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

TL;DR

This paper investigates how doubly special relativity modifies black-hole temperature near the horizon, revealing a universal temperature scaling dependent on an operational energy scale and exploring model-specific effects.

Contribution

It compares two approaches to DSR in black-hole thermodynamics, establishing a universal near-horizon temperature scaling and analyzing effects of generalized DSR parameters.

Findings

Near-horizon temperature scales with the ratio g/f of rainbow functions.

For specific DSR models, the temperature remains unchanged or is slightly modified.

Corrections are negligible for macroscopic black holes, significant near the Planck 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.