Covariant phase space with null boundaries

Kai Shi,Xuan Wang,Yihong Xiu,Hongbao Zhang

Figure 1. The foliation of the null hypersurface ${ \mathcal N }$ by the cross-section ${ \mathcal S }$ and the null generators la gives rise to a natural coordinate system {λ, xi} on ${ \mathcal N }$, with the induced volume ${\underline{{\boldsymbol{\epsilon }}}}^{(d-1)}=\sqrt{q}{\rm{d}}\lambda {{\rm{d}}}^{d-2}x$ on ${ \mathcal N }$ and ${\underline{{\boldsymbol{\epsilon }}}}^{(d-2)}=\sqrt{q}{{\rm{d}}}^{d-2}x$ on ${ \mathcal S }$.