appeal to mathematical analysis. However, we shall attempt to proceed in a very crude way without introducing mathematical symbols to any great extent.
There exist two major solutions for the finite universe. The first is the Spherical, Elliptical, or, more rigorously, the Hyperbolical universe of de Sitter; the second is the Cylindrical universe championed by Einstein. We will consider de Sitter’s universe first. But before discussing these difficult subjects, there are certain peculiarities about the laws of nature which it will be necessary to mention.
In the majority of cases, the laws of nature are expressed in the form of laws of contiguous variations in space and in time. Mathematically, this is equivalent to saying that the laws are expressed by differential equations. Now differential equations give us the law of variation of magnitudes from place to place or from time to time, but they do not yield us the precise values of these magnitudes.
Consider, for example, a differential assertion such as the following: “A train is moving away from us with a constant acceleration of one mile per second.” Now this differential law, though it specifies the nature of the motion, does not enable us to deduce the position the train will occupy at any specific instant. To solve the problem we must be informed of the train’s position, and also of its velocity, at some initial instant. This illustration gives us an idea of the limitations of differential assertions or equations. In order to render them determinate, we must always supplement them with additional information by stating the value of the variable quantity at some definite point or time. In particular, when we consider a magnitude which may vary according to some differential law from point to point throughout a region of three-dimensional space, we must state the value and other characteristics of the magnitude, or variable, over a closed surface bounding this region of space. In mathematical terminology, we must know the boundary conditions before the law can be regarded as perfectly determinate.
Let us apply these considerations to the differential law which expresses the Newtonian law of attraction. It is given, as we have seen, by Poisson’s equation
, and expresses the law of distribution of
(the Newtonian potential) in a gravitational field around matter and in the interior of matter (where
