Free-fall frame black hole in gravity’s rainbow

Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use “gravity’s rainbow,” where the spacetime background felt by a test particle would depend on its energy. In this scenario, one could rewrite the rainbow metric $g_{\mu \nu }(E)$ in terms of some orthonormal frame fields and use the modified equivalence principle to determine the energy dependence of $g_{\mu \nu }(E)$. Obviously, the form of $g_{\mu \nu }(E)$ depends on the choice of the orthonormal frame. For a static black hole, there are two natural orthonormal frames: the static one hovering above it and the freely falling one along geodesics. The cases with the static orthonormal frame have been extensively studied by many authors. The aim of this paper is to investigate properties of rainbow black holes in the scenario with the free-fall orthonormal frame. We first derive the metric of rainbow black holes and their Hawking temperatures in this free-fall scenario. Then, the thermodynamics of a rainbow Schwarzschild black hole is studied. Finally, we use the brick wall model to compute the thermal entropy of a massless scalar field near the horizon of a Schwarzschild rainbow black hole in this free-fall scenario.

Figure 1

Plot of the temperature $T_{\mathrm{BH}}/m_p$ against the mass $M/m_p$ for a FF rainbow Schwarzschild black hole. All the lines asymptotically approach $T_{\mathrm{BH}}=0$ as $M/m_p\to \infty$. The blue line is the usual case, where $T_{\mathrm{BH}}$ blows up as $M\to 0$. The red dot is the end of the red solid line, where the black hole has a remnant $M_{cr}=\frac{3^{\frac{3}{2}}}{4}{m}_p$. In this case, $T_{\mathrm{BH}}$ does not blow up as $M\to M_{cr}$. The black dotted line is the asymptotic line of the red dashed line as $M\to M_{cr}=0.5m_p$, which is the black hole’s remnant. In this case, $T_{\mathrm{BH}}$ blows up as $M\to M_{cr}$.

Figure 2

Plot of the entropy $S_{\mathrm{BH}}$ against the mass $M/m_p$ for a FF rainbow Schwarzschild black hole.