Renormalization of black hole entropy and of the gravitational coupling constant

Abstract

The quantum corrections to black hole entropy, variously defined, suffer quadratic divergences reminiscent of the ones found in the renormalization of the gravitational coupling constant (Newton constant). We consider the suggestion, due to Susskind and Uglum, that these divergences are proportional, and attempt to clarify its precise meaning. We argue that if the black hole entropy is identified using a Euclidean formulation, including the necessary surface term as proposed by Gibbons and Hawking, then the proportionality is, up to small identifiable corrections, a fairly immediate consequence of basic principles - a low-energy theorem. Thus in this framework renormalizing the Newton constant renders the entropy finite, and equal, in the limit of large mass, to its semiclassical value. As a partial check on our formal arguments we compare the one loop determinants, calculated using heat kernel regularization. An alternative definition of black hole entropy relates it to behavior at conical singularities in two dimensions, and thus to a suitable definition of geometric entropy. A definition of geometric entropy, natural from the point of view of heat kernel regularization, permits the same renormalization, but it does not yield an intrinsically positive quantity. The possibility, for scalar fields, of non-minimal coupling to background curvature allows one to consider test the framework in a continuous family of theories, and crucially involves a subtle sensitivity of geometric entropy to curved space couplings. Fermions and gauge fields are considered as well. Their functional determinants are closely related to the determinants for non-minimally coupled scalar fields with specific values for the curvature coupling, and pose no further difficulties.