Regular Black Holes: Nonsingular Spacetimes

• Regular black holes are spacetime geometries with an event horizon and a nonsingular core where all curvature invariants remain finite.
• They are constructed by modifying gravitational dynamics or matter couplings, leading to de Sitter, Minkowski, or other smooth interior solutions.
• Their models offer testable insights into gravitational collapse, thermodynamics, and observational signatures like tidal Love numbers and altered black hole shadows.

A regular black hole is a spacetime characterized by the presence of an event horizon and a globally regular (nonsingular) geometry, such that all curvature invariants remain finite and geodesics are complete everywhere, including at the core. This class of solutions—unlike the classical Kerr or Schwarzschild black holes of general relativity—lack a central singularity, instead realizing a de Sitter, Minkowski, or otherwise smooth interior through physically motivated modifications to gravitational dynamics or matter couplings. Regular black holes have become essential phenomenological laboratories for exploring singularity resolution, quantum gravity effects, and their potential observational imprints in strong-field astrophysics.

1. Fundamental Metric Structure and Classification

The canonical static, spherically symmetric regular black hole metric adopts the curvature-coordinate form

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

with the function $f(r)$ chosen such that $f(r) = 1 - 2m(r)/r$, where $m(r)$ is the cumulative mass profile. The requirement of regularity, i.e., boundedness of curvature invariants (e.g., Ricci scalar $R$, Kretschmann scalar $K$), and extension of all causal geodesics to arbitrary affine parameter, imposes that near the center, $m(r)\sim r^3$, corresponding to a core geometry that is either de Sitter (if $f(r)\sim1- r^2/\ell^2$), Minkowski ($f(r)\to 1$), or more exotic, but always smooth (Bambi, 2023, Neves, 2017, Lan et al., 2023).

Classical exemplars include:

• Bardeen: $m(r) = M r^3/(r^2+g^2)^{3/2}$, a realization via nonlinear magnetic monopole electrodynamics.
• Hayward: $m(r) = M r^3/(r^3 + 2M \ell^2)$, a generic de Sitter–core extension consistent with semiquantum gravity modifications.
• Dymnikova: $m(r) = M [1-\exp(-r^3/r_0^3)]$, exponentially integrable core structure.

Extensions encompass "hollow" models with asymptotically Minkowski cores (e.g., $m(r)=M e^{-a/r}$), sub-Planckian curvature models, and phenomenological families constructed using varying anisotropic fluid equations of state (Simpson et al., 2019, Ling et al., 2021, Santos, 27 Nov 2024).

2. Curvature Regularity, Energy Conditions, and Matter Models

To ensure absence of singularities, leading curvature invariants must remain finite for all $r$, requiring $f(r)$ admit a series expansion about $r=0$ without terms diverging as negative powers of $r$. For the Bardeen and Hayward metrics, the Kretschmann scalar $K\sim \text{const}$ as $r\to0$; for exponential Minkowski-core models, $K\to 0$ (Lan et al., 2023, Santos, 27 Nov 2024, Simpson et al., 2019).

Typically, such regularity is supported by nontrivial energy–momentum sources:

• Anisotropic effective fluids with $\rho = -p_r$, generically violating the strong energy condition (SEC) near the core ($\rho+3p<0$), essential to evade the consequences of Penrose’s theorem.
• Nonlinear electrodynamics (NLED): Realizations of the Bardeen and Hayward black holes as solutions to Einstein–NLED, where matter Lagrangians are engineered to reproduce desired mass profiles and cure central singularities. Regularity constrains the parameters of the theory and, crucially, fixes integration constants to fine-tuned values, so that regular black holes are measure-zero members within a more generic singular family (Huang et al., 17 Mar 2025).
• Kiselev fluids with radially varying equation of state: New regular black holes arise by tuning the equation of state parameter $w(r)$, interpolating between known regular metrics and generating a large class of solutions with controlled energy condition violations (Santos, 27 Nov 2024).

3. Formation and Collapse Dynamics

Regular black holes can arise as the endpoint of several collapse scenarios, contrasting sharply with the singular endpoints in standard Oppenheimer-Snyder or generic gravitational collapse:

• Oppenheimer-Snyder collapse in higher-curvature gravity: In actions with an infinite tower of quasi-topological corrections (in $D\ge5$), the result of gravitational dust collapse is generically a regular black hole geometry where the collapsing shell bounces at finite radius and reemerges through a white hole, avoiding all singularities (Bueno et al., 14 May 2025, Bueno et al., 3 Dec 2024).
• Perfect-fluid collapse with smooth initial data: There exist classes of initial conditions yielding a geodesically complete, regular equilibrium final state with an event horizon, no central singularity, and a trapped region disconnected from the external universe (Mosani et al., 2023).
• Loop quantum gravity (LQG): Quantization of the Schwarzschild interior replaces the singular core by a quantum transition surface; classical singularity is replaced by a region of nonzero minimal area, resulting in a deterministic bounce (Ashtekar et al., 2023).

The general mechanism in all these constructions is the emergence of a limiting curvature scale, whether set dynamically by effective quantum corrections or built-in via a physically motivated matter sector.

4. Horizon Structure, Penrose Diagrams, and Stability

Regular black holes generically exhibit two or more horizons: an outer event horizon and a Cauchy (inner) horizon. For certain parameter regimes (related to mass, charge, or core structure), these may coalesce into a degenerate (extremal) horizon. The global causal structures closely resemble that of Reissner-Nordström, but with a regular core instead of a singularity, and often an infinite tower of asymptotically flat regions in the maximal extension (Bambi, 2023, Bonanno et al., 2020, Bonanno et al., 2022).

A persistent challenge is the stability of the inner horizon. In classical spacetimes, mass inflation, driven by exponentially amplified blue-shifting of perturbations at the Cauchy horizon (with growth rate set by its surface gravity $\kappa_-$), can render the interior physically unacceptable. Several regular black hole models, notably inner-extremal constructions with $\kappa_-=0$, or those with de Sitter-like attractors at late times, evade this instability, leading to bounded curvature growth or only polynomially mild divergences pushed off to infinite advanced time (Carballo-Rubio et al., 2022, Bonanno et al., 2020, Bonanno et al., 2022).

5. Thermodynamics and Phase-Space Constraints

Thermodynamic properties of regular black holes retain the essential features of classical black holes but display unique quantum-corrected phenomenology:

• Surface gravity and Hawking temperature: $T_H = \frac{1}{4\pi}f'(r_+)$ vanishes as extremal configurations are approached (remnant phase), preventing runaway evaporation (Bambi, 2023, Bonanno et al., 2022).
• Bekenstein-Hawking entropy: The area law, $S = \pi r_+^2$, continues to hold in leading semiclassical treatments. Quantum gravity completions and higher-curvature corrections induce subleading logarithmic or inverse-area corrections.
• First law and phase-space reduction: Due to regularity constraints among parameters (such as a fine-tuned relation between mass and charge in NLED models), the naive first law is not directly applicable. Consistent thermodynamics requires formulating the first law in the full unconstrained parameter space, then pulling back to the regular solution locus. Only by this procedure does the geometric (surface gravity) and thermodynamic temperatures coincide, resolving apparent inconsistencies found in regular black hole literature (Ma et al., 13 Jul 2025).

6. Observational Signatures and Theoretical Implications

Regular black holes present a testable interface between semiclassical and quantum gravity and are subject to increasing observational constraint:

• Tidal Love numbers (TLNs): Unlike Schwarzschild or Kerr, which have identically vanishing TLNs, regular black holes generically possess nonzero, model-dependent TLNs. These are sensitive to the core geometry (de Sitter vs. Minkowski), exhibit logarithmic scale dependence ("running" analogous to quantum field theory RG flow), and could be constrained by gravitational-wave inspiral phasings in next-generation detectors (Wang et al., 5 Dec 2025).
• Particle acceleration and geodesic structure: Near-horizon dynamics, ISCO/CPO/MBCO properties, and the possibility of Planck-scale particle acceleration in extremal limits are encoded in the modified potential and specifically depend on regularization parameters (Pradhan, 2014).
• Shadows, lensing, and quasi-normal modes: While the exterior spacetime often mimics Schwarzschild up to small corrections near the horizon, small deviations in photon sphere location and QNM spectra can serve as probes for departures from singular general relativity (Bambi, 2023, Wang et al., 5 Dec 2025).
• Evaporation and remnants: Regular black holes with extremal configurations naturally halt evaporation at finite mass and radius, suggesting a solution to the information paradox without remnants of macroscopic entropy or mass, as expected in LQG and other quantum-corrected scenarios (Ashtekar et al., 2023).

7. Generalizations, Outstanding Issues, and Research Directions

A proliferation of regular black hole classes has emerged:

• Generic matter models (Kiselev fluid, non-minimal couplings) and external frameworks (Verlinde's emergent gravity), allowing construction of both Minkowski- and de Sitter-core solutions, with—or without—apparent dark matter contributions and solid-angle deficits (Jusufi, 2022, Santos, 27 Nov 2024).
• Higher-dimensional regular solutions from quasi-topological gravity, constructed via reduction to effective Horndeski-type 2D actions, and shown to be the unique static, spherically symmetric endpoints of collapse in these models (Bueno et al., 3 Dec 2024, Bueno et al., 14 May 2025).
• Rotating and charged extensions: Partial results (e.g., via the Newman–Janis–Azreg–Aïnou algorithm) indicate that regular black hole ideas can be carried over to rotating metrics, but with careful attention to regularity conditions and potential pathologies (Jusufi, 2022).

Open issues include:

• The dynamical formation of such geometries (nonlinear collapse, stability under generic perturbations, extension to generic matter).
• The microstate origin and statistical mechanics of regular black hole entropy.
• The precise mapping to candidate quantum gravity completions (LQG, asymptotic safety, stringy corrections).
• Constraining regularization parameters via observations of black hole shadows, gravitational waves, and potential Planck-scale departures in extreme astrophysical environments.

In summary, regular black holes establish a robust, technically consistent framework for exploring singularity resolution, high-curvature gravity, and the interplay between quantum and classical spacetime physics, while yielding testable predictions across black hole thermodynamics, gravitational-wave signatures, and strong-field gravity phenomenology (Bambi, 2023, Ling et al., 2021, Simpson et al., 2019, Bonanno et al., 2020, Bueno et al., 14 May 2025, Wang et al., 5 Dec 2025, Ma et al., 13 Jul 2025, Ashtekar et al., 2023, Bueno et al., 3 Dec 2024, Santos, 27 Nov 2024).