Generalized Dymnikova Black Hole

• Generalized Dymnikova black hole is a regular solution with a de Sitter core near the origin and Schwarzschild-like behavior at large radii.
• Extensions introduce quintessence, perfect fluid dark matter, string clouds, rotation, higher dimensions, and quantum corrections, enriching its theoretical framework.
• Analyses of horizon structure, thermodynamics, photon shadows, and quasinormal modes provide insights into stability and observational signatures.

The generalized Dymnikova black hole is a class of regular black-hole geometries built around the Dymnikova core structure: a spacetime that interpolates between a de Sitter core near the origin and Schwarzschild or Schwarzschild-like behavior at large radius. In recent work, this regular-center architecture has been extended to black holes surrounded by quintessence, immersed in perfect fluid dark matter and a cloud of strings, generalized to rotating and higher-dimensional settings, derived from renormalization-group improvement, modified by Generalized Uncertainty Principle and Einstein-Gauss-Bonnet corrections, embedded in an infinite tower of higher-curvature terms, and reinterpreted in unimodular gravity with Maxwell sources (Macêdo et al., 4 Jul 2025, Ahmed et al., 25 Feb 2026, Ghosh et al., 2020, Macêdo et al., 2024, Platania, 2019, Alencar et al., 2023, Errehymy et al., 22 Sep 2025, Konoplya et al., 2024, Alencar et al., 14 May 2026).

1. Metric architecture and regular core

Across the literature, the static sector is typically written in Schwarzschild-like coordinates as

$$ds^2=-f(r)\,dt^2+f(r)^{-1}\,dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2).$$

In the quintessence extension,

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$

with $\omega\in(-1,-1/3)$ and $c>0$. As $r\to0$, $\exp(-r^3/r_*^3)\approx1$, so $f(r)\approx1-r^2/r_0^2\to$ de Sitter, while for $r\gtrsim r_*$ one has $f(r)\approx1-r_g/r\to$ Schwarzschild (Macêdo et al., 4 Jul 2025).

The same static ansatz reappears in other generalizations, but with different lapse functions. For a Dymnikova black hole immersed in perfect fluid dark matter and a cloud of strings,

$$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$

Here $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$0 is the PFDM parameter and $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$1 is the string-cloud parameter. Near $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$2 the geometry tends to de Sitter with $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$3; the large-$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$4 sector is asymptotically bounded because $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$5 (Ahmed et al., 25 Feb 2026).

Higher-dimensional extensions preserve the same qualitative interpolation. In $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$6 dimensions,

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$7

so that $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$8 at the center and approaches the Schwarzschild-Tangherlini form at large $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$9 (Macêdo et al., 2024).

Rotating counterparts are constructed by promoting the Dymnikova mass profile to a Kerr-like geometry. One representative form uses

$\omega\in(-1,-1/3)$0

in the Boyer-Lindquist line element. In the limit $\omega\in(-1,-1/3)$1 one recovers exactly the Kerr metric (Ghosh et al., 2020).

2. Horizon structure, extremality, and causal organization

The horizon equation remains model dependent but follows the same logic: horizons are the positive roots of the lapse function. In the quintessence case,

$\omega\in(-1,-1/3)$2

Merging horizons are located by the simultaneous conditions

$\omega\in(-1,-1/3)$3

which determine a critical pair $\omega\in(-1,-1/3)$4. Increasing $\omega\in(-1,-1/3)$5 or making $\omega\in(-1,-1/3)$6 less negative tends to push the outer, cosmological-like horizon inward, while the inner black-hole horizon is only mildly affected by $\omega\in(-1,-1/3)$7 but strongly by $\omega\in(-1,-1/3)$8. For $\omega\in(-1,-1/3)$9, $c>0$0, $c>0$1, two horizons coalesce at $c>0$2, and for $c>0$3 no black-hole horizon exists (Macêdo et al., 4 Jul 2025).

In rotating Dymnikova-type solutions, horizons are the roots of

$c>0$4

There are generally two roots, $c>0$5 and $c>0$6, representing the Cauchy and event horizons; at $c>0$7 they coincide, and for $c>0$8 no horizon forms (Ghosh et al., 2020). In higher-curvature Dymnikova constructions, an analogous pattern appears: for large $c>0$9 there are two real positive roots, at a critical $r\to0$0 they merge, and below that threshold the geometry is regular but horizonless (Konoplya et al., 2024).

Regularity claims are central to the Dymnikova program. In the renormalization-group improved geometry,

$r\to0$1

the small-$r\to0$2 expansion gives

$r\to0$3

so all curvature scalars remain finite at the center (Platania, 2019). In Maeda’s assessment of regular-center models, the rotating Dymnikova black hole is free from scalar polynomial curvature singularities and closed timelike curves, while the absence of parallelly propagated curvature singularities remains an open question (Maeda, 2021).

Some quantum-corrected generalizations modify the inner structure more radically. In the GUP-corrected solution, the physical domain is $r\to0$4, with a wormhole throat at $r\to0$5; for sufficiently small $r\to0$6 the spacetime can become a no-horizon, traversable wormhole rather than a black hole (Alencar et al., 2023).

3. Thermodynamics and phase structure

The Hawking temperature is generally obtained from the surface gravity. For the quintessence-surrounded Dymnikova black hole,

$r\to0$7

so

$r\to0$8

In the limit $r\to0$9, this reproduces the pure Dymnikova temperature. The specific heat,

$\exp(-r^3/r_*^3)\approx1$0

has zeros and divergences marking thermodynamic phase transitions (“Davies points”); nonzero $\exp(-r^3/r_*^3)\approx1$1 and $\exp(-r^3/r_*^3)\approx1$2 shift the critical radius and can introduce additional sign changes in $\exp(-r^3/r_*^3)\approx1$3, altering local stability (Macêdo et al., 4 Jul 2025).

The PFDM plus string-cloud generalization makes the asymptotic normalization explicit:

$\exp(-r^3/r_*^3)\approx1$4

and therefore

$\exp(-r^3/r_*^3)\approx1$5

Its heat capacity diverges where $\exp(-r^3/r_*^3)\approx1$6. The model exhibits a maximum temperature at $\exp(-r^3/r_*^3)\approx1$7, with

$\exp(-r^3/r_*^3)\approx1$8

Increasing $\exp(-r^3/r_*^3)\approx1$9 raises the maximum $f(r)\approx1-r^2/r_0^2\to$0 and shifts $f(r)\approx1-r^2/r_0^2\to$1 slightly, whereas increasing $f(r)\approx1-r^2/r_0^2\to$2 lowers $f(r)\approx1-r^2/r_0^2\to$3 and moves $f(r)\approx1-r^2/r_0^2\to$4 more substantially (Ahmed et al., 25 Feb 2026).

Other generalizations preserve the same thermodynamic pattern while modifying the detailed phase portrait. In the regular Dymnikova-Letelier spacetime, the heat capacity diverges when $f(r)\approx1-r^2/r_0^2\to$5, signaling second-order phase transitions, and the pole in $f(r)\approx1-r^2/r_0^2\to$6 shifts to larger $f(r)\approx1-r^2/r_0^2\to$7 as the string parameter $f(r)\approx1-r^2/r_0^2\to$8 increases; the Gibbs free energy

$f(r)\approx1-r^2/r_0^2\to$9

remains continuous across the transition (Santos et al., 25 May 2026). In higher dimensions, increasing $r\gtrsim r_*$0 shifts the temperature maximum to smaller $r\gtrsim r_*$1, yielding smaller remnant radii and masses (Macêdo et al., 2024).

Quantum-corrected Einstein-Gauss-Bonnet realizations add a non-area entropy term,

$r\gtrsim r_*$2

In that setting, the heat capacity changes sign where $r\gtrsim r_*$3; for $r\gtrsim r_*$4 the hole is thermally unstable, for $r\gtrsim r_*$5 stable, and at $r\gtrsim r_*$6 one has $r\gtrsim r_*$7 and a zero-temperature remnant (Errehymy et al., 22 Sep 2025).

4. Null geodesics, photon spheres, and shadow observables

For static spherical Dymnikova geometries, null geodesics are controlled by the conserved quantities

$r\gtrsim r_*$8

and by the radial equation

$r\gtrsim r_*$9

Unstable circular photon orbits satisfy

$f(r)\approx1-r_g/r\to$0

with critical impact parameter

$f(r)\approx1-r_g/r\to$1

For a distant observer, the shadow radius is $f(r)\approx1-r_g/r\to$2 or $f(r)\approx1-r_g/r\to$3 in unit-distance geometry (Macêdo et al., 4 Jul 2025).

In the quintessence model, the photon-sphere radius and shadow are highly sensitive to $f(r)\approx1-r_g/r\to$4. As $f(r)\approx1-r_g/r\to$5 or $f(r)\approx1-r_g/r\to$6, one finds

$f(r)\approx1-r_g/r\to$7

so the shadow grows. The shadow remains circular because of spherical symmetry, but its angular scale depends sensitively on $f(r)\approx1-r_g/r\to$8. With a thin equatorial accretion disk, stronger quintessence produces narrower photon rings and displaced brightness maxima (Macêdo et al., 4 Jul 2025).

The PFDM and string-cloud generalization leads to the photon-sphere condition

$f(r)\approx1-r_g/r\to$9

with

$$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$0

Increasing $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$1 decreases $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$2 and shrinks the shadow, while increasing $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$3 typically reduces $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$4 and hence $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$5 (Ahmed et al., 25 Feb 2026).

In rotating Dymnikova-type spacetimes, the shadow boundary is determined by spherical photon orbits obeying $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$6 and $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$7, with celestial coordinates

$$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$8

At fixed $$f(r)=1-a-\frac{r_g}{r}[1-e^{-r^3/r_*^3}]+\frac{k}{r}\ln(r/|k|).$$9, increasing the magnetic-charge parameter $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$00 enlarges the shadow and decreases the distortion parameter, so the shadow is larger and more circular than Kerr’s (Ghosh et al., 2020).

Higher-dimensional Dymnikova black holes also possess unstable circular photon orbits and a well-defined shadow. The shadow size grows with the black-hole scale but decreases slightly as the number of dimensions increases (Errehymy et al., 10 Jan 2026).

5. Perturbations, quasinormal modes, and dynamical response

A minimally coupled scalar perturbation in the static sector is reduced to the Schrödinger-type equation

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$01

with effective potential

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$02

in four-dimensional static models. The WKB approximation gives, at leading order,

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$03

where $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$04 is the peak of the potential (Macêdo et al., 4 Jul 2025).

Different generalizations deform the spectrum in different ways. In the quintessence-surrounded Dymnikova geometry, increasing $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$05 or making $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$06 less negative decreases $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$07 and increases $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$08, so scalar modes damp faster, indicating greater linear stability under scalar perturbations (Macêdo et al., 4 Jul 2025). In the Dymnikova-Letelier spacetime, by contrast, increasing the string parameter $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$09 raises $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$10 while decreasing $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$11; for all tested combinations, $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$12, so the perturbations decay in time (Santos et al., 25 May 2026).

Quantum and higher-curvature deformations also leave characteristic spectral fingerprints. In the renormalization-group improved Dymnikova black hole, the fundamental mode shifts only slightly, whereas overtones are strongly sensitive to the near-horizon deformation; accurate frequencies were obtained by combining a Rezzolla-Zhidenko rational approximation with the Leaver continued-fraction method (Konoplya et al., 2023). In the quantum-corrected Einstein-Gauss-Bonnet model, increasing $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$13 and $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$14 raises $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$15 slightly and reduces $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$16, corresponding to faster oscillations with slower damping (Errehymy et al., 22 Sep 2025). In the infinite-tower higher-curvature construction, dominant quasinormal frequencies were computed using both the Bernstein polynomial method and the 13th-order WKB method with Padé approximants, with a high degree of agreement between the two methods; as $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$17 grows, both $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$18 and $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$19 decrease (Konoplya et al., 2024).

Higher dimensions modify the effective potential itself. For scalar perturbations in $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$20 dimensions,

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$21

The reported trend is that increasing $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$22 raises both $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$23 and $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$24, producing higher-frequency, faster-damped ringdown, while $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$25 continues to indicate linear stability (Macêdo et al., 2024).

6. Theoretical realizations, observational interfaces, and open issues

Several papers reinterpret the Dymnikova profile as the endpoint of more fundamental constructions. In the asymptotic-safety program, iterative renormalization-group improvement of the Schwarzschild solution yields

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$26

so that the final lapse is exactly of Dymnikova type. In that approach, regularity requires a nonzero, finite UV fixed point $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$27; if $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$28 one gets $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$29, while if $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$30 one recovers the classical singularity (Platania, 2019).

A distinct nonperturbative route is provided by an infinite tower of higher-curvature corrections. There, the reduced field equations are encoded in a function $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$31 with a Lambert-$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$32 structure, and the resulting theory is explicitly described as genuinely non-perturbative in the higher-curvature couplings: no finite truncation of the curvature series can reproduce the Dymnikova generalization (Konoplya et al., 2024). In unimodular gravity, the same geometry can be sourced by standard Maxwell electrodynamics together with a radial-dependent vacuum contribution,

$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$33

and the electric field is everywhere regular with localized charge profile $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$34 satisfying $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$35 (Alencar et al., 14 May 2026).

Observationally, the main probes are shadow imaging, ring structure, and orbital dynamics. Quintessence shifts shadow size and photon-ring structure, offering a potential way to constrain $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$36 through future high-precision imaging (Macêdo et al., 4 Jul 2025). In the PFDM plus string-cloud model, the deformed orbital and epicyclic frequencies enter relativistic-precession and warped-disk QPO identifications, and fitting twin-peak QPO data for XTE J1550–564, GRO J1655–40, GRS 1915+105, and M82 X-1 via $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$37 or MCMC yields competitive constraints on $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$38 at the $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$39–$$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$40 level (Ahmed et al., 25 Feb 2026). In higher-dimensional Dymnikova-type models, EHT data for M87* and Sgr A* constrain the core scale and effective Schwarzschild radius; for $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$41, the reported $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$42 intervals are $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$43, $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$44 for Sgr A*, and $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$45, $$f(r)=1-\frac{r_g}{r}\left[1-\exp\left(-\frac{r^3}{r_*^3}\right)\right]-\frac{r_g c}{r^{3\omega+1}}, \qquad r_g\equiv 2M,\qquad r_*^3=r_g r_0^2,$$46 for M87* (Errehymy et al., 10 Jan 2026).

Two unresolved themes recur. First, the formation mechanism of the de Sitter core is still unsettled: in collapse-based models, the transition of baryonic matter into the exotic matter required to form the core remains an open question, even though explicit collapse scenarios can be engineered to end in a Dymnikova mass profile and an associated radiation burst (Vertogradov, 27 Apr 2025). Second, energy-condition statements are not uniform across the literature. One assessment concludes that the Dymnikova black hole respects the dominant energy condition everywhere, whereas the GUP-corrected analysis states that the violation of the strong, weak, and null energy conditions characteristic of the pure Dymnikova case does not occur at Planckian scales (Maeda, 2021, Alencar et al., 2023). This suggests that energy-condition behavior is construction-dependent rather than universal.