# Relativistic treatment of Verlinde's emergent force in Tsallis' statistics

## TL;DR

This paper explores a relativistic classical approach within Tsallis' statistics to derive Newtonian gravity as an emergent entropic force, highlighting the limitations of using Renyi's distribution for this purpose.

## Contribution

It introduces a relativistic Hamiltonian framework in Tsallis' statistics to derive gravitational force behavior, extending Verlinde's emergent gravity conjecture.

## Key findings

- Tsallis' relativistic distribution reproduces Newton's inverse-square law.
- Using Renyi's distribution does not successfully reproduce gravitational behavior.
- The approach aligns with classical limits at small velocities.

## Abstract

Following Chakrabarti,Chandrasekhar, and Naina [Physica A {\bf 389} (2010) 1571], we attempt a classical relativistic treatment of Verlinde's emergent entropic force conjecture by appealing to a relativistic Hamiltonian in the context of Tsalli's statistics. The ensuing partition function becomes the classical one for small velocities. We show that Tsallis' relativistic (classical) free particle distribution at temperature $T$ can generate Newton's gravitational force's $r^{-2}$ {\it distance's dependence}. If we want to repeat the concomitant argument by appealing to Renyi's distribution, the attempt fails and one needs to modify the conjecture.