Spherically symmetric gravity coupled to a scalar field with a local   Hamiltonian: the complete initial-boundary value problem using metric   variables
  We discuss a gauge fixing of gravity coupled to a scalar field in spherical symmetry such that the Hamiltonian is an integral over space of a local density. In a previous paper we had presented it using Ashtekar's new variables. Here we study it in metric variables. We specify completely the initial-boundary value problem for ingoing Gaussian pulses. 
Rodolfo Gambini and Jorge Pullin
gr-qc hep-th