Maxwell–\(\Lambda (x)\)-supported regular black holes in unimodular gravity

Abstract

Recent results [1] show that, when the Einstein Equivalence Principle (EEP) is taken as a fundamental postulate in vacuum, the resulting geometric constraints lead to the condition \(g=-1\), corresponding to the spacetime volume element of the Minkowski vacuum. This provides a motivation for considering the unimodular framework, in which the cosmological constant arises as an integration constant in the conservative case and can be generalized to a spacetime-dependent integration function \(\Lambda (x)\) when energy–momentum conservation is relaxed. Within this setting, regular black-hole geometries can be supported by significantly simpler matter sources, as part of the effective stress–energy tensor is absorbed into a spacetime-dependent vacuum contribution \(\Lambda (r)\), allowing for Maxwell realizations in a broad class of cases. A global Maxwell description exists if and only if a geometric function H(r) remains non-negative; violations of this condition signal the necessity of a nonlinear electromagnetic sector in part of the spacetime. These results suggest that, within the phenomenological non-conservative unimodular framework considered here, black-hole regularization can be partially reinterpreted as an interplay between the electromagnetic and spacetime-dependent \(\Lambda (r)\) sectors.