Null shells: general matching across null boundaries and connection with cut-and-paste formalism

TL;DR

This paper establishes the conditions for matching spacetimes across null boundaries, explores the degrees of freedom involved, and connects the formalism with Penrose's cut-and-paste method, providing a comprehensive framework for null shells.

Contribution

It derives necessary and sufficient conditions for null boundary matching, analyzes the associated degrees of freedom, and links the formalism with Penrose's cut-and-paste construction.

Findings

Conditions for null boundary matching depend on a diffeomorphism and a scalar step function.

In some cases, multiple matchings are possible due to boundary freedom.

The most general null shell with non-zero energy and flux is characterized and connected to Penrose's method.

Abstract

Null shells are a useful geometric construction to study the propagation of infinitesimally thin concentrations of massless particles or impulsive waves. In this paper, we determine and study the necessary and sufficient conditions for the matching of two spacetimes with respective null embedded hypersurfaces as boundaries. Whenever the matching is possible, it is shown to depend on a diffeomorphism between the set of null generators in each boundary and a scalar function, called step function, that determines a shift of points along the null generators. Generically there exists at most one possible matching but in some circumstances this is not so. When the null boundaries are totally geodesic, the point-to-point identification between them introduces a freedom whose nature and consequences are analyzed in detail. The expression for the energy-momentum tensor of a general null shell is also derived. Finally, we find the most general shell (with non-zero energy, energy flux and pressure) that can be generated by matching two Minkowski regions across a null hyperplane. This allows us to show how the original Penrose's cut-and-paste construction fits and connects with the standard matching formalism.