Thermal relativity, modified Hawking radiation, and the generalized uncertainty principle

After a brief review of the thermal relativistic corrections to the Schwarzschild black hole entropy, it is shown how the Stefan-Boltzman law furnishes large modifications to the evaporation times of Planck-size mini-black holes, and which might furnish important clues to the nature of dark matter and dark energy since one of the novel consequences of thermal relativity is that black holes do not completely evaporate but leave a Planck size remnant. Equating the expression for the modified entropy (due to thermal relativity corrections) with Wald's entropy should, in principle, determine the functional form of the modified gravitational Lagrangian L(R-abcd). We proceed to derive the generalized uncertainty relation which corresponds to the effective temperature T-eff = T-H(1 - T-H(2)/T-P(2))-1/2 associated with thermal relativity and given in terms of the Hawking (T-H) and Planck (T-P) temperature, respectively. Such modified uncertainty relation agrees with the one provided by string theory up to first order in the expansion in powers of (delta(p))(2)/M-P(2). Both lead to a minimal length (Planck size) uncertainty. Finally, an explicit analytical expression is found for the modifications to the purely thermal spectrum of Hawking radiation which could cast some light into the resolution of the black hole information paradox.