Second Order Post Newtonian Equations of Light Propagation in Multiple Systems

TL;DR

This paper extends the first order post-Newtonian scheme to second order for light propagation in multiple systems, providing more precise equations useful for future space mission observations.

Contribution

It develops a second order post-Newtonian framework for light propagation, including metric tensor extensions and parametrized equations for improved accuracy.

Findings

Derived second order equations of light rays using iterative methods.

Extended metric tensor components to second order in global and local coordinates.

Presented parametrized 2PN equations for potential use in space observations.

Abstract

The first order post Newtonian scheme in multiple systems presented by Damour-Soffel-Xu is extended to the second order one for light propagation without changing the advantage of the scheme on the linear partial differential equations of potential and vector potential. The spatial components of the metric tensor are extended to the second order level both in the global coordinates ($q_{ij}/c^4$ term) and in a local coordinates ($Q_{ab}/c^4$ term). The equations of $q_{ij}$ (or $Q_{ab}$) are deduced from Einstein field equations. The linear relationship between $q_{ij}$ and $Q_{ab}$ are presented also. The 2PN equations of light ray based on the extended scheme are deduced by means of the iterative method. We also use parametrized second post Newtonian metric tensor to substitute into the null geodetic equations to obtain the parametrized second order equations of light ray which might be useful in the observation and measurement in the future space missions.