Dymnikova-Schwinger Black Holes

• Dymnikova-Schwinger black holes are regular spacetimes where a de Sitter core replaces the singularity using an exponential, Schwinger-effect inspired source profile.
• Quantum corrections, such as those from the generalized uncertainty principle and Einstein-Gauss-Bonnet terms, modify horizon structure, remnant formation, and stability features.
• Extensions to higher dimensions, unimodular gravity, and environmental deformations reveal diverse thermodynamic behaviors and observational signatures in these regular geometries.

Dymnikova-Schwinger black holes are regular black-hole geometries in which the central singularity is replaced by a de Sitter core, while the source profile is motivated by the gravitational analogue of the Schwinger effect. In their standard static, spherically symmetric form, they interpolate between de Sitter behavior near the origin and Schwarzschild-like behavior at large radius, and they have since been extended by generalized uncertainty principle (GUP) corrections, Einstein-Gauss-Bonnet (EGB) terms, higher-dimensional generalizations, asymptotic-safety-inspired running couplings, and alternative matter sectors (Alencar et al., 2023, Errehymy et al., 22 Sep 2025).

1. Foundational definition and Schwinger interpretation

The four-dimensional Dymnikova geometry is commonly written as

$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$

with lapse function

$$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$

or, equivalently,

$$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$

These forms encode the same qualitative structure: a finite-curvature de Sitter core at small $r$ and Schwarzschild asymptotics at large $r$ (Macêdo et al., 4 Jul 2025, Skvortsova, 7 Jun 2026).

The Schwinger interpretation enters through the matter profile. In the classical Dymnikova vacuum, the energy density is taken as

$\rho(r)=\rho_0 \exp\left(-\frac{r^3}{a^3}\right),$

and several later works interpret this as the gravitational analogue of Schwinger pair production in a strong field. In four dimensions, this analogy is made explicit by associating the relevant field strength with curvature and then modifying the pair-creation rate through a minimal-length deformation (Alencar et al., 2023, Estrada et al., 2023).

Within this terminology, “Dymnikova-Schwinger” therefore refers less to a single exact metric than to a class of regular black-hole constructions in which exponential core suppression, Schwinger-like vacuum polarization, and de Sitter regularization are combined. This suggests a unifying label for a family of regular spacetimes rather than a uniquely fixed solution.

2. Geometric construction and quantum-corrected source profiles

A prominent quantum-corrected realization is the EGB construction with static, spherically symmetric ansatz

$$ds^2=-h(r)\,dt^2+\frac{dr^2}{h(r)}+r^2 d\Omega_{D-2}^2,$$

derived from the action

$$\mathcal{S}=\frac{1}{2k_D^2}\int d^D x \sqrt{-g}\left[R+\alpha \mathcal{L}_{GB}\right],$$

with

$$\mathcal{L}_{GB}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2.$$

In the four-dimensional implementation described in the literature, the source is engineered to parallel the Schwinger effect and to incorporate a minimal length $\sqrt{\beta}$ through the GUP, with energy density

$$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$0

The resulting metric function is taken on the minus branch,

$$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$1

where $$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$2 contains the cumulative mass profile and the GUP-dependent exponential-integral terms (Errehymy et al., 22 Sep 2025).

A distinct GUP construction starts from the Schwinger-corrected density and, for large $$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$3, uses the approximate profile

$$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$4

At short distance, the exact GUP-corrected density is

$$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$5

with $$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$6. This profile is defined only for $$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$7, so the minimal radius is built into the construction itself (Alencar et al., 2023).

A third route arises in asymptotically safe gravity, where the Dymnikova lapse follows from a running Newton coupling and takes the form

$$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$8

Here $$f(r)=1-\frac{r_g}{r}\left(1-e^{-r^3/r_*^3}\right), \qquad r_*^3=r_g r_0^2,$$9 is the quantum scale at which near-horizon corrections become relevant (Konoplya et al., 2023, Lütfüoğlu et al., 29 Sep 2025).

3. Horizons, regularity, and energy conditions

The horizon structure is strongly parameter dependent. In the EGB-GUP model, one finds configurations with two horizons, a single degenerate extremal horizon, or no horizon, depending on the Gauss-Bonnet coupling $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$0 and the GUP parameter $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$1; larger $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$2 and $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$3 shrink the event horizon and make the black hole more compact (Errehymy et al., 22 Sep 2025).

Regularity is the central structural property. In the Dymnikova metric, the de Sitter core removes the Schwarzschild singularity, and in the GUP-corrected versions the Kretschmann scalar remains finite for all admissible radii. In the exact GUP construction, the geometry terminates at a minimal sphere $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$4, which is interpreted as a wormhole throat or bounce surface rather than a singular center. For macroscopic objects the GUP correction is confined to the innermost region, whereas for Planckian and sub-Planckian regimes the throat can dominate the interior structure and horizons may be absent (Alencar et al., 2023).

The status of the energy conditions is model dependent. In the pure Dymnikova case, violations of the strong, weak, and null energy conditions are characteristic of the core region. In contrast, the GUP-corrected solutions suppress these violations near the GUP scale; when the de Sitter radius and minimal length are comparable, the null, weak, and strong energy conditions can be satisfied everywhere or at least in the innermost region (Alencar et al., 2023). In traversable wormhole generalizations based on the same density profile, the fluid is phantom-like near the throat for $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$5, while the GUP correction attenuates weak and null energy-condition violations and can restore non-phantom behavior at the Planck scale (Estrada et al., 2023).

A recurrent misconception is that regularity necessarily entails a repulsive interior. An invariant analysis based on the Ricci scalar and Riemann-tensor eigenvalues finds that the Dymnikova spacetime does not exhibit regions in which gravity changes sign; neither the Ricci scalar nor the relevant eigenvalues cross zero for physically reasonable parameters (Luongo et al., 2023).

4. Thermodynamics, evaporation, and remnant formation

Thermodynamic analysis consistently points toward remnant formation. In the EGB-GUP setting, the Hawking temperature is defined by

$$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$6

and the entropy acquires the logarithmic Gauss-Bonnet correction

$$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$7

The temperature falls to zero at a finite nonzero horizon radius, the heat capacity diverges at a critical point and vanishes at the remnant radius, and the remnant mass and size increase with both $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$8 and $$f(r)=1-\frac{2M}{r}\left[1-\exp\left(-\frac{r^3}{2Mr_0^2}\right)\right].$$9 (Errehymy et al., 22 Sep 2025).

The same qualitative picture appears in higher-dimensional Einstein gravity. For the $r$0-dimensional Dymnikova metric,

$r$1

the Hawking temperature vanishes at finite $r$2, indicating remnant formation, while the Davies point and remnant radius shift to smaller $r$3 as the number of dimensions increases (Macêdo et al., 2024).

Detailed Hawking-radiation studies sharpen this picture. Numerical greybody calculations for photons, massless Dirac fermions, gravitons, and Standard Model test fields show that the main effect near the endpoint is not a large change in transmission probabilities. Instead, the rapid decrease of the Hawking temperature strongly suppresses the total luminosity. The photon channel is suppressed more efficiently than the light-fermion channel, the graviton contribution remains subdominant, and the approach to the cold remnant is asymptotic rather than a finite-time complete evaporation (Skvortsova, 7 Jun 2026).

The remnant scenario is frequently connected to the information-loss problem. In the EGB-GUP analysis, the absence of complete evaporation and the persistence of a nonsingular finite-mass endpoint are presented as a mechanism that can mitigate the standard information-loss argument (Errehymy et al., 22 Sep 2025).

5. Perturbations, quasinormal modes, and greybody factors

Perturbative analyses have established linear stability across a wide range of Dymnikova-Schwinger constructions. In the EGB-GUP model, scalar perturbations are treated through a Schrödinger-like wave equation and the WKB approximation, yielding quasinormal frequencies

$r$4

The real part increases slightly with $r$5 and $r$6, while $r$7 decreases as $r$8 increases, so the ringdown becomes longer lived; throughout the examined parameter range, $r$9, confirming stability (Errehymy et al., 22 Sep 2025).

For the asymptotically-safe or renormalization-group-improved Dymnikova geometry, the fundamental mode is only slightly affected by the quantum correction, whereas higher overtones change at a much stronger rate because the deformation is concentrated near the event horizon. This near-horizon sensitivity motivated a continued-fraction parametrization that makes Leaver’s method applicable even when the exact metric function is non-rational (Konoplya et al., 2023).

Axial gravitational perturbations produce a complementary result. In the $r$0-deformed Dymnikova metric, the effective potential differs from Regge–Wheeler mainly near the horizon. Greybody factors and absorption cross-sections therefore show only minor deviations from the Schwarzschild case, and the quasinormal-mode/greybody-factor correspondence holds with high accuracy for $r$1 (Dubinsky, 14 Sep 2025). Direct time-domain and WKB-Padé analyses likewise find systematic shifts in the dominant gravitational quasinormal frequencies: as $r$2 increases, the real oscillation frequency decreases and the damping rate becomes smaller, implying longer-lived modes (Lütfüoğlu et al., 29 Sep 2025).

Environmental deformations can alter these conclusions. Surrounding the Dymnikova black hole with quintessence modifies the metric by adding a Kiselev-type term, changes the Hawking temperature and specific heat, and enhances scalar-mode damping; however, this deformation also breaks regularity, since the Kretschmann scalar diverges at $r$3 (Macêdo et al., 4 Jul 2025).

6. Extensions, embeddings, and unresolved problems

Dymnikova-Schwinger black holes now appear in several broader theoretical settings. One of the strongest existence results is the derivation of the $r$4-dimensional Dymnikova geometry from an infinite tower of higher-curvature corrections. In that framework,

$r$5

and the solution is intrinsically nonperturbative in the coupling parameter: it cannot be approximated by any finite number of curvature powers (Konoplya et al., 2024).

A different reinterpretation is provided by unimodular gravity. There the same Dymnikova geometry can be generated with standard Maxwell electrodynamics, provided the effective vacuum sector becomes dynamical through a radial cosmological contribution $r$6. In this construction, the electric field is everywhere regular, the charge distribution is localized, and the asymptotic charge vanishes, so the spacetime does not behave as an asymptotically charged object (Alencar et al., 14 May 2026).

Astrophysical and cosmological environments have also been explored. Perfect fluid dark matter and a cloud of strings modify the lapse function, Hawking temperature, phase structure, shadow radius, circular orbits, and epicyclic frequencies, thereby changing QPO phenomenology in a parameter-dependent way (Ahmed et al., 25 Feb 2026). Quintessence changes null geodesics, shadows, and scalar quasinormal modes, but at the cost of destroying the original regularity (Macêdo et al., 4 Jul 2025).

The dynamical origin of the de Sitter core remains an open problem. A collapse model in which baryonic matter undergoes a phase transition into exotic matter, with simultaneous emission of electromagnetic radiation, shows how Dymnikova- and Hayward-type regular black holes can emerge in a generalized dynamical framework. In that scenario, the radiation profile is model dependent and may serve as an observational fingerprint, but the basic issue remains explicit: ordinary baryonic matter does not naturally transition into the exotic matter required to form a de Sitter core (Vertogradov, 27 Apr 2025).

Related traversable wormholes underscore the breadth of the framework. Generalized Dymnikova-Schwinger densities in $r$7 dimensions yield asymptotically flat wormholes, and GUP corrections increase the slope of the embedding diagram near the throat while attenuating weak and null energy-condition violation (Estrada et al., 2023). This suggests that Dymnikova-Schwinger constructions occupy a broader category of nonsingular compact-object and black-bounce geometries rather than a single isolated regular black-hole model.