Boundary Schrödinger equation in quantum geometrodynamics

We consider the quantum geometrodynamical propagator for a spatially bounded system. We formally derive a geometrodynamical Schrödinger equation which is expected to govern the time evolution of observables as measured on the spatial boundary of the system. We show that in the weak gravity limit the boundary Schrödinger equation reduces to the usual functional Schrödinger equation for the matter fields.