Thermodynamics of Brans–Dicke–Maxwell black holes in a (2+1)-dimensional spacetime

We study thermodynamics and thermal stability of three-dimensional (3D) charged black holes (BHs) in the Brans–Dicke (BD) gravity theory. The BD field equations, which are written in the Jordan frame, are nonlinear and too complex for direct solving. Thus, by utilizing the conformal transformations (CT), we translated them to the Einstein frame, where the equations are solved easier. In the Einstein–Maxwell-dilaton (EMd) theory, the number of unknowns is one more than the unique equations. Thus, we solved this problem by using an exponential ansatz function and introduced four novel classes of 3D EMd BHs with unusual asymptotic behavior. We proved validity of the first law of BH thermodynamics (FLT), and explored thermal stability of the EMd BHs by use of the canonical ensemble method. We introduced 3D BD-Maxwell BHs by applying the inverse CT on the Einstein frame counterparts. By using the plots, we showed that our solutions can produce BD BHs without horizon, with one or two horizons. By direct calculations, we proved that thermodynamic quantities are conformal-invariant and FLT is valid for the BD BHs too. Moreover, stability properties of the BD BHs are just like those of EMd ones.