## Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

March 31, 2019

Abstract

NB: This documentation is taken from the Extract thorn on which WaveExtractL is based. There may be some diﬀerences between WaveExtractL and Extract, which are not documented here.

### 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 ${h}_{\alpha \beta }$:

 ${g}_{\alpha \beta }={g}_{\alpha \beta }^{Schwarz}+{h}_{\alpha \beta }$ (1)

with

 $\left\{{g}_{\alpha \beta }^{Schwarz}\right\}\left(t,r,𝜃,\varphi \right)=\left(\begin{array}{cccc}\hfill -S\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {S}^{-1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}{sin}^{2}𝜃\hfill \end{array}\right)\phantom{\rule{2em}{0ex}}S\left(r\right)=1-\frac{2M}{r}$ (2)

The 3-metric perturbations ${\gamma }_{ij}$ can be decomposed using tensor harmonics into ${\gamma }_{ij}^{lm}\left(t,r\right)$ where

${\gamma }_{ij}\left(t,r,𝜃,\varphi \right)=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}{\gamma }_{ij}^{lm}\left(t,r\right)$

and

${\gamma }_{ij}\left(t,r,𝜃,\varphi \right)=\sum _{k=0}^{6}{p}_{k}\left(t,r\right){V}_{k}\left(𝜃,\varphi \right)$

where $\left\{{V}_{k}\right\}$ 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 $\left\{{c}_{1}^{×lm},{c}_{2}^{×lm},{h}_{1}^{+lm},{H}_{2}^{+lm},{K}^{+lm},{G}^{+lm}\right\}$ . Where each of the functions is either odd ($×$) or even ($+$) parity. The decomposition is then written

$\begin{array}{rcll}{\gamma }_{ij}^{lm}& =& {c}_{1}^{×lm}{\left({ê}_{1}\right)}_{ij}^{lm}+{c}_{2}^{×lm}{\left({ê}_{2}\right)}_{ij}^{lm}& \text{}\\ & +& {h}_{1}^{+lm}{\left({\stackrel{̂}{f}}_{1}\right)}_{ij}^{lm}+{A}^{2}{H}_{2}^{+lm}{\left({\stackrel{̂}{f}}_{2}\right)}_{ij}^{lm}+{R}^{2}{K}^{+lm}{\left({\stackrel{̂}{f}}_{3}\right)}_{ij}^{lm}+{R}^{2}{G}^{+lm}{\left({\stackrel{̂}{f}}_{4}\right)}_{ij}^{lm}& \text{(3)}\text{}\text{}\end{array}$

which we can write in an expanded form as

$\begin{array}{rcll}{\gamma }_{rr}^{lm}& =& {A}^{2}{H}_{2}^{+lm}{Y}_{lm}& \text{(4)}\text{}\text{}\\ {\gamma }_{r𝜃}^{lm}& =& -{c}_{1}^{×lm}\frac{1}{sin𝜃}{Y}_{lm,\varphi }+{h}_{1}^{+lm}{Y}_{lm,𝜃}& \text{(5)}\text{}\text{}\\ {\gamma }_{r\varphi }^{lm}& =& {c}_{1}^{×lm}sin𝜃{Y}_{lm,𝜃}+{h}_{1}^{+lm}{Y}_{lm,\varphi }& \text{(6)}\text{}\text{}\\ {\gamma }_{𝜃𝜃}^{lm}& =& {c}_{2}^{×lm}\frac{1}{sin𝜃}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)+{R}^{2}{K}^{+lm}{Y}_{lm}+{R}^{2}{G}^{+lm}{Y}_{lm,𝜃𝜃}& \text{(7)}\text{}\text{}\\ {\gamma }_{𝜃\varphi }^{lm}& =& -{c}_{2}^{×lm}sin𝜃\frac{1}{2}\left({Y}_{lm,𝜃𝜃}-cot𝜃{Y}_{lm,𝜃}-\frac{1}{{sin}^{2}𝜃}{Y}_{lm}\right)+{R}^{2}{G}^{+lm}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)& \text{(8)}\text{}\text{}\\ {\gamma }_{\varphi \varphi }^{lm}& =& -sin𝜃{c}_{2}^{×lm}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)+{R}^{2}{K}^{+lm}{sin}^{2}𝜃{Y}_{lm}+{R}^{2}{G}^{+lm}\left({Y}_{lm,\varphi \varphi }+sin𝜃cos𝜃{Y}_{lm,𝜃}\right)& \text{(9)}\text{}\text{}\end{array}$

A similar decomposition allows the four gauge components of the 4-metric to be written in terms of three even-parity variables $\left\{{H}_{0},{H}_{1},{h}_{0}\right\}$ and the one odd-parity variable $\left\{{c}_{0}\right\}$

$\begin{array}{rcll}{g}_{tt}^{lm}& =& {N}^{2}{H}_{0}^{+lm}{Y}_{lm}& \text{(10)}\text{}\text{}\\ {g}_{tr}^{lm}& =& {H}_{1}^{+lm}{Y}_{lm}& \text{(11)}\text{}\text{}\\ {g}_{t𝜃}^{lm}& =& {h}_{0}^{+lm}{Y}_{lm,𝜃}-{c}_{0}^{×lm}\frac{1}{sin𝜃}{Y}_{lm,\varphi }& \text{(12)}\text{}\text{}\\ {g}_{t\varphi }^{lm}& =& {h}_{0}^{+lm}{Y}_{lm,\varphi }+{c}_{0}^{×lm}sin𝜃{Y}_{lm,𝜃}& \text{(13)}\text{}\text{}\end{array}$

Also from ${g}_{tt}=-{\alpha }^{2}+{\beta }_{i}{\beta }^{i}$ we have

 ${\alpha }^{lm}=-\frac{1}{2}N{H}_{0}^{+lm}{Y}_{lm}$ (14)

It is useful to also write this with the perturbation split into even and odd parity parts:

${g}_{\alpha \beta }={g}_{\alpha \beta }^{background}+\sum _{l,m}{g}_{\alpha \beta }^{lm,odd}+\sum _{l,m}{g}_{\alpha \beta }^{lm,even}$

where (dropping some superscripts)

$\begin{array}{rcll}\left\{{g}_{\alpha \beta }^{odd}\right\}& =& \left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill -{c}_{0}\frac{1}{sin𝜃}{Y}_{lm,\varphi }\hfill & \hfill {c}_{0}sin𝜃{Y}_{lm,𝜃}\hfill \\ \hfill .\hfill & \hfill 0\hfill & \hfill -{c}_{1}\frac{1}{sin𝜃}{Y}_{lm,\varphi }\hfill & \hfill {c}_{1}sin𝜃{Y}_{lm,𝜃}\hfill \\ \hfill .\hfill & \hfill .\hfill & \hfill {c}_{2}\frac{1}{sin𝜃}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)\hfill & \hfill {c}_{2}\frac{1}{2}\left(\frac{1}{sin𝜃}{Y}_{lm,\varphi \varphi }+cos𝜃{Y}_{lm,𝜃}-sin𝜃{Y}_{lm,𝜃𝜃}\right)\hfill \\ \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill {c}_{2}\left(-sin𝜃{Y}_{lm,𝜃\varphi }+cos𝜃{Y}_{lm,\varphi }\right)\hfill \end{array}\right)& \text{}\\ \left\{{g}_{\alpha \beta }^{even}\right\}& =& \left(\begin{array}{cccc}\hfill {N}^{2}{H}_{0}{Y}_{lm}\hfill & \hfill {H}_{1}{Y}_{lm}\hfill & \hfill {h}_{0}{Y}_{lm,𝜃}\hfill & \hfill {h}_{0}{Y}_{lm,\varphi }\hfill \\ \hfill .\hfill & \hfill {A}^{2}{H}_{2}{Y}_{lm}\hfill & \hfill {h}_{1}{Y}_{lm,𝜃}\hfill & \hfill {h}_{1}{Y}_{lm,\varphi }\hfill \\ \hfill .\hfill & \hfill .\hfill & \hfill {R}^{2}K{Y}_{lm}+{r}^{2}G{Y}_{lm,𝜃𝜃}\hfill & \hfill {R}^{2}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)\hfill \\ \hfill .\hfill & \hfill .\hfill & \hfill .\hfill & \hfill {R}^{2}K{sin}^{2}𝜃{Y}_{lm}+{R}^{2}G\left({Y}_{lm,\varphi \varphi }+sin𝜃cos𝜃{Y}_{lm,𝜃}\right)\hfill \end{array}\right)& \text{}\end{array}$

Now, for such a Schwarzschild background we can deﬁne two (and only two) unconstrained gauge invariant quantities ${Q}_{lm}^{×}={Q}_{lm}^{×}\left({c}_{1}^{×lm},{c}_{2}^{×lm}\right)$ and ${Q}_{lm}^{+}={Q}_{lm}^{+}\left({K}^{+lm},{G}^{+lm},{H}_{2}^{+lm},{h}_{1}^{+lm}\right)$, which from  are

$\begin{array}{rcll}{Q}_{lm}^{×}& =& \sqrt{\frac{2\left(l+2\right)!}{\left(l-2\right)!}}\left[{c}_{1}^{×lm}+\frac{1}{2}\left({\partial }_{r}{c}_{2}^{×lm}-\frac{2}{r}{c}_{2}^{×lm}\right)\right]\frac{S}{r}& \text{(15)}\text{}\text{}\\ {Q}_{lm}^{+}& =& \frac{1}{\Lambda }\sqrt{\frac{2\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}}\left(4r{S}^{2}{k}_{2}+l\left(l+1\right)r{k}_{1}\right)& \text{(16)}\text{}\text{}\\ & \equiv & \frac{1}{\Lambda }\sqrt{\frac{2\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}}\left(l\left(l+1\right)S\left({r}^{2}{\partial }_{r}{G}^{+lm}-2{h}_{1}^{+lm}\right)+2rS\left({H}_{2}^{+lm}-r{\partial }_{r}{K}^{+lm}\right)+\Lambda r{K}^{+lm}\right)& \text{(17)}\text{}\text{}\end{array}$

where

$\begin{array}{rcll}{k}_{1}& =& {K}^{+lm}+\frac{S}{r}\left({r}^{2}{\partial }_{r}{G}^{+lm}-2{h}_{1}^{+lm}\right)& \text{(18)}\text{}\text{}\\ {k}_{2}& =& \frac{1}{2S}\left[{H}_{2}^{+lm}-r{\partial }_{r}{k}_{1}-\left(1-\frac{M}{rS}\right){k}_{1}+{S}^{1∕2}{\partial }_{r}\left({r}^{2}{S}^{1∕2}{\partial }_{r}{G}^{+lm}-2{S}^{1∕2}{h}_{1}^{+lm}\right)\right]& \text{(19)}\text{}\text{}\\ & \equiv & \frac{1}{2S}\left[{H}_{2}-r{K}_{,r}-\frac{r-3M}{r-2M}K\right]& \text{(20)}\text{}\text{}\end{array}$

NOTE: These quantities compare with those in Moncrief  by

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

$\begin{array}{rcll}& & \left({\partial }_{t}^{2}-{\partial }_{{r}^{\ast }}^{2}\right){Q}_{lm}^{×}+S\left[\frac{l\left(l+1\right)}{{r}^{2}}-\frac{6M}{{r}^{3}}\right]{Q}_{lm}^{×}=0& \text{}\\ & & \left({\partial }_{t}^{2}-{\partial }_{{r}^{\ast }}^{2}\right){Q}_{lm}^{+}+S\left[\frac{1}{{\Lambda }^{2}}\left(\frac{72{M}^{3}}{{r}^{5}}-\frac{12M}{{r}^{3}}\left(l-1\right)\left(l+2\right)\left(1-\frac{3M}{r}\right)\right)+\frac{l\left(l-1\right)\left(l+1\right)\left(l+2\right)}{{r}^{2}\Lambda }\right]{Q}_{lm}^{+}=0& \text{}\end{array}$

where

$\begin{array}{rcll}\Lambda & =& \left(l-1\right)\left(l+2\right)+6M∕r& \text{}\\ {r}^{\ast }& =& r+2Mln\left(r∕2M-1\right)& \text{}\end{array}$

### 3 Numerical Implementation

The implementation assumes that the numerical solution, on a Cartesian grid, is approximately Schwarzshild on the spheres of constant $r=\sqrt{\left(}{x}^{2}+{y}^{2}+{z}^{2}\right)$ 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.

#### 3.2 Calculate Radial Transformation

The areal coordinate $\stackrel{̂}{r}$ of each sphere is calculated by

 $\stackrel{̂}{r}=\stackrel{̂}{r}\left(r\right)={\left[\frac{1}{4\pi }\int \sqrt{{\gamma }_{𝜃𝜃}{\gamma }_{\varphi \varphi }}d𝜃d\varphi \right]}^{1∕2}$ (21)

from which

 $\frac{d\stackrel{̂}{r}}{d\eta }=\frac{1}{16\pi \stackrel{̂}{r}}\int \frac{{\gamma }_{𝜃𝜃,\eta }{\gamma }_{\varphi \varphi }+{\gamma }_{𝜃𝜃}{\gamma }_{\varphi \varphi ,\eta }}{\sqrt{{\gamma }_{𝜃𝜃}{\gamma }_{\varphi \varphi }}}\phantom{\rule{1em}{0ex}}d𝜃d\varphi$ (22)

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

 $S\left(\stackrel{̂}{r}\right)={\left(\frac{\partial \stackrel{̂}{r}}{\partial r}\right)}^{2}\int {\gamma }_{rr}\phantom{\rule{1em}{0ex}}d𝜃d\varphi$ (23)
 $M\left(\stackrel{̂}{r}\right)=\stackrel{̂}{r}\frac{1-S}{2}$ (24)

#### 3.4 Calculate Regge-Wheeler Variables

$\begin{array}{rcll}{c}_{1}^{×lm}& =& \frac{1}{l\left(l+1\right)}\int \frac{{\gamma }_{\stackrel{̂}{r}\varphi }{Y}_{lm,𝜃}^{\ast }-{\gamma }_{\stackrel{̂}{r}𝜃}{Y}_{lm,\varphi }^{\ast }}{sin𝜃}d\Omega & \text{}\\ {c}_{2}^{×lm}& =& -\frac{2}{l\left(l+1\right)\left(l-1\right)\left(l+2\right)}\int \left\{\left(-\frac{1}{{sin}^{2}𝜃}{\gamma }_{𝜃𝜃}+\frac{1}{{sin}^{4}𝜃}{\gamma }_{\varphi \varphi }\right)\left(sin𝜃{Y}_{lm,𝜃\varphi }^{\ast }-cos𝜃{Y}_{lm,\varphi }^{\ast }\right)\right\& \text{}\\ & & +\frac{1}{sin𝜃}{\gamma }_{𝜃\varphi }\left({Y}_{lm,𝜃𝜃}^{\ast }-cot𝜃{Y}_{lm,𝜃}^{\ast }-\frac{1}{{sin}^{2}𝜃}{Y}_{lm,\varphi \varphi }^{\ast }\right)}d\Omega & \text{}\\ {h}_{1}^{+lm}& =& \frac{1}{l\left(l+1\right)}\int \left\{{\gamma }_{\stackrel{̂}{r}𝜃}{Y}_{lm,𝜃}^{\ast }+\frac{1}{{sin}^{2}𝜃}{\gamma }_{\stackrel{̂}{r}\varphi }{Y}_{lm,\varphi }^{\ast }\right\}d\Omega & \text{}\\ {H}_{2}^{+lm}& =& S\int {\gamma }_{\stackrel{̂}{r}\stackrel{̂}{r}}{Y}_{lm}^{\ast }d\Omega & \text{}\\ {K}^{+lm}& =& \frac{1}{2{\stackrel{̂}{r}}^{2}}\int \left({\gamma }_{𝜃𝜃}+\frac{1}{{sin}^{2}𝜃}{\gamma }_{\varphi \varphi }\right){Y}_{lm}^{\ast }d\Omega & \text{}\\ & & +\frac{1}{2{\stackrel{̂}{r}}^{2}\left(l-1\right)\left(l+2\right)}\int \left\{\left({\gamma }_{𝜃𝜃}-\frac{{\gamma }_{\varphi \varphi }}{{sin}^{2}𝜃}\right)\left({Y}_{lm,𝜃𝜃}^{\ast }-cot𝜃{Y}_{lm,𝜃}^{\ast }-\frac{1}{{sin}^{2}𝜃}{Y}_{lm,\varphi \varphi }^{\ast }\right)\right\& \text{}\\ & & +\frac{4}{{sin}^{2}𝜃}{\gamma }_{𝜃\varphi }\left({Y}_{lm,𝜃\varphi }^{\ast }-cot𝜃{Y}_{lm,\varphi }^{\ast }\right)}d\Omega & \text{}\\ {G}^{+lm}& =& \frac{1}{{\stackrel{̂}{r}}^{2}l\left(l+1\right)\left(l-1\right)\left(l+2\right)}\int \left\{\left({\gamma }_{𝜃𝜃}-\frac{{\gamma }_{\varphi \varphi }}{{sin}^{2}𝜃}\right)\left({Y}_{lm,𝜃𝜃}^{\ast }-cot𝜃{Y}_{lm,𝜃}^{\ast }-\frac{1}{{sin}^{2}𝜃}{Y}_{lm,\varphi \varphi }^{\ast }\right)\right\& \text{}\\ & & +\frac{4}{{sin}^{2}𝜃}{\gamma }_{𝜃\varphi }\left({Y}_{lm,𝜃\varphi }^{\ast }-cot𝜃{Y}_{lm,\varphi }^{\ast }\right)}d\Omega & \text{}\end{array}$

where

$\begin{array}{rcll}{\gamma }_{\stackrel{̂}{r}\stackrel{̂}{r}}& =& \frac{\partial r}{\partial \stackrel{̂}{r}}\frac{\partial r}{\partial \stackrel{̂}{r}}{\gamma }_{rr}& \text{(25)}\text{}\text{}\\ {\gamma }_{\stackrel{̂}{r}𝜃}& =& \frac{\partial r}{\partial \stackrel{̂}{r}}{\gamma }_{r𝜃}& \text{(26)}\text{}\text{}\\ {\gamma }_{\stackrel{̂}{r}\varphi }& =& \frac{\partial r}{\partial \stackrel{̂}{r}}{\gamma }_{r\varphi }& \text{(27)}\text{}\text{}\end{array}$

#### 3.5 Calculate Gauge Invariant Quantities

$\begin{array}{rcll}{Q}_{lm}^{×}& =& \sqrt{\frac{2\left(l+2\right)!}{\left(l-2\right)!}}\left[{c}_{1}^{×lm}+\frac{1}{2}\left({\partial }_{\stackrel{̂}{r}}{c}_{2}^{×lm}-\frac{2}{\stackrel{̂}{r}}{c}_{2}^{×lm}\right)\right]\frac{S}{\stackrel{̂}{r}}& \text{(28)}\text{}\text{}\\ {Q}_{lm}^{+}& =& \frac{1}{\left(l-1\right)\left(l+2\right)+6M∕\stackrel{̂}{r}}\sqrt{\frac{2\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}}\left(4\stackrel{̂}{r}{S}^{2}{k}_{2}+l\left(l+1\right)\stackrel{̂}{r}{k}_{1}\right)& \text{(29)}\text{}\text{}\end{array}$

where

$\begin{array}{rcll}{k}_{1}& =& {K}^{+lm}+\frac{S}{\stackrel{̂}{r}}\left({\stackrel{̂}{r}}^{2}{\partial }_{\stackrel{̂}{r}}{G}^{+lm}-2{h}_{1}^{+lm}\right)& \text{(30)}\text{}\text{}\\ {k}_{2}& =& \frac{1}{2S}\left[{H}_{2}^{+lm}-\stackrel{̂}{r}{\partial }_{\stackrel{̂}{r}}{k}_{1}-\left(1-\frac{M}{\stackrel{̂}{r}S}\right){k}_{1}+{S}^{1∕2}{\partial }_{\stackrel{̂}{r}}\left({\stackrel{̂}{r}}^{2}{S}^{1∕2}{\partial }_{\stackrel{̂}{r}}{G}^{+lm}-2{S}^{1∕2}{h}_{1}^{+lm}& \text{(31)}\text{}\text{}\end{array}$

### 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 (u̇l) 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.

• ADMmass_R?.[tu]l

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

$\begin{array}{rcll}{\left({ê}_{1}\right)}^{lm}& =& \left(\begin{array}{ccc}\hfill 0\hfill & \hfill -\frac{1}{sin𝜃}{Y}_{lm,\varphi }\hfill & \hfill sin𝜃{Y}_{lm,𝜃}\hfill \\ \hfill .\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill .\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)& \text{}\\ {\left({ê}_{2}\right)}^{lm}& =& \left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{sin𝜃}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)\hfill & \hfill .\hfill \\ \hfill 0\hfill & \hfill -\frac{sin𝜃}{2}\left[{Y}_{lm,𝜃𝜃}-cot𝜃{Y}_{lm,𝜃}-\frac{1}{{sin}^{2}𝜃}{Y}_{lm,\varphi \varphi }\right]\hfill & \hfill -sin𝜃\left[{Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right]\hfill \end{array}\right)& \text{}\\ {\left({\stackrel{̂}{f}}_{1}\right)}^{lm}& =& \left(\begin{array}{ccc}\hfill 0\hfill & \hfill {Y}_{lm,𝜃}\hfill & \hfill {Y}_{lm,\varphi }\hfill \\ \hfill .\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill .\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)& \text{}\\ {\left({\stackrel{̂}{f}}_{2}\right)}^{lm}& =& \left(\begin{array}{ccc}\hfill {Y}_{lm}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)& \text{}\\ {\left({\stackrel{̂}{f}}_{3}\right)}^{lm}& =& \left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {Y}_{lm}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {sin}^{2}𝜃{Y}_{lm}\hfill \end{array}\right)& \text{}\\ {\left({\stackrel{̂}{f}}_{4}\right)}^{lm}& =& \left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {Y}_{lm,𝜃𝜃}\hfill & \hfill .\hfill \\ \hfill 0\hfill & \hfill {Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\hfill & \hfill {Y}_{lm,\varphi \varphi }+sin𝜃cos𝜃{Y}_{lm,𝜃}\hfill \end{array}\right)& \text{}\end{array}$

### 7 Appendix: Transformation Between Cartesian and Spherical Coordinates

First, the transformations between metric components in $\left(x,y,z\right)$ and $\left(r,𝜃,\varphi \right)$ coordinates. Here, $\rho =\sqrt{{x}^{2}+{y}^{2}}=rsin𝜃$,

$\begin{array}{rcll}\frac{\partial x}{\partial r}& =& sin𝜃cos\varphi =\frac{x}{r}& \text{}\\ \frac{\partial y}{\partial r}& =& sin𝜃sin\varphi =\frac{y}{r}& \text{}\\ \frac{\partial z}{\partial r}& =& cos𝜃=\frac{z}{r}& \text{}\\ \frac{\partial x}{\partial 𝜃}& =& rcos𝜃cos\varphi =\frac{xz}{\rho }& \text{}\\ \frac{\partial y}{\partial 𝜃}& =& rcos𝜃sin\varphi =\frac{yz}{\rho }& \text{}\\ \frac{\partial z}{\partial 𝜃}& =& -rsin𝜃=-\rho & \text{}\\ \frac{\partial x}{\partial \varphi }& =& -rsin𝜃sin\varphi =-y& \text{}\\ \frac{\partial y}{\partial \varphi }& =& rsin𝜃cos\varphi =x& \text{}\\ \frac{\partial z}{\partial \varphi }& =& 0& \text{}\end{array}$

$\begin{array}{rcll}{\gamma }_{rr}& =& \frac{1}{{r}^{2}}\left({x}^{2}{\gamma }_{xx}+{y}^{2}{\gamma }_{yy}+{z}^{2}{\gamma }_{zz}+2xy{\gamma }_{xy}+2xz{\gamma }_{xz}+2yz{\gamma }_{yz}\right)& \text{}\\ {\gamma }_{r𝜃}& =& \frac{1}{r\rho }\left({x}^{2}z{\gamma }_{xx}+{y}^{2}z{\gamma }_{yy}-z{\rho }^{2}{\gamma }_{zz}+2xyz{\gamma }_{xy}+x\left({z}^{2}-{\rho }^{2}\right){\gamma }_{xz}+y\left({z}^{2}-{\rho }^{2}\right){\gamma }_{yz}\right)& \text{}\\ {\gamma }_{r\varphi }& =& \frac{1}{r}\left(-xy{\gamma }_{xx}+xy{\gamma }_{yy}+\left({x}^{2}-{y}^{2}\right){\gamma }_{xy}-yz{\gamma }_{xz}+xz{\gamma }_{yz}\right)& \text{}\\ {\gamma }_{𝜃𝜃}& =& \frac{1}{{\rho }^{2}}\left({x}^{2}{z}^{2}{\gamma }_{xx}+2xy{z}^{2}{\gamma }_{xy}-2xz{\rho }^{2}{\gamma }_{xz}+{y}^{2}{z}^{2}{\gamma }_{yy}-2yz{\rho }^{2}{\gamma }_{yz}+{\rho }^{4}{\gamma }_{zz}\right)& \text{}\\ {\gamma }_{𝜃\varphi }& =& \frac{1}{\rho }\left(-xyz{\gamma }_{xx}+\left({x}^{2}-{y}^{2}\right)z{\gamma }_{xy}+{\rho }^{2}y{\gamma }_{xz}+xyz{\gamma }_{yy}-{\rho }^{2}x{\gamma }_{yz}\right)& \text{}\\ {\gamma }_{\varphi \varphi }& =& {y}^{2}{\gamma }_{xx}-2xy{\gamma }_{xy}+{x}^{2}{\gamma }_{yy}& \text{}\end{array}$

or,

$\begin{array}{rcll}{\gamma }_{rr}& =& {sin}^{2}𝜃{cos}^{2}\varphi {\gamma }_{xx}+{sin}^{2}𝜃{sin}^{2}\varphi {\gamma }_{yy}+{cos}^{2}𝜃{\gamma }_{zz}+2{sin}^{2}𝜃cos\varphi sin\varphi {\gamma }_{xy}+2sin𝜃cos𝜃cos\varphi {\gamma }_{xz}& \text{}\\ & & +2sin𝜃cos𝜃sin\varphi {\gamma }_{yz}& \text{}\\ {\gamma }_{r𝜃}& =& r\left(sin𝜃cos𝜃{cos}^{2}\varphi {\gamma }_{xx}+2\ast sin𝜃cos𝜃sin\varphi cos\varphi {\gamma }_{xy}+\left({cos}^{2}𝜃-{sin}^{2}𝜃\right)cos\varphi {\gamma }_{xz}+sin𝜃cos𝜃{sin}^{2}\varphi {\gamma }_{yy}& \text{}\\ & & +\left({cos}^{2}𝜃-{sin}^{2}𝜃\right)sin\varphi {\gamma }_{yz}-sin𝜃cos𝜃{\gamma }_{zz}\right)& \text{}\\ {\gamma }_{r\varphi }& =& rsin𝜃\left(-sin𝜃sin\varphi cos\varphi {\gamma }_{xx}-sin𝜃\left({sin}^{2}\varphi -{cos}^{2}\varphi \right){\gamma }_{xy}-cos𝜃sin\varphi {\gamma }_{xz}+sin𝜃sin\varphi cos\varphi {\gamma }_{yy}& \text{}\\ & & +cos𝜃cos\varphi {\gamma }_{yz}\right)& \text{}\\ {\gamma }_{𝜃𝜃}& =& {r}^{2}\left({cos}^{2}𝜃{cos}^{2}\varphi {\gamma }_{xx}+2{cos}^{2}𝜃sin\varphi cos\varphi {\gamma }_{xy}-2sin𝜃cos𝜃cos\varphi {\gamma }_{xz}+{cos}^{2}𝜃{sin}^{2}\varphi {\gamma }_{yy}& \text{}\\ & & -2sin𝜃cos𝜃sin\varphi {\gamma }_{yz}+{sin}^{2}𝜃{\gamma }_{zz}\right)& \text{}\\ {\gamma }_{𝜃\varphi }& =& {r}^{2}sin𝜃\left(-cos𝜃sin\varphi cos\varphi {\gamma }_{xx}-cos𝜃\left({sin}^{2}\varphi -{cos}^{2}\varphi \right){\gamma }_{xy}+sin𝜃sin\varphi {\gamma }_{xz}+cos𝜃sin\varphi cos\varphi {\gamma }_{yy}& \text{}\\ & & -sin𝜃cos\varphi {\gamma }_{yz}\right)& \text{}\\ {\gamma }_{\varphi \varphi }& =& {r}^{2}{sin}^{2}𝜃\left({sin}^{2}\varphi {\gamma }_{xx}-2sin\varphi cos\varphi {\gamma }_{xy}+{cos}^{2}\varphi {\gamma }_{yy}\right)& \text{}\end{array}$

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

$\begin{array}{rcll}{\gamma }_{rr,\eta }& =& {sin}^{2}𝜃{cos}^{2}\varphi {\gamma }_{xx,\eta }+{sin}^{2}𝜃{sin}^{2}\varphi {\gamma }_{yy,\eta }+{cos}^{2}𝜃{\gamma }_{zz,\eta }+2{sin}^{2}𝜃cos\varphi sin\varphi {\gamma }_{xy,\eta }& \text{}\\ & & +2sin𝜃cos𝜃cos\varphi {\gamma }_{xz,\eta }+2sin𝜃cos𝜃sin\varphi {\gamma }_{yz,\eta }& \text{}\\ {\gamma }_{r𝜃,\eta }& =& \frac{1}{r}{\gamma }_{r𝜃}+r\left(sin𝜃cos𝜃{cos}^{2}\varphi {\gamma }_{xx,\eta }+sin𝜃cos𝜃sin\varphi cos\varphi {\gamma }_{xy,\eta }+\left({cos}^{2}𝜃-{sin}^{2}𝜃\right)cos\varphi {\gamma }_{xz,\eta }& \text{}\\ & & +sin𝜃cos𝜃{sin}^{2}\varphi {\gamma }_{yy,\eta }+\left({cos}^{2}𝜃-{sin}^{2}𝜃\right)sin\varphi {\gamma }_{yz,\eta }-sin𝜃cos𝜃{\gamma }_{zz,\eta }\right)& \text{}\\ {\gamma }_{r\varphi ,\eta }& =& \frac{1}{r}{\gamma }_{r\varphi }+rsin𝜃\left(-sin𝜃sin\varphi cos\varphi {\gamma }_{xx,\eta }-sin𝜃\left({sin}^{2}\varphi -{cos}^{2}\varphi \right){\gamma }_{xy,\eta }-cos𝜃sin\varphi {\gamma }_{xz,\eta }& \text{}\\ & & +sin𝜃sin\varphi cos\varphi {\gamma }_{yy,\eta }+cos𝜃cos\varphi {\gamma }_{yz,\eta }\right)& \text{}\\ {\gamma }_{𝜃𝜃,\eta }& =& \frac{2}{r}{\gamma }_{𝜃𝜃}+{r}^{2}\left({cos}^{2}𝜃{cos}^{2}\varphi {\gamma }_{xx,\eta }+2{cos}^{2}𝜃sin\varphi cos\varphi {\gamma }_{xy,\eta }-2sin𝜃cos𝜃cos\varphi {\gamma }_{xz,\eta }& \text{}\\ & & +{cos}^{2}𝜃{sin}^{2}\varphi {\gamma }_{yy,\eta }-2sin𝜃cos𝜃sin\varphi {\gamma }_{yz,\eta }+{sin}^{2}𝜃{\gamma }_{zz,\eta }\right)& \text{}\\ {\gamma }_{𝜃\varphi ,\eta }& =& \frac{2}{r}{\gamma }_{𝜃\varphi }+{r}^{2}sin𝜃\left(-cos𝜃sin\varphi cos\varphi {\gamma }_{xx,\eta }-cos𝜃\left({sin}^{2}\varphi -{cos}^{2}\varphi \right){\gamma }_{xy,\eta }+sin𝜃sin\varphi {\gamma }_{xz,\eta }& \text{}\\ & & +cos𝜃sin\varphi cos\varphi {\gamma }_{yy,\eta }-sin𝜃cos\varphi {\gamma }_{yz,\eta }\right)& \text{}\\ {\gamma }_{\varphi \varphi ,\eta }& =& \frac{2}{r}{\gamma }_{\varphi \varphi }+{r}^{2}{sin}^{2}𝜃\left({sin}^{2}\varphi {\gamma }_{xx,\eta }-2sin\varphi cos\varphi {\gamma }_{xy,\eta }+{cos}^{2}\varphi {\gamma }_{yy,\eta }\right)& \text{}\end{array}$

### 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

 ${\int }_{{x}_{1}}^{{x}_{2}}f\left(x\right)dx=\frac{h}{3}\left[{f}_{1}+4{f}_{2}+2{f}_{3}+4{f}_{4}+\dots +2{f}_{N-2}+4{f}_{N-1}+{f}_{N}\right]+O\left(1∕{N}^{4}\right)$ (32)

$N$ must be an odd number.

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