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REVIEW 1 major objections 6 minor 101 references

Observable thin accretion disk around a self-dual black hole in loop quantum gravity

T0 review · 1 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Orbital data limit the LQG quantum parameter P to ≤4.3×10⁻⁵, and the resulting self-dual black hole would appear as a slightly smaller, brighter accretion disk than Schwarzschild.

desk verdict Routine but competent application of standard disk imaging to the self-dual LQG metric; the constraints reproduce known bounds, and the headline observable differences are illustrated at P values the paper's own analysis excludes. read the letter →

arxiv 2509.10953 v1 pith:YJWIUA4J submitted 2025-09-13 gr-qc

Tursunali Xamidov , Javokhir Sharipov , Sanjar Shaymatov This is my paper
classification gr-qc
keywords loopquantumgravityself-dualblackholepolymericfunctionaccretiondiskphotonsphereperihelionshiftS2starNovikov-Thornemodel
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 argues that the self-dual black hole of loop quantum gravity leaves observable imprints on orbital motion and accretion-disk images. Using Mercury's perihelion shift and the S2 star's orbit around Sgr A*, it pins down the polymeric parameter P to P≤0.000043 and P≤0.067419. It then shows that for larger P the photon sphere shrinks, light is deflected less, and the thin accretion disk appears smaller and brighter than in Schwarzschild. A distant observer would see roughly 25% more flux from a P=0.1 self-dual BH than from a Schwarzschild BH at an 85° inclination. The paper's point is that loop quantum gravity effects are not necessarily hidden.

What carries the argument

The central object is the self-dual spacetime metric of LQG, a quantum-corrected Schwarzschild geometry expressed in terms of the polymeric function P = (√(1+ε²)−1)/(√(1+ε²)+1) with ε = γδ, where γ is the Barbero-Immirzi parameter and δ the LQG polymeric parameter. The argument proceeds from the geodesic equations of this metric: a linearized perihelion-shift formula yields the P bounds, and numerical ray tracing of the null geodesic equation for the impact parameter b, combined with the Novikov-Thorne radiation flux formula, produces the predicted disk images and fluxes.

What would settle it

Measure Sgr A*'s photon-ring diameter to about 8% precision: the self-dual BH with P=0.03 predicts a critical impact parameter of 4.803 versus 5.196 for Schwarzschild, a 7.6% smaller ring; a ring diameter consistent with Schwarzschild at that precision would falsify the large-P prediction of this µ0-scheme metric.

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Extended reading notes

Core claim

For the self-dual LQG black hole metric obtained in the µ0-scheme, the polymeric function P controls the deviation from Schwarzschild: it moves the horizons, shrinks the photon sphere (the critical impact parameter falls from 5.196 to 4.803 as P goes from 0 to 0.03), and weakens the gravitational deflection of light. The paper derives the P-dependence of the perihelion precession, uses Mercury and S2 data to bound P, and shows via Novikov-Thorne modeling that the accretion disk around the self-dual BH is smaller and brighter, with slightly smaller redshifts, than around Schwarzschild.

Load-bearing premise

The predictions rest on the assumption that the LQG-corrected Schwarzschild spacetime is the one obtained by fixing the polymer parameters δ_b and δ_c as constants (the µ0-scheme) and by neglecting the minimal-area term a0; if loop quantum gravity instead picks a different quantization scheme, the effective metric, the parameter P, and all derived signals change.

Editorial extensions

If this is right

  • Mercury data constrain P to ≤4.3×10⁻⁵ and S2 to ≤0.067, forcing LQG corrections to be tiny at solar-system scales.
  • For larger P, the photon sphere and shadow shrink: the critical impact parameter decreases from 5.196 for Schwarzschild to 4.803 for P=0.03.
  • Both direct and secondary disk images shrink as P increases, with secondary images shrinking slightly faster than direct ones.
  • The observed flux brightens: at 85° inclination, the self-dual BH with P=0.1 is about 25% brighter than Schwarzschild.
  • The redshift is slightly weaker: z_max is about 0.95 for P=0.1 versus 1.12 for Schwarzschild at the same inclination.

Reading between the lines

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

  • If future high-resolution observations measure the Sgr A* ring diameter to about 8% precision, they could directly test the µ0-scheme prediction without relying on orbital dynamics.
  • The paper's disk images at P=0.05–0.1 use values far above the Mercury bound (P≤4.3×10⁻⁵), so the realistic brightening at currently allowed P is likely much smaller unless alternative LQG schemes permit larger strong-field deviations.
  • Because the metric reduces to Schwarzschild when a0=0 and P=0, the predictions form a one-parameter family; measuring both the shadow size and the disk flux could break degeneracies with spin in rotating generalizations.
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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

1 major / 6 minor

Summary. The paper studies timelike and null geodesics in the self-dual (quantum-corrected Schwarzschild) black hole spacetime of loop quantum gravity in the mu0-scheme. It derives constraints on the polymeric function P from the Mercury perihelion shift and the S2-star orbit around Sgr A*, finding P<=4.3e-5 and P<=6.74e-2, respectively. It then uses the Novikov-Thorne thin-disk model to compute direct and secondary images, the observed energy flux, and the redshift distribution for P=0, 0.05, and 0.1 at inclinations 17, 53, and 85 degrees. The central claim is that, compared with Schwarzschild, the self-dual black hole appears smaller and brighter and that these differences may serve as observational signatures of LQG.

Significance. If the adopted metric and the computations are taken at face value, the paper provides a concrete set of predictions for one specific LQG-inspired regular black hole model. The perihelion-shift and S2 constraints are derived from independent astronomical data in a transparent manner, and no circularity is present in the constraint-to-prediction logic: the disk images are obtained from the input metric, not fitted to the data. The main weakness is that the parameter values used in the central observational comparison (P=0.05 and, especially, P=0.1) are excluded by the paper's own Mercury and S2 bounds, so the headline 'smaller and brighter' claim is not supported within the allowed parameter region.

major comments (1)
  1. [Sec. IV; Fig. 5; Sec. V] The conclusion that these distinctions 'may provide new insights... in future observations' is an overreach given that no estimate is provided of the magnitude of the effect at the allowed P<=4.3e-5. The only allowed-P calculation shown is the perihelion constraint itself; the disk images use excluded values. A quantitative statement about detectability, or at least about the trend as P approaches the allowed upper bound, is needed before the observational-signature claim can be assessed.
minor comments (6)
  1. [Fig. 6 caption] The caption says 'From top to bottom, the columns represent inclination angles' and 'from left to right, the rows correspond to P values'; the words 'columns' and 'rows' appear to be swapped.
  2. [Eqs. (25) and (28)] The bounds P<=0.000043 and P<=0.067419 are quoted without specifying the confidence level or the propagation of the observational uncertainties; please state whether these are 1-sigma, 2-sigma, or worst-case limits.
  3. [Sec. II, Eq. (6)] The notation is confusing: P is called the 'polymeric function' but is defined in terms of epsilon=delta*gamma, while delta is called the 'polymeric parameter'. Clarify the relation between P, delta, and the quantities fixed in the mu0-scheme.
  4. [Sec. II, after Eq. (5)] There is a typo: 'Planck length l_P l' should read 'Planck length l_P' or similar. Also, the sentence about a0=0 is an important approximation and should be stated as an explicit assumption in the conclusions.
  5. [Sec. II.B, Eqs. (23) and (26)] The units of the perihelion shift are not uniform: Eq. (23) is in rad/revolution, while Eq. (26) is in arcsec/year. State both units explicitly to avoid confusion.
  6. [Sec. IV, Eq. (54)] The symbol b appears in the redshift factor without being redefined in this section; it is the impact parameter introduced in Sec. III, but this should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the disk images and fluxes are direct consequences of the input LQG metric, and the P constraints come from independent Mercury and S2 data.

full rationale

The paper's derivation chain is: start with a known effective LQG self-dual Schwarzschild metric (with free parameter P), compute timelike geodesics, fit/constrain P using independent observed perihelion shifts (Mercury and S2), then use the same metric to compute null geodesics, accretion disk images, redshift, and flux. No fitted parameter is renamed as a prediction: the disk calculations use hand-picked P values (0.05, 0.1) as an illustration, not as values inferred from disk observations. The constraints on P from Mercury and S2 are external empirical inputs, not outputs of the disk model. The self-citations (e.g., refs. [36], [70], [72], [77], [88], [99]) are used for standard geodesic/flux formulas or for observational data compilations and are not load-bearing; the core metric comes from Modesto [12], an external source. The use of P=0.05 and P=0.1 in the disk figures, despite the paper's own Mercury bound P<=4.3e-5, is an internal consistency/overinterpretation concern, not circularity, because the disk observables are not used to infer or validate P. Therefore the central derivations are self-contained with respect to their inputs, and there is no circular step that reduces a prediction to an input by construction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The paper's central results rest on the LQG metric construction, the small-P expansion, the Novikov-Thorne disk model, and the adopted observational data. No machine-checked proofs or shipped code/data exist to verify any of these, so the ledger is dominated by domain assumptions from the LQG and accretion disk literature.

free parameters (4)
  • Polymeric function P = Constraints: P <= 4.3e-5 (Mercury), P <= 0.067 (S2); disk figures use P = 0, 0.01, 0.03, 0.05, 0.1
    P is the central model parameter; the disk image and flux comparisons are parameter scans over hand-chosen values, some outside the paper's own bounds.
  • Mass accretion rate Mdot0
    Flux F(r) in Eq. (49) scales linearly with Mdot0; the axis in Fig. 7 is in units of 1e-6 with Mdot0 not specified, so absolute flux is undetermined.
  • Observer inclination angles = 17 deg, 53 deg, 85 deg
    Chosen in Figs. 6, 8, 9 to illustrate viewing-angle dependence; not fitted to data.
  • Disk outer radius = R = 25 (also 10, 15, 20 in Fig. 5)
    The integration upper limit for disk images is chosen by hand; flux integrals extend from ISCO to R=25.
assumptions (7)
  • domain assumption The self-dual LQG metric (Eqs. 2-5), taken from Modesto 2010, is the correct effective description of a quantum-corrected Schwarzschild black hole.
    Invoked at the start of Sec. II as the spacetime under study; all results depend on this metric.
  • domain assumption The mu0-scheme fixes delta_b and delta_c as constants, with P defined by Eq. (6).
    Sec. II states this construction choice; if replaced by the mu-scheme, the metric changes.
  • domain assumption The Planck-area term a0 can be set to zero.
    Sec. II after Eq. (6): 'its contribution can be regarded as negligible as it is proportional to the Planck length. Consequently a0=0 can be set'.
  • standard math P is small enough to truncate the geodesic equation at linear order.
    Eq. (18): 'Since P is a small parameter ... we expand ... keep only the linear order'.
  • domain assumption The Novikov-Thorne model describes the accretion disk.
    Sec. IV: disk treated as optically thick, geometrically thin; Eq. (49) is the NT flux.
  • domain assumption The observed Mercury perihelion shift and S2-star precession values are accurate and applicable.
    Eqs. (24) and (27) adopt values from Refs. [85-89].
  • standard math The Luminet redshift formula (54) correctly maps emitted to observed flux.
    Sec. IV, Eq. (54), following Ref. [101].

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Pith. "Pith review of Observable thin accretion disk around a self-dual black hole in loop quantum gravity." pith.science (2026). https://pith.science/paper/YJWIUA4J

@misc{pith2026250910953,
  author       = {Pith},
  title        = {Pith review of: Observable thin accretion disk around a self-dual black hole in loop quantum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJWIUA4J}},
  note         = {Machine review of arXiv:2509.10953}
}
abstract

In this paper, we study a self-dual black hole (BH) in Loop Quantum Gravity (LQG), analyzing both timelike and null geodesics. Using observational data from Mercury's perihelion shift and the orbit of the S2 star around Sagittarius A$^{\star}$ (Sgr A$^{\star}$), we derive constraints on the polymeric function $P$. We further investigate photon trajectories near the self-dual BH under various scenarios to explore their observational relevance. Finally, we examine the properties of accretion disks around the self-dual BH in LQG, including their direct and secondary images, and study the redshift and the observed energy flux distribution across the accretion disk as measured by distant observers for different inclination angles. Our findings provide new insights into the physical nature and accretion properties of self-dual BHs in LQG and their possible observational consequences.

Figures

Figures reproduced from arXiv: 2509.10953 by the authors.

Figure 1
Figure 1. FIG. 1. The trajectories of photons are shown in separate plots for different values of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The trajectories of photons as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The plot shows the number of photon orbits [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic representation of the coordinate system used to construct the accretion disk image. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The image formation diagram for various values of the quantum correction parameter [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Direct and secondary image of the thin accretion disk around a self-dual BH in LQG. From top to bottom, the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The energy flux [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The distribution of the observed flux [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The distribution of the redshift factor [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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