CHAPTER XXXIV
THE FINITENESS OF THE UNIVERSE
IN classical science, space, the fundamental continuum, was thought to be Euclidean and flat throughout, hence infinite in extent. As soon, however, as Einstein was led to believe that the fundamental space-time continuum of the universe was not rigorously flat, but manifested various degrees of curvature from place to place around matter, the possibility of its turning out to be curled round on itself, and hence finite, had to be investigated further before any definite opinion could be expressed. True, this fundamental continuum was no longer three-dimensional space; it was four-dimensional space-time. But, apart from this difference, the same major problem still presented itself, and we were permitted to ask: “Is the universe of space-time infinite or is it finite?”
More precisely, if we appeal to a two-dimensional analogy, must we regard the universe of space-time as curved locally around the individual masses of the universe, but as flat or quasi-Euclidean on the whole over an infinite area, much like the rippled surface of an infinite pond of water? Or must we suppose that the rippled surface of the water is more akin to that of the rippled surface of a liquid sphere closing round on itself?
Now, if we consider the theory in its present state, we must agree that a finite universe is excluded. We can understand this point by considering the law of gravitation, which in Einstein’s theory is also a law of space-time curvature. We remember that the gravitational equations were given by
where the
’s represented the curvatures of space-time at a point and where
represented the characteristics of matter at the same point. Outside matter