Black hole shadow and wandering null geodesics

The role of the wandering null geodesic is studied in a black hole spacetime. Based on the continuity of the solution of the geodesic equation, the wandering null geodesics commonly exist and explain the typical phenomena of the optical observation of event horizons. Moreover, a new concept of “black room” is investigated relating the wandering null geodesic to the black hole shadow more closely.

Figure 1

The conformal diagram of a part of an asymptotically flat black hole spacetime: $S_{\mathrm{in}}[t]$, $S_{\mathrm{out}}[t,u]$ on $\mathcal{C}[t]$ are drawn.

Figure 2

In the Schwarzschild spacetime, the photon sphere exists on $r=3m$. A null geodesic from the point on the photon sphere will go inside and outside of the sphere $r=3m$, or be still on this photon sphere. They make the wavefront of the null geodesics topologically complicated after sufficient time has gone by.

Figure 3

Singular congruences are considered around a past wandering null geodesic. While there are an infinite number of conjugate points on the past wandering null geodesic, there are a finite number of conjugate points on a past directed null geodesic starting from $p$. The final (oldest) conjugate point $q_f$ determines a past set $I^{-}(q_f)$ and a segment of the null geodesic from $q_f$ should be a generator of the boundary of $I^{-}(q_f)$. In the star, the sectional curvature is positive and the null geodesic congruence would expand in past direction. That means we cannot receive the information of the spacetime point very close to the event horizon.