Can rigidly rotating polytropes be sources of the Kerr metric?

TL;DR

This paper investigates whether rigidly rotating polytropes can serve as sources for the Kerr metric, using approximate solutions to Einstein's equations and analyzing their multipole moments.

Contribution

It provides an approximate solution to the gravitational field of rotating polytropes and demonstrates their multipole moments cannot match those of the Kerr metric.

Findings

Polytropes' multipole moments do not match Kerr's.

Approximate solutions include second-order rotational corrections.

The metric is asymptotically flat in harmonic coordinates.

Abstract

We use a recent result by Cabezas et al. to build up an approximate solution to the gravitational field created by a rigidly rotating polytrope. We solve the linearized Einstein equations inside and outside the surface of zero pressure including second-order corrections due to rotational motion to get an asymptotically flat metric in a global harmonic coordinate system. We prove that if the metric and their first derivatives are continuous on the matching surface up to this order of approximation, the multipole moments of this metric cannot be fitted to those of the Kerr metric.