Hamiltonian formalism for the Oppenheimer-Snyder model

An effective action in Hamiltonian form is derived for a self-gravitating sphere of isotropic homogeneous dust. Starting from the Einstein-Hilbert action for baryotropic perfect fluids and making use of the symmetry and equation of state of the matter distribution we obtain a family of 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. By replacing the (fixed) geodesic energy with its expression in terms of the Schwarzschild time at the surface of the sphere and treating the latter as a further canonical variable we finally obtain an extended action which covers the full space of solutions. Generalization to the (inhomogeneous) Tolman model is shown to be straightforward. Quantization is also discussed.