Light deflection in the post-linear gravitational field of bounded point-like masses

Abstract

Light deflection in the post-linear gravitational field of two bounded point-like masses is treated. Both the light source and the observer are assumed to be located at infinity in an asymptotically flat space. The equations of light propagation are explicitly integrated to the second order in $G/c^2$. Some of the integrals are evaluated by making use of an expansion in powers of the ratio of the relative separation distance to the impact parameter $(r_{12}/\xi)$. A discussion of which orders must be retained to be consistent with the expansion in terms of $G/c^2$ is given. It is shown that the expression obtained in this paper for the angle of light deflection is fully equivalent to the expression obtained by Kopeikin and Sch\"afer up to the order given there. The deflection angle takes a particularly simple form for a light ray originally propagating orthogonal to the orbital plane of a binary with equal masses. Application of the formulae for the deflection angle to the double pulsar PSR J0737-3039 for an impact parameter five times greater than the relative separation distance of the binary's components shows that the corrections to the Epstein-Shapiro light deflection angle of about $10^{-6}$ arcsec lie between $10^{-7}$ and $10^{-8}$ arcsec.