Optical Signatures of Black Holes Surrounded by a Generalized Cloud of Strings

Carlos F. S. Pereira; Diego S\'aez-Chill\'on G\'omez; Ednaldo L. B. Junior; Marcos V. de S. Silva

REVIEW

Optical Signatures of Black Holes Surrounded by a Generalized Cloud of Strings

Marcos V. de S. Silva, Ednaldo L. B. Junior, Carlos F. S. Pereira, Diego Sáez-Chillón Gómez

General Relativity and Quantum Cosmology · Submitted 3 Oct 2026

Pith's verdict: Solid, incremental shadow phenomenology with one real internal inconsistency: the claimed Letelier limit in Eq. (35) doesn't follow from Eq. (34), and the paper's validation statement needs correcting.

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arXiv ID2610.04173 (version 2)

Pith IDSVL4KJEI, Pith's permanent record of this paper and its review

CitationCarlos F. S. Pereira, Diego S\'aez-Chill\'on G\'omez, Ednaldo L. B. Junior, Marcos V. de S. Silva. "Optical Signatures of Black Holes Surrounded by a Generalized Cloud of Strings." arXiv:2610.04173 (2026). https://pith.science/paper/SVL4KJEI

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Editor's note

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Colleague,

The useful news first: the central physical claim is real. The paper shows, via transfer functions and backward ray tracing, that the Letelier–Alencar cloud-of-strings metric produces lensing images that are not a simple rescaling of Schwarzschild — configurations can have a smaller shadow yet an outward-displaced Einstein ring, because the shadow boundary is set by the critical orbit while the ring position is set by the full lens map. That decoupling argument is physically clear and rests on independent evidence. The weak-field deflection is computed two ways, Gauss–Bonnet and direct geodesic expansion, and the two agree (Eqs. 34 and 37); that is a genuine consistency check. The Shapiro delay, the transfer functions, and the care taken to distinguish critical impact parameter from physical shadow radius in this non-asymptotically-flat geometry are all competently handled. This is solid, incremental work for the shadow-phenomenology program.

The soft spot is real, localized, and lands on the paper's own validation claim. In Sec. V the authors say the ls→0 (Letelier) limit of the Gauss–Bonnet deflection recovers the known result, quoting Eq. (35) with a mass term proportional to 4M/[b(1−gs²)^{3/2}]. But setting ls→0 in Eq. (34) leaves (1−gs²)^{-2}, since β is independent of ls. Eq. (37), the independent geodesic result, also carries (1−gs²)^{-2}. Two of the three formulas agree; (35) is the outlier. Either (35) is wrong — in which case the Summary's claim that the Letelier result is recovered, and the abstract's 'consistent results,' are overstated as written — or the global-monopole literature they cite actually supports the −3/2 power, in which case (34) and (37) both need correcting and the two-method consistency check loses its validating force. Either way there is a concrete internal mismatch in a headline-adjacent claim, and the authors do not flag it. It does not touch the decoupling result, but it must be resolved.

Other caveats are proportionate: no code or data is shipped, the (gs, ls) bounds are fits of free parameters to the EHT windows rather than predictions, and a static spherical metric constrains the rotating Sgr A* and M87* shadows. The authors do acknowledge the rotating-extension gap in their conclusions, so that point is stated, not hidden.

Who gets value: the black-hole-shadow and modified-gravity lensing community. It deserves a serious referee — send it to review, ask for the (34)/(35) inconsistency to be resolved and the relevant Summary/abstract sentences corrected. With that, a conditional accept is reasonable.

Referee report

Significance

If the results stand, the paper provides a useful extension of the shadow/lensing literature to the Letelier–Alencar spacetime. Its genuine strengths are: two independent weak-field computations that agree at the order considered (Eqs. (34) and (37)); an explicit finite-distance shadow formula (Eq. (16)) that cleanly separates the critical impact parameter from the screen observable; and a concrete, falsifiable decoupling claim (smaller shadow coexisting with an outward-displaced Einstein ring) supported by a stated mechanistic explanation (fixed conical pre-bending reduces the mass deflection needed to form the ring). The ray-tracing and transfer-function analysis is standard but competently executed. The main liability is the internal inconsistency in the weak-field section between Eq. (35) and Eqs. (34)/(37), which affects the claim that the Letelier limit is recovered; this must be settled before the abstract and summary statements can be taken at face value.

Major objections (2)

[§V, Eqs. (34), (35), (37)] Eq. (35) is not the l_s→0 limit of Eq. (34). Since β=√(1−g_s²) is independent of l_s, taking l_s→0 in (34) leaves a mass term 4M/[b(1−g_s²)²], while Eq. (35) carries 4M/[b(1−g_s²)^{3/2}]. I checked the Gauss–Bonnet setup: with K≈2(g_s²−1)(M−g_s²l_sΓ(3/4)²/(2√π))/r³, dS≈r dr dϕ/β³, and r(ϕ)=βb/sin(βϕ), the integral in (33) gives exactly the −2 power, i.e. Eq. (34) follows from its own derivation. The independent geodesic expansion (37) also gives the −2 power. Hence (34) and (37) agree with each other but not with (35). The authors must state which branch is correct: if (35) is wrong, then the Summary's claim ("The Schwarzschild and original Letelier results are recovered in the limits g_s→0 and l_s→0, respectively") is false as written and the abstract's "consistent results" overstates the situation as presented; if the literature result (e.g., the topological-charge/global-monopole form attributed to Ref. [75]) supports the −3/2 power, then Eqs. (34) and (37) would need correction and the two-method consistency check loses its validating force. Either way this is a load-bearing, localized discrepancy that must be reconciled (a full derivation of the Letelier-limit deflection from the geodesic/GB setup, with a clear statement of which impact parameter — turning-point value b=r₀/√f(r₀) or an asymptotic one — is being used).
[§IV, Figs. 6–7] The constraint regions in Figs. 6–7 are obtained by comparing the static, spherically symmetric shadow formula (16) with the EHT-inferred shadow radii of Sgr A* and M87*, which are rotating objects. The mapping between a static-model shadow radius and a Kerr-like shadow radius is asserted rather than established, and the derived (g_s, l_s) ranges inherit this transfer. Since this is presented as a headline result ("we use this distinction to derive constraints on the model parameters"), the text should state explicitly in §IV that the bounds are conditional on this static-to-rotating identification, even though a rotating extension is listed as future work in §VII. Without such a caveat the allowed regions in Figs. 6–7 overstate their current applicability.

Minor objections (5)

[§V.C, Eq. (43) and Table I] The unit of ΔT_Sch for M87* is written as "1.41968×10^6 s²" — the superscript 2 on the seconds appears to be a typo (the quantity is a time, not a squared time). Also the sentence beginning "First, the time delay for Schwarzschild to Sgr A*" should read "for Sgr A*".
[§V.A, Eq. (34)] The intermediate display line of Eq. (34) is garbled: the l_s term appears without its denominator and the mass term is written with a minus sign ("−4√πM/(b√πβ⁴)") that flips to a plus in the final line. Please streamline the derivation display so the algebra can be followed term by term.
[§VI.B, Fig. 13] The headline claim (smaller shadow with an outward-displaced Einstein ring) is supported only qualitatively by visual inspection of Fig. 13 for a single observer position (r_O=15M). A quantitative plot of the ring's angular radius (or of the lensed image position) against g_s and l_s for fixed shadow radius would materially strengthen the claim, and would also exclude an artifact of the finite observer distance.
[§V.A, text after Eq. (35)] The reference to the topological-charge deflection is spelled "Kimet Jusufi et al." — the first author's name should be "Kimet Jusufi" consistently, and the actual result of Ref. [75] (and of the finite-distance analysis in Ref. [80]) should be quoted explicitly so the reader can compare the mass-term power with Eqs. (34) and (37).
[§IV, Eqs. (18)–(19)] The shadow-radius constraints use Keck+VLTI data for Sgr A* and EHT data for M87*, citing Refs. [11,72,73]. Since EHT's own Sgr A* results (Ref. [14]) are listed among the references, a sentence justifying the choice of the Keck+VLTI combination for Sgr A* would help the reader weigh the derived parameter ranges.

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The paper

The paper claims

This paper claims that the optical images of a black hole wrapped in a generalized cloud of strings cannot be reduced to any single rescaled Schwarzschild picture, because the shadow boundary and the Einstein ring position respond to the…

The reading

This paper tries to establish that a black hole surrounded by a generalized cloud of strings, a continuous distribution of one-dimensional defects, produces optical images that no single rescaling of the Schwarzschild picture can reproduce. It claims that the string parameters $g_s$ and $l_s$ can either enlarge or shrink the photon sphere, and that because the spacetime is not asymptotically flat, the critical impact parameter must be distinguished from the physical shadow radius. Using that distinction, the paper derives allowed regions for $(g_s, l_s)$ from the shadow-radius bounds of Sgr A* and M87*, calculates deflection and Shapiro delay with a conical-background term, and ray-traces thin-disk and celestial-sphere images. The key result is that a configuration with a smaller shadow than Schwarzschild can still displace the main Einstein ring outward, so shadow size, ring position, deflection angle, and time delay each probe a different aspect of photon propagation.

What carries the argument

The load-bearing mechanism is the Letelier-Alencar metric function, $f(r) = 1 - \frac{2M}{r} + \frac{g_s^2 l_s^2}{r^2}\,{}_2F_1\!\left(-\frac{1}{2},-\frac{1}{4},\frac{3}{4},-\frac{r^4}{l_s^4}\right)$, whose asymptotic limit is $f(\infty) = 1 - g_s^2$ rather than 1. That single fact does most of the work: it makes the asymptotic optical geometry conical, so the paper replaces the straight-line ray trajectory with $r(\phi) = b\beta/\sin(\beta\phi)$ with $\beta = \sqrt{1 - g_s^2}$, and it inserts the factor $\sqrt{f(r_O)}$ into the shadow radius $r_{sh} = r_c \sqrt{f(r_O)/f(r_c)}$, separating $r_{sh}$ from the critical impact parameter $b_c$. The photon-sphere condition $2f(r_c) - r_c f'(r_c) = 0$ together with these two corrections carries the shadow, deflection, and time-delay calculations, while the transfer functions $r_n(b)$ carry the ray-traced image analysis.

Extended reading notes

Core claim

The central claim is stated in the paper's conclusion: the optical signatures of the Letelier-Alencar geometry cannot be characterized by a single length scale or by a simple deformation of the Schwarzschild shadow. The discovery is that the two string parameters act in opposite directions depending on their sizes, with small $l_s$ making the photon sphere and critical impact parameter grow as $g_s$ increases, while large $l_s$ reverses that trend, and that a smaller shadow does not imply an inward-shifted Einstein ring. The paper demonstrates this through backward ray tracing on a celestial sphere, finding configurations with $l_s/M = 2$ and $g_s = 0.5$ where the shadow is smaller than in Schwarzschild but the main lensed image of a background source still appears farther from the center. Because the asymptotic metric converges to $f(r) \to 1 - g_s^2$, the physical shadow radius $r_{sh}$ gains a factor of $\sqrt{f(r_O)}$ relative to the critical impact parameter $b_c$, which decouples the two observables. The paper concludes that shadow boundary, lensed-image position, deflection angle, and Shapiro time delay carry complementary information about the string environment.

Load-bearing premise

The load-bearing premise is that the shadow of a static, spherically symmetric model can be matched to the observed shadows of the rotating objects Sgr A* and M87*; the paper imports the geometry from a cited derivation and does not establish this static-to-rotating transfer, so the derived bounds inherit that unverified step.

Editorial extensions

If this is right

A single shadow-radius measurement cannot constrain $g_s$ and $l_s$ independently; the paper's comparison leaves a substantial compatible region from Sgr A* data and an even larger one from M87* data.
Precision timing of light propagation could isolate $g_s$ through the distance-dependent Shapiro delay term $2g_s^2(r_1 + r_2)$, a contribution that is essentially insensitive to $l_s$ in realistic configurations.
The non-uniform transfer functions imply that the direct, lensed, and photon-ring contributions to a disk image shift by different amounts as the string parameters change, so future high-resolution images should show these orders moving apart.
A configuration with a smaller shadow and an outwardly displaced Einstein ring is a concrete signature that can be searched for in EHT-like images of supermassive black holes.

Reading between the lines

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

An observational program could test the decoupling directly: if the shadow boundary and the Einstein ring position always scale together across many sources, the paper's central claim would fail; if they drift apart, that supports a non-asymptotically flat environment around the black hole.
The same conical-asymptotic mechanism likely operates in other spacetimes with global monopoles or topological charges, so the shadow-versus-ring decoupling may be a general marker of such geometries rather than a peculiarity of the string cloud.
Because the paper matches a static model to the shadows of rotating objects, a full Kerr-like treatment could either tighten or dissolve the derived $(g_s, l_s)$ bounds; that extension is the natural next step to judge whether the predicted signatures can survive contact with realistic astrophysics.

Original abstract

In this work, we investigate the optical signatures of black holes surrounded by a generalized cloud of strings, described by the Letelier--Alencar spacetime. We first analyze null geodesics, showing that the string parameters can either increase or decrease the photon sphere radius and the critical impact parameter relative to their Schwarzschild values. Since the spacetime is not asymptotically flat, the critical impact parameter does not directly coincide with the physical shadow radius. We use this distinction to derive constraints on the model parameters from the shadow radius bounds of Sgr A* and M87*. In the weak-field regime, we calculate the light deflection angle using both the Gauss--Bonnet method and a perturbative expansion of the null geodesic equations, obtaining consistent results that contain a contribution from the asymptotically conical geometry in addition to the local gravitational deflection. We also examine the Shapiro time delay and identify a distance dependent contribution associated with the non Minkowskian asymptotic background. Finally, we study the optical appearance of geometrically and optically thin accretion disks and construct celestial sphere images through backward ray tracing. The transfer functions and lensing images show that the generalized cloud of strings modifies the null geodesic mapping in a nonuniform manner rather than producing a simple rescaling of the Schwarzschild image. In particular, configurations with a smaller shadow may still produce an outward displacement of the main Einstein ring image, demonstrating that different optical observables probe distinct aspects of photon propagation in this geometry.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-06: 1 request, 7,085 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.47 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this work, we investigate the optical signatures of black holes surrounded by a generalized cloud of strings, described by the Letelier--Alencar spacetime. We first analyze null geodesics, showing that the string parameters can either increase or decrease the photon sphere radius and the critical impact parameter relative to their Schwarzschild values. Since the spacetime is not asymptotically flat, the critical impact parameter does not directly coincide with the physical shadow radius. We use this distinction to derive constraints on the model parameters from the shadow radius bounds of Sgr A* and M87*. In the weak-field regime, we calculate the light deflection angle using both the Gauss--Bonnet method and a perturbative expansion of the null geodesic equations, obtaining consistent results that contain a contribution from the asymptotically conical geometry in addition to the local gravitational deflection. We also examine the Shapiro time delay and identify a distance dependent contribution associated with the non Minkowskian asymptotic background. Finally, we study the optical appearance of geometrically and optically thin accretion disks and construct celestial sphere images through backward ray tracing. The transfer functions and lensing images show that the generalized cloud of strings modifies the null geodesic mapping in a nonuniform manner rather than producing a simple rescaling of the Schwarzschild image. In particular, configurations with a smaller shadow may still produce an outward displacement of the main Einstein ring image, demonstrating that different optical observables probe distinct aspects of photon propagation in this geometry.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work, we investigate the optical signatures of black holes surrounded by a generalized cloud of strings, described by the Letelier--Alencar spacetime.	contribution 0.98	0.79	0.04	asserted 0.84
s1	We first analyze null geodesics, showing that the string parameters can either increase or decrease the photon sphere radius and the critical impact parameter relative to their Schwarzschild values.	result 0.90	0.95	0.09	asserted 0.82
s2	Since the spacetime is not asymptotically flat, the critical impact parameter does not directly coincide with the physical shadow radius.	background 0.45	0.25	0.04	asserted 1.00
s3	We use this distinction to derive constraints on the model parameters from the shadow radius bounds of Sgr A* and M87*.	method 0.49	0.95	0.09	asserted 0.99
s4	In the weak-field regime, we calculate the light deflection angle using both the Gauss--Bonnet method and a perturbative expansion of the null geodesic equations, obtaining consistent results that contain a contribution from the asymptotically conical geometry in addition to the local gravitational deflection.	method 0.48	0.96	0.07	asserted 1.00
s5	We also examine the Shapiro time delay and identify a distance dependent contribution associated with the non Minkowskian asymptotic background.	result 0.70	0.96	0.04	asserted 1.00
s6	Finally, we study the optical appearance of geometrically and optically thin accretion disks and construct celestial sphere images through backward ray tracing.	contribution 0.51	0.96	0.05	asserted 0.99
s7	The transfer functions and lensing images show that the generalized cloud of strings modifies the null geodesic mapping in a nonuniform manner rather than producing a simple rescaling of the Schwarzschild image.	result 0.93	0.95	0.38	asserted 1.00
s8	In particular, configurations with a smaller shadow may still produce an outward displacement of the main Einstein ring image, demonstrating that different optical observables probe distinct aspects of photon propagation in this geometry.	interpretation 0.67	0.93	0.42	asserted 0.59

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Behavior of the effective potential for photons as a function of the radial coordinate. In the left panel, we fix

**Figure 2.** Behavior of the photon sphere radius for different values of the cloud of strings parameters. In the left panel, we fix

**Figure 3.** Behavior of the critical impact parameter for different values of the cloud of strings parameters. In the upper panel,

**Figure 4.** Behavior of the shape formed by the critical impact parameter for the Letelier–Alencar spacetime for different values

**Figure 5.** Photon trajectories in the Letelier–Alencar spacetime for different values of the cloud of strings parameters. In the

**Figure 6.** Shadow radius of the black hole for the Letelier–Alencar spacetime. The red and blue regions represent the intervals

**Figure 7.** Shadow radius of the black hole for the Letelier-Alencar spacetime. The red and blue regions represent the intervals

**Figure 8.** Behavior of the intensity profiles according to

**Figure 9.** Behavior of the transfer function as a function of

**Figure 10.** Observed intensity profiles (left panels) and optical appearances (right panels) for the Schwarzschild case. From

**Figure 11.** Observed intensity profiles (left panels) and optical appearances (right panels) for the Letelier–Alencar case with

**Figure 12.** Observed intensity profiles (left panels) and optical appearances (right panels) for the Letelier–Alencar case with

**Figure 13.** Celestial sphere images obtained through back

Figures reproduced from arXiv:2610.04173 by the authors.

References

Discussion

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