Recent progress on the notion of global hyperbolicity

TL;DR

This paper reviews classical and recent approaches to global hyperbolicity in Mathematical Relativity, covering concepts like Cauchy hypersurfaces, causal boundaries, and criteria for checking hyperbolicity.

Contribution

It synthesizes recent advances and structural results on globally hyperbolic spacetimes, including embeddability and boundary notions, providing a comprehensive overview.

Findings

Structural results on globally hyperbolic spacetimes

Revised notions of causal and conformal boundaries

Criteria for checking global hyperbolicity in various spacetimes

Abstract

Global hyperbolicity is a central concept in Mathematical Relativity. Here, we review the different approaches to this concept explaining both, classical approaches and recent results. The former includes Cauchy hypersurfaces, naked singularities, and the space of the causal curves connecting two events. The latter includes structural results on globally hyperbolic spacetimes, their embeddability in Lorentz-Minkowski, and the recently revised notions of both, causal and conformal boundaries. Moreover, two criteria for checking global hyperbolicity are reviewed. The first one applies to general splitting spacetimes. The second one characterizes accurately global hyperbolicity and spacelike Cauchy hypersurfaces for standard stationary spacetimes, in terms of a naturally associated Finsler metric.