Breathing Black Hole Shadows in Modified Gravity (MOG)

In this paper, we investigate the dynamic phenomenological signatures of a Schwarzschild-MOG black hole perturbed by passing gravitational waves. By integrating the perturbed null geodesic equations, we show that the additional scalar and vector degrees of freedom in Scalar-Tensor-Vector Gravity modify photon trajectories in a time-dependent manner, breaking degeneracies with standard General Relativity. We identify two distinct dynamical signatures that operate in a strictly chronological sequence. Initially, the massless scalar field induces a breathing-mode polarization, leading to a periodic modulation of the apparent shadow area. Subsequently, the massive vector field undergoes dispersive propagation, producing a delayed contribution that generates secondary longitudinal metric perturbations. This delayed response manifests as a transient, asymmetric displacement of the shadow on the screen of the observer. While pure scalar-tensor theories, such as Horndeski gravity, predict similar initial breathing modes, they lack the massive Proca field required to produce the delayed asymmetric response. Consequently, this two-stage dynamic sequence serves as a distinctive signature unique to Scalar-Tensor-Vector Gravity. Within the strong-field regime of an extreme mass ratio inspiral, these effects produce fractional deviations in observable shadow properties at the level of [Formula: see text]. Detecting this chronological separation imposes extreme requirements on both temporal sampling and angular resolution, rendering observational prospects challenging due to the constraints of signal-to-noise ratios and intrinsic astrophysical variability in the emission region.