User: jfaber Date: 2012/03/13 02:10 PM

Modified: / ET.tex

Log: Modified stress-energy source terms in initial data, vacuum evolution, anc constraint treatment to be in uniform notation. PLEASE DOUBLECHECK THE REVISED BSSN EQUATIONS BEFORE SUBMITTING!

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Directory: / ============

File [modified]: ET.tex Delta lines: +17 -15 =================================================================== --- ET.tex 2012-03-13 18:18:06 UTC (rev 311) +++ ET.tex 2012-03-13 19:10:07 UTC (rev 312) @@ -1214,20 +1214,20 @@ \codename{Meudon_Bin_NS} handles binary NS data described in~\cite{Gourgoulhon:2000nn}, which represent solutions of the equations \begin{eqnarray*} -&&\nabla^2\nu_{(m)} = 4\pi\psi^4(\hat{E}_{(m)}+\hat{S}_{(m)})+\psi^4 K_{ij}K^{ij}_{(m)}-\nabla_i\nu_{(m)}\nabla^i\beta\ -&&\nabla^2\beta_{(m)}=4\pi\psi^4\hat{S}_{(m)}+\frac{3}{4}\psi^4 K_{ij}K^{ij}_{(m)}-\frac{1}{2}(\nabla_i\nu_{(m)}\nabla^i\nu+\nabla_i\beta_{(m)}\nabla^i\beta)\ -&&\nabla^2\beta^i_{(m)}+\frac{1}{3}\nabla^i\nabla_j \beta^j_{(m)} = -16\pi\alpha\psi^4(\hat{E}_{(m)}+P_{(m)})v^i_{(m)}+2\alpha\psi^4K^{ij}_{(m)}\nabla_j(3\beta-4\nu) +&&\nabla^2\nu_{(m)} = 4\pi\psi^4(E_{(m)}+S_{(m)})+\psi^4 K_{ij}K^{ij}_{(m)}-\nabla_i\nu_{(m)}\nabla^i\beta\ +&&\nabla^2\beta_{(m)}=4\pi\psi^4S_{(m)}+\frac{3}{4}\psi^4 K_{ij}K^{ij}_{(m)}-\frac{1}{2}(\nabla_i\nu_{(m)}\nabla^i\nu+\nabla_i\beta_{(m)}\nabla^i\beta)\ +&&\nabla^2\beta^i_{(m)}+\frac{1}{3}\nabla^i\nabla_j \beta^j_{(m)} = -16\pi\alpha\psi^4(E_{(m)}+P_{(m)})v^i_{(m)}+2\alpha\psi^4K^{ij}_{(m)}\nabla_j(3\beta-4\nu) \end{eqnarray*} where $\nu$ and $\beta$ are defined as \begin{equation} \nu\equiv \log\alpha;~~\beta\equiv\ln\alpha\psi^2. \end{equation} These equations are merely convenient reparameterizations of the ones used to generate binary BH data, with the matter source terms included. The extrinsic curvature is computed using~\ref{eq:lorextrcurv}, with both $K^{ij}$ and $\beta^i$ replaced by the split versions. -The matter sources terms are, respectively, -\begin{eqnarray} -\hat{E}&=&\alpha^2(u^0)^2 \rho h-P\ -\hat{S}&=&3P+(\hat{E}+P)\frac{\alpha^2(u^0)^2 - 1}{\alpha^2(u^0)^2} -\end{eqnarray} +The matter sources terms $E$ and $S$, representing projections of the stress-energy tensor, are defined in~\ref{eq:e_tmunu} and \ref{eq:s_tmunu} below. +%\begin{eqnarray} +%e&=&n_\alpha n_\beta T^{\alpha\beta}=\alpha^2(u^0)^2 \rho h-P\ +%S&=&3P+(\hat{E}+P)\frac{\alpha^2(u^0)^2 - 1}{\alpha^2(u^0)^2} +%\end{eqnarray} Lorene allows for two different NS spin states, either irrotational or synchronized. In the synchronized case, the velocity may be specified as a function of position once the orbital velocity is determined, while the irrotational case yields a rather complicated differential equation for the velocity potential which may then be used to determine the corresponding 3-velocity (see Equation~38 of \cite{Gourgoulhon:2000nn}).

\codename{Meudon_Mag_NS} @@ -1277,7 +1277,7 @@ \ \partial_0 K & = & -e^{-4\phi} \left[ \tilde{D}^i \tilde{D}_i \alpha + 2 \partial_i \phi \cdot \tilde{D}^i \alpha \right] + \alpha - \left( \tilde{A}^{ij} \tilde{A}_{ij} + \frac{1}{3} K^2 \right) - \alpha S + \left( \tilde{A}^{ij} \tilde{A}_{ij} + \frac{1}{3} K^2 \right) + \frac{\alpha}{2}(E+ S) \ \partial_0 \beta^i & = & \alpha^2 G(\alpha,\phi,x^\mu) B^i \ @@ -1298,7 +1298,7 @@ & & {} + \alpha K\tilde{A}_{ij} - 2\alpha\tilde{A}_{ik}\tilde{A}^k_{; j} + 2\tilde{A}_{k(i}\partial_{j)}\beta^k - \frac{2}{3}\tilde{A}_{ij}\partial_k\beta^k - - \alpha e^{-4\phi} \hat{S}_{ij} + - \alpha e^{-4\phi} S_{ij}^{TF} \ \partial_0\tilde{\Gamma}^i & = & \tilde{\gamma}^{kl}\partial_k\partial_l\beta^i @@ -1310,18 +1310,20 @@ + 2\alpha\left[ (m-1)\partial_k\tilde{A}^{ki} - \frac{2m}{3}\tilde{D}^i K \right. \nonumber \ & & {} + m(\tilde{\Gamma}^i_{; kl}\tilde{A}^{kl} + - 6\tilde{A}^{ij}\partial_j\phi) \Bigg] - S^i, + 6\tilde{A}^{ij}\partial_j\phi) \Bigg] -2\alpha \tilde{\gamma}^{ij} S_j \end{eqnarray} \end{widetext} with the momentum constraint damping constant set to $m=1$. The stress energy tensor $T_{\mu\nu}$ is incorporated via the projections \begin{eqnarray} - E & \equiv & \frac{1}{\alpha^2} \left( T_{00} - 2 \beta^i T_{0i} + - \beta^i \beta^j T^{ij} \right) + E & \equiv & n_\alpha n_\beta T^{\alpha\beta} = \frac{1}{\alpha^2} \left( T_{00} - 2 \beta^i T_{0i} + + \beta^i \beta^j T^{ij} \right)\label{eq:e_tmunu} + \ + S_{ij} & \equiv & \gamma_{i\alpha} \gamma_{j\beta} T^{\alpha\beta} \ - S & \equiv & g^{ij} T_{ij} + S&\equiv&S^i_i = \gamma^{ij} S_{ij} \label{eq:s_tmunu} \ - S_i & \equiv & - \frac{1}{\alpha} \left( T_{0i} - \beta^j T_{ij} \right) . + S_i & \equiv &-\gamma_{i\alpha} n_\beta T^{\alpha\beta}= - \frac{1}{\alpha} \left( T_{0i} - \beta^j T_{ij} \right) . \end{eqnarray} We have introduced the notation $\partial_0 = \partial_t - \beta^j\partial_j$. All quantities with a tilde involve