$\displaystyle=\mathscr{O}_{M}(1;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))^{c}_{ab}\text{.}$

###### Proof.

See Appendix 7.7. ∎

The following two propositions contain our main wave-transport system. In
particular, the key step in the proof of our main result is a unique
continuation result on this system.

######  Proposition 3.13.

Let $(\mathscr{M},g)$ and $(\mathscr{M},\check{g})$ denote two vacuum FG-aAdS
segments. Then,

(3.16) $\displaystyle\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}})$
$\displaystyle=\mathsf{m}-\check{\mathsf{m}}\text{,}$
$\displaystyle\mathscr{L}_{\rho}\mathsf{Q}$
$\displaystyle=\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(\rho;\mathsf{Q})\text{,}$
$\displaystyle\mathscr{L}_{\rho}[\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})]$
$\displaystyle=-2\rho^{-1}\mathsf{W}^{2}+\mathscr{O}_{M-3}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(1;\mathsf{Q})+\mathscr{O}_{M-2}(1;\mathsf{m}-\check{\mathsf{m}})\text{,}$
$\displaystyle\mathscr{L}_{\rho}\mathsf{B}$
$\displaystyle=\mathscr{S}(\mathsf{W}^{1})+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(\rho;\mathsf{Q})+\mathscr{O}_{M-1}(1;\mathsf{m}-\check{\mathsf{m}})+\mathscr{O}_{M-2}(\rho;\mathsf{B})\text{.}$

In addition, the following derivative transport equations hold:

(3.17)
$\displaystyle\mathscr{L}_{\rho}\mathsf{D}(\mathsf{g}-\check{\mathsf{g}})$
$\displaystyle=\mathsf{D}(\mathsf{m}-\check{\mathsf{m}})+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})\text{,}$
$\displaystyle\mathscr{L}_{\rho}\mathsf{D}\mathsf{Q}$
$\displaystyle=\mathscr{O}_{M-2}(\rho;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))+\mathscr{O}_{M-2}(\rho;\mathsf{D}\mathsf{Q})+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(\rho;\mathsf{Q})\text{,}$
$\displaystyle\mathscr{L}_{\rho}\mathsf{D}[\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})]$
$\displaystyle=-2\rho^{-1}\mathsf{D}\mathsf{W}^{2}+\mathscr{O}_{M-3}(1;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))+\mathscr{O}_{M-3}(1;\mathsf{D}\mathsf{Q})+\mathscr{O}_{M-2}(1;\mathsf{D}(\mathsf{m}-\check{\mathsf{m}}))$
$\displaystyle\qquad+\mathscr{O}_{M-4}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-4}(1;\mathsf{Q})+\mathscr{O}_{M-3}(1;\mathsf{m}-\check{\mathsf{m}})\text{,}$
$\displaystyle\mathscr{L}_{\rho}\mathsf{D}\mathsf{B}$
$\displaystyle=\mathscr{S}(\mathsf{D}\mathsf{W}^{1})+\mathscr{O}_{M-3}(\rho;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))+\mathscr{O}_{M-3}(\rho;\mathsf{D}\mathsf{Q})+\mathscr{O}_{M-1}(1;\mathsf{D}(\mathsf{m}-\check{\mathsf{m}}))$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho;\mathsf{D}\mathsf{B})+\mathscr{O}_{M-4}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-4}(\rho;\mathsf{Q})$
$\displaystyle\qquad+\mathscr{O}_{M-2}(1;\mathsf{m}-\check{\mathsf{m}})+\mathscr{O}_{M-3}(\rho;\mathsf{B})\text{.}$

###### Proof.

See Appendix 7.8. ∎

######  Proposition 3.14.

Let $(\mathscr{M},g)$ and $(\mathscr{M},\check{g})$ denote two vacuum FG-aAdS
segments. Then,

(3.18) $\displaystyle\bar{\Box}\mathsf{W}^{2}+2(n-2)\mathsf{W}^{2}$
$\displaystyle=\mathsf{F}^{2}\text{,}$
$\displaystyle\bar{\Box}\mathsf{W}^{1}+(n-1)\mathsf{W}^{1}$
$\displaystyle=\mathsf{F}^{1}\text{,}$
$\displaystyle\bar{\Box}\mathsf{W}^{\star}$
$\displaystyle=\mathsf{F}^{\star}\text{,}$

where each
$\mathsf{F}\in\\{\mathsf{F}^{2},\mathsf{F}^{1},\mathsf{F}^{\star}\\}$ is
schematically of the form

(3.19) $\displaystyle\mathsf{F}$
$\displaystyle=\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{Q})+\mathscr{O}_{M-3}(\rho;\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{D}(\mathsf{m}-\check{\mathsf{m}}))_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{2};\mathsf{D}\mathsf{B})_{\bar{a}}+\sum_{\mathsf{V}\in\\{\mathsf{W}^{\star},\mathsf{W}^{1},\mathsf{W}^{2}\\}}[\mathscr{O}_{M-3}(\rho^{2};\mathsf{V})+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}\mathsf{V})]_{\bar{a}}\text{.}$

###### Proof.

See Appendix 7.9. ∎

Finally, we collect here convenient forms for the differences of the Bianchi
equations (3.5). These will be needed in another step in the proof of our main
result—showing that the quantities relating to the differences of two
geometries vanish to arbitrarily high order.

######  Proposition 3.15.

Let $(\mathscr{M},g)$ and $(\mathscr{M},\check{g})$ denote two vacuum FG-aAdS
segments. Then,

(3.20) $\displaystyle\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}})$
$\displaystyle=\mathsf{m}-\check{\mathsf{m}}\text{,}$
$\displaystyle\mathscr{L}_{\rho}[\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})]$
$\displaystyle=-2\rho^{-1}(\mathsf{w}^{2}-\check{\mathsf{w}}^{2})+\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(1;\mathsf{m}-\check{\mathsf{m}})\text{,}$
$\displaystyle\mathscr{L}_{\rho}[\rho^{2-n}(\mathsf{w}^{2}-\check{\mathsf{w}}^{2})]$
$\displaystyle=\mathscr{O}_{M}(\rho^{2-n};\mathsf{D}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1}))+\mathscr{O}_{M-4}(\rho^{3-n};\mathsf{g}-\check{\mathsf{g}})$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{2-n};\mathsf{m}-\check{\mathsf{m}})+\mathscr{O}_{M-3}(\rho^{3-n};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{3-n};\mathsf{w}^{0}-\check{\mathsf{w}}^{0})+\mathscr{O}_{M-2}(\rho^{3-n};\mathsf{w}^{2}-\check{\mathsf{w}}^{2})\text{,}$
$\displaystyle\mathscr{L}_{\rho}[\rho^{-1}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})]$
$\displaystyle=\rho^{-1}\,\mathscr{S}(\mathsf{D}(\mathsf{w}^{2}-\check{\mathsf{w}}^{2}))+\mathscr{O}_{M-3}(1;\mathsf{g}-\check{\mathsf{g}})$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{-1};\mathsf{m}-\check{\mathsf{m}})+\mathscr{O}_{M-3}(1;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))$
$\displaystyle\qquad+\mathscr{O}_{M-1}(1;\mathsf{w}^{1}-\check{\mathsf{w}}^{1})\text{,}$
$\displaystyle\mathscr{L}_{\rho}(\mathsf{w}^{0}-\check{\mathsf{w}}^{0})$
$\displaystyle=\mathscr{S}(\mathsf{D}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1}))+\mathscr{O}_{M-2}(\rho^{-1};\mathsf{w}^{2}-\check{\mathsf{w}}^{2})$
$\displaystyle\qquad+\mathscr{O}_{M-3}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(1;\mathsf{m}-\check{\mathsf{m}})$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))+\mathscr{O}_{M-2}(\rho;\mathsf{w}^{0}-\check{\mathsf{w}}^{0})\text{.}$

###### Proof.

See Appendix 7.10. ∎

######  Remark 3.16.

The system (3.20) is sufficient to obtain higher-order vanishing of the
differences of geometric quantities at the boundary (see Proposition 5.3).
However, we will need to work with the larger system (3.16)–(3.18)—containing
renormalized quantities $\mathsf{Q}$, $\mathsf{B}$, $\mathsf{W}^{\star}$,
$\mathsf{W}^{1}$, $\mathsf{W}^{2}$—in order to close the Carleman estimates,
within which we cannot afford a loss in derivatives.

##  4\. The Carleman Estimates

In this section, we state the two Carleman estimates for vertical tensor
fields that constitute the main analytic ingredients for the proof of our main
results.

###  4.1. The Wave Carleman Estimate

The first Carleman estimate we discuss is that for wave equations—namely, the
main results obtained in [18, 30, 31, 45]. We begin by discussing the best-
known conditions needed on the conformal boundary for such an estimate to
hold.

######  Definition 4.1.

Let $(\mathscr{M},g)$ be an FG-aAdS segment, and let $\mathfrak{h}$ be a
Riemannian metric on $\mathscr{I}$.

  * •

We can also view $\mathfrak{h}$ as a $\rho$-independent vertical Riemannian
metric.424242See Remark 2.3.

  * •

For a vertical tensor field $\mathsf{A}$, we write
$|\mathsf{A}|_{\mathfrak{h}}$ to denote its pointwise $\mathfrak{h}$-norm. In
other words, if $\mathsf{A}$ has rank $(k,l)$, then with respect to any
coordinate system $(U,\varphi)$ on $\mathscr{I}$, we have

(4.1)
$|\mathsf{A}|_{\mathfrak{h}}^{2}=\mathfrak{h}_{a_{1}c_{1}}\dots\mathfrak{h}_{a_{k}c_{k}}\mathfrak{h}^{b_{1}d_{1}}\dots\mathfrak{h}^{b_{l}d_{l}}\mathsf{A}^{a_{1}\dots
a_{k}}_{b_{1}\dots b_{l}}\mathsf{A}^{c_{1}\dots c_{k}}_{d_{1}\dots
d_{l}}\text{.}$

######  Remark 4.2.

The metric $\mathfrak{h}$ is only used as a coordinate-independent way to
measure the sizes of vertical tensor fields. Our main results will not depend
on a particular choice of $\mathfrak{h}$.

######  Definition 4.3 (Definition 3.1 of [18]).

Let $(\mathscr{M},g)$ be a vacuum FG-aAdS segment, let $\mathfrak{h}$ be a
Riemannian metric on $\mathscr{I}$, and let $\mathscr{D}\subset\mathscr{I}$ be
open with compact closure. We say $(\mathscr{D},\mathfrak{g})$ satisfies the
_generalized null convexity criterion_ (or _GNCC_) iff there exist $\eta\in
C^{4}(\bar{\mathscr{D}})$ and $c>0$ satisfying

(4.2)
$\displaystyle\begin{cases}\left(\mathfrak{D}^{2}\eta+\frac{1}{n-2}\,\eta\,\mathfrak{Rc}\right)(\mathfrak{Z},\mathfrak{Z})>c\eta\,\mathfrak{h}(\mathfrak{Z},\mathfrak{Z})&\text{in
}\mathscr{D}\text{,}\\\ \eta>0&\text{in }\mathscr{D}\text{,}\\\
\eta=0&\text{on }\partial\mathscr{D}\text{,}\end{cases}$

for all vectors fields $\mathfrak{Z}$ on $\mathscr{D}$ satisfying
$\mathfrak{g}(\mathfrak{Z},\mathfrak{Z})=0$.

######  Remark 4.4.

In Definition 4.3, we specialized to vacuum FG-aAdS segments. However, the
GNCC can be directly extended, as in [18], to strongly FG-aAdS segments that
are not necessarily vacuum. In the more general setting, the Ricci curvature
$\mathfrak{Rc}$ in (4.2) is replaced by
$-\mathfrak{g}^{\scriptscriptstyle(2)}$.434343Note that for vacuum FG-aAdS
segments, (2.14) implies
$-\mathfrak{g}^{\scriptscriptstyle(2)}(\mathfrak{Z},\mathfrak{Z})=\frac{1}{n-2}\,\mathfrak{Rc}(\mathfrak{Z},\mathfrak{Z})$
for any $\mathfrak{g}$-null vector field $\mathfrak{Z}$.

Next, we recall some quantities that will be essential to our Carleman
estimates:

######  Definition 4.5.

Assume the setting of Definition 4.3—in particular, let
$(\mathscr{D},\mathfrak{g})$ satisfy the GNCC, with $\eta$ satisfying (4.2).
Within this setting, we define the following additional quantities:

  * •

Let $f_{\eta}:=f\in C^{4}((0,\rho_{0}]\times\mathscr{D})$ denote the function

(4.3) $f:=\rho\eta^{-1}\text{.}$

  * •

In addition, given $f_{\star}>0$, we define the domain

(4.4) $\Omega_{f_{\star}}:=\\{z\in\mathscr{M}\mid f(z)<f_{\star}\\}\text{.}$

We can now state the precise form of the Carleman estimate for wave equations
from [18]. Here, to slightly simplify the presentation, we express this in a
less general form than in [18].

######  Theorem 4.6 (Theorem 5.11 of [18]).

Let $(\mathscr{M},g)$ be a vacuum FG-aAdS segment. In addition:

  * •

Let $\mathfrak{h}$ be a Riemannian metric on $\mathscr{I}$, and
$\mathscr{D}\subset\mathscr{I}$ be open with compact closure.

  * •

Assume $(\mathscr{D},\mathfrak{g})$ satisfies the GNCC, with $\eta\in
C^{4}(\bar{\mathscr{D}})$ as in (4.2) and $\mathfrak{h}$ as above.

  * •

Fix integers $k,l\geq 0$ and a constant $\sigma\in\mathbb{R}$.

Then, there exist $\mathcal{C}_{0}\geq 0$ and $\mathcal{C},\mathcal{C}_{b}>0$
(depending on $\mathsf{g}$, $\mathfrak{h}$, $\mathscr{D}$, $k$, $l$) such that
for any $\kappa\in\mathbb{R}$ with

(4.5) $2\kappa\geq
n-1+\mathcal{C}_{0}\text{,}\qquad\kappa^{2}-(n-2)\kappa+\sigma-(n-1)-\mathcal{C}_{0}\geq
0\text{,}$

and for any constants $f_{\star},\lambda,p>0$ with

(4.6)
$0<f_{\star}\ll_{\mathsf{g},\mathfrak{h},\mathscr{D},k,l}1\text{,}\qquad\lambda\gg_{\mathsf{g},\mathfrak{h},\mathscr{D},k,l}|\kappa|+|\sigma|\text{,}\qquad
0<2p<1\text{,}$

the following Carleman estimate holds for any vertical tensor field
$\mathsf{\Phi}$ on $\mathscr{M}$ of rank $(k,l)$ such that both
$\mathsf{\Phi}$ and $\smash{\bar{\nabla}}\mathsf{\Phi}$ vanish identically on
$\\{f=f_{\star}\\}$:444444For notational convenience, we replaced the
parameter $\lambda$ in [18] by $\lambda/2$ here.

(4.7) $\displaystyle\int_{\Omega_{f_{\star}}}e^{-\lambda
p^{-1}f^{p}}f^{n-2-p-2\kappa}|(\bar{\Box}+\sigma)\mathsf{\Phi}|_{\mathfrak{h}}^{2}\,d\mu_{g}$
$\displaystyle\quad\qquad+\mathcal{C}_{b}\lambda^{3}\limsup_{\rho_{\star}\searrow
0}\int_{\Omega_{f_{\star}}\cap\\{\rho=\rho_{\star}\\}}[|\bar{\mathsf{D}}_{\partial_{\rho}}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+|\mathsf{D}(\rho^{-\kappa}\mathsf{\Phi})|_{\mathfrak{h}}^{2}+|\rho^{-\kappa-1}\mathsf{\Phi}|_{\mathfrak{h}}^{2}]\,d\mu_{\mathsf{g}}$
$\displaystyle\quad\geq\mathcal{C}\lambda\int_{\Omega_{f_{\star}}}e^{-\lambda
p^{-1}f^{p}}f^{n-2-2\kappa}(\rho^{4}|\bar{\mathsf{D}}_{\partial_{\rho}}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathsf{D}\mathsf{\Phi}|_{\mathfrak{h}}^{2}+f^{2p}|\mathsf{\Phi}|_{\mathfrak{h}}^{2})\,d\mu_{g}\text{.}$

Here, $d\mu_{g}$ denotes the volume form on $\mathscr{M}$ induced by the
spacetime metric $g$, while $d\mu_{\mathsf{g}}$ denotes the volume forms on
the level sets of $\rho$ induced by the vertical metric $\mathsf{g}$.

######  Remark 4.7.

We note that Theorem 4.6 only considers vacuum FG-aAdS segments, whereas the
more general [18, Theorem 5.11] also allows for some non-vacuum FG-aAdS
segments (under a more general GNCC). Moreover, [18, Theorem 5.11] allows for
an additional first-order term in the wave equation that is vanishing at a
slower “critical” rate toward the conformal boundary.454545See the quantity
$X$ in [18, Theorem 5.11]; however, we will not need this extra generality
here.

######  Remark 4.8.

If $k=l=0$ (i.e. $\mathsf{\Phi}$ is scalar), then Theorem 4.6 holds with
$\mathcal{C}_{0}=0$; see [18].

###  4.2. The Transport Carleman Estimate

Next, we prove a simple Carleman estimate for transport equations, in the same
setting and with the same weights as in Theorem 4.6.

######  Proposition 4.9.

Let $(\mathscr{M},g)$ be a vacuum FG-aAdS segment. In addition:

  * •

Let $\mathfrak{h}$ be a Riemannian metric on $\mathscr{I}$, and
$\mathscr{D}\subset\mathscr{I}$ be open with compact closure.

  * •

Assume $(\mathscr{D},\mathfrak{g})$ satisfies the GNCC, with $\eta\in
C^{4}(\bar{\mathscr{D}})$ as in (4.2) and $\mathfrak{h}$ as above.

Then, for any $s\geq 0$, $\kappa\in\mathbb{R}$, and $\lambda,f_{\star},p>0$
satisfying

(4.8) $2\kappa\geq\max(n-2,s-3)\text{,}\qquad 0<2p<1\text{,}\qquad
0<f_{\ast}\ll_{\mathsf{g},\mathscr{D}}1\text{,}$

there exist $\mathcal{C}^{\prime},\mathcal{C}_{b}^{\prime}>0$ (depending on
$\mathsf{g}$, $\mathfrak{h}$, $\mathscr{D}$) such that for every vertical
tensor field $\Psi$ on $\mathscr{M}$,

(4.9) $\displaystyle\int_{\Omega_{f_{\star}}}e^{-\lambda
p^{-1}f^{p}}f^{n-2-p-2\kappa}\rho^{s+2}|\mathscr{L}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\,d\mu_{g}+\mathcal{C}_{b}^{\prime}\lambda\limsup_{\rho_{\star}\searrow
0}\int_{\Omega_{f_{\star}}\cap\\{\rho=\rho_{\ast}\\}}\rho^{s}|\rho^{-\kappa-1}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\,d\mu_{\mathsf{g}}$
$\displaystyle\quad\geq\mathcal{C}\lambda\int_{\Omega_{f_{\star}}}e^{-\lambda
p^{-1}f^{p}}f^{n-2-2\kappa}\rho^{s}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\,d\mu_{g}\text{.}$

###### Proof.

Using that both $\mathfrak{h}$ and $\eta$ are $\rho$-independent, and
recalling (4.3), we obtain

(4.10) $\displaystyle\mathscr{L}_{\rho}\left(e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)+(2\kappa+2-s+\lambda
f^{p})e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n-1}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\quad\leq e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}|\mathscr{L}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}\text{.}$

Applying the Cauchy-Schwarz inequality to the right-hand side of the above and
rearranging yields

(4.11) $\displaystyle(2\kappa+2-s)e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n-1}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}$
$\displaystyle\leq-\mathscr{L}_{\rho}\left(e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)$
$\displaystyle\qquad+\frac{1}{4}\lambda^{-1}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\rho^{s-n+1}|\mathscr{L}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{.}$

We now integrate the above inequality over the region
$\Omega_{f_{\star},\rho_{\star}}:=\Omega_{f_{\star}}\cap\\{\rho>\rho_{\star}\\}$
for an arbitrary $0<\rho_{\star}\ll f_{\star}$, first over level sets of
$\rho$ with respect to the $\rho$-independent volume forms
$d\mu_{\mathfrak{h}}$ induced by $\mathfrak{h}$, and then over $\rho$. This
yields the estimate

$\displaystyle(2\kappa+2-s)\lambda\int_{\Omega_{f_{\star},\rho_{\star}}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n-1}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\
d\mu_{\mathfrak{h}}d\rho$
$\displaystyle\quad\leq\int_{\Omega_{f_{\star},\rho_{\star}}}\left[\frac{1}{4}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\rho^{s-n+1}|\mathscr{L}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}-\lambda\mathscr{L}_{\rho}\left(e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\right)\right]d\mu_{\mathfrak{h}}d\rho$
$\displaystyle\quad\leq\frac{1}{4}\int_{\Omega_{f_{\star},\rho_{\star}}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\rho^{s-n+1}|\mathscr{L}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\,d\mu_{\mathfrak{h}}d\rho+\lambda\int_{\Omega_{f_{\star}}\cap\\{\rho=\rho_{\star}\\}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}d\mu_{\mathfrak{h}}\text{,}$

where in the last step, we applied the fundamental theorem of calculus; note
in particular that the ensuing boundary term on $\\{f=f_{\star}\\}$ is
negative and can hence be neglected. The above now yields, for some constants
$\mathcal{C}^{\prime},\mathcal{C}_{b,0}^{\prime}>0$ (depending on
$\mathsf{g}$, $\mathfrak{h}$, $\mathscr{D}$),

(4.12)
$\displaystyle\mathcal{C}^{\prime}\lambda\int_{\Omega_{f_{\star},\rho_{\star}}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}\ d\mu_{g}$
$\displaystyle\leq\frac{1}{4}\int_{\Omega_{f_{\star},\rho_{\star}}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\rho^{s+2}|\mathscr{L}_{\rho}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\,d\mu_{g}$
$\displaystyle\qquad+\mathcal{C}_{b,0}^{\prime}\lambda\int_{\Omega_{f_{\star}}\cap\\{\rho=\rho_{\star}\\}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}d\mu_{\mathsf{g}}\text{,}$

since $d\mu_{\mathfrak{h}}$ and $d\mu_{\mathsf{g}}$ are comparable due to the
compactness of $\bar{\mathscr{D}}$, and since (2.7) implies

$d\mu_{g}=\rho^{-n-1}\,d\mu_{\mathsf{g}}d\rho\text{.}$

Finally, letting $\rho_{\star}\searrow 0$ in (4.12) yields the desired (4.9),
since

$e^{-\lambda p^{-1}f^{p}}\leq 1\text{,}\qquad
f^{n-2-2\kappa}\rho^{s-n}|\mathsf{\Psi}|_{\mathfrak{h}}^{2}=\eta^{2\kappa-(n-2)}\,\rho^{s}|\rho^{-\kappa-1}\mathsf{\Psi}|_{\mathfrak{h}}^{2}\text{,}$

and since the exponent $2\kappa-(n-2)\geq 0$ by (4.8). ∎

######  Remark 4.10.

Note that the GNCC is not required for Proposition 4.9 and its proof. However,
it is convenient to include this in the statement Proposition 4.9, as the
transport Carleman estimate will be applied concurrently with and in the same
setting as the wave Carleman estimate.

##  5\. Unique Continuation

In this section, we turn toward establishing unique continuation for the
Einstein-vacuum equations. In particular, we prove the following theorem,
which can be seen as a precise statement of most of Theorem 1.5—the special
case in which the pair of holographic data are equal.464646The remainder of
Theorem 1.5, which deals with gauge covariance issues, is treated in Section
6.1 below.

######  Theorem 5.1.

Let $n>2$, and let $(\mathscr{M},g)$, $(\mathscr{M},\check{g})$ be vacuum FG-
aAdS segments (on a common aAdS region
$\mathscr{M}:=(0,\rho_{0}]\times\mathscr{I}$),474747This is the same setup as
in Section 3.2; the reader is referred there for further notational details.
with associated holographic data
$(\mathscr{I},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$,
$(\mathscr{I},\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
(respectively). In addition, let $\mathscr{D}\subset\mathscr{I}$ be open with
compact closure, and assume:

  * •

$(\mathscr{M},g)$ and $(\mathscr{M},\check{g})$ are regular to some large
enough order $M_{0}$ (depending on $\mathsf{g}$, $\check{\mathsf{g}}$,
$\mathscr{D}$).

  * •

The holographic data coincide on $\mathscr{D}$:

(5.1)
$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})|_{\mathscr{D}}=(\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})|_{\mathscr{D}}\text{.}$

  * •

$(\mathscr{D},\mathfrak{g}^{\scriptscriptstyle(0)})$ (or equivalently,
$(\mathscr{D},\check{\mathfrak{g}}^{\scriptscriptstyle(0)})$) satisfies the
GNCC.

Then, $g=\check{g}$ on a neighbourhood of $\\{0\\}\times\mathscr{D}$ (viewed
as part of the conformal boundary)—more specifically, there exists some
sufficiently small $f_{\star}>0$ such that $g=\check{g}$ on the region
$\Omega_{f_{\star}}$.484848See Definition 4.5 for the definition of
$\Omega_{f_{\star}}$.

######  Remark 5.2.

Note that Theorem 5.1 offers a more precise conclusion than Theorem 1.5—in the
special case (5.1), the isometry in Theorem 1.5 is simply the identity map.

The proof of Theorem 5.1 is given in the remainder of this section.
Throughout, we assume the hypotheses of Theorem 5.1 hold. Furthermore, we
adopt the notational conventions of Section 3.2 regarding quantities defined
with respect to $g$ and $\check{g}$.

###  5.1. Deducing Higher-Order Vanishing

For convenience, we can assume, without loss of generality, that $M_{0}-n$ is
an even natural number. The first step is to derive a sufficiently high order
of vanishing for $\mathsf{g}-\check{\mathsf{g}}$,
$\mathsf{m}-\check{\mathsf{m}}$, $\mathsf{w}^{2}-\check{\mathsf{w}}^{2}$,
$\mathsf{w}^{1}-\check{\mathsf{w}}^{1}$, and
$\mathsf{w}^{0}-\check{\mathsf{w}}^{0}$. This will ensure that we can apply
the Carleman estimates to (variants of) these quantities as needed later in
the proof.

The key is to use the equations from Proposition 3.15 satisfied by these
quantities to exchange regularity in the vertical directions for higher orders
of vanishing. More specifically, the following proposition is a quantitative
statement of the fact that if the our spacetime is regular to high enough
order, then we can achieve a corresponding order of vanishing for the above
quantities as $\rho\searrow 0$:

######  Proposition 5.3.

Consider the quantities $\mathsf{g}-\check{\mathsf{g}}$,
$\mathsf{m}-\check{\mathsf{m}}$, $\mathsf{w}^{2}-\check{\mathsf{w}}^{2}$,
$\mathsf{w}^{1}-\check{\mathsf{w}}^{1}$,
$\mathsf{w}^{0}-\check{\mathsf{w}}^{0}$. Then:

  * •

There exist vertical tensor fields $\mathsf{r}_{\mathsf{g}}$,
$\mathsf{r}_{\mathsf{m}}$ such that

(5.2)
$\displaystyle\mathsf{g}-\check{\mathsf{g}}=\rho^{M_{0}-2}\,\mathsf{r}_{\mathsf{g}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{g}}\rightarrow^{2}0\text{,}$
$\displaystyle\mathsf{m}-\check{\mathsf{m}}=\rho^{M_{0}-3}\,\mathsf{r}_{\mathsf{m}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{m}}\rightarrow^{2}0\text{.}$

  * •

There exist vertical tensor fields $\mathsf{r}_{\mathsf{w}^{2}}$,
$\mathsf{r}_{\mathsf{w}^{1}}$, $\mathsf{r}_{\mathsf{w}^{0}}$ such that

(5.3)
$\displaystyle\mathsf{w}^{2}-\check{\mathsf{w}}^{2}=\rho^{M_{0}-4}\,\mathsf{r}_{\mathsf{w}^{2}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{w}^{2}}\rightarrow^{2}0\text{,}$
$\displaystyle\mathsf{w}^{1}-\check{\mathsf{w}}^{1}=\rho^{M_{0}-3}\,\mathsf{r}_{\mathsf{w}^{1}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{w}^{1}}\rightarrow^{1}0\text{,}$
$\displaystyle\mathsf{w}^{0}-\check{\mathsf{w}}^{0}=\rho^{M_{0}-4}\,\mathsf{r}_{\mathsf{w}^{0}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{w}^{0}}\rightarrow^{2}0\text{.}$

###### Proof.

First, note that since
$\mathfrak{g}^{\scriptscriptstyle(0)}=\check{\mathfrak{g}}^{\scriptscriptstyle(0)}$
and
$\mathfrak{g}^{\scriptscriptstyle(n)}=\check{\mathfrak{g}}^{\scriptscriptstyle(n)}$,
then Theorem 2.16 implies

(5.4)
$\mathfrak{g}^{\scriptscriptstyle(k)}=\check{\mathfrak{g}}^{\scriptscriptstyle(k)}\text{,}\qquad\mathfrak{g}^{\scriptscriptstyle(\star)}=\check{\mathfrak{g}}^{\scriptscriptstyle(\star)}\text{,}\qquad
0\leq k\leq n\text{,}$

where the Fefferman-Graham coefficients
$\mathfrak{g}^{\scriptscriptstyle(2)},\dots,\mathfrak{g}^{\scriptscriptstyle(\star)},\mathfrak{g}^{\scriptscriptstyle(n)}$
and
$\check{\mathfrak{g}}^{\scriptscriptstyle(2)},\dots,\check{\mathfrak{g}}^{\scriptscriptstyle(\star)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)}$
are defined as in Theorem 2.16 and Corollary 2.18—with respect to $g$ and
$\check{g}$, respectively. The above and Corollary 2.18 then imply that there
are vertical tensor fields $\mathsf{r}_{\mathsf{g}}$,
$\mathsf{r}_{\mathsf{m}}$ satisfying

(5.5)
$\displaystyle\mathsf{g}-\check{\mathsf{g}}=\rho^{n}\,\mathsf{r}_{\mathsf{g}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{g}}\rightarrow^{M_{0}-n}0\text{,}$
$\displaystyle\mathsf{m}-\check{\mathsf{m}}=\rho^{n-1}\,\mathsf{r}_{\mathsf{m}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{m}}\rightarrow^{M_{0}-n}0\text{.}$

Moreover, the equations (3.3), along with (5.5), yield vertical tensor fields
$\mathsf{r}_{\mathsf{w}^{2}}$, $\mathsf{r}_{\mathsf{w}^{1}}$,
$\mathsf{r}_{\mathsf{w}^{0}}$ with

(5.6)
$\displaystyle\mathsf{w}^{2}-\check{\mathsf{w}}^{2}=\rho^{n-2}\,\mathsf{r}_{\mathsf{w}^{2}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{w}^{2}}\rightarrow^{M_{0}-n}0\text{,}$
$\displaystyle\mathsf{w}^{1}-\check{\mathsf{w}}^{1}=\rho^{n-1}\,\mathsf{r}_{\mathsf{w}^{1}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{w}^{1}}\rightarrow^{M_{0}-n-1}0\text{,}$
$\displaystyle\mathsf{w}^{0}-\check{\mathsf{w}}^{0}=\rho^{n-2}\,\mathsf{r}_{\mathsf{w}^{0}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{w}^{0}}\rightarrow^{M_{0}-n}0\text{.}$

The idea now is to use the Bianchi and transport system (3.20) to inductively
improve the initial orders of vanishing from (5.5) and (5.6). Observe that by
(5.5) and (5.6), the quantities

$\rho^{-n+2}(\mathsf{w}^{2}-\check{\mathsf{w}}^{2})\text{,}\qquad\rho^{-1}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})\text{,}\qquad\mathsf{w}^{0}-\check{\mathsf{w}}^{0}\text{,}\qquad\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})\text{,}\qquad\mathsf{g}-\check{\mathsf{g}}$

all vanish in the boundary limit $\rho\searrow 0$. Thus, we can now integrate
all the equations in (3.20) successively from $\rho=0$, without obtaining any
boundary terms as $\rho\searrow 0$.

Integrating the third equation in (3.20) and using (5.5)–(5.6), we deduce the
existence of a (new) vertical tensor field $\mathsf{r}_{\mathsf{w}^{2}}$ (for
brevity, we use the same notation as before) such that

(5.7)
$\mathsf{w}^{2}-\check{\mathsf{w}}^{2}=\rho^{n}\,\mathsf{r}_{\mathsf{w}^{2}}\text{,}\qquad\mathsf{r}_{\mathsf{w}^{2}}\rightarrow^{M_{0}-n-2}0\text{.}$

Integrating next the fifth part of (3.20) and using (5.5), (5.6), and the
improved asymptotics for $\mathsf{w}^{2}-\check{\mathsf{w}}^{2}$ in (5.7)
yields a vertical tensor field $\mathsf{r}_{\mathsf{w}^{0}}$ such that

(5.8)
$\mathsf{w}^{0}-\check{\mathsf{w}}^{0}=\rho^{n}\,\mathsf{r}_{\mathsf{w}^{0}}\text{,}\qquad\mathsf{r}_{\mathsf{w}^{0}}\rightarrow^{M_{0}-n-2}0\text{.}$

The next quantity in the hierarchy is $\mathsf{m}-\check{\mathsf{m}}$;
integrating the second identity in (3.20) and using (5.5) and (5.7), we deduce
the existence of a vertical tensor field $\mathsf{r}_{\mathsf{m}}$ such that

(5.9)
$\mathsf{m}-\check{\mathsf{m}}=\rho^{n+1}\mathsf{r}_{\mathsf{m}}\text{,}\qquad\mathsf{r}_{\mathsf{m}}\rightarrow^{M_{0}-n-2}0\text{.}$

Then, integrating the first part of (3.20) and applying (5.9) yields an
$\mathsf{r}_{\mathsf{g}}$ with

(5.10)
$\mathsf{g}-\check{\mathsf{g}}=\rho^{n+2}\,\mathsf{r}_{\mathsf{g}}\text{,}\qquad\mathsf{r}_{\mathsf{g}}\rightarrow^{M_{0}-n-2}0\text{.}$

Integrating finally the fourth identity of (3.20) and recalling (5.6)–(5.10)
results in an $\mathsf{r}_{\mathsf{w}^{1}}$ with

(5.11)
$\mathsf{w}^{1}-\check{\mathsf{w}}^{1}=\rho^{n+1}\,\mathsf{r}_{\mathsf{w}^{1}}\text{,}\qquad\mathsf{r}_{\mathsf{w}^{1}}\rightarrow^{M_{0}-n-3}0\text{.}$

At this point, the vanishing of all quantities has been improved by two powers
of $\rho$ at the cost of two derivatives of regularity, compared to the
initial asymptotics (5.5)–(5.6). By iterating this process (now inserting the
improved asymptotics into the right-hand sides of (3.20)), we can repeatedly
trade two derivatives of regularity for two powers of $\rho$. Since $M_{0}-n$
is even by assumption, the proof is completed after iterating this process
$(M_{0}-n-2)/2$ times. ∎

######  Corollary 5.4.

Let the renormalized quantities $\mathsf{Q}$, $\mathsf{B}$, $\mathsf{W}^{2}$,
$\mathsf{W}^{1}$, $\mathsf{W}^{\star}$ be as in Definition 3.11. Then, there
exist vertical tensor fields $\mathsf{r}_{\mathsf{Q}}$,
$\mathsf{r}_{\mathsf{B}}$, $\mathsf{r}_{\mathsf{W}^{2}}$,
$\mathsf{r}_{\mathsf{W}^{1}}$, $\mathsf{r}_{\mathsf{W}^{\star}}$ such that

(5.12) $\displaystyle\mathsf{Q}=\rho^{M_{0}}\,\mathsf{r}_{\mathsf{Q}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{Q}}\rightarrow^{2}0\text{,}$
$\displaystyle\mathsf{B}=\rho^{M_{0}-2}\,\mathsf{r}_{\mathsf{B}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{B}}\rightarrow^{1}0\text{,}$
$\displaystyle\mathsf{W}^{2}=\rho^{M_{0}-4}\,\mathsf{r}_{\mathsf{W}^{2}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{W}^{2}}\rightarrow^{2}0\text{,}$
$\displaystyle\mathsf{W}^{1}=\rho^{M_{0}-3}\,\mathsf{r}_{\mathsf{W}^{1}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{W}^{1}}\rightarrow^{1}0\text{,}$
$\displaystyle\mathsf{W}^{\star}=\rho^{M_{0}-4}\,\mathsf{r}_{\mathsf{W}^{\star}}\text{,}$
$\displaystyle\qquad\mathsf{r}_{\mathsf{W}^{\star}}\rightarrow^{2}0\text{.}$

###### Proof.

This is an immediate consequence of (3.2), Definition 3.11, and Proposition
5.3. ∎

###  5.2. Applying the Carleman Estimates

Fix any Riemannian metric $\mathfrak{h}$ on $\mathscr{I}$, and fix additional
constants $p,f_{\star}>0$ (whose precise values will be determined later) such
that

(5.13) $0<2p<1\text{,}\qquad f_{\star}\ll_{\mathsf{g},\mathscr{D}}1\text{.}$

Furthermore, as long as $M_{0}$ is sufficiently large (depending on
$\mathsf{g}$, $\mathscr{D}$), we can find a sufficiently large
$\kappa\in\mathbb{R}$ (again, the precise value will be determined later)
satisfying

(5.14) $1\ll_{n,\mathsf{g},\mathscr{D}}\kappa\leq M_{0}-5\text{.}$

######  Remark 5.5.

In particular, $p,f_{\star},\kappa$ are chosen so that we can apply the
Carleman estimates (4.7) and (4.9) as needed. Also, although all the above
parameters depend on $\mathfrak{h}$, we can instead view this as a dependence
on $\mathscr{D}$, as any choice of a Riemannian metric $\mathfrak{h}$ on
$\bar{\mathscr{D}}$ suffices.

In addition, let $f$ be as in (4.3), where $\eta$ is given from the GNCC
assumption. We also let

(5.15) $\chi:=\bar{\chi}\circ f\text{,}$

where $\bar{\chi}:\mathbb{R}\rightarrow[0,1]$ is a smooth cut-off function
satisfying

(5.16) $\bar{\chi}(s)=\begin{cases}1&s\leq f_{i}\text{,}\\\ 0&s\geq
f_{e}\text{,}\end{cases}\qquad 0<f_{i}<f_{e}<f_{\star}\text{,}$

and we define the following associated subregions of $\Omega_{f_{\star}}$ (see
(4.4)):

(5.17)
$\displaystyle\Omega_{i}:=\\{f<f_{i}\\}\text{,}\qquad\Omega_{e}:=\\{f_{i}\leq
f<f_{e}\\}\text{.}$

Lastly, for convenience, we define the following collections of quantities:

(5.18)
$\Xi^{\star}:=\\{\mathsf{W}^{\star},\mathsf{W}^{1},\mathsf{W}^{2}\\}\text{,}\qquad\Upsilon^{\star}:=\\{\mathsf{g}-\check{\mathsf{g}},\mathsf{Q},\mathsf{m}-\check{\mathsf{m}},\mathsf{B}\\}\text{.}$

We now apply Theorem 4.6 to $\chi\mathsf{V}$, for each
$\mathsf{V}\in\Xi^{\star}$, on the region $\Omega_{f_{\star}}$ (since
$\chi\mathsf{V}$ vanishes near $\\{f=f_{\star}\\}$). Restricting the right-
hand side of (4.7) to $\Omega_{i}$, on which $\chi\equiv 1$, we then obtain

(5.19) $\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{2p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})\,d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}\cup\Omega_{e}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\sum_{\mathsf{V}\in\Xi^{\star}}|(\bar{\Box}+\sigma_{\mathsf{V}})(\chi\mathsf{V})|_{\mathfrak{h}}^{2}\,d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}\cup\Omega_{e}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{-p}\sum_{\mathsf{V}\in\Xi^{\star}}(|\mathcal{F}_{\mathsf{V}}|_{\mathfrak{h}}^{2}+|\mathcal{G}_{\mathsf{V}}|_{\mathfrak{h}}^{2})\,d\mu_{g}$

for any $\lambda\gg_{\mathsf{g},\mathscr{D}}1$, where
$(\sigma_{\mathsf{W}^{2}},\sigma_{\mathsf{W}^{1}},\sigma_{\mathsf{W}^{\star}})=(2n-4,n-1,0)$,
and where

(5.20)
$\mathcal{F}_{\mathsf{V}}:=\chi(\bar{\Box}+\sigma_{\mathsf{V}})\mathsf{V}\text{,}\qquad\mathcal{G}_{\mathsf{V}}:=(\bar{\Box}+\sigma_{\mathsf{V}})(\chi\mathsf{V})-\mathcal{F}_{\mathsf{V}}\text{,}\qquad\mathsf{V}\in\Xi^{\star}\text{.}$

Note also in the last step of (5.19), we used that $\rho\lesssim f$ by
definition.

Observe the boundary limit as $\rho_{\star}\searrow 0$ in (4.7) vanishes due
to Corollary 5.4 and (5.14). Moreover, note that since every term in
$\mathcal{G}_{\mathsf{V}}$ involves at least one derivative of $\chi$, then it
is supported in $\Omega_{e}$. Expanding $\mathcal{F}_{\mathsf{V}}$ on
$\Omega_{i}$ using Proposition 3.14, then (5.19) becomes

(5.21) $\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{2p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})\,d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\left[\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{4-p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{6-p}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})+\sum_{\mathsf{U}\in\Upsilon^{\star}}(\rho^{2-p}|\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{4-p}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2})\right]d\mu_{g}$
$\displaystyle\quad\qquad+\int_{\Omega_{e}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{-p}\sum_{\mathsf{V}\in\Xi^{\star}}(|\mathcal{F}_{\mathsf{V}}|_{\mathfrak{h}}^{2}+|\mathcal{G}_{\mathsf{V}}|_{\mathfrak{h}}^{2})\,d\mu_{g}\text{.}$

Next, we can similarly apply the transport Carleman estimate of Proposition
4.9 to each $\mathsf{U}\in\Upsilon^{\ast}$ on $\Omega_{i}$. In particular,
applying (4.9) with $s=2$ (for $\mathsf{U}=\mathsf{m}-\check{\mathsf{m}}$) and
$s=0$ (otherwise) yields

(5.22) $\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\sum_{\mathsf{U}\in\Upsilon^{\star}}|\mathsf{U}|_{\mathfrak{h}}^{2}\,d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-p-2\kappa}\left[\sum_{\mathsf{U}\in\Upsilon^{\star}\setminus\\{\mathsf{m}-\check{\mathsf{m}}\\}}\rho^{2}|\mathscr{L}_{\rho}\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathscr{L}_{\rho}[\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})]|_{\mathfrak{h}}^{2}\right]d\mu_{g}\text{,}$

where $\lambda$ is as before. (The boundary term as $\rho_{\star}\searrow 0$
in (4.9) vanishes by Proposition 5.3, Corollary 5.4, and (5.14). Also, note
the cutoff $\chi$ is not needed here, since Proposition 4.9 does not require
$\Psi$ to vanish near $\\{f=f_{\ast}\\}$.) Applying (3.16) to the right-hand
side of (5.22), we obtain that

(5.23) $\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\sum_{\mathsf{U}\in\Upsilon^{\star}}|\mathsf{U}|_{\mathfrak{h}}^{2}\,d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\left[\sum_{\mathsf{V}\in\Xi^{\star}}\rho^{2-p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\sum_{\mathsf{U}\in\Upsilon^{\star}}\rho^{2-p}|\mathsf{U}|_{\mathfrak{h}}^{2}\right]d\mu_{g}\text{.}$

We can also obtain analogous bounds for $\mathsf{D}\mathsf{U}$, for each
$\mathsf{U}\in\Upsilon^{\ast}$. In particular, we apply Proposition 4.9 to the
equations (3.17) (with $s=5$ for $\mathsf{U}=\mathsf{m}-\check{\mathsf{m}}$
and $s=3$ otherwise), which yields

(5.24) $\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\sum_{\mathsf{U}\in\Upsilon^{\star}}\rho^{3}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2}\,d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\left[\sum_{\mathsf{V}\in\Xi^{\star}}\rho^{5-p}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2}+\sum_{\mathsf{U}\in\Upsilon^{\star}}(\rho^{5-p}|\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{5-p}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2})\right]d\mu_{g}\text{.}$

Summing the estimates (5.21), (5.23), and (5.24) yields

$\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\left[\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{2p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})+\sum_{\mathsf{U}\in\Upsilon^{\star}}(|\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{3}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2})\right]d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{i}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\left[\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{2-p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{5-p}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})+\sum_{\mathsf{U}\in\Upsilon^{\star}}(\rho^{2-p}|\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{4-p}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2})\right]d\mu_{g}$
$\displaystyle\quad\qquad+\int_{\Omega_{e}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{-p}\sum_{\mathsf{V}\in\Xi^{\star}}(|\mathcal{F}_{\mathsf{V}}|_{\mathfrak{h}}^{2}+|\mathcal{G}_{\mathsf{V}}|_{\mathfrak{h}}^{2})\,d\mu_{g}\text{.}$

Taking $\lambda$ sufficiently large in the above, the integral over
$\Omega_{i}$ in the right-hand side can be absorbed into the left-hand side,
since the quantities in the right contain higher powers of $\rho$ than those
in the left (and since $\rho$ is bounded on $\Omega_{i}$). As a result, we
have

(5.25) $\displaystyle\lambda\int_{\Omega_{i}} e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\left[\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{2p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})+\sum_{\mathsf{U}\in\Upsilon^{\star}}(|\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{3}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2})\right]d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{e}}e^{-\frac{\lambda
f^{p}}{p}}f^{n-2-2\kappa}\rho^{-p}\sum_{\mathsf{V}\in\Xi^{\star}}(|\mathcal{F}_{\mathsf{V}}|_{\mathfrak{h}}^{2}+|\mathcal{G}_{\mathsf{V}}|_{\mathfrak{h}}^{2})\,d\mu_{g}\text{.}$

Finally, using that the Carleman weight

$w_{\lambda}(f):=e^{-2\lambda p^{-1}f^{p}}f^{n-2-2\kappa-p}$

satisfies $w_{\lambda}(f)\leq w_{\lambda}(f_{i})$ on $\Omega_{e}$ and
$w_{\lambda}(f)\geq w_{\lambda}(f_{i})$ on $\Omega_{i}$, we can eliminate
$w_{\lambda}(f)$ from (5.25):

$\displaystyle\lambda\int_{\Omega_{i}}
\left[\sum_{\mathsf{V}\in\Xi^{\star}}(\rho^{2p}|\mathsf{V}|_{\mathfrak{h}}^{2}+\rho^{4}|\mathsf{D}\mathsf{V}|_{\mathfrak{h}}^{2})+\sum_{\mathsf{U}\in\Upsilon^{\star}}(|\mathsf{U}|_{\mathfrak{h}}^{2}+\rho^{3}|\mathsf{D}\mathsf{U}|_{\mathfrak{h}}^{2})\right]d\mu_{g}$
$\displaystyle\quad\lesssim\int_{\Omega_{e}}\rho^{-p}\sum_{\mathsf{V}\in\Xi^{\star}}(|\mathcal{F}_{\mathsf{V}}|_{\mathfrak{h}}^{2}+|\mathcal{G}_{\mathsf{V}}|_{\mathfrak{h}}^{2})\,d\mu_{g}\text{.}$

Since the right-hand side of the above is finite by our vanishing assumptions,
we can divide by $\lambda$ and take the limit $\lambda\rightarrow\infty$ to
deduce in particular that $\mathsf{g}-\check{\mathsf{g}}\equiv 0$ in
$\Omega_{i}$.

######  Remark 5.6.

Note that by taking $f_{i}$ to be arbitrarily close to $f_{\star}$, the above
argument yields that $\mathsf{g}-\check{\mathsf{g}}$ vanishes (along with all
the other associated difference quantities) on $\Omega_{f_{\star}}$.

Since $g$ and $\check{g}$ are assumed to be in the FG gauge (2.7), and with
the same radial coordinate $\rho$, the above immediately yields $g=\check{g}$
on $\Omega_{i}$. This finishes the proof of Theorem 5.1.

##  6\. The Main Results

In this section, we state and prove precise versions of the main results of
this paper:

  * •

Section 6.1 is dedicated to the precise statement and proof of Theorem 1.5.

  * •

Section 6.2 is dedicated to the precise statements and proofs of Theorems 1.7
and 1.10.

###  6.1. Gauge Covariance

In Theorem 5.1, we had assumed the two spacetimes in question had identical
boundary data. Here, we extend Theorem 5.1—we establish the same conclusions,
but we weaken the assumptions so that the spacetimes have _gauge-equivalent_
boundary data. The key idea is to apply a specific change of coordinates
(preserving the Fefferman-Graham gauge condition) on one spacetime, so that
the boundary data for the two spacetimes become equal.

Before discussing the precise result, let us first give a more detailed
description of gauge transformations in our current context of FG-aAdS
segments.

######  Definition 6.1.

Let $(\mathscr{M},g)$ be an FG-aAdS segment, and let
$\mathscr{V}\subset\mathscr{M}$ be an open neighbourhood of the conformal
boundary.494949More precisely, for any $p\in\mathscr{I}$, there exists
$\rho_{p}>0$ such that $\mathscr{V}$ contains $(0,\rho_{p})\times\\{p\\}$. A
function $\check{\rho}\in C^{\infty}(\mathscr{V})$ is called an _FG radius_
for $(\mathscr{V},g)$ iff

(6.1)
$\check{\rho}>0\text{,}\qquad\check{\rho}^{-2}g^{-1}(d\check{\rho},d\check{\rho})=1\text{.}$

######  Remark 6.2.

Note that $\rho$ itself is an FG radius for $(\mathscr{M},g)$.

Informally, we can view such an FG radius as a change of boundary defining
function (from $\rho$ to $\check{\rho}$) that preserves the Fefferman-Graham
gauge condition (2.7). A more detailed justification, in terms of our language
of FG-aAdS segments, arises from the following.

Consider a FG radius $\check{\rho}$ for $(\mathscr{V},g)$ satisfying (6.1).
Given $\sigma>0$ and $p\in\mathscr{I}$, we identify the pair $(\sigma,p)$ with
$\gamma_{p}(\sigma)\in\mathscr{V}$, where $\gamma_{p}$ is the integral curve
of the $\check{\rho}^{2}g$-gradient of $\check{\rho}$ that satisfies505050In
other words, $\gamma_{p}$ emanates from the point $p$ on the conformal
boundary.

$\lim_{\sigma\searrow 0}\gamma_{p}(\sigma)=(0,p)\text{.}$

(when all the above quantities exist). Then, given any
$\mathscr{J}\subset\mathscr{I}$ with $\bar{\mathscr{J}}\subseteq\mathscr{I}$,
the above identifies $(0,\check{\rho}_{0}]\times\mathscr{J}$, for some
$\check{\rho}_{0}>0$, with an open submanifold
$\smash{\check{\mathscr{M}}}\subseteq\mathscr{M}$. Moreover, by the second
part of (6.1), the projection onto the $(0,\check{\rho}_{0}]$-component is
simply $\check{\rho}$. As this gradient is ($g$-)normal to the level sets of
$\check{\rho}$, then $g$ is given—in terms of $\check{\rho}$ and frames
transported along this gradient—by

(6.2) $g=\check{\rho}^{-2}(d\check{\rho}^{2}+\mathsf{g}^{\ast})\text{,}$

for some ($\check{\rho}$-)vertical tensor field $\mathsf{g}^{\ast}$. In other
words, $\tilde{\rho}$ generates an FG-aAdS segment, characterized by (6.2),
that is isometric to (part of) the original FG-aAdS segment $(\mathscr{M},g)$
defined from $\rho$.

It is well-known in the physics literature (see, e.g., [20, 35]) that these
transformations preserving the FG gauge can be characterized in terms of
corresponding transformations of the boundary data. Below, we present this in
a more rigorous form, adapted to the setting of this paper.

######  Proposition 6.3.

Let $(\mathscr{M},g)$ be a vacuum FG-aAdS segment, and fix $\mathfrak{a}\in
C^{\infty}(\mathscr{I})$. Then, there is a neighbourhood
$\mathscr{V}\subset\mathscr{M}$ of the conformal boundary and a unique FG
radius $\check{\rho}$ on $(\mathscr{V},g)$ such that

(6.3) $\frac{\check{\rho}}{\rho}\rightarrow^{0}e^{\mathfrak{a}}\text{.}$

Furthermore, let
$(\mathscr{I},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$
and
$(\mathscr{I},\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
denote the holographic data associated to the FG-aAdS segments constructed
from $(\mathscr{M},g)$ with respect to $\rho$ and $\check{\rho}$,
respectively. Then, there exists a universal algebraic function $\mathscr{F}$
(i.e., independent of $\mathscr{M},g,\check{g},\rho,\check{\rho}$) such that

(6.4)
$\check{\mathfrak{g}}^{\scriptscriptstyle(0)}=e^{2\mathfrak{a}}\mathfrak{g}^{\scriptscriptstyle(0)}\text{,}\qquad\check{\mathfrak{g}}^{\scriptscriptstyle(n)}=\mathscr{F}(\mathfrak{g}^{\scriptscriptstyle(n)},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{R},\dots,\mathfrak{D}^{n-2}\mathfrak{R},\mathfrak{a},\mathfrak{D}\mathfrak{a},\dots,\mathfrak{D}^{n}\mathfrak{a})\text{,}$

where $\mathfrak{D}$ and $\mathfrak{R}$ denotes the Levi-Civita connection and
Riemann curvature for $\mathfrak{g}^{\scriptscriptstyle(0)}$, respectively.

###### Proof sketch of Proposition 6.3..

To obtain $\check{\rho}$, we adapt an argument inspired by the constructions
from [25]. First, we consider the following ansatz for our desired FG radius:

(6.5) $\check{\rho}:=e^{\mathsf{a}}\rho\text{,}\qquad\mathsf{a}\in
C^{\infty}(\mathscr{M})\text{.}$

Then, the defining equation (6.1) for $\check{\rho}$ expands as

$1=\rho^{-2}g^{-1}(d\rho,d\rho)+2\rho\cdot\rho^{-2}g^{-1}(d\rho,d\mathsf{a})+\rho^{2}\cdot\rho^{-2}g^{-1}(d\mathsf{a},d\mathsf{a})\text{.}$

Since $g$ and $\rho$ satisfy (2.7), the above becomes

(6.6) $\displaystyle 0$
$\displaystyle=2\rho^{-2}g^{-1}(d\rho,d\mathsf{a})+\rho\cdot\rho^{-2}g^{-1}(d\mathsf{a},d\mathsf{a})$
$\displaystyle=(2+\rho\mathscr{L}_{\rho}\mathsf{a})\mathscr{L}_{\rho}\mathsf{a}+\rho\,\mathsf{g}^{-1}(\mathsf{D}\mathsf{a},\mathsf{D}\mathsf{a})\text{.}$

Observe (6.6) yields a fully nonlinear equation for $\mathsf{a}$, which can be
uniquely solved via the method of characteristics on a neighbourhood
$\mathscr{V}$ of the conformal boundary given initial data

$\lim_{\sigma\searrow 0}\mathsf{a}|_{\sigma}=\mathfrak{a}\text{.}$

Then, $\check{\rho}=e^{\mathsf{a}}\rho$ yields the FG radius on $\mathscr{V}$
which satisfies (6.3).

The relations between
$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$
and
$(\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$,
as well as their derivations, are standard in physics literature—see, for
instance, [20, 35]. A detailed derivation of the first relation in (6.4), in
the context of FG-aAdS segments, can be found in [18, Proposition 3.4]. ∎

######  Remark 6.4.

Using (6.6), along with the techniques of [50], one can derive boundary limits
for $\rho$-derivatives of both $\mathsf{a}$ and $\check{\rho}$ in Proposition
6.3. From these limits, one then obtains partial series expansions for
$\mathsf{a}$ and $\check{\rho}$ in terms of powers of $\rho$, similar to those
in Corollary 2.18. In particular, these expansions justify the ansatz for FG
gauge transforms used in [18, Equation (3.4)].

######  Remark 6.5.

In addition to (6.4), there are similar transformation formulas for the other
coefficients
$\mathfrak{g}^{\scriptscriptstyle(2)},\mathfrak{g}^{\scriptscriptstyle(4)},\dots,\mathfrak{g}^{\scriptscriptstyle(\star)}$
of the Fefferman-Graham expansion. (For example,
$-\mathfrak{g}^{\scriptscriptstyle(2)}$ transforms like the Schouten tensor
under conformal rescalings—see [18, Proposition 3.4] for a detailed
derivation.)

In particular, Proposition 6.3 implies that two holographic data
$(\mathscr{I},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$,
$(\mathscr{I},\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
that are related via (6.4) are associated to the _same_ aAdS spacetime, via
two different FG gauges on this spacetime. This motivates the following
definition:

######  Definition 6.6.

Let $\mathscr{I}$ be an $n$-dimensional manifold, let
$\mathfrak{g}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(0)}$
be two Lorentzian metrics on $\mathscr{I}$, and let
$\mathfrak{g}^{\scriptscriptstyle(n)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)}$
be two symmetric rank-$(0,2)$ tensor fields on $\mathscr{I}$. We say
$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$
and
$(\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
are _gauge-equivalent_ on $\mathscr{D}\subset\mathscr{I}$ iff these quantities
satisfy (6.4) on $\mathscr{D}$ for some $\mathfrak{a}\in
C^{\infty}(\mathscr{I})$.

We can now state the precise version of Theorem 1.5, which combines the
results of Theorem 5.1 while also taking into account the above gauge
covariance:

######  Theorem 6.7.

Let $n>2$, and let $(\mathscr{M},g)$, $(\mathscr{M},\check{g})$ be vacuum FG-
aAdS segments (on a common aAdS region $\mathscr{M}$), with associated
holographic data
$(\mathscr{I},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$,
$(\mathscr{I},\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
(respectively). In addition, let $\mathscr{D}\subset\mathscr{I}$ be open with
compact closure, and assume:

  * •

$(\mathscr{M},g)$ and $(\mathscr{M},\check{g})$ are regular to some large
enough order $M_{0}$ (depending on $\mathsf{g}$, $\check{\mathsf{g}}$,
$\mathscr{D}$).

  * •

$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$
and
$(\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
are gauge-equivalent on $\mathscr{D}$.

  * •

$(\mathscr{D},\mathfrak{g}^{\scriptscriptstyle(0)})$ (or equivalently,
$(\mathscr{D},\check{\mathfrak{g}}^{\scriptscriptstyle(0)})$) satisfies the
GNCC.

Then, $g$ and $\check{g}$ are isometric near $\\{0\\}\times\mathscr{D}$
(viewed as part of the conformal boundary). To be more precise, there exists a
sufficiently small $f_{\star}>0$ and some
$\Psi:\Omega_{f_{\star}}\rightarrow\mathscr{M}$ such that515151In other words,
$\Psi$ is an isometry that fixes the conformal boundary.

(6.7) $\Psi_{\ast}\check{g}=g\text{,}\qquad\lim_{\sigma\searrow
0}\Psi(\sigma,p)=(0,p)\text{,}\quad p\in\mathscr{D}\text{.}$

###### Proof.

By gauge equivalence, there exists $\mathfrak{a}\in C^{\infty}(\mathscr{I})$
so that (6.4) holds on $\mathscr{D}$. Applying Proposition 6.3, we obtain an
FG radius $\check{\rho}$ such (6.3) holds. Using $\check{\rho}$ (see the
discussion below Definition 6.1), we can identify $(\mathscr{M},g)$ with
another FG-aAdS segment $(\mathscr{N},g)$, for which the induced holographic
data is
$(\mathscr{I},\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$.
Moreover, applying another isometry that identifies the FG gauges for
$(\mathscr{N},g)$ and $(\mathscr{M},\check{g})$,525252Here, one maps the
integral curves of the gradient of $\tilde{\rho}$ in $\mathscr{N}$ to the
integral curves of the gradient of $\rho$ in $\mathscr{M}$. we then arrive at
the setting of Theorem 5.1—two FG-aAdS segments on a common manifold
$\mathscr{M}$, with identical data
$(\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
on $\mathscr{D}$.

The result now follows by applying Theorem 5.1; the desired isometry $\Psi$ is
then constructed by unwinding all the identifications made in the preceding
discussion. ∎

######  Remark 6.8.

Recall the GNCC is gauge-invariant, so that in the setting of Theorem 6.7, we
have that the GNCC holds for
$(\mathscr{D},\mathfrak{g}^{\scriptscriptstyle(0)})$ if and only if it holds
for $(\mathscr{D},\check{\mathfrak{g}}^{\scriptscriptstyle(0)})$; see [18,
Proposition 3.6] for a proof of this statement. Therefore, it suffices to
assume the GNCC only for $\mathfrak{g}^{\scriptscriptstyle(0)}$.

###  6.2. Extension of Symmetries

One application of Theorem 5.1 is that it immediately implies that holographic
symmetries on the conformal boundary must be inherited in the bulk spacetime.
Here, we prove two versions of this, one for discrete and the other for
continuous symmetries.

The first result is the precise analogue of Theorem 1.10 from the
introduction:

######  Theorem 6.9.

Let $n>2$, and let $(\mathscr{M},g)$ be a vacuum FG-aAdS segment, with
associated holographic data
$(\mathscr{I},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$.
Consider also a smooth, invertible function
$\phi:\bar{\mathscr{D}}\rightarrow\mathscr{I}$,535353More accurately,
$\phi:\mathscr{J}\rightarrow\mathscr{I}$ for some open
$\mathscr{J}\subseteq\mathscr{I}$ that contains $\bar{\mathscr{D}}$. where
$\mathscr{D}\subset\mathscr{I}$ is open with compact closure. In addition,
assume that the following hold:

  * •

$(\mathscr{M},g)$ is regular to some large enough order $M_{0}$ (depending on
$\mathsf{g}$, $\mathscr{D}$).

  * •

$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$
and
$(\phi_{\ast}\mathfrak{g}^{\scriptscriptstyle(0)},\phi_{\ast}\mathfrak{g}^{\scriptscriptstyle(n)})$
are gauge-equivalent on $\mathscr{D}$.

  * •

$(\mathscr{D},\mathfrak{g}^{\scriptscriptstyle(0)})$ satisfies the GNCC.

Then, $\phi$ extends to an isometry of $g$ near $\\{0\\}\times\mathscr{D}$
(viewed as part of the conformal boundary). To be more precise, there exists a
sufficiently small $f_{\star}>0$ and some
$\Phi:\Omega_{f_{\star}}\rightarrow\mathscr{M}$ such that545454In other words,
$\Phi$ is an isometry that asymptotes to $\phi$ at the conformal boundary.

(6.8) $\Phi_{\ast}g=g\text{,}\qquad\lim_{\sigma\searrow
0}\Phi(\sigma,p)=(0,\phi(p))\text{,}\quad p\in\mathscr{D}\text{.}$

Furthermore, if $\phi$ is a _holographic isometry_ on $\mathscr{D}$, i.e.,

(6.9)
$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})|_{\mathscr{D}}=(\phi_{\ast}\mathfrak{g}^{\scriptscriptstyle(0)},\phi_{\ast}\mathfrak{g}^{\scriptscriptstyle(n)})|_{\mathscr{D}}\text{,}$

then the bulk isometry $\Phi$ is given explicitly as

(6.10)
$\Phi(\sigma,p)=(\sigma,\phi(p))\text{,}\qquad(\sigma,p)\in\Omega_{f_{\ast}}\text{.}$

###### Proof.

First, applying a gauge transformation as in the proof of Theorem 6.7, we can
assume (6.9) holds. Thus, it suffices to show that the map $\Phi$ given by
(6.10) is an isometry.

Observe that (6.10) implies the pullback $\Phi_{\ast}g$ satisfies

$\Phi_{\ast}g=\rho^{-2}(d\rho^{2}+\phi^{\ast}\mathsf{g})\text{,}$

where $\phi_{\ast}\mathsf{g}$ is defined to the the pullback through $\phi$ on
each level set of $\rho$:

$(\phi_{\ast}\mathsf{g})|_{\sigma}:=\phi_{\ast}(\mathsf{g}|_{\sigma})\text{,}\qquad\sigma\in(0,\rho_{0}]\text{.}$

In particular, $\Phi_{\ast}g$ defines a vacuum FG-aAdS segment (with the same
$\rho$ as before), whose associated boundary data is
$(\phi^{-1}(\mathscr{I}),\phi_{\ast}\mathfrak{g}^{\scriptscriptstyle(0)},\phi_{\ast}\mathfrak{g}^{\scriptscriptstyle(n)})$.
From (6.9), we see that $g$ and $\Phi_{\ast}g$ have the same data on
$\mathscr{D}$, hence Theorem 5.1 yields $g=\Phi_{\ast}g$ on some
$\Omega_{f_{\star}}$, for $f_{\star}>0$ sufficiently small. ∎

######  Remark 6.10.

Note that in general, $\phi|_{\mathscr{D}}$ in Theorem 6.9 is a conformal
isometry of $\mathfrak{g}$. Furthermore, the extra condition (6.9) implies
that $\phi|_{\mathscr{D}}$ is a full isometry of $\mathfrak{g}$.

Theorem 6.9 implies the following extension result for Killing vector fields,
which is, in addition, the precise analogue of Theorem 1.7 from the
introduction:

######  Theorem 6.11.

Let $n>2$, and let $(\mathscr{M},g)$ be a vacuum FG-aAdS segment, with
holographic data
$(\mathscr{I},\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$.
Fix also a smooth vector field $\mathfrak{K}$ on
$\bar{\mathscr{D}}$,555555More precisely, $\mathfrak{K}$ is a vector field on
some open $\mathscr{J}\subseteq\mathscr{I}$ that contains $\bar{\mathscr{D}}$.
with $\mathscr{D}\subset\mathscr{I}$ and $\bar{\mathscr{D}}$ compact, and
assume:

  * •

$(\mathscr{M},g)$ is regular to some large enough order $M_{0}$ (depending on
$\mathsf{g}$, $\mathscr{D}$).

  * •

$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})$
is gauge-equivalent on $\mathscr{D}$ to some
$(\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\check{\mathfrak{g}}^{\scriptscriptstyle(n)})$
satisfying

(6.11)
$(\mathscr{L}_{\mathfrak{K}}\check{\mathfrak{g}}^{\scriptscriptstyle(0)},\mathscr{L}_{\mathfrak{K}}\check{\mathfrak{g}}^{\scriptscriptstyle(n)})|_{\mathscr{D}}=0\text{.}$

  * •

$(\mathscr{D},\mathfrak{g}^{\scriptscriptstyle(0)})$ satisfies the GNCC.

Then, $\mathfrak{K}$ extends to a Killing vector field $K$ for $g$ near
$\\{0\\}\times\mathscr{D}$ (viewed as part of the conformal boundary). More
precisely, there exist a sufficiently small $f_{\star}>0$ and vector field

(6.12) $K:=K^{\rho}\partial_{\rho}+\mathsf{K}\text{,}$

where $K^{\rho}\in C^{\infty}(\Omega_{f_{\star}})$ and $\mathsf{K}$ is a
vertical vector field on $\Omega_{f_{\star}}$, such that565656In other words,
$K$ is a $g$-Killing vector field that asymptotes to $\mathfrak{K}$ at the
conformal boundary.

(6.13) $\mathscr{L}_{K}g=0\text{,}\qquad
K^{\rho}\rightarrow^{0}0\text{,}\qquad\mathsf{K}\rightarrow^{0}\mathfrak{K}\text{.}$

Furthermore, if $\mathfrak{K}$ is a _holographic Killing field_ on
$\mathscr{D}$, i.e.,

(6.14)
$(\mathscr{L}_{\mathfrak{K}}\mathfrak{g}^{\scriptscriptstyle(0)},\mathscr{L}_{\mathfrak{K}}\mathfrak{g}^{\scriptscriptstyle(n)})|_{\mathscr{D}}=0\text{.}$

then $K$ is vertical ($K^{\rho}\equiv 0$) and can be explicitly described via
the relation

(6.15)
$\mathscr{L}_{\rho}\mathsf{K}=0\text{,}\qquad\mathsf{K}|_{\sigma}\rightarrow^{0}\mathfrak{K}\text{.}$

###### Proof.

Once again, by a gauge transformation, it suffices to consider the special
case in which (6.14) holds. Let $\phi_{s}$, for $s\in\mathbb{R}$ small enough,
denote transport along the integral curves of $\mathfrak{K}$ by parameter $s$.
By definition, each $\phi_{s}$ is a holographic isometry on $\mathscr{D}$,

$(\mathfrak{g}^{\scriptscriptstyle(0)},\mathfrak{g}^{\scriptscriptstyle(n)})|_{\mathscr{D}}=(\phi_{s\ast}\mathfrak{g}^{\scriptscriptstyle(0)},\phi_{s\ast}\mathfrak{g}^{\scriptscriptstyle(n)})|_{\mathscr{D}}\text{,}$

so by Theorem 6.9, it extends to a bulk isometry $\Phi_{s}$ on some
$\Omega_{f_{\star}}$, $f_{\star}>0$, with

(6.16) $\Phi_{s}(\sigma,p):=(\sigma,\phi_{s}(p))\text{,}$

and with $\Phi_{s\ast}g=g$ near $\\{0\\}\times\mathscr{D}$. Let $K$ be the
generator of the family $\\{\Phi_{s}\\}_{s\in\mathbb{R}}$; note that by
definition, $K$ satisfies the first part of (6.13). Finally, one directly
deduces from (6.16) that $K$ must be vertical, and that $K:=\mathsf{K}$
satisfies the transport relation (6.15). ∎

##  7\. Additional Details and Computations

This appendix contains additional proofs and computational details for
readers’ convenience. We begin with some preliminary formulas that will be
useful in later sections.

###  7.1. Proof of Proposition 2.33

Throughout, we assume indices are with respect to arbitrary coordinates
$(U,\varphi)$ on $\mathscr{I}$, and we let $(k,l)$ be the rank of
$\mathsf{A}$. The first part of (2.33) is a consequence of (2.32) and the
following commutation formula from [50, Proposition 2.27]:

$\displaystyle\mathscr{L}_{\rho}\mathsf{D}_{c}\mathsf{A}^{\bar{a}}{}_{\bar{b}}-\mathsf{D}_{c}(\mathscr{L}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{k}\mathsf{g}^{a_{i}d}(\mathsf{D}_{c}\mathscr{L}_{\rho}\mathsf{g}_{de}+\mathsf{D}_{e}\mathscr{L}_{\rho}\mathsf{g}_{dc}-\mathsf{D}_{d}\mathscr{L}_{\rho}\mathsf{g}_{ce})\mathsf{A}^{\hat{a}_{i}\\![e]}{}_{\bar{b}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{j=1}^{l}\mathsf{g}^{de}(\mathsf{D}_{c}\mathscr{L}_{\rho}\mathsf{g}_{db_{j}}+\mathsf{D}_{b_{j}}\mathscr{L}_{\rho}\mathsf{g}_{dc}-\mathsf{D}_{d}\mathscr{L}_{\rho}\mathsf{g}_{cb_{j}})\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![e]}\text{,}$

Next, recalling the second formula in (2.20), we obtain that

$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{D}_{c}\mathsf{A}^{\bar{a}}{}_{\bar{b}}$
$\displaystyle=\mathscr{L}_{\rho}\mathsf{D}_{c}\mathsf{A}^{\bar{a}}{}_{\bar{b}}-\frac{1}{2}\mathsf{g}^{de}\mathscr{L}_{\rho}\mathsf{g}_{cd}\,\mathsf{D}_{e}\mathsf{A}^{\bar{a}}{}_{\bar{b}}+\frac{1}{2}\sum_{i=1}^{k}\mathsf{g}^{a_{i}d}\mathscr{L}_{\rho}\mathsf{g}_{de}\,\mathsf{D}_{c}\mathsf{A}^{\hat{a}_{i}\\![e]}{}_{\bar{b}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{j=1}^{l}\mathsf{g}^{de}\mathscr{L}_{\rho}\mathsf{g}_{b_{j}d}\,\mathsf{D}_{c}\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![e]}\text{,}$
$\displaystyle\mathsf{D}_{c}(\bar{\mathsf{D}}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}$
$\displaystyle=\mathsf{D}_{c}(\mathscr{L}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}+\frac{1}{2}\sum_{i=1}^{k}\mathsf{D}_{c}(\mathsf{g}^{a_{i}d}\mathscr{L}_{\rho}\mathsf{g}_{de}\,\mathsf{A}^{\hat{a}_{i}\\![e]}{}_{\bar{b}})-\frac{1}{2}\sum_{j=1}^{l}\mathsf{D}_{c}(\mathsf{g}^{de}\mathscr{L}_{\rho}\mathsf{g}_{b_{j}d}\,\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![e]})\text{.}$

Subtracting the above two equations and recalling the first part of (2.33)
yields

$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{D}_{c}\mathsf{A}^{\bar{a}}{}_{\bar{b}}$
$\displaystyle=\mathsf{D}_{c}(\bar{\mathsf{D}}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}+\mathscr{L}_{\rho}\mathsf{D}_{c}\mathsf{A}^{\bar{a}}{}_{\bar{b}}-\mathsf{D}_{c}(\mathscr{L}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\,\mathsf{D}_{e}\mathsf{A}^{\bar{a}}{}_{\bar{b}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{i=1}^{k}\mathsf{g}^{a_{i}d}\mathsf{D}_{c}\mathsf{m}_{de}\,\mathsf{A}^{\hat{a}_{i}\\![e]}{}_{\bar{b}}+\frac{1}{2}\sum_{j=1}^{l}\mathsf{g}^{de}\mathsf{D}_{c}\mathsf{m}_{b_{j}d}\,\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![e]}$
$\displaystyle=\mathsf{D}_{c}(\bar{\mathsf{D}}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\,\mathsf{D}_{e}\mathsf{A}^{\bar{a}}{}_{\bar{b}}+\frac{1}{2}\sum_{i=1}^{k}\mathsf{g}^{a_{i}d}(\mathsf{D}_{e}\mathsf{m}_{cd}-\mathsf{D}_{d}\mathsf{m}_{ce})\mathsf{A}^{\hat{a}_{i}\\![e]}{}_{\bar{b}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{j=1}^{l}\mathsf{g}^{de}(\mathsf{D}_{b_{j}}\mathsf{m}_{cd}-\mathsf{D}_{d}\mathsf{m}_{cb_{j}})\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![e]}\text{,}$

from which the second identity in (2.33) follows.

Next, recalling Definition 2.27 for $\bar{\Box}$, we expand (partially in
$\varphi_{\rho}$-coordinates)

(7.1) $\displaystyle\bar{\Box}(\rho^{p}\mathsf{A})$
$\displaystyle=\mathsf{g}^{\alpha\beta}\bar{\nabla}_{\alpha}(\rho^{p}\bar{\nabla}_{\beta}\mathsf{A}+p\rho^{p-1}\bar{\nabla}_{\beta}\rho\cdot\mathsf{A})$
$\displaystyle=\rho^{p}\bar{\Box}\mathsf{A}+2p\rho^{p-1}g^{\alpha\beta}\nabla_{\alpha}\rho\,\bar{\mathsf{D}}_{\beta}\mathsf{A}+p(p-1)\rho^{p-2}g^{\alpha\beta}\nabla_{\alpha}\rho\nabla_{\beta}\rho\,\mathsf{A}+p\rho^{p-1}\Box\rho\,\mathsf{A}$
$\displaystyle=\rho^{p}\bar{\Box}\mathsf{A}+2p\rho^{p+1}\,\bar{\mathsf{D}}_{\rho}\mathsf{A}+p(p-1)\rho^{p}\,\mathsf{A}+p\rho^{p-1}\Box\rho\,\mathsf{A}\text{,}$

where we also recalled (2.7) in the last step. By (2.7), (2.28), and (2.32),
we have

$\displaystyle\Box\rho$
$\displaystyle=-\rho^{2}\Gamma^{\rho}_{\rho\rho}-\rho^{2}\mathsf{g}^{ab}\Gamma^{\rho}_{ab}$
$\displaystyle=-(n-1)\rho+\frac{1}{2}\rho^{2}\mathsf{g}^{ab}\mathsf{m}_{ab}\text{.}$

Thus, combining (7.1) with the above yields

$\bar{\Box}(\rho^{p}\mathsf{A})=\rho^{p}\bar{\Box}\mathsf{A}+2p\rho^{p+1}\,\bar{\mathsf{D}}_{\rho}\mathsf{A}-p(n-p)\rho^{p}\,\mathsf{A}+\frac{1}{2}p\rho^{p+1}\mathsf{g}^{ab}\mathsf{m}_{ab}\,\mathsf{A}\text{,}$

which immediately imples (2.34).

Finally, for (2.35), the definitions of $\bar{\nabla}^{2}_{\rho\rho}$ and
$\bar{\mathsf{D}}_{\rho}^{2}$, along with Proposition 2.29, yield:

$\displaystyle\bar{\nabla}_{\rho\rho}\mathsf{A}^{\bar{a}}{}_{\bar{b}}$
$\displaystyle=\partial_{\rho}(\bar{\mathsf{D}}_{\rho}\mathsf{A}^{\bar{a}}{}_{\bar{b}})-\Gamma^{\rho}_{\rho\rho}\bar{\mathsf{D}}_{\rho}\mathsf{A}^{\bar{a}}{}_{\bar{b}}+\sum_{i=1}^{k}\mathsf{\Gamma}^{a_{i}}_{\rho
d}\mathsf{A}^{\hat{a}_{i}\\![d]}{}_{\bar{b}}-\sum_{j=1}^{l}\mathsf{\Gamma}^{d}_{\rho
b_{j}}\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![d]}\text{,}$
$\displaystyle\bar{\mathsf{D}}_{\rho}(\bar{\mathsf{D}}_{\rho}\mathsf{A})^{\bar{a}}{}_{\bar{b}}$
$\displaystyle=\partial_{\rho}(\bar{\mathsf{D}}_{\rho}\mathsf{A}^{\bar{a}}{}_{\bar{b}})+\sum_{i=1}^{k}\mathsf{\Gamma}^{a_{i}}_{\rho
d}\mathsf{A}^{\hat{a}_{i}\\![d]}{}_{\bar{b}}-\sum_{j=1}^{l}\mathsf{\Gamma}^{d}_{\rho
b_{j}}\mathsf{A}^{\bar{a}}{}_{\hat{b}_{j}\\![d]}\text{.}$

Subtracting the above two equations and recalling the first part of (2.28)
yields

$\bar{\nabla}_{\rho\rho}\mathsf{A}=\bar{\mathsf{D}}_{\rho}(\bar{\mathsf{D}}_{\rho}\mathsf{A})+\rho^{-1}\bar{\mathsf{D}}_{\rho}\mathsf{A}\text{.}$

Furthermore, relating $\bar{\mathsf{D}}_{\rho}$ and $\mathscr{L}_{\rho}$ using
(2.20), the above then becomes575757Here, we also used the standard identity
$\mathscr{L}_{\rho}\mathsf{g}^{-1}=\mathscr{S}(\mathsf{g}^{-2},\mathsf{m})$.

(7.2) $\displaystyle\bar{\nabla}_{\rho\rho}\mathsf{A}$
$\displaystyle=\mathscr{L}_{\rho}(\bar{\mathsf{D}}_{\rho}\mathsf{A})+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\bar{\mathsf{D}}_{\rho}\mathsf{A})+\rho^{-1}\mathscr{L}_{\rho}\mathsf{A}+\rho^{-1}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{A})$
$\displaystyle=\mathscr{L}_{\rho}^{2}\mathsf{A}+\rho^{-1}\mathscr{L}_{\rho}\mathsf{A}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathscr{L}_{\rho}\mathsf{A})+\rho^{-1}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{A})$
$\displaystyle\qquad+\mathscr{S}(\mathsf{g}^{-1},\mathscr{L}_{\rho}\mathsf{m},\mathsf{A})+\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{A})\text{.}$

Next, a similar decomposition of mixed and vertical derivatives in vertical
components yields

$\displaystyle\bar{\nabla}_{ab}\mathsf{A}$
$\displaystyle=\mathsf{D}_{ab}\mathsf{A}-\Gamma^{\rho}_{ab}\bar{\mathsf{D}}_{\rho}\mathsf{A}$
$\displaystyle=\mathsf{D}_{ab}\mathsf{A}-\rho^{-1}\mathsf{g}_{ab}\bar{\mathsf{D}}_{\rho}\mathsf{A}+\mathscr{S}(\mathsf{m},\bar{\mathsf{D}}_{\rho}\mathsf{A})_{ab}$
$\displaystyle=\mathsf{D}_{ab}\mathsf{A}-\rho^{-1}\mathsf{g}_{ab}\mathscr{L}_{\rho}\mathsf{A}+\rho^{-1}\mathsf{g}_{ab}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{A})+\mathscr{S}(\mathsf{m},\mathscr{L}_{\rho}\mathsf{A})_{ab}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{A})_{ab}\text{.}$

Therefore, applying (2.7), Definition 2.27, (7.2), and the above, we conclude
that

$\displaystyle\bar{\Box}\mathsf{A}$
$\displaystyle=\rho^{2}\bar{\nabla}_{\rho\rho}\mathsf{A}+\rho^{2}\mathsf{g}^{ab}\bar{\nabla}_{ab}\mathsf{A}$
$\displaystyle=\rho^{2}\mathscr{L}_{\rho}^{2}\mathsf{A}-(n-1)\rho\mathscr{L}_{\rho}\mathsf{A}+\rho^{2}\mathsf{g}^{ab}\mathsf{D}_{ab}\mathsf{A}+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathscr{L}_{\rho}\mathsf{A})+\rho\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{A})$
$\displaystyle\qquad+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathscr{L}_{\rho}\mathsf{m},\mathsf{A})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{A})\text{,}$

which is precisely the last part of (2.35).

###  7.2. Proof of Proposition 2.34

We assume all indices are with respect to $\varphi$\- and
$\varphi_{\rho}$-coordinates, and we let $\Gamma$ and $\mathsf{\Gamma}$ be the
corresponding Christoffel symbols, as defined in Proposition 2.29. For future
convenience, we also set $l:=l_{1}+l_{2}$, and we define (via indices) the
vertical tensor fields

(7.3)
$\mathsf{k}_{ac}:=\rho^{-1}\mathsf{g}_{ac}-\frac{1}{2}\mathsf{m}_{ac}\text{,}\qquad\mathsf{k}^{b}{}_{a}:=\rho^{-1}\delta^{b}{}_{a}-\frac{1}{2}\mathsf{g}^{bc}\mathsf{m}_{ac}\text{.}$

By the definitions of $\nabla$ and $\bar{\mathsf{D}}$ (see Proposition 2.22
and (2.29)), along with (2.28), we have

$\nabla_{\rho}F_{\bar{\rho}\bar{a}}=\partial_{\rho}(F_{\bar{\rho}\bar{a}})-\sum_{i=1}^{l_{1}}\Gamma^{\rho}_{\rho\rho}\,F_{\bar{\rho}\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma^{b}_{\rho
a_{j}}\,F_{\bar{\rho}\hat{a}_{j}\\![b]}\text{,}\qquad\bar{\mathsf{D}}_{\rho}\mathsf{f}_{\bar{a}}=\partial_{\rho}(\mathsf{f}_{\bar{a}})-\sum_{j=1}^{l_{2}}\mathsf{\Gamma}^{b}_{\rho
a_{j}}\mathsf{f}_{\hat{a}_{j}\\![b]}\text{.}$

Subtracting the two equations above and applying (2.28) and (2.36) yields

$\displaystyle\nabla_{\rho}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\bar{\mathsf{D}}_{\rho}\mathsf{f}_{\bar{a}}-\sum_{i=1}^{l_{1}}\Gamma^{\rho}_{\rho\rho}\,\mathsf{f}_{\bar{a}}-\sum_{j=1}^{l_{2}}(\Gamma^{b}_{\rho
a_{j}}-\mathsf{\Gamma}^{b}_{\rho a_{j}})\mathsf{f}_{\hat{a}_{j}\\![b]}$
$\displaystyle=\bar{\mathsf{D}}_{\rho}\mathsf{f}_{\bar{a}}+l_{1}\rho^{-1}\,\mathsf{f}_{\bar{a}}+l_{2}\rho^{-1}\,\mathsf{f}_{\bar{a}}$
$\displaystyle=\bar{\mathsf{D}}_{\rho}\mathsf{f}_{\bar{a}}+(l_{1}+l_{2})\rho^{-1}\,\mathsf{f}_{\bar{a}}\text{,}$

from which the first equation in (2.37) follows.

Similarly, the definitions of $\nabla$ and $\bar{\mathsf{D}}$ imply

$\nabla_{c}F_{\bar{\rho}\bar{a}}=\partial_{c}(F_{\bar{\rho}\bar{a}})-\sum_{i=1}^{l_{1}}\Gamma_{c\rho}^{b}\,F_{\hat{\rho}_{i}\\![b]\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma_{ca_{j}}^{\beta}\,F_{\bar{\rho}\hat{a}_{j}\\![\beta]}\text{,}\qquad\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}}=\partial_{c}(\mathsf{f}_{\bar{a}})-\sum_{j=1}^{l_{2}}\mathsf{\Gamma}_{ca_{j}}^{b}\,\mathsf{f}_{\hat{a}_{j}\\![b]}\text{.}$

Subtracting the above equations and recalling (2.28), (2.38), and (2.39), we
obtain

(7.4) $\displaystyle\nabla_{c}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}}-\sum_{i=1}^{l_{1}}\Gamma_{c\rho}^{b}\,F_{\hat{\rho}_{i}\\![b]\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma_{ca_{j}}^{\rho}\,F_{\bar{\rho}\hat{a}_{j}\\![\rho]}$
$\displaystyle=\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}}+\sum_{i=1}^{l_{1}}\mathsf{k}^{b}{}_{c}\,(\mathsf{f}^{\rho}_{i})_{b\bar{a}}-\sum_{j=1}^{l_{2}}\mathsf{k}_{ca_{j}}\,(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}\text{.}$

Combining the above with (7.3) yields the second part of (2.37).

For the last identity in (2.37), we begin with $\rho$-derivatives. First, by
Proposition 2.29,

(7.5) $\displaystyle\nabla_{\rho\rho}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\partial_{\rho}(\nabla_{\rho}F_{\bar{\rho}\bar{a}})-\Gamma^{\rho}_{\rho\rho}\,\nabla_{\rho}F_{\bar{\rho}\bar{a}}-\sum_{i=1}^{l_{1}}\Gamma^{\rho}_{\rho\rho}\,\nabla_{\rho}F_{\bar{\rho}\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma^{b}_{\rho
a_{j}}\,\nabla_{\rho}F_{\bar{\rho}\hat{a}_{j}\\![b]}\text{,}$
$\displaystyle=\partial_{\rho}(\nabla_{\rho}F_{\bar{\rho}\bar{a}})+(l_{1}+1)\rho^{-1}\,\nabla_{\rho}F_{\bar{\rho}\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma^{b}_{\rho
a_{j}}\,\nabla_{\rho}F_{\bar{\rho}\hat{a}_{j}\\![b]}\text{,}$

Similarly, for the corresponding mixed derivatives, we apply (2.29) and
compute

(7.6)
$\displaystyle\rho^{-l}\bar{\nabla}_{\rho\rho}(\rho^{l}\mathsf{f})_{\bar{a}}$
$\displaystyle=\rho^{-l}\partial_{\rho}[\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}]-\Gamma^{\rho}_{\rho\rho}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}-\sum_{j=1}^{l_{2}}\mathsf{\Gamma}^{b}_{\rho
a_{j}}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\hat{a}_{j}\\![b]}$
$\displaystyle=\partial_{\rho}[\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}]+(l+1)\rho^{-1}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}-\sum_{j=1}^{l_{2}}\mathsf{\Gamma}^{b}_{\rho
a_{j}}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\hat{a}_{j}\\![b]}\text{,}$

where we also recalled (2.28) and the basic properties from Proposition 2.26.
Subtracting (7.6) from (7.5), while applying both (2.28) and the first part of
(2.37), we obtain that

(7.7) $\displaystyle\nabla_{\rho\rho}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\rho^{-l}\bar{\nabla}_{\rho\rho}(\rho^{l}\mathsf{f})_{\bar{a}}+(l_{1}-l)\rho^{-1}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}-\sum_{j=1}^{l_{2}}(\Gamma^{b}_{\rho
a_{j}}-\mathsf{\Gamma}^{b}_{\rho
a_{j}})\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\hat{a}_{j}\\![b]}$
$\displaystyle=\rho^{-l}\bar{\nabla}_{\rho\rho}(\rho^{l}\mathsf{f})_{\bar{a}}+(l_{1}-l)\rho^{-1}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}-l_{2}\rho^{-1}\,\rho^{-l}\bar{\mathsf{D}}_{\rho}(\rho^{l}\mathsf{f})_{\bar{a}}$
$\displaystyle=\rho^{-l}\bar{\nabla}_{\rho\rho}(\rho^{l}\mathsf{f})_{\bar{a}}\text{.}$

Next, applying again (2.29), we compute

$\displaystyle\nabla_{bc}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\partial_{b}(\nabla_{c}F_{\bar{\rho}\bar{a}})-\Gamma^{\alpha}_{bc}\,\nabla_{\alpha}F_{\bar{\rho}\bar{a}}-\sum_{i=1}^{l_{1}}\Gamma^{d}_{b\rho}\,\nabla_{c}F_{\hat{\rho}_{i}\\![d]\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma^{\delta}_{ba_{j}}\,\nabla_{c}F_{\bar{\rho}\hat{a}_{j}\\![\delta]}\text{,}$
$\displaystyle\rho^{-l}\bar{\nabla}_{bc}(\rho^{l}\mathsf{f})_{\bar{a}}$
$\displaystyle=\partial_{b}(\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}})-\Gamma^{\alpha}_{bc}\,\rho^{-l}\bar{\mathsf{D}}_{\alpha}(\rho^{l}\mathsf{f})_{\bar{a}}-\sum_{j=1}^{l_{2}}\mathsf{\Gamma}^{d}_{ba_{j}}\,\bar{\mathsf{D}}_{c}\mathsf{f}_{\hat{a}_{j}\\![d]}\text{.}$

Subtracting the two equations and recalling (2.28) then yields

(7.8) $\displaystyle\nabla_{bc}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\rho^{-l}\bar{\nabla}_{bc}(\rho^{l}\mathsf{f})_{\bar{a}}+\partial_{b}(\nabla_{c}F_{\bar{\rho}\bar{a}}-\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}})-\Gamma^{\alpha}_{bc}[\nabla_{\alpha}F_{\bar{\rho}\bar{a}}-\rho^{-l}\bar{\mathsf{D}}_{\alpha}(\rho^{l}\mathsf{f})_{\bar{a}}]$
$\displaystyle\qquad-\sum_{i=1}^{l_{1}}\Gamma^{d}_{b\rho}\,\nabla_{c}F_{\hat{\rho}_{i}\\![d]\bar{a}}-\sum_{j=1}^{l_{2}}\Gamma^{\rho}_{ba_{j}}\,\nabla_{c}F_{\bar{\rho}\hat{a}_{j}\\![\rho]}-\sum_{j=1}^{l_{2}}\mathsf{\Gamma}^{d}_{ba_{j}}\,(\nabla_{c}F_{\bar{\rho}\bar{a}}-\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}})$
$\displaystyle:=\rho^{-l}\bar{\nabla}_{bc}(\rho^{l}\mathsf{f})_{\bar{a}}+I_{1}+I_{2}+I_{3}+I_{4}+I_{5}\text{.}$

To simplify the upcoming computations, we define, for all $1\leq i\leq l_{1}$
and $1\leq j\leq l_{2}$, the vertical tensor fields $\mathsf{z}$,
$\mathsf{z}^{\rho}_{i}$, $\mathsf{z}^{v}_{j}$—of ranks $(0,l_{2}+1)$,
$(0,l_{2}+2)$, $(0,l_{2})$, respectively—via the index formulas

(7.9) $\displaystyle\mathsf{z}_{c\bar{a}}$
$\displaystyle:=\nabla_{c}F_{\bar{\rho}\bar{a}}-\bar{\mathsf{D}}_{c}\mathsf{f}_{\bar{a}}\text{,}$
$\displaystyle(\mathsf{z}^{\rho}_{i})_{cb\bar{a}}$
$\displaystyle:=\nabla_{c}F_{\hat{\rho}_{i}\\![b]\bar{a}}-\bar{\mathsf{D}}_{c}(\mathsf{f}^{\rho}_{i})_{b\bar{a}}\text{,}$
$\displaystyle(\mathsf{z}^{v}_{j})_{c\hat{a}_{j}}$
$\displaystyle:=\nabla_{c}F_{\bar{\rho}\hat{a}_{j}\\![\rho]}-\bar{\mathsf{D}}_{c}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}\text{.}$

Applying (2.28), the first part of (2.37), and (7.9) to the term $I_{2}$ from
(7.8), we obtain

$I_{2}=-\Gamma^{d}_{bc}(\nabla_{d}F_{\bar{\rho}\bar{a}}-\bar{\mathsf{D}}_{d}\mathsf{f}_{\bar{a}})\text{.}$

From (7.8), the first part of (7.9), and the above, we see that

(7.10) $I_{1}+I_{2}+I_{5}=\bar{\mathsf{D}}_{b}\mathsf{z}_{c\bar{a}}\text{.}$

Similarly, for $I_{3}$ and $I_{4}$, we again apply (2.28) and (7.9):

(7.11) $\displaystyle I_{3}$
$\displaystyle=\sum_{i=1}^{l_{1}}\mathsf{k}^{d}{}_{b}\,\bar{\mathsf{D}}_{c}(\mathsf{f}^{\rho}_{i})_{d\bar{a}}+\sum_{i=1}^{l_{1}}\mathsf{k}^{d}{}_{b}\,(\mathsf{z}^{\rho}_{i})_{cd\bar{a}}\text{,}$
$\displaystyle I_{4}$
$\displaystyle=-\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}b}\,\bar{\mathsf{D}}_{c}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}-\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}b}\,(\mathsf{z}^{v}_{j})_{c\hat{a}_{j}}\text{.}$

Now, recalling (7.4), along with (7.9), we deduce

(7.12) $\displaystyle\mathsf{z}_{c\bar{a}}$
$\displaystyle=\sum_{i=1}^{l_{1}}\mathsf{k}^{e}{}_{c}\,(\mathsf{f}^{\rho}_{i})_{e\bar{a}}-\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}c}\,(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}\text{,}$
$\displaystyle(\mathsf{z}^{\rho}_{i})_{cd\bar{a}}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq j\leq l_{1}\\\ j\neq
i\end{subarray}}\mathsf{k}^{e}{}_{c}\,(\mathsf{f}^{\rho,\rho}_{i,j})_{ed\bar{a}}-\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}c}\,(\mathsf{f}^{\rho,v}_{i,j})_{d\hat{a}_{j}}-\mathsf{k}_{bc}\,\mathsf{f}_{\bar{a}}\text{,}$
$\displaystyle(\mathsf{z}^{v}_{j})_{c\hat{a}_{j}}$
$\displaystyle=\sum_{i=1}^{l_{1}}\mathsf{k}^{e}{}_{c}\,(\mathsf{f}^{\rho,v}_{i,j})_{e\hat{a}_{j}}+\mathsf{k}^{e}{}_{c}\,\mathsf{f}_{\hat{a}_{j}\\![e]}-\sum_{\begin{subarray}{c}1\leq
i\leq l_{2}\\\ i\neq
j\end{subarray}}\mathsf{k}_{a_{i}c}\,(\mathsf{f}^{v,v}_{i,j})_{\hat{a}_{i,j}}\text{.}$

Combining (7.3), (7.10), and the above, we conclude that

(7.13) $\displaystyle I_{1}+I_{2}+I_{5}$
$\displaystyle=\sum_{i=1}^{l_{1}}[\mathsf{k}^{e}{}_{c}\,\bar{\mathsf{D}}_{b}(\mathsf{f}^{\rho}_{i})_{e\bar{a}}+\bar{\mathsf{D}}_{b}\mathsf{k}^{d}{}_{c}\,(\mathsf{f}^{\rho}_{i})_{e\bar{a}}]-\sum_{j=1}^{l_{2}}[\mathsf{k}_{a_{j}c}\,\bar{\mathsf{D}}_{b}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}+\bar{\mathsf{D}}_{b}\mathsf{k}_{a_{j}c}\,(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}]$
$\displaystyle=\rho^{-1}\sum_{i=1}^{l_{1}}\bar{\mathsf{D}}_{b}(\mathsf{f}^{\rho}_{i})_{c\bar{a}}-\rho^{-1}\sum_{j=1}^{l_{2}}\mathsf{g}_{a_{j}c}\,\bar{\mathsf{D}}_{b}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}+\sum_{i=1}^{l_{1}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\bar{\mathsf{D}}f^{\rho}_{i})_{cb\bar{a}}$
$\displaystyle\qquad+\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{m},\bar{\mathsf{D}}\mathsf{f}^{v}_{j})_{cb\bar{a}}+\sum_{i=1}^{l_{1}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{D}\mathsf{m},\mathsf{f}^{\rho}_{i})_{cb\bar{a}}+\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{D}\mathsf{m},\mathsf{f}^{v}_{j})_{cb\bar{a}}\text{.}$

Similar computations using (7.11) also yield

$\displaystyle I_{3}$
$\displaystyle=\sum_{i=1}^{l_{1}}\mathsf{k}^{d}{}_{b}\,\bar{\mathsf{D}}_{c}(\mathsf{f}^{\rho}_{i})_{d\bar{a}}+2\sum_{1\leq
i<j\leq
l_{1}}\mathsf{k}^{d}{}_{b}\mathsf{k}^{e}{}_{c}\,(\mathsf{f}^{\rho,\rho}_{i,j})_{ed\bar{a}}-\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathsf{k}^{d}{}_{b}\mathsf{k}_{a_{j}c}\,(\mathsf{f}^{\rho,v}_{i,j})_{d\hat{a}_{j}}-l_{1}\mathsf{k}^{d}{}_{b}\mathsf{k}_{dc}\,\mathsf{f}_{\bar{a}}$
$\displaystyle=\rho^{-1}\sum_{i=1}^{l_{1}}\bar{\mathsf{D}}_{c}(\mathsf{f}^{\rho}_{i})_{b\bar{a}}-l_{1}\rho^{-2}\mathsf{g}_{bc}\,\mathsf{f}_{\bar{a}}+2\rho^{-2}\sum_{1\leq
i<j\leq
l_{1}}(\mathsf{f}^{\rho,\rho}_{i,j})_{cb\bar{a}}-\rho^{-2}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathsf{g}_{a_{j}c}\,(\mathsf{f}^{\rho,v}_{i,j})_{b\hat{a}_{j}}\text{,}$
$\displaystyle\qquad+\sum_{i=1}^{l_{1}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\bar{\mathsf{D}}\mathsf{f}^{\rho}_{i})_{cb\bar{a}}+\rho^{-1}\mathscr{S}(\mathsf{m},\mathsf{f})_{cb\bar{a}}+\rho^{-1}\sum_{1\leq
i<j\leq
l_{1}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{f}^{\rho,\rho}_{i,j})_{cb\bar{a}}$
$\displaystyle\qquad+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{f})_{cb\bar{a}}+\sum_{1\leq
i<j\leq
l_{1}}\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{f}^{\rho,\rho}_{i,j})_{cb\bar{a}}+\rho^{-1}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{cb\bar{a}}$
$\displaystyle\qquad+\rho^{-1}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g},\mathsf{g}^{-1},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{cb\bar{a}}+\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{cb\bar{a}}\text{,}$
$\displaystyle I_{4}$
$\displaystyle=-\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}b}\,\bar{\mathsf{D}}_{c}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}-\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}b}\mathsf{k}^{e}{}_{c}\,(\mathsf{f}^{\rho,v}_{i,j})_{e\hat{a}_{j}}-\sum_{j=1}^{l_{2}}\mathsf{k}_{a_{j}b}\mathsf{k}^{e}{}_{c}\,\mathsf{f}_{\hat{a}_{j}\\![e]}$
$\displaystyle\qquad+2\sum_{1\leq i<j\leq
l_{2}}\mathsf{k}_{a_{i}c}\mathsf{k}_{a_{j}b}\,(\mathsf{f}^{v,v}_{i,j})_{\hat{a}_{i,j}}$
$\displaystyle=-\rho^{-1}\sum_{j=1}^{l_{2}}\mathsf{g}_{a_{j}b}\,\bar{\mathsf{D}}_{c}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}-\rho^{-2}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathsf{g}_{a_{j}b}\,(\mathsf{f}^{\rho,v}_{i,j})_{c\hat{a}_{j}}-\rho^{-2}\sum_{j=1}^{l_{2}}\mathsf{g}_{a_{j}b}\,\mathsf{f}_{\hat{a}_{j}\\![c]}$
$\displaystyle\qquad+2\rho^{-2}\sum_{1\leq i<j\leq
l_{2}}\mathsf{g}_{a_{i}c}\mathsf{g}_{a_{j}b}\,(\mathsf{f}^{v,v}_{i,j})_{\hat{a}_{i,j}}+\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{m},\mathsf{D}\mathsf{f}^{v}_{j})_{cb\bar{a}}+\rho^{-1}\mathscr{S}(\mathsf{m},\mathsf{f})_{cb\bar{a}}$
$\displaystyle\qquad+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{f})_{cb\bar{a}}+\rho^{-1}\sum_{1\leq
i<j\leq
l_{2}}\mathscr{S}(\mathsf{g},\mathsf{m},\mathsf{f}^{v,v}_{i,j})_{cb\bar{a}}+\sum_{1\leq
i<j\leq
l_{2}}\mathscr{S}(\mathsf{m},\mathsf{m},\mathsf{f}^{v,v}_{i,j})_{cb\bar{a}}$
$\displaystyle\qquad+\rho^{-1}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{cb\bar{a}}+\rho^{-1}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g},\mathsf{g}^{-1},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{cb\bar{a}}$
$\displaystyle\qquad+\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{cb\bar{a}}\text{.}$

Finally, combining (7.8), (7.13), and the above, we obtain

$\displaystyle\mathsf{g}^{bc}\,\nabla_{bc}F_{\bar{\rho}\bar{a}}$
$\displaystyle=\rho^{-l}\mathsf{g}^{bc}\bar{\nabla}_{bc}(\rho^{l}\mathsf{f})_{\bar{a}}+2\rho^{-1}\left[\sum_{i=1}^{l_{1}}\mathsf{g}^{bc}\bar{\mathsf{D}}_{b}(\mathsf{f}^{\rho}_{i})_{c\bar{a}}-\sum_{j=1}^{l_{2}}\bar{\mathsf{D}}_{a_{j}}(\mathsf{f}^{v}_{j})_{\hat{a}_{j}}-(nl_{1}+l_{2})\rho^{-2}\,\mathsf{f}_{\bar{a}}\right]$
$\displaystyle\qquad+2\rho^{-2}\left[\sum_{1\leq i<j\leq
l_{1}}\mathsf{g}^{bc}\,(\mathsf{f}^{\rho,\rho}_{i,j})_{cb\bar{a}}-\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}(\mathsf{f}^{\rho,v}_{i,j})_{a_{j}\hat{a}_{j}}+\sum_{1\leq
i<j\leq
l_{2}}\mathsf{g}_{a_{i}a_{j}}\,(\mathsf{f}^{v,v}_{i,j})_{\hat{a}_{i,j}}\right]$
$\displaystyle\qquad+\sum_{i=1}^{l_{1}}\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\bar{\mathsf{D}}f^{\rho}_{i})_{\bar{a}}+\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\bar{\mathsf{D}}f^{v}_{j})_{\bar{a}}+\sum_{i=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-2},\mathsf{D}\mathsf{m},f^{\rho}_{i})_{\bar{a}}$
$\displaystyle\qquad+\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{D}\mathsf{m},f^{v}_{j})_{\bar{a}}+\rho^{-1}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{f})_{\bar{a}}+\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{f})_{\bar{a}}$
$\displaystyle\qquad+\rho^{-1}\sum_{1\leq i<j\leq
l_{1}}\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{f}^{\rho,\rho}_{i,j})_{\bar{a}}+\sum_{1\leq
i<j\leq
l_{1}}\mathscr{S}(\mathsf{g}^{-3},\mathsf{m},\mathsf{m},\mathsf{f}^{\rho,\rho}_{i,j})_{\bar{a}}$
$\displaystyle\qquad+\rho^{-1}\sum_{1\leq i<j\leq
l_{2}}\mathscr{S}(\mathsf{g},\mathsf{g}^{-1},\mathsf{m},\mathsf{f}^{v,v}_{i,j})_{\bar{a}}+\sum_{1\leq
i<j\leq
l_{2}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{f}^{v,v}_{i,j})_{\bar{a}}$
$\displaystyle\qquad+\rho^{-1}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{\bar{a}}+\rho^{-1}\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g},\mathsf{g}^{-2},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{\bar{a}}$
$\displaystyle\qquad+\sum_{i=1}^{l_{1}}\sum_{j=1}^{l_{2}}\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{f}^{\rho,v}_{i,j})_{\bar{a}}\text{.}$

The last formula in (2.37) now follows from (7.7), the above, and the fact
that

$\Box
F=\rho^{2}(\nabla_{\rho\rho}F+\mathsf{g}^{bc}\nabla_{bc}F)\text{,}\qquad\rho^{-1}\bar{\Box}(\rho^{l}\mathsf{f})=\rho^{2}[\rho^{-l}\bar{\nabla}_{\rho\rho}(\rho^{l}\mathsf{f})+\mathsf{g}^{bc}\rho^{-l}\bar{\nabla}_{bc}(\rho^{l}\mathsf{f})]\text{.}$

###  7.3. Proof of Proposition 3.4

Throughout this proof, we assume all indices are raised and lowered using $g$.
First, the Bianchi identity and Proposition 2.13 yield

(7.14)
$\displaystyle\nabla_{\mu}W_{\alpha\beta\gamma\delta}+\nabla_{\gamma}W_{\alpha\beta\delta\mu}+\nabla_{\delta}W_{\alpha\beta\mu\gamma}$
$\displaystyle=0\text{,}$
$\displaystyle\nabla^{\mu}W_{\mu\beta\gamma\delta}=\nabla^{\mu}W_{\mu\beta\gamma\delta}+\nabla_{\gamma}W_{\mu\beta\delta}{}^{\mu}+\nabla_{\delta}W_{\mu\beta}{}^{\mu}{}_{\gamma}$
$\displaystyle=0\text{,}$

which immediately proves the first identity in (3.4).

Next, taking a divergence of the first part of (7.14) (in the “$\mu$”
component) yields

(7.15)
$0=\Box\mathcal{W}_{\alpha\beta\gamma\delta}+\nabla^{\mu}{}_{\gamma}W_{\alpha\beta\delta\mu}+\nabla^{\mu}{}_{\delta}W_{\alpha\beta\mu\gamma}\text{.}$

Commuting derivatives and then applying the second part of (7.14), we obtain

(7.16) $\displaystyle\nabla^{\mu}{}_{\gamma}W_{\alpha\beta\delta\mu}$
$\displaystyle=-R^{\lambda}{}_{\alpha}{}^{\mu}{}_{\gamma}W_{\lambda\beta\delta\mu}-R^{\lambda}{}_{\beta}{}^{\mu}{}_{\gamma}W_{\alpha\lambda\delta\mu}-R^{\lambda}{}_{\delta}{}^{\mu}{}_{\gamma}W_{\alpha\beta\lambda\mu}-R^{\lambda}{}_{\mu}{}^{\mu}{}_{\gamma}W_{\alpha\beta\delta\lambda}$
$\displaystyle:=I_{1,1}+I_{1,2}+I_{1,3}+I_{1,4}\text{,}$
$\displaystyle\nabla^{\mu}{}_{\delta}W_{\alpha\beta\mu\gamma}$
$\displaystyle=-R^{\lambda}{}_{\alpha}{}^{\mu}{}_{\delta}W_{\lambda\beta\mu\gamma}-R^{\lambda}{}_{\beta}{}^{\mu}{}_{\delta}W_{\alpha\lambda\mu\gamma}-R^{\lambda}{}_{\mu}{}^{\mu}{}_{\delta}W_{\alpha\beta\lambda\gamma}-R^{\lambda}{}_{\gamma}{}^{\mu}{}_{\delta}W_{\alpha\beta\mu\lambda}$
$\displaystyle:=I_{2,1}+I_{2,2}+I_{2,3}+I_{2,4}\text{.}$

Now, using Proposition 2.13, we expand

$\displaystyle I_{1,1}+I_{2,2}$
$\displaystyle=-(W^{\lambda}{}_{\alpha}{}^{\mu}{}_{\gamma}+g^{\lambda}{}_{\gamma}g_{\alpha}{}^{\mu})W_{\lambda\beta\delta\mu}-(W^{\lambda}{}_{\beta}{}^{\mu}{}_{\delta}+g^{\lambda}{}_{\delta}g_{\beta}{}^{\mu})W_{\alpha\lambda\mu\gamma}$
$\displaystyle=-2W_{\alpha\delta\beta\gamma}+W^{\lambda}{}_{\alpha}{}^{\mu}{}_{\gamma}W_{\lambda\beta\mu\delta}+W^{\lambda}{}_{\beta}{}^{\mu}{}_{\delta}W_{\lambda\alpha\mu\gamma}\text{,}$
$\displaystyle I_{1,2}+I_{2,1}$
$\displaystyle=-(W^{\lambda}{}_{\beta}{}^{\mu}{}_{\gamma}+g^{\lambda}{}_{\gamma}g_{\beta}{}^{\mu})W_{\alpha\lambda\delta\mu}-(W^{\lambda}{}_{\alpha}{}^{\mu}{}_{\delta}+g^{\lambda}{}_{\delta}g_{\alpha}{}^{\mu})W_{\lambda\beta\mu\gamma}$
$\displaystyle=-2W_{\alpha\gamma\delta\beta}-W^{\lambda}{}_{\beta}{}^{\mu}{}_{\gamma}W_{\lambda\alpha\mu\delta}-W^{\lambda}{}_{\alpha}{}^{\mu}{}_{\delta}W_{\lambda\beta\mu\gamma}\text{,}$
$\displaystyle I_{1,3}+I_{2,4}$
$\displaystyle=-(W^{\lambda}{}_{\delta}{}^{\mu}{}_{\gamma}+g^{\lambda}{}_{\gamma}g_{\delta}{}^{\mu})W_{\alpha\beta\lambda\mu}-(W^{\lambda}{}_{\gamma}{}^{\mu}{}_{\delta}+g^{\lambda}{}_{\delta}g_{\gamma}{}^{\mu})W_{\alpha\beta\mu\lambda}$
$\displaystyle=-2W_{\alpha\beta\gamma\delta}-W^{\lambda}{}_{\delta}{}^{\mu}{}_{\gamma}W_{\alpha\beta\lambda\mu}-W^{\lambda}{}_{\gamma}{}^{\mu}{}_{\delta}W_{\alpha\beta\mu\lambda}\text{,}$
$\displaystyle I_{1,4}+I_{2,3}$
$\displaystyle=2nW_{\alpha\beta\gamma\delta}\text{.}$

Summing the above and recalling the symmetries of $W$, we then obtain

$\displaystyle\sum_{i=1}^{2}\sum_{j=1}^{4}I_{i,j}$
$\displaystyle=2nW_{\alpha\beta\gamma\delta}+W^{\lambda}{}_{\alpha}{}^{\mu}{}_{\gamma}W_{\lambda\beta\mu\delta}+W^{\lambda}{}_{\beta}{}^{\mu}{}_{\delta}W_{\lambda\alpha\mu\gamma}$
$\displaystyle\qquad-W^{\lambda}{}_{\beta}{}^{\mu}{}_{\gamma}W_{\lambda\alpha\mu\delta}-W^{\lambda}{}_{\alpha}{}^{\mu}{}_{\delta}W_{\lambda\beta\mu\gamma}+W^{\lambda\mu}{}_{\gamma\delta}W_{\alpha\beta\lambda\mu}\text{.}$

Combining (7.15), (7.16), and the above results in the remaining equation of
(3.4).

###  7.4. Proof of Proposition 3.5

Let us begin with the first two identities in (3.5). Applying the first
equation in (2.37) to $F:=W$ (with permuted indices) yields

$\displaystyle\rho^{2}\nabla_{\rho}W_{\rho a\rho b}$
$\displaystyle=\rho^{-2}\bar{\mathsf{D}}_{\rho}(\rho^{2}\mathsf{w}^{2}_{ab})$
$\displaystyle=\bar{\mathsf{D}}_{\rho}\mathsf{w}^{2}_{ab}+2\rho^{-1}\mathsf{w}^{2}_{ab}\text{,}$
$\displaystyle\rho^{2}\nabla_{\rho}W_{\rho abc}$
$\displaystyle=\rho^{-2}\bar{\mathsf{D}}_{\rho}(\rho^{2}\mathsf{w}^{1}_{abc})$
$\displaystyle=\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+2\rho^{-1}\mathsf{w}^{1}_{abc}\text{.}$

Applying (2.7) and the first identity in (3.4) to the left-hand sides of the
above yields

(7.17)
$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{2}_{ab}+2\rho^{-1}\mathsf{w}^{2}_{ab}$
$\displaystyle=\mathsf{g}^{cd}\nabla_{c}(\rho^{2}W)_{\rho bad}\text{,}$
$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+2\rho^{-1}\mathsf{w}^{1}_{abc}$
$\displaystyle=-\mathsf{g}^{de}\nabla_{d}(\rho^{2}W)_{eabc}\text{.}$

Applying the second identity of (2.37) to the first equation in (7.17), we
have

(7.18)
$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{2}_{ab}+2\rho^{-1}\mathsf{w}^{2}_{ab}$
$\displaystyle=\mathsf{g}^{cd}\mathsf{D}_{c}\mathsf{w}^{1}_{bad}-\rho^{-1}\mathsf{g}^{cd}\,\mathsf{w}^{0}_{dacb}+\rho^{-1}\mathsf{g}^{cd}\mathsf{g}_{cd}\,\mathsf{w}^{2}_{ab}-\rho^{-1}\mathsf{g}^{cd}\mathsf{g}_{ca}\,\mathsf{w}^{2}_{db}$
$\displaystyle\qquad+\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{0})_{ab}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})_{ab}$
$\displaystyle=\mathsf{g}^{cd}\mathsf{D}_{c}\mathsf{w}^{1}_{bad}+n\rho^{-1}\mathsf{w}^{2}_{ab}+\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{0})_{ab}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})_{ab}\text{,}$

where in the last step, we also used Definition 3.1 and the trace-free
property of $W$ to obtain

$-\mathsf{g}^{cd}\,\mathsf{w}^{0}_{dacb}=\mathsf{w}^{2}_{ab}\text{.}$

The first part of (3.5) follows immediately from (7.18).

Similarly, applying (2.37) to the second part of (7.17) yields

(7.19)
$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+2\rho^{-1}\mathsf{w}^{1}_{abc}$
$\displaystyle=-\mathsf{g}^{de}\mathsf{D}_{d}\mathsf{w}^{0}_{eabc}+\rho^{-1}\mathsf{g}^{de}\mathsf{g}_{de}\,\mathsf{w}^{1}_{abc}-\rho^{-1}\mathsf{g}^{de}\mathsf{g}_{da}\,\mathsf{w}^{1}_{ebc}$
$\displaystyle\qquad+\rho^{-1}\mathsf{g}^{de}\mathsf{g}_{db}\,\mathsf{w}^{1}_{cea}-\rho^{-1}\mathsf{g}^{de}\mathsf{g}_{dc}\,\mathsf{w}^{1}_{bea}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{1})_{abc}$
$\displaystyle=-\mathsf{g}^{de}\mathsf{D}_{d}\mathsf{w}^{0}_{eabc}+n\rho^{-1}\,\mathsf{w}^{1}_{abc}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{1})_{abc}\text{,}$

where in the last step, we noted from Definition 3.1 and the symmetries of $W$
that

$-\mathsf{w}^{1}_{abc}+\mathsf{w}^{1}_{cba}-\mathsf{w}^{1}_{bca}=0\text{.}$

The second identity in (3.5) now follows from (7.19).

For the two remaining parts of (3.5), we again start with the first part of
(2.37):

$\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+2\rho^{-1}\mathsf{w}^{1}_{abc}=\rho^{2}\nabla_{\rho}W_{\rho
abc}\text{,}\qquad\bar{\mathsf{D}}_{\rho}\mathsf{w}^{0}_{abcd}+2\rho^{-1}\mathsf{w}^{0}_{abcd}=\rho^{2}\nabla_{\rho}W_{abcd}\text{.}$

The right-hand sides of the above can then be expanded using the Bianchi
identity for $W$:

$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+2\rho^{-1}\mathsf{w}^{1}_{abc}$
$\displaystyle=\nabla_{b}(\rho^{2}W)_{\rho a\rho
c}-\nabla_{c}(\rho^{2}W)_{\rho a\rho b}\text{,}$
$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{0}_{abcd}+2\rho^{-1}\mathsf{w}^{0}_{abcd}$
$\displaystyle=\nabla_{a}(\rho^{2}W)_{\rho bcd}-\nabla_{b}(\rho^{2}W)_{\rho
acd}\text{.}$

Applying (2.37) to each term on the right-hand side of the above, we then
obtain

$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+2\rho^{-1}\mathsf{w}^{1}_{abc}$
$\displaystyle=\mathsf{D}_{b}\mathsf{w}^{2}_{ac}+\rho^{-1}\mathsf{w}^{1}_{cba}+\rho^{-1}\mathsf{w}^{1}_{abc}-\mathsf{D}_{c}\mathsf{w}^{2}_{ab}-\rho^{-1}\mathsf{w}^{1}_{bca}-\rho^{-1}\mathsf{w}^{1}_{acb}$
$\displaystyle\qquad+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{1})_{abc}$
$\displaystyle=\mathsf{D}_{b}\mathsf{w}^{2}_{ac}-\mathsf{D}_{c}\mathsf{w}^{2}_{ab}+3\rho^{-1}\mathsf{w}^{1}_{abc}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{1})_{abc}\text{,}$
$\displaystyle\bar{\mathsf{D}}_{\rho}\mathsf{w}^{0}_{abcd}+2\rho^{-1}\mathsf{w}^{0}_{abcd}$
$\displaystyle=\mathsf{D}_{a}\mathsf{w}^{1}_{bcd}+\rho^{-1}\mathsf{w}^{0}_{abcd}-\rho^{-1}\mathsf{g}_{ac}\,\mathsf{w}^{2}_{bd}+\rho^{-1}\mathsf{g}_{ad}\,\mathsf{w}^{2}_{bc}-\mathsf{D}_{b}\mathsf{w}^{1}_{acd}-\rho^{-1}\mathsf{w}^{0}_{bacd}$
$\displaystyle\qquad+\rho^{-1}\mathsf{g}_{bc}\,\mathsf{w}^{2}_{ad}-\rho^{-1}\mathsf{g}_{bd}\,\mathsf{w}^{2}_{ac}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{0})_{abcd}+\mathscr{S}(\mathsf{m},\mathsf{w}^{2})_{abcd}$
$\displaystyle=\mathsf{D}_{a}\mathsf{w}^{1}_{bcd}-\mathsf{D}_{b}\mathsf{w}^{1}_{acd}+2\rho^{-1}\mathsf{w}^{0}_{abcd}+\rho^{-1}\mathsf{g}_{ad}\,\mathsf{w}^{2}_{bc}+\rho^{-1}\mathsf{g}_{bc}\,\mathsf{w}^{2}_{ad}$
$\displaystyle\qquad-\rho^{-1}\mathsf{g}_{ac}\,\mathsf{w}^{2}_{bd}-\rho^{-1}\mathsf{g}_{bd}\,\mathsf{w}^{2}_{ac}+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{0})_{abcd}+\mathscr{S}(\mathsf{m},\mathsf{w}^{2})_{abcd}\text{.}$

The last two identities of (3.5) are now immediate consequences of the above.

###  7.5. Proof of Proposition 3.6

Throughout, we assume all indices are with respect to $\varphi$ and
$\varphi_{\rho}$, for some arbitrary coordinates $(U,\varphi)$ on
$\mathscr{I}$. Also, for convenience, we define $Q$ by

(7.20)
$Q_{\alpha\beta\gamma\delta}:=g^{\lambda\kappa}g^{\mu\nu}(2W_{\lambda\alpha\mu\delta}W_{\kappa\beta\nu\gamma}-2W_{\lambda\alpha\mu\gamma}W_{\kappa\beta\nu\delta}-W_{\lambda\mu\gamma\delta}W_{\alpha\beta\kappa\nu})\text{,}$

that is, the right-hand side of the wave equation in (3.4). Then, by (2.7) and
(3.1), we have

(7.21) $\displaystyle Q_{\rho b\rho d}$
$\displaystyle=g^{\lambda\kappa}g^{\mu\nu}(2W_{\lambda\rho\mu d}W_{\kappa
b\nu\rho}-2W_{\lambda\rho\mu\rho}W_{\kappa b\nu d}-W_{\lambda\mu\rho d}W_{\rho
b\kappa\nu})$
$\displaystyle=\mathscr{S}(\mathsf{g}^{-1},\mathsf{w}^{2},\mathsf{w}^{2})+\mathscr{S}(\mathsf{g}^{-2},\mathsf{w}^{1},\mathsf{w}^{1})+\mathscr{S}(\mathsf{g}^{-2},\mathsf{w}^{0},\mathsf{w}^{2})\text{,}$
$\displaystyle Q_{\rho bcd}$
$\displaystyle=g^{\lambda\kappa}g^{\mu\nu}(2W_{\lambda\rho\mu d}W_{\kappa b\nu
c}-2W_{\lambda\rho\mu c}W_{\kappa b\nu d}-W_{\lambda\mu cd}W_{\rho
b\kappa\nu})$
$\displaystyle=\mathscr{S}(\mathsf{g}^{-1},\mathsf{w}^{1},\mathsf{w}^{2})+\mathscr{S}(\mathsf{g}^{-2},\mathsf{w}^{0},\mathsf{w}^{1})\text{,}$
$\displaystyle Q_{abcd}$
$\displaystyle=g^{\lambda\kappa}g^{\mu\nu}(2W_{\lambda a\mu d}W_{\kappa b\nu
c}-2W_{\lambda a\mu c}W_{\kappa b\nu d}-W_{\lambda\mu cd}W_{ab\kappa\nu})$
$\displaystyle=\mathscr{S}(\mathsf{w}^{2},\mathsf{w}^{2})+\mathscr{S}(\mathsf{g}^{-1},\mathsf{w}^{1},\mathsf{w}^{1})+\mathscr{S}(\mathsf{g}^{-2},\mathsf{w}^{0},\mathsf{w}^{0})\text{.}$

We now apply (3.1) and the last equation in (2.37) to $W$ to obtain

(7.22) $\displaystyle\rho^{2}\Box W_{\rho a\rho b}$
$\displaystyle=\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{2})_{ab}+2\rho\mathsf{g}^{cd}(\mathsf{D}_{c}\mathsf{w}^{1}_{bda}+\mathsf{D}_{c}\mathsf{w}^{1}_{adb})-0-(2n+2)\,\mathsf{w}^{2}_{ab}$
$\displaystyle\qquad+4\,\mathsf{w}^{2}_{ab}+2\mathsf{g}^{cd}\,\mathsf{w}^{0}_{cadb}+0+\mathsf{E}^{2}_{ab}$
$\displaystyle=\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{2})_{ab}+2\rho\mathsf{g}^{cd}(\mathsf{D}_{c}\mathsf{w}^{1}_{bda}+\mathsf{D}_{c}\mathsf{w}^{1}_{adb})-2n\,\mathsf{w}^{2}_{ab}+\mathsf{E}^{2}_{ab}\text{,}$

where we used that $-\mathsf{w}^{2}$ is the $\mathsf{g}$-trace of
$\mathsf{w}^{0}$, and where the error terms $\mathsf{E}^{2}$ are given by

(7.23) $\displaystyle\mathsf{E}^{2}$
$\displaystyle=\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{D}\mathsf{w}^{1})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{D}\mathsf{m},\mathsf{w}^{1})+\rho\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})+\rho\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{0})$
$\displaystyle\qquad+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{w}^{2})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-3},\mathsf{m},\mathsf{m},\mathsf{w}^{0})+\rho\,\mathscr{S}(\mathsf{g},\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{2})\text{,}$

Applying the second part of (3.4) and the first part of (3.5) to (7.22) yields

$\displaystyle\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{2})_{ab}+4(n-2)\mathsf{w}^{2}_{ab}$
$\displaystyle=\rho^{2}\Box W_{\rho a\rho
b}+4\rho\,\bar{\mathsf{D}}_{\rho}\mathsf{w}^{2}_{ab}+2n\,\mathsf{w}^{2}_{ab}+\mathsf{E}^{2}_{ab}$
$\displaystyle=\rho^{2}Q_{\rho a\rho
b}+4\rho\,\bar{\mathsf{D}}_{\rho}\mathsf{w}^{2}_{ab}+\mathsf{E}^{2}_{ab}\text{,}$

Moreover, expanding the left-hand side using (2.34) (with $p=2$), we see that

(7.24)
$\bar{\Box}\mathsf{w}^{2}_{ab}+2(n-2)\mathsf{w}^{2}_{ab}=\mathsf{E}^{2}_{ab}+\rho^{2}Q_{\rho
a\rho b}\text{,}$

The first equation in (3.6) now follows from (7.21), (7.23), and the above.

The remaining parts of (3.6) are treated similarly. Again, we first apply the
last part of (2.37):

(7.25) $\displaystyle\rho^{2}\Box W_{\rho abc}$
$\displaystyle=\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{1})_{abc}+2\rho\mathsf{g}^{de}\,\mathsf{D}_{d}\mathsf{w}^{0}_{eabc}+2\rho\,(\mathsf{D}_{c}\mathsf{w}^{2}_{ab}-\mathsf{D}_{b}\mathsf{w}^{2}_{ac})$
$\displaystyle\qquad-(n+3)\,\mathsf{w}^{1}_{abc}+\mathsf{E}^{1}_{abc}\text{,}$
$\displaystyle\rho^{2}\Box W_{abcd}$
$\displaystyle=\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{0})_{abcd}-2\rho\,(\mathsf{D}_{a}\mathsf{w}^{1}_{bcd}-\mathsf{D}_{b}\mathsf{w}^{1}_{acd}+\mathsf{D}_{c}\mathsf{w}^{1}_{dab}-\mathsf{D}_{d}\mathsf{w}^{1}_{cab})$
$\displaystyle\qquad-4\,\mathsf{w}^{0}_{abcd}+2(\mathsf{g}_{ac}\,\mathsf{w}^{2}_{bd}-\mathsf{g}_{ad}\,\mathsf{w}^{2}_{bc}-\mathsf{g}_{bc}\,\mathsf{w}^{2}_{ad}+\mathsf{g}_{bd}\,\mathsf{w}^{2}_{ac})+\mathsf{E}^{0}_{abcd}\text{,}$

where we also made use of the symmetries of $W$, and where the corresponding
error terms are

(7.26) $\displaystyle\mathsf{E}^{1}$
$\displaystyle=\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{D}\mathsf{w}^{0})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{D}\mathsf{w}^{2})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{D}\mathsf{m},\mathsf{w}^{0})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{D}\mathsf{m},\mathsf{w}^{2})$
$\displaystyle\qquad+\rho\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{1})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{w}^{1})+\rho\,\mathscr{S}(\mathsf{g},\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{1})\text{,}$
$\displaystyle\mathsf{E}^{0}$
$\displaystyle=\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{D}\mathsf{w}^{1})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{D}\mathsf{m},\mathsf{w}^{1})+\rho\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{0})$
$\displaystyle\qquad+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\mathsf{w}^{0})+\rho\,\mathscr{S}(\mathsf{g},\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})+\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{m},\mathsf{w}^{2})\text{.}$

The first-order terms in the right-hand sides of (7.25) can be further
expanded using (3.5), and the terms involving $\bar{\Box}$ in (7.27) can be
expanded using (2.34):

(7.27) $\displaystyle\rho^{2}\Box W_{\rho abc}$
$\displaystyle=\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{1})_{abc}-4\rho\,\bar{\mathsf{D}}_{\rho}\mathsf{w}^{1}_{abc}+(n-5)\,\mathsf{w}^{1}_{abc}+\mathsf{E}^{1}_{abc}$
$\displaystyle=\bar{\Box}\mathsf{w}^{1}_{abc}-(n+1)\,\mathsf{w}^{1}_{abc}+\mathsf{E}^{1}_{abc}\text{,}$
$\displaystyle\rho^{2}\Box W_{abcd}$
$\displaystyle=\rho^{-2}\bar{\Box}(\rho^{2}\mathsf{w}^{0})_{abcd}-4\rho\,\bar{\mathsf{D}}_{\rho}\mathsf{w}^{0}_{abcd}-4\,\mathsf{w}^{0}_{abcd}$
$\displaystyle\qquad+2(\mathsf{g}_{ad}\,\mathsf{w}^{2}_{bc}+\mathsf{g}_{bc}\,\mathsf{w}^{2}_{ad}-\mathsf{g}_{ac}\,\mathsf{w}^{2}_{bd}-\mathsf{g}_{bd}\,\mathsf{w}^{2}_{ac})+\mathsf{E}^{0}_{abcd}$
$\displaystyle=\bar{\Box}\mathsf{w}^{0}_{abcd}-2n\,\mathsf{w}^{0}_{abcd}+2(\mathsf{g}_{ad}\,\mathsf{w}^{2}_{bc}+\mathsf{g}_{bc}\,\mathsf{w}^{2}_{ad}-\mathsf{g}_{ac}\,\mathsf{w}^{2}_{bd}-\mathsf{g}_{bd}\,\mathsf{w}^{2}_{ac})+\mathsf{E}^{0}_{abcd}\text{.}$

The left-hand sides of (7.27) can be expanded using (3.4), (7.20), and (7.21):

$\displaystyle\rho^{2}\Box W_{\rho abc}$
$\displaystyle=-2n\,\mathsf{w}^{1}_{abc}+\rho^{2}\,Q_{\rho abc}\text{,}$
$\displaystyle\rho^{2}\Box W_{abcd}$
$\displaystyle=-2n\,\mathsf{w}^{0}_{abcd}+\rho^{2}\,Q_{abcd}\text{.}$

Combining the above with (7.21) and (7.27) results in the second equation in
(3.6), as well as

$\bar{\Box}\mathsf{w}^{0}_{abcd}=-2(\mathsf{g}_{ad}\,\mathsf{w}^{2}_{bc}+\mathsf{g}_{bc}\,\mathsf{w}^{2}_{ad}-\mathsf{g}_{ac}\,\mathsf{w}^{2}_{bd}-\mathsf{g}_{bd}\,\mathsf{w}^{2}_{ac})+\mathsf{E}^{0}_{abcd}+\rho^{2}Q_{abcd}\text{.}$

Recalling (3.2), (7.24), and the above, we then see that

$\displaystyle\bar{\Box}\mathsf{w}^{\star}_{abcd}$
$\displaystyle=\mathsf{E}^{0}_{abcd}-\frac{1}{n-2}(\mathsf{g}_{ad}\mathsf{E}^{2}_{bc}+\mathsf{g}_{bc}\mathsf{E}^{2}_{ad}-\mathsf{g}_{ac}\mathsf{E}^{2}_{bd}-\mathsf{g}_{bd}\mathsf{E}_{ac})+\rho^{2}Q_{abcd}$
$\displaystyle\qquad-\frac{\rho^{2}}{n-2}(\mathsf{g}_{ad}Q_{\rho b\rho
c}+\mathsf{g}_{bc}Q_{\rho a\rho d}-\mathsf{g}_{ac}Q_{\rho b\rho
d}-\mathsf{g}_{bd}Q_{\rho a\rho c})\text{.}$

The last equation in (3.6) now follows from (7.21), (7.23), (7.26), and the
above.

###  7.6. Proof of Proposition 3.8

The first two parts of (3.8) follow immediately from Theorem 2.16, in
particular from the boundary limits for $\mathsf{g}$ and $\mathsf{g}^{-1}$.
The last two parts of (3.8) also follow from Theorem 2.16, using the boundary
limits for $\mathscr{L}_{\rho}\mathsf{g}$ and
$\mathscr{L}_{\rho}^{2}\mathsf{g}$, along with Taylor’s theorem.

Next, for the Weyl curvature components, the key is the formulation (3.3) of
the Einstein-vacuum equations. From the boundary limits in Theorem 2.16, we
see that

(7.28)
$\rho^{-1}\mathscr{L}_{\rho}\mathsf{g}=\mathscr{O}_{M-2}(1)\text{,}\qquad-\mathscr{L}^{2}_{\rho}\mathsf{g}+\rho^{-1}\mathscr{L}_{\rho}\mathsf{g}=\mathscr{O}_{M-2}(1)=\mathscr{O}_{M-3}(\rho)\text{.}$

Moreover, writing the components of $\mathsf{R}$ in terms of coordinate
derivatives of $\mathsf{g}$ yields

(7.29) $\mathsf{R}=\mathscr{O}_{M-2}(1)\text{.}$

Combining (3.3), (3.8), (7.28), and (7.29) results in the asymptotics for
$\mathsf{w}^{0}$, $\mathsf{w}^{1}$, and $\mathsf{w}^{2}$ in (3.9). The
asymptotics for $\mathsf{w}^{\star}$ in (3.9) then follow from (3.2) and the
above.

The asymptotics for $\mathscr{L}_{\rho}\mathsf{w}^{0}$,
$\mathscr{L}_{\rho}\mathsf{w}^{1}$, and $\mathscr{L}_{\rho}\mathsf{w}^{2}$ are
proved similarly—we apply $\mathscr{L}_{\rho}$ to both sides of (3.3), and we
then apply the boundary limits in Theorem 2.16 to each term on the right-hand
side. In addition, we make use of the following observations:

  * •

For $\mathscr{L}_{\rho}\mathsf{w}^{1}$, we also use (2.33) to commute
$\mathscr{L}_{\rho}$ with $\mathsf{D}$.

  * •

For $\mathscr{L}_{\rho}\mathsf{w}^{0}$, we express the components of
$\mathsf{R}$ in terms of derivatives of $\mathsf{g}$, which yields

$\mathscr{L}_{\rho}\mathsf{R}=\mathscr{O}_{M-3}(1)\text{.}$

  * •

For both $\mathscr{L}_{\rho}\mathsf{w}^{0}$ and
$\mathscr{L}_{\rho}\mathsf{w}^{2}$, we also note that

$\mathscr{L}_{\rho}(\rho^{-1}\mathsf{m})=\mathscr{O}_{M-3}(1)\text{.}$

Finally, the asymptotics for $\mathscr{L}_{\rho}\mathsf{w}^{\star}$ once again
follow from (3.2) and the above.

###  7.7. Proof of Proposition 3.12

The first part of (3.14) is a consequence of the following identity, which
holds with respect to any coordinate system $(U,\varphi)$ on $\mathscr{I}$:

$\mathsf{g}^{ab}-\check{\mathsf{g}}^{ab}=-\mathsf{g}^{ac}(\mathsf{g}_{cd}-\check{\mathsf{g}}_{cd})\check{\mathsf{g}}^{bd}\text{,}$

Next, we can view (3.11) as a system of differential equations for the
components of $\mathsf{Q}$, with coefficients and sources given by
$\mathsf{g}^{-1}$, $\mathsf{g}-\check{\mathsf{g}}$, and $\mathsf{m}$. By
solving these equations, we can write the components of
$\mathsf{Q}|_{\sigma}$, $\sigma\in(0,\rho_{0}]$, as integrals in $\rho$ (from
$0$ to $\sigma$) of some combination of $\mathsf{g}^{-1}$,
$\mathsf{g}-\check{\mathsf{g}}$, and $\mathsf{m}$, along with a matrix
exponential factor. Combining the above with (3.8) results in the second part
of (3.14). (In particular, we obtain one extra power of $\rho$ from the
integral with respect to $\rho$.) The third part of (3.14) then follows from
(3.8), (3.12), and the second part of (3.14).

It remains to prove (3.15). The first part of (3.15) is immediate, since it is
clear from the first two parts of (2.28) that
$\smash{\Gamma^{\alpha}_{\rho\rho},\check{\Gamma}^{\alpha}_{\rho\rho}}$ and
$\smash{\Gamma^{\rho}_{\alpha\rho},\check{\Gamma}^{\rho}_{\alpha\rho}}$ are
identical. Moreover, the second identity in (3.15) follows immediately from
applying the fifth part of (2.28). Similarly, for the third part of (3.15), we
use the third and fourth parts of (2.28) in order to obtain

$(\Gamma-\check{\Gamma})^{a}_{\rho
b}=(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{a}_{\rho
b}=\frac{1}{2}(\mathsf{g}^{cd}\mathsf{m}_{ad}-\check{\mathsf{g}}^{cd}\check{\mathsf{m}}_{ad})\text{.}$

Recalling the asymptotics of (3.8) and the first part of (3.14) yields, as
desired,

$\displaystyle(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{a}_{\rho b}$
$\displaystyle=\frac{1}{2}(\mathsf{g}^{cd}-\check{\mathsf{g}}^{cd})\mathsf{m}_{ad}+\frac{1}{2}\check{\mathsf{g}}^{cd}(\mathsf{m}_{ad}-\check{\mathsf{m}}_{ad})$
$\displaystyle=\mathscr{O}_{M-2}(\rho;\mathsf{g}^{-1}-\check{\mathsf{g}}^{-1})^{a}{}_{b}+\mathscr{O}_{M}(1;\mathsf{m}-\check{\mathsf{m}})^{a}{}_{b}$
$\displaystyle=\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})^{a}{}_{b}+\mathscr{O}_{M}(1;\mathsf{m}-\check{\mathsf{m}})^{a}{}_{b}\text{.}$

For the last part of (3.15), note the last part of (2.28) implies

$(\Gamma-\check{\Gamma})^{c}_{ab}=(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{c}_{ab}\text{.}$

Expanding the Christoffel symbols, we then obtain, via a direct computation,

$\displaystyle(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{c}_{ab}$
$\displaystyle=\frac{1}{2}\mathsf{g}^{cd}(\partial_{a}\mathsf{g}_{db}+\partial_{b}\mathsf{g}_{da}-\partial_{d}\mathsf{g}_{ab})-\frac{1}{2}\check{\mathsf{g}}^{cd}(\partial_{a}\check{\mathsf{g}}_{db}+\partial_{b}\check{\mathsf{g}}_{da}-\partial_{d}\check{\mathsf{g}}_{ab})$
$\displaystyle=\frac{1}{2}\check{\mathsf{g}}^{cd}[\mathsf{D}_{a}(\mathsf{g}-\check{\mathsf{g}})_{db}+\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}})_{da}-\mathsf{D}_{d}(\mathsf{g}-\check{\mathsf{g}})_{ab}]\text{,}$

as desired (see also [53, Appendix D]). The final equality in (3.15) now
follows from (3.8).

###  7.8. Proof of Proposition 3.13

The first part of (3.16) follows from the definitions (2.32) of $\mathsf{m}$
and $\check{\mathsf{m}}$. For the second part of (3.16), we combine the
asymptotics (3.8) with the definition (3.11).

Next, we take the difference of two applications of the third part of (3.3) to
obtain

$\mathscr{L}_{\rho}(\mathsf{m}-\check{\mathsf{m}})_{ab}=\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})_{ab}-2(\mathsf{w}^{2}-\check{\mathsf{w}}^{2})_{ab}+\frac{1}{4}(\mathsf{g}^{cd}\mathsf{m}_{ad}\mathsf{m}_{bc}-\check{\mathsf{g}}^{cd}\check{\mathsf{m}}_{ad}\check{\mathsf{m}}_{bc})\text{.}$

The last term can be treated using (3.8) and the first part of (3.14), and we
obtain

$\mathscr{L}_{\rho}(\mathsf{m}-\check{\mathsf{m}})=\rho^{-1}(\mathsf{m}-\check{\mathsf{m}})-2(\mathsf{w}^{2}-\check{\mathsf{w}}^{2})+\mathscr{O}_{M-2}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(\rho;\mathsf{m}-\check{\mathsf{m}})\text{.}$

The third part of (3.16) follows immediately from the above, since (3.9) and
(3.13) imply

$\mathsf{w}^{2}-\check{\mathsf{w}}^{2}=\mathsf{W}^{2}+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(\rho;\mathsf{Q})\text{.}$

For the final identity in (3.16), we begin by applying $\mathscr{L}_{\rho}$ to
(3.12), and then commuting $\mathsf{D}$ and $\mathscr{L}_{\rho}$ via (2.33).
The error term in (2.33) can then be expressed as asymptotic terms using
(3.8):

$\displaystyle\mathscr{L}_{\rho}\mathsf{B}_{cab}$
$\displaystyle=\mathsf{D}_{c}\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}})_{ab}-\mathsf{D}_{a}\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}})_{cb}-\mathsf{D}_{b}\mathscr{L}_{\rho}\mathsf{Q}_{ca}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}\text{.}$

For the first three terms on the right-hand side, we apply (2.32), (3.8), and
(3.11) to obtain

$\displaystyle\mathscr{L}_{\rho}\mathsf{B}_{cab}$
$\displaystyle=\mathsf{D}_{c}(\mathsf{m}-\check{\mathsf{m}})_{ab}-\mathsf{D}_{a}(\mathsf{m}-\check{\mathsf{m}})_{cb}-\frac{1}{2}\mathsf{g}^{de}\mathsf{D}_{b}[\mathsf{m}_{ad}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{ce}]$
$\displaystyle\qquad+\frac{1}{2}\mathsf{g}^{de}\mathsf{D}_{b}[\mathsf{m}_{cd}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{ae}]+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}$
$\displaystyle=(\mathsf{D}_{c}\mathsf{m}_{ab}-\mathsf{D}_{a}\mathsf{m}_{cb})-(\mathsf{D}_{c}\check{\mathsf{m}}_{ab}-\mathsf{D}_{a}\check{\mathsf{m}}_{cb})-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{ad}\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{ce}$
$\displaystyle\qquad+\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{ae}+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}\text{.}$

Applying the first part of (3.3), we then have

(7.30) $\displaystyle\mathscr{L}_{\rho}\mathsf{B}_{cab}$
$\displaystyle=2(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})_{bac}-(\mathsf{D}-\check{\mathsf{D}})_{c}\check{\mathsf{m}}_{ab}+(\mathsf{D}-\check{\mathsf{D}})_{a}\check{\mathsf{m}}_{cb}-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{ad}\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{ce}$
$\displaystyle\qquad+\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{ae}+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}\text{.}$

For the terms in (7.30) involving $\mathsf{D}-\check{\mathsf{D}}$, we expand
using the last part of (2.28):

$\displaystyle-(\mathsf{D}-\check{\mathsf{D}})_{c}\check{\mathsf{m}}_{ab}+(\mathsf{D}-\check{\mathsf{D}})_{a}\check{\mathsf{m}}_{cb}$
$\displaystyle=(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{d}_{ca}\check{\mathsf{m}}_{db}+(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{d}_{cb}\check{\mathsf{m}}_{ad}-(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{d}_{ac}\check{\mathsf{m}}_{db}-(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{d}_{ab}\check{\mathsf{m}}_{cd}$
$\displaystyle=\frac{1}{2}\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{ad}[\mathsf{D}_{c}(\mathsf{g}-\check{\mathsf{g}})_{eb}+\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}})_{ec}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{bc}]$
$\displaystyle\qquad-\frac{1}{2}\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{cd}[\mathsf{D}_{a}(\mathsf{g}-\check{\mathsf{g}})_{eb}+\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}})_{ea}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{ba}]\text{.}$

Combining the above with (7.30), we conclude that

(7.31) $\displaystyle\mathscr{L}_{\rho}\mathsf{B}_{cab}$
$\displaystyle=2(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})_{bac}+\frac{1}{2}(\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{ad}-\mathsf{g}^{de}\mathsf{m}_{ad})\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}})_{ce}$
$\displaystyle\qquad-\frac{1}{2}(\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{cd}-\mathsf{g}^{de}\mathsf{m}_{cd})\mathsf{D}_{b}(\mathsf{g}-\check{\mathsf{g}})_{ae}+\frac{1}{2}\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{ad}[\mathsf{D}_{c}(\mathsf{g}-\check{\mathsf{g}})_{eb}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{bc}]$
$\displaystyle\qquad-\frac{1}{2}\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{cd}[\mathsf{D}_{a}(\mathsf{g}-\check{\mathsf{g}})_{eb}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{ba}]-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{ad}\mathsf{D}_{b}\mathsf{Q}_{ce}+\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\mathsf{D}_{b}\mathsf{Q}_{ae}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}$
$\displaystyle=2(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})_{bac}+\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{ad}\mathsf{D}_{b}\mathsf{Q}_{ce}+\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\mathsf{D}_{b}\mathsf{Q}_{ae}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}\text{.}$

For $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$, we take as “main terms” the
differences $\mathsf{g}^{-1}-\check{\mathsf{g}}^{-1}$ and
$\mathsf{m}-\check{\mathsf{m}}$. (In particular, we treat
$\mathsf{D}\mathsf{g}$ and $\mathsf{D}\check{\mathsf{g}}$ as coefficients of
the form $\mathscr{O}_{M-1}(1)$.) Recalling (3.8) and (3.14) then yields

(7.32) $\displaystyle\mathcal{I}_{1}$
$\displaystyle=\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-1}(1;\mathsf{m}-\check{\mathsf{m}})_{cab}\text{,}$
$\displaystyle\mathcal{I}_{2}$
$\displaystyle=\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-1}(1;\mathsf{m}-\check{\mathsf{m}})_{cab}\text{.}$

By similar computations, we also obtain

(7.33) $\displaystyle\mathcal{I}_{3}$
$\displaystyle=\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{ad}[\mathsf{D}_{c}(\mathsf{g}-\check{\mathsf{g}})_{eb}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{bc}]+\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-1}(1;\mathsf{m}-\check{\mathsf{m}})_{cab}\text{,}$
$\displaystyle\mathcal{I}_{4}$
$\displaystyle=-\frac{1}{2}\check{\mathsf{g}}^{de}\check{\mathsf{m}}_{cd}[\mathsf{D}_{a}(\mathsf{g}-\check{\mathsf{g}})_{eb}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{ba}]+\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}+\mathscr{O}_{M-1}(1;\mathsf{m}-\check{\mathsf{m}})_{cab}\text{.}$

Combining (7.31)–(7.33) and recalling (3.12) results in

$\displaystyle\mathscr{L}_{\rho}\mathsf{B}_{cab}$
$\displaystyle=2(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})_{bac}+\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{ad}\mathsf{B}_{ceb}-\frac{1}{2}\mathsf{g}^{de}\mathsf{m}_{cd}\mathsf{B}_{aeb}+\mathscr{O}_{M-3}(\rho;\mathsf{g}-\check{\mathsf{g}})_{cab}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho;\mathsf{Q})_{cab}+\mathscr{O}_{M-1}(1;\mathsf{m}-\check{\mathsf{m}})_{cab}\text{.}$

The above, along with Proposition 3.8 and (3.13), immediately yield the last
part of (3.16).

It remains to prove the derivative transport equations (3.17). These are in
fact immediate consequences of the preceding (3.16), since (2.33) and (3.8)
imply that

$\mathscr{L}_{\rho}\mathsf{D}\mathsf{A}=\mathsf{D}\mathscr{L}_{\rho}\mathsf{D}\mathsf{A}+\mathscr{O}_{M-3}(\rho;\mathsf{A})$

holds for any vertical tensor field $\mathsf{A}$.

###  7.9. Proof of Proposition 3.14

We prove all three formulas at once. Throughout, we fix

$(\mathsf{w},\check{\mathsf{w}},\mathsf{W})\in\\{(\mathsf{w}^{2},\check{\mathsf{w}}^{2},\mathsf{W}^{2}),(\mathsf{w}^{1},\check{\mathsf{w}}^{1},\mathsf{W}^{1}),(\mathsf{w}^{\star},\check{\mathsf{w}}^{1},\mathsf{W}^{\star})\\}\text{,}$

as well as the corresponding constant $\sigma_{\mathsf{w}}$:

$\sigma_{\mathsf{w}^{2}}:=2(n-2)\text{,}\qquad\sigma_{\mathsf{w}^{1}}:=n-1\text{,}\qquad\sigma_{\mathsf{w}^{\star}}:=0\text{.}$

Then, from (3.13), we see that $\mathsf{W}$ satisfies the following wave
equation:

(7.34) $\displaystyle(\bar{\Box}+\sigma_{\mathsf{w}})\mathsf{W}_{\bar{a}}$
$\displaystyle=[(\bar{\Box}+\sigma_{\mathsf{w}})\mathsf{w}-(\check{\bar{\Box}}+\sigma_{\mathsf{w}})\check{\mathsf{w}}]_{\bar{a}}-(\bar{\Box}-\check{\bar{\Box}})\check{\mathsf{w}}_{\bar{a}}$
$\displaystyle\qquad-\mathsf{g}^{bc}\sum_{j=1}^{d}(\bar{\Box}+\sigma_{\mathsf{w}})[\check{\mathsf{w}}_{\bar{a}_{j}[b]}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}c}]$
$\displaystyle=\mathscr{A}_{0}+\mathscr{A}_{1}+\mathscr{A}_{2}\text{.}$

Here and below, we index with respect to an arbitrary coordinate system
$(U,\varphi)$ on $\mathscr{I}$.

To expand $\mathscr{A}_{0}$, we use that both $\mathsf{w}$ and
$\check{\mathsf{w}}$ solve wave equations of the form (3.6), and we take the
difference of these two equations. In particular, we take the difference of
each pair of corresponding terms on the right-hand side of the relevant
equation in (3.6). Like in earlier proofs, each of these can be written as a
sum, with each term involving a geometric difference quantity along with other
coefficients that can be controlled using Proposition 3.8 and (3.14). For
example, we can treat585858On the left-hand side, the two schematic terms have
the same algebraic form.

$\displaystyle\rho\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})-\rho\,\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},\check{\mathsf{w}}^{2})$
$\displaystyle=\rho\,\mathscr{S}(\mathsf{g}^{-1}-\check{\mathsf{g}}^{-1},\mathsf{m},\mathsf{w}^{2})+\rho\,\mathscr{S}(\check{\mathsf{g}}^{-1},\mathsf{m}-\check{\mathsf{m}},\mathsf{w}^{2})$
$\displaystyle\qquad+\rho\,\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},\mathsf{w}^{2}-\check{\mathsf{w}}^{2})$
$\displaystyle=\mathscr{O}_{M-2}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(\rho;\mathsf{m}-\check{\mathsf{m}})+\mathscr{O}_{M-2}(\rho^{2};\mathsf{w}^{2}-\check{\mathsf{w}}^{2})\text{.}$

For differences involving a vertical derivative, one must also take into
account the difference $\mathsf{D}-\check{\mathsf{D}}$ of connections; here,
the key is to write this in terms of differences of Christoffel symbols and
then apply the last identitiy in (3.15). One example of this is the following:

$\displaystyle\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{D}\mathsf{w}^{1})-\rho^{2}\,\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},\check{\mathsf{D}}\check{\mathsf{w}}^{1})$
$\displaystyle=\rho^{2}\,\mathscr{S}(\mathsf{g}^{-1}-\check{\mathsf{g}}^{-1},\mathsf{m},\mathsf{D}\mathsf{w}^{1})+\rho^{2}\,\mathscr{S}(\check{\mathsf{g}}^{-1},\mathsf{m}-\check{\mathsf{m}},\mathsf{D}\mathsf{w}^{1})$
$\displaystyle\qquad+\rho^{2}\,\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},\mathsf{D}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1}))$
$\displaystyle\qquad+\rho^{2}\,\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},(\mathsf{D}-\check{\mathsf{D}})\check{\mathsf{w}}^{1})$
$\displaystyle=\mathscr{O}_{M-3}(\rho^{3};\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(\rho^{2};\mathsf{m}-\check{\mathsf{m}})$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1}))+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))\text{.}$

After a diligent analysis of all such difference terms, we obtain

(7.35) $\displaystyle\mathscr{A}_{0}$
$\displaystyle=\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho;\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{2};\mathsf{D}(\mathsf{m}-\check{\mathsf{m}}))_{\bar{a}}+\sum_{\mathsf{v}\in\\{\mathsf{w}^{0},\mathsf{w}^{1},\mathsf{w}^{2}\\}}[\mathscr{O}_{M-3}(\rho^{2};\mathsf{v}-\check{\mathsf{v}})+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}(\mathsf{v}-\check{\mathsf{v}}))]_{\bar{a}}\text{.}$

In addition, using Proposition 3.8 and (3.13), we can then express, for
$k\in\\{1,2\\}$,

(7.36) $\displaystyle\mathsf{w}^{k}-\check{\mathsf{w}}^{k}$
$\displaystyle=\mathsf{W}^{k}+\mathscr{O}_{M-2}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(1;\mathsf{Q})\text{,}$
$\displaystyle\mathsf{D}(\mathsf{w}^{k}-\check{\mathsf{w}}^{k})$
$\displaystyle=\mathsf{D}\mathsf{W}^{k}+\mathscr{O}_{M-3}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(1;\mathsf{Q})$
$\displaystyle\qquad+\mathscr{O}_{M-2}(1;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))+\mathscr{O}_{M-2}(1;\mathsf{D}\mathsf{Q})\text{.}$

A similar computation can be done for $\mathsf{w}^{0}-\check{\mathsf{w}}^{0}$
and $\mathsf{W}^{\star}$, but we also recall (3.2) and (7.36):

(7.37) $\displaystyle\mathsf{w}^{0}-\check{\mathsf{w}}^{0}$
$\displaystyle=(\mathsf{w}^{\star}-\check{\mathsf{w}}^{\star})+\mathscr{S}(\mathsf{g}-\check{\mathsf{g}},\mathsf{w}^{2})+\mathscr{S}(\check{\mathsf{g}}^{-1},\mathsf{w}^{2}-\check{\mathsf{w}}^{2})$
$\displaystyle=\mathsf{W}^{\star}+\mathscr{O}_{M-2}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(1;\mathsf{Q})+\mathscr{O}_{M}(1;\mathsf{W}^{2})\text{,}$
$\displaystyle\mathsf{D}(\mathsf{w}^{0}-\check{\mathsf{w}}^{0})$
$\displaystyle=\mathsf{D}\mathsf{W}^{\star}+\mathscr{O}_{M-3}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(1;\mathsf{Q})+\mathscr{O}_{M-2}(1;\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))$
$\displaystyle\qquad+\mathscr{O}_{M-2}(1;\mathsf{D}\mathsf{Q})+\mathscr{O}_{M-1}(1;\mathsf{W}^{2})+\mathscr{O}_{M}(1;\mathsf{D}\mathsf{W}^{2})\text{.}$

Combining (7.35)–(7.37), we conclude that

(7.38) $\displaystyle\mathscr{A}_{0}$
$\displaystyle=\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{Q})+\mathscr{O}_{M-3}(\rho;\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{D}(\mathsf{m}-\check{\mathsf{m}}))_{\bar{a}}$
$\displaystyle\qquad+\sum_{\mathsf{V}\in\\{\mathsf{W}^{\star},\mathsf{W}^{1},\mathsf{W}^{2}\\}}[\mathscr{O}_{M-3}(\rho^{2};\mathsf{V})+\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}\mathsf{V})]_{\bar{a}}\text{.}$

Next, for $\mathscr{A}_{1}$, we apply (2.35) twice—once with $\bar{\Box}$ and
once with $\check{\bar{\Box}}$—and subtract the resulting equations in order
to obtain (note that $\mathscr{L}_{\rho}$ is independent of the metric)

$\displaystyle\mathscr{A}_{1}$
$\displaystyle=-\rho^{2}[\mathsf{g}^{bc}\mathsf{D}_{bc}-\check{\mathsf{g}}^{bc}\check{\mathsf{D}}_{bc}]\check{\mathsf{w}}_{\bar{a}}+\rho^{2}\,[\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathscr{L}_{\rho}\check{\mathsf{w}})-\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},\mathscr{L}_{\rho}\check{\mathsf{w}})]_{\bar{a}}$
$\displaystyle\qquad+\rho\,[\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\check{\mathsf{w}})-\mathscr{S}(\check{\mathsf{g}}^{-1},\check{\mathsf{m}},\check{\mathsf{w}})]_{\bar{a}}+\rho^{2}\,[\mathscr{S}(\mathsf{g}^{-1},\mathscr{L}_{\rho}\mathsf{m},\check{\mathsf{w}})-\mathscr{S}(\check{\mathsf{g}}^{-1},\mathscr{L}_{\rho}\check{\mathsf{m}},\check{\mathsf{w}})]_{\bar{a}}$
$\displaystyle\qquad+\rho^{2}\,[\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{m},\check{\mathsf{w}})-\mathscr{S}(\check{\mathsf{g}}^{-2},\check{\mathsf{m}},\check{\mathsf{m}},\check{\mathsf{w}})]_{\bar{a}}\text{,}$

where each matching pair of schematic terms has the same algebraic form.
Similar to the previous treatment of $\mathscr{A}_{0}$, each schematic
difference on the right-hand side of the above can be expanded and then
controlled using Proposition 3.8 and (3.14); this then yields

$\displaystyle\mathscr{A}_{1}$
$\displaystyle=-\rho^{2}[\mathsf{g}^{bc}\mathsf{D}_{bc}-\check{\mathsf{g}}^{bc}\check{\mathsf{D}}_{bc}]\check{\mathsf{w}}_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho,\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{2};\mathscr{L}_{\rho}(\mathsf{m}-\check{\mathsf{m}}))_{\bar{a}}\text{.}$

In addition, we expand the last term of the above using the third identity in
(3.16):

(7.39) $\displaystyle\mathscr{A}_{1}$
$\displaystyle=-\rho^{2}[\mathsf{g}^{bc}\mathsf{D}_{bc}-\check{\mathsf{g}}^{bc}\check{\mathsf{D}}_{bc}]\check{\mathsf{w}}_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{3};\mathsf{Q})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho,\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{W}^{2})_{\bar{a}}$
$\displaystyle=\mathscr{A}_{1,v}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{3};\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-3}(\rho,\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{W}^{2})_{\bar{a}}\text{.}$

To handle $\mathscr{A}_{1,v}$, we begin by expanding

$\displaystyle\mathscr{A}_{1,v}$
$\displaystyle=-\rho^{2}\mathsf{g}^{bc}(\mathsf{D}_{bc}-\check{\mathsf{D}}_{bc})\check{\mathsf{w}}_{\bar{a}}-\rho^{2}(\mathsf{g}^{bc}-\check{\mathsf{g}}^{bc})\check{\mathsf{D}}_{bc}\check{\mathsf{w}}_{\bar{a}}$
$\displaystyle=-\rho^{2}\mathsf{g}^{bc}\mathsf{D}_{b}(\mathsf{D}-\check{\mathsf{D}})_{c}\check{\mathsf{w}}_{\bar{a}}-\rho^{2}\mathsf{g}^{bc}(\mathsf{D}-\check{\mathsf{D}})_{b}\check{\mathsf{D}}_{c}\check{\mathsf{w}}_{\bar{a}}-\rho^{2}(\mathsf{g}^{bc}-\check{\mathsf{g}}^{bc})\check{\mathsf{D}}_{bc}\check{\mathsf{w}}_{\bar{a}}\text{.}$

The operator $\mathsf{D}-\check{\mathsf{D}}$ can be expressed in terms of
differences of Christoffel symbols. Applying again Proposition 3.8, (3.14),
and the last part of (3.15), the above becomes

(7.40) $\displaystyle\mathscr{A}_{1,v}$
$\displaystyle=\rho^{2}\mathsf{g}^{bc}\sum_{j=1}^{d}\mathsf{D}_{b}[(\mathsf{\Gamma}-\check{\mathsf{\Gamma}})^{d}_{ca_{j}}\check{\mathsf{w}}_{\bar{a}_{j}[d]}]+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}+\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}$
$\displaystyle=\rho^{2}\mathsf{g}^{bc}\sum_{j=1}^{d}\mathsf{D}_{b}\\{\check{\mathsf{g}}^{de}[\mathsf{D}_{c}(\mathsf{g}-\check{\mathsf{g}})_{ea_{j}}+\mathsf{D}_{a_{j}}(\mathsf{g}-\check{\mathsf{g}})_{ec}-\mathsf{D}_{e}(\mathsf{g}-\check{\mathsf{g}})_{ca_{j}}]\check{\mathsf{w}}_{\bar{a}_{j}[d]}\\}$
$\displaystyle\qquad+\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}$
$\displaystyle=\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}[\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}})_{ea_{j}}+\mathsf{D}_{ba_{j}}(\mathsf{g}-\check{\mathsf{g}})_{ec}-\mathsf{D}_{be}(\mathsf{g}-\check{\mathsf{g}})_{ca_{j}}]\check{\mathsf{w}}_{\bar{a}_{j}[d]}$
$\displaystyle\qquad+\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}\text{.}$

(Notice in the last step, any term having less than two derivatives of
$\mathsf{g}-\check{\mathsf{g}}$ can be included with the schematic terms. In
addition, $\check{\mathsf{g}}^{de}$ can be converted to $\mathsf{g}^{de}$ at
the cost of an extra term involving $\mathsf{g}-\check{\mathsf{g}}$, which can
also be absorbed into the schematic terms.)

Recalling the definitions (3.11) and (3.12), then (7.40) can be rewritten as

$\displaystyle\mathscr{A}_{1,v}$
$\displaystyle=\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}[\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}})_{ea_{j}}+\mathsf{D}_{bc}\mathsf{Q}_{a_{j}e}+\mathsf{D}_{b}\mathsf{B}_{a_{j}ec}]\check{\mathsf{w}}_{\bar{a}_{j}[d]}$
$\displaystyle\qquad+\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}$
$\displaystyle=\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}\check{\mathsf{w}}_{\bar{a}_{j}[d]}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{D}\mathsf{B})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}\text{.}$

Combining (7.39) and the above, we conclude that

(7.41) $\displaystyle\mathscr{A}_{1}$
$\displaystyle=\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}\check{\mathsf{w}}_{\bar{a}_{j}[d]}+\mathscr{O}_{M-4}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}$