From cab79658b8e19947c3b4058f75961f01a055ef2e Mon Sep 17 00:00:00 2001 From: allen Date: Mon, 1 Nov 1999 11:27:16 +0000 Subject: This commit was generated by cvs2svn to compensate for changes in r2, which included commits to RCS files with non-trunk default branches. git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/IDAxiBrillBH/trunk@3 0a4070d5-58f5-498f-b6c0-2693e757fa0f --- doc/IDAxiBrillBH.tex | 160 +++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 160 insertions(+) create mode 100644 doc/IDAxiBrillBH.tex (limited to 'doc') diff --git a/doc/IDAxiBrillBH.tex b/doc/IDAxiBrillBH.tex new file mode 100644 index 0000000..0bbff8e --- /dev/null +++ b/doc/IDAxiBrillBH.tex @@ -0,0 +1,160 @@ +% Thorn documentation template +\documentclass{article} +\begin{document} + +\title{IDAxiBrillBH} +\author{Paul Walker, Steve Brand} +\date{1 September 1999} +\maketitle + +\abstract{Thorn IDAixBrillBH provides analytic initial data for 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.} + +\section{Purpose} + +The pioneer, Bernstein, studied 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 +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. + +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 +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 +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 +spherical-polar coordinates, one can write the 3-metric, +\begin{equation} +\label{eqn:sph-cood} +ds^2 = \psi^4 (e^{2q} (dr^2 + r^2 d \theta^2) + r^2 \sin \theta d +\phi^2), +\end{equation} +where $q$ is the Brill ``packet'' which takes some functional form. +Using this ansatz with (\ref{eqn:conformal_hamiltonian}) leads to +an elliptic equation for $\psi$ which must be solved +numerically. Applying the isometry condition on $\psi$ at a finite +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 +Schwarzschild solution. The typical $q$ function used in +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 +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 +logarithmic radial coordinate + +\begin{equation} +\label{eta_coord} +\eta = \mbox{ln} (\frac{2r}{m}). +\end{equation} + +The scale parameter $m$ is equal to the mass of the Schwarzschild +black hole, if $q=0$. In this coordinate, the 3-metric is +\begin{equation} +\label{eqn:metric_brill_eta} +ds^2 = \tilde{\psi}^4 (e^{2q} (d\eta^2+d\theta^2)+\sin^2 \theta +d\phi^2), +\end{equation} +and the Schwarzschild solution is +\begin{equation} +\label{eqn:psi} +\tilde{\psi} = \sqrt{2M} \cosh (\frac{\eta}{2}). +\end{equation} +We also change the notation of $\psi$ for the conformal factor is same +as $\tilde{\psi}$ \cite{Camarda97a}, for the $\eta$ coordinate has the +factor $r^{1/2}$ in the conformal factor. Clearly $\psi(\eta)$ and +$\psi$ differ by a factor of $\sqrt{r}$. The Hamiltonian +constraint is +\begin{equation} +\label{eqn:ham} +\frac{\partial^2 \tilde{\psi}}{\partial \eta^2} + \frac{\partial^2 + \tilde{\psi}}{\partial \theta^2} + \cot \theta \frac{\partial + \tilde{\psi}}{\partial \theta} = - \frac{1}{4} \tilde{\psi} +(\frac{\partial^2 q}{\partial \eta^2} + \frac{\partial^2 q}{\partial + \theta^2} -1). +\end{equation} + +For solving this Hamiltonian constraint numerically. At first +we substitute +\begin{eqnarray} +\delta \tilde{\psi} & = & \tilde{\psi}+\tilde{\psi}_0 \\ + & = & \tilde{\psi}-\sqrt{2m} \cosh(\frac{\eta}{2}). +\end{eqnarray} +to the equation~(\ref{eqn:ham}), then we can linearize it as +\begin{equation} +\frac{\partial^2 \delta\tilde{\psi}}{\partial \eta^2} + \frac{\partial^2 + \delta\tilde{\psi}}{\partial \theta^2} + \cot \theta \frac{\partial + \delta\tilde{\psi}}{\partial \theta} = - \frac{1}{4} +(\delta\tilde{\psi} + \tilde{\psi}_0) (\frac{\partial^2 q}{\partial + \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: +\begin{equation} +\frac{\partial \tilde{\psi}}{\partial \eta}|_{\eta = 0} = 0, +\end{equation} +and outer boundary condition, which is 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} + +\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. + +% Automatically created from the ccl files +% Do not worry for now. +\include{interface} +\include{param} +\include{schedule} +\bibliographystyle{prsty} +\begin{thebibliography}{10} +\bibitem{Bernstein93a} + D. Bernstein, Ph.D thesis University of Illinois Urbana-Champaign, + (1993) +\bibitem{Brill59} + D. S. Brill,Ann. Phys.{\bf 7}, 466 (1959) +\bibitem{Camarda97a} + K. Camarda, Ph.D thesis University of Illinois Urbana-Champaign, (1998) +\end{thebibliography} +\end{document} -- cgit v1.2.3