Higher-Dimensional Regular Black Holes

• Higher-Dimensional Regular Black Holes are static, spherically symmetric solutions in D≥5 that use an infinite tower of curvature corrections to eliminate singularities.
• They are derived from modified gravitational actions with a master algebraic equation that ensures a unique, regular metric and a de Sitter-like core.
• These models exhibit distinctive horizon structures, quasinormal mode spectra, and observable imprints in gravitational wave signals and black hole shadows.

Higher-dimensional @@@@1@@@@ are static, spherically symmetric solutions of gravity theories in $D \geq 5$ spacetime dimensions that incorporate infinite towers of higher-curvature corrections to the Einstein-Hilbert action, engineered to resolve the curvature singularities present in classical black holes. These solutions possess a nonsingular de Sitter-like core, satisfy generic regularity criteria for all curvature invariants, and can be realized as unique vacuum solutions of broad gravitational actions without the need for additional matter fields. They are now central to gravitational effective field theory, quantum gravity phenomenology, and the theoretical interpretation of gravitational wave and black hole shadow data.

1. Gravitational Action and Field Equations

Regular black holes in higher dimensions arise from actions of the form

$$S = \frac{1}{16\pi G_D} \int d^D x \, \sqrt{-g} \Big[ R + \sum_{n=2}^\infty \alpha_n \, \mathcal{Z}_n \Big],$$

where $\mathcal{Z}_n$ are quasi-topological curvature invariants of order $R^n$ constructed so that the static, spherically symmetric field equations remain second order, and the coupling constants $\alpha_n$ satisfy positivity and convergence conditions (e.g., $\alpha_n \geq 0$, $\limsup_{n\to\infty} (\alpha_n)^{1/n} = 1/C > 0$) (Bueno et al., 2024, Arbelaez, 29 Sep 2025). This structure enables a closed algebraic reduction of the field equations for metrics of the form

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

The metric function $f(r)$ is determined by the auxiliary function $\psi(r)$ through $f(r) = 1 - r^2 \psi(r)$. The master algebraic equation is

$$h(\psi) \equiv \psi + \sum_{n=2}^\infty \alpha_n \psi^n = \frac{m}{r^{D-1}},$$

where $m$ is proportional to the ADM mass. For each $D$, and each mass parameter, the solution is unique and regular, implementing a higher-dimensional Birkhoff theorem (Bueno et al., 2024, Bueno et al., 2024).

2. Families of Regular Black Hole Solutions

Several infinite sequences $\{\alpha_n\}$ yield distinct analytic families of regular metrics, reproducing and generalizing known lower-dimensional solutions:

Model label	Coupling sequence $\{\alpha_n\}$	Metric function $f(r)$
(a) Hayward	$\alpha_n = \alpha^{n-1}$	$1 - m r^2 / ( r^{D-1} + \alpha m )$
(c) Exponential (Dymnikova)	$\alpha_n = \alpha^{n-1}/n$	$$1 - \frac{r^2}{\alpha} [ 1 - \exp(-\alpha m / r^{D-1}) ]$$
(f) Square root	$$\alpha_n = \frac{\Gamma(2n-1)\alpha^{n-1}}{4^{n-1}\Gamma(n)^2 }$$	$$1 - \frac{2 m r^2}{m\alpha + \sqrt{4 r^{2(D-1)} + m^2\alpha^2}}$$

All these solutions converge to Schwarzschild–Tangherlini ($f \to 1 - m/r^{D-3}$) for $\alpha_n\to0$ and admit exact or invertible analytic forms for $f(r)$ in terms of $h(\psi)$ and its inverse $h^{-1}(x)$ (Bueno et al., 2024, Arbelaez, 29 Sep 2025).

3. Regularity and Core Structure

A defining property is the nonsingular behavior of curvature invariants. As $r \to 0$, the solution exhibits a de Sitter core: $$\psi(r) \to \Lambda \equiv \lim_{n\to\infty} \left( \frac{m}{\alpha_n} \right)^{1/n}, \quad f(r) \simeq 1 - \Lambda r^2.$$ This yields finite values for all invariants: $$R \simeq D(D-1)\Lambda, \quad R_{\mu\nu}R^{\mu\nu} \simeq D(D-1)^2\Lambda^2, \quad R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \simeq 2D(D-1)\Lambda^2.$$ Truncating the sum at finite $n_\text{max}$ leads to divergent invariants at $r=0$, underscoring the necessity of the infinite tower (Arbelaez, 29 Sep 2025, Konoplya et al., 2024). In models inspired by Dymnikova or analogues of Schwinger pair production, the energy density or the effective mass function decays exponentially with radius, enforcing regularity (Errehymy et al., 10 Jan 2026, Estrada et al., 2024).

4. Horizon Structure and Global Properties

The horizon radii $r_h$ solve $f(r_h)=0$, or equivalently $h(1/r_h^2) = m / r_h^{D-1}$. Regular solutions generically display:

• Two horizons for $m$ above a critical threshold --- an outer event horizon $r_+$ near the Schwarzschild radius and an inner Cauchy horizon $r_- \sim$ the de Sitter core scale.
• A single degenerate (extremal) horizon at the threshold.
• No horizons (naked-core geometry, or “regular soliton”) for smaller $m$ (Bueno et al., 2024, Arbelaez, 29 Sep 2025, Bueno et al., 2024).

In pure Lovelock regular black holes, the global structure depends sensitively on the Lovelock order $n$ and $(d=2n+1)$ versus $d>2n+1$: for specific combinations, only event and cosmological horizons are present with no Cauchy horizon (Estrada et al., 2024).

5. Dynamical Formation and Limiting Curvature

Recent studies show that regular higher-dimensional black holes naturally form as the endpoint of gravitational collapse. For the Oppenheimer–Snyder and thin-shell scenarios, generalized Israel junction conditions ensure that the matter shells (dust or thin shells) collapse inward until minimum radius inside the inner horizon, bounce (due to the regular core), and re-emerge into a new asymptotic region as a white hole (Bueno et al., 14 May 2025, Bueno et al., 2024). This process is periodic in proper time for dust, or can connect successive universes.

A universal, solution-independent upper bound on the spacetime curvature is established: $K(r) = R_{abcd}R^{abcd} \leq 2D(D-1)\psi_0^2,$ where $\psi_0$ is the value at which the characteristic series $h(\psi)$ diverges, realizing the Markov limiting curvature hypothesis (Bueno et al., 2024).

Cosmological analogues, where FLRW big bang and crunch singularities are replaced by regular bounces at minimum scale factor set by the core scale, are a structural property of the class of theories admitting regular black holes (Bueno et al., 14 May 2025).

6. Quasinormal Modes and Ringdown

Perturbations of regular higher-dimensional black holes obey a Schrödinger-like equation for the master variable $\psi$: $$\frac{d^2\psi}{dr_*^2} + [\omega^2 - V_\text{eff}(r)]\psi = 0,\quad dr_*/dr = 1/f(r),$$ with effective potential structure varying by field spin. The Wentzel–Kramers–Brillouin (WKB) method and eikonal (large $\ell$) limit yield for the quasinormal mode (QNM) spectrum (Arbelaez, 29 Sep 2025, Konoplya et al., 2024): $$\omega_n \approx \ell \Omega_c - i(n+1/2)|\Lambda_c|,$$ where $\Omega_c$ and $\Lambda_c$ are determined from the properties at the photon sphere. Increasing the higher-curvature couplings $\alpha$ (making the core more pronounced) systematically lowers both real and imaginary parts of the QNM frequencies relative to Schwarzschild–Tangherlini. This effect persists as $D$ increases.

High overtones are particularly sensitive to the near-horizon geometry, sometimes yielding vanishingly small real parts (unconventional modes) for moderate $\alpha_n$ (Konoplya et al., 2024). Truncating the series $\{\alpha_n\}$ at even modest $n$ rapidly converges QNM frequencies to the infinite limit within $<1\%$ (Konoplya et al., 2024).

Late-time tails of perturbations coincide with those of standard black holes: $\Psi \sim t^{-(2\ell+D-2)}$ for odd $D$, $\Psi \sim t^{-(2\ell+3D-8)}$ for even $D$.

7. Thermodynamics, Shadows, and Observational Features

Thermodynamic quantities, including the Hawking temperature and Wald entropy, are computable in closed form for any solution: $$T = \frac{1}{4\pi r_+}\left[\frac{(D-1) r_+^2 h(\psi_+)}{h'(\psi_+)} - 2\right],\quad S = -\frac{(D-2)\Omega_{D-2}}{8 G_D} \int^{\psi_+}\frac{h'(\psi)}{\psi^{D/2}} d\psi.$$ Event Horizon Telescope (EHT)–scale observations constrain regular black hole parameters: higher-dimensional regular black holes of Dymnikova-type produce shadow sizes that decrease slowly with increasing $D$ and grow with black hole scale $r_s$ but are weakly dependent on the de Sitter core size $r_0$. The Hawking temperature increases rapidly with $D$, and the emission power is higher for large $D$, though increasing $r_0$ suppresses emission (Errehymy et al., 10 Jan 2026).

Tables of 1$\sigma$ intervals for $r_0$ and $r_s$ compatible with the observed shadow sizes are provided for SgrA$^*$ and M87$^*$, with $D=5$ as the minimal higher-dimensional extension compatible with observations (Errehymy et al., 10 Jan 2026).

An important limitation is that, when coupled to linear electrodynamics and noncommutative-inspired matter profiles, stable regular Reissner–Nordström black holes exist only in four dimensions, and attempts to generalize these constructions to $n\ge5$ do not yield stable, regular, charged solutions (Wu et al., 2018).

8. Pure Lovelock and Quantum-Analogue Models

In pure Lovelock gravity of order $n$ (where the action retains only the $n$th Lovelock term), regular black hole models can be constructed by setting the local energy density as an exponential in the vacuum gravitational tension (proportional to the square root of the Kretschmann invariant), inspired by the form $\rho(r)\sim\exp[-F_c/F(r)]$. Critical properties include:

• For odd $n$ and $d>2n+1$, the black hole displays both inner and outer horizons, with an endpoint thermodynamically stable zero-temperature remnant as horizons coalesce.
• For even $n$, the transverse geometry is hyperbolic, inner horizons are absent for $d=2n+1$, and the remnant is reached via coincidence of event and cosmological horizons.
• The analogy to the Schwinger effect suggests the energy density near the core encodes quantum polarization effects (Estrada et al., 2024).

These models further demonstrate that regularity can be realized independent of matter content, purely via higher-curvature gravitational dynamics.

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Higher-dimensional regular black holes provide a technically complete, analytic resolution of classical singularities within a broad sector of gravitational effective field theory, are robust under dynamical formation, and admit observable imprints in gravitational wave and black hole imaging data, especially through deviations in ringdown frequencies and shadow radii. The structure and implications of these solutions are central to linking gravitational theory, quantum gravity phenomenology, and astrophysical observations (Bueno et al., 2024, Arbelaez, 29 Sep 2025, Errehymy et al., 10 Jan 2026, Estrada et al., 2024, Bueno et al., 2024, Bueno et al., 14 May 2025, Konoplya et al., 2024).