Photon-Return and Polar-Focusing Crossovers near Extremal Kerr

Abstract

How precisely must a photon's impact parameter be tuned for it to complete $n$ windings around a nearly extremal Kerr black hole before escaping, and how does the same near extremal scaling control the focusing of neighboring rays? For equatorial prograde returns we find, with $κ=\sqrt{1-a^2/M^2}$ and $X=nκ$, $ε_n=\sqrt{3}\,κ/(e^{2πX}-1) +O(κ^2|\lnκ|)$. This single law connects the extremal inverse-winding hierarchy to subextremal exponential accumulation. Perturbing the return family quasi-equatorially, we derive a second scaling law for the polar Jacobi phase, $Ψ_θ=2\ln[\sinh(πX)/κ]+C_s+o(1)$, and show that its focusing points cross from geometric spacing in winding number to asymptotically uniform spacing $Δ\mathcal N\simeq(2κ)^{-1}$. Numerical integrations of the exact Kerr radial dynamics and linear polar equation verify both scaling collapses. Thus near extremal Kerr exhibits parametrically distinct winding and polar-focusing scales, rather than a discontinuous jump between extremal and subextremal lensing hierarchies.