A comment on black hole entropy or does Nature abhor a logarithm?

TL;DR

The paper investigates the leading quantum correction to black hole entropy, traditionally modeled as a logarithmic term $S_{bh}=S_{BH}+b\ln[S_{BH}]+\dots$, and seeks to constrain the logarithmic prefactor $b$ using horizon-area quantization and statistical arguments. It identifies two independent sources of the log term: microcanonical corrections (negative) and canonical corrections from thermal fluctuations (positive), proposing a decomposition $b=b_{mc}+b_{C}$. By combining estimates from loop quantum gravity and grand-canonical analyses, the author argues that these contributions cancel to yield $b=0$, a result that is consistent with Hod's requirement that $b$ be a non-negative integer and with holographic bounds. Addenda discuss potential revisions to this conclusion in light of recent loop quantum gravity developments (which could shift $b$ to $+1$) and critique the grand-canonical approach, indicating that the precise value of $b$ remains contingent on the underlying quantum gravity framework and dimensionality.

Abstract

There has been substantial interest, as of late, in the quantum-corrected form of the Bekenstein-Hawking black hole entropy. The consensus viewpoint is that the leading-order correction should be a logarithm of the horizon area; however, the value of the logarithmic prefactor remains a point of notable controversy. Very recently, Hod has employed statistical arguments that constrain this prefactor to be a non-negative integer. In the current paper, we invoke some independent considerations to argue that the "best guess" for the prefactor might simply be zero. Significantly, this value complies with the prior prediction and, moreover, seems suggestive of some fundamental symmetry.