Superrotation Charge and Supertranslation Hair on Black Holes

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.