Three-Dimensional Geometry as Carrier of Information about Time

Abstract

A geometry of curved empty space which evolves in time in accordance with Einstein's field equations may be termed a "geometrodynamical history." It is known that such a history can be specified by giving on a 3-dimensional space-like hypersurface ("initial surface") (1) the geometry intrinsic to this surface and (2) the extrinsic curvature of this surface (having to do with how the surface is imbedded, or is to be imbedded, in a yet-to-be-constructed 4-dimensional manifold). However, the intrinsic and extrinsic curvatures of the surface cannot be specified independently, but have to satisfy the initial value equations of Foures and Lichnerowicz (analogous to div $E=0\mathrm{}$ and div $B=0\mathrm{}$ in electromagnetism). An alternative way of specifying a history is outlined here in which the intrinsic geometry is given freely on each of two hyper-surfaces, and nothing is specified as to the extrinsic curvature of either. In the special case in which the two so-specified 3-geometries are nearly alike—in a sense specified more precisely in the text—a procedure is outlined in order to find the following from Einstein's equations: (1) the invariant space-time interval between an arbitrary point on one surface and a nearby point on the other surface (and thus the 4-geometry interior to the thin sandwich); (2) the extrinsic curvature of the sandwich; hence (via the rest of Einstein's equations) (3) the entire enveloping 4-geometry or geometrodynamical history); and thus finally (4) the time-like separation of the original surfaces and their location in spacetime. In this sense two 3-geometries carry latent information about time.