Hawking, Fiducial, and Free-fall Temperatures of Black Hole on Gravity's Rainbow

TL;DR

This paper investigates how gravity's rainbow modifies black hole temperatures, revealing that fiducial temperature diverges at the horizon while free-fall temperature remains finite and vanishes there, supporting black hole complementarity.

Contribution

It introduces modified definitions of fiducial and free-fall temperatures in gravity's rainbow and analyzes their behaviors near the black hole horizon.

Findings

Fiducial temperature is blue-shifted and diverges at the horizon.

Free-fall temperature remains finite and vanishes at the horizon.

Black hole complementarity still holds in gravity's rainbow.

Abstract

On gravity's rainbow, the energy of particles affects the geometry of black hole in such a way that the Hawking temperature of black hole is appropriately modified. It means that the fiducial and free-fall temperatures on the background of black hole should also be modified according to deformation of the geometry. We find that the fiducial temperature takes the blue-shifted modified Hawking temperature so that it is divergent at the horizon, while the free-fall temperature defined by employing the thermodynamic Stefan-Boltzmann relation on gravity's rainbow is finite everywhere without the blue-shift effect; in particular, it is vanishing at the horizon. In this respect, the behaviors of the two-different temperatures at the horizon show that black hole complementarity is still required on gravity's rainbow.