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

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 $$g=-1$$ 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 Λ (x) Λ ( 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 Λ (r) Λ ( 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 Λ (r) Λ ( r ) sectors.