Hi Andre et al., just in case you have a wheel mouse, could you please check why the cursor "hangs" in the math formula? Load the attached file, move the cursor to the beginning of the document and scroll down with the wheel of your mouse. Another interesting fact: You cannot exit LyX if you are in mathed. Michael -- ====================================================================== Michael Schmitt phone: +49 451 500 3725 Institute for Telematics secretary: +49 451 500 3721 Medical University of Luebeck fax: +49 451 500 3722 Ratzeburger Allee 160 eMail: [EMAIL PROTECTED] D-23538 Luebeck, Germany WWW: http://www.itm.mu-luebeck.de ======================================================================
#LyX 1.2 created this file. For more info see http://www.lyx.org/ \lyxformat 220 \textclass article \begin_preamble \usepackage{amsfonts} \end_preamble \language english \inputencoding latin1 \fontscheme default \graphics default \paperfontsize default \spacing single \papersize Default \paperpackage a4 \use_geometry 0 \use_amsmath 1 \use_natbib 0 \use_numerical_citations 0 \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \quotes_times 2 \papercolumns 1 \papersides 1 \paperpagestyle default \layout Title An Efficient Method for Fully Relativistic Simulations of Coalescing Binary Neutron Stars \layout Author Walter Landry \newline Physics Dept., University of Utah, SLC, UT 84112 \layout Date \SpecialChar ~ \layout Standard These are all the equations from Walter's PhD thesis which he prepared using LyX. \layout Standard \begin_inset Formula \begin{equation} ds^{2} =g_{\mu \nu } dx^{\mu } dx^{\nu } =-\alpha ^{2} dt^{2} +\gamma _{ij} (dx^{i} +\beta ^{i} dt)(dx^{j} +\beta ^{j} dt).\label{first}\end{equation} \end_inset \begin_inset Formula \begin{equation} \frac{\partial \gamma _{ij} }{\partial t}=-2\alpha K_{ij} +\nabla _{i} \beta _{j} +\nabla _{j} \beta _{i} ,\label{g dot}\end{equation} \end_inset \layout Standard \begin_inset Formula \begin{eqnarray} \frac{\partial K_{ij} }{\partial t}=-\nabla _{i} \nabla _{j} \alpha +K_{lj} \nabla _{i} \beta ^{l} +K_{il} \nabla _{j} \beta ^{l} +\beta ^{l} \nabla _{l} K_{ij} & & \nonumber \\ +\alpha \left[ R_{ij} -2K_{il} K^{l}_{j} +KK_{ij} -S_{ij} -\frac{1}{2}\gamma _{ij} +(\rho -S)\right] . & & \label{K dot}\end{eqnarray} \end_inset \begin_inset Formula \begin{eqnarray} R_{ij} =\frac{1}{2}\gamma ^{kl} & \left[ \gamma _{kj,il} +\gamma _{il,kj} -\gamma _{kl,ij} -\gamma _{ij,kl} \right. & \nonumber \\ & \left. +2\left( \Gamma ^{m}_{il} \Gamma _{mkj} -\Gamma ^{m}_{ij} \Gamma _{mkl} \right) \right] . & \label{Ricci}\end{eqnarray} \end_inset \begin_inset Formula \[ n_{\mu } =(-\alpha ,0,0,0).\] \end_inset \begin_inset Formula \begin{eqnarray*} \rho & = & 8\pi n_{\mu } n_{\nu } T^{\mu \nu } =8\pi \alpha ^{2} T^{tt} , \\ J^{i} & = & -8\pi n_{\mu } \gamma ^{i}_{j} T^{\mu j} , \\ S_{ij} & = & 8\pi \gamma _{ik} \gamma _{jl} T^{kl} ,\end{eqnarray*} \end_inset \begin_inset Formula \[ S=\gamma ^{ij} S_{ij} .\] \end_inset \layout Standard \begin_inset Formula \begin{equation} R+K^{2} -K_{ij} K^{ij} =2\rho ,\label{Energy}\end{equation} \end_inset \begin_inset Formula \begin{equation} \nabla _{j} \left( K^{ij} -\gamma ^{ij} K\right) =J^{i} .\label{Momentum}\end{equation} \end_inset \layout Standard \begin_inset Formula \[ K^{ij} =\psi ^{-10} \left( \widetilde{A}^{ij} +\left( lX\right) ^{ij} \right) +\frac{1}{3}\psi ^{-4} \widetilde{\gamma }^{ik} \mathrm{T}\mathrm{r}K,\] \end_inset \begin_inset Formula \[ \left( lX\right) ^{ij} =\widetilde{\nabla }^{i} X^{j} +\widetilde{\nabla }^{j} X^{i} -\frac{2}{3}\widetilde{\gamma }^{ij} \widetilde{\nabla }_{k} X^{k} ,\] \end_inset \begin_inset Formula \begin{eqnarray} -8\widetilde{\nabla }^{2} \psi & = & -\widetilde{R}\psi -\frac{2}{3}\left( trK\right) ^{2} \psi ^{5} \nonumber \\ & & +\left( \widetilde{A}^{ij} +\left( lX\right) ^{ij} \right) ^{2} \psi ^{-7} +2\rho \psi ^{5} ,\label{phi constraint} \\ \widetilde{\nabla }^{2} X^{i} & + & \frac{1}{3}\widetilde{\nabla }^{i} \widetilde{\nabla }_{j} X^{j} +\widetilde{R}^{i}_{j} X^{j} \nonumber \\ & = & J^{i} \psi ^{10} -\widetilde{\nabla }_{j} \widetilde{A}^{ij} +\frac{2}{3}\psi ^{6} \widetilde{\nabla }^{i} trK,\label{X constraint}\end{eqnarray} \end_inset \begin_inset Formula \begin{eqnarray} -8\widetilde{\nabla }^{2} (\psi _{0} +\delta \psi )=-\widetilde{R}(\psi _{0} +\delta \psi )-\frac{2}{3}\left( trK\right) ^{2} \psi ^{4}_{0} (\psi _{0} +5\delta \psi ) & & \nonumber \\ +\left( \widetilde{A}^{ij} +\left( lX_{0} \right) ^{ij} \right) ^{2} \psi ^{-8}_{0} +(\psi _{0} -7\delta \psi )+2\rho \psi ^{4}_{0} (\psi _{0} +5\delta \psi ), & & +\label{phi linear}\end{eqnarray} \end_inset \begin_inset Formula \begin{eqnarray} \nabla ^{2} (X^{i}_{0} +\delta X^{i} )+\frac{1}{3}\widetilde{\nabla }^{i} \widetilde{\nabla }_{j} (X^{j}_{0} +\delta X)+\widetilde{R}^{i}_{j} (X^{j}_{0} +\delta X^{i} ) & & \nonumber \\ =J^{i} \psi ^{10}_{0} -\widetilde{\nabla }_{j} \widetilde{A}^{ij} +\frac{2}{3}\psi ^{6}_{0} \widetilde{\nabla }^{i} trK. & & \label{X linear}\end{eqnarray} \end_inset \layout Standard \begin_inset Formula \begin{equation} x^{\mu } \rightarrow x^{\mu } +\xi ^{\mu } .\label{gauge perturbation}\end{equation} \end_inset \layout Standard \begin_inset Formula \begin{equation} \Box x^{\mu } =0.\label{gauge}\end{equation} \end_inset \begin_inset Formula \begin{equation} \Box \xi ^{\mu } =0.\label{perturbation}\end{equation} \end_inset \begin_inset Formula \begin{equation} \frac{\partial g_{tt} }{\partial t}=\left( \gamma ^{ij} \alpha ^{2} -\beta ^{i} \beta ^{j} \right) \left( -\gamma _{ij,t} +2\beta _{i,j} \right) +2\beta ^{i} g_{tt,i} \label{g_{t}t dot}\end{equation} \end_inset \begin_inset Formula \begin{eqnarray} \frac{\partial \beta _{k} }{\partial t} & = & 2\beta ^{i} \left( \gamma _{ki,t} -\beta _{i,k} +\beta _{k,i} \right) \nonumber \\ & & -\left( \gamma ^{ij} \alpha ^{2} -\beta ^{i} \beta ^{j} \right) \left( \gamma _{ij,k} -2\gamma _{kj,i} \right) +g_{tt,k} .\label{Shift dot}\end{eqnarray} \end_inset \layout Standard \begin_inset Formula \[ \frac{\partial (\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e})}{\partial t}+\partial _{i} (\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x})^{i} =(\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}),\] \end_inset \begin_inset Formula \[ \frac{\partial Q}{\partial t}=F(Q)\] \end_inset \begin_inset Formula \[ Q_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}} =Q_{t} +\frac{\Delta t}{2}F(Q_{t} ),\] \end_inset \begin_inset Formula \[ Q_{t+\Delta t} =Q_{t} +\Delta t\, F(Q_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{e}} ).\] \end_inset \layout Standard \begin_inset Formula \[ Q(t,r)=\frac{G(\alpha t-(\det \gamma )^{\frac{1}{6}} r)}{r},\] \end_inset \begin_inset Formula \begin{equation} \frac{\partial \gamma _{ij} }{\partial t}=-2\alpha K_{ij} +\nabla _{i} \beta _{j} +\nabla _{j} \beta _{i} -q\left( \Delta x\right) ^{3} \nabla ^{4} \gamma _{ij} ,\label{g diffuse}\end{equation} \end_inset \begin_inset Formula \[ \gamma _{\mathrm{n}\mathrm{e}\mathrm{w}} =\gamma _{\mathrm{o}\mathrm{l}\mathrm{d}} +\Delta t(RHS),\] \end_inset \begin_inset Formula \[ \gamma _{\mathrm{n}\mathrm{e}\mathrm{w}} =\gamma _{\mathrm{o}\mathrm{l}\mathrm{d}} +\Delta t\lbrace RHS-q\Delta x^{3} \nabla ^{4} [\gamma _{\mathrm{o}\mathrm{l}\mathrm{d}} +\Delta t\left( RHS\right) ]\rbrace .\] \end_inset \begin_inset Formula \begin{eqnarray} h_{+} & = & \frac{1}{2}\left( \gamma _{xx} -\gamma _{yy} \right) ,\label{h_{p}lus} \\ h_{\times } & = & \gamma _{xy} .\label{h_{c}ross}\end{eqnarray} \end_inset \begin_inset Formula \[ T_{\mu \nu } =\frac{1}{2}\left\langle h_{ij,\mu } h_{ij,\nu } \right\rangle ,\] \end_inset \begin_inset Formula \[ L\sim 4\pi \cdot \frac{1}{2}\frac{\left( h^{2}_{+} +h^{2}_{\times } \right) \left( 4R_{*} \right) ^{2} }{\left( 10R_{*} \right) ^{2} }=2\cdot 10^{-4} .\] \end_inset \layout Standard \begin_inset Formula \[ L=\frac{32}{5}\frac{\mu ^{3} M^{2} }{a^{5} }\sim 3\cdot 10^{-7} ,\] \end_inset \the_end
