Spherically symmetric gravitating shells as a reparametrization invariant system
Abstract
Spherically symmetric thin shells are considered that are made of barotropic ideal fluid and move 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 starting point of the investigation is a general super-Hamiltonian derived in a previous paper 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 variational principle obtained by this method depends only on shell variables, is reparametrization invariant, and contains both first- and second-class constraints. 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. Other equivalent forms of the constrained system are derived. 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 one of the super-Hamiltonians is extended to this manifold by continuity.Keywords
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