Lagrangian formulation of the Tsallis entropy

TL;DR

This paper derives the Tsallis entropy from a modified gravity Lagrangian using Wald's formalism, linking nonadditive entropy to gravitational theories and exploring implications for black hole thermodynamics and cosmology.

Contribution

It presents a novel derivation of Tsallis entropy from a generalized Lagrangian in $f(R)$ gravity, connecting nonadditive entropy to gravitational modifications.

Findings

Tsallis entropy can be derived from a Lagrangian proportional to $R^{1+ extepsilon}$.

The approach links nonadditive entropy to deviations from general relativity.

Implications for black hole thermodynamics and cosmological observations are discussed.

Abstract

We investigate the gravitational origin of the Tsallis entropy, characterized by the nonadditive index $\delta$. Utilizing Wald's formalism within the framework of $f(R)$ modified theories of gravity, we evaluate the entropy on the black hole horizon for constant curvature solutions to the spherically symmetric vacuum field equations. In so doing, we demonstrate that the Tsallis entropy can be effectively derived from a generalization of the Lagrangian $\mathcal{L}\propto R^{1+\epsilon}$, where $\epsilon=\delta-1$ quantifies small deviations from general relativity. In conclusion, we examine the physical implications of our findings in light of cosmological observations and elaborate on the possibility of solving the thermodynamic instability of Schwarzschild black holes.