A Note on Non-compact Cauchy surface

TL;DR

This paper demonstrates that for space-times with non-compact Cauchy surfaces, their entire topological, differentiable, and causal structures can be encoded through specific classes of compact subsets of the Cauchy surface, linking surface properties to the overall space-time structure.

Contribution

It introduces a method to determine the full structure of space-time from compact subsets of its non-compact Cauchy surface, providing a new way to encode space-time geometry.

Findings

Causal structure determines topological and conformal structures.

Full space-time structure can be reconstructed from Cauchy surface subsets.

A natural encoding of space-time structures via Cauchy surface subsets.

Abstract

It is shown that if a space-time has non-compact Cauchy surface, then its topological, differentiable, and causal structure are completely determined by a class of compact subsets of its Cauchy surface. Since causal structure determines its topological, differentiable, and conformal structure of space-time, this gives a natural way to encode the corresponding structures into its Cauchy surface.