Quantum-Corrected OS Black Hole

Updated 1 January 2026

Quantum Oppenheimer-Snyder Black Holes integrate quantum gravity corrections into classical dust collapse, replacing singularities with bounces and modified horizon structures.
They employ effective field theory, loop quantum cosmology, and coherent state quantization to adjust both interior FLRW dynamics and the exterior Schwarzschild metric.
Observable implications include altered black hole shadows, shifted quasinormal modes, and the potential for stable remnants that inform the black hole information paradox.

A Quantum Oppenheimer-Snyder (OS) Black Hole generalizes the classical OS collapse by incorporating quantum gravity corrections—most notably from effective field theory, loop quantum cosmology, and path-integral (coherent state) quantization—into the gravitational collapse of a homogeneous dust ball. The resulting models feature a quantum-corrected metric, singularity avoidance via interior bounces, new thermodynamic and radiative effects, and the possibility of quantum "hair" on the exterior metric encoding information about the collapsing matter's microstate.

1. Quantum-Corrected Oppenheimer-Snyder Collapse: Formulation

The classical OS model describes the collapse of a homogeneous pressureless dust sphere (with interior a closed FLRW region and Schwarzschild exterior), forming a black hole with a central singularity. Quantum gravitational corrections modify both the interior evolution and the exterior geometry. In the effective approach relevant for semiclassical and loop quantum cosmology corrections, the corrected metrics are as follows:

Interior (Quantum FLRW Dust Ball):

$ds^2_{\mathrm{in}} = -d\tau^2 + a(\tau)^2 \big[dr^2 + r^2 d\Omega^2\big]$

with the quantum-corrected Friedmann equation:

$H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho \left(1 - \frac{\rho}{\rho_c}\right)$

where $\rho_c$ is a critical density reflecting quantum geometry effects (%%%%1%%%% Planck scale).

Exterior (Quantum-Corrected Schwarzschild):

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

where the lapse function generically takes the form

$f(r) = 1 - \frac{2GM}{r} + \alpha \frac{G^2 M^2}{r^4}$

with $\alpha$ a parameter determined by the underlying quantum theory (e.g., $\alpha = 16\sqrt{3}\pi\gamma^3 \ell_p^2$ in LQC, with $\gamma$ the Barbero–Immirzi parameter) (Lewandowski et al., 2022, Tan, 9 Feb 2025, Tan et al., 30 Dec 2025).

For more general corrections (Lorentz-term, nonlinear electrodynamics, higher dimensions), $f(r)$ is further deformed, but the $1/r^4$ modification is universal in 4d LQC models (Ou et al., 2 Aug 2025, Mazharimousavi, 12 Feb 2025, Shi et al., 2024).

The junction between the interior and exterior is implemented using the Darmois–Israel conditions, ensuring continuity of both induced metric and extrinsic curvature at the moving dust boundary (Lewandowski et al., 2022, Fazzini et al., 2023).

2. Structure, Horizons, and Non-Singularity

Bounce and Singularity Resolution

Quantum corrections replace the classical singularity by a bounce: the interior scale factor $a(\tau)$ shrinks to a nonzero minimum (set by $\rho_c$ ), then expands, corresponding to the "time-reversed" white hole branch. The minimal radius (bounce radius) is

$r_b = (\alpha GM/2)^{1/3}$

This singularity avoidance is robust to quantization choices in LQC formulations and coherent-state quantizations, with curvature invariants (e.g., Kretschmann scalar) remaining finite and bounded by the Planck scale (Lewandowski et al., 2022, Góźdź et al., 2023, Bobula, 2024).

Horizon Structure and Mass Gap

The modified exterior admits two regular horizons for $M > M_{\min}$ :

Outer ("event") horizon: $r_+$
Inner (Cauchy) horizon: $r_-$

The horizon radii are roots of $f(r)=0$ . The two-horizon structure is analogous to Reissner–Nordström, but with all parameters set by the quantum-corrected matching (Lewandowski et al., 2022, Yang et al., 2022, Mazharimousavi, 12 Feb 2025). A nonzero minimal mass ("mass gap") exists, given by

$M_{\min} = \frac{4}{3\sqrt{3}G} \sqrt{\alpha}$

No black hole forms for masses below this threshold (Lewandowski et al., 2022).

Causal Structure and Maximal Extension

The Penrose diagram of the quantum OS black hole mirrors that of non-extremal Reissner–Nordström: a sequence of asymptotically flat, trapped, and "wormhole throat" regions. The dust surface begins in the external region, enters the trapped interior, bounces, and reemerges as a white hole (Lewandowski et al., 2022, Fazzini et al., 2023).

3. Quantum Gravitational "Hair" and Information Encoding

At leading perturbative order, the effective field equations yield time-dependent corrections ( $\delta f, \delta g$ ) to the exterior metric that depend on the interior dust configuration (initial size, density, slow time-derivatives). These "quantum hair" corrections persist through the horizon formation and encode multipole-like coefficients ( $\mathcal Q \propto r_s^3$ ), in principle accessible via metric expansion at large $R$ : $g_{tt} = -1 + \frac{2GM}{R} + \frac{\mathcal Q}{R^5} + \cdots$ This breaks the classical no-hair theorem at loop level and offers a potential information channel for the recovery of interior microstates by external observers (Calmet et al., 2023).

4. Observational Phenomenology

Black Hole Shadows and Photon Sphere

The photon-sphere (radius $r_{\mathrm{ph}}$ ) and shadow radius ( $b_c$ ) are shifted compared to Schwarzschild: $r_{\rm ph}(\alpha) \approx \frac{3}{2}R_s - \frac{2\alpha}{9R_s} + \cdots,\qquad b_c = \frac{r_{\rm ph}}{\sqrt{f(r_{\rm ph})}}$ Quantum corrections generally reduce the shadow size ( $b_c$ ) by 1–2% for $\alpha = \mathcal{O}(0.1)$ (Yang et al., 2022, Luo et al., 2024). Additional deformations arise if the exterior is further generalized (e.g., with quintessence, string clouds, NED) (Ahmed et al., 5 Aug 2025).

Gravitational Wave Ringdown and Quasinormal Modes

Scalar, electromagnetic, and neutrino perturbations obey modified Regge–Wheeler equations on the quantum-corrected geometry. Quantum corrections shift the real QNM frequencies slightly (few percent), but substantially suppress the damping rate (imaginary part). For example, as the quantum parameter increases:

$\Re(\omega)$ up by $\sim$ 2–3%
$|\Im(\omega)|$ down by up to 13% (longer-lived ringdown) (Skvortsova, 2024, Ou et al., 2 Aug 2025) This signals a distinctive slow ringdown in gravitational wave data—potentially detectable by precision GW observatories (Yang et al., 29 Sep 2025).

Hawking Radiation and Black Hole Remnants

Calculations using the Parikh–Wilczek tunneling framework incorporate the quantum-corrected metric and self-gravitation, yielding:

Modified emission rate
Logarithmic corrections to the Bekenstein–Hawking entropy:

$S_{qOS} = \frac{A}{4\ell_p^2} + (\pi \alpha/2) \ln(A/\ell_p^2) + \cdots$

At late stages, the evaporation process slows and terminates at a mass $M_{\mathrm{final}} > M_{\mathrm{min}}$ , resulting in a cold, stable, remnant (Tan et al., 30 Dec 2025). The stability of these remnants is confirmed by QNM analysis (all $\Im\omega < 0$ ).

Precision Constraints via Extreme Mass-Ratio Inspirals

Observations of eccentric extreme mass-ratio inspirals (EMRIs) provide the most stringent bounds on the quantum parameter, with projected LISA constraints at $\alpha \lesssim 10^{-5}$ —far exceeding current shadow-based bounds ( $\alpha \lesssim \mathcal{O}(1)$ ) (Yang et al., 29 Sep 2025).

5. Extensions: Charges, Exotic Fields, Higher-Dimensional Generalizations

Charged Quantum OS Black Holes

Including electric/magnetic charge via nonlinear electrodynamics, the exterior metric retains an $M^2/r^4$ -type correction, with equilibrium of a charged thin shell at

$R_{\mathrm{eq}} = \left[3(M_{\mathrm{APS}} m + \alpha Q^2)\right]^{1/3}$

Such models admit two horizons and support small-amplitude oscillations of the shell; the inner Cauchy horizon remains unstable to mass inflation (Mazharimousavi, 12 Feb 2025).

Influence of Quintessence and String Clouds

Quantum OS black holes embedded in backgrounds with cloud-of-strings and quintessence fields exhibit a spectrum of horizon, shadow, QNM, and thermodynamic behaviors, including exotic phenomena (negative temperature, multi-phase transitions) (Ahmed et al., 5 Aug 2025).

Higher-Dimensional Models

Generalizing to $(d+1)$ dimensions, quantum bounces universally occur, singularities are avoided, and the exterior is uniquely determined by junction conditions. The QNM and thermodynamic spectrum features dimension-dependent quantum corrections, with logarithmic entropy corrections in 4d and polynomial in higher dimensions (Shi et al., 2024).

6. Foundational Aspects: Quantization Approaches and Coordinate Pathologies

Affine and Integral Coherent-State Quantization

Alternative quantizations (e.g., affine coherent states, integral quantization) yield globally bounded curvature, generic bounce scenarios, and spectrum quantization for the black hole mass. However, ensuring the bounce occurs inside the photon sphere (i.e., behind a genuine trapping region) typically requires tuning quantization ambiguities (Piechocki et al., 2020, Góźdź et al., 2023).

Coordinate Dependence and Continuity Across the Bounce

Attempts to write the quantum-corrected geometry in a single coordinate chart across the bounce (e.g., Painlevé–Gullstrand slicing) encounter apparent discontinuities, which are coordinate artifacts. The physical geometry, as verified in matching (e.g., Schwarzschild-like time), remains smooth and shock-free through the bounce, transitioning into the expanding white hole branch (Fazzini et al., 2023).

7. Physical Implications and Uniqueness

Violation of No-Hair: The quantum-corrected metric carries information about the interior and initial dust profile, breaking the classical no-hair property and potentially resolving aspects of the black hole information paradox (Calmet et al., 2023).
Remnants and Unitarity: The persistence of a stable quantum remnant implies a possible endpoint for evaporation, consistent with unitary evolution, as verified in tunneling and dynamical models (Tan et al., 30 Dec 2025, Corda et al., 2021).
Minimum Mass: No black hole forms for $M < M_{\min}$ , providing a Planck-scale cutoff for Hawking evaporation (Lewandowski et al., 2022).

The quantum Oppenheimer–Snyder black hole construction provides a well-controlled framework—grounded in LQC, effective field theory, and canonical quantization—for nonsingular gravitational collapse, quantum-induced horizon and thermodynamic corrections, and new observables for quantum gravity phenomenology in black holes (Lewandowski et al., 2022, Yang et al., 2022, Calmet et al., 2023, Mazharimousavi, 12 Feb 2025, Bobula, 2024, Tan, 9 Feb 2025, Tan et al., 30 Dec 2025).