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

# Dymnikova Regular Black Hole

Dymnikova Regular Black Hole

A Dymnikova regular black hole is a static, spherically symmetric solution of the gravitational field equations that replaces the central singularity of the standard Schwarzschild solution with a smooth de Sitter-like core. The Dymnikova profile is realized via a smooth energy-density distribution and produces a spacetime with two horizons—an inner (Cauchy) and an outer (event) horizon. Recent developments generalize the Dymnikova construction to arbitrary spacetime dimensions, analyze its thermodynamic properties, photon trajectories, observational signatures, and confront the model with Event Horizon Telescope (EHT) data to yield empirical constraints [2601.06711].

## 1. Spacetime Metric and Defining Parameters

The Dymnikova-type metric in $D$ spacetime dimensions is
\[
ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2
\]
with the lapse function
\[
f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]
\]
where the parameters are defined as:
\[
r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}
\]
Here, $r_0$ encodes the de Sitter core length scale, $\rho_0$ is the central energy density, $r_s$ is the would-be Schwarzschild radius in $D$ dimensions, $M$ is the ADM mass, and $\Omega_{D-2} = 2\pi^{(D-1)/2}/\Gamma(\frac{D-1}{2})$ gives the area of the unit $(D-2)$-sphere.

The exponential profile for $f(r)$ ensures the metric is regular at $r=0$, with a finite curvature invariant.

## 2. Horizon Structure

Dymnikova regular black holes typically exhibit two horizons, determined by real positive roots $\{ r_-,\,r_+\}$ of $f(r_\mathrm{H})=0$:
\[
f(r_H) = 0 \quad \Longrightarrow \quad \frac{r_s^{D-3}}{r_H^{D-3}} \left[1 - e^{-r_H^{D-1}/r_\star^{D-1}}\right] = 1
\]
Under physical conditions $r_s \gg r_0$, the roots are:
\[
r_- \approx r_0 \left[1 - \mathcal{O}\left(e^{-r_0/r_s}\right)\right]\,, \quad r_+ \approx r_s \left[1 - \mathcal{O}\left(e^{-r_s^2/r_0^2}\right)\right]
\]
The solution thus interpolates between a de Sitter core near $r\approx 0$ and an asymptotically Schwarzschild geometry at large $r$.

## 3. Photon Dynamics and Black Hole Shadow

Null geodesics in the equatorial plane $(\theta = \pi/2)$ with energy $E$ and angular momentum $L$ are governed by
\[
\dot r^2 + V_{\mathrm{eff}}(r) = 0\,, \qquad V_{\mathrm{eff}}(r) = f(r)\frac{L^2}{r^2} - E^2
\]
The photon sphere is located at $r_\mathrm{ph}$ such that
\[
V_{\mathrm{eff}}(r_\mathrm{ph}) = 0\,, \quad \frac{dV_{\mathrm{eff}}}{dr}\Big|_{r_\mathrm{ph}} = 0 \quad \Longrightarrow \quad r_\mathrm{ph}\,f'(r_\mathrm{ph}) - 2f(r_\mathrm{ph}) = 0
\]
The shadow radius observed at infinity is
\[
R_s = \frac{r_\mathrm{ph}}{\sqrt{f(r_\mathrm{ph})}}
\]
Numerical analysis reveals $R_s$ decreases as $D$ increases at fixed mass and $r_0$ (the gravitational potential "dilutes" in higher $D$), but grows with both mass $M$ and Schwarzschild radius $r_s$ [2601.06711]. 

## 4. Thermodynamic Properties

### Hawking Temperature

The Hawking temperature is determined by the surface gravity at the outer horizon $r_+$:
\[
T_H = \frac{1}{4\pi} f'(r_+) = \frac{1}{4\pi} \left[ \frac{D-3}{r_+} + \frac{D-1}{r_0^2 r_s^{D-3} r_+} \exp\left(-\frac{r_+^{D-1}}{r_\star^{D-1}}\right) \right]
\]
$T_H$ is positive-definite, increases with $r_0$, $r_s$, and $D$.

### Entropy and Energy Emission

The Bekenstein–Hawking entropy is
\[
S = \frac{A}{4} = \frac{\Omega_{D-2}}{4} r_+^{D-2}
\]
The spectral energy emission rate in $D$ dimensions, via summing absorption cross-sections or greybody factors, is given by
\[
\frac{d^2 E}{d\omega\,dt} = \sum_{s,\ell} N_s\,\frac{\Gamma^s_\ell(\omega)}{e^{\omega/T_H} - (-1)^{2s}}\,\frac{\omega^{D-2}}{(2\pi)^{D-1}\,\Omega_{D-2}}
\]
or, equivalently, via the absorption cross-section
\[
\frac{d^2 E}{d\omega\,dt} = \sum_s N_s\,\frac{\sigma^s_{\mathrm{abs}}(\omega)\,\Omega_{D-2}}{(2\pi)^{D-1}}\,\frac{\omega^{D-1}}{e^{\omega/T_H} - (-1)^{2s}}
\]
Increasing $r_0$ suppresses emission; increasing $r_s$ or $D$ enhances it.

## 5. Observational Signatures and EHT Constraints

By matching the predicted shadow radius to EHT observations (specifically M87* and Sgr A*), empirical bounds on $r_0$ and $r_s$ are obtained:
- For Sgr A* (1$\sigma$): $r_0 \in [0.0062,\,0.119]$, $r_s \in [0.0048,\,0.102]$
- For M87* (1$\sigma$): $r_0 \in [0.035,\,0.084]$, $r_s \in [0.029,\,0.077]$
[2601.06711]

These intervals require $r_0$ to be much less than the horizon size for astrophysical black holes. The suppression of the shadow radius at higher $D$ implies reduced projected diameters in extra-dimensional scenarios.

## 6. Physical Interpretation and Theoretical Context

The Dymnikova construction realizes a regular black hole by exponentiating curvature corrections, effectively modeling a "core removal" via a nonsingular, de Sitter-like region with finite central energy density. The core scale $r_0$ sets a transition between quantum-modified (regular) and classical (Schwarzschild-like) geometry. In the limit $r_0 \rightarrow 0$, the standard singular solution is recovered. In contrast to four-dimensional regular black holes, which often require nonstandard matter or nonlinear electrodynamics, the higher-dimensional Dymnikova solution can be embedded within the framework of effective theories with infinite towers of higher-curvature terms [2403.04827].

Higher $D$ alters both the causal structure (horizon configuration) and observational characteristics (shadow, thermodynamic observables). The presence of two horizons (event and Cauchy) is generic for $r_s \gg r_0$, with the Cauchy horizon located near the core scale and the outer horizon close to the Schwarzschild radius.

Thermodynamically, the presence of a de Sitter-like core modifies evaporation: $T_H$ remains positive and nonzero at extremality, and the emission spectrum is regularized; this opens a window for possible quantum gravity signatures in black hole observations.

## 7. Relation to Broader Regular Black Hole Research

The Dymnikova profile sits within a broader class of regular black holes, including Hayward and Bardeen-type metrics, and can be realized as a special case of the exponential resummation of higher-curvature corrections in “quasi-topological” gravities [2403.04827, 2403.07848, 2509.25141]. These constructions generically provide: (i) complete resolution of the central singularity ($R$, $K$ finite at $r=0$), (ii) two-horizon structure, (iii) smooth matching to Schwarzschild/Tangherlini at large $r$, and (iv) distinctive gravitational wave and shadow signatures potentially detectable in current and future observations.

The main physical upshot of the Dymnikova construction is the consistent realization of singularity-free black holes that are sharply constrained by EHT data—offering both a phenomenological tool for probing Planck-scale regularization and a theoretical template for embedding non-singular objects into higher-dimensional gravitational models.

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For explicit technical details, analytic formulas for horizons, shadows, thermodynamics, and empirical constraint intervals, see [2601.06711]. The integration and comparative context with other higher-dimensional regular black hole models are presented in [2403.04827, 2403.07848, 2509.25141].

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