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.