Generalization of Photon Sphere by Causal Structure--- In Order to See Dynamical Black Hole Shadow

Abstract

A generalization of the photon sphere for general black hole spacetimes is investigated based on their causal structure. We define a `wandering null geodesic' as a complete null geodesic accompanied by repetitive conjugate points, which corresponds to a null geodesic on the photon sphere in Schwarzschild spacetime. We also define a `wandering set', that is, a set of wandering null geodesics as a generalization of the photon sphere, and moreover, a truncated wandering set to discuss its formation related to the black hole formation. Then we examine the existence of a wandering null geodesic in general black hole spacetimes mainly in terms of Weyl focusing. We will see the essence of the generalized photon sphere is not the stationary cycling of the photon orbits, which is the concept only available in a stationary spacetime, but their accumulation. With any wandering null geodesic, it is implied that this accumulation will be occur anywhere in an asymptotically flat spacetime.