Regular Black Hole Models

• Regular black hole models are nonsingular spacetime solutions featuring a de Sitter or Minkowski core with similar external horizons to Schwarzschild/Kerr metrics.
• They utilize modified mass functions, nonlinear electrodynamics, and higher-curvature or nonlocal gravity to ensure finite curvature invariants across the manifold.
• These models predict observable differences in shadow size, accretion disk dynamics, and horizon structure that can be tested with current EHT data and future astrophysical observations.

Regular black hole models describe spacetimes which, unlike the traditional Schwarzschild or Kerr solutions of general relativity, possess no curvature singularities anywhere in their manifold. The central motivation is the resolution of the infinite curvature and breakdown of classical predictability at $r=0$ inherent in standard black holes, while retaining the key external features (horizons, asymptotic flatness) that correspond to observable astrophysical black holes. The construction of such models employs modifications of the matter sector, nonlinear electrodynamics couplings, higher-curvature or nonlocal gravity, or nontrivial dynamics during gravitational collapse. Regular black holes are characterized by having at least one event horizon, a regular (often de Sitter or locally Minkowski) core, and parametric continuity with Schwarzschild/Kerr in the appropriate limits.

1. Fundamental Geometric Structure and Regularity Criteria

Generic regular black hole (RBH) metrics adopt a static, spherically symmetric line element,

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

with $f(r)$ modified from the Schwarzschild lapse to regularize the core. The archetype is $f(r) = 1 - 2m(r)/r$, where the mass function $m(r)$ is constructed such that (i) $m(r)\sim r^3$ as $r \to 0$ (yielding $f(r)\sim 1 - \Lambda_\text{eff} r^2/3$ and a de Sitter-like regular center), and (ii) $m(r)\to M$ as $r\to\infty$ guaranteeing asymptotic flatness. All scalar curvature invariants, such as the Ricci scalar $R$, Kretschmann scalar $K = R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$, and the Ricci tensor square are constructed to be manifestly finite throughout the manifold—provided the leading core behavior of $m(r)\sim r^3$ is respected (Bambi, 2023, Casadio et al., 19 Feb 2025).

Table 1 lists common static RBH models:

Model	$m(r)$ profile	Regularity mechanism
Bardeen	$M\, r^3/(r^2+g^2)^{3/2}$	NED magnetic monopole
Hayward	$M\, r^3/(r^3+2M\ell^2)$	Phenomenological (vacuum pol.)
Dymnikova	$M[1 - e^{-r^3/r_0^3}]$	Exponential NED core
Exponential	$M\,e^{-a/r}$	Asymptotic Minkowski core

All these models produce an effective cosmological constant $\Lambda_\text{eff}$ in the core and eliminate the $r=0$ singularity (Bambi, 2023, Simpson et al., 2019).

2. Matter Sources, Effective Energy Conditions, and Field-Theoretic Realizations

RBHs are not vacuum Einstein solutions in the interior; their stress-energy tensors $T_{\mu\nu}$ violate the strong energy condition (SEC) near the core, enabling singularity avoidance per the Hawking-Penrose theorems (Bambi, 2023, Casadio et al., 19 Feb 2025). The supporting matter is typically either:

• Anisotropic fluids with $p_r = -\epsilon$ and $p_\perp \neq p_r$ yielding the required $m(r)\sim r^3$ behavior.
• Nonlinear electrodynamics (NED)—notably, Bardeen and Dymnikova models provide explicit NED Lagrangians, e.g., $\mathcal{L}(F) = (3/sb^2)\exp[-(2/(b^2 F))^{3/4}]$ for Dymnikova's case (Ghosh et al., 2020).
• Effective equations of state such as $P(r) = k(r)\epsilon(r)$ with $k(r)\to -1$ near the center (mimicking vacuum/dark energy) (Vertogradov et al., 4 Aug 2024).
• Higher-derivative, nonlocal, or quasitopological gravity actions; the regular core is realized as a direct result of infinite-order curvature corrections, requiring no exotic matter (Soto, 17 Nov 2025, Bueno et al., 14 May 2025).
• Interpenetrating fluids (e.g., baryonic and quark-matter with inhomogeneous conversion rates) generating finite-density de Sitter-like cores (Vertogradov et al., 10 Apr 2025).

The weak energy condition (WEC) is often satisfied globally except for the SEC-violating core. Dominant energy condition (DEC) violations may appear only within a finite inner region or be avoided by suitable matter-phase transitions (Vertogradov et al., 1 Feb 2025).

3. Horizon Structure, Extremality, and Causal Features

RBH spacetimes typically exhibit two horizons—the standard event (outer) horizon $r_+$ and a Cauchy (inner) horizon $r_-$—analogous to Reissner–Nordström, but the central singularity is replaced by a regular de Sitter or Minkowski-like region. The horizon structure is governed by the roots of $f(r)=0$; detailed analysis in models such as

$$f(r) = 1 - 2M\frac{r^3}{(r^2+g^2)^{3/2}}/r \quad \text{(Bardeen)},\qquad f(r) = 1 - 2M / r \cdot e^{-a/r} \quad \text{(Exponential)}$$

shows two real, positive horizons for regulator parameter below a critical value (e.g., $g<g_*$, $a<2M/e$), merging into an extremal configuration at the threshold (Simpson et al., 2019, Bambi, 2023). Beyond this, the spacetime is regular and horizonless, describing an ultracompact object.

In dynamical collapse models, the horizon formation can be postponed by fluid phase transitions or energy exchange rates, allowing for compression to high densities before an apparent horizon forms—regularizing the late-stage gravitational collapse (Vertogradov, 23 Jan 2025, Vertogradov et al., 1 Feb 2025, Vertogradov et al., 10 Apr 2025).

The Penrose diagram of a regular black hole replaces the singularity with a regular core, and the continuation through the Cauchy horizon can lead to cyclic or bouncing universes in some extensions (Casadio et al., 19 Feb 2025, Bueno et al., 14 May 2025).

4. Rotating Regular Black Holes and Generalizations

The extension to rotating regular black holes utilizes the Newman–Janis algorithm, applied either directly to a regular static “seed” or to explicit NED-sourced RBH solutions (Ghosh et al., 2020, Kamenshchik et al., 2023, Johannsen, 2015). The resulting metrics are deformed Kerr-like spacetimes with three constants of motion if the construction preserves separability. Prototype forms include

$$ds^2 = -\frac{\Delta-a^2\sin^2\theta}{\Sigma}dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}dt\,d\varphi + \cdots,$$

with deformation functions in $\Delta(r)$ and the mass profile $M(r)$ encoding regularity (e.g., $M(r)=M[1-\exp(-r^3/b^3)]$ for Dymnikova, or explicit regularizing parameters in the scalar sector). The presence of the regularizing scale modifies the location of both event and Cauchy horizons, the ergoregion, and the photon sphere.

Curvature invariants are again finite everywhere for nonzero regulator, even in the rotating configurations (Ghosh et al., 2020, Kamenshchik et al., 2023).

5. Observational Phenomenology: Shadows, Accretion, and Astrophysical Constraints

RBH models preserve Schwarzschild or Kerr geometry at large $r$, and observational differences partly manifest in the photon sphere, black hole shadow, and inner accretion flow. Typical deviations are at percent or sub-percent levels for $g/M, \ell/M, a/M$ parameters lying within the current observational bounds of the Event Horizon Telescope (EHT) (Kumar et al., 2020, Jafarzade et al., 2021, Fauzi et al., 25 Nov 2024).

Main effects:

• The shadow radius may be modestly reduced (Hayward/Bardeen) or increased (NED-charged rotating RBH), with the shape becoming more circular as the regulator parameter increases (Ghosh et al., 2020, Jafarzade et al., 2021). Acceptable parameter values correspond to less than 10% deviation from the Kerr silhouette.
• The innermost stable circular orbit (ISCO) and efficiency of accretion disks shift inwards, enabling regular RBHs to mimic slowly rotating Kerr holes in ISCO-based luminosity, temperature, and conversion efficiency signatures (Akbarieh et al., 2023).
• Dynamical collapse RBHs also predict characteristic delays or signatures in horizon formation and a dependence of photon sphere/shadow radius on local equation of state, potentially testable with future EHT precision (Vertogradov et al., 1 Feb 2025, Vertogradov et al., 10 Apr 2025).

Current EHT data restrict $g/M\lesssim 0.25-0.65$ depending on the model (Kumar et al., 2020), and similar upper bounds exist for the regularizing parameters in Minkowski-core and bounce models (Jafarzade et al., 2021).

6. Physical Formation Scenarios and Theoretical Implications

RBHs can arise through several physical mechanisms:

• NED-sourced field configurations with suitable magnetic monopole or nonlinear backgrounds (Bambi, 2023, Ghosh et al., 2020).
• Regularizing phase transitions between baryonic and quark matter, leading naturally to de Sitter-like cores via inhomogeneous conversion rates (Vertogradov et al., 10 Apr 2025).
• Gravitational collapse of ordinary matter with energy conversion (e.g., dust+radiation, baryon-to-quark transitions, or polytropic anisotropic fluids), producing regular cores dynamically (Vertogradov, 23 Jan 2025, Sajadi et al., 2023).
• Infinite-derivative, nonlocal, or quasitopological higher-curvature gravitational corrections regularizing the metric at short scales and ensuring a finite density and absence of singularity, even for realistic dynamical collapse (Bueno et al., 14 May 2025, Soto, 17 Nov 2025).

In all cases, removal of singularities requires violation of the strong energy condition and often a physical process (phase transition, energy exchange) concentrated at high density/central core.

7. Extensions: Multi-shell and Finite-Boundary Models

The generalization to smoothed mass distributions or core–multi-shell geometries provides models with multiple horizons and a richer phenomenology, realized by replacing the central Dirac delta profile by some smooth density $\mathcal{D}(r,\theta)$ (Azreg-Aïnou, 2017). These allow explicit construction of noncommutative-inspired and step-function core regular black holes. Moreover, introducing a finite boundary to the mass profile (as in horizonless ultracompact models) further modifies photon sphere structure and may provide discriminants for distinguishing classical black holes from horizonless compact stars (Fauzi et al., 25 Nov 2024).

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References:

• (Bambi, 2023) Regular Black Holes: Towards a New Paradigm of Gravitational Collapse
• (Casadio et al., 19 Feb 2025) Regular Schwarzschild black holes and cosmological models
• (Ghosh et al., 2020) Ergosphere and shadow of a rotating regular black hole
• (Kumar et al., 2020) Testing Rotating Regular Metrics as Candidates for Astrophysical Black Holes
• (Simpson et al., 2019) Regular black holes with asymptotically Minkowski cores
• (Soto, 17 Nov 2025) Modified Gravity and Regular Black Hole Models
• (Mosani et al., 2023) Regular black hole from regular initial data
• (Vertogradov, 23 Jan 2025) Regular Black Hole from gravitational collapse of dust and radiation
• (Vertogradov et al., 1 Feb 2025) Formation of regular black hole from baryonic matter
• (Vertogradov et al., 10 Apr 2025) Regular Black Hole Models in the Transition from Baryonic Matter to Quark Matter
• (Fauzi et al., 25 Nov 2024) Shadow images of regular black hole with finite boundary
• (Sajadi et al., 2023) Anisotropic Generalized Polytropic Spheres: Regular 3D Black Holes
• (Azreg-Aïnou, 2017) Smoothed one-core and core--multi-shell regular black holes
• (Akbarieh et al., 2023) Accretion disk around regular black holes
• (Hu et al., 2023) A regular black hole as the final state of evolution of a singular black hole
• (Vertogradov et al., 4 Aug 2024) Exact Regular Black Hole Solutions with de Sitter Cores and Hagedorn Fluid
• (Kamenshchik et al., 2023) Newman-Janis algorithm's application to regular black hole models
• (Johannsen, 2015) Regular Black Hole Metric with Three Constants of Motion
• (Jafarzade et al., 2021) Observational optical constraints of regular black holes
• (Bueno et al., 14 May 2025) Regular black holes from Oppenheimer-Snyder collapse