Breathing Black Hole Shadows in Modified Gravity (MOG)

Abstract

In this paper, we investigate the dynamic phenomenological signatures of a Schwarzschild-MOG black hole shadow perturbed by passing gravitational waves. By perturbing the Hamilton-Jacobi equation for photon null geodesics, we demonstrate that the unique field content of MOG breaks the observational degeneracy with standard General Relativity. We mathematically prove two distinct, time-dependent signatures. First, the massless MOG scalar field induces a volumetric ``breathing mode'' polarization, causing the total apparent area of the shadow to rhythmically expand and contract. Second, the massive MOG vector field undergoes quantum vacuum dispersion, arriving at the observer with a predictable time delay. This delayed massive wave sources secondary longitudinal metric perturbations that manifest as a sudden, asymmetric translational wobble of the shadow on the celestial screen. These dynamic geometric shifts offer a robust observational template for next-generation interferometry to strictly test the existence of massive force carriers and scalar fields in gravity.

Figures (2)

• Figure 1: Simultaneous evolution of the black hole shadow radius $R_{\text{sh}}$ (left axis) and the total apparent area fluctuation $\delta A(t)$ (right axis) under the influence of a transient scalar gravitational wave. The navy dashed curves represent the Schwarzschild (GR) baseline, while the crimson and green solid curves depict the enhanced Schwarzschild-MOG response for a deformation parameter $\alpha = 0.2$. Both models exhibit a characteristic "breathing mode" triggered at the wave arrival time $t_0 = 5.0$, characterized by an exponentially damped oscillation ($\tau = 8.0$) at the quasi-normal mode frequency $\omega_b = 1.5$. Note that the MOG framework yields a larger static baseline and a higher amplitude of area fluctuation, providing a distinct observational signature for scalar-tensor-vector gravity in the strong-field regime.
• Figure 2: The translational displacement (wobble) of the black hole shadow center in the celestial $X$ and $Y$ coordinates. While the primary massless waves arrive at $t_0$, the massive vector field arrives at $t_v$ due to dispersive time delay $\Delta t$. This causes a sudden, asymmetric translational shift of the shadow center, followed by a damped relaxation. In standard General Relativity, these longitudinal perturbations are absent, and the shadow center remains stationary at the origin.