Some initial documentation for extract.

git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinAnalysis/Extract/trunk@64 5301f0c2-dbc4-4cee-b2f5-8d7afba4d129

1 files changed, 833 insertions, 8 deletions

diff --git a/doc/documentation.tex b/doc/documentation.tex
index b097bbe..aac6987 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -74,10 +74,10 @@
 \begin{document}
 
 % The author of the documentation
-\author{} 
+\author{Gabrielle Allen} 
 
 % The title of the document (not necessarily the name of the Thorn)
-\title{}
+\title{Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes}
 
 % the date your document was last changed, if your document is in CVS, 
 % please us:
@@ -91,7 +91,22 @@
 
 % Add all definitions used in this documentation here 
 %   \def\mydef etc
-
+\def\a   {\alpha}
+\def\b   {\beta}
+\def\p   {\phi}
+\def\t   {\theta}
+\def\Y   {Y_{lm}}
+\def\Ys  {Y^*_{lm}}
+\def\Yt  {Y_{lm,\theta}}
+\def\Ytt {Y_{lm,\theta\theta}}
+\def\Ytp {Y_{lm,\theta\phi}}
+\def\Yp  {Y_{lm,\phi}}
+\def\Ypp {Y_{lm,\phi\phi}}
+\def\Yz  {Y_{l0}}
+\def\Yzt {Y_{l0,\theta}}
+\def\Yztt{Y_{l0,\theta\theta}}
+\def\c   {\cos\theta}
+\def\s   {\sin\theta}
 % Add an abstract for this thorn's documentation
 \begin{abstract}
 
@@ -102,33 +117,843 @@
 
 \section{Introduction}
 
+Thorn Extract calculates first order gauge invariant waveforms from a numerical spacetime, under the basic assumption that, at the spheres of extract the spacetime is approximately Schwarzschild. In addition, other quantities such as mass, angular momentum and spin can be determined.
+
+This thorn should not be used blindly, it will always return some waveform, however it is up to the user to determine whether this is the appropriate expected first order gauge invariant waveform.
+
 \section{Physical System}
 
+\subsection{Wave Forms}
+
+Assume a spacetime $g_{\alpha\beta}$ which can be written as a Schwarzschild 
+background $g_{\alpha\beta}^{Schwarz}$ with perturbations $h_{\alpha\beta}$:
+\begin{equation}
+g_{\alpha\beta} = g^{Schwarz}_{\alpha\beta} + h_{\alpha\beta}
+\end{equation}
+with
+$$
+\{g^{Schwarz}_{\alpha\beta}\}(t,r,\theta,\phi) = 
+\left( \begin{array}{cccc}
+ -S & 0      & 0   & 0                \\
+ 0  & S^{-1} & 0   & 0                \\
+ 0  & 0      & r^2 & 0                \\
+ 0  & 0      & 0   & r^2 \sin^2\theta
+\end{array}\right)
+\qquad
+S(r)=1-\frac{2M}{r}
+$$
+%
+The 3-metric perturbations $\gamma_{ij}$ can be decomposed using tensor
+harmonics into  $\gamma_{ij}^{lm}(t,r)$ where
+$$
+  \gamma_{ij}(t,r,\theta,\phi)=\sum_{l=0}^\infty \sum_{m=-l}^l
+                       \gamma_{ij}^{lm}(t,r)
+$$
+%
+and
+%
+$$
+  \gamma_{ij}(t,r,\t,\p) = \sum_{k=0}^6 p_k(t,r) {\bf V}_k(\t,\p)
+$$
+where $\{{\bf V}_k\}$ is some basis for tensors on a 2-sphere
+in 3-D Euclidean space.
+%
+%
+%
+Working with the Regge-Wheeler basis (see Section~\ref{reggewheeler})
+the 3-metric is then expanded in terms of the (six) standard
+Regge-Wheeler functions $\{c_1^{\times lm}, c_2^{\times lm},
+h_1^{+lm}, H_2^{+lm}, K^{+lm}, G^{+lm}\}$
+\cite{regge},
+\cite{moncrief74}. 
+Where each of the functions is either  {\it odd} 
+  ($\times$) or 
+  {\it even} 
+  ($+$) 
+parity. The decomposition is then written 
+\begin{eqnarray}
+\gamma_{ij}^{lm} & = & c_1^{\times lm}(\hat{e}_1)_{ij}^{lm}
+                   +   c_2^{\times lm}(\hat{e}_2)_{ij}^{lm} 
+\nonumber\\
+                 & + & h_1^{+lm}(\hat{f}_1)_{ij}^{lm} 
+                   +   A^2 H_2^{+lm}(\hat{f}_2)_{ij}^{lm}
+                   +   R^2 K^{+lm}(\hat{f}_3)_{ij}^{lm}
+                   +   R^2 G^{+lm}(\hat{f}_4)_{ij}^{lm}
+\end{eqnarray}
+which we can write in an expanded form as 
+\begin{eqnarray}
+\gamma_{rr}^{lm} 
+  & = & A^2 H_2^{+lm} \Y 
+\\
+\gamma_{r\t}^{lm} 
+  & = & - c_1^{\times lm} \frac{1}{\s} \Yp+h_1^{+lm}\Yt 
+\\
+\gamma_{r\p}^{lm} 
+  & = & c_1^{\times lm} \s \Yt+ h_1^{+lm}\Yp 
+\\
+\gamma_{\t\t}^{lm} 
+  & = & c_2^{\times lm}\frac{1}{\s}(\Ytp-\cot\t \Yp) 
+      + R^2 K^{+lm}\Y + R^2 G^{+lm}    \Ytt 
+\\
+\gamma_{\t\p}^{lm} 
+  & = & -c_2^{\times lm}\s \frac{1}{2} 
+  \left(
+  \Ytt-\cot\t \Yt-\frac{1}{\sin^2\theta}\Y \right)
+  + R^2 G^{+lm}(\Ytp-\cot\t \Yp)
+\\
+\gamma_{\p\p}^{lm}
+  & = &  -\s c_2^{\times lm} (\Ytp - \cot\t \Yp)
+        +R^2 K^{+lm}\sin^2\t \Y
+        +R^2 G^{+lm} (\Ypp+\s\c \Yt)
+\end{eqnarray}
+
+A similar decomposition allows the four gauge components of the 4-metric to 
+be written in terms of  {\it three} even-parity variables 
+  $\{H_0,H_1,h_0\}$
+and the {\it one} odd-parity variable 
+  $\{c_0\}$ 
+\begin{eqnarray}
+g_{tt}^{lm} 
+  & = & N^2 H_0^{+lm} \Y 
+\\
+g_{tr}^{lm}
+  & = & H_1^{+lm} \Y
+\\
+g_{t\t}^{lm}
+  & = & h_0^{+lm} \Yt - c_0^{\times lm}\frac{1}{\s}\Yp
+\\
+g_{t\p}^{lm}
+  & = & h_0^{+lm} \Yp + c_0^{\times lm} \s \Yt
+\end{eqnarray}         
+Also from $g_{tt}=-\alpha^2+\beta_i\beta^i$ we have
+$$
+  \alpha^{lm} = -\frac{1}{2}NH_0^{+lm}Y_{lm}
+$$
+
+
+It is useful to also write this with the perturbation split into even and
+odd parity parts:
+$$
+g_{\alpha\beta} = {g}^{background}_{\alpha\beta} +
+   \sum_{l,m} g^{lm,odd}_{\alpha\beta}
++\sum_{l,m} g^{lm,even}_{\alpha\beta}
+$$
+where (dropping some superscripts)
+\begin{eqnarray*}
+\{g_{\alpha\beta}^{odd}\}
+&=&
+\left( 
+\begin{array}{cccc}
+0 & 0 &  - c_0\frac{1}{\s}\Yp
+    & c_0 \s \Yt
+\\
+. & 0 & - c_1\frac{1}{\s} \Yp
+  & c_1 \s \Yt
+\\
+. & . & c_2\frac{1}{\s}(\Ytp-\cot\t \Yp)  
+  & c_2\frac{1}{2} \left(\frac{1}{\s}
+          \Ypp+\c\Yt-\s\Ytt\right)
+\\
+.&.&.&c_2 (-\s \Ytp+\c \Yp)
+\end{array}
+\right)
+\\
+\{g_{\alpha\beta}^{even}\}
+&=&
+\left( 
+\begin{array}{cccc}
+N^2 H_0\Y & H_1\Y       & h_0\Yt          & h_0 \Yp             \\ 
+.       & A^2H_2\Y & h_1\Yt          & h_1 \Yp             \\
+.       & .           & R^2K\Y+r^2G\Ytt & R^2(\Ytp-\cot\t\Yp) \\
+.       & .           & .                & R^2 K\sin^2\t\Y+R^2G(\Ypp+\s\c\Yt)
+\end{array}
+\right)
+\end{eqnarray*}
+
+Now, for such a Schwarzschild background we can define two (and only two)
+unconstrained gauge invariant quantities 
+  $Q^{\times}_{lm}=Q^{\times}_{lm}(c_1^{\times lm},c_2^{\times lm})$ 
+and
+  $Q^{+}_{lm}=Q^{+}_{lm}(K^{+ lm},G^{+ lm},H_2^{+lm},h_1^{+lm})$, 
+which from
+\cite{abrahams96a} are
+\begin{eqnarray}
+Q^{\times}_{lm} 
+  & = & \sqrt{\frac{2(l+2)!}{(l-2)!}}\left[c_1^{\times lm}
+        + \frac{1}{2}\left(\partial_r c_2^{\times lm} - \frac{2}{r}
+        c_2^{\times lm}\right)\right] \frac{S}{r}
+\\
+Q^{+}_{lm}
+  & = & \frac{1}{\Lambda}\sqrt{\frac{2(l-1)(l+2)}{l(l+1)}}
+        (4rS^2 k_2+l(l+1)r k_1) 
+\\
+  & \equiv &
+        \frac{1}{\Lambda}\sqrt{\frac{2(l-1)(l+2)}{l(l+1)}}
+        \left(l(l+1)S(r^2\partial_r G^{+lm}-2h_1^{+lm})+
+        2rS(H_2^{+lm}-r\partial_r K^{+lm})+\Lambda r K^{+lm}\right)
+\end{eqnarray}
+where
+\begin{eqnarray}
+k_1 & = & K^{+lm} + \frac{S}{r}(r^2\partial_r G^{+lm} - 2h^{+lm}_1) \\
+k_2 & = & \frac{1}{2S}
+          \left[H^{+lm}_2-r\partial_r k_1-\left(1-\frac{M}{rS}\right) 
+            k_1 + S^{1/2}\partial_r
+          (r^2 S^{1/2} \partial_r G^{+lm}-2S^{1/2}h_1^{+lm})\right]
+\\
+&\equiv& \frac{1}{2S}\left[H_2-rK_{,r}-\frac{r-3M}{r-2M}K\right]
+\end{eqnarray}
+
+\noindent
+NOTE: These quantities compare with those in Moncrief \cite{moncrief74} by
+\begin{eqnarray*}
+\mbox{Moncriefs odd parity Q: }\qquad Q^\times_{lm} &=&
+ \sqrt{\frac{2(l+2)!}{(l-2)!}}Q
+ \\
+\mbox{Moncriefs even parity Q: } \qquad Q^+_{lm} &=&
+ \sqrt{\frac{2(l-1)(l+2)}{l(l+1)}}Q
+\end{eqnarray*}
+
+Note that these quantities only depend on the purely spatial 
+Regge-Wheeler functions, and not the gauge parts. (In the Regge-Wheeler 
+and Zerilli gauges, these are just respectively (up to a rescaling)
+ the Regge-Wheeler 
+and Zerilli functions).
+These quantities satisfy the wave equations
+\begin{eqnarray*}
+  &&(\partial^2_t-\partial^2_{r^*})Q^\times_{lm}+S\left[\frac{l(l+1)}{r^2}-\frac{6M}{r^3}
+  \right]Q^{\times}_{lm}  =  0 
+  \\
+  &&(\partial^2_t-\partial^2_{r^*})Q^+_{lm}+S\left[
+    \frac{1}{\Lambda^2}\left(\frac{72M^3}{r^5}-\frac{12M}{r^3}(l-1)(l+2)\left(1-\frac{3M}{r}\right)
+    \right)+\frac{l(l-1)(l+1)(l+2)}{r^2\Lambda}\right]Q^+_{lm}=0
+\end{eqnarray*}
+where
+\begin{eqnarray*}
+  \Lambda &=& (l-1)(l+2)+6M/r \\
+  r^*     &=& r+2M\ln(r/2M-1)
+\end{eqnarray*}
+ 
+
+
+
 \section{Numerical Implementation}
 
+The implementation assumes that the numerical solution, on a Cartesian grid, 
+is approximately Schwarzshild on the spheres of constant $r=\sqrt(x^2+y^2+z^2)$
+where the waveforms are extracted. The general procedure is then:
+
+\begin{itemize}
+
+\item Project the required metric components, and radial derivatives of
+	metric components,
+      onto spheres of constant coordinate radius
+      (these spheres are chosen via parameters). 
+
+\item Transform the metric components and there derivatives on the 2-spheres 
+	from Cartesian coordinates 
+	into a spherical coordinate system.
+
+\item Calculate the physical metric on these spheres if a conformal factor 
+	is being used.
+
+\item Calculate the transformation from the coordinate radius to an areal radius for each 
+	sphere.
+
+\item Calculate the $S$ factor on each sphere. Combined with the areal radius This also produces an estimate of the mass.
+
+\item Calculate the six Regge-Wheeler variables, and required radial derivatives, on these spheres by integration
+	of combinations of the metric components over each sphere. 
+
+\item Contruct the gauge invariant quantities from these Regge-Wheeler variables.
+
+\end{itemize}
+
+\subsection{Project onto Spheres of Constant Radius}
+
+This is performed by interpolating the metric components, and if needed the conformal factor, onto the spheres. Although 2-spheres are hardcoded, the source code could easily be changed here to project onto e.g. 2-ellipsoids.
+
+\subsection{Calculate Radial Transformation}
+
+The areal coordinate $\hat{r}$ of each sphere is calculated by
+\begin{equation}
+  \hat{r}    =  \hat{r}(r) = \left[
+            \frac{1}{4\pi}
+            \int\sqrt{\gamma_{\t\t}
+            \gamma_{\p\p}}d\t d\p \right]^{1/2}
+\end{equation}
+from which
+\begin{equation}
+\frac{d\hat{r}}{d\eta} = \frac{1}{16\pi \hat{r}}
+  \int\frac{\gamma_{\t\t,\eta}\gamma_{\p\p}+\gamma_{\t\t}\gamma_{\p\p,\eta}}
+  {\sqrt{\gamma_{\t\t}\gamma_{\p\p}}} \ d\t d\p
+\end{equation}
+
+Note that this is not the only way to combine metric components to get
+the areal radius, but this one was used because it gave better values 
+for extracting close to the event horizon for perturbations of black holes.
+
+\subsection{Calculate $S$ factor and Mass Estimate}
+
+\begin{equation}
+S(\hat{r}) = \left(\frac{\partial\hat{r}}{\partial r}\right)^2 \int g_{tt} \ d\t d\p
+\end{equation}
+
+\begin{equation}
+M(\hat{r}) = \hat{r}\frac{1-S}{2}
+\end{equation}
+
+\subsection{Calculate Regge-Wheeler Variables}
+
+\begin{eqnarray*}
+c_1^{\times lm}  &=&  \frac{1}{l(l+1)}
+                      \int \frac{\gamma_{\hat{r}\p}Y^*_{lm,\t}
+                                -\gamma_{\hat{r}\t} Y^*_{lm,\p} }
+                     {\s}d\Omega
+\\
+c_2^{\times lm} & = & -\frac{2}{l(l+1)(l-1)(l+2)}
+                      \int\left\{
+           \left(-\frac{1}{\sin^2\t}\gamma_{\t\t}+\frac{1}
+           {\sin^4\t}\gamma_{\p\p}\right)
+           (\s Y^*_{lm,\t\p}-\c Y^*_{lm,\p})
+\right.
+\\
+&&\left.
+           + \frac{1}{\s} \gamma_{\t\p}
+           (Y^*_{lm,\t\t}-\cot\t Y^*_{lm,\t}
+           -\frac{1}{\sin^2\t}Y^*_{lm,\p\p}) \right\}d\Omega
+\\
+h_1^{+lm} &=& \frac{1}{l(l+1)}
+            \int \left\{
+                \gamma_{\hat{r}\t} Y^*_{lm,\t} + \frac{1}{\sin^2\t}
+                \gamma_{\hat{r}\p}Y^*_{lm,\p}\right\} d\Omega
+\\
+H_2^{+lm} &=& S  \int \gamma_{\hat{r}\hat{r}} \Ys d\Omega
+\\
+K^{+lm}   &=& \frac{1}{2\hat{r}^2} \int \left(\gamma_{\t\t}+
+           \frac{1}{\sin^2\t}\gamma_{\p\p}\right)\Ys
+           d\Omega
+\\
+          &&+\frac{1}{2\hat{r}^2(l-1)(l+2)}
+\int \left\{
+  \left(\gamma_{\t\t}-\frac{\gamma_{\p\p}}{\sin^2\t}\right)
+    \left(Y^*_{lm,\t\t}-\cot\t Y^*_{lm,\t}-\frac{1}{\sin^2\t}
+    Y^*_{lm,\p\p}\right) 
+\right.
+\\
+&&\left.
+   + \frac{4}{\sin^2\t}\gamma_{\t\p}(Y^*_{lm,\t\p}-\cot\t
+     Y^*_{lm,\p})
+     \right \} d\Omega
+\\
+G^{+lm}  &=& \frac{1}{\hat{r}^2 l(l+1)(l-1)(l+2)}
+  \int \left\{
+  \left(\gamma_{\t\t}-\frac{\gamma_{\p\p}}{\sin^2\t}\right)
+    \left(Y^*_{lm,\t\t}-\cot\t Y^*_{lm,\t}-\frac{1}{\sin^2\t}
+    Y^*_{lm,\p\p}\right) 
+\right.
+\\
+&&\left.
+   +\frac{4}{\sin^2\t}\gamma_{\t\p}(Y^*_{lm,\t\p}-\cot\t
+   Y^*_{lm,\p})
+   \right\}d\Omega
+\end{eqnarray*}
+where
+\begin{eqnarray}
+\gamma_{\hat{r}\hat{r}}      & = & \frac{\partial r}{\partial \hat{r}}
+                       \frac{\partial r}{\partial \hat{r}}
+                       \gamma_{rr} 
+\\
+\gamma_{\hat{r}\t} & = & \frac{\partial r}{\partial \hat{r}}
+                       \gamma_{r\t} 
+\\
+\gamma_{\hat{r}\p}   & = & \frac{\partial r}{\partial \hat{r}}
+                       \gamma_{r\p}
+\end{eqnarray}
+
+\subsection{Calculate Gauge Invariant Quantities}
+
+\begin{eqnarray}
+Q^{\times}_{lm} 
+  & = & \sqrt{\frac{2(l+2)!}{(l-2)!}}\left[c_1^{\times lm}
+        + \frac{1}{2}\left(\partial_{\hat{r}} c_2^{\times lm} - \frac{2}{\hat{r}}
+        c_2^{\times lm}\right)\right] \frac{S}{\hat{r}}
+\\
+Q^{+}_{lm}
+  & = & \frac{1}{(l-1)(l+2)+6M/\hat{r}}\sqrt{\frac{2(l-1)(l+2)}{l(l+1)}}
+        (4\hat{r}S^2 k_2+l(l+1)\hat{r} k_1) 
+\end{eqnarray}
+where
+\begin{eqnarray}
+k_1 & = & K^{+lm} + \frac{S}{\hat{r}}(\hat{r}^2\partial_{\hat{r}} G^{+lm} - 2h^{+lm}_1) \\
+k_2 & = & \frac{1}{2S}
+          [H^{+lm}_2-\hat{r}\partial_{\hat{r}} k_1-(1-\frac{M}{\hat{r}S}) k_1 + S^{1/2}\partial_{\hat{r}}
+          (\hat{r}^2 S^{1/2} \partial_{\hat{r}} G^{+lm}-2S^{1/2}h_1^{+lm}
+\end{eqnarray}
+
 \section{Using This Thorn}
 
+Use this thorn very carefully. Check the validity of the waveforms by running
+tests with different resolutions, different outer boundary conditions, etc
+to check that the waveforms are consistent.
+
 \subsection{Obtaining This Thorn}
 
+This thorn is part of the CactusEinstein arrangement.
+
 \subsection{Basic Usage}
 
 \subsection{Special Behaviour}
 
 \subsection{Interaction With Other Thorns}
 
-\subsection{Support and Feedback}
-
 \section{History}
 
 \subsection{Thorn Source Code}
 
-\subsection{Thorn Documentation}
+Much of the source code for Extract comes from a code written outside
+of Cactus for extracting waveforms from data generated by the NCSA
+G-Code for compare with linear evolutions of waveforms extracted from
+the Cauchy initial data.  This work was carried out in collaboration
+with Karen Camarda and Ed Seidel.
 
-\subsection{Acknowledgements}
 
+\section{Appendix: Regge-Wheeler Harmonics}
 
-\begin{thebibliography}{9}
+\label{reggewheeler}
+
+\begin{eqnarray*}
+(\hat{e}_1)^{lm} &=& 
+\left( \begin{array}{ccc}
+0  & -\frac{1}{\s}\Yp & \s \Yt \\
+.  & 0                                & 0                        \\
+.  & 0                                & 0 
+\end{array}\right)
+\\
+(\hat{e}_2)^{lm} &=& 
+\left( \begin{array}{ccc} 
+0 & 0 & 0 \\
+0 & \frac{1}{\s}(\Ytp-\cot\t \Yp) & . \\
+0 & -\frac{\s}{2}[\Ytt-\cot\t 
+    \Yt-\frac{1}{\sin^2\t}\Ypp]           & 
+            -\s [\Ytp-\cot\t \Yp]
+\end{array}\right)
+\\
+(\hat{f}_1)^{lm} &=& 
+\left( \begin{array}{ccc}
+  0 & \Yt & \Yp \\
+  . & 0   & 0           \\
+  . & 0   & 0 
+\end{array}\right)
+\\
+(\hat{f}_2)^{lm} &=& 
+\left( \begin{array}{ccc}
+\Y & 0 & 0 \\
+0      & 0 & 0 \\
+0      & 0 & 0 
+\end{array}\right)
+\\
+(\hat{f}_3)^{lm} &=& 
+\left( \begin{array}{ccc}
+0 & 0  & 0                  \\
+0 & \Y & 0                  \\
+0 & 0  & \sin^2\t \Y 
+\end{array}\right)
+\\
+(\hat{f}_4)^{lm} &=& 
+\left( \begin{array}{ccc}
+0 & 0                   & 0 \\
+0 & \Ytt & . \\
+0 & \Ytp-\cot \t \Yp & \Ypp+ \s \c \Yt
+\end{array}\right)
+\end{eqnarray*}
+
+\section{Appendix: Transformation Between Cartesian and Spherical Coordinates}
+
+First, the transformations between metric components in $(x,y,z)$ and $(r,\t,\p)$ coordinates. Here, $\rho=\sqrt{x^2+y^2}=r\s$,
+\begin{eqnarray*}
+  \frac{\partial x}{\partial r}
+  &=&
+  \sin\t\cos\p 
+  =
+  \frac{x}{r}
+\\
+  \frac{\partial y}{\partial r}
+  &=&
+  \sin\t\sin\p 
+  =
+  \frac{y}{r}
+\\
+  \frac{\partial z}{\partial r}
+  &=&
+  \cos\t 
+  =
+  \frac{z}{r}
+\\
+  \frac{\partial x}{\partial \t}
+  &=&
+  r\cos\t\cos\p 
+  =
+  \frac{xz}{\rho}
+\\
+  \frac{\partial y}{\partial \t}
+  &=&
+  r\cos\t\sin\p 
+  =
+  \frac{yz}{\rho}
+\\
+  \frac{\partial z}{\partial \t}
+  &=&
+  -r\sin\t 
+  =
+  -\rho
+\\
+  \frac{\partial x}{\partial \p}
+  &=&
+  -r\sin\t\sin\p
+  =
+  -y
+\\
+  \frac{\partial y}{\partial \p}
+  &=&
+  r\sin\t\cos\p 
+  =
+  x
+\\
+  \frac{\partial z}{\partial \p}
+  &=&
+  0
+\end{eqnarray*}
+
+
+\begin{eqnarray*}
+  \gamma_{rr} &=&
+  \frac{1}{r^2}
+     (x^2\gamma_{xx}+
+      y^2\gamma_{yy}+
+      z^2\gamma_{zz}+
+      2xy\gamma_{xy}+
+      2xz\gamma_{xz}+
+      2yz\gamma_{yz})
+\\
+  \gamma_{r\t} &=&
+  \frac{1}{r\rho}
+     (x^2 z \gamma_{xx}
+     +y^2 z \gamma_{yy}
+     -z \rho^2 \gamma_{zz}
+     +2xyz \gamma_{xy}
+     +x(z^2-\rho^2)\gamma_{xz}
+     +y(z^2-\rho^2)\gamma_{yz})
+\\
+  \gamma_{r\p} &=&
+  \frac{1}{r}
+     (-xy\gamma_{xx}
+      +xy\gamma_{yy}
+      +(x^2-y^2)\gamma_{xy}
+      -yz \gamma_{xz}
+      +xz\gamma_{yz})
+\\
+  \gamma_{\t\t} &=&
+  \frac{1}{\rho^2}
+  (x^2z^2\gamma_{xx}
+  +2xyz^2\gamma_{xy}
+  -2xz\rho^2\gamma_{xz}
+  +y^2z^2\gamma_{yy}
+  -2yz\rho^2\gamma_{yz}
+  +\rho^4\gamma_{zz})
+\\
+  \gamma_{\t\p} &=&
+  \frac{1}{\rho}
+  (-xyz\gamma_{xx}
+   +(x^2-y^2)z\gamma_{xy}
+   +\rho^2 y \gamma_{xz}
+   +xyz\gamma_{yy}
+   -\rho^2 x \gamma_{yz})
+\\
+  \gamma_{\p\p} &=&
+   y^2\gamma_{xx}
+   -2xy\gamma_{xy}
+   +x^2\gamma_{yy}
+\end{eqnarray*}   
+or,
+\begin{eqnarray*}
+\gamma_{rr}&=&
+\sin^2\t\cos^2\p\gamma_{xx}
++\sin^2\t\sin^2\p\gamma_{yy}
++\cos^2\t\gamma_{zz}
++2\sin^2\theta\cos\p\sin\p\gamma_{xy}
++2\sin\t\cos\t\cos\p\gamma_{xz}
+\\
+&&
++2\s\c\sin\p\gamma_{yz}
+\\
+\gamma_{r\t}&=&
+r(\s\c\cos^2\phi\gamma_{xx}
++2*\s\c\sin\p\cos\p\gamma_{xy}
++(\cos^2\t-\sin^2\t)\cos\p\gamma_{xz}
++\s\c\sin^2\p\gamma_{yy}
+\\
+&&
++(\cos^2\t-\sin^2\t)\sin\p\gamma_{yz}
+-\s\c\gamma_{zz})
+\\
+\gamma_{r\p}&=&
+r\s(-\s\sin\p\cos\p\gamma_{xx}
+-\s(\sin^2\p-\cos^2\p)\gamma_{xy}
+-\c\sin\p\gamma_{xz}
++\s\sin\p\cos\p\gamma_{yy}
+\\
+&&
++\c\cos\p\gamma_{yz})
+\\
+\gamma_{\t\t}&=&
+r^2(\cos^2\t\cos^2\p\gamma_{xx}
++2\cos^2\t\sin\p\cos\p\gamma_{xy}
+-2\s\c\cos\p\gamma_{xz}
++\cos^2\t\sin^2\p\gamma_{yy}
+\\
+&&
+-2\s\c\sin\p\gamma_{yz}
++\sin^2\t\gamma_{zz})
+\\
+\gamma_{\t\p}&=&
+r^2\s(-\c\sin\p\cos\p\gamma_{xx}
+-\c(\sin^2\p-\cos^2\p)\gamma_{xy}
++\s\sin\p\gamma_{xz}
++\c\sin\p\cos\p\gamma_{yy}
+\\
+&&
+-\s\cos\p\gamma_{yz})
+\\
+\gamma_{\p\p}&=&
+r^2\sin^2\t(\sin^2\p\gamma_{xx}
+-2\sin\p\cos\p\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\t\cos^2\p\gamma_{xx,\eta}
++\sin^2\t\sin^2\p\gamma_{yy,\eta}
++\cos^2\t\gamma_{zz,\eta}
++2\sin^2\theta\cos\p\sin\p\gamma_{xy,\eta}
+\\
+&&
++2\sin\t\cos\t\cos\p\gamma_{xz,\eta}
++2\s\c\sin\p\gamma_{yz,\eta}
+\\
+\gamma_{r\t,\eta}&=& 
+\frac{1}{r}\gamma_{r\t}+
+r(\s\c\cos^2\phi\gamma_{xx,\eta}
++\s\c\sin\p\cos\p\gamma_{xy,\eta}
++(\cos^2\t-\sin^2\t)\cos\p\gamma_{xz,\eta}
+\\
+&&
++\s\c\sin^2\p\gamma_{yy,\eta}
++(\cos^2\t-\sin^2\t)\sin\p\gamma_{yz,\eta}
+-\s\c\gamma_{zz,\eta})
+\\
+\gamma_{r\p,\eta}&=&
+\frac{1}{r}\gamma_{r\p}+
+r\s(-\s\sin\p\cos\p\gamma_{xx,\eta}
+-\s(\sin^2\p-\cos^2\p)\gamma_{xy,\eta}
+-\c\sin\p\gamma_{xz,\eta}
+\\
+&&
++\s\sin\p\cos\p\gamma_{yy,\eta}
++\c\cos\p\gamma_{yz,\eta})
+\\
+\gamma_{\t\t,\eta}&=&
+\frac{2}{r}\gamma_{\t\t}+
+r^2(\cos^2\t\cos^2\p\gamma_{xx,\eta}
++2\cos^2\t\sin\p\cos\p\gamma_{xy,\eta}
+-2\s\c\cos\p\gamma_{xz,\eta}
+\\
+&&
++\cos^2\t\sin^2\p\gamma_{yy,\eta}
+-2\s\c\sin\p\gamma_{yz,\eta}
++\sin^2\t\gamma_{zz,\eta})
+\\
+\gamma_{\t\p,\eta}&=&
+\frac{2}{r}\gamma_{\t\p}+
+r^2\s(-\c\sin\p\cos\p\gamma_{xx,\eta}
+-\c(\sin^2\p-\cos^2\p)\gamma_{xy,\eta}
++\s\sin\p\gamma_{xz,\eta}
+\\
+&&
++\c\sin\p\cos\p\gamma_{yy,\eta}
+-\s\cos\p\gamma_{yz,\eta})
+\\
+\gamma_{\p\p,\eta}&=&
+\frac{2}{r}\gamma_{\p\p}+
+r^2\sin^2\t(\sin^2\p\gamma_{xx,\eta}
+-2\sin\p\cos\p\gamma_{xy,\eta}
++\cos^2\phi\gamma_{yy,\eta})
+\end{eqnarray*}
+
+\section{Appendix: Integrations Over the 2-Spheres}
+
+
+This is done by using Simpson's rule twice. Once in each coordinate 
+direction. Simpson's rule is
+\begin{equation}
+\int^{x_2}_{x_1} f(x) dx = 
+  \frac{h}{3} [f_1+4f_2+2f_3+4f_4+\ldots+2f_{N-2}+4 f_{N-1}+f_N]
+  +O(1/N^4)
+\end{equation}
+$N$ must be an odd number.
+
+
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