The deformed figures of the Dedekind ellipsoids in the post-Newtonian approximation to general relativity

Abstract

The effects of general relativity, in the post-Newtonian approximation, on the Dedekind figures of equilibrium of homogeneous masses are determined. It is shown how the post-Newtonian figures can be obtained by first altering the velocity field in the Dedekind ellipsoid appropriately, and then subjecting it to a suitable Lagrangian displacement cubic in the coordinates. The solution exhibits a singularity at a point where the axes of the Dedekind ellipsoid are in the ratios 1:0.6158:0.4412. However, in contrast to what happens along the Jacobian sequence, the occurrence of the singularity along the Dedekind sequence is not associated with the onset of any instability at that point by a strict Newtonian-like dynamic perturbation.