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. Using a Euclidean formulation the proportionality is 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 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. Geometric entropy permits the same renormalization, but it does not yield an intrinsically positive quantity. For scalar fields geometric entropy is subtly sensitive to curved space couplings, even in the limit of flat space. 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.