The Post-Newtonian Maclaurin Spheroids to Arbitrary Order

TL;DR

This paper introduces an iterative method to compute high-order post-Newtonian corrections for relativistic Maclaurin spheroids, enabling detailed analysis near bifurcation points and validating results against numerical data.

Contribution

It presents a novel iterative scheme for calculating arbitrary post-Newtonian orders of relativistic spheroids, extending previous methods and providing explicit solutions up to the fourth order.

Findings

Explicit post-Newtonian expansion up to fourth order.

Analysis of solution structure near bifurcation points.

Good agreement with numerical results confirms accuracy.

Abstract

In this paper, we develop an iterative scheme to enable the explicit calculation of an arbitrary post-Newtonian order for a relativistic body that reduces to the Maclaurin spheroid in the appropriate limit. This scheme allows for an analysis of the structure of the solution in the vicinity of bifurcation points along the Maclaurin sequence. The post-Newtonian expansion is solved explicitly to the fourth order and its accuracy and convergence are studied by comparing it to highly accurate numerical results.