Spherically symmetric gravitating shell as a reparametrization invariant system

TL;DR

This paper reformulates the dynamics of spherically symmetric gravitating shells into a reparametrization invariant system, simplifying the action and analyzing the structure of constraints and phase space, especially near black hole horizons.

Contribution

It derives a simplified, reparametrization invariant variational principle for gravitating shells, extending the phase space to include horizon regions and analyzing the constraint structure.

Findings

The system depends only on shell variables after transformation.

The phase space includes 16 sectors connected into a single symplectic manifold.

Poisson brackets vanish at intersections of horizons.

Abstract

The subject of this paper are spherically symmetric thin shells made of barotropic ideal fluid and moving under the influence of their own gravitational field as well as that of a central black hole; the cosmological constant is assumed to be zero. The general super-Hamiltonian derived in a previous paper is rewritten for this spherically symmetric special case. The dependence of the resulting action on the gravitational variables is trivialized by a transformation due to Kucha\v{r}. The resulting variational principle depends only on shell variables, is reparametrization invariant, and includes both first- and second-class constraints. Several equivalent forms of the constrained system are written down. Exclusion of the second-class constraints leads to a super-Hamiltonian which appears to overlap with that by Ansoldi et al. in a quarter of the phase space. As Kucha\v{r}' variables are singular at the horizons of both Schwarzschild spacetimes inside and outside the shell, the dynamics is first well-defined only inside of 16 disjoint sectors. The 16 sectors are, however, shown to be contained in a single, connected symplectic manifold and the constraints are extended to this manifold by continuity. Poisson bracket between no two independent spacetime coordinates of the shell vanish at any intersection of two horizons.