Lensing by Kerr Black Holes. II: Analytical Study of Quasi-Equatorial Lensing Observables

TL;DR

This paper develops an analytical third-order perturbative theory for quasi-equatorial gravitational lensing by Kerr black holes, providing new formulas for observables including bending angles, image positions, magnifications, and time delays, with displacement effects included.

Contribution

It introduces a third-order perturbative framework for Kerr black hole lensing observables, extending previous weak-deflection models and including displacement effects.

Findings

Derived new formulas for bending angles and image positions to third order in epsilon.

Quantified the impact of displacement on lensing observables, appearing at second order.

Provided results for Schwarzschild lensing with displacement when spin is zero.

Abstract

In this second paper, we develop an analytical theory of quasi-equatorial lensing by Kerr black holes. In this setting we solve perturbatively our general lens equation with displacement given in Paper I, going beyond weak-deflection Kerr lensing to third order in our expansion parameter epsilon, which is the ratio of the angular gravitational radius to the angular Einstein radius. We obtain new formulas and results for the bending angle, image positions, image magnifications, total unsigned magnification, and centroid, all to third order in epsilon and including the displacement. New results on the time delay between images are also given to second order in epsilon, again including displacement. For all lensing observables we show that the displacement begins to appear only at second order in epsilon. When there is no spin, we obtain new results on the lensing observables for Schwarzschild lensing with displacement.