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

Abstract

The null geodesic equations that describe motion of photons in Kerr spacetime are solved exactly in the presence of the cosmological constant Λ. 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 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 non-zero cosmological constant. *

Shadow of a rotating braneworld black hole

“…Non-rotating braneworld black holes were studied as gravitational lenses [14] as well. Kerr black hole lenses were considered by several authors [15][16][17][18]. Rotating black holes present apparent shapes or shadows with an optical deformation due to the spin [19,20], instead of being circles as in the case of non-rotating ones.…”

Shadow of a Kaluza-Klein rotating dilaton black hole

“…Nonrotating black holes coming from alternative theories were considered as gravitational lenses as well [6,7]. Kerr black holes were analyzed as lenses by several researchers [8][9][10][11]; in particular, the strong deflection limit was extended to these objects [8,10]. The shadows (or apparent shapes) of nonrotating black holes are circular, but rotating ones present a deformation caused by the spin [12,13].…”

Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

“…in order to treat some hyperelliptic integrals: see [12] or [13]. Moreover the cosmologist Kraniotis [14,15] recurred to Lauricella functions in order to treat Kerr and Kerr-(anti) de Sitter black holes, and the bending of light in Kerr and Kerr-(anti) de Sitter spacetimes. Nevertheless their employ presently is still not so popular.…”