Extending the First-Order Post-Newtonian Scheme in Multiple Systems to the Second-Order Contributions to Light Propagation

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 virtue of the scheme on the linear partial differential equations of the potential and vector potential. The spatial components of the metric are extended to the second-order level both in a global coordinates (q_ij/c⁴) and a local coordinates (Q_ab/c⁴). The equations of q_ij(or Q_ab) are obtained from the field equations. The relationship between q_ij and Q_ab are also presented. In the special case of the solar system (isotropic condition is applied (q_ij
= δ_ijq)), we obtain the solution of q. Finally, a further extension of the second-order contributions in the parametrized post-Newtonian formalism is discussed.