Superrotation Charge and Supertranslation Hair on Black Holes

TL;DR

This work shows that classical GR in asymptotically flat spacetimes harbors an infinite set of conserved charges from supertranslation and superrotation symmetries, implying black holes possess an infinite head of soft hair that is distinguished by these charges. It provides a concrete Bondi-gauge construction of Schwarzschild supertranslations, demonstrates how asymmetric null shocks implant hair, and derives horizon-specific charges that generate horizon supertranslations, linking boundary soft hair to interior horizon dynamics. The paper also computes how supertranslated black holes carry nontrivial superrotation charges, offering a practical diagnostic of hair, and develops a covariant phase-space framework that ties horizon data to asymptotic charges, with implications for the quantum theory. Overall, the results illuminate how classical information about a black hole’s soft hair is encoded in horizon and boundary charges and how this structure could inform quantum gravitational questions about information and state counting.

Abstract

It is shown that black hole spacetimes in classical Einstein gravity are characterized by, in addition to their ADM mass $M$, momentum $\vec P$, angular momentum $\vec J$ and boost charge $\vec K$, an infinite head of supertranslation hair. The distinct black holes are distinguished by classical superrotation charges measured at infinity. Solutions with supertranslation hair are diffeomorphic to the Schwarzschild spacetime, but the diffeomorphisms are part of the BMS subgroup and act nontrivially on the physical phase space. It is shown that a black hole can be supertranslated by throwing in an asymmetric shock wave. A leading-order Bondi-gauge expression is derived for the linearized horizon supertranslation charge and shown to generate, via the Dirac bracket, supertranslations on the linearized phase space of gravitational excitations of the horizon. The considerations of this paper are largely classical augmented by comments on their implications for the quantum theory.