The quadrupole moment of slowly rotating fluid balls

TL;DR

This paper investigates the quadrupole moments of slowly rotating fluid stars using second order Hartle formalism, numerical solutions for various equations of state, and analytical post-Minkowskian limits, revealing deviations from Kerr metric predictions.

Contribution

It provides a detailed numerical and analytical study of quadrupole moments in rotating fluid stars, highlighting differences from Kerr black holes and illustrating multipole moment calculations.

Findings

Quadrupole moments differ from Kerr metric predictions.

Numerical solutions depend on the chosen equation of state.

Analytical post-Minkowskian limit supports numerical results.

Abstract

In this paper we use the second order formalism of Hartle to study slowly and rigidly rotating stars with focus on the quadrupole moment of the object. The second order field equations for the interior fluid are solved numerically for different classes of possible equations of state and these solutions are then matched to a vacuum solution that includes the general asymptotically flat axisymmetric metric to second order, using the Darmois-Israel procedure. For these solutions we find that the quadrupole moment differs from that of the Kerr metric, as has also been found for some equations of state in other studies. Further we consider the post-Minkowskian limit analytically. In the paper we also illustrate how the relativistic multipole moments can be calculated from a complex gravitational potential.