$\displaystyle\frac{V}{r}=$
$\displaystyle\frac{\Lambda}{3}e^{2\beta_{0}}r^{2}-r(l+D_{A}U^{A}_{0})$
(4.2.23)
$\displaystyle-e^{2\beta_{0}}\Big{[}\frac{1}{2}\Big{(}R[q]+\frac{\Lambda}{8}C_{AB}C^{AB}\Big{)}+2D_{A}\partial^{A}\beta_{0}+4\partial_{A}\beta_{0}\partial^{A}\beta_{0}\Big{]}+\frac{2M}{r}+\mathcal{O}(r^{-1})$

where $M(u,x^{A})$ is an integration “constant” which, in flat asymptotics, is
recognized as the Bondi mass aspect.

Afterwards, we solve (4.2.8) order by order, which provides us the constraints
imposed on each independent order of $g_{AB}$. The leading
$\mathcal{O}(r^{-1})$ order of that equation yields

$\frac{\Lambda}{3}C_{AB}=e^{-2\beta_{0}}\Big{[}(\partial_{u}-l)q_{AB}+2D_{(A}U^{0}_{B)}-D^{C}U^{0}_{C}q_{AB}\Big{]}.$
(4.2.24)

This result shows that there is a discrete bifurcation between the
asymptotically flat case and the case $\Lambda\neq 0$. Indeed, when
$\Lambda=0$, the left-hand-side vanishes, which leads to a constraint on the
time-dependence of $q_{AB}$, see (2.1.34). As a consequence, the field
$q_{AB}$ is constrained while $C_{AB}$ is completely free and interpreted as
the shear. For (A)dS4 asymptotics, $C_{AB}$ is entirely determined by $q_{AB}$
and $U^{A}_{0}$, while the boundary metric $q_{AB}$ is left completely
undetermined by the equations of motion. This is consistent with previous
analyses [136, 291, 147, 149, 150].

Going to $\mathcal{O}(r^{-2})$, we get

$\frac{\Lambda}{3}\mathcal{D}_{AB}=0,$ (4.2.25)

which removes the logarithmic term in (4.2.19) for $\Lambda\neq 0$, but not
for $\Lambda=0$. The condition at next $\mathcal{O}(r^{-3})$ order

$\partial_{u}\mathcal{D}_{AB}+U_{0}^{C}D_{C}\mathcal{D}_{AB}+2\mathcal{D}_{C(A}D_{B)}U_{0}^{C}=0,$
(4.2.26)

is thus trivial for $\Lambda\neq 0$, but reduces to
$\partial_{u}\mathcal{D}_{AB}=0$ in the flat limit, consistently with previous
results [46, 237].

Using an iterative argument as in [150], we now make the following
observation. If we decompose $g_{AB}=r^{2}\sum_{n\geq 0}g_{AB}^{(n)}r^{-n}$,
we see that the iterative solution of (4.2.8) is organized as $\Lambda
g_{AB}^{(n)}=\partial_{u}g_{AB}^{(n-1)}+(...)$ at order $\mathcal{O}(r^{-n})$,
$n\in\mathbb{N}_{0}$. Accordingly, the form of $\mathcal{E}_{AB}$ should have
been fixed by the equation (4.2.26) found at $\mathcal{O}(r^{-3})$, but it is
not the case, since both contributions of $\mathcal{E}_{AB}$ cancel between
$G_{AB}$ and $\Lambda g_{AB}$. Moreover, the equation $\Lambda
g_{AB}^{(4)}=\partial_{u}g_{AB}^{(3)}+(...)$ at next order turns out to be a
constraint for $g_{AB}^{(4)}\sim\mathcal{F}_{AB}$, determined with other
subleading data such as $C_{AB}$ or
$\partial_{u}g_{AB}^{(3)}\sim\partial_{u}\mathcal{E}_{AB}$. It shows that
$\mathcal{E}_{AB}$ is a set of two free data on the boundary, built up from
two arbitrary functions of $(u,x^{A})$. It shows moreover that there are no
additional data to be uncovered for $\Lambda\neq 0$.

As a conclusion, Einstein’s equations $(r,r)$, $(r,A)$, $(r,u)$ and $(A,B)$
can be solved iteratively in the asymptotic expansion for $\Lambda\neq 0$. We
identified 11 independent functions $\\{\beta_{0}(u,x^{A})$,
$U^{A}_{0}(u,x^{B})$, $q_{AB}(u,x^{C})$, $M(u,x^{C})$, $N_{A}(u,x^{C})$,
$\mathcal{E}_{AB}(u,x^{C})\\}$ that determine the asymptotic solution. We will
see below that the remaining equations are equivalent to evolution equations
for $M(u,x^{A})$ and $N_{A}(u,x^{B})$. Hence the solution space is

$\mathcal{S}_{\Lambda}=\left\\{g_{\mu\nu}\left[\beta_{0},U^{A}_{0},q_{AB},M,N_{A},\mathcal{D}_{AB},\mathcal{E}_{AB}\right]\
\Big{|}\ G_{\mu\nu}[g]+\Lambda g_{\mu\nu}=0\right\\}.$ (4.2.27)

This contrasts with the asymptotically flat case $\Lambda=0$ where an infinite
series of functions appear in the radial expansion of $g_{AB}$, see (2.1.37).
We will give more details about this discrepancy while discussing the flat
limit.

##### 4.2.1.4 $\Lambda$-BMS4 boundary gauge fixing

At this point, we solved all of the algebraic relations among the Einstein
equations. As we said, it remains to give the dynamical equations for the
Bondi mass and angular momentum aspects. In full generality, these constraint
equations look like a complete mess due to the unbecoming presence of the
boundary fields $\beta_{0}$ and $U^{A}_{0}$. In that case, why do we not
simplify our analysis by imposing a $\Lambda$-BMS type of boundary gauge
fixing in order to get the expressions tractable and readable? As we stressed
earlier, there is a more fundamental reason for which this reduction is
appealing: the non-trivial charges will obey flux-balance laws at conformal
infinity and the natural conjugated diffeomorphism parameters are codimension
2 functions on the angles, which appear as integration constants of the
constraint equations (4.1.2) recast in the Bondi gauge as (4.2.63)–(4.2.65).

Let us see how the boundary gauge fixing acts on Bondi variables. The pullback
of the most general Bondi metric satisfying (4.2.2) to the conformal boundary
gives

$\left.\text{d}s^{2}\right|_{\mathscr{I}}=\Big{[}\frac{\Lambda}{3}e^{4\beta_{0}}+U_{0}^{A}U^{0}_{A}\Big{]}\text{d}u^{2}-2U_{A}^{0}\text{d}u\text{d}x^{A}+q_{AB}\text{d}x^{A}\text{d}x^{B}.$
(4.2.28)

Defining a set of Gaussian normal coordinates foliated along $u$ and fixing
the transverse volume thanks to a Weyl rescaling on the boundary as in (4.1.1)
yield

$\beta_{0}=0,\quad U^{A}_{0}=0,\quad\sqrt{q}=\sqrt{\bar{q}}$ (4.2.29)

where $\sqrt{\bar{q}}$ is an arbitrary measure on the sphere. Generically, it
may depend on time, but it satisfies $\delta\sqrt{\bar{q}}$ for any linearized
variation. The gauge (4.2.29) is a temporal boundary gauge for $\Lambda<0$, a
radial boundary gauge for $\Lambda>0$ and a null boundary gauge for
$\Lambda=0$ with $g_{ur}=-1+\mathcal{O}(r^{-1})$ in (2.1.8). But is this
always reachable for any value of $\Lambda$ in the Bondi gauge? Computing the
Lie derivative on the Bondi metric on-shell and retaining only the leading
$\mathcal{O}(r^{2})$ terms, we get the transformation laws of the boundary
fields $q_{AB}$, $\beta_{0}$ and $U^{A}_{0}$ under the set of residual gauge
transformations (2.1.12):

$\displaystyle\delta_{\xi}q_{AB}$
$\displaystyle=f(\partial_{u}-l)q_{AB}+(\mathcal{L}_{Y}-D_{C}Y^{C}+2\omega)q_{AB}$
$\displaystyle\quad-2(U_{(A}^{0}\partial_{B)}f-\frac{1}{2}q_{AB}U^{C}_{0}\partial_{C}f),$
(4.2.30) $\displaystyle\delta_{\xi}\beta_{0}$
$\displaystyle=(f\partial_{u}+\mathcal{L}_{Y})\beta_{0}+\frac{1}{2}\Big{[}\partial_{u}-\frac{1}{2}l+\frac{3}{2}U^{A}_{0}\partial_{A}\Big{]}f-\frac{1}{4}(D_{A}Y^{A}-2\omega),$
(4.2.31) $\displaystyle\delta_{\xi}U^{A}_{0}$
$\displaystyle=(f\partial_{u}+\mathcal{L}_{Y})U_{0}^{A}-\Big{[}\partial_{u}Y^{A}-\frac{1}{\ell^{2}}e^{4\beta_{0}}q^{AB}\partial_{B}f\Big{]}+U_{0}^{A}(\partial_{u}f+U_{0}^{B}\partial_{B}f).$
(4.2.32)

The first equation implies that $q^{AB}\delta_{\xi}q_{AB}=4\omega$. We can
therefore adjust the Weyl generator $\omega$ in order to reach the gauge
$\sqrt{q}=\sqrt{\bar{q}}$. This is similar to the boundary gauge fixing
(4.1.1) in the SFG gauge where one has to use the bulk information of the
boundary diffeomorphism to fix the area of the transverse spaces. Although the
parameters are completely redistributed in the Bondi gauge, the establishment
of a dictionary in section 4.2.2 will show that both boundary gauge fixings
are identical. The form of the infinitesimal transformations (4.2.31)-(4.2.32)
involves $\partial_{u}f$ and $\partial_{u}Y^{A}$. This ensures that a finite
gauge transformation labeled by $f,Y^{A}$ can be found by integration over $u$
in order to reach $\beta_{0}=0$, $U^{A}_{0}=0$, at least in a local patch. As
a result, the conditions (4.2.29) can always be reached by gauge fixing, at
least locally. The vanishing of the inhomogeneous contributions in the
transformation laws (4.2.31)-(4.2.32) constrains the parameters $f,Y^{A}$ and
reduces the set of allowed vectors as it has been observed in the SFG gauge.
When $\Lambda\neq 0$, the residual diffeomorphisms constitute the
$\Lambda$-BMS4 algebroid as we will review below. When $\Lambda=0$, these
residual diffeomorphisms define the Generalized BMS4 algebra. This is
corollary of the results presented in section 4.1.2, up to the subtlety that
here the flat limit is also well-defined at the level of the solution space
and not at the level of the symmetries only. A more precise discussion on the
properties of the $\Lambda$-BMS4 algebroid as well as the explicit flat limit
of the symmetry parameters is postponed to section 4.2.4.

##### 4.2.1.5 Constraint equations as Bondi evolution equations

Assuming the gauge fixing conditions (4.2.29), we are now ready to present the
evolution equations that follow from the remaining Einstein equations. As
justified before, the $\mathcal{O}(r^{0})$ part of $r^{2}(R_{uA}-\Lambda
g_{uA})=0$ fixes the temporal evolution of $N_{A}$. From the Christoffel
symbols found in [46], we can develop the first term as

$\displaystyle R_{uA}=$
$\displaystyle-(\partial_{u}-l)\partial_{A}\beta-\partial_{A}l-(\partial_{u}+l)n_{A}$
(4.2.33)
$\displaystyle+n_{B}\mathcal{D}^{B}U_{A}-\partial_{B}\beta\mathcal{D}_{A}U^{B}+2U^{B}(\partial_{A}\beta\partial_{B}\beta+n_{A}n_{B})$
$\displaystyle+\mathcal{D}_{B}\Big{[}l^{B}_{A}+\frac{1}{2}(\mathcal{D}^{B}U_{A}-\mathcal{D}_{A}U^{B})+U^{B}(\partial_{A}\beta-
n_{A})\Big{]}+2n_{B}l^{B}_{A}$
$\displaystyle-\frac{1}{2}(\partial_{r}+2\partial_{r}\beta+\frac{2}{r})\partial_{A}\frac{V}{r}-\frac{V}{r}(\partial_{r}+\frac{2}{r})n_{A}+k_{A}^{B}(\partial_{B}\frac{V}{r}+2\frac{V}{r}n_{B})$
$\displaystyle-e^{-2\beta}(\partial_{r}+\frac{2}{r})\Big{[}U^{B}(l_{AB}+\frac{V}{r}k_{AB}+\mathcal{D}_{(A}U_{B)})\Big{]}$
$\displaystyle-e^{-2\beta}U^{B}\Big{[}(\partial_{u}+l)k_{AB}-4l^{C}_{(A}k_{B)C}-2k^{C}_{A}k_{BC}\frac{V}{r}+\mathcal{D}_{C}(k_{AB}U^{C})-2k_{C(A}\mathcal{D}^{C}U_{B)}\Big{]}.$

Let us emphasize that the $r$-dependence of the fields is not yet explicit in
this expression: the upper case Latin indices are lowered and raised by the
full metric $g_{AB}$ and its inverse. Expanding all fields in power series of
$1/r$ in $R_{uA}$ and $\Lambda g_{uA}$, and selecting the $1/r^{2}$ terms
yields

$(\partial_{u}+l)N_{A}^{(\Lambda)}-\partial_{A}M^{(\Lambda)}-\frac{\Lambda}{2}D^{B}J_{AB}=0.$
(4.2.34)

Here, we defined with hindsight the Bondi mass and angular momentum aspects
for $\Lambda\neq 0$ as

$\displaystyle M^{(\Lambda)}$
$\displaystyle=M+\frac{1}{16}(\partial_{u}+l)(C_{CD}C^{CD}),$ (4.2.35)
$\displaystyle N^{(\Lambda)}_{A}$
$\displaystyle=N_{A}-\frac{3}{2\Lambda}D^{B}(N_{AB}-\frac{1}{2}lC_{AB})-\frac{3}{4}\partial_{A}(\frac{1}{\Lambda}R[q]-\frac{3}{8}C_{CD}C^{CD}),$
(4.2.36)

and the traceless symmetric tensor $J_{AB}$ ($q^{AB}J_{AB}=0$) as

$\displaystyle J_{AB}=$
$\displaystyle-\mathcal{E}_{AB}-\frac{3}{\Lambda^{2}}\Big{[}\partial_{u}(N_{AB}-\frac{1}{2}lC_{AB})-\frac{\Lambda}{2}q_{AB}C^{CD}(N_{CD}-\frac{1}{2}lC_{CD})\Big{]}$
$\displaystyle\quad+\frac{3}{\Lambda^{2}}(D_{A}D_{B}l-\frac{1}{2}q_{AB}D_{C}D^{C}l)$
$\displaystyle\quad-\frac{1}{\Lambda}(D_{(A}D^{C}C_{B)C}-\frac{1}{2}q_{AB}D^{C}D^{D}C_{CD})$
$\displaystyle\quad+C_{AB}\Big{[}\frac{5}{16}C_{CD}C^{CD}+\frac{1}{2\Lambda}R[q]\Big{]}.$
(4.2.37)

We used the notation $N_{AB}\equiv\partial_{u}C_{AB}$. This tensor is
symmetric and obeys $q^{AB}N_{AB}=\frac{\Lambda}{3}C^{AB}C_{AB}$. When
$\Lambda=0$, $N_{AB}$ is thus traceless and represents the Bondi news tensor.
We will justify the definitions (4.2.35)-(4.2.36) in section 4.2.2. Notice
also that $\partial_{u}q_{AB}$ has been eliminated using (4.2.24). The
transformations of these fields under the residual gauge symmetries $\xi$
preserving the Bondi gauge (2.1.8) and the boundary gauge (4.2.29) are given
by

$\displaystyle\delta_{\xi}M^{(\Lambda)}$
$\displaystyle=[f\partial_{u}+\mathcal{L}_{Y}+\frac{3}{2}(D_{A}Y^{A}+fl-2\omega)]M^{(\Lambda)}-\frac{\Lambda}{3}N_{A}^{(\Lambda)}\partial^{A}f,$
(4.2.38) $\displaystyle\delta_{\xi}N_{A}^{(\Lambda)}$
$\displaystyle=[f\partial_{u}+\mathcal{L}_{Y}+D_{B}Y^{B}+fl-2\omega]N_{A}^{(\Lambda)}+3M^{(\Lambda)}\partial_{A}f+\frac{\Lambda}{2}J_{AB}\partial^{B}f,$
(4.2.39) $\displaystyle\delta_{\xi}J_{AB}$
$\displaystyle=[f\partial_{u}+\mathcal{L}_{Y}+\frac{1}{2}(D_{C}Y^{C}+fl-2\omega)]J_{AB}$
$\displaystyle\quad\,-\frac{4}{3}(N_{(A}^{(\Lambda)}\partial_{B)}f-\frac{1}{2}N_{C}^{(\Lambda)}\partial^{C}fq_{AB}).$
(4.2.40)

Let us now derive the temporal evolution of $M$, encoded in the
$r$-independent part of $r^{2}(R_{uu}-\Lambda g_{uu})=0$. The first term is
worked out to be

$\begin{split}R_{uu}=\hskip
5.69046pt&(\partial_{u}+2\partial_{u}\beta+l)\Gamma^{u}_{uu}+(\partial_{r}+2\partial_{r}\beta+\frac{2}{r})\Gamma^{r}_{uu}+(\mathcal{D}_{A}+2\partial_{A}\beta)\Gamma^{A}_{uu}\\\
&-2\partial_{u}^{2}\beta-\partial_{u}l-(\Gamma^{u}_{uu})^{2}-2\Gamma^{u}_{uA}\Gamma^{A}_{uu}-(\Gamma^{r}_{ur})^{2}-2\Gamma^{r}_{uA}\Gamma^{A}_{ur}-\Gamma^{A}_{uB}\Gamma^{B}_{uA}\end{split}$
(4.2.41)

where all Christoffel symbols can be found in page 26 of [46]. We finally get

$(\partial_{u}+\frac{3}{2}l)M^{(\Lambda)}+\frac{\Lambda}{6}D^{A}N^{(\Lambda)}_{A}+\frac{\Lambda^{2}}{24}C_{AB}J^{AB}=0.$
(4.2.42)

As a conclusion, in the Bondi gauge (2.1.8), assuming the fall-off condition
(4.2.2) and the boundary gauge fixing (4.2.29), the general solution to
Einstein’s equations with $\Lambda\neq 0$ is entirely determined by the
following 7 free functions of $(u,x^{A})$: $q_{AB}$ with fixed area
$\sqrt{\bar{q}}$, $M$, $N_{A}$ and $J_{AB}$ tracefree, $M$ and $N_{A}$ being
constrained by the evolution equations (4.2.42)-(4.2.34), i.e.

$\bar{\mathcal{S}}_{\Lambda}=\left\\{g_{\mu\nu}\left[q_{AB},M,N_{A},J_{AB}\right]\
\Big{|}\ G_{\mu\nu}[g]+\Lambda
g_{\mu\nu}=0\,;\bm{T},\sqrt{q}=\sqrt{\bar{q}}\right\\},$ (4.2.43)

where we indicate clearly the boundary background structure associated with
the solution space, namely the foliation $\bm{T}$ and the codimension 2 volume
$\sqrt{\bar{q}}$. Remark that in general, such a solution space does not
contain the (A)dS4 global vacuum, since the boundary volume is not necessarily
the unit-round sphere volume $\sqrt{\mathring{q}}$. In the following, as in
the flat case, we will mainly be interested in the solution spaces

$\mathring{\mathcal{S}}_{\Lambda}=\left\\{g_{\mu\nu}\left[q_{AB},M,N_{A},J_{AB}\right]\
\Big{|}\ G_{\mu\nu}[g]+\Lambda
g_{\mu\nu}=0\,;\bm{T},\sqrt{q}=\sqrt{\mathring{q}}\right\\},$ (4.2.44)

which all contain (A)dS4 as particular vacuum solution for any value of
$\Lambda$. For the purpose to match these solution spaces to with our previous
analysis in the SFG gauge, we have to study how to relate the Bondi parameters
of $\bar{\mathcal{S}}_{\Lambda}$ to the SFG variables
$(g_{ab}^{(0)},T^{ab}_{[3]})$.

#### 4.2.2 Dictionary between Starobinsky/Fefferman-Graham and Bondi gauges

In this section, we find the explicit change of coordinates that maps a
general vacuum solution of asymptotically locally (A)dS4 gravity in the Bondi
gauge to the SFG gauge. This procedure will lead to the explicit dictionary
linking the free functions (4.2.27) defined in the Bondi gauge to the
holographic functions defined in SFG gauge, namely the boundary metric
$g_{ab}^{(0)}$ and the boundary stress-tensor

$T^{[3]}_{ab}=\eta\frac{\sqrt{3|\Lambda|}}{16\pi G}g^{(3)}_{ab}$ (4.2.45)

which particularizes (3.1.12) for $d=3$.

We follow and further develop the procedure introduced in [150]. The
intermediate steps are presented for the AdS case ($\Lambda<0$) for the sake
of conciseness, but we restore $\Lambda$ with an indeterminate sign in the
final expressions, providing thus the change of coordinate for both dS and AdS
cases. We first note that one can map the AdS4 vacuum metric in Bondi
coordinates

$\text{d}s^{2}_{\text{AdS}_{4}}=-\left(\frac{r^{2}}{\ell^{2}}+1\right)\text{d}u^{2}-2\text{d}u\text{d}r+r^{2}\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}$
(4.2.46)

to the global static patch (3.1.6) $(t_{\star},r_{\star},x^{A}_{\star})$ by
using $u=t_{\star}-r_{\star}$ where the tortoise coordinate is
$r_{\star}\equiv\ell[\arctan\left(\frac{r}{\ell}\right)-\frac{\pi}{2}]$. This
maps the conformal infinity $\mathscr{I}$ situated at $r=\infty$ to
$r_{\star}=0$, which is very convenient since this also matches immediately
the location of $\mathscr{I}$ at the roots of the holographic coordinate
$\rho$ in the SFG coordinate system. Therefore the change of coordinates from
$(t_{\star},r_{\star},x^{A}_{\star})$ to SFG gauge $(t,\rho,x^{A})$ can be
performed perturbatively in series of $\rho$ around $\rho=0$, identified with
$r_{\star}=0$. At leading order, the time $t_{\star}$ of the global patch is
equivalent to the SFG time $t$, which justifies a bit more to use the global
coordinates as an intermediate coordinate system trading the retarded time $u$
for $t$. The general algorithm is thus the following:

  1. 1.

Starting from any asymptotically locally AdS4 solution formulated in the Bondi
gauge $(u,r,x^{A})$, we perform the preliminary change to the coordinate
system $(t_{\star},r_{\star},x^{A}_{\star})$,

$\begin{split}u&\to t_{\star}-r_{\star},\quad x^{A}\to x^{A}_{\star},\\\
r&\to\ell\tan\Big{[}\frac{r_{\star}}{\ell}+\frac{\pi}{2}\Big{]}=-\frac{\ell^{2}}{r_{\star}}+\frac{r_{\star}}{3}+\frac{r_{\star}^{3}}{45\ell^{2}}+\mathcal{O}(r_{\star}^{5})\\\
&\phantom{\to\ell\tan\Big{[}\frac{r_{\star}}{\ell}+\frac{\pi}{2}\Big{]}}=\frac{3}{\Lambda}\frac{1}{r_{\star}}+\frac{r_{\star}}{3}-\frac{\Lambda}{135}r_{\star}^{3}+\mathcal{O}(r_{\star}^{5})\end{split}$
(4.2.47)

where the last equality uses $\Lambda=-3/\ell^{2}$. This restores $\Lambda$
instead of $\ell$ and one can check that this perturbative change of
coordinate $r\to r_{\star}$ transforms both dS and AdS global vacua from the
Bondi gauge to the line elements (3.1.3) and (3.1.6) respectively. Hence the
expression where $\Lambda$ replaces $\sim\ell^{-2}$ is valid for both signs of
the cosmological constant. The exact change for $\Lambda>0$ is
$r\to\ell\coth(r_{\star}/\ell)$ whose asymptotic series for $r_{\star}\to 0$
is precisely (4.2.47) when $\ell$ is traded for $\Lambda$.

  2. 2.

In a second time, we reach perturbatively the SFG gauge up to order
$N\in\mathbb{N}$,

$g_{\rho\rho}=-\frac{3}{\Lambda}\frac{1}{\rho^{2}}\Big{(}1+\mathcal{O}(\rho^{N+1})\Big{)},\quad
g_{\rho t}=\frac{1}{\rho^{2}}\mathcal{O}(\rho^{N+1}),\quad g_{\rho
A}=\frac{1}{\rho^{2}}\mathcal{O}(\rho^{N+1}),$ (4.2.48)

thanks to a second change of coordinates,

$\begin{split}r_{\star}&\to\sum_{n=1}^{N+1}R_{n}(t,x^{A})\rho^{n},\\\
t_{\star}&\to t+\sum_{n=1}^{N+1}T_{n}(t,x^{A})\rho^{n},\\\ x^{A}_{\star}&\to
x^{A}+\sum_{n=1}^{N+1}X^{A}_{n}(t,x^{B})\rho^{n}.\end{split}$ (4.2.49)

In order to obtain all of the free functions in $\gamma_{ab}$ for $d=3$, we
need to proceed up to order $N=3$. For each $n$, each gauge condition (4.2.48)
can be solved separately and determines algebraically $R_{n}$, $T_{n}$ and
$X_{n}^{A}$ respectively. Only the function $R_{1}(t,x^{A})$ remains
unconstrained by these conditions, since it represents a Weyl transformation
on the boundary metric that is allowed within the SFG gauge. This Weyl
transformation is constrained however by the choice of the luminosity distance
$r$ in Bondi coordinates which ensures that $g_{AB}^{(0)}=q_{AB}$. The
explicit change of coordinates (4.2.49) can be found in appendix D, here we
just plan to present the final result of the diffeomorphism on the free
functions of the solution space.

The boundary metric in the SFG gauge is related to the functions in the Bondi
gauge through

$g_{tt}^{(0)}=\frac{\Lambda}{3}e^{4\beta_{0}}+U_{0}^{C}U_{C}^{0},\qquad
g_{tA}^{(0)}=-U_{A}^{0},\qquad g_{AB}^{(0)}=q_{AB},$ (4.2.50)

in accordance with (4.2.28), where all quantities on the right-hand sides are
now evaluated as functions of $(t,x^{A})$. The parameters
$\\{\sigma,\bar{\xi}^{t},\bar{\xi}^{A}\\}$ of the residual gauge
diffeomorphisms in the SFG gauge (3.1.46) and (3.1.48) can be related to those
of the Bondi gauge appearing in (2.1.12) through

$\begin{split}\bar{\xi}^{t}&=f,\\\ \bar{\xi}^{A}&=Y^{A},\\\
\sigma&=\frac{1}{2}(D_{A}Y^{A}+fl-U_{0}^{A}\partial_{A}f-2\omega),\end{split}$
(4.2.51)

where all parameters on the right-hand sides are also evaluated as functions
of $(t,x^{A})$. Considering (4.2.50), the boundary gauge fixings (4.1.1)
(particularized here for $d=3$) and (4.2.29) are equivalent: for $\Lambda<0$
(resp. $\Lambda>0$), there are exactly the temporal (resp. radial) gauge for
the boundary metric, with a fixed area form on the 2-dimensional transverse
spaces.

Pursuing the change of coordinates to SFG gauge up to $N=3$, it can be shown
that the holographic stress tensor
$T_{ab}^{[3]}\stackrel{{\scriptstyle\text{not}}}{{=}}T_{ab}$ is given, in
terms of Bondi variables, by

$T_{ab}=\frac{\sqrt{3|\Lambda|}}{16\pi
G}\left[\begin{array}[]{cc}-\frac{4}{3}M^{(\Lambda)}&-\frac{2}{3}N^{(\Lambda)}_{B}\\\
-\frac{2}{3}N^{(\Lambda)}_{A}&J_{AB}+\frac{2}{\Lambda}M^{(\Lambda)}q_{AB}\end{array}\right]$
(4.2.52)

where $M^{(\Lambda)}(t,x^{A})$ and $N^{(\Lambda)}_{A}(t,x^{B})$ are the
boundary fields defined as (4.2.35)-(4.2.36) and $J_{AB}$ is precisely the
tensor (4.2.37), all evaluated as functions of $t$ instead of $u$. In order to
complete the picture, let us see how the conservation equation $D_{a}T^{ab}=0$
as well as the trace condition $g^{ab}_{(0)}T_{ab}=0$ are translated in the
Bondi gauge. Owing to the $\Lambda$-BMS boundary gauge fixing (4.1.1) for
$d=3$, the latter reads as

$q^{AB}T_{AB}=-\frac{3}{\Lambda}T_{tt}\Leftrightarrow q^{AB}J_{AB}=0.$
(4.2.53)

We define $T^{TF}_{AB}$ as the tracefree part of $T_{AB}$, i.e.
$T_{AB}=T^{TF}_{AB}-\frac{3}{2\Lambda}T_{tt}q_{AB}$. The conservation equation
yields

$\begin{split}(\partial_{t}+\frac{3}{2}l)T_{tt}+\frac{\Lambda}{3}D^{A}T_{tA}-\frac{\Lambda}{6}\partial_{t}q_{AB}T^{AB}_{TF}&=0,\\\
(\partial_{t}+l)T_{tA}-\frac{1}{2}\partial_{A}T_{tt}+\frac{\Lambda}{3}D^{B}T^{TF}_{AB}&=0,\end{split}$
(4.2.54)

which can be seen as time-evolution constraints for $T_{tt}$ and $T_{tA}$
respectively. The parallel is pretty clear: the conservation equations
(4.2.54) are in fact equivalent to (4.2.42) and (4.2.34). The proof amounts to
using the dictionary (4.2.52) and solve $\partial_{t}q_{AB}$ in terms of
$C_{AB}$ using (4.2.24). Morever, we checked that the transformation laws
(4.2.38)-(4.2.40) are equivalent to the variation

$\delta_{\xi}T_{ab}=\mathcal{L}_{\bar{\xi}}T_{ab}+\sigma\,T_{ab},$ (4.2.55)

which is (3.1.63) particularized for $d=3$. We therefore identified the Bondi
mass aspect $M^{(\Lambda)}$ and the Bondi angular momentum aspect
$N^{(\Lambda)}_{A}$ as the components $T_{tt}$ and $T_{tA}$ of the holographic
stress-tensor, up to a normalization constant.

#### 4.2.3 Bondi news and Bondi mass in (A)dS4

Now that the matching of the solution space parameters has been executed, let
us understand what happens at the level of the presymplectic structure coming
from $\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]$. The latter
has been computed in the SFG gauge: we thus need to invert (at least at
leading order) the change of coordinates constructed in section 4.2.2 to
translate this expression into the Bondi gauge. Assuming the $\Lambda$-BMS4
gauge fixing, from (4.2.49) and the expressions
$R_{1}(t,x^{A})=-\frac{3}{\Lambda}$, $T_{1}(t,x^{A})=X_{1}^{A}(t,x^{B})=0$
given in appendix D, we have

$\rho=-\frac{\Lambda}{3}r_{\star}+\mathcal{O}(r_{\star}^{3}),\qquad
t=t_{\star}+\mathcal{O}(r_{\star}),\qquad
x^{A}=x^{A}_{\star}+\mathcal{O}(r_{\star}).$ (4.2.56)

Provided that the field-dependence involved in (4.2.49) appears only at
subleading orders in $\rho$ (or equivalently in $r_{\star}$), we can observe
that the variational operator $\delta$ is not modified at leading order.
Coupling both pieces of information, we see that the leading contributions in
$\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]$ are not affected
by the change of coordinates, hence
$\Theta^{r_{\star}}_{\Lambda\text{-BMS}_{4}}=\Theta^{\rho}_{\Lambda\text{-BMS}_{4}}+\mathcal{O}(r_{\star})$,
$\Theta^{a}_{\Lambda\text{-BMS}_{4}}=(\Theta^{t_{\star}}_{\Lambda\text{-BMS}_{4}},\Theta^{A_{\star}}_{\Lambda\text{-BMS}_{4}})+\mathcal{O}(r_{\star})$.
We compose this transformation with the inverse of (4.2.47) to reach the Bondi
gauge $(u,r,x^{A})$. We have

$r_{\star}=\frac{3}{\Lambda}\frac{1}{r}+\mathcal{O}(r^{-3}),\quad
t_{\star}=u+\mathcal{O}(r^{-1}),\quad x^{A}_{\star}=x^{A},$ (4.2.57)

which implies finally that

$\rho=-\frac{1}{r}+\mathcal{O}(r^{-3}),\quad t=u+\mathcal{O}(r^{-1}),\quad
x^{A}=x^{A}+\mathcal{O}(r^{-1}).$ (4.2.58)

As a result,
$\Theta^{r}_{\Lambda\text{-BMS}_{4}}=\Theta^{\rho}_{\Lambda\text{-BMS}_{4}}+\mathcal{O}(\rho)$.
Particularizing (3.2.34) for $d=3$ and using (4.2.29) as well as the
parametrization (4.2.52), we get the analog of (4.1.12) in Bondi variables,
that is

$\boxed{\Theta_{\Lambda\text{-BMS}_{4}}^{r}[\phi;\delta\phi]=\frac{\Lambda}{2}\frac{\sqrt{\mathring{q}}}{16\pi
G}J^{AB}\delta q_{AB}+\mathcal{O}(r^{-1}).}$ (4.2.59)

Since all dynamical quantities can be computed from this presymplectic
potential, it shows the equivalence between both gauges at the level of the
dynamics.

The action principle (3.2.35) becomes

$\delta S_{ren}^{\Lambda\text{-BMS}_{4}}[\phi]=\frac{\Lambda}{2}\frac{1}{16\pi
G}\int_{\mathscr{I}}(\text{d}^{3}x)\sqrt{\mathring{q}}\,J^{AB}\delta q_{AB}.$
(4.2.60)

The sources for the leaks are identified to be $q_{AB}$ and $J^{AB}$. The
presymplectic form associated with (4.2.59) is

$\boxed{\omega_{\Lambda\text{-BMS}_{4}}^{r}[\phi;\delta_{1}\phi,\delta_{2}\phi]=\frac{\Lambda}{2}\frac{\sqrt{\mathring{q}}}{16\pi
G}\delta_{1}J^{AB}\wedge\delta_{2}q_{AB}+\mathcal{O}(r^{-1}).}$ (4.2.61)

Again thanks to the fundamental theorem (2.2.57), or (3.2.38), (4.2.61) shows
that the information on gravitational energy fluxes passing through
$\mathscr{I}$ is entirely contained in the couple $(J^{AB},q_{AB})$. We will
therefore call this couple of functions the news for asymptotically locally
(A)dS4 spacetimes. The expression (4.2.61) is perfectly reminiscent to the
flat presymplectic form (2.2.124) which is proportional to $\delta
N^{AB}\wedge\delta C_{AB}$ when evaluated for $\delta q_{AB}=0$. This is in
fact the direct analogue in asymptotically locally (A)dS4 spacetimes of the
asymptotically flat expression of the presymplectic flux. The role of Bondi
news and shear $(N^{AB},C_{AB})$ is now played by $(J^{AB},q_{AB})$. Note that
while $N_{AB}=\partial_{u}C_{AB}$, there is no such relationship between
$J^{AB}$ and $q_{AB}$ in asymptotically locally dS4 spacetimes since otherwise
the Cauchy data would be constrained. In asymptotically locally AdS4
spacetimes, a well-defined Cauchy problem will require a condition at
conformal infinity and traditionally for conservative (reflexive) boundary
conditions, we would like to cancel the flux induced by (4.2.61). These
conservative subsectors of the $\Lambda$-BMS4 phase space will be treated in
section 4.2.5.

Let us for the moment delay the determination of the surface charges induced
by (4.2.61) (see section 4.2.4.3) but discuss qualitatively the physical
content of this symplectic form. For that purpose, we recall that for standard
asymptotically flat spacetimes, the Bondi mass is given by
$\mathcal{M}=\oint_{S_{\infty}}(\text{d}^{2}x)\,\sqrt{\mathring{q}}\,M$ and
obeys the Bondi mass loss formula (2.2.133), implying that $\mathcal{M}(u)$ is
a decreasing function of $u$. The integrations are performed on a transverse
2-surface $S_{\infty}$ at $\mathscr{I}^{+}$. The definition of Bondi mass for
$\Lambda\neq 0$ is instead
$\mathcal{M}^{(\Lambda)}=\oint_{S_{\infty}}(\text{d}^{2}x)\,\sqrt{\mathring{q}}\,M^{(\Lambda)}$
and it evolves as

$\frac{\text{d}}{\text{d}u}\mathcal{M}^{(\Lambda)}=-\frac{\Lambda^{2}}{24}\oint_{S_{\infty}}(\text{d}^{2}x)\,\sqrt{\mathring{q}}\,J^{AB}C_{AB}=-\frac{\Lambda}{8}\oint_{S_{\infty}}(\text{d}^{2}x)\,\sqrt{\mathring{q}}\,J^{AB}\partial_{u}q_{AB},$
(4.2.62)

after using (4.2.24) and (4.2.29). The major subtlety here is that the right-
hand side of (4.2.62) is not manifestly non-positive! It remains an important
problem to determinate whether the Bondi mass $\mathcal{M}^{(\Lambda)}$
decreases with $u$, what is currently under investigation.

#### 4.2.4 $\Lambda$-BMS4 algebroid in Bondi gauge

In this section, we use the dictionary provided in section 4.2.2 to repeat the
analysis of the set of residual gauge transformations that preserve the
boundary gauge fixing (4.2.29) in the Bondi gauge and form the $\Lambda$-BMS4
algebroid, as a corollary of the various definitions given in section 4.1.2.
We give some explicit solutions for the $\Lambda$-BMS4 generators around the
unit-round sphere metric, where the constraint equations can be solved
analytically in terms of infinite expansions in spherical harmonics. We also
comment on the direct link between the codimension 2 functions appearing in
these generators and the BMS4 supertranslations and super-Lorentz parameters
in the flat limit. Thereafter, we give some additional comments on the
$\Lambda$-BMS4 charge algebra, particularizing the results of the above
section 4.1.3, and the associated flux-balance laws at conformal infinity.

##### 4.2.4.1 Bondi gauge parameters and constraints

As announced, we study here the consequences of the boundary gauge fixing
(4.2.29) on the residual gauge transformations intrinsically in the Bondi
gauge. We start from (2.1.12) and impose (4.2.29), which is (4.1.1) specified
for $d=3$ and interpreted in the Bondi gauge thanks to the diffeomorphism
constructed in the previous section 4.2.2. Doing so, we should like to make a
slight generalization, considering any fixed boundary volume form instead of
$\sqrt{\mathring{q}}$ of the unit-round sphere. Using (4.2.30)-(4.2.32), the
condition $\delta_{\xi}\sqrt{q}=0$ leads to

$\omega=0.$ (4.2.63)

The condition $\delta_{\xi}\beta_{0}=0$ leads to

$\Big{(}\partial_{u}-\frac{1}{2}l\Big{)}f=\frac{1}{2}D_{A}Y^{A},$ (4.2.64)

while $\delta_{\xi}U_{0}^{A}=0$ gives

$\partial_{u}Y^{A}=-\frac{\Lambda}{3}q^{AB}\partial_{B}f.$ (4.2.65)

The solution to (4.2.64)-(4.2.65) admits three integration “constants”
$T(x^{A}),V^{A}(x^{B})$, though the aforementioned constraints cannot be
explicitly integrated for an arbitrary transverse metric $q_{AB}$ in terms of
these functions. In the SFG notation, the equations (4.2.63)-(4.2.65) are
equivalent to

$\displaystyle\sigma$ $\displaystyle=\frac{1}{2}(D_{A}^{(0)}\xi^{A}_{0}+fl),$
(4.2.66) $\displaystyle\Big{(}\partial_{t}-\frac{1}{2}l\Big{)}\xi^{t}_{0}$
$\displaystyle=\frac{1}{2}D_{A}^{(0)}\xi^{A}_{0},$ (4.2.67)
$\displaystyle\partial_{t}\xi^{A}_{0}$
$\displaystyle=-\frac{\Lambda}{3}g^{AB}_{(0)}\partial_{B}\xi^{t}_{(0)}.$
(4.2.68)

Up to the incorporation of the $l$ contributions appearing only when
$\partial_{u}\sqrt{\bar{q}}\neq 0$, we find exactly the same differential
structure as in (4.1.2). The generators $f,Y^{A}$ span the $\Lambda$-BMS4
algebroid written in the Bondi gauge. They satisfy the commutation relations

$[\bar{\xi}_{1},\bar{\xi}_{2}]_{\star}=\hat{\bar{\xi}},$ (4.2.69)

where $\hat{\bar{\xi}}=\hat{f}\partial_{u}+\hat{Y}^{A}\partial_{A}$ with

$\displaystyle\hat{f}$
$\displaystyle=Y_{1}^{A}\partial_{A}f_{2}+\frac{1}{2}f_{1}D_{A}Y_{2}^{A}-(1\leftrightarrow
2),$ (4.2.70) $\displaystyle\hat{Y}^{A}$
$\displaystyle=Y^{B}_{1}\partial_{B}Y_{2}^{A}-\frac{\Lambda}{3}f_{1}q^{AB}\partial_{B}f_{2}-(1\leftrightarrow
2).$ (4.2.71)

as a corollary of (4.1.3)-(4.1.4). In the $4d$ case, the analogy with the
standard (Generalized) BMS4 algebra, reviewed in section 2.1.5, is very
stringent and we may call the vectors generated by $T(x^{A})$ and
$V^{A}(x^{B})$ the $\Lambda$-supertranslation and the $\Lambda$-super-Lorentz
generators, respectively. In the asymptotically flat limit $\Lambda=0$ and for
time-independent transverse metric (i.e.
$\sqrt{\bar{q}}=\sqrt{\mathring{q}}$), the functions $Y^{A}$, $f$ reduce to
$Y^{A}=V^{A}(x^{B})$, $f=T(x^{A})+\frac{u}{2}D_{A}V^{A}$ and the structure
constants reduce to those of the Generalized BMS4 algebra. For $\Lambda\neq
0$, the fragmentation between “supertranslations” and “super-Lorentz”
transformations is not clear and this terminology has to be intended more at
the level of the pedagogical analogy than at a more fundamental level. In
particular, the $\Lambda$-supertranslations do not commute and the structure
constants depend explicitly on $q_{AB}$. As before, when the transverse metric
$q_{AB}$ is equal to the unit-round sphere metric $\mathring{q}_{AB}$ (i.e.
for Dirichlet boundary conditions), the $\Lambda-$BMS4 algebra contains the
$SO(3,2)$ algebra for $\Lambda<0$ and the $SO(1,4)$ algebra for $\Lambda>0$,
see [222]. For the interested reader, we review the explicit expressions of
the $SO(3,2)$ Killing vectors of AdS4 in Appendix C.

##### 4.2.4.2 Explicit solutions and flat limit

We mentioned in the previous paragraph that the joint solution of (4.2.64) and
(4.2.65) in terms of codimension 2 fields $T(x^{A})$ and $V^{A}(x^{B})$ for
arbitrary background fields $q_{AB},l$ is yet unknown to us. In what follows,
we show how to get a flavor of the analytical structure of $f$ and $Y^{A}$ in
terms of codimension 2 functions by attempting to integrate this differential
system around the unit-round sphere $q_{AB}=\mathring{q}_{AB}$, $l=0$. We
believe that a argument by covariance could allow to obtain the solution for
the generic case $q_{AB}\neq\mathring{q}_{AB}$, $l\neq 0$ from the results we
present here, but we do not plan to discuss it in details. Instead, we close
the section by presenting a beautiful construction of the flat limit
straightforwardly taken on the explicit solutions. Note that our derivation is
here performed in the Bondi gauge. The exact analogous procedure in the SFG
gauge can be found in the original article [37].

We set $\ell^{2}=-3\eta/\Lambda$ as usual, $\sqrt{q}=\sqrt{\mathring{q}}$ and
$l=0$, but we conserve $q_{AB}$ general until we need to particularize to the
unit-round sphere metric. The first goal is to obtain scalar differential
equations for the parameters. Taking a second time derivative on (4.2.64) and
using (4.2.65), we obtain that

$\partial_{u}^{2}f-\frac{\eta}{2\ell^{2}}D_{A}D^{A}f=0,$ (4.2.72)

i.e. $f$ is governed by a wave equation whose group velocity is commensurable
to $\Lambda$. We further separate $Y^{A}$ into a scalar field $\Phi$ and a
pseudo-scalar field $\Psi$ thanks to the Helmholtz decomposition
$Y^{A}=q^{AB}\partial_{B}\Phi(u,x^{C})+\epsilon^{AB}\partial_{B}\Psi(u,x^{C})$,
where
$\epsilon^{AB}=\sqrt{\mathring{q}}\varepsilon^{AB}=\mathring{\epsilon}^{AB}$
is the Levi-Civita tensor on the unit-round sphere. Since the evolution
equation for $Y^{A}$ is a vector constraint on the sphere, we can solve it for
its curl and its divergence separately. This yields respectively

$\displaystyle D_{A}D^{A}(\partial_{u}\Psi)=\varepsilon_{AB}D^{A}W^{B},$
(4.2.73) $\displaystyle
D_{A}D^{A}\left(\partial_{u}\Phi-\frac{\eta}{\ell^{2}}f\right)=-D_{A}W^{A},$
(4.2.74)

where $W^{A}\equiv\partial_{u}q^{AB}\partial_{B}\Phi$ (recall that, at this
stage, we have not yet imposed $q_{AB}=\mathring{q}_{AB}$). It is worth
noticing that the wave equation (4.2.72) does not fully determine $f$, since
this is a second order equation with respect to $u$, while the original
constraint on $f$ (4.2.64) is a first order equation. The information we have
lost by applying the second derivative is encoded in the remaining condition

$\partial_{u}f=\frac{1}{2}D_{A}D^{A}\Phi$ (4.2.75)

that we need to take into account. Hence the system is rephrased as
(4.2.72)–(4.2.75).

The equations for $\partial_{u}f$, $\partial_{u}\Phi$ and $\partial_{u}\Psi$
determine the fields $f$, $\Phi$ and $\Psi$ up to three arbitrary function of
the angles $(\hat{f},\hat{\Phi},\hat{\Psi})$. Let us show that these functions
are constrained. We obtain $D_{A}D^{A}\hat{f}=0$ directly from (4.2.72) and
$D_{A}D^{A}\hat{\Phi}=0$ from (4.2.75). Recalling that only constants are
harmonic functions on compact manifolds, we can set $\hat{f}=f_{0}$
($f_{0}\in\mathbb{R}$) and $\hat{\Phi}=0$, since a constant value of $\Phi$
does not appear in the vector $Y^{A}$. Hence we see that the solution for
$\partial_{u}f$ and $\partial_{u}\Phi$ fully determines $f$ (up to a residual
constant) and $\Phi$. However, we also observe that nothing constrains
$\hat{\Psi}$, showing that it will remain an arbitrary function on the angles
in $\Psi$. Finally the wave equation involving $f$ gives rise to two arbitrary
functions of the angles as integration constants, which will be brought to
$\Phi$ thanks to (4.2.74). Hence the number of arbitrary functions of $x^{A}$
is shown to be three, as expected!

We now set $q_{AB}\equiv\mathring{q}_{AB}$ and $W^{A}$ vanishes identically.
The solution of (4.2.73) is $\partial_{u}\Psi=c$ for some real constant $c$.
This constant can be removed as an ambiguity in defining the Helmholtz fields,
since it is responsible for a linear term $cu$ in $\Psi$ which will never
contribute to the actual vector $Y^{A}$. The solution for $\Psi$ is thus
simply $\Psi=\Psi(x^{A})$. We can directly solve (4.2.72) by using Fourier
transform methods. We obtain

$f(u,x^{A})=\int_{0}^{\infty}\text{d}\omega\,\left[f_{E}(x^{A})\cos\left(\frac{\omega\,u}{\ell}\right)+f_{O}(x^{A})\sin\left(\frac{\omega\,u}{\ell}\right)\right]$
(4.2.76)

if $\Lambda<0$ and

$f(u,x^{A})=\int_{0}^{\infty}\text{d}\omega\,\left[f_{E}(x^{A})\cosh\left(\frac{\omega\,u}{\ell}\right)+f_{O}(x^{A})\sinh\left(\frac{\omega\,u}{\ell}\right)\right]$
(4.2.77)

if $\Lambda>0$. In both cases, the Fourier coefficients are constrained as
follows:

$\mathring{D}_{A}\mathring{D}^{A}f_{E}=-2\omega^{2}f_{E},\quad\mathring{D}_{A}\mathring{D}^{A}f_{O}=-2\omega^{2}f_{O}.$
(4.2.78)

These equations select discrete values for $\omega$ which satisfy
$\omega_{\texttt{{l}}}^{2}=\frac{1}{2}\texttt{{l}}(\texttt{{l}}+1)$ with
$\texttt{{l}}\in\mathbb{N}$ and the solution for $f$ is given in terms of the
real spherical harmonics $Y_{\texttt{{lm}}}(x^{A})$
($\texttt{{m}}\in\mathbb{N}$, $|\texttt{{m}}|\leq\texttt{{l}}$). Indeed,

$f(u,x^{A})=\left\\{\begin{aligned}
\,&\sum_{\texttt{{l,m}}}\left[a_{\texttt{{lm}}}\cos\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)+b_{\texttt{{lm}}}\sin\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)\right]Y_{\texttt{{lm}}}(x^{A})\text{
if }\Lambda<0,\\\
\,&\sum_{\texttt{{l,m}}}\left[a_{\texttt{{lm}}}\cosh\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)+b_{\texttt{{lm}}}\sinh\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)\right]Y_{\texttt{{lm}}}(x^{A})\text{
if }\Lambda>0,\end{aligned}\right.$ (4.2.79)

where $\\{a_{\texttt{{lm}}}\\}$ and $\\{b_{\texttt{{lm}}}\\}$ are two sets of
real constants. From (4.2.74), we can deduce that

$\partial_{u}\Phi-\frac{\eta}{\ell^{2}}f=0$ (4.2.80)

up to some real constant that can again be removed as an ambiguity in defining
$\Phi$. Taking one more derivative with respect to $u$ and recalling that
(4.2.75) holds, we obtain that $\Phi$ satisfies the same wave equation as $f$,

$\partial_{u}^{2}\Phi-\frac{\eta}{2\ell^{2}}\mathring{D}_{A}\mathring{D}^{A}\Phi=0.$
(4.2.81)

The general solution has already been derived (see (4.2.79)) and reads as

$\boxed{\Phi(u,x^{A})=\left\\{\begin{aligned}
\,&\sum_{\texttt{{l,m}}}\left[A_{\texttt{{lm}}}\cos\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)+\frac{1}{\ell}\,B_{\texttt{{lm}}}\sin\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)\right]Y_{\texttt{{lm}}}(x^{A})\text{
if }\Lambda<0,\\\
\,&\sum_{\texttt{{l,m}}}\left[A_{\texttt{{lm}}}\cosh\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)+\frac{1}{\ell}\,B_{\texttt{{lm}}}\sinh\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)\right]Y_{\texttt{{lm}}}(x^{A})\text{
if }\Lambda>0,\end{aligned}\right.}$ (4.2.82)

involving two new sets of real constants $\\{A_{\texttt{{lm}}}\\}$ and
$\\{B_{\texttt{{lm}}}\\}$. The $1/\ell$ factor in front of the $B$’s is for
now purely conventional, but will ensure later that the coefficients $A_{lm}$
and $B_{lm}$ do not depend on $\ell$ in the flat limit process
$\ell\to\infty$. These new sets of constants are not independent of
$\\{a_{lm}\\}$ and $\\{b_{lm}\\}$, since we must require that the remaining
constraints (4.2.75) and (4.2.80) hold. The first one has been implemented in
the derivation of the wave equation (4.2.81), so we just have to impose the
second one. This yields

$a_{\texttt{{lm}}}=\eta\,B_{\texttt{{lm}}}\,\omega_{\texttt{{l}}},\quad
b_{\texttt{{lm}}}=-\ell\,A_{\texttt{{lm}}}\,\omega_{\texttt{{l}}}.$ (4.2.83)

Hence, the gauge parameter $f$ reads as

$\boxed{f(u,x^{A})=\left\\{\begin{aligned}
\,&\sum_{\texttt{{l,m}}}\left[B_{\texttt{{lm}}}\cos\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)-\ell\,A_{\texttt{{lm}}}\sin\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)\right]\omega_{\texttt{{l}}}\,Y_{\texttt{{lm}}}(x^{A})\text{
if }\Lambda<0,\\\
\,&-\sum_{\texttt{{l,m}}}\left[B_{\texttt{{lm}}}\cosh\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)+\ell\,A_{\texttt{{lm}}}\sinh\left(\frac{\omega_{\texttt{{l}}}\,u}{\ell}\right)\right]\omega_{\texttt{{l}}}\,Y_{\texttt{{lm}}}(x^{A})\text{
if }\Lambda>0,\end{aligned}\right.}$ (4.2.84)

The exact isometries of global (A)dS4, reviewed in appendix C, are recovered
if we consider the lowest modes $\texttt{{l}}=0$ and $\texttt{{l}}=1$. Indeed,
if we further require that $\delta_{\xi}q_{AB}=0$, it comes

$\mathring{D}_{A}\xi^{(0)}_{B}+\mathring{D}_{B}\xi^{(0)}_{A}-\mathring{D}_{C}\xi_{(0)}^{C}\mathring{q}_{AB}=0\Leftrightarrow\left\\{\begin{aligned}
(\mathring{D}_{A}\mathring{D}_{B}\Phi)^{TF}&=0,\\\
\varepsilon_{C(A}\mathring{D}_{B)}\partial^{C}\Psi&=0.\end{aligned}\right.$
(4.2.85)

In stereographic coordinates $(z,\bar{z})$, if we introduce the auxiliary
fields $\phi\equiv(1+z\bar{z})\Phi$ and $\psi\equiv(1+z\bar{z})\Psi$, these
equations become simply $\partial_{z}^{2}\phi=0=\partial_{\bar{z}}^{2}\phi$
and $\partial_{z}^{2}\psi=0=\partial_{\bar{z}}^{2}\psi$. Hence $\phi$ and
$\psi$ are at most linear in $z$ and $\bar{z}$, the only non-linear piece that
can appear being the squared modulus $z\bar{z}$. In conclusion, the solution
to the conformal Killing equation developed for the Helmholtz fields $\Phi$
and $\Psi$ only involve the lowest (real) spherical harmonics with
$\texttt{{l}}=0,1$.

Finally, let us show that the flat limit obtained by taking directly
$\ell\to\infty$ in (4.2.82) and (4.2.84) reproduces the expression of the
Generalized BMS4 generators, which are solutions of (2.1.68). The flat limit
of (4.2.82) is immediate and gives

$\Phi(u,x^{A})=\sum_{\texttt{{l,m}}}A_{\texttt{{lm}}}Y_{\texttt{{lm}}}(x^{A})=\Phi(x^{A})$
(4.2.86)

as expected. The two Helmholtz fields $\Phi$ and $\Psi$ are arbitrary and
define a general super-Lorentz transformation $Y^{A}$ around
$q_{AB}=\mathring{q}_{AB}$. The same limit of (4.2.84) leads to

$\begin{split}f(u,x^{A})&=\sum_{\texttt{{l,m}}}\left[\eta
B_{\texttt{{lm}}}-\omega_{\texttt{{l}}}\,u\,A_{\texttt{{lm}}}\right]\omega_{\texttt{{l}}}\,Y_{\texttt{{lm}}}(x^{A})\\\
&=\sum_{\texttt{{l,m}}}\left[\tilde{B}_{\texttt{{lm}}}\,Y_{\texttt{{lm}}}(x^{A})-\omega_{\texttt{{l}}}^{2}\,u\,A_{\texttt{{lm}}}\,Y_{\texttt{{lm}}}(x^{A})\right],\,\left(\tilde{B}_{\texttt{{lm}}}\equiv\eta\,\omega_{\texttt{{l}}}B_{\texttt{{lm}}}\right)\\\
&=\sum_{\texttt{{l,m}}}\tilde{B}_{\texttt{{lm}}}\,Y_{\texttt{{lm}}}(x^{A})+\frac{u}{2}\sum_{\texttt{{l,m}}}A_{\texttt{{lm}}}\mathring{D}_{B}\mathring{D}^{B}Y_{\texttt{{lm}}}(x^{A})\\\
&=T(x^{A})+\frac{u}{2}\mathring{D}^{B}\mathring{D}_{B}\Phi(x^{A})\\\
&=T(x^{A})+\frac{u}{2}\mathring{D}_{B}Y^{B}(x^{A}),\end{split}$ (4.2.87)

where $T(x^{A})$ is an arbitrary scalar field on the celestial sphere.

##### 4.2.4.3 $\Lambda$-BMS4 charges and algebra

Before proceeding further, we want to particularize the results of section
4.1.3 in this paragraph. We are back to consider the charges (4.1.14) for the
split (4.1.15). For $d=3$, the Weyl non-integrable charge vanishes. Using
$\text{d}^{2}\Omega=\sqrt{\mathring{q}}(\text{d}^{2}x)$ and the dictionary
elaborated in section 4.2.2 to express the charges with Bondi quantities, we
find

$\displaystyle\delta/H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]$
$\displaystyle=\delta
H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]+\Xi_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]$
(4.2.88)

with

$\displaystyle H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]$
$\displaystyle=\frac{\eta}{16\pi G}\oint_{S_{\infty}}\text{d}^{2}\Omega\
\left[4\,f\,M^{(\Lambda)}+2\,Y^{A}\,N_{A}^{(\Lambda)}\right],$ (4.2.89)
$\displaystyle\Xi_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]$
$\displaystyle=\frac{\Lambda}{32\pi G}\oint_{S_{\infty}}\text{d}^{2}\Omega\
\left(fJ_{AB}\delta
q^{AB}\right)-H_{\delta\xi}^{\Lambda\text{-BMS}_{4}}[\phi]$ (4.2.90)

and all the involved quantities are intended as functions of $(t,x^{A})$ in
the SFG gauge. This is the last incursion of this coordinate system in our
analysis but we prefer keep it while discussing the charges for the sake of
clarity. Written in that way, the $\Lambda$-BMS4 charges are reminiscent of
the BMS4 charges for asymptotically Minkowskian spacetimes [31, 46], see
(2.2.127)-(2.2.128) with fixed boundary metric $\delta q^{AB}=0$. In
particular, the non-integrable part involves the symplectic pairing
$(J_{AB},q^{AB})$ as defined in section 4.2.3. As a corollary of (4.1.18) and
(4.1.19) for $d=3$, the algebra of charges (4.2.88) represent the
$\Lambda$-BMS4 algebroid without any central extension to the Barnich-
Troessaert bracket:

$\boxed{\\{H_{\xi_{1}}^{\Lambda\text{-BMS}_{4}}[\phi],H^{\Lambda\text{-BMS}_{4}}_{\xi_{2}}[\phi]\\}_{\star}=H^{\Lambda\text{-BMS}_{4}}_{[\xi_{1},\xi_{2}]_{\star}}[\phi].}$
(4.2.91)

It is interesting to observe that there is no 2-cocycle at the level of the
charge algebra for the immediate extension of the Generalized BMS4 symmetries
for $\Lambda\neq 0$ although the flat limit of (4.2.91) is supposed to exhibit
the 2-cocycle previously calculated as (2.3.24). Any shortcut reasoning of
this kind will be pondered by the explicit flat limit of the $\Lambda$-BMS4
phase space described in section 4.3.3, because this process is not well-
defined for dynamical quantities such as the presymplectic structure or the
charges. Both of them contain poles in $\Lambda^{-1}$ that will require
further renormalization of the action principle before sending $\Lambda$ to
zero.

As a crucial remark, let us mention that (4.2.91) controls the non-
conservation of the $\Lambda$-BMS4 charges. The proof of this statement has
already been elaborated for the Generalized BMS4 charges in section 2.3.3, and
reads as follows. The total evolution of the Hamiltonian associated with
$\xi$, considered as the integrable part selected in (4.2.88), is governed by
the derivative along the foliation $\bm{T}$ on $\mathscr{I}$, given by
$\frac{\text{d}}{\text{d}u}\equiv\frac{\text{d}}{\text{d}t}+\mathcal{O}(r^{-1})$.
By definition, we have

$\frac{\text{d}}{\text{d}u}H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]=\frac{\partial}{\partial
u}H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]+\delta_{\partial_{u}}H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]-H_{\delta_{\partial_{u}}\xi}^{\Lambda\text{-BMS}_{4}}[\phi]$
(4.2.92)

which constitutes a slight generalization of (2.3.25) while dealing with
field-dependent diffeomorphism parameters. The third term is indeed zero for
asymptotically flat configurations since $\delta_{\xi}f=0=\delta_{\xi}Y^{A}$.
We have

$\frac{\partial}{\partial
u}H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]-H_{\delta_{\partial_{u}}\xi}^{\Lambda\text{-BMS}_{4}}[\phi]=H_{\partial_{u}\xi-\delta_{\partial_{u}}\xi}^{\Lambda\text{-BMS}_{4}}[\phi]=-H_{[\xi,\partial_{u}]_{\star}}^{\Lambda\text{-BMS}_{4}}[\phi]$
(4.2.93)

because the charges depend linearly upon the gauge parameter. The conclusion
of the proof is similar to what we did before. Using the algebra (4.2.91), we
obtain easily

$\frac{\text{d}}{\text{d}u}H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]=-\Xi_{\partial_{u}}^{\Lambda\text{-BMS}_{4}}[\phi;\delta_{\xi}\phi]=-\oint_{S_{\infty}}\iota_{\partial_{u}}\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}[\phi;\delta_{\xi}\phi],$
(4.2.94)

recognizing the renormalized presymplectic flux (4.2.60) in the non-integrable
part (4.2.90). This flux formula is a particular case of Eq. (4.9) of [134]
for $\Lambda>0$ and is the analogue of the flux formula (2.3.28) in
asymptotically locally flat spacetimes [31, 14].

The computation of the charges (4.2.88) allows us to conclude by giving a
striking argument in favor of the $\Lambda$-BMS4 algebroid as candidate for
the role of natural asymptotic symmetries of Al(A)dS4 gravity. While
experiencing some leaks through the conformal boundary, the fundamental
theorem (2.2.57) indicates that each charge is associated with a flux-balance
law controlling its non-conservation. By a counting of boundary degrees of
freedom, we see that there are three ordinary differential equations in time
at the boundary, which are given by (4.2.54) as a consequence of the
conservation of the holographic stress tensor on $\mathscr{I}$. Written in
Bondi coordinates, these equations are (4.2.42)-(4.2.34) and give thus three
flux-balance laws for the Bondi mass as well as the Bondi angular momentum for
$\Lambda\neq 0$. It can be shown that they control the time-derivative of the
asymptotic charges (4.2.89) in accordance with (4.2.94). We conclude that the
$\Lambda$-BMS4 algebroid, whose elements are parametrized by three functions
$f,Y^{A}$ on $\mathscr{I}$, is the non-trivial sub-algebra of
Diff($\mathscr{I}$) which encapsulates the three towers of non-trivial charges
that obey the right number of flux-balance laws (or Ward identities at quantum
level, see section 5.3.4.1). Assorted with the argument according to which
(4.2.29) is a boundary gauge fixing which does not reduce the solution space,
we have in fact proven that the $\Lambda$-BMS4 phase space is complete in the
sense that it arises from the most stringent gauge fixing that does not rule
out any solution while admitting as many asymptotic generators as necessary to
reproduce the three flux-balance laws naturally present in the solution space.
In other words, this is the canonical subset of Diff($\mathscr{I}$) associated
with the physical flux given locally by (4.2.42) and (4.2.34).

Two last remarks should be made at this stage:

  * $\rhd$

Changes of foliation $\bm{T}$ span an equivalence class of boundary gauge
fixings and $\Lambda$-BMS4 algebroids: this amounts to saying that we can
select Gaussian normal coordinates at will on $\mathscr{I}$ when $\Lambda\neq
0$. However, in flat space, the null direction on $\mathscr{I}^{+}$ is given
unambiguously. As we will observe in section 4.3, only the consistency with
the flat limit process breaks this equivalence relation by selecting a
preferred direction on $\mathscr{I}$ that becomes the null direction when
$\Lambda\to 0$.

  * $\rhd$

Also in flat case, the Generalized BMS4 symmetries are complete only at second
order in the Bondi expansion, in the sense that no other evolution equation
than (2.1.44) and (2.1.45) arises from Einstein’s equations up to that order.
But there is an infinite tower of subleading flux-balance laws beyond that
order, related to $\mathcal{E}_{AB}$, $\mathcal{F}_{AB}$ etc. [46], see
section 2.1.5.2. This is in contrast with what happens here, where (4.2.42)
and (4.2.34) are the only flux-balance laws appearing in the Bondi expansion
at large radii.

#### 4.2.5 Conservative subsectors and stationary solutions

In this short section, we investigate a bit some conservative subsectors of
the leaky boundary conditions leading to the $\Lambda$-BMS4 phase space.
Irrelevant for dS4 asymptotics, they ensure the stationarity of the
variational problem as well as the well-definiteness of the Cauchy problem in
the case of asymptotically locally AdS4 spacetimes. They need a supplementary
boundary condition to be implemented at the spatial boundary
$\mathscr{I}_{\text{AdS}}$ of AdS4 [175] and amounts to requiring that the
symplectic flux (4.2.60) at spatial infinity is identically zero. Along with
the famous Dirichlet boundary conditions [62, 159, 160, 64, 63, 161, 166] that
are the natural conservative subcase of the $\Lambda$-BMS4 boundary conditions
fixing completely the boundary structure, we derive new boundary conditions
for asymptotically locally AdS4 spacetimes that admit the Schwarzschild-AdS
black hole and new stationary rotating solutions distinct from the Kerr-AdS
black hole [294] (see also [295, 296] for the higher dimensional
generalization). The asymptotic symmetry group is shown to be a subgroup of
the $\Lambda$-BMS4 group consisting of time translations and area-preserving
diffeomorphisms.

##### 4.2.5.1 Mixed Neumann conservative boundary conditions

A vanishing symplectic flux in (4.2.60) is observed when $\delta q_{AB}=0$ for
$J^{AB}$ completely free. This subsector of the $\Lambda$-BMS4 phase space
consists of solutions with Dirichlet boundary conditions defined in section
3.4.1. The fixation of the boundary metric reduces the asymptotic symmetry
group to the group of exact symmetries of AdS4, namely $SO(3,2)$, see section
4.2.4. Neumann boundary conditions defined in section 3.4.2 freeze the
components of the holographic stress-tensor while leaving the whole boundary
metric free to fluctuate. The resulting asymptotic symmetry group is empty:
all residual gauge transformations have vanishing charges. This is due to the
fact that there are kinematical degrees of freedom in the fluctuations of the
boundary metric that should be fixed by some boundary gauge fixing, allowing
the symplectic conjugate parts of the stress tensor to be non-zero while
enforcing a vanishing symplectic flux. This is precisely what the
$\Lambda$-BMS4 boundary gauge fixing (4.2.29) does: the equation (4.2.59)
shows that we can set partial Neumann boundary conditions

$J^{AB}=0$ (4.2.95)

to kill the symplectic flux without constraining the momentum components
$T^{tt}$, $T^{tA}$ of the holographic stress-tensor, which contain the mass
and angular momentum aspects, see (4.2.52). These are conservative boundary
conditions compatible with a fluctuating boundary structure. More concretely,
(4.2.95) coupled with (4.2.29) provides new mixed Dirichlet-Neumann boundary
conditions, where the latter condition is a Dirichlet boundary condition on
part of the boundary metric, but which the good property of being always
reachable by gauge fixing. The additional fixation (4.2.95) restricts the
phase space of solutions as well as the asymptotic symmetries. Let us find out
what remains of the $\Lambda$-BMS4 algebroid when (4.2.95) holds.

##### 4.2.5.2 Asymptotic symmetry algebra

We derive the asymptotic symmetries preserving the boundary conditions and
compute the associated charge algebra. We show that the subgroup of asymptotic
symmetries preserving (4.2.95) is the direct product
$\mathbb{R}\times\mathcal{A}$ where $\mathbb{R}$ are the time translations on
the boundary (along the foliation vector $\bm{T}$) and $\mathcal{A}$ is the
group of 2-dimensional area-preserving diffeomorphisms on the transverse
2-spheres. As expected for a set of conservative boundary conditions, the
associated surface charges are finite, integrable, conserved and generically
non-vanishing on the phase space.

Everything starts from the transformation law of $J_{AB}$. Using (4.2.52),
$\delta_{\xi}J_{AB}$ can be extracted in the SFG gauge from the variation
(4.2.55), then reinterpreted in the Bondi gauge with the dictionary (4.2.51).
The result is

$\delta_{\xi}{J}_{AB}=(f\partial_{u}+\mathcal{L}_{Y}+\frac{1}{2}D_{C}Y^{C}){J}_{AB}-\frac{4}{3}\Big{[}N_{(A}\partial_{B)}f-\frac{1}{2}N_{C}\partial^{C}fq_{AB}\Big{]}.$
(4.2.96)

Imposing $\delta_{\xi}J_{AB}=0$ leads to the following constraint on the
residual gauge diffeomorphisms:

$\partial_{A}f=0.$ (4.2.97)

This is the minimal field-independent condition to cancel the inhomogeneous
part of the transformation (4.2.96). Therefore, the asymptotic symmetry
generators satisfy the relations

$\partial_{u}f=\frac{1}{2}D_{A}Y^{A},\qquad\partial_{u}Y^{A}=0.$ (4.2.98)

as a corollary of (4.2.64)-(4.2.65). The second equation implies
$Y^{A}=Y^{A}(x^{B})$, while the first gives

$f=T+\frac{u}{2}D_{A}Y^{A},$ (4.2.99)

where $T$ is a constant by virtue of (4.2.97) and $D_{A}Y^{A}\equiv c$ where
$c$ is also a constant. The structure of $f$ and $Y^{A}$ is reminiscent of the
Generalized BMS4 algebra in the flat case. We use again the Helmholtz
decomposition for the vector $Y^{A}$ into a divergence-free and a curl-free
part, $Y^{A}=\epsilon^{AB}\partial_{B}\Psi+q^{AB}\partial_{B}\Phi$ where
$\Psi$ and $\Phi$ are functions of $x^{C}$. We obtain thus that $\Psi$ is
unconstrained while $D_{A}D^{A}\Phi=c$. On the sphere, this last equation
admits a solution if and only if $c=0$ which is given by $\Phi=0$. Therefore,
the asymptotic symmetry generators are given by

$f=T,\qquad Y^{A}=\epsilon^{AB}\partial_{B}\Psi(x^{C}),$ (4.2.100)

where $T$ is a constant and $\Psi(x^{C})$ is arbitrary and defines an area-
preserving diffeomorphism on the 2-sphere. Indeed, we have
$\mathcal{L}_{Y}\sqrt{q}=\partial_{A}(\sqrt{q}Y^{A})=\sqrt{q}D_{A}Y^{A}=0$.
Writing the boundary diffeomorphism as
$\bar{\xi}=\bar{\xi}^{a}\partial_{a}=T\partial_{t}+\epsilon^{AB}\partial_{B}\Psi\partial_{A}$,
we have
$[\bar{\xi}_{1},\bar{\xi}_{2}]_{\star}=[\bar{\xi}_{1},\bar{\xi}_{2}]=\hat{\bar{\xi}}$
where

$\hat{T}=0,\qquad\hat{\Psi}=\epsilon^{AB}\partial_{A}\Psi_{2}\partial_{B}\Psi_{1}.$
(4.2.101)

Hence, after imposing the boundary condition (4.2.95), the $\Lambda$-BMS4
algebroid reduces to the $\mathbb{R}\oplus\mathcal{A}$ algebra where
$\mathbb{R}$ denotes the abelian time translations and $\mathcal{A}$ is the
algebra of 2$d$ area-preserving diffeomorphisms. The latter symmetries are an
infinite-dimensional extension of the $SO(3)$ rotations. They are in addition
field-independent, since (4.2.97) has decoupled the constraint (4.2.65) in
which the boundary metric $q_{AB}$ appeared naturally.

Let us now obtain the associated charges. Again, the computation is more
practical in SFG gauge, but we can translate the results into the Bondi gauge
afterwards. We particularize the charge formula (4.1.14) with (4.1.15) to
$d=3$. The Weyl part $W^{[3]}_{\sigma}$ is absent and the first non-integrable
term vanishes because $T_{AB}^{TF}\propto J_{AB}=0$. Hence the charges are
integrable and read as

$Q_{\xi(T,\Psi)}[g]=\oint_{S_{\infty}}\text{d}^{2}\Omega\,\sqrt{\bar{q}}\,[T^{t}_{\;\,\,t}\,T+T^{t}_{\;\,\,A}\epsilon^{AB}\partial_{B}\Psi].$
(4.2.102)

Translating this result in the Bondi gauge thanks to (4.2.52), we get

$Q_{\xi(T,\Psi)}[g]=\frac{\ell}{16\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\,\sqrt{\bar{q}}\,\left[4\,T\,M^{(\Lambda)}+2\epsilon^{AB}N_{A}^{(\Lambda)}\partial_{B}\Psi\right].$
(4.2.103)

From this expression, we see that the charges associated with the symmetry
$\mathbb{R}\oplus\mathcal{A}$ are generically non-vanishing. Taking $T=1$ and
$\Psi=0$ gives the total energy. The first harmonic modes of $\Psi$ give three
components of the total angular momentum, while the higher modes give an
infinite tower of charges. Using the commutation relations (4.2.101) and the
equations of motion (4.2.54), a simple computation shows that the charges
(4.2.102) satisfy the algebra

$-\delta_{\xi(T_{1},\Psi_{1})}Q_{\xi(T_{2},\Psi_{2})}=Q_{\xi(\hat{T},\hat{\Psi})}.$
(4.2.104)

The charges $Q_{\xi(T,\Psi)}[g]$ form a representation of
$\mathbb{R}\oplus\mathcal{A}$ without central extension.

##### 4.2.5.3 Stationary solutions

Let us study the stationary sector of the phase space associated with the
boundary condition (4.2.95). For these solutions, we have successively
$\partial_{t}q_{AB}=0$ or equivalently $C_{AB}=0$ and
$\partial_{t}M^{(\Lambda)}=0$, $\partial_{t}N_{A}^{(\Lambda)}$. Using (4.2.34)
and (4.2.42) we get

$D_{A}N^{A}_{(\Lambda)}=0\Leftrightarrow
N^{A}_{(\Lambda)}=\epsilon^{AB}D_{B}\alpha(x^{C}),\quad\partial_{A}M^{(\Lambda)}=0$
(4.2.105)

for $\alpha(x^{C})$ an arbitrary function of the angles. As a result, even for
a stationary class of solutions, we see that the charges (4.2.103) associated
with the area-preserving diffeomorphisms are generically non-vanishing. It
would be interesting to study the regularity of the general solutions
(4.2.105) in the bulk of spacetime, but answering this question is definitely
beyond the scope of our analysis.

We now discuss some prototypical examples of stationary solutions: the family
of Kerr-AdS4 metric [294], describing celestial bodies in stationary rotation
with negative cosmological constant. Clearly the static Schwarzschild-AdS4
solution is included in the phase space with $J_{AB}=0$. Indeed, it can be set
in the SFG gauge, which allows to identify $q_{AB}=\mathring{q}_{AB}$ the unit
metric on the sphere, as well as ${T^{t}}_{t}=\frac{M}{4\pi G}$, $T_{tA}=0$
and $T_{AB}=0$, which finally implies $J_{AB}=0$. This is a good point. Now
let us introduce rotation of the central body. The boundary metric and
holographic stress-tensor of Kerr-AdS4 are given in the conformally flat frame
by [297, 298, 299]

$\displaystyle g^{(0)}_{ab}\text{d}x^{a}\text{d}x^{b}$
$\displaystyle=-\ell^{-2}\text{d}t^{2}+\text{d}\theta^{2}+\sin^{2}\theta\text{d}\phi^{2},$
(4.2.106) $\displaystyle T^{ab}$
$\displaystyle=T_{\text{Kerr}}^{ab}\equiv-\frac{m\gamma^{3}\ell}{8\pi}(3u^{a}u^{b}+g_{(0)}^{ab}),$
(4.2.107)

where $\Xi=1-a^{2}\ell^{-2}$ and

$u^{a}\partial_{a}=\gamma\ell(\partial_{t}+\frac{a}{\ell^{2}}\partial_{\phi}),\qquad\gamma^{-1}\equiv\sqrt{1-\frac{a^{2}}{\ell^{2}}\sin^{2}\theta}.$
(4.2.108)

The total mass and angular momentum are
$\mathcal{M}=\int\sqrt{\bar{q}}\,T^{t}_{\,\,t}=\frac{m}{\Xi^{2}}$,
$\mathcal{J}=Ma=-\int\sqrt{\bar{q}}\,T^{t}_{\,\,\phi}=\frac{ma}{\Xi^{2}}$. We
observe that $J_{AB}\neq 0$. Therefore, the Kerr-AdS4 solution is not included
in the phase space with $J_{AB}=0$ (although it is contained in the phase
space with Dirichlet boundary conditions). However, it is possible to obtain a
stationary axisymmetric solution with $J_{AB}=0$ as follows. The most general
diagonal traceless and divergence free stationary $T^{ab}$ is given by

$\begin{split}&T^{tt}_{\text{corr}}=\ell^{2}[2T^{\theta\theta}(\theta)+\tan\theta\leavevmode\nobreak\
{T^{\theta\theta}}^{\prime}(\theta)],\quad
T^{\theta\theta}_{\text{corr}}=T^{\theta\theta}(\theta),\\\
&T^{\phi\phi}_{\text{corr}}=\frac{1}{\sin^{2}\theta}[T^{\theta\theta}(\theta)+\tan\theta\leavevmode\nobreak\
{T^{\theta\theta}}^{\prime}(\theta)]\end{split}$ (4.2.109)

and the other components are set to zero. We consider the sum of
$T_{\text{Kerr}}+T_{\text{corr}}$. We solve for $T^{\theta\theta}(\theta)$ to
set $J^{AB}=0$. The regular solution at $\mathscr{I}$ is unique and given by

$T^{tt}=-\frac{m\ell^{3}}{4\pi},\qquad
T^{t\phi}=-\frac{3am\ell\gamma^{5}}{8\pi},\qquad
T^{AB}=-\frac{m\ell}{8\pi}q^{AB}.$ (4.2.110)

The total mass and angular momentum of the new solution are
$\mathcal{M}=\int\sqrt{-g_{(0)}}\,T^{t}_{\,\,t}=m$,
$\mathcal{J}=-\int\sqrt{-g_{(0)}}\,T^{t}_{\,\,\phi}=\frac{ma}{\Xi^{2}}$.
Again, to know whether this solution is regular in the bulk of spacetime
remains an open question. This ends our discussion of some interesting aspects
of the $\Lambda$-BMS4 phase space.

###  4.3 Flat limit of the $\Lambda$-BMS4 phase space

The correspondence between both formulations of the $\Lambda$-BMS4 phase space
formulated in the SFG and the Bondi gauges, as well as the straightforward
link between $\Lambda$-BMS4 and Generalized BMS4 algebras at flat limit
$\Lambda\to 0$, give us confidence to build a rigorous and meaningful flat
limit process to map the leaky phase spaces in the presence of $\Lambda\neq 0$
discussed thus far onto the asymptotically flat phase space discussed in
chapter 2 when sending $\Lambda$ to zero. One of the major advantages is to
exploit the better control on renormalization that we have for (A)dS
asymptotics to asymptotically flat spacetimes. For instance, this would
provide a firm base for the Iyer-Wald counter-term (2.2.116)-(2.2.117)
postulated when attempting to define a finite action principle for the
Generalized BMS4 phase space. We proceed step by step: we first comment on the
various subtleties of the flat limit at a conceptual level and how we treat
them; next, we prove that the solution space admits a well-defined flat;
finally, we analyze the same limit at the level of the presymplectic
structure, which needs a further regularization by a corner-term subtraction.

#### 4.3.1 The philosophy of the flat limit

The fact that the Bondi coordinates (2.1.8) can be defined for any value of
the cosmological constant is not the only reason for using them to define the
flat limit procedure. The choice is motivated by the intuition according to
which null-type coordinates are more adapted than others to take the flat
limit and the naive argument is as follows. Let us consider an AlAdS4
spacetime. It is well-known that the cosmological constant is manifesting
itself at cosmological scales, i.e. for phenomena occurring at relative
distance $r\sim\ell$, where $\ell\sim\sqrt{1/\Lambda}$ is the AdS radius
defined previously. On astrophysical scales $r\ll\ell$, the gravitation field
can be described in good approximation with $\Lambda\approx 0$. We can thus
imagine a little causal diamond representing an asymptotically flat spacetime
sitting far deep in the bulk of the AlAdS4 spacetime under consideration, as
represented in Figure 4.2.

$\mathscr{I}^{+}_{\text{dS}}$..$\mathscr{I}^{+}$ Asymptotically flat
($r\ll\Lambda^{-1}$) ..$\mathscr{I}^{-}$$i^{0}$

(a)

$\mathscr{I}_{\text{AdS}}$..$\mathscr{I}^{+}$..$\mathscr{I}^{-}$$i^{0}$
Asymptotically flat ($r\ll|\Lambda|^{-1}$)

(b)

Figure 4.2: Energy flow in the bulk and flat limit of (A)dS asymptotics.

We further assume that the Bondi coordinate system continues to be valid far
enough from the conformal boundary $\mathscr{I}_{\text{AdS}}$ in order to
cover this little central diamond. This pictorial description shows that the
outgoing null rays naturally map the “points” of $\mathscr{I}^{+}$ of the
asymptotically flat spacetime in the deep bulk to the “points” of the
conformal infinity $\mathscr{I}_{\text{AdS}}$. Following the flow of null
radiation shows that in for any compact submanifolds of $\mathscr{I}^{+}$ and
$\mathscr{I}_{\text{AdS}}$, this relation is bijective if we assume that all
of the sources of gravitational radiation are beyond $\mathscr{I}^{+}$ in the
bulk. This is precisely why there is a link between the dynamical quantities,
such as the Bondi mass and angular momentum aspects, defined for the
$\Lambda$-BMS4 phase space as codimension 1 functions living on
$\mathscr{I}_{\text{AdS}}$ and the homologuous functions in the context of the
Generalized BMS4 phase space. Considering the asymptotically flat limit as a
zoom on the central diamond also allows to understand why the symmetries have
a well-defined flat limit, but the dynamical quantities have not. Indeed,
symmetries are not only defined on the boundary but extends to the bulk, see
(2.1.12). Assuming that the power series in $1/r$ still holds for values of
$r\ll\ell$, the action of a symmetry at the vicinity of
$\mathscr{I}_{\text{AdS}}$ is transmitted along the outgoing null rays (that
are immutable because they define the Bondi coordinate $u$) to
$\mathscr{I}^{+}$. However, the dynamical quantities are not expected to have
a well-defined flat limit since energy flows along the null lines from
$\mathscr{I}^{+}$ to $\mathscr{I}_{\text{AdS}}$. This observation also applies
for the symplectic structure, the surface charges and their algebra: their
bare expressions are all singular in the flat because of divergences in the
limit $\Lambda\to 0$. The construction of a full flat limit process must
therefore be assorted with some further renormalization of the presymplectic
structure, this time not aimed at curing radial divergences, but parametric
poles in $\ell\sim 1/\Lambda$ in the limit $\Lambda\to 0$ or $\ell\to+\infty$.
Note that all of these features can be discussed for AldS4 spacetimes with
perfectly equivalent conclusions.

We are done with the naive picture that can merely give a flavor of what we
can expect from the flat limit. The major technical obstruction to consider
the flat limit as a zoom in the far bulk is the necessity to understand the
link between the data in the Bondi gauge defined around the conformal infinity
$\mathscr{I}$ of the host spacetime and the data in the radically different
Bondi gauge describing the gravitational field around the null hypersurface
$\mathscr{I}^{+}$ delimiting the central diamond at finite distance. This
would demand to resum the Bondi expansion in the far bulk of the Al(A)dS4
spacetime and again perform a matching between gauges, which is something we
do not want to do and which is probably impossible in general. What we can do
is to consider the $\Lambda$-BMS4 solution spaces,
$\bar{\mathcal{S}}_{\Lambda}$, defined in (4.2.43), as a one-parameter family
of solution spaces whose $\bar{\mathcal{S}}_{0}$ defined in (2.1.37) would be
an adherence point in the limit $\Lambda\to 0$, at the price, of course, of
performing some field redefinitions in the parameters of
$\bar{\mathcal{S}}_{\Lambda}$ to ensure that the limit is not singular. The
“future of light” $\mathscr{I}$ for solutions in $\bar{\mathcal{S}}_{\Lambda}$
is mapped to the “future of light” $\mathscr{I}^{+}$ for solutions in
$\bar{\mathcal{S}}_{0}$ while keeping the same boundary structure. For
instance, the norm of the boundary foliation vector $\bm{T}$ is proportional
to $\Lambda$: $\bm{T}$ is timelike for AdS asymptotics, spacelike for dS
asymptotics and the gap between both cases is filled by the regular limit
$\Lambda=0$ where $\bm{T}$ is null, as expected. The $u$ coordinate evolving
on the flow of $\bm{T}$ changes its nature, from the intrinsic point of view
of the boundary, but remains a null coordinate with respect to the bulk (by
definition of the Bondi gauge) and the limit is smooth. The other coordinates
$r$ and $x^{A}$ are not touched by the flat limit, since the background
structure related to them (a radial null foliation in the bulk and a
parametrization of the codimension 2 spaces transverse to $\bm{T}$) is
invariant under modifications of $\Lambda$. This is also the point of view of
[40], where a similar process was successfully performed in 3$d$ gravity. In
summary, with the parametric limit $\Lambda\to 0$ we perform below, we just
map boundaries on boundaries, which is a complementary picture of the naive
“zoom” setup, but offers a better analytical control. Let us explain how to
implement this process first at the level of the solution space.

#### 4.3.2 Flat limit of the solution space

For any non-vanishing value of the cosmological constant, the mere hypothesis
of conformal compactification with analyticity (4.2.2) leads to the most
general solution space in the Bondi gauge by virtue of the Fefferman-Graham
theorem [200]. As announced, the analyticity requirement breaks down when in
general $\Lambda=0$ but was maintained in our presentation of the Generalized
BMS4 solution space in section 2.1.5.2. The flat limit of any Al(A)dS4
solution space can thus lead only to the subsector of asymptotically (locally)
flat gravity for which the logarithmic branch has been neglected. Since we
have been only concerned in the analytic branch in the flat case, this is not
a restriction for our analysis.

The cornerstone of the flat limit process is the constraint equation (4.2.24)
which reads as

$(\partial_{u}-l)q_{AB}=\frac{\Lambda}{3}C_{AB}$ (4.3.1)

after the boundary gauge fixing (4.2.29) in $\bar{\mathcal{S}}_{\Lambda}$.
Like all of the analytical constraints brought by Einstein’s equations
(4.2.8), it acts as a switch leading to radically different conclusions
depending on whether $\Lambda$ is vanishing or not.

  * $\rhd$

When $\Lambda\neq 0$, the equation (4.3.1) determines algebraically $C_{AB}$
in terms of the time evolution of the boundary metric, which is completely
unconstrained.

  * $\rhd$

When $\Lambda=0$, the meaning of (4.3.1) drastically changes. It imposes a
strong constraint on the boundary metric, prescribing its time evolution as
(2.1.34), while keeping the shear $C_{AB}$ completely free.

We see that imposing this equation of motion does not commute with the flat
limit $\Lambda\to 0$, simply because, depending upon the way we solve it (i.e.
for $\partial_{u}q_{AB}$ or $C_{AB}$), the limit $\Lambda\to 0$ can be either
well-defined or ill-defined. Our prescription for the flat limit process is to
conserve the well-behaved field $C_{AB}$ explicit and trade, thanks to
(4.3.1), any time derivative of the boundary metric for the expression
$\Lambda C_{AB}$ which is vanishing in the flat limit. This allows us to be
systematic and track down any contribution in $\Lambda$ coupled to well-
behaved fields like $C_{AB}$. Doing so, we assume that the latter does not
depend upon $\Lambda$, in order to end up with expressions in which any
dependency in $\Lambda$ is clear, and take the limit $\Lambda\to 0$
analytically afterwards, as we could do for numerical functions $f(\Lambda)$.
This also amounts to extracting explicitly the trace of
$N_{AB}=\partial_{u}C_{AB}$, which contains an hidden contribution in
$\Lambda$. Defining

$N_{AB}^{TF}\equiv N_{AB}-\frac{\Lambda}{6}q_{AB}C_{CD}C^{CD},\quad
q^{AB}N_{AB}^{TF}=0,$ (4.3.2)

we get a tensor whose limit for $\Lambda\to 0$ is smooth: it is assimilated to
the Bondi news tensor in the flat case.

A second switch, less crucial although nevertheless important, involves
$\mathcal{D}_{AB}$. The relevant equations of motion are (4.2.25), (4.2.26)
and $D_{B}\mathcal{D}^{AB}=0$ by virtue of the Fefferman-Graham theorem [199,
200, 292] which forbids the appearance of a logarithmic term in (4.2.19). In
the boundary gauge fixing (4.2.29), the second equation states nothing but
$\partial_{u}\mathcal{D}_{AB}=0$ which has a trivial flat limit. The
divergence-free condition also admits a direct flat limit and is in accordance
with (2.1.32). However, the condition $\Lambda\mathcal{D}_{AB}=0$ indicates
that $\mathcal{D}_{AB}$ itself has a poor behavior in the flat limit, since it
can be non-zero only if $\Lambda=0$. Again, imposing that the product
$\Lambda\mathcal{D}_{AB}$ is zero does not commute with the flat limit process
$\Lambda\to 0$, because if we impose it, for instance, before sending
$\Lambda$ to zero, we would get only the subsector of the flat theory with
$\mathcal{D}_{AB}=0$. The trick is to write all the equations with an apparent
time-independent divergence-free $\mathcal{D}_{AB}$ for any value of $\Lambda$
and impose the additional constraint $\Lambda\mathcal{D}_{AB}=0$ only if
$\Lambda$ is fixed. However, the flat limit $\Lambda\to 0$ has to be
understood as the limit of the expressions with the apparent
$\mathcal{D}_{AB}$ while maintaining the constraint
$\Lambda\mathcal{D}_{AB}=0$ without solving it, in such a way that
$\mathcal{D}_{AB}$ is present in the flat limit and unconstrained by the
equations of motion. Denoting again $g_{AB}=r^{2}\sum_{n\geq
0}g^{(n)}_{AB}r^{-n}$, the switch conditions involving
$\mathcal{E}_{AB},\mathcal{F}_{AB},\dots$ formally read as $\Lambda
g^{(n)}_{AB}=\partial_{u}g_{AB}^{(n-1)}+(\dots)$ at subleading orders. In the
flat limit, they degenerate when into the evolution equations
$\partial_{u}g_{AB}^{(n-1)}+(\dots)=0$. Hence the symmetric traceless tensor
fields $\mathcal{E}_{AB},\mathcal{F}_{AB},\dots$ are no longer algebraically
constrained by the fields of lower order: only their time derivative is kept
constrained, in accordance with the discussion below the equation (2.1.34). We
obtain a infinite-dimensional countable tower of fields determined by a
function of the angles at some retarded time $u=u_{0}$ and whose value of
$\mathscr{I}^{+}$ is obtained by the integration of the associated evolution
equations in $u$ [46, 237, 242].

With this knowledge, the flat limit of the other algebraic equations is
immediate. Indeed, imposing the boundary gauge fixing (4.2.29) in (4.2.14)
rules out the leading $\beta_{0}$ field and nothing depends upon $\Lambda$
except the constraint (4.2.25) that we do not impose before taking the limit
in $\Lambda$. Hence the flattened version of (4.2.14) is (2.1.30) as it
should. Working in the same way on (4.2.19) gives straightforwardly (2.1.31)
in the flat limit without the logarithmic term since $D_{B}\mathcal{D}^{AB}=0$
has been maintained during the process. Finally (4.2.23) is sent on (2.1.33)
since $\beta_{0}$ and $U_{0}^{A}$ disappear when the conditions (4.2.29) hold
and the linear pieces in $\Lambda$ at $\mathcal{O}(r^{2})$ and
$\mathcal{O}(r^{0})$ orders are consistently brought to zero. The
asymptotically flat limit of the evolution equations (4.2.34) and (4.2.42)
requires a bit more care, provided that the quantities (4.2.35), (4.2.36) and
(4.2.37) defined intrinsically for $\Lambda\neq 0$ configurations are not
meant to be well-defined in the flat limit. We start by collecting the terms
in $1/\Lambda$ hidden in (4.2.34):

$\begin{split}-\frac{3}{2\Lambda}\Big{[}&(\partial_{u}+l)D^{B}(\partial_{u}C_{AB}-\frac{1}{2}lC_{AB})-D^{B}\partial_{u}(\partial_{u}C_{AB}-\frac{1}{2}lC_{AB})\\\
&\quad+\frac{1}{2}(\partial_{u}+l)\partial_{A}R[q]+D^{B}(D_{A}D_{B}l-\frac{1}{2}D_{C}D^{C}lq_{AB})\Big{]}.\end{split}$
(4.3.3)

We recall that there are two subtle steps in the massaging of the evolution
equation needed before taking the limit $\Lambda\to 0$. First, we have to
develop the remaining derivatives in $u$ acting on covariant derivatives and
solve the switch constraint (4.3.1) for $\partial_{u}q_{AB}$ to highlight all
the factors in $\Lambda$. Second, we need to extract the trace of $N_{AB}$,
which also contains a residual contribution $\sim\Lambda$. We end up with

$\begin{split}\eqref{eq:ProblematicTerms}=\,\,&\frac{1}{2}D_{C}(N_{AB}^{TF}C^{BC})+\frac{1}{4}N_{BC}^{TF}D_{A}C^{BC}-\frac{1}{4}D_{A}D_{B}D_{C}C^{BC}\\\
&+\frac{1}{8}C^{B}_{C}C^{C}_{B}\partial_{A}l-\frac{3}{16}l\partial_{A}(C^{B}_{C}C^{C}_{B}).\end{split}$
(4.3.4)

The following identities turn out to be useful for the computation:

$\begin{split}(\partial_{u}+2l)H^{AB}&=q^{AC}\partial_{u}H_{CD}q^{BD}+\frac{2\Lambda}{3}C^{C(A}H^{B)}_{C},\\\
(\partial_{u}+l)(D^{B}H_{AB})&=D^{B}\partial_{u}H_{AB}-\frac{1}{2}q^{CD}H_{CD}\partial_{A}l\\\
&\quad\,-\frac{\Lambda}{3}\Big{[}D_{B}(C^{BC}H_{AC})+\frac{1}{2}H^{BC}D_{A}C_{BC}\Big{]},\\\
(\partial_{u}+l)C^{AB}C_{AB}&=2N^{AB}C_{AB}-lC_{AB}C^{AB},\\\
(\partial_{u}+l)\partial_{A}R[q]&=-(D^{B}D_{B}+\frac{1}{2}R[q])\partial_{A}l+\frac{\Lambda}{3}D_{A}D_{B}D_{C}C^{BC}\end{split}$
(4.3.5)

for an arbitrary symmetric rank 2 transverse tensor $H_{AB}(u,x^{C})$. Using
(E.11), the first term of (4.3.4) can be rewritten as

$\frac{1}{2}D_{C}(N_{AB}^{TF}C^{BC})=\frac{1}{4}D_{B}(N_{AC}^{TF}C^{BC}-C_{AC}N^{BC}_{TF})+\frac{1}{4}\partial_{A}(C_{BD}N^{BD}_{TF}).$
(4.3.6)

We can now present (4.2.34) in a way that makes terms in $\Lambda$ explicit:

$\displaystyle(\partial_{u}+l)N_{A}$
$\displaystyle-\partial_{A}M-\frac{1}{4}C_{AB}\partial^{B}R[q]-\frac{1}{16}\partial_{A}(N_{BC}^{TF}C^{BC})$
(4.3.7)
$\displaystyle-\frac{1}{32}l\partial_{A}(C_{BC}C^{BC})+\frac{1}{4}N_{BC}^{TF}D_{A}C^{BC}+\frac{1}{4}D_{B}(C^{BC}N_{AC}^{TF}-N^{BC}_{TF}C_{AC})$
$\displaystyle+\frac{1}{4}D_{B}(D^{B}D^{C}C_{AC}-D_{A}D_{C}C^{BC})+\frac{\Lambda}{2}D^{B}(\mathcal{E}_{AB}-\frac{7}{96}C^{C}_{D}C^{D}_{C}C_{AB})=0.$

As a result, the asymptotically flat limit can be safely taken and (4.3.7)
reduces to

$\displaystyle(\partial_{u}+l)N_{A}$
$\displaystyle-\partial_{A}M-\frac{1}{4}C_{AB}\partial^{B}R[q]-\frac{1}{16}\partial_{A}(N_{BC}C^{BC})$
$\displaystyle-\frac{1}{32}l\partial_{A}(C_{BC}C^{BC})+\frac{1}{4}N_{BC}D_{A}C^{BC}+\frac{1}{4}D_{B}(C^{BC}N_{AC}-N^{BC}C_{AC})$
$\displaystyle+\frac{1}{4}D_{B}(D^{B}D^{C}C_{AC}-D_{A}D_{C}C^{BC})=0,$ (4.3.8)

where we identified $N_{AB}=N_{AB}^{TF}$ when $\Lambda=0$. This equation fully
agrees with (2.1.36). Finally, the asymptotically flat limit of the second
evolution equation (4.2.42) is straightforward and gives

$\begin{split}(\partial_{u}+\frac{3}{2}l)M+\frac{1}{8}N_{AB}N^{AB}-\frac{1}{8}lN_{AB}C^{AB}+\frac{1}{32}l^{2}C_{AB}C^{AB}-\frac{1}{8}D_{A}D^{A}R[q]\\\
-\frac{1}{4}D_{A}D_{B}N^{AB}+\frac{1}{4}C^{AB}D_{A}D_{B}l+\frac{1}{4}\partial_{(A}lD_{B)}C^{AB}+\frac{1}{8}lD_{A}D_{B}C^{AB}=0,\end{split}$
(4.3.9)

in perfect agreement with (2.1.35). Note that (4.3.8) and (4.3.9) are also in
agreement with (4.49) and (4.50) of [46] respectively, after a change of
conventions: the Bondi news tensor is defined in [46] as
$N_{AB}^{\text{\cite[cite]{[\@@bibref{}{Barnich:2010eb}{}{}]}}}=\partial_{u}C_{AB}-lC_{AB}$
while we always define $N_{AB}=\partial_{u}C_{AB}$.

Summarizing the results, we find that

$\lim_{\Lambda\to 0}\bar{\mathcal{S}}_{\Lambda}=\bar{\mathcal{S}}_{0}$
(4.3.10)

where the solution spaces are defined in (4.2.43) and (2.1.37) respectively
and the limit $\Lambda\to 0$ is defined formally with the prescriptions
enunciated at the beginning of the current section. Fixing the boundary volume
form to include the global vacua (A)dS4 and Minkowski in the solution spaces –
which amounts to selecting the $\Lambda$-BMS4 orbit for which
$\sqrt{q}=\sqrt{\mathring{q}}$, we obtain

$\boxed{\lim_{\Lambda\to
0}\mathring{\mathcal{S}}_{\Lambda}=\mathring{\mathcal{S}}_{0}.}$ (4.3.11)

Although not surprising since the Bondi gauge is defined for any value of
$\Lambda$ and the derivation of the respective solution spaces when
$\Lambda=0$ and $\Lambda\neq 0$ is similar, it is yet remarkable to observe
that a straightforward flat limit process (up to some purely analytical
subtleties) maps the evolution equations of Bondi aspects in Al(A)dS4
spacetimes to the standard time evolution constraints of the Bondi aspects
defined for flat asymptotics. Provided that the evolution of mass and angular
momentum for $\Lambda\neq 0$ is governed by the (covariant) conservation of a
boundary stress-tensor, this fact can help as a building block for the
understanding of a putative 4$d$-3$d$ formulation of flat holography, where
$M$ and $N_{A}$ would evolve as momenta on the codimension 1 null (or
Carrollian) manifold at the boundary (see e.g. [267, 300, 301, 302, 303,
304]). Note also that the flat limit process is intimately related to the
choice of a preferred boundary foliation $\bm{T}$. When $\Lambda\neq 0$, as
noted previously, the fixation of $\bm{T}$ is merely a choice of Gaussian
normal coordinates on $\mathscr{I}$ which can always be made without ruling
out any solution in the phase space. However, when taking the limit
$\Lambda\to 0$ while keeping $\bm{T}$ untouched, this vector determines the
(degenerate) null direction along which the flat $\mathscr{I}^{+}$ will be
aligned. As a result, considering a particular foliation $\bm{T}$ gives one
particular flat limit for which the hypersurfaces at constant retarded time
have been preserved and $\bm{T}$ has becomed null on the boundary.

#### 4.3.3 Flat limit of the action and corner terms

Finally we turn our interest on the flat limit of the symplectic structure on
the $\Lambda$-BMS4 phase space. The most rigorous way to proceed is to
reconsider the on-shell renormalized variational principle (4.2.60), isolate
the explicit dependency in $\Lambda$ and see if the flat limit $\Lambda\to 0$
is readily possible or not. If a whole section has been devoted to discussing
this process, one can already expect that it is neither direct nor even well-
defined a priori. Indeed, we prove here that a further renormalization of the
on-shell action principle, although finite in $r$, is mandatory in order to
obtain a meaningful asymptotically flat limit.

##### 4.3.3.1 Corner terms and prescription for renormalization

We reformulate the $\Lambda$-BMS4 variational principle (4.2.60) in a way
similar to the asymptotically flat variational principle discussed in section
2.2.4.1, except that the radial regulation in $r$ has already been taken into
account by holographic renormalization, see section 3.2.1.1. Recall that the
construction of the $\Lambda$-BMS4 phase space assumes the existence of a
foliation $\bm{T}$ on the conformal boundary $\mathscr{I}$ with topological
spheres as orthogonal surfaces of measure $\sqrt{\mathring{q}}$. Considering a
portion of $\mathscr{I}$, the initial and final values of the foliation
parametrized by $u$ define the “corner” boundaries denoted as
$\partial\\!\mathscr{I}_{+}$ (at $u=u^{+}$) and $\partial\\!\mathscr{I}_{-}$
(at $u=u^{-}$), see Figure 4.3.

Ignoring for the moment the boundary terms at fixed $u=u^{\pm}$, the variation
of the renormalized action (3.2.1) is given on-shell by

$\delta
S_{ren}^{\Lambda\text{-BMS}_{4}}[\phi]=\int_{\mathscr{I}}\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]=\frac{\Lambda}{32\pi
G}\int_{\mathscr{I}}(\text{d}^{3}x)\sqrt{\mathring{q}}\,J_{(tot)}^{AB}\delta
q_{AB},$ (4.3.12)

assuming the boundary gauge fixing (4.2.29) and incorporating with hindsight
the contribution of some $\bm{L}_{\circ}$ to the stress-tensor, written as in
(3.2.27) in the SFG gauge but easily transposed in the Bondi gauge with the
algorithm described in section 4.2.2. We have
$J_{AB}^{(tot)}=J_{AB}+J_{AB}^{\circ}$ where $J_{AB}$ is still defined in
(4.2.52) and
$J_{AB}^{\circ}=T_{AB}^{\circ}-\frac{1}{2}q_{AB}q^{CD}T_{CD}^{\circ}$.

$\partial\mathscr{I}_{+}$$\partial\mathscr{I}_{-}$$\mathscr{I}$$u=\text{Cst}$$\bm{N}$$\bm{T}\qquad$$\bm{N}$$\bm{T}\qquad$$\bm{N}$$\bm{T}\qquad$$\bm{N}$$\bm{T}\qquad$$\bm{N}$$\bm{T}\qquad$
(a) _AlAdS 4_ case.

South poleNorth poleCosmological
horizon$u=\text{Cst}$$\mathscr{I}$$\partial\mathscr{I}_{+}$$\partial\mathscr{I}_{-}$$\bm{N}$$\bm{T}$
(b) _AldS 4_ case.

Figure 4.3: Geometry of Al(A)dS4 boundaries with background structure.

As a boundary Lagrangian for the gravitational field, we assume that

$\bm{L}_{\circ}=L_{\circ}[q_{AB},\bm{T},\mathring{q}]\,(\text{d}^{2}x)$
(4.3.13)

is only built from the boundary background structure $\bm{T}$,
$\sqrt{\mathring{q}}$ and the metric on the transverse space $q_{AB}$. Using
these new assumptions and the substitution
$\sqrt{\mathring{q}}(\text{d}^{3}x)=\text{d}u\,\text{d}^{2}\Omega$ we have,
equivalently,

$\delta
S_{ren}^{\Lambda\text{-BMS}_{4}}[\phi]=\int_{\mathscr{I}}\bm{\Theta}^{\Lambda\text{-BMS}_{4}}_{ren}[\phi;\delta\phi]=\int_{\mathscr{I}}\text{d}u\,\text{d}^{2}\Omega\,\left(-\frac{\Lambda}{32\pi
G}J_{AB}+\frac{1}{\sqrt{\mathring{q}}}\frac{\delta L_{\circ}}{\delta
q^{AB}}\right)\delta q^{AB}.$ (4.3.14)

The flat limit $\Lambda\to 0$ is obtained by massaging the first term of
(4.3.14) in a way analogous to the procedure already described in section
4.3.2. Thanks to (4.2.37), we develop $J_{AB}$ in terms of $C_{AB}$, $q_{AB}$,
$N_{AB}^{TF}$, $M$ and $N_{A}$ where $N_{AB}^{TF}$ is the trace-free part of
the Bondi news tensor $N_{AB}$. In particular, all derivatives of the boundary
metric $\partial_{u}q_{AB}$ need to be expressed in terms of $C_{AB}$ using
(4.3.1). The computation requires a bit of dexterity while dealing with
variations of the fields $q_{AB}$, $C_{AB}$, $N_{AB}^{TF}$…It will not be
reproduced here but the reader interested in verifying the algebraic
development would find a list of useful relations for that task in appendix E.
Using the form notation
$\bm{L}_{\circ}=\text{d}u\frac{\text{d}^{2}\Omega}{\sqrt{\mathring{q}}}L_{\circ}$,
we obtain

$\displaystyle\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]\Big{|}_{\mathscr{I}}$
$\displaystyle=\frac{\text{d}u\,\text{d}^{2}\Omega}{16\pi
G}\left[\frac{3}{2\Lambda}\partial_{u}N_{AB}^{TF}-D_{(A}\mathcal{U}_{B)}-\frac{1}{4}R[q]C_{AB}\right]\delta
q^{AB}$ $\displaystyle\quad+\frac{\delta\bm{L}_{\circ}}{\delta q^{AB}}\delta
q^{AB}+\mathcal{O}(\Lambda),$ (4.3.15)

where $\mathcal{O}(\Lambda)$ denotes terms that vanish in the $\Lambda\to 0$
limit. We observe that there is a pole $\sim\Lambda^{-1}$ and the flat limit
is not well-defined. However, if we write (4.3.15) in the equivalent form

$\displaystyle\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]\Big{|}_{\mathscr{I}}$
$\displaystyle=\frac{\text{d}u\,\text{d}^{2}\Omega}{16\pi
G}\left[\frac{3}{2\Lambda}\partial_{u}(N_{AB}^{TF}\delta
q^{AB})+\frac{1}{2}\Big{(}N^{AB}_{TF}+\frac{1}{2}R[q]q^{AB}\Big{)}\delta
C_{AB}-D_{(A}\mathcal{U}_{B)}\delta q^{AB}\right]$
$\displaystyle\quad+\frac{\delta\bm{L}_{\circ}}{\delta q^{AB}}\delta
q^{AB}+\mathcal{O}(\Lambda),$ (4.3.16)

it becomes clear that the divergence in the variation of the action is a
corner term defined at the boundaries $\partial\\!\mathscr{I}_{+}$,
$\partial\\!\mathscr{I}_{-}$ of the conformal boundary $\mathscr{I}$. In order
to make these corner terms explicit, we consider the total derivative boundary
Lagrangian

$L_{\circ}=\partial_{u}L_{C}[q_{AB},\bm{T},\mathring{q}],$ (4.3.17)

where the corner Lagrangian is

$L_{C}[q_{AB},\bm{T},\mathring{q}]\equiv\frac{\sqrt{\mathring{q}}}{64\pi
G}C^{AB}C_{AB}=\frac{\sqrt{\mathring{q}}\ell^{4}}{64\pi
G}\partial_{u}q_{AB}q^{AC}q^{BD}\partial_{u}q_{CD}.$ (4.3.18)

We claim that the Lagrangian $L_{\circ}$ is of the form considered in section
3.2.1. It is covariant on $\mathscr{I}$ since we can rewrite it as

$\boxed{L_{\circ}=\frac{\sqrt{\mathring{q}}\ell}{64\pi
G}\mathcal{L}_{\bm{T}}\left[\mathcal{L}_{\bm{T}}g^{(0)}_{ab}P^{ac}P^{bd}\mathcal{L}_{\bm{T}}g^{(0)}_{cd}\right].}$
(4.3.19)

Here $P_{ab}=g_{ab}^{(0)}+\eta T_{a}T_{b}$ is the projector onto surfaces
orthogonal to $\bm{T}$ and (4.3.19) is indeed a tensorial expression on the
conformal boundary. Since $\bm{L}_{\circ}=\text{d}\bm{L}_{C}$ where
$\bm{L}_{C}=\frac{\text{d}^{2}\Omega}{\sqrt{\mathring{q}}}L_{C}\equiv\text{d}^{2}xL_{C}$,
its stress-energy tensor (3.2.27) vanishes, $T_{\circ}^{ab}=0$, which implies
that the last term of (4.3.16) vanishes. In particular, it obeys trivially our
hypotheses (3.2.28). However, this peculiarity indicates that boundary
Lagrangians like $\bm{L}_{\circ}=\text{d}\bm{L}_{C}$ have no contributions in
(3.2.33), hence this prescription fails to be helpful for corner Lagrangians.

Let us see how to overcome this difficulty. In the action (3.2.1), the term
associated with the boundary Lagrangian (4.3.17) can be written as

$\int_{\mathscr{I}}\text{d}^{2}x\,\text{d}u\,L_{\circ}=\int_{\partial\\!\mathscr{I}_{+}}\text{d}^{2}x\,L_{C}-\int_{\partial\\!\mathscr{I}_{-}}\text{d}^{2}x\,L_{C}.$
(4.3.20)

It is manifestly the difference of two corner terms at
$\partial\\!\mathscr{I}_{+}$ and $\partial\\!\mathscr{I}_{-}$. The variation
of the Lagrangian $L_{\circ}$ gives

$\displaystyle\delta L_{\circ}$ $\displaystyle=\partial_{u}\left(\frac{\delta
L_{C}}{\delta q_{AB}}\delta q_{AB}\right)+\partial_{u}\Theta^{C}_{\circ},$
(4.3.21) $\displaystyle\frac{\delta L_{C}}{\delta q_{AB}}$
$\displaystyle=-\frac{3\sqrt{\mathring{q}}}{32\pi
G\Lambda}N_{TF}^{AB},\qquad\Theta^{C}_{\circ}=\frac{3\sqrt{\mathring{q}}}{32\pi
G\Lambda}\partial_{u}\left(C^{AB}\delta q_{AB}\right).$ (4.3.22)

To obtain this result, we used (4.3.1) and $\delta\sqrt{\mathring{q}}=0$ (see
also the useful relations of Appendix E). With these definitions, we can now
rewrite the divergent term in (4.3.16) as

$\frac{3\sqrt{\mathring{q}}}{32\pi G\Lambda}\partial_{u}(N_{AB}^{TF}\delta
q^{AB})=\partial_{u}\left(\frac{\delta L_{C}}{\delta q_{AB}}\delta
q_{AB}\right)=\delta L_{\circ}-\partial_{u}\Theta^{C}_{\circ}.$ (4.3.23)

Note that despite the notation, $\Theta^{C}_{\circ}$ appearing in (4.3.22) is
not the presymplectic potential associated with $L_{\circ}$, the latter being
not defined since the sphere has no boundary. This is a pure corner term
living on each section of $\mathscr{I}$, in particular on
$\partial\\!\mathscr{I}_{-}$ and $\partial\\!\mathscr{I}_{+}$. From the datum
of $\Theta^{C}_{\circ}$, we build the _corner presymplectic potential_ and
_corner presymplectic form_ as

$\bm{\Theta}^{C}_{\circ}=\text{d}^{2}x\,\Theta^{C}_{\circ},\qquad\bm{\omega}_{\circ}^{C}(\delta_{1}q_{AB},\delta_{2}q_{AB};q_{AB})=\delta_{1}\bm{\Theta}^{C}_{\circ}(\delta_{2}q_{AB};q_{AB})-(1\leftrightarrow
2).$ (4.3.24)

We emphasize that they are defined on each sphere of the boundary
$\mathscr{I}$, not only at $\partial\\!\mathscr{I}_{\pm}$. The presymplectic
potential $\bm{\Theta}_{\circ}$ of $L_{\circ}$ defined in (3.2.29) is given by
the sum $\frac{\delta\bm{L}_{C}}{\delta q_{AB}}\delta
q_{AB}+\bm{\Theta}^{C}_{\circ}$. We observe that
$-\delta\bm{L}_{\circ}+\text{d}\bm{\Theta}_{\circ}=0$ hence there is no
contribution brought to the prescription (3.2.33) for renormalizing the
presymplectic potential. This is consistent with the discussion below
(4.3.19). Inspired by the presence of the corner terms in (4.3.23), we supply
this prescription with

$\boxed{\bm{\Theta}_{ren,tot}^{\Lambda\text{-BMS}_{4}}\Big{|}_{\mathscr{I}}=\bm{\Theta}_{ren}^{\Lambda\text{-BMS}_{4}}\Big{|}_{\mathscr{I}}-\delta\text{d}\bm{L}_{C}+\text{d}\bm{\Theta}^{C}_{\circ}.}$
(4.3.25)

As already discussed in section 3.2.2.2, such a prescription fixes the two
standard ambiguities that arise in the covariant phase space formalism, here
for codimension 2 objects on sections of $\mathscr{I}$. Taking a variation on
(4.3.25), we obtain the associated presymplectic form as

$\bm{\omega}_{ren,tot}^{\Lambda\text{-BMS}_{4}}\Big{|}_{\mathscr{I}}=\bm{\omega}_{ren}^{\Lambda\text{-BMS}_{4}}\Big{|}_{\mathscr{I}}+\text{d}\bm{\omega}^{C}_{\circ},$
(4.3.26)

where $\bm{\omega}_{ren}^{\Lambda\text{-BMS}_{4}}$ is given by (3.2.36) with
the $\Lambda$-BMS4 gauge fixing and its pull-back to $\mathscr{I}$ is the
leading order of (4.2.61). Our construction, in particular (4.3.23), ensures
that the inclusion of the corner terms according to (4.3.25) in the
renormalization process of the presymplectic potential are sufficient to
remove the pole in $\Lambda^{-1}$ in (4.3.16): as a result, the asymptotically
flat limit $\Lambda\to 0$ can now be taken safely in (4.3.26), which provides
the most important quantity in the definition of the covariant phase space.

##### 4.3.3.2 Corner contributions to the charge algebra

Before taking the flat limit, let us study how the incorporation of these
corner terms impacts the $\Lambda$-BMS4 charge algebra (4.2.91). The
codimension $2$ form
$\bm{k}_{\xi,(tot)}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]$ including the
contribution of the corner presymplectic form verifies

$\text{d}\bm{k}_{\xi,(tot)}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]\Big{|}_{\mathscr{I}}=\bm{\omega}_{ren,tot}^{\Lambda\text{-BMS}_{4}}[\phi;\delta_{\xi}\phi,\delta\phi]\Big{|}_{\mathscr{I}},$
(4.3.27)

from which we deduce

$\delta/H_{\xi,\text{tot}}^{\Lambda\text{-BMS}_{4}}[\phi]=\delta
H_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi]+\Xi_{\xi}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]+\int_{S^{2}_{\infty}}\text{d}^{2}\Omega\,\,\omega_{\circ}^{C}[\delta_{\xi}q_{AB},\delta
q_{AB}].$ (4.3.28)

because of (4.3.26). We dropped the dependence on $q_{AB}$ in
$\omega_{\circ}^{C}$ because it involves only its variation $\delta q_{AB}$,
see (4.3.22) and (4.3.24). Since the corner presymplectic structure term is
not integrable, we keep our definition (4.2.89) of the Hamiltonian and add
this term to the non-integrable term (4.2.90). Again, the computation of the
charge algebra under the Barnich-Troessaert bracket gives

$\begin{split}&\left\\{H_{\xi_{1}}^{\Lambda\text{-BMS}_{4}}[\phi],H_{\xi_{2}}^{\Lambda\text{-BMS}_{4}}[\phi]\right\\}_{\star,(tot)}\\\
&\qquad=\delta_{\xi_{2}}H_{\xi_{1}}^{\Lambda\text{-BMS}_{4}}[\phi]+\Xi_{\xi_{2}}^{\Lambda\text{-BMS}_{4}}[\delta_{\xi_{1}}\phi;\phi]+\int_{S^{2}_{\infty}}\text{d}^{2}\Omega\,\omega_{\circ}^{C}[\delta_{\xi_{2}}q_{AB},\delta_{\xi_{1}}q_{AB}]\\\
&\qquad\equiv
H^{\Lambda\text{-BMS}_{4}}_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+K^{\Lambda\text{-BMS}_{4},(tot)}_{\xi_{1},\xi_{2}}[q_{AB}]\end{split}$
(4.3.29)

as a corollary of (4.2.91). The additional term

$K^{\Lambda\text{-BMS}_{4},(tot)}_{\xi_{1},\xi_{2}}[q_{AB}]=-\int_{S^{2}_{\infty}}\text{d}^{2}\Omega\,\omega_{\circ}^{C}[\delta_{\xi_{1}}q_{AB},\delta_{\xi_{2}}q_{AB}]$
(4.3.30)

is obviously antisymmetric under the exchange $\xi_{1}\leftrightarrow\xi_{2}$
and satisfies the cocycle condition (2.3.15). Hence, the corner terms
naturally give rise to a field-dependent 2-cocycle in the right-hand side of
the $\Lambda$-BMS4 charge algebra.

##### 4.3.3.3 Symplectic structure at flat limit and discussion

Finally, let us consider the flat limit of the symplectic structure. Taking
$\Lambda\to 0$, the equations (4.3.25) and (4.3.26) give respectively

$\lim_{\Lambda\to
0}\bm{\Theta}_{ren,tot}^{\Lambda\text{-BMS}_{4}}[\phi;\delta\phi]\Big{|}_{\mathscr{I}}=\frac{\text{d}u\,\text{d}^{2}\Omega}{16\pi
G}\left[\frac{1}{2}\Big{(}N^{AB}_{TF}+\frac{1}{2}R[q]q^{AB}\Big{)}\delta
C_{AB}-D_{(A}\mathcal{U}_{B)}\delta q^{AB}\right]$ (4.3.31)

and more importantly

$\boxed{\begin{aligned} &\lim_{\Lambda\to
0}\bm{\omega}_{ren,tot}^{\Lambda\text{-BMS}_{4}}[\phi;\delta_{1}\phi,\delta_{2}\phi]\Big{|}_{\mathscr{I}}\\\
&\qquad=\frac{\text{d}u\,\text{d}^{2}\Omega}{16\pi
G}\left[\frac{1}{2}\delta_{1}\left(N^{AB}+\frac{1}{2}R[q]q^{AB}\right)\wedge\delta_{2}C_{AB}-\delta_{1}\left(D_{(A}\mathcal{U}_{B)}\right)\wedge\delta_{2}q^{AB}\right].\end{aligned}}$
(4.3.32)

The presymplectic current (4.3.32) is the Generalized BMS4 presymplectic
structure after pullback on the future null infinity
$\mathscr{I}=\mathscr{I}^{+}$ in asymptotically locally flat spacetimes. This
expression exactly matches with the presymplectic structure obtained by
regularization methods for asymptotically locally flat spacetimes as an
example of covariant phase space methods in section 2.2.4. More precisely,
(4.3.32) and (2.2.124) are stricly identical, which is a highly non-trivial
agreement! Recall that the regularization procedure used in sections 2.2.4.1
and 2.2.4.2 [45] was not explicitly covariant [60]: the corner counterterms
(2.2.116)-(2.2.117) were indeed given in a coordinate dependent way ($r$ is
clearly apparent) and we did not give a proof that they were part of a
covariant Iyer-Wald counterterm $\bm{Y}$ as we introduced to renormalize the
presymplectic potential. But since the procedure used here is explicitly
covariant (in terms of the boundary structure for $\Lambda\neq 0$) and the
flat limit process is consistent either at the level of the solution space,
the symmetries or the phase space, this therefore justifies a posteriori the
counterterm prescription for subtracting a radial divergence used in [45] and
reviewed in chapter 2. We find curious that the radially diverging term
encountered in the flat case, namely (2.2.116), is structurally similar to the
$\Lambda\to 0$ diverging term found here (4.3.22). Here, using the
prescription (4.3.25), we are able to track the origin of this term coming
from a corner Lagrangian (4.3.18). The latter is a kinetic action for the
boundary metric of Al(A)dS4 spacetimes. Let us conclude by making a few
comments.

  * $\rhd$

Even though the flat limit of the action only requires to consider the corner
action $\int_{\partial\mathscr{I}_{+}}L_{C}$ $-$
$\int_{\partial\mathscr{I}_{-}}L_{C}$ at $u=u^{\pm}$, the total presymplectic
potential (4.3.25) and current (4.3.26) are defined from the corner action at
any $u$ along $\mathscr{I}$, which also shifts the surface charges at any $u$.
In particular, this brings a non-zero field-dependent 2-cocycle (4.3.30) in
the surface charge algebra. The link between this modified $\Lambda$-BMS4
charge algebra (4.3.29) that takes the presence of corner terms into account
and the Generalized BMS4 charge algebra discussed in section 2.3.3 is not
direct. Indeed, due to the subleading field-dependence of the diffeomorphism
between SFG and Bondi gauges, the asymptotic Killing vectors do not transform
as simple vectors (see e.g. Eq. (70) of [185]). This implies that the surface
charge codimension 2 form transforms non-trivially, which leads to a shift of
the objects appearing in the charge algebra that is hard to track. For
example, the 2-cocycle in (4.3.29) does not admit a well-defined flat limit
and is therefore not directly related to the 2-cocycle (2.3.24) obtained in
the flat case. It is simpler to take the flat limit at the level of the
symplectic structure that determines all dynamical quantities rather than at
the level of the charge algebra. Since the flat limit of the renormalized
$\Lambda$-BMS4 presymplectic structure is the Generalized BMS4 presymplectic
structure, this proves indirectly that the $\Lambda$-BMS4 charges (and charge
algebra) are flattened into the Generalized BMS4 charges (and charge algebra)
even though we do not present the intricate computation making the link
explicit. But since all dynamical quantities are derived from the
presymplectic structure, understanding the flat limit at the level of it is
necessary and sufficient to consider the flat limit of the phase space.

  * $\rhd$

The next observation has already been mentioned but will be rephrased here for
summarizing perhaps the most important result of this section. The Compère-
Marolf prescription (3.2.33) defined in [85] fixes the usual Iyer-Wald
ambiguities in the definition of the presymplectic potential [77, 79, 78]
using as an input the boundary counterterms defined at $\mathscr{I}$ (see also
[166]). Such a prescription fails to give a renormalized symplectic structure
in the flat limit because there subsists a pole in $\Lambda^{-1}$ in the
presymplectic potential which can only be subtracted by a corner counterterm
living on codimension 2 sections of $\mathscr{I}$. We argued that the
existence of a corner Lagrangian defined for each sphere on $\mathscr{I}$
naturally leads to the additional prescription (4.3.25) which gives a well-
defined symplectic structure in the flat limit.

  * $\rhd$

Finally, we already observed e.g. in section 4.2.4.1 that we need the
$\Lambda$-BMS4 algebroid to recover the (Generalized) BMS4 group at flat
limit. These symmetries are related to leaky boundary conditions putting some
non-vanishing symplectic flux through the conformal boundary $\mathscr{I}$,
while conservative boundary conditions with non-trivial asymptotic group such
as Dirichlet boundary conditions $\delta q_{AB}=0$ only give a finite-
dimensional asymptotic group in the flat limit, the exact Poincaré group (see
(4.1.11)). Here we have seen that the leaks through $\mathscr{I}$ allowed by
the $\Lambda$-BMS4 boundary conditions are essential to recover the
presymplectic form (4.3.32) which must be non-zero even when considering
strict Minkowskian asymptotics with $\delta q_{AB}=0$. The particular
structure of the conformal boundary of (A)dS4 implies that one cannot extend
the finite symmetry groups $SO(3,2)$ or $SO(4,1)$ when $\Lambda\neq 0$ with
some avatar of “supertranslations” without gaining “superrotations” for free,
which is the take-away message from the solution of (4.2.64)-(4.2.65). As a
result, the flat limit of the $\Lambda$-BMS4 phase space does naturally
include the super-Lorentz transformations introduced in section 2.1.5.3 in a
natural way beside the traditional supertranslations. This is another striking
evidence that “superrotations” are as legitimate asymptotic symmetries than
the supertranslations in the flat case and can be interestingly included in
the asymptotically flat phase space!

This last observation puts an end to our fruitful journey at the boundary of
(A)dS spacetimes in Einstein’s gravity. Besides the results intrinsically
interesting for radiative solutions with cosmological constant, we have
acquired a fundamental and highly non-trivial theoretical confirmation of our
heuristic renormalization in the asymptotically locally flat phase space about
which it remains a lot to be said. The next chapter is thus an invitation to
go back to the flat land to pursue the analysis started in the chapter 2 as a
motivating example for the covariant phase space formalism. $\blacksquare$

##  Chapter 5 Super-Lorentz transformations and Gravitational Memories

After our captivating peregrinations in Al(A)dS spacetimes concluded by a
controlled landing in the form of a flat limit process, Chapter 5 recovers the
well-known asymptotically flat landscapes about which several points require
clarification. Section 5.1 derives a closed-form expression of the orbit of
Minkowski spacetime under Generalized BMS4 transformations, i.e. arbitrary
Diff$(S^{2})$ super-Lorentz transformations and smooth supertranslations. The
resulting metrics are labelled by the superboost, superrotation and
supertranslation fields: they determine the vacuum structure of asymptotically
locally flat spacetimes. Section 5.2 then discusses the associated
gravitational memory effects and details physical processes that can lead to
transitions among the above-mentioned vacua. Impulsive vacuum transitions
driven by overleading cosmic events are related to the refraction memory
effect and the usual displacement memory effect, which also receives a
detailed review in this chapter. Finally, in section 5.3, we conclude the
discussion about the radiative phase spaces admitting the whole Generalized
BMS4 group as the asymptotic symmetry group that we initiated in section 2.2.4
by providing a set of meaningful physical prescriptions to isolate finite
Hamiltonian canonically conjugated to these symmetries. We show that our final
surface charge expressions are consistent with the leading and subleading soft
graviton theorems and we contrast the leading infrared triangle structure to a
new mixed overleading/subleading square structure.

This chapter mixes the parts of [45] that were not presented in chapter 2 with
refined results given in [37] concerning the Generalized BMS4 charge algebra
as well as the suitable choice for the finite Hamiltonians.

###  5.1 Vacua of the gravitational field

Let $g_{\mu\nu}$ be a state of the gravitational field that belongs to the
solution space (2.1.40) in the Bondi gauge. It is said to be a vacuum state if
it is diffeomorphic to the Minkowski spacetime

$\text{d}s^{2}=-\text{d}u^{2}+2\text{d}u\text{d}r+r^{2}\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}$
(5.1.1)

that, written in retarded coordinates, is trivially part of (2.1.40). This
metric is Riemann-flat $R_{\mu\nu\alpha\beta}=0$ by construction and the Weyl
tensor is also vanishing, $W_{\mu\nu\alpha\beta}=0$, which is expected from a
solution without gravitation. The statement of vanishing Weyl is a tensorial
proposition that is untouched by the action of the allowed residual gauge
diffeomorphisms preserving (2.1.40), which constitutes the orbit of the
Generalized BMS4 group for which (2.1.27) is obeyed with
$\sqrt{\bar{q}}=\sqrt{\mathring{q}}$. This is the only orbit that contains
(5.1.1) as particular solution for the diffeomorphism generators $f=0$ and
$Y^{A}=0$. The whole Generalized BMS4 orbit of vacua is obtained in practice
by exponentiating a general BMS4 transformation (generated by vectors (2.1.12)
parametrized by $f$, $Y^{A}$, $\omega=0$, $\sqrt{q}=\sqrt{\mathring{q}}$)
acting on the Minkowski spacetime (5.1.1). The subset of this orbit for which
only supertranslations are turned on contains the non-equivalent vacua of
asymptotically flat spacetimes, already covered in section 2.1.5.3: they are
labeled by a single fundamental field labeling inequivalent vacua: the
supertranslation field $C(x^{A})$ defined in (2.1.63)-(2.1.64). Here we derive
the orbit of Minkowski under supertranslations as well as super-Lorentz
transformations that we exponentiate following the method developed in [99].
This leads to the appearance of new boundary fields beside of $C$ that are
coined as the super-Lorentz fields.

#### 5.1.1 Generation of the vacua

The exponentiation leading to finite Generalized BMS4 transformations
simplifies considerably if we start from the complex plane coordinates
$(u_{C},r_{C},z_{C},\bar{z}_{C})$ in which the flat metric reads as

$\text{d}s^{2}=-2\text{d}u_{C}\text{d}r_{C}+2r_{C}^{2}\text{d}z_{C}\text{d}\bar{z}_{C}.$
(5.1.2)

One obtains this line element from (5.1.1) by the following change of
coordinates [99]

$\begin{split}r_{C}&=\frac{\sqrt{2}r}{1+z\bar{z}}+\frac{1}{\sqrt{2}}u\,,\quad
u_{C}=\frac{1+z\bar{z}}{\sqrt{2}}u-\frac{z\bar{z}}{2r_{C}}u^{2}\,,\quad
z_{C}=z-\frac{zu}{\sqrt{2}r_{C}}u\,,\quad\bar{z}_{C}=z_{C}^{*},\end{split}$
(5.1.3)

where $(z,\bar{z})$ are the stereographic coordinates on the celestial sphere.
We define the background structures

$\gamma_{\underline{A}\underline{B}}=\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right],\quad\epsilon_{\underline{A}\underline{B}}=\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right]$ (5.1.4)

with inverse
$\gamma^{\underline{A}\underline{B}}=\gamma_{\underline{A}\underline{B}}$,
$\epsilon^{\underline{A}\underline{B}}=\epsilon_{\underline{A}\underline{B}}$.
In the following, all underlined indices are related to the complex plane
metric $\gamma_{\underline{A}\underline{B}}$ on the codimension 2 surfaces of
constant $u_{C},r_{C}$. The goal is to introduce a diffeomorphism to Bondi
gauge $(u_{C},r_{C},z_{C},\bar{z}_{C})\to(u,r,z,\bar{z})$ that exponentiates
$\text{Diff}(S^{2})$ super-Lorentz transformations and supertranslations.
Requiring that $(u,r,z,\bar{z})$ are Bondi coordinates leads to the conditions
(2.1.7). The algebraic conditions $g_{rr}=0$, $g_{rA}=0$ are solved if

$\displaystyle r_{C}$ $\displaystyle=r_{C}(r,u,z,\bar{z}),$ (5.1.5)
$\displaystyle u_{C}$
$\displaystyle=W(u,z,\bar{z})-\frac{1}{r_{C}}\gamma_{\underline{A}\underline{B}}H^{\underline{A}}(u,z,\bar{z})H^{\underline{B}}(u,z,\bar{z}),$
(5.1.6) $\displaystyle z_{C}^{\underline{A}}$
$\displaystyle=G^{\underline{A}}(z,\bar{z})-\frac{1}{r_{C}}H^{\underline{A}}(u,z,\bar{z}),\quad
H^{\underline{A}}(u,z,\bar{z})=-\frac{1}{D_{G}}\epsilon^{\underline{A}\underline{B}}\gamma_{\underline{B}\underline{C}}\epsilon^{AB}\partial_{A}W\partial_{B}G^{\underline{C}}$
(5.1.7)

where
$D_{G}=\det(\partial_{A}G^{\underline{B}})=\frac{1}{2!}\epsilon_{\underline{A}\underline{B}}\epsilon^{AB}\partial_{A}G^{\underline{A}}\partial_{B}G^{\underline{B}}$.
The determinant condition $\partial_{r}(r^{-4}\det g_{AB})=0$ fixes the
functional dependence of $r_{C}$ as [99]

$\displaystyle r_{C}(r,u,z,\bar{z})$
$\displaystyle=R_{0}(u,z,\bar{z})+\sqrt{\frac{r^{2}}{(\partial_{u}W)^{2}}+R_{1}(u,z,\bar{z})}.$
(5.1.8)

The function $r_{C}$ admits a well-defined series in inverse powers of $r$:

$r_{C}(r,u,z^{c})=\frac{r}{(\partial_{u}W)^{2}}+R_{0}(u,z,\bar{z})+\frac{1}{2}(\partial_{u}W)^{2}R_{1}(u,z,\bar{z})\frac{1}{r}+\mathcal{O}(r^{-3}).$
(5.1.9)

The expressions of the unknown codimension 1 functions $R_{0}$ and $R_{1}$ are
respectively obtained by requiring that the determinant condition is obeyed up
to second and third order in $1/r$. The information is propagated at
subleading orders with the power expansion (5.1.9) of the functional
dependence in $r$.

Acting with the change of coordinates (5.1.5)-(5.1.7) on (5.1.2) gives the
Minkowski metric in the Bondi gauge up to a finite Generalized BMS4
transformation. Expanding the resulting $g_{AB}$ in powers of $r$ as in
(2.1.28), we can read the boundary metric $q_{AB}$ in the $\mathcal{O}(r^{2})$
contribution. The boundary conditions (2.1.25) and (2.1.39) leading to the
Generalized BMS4 solution space (2.1.40) require that $\partial_{u}q_{AB}=0$
and $g_{uu}$ is finite in $r$ by virtue of (2.1.42) and (2.1.43). This yields
$\partial^{2}_{u}W=0$, so $W$ is at most linear in $u$. Moreover, regularity
of the finite diffeomorphism (5.1.8) implies that $\partial_{u}W$ is nowhere
vanishing. Therefore,

$W(u,z^{c})=\exp\left[\frac{1}{2}\Phi(z,\bar{z})\right](u+C(z,\bar{z})).$
(5.1.10)

and the boundary metric reads as

$q_{AB}=q_{AB}^{vac}\equiv
e^{-\Phi}\partial_{A}G^{\underline{A}}\partial_{B}G^{\underline{B}}\gamma_{\underline{A}\underline{B}}.$
(5.1.11)

It is indeed the result of a large diffeomorphism and a Weyl transformation.
The latter is obviously constrained since $\det q_{AB}=\mathring{q}$ or
$\Phi=\ln D_{G}-\ln\sqrt{\mathring{q}}$ using (5.1.11). The asymptotic shear
$C_{AB}$ appearing at $\mathcal{O}(r)$ in $g_{AB}$ is found to be the trace-
free part ($TF$) of the following tensor

$C_{AB}=C_{AB}^{vac}\equiv\left[\frac{2}{(\partial_{u}W)^{2}}\partial_{u}\left(D_{A}WD_{B}W\right)-\frac{2}{\partial_{u}W}D_{A}D_{B}W\right]^{TF}.$
(5.1.12)

Introducing (5.1.10), it comes

$C^{vac}_{AB}[\Phi,C]=(u+C)N^{vac}_{AB}+C^{(0)}_{AB},\quad\left\\{\begin{array}[]{ccl}N^{vac}_{AB}&=&\left[\frac{1}{2}D_{A}\Phi
D_{B}\Phi-D_{A}D_{B}\Phi\right]^{TF}\\\
C^{(0)}_{AB}&=&-2D_{A}D_{B}C+q_{AB}D^{2}C.\end{array}\right.$ (5.1.13)

We find that all explicit reference on $\gamma_{\underline{A}\underline{B}}$
or $G^{\underline{A}}$ disappeared. The tensor $C_{AB}^{(0)}$ is the standard
asymptotic shear for the BMS4 vacua. Moreover, the news tensor of the vacua
$N^{vac}_{AB}$ is only built up with $\Phi$. It can be checked that the
boundary Ricci scalar is given in terms of $\Phi$ as

$R[q]=D^{2}\Phi,$ (5.1.14)

which implies

$D_{A}N_{vac}^{AB}=-\frac{1}{2}D^{B}R[q].$ (5.1.15)

We can therefore add a trace to $N^{vac}_{AB}$ to form the conserved stress-
tensor

$T_{AB}[\Phi]=\frac{1}{2}D_{A}\Phi D_{B}\Phi-
D_{A}D_{B}\Phi+\frac{1}{2}q_{AB}\left(2D^{2}\Phi-\frac{1}{2}D^{C}\Phi
D_{C}\Phi\right).$ (5.1.16)

Its trace is equal to $D^{2}\Phi$. The tensor $T_{AB}$ is precisely the
stress-tensor of Euclidean Liouville theory [305, 306, 307, 308, 309]

$L[\Phi;q_{AB}]=\sqrt{q}\left(\frac{1}{2}D^{A}\Phi D_{A}\Phi+\alpha
e^{\Phi}+R[q]\Phi\right)$ (5.1.17)

for the field $\Phi$, where the parameter $\alpha$ is zero in order to satisfy
(5.1.14). Note that in order to derive the stress-tensor (5.1.16) from the
Lagrangian (5.1.17), one needs to consider the Liouville field $\Phi$ off-
shell but fix the metric $q_{AB}$ as a background field. The equation of
motion for $\Phi$ derived from the variational principle (5.1.17) on a fixed
background field $q_{AB}$ is (5.1.14), coined as the Liouville equation. Under
a super-Lorentz transformation

$\delta_{Y}(D^{2}\Phi-R[q])=(\mathcal{L}_{Y}+D_{A}Y^{A})(D^{2}\Phi-R[q]).$
(5.1.18)

Therefore, imposing the Liouville equation is consistent with the action of
super-Lorentz transformations and we can demand that it holds for any metric
$q_{AB}$ obtained by super-Lorentz transformation from the sphere metric
$\mathring{q}_{AB}$. Note also that aside from its divergence, see (5.1.15),
the curl of $N_{AB}^{vac}$ is also fixed. Indeed, one can show that

$D_{[A}N^{vac}_{B]C}=-\frac{1}{2}D_{[A}R[q]\,q_{B]C}$ (5.1.19)

is equivalent to (5.1.14), see e.g. [194]. In [12], Geroch showed that there
exists a unique curl-free tensor $\rho_{AB}$ such that $q^{AB}\rho_{AB}=R[q]$.
The condition (5.1.19) implies that
$\rho_{AB}=\frac{1}{2}R[q]q_{AB}+N_{AB}^{vac}$, hence $N_{AB}^{vac}$ is
uniquely defined when the super-Lorentz frame $q_{AB}$ is fixed. In
particular, when $q_{AB}=\mathring{q}_{AB}$, $N_{AB}^{vac}$ is identically
vanishing. A second thing that has to be mentioned about $N_{AB}^{vac}$ and
particularly about $\Phi$, is that the latter can always be modified by a
conformal isometry $Y^{A}$ of $q_{AB}$ without changing the vacuum news
tensor. Indeed, using $\mathcal{L}_{Y}q_{AB}=D_{C}Y^{C}q_{AB}$, it can be
shown that $\Phi$ and $\Phi+D_{A}Y^{A}$ lead to the same $N_{AB}^{vac}$
because the differential operator in (5.1.13) is blind to such a shift in
$\Phi$. This ambiguity is irrelevant once the reference metric
$\gamma_{\underline{A}\underline{B}}$ is fixed in (5.1.11). Moreover, $\Phi$
never appears alone in the following but only through the differential
operator defining $N_{AB}^{vac}[\Phi]$. This expresses the fact that the
Minkowski vacuum (5.1.1) is preserved by the Lorentz transformations and
$N_{AB}^{vac}$ is only turned up by pure superboost transformations.

Using the boundary metric (5.1.11) and shear (5.1.12), one can work out the
covariant expressions for $R_{0}$ and $R_{1}$ in (5.1.8). They are given by

$R_{0}=\frac{1}{2}e^{-\Phi}D^{A}D_{A}W\qquad\text{and}\qquad
R_{1}=\frac{1}{8}e^{-\Phi}C_{AB}C^{AB}.$ (5.1.20)

Finally, after some algebra, one can write the full metric as

$\boxed{\text{d}s^{2}_{vac}=-\frac{R[q]}{2}\text{d}u^{2}-2\text{d}\rho\text{d}u+(\rho^{2}q_{AB}+\rho
C^{vac}_{AB}+\frac{1}{8}C_{vac}^{2}q_{AB})\text{d}x^{A}\text{d}x^{B}+D^{B}C^{vac}_{AB}\text{d}x^{A}\text{d}u}$
(5.1.21)

where $\rho=\sqrt{r^{2}+\frac{1}{8}C_{vac}^{2}}$, $C_{vac}^{2}\equiv
C_{AB}^{vac}C^{AB}_{vac}$, is a derived quantity in terms of the Bondi radius
$r$ which is nothing but the radial Newman-Unti coordinate (see section
2.1.1.2 for a definition). We observe that the metric of the vacua is thus
more natural in the Newman-Unti gauge $(u,\rho,z^{A})$ where
$g_{\rho\mu}=-\delta_{\mu}^{u}$.

Let us also comment on the meromorphic extension of the Lorentz group. When
super-Lorentz transformations reduce to local conformal Killing vectors on
$S^{2}$ i.e. $G^{z}=G(z)$ and $G^{\bar{z}}\equiv\bar{G}(\bar{z})$, the
boundary metric after a diffeomorphism is the unit round metric on the sphere

$\displaystyle\mathring{q}_{AB}\text{d}z^{A}\text{d}z^{B}=2\mathring{q}\text{d}z\text{d}\bar{z},\qquad\mathring{q}=\frac{2}{(1+z\bar{z})^{2}}$
(5.1.22)

(and $R[q]=\mathring{R}=2$) _except at the singular points of_ $G(z)$. The
Liouville field reduces to the sum of a meromorphic and an anti-meromorphic
part minus the unit sphere factor

$\Phi=\phi(z)+\bar{\phi}(\bar{z})-\ln\sqrt{\mathring{q}}.$ (5.1.23)

The metric (5.1.21) then exactly reproduces the expression of [99] with the
substitution
$T^{\text{\cite[cite]{[\@@bibref{}{Compere:2016jwb}{}{}]}}}_{AB}=\frac{1}{2}N^{vac}_{AB}$.
We have therefore found the generalization of the metric of the vacua for
arbitrary $\text{Diff}(S^{2})$ super-Lorentz transformations together with
arbitrary supertranslations.

#### 5.1.2 The superboost, superrotation and supertranslation fields

A general vacuum metric is parametrized by a boundary metric $q^{vac}_{AB}$,
the field $C$ that we call the _supertranslation field_ and $\Phi$ that we
will call either the _Liouville field_ or the _superboost field_. Under a BMS
transformation, the bulk metric transforms into itself, with the following
transformation law of its boundary fields,

$\displaystyle\delta_{T,Y}q^{vac}_{AB}$
$\displaystyle=D_{A}Y_{B}+D_{B}Y_{A}-q^{vac}_{AB}D_{C}Y^{C},$ (5.1.24)
$\displaystyle\delta_{T,Y}\Phi$
$\displaystyle=Y^{A}\partial_{A}\Phi+D_{A}Y^{A},$ (5.1.25)
$\displaystyle\delta_{T,Y}C$
$\displaystyle=T+Y^{A}\partial_{A}C-\frac{1}{2}CD_{A}Y^{A}.$ (5.1.26)

Only the divergence of a general super-Lorentz transformation sources the
Liouville field. Since rotations are divergence-free but boosts are not, we
call $\Phi$ the _superboost field_. In general, one can decompose a vector on
the 2-sphere as a divergence and a curl part. For a generic superrotation,
there should be a field that is sourced by the curl of $Y^{A}$. We call this
field the _superrotation field_ $\widetilde{\Phi}$ and we postulate its
transformation law

$\displaystyle\delta_{T,Y}\widetilde{\Phi}$ $\displaystyle=$ $\displaystyle
Y^{A}\partial_{A}\widetilde{\Phi}+\epsilon^{AB}D_{A}Y_{B}.$ (5.1.27)

Where is that field in (5.1.21)? In fact, the boundary metric $q^{vac}_{AB}$
is not a fundamental field. It depends upon the Liouville field $\Phi$ and the
background metric $\gamma_{\underline{A}\underline{B}}$. Since it transforms
under superrotations, the metric (5.1.11) should also depend upon the
superrotation field $\widetilde{\Phi}$. The explicit form
$q^{vac}_{AB}[\gamma_{\underline{A}\underline{B}},\Phi,\widetilde{\Phi}]$ is
not known to us and will not be needed in the following. We will call the set
of boundary fields $(\Phi,\widetilde{\Phi})$ the super-Lorentz fields.

Under a Generalized BMS4 transformation, the news of the vacua $N^{vac}_{AB}$
and the tensor $C^{(0)}_{AB}$ transform inhomogenously as

$\displaystyle\delta_{T,Y}N^{vac}_{AB}$
$\displaystyle=\mathcal{L}_{Y}N^{vac}_{AB}-D_{A}D_{B}D_{C}Y^{C}+\frac{1}{2}q_{AB}D_{C}D^{C}D_{D}Y^{D},$
(5.1.28) $\displaystyle\delta_{T,Y}C^{(0)}_{AB}$
$\displaystyle=\mathcal{L}_{Y}C^{(0)}_{AB}-\frac{1}{2}D_{C}Y^{C}C^{(0)}_{AB}-2D_{A}D_{B}T+q_{AB}D_{C}D^{C}T.$
(5.1.29)

From (5.1.21), one can read off the explicit expressions of the Bondi mass and
angular momentum aspects of the vacua

$\begin{split}M&=-\frac{1}{8}N^{vac}_{AB}C_{vac}^{AB},\\\
N_{A}&=-\frac{3}{32}D_{A}(C^{vac}_{BC}C_{vac}^{BC})-\frac{1}{4}C^{vac}_{AB}D_{C}C_{vac}^{BC}.\end{split}$
(5.1.30)

The Bondi mass is time-dependent and its spectrum is not bounded from below
because $\partial_{u}M=-\frac{1}{8}N_{AB}^{vac}N^{AB}_{vac}$ as observed in
[99]. Yet, the Weyl tensor is identically zero and the standard Newtonian
potential vanishes. This indicates that the physical mass should be
identically zero, or alternatively, that the Bondi mass aspect should be
refined in some sense, provided that $M$ is also diverging in $u$ at spatial
infinity $u\to-\infty$. For asymptotically Minkowskian spacetimes, the
integral of $M$ on the asymptotic sphere $u\to-\infty$ is identified with the
total (ADM) mass of the spacetime computed from spatial infinity. Keeping $M$
as the faithful mass aspect would yield an divergent value for the total mass
of spacetime even without bulk sources, which is not suitable. This discussion
is overlapping the question of the finiteness of the charges as well as the
behavior in $u$ of the various Bondi fields around the corners $u\to-\infty$
(spatial infinity) and $u\to+\infty$ (future timelike infinity) that we will
address in section 5.3.1.

###  5.2 Gravitational memory effects

Like any gauge theory, Einstein’s gravity possesses an infinite-dimensional
class of degenerate though unequivalent vacua [112]. They are represented by
the action of finite residual gauge diffeomorphisms on the global Minkowski
vacuum, in any kind of gauge fixing deleting the kinematical redundancies in
order to single out the only dynamic degrees of freedom in the gauge field
(here, the metric tensor). In this point of view, the vacua derived in the
previous section are simply stationary configurations without zero modes such
as mass, linear or angular momenta. For localized radiative phenomena which
occur only in a bounded range of (retarded) time $u_{i}<u<u_{f}$, one can
expect that the gravitational field, coming from some vacuum configuration for
$u<u_{i}$ relaxes back to a vacuum state for $u>u_{f}$ when the source for
radiation has been turned off (modulo some zero modes like mass or angular
momentum). For generic non-stationary phases, there is no reason for the early
and late vacua to be equivalent: the radiative process is responsible for a
vacuum transition. The existence of non-equivalent vacua related by
diffeomorphism indicates that the final vacuum is not arbitrary and should be
related by a residual gauge transformation to the original vacuum.
Incidentally, this gives one supplementary motivation to study infinite-
dimensional extensions of the natural asymptotic group which would be, in the
case of interest, the Poincaré group. The observation of the vacuum transition
between “early” and “late” times, i.e. $u<u_{i}$ and $u>u_{f}$, as a permanent
relic of the radiative phenomenon occured in the time interval
$u_{i}<u<u_{f}$, is possible thanks to (at least conceptually) simple
experimental devices that we describe below. This is called a gravitational
memory effect [97, 96, 193, 15, 16]. The concern of the present section is to
derive the memory effects associated with Generalized and Extended BMS4
symmetries and identify their possible sources in the evolution equations
governing the transitions.

#### 5.2.1 Axiomatic definition and relation to gauge symmetries

Before considering concrete examples, we want to give an axiomatic definition
of a gravitational memory effect that would include all of the particular
cases discussed later in this section and also other known effects in the
literature. This definition is taken from the inspirational essay [310].

Let us consider a class of asymptotically locally flat spacetimes for which
$\mathscr{I}^{+}$ exists, assorted with a class of observables
$O(\theta,\phi)$ defined on $\mathscr{I}^{+}$ and required to be
diffeomorphism-invariant. $O(\theta,\phi)$ arises from the integration on the
whole boundary $\mathscr{I}^{+}$ of some functional of gauge-invariant fields
living in the asymptotic region (a canonical example is the Weyl tensor for
pure gravity). The boundaries of $\mathscr{I}^{+}$ are $\mathscr{I}^{+}_{+}$
in the future (future timelike infinity) and $\mathscr{I}^{+}_{-}$ (spatial
infinity). One says that $O$ is a memory observable if and only if it can be
written as the difference between a local field $\varphi_{+}$ defined at
$\mathscr{I}^{+}_{+}$ and a local field $\varphi_{-}$ at
$\mathscr{I}^{+}_{-}$, i.e. $O=\varphi_{+}-\varphi_{-}$, in some coordinate
system. The fields $\varphi_{+}$ and $\varphi_{-}$ might not be defined
locally in any gauge but there must exists a particular gauge where this is
the case. In the (possibly unique) gauge $(x^{\mu})$ where these fields are
local functions of the boundary coordinates, they are coined as the memory
fields.

As mentioned in the introduction of the section, we are interested in
dynamical processes that occur in a finite time lapse. Hence, we restrict our
analysis to the class of spacetime manifolds relaxing back to some stationary
configurations at “early time” (i.e. around $\mathscr{I}^{+}_{-}$) and at
“late time” (i.e. around $\mathscr{I}^{+}_{+}$). These stationary
configurations are vacua possibly endowed with zero modes like constant mass
or angular momentum. At early and late times, the diffeomorphism-invariant
quantities are expected to be zero in the absence of radiation, but the memory
fields $\varphi_{\pm}$ can generically be non-zero. The gauge-invariant fields
(keep the example of the Weyl tensor in mind) decay to zero in the stationary
zones around $\mathscr{I}^{+}_{-}$ and $\mathscr{I}^{+}_{+}$. Therefore, there
must exist a residual gauge transformation $\xi$ to shift the memory fields
back to zero for early and late times. Let us give an illustrative example:
let us assume for instance that the spacetime asymptotes to Minkowski at late
times. Hence there exists at least one coordinate system around
$\mathscr{I}^{+}_{-}$ in which the memory field $\varphi_{+}$, absent by
design in Minkowski, is identically vanishing. So the stationary line element
at late time is diffeomorphic to Minkowski, and the corresponding
diffeomorphism verifies $\delta_{\xi}\varphi_{+}=0$. The same reasoning can be
applied in the presence of stationary zero modes such as constant mass or
angular momentum (more relevant at early time to allow i.e. for total (ADM)
charges).

Recalling now the gauge invariance of $O$, i.e. $\delta_{\xi}O=0$, we have
$\delta_{\xi}\varphi_{+}=\delta_{\xi}\varphi_{-}$ and $\xi$ shifts the memory
fields identically. This is not a surprise since residual gauge parameters
surviving simultaneously the gauge fixing and the imposition of some boundary
conditions are codimension 2 fields that depend only upon the angles and no
longer on time. For example, the solution space $\mathring{\mathcal{S}}_{0}$
of Einstein’s gravity in the Bondi gauge with asymptotically locally boundary
conditions admits the Generalized BMS4 transformations as residual
diffeomorphisms parametrized by $T(\theta,\phi)$ and $Y^{A}(\theta,\phi)$. A
Generalied BMS4 transformation thus acts globally on $\mathscr{I}^{+}$,
regardless of the local value of the retarded time $u$. More generally, once
the gauge has been fixed, we have the following result:

The set of memory fields is in one-to-one correspondence with the codimension
2 parameters of the set of residual gauge diffeomorphisms.