# An approximate global solution of Einstein's equations for differentially rotating compact body

## TL;DR

This paper develops an approximate global solution to Einstein's equations for a differentially rotating perfect fluid star using weak field and slow rotation approximations, matching interior and exterior solutions.

## Contribution

It introduces a method to model differentially rotating stars in general relativity with an approximate global metric based on perturbation theory.

## Key findings

- Derived interior and exterior solutions in harmonic coordinates.
- Expressed physical constants like mass and angular momentum in terms of model parameters.
- Provided a framework for modeling rotating stars with differential rotation in GR.

## Abstract

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be thought as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. 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 coordinates. Interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to get a global solution. The resulting metric depends on four arbitrary constants: mass density, rotational velocity at $r=0$, a parameter which takes into account of the change in the rotational velocity through the star and the star radius in the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined in terms of these four parameters.