Black-hole thermodynamics in doubly special relativity: near-horizon g/f temperature scaling under a shared operational scale

Abstract

Doubly Special Relativity (DSR) deforms special-relativistic kinematics by introducing an invariant Planck energy scale $E_{\mathrm{Pl}}$ alongside the speed of light, while preserving the relativity principle. A key issue in curved spacetimes, particularly black-hole thermodynamics, is the operational meaning of the ``energy'' in modified dispersion relations (MDRs). We compare two common implementations in a controlled static black-hole spacetime: (i) MDRs in local orthonormal frames on a fixed background geometry, and (ii) the rainbow-metric approach with an energy-dependent family of effective metrics. For static, spherically symmetric horizons and using a consistent finite operational energy scale $E_\star$ for emitted quanta, both yield the same near-horizon temperature rescaling \[ T(E_\star)=T_0\,\frac{g(E_\star/E_{\mathrm{Pl}})}{f(E_\star/E_{\mathrm{Pl}})}, \quad T_0=\kappa_0/(2\pi), \] where $f$ and $g$ are the standard rainbow/MDR functions. This establishes a universality of the tunneling/surface-gravity temperature, with deformation entering solely via the ratio $g/f$. We illustrate for Amelino-Camelia MDR and Magueijo-Smolin DSR (where $f=g$, implying $T(E_\star)=T_0$). Extending to a two-parameter generalized DSR (G-DSR) with leading parameters $(\alpha_2, \Delta\alpha)$, we obtain \[ T_{\mathrm{GDRS}}(E_\star) = T_0 \sqrt{\frac{1-2\Delta\alpha\,(E_\star/E_{\mathrm{Pl}})}{1-2\alpha_2\,(E_\star/E_{\mathrm{Pl}})}} \simeq T_0 [1 - (\Delta\alpha - \alpha_2) E_\star/E_{\mathrm{Pl}}]. \] We discuss the role of $\Delta\alpha - \alpha_2$ (vanishing correction for the symmetric $\Delta\alpha=\alpha_2$ subfamily) and note that further model dependence arises from phase-space measures, greybody factors, and non-linear composition laws. Corrections are strongly suppressed for macroscopic black holes and become relevant only near the Planck regime.