meaning and derivation of pure gauge solutions.

####  6.1.1 Coordinate pure gauge solutions

Consider the smooth scalar functions $(h_{1},\Omega^{2}_{M}h_{2})$ and
$\mathbb{S}^{2}_{u,v}$ vector field $h$ on
$\mathcal{M}\setminus\mathcal{H}^{+}$ which extend regularly to
$\mathcal{H}^{+}$ and satisfy the system of ODEs242424The notation
$h^{\theta^{A}}$ denotes the $\theta^{A}$-(coordinate) component of $h$. The
reader should note the difference with the frame index notation
$h^{A}=h(\not{g}^{AB}e_{B}^{\text{as}})$.

$\displaystyle\operatorname{\not{\nabla}}_{4}(\Omega^{2}_{M}h_{2})-\frac{2M}{r^{2}}(\Omega^{2}_{M}h_{2})$
$\displaystyle=0\,,$ (80)
$\displaystyle\operatorname{\not{\nabla}}_{4}^{2}h_{1}+\frac{2M}{r^{2}}(\operatorname{\not{\nabla}}_{4}h_{1})+\frac{4M}{r^{3}}(\Omega^{2}_{M}h_{2}-\Omega^{2}_{M}h_{1})$
$\displaystyle=0\,,$ (81)
$\displaystyle\operatorname{\not{\nabla}}_{4}^{2}h^{\theta^{A}}+2\left(\frac{r-3M}{r^{2}}\right)\operatorname{\not{\nabla}}_{4}h^{\theta^{A}}$
$\displaystyle=0\,.$ (82)

######  Definition 6.5 (Coordinate pure gauge solutions).

We define _coordinate pure gauge solutions_ as the family of solutions
$\mathsf{CG}(h_{1},h_{2},h)$ to the linearised system of equations with
linearised frame coefficients and induced metric

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}$
$\displaystyle=-\operatorname{\not{\nabla}}(\Omega^{2}_{M}h_{2})+\frac{1}{2}\,(\operatorname{\not{\nabla}}_{4}h)_{\flat}-\frac{\Omega^{2}_{M}}{2\,r}\,h_{\flat}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}$
$\displaystyle=-\operatorname{\not{\nabla}}h_{1}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}$
$\displaystyle=-\operatorname{\not{\nabla}}_{3}(\Omega^{2}_{M}h_{2})-\operatorname{\not{\nabla}}_{4}h_{1}-\frac{2M}{r^{2}}\,h_{1}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}$
$\displaystyle=-\operatorname{\not{\nabla}}_{3}h_{1}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}$
$\displaystyle=-(\operatorname{\not{\nabla}}_{3}h)_{\flat}-\frac{1}{r}\,h_{\flat}\,,$
$\displaystyle(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})$
$\displaystyle=\frac{4\Omega_{M}^{2}}{r}(h_{1}-h_{2})+2\,(\not{\textup{div}}\,h)\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}$
$\displaystyle=-4\,(\not{\mathcal{D}}_{2}^{\star}\,h)_{\flat}\,.$

The linearised connection coefficients and curvature components of coordinate
pure gauge solutions can be determined through the linearised system of
equations. We note that

$\displaystyle(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})$
$\displaystyle=-2\,\frac{\Omega_{M}^{2}(r-4M)}{r^{3}}(h_{1}-h_{2})+\frac{2\,\Omega_{M}^{2}}{r}\operatorname{\not{\nabla}}_{4}h_{1}+\not{\textup{div}}(\operatorname{\not{\nabla}}_{4}\,h)_{\flat}-\frac{\Omega_{M}^{2}}{r}\,(\not{\textup{div}}\,h)\,,$
$\displaystyle(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})$
$\displaystyle=\frac{2\,\Omega_{M}^{2}}{r^{2}}\,(h_{1}-h_{2})+\frac{2}{r}\,(\operatorname{\not{\nabla}}_{4}h_{1})\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}$
$\displaystyle=-\operatorname{\not{\nabla}}_{3}\operatorname{\not{\nabla}}_{4}h_{1}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})$
$\displaystyle=\operatorname{\not{\nabla}}_{4}(\not{\textup{curl}}\,h)\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})$
$\displaystyle=0\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}$
$\displaystyle=-2\,\not{\mathcal{D}}_{2}^{\star}(\operatorname{\not{\nabla}}_{4}h)_{\flat}+\frac{2\,\Omega^{2}_{M}}{r}(\not{\mathcal{D}}_{2}^{\star}\,h)_{\flat}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}$
$\displaystyle=-\operatorname{\not{\nabla}}_{3}(\not{\mathcal{D}}_{2}^{\star}\,h)_{\flat}$

and

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$
$\displaystyle=0\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$
$\displaystyle=0\,.$

On the event horizon $\mathcal{H}^{+}$, we have

$\displaystyle(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})(\infty,v)=$
$\displaystyle\,\frac{1}{2}\operatorname{\not{\nabla}}_{3}(\not{\textup{div}}(\operatorname{\not{\nabla}}_{4}\,h)_{\flat})(\infty,v)-\frac{1}{4M}\,\not{\textup{div}}(\operatorname{\not{\nabla}}_{4}\,h)_{\flat}(\infty,v)+\frac{1}{8M^{2}}\,(\not{\textup{div}}\,h)(\infty,v)$
$\displaystyle-\frac{3}{2M^{3}}\,(\Omega_{M}^{2}h_{2})(\infty,v)\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}(\infty,v)=$
$\displaystyle\,-\frac{1}{4\,M}\,\not{\textup{div}}(\operatorname{\not{\nabla}}_{4}h)_{\flat}(\infty,v)-\frac{3}{8\,M^{3}}\,(\Omega_{M}^{2}h_{2})(\infty,v)\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}(\infty,v)=$
$\displaystyle\,-\frac{1}{4\,M}\,\not{\textup{curl}}(\operatorname{\not{\nabla}}_{4}h)_{\flat}(\infty,v)\,.$

######  Remark 6.6.

The fact that coordinate pure gauge solutions, as defined in Definition 6.5,
are indeed solutions to the linearised system of equations is a check left to
the reader.

####  6.1.2 Frame pure gauge solutions

Consider the $\mathbb{S}^{2}_{u,v}$ vector field $k$ on
$\mathcal{M}\setminus\mathcal{H}^{+}$ which extends regularly to
$\mathcal{H}^{+}$ and satisfies the system of ODEs

$\operatorname{\not{\nabla}}_{4}k^{A}+\frac{2M}{r^{2}}\,k^{A}=0\,.$ (83)

######  Definition 6.7 (Frame pure gauge solutions).

We define _frame pure gauge solutions_ as the family of solutions
$\mathsf{FG}(k)$ to the linearised system of equations such that

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}$
$\displaystyle=k_{\flat}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}$
$\displaystyle=\frac{1}{2}\,k_{\flat}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}$
$\displaystyle=\frac{\Omega^{2}_{M}}{2\,r}\,k_{\flat}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}$
$\displaystyle=\frac{r-4M}{2\,r^{2}}\,k_{\flat}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}$
$\displaystyle=\frac{1}{2}\,(\operatorname{\not{\nabla}}_{3}k)_{\flat}-\frac{1}{2\,r}\,k_{\flat}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}$
$\displaystyle=-(\not{\mathcal{D}}_{2}^{\star}\,k)_{\flat}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\operatorname{\underline{\chi}}})$
$\displaystyle=\not{\textup{div}}\,k\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})$
$\displaystyle=\not{\textup{curl}}\,k\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}$
$\displaystyle=-\frac{\Omega_{M}^{2}}{2\,r}\,\not{\textup{div}}\,k\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}$
$\displaystyle=-\not{\textup{div}}\,(\not{\mathcal{D}}_{2}^{\star}\,k)_{\flat}-\frac{1}{2}\operatorname{\not{\nabla}}\not{\textup{div}}\,k+\frac{1}{2}{}^{\star}\operatorname{\not{\nabla}}\not{\textup{curl}}\,k-\frac{r-3M}{r^{3}}\,k_{\flat}$

and all the remaining linearised frame coefficients, connection coefficients
and curvature components identically vanishing.

######  Remark 6.8.

For any frame pure gauge solution, we have

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$
$\displaystyle=0\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$
$\displaystyle=0\,,$
$\displaystyle(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})(\infty,v)$
$\displaystyle=0\,.$

######  Remark 6.9.

The fact that frame pure gauge solutions, as defined in Definition 6.7, are
indeed solutions to the linearised system of equations is a check left to the
reader.

####  6.1.3 Achieving the initial-data gauge normalisation

We prove the following key proposition.

######  Proposition 6.10.

For any solution $\mathfrak{S}$ to the linearised system of equations arising
from smooth, asymptotically flat (to sufficiently high order) seed initial
data, there exist a coordinate pure gauge solution
$\mathsf{CG}(h_{1},h_{2},h)$ and a frame pure gauge solution $\mathsf{FG}(k)$
such that the solution

$\mathfrak{S}_{N}=\mathfrak{S}+\mathsf{CG}(h_{1},h_{2},h)+\mathsf{FG}(k)$ (84)

is an initial-data normalised solution to the linearised system of equations.
The solution $\mathfrak{S}_{N}$ arises from smooth, asymptotically flat seed
initial data.

###### Proof.

We consider a solution $\mathfrak{S}$ to the linearised system of equations.
To achieve the initial-data normalisation, we need to suitably construct the
quantities $h_{1}$, $h_{2}$, $h$ and $k$ on $\mathcal{M}$. We note that the
ODEs (80), (81), (82) and (83) only allow to choose

$h_{1}\,,\,\Omega^{2}_{M}h_{2}\,,\,h^{\theta^{A}}\,,\,\operatorname{\not{\nabla}}_{4}h_{1}\,,\,\operatorname{\not{\nabla}}_{4}h^{\theta^{A}}\,,\,k^{A}$
(85)

freely on $\underline{C}_{v_{0}}$. Our proof constructs the quantities (85) on
$\underline{C}_{v_{0}}$. The fact that the quantities constructed achieve (84)
is a check left to the reader.

We divide the construction into three parts, each corresponding to the
construction of the $\ell=0$, $\ell=1$ and $\ell\geq 2$-spherical modes of the
quantities (85) respectively. The quantities $h$,
$\operatorname{\not{\nabla}}_{4}h$ and $k$ will be treated as
$\mathbb{S}^{2}_{u,v}$ one-tensors (denoted $h_{\flat}$,
$\operatorname{\not{\nabla}}_{4}h_{\flat}$ and $k_{\flat}$), and thus are
supported on $\ell\geq 1$-spherical modes.

$\ell=0$-spherical modes: We choose $(h_{1})|_{\ell=0}$,
$(\Omega^{2}_{M}h_{2})|_{\ell=0}$ and
$(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=0}$ on $\underline{C}_{v_{0}}$
as solutions to the system of ODEs in the $e_{3}^{\text{as}}$-direction

$\displaystyle\operatorname{\not{\nabla}}_{3}(h_{1})|_{\ell=0}=$
$\displaystyle\,(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3})_{\mathfrak{S}}|_{\ell=0}\,,$
$\displaystyle\operatorname{\not{\nabla}}_{3}(\Omega^{2}_{M}h_{2})|_{\ell=0}=$
$\displaystyle\,-(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=0}-\frac{2M}{r^{2}}\,(h_{1})|_{\ell=0}+(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=0}$
$\displaystyle+4M^{2}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}_{\mathfrak{S}}|_{\ell=0}-\frac{3}{8\,M^{3}}\,(\Omega_{M}^{2}h_{2})|_{\ell=0})(\infty,v_{0})\,,$
$\displaystyle\operatorname{\not{\nabla}}_{3}(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=0}=$
$\displaystyle\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}})_{\mathfrak{S}}|_{\ell=0}$

with initial conditions on $\mathbb{S}^{2}_{\infty,v_{0}}$ such that

$\displaystyle(\Omega_{M}^{2}h_{2})|_{\ell=0}(\infty,v_{0})$
$\displaystyle=2\,M^{2}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})_{\mathfrak{S}}|_{\ell=0}(\infty,v_{0})\,,$
$\displaystyle(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=0}(\infty,v_{0})$
$\displaystyle=4M^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}_{\mathfrak{S}}|_{\ell=0}(\infty,v_{0})-2\,M\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})_{\mathfrak{S}}|_{\ell=0}(\infty,v_{0})-M\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})_{\mathfrak{S}}|_{\ell=0}(\infty,v_{0})\,.$

One can choose $(h_{1})|_{\ell=0}(\infty,v_{0})$ arbitrarily.

$\ell=1$-spherical modes: We choose $(h_{1})|_{\ell=1}$,
$(\Omega^{2}_{M}h_{2})|_{\ell=1}$ and
$(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=1}$ on $\underline{C}_{v_{0}}$
as solutions to the system of ODEs in the $e_{3}^{\text{as}}$-direction

$\displaystyle\operatorname{\not{\nabla}}_{3}(h_{1})|_{\ell=1}$
$\displaystyle=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3})_{\mathfrak{S}}|_{\ell=1}\,,$
$\displaystyle\operatorname{\not{\nabla}}_{3}(\Omega^{2}_{M}h_{2})|_{\ell=1}$
$\displaystyle=-(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=1}-\frac{2M}{r^{2}}\,(h_{1})|_{\ell=1}+(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1}\,,$
$\displaystyle\operatorname{\not{\nabla}}_{3}(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=1}$
$\displaystyle=(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}})_{\mathfrak{S}}|_{\ell=1}$

with initial conditions on $\mathbb{S}^{2}_{\infty,v_{0}}$ solving the
elliptic system

$\displaystyle
2\,\not{\Delta}(\Omega^{2}_{M}h_{2})|_{\ell=1}(\infty,v_{0})-\frac{1}{2\,M^{2}}(\Omega_{M}^{2}h_{2})|_{\ell=1}(\infty,v_{0})=$
$\displaystyle\,2\,(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})$
$\displaystyle-(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})\,,$
$\displaystyle
2\,\not{\Delta}\,h_{1}|_{\ell=1}(\infty,v_{0})-\frac{1}{2\,M^{2}}\,(\Omega_{M}^{2}h_{2})|_{\ell=1}(\infty,v_{0})+\frac{1}{M}\,(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=1}(\infty,v_{0})=$
$\displaystyle\,2\,(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})$
$\displaystyle-(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})\,,$
$\displaystyle\not{\Delta}(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell=1}(\infty,v_{0})+\frac{1}{2M}\not{\Delta}\,h_{1}|_{\ell=1}$
$\displaystyle(\infty,v_{0})+\frac{3}{2M^{3}}\,(\Omega_{M}^{2}h_{2})|_{\ell=1}(\infty,v_{0})$
$\displaystyle=$
$\displaystyle\,-\operatorname{\not{\nabla}}_{3}(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})+\frac{1}{2M}(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})$
$\displaystyle+\not{\Delta}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})+(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})\,.$

We then choose $k_{\flat}|_{\ell=1}$ on $\underline{C}_{v_{0}}$ such that

$\displaystyle(k_{\flat})|_{\ell=1}=$
$\displaystyle\,2\,\operatorname{\not{\nabla}}(h_{1})|_{\ell=1}$
$\displaystyle-2\,(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})_{\mathfrak{S}}|_{\ell=1}+\frac{8}{3}\,M^{2}\left(r+M\right){}^{\star}\operatorname{\not{\nabla}}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}_{\mathfrak{S}}|_{\ell=1}+\frac{1}{2\,M}\,(\not{\textup{curl}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1})(\infty,v_{0})\,.$

We choose $(\not{\textup{div}}\,h)|_{\ell=1}$ and
$(\not{\textup{curl}}\,h)|_{\ell=1}$ on $\underline{C}_{v_{0}}$ such that

$\displaystyle\operatorname{\not{\nabla}}_{3}(\not{\textup{curl}}\,h)|_{\ell=1}=$
$\displaystyle\,(\not{\textup{curl}}\,k)|_{\ell=1}$
$\displaystyle+(\not{\textup{curl}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}})_{\mathfrak{S}}|_{\ell=1}+\frac{8}{3}\,M(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}_{\mathfrak{S}}|_{\ell=1}+\frac{1}{2\,M}\,(\not{\textup{curl}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1})(\infty,v_{0})$

and

$(\not{\textup{div}}\,h)|_{\ell=1}(\infty,v_{0})=\frac{1}{M}(\Omega_{M}^{2}h_{2})|_{\ell=1}(\infty,v_{0})-\frac{1}{2}\,(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})_{\mathfrak{S}}|_{\ell=1}(\infty,v_{0})\,.$

One can arbitrarily choose $(\not{\textup{curl}}\,h)|_{\ell=1}(\infty,v_{0})$
and $(\not{\textup{div}}\,h)|_{\ell=1}$ away from
$\mathbb{S}^{2}_{\infty,v_{0}}$. We conclude by choosing
$(\operatorname{\not{\nabla}}_{4}h)_{\flat}|_{\ell=1}$ such that

$\displaystyle(\operatorname{\not{\nabla}}_{4}h)_{\flat}|_{\ell=1}=$
$\displaystyle\,\frac{\Omega^{2}_{M}}{r}\,(h_{\flat})|_{\ell=1}+2\,\operatorname{\not{\nabla}}(\Omega^{2}_{M}h_{2})|_{\ell=1}$
$\displaystyle-2\,(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1}+\frac{8}{3}\,\Omega^{2}_{M}\,M^{3}\,{}^{\star}\operatorname{\not{\nabla}}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}_{\mathfrak{S}}|_{\ell=1}+\frac{1}{2\,M}\,(\not{\textup{curl}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell=1})(\infty,v_{0})\,.$

$\ell\geq 2$-spherical modes: We set $h_{1}|_{\ell\geq 2}\equiv 0$ on
$\underline{C}_{v_{0}}$. We choose $(\Omega^{2}_{M}h_{2})|_{\ell\geq 2}$ on
$\underline{C}_{v_{0}}$ as the solution to the ODE in the
$e_{3}^{\text{as}}$-direction

$\operatorname{\not{\nabla}}_{3}(\Omega^{2}_{M}h_{2})|_{\ell\geq
2}+(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell\geq
2}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}$

with initial condition on $\mathbb{S}^{2}_{\infty,v_{0}}$ such that

$\displaystyle 2\,\not{\Delta}(\Omega^{2}_{M}h_{2})|_{\ell\geq
2}(\infty,v_{0})-\frac{1}{2\,M^{2}}(\Omega_{M}^{2}h_{2})|_{\ell\geq
2}(\infty,v_{0})=$
$\displaystyle\,2\,(\not{\text{div}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})$
$\displaystyle-(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})$

and $(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell\geq
2}=(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell\geq 2}(\theta^{A})$ on
$\underline{C}_{v_{0}}$ such that

$\displaystyle\not{\Delta}(\operatorname{\not{\nabla}}_{4}h_{1})|_{\ell\geq
2}+\frac{3}{2M^{3}}\,(\Omega_{M}^{2}h_{2})|_{\ell\geq 2}(\infty,v_{0})=$
$\displaystyle\,-\operatorname{\not{\nabla}}_{3}(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})+\frac{1}{2M}(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})$
$\displaystyle+\not{\Delta}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})+(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})\,.$

We choose $h_{\flat}|_{\ell\geq 2}=h_{\flat}|_{\ell\geq 2}(\theta^{A})$ on
$\underline{C}_{v_{0}}$ such that252525By considering the unique
representation $h=\not{\mathcal{D}}_{1}^{\star}(q_{1},q_{2})$ for scalar
functions $q_{i}=q_{i}(\theta^{A})$, one can check that the equations (86) and
(87) form an elliptic system on spheres for $q_{1}$ and $q_{2}$ which can
indeed be solved.

$(\not{\textup{div}}\,h)|_{\ell\geq
2}=\frac{1}{M}\,(\Omega_{M}^{2}h_{2})|_{\ell\geq
2}(\infty,v_{0})-\frac{1}{2}\,(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})$ (86)

and

$\displaystyle\not{\text{curl}}\,\not{\text{div}}\,\not{\mathcal{D}}_{2}^{\star}\,h_{\flat}=$
$\displaystyle\,\frac{1}{4}\,(\not{\text{curl}}\,\not{\text{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}})_{\mathfrak{S}}(\infty,v_{0})-\frac{1}{2}\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v_{0})$ (87)
$\displaystyle-\frac{1}{4M}\,(\not{\text{curl}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}(\infty,v)\,.$

We then chose
$(\not{\text{div}}\,(\operatorname{\not{\nabla}}_{4}h))|_{\ell\geq 2}$ and
$(\not{\text{curl}}\,(\operatorname{\not{\nabla}}_{4}h))|_{\ell\geq 2}$ on
$\underline{C}_{v_{0}}$ such that

$(\not{\text{div}}\,(\operatorname{\not{\nabla}}_{4}h))|_{\ell\geq
2}=2\,\not{\Delta}(\Omega^{2}_{M}h_{2})|_{\ell\geq
2}+\frac{\Omega^{2}_{M}}{r}\,(\not{\text{div}}\,h)|_{\ell\geq
2}-2\,(\not{\text{div}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}$

and

$(\not{\text{curl}}\,(\operatorname{\not{\nabla}}_{4}h))|_{\ell\geq
2}=\frac{\Omega^{2}_{M}}{r}\,(\not{\text{curl}}\,h)|_{\ell\geq
2}-2\,(\not{\text{curl}}\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4})_{\mathfrak{S}}|_{\ell\geq
2}\,.$

One can arbitrarily choose $k_{\flat}|_{\ell\geq 2}$ on
$\underline{C}_{v_{0}}$. ∎

######  Remark 6.11.

As noted in the proof of Proposition 6.10, given a solution $\mathfrak{S}$ to
the linearised system of equations, one has some freedom in constructing the
geometric quantities $(h_{1},h_{2},h,k)$ which achieve (84). In other words,
given a solution $\mathfrak{S}$, the linear combination of coordinate and
frame pure gauge solutions which achieves (84) is not unique. This is a
manifestation of the fact that our Definition 6.1 of initial-data normalised
solutions does not fix the residual gauge freedom completely. In fact, one can
enforce the uniqueness of the pure gauge solutions in Proposition 6.10 by
imposing extra normalisation identities in Definition 6.1, which would not
however play any other relevant role in the problem. This aspect may be
compared with Theorem 9.1 in [5], where the pure gauge solution employed to
achieve the initial-data normalisation is unique.

######  Remark 6.12.

One can check that if the smooth seed initial data relative to the given
solution $\mathfrak{S}$ are asymptotically flat with weight $s$ and to order
$n$, then the smooth seed initial data relative to the initial-data normalised
solution $\mathfrak{S}_{N}$ are asymptotically flat with weight $s$ and to
order $n-2$.

###  6.2 Reference linearised solutions

In this section, we define the _reference linearised solutions_ to the
linearised system of equations. See Section 2.3.2 of the overview for a
discussion about the meaning and derivation of these solutions.

######  Definition 6.13 (Reference linearised Schwarzschild solutions).

Let $\mathfrak{m}\in\mathbb{R}$. We define the _reference linearised
Schwarzschild solutions_ as the one-parameter family of solutions
$\mathsf{S}(\mathfrak{m})$ to the linearised system of equations such that

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}$
$\displaystyle=\mathfrak{m}\,,$
$\displaystyle(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})$
$\displaystyle=-2\,\mathfrak{m}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\operatorname{\underline{\chi}}})$
$\displaystyle=(\textup{tr}\operatorname{\underline{\chi}})_{M}\,\mathfrak{m}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}$
$\displaystyle=\rho_{M}\,\mathfrak{m}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}$
$\displaystyle=\not{K}_{M}\,\mathfrak{m}$

on $\mathcal{R}_{u_{0},v_{0}}$ and all the remaining linearised frame
coefficients, connection coefficients and curvature components identically
vanishing.

######  Definition 6.14 ($\ell=1$-reference linearised Kerr solutions).

Let $\mathfrak{a}\in\mathbb{R}$ and $m=-1,0,1$. We define the _$\ell=1$
-reference linearised Kerr solutions_ as the two-parameter family of solutions
$\mathsf{K}(\mathfrak{a},m)$ to the linearised system of equations such that

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4,m}$
$\displaystyle=\frac{\mathfrak{a}\Omega^{2}_{M}}{2r}\,r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3,m}$
$\displaystyle=\frac{\mathfrak{a}}{2Mr}\left(r+M\right)r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{m}$
$\displaystyle=-\frac{\mathfrak{a}r}{2M^{2}}\,r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})_{m}$
$\displaystyle=\frac{2\mathfrak{a}\Omega^{2}_{M}}{r^{2}}\,Y_{m}^{\ell=1}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})_{m}$
$\displaystyle=\frac{2\mathfrak{a}}{Mr^{2}}\left(r+M\right)Y_{m}^{\ell=1}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}_{m}$
$\displaystyle=\frac{\mathfrak{a}}{2Mr}\,r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}_{m}$
$\displaystyle=-\frac{\mathfrak{a}}{2Mr}\,r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}_{m}$
$\displaystyle=\frac{\mathfrak{a}\Omega^{2}_{M}}{2Mr}\,r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}_{m}$
$\displaystyle=\frac{3\mathfrak{a}}{r^{3}}\,r\,{}^{\star}\operatorname{\not{\nabla}}Y_{m}^{\ell=1}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}_{m}$
$\displaystyle=\frac{6\mathfrak{a}M}{r^{4}}\,Y_{m}^{\ell=1}$

on $\mathcal{R}_{u_{0},v_{0}}$ and all the remaining linearised frame
coefficients, connection coefficients and curvature components identically
vanishing.

######  Remark 6.15.

We note that, in particular, both
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$
identically vanish for any reference linearised Schwarzschild and
$\ell=1$-reference linearised Kerr solution.

######  Remark 6.16.

The fact that the reference linearised Schwarzschild and $\ell=1$-reference
linearised Kerr solutions, as defined in Definitions 6.13 and 6.14, are indeed
solutions to the system of equations is a check left to the reader.

######  Remark 6.17.

The reference linearised Schwarzschild and $\ell=1$-reference linearised Kerr
solutions are initial-data normalised. See Definition 6.1.

We conclude with the following definition and remark.

######  Definition 6.18 (Reference linearised Kerr solutions).

Let
$\mathfrak{m},\mathfrak{s}_{-1},\mathfrak{s}_{0},\mathfrak{s}_{-1}\in\mathbb{R}$.
We define the _reference linearised Kerr solution_
$\mathcal{K}_{\mathfrak{m},\mathfrak{s}_{m}}$ with parameters
$(\mathfrak{m},\mathfrak{s}_{-1},\mathfrak{s}_{0},\mathfrak{s}_{-1})$ as the
sum of the reference linearised Schwarzschild solution
$\mathsf{S}(\mathfrak{m})$ with a linear combination of $\ell=1$-reference
linearised Kerr solutions $\mathsf{K}(\mathfrak{a},m)$ such that

$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}[\mathcal{K}_{\mathfrak{m},\mathfrak{s}_{m}}]=\sum_{m}\mathfrak{s}_{m}\cdot\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}_{m}[\mathsf{K}(\mathfrak{a},m)]\,.$

######  Remark 6.19.

A reference linearised Kerr solution
$\mathcal{K}_{\mathfrak{m},\mathfrak{s}_{m}}$ is only supported on
$\ell=0,1$-spherical modes.

###  6.3 Properties of initial-data normalised solutions

In this section, we state the main global properties of initial-data
normalised solutions. Propositions 6.20 and 6.21 can be easily checked using
the system of linearised equations.

######  Proposition 6.20 (Global properties $\ell=0,1$-spherical modes).

Consider an initial-data normalised solution $\mathfrak{S}_{N}$. Then,
$\mathfrak{S}_{N}$ satisfies the following identities:

  * •

On $\mathcal{R}_{u_{0},v_{0}}$ (including $\mathcal{H}^{+}_{v\geq v_{0}}$):
The identities

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=1}$
$\displaystyle=0\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=0}$
$\displaystyle=0\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})|_{\ell=0}$
$\displaystyle=0\,.$

  * •

On $\mathcal{H}_{v\geq v_{0}}^{+}$: The identities

$\displaystyle(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=1}$
$\displaystyle=0\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi})|_{\ell=0,1}$
$\displaystyle=0\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=1}$
$\displaystyle=0\,,$
$\displaystyle(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})|_{\ell=1}$
$\displaystyle=0\,.$

######  Proposition 6.21 (Global properties $\ell\geq 2$-spherical modes).

Consider an initial-data normalised solution $\mathfrak{S}_{N}$. Then,
$\mathfrak{S}_{N}$ satisfies the following identities:

  * •

On $\mathcal{R}_{u_{0},v_{0}}$ (including $\mathcal{H}^{+}_{v\geq v_{0}}$):
The identities

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell\geq
2}$ $\displaystyle=0\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}|_{\ell\geq
2}$ $\displaystyle=0\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|_{\ell\geq
2}$
$\displaystyle=\frac{r^{2}}{2M}\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}})|_{\ell\geq
2}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell\geq
2}$ $\displaystyle=0\,.$

  * •

On $\mathcal{H}^{+}_{v\geq v_{0}}$: The identities

$\displaystyle(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell\geq
2}$ $\displaystyle=0\,,$
$\displaystyle\not{\textup{curl}}\,\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}$
$\displaystyle=2\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell\geq
2}\,,$
$\displaystyle(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi})|_{\ell\geq
2}$ $\displaystyle=0\,,$
$\displaystyle(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})|_{\ell\geq
2}$ $\displaystyle=0\,,$
$\displaystyle(\not{\textup{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|_{\ell\geq
2}$
$\displaystyle=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell\geq
2}\,.$

######  Proposition 6.22.

Consider an initial-data normalised solution $\mathfrak{S}_{N}$. Then, the
linearised quantities

$\displaystyle\Omega^{-4}_{M}\,(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi})|_{\ell\geq
1}\,,$
$\displaystyle\Omega^{-4}_{M}\,\mathcal{A}^{[n]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi})$

are uniformly bounded on $\underline{C}_{v_{0}}$.

###### Proof.

This fact can be deduced from the vanishing of both
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell\geq
1}$ and
$\operatorname{\not{\nabla}}_{3}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell\geq
1}$ on $\mathbb{S}^{2}_{\infty,v_{0}}$, the latter following from equation
(55) and the properties
$(\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})|_{\ell\geq
1}=0$ and
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=1}=0$
on $\mathbb{S}^{2}_{\infty,v_{0}}$. ∎

We conclude the section with a key proposition.

######  Proposition 6.23.

Consider an initial-data normalised solution $\mathfrak{S}_{N}$. Then, the
$\ell=0,1$-spherical projection of $\mathfrak{S}_{N}$ is a reference
linearised Kerr solution $\mathcal{K}_{\mathfrak{m},\mathfrak{s}_{m}}$ with
parameters $(\mathfrak{m},\mathfrak{s}_{m})$ such that

$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}[\mathfrak{S}_{N}]|_{\ell=0}(\infty,v_{0})$
$\displaystyle=-\frac{\mathfrak{m}}{4M^{2}}\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}[\mathfrak{S}_{N}]|_{\ell=1}(\infty,v_{0})$
$\displaystyle=\sum_{m}\mathfrak{s}_{m}Y_{m}^{\ell=1}\,.$

###### Proof.

The symmetric traceless $\mathbb{S}^{2}_{u,v}$ two-tensors

$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}\,,$

as well as the scalar functions obtained by taking their
$\not{\text{div}}\,\not{\text{div}}$ and
$\not{\text{curl}}\,\not{\text{div}}$, are supported on $\ell\geq 2$-spherical
modes.

We treat the $\ell=0$ and $\ell=1$-spherical modes for the remaining
linearised quantities separately. Before we start, it is useful to recall the
identity

$2\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}=-\frac{1}{2}\,\not{\Delta}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})+\not{\text{div}}\,\not{\text{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}-\frac{1}{r^{2}}\,(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})\,.$
(88)

$\ell=0$-spherical modes: It suffices to prove

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=0}=\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=0}=(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=0}=0\,,$
$\displaystyle(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell=0}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=0}=(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=0}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})|_{\ell=0}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=0}=0\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=0}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=0}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell=0}=0$

on $\mathcal{R}_{u_{0},v_{0}}$ for any initial-data normalised solution
$\mathfrak{S}_{N}$ such that

$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}[\mathfrak{S}_{N}]|_{\ell=0}(\infty,v_{0})=0\,.$
(89)

We note that one can immediately conclude
$(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=0}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})|_{\ell=0}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell=0}=0$
on $\mathcal{R}_{u_{0},v_{0}}$ from the equations (42), (43) and (65).

Using Definition 6.1, the identity (89) implies
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=0}=0$
on $\underline{C}_{v_{0}}$ and
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=0}=0$
on $\mathbb{S}^{2}_{\infty,v_{0}}$. We also recall that, by Definition 6.1, we
have
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=0}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=0}=0$
on $\underline{C}_{v_{0}}$. Using equation (51), one has
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=0}=0$
on $\underline{C}_{v_{0}}$. Then,
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=0}=0$ on
$\underline{C}_{v_{0}}$ by equation (74). By equation (55),
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell=0}$
on $\underline{C}_{v_{0}}$. We also have
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=0}=0$ on
$\underline{C}_{v_{0}}$ by equation (68), which implies
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=0}=0$ on
$\underline{C}_{v_{0}}$ by the identity (88). This concludes the vanishing of
all quantities on $\underline{C}_{v_{0}}$.

Using equation (35) and (50), one has
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=0}=(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell=0}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equation (72) then gives
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=0}=0$ on
$\mathcal{R}_{u_{0},v_{0}}$ and equation (46) gives
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=0}=0$ on
$\mathcal{R}_{u_{0},v_{0}}$. Equation (54) implies
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=0}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equation (68) gives
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=0}=0$ on
$\mathcal{R}_{u_{0},v_{0}}$. Equations (36) and (62) now form an homogeneous
system of coupled ODEs in the $e_{4}^{\text{as}}$-direction for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=0}$
and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=0}$.
Since the quantities identically vanish on $\underline{C}_{v_{0}}$, one can
conclude
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=0}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=0}=0$
on $\mathcal{R}_{u_{0},v_{0}}$.

$\ell=1$-spherical modes: It suffices to prove

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=1}=\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=1}=(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=1}=0\,,$
$\displaystyle(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell=1}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=1}=(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=1}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=1}=0\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell=1}=0$

and

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}|_{\ell=1}=\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}|_{\ell=1}=\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|_{\ell=1}=0\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}=0\,,$
$\displaystyle\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}=0$

on $\mathcal{R}_{u_{0},v_{0}}$ for any initial-data normalised solution
$\mathfrak{S}_{N}$ such that

$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}[\mathfrak{S}_{N}]|_{\ell=1}(\infty,v_{0})=0\,.$
(90)

Using Definition 6.1, the identity (90) implies
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}|_{\ell=1}=\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|_{\ell=1}=(\not{\textup{curl}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}})|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. We also recall that, by Definition 6.1, we have
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=1}=\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Equations (42) and (43) imply
$(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=1}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$, while equation (41) implies

$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}$
(91)

on $\underline{C}_{v_{0}}$. Equation (39) yields
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Using equations (59) and (65), we have
$\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}=0$
on $\mathbb{S}^{2}_{\infty,v_{0}}$. Using the identity (88) for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=1}$ and
the normalisation identity
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=1}(\infty,v_{0})=0$,
we have
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=1}=0$ on
$\mathbb{S}^{2}_{\infty,v_{0}}$. By applying equation (68), we get
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=1}=0$ on
$\mathbb{S}^{2}_{\infty,v_{0}}$. Using the normalisation identity
$(\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}+\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}})|_{\ell=1}(\infty,v_{0})=0$,
we get
$\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}=0$
on $\mathbb{S}^{2}_{\infty,v_{0}}$. Using the equation (66) and (67), we have
$\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}=0$
on $\mathbb{S}^{2}_{\infty,v_{0}}$. This concludes the vanishing of all
quantities on $\mathbb{S}^{2}_{\infty,v_{0}}$.

Equation (77) gives
$\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Equation (64), together with the identity (91),
give
$\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}=\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Equation (51) gives
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Equations (74) and (75) give
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Equation (55) then gives
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$, which, through equation (66), yields
$\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. Equation (68) gives
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=1}=0$ on
$\underline{C}_{v_{0}}$, which then implies
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=1}=0$ on
$\underline{C}_{v_{0}}$ by the identity (88). Equation (47) gives
$(\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}})|_{\ell=1}=0$
on $\underline{C}_{v_{0}}$. This concludes the vanishing of all quantities on
$\underline{C}_{v_{0}}$.

Equation (35) gives
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{4}|_{\ell=1}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equations (38), (50) and (56) give
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}|_{\ell=1}=0$
and
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})|_{\ell=1}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\chi})|_{\ell=1}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equation (46) then gives
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell=1}=0$ on
$\mathcal{R}_{u_{0},v_{0}}$. Equation (70) gives
$\not{\text{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=\not{\text{curl}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|_{\ell=1}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equations (72) and (73) then give
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}|_{\ell=1}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equations (54) and (58) give
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})|_{\ell=1}=(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})|_{\ell=1}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. Equation (60) yields
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|_{\ell=1}=0$.
Then, equation (37) gives
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}|_{\ell=1}=0$
on $\mathcal{R}_{u_{0},v_{0}}$. We have
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widetilde{\not{K}}}|_{\ell=1}=0$ on
$\mathcal{R}_{u_{0},v_{0}}$ from equation (68). Equations (36), (40), (61),
(62), (63) and (76) now form an homogeneous system of coupled ODEs in the
$e_{4}^{\text{as}}$-direction for

$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=1}\,,\,\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|_{\ell=1}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}|_{\ell=1}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=1}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}\,,\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}\,.$

Since all the quantities identically vanish on $\underline{C}_{v_{0}}$, one
can conclude

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}|_{\ell=1}$
$\displaystyle=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|_{\ell=1}=0\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|_{\ell=1}$
$\displaystyle=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}|_{\ell=1}=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}|_{\ell=1}=0$

on $\mathcal{R}_{u_{0},v_{0}}$. ∎

##  7 The main theorem

In this section, we state the main theorem of the present work.

###  7.1 Energies and norms

We define the initial energy fluxes for the (derived) gauge invariant
linearised quantities262626The definition of the initial energy fluxes for the
(derived) gauge invariant linearised quantities is analogous to the one
presented in Sections 10.1.1 and 10.2.1 of [5], but with the regularising
weights for the linearised quantities removed.

$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]:=$
$\displaystyle\,\sum_{i+j\leq
n}\lVert\operatorname{\not{\nabla}}_{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
r\operatorname{\not{\nabla}}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
(92) $\displaystyle+\lVert
r\operatorname{\not{\nabla}}_{4}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert\operatorname{\not{\nabla}}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert
r^{-1}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi})\rVert_{L^{2}(C_{u_{0}})}^{2}\,,$
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}]:=$
$\displaystyle\,\sum_{i+j\leq
n}\lVert\operatorname{\not{\nabla}}_{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
r\operatorname{\not{\nabla}}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
(93) $\displaystyle+\lVert
r\operatorname{\not{\nabla}}_{4}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert\operatorname{\not{\nabla}}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert
r^{-1}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}})\rVert_{L^{2}(C_{u_{0}})}^{2}\,,$
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]:=$
$\displaystyle\,\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]+\sum_{i+j\leq
n}\lVert
r^{4-\delta}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})\rVert_{L^{2}(C_{u_{0}})}^{2}\,,$
(94)
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}]:=$
$\displaystyle\,\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}]+\sum_{i+j\leq
n}\lVert
r^{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}\,,$
(95)
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]:=$
$\displaystyle\,\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]+\sum_{i+j\leq
n}\lVert
r^{3-\delta}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha})\rVert_{L^{2}(C_{u_{0}})}^{2}\,,$
(96)
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]:=$
$\displaystyle\,\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}]+\sum_{i+j\leq
n}\lVert
T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\alpha}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}\,,$
(97)
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]:=$
$\displaystyle\,\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
(98) $\displaystyle+\sum_{i+j\leq n}\lVert r^{4-\delta}\cdot
r\operatorname{\not{\nabla}}_{4}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert
r^{4-\delta}\operatorname{\not{\nabla}}_{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})\rVert_{L^{2}(C_{u_{0}})}^{2}$
$\displaystyle+\lVert r^{4-\delta}\cdot
r\operatorname{\not{\nabla}}_{A}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert
r^{3-\delta}\cdot
r\operatorname{\not{\nabla}}_{4}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha})\rVert_{L^{2}(C_{u_{0}})}^{2}$
$\displaystyle+\lVert
r^{3-\delta}\operatorname{\not{\nabla}}_{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha})\rVert_{L^{2}(C_{u_{0}})}^{2}+\lVert
r^{3-\delta}\cdot
r\operatorname{\not{\nabla}}_{A}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha})\rVert_{L^{2}(C_{u_{0}})}^{2}\,,$
$\displaystyle\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]:=$
$\displaystyle\,\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]$
(99) $\displaystyle+\sum_{i+j\leq n}\lVert r^{3}\cdot
r\operatorname{\not{\nabla}}_{4}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
r^{3}\operatorname{\not{\nabla}}_{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
$\displaystyle+\lVert r^{3}\cdot
r\operatorname{\not{\nabla}}_{A}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
r\operatorname{\not{\nabla}}_{4}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\alpha}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
$\displaystyle+\lVert\operatorname{\not{\nabla}}_{3}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\alpha}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert
r\operatorname{\not{\nabla}}_{A}(T^{i}(r\operatorname{\not{\nabla}}_{A})^{j}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\alpha}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}\,,$

with $T$ the stationary Killing vector field, $n\in\mathbb{N}_{0}$,
$\delta\geq 0$ and the linearised quantities
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}$,
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}$,
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}$ and
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$ defined in Section
8. We define the initial energy flux for the gauge dependent linearised
quantities

$\displaystyle\mathbb{D}_{0}:=$
$\displaystyle\,\lVert\operatorname{\not{\nabla}}_{3}^{2}(\mathcal{A}^{[3]}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
(100)
$\displaystyle+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\Omega^{-4}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}}))\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
$\displaystyle+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\not{\textup{div}}\,\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
$\displaystyle+\lVert\not{\textup{div}}\,\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\mathcal{A}^{[3]}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\Omega^{-2}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}}))\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}})\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}\,.$

We define the initial energy

$\mathbb{E}_{0}:=\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]+\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]+\mathbb{D}_{0}\,.$
(101)

We define the spacetime energies

$\displaystyle\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\mathcal{I},\delta}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]\,,\,\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\textup{deg}}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]\,,\,\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\textup{master}}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,,$
$\displaystyle\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\mathcal{I},\delta}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}]\,,\,\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\textup{deg}}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}]\,,\,\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\textup{master}}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]$

as defined in Sections 10.1.1 and 10.2.1 of [5], but with the regularising
weights for the linearised quantities removed.

###  7.2 The linear stability statement

The following is the main result of the paper.

######  Theorem 7.1 (Linear stability of the Schwarzschild solution).

Consider an initial-data normalised solution $\mathfrak{S}_{N}$ to the system
of linearised equations arising from smooth, asymptotically flat seed initial
data on $\mathcal{S}_{u_{0},v_{0}}$. Assume that the initial energy
$\mathbb{E}_{0}$ is finite. Then,

  * •

The uniform boundedness estimates for the linearised curvature components

$\displaystyle\sup_{u\geq
u_{0}}\int_{v_{0}}^{v}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{1}{r^{2}}\,|\mathcal{A}^{[5]}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (102) $\displaystyle\sup_{v\geq
v_{0}}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[5]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (103) $\displaystyle\sup_{u\geq
u_{0}}\int_{v_{0}}^{v}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{1}{r^{2}}\,|\mathcal{A}^{[4]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (104) $\displaystyle\sup_{v\geq
v_{0}}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[4]}r\not{\mathcal{D}}_{2}^{\star}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}$ (105)

and

$\displaystyle\sup_{u\geq
u_{0}}\int_{v_{0}}^{v}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{1}{r^{2}}\,|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (106) $\displaystyle\sup_{v\geq
v_{0}}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}\,|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}$ (107)

hold.

  * •

Uniform boundedness estimates (in terms of $\mathbb{E}_{0}$) for weighted
$L^{2}$-fluxes on null cones of up to five angular derivatives of all
linearised connection coefficients, frame coefficients and induced metric
hold.

  * •

For any $\delta>0$, $v\geq v_{0}$ and $u\geq u_{0}$, the integrated energy
decay estimates for the linearised curvature components

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[5]}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (108)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[5]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (109)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[4]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (110)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[4]}r\not{\mathcal{D}}_{2}^{\star}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (111)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}$ (112)

hold. In particular, analogous integrated energy decay estimates, but without
degenerating factor, hold for one less angular derivative of all linearised
curvature components.

  * •

For any $\delta>0$, $v\geq v_{0}$ and $u\geq u_{0}$, the integrated energy
decay estimates for the linearised connection coefficients

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}|\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (113)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (114)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r^{2}\Omega^{-4}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi}))|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (115)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\operatorname{\underline{\chi}}}))|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (116)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}}))|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (117)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r\,\not{\textup{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (118)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (119)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (120)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (121)

for the linearised frame coefficients

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (122)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{-1}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (123)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r\,\not{\textup{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r^{-1}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3})|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}$ (124)

and for the linearised induced metric

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\Omega^{-2}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}}))|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}\,,$ (125)
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2}$
$\displaystyle\lesssim\mathbb{E}_{0}$ (126)

hold. Moreover, the estimates (113)-(126) can be upgraded to (possibly non-
degenerate) integrated energy decay estimates (in terms of $\mathbb{E}_{0}$)
for up to five angular derivatives of all linearised connection coefficients,
frame coefficients and induced metric.

######  Remark 7.2.

In the present work, we prove the estimates (108)-(112) and (113)-(126). The
proof of the estimates (102)-(107) can be easily reconstructed by the reader
from our proof of the decay estimates (108)-(112). The procedure to upgrade
the estimates (113)-(126) to estimates for higher angular derivatives of the
linearised quantities is standard (see, for instance, Propositions 9.24 and
9.26). We also remark that some of the estimates stated in our main theorem
are not expected to be optimal near infinity, meaning that analogous estimates
with stronger weights in $r$ can be proven.

######  Remark 7.3.

Theorem 7.1 controls the $\ell\geq 2$-spherical projection of all linearised
quantities in the linearised system of equations. In view of Proposition 6.23,
the theorem can be interpreted as the full linear stability result.

######  Remark 7.4.

For smooth seed initial data which are asymptotically flat with suitable
weight $s$ and order $n$ (see Definition 5.3), the initial energy
$\mathbb{E}_{0}$ is, in fact, finite. We note, in particular, that Proposition
6.22 and the horizon normalisation identities of Proposition 6.1 guarantee the
finiteness of the initial energy fluxes in (100) containing a growing factor
towards the event horizon.

######  Remark 7.5.

The top-order term in the initial energy $\mathbb{E}_{0}$ is at the level of
five derivatives of linearised curvature components, and thus the (degenerate)
integrated decay estimates for the linearised curvature components in Theorem
7.1 do _not_ lose derivatives.

######  Remark 7.6.

In the initial energy flux for the gauge dependent linearised quantities, some
of the terms can be dropped by using the linearised system of equations. The
form (100) is however more convenient to read off the initial fluxes needed
for the proof of Theorem 7.1. We also note that the initial energy
$\mathbb{E}_{0}$ vanishes for smooth seed initial data corresponding to a
reference linearised Kerr solution.

##  8 Analysis of the gauge invariant linearised quantities

The analysis of the gauge invariant linearised quantities in the system relies
on Sections 7, 11 and 12 of [5]. In this section, we explain how the results
of [5] can be applied to our setting.

We define the (regular) symmetric traceless $\mathbb{S}^{2}_{u,v}$ two-tensors

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}$
$\displaystyle:=-\frac{1}{2\,r}\operatorname{\not{\nabla}}_{3}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}})\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{P}$
$\displaystyle:=\frac{1}{r^{3}}\operatorname{\not{\nabla}}_{3}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}r^{3})\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}$
$\displaystyle:=r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}$
$\displaystyle:=\frac{1}{2\,r\,\Omega^{4}_{M}}\operatorname{\not{\nabla}}_{4}(r\,\Omega^{4}_{M}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}})\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}$
$\displaystyle:=-\frac{1}{r^{3}\Omega^{2}}\operatorname{\not{\nabla}}_{4}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}r^{3}\Omega)\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$
$\displaystyle:=r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}\,.$

The _Teukolsky equation of spin $+2$_ for a symmetric traceless
$\mathbb{S}^{2}_{u,v}$ two-tensor is the equation (173) in [5]. The _Teukolsky
equation of spin $-2$_ for a symmetric traceless $\mathbb{S}^{2}_{u,v}$ two-
tensor is the equation (174) in [5]. The _Regge–Wheeler equation_ for a
symmetric traceless $\mathbb{S}^{2}_{u,v}$ two-tensor is the equation (175) in
[5].

The following proposition is the analogue of Proposition 7.4.1 of [5] in our
setting. The proof is a check left to the reader.

######  Proposition 8.1.

Consider a solution $\mathfrak{S}$ to the linearised system of equations.
Then, the linearised curvature components
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$
satisfy the Teukolsky equations of spin $+2$ and $-2$ respectively, the
quantities $\overset{\text{\scalebox{0.6}{$(1)$}}}{P}$ and
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}$ satisfy the
Regge–Wheeler equation and the identities

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}+\frac{3}{2}\,\rho\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}-\frac{3}{2}\,\rho\,\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}$

and

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{P}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3}{4}\,\rho\,(\textup{tr}\operatorname{\underline{\chi}})\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}-\frac{3}{4}\,\rho\,(\textup{tr}\chi)\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}+(\operatorname{\not{\nabla}}_{3}\rho)\,\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}+(\operatorname{\not{\nabla}}_{4}\rho)\,\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3}{4}\,\rho\,(\textup{tr}\operatorname{\underline{\chi}})\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}-\frac{3}{4}\,\rho\,(\textup{tr}\chi)\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}+(\operatorname{\not{\nabla}}_{3}\rho)\,\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{4}+(\operatorname{\not{\nabla}}_{4}\rho)\,\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}$

hold.

######  Remark 8.2.

Consider an initial-data normalised solution $\mathfrak{S}_{N}$ to the system
of linearised equations. Then,

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{P}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3}{4}\,\rho\,(\textup{tr}\operatorname{\underline{\chi}})\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}-\frac{3}{4}\,\rho\,(\textup{tr}\chi)\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}+(\operatorname{\not{\nabla}}_{4}\rho)\,\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3}{4}\,\rho\,(\textup{tr}\operatorname{\underline{\chi}})\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}-\frac{3}{4}\,\rho\,(\textup{tr}\chi)\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}+(\operatorname{\not{\nabla}}_{4}\rho)\,\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}\,.$

Note that the last term in both identities does not appear in the
corresponding identities (183) and (184) in [5]. However, we have

$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{P}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3}{4}\,\rho\,(\textup{tr}\operatorname{\underline{\chi}})\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\,,$
$\displaystyle\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}$
$\displaystyle=\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3}{4}\,\rho\,(\textup{tr}\operatorname{\underline{\chi}})\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}$

on $\mathcal{H}^{+}_{v\geq v_{0}}$, which do coincide with the identities
(183) and (184) in [5] restricted to the event horizon. This fact will be
exploited to repeat in our setting the analysis for the horizon fluxes of
Section 13.2 in [5].

The following proposition states integrated energy decay for the gauge
invariant linearised quantities.

######  Proposition 8.3 (Integrated decay for the gauge invariant linearised
quantities).

Consider a solution $\mathfrak{S}$ to the linearised system of equations.
Assume that the initial energy fluxes (92) and (98) are finite. Then, for any
$\delta>0$,

  * •

Theorem 1 of [5] holds for $\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}$ and
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$. In particular, we
have the integrated energy decay estimate

$\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\mathcal{I},\delta}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]+\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\textup{deg}}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]$
(127)

for any $n\in\mathbb{N}_{0}$. An analogous integrated energy decay estimate
holds for $\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$.

  * •

Theorem 2 of [5] holds for $\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}$,
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}$,
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$ and
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$,
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}$,
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$.
In particular, we have

$\mathbb{I}^{n,T,\operatorname{\not{\nabla}}}_{\textup{master}}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
(128)

for any $n\in\mathbb{N}_{0}$. An analogous integrated energy decay estimate
holds for $\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$,
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}$,
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$.

The two following statements are corollaries of Proposition 8.3 and collect
the integrated energy decay estimates for the gauge invariant linearised
quantities which are necessary for our later gauge dependent analysis.

######  Corollary 8.4.

Consider a solution $\mathfrak{S}$ to the linearised system of equations.
Assume that the initial energy flux (92) is finite. Then, for any $\delta>0$,
the estimate (127) implies the integrated energy decay estimates

$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[n]}(r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P})|^{2}\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]\,,$
(129)
$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[n+1]}(r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P})|^{2}\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]$
(130)

for any $n\in\mathbb{N}_{0}$. Analogous integrated energy decay estimates hold
for $\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{P}}$,
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}}$. In particular, the
initial energy flux
$\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi}]$
controls $\mathcal{A}^{[2]}(r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P})$
without degeneration (at $r=3M$) and
$\mathcal{A}^{[3]}(r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P})$ with
degeneration (at $r=3M$).

######  Corollary 8.5.

Consider a solution $\mathfrak{S}$ to the linearised system of equations.
Assume that the initial energy fluxes (94)-(99) are finite. Then, for any
$\delta>0$, we have the integrated energy decay estimates

$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}|^{2}\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]\,,$
(131)
$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r^{3}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}|^{2}\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}]$
(132)

and

$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}|^{2}\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,,$
(133)
$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}|^{2}\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]\,.$
(134)

The integrated energy decay estimate (128) implies the integrated energy decay
estimates

$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,(|\mathcal{A}^{[n]}(r^{4}(\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}))|^{2}+|\mathcal{A}^{[n]}(r^{3}(\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}))|^{2})\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,,$
(135)
$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,(|\mathcal{A}^{[n]}(r^{3}(\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}))|^{2}+|\mathcal{A}^{[n]}(r(\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}))|^{2})\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]$
(136)

and the degenerate integrated energy decay estimates

$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[n+1]}(r^{4}(\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}))|^{2}\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,,$
(137)
$\displaystyle\int_{u_{0}}^{\infty}d\bar{u}\int_{v_{0}}^{\infty}d\bar{v}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[n+1]}(r^{3}(\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}}))|^{2}\lesssim\mathbb{F}^{n,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]\,.$
(138)

In particular, the initial energy flux
$\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
controls
$\mathcal{A}^{[3]}(r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})$ without
degeneration (at $r=3M$) and
$\mathcal{A}^{[4]}(r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})$ with
degeneration (at $r=3M$). An analogous statement holds for the initial energy
flux
$\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]$.

We state the following top-order integrated energy decay estimates for angular
derivatives of $\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$
and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$,
corresponding to Proposition 12.3.2 of [5] with $i=5$.

######  Proposition 8.6 (Top-order integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}$).

Consider a solution $\mathfrak{S}$ to the linearised system of equations.
Assume that the initial energy fluxes (98) and (99) are finite. Then, for any
$\delta>0$, $v\geq v_{0}$ and $u\geq u_{0}$, we have the integrated energy
decay estimates

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[5]}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}})|^{2}$
$\displaystyle\lesssim\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,,$
$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[5]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}})|^{2}$
$\displaystyle\lesssim\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\Psi}},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\alpha}}}}]\,.$

##  9 Analysis of the gauge dependent linearised quantities

In this section, we present the analysis of the gauge dependent linearised
quantities in the system.

###  9.1 Horizon energy fluxes

The following propositions state the necessary estimates for the horizon
energy fluxes.

######  Proposition 9.1.

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy flux (94) is finite. Then,

$\int_{\mathcal{H}^{+}(v_{0},\infty)}dv\sin\theta\,d\theta\,d\phi\,\left(\frac{9}{4}\,\rho^{2}|\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}|^{2}+|\not{\mathcal{D}}_{2}^{\star}\,\not{\textup{div}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}|+|\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|^{2}+\frac{6M}{r^{3}}\,|\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}|^{2}\right)\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]$
(139)

and

$\int_{\mathcal{H}^{+}(v_{0},\infty)}dv\sin\theta\,d\theta\,d\phi\,(|\operatorname{\not{\nabla}}_{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}|^{2}+|\not{\mathcal{D}}_{2}^{\star}\,\not{\textup{div}}\operatorname{\not{\nabla}}_{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}|+|\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|^{2}+|\not{\textup{div}}\,\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|^{2}+|\not{\mathcal{D}}_{2}^{\star}\,\not{\textup{div}}\,\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}|^{2})\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]\,.$
(140)

###### Proof.

In view of the initial-data normalisation identities of Definition 6.1, the
proof is completely analogous to that of Propositions 13.2.1 and 13.2.2 in
[5]. ∎

######  Proposition 9.2.

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy flux (94) is finite. Then,

$\int_{\mathcal{H}^{+}(v_{0},\infty)}dv\sin\theta\,d\theta\,d\phi\,(|\not{\textup{div}}\,\not{\mathcal{D}}^{\star}_{2}\operatorname{\not{\nabla}}\not{\textup{div}}\,\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2}+|\not{\textup{div}}\,\not{\mathcal{D}}^{\star}_{2}{}^{\star}\operatorname{\not{\nabla}}\not{\textup{curl}}\,\not{\textup{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2})\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]\,.$
(141)

###### Proof.

From the estimate (139) and Proposition 8.3, we control the horizon flux

$\int_{\mathcal{H}^{+}(v_{0},\infty)}dv\sin\theta\,d\theta\,d\phi\,|\not{\text{div}}\,\not{\mathcal{D}}^{\star}_{2}\not{\mathcal{D}}^{\star}_{1}(-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})|^{2}\lesssim\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi}]$

(see the analogous estimate (335) of Proposition 13.2.3 in [5]). We recall
that $(\text{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})|_{\ell\geq
1}=0$ on $\mathcal{H}^{+}$, and thus the equation (68) on $\mathcal{H}^{+}$
yields

$\not{\text{div}}\,\not{\mathcal{D}}^{\star}_{2}\operatorname{\not{\nabla}}\not{\text{div}}\,\not{\text{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}=-2\,\not{\text{div}}\,\not{\mathcal{D}}^{\star}_{2}\operatorname{\not{\nabla}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}\,,$

which allows to control the first term in (141). For the second term in (141),
we exploit the identity

$\not{\text{div}}\,\not{\mathcal{D}}^{\star}_{2}{}^{\star}\operatorname{\not{\nabla}}\not{\text{curl}}\,\not{\text{div}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}=2\,\not{\text{div}}\,\not{\mathcal{D}}^{\star}_{2}{}^{\star}\operatorname{\not{\nabla}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}$

holding on $\mathcal{H}^{+}$ for any initial-data normalised solution
$\mathfrak{S}_{N}$. ∎

######  Remark 9.3.

Higher order horizon fluxes of the form of Proposition 13.2.3 in [5] can also
be obtained.

######  Remark 9.4.

At this stage, one can already establish polynomial decay in $v$ along the
event horizon $\mathcal{H}^{+}$ for some of the horizon fluxes.

###  9.2 Integrated energy decay for the linearised outgoing shear

In this section, we state integrated energy decay for the linearised outgoing
shear. The proof can be obtained by following the proof of Propositions 13.3.1
and 13.3.3 in [5].

######  Proposition 9.5 (Integrated decay for the linearised outgoing shear).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy flux (96) and energy (101) are finite. Then, for any
$\delta>0$, $v\geq v_{0}$ and $u\geq u_{0}$, we have

$\displaystyle\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(|\operatorname{\not{\nabla}}_{3}^{2}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})|^{2}+|\operatorname{\not{\nabla}}_{3}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})|^{2}+|r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}|^{2}\right)\lesssim$
$\displaystyle\,\lVert\operatorname{\not{\nabla}}_{3}^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}$
(142)
$\displaystyle+\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$

and

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(|\operatorname{\not{\nabla}}_{3}^{2}(\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}))|^{2}+|\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}))|^{2}+|\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})|^{2}\right)\lesssim\mathbb{E}_{0}\,.$
(143)

###  9.3 Integrated energy decay for the linearised ingoing shear

In this section, we state and prove integrated energy decay for the linearised
ingoing shear.

######  Proposition 9.6 (Integrated decay for the linearised ingoing shear).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy flux (96) and energy (101) are finite. Then, for any
$\delta>0$, $v\geq v_{0}$ and $u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}\lesssim\lVert\operatorname{\not{\nabla}}_{3}^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
(144)

and

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(145)

###### Proof.

Let $r_{2}=10M$. The red-shift estimate

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}(r_{2},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}|\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}\lesssim\lVert\operatorname{\not{\nabla}}_{3}^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
(146)

is immediate from equation (52) written in the form

$\operatorname{\not{\nabla}}_{4}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})+\operatorname{\hat{\omega}}\,(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})=\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\,.$

To establish (9.3), one contracts the equation with
$r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}$,
applies the Cauchy-Schwartz inequality, integrates over
$[u_{0}(r_{2},v),u]\times[v_{0},v]\times\mathbb{S}^{2}_{u,v}$ and, using
Proposition 9.5, obtains the estimate

$\displaystyle\sup_{\bar{v}\in(v_{0},v)}$
$\displaystyle\int_{u_{0}(r_{2},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}|r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}$
$\displaystyle+\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}(r_{2},\bar{v})}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}|r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}\lesssim\lVert\operatorname{\not{\nabla}}_{3}^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,.$

The first term on the left hand side and the weights in $r$ can be dropped.

We now wish to extend (9.3) to a global (in $r$) estimate on
$\mathcal{R}_{u_{0},v_{0}}$. We write equation (52) as

$\operatorname{\not{\nabla}}_{4}(\Omega^{2}r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})=\frac{\Omega^{2}}{r^{2}}\,(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})\,.$

Let $r_{1}=5M$ and $\xi=\xi(r)$ a cut-off function such that $\xi\equiv 0$ for
$2M\leq r\leq r_{1}$ and $\xi=1$ for $r\geq r_{2}$. We contract the equation
with
$\Omega^{2}r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}$,
apply the Cauchy-Schwartz inequality ($\epsilon>0$), take an overall
multiplication by $r^{-\delta}\xi$ and obtain

$\frac{1}{2}\,\operatorname{\not{\nabla}}_{4}(r^{-\delta}\xi\cdot|\Omega^{2}r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2})-r^{-\delta}(\operatorname{\not{\nabla}}_{4}\xi)\cdot|\Omega^{2}r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}+\delta\,\frac{\Omega^{2}}{r^{1+\delta}}\,\xi\cdot|\Omega^{2}r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}-\epsilon\,\frac{\Omega^{2}}{2\,r^{2+\delta}}\,\xi\cdot|\Omega^{2}r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}\leq\frac{\Omega^{2}}{2\,\epsilon\,r^{2+\delta}}\,\xi\cdot|r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}|^{2}\,.$

For any $\delta>0$, the constant $\epsilon$ can be chosen sufficiently small
so that the fourth term on the left hand side can be absorbed by the third
one. The second term is only supported on $r_{1}\leq r\leq r_{2}$ and, upon
integration, can be controlled by the estimate (9.3). We integrate over
$[u_{0},u(r_{1},v)]\times[v_{0},v]\times\mathbb{S}^{2}_{u,v}$ and, using the
fact that $\Omega^{2}$ is bounded below away from $\mathcal{H}^{+}$ and again
the estimate (9.3) to bound the fixed-$r$ boundary term arising, we obtain the
estimate

$\displaystyle\sup_{\bar{v}\in(v_{0},v)}\int_{u_{0}}^{u(r_{1},\bar{v})}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}r^{-\delta}\cdot|r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}$
(147)
$\displaystyle+\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u(r_{1},\bar{v})}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\Omega^{2}r^{-1-\delta}\cdot|r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}|^{2}\lesssim\lVert\operatorname{\not{\nabla}}_{3}^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\lVert\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}}\rVert_{L^{2}(\underline{C}_{v_{0}})}^{2}+\mathbb{F}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]\,.$

Combining the estimates (9.3) and (147) yields the final estimate (144).

To obtain the higher order estimate (145), one simply commutes equation (52)
with the angular operator $\mathcal{A}^{[3]}$ and repeats the argument above.
∎

###  9.4 Integrated energy decay for the remaining linearised quantities

In this section, we state and prove integrated energy decay for all the
remaining linearised quantities in the system. Interestingly, our scheme never
involves integrating an ingoing transport equation for a gauge dependent
linearised quantity in the ingoing direction. We remark that all the
linearised quantities that we estimate are supported on $\ell\geq 2$-spherical
modes.

######  Proposition 9.7 (Integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,(|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})|^{2}+|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}})|^{2})\lesssim\mathbb{E}_{0}\,.$
(148)

###### Proof.

The proposition follows from the identities

$\displaystyle\mathcal{A}^{[3]}(r^{4}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi})$
$\displaystyle=\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}})-\frac{3M}{r}\,\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})\,,$
$\displaystyle\mathcal{A}^{[3]}(r^{3}\overset{\text{\scalebox{0.6}{$(1)$}}}{\underline{\psi}})$
$\displaystyle=\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}})+\frac{3M}{r}\,\mathcal{A}^{[3]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})$

and the combination of Propositions 9.5 and 9.6 with Proposition 8.3. ∎

######  Proposition 9.8 (Integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(149)

###### Proof.

One exploits the $\mathcal{A}^{[3]}$-commuted equation (53) and Propositions
9.5 and 9.6. ∎

######  Proposition 9.9 (Integrated decay for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}|^{2}\lesssim\mathbb{E}_{0}\,.$
(150)

###### Proof.

We commute equation (40) with
$\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}$ and use the initial-data
normalisation identity

$\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3}=\frac{r^{2}}{2M}(\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}-\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}})$
(151)

to obtain the red-shifted transport equation

$\operatorname{\not{\nabla}}_{4}(\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})+\frac{1}{2}\,(\text{tr}\chi)(\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})+\operatorname{\hat{\omega}}\,(\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})=\frac{1}{r}\,\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})\,.$

The proof then follows the scheme of the proof of Proposition 9.6 and exploits
Proposition 9.8. The proof of the red-shift estimate is completely analogous.
For integrated decay in the large-$r$ region, one writes the equation as

$\operatorname{\not{\nabla}}_{4}(\Omega^{2}\,\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})+\frac{\Omega^{2}}{r}\,(\Omega^{2}\,\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})=\frac{1}{r}\,\Omega^{2}\,\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})$

and follows the same procedure adopted to prove the estimate (147). ∎

######  Proposition 9.10 (Integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(152)

###### Proof.

One uses the $\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}$-commuted red-
shifted equation (63) combined with Propositions 9.7 and 9.9. The scheme of
the proof is analogous to that of Proposition 9.6. ∎

######  Remark 9.11.

Exploiting the identity (151) and Proposition 9.9, one can immediately show
that integrated energy decay for (angular derivatives of) the difference
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}-\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}$
gains a power of $r$ compared to integrated energy decay for (angular
derivatives of) $\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}}$
(Proposition 9.8) and (angular derivatives of)
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\zeta}}$ (Proposition
9.10) separately.

######  Proposition 9.12 (Top-order integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(153)

###### Proof.

The estimate can be proven by direct application of the identity

$\mathcal{A}^{[3]}(r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P})=\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\not{\mathcal{D}}_{1}^{\star}(-r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}},r^{3}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}})-\frac{3M}{r}\,\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}})+3M\,\Omega^{2}\,(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})+6M\,\Omega^{2}\,(r\not{\mathcal{D}}_{2}^{\star}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\not{\mathfrak{f}}}}_{3})$

and Propositions 8.3, 9.5, 9.6 and 9.9. We recall that the initial energy flux
$\mathbb{F}^{2,T,\operatorname{\not{\nabla}}}_{0}[\overset{\text{\scalebox{0.6}{$(1)$}}}{\Psi},\mathfrak{D}\overset{\text{\scalebox{0.6}{$(1)$}}}{\psi},\mathfrak{D}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\alpha}}]$
only controls
$\mathcal{A}^{[3]}(r^{5}\overset{\text{\scalebox{0.6}{$(1)$}}}{P})$ with
degeneration at trapping ($r=3M$). ∎

######  Remark 9.13.

In contrast with Proposition 9.7, the (degenerate) integrated decay estimate
from Proposition 9.12 already controls the linearised curvature components
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\rho}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\sigma}}$ at top order,
without losing derivatives. We will establish an analogous top-order estimate
for $\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\beta}}$ and
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}}$
in Proposition 9.27.

######  Proposition 9.14 (Integrated decay for
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}})$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r^{2}\Omega^{-4}(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\chi}))|^{2}\lesssim\mathbb{E}_{0}\,.$
(154)

###### Proof.

We write equation (50) as the red-shifted equation

$\operatorname{\not{\nabla}}_{4}(\Omega^{-4}\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}}))+(\text{tr}\chi)(\Omega^{-4}\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}}))+\operatorname{\hat{\omega}}\,(\Omega^{-4}\,(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}}))=0$

and recall Proposition 6.22. One can then commute with
$\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}$
and apply the scheme of the proof of Proposition 9.6. ∎

######  Remark 9.15.

Proposition 9.14 can, in principle, be proven at the very start of our scheme
to control the gauge dependent linearised quantities. However, the proposition
is only needed at this stage of the scheme, when one wants to prove the
integrated energy decay estimate of Proposition 9.16.

######  Proposition 9.16 (Integrated decay for
$(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}})$
and
$(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}})$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\left(1-\frac{3M}{r}\right)^{2}(|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r(\overset{\text{\scalebox{0.6}{$(1)$}}}{\textup{tr}\operatorname{\underline{\chi}}}))|^{2}+|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}}))|^{2})\lesssim\mathbb{E}_{0}\,.$
(155)

###### Proof.

We use the
$\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}$-commuted
red-shifted equations (54) and (58) and Propositions 9.14 and 9.12. We note
that, in particular, the degeneration at trapping ($r=3M$) is introduced by
Proposition 9.12. ∎

######  Proposition 9.17 (Integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}}|^{2}\lesssim\mathbb{E}_{0}\,.$
(156)

###### Proof.

We use the $\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}$-commuted red-
shifted equation (61) and Propositions 9.8 and 9.7. ∎

######  Proposition 9.18 (Integrated decay for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}(r^{-1}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(157)

###### Proof.

We use the $\mathcal{A}^{[3]}r\not{\mathcal{D}}_{2}^{\star}$-commuted red-
shifted equation (37) and Proposition 9.8. ∎

######  Proposition 9.19 (Integrated decay for
$\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r\,\not{\textup{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\hat{\operatorname{\underline{\omega}}}}}|^{2}\lesssim\mathbb{E}_{0}\,.$
(158)

###### Proof.

We exploit the
$r\,\not{\text{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}$-commuted
red-shifted equation (62). We note the auxiliary identity

$\displaystyle\Omega^{2}\,(r\,\not{\text{div}}\,r^{2}\,\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3})=$
$\displaystyle\,-\Omega^{2}\,r\,(r\,\not{\text{div}}\,r\not{\mathcal{D}}_{2}^{\star}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\xi}}}})+2\,\mathcal{A}^{[3]}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\underline{\chi}}}}})-r\,\not{\text{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\operatorname{\underline{\chi}}}}))$
(159)
$\displaystyle+r\,\not{\text{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}{}^{\star}\operatorname{\not{\nabla}}(r(\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{\varepsilon}\cdot\operatorname{\underline{\chi}}}))-r\,\not{\text{div}}\,r\not{\mathcal{D}}_{2}^{\star}(r\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\eta}})-2\,r\,\not{\text{div}}\,r\not{\mathcal{D}}_{2}^{\star}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\underline{\beta}}}})$

to replace
$(\operatorname{\not{\nabla}}_{4}\operatorname{\hat{\omega}})(r\,\not{\text{div}}\,r^{2}\,\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3})$
on the right hand side. The identity (159) is obtained by taking the
$r\,\not{\text{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}$-derivative of
equation (67) and then commuting it by a
$\operatorname{\not{\nabla}}_{3}$-derivative. ∎

######  Proposition 9.20 (Integrated decay for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|r\,\not{\textup{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r^{-1}\overset{\text{\scalebox{0.6}{$(1)$}}}{\mathfrak{\underline{\mathfrak{f}}}}_{3})|^{2}\lesssim\mathbb{E}_{0}\,.$
(160)

###### Proof.

We exploit the
$r\,\not{\text{div}}\,r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}$-commuted
red-shifted equation (36) and Proposition 9.19. ∎

######  Proposition 9.21 (Integrated decay for
$(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\Omega^{-2}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}}))|^{2}\lesssim\mathbb{E}_{0}\,.$
(161)

###### Proof.

We exploit the
$\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}$-commuted
red-shifted equation

$\operatorname{\not{\nabla}}_{4}(\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\Omega^{-2}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})))+\operatorname{\hat{\omega}}\,(\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(\Omega^{-2}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})))=\frac{2\Omega^{2}}{r^{2}}\,\mathcal{A}^{[3]}r^{2}\not{\mathcal{D}}_{2}^{\star}\operatorname{\not{\nabla}}(r^{2}\,\Omega^{-4}(\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\text{tr}\chi}}))$

obtained from equation (46) and Proposition 9.14. We note that (angular
derivatives of)
$r\operatorname{\not{\nabla}}(\Omega^{-2}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}}))$
are uniformly bounded on $\underline{C}_{v_{0}}$ for smooth seed initial data
by the initial-data normalisation
$\operatorname{\not{\nabla}}(\textup{tr}\overset{\text{\scalebox{0.6}{$(1)$}}}{\not{g}})=0$
on $\mathbb{S}^{2}_{\infty,v_{0}}$. ∎

######  Proposition 9.22 (Integrated decay for
$\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}$).

Let $\mathfrak{S}_{N}$ be an initial-data normalised solution. Assume that the
initial energy (101) is finite. Then, for any $\delta>0$, $v\geq v_{0}$ and
$u\geq u_{0}$, we have

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}|^{2}\lesssim\mathbb{E}_{0}\,.$
(162)

###### Proof.

We exploit the $\operatorname{\not{\nabla}}_{3}\mathcal{A}^{[3]}$-commuted
red-shifted equation

$\operatorname{\not{\nabla}}_{4}(\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}))+\operatorname{\hat{\omega}}\,(\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}}))=\frac{2}{r^{2}}\,\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}))+\frac{4}{r^{3}}\,(\mathcal{A}^{[3]}(r^{2}\operatorname{\overset{\text{\scalebox{0.6}{$(1)$}}}{\operatorname{\widehat{\chi}}}}))$
(163)

obtained from commuting equation (44) and Proposition 9.5 to obtain the
integrated energy decay estimate

$\int_{v_{0}}^{v}d\bar{v}\int_{u_{0}}^{u}d\bar{u}\int_{\mathbb{S}^{2}_{\bar{u},\bar{v}}}\sin\theta\,d\theta\,d\phi\,\frac{\Omega^{2}}{r^{1+\delta}}\,|\operatorname{\not{\nabla}}_{3}(\mathcal{A}^{[3]}\overset{\text{\scalebox{0.6}{$(1)$}}}{\widehat{\not{g}}})|^{2}\lesssim\mathbb{E}_{0}\,.$
(164)

Let $r_{1}=10M$. By applying an argument as in Lemma 13.3.1 of [5], one
obtains the estimate