Page:Eddington A. Space Time and Gravitation. 1920.djvu/164

Action is the curvature of the world. It is scarcely possible to visualise this statement, because our notion of curvature is derived from surfaces of two dimensions in a three-dimensional space, and this gives too limited an idea of the possibilities of a four-dimensional surface in space of five or more dimensions. In two dimensions there is just one total curvature, and if that vanishes the surface is flat or at least can be unrolled into a plane. In four dimensions there are many coefficients of curvature; but there is one curvature par excellence, which is, of course, an invariant independent of our mesh-system. It is the quantity we have denoted by $G$. It does not follow that if the curvature vanishes space-time is flat; we have seen in fact that in a natural gravitational field space-time is not flat although there may be no mass or energy and therefore no action or curvature.

Wherever there is matter^[1] there is action and therefore curvature; and it is interesting to notice that in ordinary matter the curvature of the space-time world is by no means insignificant. For example, in water of ordinary density the curvature is the same as that of space in the form of a sphere of radius 570,000,000 kilometres. The result is even more surprising if expressed in time units; the radius is about half-an-hour.

It is difficult to picture quite what this means; but at least we can predict that a globe of water of 570,000,000 km. radius would have extraordinary properties. Presumably there must be an upper limit to the possible size of a globe of water. So far as I can make out a homogeneous mass of water of about this size (and no larger) could exist. It would have no centre, and no boundary, every point of it being in the same position with respect to the whole mass as every other point of it—like points on the surface of a sphere with respect to the surface. Any ray of light after travelling for an hour or two would come back to the starting point. Nothing could enter or leave the mass, because there is no boundary to enter or leave by; in fact, it is coextensive with space. There could not be any other world anywhere else, because there isn't an "anywhere else."

The mass of this volume of water is not so great as the most

1. It is rather curious that there is no action in space containing only light. Light has mass ($M$) of the ordinary kind; but the invariant mass ($m$) vanishes.