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

Abstract

Following Chakrabarti, et al. [R. Chakrabarti, R. Chandrashekar and S. S. Naina Mohammed, Physica A 389, 1571 (2010)], 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 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.