An approximate global solution of Einstein's equations for a finite body

TL;DR

This paper develops an approximate global solution to Einstein's equations for a rotating, constant-density fluid sphere, providing insights into its gravitational field and limitations regarding Kerr metric sources.

Contribution

It introduces a novel approximate solution for a rotating fluid star using post-Minkowskian and slow-rotation approximations, matching interior and exterior metrics.

Findings

Derived interior and exterior solutions in harmonic coordinates.

Connected physical parameters to exterior metric constants.

Showed the fluid model cannot produce the Kerr metric.

Abstract

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to show that this type of fluid cannot be a source of the Kerr metric