Gravity and BF theory defined in bounded regions

Abstract

We study Einstein gravity in a finite spatial region. By requiring a well-defined variational principle, we identify all local boundary conditions, derive surface observables, and compute their algebra. The observables arise as induced surface terms, which contribute to a non-vanishing Hamiltonian. Unlike the asymptotically flat case, we find that there are an infinite number of surface observables. We give a similar analysis for SU(2) BF theory.