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arxiv: 2604.05686 · v1 · submitted 2026-04-07 · 🌀 gr-qc

Strong Lensing and Quasinormal modes of black hole around global monopole

Irengbam Roshila Devi , Ningthoujam Media , Yenshembam Priyobarta Singh , Telem Ibungochouba Singh This is my paper

Pith reviewed 2026-05-10 19:51 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black holeglobal monopolestrong lensingquasinormal modesblack hole shadowinnermost stable circular orbitelectromagnetic perturbation
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The pith

Increasing the global monopole parameter around a black hole produces gravitational waves with slower damping oscillations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a static spherically symmetric black hole that contains a global monopole characterized by a parameter eta. It computes the deflection of massive particles in the strong-field limit, the size of the black hole shadow, the radii of stable circular orbits including the ISCO, and the quasinormal modes excited by electromagnetic perturbations. The central result is that larger values of the monopole parameter increase the shadow radius, raise the ISCO radius, and reduce the damping rate of the quasinormal ringing. A sympathetic reader would care because these changes alter both the lensing signatures and the ringdown waveforms that could be observed from real black holes if global monopoles exist.

Core claim

In the spacetime metric of a black hole with a global monopole, the quasinormal frequencies for electromagnetic perturbations are computed as a function of the monopole parameter eta. The results show that increasing eta produces modes with smaller imaginary parts, corresponding to gravitational waves that damp more slowly. Time-domain evolution of the perturbations confirms the characteristic quasinormal ringing followed by late-time power-law tails, validating the frequency-domain spectrum.

What carries the argument

The static spherically symmetric metric containing a global monopole, used to derive deflection angles, shadow radii, ISCO locations, and quasinormal mode spectra via standard perturbation methods.

If this is right

  • Larger monopole parameter produces larger black hole shadow radii.
  • The radius of the innermost stable circular orbit increases monotonically with the monopole parameter.
  • Deflection angle of massive particles grows when the monopole parameters increase.
  • Stability analysis via Lyapunov exponent indicates changes in timelike geodesic behavior with eta.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Observed black hole shadows could be used to place upper limits on possible monopole strength if the metric form holds.
  • Slower damping could extend the duration of detectable ringdown signals in gravitational wave data.
  • The late-time power-law tails remain present, so any monopole effect would appear mainly in the decay rate rather than the tail exponent.

Load-bearing premise

The spacetime is exactly described by the given static spherically symmetric metric with a global monopole at the center, taken without re-derivation or stability check.

What would settle it

A direct computation or numerical evolution showing that the imaginary part of the quasinormal frequency increases or remains unchanged as the monopole parameter rises.

Figures

Figures reproduced from arXiv: 2604.05686 by Irengbam Roshila Devi, Ningthoujam Media, Telem Ibungochouba Singh, Yenshembam Priyobarta Singh.

Figure 1
Figure 1. Figure 1: Variation of the metric function as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The plot shows α(rm) changes with b for various η (left panel) and λ (right panel). The colored points on the horizontal axis indicate bc, where the deflection angle diverges. 2.1. Lens observation In this section, we will analyse the effect of lensing observables by the parameters λ and η in the strong field limit. Assume that the observer and the source are located in a flat region of spacetime, situated… view at source ↗
Figure 3
Figure 3. Figure 3: Variation of the strong lensing observable [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Variation of the strong lensing observable [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of the strong lensing observable [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Lensing coefficient ¯a in the strong field limit versus the parameters η (left) and λ (right). 0.00 0.05 0.10 0.15 -1 0 1 2 3 4 5 η b (a) Here, M = 1,G = 1 and λ = 105 100 105 110 115 120 125 130 -0.02 0.00 0.02 0.04 λ b (b) Here, M = 1,G = 1 and η = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Lensing coefficient b¯ in the strong field limit versus the parameters η (left) and λ (right). 11 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plot of the outermost Einstein radius for di [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Behavior of the Einstein ring θ E 1 in strong field limit with respect to the parameters η (left) and λ (right). 12 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Behavior of the time deley ∆T s 2,1 in strong field limit with respect to the parameters η (left) and λ (right). 2.4. Shadow radius of black hole The black hole’s gravitational field deflects light coming from faraway celestial bodies. Some of the photons which move along particular unstable circular orbits are trapped by the gravitational field, creating a photon sphere. In this section, we examine the e… view at source ↗
Figure 11
Figure 11. Figure 11: Plot of photon sphere (left) and the shadow radius (right) for varying [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Plot of the photon sphere (left) and the shadow radius (right) for di [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Variation of the effective potential for different values of (a) η with λ = 105, L = 5, G = 1, M = 1 and (b) λ with L = 5, M = 1, G = 1 and η = 0.05. In Figs. 13 and 14, we illustrate the general behaviour of the effective potential Ve f f(r) as a function of the radial coordinate r. The unstable and stable circular orbits are determined by the maximum and minimum values of Ve f f(r). So, the unstable cir… view at source ↗
Figure 14
Figure 14. Figure 14: Variation of the effective potential for varying L with M = 1, λ = 105, η = 0.05, G = 1. notice that the radius of the unstable circular orbit decreases with increasing angular momentum L, while the radius of the stable circular orbit increases. For L = 3, the unstable circular orbit occurs at rU = 1.2483, the stable circular orbit at rS = 15.6586 and for L = 2, rU1 = 1.4179 and rS 1 = 6.1165. The ISCO co… view at source ↗
Figure 15
Figure 15. Figure 15: Radial profile of the energy E of timelike particles with respect to η (left) and λ (right). Here, (a) M = 1, λ = 105, G = 1 and (b) M = 1, η = 0.05, G = 1. η=0 η=0.04 η=0.08 1 2 3 4 5 6 1.0 1.5 2.0 2.5 3.0 3.5 r ℒ (a) λ=105 λ=120 λ=140 1 2 3 4 5 6 1.0 1.5 2.0 2.5 3.0 3.5 r ℒ (b) [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Radial profile of the angular momentum L of timelike particles with respect to η (left) and λ (right). Here, (a) M = 1, λ = 105, G = 1 and (b) M = 1, η = 0.05, G = 1. radius of the ISCO. Therefore, from Figs. 15 and 16, we notice that the radius of the ISCO enlarges with the increase in the global monopole parameter η and the constant parameter λ. The minimum radius at which stable circular orbits can exi… view at source ↗
Figure 17
Figure 17. Figure 17: Variation of the radius of the ISCO (rISCO) with respect to η (left) and λ (right). Here, (a) M = 1, λ = 105, G = 1 and (b) M = 1, η = 0.05, G = 1. 3.2. Stability of the timelike particles We will analyze the stability or instability of timelike geodesics in the spacetime using the Lyapunov exponents. The Lyapunov exponent evaluates how fast trajectories in the close vicinity of a spacetime either come to… view at source ↗
Figure 18
Figure 18. Figure 18: Plot of the specific energy EISCO at ISCO with respect to η (left) and λ (right). Here, (a) M = 1, λ = 105, G = 1 and (b) M = 1, η = 0.05, G = 1. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 1.4 η 1.6 1.8 2.0 2.2 2.4 2.6 ℒISCO (a) 100 110 120 130 140 150 160 1.50 λ 1.55 1.60 1.65 1.70 1.75 1.80 ℒISCO (b) [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Plot of the specific energy LISCO at ISCO with respect to η (left) and λ (right). Here, (a) M = 1, λ = 105, G = 1 and (b) M = 1, η = 0.05, G = 1. (converge) or move apart (diverge) over time [90]. Here, we will find the Lyapunov exponents of massive particles in an unstable circular orbit on the black hole’s equatorial plane. The Lyapunov exponent λL is defined as [88] λL = s − 1 2t˙ 2 ∂ 2Ve f f(r) ∂r 2 .… view at source ↗
Figure 20
Figure 20. Figure 20: Illustration of the Lyapunov exponent λL for varying η (left) and λ (right). Here, (a) M = 1, λ = 105, L = 1, G = 1 and (b) M = 1, η = 0.05, L = 1, G = 1. The analysis of circular orbits and their stability provides important information about particle dynamics in the given spacetime. To further probe the physical properties of the black hole, it is crucial to examine its response under external perturbat… view at source ↗
Figure 21
Figure 21. Figure 21: Effective potential graph as a function of the parameters η (left) and λ (right).The physical parameters are chosen as (a) M = 1, λ = 105, G = 1 and (b) M = 1, η = 0.05, G = 1. where n denotes the overtone number and the subscript 0 denotes evaluation at r∗(r0), the location where the effective potential attains its peak. V ′′ 0 is the second derivative of the potential with respect to r∗ evaluated at r0.… view at source ↗
Figure 22
Figure 22. Figure 22: Time-domain profiles of |ψ(t)| (log scale) showing quasinormal ringing and late-time tails for different values of η (left) and λ (right).The physical parameters are chosen as (a) M = 1, λ = 150, G = 1 and (b) M = 1, η = 0.04, G = 1. 7. Summary and Conclusion In this paper, we investigate a static and spherically symmetric spacetime with a global monopole, exploring the strong gravitational lensing effect… view at source ↗
read the original abstract

In this paper, we investigate various key aspects of a static and spherically symmetric black hole with global monopole. Firstly, we analyze the deflection angle in the strong field limit of massive particle by the global monopole. It shows that the angle of deflection increases when the two characteristic parameters for monopole configuration increase. The influence of the global monopole parameter on the lensing observables and the black hole shadow are studied. This shows that larger monopole parameter corresponds to larger shadow radii. The dynamics of timelike geodesics is also investigated in the spacetime. General circular orbits and the innermost stable circular orbits (ISCO) of timelike particles are discussed, highlighting that the monopole parameter significantly affects the circular orbits and the ISCO. In particular, it is observed that the radius of ISCO rises monotonically with $\eta$. In addition, the Lyapunov exponent is used to analyze the stability of timelike geodesics. The quasinormal modes for electromagnetic perturbation of the black hole with varying $\eta$ is also investigated. Our findings indicate that increasing the monopole parameter gives rise to gravitational waves with slower damping oscillations. To further validate the derived quasinormal mode spectrum, we discuss the evolution of electromagnetic perturbations in the time domain profile, confirming the presence of the characteristic quasinormal ringing followed by late-time power-law tails.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines a static spherically symmetric black hole spacetime containing a global monopole, computing the strong-field deflection angle of massive particles, lensing observables and shadow radius, properties of timelike geodesics including circular orbits and ISCO radius, Lyapunov exponents for stability, and quasinormal frequencies of electromagnetic perturbations. It reports that the monopole parameter increases deflection angles and shadow/ISCO radii while producing slower damping in the QNMs, with time-domain evolution used to confirm the ringing phase and late-time tails.

Significance. If the reported monotonic trends hold under scrutiny, the work adds concrete analytic and numerical results on how global monopoles modify lensing and gravitational-wave signatures, which may be relevant for distinguishing such spacetimes observationally. The combination of strong-lensing formulas, Lyapunov analysis, and time-domain validation of QNMs provides a reasonably self-contained set of checks within the adopted metric.

major comments (2)
  1. Quasinormal modes section: the central claim that increasing the monopole parameter produces slower damping oscillations is presented without error bars, resolution studies, or convergence tests on the numerical frequencies; this omission directly affects in the reported monotonic trend.
  2. Metric and geodesic sections: the explicit line element is not written out, so the effective potentials for geodesics and the electromagnetic perturbation equations cannot be independently reconstructed from the text alone.
minor comments (2)
  1. Abstract: the phrase 'two characteristic parameters for monopole configuration' is used, yet only a single parameter η appears in the reported trends; the second parameter should be identified and its role clarified.
  2. Time-domain discussion: the late-time power-law tails are stated to be present but no comparison is made to the expected analytic decay exponents for electromagnetic perturbations on this background.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment and recommendation for minor revision. We address each major comment below and will incorporate the suggested improvements in the revised version.

read point-by-point responses

  1. Referee: Quasinormal modes section: the central claim that increasing the monopole parameter produces slower damping oscillations is presented without error bars, resolution studies, or convergence tests on the numerical frequencies; this omission directly affects in the reported monotonic trend.

    Authors: We thank the referee for highlighting this point. The quasinormal frequencies were computed using a standard numerical approach for the electromagnetic perturbations, and the time-domain evolution already serves as an independent check confirming the ringing phase and the trend of slower damping with larger monopole parameter. However, we agree that explicit convergence tests and error estimates would strengthen the claim. In the revised manuscript we will add a brief discussion of the numerical method, results from resolution studies (varying grid size and time step), and approximate error bars on the imaginary parts of the frequencies to support the reported monotonic trend. revision: yes

  2. Referee: Metric and geodesic sections: the explicit line element is not written out, so the effective potentials for geodesics and the electromagnetic perturbation equations cannot be independently reconstructed from the text alone.

    Authors: We agree that explicitly stating the line element improves readability and allows independent verification. Although the metric is the standard static spherically symmetric form with the global monopole parameter, we will insert the full line element at the start of the relevant sections in the revised manuscript, followed by the derived effective potentials and perturbation equations for completeness. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from the standard static spherically symmetric metric with global monopole taken from prior literature (not self-authored in a load-bearing way), then applies standard geodesic equations, strong-lensing formulas, and the Regge-Wheeler/Zerilli perturbation equations for electromagnetic modes. All reported results—deflection angles, shadow radii, ISCO radii, Lyapunov exponents, and QNM frequencies/damping times—are obtained by direct numerical integration or analytic approximation of these equations as functions of the monopole parameter η. No parameters are fitted to the target observables, no self-referential definitions appear, and the central claim (slower damping with larger η) is a computed output rather than an input. The paper is therefore self-contained against external benchmarks with no reduction of predictions to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of an exact static spherical solution containing a global monopole (standard in the literature) and on the validity of linear perturbation theory for electromagnetic fields. No new entities are postulated and no parameters are fitted to data.

axioms (2)
  • domain assumption The spacetime is exactly described by the static spherically symmetric line element containing a global monopole.
    Invoked at the outset of every calculation; taken from prior literature without re-derivation.
  • standard math Linear perturbation theory applies to electromagnetic fields on this background.
    Used to obtain the quasinormal-mode spectrum and time-domain evolution.

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Reference graph

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