Black-hole thermodynamics in doubly special relativity: near-horizon g/f temperature scaling under a shared operational scale
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2026
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| author | Boumali, Abdelmalek Jafari, Nosratollah |
| author_facet | Boumali, Abdelmalek Jafari, Nosratollah |
| contents | 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 \[ T(E_\star)=T_0\,\frac{g(E_\star/E_{\mathrm{Pl}})}{f(E_\star/E_{\mathrm{Pl}})}, \quad T_0=κ_0/(2π), \] 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 $(α_2, Δα)$, we obtain \[ T_{\mathrm{GDRS}}(E_\star) = T_0 \sqrt{\frac{1-2Δα\,(E_\star/E_{\mathrm{Pl}})}{1-2α_2\,(E_\star/E_{\mathrm{Pl}})}} \simeq T_0 [1 - (Δα- α_2) E_\star/E_{\mathrm{Pl}}]. \]
We discuss the role of $Δα- α_2$ (vanishing correction for the symmetric $Δα=α_2$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_15216 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Black-hole thermodynamics in doubly special relativity: near-horizon g/f temperature scaling under a shared operational scale Boumali, Abdelmalek Jafari, Nosratollah General Relativity and Quantum Cosmology High Energy Physics - Theory 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 \[ T(E_\star)=T_0\,\frac{g(E_\star/E_{\mathrm{Pl}})}{f(E_\star/E_{\mathrm{Pl}})}, \quad T_0=κ_0/(2π), \] 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 $(α_2, Δα)$, we obtain \[ T_{\mathrm{GDRS}}(E_\star) = T_0 \sqrt{\frac{1-2Δα\,(E_\star/E_{\mathrm{Pl}})}{1-2α_2\,(E_\star/E_{\mathrm{Pl}})}} \simeq T_0 [1 - (Δα- α_2) E_\star/E_{\mathrm{Pl}}]. \] We discuss the role of $Δα- α_2$ (vanishing correction for the symmetric $Δα=α_2$ 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. |
| title | Black-hole thermodynamics in doubly special relativity: near-horizon g/f temperature scaling under a shared operational scale |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2602.15216 |