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.

References

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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 IOSTRING

10 Interfaces

General

Implements:

extract

Inherits:

grid

admbase

staticconformal

io

Grid Variables

10.0.1 PRIVATE GROUPS

  Group Names     Variable Names   Details    
temps compact 0
temp3d dimensions 3
g00 distribution DEFAULT
group type GF
timelevels 1
variable type REAL

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