Strong Lensing and Quasinormal modes of black hole around global monopole

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 $η$. 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 $η$ 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.

• The paper demonstrates that global monopoles enhance deflection angles and expand Einstein ring radii using both analytic and numerical methods.
• It quantifies how increased monopole parameters shift ISCOs, enlarge shadow radii, and modify photon sphere properties.
• The QNM analysis reveals that higher monopole strength lowers oscillation frequencies and prolongs ringdown signals, affecting observational signatures.

Strong Gravitational Lensing and Quasinormal Modes of Black Holes with Global Monopoles

Introduction

The paper "Strong Lensing and Quasinormal modes of black hole around global monopole" (2604.05686) presents a comprehensive study of the spacetime structure and observational signatures of static, spherically symmetric black holes endowed with global monopoles. The analysis encompasses strong field gravitational lensing, shadow formation, timelike geodesic motion, and the spectrum of electromagnetic quasinormal modes (QNMs). Notably, the impact of the global monopole parameter $\eta$ and the self-coupling constant $\lambda$ on lensing observables, shadow radius, ISCO, Lyapunov exponent, and QNM frequencies is rigorously quantified using both analytical and numerical techniques.

Black Hole Metric and Horizon Structure

The monopole-modified metric employs a spherically symmetric ansatz, with the line element incorporating the parameters $\eta$ (symmetry-breaking scale) and $\lambda$ (self-coupling). Analysis of the metric function reveals that, for sufficient values of $M$ and $\lambda$, the spacetime admits two horizons: a Cauchy and an event horizon, whose separation and radii depend sensitively on $\eta$ and $\lambda$.

Figure 1: Variation of the metric function as a function of $\eta$ (left) and $\lambda$ (right). Enhanced monopole effects increase the event horizon radius and separation between horizons.

Strong Gravitational Lensing: Deflection Angles and Observables

Application of the strong deflection limit formalism, following Bozza and related works, leads to analytic expressions for the deflection angle $\lambda$0, lensing coefficients $\lambda$1 and $\lambda$2, and critical impact parameter $\lambda$3 for photon orbits near the black hole. The results demonstrate a monotonic increase in the deflection angle and divergence at larger $\lambda$4 as either $\lambda$5 or $\lambda$6 increases. This effect translates to notable changes in strong field lensing observables, including the limiting angular position $\lambda$7, angular separation $\lambda$8, and the flux ratio $\lambda$9.

Figure 2: The plot shows $\eta$0 as a function of the impact parameter $\eta$1 for various $\eta$2 (left) and $\eta$3 (right); divergence reflects the photon sphere's role.

Figure 3: $\eta$4 increases as either $\eta$5 or $\eta$6 is raised, indicating a larger angular radius for the limiting set of unresolved images.

Figure 4: The separation $\eta$7 grows with $\eta$8 and decreases with $\eta$9, reflecting shifts in the location of relativistic images.

Figure 5: The flux ratio $\lambda$0's non-monotonic dependence on $\lambda$1 and monotonic growth with $\lambda$2 affects detectability of relativistic images.

The paper further shows analytic and numerical evidence for monotonic growth in the Einstein ring radius with $\lambda$3 and $\lambda$4, directly linking strong field lensing to topological defect parameters intrinsic to the spacetime.

Figure 6: The radius of the outermost Einstein ring grows with rising $\lambda$5 and $\lambda$6, directly tied to spacetime modifications induced by the monopole field.

Figure 7: Behaviour of the Einstein ring $\lambda$7 highlights increased ring size for larger $\lambda$8 and $\lambda$9.

Additionally, the relative time delays between differentiated relativistic images increase with both parameters, indicating a potentially measurable strong lensing signature in time-domain multi-image observations.

Figure 8: Time delay $M$0 between the first and second relativistic images as a function of $M$1 (left) and $M$2 (right).

Black Hole Shadow and Photon Sphere

The structure of photon orbits and corresponding critical impact parameter leads to quantitative predictions for the shadow radius as seen by distant observers. The photon sphere and shadow radius both exhibit strong sensitivity to $M$3 and $M$4:

Figure 9: Photon sphere radius and shadow radius both increase with $M$5, expanding the region accessible to unstable photon trajectories.

Figure 10: Photon sphere and shadow radius growth with $M$6 displays a saturating trend at high coupling.

Tabulated results confirm that the shadow size is more responsive to $M$7, implying that detection of unusually large black hole shadows would be indicative of nontrivial monopole structure or other topological defects.

Timelike Geodesics, ISCO, and Stability

The detailed analysis of the effective potential for timelike geodesics yields explicit dependence of circular orbit structure—including $M$8, $M$9, and $\lambda$0—on both black hole parameters. The ISCO radius monotonically increases with both $\lambda$1 and $\lambda$2, with enhanced $\lambda$3 inducing a steeper increase and indicating a repulsive shift due to the topological defect. This effectively moves the zone of stable accretion disk orbits outward.

Figure 11: Radius of the ISCO as a function of $\lambda$4 (left) and $\lambda$5 (right); robust increase with both parameters extends the stable orbital region outward.

The Lyapunov exponent $\lambda$6, quantifying instability of circular timelike orbits, decreases with increasing $\lambda$7 (more stability) and increases with larger $\lambda$8 (more unstable orbits), although the effect of $\lambda$9 is dominant.

Figure 12: Lyapunov exponent $\eta$0 for varying $\eta$1 (left) and $\eta$2 (right); larger $\eta$3 yields more stable orbits.

Electromagnetic Perturbations and Quasinormal Mode Spectrum

The electromagnetic QNM spectrum is computed via both WKB-Padé and AIM methods, confirming excellent mutual agreement. A primary result of the paper is that increasing $\eta$4 suppresses the real and imaginary parts of QNM frequencies, lowering the oscillation frequency and yielding slower damping—that is, longer-lived ringdown signals. In contrast, increasing $\eta$5 produces a subtler, non-monotonic effect in the imaginary part, especially at low $\eta$6 and $\eta$7.

Figure 13: Time-domain profiles of $\eta$8 show that increasing $\eta$9 yields slower decaying quasinormal ringing; increasing $\lambda$0 prompts faster decay.

This behaviour is traced to modifications of the effective potential barrier (height reduction and narrowing with higher $\lambda$1 and $\lambda$2), allowing wave leakage and affecting mode lifetimes.

Implications and Outlook

The paper establishes that the presence of a global monopole introduces observable signatures in strong gravitational lensing (enhanced deflection angles, increased shadow size, enlarged Einstein rings, and longer time delays), geodesic orbital structure (outward-shifted ISCO and photon sphere), and in the stability and mode content of electromagnetic perturbations (longer-lived ringdown). These shifts provide feasible targets for precision lensing and multi-messenger gravitational wave observations, thus opening a pathway toward direct or indirect detection of topological defects through black hole phenomenology.

Future directions include:

• Extension to spinning and charged monopole black holes;
• Analysis of non-spherically symmetric lensing and shadow shapes;
• Application to realistic astrophysical systems where ISCO/ringdown frequency mismatches may signal underlying monopole structure;
• Inclusion of plasma and quantum corrections in the computation of observable shadows and lensing coefficients.

Conclusion

This work provides an in-depth and rigorous characterization of the influence of global monopoles on observable properties of static black holes. By connecting modifications of fundamental spacetime structure to measurable lensing, shadow, and oscillation parameters, the paper delineates a robust theoretical framework for testing the presence of topological defects via black hole astrophysics. The analytic and numeric methodology extends naturally to further settings in alternative gravity and topological defect cosmology, reinforcing the importance of precise strong-field astrophysical signatures for fundamental physics.