Black Hole Memory

The memory effect at null infinity, $\mathcal{I}^+$, can be defined in terms of the permanent relative displacement of test particles (at leading order in $1/r$) resulting from the passage of a burst of gravitational radiation. In $D=4$ spacetime dimensions, the memory effect can be characterized by the supertranslation relating the "good cuts" of $\mathcal{I}^+$ in the stationary eras at early and late retarded times. It also can be characterized in terms of charges and fluxes associated with supertranslations. Black hole event horizons are in many ways analogous to $\mathcal{I}^+$. We consider here analogous definitions of memory for a black hole, assuming that the black hole is approximately stationary at early and late advanced times, so that its event horizon is described by a Killing horizon (assumed nonextremal) at early and late times. We give prescriptions for defining preferred foliations of nonextremal Killing horizons. We give a definition of the memory tensor for a black hole in terms of the "permanent relative displacement" of the null geodesic generators of the event horizon between the early and late time stationary eras. We show that preferred foliations of the event horizon in the early and late time eras are related by a Chandrasekaran-Flanagan-Prabhu (CFP) supertranslation. However, we find that the memory tensor for a black hole horizon does not appear to be related to the CFP symmetries or their charges and fluxes in a manner similar to that occurring at $\mathcal{I}^+$.