Schwarzschild Solution for Planetary Motion: Find x'i from xi

[Philosophaie]

Schwarzschild solution for Planetary Motion:

$g_{ij}= \left( \begin{array}{cccc} \frac{1}{(1-(\frac{2*m}{r}))} & 0 & 0 & 0 \\ 0 & r^2 & 0 & 0 \\ 0 & 0 & r^2*(sin\theta)^2 & 0 \\ 0 & 0 & 0 & c^2*(1-\frac{2*m}{r}) \end{array} \right)$


where $m=\frac{G*(Mass of Sun)}{c^2}$

My question is how do you find the Resultant Contravarient Position Vector.

$x^{'i} = \left( \begin{array}{c} r' \\ \theta' \\ \phi' \\ t' \end{array} \right)$
given the Contravarient Position Vector.

$x^{i} = \left( \begin{array}{c} r \\ \theta \\ \phi \\ t \end{array} \right)$

from the Schwarzschild Metric Tensor.

 

[Dale]

Just solve the geodesic equation. The simplest case will be for constant r and $\phi$.
http://en.wikipedia.org/wiki/Schwarzschild_geodesics