with $\widetilde{\Psi}$, $\not{\nabla}_{v}\widetilde{\Psi}$ vanishing at
$v_{1}$. It is clear that

$\displaystyle\mathrm{RW}[\widetilde{\Psi}]|_{{\underline{\mathcal{C}}}}=0.$
(6.36)

Note that

$\displaystyle\not{\nabla}_{v}\mathrm{RW}[\widetilde{\Psi}]=\psi+\frac{r^{2}}{\Omega^{2}}(5-3\Omega^{2}-\mathring{\not{\Delta}})\not{\nabla}_{v}\widetilde{\Psi}+6M\widetilde{\Psi}.$

Since $\not{\nabla}_{u}(r\Omega^{2}\alpha)=0$ at ${\underline{\mathcal{C}}}$,
we also have

$\displaystyle\not{\nabla}_{v}\mathrm{RW}[\widetilde{\Psi}]|_{{\underline{\mathcal{C}}}}=0.$

Therefore, we have by (3.52) from Lemma 3.1 that $\widetilde{\Psi}$ satisfies
the Regge–Wheeler equation (RW).

Now, by definition, we have

$\displaystyle\not{\nabla}_{v}\widetilde{\Psi}=\frac{\Omega^{2}}{r^{2}}(u,v)\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}r(u,\bar{v})^{3}\alpha(u,\bar{v},\theta^{A}).$
(6.37)

We make use of the fact that $r\alpha$ decays pointwise towards
$\mathcal{I}^{-}$ to estimate

$\displaystyle\begin{split}\int_{\mathbb{S}^{2}}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\,\left|\frac{\Omega^{2}}{r^{2}}\int_{v_{1}}^{v}r^{3}\alpha\mathop{}\\!\mathrm{d}\bar{v}\right|^{2}&=\frac{\Omega^{4}}{r^{4}}\int_{\mathbb{S}^{2}}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\,\left[\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}\,\frac{r^{2}}{\Omega^{2}}\int_{-\infty}^{u_{0}}\mathop{}\\!\mathrm{d}\bar{u}\,\frac{\Omega^{2}}{r^{2}}\psi\right]^{2}\\\
&\lesssim\int_{\mathbb{S}^{2}}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\,\left[\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}\,\int_{-\infty}^{u_{0}}\mathop{}\\!\mathrm{d}\bar{u}\,\frac{\Omega^{2}}{r^{2}}|\psi|\right]^{2}\\\
&\lesssim\frac{(v_{2}-v_{1})}{r(u,v_{1})}\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}\,\int_{-\infty}^{u_{0}}\mathop{}\\!\mathrm{d}\bar{u}\int_{\mathbb{S}^{2}}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\,\frac{\Omega^{2}}{r^{2}}|\psi|^{2}\\\
&\lesssim\frac{(v_{2}-v_{1})}{r(u,v_{1})}\times\Big{(}\int_{v_{1}}^{v_{2}}\int_{\mathbb{S}^{2}}\mathop{}\\!\mathrm{d}\bar{v}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\,\left[|r\Omega^{2}\alpha|_{u_{0}}|^{2}+|\psi|_{u_{0}}|^{2}\right]\\\
&\qquad\qquad\qquad\qquad+E_{u_{0}}[\Psi](v_{1},v_{2})+\underline{E}_{v_{2}}[\Psi](-\infty,u_{0})\Big{)},\end{split}$
(6.38)

where we used (6.34) in the last step. Thus
$\|\not{\nabla}_{v}\widetilde{\Psi}(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})}\to
0$ as $u\to-\infty$. Combining this with
$\widetilde{\Psi}|_{{\underline{\mathcal{C}}}}=0$, Proposition 6.1 implies
$\widetilde{\Psi}=0$. Thus $\alpha=0$. ∎

The argument above allows us to conclude the following result, which will
later on allow us to conclude uniqueness for the full system (3.9)–(3.32).

######  Corollary 6.2.

Let $\alpha$ be a solution to (Teuk) on
$(u,v,\theta^{A})\in(-\infty,u_{0}]\times[v_{1},v_{2}]\times\mathbb{S}^{2}$
such that $\underline{E}_{v_{2}}[\Psi](-\infty,u_{0})<\infty$, where
$\Psi=\left(r^{2}\Omega^{-2}\not{\nabla}_{u}\right)^{2}r\Omega^{2}\alpha$. If
$\alpha$ satisfies

$\displaystyle\lim_{u\to-\infty}\frac{1}{r(u,v_{1})}\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}\,r(u,\bar{v})^{2}\alpha(u,\bar{v},\theta^{A})=0,\qquad\not{\nabla}_{u}(r\Omega^{2}\alpha)|_{\mathcal{C}}=0,$
(6.39)

then $\alpha=0$.

###### Proof.

Let $\widetilde{\Psi}$ be as in the proof of Proposition 6.4. We compute
$E_{u}[\widetilde{\Psi}](v_{1},v_{2})$ by directly estimating

$\|\not{\nabla}_{v}\widetilde{\Psi}(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})}^{2}\lesssim\int_{v_{1}}^{v_{2}}\mathop{}\\!\mathrm{d}v\int_{\mathbb{S}^{2}}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\left(\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}v^{\prime}r^{3}\alpha\right)^{2}\\\
\lesssim(v_{2}-v_{1})^{2}\int_{v_{1}}^{v_{2}}\mathop{}\\!\mathrm{d}v\int_{\mathbb{S}^{2}}\sin\theta\mathop{}\\!\mathrm{d}\theta\mathop{}\\!\mathrm{d}\varphi\,\,(r\alpha)^{2}\lesssim(v_{2}-v_{1})^{2}\|r\alpha(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})}^{2},$
(6.40)

and by similarly estimating

$\displaystyle\|r^{-1}\widetilde{\Psi}(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})}$
$\displaystyle\lesssim\frac{(v_{2}-v_{1})^{\frac{3}{2}}}{r(u,v_{1})}\|r\alpha(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})},$
(6.41)
$\displaystyle\|r^{-1}\mathring{\not{\nabla}}\widetilde{\Psi}(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})}$
$\displaystyle\lesssim\frac{(v_{2}-v_{1})^{\frac{3}{2}}}{r(u,v_{1})}\|\mathring{\not{\nabla}}r\alpha(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})},$
(6.42)

Combining (6.34) with (6.41), we see that
$\|r^{-1}\widetilde{\Psi}(u,\cdot,\cdot)\|_{L^{2}([v_{1},v_{2}]\times\mathbb{S}^{2})}$
decays to 0 as $u\to-\infty$. Thus, $r^{-1}\Psi$ decays in measure as
$u\to-\infty$ by Vitali’s convergence theorem. Furthermore, the assumption
that $\Psi$ is of finite energy at $v_{2}$ means the same applies to
$\widetilde{\Psi}$ by combining (6.34) with (6.40), (6.41), (6.42). A
computation shows

$\displaystyle\frac{1}{r(u,v_{1})}\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}\,r(u,\bar{v})^{2}\alpha(u,\bar{v},\theta^{A})=\left[\frac{1}{\Omega^{2}}\not{\nabla}_{v}\widetilde{\Psi}+\frac{1}{r}\widetilde{\Psi}\right]\Bigg{|}_{(u,v,\theta^{A})},$
(6.43)

hence $\not{\nabla}_{v}\widetilde{\Psi}$, too, decays in measure as
$u\to-\infty$. By Corollary 6.1, we then deduce that $\widetilde{\Psi}=0$. It
follows that $\alpha=0$. ∎

###  7 Construction of the (unique) scattering solution to (3.9)–(3.32)

In this section, we present the full proof of Theorem 5.1. First, we prove
uniqueness of scattering solutions in §7.1. We then present the construction
of the $\ell\geq 2$-part of the scattering solution in §7.2 and of the
$\ell<2$-part in §7.3. The latter will be essentially trivial, all the
difficulty is contained in $\ell\geq 2$.

####  7.1 Uniqueness of solutions to the scattering problem

We now prove the uniqueness clause of Theorem 5.1:

######  Proposition 7.1.

Assume that $\mathfrak{S}$ is a solution to (3.9)–(3.32) realising a vanishing
seed scattering data set $\mathfrak{D}$. Assume moreover that

$\displaystyle\underline{E}_{v}[\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}](-\infty,u_{0})<\infty$
(7.1)

for all $v\geq v_{1}$. Then $\mathfrak{S}$ vanishes identically.

###### Proof.

Let $\mathfrak{S}$ be a scattering solution realising $\mathfrak{D}$. Since
$\mathfrak{D}$ is trivial, we deduce from Prop. 5.1 that, in particular, the
restriction to ${\underline{\mathcal{C}}}$ of
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$, $K$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ belonging to
$\mathfrak{S}$ vanishes as well. By (3.32), we infer that
$\not{\nabla}_{u}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}})$
vanishes along ${\underline{\mathcal{C}}}$ as well. The condition (7.1),
together with

$\displaystyle\lim_{u\to-\infty}\frac{1}{r(u,v)}\int_{v_{1}}^{v}\mathop{}\\!\mathrm{d}\bar{v}\,r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}(u,\bar{v},\theta^{A})$
(7.2) $\displaystyle=-$
$\displaystyle\lim_{u\to-\infty}\left(\frac{1}{r(u,v)}\left(\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}{\Omega}(u,v,\theta^{A})-\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}{\Omega}(u,v_{1},\theta^{A})\right)\right)=0$
(7.3)

implies by Corollary 6.2 that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}\equiv 0.$ It follows
that $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}\equiv 0$.

Consecutively integrating (3.17b), (3.31b) as well as (3.30b) from
${\underline{\mathcal{C}}}$ then yields that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\equiv
0\equiv\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ as well as
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}\equiv 0$. It follows
from (3.38) that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}\equiv 0$, and
thus $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}\equiv 0$
(cf. Cor. 5.2).

With $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ both
vanishing identically, we now apply Proposition 4.7 to deduce that the
$\ell\geq 2$ component of $\mathfrak{S}$ is pure gauge, i.e.
$\mathfrak{S}_{\ell\geq 2}=\mathfrak{S}_{f_{\ell\geq
2}}+\mathfrak{S}_{\underline{f}_{\ell\geq
2}}+\mathfrak{S}_{(q_{1},q_{2})_{\ell\geq 2}}$ for some functions $f_{\ell\geq
2}$, $\underline{f}_{\ell\geq 2}$ and $(q_{1},q_{2})_{\ell\geq 2}$.

We now show that these functions vanish: Firstly, the vanishing of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ forces
$\underline{f}_{\ell\geq 2}=0$. (Recall that the kernel of
$\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}$
is spanned by the $\ell=0,1$-modes.) Secondly, the vanishing of
$\lim_{u\to-\infty}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$
implies that the outgoing gauge solution must have $\partial_{v}f_{\ell\geq
2}=0$, and the vanishing of, say,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}|_{{\underline{\mathcal{C}}}}$,
implies that $f_{\ell\geq 2}=0$. Thirdly, the vanishing of
$\lim_{u\to-\infty}r^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$
implies that $\partial_{v}{q_{1}}_{\ell\geq 2}$,
$\partial_{v}{q_{2}}_{\ell\geq 2}$ vanish, and the fact that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$ is trivial on
${\underline{\mathcal{C}}}$ leads to ${q_{1}}_{\ell\geq 2}={q_{2}}_{\ell\geq
2}=0$.

At this point, we know that $\mathfrak{S}$ is supported only on $\ell=0,1$, so
it is a linear combination of a pure gauge solution, a linearised nearby
Schwarzschild solution, and a linearised Kerr solution by Prop. 4.6.

From the fact that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}|_{{\underline{\mathcal{C}}}}=0$,
we can infer that the linearised Kerr solution must vanish.

In order to also show that the linearised nearby Schwarzschild solution
vanishes, we assume that it doesn’t: Then this solution generates nonvanishing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}_{\mathfrak{m}}=-\frac{2M\mathfrak{m}}{r^{3}}=-\frac{2}{r}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}_{\mathfrak{m}}$
along ${\underline{\mathcal{C}}}$, supported on $\ell=0$, by Prop. 4.1. But
since $\mathfrak{S}$ has
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}|_{{\underline{\mathcal{C}}}}=0=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}|_{{\underline{\mathcal{C}}}}=0$,
it must now be possible to find pure gauge solutions supported on $\ell=0$ to
kill off $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}_{\mathfrak{m}}$
and $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}_{\mathfrak{m}}$ along
${\underline{\mathcal{C}}}$. An inspection of the expressions given in
Propositions 4.3–4.4 shows that such solutions do not exist: Since the
outgoing gauge solution decays too fast near $\mathcal{I}^{-}$
($\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}_{f}\sim r^{-4}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}\sim r^{-3}$), the only
candidate to kill off the leading order behaviour of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}_{\mathfrak{m}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}_{\mathfrak{m}}$ is the ingoing
gauge solution, but this has
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}_{\underline{f}}=-\tfrac{3}{r}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}_{\underline{f}}$.
Thus the linearised nearby Schwarzschild solution also vanishes.

We now know that $\mathfrak{S}$ is a pure gauge solution supported on
$\ell=0,1$, i.e. $\mathfrak{S}=\mathfrak{S}_{f_{\ell\leq
1}}+\mathfrak{S}_{\underline{f}_{\ell\leq
1}}+\mathfrak{S}_{(q_{1},q_{2})_{\ell\leq 1}}$. The condition that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}|_{{\underline{\mathcal{C}}}}=0$
implies that $\underline{f}_{\ell\leq 1}(u,\theta^{A})=f_{\ell\leq
1}(v=v_{1},\theta^{A})$, so $\underline{f}_{\ell\leq 1}$ is independent of
$u$. The condition that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}\to 0$ as $u\to-\infty$
then implies that $\partial_{v}f_{\ell\leq 1}=0$, so we have that
$\underline{f}_{\ell\leq 1}(u,\theta^{A})=\tilde{f}_{\ell\leq
1}(\theta^{A})=f_{\ell\leq 1}(v,\theta^{A})$ for some $\tilde{f}$. Comparing
now the expressions for
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$
generated by each of these solutions along ${\underline{\mathcal{C}}}$, we
deduce that $\tilde{f}_{\ell=1}=0$. (The expression for
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$
generated by $(q_{1},q_{2})$ comes with a different $r$-weight.) We thus have
that $\underline{f}_{\ell\leq 1}(u,\theta^{A})=C=f_{\ell\leq 1}(v,\theta^{A})$
for some constant $C$, and hence, in view of Remark 4.1,
$\mathfrak{S}_{f_{\ell\leq 1}}+\mathfrak{S}_{\underline{f}_{\ell\leq 1}}$=0.

Thus, we have that $\mathfrak{S}=\mathfrak{S}_{(q_{1},q_{2})_{\ell\leq 1}}$.
Since $r^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}\to 0$ as
$u\to-\infty$, and since
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ vanishes
along ${\underline{\mathcal{C}}}$, it finally follows that $\mathfrak{S}=0$. ∎

####  7.2 Construction of the $\ell\geq 2$-part of the solution to the
scattering problem

We now present an explicit construction of a scattering solution
$\mathfrak{S}$ to (3.9)–(3.32) realising a given smooth seed scattering data
set $\mathfrak{D}$ such that $\mathfrak{S}$ and $\mathfrak{D}$ are related via
Definition 5.2.

######  Proposition 7.2.

Given a smooth seed scattering data set $\mathfrak{D}_{\ell\geq 2}$ supported
on $\ell\geq 2$ and satisfying the assumptions of Theorem 5.1, there exists a
scattering solution $\mathfrak{S}_{\ell\geq 2}$ realising
$\mathfrak{D}_{\ell\geq 2}$. By the previous Prop. 7.1, this is the unique
scattering solution realising $\mathfrak{D}_{\ell\geq 2}$.

Since the proof of Prop. 7.2 is quite long, we first give a detailed overview.

#####  7.2.1 Overview of the construction

The construction is organised into the following steps:

  1. (I)

Out of
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}$
and data on ${\underline{\mathcal{C}}}$ (from Prop. 5.1), we define various
tensor fields such as
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}$
and
${\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}$
at $\mathcal{I}^{-}$. (These are to be thought of as the pointwise limits of
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$,
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ and
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ of the
eventual solution.)

  2. (II)

We now construct the unique scattering solution
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ to (RW) such that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ restricts to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}}_{{\underline{\mathcal{C}}}}$
and $\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$
tends to
${\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}$
at $\mathcal{I}^{-}$ via Theorem 6.1, where
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}}_{{\underline{\mathcal{C}}}}$
has been constructed in Cor. 5.2. Using (6.26) then gives us the bound
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}\lesssim r^{1/2}$

  3. (III)

We then define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ by
integrating
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=(\tfrac{r^{2}}{\Omega^{2}}\not{\nabla}_{u})^{2}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}})$
(cf. (3.37)) twice from $\mathcal{I}^{-}$, defining the boundary terms at
$\mathcal{I}^{-}$ by using the definitions of
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}$
from step (II) and demanding (3.32) to hold at $\mathcal{I}^{-}$. We then show
that $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ satisfies (Teuk)
by virtue of Lemma 3.1 and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ satisfying (RW).

  4. (IV)

Having constructed $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ in
the previous step, we now define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ by integrating (3.17b)
and (3.31b) from ${\underline{\mathcal{C}}}$, respectively, with the boundary
terms
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{{\underline{\mathcal{C}}}}$
defined in Prop. 5.1. The fact that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ satisfies (Teuk)
implies that (3.32) is satisfied. We then show that
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ converges to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}$
as $u\to-\infty$, and similarly for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$.

  5. (V)

We define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ by
integrating (3.30b) from ${\underline{\mathcal{C}}}$, with boundary term
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}}_{{\underline{\mathcal{C}}}}$
as in Prop. 5.1. We then prove that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ satisfies the scalar
Regge–Wheeler equation (RW-scalar). This is done as follows:

     * •

We show that the $\mathring{\not{\mathrm{curl}}}\,$ of (3.22), replacing
$\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$
with $r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$, is satisfied:

$\displaystyle\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,r^{2}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}=r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-\mathring{\not{\mathrm{curl}}}\,\,r^{3}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}.$
(7.4)

This is proved by showing that
$\not{\nabla}_{v}\eqref{eq:construct:sigmatrick1}$ is satisfied using (3.17b),
(3.31b) and (3.30b) and using that (7.4) holds along
${\underline{\mathcal{C}}}$ by Prop. 5.1.

     * •

We then show that the $\mathring{\not{\mathrm{curl}}}\,$ of (3.28b) is
satisfied:

$\displaystyle\not{\nabla}_{u}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}=-\mathring{\not{\Delta}}r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-6M\Omega^{2}\,\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}.$
(7.5)

This again follows by virtue of (7.5) being satisfied along
${\underline{\mathcal{C}}}$ and by showing that
$\not{\nabla}_{v}\frac{r^{2}}{\Omega^{2}}\eqref{eq:construct:sigmatrick2}$ is
satisfied using (3.31b), (3.30b), (3.17b) as well as (7.4).

Equations (3.30b) and (7.5) together imply that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ solves (RW-scalar).

  6. (VI)

Next, we define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}+4\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})$.
Then $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$ satisfies
the Regge–Wheeler equation (RW). From here one, we can to some extent mirror
the previous approach, with a few extra difficulties that we will highlight
below: We define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ by
integrating
$(\tfrac{r^{2}}{\Omega^{2}}\not{\nabla}_{v})^{2}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}})$
twice from ${\underline{\mathcal{C}}}$ with data terms coming from Prop. 5.1.
The fact that $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$
satisfies (RW) is then used to show that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ satisfies
(Teuk) via Lemma 3.1.

  7. (VII)

We would like to define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ by
integrating (3.18a) from $\mathcal{I}^{-}$. Now, in general, we will have
decay no better than
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}=\mathcal{O}(|u|^{-\frac{5}{2}-})$
(cf. (5.10)), (3.18a), so
$\not{\nabla}_{u}(\Omega^{-1}r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}})=-r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$
will not be integrable. This problem is resolved by commuting (3.18a) with
$\not{\nabla}_{v}$, using that
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$
decays one power faster than
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$: We define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ by
integrating

$\displaystyle\not{\nabla}_{u}\not{\nabla}_{v}\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}=-\left(2-\frac{1}{\Omega^{2}}\right)r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}-\frac{r}{\Omega^{2}}\not{\nabla}_{v}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}})$
(7.6)

first in $u$ from $\mathcal{I}^{-}$ and then in $v$ from
${\underline{\mathcal{C}}}$. We can then define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$ via (3.27)
and deduce that (3.28a) holds by virtue of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ satisfying
(Teuk).

  8. (VIII)

The difficulty at this point is that we have no way of directly verifying
(3.30a)
($\partial_{u}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})=-\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$).
The work-around to this problem is to define a different
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ that
satisfies (3.30a), and to then show that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$
as follows:

     * •

Define $r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$
as the solution to $\mathring{\not{\mathrm{curl}}}\,\eqref{eq:lin:divxhb}$
(cf. (7.4)):

$\displaystyle
r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}:=\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}-\mathring{\not{\mathrm{curl}}}\,\frac{r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}{\Omega}.$
(7.7)

We directly deduce from the definition that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ satisfies
(3.30a).

     * •

As in step (V), we can then prove that
$\mathring{\not{\mathrm{curl}}}\,\eqref{eq:lin:beb4}$ holds with
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ replaced by
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$, from which
we can infer that
$r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$
satisfies (RW-scalar). We then show that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ attain the
same data on ${\underline{\mathcal{C}}}\cup\mathcal{I}^{-}_{v\geq v_{1}}$ and
appeal to the uniqueness clause of Theorem 6.1 to deduce
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$.

  9. (IX)

We define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$ via (3.17a)
and (3.18b), respectively. It is then a simple computation to confirm that
(3.19b) and (3.20a) hold as well. Then, using, in particular, equations
(3.30a) and (3.30b), we deduce (3.24). Since thus
$\mathring{\not{\mathrm{curl}}}\,(\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}})=0$,
we can define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ as
solution to (3.12). From
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$, we define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$ via (3.11),
and (3.20b), (3.19a) immediately follow from (3.20a), (3.19b).

  10. (X)

Similarly, since we have already proved that the
$\mathring{\not{\mathrm{curl}}}\,$-parts of (3.23) and (3.22) vanish in (7.4),
(7.7), we can define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$
via (3.22) and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$
via (3.23). Directly from this definition, we infer (3.16) by appropriately
differentiating (3.22) in $v$ and (3.23) in $u$.

  11. (XI)

As the penultimate step, we define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$ and prove that all
remaining equations of (3.9)–(3.32) featuring
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$ are satisfied (except the
Gauss equation, since we have not yet defined the metric components
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$ and
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$). The
easiest way to do this is as follows:

     * •

Prove that
$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\eqref{eq:lin:al33}$
is satisfied. This allows to define
$\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho},0)$,
and thus $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$, via (3.34), i.e.
as solution to

$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho},-r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})+6M(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}-r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}).$
(7.8)

     * •

By acting with $\not{\nabla}_{u}$ or $\not{\nabla}_{v}$ on (7.8) and using all
the previous equations, we can deduce that (3.29a), (3.29b) are satisfied. We
can then deduce the remaining Bianchi equations (3.31a), (3.28b).

     * •

We deduce (3.14) by multiplying (3.22) by $r\Omega$ and then acting with
$\not{\nabla}_{u}$. We similarly prove (3.15).

     * •

We finally prove (3.21) by acting on the definition (3.12) of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ with
$\not{\nabla}_{u}\not{\nabla}_{v}$ and using the already established
equations.

  12. (XII)

We define the remaining metric coefficients
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$ and
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ via
integration of (3.13) from $\mathcal{I}^{-}$ with
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}$
as data, and integration of (3.10b) and (3.9b) from
${\underline{\mathcal{C}}}$ with
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\underline{\mathcal{C}}}}$
(defined in Prop. 5.1) as data. By further taking the
$\not{\nabla}_{v}$-derivative of (3.9a) and (3.10a), we can establish that
(3.9b) and (3.10b) hold everywhere since they hold along
${\underline{\mathcal{C}}}$. The final thing left to do is to prove the Gauss
equation (3.25). For this, we prove that
$\not{\nabla}_{v}(r^{2}\eqref{eq:lin:K})$ holds and that, by construction,
(3.25) holds along ${\underline{\mathcal{C}}}$. We have now constructed a
solution to (3.9)–(3.32). By construction and Prop. 5.1, this solution
realises the prescribed seed data. This completes the proof.

#####  7.2.2 The full details of the construction

###### Proof of Proposition 7.2.

We now present the full details of the construction. The reader already
convinced by the overview should feel free to skip this section.

##### Step (I): Defining scattering data at $\mathcal{I}^{-}$

We first define a number of fields at $\mathcal{I}^{-}$ that will play the
role of data at $\mathcal{I}^{-}$ in the construction carried out in the
subsequent sections:

######  Definition 7.1.

Given a smooth seed scattering data set $\mathfrak{D}$, we define the
following fields at $\mathcal{I}^{-}$:

$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}=\mathring{\not{\mathrm{div}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}-\mathring{\not{\nabla}}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}},$
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}=-\not{\nabla}_{v}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}},$
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}}_{\mathcal{I}^{-}}=-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}},$
(7.9)

where
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
has been defined in (5.6), as well as

$\displaystyle\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}=-4\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\nabla}}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}+\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}},$
$\displaystyle{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}=-2(\mathring{\not{\Delta}}-4)\mathring{\not{\mathcal{D}}}^{\ast}_{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}_{\mathcal{I}^{-}}-6M\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}.$
(7.10)

######  Remark 7.1.

The above definitions will be used as data at $\mathcal{I}^{-}$ for
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$,
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}$,
$\not{\nabla}_{v}(r^{2}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}})$,
and $\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$,
respectively.

The definition above will be used explicitly to define the scattering solution
over the next few pages. On the other hand, we will now define a number of
fields at $\mathcal{I}^{-}$ for which it will later follow that they are
attained as limits of the constructed solution.

######  Definition 7.2.

Given a smooth seed scattering data set $\mathfrak{D}$, we define the
following fields at $\mathcal{I}^{-}$:

$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}}_{\mathcal{I}^{-}}(v)=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}}_{{\mathcal{S}_{\infty}}}+\int_{v_{1}}^{v}2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}(\bar{v})\mathop{}\\!\mathrm{d}\bar{v},$
$\displaystyle\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{\mathcal{I}^{-}}(v)=\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}-\int_{v_{1}}^{v}\mathring{\not{\mathrm{div}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}(\bar{v})\mathop{}\\!\mathrm{d}\bar{v},$
(7.11)
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}}_{\mathcal{I}^{-}}=\partial_{v}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}},$
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}}_{\mathcal{I}^{-}}=2\mathring{\not{\nabla}}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}},$
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{\mathcal{I}^{-}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}},$
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}}_{\mathcal{I}^{-}}=2\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{\mathcal{I}^{-}}+4\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}.$

######  Remark 7.2.

We will show in our construction that the fields above are attained as limits
of the quantities
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$,
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}$,
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$,
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}$ and
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$.
On the other hand, it will follow from our construction that the limits of the
following quantities vanish as $u\to-\infty$:
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$,
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$,
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$,
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$,
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$,
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$,
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$,
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$.

##### Step (II): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ using scattering theory
for (RW)

We now construct $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ via the
scattering theory for the Regge–Wheeler equation (RW) given by Theorem 6.1.

######  Proposition 7.3.

For a smooth seed scattering data $\mathfrak{D}$, there exists a unique smooth
finite energy solution $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$
to (RW) such that, for any $v\geq v_{1}$:

$\displaystyle\lim_{u\to-\infty}\|\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}-{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}\|_{L^{2}([v_{1},v]\times\mathbb{S}^{2})}=0,\qquad\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}}_{{\underline{\mathcal{C}}}},$
(7.12)

where
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}}_{{\underline{\mathcal{C}}}}$
is defined in Cor. 5.2 and
${\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}$
is defined in Def. 7.1. Moreover, we have the estimate

$|\mathring{\not{\nabla}}^{n}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}|\leq
C_{n}(v)r^{1/2},\qquad\forall n\in\mathbb{N}$ (7.13)

for some $C_{n}(v)$ depending continuously on $v$, and for any
$n\in\mathbb{N}$, $m\in\mathbb{N}_{\geq 1}$, we have

$\lim_{u\to-\infty}\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{n}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=\not{\nabla}_{v}^{m-1}\mathring{\not{\nabla}}^{n}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}$
(7.14)

as $u\to-\infty$, uniformly on $[v_{1},v]$ for any $v\geq v_{1}$.

###### Proof.

By the decay rate in Cor. 5.2, we deduce that
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}}_{{\underline{\mathcal{C}}}}$
has finite energy along ${\underline{\mathcal{C}}}$. The result thus follows
from the scattering theory for (RW) (Thm. 6.1) as well as (6.26) and (6.27). ∎

##### Step (III): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$

Next, we construct $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ by
integrating (3.34) from $\mathcal{I}^{-}$:

######  Definition 7.3.

For a smooth seed scattering data $\mathfrak{D}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ arising via Proposition
7.3, define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}$ by
integrating

$\displaystyle\frac{\Omega^{2}}{r^{2}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=\not{\nabla}_{u}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}$
(7.15)

in $u$ from $\mathcal{I}^{-}$ with data
$-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}}_{\mathcal{I}^{-}}$.
Similarly, we define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$
by integrating

$\displaystyle\frac{\Omega^{2}}{r^{2}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}=\not{\nabla}_{u}r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$
(7.16)

with data
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}$
at $\mathcal{I}^{-}$ (defined in Def. 7.1). By construction,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{{\underline{\mathcal{C}}}}$.

Note that these definitions are well-defined in view of estimate (7.13).

We now show that $\not{\nabla}_{v},\mathring{\not{\nabla}}$-derivatives of
$r^{3}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}$ converge to
$\not{\nabla}_{v},\mathring{\not{\nabla}}$-derivatives of
$-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$
as $u\to-\infty$, and a similar statement for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$:

######  Lemma 7.1.

For a smooth seed scattering data $\mathfrak{D}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}},\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$
arising via Definition 7.3, we have, for any $n,m\in\mathbb{N}$,

$\displaystyle\mathring{\not{\nabla}}^{n}\not{\nabla}_{v}^{m}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}(u,v,\theta^{A})\to\mathring{\not{\nabla}}^{n}\not{\nabla}_{v}^{m}\left(-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}\right)(v,\theta^{A}),$
(7.17)
$\displaystyle\mathring{\not{\nabla}}^{n}\not{\nabla}_{v}^{m}r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}(u,v,\theta^{A})\to-\mathring{\not{\nabla}}^{n}\not{\nabla}_{v}^{m+1}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}(v,\theta^{A})$
(7.18)

as $u\to-\infty$. For any $v\geq v_{1}$, this convergence is uniform on
$[v_{1},v]$.

###### Proof.

We compute that

$\displaystyle\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}=\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{v}\not{\nabla}_{u}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}=\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}-\frac{3\Omega^{2}-1}{r}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}.$
(7.19)

Integrating in $u$ and using (7.13) as well as (7.14), we see that
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}$
converges to a limit as $u\to-\infty$ and that this convergence is uniform on
$[v_{1},v]$ for any $v\geq v_{1}$. Since
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}\to-\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$,
it follows that
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}\to-\not{\nabla}_{v}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$.
The result for $\not{\nabla}_{v}$\- and $\mathring{\not{\nabla}}$-derivatives
follows by straight-forward commutation.

The result for $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ follows
analogously. ∎

Next, we want to show that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ solves the Teukolsky
equation (Teuk). For this, we first show:

######  Lemma 7.2.

For a smooth seed scattering data $\mathfrak{D}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}},\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$
arising via Definition 7.3, we have that

$\displaystyle\lim_{u\to-\infty}\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]=\lim_{u\to-\infty}\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]=0$
(7.20)

for any $v\geq v_{1}$, where the operator $\mathrm{Teuk}^{+2}$ was defined in
Def. 3.2.

###### Proof.

We can re-write the operators above as follows:

$\displaystyle\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]$
$\displaystyle=\frac{3\Omega^{2}-1}{r}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}+\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}+\left(-\mathring{\not{\Delta}}+2+\frac{6M}{r}\right)r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}},$
(7.21)
$\displaystyle\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]$
$\displaystyle=\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}-\left(\mathring{\not{\Delta}}-3\Omega^{2}-1\right)\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}+6Mr\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}.$
(7.22)

Therefore, using the previous Lemma 7.1:

$\displaystyle\lim_{u\to-\infty}\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]=-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\not{\nabla}_{v}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}-\left(\mathring{\not{\Delta}}-2\right)\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}.$
(7.23)
$\displaystyle\lim_{u\to-\infty}\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]={\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}+2\left(\mathring{\not{\Delta}}-4\right)\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}-6M\not{\nabla}_{v}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}.$
(7.24)

The right hand sides above both vanish by construction (cf. Def. 7.1). ∎

We may now infer:

######  Corollary 7.1.

For a smooth seed scattering data $\mathfrak{D}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ defined in Definition
7.3, we have that $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$
satisfies the Teukolsky equation (Teuk).

###### Proof.

Using Lemma 3.1, we have that

$\displaystyle\left(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}\right)^{2}\mathrm{Teuk}^{+2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}]=\mathrm{RW}[\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}]=0$
(7.25)

since $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ satisfies the
Regge–Wheeler equation (RW). We integrate this equation twice from
$\mathcal{I}^{-}$, where the boundary terms vanish by Lemma 7.2. ∎

##### Step (IV): Constructing of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$

We now construct $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ and show that (3.32) is
satisfied:

######  Definition 7.4.

With $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ defined in Def.
7.3, define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ to be the unique
solutions to (3.17b), (3.31b) with data
$\frac{r^{2}}{\Omega}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}$,
$\frac{r^{4}}{\Omega}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{{\underline{\mathcal{C}}}}$
respectively.

######  Lemma 7.3.

The quantities $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ satisfy eq. (3.32)

###### Proof.

Def. 7.4, together with the Teukolsky equation (Teuk) satisfied by
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$, implies that

$\displaystyle\not{\nabla}_{v}\left[\frac{r^{4}}{\Omega^{4}}\not{\nabla}_{u}r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}+2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\frac{r^{4}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}{\Omega}-6M\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}{\Omega}\right]=0.$
(7.26)

Integrating this in $v$ from ${\underline{\mathcal{C}}}$, and using that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}_{{\underline{\mathcal{C}}}}$
is related to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{{\underline{\mathcal{C}}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}_{{\underline{\mathcal{C}}}}$
via (3.32) by Proposition 5.1 proves the lemma. ∎

For later purposes, we will also need to verify that
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ and
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ realise
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$
(which we have used to define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$) as their limits at
$\mathcal{I}^{-}$:

######  Corollary 7.2.

For $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ constructed in
Definition 7.4, we have that for any $m,n\in\mathbb{N}$,

$\displaystyle\lim_{u\to-\infty}\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{m}r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}=\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{m}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}},$
$\displaystyle\lim_{u\to-\infty}\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{m}r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}=\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{m}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}},$
(7.27)

the convergence being uniform in $v$ in compact $v$-intervals.

###### Proof.

We will prove the statement for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$; the proof for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ being similar: First of
all, we have, by definition,

$\frac{1}{r}\left(\frac{r^{2}}{\Omega}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}(u,v)-\frac{r^{2}}{\Omega}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}(u)\right)=-\frac{1}{r}\int_{v_{1}}^{v}r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}(u,\bar{v})\mathop{}\\!\mathrm{d}\bar{v}.$
(7.28)

Since $r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}(u,v)$ converges
uniformly to
$-\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}$
on $[v_{1},v]$ for any fixed $v\geq v_{1}$ by Lemma 7.1, and since the ratio
$\frac{r(u,\bar{v})}{r(u,v)}$ tends to 1 uniformly in on $[v_{1},v]$ for any
fixed $v$, we have that

$\displaystyle-\frac{1}{r(u,v)}\int_{v_{1}}^{v}r(u,\bar{v})^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}(u,\bar{v})\mathop{}\\!\mathrm{d}\bar{v}\to-\int_{v_{1}}^{v}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}(\bar{v})\mathop{}\\!\mathrm{d}\bar{v}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}(v)-\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}(v_{1})$
(7.29)

as $u\to-\infty$ for any $v\geq v_{1}$. Using that
$r\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}\to\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}(v=v_{1})$
by construction (Prop. 5.1), it follows that
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}\to\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}$
as claimed. The proof for higher derivatives follows from Lemma 7.1 and
commuting. ∎

##### Step (V): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$

######  Definition 7.5.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ to be the
unique solution to (3.30b) with data
$r^{3}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}}_{{\underline{\mathcal{C}}}}$
on ${\underline{\mathcal{C}}}$.

######  Corollary 7.3.

We have for any $n\in\mathbb{N}$ and some $C_{n}(v)$ depending continuously on
$v$ that

$|\mathring{\not{\nabla}}^{n}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})|\leq
C_{n}(v)r^{\frac{1}{2}}.$ (7.30)

In addition, the following convergence is uniform in $v$ on any compact
$v$-interval:

$\lim_{u\to-\infty}\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{n}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}=-\not{\nabla}_{v}^{m-1}\mathring{\not{\nabla}}^{n}\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}},\qquad\text{for
all }m\in\mathbb{N}_{\geq 1},\,n\in\mathbb{N}.$ (7.31)

###### Proof.

The convergence (7.31) follows from the definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ and Cor. 7.2. The
upper bound (7.30) follows from the bound (7.13), the definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ and the rates (5.10)
for
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}}_{{\underline{\mathcal{C}}}}$.
∎

######  Lemma 7.4.

The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ defined
in Definition 7.5 satisfies
$\mathring{\not{\mathrm{curl}}}\,\eqref{eq:lin:divxh}$ as well as
$\mathring{\not{\mathrm{curl}}}\,\eqref{eq:lin:be3}$:

$\displaystyle\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}=r^{2}\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}},$
(7.32)
$\displaystyle\partial_{u}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}=-\mathring{\not{\Delta}}\,r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-6M\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}.$
(7.33)

###### Proof.

Equations (3.31b), (3.30b) imply

$\displaystyle\partial_{v}\,\left(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-\mathring{\not{\mathrm{curl}}}\,r^{3}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)=-\mathring{\not{\mathrm{curl}}}\,r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}=\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}{\Omega}$
(7.34)

and, since $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$ satisfy (7.32) at
${\underline{\mathcal{C}}}$, equation (7.32) propagates for $v\geq v_{1}$.

The proof for (7.33) is similar, but involves more computations: We first
multiply the LHS of (7.33) and differentiate in $v$:

$\displaystyle\partial_{v}\left(\frac{r^{2}}{\Omega^{2}}\,\partial_{u}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)$
$\displaystyle=\partial_{v}\left(\frac{r^{2}}{\Omega^{2}}\right)\partial_{u}\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+\partial_{u}\mathring{\not{\mathrm{curl}}}\,\not{\nabla}_{v}(r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}})$
(7.35)
$\displaystyle=\partial_{u}\,\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}+\frac{3\Omega^{2}-1}{r}\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}-2(3\Omega^{2}-2)\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$
$\displaystyle=\,\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\left[-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+6M\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}\right]-2(3\Omega^{2}-2)\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}},$

where we used (3.31b) in the second line and (3.32) in the third line. We re-
write the angular operator (recall the notation
$\mathring{\not{\mathrm{div}}}\,=\mathring{\not{\mathcal{D}}}_{2}$) using eq.
(2.30):

$\displaystyle\mathring{\not{\mathcal{D}}}_{1}\left[-2\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\right]=\mathring{\not{\mathcal{D}}}_{1}\left[\mathring{\not{\Delta}}+1\right]=\mathring{\not{\mathcal{D}}}_{1}\left[2-\mathring{\not{\mathcal{D}}}^{\ast}_{1}\mathring{\not{\mathcal{D}}}_{1}\right]=\left[2-\mathring{\not{\mathcal{D}}}_{1}\mathring{\not{\mathcal{D}}}^{\ast}_{1}\right]\mathring{\not{\mathcal{D}}}_{1}=\left[\mathring{\not{\Delta}}+2\right]\mathring{\not{\mathcal{D}}}_{1},$

so we have, in particular, that

$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\left(-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)=\left(\mathring{\not{\Delta}}+2\right)\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}},$

and we can write

$\displaystyle\partial_{v}\left(\,\frac{r^{2}}{\Omega^{2}}\,\partial_{u}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)=\left(\mathring{\not{\Delta}}+\frac{12M}{r}\right)\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+6M\,\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}.$

Finally, we apply (7.32) to this:

$\partial_{v}\left(\,\frac{r^{2}}{\Omega^{2}}\,\partial_{u}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)=\left(\mathring{\not{\Delta}}+\frac{12M}{r}\right)\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+6M\left(\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-\mathring{\not{\mathrm{curl}}}\,r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right).$
(7.36)

Next, we apply the operator $\partial_{v}\frac{r^{2}}{\Omega^{2}}$ to the
right hand side of (7.33):

$\partial_{v}\left(\mathring{\not{\Delta}}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}+6Mr^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}\right)=-\mathring{\not{\Delta}}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}-6Mr\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-6M\,\mathring{\not{\mathrm{curl}}}\,r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}.$
(7.37)

Combining (7.36) and (7.37), we deduce that

$\displaystyle\partial_{v}\left(\frac{r^{2}}{\Omega^{2}}\partial_{u}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+\mathring{\not{\Delta}}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}+6Mr^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}\right)=0.$
(7.38)

But since (7.33) holds along ${\underline{\mathcal{C}}}$, we can now integrate
the above from ${\underline{\mathcal{C}}}$ to deduce that it holds everywhere.
∎

######  Corollary 7.4.

The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$
constructed in Definition 7.5 satisfies the scalar Regge–Wheeler equation (RW-
scalar).

###### Proof.

This follows from (3.30b) and (7.33). ∎

##### Step (VI): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$.

We are now in position to define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$ and thus
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$:

######  Definition 7.6.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$ via

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}:=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}+2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}).$
(7.39)

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}$ to be
the unique solution to

$\displaystyle\frac{\Omega^{2}}{r^{2}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}=\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}},$
(7.40)

with data
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\psi}}}_{{\underline{\mathcal{C}}}}=2\mathring{\not{\mathcal{D}}}^{\ast}_{2}r^{2}\Omega\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\underline{\mathcal{C}}}}+6M\Omega\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}$
at ${\underline{\mathcal{C}}}$. Define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ to be the
unique solution to

$\displaystyle\frac{\Omega^{2}}{r^{2}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}=\not{\nabla}_{v}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}),$
(7.41)

with data
$r\Omega^{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}}_{{\underline{\mathcal{C}}}}$
at ${\underline{\mathcal{C}}}$.

######  Corollary 7.5.

Since $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ and
$\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})$
satisfy the Regge–Wheeler equation (RW),
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$ also satisfies
(RW). Moreover, we have for any $n\in\mathbb{N}$ and for some $C_{n}(v)$
depending continuously on $v$ the estimates

$\displaystyle|\mathring{\not{\nabla}}^{n}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}|\leq
C_{n}(v)r^{\frac{1}{2}},$
$\displaystyle|\mathring{\not{\nabla}}^{n}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}|\leq
C_{n}(v)r^{-\frac{1}{2}},$
$\displaystyle|\mathring{\not{\nabla}}^{n}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}|\leq
C_{n}(v)r^{-5/2},$ (7.42)

as well as the uniform convergence on compact $v$-intervals of

$\lim_{u\to-\infty}\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{n}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}=\not{\nabla}_{v}^{m-1}\mathring{\not{\nabla}}^{n}(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}}_{\mathcal{I}^{-}}-\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}})),\qquad\text{for
all }m\in\mathbb{N}_{\geq 1},\,n\in\mathbb{N}.$ (7.43)

###### Proof.

The first of (7.42) follows from the estimates (7.13), (7.30). Then using the
definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ together
with the rates (5.10) and (5.12), the other two rates follow as well.

The convergence (7.43) follows directly from (7.14) and (7.31). ∎

As we did for $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$, we now
show that $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$
satisfies (Teuk) by appealing to equation (3.51) from Lemma 3.1.

######  Corollary 7.6.

The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$
constructed in Def. 7.6 satisfies the Teukolsky equation (Teuk).

###### Proof.

Since $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}$
satisfies the Regge–Wheeler equation (RW), we have by Lemma 3.1 that

$\displaystyle\left(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{v}\right)^{2}\mathrm{Teuk}^{-2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}]=0.$
(7.44)

A computation analogous to that in the proof of Lemma 7.2 shows that
$\mathrm{Teuk}^{-2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}]$,
$\not{\nabla}_{v}\mathrm{Teuk}^{-2}[r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}]$
vanish at ${\underline{\mathcal{C}}}$. ∎

##### Step (VII): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$

We now construct
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$. Note that
since
$|\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}|\lesssim
C(v)r^{-\frac{5}{2}}$, we cannot integrate (3.18a) in $u$ from
$\mathcal{I}^{-}$. However, since
$|\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}|\lesssim
C(v)r^{-7/2}$ (by (7.42)), we can instead integrate the
$\not{\nabla}_{v}$-commuted equation:

######  Definition 7.7.

Define $\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}$
to be the unique solution to

$\not{\nabla}_{u}\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}=-r^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}-\left(2-\frac{1}{\Omega^{2}}\right)r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}},$
(7.45)

with data
$\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}$
at $\mathcal{I}^{-}$ defined in Def. 7.1. Define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ to be
the unique solution to

$\not{\nabla}_{v}\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}=\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}},$
(7.46)

with data
$r^{2}\Omega^{-1}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}$
at ${\underline{\mathcal{C}}}$.

######  Lemma 7.5.

The quantity
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$
constructed in Definition 7.7 satisfies

$\not{\nabla}_{u}\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}=-r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}.$
(7.47)

Moreover, we have the following uniform convergence along compact
$v$-intervals:

$\lim_{u\to-\infty}\not{\nabla}_{v}^{m}\mathring{\not{\nabla}}^{n}r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}=\not{\nabla}_{v}^{m-1}\mathring{\not{\nabla}}^{n}\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}},\qquad\text{for
all }m\in\mathbb{N}_{\geq 1},\,n\in\mathbb{N}.$ (7.48)

###### Proof.

The first statement follows by integrating the identity

$\displaystyle\not{\nabla}_{v}(r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}})=r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}+\left(2-\frac{1}{\Omega^{2}}\right)r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}=-\not{\nabla}_{u}\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}=-\not{\nabla}_{v}\not{\nabla}_{u}\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}$
(7.49)

in $v$ from ${\underline{\mathcal{C}}}$, using that (7.47) is satisfied along
${\underline{\mathcal{C}}}$ by
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}_{{\underline{\mathcal{C}}}}$
as a consequence of Prop. 5.1.

The proof of (7.48) proceeds exactly as the proof of Lemma 7.1, using the
rates (7.42). ∎

######  Definition 7.8.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$ to be
the unique solution to (3.27), with
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ defined
as in Definition 7.7 and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$ defined as
in Definition 7.6. By construction and Prop. 5.1,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\underline{\mathcal{C}}}}$.

######  Corollary 7.7.

The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$
constructed in Definition 7.8 satisfies (3.28a).

###### Proof.

This follows by computing
$\not{\nabla}_{u}(\tfrac{r^{4}}{\Omega^{4}}\eqref{eq:lin:alb4})$, using
(3.18a) for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ as well
as the Teukolsky equation (Teuk) for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$. (Recall
that the operator $\mathring{\not{\mathcal{D}}}^{\ast}_{2}$ is invertible on
$\ell\geq 2$.) ∎

##### Step (VIII): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ and proving
that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$

Recall the difficulties addressed in the overview of Step (VIII) in §7.2.1:
Since we have no direct way of inferring (3.30b) at this point, we instead
define a different
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$:

######  Definition 7.9.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ via
the $\mathring{\not{\mathrm{curl}}}\,$ of (3.23):

$\displaystyle
r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}:=\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,r^{2}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}-\mathring{\not{\mathrm{curl}}}\,r^{3}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}.$
(7.50)

By construction and Prop. 5.1,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ restricts to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}}_{{\underline{\mathcal{C}}}}$
on $\mathcal{C}$.

######  Lemma 7.6.

The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$
satisfies

$\displaystyle\partial_{u}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime})=\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}.$
(7.51)

###### Proof.

This follows by taking the $\partial_{u}$-derivative of the LHS of (7.50)
using the already established equations (3.18a), (3.28a). ∎

In order to infer that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$,
we first show that their radiation fields at $\mathcal{I}^{-}$ are identical:

######  Lemma 7.7.

For $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$, we have

$\displaystyle\lim_{u\to-\infty}\partial_{v}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime})=-\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}=\lim_{u\to-\infty}\partial_{v}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}).$
(7.52)

This convergence is uniform in $v$ on compact $v$-intervals.

###### Proof.

Note that we have already shown this for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$. For
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$, we first
take the $\partial_{v}$-derivative of (7.50) and use (7.48) to find

$\lim_{u\to-\infty}\partial_{v}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime})=\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}-\lim_{u\to-\infty}\mathring{\not{\mathrm{curl}}}\,\not{\nabla}_{v}(\tfrac{r^{3}}{\Omega}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}).$
(7.53)

In order to evaluate the limit on the RHS, we compute, using (3.27):

$\displaystyle\lim_{u\to-\infty}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\,\not{\nabla}_{v}(r^{3}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}})=\lim_{u\to-\infty}(\not{\nabla}_{v}(r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}})-6M\not{\nabla}_{v}(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}})).$
(7.54)

But

$\not{\nabla}_{v}(r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}})=r^{-1}\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}+\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\underline{\psi}}}$

tends to 0 as $u\to-\infty$ by (7.42) and (7.43), and so does
$\not{\nabla}_{v}(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}})$
by (7.48). It thus follows that
$\lim_{u\to-\infty}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\not{\nabla}_{v}r^{3}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}=0$.
In order to deduce that
$\lim_{u\to-\infty}\mathring{\not{\mathrm{curl}}}\,r^{3}\Omega^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}=0$
as well, we simply apply
$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathcal{D}}}_{2}$: To be
precise, we use that for any 1-form $\beta$:

$2\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\beta=-(\mathring{\not{\Delta}}+2)\mathring{\not{\mathrm{curl}}}\,\beta.$
(7.55)

It follows that

$\lim_{u\to-\infty}-(\mathring{\not{\Delta}}+2)\partial_{v}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime})=-(\mathring{\not{\Delta}}+2)\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}-\lim_{u\to-\infty}2\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\not{\nabla}_{v}(\tfrac{r^{3}}{\Omega}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}),$
(7.56)

The term on the RHS now vanishes by the same argument as above, and, by the
invertibility of $\mathring{\not{\Delta}}+2$ on $\ell\geq 2$, a Poincaré
inequality and Sobolev embedding:

$\lim_{u\to-\infty}\partial_{v}(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime})=\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}=\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}=\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}},$
(7.57)

where the last two equalities follow from definitions of
$\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}$
in Def. 7.1 (and the fact that
$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\nabla}}f=0=\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\nabla}}f$
for any $f\in\Gamma^{\infty}{(\mathbb{S}^{2})}$). ∎

We can now show that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ also
satisfies (RW-scalar):

######  Lemma 7.8.

The quantities
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ satisfy the
$\mathring{\not{\mathrm{curl}}}\,$ of (3.28a), i.e.:

$\displaystyle\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}=-\mathring{\not{\Delta}}r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}-6M\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}.$
(7.58)

In particular, in view of Lemma 7.6,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ satisfies
(RW-scalar).

###### Proof.

This is similar to the proof of (7.33). In the same way in which we proved
(7.38), we first show that

$\displaystyle\partial_{u}\left[\frac{r^{2}}{\Omega^{2}}\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}-\mathring{\not{\Delta}}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}-6Mr^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}\right]=0.$
(7.59)

Notice that we cannot integrate this from $\mathcal{I}^{-}$, because the
object on which $\partial_{u}$ is acting does not converge near
$\mathcal{I}^{-}$. We again resolve this problem by considering the
$\partial_{v}$-derivative:

$\displaystyle\partial_{u}\partial_{v}\left[\frac{r^{2}}{\Omega^{2}}\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}-\mathring{\not{\Delta}}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}-6Mr^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}\right]=0.$
(7.60)

We integrate this in $u$, using that the boundary term at $\mathcal{I}^{-}$
vanishes. Indeed:

######  Sublemma 1.

$\displaystyle\lim_{u\to-\infty}\partial_{v}\left(\frac{r^{2}}{\Omega^{2}}\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}})\right)=\mathring{\not{\Delta}}\,\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}=\lim_{u\to-\infty}\mathring{\not{\Delta}}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}.$
(7.61)

###### Proof of the sublemma..

As in the proof of Lemma 7.7, we use (3.27) to compute

$\displaystyle
2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\not{\nabla}_{v}\left(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{v}(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}})\right)=\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}-6M\not{\nabla}_{v}^{2}\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}+6M\frac{(3\Omega^{2}-1)}{r}\not{\nabla}_{v}\frac{r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}}{\Omega}-12M(3\Omega^{2}-2)\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}};$
(7.62)

so we have

$\lim_{u\to-\infty}2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\not{\nabla}_{v}\left(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{v}(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}})\right)=\lim_{u\to-\infty}\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}-6M\not{\nabla}_{v}\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}.$
(7.63)

In view of (7.43), we have

$\displaystyle\lim_{u\to-\infty}\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}={\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}-4\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}})$
$\displaystyle=-2(\mathring{\not{\Delta}}-4)\mathring{\not{\mathcal{D}}}^{\ast}_{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}-4\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}})-6M\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}$
(7.64)
$\displaystyle=2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}\left(\mathring{\not{\mathrm{div}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}},-\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}\right)+6M\not{\nabla}_{v}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}},$

where we also used (2.30). Acting now on this with
$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathcal{D}}}_{2}$, we obtain

$\displaystyle\lim_{u\to-\infty}2\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\not{\nabla}_{v}\left(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{v}(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}})\right)=-(\mathring{\not{\Delta}}+2)(\mathring{\not{\Delta}})\mathring{\not{\mathrm{curl}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}}_{\mathcal{I}^{-}}.$
(7.65)

The result then follows using (7.55) (and that we are supported on $\ell\geq
2$). ∎

Using this sublemma, we can now integrate (7.60) from $\mathcal{I}^{-}$ to
deduce that

$\displaystyle\partial_{v}\left[\frac{r^{2}}{\Omega^{2}}\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}-\mathring{\not{\Delta}}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}-6Mr^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}\right]=0.$
(7.66)

In turn, integrating this from ${\underline{\mathcal{C}}}$, and verifying that
$\partial_{v}\,\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}-\mathring{\not{\Delta}}r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}-6M\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=0$
at ${\underline{\mathcal{C}}}$ by computing the $\not{\nabla}_{v}$ derivative
of $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$ in
spacetime via (3.27), restricting to ${\underline{\mathcal{C}}}$ and using
Proposition 5.1, we conclude the proof of (7.58).

Acting now with (7.58) on (7.51), we infer that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ satisfies
(RW-scalar). ∎

######  Corollary 7.8.

With $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ defined
in Def. 7.9 and $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$
defined in Def. 7.5, we have
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$.

###### Proof.

Since
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$
on ${\underline{\mathcal{C}}}$ and
$\lim_{u\to-\infty}\partial_{v}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}=\lim_{u\to-\infty}\partial_{v}r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$
uniformly in $v$, the uniqueness clause of Theorem 6.1 implies
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$
by virtue of both $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}$ satisfying
(RW-scalar). ∎

##### Step (IX): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$ as well as
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$

######  Definition 7.10.

Recall that the kernel of $\mathring{\not{\mathcal{D}}}^{\ast}_{2}$ is spanned
by $\ell=0,1$. Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$ to be the
unique solutions to (3.17a) and (3.18b).

$\displaystyle-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=\not{\nabla}_{u}(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}})+\Omega^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}},\qquad-2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}=\not{\nabla}_{v}(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}})-\Omega^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}.$
(7.67)

######  Corollary 7.9.

The one-forms $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$ satisfy
(3.20a), (3.19b) and (3.24). Moreover, we have
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}}_{{\underline{\mathcal{C}}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}}_{{\underline{\mathcal{C}}}}$
as well as

$\displaystyle\lim_{u\to-\infty}r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=0,$
$\displaystyle\lim_{u\to-\infty}r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}}_{\mathcal{I}^{-}},$
(7.68)

where
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}}_{\mathcal{I}^{-}}$
is defined in Def. 7.2. This convergence is uniform on compact $v$-intervals
and commutes with $\mathring{\not{\nabla}}$ and
$\not{\nabla}_{v}$-derivatives.

###### Proof.

We prove (3.19b) by multiplying the first of (7.67) (i.e. (3.17a)) with
$\Omega^{-2}r$ and then acting with $\not{\nabla}_{v}$, using (3.17b), (3.32)
as well as the second of (7.67) (i.e. (3.18b)).

The proof of (3.20a) is analogous.

In order to prove (3.24), we take the
$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,$ of (3.17a).
From there, using (7.32), (7.50) and Corollary 7.8, we get

$\displaystyle\partial_{u}\left(r^{2}\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}-\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)=\left(\mathring{\not{\Delta}}+2\right)\,\mathring{\not{\mathrm{curl}}}\,\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}-\frac{\Omega^{2}}{r}\left(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}+\mathring{\not{\mathrm{curl}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}\right).$
(7.69)

We then utilise (3.30a) and (7.33) to obtain

$\displaystyle\left(\mathring{\not{\Delta}}+2\right)\mathring{\not{\mathrm{curl}}}\,r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=\left(\mathring{\not{\Delta}}+2\right)r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}},$
(7.70)

which implies
$\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}$
since $\mathring{\not{\Delta}}+2$ is invertible on $\ell\geq 2$.

The claim for
$\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$
follows in the same way, using equations (3.18b), (7.50), (3.30b), Lemma 7.8
and Corollary 7.8.

The claim that $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$ restricts
to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}}_{{\underline{\mathcal{C}}}}$
follows from the definition, the analogous statements for
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}$ as well
the fact that the first of (7.67) is satisfied along
${\underline{\mathcal{C}}}$ by Prop. 5.1. Furthermore, it is easy to see that
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$ tends to 0 as
$u\to-\infty$.

The uniform convergence of
$r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$ follows from
its definition and the uniform convergence of
$\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}$ to
$\EndsWiths{\underline{\stackrel{{\scriptstyle\mbox{\scalebox{0.4}{(1)}}}}{{S}}}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}$,
respectively. By integrating (3.20a), we then infer that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}}_{{\underline{\mathcal{C}}}}$.
∎

In particular, since we have now shown that
$\mathring{\not{\mathrm{curl}}}\,(\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}})=0$,
the following definition is well-defined:

######  Definition 7.11.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ via
$2\mathring{\not{\nabla}}\left(\frac{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}{\Omega}\right)=r(\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}})$,
and define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}=\partial_{v}\left(\frac{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}{\Omega}\right)$,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}=\partial_{u}\left(\frac{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}{\Omega}\right)$.

######  Corollary 7.10.

Equations (3.19a) and (3.20b) are satisfied. Moreover,
$\lim_{u\to-\infty}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$
(this convergence being uniform on compact $v$-intervals and commuting with
$\mathring{\not{\nabla}}^{n}$ and $\not{\nabla}_{v}^{m}$), and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}}_{{\underline{\mathcal{C}}}}$.

###### Proof.

Equation (3.19a) follows directly from (3.20a), and the definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$. Equation
(3.20b) follows similarly. The convergence of
$\mathring{\not{\nabla}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$
to
$\mathring{\not{\nabla}}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$
follows from the limiting behaviour of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$, and the
convergence of $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$
then follows straight-forwardly (using that we are supported on $\ell\geq 2$).
The restriction of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$ to
${\underline{\mathcal{C}}}$ follows from
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ restricting to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{{\underline{\mathcal{C}}}}$
by definition and Prop. 5.1. ∎

##### Step (X): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$
and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$.

######  Definition 7.12.

We define
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$
and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$
according to (3.22) and (3.23), i.e.

$\displaystyle\frac{1}{2\Omega}\mathring{\not{\nabla}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}=\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}+\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}+r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}},$
$\displaystyle\frac{1}{2\Omega}\mathring{\not{\nabla}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}=\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}-\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}.$
(7.71)

By construction and Prop. 5.1,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$
and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$
restrict to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}}_{{\underline{\mathcal{C}}}}$
along ${\underline{\mathcal{C}}}$, respectively.

######  Remark 7.3.

These definitions are well-defined since we already know that the
$\mathring{\not{\mathrm{curl}}}\,$ of the respective RHS’s vanish by (7.32),
(7.50) and
$\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}=r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}^{\prime}=-\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$.

######  Corollary 7.11.

The quantities
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$
and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$
defined above satisfy equations (3.16), and
$\lim_{u\to-\infty}r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}=0$,
$\lim_{u\to-\infty}r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}}_{\mathcal{I}^{-}}$,
these convergences being uniform on compact $v$-intervals and commuting with
$\not{\nabla}_{v}$\- and $\mathring{\not{\nabla}}$-derivatives.

###### Proof.

Equation (3.16a) follows by acting on the first of (7.71) with
$\not{\nabla}_{u}(\frac{r^{2}}{\Omega}\cdot)$ and using (3.17a), (3.20a) as
well as (3.31a). Equation (3.16b) follows analogously.

The limiting behaviour follows from the definitions and the previous
estimates. ∎

##### Step (XI): Constructing
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$ and proving all equations
featuring $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$

Before we can define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$, we
need to prove the following

######  Lemma 7.9.

We have

$\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=2\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(0,-r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})+6M\mathring{\not{\mathrm{curl}}}\,\mathring{\not{\mathrm{div}}}\,(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}-r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}).$
(7.72)

###### Proof.

First, note that this equation holds along ${\underline{\mathcal{C}}}$ by
construction (cf. Proposition 5.1 and Corollary 5.2). Next, we confirm that
$\not{\nabla}_{v}\eqref{eq:construct:pedanticsrho}$ holds by computing the
$\not{\nabla}_{v}$-derivative of the LHS (7.72) using the definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ and the Teukolsky
equation (Teuk) (the result is (3.39)), and using (3.30b), (3.17b) and (3.18b)
for the RHS. ∎

The above lemma enables the following definition:

######  Definition 7.13.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$ to be the unique
solution to

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}\left(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho},-r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}\right)+6M\left(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}-r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}\right).$
(7.73)

By construction and by Prop. 5.1,
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}|_{{\underline{\mathcal{C}}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$.

######  Remark 7.4.

Definition 7.13 implies, via (3.38):

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}=2\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}\left(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho},r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}\right)+6M\left(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}-r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}\right).$
(7.74)

######  Lemma 7.10.

The quantity $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}$ defined above
satisfies equations (3.31a), (3.28b) as well as (3.29).

###### Proof.

We first prove (3.31a): We take the definition (7.73), and rewrite its LHS as
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}=(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u})^{2}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}})$.
We now rewrite this as the
$\tfrac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}$-derivative of the RHS of (3.33).
Equation $\mathring{\not{\mathcal{D}}}^{\ast}_{2}\eqref{eq:lin:be3}$ then
follows by taking into account the already established equation (3.17a), and
the result follows from the invertibility of
$\mathring{\not{\mathcal{D}}}^{\ast}_{2}$.

Equation (3.28b) follows similarly, starting from (7.74).

Next, we prove (3.29b). Using (3.31b) and (3.32), we compute:

$\displaystyle\not{\nabla}_{v}\left(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}\right)=-2\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\,(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}})+6M\mathring{\not{\mathrm{div}}}\,\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}-2(3\Omega^{2}-2)r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}.$
(7.75)

Therefore, by (3.31a) and (3.19b) we have

$\displaystyle\mathring{\not{\mathcal{D}}}^{\ast}_{1}\partial_{v}\left(-r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho},r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}\right)=\left(\mathring{\not{\Delta}}+1\right)r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+6M\,\mathring{\not{\mathrm{div}}}\,\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}-2(3\Omega^{2}-2)\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+6M\left(r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}-\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}\right).$
(7.76)

Taking the divergence of both sides and using
$\mathring{\not{\mathrm{div}}}\,\left(\mathring{\not{\Delta}}+1\right)\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}=\left(\mathring{\not{\Delta}}+2\right)\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}$
and equation (3.22), we get

$\displaystyle\mathring{\not{\Delta}}\partial_{v}\,(r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho})=\mathring{\not{\Delta}}\,\mathring{\not{\mathrm{div}}}\,r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}}+3M\mathring{\not{\Delta}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}.$
(7.77)

This shows that (3.29b) is satisfied by the invertibility of
$\mathring{\not{\Delta}}$ on $\ell\geq 2$.

The argument to prove (3.29a) is analogous. ∎

######  Lemma 7.11.

The equations (3.14) and (3.15) are satisfied by
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$
and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}$.

###### Proof.

Let’s first prove (3.14): We differentiate the definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}$,
(3.22):

$\frac{1}{2}\mathring{\not{\nabla}}\not{\nabla}_{u}(r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)})=\not{\nabla}_{u}(\mathring{\not{\mathrm{div}}}\,r\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}})+\not{\nabla}_{u}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}})+\not{\nabla}_{u}(r^{2}\Omega\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\beta}})\\\
=-2\Omega^{2}\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}-\Omega^{3}\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\hat{\chi}}}+\not{\nabla}_{u}(\Omega^{2})r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}+r\Omega^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}-\Omega^{4}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+\mathring{\not{\mathcal{D}}}^{\ast}_{1}(-r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho},r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}})-\frac{6M\Omega^{2}}{r}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}},$
(7.78)

where we used (3.17a), (3.20a) and (3.31a). We now use the fact that

$-2\Omega^{2}\mathring{\not{\mathcal{D}}}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=-\Omega^{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}\mathring{\not{\mathcal{D}}}_{1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+2\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}=\Omega^{2}(\mathring{\not{\nabla}}\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}-{}^{\ast}\mathring{\not{\nabla}}\mathring{\not{\mathrm{curl}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}+2\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}})=\Omega^{2}(\mathring{\not{\nabla}}\mathring{\not{\mathrm{div}}}\,\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}-{}^{\ast}\mathring{\not{\nabla}}r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\sigma}}+2\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}),$

as well as equations (3.12) and (3.23) to conclude that
$\mathring{\not{\nabla}}\eqref{eq:lin:trx3}$ holds. Eq. (3.15) follows
similarly. ∎

######  Lemma 7.12.

The equations (3.21) are satisfied by
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$,
respectively.

###### Proof.

We compute
$\not{\nabla}_{u}\mathring{\not{\nabla}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\omega}=\not{\nabla}_{u}\not{\nabla}_{v}\mathring{\not{\nabla}}\left(\frac{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}{\Omega}\right)=\not{\nabla}_{v}\mathring{\not{\nabla}}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}$
by using (3.12), (3.20a), (3.19b) to compute
$\mathring{\not{\nabla}}\partial_{v}\left(\frac{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}{\Omega}\right)$,
and then using (3.28b), (3.31a), (3.20a) and (3.19b) to compute
$\not{\nabla}_{u}\not{\nabla}_{v}\mathring{\not{\nabla}}\left(\frac{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}{\Omega}\right)$.
∎

##### Step (XII): Constructing the remaining metric components
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$,
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$

######  Definition 7.14.

Define $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$ to be the solution to
$\not{\nabla}_{u}(r^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b})=2r^{-1}\Omega^{2}(\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}-\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}})$
(i.e. (3.13)) with data
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}$
for $r^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$ at
$\mathcal{I}^{-}$.

######  Definition 7.15.

Define ${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$
and $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$ as solutions
to (3.9b) and (3.10b) with
$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}}_{{\underline{\mathcal{C}}}}$
as data along ${\underline{\mathcal{C}}}$.

######  Lemma 7.13.

The equations (3.9a) and (3.10a) are satisfied by
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$.

###### Proof.

We present the proof for (3.9a); the proof for (3.10a) is similar. Since
(3.9a) is satisfied along ${\underline{\mathcal{C}}}$, it suffices to prove
that $\partial_{v}\eqref{eq:lin:trg3}$ holds. We prove the latter by computing
$\partial_{v}\partial_{u}{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}=\partial_{u}\partial_{v}{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$
via (3.14) and (3.13), and by computing
$\partial_{v}(2\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)})$
via (3.15). The result then follows. ∎

######  Corollary 7.12.

The following convergence is uniform in $v$ on compact $v$-intervals:

$\displaystyle\lim_{u\to-\infty}{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}=\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{\mathcal{I}^{-}},$
$\displaystyle\lim_{u\to-\infty}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}}_{\mathcal{I}^{-}},$
$\displaystyle\lim_{u\to-\infty}r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{\mathcal{I}^{-}}.$
(7.79)

These limits commute with $\not{\nabla}_{v}$-and
$\mathring{\not{\nabla}}$-derivatives.

From the definition of $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$ and
the decay rates of $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\eta}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\eta}}$, it is easy to
see that $r^{-1}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$ converges
uniformly in $v$ to $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}$ as
$u\to-\infty$, and it thus follows that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}$,
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ as well
as $r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}$ tend to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\hat{\not{g}}}}_{\mathcal{I}^{-}}$,
$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{\mathcal{I}^{-}}$,
respectively.

The final equation of (3.9)–(3.32) that we have to prove is the Gauss
equation:

######  Lemma 7.14.

The Gauss equation (3.25) is satisfied, with
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}$ defined in (3.26).

###### Proof.

We compute the $\partial_{v}$-derivative of the Gauss equation using (3.9b),
(3.10b), (3.29b), (3.16b), (3.15) and (3.11b). This shows that
$\partial_{v}\eqref{eq:lin:K}$ is satisfied. The result now follows, as (3.25)
is satisfied along ${\underline{\mathcal{C}}}$ by Prop 5.1. ∎

We have now constructed a solution to (3.9)–(3.32), and we have shown that it
satisfies Definition 5.2. This concludes the proof of Prop. 7.2. ∎

####  7.3 Construction of the $\ell<2$-part of the solution to the scattering
problem

We now present the construction of the $\ell<2$-part of the solution: This
construction will be entirely explicit:

######  Proposition 7.4.

Given a smooth seed scattering data set $\mathfrak{D}_{\ell=0}$ supported on
$\ell=0$, let
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
be as in Prop. 5.1 (see (5.6)), and let
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}$
be as in Cor. 5.3. Then
$\mathfrak{S}_{\mathfrak{m}}+\mathfrak{S}_{f}+\mathfrak{S}_{\underline{f}}$ is
a scattering solution realising $\mathfrak{D}_{\ell=0}$, with

$\displaystyle\mathfrak{m}$
$\displaystyle=\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{M}+3\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
(7.80) $\displaystyle\underline{f}$
$\displaystyle=\frac{r}{6M\Omega^{2}}(-r^{3}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}-2\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}+6M\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}})$
$\displaystyle f$
$\displaystyle=\int_{v_{1}}^{v}2(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}+\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}+\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{4M})\mathop{}\\!\mathrm{d}v^{\prime}.$

The decomposition
$\mathfrak{S}_{\mathfrak{m}}+\mathfrak{S}_{f}+\mathfrak{S}_{\underline{f}}$ is
unique up to the ambiguity addressed in Remark 4.1.

###### Proof.

It is easy to directly verify that this is a scattering solution realising
$\mathfrak{D}_{\ell=0}$. Still, we find it insightful to provide the
construction: We begin by realising that only the ingoing gauge solution
$\mathfrak{S}_{\underline{f}}$ and the nearby Schwarzschild solution
$\mathfrak{S}_{\mathfrak{m}}$ have non-trivial limits for
$r^{3}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}|_{{\underline{\mathcal{C}}}}$
and
$r^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}|_{{\underline{\mathcal{C}}}}$.
If we write $\lim_{u\to-\infty}r^{-1}\underline{f}=\underline{f}_{0}$, we then
find the system of equations

$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}=-6M\underline{f}_{0}-2M\mathfrak{m},$
$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}=2\underline{f}_{0}+\mathfrak{m},$
(7.81)

which is solved by
$\mathfrak{m}=\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{M}+3\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$,
$\underline{f}_{0}=-\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}-\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{2M}$.
We have thus already determined $\mathfrak{S}_{\mathfrak{m}}$.

Next, we fully fix $\underline{f}$ by demanding that

$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}=-\frac{6M\Omega^{2}}{r^{4}}\underline{f}-\frac{2M\mathfrak{m}}{r^{3}}=-\frac{6M\Omega^{2}}{r^{4}}\underline{f}-\frac{2\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{r^{3}}-\frac{6M\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}}{r^{3}}.$
(7.82)

Finally, we fix $f$ by noticing that both $\mathfrak{m}$ and $\underline{f}$
generate a limit of $\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}$ at
$\mathcal{I}^{-}$, by demanding that $f|_{{\underline{\mathcal{C}}}}=0$ and by

$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}=\tfrac{1}{2}\partial_{v}f+\tfrac{1}{2}\left(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}+\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{2M}\right)-\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{2M}-\frac{3\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}}{2}.$
(7.83)

It is left to show that this is indeed a scattering solution realising
$\mathfrak{D}_{\ell=0}$: By Corollary 5.3, it suffices to show that the
constructed solution realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}$,
$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\chi\right)}}_{{\mathcal{S}_{1}}}$:
Now, by construction, the solution
$\mathfrak{S}_{\mathfrak{m}}+\mathfrak{S}_{f}+\mathfrak{S}_{\underline{f}}$
realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}=-\frac{1}{2}\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}$
as well as
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$.
Moreover, since it realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$,
we can deduce from (3.29a) that it also realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}}_{{\underline{\mathcal{C}}}}$
(and thus
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}}_{{\mathcal{S}_{1}}}$)
by Prop. 5.1. Finally, we deduce from (3.16a) that
$\mathfrak{S}_{\mathfrak{m}}+\mathfrak{S}_{f}+\mathfrak{S}_{\underline{f}}$
also realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}}_{{\underline{\mathcal{C}}}}$.
∎

######  Proposition 7.5.

Given a smooth seed scattering data set $\mathfrak{D}_{\ell=1}$ supported on
$\ell=1$, let
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
be as in Prop. 5.1, and let
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}}$
be as in Cor. 5.3. Then
$\mathfrak{S}_{\mathfrak{a}}+\mathfrak{S}_{f}+\mathfrak{S}_{\underline{f}}+\mathfrak{S}_{(q_{1},q_{2})}$
is a scattering solution realising $\mathfrak{D}_{\ell=1}$ for

$\displaystyle\mathfrak{a}_{m}$
$\displaystyle=-3M\sqrt{2}(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}})^{\mathrm{H},1}_{\ell=1,m},$
(7.84) $\displaystyle(q_{1},q_{2})_{\ell=1,m}$
$\displaystyle=\left(\frac{1}{4}\left(\frac{3}{2M}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}-\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}\right),0\right))_{\ell=1,m}+\int_{v_{1}}^{v}\sqrt{2}((\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}})^{\mathrm{E},1}_{1,m},(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}})^{\mathrm{H},1}_{1,m})\mathop{}\\!\mathrm{d}v^{\prime},$
$\displaystyle f_{\ell=1,m}$
$\displaystyle=-\frac{1}{6M\sqrt{2}}(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}})^{\mathrm{E},1}_{\ell=1,m}+\int_{v_{1}}^{v}2\left(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}-\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{6M}\right)_{\ell=1,m}\mathop{}\\!\mathrm{d}v^{\prime},$
$\displaystyle\underline{f}_{\ell=1,m}$
$\displaystyle=-\frac{r^{4}}{6M\Omega^{2}}(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}})_{\ell=1,m}+\frac{1}{\sqrt{2}\cdot
6M}(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}})^{\mathrm{E},1}_{\ell=1,m}.$

###### Proof.

We uniquely determine $\mathfrak{a}$ directly from Prop. 4.2 and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}}$.
(Note that none of the pure gauge solutions affect the magnetic part of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}$.)

Now, we choose a preliminary definition of $\underline{f}_{p}$ by demanding
that $\mathfrak{S}_{\underline{f}_{p}}$ restricts to
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$:

$\underline{f}_{p}=-\frac{r^{4}}{6M\Omega^{2}}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}+\tilde{\underline{f}}(\theta^{A}).$
(7.85)

We will determine $\tilde{\underline{f}}(\theta^{A})$ later. This choice for
the ingoing gauge function induces a limit for
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ at
$\mathcal{I}^{-}$. This limit is independent of
$\tilde{\underline{f}}(\theta^{A})$. Since the outgoing gauge function always
induces a vanishing limit for
${\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}$ at
$\mathcal{I}^{-}$, we can thus fix $q_{1}(v=v_{1},\theta^{A})$ by demanding
that $\mathfrak{S}_{\underline{f}_{p}}+\mathfrak{S}_{(q_{1},q_{2})}$ restrict
to
$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}$:

$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}=\frac{3}{2M}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}-4q_{1}(v=v_{1},\theta^{A}).$
(7.86)

We now fully fix $(q_{1},q_{2})$ by demanding that
$q_{2}(v=v_{1},\theta_{A})=0$ and that
$\mathring{\not{\mathcal{D}}}^{\ast}_{1}(\partial_{v}q_{1},\partial_{v}q_{2})=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}$.

Next, we fix $\partial_{v}f$ by demanding that the limit
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$
is met:

$\frac{\partial_{v}f}{2}=\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}-\frac{\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\mathcal{S}_{\infty}}}}{12M},$
(7.87)

where the second term comes from $\underline{f}_{p}$. We fully fix $f$ by
demanding that the limit
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}}$
is attained, i.e.:

$f_{\ell=1,m}(v=v_{1})=-\frac{1}{6M\sqrt{2}}(\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}})^{\mathrm{E},1}_{\ell=1,m}.$
(7.88)

Finally, we fix $\tilde{\underline{f}}$ by demanding that
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$
is attained.

It is left to show that the constructed solution realises
$\mathfrak{D}_{\ell=1}$. By construction,
$\mathfrak{S}_{\mathfrak{a}}+\mathfrak{S}_{f}+\mathfrak{S}_{\underline{f}}+\mathfrak{S}_{(q_{1},q_{2})}$
realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}$
as limits at $\mathcal{I}^{-}$, and it realises
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}}_{{\mathcal{S}_{\infty}}}$,
$\EndsWiths{{\mathrm{tr}{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\not{g}}}}}_{{\mathcal{S}_{\infty}}}$
as well as
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\rho}}_{{\underline{\mathcal{C}}}}$.
Moreover, in view of
$\not{\nabla}_{u}\frac{r^{4}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\beta}}_{\ell=1}}{\Omega}=0$,
it follows that
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\omega}}}_{{\underline{\mathcal{C}}}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\left(\Omega\mathrm{tr}\underline{\chi}\right)}}_{{\mathcal{S}_{1}}}$
are also attained by (3.29a) and (3.16a). The result now follows from
Corollary 5.3 (as in the case $\ell=0$). ∎

###  8 The physical scattering data

The previous sections featured a general discussion of how to set up and solve
the scattering problem for a general class of seed scattering data.

In the present section, we specify this discussion to physically motivated
seed data. We will first define the condition of no incoming radiation, and
then give and physically justify various definition of seed scattering data
describing the far-region of a system of $N$ infalling bodies, and finally
collect a few basic consequences arising from these definitions that will be
useful for the analysis in the later parts of this paper.

####  8.1 The condition of no incoming radiation

We first define the condition of no incoming radiation from $\mathcal{I}^{-}$
(this definition is stated without any prior gauge fixing!):

######  Definition 8.1.

A set of smooth seed scattering data $\mathfrak{D}$ is said to satisfy the _no
incoming radiation condition_ if the following condition is satisfied:

$\mathring{\not{\mathrm{div}}}\,\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}=\mathring{\not{\nabla}}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}},$
(8.1)

where
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
is defined in (5.6). This is equivalent to demanding that

$\displaystyle\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\hat{\chi}}}}_{\mathcal{I}^{-}}=\mathring{\not{\mathcal{D}}}^{\ast}_{2}\mathring{\not{\mathcal{D}}}^{\ast}_{1}(f_{x},0),$
$\displaystyle(\mathring{\not{\Delta}}+2)f_{x}=2\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
(8.2)

for some $f_{x}\in\Gamma^{\infty}(\mathcal{I}^{-}_{v\geq v_{1}})$ with
$\partial_{v}f_{x}=0$.

The property of a seed scattering data set to have no incoming radiation
cannot be changed by addition of a pure gauge solution (Prop. 4.3–Prop. 4.5).
We are therefore justified in calling it a gauge invariant condition.

Conversely, we recall from §5.4 that the other quantities of $\mathfrak{D}$ at
$\mathcal{I}^{-}$,
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{b}}_{\mathcal{I}^{-}}$
and
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\Omega}}_{\mathcal{I}^{-}}$,
can be set to 0 by addition of pure gauge solutions. Since
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}$
can also be set to zero, the no incoming radiation condition therefore
completely eliminates the radiative, physical degrees of freedom at
$\mathcal{I}^{-}$.

Note that, in particular, if $\mathfrak{D}$ is as in Thm. 5.1 and satisfies
the no incoming radiation condition, then all the radiation fields defined in
Def. 7.1 vanish. For later reference, we highlight that this in particular
implies that $r\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$,
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$ and
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\Psi}}=\not{\nabla}_{v}(\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{v})^{2}(r\Omega^{2}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}})$
all attain vanishing limits at $\mathcal{I}^{-}$. Note that the no incoming
radiation condition is strictly stronger than demanding that the limit of
$\not{\nabla}_{v}\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\Psi}}$
vanishes.

######  Remark 8.1.

The definition of no incoming radiation here is written down in such a way as
to be gauge invariant. Most readers will be more familiar with a version of
the no incoming radiation condition in the nonlinear theory where a gauge has
already been fixed: provided one works in a gauge where the spheres at
$\mathcal{I}^{-}$ are round (i.e. $\bm{r}^{2}\bm{K}\to 1$ as
$\bm{u}\to-\infty$), the no incoming radiation condition is captured by the
requirement that the Bondi mass along $\mathcal{I}^{-}$ be constant. In view
of the mass growth formula (this is simply the time reversal of the Bondi mass
loss formula [BVdBM62, Sac62b, CK93]),

$\frac{\mathop{}\\!\mathrm{d}\bm{M}_{\mathrm{Bondi}}}{\mathop{}\\!\mathrm{d}\bm{v}}=\lim_{\bm{u}\to-\infty}\frac{1}{{16\pi}}\int_{\mathbb{S}^{2}}|\bm{r}\hat{\bm{\chi}}|^{2}\mathop{}\\!\mathrm{d}\bm{\Omega},$
(8.3)

this clearly requires $\bm{r}\hat{\bm{\chi}}=0$ at $\mathcal{I}^{-}$. This is
exactly what our condition restricts to in the case of Bondi-normalised data
(which have
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{K}}_{{\mathcal{S}_{\infty}}}=0$).

######  Remark 8.2.

Clearly, it would not be enough to replace the condition of no incoming
radiation (8.1) with a weaker condition such as demanding that
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}=0$
along $\mathcal{I}^{-}$, as this would be consistent with a constant
${\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\mathtt{P}}}_{\mathcal{I}^{-}}$
at past null infinity, which would correspond to a constant growth in energy.
However, it is interesting to note that condition (8.1) is, in fact,
equivalent to demanding that
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{\mathcal{I}^{-}}=0$
and
$\lim_{u\to-\infty}\frac{r^{2}}{\Omega^{2}}\not{\nabla}_{u}(r\Omega^{2}\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{{\underline{\mathcal{C}}}})=0$,
as can be seen from (3.32) and Definition 7.1. In other words, the no incoming
radiation condition forces
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{{\underline{\mathcal{C}}}}=o(r^{-2})$
along ${\underline{\mathcal{C}}}$. The additional requirement of finite
Regge–Wheeler energy then means that we must have
$\EndsWiths{\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}}_{{\underline{\mathcal{C}}}}=o(r^{-5/2})$.

####  8.2 Seed data describing the far-field region of the hyperbolic $N$-body
problem

The link to the physical scenario of $N$ infalling masses (with negligible
internal structure, i.e. point masses) moving along approximately hyperbolic
orbits in the infinite past is made by introducing Post-Newtonian
approximations to the Post-Minkowskian multipolar expansion framework; this
has been described in detail in §2 of [Keh24]. In short, the argument in the
Post-Minkowskian setting consists of writing down a general expression of an
outgoing vacuum solution to the Einstein equations, which takes the following
form for the curvature components of
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}$ and
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}$:121212Note
that this is exactly analogous to the statement that the general fixed-angular
frequency solution $\phi_{\ell}$ to the Minkowskian wave equation
$\Box_{\eta}\phi=0$ takes the form
$\phi_{\ell}=r^{\ell}\sum_{m=-\ell}^{\ell}\partial_{u}^{\ell}\left(\frac{f_{\ell,m}(u)Y_{\ell,m}}{r^{\ell+1}}\right)+\partial_{v}^{\ell}\left(\frac{\bar{f}_{\ell,m}(v)Y_{\ell,m}}{r^{\ell+1}}\right),$
(8.4) with the $\partial_{v}$-part of the solution vanishing because it is
demanded that the solution be of purely outgoing type.

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}_{\ell}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}r^{\ell-2}\sum_{m=-\ell}^{\ell}\not{\nabla}_{u}^{\ell-2}\left(\frac{I_{\ell,m}(u)Y_{\ell,m}^{\mathrm{E},2}-S_{\ell,m}(u)Y_{\ell,m}^{\mathrm{H},2}}{r^{3+\ell}}\right),$
(8.5)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{\underline{\alpha}}_{\ell}$
$\displaystyle=r^{\ell-2}\sum_{m=-\ell}^{\ell}\not{\nabla}_{u}^{\ell+2}\left(\frac{I_{\ell,m}(u)Y_{\ell,m}^{\mathrm{E},2}+S_{\ell,m}(u)Y_{\ell,m}^{\mathrm{H},2}}{r^{\ell-1}}\right).$
(8.6)

for some real-valued functions $I_{\ell,m}$, $S_{\ell,m}$ depending only on
$u$. Notice that (8.5) and (8.6) are exactly related via the
Teukolsky–Starobinsky identity (3.43) with $M=0$.

These expressions, which are valid in the vacuum region of spacetime, are then
matched to expressions for the matter region of spacetime (i.e. the region of
the $N$ bodies), which, in turn, are derived under a number of approximations
and matching to the Newtonian theory. The result is that the $I_{\ell,m}$
acquire the interpretation of the $\ell$-th Newtonian mass multipole moment,
and the $S_{\ell,m}$ acquire the interpretation of the $\ell$-th Newtonian
current multipole moment. Finally, these moments are computed using the
Newtonian theory (or perturbations thereof). Since, schematically, the
$\ell$-th mass multipole moment goes like $m\cdot r^{\ell}$, where $m$ and $r$
denote mass and size of the system, and since, for a system of masses
following hyperbolic orbits, the size grows linearly in time, we thus find
that $I_{\ell,m}(u)\sim|u|^{\ell}$ as $u\to-\infty$. The same rate is found
for $S_{\ell,m}$, and it can thus be seen that
$\accentset{\scalebox{0.6}{\mbox{\tiny(1)}}}{{\alpha}}_{\ell}$, within this
framework, is predicted to decay like $|u|^{2}/r^{5}\sim r^{-3}$ near past