Off-equatorial deflections and gravitational lensing. II. In general stationary and axisymmetric spacetimes

TL;DR

This paper develops a perturbative method to calculate off-equatorial gravitational lensing in stationary, axisymmetric spacetimes, accounting for finite source and detector distances, and applies it to various black hole models to analyze the effects of spin, charge, and source altitude.

Contribution

It introduces a general perturbative approach for off-equatorial lensing in complex spacetimes, extending previous methods and applying to multiple black hole solutions.

Findings

Spin and charge affect deflections at second order.

Source altitude influences deflection at leading order.

Effects of spin and charge are difficult to measure from image positions.

Abstract

In this work, we develop a general perturbative procedure to find the off-equatorial plane deflections in the weak deflection limit in general stationary and axisymmetric spacetimes, allowing the existence of the generalized Carter constant. Deflections of both null and timelike rays, with the finite distance effect of the source and detector taken into account, are obtained as dual series of $M/r_0$ and $r_0/r_{s,d}$. These deflections allow a set of exact gravitational lensing equations from which the images' apparent angular positions are solved. The method and general results are then applied to the Kerr-Newmann, Kerr-Sen, and rotating Simpson-Visser spacetimes to study the effect of the spin and characteristic (effective) charge of the spacetimes and the source altitude on the deflection angles and image apparent angles. It is found that, in general, both the spacetime spin and charge only affect the deflections from the second non-trivial order, while the source altitude influences the deflection from the leading order. Because of this, it is found that, in gravitational lensing in realistic situations, it is hard to measure the effects of the spacetime spin and charge from the images' apparent locations. We also presented the off-equatorial deflections in the rotating Bardeen, Hayward, Ghosh, and Tinchev black hole spacetimes.