Weyl vacuum solutions in spherical coordinates

The Weyl metric in spherical coordinates is given by:

(1)

Calculating the Ricci curvature tensor shows that the following components are identically zero:

Additionally, the components and are related by

Thus, Einstein's vacuum equations take the form

From the first equation, it is clear that has to be

where are the Legendre polynomials. Further manipulation reduces the remaining 3 equations into

Solving the system of linear equations formed by the two last expressions yields

where the derivatives of are given by

with the abbreviations

Hence, is

By using the recursion properties of the Legendre polynomials

can rewritten in the simpler form