Observable signatures of Black Holes with Hernquist Dark Matter Halo having a cloud of strings: From Geodesics to Shadow

Ahmad Al-Badawi; Faizuddin Ahmed; \.Izzet Sakall{\i}

REVIEW 3 major objections 6 minor 2 cited by

Observable signatures of Black Holes with Hernquist Dark Matter Halo having a cloud of strings: From Geodesics to Shadow

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A cosmic string cloud enlarges a black hole's shadow while its dark matter halo shrinks it.

desk verdict The halo-shrinks-shadow claim is a sign error: Eq. (75) contradicts the paper's own photon-sphere equation, so the headline observable is an artifact. read the letter →

arxiv 2506.18457 v1 pith:65AVBLWB submitted 2025-06-23 gr-qc hep-th

Faizuddin Ahmed , Ahmad Al-Badawi , İzzet Sakall{i} This is my paper · ORCID

classification gr-qchep-th PACS 04.70.-s95.35.+d

keywords ModifiedgravityBlackholesHernquistdarkmatterhaloCloudofstringsGeodesicsanalysisholeperturbationsshadowPhotonsphere

topics Dark Matter

open problems Dark Matter

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 builds a static, spherically symmetric metric for a Schwarzschild black hole embedded in a Hernquist dark matter halo and surrounded by a cloud of cosmic strings, with metric function $f(r)=1-\alpha-2M/r-b/(r+r_s)+r^2/\ell_p^2$, $b=4\pi\rho_s r_s^3$. The aim is to show that this one configuration puts a realistic environment around a black hole and leaves measurable traces. From the metric the paper derives null and timelike geodesics, effective potentials for scalar, electromagnetic, and Dirac fields, and the photon-sphere condition $r f'(r)=2f(r)$. The central quantitative claim is that the string parameter $\alpha$ enlarges the photon sphere and shadow radius, while halo density $\rho_s$ and core radius $r_s$ shrink them, in combinations that distinguish the configuration from a vacuum Schwarzschild black hole. If correct, black hole imaging and orbital data could in principle separate halo and string contributions.

What carries the argument

The load-bearing object is the metric function $f(r)=1-\alpha-2M/r-b/(r+r_s)+r^2/\ell_p^2$ with $b=4\pi\rho_s r_s^3$, built by superposing a Schwarzschild term, a Hernquist-halo term (from the tangential-velocity relation $A_1=B_1=\exp(-4\pi\rho_s r_s^3/(r+r_s))\approx 1-b/(r+r_s)$), and a string-cloud term $\alpha$, on an anti-de Sitter background. This single function carries the argument: it determines the effective potentials for null and timelike geodesics, the photon-sphere condition $r f'(r)=2f(r)$, the critical impact parameter and shadow radius $R_s=r_{\rm ph}/\sqrt{f(r_{\rm ph})}$, the perturbative potentials for spin-0, spin-1, and spin-1/2 fields, and the orbital and precession frequencies. All parameter dependence flows through $f(r)$, so the sign of the shadow response is read directly from how $\alpha$, $\rho_s$, and $r_s$ enter $f$.

What would settle it

Compute the full Einstein tensor of the metric (18) and compare every component with the sum of the Hernquist-halo and string-cloud energy-momentum tensors; the angular components are the decisive test because the paper only matches the time-time and radial components. If those fail, the metric is not an exact solution with the stated sources.

Watch

Extended reading notes

Core claim

The central claim is that the combined spacetime with $f(r)=1-\alpha-2M/r-b/(r+r_s)+r^2/\ell_p^2$, where $b=4\pi\rho_s r_s^3$, describes a black hole whose observable shadow and orbital dynamics encode both dark matter and cosmic string parameters. The photon sphere radius $r_{\rm ph}$ is fixed by $r f'(r)=2f(r)$, and the shadow radius is $R_s=r_{\rm ph}/\sqrt{f(r_{\rm ph})}$. Numerical evaluation shows $R_s$ rises with the string parameter $\alpha$ and falls with halo density $\rho_s$ and core radius $r_s$; at larger $\alpha$ the shadow is enlarged substantially, while denser or more extended halos pull it inward. The paper also finds that the halo and strings jointly lower the null and timelike effective potentials, make the effective force on photons more attractive at large radius, and reduce the Lyapunov exponent and geodesic precession relative to the string-cloud AdS limit. The conclusion is that the configuration leaves distinct imprints in shadow size, orbital dynamics, and perturbation barrier shapes that distinguish it from a vacuum black hole and from a black hole with only one of the two matter components.

Load-bearing premise

The construction assumes the halo contributes to the metric as $1-b/(r+r_s)$, obtained by a first-order exponential approximation and the tangential-velocity relation, and that combining the halo with a black hole and a string cloud by matching only the time-time and radial components of the Einstein equations is valid.

Editorial extensions

If this is right

A larger $\alpha$ moves the photon sphere outward and increases $R_s$; a denser or more extended halo moves it inward, so any measured shadow size carries a degeneracy between string tension and halo parameters.
The shift in the critical impact parameter $\beta_c$ directly changes the photon capture cross-section, meaning the same parameters affect how many photons reach a distant observer.
Because the Lyapunov exponent and geodesic angular velocity both decrease in the presence of the halo, the configuration should show slower photon orbital motion and altered ringdown damping relative to vacuum Schwarzschild.
The computed effective potentials for scalar, electromagnetic, and Dirac fields give concrete inputs for quasinormal-mode and stability calculations in this background.

Reading between the lines

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

The paper does not compare its shadow formula with the measured diameters of the two supermassive black holes imaged to date; a direct fit would turn the predicted $\alpha$-versus-$(\rho_s,r_s)$ trade-off into constraints on string tension and halo density.
The growing attractive force on photons at large radius implies that gravitational lensing by this black hole should deviate from Schwarzschild lensing at large impact parameters; computing the deflection angle would be a direct extension.
The 'inner BH geometry' regime where $f(r)<0$ at moderate densities hints at horizonless or naked-singularity configurations, but the paper does not analyze the global causal structure there; that is a concrete open question.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the headline claim is wrong. The paper says the Hernquist dark matter halo shrinks the shadow. Its own photon-sphere condition, Eq. (38), says the opposite. In the shadow section the authors rearrange it as Eq. (75) and flip the sign of the halo term; Table 1 follows the wrong sign. So the central observable signature is an algebraic artifact.

There is something worth keeping. The metric (18) is a sum of a Letelier string cloud, a Hernquist-style halo term, and a cosmological term, and for metrics of this form the Einstein tensor is linear in f, so the superposition is an exact solution for the sum of the three sources. The paper does not demonstrate this cleanly—it matches tt and rr components and never checks angular ones—but the construction is more defensible than it looks. The limit checks to Letelier and Schwarzschild-AdS are there, and the geodesic, Lyapunov, precession, and perturbation calculations are standard and mostly clearly written.

Soft spots, in proportion. The sign error is load-bearing: it reverses the paper's raison d'être. If Eq. (75) is fixed, the halo enlarges the shadow, so the abstract, conclusions, figures, and Table 1 all have to change. Second, the perturbation section has smaller typos (for example, the dimensionless scalar potential has +2k where it should be +2k^2, and the halo term in the second factor is off by a factor of two from the stated 8π=1 convention). Those are secondary but suggest the numerics were not checked against the formulas. Third, the parameter ranges are not tied to realistic halo densities or to M87* and Sgr A*; the claim of observable signatures is asserted rather than calibrated.

Who is this for? A specialist collecting exact shadow and perturbation formulas for modified environments might use the corrected version, but not this one as it stands. For the usual general-relativity audience, the error is checkable in ten minutes and it kills the main result.

Recommendation: desk reject. No referee time needed for the version under consideration; if the authors fix the sign and recalibrate, it becomes a different, much more modest paper—halo enlarges shadow—that would need re-review from scratch.

Referee Report

3 major / 6 minor

Summary. The manuscript constructs a static spherically symmetric spacetime (Eq. 18) describing a Schwarzschild-AdS black hole surrounded by a Hernquist dark-matter halo and a cloud of strings, with metric function f(r)=1−α−2M/r−b/(r+r_s)+r^2/ℓ_p^2 and b=4πρ_s r_s^3. It then studies null and timelike geodesics, derives effective potentials, photon-sphere and shadow radii, and analyzes scalar, electromagnetic, and Dirac perturbations. The central observational claim is that the string parameter α enlarges the shadow while the halo density ρ_s and core radius r_s shrink it, producing signatures detectable by current and future instruments.

Significance. If the central results were correct, the paper would complement the existing literature on black holes embedded in dark-matter halos and string clouds, and its explicit formulas for effective potentials, photon-sphere radii, Lyapunov exponents, and perturbation potentials could serve as useful references for numerical and observational follow-up work. However, the headline shadow prediction is invalidated by an algebraic sign error, and the combined metric is not shown to solve the full Einstein equations with the stated sources. As a result, the claimed observable signatures are not currently established, which substantially reduces the paper's significance.

major comments (3)

[§5, Eqs. (38) and (75)] Equation (75), which underpins Table 1 and Figs. 13–14, has the sign of the halo term reversed relative to the photon-sphere condition derived earlier in Eq. (38). Expanding the correct condition rf'(r)=2f(r) gives 3M(r+r_s)^2 + (α−1)r(r+r_s)^2 + (b/2)r(3r+2r_s) = 0, whereas Eq. (75) contains −(b/2)r(3r+2r_s). Since b=4πρ_s r_s^3 > 0, the correct equation drives r_ph outward and R_s = r_ph/sqrt(f(r_ph)) upward as ρ_s or r_s increases, while Eq. (75) drives them downward. The trend displayed in Table 1 and Figs. 13–14 therefore has the opposite sign to the prediction of the paper's own photon-sphere condition, directly contradicting the abstract's claim that HDMH properties tend to shrink the shadow. Relatedly, the first condition in Eq. (74) sets V_eff=0 at the photon sphere, which is not the photon-orbit condition; the derivative condition is the operative one, but the sign error in the resulting algebraic equation remains the decisive defect.
[§2, Eqs. (10)–(14)] The construction of the combined metric is not shown to satisfy the full Einstein equations. The matching procedure in Eq. (12) equates only the tt and rr components of the Einstein tensor for the pure-halo and combined metrics; the θθ and φφ components, which are sensitive to the anisotropic pressures of the string cloud (Eq. 15) and of the Hernquist ansatz, are never checked. The paper therefore has not demonstrated that metric (18) is a solution of Eq. (11) with the stated sources. In addition, the halo metric function in Eq. (6) is obtained from the tangential-velocity relation and the approximation exp(−b/(r+r_s)) ≈ 1−b/(r+r_s), which differs from the standard general-relativistic relation g^{rr}=1−2M_H(r)/r with M_H(r) from Eq. (3); the two prescriptions do not coincide, since 2M_H/r = b r/(r+r_s)^2 rather than b/(r+r_s). Because all later geodesic, perturbation, and shadow results inherit metric (18), this gap is load-bearing for the paper's central claim.
[Table 1] The row with α=0.5, r_s=0.6, ρ_s=0.6 lists r_ph=0.617369 and R_s='x', which is not a physical photon sphere. For these parameters f(r) is negative at that radius, so the point lies inside the horizon and no shadow exists. This unphysical entry follows from the incorrect photon-sphere equation and should be removed or explicitly flagged; its presence indicates that the numerical roots reported in Table 1 are not reliable as observational predictions.

minor comments (6)

[Abstract and Section 6] The claims that the configuration yields 'observational imprints detectable by present and forthcoming astrophysical instruments' are not supported by any quantitative comparison with, for example, EHT angular-diameter measurements for M87* or Sgr A*; the observational language should be tempered or supplemented with such estimates.
[§4.1 and §4.2] The text describes the background as 'incorporating a quintessence field,' but the metric (18) contains no quintessence term; this wording should be corrected to refer to the Hernquist dark-matter halo and the cloud of strings.
[§4.1, Eq. (59)] After the tortoise-coordinate transformation, the text refers to 'the line-element Eq. (23)', but Eq. (23) is the Lagrangian density; the cross-reference is incorrect and should point to the metric (17) or (18).
[§4.3, Eq. (73)] The displayed expression for the Dirac effective potential V_{±1/2} has an unbalanced bracket, making the formula ambiguous; the authors should present it with clearly matched parentheses.
[§5, Eq. (76)] The analytical expression for r_ph is extremely cumbersome and its stated reduction to r_ph=3M in the limit α=0, ρ_s=0 is not demonstrated; a simpler presentation or a numerical root-finding description would improve readability and verifiability.
[Figures 2–5 and 9–11] The conventions for the AdS scale are inconsistent across plots and captions (ℓ_p=10 in Figs. 1–2, ℓ_p=100 in Fig. 3, and k=M sqrt(−Λ/3)=0.1 in later figures); the parameters should be defined once in a table or at first use and kept consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geodesic, perturbation, and shadow results are algebraic consequences of the assumed metric, and the paper's self-citations are not load-bearing.

full rationale

The paper's central object is the metric (18), f(r)=1-alpha-2M/r-b/(r+r_s)+r^2/l_p^2, which is adopted as the starting point: the Hernquist contribution is imported from Ref. [56] and the string-cloud contribution from Letelier's solution, with the two assumed not to interact. Every subsequent result--effective potentials, photon-sphere condition, shadow radius R_s=r_ph/sqrt(f(r_ph)), Lyapunov exponent, orbital speeds, and perturbation potentials--is obtained by direct differentiation, substitution, and root-solving from this assumed metric. No parameter is fitted to data, and no 'prediction' is set by an input value; Table 1 and the figures are numerical evaluations of the paper's own algebraic expressions. The derivation of the halo term in Sec. 2 uses the tangential-velocity relation (5), assumes A1=B1, and retains leading-order terms, so the combined line element is more accurately described as an ansatz than as a fully verified solution of the Einstein equations; however, that is a modeling or correctness concern, not circular reasoning. Many citations are to the authors' own prior papers, but they support standard techniques (geodesic Lagrangians, field-perturbation equations, celestial coordinates) and do not carry the load of any central claim. No uniqueness theorem, fitted constant, or self-referential definition is invoked to force a conclusion. The apparent discrepancy between the photon-sphere conditions (38) and (75) is an internal algebraic sign issue, not a reduction of a prediction to an input. Therefore, no circular step can be exhibited, and the appropriate score is 0.

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

No new particles or fields are introduced. The free parameters ρ_s, r_s, α are model inputs scanned over, not fitted. The main axioms are the Hernquist profile, the exponential approximation for the halo metric, the non-interaction assumption, and the incomplete matching of Einstein equations.

free parameters (3)

ρs (halo central density)
Free input parameter entering the metric through b=4πρ_s r_s^3; all shadow and geodesic results are quoted as functions of it.
rs (halo core radius)
Free input parameter entering the metric through b=4πρ_s r_s^3; the paper scans over it.
α (string cloud parameter)
Free dimensionless parameter from the Letelier cloud of strings; controls the conical deficit.

assumptions (5)

domain assumption Hernquist density profile ρ(r)=ρ_s (r/r_s)^{-1}(1+r/r_s)^{-3} and the mass profile (3).
Adopted from Hernquist (1990) and Dehnen (1993) as the dark matter model; not derived in the paper.
ad hoc to paper The halo metric function is A1=B1=exp(-4πρ_s r_s^3/(r+r_s))≈1-b/(r+r_s) via the tangential velocity relation (5).
The exponential approximation and the equality A1=B1 are modeling choices; they are not exact solutions for the Hernquist mass profile in GR.
ad hoc to paper The dark matter halo and the string cloud do not interact, so their energy-momentum tensors add and the metric functions add linearly.
Stated in Section 2 before Eq. (17); no physical justification is given for linear superposition.
ad hoc to paper Matching only the tt and rr components of the Einstein tensor (Eq. 12) is sufficient to determine the combined metric.
The angular θθ and φφ components are never checked; the field equations require all components to match.
domain assumption The shadow radius for a distant observer is R_s=r_ph/sqrt(f(r_ph)), with no correction for the cosmic string deficit angle or for AdS asymptotics.
Standard in asymptotically flat spacetimes; the paper applies it without discussion to an AdS background with a conical deficit.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Observable signatures of Black Holes with Hernquist Dark Matter Halo having a cloud of strings: From Geodesics to Shadow." pith.science (2026). https://pith.science/paper/65AVBLWB

@misc{pith2026250618457,
  author       = {Pith},
  title        = {Pith review of: Observable signatures of Black Holes with Hernquist Dark Matter Halo having a cloud of strings: From Geodesics to Shadow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/65AVBLWB}},
  note         = {Machine review of arXiv:2506.18457}
}

read the original abstract

We present a comprehensive theoretical investigation of a novel black hole (BH) spacetime: a Schwarzschild BH embedded in a Hernquist-type dark matter halo (HDMH) and surrounded by a cloud of cosmic strings (CSs) -- collectively termed the Schwarzschild-HDMH with CS (SHDMHCS) configuration. By analyzing the spacetime geometry, we explore how key parameters -- the core radius and halo density of the dark matter, along with the string tension -- affect the geodesic motion of both massless and massive particles. Our results reveal that the combined influence of HDMH and CSs modifies the effective potentials for null and timelike geodesics, leading to distinct dynamical behavior compared to standard Schwarzschild geometry. We perform a perturbative analysis for scalar (spin-0), electromagnetic (spin-1), and Dirac (spin-1/2) fields, deriving the associated effective potentials and showing how both the dark halo and CSs alter field propagation and potential barriers. The shadow of the BH is studied in detail: we derive analytical expressions for photon sphere and shadow radii, finding that CSs tend to enlarge the shadow, while HDMH properties tend to shrink it. The combined effects of these parameters significantly influence the shadow's shape and size, producing potentially observable signatures. Our results establish that the SHDMHCS configuration yields distinct observational imprints detectable by present and forthcoming astrophysical instruments. This framework provides new tools for probing exotic matter distributions via gravitational wave observations, orbital dynamics, and high-resolution black hole imaging, offering a pathway to distinguish such configurations from simpler BH models in realistic environments.

Figures

Figures reproduced from arXiv: 2506.18457 by the authors.

**Figure 1.** Embedding diagrams of the SBH for various HDMH values. The BH mass is set to M = 1. Plots are governed by the metric function (14). Recently, a static and spherically symmetric Schwarzschild BH immersed in HDMH was introduced in Ref. [56]. The density for the Hernquist profile takes the form given in Eq. (2). With this, the metric for the combined system of BH and DM halo becomes [56] ds2 = −f(r) dt2 + f(r) −1 dr2 +… view at source ↗

**Figure 2.** Embedding diagrams of the SBH for various HDMH and CS values. The BH mass is set to M = 1. Plots are governed by the metric function (18). From the above expressions (19)–(21), one can easily show that at r = 0 and large distances, r → ∞, limr→0 R = ∞, limr→0 R µν Rµν = ∞, limr→0 R µνρσ Rµνρσ = ∞, limr→∞ R = 4 Λ, limr→∞ R µν Rµν = 4 Λ2 , limr→∞ R µνρσ Rµνρσ = 8 Λ2 3 . (22) From the above scalar curvature analysis, w… view at source ↗

**Figure 3.** The behavior of the metric function f(r) is illustrated as a function of the radial coordinate r, considering the individual effects of varying (a) the halo density ρs, (b) the core radius rs, and (c) the string parameter α. In all cases, the BH mass is fixed at M = 1, and the curvature radius is set to ℓp = 100. spacetime, two Killing vectors intrinsic to the metric could be found those are associated closely with … view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Behavior of the effective potential for null geodesics is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Subfigures illustrate the individual effects of (a) α, (b) rs, and (c) ρs M2 , as well as the combined effects …

**Figure 5.** Figure 5: Behavior of the force on the photon particles is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Sub-figures illustrate the individual effects of (a) α, (b) rs, and (c) ρs M2 , as well as the combined effects of (d) α…

**Figure 6.** Figure 6: Behavior of the effective potential for timelike is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Sub-figures illustrate the individual effects of (a) α, (b) rs, and (c) ρs M2 , as well as the combined effects of (d…

**Figure 7.** Figure 7: Behavior of the specific angular momentum per unit mass (L/M) of time-like particles orbiting in circular paths is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Sub-figures illustrate the individual effects of (a) α…

**Figure 8.** Figure 8: Behavior of the specific energy (E) of time-like particles orbiting in circular paths is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Sub-figures illustrate the individual effects of (a) α, (b) rs, and (c) ρs M2 , …

**Figure 9.** Figure 9: Behavior of the scalar perturbative potential is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Sub-figures illustrate the individual effects of (a) α, (b) rs, and (c) ρs M2 , as well as the combined effects of (d) α…

**Figure 10.** Figure 10: Three-dimensional plot of M2 V0: the qualitative features of spin-zero scalar perturbative potential for the dominant multipole number ℓ = 0 is shown by varying values of the string parameter α. Here, we set the dimensionless parameter k = M q − Λ 3 = 0.1. Orange curv…

**Figure 11.** Figure 11: Behavior of the electromagnetic perturbative potential M2 Vem is shown for varying values of the string parameter α, the core radius rs, and the DM halo density ρs. Sub-figures illustrate the individual effects of (a) α, (b) rs, and (c) ρs M2 , as well as the combined…

**Figure 12.** Figure 12: Comparison of the scalar and electromagnetic potential under different BHs scenario. Here, we set the string parameter α = 0.1, the dimensionless parameters M2 ρs = 2 and M q − Λ 3 = 0.1. In [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: shows three-dimensional visualizations of the photon sphere radius (left panel) and shadow radius (right panel) influenced by core density, core radius, and CS parameter. Our analysis shows that raising α expands both radii (rph, Rs). In contrast, core density and rad…

**Figure 14.** Figure 14: The geometrical shape of the shadow radius in terms of celestial coordinates for several values of [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Spontaneous scalarization around a black hole in a dark matter halo
gr-qc 2026-07 reject novelty 4.0 of 10

A Hernquist dark-matter halo is claimed to scalarize a central black hole, with the coupling quantized as η_n = −(n+1/2)²π³/(2C), but the mechanism requires ρ−p>0 while the paper's own formulas give ρ−p<0.
Particle Dynamics and Thermal Properties in Kalb-Ramond ModMax Black Holes: Theoretical Predictions for Observational Tests of Exotic Physics
gr-qc 2025-08 reject novelty 3.0 of 10

A parameter scan of geodesic, thermal, shadow, and lensing observables for Kalb-Ramond ModMax black holes, with sign errors in the temperature, specific heat, and deflection angle formulas.

Reference graph

Works this paper leans on

90 extracted references · 52 canonical work pages · cited by 2 Pith papers

[1]

Bertone, D

G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279 (2005)

2005
[2]

Persic, P

M. Persic, P. Salucci, and F. Stel, Mon. Not. R. Astron. Soc. 281, 27 (1996)

1996
[3]

Aghanim et al

N. Aghanim et al. (Planck collaboration), Astron. Astrophys. 641, A6 (2020)

2020
[4]

Hernquist, Astrophys

L. Hernquist, Astrophys. J. 356, 359 (1990)

1990
[5]

Dehnen, Mon

W. Dehnen, Mon. Not. R. Astron. Soc. 265, 250 (1993)

1993
[6]

J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997)

1997
[7]

Klypin, A

A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada, Astrophys. J. 522, 82 (1999)

1999
[8]

Akiyama et al

K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L1 (2019). 26

2019

Show all 90 references

[9]

Akiyama et al

K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, L12 (2022)

2022
[10]

Jusufi, M

K. Jusufi, M. Jamil, P. Salucci, T. Zhu, and S. Haroon, Phys. Rev. D 100, 044012 (2019)

2019
[11]

R. A. Konoplya, Phys. Lett. B 795, 1 (2019)

2019
[12]

X. Hou, Z. Xu, M. Zhou, and J. Wang, JCAP 07, 015 (2018)

2018
[13]

Al-Badawi, S

A. Al-Badawi, S. Shaymatov and Y. Sekhmani, JCAP 02, 14 (2025)

2025
[14]

Al-Badawi and S

A. Al-Badawi and S. Shaymatov, Chin. Phys. C 49, 055101 (2025)

2025
[15]

Al-Badawi and S

A. Al-Badawi and S. Shaymatov Commun. Theor. Phys. 77 035402 (2025)

2025
[16]

Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983)

S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983)

1983
[17]

K. S. Thorne, Astrophys. J. 191, 507 (1974)

1974
[18]

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973)

1973
[19]

Bozza, Phys

V. Bozza, Phys. Rev. D 66, 103001 (2002)

2002
[20]

K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000)

2000
[21]

Koga and T

Y. Koga and T. Harada, Phys. Rev. D 100, 064040 (2019)

2019
[22]

Falcke, F

H. Falcke, F. Melia, and E. Agol, Astrophys. J. Lett. 528, L13 (2000)

2000
[23]

S. A. Teukolsky, Astrophys. J. 185, 635 (1973)

1973
[24]

W. H. Press and S. A. Teukolsky, Astrophys. J. 185, 649 (1973)

1973
[25]

Regge and J

T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957)

1957
[26]

F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970)

1970
[27]

K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999)

1999
[28]

Sakalli and S

˙I. Sakalli and S. Kanzi, Turk. J. Phys. 46, 51 (2022)

2022
[29]

Cardoso and J

V. Cardoso and J. P. S. Lemos, Phys. Rev. D 64, 084017 (2001)

2001
[30]

Berti, V

E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009)

2009
[31]

Sakalli, Phys

I. Sakalli, Phys. Rev. D 94, 084040 (2016)

2016
[32]

G¨ ursel and˙I

H. G¨ ursel and˙I. Sakallı, Adv. High Energy Phys. 2018, 8504894 (2018)

2018
[33]

Sakallı, K

˙I. Sakallı, K. Jusufi and A. ¨Ovg¨ un, Gen. Rel. Grav.50, 125 (2018)

2018
[34]

Detweiler, Phys

S. Detweiler, Phys. Rev. D 22, 2323 (1980)

1980
[35]

P. L. Chrzanowski, Phys. Rev. D 11, 2042 (1975)

1975
[36]

de Vries, Class

A. de Vries, Class. Quant. Grav. 17, 123 (1999)

1999
[37]

Takahashi, Astrophys

R. Takahashi, Astrophys. J. 611, 996 (2004)

2004
[38]

Akiyama et al

K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L4 (2019)

2019
[39]

Banerjee, A

A. Banerjee, A. Pradhan, I. Sakalli and A. Dixit, Class. Quant. Grav. 42, 025008 (2025)

2025
[40]

Tangphati, I

T. Tangphati, I. Sakalli, A. Banerjee and A. Pradhan, Chin. Phys. C 49, 025110 (2025)

2025
[41]

Akiyama et al

K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, L14 (2022)

2022
[42]

Haroon, M

S. Haroon, M. Jamil, K. Jusufi, K. Lin, and R. B. Mann, Phys. Rev. D 99, 044015 (2019)

2019
[43]

¨Ovg¨ un,˙I

A. ¨Ovg¨ un,˙I. Sakallı and J. Saavedra, JCAP 10, 041 (2018)

2018
[44]

R. C. Pantig and A. ¨Ovg¨ un, Eur. Phys. J. C82, 391 (2022)

2022
[45]

Vilenkin and E

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 1994)

1994
[46]

T. W. B. Kibble, J. Phys. A 9, 1387 (1976)

1976
[47]

P. S. Letelier, Phys. Rev. D 20, 1294 (1979)

1979
[48]

Ahmed, A

F. Ahmed, A. Al-Badawi, and ˙I. Sakallı, Eur. Phys. J. C 85, 554 (2025). 27

2025
[49]

Ahmed, A

F. Ahmed, A. Al-Badawi, and ˙I Sakallı, Nucl. Phys. B 1017, 116961 (2025)

2025
[50]

Luis C. N. Santos, Phys. Rev. D 111 , 064032 (2025)

2025
[51]

Zafar, K

U. Zafar, K. Bamba, T. Rasheed and K. Bhattacharya, Phys. Lett. B 864 , 139446 (2025)

2025
[52]

Hamil and B

B. Hamil and B. C. L¨ utf¨ uo˘ glu, Chin. Phys.C 49 , 025107 (2025)

2025
[53]

H. K. Sudhanshu, D. V. Singh, S. Upadhyay, Y. Myrzakulov and K. Myrzakulov, Phys. Dark Univ. 46, 101648 (2024)

2024
[54]

J. Mo, F. Hoekstra, S. J. Neundom¨ orfer and J. van de Weygaert, Mon. Not. R. Astron. Soc. 336, 112 (2002)

2002
[55]

S. P. Bamford, R. C. Nichol, et al. , Mon. Not. Roy. Astron. Soc. 393, 1324 (2009)

2009
[56]

S. K. Jha, arXiv:2503.19938 [gr-qc]
[57]

Padmanabhan, Gravitation: Foundations and Frontiers (Cambridge University Press, Cambridge, 2010)

T. Padmanabhan, Gravitation: Foundations and Frontiers (Cambridge University Press, Cambridge, 2010)

2010
[58]

S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison Wesley, San Francisco, 2004)

2004
[59]

R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)

1984
[60]

Hawking and G

S. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)

1973
[61]

Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP Press, Woodbury, 1995)

M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP Press, Woodbury, 1995)

1995
[62]

Ashtekar and J

A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21, R53 (2004)

2004
[63]

S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983)

1983
[64]

Witten, Adv

E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)

1998
[65]

E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)

2006
[66]

G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977)

1977
[67]

J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986)

1986
[68]

V. P. Frolov and I. D. Novikov, Black Hole Physics: Basic Concepts and New Developments (Kluwer Academic Publishers, Dordrecht, 1998)

1998
[69]

Ahmed, Int

F. Ahmed, Int. J. Geom. Meths. Mod. Phys. 21, 2450187 (2024)

2024
[70]

Ahmed, Int

F. Ahmed, Int. J. Geom. Meths. Mod. Phys. 22, 2450336 (2025)

2025
[71]

Ahmed, ˙I

F. Ahmed, ˙I. Sakallı and A. Al-Badawi, Phys. Lett. B 864 , 139448 (2025)

2025
[72]

Hamil, B

B. Hamil, B. C. Lutfuoglu, F. Ahmed and Z. Yousaf, Nucl. Phys. B 1014, 116861 (2025)

2025
[73]

Cardoso, A

V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79 , 064016 (2009)

2009
[74]

Saadati, F

R. Saadati, F. Shojai, Class. Quantum Gravity 41, 015032 (2024)

2024
[75]

B. P. Abbott, et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 119, 161101 (2017)

2017
[76]

Ciftci, R.L

H. Ciftci, R.L. Hall, N. Saad, J. Phys. A: Math. Gen. 36, 11807 (2003)

2003
[77]

H. T. Cho, A. S. Cornell, J. Doukas, T. R. Huang and W. Naylor, Adv. Math. Phys. 2012, 281705 (2012)

2012
[78]

Ahmed, A

F. Ahmed, A. Al-Badawi, ˙I. Sakallı and A. Bouzenada, Nucl. Phys. B 1011, 116806 (2025)

2025
[79]

Ahmed, A

F. Ahmed, A. Al-Badawi, ˙I. Sakallı and S. Kanzi, Phys. Dark Univ. 48, 101907 (2025)

2025
[80]

Al-Badawi and F

A. Al-Badawi and F. Ahmed, Chin. J. Phys. 94, 185 (2025)

2025
[81]

Ahmed, J

F. Ahmed, J. Goswami and A. Bouzenada, Eur. Phys. J C 85, 110 (2025)

2025
[82]

Ahmed, A

F. Ahmed, A. Al-Badawi, ˙I. Sakallı, Eur. Phys. J. C 85, 554 (2025)

2025
[83]

Ahmed, A

F. Ahmed, A. Al-Badawi, and ˙I Sakallı, Phys. Dark Univ. 48, 101925 (2025)

2025
[84]

Al-Badawi, F

A. Al-Badawi, F. Ahmed and ˙I. Sakallı, Eur. Phys. J. C 85, 660 (2025)

2025
[85]

F. Ahmed. A. Al-Badawi and ˙I. Sakallı, Nucl. Phys. B 1017, 116951 (2025)

2025
[86]

Zhang, M

X. Zhang, M. Wang, and J. Jing, Sci. China-Phys. Mech. Astron. 64, 250411 (2021)

2021
[87]

J. A. Wheeler, Geometrodynamics (Academic Press, New York, 1973)

1973
[88]

A. R. Ruffini, Black Holes: Les Astres Occlus , (Gordon and Breach Science Publishers, New York, 1973)

1973
[89]

D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. 29, 465 (1957)

1957
[90]

S. E. Vazquez and E. P. Esteban, Nuovo Cimento B 119, 489 (2004). 28

2004

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Bertone, D

G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279 (2005)

2005

[2] [2]

Persic, P

M. Persic, P. Salucci, and F. Stel, Mon. Not. R. Astron. Soc. 281, 27 (1996)

1996

[3] [3]

Aghanim et al

N. Aghanim et al. (Planck collaboration), Astron. Astrophys. 641, A6 (2020)

2020

[4] [4]

Hernquist, Astrophys

L. Hernquist, Astrophys. J. 356, 359 (1990)

1990

[5] [5]

Dehnen, Mon

W. Dehnen, Mon. Not. R. Astron. Soc. 265, 250 (1993)

1993

[6] [6]

J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997)

1997

[7] [7]

Klypin, A

A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada, Astrophys. J. 522, 82 (1999)

1999

[8] [8]

Akiyama et al

K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L1 (2019). 26

2019