Appendix: Transformation Between Cartesian and Spherical Coordinates

First, the transformations between metric components in $(x,y,z)$ and $(r,\theta,\phi)$ coordinates. Here, $\rho=\sqrt{x^2+y^2}=r\sin\theta$,

$$\begin{eqnarray*} \frac{\partial x}{\partial r} &=& \sin\theta \cos\phi = ... ...a \cos\phi = x \\ \frac{\partial z}{\partial \phi } &=& 0 \end{eqnarray*}$$

$$\begin{eqnarray*} \gamma_{rr} &=& \frac{1}{r^2} (x^2\gamma_{xx}+ y^2\gamma_{... ...phi \phi } &=& y^2\gamma_{xx} -2xy\gamma_{xy} +x^2\gamma_{yy} \end{eqnarray*}$$

or,

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

We also need the transformation for the radial derivative of the metric components:

$$\begin{eqnarray*} \gamma_{rr,\eta}&=& \sin^2\theta \cos^2\phi \gamma_{xx,\eta} +... ...2\sin\phi \cos\phi \gamma_{xy,\eta} +\cos^2\phi\gamma_{yy,\eta}) \end{eqnarray*}$$