Rotating Holonomy-Corrected Black Holes

• Rotating Holonomy-Corrected Black Holes are Kerr-like spacetimes modified by loop quantum gravity holonomy corrections, replacing the Kerr ring singularity with a regular transition surface.
• They are derived from static quantum-corrected seeds using modified Newman–Janis algorithms, introducing a quantum parameter that affects horizon structure and photon dynamics.
• Observational tests, including Event Horizon Telescope data, constrain the holonomy parameter by revealing changes in shadow size, distortion, and the stability of horizons.

Rotating Holonomy-Corrected Black Holes (RHCBHs) are Kerr-like rotating black-hole spacetimes in which the departure from classical Kerr is controlled by a parameter descending from loop-quantum-gravity holonomy corrections. In current usage, the term appears explicitly for a rotating holonomy-corrected black hole constrained with Event Horizon Telescope data, while closely related work studies a “nonsingular rotating quantum-corrected black hole” built from a loop-quantum-gravity seed metric and identifies its quantum parameter as originating from holonomy modifications; together these works define the main RHCBH landscape, although adjacent literature often uses broader labels such as rotating quantum-corrected or loop-inspired rotating black holes (Ali et al., 23 May 2026, Yang et al., 2022, Ali et al., 2024).

1. Terminology and scope

In the strict sense, RHCBH denotes the Kerr-like rotating spacetime studied as a loop-quantum-gravity-motivated deformation of Kerr with a holonomy correction parameter $b$, constructed from a static holonomy-corrected Schwarzschild-like seed and analyzed through null geodesics, shadow observables, and EHT constraints (Ali et al., 23 May 2026). In a closely related but not identically named construction, the rotating metric of Brahma, Chen, and Yeom is described as a “nonsingular rotating black hole in loop quantum gravity” or “nonsingular rotating quantum-corrected black hole,” and its parameter $A_\lambda$ is tied to $\lambda_k$, which “originates from holonomy modifications” and is directly related to the fundamental area gap of loop quantum gravity (Yang et al., 2022).

This terminological split matters because the rotating sector is not uniform across the literature. Some rotating metrics are presented as RHCBHs in the narrow sense, whereas others are best classified as effective quantum-gravity or loop-inspired analogs obtained from static seeds by a modified or revised Newman–Janis algorithm. A directly relevant example is the study of two rotating effective quantum-gravity black holes with parameter $\zeta$, where the seed spacetimes are “covariant spherically symmetric BH spacetimes within effective quantum gravity,” but the rotating solutions are explicitly described as close analogs of RHCBHs rather than strict holonomy-derived rotating solutions (Ban et al., 2024).

The resulting conceptual boundary is therefore precise but not rigid. RHCBHs in the narrow sense are rotating black holes whose quantum deformation is explicitly interpreted as holonomy-induced. In the broader phenomenological sense, the label is often extended to rotating LQG-inspired effective metrics that share the same operational features: a Kerr-like line element, an additional quantum parameter, modified horizon structure, modified photon dynamics, and observational consequences in shadows or ringdown.

2. Metric constructions and quantum parameters

A principal RHCBH-type construction begins from a static spherically symmetric black hole in loop quantum gravity and applies the revised Newman–Janis algorithm. The resulting rotating metric, in Boyer–Lindquist-type coordinates, is written as

$$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$

with

$$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$

Here $b=b(r)$, $\tilde a=\tilde a(r)$, and the dimensionless quantum parameter $A_\lambda\ge 0$ is related to $\lambda_k$, with $A_\lambda$0 originating from holonomy modifications. In the limit $A_\lambda$1, this spacetime reduces to the static spherically symmetric LQG black hole; in the limit $A_\lambda$2, it reduces to Kerr. The geometry is regular everywhere for $A_\lambda$3, the Kerr ring singularity is replaced by a timelike transition surface at $A_\lambda$4, and the full spacetime is interpreted as a symmetric bounce connecting black-hole and white-hole regions (Yang et al., 2022).

The explicitly named RHCBH construction studied with EHT data starts instead from the static seed

$A_\lambda$5

where $A_\lambda$6 is the holonomy correction parameter and

$A_\lambda$7

Using the modified Newman–Janis algorithm in Azreg-Aïnou’s non-complexification form, the rotating counterpart is

$A_\lambda$8

with the key Kerr-like deformation encoded in

$A_\lambda$9

The paper notes that its typesetting is imperfect in the $\lambda_k$0 sector, but the intended interpretation is that $\lambda_k$1 is an $\lambda_k$2-dependent deformation of the Kerr quantity $\lambda_k$3, while the geodesic and shadow analysis proceeds through $\lambda_k$4. This RHCBH reduces to Kerr when $\lambda_k$5 and to the static holonomy-corrected Schwarzschild black hole when $\lambda_k$6 (Ali et al., 23 May 2026).

These constructions show two recurrent RHCBH features. First, the rotating geometry is not introduced ad hoc: it is inherited from a quantum-corrected static seed. Second, the quantum parameter enters through the radial structure that replaces the Kerr mass term, so that modifications of horizons, photon regions, and scattering follow from a deformed $\lambda_k$7.

3. Horizon structure, extremality, and causal sectors

For the rotating nonsingular LQG black hole, horizons are determined by the Kerr-like condition

$\lambda_k$8

The metric describes a black hole for $\lambda_k$9 and no horizon for $\zeta$0. Extremality remains

$\zeta$1

but the extremal horizon radius becomes

$\zeta$2

with the physically relevant two-horizon sector requiring

$\zeta$3

The parameter-space analysis divides $\zeta$4 into four regions: rotating wormhole with no horizon, black hole with one horizon, black hole with two horizons, and a no-horizon sector for sufficiently large spin. The authors focus on the two-horizon regime as the main rotating black-hole sector and emphasize that increasing $\zeta$5 shifts both inner and outer horizons relative to Kerr and shrinks the extremal horizon (Yang et al., 2022).

For the explicitly named RHCBH, the horizon equation is

$\zeta$6

with roots

$\zeta$7

Horizons exist when

$\zeta$8

The extremal case satisfies

$\zeta$9

The same paper writes the extremality boundary in the $$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$0 plane as

$$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$1

Because the seed construction restricts $$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$2, the physically relevant black-hole branch is effectively $$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$3, denoted BH-I. The interval

$$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$4

is a no-horizon region describing a regular horizonless compact spacetime. The paper then introduces two thresholds: $$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$5, where the event and Cauchy horizons merge, and $$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$6, above which unstable photon orbits disappear and a closed photon ring is lost. This establishes a central RHCBH claim: closed shadows are governed by unstable photon orbits, not strictly by the existence of an event horizon (Ali et al., 23 May 2026).

A broader rotating-quantum-black-hole question concerns the fate of the Cauchy horizon. An effective quantum-matter analysis argued that modifying only the mass profile $$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$7 is not enough to remove the Kerr-like inner horizon; one must also relax constant specific angular momentum and require

$$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$8

so that the interior may have an integrable singularity and no Cauchy horizon (Casadio et al., 2023). This suggests that RHCBH interiors retaining a Kerr-like constant spin parameter near the center will generically preserve an inner horizon, whereas models with a softened local spin profile may not.

4. Photon dynamics, shadow formation, and EHT tests

The RHCBH shadow analysis is formulated through separable null geodesics. With Hamilton–Jacobi action

$$ds^2=-\left(1-\frac{2Mb}{\rho^2}\right)dt^2-\frac{4aMb\sin^2\theta}{\rho^2}\,dt\,d\phi +\rho^2 d\theta^2+\frac{\rho^2}{\Delta}dr^2+\frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2,$$9

the radial and angular potentials are

$$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$0

$$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$1

For unstable spherical photon orbits at $$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$2,

$$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$3

The critical impact parameters are

$$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$4

For a distant observer,

$$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$5

The shadow boundary follows by varying the spherical-orbit radius $$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$6. The main optical results are that increasing $$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$7 increases asymmetry and flattens the prograde edge, increasing $$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$8 enlarges the overall shadow, and the viewing angle $$\rho^2=b^2+a^2\cos^2\theta, \qquad \Sigma=(b^2+a^2)^2-a^2\Delta\sin^2\theta.$$9 controls the degree of distortion. In equatorial photon dynamics, $b=b(r)$0 remains fixed at $b=b(r)$1, while $b=b(r)$2 increases with $b=b(r)$3, which the authors interpret as a weaker effective gravitational pull near the center. Most notably, RHCBH spacetimes can produce closed shadow rings even in the absence of an event horizon for

$b=b(r)$4

because unstable circular photon orbits persist in that interval (Ali et al., 23 May 2026).

Parameter inference in this framework uses the Kumar–Ghosh observables

$b=b(r)$5

Contours of constant shadow area $b=b(r)$6 and oblateness $b=b(r)$7 intersect uniquely in the $b=b(r)$8 plane, allowing simultaneous extraction of spin and holonomy parameter. Applying EHT angular-diameter bounds, the paper finds for M87* at $b=b(r)$9,

$\tilde a=\tilde a(r)$0

and for Sgr A* at $\tilde a=\tilde a(r)$1,

$\tilde a=\tilde a(r)$2

Current EHT data therefore allow nonzero holonomy correction over a substantial part of parameter space (Ali et al., 23 May 2026).

The broader rotating quantum-corrected literature shows that shadow response is model-dependent. In one LQG-inspired rotating quantum-corrected metric with parameter $\tilde a=\tilde a(r)$3, the shadow becomes larger and less distorted than Kerr over viable black-hole regions, while current EHT constraints still leave RQCBHs and Kerr observationally degenerate over much of the allowed space (Ali et al., 2024). By contrast, in two effective quantum-gravity rotating models with parameter $\tilde a=\tilde a(r)$4, the dominant non-extremal effect is a change in shadow size, $\tilde a=\tilde a(r)$5 decreases as $\tilde a=\tilde a(r)$6 increases, and near extremality the shadow can develop a cuspy edge with increasing distortion (Ban et al., 2024). This suggests that RHCBH shadow phenomenology cannot be reduced to a universal “quantum correction enlarges the shadow” rule; the sign and morphology of the deviation depend on the specific rotating effective metric.

5. Overspinning, scalar scattering, and cosmic censorship

The rotating nonsingular LQG black hole has also been studied through the Wald/Jacobson–Sotiriou overspinning thought experiment. For a test particle obeying

$\tilde a=\tilde a(r)$7

with conserved quantities

$\tilde a=\tilde a(r)$8

the horizon-crossing condition implies

$\tilde a=\tilde a(r)$9

Destruction of the horizon requires

$A_\lambda\ge 0$0

or

$A_\lambda\ge 0$1

Overspinning is possible only if

$A_\lambda\ge 0$2

For an extremal initial black hole, $A_\lambda\ge 0$3 and $A_\lambda\ge 0$4, while

$A_\lambda\ge 0$5

Hence

$A_\lambda\ge 0$6

with equality only at $A_\lambda\ge 0$7, so

$A_\lambda\ge 0$8

There is no allowed interval: an extremal rotating LQG black hole cannot be overspun by a test particle. In the near-extremal case, using

$A_\lambda\ge 0$9

the expansion

$\lambda_k$0

shows that the negative shift from the quantum parameter destroys the test-body overlap window that exists in near-extremal Kerr. The paper summarizes this by saying that the quantum parameter acts as a protector: larger $\lambda_k$1 makes overspinning by particles harder (Yang et al., 2022).

The scalar-field channel behaves differently. For a minimally coupled massive scalar,

$\lambda_k$2

the near-horizon radial equation becomes

$\lambda_k$3

The horizon fluxes are

$\lambda_k$4

A crucial shift occurs at extremality: $\lambda_k$5 with equality only for Kerr. The final-state criterion can then be written as

$\lambda_k$6

For an extremal black hole, the dangerous mode window is

$\lambda_k$7

Within this interval, the horizon is destroyed in the test-field approximation. The same qualitative conclusion extends to near-extremal configurations. Thus the same quantum correction that protects the horizon against particles opens a destructive frequency window for scalar waves (Yang et al., 2022).

Because this black hole is nonsingular, destroying the horizon does not expose a curvature singularity. The authors therefore argue that the result need not conflict with weak cosmic censorship in the usual predictability sense; rather, it could make the regular high-curvature quantum region, interpreted as a black-hole/white-hole transition region, accessible in principle.

6. Perturbations, quasinormal modes, and ringdown inference

A closely adjacent literature extends RHCBH-type phenomenology from imaging to ringdown. One rotating quantum-corrected black hole, generated from the static metric

$\lambda_k$8

has rotating line element

$\lambda_k$9

with

$A_\lambda$00

The static seed is explicitly tied to LQG/LQC-inspired effective dynamics through

$A_\lambda$01

Although the paper does not use the RHCBH label, it describes the model as closely related to the same phenomenological family (Chen et al., 31 Oct 2025).

The perturbation problem is posed for a massless scalar field in a hyperboloidal compactified framework. After the coordinate transformation

$A_\lambda$02

with

$A_\lambda$03

the quasinormal-mode problem becomes a two-dimensional eigenvalue problem in $A_\lambda$04, $A_\lambda$05,

$A_\lambda$06

solved with a tensor-product Chebyshev–Lobatto pseudo-spectral grid. The spectra are close to Kerr for small $A_\lambda$07, but the extra parameter induces systematic shifts in both oscillation frequency and damping rate. For inference, each mode is fitted by a bivariate rational function,

$A_\lambda$08

and a ringdown pipeline is implemented in pyRing. The main result is not a sharp bound on $A_\lambda$09, but the demonstration that strong correlation between the beyond-Kerr parameter and the intrinsic spin can significantly distort posterior distributions of $A_\lambda$10 and $A_\lambda$11, even when the spectrum deviates from Kerr only slightly (Chen et al., 31 Oct 2025).

The paper also states a major caveat: it computes scalar $A_\lambda$12 QNMs, whereas pyRing templates are built for gravitational $A_\lambda$13 perturbations. The substitution

$A_\lambda$14

is therefore only illustrative. For RHCBH studies, the implication is methodological: rotating loop-inspired beyond-Kerr parameters may couple strongly to spin in ringdown inference, but definitive constraints require the true $A_\lambda$15 quasinormal spectrum on the rotating quantum-corrected background.

7. Methodological caveats and broader context

A recurrent issue in RHCBH research is derivation versus construction. Several rotating quantum-corrected metrics are generated from static seeds by a modified or revised Newman–Janis algorithm rather than by direct polymerization of a rotating Hamiltonian. In one representative study, the resulting rotating metrics contain an undetermined overall conformal factor $A_\lambda$16, but the authors emphasize that neither the causal structure nor the null geodesics depend on $A_\lambda$17, and hence shadow size and shape do not depend on that choice (Ban et al., 2024). This shows that some observational predictions can be robust even when the rotating field equations are not fully specified.

Another frequent misconception is that every rotating quantum-corrected black hole is an RHCBH. This is not the case. A quantum-improved rotating black hole motivated by asymptotic safety modifies Kerr through a running Newton coupling,

$A_\lambda$18

and is therefore not holonomy-corrected, even though it shares features such as smaller event-horizon and ISCO radii than Kerr or Schwarzschild (Ladino et al., 2023). Likewise, a different perturbative framework based on string-theoretic $A_\lambda$19 corrections and a dilaton yields slowly rotating $A_\lambda$20-corrected black holes with explicit mixed terms of the form $A_\lambda$21, again outside the RHCBH category despite the close structural analogy of a corrected dragging sector (Agurto-Sepúlveda et al., 2022).

The present observational status is correspondingly cautious. In the explicitly named RHCBH model, nonzero holonomy correction remains consistent with current EHT data, and the paper stresses that RHCBHs provide viable alternatives to Kerr in the strong-field regime (Ali et al., 23 May 2026). This suggests that the decisive near-term problem is not merely the existence of RHCBH solutions, but discrimination among multiple non-Kerr families that can reproduce similar shadow observables while differing in their interior structure, extremality bounds, and perturbative spectra.