Dynamic shadow of a black hole with a self-interacting massive complex scalar hair

Abstract: We research dynamic shadows of a black hole with a self-interacting massive complex scalar hair. The complex scalar field $\psi$ evolves with time $t$, whose magnitude on apparent horizon $|\psi_{h}|$ starts from zero, undergoes a sharp rise followed by rapid oscillations, and eventually converges to a constant value. The variation of the photon sphere radius $r_{ps}$ is similar to that of the magnitude $|\psi_{h}|$. The emergence of the complex scalar hair $\psi$ causes the apparent horizon radius $r_{h}$ to start increasing sharply, and smoothly approaches a stable value eventually. The shadow radius $R_{sh}$ of the black hole with an accretion disk increases with $t_{o}$ the time at the observer's position. In the absence of an accretion disk, the shadow radius $R_{sh}$ is larger and also increases as $t_{o}$ increases. Furthermore, we slice the dynamical spacetime into spacelike hypersurfaces for all time points $t$. For the case with an accretion disk, the variation of $R_{sh}$ is similar to that of the apparent horizon $r_{h}$, because the inner edge of the accretion disk extends to the apparent horizon. In the absence of an accretion disk, the variation of $R_{sh}$ is similar to that of the photon sphere $r_{ps}$, because the black hole shadow boundary is determined by the photon sphere. Since the variation of $r_{ps}$ is induced by $\psi$, it can be stated that the variation in the size of the shadow is likewise caused by the change of $\psi$. Regardless of the presence or absence of the accretion disk, the emergence of the complex scalar hair $\psi$ causes the radius $R_{sh}$ of the shadow to start changing. Moreover, we investigate the time delay $\Delta t$ of lights propagating from light sources to observer. These findings not only enrich the theoretical models of dynamic black hole shadows but also provide a foundation for testing black hole spacetime dynamics.