Geometry of crossing null shells

TL;DR

This paper introduces new geometric objects on null thin layers, reformulates the Barrabès–Israel equations, and proves their continuity properties, providing insights into crossing null shells with applications to spherical symmetry.

Contribution

It presents a novel geometric framework for null shells, reformulates key equations, and proves their continuity properties, advancing the understanding of crossing null-like shells.

Findings

New geometric objects on null layers are introduced.

Barrabès–Israel equations are reformulated in a new geometric form.

Continuity properties of these objects are proved, leading to the Dray–t'Hooft–Redmount formula in spherical symmetry.

Abstract

New geometric objects on null thin layers are introduced and their importance for crossing null-like shells are discussed. The Barrab\`es--Israel equations are represented in a new geometric form and they split into decoupled system of equations for two different geometric objects: tensor density ${\bf G}^a{_b}$ and vector field $I$. Continuity properties of these objects through a crossing sphere are proved. In the case of spherical symmetry Dray--t'Hooft--Redmount formula results from continuity property of the corresponding object.