Analysis of a Shell System in General Relativity

Abstract

This Thesis concerns a thin fluid shell embedded in its own gravitational field. The starting point is a work of Hajicek and Kijowski, where the hamiltonian formalism for shell(s) (with no symmetry) in Einstein gravity is developed. An open problem at the end of that paper is to show how the hamiltonian formalism defines a regular constrained system: the hamiltonian and the constraints must be differentiable functionals on the phase space, so that their Poisson Brackets are well defined objects. On the contrary, some constraints at the shell result to be non differentiable functionals on the phase space. This problem is tackled, in the present thesis, by following the reduction procedure suggested by Teitelboim and Henneaux: the singular constraints are solved and the solution is substituted back into the hamiltonian. The resulting hamiltonian is shown to lead to equivalent dynamics, without singular constraints. Besides, the final reduced system (hamiltonian plus canonical constraints) is shown to be fully differentiable on the reduced phase space.