Extending the first-order post-Newtonian scheme in multiple systems to the second-order contributions to light propagation

TL;DR

This paper extends the post-Newtonian scheme to second-order for light propagation in multiple systems, providing new equations and solutions relevant for high-precision gravitational modeling.

Contribution

It introduces second-order contributions to the metric in the post-Newtonian scheme without altering the linear PDE framework, and explores their implications in the PPN formalism.

Findings

Derived second-order metric components for multiple systems.

Established relations between global and local coordinates.

Obtained solutions under isotropic conditions for the solar system.

Abstract

In this paper, we extend the first-order post-Newtonian scheme in multiple systems presented by Damour-Soffel-Xu to the second-order contribution to light propagation without changing the virtueof the scheme on the linear partial differential equations of the potential and vector potential. The spatial components of the metric are extended to second order level both in a global coordinates ($q_{ij}/ c^4$) and a local coordinates ($Q_{ab}/ c^4$). The equations of $q_{ij}$ (or $Q_{ab}$) are obtained from the field equations.The relationship between $q_{ij}$ and $Q_{ab}$ are presented in this paper also. In special case of the solar system (isotropic condition is applied ($q_{ij} = \delta_{ij} q $)), we obtain the solution of $q$. Finally, a further extension of the second-order contributions in the parametrized post-Newtonian formalism is discussed.