The Schwarzschild black hole as a gravitational mirror

The gravitational field outside of a nonrotating black hole is described using the Schwarzschild metric. The geodesic equations of the Schwarzschild metric are derived and those describing null and circular timelike orbits are discussed. Some numerical solutions of the null geodesic equations are shown. These depict photon trajectories which circle the black hole one or two times and then terminate at their emission points. Thus a sequence of ring-shaped mirror images is produced. An equation which gives the angle between the photon’s trajectory and the radial direction at the emitter is derived and applied to the numerical solutions. These results serve to illustrate how an observer ‘‘passes through’’ his or her mirror image at r=3 MG/c2, as he or she moves toward a Schwarzschild black hole.