From 01241eb266c096351f9613ae2d0475782983f547 Mon Sep 17 00:00:00 2001 From: allen Date: Mon, 29 Apr 2002 17:12:23 +0000 Subject: New doc from Denis git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/IDAnalyticBH/trunk@103 6a3ddf76-46e1-4315-99d9-bc56cac1ef84 --- doc/ThornGuide.tex | 132 -------------- doc/documentation.tex | 486 ++++++++++++++++++++++++++++++++++++++++---------- 2 files changed, 396 insertions(+), 222 deletions(-) delete mode 100644 doc/ThornGuide.tex (limited to 'doc') diff --git a/doc/ThornGuide.tex b/doc/ThornGuide.tex deleted file mode 100644 index c03e09b..0000000 --- a/doc/ThornGuide.tex +++ /dev/null @@ -1,132 +0,0 @@ -\documentstyle{report} -\newcommand{\parameter}[1]{{\it #1}} - -\begin{document} - -\chapter{IDAnalyticBH} - -\begin{tabular}{@{}ll} -Code Authors & Joan Masso, Paul Walker, Ed Seidel. Gabrielle Allen \\ -Maintained by & Cactus Developers \\ -Documentation Authors & -\end{tabular} - -\section{Introduction} - -\subsection{Purpose of Thorn} - -Thorn IDAnalyticBH provides analytic initial data for vacuum black -hole spacetimes. Initial data is provided for the 3-metric, extrinsic -curvature, and if appropriate the conformal factor and it's spatial -derivatives. The current initial data sets are for a single (Schwarzschild) -black hole in isotropic coordinates, up to four Brill-Lindquist black -holes, and any number of Misner-type black holes. - -\subsection{Technical Specification} - -\begin{itemize} - -\item{Implements} einsteinID -\item{Inherits from} einstein -\item{Tested with thorns} Einstein - -\end{itemize} - -\section{Theoretical Background} - - -\section{Algorithmic and Implementation Details} - -This thorn uses no special numerical methods, however two points -are worth noting - -\begin{enumerate} - -\item{} The solution for Misner is obtained by summing a sequence - -\item{} The spatial derivatives of the conformal metric (when required) - are calculated accurately using finite differencing of the - exact solution by a very small spacing - -\end{enumerate} - -\section{Using the Thorn} - -This thorn can provide either the physical metric (use\_conformal=''no'') -or the conformal metric and a conformal factor (and its spatial derivatives) -(use\_conformal=''yes''). In general, the option use\_conformal=''yes'' should -be used, since ????. - -\section{Parameters} - -\subsection{Extended Parameters} -\begin{tabular}{l|l|l|l} -&&&\\ -einstein &&&\\ -\hline -\parameter{initial\_data} & KEYWORD & schwarzschild & One Schwarzschild black hole \\ -& & bl\_bh & Brill Lindquist black holes \\ -& & misner\_bh &Misner black holes \\ -& & multiple\_misner\_bh & Multiple Misner black holes \\ -\parameter{initial\_lapse} & KEYWORD & schwarz & Set lapse to schwarzschild \\ -\end{tabular} - -\subsection{Private Parameters} - -\begin{tabular}{l|l|l|l|l} -&&&&\\ -Schwarzschild & & & & \\ -\hline -\parameter{mass} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 2.0 & Mass of black hole \\ -&&&&\\ -Multiple Misner & & & & \\ -\hline -\parameter{mu} & {\t CCTK\_REAL} & $[0,\infty)$ & 1.2 & Misner $\mu$ value \\ -\parameter{nmax} & {\t CCTK\_INT} & $[0,\infty)$ & 30 & Numer of terns to include for Misner series \\ -\parameter{misner\_nmh} & {\t CCTK\_INT} & $[0,10]$ & 1 & Number of Misner black holes \\ -&&&&\\ -Brill Lindquist & & & & \\ -\hline -\parameter{bl\_nbh} & {\t CCTK\_INT} & $[1,4]$ & 1 & Number of Brill Lindquist black holes\\ -\parameter{bl\_x0\_1} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & x-position of first BL hole\\ -\parameter{bl\_y0\_1} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & y-position of first BL hole\\ -\parameter{bl\_z0\_1} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & z-position of first BL hole\\ -\parameter{bl\_M\_1} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 1.0 & mass of first BL hole\\ -\parameter{bl\_x0\_2} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & x-position of second BL hole\\ -\parameter{bl\_y0\_2} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & y-position of second BL hole\\ -\parameter{bl\_z0\_2} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & z-position of second BL hole\\ -\parameter{bl\_M\_2} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 1.0 & mass of second BL hole\\ -\parameter{bl\_x0\_3} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & x-position of third BL hole\\ -\parameter{bl\_y0\_3} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & y-position of third BL hole\\ -\parameter{bl\_z0\_3} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & z-position of third BL hole\\ -\parameter{bl\_M\_3} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 1.0 & mass of third BL hole\\ -\parameter{bl\_x0\_4} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & x-position of fourth BL hole\\ -\parameter{bl\_y0\_4} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & y-position of fourth BL hole\\ -\parameter{bl\_z0\_4} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 0.0 & z-position of fourth BL hole\\ -\parameter{bl\_M\_4} & {\t CCTK\_REAL} & $(-\infty,\infty)$ & 1.0 & mass of fourth BL hole\\ -\end{tabular} - -\subsection{Discussion} - - - -\section{Interaction with Other Thorns} - -It is still to be decided how initial data should be supplied to -for systems of evolution equations which use more or different -variables to those in the einstein implementation - -\section{Future Development} - -Initial data sets which are missing from this thorn include - -\begin{itemize} - -\item{} Boosted single black hole -\item{} Single black hole in harmonic spatial coordinates -\item{} Single black hole in Eddington-Finkelstein coordinates - -\end{itemize} - - -\end{document} diff --git a/doc/documentation.tex b/doc/documentation.tex index b46dbd4..9c2e870 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -1,97 +1,403 @@ +% /*@@ +% @file documentation.tex +% @date 28 April 2002 +% @author Denis Pollney +% @desc +% IDAnalyticBH users guide. +% @enddesc +% @version $Header$ +% @@*/ + +% +% FIXME: Is the non-conformal schwarzschild correct in the source +% FIXME: code??? +% FIXME: Add a reference to the quasi-isotropic Kerr coordinates? +% FIXME: What is the Cadez lapse for Misner? +% FIXME: The distance calculation (tmp1) seems wrong in the +% FIXME: brill-lindquist code. +% FIXME: Check that figure was included for Cartoon docs. + \documentclass{article} +\usepackage{epsfig} + +\parskip = 0 pt +\parindent = 0pt +\oddsidemargin = 0 cm +\textwidth = 16 cm +\topmargin = -1 cm +\textheight = 24 cm + \begin{document} +\title{Using \texttt{IDAnalyticBH} (OLD EINSTEIN!!)} +\author{Denis Pollney} +\date{April 2002} -\title{IDAnalyticBH} -\author{Steve Brandt, Carsten Gundlach, Joan Masso, Ed Seidel, Paul Walker} -\date{1997-1998} \maketitle -\abstract{Analytic initial data for vacuum black hole spacetimes} - -\section{Purpose} - Initial data is provided for the 3-metric, extrinsic curvature, and - if appropriate the conformal factor and it's spatial derivatives as - well as the lapse and shift. The current initial data sets are: - \begin{enumerate} - - \item single (Schwarzschild) black hole in isotropic - coordinates $$ g_{ij} = $$ - - \item up to four Brill-Lindquist black holes, - - \item Any number of Misner-type black holes. - - \item A single Kerr black hole - - \end{enumerate} - -\section{Solutions} - -\subsection{Schwarzschild} - -\noindent{\bf 3-Metric} - -\begin{eqnarray*} -g_{xx} &=& 1 + \frac{1}{2Mr^4}\\ -g_{yy} &=& 1 + \frac{1}{2Mr^4}\\ -g_{zz} &=& 1 + \frac{1}{2Mr^4}\\ -g_{xy} &=& 0\\ -g_{xz} &=& 0\\ -g_{yz} &=& 0 -\end{eqnarray*} - -\noindent{\bf Conformal 3-Metric} - -\begin{eqnarray*} -g_{xx} &=& 1 \\ -g_{yy} &=& 1 \\ -g_{zz} &=& 1 \\ -g_{xy} &=& 0\\ -g_{xz} &=& 0\\ -g_{yz} &=& 0 -\end{eqnarray*} - -\noindent{\bf Conformal Factor} - -$$ -\phi = 1+\frac{1}{2r} -$$ - -\noindent{\bf Conformal Factor} - -\begin{eqnarray*} -K_{xx} &=& 0\\ -K_{yy} &=& 0\\ -K_{zz} &=& 0\\ -K_{xy} &=& 0\\ -K_{xz} &=& 0\\ -K_{yz} &=& 0 -\end{eqnarray*} - -\noindent{\bf Lapse} - -$$ -\alpha = \frac{2r-M}{2r+M} -$$ - - -\section{Comments} -This thorn uses no special numerical methods, however two points -are worth noting -\begin{itemize} - \item The solution for Misner is obtained by summing a sequence - \item The spatial derivatives of the conformal metric (when required) - are calculated accurately using finite differencing of the - exact solution by a very small spacing -\end{itemize} - -It is still to be decided how initial data should be supplied to -for systems of evolution equations which use more or different -variables to those in the einstein implementation - -% Automatically created from the ccl files by using gmake thorndoc -\include{interface} -\include{param} -\include{schedule} +\abstract{The \texttt{IDAnalyticBH} thorn contains a number of initial data +sets for black-hole evolutions which can be specified analytically +as metric and extrinsic curvature components. The initial data which +is included in this thorn include single Schwarzschild and Kerr black +holes, and multiple black hole Misner and Brill-Lindquist solutions. +} + + +\section{Background} + +The \texttt{IDAnalyticBH} thorn exists as a central location to place any +initial dataset for black hole evolution that can be specified +analytically in terms of the metric, $g_{ab}$, and extrinsic +curvature, $K_{ab}$. + +The thorn extends the \texttt{einstein::initial\_data} parameter by +adding the following datasets: +\begin{description} + \item[\texttt{schwarzschild}] Schwarzschild, in isotropic + coordinates; + \item[\texttt{kerr}] Kerr, in Boyer-Lindquist coordinates; + \item[\texttt{misner}] Multiple Misner black holes; + \item[\texttt{bl\_bh}] Multiple Brill-Lindquist black holes. +\end{description} +Initial data for lapse and shift can also be specified in +this thorn.\\ + +The Cactus grid-functions corresponding to the initial data are +inherited from the thorn \texttt{CactusEinstein/Einstein}, along with +the conformal factor grid-function, \texttt{psi}, and its derivatives +which are optionally set based on the value of the parameter +\texttt{einstein::use\_conformal}.\\ + +The \texttt{IDAnalyticBH} has been written and augmented over an number of +years by many Cactus authors. These include John Baker, Steve Brandt, +Carsten Gundlach, Joan Masso, Ed Seidel, and Paul Walker. The +following sections describe each of the initial datasets and their +associated parameters in turn. + +\section{Schwarzschild} + +The Schwarzschild metric corresponds to a single, static, black hole. +If the Cactus metric is specified as a conformal metric (by setting +\texttt{einstein::use\_conformal="yes"}), then the metric is +set using isotropic coordinates \cite{mtw-isotropic}: +\begin{equation} + ds^2 = -\left(\frac{2r - M}{2r + M}\right)^2 + + \left(1 + \frac{M}{2r}\right)^4 \left(dr^2 + r^2(d\theta^2 + + \sin^2\theta d\phi^2)\right), +\end{equation} +with the Schwarzschild mass given by the single free parameter $M$. +Thus, the three metric and extrinsic curvature have the values: +\begin{eqnarray} + \hat{g}_{ab} & = & \psi^4 \delta_{ab}, \\ + \psi & = & (1 + \frac{M}{2r}), \\ + K_{ab} & = & 0. +\end{eqnarray} + +The mass is specified using the parameter +\texttt{idanalyticbh::mass}. The black hole is assumed to reside at +the origin of the grid, corresponding to the location $x=y=z=0$.\\ + +If the \texttt{einstein::use\_conformal} parameter has been set, then +the metric grid-functions (\texttt{einstein::gxx}, $\ldots$, +\texttt{einstein::gzz}) are given as $\delta_{ab}$, and the conformal +factor \texttt{einstein::psi} is set to the value specified +above. The derivatives of the conformal factor +(\texttt{einstein::psix}, etc.) are determined analytically. + +In order to give the lapse an initial profile which corresponds to +isotropic lapse of the $4$-metric specified above, use the parameter +\begin{verbatim} + idanalyticbh::initial_lapse = "schwarz" +\end{verbatim} +This will cause the \texttt{einstein::alp} grid-function to be +initialised to the value: +\begin{equation} + \alpha = \frac{2r - M}{2r + M}. +\end{equation}\\ + + +Note that the Schwarzschild data has the following non-standard +behaviour in response to the \texttt{einstein::use\_conformal} +parameter. If the \emph{physical} metric is requested +(ie. \texttt{use\_conformal} is set to \texttt{"no"}) then a +\emph{different} form of the Schwarzschild metric is set: +Schwarzschild coordinates are set instead of the isotropic +coordinates: +\begin{equation} + g_{xx} = g_{yy} = g_{zz} = 1 + 2M/r. +\end{equation}\\ + + +In order to carry out an evolution of a single Schwarzschild +black hole of mass $m=1$, using an initial lapse of $\alpha=1$, you +could modify your parameter file as follows: + +\begin{verbatim} + ActiveThorns = "... Einstein IDAnalyticBH ..." + + einstein::use_conformal = "yes" + + einstein::initial_data = "schwarzschild" + einstein::initial_lapse = "one" # or "schwarz" for isotropic lapse + + idanalyticbh::mass = 1.0 +\end{verbatim} + + +\section{Kerr} + +Kerr initial data for an isolated rotating black hole is specified +using the ``quasi-isotropic'' coordinates: +\begin{equation} + ds^2 = \psi^4 (dr^2 + r^2(d\theta^2 + \chi^2\sin^2\theta d\phi^2)), +\end{equation} +where +\begin{eqnarray} + \psi^4 & = & - 2\frac{a^2}{r^2}\cos\theta\sin\theta, \\ + \chi^2 & = & p^2 / \Sigma, \\ + p^2 & = & a^2 + {r_k}^2 - a B_\phi, \\ + r_k & = & r + M + \frac{M^2 - a^2}{4r}, \\ + B_\phi & = & -2 M r_k a \sin^2\theta / \Sigma,\\ + \Sigma & = & {r_k}^2 + a^2 \cos^2\theta. +\end{eqnarray} +The two free parameters are the Kerr mass, $M$, and angular momentum, +$a$. These are specified using the parameters +\texttt{idanalyticbh::mass} and \texttt{idanalyticbh::a\_kerr} +respectively. \emph{(Note that the default values for these parameters +are $M=2$ and $a=0.1$.)} The black hole is assumed to reside at the +centre of the coordinate system, at $x=y=z=0$. + +The \texttt{einstein::use\_conformal} parameter can be used to specify +whether the metric should be conformal or not. If the metric is +conformal, then $\psi$ is initialised as a separate grid function, and +it's first and second derivatives are calculated analytically and also +stored as grid functions. Otherwise, the conformal factor is +multiplied through in the expression for the 3-metric before the +values of the \texttt{einstein::metric} variables are set. The +extrinsic curvature is also determined analytically. + +The gauge can be set to the Kerr lapse and shift with the parameters +\begin{verbatim} + idanalyticbh::initial_lapse = "kerr" + idanalyticbh::initial_shift = "kerr" +\end{verbatim} +in which case the formulas +\begin{eqnarray} + \alpha & = &\sqrt{\frac{\Delta}{p^2}}, \\ + \beta^\phi & = & -2 m r_k a / p^2, +\end{eqnarray} +where +\begin{equation} + \sqrt{\Delta} = r - \frac{m^2 - a^2}{4r}. +\end{equation} + +A set of parameters which initialise an evolution to use the Kerr +intial data with mass $M=1$ and angular momentum $a=0.3$ are: +\begin{verbatim} + ActiveThorns = "... Einstein IDAnalyticBH ..." + + einstein::use_conformal = "yes" + + einstein::initial_data = "kerr" + einstein::initial_lapse = "kerr" + einstein::initial_shift = "kerr" + + idanalyticbh::mass = 1.0 + idanalyticbh::a_kerr = 0.3 +\end{verbatim} + +\section{Misner} + +The earliest suggestion for initial data that might be said to +corresponding to multiple black holes was given by Misner in 1960 +\cite{misner:1960}. He provided a prescription for writing a metric +connecting a pair of massive bodies, instaneously at rest, whose +throats are connected by a wormhole. Using the method of images, this +solution was generalised to describe any number of black holes whose +throats connect two identical asymptotically flat spacetimes +\cite{misner:1963}. +\begin{figure} + \centering + \epsfig{file=misner.eps, height=40mm} + \caption{The topology of the Misner spacetime is that of a pair of + asymptotically flat sheets connected by a number of Einstein-Rosen + bridges. By construction, an exact isometry exists between the upper + and lower sheet across the throats. The parameter $\mu_0$ is a + measure of the distance of a loop in the surface, passing through + one throat and out the other.} +\end{figure} + +Two implementations of the Misner data are available. The first of +these, ``\texttt{misner\_bh}'', is due to Joan Masso, Ed Seidel and +Karen Camarda, and implements the original two-throat solution. The +more general solution was implemented by Steve Brandt and Carsten +Gundlach, and is available as ``\texttt{multiple\_misner\_bh}''. + +\subsection{Two-throat Misner data} + +The \texttt{misner\_bh} initial data generates a metric of the form +\begin{equation} + ds^2 = -dt^2 + \psi^4 (dx^2 + dy^2 + dz^2), +\end{equation} +where the conformal factor $\psi$ is given by +\begin{equation} + \psi = \sum^N_{n=-N} + \frac{1}{\sinh(\mu_0 n)} + \frac{1}{\sqrt{x^2 + y^2 + (z + \coth(\mu_0 n))^2}}. +\end{equation} + +The parameter $\mu_0$ is a measure of the ration of mass to separation +of the throats, and is set using the parameter +\texttt{idanalyticbh::mu}. For values less than $\mu\simeq 1.8$, the +throats will have a single event horizon. + +The summation limit $N$ can be set using the parameter +\texttt{idanalyticbh::nmax}. Ideally, it should tend to infinity, but +in practice the default value of $N=30$ works well enough for the +applications that have been tested. The \texttt{misner\_nbh} parameter +is only used for the \texttt{multiple\_misner\_bh} multi-throat data, +and will be ignored for the \texttt{misner\_bh} initial data, which +assumes two throats. + +For the given metric, the ADM mass of the system is determined via +\begin{equation} + m = 4 \sum^N_{n=1} \frac{1}{\sinh(\mu_0 n)}. +\end{equation} +This quantity is determined automatically and written to standard +output. + +If the conformal form of the metric is used (via the +\texttt{einstein::use\_conformal} parameter), then derivatives of the +conformal factor are computed analytically from the derivatives of the +above expression for $\psi$. + +To make use of the two black hole initial data, a variation of the +following set of parameters can be used: +\begin{verbatim} + ActiveThorns = "... Einstein IDAnalyticBH ..." + + einstein::use_conformal = "yes" + + einstein::initial_data = "misner_bh" + idanalyticbh::mu = 2.2 +\end{verbatim} + + +\section{Multiple-throat Misner data} + +The generalisation of the above form of data to multiple black holes +is available as the \texttt{multiple\_misner\_bh} initial data set. The +conformal factor is determined by recursively applying a Misner +isometry condition to each of the black holes relative to the others. + +The black holes are arranged at equal-spaced angles on a circle around +the origin in the $xy$-plane. The radius of the circle is $\coth\mu_0$, +where $\mu_0$ is given by the \texttt{idanalyticbh::mu} parameter, and +the first black hole lies on the $x$-axis (as in Figure +\ref{fig:multi_misner}). +\begin{figure} + \centering + \label{fig:multi_misner} + \epsfig{file=multi_misner.eps, height=40mm} + \caption{Configuration for three Misner throats using the + \texttt{multiple\_misner\_bh} initial data.} +\end{figure} + +The number of throats is given by the parameter +\texttt{idanalyticbh::misner\_nbh}, which defaults to 1 and has a +hard-coded upper limit of 10. The number of terms used in the Misner +expansion is controlled by the parameter +\texttt{idanalyticbh::nmax}, which has a default value of 30. + +For this version of the Misner data, derivatives of the conformal +factor $\psi$ are determined numerically by finite differencing, +using values of $\psi$ calculated at small distances from the point at +which the derivative is to be evaluated. The size of the numerical +stencil is hardcoded at $dx=10^-6$. + +As an example, a parameter file implementing 3 Misner black holes on a +circle of radius $\cosh 4$ would use the following parameters: +\begin{verbatim} + ActiveThorns = "... Einstein IDAnalyticBH ..." + + einstein::use_conformal = "yes" + + einstein::initial_data = "multiple_misner_bh" + + idanalyticbh::misner_nbh = 3 + idanalyticbh::mu = 4 +\end{verbatim} + + +\section{Brill-Lindquist} + +The Brill-Lindquist initial data is an alternate form of multi-throat +data which differs from the Misner data mainly in its choice of +spacetime topology. Whereas the Misner data presumes that the throats +connect a pair of asymptotically flat spacetimes which are identical +to each other, the Brill-Lindquist data connects each throat to a +separate asymptotically flat region \cite{brill-lindquist:1963}. +\begin{figure} + \centering + \epsfig{file=brill_lindquist.eps, height=40mm} + \caption{Two Brill-Lindquist throats connecting separate + asymptotically flat regions.} +\end{figure} +The form of the conformal factor is: +\begin{equation} + \psi = 1 + \sum_{i=1}^N \frac{m_i}{2r_i}, +\end{equation} +where the $m_i$ and $r_i$ are the masses and positions of the $i$ +particles. + +The parameter specifying the number of black holes is +\texttt{idanalyticbh::bl\_nbh}. A maximum of four black holes can be +specified. The mass and $(x,y,z)$ position of the first black hole is +given by \texttt{bl\_M\_1}, \texttt{bl\_x0\_1}, \texttt{bl\_y0\_1}, +\texttt{bl\_z0\_1}, with corresponding parameters for the second to +fourth black holes. Note that the default values for each of the +position coordinates are $0.0$, so that only the coordinates off +of the axes must be specified. + +If the conformal metric is used, then derivatives of the conformal +factor are calculated from the analytic derivatives of the above +expression for the conformal factor. + +To initialise a run with a pair of Brill-Lindquist black holes with +masses $1$ and $2$ and located at $\pm 1$ on the $y$-axis, a set of +parameters such as the following could be used: +\begin{verbatim} + ActiveThorns = "... Einstein IDAnalyticBH ..." + + einstein::use_conformal = "yes" + + einstein::initial_data = "bl_bh" + + idanalyticbh::bl_nbh = 2 + + idanalyticbh::bl_M_1 = 1.0 + idanalyticbh::bl_y0_1 = 1.0 + + idanalyticbh::bl_M_2 = 2.0 + idanalyticbh::bl_y0_2 = -1.0 +\end{verbatim} + +\begin{thebibliography}{9} + \bibitem{mtw-isotropic} + See, for instance, p. 840 of: + Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973) + \emph{Gravitation}, W. H. Freeman, San Francisco. + \bibitem{misner:1960} + Misner, Charles W. (1960) + \emph{Wormhole Initial Conditions}, + Phys. Rev., \textbf{118}, 1110--1111. + \bibitem{misner:1963} + Misner, Charles W. (1963) + \emph{The Method of Images in Geometrostatics}, + Ann. Phys., \textbf{24}, 102--117. + \bibitem{brill-lindquist:1963} + Brill, Dieter R., and Lindquist, Richard W. (1963) + \emph{Interaction Energy in Geometrostatics} + Phys. Rev., \textbf{131}, 471--476. +\end{thebibliography} \end{document} -- cgit v1.2.3