On the variational principle for dust shells in General Relativity

TL;DR

This paper develops a variational principle for thin dust shells in General Relativity, enabling the derivation of boundary conditions and effective actions from different observational perspectives, and explores their canonical equivalence.

Contribution

It introduces a consistent variational framework for dust shells, deriving natural boundary conditions and effective actions for interior and exterior observers, and demonstrates their canonical equivalence.

Findings

Derived natural boundary conditions for dust shells

Formulated two variants of effective action for different observers

Proved canonical equivalence of the systems in extended phase space

Abstract

The variational principle for a thin dust shell in General Relativity is constructed. The principle is compatible with the boundary-value problem of the corresponding Euler-Lagrange equations, and leads to ``natural boundary conditions'' on the shell. These conditions and the gravitational field equations which follow from an initial variational principle, are used for elimination of the gravitational degrees of freedom. The transformation of the variational formula for spherically-symmetric systems leads to two natural variants of the effective action. One of these variants describes the shell from a stationary interior observer's point of view, another from the exterior one. The conditions of isometry of the exterior and interior faces of the shell lead to the momentum and Hamiltonian constraints. The canonical equivalence of the mentioned systems is shown in the extended phase space. Some particular cases are considered.