which can be constructed from two copies of $S=1$ as given by (4.27). This
decomposition is the reason we can obtain the graviton amplitudes from $S=1$
amplitudes.

The double copy for massive particles – as we will formally discuss in chapter
6 – can be constructed using analogous KLT formulas for their massless
counterparts. This is due to the fact that for a single matter line emitting
gravitons/dilatons/axions, massive double copies can be obtained from
compactifications of their massless higher dimensional analogs. In this
section we will concentrate on the double copy for our favorite amplitudes
$A_{n}$, $n=3,4$, involving one spinning matter line emitting one and two
gravitons, respectively. This will be sufficient to also induce a double copy
formula for $M_{4}$ and $M_{5}$, due to the factorization (2).

We will use the $\odot$ symmetric product to denote the double copy of the
amplitudes written in a spin-multipole expansion. To begin, let us consider
then the photon emission amplitudes for $s\in\\{0,\frac{1}{2},1\\}$, as given
in the previous sections, and define their double copy. From two multipole
operators $X$ and $X^{\prime}$ acting on spin-$s$ states, we introduce an
operator $X\odot X^{\prime}$ acting on spin-$2s$ as

$X\odot X^{\prime}=\left\\{\begin{matrix}X\times X^{\prime},\,\,\,2s=0\\\
2^{-\left\lfloor D/2\right\rfloor}{\rm
tr}(X\not{\varepsilon}_{1}\tilde{X}^{\prime}\not{\varepsilon}_{2})\,,\quad
2s=1,\\\
\phi_{1\mu_{1}\nu_{1}}\left(X_{\,\mu_{2}}^{\mu_{1}}X^{\prime}{}_{\,\nu_{2}}^{\nu_{1}}\right)\phi_{2}^{\mu_{2}\nu_{2}}\,,\quad
2s=2,\end{matrix}\right.$ (4.28)

where $\varepsilon$ and $\phi$ are the respective massive polarizations and
$\tilde{X}$ denotes charge conjugation. In chapter 6 we will prove this
multipole operation naturally arises from the KLT double copy, and show
furthermore it can be used to obtain scattering amplitudes in a gravity theory
of a massive spin-$2s$ field. Let us for simplicity in this chapter assume it
as a valid double copy operation.

For our favorite amplitudes, the double copy formula will read

$K^{-1}_{n}A_{n}^{{\rm gr},s+\tilde{s}}=A_{n}^{{\rm ph},s}\odot A_{n}^{{\rm
ph},\tilde{s}}\,,\quad n=3,4\,.$ (4.29)

where $K_{n}$ corresponds to the massive KLT kernel for the n-point amplitude
which is simply given by the inverse of the biadjoint amplitude involving two
massive scalars of the same species ( see e.g. [195] for details on this
theory ).

The case $s=0$, $\tilde{s}\neq 0$ was introduced by Holstein et al. [190,
192]. It was used to argue that the gyromagnetic ratios of both $A_{n}^{{\rm
ph},1}$ and $A_{n}^{{\rm gr},1}$ must coincide, setting $g=2$ as a natural
value [190, 103]. We introduce the case $s,\tilde{s}\neq 0$ as a further
universality condition, and find it imposes strong restrictions on
$A_{n}^{h,s}$ for higher spins. More importantly, it can be used to directly
obtain multipoles in the classical gravitational theory.

For (4.29) to hold we need to put $A_{n}^{h,s}$ into the form (4.5), which we
have done in the first part of the present chapter (although we will lift this
restriction chapter 6). The coefficients $\omega^{(2j)}$ are universal once we
consider minimal-coupling amplitudes, which are obtained from QED at
$s=\frac{1}{2}$ and from the $W^{\pm}$-boson model at $s=1$ [190], as seen
above. In a diagramatic notation, the operation (4.28) gives the rules

$1_{s}\odot 1_{s}=1_{2s}\,,\quad
1_{s}\odot\ytableausetup{mathmode,boxsize=0.8em}\ydiagram{1,1}_{s}=\frac{1}{2}\ydiagram{1,1}_{2s}\,,$
(4.30)
$\ytableausetup{mathmode,boxsize=0.8em}\ydiagram{1,1}_{s}\odot\ydiagram{1,1}_{s}=\ydiagram{2,2}_{2s}+\ydiagram{2}_{2s}+\hat{1}_{2s}\,,$
(4.31)

which are a subset of the irreducible representations allowed by the Clebsch-
Gordan decomposition. Rule (4.31) is nothing but the Ricci decomposition of
the symmetric product of two multipole operators $J^{\mu\nu}_{s}$, and is
analog to the decomposition (4.20) above, but with non-vanishing Weyl
component. Indeed, the first term we denote by $\Sigma^{\mu\nu\rho\sigma}$ and
has the symmetries of a Weyl tensor, i.e. is the traceless part of
$\\{J^{\mu\nu},J^{\rho\sigma}\\}$. For instance, the $s=2$, 3-point
gravitational amplitude as obtained from (4.29), and using two copies of the
3-point seed (4.7), and the double copy operation (4.30) and (4.31), results
into

$A_{3}^{{\rm
gr},2}=\kappa\left(\epsilon{\cdot}p_{1}\right)^{2}\phi_{2}{\cdot}\left(\mathbb{I}{+}\frac{\epsilon_{\mu}q_{\nu}J^{\mu\nu}}{\epsilon{\cdot}p_{1}}{+}\frac{W_{\mu\nu\alpha\beta}}{4(\epsilon{\cdot}p_{1})^{2}}\Sigma^{\mu\nu\alpha\beta}\right){\cdot}\phi_{1}\,,$
(4.32)

where
$W_{\mu\nu\alpha\beta}:=q_{[\mu}\epsilon_{\nu]}q_{[\alpha}\epsilon_{\beta]}$
is the Weyl tensor of the graviton. We have also used
$K_{3}=\frac{\kappa}{4e^{2}}$ for the 3-point KLT kernel, with
$\kappa=\sqrt{32\pi G}$ and $G$ the Newton Constant. $\phi_{i}^{\alpha\beta}$
are massive spin-2 polarization tensors, built from spin-1 polarization
vectors; in $D=4$ they have $5$-independent components. One can show the same
amplitude can be computed from the covariantization of the Fierz-Pauli
Lagrangian [196]. The classical limit of this amplitude reproduces the
expected Weyl-quadrupole coupling [197, 198, 199, 200, 103], as we will
discuss in section 4.3.1.

To deeper understand these results, let us demand $A_{3}^{{\rm gr},s}$ to be
constructible from the double copy (4.29) for any spin:

$A_{3}^{{\rm gr,}s+\tilde{s}}(J^{\mu\nu}\oplus\tilde{J}^{\mu\nu})=A_{3}^{{\rm
ph},s}(J^{\mu\nu})\odot A_{3}^{{\rm ph},\tilde{s}}(\tilde{J}^{\mu\nu})\,,$
(4.33)

where $J^{\mu\nu}\oplus\tilde{J}^{\mu\nu}$ is the generator acting on a spin
$s+\tilde{s}$ representation. This relation yields the condition
$A_{3}^{1,s}A_{3}^{1,\tilde{s}}=A_{3}^{1,s+\tilde{s}}A_{3}^{1,0}$ on the
$J^{\mu\nu}$ operators. Using that $[J,\tilde{J}]=0$ and assuming the
coefficients in (4.5) to be independent of the spin leads to

$\bar{A}_{3}^{h,s}(J)=g_{h}\left({\rm\epsilon\cdot}p_{1}\right)^{h}\times
e^{\omega_{\mu\nu}J^{\mu\nu}}\,,\quad h=1,2$ (4.34)

with $\omega_{\mu\nu}=\frac{q_{[\mu}\epsilon_{\nu]}}{\epsilon\cdot p_{1}}$ and
$\mathcal{H}_{3}=\left(\epsilon{\cdot}p_{1}\right)^{h}$ fixed by the previous
examples. Here $g_{1}=4e^{2}$ and $g_{2}=\kappa$. This easily recovers such
cases and matches the Lagrangian derivation [196] for
$s\in\\{\frac{1}{2},1,2\\}$ in any dimension $D$. After some algebra, (4.34)
leads to the $D{=}4$ photon-current derived in [201, 202] for arbitrary spin
via completely different arguments. On the gravity side, its classical limit
matches the Kerr stress-energy tensor derived in [81], as we will discuss in
chapter 7, together with its spinor-helicity form found in [58], as we show in
appendix B. For $s>h$ and $D>4$, (4.34) contains a pole in $\epsilon{\cdot}p$
which reflects such interactions being non elementary [68]. In section 4.3.1
we show such pole cancels for the classical multipoles and provide a local
form of (4.34).

Now, the full quantum double copy amplitude $A_{4}^{\text{gr},2}$ for $s=2$
can be obtained by simply squaring (4.12) for $s=1$, and using the rule given
in the third line of (4.28), which similar $A_{3}^{\text{gr},2}$, can be
obtained from the covariant Fierz-Pauli Lagrangian [196]. In a multipole
decomposition, it will contain terms up to hexadecapole order. However, in
general there is not a known prescription to obtain the spin multipole
expansion in terms of irreducible representations of SO($D-,1,1$), and we
therefore write the full amplitude in terms of polarization vectors (i.e. not
in an operator language), as follows

$A_{4}^{\text{gr},2}=K_{4}^{-1}\mathcal{H}_{4}^{2}\left[{\omega_{4}^{(0)}}\varepsilon_{1}{\cdot}\varepsilon_{4}^{*}{+}\omega_{4\,\mu\nu}^{(1)}\varepsilon_{1}{\cdot}J_{1}^{(1)\mu\nu}{\cdot}\varepsilon_{4}^{*}{+}\omega_{4\,\mu\nu\rho\sigma}^{(2)}\varepsilon_{1}{\cdot}J_{1}^{(2)\mu\nu\rho\sigma}{\cdot}\varepsilon_{4}^{*}\right]^{2}\,,$
(4.35)

where we need to use the KLT kernel

$K_{4}{=}\frac{2e^{2}}{\kappa^{2}}\frac{k_{2}{\cdot}k_{3}}{p_{1}{\cdot}k_{2}\,p_{1}{\cdot}k_{3}}\,,$
(4.36)

and the polarization tensor for the massive spin $2$ states will be given by
$\varepsilon_{i}^{\mu\nu}=\varepsilon_{i}^{\mu}\varepsilon_{i}^{\nu}$, and
satisfy $\varepsilon_{i}^{\mu\nu}p_{i,\mu}=0$, for $i=1,4$, and the traceless
condition $\varepsilon_{i\,\mu}^{\mu}=0$. For the massless gravitons we have
$\epsilon_{i}^{\mu\nu}=\epsilon_{i}^{\mu}\epsilon_{i}^{\nu}$, with analog
properties to the massive polarizations. Here we use the contractions
$\varepsilon_{1}{\cdot}J_{1}^{\mu\nu}{\cdot}\varepsilon_{4}^{*}=\varepsilon_{1\alpha}(J_{1}^{\mu\nu})^{\alpha}_{\beta}\varepsilon_{4}^{*\beta}$,
and analog for
$\varepsilon_{1}{\cdot}J_{1}^{(2)\mu\nu\rho\sigma}{\cdot}\varepsilon_{4}^{*}=\varepsilon_{1,\alpha}\\{(J_{1}^{\mu\nu})^{\alpha}_{\delta},(J_{s}^{\rho\sigma})^{\delta}_{\beta}\\}\varepsilon_{4}^{*\beta}$.

Notice when written in terms of (4.11), (4.13) and (4.14), the spin 2
gravitational Compton amplitude (4.35), becomes dimension-independent. In
fact, if we think of $p_{1}$ and $p_{4}$ as massless momenta, with
$\varepsilon_{i}^{\mu\nu}$ their correspondent massless polarization tensors,
one recovers the 4-point amplitude for the scattering of 4 gravitons in
General Relativity. This is of course not a coincidence, and is a consequence
of the fact this amplitude can be written from a compactification of its
massless higher dimensional counterpart, as we will show explicitly in chapter
6.

In this chapter we are interested in computing the spin quadrupole radiation
amplitude and for that we can simply take the double copy of two spin-$1/2$
amplitudes and apply the second line of (4.31). In a multipolar expansion, and
up to quadrupolar order, the gravitational amplitude is given then by:

  * •

The scalar piece

$\bar{A}_{4}^{(0)\text{gr}}=\frac{\kappa^{2}}{8}\frac{\omega^{(0)}\omega^{(0)}}{k_{2}{\cdot}k_{3}\left(p_{1}{\cdot}k_{2}\right)\left(p_{1}{\cdot}k_{3}\right)}$
(4.37)

  * •

The spin dipole piece

$\bar{A}_{4}^{(\frac{1}{2})\text{gr}}=\frac{\kappa^{2}}{8}\frac{\omega^{(0)}\omega_{\mu\nu}^{(1)}J_{1/2}^{\mu\nu}}{k_{2}{\cdot}k_{3}\left(p_{1}{\cdot}k_{1}\right)\left(p_{1}{\cdot}k_{2}\right)}$
(4.38)

  * •

And finally quadrupolar piece

$\begin{split}\bar{A}_{4}^{(1)\text{gr}}=&\frac{\kappa^{2}}{8}\frac{\frac{1}{2}\omega^{(0)}[\omega^{(2)}]\hat{1}_{1}}{k_{2}{\cdot}k_{3}\left(p_{1}{\cdot}k_{2}\right)\left(p_{1}{\cdot}k_{3}\right)}\\\
&+\frac{\kappa^{2}}{8}\frac{\omega_{\mu\nu}^{(1)}\omega_{\rho\sigma}^{(1)}}{k_{2}{\cdot}k_{3}\left(p_{1}{\cdot}k_{2}\right)\left(p_{1}{\cdot}k_{3}\right)}\Big{[}\frac{1}{4}\Sigma_{1}^{\mu\nu\rho\sigma}{+}\frac{1}{D{-}2}\eta^{[\sigma[\nu}Q_{1}^{\mu]\rho]}{+}\frac{1}{2D(D{-}1)}\eta^{\sigma[\nu}\eta^{\mu]\rho}\hat{1}_{1}\Big{]}\end{split}$
(4.39)

Here we have omitted the contribution from  mathmode,boxsize=0.2em1,1,1,1 ,
since it does not contribute to the classical amplitude.

We have used that (4.31) reads

$\begin{split}J_{s}^{\mu\nu}\odot
J_{s}^{\rho\sigma}&{=}\frac{1}{4}\Sigma_{2s}^{\mu\nu\rho\sigma}{+}\frac{\alpha_{D}}{D{-}2}\eta^{[\sigma[\nu}Q_{2s}^{\mu]\rho]}{+}\frac{\beta_{D}}{2D(D{-}1)}\eta^{\sigma[\nu}\eta^{\mu]\rho}\hat{1}_{2s}\,.\end{split}$
(4.40)

The normalizations $\alpha_{D},\beta_{D}$ depend solely on $D$. However, it
cancels out in the full computation and hence we set
$\alpha_{D}{=}\beta_{D}{=}1$ hereafter. Similarly, the condition $A_{4}^{{\rm
ph},\frac{1}{2}}A_{4}^{{\rm ph},\frac{1}{2}}{=}A_{4}^{{\rm ph},0}A_{4}^{{\rm
ph},1}$, as implied by (4.29), for the quadrupole tensor $Q^{\mu\nu}_{2s}$,
can be traced at this order to
$[\omega_{4}^{(1)}\omega_{4}^{(1)}]_{\mu\nu}=[\omega_{4}^{(2)}]_{\mu\nu}\omega_{4}^{(0)}$,
which shows the universality of the quadrupole, and holds up to terms
subleading in the double soft limit.

Notice when doing the double copy of the massless spin-$1$ polarization
vectors we have always projected the graviton component via
$\epsilon_{i}^{\mu\nu}=\epsilon_{i}^{\mu}\epsilon_{i}^{\nu}$. Then, the
removal of the dilaton and axion component is trivial and does not require
additional subtraction schemes, or the introduction of ghost particles [203].
This is a general feature of amplitudes for one matter line emitting
$n$-massless particles, as we will see in chapter 6. In addition, since
$M_{4}^{\text{gr}}$ and $M_{5}^{\text{gr}}$ are built from these amplitudes
via the factorization (2), the removal of the dilaton and axion states from
these amplitudes is also straightforwards by using the graviton propagator.

####  4.3.1  From $\rm{SO}(D-1,1)$ to $\rm{SO}(D-1)$ multipoles

As mentioned in section 4.2.3, in order to compare spinning amplitudes with
classical results for spinning bodies it is sometimes necessary to choose a
frame through the SSC. Let us show how this arises from our setup, and make
more formal the discussion introduced by the end of section 4.2.3.

We have shown that the spin multipoles correspond to finite $\rm{SO}(D-1,1)$
transformations which map $p_{1}$ $\rightarrow$ $p_{2}$. Such Lorentz
transformations are composed of both a boost and a $\rm{SO}(D-1)$ Wigner
rotation. Spin multipoles of a massive spinning body are defined with respect
to a reference time-like direction and form irreps. of $\rm{SO}(D-1)$ acting
on the transverse directions [200, 204]. Hence, it is natural to identify such
action with Wigner rotations of the massive states entering our amplitude. A
simple choice for the time-like direction is the average momentum
$u=\frac{p}{m}=\frac{p_{1}+p_{2}}{2m}$. In this frame boosts are obtained as
$K^{\nu}=u_{\nu}J^{\mu\nu}$ whereas Wigner rotations read
$S^{\mu\nu}=J^{\mu\nu}-2u^{[\mu}K^{\nu]}$. Adopting $S^{\mu\nu}$ as classical
spin tensor then corresponds to the covariant SSC, i.e. $u_{\nu}S^{\nu\mu}=0$
[205, 206, 81] (another choice was used in [103, 207]). The momenta $p_{1}$
and $p_{2}$ can be aligned canonically to $p$ through the boost,

$p_{1}=e^{\frac{q}{2m}\cdot K}p\,,\quad p_{2}=e^{-\frac{q}{2m}\cdot K}p\,,$
(4.41)

which defines canonical polarization vectors $\varepsilon$,
$\tilde{\varepsilon}$ for $p$ through (recall $p_{2}$ is outgoing):

$\varepsilon_{1}{=}e^{\frac{q}{2m}\cdot
K}\,\varepsilon\,\,,\quad\varepsilon_{2}{=}\tilde{\varepsilon}\,e^{\frac{q}{2m}\cdot
K}\,.$ (4.42)

This replacement can then be applied to the multipole expansion (4.5),
yielding an extra power of $q$ for each power of $J$, hence preserving the
$\hbar$-scaling. We find

$\displaystyle\varepsilon_{1}{\cdot}\varepsilon_{2}$ $\displaystyle{=}$
$\displaystyle\varepsilon{\cdot}\tilde{\varepsilon}{+}\frac{1}{m}q_{\mu}\varepsilon
K^{\mu}\tilde{\varepsilon}{+}\mathcal{O}(K^{2})\,,$ (4.43)
$\displaystyle\varepsilon_{1}J^{\mu\nu}\varepsilon_{2}$ $\displaystyle{=}$
$\displaystyle\varepsilon S^{\mu\nu}\tilde{\varepsilon}{+}2u^{[\mu}\varepsilon
K^{\nu]}\tilde{\varepsilon}{+}$ (4.44)
$\displaystyle\frac{q_{\alpha}}{m}\varepsilon\\{K^{\alpha}{,}S^{\mu\nu}\\}\tilde{\varepsilon}{+}\mathcal{O}(K^{2})\,,$
$\displaystyle\varepsilon_{1}\\{J^{\mu\nu}{,}J^{\rho\sigma}\\}\varepsilon_{2}$
$\displaystyle{=}$
$\displaystyle\varepsilon\\{S^{\mu\nu}{,}S^{\rho\sigma}\\}\tilde{\varepsilon}{+}\mathcal{O}(K)\,,$
(4.45)

(for generic spin $K$ and $S$ are independent). In terms of irreducible
representations this decomposition can be thought of as branching
$\rm{SO}(D-1,1)$ into $\rm{SO}(D-1)$ [194]. For instance, the dipole branches
as
$\ytableausetup{mathmode,boxsize=0.6em}\ydiagram{1,1}\rightarrow\ydiagram{1,1}+\ydiagram{1}$,
which is a transverse dipole plus a transverse vector irrep, $K^{\mu}$. In the
same way, in general the
$\ytableausetup{mathmode,boxsize=0.6em}\ydiagram{2,2}$ irrep. of
$\rm{SO}(D-1,1)$ also contains a
$\ytableausetup{mathmode,boxsize=0.6em}\ydiagram{2}$ piece for $\rm{SO}(D-1)$.
This is the reason we can extract a quadrupole from Weyl piece in (4.31),
namely by combining (4.45) with the replacement rule

$\\{S^{\mu\nu},S^{\rho\sigma}\\}=\frac{2}{D{-}3}\left(\bar{\eta}^{\sigma[\mu}\bar{Q}^{\nu]\rho}{-}\bar{\eta}^{\rho[\mu}\bar{Q}^{\nu]\sigma}\right)+{\rm
other}\,\,{\rm irreps}\,$ (4.46)

where $\bar{\eta}^{\mu\nu}=\eta^{\mu\nu}-u^{\mu}u^{\nu}$. Thus we have the
identity (c.f. [208, 209])

$\begin{split}\omega_{\mu\nu\rho\sigma}\Sigma^{\mu\nu\rho\sigma}&=[\omega]_{\mu\nu\rho\sigma}^{\ytableausetup{mathmode,boxsize=0.15em}\ydiagram{2,2}}\langle\varepsilon_{1}|\\{J^{\mu\nu}{,}J^{\rho\sigma}\\}|\varepsilon_{2}\rangle\,,\\\
&=\frac{4}{D-3}[\omega]_{\mu\nu\rho\sigma}^{\ytableausetup{mathmode,boxsize=0.15em}\ydiagram{2,2}}u^{\nu}\bar{Q}^{\mu\rho}u^{\sigma}+O(K)\,.\end{split}$
(4.47)

For instance, we extract a quadrupole contribution from $A_{3}^{h,s}$ in
(4.32):

$A_{3}^{h,s}|_{\bar{Q}}{=}\frac{1}{4}\left(\epsilon\cdot
p_{1}\right)^{h}\frac{q\cdot\bar{Q}\cdot q}{D-3}\,.$ (4.48)

Of course, the $\rm{SO}(D-1,1)$ quadrupole present in $A_{4}^{h,s}$ also
contains a $\rm{SO}(D-1)$ quadrupole. It follows from (4.45). Similarly, the
$\hat{1}$ piece in (4.31) also have a contribution proportional to $S^{2}$. We
can summarize the map in the following way

$\begin{split}J_{1}^{\mu\nu}&\rightarrow S^{\mu\nu}\\\
Q_{1}^{\mu\nu}&\rightarrow\bar{Q}^{\mu\nu}+\frac{1}{D-1}\bar{\eta}^{\mu\nu}\hat{1}_{1}-\frac{1}{D}\hat{1}_{1}\\\
\Sigma^{\mu\nu\rho\sigma}&\rightarrow\frac{4}{D-3}u^{\nu}\bar{Q}^{\mu\rho}u^{\sigma}\\\
\hat{1}_{1}&\to\frac{1}{2}S_{\mu\nu}S^{\mu\nu},\end{split}$ (4.49)

In general the $\rm{SO}(D-1)$ multipoles defined through the covariant SSC are
given directly from the $\rm{SO}(D-1,1)$ ones, up to $O(K)$ terms. Due to
unitarity, one expects the latter to drop from the amplitude, at least for
$A_{3}$. Let us show explicitly how this happens. Note that 3-pt. kinematics
implies $[q{\cdot}K,q{\cdot}J{\cdot}\epsilon]=0$ and hence the spin piece of
the 3-pt. amplitude (4.34) reads

$\begin{split}\varepsilon_{1}e^{\frac{q\cdot J\cdot\epsilon}{\epsilon\cdot
p}}\varepsilon_{2}&{=}\tilde{\varepsilon}\exp\left(\frac{q_{\mu}\epsilon_{\nu}J^{\mu\nu}}{\epsilon\cdot
p}{+}\frac{q_{\mu}K^{\mu}}{m}\right)\varepsilon\,=\tilde{\varepsilon}e^{\mathcal{S}}\varepsilon\,\\\
&=\sum_{n=0}^{\infty}\frac{1}{n!}\tilde{\varepsilon}\left(\frac{q_{\mu}\epsilon_{\nu}S^{\mu\nu}}{\epsilon\cdot
p}\right)^{n}\varepsilon\,,\end{split}$ (4.50)

where one can check that the sum truncates at order $2s$. Thus the boost
(4.41) is effectively subtracted from the finite Lorentz transformation
leading to the interpretation of the 3-point formula as a little-group
rotation induced via photon/graviton emission. We end with a comment on the
case $s>h$ and $D>4$: Note that the pole $\epsilon\cdot p$ cancels in (4.48)
for any dimension. This means we can provide a local form of the 3-pt.
amplitude which contains the same multipoles as the exponential. For instance,

$\begin{split}A_{3}^{{\rm
ph},2}{=}\left(2e\epsilon{\cdot}p\right)\phi_{2}{\cdot}\left(\mathbb{I}{+}\frac{\epsilon_{\mu}q_{\nu}J^{\mu\nu}}{\epsilon{\cdot}p}{+}\frac{q_{\mu}q_{\rho}}{4m^{2}\,\epsilon{\cdot}p}\times\right.\left.\left[\epsilon_{\nu}p_{\sigma}{+}\epsilon_{\sigma}p_{\nu}{-}\frac{\eta_{\nu\sigma}\left(\epsilon{\cdot}p\right)}{D{-}3}\right]\\{J^{\mu\nu}{,}J^{\rho\sigma}\\}\right){\cdot}\phi_{1}\,,\end{split}$
(4.51)

also yields (4.48) and reduces to (4.34) in $D=4$.(Recall
$\phi_{i}^{\alpha\beta}$ are polarization tensors for the spin 2 matter
fields.) In general the $2^{n}$-poles [204, 81] of (4.50) are obtained by
performing $\left\lfloor\frac{n}{2}\right\rfloor$ traces with the spatial
metric $\bar{\eta}^{\alpha\beta}$ appearing in (4.46). The result takes the
local form

$\begin{split}\left.\bar{A}_{3}^{h,s}\right|_{2^{n}-{\rm
poles}}{=}g_{h}\left(\epsilon{\cdot}p\right)^{h}\sum_{n=0}^{\infty}\left(\alpha_{n}{+}\beta_{n}\frac{q_{\mu}\epsilon_{\nu}S^{\mu\nu}}{\epsilon{\cdot}p}\right)\times\bar{Q}_{\mu_{1}{\ldots}\mu_{2n}}^{(n)}q^{\mu_{1}}{\cdots}q^{\mu_{2n}}\,,\end{split}$
(4.52)

where $\alpha_{n}$, $\beta_{n}$ depend on the dimension $D$, and
$\bar{Q}{}^{(n)}$ are the transverse multipoles. In four dimensions we find
$\bar{Q}^{(n)}$ to be a tensor product of the Pauli-Lubanski vector $S^{\mu}$
[204, 103], and $\alpha_{n}=\frac{m^{-2n}}{(2n)!},$
$\beta_{n}=\frac{m^{-2n}}{(2n+1)!}$.

Spin-multipoles for D=4.

Let us finish this section by extracting the classical limit of the
gravitational Compton amplitude, up to quadratic order in spin. Now, since we
are interested in making contact with the scattering of waves off the Kerr BH,
as we will see in chapter 7, we specify the spin multipoles for the $D=4$
scenario. As already pointed out, the spin dipole can be written in terms of
the Pauli-Lubanski vector, via
$S^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}p_{1\rho}a_{\sigma}$, where $a^{\mu}$
corresponds to the radius of the Kerr ring singularity. We will expand on this
in chapter 7, for the moment we can think of it as a classical spin tenor
representing the intrinsic rotation of the K BH. In the same way, the spin
quadrupole can be put in terms of $a^{\mu}$ via

$\begin{split}\bar{Q}^{\mu\nu}&=m^{2}(a^{\mu}a^{\nu}-\frac{1}{3}\bar{\eta}^{\mu\nu}a^{2}),\end{split}$
(4.53)

where now the SSC is satisfied by the spin vector $p_{1\mu}a^{\mu}=0$. Finally
we have to do the usual $\hbar$ scaling of the massless momenta,
$k_{i}\to\hbar k_{1}$, and in the same way for the spin vector $a^{\mu}\to
a^{\mu}/\hbar$. Where we have also identified $u_{\mu}$ with the incoming
massive object’s four-velocity. In that form, one can explicitly check that
the final classical amplitude up to quadratic order in spin can be written as:

$\boxed{\langle
A_{4}^{\text{gr}}\rangle=\frac{\kappa^{2}}{8}\frac{\langle\omega_{4}^{(0)}\rangle}{k_{2}{\cdot}k_{3}(p_{1}{\cdot}k_{2})^{2}}\left[\langle\omega^{(0)}\rangle+\langle\omega_{4}^{(1)\mu\nu}\rangle\epsilon_{\mu\nu\rho\sigma}p_{1}^{\rho}a^{\sigma}+\langle\overline{\omega}^{(2)}_{4\,\alpha\beta}\rangle
a^{\alpha}a^{\beta}\right]+\mathcal{O}(a^{3})\,,}$ (4.54)

where the angles indicate the classical limit of the corresponding multipole
coefficients given in (4.11) and (4.13). We have also identify the classical
multipole coefficient for the quadratic in spin amplitude in classical
electromagnetism as

$\displaystyle\langle\overline{\omega}_{4}^{(2)\alpha\beta}\rangle$
$\displaystyle=\Big{[}p_{1}{\cdot}F_{2}{\cdot}F_{3}{\cdot}p_{1}(k_{1}{-}k_{2})_{\mu}\bar{\mathcal{P}}^{\mu\nu\alpha\beta}(k_{1}{-}k_{2})_{\nu}+\frac{k_{1}{\cdot}k_{2}m^{2}}{2}\Big{(}\bar{\mathcal{P}}^{\mu\nu\alpha\beta}+\frac{\eta^{\mu\nu}\eta^{\alpha\beta}}{2}\Big{)}F_{2}^{(\mu|\delta}F_{3}^{\gamma|\nu)}\eta_{\gamma\delta}\Big{]}\,,$
(4.55) $\displaystyle\bar{\mathcal{P}}^{\mu\nu\alpha\beta}$
$\displaystyle=\frac{\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\alpha}\eta^{\nu\beta}}{2}-\eta^{\mu\nu}\eta^{\alpha\beta}\,.$
(4.56)

Notice remarkably that after summing up all the rotation spin multiple
contributions, the classical GR amplitude has the factorization form
$A^{\text{gr}}=\langle A_{0}^{\text{ph}}\rangle\times\langle
A_{s}^{\text{ph}}\rangle$, as suggested by the authors in [54], and reflecting
the universality of the coupling of matter to the graviton. The quadratic in
spin contribution and its unitarity gluing with the 3-point amplitude,
recovers the quadratic in spin two-body radiation amplitude obtained in [120],
as we will explicitly show in section 4.3.2.

####  4.3.2 Spin multipolar expansion of $M_{4}$ and $M_{5}$

Let us in the remaining of this chapter compute the gravitational $M_{4}$ and
$M_{5}$ amplitudes up to the spin quadrupole level from the double copy. Let
us recall at leading order in the perturbative expansion, the $\hbar{\to}0$
limit of the amplitudes is captured by the cuts of $M_{4}$ and $M_{5}$ given
in (2). $M_{4}$ can then be computed from the formula (2.40), whereas for
$M_{5}$, the key point is to introduce the average momentum transfer
$q=\frac{q_{1}-q_{2}}{2}$, as done for the electromagnetic case section 2.3,
after which one expects the same construction to apply. This leave us then
with the formula (2.45), which we now use in the gravitational context.

##### Classical Double Copy

As the numerators in eqs. 2.40 and 2.45 correspond to $A_{n}^{h,s}$
amplitudes, the multipole double copy can be directly promoted to $\langle
M_{4}\rangle$ and $\langle M_{5}\rangle$. From a classical perspective, the
factorization of (2) implies that the photon numerators can always be written
as $n_{{\rm ph}}=t_{a\mu}t_{b}^{\mu}$ where $t_{a}$ and $t_{b}$ only depend on
kinematics for particle 1 and $2$ respectively. The simplest example is the
scalar piece in $\langle M_{4}^{{\rm ph}}\rangle$, where $t_{a}=p_{1}$ and
$t_{b}=p_{2}$. The KLT formula (4.29) translates to

$\boxed{n_{{\rm gr}}=n_{{\rm ph}}\odot n_{{\rm ph}}-{\rm tr}(n_{{\rm ph}}\odot
n_{{\rm ph}})\,}$ (4.57)

where we defined the trace operation as ${\rm tr}(n\odot
n)=\frac{(t_{a\mu}\odot t_{a}^{\mu})(t_{b\mu}\odot t_{b}^{\mu})}{D-2}$. By
combining (4.57) with eqs. 2.40 and 2.45, this establishes for the first time
a classical double-copy formula that can be directly proved from the standard
BCJ construction as we will see in chapter 6. Moreover, up to this order it
only requires as input Maxwell radiation as opposed to gluon color-radiation
[80, 32] and contains no Dilaton/Axion states [203, 6, 32], the latter of
which are removed by the trace subtraction in (4.57). Since the formula for
the radiative amplitude for the gravitational case (2.45) follows from the
gravitational Compton amplitude, the spurious pole $\frac{1}{q{\cdot}k}$
arises from the $t$-channel of the Compton amplitude, and its cancellation
from the final result provides an strong check of our double copy formula.

Let us in the following provide some examples of how to use this double copy
formula (4.57). Starting with $\langle M_{4}\rangle$: The simplest example is
the scalar amplitude, for which, as we already mention
$n_{0}^{\text{ph}}=p_{1}{\cdot}p_{2}$, then $t_{a}=p_{1}$ and $t_{b}=p_{3}$,
so that $tr(n_{\text{ph}}\odot n_{\text{ph}})=\frac{m_{1}^{2}m_{2}^{2}}{D-2}$.
This then leads to the scalar gravitational amplitude

$\langle M_{4}^{{(0),\rm gr}}\rangle=\frac{n_{\rm gr}}{q^{2}}=\frac{32\pi
G}{q^{2}}\left[(p_{1}{\cdot}p_{2})^{2}-\frac{m_{1}^{2}m_{2}^{2}}{D-2}\right],$
(4.58)

where the factor of $D-2$ arises from the graviton propagator. In $D=4$ we can
evaluate (1.12) to recover the 1PM scattering angle as in [48], first derived
in the classical context by Portilla [210, 211].

Next we can consider the quadratic in spin contribution. To keep notation
simple consider only particle $a$ to have spin. From (4.7) we find that at the
dipole level the numerator for $\langle M_{4}^{{\rm ph}}\rangle$ is
$n_{\frac{1}{2}}^{{\rm ph}}=n_{{\rm 0}}^{{\rm ph}}+p_{3}{\cdot}J_{a}{\cdot}q$.
The gravity result follows from (4.57) by dropping contact terms in $q^{2}$.
The rules (4.30) readily scalar as we just saw, as well as the dipole parts.
let us compute the more interesting quadrupole part; rule (4.40) gives

$\frac{(p_{3}{\cdot}J_{a}{\cdot}q)\odot(p_{3}{\cdot}J_{a}{\cdot}q)-{\rm
tr(\cdots)}}{q^{2}}=\frac{1}{4}\frac{p_{3\mu}q_{\nu}p_{3\alpha}q_{\beta}\Sigma_{a}^{\mu\nu\alpha\beta}}{q^{2}}\,,$
(4.59)

Using (4.47), the $\rm{SO}(D-1)$ quadrupole [198, 212, 200] reads

$\frac{1}{4}\frac{p_{3\mu}q_{\nu}p_{3\alpha}q_{\beta}\Sigma_{a}^{\mu\nu\alpha\beta}}{q^{2}}\rightarrow\left((p_{1}{\cdot}p_{2})^{2}{-}\frac{m^{2}_{1}m^{2}_{2}}{D{-}2}\right)\frac{q\cdot\bar{Q}_{a}\cdot
q}{2(D{-}3)q^{2}m_{a}^{2}}.$ (4.60)

Up to this order this agrees with the $D=4$ computation [81, 58, 213].
Agreement to all orders in spin is obtained from the formula (4.52).

Let us now move to compute the classical numerators for the gravitational
$\langle M_{5}\rangle$ amplitude. We start with the scalar case, for which the
gravitational numerators can be computed from putting the photon numerators
(2.46) into the double copy formula (4.57)

$\begin{split}n^{(a)}_{0,\rm{gr}}&=\frac{\kappa^{3}}{4}\left[\left(p_{1}{\cdot}p_{2}F_{1q}-p_{1}{\cdot}kF_{p}\right)^{2}-\frac{m_{1}^{2}m_{2}^{2}}{D-2}F_{1q}^{2}\right],\\\
n^{(b)}_{0,\rm{gr}}&=-\frac{\kappa^{3}}{4}\left[\left(p_{1}{\cdot}p_{2}F_{2q}+p_{2}{\cdot}kF_{p}\right)^{2}-\frac{m_{1}^{2}m_{2}^{2}}{D-2}F_{2q}^{2}\right],\end{split}$
(4.61)

where we have used  $F_{iq}=\eta_{i}(p_{i}{\cdot}F{\cdot}q)$, and
$F_{p}=p_{1}{\cdot}F{\cdot}p_{2}$. Analogous to the electromagnetic
numerators, this can also be obtained from the gluing of the scalar, 3-pt and
4-pt amplitudes through the graviton propagator. These numerators can be
introduced in the general formula (2.45), to recover the result for the
classical limit of the gravitational amplitude for scalar particles [80, 6,
61].

In analogy to the electromagnetic case, the gravitational 5-point amplitude
for scalar sources can also be put in an all order exponential soft expansion.
This is a consequence of the soft exponentiation of the gravitational Compton
amplitude, in analogy to its electromagnetic counterpart (2.23),

$\begin{split}A_{4}^{{\rm
gr}}{=}\sum_{p_{a}{=}p_{1},p_{2},k_{2}}\frac{1}{2}\frac{(\epsilon_{3}\cdot
p_{a})^{2}}{k_{3}\cdot p_{a}}e^{\frac{2F_{3}\cdot J_{a}}{\epsilon_{3}\cdot
p_{a}}}A_{3}^{{\rm
gr}}=\frac{1}{2k_{2}{\cdot}k_{3}}\times\left[\frac{(p_{1}{\cdot}\epsilon_{2})^{2}}{p_{1}{\cdot}k_{3}p_{4}{\cdot}k_{3}}F_{k}^{2}-2\frac{p_{1}{\cdot}\epsilon_{2}}{p_{1}{\cdot}k_{3}}F_{k}F_{\epsilon}+\frac{p_{4}{\cdot}k_{3}}{p_{1}{\cdot}k_{3}}F_{\epsilon}^{2}\right].\end{split}$
(4.62)

which induces the exponentiation for the scalar numerators (4.61)

$n_{{\rm
gr}}^{(a)}=\frac{F_{1q}^{2}}{2}e^{-\frac{F_{p}}{F_{1q}}(p_{1}{\cdot}k)\frac{\partial}{\partial(p_{1}{\cdot}p_{2})}}\left[(p_{1}{\cdot}p_{2})^{2}-\frac{m_{1}^{2}m_{2}^{2}}{D-2}\right]\,.$
(4.63)

Further writing $\frac{1}{q^{2}\\\ ac{PM}q\cdot k}=e^{\\\ ac{PM}q\cdot
k\frac{\partial}{\partial q^{2}}}\frac{1}{q^{2}}$ turns (2.45) into

$\boxed{\langle M_{5}^{\rm
gr}\rangle=\sum_{i=1,3}\mathcal{S}^{\text{gr}}_{i}e^{\eta_{i}\left(F_{p}\frac{p_{i}{\cdot}k}{F_{iq}}\frac{\partial}{\partial(p_{1}{\cdot}p_{2})}+q{\cdot}k\frac{\partial}{\partial
q^{2}}\right)}\langle M_{4}^{\rm gr}\rangle\,}$ (4.64)

where now the soft factor are
$\mathcal{S}^{\text{gr}}_{i}{=}\frac{\eta_{i}}{2}\frac{F_{iq}^{2}}{(p_{i}{\cdot}k)^{2}q{\cdot}k}$.
This is the gravitational analog of (2.49). Similarly to the electromagnetic
case (2.51), we can show to leading order in the soft expansion, amplitude
(4.64), when used into (1.15) leads to the memory waveform. That is

$\int\frac{d^{D}q}{(2\pi)^{D-2}}\delta(2q\cdot p_{1})\delta(2q\cdot
p_{2})e^{iq\cdot(b_{1}-b_{2})}\left(\sum_{i=1,3}\mathcal{S}_{i}\right)\langle
M_{4}^{{\rm gr}}\rangle$

as $k\to 0$. Evaluating the sum using $\Delta p_{1}=-\Delta p_{2}$ we obtain

$\epsilon_{\mu\nu}T^{\mu\nu}=\frac{F_{p}/2}{p_{1}{\cdot}kp_{2}{\cdot}k}\left(\frac{p_{1}}{p_{1}{\cdot}k}{+}\frac{p_{2}}{p_{2}{\cdot}k}\right){\cdot}F{\cdot}\Delta
p{+}\mathcal{O}(k^{0}),$ (4.65)

which at leading order in $\Delta p$ (or $G$, if restored) becomes

$T^{\mu\nu}(k)=\sqrt{8\pi
G}\times\Delta\left[\frac{p_{1}^{\mu}p_{1}^{\nu}}{p_{1}{\cdot}k}+\frac{p_{2}^{\mu}p_{2}^{\nu}}{p_{2}{\cdot}k}\right]^{{\rm
TT}}\,.$ (4.66)

In position space this gives the burst memory wave derived by Braginsky and
Thorne [110] in $D=4$ (a $\frac{1}{4\pi R}$ factor arises from the ret.
propagator as $R{\to}\infty$, see section 1.2 and [80, 214, 133]), see also
[215, 216, 217] for $D>4$. Here we have provided a direct connection with the
Soft Theorem in the gravitational case, alternative to the expectation-value
arguments of [218, 219]. This can also be seen as the Black Hole
Bremsstrahlung of [220, 136] generalized to consistently include the dynamics
of the sources.

Next, the gravitational numerators to linear-order in spin can analogously be
computed. For that we use a copy of the scalar numerators (2.46), and one of
the linear in spin numerators (4.25), into our double copy formula (4.57). We
get

$\begin{split}n^{(a)}_{\frac{1}{2},\rm{gr}}&=\frac{\kappa^{3}}{8}\Big{\\{}\left(p_{1}{\cdot}p_{2}F_{1q}-p_{1}{\cdot}kF_{p}\right)\left[\left(p_{1}{\cdot}p_{2}\,q{\cdot}k+p_{1}{\cdot}k\,p_{2}{\cdot}k\right)F{\cdot}J_{2s,1}{-}F_{1q}R_{2}{\cdot}J_{2s,1}{+}p_{1}{\cdot}k\,[F,R_{2}]{\cdot}J_{2s,1}\right]\\\
&\hskip
42.67912pt+\frac{m_{2}^{2}F_{1q}}{D-2}\left[F_{1q}(2q{-}k){\cdot}J_{2s,1}{\cdot}p_{1}{-}m_{1}^{2}q{\cdot}k\,F{\cdot}J_{2s,1}+p_{1}{\cdot}k(2q{-}k){\cdot}F{\cdot}J_{2s,1}{\cdot}p_{1}\right]\Big{\\}},\\\
n^{(b)}_{\frac{1}{2},\rm{gr}}&=-\frac{\kappa^{3}}{8}\Big{\\{}\left(p_{1}{\cdot}p_{2}F_{2q}+p_{2}{\cdot}kF_{p}\right)\left(F_{2q}(2q+k){\cdot}J_{2s,1}{\cdot}p_{2}-p_{2}{\cdot}k\,p_{2}{\cdot}F{\cdot}J_{2s,1}{\cdot}(2q+k)\right)\\\
&\hskip
42.67912pt+\frac{m_{2}^{2}F_{2q}^{2}}{D-2}(2q{+}k){\cdot}J_{2s,1}{\cdot}p_{1}\Big{\\}}.\end{split}$
(4.67)

Here we remark the generators $J_{2s,1}$ act in the gravitational theory
rather than in the electromagnetic counterpart. Similarly to the scalar case,
these numerators can be placed in (2.45) to recover the corresponding
gravitational amplitude. To obtain the full amplitude for both particles with
spin, we utilize the symmetrization mappings

$\displaystyle m_{1}\leftrightarrow m_{2},$ $\displaystyle
p_{1}\leftrightarrow p_{2},$ $\displaystyle q\rightarrow-q,$ $\displaystyle
J_{2s,1}\rightarrow J_{2s,2},$ (4.68)

in the final formula. The resulting amplitude recovers the spinning amplitude
in dilaton gravity computed in [117] for classical spinning sources, once we
remove the terms proportional to $m_{i}$ in the numerators in (4.67), which
arise from the graviton projection, and branch $J^{\mu\nu}\to S^{\mu\nu}$.
This provides a strong cross-check of our method.

Using (4.40) we can also compute the quadrupolar order. For instance, the
$Q^{\mu\nu}$ piece can be computed from two copies of the linear in spin
numerators (4.25). Let us again for simplicity consider only particle $a$ with
spin.

$\begin{split}\frac{n^{(a)}|_{Q}}{q{\cdot}k}{=}&\frac{(32\pi
G)^{\frac{3}{2}}}{8(D{-}2)}\biggl{[}\left(p_{1}{\cdot}p_{2}F_{1q}{-}p_{1}{\cdot}k\,F_{p}\right)\\{R_{2}{,}F\\}{\cdot}Q_{1}+\frac{m_{2}^{2}}{(D{-}2)}\left(F_{1q}\\{F{,}Y\\}{\cdot}Q_{1}{-}2p_{1}{\cdot}k\,p_{1}{\cdot}F{\cdot}Q_{1}{\cdot}F{\cdot}q\right)\biggr{]},\end{split}$
(4.69)

with $Y^{\mu\nu}=p_{1}^{[\mu}(2q{-}k)^{\nu]}$, whereas $n^{(b)}|_{Q}=0$. As
before, we have dropped contact terms in $q^{2}$ and used the support of
$\delta(p_{i}{\cdot}q_{i})$. This result can be shown to agree with a much
more lengthy computation of the full $M_{5}^{{\rm gr}}$ using Feynman
diagrams. At this order, $M_{5}^{{\rm gr}}$ contains classical quadrupole
pieces and quantum scalar and dipole pieces. Interestingly, while the scalar
part is trivial to identify, we have found that the dipole part can be
cancelled by adding the spin-1 spin-0 interaction
$(B_{\mu}\partial^{\mu}\phi)^{2}$ to the Lagrangian, which signals its quantum
nature.

Let us stress the numerators (4.67) and (4.69) are written in terms of the
SO($D-1,1$) multipole operators. Furthermore, for the quadrupolar
contribution, the full amplitude includes the additional irreps. ( Weyl and
trace pieces in (4.31)). In order to use the radiative amplitude in a more
realistic context, as for instance for two coalescing KBHs, as we will study
in chapter 5, the amplitude needs to be computed in terms of the spin
multipoles of the rotation group SO($3$), in $D=4$, where all the irreps. can
be written in powers of the Pauli-Lubanski vector $s^{\mu}=m\times a^{\mu}$.
This is achieved through the map (4.49), where the rotation quadrupole moment
in 4-dimensions is given in (4.53).

In $D=4$, at the quadrupole level we can however take an alternative route in
the computation of the full $M_{5}^{\text{gr}}$. Since we already have the
classical gravitational Compton amplitude written in terms of $a^{\mu}$ as
given in (4.53), we can glue it to the classical 3-point amplitude (4.52) as
given by the unitary prescription of eq. (2). This in turn will allow us to
identify the gravitational numerators entering in (2.45) in a simpler way. We
arrive at the following numerators given in (C.3) and (C.4) in appendix C They
agree with the more involving computation of the irreps in (4.31). In chapter
5, we will use numerators (4.61), (4.67), (C.3) and (C.4) to compute the
gravitational waveform at leading and subleading order in the velocity
expansion for the coalescing of two KBHs in general closed orbits, whose spins
are aligned with the angular momentum of the system. We will then specialize
to circular orbits where the waveform can be computed to all orders in the
BHs’ spins.

###  4.4 Outlook of the chapter

We have shown that key techniques of Scattering Amplitudes such as soft
theorems and double copy can be promoted directly to study classical phenomena
arising in GWs . These techniques drastically streamline the computation of
radiation and spin effects; both are phenomenologically important for Black
Holes, which are believed to be extremely spinning in nature [221, 222]. In
that direction, one could for instance apply our formalism to derive the
hexadecapole ($s=2$) order in radiation [223, 224] to LO in $G$ but all orders
in $1/c$.

Soft Theorem/Memory Effect: It would be interesting to understand the meaning
of the higher orders of (2.49), considering for instance the Spin Memory
Effect [225, 226]. Motivated by the infinite soft theorems of [227, 228] one
could expect the corrections are related to a hierarchy of symmetries. One may
also incorporate spin contributions and study their interplay with such orders
[214]. In the applications side, it is desirable to further investigate (2.49)
at loop level [229, 230], which could lead to a simple way of obtaining
$\langle M_{5}\rangle$ from $\langle M_{4}\rangle$.

Generic Orbits: For orbits more general than scattering $\mathcal{J}(k)$ does
not have the support of $\delta(2p_{i}{\cdot}q_{i})$, as will become clear in
chapter 5 [31, 166]. In fact, for bounded orbits it contains the subleading
terms $p_{i}{\cdot}q_{i}\sim\omega$. Very nicely, by keeping such terms in the
classical calculation we have checked they match with eqs. (2.46),(4.25),
which in turn arise from the form in (4.14) via a natural "$F{\to}R$
replacement". As we will show in chapter 5, one can use the amplitudes
computed in the present chapter to approach the two-body problem in General
Relativity at lower orders in the velovoty expansion, where the terms removed
by the on-shell conditions do not contribute to the waveform.

##  Chapter 5 Bounded systems and waveforms from Spinning Amplitudes

###  5.1 Introduction

So far we have been concerned with the computation of classical observables
for bodies moving in scattering orbits. However, more realistic scenarios, as
for instance, the coalescing of compact objects observed by the LIGO/Virgo
Collaboration [9], require the study of observables for bodies moving in
general closed orbits. The first approach to this problem was done in the
early days of general relativity, by Einstein predicting the existence of
gravitational waves [8] and cast the emission from a compact system into the,
now famous, Quadrupole formula for gravitational radiation. A little while
later, in a spectacular breakthrough the LIGO/Virgo Collaboration [9]
confirmed Einstein’s prediction by directly detecting the gravitational waves
emitted from a BBH. Higher order corrections to Einstein’s Quadrupole formula
in the context of the quasi-circular orbit general relativistic two-body
problem – needed to enable such detections – have traditionally been obtained
in the PN [23, 24] formalism, within numerical relativity [20] and black hole
perturbation theory [21, 22], as well as models combining these approaches
[25, 26, 27]. More recently, however, efforts have been focused on the BBH
scattering problem, in order to connect classical computations performed in
the context of the PM theory [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42], with those approaches based on the classical limit of QFT scattering
amplitudes [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 7, 57, 58,
59, 60, 61, 62, 63].

Until recently, the scattering amplitudes approach to the two-body scattering
problem had mostly focused it’s efforts in the conservative sector, although
in this work we have shown how the radiative sector to leading orders in
perturbation theory can be similarly approach from scattering amplitudes. In
addition, soft theorems [178, 177, 181] suggest that the full radiative sector
can be approached from the classical limit of a 5-point scattering amplitude,
as we have seen in previous chapters. The introduction of the KMOC formalism
[78], enabling the computation of classical observables directly from the
scattering amplitude, proved to be extremely useful in determining radiative
observables as extensively exemplified in during the body of this thesis,
ranging from the leading in $G$ memory waveform from hyperbolic, soft
encounters was presented chapter 4 , to the prediction of the waveform to all
orders in perturbation theory, but to leading order in the soft expansion. In
this same formalism, the computation of the full leading PM order radiated
four-momentum was recently presented in [93, 92]; these results were
subsequently confirmed by other methods in [77, 231, 232]. Simultaneously,
using a worldline-QFT formalism [233], the computation of the gravitational
waveform valid for all values for the momentum of the emitted graviton, was
computed in [234] (see also [235]), and extended to include spin effects in
[120]. Analogously, the scattering amplitudes approach has been employed to
study radiation scattering off of a single massive source [131, 236], where a
novel connection between scattering amplitudes and black hole perturbation
theory has emerged [84], shedding light on how to obtaining the higher-spin
gravitational Compton amplitude [237], as we will expand in chapter 7 (see
also [238, 239]).

Even with the powerful scattering amplitudes techniques at hand, so far,
radiative information from bodies moving on bounded orbits has been obtained
only via analytic continuation [37, 36] of radiation observables of scattering
bodies [93, 184, 76] (applying mainly in the large eccentricity limit).
However, the almost $40$ year old derivation of the Einstein quadrupole
formula from a Feynman diagrammatic perspective by Hari Dass and Soni [121],
and the more recent derivation by Goldberger and Ridgway using the classical
double copy [31], suggest that scattering amplitudes can indeed be used to
derive gravitational radiation emitted from objects moving on general closed
orbits (including the zero eccentricity limit, i.e., quasi-circular orbits).
In this chapter we follow this philosophy to compute the gravitational
waveform emitted from an aligned spin BBH on general and quasi-circular
orbits, up to quadratic order in the constituents spin at the leading order in
the velocity expansion and to sub-leading order in the no-spin limit, from the
classical 5-point scattering amplitude derived in chapter 4. We contrast and
compare these results to the analogous classical derivation of the corrections
to the Einstein quadrupole formula using the well-established multipolar post-
Minkowskian formalism [122, 123, 124, 125, 23].

We find perfect agreement between the classical and the scattering amplitudes
derivation of all radiative observables we consider, to the respective orders
in the spin and velocity expansions. Furthermore, we show that at leading
order in the BBH velocities, there is a one-to-one correspondence between the
BBH source’s mass and current multipole moments, and the scalar and linear-in-
spin 5-point scattering amplitude, respectively. At quadratic order in the
spin of the black holes, we demonstrate explicitly that the corresponding
contribution from the quadratic-in-spin scattering amplitude does not provide
additional spin information at the level of the waveform; hence, we conjecture
this to hold for higher-spin amplitudes as well, based on the aforementioned
correspondence. Then, the leading in velocity, all orders-in-spin waveform, is
obtained purely through the solutions to the EoM of the conservative sector of
the BBH. Furthermore, the gauge dependence of gravitational radiation
information at future null infinity is a potential source of difficulty when
comparing results obtained by different approaches. In this work, we provide
evidence that gauge freedom partially manifests itself in the integration
procedure appearing in the computation of the waveform directly from the
scattering amplitude. For quasi-circular orbits, the orbit’s kinematic
variables are subject to certain relations, such that the gravitational
waveform can take different forms without affecting the gauge invariant
information contained in the total instantaneous gravitational wave energy
flux.

This chapter is organized as follows: In section 5.2, we begin by reviewing
the classical derivation of the conservative sector of the spinning BBH to all
orders in the spins at leading PN. In section 5.2.2, we derive the associated
gravitational wave emission from this system to all order in the BHs’ spins.
We then proceed with the scattering amplitudes derivation of the waveform in
section 5.3, with the general formalism outlined in section 5.3.1. In section
5.3.2 we us the classical spinning amplitudes $M_{4}$ and $M_{5}$ obtained in
chapter 4 for explicitly determining the waveform. In section section 5.3.3,
we briefly discuss the computation of the gauge invariant energy flux, and
comment on the manifestation of the gauge freedom. We conclude with a outlook
of the chapter in section 5.4. In this chapter we use Greek letters
$\alpha,\beta\dots$ for spacetime indices and Latin letters $i,j\dots$ for
purely spatial indices. Furthermore, we use $G=c=1$ units throughout, assume
$\varepsilon_{0123}=1$, and use the
$2\nabla_{[\alpha}\nabla_{\beta]}\omega_{\mu}=R_{\alpha\beta\mu}{}^{\nu}\omega_{\nu}$
Riemann tensor sign convention.

This chapter is based on the work by the author [101].

###  5.2 Classical derivation

In order to approach the bound orbit from a classical point of view, we
utilize an effective worldline action [198, 199, 205, 200, 212, 30, 204],
parametrizing the complete set of spin-induced interactions of the two
spinning BHs in the weak-field regime, at linear order in the gravitational
constant, i.e. at PM order. As we are interested in bound, as opposed to
unbound, orbits, we will be focusing on the leading PN contribution to the 1PM
conservative sector at each order in the BHs’ spins. In the following, we
first briefly summarize the necessary conservative results established in
Refs. [240, 241, 242, 118, 243, 244, 245, 246, 247, 248, 212, 196, 223, 249,
224]. Using these results, we then tackle the radiative sector, utilizing the
multipolar post-Minkowskian formalism [122, 123, 124, 125, 24] (see also Ref.
[23] and references therein). We derive the transverse-traceless (TT) pieces
of the linear metric perturbations, $h_{\mu\nu}^{\text{TT}}$, and the total
instantaneous gravitational wave power, $\mathcal{F}$, radiated by this source
to future null infinity. We achieve this, considering all orders in the spins,
both for an aligned spin system on general orbits at leading order in
velocities, as well as specialize to quasi-circular orbits at leading and
first sub-leading orders in velocities. In this section we work in the $-+++$
signature for the flat metric.

####  5.2.1 Classical spinning Binary black hole

Let us begin by briefly reviewing the approach to the conservative sector of
the BBH dynamics at the respective orders in the weak-field and low-velocity
regimes using an effective worldline action. We start by presenting the
necessary spin-interactions to describe a rotating BH, and then move to review
how an effective spinning BBH action, needed for the computation of the
radiation field, can be derived.

##### Effective binary black hole action

An effective description of a rotating BH, obeying the no-hair theorems, as a
point particle with suitable multipolar structure in the weak-field regime
rests solely on its worldline and spin degrees of freedom [30, 199, 250, 223,
200]. The former are given by a worldline $z^{\mu}(\lambda)$ of mass $m$, with
4-velocity $u^{\mu}=dz^{\mu}/d\lambda$, while the latter are encoded in the
BH’s (mass-rescaled) angular momentum vector111This angular momentum vector
$a^{\mu}=\varepsilon^{\mu}{}_{\nu\alpha\beta}u^{\nu}S^{\alpha\beta}/(2m)$
emerges from the spin tensor $S^{\alpha\beta}$ assuming the covariant spin
supplementary condition, $p_{\mu}S^{\mu\nu}=0$, and a local body-fixed frame
$e_{A}^{\mu}(\lambda)$. See, for instance, Ref. [250] for details. $a^{\mu}$,
and local frame $e_{A}^{\mu}(\lambda)$. An effective worldline action, $S$,
that entails the dynamics of such a BH (or, more generally, a compact object)
in the weak-field regime was developed in Refs. [198, 199, 205, 200, 212]; see
Ref. [204] for further details. This action $S[h,\mathcal{K}]$, describing a
rotating compact object, is built considering all possible couplings of
gravitational, $h=\\{h_{\mu\nu}\\}$, and object specific degrees of freedom,
$\mathcal{K}=\\{z^{\mu},u^{\mu},a^{\mu},e^{\mu}_{A}\\}$, requiring covariance,
as well as reparameterization and parity invariance [251, 252, 200, 204]. At
the 1PM level, a matching procedure between the linearized Kerr metric [253,
81, 249] and the gravitational field $h_{\mu\nu}$, emanating from a generic
compact object described by $S[h,\mathcal{K}]$, leads to a unique set of non-
minimal couplings between $h$ and $\mathcal{K}$. This ultimately results in an
effective 1PM BH worldline action $S_{\text{BH}}[h,\mathcal{K}]$. This action
can be extended to higher orders in $G$ in spins (see for instance Refs. [254,
255, 256, 257]).

It was shown in Ref. [81] that for a harmonic gauge linearized Kerr BH the
infinite set of spin-couplings present in the 1PM effective worldline action
$S_{\text{BH}}[h,\mathcal{K}]$ can be resummed into an exponential function.
In a linear setup, a BH of mass $m$ traveling along the worldline
$z^{\mu}(\lambda)$, sources the gravitational field,
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}^{\text{Kerr}}+\mathcal{O}(h^{2})$, with
[81]

$\displaystyle
h_{\mu\nu}^{\text{Kerr}}=4\mathcal{P}_{\mu\nu}{}^{\alpha\beta}\hat{\mathcal{T}}^{\text{Kerr}}_{\alpha\beta}\frac{1}{\hat{r}},$
$\displaystyle\hat{\mathcal{T}}_{\mu\nu}^{\text{Kerr}}=m\exp(a*\partial)_{(\mu}{}^{\rho}u_{\nu)}u_{\rho}.$
(5.1)

Here we define
$(a*\partial)^{\mu}{}_{\nu}=\epsilon^{\mu}{}_{\nu\alpha\beta}a^{\alpha}\partial^{\beta}$
and introduced the trace reverser
$\mathcal{P}_{\mu\nu\alpha\beta}=(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\nu\alpha}\eta_{\mu\beta}-\eta_{\mu\nu}\eta_{\alpha\beta})/2$
222This is not to be confused with the $\bar{\mathcal{P}}^{\mu\nu\alpha\beta}$
tensor defined in (4.56) .. Additionally, $\hat{r}$ labels the proper distance
between the spacetime point $x$ and the worldline $z^{\mu}(\lambda)$, within
the slice orthogonal to $u^{\mu}$ [81]. In the following, we restrict
ourselves to the leading PN part of the 1PM ansatz, since this is the natural
setting for closed orbits in the weak field regime. However, while we are
expanding in $\epsilon_{\text{PN}}\sim v^{2}/c^{2}\sim GM/rc^{2}$, we consider
all orders in the spins, i.e., consider $\epsilon_{\text{spin}}\sim\chi
GM/rc^{2}$ non-perturbatively (here, $\chi$ the black hole’s dimensionless
spin parameter). To that end, we choose the Minkowski coordinate time $t$ to
parameterize the worldline $z^{\mu}$, i.e., $\lambda\rightarrow t$, and expand
the 4-velocity $u^{\mu}=(1,\boldsymbol{v})^{\mu}+\mathcal{O}(v^{2})$, with
$\dot{z}^{i}=dz^{i}/dt=v^{i}$. Given this and utilizing the three-dimensional
product $(a\times\partial)_{i}=\varepsilon_{ijk}a^{j}\partial^{k}$, the metric
(5.1) reduces to its leading PN form:

$\displaystyle\begin{aligned} h^{\text{Kerr}}_{00}&\
=\left(2\cosh(a\times\partial)-4v_{i}\sinh(a\times\partial)^{i}\right)\frac{m}{\hat{r}}+\mathcal{O}(v^{2}),\\\
h^{\text{Kerr}}_{0i}&\
=\left(4v_{i}\cosh(a\times\partial)-2\sinh(a\times\partial)_{i}\right)\frac{m}{\hat{r}}+\mathcal{O}(v^{2}),\\\
h^{\text{Kerr}}_{ij}&\
=\left(2\delta_{ij}\cosh(a\times\partial)-4v_{(i}\sinh(a\times\partial)_{j)}\right)\frac{m}{\hat{r}}+\mathcal{O}(v^{2}).\end{aligned}$
(5.2)

Note that even at zeroth order in velocity, the solution contains non-trivial
gravito-magnetic contributions, $h^{\text{Kerr}}_{0i}$, due to the presence of
the BH spin. Conversely, an effective stress-energy distribution $T_{\mu\nu}$
can be derived that yields (5.2) via the linearized Einstein equations333At
leading PN order, the spacetime effectively decomposes into space and time
parts, yielding a simplification of the linearized Einstein equations:
$\square_{\text{ret.}}^{-1}T_{\mu\nu}\rightarrow\Delta^{-1}T_{\mu\nu}$ (see
Ref. [23] for details). $\square
h_{\mu\nu}^{\text{Kerr}}=-16\pi\mathcal{P}_{\mu\nu}{}^{\alpha\beta}T_{\alpha\beta}$.
This distribution has support only on the worldline $z^{i}(t)$ and, with the
above parameterization, is given by

$\displaystyle
T_{\mu\nu}(t,x^{i})=\hat{\mathcal{T}}_{\mu\nu}^{\text{Kerr}}\delta^{3}(\boldsymbol{x}-\hat{\boldsymbol{z}}(t))+\mathcal{O}(\hat{v}^{2}).$
(5.3)

Collecting these within the worldline action, we can construct an effective
binary BBH $S_{\text{BBH}}$ that encodes the conservative dynamics with the
complete spin information at the leading PM level [81] or leading PN level
[249, 224]. That is, given two worldlines $z_{1,2}^{\mu}$, with velocities
$u^{\mu}_{1,2}$, masses $m_{1,2}$, and two spin vectors $a^{\mu}_{1,2}$ –
conveniently collected in the sets $\mathcal{K}_{1,2}$ – the spin interactions
within the binary are obtained by integrating out the gravitational field in a
Fokker-type approach [258]. Following [81, 224], in practice, the effective
action for the second BH $S_{\text{BH}}[h,\mathcal{K}_{2}]$ (containing this
BH’s degrees of freedom $\mathcal{K}_{2}$) is evaluated at the metric of the
first BH $h\rightarrow h_{1}$, such that
$S_{\text{BH}}[h,\mathcal{K}_{2}]\rightarrow
S_{\text{BH}}[h_{1},\mathcal{K}_{2}]$. However, since the metric $h_{1}$,
explicitly given in (5.1), is effectively a map from the gravitational degrees
for freedom into that BHs’ degrees of freedom, i.e.,
$h_{1}\rightarrow\mathcal{K}_{1}$, the BBH action
$S_{\text{BH}}[h_{1}\rightarrow\mathcal{K}_{1},\mathcal{K}_{2}]\rightarrow
S_{\text{BBH}}[\mathcal{K}_{1},\mathcal{K}_{2}]$, solely depends on the BHs’
degrees of freedom.

##### Conservative dynamics

In order to write out the effective BBH action
$S_{\text{BBH}}[\mathcal{K}_{1},\mathcal{K}_{2}]$ explicitly, let us define
the spatial separation $r^{i}=z^{i}_{1}-z^{i}_{2}$, with $r=|\boldsymbol{r}|$,
between the two worldlines, as well as the spin sums
$a^{i}_{+}=a_{1}^{i}+a_{2}^{i}$ and $a^{i}_{-}=a_{1}^{i}-a_{2}^{i}$. The
angular velocity444The angular velocity tensor $\Omega^{\mu\nu}=e^{\mu}\cdot
De^{\nu}/d\lambda$ is defined by means of the body fixed frame
$e_{A}^{\mu}(\lambda)$ along the worldline. The corresponding angular velocity
vector is then given by $\Omega^{i}=\varepsilon^{i}{}_{jk}\Omega^{jk}/2$. See,
for instance, Ref. [250] for details. 3-vectors $\Omega^{i}_{1,2}$ are
introduced for completeness, however, the aligned-spin dynamics are
independent of $\Omega_{1,2}^{i}$. Finally, we define the center of mass frame
velocity $v^{i}=\dot{r}^{i}=v_{1}^{i}-v_{2}^{i}$. In Refs. [249, 224] it was
shown that after integrating out the gravitational degrees of freedom, as
described in the previous section, the effective BBH action $S_{\text{BBH}}$
reduces to the two-body Lagrangian

$\displaystyle\begin{aligned}
\mathcal{L}_{\text{BBH}}=\bigg{[}\frac{m_{1}}{2}v_{1}^{2}+\frac{m_{1}}{2}\varepsilon_{ijk}a_{1}^{i}v^{j}_{1}\dot{v}_{1}^{k}+m_{1}a^{i}_{1}\Omega_{1,i}+(1\leftrightarrow
2)\bigg{]}+\bigg{[}\cosh(a_{+}\times\partial)+2v_{i}\sinh(a_{+}\times\partial)^{i}\bigg{]}\frac{m_{1}m_{2}}{r},\end{aligned}$
(5.4)

at the leading PN level. Note that here and in the remainder of this section
$\partial_{i}r^{-1}=\partial r^{-1}/\partial z^{i}_{1}=-\partial
r^{-1}/\partial z_{2}^{i}$. So far, we have assumed a leading PN treatment at
each order in spin, but kept the dynamics unrestricted. In the following we
assume that the spin degrees of freedom are fixed, i.e., the spin vectors are
independent of time, $\dot{a}_{1,2}=0$, and aligned with the orbital angular
momentum of the system: $a_{1,2}^{i}\propto L^{i}$; hence, the motion is
confined to the plane orthogonal to $L^{i}$. For later convenience, we define
the unit vector $\ell^{i}$, such that $L^{i}=|\boldsymbol{L}|\ell^{i}$.
Varying this action with respect to the worldline $z_{1}^{i}$, the classical
EoM of the system are555The corresponding equation for $-\dot{v}^{i}_{2}$
emerges from the right hand side of (5.5) under the replacement
$a_{1}^{i}\leftrightarrow a_{2}^{i}$. [224, 249]

$\displaystyle\begin{aligned} \dot{v}_{1}^{i}=\Big{(}\partial^{i}-&\
\varepsilon^{i}{}_{jk}a_{1}^{k}v^{l}\partial_{l}\partial^{j}\Big{)}\cosh(a_{+}\times\partial)\frac{m_{2}}{r}\
+2\left(v_{j}\partial^{i}-\delta^{i}_{j}v^{k}\partial_{k}\right)\sinh(a_{+}\times\partial)^{j}\frac{m_{2}}{r}+\mathcal{O}(v^{2}).\end{aligned}$
(5.5)

A geometric approach using oblate spheroidal coordinates [249, 253, 81] or an
algebraic approach, exploiting properties of the Legendre polynomials [224],
under the assumption that the motion takes place in the plane orthogonal to
the spin vectors, can be used to resum the series of differential operators in
(5.5).

In order to present the contribution of the conservative sector needed for the
radiative dynamics, we specialize to the center of mass frame for the rest of
this section. The transformation into the center of mass variables $r^{i}$
based on (5.4) (and using the total mass $M=m_{1}+m_{2}$), is corrected by the
presence of the spins only at sub-leading orders in velocities:

$\displaystyle\begin{aligned}
\hfil\displaystyle\begin{split}z^{i}_{1}=\frac{m_{2}}{M}r^{i}-b^{i},\qquad
z^{i}_{2}=-\frac{m_{1}}{M}r^{i}-b^{i},\quad
b^{i}:=\frac{1}{M}\varepsilon^{i}{}_{jk}(v^{j}_{1}S^{k}_{1}+v^{j}_{2}S^{k}_{2}).\end{split}\end{aligned}$
(5.6)

In this center of mass frame, the EoM are readily solved for quasi-circular
motion. In that scenario, the separation $r^{i}$ is related to its
acceleration $\ddot{r}^{i}$ by $r^{i}=-\ddot{r}^{i}/\omega^{2}$, where
$\omega$ is the system’s orbital frequency. This ansatz picks out the quasi-
circular orbits allowed by the BBH EoM (5.5) and is equivalent to finding a
relation between the frequency $x=(M\omega)^{2/3}$, the BHs spins
$a^{i}_{1,2}$, and the separation of the binary $r$. This relation, at the
leading PN order at each order in the BHs’ spins, is given by [224]

$\displaystyle
r(x)=\sqrt{\frac{M^{2}}{x^{2}}+\bar{a}_{+}^{2}}\left(1-\frac{x^{3/2}M}{3}\frac{\bar{\sigma}^{*}+2\bar{a}_{+}}{M^{2}+x^{2}\bar{a}_{+}^{2}}\right),$
(5.7)

where $\bar{\sigma}^{*}=(m_{2}\bar{a}_{1}+m_{1}\bar{a}_{2})/M$ and we defined
$\bar{a}_{1,2}=a_{1,2}^{i}\ell_{i}$. It should be emphasized that the even-in-
spin part of (5.7) contains only $\mathcal{O}(v^{0})$ information, while the
odd-in-spin pieces are non-zero only at first sub-leading order in velocities,
at $\mathcal{O}(v^{1})$. This solution can then be used to compute gauge
invariant quantities of the conservative sector, such as the total binding
energy and angular momentum [224].

####  5.2.2 Linearized metric perturbations at null infinity

With the conservative results in hand, in this subsection, we compute the
gravitational waves from the BBH system at future null infinity. We start by
briefly reviewing the general approach of mapping the source’s multipole
moments into the radiation field, and then move to the derivation of the TT
part of the linear metric perturbations (the gravitational waves) at null
infinity utilizing this mapping.

##### General approach

A natural choice of gauge invariant quantity capturing the radiative dynamics
at null infinity is the Newman-Penrose Weyl scalar $\Psi_{4}$. This contains
both polarization states, $h_{+}$ and $h_{\times}$, of the emitted waves,
which are the observables measured by gravitational wave detectors. Upon
choosing a suitable null tetrad, the TT part of the gravitational field,
$h_{\mu\nu}^{\text{TT}}$, can be related to $\Psi_{4}$:

$\displaystyle\Psi_{4}\sim\ddot{h}_{+}-i\ddot{h}_{\times}=\bar{m}^{\mu}\bar{m}^{\nu}\ddot{h}^{\text{TT}}_{\mu\nu}.$
(5.8)

The complex conjugate pair $\\{m^{\alpha},\bar{m}^{\alpha}\\}$ is typically
defined with respect to the flat spherically symmetric angular coordinate
directions $m=(\boldsymbol{\Theta}+i\boldsymbol{\Phi})/\sqrt{2}$. With this
choice in place, we restrict our attention to the spatial components
$h^{\text{TT}}_{ij}$, as these contain the full information of $\Psi_{4}$,
i.e., the radiative, non-stationary, degrees of freedom666As we will see
below, this choice of purely spatial $m^{\alpha}$ is equivalent to choosing a
gauge, in which the graviton polarization tensor is also purely spatial..

In the previous section, we summarized the leading PN conservative dynamics of
a spinning BBH to all orders in their spins. Given this, the well-established
multipolar post-Minkowskian formalism [122, 123, 124, 125, 23] is ideally
suited to determine the time-dependent metric perturbations at null infinity.
Within this framework, the stress energy distribution of the source,
$T_{\mu\nu}^{\text{source}}$, is mapped into a set of mass and current
symmetric and trace free (STF) source multipole moments
$\mathcal{I}_{i_{1}\dots i_{\ell}}(t)$ and $\mathcal{J}_{i_{1}\dots
i_{\ell}}(t)$. We denote $\langle i_{1}\dots i_{\ell}\rangle$ as the STF
projections of the indices $i_{1}\dots i_{\ell}$. Then the STF multipole
moments evaluated at the retarded time $T_{R}=t-R$ are defined by [23]

$\displaystyle\begin{aligned} \mathcal{I}_{i_{1}\dots i_{\ell}}=&\ \int
d\mu\left(\delta_{\ell}x_{\langle i_{1}\dots i_{\ell}\rangle}\Sigma-
f_{1,\ell}\delta_{\ell+1}x_{\langle ii_{1}\dots
i_{\ell}\rangle}\dot{\Sigma}^{i}+f_{2,\ell}\delta_{\ell+2}x_{\langle
iji_{1}\dots
i_{\ell}\rangle}\ddot{\Sigma}^{ij}\right)(\boldsymbol{x},T_{R}+zr),\\\
\mathcal{J}_{i_{1}\dots i_{\ell}}=&\ \int d\mu\ \varepsilon_{ab\langle
i_{\ell}}\left(\delta_{\ell}x_{i_{1}\dots
i_{\ell-1}\rangle}{}^{a}\Sigma^{b}-g_{1,\ell}\delta_{\ell+1}x_{i_{1}\dots
i_{\ell-1}\rangle
c}{}^{a}\dot{\Sigma}^{bc}\right)(\boldsymbol{x},T_{R}+zr),\end{aligned}$ (5.9)

where $x_{i_{1}\dots i_{\ell}}=x_{i_{1}}\dots x_{i_{\ell}}$,

$\displaystyle f_{1,\ell}=\frac{4(2\ell+1)}{(\ell+1)(2\ell+3)},$
$\displaystyle f_{2,\ell}=\frac{2(2\ell+1)}{(\ell+1)(\ell+2)(2\ell+5)},$
$\displaystyle g_{1,\ell}=\frac{2\ell+1}{(\ell+2)(2\ell+3)},$ (5.10)

and the integration measure $\int d\mu=\text{FP}\int
d^{3}\boldsymbol{x}\int_{-1}^{1}dz$. The source energy-momentum distribution
enters in $\Sigma$, via (valid only at leading PN orders)777At sub-leading PN
orders, the stress energy of the emitted gravitational waves contributes to
$\Sigma$.

$\displaystyle\Sigma=T^{00}+T^{ij}\delta_{ij},$
$\displaystyle\Sigma^{i}=T^{0i},$ $\displaystyle\Sigma^{ij}=T^{ij}.$ (5.11)

The source’s finite-size retardation effects are contained in the $z$-integral
with $\delta_{\ell}=\delta_{\ell}(z)$ in (5.9), which are given explicitly in
eq. (120) of Ref. [23]. At the orders considered in this work, at the leading
PN orders, finite size-retardation effects vanish and the $z$-integral
trivializes: $\int_{-1}^{1}dz\
\delta_{\ell}(z)f(\boldsymbol{x},T_{R}+rz)=f(\boldsymbol{x},T_{R})+\mathcal{O}(v^{2})$.
We discuss in section 5.3.1, how a similar structure as in (5.9) appears in
the scattering amplitudes approach, as well as what precisely encapsulates the
“finite size" retardation effects in that context. The lowest order moments
$\mathcal{I}$, $\mathcal{I}_{i}$, and $\mathcal{J}_{i}$ are constants of
motion representing the total conserved energy, center of mass position and
total angular momentum, respectively. Only for $\ell\geq 2$, do the multipoles
contribute non-trivially.

A matching scheme enables to directly relate these functionals for the
source’s stress-energy distribution, to the radiation field at null infinity
(at 1PM order)888Beyond linear theory, corrections to these multipole moments
are necessary [23]. [23]

$\displaystyle
h_{ij}=-4\sum_{\ell=2}^{\infty}\frac{(-1)^{\ell}}{\ell!}\left[\partial_{i_{1}\dots
i_{\ell-2}}\ddot{\mathcal{I}}_{ij}{}^{i_{1}\dots
i_{\ell-2}}R^{-1}+\frac{2\ell}{\ell+1}\partial_{ai_{1}\dots
i_{\ell-2}}\varepsilon^{ab}{}_{(i}\dot{\mathcal{J}}_{j)b}{}^{i_{1}\dots
i_{\ell-2}}R^{-1}\right].$ (5.12)

Here $\partial_{a}R^{-1}=-N_{a}/R^{2}$ is to be understood as the derivative
in the background Minkowski spacetime, where $N_{a}$ is radially outwards
pointing from the source to spatial infinity, with $N_{a}N^{a}=1$. To solely
focus on the radiation at null infinity, we work to leading order in the
expansion in $R^{-1}$. Therefore, the spatial derivatives in (5.12) act purely
on the source multipole moments, and there, can be traded for time
derivatives: $\partial_{a}f(t-R)=-\dot{f}N_{a}$. Similarly, the total
instantaneous gravitational wave energy flux $\mathcal{F}$ can be derived
directly from the source multipole moments [23].

##### Gravitational radiation from spinning binary black hole

At the 1PM level, non-linear effects vanish such that the energy-momentum of
the BBH is simply the superposition of two linearized Kerr BHs’ energy
momentum distributions (5.3),
$T_{\mu\nu}^{\text{source}}=T_{\mu\nu}^{\text{Kerr},1}+T_{\mu\nu}^{\text{Kerr},2}$.
This superposition holds in the conservative sector, while the radiative
dynamics are derived directly from derivatives acting on
$T_{\mu\nu}^{\text{source}}$ in the manner described in the previous section.
From the scattering amplitudes perspective, this superposition is reflected in
the only two-channel factorization of the classical 5-point amplitude, into
the product of a 3-pt amplitude and the gravitational Compton amplitude as
given in (2).

Leading order in velocities – As the radiative quantities $h_{ij}^{\text{TT}}$
and $\mathcal{F}$ depend on time derivatives of the source multipole moments,
we focus on time-dependent terms after fixing the angular momentum dynamics.
For the case of the above spinning BBH with aligned spins, at the leading PN
order, we expand the source $T_{\mu\nu}^{\text{source}}$ analogously to (5.2).
Given this, the resulting leading-in-velocity contributions to the source
multipole moments, utilizing (5.9), are [259, 223, 224]

$\displaystyle\mathcal{I}^{ij}_{(0)}=m_{1}z_{1}^{\langle
ij\rangle}+(1\leftrightarrow 2),$
$\displaystyle\mathcal{J}^{ij}_{(0)}=\frac{3}{2}S_{1}^{\langle
i}z_{1}^{j\rangle}+(1\leftrightarrow 2),$ (5.13)

where $(0)$ indicates the order in velocities. It should be stressed that
these are all the multipoles needed for the gravitational waveform to all
orders in the BHs’ spins, at leading order in velocity [224]. From the
amplitudes perspective, this will be reflected in the need for only the scalar
and linear-in-spin scattering amplitudes at the leading orders in velocities.
While all higher-order spin terms in the source multipole moments vanish
identically, spin contributions to the waveform at arbitrary order in the spin
expansion could enter through the solution to the EoM (5.7). We see below that
this solution to the classical EoM (5.5) introduces non-zero contributions at
arbitrary orders in the BHs’ spins for quasi-circular orbits.

Given (5.13), the metric perturbation at null infinity, for general orbits at
zeroth order in velocities, assuming aligned spins, is

$\displaystyle
h^{(0)ij}_{S^{\infty}}(T_{R},R,\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{2m_{1}}{R}\bigg{\\{}\frac{d^{2}}{dt^{2}}\left[z_{1}^{i}z_{1}^{j}\right]+\varepsilon_{pq}{}^{(i}\left(a_{1}^{j)}\dot{v}_{1}^{p}+\dot{v}_{1}^{j)}a_{1}^{p}\right)N^{q}\bigg{\\}}\bigg{|}_{t=T_{R}}+(1\leftrightarrow
2),$ (5.14)

i.e. the Einstein Quadrupole formula with spinning corrections for a binary
system. We specialize to quasi-circular orbits by introducing the orthogonal
unit vectors

$\displaystyle n^{i}=r^{i}/r=(\cos\omega t,\sin\omega t,0)^{i},$
$\displaystyle\lambda^{i}=v^{i}/v=(-\sin\omega t,\cos\omega t,0)^{i},$ (5.15)

in the center of mass frame that rotate with frequency $\omega$ in the orbital
plane. The spin vectors $a_{1,2}^{i}\propto\ell^{i}$ are aligned orthogonal to
the orbital plane, $n^{i}\lambda^{j}\varepsilon_{ij}{}^{k}=\ell^{k}$, such
that $\ell^{i}=(0,0,1)^{i}$. Furthermore, the TT projector

$\displaystyle\Pi^{ij}{}_{kl}=P^{i}{}_{k}P^{j}{}_{l}-\frac{1}{2}P^{ij}P_{kl}$
(5.16)

is defined relative to $N_{a}$, where $P_{ij}=\delta_{ij}-N_{i}N_{j}$.
Utilizing (5.6), together with the solution (5.7) to the EoM, as well as
(5.13), the gravitational waves emitted by the spinning BBH to all orders in
the BHs’ spins is conveniently written as

$\displaystyle h^{\text{TT}}_{ij}(T_{R})=\frac{2\mu}{R}\
\Pi_{ij}{}^{ab}\hat{h}_{ab}\Big{|}_{t=T_{R}},$ (5.17)

where at leading order in velocities, we have
$\hat{h}^{(0)}_{ab}=\hat{h}^{(0),\text{I2}}_{ab}+\hat{h}^{(0),\text{J2}}_{ab}$,
with

$\displaystyle\begin{aligned} \hat{h}^{(0),\text{I2}}_{ab}=&\
-2x\left(1+\frac{\bar{a}_{+}^{2}x^{2}}{M^{2}}\right)(n_{a}n_{b}-\lambda_{a}\lambda_{b})\\\
\hat{h}^{(0),\text{J2}}_{ab}=&\
-\frac{x^{2}\bar{a}_{-}}{M}\sqrt{1+\frac{\bar{a}_{+}^{2}x^{2}}{M^{2}}},\
\varepsilon_{kl(a}(\ell_{b)}n^{k}+n_{b)}\ell^{k})N^{l}.\end{aligned}$ (5.18)

Notice that here, the odd-in-spin contribution,
$\hat{h}^{(0),\text{J2}}_{ab}$, is a series that has non-zero coefficients at
arbitrary orders in spin, arising from the odd part of the solution (5.7),
while, on the other hand, the even-in-spin part,
$\hat{h}^{(0),\text{I2}}_{ab}$, provides coefficients that vanish for
$\mathcal{O}(a^{\ell\geq 3})$. This is analogous to the cancellations observed
in the conservative and radiative sectors reported in Ref. [224]. We find
agreement with the results reported in Refs. [260, 261, 259] to the respective
finite order in spin. To check for consistency to all orders in spin, the
gravitational wave modes are extracted from the spatial part of the metric
perturbations, in (5.18), by projecting onto a suitably defined basis of spin-
weighted spherical harmonics, ${}_{-2}Y_{\ell m}(\Theta,\Phi)$. Explicitly,
the gravitational wave modes $h^{\ell m}$ are defined to be $h^{\ell m}=\int
d\Omega\ {}_{-2}\bar{Y}_{\ell
m}(\Theta,\Phi)\bar{m}^{\mu}\bar{m}^{\nu}h^{\text{TT}}_{\mu\nu}$. These modes,
obtained from (5.18) in conjunction with the above defined polarization tensor
$\bar{m}^{\alpha}\bar{m}^{\beta}$, agree with the results in Ref. [224] to all
orders in the BHs’ spins at leading order in their velocities.

Sub-leading order in velocities – The sub-leading corrections to the above
radiation field are obtained in much the same way. The additional
contributions to the source multipole moments, beyond the leading pieces
(5.13), at sub-leading orders in velocities are [259, 223, 224]

$\displaystyle\begin{aligned} \mathcal{I}^{ij}_{(1)}=&\
\frac{4}{3}\left(2v_{1}^{a}S^{b}_{1}\varepsilon_{ab}{}^{\langle
i}z_{1}^{j\rangle}-z_{1}^{a}S_{1}^{b}\varepsilon_{ab}{}^{\langle
i}v^{j\rangle}\right)+(1\leftrightarrow 2),\\\ \mathcal{I}^{ijk}_{(1)}=&\
m_{1}z_{1}^{\langle ijk\rangle}-\frac{3}{m_{1}}S_{1}^{\langle
i}S_{1}^{j}z_{1}^{k\rangle}+(1\leftrightarrow 2),\\\ \mathcal{J}^{ij}_{(1)}=&\
m_{1}z_{1}^{a}v_{1}^{b}\varepsilon_{ab}{}^{\langle
i}z_{1}^{j\rangle}+\frac{1}{m_{1}}v_{1}^{a}S_{1}^{b}\varepsilon_{ab}{}^{\langle
i}S_{1}^{j\rangle}+(1\leftrightarrow 2),\\\ \mathcal{J}^{ijk}_{(1)}=&\
2S_{1}^{\langle i}z_{1}^{jk\rangle}+(1\leftrightarrow 2).\end{aligned}$ (5.19)

Also here, we focused only on those pieces that are time-dependent, i.e., that
will contribute non-vanishing terms in $h_{ij}^{(1)\text{TT}}$. Additionally,
as pointed out above, these are all necessary contributions for the full all
orders-in-spin information at sub-leading orders in velocities (at leading PN
order) [224]. Using this, together with the mapping (5.12), the decomposition
(5.17), and
$\hat{h}^{(1)}_{ab}=\hat{h}^{(1),\text{I2}}_{ab}+\hat{h}^{(1),\text{J2}}_{ab}+\hat{h}^{(1),\text{I3}}_{ab}+\hat{h}^{(1),\text{J3}}_{ab}$,
the sub-leading contribution $h_{ij}^{(1)\text{TT}}$ to all orders in spin
from a spinning binary black hole on quasi-circular orbits are

$\displaystyle\begin{aligned} \hat{h}^{(1),\text{I2}}_{ab}=&\
\frac{4x^{5/2}}{3M^{3}}\left(2\bar{a}_{+}M^{2}+\left(M^{2}-2r_{e}^{2}x^{2}\right)\bar{\sigma}^{*}\right)\left(n_{a}n_{b}-\lambda_{a}\lambda_{b}\right),\\\
\hat{h}^{(1),\text{J2}}_{ab}=&\
\frac{x^{5/2}}{3M^{4}r_{e}}\left[2r_{e}^{4}x^{2}\delta
m+\bar{a}_{-}M\left(2\bar{a}_{+}(M^{2}-r_{e}^{2}x^{2})+3r_{e}^{2}x^{2}\bar{\sigma}+M^{2}\bar{\sigma}^{*}\right)\right]\\\
&\ \times\varepsilon_{pq(a}N^{q}\left(n^{p}\ell_{b)}+n_{b)}\ell^{p}\right),\\\
\hat{h}^{(1),\text{I3}}_{ab}=&\
\frac{r_{e}x^{9/2}}{15M^{4}}\left[15\bar{a}_{+}\bar{a}_{-}M\ell_{\langle
a}\ell_{b}\lambda_{k\rangle}-r_{e}^{2}\delta m\left(30\lambda_{\langle
a}\lambda_{b}\lambda_{k\rangle}-105n_{\langle
a}n_{b}\lambda_{k\rangle}\right)\right]N^{k},\\\
\hat{h}^{(1),\text{J3}}_{ab}=&\
-\frac{48r_{e}^{2}x^{9/2}\bar{\sigma}^{*}}{6M^{3}}\epsilon_{pq(a}\delta_{b)k}n^{(k}\lambda^{p}\ell^{e)}N^{q}N_{e}.\end{aligned}$
(5.20)

Here $r_{e}=(\bar{a}_{+}^{2}+M^{2}/x^{2})^{1/2}$, which is just the leading-
in-velocities (even-in-spin) solution to the classical EoM (5.7) for quasi-
circular orbits. We check the gravitational wave modes obtained from (5.20)
with those presented in Ref. [224] and find agreement to all orders in spin.
Additionally, we compute the gauge invariant gravitational wave energy flux
with the above result together with the leading-in-velocities radiation field
and find agreement with results reported in [223, 224] (see also a detailed
discussion in section 5.3.3 below). Finally, in order to compare to the scalar
amplitude at first sub-leading orders in the BHs’ velocities in section 5.3.2,
we also present the radiation field of a non-spinning BBH system on general
orbits, to sub-leading order in velocities:

$\displaystyle
h^{(1),ij}_{S^{0},\text{TT}}=\frac{2m_{1}}{3R}\Pi^{ij}{}_{ab}\left[4\varepsilon_{pq}{}^{(a}\Big{\\{}\partial_{t}^{2}(\varepsilon_{cd}{}^{e)}z_{1}^{c}v_{1}^{d}\delta_{e}{}^{\langle
b}z_{1}^{p\rangle})\Big{\\}}N^{q}+N_{k}\partial_{t}^{3}(z_{1}^{\langle
a}z_{1}^{b}z_{1}^{k\rangle})\right]+(1\leftrightarrow 2).$ (5.21)

###  5.3 Scattering Amplitudes derivation

In the previous sections, we obtained the form of the gravitational waves
emitted from a spinning BBH on general closed orbits with aligned spins, to
leading order in the BHs’ velocities (5.14) [and on quasi-circular orbits
given in (5.18)], whereas at sub-leading order in $v$, and for quasi-circular
orbits, we derived (5.20), at each order (and to all orders) in the BHs’
spins. In the following, we show that these results follow directly from the
classical limit of the spinning 5-point scattering amplitudes derived in
chapter 4. More precisely, at leading order in velocity there is a one-to-one
correspondence between the source’s mass and current multipole moments (5.13),
and the scalar and linear-in-spin contribution to the scattering amplitude,
respectively. This correspondence allow us to derive the linear in spin,
general orbit result for the radiated gravitational field (5.14), from an
amplitudes perspective. At quadratic order in the BHs’ spins, and for quasi-
circular orbits, we demonstrate that the contribution from the quadratic in
spin amplitude is canceled by the contribution of the scalar amplitude in
conjunction with the $\mathcal{O}(S^{2})$-piece of the EoM (5.5). This leaves
only the quadrupole field, (5.44), supplemented with the solution to the EoM
(5.7), to enter at quadratic order in spin. Although we explicitly demonstrate
the cancellation for quasi-circular orbits and up to quadratic order in spin
only, we expect this theme to continue to hold for more complicated bound
orbits, as well as to higher spin orders in the 5-point scattering amplitude,
as suggested by the classical multipole moments (5.13). At sub-leading orders
in the BHs’ velocities the situation becomes more complicated; there, we
demonstrate the matching of the amplitudes to the classical computation in the
spin-less limit for quasi-circular orbits, and briefly comment on extensions
to higher orders in spin. In this section, we use the mostly minus signature
convention for the flat metric $\eta_{\mu\nu}=\rm{diag}(1,-1,-1,-1)$.

####  5.3.1 General approach

To compute the radiated field at future null infinity from the BBH system we
follow the approach used by Goldberger and Ridgway in [31] to derive the
Quadrupole formula, and extend it to include relativistic and spin effects.
This approach is based on the classical EoM for the orbiting objects in
combination with the corresponding 5-point (spinning) scattering amplitude
(see Figure 2.3). It is valid for BBHs whose components have Schwarzschild
radii $r_{1,2}=2m_{1,2}$ much smaller than their spatial separation $r$, i.e.,
$r_{1,2}\ll r$, while the radiation field wavelength is much bigger than the
size of the individual components $\lambda\gg m_{1,2}$, as well as the size of
the system $\lambda\gg r$999In the long distance separation regime, radiation
reaction effects can be neglected, since they become important only when the
separation of the two bodies is comparable to the system’s gravitational
radius [262] eq. $36.11$.. Therefore, we expect our results to be situated in
the PN regime of the binary inspiral101010We stress that even though we
concentrate mostly in the computation of gravitational waveform, an analogous
derivation follows for electromagnetic radiation, as already pointed out in
[31]..

Let us start by noting that in the limit in which $R\rightarrow\infty$, where
$R$ is the distance from the source to the observer (i.e., the radial
coordinate in Bondi-Sachs gauge) as defined above, the time-domain waveform at
retarded time $T_{R}$, has the asymptotic form [263] (see also (1.18))

$h_{\text{TT}}^{ij}(T_{R},R,\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{\kappa}{16\pi
R}\Pi^{ij}{}_{ab}\int
d\bar{\omega}e^{-i\bar{\omega}\,T_{R}}T^{ab}(\bar{\omega},\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2}).$
(5.22)

Here $\kappa^{2}=32\pi$ (recall we set $G=1$), $\bar{\omega}$ is the frequency
of the radiated wave with four momentum
$k^{\mu}=\bar{\omega}N^{\mu}=\bar{\omega}(1,\boldsymbol{N})^{\mu}$, and
$\Pi^{ij}{}_{ab}$ is the TT-projector defined in (5.16). As above, the
locations of the binary’s components are denoted by $z^{i}_{1,2}$. Analogous
to the previous section, we focus only on the spatial components of
$h^{\mu\nu}$, which contain all the radiative degrees of freedom. In what
follows we also simplify the notation for the source
$T^{ab}(\bar{\omega},\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2})\to
T^{ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})$, where it is understood that
$k^{\mu}$ has implicit the dependence in both, $\bar{\omega}$ and
$\boldsymbol{N}$.

The source $T^{ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})$, is related
directly to the 5-point scattering amplitude in Figure 2.3; therefore, in
order to focus on the spatial components, it is sufficient to work in a gauge
in which the graviton polarization tensor
$\epsilon^{\mu\nu}=\epsilon^{\mu}\epsilon^{\nu}$, is the tensor product of two
purely spatial polarization vectors $\epsilon^{\nu}$. From the classical
perspective, this choice of gauge is analogous to the conjugate pair
$\\{m^{\alpha},\bar{m}^{\alpha}\\}$ (defined in section 5.2.2) to be purely
spatial. Notice, however, the radiation field computed from a 5-point
scattering amplitude, and the corresponding field computed classically in the
previous section, can in general differ by a time independent constant, for
which, gravitational observables such as the gravitational wave energy flux,
or the radiation scalar will be insensitive to, since they are computed from
one or two time derivatives of the waveform. As shown below, this is directly
related to a freedom in choice of an integration by parts (IBP) prescription
in (5.22).

We proceed by writing the explicit form of the source
$T^{ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})$ in terms of the classical
5-point scattering amplitude. In the classical computation, $T^{ij}$
corresponds to the source entering on the right hand side of field equations,
at a given order in perturbation theory. To leading order, for scalar
particles, it was shown in [31] that the source can be rearranged in such a
way, so that the scalar 5-point amplitude can be identified as the main
kinematic object entering the graviton phase space integration, as well as the
integration over the particles proper times (which account for the particles
history). In this thesis we propose that formula to also hold for spinning
particles. That is,

$T^{ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{i}{m_{1}m_{2}}\int
d\tau_{1}d\tau_{2}\hat{d}^{4}q_{1}\hat{d}^{4}q_{2}\hat{\delta}^{4}\left(k-q_{1}-q_{2}\right)e^{iq_{1}{\cdot}z_{1}}e^{iq_{2}{\cdot}z_{2}}\langle
M_{5}^{ij}(q_{1},q_{2},k)\rangle.$ (5.23)

Here $\langle M_{5}^{ij}\rangle$ is the classical 5-point amplitude .
Conventions for the particles’ momenta and the spins are shown in Figure 2.3,
with the condition for momentum conservation $q_{1}+q_{2}=k$. We have used the
notation $\hat{d}^{4}q_{i}=\frac{d^{4}q_{i}}{(2\pi)^{4}}$, and similarly for
the momentum-conserving delta function
$\hat{\delta}^{4}(p)=(2\pi)^{4}\delta^{4}(p)$, in analogy to the notation used
in section 1.2.

This notation was in fact selected for a good reason. To motivate this
formula, although this is by no means a formal derivation, as already observed
in [31] for the tree-level amplitude, we can take the expression for the
linear in amplitude contribution to the radiation field in KMOC form for
scattering scenarios (1.16) for the gravitational case

$\mathcal{J}=\lim_{\hbar\to
0}\frac{1}{m_{1}m_{2}}\Big{\langle}\int\prod_{i=1}^{2}\left[\hat{d}^{4}q_{i}\hat{\delta}(v_{i}{\cdot}q_{i}-q_{i}^{2}/(2m_{i}))e^{ib_{i}{\cdot}q_{i}}\right]\hat{\delta}^{4}(k-q_{1}-q_{2})M_{5}\Big{\rangle},$
(5.24)

and use the integral representation for the on-shell delta functions
$\delta(x)\sim\int dye^{ixy}$. Identifying the asymptotic trajectories for the
particles $z_{i}(\tau_{i})=b_{i}^{\mu}+v_{i}^{\mu}\tau_{i}$, plus a quantum
correction $z_{Q}^{\mu}(\tau_{i})=-\frac{q_{i}}{2m_{i}}\tau_{i}$, and upon
restoring the $\hbar$-counting in the exponential, the radiation kernel can be
rewritten as

$\mathcal{J}=\lim_{\hbar\to
0}\frac{1}{m_{1}m_{2}}\Big{\langle}\int\prod_{i=1}^{2}\left[d\tau_{i}\hat{d}^{4}q_{i}e^{iq_{i}{\cdot}(z_{i}(\tau_{i})+\hbar
z_{Q}(\tau_{i}))}\right]\hat{\delta}^{4}(k-q_{1}-q_{2})M_{5}\Big{\rangle}.$
(5.25)

In the classical limit, and to leading order in perturbation theory, we can
simply drop quantum correction to the particles trajectories
$z_{Q}^{\mu}(\tau_{i})$, and recover the formula (5.23) upon promoting
$z_{i}(\tau_{i})$ to be valid for generic time dependent orbits. A similar
argument can be given to derive the BHs’ EoM directly from the amplitude, in
this case, an instantaneus impulse, starting from the linear in amplitude
contribution to the linear impulse (1.12).

$\Delta
p_{1}^{\mu}=\frac{1}{4m_{1}m_{2}}\int\hat{d}^{4}q\,d\tau_{1}\,d\tau_{2}iq^{\mu}e^{-i(z_{2}(\tau_{2})-z_{1}(\tau_{1})){\cdot}q}\langle
M_{4}\rangle\,.$ (5.26)

Below we will show how to use this formula to obtain the BHs EoM to leading
order in the velocity expansion.

The position vectors are $z^{\mu}_{A}=(\tau_{A},\boldsymbol{z}_{A})^{\mu}$,
with $A=1,2$, as described in section 5.2.1, where the proper times
$\tau_{A}$, parametrize the BHs’ trajectories. Here the product of the
exponential functions, $\prod_{A}e^{iq_{A}\cdot z_{A}}$, represents the two-
particles initial state where each particle is taken to be in a plane-wave
state. This is nothing but the Born approximation in Quantum Mechanics (See
also [121]). Here we emphasize formulas (5.23) and (5.26) are valid up to
subleading order in the velocity expansion, as we are dropping the quantum
corrections $z_{Q}^{\mu}(\tau_{i})$. In addition, the notice the delta
functions $\hat{\delta}(v_{i}{\cdot}q_{i}-q_{i}^{2}/(2m_{i}))$ effectively
impose the on-shell condition for particles in a scattering scenario. For
general trajectories, outgoing particles are no longer on-shell and therefore
the on-shell condition cannot be imposed in the amplitude.

We have striped away the graviton polarization tensor in (5.23), assuming
there exists the aforementioned gauge fixing for which the graviton
polarization tensor is purely spatial. We can further rewrite the source using
the symmetric variable $q=(q_{1}-q_{2})/2$, as well as exploiting the momentum
conserving delta function to remove one of the $q_{i}$-integrals. The result
reduces to

$T^{ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{i}{m_{1}m_{2}}\int
d\tau_{1}d\tau_{2}\hat{d}^{4}qe^{ik\cdot\tilde{z}}e^{-iq\cdot z_{21}}\langle
M_{5}^{ij}(q,k)\rangle,$ (5.27)

where $\tilde{z}=(z_{1}+z_{2})/2$ and $z_{BA}=z_{B}-z_{A}$. Since we are
interested in the bound-orbit problem, we take the slow-motion limit.
Therefore, we can write the momenta of the BHs moving on closed orbits in the
form $p_{1,2}^{\mu}=m_{1,2}v^{\mu}_{1,2}$. As noted above, we choose the frame
in which
$v_{1,2}^{\mu}=(1,\boldsymbol{v}_{1,2})^{\mu}+\mathcal{O}(v_{1,2}^{2})$, where
$v^{i}_{1,2}=dz^{i}_{1,2}/dt$, i.e. with the proper times $\tau_{1,2}$
replaced by the coordinate time (see details below). On the other hand, in the
closed orbits scenario the typical frequency of the orbit $\omega$, scales
with $v$ as $\omega\sim v/r$, where $\omega=v/r$ for quasi-circular orbits
(see also (5.15)). In this bound-orbits case, the integration in $q$ is
restricted to the potential region (technically, as an expansion in powers of
$q^{0}/|\mathbf{q}|$), where the internal graviton momentum has the scaling
$q\sim(v/r,1/r)$, while the radiated graviton momentum scaling is
$k\sim(v/r,v/r)=\bar{\omega}(1,\boldsymbol{N})$ (with
$\bar{\omega}\sim\omega$). Integration in the potential region ensures that
from the retarded propagators,

$\frac{1}{(q_{0}+i0)^{2}-\boldsymbol{q}^{2}}\rightarrow\frac{1}{v^{2}(q_{0}+i0)^{2}-\boldsymbol{q}^{2}}\approx-\frac{1}{\boldsymbol{q}^{2}}+\mathcal{O}(v^{2}),$
(5.28)

entering in the scattering amplitude, retardation effects only become
important at order $\mathcal{O}(v^{2})$, which we do not consider here. At
subleading orders in velocities, the amplitude $\langle
M_{5}^{ij}(q,k)\rangle$ has no explicit dependence on $q^{0}$. This takes care
of the $q^{0}$-integration in (5.27), which results in the delta function
$\delta(t_{2}-t_{1})$; this can be used to trivialize one of the time
integrals111111As a connection with the classical computation, the source
multipole moments [given in (5.13)] contain the finite size and retardation
effects of the binary, though, at leading and sub-leading orders in
velocities, these effects vanish (see e.g., [23]), which is equivalent to the
replacement $\tau_{1,2}\rightarrow t$ above.. With all these simplifications
in hand, the source (5.27) becomes

$T^{(0)\,ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2}){=}\frac{i}{m_{1}m_{2}}\int
dt\hat{d}^{3}\boldsymbol{q}e^{i\bar{\omega}\,t{-}i\boldsymbol{q}{\cdot}\boldsymbol{z}_{21}}\langle
M_{5,{\rm S}^{0}}^{(0)\,ij}(\boldsymbol{q},\bar{\omega}){+}M_{5,{\rm
S}^{1}}^{(0)\,ij}(\boldsymbol{q},\bar{\omega}){+}M_{5,{\rm
S}^{2}}^{(0)\,ij}(\boldsymbol{q},\bar{\omega})\rangle{+}\cdots,$ (5.29)

where the amplitude was written in a spin-multipole decomposition. The
superscript (0) indicates that we restrict these to the leading-in-$v$
contribution to the scattering amplitude (See section 5.3.2 for the
computation at the first sub-leading order in velocities contribution, for
spinless BHs ).

##### Instantaneous impulse and particles EoM

In the seminal work of Dass and Soni [121], it was claimed the conservative
4-point amplitude can be used to reproduce the particles EoM for scalar
sources. In this section we will show that indeed, the instantaneous impulse
formula (5.26) can be used to reproduce the particles EoM (5.5), to leading
order in velocity, but to all spin orders from the conservative amplitude. For
that we will first have to compute the classical conservative $\langle
M_{4}^{\text{gr}}\rangle$ amplitude to all orders in spin. At leading order in
perturbation theory this is just given by the t-channel cut as indicated in
(2). We will then need the classical limit of the 3-point amplitud to all
orders in spin. In chapter 7 we will show the spin exponentiation of the
classical gravitational 3-point amplitude (4.34), in 4-dimensions in terms of
the Kerr BH spin vector $a^{\mu}$ takes the form

$\langle
A_{3}^{a\,\text{gr}}(p_{1},q^{\pm})\rangle=\kappa(p_{1}{\cdot}\epsilon^{\pm})^{2}e^{\pm
a{\cdot}q}\,,$ (5.30)

where we have included explicitly the helicity of the emitted graviton. The
classical 4-point amplitude can be computed then following the theme of [57,
58].

$\displaystyle\langle M_{4}^{\text{gr}}\rangle$
$\displaystyle=\frac{1}{q^{2}}\Big{[}\langle
A_{3}^{a_{1}\,\text{gr}}(p_{1},q^{-})\rangle\times\langle
A_{3}^{a_{2}\,\text{gr}}(p_{2},-q^{+})\rangle+\langle
A_{3}^{a_{1}\,\text{gr}}(p_{1},q^{+})\rangle\times\langle
A_{3}^{a_{2}\,\text{gr}}(p_{2},-q^{-})\rangle\Big{]}$ (5.31)
$\displaystyle=\frac{m_{1}m_{2}}{q^{2}}\Big{[}\frac{x_{1}^{2}}{x_{2}^{2}}e^{q{\cdot}a_{+}}+\frac{x_{2}^{2}}{x_{1}^{2}}e^{-q{\cdot}a_{+}}\Big{]}$
(5.32)

where we have summed over the helicities of the exchanged graviton, and set
$a_{+}=a_{1}+a_{2}$. We have also used the well known $x_{i}$-helicity
variables from the spinor helicity formalism [68], by fixing the little group
rescaling of the internal graviton as follows [57]

$x_{2}=\sqrt{2}\frac{p_{2}{\cdot}\epsilon^{-}(-q)}{m_{2}}=-\sqrt{2}\frac{p_{2}{\cdot}\epsilon^{-}}{m_{2}}=1$
(5.33)

which implies

$x_{1}^{-1}=-\sqrt{2}\frac{p_{1}{\cdot}\epsilon^{+}}{m_{1}}=\gamma(1-v)\,,\quad
x_{1}=-\sqrt{2}\frac{p_{1}{\cdot}\epsilon^{-}}{m_{1}}=\gamma(1+v),\,,$ (5.34)

where $\gamma=\frac{1}{\sqrt{1-v^{2}}}=\frac{p_{1}{\cdot}p_{2}}{m_{1}m_{2}}$.
Using the on-shell identity
$i\epsilon_{\mu\nu\rho\sigma}p_{1}^{\mu}p_{2}^{\nu}q^{\rho}a^{\sigma}=m_{1}m_{2}\sqrt{\gamma^{2}-1}q{\cdot}a$,
and defining $\hat{\boldsymbol{p}}$ as a unit vector in the direction of the
relative momentum in the acCoM, the classical amplitude simply becomes

$\displaystyle\langle
M_{4}^{\text{gr}}\rangle=\frac{\kappa^{2}m_{1}^{2}m_{2}^{2}}{2q^{2}}\gamma^{2}\sum_{\pm}(1\pm
v)^{2}e^{\pm i\boldsymbol{q}\times\boldsymbol{p}{\cdot}\boldsymbol{a}_{+}}\,,$
(5.35) $\displaystyle\langle
M_{4}^{\text{gr}}\rangle=\frac{\kappa^{2}m_{1}^{2}m_{2}^{2}}{2q^{2}}\gamma^{2}\sum_{\pm}(1\pm
v)^{2}e^{\pm iq_{i}\times a^{i}_{+}}\,.$ (5.36)

where in the second line we have specialized to the aligned spin scenario. In
the Center of Mass (CoM) frame, $q$ is purely spatial. We can then use (5.35)
into (5.26), the $q^{0}$ integral results into the delta function
$\delta(t_{1}-t_{2})$, reflecting non-retardation effects in the conservative
sector at this order in perturbation theory. Finally, the particles EoM result
from $\boldsymbol{q}$-integration, and to leading order in the velocity
expansion, and after using the fundamental theorem of calculus121212Here we
assume boundary terms do not contribute. At leading order in the velocity
expansion seems to be a valid assumption since the amplitudes recover the
correct results for the EoM., with $\frac{\Delta p_{1}}{\Delta
t}=m_{1}\dot{v}_{1}$, results into

$\dot{v}^{i}_{1}=\partial^{i}\cosh(a_{+}\times\partial)\frac{m_{2}}{r}\,$
(5.37)

where we used $|z_{21}|=r$. This recovers the leading in velocity contribution
to the particles EoM (5.5), directly from the scattering amplitude and the
instantaneous impulse.

####  5.3.2 Computation of the radiated field

In the previous sections, we built up the 5-point gravitational spinning
scattering amplitude up to quadratic order in the BHs’ spins. With this, we
can now return to (5.22) to successively construct the emitted classical
gravitational radiation from the spinning BBH at increasing PN order. First,
we compute the gravitational waveform to leading order in velocity up to
quadratic order in the BHs’ spins, while turning to the computation of the
waveform at sub-leading order in the BHs’ velocities in the spin-less limit in
section 5.3.2.

##### Scalar waveform

The derivation of the Einstein quadrupole formula from scattering amplitudes
was first done by Hari Dass and Soni in [121]; more recently, it was derived
by Goldberger’s and Ridgway’s classical double copy approach [31]. In the
following, we re-derive the scalar term of the waveform in the Goldberger and
Ridgway setup, for completeness. This, in turn, will outline the formalism
used throughout the remaining sections to arrive at the corrections to the
quadrupole formula. Expanding the scalar amplitude – obtained by replacing the
scalar numerators (4.61) into the general formula (2.45) – to leading order in
velocities $v$, we find (one can check that actually the non-relativistic
limit of the leading order in the soft expansion produce the same result)

$\langle
M_{5,{\rm{S}^{0}}}^{(0)\,ab}(\boldsymbol{q},\bar{\omega})\rangle=-i\frac{m_{1}^{2}m_{2}^{2}}{4}\kappa^{3}\left[2\frac{q^{a}q^{b}}{\boldsymbol{q}^{4}}+\frac{1}{\bar{\omega}\boldsymbol{q}^{2}}\left(q^{a}v_{12}^{b}+q^{b}v_{12}^{a}\right)\right],$
(5.38)

where $v_{AB}=v_{A}-v_{B}$. Substituting this amplitude into the scalar source
(5.29), and integrating over $\boldsymbol{q}$ using (D.3), the non-spinning
source reduces to

$T_{\rm{S}^{0}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2}){=}{-}\int
dte^{i\bar{\omega}\,t}\frac{\kappa^{3}}{32\pi}\sum_{A,B}\frac{m_{A}m_{B}}{r^{3}}\left[\left(z_{AB}^{a}z_{AB}^{a}{-}r^{2}\delta^{ab}\right)+\frac{2i}{\bar{\omega}}\left(z_{AB}^{a}v_{A}^{b}{+}z_{AB}^{a}v_{A}^{a}\right)\right].$
(5.39)

Here, and in the following, single label sums are understood to run over the
two massive particle labels, $\sum_{A}:=\sum_{A=1}^{2}$, while the double sum
is performed imposing the constraint $A\neq B$: $\sum_{A,B}:=\sum_{A\neq
B;A,B=1}^{2}$.

Notice that the term proportional to $\delta^{ab}$ in (5.39) vanishes under
the action of the TT-projector in (5.22). Therefore, in the following, we
remove this term from the source and focus only on those parts contributing
non-trivially to the TT radiated field. Now, we use the non-spinning part of
the EoM (5.5) to rewrite the second term in the square bracket of (5.39):

$T_{\rm{S}^{0}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=-\kappa\int
dte^{i\bar{\omega}\,t}\left[\sum_{A,B}\frac{\kappa^{2}m_{A}m_{B}}{32\pi}\frac{z_{AB}^{a}z_{AB}^{a}}{r^{3}}-\frac{2i}{\bar{\omega}}\sum_{A}m_{A}\left(v_{A}^{b}\dot{v}_{A}^{a}+v_{A}^{a}\dot{v}_{A}^{b}\right)\right].$
(5.40)

The second term of this expression can be further integrated, since
$v_{A}^{b}\dot{v}_{A}^{a}+v_{A}^{a}\dot{v}_{A}^{b}=\frac{d}{dt}\left(v_{A}^{a}v_{A}^{b}\right)$.
As for the first term, this can be rewritten using

$\frac{\kappa^{2}}{32\pi}\sum_{A,B}m_{A}m_{B}\frac{z_{AB}^{a}z_{AB}^{a}}{r^{3}}=-\sum_{A}m_{A}\left(\ddot{z}_{A}^{a}z_{A}^{b}+z_{A}^{a}\ddot{z}_{A}^{b}\right),$
(5.41)

derived from the scalar EoM. Putting these ingredients together into (5.40),
we find the scalar source to be

$T_{\rm{S}^{0}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\kappa\int
dte^{i\bar{\omega}\,t}\sum_{A}m_{A}\left(\ddot{z}_{A}^{a}z_{A}^{b}+z_{A}^{a}\ddot{z}_{A}^{b}+2v_{A}^{a}v_{A}^{b}\right).$
(5.42)

Using the relation
$2v_{A}^{a}v_{A}^{b}=\frac{d^{2}}{dt^{2}}\left(z_{A}^{a}z_{A}^{b}\right)-\left(\ddot{z}_{A}^{a}z_{A}^{b}+z_{A}^{a}\ddot{z}_{A}^{b}\right)$,
the above expression can be put into the more compact form

$T_{\rm{S}^{0}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\kappa\int
dte^{i\bar{\omega}\,t}\sum_{A}m_{A}\frac{d^{2}}{dt^{2}}\left(z_{A}^{a}z_{A}^{b}\right),$
(5.43)

which in turn implies that the radiated field (5.22) for a non-spinning BBH
takes the familiar Einstein quadrupolar form:

$\boxed{h_{TT,\,\rm{S}^{0}}^{(0)\,ij}(T_{R},R,\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{\kappa^{2}}{16\pi
R}\Pi^{ij}{}_{ab}\sum_{A}m_{A}\left[\frac{d^{2}}{dt^{2}}\left(z_{A}^{a}z_{A}^{b}\right)\right]_{t=T_{R}}.}$
(5.44)

The sequence of Fourier transforms in the source (5.43) and (5.22) leads to
the evaluation of the emitted gravitational radiation at retarded time
$T_{R}$, therefore, recovering the classical result (5.14) in the no-spin-
limit. As a quick remark, notice when restoring Newton’s constant $G$ the
quadrupole radiation is linear in $G$, as opposed to gravitational
Bremsstrahlung, which is quadratic [264, 265, 266, 267]. This is of course
just a feature of using the EoM to rewrite the source.

##### Linear-in-spin waveform

In the previous section, the main components of the derivation of the
gravitational waveform from a compact binary system were outlined. In
particular, we have seen that the classical EoM play an important role in
recovering the quadrupole formula. Going beyond this, at linear order in the
BHs’ spins, there are two contributions to the waveform. First, the scalar
amplitude could be iterated with the linear-in-spin part of the classical EoM
(5.5); this contribution, however, is sub-leading in velocity as made explicit
in (5.5). Secondly, the linear-in-spin amplitude, in conjunction with the non-
spinning part of the EoM gives rise to a leading in BHs’ velocities and
linear-in-their spins contribution to the waveform. To determine the latter,
we start from the linear-in-spin amplitude obtained by replacing the linear in
spin numerators (4.67) into the general formula (2.45), setting $J^{\mu\nu}\to
S^{\mu\nu}=\frac{1}{2m}\epsilon^{\mu\nu\rho\sigma}p_{1}^{\rho}S_{1}^{\sigma}$,
where the leading-in-$v$ contribution is given by

$\langle M_{5,{\rm
S}^{1}}^{(0)\,ab}(\boldsymbol{q},\bar{\omega})\rangle=-\frac{m_{1}m_{2}\kappa^{3}}{8}\varepsilon_{efk}\left(m_{2}S_{1}^{k}{-}m_{1}S_{2}^{k}\right)N^{[e}\left(\delta^{f]a}\delta^{bc}{+}\delta^{f]b}\delta^{ac}\right)\frac{q^{c}}{\boldsymbol{q}^{2}}.$
(5.45)

Analogous to the scalar case, we can substitute this amplitude into (5.29) to
get the linear-in-spin source $T_{{\rm{S}^{1}}}^{(0)\,ab}$. After integrating
over $\boldsymbol{q}$, utilizing (D.3), this source simplifies to

$T_{{\rm{S}^{1}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{\kappa^{3}}{32\pi}\varepsilon_{efk}\left(m_{2}S_{1}^{k}{-}m_{1}S_{2}^{k}\right)N^{[e}\left(\delta^{f]a}\delta^{bc}{+}\delta^{f]b}\delta^{ac}\right)\int
dte^{i\bar{\omega}\,t}\frac{z_{21}^{c}}{r^{3}}.$ (5.46)

Powers of $r$ in the denominator can be removed by using the scalar limit of
the classical EoM (5.5). Then, analogous to the scalar computation, the
linear-in-spin source is

$T_{{\rm{S}^{1}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\kappa\varepsilon_{efk}S_{1}^{k}N^{[e}\left(\delta^{f]a}\delta^{bc}{+}\delta^{f]b}\delta^{ac}\right)\int
dte^{i\bar{\omega}\,t}\dot{v}_{1}^{c}+(1\leftrightarrow 2).$ (5.47)

Finally, the linear in spin corrections to the Einstein quadrupole formula,
derived from the above amplitude, obtained from (5.47), together with (5.22),
are

$\boxed{h_{TT,\,{\rm{S}^{1}}}^{(0)\,ij}(T_{R},R,\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{\kappa^{2}}{16\pi
R}\Pi^{ij}{}_{ab}\varepsilon_{efk}\sum_{A}S_{A}^{k}\left[N^{[e}\left(\delta^{f]a}\delta^{bc}{+}\delta^{f]b}\delta^{ac}\right)\dot{v}_{A}^{c}\right]\Big{|}_{T_{R}}.}$
(5.48)

At this stage, this correction is valid, similar to the quadrupole formula,
for general closed orbits. We find a perfect match of these spinning
corrections at linear order in the objects’ spins, with the classical
derivation, (5.14), using the identity (D.1). The linear-in-spin scattering
amplitude is universal [192, 102], therefore, so is the radiated gravitational
field (5.48). Equivalently, the classical spin dipole of a point particle is
universal, describing any spinning compact object at leading order. Therefore,
non-universality of the waveform at higher spin orders may enter only through
a solution to the classical EoM for a particular compact binary system. We
showed in section 5.2.2 that the closed orbits waveform (5.14) contains all
possible spin effects at leading order in the BHs’ velocities, before
specializing the constituents’ trajectories; i.e., $h^{(0),ij}_{TT,S^{\ell\geq
2}}=0$. Therefore, we expect to find cancellations at higher orders in spins
at the level of the scattering amplitude for $\ell>1$. Finally, as claimed
above, there exists a one-to-one correspondence between source multipole
moments and spinning scattering amplitudes:
$\mathcal{I}_{ij}\leftrightarrow\langle M_{5,{\rm S}^{0}}^{(0)\,ab}\rangle$
and $\mathcal{J}_{ij}\leftrightarrow\langle M_{5,{\rm
S}^{1}}^{(0)\,ab}\rangle$. This holds in the sense that both
$\mathcal{I}_{ij}$ and $\langle M_{5,{\rm S}^{0}}^{(0)\,ab}\rangle$ produce
the quadrupole formula (and similarly for the linear-in-spin waveform).

##### Cancellations at quadratic order in spin

In the previous section, we showed that the gravitational waveform emitted
from a spinning BBH at leading order in its velocities is entirely contained
in the linear-in-spin radiation field (5.14). Equivalently, this waveform is
obtained only using the scalar and linear-in-spin amplitude. The remaining all
orders in spin result (5.18) emerges solely from the solution (5.7) for quasi-
circular orbits. To confirm this from the scattering amplitudes perspective,
we are left to show that higher spin amplitudes do not provide additional non-
trivial contributions to the general closed orbit results presented above. In
this section, we demonstrate the cancellation at the quadratic order in the
BHs’ spins, by specializing to circular orbits and by focusing on the
$S_{1}\neq 0,S_{2}\rightarrow 0$ limit.

At leading order in the BHs’ velocities, there are two distinct contributions
to the radiated field from our approach. There is the quadratic-in-spin part
of the amplitude on the one hand – obtained from replacing the classical
numerators (C.3) into the general formula (2.45) – leading to
$T_{1,{\rm{S}^{2}}}^{(0)\,ij}$, and the scalar part (5.38) in conjunction with
the quadratic-in-spin part of the classical EoM (5.5), yielding
$T_{2,{\rm{S}^{2}}}^{(0)\,ij}$, on the other hand131313Notice, the linear-in-
spin part of the EoM is sub-leading in $v$, and therefore, when convoluted
with the linear-in-spin amplitude, the resulting quadratic in spin
contribution is pushed to sub-leading order in velocities.; both combine as

$\displaystyle
T_{\rm{S}^{2}}^{(0)\,ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=T_{1,{\rm{S}^{2}}}^{(0)\,ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})+T_{2,{\rm{S}^{2}}}^{(0)\,ij}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2}).$
(5.49)

Focusing first on the contribution from the quadratic-in-spin part of the
amplitude, to leading order in $v$ it reads

$\langle M_{5,{\rm
S}^{2}}^{(0)\,ab}(\boldsymbol{q},\bar{\omega})\rangle={\color[rgb]{0,0,0}\frac{1}{4}\,}im_{2}^{2}\kappa^{3}S_{1}^{k}S_{1}^{l}\left[V_{kl,df}^{ab}\frac{q^{d}q^{f}}{\boldsymbol{q}^{2}}+C_{kl}^{ab}\right],$
(5.50)

where we have defined the tensor
$V_{kl,df}^{ab}=\delta_{kl}\delta_{d}^{a}\delta_{f}^{b}-\frac{1}{2}\delta_{kd}\left(\delta_{f}^{a}\delta_{l}^{b}+\delta^{fb}\delta_{l}^{a}\right)$,
and $C_{kl}^{ab}$ is a contact term, which we discard, as it is irrelevant for
the gravitational waveform. As before, we insert this amplitude into the
source (5.29), and perform the $\boldsymbol{q}$-integrals aided by (D.3). The
first contribution to the source $T_{\rm{S}^{2}}^{(0)\,ij}$ is then

$T_{1,{\rm{S}^{2}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=-{\color[rgb]{0,0,0}\frac{1}{4}\,}\frac{m_{2}\kappa^{3}}{m_{1}4\pi}S_{1}^{k}S_{1}^{l}V_{kl,df}^{ab}\int
dte^{i\bar{\omega}\,t}\frac{1}{r^{5}}\left[r^{2}\delta^{df}-3z_{21}^{d}z_{21}^{f}\right].$
(5.51)

Using the scalar part of the EoM (5.5) to remove three powers of $r$ in the
denominator, the above reduces to

$T_{{1,\rm{S}^{2}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=-3\frac{m_{2}}{m_{1}}\text{$\kappa$}S_{1}^{k}S_{1}^{l}V_{kl,df}^{ab}\int
dte^{i\bar{\omega}\,t}\frac{1}{r^{2}}\left[\left(\frac{\dot{v}_{2}{\cdot}z_{12}}{m_{1}}{+}\frac{\dot{v}_{1}{\cdot}z_{21}}{m_{2}}\right)\frac{\delta^{df}}{3}{-}\left(\frac{\dot{v}_{2}^{(d}z_{12}^{f)}}{m_{1}}{+}\frac{\dot{v}_{1}^{(d}z_{21}^{f)}}{m_{2}}\right)\right],$
(5.52)

which, for quasi-circular orbits (5.15), reads

$T_{1,{\rm{S}^{2}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})\Big{|}_{\rm{circular}}={\color[rgb]{0,0,0}-2}\kappa\bar{\omega}^{2}\mu\bar{a}_{1}^{2}\int
dte^{i\bar{\omega}t}\left[2n^{a}n^{b}-\lambda^{a}\lambda^{b}\right].$ (5.53)

Recall the definition for the symmetric mass ratio $\mu=m_{1}m_{2}/M$, and
$\bar{a}_{1}=S_{1}^{i}\ell_{i}/m_{1}$, with $\ell^{i}$ perpendicular to both
$n^{i}$ and $\lambda^{i}$. Note, the solution to the classical EoM, $r(x)$, in
the numerator, cancels with the two powers of $r$ in the denominator.

We now turn to the second contribution to the source:
$T_{2,\rm{S}^{2}}^{(0)\,ij}$. To that end, we first rewrite (5.39) by
expanding the sums and removing those terms that vanish under the TT
projection:

$T_{{2,\rm{S}^{2}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=-\kappa^{3}\int
dte^{i\bar{\omega}t}\frac{m_{1}m_{2}z_{12}^{c}}{16\pi
r^{3}}\left[\delta^{c(a}z_{12}^{b)}+\frac{2i}{\bar{\omega}}\delta^{c(a}v_{12}^{b)}\right].$
(5.54)

Next we use the classical EoM to quadratic order in spin, which can be written
in the following form (see section D.2)

$\dot{v}_{1}^{l}={-}\frac{m_{2}\kappa^{2}}{32\pi}\frac{z_{12}^{l}}{r^{3}}{{\color[rgb]{0,0,0}+}}\frac{3}{4}\frac{S_{1}^{i}S_{1}^{j}}{m_{1}^{2}r^{2}}\left[\left(\delta_{ij}{-}\frac{5z_{12,i}z_{12,j}}{r^{2}}\right)\left(\dot{v}_{1}^{l}{-}\frac{m_{2}}{m_{1}}\dot{v}_{2}^{l}\right){+}2\delta^{l}_{(i}\left(\dot{v}_{1,j)}{-}\frac{m_{2}}{m_{1}}\dot{v}_{2,j)}\right)\right].$
(5.55)

Combining this with (5.54), the scalar part will recover the Einstein
quadrupole radiation formula (5.44). We stress that although the quadrupole
formula appears to be spin-independent for general orbits, spin information
arises through a specific solution to the EoM, as pointed out above. In
particular, for quasi-circular orbits the Einstein quadrupole formula provides
the quadratic-in-spin result (5.18). Let us, therefore, focus in the remaining
contribution of (5.55), which is

$\begin{split}T_{{2,{\rm{S}^{2}}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})&=-\frac{{\color[rgb]{0,0,0}3}}{4}\kappa
S_{1}^{k}S_{1}^{l}\int
dte^{i\bar{\omega}t}\left[\delta^{c(a}z_{12}^{b)}+\frac{2i}{\bar{\omega}}\delta^{c(a}v_{12}^{b)}\right]\times\frac{1}{m_{1}r^{2}}\\\
&\left[\left(\delta_{kl}{-}\frac{5z_{12,k}z_{12,l}}{r^{2}}\right)\left(\dot{v}_{1}^{c}{-}\frac{m_{2}}{m_{1}}\dot{v}_{2}^{c}\right){+}2\delta_{(k}^{c}\left(\dot{v}_{1,l)}{-}\frac{m_{2}}{m_{1}}\dot{v}_{2,l)}\right){+}\frac{m_{2}}{m_{1}}(1\leftrightarrow
2)\right].\end{split}$ (5.56)

Using the center of mass parametrization141414Note, the linear-in-spin
corrections of this parametrization is sub-leading in velocities. (5.6), the
quasi-circular orbits condition $\ddot{r}=-\bar{\omega}r$, and the unit
vectors (5.15), the source reduces to

$T_{{2,{\rm{S}^{2}}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})\Big{|}_{\rm{circular}}={\color[rgb]{0,0,0}3}\kappa\bar{\omega}^{2}\mu\bar{a}_{1}^{2}\int
dte^{i\bar{\omega}t}\left[n^{a}n^{b}+i(\lambda^{a}n^{b}+\lambda^{b}n^{a})\right],$
(5.57)

In order to remove the imaginary part of the source, we proceed as before and
use an IBP prescription. Notice, since
$(\lambda^{a}n^{b}+\lambda^{b}n^{a})=-\frac{1}{\omega}\frac{d}{dt}(\lambda^{a}\lambda^{b})$,
the IBP yields

$T_{{2,{\rm{S}^{2}}}}^{(0)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})\Big{|}_{\rm{circular}}={\color[rgb]{0,0,0}3}\kappa\bar{\omega}^{2}\mu\bar{a}_{1}^{2}\int
dte^{i\bar{\omega}t}\left[n^{a}n^{b}-\lambda^{a}\lambda^{b}\right].$ (5.58)

This has the familiar form found in (5.18). Unlike this form, in (5.53) an
extra factor of two appears in the $n^{a}n^{b}$ term. This obscures the
desired cancellation between (5.58) and (5.53) in $T_{\rm{S}^{2}}^{(0)\,ij}$.
To address this subtlety, we emphasize the degeneracy in choice of the IBP
prescription. For instance, the relations of the kinematic variables in the
center of mass frame results in
$-\frac{d}{dt}(\lambda^{a}\lambda^{b})=\frac{d}{dt}(n^{a}n^{b})=\omega(\lambda^{a}n^{b}+\lambda^{b}n^{a})$.
Using the latter equality, the IBP performed in (5.58) results in
$2n^{a}n^{b}$, instead of $n^{a}n^{b}-\lambda^{a}\lambda^{b}$. A priori,
neither of these two choices are preferred. The solution is to notice that the
freedom in the choice of the IBP prescription is a manifestation of the gauge
redundancy of the gravitational waveform at null infinity. That is, below in
section 5.3.3 we show that either choice yields the same result for the gauge
invariant gravitational wave energy flux. For now, we note only that at the
level of the gauge invariant energy flux, one factor of $n^{a}n^{b}$ in (5.53)
is equivalent to
$n^{a}n^{b}\to\frac{1}{2}(n^{a}n^{b}-\lambda^{a}\lambda^{b})$, and postpone
the justification to section 5.3.3. Therefore, both (5.53) and (5.58) yield
the same result, but with opposite sign. This implies the desired cancellation
of the waveform contributions at the quadratic order in BHs’ spins.
Equivalently, using the waveform derived from (5.53) and (5.58) to determine
the energy flux from each contribution, we see that both contributions are
identical up to an overall sign, hence, cancelling at the level of the gauge
invariant gravitational wave energy flux as well (more on this below).

##### Scalar waveform at sub-leading order in velocities

So far we have dealt with leading in BHs’ velocities spinning corrections to
the Einstein quadrupole formula (5.44). In this section, we go beyond this
restriction and consider a non-spinning BBH at the first sub-leading order in
velocities, therefore, demonstrating the applicability of our approach (5.22)
to determine the radiated gravitational waves also in this regime. At this
order, the scalar 5-point amplitude is also independent of $q^{0}$, therefore,
arguments made above in section 5.3.1 concerning the time integration still
holds. In this case, however, the first relativistic correction, in our Born
approximation, as coming from the product of the plane wave functions in
(5.27), appears in the source through the kinematic exponential
$e^{-i\boldsymbol{k}\cdot\tilde{\boldsymbol{z}}}$, and therefore contributes
to the sub-leading source $T_{S^{0}}^{(1)\,ab}$, due to the scaling
$\bar{\omega}\sim v$. That is, after time integration, the exponential
function reduces as $e^{i\boldsymbol{k}\cdot\tilde{\boldsymbol{z}}}\rightarrow
1-i\bar{\omega}\boldsymbol{N}{\cdot}\tilde{\boldsymbol{z}}+\mathcal{O}(v^{2})=1-\frac{i}{2}\bar{\omega}\boldsymbol{N}{\cdot}(\boldsymbol{z}_{1}+\boldsymbol{z}_{2})+\mathcal{O}(v^{2})$;
hence, the source is built from the order-$v^{0}$ non-spinning scattering
amplitude, $T_{2,S^{0}}^{(1)\,ab}$, as well as from the $v^{1}$-amplitude,
$T_{1,S^{0}}^{(1)\,ab}$. More concretely, the sub-leading source decomposes
as151515In principle, the classical EoM (5.5) also contain higher-order-in-$v$
corrections, which could be used in an iterative manner, starting purely from
the leading in $v$-scalar amplitude. However, these velocity corrections
vanish in the no-spin limit considered in this section.

$T_{S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=T_{1,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})+T_{2,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2}),$
(5.59)

where

$T_{1,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{i}{m_{1}m_{2}}\int
dte^{i\bar{\omega}\,t}\int\frac{d^{3}\boldsymbol{q}}{\left(2\pi\right)^{3}}e^{-i\boldsymbol{q}{\cdot}\boldsymbol{z}_{21}}\langle
M_{5,{\rm S}^{0}}^{(1)\,ab}(\boldsymbol{q},\bar{\omega})\rangle,$ (5.60)

and

$T_{2,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=\frac{1}{m_{1}m_{2}}\int
dte^{i\bar{\omega}\,t}\int\frac{d^{3}\boldsymbol{q}}{\left(2\pi\right)^{3}}e^{-i\boldsymbol{q}{\cdot}\boldsymbol{z}_{21}}\frac{\bar{\omega}}{2}\boldsymbol{N}{\cdot}(\boldsymbol{z}_{1}+\boldsymbol{z}_{2})\langle
M_{5,{\rm S}^{0}}^{(0)\,ab}(\boldsymbol{q},\bar{\omega})\rangle.$ (5.61)

Notice the superscripts in the amplitude. First, we focus on the
relativistically corrected scalar amplitude. Analogous to before, we insert
(4.61) into (2.45), but now keep the non-trivial order $\mathcal{O}(v^{1})$
contributions to the 5-point amplitude:

$\begin{split}\langle M_{5,{\rm
S^{0}}}^{(1)\,ab}(\boldsymbol{q},\bar{\omega})\rangle&=-\frac{im_{1}^{2}m_{2}^{2}\kappa^{3}}{2}N_{l}\Bigg{[}\frac{q^{l}q^{m}}{\boldsymbol{q}^{4}}\delta_{m}^{(a}(v_{1}{+}v_{2})^{b)}+\frac{q^{l}}{2\bar{\omega}\boldsymbol{q}^{2}}\left(v_{1}^{a}v_{1}^{b}-v_{2}^{a}v_{2}^{b}\right)\\\
&\hskip
85.35826pt+\frac{q^{m}}{\bar{\omega}\boldsymbol{q}^{2}}\left(v_{1}^{l}v_{1}^{(a}\delta_{m}^{b)}-v_{2}^{l}v_{2}^{(a}\delta_{m}^{b)}\right)\Bigg{]}.\end{split}$
(5.62)

Subsequently, the source (5.60), after the $\boldsymbol{q}$-integration, takes
the form

$\begin{split}T_{1,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})&=\frac{\kappa^{3}}{16\pi}N_{l}\int
dte^{i\bar{\omega}\,t}\sum_{A,B}\frac{m_{A}m_{B}}{r^{3}}\Bigg{[}\frac{1}{2}\left(\boldsymbol{z}_{AB}^{2}\delta^{lm}{-}z_{AB}^{l}z_{AB}^{m}\right)\delta_{m}^{(a}(v_{A}{+}v_{B})^{b)}\\\
&\hskip
85.35826pt{-}\frac{i}{\bar{\omega}}z_{AB}^{n}\left(\delta_{n}^{l}v_{A}^{a}v_{A}^{b}{+}2\delta_{n}^{m}v_{A}^{l}v_{A}^{(a}\delta_{m}^{b)}\right)\Bigg{]}.\end{split}$
(5.63)

In order to remove the powers of $z_{AB}\sim r$ in the denominator, we use the
scalar part of the EoM (5.5), to obtain

$\displaystyle\begin{split}T_{1,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})&=2\kappa
N_{l}\int
dte^{i\bar{\omega}\,t}\Bigg{[}-\frac{1}{2}\sum_{A,B}m_{A}\left(\dot{\boldsymbol{v}}_{A}{\cdot}\boldsymbol{z}_{AB}\delta^{lm}{-}\dot{v}_{A}^{m}z_{AB}^{l}\right)\delta_{m}^{(a}(v_{A}{+}v_{B})^{b)}\\\
&\hskip
85.35826pt{+}\frac{i}{\bar{\omega}}\sum_{A}m_{A}\dot{v}_{A}^{n}\left(\delta_{n}^{l}v_{A}^{a}v_{A}^{b}{+}2\delta_{n}^{m}v_{A}^{l}v_{A}^{(a}\delta_{m}^{b)}\right)\Bigg{]}.\end{split}$
(5.64)

The term in the second line can be integrated utilizing the relation
$\dot{v}_{A}^{n}\left(\delta_{n}^{l}v_{A}^{i}v_{A}^{j}{+}2\delta_{n}^{m}v_{A}^{l}v_{A}^{(i}\delta_{m}^{j)}\right)=\frac{d}{dt}(v_{A}^{i}v_{A}^{j}v_{A}^{l})$.
With this, this piece of the sub-leading scalar source simplifies to

$\displaystyle\begin{split}T_{1,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})&=-2\kappa
N_{l}\int
dte^{i\bar{\omega}\,t}\Bigg{[}\sum_{A,B}\frac{m_{A}}{2}\left(\dot{\boldsymbol{v}}_{A}{\cdot}\boldsymbol{z}_{AB}\delta^{lm}{-}\dot{v}_{A}^{m}z_{AB}^{l}\right)\delta_{m}^{(a}(v_{A}{+}v_{B})^{b)}{-}\sum_{A}m_{A}v_{A}^{l}v_{A}^{a}v_{A}^{b}\Bigg{]}.\end{split}$
(5.65)

We now address the second term in (5.59) – the computation of the second
contribution (5.61) to the sub-leading scalar source. The
$\boldsymbol{q}$-integration is identical to the one used leading up to
(5.40). Starting from the latter, using the relation (5.41), and multiplying
the sub-leading prefactor
$-\frac{i}{2}\bar{\omega}\boldsymbol{N}{\cdot}(\boldsymbol{z}_{1}+\boldsymbol{z}_{2})$
we arrive at

$T_{2,S^{0}}^{(1)\,ab}(k,\boldsymbol{z}_{1},\boldsymbol{z}_{2})=-\kappa\int
dte^{i\bar{\omega}\,t}\boldsymbol{N}{\cdot}(\boldsymbol{z}_{1}+\boldsymbol{z}_{2})\sum_{A}m_{A}\left[i\bar{\omega}\ddot{z}_{A}^{(a}z_{A}^{b)}-2v_{A}^{(a}\dot{v}_{A}^{b)}\right].$
(5.66)

Lastly, with the replacement $\bar{\omega}\rightarrow i\frac{d}{dt}$ the first
term is integrated. The gravitational radiation field is then determined by
putting the two sources together in (5.59), and substituting this into (5.22),
to end up at the first sub-leading in BH velocities non-spinning correction to
the Einstein quadrupole formula:

$\begin{split}h_{\text{TT},S^{0}}^{(1)\,ij}(T_{R},R,\boldsymbol{N},\boldsymbol{z}_{1},\boldsymbol{z}_{2})&=-\frac{\kappa^{2}m_{1}}{8\pi
R}\Pi^{ij}{}_{ab}N_{l}\Bigg{[}\frac{1}{2}\left(\dot{\boldsymbol{v}}_{1}{\cdot}\boldsymbol{z}_{12}\delta^{lm}{-}\dot{v}_{1}^{m}z_{12}^{l}\right)\delta_{m}^{(a}(v_{1}+v_{2})^{b)}{-}v_{1}^{l}v_{1}^{a}v_{1}^{b}\\\
&-\frac{1}{2}\left(\frac{d}{dt}\left(\ddot{z}_{1}^{(a}z_{1}^{b)}(z_{1}+z_{2})^{l}\right)+2(z_{1}+z_{2})^{l}v_{1}^{(a}\dot{v}_{1}^{b)}\right)\Bigg{]}+(1\leftrightarrow
2).\end{split}$ (5.67)

Based on our derivation, this result is valid for generic closed orbits,
provided the corresponding EoM. However, the form of this waveform is
different from the compact classical result in (5.21). This is not surprising
since, as illustrated above, there is always the freedom of choice of IBP
prescription, which casts the waveform into different forms. Finding the
prescription, for which both the amplitude’s and the classical waveforms
match, could be cumbersome for generic closed orbits. Therefore, we specialize
to the quasi-circular setting (5.15); In the latter, we find perfect agreement
between (5.67) and (5.21). We close with a remark on the correspondence
between the classical source multipole moments leading to the gravitational
radiation via the multipolar post-Minkowskian approach, and our ansatz to
compute the associate gravitational waves using spinning scattering
amplitudes. We saw in section 5.2.2, the sub-leading order result (5.20) is
built from both $\mathcal{I}_{ijk}$ and $\mathcal{J}_{ij}$. While at leading
order in the BHs’ velocities (see section 5.3.2), there exists a certain one-
to-one correspondence between the source multipole moments, at sub-leading
orders in velocities, no trivial correspondence can be extracted from our
results.

####  5.3.3 Radiated gravitational wave energy flux

In the previous sections, we showed explicitly that the radiated gravitational
field, $h^{\text{TT}}_{ij}$, computed using a classical approach and utilizing
a 5-point spinning scattering amplitude, agree in the aligned spin, general
(and quasi-circular) orbit setup at the considered orders in the velocity and
spin expansions. These are the gravitational waves emitted at an instant in
the binary’s evolution. Information about the frequency dynamics of the
radiation is contained in the emitted gauge invariant gravitational wave
energy flux. The latter is ultimately responsible for the inspiral of the two
BHs and for the characteristic increase in gravitational wave frequency
towards the merger, therefore, a crucial ingredient for gravitational wave
search strategies.

In this section, we derive the instantaneous gravitational wave energy flux
$\mathcal{F}$ using the TT metric perturbations at null infinity computed in
the previous subsections to the respective orders in the spin and velocity
expansions. In general, the total instantaneous energy loss $\mathcal{F}$ can
be obtained with

$\displaystyle\mathcal{F}=\frac{R^{2}}{32\pi}\int_{S^{2}}d\Omega\
\dot{h}^{\text{TT}}_{ij}\dot{h}^{\text{TT},ij}.$ (5.68)

Let us return here to the justification for the replacements and claims made
in section 5.3.2. The time dependence of $h^{\text{TT}}_{ij}$ is solely
contained in the center of mass variables $n^{a}$ and $\lambda^{a}$, which, in
the center of mass frame and for circular orbits, are related by
$\frac{d}{dt}(n^{a}n^{b})=\frac{1}{2}\frac{d}{dt}(n^{a}n^{b}-\lambda^{a}\lambda^{b})$.
Since only the time derivative of the radiated field,
$\dot{h}^{\text{TT}}_{ij}$, enters in (5.68), this justifies the replacement
$n^{a}n^{b}\rightarrow\frac{1}{2}(n^{a}n^{b}-\lambda^{a}\lambda^{b})$ made in
section 5.3.2 at the level of the radiated field. Furthermore, this also shows
that the gauge invariant energy flux is, in fact, independent of the IBP
prescription discussed in section 5.3.2. Therefore, the latter can be viewed
as a manifestation of the gauge freedom in the emitted waveform. Indeed, this
extends to the Newman-Penrose scalar
$\Psi_{4}\sim\bar{m}^{\mu}\bar{m}^{\nu}\ddot{h}^{\text{TT}}_{\mu\nu}$ in an
identical fashion. Exploiting this, the gravitational wave energy flux is
obtained by combining the scalar, (5.44), and linear-in-spin, (5.48), metric
perturbations $h^{\text{TT}}_{ij}$ at leading order in the BHs’ velocities, in
(5.68). For quasi-circular orbits (5.7), together with (D.2), we find the
energy loss

$\displaystyle\mathcal{F}^{(0)}_{\text{circular}}=\frac{32}{5}\frac{\mu^{2}x^{5}}{M^{2}}+\frac{2}{5}\frac{\mu^{2}x^{7}}{M^{2}}(32a_{+}^{2}+a_{-}^{2})+\mathcal{O}(a^{3}_{1,2},a_{1}a_{2}).$
(5.69)

Recall from above that $x=(M\omega)^{2/3}$. This matches perfectly with the
results reported in Refs. [245, 260, 261, 259, 223, 224] to the respective
orders in spin. In addition to this match at leading order in the black holes
velocities, the metric perturbations computed in (5.67) and (5.21) specialized
to circular orbits reproduce the correct no-spin gravitational wave energy
flux $\mathcal{F}^{(1)}_{\text{circular}}=0$ at the first sub-leading order in
velocities; this is, again, consistent with the leading no-spin PN
gravitational wave power (see, e.g., [261]). Notice, we explicitly computed
the quadratic-in-spin contributions only for one BH with spin: $S_{1}\neq
0,S_{2}\rightarrow 0$. However, as noted above, the classical derivation in
section 5.2.2 revealed that the high-order-in-spin contributions to the
circular orbit $h^{\text{TT}}_{ij}$ emerge solely from the solution to the
EoM, indicating that (5.69) already contains the $a_{1}a_{2}$-type
interactions; this is the case, as can be seen in, for instance, [223, 224],
or from using (5.17) together with (5.68). At the level of the transverse
traceless metric perturbations $h^{\text{TT}}_{ij}$, the classical derivation
showed that (5.14) contains the complete all orders-in-spin information at
leading order in velocities, since the remaining contributions to the
radiation field – i.e. $h_{ij,S^{\ell\geq 2}}^{(0)\text{TT}}=0$ – vanish,
without a specific solution to the EoM. In the scattering amplitudes setting,
we confirmed this explicitly up to $\ell=1$, since (5.44) and (5.48) agree
with (5.14) (exploiting (D.1)), and we showed the necessary cancellation for
$\ell=2$ in section 5.3.2. Therefore, we conjecture such cancellations to
occur at arbitrary order in spin, such that the solution to the EoM provides
the remaining spin-information, at leading order in velocities. The complete
all-orders in spin gravitational power result partially presented in (5.69)
was determined in [224].

###  5.4 Outlook of the chapter

In this chapter we studied the relationship between the radiative dynamics of
an aligned-spin spinning binary black hole from both, a classical, and a
scattering amplitude perspective. For the former we employed the multipolar
post-Minkowskian formalism, whereas for the latter we proposed a dictionary
built from the 5-point QFT scattering amplitude extensively studied in chapter
4. More precisely, the dictionary maps the classical limit of the 5-pt
scattering amplitude of two massive spinning particles exchanging and emitting
a graviton, to the source entering in Einstein’s equation. Furthermore, we
included information of the conservative dynamics using the classical
equations of motion, which we obtained from the instantaneous impulse formula
which takes as main input the conservative two-body amplitude. We worked in
linearized gravity, i.e., at tree-level, and to leading order in the black
holes’ velocities, but to all orders in their spin, as well as present
preliminary results at sub-leading orders in velocities (in the no-spin
limit). To leading order in the system’s velocities, we showed that there
exists a one-to-one correspondence between the source’s multipole moments, and
the scattering amplitudes. That is, the mass quadrupole in (5.13) corresponds
to the scalar amplitude (5.38), while similarly, the current quadrupole in
(5.13) is associated with the linear-in-spin amplitude (5.45). This
correspondence was made explicit in the computation of the transverse-
traceless part of the linear metric perturbations emitted to null infinity, as
well as on the gauge invariant gravitational wave energy flux. The latter
agrees for quasi-circular orbits with the existing literature [260, 261, 223,
224], both at the considered leading and sub-leading orders in the black
holes’ velocities. Therefore, gravitational waveforms and gauge invariant
powers needed for detecting gravitational waves from inspiraling black holes
can be consistently computed from the classical limit of quantum scattering
amplitudes.

The gravitational waveform is, in general, a gauge-dependent object, which
makes a comparison between the classical and the scattering amplitude’s
derivations potentially difficult. In particular, and especially for general
orbits and with spin effects, finding the corresponding gauge to undertake
such comparisons can become cumbersome. In this chapter, we found evidence
that such gauge freedom is related to the integration procedure used in the
source for Einstein’s equation, within the scattering amplitudes derivation.
We demonstrated this explicitly for quasi-circular orbits, as this restriction
simplifies the problem drastically. Importantly, we find that while the form
of the gravitational radiation field is dependent upon the integration
procedure used, the gauge-invariant gravitational wave power is independent of
such a prescription – as desired.

In this chapter, we focused entirely on the derivation of radiative degrees of
freedom from the 5-point scattering amplitude, and show classical EoM for the
system follow directly from the conservative amplitude, providing then a self-
contained amplitudes derivation for the radiated field at leading and
subleading order in the velocity expansion, but to all orders in spin. This
expand the claims for scalar sources made by Dass and Soni in [121].

The amplitudes-based construction of the radiated field (5.22), provided in
this chapter, has implicitly used the on-shell condition for the outgoing
massive particles $\delta(p_{i}{\cdot}q_{i})$, which discards terms quadratic
in the velocities as indicated by the quantum corrections to the particles
trajectories $z_{Q}(\tau_{i})$ in (5.25). These corrections can become
important if convoluted with superclassical terms coming from loop amplitudes.
This then hints that at higher orders in perturbation theory, a subtraction
scheme would be needed to cancel those superclassical contributions at the
level of the gauge invariant observable, which in this case corresponds to the
radiated energy flux $\mathcal{F}\sim\int d\omega\dot{h}^{ij}\dot{h}_{ij}$; in
addition, it would be desirable to study the connection of our approach and
that of analytic continuation methods of scattering observables [37, 36, 93] .

Besides, exploring gauge fixing procedures that allow to match the general
orbit result (5.67) to the classical result (5.21), as well as the inclusion
of spin effects at sub-leading order in velocity is left for future work.
Furthermore, in the context of scattering amplitudes, higher orders in
velocities are naturally included. However, for closed orbits, these
corrections are consistent only – by virtue of the virial theorem – when also