Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision

TL;DR

This paper reviews recent advances in the smoothness of time functions and Cauchy hypersurfaces in causality theory, providing new proofs and clarifying the hierarchy of spacetime causality conditions.

Contribution

It offers a comprehensive revision of the causal hierarchy, including self-contained proofs for key properties of causally continuous, stably causal, and globally hyperbolic spacetimes.

Findings

Resolved folk questions on smoothability of time functions

Provided self-contained proofs for causally continuous spacetimes

Clarified the causal hierarchy and its implications

Abstract

After the heroic epoch of Causality Theory, problems concerning the smoothability of time functions and Cauchy hypersurfaces remained as unanswered folk questions. Just recently solved, our aim is to discuss the state of the art on this topic, including self-contained proofs for questions on causally continuous, stably causal and globally hyperbolic spacetimes.