Covariant phase space with null boundaries

TL;DR

This paper investigates the covariant phase space with null boundaries, finding that the Harlow-Wu algorithm applies only to a specific submanifold characterized by expansion and shear free null hypersurfaces, and successfully reproduces known Hamiltonians.

Contribution

It identifies the limitations of Harlow-Wu's algorithm in the full covariant phase space and adapts it to a submanifold to recover established Hamiltonians with correct boundary terms.

Findings

Harlow-Wu's algorithm fails in the full covariant phase space with null boundaries.

The algorithm works in a submanifold with expansion and shear free null hypersurfaces.

Reproduces Hamiltonians consistent with Wald-Zoupas prescription.

Abstract

By imposing the boundary condition associated with the boundary structure of the null boundaries rather than the usual one, we find that the key requirement in Harlow-Wu's algorithm fails to be met in the whole covariant phase space. Instead, it can be satisfied in its submanifold with the null boundaries given by the expansion free and shear free hypersurfaces in Einstein's gravity, which can be regarded as the origin of the non-triviality of null boundaries in terms of Wald-Zoupas's prescription. But nevertheless, by sticking to the variational principle as our guiding principle and adapting Harlow-Wu's algorithm to the aforementioned submanifold, we successfully reproduce the Hamiltonians obtained previously by Wald-Zoupas' prescription, where not only are we endowed with the expansion free and shear free null boundary as the natural stand point for the definition of the Hamiltonian in the whole covariant phase space, but also led naturally to the correct boundary term for such a definition.