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Fingerprints of Loop Quantum Gravity Black Holes with Quintessence Field

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that combining loop-quantum-gravity corrections with a quintessence term in a single metric function produces a triple-horizon black hole with parameter-dependent shadows, ringing, and lensing.

desk verdict A workmanlike phenomenological study of a new LQG+quintessence toy metric, undermined by an unproven ansatz and internal contradictions in the QNM and deflection sections. read the letter →

arxiv 2505.12291 v1 pith:CFQS5N5T submitted 2025-05-18 gr-qc hep-th

Ahmad Al-Badawi , Faizuddin Ahmed , İzzet Sakall{i} This is my paper · ORCID
classification gr-qchep-th
keywords loopquantumgravityquintessencetriplehorizonblackholeshadowquasinormalmodesgravitationaldeflectionangleGauss-Bonnettheoremphotonsphere
topics Quantum Gravity
open problems Quantum Gravity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that a static, spherically symmetric black hole whose metric function combines a loop-quantum-gravity correction with a quintessence term can have three horizons for a specific parameter choice, and that this spacetime leaves measurable fingerprints. For $M=1$, $w=-2/3$, $c=0.06$, $\alpha=10^{-77}$, and $B=6\times 10^{38}$, the zeros of the metric function lie at approximately $r=0.66$, $1.58$, and $14.43$. The paper then traces these parameters through photon spheres, shadow radii, quasinormal-mode frequencies, and gravitational deflection angles, concluding that the combined model is stable to scalar and electromagnetic perturbations and that its lensing signature separates into classical, quintessence, and quantum contributions at different impact parameters.

What carries the argument

The load-bearing object is the single metric function $f(r)$ in Eq. (4), formed by adding the quintessence term $-c/r^{3w+1}$ to a loop-quantum-gravity-corrected Schwarzschild metric whose correction is $\frac{\alpha}{r^2}\left(\frac{B}{2}+\frac{M}{r}\right)^2$. Every subsequent result is read off from this function: horizons are its zeros, the photon sphere follows from the condition $r f'(r)=2 f(r)$, the scalar and electromagnetic perturbation potentials are built from $f(r)$ and its derivative, and the deflection angle comes from expanding the Gaussian optical curvature of $f(r)$ in powers of $1/r$. The WKB approximation and the Gauss-Bonnet method are the tools that convert $f(r)$ into quasinormal frequencies and a closed-form deflection angle.

What would settle it

A decisive calculation would be to write down an explicit action or effective field-equation system for loop quantum gravity with quintessence as a source and check whether Eq. (4) solves it. Observationally, the predicted monotonic shadow behavior (smaller with larger $\alpha$ or $B$, larger with larger $c$) can be checked against high-resolution black-hole shadow measurements; a measured shadow that moves opposite to these trends for fixed $M$ would rule out the parameter dependence.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the metric function $f(r)=1-\frac{2M}{r}-\frac{c}{r^{3w+1}}+\frac{\alpha}{r^2}\left(\frac{B}{2}+\frac{M}{r}\right)^2$ defines the spacetime of a loop-quantum-gravity black hole surrounded by a quintessence field. The main discovery emphasized is the triple-horizon configuration: with $w=-2/3$, $c=0.06$, $\alpha=10^{-77}$, $B=6\times 10^{38}$, and $M=1$, the function $f(r)$ vanishes at three radii, producing an inner horizon, a black-hole horizon, and a distant cosmological-like horizon, with a region between the second and third horizons where the radial coordinate behaves as time. The shadow radius follows from the photon-sphere condition and decreases with $\alpha$ and $B$ while increasing with $c$. The quasinormal-mode analysis reports negative imaginary frequencies, indicating stability within the parameter ranges studied, with the real frequency rising as $\alpha$ grows and falling as $c$ grows. The deflection angle derived by the Gauss-Bonnet method is $\hat{\alpha}_{\rm def}\simeq \frac{4M}{b}+\frac{c\pi(3w+2)}{2b^{3w+1}}+\frac{\pi\alpha B^2}{4b^2}+\frac{2\pi\alpha B M}{b^3}+\frac{15\pi\alpha M^2}{4b^4}$, displaying separated classical, quintessence, and quantum length scales.

Load-bearing premise

The load-bearing premise is that the loop-quantum-gravity correction and the quintessence term can simply be added inside a single metric function, without showing that the sum solves the field equations of a theory containing both ingredients; if that addition is not physically valid, the subsequent predictions describe a metric that no known theory generates.

Editorial extensions

If this is right

  • For the triple-horizon parameters, the spacetime contains a finite region between the second and third horizons where $f(r)>0$ and the radial coordinate behaves like time, so the causal structure differs from that of a Schwarzschild black hole.
  • If the metric is taken as physical, the shadow radius depends monotonically on the parameters: it decreases with $\alpha$ and $B$ and increases with the quintessence normalization $c$, offering a route to constrain both quantum and dark-energy parameters from a single black-hole image.
  • The quasinormal spectrum is stable for the parameter ranges checked, and the oscillation frequency and damping rate move in opposite directions with $\alpha$ and $c$, so ring-down observations could distinguish loop-quantum-gravity effects from quintessence effects.
  • Weak-field gravitational lensing separates into a classical $1/b$ term, a quintessence term with state-parameter-dependent falloff, and quantum terms starting at $1/b^2$, meaning measurements at different impact parameters probe different physics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension would be to construct a rotating version of the metric; the shadow would become non-circular, and the triple-horizon structure might survive only for a restricted range of spin and inclination.
  • The hierarchical deflection formula suggests a practical lensing test: measure the angle at two or three widely separated impact parameters and check whether the residuals follow the $b^{-(3w+1)}$, $b^{-2}$, $b^{-3}$ pattern rather than a single power law.
  • Deriving Eq. (4) from an explicit action or effective field equations would settle whether the triple-horizon spacetime is a genuine solution of the combined theory; this is the immediate open problem raised by the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a static, spherically symmetric black-hole metric that adds Kiselev's quintessence term to the Ashtekar-Olmedo-Singh loop-quantum-gravity effective metric, f(r)=1-2M/r-c/r^(3w+1)+(alpha/r^2)(B/2+M/r)^2. Using this metric as the sole input, the authors compute horizon structures (Table 1), embedding diagrams, null and timelike geodesics, photon-sphere and shadow radii (Table 2 and Figures 8-10), scalar and electromagnetic quasinormal modes via sixth-order WKB (Tables 3-5 and Figures 11-14), and gravitational deflection angles via the Gauss-Bonnet method (Section 6). The headline result is a triple-horizon configuration for w=-2/3, c=0.06, alpha=1e-77, B=6e38, M=1, with horizons near r=0.66, 1.58, and 14.43. The paper also claims distinct, potentially observable parameter dependences for shadows, QNM frequencies, and deflection angles.

Significance. If the central metric were a genuine solution, the paper would provide a useful phenomenological catalogue of how LQG and quintessence corrections modify black-hole observables, and some of the basic algebra is internally consistent: the horizon roots in Table 1 are reproducible, the photon-sphere condition in Eq. (45) follows from Eq. (19), and the null trajectory equation (25) is correctly derived. The paper also computes its results directly from the stated metric and does not fit data, so there is no circularity of the 'definition in terms of fitted parameters' kind. However, the central metric is introduced without any derivation from field equations, and several independent internal contradictions affect the main claims. The QNM tables contradict their own figures on the sign of the parameter trends, and the deflection-angle formula is not valid for the w=-2/3 case that the paper emphasizes. These problems are load-bearing: the 'fingerprints' are not reliable as presented, and the manuscript currently functions as a collection of calculations on an unvalidated ansatz rather than a physical model with robust predictions.

major comments (4)
  1. [Section 2, Eq. (4)] The central metric is introduced by declaration as an additive superposition of the Kiselev quintessence solution and the AOS LQG effective metric. No field equations, effective action, or LQG effective dynamics with a matter source are provided to show that Eq. (4) is a solution of any coupled system; the FLRW junction conditions in Eqs. (5)-(6) are given for the pure LQG metric and are not extended to include the quintessence source. Since Eq. (4) is the sole input for the horizon, geodesic, shadow, QNM, and deflection results in Sections 3-6, the physical status of every subsequent claim is undetermined. This is load-bearing and requires a derivation of the superposition, or a substantial reframing of the work as a phenomenological study of an ansatz metric rather than an LQG black hole surrounded by quintessence.
  2. [Section 5.1, Tables 3-5 and Figures 12-14] The numerical QNM tables and their corresponding figures directly contradict each other on the sign of the parameter trends. Table 3 lists Re(omega) decreasing from 0.493776 to 0.278834 as alpha increases from 0 to 0.08, while Figure 12 states and plots that Re(omega) increases with alpha. Table 4 lists Re(omega) increasing from 0.436613 to 0.457135 as c increases from 0 to 0.8, while Figure 13 states and plots that Re(omega) decreases with c. These sign inconsistencies concern exactly the 'distinctive spectral characteristics' advertised in the abstract, so the QNM fingerprint claim is unreliable as presented.
  3. [Section 6, Eqs. (74) and (77)] The Gauss-Bonnet deflection derivation contains internal inconsistencies. Equation (74) states alpha_def = - integral K dS - pi; substituting the Schwarzschild K=2M/r^3 into this expression gives approximately -4M/b - pi, not the 4M/b quoted in Eq. (77), so the extra -pi is silently dropped. More importantly, for w=-2/3, the value used throughout Section 6 and in Figures 8-15, the quintessence coefficient in Eq. (77), c pi (3w+2)/(2 b^(3w+1)), vanishes identically, while the optical-curvature contribution from quintessence in Eq. (70) does not converge because f(r) -> 1 - c r at infinity. The spacetime is not asymptotically flat for w=-2/3, so the boundary at infinity used in Eq. (71) and the deflection formula Eq. (77) are not defined for this case, contrary to the claims in Figure 15.
  4. [Section 2, Table 1, and Section 7] The text states that for alpha=0 the black hole maintains a single horizon regardless of the value of w, but Table 1 contains the row w=-2/3, c=0.06, alpha=0 with two horizons at r=2.324 and r=14.343. The concluding section repeats the same false statement, saying that classical black holes (alpha=0) maintain a single horizon regardless of quintessence parameters. This undermines the paper's framing of the triple-horizon structure as the novel feature unique to the combined LQG-plus-quintessence model, since the outer horizon already exists for quintessence alone in that parameter row.
minor comments (4)
  1. [Section 5] The sentence 'From expression given in Eq. (57), it becomes evident that the perturbative potential is influenced...' should refer to Eq. (56), not Eq. (57), since Eq. (57) is the specialized w=-2/3 version introduced later.
  2. [Section 3] The text near Eq. (12) mentions 'the parameter xi' and calls alpha a 'cosmic string parameter', but no xi is ever defined and alpha is elsewhere described as the LQG Planck-length parameter. This is confusing and should be corrected.
  3. [Figures 3-7 and 11-14] The parameter values used in most plots (alpha ~ 0.01-1, B ~ 0.1-2) differ by dozens of orders of magnitude from the physical values used in Table 1 and Section 2 (alpha ~ 1e-77, B ~ 6e38). The paper does not explain this rescaling or state that the plots are in arbitrary units, making it impossible to connect the qualitative trends to the claimed astrophysical regime.
  4. [Throughout] There are several typographical and wording issues, including 'contrvariant' in Section 5, 'served in Table 1' in the Figure 1 caption, and 'w. r. t.' for 'with respect to'. These do not affect the physics but should be cleaned up in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's results are direct consequences of the stated metric Eq. (4); the unproven superposition ansatz is a physical-validity concern, not a self-referential reduction.

full rationale

The derivation chain in this paper is: adopt the metric function f(r) in Eq. (4), which additively combines the Kiselev quintessence term with the AOS loop-quantum-gravity correction; then solve f(r)=0 for horizons, compute photon spheres and shadows, evaluate scalar and electromagnetic QNMs from the standard effective potential, and apply the Gauss-Bonnet method to the optical metric. Every reported quantity is a direct mathematical consequence of the explicitly stated metric, with numerical parameters supplied a priori (M=1, B=6e38, alpha=1e-77, c=0.06, w=-2/3). No parameter is fitted to a subset of data and then relabeled as a prediction; no derived quantity is defined in terms of the output it is said to predict; and no uniqueness theorem or prior result by the same authors is invoked to force the central choice. The self-citations (e.g., Refs. [43-49,55]) are used for standard Lagrangian, perturbation, and deflection-angle techniques and are not load-bearing. The LQG metric [39] and Kiselev solution [38] are external citations, not self-citations. The serious scientific concern is that Eq. (4) is introduced by declaration rather than derived from coupled field equations, so the physical status of the combined spacetime is undetermined. But an unsupported ansatz is a correctness or model-validity issue, not circularity under the definitions used here: there is no quoted equation that reduces to its own input by construction, no fitted parameter renamed as a prediction, and no load-bearing self-citation chain. A non-finding on circularity is therefore the appropriate outcome.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central calculation rests on an unproven ansatz metric built by adding two known solutions, plus standard approximation methods. No new particles or fields are introduced, and no data are fitted.

free parameters (4)
  • alpha (LQG parameter) = varied from 1e-77 (Table 1) to 0.8 (Figures 3-5)
    Chosen by hand in the plots; the LQG value from Eq. (7) is about 1e-77 for M=1, but most phenomenology uses alpha values orders of magnitude larger.
  • B (LQG coupling parameter) = 6e38 in horizon/shadow sections; 0.1 to 2 in geodesic and QNM plots
    Chosen by hand; the large value is motivated by junction conditions, but the small values used in later sections are not tied to that motivation.
  • c (quintessence normalization) = 0.06 in Table 1; 0.01 to 0.8 in later sections
    Chosen by hand to illustrate effects and to produce the triple-horizon case.
  • w (quintessence state parameter) = -2/3 for most calculations, with 0 and -1/3 also used
    The triple-horizon result relies on the choice w=-2/3, which makes the quintessence term behave like a long-range -c r contribution.
assumptions (3)
  • domain assumption The AOS LQG effective metric correctly describes quantum-corrected black holes.
    Eq. (4) is built from the AOS metric cited in Ref. [39]; the paper does not re-derive or justify this starting point.
  • ad hoc to paper Quintessence can be added linearly to the LQG metric function.
    No field equations, action, or junction conditions are shown that would make the additive combination in Eq. (4) a solution; this is the central unproved step.
  • domain assumption Sixth-order WKB approximation is valid for the quasinormal mode calculations.
    The WKB method is used for l=2, n=0 with no accuracy check or comparison to other methods, and the spacetime may require cosmological-horizon boundary conditions that WKB does not automatically handle.

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Pith. "Pith review of Fingerprints of Loop Quantum Gravity Black Holes with Quintessence Field." pith.science (2026). https://pith.science/paper/CFQS5N5T

@misc{pith2026250512291,
  author       = {Pith},
  title        = {Pith review of: Fingerprints of Loop Quantum Gravity Black Holes with Quintessence Field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFQS5N5T}},
  note         = {Machine review of arXiv:2505.12291}
}
abstract

In this study, we investigate a static, spherically symmetric black hole (BH) within the framework of Loop Quantum Gravity (LQG) surrounded by quintessence field. Our comprehensive analysis shows that the interplay between quantum corrections and exotic matter produces unique spacetime features, most notably a triple-horizon structure for specific parameter combinations. We derive the metric function incorporating both LQG parameters ($\alpha$, $B$) and quintessence parameters ($c$, $w$), analyzing its implications for horizon structure through embedding diagrams. We examine null and timelike geodesics, calculating photon spheres, effective potentials, and orbital dynamics. Our study demonstrates how quantum and quintessence parameters affect BH shadow size and shape, offering potential observational signatures. Through scalar perturbation analysis, we compute quasinormal modes (QNMs) frequencies, confirming the stability of these hybrid BHs while identifying distinctive spectral characteristics. Finally, using the Gauss-Bonnet (GB) theorem modified approach, we derive an analytical expression for gravitational deflection angles, showing a hierarchical structure of contributions from classical, quintessence, and quantum effects at different distance scales.

Figures

Figures reproduced from arXiv: 2505.12291 by the authors.

Figure 1
Figure 1. Embedding diagrams of the LQG BH for various parameter values of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Plot of f(r) curve where parameters are set as M = 1, c = 0.06, α = 10−77 and B = 6 × 1038. See also [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The behavior of the effective potential for null geodesics as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The behavior of the force acting on photon particles as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The behavior of the Lyapunov exponent κ = [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: The behavior of the effective potential for time-like geodesics as a function of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The behavior of the specific angular momentum (L) and energy (E) of time-like particles for different [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Variation of photon sphere radius for various values of BH parameters [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Variation of shadow radius for various values of BH parameters [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: shows how the shadow varies with the BH parameters. It shows that the presence of quintessence causes the shadow size to increase while decreasing with the parameters α and B. -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 X Y α=10-77 α=2×10-77 α=3×10-77 -6 -4 -2 0 2 4 6 -6 -4 -2 …
Figure 11
Figure 11. Figure 11: The behavior of the scalar perturbative potential as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Variation of amplitude and damping of QNMs with respect to the Planck length parameter for scalar [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Variation of amplitude and damping of QNMs with respect to the normalization QF constant for [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Variation of amplitude and damping of QNMs with respect to the coupling parameter for scalar and [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]
Figure 15
Figure 15. Figure 15: Deflection angle ˆαdef as a function of impact parameter b for LQG BHs surrounded by QF. The main plot shows the behavior for a wide range of impact parameters (10 ≤ b ≤ 50), while the inset provides a magnified view of the deflection angle at large impact parameters …

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.