Reduced phase space formalism for spherically symmetric geometry with a massive dust shell

We perform a Hamiltonian reduction of spherically symmetric Einstein gravity with a thin dust shell of positive rest mass. Three spatial topologies are considered: Euclidean $\left(R^3\right)$, Kruskal ${(S}^2\times R)$, and the spatial topology of a diametrically identified Kruskal $({\mathrm{RP}}^3\backslash$ $\{$a point at infinity$\})$. For the Kruskal and ${\mathrm{RP}}^3$ topologies the reduced phase space is four dimensional, with one canonical pair associated with the shell and the other with the geometry; the latter pair disappears if one prescribes the value of the Schwarzschild mass at an asymptopia or at a throat. For the Euclidean topology the reduced phase space is necessarily two dimensional, with only the canonical pair associated with the shell surviving. A time reparametrization on a two-dimensional phase space is introduced and used to bring the shell Hamiltonians to a simpler (and known) form associated with the proper time of the shell. An alternative reparametrization yields a square-root Hamiltonian that generalizes the Hamiltonian of a test shell in Minkowski space with respect to Minkowski time. Quantization is briefly discussed. The discrete mass spectrum that characterizes natural minisuperspace quantizations of vacuum wormholes and ${\mathrm{RP}}^3$ geons appears to persist as the geometrical part of the mass spectrum when the additional matter degree of freedom is added.