Black-hole thermodynamics in doubly special relativity: local-frame MDRs and rainbow metrics

Abstract

Doubly Special Relativity (DSR) deforms special-relativistic kinematics while preserving the relativity principle by introducing a second invariant scale, typically the Planck energy $E_{\rm Pl}$. Extending DSR-inspired modified dispersion relations (MDRs) to curved spacetimes is challenging, as ambiguous definitions of the deformation energy risk reintroducing preferred frames. We review three common extensions beyond flat spacetime: (i) MDRs in local orthonormal frames on fixed backgrounds, (ii) phase-space/Hamiltonian geometry with relative locality, and (iii) rainbow metrics. Using black-hole thermodynamics for static spherically symmetric horizons, we compare two implementations: (A) energy-independent background with local-frame MDR, and (B) energy-dependent rainbow metric. When the same prescription is used for the deformation energy scale $E_\star$, both approaches yield identical Hawking temperatures: \begin{equation} T(E_\star)=T_0\,\frac{g(E_\star/E_{\rm Pl})}{f(E_\star/E_{\rm Pl})}\,,\qquad T_0=\frac{κ_0}{2π}\,, \end{equation} where $κ_0$ is the classical surface gravity. This $g/f$ scaling holds for examples such as Amelino--Camelia-type MDRs (leading correction $\propto E p^2/E_{\rm Pl}$, giving $T(E_\star)\simeq T_0(1-\fracη{2}E_\star/E_{\rm Pl})$ for $η>0$) and the Magueijo--Smolin invariant ($f=g$, so $T(E_\star)=T_0$). Further DSR effects on evaporation (thresholds, phase space, greybody factors, composition laws) are discussed. Discrepancies in the literature arise mainly from different choices of $E_\star$ (energy at infinity vs.\ local frame). For macroscopic black holes, corrections are suppressed by $T_0/E_{\rm Pl}$ and become relevant only near the Planck regime, where full quantum gravity dominates.