Structure of globally hyperbolic spacetimes with timelike boundary

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.

Added Appendix B and four references, several improvements in section 2.4 and other minor modifications. To appear in Rev. Mat. Iberoamericana