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REVIEW 4 major objections 4 minor 76 references

Polymer Black Hole Surrounded by Quintessence

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proposes that a polymer (loop-quantum-gravity) black hole surrounded by a quintessence fluid reproduces the observed Sgr A* shadow for a wide range of parameters, and derives the accompanying thermodynamic and photon-emission…

desk verdict New metric combination, but it is an unverified ansatz and the EHT match is a three-parameter scan; worth reviewing, not accepting as is. read the letter →

arxiv 2507.07372 v1 pith:K4QQ4FGC submitted 2025-07-10 gr-qc

M. C. Araújo , J. G. Lima , C. R. Muniz , J. Furtado This is my paper
classification gr-qc MSC 83C5783C45
keywords polymerblackholequintessenceloopquantumgravitythermodynamicsHawkingtemperaturegreybodyfactorshadowSgrA*
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 a polymer black hole, an effective loop-quantum-gravity deformation of Schwarzschild, can be put in a quintessence background by adding the term $-c/r^{3\omega+1}$ to its lapse function, following Kiselev's approach. With that metric ansatz, it derives the Hawking temperature, entropy, heat capacity, lower bounds on electromagnetic greybody factors, photon emission rates, light-like geodesics, and black hole shadows. Its central result is that for a wide range of the couplings $c$, $\lambda$, and the equation-of-state parameter $\omega$, the shadow radius falls inside the EHT range $4.55 \leq r_{\rm sh}/M \leq 5.22$ for Sgr A*, so a non-spinning polymer black hole with quintessence can match the observed shadow size. The same analysis shows that both quintessence and polymer effects cool the black hole, suppress emission, and shift the spectral peak to lower energies, while the heat capacity changes sign at a maximum of the Hawking temperature.

What carries the argument

The central object is the lapse function of Eq. (12), $\tilde A(r) = \frac{4(\lambda M)^{2/3} - 2M\sqrt{4(\lambda M)^{2/3} + r^2} + r^2}{(\lambda M)^{2/3}+r^2} - \frac{c}{r^{3\omega+1}}$, combined with the polymer radial function $B(r)=r^2+M^2(\lambda/M^2)^{2/3}$. The first term is the polymer black hole lapse from loop-quantum-gravity polymerization; the second is the quintessence term added by hand following Kiselev's construction. Every derived quantity—horizons, Kretschmann scalar, Hawking temperature, entropy, heat capacity, effective potential for null geodesics, greybody factor bound, and shadow radius—is computed from this pair of functions.

What would settle it

Compute the Einstein tensor of the metric (5) with (12) and check whether the energy-momentum components satisfy the Kiselev relation $T^\phi_\phi = T^\theta_\theta = -\frac{1}{2}(3\omega+1)T^r_r = -\frac{1}{2}(3\omega+1)T^t_t$; if the relation fails, the spacetime does not describe a polymer black hole surrounded by quintessence in the sense assumed, and all derived thermodynamic, emission, and shadow results inherit the failure.

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Extended reading notes

Core claim

On the paper's own terms, the line element (5) with the modified lapse (12), namely $\tilde A(r) = \frac{4(\lambda M)^{2/3} - 2M\sqrt{4(\lambda M)^{2/3}+r^2}+r^2}{(\lambda M)^{2/3}+r^2} - \frac{c}{r^{3\omega+1}}$ together with the polymer sphere radius $B(r)=r^2+M^2(\lambda/M^2)^{2/3}$, is a valid static, spherically symmetric spacetime describing a polymer black hole surrounded by quintessence. This spacetime has an event horizon and a cosmological horizon, and the interplay of the polymer parameter $\lambda$ and the quintessence coupling $c$ controls the horizon radii, the temperature maximum, the entropy range, and the photon-ring and shadow radii. The quantitative claim is that the shadow radius grows with $c$, shrinks with $A_\lambda$, and can be tuned into the Sgr A* window; the paper states that spin and inclination affect shadow shape and size only weakly, so the axial, non-spinning computation is a fair comparison with EHT data. A direct corollary of the paper's computation is that both $c$ and $\lambda$ reduce the Hawking temperature and the photon emission rate.

Load-bearing premise

The paper's results rest on treating the line element (12) as a physical spacetime obtained by adding $-c/r^{3\omega+1}$ to the polymer lapse function, without deriving this from a coupled action or verifying that the resulting stress-energy satisfies Kiselev's condition (11).

Editorial extensions

If this is right

  • If the ansatz is physical, a non-spinning polymer black hole in a quintessence background can account for the EHT Sgr A* shadow size, so the observation does not by itself require rotation for this class of models.
  • Because the shadow radius increases with the quintessence coupling $c$ and decreases with the polymer parameter $A_\lambda$, future EHT measurements can jointly constrain these two parameters.
  • The reduction of the Hawking temperature and emission rate by both $c$ and $\lambda$ implies that a polymer-quintessence black hole evaporates more slowly and appears dimmer and more redshifted than a Schwarzschild black hole of the same mass.
  • The sign change of the heat capacity at the temperature maximum indicates second-order phase transitions, which would mark a finite-size endpoint or change of regime in the evaporation history.

Reading between the lines

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

  • The EHT match is really a test of the combined ansatz (12), not of the polymer black hole alone; if the Einstein tensor of (12) does not satisfy the Kiselev condition (11), the shadow match would be an artifact of the added term.
  • The paper asserts, but does not demonstrate, that spin and inclination are negligible; a rotating polymer-quintessence solution could produce a shadow several percent larger or smaller, which would matter inside the narrow Sgr A* window.
  • The combined spacetime is singular at $r=0$ for $c\neq 0$ (the paper's Kretschmann expression diverges there), so the model cannot be read as a singularity-resolution statement even though the pure polymer black hole is regular.
  • The entropy defined by $S=\int dM/T_H$ deserves a consistency check against the first law, since the non-monotonic temperature makes the integral prescription potentially ambiguous for this metric.
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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 studies a static spherically symmetric line element obtained by adding a Kiselev quintessence term -c/r^{3\omega+1} to the lapse function of a polymer black hole while keeping the angular part B(r) unchanged. From this metric it computes thermodynamic quantities (Hawking temperature, entropy, heat capacity), lower bounds on electromagnetic greybody factors and photon emission rates, null geodesics and shadows, and compares shadow radii to the EHT Sgr A* range. The central claimed result is that a wide range of (c, A_lambda, omega) yields shadow radii matching Sgr A*.

Significance. The paper is broad and applies standard black-hole toolkit methods to an interesting combination of polymer quantum effects and dark energy. Its concrete assets include the explicit horizon condition (8), a long-form Kretschmann scalar in Sec. II, and many parameter studies in Figs. 3-10. If the metric (12) were a genuine solution of a coupled system, this would be a useful phenomenological survey. However, the metric is introduced as an ansatz and the Kiselev condition (11) is never checked; because B(r) is not r^2, the superposed line element is not guaranteed to have a quintessence energy-momentum tensor. The subsequent thermodynamics and shadows are standard calculations from that ansatz rather than independent tests, and the Sgr A* comparison is a consistency scan over three free parameters rather than a prediction. No machine-checked proofs or reproducibility package are included, and key formulas (T_H, shadow radius definition, entropy integral) are absent from the text.

major comments (4)
  1. [III, Eqs. (11)-(12)] The combined metric is introduced by adding -c/r^{3\omega+1} to the polymer lapse function, with no derivation from an action or field equations and no verification of the Kiselev condition (11). In the Kiselev solution the angular part is exactly r^2, whereas here B(r)=r^2+M^2(\lambda/M^2)^{2/3}. For this metric the Einstein tensor components G^r_r and G^t_t depend on B'(r) and B''(r) in a way that does not automatically enforce T^r_r=T^t_t or the full set of equalities in (11). The paper should either derive Eq. (12) from a coupled matter model or compute the Einstein tensor and report the exact energy-momentum tensor required for (11) to hold. Since the thermodynamic, greybody, and shadow results all inherit this ansatz, this missing check is load-bearing.
  2. [IV, Eqs. (14)-(15), Figs. 3-5] The Hawking temperature is only presented through plots; no expression for T_H in terms of \tilde A'(r_h) or the surface gravity is given. The spacetime is not asymptotically flat for \omega<-1/3, so the normalization of the timelike Killing vector and hence T_H is not fixed by flat infinity and must be stated. In addition, S is defined by S=\int dM/T_H, but the paper does not specify the integration path or whether M is solved from \tilde A(r_h)=0 for fixed c, \lambda, \omega. This makes the reported 'not well-defined' entropy and the heat-capacity phase transitions impossible to verify. Please provide explicit analytic or numerical definitions behind Figs. 3-5.
  3. [V, Eq. (20)] Equation (20), the lower bound on the electromagnetic greybody factor, does not follow from (17)-(18). With B(r) from (7), \int_{r_h}^\infty dr/B(r) = (1/a)[\pi/2 - \arctan(r_h/a)] with a=M(\lambda/M^2)^{1/3}, not the printed combination involving 1/(2\pi)\sqrt{1/B(r_h)} and \arctan(r_h\sqrt{B(r_h)})/\sqrt{B(r_h)}; as written the arctangent argument has dimensions of length squared. Moreover, for \omega<-1/3 the line element has a cosmological horizon, so the scattering problem should be formulated between the two horizons rather than from r_h to infinity. The greybody and emission-rate results need to be recomputed after correcting Eq. (20).
  4. [VI, Table I, Fig. 10] The shadow radius r_sh is reported in Table I and used in the Sgr A* comparison without any defining equation. For \omega<-1/3 the quintessence term makes \tilde A(r) grow with r, so the spacetime is not asymptotically flat and an observer at infinity is not available; the shadow radius must instead be defined from the photon-ring impact parameter at a specified observer radius between the event and cosmological horizons. The paper neither gives this formula nor quantifies the asserted weak dependence of the shadow on spin and inclination for this model. Consequently, the statement in Sec. VI that a wide range of (c, A_\lambda, \omega) matches the EHT range is a parameter scan rather than a test of the model. Please provide the shadow formula, the observer-location convention, and a justification for the M normalization used in Eq. (29).
minor comments (4)
  1. [V] The symbol \omega is used for both the quintessence equation-of-state parameter and the photon frequency; Eq. (16) introduces \omega_* but Eq. (20) writes a bare \omega in the prefactor. Please use distinct symbols throughout.
  2. [Front matter and Fig. 10] There are several typos: 'Estarual' should be 'Estadual' in the affiliation, 'Comparisson' should be 'Comparison' in Fig. 10, and 'tho highlight' should be 'to highlight' in Sec. VI.
  3. [VI, Table I] Table I is said to correspond to 'the panel in Fig. 8', but Fig. 8 is a 4x4 grid with varying A_\lambda and c; please identify the exact parameter set and fixed \omega and M in the table caption.
  4. [References [39], [76]] References [39] and [76] list the same article (Vagnozzi et al., Class. Quantum Grav. 40, 165007 (2023)); please consolidate to avoid duplication.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the thermodynamic, greybody, and shadow results are derived from an explicitly stated metric ansatz rather than being used to define or fit that ansatz.

full rationale

The paper's results are consequences, not circular redefinitions, of its starting line element. Equation (12) is introduced explicitly as a starting point: "Following Kiselev's approach, the effects of quintessence can be incorporated through a redefinition of the lapse function of the metric, achieved by adding a term proportional to r^{-3\omega-1}. Our starting point, therefore, will be a line element of the form given in Eq. (5), with a modified lapse function defined by..." The Hawking temperature, entropy, heat capacity, greybody bound (Eq. (20)), and shadow radius are computed from this metric with standard formulas; none of these outputs is fed back to define the metric or to fit its parameters. The EHT comparison does not fit a parameter to data; it scans c, lambda, and omega and reports that a range reproduces the observed shadow interval, which is a consistency statement rather than a fitted-input-called-prediction. The cited polymer metric comes from external prior work (Bodendorfer, Mele and Muench; Tu, Zhu and Wang), not from the present authors, so no load-bearing self-citation is present. The paper does not check Kiselev's condition (11) for the combined metric, which is a physical-correctness concern about whether Eq. (12) describes a genuine quintessence source, but that is an unproven assumption, not a circular step. Because every downstream quantity is a straightforward functional of the openly stated line element, the derivation is self-contained and non-circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The analysis rests on one ad hoc metric ansatz (Eq. 12), one imported effective model (polymer black hole), standard semiclassical thermodynamics and scattering formulas, and the EHT Sgr A* shadow range. There are three scanned free parameters (c, lambda, omega) and no new particles or fields invented by this paper.

free parameters (3)
  • quintessence coupling c = varied from 0.02 to 0.2 in the plots
    Effective coupling of the quintessence field to the black hole, introduced by hand in Eq. (12); controls horizon positions, temperatures, emission rates, and shadow radii.
  • polymer parameter lambda = varied from 0.1 to 1.0, equivalently A_lambda from 0.2 to 0.5 at M=1
    LQG holonomy parameter imported from Refs. [34,35]; not fixed by data, scanned across the allowed range 0 <= lambda <= M^2.
  • quintessence equation-of-state parameter omega = chosen values include -1/2, -2/3, -4/5
    Only constrained to lie in (-1,-1/3) by cosmology; the paper chooses representative values rather than deriving them.
assumptions (5)
  • ad hoc to paper Metric (12): polymer lapse plus -c/r^{3*omega+1} with Kiselev condition (11) describes the combined polymer-quintessence spacetime
    Introduced in Sec. III as 'our starting point' without solving coupled field equations; all later results depend on it.
  • domain assumption Polymer black hole with symmetric bounce M_B = M_W = M (Eqs. 5-7) and 0 <= lambda <= M^2
    Imported from Refs. [34,35]; restricts to equal black and white hole masses and a single event horizon in the pure case.
  • standard math Semiclassical thermodynamics: Hawking temperature from surface gravity, entropy S = integral dM / T_H (Eq. 14), and heat capacity C = T dS/dT (Eq. 15)
    Standard formulas assumed to apply to the effective metric without derivation in Sec. IV.
  • domain assumption EHT Sgr A* Keck shadow range 4.55 <= r_sh/M <= 5.22 and weak spin/inclination dependence
    Used in Sec. VI; the range is from cited EHT and Vagnozzi works, but the weak spin/inclination dependence is asserted rather than demonstrated.
  • standard math Greybody lower bound formula (Eq. 18) and electromagnetic effective potential (Eq. 17) for mode l = 1
    Standard scattering results from Refs. [71-74], applied without rederivation in Sec. V.

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Cite this review

Pith. "Pith review of Polymer Black Hole Surrounded by Quintessence." pith.science (2026). https://pith.science/paper/K4QQ4FGC

@misc{pith2026250707372,
  author       = {Pith},
  title        = {Pith review of: Polymer Black Hole Surrounded by Quintessence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K4QQ4FGC}},
  note         = {Machine review of arXiv:2507.07372}
}
read the original abstract

In this paper, we study the polymer black hole solution surrounded by a quintessence field. The influence of quintessence on the polymer black hole is investigated through its thermodynamic properties, such as the Hawking temperature, entropy, and specific heat, which allow us to address the question of thermodynamic stability. We then calculate bounds on the electromagnetic greybody factors and photon emission rates of the black hole, highlighting the interplay between quintessence and quantum gravity effects in determining these phenomena. We also examine the effects of quintessence and quantum gravity on the geodesics and shadows of massless particles around the black hole. Our results are further compared with observational data of the Sagittarius A black hole from the Event Horizon Telescope (EHT) collaboration.

Figures

Figures reproduced from arXiv: 2507.07372 by the authors.

Figure 1
Figure 1. Lapse function plotted against the radial coordinate [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Lapse function plotted against the radial coordinate [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Plot of the Hawking temperature as a function of the event horizon radius for several values of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Plot of the entropy as a function of the event horizon radius for several values of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Plot of the heat capacity as a function of the event horizon radius for several values of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Photon emission rate as a function of the particle energy [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Geodesics for the light trajectory around a Polymer Black Hole surrounded by a Quintessence field where we aim to [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Geodesics for the light trajectory around a Polymer Black Hole surrounded by a Quintessence field. We have [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: Shadows plots for ω = −2/3 and c = 0.001 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Comparisson of the shadow radius with the data from the Sagittarius A* black hole. Notice that there is a wide [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

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