An accurate equation for the gravitational bending of light by a static massive object

TL;DR

This paper derives an exact analytical formula for gravitational light bending that accounts for finite distances of source and observer, improving upon classical models and relevant for high-precision astrometry.

Contribution

The authors present a novel, exact formula for gravitational light bending that generalizes Darwin's equation to finite source and observer distances, enhancing accuracy over previous approximations.

Findings

The new formula aligns well with PPN results up to first order.

Discrepancies of 6.6 mas for sun-grazing beams from Mercury.

Differences up to 2 mas for distant starlight.

Abstract

An exact analytical expression for the bending angle of light due to a non-rotating massive object, considering the actual distances from source and observer to the gravitational mass, is derived. Our novel formula generalizes Darwin well-known equation for gravitational light bending [Proc. R. Soc. London A 263, 39-50 (1961)], where both source and observer are placed at infinite distance from the lensing mass, and provides excellent results in comparison with the post Newtonian (PPN) formalism up to first order. As a result, the discrepancy between our recent expression and the PPN approach is 6.6 mas for sun-grazing beams coming from planet Mercury, with significant differences up to 2 mas for distant starlight. Our findings suggest that these considerations should not be dismissed for both solar system objects and extragalactic sources, where non-negligible errors might be present in ultraprecise astrometry calculations.