Thermodynamics of Brans–Dicke–BTZ black holes coupled to conformal-invariant electrodynamics

Abstract

The field equations of Brans–Dicke conformal-invariant theory in (2+1)-dimensions are highly nonlinear and difficult to solve directly. They are related to those of Einstein–dilaton theory, where the solutions can be obtained easily, by use of a mathematical tool known as the conformal transformation. The exact solutions of three-dimensional Brans–Dicke theory, which are obtained from their Einstein-dilaton counterparts, give two novel classes of conformal-invariant black holes. When the scalar potential is absent (or is considered constant) in the action, it has been shown that the exact solution of this theory is just the conformal-invariant BTZ black hole with a trivial constant scalar field. This issue corresponds to the four-dimensional Brans–Dicke–Maxwell theory discussed in Ref. [R.-G. Cai, Y. S. Myung, Phys. Rev. D 56, 3466 (1997)]. The Brans–Dicke conformal-invariant black holes’ thermodynamic quantities have been calculated by use of the appropriator methods, and it has been shown that they satisfy the first law of black hole thermodynamics in its standard form. The thermal stability of Brans–Dicke black holes has been studied by use of the canonical ensemble method and noting the signature of the black holes’ heat capacity.