Moved IDAxiBrillBH.tex to documentation.tex, so that the ThornGuide

will recognize it.

Did some minor editing and clean up.


git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/IDAxiBrillBH/trunk@37 0a4070d5-58f5-498f-b6c0-2693e757fa0f

1 files changed, 37 insertions, 29 deletions

diff --git a/doc/IDAxiBrillBH.tex b/doc/documentation.tex
index 0bbff8e..32e1678 100644
--- a/doc/IDAxiBrillBH.tex
+++ b/doc/documentation.tex
@@ -1,43 +1,49 @@
-% Thorn documentation template
 \documentclass{article}
 \begin{document}
 
 \title{IDAxiBrillBH}
-\author{Paul Walker, Steve Brand}
+\author{Paul Walker, Steve Brandt}
 \date{1 September 1999}
 \maketitle
 
-\abstract{Thorn IDAixBrillBH provides analytic initial data for vacuum
+\abstract{Thorn IDAxiBrillBH provides analytic initial data for a vacuum
   black hole spacetime:  a single Schwarzschild black hole in
   isotropic coordinates plus Brill wave.  This initial data is
   provided for the 3-conformal metric, it's spatial derivatives, and
   extrinsic curvature.}
 
+% The above author & date lines seem to be ignored, so for now I'll try to work
+% around this
+
+\section{Authors \& Date}
+Paul Walker and Steve Brandt\\
+1 September 1999
+
 \section{Purpose}
 
-The pioneer, Bernstein, studied single black hole which is
+The pioneer, Bernstein, studied a single black hole which is
 non-rotating and distorted in azimuthal line symmetry of 2 dimensional 
-case \cite{Bernstein93a}. In this non-rotating case, one choose the
+case \cite{Bernstein93a}. In this non-rotating case, one chooses the
 condition, $K_{ij} = 0$,  and 
 \begin{equation}
 \gamma_{ab} = \psi^4 \hat \gamma_{ab},
 \end{equation}
 where $\gamma_{ab}$ is the physical three metric and
-$\hat{\gamma}_{ab}$ is some chosen conformal three metric.   
+$\hat{\gamma}_{ab}$ is some chosen conformal three metric.
 
 The Hamiltonian constraint reduces to
 \begin{equation}
 \hat \Delta \psi = \frac{1}{8}\psi \hat R,
 \label{eqn:conformal_hamiltonian}
 \end{equation}
-where $\hat \Delta$ is the covariant Laplacian  and $\hat R$ is Ricci
+where $\hat \Delta$ is the covariant Laplacian and $\hat R$ is Ricci
 tensor for the conformal three metric. This form allows
 us to choose an arbitrary conformal three metric, and then solve an
-elliptic equation for the conformal factor, therefore, satisfying the
+elliptic equation for the conformal factor, therefore satisfying the
 constraint equations ($K_{ij} = 0$ trivially satisfies the momentum
 constraints in vacuum).  This approach was used to create
-``Brill wave'' in spacetime without black holes \cite{Brill59}.
-Bernstein extended to the black hole spacetime.  Using
+``Brill waves'' in a spacetime without black holes \cite{Brill59}.
+Bernstein extended this to the black hole spacetime.  Using
 spherical-polar coordinates, one can write the 3-metric,
 \begin{equation}
 \label{eqn:sph-cood}
@@ -52,29 +58,30 @@ radius, and applying $M/2r$ falloff conditions on $\psi$ at the
 outer boundary (the ``Robin'' condition), along with a packet which
 obeys the appropriate symmetries (including being invariant under the
 isometry operator), will make this solution describe a black hole with
-an incident gravitational wave.  The choice of $q=0$ produces
+an incident gravitational wave.  The choice of $q=0$ produces the
 Schwarzschild solution.   The typical $q$ function used in
-axisymmetry, and considered here in the non-rotating case is 
+axisymmetry, and considered here in the non-rotating case, is 
 \begin{equation}
 q = Q_0 \sin \theta^n \left [ \exp\left(\frac{\eta -
       \eta_0^2}{\sigma^2}\right ) + \exp\left(\frac{\eta +
       \eta_0^2}{\sigma^2}\right ) \right ].
 \end{equation}
 Note regularity along the axis requires that the exponent $n$ must be
-even.  Choosing a logarithmic radial coordinate $\eta$, which related
-asymptotic flat coordinate $r$ by $\eta = ln (2r/m)$, where m is a
+even.  Choose a logarithmic radial coordinate $\eta$, which is
+related to the
+asymptoticlly flat coordinate $r$ by $\eta = ln (2r/m)$, where $m$ is a
 scale parameter.  One can rewrite (\ref{eqn:sph-cood}) as
 \begin{equation}
 ds^2 = \psi(\eta)^4 [ e^{2 q} (d \eta^2 +  d\theta^2) +  \sin^2
 \theta d\phi^2].
 \end{equation}
 
-In the previous Breinstein work, the above $r$ are transformed to a
+In the previous Breinstein work, the above $r$ is transformed to a
 logarithmic radial coordinate
 
 \begin{equation}
 \label{eta_coord}
-\eta = \mbox{ln} (\frac{2r}{m}).
+\eta = \ln{\frac{2r}{m}}.
 \end{equation}
 
 The scale parameter $m$ is equal to the mass of the Schwarzschild
@@ -118,35 +125,36 @@ to the equation~(\ref{eqn:ham}), then we can linearize it as
   \eta^2} + \frac{\partial^2 q}{\partial \theta^2} -1).
 \label{eqn:ham_linear}
 \end{equation}
-For the boundary conditions, we use for inner boundary condition,
-which is isometry condition:
+For the boundary conditions, we use for the inner boundary condition
+an isometry condition:
 \begin{equation}
 \frac{\partial \tilde{\psi}}{\partial \eta}|_{\eta = 0} = 0,
 \end{equation}
-and outer boundary condition, which is Robin condition:
+and outer boundary condition, a Robin condition:
 \begin{equation}
 (\frac{\partial \tilde{\psi}}{\partial \eta} + \frac{1}{2}
 \tilde{\psi})|_{\eta=\eta_{max}} = 0.
 \end{equation}
 
-
-This thorn provides
- \begin{enumerate}
-  \item CactusEinstein
- \end{enumerate}
+%  [[ DPR: What is this: ?? ]]
+%This thorn provides
+% \begin{enumerate}
+%  \item CactusEinstein
+% \end{enumerate}
 
 \section{Comments}
 
 We calculate equation~(\ref{eqn:ham_linear}) with spherical
-coordinate.  However, Cactus needs Cartesian coordinate.  Then, we
-interpolate $\psi$ to the Cartesian grid by using interpolator.  Note
-interpolator has linear, quadratic, and cubic interpolation.  
+coordinates.  However, Cactus needs Cartesian coordinates.  Therefore,
+we interpolate $\psi$ to the Cartesian grid by using an interpolator.
+Note that the interpolator has linear, quadratic, and cubic
+interpolation.
 
-% Automatically created from the ccl files 
-% Do not worry for now.
+% Automatically created from the ccl files:
 \include{interface}
 \include{param}
 \include{schedule}
+
 \bibliographystyle{prsty}
 \begin{thebibliography}{10}
 \bibitem{Bernstein93a}