Gravitational Metric of a Star

Solving the classical equations of motion in general relativity recursively, we consider the metric of a spatially localized and stationary source of matter. Having in mind a star of general composition, we characterize it by means of its infinite set of mass and current multipoles. Specializing to de Donder gauge we set up the recursive equations that produce the metric outside the star to any desired order in perturbation theory, expanded both in Newton's constant and in the order of multipoles. Up to second post-Minkowskian order we express the result to any order in the multipole expansion in terms of generalized (tensor) bubble integrals in momentum space and a corresponding simple expansion in inverse distances. In a special corner of the space of multipoles we recover the Kerr black hole solution to the given order. By tweaking just slightly the multipoles away from the Kerr limit the metric will describe stars that are Kerr-like and yet are not black holes. A subtlety with respect to the gauge ambiguity of de Donder gauge is also pointed out.