On the simplest static and stationary vacuum quadrupolar metrics

TL;DR

This paper introduces a simple static and stationary vacuum quadrupolar metric derived from the Bach-Weyl solution, extending it to complex parameters, and compares it with the Zipoy-Voorhees metric, also exploring the addition of angular momentum.

Contribution

It presents a new 2-parameter static quadrupolar model and extends it to a 3-parameter stationary spacetime, analyzing its properties and relativistic multipole moments.

Findings

The model allows arbitrary quadrupole parameters without restrictions.

Comparison shows the Zipoy-Voorhees metric cannot have arbitrary negative quadrupole moments.

Adding angular momentum yields a stationary solitonic spacetime with well-defined multipole moments.

Abstract

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 2-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 $\delta$-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 3-parameter stationary solitonic spacetime, which permits us to introduce the notion of the Fodor-Hoenselaers-Perj\'es relativistic multipole moments.