Kerr black hole lensing for generic observers in the strong deflection limit

Abstract

We generalize our previous work on gravitational lensing by a Kerr black hole in the strong deflection limit, removing the restriction to observers on the equatorial plane. Starting from the Schwarzschild solution and adding corrections up to the second order in the black hole spin, we perform a complete analytical study of the lens equation for relativistic images created by photons passing very close to a Kerr black hole. We find out that, to the lowest order, all observables (including shape and shift of the black hole shadow, caustic drift and size, images position and magnification) depend on the projection of the spin on a plane orthogonal to the line of sight. In order to break the degeneracy between the black hole spin and its inclination relative to the observer, it is necessary to push the expansion to higher orders. In terms of future VLBI observations, this implies that very accurate measures are needed to determine these two parameters separately.

Analytical Kerr black hole lensing in the weak deflection limit

“…The finite size of the caustic should show up at the next order due to terms ∝ a 2 r 2 Sch as suggested by numerical results [8]. It could be of interest to draw some comparison with the case of the strong deflection limit [9,10,11]. Such a limit has been treated considering small values of the angular momentum and including corrections proportional to a 2 .…”

“…We remark as the integral in Eq. (B1) can be more easily solved in terms of the integration variable µ/µ + [5,11,16]. The function in Eq.…”

“…The two components of the deflection angle can then be evaluated considering the geodesics for source and observer Eqs. (11,12) at r o , r s → ∞, through the equation…”

“…Rauch and Blandford [8] provided a detailed analysis of the optical structure of the primary caustic surface. The analytical extension of the strong deflection limit methodology to the Kerr black hole has been performed as well in the limit of small values of the angular momentum and for sources in the asymptotically flat region of the space-time [9,10,11]. A full description of caustics and the inversion of lens mapping for sources near them has been performed up to the second order in a.…”

Strong Gravitational Lensing in the Einstein-Proca Theory

“…(15), (20)(21)(22), we gained some parameters of a spherically symmetric spacetime in the Einstein-Proca theory as …”