Global hyperbolicity for spacetimes with continuous metrics

TL;DR

This paper extends the concept of global hyperbolicity to spacetimes with continuous metrics, showing that key properties and equivalences hold even without smoothness, thus broadening the scope of causal structure analysis.

Contribution

It demonstrates that the classical equivalences of global hyperbolicity remain valid for continuous metrics, expanding the theoretical framework of causal spacetime analysis.

Findings

Global hyperbolicity is equivalent to the compactness of causal curves.

Existence of a Cauchy hypersurface is guaranteed under continuous metrics.

Global hyperbolicity implies causal simplicity and stable causality.

Abstract

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the compactness of the space of causal curves and to the existence of a Cauchy hypersurface. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points.