Logarithmic correction to BH entropy as Noether charge

TL;DR

This paper derives a universal logarithmic correction to black hole entropy from the type-A trace anomaly using Wald's formalism, confirmed by explicit calculations for Schwarzschild and topological black holes.

Contribution

It introduces a novel correction to black hole entropy linked to the trace anomaly, connecting quantum effects with geometric properties of horizons.

Findings

Logarithmic correction formula involving trace anomaly coefficient and horizon topology

Agreement with one-loop quantum corrections for Schwarzschild black holes

Application to topological black holes confirms universality of correction

Abstract

We consider the role of the type-A trace anomaly in static black hole solutions to semiclassical Einstein equation in four dimensions. Via Wald's Noether charge formalism, we compute the contribution to the entropy coming from the anomaly induced effective action and unveil a logarithmic correction to the Bekenstein-Hawking area law. The corrected entropy is given by a seemingly universal formula involving the coefficient of the type-A trace anomaly, the Euler characteristic of the horizon and the value at the horizon of the solution to the uniformization problem for Q-curvature. Two instances are examined in detail: Schwarzschild and a four-dimensional massless topological black hole. We also find agreement with the logarithmic correction due to one-loop contribution of conformal fields in the Schwarzschild background.