Variational formalism for generic shells in general relativity

TL;DR

This paper develops a unified variational framework for describing thin shells of arbitrary signature in general relativity, generalizing previous separate treatments for timelike and null shells, and reproduces known results through regularization schemes.

Contribution

It introduces a unified variational principle for generic shells in GR, including regularization methods, and derives shell equations consistent with prior distribution-based results.

Findings

Unified shell equations for arbitrary signatures derived

Regularization schemes shown to be equivalent

Results consistent with previous distribution theory approaches

Abstract

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equations derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.