User: tbode
Date: 2011/04/01 09:03 PM
Modified:
/
ET.tex
Log:
Initial rewrite/expansion: Analysis
File Changes:
Directory: /
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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}
