Optical Appearance and Ringdown of Black Holes in a Kalb Ramond Field Coupled to Perfect Fluid Dark Matter

Optical Appearance and Ringdown of Black Holes in a Kalb–Ramond Field Coupled to Perfect Fluid Dark Matter: An In-Depth Analysis

Theoretical Framework and Black Hole Solutions

The study presents a comprehensive analysis of static, spherically symmetric black holes in the background of a Kalb–Ramond (KR) field nonminimally coupled to perfect fluid dark matter (PFDM). This framework extends general relativity by introducing a Lorentz-violating antisymmetric tensor field characterized by a parameter $\alpha$ and includes PFDM via the parameter $\lambda$, both altering the gravitational sector. The resulting metric,

$$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$

with $f(r)$ explicitly depending on both $\alpha$ and $\lambda$, embodies nontrivial IR and UV modifications. These parameters induce significant modifications of horizon structure, photon sphere, and geodesic properties, with reductions in the horizon radius $r_h$, photon sphere radius $r_{\rm ph}$, critical impact parameter $b_c$, and ISCO radius $r_{\rm isco}$ as either $\lambda$0 or $\lambda$1 increases.

Photon Trajectories, Black Hole Shadow, and Photon Rings

The propagation of null geodesics is governed by effective potentials encoding the parametric deformations from the KR field and PFDM. The paper details the classification of photon trajectories via impact parameter $\lambda$2, with critical values distinguishing between direct, lensed, and photon-ring (multi-orbit) rays. The function $\lambda$3, as defined by the transformed radial equation, encapsulates the allowed and forbidden regions for photon motion.

Figure 1: The function $\lambda$4 delineates the permitted and forbidden domains in photon propagation for $\lambda$5 and $\lambda$6, crucial for shadow formation.

The relationship between the number of photon orbits $\lambda$7 and the impact parameter $\lambda$8 captures the transition from direct escape to capture and supports an explicit multi-image structure, essential for interpreting high-resolution images of black hole environments.

Figure 2: Mapping of $\lambda$9 and associated photon trajectories demonstrates the onset of ring formation as $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$0 approaches $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$1.

The shadow boundary—the observable photon ring—shrinks monotonically as $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$2 or $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$3 increases, which has direct implications for interpreting EHT-like observations in environments where exotic fields and nonbaryonic matter are present.

Thin Accretion Disk Models and Optical Appearance

Three thin-disk emission models are thoroughly examined, each with distinct emissivity prescriptions. The transfer functions $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$4, determined by repeated photon-disk intersection points, encode the structure of direct, lensing, and photon rings present in the observed intensity.

Figure 3: The leading transfer functions $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$5 showcase the diminishing contribution of higher-order disk images to the total observed intensity.

Images of the accretion disk exhibit a marked dependence on the KR and PFDM parameters. Direct emission features single-peaked profiles, but strong-field lensing generates multi-peak observed intensity profiles, with the separation and prominence of photon rings modulated by $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$6 and $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$7.

Figure 4: Synthetic disk images for varying $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$8 (with fixed $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2),$$9) illustrate the compressive effect on both emission and observed intensity maps.

Figure 5: Disk morphology for varying $f(r)$0 (with fixed $f(r)$1) further corroborates the systematic contraction of bright rings and the photon ring radius.

The impact of varying $f(r)$2 and $f(r)$3 reveals that enhanced Lorentz violation or dark matter density compresses all detectable disk and shadow features toward smaller angular radii, an effect of direct observational interest.

Black Hole Ringdown and Quasinormal Mode Analysis

The study investigates scalar, electromagnetic, and axial gravitational perturbations, each governed by Regge-Wheeler–like wave equations with effective potentials $f(r)$4 exhibiting parameter-sensitive peak heights and widths.

Figure 6: Effective potential profiles for different perturbation spins manifest the impact of $f(r)$5 on mode trapping and transmission.

Figure 7: Increasing $f(r)$6 elevates barrier height and sharpens confinement of the perturbation field.

For each spin, the real and imaginary parts of the QNM frequencies ($f(r)$7) are computed via both 6th-order WKB and time-domain (Prony) methods. As $f(r)$8 or $f(r)$9 increases, both $\alpha$0 and $\alpha$1 grow, signifying more rapid oscillations and faster ringdown decay. Notably, scalar modes have the largest $\alpha$2 and the fastest decay, while axial gravitational modes are the slowest to damp.

Figure 8: Time-domain profiles highlight differential damping behaviors across field spins for fixed $\alpha$3.

Figure 9: Variations in ringdown persistence and frequency with increasing $\alpha$4 are directly visible in the time-evolution data.

At high-$\alpha$5 (eikonal) limit, the QNM frequencies align with the orbital frequency and instability exponent of the photon sphere, demonstrating a direct correspondence between optical and dynamical signatures, with decreasing relative error as $\alpha$6. This provides a robust theoretical link relevant to multi-messenger astrophysical tests.

Implications and Future Directions

The results establish that modifications from Kalb–Ramond fields and PFDM yield measurable shifts in both optical and gravitational-wave signatures around black holes. Observed shadow size, photon ring location, peak disk emission, and QNM spectrum are all systematically reduced and tightened with increasing Lorentz-violation or dark-matter background. This has two direct implications:

1. Observational constraints: Precise VLBI imaging and ringdown spectroscopy can jointly constrain $\alpha$7 in real astrophysical scenarios, distinguishing modified gravity and exotic matter models.
2. Model discrimination: Departure from the standard Kerr paradigm in horizon-scale and GW data—especially when multi-modal QNM spectra are available—can be used to test the presence of nontrivial Lorentz violation and/or dark matter halos.

Theoretical extensions may involve inclusion of black hole spin, non-spherical accretion, or polarized/mixed perturbation analysis. Further, non-eikonal QNM modes and the regime of strong gravitational lensing with plasma or magnetic fields require detailed study for direct application to upcoming observational campaigns.

Conclusion

This investigation clarifies how the combined effects of a Kalb–Ramond field and perfect fluid dark matter alter the optical structures and dynamical ringdown signals of black holes. The explicit connection between deformation parameters $\alpha$8 and both multimodal observational features provides a concrete avenue for future constraints and theoretical exploration in both electromagnetic and gravitational wave observations. The parametric trends identified herein will be critical for assessing alternative gravitational phenomenology in astrophysical tests across the next generation of observational platforms.