Abstract Formulation of the Spacetime Matching Problem and Null Thin Shells

TL;DR

This paper develops an abstract formalism for matching spacetimes, especially null boundaries, providing explicit formulas for the energy content of null thin shells and extending to multiple matchings with applications to (anti-)de Sitter and Minkowski spacetimes.

Contribution

It introduces a fully abstract hypersurface data formalism for spacetime matching, including null shells, with explicit energy expressions and multiple matching scenarios.

Findings

Explicit energy formulas for null thin shells.

Formalism applies to arbitrary topology.

Includes examples with (anti-)de Sitter and Minkowski spacetimes.

Abstract

The formalism of hypersurface data is a framework to study hypersurfaces of any causal character abstractly (i.e. without the need of viewing them as embedded in an ambient space). In this paper we exploit this formalism to study the general problem of matching two spacetimes in a fully abstract manner, as this turns out to be advantageous over other approaches in several respects. We then concentrate on the case when the boundaries are null and prove that the whole matching is determined by a diffeomorphism $\varphi$ on the abstract data set. By exploiting the gauge structure of the formalism we find explicit expressions for the gravitational/matter-energy content of any null thin shell. The results hold for arbitrary topology. A particular case of interest is when more than one matching is allowed. Assuming that one such matchings has already been solved, we provide explicit expressions for the gravitational/matter-energy content of any other shell in terms of the known one. This situation covers, in particular, all cut-and-paste constructions, where one can simply take as known matching the trivial re-attachment of the two regions. We include, as an example, the most general matching of two regions of the (anti-)de Sitter or Minkowski spacetime across a totally geodesic null hypersurface.