# Light propagation in 2PN approximation in the field of one moving monopole II. Boundary value problem

## TL;DR

This paper develops analytical solutions for light propagation in the gravitational field of a moving monopole body at the second post-Newtonian level, with implications for ultra-precise astrometry at the nano-arcsecond scale.

## Contribution

It provides explicit analytical transformations and assesses the significance of higher-order terms, including enhanced 3PN and 4PN effects, for high-precision light deflection measurements.

## Key findings

- Analytical solutions for boundary value problem in 2PN approximation.
- Simplified transformations for 1 nano-arcsecond accuracy.
- Enhanced 3PN terms are relevant for ultra-precise astrometry.

## Abstract

In this investigation the boundary value problem of light propagation in the gravitational field of one arbitrarily moving body with monopole structure is considered in the second post-Newtonian approximation. The solution of the boundary value problem comprises a set of altogether three transformations: k -> sigma and sigma -> n and k -> n. Analytical solutions of these transformations are given and the upper limit of each individual term is determined. Based on these results, simplified transformations are obtained by keeping only those terms relevant for the given goal accuracy of 1 nano-arcsecond in light deflection. Like in case of light propagation in the gravitational field of one body at rest, there are so-called enhanced terms which are of second post-Newtonian order but contain one and the same typical large numerical factor. Finally, the impact of enhanced terms beyond 2PN approximation is considered. It is found that enhanced 3PN terms are relevant for astrometry on the level of 1 nano-arcsecond in light deflection, while enhanced 4PN terms are negligible, except for grazing rays at the Sun.