Two-Point Pad\'{e} Approximants for the Deflection of Light in the Schwarzschild Black Hole Metric

TL;DR

This paper introduces Padé 2-point approximants to accurately model the deflection angle of light near a Schwarzschild black hole across all impact parameters.

Contribution

The paper develops and validates Padé approximants that improve the approximation of light deflection angles over previous methods.

Findings

Padé [2,2] approximants accurately model deflection angles for all impact parameters above the critical value.

A simpler quadratic approximation performs well in the mid-range but less so at extremes.

The approximants relate impact parameters to deflection angles via exponential functions, covering the full range.

Abstract

The deflection angle of a light ray passing the Schwarzschild (spherically symmetric vacuum) black hole was calculated by Charles Galton Darwin in 1959 in terms of the elliptic integral of the first kind. This calculation has been repeated many times and has also been given approximately in terms of elementary functions for impact parameters that either are not too small or are close to the critical impact parameter. Here I present Pad\'{e} 2-point approximants of order [2,2] (quadratic numerators and denominators), relating the critical impact parameter divided by the actual impact parameter to the exponential of the negative of the deflection angle, that fairly accurately cover the full range of impact parameters greater than the critical impact parameter, which is the case for all photon trajectories that remain outside the black hole. I also present a simpler quadratic approximation that works as well in the middle of the range but not so well at the extremes.