Photon Boomerang in a Nearly Extreme Kerr Metric

The Kerr rotating black hole metric has unstable photon orbits that orbit around the hole at fixed values of the Boyer-Lindquist coordinate $r$ that depend on the axial angular momentum of the orbit, as well as on the parameters of the hole. For zero orbital axial angular momentum, these orbits cross the rotational axes at a fixed value of $r$ that depends on the mass $M$ and angular momentum $J$ of the black hole. Nonzero angular momentum of the hole causes the photon orbit to rotate so that its direction when crossing the north polar axis changes from one crossing to the next by an angle I shall call $\Delta\phi$, which depends on the black hole dimensionless rotation parameter $a/M = cJ/(GM^2)$ by an equation involving a complete elliptic integral of the first kind. When the black hole has $a/M \approx 0.994\,341\,179\,923\,26$, which is nearly maximally rotating, a photon sent out in a constant-$r$ direction from the north polar axis at $r \approx 2.423\,776\,210\,035\,73\, GM/c^2$ returns to the north polar axis in precisely the opposite direction (in a frame nonrotating with respect to the distant stars), a photon boomerang.