Generalizing the Penrose cut-and-paste method: Null shells with pressure and energy flux

TL;DR

This paper extends Penrose's cut-and-paste method to construct null shells with arbitrary matter content, including pressure and energy flux, by deriving a generalized metric and providing an explicit example in Minkowski space.

Contribution

It introduces a generalized cut-and-paste technique for null shells with diverse matter properties, broadening the method's applicability beyond pure gravitational or null-dust shells.

Findings

Derived a Lipschitz continuous metric form for general null shell matching.

Obtained coordinate transformations for the generalized cut-and-paste metric.

Provided an explicit example of a null shell with energy flux and pressure in Minkowski space.

Abstract

The cut-and-paste method is a procedure for constructing null thin shells by matching two regions of the same spacetime across a null hypersurface. Originally proposed by Penrose, it has so far allowed to describe purely gravitational and null-dust shells in constant-curvature backgrounds. In this paper, we extend the cut-and-paste method to null shells with arbitrary gravitational/matter content. To that aim, we first derive a locally Lipschitz continuous form of the metric of the spacetime resulting from the most general matching of two constant-curvature spacetimes with totally geodesic null boundaries, and then obtain the coordinate transformation that turns this metric into the cut-and-paste form with a Dirac-delta term. The paper includes an example of a null shell with non-trivial energy density, energy flux and pressure in Minkowski space.