Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

TL;DR

This paper surveys the relationship between Cauchy slicings and causal boundaries in globally hyperbolic spacetimes, introduces counterexamples illustrating key properties, and refines understanding of the structure of such metrics.

Contribution

It provides new counterexamples and insights into the properties of Cauchy slicings, causal boundaries, and the structure of globally hyperbolic metrics on product manifolds.

Findings

Counterexamples show independence between slice completeness and hyperbolicity

Uniform bounds on slicings are necessary but not sufficient for causal boundary computation

The space of globally hyperbolic metrics is non-convex but path-connected

Abstract

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed and revisited, starting at the seminal conformal boundary constructions by R. Penrose. Our study covers: (1) adaptive possibilities and techniques for their Cauchy slicings, (2) global hyperbolicity of sliced spacetimes, (3) critical review on the conformal and causal boundaries for a globally hyperbolic spacetime, and (4) procedures to compute the causal boundary of a Cauchy temporal splitting by using isocausal comparison with a static product. New simple counterexamples on $\mathbb{R}^2$ illustrate a variety of possibilities related to these splittings, such as the logical independence (for normalized sliced spacetimes) between the completeness of the slices and global hyperbolicity, the necessity of uniform bounds on the slicings in order to ensure global hyperbolicity, or the insufficience of these bounds for the computation of the causal boundary. A refinement of one of these examples shows that the space of all the (normalized, conformal classes of) globally hyperbolic metrics on a smooth product manifold $\mathbb{R}\times S$ is not convex, even though it is path connected by means of piecewise convex combinations.