Nonextensive Black Hole Thermodynamics from Generalized Euclidean Path Integral and Wick's Rotation

TL;DR

This paper develops a generalized Euclidean path integral framework incorporating nonextensive statistical mechanics, enabling the study of black hole thermodynamics under nonextensive conditions and exploring implications for AdS/CFT correspondence.

Contribution

It introduces a generalized Wick's rotation for nonextensive statistics and applies it to black hole thermodynamics, bridging extensive and nonextensive formalisms.

Findings

Generalized Wick's rotation for nonextensive statistics

Unified treatment of extensive and nonextensive black hole thermodynamics

Strengthened equivalence between different black hole statistics and potential implications for AdS/CFT

Abstract

This paper extends the Euclidean path integral formalism to account for nonextensive statistical mechanics. Concretely, we introduce a generalized Wick's rotation from real time $t$ to imaginary time $\tau$ such that, $t\rightarrow-i f_\alpha(\tau)$, where $f_\alpha$ a differentiable function and $\alpha$ is a parameter related to nonextensivity. The standard extensive formalism is recovered in the limit $\alpha\rightarrow0$ and $f_0(\tau)=\tau$. Furthermore, we apply this generalized Euclidean path integral to black hole thermodynamics and derive the generalized Wick's rotations given the nonextensive statistics. The proposed formulation enables the treatment of nonextensive statistics on the same footing as extensive Gibbs-Boltzmann statistics. Moreover, we define a universal measure, $\eta$, for the nonextensivity character of statistics. Lastly, based on the present formalism, we strengthen the equivalence between the AdS-Schwarzschild black hole in Gibbs-Boltzmann statistics and the flat-Schwarzschild black hole within R\'enyi statistics and suggest a potential reformulation of the $AdS_5$/$CFT_4$ duality.