SPHERICALLY SYMMETRICAL GRAVITATING SHELL AS A REPARAMETRIZATION-INVARIANT SYSTEM

The subject of this paper is spherically symmetric thin shells made of a baryotropic ideal fluid which moves under the influence of its own gravitational field as well as that of a central black hole; the cosmo logical constant is assumed to be zero. The general super-Hamiltonian derived in a previous paper is rewritten for this spherically symmetri c special case. The dependence of the resulting action on the gravitat ional variables is trivialized by a transformation due to Kuchar. The resulting variational principle depends only on shell variables, is re parametrization invariant, and includes both first-and second-class co nstraints. Several equivalent forms of the constrained system are writ ten down. The exclusion of the second-class constraints leads to a sup er-Hamiltonian which appears to overlap with that by Ansoldi et al. in a quarter of the phase space. As the Kuchar 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 s ectors. The 16 sectors are, however, shown to be contained in a single , connected symplectic manifold and the constraints are extended to th is manifold by continuity. Poisson brackets between no two independent spacetime coordinates of the shell vanish at any intersection of two horizons.