The well known Schwarzschild solution is given by
\begin{equation} (ds)^2 = \left( 1 - \frac{2GM}{c^2r} \right) dt^2 - \left( 1 - \frac{2GM}{c^2r} \right)^{-1} dr^2 - r^2 d\Omega^2. \end{equation} It has been on my mind a lot about attempting to represent an aspherical body in general relativity to model the geometry of the Earth. In Newtonian mechanics a more accurate description of the Earth's oblate geometry can be obtained by expanding the Newtonian potential using spherical harmonics. We can clearly identify the Newtonian potential in the Schwarzschild solution above.
There is obviously a reason stopping the spherical harmonic expansion into the above equation. However, I don't know why? Any suggestions?