Defining Spatial geometry via Spacetime Causal Structure

Abstract

We  find an  approximation of  the induced spatial distance on a Cauchy hypersurface using only the causal structure and local volume element. The approximation can be made arbitrarily precise for a globally hyperbolic spacetime with compact Cauchy hypersurfaces. This prescription has a discrete analog which we use to evaluate the induced spatial distance in a continuum-like causal set.  As expected, because of  discrete asymptotic silence, this gives a poor approximation to the continuum distance in the "UV".  For larger distances, on the other hand, the approximation works very well, as long as the discreteness scale is significantly smaller than the extrinsic and intrinsic curvature scale of the spatial hypersurface. This paves the way for obtaining detailed spatial geometric information from the causal structure, which has implications for the continuum approximation of causal set theory.