⚛️ general relativity

Black hole shadow parameters and quasi-normal modes for Weyl-incorporated gravity

This paper investigates the shadow parameters and fundamental quasi-normal mode frequencies of black holes within the Modified Relativistic Dynamics (MORD) framework of Weyl-interaction gravity, analyzing their dependence on the coupling parameter $\lambda$ under conditions of constant-density baryonic matter halos and homogeneous plasma backgrounds.

Original authors: Noraiz Tahir, Mubasher Jamil, Kaynat Fatima, Tajammal Hussain Khokhar

Published 2026-07-28

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CC BY 4.0

Original authors: Noraiz Tahir, Mubasher Jamil, Kaynat Fatima, Tajammal Hussain Khokhar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible trampoline. In the 1900s, a genius named Einstein figured out that heavy objects, like stars, don't just sit on this trampoline; they warp it, creating dips and curves. This warping is what we feel as gravity. But for decades, scientists noticed something weird: galaxies spin so fast that, according to Einstein's rules, they should fly apart. To fix this, we invented "dark matter," a ghostly substance we can't see but must be there to hold the galaxies together.

Now, a few brave thinkers asked: "What if we don't need ghost matter? What if Einstein's rules just need a tiny tweak?" This paper dives into one of those tweaks, called "Weyl-incorporated gravity." Think of it like this: in the standard story, matter and gravity are polite neighbors who only talk through the trampoline fabric itself. But in this new story, they have a secret, direct phone line. The math adds a special "interaction term" that lets matter talk directly to the pure, radiating part of the gravitational field. It's like if the trampoline could whisper directly to the bowling ball sitting on it, changing how they move together. Scientists are obsessed with this because if it's true, it could explain the spinning galaxies without needing any invisible dark matter at all.

So, what did the authors of this paper actually do? They decided to test this "secret phone line" theory on the most extreme objects in the universe: black holes. A black hole is a place where gravity is so strong that not even light can escape, creating a dark circle in the sky called a "shadow." The team wanted to see if this new gravity theory changes the size or shape of that shadow.

They started by building a mathematical model of a black hole sitting inside a cloud of regular matter (like the missing bits of stars we call "baryonic matter"). They assumed this cloud had a constant density, like a perfectly uniform fog. Then, they crunched the numbers to see how the "Weyl coupling" (the strength of that secret phone line) changed the black hole's behavior. They found that if this theory is true, the black hole's shadow isn't exactly the same as Einstein predicted. The size of the shadow depends on a specific number, called $\lambda$ , which describes how strong that matter-gravity connection is. For the black holes they modeled, they found that $\lambda$ needs to be somewhere between 3.5 and 3.9 to keep the black hole looking like a black hole.

But they didn't stop at just looking at the shadow. They also asked: "If we poke this black hole, how does it sing?" When a black hole is disturbed, it vibrates like a bell, sending out ripples called "quasi-normal modes." The authors simulated these vibrations using two different math tricks (WKB and AIM) to see how long the black hole would ring. Their results suggest that in this new theory, the black hole's "ringing" lasts a very, very long time—thousands of cycles before fading away. This is much longer than what we'd expect from Einstein's standard rules. They also checked what happens if the black hole is surrounded by a "plasma" (a hot, electric gas), and found that this gas makes the shadow slightly smaller, but the long-lasting ringing remains.

The authors are careful to tell us that this is a theoretical exploration, not a final proof. They used simplified models (like the uniform fog) to isolate the effects of the new theory. They aren't saying, "This is definitely how the universe works," but rather, "If this theory is real, here is what we should see." They suggest that future telescopes, like the Event Horizon Telescope which took the first picture of a black hole, might be able to spot these differences. If we see a black hole shadow that's slightly off or hear it ring for an unusually long time, it could be the first clue that gravity has a secret phone line we never knew about. Until then, it remains a fascinating "what if" that challenges our understanding of the cosmos.

Technical Summary: Black Hole Shadow Parameters and Quasi-Normal Modes for Weyl-Incorporated Gravity

Problem Statement and Theoretical Framework
This paper investigates the phenomenology of "Weyl-incorporated gravity" (WIG), a modified gravity theory proposed as a minimal extension to General Relativity (GR). The theoretical motivation for WIG stems from an analogy with quantum field theory, where the stress-energy tensor $T_{\mu\nu}$ couples directly to the pure radiative part of the gravitational field, represented by the Weyl tensor $C_{\alpha\mu\beta\nu}$ . This interaction is introduced via an additional term, $\lambda C_{\alpha\mu\beta\nu}T^{\alpha\beta}T^{\mu\nu}$ , in the Einstein-Hilbert Lagrangian, where $\lambda$ is a coupling constant. The authors aim to explore the strong-field regime of this theory by computing black hole shadow parameters and quasi-normal mode (QNM) frequencies, specifically examining whether the coupling parameter $\lambda$ exhibits dependence on black hole mass scales and how it alters observable signatures compared to standard GR.

Methodology
The study employs a spherically symmetric static metric ansatz, $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2d\Omega^2$ , within the WIG field equations. The analysis proceeds through several stages:

Metric Derivation: The authors assume the black hole is embedded in a constant-density baryonic matter halo ( $\rho_c$ ). They derive the lapse function $f(r)$ by solving the coupled field equations, which include a nonlinear interaction term $I_{\mu\nu}$ arising from the Weyl-matter coupling. The solution is obtained via a perturbative order-by-order approximation, treating the constant density halo as a leading-order background.
Shadow Parameter Calculation: Using the derived lapse function, the authors calculate the effective potential for photon orbits. They determine the photon sphere radius ( $r_p$ ) and the critical impact parameter ( $b = \sqrt{\eta + \zeta^2}$ ) to estimate the black hole shadow radius ( $R_s$ ). This analysis is performed for various black hole masses ( $m$ ) to identify the specific values of $\lambda$ required to maintain a valid event horizon structure.
Plasma Effects: The analysis is extended to include a homogeneous plasma background. The refractive index of the plasma is incorporated into the Hamilton-Jacobi equation to determine its impact on the null geodesics and the resulting shadow size.
Quasi-Normal Modes (QNMs): The stability of the spacetime is probed by analyzing massive scalar field perturbations. The authors employ the test-field approximation, keeping the background metric and matter fields fixed while perturbing the scalar field. Two independent semi-analytical methods are used to compute QNM frequencies ( $\omega = \omega_R + i\omega_I$ $ω = ω_{R} + i ω_{I}$ ):
- The 6th-order WKB approximation.
- The Asymptotic Iteration Method (AIM).
- Additionally, a time-domain integration using finite difference methods in characteristic null coordinates is performed to study the temporal evolution of the perturbations.

Key Results

Shadow Parameters: For a constant-density halo ( $\rho_c = 0.5$ ), the authors find that the coupling constant $\lambda$ must take values in the range $\lambda \sim 3.5 - 3.9$ to satisfy the horizon condition for black holes of varying masses. The shadow radius ( $R_s$ ) shows a positive correlation with black hole mass. The presence of the Weyl coupling modifies the effective potential, slightly elevating the photon sphere and shifting the shadow size relative to the Schwarzschild limit.
Plasma Influence: The introduction of a homogeneous plasma background reduces the shadow size. The reduction is more pronounced for higher plasma densities ( $\rho_p = 0.4$ compared to $\rho_p = 0.2$ ).
Quasi-Normal Modes: The QNM analysis reveals that all computed modes possess negative imaginary parts ( $\omega_I < 0$ $ω_{I} < 0$ ), indicating exponential decay and stability of the scalar perturbations in the test-field approximation.
- For massive scalar fields ( $\mu > 0$ ), the quality factors ( $Q = \omega_R / |\omega_I|$ ) are exceptionally high, reaching values of $O(10^3 - 10^4)$ . This implies that perturbations undergo thousands of oscillation cycles before decaying.
- The WKB and AIM methods show strong agreement (relative discrepancy < 1%) for $\mu > 1$ , though a larger variance (~30%) is observed for the fundamental massless mode ( $\ell=0, \mu=0$ ), consistent with known WKB limitations.
Time-Domain Evolution: Numerical integration of the wave equation shows that while early-time waveforms are similar to the Schwarzschild case, the WIG spacetime exhibits prolonged quasinormal ringing and slower decay in the intermediate and late-time regimes, particularly for low and intermediate multipoles ( $\ell=1, 2$ ).

Significance and Claims
The paper positions itself as a theoretical exploration of the WIG framework under simplified assumptions. The authors explicitly state that their results are not observational constraints but rather predictions of the theory's behavior in a specific non-GR sector.

Theoretical Consistency: The study demonstrates that WIG can support black hole solutions with consistent horizon structures in the presence of matter halos, provided specific values for the coupling constant $\lambda$ are adopted.
Distinctive Signatures: The primary phenomenological contribution is the prediction of "exceptionally long-lived" scalar perturbations (high $Q$ -factors) and modified shadow radii. These features arise from the effective potential modifications induced by the Weyl-matter coupling.
Limitations and Future Work: The authors are modest regarding the astrophysical applicability of their current results. They acknowledge that the constant-density halo assumption and the test-field approximation (which does not perturb the metric or interaction terms) are pedagogical simplifications. They note that degeneracies exist between modified gravity parameters and astrophysical uncertainties (e.g., accretion flow geometry). Consequently, the paper concludes that while WIG offers interesting deviations from GR, distinguishing it observationally requires future work involving realistic halo profiles, rotating black hole solutions, and a full perturbative analysis of the coupled metric-matter system.

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