Hamiltonian formalism for the Oppenheimer-Snyder model

TL;DR

This paper derives a Hamiltonian formalism for the Oppenheimer-Snyder model of gravitational collapse, providing a framework for analyzing the dynamics and potential quantization of a dust sphere in general relativity.

Contribution

It introduces a Hamiltonian approach for the Oppenheimer-Snyder model, enabling a canonical analysis and setting the stage for quantization of gravitational collapse.

Findings

Derived effective actions for dust sphere dynamics

Identified conserved ADM energy as a key variable

Outlined potential for quantization of the model

Abstract

A family of effective actions in Hamiltonian form is derived for a self-gravitating sphere of isotropic homogeneous dust. Starting from the Einstein-Hilbert action for barotropic perfect fluids and making use of the symmetry and equation of state of the matter distribution we obtain reduced actions for two canonical variables, namely the radius of the sphere and its ADM energy, the latter being conserved along trajectories of the former. These actions differ by the value of the (conserved) geodesic energy of the radius of the sphere which defines (disconnected) classes of solutions in correspondence to the inner geometry and proper volume of the sphere. Each class is thus treated as one constrained dynamical system and the union of all classes covers the full phase space of the model. Generalization to the (inhomogeneous) Tolman model is shown to be straightforward. Quantization is also discussed.