Thermodynamics of Euler-Heisenberg black holes in three-dimensional Brans-Dicke gravity

We obtained the field equations of Brans-Dicke (BD) theory in the presence of Euler-Heisenberg (EH) electrodynamics, by using the variational principle. They are strongly coupled and the analytic solutions cannot be obtained, easily. Conformal transformations is a mathematical tool by use of which one can translate the BD action, which is written in the Jordan frame, to the Einstein-dilaton action, in the well-known Einstein frame. Through this process, the scalar-coupled EH electrodynamics was obtained and after solving the Einstein frame field equations we found that so long as the nonlinearity parameter is treated as a conformal-invariant quantity, the first law of black hole (BH) thermodynamics will be violated. By assuming an appropriate transformation relation for the EH nonlinearity parameter, the exact BH solutions were obtained without any theoretical problems, and after calculating thermodynamic quantities validity of the thermodynamical first law was proved. Then thermal stability of the Einstein-dilaton BHs was analyzed in the canonical ensemble method. In the next stage, we obtained the Jordan frame exact solutions by applying inverse transformations on their Einstein frame corresponding ones. The BD-EH BHs were introduced which are asymptotically unusual and show BHs with one, two, and three horizons. After calculating the thermodynamic quantities, we proved that the first law of BH thermodynamics is valid for the BD BHs too. We analyzed the thermal stability of the BD BHs by using the canonical ensemble method. Because of the transformation properties of the EH nonlinearity parameter, the stability properties of BD BHs are slightly different from those of Einstein-dilaton.