General matching across Killing horizons of zero order

TL;DR

This paper investigates the geometric conditions for matching spacetimes across Killing horizons of zero order, analyzing different boundary types and symmetries, and explicitly describing the associated shells and their energy-momentum tensors.

Contribution

It provides a comprehensive analysis of matching conditions across Killing horizons of zero order, including explicit forms of the step function and shell properties in various symmetry cases.

Findings

Different matching conditions for non-degenerate and degenerate boundaries.

Explicit form of the step function in each case.

Complete description of shells on bifurcation surfaces.

Abstract

Null shells are a useful geometric construction to study the propagation of infinitesimally thin concentrations of massless particles or impulsive waves. After recalling the necessary and sufficient conditions obtained in [28] that allow for the matching of two spacetimes with null embedded hypersurfaces as boundaries, we will address the problem of matching across Killing horizons of zero order in the case when the symmetry generators are to be identified. The results are substantially different depending on whether the boundaries are non-degenerate or degenerate, and contain or not fixed points (in particular, in the former case the shells have zero pressure but non-vanishing energy density and energy flux in general). We will present the explicit form of the so-called step function in each situation. We will then concentrate on the case of actual Killing horizons admitting a bifurcation surface, where a complete description of the shell and its energy-momentum tensor can be obtained. We will conclude particularizing to the matching of two spacetimes with spherical, plane or hyperbolic symmetry without imposing this symmetry on the shell itself.