Covariant phase space with null boundaries

Kai Shi,Xuan Wang,Yihong Xiu,Hongbao Zhang

Figure 3. ${{\mathscr{F}}}_{{\mathfrak{p}}}$ is specified as the configuration space associated with the null boundary structure, where ${\bar{{\mathscr{F}}}}_{{\mathfrak{p}}}$ represents the covariant phase space, while ${{\mathscr{F}}}_{{\mathfrak{f}}}$ denotes its submanifold with null boundaries given by free horizons. The intersection of ${\bar{{\mathscr{F}}}}_{{\mathfrak{p}}}$ and ${{\mathscr{F}}}_{{\mathfrak{f}}}$ gives rise to the submanifold as the stand point for the definition of the Hamiltonian.