Covariant phase space with null boundaries

By imposing the boundary condition associated with the boundary structure of the null boundaries rather than the usual one,we find for Einstein’s gravity that the variational principle works only in its submanifold with the null boundaries given by the expansion-free and shearfree hypersurfaces rather than in the whole covariant phase space.This implies that the key requirement in Harlow-Wu’s algorithm for the timelike boundaries is too restrictive for the null ones.To incorporate more generic situations into Harlow-Wu’s algorithm,we relax such a requirement.As a result,we successfully reproduce the Hamiltonian obtained previously by Wald-Zoupas’prescription for Einstein’s gravity.