REVIEW 3 major objections 3 minor
Study on deflection angle, shadow, quasinormal modes, and greybody factor of the black hole surrounded by quintessence in Rastall gravity
T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read A quintessence field around a charged Rastall-gravity black hole raises the weak-field deflection angle above the Schwarzschild and Reissner–Nordström values, and all results reduce to those limits when the quintessence parameter vanishes.
desk verdict Standard-application black-hole observables paper; direction of the quintessence effect is unverifiable from the abstract and may be sign-flipped. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine of the analysis is an assumed static, spherically symmetric, charged black hole solution of Rastall's field equations with a quintessence energy-momentum tensor as the source—this is the metric from which every observable is derived, although the abstract does not display it. Four standard tools are then applied to that metric: the Gauss–Bonnet theorem to integrate the photon trajectory in the weak-field regime, the null-geodesic condition to locate the photon sphere and hence the shadow radius, the eikonal correspondence that identifies the real part of quasinormal frequencies with the angular velocity of the unstable null circular geodesic and the imaginary part with its Lyapuno
What would settle it
Take the metric the authors assume and substitute it directly into Rastall's field equations with the quintessence stress-energy tensor; if the field equations are not satisfied identically, the observable predictions are void. A purely observational check would be a lensing or shadow measurement of a black hole with independently known mass and distance that finds a deflection angle smaller than the Schwarzschild value, which would contradict the claimed increase for positive Nq.
Extended reading notes
Core claim
The central claim is that the quintessence background in Rastall gravity does not merely decorate the spacetime; it changes measurable gravitational observables in a systematic, parameter-dependent way. Specifically, for the charged Rastall–quintessence black hole, the weak-field deflection angle computed by the Gauss–Bonnet method is larger than the corresponding angle for Schwarzschild or Reissner–Nordström black holes when the quintessence parameter Nq is positive, and it falls off with impact parameter b and charge Q while rising with mass m. The photon sphere and shadow radius shrink with Q and grow with m; the quasinormal mode frequency in the eikonal limit is linked to the coordinate-
Load-bearing premise
The entire set of predictions rests on the unstated premise that the charged, quintessence-surrounded metric used at the start is a genuine solution of Rastall's field equations; if that metric does not satisfy the field equations, every derived deflection, shadow, quasinormal-mode, and greybody result collapses.
Editorial extensions
If this is right
- Measurements of gravitational lensing around a black hole whose mass and distance are known could distinguish this Rastall–quintessence model from Schwarzschild or Reissner–Nordström, because positive Nq raises the deflection angle above the GR values.
- Shadow imaging would see a smaller shadow for larger charge and a larger shadow for larger mass; matching both trends fixes Q and m jointly.
- Ringdown observations could test the eikonal relation: the quasinormal frequency and damping time should be exactly the angular frequency and Lyapunov timescale of the null circular orbit.
- The greybody-factor bound controls how much Hawking radiation escapes, so the predicted decrease with Q and increase with m feeds into evaporation-rate estimates for this class of black holes.
- The recovery of Reissner–Nordström and Schwarzschild limits at Nq=0 and Nq=Q=0 means the model is calibrated against known spacetimes, making the new quintessence terms cleanly identifiable.
Reading between the lines
- If the four observables are measured independently for the same object, they overdetermine the three parameters (m, Q, Nq); a consistent inversion would strengthen the model, and any inconsistency would falsify the metric at the level of its observable predictions.
- The abstract does not give numerical magnitudes for the quintessence effect, so astrophysical detectability remains open; bounding Nq with current lensing or shadow data would settle whether the predicted boost in deflection is observable with existing instruments.
- The natural next step, left implicit here, is a rotating generalization: spin would break shadow circularity and make the quintessence signature even sharper than the static case treated in the paper.
- Because the whole chain hangs on a single assumed metric, an independent check that the metric satisfies the Rastall field equations rather than merely reproducing known limits is the cheapest way to validate the computation.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper investigates the weak-field deflection angle, photon shadow, quasinormal modes, and greybody factor for a charged black hole surrounded by quintessence in Rastall gravity. The abstract claims that the deflection angle decreases with impact parameter and charge, increases with mass, and that positive quintessence parameter Nq increases the deflection angle relative to Schwarzschild or Reissner-Nordström; that the shadow shrinks with charge and expands with mass; that quasinormal modes in the eikonal limit are expressible via null circular geodesics; and that the greybody bound decreases with charge and increases with mass. The abstract also states that all results reduce to known limits for Nq=0 and Nq=Q=0. This review is based solely on the abstract, as the full manuscript was not provided.
Significance. If the derivations are correct, the paper would provide a useful catalogue of observational signatures for a black hole spacetime in Rastall gravity with quintessence, using established methods (Gauss-Bonnet deflection, photon-sphere/eikonal correspondence, greybody bound). The explicit limit checks to Reissner-Nordström and Schwarzschild are a positive sign of internal consistency. However, because the full text is unavailable, the central claims cannot be verified, and one directional claim in the abstract appears sensitive to a metric sign convention that is not stated.
major comments (3)
- [Abstract (central claim)] The statement that 'positive Nq' leads to a higher deflection angle is not evaluable without specifying the metric sign and the equation-of-state parameter ω. In the standard Kiselev-like term g_tt = 1 - 2M/r + Q^2/r^2 - Nq/r^{3ω+1}, choosing ω = -2/3 gives a term -Nq r. A positive Nq then produces a repulsive contribution in the weak-field limit, which would decrease the deflection angle rather than increase it. If the metric instead uses +Nq r, the effective source has negative energy density and is not standard quintessence. The abstract should state the metric convention, the value of ω, and the relation of Nq to the Rastall coupling; otherwise the headline directional claim is ambiguous. Since the deflection angle sign propagates to the shadow and other observables, this is a load-bearing point.
- [Full text (not supplied)] The claimed reductions to Reissner-Nordström and Schwarzschild limits are plausible but cannot be checked. In particular, the assumed metric must be shown to solve the Rastall field equations with a properly defined quintessence energy-momentum tensor. The abstract does not explain how Nq arises from the Rastall coupling constants or how the quintessence background is constructed. Without this, all derived observables are conditional on an unverified background. The authors should provide the metric, the field equations, and the energy-momentum tensor in the manuscript.
- [Eikonal correspondence/greybody bound] The abstract states that the Lyapunov-exponent method gives the quasinormal modes in the eikonal limit and that a lower bound on the greybody factor is derived. In modified-gravity or non-Einstein settings, the usual null-geodesic correspondence can require verification; the abstract provides no details of the effective potential or the bounds used. This is not necessarily an error, but it is a necessary check that the paper should present explicitly.
minor comments (3)
- [Abstract notation] The symbol Nq is used without definition. If it is the quintessence state parameter, it should be named and its allowed range stated (e.g., physically customary ω values).
- [Abstract notation] The quantity '{\lambda}c' (coordinate time Lyapunov exponent) is introduced without a clear definition. Please define it and distinguish it from proper-time Lyapunov exponents.
- [Abstract wording] The phrase 'gradually increase with increasing black hole mass m' is qualitative. For a quantitative claim, specify the regime (e.g., weak-field, small Nq) and perhaps the leading-order term.
Circularity Check
No circularity found in the abstract-level derivation chain; the observables are computed from an assumed metric via standard formulas and only checked against known limits.
full rationale
This abstract-only review identifies no circular steps. The four observables (deflection angle, shadow, quasinormal modes, greybody factor) are computed from an assumed Rastall-quintessence metric using standard methods: the Gauss-Bonnet weak-deflection formula, photon-sphere/shadow relations, the eikonal Lyapunov/QNM correspondence, and a greybody-factor lower bound. The statements that results reduce to Reissner-Nordstrom for Nq=0 and to Schwarzschild for Nq=Q=0 are consistency checks, not fitted inputs. No parameter is fitted to the predicted quantities; no self-citation or uniqueness theorem is invoked; no quantity is defined in terms of another predicted quantity. The underlying metric assumption and the sign convention for Nq are physical-validity concerns rather than circularity, because the paper does not derive the metric from the quantities it later predicts. Accordingly, no specific circular step can be quoted or exhibited, and the appropriate score is 0.
Assumptions & free parameters
free parameters (1)
- Nq (quintessence state parameter)
assumptions (4)
- domain assumption The metric/field configuration for a charged black hole surrounded by quintessence in Rastall gravity is the correct solution of the Rastall field equations.
- domain assumption The Gauss-Bonnet theorem is applicable to the optical metric for weak-field deflection angle calculation.
- domain assumption In the eikonal limit, quasinormal mode frequencies and instability timescales are determined by the unstable null circular geodesics via Lyapunov exponents.
- domain assumption The greybody factor lower bound is computed via the standard integral method for spherically symmetric black holes.
Cite this review
Pith. "Pith review of Study on deflection angle, shadow, quasinormal modes, and greybody factor of the black hole surrounded by quintessence in Rastall gravity." pith.science (2026). https://pith.science/paper/BDN46EWS
@misc{pith2026250809221,
author = {Pith},
title = {Pith review of: Study on deflection angle, shadow, quasinormal modes, and greybody factor of the black hole surrounded by quintessence in Rastall gravity},
year = {2026},
howpublished = {\url{https://pith.science/paper/BDN46EWS}},
note = {Machine review of arXiv:2508.09221}
}
read the original abstract
The present study focuses on investigating the deflection angle in the weak-field approximation, shadow, quasinormal modes using Lyapunov exponents, and lower bound of the greybody factor for a charged black hole surrounded by a quintessence field in Rastall gravity. The weak deflection angles are calculated using the Gauss-Bonnet method. They decrease with increasing impact parameter b and charge Q, but gradually increase with increasing black hole mass m. Notably, the presence of a surrounding quintessence field in Rastall gravity leads to a higher deflection angle compared to Schwarzschild or Reissner-Nordstrom black holes with positive Nq. The photon sphere and shadow of the black hole are analysed concerning the charge Q and mass m; they shrink as Q increases and expand with increasing m. We further analyse the quasinormal modes of the black hole, explicitly derive the coordinate time Lyapunov exponent {\lambda}c and the quasinormal frequency {\omega}. In the eikonal limit, the Lyapunov exponent ensures that the real and imaginary parts of the quasinormal modes can be expressed by the frequency and instability time scale of the unstable null circular geodesics. Additionally, we derive the lower bounds of the greybody factor Gb; it decreases for increasing charge Q, while the increasing mass m enhances it. Importantly, all the findings reduce to those of the Reissner-Nordstrom black hole for Nq=0 and to the Schwarzschild black hole for Nq=Q=0.
Reviewed August 5, 2026 · model on record in the stance chip above.
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