Symmetries and charges of general relativity at null boundaries

Abstract

We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger.

T-Witts from the horizon

“…Many previous near horizon studies invoked Gaussian null coordinates, see e.g. [3][4][5]27]. It is one of the purposes of this subsection to show that there is loss of generality in assuming Gaussian null coordinates, because the coordinate transformations required to transform to these coordinates are not proper ones within our more general set of boundary conditions (6.3).…”

“…The only difference to the original BMS 4 algebra [23,89,90] is the lower spin of the supertranslations, see the discussion in the appendix of [7]. 5. The algebra discussed in [4,5,27] is a subalgebra generated by L p ,L p , R p,q −1 and R p,q 0 .…”

“…5. The algebra discussed in [4,5,27] is a subalgebra generated by L p ,L p , R p,q −1 and R p,q 0 . Upon changing the basis 7 R p,q 0 → R p,q 0 + T p,q 0 this subalgebra arises as a special case discussed in section 7 where surface gravity is constant and Gaussian null coordinates are used (with loss of generality), see the next item.…”

Angular momentum at null infinity in Einstein-Maxwell theory

“…The Wald-Zoupas prescription can also be applied to finite null surfaces in vacuum GR [34]. For Einstein-Maxwell theory at finite null surfaces, we expect that there is a similar contribution to the charges and fluxes associated with finite null boundary symmetries considered in [34] that arises from the Maxwell fields.…”

“…The Wald-Zoupas prescription can also be applied to finite null surfaces in vacuum GR [34]. For Einstein-Maxwell theory at finite null surfaces, we expect that there is a similar contribution to the charges and fluxes associated with finite null boundary symmetries considered in [34] that arises from the Maxwell fields. Such an analysis could also be useful in deriving conservation laws in Einstein-Maxwell theory through local regions bounded by a causal diamond similar to those in vacuum GR [35].…”