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

The research, carried out by A. A. M. Silva, M. H. Macêdo and R. R. Landim of the Federal University of Ceará in Brazil, focuses on the generalized Bardeen black hole, a two-parameter spacetime introduced by J. C. S. Neves and A. Saa that contains several celebrated singularity-free solutions as special cases. By tuning the parameters alpha and beta, the same metric reproduces the Bardeen black hole, the Hayward black hole, and a Simpson–Visser-type geometry, all of which replace the pathological central point of the Schwarzschild solution with a smooth, de Sitter-like core of finite curvature. The team’s central question was deceptively simple: do these different regular black holes belong to fundamentally distinct thermodynamic classes, or are they merely variations on a single topological theme?

To answer it, the authors employed a technique known as the off-shell generalized Helmholtz free energy method, a framework in which black hole thermodynamics is recast as the study of zeros of a carefully constructed vector field. In ordinary black hole thermodynamics, following the foundational work of Bekenstein and Hawking, the entropy of a black hole is proportional to the area of its event horizon and its temperature is proportional to its surface gravity. But local quantities like heat capacity can be cumbersome to work with, particularly for regular black holes where extra terms complicate the first law of thermodynamics. The topological approach sidesteps some of these difficulties by encoding stability information in winding numbers, integer-valued quantities that count how many times the vector field wraps around each of its zeros.

The machinery works as follows. The researchers defined a vector field whose first component is the derivative of the generalized Helmholtz free energy with respect to the horizon radius, evaluated at a Euclidean time period tau that acts as an inverse temperature. The zeros of this field correspond to thermodynamic equilibrium states of the black hole, and evaluating the on-shell condition tau equals one over T exactly recovers the Hawking temperature, a consistency check that the method passes for every geometry in the family. Each isolated, non-degenerate zero carries a winding number determined by the sign of the Jacobian of the mapping, which for this construction reduces to the sign of the second derivative of the free energy. Positive winding numbers mark locally stable branches with positive heat capacity; negative ones mark unstable branches.

The results are striking in their unity. Whenever two regular zeros exist simultaneously on the physical outer-horizon branch, they come as a pair: one smaller-radius defect with winding number plus one, corresponding to a locally stable configuration, and one larger-radius defect with winding number minus one, corresponding to an unstable one. Adding the charges gives a total topological number W equal to zero for the Bardeen case, the Hayward case, and the Simpson–Visser-type case alike. As the control parameter tau is lowered toward a critical value, the two defects slide toward each other along the inverse-temperature curve and coalesce at a critical radius, where the Jacobian vanishes and the black hole undergoes a phase transition signaled by a divergence and sign change of the heat capacity. Below the critical value, no physical zeros remain at all.

Crucially, the parameters alpha and beta, which encode how the singularity is regularized, do not change this global verdict. What they do control is everything local: the extremal bound below which no physical outer horizon exists, the exact location of the critical radius, the critical inverse temperature, the maximum Hawking temperature, and the width of the locally stable window. For the Bardeen case, with alpha equal to three and beta equal to two, the critical radius sits at roughly 2.697 times the length parameter a, while the Hayward case, with alpha and beta both equal to three, places it at approximately 2.168, and the Simpson–Visser-type case at exactly 1. The Simpson–Visser-type configuration also reaches the highest maximum temperature of the three, followed by Hayward and then Bardeen, demonstrating that the regularization mechanism directly shapes the thermal behavior even when the topological classification is unchanged.

The Schwarzschild limit tells a different story. As the length parameter a goes to zero, the regular core disappears, the singularity returns, and the inverse-temperature curve becomes monotonic, with no finite critical point. Only a single defect survives, carrying winding number minus one and yielding a total topological number W equal to minus one, in perfect agreement with the well-known thermodynamic instability of the Schwarzschild black hole, whose heat capacity is always negative. Placed within the recently developed universal thermodynamic-topological classification, the regular cases fall into the established W0-plus class, characterized by the stable-to-unstable ordering of their defects, while the Schwarzschild limit belongs to the W1-minus class. The authors emphasize that their contribution is not the discovery of a new topological class but a unified, family-wide analytical demonstration that varying the regularization parameters shifts local thermodynamic data without ever moving a configuration out of its global class.

Physically, the findings touch on one of the most tantalizing possibilities in black hole physics: remnants. Because the Hawking temperature of these regular geometries vanishes exactly at the extremal radius, evaporation halts at a finite mass, leaving behind a stable, non-radiating object. Such remnants have even been proposed as candidates for dark matter, and the new topological framework offers a systematic way to assess their thermodynamic viability across an entire family of models rather than case by case. The study also draws an intriguing comparison with recent work on regular black holes built from pure gravity through infinite towers of higher-curvature corrections, which, despite a completely different gravitational origin and entropy prescription, land in the very same W0-plus class, hinting at a universality that transcends the details of how regularity is achieved.

The authors are careful to note the limits of their analysis. The entire classification is performed in a fixed-parameter ensemble in which the length parameter a, along with alpha and beta, is held fixed during thermodynamic variations; promoting these quantities to thermodynamic variables would modify the first law and could, in principle, alter the topological structure. Future extensions to charged, rotating, or higher-dimensional generalizations remain open territory. Nevertheless, the work delivers a crisp analytical criterion separating parameter-dependent local stability from the global topological identity of regular black holes, and it reinforces a growing realization in gravitational physics: that the deepest questions about spacetime, from the fate of singularities to the stability of horizons, may be answered not by looking at any single solution, but by reading the topological signatures written into the thermodynamics of entire families of them.

Subject of Research: Topological thermodynamics and stability classification of regular black holes in the generalized Bardeen spacetime family

Article Title: Topological thermodynamics of generalized Bardeen black hole

Article References: Silva, A. A. M., Macêdo, M. H., & Landim, R. R. (2026). Topological thermodynamics of generalized Bardeen black hole. The European Physical Journal C, 86(9), Article 1086. https://doi.org/10.1140/epjc/s10052-026-16379-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16379-4

Keywords: black hole thermodynamics, topological classification, regular black holes, Bardeen black hole, Hayward black hole, Simpson–Visser spacetime, winding number, Helmholtz free energy, Hawking temperature, phase transition, heat capacity, black hole remnants