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

TL;DR

This paper addresses how a self-interacting massive complex scalar hair dynamically reshapes black hole shadows. It adopts a dynamical Einstein–Maxwell framework with a scalar potential $V(\psi)$ and solves the system in Painlevé–Gullstrand coordinates, extracting the time evolution of the scalar field $\psi(t,r)$, the metric functions, and the gauge field; shadows are computed via backward ray-tracing for a face-on thin disk, with and without disk presence. The main findings are that the scalar hair drives time-dependent changes in the photon sphere radius $r_{ps}$ and the apparent horizon radius $r_h$, which in turn modulate the shadow radius $R_{sh}$—increasing with observer time $t_o$, with disk cases tying the shadow boundary to $r_h$ and disk-free cases tying it to $r_{ps}$—and that time delays $\Delta t$ vary across direct, lensing, and photon-ring paths. Collectively, the results provide a dynamical framework for interpreting evolving black hole shadows and offer potential observational signatures for testing black hole spacetime dynamics with future high-resolution imaging missions.

Abstract

We investigate the dynamic shadows of a black hole with a self-interacting massive complex scalar hair. The complex scalar field ψevolves with time t, and its magnitude on the apparent horizon |ψ_{h}| starts from zero, undergoes a sharp rise followed by rapid oscillations, and eventually converges to a constant value. The variation in the photon sphere radius r_{ps} is similar to that of the magnitude |ψ_{h}|. Owing to the emergence of the complex scalar hair ψ, the apparent horizon radius r_{h} starts increasing sharply and then smoothly approaches a stable value eventually. The shadow radius R_{sh} of the black hole with an accretion disk increases with time t_{o} 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 in R_{sh} is similar to that in 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 in R_{sh} is similar to that in the photon sphere radius r_{ps}, because the black hole shadow boundary is determined by the photon sphere. As the variation in r_{ps} is induced by ψ, it can be stated that the variation in the size of the shadow is similarly caused by the change in ψ. Regardless of the presence or absence of the accretion disk, the emergence of the complex scalar hair ψcauses the radius R_{sh} of the shadow to start changing. Moreover, we investigate the time delay Δt of lights propagating from light sources to the 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.

Figures (15)

• Figure 1: Contour plot of $g_{tt}$ as a function of time $t$ and radial coordinate $r$. This parameter asymptotically approaches $-1$ as $r\rightarrow\infty$. The contour line of $g_{tt}=0$ corresponds to the position of the apparent horizon $r_{h}$.
• Figure 2: Contour plot of $g_{tr}$ as a function of time $t$ and radial coordinate $r$. This parameter asymptotically approaches $0$ as $r\rightarrow\infty$.
• Figure 3: Variation in the magnitude of the complex scalar field on the horizon $|\psi_{h}|$ with time $t$.
• Figure 4: Variations in the apparent horizon $r_{h}$ (marked by a black dashed line) and photon sphere $r_{ps}$ (marked by a red solid line) with respect to the variable $t$.
• Figure 5: Light rays corresponding to shadow (black), direct emission (blue), lensing rings (orange), and photon rings (red). The black disk represents the black hole, and the observer is positioned to its right at a distance of $r_{o}=50$. The accretion disk is viewed face-on by the observer.