Regular de Sitter Core Black Holes

• Regular de Sitter core black holes are spacetimes where the central singularity is replaced by a de Sitter region with a vacuum equation p = -ρ, maintaining finite curvature invariants.
• They employ a core-envelope structure with an interpolated metric that matches a de Sitter inner core to an asymptotic black hole exterior via smooth junction conditions.
• These models extend across various gravity theories and present modified horizon structures, thermodynamic properties, and observational signatures such as altered photon spheres.

A regular de Sitter core black hole is a spacetime solution in general relativity (or extended frameworks) in which the singularity at the classical black hole center is dynamically replaced by a de Sitter (dS) region—characterized by a vacuum equation of state $p = -\rho$—ensuring that all curvature invariants remain finite throughout the manifold. These models encompass a wide class of solutions with diverse matter content, horizon structures, matching conditions, and physical interpretations, and they have been constructed in Einstein gravity, Einstein–Maxwell theory, Lovelock gravity, %%%%1%%%% theories, and higher-dimensional or brane world frameworks. The central motivation is to realize black holes that are geodesically complete and physically regular, circumventing standard singularity theorems by controlled violation of critical energy or convergence conditions.

1. Construction Principles and Metric Structure

Regular de Sitter core black holes employ a core-envelope structure: the deep interior ($r\to0$) is modeled by a dS spacetime, while the exterior reproduces standard asymptotic (Schwarzschild, Reissner–Nordström, Kerr, or Kiselev) behavior. This is typically accomplished by prescribing a mass function $m(r)$ or a redshift function $f(r)$ that interpolates between de Sitter at small $r$ and black hole solutions at large $r$, guaranteeing both differentiability and physical acceptability: $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2d\Omega^2,$ where, generically, \begin{align*} f(r) &\sim 1 - \frac{r^2}{h^2}, \quad r\to0 \quad \text{(de Sitter core)} \ f(r) &\to 1 - \frac{2M}{r} + \cdots, \quad r\to\infty \quad \text{(vacuum or asymptotically (A)dS)} \end{align*} A class of exact solutions is given by (Ghosh et al., 27 Mar 2025), with

$F(r) = 1 - \frac{2mr^2}{(r+l)^3},$

so that for $r\to0$, $F(r)\to 1-\frac{1}{3}\Lambda r^2$ with $\Lambda=6m/l^3$ (de Sitter), and for $r\to\infty$, $F(r)\to 1-2m/r$ (Schwarzschild). The radius $l$ sets the scale of the dS core. The event horizon location is set by $F(r_h)=0$. Similar matching appears in models with charged matter (Lemos et al., 2011), rotating/axisymmetric generalizations (Ilijic et al., 2023), and multi-shell or smeared-matter configurations (Azreg-Aïnou, 2017).

2. Matter Content, Equations of State, and Matching

The regular core is realized by matter, fluid, or effective stress tensors obeying appropriate (often variable) equations of state yielding the dS relation near $r=0$. The canonical choice is $p=-\rho$, recovered via false vacuum models, anisotropic fluids, or phenomenological equations, $P=k(r)\rho$ with $k(r)\to -1$ at small $r$ (Vertogradov et al., 4 Aug 2024). Charged extensions are supported by prescribed charge distributions $Q(r)$, either smeared (power laws) or shelled, the latter leading to solutions with charged matter layers at $r=a$ (Lemos et al., 2011, Masa et al., 2020, Masa et al., 2020).

Matching conditions at the interface between the dS core and external vacuum (or electrovacuum) region are enforced via the Israel or Darmois–Israel junction conditions, frequently requiring continuity of the induced metric and controlled jumps in extrinsic curvature, corresponding to the physical shell. The matter content at the shell can be described by a surface energy density $\sigma$ and pressure $\mathcal{P}$, with barotropic relations $\mathcal{P} = \omega\sigma$ and $\omega$ adjusted to ensure stability or energy condition satisfaction (Masa et al., 2020, Saadati et al., 2020, Masa et al., 2020).

3. Horizon Structure, Regularity, and Geodesic Completeness

The horizon structure in regular dS core black holes is determined by the form of $f(r)$ and the matching conditions. Depending on parameter choices (ADM mass, charge, cosmological constant, core scale), various configurations arise:

• Single-horizon, Schwarzschild-like regular black holes (Ghosh et al., 27 Mar 2025).
• Two-horizon structures: an outer event horizon and an inner Cauchy (or degenerate, extremal) horizon (Casadio et al., 19 Feb 2025).
• Multihorizon spacetimes when the mass distribution admits multiple inflection points (Azreg-Aïnou, 2017).
• Shell configurations allow for regular overcharged stars or gravastars if the shell lies outside any horizon (Lemos et al., 2011, Masa et al., 2020, Masa et al., 2020).

Regularity is certified by demonstrating that all curvature invariants (Ricci, Kretschmann) are finite everywhere, particularly at $r=0$, and that geodesics can be analytically extended to arbitrary affine parameter without encountering a true singularity (Ghosh et al., 27 Mar 2025, Casadio et al., 19 Feb 2025). The expansion scalar entering the Raychaudhuri equation remains finite at the center, and its derivative vanishes, precluding caustic formation.

4. Energy and Convergence Conditions

A distinctive property of regular de Sitter core black holes is the controlled violation of energy or convergence conditions necessary to circumvent the Hawking–Penrose singularity theorem (Borissova et al., 10 Sep 2025). For spherically symmetric metrics,

• The Weak, Null, and Dominant Energy Conditions (WEC, NEC, DEC) can be enforced globally or across the majority of the spacetime, e.g., $$[R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}]t^{\mu}t^{\nu} \geq 0$$ for comoving observers (Ghosh et al., 27 Mar 2025).
• The Strong Energy Condition (SEC) is violated in the neighborhood of the core, in accordance with Zaslavskii's criterion which demands only $r \geq r_h/2$ for validity (Ghosh et al., 27 Mar 2025).

In terms of the Misner–Sharp mass function $m(r)$, the decomposition of the Timelike Convergence Condition (TCC) leads to three inequalities, of which $\text{(a) } -m''(r) \geq 0$ cannot hold near the dS core where $m(r)\sim m_3 r^3$ and $-m'' \sim -6 m_3 < 0$ for $m_3>0$ (Borissova et al., 10 Sep 2025). This explicit TCC violation near $r=0$ is the mechanism by which regular black holes avoid the conclusions of classical singularity theorems, while still fulfilling the NCC globally. In anti–de Sitter core models, the location and type of convergence condition violation differ, with the TCC typically satisfied at the core but violated at finite $r$ (Borissova et al., 10 Sep 2025).

5. Extensions: Gravity Theories, Rotating Solutions, and Multidimensional Models

While classical constructions are in Einstein gravity, regular dS core black holes are also realized in modified gravity scenarios:

• $f(\mathcal{R})$ gravity: The Ricci scalar $\mathcal{R}$ is determined from the chosen metric, and an effective action $f(\mathcal{R})$ is constructed (numerically and via Padé approximants) so that the field equations are satisfied with the regular core. The leading term recovers general relativity, while higher-order corrections regularize the strong-curvature domain (Ghosh et al., 27 Mar 2025).
• Lovelock gravity admits analogous finite-mass-density regular black holes, ensuring de Sitter cores in any dimension and nontrivial phase structure in the presence of a negative cosmological constant (Aros et al., 2019).
• Braneworld and higher-dimensional constructions yield analytic solutions which induce regular dS core black holes on the 4D brane worldvolume (Neves, 2021).

Rotating generalizations employ a $C^n$ transition from a de Sitter core to a Kerr exterior, with smooth step functions ensuring analytic matching and the absence of $\delta$-function sources in the transition region (Ilijic et al., 2023). Topological changes, such as the ergoregion evolving from $S^2$ to $S^1\times S^1$, are realized in these mimicker models.

6. Physical Properties, Phenomenological Features, and Observability

These black holes are designed to avoid central singularities—regularity is enforced by de Sitter-like cores, finite curvature, and geodesic completeness. Thermodynamic quantities and quasinormal mode spectra are modified by corrections to the lapse function, with observable consequences in the spectra of emitted Hawking radiation and ringdown (Heidari et al., 6 Dec 2024).

The presence of a de Sitter core and the associated modification to the effective potential lead to altered photon sphere and shadow properties. For example, in regular dS core models, the black hole shadow can be smaller and the relative contribution of higher-order photon rings enhanced compared to standard Schwarzschild, offering potential tests using high-resolution imaging (e.g., Event Horizon Telescope measurements of Sgr A*) (Martino et al., 2023, Vertogradov et al., 4 Aug 2024).

Parameter ranges are subject to astrophysical constraints, such as the observed size of the photon sphere, mass-to-radius ratios, and the absence/presence of extremal configurations (Azreg-Aïnou, 2017, Casadio et al., 19 Feb 2025, Vertogradov et al., 4 Aug 2024). Regular black hole solutions encompass both quasi-extremal scenarios and objects without horizons (overcharged stars, gravastars), depending on the matter content and matching radius (Lemos et al., 2011, Masa et al., 2020, Masa et al., 2020).

7. Implications for Gravitational Collapse, Cosmology, and the No-Hair Theorem

Regular de Sitter core black hole models provide a route to resolve the endpoint problem of gravitational collapse within classical or semi-classical gravity, demonstrating that singularities are not inevitable if energy/convergence condition violations are permitted in limited domains. Dynamical solutions exhibit transitions between regular and singular phases, with oscillatory behavior possible under suitable evolution of mass parameters (Vertogradov et al., 4 Aug 2024). In certain interior regions between horizons, the metric describes a non-singular Kantowski–Sachs cosmology, offering intriguing cosmological interpretations (Casadio et al., 19 Feb 2025).

These spacetimes can be constructed so that all macroscopic parameters are tied to the ADM mass, in harmony with the no-hair theorem for nonrotating, neutral black holes, and can be embedded in $f(\mathcal{R})$ frameworks consistent with observations at low curvature and regularization at high curvature (Ghosh et al., 27 Mar 2025).

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In sum, regular de Sitter core black holes constitute a class of physically viable, mathematical solutions in general relativity and its extensions. They are typified by smooth, non-singular interiors, finite curvature invariants, and geodesically complete geometries. Their explicit construction and analysis require an overview of matter modeling, junction conditions, energy/convergence condition diagnostics, and a careful accounting of global properties, with direct implications for black hole physics, potential observational signatures, and the foundational limits of classical gravity theory.