Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

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.

arXiv comment51 pages, 9 figures. Final version with minor modifications to appear in the Springer Topical Collection Singularity theorems, causality, and all that (SCRI21)

References in corpus (43)

• The generally covariant locality principle -- A new paradigm for local quantum physics
• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes
• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Causality theory for closed cone structures with applications
• On the definition and examples of cones and Finsler spacetimes
• Problems which are well-posed in a generalized sense with applications to the Einstein equations
• Causal Relationship: a new tool for the causal characterization of Lorentzian manifolds
• On the interplay between Lorentzian Causality and Finsler metrics of Randers type
• Topology of the Future Chronological Boundary: Universality for Spacelike Boundaries
• An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary
• Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions
• Topological censorship for Kaluza-Klein space-times
• Quantum charges and spacetime topology: The emergence of new superselection sectors
• The Causal Boundary of spacetimes revisited
• Universality of the Future Chronological Boundary
• On differentiability of volume time functions
• Time functions as utilities
• Globally hyperbolic spacetimes can be defined without the 'causal' condition
• Global hyperbolicity is stable in the interval topology
• Topology of the Causal Boundary for Standard Static Spacetimes
• The causal boundary of product spacetimes
• Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary
• Structure of globally hyperbolic spacetimes with timelike boundary
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples
• Cosmic censorship of smooth structures
• Boundary Conditions for the Gravitational Field
• Connecting solutions of the Lorentz force equation do exist
• Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case
• Isocausal spacetimes may have different causal boundaries
• Geometrization of metric boundary data for Einstein's equations
• On the existence of smooth Cauchy steep time functions
• A note on invariant temporal functions
• Computability of the causal boundary by using isocausality
• Determining an asymptotically AdS spacetime from data on its conformal boundary
• A note on the causal homotopy classes of a globally hyperbolic spacetime
• Killing boundary data for anti-de Sitter-like spacetimes
• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
• On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
• On the global "two-sided" characteristic Cauchy problem for linear wave equations on manifolds
• Cauchy surfaces and diffeomorphism types of globally hyperbolic spacetimes

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• Pseudo-timelike loops in signature changing semi-Riemannian manifolds with a transverse radical
• Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds
• Global Hyperbolicity and Self-adjointness
• A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes
• On complete trapped submanifolds in globally hyperbolic spacetimes