Post-linear metric of a compact source of matter

The multipolar post-Minkowskian (MPM) formalism represents an approach for determining the metric density in the exterior of a compact source of matter. In the MPM formalism the metric density is given in harmonic coordinates and in terms of symmetric tracefree (STF) multipoles. In this investigation, the post-linear metric density of this formalism is used in order to determine the post-linear metric tensor in the exterior of a compact source of matter. The metric tensor is given in harmonic coordinates and in terms of STF multipoles. The post-linear metric coefficients are associated with an integration procedure. The integration of these post-linear metric coefficients is performed explicitly for the case of a stationary source, where the first multipoles (monopole and quadrupole) of the source are taken into account. These studies are a requirement for further investigations in the theory of light propagation aiming at highly precise astrometric measurements in the solar system, where the post-linear coefficients of the metric tensor of solar system bodies become relevant.