The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. The Deformed Figures of the Maclaurin Spheroids
The equations of post-Newtonian hydrodynamics are solved appropriately for a uniformly rotating homogeneous mass with symmetry about the axis of rotation. The post-Newtonian figure is obtained as a deformation of the Newtonian Maclaurin spheroid (with semi-axes and , say) by a Lagrangian displacement proportional to (2) =a'12( 2Xl 2x2, - ), where w denotes the distance from the axis of rotation and x1 and x2 are the Cartesian coordinates in the equatorial plane. It is shown that the equation defining the boundary of the post-Newtonian configuration is of the form 2 2 R 5 4 2+-2-t-2S2t(e)--4-22 =0, ai a3 a1 a1 a1 a3 (e) is a determinate function of the eccentricity e of the Maclaurin spheroid and R5 (= 2CM/c2) is the Schwarzschild radius. The function S t(e) is tabulated in the paper. Further, the angular velocity of rotation of the post-Newtonian configuration differs from that of the Maclaurin spheroid by an amount which is also tabulated. The solution of the post-Newtonian equations exhibits a singularity at a certain eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin of this singularity is that at e* the Maclaurin spheroid allows an infinitesimal neutral deformation by a displacement proportional to (2); and the Newtonian instability of the Maclaurin spheroid at e* is excited by the post-Newtonian effects of general relativity.