Black Hole Thermodynamics via Tsallis Statistical Mechanics and Phase Transitions Probed by Optical Characteristics

TL;DR

This paper introduces a non-extensive thermodynamic framework for Reissner--Nordstr"om black holes using Tsallis statistics, revealing phase transitions and optical signatures that mirror thermodynamic behavior.

Contribution

It develops a generalized entropy consistent with the area law, uncovers Van der Waals-like phase transitions, and links optical properties to thermodynamic critical phenomena.

Findings

Derived a generalized Bekenstein--Hawking entropy within Tsallis statistics.

Identified Van der Waals-like phase transitions with mean-field critical exponents.

Linked photon-sphere observables to thermodynamic variables, tracking critical behavior.

Abstract

We develop a non-extensive thermodynamic framework for Reissner--Nordstr\"om black holes based on a near-horizon photon-gas model within Tsallis statistics. We derive the generalized Bekenstein--Hawking entropy based on such an approach, consistent with the Bekenstein--Hawking area law in the extensive limit, $q \rightarrow 1$. The induced deformation gives rise to a rich thermodynamic structure consisting of small, intermediate, and large black-hole branches, exhibiting Van der Waals-like phase transitions characterized by mean-field critical exponents. We further establish an optical--thermodynamic analogy by relating photon-sphere observables, including orbital periods and Lyapunov exponents, to thermodynamic variables. These optical signatures qualitatively track the thermodynamic critical behavior and phase structure, suggesting their potential relevance as observational probes in future high-resolution measurements. These results may shed light on a conceptual connection between non-extensive entropy, black-hole critical phenomena, and strong-gravity optics.