Phase Transition of Black Holes in Brans–Dicke Born–Infeld Gravity through Geometrical Thermodynamics

Received Mar 10, 2017; Accepted Jul 13, 2017

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1. Introduction

General relativity is accepted as a standard theory of gravitation and is able to pass more observational tests [1]. Although this theory is successful in various domains, it cannot describe some experimental evidences such as the accelerating expansion of the Universe [2–4]. Moreover, the general relativity theory is not consistent with Mach’s principle nor Dirac’s large number hypothesis [5, 6]. In addition, one needs further accurate observations to fully confirm (or disprove) the validity of general relativity in the high curvature regime such as black hole systems and other massive objects. Therefore, in recent years, more attentions have been focused on alternative theories of gravity. The most considerable alternative theories of gravity are the scalar-tensor theories. One of the good examples of these theories is Brans–Dicke (BD) theory which was introduced in 1961 to combine Mach’s principle with Einstein’s theory of gravity [7]. It is worthwhile to mention that BD theory is one of the modified theories of general relativity which can be used for several cosmological problems like inflation, cosmic acceleration, and dark energy modeling [8–10]. Also, it has a customizable parameter (ω) which indicates the strength of coupling between the matter and scalar fields. The action of 4-dimensional BD theory can be written as(1)S=116π∫d4x-gΦR-ωΦ∇Φ2,where R and Φ are, respectively, the Ricci scalar and self-gravitating scalar field. It is interesting to note that 4-dimensional stationary vacuum BD solution is just the Kerr solution with a trivial scalar field [11]. In addition, Cai and Myung proved that 4-dimensional solution of...