Hamiltonian spacetime dynamics with a spherical null-dust shell

Abstract

We consider the Hamiltonian dynamics of spherically symmetric Einstein gravity with a thin null-dust shell, under boundary conditions that fix the evolution of the spatial hypersurfaces at the two asymptotically flat infinities of a Kruskal-like manifold. The constraints are eliminated via a Kuchar-type canonical transformation and Hamiltonian reduction. The reduced phase space $\tilde\Gamma$ consists of two disconnected copies of $R^4$, each associated with one direction of the shell motion. The right-moving and left-moving test shell limits can be attached to the respective components of $\tilde\Gamma$ as smooth boundaries with topology $R^3$. Choosing the right-hand-side and left-hand-side masses as configuration variables provides a global canonical chart on each component of $\tilde\Gamma$, and renders the Hamiltonian simple, but encodes the shell dynamics in the momenta in a convoluted way. Choosing the shell curvature radius and the "interior" mass as configuration variables renders the shell dynamics transparent in an arbitrarily specifiable stationary gauge "exterior" to the shell, but the resulting local canonical charts do not cover the three-dimensional subset of $\tilde\Gamma$ that corresponds to a horizon-straddling shell. When the evolution at the infinities is freed by introducing parametrization clocks, we find on the unreduced phase space a global canonical chart that completely decouples the physical degrees of freedom from the pure gauge degrees of freedom. Replacing one infinity by a flat interior leads to analogous results, but with the reduced phase space $R^2 \cup R^2$. The utility of the results for quantization is discussed.