Null boundary phase space: slicings, news & memory

Abstract

We construct the boundary phase space in D-dimensional Einstein gravity with a generic given co-dimension one null surface $$ \mathcal{N} $$ N as the boundary. The associated boundary symmetry algebra is a semi-direct sum of diffeomorphisms of $$ \mathcal{N} $$ N and Weyl rescalings. It is generated by D towers of surface charges that are generic functions over $$ \mathcal{N} $$ N . These surface charges can be rendered integrable for appropriate slicings of the phase space, provided there is no graviton flux through $$ \mathcal{N} $$ N . In one particular slicing of this type, the charge algebra is the direct sum of the Heisenberg algebra and diffeomorphisms of the transverse space, $$ \mathcal{N} $$ N v for any fixed value of the advanced time v. Finally, we introduce null surface expansion- and spin-memories, and discuss associated memory effects that encode the passage of gravitational waves through $$ \mathcal{N} $$ N , imprinted in a change of the surface charges.

Near horizon gravitational charges

“…We first compute the surface charge from the Palatini action. Our result is consistent with [35][36][37]. Then we derive the surface charge from the Holst term.…”

“…This charge matches the one in [36] explicitly using the relations by the end of Section 2. Note that fixing η " 1 requires that W " 2B v T in [36] which also matches our near horizon symmetry generators in Section 3.…”

“…In the next Section, we derive the near horizon solution space for the NP equations with suitable gauge and boundary conditions. We compare our results with the ones in [35] and [36]. Section 3 is devoted to the near horizon symmetry and the transformation law of the near horizon fields.…”

“…The solution space we obtained is larger than that in [35] where the authors considered a non-expanding horizon, i.e., µ 0 " 0. While our solution space is smaller than that in [36] where the authors consider the case with the shear of the horizon generator which can also characterize flux going through the horizon. To recover our solution space, one just needs to set the fields in [36] to 4…”

“…The reason is that we want to fix the r " 0 null hypersurface to be the boundary. Somehow this can be understood as the fact that the existence of a boundary at r " 0 breaks the translational invariance along r direction [35,36].…”