E. L. B. Junior, J. T. S. S. Junior, F. S. N. Lobo, J. A. A. Ramos, M. E. Rodrigues, D. Rubiera-Garcia, L. Silva, H. A. Vieira
Abstract
We reconstruct a static and spherically symmetric black hole geometry originally proposed as an effective metric by identifying a consistent matter source derived from a fundamental action. The key new result is the demonstration that a previously phenomenological geometry can be obtained from a well-defined variational principle, with the deformation parameters consistently identified as a magnetic charge, thereby providing a concrete theoretical foundation for the effective metric. The space-time is supported by a magnetically charged nonlinear electrodynamics (NED) field non-minimally coupled to a scalar field. Dimensional consistency reduces the parameter space to a single magnetic charge, and the inverse construction formalism yields a one-parameter family of electromagnetic Lagrangians L(F)=Fn+1/(n+1), encompassing both linear and nonlinear electrodynamics. We analyze the horizon structure and determine the critical magnetic charge separating black hole and horizonless configurations. The photon sphere and the corresponding shadow radius are computed, and observational bounds from the Event Horizon Telescope for Sagittarius A* constrain the allowed range of the magnetic charge. In the extended phase space thermodynamics, the solution satisfies the first law and the Smarr relation, exhibits a Hawking--Page phase transition, and presents a single change in stability without van der Waals--type critical behavior. We also investigate the topological properties of both the photon sphere and the thermodynamic parameter space. The photon sphere carries a total topological charge Qtot=-1, while the thermodynamic vector field yields a global winding number W=0, placing the solution in the same topological class as the one of the Reissner--Nordström black hole. We further discuss the universal thermodynamic topological classification of black hole solutions and the role of different thermodynamic ensembles, placing our results within the broader framework of topological black hole thermodynamics. We finally discuss the versatility of this non-minimal coupling framework in both providing theoretical support to previously introduced solution and also to connect them to observational settings within strong-field gravity.
Keywords
Scalar-electromagnetic couplings / Deformed black holes / Shadows / Thermodynamic topology
Physics of the Dark Universe
Volume 53, Number 102433
2026 September
