Universal Relations with the Non-Extensive Entropy Perspective

TL;DR

This work investigates whether a universal extremality–entropy relation, $\\frac{\\partial M_{ext}(\\vec{\\mathcal{Q}},\\varepsilon)}{\\partial \\varepsilon} = \\lim_{M \\rightarrow M_{ext}} -T \\left(\\frac{\\partial S(M,\\vec{\\mathcal{Q}},\\varepsilon)}{\\partial \\varepsilon}\\right)_{M,\\vec{\\mathcal{Q}}}$, holds for charged AdS black holes when non-extensive entropy generalizations are used. The authors analyze Barrow, Kaniadakis, Rényi, Sharma–Mittal, and Tsallis–Cirto entropies by computing the mass, temperature, and other thermodynamic quantities under a small cosmological-constant perturbation $\\varepsilon$, find that the original universality relation fails in each non-extensive case, and then restore compatibility by introducing proposal-specific entropy-derivative corrections, such as $\\frac{\\partial S}{\\partial S} = f(S)$ (e.g., $\\sqrt{1+(K S_K)^2}$, $e^{-\\lambda S_R}$, $(\\rho S_{SM}+1)^{\\frac{\\lambda}{\\rho}-1}$, etc.). The results indicate that universality robustly holds only for the standard Bekenstein–Hawking entropy, while non-extensive generalizations require tailored adjustments, with implications for holography and quantum gravity frameworks that rely on universal extremality relations.

Abstract

Recent advancements in black hole thermodynamics have introduced corrections to elucidate the relationship between entropy and extremality bound of black holes. Traditionally, this relationship has been studied in the context of black holes characterized by Bekenstein-Hawking entropy. However, this study extends the investigation to encompass non-extensive generalizations of entropy. We introduce a minor constant correction, denoted as $(\varepsilon)$, and examine the universal relations for a charged Anti-de Sitter (AdS) black hole. Our findings indicate that these universal relations do not hold for the charged AdS black hole when described by the non-extensive generalizations of entropy. Of course, with some adjustments, the universality relations are met. In contrast, the universal relations remain compatible when the black hole is described by Bekenstein-Hawking entropy.