Globally hyperbolic spacetimes can be defined without the 'causal' condition

TL;DR

This paper demonstrates that in reasonable spacetimes, global hyperbolicity can be characterized solely by the compactness of causal diamonds, eliminating the need for an explicit causality condition, and extends the concept to less regular structures.

Contribution

It proves that causality can be deduced from the compactness of causal diamonds in reasonable spacetimes and extends the definition of global hyperbolicity to non-regular and general cone structures.

Findings

Causal diamonds are compact iff the spacetime is globally hyperbolic.

Causality can be deduced from geometric properties without explicit assumptions.

Results apply to non-regular and semi-continuous cone structures.

Abstract

Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non $C^{1,1}$ metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity.