Thermodynamics of charged three-dimensional scalar-tensor black holes: the problem of mathematical inconsistency and its solution

Abstract

By applying conformal transformations on the action of scalar–tensor-Euler–Heisenberg theory, we obtain the exact black hole (BH) solutions in its conformal related, the well-known Einstein frame. Through imposing the conditions of (a) vanishing the electric potential at large distance from the source and (b) validity of the first law of BH thermodynamics, we obtain a set of three requirements which are not consistent, mathematically. For solving this problem we assume that under conformal transformations the nonlinearity parameter of Euler–Heisenberg (EH) electrodynamics must transform as a→ae4αϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\rightarrow ae^{4\\alpha \\phi }$$\\end{document}. Then, we obtain the exact solutions of this theory, in both of Einstein and Jordan frames, without any mathematical problems. After calculating thermodynamic quantities, we investigate validity of the thermodynamical first law (TFL) and thermal stability of the EH-BTZ BHs in both of Jordan and Einstein frames, separately.