Equations for general shells

TL;DR

This paper derives comprehensive equations for shells of arbitrary causal character in any dimension, revealing new relations and extending classical results like Israel equations, with broad applications in gravitational physics.

Contribution

It introduces new equations for shells of arbitrary causal character, including cases with changing causal nature, expanding the theoretical framework beyond traditional null or non-null shells.

Findings

New equations for shells with changing causal character

Recovery of Israel equations for non-null shells

Additional relations involving the Weyl tensor in null shells

Abstract

The complete set of (field) equations for shells of arbitrary, even changing, causal character are derived in arbitrary dimension. New equations that seem to have never been considered in the literature emerge, even in the traditional cases of everywhere non-null, or everywhere null, shells. In the latter case there arise field equations for some degrees of freedom encoded exclusively in the distributional part of the Weyl tensor. For non-null shells the standard Israel equations are recovered but not only, the additional relations containing also relevant information. The results are applicable to a widespread literature on domain walls, branes and braneworlds, gravitational layers, impulsive gravitational waves, and the like. Moreover, they are of a geometric nature, and thus they can be used in any theory based on a Lorentzian manifold.