A Temporal Splitting Theorem for Chronological Spaces

Abstract

We obtain a generalization of Geroch's Theorem on temporal foliations in a globally hyperbolic space-time to chronological spaces under suitable topological and order-theoretic requirements. The foliation can be chosen to contain a given relatively compact antichain in one slice. We first prove the corresponding extension result for countable causal sets, and then give a direct proof for chronological spaces using past sets. Furthermore, we show that any chronological space-time admits a generalized Cauchy hypersurface containing a prescribed compact achronal set: it is crossed once by every inextendible timelike curve which is also a maximal chronological chain.

arXiv comment24 pages, 3 figures. Substantially revised and expanded. New proof of the discrete antichain-extension result; the earlier compatibility proposition is omitted. The continuum construction is replaced by a direct past-set argument. Includes temporal splitting for chronological spaces under explicit hypotheses and a separate generalized Cauchy-hypersurface theorem