Uniform Strong-Deflection Lensing in Near-Extremal Kerr

Abstract

Strong-deflection lensing by a subextremal Kerr black hole exhibits a logarithmic critical divergence, whereas the exactly extremal prograde problem develops stronger power-law behavior. We construct a uniform asymptotic description of the transition between these regimes for prograde equatorial null geodesics. The distinguished joint near-critical and near-extremal limit occurs when the impact-parameter offset from the exact critical value is of order \(Mκ\), where \(κ=\sqrt{1-a^2/M^2}\). In this limit, the Kerr radial potential reduces to a quadratic normal form, yielding an analytic crossover function that connects the subextremal logarithmic and extremal power-law regimes. We derive a matched next-to-leading-order correction and validate it against high-precision integrations of the exact Kerr geodesic equations, finding improvement for all 55 trajectories tested. The same asymptotic structure yields a higher-order image-branch law interpolating between algebraic and exponential accumulation, with crossover order scaling as \(κ^{-1}\). We also derive and numerically validate the corresponding image-branch coordinate-time delay law, whose high-order extremal-like limit approaches the horizon-period spacing \(4πM\).