Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

Gabrielle Allen

Abstract

1 Introduction

Thorn Extract calculates first order gauge invariant waveforms from a numerical spacetime, under the basic assumption that, at the spheres of extract the spacetime is approximately Schwarzschild. In addition, other quantities such as mass, angular momentum and spin can be determined.

This thorn should not be used blindly, it will always return some waveform, however it is up to the user to determine whether this is the appropriate expected first order gauge invariant waveform.

2 Physical System

2.1 Wave Forms

Assume a spacetime \(g_{\alpha \beta }\) which can be written as a Schwarzschild background \(g_{\alpha \beta }^{Schwarz}\) with perturbations \(h_{\alpha \beta }\): \begin {equation} g_{\alpha \beta } = g^{Schwarz}_{\alpha \beta } + h_{\alpha \beta } \end {equation} with \begin {equation} \{g^{Schwarz}_{\alpha \beta }\}(t,r,\theta ,\phi ) = \left ( \begin {array}{cccc} -S & 0 & 0 & 0 \\ 0 & S^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin ^2\theta \end {array}\right ) \qquad S(r)=1-\frac {2M}{r} \end {equation} The 3-metric perturbations \(\gamma _{ij}\) can be decomposed using tensor harmonics into \(\gamma _{ij}^{lm}(t,r)\) where

             ∞   l
γ (t,r,𝜃,ϕ) = ∑  ∑   γlm (t,r)
 ij          l=0m= −l ij

and

             6
γ (t,r,𝜃,ϕ) = ∑ p (t,r)V  (𝜃,ϕ )
 ij          k=0 k      k

where \(\{{\bf V}_k\}\) is some basis for tensors on a 2-sphere in 3-D Euclidean space. Working with the Regge-Wheeler basis (see Section 6) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions \(\{c_1^{\times lm}, c_2^{\times lm}, h_1^{+lm}, H_2^{+lm}, K^{+lm}, G^{+lm}\}\) [19][16]. Where each of the functions is either odd (\(\times \)) or even (\(+\)) parity. The decomposition is then written

\begin {eqnarray} \gamma _{ij}^{lm} & = & c_1^{\times lm}(\hat {e}_1)_{ij}^{lm} + c_2^{\times lm}(\hat {e}_2)_{ij}^{lm} \nonumber \\ & + & h_1^{+lm}(\hat {f}_1)_{ij}^{lm} + A^2 H_2^{+lm}(\hat {f}_2)_{ij}^{lm} + R^2 K^{+lm}(\hat {f}_3)_{ij}^{lm} + R^2 G^{+lm}(\hat {f}_4)_{ij}^{lm} \end {eqnarray}

which we can write in an expanded form as

\begin {eqnarray} \gamma _{rr}^{lm} & = & A^2 H_2^{+lm} \Y \\ \gamma _{r\t }^{lm} & = & - c_1^{\times lm} \frac {1}{\s } \Yp +h_1^{+lm}\Yt \\ \gamma _{r\p }^{lm} & = & c_1^{\times lm} \s \Yt + h_1^{+lm}\Yp \\ \gamma _{\t \t }^{lm} & = & c_2^{\times lm}\frac {1}{\s }(\Ytp -\cot \t \Yp ) + R^2 K^{+lm}\Y + R^2 G^{+lm} \Ytt \\ \gamma _{\t \p }^{lm} & = & -c_2^{\times lm}\s \frac {1}{2} \left ( \Ytt -\cot \t \Yt -\frac {1}{\sin ^2\theta }\Y \right ) + R^2 G^{+lm}(\Ytp -\cot \t \Yp ) \\ \gamma _{\p \p }^{lm} & = & -\s c_2^{\times lm} (\Ytp - \cot \t \Yp ) +R^2 K^{+lm}\sin ^2\t \Y +R^2 G^{+lm} (\Ypp +\s \c \Yt ) \end {eqnarray}

A similar decomposition allows the four gauge components of the 4-metric to be written in terms of three even-parity variables \(\{H_0,H_1,h_0\}\) and the one odd-parity variable \(\{c_0\}\)

\begin {eqnarray} g_{tt}^{lm} & = & N^2 H_0^{+lm} \Y \\ g_{tr}^{lm} & = & H_1^{+lm} \Y \\ g_{t\t }^{lm} & = & h_0^{+lm} \Yt - c_0^{\times lm}\frac {1}{\s }\Yp \\ g_{t\p }^{lm} & = & h_0^{+lm} \Yp + c_0^{\times lm} \s \Yt \end {eqnarray}

Also from \(g_{tt}=-\alpha ^2+\beta _i\beta ^i\) we have \begin {equation} \alpha ^{lm} = -\frac {1}{2}NH_0^{+lm}Y_{lm} \end {equation} It is useful to also write this with the perturbation split into even and odd parity parts:

       background  ∑   lm,odd  ∑   lm,even
gαβ = gαβ      +    gαβ   +    gαβ
                 l,m          l,m

where (dropping some superscripts)

\begin {eqnarray*} \{g_{\alpha \beta }^{odd}\} &=& \left ( \begin {array}{cccc} 0 & 0 & - c_0\frac {1}{\s }\Yp & c_0 \s \Yt \\ . & 0 & - c_1\frac {1}{\s } \Yp & c_1 \s \Yt \\ . & . & c_2\frac {1}{\s }(\Ytp -\cot \t \Yp ) & c_2\frac {1}{2} \left (\frac {1}{\s } \Ypp +\c \Yt -\s \Ytt \right ) \\ .&.&.&c_2 (-\s \Ytp +\c \Yp ) \end {array} \right ) \\ \{g_{\alpha \beta }^{even}\} &=& \left ( \begin {array}{cccc} N^2 H_0\Y & H_1\Y & h_0\Yt & h_0 \Yp \\ . & A^2H_2\Y & h_1\Yt & h_1 \Yp \\ . & . & R^2K\Y +r^2G\Ytt & R^2(\Ytp -\cot \t \Yp ) \\ . & . & . & R^2 K\sin ^2\t \Y +R^2G(\Ypp +\s \c \Yt ) \end {array} \right ) \end {eqnarray*}

Now, for such a Schwarzschild background we can define two (and only two) unconstrained gauge invariant quantities \(Q^{\times }_{lm}=Q^{\times }_{lm}(c_1^{\times lm},c_2^{\times lm})\) and \(Q^{+}_{lm}=Q^{+}_{lm}(K^{+ lm},G^{+ lm},H_2^{+lm},h_1^{+lm})\), which from [3] are

\begin {eqnarray} Q^{\times }_{lm} & = & \sqrt {\frac {2(l+2)!}{(l-2)!}}\left [c_1^{\times lm} + \frac {1}{2}\left (\partial _r c_2^{\times lm} - \frac {2}{r} c_2^{\times lm}\right )\right ] \frac {S}{r} \\ Q^{+}_{lm} & = & \frac {1}{\Lambda }\sqrt {\frac {2(l-1)(l+2)}{l(l+1)}} (4rS^2 k_2+l(l+1)r k_1) \\ & \equiv & \frac {1}{\Lambda }\sqrt {\frac {2(l-1)(l+2)}{l(l+1)}} \left (l(l+1)S(r^2\partial _r G^{+lm}-2h_1^{+lm})+ 2rS(H_2^{+lm}-r\partial _r K^{+lm})+\Lambda r K^{+lm}\right ) \end {eqnarray}

where

\begin {eqnarray} k_1 & = & K^{+lm} + \frac {S}{r}(r^2\partial _r G^{+lm} - 2h^{+lm}_1) \\ k_2 & = & \frac {1}{2S} \left [H^{+lm}_2-r\partial _r k_1-\left (1-\frac {M}{rS}\right ) k_1 + S^{1/2}\partial _r (r^2 S^{1/2} \partial _r G^{+lm}-2S^{1/2}h_1^{+lm})\right ] \\ &\equiv & \frac {1}{2S}\left [H_2-rK_{,r}-\frac {r-3M}{r-2M}K\right ] \end {eqnarray}

NOTE: These quantities compare with those in Moncrief [16] by

\begin {eqnarray*} \mbox {Moncriefs odd parity Q: }\qquad Q^\times _{lm} &=& \sqrt {\frac {2(l+2)!}{(l-2)!}}Q \\ \mbox {Moncriefs even parity Q: } \qquad Q^+_{lm} &=& \sqrt {\frac {2(l-1)(l+2)}{l(l+1)}}Q \end {eqnarray*}

Note that these quantities only depend on the purely spatial Regge-Wheeler functions, and not the gauge parts. (In the Regge-Wheeler and Zerilli gauges, these are just respectively (up to a rescaling) the Regge-Wheeler and Zerilli functions). These quantities satisfy the wave equations

\begin {eqnarray*} &&(\partial ^2_t-\partial ^2_{r^*})Q^\times _{lm}+S\left [\frac {l(l+1)}{r^2}-\frac {6M}{r^3} \right ]Q^{\times }_{lm} = 0 \\ &&(\partial ^2_t-\partial ^2_{r^*})Q^+_{lm}+S\left [ \frac {1}{\Lambda ^2}\left (\frac {72M^3}{r^5}-\frac {12M}{r^3}(l-1)(l+2)\left (1-\frac {3M}{r}\right ) \right )+\frac {l(l-1)(l+1)(l+2)}{r^2\Lambda }\right ]Q^+_{lm}=0 \end {eqnarray*}

where

\begin {eqnarray*} \Lambda &=& (l-1)(l+2)+6M/r \\ r^* &=& r+2M\ln (r/2M-1) \end {eqnarray*}

3 Numerical Implementation

The implementation assumes that the numerical solution, on a Cartesian grid, is approximately Schwarzshild on the spheres of constant \(r=\sqrt (x^2+y^2+z^2)\) where the waveforms are extracted. The general procedure is then:

3.1 Project onto Spheres of Constant Radius

This is performed by interpolating the metric components, and if needed the conformal factor, onto the spheres. Although 2-spheres are hardcoded, the source code could easily be changed here to project onto e.g. 2-ellipsoids.

3.2 Calculate Radial Transformation

The areal coordinate \(\hat {r}\) of each sphere is calculated by \begin {equation} \hat {r} = \hat {r}(r) = \left [ \frac {1}{4\pi } \int \sqrt {\gamma _{\t \t } \gamma _{\p \p }}d\t d\p \right ]^{1/2} \end {equation} from which \begin {equation} \frac {d\hat {r}}{d\eta } = \frac {1}{16\pi \hat {r}} \int \frac {\gamma _{\t \t ,\eta }\gamma _{\p \p }+\gamma _{\t \t }\gamma _{\p \p ,\eta }} {\sqrt {\gamma _{\t \t }\gamma _{\p \p }}} \ d\t d\p \end {equation} Note that this is not the only way to combine metric components to get the areal radius, but this one was used because it gave better values for extracting close to the event horizon for perturbations of black holes.

3.3 Calculate \(S\) factor and Mass Estimate

\begin {equation} S(\hat {r}) = \left (\frac {\partial \hat {r}}{\partial r}\right )^2 \int \gamma _{rr} \ d\t d\p \end {equation}

\begin {equation} M(\hat {r}) = \hat {r}\frac {1-S}{2} \end {equation}

3.4 Calculate Regge-Wheeler Variables

\begin {eqnarray*} c_1^{\times lm} &=& \frac {1}{l(l+1)} \int \frac {\gamma _{\hat {r}\p }Y^*_{lm,\t } -\gamma _{\hat {r}\t } Y^*_{lm,\p } } {\s }d\Omega \\ c_2^{\times lm} & = & -\frac {2}{l(l+1)(l-1)(l+2)} \int \left \{ \left (-\frac {1}{\sin ^2\t }\gamma _{\t \t }+\frac {1} {\sin ^4\t }\gamma _{\p \p }\right ) (\s Y^*_{lm,\t \p }-\c Y^*_{lm,\p }) \right . \\ &&\left . + \frac {1}{\s } \gamma _{\t \p } (Y^*_{lm,\t \t }-\cot \t Y^*_{lm,\t } -\frac {1}{\sin ^2\t }Y^*_{lm,\p \p }) \right \}d\Omega \\ h_1^{+lm} &=& \frac {1}{l(l+1)} \int \left \{ \gamma _{\hat {r}\t } Y^*_{lm,\t } + \frac {1}{\sin ^2\t } \gamma _{\hat {r}\p }Y^*_{lm,\p }\right \} d\Omega \\ H_2^{+lm} &=& S \int \gamma _{\hat {r}\hat {r}} \Ys d\Omega \\ K^{+lm} &=& \frac {1}{2\hat {r}^2} \int \left (\gamma _{\t \t }+ \frac {1}{\sin ^2\t }\gamma _{\p \p }\right )\Ys d\Omega \\ &&+\frac {1}{2\hat {r}^2(l-1)(l+2)} \int \left \{ \left (\gamma _{\t \t }-\frac {\gamma _{\p \p }}{\sin ^2\t }\right ) \left (Y^*_{lm,\t \t }-\cot \t Y^*_{lm,\t }-\frac {1}{\sin ^2\t } Y^*_{lm,\p \p }\right ) \right . \\ &&\left . + \frac {4}{\sin ^2\t }\gamma _{\t \p }(Y^*_{lm,\t \p }-\cot \t Y^*_{lm,\p }) \right \} d\Omega \\ G^{+lm} &=& \frac {1}{\hat {r}^2 l(l+1)(l-1)(l+2)} \int \left \{ \left (\gamma _{\t \t }-\frac {\gamma _{\p \p }}{\sin ^2\t }\right ) \left (Y^*_{lm,\t \t }-\cot \t Y^*_{lm,\t }-\frac {1}{\sin ^2\t } Y^*_{lm,\p \p }\right ) \right . \\ &&\left . +\frac {4}{\sin ^2\t }\gamma _{\t \p }(Y^*_{lm,\t \p }-\cot \t Y^*_{lm,\p }) \right \}d\Omega \end {eqnarray*}

where

\begin {eqnarray} \gamma _{\hat {r}\hat {r}} & = & \frac {\partial r}{\partial \hat {r}} \frac {\partial r}{\partial \hat {r}} \gamma _{rr} \\ \gamma _{\hat {r}\t } & = & \frac {\partial r}{\partial \hat {r}} \gamma _{r\t } \\ \gamma _{\hat {r}\p } & = & \frac {\partial r}{\partial \hat {r}} \gamma _{r\p } \end {eqnarray}

3.5 Calculate Gauge Invariant Quantities

\begin {eqnarray} Q^{\times }_{lm} & = & \sqrt {\frac {2(l+2)!}{(l-2)!}}\left [c_1^{\times lm} + \frac {1}{2}\left (\partial _{\hat {r}} c_2^{\times lm} - \frac {2}{\hat {r}} c_2^{\times lm}\right )\right ] \frac {S}{\hat {r}} \\ Q^{+}_{lm} & = & \frac {1}{(l-1)(l+2)+6M/\hat {r}}\sqrt {\frac {2(l-1)(l+2)}{l(l+1)}} (4\hat {r}S^2 k_2+l(l+1)\hat {r} k_1) \end {eqnarray}

where

\begin {eqnarray} k_1 & = & K^{+lm} + \frac {S}{\hat {r}}(\hat {r}^2\partial _{\hat {r}} G^{+lm} - 2h^{+lm}_1) \\ k_2 & = & \frac {1}{2S} [H^{+lm}_2-\hat {r}\partial _{\hat {r}} k_1-(1-\frac {M}{\hat {r}S}) k_1 + S^{1/2}\partial _{\hat {r}} (\hat {r}^2 S^{1/2} \partial _{\hat {r}} G^{+lm}-2S^{1/2}h_1^{+lm} \end {eqnarray}

4 Using This Thorn

Use this thorn very carefully. Check the validity of the waveforms by running tests with different resolutions, different outer boundary conditions, etc to check that the waveforms are consistent.

4.1 Basic Usage

4.2 Output Files

Although Extract is really an ANALYSIS thorn, at the moment it is scheduled at POSTSTEP, with the iterations at which output is performed determined by the parameter itout. Output files from Extract are always placed in the main output directory defined by CactusBase/IOUtil.

Output files are generated for each detector (2-sphere) used, and these detectors are identified in the name of each output file by R1, R2, ….

The extension denotes whether coordinate time (ṫl) or proper time (˙u  l) is used for the first column.

5 History

Much of the source code for Extract comes from a code written outside of Cactus for extracting waveforms from data generated by the NCSA G-Code for compare with linear evolutions of waveforms extracted from the Cauchy initial data. This work was carried out in collaboration with Karen Camarda and Ed Seidel.

6 Appendix: Regge-Wheeler Harmonics

\begin {eqnarray*} (\hat {e}_1)^{lm} &=& \left ( \begin {array}{ccc} 0 & -\frac {1}{\s }\Yp & \s \Yt \\ . & 0 & 0 \\ . & 0 & 0 \end {array}\right ) \\ (\hat {e}_2)^{lm} &=& \left ( \begin {array}{ccc} 0 & 0 & 0 \\ 0 & \frac {1}{\s }(\Ytp -\cot \t \Yp ) & . \\ 0 & -\frac {\s }{2}[\Ytt -\cot \t \Yt -\frac {1}{\sin ^2\t }\Ypp ] & -\s [\Ytp -\cot \t \Yp ] \end {array}\right ) \\ (\hat {f}_1)^{lm} &=& \left ( \begin {array}{ccc} 0 & \Yt & \Yp \\ . & 0 & 0 \\ . & 0 & 0 \end {array}\right ) \\ (\hat {f}_2)^{lm} &=& \left ( \begin {array}{ccc} \Y & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end {array}\right ) \\ (\hat {f}_3)^{lm} &=& \left ( \begin {array}{ccc} 0 & 0 & 0 \\ 0 & \Y & 0 \\ 0 & 0 & \sin ^2\t \Y \end {array}\right ) \\ (\hat {f}_4)^{lm} &=& \left ( \begin {array}{ccc} 0 & 0 & 0 \\ 0 & \Ytt & . \\ 0 & \Ytp -\cot \t \Yp & \Ypp + \s \c \Yt \end {array}\right ) \end {eqnarray*}

7 Appendix: Transformation Between Cartesian and Spherical Coordinates

First, the transformations between metric components in \((x,y,z)\) and \((r,\t ,\p )\) coordinates. Here, \(\rho =\sqrt {x^2+y^2}=r\s \),

\begin {eqnarray*} \frac {\partial x}{\partial r} &=& \sin \t \cos \p = \frac {x}{r} \\ \frac {\partial y}{\partial r} &=& \sin \t \sin \p = \frac {y}{r} \\ \frac {\partial z}{\partial r} &=& \cos \t = \frac {z}{r} \\ \frac {\partial x}{\partial \t } &=& r\cos \t \cos \p = \frac {xz}{\rho } \\ \frac {\partial y}{\partial \t } &=& r\cos \t \sin \p = \frac {yz}{\rho } \\ \frac {\partial z}{\partial \t } &=& -r\sin \t = -\rho \\ \frac {\partial x}{\partial \p } &=& -r\sin \t \sin \p = -y \\ \frac {\partial y}{\partial \p } &=& r\sin \t \cos \p = x \\ \frac {\partial z}{\partial \p } &=& 0 \end {eqnarray*}
\begin {eqnarray*} \gamma _{rr} &=& \frac {1}{r^2} (x^2\gamma _{xx}+ y^2\gamma _{yy}+ z^2\gamma _{zz}+ 2xy\gamma _{xy}+ 2xz\gamma _{xz}+ 2yz\gamma _{yz}) \\ \gamma _{r\t } &=& \frac {1}{r\rho } (x^2 z \gamma _{xx} +y^2 z \gamma _{yy} -z \rho ^2 \gamma _{zz} +2xyz \gamma _{xy} +x(z^2-\rho ^2)\gamma _{xz} +y(z^2-\rho ^2)\gamma _{yz}) \\ \gamma _{r\p } &=& \frac {1}{r} (-xy\gamma _{xx} +xy\gamma _{yy} +(x^2-y^2)\gamma _{xy} -yz \gamma _{xz} +xz\gamma _{yz}) \\ \gamma _{\t \t } &=& \frac {1}{\rho ^2} (x^2z^2\gamma _{xx} +2xyz^2\gamma _{xy} -2xz\rho ^2\gamma _{xz} +y^2z^2\gamma _{yy} -2yz\rho ^2\gamma _{yz} +\rho ^4\gamma _{zz}) \\ \gamma _{\t \p } &=& \frac {1}{\rho } (-xyz\gamma _{xx} +(x^2-y^2)z\gamma _{xy} +\rho ^2 y \gamma _{xz} +xyz\gamma _{yy} -\rho ^2 x \gamma _{yz}) \\ \gamma _{\p \p } &=& y^2\gamma _{xx} -2xy\gamma _{xy} +x^2\gamma _{yy} \end {eqnarray*}

or,

\begin {eqnarray*} \gamma _{rr}&=& \sin ^2\t \cos ^2\p \gamma _{xx} +\sin ^2\t \sin ^2\p \gamma _{yy} +\cos ^2\t \gamma _{zz} +2\sin ^2\theta \cos \p \sin \p \gamma _{xy} +2\sin \t \cos \t \cos \p \gamma _{xz} \\ && +2\s \c \sin \p \gamma _{yz} \\ \gamma _{r\t }&=& r(\s \c \cos ^2\phi \gamma _{xx} +2*\s \c \sin \p \cos \p \gamma _{xy} +(\cos ^2\t -\sin ^2\t )\cos \p \gamma _{xz} +\s \c \sin ^2\p \gamma _{yy} \\ && +(\cos ^2\t -\sin ^2\t )\sin \p \gamma _{yz} -\s \c \gamma _{zz}) \\ \gamma _{r\p }&=& r\s (-\s \sin \p \cos \p \gamma _{xx} -\s (\sin ^2\p -\cos ^2\p )\gamma _{xy} -\c \sin \p \gamma _{xz} +\s \sin \p \cos \p \gamma _{yy} \\ && +\c \cos \p \gamma _{yz}) \\ \gamma _{\t \t }&=& r^2(\cos ^2\t \cos ^2\p \gamma _{xx} +2\cos ^2\t \sin \p \cos \p \gamma _{xy} -2\s \c \cos \p \gamma _{xz} +\cos ^2\t \sin ^2\p \gamma _{yy} \\ && -2\s \c \sin \p \gamma _{yz} +\sin ^2\t \gamma _{zz}) \\ \gamma _{\t \p }&=& r^2\s (-\c \sin \p \cos \p \gamma _{xx} -\c (\sin ^2\p -\cos ^2\p )\gamma _{xy} +\s \sin \p \gamma _{xz} +\c \sin \p \cos \p \gamma _{yy} \\ && -\s \cos \p \gamma _{yz}) \\ \gamma _{\p \p }&=& r^2\sin ^2\t (\sin ^2\p \gamma _{xx} -2\sin \p \cos \p \gamma _{xy} +\cos ^2\phi \gamma _{yy}) \end {eqnarray*}

We also need the transformation for the radial derivative of the metric components:

\begin {eqnarray*} \gamma _{rr,\eta }&=& \sin ^2\t \cos ^2\p \gamma _{xx,\eta } +\sin ^2\t \sin ^2\p \gamma _{yy,\eta } +\cos ^2\t \gamma _{zz,\eta } +2\sin ^2\theta \cos \p \sin \p \gamma _{xy,\eta } \\ && +2\sin \t \cos \t \cos \p \gamma _{xz,\eta } +2\s \c \sin \p \gamma _{yz,\eta } \\ \gamma _{r\t ,\eta }&=& \frac {1}{r}\gamma _{r\t }+ r(\s \c \cos ^2\phi \gamma _{xx,\eta } +\s \c \sin \p \cos \p \gamma _{xy,\eta } +(\cos ^2\t -\sin ^2\t )\cos \p \gamma _{xz,\eta } \\ && +\s \c \sin ^2\p \gamma _{yy,\eta } +(\cos ^2\t -\sin ^2\t )\sin \p \gamma _{yz,\eta } -\s \c \gamma _{zz,\eta }) \\ \gamma _{r\p ,\eta }&=& \frac {1}{r}\gamma _{r\p }+ r\s (-\s \sin \p \cos \p \gamma _{xx,\eta } -\s (\sin ^2\p -\cos ^2\p )\gamma _{xy,\eta } -\c \sin \p \gamma _{xz,\eta } \\ && +\s \sin \p \cos \p \gamma _{yy,\eta } +\c \cos \p \gamma _{yz,\eta }) \\ \gamma _{\t \t ,\eta }&=& \frac {2}{r}\gamma _{\t \t }+ r^2(\cos ^2\t \cos ^2\p \gamma _{xx,\eta } +2\cos ^2\t \sin \p \cos \p \gamma _{xy,\eta } -2\s \c \cos \p \gamma _{xz,\eta } \\ && +\cos ^2\t \sin ^2\p \gamma _{yy,\eta } -2\s \c \sin \p \gamma _{yz,\eta } +\sin ^2\t \gamma _{zz,\eta }) \\ \gamma _{\t \p ,\eta }&=& \frac {2}{r}\gamma _{\t \p }+ r^2\s (-\c \sin \p \cos \p \gamma _{xx,\eta } -\c (\sin ^2\p -\cos ^2\p )\gamma _{xy,\eta } +\s \sin \p \gamma _{xz,\eta } \\ && +\c \sin \p \cos \p \gamma _{yy,\eta } -\s \cos \p \gamma _{yz,\eta }) \\ \gamma _{\p \p ,\eta }&=& \frac {2}{r}\gamma _{\p \p }+ r^2\sin ^2\t (\sin ^2\p \gamma _{xx,\eta } -2\sin \p \cos \p \gamma _{xy,\eta } +\cos ^2\phi \gamma _{yy,\eta }) \end {eqnarray*}

8 Appendix: Integrations Over the 2-Spheres

This is done by using Simpson’s rule twice. Once in each coordinate direction. Simpson’s rule is \begin {equation} \int ^{x_2}_{x_1} f(x) dx = \frac {h}{3} [f_1+4f_2+2f_3+4f_4+\ldots +2f_{N-2}+4 f_{N-1}+f_N] +O(1/N^4) \end {equation} \(N\) must be an odd number.

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9 Parameters




all_modes
Scope: private  BOOLEAN



Description: Extract: all l,m modes up to l



  Default: yes






cauchy
Scope: private  BOOLEAN



Description: Do Cauchy data extraction at given timestep



  Default: no






cauchy_dr
Scope: private  REAL



Description: Gridspacing for Cauchy data extraction



Range   Default: 0.2
*:*






cauchy_r1
Scope: private  REAL



Description: First radius for Cauchy data extraction



Range   Default: 1.0
*:*






cauchy_timestep
Scope: private  INT



Description: Timestep for Cauchy data extraction



Range   Default: (none)
0:*






detector1
Scope: private  REAL



Description: Coordinate radius of detector 1



Range   Default: 5.0
0:*






detector2
Scope: private  REAL



Description: Coordinate radius of detector 2



Range   Default: 5.0
0:*






detector3
Scope: private  REAL



Description: Coordinate radius of detector 3



Range   Default: 5.0
0:*






detector4
Scope: private  REAL



Description: Coordinate radius of detector 4



Range   Default: 5.0
0:*






detector5
Scope: private  REAL



Description: Coordinate radius of detector 5



Range   Default: 5.0
0:*






detector6
Scope: private  REAL



Description: Coordinate radius of detector 6



Range   Default: 5.0
0:*






detector7
Scope: private  REAL



Description: Coordinate radius of detector 7



Range   Default: 5.0
0:*






detector8
Scope: private  REAL



Description: Coordinate radius of detector 8



Range   Default: 5.0
0:*






detector9
Scope: private  REAL



Description: Coordinate radius of detector 9



Range   Default: 5.0
0:*






do_momentum
Scope: private  BOOLEAN



Description: Calculate momentum at extraction radii



  Default: no






do_spin
Scope: private  BOOLEAN



Description: Calculate spin at extraction radii



  Default: no






doadmmass
Scope: private  BOOLEAN



Description: Calculate ADM mass at extraction radii



  Default: no






interpolation_operator
Scope: private  STRING



Description: Interpolation operator to use (check LocalInterp)



Range   Default: uniform cartesian
.+






interpolation_order
Scope: private  INT



Description: Order for interpolation



Range   Default: 1
1:4
Choose between first and forth order interpolation






itout
Scope: private  INT



Description: How often to extract, in iterations



Range   Default: 1
0:*






l_mode
Scope: private  INT



Description: l mode



Range   Default: 2
0:*






m_mode
Scope: private  INT



Description: m mode (ignore if extracting all modes



Range   Default: (none)
0:*






np
Scope: private  INT



Description: Number of phi divisions



Range   Default: 100
0:*






nt
Scope: private  INT



Description: Number of theta divisions



Range   Default: 100
0:*






num_detectors
Scope: private  INT



Description: Number of detectors



Range   Default: (none)
0:*






origin_x
Scope: private  REAL



Description: x-origin to extract about



Range   Default: 0.0
*:*






origin_y
Scope: private  REAL



Description: y-origin to extract about



Range   Default: 0.0
*:*






origin_z
Scope: private  REAL



Description: z-origin to extract about



Range   Default: 0.0
*:*






timecoord
Scope: private  KEYWORD



Description: Which time coordinate to use



Range   Default: both
proper
coordinate
both






verbose
Scope: private  BOOLEAN



Description: Say what is happening



  Default: no






out_dir
Scope: shared from IO STRING



10 Interfaces

General

Implements:

extract

Inherits:

grid

admbase

staticconformal

io

Grid Variables

10.0.1 PRIVATE GROUPS




  Group Names    Variable Names    Details   




temps   compact0
temp3d   dimensions3
g00   distributionDEFAULT
  group typeGF
  timelevels1
 variable typeREAL




11 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinAnalysis/Extract. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

NONE

Scheduled Functions

CCTK_PARAMCHECK

  extract_paramcheck

  check parameters

 

 Language:c
 Options: global
 Type: function

CCTK_POSTSTEP

  extract

  extract waveforms

 

 Language:fortran
 Storage: temps
 Type: function