From a8d3a7248bf04b1a18033d37784659d197106c1e Mon Sep 17 00:00:00 2001 From: schnetter Date: Sat, 30 Apr 2005 15:09:38 +0000 Subject: Describe "Minkowski/shifted gauge wave" and "Schwarzschild/BL". git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/Exact/trunk@223 e296648e-0e4f-0410-bd07-d597d9acff87 --- doc/documentation.tex | 87 +++++++++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 84 insertions(+), 3 deletions(-) (limited to 'doc') diff --git a/doc/documentation.tex b/doc/documentation.tex index ad6d338..3bc1950 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -80,7 +80,8 @@ \author{Original code by Carsten Gundlach and Miguel Alcubierre, \\ exact solutions added by many other people, \\ - this documentation by Jonathan Thornburg} + this documentation by Jonathan Thornburg, + and by other people} % The title of the document (not necessarily the name of the Thorn) \title{Thorn Guide for the {\bf Exact} Thorn} @@ -212,6 +213,8 @@ Model Name & -- & Minkowski spacetime in non-trivial spatial coordinates\\ {\tt "Minkowski/gauge wave"} & -- & Minkowski spacetime in gauge-wave coordinates \\ +{\tt "Minkowski/shifted gauge wave"} + & -- & Minkowski spacetime in shifted gauge-wave coordinates \\ {\tt "Minkowski/conf wave"} & -- & Minkowski spacetime with $\sin$ in conformal factor \\[1ex] % @@ -222,6 +225,8 @@ Model Name {\tt "Schwarzschild/PG"} & -- & Schwarzschild spacetime in Painlev\'{e}-Gullstrand coordinates (these have a flat 3-metric) \\ +{\tt "Schwarzschild/BL"} + & -- & Schwarzschild spacetime in Brill-Lindquist coordinates\\ {\tt "Schwarzschild/Novikov"} & -- & Schwarzschild spacetime in Novikov coordinates \\ {\tt "Kerr/Boyer-Lindquist"} @@ -599,8 +604,9 @@ The line element is \begin{equation} ds^2=-H dt^2 +Hdx^2+dy^2+dz^2, \end{equation} -where $H=H(x-t)$, for instance $H=1-a\sin\left((x-t)/\Lambda\right)$. -This is flat space but the slice is a planar wave travelling along the x axis. +where $H=H(x-t)$, for instance $H=1-A\sin\left((x-t)/\Lambda\right)$. +This is a flat spacetime, but the slice is a planar wave travelling +along the x axis. This thorn implements several possible choices for the $H$ function, controlled by the \verb|Minkowski_gauge_wave__what_fn| parameter: @@ -634,6 +640,51 @@ periodic boundary wouldn't make sence), especially in the diagonal case. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Minkowski Spacetime in shifted gauge-wave coordinates} + +\verb|Exact::exact_model = "Minkowski/shifted gauge wave"| specifies Minkowski +spacetime with the ``shifted gauge-wave'' coordinates suggested by +Jeff Winicour: +The line element is +\begin{equation} +ds^2 = (H-1)\, dt^2 + (H+1)\, dx^2 - 2H\, dt\, dx + dy^2 + dz^2 +\end{equation} +where $H=H(x-t)$, for instance $H=A\sin\left((x-t)/\Lambda\right)$. +This is a flat spacetime, but the slice is a planar wave travelling +along the x axis. + +This thorn implements one choice for the $H$ function, controlled by +the \verb|Minkowski_gauge_wave__what_fn| parameter: +\begin{eqnarray} +H(x-t) &=& 1 - A \sin \left(\frac{x-\omega t}{\Lambda} - \delta\right) +\end{eqnarray} + +The parameters are +\begin{itemize} +\item $A = \verb|Minkowski_gauge_wave__amplitude|$, the amplitude +\item $\omega = \verb|Minkowski_gauge_wave__omega|$, the angular frequency +\item $\lambda = \verb|Minkowski_gauge_wave__lambda|$, the wavelength +\item $\delta = \verb|Minkowski_gauge_wave__phase|$, the phase shift +\end{itemize} +A plane wave has $\omega = \pm \lambda$ for a wave that travels in the +$x$ direction, and $\omega = \pm \lambda \sqrt{2}$ for a diagonal +wave. + +If the Boolean parameter \verb|Minkowski_gauge_wave__diagonal| is +true, then we make the gauge wave travel diagonally across the grid by +the coordinate transformation +\begin{eqnarray} +x &=& \frac{1}{\sqrt{2}}(x^\prime - y^\prime) \\ +y &=& \frac{1}{\sqrt{2}}(x^\prime + y^\prime) %%%\\ +\end{eqnarray} +For code testing, the idea is to test evolving this with periodic +boundary conditions, to see whether the code is able to cope with +that. The tricky part is to make the wave fit the grid exactly +(otherwise the periodic boundary wouldn't make sence), especially in +the diagonal case. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \subsection{Minkowski Spacetime with $\sin$ term in conformal factor} \verb|Exact::exact_model = "Minkowski/conf wave"| specifies Minkowski @@ -787,6 +838,36 @@ g_{ab} = \left[ \right] \end{equation} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Schwarzschild spacetime in Brill--Lindquist coordinates} + +\verb|Exact::exact_model = "Schwarzschild/BL"| specifies Schwarzschild +spacetime in Brill--Lindquist coordinates. These coordinates have the +interesting property that the spatial metric is conformally flat and +time-symmetric for the initial data. The only physics parameter is +the black hole mass $m = \verb|Schwarzschild_BL__mass|$. + +There is also a numerical parameter \verb|Schwarzschild_BL__epsilon| +which is used to avoid division by zero if a grid point falls exactly +at the origin; the default setting should be ok for most purposes. + +In the usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates, the +4-metric is given by +\begin{eqnarray} + \alpha & = & 1 +\\ + \beta^i & = & 0 +\\ + \gamma_{ij} & = & \Psi^4\, \delta_{ij} +\end{eqnarray} +with the conformal factor +\begin{eqnarray} + \Psi & = & 1 + \frac{m}{2r} +\end{eqnarray} +where $r$ is the coordinate radius. + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Schwarzschild Spacetime in Novikov coordinates} -- cgit v1.2.3