Photon boomerang in a nearly extreme Kerr metric

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