Dymnikova Black Holes in Unimodular Gravity: Maxwell Sources and Vacuum Contributions

Abstract: In this work, we investigate the Dymnikova regular black hole within the framework of unimodular gravity, emphasizing the role of the effective vacuum sector in the regularization of the geometry. By allowing a controlled violation of the covariant conservation of the energy--momentum tensor, the cosmological contribution emerges dynamically as a radial-dependent function, $Λ=Λ(r)$. We first reinterpret the Dymnikova spacetime as a charged configuration supported by nonlinear electrodynamics and derive the corresponding electric and magnetic sources. Subsequently, we demonstrate that the same geometry can be consistently generated by standard Maxwell electrodynamics in unimodular gravity. In this construction, the resulting electric field is everywhere regular and corresponds to a localized charge distribution with vanishing asymptotic charge, indicating that the spacetime does not behave as an asymptotically charged object.

• The paper shows a novel realization of Dymnikova regular black holes using standard Maxwell electrodynamics within a unimodular gravity framework.
• It reveals that a geometry-dependent cosmological term, emerging from non-conserved energy-momentum, regularizes the black hole core and avoids singularities.
• Numerical results confirm that the electric field is regular and localized, effectively separating classical matter behavior from vacuum-induced energy condition violations.

Dymnikova Regular Black Holes in Unimodular Gravity: Maxwell Sources and Vacuum Contributions

Introduction

The paper "Dymnikova Black Holes in Unimodular Gravity: Maxwell Sources and Vacuum Contributions" (2605.15255) systematically investigates the construction and interpretation of the Dymnikova regular black hole solution within the unimodular gravity framework. By explicitly treating the energy-momentum tensor's non-conservation allowed by unimodular gravity, the work demonstrates how a dynamically emergent, geometry-dependent cosmological function regularizes the black hole spacetime. The authors revisit nonlinear electrodynamic (NED) descriptions and present, for the first time, a fully consistent realization with standard Maxwell electrodynamics in unimodular gravity, revealing novel implications regarding the matter sector and vacuum contributions.

Background: Regular Black Hole Solutions and Unimodular Gravity

Regular black hole (RBH) metrics replace the central singularity with a regular core, typically modeled by matter distributions that violate classical energy conditions. The Dymnikova solution [Dymnikova:1992ux] features an exponential energy density profile $\rho_0\,e^{-r^3/r_*^3}$ with a de Sitter core and Schwarzschild asymptotics. Previous studies attribute RBHs to various sources, especially NED, which enables regular geometries via modified electromagnetic Lagrangians that permit energy condition violations in a controlled fashion [Bronnikov2023, Ayón-Beato & Garcia:2000].

Unimodular gravity imposes the constraint $\sqrt{-g}=g_0$ on the metric determinant. While classically equivalent to general relativity (GR), unimodular gravity alters the cosmological constant's status to an integration function and relaxes strict energy-momentum conservation. This facilitates matter models with emergent, position-dependent vacuum contributions, a property leveraged for regularization in compact object solutions [Jirousek:2023gzr, Smolin:2009ti].

Nonlinear Electrodynamics Sources for the Dymnikova Solution

The authors first recast the Dymnikova metric as a NED-supported geometry. The solution admits magnetic and electric origins via spherically symmetric field configurations:

• Magnetic Source: The magnetic field profile yields a NED Lagrangian $$L(F)=-\frac{12m}{P^3} e^{-\left(\frac{8}{P^6}F^3\right)^{1/4}}$$, which approaches a constant near $r=0$ and zero for $r\to\infty$. This is in line with Bronnikov's necessary conditions for regular metrics in NED [Bronnikov2023].
• Electric Source: The electric field $E(r)$ is regular everywhere and localizes charge, with vanishing asymptotic flux. The Lagrangian $L(r)$ and $L_F(r)$ cannot be inverted in closed form, but their local behavior matches the magnetic case.

Both configurations confirm that regular RBHs can be phenomenologically realized by NED, but require the matter sector itself to violate classical energy conditions—a limitation from a physical interpretation perspective.

Maxwell Realization in Unimodular Gravity

The novel contribution of the paper is the construction of the Dymnikova metric as a solution of standard Maxwell electrodynamics within unimodular gravity. In this framework:

• The energy-momentum tensor is not strictly conserved but instead allows the cosmological term $\Lambda(r)$ to dynamically arise from the geometry.
• The Maxwell sector is completely regular, satisfying all classical energy conditions, with the ordinary electric field $E(r)$ localized and vanishing asymptotically: $\sqrt{-g}=g_0$0.
• The regularization of the spacetime—specifically, the de Sitter core and global regularity—is achieved by the emergent, radial-dependent vacuum term $\sqrt{-g}=g_0$1, which interpolates between a constant at the origin and zero at infinity.

The Maxwell field remains physically consistent everywhere ($\sqrt{-g}=g_0$2 for all $\sqrt{-g}=g_0$3), and the vacuum contribution responsible for regularity is exclusively geometrically determined, $\sqrt{-g}=g_0$4 in the traceless sector.

Numerical Results and Contradictory Claims

A strong result is the explicit demonstration that standard Maxwell electrodynamics can support the Dymnikova geometry in unimodular gravity, with the violations needed for regularity entirely attributable to the dynamically emergent vacuum sector, not the ordinary matter sector. This is contradictory to previous presumptions in regular black hole phenomenology, where violations are traditionally assigned to the electromagnetic field's nonlinearity.

The electric field is everywhere regular and the charge distribution is localized, with vanishing global charge, a distinct feature from typical charged RBH solutions.

Implications and Prospects in Theoretical Physics

This mechanism offers a clean sectoral separation: standard Maxwell matter can be used, relegating all energy condition violations to the vacuum sector governed by geometry. This enhances physical interpretability and paves the way for regular compact object models that are compatible with classical matter fields. It also provides a concrete realization of the original Dymnikova interpretation: regular black holes as vacuum-induced configurations, possibly related to gravitational vacuum polarization.

Practically, these findings underpin new classes of compact objects in unimodular gravity (including black strings and wormholes [3137491, Alencar:2026oxy]), with ramifications for quantum gravity, gravitational wave signatures, and observational tests via Event Horizon Telescope imaging.

Future developments could include dynamical or quantum corrections to unimodular gravity, higher-dimensional extensions, and applications to cosmological inhomogeneities where vacuum contributions are spatially modulated.

Conclusion

The authors have rigorously established that the Dymnikova solution is consistently realizable with physical Maxwell sources in unimodular gravity, with regularization fully encoded in a geometry-dependent vacuum sector. This clarifies the matter-vacuum separation, extends previous NED approaches, and motivates further research into unimodular gravity’s phenomenology and its physical ramifications in the context of regular black holes and related compact objects.