Relation between the guessed and the derived super-Hamiltonians for spherically symmetric shells

TL;DR

This paper explores the relationship between two different Hamiltonian formulations of spherically symmetric thin shells in general relativity, revealing their structural differences and implications for quantum extensions.

Contribution

It establishes the connection between two constraint dynamical systems for shell dynamics, introduces simplifying variables, and analyzes their symmetry groups and dimensionalities.

Findings

Gamma_1 is three-dimensional, Gamma_2 is five-dimensional

Gamma_1's description is incomplete for predicting measurable properties

Gamma_1 is locally equivalent to a subsystem of Gamma_2

Abstract

The Hamiltonian dynamics of spherically symmetric massive thin shells in the general relativity is studied. Two different constraint dynamical systems representing this dynamics have been described recently; the relation of these two systems is investigated. The symmetry groups of both systems are found. New variables are used, which among other things simplify the complicated system a great deal. The systems are reduced to presymplectic manifolds Gamma_1 and Gamma_2, lest non-physical aspects like gauge fixings or embeddings in extended phase spaces complicate the line of reasoning. The following facts are shown. Gamma_1 is three- and Gamma_2 is five-dimensional; the description of the shell dynamics by Gamma_1 is incomplete so that some measurable properties of the shell cannot be predicted. Gamma_1 is locally equivalent to a subsystem of Gamma_2 and the corresponding local morphisms are not unique, due to the large symmetry group of Gamma_2. Some consequences for the recent extensions of the quantum shell dynamics through the singularity are discussed.