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=√(1−a2/M2) and X=nκ, εn=√3 κ/(e2πX−1) +O(κ2|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, Ψθ=2ln[sinh(πX)/κ]+Cs+o(1), and show that its focusing points cross from geometric spacing in winding number to asymptotically uniform spacing Δ𝒩≃(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.
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BibTeX
@misc{gonzalezmoya2026photonreturn,
title = {{Photon-Return and Polar-Focusing Crossovers near Extremal Kerr}},
author = {Valentina Gonzalez-Moya and Ali Ovgun and Joel Saavedra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21848},
eprint = {2610.21848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21848},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Photon-Return and Polar-Focusing Crossovers near Extremal Kerr
AU - Valentina Gonzalez-Moya
AU - Ali Ovgun
AU - Joel Saavedra
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21848
UR - https://arxiv.org/abs/2610.21848
AB - 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=√(1−a^2/M^2) and X=nκ, ε_n=√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=2ln[sinh(πX)/κ]+C_s+o(1), and show that its focusing points cross from geometric spacing in winding number to asymptotically uniform spacing Δ𝒩≃(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.
ER -