Effective metric description of charged black holes

Abstract Charged black holes arise as solutions of General Relativity (GR) coupled to Maxwell theory. As functions of the mass and charge, they can exhibit extremal behavior, in which case they are stable against thermal decay. (Quantum) corrections to GR are expected to alter the classical features of these objects, especially near extremality. To capture such effects in a model-independent way, we extend the Effective Metric Description (EMD) previously introduced in (Del Piano et al. in Phys Rev D 109(2):024045, 2024, https://doi.org/10.1103/PhysRevD.109.024045 , arXiv:2307.13489 [gr-qc]; Eur Phys J C 84(12):1273, 2024, https://doi.org/10.1140/epjc/s10052-024-13609-5 , arXiv:2403.12679 [gr-qc]) for spherically symmetric and static black holes. The EMD parametrizes deformations of the metric in terms of physical quantities, such as the radial spatial distance to the event horizon. While the latter is still viable for non-extremal charged black holes, we argue that the proper time of a free-falling observer is better suited in the extremal case: we derive the necessary conditions for the parameters of such an EMD for constructing a consistent space-time in the vicinity of the (extremal) horizon. Finally, we illustrate our framework through a concrete example, and mention implications of the Weak Gravity Conjecture on the effective metric parameters.