Thermodynamics of Black Holes with R\'enyi Entropy from Classical Gravity

TL;DR

This paper extends black hole thermodynamics by incorporating Rényi entropy, deriving a generalized Smarr formula and temperature, and demonstrating the approach's broad applicability within classical gravity frameworks.

Contribution

It introduces treating the nonextensive Rényi entropy parameter as a thermodynamic variable, generalizing black hole thermodynamics and connecting geometric relations with nonextensive entropy.

Findings

Derived a generalized Smarr formula for black holes with Rényi entropy.

Defined Rényi temperature consistent with existing literature.

Showed the approach's applicability to various black hole types.

Abstract

The nonextensive nature of black holes is one of the most intriguing discoveries. In fact, the black hole entropy is a nonextensive quantity that scales by its surface area at the event horizon. In our work, we extend the thermodynamic phase space of black holes by treating the nonextensive parameter of the R\'enyi entropy as the thermodynamic variable. Using Euler's theorem for a homogeneous function of the black holes' mass, the compatible Smarr formula and the first law of black hole thermodynamics can be obtained. It is also demonstrated that, by keeping the same form of the black hole mass, the R\'enyi temperature is straightforwardly defined as proposed in the literature. Since many different types of black holes can indeed be successfully treated with such a procedure, our consideration is fairly general. It is worthwhile to argue that the black hole thermodynamics in R\'enyi statistics is rooted from the relation among geometric quantities in the same way as the standard approach corresponding to the Gibbs-Boltzmann statistics. Even though our results are based on classical gravity, they may pave the way to derive the R\'enyi temperature using the notion of quantum field in curved spacetime.