Dymnikova Regular Black Hole

• Dymnikova regular black holes are static, spherically symmetric solutions that replace the central singularity with a smooth de Sitter-like core and exhibit an inner and outer horizon.
• The model’s dual-horizon structure and modified thermodynamic properties, including a positive Hawking temperature, offer insights into emission spectra and quantum gravity effects.
• Integrating higher-dimensional analyses, the construction provides empirical constraints from EHT observations and connects broader regular black hole research through its non-singular behavior.

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 (Errehymy et al., 10 Jan 2026).

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 $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$0-sphere.

The exponential profile for $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$1 ensures the metric is regular at $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$2, with a finite curvature invariant.

2. Horizon Structure

Dymnikova regular black holes typically exhibit two horizons, determined by real positive roots $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$3 of $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$4: $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$5 Under physical conditions $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$6, the roots are: $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$7 The solution thus interpolates between a de Sitter core near $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$8 and an asymptotically Schwarzschild geometry at large $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega_{D-2}^2$$9.

3. Photon Dynamics and Black Hole Shadow

Null geodesics in the equatorial plane $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$0 with energy $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$1 and angular momentum $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$2 are governed by

$$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$3

The photon sphere is located at $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$4 such that

$$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$5

The shadow radius observed at infinity is

$$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$6

Numerical analysis reveals $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$7 decreases as $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$8 increases at fixed mass and $$f(r) = 1 - \frac{r_s^{D-3}}{r^{D-3}} \left[1 - \exp\left(-\frac{r^{D-1}}{r_\star^{D-1}}\right)\right]$$9 (the gravitational potential "dilutes" in higher $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$0), but grows with both mass $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$1 and Schwarzschild radius $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$2 (Errehymy et al., 10 Jan 2026).

4. Thermodynamic Properties

Hawking Temperature

The Hawking temperature is determined by the surface gravity at the outer horizon $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$3: $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$4 $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$5 is positive-definite, increases with $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$6, $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$7, and $$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$8.

Entropy and Energy Emission

The Bekenstein–Hawking entropy is

$$r_\star^{D-1} = r_0^2\,r_s^{D-3}\,, \qquad r_0^2 = \frac{(D-1)(D-2)}{16\pi\rho_0}\,, \qquad r_s^{D-3} = \frac{16\pi M}{(D-2)\,\Omega_{D-2}}$$9

The spectral energy emission rate in $r_0$0 dimensions, via summing absorption cross-sections or greybody factors, is given by

$r_0$1

or, equivalently, via the absorption cross-section

$r_0$2

Increasing $r_0$3 suppresses emission; increasing $r_0$4 or $r_0$5 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$6 and $r_0$7 are obtained:

• For Sgr A* (1$r_0$8): $r_0$9, $\rho_0$0
• For M87* (1$\rho_0$1): $\rho_0$2, $\rho_0$3 (Errehymy et al., 10 Jan 2026)

These intervals require $\rho_0$4 to be much less than the horizon size for astrophysical black holes. The suppression of the shadow radius at higher $\rho_0$5 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 $\rho_0$6 sets a transition between quantum-modified (regular) and classical (Schwarzschild-like) geometry. In the limit $\rho_0$7, 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 (Bueno et al., 2024).

Higher $\rho_0$8 alters both the causal structure (horizon configuration) and observational characteristics (shadow, thermodynamic observables). The presence of two horizons (event and Cauchy) is generic for $\rho_0$9, 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: $r_s$0 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 (Bueno et al., 2024, Konoplya et al., 2024, Arbelaez, 29 Sep 2025). These constructions generically provide: (i) complete resolution of the central singularity ($r_s$1, $r_s$2 finite at $r_s$3), (ii) two-horizon structure, (iii) smooth matching to Schwarzschild/Tangherlini at large $r_s$4, 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 (Errehymy et al., 10 Jan 2026). The integration and comparative context with other higher-dimensional regular black hole models are presented in (Bueno et al., 2024, Konoplya et al., 2024, Arbelaez, 29 Sep 2025).