The Nonextensive Thermodynamics of Q-Deformed Spacetime in General Relativity

Abstract

Recently we discussed several models of apriori q-deformed spacetimes and spacetime metrics, arguing that the resulting q-diffusion tensors of the propagation were also the (inverse) q-strain or when unreferenced the metric tensors in 3D and q-(fundamental) metric in 4D deformed Minkowski spacetime. The topology of q-deformed tensors in the usual way resulted in non-zero Riemann tensors, Ricci tensors upon contraction and non-zero q-scalar 'curvature' like terms that modified otherwise flat spacetime. We stated that these results obtained gravitational-like metric tensors, Ricci tensors and then Einstein tensors and equations relating q-deformed geometry to non-zero stress-energy tensors. In this letter we apply thermodynamics to the 4D (nD generally) q-deformed spacetime(s) as we have represented them in the above q-deformed topology. Specifically we will discuss the relation between entropy and stress-strain and internal energy. And as entropy can be written as an extensive composition e.x. the Gibbs-Boltzmann or Von Neumann logarithmic form or as a nonextensive composition such as the Tsallis q-power law type, we seek to also connect to the idea of q-deformed (general) relativistic spacetime to nonextensive entropy and thermodynamics.