Generalizing three-dimensional Born-Infeld black holes to the Brans-Dicke gravity

Analytically solving the strongly coupled field equations for Born-Infeld (BI) black holes (BHs), derived from the variation of the Brans-Dicke (BD) theory action, poses a significant challenge. To overcome this, we employ conformal transformations (CTs) to map the Jordan frame action into the Einstein frame. This simplifies the theory to Einstein-dilaton gravity, facilitating more tractable analytical solutions. Through these transformations, we also derive a scalar-coupled BI Lagrangian by allowing the BI nonlinearity parameter to vary naturally. First, we obtain new exact BH solutions within the Einstein-dilaton framework and verify the validity of the first law of thermodynamics. We then analyze their thermal stability using the canonical ensemble method. Subsequently, we reconstruct the exact Jordan frame solutions by inverting both the BI parameter and the CTs from their Einstein frame counterparts. Finally, we compute the thermodynamic quantities of the BD BI BHs, confirm the first law's validity, and examine thermal stability via the canonical ensemble method.