Precise analytic treatment of Kerr and Kerr-(anti) de Sitter black holes as gravitational lenses

TL;DR

This paper provides exact analytical solutions for photon trajectories and gravitational lensing effects around Kerr and Kerr-(anti) de Sitter black holes, including deflection angles, image positions, and magnifications, using hypergeometric functions.

Contribution

It introduces closed-form solutions for photon orbits and lensing phenomena in Kerr spacetimes with a cosmological constant, advancing analytical understanding of black hole gravitational lensing.

Findings

Exact solutions for photon deflection angles in Kerr spacetime.

Closed-form expressions for black hole lensing image positions.

Analytic calculation of magnification factors for generic photon orbits.

Abstract

The null geodesic equations that describe motion of photons in Kerr spacetime are solved exactly in the presence of the cosmological constant $\Lambda$. The exact solution for the deflection angle for generic light orbits (i.e. non-polar, non-equatorial) is calculated in terms of the generalized hypergeometric functions of Appell and Lauricella. We then consider the more involved issue in which the black hole acts as a `gravitational lens'. The constructed Kerr black hole gravitational lens geometry consists of an observer and a source located far away and placed at arbitrary inclination with respect to the black hole's equatorial plane. The resulting lens equations are solved elegantly in terms of Appell-Lauricella hypergeometric functions and the Weierstrass elliptic function. We then, systematically, apply our closed form solutions for calculating the image and source positions of generic photon orbits that solve the lens equations and reach an observer located at various values of the polar angle for various values of the Kerr parameter and the first integrals of motion. In this framework, the magnification factors for generic orbits are calculated in closed analytic form for the first time. The exercise is repeated with the appropriate modifications for the case of a non-zero cosmological constant.