Free-fall Frame Black Hole in Gravity's Rainbow

TL;DR

This paper explores the properties and thermodynamics of black holes in gravity's rainbow framework using a free-fall orthonormal frame, deriving metrics, temperatures, and entropy to extend previous static frame studies.

Contribution

It introduces a novel analysis of rainbow black holes from a free-fall frame perspective, including metric derivation and thermodynamic properties, expanding understanding beyond static frame models.

Findings

Derived the rainbow black hole metric in free-fall frame

Calculated Hawking temperature in the free-fall scenario

Computed thermal entropy near the horizon using the brick wall model

Abstract

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}\left( E\right) $ in terms of some orthonormal frame fields and use the modified equivalence principle to determine the energy dependence of $g_{\mu\nu}\left( E\right) $. Obviously, the form of $g_{\mu\nu}\left( E\right) $ 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 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.