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Description
We construct the exterior metric of a stationary compact source within the multipolar post-Minkowskian (MPM) formalism and develop a recursive momentum-space reformulation of the construction. The MPM framework solves the Einstein equations outside a compact source perturbatively in Newton's constant, decomposing the metric into symmetric trace-free mass and current multipole moments. In this momentum space, the multipolar structure emerges naturally from the recursive field equations, simplifying tensor structures and clarifying how multipole moments enter at successive PM orders. Crucially, the nonlinearities of the Einstein equations reduce in this picture to the iterative evaluation of bubble integrals, making explicit the nonlinear multipolar interactions, including monopole–quadrupole, quadrupole-quadrupole, and higher-order couplings. As a consistency check, we match the generic multipolar solution to the Kerr metric generated independently by a similar recursive formulation, verifying that the construction reproduces the correct Kerr multipole structure, which includes its infinite tower of moments, and thus captures the nonlinear structure of rotating vacuum solutions.