Holonomy-Corrected Schwarzschild Black Hole

• Holonomy-corrected Schwarzschild black hole is a loop quantum gravity-based geometry where polymerization of canonical variables replaces the classical singularity with a minimal spacelike hypersurface.
• It utilizes both point holonomy modifications and anomaly-free canonical transformations to generate deformed constraint algebras and yield covariant, nonsingular effective metrics.
• These corrections influence observable features such as reduced Hawking temperature, modified lensing properties, and distinctive quasinormal mode spectra while recovering Schwarzschild in the classical limit.

A holonomy-corrected Schwarzschild black hole is a loop-quantum-gravity-inspired effective black-hole geometry in which the spherically symmetric Schwarzschild sector is modified by polymerization or holonomy corrections to the canonical variables. In the literature, the term covers two closely related but technically distinct constructions. One line of work formulates point holonomy corrections directly in the spherically symmetric Hamiltonian system and finds a first-class but deformed hypersurface-deformation algebra, no nontrivial static vacuum solution, and a characteristic signature-change mechanism in the deep quantum regime (Tibrewala, 2012). A second line implements holonomy corrections through an anomaly-free canonical transformation and a regularized recombination of constraints, yielding a fully covariant effective spacetime with a static exterior metric, a minimal spacelike hypersurface at $r=r_0$, two asymptotically flat exteriors of equal mass, and a nonsingular black-hole/white-hole global structure that reduces to Schwarzschild as the polymerization parameter vanishes (Alonso-Bardaji et al., 2021, Alonso-Bardaji et al., 2022).

1. Canonical origin in spherically symmetric loop quantum gravity

The canonical starting point is spherical symmetry in Ashtekar–Barbero variables, with two triad components $E^x(x),E^\phi(x)$ and two extrinsic-curvature components $K_x(x),K_\phi(x)$. The basic Poisson brackets are

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$

and the spatial line element reconstructed from the triads is

$$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$

With lapse $N$ and radial shift $N^x$, the corresponding four-dimensional line element is

$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$

In vacuum, the diffeomorphism constraint is

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$

while the gravitational Hamiltonian constraint is

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$

with $E^x(x),E^\phi(x)$0. Classically, the constraints are first class and satisfy

$E^x(x),E^\phi(x)$1

These relations supply the benchmark against which holonomy modifications are assessed (Tibrewala, 2012).

Point holonomy corrections polymerize the angular extrinsic-curvature component $E^x(x),E^\phi(x)$2. In the formulation of (Tibrewala, 2012), the Hamiltonian is modified by replacing the quadratic $E^x(x),E^\phi(x)$3 term by $E^x(x),E^\phi(x)$4 and the linear $E^x(x),E^\phi(x)$5 term by $E^x(x),E^\phi(x)$6, while $E^x(x),E^\phi(x)$7 is kept classical. Off-shell closure requires

$E^x(x),E^\phi(x)$8

A consistent phase-space-independent choice is

$E^x(x),E^\phi(x)$9

and a more general family is

$K_x(x),K_\phi(x)$0

The same paper also considers phase-space-dependent polymerization scales $K_x(x),K_\phi(x)$1, $K_x(x),K_\phi(x)$2, leading to a modified closure condition and to explicit $K_x(x),K_\phi(x)$3-dependent solutions for $K_x(x),K_\phi(x)$4 (Tibrewala, 2012).

A distinct anomaly-free implementation appears in the effective theory of (Alonso-Bardaji et al., 2021) and its covariant development (Alonso-Bardaji et al., 2022). There the holonomy map is realized as the canonical transformation

$K_x(x),K_\phi(x)$5

together with a regularized linear combination of the Hamiltonian and diffeomorphism constraints. The resulting on-shell constant of motion is

$K_x(x),K_\phi(x)$6

and the covariant holonomy scale is encoded in

$K_x(x),K_\phi(x)$7

In this construction, the surfaces $K_x(x),K_\phi(x)$8 are covariantly identified with $K_x(x),K_\phi(x)$9 (Alonso-Bardaji et al., 2021, Alonso-Bardaji et al., 2022).

2. Constraint algebra, covariance, and signature change

In the direct point-holonomy treatment, the modified Hamiltonian remains first class but the $$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$0 bracket is deformed. For phase-space-independent polymerization one obtains

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$1

For

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$2

the deformation is $$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$3; for phase-space-dependent schemes,

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$4

Signature change occurs whenever $$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$5, and for the phase-space-dependent stationary Schwarzschild-like solution the deformation becomes

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$6

which is negative in the deep quantum regime

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$7

The paper therefore associates the deformed algebra with an Euclidean core or modified causal structure in the interior (Tibrewala, 2012).

A central consequence of the deformed algebra is that gauge transformations generated by $$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$8 are no longer equivalent to classical coordinate transformations. In the same analysis, the standard identification

$$\{K_x(x),E^x(y)\}=2G\delta(x-y),\qquad \{K_\phi(x),E^\phi(y)\}=G\delta(x-y),$$9

is said to be covariant only when the classical hypersurface-deformation algebra holds. For holonomy corrections alone, the algebra is necessarily deformed, the paper finds it difficult to construct a covariant metric, and coordinate transformations of naive metric ansätze do not map solutions to solutions. A specific difficulty is a large degeneracy of stationary solutions: different choices of the lapse $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$0, all with the correct classical limit, lead to inequivalent $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$1 profiles through the same effective field equation (Tibrewala, 2012).

The same work also shows that holonomy corrections alone preclude nontrivial static vacuum solutions. In the static gauge $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$2 and $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$3, the $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$4-equation forces $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$5, which in turn implies $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$6, sending the system back to the classical branch $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$7. Time-independent but stationary solutions do exist, however. In areal gauge $$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$8 and under time independence, the equations imply

$$q_{xx}=\frac{(E^\phi)^2}{|E^x|},\qquad q_{\theta\theta}=|E^x|.$$9

while the Hamiltonian constraint reduces to

$N$0

Two explicit stationary families are then exhibited: a Schwarzschild-like gauge with

$N$1

and a Painlevé–Gullstrand-like gauge with $N$2 and $N$3 (Tibrewala, 2012).

By contrast, the anomaly-free model of (Alonso-Bardaji et al., 2021) and (Alonso-Bardaji et al., 2022) is constructed so that gauge transformations on phase space correspond to coordinate changes on spacetime. Its deformed Hamiltonian satisfies

$N$4

with non-negative structure function

$N$5

The metric is then defined by

$N$6

which gives a covariant spacetime reconstruction in that framework (Alonso-Bardaji et al., 2021, Alonso-Bardaji et al., 2022).

3. Covariant effective geometry and nonsingular Schwarzschild sector

The best-known static holonomy-corrected Schwarzschild metric in the later phenomenological literature is the exterior solution of the covariant anomaly-free model: $N$7 Here

$N$8

with $N$9 the dimensionless polymerization parameter. The lapse function is Schwarzschild-like, while the radial coefficient carries the holonomy correction. The event horizon remains at

$N^x$0

and the classical singularity is replaced by a minimal spacelike hypersurface at

$N^x$1

The same geometry is also written in later papers as

$N^x$2

or equivalently

$N^x$3

with $N^x$4 or $N^x$5 playing the role of the holonomy length scale (Alonso-Bardaji et al., 2022, Soares et al., 2023, Junior et al., 2023).

A compact notation correspondence used in the literature is:

Source notation	Static radial factor	Polymer relation
$N^x$6	$N^x$7	$N^x$8
$N^x$9	$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$0	$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$1
$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$2	$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$3	$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$4 is the holonomy-correction parameter

The covariant model also supplies interior and global charts. In a homogeneous interior gauge with $ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$5, the line element is

$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$6

A global chart $ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$7 covering two exteriors and the whole interior is

$ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$8

where $ds^2=-N^2dt^2+q_{xx}(dx+N^xdt)^2+|E^x|\,d\Omega^2.$9 is smooth and even, with

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$0

The resulting maximal analytic extension contains two asymptotically flat exterior regions, a black-hole region, a white-hole region, and a minimal surface at $$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$1; the spacetime is geodesically complete in the sense stated in the model papers (Alonso-Bardaji et al., 2021, Alonso-Bardaji et al., 2022).

The curvature invariants remain finite everywhere in this construction. The Ricci scalar is

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$2

and the Kretschmann scalar is

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$3

The mass notions are also split: the Komar mass is

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$4

the Hawking or Misner–Sharp mass is

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$5

and the ADM mass on static slices is

$$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$6

These formulas show that the parameter $$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$7 is the Komar mass at infinity, whereas the ADM mass on the static exterior slices is shifted by $$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$8 (Alonso-Bardaji et al., 2022).

4. Classical limit, horizons, photon sphere, and thermodynamics

All effective constructions considered here recover Schwarzschild in the appropriate classical limit. In the direct point-holonomy analysis, the limit $$D[N^x]=\frac{1}{2G}\int dx\,N^x\big(2E^\phi K'_\phi-K_x(E^x)'\big),$$9 or large $$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$0 gives

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$1

and the stationary solutions reduce to the standard Schwarzschild geometry (Tibrewala, 2012). In the covariant static model, $$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$2 implies $$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$3, so

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$4

(Alonso-Bardaji et al., 2021, Alonso-Bardaji et al., 2022).

A recurring structural feature of the static holonomy-corrected metric is that the photon sphere is unchanged. Since

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$5

the photon-sphere condition

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$6

still gives

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$7

and the critical impact parameter remains

$$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$8

This is emphasized in both weak- and strong-lensing studies of the static metric (Junior et al., 2023, Soares et al., 2023). It also underlies the axial perturbation analysis, where the eikonal light ring remains at $$H_{\rm grav}[N]=-\frac{1}{2G}\int dx\,N|E^x|^{-1/2}\left[K_\phi^2E^\phi+2K_\phi K_xE^x+(1-\Gamma_\phi^2)E^\phi+2\Gamma'_\phi E^x\right],$$9 because $E^x(x),E^\phi(x)$00 is Schwarzschild-like (Yang et al., 2024).

The horizon structure of the static covariant metric is correspondingly simple in the exterior region. The outer event horizon stays at $E^x(x),E^\phi(x)$01, while the second zero of the radial factor lies at $E^x(x),E^\phi(x)$02 or its notation-equivalent $E^x(x),E^\phi(x)$03, inside the event horizon in the exterior chart (Alonso-Bardaji et al., 2022, Moreira et al., 2023, Lütfüoğlu et al., 27 May 2026). The Hawking temperature is reduced. In the notation

$E^x(x),E^\phi(x)$04

the static line element can be written as

$E^x(x),E^\phi(x)$05

with surface gravity

$E^x(x),E^\phi(x)$06

The same reduction appears in the covariant model as

$E^x(x),E^\phi(x)$07

Thus the exterior lapse stays classical, but the surface gravity and Hawking temperature decrease monotonically as the holonomy scale grows (Alonso-Bardaji et al., 2022, Lütfüoğlu et al., 27 May 2026).

A common misconception is that “holonomy-corrected Schwarzschild” always denotes a unique static metric. The literature does not support that simplification. In (Tibrewala, 2012), holonomy corrections implemented directly at the level of point holonomies yield no nontrivial static vacuum solution and do not lead to a robust covariant metric. In (Alonso-Bardaji et al., 2021) and (Alonso-Bardaji et al., 2022), the anomaly-free canonical transformation and regularized constraint recombination instead produce a static, covariant, nonsingular exterior geometry. The two constructions address different effective realizations of holonomy corrections.

5. Perturbations, quasinormal modes, greybody factors, and Hawking radiation

For the static covariant metric, minimally coupled perturbations satisfy Schrödinger-like wave equations in the tortoise coordinate

$E^x(x),E^\phi(x)$08

or, in the dimensionless notation of (Lütfüoğlu et al., 27 May 2026),

$E^x(x),E^\phi(x)$09

For scalar perturbations, the effective potential is

$E^x(x),E^\phi(x)$10

The scalar quasinormal spectrum has several distinctive features. As $E^x(x),E^\phi(x)$11 increases, the damping generally decreases; for $E^x(x),E^\phi(x)$12 and $E^x(x),E^\phi(x)$13, the frequencies trace self-intersecting spirals in the complex plane and accumulate toward an extremal value as $E^x(x),E^\phi(x)$14. The paper also reports an oscillatory pattern in $E^x(x),E^\phi(x)$15 across overtones, with near-purely damped modes appearing in the numerical spectrum, while the late-time tail remains

$E^x(x),E^\phi(x)$16

unchanged from Schwarzschild (Moreira et al., 2023).

Axial gravitational perturbations have likewise been derived for the same static geometry. The master equation uses the effective Regge–Wheeler-type potential

$E^x(x),E^\phi(x)$17

with

$E^x(x),E^\phi(x)$18

In the reported $E^x(x),E^\phi(x)$19 modes, $E^x(x),E^\phi(x)$20 increases slightly with $E^x(x),E^\phi(x)$21, while $E^x(x),E^\phi(x)$22 decreases, so the ringdown becomes longer lived. The same study formulates a parameterized quasinormal-frequency expansion around Schwarzschild and concludes that

$E^x(x),E^\phi(x)$23

is a necessary condition for the parameterized approximation to remain valid (Yang et al., 2024).

Greybody factors and Hawking radiation exhibit marked spin dependence. For scalar, electromagnetic, and massless Dirac fields, the reduced radial equations take the form

$E^x(x),E^\phi(x)$24

The dominant scalar mode becomes more transparent as the holonomy parameter increases, the dominant electromagnetic mode shifts slightly upward in threshold, and the dominant Dirac mode is only mildly modified. The scalar absorption cross section preserves the universal low-frequency limit

$E^x(x),E^\phi(x)$25

whereas the electromagnetic cross section is mainly altered in the infrared and the Dirac cross section develops a strongly suppressed low-frequency tail. When thermal weighting is included, the temperature decrease dominates the emission: integrated power drops strongly with increasing $E^x(x),E^\phi(x)$26, the electromagnetic sector is most strongly quenched, and the fermionic sector becomes dominant in the Page-style aggregate once $E^x(x),E^\phi(x)$27 is appreciable (Lütfüoğlu et al., 27 May 2026).

6. Lensing, shadow, and observational signatures

The static holonomy-corrected Schwarzschild metric has been studied extensively through gravitational lensing. In the weak-field regime, one analysis writes the deflection angle as

$E^x(x),E^\phi(x)$28

for the metric with radial function

$E^x(x),E^\phi(x)$29

The leading Schwarzschild term is therefore augmented by an $E^x(x),E^\phi(x)$30 correction, plus mixed $E^x(x),E^\phi(x)$31 and pure $E^x(x),E^\phi(x)$32 terms (Soares et al., 2023). A separate treatment based on the polymerization parameter $E^x(x),E^\phi(x)$33 expands the weak-field deflection as

$E^x(x),E^\phi(x)$34

with

$E^x(x),E^\phi(x)$35

and higher coefficients given explicitly up to $E^x(x),E^\phi(x)$36. All $E^x(x),E^\phi(x)$37 increase with $E^x(x),E^\phi(x)$38, so the bending angle increases at fixed $E^x(x),E^\phi(x)$39 (Junior et al., 2023).

In the strong-deflection regime, the static metric still has

$E^x(x),E^\phi(x)$40

but the logarithmic coefficient is modified. In the $E^x(x),E^\phi(x)$41-notation,

$E^x(x),E^\phi(x)$42

and in the $E^x(x),E^\phi(x)$43-notation,

$E^x(x),E^\phi(x)$44

Since $E^x(x),E^\phi(x)$45 or $E^x(x),E^\phi(x)$46 for positive correction parameter, the logarithmic divergence is steeper than in Schwarzschild. The principal strong-field observables are

$E^x(x),E^\phi(x)$47

or equivalently $E^x(x),E^\phi(x)$48. Increasing the holonomy correction increases the angular separation $E^x(x),E^\phi(x)$49 and decreases the flux ratio, so the first relativistic image becomes less dominant relative to the others (Soares et al., 2023, Junior et al., 2023).

The weak-field Einstein-ring scale is also enlarged. In the $E^x(x),E^\phi(x)$50-parameterization,

$E^x(x),E^\phi(x)$51

while in the $E^x(x),E^\phi(x)$52-expansion the modified coefficients enter the image positions, magnifications, centroid, and time delay through the Virbhadra–Ellis lens equation and its perturbative solution (Soares et al., 2023, Junior et al., 2023). These analyses consistently report that the holonomy parameter increases the deflection angle, increases the separation between relativistic images, and decreases the brightness contrast of the first image.

Shadow phenomenology depends on which model is being used. In the static metric studied in lensing papers, the unchanged photon sphere implies an unchanged critical impact parameter, so the asymptotic strong-lensing accumulation angle $E^x(x),E^\phi(x)$53 is identical to Schwarzschild (Soares et al., 2023, Junior et al., 2023). In the non-rotating limit of the rotating holonomy-corrected black hole, the seed metric is

$E^x(x),E^\phi(x)$54

and again

$E^x(x),E^\phi(x)$55

so the shadow at $E^x(x),E^\phi(x)$56 is exactly Schwarzschild-sized in that family (Ali et al., 23 May 2026). Current Event Horizon Telescope constraints in that work therefore operate primarily through the rotating sector rather than through the non-rotating seed itself.

Taken together, the observational literature identifies a characteristic pattern for the static holonomy-corrected Schwarzschild geometry: the outer horizon and photon sphere remain at their Schwarzschild radii; the radial metric function carries the correction; weak and strong lensing are enhanced; the first relativistic image becomes less dominant; the Hawking temperature is lowered; scalar ringdown becomes less damped; and the global structure is replaced, in the covariant anomaly-free model, by a regular black-hole/white-hole spacetime with a minimal spacelike hypersurface (Alonso-Bardaji et al., 2022, Moreira et al., 2023, Lütfüoğlu et al., 27 May 2026).