Further Insights into Thermal Relativity Theory and Black Hole Thermodynamics

We continue to explore the consequences of Thermal Relativity Theory to the physics of black holes. The thermal analog of Lorentz transformations in the $tangent$ space of the thermodynamic manifold are studied in connection to the Hawking evaporation of Schwarzschild black holes and one finds that there is $no$ bound to the thermal analog of proper accelerations despite the maximal bound on the thermal analog of velocity given by the Planck temperature. The proper entropic infinitesimal interval corresponding to the Kerr-Newman black hole involves a $ 3 \times 3 $ non-Hessian metric with diagonal and off-diagonal terms of the form $ ( d{\bf s} )^2 = g_{ ab } ( M, Q, J ) d Z^a dZ^b$, where $ Z^a = M, Q, J $ are the mass, charge and angular momentum, respectively. Black holes in asymptotically Anti de Sitter (de Sitter) spacetimes are more subtle to study since the mass turns out to be related to the $enthalpy$ rather that the internal energy. We finalize with some remarks about the thermal-relativistic analog of proper force, the need to extend our analysis of Gibbs-Boltzmann entropy to the case of Reny and Tsallis entropies, and to complexify spacetime.