Structure of globally hyperbolic spacetimes with timelike boundary

TL;DR

This paper studies the structure of globally hyperbolic spacetimes with timelike boundary, proving their orthogonal splitting and embedding properties, which extend known results to spacetimes with boundary and naked singularities.

Contribution

It establishes the orthogonal splitting of such spacetimes and constructs a Cauchy temporal function tangent to the boundary, extending properties of boundaryless spacetimes.

Findings

Globally hyperbolic spacetimes with timelike boundary split orthogonally as ${ m R} imes ar{ m abla}$.

Existence of a Cauchy temporal function with gradient tangent to the boundary.

Interior regions embed isometrically into ${ m L}^N$, extending properties to causally continuous spacetimes.

Abstract

Globally hyperbolic spacetimes with timelike boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\overline{M}$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $\partial M$, the splitting of any globally hyperbolic $(\overline{M},g)$ as an orthogonal product ${\mathbb R}\times \bar{\Sigma}$ with Cauchy slices with boundary $\{t\}\times \bar{\Sigma}$ is proved. This is obtained by constructing a Cauchy temporal function $\tau$ with gradient $\nabla \tau$ tangent to $\partial M$ on the boundary. To construct such a $\tau$, results on stability of both, global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by $\partial M$. As a consequence, the interior $M$ both, splits orthogonally and can be embedded isometrically in ${\mathbb L}^N$, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.