Probing Rotating Einstein-Power-Yang-Mills Black Holes through Shadows and Quasinormal Modes: Prospects for Event Horizon Telescope Constraints

Technical Summary: Probing Rotating Einstein–Power–Yang–Mills Black Holes through Shadows and Quasinormal Modes

Problem Statement
While the Event Horizon Telescope (EHT) has successfully imaged the shadows of M87* and Sgr A*, moving strong-field gravity into the observational regime, theoretical models often rely on static metrics or specific limits (e.g., Reissner-Nordström). The Einstein–Power–Yang–Mills (EPYM) theory, which generalizes Maxwell and Yang-Mills fields via a power exponent $q$, has been studied extensively in static, spherically symmetric configurations. However, a rotating EPYM black hole solution, free of additional couplings like Gauss-Bonnet terms, had not been constructed and constrained against EHT data. Furthermore, the interplay between the spin $a$, the magnetic Yang-Mills charge $Q$, and the power parameter $q$ on both the shadow geometry and the quasinormal mode (QNM) spectrum in a single rotating framework remained unexplored.

Methodology
The authors address this gap through a three-step theoretical and phenomenological approach:

1. Metric Construction: Starting from the static, spherically symmetric EPYM seed metric, the authors generate a rotating solution using the complexification-free Newman-Janis algorithm (Azreg-Aïnou version). This method avoids the ambiguities of standard complexification procedures for non-vacuum sources, yielding a Kerr-like metric in Boyer-Lindquist coordinates controlled by mass $M$, spin $a$, magnetic charge $Q$, and power exponent $q$.
2. Shadow and Photon Region Analysis: Utilizing the separability of the Hamilton-Jacobi equation for this metric, the authors derive the equations for null geodesics. They compute the unstable photon region and the resulting shadow silhouette as seen by a distant observer. Key observables include the areal shadow radius $R_s$, the oblateness $D$, and the distortion parameter $\delta_s$. These are used to calculate the angular diameter $\theta_d$ and the fractional Schwarzschild deviation $\delta$.
3. Quasinormal Mode Calculation: The authors analyze the stability and ringdown of the spacetime by studying massless scalar perturbations. They employ the leading-order Wentzel-Kramers-Brillouin (WKB) approximation on the Teukolsky-type radial potential to compute the complex quasinormal frequencies $\omega = \omega_R + i\omega_I$.

Key Results

• Horizon Structure and Ergoregion: The rotating EPYM black hole possesses an event horizon and a Cauchy horizon for sub-extremal spins. The extremal spin limit $a_{ext}$ depends on $Q$ and $q$. The charge $Q$ acts repulsively, shrinking the event horizon as it increases, while the power parameter $q$ modulates the strength of this effect. The ergoregion exists for all $a > 0$ and its extent is sensitive to both $Q$ and $q$.
• Shadow Constraints from EHT:
  • The Yang-Mills charge $Q$ is repulsive for all admitted $q$ ($3/4 < q < 3/2$), causing the shadow to contract as $Q$ increases.
  • The power parameter $q$ dictates the sensitivity of the shadow to the charge. Near the lower bound ($q \to 3/4$), the shadow contracts sharply with increasing $Q$, leading to tight constraints. Near the upper bound ($q \to 3/2$), the shadow becomes nearly insensitive to $Q$, weakening the constraints derived from angular size alone.
  • Quantitative Bounds: Using EHT measurements for M87* and Sgr A*, the authors constrain the parameter space. At the Maxwell point ($q=1$) and fixed spin $a=0.7M$, the M87* angular size restricts $Q \lesssim 0.52 M$ (1$\sigma$). A spin-marginalized joint likelihood analysis combining both sources tightens this to $Q \lesssim 0.26 M$ at 1$\sigma$. The deviation parameter $\delta$ provides complementary constraints, particularly effective near $q=3/2$ where size constraints weaken.
• Energy Emission: The shadow radius determines the high-frequency limiting absorption cross-section. The authors find that increasing $Q$ suppresses the energy emission rate and extends the black hole's lifetime, whereas increasing $q$ enhances emission. Near-extremal spin suppresses emission entirely due to vanishing Hawking temperature.
• Quasinormal Modes:
  • The oscillation frequency ($\omega_R$) increases with both charge $Q$ and spin $a$, while the damping rate ($-\omega_I$) decreases.
  • This confirms an inverse correlation between shadow size and ringdown frequency, consistent with the eikonal limit correspondence.
  • The power parameter $q$ leaves an imprint on the QNM spectrum, but the shift is small (a few percent for moderate charges) and currently lies below the resolution of gravitational-wave detectors like LISA, though it is theoretically observable in principle.

Significance and Claims
The paper claims to provide the first rotating EPYM black hole solution free of Gauss-Bonnet terms and to confront this specific geometry with EHT data. The primary significance lies in demonstrating that the shadow size and QNM spectrum offer a dual probe to disentangle the effects of the Yang-Mills charge $Q$ and the power exponent $q$. The authors emphasize that while current data is consistent with the standard Kerr black hole (vanishing $Q$), the framework allows for independent constraints on these parameters. The study highlights that the sensitivity of observables to the charge is not uniform but is "graded" by the power parameter $q$, being strongest near $q=3/4$ and weakest near $q=3/2$. The work establishes a phenomenological baseline for testing nonlinear gauge field theories against future high-precision black hole imaging and gravitational wave observations.