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

摘要

We perform a Hamiltonian reduction of spherically symmetric Einstein gravitywith a thin dust shell of positive rest mass. Three spatial topologies areconsidered: Euclidean (R^3), Kruskal (S^2 x R), and the spatial topology of adiametrically identified Kruskal (RP^3 - {a point at infinity}). For theKruskal and RP^3 topologies the reduced phase space is four-dimensional, withone canonical pair associated with the shell and the other with the geometry;the latter pair disappears if one prescribes the value of the Schwarzschildmass at an asymptopia or at a throat. For the Euclidean topology the reducedphase space is necessarily two-dimensional, with only the canonical pairassociated with the shell surviving. A time-reparametrization on atwo-dimensional phase space is introduced and used to bring the shellHamiltonians to a simpler (and known) form associated with the proper time ofthe shell. An alternative reparametrization yields a square-root Hamiltonianthat generalizes the Hamiltonian of a test shell in Minkowski space withrespect to Minkowski time. Quantization is briefly discussed. The discrete massspectrum that characterizes natural minisuperspace quantizations of vacuumwormholes and RP^3-geons appears to persist as the geometrical part of the massspectrum when the additional matter degree of freedom is added.