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Free-fall Rainbow BTZ Black Hole

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. For a black hole, there are two natural orthonormal frames, the stationary one hovering above it and freely falling one along geodesics. Since the rainbow metric is the metric that the radiated particles "see", a more natural orthonormal frame is the one anchored to the particles. And the cases with the stationary orthonormal frame have been extensively studied in the literature. In this paper, we investigate properties of rainbow BTZ black holes in the scenario with the free-fall orthonormal frame. We first review the thermodynamic properties of a BTZ black hole. Furthermore, we obtain the Free-fall (FF) rainbow BTZ black hole and then calculate its Hawking temperature via the Hamilton-Jacobi method. Finally, we discuss the thermodynamic properties of a FF stationary rainbow BTZ black hole .