Post-linear Schwarzschild solution in harmonic coordinates: Elimination of structure-dependent terms

Abstract

The paper deals with a special kind of problems that appear in solutions of Einstein's field equations for extended bodies: many structure-dependent terms appear in intermediate calculations that cancel exactly in virtue of the local equations of motion or can be eliminated by appropriate gauge transformations. For a single body at rest these problems are well understood for both the post-Newtonian and the post-Minkowskian cases. However, the situation is still unclear for approximations of higher orders. This paper discusses this problem for a "body" of spherical symmetry to post-linear order. We explicitly demonstrate how the usual Schwarzschild field can be derived directly from the field equations in the post-linear approximation in the harmonic gauge and for an arbitrary spherically symmetric matter distribution. Both external and internal solutions are considered. The case of static incompressible fluid is then compared to the well-known results from the literature. The results of this paper can be applied to generalize the well-known post-Newtonian and post-Minkowskian multipole expansions of the metric in the post-linear approximation.