Post-Newtonian approximation of the rigidly rotating disc of dust to arbitrary order

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 ${12}^{\mathrm{th}}$ PN level and are listed in this paper up to the $4^{\mathrm{th}}$ 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.