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.

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

Cited by in corpus (15)

• On the causal hierarchy of Lorentzian length spaces
• Covariant canonical formulations of classical field theories
• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes
• Evolution of confined quantum scalar fields in curved spacetime. Part II
• On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
• Classification theorem and properties of singular solutions to the Tolman-Oppenheimer-Volkoff equation
• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
• Renormalisation in maximally symmetric spaces and semiclassical gravity in Anti-de Sitter spacetime
• Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions
• The space of light rays: Causality and $L$-boundary
• The codimension 2 null cut locus with applications to spacetime topology
• Spontaneous particle creation by oscillating compact stars
• Quantum fields on semi-globally hyperbolic space-times
• On the Cauchy problem for the Fadaray tensor on globally hyperbolic manifolds with timelike boundary