User: tbode Date: 2011/04/01 09:03 PM
Modified: / ET.tex
Log: Initial rewrite/expansion: Analysis
File Changes:
Directory: / ============
File [modified]: ET.tex Delta lines: +293 -31 =================================================================== --- ET.tex 2011-03-28 14:56:57 UTC (rev 41) +++ ET.tex 2011-04-02 02:03:03 UTC (rev 42) @@ -28,6 +28,7 @@ \def\bh#1{black hole#1 (BH#1)\gdef\bh{BH}} \def\ns#1{neutron star#1 (NS#1)\gdef\ns{NS}} \def\gt#1{Georgia Tech#1 (GaTech#1)\gdef\gt{GaTech}} +\def\ahz#1{apparent horizon#1 (AH#1)\gdef\ahz{AH}}
\newcommand{\codename}[1]{\texttt{#1}} \newcommand{\todo}[1]{{\color{red}$\blacksquare$~\textsf{[TODO: #1]}}} @@ -836,42 +837,303 @@
\subsection{Analysis\pages{4 Tanja}} +It is beneficial to evaluate common analysis quantities online +rather than offline, in time-consuming post-processing procedures. +Beyond extracting physics, these quantities are often used as measures +of how accurately the simulation is progressing. Below we detail the +quantities available within Einstein Toolkit modules and the +assumptions and equations used by each. The analysis capabilities of +the Einstein Toolkit broadly fall into three categories: horizons, +masses and momenta, and gravitational waves. Many modules bridge these +these categories and some fall outside them. The latter are described +in the last subsection, including constraint monitoring and tools for +commonly required derived spacetime quantities. The following discussion +is meant as an overview of the most common tools rather than an +exhaustive list of functionality. In most cases, the analysis modules +work on the variables stored in the base modules discussed in +Sec.~\ref{sec:base_modules} (\codename{ADMBase}, \codename{TmunuBase}, +and \codename{HydroBase}) to create as portable a tool as possible.
-Any simulated system may have many applicable quantities for analysis which -are most efficiently calculated during evolution. The Einstein Toolkit -currently includes modules to facilitate the calculation of the most common -analysis quantities: masses, horizons, and gravitational waves. In addition -to these, detailed below, the \codename{ADMConstraints} module calculates the -constraints required, but not imposed by, the formulation for assessing the -simulation's accuracy and stability. Finally, \codename{ADMAnalysis} calculates -several quantities often needed in analyzing the spacetime evolution: the trace -of the extrinsic curvature $K$, the determinant of the 3-metric, the spatial -metric and extrinsic curvature in spherical coordinates, and the spatial Ricci -tensor and trace. +\subsubsection{Horizons} +For spacetimes which contain a \bh{,} the Einstein Toolkit provides +one module (\codename{EHFinder}) for finding event horizons and +two modules for finding \ahz{s} (\codename{AHFinder} +and \codename{AHFinderDirect}).
-\subsubsection*{Horizons} For spacetimes which contain a \bh{,} the Toolkit -contains modules to find both the event and apparent horizons and their -properties. The \codename{EHFinder} module traces null surfaces backwards in -time through a pre-evolved spacetime. The Toolkit also includes two separate -apparent horizon finders: Thornburg's \codename{AHFinderDirect} and -\codename{AHFinder}. +% Event horizon +The event horizon module \codename{EHFinder}~\cite{Diener:2003jc} +evolves a null surface backwards in time given an initial guess (e.g. +the last apparent horizon) which will, in the vicinity of an event +horizon, converge exponentially to that event horizon. This has to be +done after a simulation has already evolved the initial data forward +in time with enough 3D data written out that the full 4-metric can be +recovered at each timestep.
-\subsubsection*{Masses} In analyzing spacetimes, it is important to calculate -the masses and momenta of the system. Several Toolkit modules include the -capability of calculating several types of masses. The apparent horizon modules -include in their output the areal masses of the found horizons. The -\codename{Extract} module calculates ADM mass and momenta when extracting -gravitational waves. Additionally, \codename{HydroAnalysis} calculates finds -the center of mass and the point of maximum matter density of a hydronamic -field. +In \codename{EHFinder}, the null surface is represented by a function +$f(t,x^i)=0$ which is required to satisfy the null condition +$g^{\alpha\beta} \partial_\alpha f \partial_\beta f = 0$. In the +standard numerical 3+1 form of the metric, this null condition can be +expanded out into an evolution equation for $f$ as +\begin{equation} + \partial_t f = \beta^i \partial_i f - \sqrt{\alpha^2 \gamma^{ij} + \partial_i f \partial_j f} +\end{equation} +where the roots are chosen to describe outgoing null geodesics. The +function $f$ is chosen such that it is negative within the initial +guess of the horizon and positive without, initially set to a distance +measure from the initial surface guess such as +$f(t_0,x^i)=\sqrt{(x^i-x^i_0)(x_i-x_{i(0)})}-r_0$. +Numerically there is a problem with the steepening of $\nabla f$ during +the evolution, so the function $f$ is regularly re-initialized during +the evolution such that $|\nabla f|\simeq1$. This is done by evolving +\begin{equation} + \frac{df}{d\lambda} = -\frac{f}{\sqrt{f^2+1}}\left(|\nabla f|-1\right) +\end{equation} +for some unphysical parameter $\lambda$ until a steady state has been +reached. As the isosurface $f=0$ converges exponentially to the event +horizon, it is useful to evolve two such null surfaces which bracket +the approximate position of the anticipated event horizon to further +narrow the region containing the event horizon.
-\subsubsection*{Gravitational Waves} Within the Toolkit are modules for extracting -gravitational waves using two formulations: the Moncrief formalism assuming -the spacetime approximates spherical symmetry \codename{Extract}) and the Weyl -scalar $\Psi_4$ constructed on a fiducial tetrad (\codename{WeylScal4}). The -Weyl scalars, as well as any other quantity, can be projected onto a basis -of spin-weighted spherical harmonics using the \codename{Multipole} module. +Unfortunately event horizons can only be found after the full spacetime +has been evolved. It is often useful, however, to know the positions +and shapes of any \bh{} on a given hypersurface for purposes such as +excision, accretion, and local measures of its mass and spin. The +Einstein Toolkit provides several algorithms of varying speed and +accuracy to find marginally trapped surfaces, of which the outermost are +\ahz{s}. All finders utilize that the outgoing null geodesics have +vanishing expansion on an \ahz{} which, in the usual 3+1 quantities, +can be written as +\begin{equation} \label{eq:ah_theta} + \Theta \equiv \nabla_i n^i + K_{ij} n^i n^j - K = 0 +\end{equation} +where $n^i$ is the unit outgoing normal to the 2-surface.
+The module \codename{AHFinder} provides two algorithms for locating +\ahz{s}. The minimization algorithm~\cite{Anninos98b} finds the local +minimum of $\oint_S (\Theta - \Theta_o )^2 d^2S$ corresponding to a +surface of constant expansion $\Theta_o$, with $\Theta_o=0$ +corresponding to the \ahz{.} For time-symmetric data, the option exists +to find instead the minimum of the surface area, which in this case +corresponds to an \ahz{.} An alternative algorithm provided by +\codename{AHFinder}, the flow algorithm~\cite{Gundlach97a}, on +which the above described \codename{EHFinder} is also based. +Defining a surface as a level set $f(x^i)=r-h(\theta,\phi)=0$, +and introducing an unphysical timelike parameter $\lambda$ to +parametrize the flow of $h$ towards a solution, Eq.~\ref{eq:ah_theta} +can be rewritten as +\begin{equation} + \partial_\lambda h = - \left( \frac{\alpha}{\ell_\mathrm{max} + (\ell_\mathrm{max}+1)} + \beta \right) \left( 1 - + \frac{\beta}{\alpha} L^2\right)^{-1} \rho \Theta +\end{equation} +where $\rho$ is a strictly positive weight, $L^2$ is the Laplacian +of the 2D metric(?), and $\alpha$,$\beta$, and $\ell_\mathrm{max}$ are +free parameters. Decomposing $h(\theta,\phi)$ onto a basis of spherical +harmonics, the coefficients $a_{\ell m}$ evolve iteratively towards a +solution as +\begin{equation} + a_{\ell m}^{(n+1)} = a_{\ell m}^{(n)} - + \frac{\alpha + \beta \ell_\mathrm{max} + \left(\ell_\mathrm{max}+1\right) } + {\ell_\mathrm{max}\left(\ell_\mathrm{max}+1\right) + \left(1+\beta\ell(\ell+1)/\alpha\right)} + \left(\rho\Theta\right)_{\ell m}^{(n)} +\end{equation} + +The \codename{AHFinderDirect} module~\cite{Thornburg:2003sf} is a +faster alternative to \codename{AHFinder}. Its approach is to view +Eq.\ref{eq:ah_theta} as an elliptic PDE for $h(\theta,\phi)$ on $S^2$ +using standard finite differencing methods. Rewriting Eq.~\ref{ah_theta} +in the form +\begin{equation} + \Theta \equiv \Theta\left(h,\partial_u h,\partial_{uv}h; +\gamma_{ij},K_{ij},\partial_k \gamma_{ij}\right) = 0,, +\end{equation} +the expansion $\Theta$ is evaluated on a trial surface, then iterates +using a Newton-method updating equations $\bf{J}\cdot\delta h=-\Theta$ +where $\bf{J}$ is the Jacobian matrix. +The drawback of this method is that it is not guaranteed to give the +outermost marginally trapped surface. + +\subsubsection{Masses and Momenta} +Two distinct measures of mass and momenta are available in relativistic +spacetimes. First, ADM mass and angular momentum evaluated as either +surface integrals at infinity or volume integrals over entire +hypersurfaces give a measure of the total energy and angular momentum +in the spacetime. The module \codename{ML_ADMQuantities} of the +McLachlan code~\cite{McLachlan:web} uses the latter method, creating +gridfunctions containing the integrand of the volume +integrals~\cite{Yo02a} +\begin{eqnarray} + M &=& \frac{1}{16\pi} \int_\Omega d^3 x \left[ e^{5 \phi} + \left( 16 \pi \rho + \tilde{A}_{ij} \tilde{A}^{ij} - \frac23 K^2 + \right) - e^\phi \tilde{R} \right] \ + J_i &=& \frac{1}{8 \pi} \epsilon_{ij}^k \int_\Omega d^3 x \left[ + e^{6\phi}\left( \tilde{A}^j_k + \frac23 x^j \tilde{D}_k K + - \frac12 x^j \tilde{A}_{\ell n} \partial_k \tilde{\gamma}^{\ell n} + + 8 \pi x^j S_k \right) \right] +\end{eqnarray} +on which the user can use the reduction functions provided by +\codename{Carpet} to perform the volume integral. We note that +\codename{ML_ADMQuantities} inherits directly from the BSSN variables +stored in \codename{McLachlan} rather than strictly from the base modules. +As the surface terms required when converting a surface integral to a +volume integral are neglected, this procedures hold under the assumption +that the integrals of the integrands $\tilde{D}^i e^\phi$ and +$e^{6\phi} \epsilon_{ij}^k x^j \tilde{A}^\ell_k$ on the boundaries of +the computational domain vanish. The ADM mass and angular momentum +can also be calculated by the module \codename{Extract}, from the +variables stored in the base modules, as surface integrals~\cite{Bowen80} +\begin{eqnarray} + M &=& - \frac{1}{2\pi} \oint \tilde{D}^i \psi d^2 S_i \ + J_i &=& \frac{1}{16\pi} \epsilon_{ijk} \oint \left( x^j K^{km} + - x^k K^{jm} \right) d^2 S_m +\end{eqnarray} +on a specified spherical surface preferrably far from the center of +the domain as these quantities are only properly interpretable when +calculated at infinity. + +Second, there are the quasi-local measures of mass and momentum. From +any \ahz{s} found during the spacetime, both +\codename{AHFinderDirect} and \codename{AHFinder} output the +corresponding mass derived from the area of the horizon $m_H = +\sqrt{A/(16\pi)}$. In addition, the module \codename{QuasiLocalMeasures} +takes a specified surface, generally a previously located \ahz{} +and calculates the mass, spin and their moments, among other quantities, +on the surface based on the isolated and dynamical horizon framework. +\todo{... is this what it does? ...} + +Finally, the module \codename{HydroAnalysis} additionally locates the +center of mass as well as the point of maximum rest mass density of a +matter field. + +\subsubsection{Gravitational Waves} +One of the main goals of numerical relativity to date is source +modelling for the production of numerical gravitational waveforms to +be used as a basis of tests and template formation by analysts working +with data from the various gravitational wave detectors around the +globe. The Einstein Toolkit includes modules for extracting +gravitational waves via either the Moncrief formalism of a perturbation +on Schwarzschild or calculation of the Weyl scalar $\Psi_4$. + +The module \codename{Extract} uses the Moncrief formalism~\cite{ +Moncrief74} to extract, given spherical surfaces of constant coordinate +radii, gauge-invariant wavefunctions $Q_{\ell m}^\times$ and $Q_{\ell +m}^+$. The spatial metric is expressed as a perturbation on +Schwarzschild and expanded onto a basis for tensors chosen here to be +the Regge-Wheeler harmonics~\cite{Regge57} described by six standard +Regge-Wheeler functions $\lbrace c_1^{\times\ell m}, c_2^{\times\ell m}, +h_1^{+\ell m}, H_2^{+\ell m},K^{+\ell m}, G^{+ \ell m} \rbrace$. From +these basis functions the gauge-invariant quantities +\begin{eqnarray} + Q_{\ell m}^\times &=& \sqrt{\frac{2(\ell+2)!}{(\ell-2)!}} + \left[c_1^{\times\ell m} + \frac12\left( \partial_r - \frac{2}{r} + \right) c_2^{\times\ell m} \right] \frac{S}{r} \ + Q_{\ell m}^+ &=& \frac{1}{\Lambda} \sqrt{ \frac{2(\ell-1)(\ell+2)} + {\ell(\ell+1)}} \Bigg( \ell(\ell+1)S(r^2 \partial_r G^{+ \ell m} + - 2 h_1^{+\ell m} ) \nonumber \ + & & + 2rS(H_2^{+\ell m}-r\partial_rK^{+\ell m}) + + \Lambda r K^{+\ell m} \Bigg) +\end{eqnarray} +are calculated where $S=1-2M/r$ and $\Lambda=(\ell-1)(\ell+2)+6M/r$. +These functions then satisfy the wave equations +\begin{eqnarray} + (\partial_t^2-\partial_{r^*}^2)Q_{\ell m}^\times &=& + - S \left[ \frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3} \right] + Q_{\ell m}^\times \ + (\partial_t^2-\partial_{r^*}^2)Q_{\ell m}^+ &=& + - S \Bigg[ \frac{1}{\Lambda^2} \left( \frac{72M^3}{r^5}-\frac{12M + (\ell-1)(\ell+2)}{r^3}\left(1-\frac{3M}{r}\right) \right) + \nonumber \ + & & + + \frac{\ell(\ell^2-1)(\ell+2)}{r^2\Lambda} \Bigg] Q_{\ell m}^+ +\end{eqnarray} +where $r^*=r+2M \ln(r/2M-1)$. +As these functions describe the 4-metric as a perturbation on +Schwarzschild, the spacetime must be approximately spherically +symmetric for the output to be properly interpretable as first order +gauge invariant waveforms. + +For more general spacetimes, the module \codename{WeylScal4} calculates +the complex Weyl scalar $\Psi_4=C_{\alpha\beta\gamma\delta},n^\alpha +\bar{m}^\beta n^\gamma \bar{m}^\delta$, which is a projection +of the Weyl tensor onto components of a null tetrad. +\codename{WeylScal4} uses the fiducial tetrad~\cite{Baker:2001sf}, +written in 3+1 decomposed form as +\begin{eqnarray} + \ell^\mu &=& \frac{1}{\sqrt{2}}\left(u^\mu+\tilde{r}^\mu\right) \ + n^\mu &=& \frac{1}{\sqrt{2}}\left(u^\mu-\tilde{r}^\mu\right) \ + m^\mu &=&\frac{1}{\sqrt{2}}\left(\tilde{\theta}^\mu+i\tilde{\phi}^\mu\right) +\end{eqnarray} +where $u^\mu$ is the unit normal to the hypersurface. The spatial +vectors $\lbrace \tilde{r}^\mu, \tilde{\theta}^\mu, \tilde{\phi}^\mu +\rbrace$ are created by initializing as $\tilde{r}^\mu = +\lbrace0,x^i\rbrace$, $\tilde{\phi}^\mu = \lbrace0,-y,x,0\rbrace$, and +$\tilde{\theta}^\mu=\lbrace0,\sqrt{\gamma} \gamma^{ik} \epsilon_{k\ell +m} \phi^\ell r^m\rbrace$, then orthonormalizing starting with +$\tilde{\phi}^i$ and invoking a Gram-Schmidt procedure at each step to +ensure the continued orthonormality of this spatial triad. + +Without loss of generality, this particular Weyl scalar is calculated +explicitly in terms of projections of the 3-Riemann tensor onto a null +tetrad as +\begin{eqnarray} + \Psi_4 &=& \mathcal{R}_{ijk\ell} n^i \bar{m}^j n^k \bar{m}^\ell + + 2 \mathcal{R}_{0jk\ell} \left( n^0 \bar{m}^j n^k \bar{m}^\ell + - \bar{m}^0 n^j n^k \bar{m}^\ell \right) \nonumber \ + &+& \mathcal{R}_{0j0\ell} \left( n^0 \bar{m}^j n^0 \bar{m}^\ell + + \bar{m}^0 n^j \bar{m}^0 n^\ell - 2n^0 \bar{m}^j \bar{m}^0 + n^\ell \right),. +\end{eqnarray} +For a suitably chosen tetrad, this scalar in the radiation zone is +related to the strain of the gravitational waves as +\begin{equation} + h = h_+ - i h_\times = - \int_{-\infty}^t dt^\prime + \int_{-\infty}^{t^\prime} \Psi_4 dt^{\prime\prime},. +\end{equation} + +While the waveforms generated by \codename{Extract} are +already decomposed on a convenient basis to separate modes, the +complex quantity $\Psi_4$ is merely provided by \codename{WeylScal4} as +a complex gridfunction. For this quantity, and any other real or +complex gridfunction, the module \codename{Multipole} interpolates +the field $u(t,r,\theta,\phi)$ onto coordinate spheres of given radii +and calculates the coefficients +\begin{equation} + C^{\ell m} \left(t,r\right) = \int {}_s Y_{\ell m}^* u(t,r,\theta,\phi) + r^2 d\Omega +\end{equation} +of a projection onto spin-weighted spherical harmonics ${}_s Y_{\ell m}$. + +\subsubsection{Other} +The remaining analysis capabilities of the Einstein Toolkit span a +variety of primarily vacuum-based functions. +First, modules are provided to calculate the Hamiltonian and Momentum +constraints which are used to monitor how well the evolved spacetime +satisfies the Einstein field equations. Two modules, +\codename{ADMConstraints} and \codename{ML_ADMConstraints} provide these +quantities. Both calculate these directly from variables stored in the +base modules described in Sec.~\ref{sec:base_modules}, explicitly written +as: +\begin{eqnarray} + H &=& R - K^i_j K^j_i + K^2 - 16 \pi \rho \ + M_i &=& \nabla_j K_i^j - \nabla_i K - 8 \pi S_i +\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 +energy tensor $T_{\mu\nu}$, as the module \codename{ADMConstraints} +uses a \todo{deprecated?} functionality which does not require storage +for $T_{\mu\nu}$. + +Finally, \codename{ADMAnalysis} calculates a variety of derived spacetime +quantities which are often useful in postprocessing such as the determinant +of the 3-metric $\det{\gamma}$, the trace of the extrinsic curvature $K$, +the 3-Ricci tensor in Cartesian coordinates $\mathcal{R}_{ij}$ and its trace +$\mathcal{R}$, as well as the 3-metric and extrinsic curvature converted to +spherical coordinates. + + \subsection{Relativity Tools\pages{3 Peter}}
\paragraph{Black Hole Excision}