Bending of Light and Inhomogeneous Picard-Fuchs Equation

TL;DR

This paper derives an analytic expansion for the gravitational bending angle of light around a Schwarzschild black hole using an inhomogeneous Picard-Fuchs equation, enabling explicit weak and strong deflection calculations.

Contribution

It introduces a novel application of inhomogeneous Picard-Fuchs equations to compute the light bending angle analytically in black hole spacetimes.

Findings

Explicit weak and strong deflection expansions of the bending angle.

Analytic expressions for Schwarzschild and Reissner-Nordström spacetimes.

Confirmation that direct integration yields the same results.

Abstract

Bending of light rays by gravitational sources is one of the first evidences of the general relativity. When the gravitational souce is a stationary massive object such as a black hole, the bending angle has an integral representation, from which various series expansions in terms of the parameters of orbit and the background spacetime has been derived. However, it is not clear that it has any analytic expansion. In this paper, we show that such an analytic expansion can be obtained for the case of a Schwarzschild black hole by solving an inhomogeneous Picard-Fuchs equation, which has been applied to compute effective superpotentials on D-branes in the Calabi-Yau manifolds. From the analytic expression of the bending angle, both weak and strong deflection expansions are explicitly obtained. We show that the result can be obtained by the direct integration approach. We also discuss how the charge of the gravitational source affects the bending angle and show that a similar analytic expression can be obtained for the extremal Reissner-Nordstroem spacetime.