General static axisymmetric solution of Einstein’s vacuum field equations in prolate spheroidal coordinates

A solution of Einstein's vacuum field equations is presented explicitly in Eqs. (22), (28), and (43). This completes our stationary metric presented previously and therefore gives the exact solution to the problem of describing the exterior field of rotating deformed bodies. The metric given here is equivalent to the general Weyl metric; however, we use prolate spheroidal coordinates for conve- nience. We investigate the Newtonian and the relativistic (Geroch-Hansen) multipole moments of the solution and conclude that it describes the exterior gravitational field of a static axisymmetric mass distribution.

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