[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 312)

[jfaber at einsteintoolkit.org]

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!

File Changes:

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