The Post-Newtonian Approximation of the Rigidly Rotating Disc of Dust to Arbitrary Order

TL;DR

This paper develops an iterative scheme to compute high-order post-Newtonian approximations of a rotating dust disc, demonstrating convergence and reliability even in relativistic regimes, and analyzing ergospheres at various approximation levels.

Contribution

It introduces a method to calculate arbitrary coefficients in the post-Newtonian expansion of a rotating dust disc, extending previous solutions to higher orders.

Findings

Coefficients calculated up to 12th PN level.

Series convergence confirmed in relativistic cases.

Ergospheres analyzed at multiple approximation orders.

Abstract

Using the analytic, global solution for the rigidly rotating disc of dust as a starting point, an iteration scheme is presented for the calculation of an arbitrary coefficient in the post-Newtonian (PN) approximation of this solution. The coefficients were explicitly calculated up to the 12th PN level and are listed in this paper up to the 4th PN level. The convergence of the series is discussed and the approximation is found to be reliable even in highly relativistic cases. Finally, the ergospheres are calculated at increasing orders of the approximation and for increasingly relativistic situations.