Horizon Foliations and Black Hole Memory*

We propose a notion of a "black hole memory effect" along the event horizon of a black hole spacetime, in analogy with the characterization of the memory effect at null infinity in terms of supertranslations. We define a "preferred foliation" of the event horizon during stationary eras by requiring that the expansion of the transverse null geodesics be constant. We prove existence and uniqueness of this foliation. We show that this foliation differs, in general, from another preferred foliation previously considered by Ashtekar, Beetle, and Lewandowski and provide a new characterization of the latter. We consider a black hole that is initially stationary, goes through a non-stationary era, and becomes stationary again at late times. Black hole memory is then defined in terms of the symmetry transformation of the horizon that takes one from the early time preferred foliation to the late time preferred foliation.