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

TL;DR

This paper develops a Hamiltonian reduction framework for spherically symmetric Einstein gravity with a massive dust shell, exploring different topologies and their phase spaces, and discusses implications for quantization and mass spectra.

Contribution

It introduces a reduced phase space formalism for spherically symmetric gravity with a dust shell, analyzing topologies and simplifying shell Hamiltonians for quantization.

Findings

Reduced phase space is four-dimensional for Kruskal and RP^3 topologies.

Euclidean topology yields a two-dimensional phase space.

The shell Hamiltonian can be simplified to a proper time form or a square-root form.

Abstract

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 (R^3), Kruskal (S^2 x R), and the spatial topology of a diametrically identified Kruskal (RP^3 - {a point at infinity}). For the Kruskal and 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 RP^3-geons appears to persist as the geometrical part of the mass spectrum when the additional matter degree of freedom is added.