On Wald entropy of black holes: logarithmic corrections and trace anomaly

TL;DR

This paper investigates the universal logarithmic corrections to black hole entropy caused by quantum conformal matter, linking these corrections to Weyl anomalies and exploring their derivation via Wald's formalism, especially for extremal black holes.

Contribution

It extends the understanding of logarithmic entropy corrections to extremal black holes using Wald's Noether charge formalism and connects these corrections to Weyl anomalies and Q-curvature problems.

Findings

Logarithmic corrections are universal and linked to Weyl anomalies.

Wald entropy can be derived from heat kernel and Euclidean methods.

Results apply to extremal black holes and relate to entanglement entropy.

Abstract

Quantum effects due to conformal matter in a black hole background result in universal logarithmic corrections to black-hole entropy. The universality resides in the connection of the log term coefficient with those of type-A and type-B Weyl anomalies, regularization-scheme independent quantities. We presently study the case of extremal black holes within Wald's Noether charge formalism. In the conformal class of flat metrics, we are again able to unveil the log term in the entropy from the horizon value of the solution to the Q-curvature uniformization problem. Beyond conformally flat backgrounds, type-B Weyl anomaly becomes an obstruction to considering flat space as the fiducial metric and the search for a metric of constant Q-curvature remains open. Notwithstanding, by a uniform scaling argument we show that the results based on heat kernel and euclidean computations (namely entropy function and conical defect) can also be derived as Wald entropy, that is, as Noether charge of the integrated anomaly or conformal index. We finally comment on the relation with entanglement entropy.