The linearised Einstein equations in a double null gauge on a Schwarzschild exterior background are the following:

* The transport equations governing the linearised metric components:

equationmetric transport in 4 direction trace
equationmetric transport in 4 direction traceless
equationmetric transport in 3 direction trace
equationmetric transport in 3 direction traceless
equationpartial_u b
∂_v = 2-2 ,                = 2Ω+2_2 ,
<ref>, <ref>

∂_u = 2 ,               = 2Ω ,
<ref>, <ref>

∂_u     = 2Ω^2(      -) ,

\begin{equation*}
\refstepcounter{equation}\latexlabel{partial v Olin}
\refstepcounter{equation}\latexlabel{partial u Olin}
\refstepcounter{equation}\latexlabel{elin eblin Olin}
\qquad\qquad\partial_v\left(\frac{\Olino}{\Omega}\right)\,=\,\olin,\qquad\qquad \partial_u\left(\frac{\Olino}{\Omega}\right)\,=\,\olinb,\qquad\qquad \elin_A+\eblin_A=2\slashednabla_A\left(\frac{\Olino}{\Omega}\right),
\tag{\ref{partial v Olin},\;\ref{partial u Olin},\;\ref{elin eblin Olin}}
\end{equation*}

* The equations governing the linearised connection coefficients:

\begin{equation} \label{D4TrChiBar}
\partial_v\; r\otxb=2\Omega^2\left(\divr\; r\,\eblin+r\,\rlin -\frac{4M}{r^2}\,\frac{\Olino}{\Omega}\right)+\Omega^2\,\otx,
\end{equation}

\begin{equation}\label{D3TrChi}
\partial_u\; r\otx=2\Omega^2\left(\divr\; r\,\elin+r\,\rlin-\frac{4M}{r^2}\,\frac{\Olino}{\Omega}\right) -\Omega^2\, \otxb,
\end{equation}

∂_v[r^2/Ω^2]=4r ,                 ∂_u[r^2/Ω^2]=-4r ,
<ref>, <ref>

r^2/Ω = -r^2 ,                       r^2/Ω = -r^2 ,
<ref>, <ref>

\begin{equation}\label{D3Chihat}
\nablau\; r\Omega\,\xlin\,=\,-2r\fancydstar_2 \,\Omega^2 \,\elin\,-\,\Omega^2 \left(\Omega\, \xblin\right),
\end{equation}

\begin{equation}\label{D4Chihatbar}
\nablav\;r\Omega\,\xblin\,=\,-2r\fancydstar_2\,\Omega^2\,\eblin\,+\,\Omega^2\left(\Omega\,\xlin\right),
\end{equation}

\begin{equation*}
\refstepcounter{equation}\latexlabel{D3etabar}
\refstepcounter{equation}\latexlabel{D4eta}
\qquad\qquad\nablau\,r\,\eblin\,=\,r\Omega\,\bblin-\,\Omega^2\elin\,,\qquad\qquad \nablav\,r\,\elin\,=\,-r\Omega\,\blin+\Omega^2\,\eblin\,,
\tag{\ref{D3etabar},\;\ref{D4eta}}
\end{equation*}

\begin{equation*}
\refstepcounter{equation}\latexlabel{D4etabar}
\refstepcounter{equation}\latexlabel{D3eta}
\qquad\qquad\nablav\,r^2\,\eblin\,=\,2r^2\slashednabla \olin\,+\,r^2\Omega\,\blin\,,\qquad\qquad \nablau\,r^2\,\elin\,=\,2r^2\slashednabla\olinb\,-\,r^2\Omega\,\bblin\,,
\tag{\ref{D4etabar},\;\ref{D3eta}}
\end{equation*}

\begin{equation}\label{transport equation of little omega}
\partial_v \overone{\underline\omega}=-\Omega^2\left(\overone{\rho}-\frac{4M}{r^3}\frac{\overone{\Omega}}{\Omega}\right)=\partial_u \overone{\omega}
\end{equation}

* The equations governing the curvature components:

rΩ^2  = -2r_2Ω^2Ω +6MΩ^2/r^2Ω  ,                  rΩ^2  = 2r_2Ω^2Ω +6MΩ^2/r^2Ω  ,
<ref>, <ref>

r^4/Ω = r r^3 ,                    r^4/Ω = -r r^3 ,
<ref>, <ref>

r^2Ω =r_1(rΩ^2 ,rΩ^2 ) + 6MΩ^2/r  ,                  r^2Ω =r_1(-rΩ^2 ,rΩ^2 ) - 6MΩ^2/r  ,
<ref>, <ref>

∂_v r^3 = r r^2Ω  + 3M  ,                 ∂_u r^3 = r r^2Ω  + 3M  ,
<ref>, <ref>

∂_v r^3 = -r r^2Ω  ,                 ∂_u r^3 = r r^2Ω  ,
<ref>, <ref>

* The linearised Codazzi equations

equationelliptic equation 1
equationelliptic equation 2
equationelliptic equation 3 and 4
= Ω/r  +  + 1/2Ω   ,      <ref>

= -Ω/r  +  + 1/2Ω   ,      <ref>

=                   = -      ,<ref>

and the linearised Gauss equation

\begin{align}\label{Gauss}
\Klin&\,=\,-\,\rlin\,-\frac{1}{2r}\left(\overone{\Omega\tr\underline\chi}-\overone{\Omega\tr\chi}\right)-\frac{2\Omega}{r^2}\;\Olino.
\end{align}

Here, $\glinh$, $\xlin$, $\xblin$, $\alin$, $\ablin$ are symmetric traceless $S^2_{u,v}$-tangent tensor fields, $\bmlin$, $\elin$, $\eblin$, $\blin$, $\bblin$ are $S^2_{u,v}$-tangent $1$-form fields, and $\glinto$, $\Olino$, $\olin$, $\olinb$, $\otx$, $\otxb$, $\rlin$, $\slin$ are scalar fields. The linearised Gaussian curvature $\overone{K}$ is given by

\begin{align}\label{Formula for K}
\Klin[\,\glin\,]:=-\left(\frac{1}{4}{\slashed\Delta}+\frac{1}{2r^2}\right)\tr_{\slashed{g}}\glin+\frac{1}{2}\slashed{div}\slashed{div}\overone{\hat{\slashed{g}}}.
\end{align}

Let $\mathcal{S}\subset \mathscr{M}$ be open. If the tuple

\begin{align}\label{tuple}
\mathfrak{S}=\left(\;\glinh\;,\;\tr\glin\;,\;\bmlin\;,\;\Olino\;,\;\xlin\;,\;\xblin\;,\;\trx\;,\;\trxbar\;,\;\elin\;,\;\eblin\;,\;\olin\;,\;\olinb\;,\;\alin\;,\;\ablin\;,\;\blin\;,\;\bblin\;,\;\rlin\;,\;\slin\;\right),
\end{align}

satisfies the equations above on $\mathcal{S}$, we say that $\mathfrak{S}$ is a solution to the linearised Einstein equations, or alternatively, a solution to linearised gravity, on $\mathcal{S}$.

We may refer to a component $\upsilon$ of a solution $\mathfrak{S}$ by $\upsilon[\mathfrak{S}]$. For example, $\xlin[\mathfrak{S}]$ is the component $\xlin$ of the tuple don't know defining the solution $\mathfrak{S}$. We will often drop the reference to $\mathfrak{S}$ when no ambiguity arises.

In defining a solution to the system above, we treat the metric, connection and curvature components as independent quantities, related by transport equations above.

§.§ The linearised equations in Kruskal coordiantes

It is necessary to cast the equations in the Kruskal frame to compensate for the degeneration of the Eddington–Finkelstein frame near the bifurcation sphere $\mathcal{B}$. When re-scaling the quantities defining a solution to as to be consistent with linearisation in the Kruskal frame,

\begin{align}\label{Regular linearised quantities}
\begin{split}
&\qquad\glinhK:=\glinh,\qquad \tr\glinK:=\tr\glin,\qquad \bmlinK:=V^{-1}\bmlin,\qquad\OlinK:=\Olin,\\[3pt]
&\qquad\qquad\qquad\alinK:=V^{-2}\Omega^2\alin,\qquad\qquad\qquad \ablinK:=U^{-2}\Omega^2\ablin, \\[5pt]
&\;\qquad\qquad\qquad\blinK:=V^{-1}\Omega\blin,\qquad\qquad\qquad\; \bblinK:=U^{-1}\Omega\bblin, \\[6pt]
&\qquad\qquad\qquad\qquad\rlinK:=\rlin,\qquad\qquad\qquad\qquad\quad \slinK:=\slin,
\end{split}
\end{align}

we obtain the following system:

\begin{align}
(2M)^{-1}\partial_V \tr\glin\;=\;2\otxK-2\;\slashed{div}\,\bmlinK,\qquad\qquad&(2M)^{-1}\slashednabla_V \glinhK\;=\;2\xlinK+2\fancydstar_2\bmlinK,\label{metric transport in 4 direction Kruskal}\\
(2M)^{-1}\partial_U \tr\glinK\;=\;-2\otxbK,\qquad\qquad &(2M)^{-1}\slashednabla_U\glinhK\;=\;-2\xblinK.\label{metric transport in 3 direction Kruskal}\\
\frac{\fr}{2M}\partial_U\bmlinK^A\;=\;2(\elinK-&\eblinK)^A.\label{partial_u b Kruskal}\\
(2M)^{-1}\partial_V\left(\OlinK\right)\;=\;\olinK,\qquad\qquad(2M)^{-1}\partial_U\left(\OlinK\right)&\;=\;-\olinbK,\qquad\qquad\elinK+\eblinK\;=\;2\slashednabla_A \left(\OlinK\right),\label{omega omegabar eta etabar Kruskal}
\end{align}

\begin{equation} \label{D4TrChiBar Kruskal}
\frac{\fr}{2M}\partial_V \otxbK=2\left(\slashed{div}\; \elinK+\rlinK-\frac{4M}{r^3}\OlinK\right) -\frac{1}{r}\left(V\otxK-U\otxbK\right)=\frac{\fr}{2M}\partial_U \otxK,
\end{equation}

\begin{equation}\label{D4TrChi Kruskal}
\end{equation}

\begin{equation}\label{D3TrChiBar Kruskal}
\end{equation}

\begin{equation}\label{D4Chihat Kruskal}
(2M)^{-1}\slashednabla_V \xlinK-\frac{U}{2M\fr}\left(1+\frac{6M}{r}\right)\xlinK=-\alinK,
\end{equation}

\begin{align}\label{D3Chiharbar Kruskal}
(2M)^{-1}\slashednabla_U \xblinK-\frac{V}{2M\fr}\left(1+\frac{6M}{r}\right)\xblinK=-\ablinK  ,
\end{align}

\begin{equation}\label{D3Chihat Kruskal}
\frac{\fr}{2M}\slashednabla_U\; r\xlinK=-2r\slashed{\mathcal{D}}^*_2 \elinK+{U}\xblinK,
\end{equation}

\begin{equation}\label{D4Chihatbar Kruskal}
\frac{\fr}{2M}\slashednabla_V\; r\xblinK=-2r\slashed{\mathcal{D}}^*_2 \eblinK-V\xlinK,
\end{equation}

\begin{equation}\label{D3etabar Kruskal}
(2M)^{-1}\slashednabla_Ur\eblinK=r\bblinK+\frac{U}{\fr}\eblinK,\qquad\qquad (2M)^{-1}\slashednabla_Vr\elinK=-r\blinK-\frac{U}{\fr}\eblinK,
\end{equation}

\begin{equation}\label{D4etabar Kruskal}
(2M)^{-1}\slashednabla_V r^2\eblinK=-2\mathring{\slashednabla}r\olinK-{r^2}\blinK,\qquad\qquad (2M)^{-1}\slashednabla_U r^2\elinK=-2\mathring{\slashednabla}r\olinbK-{r^2}\bblinK,
\end{equation}

\begin{equation}\label{transport equation of little omega Kruskal}
(2M)^{-1}\partial_V\olinbK\,=\,\frac{1}{\fr}\left(\rlinK-\frac{4M}{r^3}\OlinK\right)=-(2M)^{-1}\partial_U \olinK,
\end{equation}

\begin{equation}\label{Bianchi +2 Kruskal}
\frac{\fr}{2M}\slashednabla_U \;r\alinK=-2\mathring{\fancydstar_2}\,\blinK+\frac{6M}{r^2}\xlinK,\qquad\qquad \frac{\fr}{2M}\slashednabla_V \;r\ablinK=2\mathring{\fancydstar_2}\,\bblinK+\frac{6M}{r^2}\xblinK,
\end{equation}

\begin{equation}\label{Bianchi +1a Kruskal}
(2M)^{-1}\slashednabla_V r^2\blinK-\frac{U}{2M\fr}\left(1+\frac{6M}{r}\right)r^2\blinK=\divo\;r\alinK,
\end{equation}

\begin{align}\label{Bianchi -1a Kruskal}
(2M)^{-1}\slashednabla_U r^2\bblinK-\frac{V}{2M\fr}\left(1+\frac{6M}{r}\right)r^2\bblinK=\divo\;r\ablinK,
\end{align}

\begin{align}
\frac{\fr}{2M}\slashednabla_U r^2\blinK=\mathring{\fancydstar_1}(-r\rlinK,r\slinK)-\frac{6M}{r}\elinK,&\qquad\qquad\frac{\fr}{2M}\slashednabla_V r^2\bblinK=-\mathring{\fancydstar_1}(r\rlinK,r\slinK)-\frac{6M}{r}\eblinK,\label{Bianchi +1b Kruskal}\\[4pt]
(2M)^{-1}\slashednabla_U\; r^3\rlinK=\divo\;r^2\bblinK-3M \otxbK,&\qquad \qquad(2M)^{-1}\slashednabla_V\; r^3\rlinK=\divo\;r^2\blinK+3M \otxK,\label{Bianchi 0 Kruskal}\\[6pt]
(2M)^{-1}\slashednabla_U\; r^3\slinK=\curlo \;r^2\bblinK,&\qquad\qquad (2M)^{-1}\slashednabla_V \; r^3\slinK=-\curlo\;r^2\blinK\label{Bianchi 0* Kruskal}
\end{align}

\begin{align}
&\slashed{{div}}\xblinK\;=\;-\frac{V}{r\fr}\elinK+\bblinK+\frac{1}{2}\slashednabla \otxbK,\label{elliptic equation 1 Kruskal}\\
&\slashed{{div}}\xlinK\;=\;\frac{U}{r\fr}\eblinK-\blinK+\frac{1}{2}\slashednabla \otxK,\label{elliptic equation 2 Kruskal}\\
\slashed{curl}& \elinK \;=\;\slinK,\qquad\qquad\qquad \slashed{curl} \eblinK \;=\;-\slinK,\label{elliptic equation 3 and 4 Kruskal}
\end{align}

\begin{align}\label{Gauss Kruskal}
\KlinK&=-\rlinK+\frac{1}{2r}\left(V\otxK-U\otxbK\right)-\frac{2\Omega^2}{r^2}\OlinK.
\end{align}

§ THE TEUKOLSKY EQUATIONS, THE TEUKOLSKY–STAROBINSKY IDENTITIES AND THE REGGE–WHEELER EQUATION

§.§ The spin $\pm2$ Teukolsky equations

Let $\alin$, $\ablin$ belong to a solution to the linearised Einstein equations . It is easy to show that the equation implies that $\alin$, $\ablin$ obey 2nd order hyperbolic equations that decouple from the rest of the system . These are the well-known Teukolsky equations:

\begin{align}\label{T+2}
\frac{\Omega^2}{r^2}\nablav \frac{r^4}{\Omega^4}\nablau r\Omega^2\alin=-2r^2\fancydstar_2\fancyd_2 r\Omega^2\alin-\frac{6M}{r}r\Omega^2\alin,
\end{align}

\begin{align}\label{T-2}
\frac{\Omega^2}{r^2}\nablau \frac{r^4}{\Omega^4}\nablav r\Omega^2\ablin=-2r^2\fancydstar_2\fancyd_2 r\Omega^2\ablin-\frac{6M}{r}r\Omega^2\ablin.
\end{align}

<Ref> is known as the Spin $+2$ Teukolsky equation, and <ref> is known as the Spin $-2$ Teukolsky equation.  For a derivation of T+2, T-2 see <cit.>.

We now state some well-posedness results concerning the Teukolsky equations T+2, T-2:

Prescribe on ${\Sigma^*_+}$ a pair of smooth symmetric traceless $S^2_{u,v}$ 2-tensor fields $(\upalpha,\upalpha')$. Then there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor field $\Omega^2\alpha$ that satisfies T+2 on $D^+({\Sigma^*_+})$, with $\Omega^2\alpha|_{{\Sigma^*_+}}=\upalpha, \slashednabla_{n_{{\Sigma^*_+}}}\Omega^2\alpha|_{{\Sigma^*_+}}=\upalpha'$.

Prescribe on ${\Sigma^*_+}$ a pair of smooth symmetric traceless $S^2_{u,v}$ 2-tensor fields $(\underline\upalpha,\underline\upalpha')$. Then there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor field $\Omega^{-2}\underline\alpha$ that satisfies T-2 on $D^+({\Sigma^*_+})$, with $\Omega^{-2}\underline\alpha|_{{\Sigma^*_+}}=\underline\upalpha, \slashednabla_{n_{{\Sigma^*_+}}}\Omega^2\underline\alpha|_{{\Sigma^*_+}}=\underline\upalpha'$.

The same applies replacing ${\Sigma^*_+}$ with any other $\mathscr{H}^+$-penetrating spacelike surface ending at $i^0$.

The degeneration of the EF frame discussed in <Ref> is inherited by T+2, T-2, and we must work with $\widetilde{\alpha}=V^{-2}\Omega^2\alpha, \widetilde{\underline\alpha}=U^2\Omega^{-2}\underline\alpha$ in order to study the Teukolsky equations with data on $\overline{\Sigma}$. The weighted quantities $\widetilde{\alpha}, \widetilde{\underline\alpha}$ satisfy the following equations:

\begin{align}\label{T+2B}
\frac{1}{\Omega^2}\nablau\nablav r\widetilde{\alpha}+\frac{1}{M}(4-3\Omega^2)\nablau r\widetilde{\alpha}-\frac{1}{r}(3\Omega^2-5)\widetilde{\alpha}-\slashed{\Delta}r\widetilde{\alpha}=0,
\end{align}

\begin{align}\label{T-2B}
\frac{1}{\Omega^2}\nablau\nablav r\widetilde{\underline\alpha}-\frac{1}{M}(4-3\Omega^2)\nablav r\widetilde{\underline\alpha}-\frac{1}{r}(3\Omega^2-5)\widetilde{\underline\alpha}-\slashed{\Delta}r\widetilde{\underline\alpha}=0.
\end{align}

Equations (<ref>) and (<ref>) do not degenerate near $\mathcal{B}$ and we can make the following well-posedness statement:

Prescribe a pair of smooth symmetric traceless $S^2_{U,V}$ 2-tensor fields $(\widetilde{\upalpha},\widetilde{\upalpha}')$ on $\overline{\Sigma}$. Then there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor field $\Omega^2{\alpha}$ that satisfies (<ref>) on $ D^+(\overline{\Sigma})$ with $V^{-2}\Omega^2\alpha|_{\overline{\Sigma}}=\widetilde{\upalpha}$ and $\slashednabla_{n_{\overline{\Sigma}}}V^{-2}\Omega^2\alpha|_{\overline{\Sigma}}=\widetilde{\upalpha}'$.

Prescribe a pair of smooth symmetric traceless $S^2_{U,V}$ 2-tensor fields $(\widetilde{\underline\upalpha},\widetilde{\underline\upalpha}')$ on $\overline{\Sigma}$. Then there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor field $\Omega^{-2}{\underline\alpha}$ that satisfies (<ref>) on $ D^+(\overline{\Sigma})$ with $V^{2}\Omega^{-2}\underline\alpha|_{\overline{\Sigma}}=\widetilde{\underline\upalpha}$ and $\slashednabla_{n_{\overline{\Sigma}}}V^{2}\Omega^{-2}\underline\alpha|_{\overline{\Sigma}}=\widetilde{\underline\upalpha}'$.

Analogous statements to the above apply to past development from $\overline{\Sigma}$ with $U,\Omega^2$ switching places with $V,\Omega^{-2}$ respectively.

Under the transformation $t\longrightarrow-t$, $u\longrightarrow -v$ and $v\longrightarrow -u$ and thus $\alpha(u,v,\theta^A)\longrightarrow\alpha(-v,-u,\theta^A)=:\invertedalpha(u,v,\theta^A)$ and $\underline\alpha(u,v,\theta^A)\longrightarrow\underline\alpha(-v,-u,\theta^A)=:\underline\invertedalpha(u,v,\theta^A)$.

It is clear $\invertedalpha(u,v,\theta^A)$ satisfies the $-2$ Teukolsky equation, i.e. the equation satisfied by $\underline\alpha$. Similarly,  $\underline\invertedalpha(u,v,\theta^A)$ satisfies the $+2$ Teukolsky equation, i.e. the equation satisfied by $\alpha$. This observation means that the asymptotics of $\alpha$ towards the future are identical to those of $\underline\alpha$ towards the past.  Therefore, determining the asymptotics of both $\underline\alpha$ and $\alpha$ towards the future is enough to determine the asymptotics of either $\alpha$ or $\underline\alpha$ in both the past and future directions. We will use this fact to obtain bijective scattering maps from studying the forward evolution of the fields $\alpha,\underline\alpha$.

§.§ Physical-space Chandrasekhar transformations and the Regge–Wheeler equation

The Regge–Wheeler equation for a symmetric traceless $S^2_{u,v}$ 2-tensor $\Psi$ is given by

\begin{align}\label{RW}
\nablav\nablau\Psi-\Omega^2\slashed{\Delta}\Psi+\frac{\Omega^2}{r^2}(3\Omega^2+1)\Psi=0.
\end{align}

In the Schwarzschild spacetime, the Teukolsky equations lead to the Regge–Wheeler equations in the following sense: define the following hierarchy of fields

r^3Ωψ   :=r^2/Ω^2rΩ^2α,        Ψ      :=r^2/Ω^2r^3Ωψ=(r^2/Ω^2)^2 rΩ^2α,

r^3Ωψ   :=r^2/Ω^2rΩ^2α,        Ψ      :=r^2/Ω^2r^3Ωψ=(r^2/Ω^2)^2 rΩ^2α.

Then we have

Assume $\alpha$ satisfies the $+2$ Teukolsky equation T+2, $\underline\alpha$ satisfies the $-2$ Teukolsky equation T-2. Then $\Psi, \underline\Psi$ of <ref> satisfy the Regge–Wheeler equation RW.

Direct computation. See <cit.> for the details.

In light of the appearance of the Regge–Wheeler equation RW as a consequence of the linearised system , we make the following definitions:

Let $\alin$, $\ablin$ belong to a solution to the system . We define

\begin{align}\label{definition of plin and pblin}
r^3\Omega\plin:=\frac{r^2}{\Omega^2}\nablau r\Omega\alin,\qquad\qquad r^3\Omega\pblin:=\frac{r^2}{\Omega^2}\nablav r\Omega\ablin,
\end{align}

\begin{align}\label{definition of Psilin and Psilinb}
\Psilin:=\frac{r^2}{\Omega^2}\nablau r^3\Omega\plin=\left(\frac{r^2}{\Omega^2}\nablau\right)^2r\Omega\alin,\qquad\qquad\Psilin:=\frac{r^2}{\Omega^2}\nablav r^3\Omega\pblin=\left(\frac{r^2}{\Omega^2}\nablav\right)^2r\Omega\ablin.
\end{align}

Let $\alin, \ablin$ belong to a solution to the linearised Einstein equations .
Define $\Psilin:=\left(\frac{r^2}{\Omega^2}\nablau\right)^2r\Omega^2\alin$, $\Psilinb:=\left(\frac{r^2}{\Omega^2}\nablav\right)^2r\Omega^2\ablin$. Then $\Psilin$, $\Psilinb$ both satisfy the Regge–Wheeler equation RW.

Let $\alin$, $\ablin$, $\Psilin$, $\Psilinb$ be as in <Ref>. Tracking the definitions definition of Psilin and Psilinb through the system , we get

\begin{align}
\Psilin&=2r^2\fancydstar_2\fancydstar_1\left(r^3\rlin,\,-r^3\slin\right)+6M\left(r\Omega\xlin-r\Omega\xblin\right),\label{expression for Psilin}\\
\Psilinb&=2r^2\fancydstar_2\fancydstar_1\left(r^3\rlin,\,r^3\slin\right)+6M\left(r\Omega\xlin-r\Omega\xblin\right).\label{expression for Psilinb}
\end{align}

We now state a standard well-posedness result for (<ref>) for data on ${\Sigma^*_+}$:

For any pair $(\uppsi,\uppsi')$ of smooth symmetric traceless $S^2_r$ 2-tensor fields on ${\Sigma^*_+}$, there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor field $\Psi$ which solves <ref> in $ D^+({\Sigma^*_+})$ such that $\Psi|_{{\Sigma^*_+}}=\uppsi$ and $\slashednabla_{n_{{\Sigma^*_+}}} \Psi|_{{\Sigma^*_+}}=\uppsi'$. The same applies when data are posed on $\Sigma$ or $\overline{\Sigma}$.

In contrast to the Teukolsky equations T+2, T-2, the Regge–Wheeler equation RW does not suffer from additional regularity issues near $\mathcal{B}$, as can be seen by rewriting <ref> in Kruskal coordinates:

\begin{align}\label{RW in Kruskal}
\slashednabla_U\slashednabla_V\Psi-{\slashed{\Delta}}\Psi+\frac{3\Omega^2+1}{r^2}\Psi=0.
\end{align}

Note that the relation between $\Psi$ and $\alpha$ in hier takes the following form in the Kruskal frame:

\begin{align}\label{Kruskal hier+}
\Psi=\left(\frac{r^2}{\Omega^2}\nablau\right)^2r\Omega^2\alpha=\left(2Mr^2\fr(r)\slashednabla_U\right)^2r\tilde{\alpha}.
\end{align}

<Ref> is valid replacing ${\Sigma^*_+}$ with $\overline{\Sigma}$ everywhere.

For backwards scattering we will need the following well-posedness statement:

Let $u_+<\infty, v_+<v_*<\infty$. Let $\widetilde{\Sigma}$ be a spacelike hypersurface connecting $\mathscr{H}^+$ at $v=v_+$ to $\mathscr{I}^+$ at $u=u_+$ and let $\underline{\mathscr{C}}=\underline{\mathscr{C}}_{v_*}\cap J^+(\widetilde{\Sigma})\cap\{t\geq0\}$. Prescribe a pair of smooth symmetric traceless $S^2_{u,v}$ 2-tensor fields:

* $\Psi_{{\mathscr{H}^+}}$ on ${\overline{\mathscr{H}^+}}\cap\{v<v_+\}$ vanishing in a neighborhood of $\widetilde{\Sigma}$,

* $\Psi_{0,in}$ on $\underline{\mathscr{C}}$ vanishing in a neighborhood of $\widetilde{\Sigma}$.

Then there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor $\Psi$ on $D^-\left(\overline{\mathscr{H}^+}\cup\widetilde{\Sigma}\cup\underline{\mathscr{C}}\right)\cap J^+(\overline{\Sigma})$ such that $\Psi|_{\overline{\mathscr{H}^+}}=\Psi_{{\mathscr{H}^+}}$, $\Psi|_{\underline{\mathscr{C}}}=\Psi_{0,in}$ and $\left(\Psi|_{\widetilde{\Sigma}},\slashednabla_{n_{\widetilde{\Sigma}}}\Psi|_{\widetilde{\Sigma}}\right)=(0,0)$.

We will also need

Let $(\uppsi,\uppsi')$ be smooth symmetric traceless $S^2_{u,v}$ 2-tensor fields on ${\Sigma^*_+}$, $\uppsi_{\mathscr{H}^+}$ be a smooth symmetric traceless $S^2_{\infty,v}$ 2-tensor field on $\overline{\mathscr{H}^+}\cap\{t^*_+\leq0\}$. Then there exists a unique smooth symmetric traceless $S^2_{u,v}$ 2-tensor field $\Psi$ on $J^-({\Sigma^*_+})$ such that $\Psi|_{\overline{\mathscr{H}^+}\cap\{t^*_+\leq0\}}=\uppsi_{\mathscr{H}^+}, \left(\Psi|_{{\Sigma^*_+}},\slashednabla_{n_{{\Sigma^*_+}}}\Psi|_{{\Sigma^*_+}}\right)=(\uppsi,\uppsi')$.

Unlike the Teukolsky equations T+2, T-2, the Regge–Wheeler equation RW is invariant under time inversion. If $\Psi(u,v)$ satisfies RW, then $\invertedpsi(u,v):=\Psi(-v,-u)$ also satisfies RW.

§.§.§ Further constraints among $\Psilin$, $\alin$ and $\Psilinb$, $\ablin$

Using that $\alin$ satisfies T+2, $\Psilin$ satisfies RW and $\Psilin$, $\alin$ are related by hier, a computation shows

\begin{align}\label{further constraint --}
\Omega\slashed{\nabla}_4\Psilin=-(3\Omega^2-1)\frac{r^2}{\Omega^2}\Omega\slashed{\nabla}_3r\Omega^2\alin-6Mr\Omega^2\alin+\mathcal{A}_2\frac{r^2}{\Omega^2}\Omega\slashed{\nabla}_3r\Omega^2\alin,
\end{align}

\begin{align}\label{further constraint -}
\frac{\Omega^2}{r^2}\nablav\frac{r^2}{\Omega^2}\nablav\Psilin=\mathcal{A}_2\left(\mathcal{A}_2-2\right)r\Omega^2\alin-6M(\nablau+\nablav)r\Omega^2\alin.
\end{align}

Similar constraints relates $\ablin$, $\Psilinb$:

\begin{align}\label{further constraint ++}
\Omega\slashed{\nabla}_3\Psilinb=-(3\Omega^2-1)\frac{r^2}{\Omega^2}\Omega\slashed{\nabla}_4r\Omega^2\ablin+6Mr\Omega^2\ablin+\mathcal{A}_2\frac{r^2}{\Omega^2}\Omega\slashed{\nabla}_4r\Omega^2\ablin,
\end{align}

\begin{align}\label{further constraint +}
\frac{\Omega^2}{r^2}\nablau\frac{r^2}{\Omega^2}\nablau\Psilinb=\mathcal{A}_2\left(\mathcal{A}_2-2\right)r\Omega^2\ablin+6M(\nablau+\nablav)r\Omega^2\ablin.
\end{align}

§.§.§ Aside: the Teukolsky–Starobinsky identities

Assume $\alin, \ablin$ belong to a solution to the linearised Einstein equations. In addition to satisfying the Teukolsky equations T+2, T-2, the quantities $\alin, \ablin$ satisfy the Teukolsky–Starobinsky identities:

\begin{align}\label{eq:TS-}
\frac{\Omega^2}{r^2}\Omega\slashed{\nabla}_3 \left(\frac{r^2}{\Omega^2}\nablau\right)^3r\Omega^2\overone\alpha=2r^4\slashed{\mathcal{D}}^*_2\slashed{\mathcal{D}}^*_1\overline{\slashed{\mathcal{D}}}_1\slashed{\mathcal{D}}_2 r\Omega^2\overone{\underline\alpha}+6M\left[\Omega\slashed{\nabla}_4+\Omega\slashed{\nabla}_3\right]r\Omega^2\overone{\underline\alpha}.
\end{align}

\begin{align}\label{eq:TS+}
\frac{\Omega^2}{r^2}\Omega\slashed{\nabla}_4 \left(\frac{r^2}{\Omega^2}\nablav\right)^3r\Omega^2\overone{\underline\alpha}=2r^4\slashed{\mathcal{D}}^*_2\slashed{\mathcal{D}}^*_1\overline{\slashed{\mathcal{D}}}_1\slashed{\mathcal{D}}_2 r\Omega^2\overone{\alpha}-6M\left[\Omega\slashed{\nabla}_4+\Omega\slashed{\nabla}_3\right]r\Omega^2\overone{\alpha}.
\end{align}

The equations above were derived from the full system in Section 3.2 of Part I <cit.>. We used the Teukolsky–Starobinsky identities in <cit.> to couple solutions to the Teukolsky equations and construct a combined scattering theory which governs the radiating degrees of freedom of the linearised Einstein equations . The results of this paper will indeed confirm that the radiation energy of the full system can be completely characterised via the system composed of the Teukolsky equations T+2, T-2 and the Teukolsky–Starobinsky identities eq:TS-, eq:TS+.

§ WELL-POSEDNESS OF THE LINEARISED EINSTEIN EQUATIONS IN A DOUBLE NULL GAUGE

In this section we study the well-posedness of the linearised Einstein equations with smooth initial data on ${\Sigma^*_+}$ and $\overline{\Sigma}$. First, we look at the constraint equations on $\Sigma^*_+$ and show how we can prescribe a set of free seed data and solve the constraint equations to obtain a full Cauchy data set. We then prove local existence and uniqueness of solutions to the linearised Einstein system starting from smooth seed data that satisfy the constraint equations.

§.§ The constraint equations

We first list the constraints implied by the system on each of ${\Sigma^*_+}, \overline{\Sigma}$. We then describe a scheme to solve the equations and obtain compactly supported Cauchy data out of compactly supported seed data.

§.§.§ The constraints on ${\Sigma^*_+}$

A solution $\mathfrak{S}$ to the linearised Einstein system satisfies the following equations on the surface ${\Sigma^*_+}$:

The linearised Einstein system induces the following constraints on ${\Sigma^*_+}$:

\begin{align}\label{Spacelike Codazzis at Sigma star}
\begin{split}
\slashednabla_N\divo{\glinh}\,-&\frac{1}{r(2-\Omega^2)}\mathring{\fancyd_2}\mathring{\fancydstar_2}\,\bmlin=\frac{8M}{r^2(2-\Omega^2)}\mathring{\slashed{d}}\left({\Olin}\right)+2N\left[\mathring{\slashed{d}}\left({\Olin}\right)\right]\\&+\frac{1}{\Omega^2(2-\Omega^2)}\partial_u \bmlin-\frac{1}{4\Omega^2}\slashednabla_{\ustar}\frac{r}{\Omega^2}\partial_u\bmlin+\frac{1}{2r}\mathring{\slashednabla}\divo\frac{\bmlin}{2-\Omega^2}+\frac{1}{2}\mathring{\slashednabla}\,\,N\tr\glin,
\end{split}
\end{align}

\begin{align}\label{Spacelike Gauss at Sigma star}
\begin{split}
&-\frac{3}{8r\Omega^2}\partial_{\ustar}\mathring{\slashed{div}}\bmlin+\frac{1}{4r}\mathring{\slashed{div}}\,N\, \bmlin+\frac{\Omega^2+3}{2r(2-\Omega^2)}\divr{\bmlin}+\frac{1}{r^2}\left(\mathring{\slashed{\Delta}}-2\right)\Olin+\frac{\Omega^4-3\Omega^2+4}{4r(2-\Omega^2)}\,N\,\tr\glin\\&+\frac{3\Omega^4-5\Omega^2-4}{4r(2-\Omega^2)}\frac{1}{2\Omega^2}\partial_{\ustar}\tr\glin-\frac{2-\Omega^2}{4\Omega^2}\partial_{\ustar}\left(N-\frac{1}{2\Omega^2}\partial_{\ustar}\right)\tr\glin-\frac{2\Omega^2}{r}\left(N-\frac{1}{2\Omega^2}\partial_{\ustar}\right)\Olin-\Klin=0
\end{split}
\end{align}

\begin{align}\label{Spacelike connection constraint at Sigma star}
\begin{split}
\end{split}
\end{align}

Note that Spacelike Codazzis at Sigma star derives from the Codazzi equations elliptic equation 1, elliptic equation 2 and the equations D3etabar, D4eta. Equation Spacelike Gauss at Sigma star follows from the Gauss equation Gauss and the equation D3TrChi (or equivalently D4TrChiBar), and either one of D4TrChi and  D3TrChibar. The equation Spacelike connection constraint at Sigma star is derived from the equations D4TrChi and D3TrChibar.

We can derive from Spacelike Gauss at Sigma star and Spacelike connection constraint at Sigma star

\begin{align}\label{constraint equation for Olin on Sigma star}
\begin{split}
\frac{32M}{r^2}N(\Olin)+&\frac{8M^2}{r^3(2-\Omega^2)}N\tr\glin=-(2-\Omega^2)\left(\frac{1}{\Omega^2}\partial_{\ustar}\right)^2\tr\glin+\frac{11-5\Omega^2}{r(2-\Omega^2)}\frac{1}{\Omega^2}\partial_{\ustar}\tr\glin+8\Klin\\&-\frac{8}{r}\frac{1}{\Omega^2}\partial_\ustar\,\Olin-\frac{8}{r}\left(\mathring{\slashed{\Delta}}-2\right)\Olin+\frac{2}{r\Omega^2}\divo\partial_\ustar\bmlin+\frac{2\Omega^4+9\Omega^2-26}{r(2-\Omega^2)^2}\divo\bmlin,
\end{split}
\end{align}

§.§.§ The constraints on $\overline{\Sigma}$

We study the constraint equations on $\overline{\Sigma}$ in the Kruskal coordinates and the coordinates $(R,\theta^A)$ induced by it on $\overline{\Sigma}$ ($R:=V-U$ was defined in T R Kruskal coordinates):

The linearised Einstein system induces the following constraints on $\overline{\Sigma}$:

\begin{align}\label{Spacelike Codazzis at Sigma bar}
\begin{split}
\slashednabla_T\divr\glinh-2M\fancyd_2\fancydstar_2\frac{\bmlin}{V}=&-\frac{2}{r}\slashednabla_T{\slashednabla}\Olin+\frac{R}{r}\left(3+\frac{r}{2M}\right)(\partial_T-\partial_R)\frac{\bmlin}{V}-\frac{f}{4M}\slashednabla_R\left(\partial_T-\partial_R\right)\frac{\bmlin}{V}+\frac{1}{2}{\slashednabla}\partial_T\tr\glin\\&+\frac{1}{2}{\slashednabla}\divr\frac{\bmlin}{V},
\end{split}
\end{align}

\begin{align}\label{Spacelike Gauss at Sigma bar}
\begin{split}
\Klin=&\frac{1}{r^2}\left(\mathring{\slashed{\Delta}}-2\right)\Olin+\frac{f}{8M}\divr\left(\partial_T-\partial_R\right)\frac{\bmlin}{V}+\frac{f}{8M^2}\partial_R\left(\partial_T-\partial_R\right)\tr\glin+\frac{R}{8M^2}\left(1+\frac{10M}{r}\right)\partial_R\tr\glin\\&-\frac{R}{8M^2}\left(1+\frac{6M}{r}\right)\partial_T\tr\glin+\frac{R}{Mr}\left(\partial_T-\partial_R\right)\Olin+\frac{1}{r}\divr\bmlin,
\end{split}
\end{align}

\begin{align}\label{Spacelike connection constraint at Sigma bar}
\begin{split}
2f\partial_R\partial_T\tr\glin+\frac{R}{2}\left(1+\frac{6M}{r}\right)\partial_T\tr\glin+{2Mf}(\partial_T+\partial_R) \divr\frac{\bmlin}{V}+MR\left(1+\frac{6M}{r}\right)\divr\frac{\bmlin}{V}=\frac{8M}{r}R\partial_T\Olin.
\end{split}
\end{align}

Subtracting Spacelike connection constraint at Sigma bar from Spacelike Gauss at Sigma bar we get

\begin{align}
\begin{split}
&2U\left(1+\frac{6M}{r}\right)\partial_T \tr\glin+\frac{R}{2}\left(1+\frac{10M}{r}\right)\partial_R \tr\glin-2f\partial_R^2\tr\glin+\frac{4MR}{r}\left(-4\partial_T+\partial_R\right)\Olin\\&-r\Omega^2(2\Omega^2-1)\divr\bmlin-2Mf\divr\,\partial_R\frac{\bmlin}{V}+16M^2\left(\frac{1}{r^2}(\mathring{\slashed{\Delta}}-2)\Olin-\Klin\right)=0
\end{split}
\end{align}

§.§ Existence and uniqueness of solutions to the linearised Einstein equations

Let $\mathcal{S}$ be a spherically symmetric subdomain of ${\Sigma^*_+}$. A smooth initial data set on $\mathcal{S}$ is a tuple ($\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$) where $\glinhs,\,\glinhs'\in\Gamma^{(2)}(\mathcal{S}),\; \bmlins,\,\bmlins'\in\Gamma^{(1)}(\mathcal{S}),$ and $\Olinos,\tr\glins,\,\Olinos',\tr\glins'\in\Gamma^{(0)}(\mathcal{S})$, such that the constraint equations Spacelike Codazzis at Sigma star, Spacelike Gauss at Sigma star, Spacelike connection constraint at Sigma star are satisfied.

We now prove that the linearised Einstein system is well-posed for smooth initial data on either ${\Sigma^*_+}$ or $\overline{\Sigma}$. In the following proposition, we use the well-posedness of the Teukolsky equations to construct $\alin, \ablin$, which are then used to source the remaining components of the solution.

Let $\mathcal{S}$ be a spherically symmetric subdomain in ${\Sigma^*_+}$ and let ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$ ) be an initial data set on $\mathcal{S}$ in the sense of <Ref>. Then there exists a unique solution

\begin{align}
\mathfrak{S}=\left(\;\glinh\;,\;\tr\glin\;,\;\bmlin\;,\;\Olino\;,\;\xlin\;,\;\xblin\;,\;\trx\;,\;\trxbar\;,\;\elin\;,\;\eblin\;,\;\olin\;,\;\olinb\;,\;\alin\;,\;\ablin\;,\;\blin\;,\;\bblin\;,\;\rlin\;,\;\slin\;\right),
\end{align}

to the linearised Einstein equations on $D^+(\mathcal{S})$ in the sense of <Ref>, such that the restriction of $(\,\glinh\;,\;\tr\glin\;,\;\Olin\;,\;\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$), and the restriction of $(\,\slashednabla_{n_{{\Sigma^*_+}}}\glinh\;,\;\slashednabla_{n_{{\Sigma^*_+}}}\tr\glin\;$, $\;\slashednabla_{n_{{\Sigma^*_+}}}\Olin\;,\;\slashednabla_{n_{{\Sigma^*_+}}}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$).

We will assume that the given initial data is supported on $\ell\geq2$ and we postpone the treatment of the $\ell=0,1$ modes until <Ref>.
Define the following quantities on $\mathcal{S}$:

\begin{align}
&\elins:=\slashednabla\Olinos+\frac{1}{4\sqrt{2-\Omega^2}}\bmlins'+\frac{1}{8\Omega^2}\partial_\ustar\bmlins,\qquad\eblins:=2\slashednabla\Olinos-\elins,\label{elin and eblin data on Sigma*}\\
&\xlins:=\frac{\sqrt{2-\Omega^2}}{2}\glinhs'-\frac{2-\Omega^2}{4\Omega^2}\slashednabla_\ustar\glinhs-\fancydstar_2\bmlins,\qquad\qquad\xblins:=\frac{1}{\sqrt{2\Omega^2}}\glinhs'+\frac{1}{2\Omega^2}\slashednabla_\ustar \glinhs,\label{xlin and xblin data on Sigma*}\\
&\alins:=2(2-\Omega^2)\fancydstar_2\elins-\frac{2\Omega^2+1}{r}\xlins+\frac{2-\Omega^2}{\Omega^2}\slashednabla_\ustar \xlins+\frac{\Omega^2(2-\Omega^2)}{r}\xblins,\label{alpha data on Sigma*}\\
&\ablins:=\frac{2}{2-\Omega^2}\fancydstar_2\eblins+\frac{5-2\Omega^2}{r(2-\Omega^2)}\xblins-\frac{1}{\Omega^2}\slashednabla_\ustar\xblins-\frac{1}{r(2-\Omega^2)}\xlins,\label{alphabar data on Sigma^*}\\
&\otxs=\frac{\sqrt{2-\Omega^2}}{2}\tr\glins'-\frac{2-\Omega^2}{4\Omega^2}\partial_\ustar\glins-\divr\bmlins,\qquad\qquad \otxbs=\frac{1}{2\sqrt{2-\Omega^2}}\tr\glins'-\frac{1}{4\Omega^2}\partial_\ustar\tr\glins,\label{otx data on Sigma*}\\
&\blins=-\divr\,\xlins-\frac{\Omega^2}{r}\eblins+\frac{1}{2}\slashednabla\otxs,\qquad\qquad \bblins=\divr\,\xblins-\frac{1}{r}\elins-\frac{1}{2}\slashednabla\otxbs,\label{beta and betabar data on Sigma*}\\
&\slins=\slashed{curl}\,\elins=-\slashed{curl}\,\eblins,\qquad\qquad \rlins=-\Klin[\glins]+\frac{1}{2r}\left(\otxs-\otxbs\right)-\frac{2\Omega^2}{r^2}\Olinos.\label{sigma and rho data on Sigma*}
\end{align}

Moreover, define

\begin{align}
&\alins'=\sqrt{2-\Omega^2}\left[-\frac{1}{2\Omega^2}\slashednabla_\ustar\alins+\frac{1}{r}\alins-2\fancydstar_2\blins+\frac{6M}{r^3}\xlins\right],\label{data for time derivative of alpha on Sigma star}\\ &\ablins'=\frac{\sqrt{2-\Omega^2}}{2\Omega^2}\slashednabla_{\ustar}\ablins-\frac{\sqrt{2-\Omega^2}}{r}\ablins+\frac{2}{\sqrt{2-\Omega^2}}\fancydstar_2\,\bblins+\frac{6M}{r^3\sqrt{2-\Omega^2}}\xblins.\label{data for time derivative of alphabar on Sigma star}
\end{align}

With data $(\alins,\alins')$ for T+2 and $(\ablins,\ablins')$ for T-2,  <Ref> allow us to construct smooth solutions $\alin,\ablin$ to the Teukolsky equations T+2, T-2, respectively, on $D^+(\mathcal{S})$.
We can immediately solve the transport equations D4Chihat, D3Chihatbar for $\xlin, \xblin$ using $\xlins, \xblins$ as data, respectively. Similarly, we can find $\blin_{\ell\geq2}, \bblin_{\ell\geq2}$ using Bianchi+1a, Bianchi-1a with beta and betabar data on Sigma* as data. The $+2$ Teukolsky equation T+2 together with equations in D4Chihat, Bianchi+1a now imply

\begin{align}
\nablav\left(-2\fancydstar_2 \frac{r^4\blin}{\Omega}+6M\frac{r^2\xlin}{\Omega}\right)=\left(\mathcal{A}_2-\frac{6M}{r}\right)r^3\alin=\nablav\left(\frac{r^4}{\Omega^4}\nablau r\Omega^2\alin\right),
\end{align}

so equation Bianchi+2 is satisfied, and a similar argument shows that equation Bianchi-2 is also satisfied.

We now define $\elin,\eblin$ up to $\ell=0,1$ via D3Chihat, D4Chihatbar:

\begin{align}
\begin{split}
\frac{1}{\Omega^2}\nablau r\Omega\xlin&=-2\fancydstar_2 \elin_{\ell\geq2}-\Omega\xblin,\\
\nablav\Omega^{-1}\xblin+\frac{1}{r}\Omega^{-1}\xblin&=-2\fancydstar_2\eblin_{\ell\geq2}+\Omega\xlin.
\end{split}
\end{align}

Define $\glinh$ via metric transport in 3 direction traceless with $\glinhs$ as data, and define $\bmlins$ up to $\ell=0,1$ via metric transport in 4 direction traceless (note that by xlin and xblin data on Sigma*, $\bmlin_{\ell\geq2}|_{\mathcal{S}}=\bmlins_{\ell\geq2}$). Note that

\begin{align}
\begin{split}
\nablau r\nablav \glinh&=-\Omega^2\nablav\glinh+r\nablav\left[2\Omega\xblin\right]=-2\Omega^2\left[\Omega\xlin-\fancydstar_2\bmlin\right]+2\nablav r\Omega\xblin-2\Omega^2\Omega\xblin.\\
\end{split}
\end{align}

On the other hand, using equation metric transport in 4 direction traceless we have

\begin{align}
\nablau r\nablav \glinh=2\nablau r\Omega\xlin+2\mathring{\fancydstar_2}\nablau \bmlin.
\end{align}

Thus D3Chihat implies

\begin{align}
\fancydstar_2\left[\nablau\bmlin_{\ell\geq2}-2\Omega^2(\elin_{\ell\geq2}-\eblin_{\ell\geq2})-\frac{\Omega^2}{r}\bmlin_{\ell\geq2}\right]=0,
\end{align}

which implies that partial_u b holds up to $\ell=0,1$. An identical argument shows that equations D3etabar, D4eta are satisfied up to $\ell=0,1$ modes, using D4Chihat, D3Chihat, and D4Chihatbar. Note that now

\begin{align}
\Omega^2\,\alin&=\frac{2-\Omega^2}{\Omega^2}\nablaubar \Omega\xlin+\frac{(2-\Omega^2)}{r}\Omega\xblin+2(2-\Omega^2)\fancydstar_2\elin_{\ell\geq2}-\frac{2\Omega^2+1}{r}\Omega\xlin,\\
\Omega^{-2}\ablin&=-\frac{1}{\Omega^2}\nablaubar \Omega^{-1}\xblin-\frac{1}{r(2-\Omega^2)}\Omega\xlin+\frac{2}{2-\Omega^2}\fancydstar_2\eblin_{\ell\geq2}+\frac{5-2\Omega^2}{r(2-\Omega^2)}\Omega^{-1}\xblin,
\end{align}

so in particular $\elin_{\ell\geq2}|_{\mathcal{S}}=\elins_{\ell\geq2}$ and $\eblin_{\ell\geq2}|_{\mathcal{S}}=\eblins_{\ell\geq2}$, and thus we have $\frac{1}{\Omega^2}\nablau\bmlin_{\ell\geq2}^A\Big|_{\mathcal{S}}=\bmlins'+\frac{1}{2\Omega^2}\nablaubar \bmlins$.

With the quantities $\alin, \ablin, \Psilin, \Psilinb, \xlin, \xblin, \blin, \bblin, \elin, \eblin$ at hand, define

\begin{align}
\end{align}

A computation using equations D4Chihat, D3Chihatbar, D3Chihat, D4Chihatbar, D3etabar, D4eta shows that the above quantities satisfy

\begin{align}\label{equation of C}
\begin{split}
\left(\nablau\nablav-\frac{\Omega^2}{r^2}\mathring{\slashed{\Delta}}\right) r^2\text{C}=\frac{1}{r^2}\left[\nablau (r^4\Omega \,\text{Codazzi}^+)+\nablav(r^4\Omega\,\text{Codazzi}^-)\right],
\end{split}
\end{align}

\begin{align}
\refstepcounter{equation}\latexlabel{equation of Codazzi+}
\refstepcounter{equation}\latexlabel{equation of Codazzi-}
\nablav\left[\frac{r^4}{\Omega}\text{Codazzi}^+\right]=-r\nablav(r^2\text{C}),\qquad\qquad \nablau\left[\frac{r^4}{\Omega}\text{Codazzi}^-\right]=r\nablau(r^2\text{C}).\tag{\ref{equation of Codazzi+},\;\ref{equation of Codazzi-}}
\end{align}

A standard local well-posedness argument using energy estimates and Grönwall's inequality shows that the system equation of C, equation of Codazzi+, equation of Codazzi- admits only the trivial solution for trivial data on $\mathcal{S}$.

We may then define $\Olino$ up to $\ell=0,1$ by $2\slashednabla_A\Olin_{\ell\geq2}=\elin_{\ell\geq2}+\eblin_{\ell\geq2}$ and we immediately have that $\Olin_{\ell\geq2}|_{\mathcal{S}}=\Olinos_{\ell\geq2}$ by elin and eblin data on Sigma*.
We now define $\otx_{\ell\geq2}, \otxb_{\ell\geq2}$ via D4TrChi using $\otxs, \otxbs$ as data. We can now establish that the Codazzi equations are satisfied: we have

\begin{align}
\divo \nablau \frac{r^2\xblin}{\Omega}=\divo(-r^2\ablin),
\end{align}

\begin{align}
\begin{split}
\nablau\frac{r^3}{\Omega}\left(\frac{\Omega}{r}\elin_{\ell\geq2}+\bblin_{\ell\geq2}+\frac{1}{2\Omega}\slashednabla\otxb_{\ell\geq2}\right)&=2r\mathring{\slashednabla}\olinb_{\ell\geq2}-r^2\Omega\bblin_{\ell\geq2}+\frac{\Omega^2}{r^2}\frac{r^4\bblin_{\ell\geq2}}{\Omega}-\frac{1}{r}\divo r^3\ablin-2\mathring{\slashednabla}r\olinb_{\ell\geq2}\\&=-\divo r^2\ablin,
\end{split}
\end{align}

and since $\bblins$ was chosen consistently with the Codazzi equation elliptic equation 2 via beta and betabar data on Sigma*, we see that elliptic equation 2 is propagated. A similar argument shows that elliptic equation 1 is also propagated.

We now define $\rlin_{\ell\geq2}, \slin_{\ell\geq2}$ by solving Bianchi-0, Bianchi-0*:

\begin{align}
\nablau r^3\rlin_{\ell\geq2}=-\divo r^2\Omega\bblin_{\ell\geq2}+3M\otxb_{\ell\geq2},\qquad\nablau r^3\slin_{\ell\geq2}=-\curlo\, r^2\Omega\bblin_{\ell\geq2},
\end{align}

with data $\rlins, \slins$. This defines $\rlin$ and $\slin$ up to $\ell=0,1$. Applying $\nablau \frac{r^2}{\Omega^2}$ to both sides Bianchi+1b and using Bianchi-2, Bianchi-1a, Bianchi-0, Bianchi-0*, D3etabar and the Codazzi equation elliptic equation 1, we get

\begin{align}
\nablau \frac{r^2}{\Omega^2}\nablav r^2\Omega\bblin_{\ell\geq2}=\nablau\left[\fancydstarring_1(r^3\rlin_{\ell\geq2},r^3\slin_{\ell\geq2})+6Mr\eblin_{\ell\geq2}\right].
\end{align}

We may check using the definitions of $\ablins, \bblins, \rlins, \slins$ and $\eblins$ on $\mathcal{S}$ and the constraint equations Spacelike Codazzis at Sigma star–Spacelike connection constraint at Sigma star that equation Bianchi+1b is satisfied by the initial data on $\mathcal{S}$, thus Bianchi+1b propagates on $D^+(\mathcal{S})$. In turn, applying $\nablav$ to elliptic equation 1 and using Bianchi+1b, D4eta, D4Chihatbar, and elliptic equation 2, we get that D4TrChiBar holds.

Note that using Bianchi+2, Bianchi+1a and D4Chihat, we have

\begin{align}\label{d4 d3 beta}
&\nablav\frac{r^2}{\Omega^2}\nablau  r^2\Omega\blin_{\ell\geq2}=-2\mathring{\fancyd_2}\mathring{\fancydstar_2}r^2\Omega\blin-2(3\Omega^2-2)r^2\Omega\blin_{\ell\geq2}+6M\divo \Omega\xlin,
%&\left(\nablav\frac{r^2}{\Omega^2}\right)^2\nablau r^2\Omega\blin=\left(-2\mathring{\fancyd_2}\mathring{\fancydstar_2}-3\Omega^2+1\right)\mathring{\fancyd_2}r^3\alin-12M r^2\Omega\blin.
\end{align}

We can also deduce from D3etabar and Bianchi+0* that

\begin{align}
\partial_u \left(\curlo\, r^2\eblin_{\ell\geq2}\right)=\curlo\, r^2\Omega\bblin_{\ell\geq2}
\end{align}

which when combined with the definition of $\slins$ in sigma and rho data on Sigma* implies elliptic equation 3 and 4, which then implies Bianchi+0*. Since Bianchi-2, Bianchi+1b and D4Chihatbar are satisfied, we have that the expression expression for Psilinb for $\Psilinb$ applies:

\begin{align}
\Psilinb=2r^2\fancydstar_2\fancydstar_1\left(r^3\rlin_{\ell\geq2},\,r^3\slin_{\ell\geq2}\right)+6M\left(r\Omega\xlin-r\Omega\xblin\right).
\end{align}

Given that $\text{A}=\text{C}=0$, this immediately implies $\curlr\elin_{\ell\geq2}=-\curlr\eblin_{\ell\geq2}=\slin_{\ell\geq2}$. Therefore, the vanishing of $\underline{\text{A}}$ implies

\begin{align}\label{Expression for Psi WP}
\Psilin=2r^2\fancydstar_2\fancydstar_1\left(r^3\rlin_{\ell\geq2},-r^3\slin_{\ell\geq2}\right)+6M\left(r\Omega\xlin-r\Omega\xblin\right).
\end{align}

A computation shows that $\left\{\nablau r^2\Omega\blin-\mathring{\fancydstar_1}(-r\Omega^2\rlin,r\Omega^2\slin)-\frac{6M}{r}\Omega^2\elin\right\}\Big|_{{\Sigma^*_+}}=0$, i.e. using the first equation in Bianchi +1a and the constraint equations Spacelike Codazzis at Sigma bar, Spacelike Gauss at Sigma bar,

\begin{align}\label{verify Psi one}
\nablaubar\, r^2\blins+\frac{\Omega^2}{2-\Omega^2}\left\{\divo r\alins-\frac{3\Omega^2-1}{r}r^2\blins\right\}=\mathring{\fancydstar_1}(-r\Omega^2\rlins,r\Omega^2\slins)-\frac{6M}{r}\Omega^2\elins,
\end{align}

and moreover, using the second equation in Bianchi 0 and the full constraint equations Spacelike Codazzis at Sigma bar, Spacelike Gauss at Sigma bar, Spacelike connection constraint at Sigma bar we get $\left\{\nablav r^3\elin-\divo r^2\Omega\blin-3M \otx\right\}\Big|_{{\Sigma^*_+}}=0$, i.e.

\begin{align}\label{verify Psi two}
-\divo r^2\bblins+3M\otxbs-\frac{1}{\Omega^2}\nablaubar r^3\rlins=\frac{1}{2-\Omega^2}\left[\divo r^2\blins+3M \otxs\right]
\end{align}

<Ref> imply that

\begin{align}
&\left(\Psilin-\Psilinb\right)\Big|_{{\Sigma^*_+}}+4\mathring{\fancydstar_2}\mathring{\fancydstar_1}(0,r^3\slins)=0,\label{slin in terms of Psilin Psilinb}\\
\end{align}

Given that $\Psilin$, $\Psilinb$, $2\mathring{\fancydstar_2}\mathring{\fancydstar_1}(0,\slin)$ all satisfy the Regge–Wheeler equation RW, we finally get Expression for Psi WP.

The above implies Bianchi-1b is satisfied. Using d4 d3 beta, we then get Bianchi+0. Equation transport equation of little omega follows from applying $\nablau$ to D4etabar and following through using Bianchi+1b, Bianchi-1b. Equation D3TrChi follows by applying $\nablau$ to D4TrChi and using D4TrChiBar and transport equation of little omega. Now  equations metric transport in 4 direction trace, metric transport in 3 direction trace and the Gauss equation Gauss follow immediately.

A similar proof goes through to show the well-posedness of the system with data on $\overline{\Sigma}$ defined as follows:

Let $\mathcal{S}$ be an spherically symmetric subdomain of $\overline{\Sigma}$. A smooth initial data set on $\mathcal{S}$ is a tuple ($\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$) where $\glinhs,\,\glinhs'\in\Gamma^{(2)}(\mathcal{S}),\; \bmlins,\,\bmlins'\in\Gamma^{(1)}(\mathcal{S}),$ and $\Olinos,\tr\glins,\,\Olinos',\tr\glins'\in\Gamma^{(0)}(\mathcal{S})$, such that the constraint equations Spacelike Codazzis at Sigma bar, Spacelike Gauss at Sigma bar, Spacelike connection constraint at Sigma bar are satisfied.

In the following, $n^+_{\overline{\Sigma}}$ is the future directed unit normal to $\overline{\Sigma}$:

Let $\mathcal{S}$ be an spherically symmetric subdomain of $\overline{\Sigma}$ and let ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$ ) be an initial data set on $\mathcal{S}$ in the sense of <Ref>. Then there exists a unique solution $\mathfrak{S}$

\begin{align}
\mathfrak{S}=\left(\;\glinhK\;,\;\tr\glinK\;,\;\bmlinK\;,\;\OlinoK\;,\;\xlinK\;,\;\xblinK\;,\;\otxK\;,\;\otxbK\;,\;\elinK\;,\;\eblinK\;,\;\olinK\;,\;\olinbK\;,\;\alinK\;,\;\ablinK\;,\;\blinK\;,\;\bblinK\;,\;\rlinK\;,\;\slinK\;\right),
\end{align}

\begin{align}
\mathfrak{S}=\left(\;\glinh\;,\;\tr\glin\;,\;\bmlin\;,\;\Olino\;,\;\xlin\;,\;\xblin\;,\;\otx\;,\;\otxb\;,\;\elin\;,\;\eblin\;,\;\olin\;,\;\olinb\;,\;\alin\;,\;\ablinK\;,\;\blin\;,\;\bblin\;,\;\rlin\;,\;\slin\;\right),
\end{align}

to the linearised Einstein equations on $D^+(\mathcal{S})$ in the sense of <Ref>, such that the restriction of $(\,\glinh\;,\;\tr\glin\;,\;\Olin\;,\;V^{-1}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$), and the restriction of $(\,\slashednabla_{n^+_{\overline{\Sigma}}}\glinh\;$, $\;\slashednabla_{n^+_{\overline{\Sigma}}}\tr\glin\;,\;\slashednabla_{n^+_{\overline{\Sigma}}}\Olin\;,\;\slashednabla_{n^+_{\overline{\Sigma}}}V^{-1}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$).

An identical argument goes for the initial value problem with solutions to evolved from data on $\Sigma^*_-$ to $J^-(\Sigma^*_-)$, or solutions to evolved on $J^-(\overline{\Sigma})$ from initial data $\overline{\Sigma}$. In the following, $n^-_{\overline{\Sigma}}=-n_{\overline{\Sigma}}$, the past directed unit normal to $\overline{\Sigma}$.

Let $\mathcal{S}$ be a spherically symmetric subdomain of ${\Sigma^*_-}$. A smooth initial data set on $\mathcal{S}$ is a tuple ($\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$) where $\glinhs,\,\glinhs'\in\Gamma^{(2)}(\mathcal{S}),\; \bmlins,\,\bmlins'\in\Gamma^{(1)}(\mathcal{S}),$ and $\Olinos,\tr\glins,\,\Olinos',\tr\glins'\in\Gamma^{(0)}(\mathcal{S})$, such that the constraint equations induced by the system on $\Sigma^*_-$ are satisfied.

Let $\mathcal{S}$ be a spherically symmetric subdomain in ${\Sigma^*_-}$ and let ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$ ) be an initial data set on $\mathcal{S}$ in the sense of <Ref>. Then there exists a unique solution $\mathfrak{S}$

\begin{align}
\mathfrak{S}=\left(\;\glinh\;,\;\tr\glin\;,\;\bmlin\;,\;\Olino\;,\;\xlin\;,\;\xblin\;,\;\trx\;,\;\trxbar\;,\;\elin\;,\;\eblin\;,\;\olin\;,\;\olinb\;,\;\alin\;,\;\ablin\;,\;\blin\;,\;\bblin\;,\;\rlin\;,\;\slin\;\right),
\end{align}

to the linearised Einstein equations on $J^-(\mathcal{S})$ in the sense of <Ref>, such that the restriction of $(\,\glinh\;,\;\tr\glin\;,\;\Olin\;,\;\Omega^{-2}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$), and the restriction of $(\,\slashednabla_{n_{{\Sigma^*_-}}}\glinh\;,\;\slashednabla_{n_{{\Sigma^*_-}}}\tr\glin\;$, $\;\slashednabla_{n_{{\Sigma^*_-}}}\Olin\;,\;\slashednabla_{n_{{\Sigma^*_-}}}\Omega^{-2}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$).

Let $\mathcal{S}$ be a spherically symmetric subdomain of $\overline{\Sigma}$ and let ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$ ) be an initial data set on $\mathcal{S}$ in the sense of <Ref>. Then there exists a unique solution $\mathfrak{S}$
to the linearised Einstein equations on $J^-(\mathcal{S})$ in the sense of <Ref>, such that the restriction of $(\,\glinh\;,\;\tr\glin\;,\;\Olin\;,\;V^{-1}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$), and the restriction of $(\,\slashednabla_{n^-_{\overline{\Sigma}}}\glinh\;,\;\slashednabla_{n^-_{\overline{\Sigma}}}\tr\glin\;,\;\slashednabla_{n^-_{\overline{\Sigma}}}\Olin\;,\;\slashednabla_{n^-_{\overline{\Sigma}}}V^{-1}\bmlin\,)$ to $\mathcal{S}$ coincides with ( $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$).

For convenience, we state and prove a well-posedness theorem for the mixed Cauchy-characteristic value problem with data on $\Sigma^*_+$ and $\mathscr{H}^+\cap\{v\leq0\}$.

Let ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$ ) be an initial data set on $\Sigma^*_+$ in the sense of <Ref> and let $\Olinos_{\mathscr{H}^+}$ be a smooth scalar field on $\overline{\mathscr{H}^+}\cap\{v\leq 0\}$, $\glinhs_{\mathscr{H}^+}$ be a smooth symmetric traceless $S^2$ 2-tensor field on $\overline{\mathscr{H}^+}\cap\{v\leq 0\}$. Then there exists a unique solution

\begin{align}
\mathfrak{S}=\left(\;\glinh\;,\;\tr\glin\;,\;\bmlin\;,\;\Olino\;,\;\xlin\;,\;\xblin\;,\;\trx\;,\;\trxbar\;,\;\elin\;,\;\eblin\;,\;\olin\;,\;\olinb\;,\;\alin\;,\;\ablin\;,\;\blin\;,\;\bblin\;,\;\rlin\;,\;\slin\;\right),
\end{align}

to the linearised Einstein equations on $D^+(\overline{\Sigma})\cap J^-(\Sigma^*_+)$ in the sense of <Ref>, such that the restriction of $(\,\glinh\;,\;\tr\glin\;,\;\Olin\;,\;\bmlin\,)$ to $\Sigma^*_+$ coincides with ( $\glinhs, \tr\glins$, $\Olinos$, $\bmlins$), the restriction of $(\,\slashednabla_{n_{{\Sigma^*_+}}}\glinh\;,\;\slashednabla_{n_{{\Sigma^*_+}}}\tr\glin\;$, $\;\slashednabla_{n_{{\Sigma^*_+}}}\Olin\;,\;\slashednabla_{n_{{\Sigma^*_+}}}\bmlin\,)$ to $\Sigma^*_+$ coincides with ( $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'$), and the restriction of $\Olino$, $\glinh$ to $\overline{\mathscr{H}^+}\cap\{v\leq 0\}$ coincides with $\Olinos_{\mathscr{H}^+}$ and $\glinhs_{\mathscr{H}^+}$ respectively.

With the given data on $\Sigma^*_+$, we may use an identical argument to that of the proof of <Ref> to construct $\mathfrak{S}$ on $D^-(\Sigma^*_+)$ restricting to the given Cauchy data on $\Sigma^*_+$. Combining this solution with the given data on $\overline{\mathscr{H}^+}\cap\{v\leq0\}$, we may adapt Theorem 8.1 of <cit.> to construct the solution $\mathfrak{S}$ in the region $J^+(\overline{\Sigma})\cap\{V\leq 1, U\leq0\}$.

§.§ Pure gauge solutions

In <cit.>, it was shown that linearising infinitesimal diffeomorphisms which preserve the double null gauge generates solutions to the linearised Einstein equations. These solutions represent the residual gauge ambiguity in choosing a double null gauge. As shown in <cit.>, these solutions can be classified into three families, which we present in this section and refer the reader to Section 5 of <cit.> for the details of their derivation.

§.§.§ Family $\mathfrak{G}_{out}$: Pure gauge solutions generated by variations along the outgoing null direction

Let $f$ be a smooth function of $v,\theta^A$. The following is a solution of the system :

\begin{align}
2\Olin &= \frac{1}{\Omega^2} \partial_v \left(f \Omega^2\right)  , & \glinh&=  \frac{4}{r} r^2 \slashed{\mathcal{D}}_2^\star\fancydstar_1(f,0)  \ , &\tr\glin &= \frac{4}{r}\left(\mathring{\slashed{\Delta}}+\Omega^2\right)f  , \nonumber \\
\bmlin &= 2r\fancydstar_1 \left[ r\partial_v \left(\frac{f}{r}\right),0\right] , & \elin &=- \frac{\Omega^2}{r^2} r \fancydstar_1(f,0)  , & \eblin &= -r\fancydstar_1\left[\frac{1}{\Omega^2}\partial_v \left(\frac{\Omega^2}{r}f\right),0 \right]   \nonumber \, ,
\nonumber \\
\Omega\xblin &= 2\frac{\Omega^2}{r^2} r^2 \slashed{\mathcal{D}}_2^\star \fancydstar_1 (f,0)  , & \otx &= 2 \partial_v \left(\frac{f \Omega^2}{r}\right)  , & \otxb &=  2\frac{\Omega^2}{r^2} \left[\mathring{\slashed{\Delta}} + 2\Omega^2-1 \right]f  , \nonumber \\
\rlin &= \frac{6M \Omega^2}{r^4} f  , & \bblin&= -\frac{6M\Omega}{r^4}  r \fancydstar_1 (f,0) , & \Klin &= -\frac{\Omega^2}{r^3}\left(\mathring{\slashed{\Delta}} + 2\right)f \nonumber
\end{align}

\[
\xlin = \alin = \ablin = 0 \ \ \ , \ \ \ \blin = 0 \ \ \ , \ \ \ \slin = 0 \nonumber \, .
\]

Given $f=f(v,\theta^A)$, define $\mathfrak{G}_{out}[f]$ to be the pure gauge solution generated by $f$ in the sense of <Ref>.

§.§.§ Family $\mathfrak{G}_{in}$ : Pure gauge solutions generated by variations in the outgoing null direction

Let $\fbar$ be a function of $(u,\theta^A)$ such that $\Omega^2\fbar$ is smooth on $J^+({\Sigma^*_+})$ including $\mathscr{H}^+_{\geq0}$. The following is a solution of the system :

\begin{align}
2\Olin &= \frac{1}{\Omega^2} \partial_u \left(\fbar \Omega^2\right), \, & \tr\glin &= -\frac{4\Omega^2 }{r}\fbar , & \bmlin &= -2\frac{\Omega^2}{r} r\fancydstar_1(\fbar,0), \nonumber \\
\Omega\xlin &= 2\frac{\Omega^2}{r^2} r^2 \slashed{\mathcal{D}}_2^\star \fancydstar_1 (\fbar,0), & \elin &= -\frac{1}{\Omega^2} r\fancydstar_1  \left[\partial_u \left(\frac{\Omega^2}{r}\fbar\right),0 \right] , & \eblin &= \frac{\Omega^2}{r^2} r \fancydstar_1( \fbar ,0), \nonumber \, \\
\otxb &= -2 \partial_u \left(\frac{ \Omega^2}{r}\fbar\right) , \ & \otx &= 2\frac{\Omega^2}{r^2} \left[\mathring{\slashed{\Delta}}+ 2\Omega^2-1\right]\fbar ,
\nonumber  \\
\blin &= \frac{6M\Omega}{r^4}  r \fancydstar_1( \fbar,0) , &  \rlin &= -\frac{6M \Omega^2}{r^4} \fbar  \, , & \Klin &= +\frac{\Omega^2}{r^3} \left(\mathring{\slashed{\Delta}} + 2 \right)\fbar \nonumber
\end{align}

\[
0 = \glinh= \xblin = \alin = \ablin = 0 \ \ \ , \ \ \ \bblin = 0 \ \ \ , \ \ \ \slin = 0 \nonumber \, .
\]

Given $\underline{f}=\underline{f}(v,\theta^A)$, define $\mathfrak{G}_{in}[\underline{f}]$ to be the pure gauge solution generated by $\underline{f}$ in the sense of <Ref>.

§.§.§ Family $\mathfrak{G}_{res}$: Residual pure gauge solutions

For any smooth functions $q_1\left(v,\theta,\phi\right)$ and $q_2\left(v,\theta,\phi\right)$ the following is a pure gauge solution of the system of gravitational perturbations:

\begin{align}
\glinh  &=2r^2 \slashed{\mathcal{D}}_2^\star \slashed{\mathcal{D}}_1^\star \left(q_1,q_2\right) , &\tr\glin &= 2r^2 \slashed{\Delta} q_1, & \bmlin &= r^2 \slashed{\mathcal{D}}_1^\star \left(\partial_v q_1, \partial_v q_2\right) \, , \nonumber
\end{align}

while the linearised
metric coefficient $\Olino$ as well as all linearised connection coefficients and  curvature components vanish.

Given $q_1=q_1(v,\theta^A), q_2=q_2(v,\theta^A)$, define $\mathfrak{G}_{res}[q_1,q_2]$ to be the pure gauge solution generated by $q_1,q_2$ in the sense of <Ref>.

A pure gauge solution is a solution to the linearised equations that can be expressed as a linear combination of the pure gauge solutions given in <Ref>

Note that if $\fbar$ of <Ref> is such that $\Omega^2\fbar$ is smooth on $J^+({\Sigma^*_+})$ including $\mathscr{H}^+_{\geq0}$, then $U\fbar$ is smooth on the whole of $\mathscr{M}$, thus by regarding $U\fbar$ instead of $\fbar$ as the generator of the family of solutions to defined in <Ref>, we can extend this family to $\mathscr{M}$ as solutions to . A similar remark goes for <Ref> by regarding $Vf$ instead of $f$ as the generator of outgoing gauge solutions.

§.§ Constructing initial data on ${\Sigma^*_\pm}, \overline{\Sigma}$

We now show how to construct an initial data set on ${\Sigma^*_+}$ which satisfies the constraints Spacelike Codazzis at Sigma star–Spacelike Gauss at Sigma bar. The $\ell=0,1$ modes of the solution to the system are determined by the linearised Kerr parameters, so we confine our attention here to the $\ell\geq2$ part and treat the $\ell=0,1$ modes in the next section.

In the following we see that having seed data for certain components of the initial data for the metric and connection components on either $\Sigma^*_+$ or $\overline{\Sigma}$  can be extended to a full initial data set by solving 2-dimensional elliptic equations on each of the spheres foliating $\Sigma^*_+$ or $\overline{\Sigma}$:

Given smooth data on ${\Sigma^*_+}$ for

\begin{align}\label{12 10 2021}
\glinh,\; \bmlin_{\ell\geq2},\;  \curlr \slashednabla_N\bmlin_{\ell\geq2}, \;   \tr\glin_{\ell\geq2},\;   \slashednabla_N \tr\glin_{\ell\geq2},\;   \Olin_{\ell\geq2},
\end{align}

there exists a unique set of smooth, compactly supported data for

\begin{align}
\slashednabla_N \glinh,\;   \slashednabla_N \Olin_{\ell\geq2},\;   \slashednabla_N \bmlin_{\ell\geq2},
\end{align}

which complete 12 10 2021 into an initial data set for the system which is supported on $\ell\geq2$ spherical harmonic modes.

With the given data, $\slashednabla_N \Olin_{\ell\geq2}$ is given by constraint equation for Olin on Sigma star. We can then solve for $\divr\slashednabla_N \bmlin$ using Spacelike connection constraint at Sigma star. Assuming $f$, $g$ are scalar functions with vanishing $\ell=0,1$ spherical harmonic modes such that $\slashednabla_N\bmlin_{\ell\geq2}=\fancydstar_1(f,g)$ (see <Ref>), we may find $f$, $g$ via

\begin{align}
{\slashed{\Delta}}f=\divr\slashednabla_N\bmlin_{\ell\geq2},\qquad  {\slashed{\Delta}}g=\curlr\slashednabla_N\bmlin_{\ell\geq2}.
\end{align}

We can finally solve for $\fancyd_2\slashednabla_N \glinh$ using Spacelike Codazzis at Sigma star. Assuming $\slashednabla_N\glinh=2\fancydstar_2\fancydstar_1(f',g')$, for $f'$, $g'$ scalar functions with vanishing $\ell=0,1$ spherical harmonic modes, we can find $f'$, $g'$ via

\begin{align}
\fancydring_1\fancydring_2\slashednabla_N\glinh=\mathring{\slashed{\Delta}}(\mathring{\slashed{\Delta}}+2)(f',g').
\end{align}

An identical argument applied to data on $\overline{\Sigma}$ gives

Given smooth, compactly supported data on $\overline{\Sigma}$ for

\begin{align}\label{12 10 2021 2}
\glinh,\; V^{-1}\bmlin_{\ell\geq2},\;  \curlr \slashednabla_TV^{-1}\bmlin_{\ell\geq2}, \;   \tr\glin_{\ell\geq2},\;   \slashednabla_T \tr\glin_{\ell\geq2},\;   \Olin_{\ell\geq2},
\end{align}

there exists a unique set of smooth, compactly supported data for

\begin{align}
\slashednabla_T \glinh,\;   \slashednabla_T \Olin_{\ell\geq2},\;  \divr \slashednabla_T V^{-1}\bmlin_{\ell\geq2},
\end{align}

which complete 12 10 2021 2 into an initial data set for the system which is supported on $\ell\geq2$ spherical harmonic modes.

In order to use the estimates of <cit.>, it is necessary to work in a gauge where
the initial data satisfy the following on ${\Sigma^*_+}\cap\mathscr{H}^+$:

\begin{align}\label{initial horizon gauge condition}
\otx|_{{\Sigma^*_+}\cap\mathscr{H}^+}=0,\qquad\qquad\left[\rlin-\rlin_{\ell=0}+\divr\elin\right]_{{\Sigma^*_+}\cap\mathscr{H}^+}=0.
\end{align}

Similarly we will need the following conditions on $\Sigma^*_-\cap\mathscr{H}^-$ to estimate solutions evolved from data on $\Sigma^*_-$ towards $\mathscr{I}^-$, $\mathscr{H}^-_{\leq0}$:

\begin{align}\label{initial horizon gauge condition past}
\otxb|_{{\Sigma^*_-}\cap\mathscr{H}^-}=0,\qquad\qquad\left[\rlin-\rlin_{\ell=0}+\divr\eblin\right]_{{\Sigma^*_-}\cap\mathscr{H}^-}=0.
\end{align}

We will also study solutions from data on $\overline{\Sigma}$ on  $J^+(\overline{\Sigma})$ as well as $J^-(\overline{\Sigma})$. In order for data on $\overline{\Sigma}$ to allow for  initial horizon gauge condition to be satisfied on $\mathscr{H}^+$ and initial horizon gauge condition past to be satisfied on $\mathscr{H}^-$ we will need the following gauge conditions on $\mathcal{B}$:

\begin{align}\label{bifurcation gauge conditions}
\divo(\elin-\eblin)\Big|_{\mathcal{B}}=0,\qquad\qquad \left[\rlin-\rlin_{\ell=0}+{\slashed{\Delta}}\Olin\right]\Big|_{\mathcal{B}}=0.
\end{align}

We can always transform an initial data set constructed via <Ref> to a gauge where initial horizon gauge condition is satisfied by the $\ell\geq2$ component:

Given an initial data set $\mathfrak{D}$ constructed in the sense of <Ref>, there exists a compactly supported function $\fbar(u,\theta^A)$ generating a pure gauge solution according to <Ref> which, when subtracted from the solution generated by $\mathfrak{D}$, gives a solution whose initial data set $\mathfrak{D}'$ is smooth, compactly supported and satisfies the conditions initial horizon gauge condition.

Let $\xi$ be a smooth cutoff function on the interval $[0,1]$ with $\xi(0)=1$, $\xi(1)=0$ and $\xi'|_{[0,\frac{1}{4}]}=\xi'|_{[\frac{3}{4},1]}=0$. Define $\fbar_1(\theta^A)$, $\fbar_2(\theta^A)$ to be the unique pair satisfying

\begin{align}
\end{align}

Define $\fbar$ by $U\fbar=\xi(U)\times\left(\fbar_1+U\fbar_2\right)$.

§.§.§ The $\Sigma^*_\pm$, $\overline{\Sigma}^{\pm}
$ gauges

We now prescribe a scheme to fix the gauge freedom on the $\ell\geq2$ component of initial data on ${\Sigma^*_+}$ which is compatible with the conditions initial horizon gauge condition on $\mathscr{H}^+_{\geq0}\cap{\Sigma^*_+}$.

Let $\mathfrak{D}$ be an initial data set on ${\Sigma^*_+}$ for the system . We say that $\mathfrak{D}$ is in the “${\Sigma^*_+}$-gauge" if, using the coordinate system $(r,\theta^A)$ on $\Sigma^*_+$, $\mathfrak{D}$ on ${\Sigma^*_+}$ satisfies

\begin{align}\label{Sigma*+ gauge conditions}
\left[(2M\partial_r)^2+3(2M\partial_{r})+2\right]\divo^2\Omega\xlin|_{\Sigma^*_+}=0,\qquad \divo^2\Omega^{-1}\xblin|_{\Sigma^*_+}=0,\qquad \glinh|_{\Sigma^*_+}=0.
\end{align}

in addition to the horizon gauge conditions initial  horizon gauge condition.

Let $\mathfrak{D}$ be a smooth initial data set on ${\Sigma^*_+}$ for the system whose components are supported on the spherical harmonics with $\ell\geq2$. There exists a unique initial data set $\mathfrak{D}'$, related to $\mathfrak{D}$ by a gauge transformation, such that $\mathfrak{D}'$ is in the ${\Sigma^*_+}$-gauge of <Ref>.

We find the unique $\fbar(u,\theta^A)$ satisfying

\begin{align}\label{instinctive dismissal}
\left[(2M\partial_r)^2+6M\partial_{r}+2\right]\mathring{\slashed{\Delta}}(\mathring{\slashed{\Delta}}+2)\Omega^2\fbar=\left[(2M\partial_r)^2+3(2M\partial_{r})+2\right]\divo^2\Omega\xlin|_{\Sigma^*_+},
\end{align}

with initial conditions $\Omega^2\fbar|_{\mathscr{H}^+}=\fbar_1$, $\partial_r\Omega^2\fbar|_{\mathscr{H}^+}=\fbar_2$ where

\begin{align}
\end{align}

We take $f(v,\theta^A)$ such that

\begin{align}
\Omega^{-1}\xblin|_{\Sigma^*_+}=2\fancydstar_2\fancydstar_1(f,0).
\end{align}

We now take $q_1, q_2$ such that

\begin{align}
\glinh=2\fancydstarring_2\fancydstarring_1\left(\frac{2f}{r}+q_1,q_2\right).
\end{align}

We modify $\mathfrak{D}$ by the gauge transformation generated by $\fbar$ according to <Ref>, $(q_1, q_2)$ according to
<Ref> and $f$ according to <Ref> to arrive at $\mathfrak{D}'$

Let $\mathfrak{D}$ be an initial data set on ${\Sigma^*_-}$ for the system . We say that $\mathfrak{D}$ is in the “${\Sigma^*_-}$-gauge" if, using the coordinate system $(r,\theta^A)$ on $\Sigma^*_-$, $\mathfrak{D}$ on ${\Sigma^*_-}$ satisfies

\begin{align}
\left[(2M\partial_r)^2+6M\partial_{r}+2\right]\divo^2\Omega\xblin|_{\Sigma^*_-}=0,\qquad \divo^2\Omega^{-1}\xlin|_{\Sigma^*_-}=0,\qquad \glinh|_{\Sigma^*_-}=0.
\end{align}

in addition to the horizon gauge conditions initial horizon gauge condition past.

A similar scheme may be devised to fix the gauge freedom in data on $\overline{\Sigma}$. Here we replace the horizon gauge conditions initial horizon gauge condition by bifurcation gauge conditions

Let $\mathfrak{D}$ be an initial data set on ${\overline{\Sigma}}$ for the system . We say that $\mathfrak{D}$ is in the “$\overline{\Sigma}_+$-gauge" if $\mathfrak{D}$ on $\overline{\Sigma}$ satisfies

\begin{align}
\left[\left(2M\partial_r\right)^2+6M\partial_r+2\right]\divo^2V\Omega^{-1}\xlin|_{\overline{\Sigma}\,}=0,\qquad \divo^2V^{-1}\Omega\xblin|_{\overline{\Sigma}\,}=0,\qquad \glinh|_{\overline{\Sigma}\,}=0,
\end{align}

in addition to the horizon gauge conditions bifurcation gauge conditions. It is said to be in the $\overline{\Sigma}_-$ gauge if it satisfies

\begin{align}
\left[\left(2M\partial_r\right)^2+6M\partial_r+2\right]\divo^2V^{-1}\Omega\xblin|_{\overline{\Sigma}\,}=0,\qquad \divo^2V\Omega^{-1}\xlin|_{\overline{\Sigma}\,}=0,\qquad \glinh|_{\overline{\Sigma}\,}=0,
\end{align}

in addition to the horizon gauge conditions bifurcation gauge conditions.

Let $\mathfrak{D}$ be a smooth, compactly supported initial data set on $\overline{\Sigma}$ for the system whose components are supported on the spherical harmonics with $\ell\geq2$. There exists a unique initial data set $\mathfrak{D}'$, related to $\mathfrak{D}$ by a gauge transformation, such that $\mathfrak{D}'$ is in the $\overline{\Sigma}_+$-gauge of <Ref>. A similar statement applies to the $\overline{\Sigma}_-$.

§.§ The $\ell=0,1$ component: a Linearised Kerr family

In Section 6.2 of <cit.>, a family of linearised Kerr solutions was defined by linearising the Kerr metric in the parameters $(M,a)$ against a Schwarzschild background $(M,a=0)$. The resulting linearised metric defines a solution to the linearised Einstein equations . In the following we define reference linearised Kerr solutions and show that any smooth solution to the system supported only on $\ell=0,1$ modes is a sum of the reference Kerr family and a pure gauge solution. We treat the cases $\ell=0$ and $\ell=1$ separately, yielding reference Schwarzschild solutions in the first case and slowly rotating Kerr solutions in the second case.

§.§.§ The $\ell=0$ mode and reference linearised Schwarzschild solutions

Given $\mathfrak{m}\in\mathbb{R}$, define the solution $\mathfrak{K}_{\ell=0}(\mathfrak{m})$ by

\begin{align}
2\Olin=-\mathfrak{m},\qquad \tr \glin=-2\mathfrak{m},\qquad \rlin=-\frac{2M}{r^3}\mathfrak{m},\qquad \Klin=\frac{\mathfrak{m}}{r^2},
\end{align}

with all remaining components vanishing.

We now show the following:

Any solution to the linearised Einstein equations which is supported on $\ell=0$ spherical harmonics is a sum of the reference linearised Schwarzschild solution defined above in <Ref> and a pure gauge solution.

For $\ell=0$ modes, only the linearised metric components $\Olin_{\ell=0}, \tr\glin_{\ell=0}$ are nontrivial, thus $\otx_{\ell=0}, \otxb_{\ell=0}, \olin_{\ell=0}, \olinb_{\ell=0}$ are the only nontrivial connection components and $\rlin_{\ell=0}$ is the only nontrivial curvature component. We may then derive the following equations among the nontrivial components:

The $\ell=0$ components of a solution $\mathfrak{S}$ to satisfy

\begin{align}\label{ell=0 diff between rlin and trglin}
\partial_v\left(r^3\rlin_{\ell=0}-\frac{3M}{2}\tr\glin_{\ell=0}\right)=\partial_u\left(r^3\rlin_{\ell=0}-\frac{3M}{2}\tr\glin_{\ell=0}\right)=0,
\end{align}

\begin{align}\label{ell=0 partial v}
\partial_v\left[\frac{r}{\Omega^2}\otx+\frac{1}{2}\tr\glin-4\Olin\right]=0,
\end{align}

\begin{align}\label{ell=0 partial u}
\partial_u\left[\frac{r}{\Omega^2}\otxb-\frac{1}{2}\tr\glin+4\Olin\right]=0,
\end{align}

\begin{align}\label{ell=0 Gauss}
\frac{1}{2r^2}\tr\glin=\rlin-\frac{1}{2r}\left[\otx-\otxb\right]+\frac{2\Omega^2}{r^2}\Olin.
\end{align}

Note that ell=0 diff between rlin and trglin implies that $r^3\rlin_{\ell=0}$ and $\frac{3M}{2}\tr\glin_{\ell=0}$ differ by a constant. Looking at the gauge solutions defined in <Ref>, <Ref>, <Ref> we see that the difference between $r^3\rlin_{\ell=0}$ and $\frac{3M}{2}\tr\glin_{\ell=0}$ is in fact gauge invariant. Inspired by the reference linearised Schwarzschild family given by <Ref>, we denote this difference by

\begin{align}\label{ell=0 diff between rlin and trglin defined}
\end{align}

for $\mathfrak{m}$ a real constant. We will now use the equations ell=0 diff between rlin and trglin–ell=0 Gauss of <Ref> to find an explicit parametrisation of all solutions to the $\ell=0$ linearised Einstein equations which exhibits the classification given by <Ref>. It suffices to deal with the case of data on $\Sigma^*_+$ as data on $\Sigma^*_-$ or $\overline{\Sigma}$ are dealt with similarly.

For any solution $\mathfrak{S}$ to the linearised Einstein equations which is supported on $\ell=0$ spherical harmonic modes in an open set $\mathcal{S}\in J^+(\Sigma^*_+)$ and which is $C^2$ away from $\mathscr{H}^+_{\geq0}$, there exists scalar functions $f_{\ell=0}(v)$, $\fbar_{\ell=0}(u)$ and a real parameter $\mathfrak{m}$ such that $\mathfrak{S}$ is given by

\begin{align}
\mathfrak{S}=\mathfrak{G}_{in}[\fbar_{\ell=0}]+\mathfrak{G}_{out}[f_{\ell=0}]+\mathfrak{K}_{\ell=0}[\mathfrak{m}].
\end{align}

Equations ell=0 partial v, ell=0 partial u imply

\begin{align}\label{ell=0 classification first step}
\frac{r}{\Omega^2}\otx+\frac{1}{2}\tr\glin-4\Olin=-2\partial_u\fbar_{\ell=0},\qquad  \frac{r}{\Omega^2}\otxb-\frac{1}{2}\tr\glin+4\Olin=2\partial_v f_{\ell=0}
\end{align}

for some scalar functions $f_{\ell=0}(v)$, $\fbar_{\ell=0}(u)$. Therefore,

\begin{align}
\partial_t\tr\glin_{\ell=0}=4\partial_t\left(\frac{\Omega^2}{r}(f-\fbar)\right),
\end{align}

which in turn implies

\begin{align}
\tr\glin_{\ell=0}=\frac{4\Omega^2}{r}(f-\fbar)+\inhom(r).
\end{align}

For some scalar function $\inhom(r)$. Using metric transport in 3 direction trace, metric transport in 4 direction trace, ell=0 diff between rlin and trglin defined and ell=0 classification first step we derive

\begin{align}
\end{align}

\begin{align}
\end{align}

\begin{align}
\otx_{\ell=0}=-\frac{2\Omega^2}{r^2}(2\Omega^2-1)(f_{\ell=0}-\fbar_{\ell=0})+\frac{2\Omega^2}{r}\partial_vf_{\ell=0}+\frac{\Omega^2}{2}(\inhom)',
\end{align}

\begin{align}
\otxb_{\ell=0}=\frac{2\Omega^2}{r^2}(2\Omega^2-1)(f_{\ell=0}-\fbar_{\ell=0})-\frac{2\Omega^2}{r}\partial_u\fbar_{\ell=0}-\frac{\Omega^2}{2}(\inhom)'.
\end{align}

where $(\inhom)'=d(\inhom)/dr$. Gauss' equation now implies

\begin{align}
\Omega^2(\inhom)'=\frac{\inhom}{r}\left(-1+\frac{4M}{r}\right)+\frac{4M\mathfrak{m}}{r^2}.
\end{align}

The general solution to the above equation is given by

\begin{align}
\inhom=\lambda\frac{\Omega^2}{r}+\frac{\Omega^2}{r}\int dr \left[\frac{4M\mathfrak{m}}{r^2\Omega^2}\times\frac{r}{\Omega^2}\right]=(\lambda+4M\mathfrak{m})\frac{\Omega^2}{r}+\frac{2\mathfrak{m}\Omega^2}{r}r_*-2\mathfrak{m}.
\end{align}

Note that the first term on the right hand side can be obtained according to <Ref> by a pure gauge solution generated by $f(v)=\lambda+4M\mathfrak{m}$, $\fbar(u)=0$, while the second term can similarly be generated according to <Ref>, <Ref> by $f(v)=\frac{\mathfrak{m}}{2}v$, $\fbar(u)=\frac{\mathfrak{m}}{2}u$.

We now indicate how to explicitly find $\mathfrak{m}$, $f_{\ell=0}$, $\fbar_{\ell=0}$ of <Ref>. Given initial data for $\Olin_{\ell=0}$, $N\Olin_{\ell=0}$, $\tr\glin_{\ell=0}$, $N\tr\glin_{\ell=0}$, we may solve for $\rlin$ via the Gauss equation ell=0 Gauss, thus $\mathfrak{m}$ can then be found via ell=0 diff between rlin and trglin defined. Subtracting a reference Schwarzschild solution with mass parameter $\mathfrak{m}$ leads to a pure gauge solution as shown in <Ref>. We may now determine $f_{\ell=0}|_{\Sigma^*_+}$ via the first relation in ell=0 classification first step: Let $f_{\ell=0, \Sigma^*_+}(r)$ be given by

\begin{align}\label{ell=0 inverse gauge transform 1}
f_{\ell=0, \Sigma^*_+}(r)=C-\int_{2M}^rd\bar{r}\left(\frac{r}{\Omega^2}{\otxb}_{\ell=0}-\frac{1}{2}\tr\glin_{\ell=0}+4\Olin_{\ell=0}\right)\Big|_{\Sigma^*}(\bar{r}),
\end{align}

where $C$ is given by

\begin{align}\label{ell=0 inverse gauge transform constant}
\end{align}

Now take $\fbar_{\ell=0, \Sigma^*_+}(r)=-\frac{r}{4}\tr\glin_{\ell=0}+\Omega^2f_{\ell=0, \Sigma^*_+}$. Then

\begin{align}\label{ell=0 inverse gauge transform 2}
&f_{\ell=0}(v)=f_{\ell=0, \Sigma^*_+}(r-2M),\qquad\qquad
\fbar_{\ell=0}(u)=\fbar_{\ell=0, \Sigma^*_+}(r(u,\vsigmap{u}))
\end{align}

give the generators of gauge solutions that comprise the remainder of $\mathfrak{S}$.

Finally, we show how to solve the constraint equations for the $\ell=0$ mode, given as initial data ${\tr\glin_{\ell=0}}|_{\Sigma^*_+}$, ${N\tr\glin_{\ell=0}}|_{\Sigma^*_+}$ on $\Sigma^*_+$.

Given smooth scalars  ${\tr\glin_{\ell=0}}|_{\Sigma^*_+}$, ${N\tr\glin_{\ell=0}}|_{\Sigma^*_+}$ on $\Sigma^*_+$, there exist smooth scalar fields $\Olino_{\ell=0}|_{\Sigma^*_+}$, $N\Olino_{\ell=0}|_{\Sigma^*_+}$ on $\Sigma^*_+$ such that $({\tr\glin_{\ell=0}}|_{\Sigma^*_+},{N\tr\glin_{\ell=0}}|_{\Sigma^*_+},\Olino_{\ell=0}|_{\Sigma^*_+},N\Olino_{\ell=0}|_{\Sigma^*_+})$ satisfy  the constraint equations Spacelike Gauss at Sigma star, Spacelike connection constraint at Sigma star, i.e. the equations

\begin{align}\label{Spacelike Gauss at Sigma star ell=0}
\begin{split}
\end{split}
\end{align}

\begin{align}\label{Spacelike connection constraint at Sigma star ell=0}
\begin{split}
\end{split}
\end{align}

For $\ell=0$ equation constraint equation for Olin on Sigma star becomes

\begin{align}\label{constraint equation for Olin on Sigma star ell=0}
\begin{split}
\end{split}
\end{align}

We can solve for ${N\Olin_{\ell=0}}\Big|_{\Sigma^*_+}$ in terms of $\partial_\ubar{\Olin_{\ell=0}}\Big|_{\Sigma^*_+}$, ${\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}$, ${N\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}$ via Spacelike connection constraint at Sigma star ell=0 to get

\begin{align}\label{28 02 2022 2}
\begin{split}
{N(\Olin)_{\ell=0}}\Big|_{\Sigma^*_+}&=-\frac{2M}{r}\frac{1}{2\Omega^2}\partial_\ustar{\Olin_{\ell=0}}\Big|_{\Sigma^*_+}-\frac{r(2-\Omega^2)}{8\Omega^2}\partial_\ustar {N\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}\\&+\frac{r(3\Omega^4-10\Omega^2+7)}{16(2-\Omega^2)}\frac{1}{\Omega^2}\partial_\ustar {\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}+\frac{3-\Omega^2}{8}{N\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}.
\end{split}
\end{align}

We can combine constraint equation for Olin on Sigma star ell=0, 28 02 2022 2 to gain an ODE for ${\Olin_{\ell=0}}\Big|_{\Sigma^*_+}$:

\begin{align}
\frac{8}{r}\left(1-\frac{4M^2}{r^2}\right)\frac{1}{\Omega^2}\partial_\ustar{\Olin_{\ell=0}}\Big|_{\Sigma^*_+}=
\end{align}

Equation Spacelike Gauss at Sigma star ell=0 at $r=2M$ becomes

\begin{align}\label{28 02 2022}
{N\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}=-\frac{2}{M}{\Olin_{\ell=0}}\Big|_{\Sigma^*_+}-\frac{1}{2\Omega^2}\partial_\ubar {\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}-\frac{2M}{\Omega^2}\partial_\ubar\left(N-\frac{1}{2\Omega^2}\partial_\ubar\right){\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}+\frac{1}{2M}{\tr\glin_{\ell=0}}\Big|_{\Sigma^*_+}=0.
\end{align}

Finally, it suffices to have as initial data the values of $\Olin_{\ell=0}|_{\Sigma^*_+}$, $\tr\glin_{\ell=0}|_{\Sigma^*_+}$, as equation constraint equation for Olin on Sigma star gives an algebraic constraint between $N\Olino_{\ell=0}|_{\Sigma^*_+}$ and   $N\tr\glin_{\ell=0}|_{\Sigma^*_+}$, which allows us to use Spacelike connection constraint at Sigma star to solve an ODE for $N\tr\glin_{\ell=0}|_{\Sigma^*_+}$ with the condition that $\otx|_{\Sigma^*_+\cap\mathscr{H}^+_{\geq0}}=0$. Given uniformly bounded data $\Olin_{\ell=0}|_{\Sigma^*_+}$, $\tr\glin_{\ell=0}|_{\Sigma^*_+}$, this process leads to uniformly bounded initial data for the $\ell=0$ modes, which then can be gauge-transformed to the reference linearised Schwarzschild solution given by <Ref>. We may estimate this gauge transformation via ell=0 inverse gauge transform 1–ell=0 inverse gauge transform 2.

§.§.§ The $\ell=1$ modes and rotating reference linearised Kerr solutions

We now turn to the $\ell=1$ modes

Let $Y^{\ell=1}_m$ be the spherical harmonics of eigenvalue $\ell=1$ and azimuthal numbers $m=-1,0,1$. For any $\bm{s}=(s_1,s_0,s_1)\in \mathbb{R}^3$, denote $Y_{\bm{s}}=\sum_{i=-1,0,1}s_{i}Y^{\ell=1}_{m=s_i}$.
The following is a family of solutions to the system : The nonvanishing metric components are

\begin{align}\label{Kerr shift}
\bmlin_A=\frac{4M}{r}\slashed{\epsilon}_A{}^B\partial_B Y_{\bm{s}}.
\end{align}

The nonvanishing connection components are

\begin{align}
\elin=\frac{3M}{r^2}\slashed{\epsilon}_A{}^B\partial_B Y_{\bm{s}},\qquad \eblin=-\elin.
\end{align}

The nonvanishing curvature components are

\begin{align}
\blin=\frac{\Omega}{r}\elin,\qquad\bblin=-\blin,\qquad \slin=\frac{6}{r^4}MY_{\bm{s}}.
\end{align}

We denote the solution above by $\mathfrak{K}_{\ell=1}(\bm{s})$.

In analogy with <Ref>, we will now show that if a solution to has $\slin_{\ell=1}=0$ then its $\ell=1$ component is a pure gauge solution.

Let $\mathfrak{S}$ be a $C^2$ solution to the linearised Einstein equations on an open subset $\mathcal{S}$ of $J^+(\Sigma^*_+)$ which is supported on $\ell=1$ only spherical harmonic modes, such that $\slin=0$. Then there exist scalar functions $f_{\ell=1}(v,\theta^A)$, ${q_1}_{\ell=1}(v,\theta^A)$, ${q_2}_{\ell=1}(v,\theta^A)$, $\fbar_{\ell=1}(u,\theta^A)$ supported on $\ell=1$ spherical harmonic modes such that

\begin{align}\label{ell=1 classification statement}
\mathfrak{S}=\mathfrak{G}_{in}[\fbar_{\ell=1}]+\mathfrak{G}_{out}[f_{\ell=1}]+\mathfrak{G}_{res}[({q_1}_{\ell=1},{q_2}_{\ell=1})].
\end{align}

We define $f_{\ell=1}$, $\fbar_{\ell=1}$ of the proposition using the equations Bianchi+1a, Bianchi-1a, that $\alin$, $\ablin$ have vanishing projections on $\ell=1$ modes, and the vanishing of $\slin_{\ell=1}$,

\begin{align}
r^2\Omega\blin=6M\fancydstarring_1\left(\frac{\Omega^2}{r^2}\fbar_{\ell=1},0\right),\qquad\qquad r^2\Omega\bblin=-6M\fancydstarring_1\left(\frac{\Omega^2}{r^2}f_{\ell=1},0\right).
\end{align}

The equations Bianchi+1b, Bianchi-1b imply

\begin{align}
\frac{\Omega^2}{r}\left(\eblin_{\ell=1}-\elin_{\ell=1}\right)=\fancydstarring_1\left[\partial_u\left(\frac{\Omega^2}{r^2}\fbar_{\ell=1}\right)-\partial_v\left(\frac{\Omega^2}{r^2}f_{\ell=1}\right),\;0\right].
\end{align}

Integrating partial_u b, there exists ${q_1}_{\ell=1}(v,\theta^A)$, ${q_2}_{\ell=1}(v,\theta^A)$ such that

\begin{align}
\bmlin_{\ell=1}=\fancydstarring_1\left[-2\frac{\Omega^2}{r}\fbar_{\ell=1}+2r\partial_v\left(\frac{f_{\ell=1}}{r}\right)+r\partial_v{q_1}_{\ell=1},\; r\partial_v{q_2}_{\ell=1}\right].
\end{align}

The Gauss equation Gauss implies

\begin{align}
\otx_{\ell=1}-\otxb_{\ell=1}=2r\rlin_{\ell=1}+\frac{4\Omega^2}{r}\Olin,
\end{align}

whereas the Codazzi equations elliptic equation 1, elliptic equation 2 imply

\begin{align}
\frac{1}{2\Omega}\slashednabla\left(\otx_{\ell=1}-\otxb_{\ell=1}\right)=\blin+\bblin+\frac{\Omega}{r}\left(\elin_{\ell=1}+\eblin_{\ell=1}\right).
\end{align}

Taking the divergence gives

\begin{align}
\otx_{\ell=1}-\otxb_{\ell=1}=-\divo\left(r\Omega\blin_{\ell=1}+r\Omega\bblin_{\ell=1}\right),
\end{align}

Thus we find

\begin{align}
\rlin_{\ell=1}=-\frac{6M\Omega^2}{r^4}\left(f_{\ell=1}-\fbar_{\ell=1}\right).
\end{align}

Now we can use the Codazzi equations elliptic equation 1, elliptic equation 2 to find $\otx_{\ell=1}+\otxb_{\ell=1}$:

\begin{align}
\otx_{\ell=1}+\otxb_{\ell=1}=2\frac{\Omega^2}{r}\left(\partial_vf_{\ell=1}-\partial_u\fbar_{\ell=1}-\frac{2}{r}(f_{\ell=1}+\fbar_{\ell=1})\right).
\end{align}

Adding the equations Bianchi 0 gives

\begin{align}
\partial_t r^3\rlin_{\ell=1}=\partial_t\left[\frac{6M\Omega^2}{r}(f_{\ell=1}-\fbar_{\ell=1})\right],
\end{align}

which implies

\begin{align}
\end{align}

for some scalar function $\inhom=\inhom(r,\theta^A)$ which is in the span of $\ell=1$ spherical harmonic modes. Subtracting the first equation of Bianchi 0 from the second we find

\begin{align}
\end{align}

and moreover,

\begin{align}
\elin_{\ell=1}=-\fancydstarring_1\left(\frac{\Omega^2}{r^2}f_{\ell=1}+\frac{1}{\Omega^2}\partial_u\left(\frac{\Omega^2}{r}\fbar_{\ell=1}\right)+\frac{1}{6Mr}\inhom,\;0\right),
\end{align}

\begin{align}
\eblin_{\ell=1}=-\fancydstarring_1\left(-\frac{\Omega^2}{r^2}\fbar_{\ell=1}+\frac{1}{\Omega^2}\partial_v\left(\frac{\Omega^2}{r}f_{\ell=1}\right)+\frac{1}{6Mr}\inhom,\;0\right),
\end{align}

Continuing this procedure we show that in fact $\mathfrak{S}$ is given by ell=1 classification statement.

The proof of <Ref> gives a recipe for finding the gauge transformation that brings the solution back to the reference form given in <Ref>.

Let $\mathfrak{m}\in\mathbb{R}$, $\bm{s}=(s_{-1},s_0,s_1)\in\mathbb{R}^3$. We denote by $\mathfrak{K}(\mathfrak{m},\bm{s})$ the sum of the solutions $\mathfrak{K}_{\ell=0}(\mathfrak{m})$ and $\mathfrak{K}_{\ell=1}(\bm{s})$, where $\mathfrak{K}_{\ell=0}$ is parametrised by $\mathfrak{m}$ as in <Ref> and $\mathfrak{K}_{\ell=1}$ is parametrised by $\bm{s}$ as in <Ref>.

§.§ Asymptotic flatness at spacelike infinity

When working with non-compactly supported initial data sets, we will use the following definition of asymptotic flatness at spacelike infinity:

Let $\digamma$ be a scalar function on $\mathbb{R}$. We say that $\digamma$ is $O_n(y^{-k})$ as $r\longrightarrow\infty$ if

\begin{align}
\begin{split}
\partial_y^i \digamma =O\left(\frac{1}{y^{k+i}}\right) \text{ for } i=0,\dots n.
\end{split}
\end{align}

We say that $\digamma$ is $O_\infty(y^{-k})$ if it is $O_n(y^{-k})$ for any $n\in\mathbb{N}$.

If $(\uppsi,\uppsi')$ is an initial data set for the Regge–Wheeler equation RW on ${\Sigma^*_+}$, $\Sigma^*_-$ or $\overline{\Sigma}$, we say that $(\uppsi,\uppsi')$ is $O_n(r^{-k})$ if $\uppsi=O_n(r^{-k})$ and $\uppsi'=O_n(r^{-(k+1)})$ in the sense of <Ref>. The same defintion applies to initial data on ${\Sigma^*_+}$, $\Sigma^*_-$ or $\overline{\Sigma}$ for the Teukolsky equations T+2, T-2.

Given the results of <Ref>, we will always evolve from data for which the $\ell=0,1$ modes are induced by the reference linearised Kerr. Thus we define the notion of asymptotic flatness at spacelike infinity as follows:

Let $\mathfrak{D}$ be an initial data set on ${\Sigma^*_+}$ for the system in the sense of <Ref>. We say that $\mathfrak{D}$ is asymptotically flat to order $(\gamma,N)$ at spacelike infinity if the $\ell=0,1$ components of $\mathfrak{D}$ arise from the restriction of $\mathfrak{K}(\mathfrak{m},\bm{s})$ given in for some real parameters $\mathfrak{m}\in\mathbb{R}$,  $\bm{s}=(s_1,s_0,s_1)\in\mathbb{R}^3$ as in <Ref>, while the $\ell\geq2$ component of $\mathfrak{D}$ satisfies

\begin{align}
&\partial_r^n \Olinos, \partial_r^n \tr\glin=O\left(\frac{1}{r^{\gamma+n}}\right),\\
&\partial_r^n \bmlins_A=O\left(\frac{1}{r^{\gamma+n-1}}\right),\\
&\partial_r^n \glinhs_{AB}=O\left(\frac{1}{r^{\gamma+n-2}}\right),
\end{align}

for all integers $ n\leq N$, with respect to any coordinate system on the unit sphere $S^2$ (see <Ref>).

A similar definition applies for initial data on $\overline{\Sigma}$.
We will refer to initial data for the linearised system simply as “asymptotically flat initial data" if it is asymptotically flat to order $(\gamma,\infty)$ with $\gamma> 1$. If $\gamma=1$, we say the initial data is “weakly asymptotically flat".

An initial data set on $\Sigma^*_+$ satisfying the conclusions of <Ref> satisfies

\begin{align}
\left|\overone{R}_{\ell\geq2}|_{\Sigma^*_+}\right|,\;\left|\overone{\Gamma}_{\ell\geq2}|_{\Sigma^*_+}\right|,\,\left|\overone{g}_{\ell\geq2}|_{\Sigma^*_+}\right|\sim O_{\infty}\left(e^{-\frac{r}{2M}}\right),
\end{align}

as $r\longrightarrow\infty$, where $\overone{R}$ stands for any linearised curvature component, $\overone{\Gamma}$ stands for any linearised connection component, and $\overone{g}$ stands for any component of the linearised metric. The same applies to initial data on $\overline{\Sigma}$ satisfying the $\overline{\Sigma}_+$-gauge and to initial data on $\Sigma^*_-$ satisfying the $\Sigma^*_-$-gauge. An initial data set on $\overline{\Sigma}$ satisfying the conditions of the $\overline{\Sigma}_-$ gauge of <Ref> has

\begin{align}
\left|\overone{R}_{\ell\geq2}|_{\overline{\Sigma}\,}\right|,\;\left|\overone{\Gamma}_{\ell\geq2}|_{\overline{\Sigma}\,}\right|,\,\left|\overone{g}_{\ell\geq2}|_{\overline{\Sigma}\,}\right|\sim O_\infty\left(\frac{r}{2M}e^{-\frac{r}{2M}}\right)
\end{align}

as $r\longrightarrow\infty$.

§ ASYMPTOTIC FLATNESS AT NULL INFINITY AND BONDI-NORMALISED COORDINATES

In studying the asymptotics of solutions to towards future (and past) null infinity, we will adopt the notion of asymptotic flatness at null infinity enabled by the stability result of <cit.>[See Section 8.3 of <cit.>]. In the course of proving Theorems 1, 2 and Corollary 1 we will develop a theory of radiation fields which allows for a stronger notion of asymptotic flatness at null infinity as required for the purposes of the scattering theory to be constructed here. We now state the definition of asymptotic flatness at null infinity to be used in this paper:

We say a solution $\mathfrak{S}$ to is   for some $0<s\leq1$ and any integer $k$ and index $\gamma$ with $k+|\gamma|\leq n$ the following holds along $\mathscr{C}\cap J^+(\Sigma)$ for any $u$:

\begin{align}\label{asym flatness boundedness of lapse at scri+}
|(\mathring{\slashednabla})^\gamma\Olin|\,+\,|(\mathring{\slashednabla})^\gamma (r\nablav)^k r^{2} \olin|\,\leq C(k,\gamma),
\end{align}

\begin{align}\label{asym flatness boundedness of shift at scri+}
|(\mathring{\slashednabla})^\gamma (r\nablav)^k r^{-1} \bmlin_A|\,\leq C(k,\gamma)
\end{align}

\begin{align}\label{asym flatness boundedness of glinh at scri+}
|(\mathring{\slashednabla})^\gamma r^{-2}\glinh_{AB}|\,+\,|(\mathring{\slashednabla})^\gamma (r\nablav)^k  \xlin_{AB}|+|(\mathring{\slashednabla})^\gamma (r\nablav)^k r^{1+s} \alin_{AB}|\,\leq C(k,\gamma).
\end{align}

We say that $\mathfrak{S}$ is  with weight $s$ to order $n$ at $\mathscr{I}^+$ if, in addition to the above, the pointwise limits of the following weighted components of $\mathfrak{S}$ as $v\longrightarrow\infty$ for any finite fixed $u$ define smooth fields on $\mathscr{I}^+$ (in the sense of <Ref>):

\begin{align}\label{ansatz of radiation fields scri+}
\glinh,\, \Olino,\, r^{-1}\bmlin,\, r^2\xlin,\, r\xblin,\, r\ablin,\, r^2\bblin,\, r^3\rlin,\, r^3\slin.
\end{align}

We say a solution $\mathfrak{S}$ to is   for some $0<s\leq1$ and any integer $k$ and index $\gamma$ with $k+|\gamma|\leq n$ the following holds along $\mathscr{C}\cap J^-(\Sigma)$ for any $v$:

\begin{align}\label{asym flatness boundedness of lapse at scri-}
|(\mathring{\slashednabla})^\gamma\Olin|\,+\,|(\mathring{\slashednabla})^\gamma (r\nablau)^k r^{2} \olinb|\,\leq C(k,\gamma),
\end{align}

\begin{align}\label{asym flatness boundedness of shift at scri-}
|(\mathring{\slashednabla})^\gamma (r\nablau)^k r^{-1} \bmlin_A|\,\leq C(k,\gamma)
\end{align}

\begin{align}\label{asym flatness boundedness of glinh at scri-}
|(\mathring{\slashednabla})^\gamma r^{-2}\glinh_{AB}|\,+\,|(\mathring{\slashednabla})^\gamma (r\nablau)^k  \xblin_{AB}|+|(\mathring{\slashednabla})^\gamma (r\nablau)^k r^{1+s} \ablin_{AB}|\,\leq C(k,\gamma).
\end{align}

We say that $\mathfrak{S}$ is  with weight $s$ to order $n$ at $\mathscr{I}^-$ if, in addition to the above, the pointwise limits of the following weighted components of $\mathfrak{S}$ as $u\longrightarrow-\infty$ for any finite fixed $v$ define smooth fields on $\mathscr{I}^-$ (in the sense of <Ref>):

\begin{align}\label{ansatz of radiation at scri-}
\glinh,\, \Olino,\, r^{-1}\bmlin,\, r^2\xblin,\, r\xlin,\, r\alin,\, r^2\blin,\, r^3\rlin,\, r^3\slin.
\end{align}

For a spacetime which is strongly asymptotically flat at $\mathscr{I}^+$ we denote the limits of the quantities listed in ansatz of radiation fields scri+ by

\begin{align}
\Olinos_{\mathscr{I}^+},\, \bmlins_{\mathscr{I}^+},\, \xlins_{\mathscr{I}^+},\, \xblins_{\mathscr{I}^+},\, \ablins_{\mathscr{I}^+},\, \bblins_{\mathscr{I}^+},\, \rlins_{\mathscr{I}^+},\, \slins_{\mathscr{I}^+}.
\end{align}

When the limits of either of $r\tr\glin$, $r\glinh$ is well-defined at $\mathscr{I}^+$, we denote it by

\begin{align}
\tr\glins_{\mathscr{I}^+},\; \glinhs_{\mathscr{I}^+}
\end{align}

respectively. An analogous notational prescription applies at $\mathscr{I}^-$.

The reference linearised Kerr family defined in <Ref> is strongly asymptotically flat in the sense of <Ref> above.

In addition to the radiation fields defined at $\mathscr{I}^+$ via <Ref> and $\mathscr{I}^-$ via <Ref>, we will sometimes refer to the following limits when they are well-defined:

\begin{align}
\\&\Phii{i}_{\mathscr{I}^+}:=\lim_{v\longrightarrow\infty}\left(\frac{r^2}{\Omega^2}\nablav\right)^{i}\Psilin, \\&\underline{\Phi}^{(i)}_{\mathscr{I}^+}:=\lim_{v\longrightarrow\infty}\left(\frac{r^2}{\Omega^2}\nablav\right)^{i}\Psilinb, \\&\mathcal{X}^{(i)}_{\mathscr{I}^+}:=\lim_{v\longrightarrow\infty}\left(\frac{r^2}{\Omega^2}\nablav\right)^{i}r^2\xlin(u,v,\theta^A),
\\&\underline{\mathcal{X}}^{(i)}_{\mathscr{I}^+}:=\lim_{v\longrightarrow\infty}\left(\frac{r^2}{\Omega^2}\nablav\right)^{i}r\xblin(u,v,\theta^A),
\\&\hat{\mathtt{g}}^{(n)}:=\lim_{v\longrightarrow\infty}\left(\frac{r^2}{\Omega^2}\nablav\right)^{i}r\glinh(u,v,\theta^A).
\end{align}

An analogous prescription applies at $\mathscr{I}^-$.

§.§ Future Bondi-normalised double null gauges and the BMS$+$ group

We have already defined the (future) Bondi normalisation in <Ref> of the introduction. We now study it in more detail and derive the restrictions it imposes on the gauge freedom given by the pure gauge solution of <Ref>, <Ref> and <Ref>.

Let $\mathfrak{S}$ be a solution to the system with linearised mass parameter $\mathfrak{m}$. We say that $\mathfrak{S}$ is in future Bondi-normalised (or that $\mathfrak{S}$ is in the $\mathscr{I}^+$ gauge) if the solution is  asymptotically flat at $\mathscr{I}^+$ in the sense of <Ref> and if in addition the following pointwise limits are attained along any $\mathscr{C}_{u},\;|u|<\infty$:

\begin{align}
&\lim_{v\longrightarrow\infty} \Olino(u,v,\theta^A)=-\frac{1}{2}\mathfrak{m},\label{Olino gauge condition at scri+}\\
%&\lim_{v\longrightarrow\infty} r^2\otx=0,\label{outgoing expansion gauge condition at scri+}\\
&\lim_{v\longrightarrow\infty}r^2\Klin_{\ell\geq2}(u,v,\theta^A)=0,\label{round sphere at scri+}
\end{align}

and in addition we have

\begin{align}\label{fixing residual freedom Bondi}
\lim_{v\longrightarrow\infty}\glinh(u,v,\theta^A)=0,
\end{align}

\begin{align}\label{fixing residual freedom Bondi trace}
\lim_{v\longrightarrow\infty}\tr\glin_{\ell=1}(u,v,\theta^A)=0.
\end{align}

While the linearised Kerr solutions $\mathfrak{K}(\mathfrak{m},\bm{s})$ are not future Bondi-normalised in the sense defined above, their renormalised versions $\mathfrak{K}^+(\mathfrak{m},\bm{s})$ satisfy the conditions Olino gauge condition at scri+–round sphere at scri+.

Assume that $\mathfrak{G}$ is a pure gauge solution which is asymptotically flat at $\mathscr{I}^+$ in the sense of <Ref> to infinite order, and assume in addition that the conditions round sphere at scri+ and

\begin{align}
\lim_{v\longrightarrow\infty} \Olino(u,v,\theta^A)=0,\label{Olino gauge condition at scri+ gauge}
\end{align}

are satisfied. Then the generator of ingoing gauge solutions $\fbar$ is of the form

\begin{align}\label{kirahvi nimelta tuike}
\fbar(u,\theta^A)=\underline{L}(\theta^A)u+\underline{T}(\theta^A),
\end{align}

where $\underline{L}$ is a linear combination of the $\ell=1$ spherical harmonics and $\underline{T}$ is any scalar function on $S^2$.
Moreover, the generator of outgoing gauge solutions $f(v,\theta^A)$ grows at most linearly in $v$. Finally, the generators of residual gauge solutions $q_1, q_2$ have smooth limits as $v\longrightarrow\infty$.

Imposing Olino gauge condition at scri+ together with asym flatness boundedness of lapse at scri+, we have

\begin{align}\label{BMS intermediate thing in Olin}
\left|\partial_v(f\Omega^2)+\Omega^2\partial_u\fbar-\frac{2M\Omega^2}{r^2}\fbar\right|\lesssim\frac{B(u)\Omega^2}{r},
\end{align}

for some constant $B$ that may depend on $u$, thus for any fixed $u$, $\partial_v(\Omega^2f)$ is bounded. As $f$ does not depend on $u$,  there exists constants $C,C'$ such that

\begin{align}
|f(v,\theta^A)|\leq Cv+C'.
\end{align}

In particular, $v^{-1}f$ is bounded. Condition Olino gauge condition at scri+ now implies that $\partial_vf(v,\theta^A)$ converges as $v\longrightarrow\infty$ and we have

\begin{align}
\partial_u\fbar=-\lim_{v\longrightarrow\infty}\partial_vf,
\end{align}

so $\partial_u\fbar$ is independent of $u$ and $\fbar$ must be affine in $u$: there exist smooth functions $L,T_u$ on $S^2$ such that

\begin{align}
\fbar(u,\theta^A)=\underline{L}(\theta^A)u+\underline{T}(\theta^A).
\end{align}

Finally, note that the boundedness of $r^2\olin$ (condition asym flatness boundedness of lapse at scri+) implies that

\begin{align}
r\partial_v f-f+r\partial_u(\fbar)\lesssim \log(2M)^{-1}r
\end{align}

as $v\longrightarrow\infty$.

Iterating the above, we obtain

\begin{align}
|(\mathring{\slashednabla})^\gamma(v\partial_v)^k v^2\partial_v^{2} f_{\ell\neq1}|+|(\mathring{\slashednabla})^\gamma(v\partial_v)^kv^2\partial_v q_1|+|(\mathring{\slashednabla})^\gamma(v\partial_v)^kv^2\partial_v q_2|\,\leq B_{k,\gamma}.
\end{align}

For $\ell\neq1$, the condition round sphere at scri+ implies $v^{-1}f_{\ell\neq1}\longrightarrow0$ as $v\longrightarrow\infty$. Since $\partial_v f_{\ell\neq1}\longrightarrow -L_{\ell\neq1}$, we must have that $L_{\ell\neq1}=0$.

The restriction round sphere at scri+ imposes the geometric condition that the induced metric on the $S^2$ cross-sections of $\mathscr{I}^+$ must be that of the round sphere. It has been proposed in the literature that the residual gauge freedom to choose coordinates on these cross-sections should be used to define an extensions of the BMS group. See for example <cit.>.

\begin{align}
\lim_{v\longrightarrow\infty}\glinh=\glinh_{\infty}\neq0,\qquad \lim_{v\longrightarrow\infty}r\xblin=\xblins_{\mathscr{I}^+},\qquad \lim_{v\longrightarrow\infty}r\bmlin=0.
\end{align}

Then we have

\begin{align}
r\nablav r(\glinh-\glinh_{\infty})=2r\Omega^2(\glinh-\glinh_{\infty})+2r^2\Omega\xlin-2\mathring{\fancydstar_2}r\bmlin.
\end{align}

In order for $r(\glinh-\glinh_{\infty})$ to converge, we must have

\begin{align}
\end{align}

as $v\longrightarrow\infty$. Similarly, for $\tr\glin$ to have a limit $(\tr\glin)_{\infty}$ as $v\longrightarrow\infty$ we need

\begin{align}
\end{align}

In <Ref>, assume in addition that condition shift condition at scri+ is satisfied. Let $q_{1,\infty}$, $q_{2,\infty}$ be the limits of $q_1$, $q_2$. Then $v(q_1-q_{1,\infty})$, $v(q_2-q_{2,\infty})$ converge as $v\longrightarrow\infty$ and

\begin{align}
\lim_{v\longrightarrow\infty} v(q_1-q_{1,\infty})=-2\fbar_{\ell\geq1}-2\lim_{v\longrightarrow\infty}f_{\ell\geq1},\qquad\lim_{v\longrightarrow\infty} v(q_2-q_{2,\infty})|_{\ell\geq1}=0.
\end{align}

The gauge conditions (<ref>) and (<ref>) comprise the Bondi gauge in the sense defined in e.g. <cit.>. Once (<ref>), (<ref>) are imposed, the residual gauge freedom allows to freely specify coordinates on the “sphere at infinity" while maintaining a standard round metric at $\mathscr{I}^+$. We fix this residual gauge freedom by requiring the condition (<ref>) be satisfied in addition to (<ref>), (<ref>) in order to obtain a scheme to fully fix the gauge in terms of conditions on scattering data at $\mathscr{I}^+$, $\mathscr{H}^+_{\geq0}$.

The function $L$ in <Ref> above encodes the action of asymptotic Lorentz boosts. Asymptotic rotations are induced by residual gauge solutions generated by $(q_1=0,q_2)$ in the sense of <Ref>, where $q_2$ is restricted to $\ell=1$ modes. Asymptotic translations are encoded in the $\ell=0,1$ modes of $T_u$.

Let $\underline{L}(\theta^A)$, $\underline{T}(\theta^A)$ be smooth scalar functions on the unit sphere with $\underline{L}$ in the span of the spherical harmonics $Y^{\ell=1}_{m}$ of azimuthal number $\ell=1$. A BMS$+$ gauge solution is a pure gauge solution made of the sum of solutions $\underline{\mathfrak{G}}$, $\mathfrak{G}$ and $\mathfrak{G}_{res}$, where $\underline{\mathfrak{G}}$ is generated by $\fbar$ given by

\begin{align}
\fbar(u,\theta^A)=\underline{L}(\theta^A)u+\underline{T}(\theta^A),
\end{align}

$\mathfrak{G}$ is generated by $f(v,\theta^A)$ grows at most linearly in $v$, and $\mathfrak{G}_{res}$ generated by the pair $(0,q_2)$ with $q_2$ in the span of the $\ell=1$ spherical harmonics. We refer to the addition of a BMS$+$ gauge solution as a BMS$+$ transformation.

§.§ Past Bondi-normalised double null gauges and the BMS$-$ group

In light of <Ref> we define the analogous past-Bondi normalisation on $\mathscr{I}^-$ as follows:

Let $\mathfrak{S}$ be a solution to the system with linearised mass parameter $\mathfrak{m}$. We say that $\mathfrak{S}$ is in future Bondi-normalised (or that $\mathfrak{S}$ is in the $\mathscr{I}^+$ gauge) if the solution is  asymptotically flat at $\mathscr{I}^+$ in the sense of <Ref> and if in addition the following pointwise limits are attained along any $\mathscr{C}_{u},\;|u|<\infty$:

\begin{align}
&\lim_{u\longrightarrow-\infty} \Olino(u,v,\theta^A)=-\frac{1}{2}\mathfrak{m},\label{Olino gauge condition at scri-}\\
%&\lim_{v\longrightarrow\infty} r^2\otx=0,\label{outgoing expansion gauge condition at scri+}\\
&\lim_{u\longrightarrow-\infty}r^2\Klin_{\ell\geq2}(u,v,\theta^A)=0,\label{round sphere at scri-}
\end{align}

and in addition we have

\begin{align}\label{fixing residual freedom Bondi past}
\lim_{u\longrightarrow-\infty}\glinh(u,v,\theta^A)=0.
\end{align}

\begin{align}\label{fixing residual freedom Bondi past bmlin}
\lim_{u\longrightarrow-\infty}\tr\glin_{\ell=1}(u,v,\theta^A)=0,
\end{align}

Following similar steps to the proof of <Ref>, we may obtain the following:

Assume that $\mathfrak{G}$ is a pure gauge solution which is asymptotically flat at $\mathscr{I}^-$ in the sense of <Ref>  to infinite order, and assume in addition that the condition Olino gauge condition at scri- is satisfied. Then the generator of outgoing gauge solutions $f$ is of the form

\begin{align}\label{kirahvi nimelta tuike}
\end{align}

where $L$ is a linear combination of the $\ell=1$ spherical harmonics and $T$ is any scalar function on $S^2$.
Moreover, the generator of outgoing gauge solutions $\fbar(u,\theta^A)$ grows at most linearly in $u$. Finally, the generators of residual gauge solutions $q_1, q_2$ are smooth functions which are independent of $v$.

Let $L(\theta^A)$, $T(\theta^A)$ be smooth scalar functions on the unit sphere with $L$ in the span of the spherical harmonics $Y^{\ell=1}_{m}$ of azimuthal number $\ell=1$. A BMS$-$ gauge solution is a pure gauge solution made of the sum of solutions $\underline{\mathfrak{G}}$, $\mathfrak{G}$ and $\mathfrak{G}_{res}$, where ${\mathfrak{G}}$ is generated by $f$ given by

\begin{align}
\end{align}

$\underline{\mathfrak{G}}$ is generated by $\fbar(u,\theta^A)$ grows at most linearly in $u$, and $\mathfrak{G}_{res}$ generated by the pair $(0,q_2)$ with $q_2$ in the span of the $\ell=1$ spherical harmonics. We refer to the addition of a BMS$-$ gauge solution as a BMS$-$ transformation.

§ THE HORIZON-NORMALISED GAUGES

In this section we introduce gauge conditions for radiation fields on $\mathscr{H}^+_{\geq0}$ or $\overline{\mathscr{H}^+}$ that will be used to fix the residual gauge freedom left by imposing the future Bondi gauge conditions Olino gauge condition at scri+–round sphere at scri+ at $\mathscr{I}^+$.

§.§ The $\mathscr{H}^+_{\geq0}$-normalised gauge

Let $\mathfrak{S}$ be a solution to the linearised Einstein equations on $D^+(\Sigma^*_+)$ with linearised Kerr parameters $(\mathfrak{m},\mathfrak{a})$. We say that $\mathfrak{S}$ is in the $\mathscr{H}^+_{\geq0}$ gauge if the conditions initial horizon gauge condition are satisfied and

Ω^-2_ℓ=0,1|_ℋ^+_≥0∩Σ^*_+=0,          ^2Ω^-1|_ℋ^+_≥0∩Σ^*_+=0,

|_ℋ^+_≥0=-1/2𝔪,          ^A|_ℋ^+_≥0=4M/rϵ^AB∑_m∈{-1,0,1}𝔞_m∂_A Y_m^ℓ=1.

We now show that combining the Bondi gauge conditions Olino gauge condition at scri+–round sphere at scri+ with the conditions of <Ref> fixes the gauge freedom completely on $D^+(\Sigma^*_+)$.

Assume that $\mathfrak{G}$ is a pure gauge solution to that is both $\mathscr{I}^+$ and $\mathscr{H}^+_{\geq0}$ normalised. Then $\mathfrak{G}$ is trivial.

Let $f(v,\theta^A)$, $\fbar(u,\theta^A)$, $q_1(v,\theta^A)$, $q_2(v,\theta^A)$ be the generators of $\mathfrak{G}$ according to <Ref> respectively. We handle the $\ell\geq2$ component first. The Bondi gauge conditions Olino gauge condition at scri+–round sphere at scri+ give the restriction $\fbar_{\ell\geq2}=T_{u,\ell\geq2}(\theta^A)$ according to <Ref>. The second condition in initial horizon gauge condition forces $T_{u,\ell\geq2}=0$, thus $\fbar_{u,\ell\geq2}=0$.

The condition $\bmlin_{\ell\geq2}[\mathfrak{G}]|_{\mathscr{H}^+_{\geq0}}=0$ means that ${q_2}_{\ell\geq2}(v,\theta^A)=C_2(\theta^A)$ is constant in $v$ and

\begin{align}
\partial_v\left(f_{\ell\geq2}+M{q_1}_{\ell\geq2}\right)=0,
\end{align}

thus ${q_1}_{\ell\geq2}=-\frac{1}{M}f_{\ell\geq2}+C_1$ for some smooth scalar functions $C_1$, $C_2$ on $S^2$. The condition that $\Olin_{\ell\geq2}|_{\mathscr{H}^+_{\geq0}}=0$ implies

\begin{align}\label{27 12 2021}
\end{align}

The condition that $\divo^2\Omega^{-1}\xblin$ vanishes on $\mathscr{H}^+_{\geq0}\cap \Sigma^*_+$ implies that $S=0$. Thus $f_{\ell\geq2}=0$ and ${q_1}_{\ell\geq2}=C_1$. The condition (<ref>) implies $C_1=C_2=0$.

Turning to the $\ell=1$ mode, the Bondi gauge conditions give the restriction $\fbar_{\ell=1}=L_{\ell=1}u+T_{u,\ell=1}$. Demanding that $\Olin$ is regular at $\mathscr{H}^+$ forces $L_{\ell=1}=0$, and the vanishing of $\rlin_{\ell=1}+\divr\elin_{\ell=1}$ on $\mathscr{H}^+_{\geq0}$ leads to $T_{u,\ell=1}=0$. Having $\Olin[\mathfrak{G}]$ vanish on $\mathscr{H}^+_{\geq0}$ restricts $f_{\ell=1}$ to the form given in 27 12 2021, and the condition $\Omega^{-2}\otx_{\ell=1}|_{\mathscr{H}^+_{\geq0}\cap\Sigma^*_+}=0$ leads to $S=0$.
Having condition Kerr shift forces ${q_1}_{\ell=1}, {q_2}_{\ell=1}$ to be constant in $v$. The condition (<ref>) leads to ${q_1}_{\ell=1}=0$. Note that $(0,q_2)$ generates a trivial gauge solution via <Ref> if $q_2$ is only supported on $\ell=1$ modes.

We finally treat the $\ell=0$ mode. The vanishing of $\Olin$ at $\mathscr{H}^+_{\geq0}$ implies

\begin{align}
\end{align}

where $C$ is a constant. The vanishing of $\Omega^{-2}\otxb$ on $\mathscr{H}^+_{\geq0}\cap\Sigma^*_+$ implies that $C=0$. Thus we can only have  $f_{\ell=0}=\fbar_{\ell=0}=T_{u,\ell=0}$, and it is easy to check that $\mathfrak{G}_{in}[T_{u,\ell=0}]+\mathfrak{G}_{out}[T_{u,\ell=0}]$ is trivial for any constant $T_{u,\ell=0}$.

\begin{align}
\mathfrak{G}_{+,\;\ell=0}=\mathfrak{G}_{in}\left[\frac{1}{2}\mathfrak{m}v\right]+\mathfrak{G}_{out}\left[\frac{1}{2}\mathfrak{m}u\right].
\end{align}

Note that $\mathfrak{K}_{\ell=0}+\mathfrak{G}_{+,\ell=0}$ is not regular near $\mathscr{H}^+_{\geq0}$, since

\begin{align}
\end{align}

In fact, we can show that a solution to supported on $\ell=0$ with $\Olin|_{\mathscr{H}^+_{\geq0}}=0$ can not be regular at $\mathscr{H}^+_{\geq0}$ unless $\mathfrak{m}$, defined via ell=0 diff between rlin and trglin defined, is vanishing:

If $\mathfrak{S}$ is a solution to the linearised Einstein equations which is spherically symmetric and future Bondi-normalised, such that $\Olin$ is finite in a neighbourhood of $\mathscr{H}^+_{\geq0}$. Then $\mathfrak{S}$ is a pure gauge solution.

By <Ref> it suffices to consider solutions of the form $\mathfrak{S}=\mathfrak{K}_{\ell=0}+\mathfrak{G}$ for a pure gauge solution $\mathfrak{G}$. With the conditions Olino gauge condition at scri+–round sphere at scri+, we can adapt the argument leading to <Ref> to show that the scalar functions $f(v)$, $\fbar(u)$ generating $\mathfrak{G}$ are restricted to

\begin{align}
\fbar(u)=Au+B,\qquad \lim_{v\longrightarrow\infty}\frac{f}{v}=\lim_{v\longrightarrow\infty}\partial_v f=\mathfrak{m}+A.
\end{align}

for real constant $A,B$. The solution $\mathfrak{G}$ is regular on $\mathscr{H}^+_{\geq0}$ iff $A=0$. We then have

\begin{align}
\end{align}

as $v\longrightarrow\infty$. Thus we must have $\mathfrak{m}=0$, and <Ref> tells us that $\mathfrak{G}$ in this case is pure gauge.

A similar statement to <Ref> applies at $\overline{{\mathscr{H}^-}}$.

Note that $\mathfrak{K}_{\ell=0}+\mathfrak{G}$ also satisfies

\begin{align}
\tr\glin_{\ell=0}=-\frac{4M\mathfrak{m}}{r}-\frac{4M\mathfrak{m}\Omega^2}{r}+8M\mathfrak{m}\frac{\Omega^2}{r}+4M\mathfrak{m}\frac{\Omega^2}{r}\log\frac{r-2M}{\sqrt{v^2+(2M)^2}}.
\end{align}

In particular we have

\begin{align}\label{ell=0 additional gauge condition}
\end{align}

and that $r^2\tr\glin_{\ell=0}$ converges pointwise as $v\longrightarrow\infty$. Imposing the condition ell=0 additional gauge condition in addition to future Bondi-normalisation, the BMS$+$ is reduced to transformations such that $T_u=T_v$.

If we combine the Bondi conditions Olino gauge condition at scri+, round sphere at scri+ with the condition that $r^2\tr\glin_{\ell=0}$ is bounded along outgoing null hypersurfaces as $v\longrightarrow\infty$, we obtain a set of gauge conditions that completely fixes the gauge for the $\ell=0$ component of the system .

Note in particular that

\begin{align}
&2\Olin-\frac{1}{2}\mathfrak{m}\log\left|\frac{r}{2M}-1\right|\longrightarrow0,\label{ell=0 Bondi normalised divergence at H+ 1}
\\&\Omega^{-2}\otxb+\mathfrak{m}\log\left|\frac{r}{2M}-1\right|\longrightarrow -\frac{\mathfrak{m}}{2M}\label{ell=0 Bondi normalised divergence at H+ 2}
\end{align}

pointwise as $u\longrightarrow\infty$. If we set ell=0 Bondi normalised divergence at H+ 1,  ell=0 Bondi normalised divergence at H+ 2 as conditions in addition to those of the Bondi-normalisation Olino gauge condition at scri+, round sphere at scri+, then the residual gauge freedom vanishes and the gauge is fixed:

A pure gauge solution which is spherically symmetric and satisfies  Olino gauge condition at scri+, round sphere at scri+ at $\mathscr{I}^+$ and

\begin{align}
\Olin_{\ell=0}|_{\mathscr{H}^+}=0,\qquad \Omega^{-2}\otxb_{\ell=0}\Big|_{\mathscr{H}^+_{\geq0}\cap\Sigma^*_+}=0
\end{align}

is trivial.

Assume $\mathfrak{G}$ is a pure gauge solution generated by $f_{\ell=0}(v)$ via <Ref> and $\fbar_{\ell=0}(u)$ via <Ref>. The Bondi gauge conditions Olino gauge condition at scri+, round sphere at scri+ imply that $\fbar_{\ell=0}=T_{u,\ell=0}$. The vanishing of $\Olin$ at $\mathscr{H}^+_{\geq0}$ implies

\begin{align}
\end{align}

where $C$ is a constant. The vanishing of $\Omega^{-2}\otxb$ on $\mathscr{H}^+_{\geq0}\cap\Sigma^*_+$ implies that $C=0$. Thus we can only have  $f_{\ell=0}=\fbar_{\ell=0}=T_{u,\ell=0}$, and it is easy to check that this generates trivial gauge solutions.

the reference linearised Kerr family defined in <Ref> is already both future and past Bondi-normalised.

§.§ The $\overline{\mathscr{H}^+}$-normalised gauge

Let $\mathfrak{S}$ be a solution to the linearised Einstein equations on $D^+(\overline{\Sigma})$ with linearised Kerr parameters $(\mathfrak{m},\mathfrak{a})$. We say that $\mathfrak{S}$ satisfies the $\overline{\mathscr{H}^+}$ gauge conditions if  bifurcation gauge conditions are satisfied and

_ℓ=0,1|_ℬ=0,           ^2|_ℬ=0,

|_ℋ^+=-1/2𝔪,           |_ℋ^+=2ϵ^AB∑_m∈{-1,0,1}𝔞_m∂_A Y_m^ℓ=1.

When discussing radiation fields at either of $\mathscr{H}^+_{\geq0}$ or $\mathscr{H}^-_{\leq0}$, we will use an analogous notation to that given in <Ref> to denote the combinations that induce regular fields at $\mathscr{H}^\pm_{\geq0}$. For instance,

\begin{align}
\xlins_{\mathscr{H}^+}:=\Omega\xlin|_{\mathscr{H}^+_{\geq0}},\qquad \xblins_{\mathscr{H}^+}:=\Omega^{-1}\xlin|_{\mathscr{H}^+_{\geq0}},
\end{align}

and so on. When discussing the radiation fields near $\mathcal{B}$ we will explicitly attach the weights in Kruskal variables that are necessary to obtain a regular quantity therein.

§.§ The $\overline{\mathscr{H}^-}$-normalised gauge

Let $\mathfrak{S}$ be a solution to the linearised Einstien equations on $D^-(\overline{\Sigma})$ with linearised Kerr parameters $(\mathfrak{m},\mathfrak{a})$. We say that $\mathfrak{S}$ satisfies the $\overline{\mathscr{H}^-}$ gauge conditions if (<ref>) are satisfied and

\begin{align}
\otxK_{\ell=0,1}\Big|_{\mathcal{B}}=0,\qquad\divo^2\xlinK\Big|_{\mathcal{B}}=0,\qquad
\Olin\Big|_{\overline{\mathscr{H}^-}}=-\frac{1}{2}\mathfrak{m}.%\qquad \bmlinK|_{\overline{\mathscr{H}^-}}=2\slashed{\epsilon}^{AB}\sum_{m\in\{-1,0,1\}}\mathfrak{a}_m\partial_A Y_m^{\ell=1}.
\end{align}

§.§ The global scattering gauge

We now define the gauge we will use to study scattering over the whole of $\overline{\mathscr{M}}$.

Let $\mathfrak{S}$ be a solution to defined over the whole of $\overline{\mathscr{M}}$ (i.e. $\mathfrak{S}$ defines a solution to over $\overline{\mathscr{M}}$) with linearised Kerr parameters $(\mathfrak{m},\mathfrak{a})$. We say $\mathfrak{S}$ is in the global scattering gauge if the conditions (<ref>), (<ref>), (<ref>) at $\mathscr{I}^+$ are satisfied, the conditions at $\mathscr{I}^-$ (<ref>), (<ref>) are satisfied, the conditions bifurcation gauge conditions at $\mathcal{B}$ are satisfied, and the condition

\begin{align}\label{auxiliary gauge conditions global BMS main body}
\end{align}

is satisfied.

Note that if a solution $\mathfrak{S}$ is both future and past Bondi-normalised, the residual gauge freedom in $u,v$ has the form

\begin{align}
u\longrightarrow u+\epsilon(\underline{L}u+\underline{T}),\qquad u\longrightarrow u+\epsilon({L}v+T).
\end{align}

The condition that $\Olino$ must vanish at both $\mathscr{I}^+$ and $\mathscr{I}^-$ forces that

\begin{align}
\underline{L}=-{L}.
\end{align}

The argument leading to <Ref> can be applied to the global scattering gauge and we can conclude the following:

Assume that $\mathfrak{S}$ is a solution to , $\mathfrak{G}$ is a pure gauge solution to and that  both $\mathfrak{S}$ and $\mathfrak{S}+\mathfrak{G}$ are in the global scattering gauge. Then $\mathfrak{G}$ is trivial.

§ STATEMENT OF RESULTS

We first define the spaces of scattering states in <Ref>. The precise statement of the main results of this paper follows in <Ref>.

§.§ Spaces of scattering states

Let $\mathfrak{D}=(\,\glinhs, \tr\glins$, $\Olinos$, $\bmlins$, $\glinhs'$, $\tr\glins'$, $\Olinos'$, $\bmlins'\,)$ be a smooth initial data set on ${\Sigma^*_+}$ for the system in the sense of <Ref>. Define

\begin{align}
\left\|\mathfrak{D}\right\|_{\mathcal{E}^{T}_{{\Sigma^*_+}}}=\left\|(\Psilin|_{{\Sigma^*_+}},\slashednabla_{n_{{\Sigma^*_+}}}\Psilin|_{{\Sigma^*_+}})\right\|_{\mathcal{E}^{T,RW}_{{\Sigma^*_+}}}.
\end{align}

Define the space $\mathcal{E}^{T}_{{\Sigma^*_+}}$ to be the completion of data $\mathfrak{D}$ which is asymptotically flat to order $(2,\infty)$, and which satisfies the conditions of the $\Sigma^*_+$ gauge, under the norm $\|\;\|_{\mathcal{E}^{T}_{{\Sigma^*_+}}}$. The space $\mathcal{E}^{T}_{\overline{\Sigma},+}$ is defined similarly (with respect to the $\overline{\Sigma}_+$-gauge).

Assume $\mathfrak{D}$ is an initial data set on ${\Sigma^*_+}$ for which is in the ${\Sigma^*_+}$-gauge and with $\|\mathfrak{D}\|_{\mathcal{E}^{T,RW}_{{\Sigma^*_+}}}=0$. Then $\mathfrak{D}$ is trivial.

The vanishing of $\|\mathfrak{D}\|_{\mathcal{E}^{T,RW}_{{\Sigma^*_+}}}$ implies that $\alin=\ablin=0$ on ${\Sigma^*_+}$ (see Remark 4.1.1 and Proposition 4.2.1 in <cit.>). Thus $\mathfrak{D}$ generates a pure gauge solution. The conditions defining the ${\Sigma^*_+}$-gauge imply that the generators of this gauge solution must all vanish.

Note that $\left\|(\Psilin|_{{\Sigma^*_+}},\slashednabla_{n_{{\Sigma^*_+}}}\Psilin|_{{\Sigma^*_+}})\right\|_{\mathcal{E}^{T,RW}_{{\Sigma^*_+}}}$ and $\left\|(\Psilinb|_{{\Sigma^*_+}},\slashednabla_{n_{{\Sigma^*_+}}}\Psilinb|_{{\Sigma^*_+}})\right\|_{\mathcal{E}^{T,RW}_{{\Sigma^*_+}}}$ are not equal, the difference between them being related to the $-$ sign discrepancy in the contribution of $\slin$ between the expressions   expression for Psilin and expression for Psilinb for $\Psilin$ and $\Psilinb$. However, the norms defined on $\Sigma$, $\overline{\Sigma}$ satisfy

\begin{align}
&\left\|(\Psilin|_{\Sigma},\slashednabla_{n_{\Sigma}}\Psilin|_{\Sigma})\right\|_{\mathcal{E}^{T,RW}_{\Sigma}}=\left\|(\Psilinb|_{\Sigma},\slashednabla_{n_{\Sigma}}\Psilinb|_{\Sigma})\right\|_{\mathcal{E}^{T,RW}_{\Sigma}},\\ &\left\|(\Psilin|_{\overline{\Sigma}},\slashednabla_{n_{\overline{\Sigma}}}\Psilin|_{\overline{\Sigma}})\right\|_{\mathcal{E}^{T,RW}_{\overline{\Sigma}}}=\left\|(\Psilinb|_{\overline{\Sigma}},\slashednabla_{n_{\overline{\Sigma}}}\Psilinb|_{\overline{\Sigma}})\right\|_{\mathcal{E}^{T,RW}_{\overline{\Sigma}}}.
\end{align}

Indeed, since

\begin{align}
\begin{split}
\int_{S^2}\dw|\Psilin|^2&=\|2\mathring{\fancydstar_2}\mathring{\fancydstar_1}(\rlin,\slin)\|_{L^2(S^2)}^2+\|6M(r\Omega\xlin-r\Omega\xblin)\|_{L^2(S^2)}^2\\&+\int_{S^2}\dw\,24M\mathring{\fancydstar_2}\mathring{\fancydstar_1}(r^3\rlin,r^3\slin)\times(r\Omega\xlin-r\Omega\xblin)\\&
=\|2\mathring{\fancydstar_2}\mathring{\fancydstar_1}(\rlin,\slin)\|_{L^2(S^2)}^2+\|6M(r\Omega\xlin-r\Omega\xblin)\|_{L^2(S^2)}^2\\&+\int_{S^2}\dw 24M\left[\rlin\times\divo^2(r\Omega\xlin-r\Omega\xblin)+\slin\times \curlo\divo(r\Omega\xlin-r\Omega\xblin)\right],
\end{split}
\end{align}

\begin{align}
\begin{split}
\int_{S^2}\dw|\Psilinb|^2&=\|2\mathring{\fancydstar_2}\mathring{\fancydstar_1}(\rlin,\slin)\|_{L^2(S^2)}^2+\|6M(r\Omega\xlin-r\Omega\xblin)\|_{L^2(S^2)}^2\\&+\int_{S^2}\dw 12M\left[\rlin\times\divo^2(r\Omega\xlin-r\Omega\xblin)-\slin\times \curlo\divo(r\Omega\xlin-r\Omega\xblin)\right],
\end{split}
\end{align}

The Codazzi equations elliptic equation 1, elliptic equation 2, Bianchi-0* and Bianchi+0* imply

\begin{align}
\curlo\,\divo(r\Omega\xlin-r\Omega\xblin)=2\partial_t r^3\slin.
\end{align}

\begin{align}
\begin{split}
\int_{S^2}\dw|\mathring{\slashednabla}\Psilin|^2&=\|2\mathring{\slashednabla}\mathring{\fancydstar_2}\mathring{\fancydstar_1}(\rlin,\slin)\|_{L^2(S^2)}^2+\|6M\mathring{\slashednabla}(r\Omega\xlin-r\Omega\xblin)\|_{L^2(S^2)}^2\\&
\end{split}
\end{align}

The cross term evaluates to

\begin{align}
\begin{split}
&-\int_{S^2}\dw 24M\left[(\mathring{\slashed{\Delta}}+4)\rlin\times\divo^2(r\Omega\xlin-r\Omega\xblin)+(\mathring{\slashed{\Delta}}+4)\slin\times \curlo\divo(r\Omega\xlin-r\Omega\xblin)\right],
\end{split}
\end{align}

and the term containing $\slin$ above becomes

\begin{align}
-24M\int_{S^2}\dw(\mathring{\slashed{\Delta}}+4)\slin\cdot 2\partial_t\slin.
\end{align}

The contribution of the cross terms in $\left\|(\Psilin|_{\overline{\Sigma}},\slashednabla_{n_{\overline{\Sigma}}}\Psilin|_{\overline{\Sigma}})\right\|_{\mathcal{E}^{T,RW}_{\overline{\Sigma}}}$ containing $\slin$ is

\begin{align}\label{cross term slin 1}
\begin{split}
48M\int_{\overline{\Sigma}}dr\dw\, &\frac{3\Omega^2+1}{r^2}\,r^3\slin \cdot \partial_t r^3\slin-\frac{1}{r^2}(\mathring{\slashed{\Delta}}+4)r^3\slin\cdot \partial_tr^3\slin\\&+\Omega^2\partial_{r}r^3\slin\cdot\partial_{r}\partial_tr^3\slin+\Omega^{-2}\partial_tr^3\slin\cdot \partial_t^2r^3\slin
\end{split}
\end{align}

Note that since $\Psilin-\Psilinb=4\mathring{\fancydstar_2}\mathring{\fancydstar_1}(0,r^3\slin)$, it is clear that $\slin$ satisfies the following equation

\begin{align}\label{wave equation for slin}
\left[\partial_t^2-\partial_{r_*}^2-\frac{\Omega^2}{r^2}\mathring{\slashed{\Delta}}-6M\frac{\Omega^2}{r^3}\right]r^3\slin=0.
\end{align}

For an asymptotically flat initial data set $\mathfrak{D}$ on $\overline{\Sigma}$, we may evaluate the contribution of cross term slin 1 integrating by parts in $r_*$ and using wave equation for slin to get

\begin{align}
\int_{\overline{\Sigma}}dr\dw\,\partial_r(|\partial_tr^3\slin|^2),
\end{align}

which produces vanishing terms at $\mathcal{B}$ (since $\partial_t$ vanishes there) and in the limit towards $i^0$ since $\mathfrak{D}$ is asymptotically flat.

Define the space of future scattering states $\mathcal{E}^{T}_{\mathscr{H}^+_{\geq0}}$ on $\mathscr{H}^+_{\geq0}$ to be the completion of $\Gamma_c^2(\mathscr{H}^+_{\geq0})$ under the norm

\begin{align}\label{scattering norm on H+}
\begin{split}
\|X\|^2_{\mathcal{E}^{T}_{\mathscr{H}^+_{\geq0}}}=&\left\|\mathcal{A}_2(\mathcal{A}_2-2)X\right\|^2_{L^2(\mathscr{H}^+_{\geq0})}+\left\|6M\partial_v X\right\|^2_{L^2(\mathscr{H}^+_{\geq0})}\\&+\int_{S^2_{\infty,0}}\sin\theta d\theta d\phi \left(\left|\mathring{\slashed{\Delta}}X\right|^2+6\left|\mathring{\slashednabla}X\right|^2+8\Big|X\Big|^2\right).
\end{split}
\end{align}

Similarly, define the space of future scattering states $\mathcal{E}^{T}_{\overline{\mathscr{H}^+}}$ via the norm

\begin{align}
&\|X\|^2_{\mathcal{E}^{T}_{\overline{\mathscr{H}^+}}}=\left\|\mathcal{A}_2(\mathcal{A}_2-2)X\right\|^2_{L^2(\overline{\mathscr{H}^+})}+\left\|6M\partial_v X\right\|^2_{L^2(\overline{\mathscr{H}^+})}.
\end{align}

Define the space of future scattering states $\mathcal{E}^{T}_{\mathscr{I}^+}$ on $\mathscr{I}^+$ to be the completion of $\Gamma_c^2(\mathscr{I}^+)$ under the norm

\begin{align}\label{scattering norm at scri+}
\|\underline{X}\|^2_{\mathcal{E}^{T}_{\mathscr{I}^+}}=&\left\|\mathcal{A}_2(\mathcal{A}_2-2)\underline{X}\right\|^2_{L^2(\mathscr{I}^+)}+\left\|6M\partial_u\underline{X}\right\|^2_{L^2(\mathscr{I}^+)}.
\end{align}

Define the space of future scattering states $\mathcal{E}^{T}_{{\overline{\mathscr{H}^-}}}$ on ${\overline{\mathscr{H}^-}}$ to be the completion of $\Gamma_c^2({\overline{\mathscr{H}^-}})$ under the norm

\begin{align}
\|\underline{X}\|^2_{\mathcal{E}^{T}_{\overline{\mathscr{H}^-}}}=\left\|\mathcal{A}_2(\mathcal{A}_2-2)\underline{X}\right\|^2_{L^2({\overline{\mathscr{H}^-}})}+\left\|6M\partial_u\underline{X}\right\|^2_{L^2({\overline{\mathscr{H}^-}})},
\end{align}

Similarly, define the space of future scattering states  $\mathcal{E}^{T}_{\overline{\mathscr{H}^-}}$ to be the completion of $\Gamma_c^2(\overline{\mathscr{H}^-})$ under the norm

\begin{align}
\end{align}

Define the space of future scattering states $\mathcal{E}^{T}_{\mathscr{I}^-}$ on $\mathscr{I}^-$ to be the completion of $\Gamma_c^2(\mathscr{I}^-)$ under the norm

\begin{align}\label{scattering norm at scri+}
\|{X}\|^2_{\mathcal{E}^{T}_{\mathscr{I}^-}}=&\left\|\mathcal{A}_2(\mathcal{A}_2-2)\partial_v{X}\right\|^2_{L^2(\mathscr{I}^-)}+\left\|6M\partial_v^2 {X}\right\|^2_{L^2(\mathscr{I}^-)}.
\end{align}

§.§ Main theorems

We now present the full statements of Theorems II, III and Corollary II. The statement of Theorem II is contained in Theorems IIA, IIB and IIC below.

§.§.§ Theorem IIA: Forward scattering to $\mathscr{I}^+$ and $\mathscr{H}^+_{\geq0}$, $\overline{\mathscr{H}^+}$

For a solution $\mathfrak{S}$ to the system defined on $J^+(\Sigma^*)$ arising from asymptotically flat initial data on $\Sigma^*_+$, we denote by $\mathbb{E}_{\Sigma^*_+\rightarrow \mathscr{I}^+}[\mathfrak{S}]$ the $L^2(\Sigma^*_+)$ quantity

\begin{align}
\begin{split}
\mathbb{E}_{{\Sigma^*_+}\rightarrow \mathscr{I}^+}[\mathfrak{S}]&:=\mathbb{F}^2_{{\Sigma^*_+}}[\Psilin]+\int_{{\Sigma^*_+}}dr\sin\theta d\theta d\phi \left[r^{\frac{13}{2}}|\Omega\plin|^2+r^{\frac{9}{2}}|\Omega^2\alin|^2\right]\\& +\int_{{\Sigma^*_+}}\Omega^2du\sin\theta d\theta d\phi\Bigg[\left|\frac{1}{\Omega}\nablagml\left(\frac{1}{\Omega}\nablagml(r^2\Omega\xlin)\right)\right|^2+\left|\frac{1}{\Omega}\nablagml(r^2\Omega\xlin)\right|^2+\frac{1}{r^2}|r^2\Omega\xlin|^2\Bigg].
\end{split}
\end{align}

For a solution $\mathfrak{S}$ to the system defined on $J^+(\Sigma^*_+)$ arising from data on $\Sigma^*_+$ which is constructed from smooth, compactly supported seed data via <Ref>, we denote by $\mathbb{E}_{\Sigma^*_+\rightarrow \mathscr{I}^+\cup \mathscr{H}^+_{\geq0}}[\mathfrak{S}]$ the quantity

\begin{align}
\begin{split}
\mathbb{E}_{\Sigma^*_+\rightarrow \mathscr{I}^+\cup \mathscr{H}^+_{\geq0}}[\mathfrak{S}]:=\mathbb{E}_{{\Sigma^*_+}\rightarrow \mathscr{I}^+}[\mathfrak{S}]+&\int_{\Sigma^*_+}dr\dw\,|r^3\Omega^{-1}\pblin|^2+|r\Omega^{-1}\ablin|^2+\int_{\Sigma^*_+\cap\mathscr{H}^+_{\geq0}}\dw\,\left|\Omega^{-1}\pblin\right|^2.
\end{split}
\end{align}

[Theorem IIA]
forward scattering full system thmTheorem IIA
Let $\mathfrak{D}$ be an initial data set for the linearised Einstein equations  on $\Sigma^*_+$ in the sense of <Ref>, and assume that $\mathfrak{D}$ is asymptotically flat in the sense of <Ref> and that it satisfies the conditions of the $\Sigma^*_+$-gauge. Then the solution $\mathfrak{S}_{\Sigma^*_+}$ to the system arising from $\mathfrak{D}$ via <Ref> gives rise to smooth radiation fields $\xblins_{\mathscr{I}^+}\in \Gamma^{(2)}(\mathscr{I}^+)\cap \mathcal{E}^T_{\mathscr{I}^+}$, $\xlins_{\mathscr{H}^+}\in \Gamma^{(2)}(\mathscr{H}^+_{\geq0})\cap \mathcal{E}^T_{\mathscr{H}^+_{\geq0}}$ via

\begin{align}\label{radiation thm scri+}
\xblins_{\mathscr{I}^+}(u,\theta^A)=\lim_{v\longrightarrow\infty}r\xblin(u,v,\theta^A),\qquad\qquad\xlins_{\mathscr{H}^+}(v,\theta^A)=\Omega\xlin|_{\mathscr{H}^+_{\geq0}},
\end{align}

and we have

\begin{align}\label{radiation thm H+}
\|\,\mathfrak{D}\,\|^2_{\mathcal{E}^T_{{\Sigma^*_+}}}=\|\,\xblins_{\mathscr{I}^+}\,\|^2_{\mathcal{E}^T_{\mathscr{I}^+}}+\|\,\xlins_{\mathscr{H}^+}\,\|^2_{\mathcal{E}^T_{\mathscr{H}^+_{\geq0}}}.
\end{align}

Furthermore, there exists a unique pair of pure gauge solutions $(\mathfrak{G}_{\mathscr{I}^+},\underline{\mathfrak{G}}_{\mathscr{H}^+_{\geq0}})$, such that $\mathfrak{S}_{\Sigma^*_+}+\mathfrak{G}_{\mathscr{I}^+}$ is future Bondi-normalised (as in <Ref>) and satisfies the horizon gauge conditions (<ref>), $\mathfrak{S}_{\Sigma^*_+}+\underline{\mathfrak{G}}_{\mathscr{H}^+_{\geq0}}$ is in the $\mathscr{H}^+_{\geq0}$-gauge (as in <Ref>), and $\mathfrak{S}_{\Sigma^*_+}+\mathfrak{G}_{\mathscr{I}^+}+\underline{\mathfrak{G}}_{\mathscr{H}^+_{\geq0}}$ satisfies both the $\mathscr{I}^+$ and $\mathscr{H}^+_{\geq0}$ gauge conditions. The radiation fields $\xblins_{\mathscr{I}^+}$ and $\xlins_{\mathscr{H}^+}$ of $\mathscr{S}_{\Sigma^*_+}$ are invariant under the the addition of $\mathfrak{G}_{\mathscr{I}^+}$ and $\underline{\mathfrak{G}}_{\mathscr{H}^+_{\geq0}}$, and the gauge solutions $\mathfrak{G}_{\mathscr{I}^+}$ and $\underline{\mathfrak{G}}_{\mathscr{H}^+_{\geq0}}$ are both bounded in terms of $\mathfrak{D}$ via the estimate as follows: the generator of $\mathfrak{G}_{\mathscr{I}^+}$, constructed via <Ref>, is controlled by the estimate

\begin{align}
\|\partial_u\outwardsgaugefunction\|^2_{L^2(\mathbb{R}\times S^2)}\lesssim \mathbb{E}_{\Sigma^*_+\rightarrow \mathscr{I}^+}[\mathfrak{S}].
\end{align}

The generators of the gauge solution $\underline{\mathfrak{G}}_{\mathscr{H}^+_{\geq0}}$, constructed via <Ref>, <Ref>, are controlled by the estimate

\begin{align}
\|\,\inwardsgaugefunction\,\|^2_{L^\infty_v H^2(S^2)}+\|\partial_v\inwardsgaugefunction\|^2_{L^2(\mathbb{R}\times S^2)}+\|(\partial_v\mathfrak{q}_1,\partial_v\mathfrak{q}_2)\|^2_{L^2(\mathbb{R}\times S^2)}\lesssim \mathbb{E}_{\Sigma^*_+\rightarrow \mathscr{I}^+\cup \mathscr{H}^+_{\geq0}}[\mathfrak{S}].
\end{align}

The above construction extends to a unitary map

\begin{align}\label{forward map}
\mathscr{F}^+:\mathcal{E}^T_{{\Sigma^*_+}}\longrightarrow \mathcal{E}^T_{\mathscr{I}^+}\oplus \mathcal{E}^T_{\mathscr{H}^+_{\geq0}}.
\end{align}

A similar statement applies to forward evolution from initial data on $\overline{\Sigma}$ and we similarly have the unitary maps

\begin{align}
&\mathscr{F}^+:\mathcal{E}^T_{\overline{\Sigma},+}\longrightarrow \mathcal{E}^T_{\mathscr{I}^+}\oplus \mathcal{E}^T_{\overline{\mathscr{H}^+}}.
\end{align}

§.§.§ Theorem IIB: Backwards scattering from $\mathscr{I}^+$ and  $\mathscr{H}^+_{\geq0}$, $\overline{\mathscr{H}^+}$

Let $\mathfrak{S}$ be a solution to which is both future horizon and Bondi-normalised, such that $\mathfrak{S}$ defines smooth, compactly supported elements of $\Gamma^{(2)}(\mathbb{R}\times S^2)$ via

\begin{align}
\xblins_{\mathscr{I}^+}:=\lim_{v\longrightarrow\infty}r\xblin(u,v,\theta^A),\qquad\qquad \xlins_{\mathscr{H}^+}:=\Omega\xlin|_{\mathscr{H}^+_{\geq0}}.
\end{align}

We denote by $\mathbb{E}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+\rightarrow \Sigma^*_+}[\mathfrak{S}]$ the quantity given by

\begin{align}
\begin{split}
\mathbb{E}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+\rightarrow \Sigma^*_+}[\mathfrak{S}]&:=e^{\frac{4}{M}v_+}\sum_{i=0,1}\left[\left\|\partial_v^i\xlins_{\mathscr{H}^+}\right\|^2_{\mathcal{E}^T_{\mathscr{H}^+_{\geq0}}}+\sum_{|\gamma|\leq 2}\left\|\mathring{\slashednabla}^\gamma \partial_u^i \xblins_{\mathscr{I}^+}\right\|^2_{\mathcal{E}^T_{\mathscr{I}^+}}\right]\\
\end{split}
\end{align}

where $v_+$ denotes the end point of the support of $\glinhs_{\mathscr{H}^+}$ at $\mathscr{H}^+_{\geq0}$.
If $\mathfrak{S}$ is defined over $D^+(\overline{\Sigma})$ we denote by $\mathbb{E}_{\overline{\mathscr{H}^+}\cup\mathscr{I}^+\rightarrow \overline{\Sigma}}[\mathfrak{S}]$ the quantity

\begin{align}
\begin{split}
\mathbb{E}_{\overline{\mathscr{H}^+}\cup\mathscr{I}^+\rightarrow \overline{\Sigma}}[\mathfrak{S}]:=&\mathbb{E}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+\rightarrow \Sigma^*_+}[\mathfrak{S}]+\int_{\overline{\mathscr{H}^+}\cap\{V\leq1\}}d\bar{v}\left[|V^{-1}\xlins_{\mathscr{H}^+}|^2+|\slashednabla_VV^{-1}\xlins_{\mathscr{H}^+}|^2+|\slashednabla_V^2V^{-1}\xlins_{\mathscr{H}^+}|^2+|\mathring{\slashednabla}\Psilinb|^2\right].
\end{split}
\end{align}

See <Ref> and <Ref> for the definitions of the quantities mentioned above.

[Theorem IIB]
backwards scattering full system thmTheorem IIB
Let $\xblins_{\mathscr{I}^+}\in\Gamma^{(2)}_c(\mathscr{I}^+)$, $\xlins_{\mathscr{H}^+}\in\Gamma^{(2)}_c(\mathscr{H}^+_{\geq0})$. There exists a unique solution $\mathfrak{S}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+}$ to the linearised Einstein equations which realises $(\xlins_{\mathscr{I}^+}$, $\xlins_{\mathscr{H}^+})$ as its radiation fields on $\mathscr{I}^+$, $\mathscr{H}^+_{\geq0}$ respectively,

\begin{align}
\xblins_{\mathscr{I}^+}(u,\theta^A)=\lim_{v\longrightarrow\infty}r\xblin(u,v,\theta^A),\qquad\qquad\xlins_{\mathscr{H}^+}(v,\theta^A)=\Omega\xlin|_{\mathscr{H}^+_{\geq0}}.
\end{align}

The solution $\mathfrak{S}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+}$ induces asymptotically flat initial data on $\Sigma^*_+$ and satisfies both the $\mathscr{I}^+$ and the $\mathscr{H}^+_{\geq0}$ gauge conditions. Furthermore, there exists a unique pure gauge solution $\mathfrak{G}_{\Sigma^*_+}$ such that the data induced by $\mathfrak{S}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+}+\mathfrak{G}_{\Sigma^*_+}$ on $\Sigma^*_+$ is asymptotically flat to order $(1,\infty)$ and satisfies the conditions of the $\Sigma^*_+$ gauge. The generators of the gauge solution $\mathfrak{G}_{\Sigma^*_+}$ are bounded in terms of the scattering data $\xblins_{\mathscr{I}^+}$, $\xlins_{\mathscr{H}^+}$ via the estimate

\begin{align}
\|\mathfrak{f}_{\Sigma^*_+}\|^2_{H^2_rL^2(S^2)}+ \|\underline{\mathfrak{f}}_{\Sigma^*_+}\|^2_{L^2(\Sigma^*_+)}+\|(\mathfrak{q}_{\Sigma^*_+,1},\mathfrak{q}_{\Sigma^*_+,2})\|^2_{L^2(\Sigma^*_+)}\lesssim \mathbb{E}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+\rightarrow \Sigma^*_+}[\mathfrak{S}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+}].
\end{align}

Finally, the initial data $\mathfrak{D}$ on ${\Sigma^*_+}$ induced by $\mathfrak{S}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+}+\mathfrak{G}_{\Sigma^*_+}$ is in $\mathcal{E}^T_{{\Sigma^*_+}}$ and we have

\begin{align}
\|\,\mathfrak{D}[\mathfrak{S}_{\mathscr{H}^+_{\geq0}\cup\mathscr{I}^+}+\mathfrak{G}_{\Sigma^*_+}]\,\|^2_{\mathcal{E}^T_{{\Sigma^*_+}}}=\|\,\xblins_{\mathscr{I}^+}\,\|^2_{\mathcal{E}^T_{\mathscr{I}^+}}+\|\,\xlins_{\mathscr{H}^+}\,\|^2_{\mathcal{E}^T_{\mathscr{H}^+_{\geq0}}}.
\end{align}

The above defines a map

\begin{align}
\mathscr{B}^-:\mathcal{E}^T_{\mathscr{I}^-}\oplus \mathcal{E}^T_{\mathscr{H}^+_{\geq0}}\longrightarrow\mathcal{E}^T_{{\Sigma^*_+}}
\end{align}

which inverts the map $\mathscr{F}^+$ of <ref>:

\begin{align}
\mathscr{B}^-\circ\mathscr{F}^+=Id_{\mathcal{E}^T_{{\Sigma^*_+}},\qquad} \mathscr{F}^+\circ\mathscr{B}^-=Id_{\mathcal{E}^T_{\mathscr{I}^+}\oplus\mathcal{E}^T_{\mathscr{H}^+_{\geq0}}}.
\end{align}

The same conclusions apply replacing $\mathscr{H}^+_{\geq0}$ with  $\overline{\mathscr{H}^+}$, ${\Sigma^*_+}$ with $\overline{\Sigma}$, and the $\Sigma^*_+$-gauge with the $\overline{\Sigma}_+$-gauge.

§.§.§ Theorem IIC: Scattering on $J^-(\overline{\Sigma})$

Let $\mathfrak{S}$ be a solution to which is both past horizon and Bondi-normalised, such that $\mathfrak{S}$ defines smooth, compactly supported elements of $\Gamma^{(2)}(\mathbb{R}\times S^2)$ via

\begin{align}