User: jfaber
Date: 2012/03/09 10:20 AM
Modified:
/
ET.tex
Log:
Updated initial data section for evenness in presentation
Longer writeup of Lorene code
Added table of key variables to aid in uniform presentation
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +157 -70
===================================================================
--- ET.tex 2012-03-08 18:26:27 UTC (rev 272)
+++ ET.tex 2012-03-09 16:20:05 UTC (rev 273)
@@ -650,6 +650,42 @@
as well as the wave extraction thorn \codename{WeylScal4}, are both generated
using \codename{Kranc}, and hence support all the above features.
+\begin{table}
+\label{table:vars}
+{\centering
+\begin{tabular}{llll}
+Symbol & Quantity & Equation & ET thorn::group \\
+\hline
+$G_{\mu\nu}$ & Einstein Tensor & \protect\ref{eq:einstein} & N/A\\
+$T_{\mu\nu}$ & Stress-Energy Tensor & \protect\ref{eq:Tmunu} & TmunuBase::stress\_energy\_tensor\\
+$g_{\mu\nu}$ & Spacetime 4-metric & \protect\ref{eq:adm} & N/A\\
+$F_{\mu\nu},~(^*F^{\mu\nu})$ & (Dual) Faraday tensor & \protect\ref{eq:Bi} & N/A\\
+$u^\mu$ & 4-velocity & N/A & N/A\\
+$\gamma_{ij}$ & Spatial 3-metric & \protect\ref{eq:adm} & ADMBase::metric\\
+$\alpha$ & Lapse function & \protect\ref{eq:adm} & ADMBase::lapse\\
+$\beta^i$ & Shift vector & \protect\ref{eq:adm} & ADMBase::shift\\
+$n^\mu$,$ n_\mu$ & Unit normal & N/A & N/A\\
+$K_{ij}$ & Extrinsic curvature & \protect\ref{eq:extrcurv} & ADMBase::curv\\
+$\rho$ & Rest mass density & \protect\ref{eq:Tmunu},\protect\ref{eq:enthalpy} & HydroBase::rho\\
+$P$ & Fluid pressure & \protect\ref{eq:enthalpy} & HydroBase::press\\
+$\epsilon$ & Internal energy density & \protect\ref{eq:enthalpy} & Hydrobase::eps\\
+$h$ & Specific enthalpy & \protect\ref{eq:enthalpy} & N/A\\
+$v^i$ & 3-velocity & \protect\ref{eq:3vel} & HydroBase::vel\\
+$B^i$ & Magnetic field vector & \protect\ref{eq:Bi} & HydroBase::Bvec\\
+$\psi$, ($\phi$) & (Logarithmic) conformal factor & \protect\ref{eq:confpsi},\protect\ref{eq:phi} & (ML\_BSSN/ML\_log\_confac) \\
+$\tilde{\gamma}_{ij}$ & Conformal 3-metric & \protect\ref{eq:tildegamma} & ML\_BSSN/ML\_metric \\
+$K$ & Trace of extrinsic curvature & \protect\ref{eq:trK} & ML\_BSSN/ML\_trace\_curv \\
+$\tilde{A}_{ij}$ & Conformal traceless extrinsic curvature & \protect\ref{eq:Atij} & ML\_BSSN/ML\_curv \\
+$\tilde{\Gamma}^i$ & Conformal connection & \protect\ref{eq:Gammai} & ML\_BSSN/ML\_Gamma \\
+$D$ & Conservative density & \protect\ref{eq:p2c1} & GRHydro::dens\\
+$S^i$ & Conservative momentum & \protect\ref{eq:p2c2} & GRHydro::scon\\
+$\tau$ & Conservative energy density & \protect\ref{eq:p2c3} & GRHydro::tau\\
+$W$ & Lorentz factor & \protect\ref{eq:Lorentz} & GRHydro::w\_lorentz\\
+\end{tabular}\\}
+\caption{Summary of key variables used throughout this paper, along with the equations that define them and the ET thorn and group name used to store the quantity, where applicable.}
+\end{table}
+
+
\section{Components}%
\label{sec:components}
The Einstein Toolkit uses the modular {\tt Cactus} framework as its underlying infrastructure.
@@ -733,7 +769,7 @@
Greek letters to index 4-dimensional spacetime quantities, with the index running from 0 to 3.
The remaining dynamical component of the spacetime is contained in the definition of the extrinsic curvature $K_{ij}$, which is defined in terms of the time derivative of the metric after incorporating a Lie derivative with respect to the shift vector:
\begin{equation}
-K_{ij}\equiv -\frac{1}{2\alpha}(\partial_t-\mathcal{L}_\beta)\gamma_{ij}.
+K_{ij}\equiv -\frac{1}{2\alpha}(\partial_t-\mathcal{L}_\beta)\gamma_{ij}.\label{eq:extrcurv}
\end{equation}
The three-metric, extrinsic curvature, lapse function, and shift vector are all
declared as variables in the \codename{ADMBase} module,
@@ -809,7 +845,7 @@
\item \verb|eps|: internal energy density $\epsilon$
\item \verb|vel[3]|: contravariant fluid three velocity $v^i$ defined as
\begin{equation}
- v^i = \frac{u^i}{\alpha u^0} + \frac{\beta^i}{\alpha}
+ v^i = \frac{u^i}{\alpha u^0} + \frac{\beta^i}{\alpha}\label{eq:3vel}
\end{equation}
in terms of the four-velocity $u^\mu$, lapse, and shift vector
.
@@ -818,7 +854,7 @@
\item \verb|entropy|: specific entropy per particle $s$
\item \verb|Bvec[3]|: contravariant magnetic field vector defined as
\begin{equation}
- B^i = \frac{1}{\sqrt{4\pi}} n_{\nu} F^{*\nu i}
+ B^i = \frac{1}{\sqrt{4\pi}} n_{\nu} F^{*\nu i}\label{eq:Bi}
\end{equation}
in terms of the dual
$F^{*\mu\nu} = \frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$
@@ -850,8 +886,11 @@
T^{\mu\nu} = \rho h u^\mu u^\nu + g^{\mu\nu} P\,\,,
\end{equation}
where $\rho$, $u^\mu$, and
-$g^{\mu\nu}$ are defined above, and $h = 1 + \epsilon + P/\rho$ is
-the relativistic specific enthalpy.
+$g^{\mu\nu}$ are defined above, and
+\begin{equation}
+h = 1 + \epsilon + P/\rho\label{eq:enthalpy}
+\end{equation}
+ is the relativistic specific enthalpy.
The thorn \codename{TmunuBase} provides grid functions for the stress-energy
tensor $T_{\mu\nu}$ as well as schedule groups to manage when $T_{\mu\nu}$ is
@@ -905,7 +944,7 @@
The conversion between the physical metric and extrinsic
curvature and conformal versions of these is handled solely within evolution modules, which are responsible
for calculating the conformally related three metric $\tilde{\gamma}_{ij}$,
-the conformal factor $\psi$, the conformal traceless extrinsic curvature
+the conformal factor $\psi$ or its logarithm $\phi$, the conformal traceless extrinsic curvature
$\tilde{A}_{ij}$ and the trace of the extrinsic curvature $K$, as well as
initializing the BSSN variable $\tilde{\Gamma}^i$ should that be the evolution
formalism chosen (see section~\ref{sec:Kevol} for definitions of these). Optionally, many initial data modules also supply values
@@ -962,30 +1001,43 @@
usually solved on-line at the beginning of an inspiral simulation. This
approach allows all information required to set up the run to be kept in the
single \codename{Cactus} parameter file.
-\codename{TwoPunctures} assumes the extrinsic curvature for each BH corresponds to
-the Bowen-York form~\cite{Bowen:1980yu}:
+\codename{TwoPunctures} assumes the conformal extrinsic curvature $\tilde{K}_{ij}$ corresponds to
+the Bowen-York form or each BH ~\cite{Bowen:1980yu}:
\begin{eqnarray}
-K_{(m)}^{ij}&=&\frac{3}{2r^2}(p_{(m)}^i\hat{N}^j+p_{(m)}^j\hat{N}^i-(\gamma^{ij}-\hat{N}^i\hat{N}^j)p_{(m)}^k\hat{N}_k))\nonumber\\
+\tilde{K}_{(m)}^{ij}&=&\frac{3}{2r^2}(p_{(m)}^i\hat{N}^j+p_{(m)}^j\hat{N}^i-(\tilde{\gamma}^{ij}-\hat{N}^i\hat{N}^j)p_{(m)}^k\hat{N}_k))\nonumber\\
&&+\frac{3}{r^3}(\varepsilon^{ikl}S^{(m)}_k\hat{N}_l\hat{N}^j+\varepsilon^{jkl}S^{(m)}_k\hat{N}_l\hat{N}^i),
\end{eqnarray}
where the sub/superscript $(m)$ refers to the contribution from BH $m=1,2$; the
-3-momentum is $p^i$; the BH spin angular momentum is $S_i$; the conformal 3-metric
-$\gamma^{ij}$ is assumed to be flat, i.e.\ $\gamma_{ij}=\eta_{ij}$, and $\hat{N}^i=x^i/r$
-is the Cartesian normal vector relative to the position of each BH in turn. This
-solution automatically satisfies the momentum constraint, and the Hamiltonian constraint
-may be written as an elliptic equation for the conformal factor, defined by the condition $g_{ij}=\psi^4\gamma_{ij}$ or equivalently $\psi\equiv (\det|g_{ij}|)^{1/12}$:
+3-momentum is $p^i$; the BH spin angular momentum is $S_i$; and $\hat{N}^i=x^i/r$
+is the Cartesian normal vector relative to the position of each BH in turn.
+The conformal extrinsic curvature
+is raised and lowered using the conformal metric $\tilde{\gamma}_{ij}$ (see~\ref{eq:tildegamma}) and is related to the physical extrinsic curvature by the conditions
\begin{equation}
-\Delta \psi+\frac{1}{8}K^{ij}K_{ij}\psi^{-7}=0
+\tilde{K}_{ij} = \psi^2K_{ij};~~\tilde{K}^{ij} = \psi^{10}K_{ij},
\end{equation}
+where the conformal $\psi$ is defined as
+\begin{equation}
+\psi\equiv |\det\gamma_{ij}|^{1/12} = e^\phi~~~~({\rm so~that}~\gamma_{ij} = \psi^4\tilde{\gamma}_{ij},~\det\tilde{\gamma}_{ij}=1)
+\label{eq:confpsi}
+\end{equation}
+and $\phi$ (the logarithmic conformal factor) is defined below (see~\ref{eq:phi}).
+In \codename{TwoPunctures},
+the conformal 3-metric
+$\tilde{\gamma}^{ij}$ is assumed to be flat, i.e.\ $\tilde{\gamma}_{ij}=\eta_{ij}$ The Bowen-York
+solution automatically satisfies the momentum constraint (see~\ref{eqn:analysis_momentum_constraint}, and the Hamiltonian constraint
+(\ref{eqn:analysis_hamiltonian_constraint}) may be written as an elliptic equation for the conformal factor:
+\begin{equation}
+\Delta \psi+\frac{1}{8}\tilde{K}^{ij}\tilde{K}_{ij}\psi^{-7}=0
+\end{equation}
Decomposing the conformal factor into a singular analytical term and a regular term $u$,
such that
\begin{equation}
\psi = \frac{m_1}{2r_1}+\frac{m_2}{2r_2}+u\equiv \frac{1}{\Psi}+u
\end{equation}
-where $m_1,~m_2$ and $r_1,~r_2$ are the mass of and distance to each BH, respectively, and $\Psi$ is defined by the equation itself, the Hamiltonian
+where $m_1,~m_2$ and $r_1,~r_2$ are the mass of and distance to each BH, respectively, and $\Psi\equiv\left( \frac{m_1}{2r_1}+\frac{m_2}{2r_2}\right)^{-1}$, the Hamiltonian
constraint may be written as
\begin{equation}
-\Delta u +\left[\frac{1}{8}\Psi^7K^{ij}K_{ij}\right](1+\Psi u)^{-7}\label{eq:twopunc_u}=0
+\Delta u +\left[\frac{1}{8}\Psi^7\tilde{K}^{ij}\tilde{K}_{ij}\right](1+\Psi u)^{-7}\label{eq:twopunc_u}=0
\end{equation}
subject to the boundary condition $u\rightarrow 1$ as $r\rightarrow\infty$. In Cartesian
coordinates, the function $u$ is infinitely differentiable everywhere except the
@@ -1014,6 +1066,51 @@
\label{fig:TP_BHNS_coordinates}
\end{figure}
+
+\subsubsection{TOVSolver}
+\label{sec:TOVSolver}
+The \codename{TOVSolver} routine in the ET solves the standard TOV equations
+\cite{Tolman:1939jz,Oppenheimer:1939ne} for the pressure, enclosed gravitational mass $M_e$, and gravitational potential $\Phi=\log\alpha$ in the interior of a spherically symmetric star in hydrostatic equilibrium, expressed using the Schwarzschild (areal)
+radius $\hat{r}$:
+\begin{eqnarray}
+ \label{eq:TOViso}
+ \frac{d P}{d \hat{r}} & = & -(e + P) \frac{M_e + 4\pi \hat{r}^3 P}{\hat{r}(\hat{r} - 2M_e)}\nonumber\\
+%
+ \frac{d M_e}{d \hat{r}} & = & 4 \pi \hat{r}^2 e\nonumber\\
+%
+ \frac{d \Phi}{d \hat{r}} & = & \frac{M_e + 4\pi \hat{r}^3 P}{\hat{r}(\hat{r} -
+ 2M_e)}.
+\end{eqnarray}
+where $e\equiv \rho(1+\epsilon)$ is the energy density of the fluid, including the internal energy contribution\footnote[1]{We note that since different application thorns may define their own local variables, the energy density is referred to as {\tt rho} within \codename{TOVSolver}, as the projected energy density $E$, defined in Sec.~\protect\ref{sec:Kevol}, is within \codename{McLachlan} and a few other thorns. Similar ambiguities exist for other commonly used variable names, particularly $\phi$ and $\alpha$.}.
+The routine also supplies the analytically known
+solution in the exterior,
+\begin{eqnarray}
+ P & = & P({\tt TOV\_atmosphere}),\nonumber \\
+ M_e & = & M, \nonumber\\
+ \Phi & = &\dfrac{1}{2} \log(1-2M / \hat{r})
+ \label{eq:TOVexterior}
+\end{eqnarray}
+where {\tt TOV\_atmosphere} is a parameter used to specify the density of the
+ambient atmosphere and a polytropic equation of state is assumed, and $M$ is the total gravitational mass of the star. Since the isotropic radius $r$ is the more
+commonly preferred choice to initiate dynamical calculations, the code then
+transforms all variables into isotropic coordinates, integrating the radius
+conversion formula
+\begin{equation}
+\label{eq:rbar}
+\frac{d (\log(r / \hat{r}))}{\partial \hat{r}} = \frac{\hat{r}^{1/2} - (\hat{r}-2M_e)^{1/2}}{\hat{r}(\hat{r}-2M_e)^{1/2}} \ .
+\end{equation}
+subject to the boundary condition that in the exterior,
+\begin{eqnarray}
+r &=& \dfrac{1}{2}\left(\sqrt{\hat{r}^2-2Mr}+\hat{r} -M\right)\nonumber \\
+\hat{r}&=&r\left(1+\dfrac{M}{2r}\right)^2 \ ,
+\end{eqnarray}
+handling with some care the potentially singular terms that appear at the origin.
+In converting the solution into the variables required for a dynamical evolution, one may assume that the metric is conformally flat, with a conformal factor given by $\psi = \sqrt{\hat{r}/r}$, or equivalently, a logarithmic conformal factor $\phi = \frac{1}{2}\log(\hat{r}/r)$.
+
+To facilitate the construction of stars in more complicated dynamical configurations,
+the code allows users to apply a uniform velocity to the NS, though this does not affect
+the ODE solution nor the resulting density profile, and thus does not represent a fully-self-conistent solution.
+
\subsubsection{Lorene-based binary data}
The Einstein Toolkit contains three routines that can read in publicly available data generated
@@ -1031,7 +1128,7 @@
equations for binary initial configurations~\cite{York:1998hy} and a single-grid
spectral method for rotating stars. For binaries, five elliptic equations for
the shift, lapse, and conformal factor are written down and the source terms
-are divided into pieces that are attributed to each of the two objects.
+are split into pieces that are attributed to each of the two objects.
Matter source terms are ideal for this split, since they are compactly supported,
while extrinsic curvature source terms are spatially extended but with sufficiently
rapid falloff at large radii to yield convergent solutions. Around each object,
@@ -1057,57 +1154,43 @@
\end{figure}
\codename{Meudon\_Bin\_BH} can read in BH-BH binary initial data described
-in~\cite{Grandclement:2001ed}, while \codename{Meudon\_Bin\_NS}
-handles binary NS data from~\cite{Gourgoulhon:2000nn}. \codename{Meudon\_Mag\_NS}
-may be used to read in magnetized isolated NS data~\cite{Lorene:web}.
-
-\subsubsection{TOVSolver}
-\label{sec:TOVSolver}
-The \codename{TOVSolver} routine in the ET solves the standard TOV equations
-\cite{Tolman:1939jz,Oppenheimer:1939ne} expressed using the Schwarzschild (areal)
-radius $r$ in the interior of a spherically symmetric star in hydrostatic equilibrium:
-\begin{eqnarray}
- \label{eq:TOViso}
- \frac{d P}{d r} & = & -(e + P) \frac{m + 4\pi r^3 P}{r(r - 2m)}\nonumber\\
-%
- \frac{d m}{d r} & = & 4 \pi r^2 e\nonumber\\
-%
- \frac{d \Phi}{d r} & = & \frac{m + 4\pi r^3 P}{r(r -
- 2m)}.
-\end{eqnarray}
-This assumes the stress energy tensor is given by
+in~\cite{Grandclement:2001ed}, representing solutions to the hamiltonian and momentum constraints, along with the trace of the spatial components of the Einstein equations, form the linked elliptic equation set:
+\begin{eqnarray*}
+&&\nabla^2\alpha_{(m)} = \alpha \psi^4 K_{ij} K^{ij}_{(m)}\\
+&&\nabla^2\psi_{(m)}=-\frac{\psi^5}{8}K_{ij}K^{ij}_{(m)}\\
+&&\nabla^2\beta^i_{(m)}+\frac{1}{3}\nabla^i\nabla_j \beta^j_{(m)} = 2K^{ij}(\nabla_j \alpha_{(m)}-\frac{6\alpha}{\psi}\nabla_j\psi_{(m)})
+\end{eqnarray*}
+where the sub/superscript $(m)=1,2$ indexes the two BHs, and the extrinsic curvature is calculated by the Killing equation assuming helicoidal invariance, which yields the condition
\begin{equation}
- \label{eq:Tmunu_e}
- T^{\mu\nu} = (e + P)u^{\mu}u^{\nu} + Pg^{\mu\nu},
+K^{ij} = \frac{1}{2\alpha\psi^4}(\nabla^i \beta^j+\nabla^j\beta^i-\frac{2}{3}\eta^{ij}\nabla_k\beta^k)\label{eq:lorextrcurv}
\end{equation}
-where $e\equiv \rho(1+\epsilon)$ is the energy density of the fluid, including the internal energy contribution\footnote[1]{We note that since different application thorns may define their own local variables, the energy density is referred to as {\tt rho} within \codename{TOVSolver}, as the projected energy density $E$, defined in Sec.~\protect\ref{sec:Kevol}, is within \codename{McLachlan} and a few other thorns. Similar ambiguities exist for other commonly used variable names, particularly $\phi$ and $\alpha$.}, $m$ is the gravitational mass inside a sphere of radius $r$, and
-$\Phi$ the logarithm of the lapse. The routine also supplies the analytically known
-solution in the exterior,
-\begin{eqnarray}
- P & = & P({\tt TOV\_atmosphere}),\nonumber \\
- m & = & M, \nonumber\\
- \Phi & = &\dfrac{1}{2} \log(1-2M / r)
- \label{eq:TOVexterior}
-\end{eqnarray}
-where {\tt TOV\_atmosphere} is a parameter used to specify the density of the
-ambient atmosphere and a polytropic equation of state is assumed. Since the isotropic radius $\bar{r}$ is the more
-commonly preferred choice to initiate dynamical calculations, the code then
-transforms all variables into isotropic coordinates, integrating the radius
-conversion formula
+The splitting of the extrinsic curvature into its two BH-based components is complicated, as are the boundary conditions imposed at the BH throats, and both are described in detail in ~\cite{Grandclement:2001ed}; in general, though, all split quantities may be summed to reconstruct a global quantity, taking into account the asymptotic values:
+\begin{eqnarray*}
+\alpha = 1+\alpha_{(1)}+\alpha_{(2)}&;&~~ \beta^i = \beta^i_{(1)}+\beta^i_{(2)};\\
+\psi =1+ \psi_{(1)}+\psi_{(2)}&;&~~K^{ij} = K^{ij}_{(1)}+K^{ij}_{(2)}.
+\end{eqnarray*}
+
+\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)
+\end{eqnarray*}
+where $\nu$ and $\beta$ are defined as
\begin{equation}
-\label{eq:rbar}
-\frac{d (\log(\bar{r} / r))}{\partial r} = \frac{r^{1/2} - (r-2m)^{1/2}}{r(r-2m)^{1/2}} \ .
+\nu\equiv \log\alpha;~~\beta\equiv\ln\alpha\psi^2.
\end{equation}
-subject to the boundary condition that in the exterior,
+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}
-\bar{r} &=& \dfrac{1}{2}\left(\sqrt{r^2-2Mr}+r -M\right)\nonumber \\
-r&=&\bar{r}\left(1+\dfrac{M}{2\bar{r}}\right)^2 \ ,
+\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}
-handling with some care the potentially singular terms that appear at the origin.
+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}).
-To facilitate the construction of stars in more complicated dynamical configurations,
-one may also apply a uniform velocity to the NS, though this does not affect
-the ODE solution nor the resulting density profile.
+ \codename{Meudon\_Mag\_NS}
+may be used to read in magnetized isolated NS data~\cite{Lorene:web,Bocquet:1995je}. The spacetime metric is determined using the same equations as for a binary NS configuration, but with magnetic contributions added to the matter source terms via their contributions to the stress-energy tensor, and no splitting required. The magnetic field 4-vector, which is assumed only to have non-zero components in the $t$ and $\phi$ directions, may be determined from two Poisson-type equations.
\subsection{Spacetime Curvature Evolution}
\label{sec:Kevol}
@@ -1124,7 +1207,7 @@
for the spacetime variables (orders 2, 4, 6 and 8 are currently implemented)
and adds a Kreiss-Oliger
dissipation term to remove high frequency noise.
-The evolved variables are the conformal factor $\Phi$ (the $W$ method
+The evolved variables are the logarithmic conformal factor $\phi$ (the $W$ method
\cite{Marronetti:2007wz} is also implemented), the conformal
3-metric $\tilde{\gamma}_{ij}$, the trace $K$ of the extrinsic curvature,
the trace free extrinsic curvature $A_{ij}$ and the conformal connection
@@ -1133,16 +1216,16 @@
curvature $K_{ij}$ by:
\begin{eqnarray}
- \phi & \equiv & \log \left[ \frac{1}{12} \det \gamma_{ij} \right]\,,
+ \phi & \equiv & \log \left[ \frac{1}{12} \det \gamma_{ij} \right]\,,\label{eq:phi}
\\
- \tilde\gamma_{ij} & \equiv & e^{-4\phi}\; \gamma_{ij}\,,
+ \tilde\gamma_{ij} & \equiv & e^{-4\phi}\; \gamma_{ij}\,,\label{eq:tildegamma}
\\
- K & \equiv & g^{ij} K_{ij}\,,
+ K & \equiv & g^{ij} K_{ij}\,,\label{eq:trK}
\\
- \tilde A_{ij} & \equiv & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K
+ \tilde A_{ij} & \equiv & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K\label{eq:Atij}
\right]\,,
\\
- \tilde\Gamma^i & \equiv & \tilde\gamma^{jk} \tilde\Gamma^i_{jk} .
+ \tilde\Gamma^i & \equiv & \tilde\gamma^{jk} \tilde\Gamma^i_{jk} .\label{eq:Gammai}
\end{eqnarray}
The evolution equations are then:
@@ -1436,8 +1519,12 @@
v^i = \frac{u^i}{W} + \frac{\beta^i}{\alpha}\,\,,
\label{eq:vel}
\end{equation}
-where $W = (1-v^i v_i)^{-1/2}$ is the Lorentz factor. The contravariant 4-velocity is then given by:
+where
\begin{equation}
+W \equiv \alpha u^0 = (1-v^i v_i)^{-1/2}\label{eq:Lorentz}
+\end{equation}
+ is the Lorentz factor. The contravariant 4-velocity is then given by:
+\begin{equation}
u^0 = \frac{W}{\alpha}\,,\qquad
u^i = W \left( v^i - \frac{\beta^i}{\alpha}\right)\,\,,
\end{equation}
@@ -2081,7 +2168,7 @@
as:
\begin{eqnarray}
H &=& R - K^i{}_j K^j{}_i + K^2 - 16 \pi E \label{eqn:analysis_hamiltonian_constraint}\\
- M_i &=& \nabla_j K_i{}^j - \nabla_i K - 8 \pi S_i
+ M_i &=& \nabla_j K_i{}^j - \nabla_i K - 8 \pi S_i \label{eqn:analysis_momentum_constraint}
\end{eqnarray}
where $S_i=-\frac{1}{\alpha} \left( T_{i0} - \beta^j T_{ij} \right)$.
The difference between these modules lies in how they access the stress
