Approximating light rays in the Schwarzschild field

TL;DR

This paper introduces a new simple formula that accurately approximates photon trajectories in the Schwarzschild gravitational field, outperforming existing methods especially at finite radii, and is useful for gravitational lensing calculations.

Contribution

A novel approximation formula for photon paths in Schwarzschild spacetime that improves accuracy over previous simple models.

Findings

The new approximation outperforms existing methods in accuracy.

It effectively models photon trajectories at finite radii.

The formula is practical for lensing and ray-deflection calculations.

Abstract

A short formula is suggested which approximates photon trajectories in the Schwarzschild field better than other simple prescriptions from the literature. We compare it with various "low-order competitors", namely with those following from exact formulas for small $M$, with one of the results based on pseudo-Newtonian potentials, with a suitably adjusted hyperbola, and with the effective and often employed approximation by Beloborodov. Our main concern is the shape of the photon trajectories at finite radii, yet asymptotic behaviour is also discussed, important for lensing. An example is attached indicating that the newly suggested approximation is usable--and very accurate--for practically solving the ray-deflection exercise.