Slowly Rotating Homogeneous Stars and the Heun Equation

TL;DR

This paper applies Hartle's slow rotation scheme to homogeneous stars, solving the key equation as a Heun equation, and provides a convergent series solution that approximates the metric effectively.

Contribution

It introduces a series solution to the Heun equation in the context of slowly rotating homogeneous stars, enabling explicit metric expressions.

Findings

Series solution converges rapidly for slow rotation

Comparison with numerical solutions validates the approximation

Explicit metric expressions derived from the series

Abstract

The scheme developed by Hartle for describing slowly rotating bodies in 1967 was applied to the simple model of constant density by Chandrasekhar and Miller in 1974. The pivotal equation one has to solve turns out to be one of Heun's equations. After a brief discussion of this equation and the chances of finding a closed form solution, a quickly converging series solution of it is presented. A comparison with numerical solutions of the full Einstein equations allows one to truncate the series at an order appropriate to the slow rotation approximation. The truncated solution is then used to provide explicit expressions for the metric.