Logarithmic correction to BH entropy as Noether charge

TL;DR

This work derives a macroscopic, Wald-based logarithmic correction to black hole entropy arising from the type-A trace anomaly of conformal matter in four dimensions. By recasting the anomaly as a local anomaly-induced action via an auxiliary field $\phi$ tied to $Q$-curvature, and applying Wald's Noether-charge formalism, the authors obtain a universal correction $S_{bh}=S_{BH}-a\chi_{\mathcal{H}}\phi_{\mathcal{H}}+\cdots$, with $\phi_{\mathcal{H}}$ encoding near-horizon geometry and leading to a logarithm in the horizon area. In Schwarzschild spacetime this reduces to $S_{bh}=S_{BH}+2a\ln S_{BH}+O(1)$, while for massless topological black holes the correction is $S_{bh}=S_{BH}-a\chi_{\mathcal{H}}\ln S_{BH}+O(1)$; backreaction analyses via the First Law confirm the log-term and its dependence on horizon topology. The results connect macroscopic entropy corrections to the universal structure of the trace anomaly and suggest a link to the geometric problem of $Q$-curvature uniformization, with several avenues for generalization and further study.

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.