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AdS Black String in a Cosmic Web: Geodesics, Shadows, and Thermodynamics

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A cylindrically symmetric AdS black string, surrounded by a cloud of strings and quintessence, shrinks its photon orbit and shadow, with Sgr A* data favoring a narrow parameter slice.

desk verdict The paper's EHT shadow constraint is built on numbers that do not follow from its own equations—Table 1 uses a quintessence constant ten times the stated values, and the closed-form shadow radii contradict Eq. (23)—while the thermodynamic sections are mostly sound. read the letter →

arxiv 2505.13833 v1 pith:I5YYXYPD submitted 2025-05-20 gr-qc

Faizuddin Ahmed , Ahmad Al-Badawi , Izzet Sakall{i} This is my paper · ORCID
classification gr-qc
keywords AdSblackstringcloudofstringsquintessencefieldholeshadowphotoncircularorbitgeodesicmotionC-energyhorizontemperature
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

This paper takes a static, cylindrically symmetric AdS black-string solution and adds a cloud of strings (parameter $\alpha$) and a quintessence field (parameters $c$ and $w$), then asks how these additions change what an observer would see and measure. Its central claim is that larger $\alpha$ and $c$ shrink both the circular photon orbit and the black string's shadow radius, and that for $w=-2/3$ with $c=0.001$ the shadow radius passes through $R_s\approx 9.97$ near $\alpha\approx 0.6$, a value compatible with the Event Horizon Telescope's $48.7\pm 7.0\,\mu$as measurement of Sagittarius A*. A sympathetic reader would care because this is a concrete route from a cylindrical spacetime model to an actual observational constraint, and because the same machinery predicts photon trajectories, particle-orbit energies, C-energy profiles, and horizon-temperature behavior that could be tested with sharper instruments.

What carries the argument

The load-bearing object is the metric function $A(r)=\alpha-2M/r+r^2/\ell_p^2+c/r^{3w+1}$ together with the effective potential $V_{\rm eff}=(-\varepsilon+L^2/r^2+\ell_p^2p_z^2/r^2)A(r)$. Photon circular orbits are located by $V_{\rm eff}'(r)=0$, which reduces to $rA'(r)-2A(r)=0$; the shadow radius is then $R_s=r_{\rm ph}/\sqrt{A(r_{\rm ph})}$. The bridge to observation is the conversion $\theta_{\rm sh}=2R_s(M/D)$ using Sgr A*'s mass-to-distance ratio, which turns the cylindrical-geometry calculation into a constraint from the EHT angular-diameter measurement.

What would settle it

Integrate null geodesics through the full metric (1) from a static observer at a finite radius with an explicit AdS boundary prescription, locate the apparent shadow edge without assuming $R_s=r_{\rm ph}/\sqrt{A(r_{\rm ph})}$, and compare the resulting image with Sgr A*; if the ray-traced shadow radius or shape disagrees with the Table 1 values, the paper's EHT constraint does not hold. A cheaper check is whether any cylindrical spacetime without a photon sphere can produce an approximately circular $48.7\,\mu$as image at all.

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Extended reading notes

Core claim

Starting from the metric $ds^2=-A(r)dt^2+dr^2/A(r)+r^2d\varphi^2+(r^2/\ell_p^2)dz^2$ with $A(r)=\alpha-2M/r+r^2/\ell_p^2+c/r^{3w+1}$, the paper derives the circular photon orbit radius $r_{\rm ph}$ from the turning-point condition on the effective potential and defines the shadow radius as $R_s=r_{\rm ph}/\sqrt{A(r_{\rm ph})}$. It obtains closed-form shadow radii for the quintessence state parameters $w=-1/3$, $-2/3$, and $-1$ (Eqs. 24-26), and its central result is that the cloud-of-strings parameter $\alpha$ and the quintessence parameters $(c,w)$ each reduce the orbit and shadow size, with the combined effect stronger than either alone. Converting the theoretical shadow radii to angular size with $\theta_{\rm sh}=2R_s(M/D)$ and comparing with Sgr A*'s EHT angular diameter, the paper reports that only a narrow region of parameter space survives: $w=-2/3$, $c=0.001$, and $\alpha$ near $0.6$ (yielding $R_s\approx 9.97$), or $w=-1$ with $\alpha$ between roughly $0.4$ and $0.6$. Alongside this, it reports that time-like circular-orbit energy rises with $\alpha$ and $c$, C-energy density falls with radial distance, the massless scalar perturbative potential grows with $r$ for the chosen parameters, and horizon temperature increases linearly with the radial parameter $r_s=2M$ with a $w$-dependent response to $\alpha$ and $c$.

Load-bearing premise

The entire Sgr A* comparison rests on treating $R_s=r_{\rm ph}/\sqrt{A(r_{\rm ph})}$, computed for a cylindrically symmetric spacetime, as the shadow radius a distant observer would measure, and on converting it with $\theta_{\rm sh}=2R_s(M/D)$; the paper's own footnote 4 concedes that cylindrical symmetry admits no standard photon sphere, and no boundary prescription for an observer at infinity in AdS is supplied.

Editorial extensions

If this is right

  • If the shadow-radius formulas (24)-(26) are correct, the Sgr A* measurement pins the cloud-of-strings parameter to $\alpha\approx 0.6$ on the $w=-2/3$, $c=0.001$ branch, with the other Table 1 combinations producing shadows too large or too small.
  • Larger $\alpha$ and $c$ shrink both the circular photon orbit and the shadow for every $w$ in the allowed range, so gravitational lensing and photon-capture cross sections are systematically smaller than for a plain AdS black string.
  • Time-like circular orbits carry higher energy as $\alpha$ and $c$ increase, which shifts the energetics and stability of accreted matter in any disk model built on this spacetime.
  • C-energy density decreases with radial distance, with $\alpha$ pushing the profile down and $c$ pushing it up, so energy-extraction and accretion estimates must use this modified radial profile rather than the unmodified black-string one.
  • Horizon temperature grows linearly with $r_s$, and the shift caused by increasing $\alpha$ and $c$ changes sign depending on $w$, giving parameter-dependent evaporation behavior.

Reading between the lines

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

  • Not in the paper: a true distant-observer image of this spacetime has not been computed; ray tracing the full metric would likely produce a shadow that is not the circular disk shown in Fig. 4 but elongated along the string axis, and that shape difference would be a direct observational discriminator from spherical black holes.
  • Not in the paper: applying the same $\theta_{\rm sh}=2R_s(M/D)$ conversion to the M87* shadow measurement would provide an independent check; if the M87* data force a disjoint $(\alpha,c,w)$ region, the single-object fit to Sgr A* loses its force.
  • Not in the paper: for $w=-1/3$ the photon-orbit radius reduces to $r_{\rm ph}=3M/(\alpha+c)$, suggesting the two matter fields act through one effective parameter for that state; future data fits could exploit this degeneracy rather than scan all three parameters separately.
  • Not in the paper: because the spacetime is asymptotically AdS and has no standard photon sphere, the shadow comparison should be re-derived with an explicit boundary prescription for an observer at finite radius; until then, the $\alpha\approx 0.6$ match is a model-level coincidence rather than a proven image prediction.
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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 / 6 minor

Summary. The manuscript studies a static, cylindrically symmetric AdS black string surrounded by a cloud of strings and a quintessence field, using the metric of Ref. [1]. It computes null and timelike geodesics, a photon-orbit/shadow radius, C-energy, massless scalar perturbations, and thermodynamic quantities such as Hawking temperature and entropy. The headline observational claim, in the abstract and Sec. 2.1.1, is that the cloud-of-strings parameter α and quintessence parameters (c,w) shrink the photon orbit and shadow, and that parameter choices such as w=-2/3, c=0.001, α≈0.6 give shadow radii most compatible with EHT Sgr A* observations.

Significance. If correct, the paper would provide a fairly complete catalogue of geodesic, perturbative, and thermodynamic properties for a recently proposed exact solution, together with a falsifiable EHT constraint. The shadow section is presented as the main new observational output. However, the numerical and algebraic checks below show that Table 1 and the closed-form shadow radii (24)-(26) are not consistent with the defining formula (23), and that the EHT compatibility paragraph relies on those erroneous numbers. In addition, no boundary prescription is supplied for an 'observer at infinity' in this AdS cylindrical spacetime, so even corrected numbers would not by themselves justify the spherical-black-hole shadow comparison. The non-shadow parts of the paper contain useful formulas, but the central observational claim is not supported as written.

major comments (4)
  1. [Table 1 and Sec. 2.1.1] Table 1 is not reproducible from Eqs. (17), (18), (20), (22), and (23). For w=-1, α=0.2, c=0.001, M=1, ℓp=300, Eq. (17) gives r_ph=3M/α=15, and Eq. (23) gives Rs=15/sqrt(0.2-2/15+225/300^2+0.001*225) ≈ 27.66, not the printed 9.84975. The printed value is obtained with c=0.01 instead of c=0.001, and the same factor-ten mismatch appears in the other rows of the table. Consequently Figures 2-4 and the EHT paragraph inherit incorrect numbers.
  2. [Eqs. (24)-(26)] The closed-form shadow radii in Eqs. (24)-(26) do not follow from Eq. (23). For w=-1/3, substituting r_ph=3M/(α+c) into Eq. (23) produces an (α+c)^3 term under the square root (or the reciprocal placement), whereas Eq. (24) has (α+c)^2; numerically, with M=1, ℓp=300, α=0.2, c=0.001, Eq. (23) gives Rs≈56.6, while Eq. (24) as printed gives Rs≈1.05. For w=-1, Eq. (26) multiplies by sqrt(A(r_ph)) instead of dividing, yielding ≈8.14 for the same row rather than ≈27.66. These are structural inconsistencies, not round-off errors.
  3. [Sec. 2.1.1 and footnote 4] The EHT comparison treats Eq. (23) as the observable shadow radius for an observer at infinity and converts it to an angular diameter via θ_sh=2Rs(M/D). No derivation of this identification is given for a cylindrical AdS spacetime, and footnote 4 explicitly concedes that there is no standard photon sphere in this geometry. The paper also does not supply a boundary-to-bulk prescription for an 'observer at infinity' in AdS. Since the abstract's central claim depends on this identification, the observational constraint is not established.
  4. [Sec. 2.1.1, EHT arithmetic] Even accepting Eq. (23) and the paper's own conversion formula, the claimed EHT compatibility is not supported by the printed numbers. With M/D≈4.3×10^-11 radians, the quoted Rs≈9.97 gives θ_sh=2×9.97×4.3×10^-11 radians ≈ 177 μas, not the EHT value 48.7±7.0 μas. The claimed compatible range would require Rs≈2.7, which is below the values in Table 1. Thus the sentence 'values of α near 0.6 yielding Rs≈9.97 would produce angular diameters most compatible with EHT observations' is numerically inconsistent with the stated formula.
minor comments (6)
  1. [Sec. 2.1, opening paragraph] The text refers to 'Eq. (5)' when introducing the metric; the metric is Eq. (1).
  2. [Introduction, notation] The symbol L is used both for the Lagrangian in Eq. (5) and for angular momentum in Eq. (7); this creates confusion in expressions such as 'ε=2L' and Eq. (10).
  3. [Introduction] The text contains 'COP radius' where 'CPO radius' is meant.
  4. [Conclusion] The conclusions cite 'Eq. (28)' for the w=-1/3 CPO radius; the correct equation is Eq. (20).
  5. [Figures and parameter values] The figures use Λ=-0.003 or Λ=-0.03 (e.g., Figs. 1, 6, 11) while Table 1 uses ℓp=300, which corresponds to Λ=-3.33×10^-5; the parameter sets in different sections should be aligned or their different choices explicitly justified.
  6. [Eqs. (34)-(47)] The perturbative parameter ε in Eq. (35) is never assigned a numerical value, and the coefficients C_i,D_i in Eqs. (45) and (47) are not determined from initial or boundary conditions; the 'analytical solution' is thus a formal truncated Fourier series rather than a predictive trajectory.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: shadow formulas and geodesic analysis are derived in-paper from an externally sourced metric; the EHT comparison is a constraint application, not a fitted prediction, although Table 1 contains severe numerical inconsistencies.

full rationale

The derivation chain is self-contained and non-circular. The metric (Eqs. 1-2) is taken from external Ref. [1], not from a self-citation, and the effective potential, CPO radius, and shadow formula are derived explicitly from the Lagrangian and the condition V'_eff=0. The closed-form shadow expressions (24)-(26) are presented as analytical solutions based on the metric, and the EHT paragraph uses the standard angular-diameter conversion to compare theoretical shadow radii with the Sgr A* measurement; this is a constraint on parameters, not a fit of a parameter to data that is then renamed a prediction. No self-citation is load-bearing: Refs. [46-50] are methodological pointers for geodesic and perturbation techniques, and the present derivations are carried out in the text. There is no uniqueness theorem imported from the authors' prior work and no ansatz smuggled in via citation. The serious problems in the paper are numerical inconsistency, not circularity: Table 1 values are not reproducible from Eqs. (18)-(23) with the stated c values (they correspond to c=0.01 instead of c=0.001), and Eqs. (24)-(26) are not algebraically equivalent to substituting r_ph into Eq. (23). These errors invalidate the claimed EHT compatibility, but they do not constitute a reduction of the derivation to its inputs, so they fall outside the circularity definition and do not raise the circularity score.

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

The model parameters α, c, w, and ℓp are inherited from the metric (1)-(2) of Ref [1] rather than fitted by this paper, but the paper's only contact with observation, the EHT shadow comparison, selects values of α, c, and w to reproduce the measured Sgr A* angular diameter, so as used here they function as fitted parameters. The table's stated c values do not reproduce the computed shadow radii: the values correspond to c ten times larger. The main ad hoc input beyond standard GR is the identification of Rs = rph/√A(rph) as the observable shadow for a cylindrical AdS spacetime (Eq. 23), and the small-ε perturbation expansion for photon trajectories (Eq. 35) with ε never specified. No new entities are introduced.

free parameters (4)
  • α (cloud-of-strings parameter) = scanned over 0.2-0.6 in Table 1; 'α near 0.6' claimed EHT-favored for w = -2/3
    Inherited from Ref [1]; in this paper it is scanned to make the shadow radius match the EHT Sgr A* angular diameter, so it functions as a fitted parameter in the constraint claim.
  • c (quintessence normalization) = stated 0.001/0.003/0.005 in Table 1, but values reproduce only with 0.01/0.03/0.05
    The table's shadow radii match rph/√A(rph) only when c is ten times the printed row value; the EHT-favored value c = 0.001 therefore refers to numbers computed with c = 0.01.
  • w (quintessence state parameter) = scanned over -1/3, -2/3, -1
    Chosen per case to close the equations (17), (81), (84), (89) and to scan shadow sizes; not derived.
  • ℓp (AdS radius) = 300 in Table 1; Λ = -0.003 (ℓp ≈ 31.6) in Figures 1, 5, 6, 7, 8; a = 0.0316 in Fig. 10
    Set by hand; the table and the figures use different values without discussion, and the EHT constraint uses ℓp = 300.
assumptions (4)
  • domain assumption The line element (1)-(2) taken from Ref [1] is an actual solution of the Einstein equations with the cloud-of-strings and quintessence energy-momentum content, with 0 < α < 1 and -1 < w < -1/3.
    Every subsequent computation (geodesics, shadows, C-energy, perturbations, thermodynamics) uses A(r) = α - 2M/r + r²/ℓp² + c/r^{3w+1} as given; the paper never re-derives or cross-checks the field equations. Invoked in Eqs. (1)-(2) and throughout.
  • domain assumption The Kiselev quintessence term c/r^{3w+1} and the Letelier cloud-of-strings parameter α admit a clean superposition inside A(r) with the stated energy conditions.
    The paper relies on the metric from Ref [1], which combines Refs [17] and [31]; it does not verify energy conditions, and it does not address known critiques of Kiselev's solution as quintessence. Invoked at Eq. (2).
  • ad hoc to paper Rs = rph/√A(rph) (Eq. 23) is the observable shadow radius for an observer at infinity in this cylindrical AdS spacetime, and θsh = 2Rs(M/D) can be compared with EHT Sgr A* data.
    Used to produce Table 1 and the EHT constraint. Cylindrical symmetry has no photon sphere in the standard sense (footnote 4), and AdS 'observer at infinity' needs a boundary prescription that is not given. This is the load-bearing observational identification.
  • ad hoc to paper The perturbative expansion u = u0 + εu1 + ... (Eq. 35) is valid for the plotted parameters, with ε small and never assigned a numerical value.
    Introduced in Sec. 2.1.1 to solve Eq. (34); the size of ε and the convergence range are never stated, and the particular solutions used are algebraically incorrect.

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Pith. "Pith review of AdS Black String in a Cosmic Web: Geodesics, Shadows, and Thermodynamics." pith.science (2026). https://pith.science/paper/I5YYXYPD

@misc{pith2026250513833,
  author       = {Pith},
  title        = {Pith review of: AdS Black String in a Cosmic Web: Geodesics, Shadows, and Thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5YYXYPD}},
  note         = {Machine review of arXiv:2505.13833}
}
abstract

In a recent article (Ref. \cite{AOP}), the authors obtained a static, cylindrically symmetric Anti-de Sitter (AdS) black string (BS) solutions, which are cylindrical generalizations of black holes (BHs), surrounded by a cloud of strings (CS) and the quintessence field (QF), and discussed its properties. In the present study, we present a comprehensive analysis of cylindrically symmetric AdS BSs surrounded by CS and QF. Our analysis yields several significant results. We demonstrate that the presence of CS (parameter $\alpha$) and QF (parameters $c$ and $w$) reduces the radius of circular photon orbits (CPO) and BH shadow size, with measurements constrained by Event Horizon Telescope (EHT) observations of Sagittarius A*. We find that time-like particle orbits show increased energy with higher $\alpha$ and $c$ values. We calculate that C-energy, representing gravitational energy within cylindrical radius, decreases with radial distance but exhibits distinct responses to CS and QF parameters. We observe that scalar perturbation potential increases $r$ for specific $\alpha$ and $c$ values, indicating stronger field-spacetime interactions away from the BS. We determine that Hawking temperature increases linearly with Schwarzschild radius, with parameter-dependent behavior varying based on QF state parameter $w$. These results demonstrate how CS and QF significantly modify the geodesic, perturbative, and thermodynamic properties of AdS BS, with potential observational implications for gravitational lensing, accretion disk dynamics, and BH evaporation signatures in future astronomical observations.

Figures

Figures reproduced from arXiv: 2505.13833 by the authors.

Figure 1
Figure 1. The effective potential for non-planar null geodesics, ˙z [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Variation of the radius of CPO with α, for different values of w (left panel) and c (right panel). Here with set M = 1. We provide analytical solutions for the shadow radius of the BS based on the metric (1) (i) for w = −1/3, Rs = 3 √ 3 M q (α + c) 2 + 27 M2 ℓ 2 p , (24) (ii) for w = −2/3, Rs = α  −1 + q 1 + 6 c M α2  c s α 3  −1 + 2q 1 + 6 c M α2  + 6 c M−2 α2  −1+q 1+ 6 c M α2  c 2 ℓ 2 p , (25) (iii) for w =… view at source ↗
Figure 3
Figure 3. Variation of the shadow radius with α, for different values of w (left panel) and c (right panel). Here we set M = 1 and ℓp = 300. As shown in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The geometrical shape of the shadow radius in terms of celestial coordinates for several values of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Force on massless photon particles as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The effective potential for non-planar time-like geodesics, ˙z [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The energy of time-like particles in non-planar geodesics, ˙z [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: The C-energy density as a function of r for BH. Here M = 1 and Λ = −0.003. In [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Scalar perturbative potential Vm as a function of r. 2.4 Thermodynamic Properties of AdS BS Spacetime In this section, we study thermodynamics of the BS solution (1) and analyze the effects of cosmic strings and QF on thermodynamic variables. The BS mass can be obtaine…
Figure 10
Figure 10. Figure 10: There-dimensional illustration of the dimensional quantity [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: The Hawking temperature TH as a function of the Schwarzschild-like radius r = rs. Here M = 1 and Λ = −0.03. w rh Mh = 1 2  α rh + r 3 h ℓ 2 p + c r 3 w h  TH = 1 2 π rh  Mh rh − c (3 w+1)/2 r 3 w+1 h + r 2 h ℓ 2 p  -1 rs α 1 2  rs + 1 α3  1 ℓ 2 p + c  r 3 s  α…

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