Do Black Holes With Generalized Entropy Violate Bekenstein Bound?

TL;DR

This work investigates whether black holes with generalized entropy violate the Bekenstein bound. It adopts the Generalized Entropy and Varying-G gravity (GEVAG) framework, which ties entropy generalization to an area-dependent effective gravitational constant $G_\\text{eff}$ and to a thermodynamic energy $E$ distinct from the ADM mass. Through explicit analysis of Tsallis-Schwarzschild and Rényi-Schwarzschild black holes, the authors show that a relaxed bound $S \\le C R E$ holds with a model-dependent coefficient $C$ that remains finite and is typically at or below $2\\pi$ in the examined cases. They further establish a general criterion $C=f(A)/(8 G_\\text{eff} M E)$ for any generalized entropy $S=f(A)/4G$, demonstrating that the Bekenstein bound remains meaningful when gravity is consistently modified, thereby supporting the viability of the GEVAG approach.

Abstract

In general yes, but also not quite. It is known that if the Bekenstein-Hawking entropy is replaced by some kind of generalized entropy, then the Bekenstein bound may be grossly violated. In this work, we show that this undesired violation can be avoided if we employ the equivalence between generalized entropy and varying-$G$ gravity (GEVAG). In this approach, modifying entropy necessarily also modifies gravity (as one should expect if gravity is indeed inherently tied to thermodynamics), which leads to an effective gravitational "constant" $G_\text{eff}$ that is area-dependent, and a thermodynamic energy that is distinct from the ADM mass. We show that a relaxed Bekenstein bound of the form $S \leqslant CRE$ is always satisfied, albeit the coefficient $C$ is no longer $2π$.

Figures (3)

• Figure 1: The Bekenstein constant $C$ for the Tsallis-Schwarzschild black hole is bounded between 0 and $4\pi$, with $0< C \leqslant 2\pi$ for $1/2 <\delta \leqslant 1$. The dashed vertical line and horizontal line are $\delta=1$ and $C=2\pi$, respectively.
• Figure 2: The ratio $E/M$ for the Rényi-Schwarzschild black hole. The horizontal axis has been re-scaled for clarity. The domain of $a$ is $0 < a < 1/16\pi \approx 0.0199$. We have $E/M=1$ when $a=0$, the GR case. On the other hand, $E/M$ tends to $\pi/2$ when $a\to 1/16\pi$.
• Figure 3: The Bekenstein constant $C$ for Rényi-Schwarzschild black hole, as a function of the dimensionless Rényi parameter $a \coloneq \alpha GM^2$. The horizontal axis has been re-scaled for clarity. The domain of $a$ is $0 < a < 1/16\pi \approx 0.0199$. We have $C \to 2\pi$ in the limit $a\to 0$, and $C \to 8 \ln 2 \approx 5.5452$ when $a\to 1/16\pi$.