Transformation to Spherical Cooordinates

The values of the metric and/or extrinsic curvature in a spherical polar coordinate system $(r,\theta,\phi)$ evaluated at each point on the computational grid are placed in the grid functions (grr, grt, grp, gtt, gtp, gpp) and (krr, krt, krp, ktt, ktp, kpp). In the spherical transformation, the $\theta$ coordinate is referred to as q and the $\phi$ as p.

The general transformation from Cartesian to Spherical for such tensors is

$$\begin{eqnarray*} A_{rr}&=& \sin^2\theta\cos^2\phi A_{xx} +\sin^2\theta\sin^2\ph... ...(\sin^2\phi A_{xx} -2\sin\phi\cos\phi A_{xy} +\cos^2\phi A_{yy}) \end{eqnarray*}$$

If the parameter normalize_dtheta_dphi is set to yes, the angular components are projected onto the vectors $(r d\theta, r \sin\theta d \phi)$ instead of the default vector $(d \theta, d\phi)$. That is,

$$\begin{eqnarray*} A_{\theta\theta} & \rightarrow & A_{\theta\theta}/r^2 \\ A_{\... ...\\ A_{\theta\phi} & \rightarrow & A_{\theta\phi}/r^2\sin\theta) \end{eqnarray*}$$