General Relativistic Gravitational Field of a Rigidly Rotating Disk of Dust: Axis Potential, Disk Metric, and Surface Mass Density

In a recent paper we described the general relativistic gravitational field of a rigidly rotating disk of dust in terms of two linear integral equations (a "small" one and a "big" one). Here we present the exact solution to the small integral equation. The Ernst potential on the symmetry axis, the disk metric, as well as the surface mass density are given up to quadratures in terms of the $\wp$ function of the Weierstrass.