## WaveExtractCPM

March 13, 2018

### 1 Introduction

The WaveExtractCPM thorn uses the Cunningham-Price-Moncrief formalism [1213] to calculate ﬁrst order gauge invariant waveforms from a numerical spacetime. It relies on the basic assumption that the region of the spacetime where the extraction spheres are located is well-modelled as a linear perturbation to a Schwarzschild black hole. In addition to waveforms, the thorn can also compute other quantities such as mass, angular momentum and spin.

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

Consider a known, background solution to the Einstein equations ${g}_{\mu \nu }$. A ﬁrst-order perturbation to that metric, ${p}_{\mu \nu }$ yields

 ${g}_{\mu \nu }={g}_{\mu \nu }+{p}_{\mu \nu }\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|{p}_{\mu \nu }|\ll |{g}_{\mu \nu }|.$ (1)

We denote covariant derivatives with respect to the background metric ${g}_{\mu \nu }$ with ${\nabla }_{\mu }$ or ${}_{|\mu }$. Standard textbook analysis yields the ﬁrst-order vacuum ﬁeld equations in an unchosen gauge (deﬁning $\square {\equiv {}^{|\alpha }}_{}\alpha$ and $p\equiv {{p}^{\alpha }}_{\alpha }$)

 $-\square {p}_{\mu \nu }-{p}_{|\mu \nu }+{{p}^{\alpha }}_{\nu |\mu \alpha }+{p}_{\mu \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|\nu \alpha }^{\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\alpha }=0.$ (2)

Or, in Lorenz gauge (${{\text{}\stackrel{̄}{p}\text{}}_{\mu \nu |}}^{\nu }=0$)

 $\square {\text{}\stackrel{̄}{p}\text{}}_{\mu \nu }+2{R}_{\alpha \mu \beta \nu }{\text{}\stackrel{̄}{p}\text{}}^{\alpha \beta }=0,$ (3)

where an overbear indicates trace-reversal: ${\text{}\stackrel{̄}{p}\text{}}_{\mu \nu }={p}_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }p$.

#### 2.1 The ${\mathsc{ℳ}}^{2}×{\mathsc{𝒮}}^{2}$ decomposition in a spherically symmetric spacetime

Now we specialize to a spherically symmetric background. In this section we introduce formalism from [?] for doing a harmonic decomposition of scalar, vectors, and tensors in such a spacetime. We specialize to Schwarzschild spacetime with Schwarzschild coordinates and decompose its metric ${g}_{\mu \nu }$ on two submanifolds, yielding ${g}_{ab}$ and ${g}_{AB}={r}^{2}{\Omega }_{AB}$. Here $a,b,\dots \in \left\{0,1\right\}$ and $A,B,\dots \in \left\{𝜃,\varphi \right\}$. The ${x}^{a}$ coordinates span the “$\left(t,r\right)$ plane” while ${x}^{A}$ are the standard two-sphere polar and azimuthal coordinates. In matrix form we have

 ${g}_{\mu \nu }\doteq \left[\begin{array}{cccc}\hfill {g}_{00}\hfill & \hfill {g}_{01}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {g}_{10}\hfill & \hfill {g}_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}{\Omega }_{𝜃𝜃}\hfill & \hfill {r}^{2}{\Omega }_{𝜃\varphi }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}{\Omega }_{\varphi 𝜃}\hfill & \hfill {r}^{2}{\Omega }_{\varphi \varphi }\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill {g}_{00}\hfill & \hfill {g}_{01}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {g}_{10}\hfill & \hfill {g}_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}{sin}^{2}𝜃\hfill \end{array}\right].$ (4)

Speciﬁcally, we are interested in an expression of the Schwarzschild metric that is covariant under two-dimensional transformations: ${x}^{a}\to {x{}^{\prime }}^{a}$. The line element can be written as

 $d{s}^{2}={g}_{ab}\phantom{\rule{1em}{0ex}}d{x}^{a}d{x}^{b}+{r}^{2}{\Omega }_{AB}\phantom{\rule{1em}{0ex}}d{x}^{A}d{x}^{B}.$ (5)

In Schwarzschild coordinates, the submanifold ${\mathsc{ℳ}}^{2}$ has a metric and inverse

 ${g}_{ab}\doteq \left[\begin{array}{cc}\hfill -f\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1∕f\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{g}^{ab}\doteq \left[\begin{array}{cc}\hfill -1∕f\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill f\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}f\equiv 1-\frac{2M}{r},$ (6)

where $M$ is the mass of the system. The unit two-sphere has a metric and inverse

 ${\Omega }_{AB}\doteq \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {sin}^{2}𝜃\hfill \end{array}\right],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\Omega }^{AB}\doteq \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1∕{sin}^{2}𝜃\hfill \end{array}\right].$ (7)

Note that in general (oﬀ the unit two-sphere) we use the metric ${g}_{AB}\equiv {r}^{2}{\Omega }_{AB}$.

#### 2.2 Even parity

Of the ten MP amplitudes, seven are in the even-parity sector. Using the decomposition of Martel and Poisson [?], they are

$\begin{array}{lllllll}\hfill {p}_{ab}\left({x}^{\mu }\right)& =\sum _{\ell ,m}{h}_{ab}^{\ell m}{Y}^{\ell m},\phantom{\rule{2em}{0ex}}& \hfill {p}_{aB}\left({x}^{\mu }\right)& =\sum _{\ell ,m}{j}_{a}^{\ell m}{Y}_{B}^{\ell m},\phantom{\rule{2em}{0ex}}& \hfill {p}_{AB}\left({x}^{\mu }\right)& ={r}^{2}\sum _{\ell ,m}\left(\right{K}^{\ell m}{\Omega }_{AB}{Y}^{\ell m}+{G}^{\ell m}{Y}_{AB}^{\ell m}\left)\right.\phantom{\rule{2em}{0ex}}& \hfill \text{(8)}\end{array}$

The even-parity scalar (${Y}^{\ell m}$), vector (${Y}_{A}^{\ell m}$), and tensor (${Y}_{AB}^{\ell m}$ and ${\Omega }_{AB}{Y}^{\ell m}$) spherical harmonics are deﬁned in [?]. Note that ${Y}_{AB}^{\ell m}$ is the trace-free tensor spherical harmonic, which diﬀers from what Regge and Wheeler used in their original work [?].

If we have the metric perturbation, we can compute the amplitudes by using the completeness of the spherical harmonics. First in the ${\mathsc{ℳ}}^{2}$ sector,

 ${h}_{ab}^{\ell m}=\int {p}_{ab}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}d\Omega$ (9)
 ${j}_{a}^{\ell m}=\frac{{r}^{2}}{\ell \left(\ell +1\right)}\int {p}_{aB}{Ȳ}_{\ell m}^{B}d\Omega .$ (10)

$\begin{array}{lll}\hfill {K}^{\ell m}& =\frac{1}{2}\int {p}_{AB}{g}^{AB}{Ȳ}^{\ell m}d\Omega .\phantom{\rule{2em}{0ex}}& \hfill \text{(11)}\end{array}$ Lastly $\begin{array}{lll}\hfill \int {p}_{AB}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}^{AB}d\Omega & ={r}^{2}\sum _{\ell ,m}\int \left(\right{K}^{\ell m}{\Omega }_{AB}{Y}^{\ell m}+{G}^{\ell m}{Y}_{AB}^{\ell m}\left)\right{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}^{AB}d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(12)}\\ \hfill & ={r}^{2}\sum _{\ell ,m}{G}^{\ell m}\int {Y}_{AB}^{\ell m}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}^{AB}d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(13)}\\ \hfill & ={r}^{2}\sum _{\ell ,m}{G}^{\ell m}\frac{1}{2{r}^{4}}\left(\ell -1\right)\ell \left(\ell +1\right)\left(\ell +2\right){\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }},\phantom{\rule{2em}{0ex}}& \hfill \text{(14)}\end{array}$

and thus

 ${G}^{\ell m}=2{r}^{2}\frac{\left(\ell -2\right)!}{\left(\ell +2\right)!}\int {p}_{AB}{Ȳ}_{\ell m}^{AB}d\Omega$ (15)

We can now expand out the sums. See Appendix ?? for the components of the vector and tensor spherical harmonics. In going from the expressions above to these, note the inverse metric ${g}^{AB}$, which provides factors of $1∕{r}^{2}$ and $1∕{sin}^{2}𝜃$.

$\begin{array}{lll}\hfill {h}_{tt}^{\ell m}=\int {p}_{tt}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{h}_{tr}^{\ell m}=\int {p}_{tr}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{h}_{rr}^{\ell m}=\int {p}_{rr}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ,& \phantom{\rule{2em}{0ex}}& \hfill \text{(16)}\end{array}$

$\begin{array}{lll}\hfill {j}_{t}^{\ell m}& =\frac{{r}^{2}}{\ell \left(\ell +1\right)}\int \left({p}_{t𝜃}{Ȳ}_{\ell m}^{𝜃}+{p}_{t\varphi }{Ȳ}_{\ell m}^{\varphi }\right)d\Omega =\frac{1}{\ell \left(\ell +1\right)}\int \left({p}_{t𝜃}{Ȳ}_{𝜃}^{\ell m}+\frac{1}{{sin}^{2}𝜃}{p}_{t\varphi }{Ȳ}_{\varphi }^{\ell m}\right)d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(17)}\\ \hfill {j}_{r}^{\ell m}& =\frac{{r}^{2}}{\ell \left(\ell +1\right)}\int \left({p}_{r𝜃}{Ȳ}_{\ell m}^{𝜃}+{p}_{r\varphi }{Ȳ}_{\ell m}^{\varphi }\right)d\Omega =\frac{1}{\ell \left(\ell +1\right)}\int \left({p}_{r𝜃}{Ȳ}_{𝜃}^{\ell m}+\frac{1}{{sin}^{2}𝜃}{p}_{r\varphi }{Ȳ}_{\varphi }^{\ell m}\right)d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(18)}\end{array}$ $\begin{array}{lll}\hfill {K}^{\ell m}& =\frac{1}{2}\int \left({p}_{𝜃𝜃}{g}^{𝜃𝜃}+{p}_{\varphi \varphi }{g}^{\varphi \varphi }\right){Ȳ}^{\ell m}d\Omega =\frac{1}{2{r}^{2}}\int \left({p}_{𝜃𝜃}+\frac{1}{{sin}^{2}𝜃}{p}_{\varphi \varphi }\right){Ȳ}^{\ell m}d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(19)}\end{array}$

$\begin{array}{lll}\hfill {G}^{\ell m}& =2{r}^{2}\frac{\left(\ell -2\right)!}{\left(\ell +2\right)!}\int \left[{p}_{𝜃𝜃}{Ȳ}_{\ell m}^{𝜃𝜃}+2{p}_{𝜃\varphi }{Ȳ}_{\ell m}^{𝜃\varphi }+{p}_{\varphi \varphi }{Ȳ}_{\ell m}^{\varphi \varphi }\right]d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(20)}\\ \hfill & =\frac{2}{{r}^{2}}\frac{\left(\ell -2\right)!}{\left(\ell +2\right)!}\int \left[{p}_{𝜃𝜃}{Ȳ}_{𝜃𝜃}^{\ell m}+\frac{2}{{sin}^{2}𝜃}{p}_{𝜃\varphi }{Ȳ}_{𝜃\varphi }^{\ell m}+\frac{1}{{sin}^{4}𝜃}{p}_{\varphi \varphi }{Ȳ}_{\varphi \varphi }^{\ell m}\right]d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(21)}\end{array}$

For the remainder of this section, we drop $\ell$ and $m$ indices for the sake of brevity.

In Schwarzschild coordinates, the amplitudes deﬁned here are related to Regge and Wheeler’s original quantities. In the “$t,r$ sector,” ${h}_{tt}=f{H}_{0}$, ${h}_{tr}={H}_{1}$, and ${h}_{rr}={H}_{2}∕f$. For the oﬀ-diagonal elements, ${j}_{t}={h}_{0}$ and ${j}_{r}={h}_{1}$. Finally, on the two-sphere ${G}_{here}={G}_{RW}$, while ${K}_{here}={K}_{RW}-\ell \left(\ell +1\right)G∕2$.

In the even-parity sector there are four gauge-invariant ﬁelds, formed from linear combinations of the metric perturbation amplitudes and their ﬁrst derivatives [?]

$\begin{array}{lll}\hfill \begin{array}{rl}{\stackrel{̃}{h}}_{tt}& ={h}_{tt}-2{\partial }_{t}{j}_{t}+\frac{2Mf}{{r}^{2}}{j}_{r}+{r}^{2}{\partial }_{t}^{2}G-Mf{\partial }_{r}G\\ {\stackrel{̃}{h}}_{tr}& ={h}_{tr}-{\partial }_{r}{j}_{t}-{\partial }_{t}{j}_{r}+\frac{2M}{f{r}^{2}}{j}_{t}+{r}^{2}{\partial }_{t}{\partial }_{r}G+\frac{r-3M}{f}{\partial }_{t}G\\ {\stackrel{̃}{h}}_{rr}& ={h}_{rr}-2{\partial }_{r}{j}_{r}-\frac{2M}{f{r}^{2}}{j}_{r}+{r}^{2}{\partial }_{r}^{2}G+\frac{2r-3M}{f}{\partial }_{r}G\\ \stackrel{̃}{K}& =K-\frac{2f}{r}{j}_{r}+rf{\partial }_{r}G+\left(\lambda +1\right)G.\end{array}& \phantom{\rule{2em}{0ex}}& \hfill \text{(22)}\end{array}$

Note that in RW gauge $G={j}^{t}={j}^{r}=0$. Examining the gauge invariant quantities, we ﬁnd

 ${\stackrel{̃}{h}}_{tt}={h}_{tt},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{̃}{h}}_{tr}={h}_{tr},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{̃}{h}}_{rr}={h}_{rr},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\stackrel{̃}{K}=K.$ (23)

Written in terms of the gauge-invariant ﬁelds, the seven vacuum ﬁeld equations (for our purposes, we are deep in the wave zone and are not concerned with the isolated source) are

$\begin{array}{lll}\hfill -{\partial }_{r}^{2}\stackrel{̃}{K}-\frac{3r-5M}{{r}^{2}f}{\partial }_{r}\stackrel{̃}{K}+\frac{f}{r}{\partial }_{r}{\stackrel{̃}{h}}_{rr}+\frac{\left(\lambda +2\right)r+2M}{{r}^{3}}{\stackrel{̃}{h}}_{rr}+\frac{\lambda }{{r}^{2}f}\stackrel{̃}{K}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(24)}\\ \hfill {\partial }_{t}{\partial }_{r}\stackrel{̃}{K}+\frac{r-3M}{{r}^{2}f}{\partial }_{t}\stackrel{̃}{K}-\frac{f}{r}{\partial }_{t}{\stackrel{̃}{h}}_{rr}-\frac{\lambda +1}{{r}^{2}}{\stackrel{̃}{h}}_{tr}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(25)}\\ \hfill -{\partial }_{t}^{2}\stackrel{̃}{K}+\frac{\left(r-M\right)f}{{r}^{2}}{\partial }_{r}\stackrel{̃}{K}+\frac{2f}{r}{\partial }_{t}{\stackrel{̃}{h}}_{tr}-\frac{f}{r}{\partial }_{r}{\stackrel{̃}{h}}_{tt}+\frac{\left(\lambda +1\right)r+2M}{{r}^{3}}{\stackrel{̃}{h}}_{tt}-\frac{{f}^{2}}{{r}^{2}}{\stackrel{̃}{h}}_{rr}-\frac{\lambda f}{{r}^{2}}\stackrel{̃}{K}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(26)}\\ \hfill {\partial }_{t}{\stackrel{̃}{h}}_{rr}-{\partial }_{r}{\stackrel{̃}{h}}_{tr}+\frac{1}{f}{\partial }_{t}\stackrel{̃}{K}-\frac{2M}{{r}^{2}f}{\stackrel{̃}{h}}_{tr}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(27)}\\ \hfill -{\partial }_{t}{\stackrel{̃}{h}}_{tr}+{\partial }_{r}{\stackrel{̃}{h}}_{tt}-f{\partial }_{r}\stackrel{̃}{K}-\frac{r-M}{{r}^{2}f}{\stackrel{̃}{h}}_{tt}+\frac{\left(r-M\right)f}{{r}^{2}}{\stackrel{̃}{h}}_{rr}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(28)}\\ \hfill \begin{array}{rl}-{\partial }_{t}^{2}{\stackrel{̃}{h}}_{rr}+2{\partial }_{t}{\partial }_{r}{\stackrel{̃}{h}}_{tr}-{\partial }_{r}^{2}{\stackrel{̃}{h}}_{tt}-\frac{1}{f}{\partial }_{t}^{2}\stackrel{̃}{K}+f{\partial }_{r}^{2}\stackrel{̃}{K}\phantom{\rule{199.16928pt}{0ex}}& \\ +\frac{2\left(r-M\right)}{{r}^{2}f}{\partial }_{t}{\stackrel{̃}{h}}_{tr}-\frac{r-3M}{{r}^{2}f}{\partial }_{r}{\stackrel{̃}{h}}_{tt}-\frac{\left(r-M\right)f}{{r}^{2}}{\partial }_{r}{\stackrel{̃}{h}}_{rr}+\frac{2\left(r-M\right)}{{r}^{2}}{\partial }_{r}\stackrel{̃}{K}\phantom{\rule{71.13188pt}{0ex}}& \\ +\frac{\left(\lambda +1\right){r}^{2}-2\left(\lambda +2\right)Mr+2{M}^{2}}{{r}^{4}{f}^{2}}{\stackrel{̃}{h}}_{tt}-\frac{\left(\lambda +1\right){r}^{2}-2\lambda Mr-2{M}^{2}}{{r}^{4}}{\stackrel{̃}{h}}_{rr}& =0,\end{array}& \phantom{\rule{2em}{0ex}}& \hfill \text{(29)}\\ \hfill \frac{1}{f}{\stackrel{̃}{h}}_{tt}-f{\stackrel{̃}{h}}_{rr}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(30)}\end{array}$

where we have introduced

 $\Lambda \left(r\right)\equiv \lambda +\frac{3M}{r},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\lambda \equiv \frac{\left(\ell +2\right)\left(\ell -1\right)}{2}.$ (31)

We use the gauge invariant Zerilli-Moncrief master function (see [??], modifying the approach of [?]), which is

 ${\Psi }_{even}\left(t,r\right)\equiv \frac{r}{\lambda +1}\left[\stackrel{̃}{K}+\frac{f}{\Lambda }\left(f{\stackrel{̃}{h}}_{rr}-r{\partial }_{r}\stackrel{̃}{K}\right)\right],$ (32)

in Schwarzschild coordinates. Plugging in the gauge invariant ﬁelds from above, one ﬁnds

 ${\Psi }_{even}\left(t,r\right)=rG-\frac{2f}{\Lambda }{j}_{r}+\frac{r}{\lambda +1}\left[K+\frac{f}{\Lambda }\left(f{h}_{rr}-r{\partial }_{r}K\right)\right]$ (33)

Conveniently, all the second-order derivatives cancel. We are also interested in the time derivative of the master function, which is used for computing energy and angular momentum ﬂuxes. We diﬀerentiate Eq. (32) with respect to time, using Eq. (25) to remove the ${\partial }_{t}{\partial }_{r}\stackrel{̃}{K}$ terms. Then, substituting in the gauge invariant ﬁelds we ﬁnd

 ${\partial }_{t}{\Psi }_{even}\left(t,r\right)=r{\partial }_{t}G+\frac{1}{\Lambda }\left[-f{h}_{tr}-\frac{2M}{{r}^{2}}{j}_{t}+f{\partial }_{r}{j}_{t}+r{\partial }_{t}K-f{\partial }_{t}{j}_{r}\right].$ (34)

Again, simpliﬁcation happens and we are left with at most ﬁrst-order derivatives of the MP amplitudes.

#### 2.3 Odd parity

The remaining three MP amplitudes belong to the odd-parity sector,

$\begin{array}{lll}\hfill {p}_{ab}\left({x}^{\mu }\right)=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{p}_{aB}\left({x}^{\mu }\right)=\sum _{\ell ,m}{h}_{a}^{\ell m}{X}_{B}^{\ell m},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{p}_{AB}\left({x}^{\mu }\right)=\sum _{\ell ,m}{h}_{2}^{\ell m}{X}_{AB}^{\ell m}.& \phantom{\rule{2em}{0ex}}& \hfill \text{(35)}\end{array}$

The vector (${X}_{B}^{\ell m}$) and tensor (${X}_{AB}^{\ell m}$) spherical harmonics are those deﬁned in [?]. Note that the tensor spherical harmonics diﬀer from those used by Regge and Wheeler by a minus sign.

If we have the metric perturbation, we can compute the amplitudes by using the completeness of the spherical harmonics

 ${h}_{a}^{\ell m}=\frac{{r}^{2}}{\ell \left(\ell +1\right)}\int {p}_{aB}{\stackrel{̄}{X}}_{\ell m}^{B}d\Omega .$ (36)
 ${h}_{2}^{\ell m}=2{r}^{4}\frac{\left(\ell -2\right)!}{\left(\ell +2\right)!}\int {p}_{AB}{\stackrel{̄}{X}}_{\ell m}^{AB}d\Omega$ (37)

We can now expand out the sums. See Appendix ?? for the components of the vector and tensor spherical harmonics. In going from the expressions above to these, note the inverse metric ${g}^{AB}$, which provides factors of $1∕{r}^{2}$ and $1∕{sin}^{2}𝜃$.

$\begin{array}{lll}\hfill {h}_{t}^{\ell m}& =\frac{{r}^{2}}{\ell \left(\ell +1\right)}\int \left({p}_{t𝜃}{\stackrel{̄}{X}}_{\ell m}^{𝜃}+{p}_{t\varphi }{\stackrel{̄}{X}}_{\ell m}^{\varphi }\right)d\Omega =\frac{1}{\ell \left(\ell +1\right)}\int \frac{1}{sin𝜃}\left(-{p}_{t𝜃}{Ȳ}_{,\varphi }^{\ell m}+{p}_{t\varphi }{Ȳ}_{,𝜃}^{\ell m}\right)d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(38)}\\ \hfill {h}_{r}^{\ell m}& =\frac{{r}^{2}}{\ell \left(\ell +1\right)}\int \left({p}_{r𝜃}{\stackrel{̄}{X}}_{\ell m}^{𝜃}+{p}_{r\varphi }{\stackrel{̄}{X}}_{\ell m}^{\varphi }\right)d\Omega =\frac{1}{\ell \left(\ell +1\right)}\int \frac{1}{sin𝜃}\left(-{p}_{r𝜃}{Ȳ}_{,\varphi }^{\ell m}+{p}_{r\varphi }{Ȳ}_{,𝜃}^{\ell m}\right)d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(39)}\end{array}$

$\begin{array}{lll}\hfill {h}_{2}^{\ell m}& =2{r}^{4}\frac{\left(\ell -2\right)!}{\left(\ell +2\right)!}\int \left({p}_{𝜃𝜃}{\stackrel{̄}{X}}_{\ell m}^{𝜃𝜃}+2{p}_{𝜃\varphi }{\stackrel{̄}{X}}_{\ell m}^{𝜃\varphi }+{p}_{\varphi \varphi }{\stackrel{̄}{X}}_{\ell m}^{\varphi \varphi }\right)d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(40)}\\ \hfill & =2\frac{\left(\ell -2\right)!}{\left(\ell +2\right)!}\int \frac{1}{sin𝜃}\left[\right{p}_{𝜃𝜃}\left(\frac{cos𝜃}{sin𝜃}{Ȳ}_{,\varphi }^{\ell m}-{Ȳ}_{,𝜃\varphi }^{\ell m}\right)-{p}_{𝜃\varphi }\left(\frac{1}{{sin}^{2}𝜃}{Ȳ}_{,\varphi \varphi }^{\ell m}+\frac{cos𝜃}{sin𝜃}{Ȳ}_{,𝜃}^{\ell m}-{Ȳ}_{,𝜃𝜃}^{\ell m}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{236.80481pt}{0ex}}+{p}_{\varphi \varphi }\left(\frac{1}{{sin}^{2}𝜃}{Ȳ}_{,\varphi 𝜃}^{\ell m}-\frac{cos𝜃}{{sin}^{3}𝜃}{Ȳ}_{,\varphi }^{\ell m}\right)\left]\rightd\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(41)}\end{array}$

For the remainder of this section, we again drop $\ell$ and $m$ indices.

These MP amplitudes are related to Regge and Wheeler’s quantities through ${h}_{t}={h}_{0}$, ${h}_{r}={h}_{1}$, and ${h}_{2}^{here}=-{h}_{2}^{RW}$.

In the odd-parity sector there are two gauge-invariant ﬁelds, formed from linear combinations of the metric perturbation amplitudes and their ﬁrst derivatives [?]

 ${\stackrel{̃}{h}}_{t}\equiv {h}_{t}-\frac{1}{2}\frac{\partial {h}_{2}}{\partial t},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{̃}{h}}_{r}\equiv {h}_{r}-\frac{1}{2}\frac{\partial {h}_{2}}{\partial r}+\frac{{h}_{2}}{r}.$ (42)

Note that in Regge-Wheeler gauge ${h}_{2}=0$ and then

 ${\stackrel{̃}{h}}_{t}\equiv {h}_{t},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{̃}{h}}_{r}\equiv {h}_{r}.$ (43)

Written in terms of the gauge-invariant ﬁelds, the three vacuum ﬁeld equations (for our purposes, we are deep in the wave zone and are not concerned with the isolated source) are

$\begin{array}{lll}\hfill -{\partial }_{t}{\partial }_{r}{\stackrel{̃}{h}}_{r}+{\partial }_{r}^{2}{\stackrel{̃}{h}}_{t}-\frac{2}{r}{\partial }_{t}{\stackrel{̃}{h}}_{r}-\frac{2\left(\lambda +1\right)r-4M}{{r}^{3}f}{\stackrel{̃}{h}}_{t}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(44)}\\ \hfill {\partial }_{t}^{2}{\stackrel{̃}{h}}_{r}-{\partial }_{t}{\partial }_{r}{\stackrel{̃}{h}}_{t}+\frac{2}{r}{\partial }_{t}{\stackrel{̃}{h}}_{t}+\frac{2\lambda f}{{r}^{2}}{\stackrel{̃}{h}}_{r}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(45)}\\ \hfill -\frac{1}{f}{\partial }_{t}{\stackrel{̃}{h}}_{t}+f{\partial }_{r}{\stackrel{̃}{h}}_{r}+\frac{2M}{{r}^{2}}{\stackrel{̃}{h}}_{r}& =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(46)}\end{array}$

where, recall that we have deﬁned

 $\lambda \equiv \frac{\left(\ell +2\right)\left(\ell -1\right)}{2}.$ (47)

In the odd-parity sector, we use the gauge-invariant Cunningham-Price-Moncrief master function [?], which in Schwarzschild coordinates is

 ${\Psi }_{odd}\left(t,r\right)\equiv \frac{r}{\lambda }\left[{\partial }_{r}{\stackrel{̃}{h}}_{t}-{\partial }_{t}{\stackrel{̃}{h}}_{r}-\frac{2}{r}{\stackrel{̃}{h}}_{t}\right].$ (48)

Plugging in the gauge invariant ﬁelds from above, we ﬁnd that all the ${h}_{2}$ terms cancel and the result is simply

 ${\Psi }_{odd}\left(t,r\right)\equiv \frac{r}{\lambda }\left[{\partial }_{r}{h}_{t}-{\partial }_{t}{h}_{r}-\frac{2}{r}{h}_{t}\right].$ (49)

We are also interested in the time derivative of the master function, which is used for computing energy and angular momentum ﬂuxes. We diﬀerentiate Eq. (48) with respect to time, using Eq. (45) to remove the ${\partial }_{t}{\partial }_{r}{\stackrel{̃}{h}}_{t}$ and ${\partial }_{t}^{2}{\stackrel{̃}{h}}_{r}$ terms. Then, substituting in the gauge invariant ﬁelds we ﬁnd

 ${\partial }_{t}{\Psi }_{odd}=\frac{f}{r}\left[2{h}_{r}+\frac{2}{r}{h}_{2}-{\partial }_{r}{h}_{2}\right].$ (50)

In RW gauge where ${h}_{2}$ vanishes it becomes clear that the CPM function ${\Psi }_{odd}$ is just twice the time integral of the original RW function ${\Psi }_{RW}=f{h}_{r}∕r$.

#### 2.4 Wave Forms

The energy and angular momentum ﬂuxes, for each $\ell ,m$ mode, can be written as [?]

$\begin{array}{lll}\hfill {Ė}_{\ell m}& =\frac{1}{64\pi }\frac{\left(\ell +2\right)!}{\left(\ell -2\right)!}\left({\left|{\stackrel{̇}{\Psi }}_{even}^{\ell m}\right|}^{2}+{\left|{\stackrel{̇}{\Psi }}_{odd}^{\ell m}\right|}^{2}\right),\phantom{\rule{2em}{0ex}}& \hfill \text{(51)}\\ \hfill {\stackrel{̇}{L}}_{\ell m}& =\frac{im}{64\pi }\frac{\left(\ell +2\right)!}{\left(\ell -2\right)!}\left({\stackrel{̇}{\Psi }}_{even}^{\ell m}{\Psi }_{\ell m}^{±\phantom{\rule{0.3em}{0ex}}\ast }+{\stackrel{̇}{\Psi }}_{odd}^{\ell m}{\Psi }_{\ell m}^{±\phantom{\rule{0.3em}{0ex}}\ast }\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(52)}\end{array}$

Here, an asterisk signiﬁes complex conjugation. Assume a spacetime ${g}_{\alpha \beta }$ can be written as a Schwarzschild background ${g}_{\alpha \beta }^{\left(0\right)}$ with perturbation ${h}_{\alpha \beta }$,

 ${g}_{\alpha \beta }={g}_{\alpha \beta }^{\left(0\right)}+{h}_{\alpha \beta }.$ (53)

In spherical coordinates, $\left(t,r,𝜃,\varphi \right)$, the background metric is given by

 ${g}^{\left(0\right)}=\left(\begin{array}{cccc}\hfill -f\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {f}^{-1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {r}^{2}{sin}^{2}𝜃\hfill \end{array}\right),\phantom{\rule{2em}{0ex}}f\left(r\right)=1-\frac{2M}{r}.$ (54)

The 3-metric perturbations ${\gamma }_{ij}$ can be decomposed using tensor spherical harmonics to obtain a set of metric perturbation amplitudes ${\gamma }_{ij}^{lm}\left(t,r\right)$, 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

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

with ${V}_{k}$ being a basis for tensors on a 2-sphere in 3-D Euclidean space.

Working with the Regge-Wheeler basis (see Appendix B) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions ${c}_{1}^{×lm}$, ${c}_{2}^{×lm}$, ${h}_{1}^{+lm}$, ${H}_{2}^{+lm}$, ${K}^{+lm}$, ${G}^{+lm}$ [1916]. Each of these functions is either odd ($×$) or even ($+$) parity. The decomposition is then written

$\begin{array}{rcll}{\gamma }_{ij}^{lm}& =& {c}_{1}^{×lm}{\left({ê}_{1}\right)}_{ij}^{lm}+{c}_{2}^{×lm}{\left({ê}_{2}\right)}_{ij}^{lm}& \text{}\\ & +& {h}_{1}^{+lm}{\left({\stackrel{̂}{f}}_{1}\right)}_{ij}^{lm}+{A}^{2}{H}_{2}^{+lm}{\left({\stackrel{̂}{f}}_{2}\right)}_{ij}^{lm}+{R}^{2}{K}^{+lm}{\left({\stackrel{̂}{f}}_{3}\right)}_{ij}^{lm}+{R}^{2}{G}^{+lm}{\left({\stackrel{̂}{f}}_{4}\right)}_{ij}^{lm},& \text{(55)}\text{}\text{}\end{array}$

which we can write in an expanded form as

$\begin{array}{lll}\hfill {\gamma }_{rr}^{lm}& ={A}^{2}{H}_{2}^{+lm}{Y}_{lm},\phantom{\rule{2em}{0ex}}& \hfill \text{(56)}\\ \hfill {\gamma }_{r𝜃}^{lm}& =-{c}_{1}^{×lm}\frac{1}{sin𝜃}{Y}_{lm,\varphi }+{h}_{1}^{+lm}{Y}_{lm,𝜃},\phantom{\rule{2em}{0ex}}& \hfill \text{(57)}\\ \hfill {\gamma }_{r\varphi }^{lm}& ={c}_{1}^{×lm}sin𝜃{Y}_{lm,𝜃}+{h}_{1}^{+lm}{Y}_{lm,\varphi },\phantom{\rule{2em}{0ex}}& \hfill \text{(58)}\\ \hfill {\gamma }_{𝜃𝜃}^{lm}& ={c}_{2}^{×lm}\frac{1}{sin𝜃}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)+{R}^{2}{K}^{+lm}{Y}_{lm}+{R}^{2}{G}^{+lm}{Y}_{lm,𝜃𝜃},\phantom{\rule{2em}{0ex}}& \hfill \text{(59)}\\ \hfill {\gamma }_{𝜃\varphi }^{lm}& =-{c}_{2}^{×lm}sin𝜃\frac{1}{2}\left({Y}_{lm,𝜃𝜃}-cot𝜃{Y}_{lm,𝜃}-\frac{1}{{sin}^{2}𝜃}{Y}_{lm}\right)+{R}^{2}{G}^{+lm}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right),\phantom{\rule{2em}{0ex}}& \hfill \text{(60)}\\ \hfill {\gamma }_{\varphi \varphi }^{lm}& =-sin𝜃{c}_{2}^{×lm}\left({Y}_{lm,𝜃\varphi }-cot𝜃{Y}_{lm,\varphi }\right)+{R}^{2}{K}^{+lm}{sin}^{2}𝜃{Y}_{lm}+{R}^{2}{G}^{+lm}\left({Y}_{lm,\varphi \varphi }+sin𝜃cos𝜃{Y}_{lm,𝜃}\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(61)}\end{array}$

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

$\begin{array}{rcll}{g}_{tt}^{lm}& =& {N}^{2}{H}_{0}^{+lm}{Y}_{lm},& \text{(62)}\text{}\text{}\\ {g}_{tr}^{lm}& =& {H}_{1}^{+lm}{Y}_{lm},& \text{(63)}\text{}\text{}\\ {g}_{t𝜃}^{lm}& =& {h}_{0}^{+lm}{Y}_{lm,𝜃}-{c}_{0}^{×lm}\frac{1}{sin𝜃}{Y}_{lm,\varphi },& \text{(64)}\text{}\text{}\\ {g}_{t\varphi }^{lm}& =& {h}_{0}^{+lm}{Y}_{lm,\varphi }+{c}_{0}^{×lm}sin𝜃{Y}_{lm,𝜃}.& \text{(65)}\text{}\text{}\end{array}$

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

 ${\alpha }^{lm}=-\frac{1}{2}N{H}_{0}^{+lm}{Y}_{lm}.$ (66)

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

${g}_{\alpha \beta }={g}_{\alpha \beta }^{\left(0\right)}+\sum _{l,m}{h}_{\alpha \beta }^{lm,odd}+\sum _{l,m}{h}_{\alpha \beta }^{lm,even}$

where (dropping some superscripts)

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

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 [3] are

$\begin{array}{rcll}{Q}_{lm}^{×}& =& \sqrt{\frac{2\left(l+2\right)!}{\left(l-2\right)!}}\left[{c}_{1}^{×lm}+\frac{1}{2}\left({\partial }_{r}{c}_{2}^{×lm}-\frac{2}{r}{c}_{2}^{×lm}\right)\right]\frac{S}{r}& \text{(67)}\text{}\text{}\\ {Q}_{lm}^{+}& =& \frac{1}{\Lambda }\sqrt{\frac{2\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}}\left(4r{S}^{2}{k}_{2}+l\left(l+1\right)r{k}_{1}\right)& \text{(68)}\text{}\text{}\\ & \equiv & \frac{1}{\Lambda }\sqrt{\frac{2\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}}\left(l\left(l+1\right)S\left({r}^{2}{\partial }_{r}{G}^{+lm}-2{h}_{1}^{+lm}\right)+2rS\left({H}_{2}^{+lm}-r{\partial }_{r}{K}^{+lm}\right)+\Lambda r{K}^{+lm}\right)& \text{(69)}\text{}\text{}\end{array}$

where

$\begin{array}{rcll}{k}_{1}& =& {K}^{+lm}+\frac{S}{r}\left({r}^{2}{\partial }_{r}{G}^{+lm}-2{h}_{1}^{+lm}\right)& \text{(70)}\text{}\text{}\\ {k}_{2}& =& \frac{1}{2S}\left[{H}_{2}^{+lm}-r{\partial }_{r}{k}_{1}-\left(1-\frac{M}{rS}\right){k}_{1}+{S}^{1∕2}{\partial }_{r}\left({r}^{2}{S}^{1∕2}{\partial }_{r}{G}^{+lm}-2{S}^{1∕2}{h}_{1}^{+lm}\right)\right]& \text{(71)}\text{}\text{}\\ & \equiv & \frac{1}{2S}\left[{H}_{2}-r{K}_{,r}-\frac{r-3M}{r-2M}K\right]& \text{(72)}\text{}\text{}\end{array}$

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

 $\stackrel{̂}{r}=\stackrel{̂}{r}\left(r\right)={\left[\frac{1}{4\pi }\int \sqrt{{\gamma }_{𝜃𝜃}{\gamma }_{\varphi \varphi }}d𝜃d\varphi \right]}^{1∕2}$ (73)

from which

 $\frac{d\stackrel{̂}{r}}{d\eta }=\frac{1}{16\pi \stackrel{̂}{r}}\int \frac{{\gamma }_{𝜃𝜃,\eta }{\gamma }_{\varphi \varphi }+{\gamma }_{𝜃𝜃}{\gamma }_{\varphi \varphi ,\eta }}{\sqrt{{\gamma }_{𝜃𝜃}{\gamma }_{\varphi \varphi }}}\phantom{\rule{1em}{0ex}}d𝜃d\varphi$ (74)

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

 $S\left(\stackrel{̂}{r}\right)={\left(\frac{\partial \stackrel{̂}{r}}{\partial r}\right)}^{2}\int {\gamma }_{rr}\phantom{\rule{1em}{0ex}}d𝜃d\varphi$ (75)
 $M\left(\stackrel{̂}{r}\right)=\stackrel{̂}{r}\frac{1-S}{2}$ (76)

#### 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

$\begin{array}{rcll}{\gamma }_{\stackrel{̂}{r}\stackrel{̂}{r}}& =& \frac{\partial r}{\partial \stackrel{̂}{r}}\frac{\partial r}{\partial \stackrel{̂}{r}}{\gamma }_{rr}& \text{(77)}\text{}\text{}\\ {\gamma }_{\stackrel{̂}{r}𝜃}& =& \frac{\partial r}{\partial \stackrel{̂}{r}}{\gamma }_{r𝜃}& \text{(78)}\text{}\text{}\\ {\gamma }_{\stackrel{̂}{r}\varphi }& =& \frac{\partial r}{\partial \stackrel{̂}{r}}{\gamma }_{r\varphi }& \text{(79)}\text{}\text{}\end{array}$

#### 3.5 Calculate Gauge Invariant Quantities

$\begin{array}{rcll}{Q}_{lm}^{×}& =& \sqrt{\frac{2\left(l+2\right)!}{\left(l-2\right)!}}\left[{c}_{1}^{×lm}+\frac{1}{2}\left({\partial }_{\stackrel{̂}{r}}{c}_{2}^{×lm}-\frac{2}{\stackrel{̂}{r}}{c}_{2}^{×lm}\right)\right]\frac{S}{\stackrel{̂}{r}}& \text{(80)}\text{}\text{}\\ {Q}_{lm}^{+}& =& \frac{1}{\left(l-1\right)\left(l+2\right)+6M∕\stackrel{̂}{r}}\sqrt{\frac{2\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}}\left(4\stackrel{̂}{r}{S}^{2}{k}_{2}+l\left(l+1\right)\stackrel{̂}{r}{k}_{1}\right)& \text{(81)}\text{}\text{}\end{array}$

where

$\begin{array}{rcll}{k}_{1}& =& {K}^{+lm}+\frac{S}{\stackrel{̂}{r}}\left({\stackrel{̂}{r}}^{2}{\partial }_{\stackrel{̂}{r}}{G}^{+lm}-2{h}_{1}^{+lm}\right)& \text{(82)}\text{}\text{}\\ {k}_{2}& =& \frac{1}{2S}\left[{H}_{2}^{+lm}-\stackrel{̂}{r}{\partial }_{\stackrel{̂}{r}}{k}_{1}-\left(1-\frac{M}{\stackrel{̂}{r}S}\right){k}_{1}+{S}^{1∕2}{\partial }_{\stackrel{̂}{r}}\left({\stackrel{̂}{r}}^{2}{S}^{1∕2}{\partial }_{\stackrel{̂}{r}}{G}^{+lm}-2{S}^{1∕2}{h}_{1}^{+lm}& \text{(83)}\text{}\text{}\end{array}$

### 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.

#### 4.1 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

This document (and the WaveExtractCPM thorn itself) are based on the Extract thorn writen by Gabrielle Allen. 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 comparison with linear evolutions of waveforms extracted from the Cauchy initial data. This work was carried out in collaboration with Karen Camarda and Ed Seidel.

### A Spherical harmonics

Now, consider spherical harmonics, starting with the scalar case. They are eigenfunctions, satisfying the equation

 $\left[\frac{1}{sin𝜃}{\partial }_{𝜃}\left(sin𝜃\cdot {\partial }_{𝜃}\right)+\frac{1}{{sin}^{2}𝜃}{\partial }_{\varphi }^{2}+\ell \left(\ell +1\right)\right]{Y}_{\ell m}\left(𝜃,\varphi \right)=0.$ (84)

Acting on a test scalar function $f$ we have

$\begin{array}{lll}\hfill {\Omega }^{AB}{D}_{A}{D}_{B}f& ={\Omega }^{AB}\left({\partial }_{A}{\partial }_{B}-{{\Gamma }^{C}}_{AB}{\partial }_{C}\right)f\phantom{\rule{2em}{0ex}}& \hfill \text{(85)}\\ \hfill & ={\partial }_{𝜃}^{2}f-{{\Gamma }^{C}}_{𝜃𝜃}{\partial }_{C}f+\frac{1}{{sin}^{2}𝜃}{\partial }_{\varphi }^{2}f-\frac{1}{{sin}^{2}𝜃}{{\Gamma }^{C}}_{\varphi \varphi }{\partial }_{C}f\phantom{\rule{2em}{0ex}}& \hfill \text{(86)}\\ \hfill & =\left(\frac{1}{sin𝜃}{\partial }_{𝜃}\left(sin𝜃\cdot {\partial }_{𝜃}\right)+\frac{1}{{sin}^{2}𝜃}{\partial }_{\varphi }^{2}\right)f.\phantom{\rule{2em}{0ex}}& \hfill \text{(87)}\end{array}$

So, we can write Eq. (84) in the compact form

 $\left[{\Omega }^{AB}{D}_{A}{D}_{B}+\ell \left(\ell +1\right)\right]{Y}_{\ell m}\left(𝜃,\varphi \right)=0.$ (88)

The solution to this equation with standard normalization [?] is

 ${Y}_{\ell m}=\sqrt{\frac{2\ell +1}{4\pi }\frac{\left(\ell -m\right)!}{\left(\ell +m\right)!}}\phantom{\rule{1em}{0ex}}{P}_{\ell }^{m}\left(cos𝜃\right){e}^{im\varphi }$ (89)

where ${P}_{\ell }^{m}$ are the associated Legendre functions. These are an orthonormal set of functions,

 $\int {Y}_{\ell m}\left(𝜃,\varphi \right){Ȳ}_{{\ell }^{\prime }{m}^{\prime }}\left(𝜃,\varphi \right)d\Omega ={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.$ (90)

Here $d\Omega =sin𝜃\phantom{\rule{1em}{0ex}}d𝜃\phantom{\rule{1em}{0ex}}d\varphi$ and the overbar represents complex conjugation.

We can use the covariant derivative ${D}_{A}$ to take derivatives of this scalar function to deﬁne vector and tensor spherical harmonics. There are even- and odd-parity vector spherical harmonics. We deﬁne the even ones as the covariant derivative of the scalar harmonics:

 ${Y}_{A}^{\ell m}\left(𝜃,\varphi \right)\equiv {D}_{A}{Y}^{\ell m}\left(𝜃,\varphi \right)\doteq \left[\begin{array}{c}\hfill {\partial }_{𝜃}{Y}_{\ell m}\hfill \\ \hfill {\partial }_{\varphi }{Y}_{\ell m}\hfill \end{array}\right].$ (91)

In order to create the odd-parity vectorial harmonics we need to deﬁne the Levi-Civita tensor on the two-sphere:

 ${𝜀}_{AB}\doteq \left[\begin{array}{cc}\hfill 0\hfill & \hfill sin𝜃\hfill \\ \hfill -sin𝜃\hfill & \hfill 0\hfill \end{array}\right].$ (92)

Using this, the odd-parity harmonics are

 ${X}_{A}^{\ell m}\left(𝜃,\varphi \right)\equiv -{{𝜀}_{A}}^{B}{D}_{B}{Y}^{\ell m}\left(𝜃,\varphi \right)=-{\Omega }^{CB}{𝜀}_{AC}{Y}_{B}^{\ell m}\left(𝜃,\varphi \right).$ (93)

Switching to matrices we can calculate the components:

$\begin{array}{lll}\hfill {X}_{A}^{\ell m}\left(𝜃,\varphi \right)& \doteq -\left[\begin{array}{cc}\hfill 0\hfill & \hfill sin𝜃\hfill \\ \hfill -sin𝜃\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1∕{sin}^{2}𝜃\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\partial }_{𝜃}{Y}_{\ell m}\hfill \\ \hfill {\partial }_{\varphi }{Y}_{\ell m}\hfill \end{array}\right]\phantom{\rule{2em}{0ex}}& \hfill \text{(94)}\\ \hfill & \doteq \left[\begin{array}{c}\hfill -{\partial }_{\varphi }{Y}_{\ell m}∕sin𝜃\hfill \\ \hfill sin𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}{Y}_{\ell m}\hfill \end{array}\right].\phantom{\rule{2em}{0ex}}& \hfill \text{(95)}\end{array}$

The tensor spherical harmonics also are either even- and odd-parity. There are two even-parity ones,

 ${Y}_{\ell m}{\Omega }_{AB}\doteq \left[\begin{array}{cc}\hfill {Y}_{\ell m}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {sin}^{2}𝜃{Y}_{\ell m}\hfill \end{array}\right]$ (96)

and the more complicated

$\begin{array}{lll}\hfill {Y}_{AB}^{\ell m}& \equiv \left[{D}_{A}{D}_{B}+\frac{1}{2}\ell \left(\ell +1\right){\Omega }_{AB}\right]{Y}_{\ell m}\phantom{\rule{2em}{0ex}}& \hfill \text{(97)}\\ \hfill & ={\partial }_{A}{\partial }_{B}{Y}_{\ell m}-{{\Gamma }^{C}}_{AB}{\partial }_{C}{Y}_{\ell m}+\frac{1}{2}\ell \left(\ell +1\right){\Omega }_{AB}{Y}_{\ell m}.\phantom{\rule{2em}{0ex}}& \hfill \text{(98)}\end{array}$

We’ve already calculated the connection coeﬃcients, so evaluating this is straightforward, leaving us with the components

 ${Y}_{AB}^{\ell m}\doteq \left[\begin{array}{cc}\hfill \left({\partial }_{𝜃}^{2}+\frac{\ell \left(\ell +1\right)}{2}\right){Y}_{\ell m}\hfill & \hfill \left({\partial }_{𝜃}{\partial }_{\varphi }-cot𝜃\phantom{\rule{1em}{0ex}}{\partial }_{\varphi }\right){Y}_{\ell m}\hfill \\ \hfill \left({\partial }_{𝜃}{\partial }_{\varphi }-cot𝜃\phantom{\rule{1em}{0ex}}{\partial }_{\varphi }\right){Y}_{\ell m}\hfill & \hfill \left({\partial }_{\varphi }^{2}+sin𝜃cos𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}+\frac{\ell \left(\ell +1\right)}{2}{sin}^{2}𝜃\right){Y}_{\ell m}\hfill \end{array}\right].$ (99)

The odd-parity tensor harmonics are

$\begin{array}{lll}\hfill {X}_{AB}^{\ell m}& =-\frac{1}{2}\left[{{𝜀}_{A}}^{C}{D}_{B}+{{𝜀}_{B}}^{C}{D}_{A}\right]{D}_{C}{Y}_{\ell m}\phantom{\rule{2em}{0ex}}& \hfill \text{(100)}\\ \hfill & =-\frac{1}{2}\left[{{𝜀}_{A}}^{𝜃}{D}_{B}{D}_{𝜃}+{{𝜀}_{A}}^{\varphi }{D}_{B}{D}_{\varphi }+{{𝜀}_{B}}^{\varphi }{D}_{A}{D}_{𝜃}+{{𝜀}_{B}}^{\varphi }{D}_{A}{D}_{\varphi }\right]{Y}_{\ell m}.\phantom{\rule{2em}{0ex}}& \hfill \text{(101)}\end{array}$

In matrix form we have

 ${X}_{AB}^{\ell m}\doteq \left[\begin{array}{cc}\hfill \left(-\frac{1}{sin𝜃}{\partial }_{𝜃}{\partial }_{\varphi }+\frac{cos𝜃}{{sin}^{2}𝜃}{\partial }_{\varphi }\right){Y}_{\ell m}\hfill & \hfill -\frac{1}{2}\left(\frac{{\partial }_{\varphi }^{2}}{sin𝜃}+cos𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}-sin𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}^{2}\right){Y}_{\ell m}\hfill \\ \hfill -\frac{1}{2}\left(\frac{{\partial }_{\varphi }^{2}}{sin𝜃}+cos𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}-sin𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}^{2}\right){Y}_{\ell m}\hfill & \hfill \left(sin𝜃\phantom{\rule{1em}{0ex}}{\partial }_{\varphi }{\partial }_{𝜃}-cos𝜃\phantom{\rule{1em}{0ex}}{\partial }_{\varphi }\right){Y}_{\ell m}\hfill \end{array}\right].$ (102)

Now we look at some identities involving these spherical harmonics. We have already seen in Eq. (90) that the scalar spherical harmonics are orthonormal. Now consider

$\begin{array}{lll}\hfill \int {Y}_{\ell m}^{A}{Ȳ}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =\frac{1}{{r}^{2}}\int {\Omega }^{AB}{D}_{A}{Y}_{\ell m}{D}_{B}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(103)}\\ \hfill & =\frac{1}{{r}^{2}}\int \left({\partial }_{𝜃}{Y}_{\ell m}{\partial }_{𝜃}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}+\frac{1}{{sin}^{2}𝜃}{\partial }_{\varphi }{Y}_{\ell m}{\partial }_{\varphi }{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}\right)sin𝜃\phantom{\rule{1em}{0ex}}d𝜃\phantom{\rule{1em}{0ex}}d\varphi .\phantom{\rule{2em}{0ex}}& \hfill \text{(104)}\end{array}$

We integrate by parts (note that surface terms vanish by periodicity as we integrate of the full $4\pi$ steradians) and ﬁnd

$\begin{array}{llll}\hfill \int {Y}_{\ell m}^{A}{Ȳ}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =\frac{1}{{r}^{2}}\int \left[\right-\frac{1}{sin𝜃}{\partial }_{𝜃}\left(sin𝜃\phantom{\rule{1em}{0ex}}{\partial }_{𝜃}{Y}_{\ell m}\right){Ȳ}_{{\ell }^{\prime }{m}^{\prime }}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{129.16626pt}{0ex}}-\frac{1}{{sin}^{2}𝜃}{\partial }_{\varphi }^{2}{Y}_{\ell m}{Ȳ}_{{\ell }^{\prime }{m}^{\prime }}\left]\rightsin𝜃\phantom{\rule{1em}{0ex}}d𝜃\phantom{\rule{1em}{0ex}}d\varphi \phantom{\rule{2em}{0ex}}& \hfill \text{(105)}\\ \hfill & =\frac{1}{{r}^{2}}\ell \left(\ell +1\right){\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.\phantom{\rule{2em}{0ex}}& \hfill \text{(106)}\end{array}$

The odd-parity equivalent is

$\begin{array}{lll}\hfill \int {X}_{\ell m}^{A}{\stackrel{̄}{X}}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =\int {{𝜀}^{A}}_{C}{Y}_{\ell m}^{C}{{𝜀}_{A}}^{B}{Ȳ}_{B}^{{\ell }^{\prime }{m}^{\prime }}d\Omega .\phantom{\rule{2em}{0ex}}& \hfill \text{(107)}\end{array}$

This 2D contraction of the Levi-Civita tensor gives the negative of the Kronecker delta, and therefore

$\begin{array}{lll}\hfill \int {X}_{\ell m}^{A}{\stackrel{̄}{X}}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =\int {{\delta }^{B}}_{C}{Y}_{\ell m}^{C}{Ȳ}_{B}^{{\ell }^{\prime }{m}^{\prime }}d\Omega =\int {Y}_{\ell m}^{A}{Ȳ}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega =\frac{1}{{r}^{2}}\ell \left(\ell +1\right){\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.\phantom{\rule{2em}{0ex}}& \hfill \text{(108)}\end{array}$

Now, when we contract the even and odd-parity vector harmonics we get

$\begin{array}{lll}\hfill \int {Y}_{\ell m}^{A}{\stackrel{̄}{X}}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =-\int {D}_{A}{Y}^{\ell m}{𝜀}^{AB}{D}_{B}{Ȳ}^{{\ell }^{\prime }{m}^{\prime }}d\Omega .\phantom{\rule{2em}{0ex}}& \hfill \text{(109)}\end{array}$

By parts integration we have

$\begin{array}{lll}\hfill \int {Y}_{\ell m}^{A}{\stackrel{̄}{X}}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =\int {𝜀}^{AB}{D}_{A}{D}_{B}{Y}^{\ell m}{Ȳ}^{{\ell }^{\prime }{m}^{\prime }}d\Omega =0=\int {Ȳ}_{\ell m}^{A}{X}_{A}^{{\ell }^{\prime }{m}^{\prime }}d\Omega ,\phantom{\rule{2em}{0ex}}& \hfill \text{(110)}\end{array}$

because of the derivatives commute while the Levi-Civita tensor is antisymmetric. Consider now ${\Omega }^{AB}{D}_{A}{D}_{B}{Y}_{C}^{\ell m}={\Omega }^{AB}{D}_{A}{D}_{B}{D}_{C}{Y}^{\ell m}.$ The two closest covariant derivatives commute, but we have to use the rule

 $\left[{D}_{A},{D}_{B}\right]{V}^{C}={{R}^{C}}_{DAB}{V}^{D}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left[{D}_{A},{D}_{B}\right]{V}_{C}={R}_{C\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}AB}^{\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}D}{V}_{D}$ (111)

to commute the outer two, and therefore

$\begin{array}{lll}\hfill {\Omega }^{AB}{D}_{A}{D}_{B}{Y}_{C}^{\ell m}& ={\Omega }^{AB}{D}_{A}{D}_{C}{D}_{B}{Y}^{\ell m}\phantom{\rule{2em}{0ex}}& \hfill \text{(112)}\\ \hfill & ={\Omega }^{AB}\left({D}_{C}{D}_{A}{D}_{B}+{R}_{B\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}AC}^{\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}D}{D}_{D}\right){Y}^{\ell m}.\phantom{\rule{2em}{0ex}}& \hfill \text{(113)}\end{array}$

Using the diﬀerential equation for the scalar harmonics, we get

$\begin{array}{lll}\hfill {\Omega }^{AB}{D}_{A}{D}_{B}{Y}_{C}^{\ell m}& =-\ell \left(\ell +1\right){Y}_{C}^{\ell m}+{\Omega }^{AB}\frac{1}{{r}^{2}}{\Omega }^{DE}\left({\Omega }_{BA}{\Omega }_{EC}-{\Omega }_{BC}{\Omega }_{EA}\right){Y}_{D}^{\ell m}\phantom{\rule{2em}{0ex}}& \hfill \text{(114)}\\ \hfill & =\left[\right1-\ell \left(\ell +1\right)\left]\right{Y}_{C}^{\ell m}.\phantom{\rule{2em}{0ex}}& \hfill \text{(115)}\end{array}$

Additionally, we have

$\begin{array}{lll}\hfill {\Omega }^{AB}{D}_{A}{D}_{B}{X}_{C}^{\ell m}=-{{𝜀}_{C}}^{D}{\Omega }^{AB}{D}_{A}{D}_{B}{Y}_{D}^{\ell m}=\left[\right1-\ell \left(\ell +1\right)\left]\right{X}_{C}^{\ell m}.& \phantom{\rule{2em}{0ex}}& \hfill \text{(116)}\end{array}$

Taking the divergence ${Y}_{\ell m}^{A}$ and ${X}_{\ell m}^{A}$ gives

$\begin{array}{lll}\hfill {D}_{A}{Y}_{\ell m}^{A}& =\frac{1}{{r}^{2}}{\Omega }^{AB}{D}_{A}{D}_{B}{Y}_{\ell m}=-\frac{\ell \left(\ell +1\right)}{{r}^{2}}{Y}_{\ell m}\phantom{\rule{2em}{0ex}}& \hfill \text{(117)}\\ \hfill {D}_{A}{X}_{\ell m}^{A}& =-\frac{1}{{r}^{2}}{\Omega }^{AB}{D}_{A}{{𝜀}_{B}}^{C}{D}_{C}{Y}_{\ell m}=-\frac{1}{{r}^{2}}{𝜀}^{AC}{D}_{A}{D}_{C}{Y}_{\ell m}=0.\phantom{\rule{2em}{0ex}}& \hfill \text{(118)}\end{array}$

Now we consider contractions of the tensor harmonics. First of all, because they are each trace free, we have

 ${\Omega }^{AB}{Y}_{AB}^{\ell m}={\Omega }^{AB}{X}_{AB}^{\ell m}=0.$ (119)

This is clear from inspecting the matrix forms of these harmonics above. Note that this implies that both ${Y}_{AB}^{\ell m}$ and ${X}_{AB}^{\ell m}$ are orthogonal to ${\Omega }_{AB}{Y}_{\ell m}$. Now, we consider

$\begin{array}{llll}\hfill & \int {Y}_{\ell m}^{AB}{Ȳ}_{AB}^{{\ell }^{\prime }{m}^{\prime }}d\Omega \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\int {g}^{AC}{g}^{BD}\left[{D}_{C}{D}_{D}+\frac{\ell \left(\ell +1\right)}{2}{\Omega }_{DC}\right]{Y}_{\ell m}\left[{D}_{A}{D}_{B}+\frac{{\ell }^{\prime }\left({\ell }^{\prime }+1\right)}{2}{\Omega }_{AB}\right]{Ȳ}^{{\ell }^{\prime }{m}^{\prime }}d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(120)}\\ \hfill & =\frac{1}{{r}^{4}}\int \left[-{\Omega }^{AC}{\Omega }^{BD}{D}_{A}{D}_{C}{D}_{D}{Y}_{\ell m}{D}_{B}{Ȳ}^{{\ell }^{\prime }{m}^{\prime }}-\frac{1}{2}{\ell }^{\prime }\left({\ell }^{\prime }+1\right)\ell \left(\ell +1\right){Y}_{\ell m}{Ȳ}^{{\ell }^{\prime }{m}^{\prime }}\right]d\Omega \phantom{\rule{2em}{0ex}}& \hfill \text{(121)}\end{array}$

So, in order to evaluate this we need the harmonic operator (${\Omega }^{AB}{D}_{A}{D}_{B}$) acting on ${Y}_{C}$, which we calculated above. Using it and the completeness of the scalar harmonics gives

$\begin{array}{llll}\hfill & \int {Y}_{\ell m}^{AB}{Ȳ}_{AB}^{{\ell }^{\prime }{m}^{\prime }}d\Omega \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{1}{{r}^{4}}\int \left[-{\Omega }^{BD}\left[\right1-\ell \left(\ell +1\right)\left]\right{D}_{D}{Y}_{\ell m}{D}_{B}{Ȳ}^{{\ell }^{\prime }{m}^{\prime }}\right]d\Omega -\frac{1}{2{r}^{4}}{\ell }^{2}{\left(\ell +1\right)}^{2}{\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}\phantom{\rule{2em}{0ex}}& \hfill \text{(122)}\\ \hfill & =\frac{1}{2{r}^{4}}\left(\ell -1\right)\ell \left(\ell +1\right)\left(\ell +2\right){\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.\phantom{\rule{2em}{0ex}}& \hfill \text{(123)}\end{array}$

A similar, though slightly longer calculation for the odd-parity case gives

$\begin{array}{lll}\hfill \int {X}_{\ell m}^{AB}{\stackrel{̄}{X}}_{AB}^{{\ell }^{\prime }{m}^{\prime }}d\Omega & =\frac{1}{2{r}^{4}}\left(\ell -1\right)\ell \left(\ell +1\right)\left(\ell +2\right){\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.\phantom{\rule{2em}{0ex}}& \hfill \text{(124)}\end{array}$

For the divergence of the tensor harmonics we ﬁrst consider the even-parity case,

$\begin{array}{lll}\hfill {D}^{B}{Y}_{AB}^{\ell m}& =\frac{1}{{r}^{2}}{\Omega }^{BC}{D}_{C}\left[{D}_{A}{D}_{B}{Y}^{\ell m}+\frac{1}{2}\ell \left(\ell +1\right){\Omega }_{AB}{Y}^{\ell m}\right],\phantom{\rule{2em}{0ex}}& \hfill \text{(125)}\\ \hfill & =\frac{1}{{r}^{2}}{\Omega }^{BC}\left({D}_{A}{D}_{C}{D}_{B}{Y}^{\ell m}+{R}_{B\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}CA}^{\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}D}{D}_{D}{Y}^{\ell m}\right)+\frac{1}{2{r}^{2}}\ell \left(\ell +1\right){D}_{A}{Y}^{\ell m},\phantom{\rule{2em}{0ex}}& \hfill \text{(126)}\\ \hfill & =\frac{1}{{r}^{2}}\left[1-\frac{1}{2}\ell \left(\ell +1\right)\right]{Y}_{A}^{\ell m}.\phantom{\rule{2em}{0ex}}& \hfill \text{(127)}\end{array}$

For the odd-parity harmonics we have

$\begin{array}{lll}\hfill {D}^{B}{X}_{AB}^{\ell m}& =\frac{1}{2}\frac{1}{{r}^{2}}{\Omega }^{BD}{D}_{D}\left[{D}_{B}{X}_{A}^{\ell m}+{D}_{A}{X}_{B}^{\ell m}\right],\phantom{\rule{2em}{0ex}}& \hfill \text{(128)}\\ \hfill & =\frac{1}{2{r}^{2}}\left[\right1-\ell \left(\ell +1\right)\left]\right{X}_{A}^{\ell m}+\frac{1}{2{r}^{2}}{\Omega }^{BD}\left({D}_{A}{D}_{D}{X}_{B}^{\ell m}+{R}_{BCDA}{X}_{\ell m}^{C}\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(129)}\end{array}$

The divergence of ${X}_{B}^{\ell m}$ vanishes, so we are left with

$\begin{array}{lll}\hfill {D}^{B}{X}_{AB}^{\ell m}& =\frac{1}{2{r}^{2}}\left[\right1-\ell \left(\ell +1\right)\left]\right{X}_{A}^{\ell m}+\frac{1}{2{r}^{2}}{\Omega }^{BD}{r}^{2}\left({\Omega }_{BD}{\Omega }_{CA}-{\Omega }_{BA}{\Omega }_{CD}\right){X}_{\ell m}^{C},\phantom{\rule{2em}{0ex}}& \hfill \text{(130)}\\ \hfill & =\frac{1}{{r}^{2}}\left[1-\frac{1}{2}\ell \left(\ell +1\right)\right]{X}_{A}^{\ell m}.\phantom{\rule{2em}{0ex}}& \hfill \text{(131)}\end{array}$

### B Regge-Wheeler Harmonics

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

### C 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}$

### D Integrations Over the 2-Spheres

This is done by using Simpson’s rule twice. Once in each coordinate direction. Simpson’s rule is

 ${\int }_{{x}_{1}}^{{x}_{2}}f\left(x\right)dx=\frac{h}{3}\left[{f}_{1}+4{f}_{2}+2{f}_{3}+4{f}_{4}+\dots +2{f}_{N-2}+4{f}_{N-1}+{f}_{N}\right]+O\left(1∕{N}^{4}\right)$ (132)

$N$ must be an odd number.

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