Simplest static and stationary vacuum quadrupolar metrics

In the present paper, we argue that a special case of the Bach-Weyl metric describing a static configuration of two Schwarzschild black holes gives rise, after extending its parameter space to complex values, to a very simple two-parameter model for the gravitational field of a static deformed mass. We compare this model, which has no restrictions on the quadrupole parameter, with the well-known Zipoy-Voorhees δ metric and show in particular that the mass quadrupole moment in the latter solution cannot take arbitrary negative values. We subsequently add an arbitrary angular momentum to our static model and study some properties of the resulting three-parameter stationary solitonic spacetime, which permits us to introduce the notion of the Fodor-Hoenselaers-Perjés relativistic multipole moments.