Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

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

and

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:

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:

• Project the required metric components, and radial derivatives of metric components, onto spheres of constant coordinate radius (these spheres are chosen via parameters).

• Transform the metric components and there derivatives on the 2-spheres from Cartesian coordinates into a spherical coordinate system.

• Calculate the physical metric on these spheres if a conformal factor is being used.

• Calculate the transformation from the coordinate radius to an areal radius for each sphere.

• Calculate the $$S$$ factor on each sphere. Combined with the areal radius This also produces an estimate of the mass.

• Calculate the six Regge-Wheeler variables, and required radial derivatives, on these spheres by integration of combinations of the metric components over each sphere.

• Contruct the gauge invariant quantities from these Regge-Wheeler variables.

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.

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.

4.2 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 (l) is used for the first column.

• rsch_R?.[tu]l

The extracted areal radius on each 2-sphere.

• mass_R?.[tu]l

Mass estimate calculated from $$g_{rr}$$ on each 2-sphere.

• Qeven_R?_??.[tu]l

The even parity gauge invariate variable (waveform) on each 2-sphere. This is a complex quantity, the 2nd column is the real part, and the third column the imaginary part.

• Qodd_R?_??.[tu]l

The odd parity gauge invariate variable (waveform) on each 2-sphere. This is a complex quantity, the 2nd column is the real part, and the third column the imaginary part.

Estimate of ADM mass enclosed within each 2-sphere. (To produce this set doADMmass = ‘‘yes’’).

• momentum_[xyz]_R?.[tu]l

Estimate of momentum at each 2-sphere. (To produce this set do_momentum = ‘‘yes’’).

• spin_[xyz]_R?.[tu]l

Estimate of momentum at each 2-sphere. (To produce this set do_spin = ‘‘yes’’).

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

[1]   Abrahams A.M. & Cook G.B. “Collisions of boosted black holes: Perturbation theory predictions of gravitational radiation” Phys. Rev. D 50 R2364-R2367 (1994).

[2]   Abrahams A.M., Shapiro S.L. & Teukolsky S.A. “Calculation of gravitational wave forms from black hole collisions and disk collapse: Applying perturbation theory to numerical spacetimes” Phys. Rev. D. 51 4295 (1995).

[3]   Abrahams A.M. & Price R.H. “Applying black hole perturbation theory to numerically generated spacetimes” Phys. Rev. D. 53 1963 (1996).

[4]   Abrahams A.M. & Price R.H. “Black-hole collisions from Brill-Lindquist initial data: Predictions of perturbation theory” Phys. Rev. D. 53 1972 (1996).

[5]   Abramowitz, M. & Stegun A. “Pocket Book of Mathematical Functions (Abridged Handbook of Mathematical Functions”, Verlag Harri Deutsch (1984).

[6]   Andrade Z., & Price R.H. “Head-on collisions of unequal mass black holes: Close-limit predictions”, preprint (1996).

[7]   Anninos P., Price R.H., Pullin J., Seidel E., and Suen W-M. “Head-on collision of two black holes: Comparison of different approaches” Phys. Rev. D. 52 4462 (1995).

[8]   Arfken, G. “Mathematical Methods for Physicists”, Academic Press (1985).

[9]   Baker J., Abrahams A., Anninos P., Brant S., Price R., Pullin J. & Seidel E. “The collision of boosted black holes” (preprint) (1996).

[10]   Baker J. & Li C.B. “The two-phase approximation for black hole collisions: Is it robust” preprint (gr-qc/9701035), (1997).

[11]   Brandt S.R. & Seidel E. “The evolution of distorted rotating black holes III: Initial data” (preprint) (1996).

[12]   Cunningham C.T., Price R.H., Moncrief V., “Radiation from collapsing relativistic stars. I. Linearized Odd-Parity Radiation” Ap. J. 224 543-667 (1978).

[13]   Cunningham C.T., Price R.H., Moncrief V., “Radiation from collapsing relativistic stars. I. Linearized Even-Parity Radiation” Ap. J. 230 870-892 (1979).

[14]   Landau L.D. & Lifschitz E.M., “The Classical Theory of Fields” (4th Edition), Pergamon Press (1980).

[15]   Mathews J. “”, J. Soc. Ind. Appl. Math. 10 768 (1962).

[16]   Moncrief V. “Gravitational perturbations of spherically symmetric systems. I. The exterior problem” Annals of Physics 88 323-342 (1974).

[17]   Press W.H., Flannery B.P., Teukolsky S.A., & Vetterling W.T., “Numerical Recipes, The Art of Scientific Computing” Cambridge University Press (1989).

[18]   Price R.H. & Pullin J. “Colliding black holes: The close limit”, Phys. Rev. Lett. 72 3297-3300 (1994).

[19]   Regge T., & Wheeler J.A. “Stability of a Schwarzschild Singularity”, Phys. Rev. D 108 1063 (1957).

[20]   Seidel E. Phys Rev D. 42 1884 (1990).

[21]   Thorne K.S., “Multipole expansions of gravitational radiation”, Rev. Mod. Phys. 52 299 (1980).

[22]   Vishveshwara C.V., “Stability of the Schwarzschild metric”, Phys. Rev. D. 1 2870, (1970).

[23]   Zerilli F.J., “Tensor harmonics in canonical form for gravitational radiation and other applications”, J. Math. Phys. 11 2203, (1970).

[24]   Zerilli F.J., “Gravitational field of a particle falling in a Schwarzschild geometry analysed in tensor harmonics”, Phys. Rev. D. 2 2141, (1970).

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

 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

Implements:

extract

Inherits:

grid

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.

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