Phase transition of charged Black Holes in Brans–Dicke theory through geometrical thermodynamics

S. H. Hendi,S. Panahiyan,Z. Armanfard,B. Eslam Panah

doi:10.1140/epjc/s10052-016-4235-1

Abstract

In this paper, we take into account black hole solutions of Brans-Dicke-Maxwell theory and investigate their stability and phase transition points. We apply the concept of geometry in thermodynamics to obtain phase transition points and compare its results with those, calculated in canonical ensemble through heat capacity. We show that these black holes enjoy second order phase transitions. We also show that there is a lower bound for the horizon radius of physical charged black holes in Brans-Dicke theory which is originated from restrictions of positivity of temperature. In addition, we find that employing specific thermodynamical metric in the context of geometrical thermodynamics, yields divergencies for thermodynamical Ricci scalar in places of the phase transitions. It will be pointed out that due to characteristics behavior of thermodynamical Ricci scalar around its divergence points, one is able to distinguish the physical limitation point from the phase transitions. In addition, the free energy of these black holes will be obtained and its behavior will be investigated. It will be shown that the behavior of the free energy in the place where the heat capacity diverges, demonstrates second order phase transition characteristics.

Highlights

Einstein’s general relativity is able to describe the dynamics of our solar system well enough
This theory predicted the existence of gravitational waves, which recently were observed by the LIGO and Virgo collaborations [1]
It is evident from the values obtained for the critical horizon radius and their corresponding critical temperatures that they coincide with the extrema in the free energy and divergencies of the heat capacity as well

Summary

Introduction

Einstein’s general relativity is able to describe the dynamics of our solar system well enough. In the past few decades, applying the thermodynamical geometry for studying the phase transition of black holes has gained a lot of attention These studies were pioneered by Weinhold [58,59] and Ruppeiner [60,61]. The metric that Ruppeiner introduced was defined as the minus second derivatives of entropy with respect to internal energy and other extensive quantities, which was conformal to Weinhold’s metric [62] Applying these treatments to the study of black hole thermodynamics caused some puzzling anomalies. 3, we introduce the approaches for studying phase transitions of these black holes in the context of the heat capacity and geometrical thermodynamics.

Black holes solutions in BDM gravity

Closing remarks