The idea that matter might shed its ordinary character at extreme densities is not new. In the 1960s, Bardeen proposed the first phenomenological black-hole solution without a central singularity, and Gliner and Sakharov independently suggested that, at sufficiently high densities, matter may undergo a transition into a vacuum-like state with an equation of state P equal to minus rho. In that regime the innermost geometry tends toward de Sitter spacetime, which is why the regular central region of such objects is usually called a de Sitter core. Regular black holes built on this principle, including the Dymnikova, Hayward, and Bardeen models, must therefore contain matter that violates the strong energy condition in their deep interiors. The trouble is observational: once a black hole has formed, an external observer has no direct access to its central region, so it becomes extraordinarily difficult to determine whether the interior holds a singularity or a regular core. Shadow studies, quasi-normal modes, and thermodynamic analyses show that regular black holes can be made compatible with Event Horizon Telescope constraints, but singular black holes can achieve similar agreement, leaving the question open.

This is where the new work, published in The European Physical Journal C by Vitalii Vertogradov, Zhanna Kuznetsova, and Yu Shi, changes the strategy. Instead of examining only the final compact object, the authors investigate the formation process itself. Ordinary massive stars contain no matter capable of violating the strong energy condition, so if a regular black hole forms, the exotic sector responsible for singularity avoidance must be generated dynamically during the collapse. At critical densities, ordinary baryonic matter may undergo a phase transition into a new state that prevents the central singularity from forming. If this transition occurs before the event horizon completely hides the relevant region, part of the released energy may escape as electromagnetic radiation. The authors estimate the density of that radiation and its corresponding bolometric luminosity, and they show that the process can produce luminosities comparable to those of gamma-ray bursts, the cosmic high-energy transients first identified by Klebesadel and colleagues.

Gamma-ray bursts provide a natural context for such a mechanism. The observed population is traditionally divided into short and long bursts with a threshold of roughly two seconds. Long bursts are usually associated with the collapse of massive stars and the formation of a relativistic jet powered by a compact central engine, an association strongly supported by the detection of SN 1998bw in connection with GRB 980425. Short bursts, in contrast, are connected with compact binary mergers, a picture confirmed by the multi-messenger event GW170817 paired with GRB 170817A. Yet the physical origin of these explosions remains incompletely understood, and recent events challenge the standard classification. The burst GRB 221009A, often called the Brightest Of All Time, reached an isotropic-equivalent energy of order ten to the fifty-fifth erg, while GRB 250702B appeared as an exceptionally long and repeating transient with activity lasting a day and energetics of at least ten to the fifty-fourth erg. Such outliers motivate the search for additional channels of high-energy emission during compact-object formation.

The technical core of the model lies in a dynamical spacetime written in Eddington-Finkelstein coordinates, where the mass function M depends on both the Eddington time and the radius. Applying the Einstein equations to this geometry yields a radiation flux density proportional to the time derivative of the mass function, together with expressions for the energy density and pressure of the collapsing matter. The authors decompose the matter source into a barotropic component, a radiative component, and the final exotic matter, imposing conservation of the total energy-momentum tensor while allowing the individual components to exchange energy. A dimensionless conversion rate, which must depend only on invariant quantities such as the density and pressure of the newly formed matter to preserve general covariance, governs how baryonic matter transforms into the new phase with radiation emission. Solving the resulting coupled equations produces a compact expression relating the radiation density directly to the density and pressure of the matter supporting the final regular black hole: the radiation is not introduced as an external phenomenological source but follows from the properties of the matter that removes the singularity.

To locate the region where this radiation exists, the authors match the collapsing exterior, described by the Kiselev solution supported by barotropic matter, with each regular interior geometry. They impose a smooth matching that goes beyond the standard Darmois-Israel junction conditions, requiring continuity of the mass function and its first and second derivatives at a critical radius where the equations of state of the exterior matter and the exotic matter coincide. At this critical radius the transition is complete and the radiation density vanishes, so radiation is produced only in the intermediate region where baryonic matter is being converted into the new phase. For the Dymnikova model, whose pressure can be rewritten as a generalized Hagedorn equation of state, the critical radius follows from a simple cubic relation. The Hayward model yields a generalized polytropic equation of state, and the Bardeen model, whose magnetic charge is supported by nonlinear electrodynamics, follows an analogous construction. In every case the matching procedure provides a physically consistent transition from a barotropic exterior to a regular de Sitter-like core.

The radiation density profiles reveal a striking dependence on the regularization scale, the parameter that controls how strongly each model deviates from the Schwarzschild geometry. For the Dymnikova metric, the maximum radiation density corresponds to the smallest de Sitter core radius considered, and the profile flattens as that radius grows. For the Hayward and Bardeen metrics, the radiation density likewise peaks for the minimal values of the regularization parameters, and the Bardeen maximum exceeds the Hayward maximum for the same parameter value. This behavior is physically natural: as the collapsing object becomes denser, the conversion rate of baryonic matter into the nonsingular phase accelerates, reaching a maximum where the radiation density also peaks, before both decline to zero at the critical radius. A more compact regular core corresponds to a stronger gravitational field region, where the conversion can release more energy.

The luminosity estimate then connects the model to observation. For an approximately radial null flux, the bolometric luminosity measured through a sphere of radius r equals four pi r squared times the physical energy density of the outgoing radiation. Restoring cgs units, the dominant dimensional factor is c to the fifth power divided by G, numerically 3.63 times ten to the fifty-ninth erg per second, a scale set purely by the fundamental constants. Observed gamma-ray bursts reach luminosities of order ten to the fifty-fifth erg per second, so the remaining dimensionless factor must be of order ten to the minus fourth. Because radiation produced below the event horizon cannot reach a distant observer, the relevant estimate must be made near the outer event horizon, whose location depends on the black-hole mass and the regularization parameter. The analysis shows that increasing the regularization parameter makes the outer horizon shrink and the inner horizon expand, until at a critical value the two merge into an extremal black hole, which bounds the mass derivative between zero and one half.

The resulting constraints are remarkably tight. For a ten solar mass black hole, the Dymnikova model requires a dimensionless regularization parameter of order ten to the minus one, while the Hayward and Bardeen models require parameters of order ten to the minus two. In all three cases the outer horizon remains very close to the Schwarzschild value of twice the mass, meaning the geometry represents only a minimal deviation from the standard black hole. This conclusion is qualitatively consistent with current shadow observations, which allow only moderate deviations from the Schwarzschild shadow size, so the same small-deviation regime that keeps the luminosity estimate under control also keeps the near-horizon geometry close to the standard picture. The importance is twofold: the formation of a regular black hole may be accompanied by an observable electromagnetic signal carrying information about the interior structure that becomes inaccessible after horizon formation, and the burst luminosity provides an additional constraint on the regularization parameters. A gamma-ray burst is not a unique signature of a regular black hole, but it tests the matter conversion during collapse rather than the final geometry, so shadow and burst observations can be combined to distinguish singular from regular black holes. The authors emphasize that their estimate is an order-of-magnitude first step: no detailed spectrum or prompt-emission model was constructed, and future work must include a fully dynamical description of the collapse, the question of whether radiation escapes before the horizon hides the transition region, and an extension to rotating regular black holes, since astrophysical collapse is expected to produce spinning compact objects.

Subject of Research: Theoretical Physics

Article Title: Regular black hole formation and gamma-ray burst from matter conversion

Article References: Vertogradov, V., Kuznetsova, Z., & Shi, Y. (2026). Regular black hole formation and gamma-ray burst from matter conversion. The European Physical Journal C, 86(10), Article 1208. https://doi.org/10.1140/epjc/s10052-026-16452-y

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16452-y

Keywords: black holes, gamma-ray bursts, gravitational collapse, singularity, regular black holes, phase transition, general relativity, de Sitter core, stellar collapse, high-energy astrophysics, Regular, black