Quantum gravity on a manifold with boundaries: Schrödinger evolution and constraints

Abstract In this work, we derive the boundary Schrödinger (functional) equation for wave function of a quantum gravity system on manifold with boundaries together new constraint defined timelike boundary. From detailed analysis condition spatial boundary, find that while lapse and shift functions are independent Lagrange multipliers bulk, these two related; namely, they not independent. Hamiltonian ADM formalism, multiplier, solving conditions involving evaluated is introduced. The classical motion associated multiplier turns out to be an identity when solution Einstein’s equations. On other hand, its counterpart gravitational degrees freedom only This has been taken into account before studying evolution manifolds boundaries.