Three-dimensional energy-dependent $C$-metric: black hole solutions

TL;DR

This paper develops a three-dimensional energy-dependent $C$-metric, derives accelerating BTZ black hole solutions in gravity's rainbow, and analyzes their thermodynamic properties and stability, revealing conditions for physical and stable black holes.

Contribution

It introduces a novel three-dimensional energy-dependent $C$-metric and explores the thermodynamics and stability of the resulting accelerating BTZ black holes.

Findings

Black holes have a critical radius where temperature and entropy change sign.

The temperature and entropy exhibit similar behavior before and after the critical radius.

Accelerating AdS BTZ black holes can be both physically valid and locally stable.

Abstract

Considering a three-dimensional $C$-metric and adding energy-dependent to this spacetime, we first create a three-dimensional energy-dependent $C$-metric. Then, we extract accelerating BTZ black hole solutions in gravity's rainbow. Besides, we show that (A)dS black holes cover by an event horizon that depends on all the parameters of this theory. Using the definition of Hawking temperature, we obtain the temperature of these black holes and study the effects of various parameters on this quantity. We find a critical radius in which the temperature is always positive (negative) before (after) it. Then, we obtain the entropy of such black holes. Our analysis indicates that there is the same behavior for entropy, similar to the temperature. Indeed, before (after) the critical radius, the entropy is positive (negative). In order to study the local stability of such black holes, we calculate the heat capacity. We find two different behaviors for the heat capacity, which depend on the cosmological energy-dependent constant. As a final result, accelerating AdS BTZ black holes can satisfy the physical condition and local stability at the same time.