Chaotic Shadows of Black Holes: A Short Review
  We give a brief review on the formation and the calculation of black hole
shadow. Firstly, we introduce the conception of black hole shadow and the
current works on a variety of black hole shadows. Secondly, we present main
methods of calculating photon sphere radius and shadow radius, and then explain
how the photon sphere affect the boundary of black hole shadow. We review the
analytical calculation for black hole shadows which have analytic expressions
for shadow boundary due to the integrable photon motion system. And we
introduce the fundamental photon orbits which can explain the patterns of black
hole shadow shape. Finally, we review the numerical calculation of black hole
shadows with the backward ray-tracing method, and introduce some chaotic black
hole shadows with self-similar fractal structures. Since the gravitational
waves from the merger of binary black holes have been detected, we introduce a
couple of shadows of binary black holes, which all have the eyebrowlike shadows
around the main shadows with the fractal structures. We discuss the invariant
phase space structures of photon motion system in black hole space-time, and
explain the formation of black hole shadow is dominated by the invariant
manifolds of certain Lyapunov orbits near the fixed points.
