Unimodular Gravity Rewrites the Recipe for Regular Black Holes

Black holes are famous for the singularities hiding at their centers, points where the equations of general relativity predict infinite curvature and the known laws of physics simply stop. For more than half a century, theorists have tried to sidestep this catastrophe by constructing so-called regular black holes, geometries that look almost identical to Schwarzschild or Reissner–Nordström from the outside but conceal a smooth, finite core instead of a singularity. The catch has always been the price of admission: in standard Einstein gravity, these well-behaved spacetimes demand exotic matter sources, typically highly nonlinear electromagnetic fields whose Lagrangians are complicated, often multivalued, and frequently lacking any clear physical interpretation. A new theoretical study published in General Relativity and Gravitation by G. Alencar of the Federal University of Ceará in Brazil proposes a strikingly different way to pay that price, and the implications could reshape how physicists think about what really holds a regular black hole together.

The key move in the new work is a shift to unimodular gravity, a variant of Einstein’s theory in which the fundamental symmetry is restricted to volume-preserving transformations of spacetime. Rather than fixing the full metric, the theory fixes only the determinant of the metric tensor, which corresponds to the spacetime volume element. This seemingly technical restriction has a profound consequence: the cosmological constant, normally inserted by hand into Einstein’s equations as a bare parameter of the action, instead emerges as an integration constant of the field equations themselves. Because it is not written into the action from the start, the usual argument that quantum-mechanical radiative corrections should drive the cosmological constant to enormous values does not apply in the same way, offering a long-discussed perspective on the worst fine-tuning problem in physics.

Alencar’s study builds on a recent result showing that if the Einstein Equivalence Principle is taken as a fundamental postulate in vacuum, the resulting geometric constraints force the metric determinant to equal minus one, precisely the volume element of the Minkowski vacuum. That observation provides an independent motivation for the unimodular framework. But the new paper goes further by relaxing the standard conservation law of energy and momentum. In ordinary general relativity, the stress–energy tensor must be covariantly conserved. In the non-conservative unimodular framework adopted here, that conservation can be violated in a controlled, integrable way, and the violation is encoded in a spacetime-dependent integration function Lambda(x) that plays the role of a dynamical vacuum contribution. The effective field equations then read as Einstein’s equations with an additional term Lambda times the metric, while the matter sector alone fails to be conserved by exactly the gradient of Lambda.

This relaxation opens a remarkable door. Regular black holes in Einstein gravity require an anisotropic fluid source with an energy density and pressures determined entirely by the geometry through a mass function M(r). In the unimodular framework, the effective source splits into two pieces: the physical matter content and the vacuum contribution Lambda(r). The combination rho plus p_r, which controls several crucial geometric properties of spherically symmetric spacetimes and governs the radial null energy condition, remains exactly invariant under this redistribution. In other words, unimodular gravity does not change the geometry or the key causal structure; it simply reshuffles which part of the gravitational source is attributed to matter and which part is attributed to the vacuum. Part of the complexity traditionally blamed on exotic matter can be silently absorbed by the spacetime-dependent cosmological term.

The framework produces a clean reconstruction machinery. Once a regular black hole geometry is specified through its mass function, the field equations determine two combinations: the product of the electromagnetic invariant F with the derivative of the Lagrangian, and the sum of the Lagrangian and Lambda. The geometry fixes only these combinations, not the individual pieces, which considerably enlarges the class of admissible matter sources. Gradients of Lambda act as an effective electromagnetic source, with a current related to the radial variation of the vacuum sector. A compatibility condition from the electromagnetic Bianchi identity requires the field two-form to be invariant under the flow generated by this current, but the analysis shows this condition is automatically satisfied for the static, spherically symmetric electric configurations considered in the work.

The magnetic and electric sectors behave in strikingly different ways. For purely magnetic monopole configurations, the consistency condition forces Lambda to be a constant, collapsing the entire framework back to the familiar Einstein gravity plus nonlinear electrodynamics with a cosmological constant. Nothing genuinely new appears. For electrically charged configurations, however, Lambda(r) remains dynamical, and this is where the framework earns its keep. The genuinely novel features of the non-conservative unimodular description arise exclusively in the electric sector, where the vacuum contribution can vary with radius and take over part of the job normally done by nonlinear electromagnetic fields.

The most dramatic result concerns ordinary Maxwell electrodynamics. In the Maxwell limit, where the Lagrangian is simply minus F over four and the stress–energy tensor is traceless, both the electric field and the vacuum contribution become completely determined by the geometry alone. The electric field squared equals twice a geometric function H(r) built from the mass function, while Lambda is fixed by derivatives of M(r). But there is a catch, and it is an elegant one: the electric field must be real, which requires H(r) to remain non-negative everywhere. This single inequality serves as a sharp criterion for whether a given regular black hole can be supported by plain Maxwell fields plus a dynamical vacuum term, or whether a genuinely nonlinear electromagnetic sector is unavoidable in part of the spacetime.

Applying the criterion to classic examples yields a fascinating split. The Bardeen black hole, historically the first regular geometry ever proposed, passes with flying colors: its geometric function is positive for all radii, so the entire spacetime admits a Maxwell electric realization, with Lambda(r) providing the de Sitter core near the origin, where it approaches minus six m over b cubed, and fading to zero at large distances. The de Sitter-core branch of the broad Fan–Wang family likewise admits a global Maxwell description, and the resulting electric field takes a far simpler form than the intricate Lagrangian required in the original nonlinear electrodynamics construction. By contrast, the Ayon–Beato–Garcia solution, a genuinely charged regular black hole with proper Coulomb asymptotics, fails the test in its deep interior, where H(r) turns negative near the center. There the Maxwell description survives only in the outer region, and nonlinear electrodynamics must return to do the heavy lifting inside a critical radius.

Alencar is careful to frame the non-conservative sector as phenomenological rather than fundamental. No microscopic derivation of the specific energy–momentum exchange encoded by Lambda(x) is assumed, and quantum-gravity effects have been discussed in the literature as a possible origin of such violations, but the work does not claim to derive them. Whether these static solutions can arise dynamically as end states of gravitational collapse requires a time-dependent analysis that lies beyond the present scope. Still, the trace equation offers a promising consistency check: for Maxwell fields the vacuum contribution is simply minus a quarter of the scalar curvature, and demanding that the same choices of Lambda yield consistent descriptions across cosmology, compact stars, and black-hole shadows could strongly constrain its admissible form.

The broader message is conceptually provocative. Black-hole regularization, long treated as a problem of finding ever more exotic matter sources, can in this framework be partially reinterpreted as an interplay between electromagnetism and the vacuum itself. The curvature that smooths out the singularity does not need to come entirely from nonlinear fields; part of it can be carried by a spacetime-dependent cosmological term that emerges naturally from the structure of unimodular gravity. With ongoing extensions to wormholes and regular black strings already underway, the stage is set for a wider reassessment of what, exactly, is holding the smoothest black holes in the universe together.

Subject of Research: Regular black hole solutions supported by Maxwell electrodynamics and a spacetime-dependent cosmological term in non-conservative unimodular gravity

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

Article References: Alencar, G. (2026). Maxwell–$$\Lambda (x)$$-supported regular black holes in unimodular gravity. General Relativity and Gravitation, 58(10), Article 112. https://doi.org/10.1007/s10714-026-03617-z

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03617-z

Keywords: unimodular gravity, regular black holes, Maxwell electrodynamics, nonlinear electrodynamics, cosmological constant, spacetime-dependent Lambda, Einstein equivalence principle, Bardeen black hole, Ayon-Beato-Garcia solution, energy-momentum non-conservation, de Sitter core, general relativity