Page:Eddington A. Space Time and Gravitation. 1920.djvu/174

158 must, so to speak, extend the world. We can imagine the world stretched out like a plane sheet; but then the stretching cause—the cause of the intervals—is relegated beyond the bounds of space and time, i.e. to infinity. This is the view objected to, though the writer does not consider that the objection has much force. An alternative way is to inflate the world from inside, as a balloon is blown out. In this case the stretching force is not relegated to infinity, and ruled outside the scope of experiment; it is acting at every point of space and time, curving the world to a sphere. We thus get the idea that space-time may have an essential curvature on a great scale independent of the small hummocks due to recognised matter.

It is not necessary to speculate whether the curvature is produced (as in the balloon) by some pressure applied from a fifth dimension. For us it will appear as an innate tendency of four-dimensional space-time to curve. It may be asked, what have we gained by substituting a natural curvature of space-time for a natural stretched condition corresponding to the inertial frame? As an explanation, nothing. But there is this difference, that the theory of the inertial frame can now be included in the differential law of gravitation instead of remaining outside and additional to the law.

It will be remembered that one clue by which we previously reached the law of gravitation was that flat space-time must be compatible with it. But if space-time is to have a small natural curvature independent of matter this condition is now altered. It is not difficult to find the necessary alteration of the law. It will contain an additional, and at present unknown, constant, which determines the size of the world.

Spherical space is not very easy to imagine. We have to think of the properties of the surface of a sphere—the two-dimensional case—and try to conceive something similar applied to three-dimensional space. Stationing ourselves at a point let us draw a series of spheres of successively greater radii. The surface of a sphere of radius $$r$$ should be proportional to $$r^2$$; but in spherical space the areas of the more distant spheres begin to fall below the proper proportion. There is not so much room out there as we expected to find. Ultimately we reach a sphere