---
title: Generalized Dymnikova Black Hole
url: https://www.emergentmind.com/topics/generalized-dymnikova-black-hole
type: topic
---

# Generalized Dymnikova Black Hole

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 [2507.03701] [2602.22264] [2006.07570] [2404.02818] [1903.10411] [2309.03920] [2509.17630] [2404.09063] [2605.15255].

## 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 [2507.03701].

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 $k$ is the PFDM parameter and $a$ is the string-cloud parameter. Near $r\ll r_*$ the geometry tends to de Sitter with $\Lambda=1/r_0^2$; the large-$r$ sector is asymptotically bounded because $f\to1-a$ [2602.22264].

Higher-dimensional extensions preserve the same qualitative interpolation. In $D\ge4$ dimensions,
$$
f(r)=1-\frac{r_s^{D-3}}{r^{D-3}}\left(1-e^{-r^{D-1}/r_*^{D-1}}\right),
\qquad
r_*^{D-1}=r_0^2\,r_s^{D-3},
$$
so that $f(r)\to1-(r/r_0)^2+O(r^4)$ at the center and approaches the Schwarzschild-Tangherlini form at large $r$ [2404.02818].

Rotating counterparts are constructed by promoting the Dymnikova mass profile to a Kerr-like geometry. One representative form uses
$$
M(r)=M[1-e^{-r^3/b^3}],\qquad
\Sigma=r^2+a^2\cos^2\theta,\qquad
\Delta=r^2+a^2-2M(r)r,
$$
in the Boyer-Lindquist line element. In the limit $b\to0$ one recovers exactly the Kerr metric [2006.07570].

## 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,
$$
f(r_h)=1-\frac{r_g}{r_h}\left[1-e^{-r_h^3/r_*^3}\right]-\frac{r_g c}{r_h^{3\omega+1}}=0.
$$
Merging horizons are located by the simultaneous conditions
$$
f(r_{\rm ext})=0,\qquad f'(r_{\rm ext})=0,
$$
which determine a critical pair $(c_{\rm ext},r_{\rm ext})$. Increasing $c$ or making $\omega$ less negative tends to push the outer, cosmological-like horizon inward, while the inner black-hole horizon is only mildly affected by $c$ but strongly by $r_0$. For $r_g=1$, $r_0=0.3$, $\omega=-2/3$, two horizons coalesce at $c\simeq0.014$, and for $c>c_{\rm ext}$ no black-hole horizon exists [2507.03701].

In rotating Dymnikova-type solutions, horizons are the roots of
$$
\Delta(r)=r^2+a^2-2Mr\,[1-e^{-r^3/b^3}]=0.
$$
There are generally two roots, $r_-$ and $r_+$, representing the Cauchy and event horizons; at $b=b_c(a,M)$ they coincide, and for $b>b_c$ no horizon forms [2006.07570]. In higher-curvature Dymnikova constructions, an analogous pattern appears: for large $\mu/\alpha$ there are two real positive roots, at a critical $\mu_{\rm crit}(\alpha,D)$ they merge, and below that threshold the geometry is regular but horizonless [2404.09063].

Regularity claims are central to the Dymnikova program. In the renormalization-group improved geometry,
$$
f(r)=1-\frac{2m(r)}{r},\qquad
m(r)=m\left[1-e^{-r^3/(r_s\ell_{cr})}\right],
$$
the small-$r$ expansion gives
$$
f(r)=1-\frac{r^2}{\ell_{cr}^2}+O(r^5),\qquad
R(r)=\frac{12}{\ell_{cr}^2}+O(r),\qquad
K(r)=\frac{24}{\ell_{cr}^4}+O(r^2),
$$
so all curvature scalars remain finite at the center [1903.10411]. 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 [2107.04791].

Some quantum-corrected generalizations modify the inner structure more radically. In the GUP-corrected solution, the physical domain is $r\ge r_c$, with a wormhole throat at $r=r_c$; for sufficiently small $r_0/\ell_{\rm GUP}$ the spacetime can become a no-horizon, traversable wormhole rather than a black hole [2309.03920].

## 3. Thermodynamics and phase structure

The Hawking temperature is generally obtained from the surface gravity. For the quintessence-surrounded Dymnikova black hole,
$$
T_H=\frac{\kappa}{2\pi},\qquad \kappa=\frac12 f'(r_h),
$$
so
$$
T_H(r_h;\omega,c)=\frac{1}{4\pi}f'(r_h).
$$
In the limit $c\to0$, this reproduces the pure Dymnikova temperature. The specific heat,
$$
C=\frac{dM}{dT_H},
$$
has zeros and divergences marking thermodynamic phase transitions (“Davies points”); nonzero $c$ and $\omega$ shift the critical radius and can introduce additional sign changes in $C$, altering local stability [2507.03701].

The PFDM plus string-cloud generalization makes the asymptotic normalization explicit:
$$
\kappa=\frac{N f'(r_h)}{2},\qquad N=\frac{1}{\sqrt{1-a}},
$$
and therefore
$$
T_H(r_h)=\frac{f'(r_h)}{4\pi\sqrt{1-a}}.
$$
Its heat capacity diverges where $dT_H/dr_h=0$. The model exhibits a maximum temperature at $r_h=r_c$, with
$$
r_h<r_c \Rightarrow C<0,\qquad r_h>r_c \Rightarrow C>0.
$$
Increasing $a$ raises the maximum $T_H$ and shifts $r_c$ slightly, whereas increasing $k$ lowers $T_H$ and moves $r_c$ more substantially [2602.22264].

Other generalizations preserve the same thermodynamic pattern while modifying the detailed phase portrait. In the regular Dymnikova-Letelier spacetime, the heat capacity diverges when $dT_H/dr_h=0$, signaling second-order phase transitions, and the pole in $C$ shifts to larger $r_h$ as the string parameter $\epsilon$ increases; the Gibbs free energy
$$
G=M-T_H S,\qquad S=\pi r_h^2,
$$
remains continuous across the transition [2605.26372]. In higher dimensions, increasing $D$ shifts the temperature maximum to smaller $r_h$, yielding smaller remnant radii and masses [2404.02818].

Quantum-corrected Einstein-Gauss-Bonnet realizations add a non-area entropy term,
$$
S(r_h)=\frac{A}{4}+2\pi\alpha\ln(A/A_0),\qquad A=4\pi r_h^2.
$$
In that setting, the heat capacity changes sign where $dT_H/dr_h=0$; for $r_h>r_a$ the hole is thermally unstable, for $r_c<r_h<r_a$ stable, and at $r_h=r_c$ one has $T_H=0$ and a zero-temperature remnant [2509.17630].

## 4. Null geodesics, photon spheres, and shadow observables

For static spherical Dymnikova geometries, null geodesics are controlled by the conserved quantities
$$
E=f(r)\frac{dt}{d\lambda},\qquad L=r^2\frac{d\phi}{d\lambda},
$$
and by the radial equation
$$
\left(\frac{dr}{d\lambda}\right)^2+V_{\rm eff}(r)=E^2,
\qquad
V_{\rm eff}(r)=f(r)\frac{L^2}{r^2}.
$$
Unstable circular photon orbits satisfy
$$
r_{\rm ph}f'(r_{\rm ph})-2f(r_{\rm ph})=0,
$$
with critical impact parameter
$$
b_c=\frac{r_{\rm ph}}{\sqrt{f(r_{\rm ph})}}.
$$
For a distant observer, the shadow radius is $R_{\rm sh}\simeq b_c$ or $\alpha_{\rm sh}\simeq b_c$ in unit-distance geometry [2507.03701].

In the quintessence model, the photon-sphere radius and shadow are highly sensitive to $(\omega,c)$. As $c\uparrow$ or $\omega\to-1/3^-$, one finds
$$
r_{\rm ph}\uparrow,\qquad b_c\uparrow,
$$
so the shadow grows. The shadow remains circular because of spherical symmetry, but its angular scale depends sensitively on $(\omega,c)$. With a thin equatorial accretion disk, stronger quintessence produces narrower photon rings and displaced brightness maxima [2507.03701].

The PFDM and string-cloud generalization leads to the photon-sphere condition
$$
\frac{d}{dr}\left[\frac{f(r)}{r^2}\right]=0
\quad\Longrightarrow\quad
2f(r_{ps})-r_{ps}f'(r_{ps})=0,
$$
with
$$
R_{\rm sh}=\beta_c\sqrt{1-a},\qquad
\beta_c=\frac{r_{ps}}{\sqrt{f(r_{ps})}}.
$$
Increasing $a$ decreases $f(\infty)=1-a$ and shrinks the shadow, while increasing $k$ typically reduces $r_{ps}$ and hence $R_{\rm sh}$ [2602.22264].

In rotating Dymnikova-type spacetimes, the shadow boundary is determined by spherical photon orbits obeying $R(r)=0$ and $dR/dr=0$, with celestial coordinates
$$
\alpha=-\xi\csc\theta_0,\qquad
\beta=\pm\sqrt{\eta+a^2\cos^2\theta_0-\xi^2\cot^2\theta_0}.
$$
At fixed $(M,a)$, increasing the magnetic-charge parameter $b$ enlarges the shadow and decreases the distortion parameter, so the shadow is larger and more circular than Kerr’s [2006.07570].

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 [2601.06711].

## 5. Perturbations, quasinormal modes, and dynamical response

A minimally coupled scalar perturbation in the static sector is reduced to the Schrödinger-type equation
$$
\frac{d^2\Phi}{dr_*^2}+[\omega^2-V_s(r)]\Phi=0,
\qquad
\frac{dr_*}{dr}=\frac{1}{f(r)},
$$
with effective potential
$$
V_s(r)=f(r)\left[\frac{\ell(\ell+1)}{r^2}+\frac{f'(r)}{r}\right]
$$
in four-dimensional static models. The WKB approximation gives, at leading order,
$$
\omega_n^2\approx V_s(r_0)-i\left(n+\tfrac12\right)\sqrt{-2V_s''(r_0)}+\cdots,
$$
where $r_0$ is the peak of the potential [2507.03701].

Different generalizations deform the spectrum in different ways. In the quintessence-surrounded Dymnikova geometry, increasing $c$ or making $\omega$ less negative decreases $\Re\,\omega_n$ and increases $|\Im\,\omega_n|$, so scalar modes damp faster, indicating greater linear stability under scalar perturbations [2507.03701]. In the Dymnikova-Letelier spacetime, by contrast, increasing the string parameter $\epsilon$ raises $\Re(\omega_n)$ while decreasing $-\Im(\omega_n)$; for all tested combinations, $\Im(\omega_n)<0$, so the perturbations decay in time [2605.26372].

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 [2303.01987]. In the quantum-corrected Einstein-Gauss-Bonnet model, increasing $\alpha$ and $\beta$ raises $\omega_{\rm Re}$ slightly and reduces $|\omega_{\rm Im}|$, corresponding to faster oscillations with slower damping [2509.17630]. 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 $\alpha$ grows, both $\Re(\omega)$ and $|\Im(\omega)|$ decrease [2404.09063].

Higher dimensions modify the effective potential itself. For scalar perturbations in $D$ dimensions,
$$
V(r)=f(r)\left[\frac{\ell(\ell+D-3)}{r^2}
+\frac{(D-2)(D-4)}{4r^2}f(r)
-\frac{D-2}{2r}\frac{df(r)}{dr}\right].
$$
The reported trend is that increasing $D$ raises both $\Re(\omega)$ and $|\Im(\omega)|$, producing higher-frequency, faster-damped ringdown, while $\Im(\omega)<0$ continues to indicate linear stability [2404.02818].

## 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
$$
G(r)=G_0\left[1-\exp\left(-\frac{r^3}{r_s\ell_{cr}}\right)\right],
$$
so that the final lapse is exactly of Dymnikova type. In that approach, regularity requires a nonzero, finite UV fixed point $g_*$; if $g_*\to0$ one gets $G(r)\equiv0$, while if $g_*\to\infty$ one recovers the classical singularity [1903.10411].

A distinct nonperturbative route is provided by an infinite tower of higher-curvature corrections. There, the reduced field equations are encoded in a function $h(\psi)$ with a Lambert-$W$ 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 [2404.09063]. In unimodular gravity, the same geometry can be sourced by standard Maxwell electrodynamics together with a radial-dependent vacuum contribution,
$$
\Lambda(r)=\frac{R}{4}
=\frac{3m}{2q^6}e^{-r^3/q^3}(4q^3-3r^3),
$$
and the electric field is everywhere regular with localized charge profile $Q(r)=r^2E(r)$ satisfying $Q_\infty=0$ [2605.15255].

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 $(\omega,c)$ through future high-precision imaging [2507.03701]. 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 $\chi^2$ or MCMC yields competitive constraints on $(a,k)$ at the $\sim10$–$20\%$ level [2602.22264]. In higher-dimensional Dymnikova-type models, EHT data for M87* and Sgr A* constrain the core scale and effective Schwarzschild radius; for $D=5$, the reported $1\sigma$ intervals are $r_0\in[0.006,0.12]$, $r_s\in[0.005,0.10]$ for Sgr A*, and $r_0\in[0.035,0.084]$, $r_s\in[0.029,0.077]$ for M87* [2601.06711].

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 [2504.19292]. 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 [2107.04791] [2309.03920]. This suggests that energy-condition behavior is construction-dependent rather than universal.

Source: https://www.emergentmind.com/topics/generalized-dymnikova-black-hole