Dymnikova Black Hole Geometry

• The Dymnikova black hole is defined by an exponential mass profile that smoothly transitions from a de Sitter core at r → 0 to a Schwarzschild exterior at large r.
• It is supported by an anisotropic fluid whose stress-energy distribution circumvents central singularities while satisfying key weak energy conditions.
• Perturbative analyses, including quasinormal modes and grey-body factors, reveal distinct near-horizon behaviors and quantum corrections that may impact observable signatures.

The Dymnikova black hole is a regular black-hole geometry in which the Schwarzschild central singularity is replaced by a de Sitter-like core while the spacetime remains asymptotically Schwarzschild at large radius. In the recent literature summarized here, the most frequently used form is a static, spherically symmetric metric with an exponentially suppressed mass profile,

$$ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,\qquad f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr],$$

or equivalently $f(r)=1-2M(r)/r$ with $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$. Related papers use equivalent parametrizations involving $l_{\rm cr}$ or $r_*^3=2Mr_0^2$, and some criteria studies also analyze alternative Dymnikova mass profiles that preserve the same regular-center construction (Vertogradov, 27 Apr 2025, Maeda, 2021).

1. Defining geometry and horizon structure

A standard Dymnikova line element is

$ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$

with

$$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$

This realizes a smooth interpolation between a Schwarzschild exterior and a de Sitter interior. As $r\to\infty$, the exponential term vanishes and one recovers $f(r)\to 1-2M/r$. As $r\to0$, the mass function behaves as $f(r)=1-2M(r)/r$0, so the metric approaches de Sitter space rather than developing a curvature singularity (Vertogradov, 27 Apr 2025).

In the $f(r)=1-2M(r)/r$1 notation used in perturbative studies,

$f(r)=1-2M(r)/r$2

and for $f(r)=1-2M(r)/r$3,

$f(r)=1-2M(r)/r$4

which is exactly the de Sitter form with effective cosmological constant $f(r)=1-2M(r)/r$5. This identifies $f(r)=1-2M(r)/r$6 as the scale controlling the regular core. The Schwarzschild limit is recovered as $f(r)=1-2M(r)/r$7 (Dubinsky, 14 Sep 2025).

The horizons are the positive roots of

$f(r)=1-2M(r)/r$8

For sufficiently small core scale relative to the mass there are two roots, an outer event horizon and an inner Cauchy horizon. In the small-$f(r)=1-2M(r)/r$9 regime,

$M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$0

As $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$1, the inner horizon shrinks to zero and the outer horizon approaches the Schwarzschild value $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$2 (Vertogradov, 27 Apr 2025).

2. Stress-energy support, regularity, and energy conditions

For the exponential Dymnikova profile, the source is an anisotropic fluid. In the collapse construction that matches onto the final static geometry, the exotic core has

$M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$3

As $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$4, $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$5 and $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$6, so the interior is exactly de Sitter. The regular-center mechanism is therefore tied to vacuum-like radial pressure in the core (Vertogradov, 27 Apr 2025).

The same anisotropic pattern also appears in renormalization-group interpretations. In the self-consistent RG construction the effective stress-energy tensor has $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$7, $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$8, and $M(r)=M[1-\exp(-r^3/(2Mr_0^2))]$9. In that setting the weak energy condition is satisfied everywhere, while the strong energy condition is violated near the core, as expected for de Sitter-like vacuum polarization. This places the regularization mechanism in direct correspondence with an effective matter sector generated by the running Newton coupling (Platania, 2019).

A distinct Dymnikova profile appears in criteria studies of regular-center geometries: $l_{\rm cr}$0 For this version,

$l_{\rm cr}$1

The null and weak energy conditions hold globally, the dominant energy condition also holds everywhere, and the strong energy condition is negative near the center. This makes clear that the label “Dymnikova black hole” in the literature encompasses more than one explicit mass profile, but the common structural feature is a regular de Sitter-like center joined to an asymptotically Schwarzschild exterior (Maeda, 2021).

3. Gravitational collapse and dynamical formation

A central open issue is whether ordinary matter can dynamically produce the de Sitter core required by a regular black hole. Vertogradov and Ōvgün address this by modeling gravitational collapse in an ingoing Vaidya-type spacetime,

$l_{\rm cr}$2

The initial matter is taken to be a barotropic fluid with equation of state $l_{\rm cr}$3, $l_{\rm cr}$4, $l_{\rm cr}$5. When a critical density is reached, this baryonic matter converts into the exotic fluid that supports the Dymnikova core, with the excess energy carried away as null electromagnetic radiation (Vertogradov, 27 Apr 2025).

The total conservation law splits into coupled continuity equations,

$l_{\rm cr}$6

where $l_{\rm cr}$7 and $l_{\rm cr}$8 is a phenomenological conversion rate. Solving these gives

$l_{\rm cr}$9

after which $r_*^3=2Mr_0^2$0 follows by substitution. The conversion function $r_*^3=2Mr_0^2$1 is then chosen so that the final total density exactly reproduces the Dymnikova profile (Vertogradov, 27 Apr 2025).

For the final static state, the required density is

$r_*^3=2Mr_0^2$2

and the corresponding conversion rate is

$r_*^3=2Mr_0^2$3

The emitted radiation density becomes

$r_*^3=2Mr_0^2$4

Its volume integral defines a luminosity $r_*^3=2Mr_0^2$5. Because the functional form of $r_*^3=2Mr_0^2$6 depends on both $r_*^3=2Mr_0^2$7 and $r_*^3=2Mr_0^2$8, the model predicts that the spectrum and time profile of the outburst could distinguish a Dymnikova collapse from a Hayward collapse (Vertogradov, 27 Apr 2025).

This construction remains explicitly phenomenological. The formation of the de Sitter core during collapse is stated to be an open question, the baryonic-to-exotic conversion is encoded entirely in $r_*^3=2Mr_0^2$9, and the weak energy condition on the baryonic component requires $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$0. A plausible implication is that observationally useful signatures may arise before a complete microphysical account of the phase conversion is available (Vertogradov, 27 Apr 2025).

4. Linear perturbations, ringdown, grey-body factors, and Hawking emission

Perturbative analyses treat the Dymnikova geometry through Schrödinger-type master equations in the tortoise coordinate $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$1, defined by $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$2. For axial gravitational perturbations,

$ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$3

The effective potential differs from the Regge–Wheeler form only near the horizon when $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$4 is small, which is why many observables remain close to Schwarzschild values (Dubinsky, 14 Sep 2025).

Quasinormal-mode calculations show a pronounced hierarchy between the fundamental mode and the overtones. For the renormalization-group improved Dymnikova black hole, the fundamental scalar mode is only slightly affected by the quantum correction, whereas the overtones change at a much stronger rate. This “outburst of overtones” is traced to a deformation of the geometry solely near the event horizon. In the axial gravitational sector, WKB-Padé and time-domain calculations find that increasing $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$5 decreases both $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$6 and $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$7, so the modes oscillate more slowly and live longer; the time-domain extraction at $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$8 agrees with the WKB-Padé result to within $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,$9 (Konoplya et al., 2023, Lütfüoğlu et al., 29 Sep 2025).

Grey-body calculations indicate a different level of sensitivity. For axial gravitational perturbations, varying $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$0 changes the grey-body factors and absorption cross-sections only by a few percent, and the deviations decrease with increasing multipole number. The Hawking radiation spectrum is therefore governed mainly by the modified Hawking temperature, with grey-body factors contributing only subleading corrections. The proposed correspondence between quasinormal frequencies and transmission coefficients remains accurate for multipoles $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$1 in the Dymnikova case (Dubinsky, 14 Sep 2025).

Massive scalar perturbations introduce genuinely new behavior. In that sector the dominant oscillation frequency grows with the field mass $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$2, while the damping rate decreases, leading to quasi-resonances at sufficiently large $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$3. The late-time waveform shows oscillatory power-law tails,

$$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$4

at intermediate times and

$$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$5

in the asymptotic regime. Increasing $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$6 also strongly suppresses grey-body factors, especially at low frequency, so massive fields provide a more sensitive probe of the near-horizon deformation than the massless grey-body spectrum does (Lütfüoğlu et al., 25 Jan 2026).

Semiclassical Hawking-radiation studies reinforce the same picture. Grey-body thresholds for photons, light fermions, and gravitons move only slightly as the geometry approaches extremality, but the rapid decrease of $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$7 strongly suppresses the total luminosity. Near the endpoint the photon channel is suppressed more efficiently than the fermionic channels, the gravitational contribution remains subdominant, and the black hole approaches a cold extremal remnant only asymptotically. In the fixed-core adiabatic model, the evaporation time to a near-extremal cutoff is much longer than the Schwarzschild lifetime for the same initial mass (Skvortsova, 7 Jun 2026).

5. Renormalization-group, unimodular, higher-curvature, and GUP constructions

One important line of research derives the Dymnikova form from a self-consistent renormalization-group improvement of Schwarzschild. Starting from the classical lapse and iteratively replacing $$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$8, Platania obtains a fixed-point solution

$$f(r)=1-\frac{2M}{r}\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr], \qquad M(r)=M\Bigl[1-\exp\!\Bigl(-\frac{r^3}{2Mr_0^2}\Bigr)\Bigr].$$9

which yields

$r\to\infty$0

The Dymnikova mass profile is therefore recovered as the fixed point of a dynamical renormalization procedure rather than inserted by hand (Platania, 2019).

A different embedding arises in unimodular gravity. There the traceless field equations and a controlled violation of covariant conservation produce a radial cosmological term $r\to\infty$1 through

$r\to\infty$2

The Dymnikova geometry can then be reconstructed both from nonlinear electrodynamics and from standard Maxwell electrodynamics. In the Maxwell realization,

$r\to\infty$3

the energy density is finite,

$r\to\infty$4

and the effective enclosed charge satisfies $r\to\infty$5 as $r\to\infty$6. The same regular core is thus generated without asymptotic electric charge (Alencar et al., 14 May 2026).

An infinite tower of higher-curvature corrections provides yet another route. In that framework, a nonperturbative choice of the master function $r\to\infty$7 yields the $r\to\infty$8-dimensional Dymnikova lapse

$r\to\infty$9

Near the center,

$f(r)\to 1-2M/r$0

so the core is exactly de Sitter with finite curvature invariants. The same construction supports a stable quasinormal spectrum computed independently by the Bernstein polynomial method and the 13th-order WKB method with Padé approximants (Konoplya et al., 2024).

Generalized uncertainty principle corrections modify the Dymnikova vacuum in a different manner. The GUP-corrected density is nonzero only for $f(r)\to 1-2M/r$1, where $f(r)\to 1-2M/r$2, and the metric remains nonsingular. The minimal radius $f(r)\to 1-2M/r$3 acts as a wormhole throat inside the de Sitter core. In this model the null and weak energy conditions hold everywhere, and the region of strong-energy-condition violation shrinks as the GUP scale approaches the core scale. This suggests that the usual energy-condition violations of regular black holes can be softened at Planckian scales by the minimal-length deformation (Alencar et al., 2023).

6. Rotating, environmental, higher-dimensional, and observational developments

The static exponential Dymnikova solution also admits a rotating generalization. In one construction, the static spacetime is obtained as an exact solution of Einstein gravity coupled to nonlinear electrodynamics with magnetic charge parameter $f(r)\to 1-2M/r$4, and the Newman–Janis procedure yields a Kerr-like metric with

$f(r)\to 1-2M/r$5

As $f(r)\to 1-2M/r$6, the Kerr limit is recovered. The rotating regular black hole has a larger ergoregion thickness at the equator as $f(r)\to 1-2M/r$7 increases, and its shadow is larger but less distorted than the Kerr shadow. The same study reports a one-to-one correspondence between ergosphere and shadow (Ghosh et al., 2020).

Environmental modifications preserve the Dymnikova core only conditionally. Adding a Kiselev quintessence term gives

$f(r)\to 1-2M/r$8

with up to three positive horizons, including a cosmological-type horizon for sufficiently large $f(r)\to 1-2M/r$9. In that model the Hawking temperature and heat capacity exhibit parameter-dependent phase transitions, and nonzero $r\to0$0 can destroy the regular core at small radius. Related constructions with perfect-fluid dark matter plus a cloud of strings, and with the Dymnikova–Letelier string-fluid deformation, also produce non-monotonic temperature behavior, Davies-type phase transitions, systematic shifts in quasinormal frequencies, and measurable changes in the photon sphere, shadow radius, and quasi-periodic oscillation frequencies (Macêdo et al., 4 Jul 2025, Ahmed et al., 25 Feb 2026, Santos et al., 25 May 2026).

Higher-dimensional extensions have been studied both theoretically and against data. In $r\to0$1 dimensions one commonly writes

$r\to0$2

The shadow size grows with the black-hole scale but decreases slightly as the number of dimensions increases. A comparison with Event Horizon Telescope measurements gives, for $r\to0$3, the $r\to0$4 intervals $r\to0$5 for Sgr A* and $r\to0$6 for M87*. The same analysis states that Dymnikova cores shift the shadow size only at the $r\to0$7–$r\to0$8 level for the extreme allowed $r\to0$9 values, while current EHT uncertainties remain at the $f(r)=1-2M(r)/r$00–$f(r)=1-2M(r)/r$01 level (Errehymy et al., 10 Jan 2026).

Across these developments, the Dymnikova black hole remains defined by a specific regular-center mechanism: the mass function vanishes sufficiently rapidly as $f(r)=1-2M(r)/r$02 to generate a de Sitter core, yet approaches a constant fast enough to preserve the Schwarzschild exterior. The main open issues identified in the surveyed literature are the microphysical origin of the exotic core during collapse, the model dependence introduced by environmental or higher-curvature deformations, and the extent to which near-horizon observables—especially overtones, early-time ringdown, shadow systematics, and the thermodynamic suppression near extremality—can distinguish the Dymnikova geometry from singular alternatives (Vertogradov, 27 Apr 2025).