Entropy and temperature of black holes in a gravity's rainbow

TL;DR

This paper explores how modifications to dispersion relations in gravity's rainbow theories affect black hole entropy and temperature, revealing scenarios where black hole temperature can vanish, impacting black hole evaporation.

Contribution

It introduces two distinct gravity's rainbow models based on doubly special relativity and derives their effects on black hole thermodynamics, including conditions for temperature vanishing.

Findings

Black hole temperature can vanish as mass approaches zero in these models.

Different rainbow realizations lead to varied thermodynamic behaviors.

Black hole evaporation outcomes are significantly affected by the modified dispersion relations.

Abstract

The linear relation between the entropy and area of a black hole can be derived from the Heisenberg principle, the energy-momentum dispersion relation of special relativity, and general considerations about black holes. There exist results in quantum gravity and related contexts suggesting the modification of the usual dispersion relation and uncertainty principle. One of these contexts is the gravity's rainbow formalism. We analyze the consequences of such a modification for black hole thermodynamics from the perspective of two distinct rainbow realizations built from doubly special relativity. One is the proposal of Magueijo and Smolin and the other is based on a canonical implementation of doubly special relativity put forward recently by the authors. In these scenarios, we obtain modified expressions for the entropy and temperature of black holes. We show that, for a family of doubly special relativity theories satisfying certain properties, the temperature can vanish in the limit of zero black hole mass. For the Magueijo and Smolin proposal, this is only possible for some restricted class of models with bounded energy and unbounded momentum. With the proposal of a canonical implementation, on the other hand, the temperature may vanish for more general theories; in particular, the momentum may also be bounded, with bounded or unbounded energy. This opens new possibilities for the outcome of black hole evaporation in the framework of a gravity's rainbow.