Spherically Symmetric Regular Black Holes

• The study demonstrates that spherically symmetric regular black holes replace classical singularities with a de Sitter-like core, ensuring finite curvature invariants.
• It employs matching conditions, such as the Israel junction conditions, to seamlessly connect the regular interior with external solutions like Schwarzschild or Reissner–Nordström.
• The work classifies solutions based on parameters, revealing nonextremal, extremal, and overcharged configurations, which offer practical insights into stability and observational signatures.

Spherically symmetric regular black holes are solutions to the Einstein equations (possibly coupled to additional fields or modified by higher-curvature corrections) that evade the formation of a central curvature singularity. They achieve this by replacing the standard singular core with a regular interior—often of de Sitter–like or similar character—while preserving essential black hole features such as horizons. The construction, classification, dynamics, and physical realization of these spacetimes have developed into a rich field at the intersection of classical general relativity, modified gravity, quantum gravity, and mathematical relativity, as evidenced by recent research.

1. Interior Structure and Core Regularization

The essential principle of these spacetimes is the removal of the central singularity through regularization. In many canonical models, the interior (for $r \leq r_0$) is modeled as a de Sitter-like core, characterized by an energy–momentum tensor with equation of state $p(r) = -\rho_m(r)$, mimicking a false vacuum configuration. The metric in this region typically takes the form: $B(r) = A^{-1}(r) = 1 - \frac{r^2}{R^2}$ where $R$ is the de Sitter radius. The corresponding energy–momentum content acts as an effective cosmological constant, ensuring that curvature invariants such as the Ricci scalar and the Kretschmann scalar remain finite throughout the core, thus enforcing regularity at $r=0$ and avoiding the classical black hole singularity (Lemos et al., 2011).

Alternative interior regularizations arise in treatments with anisotropic fluids whose energy density and pressures are related via $\rho + p_r = 0$ (Kyriakopoulos, 2016), or in the context of "limiting curvature" gravity theories where dynamical constraints cap curvature invariants at universal, theory-specified maxima (Frolov et al., 2021). The general behavior is that near the origin, the metric function expands as $f(r)\approx 1 - y r^2$, with $y>0$, representing a (anti-)de Sitter or Nariai/Bertotti–Robinson-type model geometry (Bargueño, 2020).

2. Junction to External Geometry and Boundary Conditions

The regular core is matched to an external solution, such as the vacuum Reissner–Nordström spacetime or the Schwarzschild solution, through a boundary at $r = r_0$. The transition is mediated by a hypersurface (the boundary), across which the Israel junction conditions are imposed to ensure the continuity of the first and second fundamental forms (metric and extrinsic curvature) (Lemos et al., 2011).

Crucially, in models involving electric (or magnetic) charge, the charge distribution is often entirely localized on the shell at $r_0$, forming an "electrically charged coat" described by a delta-function in the charge density: $$Q(r) = \begin{cases} 0 & r < r_0 \ q & r \geq r_0 \end{cases}$$ and $8\pi\rho_m(r) = 3/R^2 - Q^2(r)/r^4$, $8\pi p(r) = -3/R^2 + Q^2(r)/r^4$. The pressure-free condition at the boundary ($p(r_0)=0$) yields explicit relations among the parameters, e.g., $R\cdot q = \sqrt{3}\, r_0^2$ (Lemos et al., 2011).

The general approach ensures that:

• The de Sitter core regularizes the geometry at small $r$.
• The exterior (for $r \geq r_0$) can be asymptotically flat (Schwarzschild or Reissner–Nordström).
• The matching preserves the regularity of both the metric and curvature invariants across the boundary.

3. Classification and Types of Regular Black Hole Solutions

By tuning the parameters (mass $m$, charge $q$, de Sitter radius $R$, and boundary radius $r_0$), one constructs a variety of regular black hole objects:

• Regular nonextremal black holes: For $r_0 < r_-$ (the inner Reissner–Nordström/Cauchy horizon), the matter region is bounded by a timelike shell—fully regular—with the boundary always inside the inner horizon. In the limit $r_0 \to r_-$, the boundary is null.
• Regular extremal black holes: The loci $r_0/q = 2/3$ correspond to the merging of inner and outer horizons ($r_+=r_-=m=q$), with the boundary at finite $r_0$ but infinite proper distance to the horizon.
• Regular overcharged stars: For $q > m$, no horizons are present; the configuration describes a regular, horizonless object—an overcharged star—with exterior overcharged Reissner–Nordström geometry.
• Multi-horizon/multi-shell solutions: Employing non-monotonic or composite mass distributions with several inflection points, the cumulative mass profiles support solutions with multiple inner and outer horizons (Azreg-Aïnou, 2017).

Key matching relations include: $$1 - \frac{r_0^2}{R^2} = 1 - \frac{2m}{r_0} + \frac{q^2}{r_0^2}$$

$$R\cdot q = \sqrt{3} r_0^2, \quad m r_0 = \frac{2}{3} q^2$$

4. Source Matter Models and Theoretical Frameworks

The stress–energy tensors sustaining regular black holes are typically "exotic", violating some standard energy conditions. The main realizations include:

• (Nonlinear) Electrodynamics: Einstein gravity coupled to a nonlinear electromagnetic field, with Lagrangians constructed to ensure finite field strengths and energy densities at the center, e.g., the Bardeen metric as a magnetic monopole in NED (Colléaux et al., 2017, Kar et al., 2023, Li et al., 2013).
• Anisotropic fluids: The matter content is modeled as fluids with $\rho + p_r = 0$ ("false vacuum"), providing the required negative pressure near the core to generate repulsive gravity and regularize the center (Kyriakopoulos, 2016).
• Limiting curvature models: Theories with Lagrange multiplier–enforced upper bounds on curvature invariants, where the usual divergence in invariants (e.g., $R_{\mu\nu\lambda\sigma} R^{\mu\nu\lambda\sigma}$) is avoided by dynamical saturation (Frolov et al., 2021).
• Multi-shell/core–shell distributions: Systems constructed with smooth or piecewise continuous mass density profiles, generalized to allow multiple shells and cores, each supporting a horizon or transition layer (Azreg-Aïnou, 2017).
• Scalar–tensor reconstructions: Regular metrics are obtained as exact solutions within scalar–tensor theories, with the scalar potential and coupling function engineered to reproduce the desired spacetime (Calzá et al., 7 May 2025).

The energy conditions are satisfied only in some regions of the spacetime; violations are necessary to evade the singularity theorems. The total charge or mass is often "pushed" toward the boundary layer rather than being distributed through the core.

5. Causal Structure, Horizons, and Instabilities

Regular black hole spacetimes generally feature rich horizon structures:

• Outer event horizon: Analogous to the Schwarzschild or Reissner–Nordström event horizon.
• Inner Cauchy horizon: Emerges from the transition to the regular core; susceptible in standard (spherical) models to the "mass inflation" instability, wherein perturbative influx of energy leads to the exponential growth of the local mass and curvature at the inner horizon (Bonanno et al., 2020, Bonanno et al., 2022).

Phenomenologies to avoid the Cauchy horizon/mass inflation instability include:

• Topology-changing cores: Constructing regular black holes with hyperbolic ($k=-1$) or toroidal ($k=0$) horizon topologies, or by deforming the areal radius such that the inner horizon is replaced by a limiting surface (Calzá et al., 7 May 2025).
• Three-dimensional topological models: Employing Seifert fiber bundle structures, the spatial geometry near the core can take forms such as $S^3$ (de Sitter), $H^3$ (anti–de Sitter), $S^1\times S^2$ (Nariai), or $\mathbb{R}\times S^2$ (Bertotti–Robinson), each correlated with different causal and topological transitions (Bargueño, 2020).
• Birkhoff-type uniqueness: In higher-dimensional gravity with infinite towers of higher-curvature corrections (quasi-topological gravity), regular black holes are the unique spherically symmetric solutions, and the endpoint of collapse features bounces and white hole ejections, avoiding both central singularities and Cauchy horizon pathologies (Bueno et al., 3 Dec 2024, Bueno et al., 14 May 2025).

In all cases, the physical nature of the horizon (timelike/null), geodesic completeness, and the presence or absence of mass inflation are determined by the functional form of the mass function and core regularization.

6. Dynamical Formation, Collapse, and End States

A key advance beyond earlier static models is the dynamical formation of regular black holes from gravitational collapse:

• Collapsing shells or dust stars governed by regular initial data (satisfying suitable mass profile choices such as $F(z)\sim z^3$ near the center) naturally evolve toward equilibrium spacetimes with nonsingular cores and marginally trapped surfaces (MTS) that causally disconnect the inside and outside, forming a black hole without a central singularity (Mosani et al., 2023).
• In infinite higher-curvature gravity, the collapse of a dust star proceeds as in Oppenheimer–Snyder, but instead of a singularity, the star contracts to a minimum radius (inside the inner horizon), bounces, and emerges into another asymptotically flat region through a white hole, leading to cyclic evolution (Bueno et al., 14 May 2025, Bueno et al., 3 Dec 2024).
• The Israel (or generalized) junction conditions in these models ensure the proper gluing of interior (FLRW or de Sitter–like) and exterior geometries, and encode the dynamical equations for the moving shell or surface (Bueno et al., 3 Dec 2024, Bueno et al., 14 May 2025).

7. Physical Implications, Observables, and Theoretical Consequences

Regular black holes provide explicit counterexamples to the classical singularity theorems; their construction demonstrates that classical general relativity or suitable modifications can, under certain matter or action choices, admit singularity-free black holes. Implications include:

• Quantum gravity models: Planck-scale charged regular black holes can serve as possible models for elementary particles or quantum black holes (Lemos et al., 2011).
• Thermodynamics and evaporation: The presence of an inner horizon or remnant structure modifies black hole thermodynamics; for regular black holes with a de Sitter core, the Hawking temperature vanishes as the mass approaches a critical value, leading to a cold remnant (Bonanno et al., 2022).
• Quasinormal modes and observational signatures: The presence of regular cores and modified potentials affect the spectrum of quasinormal modes, potentially leaving traceable imprints in the ringdown phase of gravitational wave signals (Li et al., 2013).
• Black hole–to–particle/soliton transitions and dark matter: Certain parameter regimes give rise to gravitational soliton solutions (regular, horizonless configurations) which may contribute to dark matter (Kyriakopoulos, 2016).
• Information paradox and entanglement entropy: The island prescription for entanglement entropy yields Page curves for regular black holes, and indicates only modest deviations from Schwarzschild behavior unless the regular core structure is pronounced (Luongo et al., 2023).
• Astrophysical modeling: The external fields of regular black holes can match observed properties (e.g., QPOs in accreting sources) and accommodate neutron stars within their parameter space (Boshkayev et al., 2023).
• Limiting curvature and universality: In some quasi-topological higher-curvature models, curvature invariants are universally bounded, providing a realization of the Markov limiting curvature hypothesis (Bueno et al., 3 Dec 2024).

8. Summary Table: Key Regular Black Hole Models

Core Type	Exterior Solution	Key Regularization Mechanism	Causal Structure
de Sitter core	Reissner–Nordström	False vacuum, charged shell at boundary	Event + Cauchy horizons
de Sitter core	Schwarzschild	p + p_r = 0 fluid, matched at r=k	Event horizon or none (soliton)
Smeared core	Schwarzschild	Smooth mass density $\mathcal{D}(r,\theta)$	Multi-horizon/multi-shell
Nonlinear NED	Asymptotically flat	Magnetic or electric monopole, NED Lagrangian	Charged event horizon
Limiting curvature	Schwarzschild-like	Dynamical capping of invariants	Bouncing anisotropic interior
Quasi-topological	Higher-D black hole	Infinite curvature invariants, algebraic reduction	Bounce and cyclic collapse

These various constructions illustrate the breadth of methodologies for generating spherically symmetric regular black holes and highlight the interplay of matter content, modified gravity, topology, and dynamics in evading singularities while preserving essential black hole phenomenology. The ongoing investigation of these solutions provides crucial insights for both theoretical physics and astrophysical applications.