In the case under interest, we expect to observe a memory effect associated
with supertranslations, encoded in the supertranslation field $C$ and a memory
effect associated with super-Lorentz transformations, encoded in the super-
Lorentz fields. We shall start by illustrating the axiomatic definition we
have just reviewed for the displacement memory effect associated with the
supertranslation symmetry, before focusing on the superboost field for which
we have derived the explicit dependency in the gravitational field.

#### 5.2.2 Linear displacement memory effect

As a warm-up, let us briefly review the first historical prediction of a
gravitational memory effect that is the linear displacement memory effect
[193, 96, 97, 15, 16] related to the supertranslation symmetry. In this
subsection, we focus on the subspace
$\mathring{\mathcal{S}}_{0}^{\text{Mink}}$ of asymptotically Minkowskian
solutions and the assorted global BMS4 symmetry group.

Radiation$i^{0}$$\mathscr{I}^{+}$$\mathscr{I}^{-}$$i^{+}$$i^{-}$$u=u_{-}$$u=u_{+}$
Figure 5.1: Model of detector for gravitational memories.

Let us consider a couple of inertial observers (that we will refer to as the
“detector”) travelling near future null infinity $\mathscr{I}^{+}$. The
detector is localized in a region with no gravitational radiation, or more
generally no null signal, at both late and early (retarded) times. Let us
declare that the radiation is turned on at $u=u_{-}$ and stops at $u=u_{+}$.
For any value of retarded time, exclusive of the interval $[u_{-},u_{+}]$, the
Bondi news tensor and the matter stress-tensor are identically zero by
hypothesis: we are in the vacuum. The metric in the regions $u<u_{-}$ and
$u>u_{+}$ is (5.1.21) (written in the more convenient Newmann-Unti
coordinates), particularized for $q_{AB}=\mathring{q}_{AB}$ and therefore
$C_{AB}=C_{AB}^{(0)}$ according to (5.1.13). Both vacua are labeled by the
supertranslation field $C_{-}$ for $u\leq u_{-}$ and $C_{+}$ for $u\geq u_{+}$
(see also section 2.1.5.3).

The detector, which moves on a timelike trajectory in the far region depicted
in green on Figure 5.1 [57], experiences null radiation only during the time
interval $\Delta u=u_{+}-u_{-}$. The two inertial observers forming the
detector follow a timelike geodesics in the vicinity of $\mathscr{I}^{+}$,
characterized by a $4$-velocity $v=v^{\mu}\partial_{\mu}$. Since their
trajectories are located near $\mathscr{I}^{+}$, we can admit that
$v^{\mu}\partial_{\mu}=\partial_{u}+\mathcal{O}(\rho^{-2})$ up to subleading
corrections necessary for $v$ to verify $v^{\mu}v_{\mu}=-1$ asymptotically.
The separation between the two geodesic trajectories is given by the deviation
vector $s=s^{\mu}\partial_{\mu}$ which is solution of the equation of geodesic
deviation :

$\nabla_{v}\nabla_{v}s^{\mu}=R^{\mu}_{\phantom{\mu}\alpha\beta\gamma}v^{\alpha}v^{\beta}s^{\gamma}$
(5.2.1)

where $\nabla_{v}=v^{\mu}\nabla_{\mu}$ is the directional derivative along
$v^{\mu}$. We suppose that both detectors move on the same celestial sphere,
so $s^{\rho}=0$, with an initial angular separation $s^{A}=s^{A}_{0}$ for
$u<u_{-}$. We get

$\rho^{2}\mathring{q}_{AB}\partial_{u}^{2}s^{B}=R_{uAuB}s^{B}\Leftrightarrow\mathring{q}_{AB}\partial_{u}^{2}s^{B}=\frac{1}{2\rho}\partial_{u}^{2}C_{AB}s^{B}+\mathcal{O}(\rho^{-2})$
(5.2.2)

after using
$R_{uAuB}=-\frac{\rho}{2}\partial_{u}^{2}C_{AB}+\mathcal{O}(\rho^{0})$. One
can check the latter relation using the metric in the Bondi gauge then
translating the result in the Newman-Unti gauge, which is easy because
$\rho=r+\mathcal{O}(r^{-1})$ when $\beta=\mathcal{O}(r^{-2})$. We introduce
some angular perturbation $s^{A}=s^{A}_{0}+\rho^{-1}\Delta
s^{A}+\mathcal{O}(\rho^{-2})$ in the deviation vector for any $u\geq u_{+}$.
Using $\Delta u\ll\rho$, we can integrate on $u$ to get [98]

$\mathring{q}_{AB}\Delta s^{B}=\frac{1}{2}\Delta
C_{AB}^{(0)}s_{0}^{B}+\mathcal{O}(\rho^{-1}).$ (5.2.3)

Therefore, if the phenomena occurring during the non-stationary phase
$[u_{-},u_{+}]$ are responsible for a shift in the asymptotic shear, the
angular separation of the inertial observers will be irreversibly shifted:
this is the displacement memory effect! The spacetime is able to keep track of
the radiative phase in terms of a DC effect. Flashing a light between these
two observers will measure the shift $\Delta s^{A}$, which is therefore
detectable. This was first observed by Zeldovich and Polnarev in 1974 [193]
and nicely reviewed in the Bondi gauge by Strominger and Zhiboedov [98]. The
figure 5.2 provides a schematic picture of the process at null infinity.

$\mathbb{R}$$S^{2}$$u=u_{-}$$u=u_{+}$$s^{A}_{0}$$\Delta s^{A}$ Gravitational
radiation Minkowski vacuum (C $=$ 0) Another vacuum (C $\neq$ 0)
Supertranslation $\bm{\delta_{T}C=T}$$\mathscr{I}^{+}$ Figure 5.2:
Displacement memory effect.

Making contact with our discussion in section 5.1, the displacement memory
effect is equivalent to a change of vacuum between the two stationary phases
$u<u_{-}$ and $u>u_{+}$, for which the supertranslation field $C_{-}$ has been
shifted into $C_{+}=C_{-}+\Delta C$. By virtue of (5.1.26), we see that there
exists a smooth supertranslation parametrized by $T(x^{A})$ such that $\Delta
C=T$. The kinematical equation (5.2.3) informs us how to “measure” the action
of a supertranslation. The presence of the infinite-dimensional extension of
the Poincaré group as BMS4 supertranslations is therefore mandatory if we
allow radiative configurations in the solution space [74, 219]. In the absence
of pure supertranslations, the only shifts allowed in the field $C$ are
annihilated by the operator $(D_{A}D_{B})^{TF}$ because the four translation
parameters $T$ precisely verify $(D_{A}D_{B}T)^{TF}=0$ by definition.
Therefore $\Delta C_{AB}^{(0)}$ would be zero for any vacuum transition and
the shift (5.2.3) would never be observed. But this statement is equivalent to
assume that there is no radiative mode in the phase space.

Let us now study the causes of the displacement (5.2.3). Any physical process
that can change the tensor $C_{AB}$ leads to a displacement memory effect. To
analyze the possible sources, let us consider the time evolution of the Bondi
mass aspect introducing some matter stress-tensor $T_{\mu\nu}^{M}$ in the bulk
of spacetime. The equation (2.1.44) is modified as

$\partial_{u}M=-\frac{1}{8}N_{AB}N^{AB}+\frac{1}{4}D_{A}D_{B}N^{AB}-4\pi
G\lim_{r\to\infty}(r^{2}T^{M}_{uu})$ (5.2.4)

if we are on the unit-round sphere metric (see e.g. [112, 240]). The
demonstration follows the reasoning presented in section (2.1.5.2) and
requires to extract the $r$-independent part of $r^{2}G_{uu}=8\pi
GT_{uu}^{M}$. Integrating between $u_{-}$ and $u_{+}$, this equation simply
yields [98]

$-\frac{1}{4}(D^{2}+2)D^{2}\Delta C=\Delta
M+\int_{u_{-}}^{u_{+}}\text{d}u\>\left[\frac{1}{8}N_{AB}N^{AB}+4\pi
G\lim_{r\rightarrow\infty}(r^{2}T_{uu}^{M})\right]$ (5.2.5)

for $D^{2}\equiv D^{A}D_{A}$ and using (5.1.13) for both values of $u=u_{-}$,
$u=u_{+}$. The shift of supertranslation field $\Delta C$ obeys a quartic
elliptic equation which is sourced by 3 qualitatively distinct terms, hence
the displacement memory detector will trigger for each of the following
causes:

  1. 1.

If the Bondi mass aspect varies between $u_{-}$ and $u_{+}$. This is sometimes
called ordinary memory [193, 95]. For example, a single massive body
containing a string that suddently separate into two parts due to a trigger
will modify the Bondi mass aspect $M$ because the mass will suddenly possess a
strong dipolar component. What Einstein gravity tells us is that a signal is
sent at null infinity with that information and the memory effect follows.

  2. 2.

If some null matter represented by $T_{uu}^{M}=\mathcal{O}(r^{-2})$ reaches
$\mathscr{I}^{+}$ between $u_{-}$ and $u_{+}$. This is sometimes called the
null memory effect. For example, electromagnetic radiation causes the
displacement memory effect.

  3. 3.

And finally, if gravitational waves, whose flux is encoded in the news tensor
$N_{AB}$, pass through $\mathscr{I}^{+}$ between $u_{-}$ and $u_{+}$. This is
called the Christodoulou effect [97, 96], which appeared also earlier in
perturbative form in post-Newtonian formalism [15, 16].

In the literature, one also find the epithet “non-linear” to qualify the
Christodoulou effect since it is produced by the gravitational field itself,
by opposition with both other effects which are qualified as “linear” because
the sources are material.

Let us finish by two concluding observations. First of all, the description of
a gravitational memory effect always comes with two equations, (5.2.3) and
(5.2.5) in the present case.

  * $\rhd$

The first equation is kinematical and tells us how the radiative phase affects
the detectors. It allows to determine what is the memory field $\varphi$ that
will be permanently shifted (i.e. $\varphi_{-}\to\varphi_{+}$) after the end
of the non-stationary phase at $u>u_{+}$. It will thus be the depositary of
the “memory” of what happened in the time interval $[u_{-},u_{+}]$. This role
is played here by the supertranslation field $C$. The inhomogeneous part
$\delta^{I}_{\xi}\varphi$ of the transformation law of the memory field gives
the asymptotic symmetry $\xi$ relating both vacua for $u<u_{-}$ and $u>u_{+}$.
Indeed, $\delta^{I}_{\xi}C=T$ from (5.1.26) and the final vacuum is the
supertranslated vacua with $T=\Delta C$. Matching with the notations of
section 5.2.1, $\varphi\equiv C$ is well a local field in Bondi (or Newman-
Unti) gauge and $O(\theta,\phi)\equiv\Delta C$. The latter is gauge-invariant:
performing a global supertranslation $\xi$ of amplitude $T_{glob}$ on the
whole solution will modify
$\delta_{\xi}\varphi_{+}=T_{glob}=\delta_{\xi}\varphi_{-}$ hence
$\delta_{\xi}\Delta C=0$. Note crucially that $T_{glob}$ is an actual
transformation on the solution space while $T$ is the mapping between early
and late stationary configurations: $T_{glob}$ represents any supertranslation
that can be performed on the whole solution obtained by gluing continuously
the three regions $u<u_{i}$, $u_{i}<u<u_{f}$ and $u>u_{f}$, while $T$ is
determined as the diffeomorphism needed to map the late region $u<u_{i}$ onto
the late region $u>u_{f}$.

  * $\rhd$

The second equation is dynamical and is used to compute the exact shift
$\Delta C$ of the memory field from the data of the sources (distribution of
mass, radiation,…). This information has to come from the equations of motion
for the gravitational field, eventually coupled with a matter stress-tensor.
The evolution of the memory field is dictated by a relevant evolution equation
implying quantities of the same nature: hence the evolution of the scalar
field $C$ is governed by the constraint equation on the Bondi mass aspect.

Note finally that the displacement memory effect has not yet been observed at
the time of writing. Nevertheless, we are hopeful that the experimental
confirmation will be obtained soon from gravitational astronomy, by means of
gravitational wave detectors [311] or pulsar timing arrays [312, 313].

#### 5.2.3 Superboost transitions

Let us now focus on gravitational memory effects related to super-Lorentz
transformations. We start by describing the constraints on the bulk data
producing generic superboost transitions that change the boundary metric and
the vacuum Bondi news tensor at leading order. We also explain why we are
mainly interested in the impulsive limit of such exotic events and
characterize their imprint on asymptotic detectors of free-falling observers.
We conclude that superboost vacuum transitions allow to deduce the velocity
kick/refraction memory effects [190, 191, 192] from a symmetry principle in
the asymptotic region. These effects latter are qualitatively distinct from
the spin memory effect [100] which has no local associated memory field in the
Bondi gauge (see also [211] for a review). It is in fact related to subleading
diffeomorphism transitions in (retarded) harmonic gauge [118]. They are also
distinct from the center-of-mass memory effect [101], also lacking of a local
memory field in Bondi coordinates and related to changes in the center-of-mass
part of the angular momentum.

##### 5.2.3.1 Penrose’s gravitational shockwaves

In the previous section, we reviewed the effect of permanent displacement of
asymptotic free-falling observers due to a radiative process leading to a
transition between two supertranslated vacua in
$\mathring{\mathcal{S}}_{0}^{\text{Mink}}$. Analogously, we define here a
superboost transition as a transition between two vacua associated with two
different superboost fields. The configuration is as follows:

  * $\rhd$

We consider a spacetime in $\mathring{\mathcal{S}}_{0}$ for which we assume
that the radiative phase has compact support in the time interval
$[u_{-},u_{+}]$.

  * $\rhd$

For $u<u_{-}$, the spacetime is Minkowski in some coordinate system where
$\Phi=\Phi_{-}(x^{A})$,
$q_{AB}=q_{AB}^{vac}[\mathring{q},\Phi_{-},\widetilde{\Phi}]$,
$N_{AB}^{vac}=N_{AB}^{vac}[\Phi_{-}]$ as initial conditions.

  * $\rhd$

For $u>u_{+}$, the spacetime is also Minkowski in another coordinate system
for $\Phi=\Phi_{+}(x^{A})$,
$q_{AB}=q_{AB}^{vac}[\mathring{q},\Phi_{+},\widetilde{\Phi}]$,
$N_{AB}^{vac}=N_{AB}^{vac}[\Phi_{+}]$ as late data.

During the non-stationary phase $u_{-}<u<u_{+}$, the metric field must vary in
time at leading order. This information, combined with the equation of motion
(2.1.42), implies that a superboost transition cannot be observed as a vacuum
solution of Einstein’s equation but needs the coupling with some matter with
exotic fall-offs in its stress-energy tensor in a way to modify the leading
structure at $\mathscr{I}^{+}$. A simple power counting in (2.1.34) gives that
$T_{AB}=\mathcal{O}(r)$ for $u_{-}<u<u_{+}$ is required to observe a
superboost transition. Note also that (2.1.34) implies that the time-
dependency of the metric can only be turned on in the vacuum by a transition
in some conformal factor in front of $q_{AB}$ for which the boundary condition
(2.1.27) cannot be obeyed, but it is definitely not a superboost transition
despite some similarities. We plan to study these generalized transitions
(probably related to Robinson-Trautman waves [195] and the Weyl symmetry [91,
46]) in future works. Instead of entering into these considerations, we will
rather discuss here superboost transitions driven by topological defects for
which the vacuum Einstein equation, in particular (2.1.42), is obeyed locally
except at some singular points. We will see that we can yet extract a lot of
information from these processes that are in fact describing gravitational
shockwaves in the bulk.

Let us briefly review the way Penrose introduced the notion of impulsive
gravitational wave in [188] thanks to the suggestively called “scissor and
paste” construction, that we translate here in our formalism. The impulsive
gravitational wave, or shockwave, is modelized by a local vacuum solution of
Einstein’s field equation obtained by cutting the spacetime along a null
hypersurface located at $u=u_{\star}$ and gluing two different patches
$\mathscr{M}_{-}=(u_{-},r_{-},z_{-},\bar{z}_{-})$ for $u<u_{\star}$ and
$\mathscr{M}_{+}=(u_{+},r_{+},z_{+},\bar{z}_{+})$ for $u>u_{\star}$ of
Minkowski spacetime $\mathscr{M}$. The construction is schematized in Figure
5.3 [188].

$u=u_{\star}$Null generator$\mathscr{M}_{-}$$\mathscr{M}_{+}$ Figure 5.3:
Penrose’s “scissor and paste” construction.

Assuming that, for $u<u_{\star}$, the spacetime is the global Minkowski vacuum
in Bondi coordinates where $x^{A}_{-}=(z_{-},\bar{z}_{-})$ are the
stereographic coordinates on the sphere, we can impose the continuity
conditions
$r_{-}^{2}q^{vac}_{AB}[\mathring{q},\Phi_{-},\widetilde{\Phi}]\text{d}x^{A}_{-}\text{d}x^{B}_{-}=r_{+}^{2}q^{vac}_{AB}[\mathring{q},\Phi_{+},\widetilde{\Phi}]\text{d}x^{A}_{+}\text{d}x^{B}_{+}$
almost everywhere on the celestial sphere. This is solved locally by the
holomorphic transformation $z_{+}=G(z_{-})$ compensated by a suitable Weyl
rescaling $r_{-}\to r_{+}$ in order to keep the sphere area preserved across
$u=u_{\star}$. By hypothesis $\Phi_{-}=-\ln\sqrt{\mathring{q}}$,
$q^{vac}_{AB}\text{d}x^{A}_{-}\text{d}x^{B}_{-}=\mathring{q}_{AB}\text{d}x^{A}_{-}\text{d}x^{B}_{-}=2r_{-}\sqrt{\mathring{q}}\text{d}z_{-}\text{d}\bar{z}_{-}$
and $N_{AB}^{vac}[\Phi_{-}]=0$. Taking the convenient choice of origin for the
retarded time such that $u_{\star}=0$, the line element resulting from the
scissor and paste construction can be written as [189]

$\text{d}s^{2}=-\text{d}u^{2}-2\text{d}\rho\text{d}u+\left[\rho^{2}q_{AB}+u\rho\Theta(u)N^{vac}_{AB}+\frac{u^{2}}{8}\Theta(u)N^{vac}_{CD}N_{vac}^{CD}q_{AB}\right]\text{d}x^{A}\text{d}x^{B}.$
(5.2.6)

in the Newmann-Unti gauge. Here
$N^{vac}_{AB}=\left[\frac{1}{2}D_{A}\Phi_{+}D_{B}\Phi_{+}-D_{A}D_{B}\Phi_{+}\right]^{TF}$
and $\Theta(u)$ is the Heaviside distribution. The vacuum news tensor
coincides with (5.1.13) after substituting
$\Phi=-\ln\sqrt{\mathring{q}}+\phi_{+}(z)+\bar{\phi}_{+}(\bar{z})=\Phi_{+}$ as
in (5.1.23). The presence of a gravitational shockwave located on the null
hypersurface $u=0$ is responsible for a superboost transition between two
vacua which differ by a meromorphic superboost transformation. By
construction, (5.2.6) is globally Riemann-flat for $u<0$ and locally for $u>0$
except at the singularities of the meromorphic function $\phi_{+}(z)$, located
at the poles $z=0,\infty$ of the celestial sphere. At these points, one should
not forget that $q_{AB}\neq\mathring{q}_{AB}$ is singular and
$g_{uu}=-1-\frac{1}{2}D^{2}\phi_{+}$ but otherwise $q_{AB}=\mathring{q}_{AB}$
and $g_{uu}=-1$. One can thus prove that (5.2.6) is Ricci-flat for any $u$
including the location of the shockwave, also except at the singular points.
At $u=0$ the Riemann-flatness, is only broken by the shockwave which
contributes to the Weyl curvature as
$W_{uAuB}=r\delta(u)N_{AB}^{vac}[\phi_{+}]$.

To summarize, the singular impulsive limit describing transitions induced by
shockwaves requires to consider singular diffeomorphisms transitions that are
meromorphic superboost transitions. The metric $q_{AB}$ is the unit sphere
metric globally for $u<0$ and locally for $u>0$, but it contains singularities
at isolated points for $u>0$. These singularities have been understood as
cosmic string decays [189, 314, 315, 187] that provide a concrete physical
process to be associated with superboost transitions. The metric (5.2.6) for
$u>0$ is thought to represent a cosmic string supported at the end points
$z=0,\infty$ if the superboost relating the two vacua is driven by a
transformation of the form $G(z)=z^{1+\varepsilon}$ [189, 187]. This is a
topological defect in the Minkowski spacetime which induces a deficit angle
linear in $\varepsilon$ along the polar axis $z=0,\infty$ (or $\theta=0,\pi$
for the standard spherical colatitude) of the null hypersurface $u=0$. Because
the string pierces null infinity, the boundary metric is singular at the
anchor points, what is in line with the analysis above. The time-reversion
$u\to-u$ of (5.2.6) is thus seen as the snapping of such a cosmic string at
$u=0$, the two free ends starting from $r=u=0$ in Minkowski space and travel
along the singular points $z=0,\infty$ at the speed of light towards null
infinity, see Figure 5.4 [189]. When the string snaps, a gravitational
shockwave is emitted and travels along the null front $u=0$. The free
extremities of the string support particles or, better, black holes in the
full non-linear theory. An impulsive superboost is thus a toy-model to analyze
the radiative consequences of a black hole pair creation in the bulk. Some
explicit computations and results in that direction can be found in [187].

$u=0$$\theta=0$$\theta=\pi$Cosmic string Figure 5.4: Gravitational shockwave
induced by cosmic string snapping.

##### 5.2.3.2 General impulsive gravitational wave transitions

In general, both the supertranslation field $C$ and the superboost field
$\Phi$ can change with hard (finite energy) processes involving null radiation
reaching $\mathscr{I}^{+}$. This null radiation can have its origin in matter
or in gravity itself. Such processes, occurring during a finite amount of time
$[u_{-},u_{+}]$, induce vacuum transitions among initial $(C_{-},\Phi_{-})$
($u<u_{-}$) and final $(C_{+},\Phi_{+})$ ($u<u_{+}$) boundary fields. The
difference between these fields can be expressed in terms of components of the
matter stress-tensor and metric potentials reaching $\mathscr{I}^{+}$. The
simplest possible transition between vacua are shockwaves (say, at $u=0$) that
carry a matter stress-tensor and a Weyl curvature proportional to a
$\delta(u)$ function, as in the original Penrose construction [188]. A
distinct vacuum lies on each side of the shockwave and the transition between
the boundary fields is dictated by the matter stress-tensor as well as the
radiative components of the Weyl tensor. This picture allows us to include
both types of transitions previously discussed in sections 5.2.2 and 5.2.3.1.
For $u<0$ and $u>0$, the line element describes a vacuum situation which reads
as (5.1.21) in Newmann-Unti coordinates.

A general shockwave takes the form

$\text{d}s^{2}=-\frac{R[q]}{2}\text{d}u^{2}-2\text{d}\rho\text{d}u+(\rho^{2}q_{AB}+\rho
C_{AB}+\frac{1}{8}C_{CD}C^{CD}q_{AB})\text{d}x^{A}\text{d}x^{B}+D^{B}C_{AB}\text{d}x^{A}\text{d}u$
(5.2.7)

where

$\displaystyle q_{AB}$
$\displaystyle=\Theta(-u)q_{AB}^{vac}[\Phi_{-}]+\Theta(u)q_{AB}^{vac}[\Phi_{+}],$
(5.2.8) $\displaystyle C_{AB}$
$\displaystyle=\Theta(-u)C_{AB}^{vac}[\Phi_{-},C_{-}]+\Theta(u)C_{AB}^{vac}[\Phi_{+},C_{+}]$
(5.2.9)

where $q_{AB}^{vac}[\Phi]$ and $C_{AB}^{vac}[\Phi,C]$ are given in (5.1.11)
and (5.1.13). The metric (5.2.6) is recovered for
$\Phi_{-}=-\ln\sqrt{\mathring{q}}$,
$\Phi_{+}=\phi_{+}(z)+\bar{\phi}_{+}(\bar{z})-\ln\sqrt{\mathring{q}}$ as in
(5.1.23) and $C_{+}=C_{-}=0$. Our goal now is to study the memory effects
associated with such a vacuum transition. Among them, there is obviously the
shift of the supertranslation field previously analyzed in section 5.2.2.

##### 5.2.3.3 Evolution of the Bondi mass aspect and the center-of-mass

In the absence of superboost transitions and assuming the standard boundary
condition $q_{AB}=\mathring{q}_{AB}$, the integral between initial $u_{-}$ and
final retarded times $u_{+}$ of the evolution equation (2.1.44) for the Bondi
mass aspect can be re-expressed as the differential equation (5.2.5)
determining the difference between the supertranslation field $\Delta
C=C_{+}-C_{-}$ between initial and final retarded times. This is one of the
two master equations governing the vacuum transition. The four lowest
spherical harmonics $\texttt{{l}}=0,1$ are zero modes of the differential
operator appearing on the left-hand side of (5.2.5). Recall that translations
precisely shift the supertranslation field as (5.1.26) or more particularly
(2.1.64). The 4 lowest harmonics of $C$ can thus be interpretated as the
center-of-mass of the asymptotically flat system. This center-of-mass is not
constrained by the conservation law (5.2.5) as observed previously.

A new feature arises in the presence of a superboost transition. The four zero
modes of the supertranslation field $C$ are now determined by the evolution
equation. This can be seen in the context of impulsive transitions (5.2.7).
For simplicity, we take $C_{-}=0$ and $\Phi_{-}=-\ln\sqrt{\mathring{q}}$
($q_{AB}[\Phi_{-}]=\mathring{q}_{AB}$). Given that the Bondi mass aspect and
the Bondi news of the vacua are non-zero (5.1.30), we first define the
“normalized” Bondi mass aspect and Bondi news as

$\displaystyle\bar{M}$ $\displaystyle=$ $\displaystyle
M+\frac{1}{8}C_{AB}N^{AB}_{vac}[\Phi_{+}],$ (5.2.10)
$\displaystyle\hat{N}_{AB}$ $\displaystyle=$ $\displaystyle
N_{AB}-\Theta(u)N_{AB}^{vac}[\Phi_{+}],$ (5.2.11)

which are zero for the vacua (5.1.21). We will give a precise meaning for
these shifted quantities in sections 5.3.1 and 5.3.2, where we will show that
$\bar{M}$ is the momentum canonically conjugated to the supertranslation
parameter $T$ in the finite Generalized BMS4 Hamiltonian at $\mathscr{I}^{+}$
while $\hat{N}_{AB}$ is the physical notion of Bondi news which is not
improperly sourced by super-Lorentz transformations. After integration over
$u$ of (2.1.44), we take the corollary (5.1.15) of the Liouville equation
(5.1.14) into account to obtain

$-\frac{1}{4}D^{2}(D^{2}+R[q])C_{+}+\frac{1}{4}N^{AB}_{vac}[\Phi_{+}]D_{A}D_{B}C_{+}+\frac{1}{8}C_{+}D_{A}D^{A}R[q]=\Delta\bar{M}+\int_{u_{-}}^{u_{+}}\text{d}u\>T_{uu}$
(5.2.12)

where

$T_{uu}=\frac{1}{8}\hat{N}^{AB}\hat{N}_{AB}+4\pi
G\lim_{r\rightarrow\infty}(r^{2}T_{uu}^{M}),$ (5.2.13)

$\Delta\bar{M}$ act as sources for $C_{+}$ and all quantities are evaluated on
the final metric $q_{AB}[\Phi_{+}]$. We have that $\Delta\bar{M}=0$ for
transitions between vacua but we included it (1) for making the comparison
with (5.2.5) more manifest and (2) for leaving the possiblity to easily adapt
the result in the presence of zero modes like some ADM mass when $u<0$. Here
is the main observatio: the lowest $\texttt{{l}}=0,1$ spherical harmonics of
the supertranslation field $C$ are not zero modes of the quartic differential
operator on the left-hand side of (5.2.12) for any inhomogeously curved
boundary metric, i.e. in an arbitrary super-Lorentz frame where neither
$q_{AB}$ nor $N_{AB}^{vac}$ are trivial. Therefore, the center-of-mass is also
determined by the equation of motion for the Bondi mass aspect. We are now
about to discuss the second aspect of the memory effect, which is the
predictions of the geodesic deviation equation.

##### 5.2.3.4 Refraction/Velocity kick memory

Let us consider a Bondi detector formed by two free-falling observers near
$\mathscr{I}^{+}$ that evolve at some finite large radius $\rho$ in the
impulsive gravitational wave spacetime (5.2.6). We continue to work in the
Newmann-Unti coordinate system and we just study pure superboost transitions
for the moment, i.e. transitions with $\Delta C=C_{+}=C_{-}=0$. We also choose
to work with observers away from the singular points on the celestial sphere
at any retarded time $u$, so that we can ignore the singularities due to the
meromorphic superboosts. By definition, the observers have a 4-velocity like
$v^{\mu}\partial_{\mu}=\partial_{u}+\mathcal{O}(\rho^{-1})$ as before. The
deviation vector $s^{\mu}$ between two neighboring geodesics obeys again
(5.2.1). As in section 5.2.2, we assume that $s^{\rho}=0$ and
$s^{A}=s^{A}_{0}$ at early times. Since we are away from the singularities,
the boundary metric can be assumed to stick at $\mathring{q}_{AB}$ and we have
$R_{uAuB}=-\frac{\rho}{2}\partial^{2}_{u}C_{AB}+\mathcal{O}(\rho^{0})$ as
before. Here $C_{AB}=u\Theta(u)N_{AB}^{vac}$ and therefore

$q_{AB}\partial_{u}^{2}s^{B}=\frac{1}{2\rho}\delta(u)N_{AB}^{vac}s^{B}+\mathcal{O}(\rho^{-2}).$
(5.2.14)

When dealing with impulsive transitions, any primitive or derivative in $u$
involving $\Theta(u)$ or $\delta(u)$ is intended in the sense of
distributions. For instance, $\partial_{u}\Theta(u)\sim\delta(u)$ and
$\partial_{u}(u\Theta(u))\sim u\delta(u)+\Theta(u)\sim\Theta(u)$ when
integrated against any numerical test function in $u$. We deduce from (5.2.14)
that the angular deviation is given by $s^{A}=s^{A}_{0}+\rho^{-1}\Delta
s^{A}(u,x^{A})+\mathcal{O}(\rho^{-2})$ and after two integrations in $u$,
$\delta(u)\to\Theta(u)\to u\Theta(u)$ so

$\Delta s^{A}=\frac{u}{2}\Theta(u)q^{AB}N_{BC}^{vac}s^{C}_{0}.$ (5.2.15)

Before the shockwave, there is no relative angular velocity between observers.
After the shockwave, this relative angular velocity is turned on at order
$\propto\rho^{-1}$, since the right-hand side of (5.2.15) is linear in $u$. We
observe a velocity kick between two such neighboring geodesics due to the
shockwave [190, 191, 192], see figure 5.5 for a schematic picture. This is a
qualitatively distinct effect from the displacement memory effect [193, 15,
97, 16, 316] reviewed in section 5.2.2, the spin effect [100] and the center-
of-mass effect [101].

$\mathbb{R}$$S^{2}$$u=u_{-}$$u=u_{+}$$s^{A}_{0}$$\Delta s^{A}$ Non-stationary
phase Minkowski vacuum ($N_{AB}^{vac}[\Phi]$ $=$ 0) Another vacuum
($N_{AB}^{vac}[\Phi]$ $\neq$ 0) Superboost transition
$\bm{\delta_{Y}\Phi=D_{A}Y^{A}}$$\mathscr{I}^{+}$ Figure 5.5: Velocity kick
memory effect.

Let us analyze if the same is true for incoming light rays traveling in the
asymptotic region. One can consider a congruence of null geodesics which
admits a constant leading angular velocity $\Omega^{A}\partial_{A}$, with
total 4-impulsion

$v^{\mu}\partial_{\mu}=\left(\sqrt{\Omega^{A}q_{AB}\Omega^{B}}+\mathcal{O}(\rho^{-1})\right)\partial_{u}+\mathcal{O}(\rho^{-1})\partial_{\rho}+\frac{1}{\rho}\left(\Omega^{A}+\mathcal{O}(\rho^{-1})\right)\partial_{A}.$
(5.2.16)

We consider again a deviation vector of the form
$s^{A}=s^{A}_{0}+\frac{1}{\rho}\Delta s^{A}(u,x^{A})+\mathcal{O}(\rho^{-2})$.
A simple computation convinces that the deviation vector obeys again (5.2.15).
Null geodesics are thus refracted by the shockwave. This is the refraction
memory effect usually described in the bulk of spacetime [190, 191, 192]. We
identified here the class of null geodesics which displays the refraction
memory effect close to future null infinity.

Let us now shortly comment on the case where the change of boundary metric is
not localized at individual points. The main point is that timelike geodesics
will now admit non-trivial deviation vector already at leading order
$\propto\,\rho^{0}$, $s^{A}=s^{A}_{lead}+\mathcal{O}(\rho^{-1})$, with

$\frac{1}{2}q_{AB}\partial_{u}^{2}s^{B}_{lead}+\frac{1}{2}\partial_{u}^{2}(q_{AB}s^{B}_{lead})=-\frac{1}{2}\partial_{u}^{2}q_{AB}s_{lead}^{B}.$
(5.2.17)

A velocity kick will therefore already occur at order $\rho^{0}$.

##### 5.2.3.5 A new non-linear displacement memory

Until now, we have considered impulsive vacuum transitions (5.2.6) without any
shift in the supertranslation field. We explain now that there is a non-linear
displacement memory induced by a superboost transition, when it is accompanied
by a supertranslation transition. This case was not considered in [190, 191,
192] where all supertranslation transitions were absent. In order to describe
the effect, we can consider either timelike or null geodesics. For
definiteness, we consider a congruence of timelike geodesics that evolve at
some finite large radius $\rho$ in the general impulsive gravitational wave
spacetime (5.2.7). For simplicity, we assume that the metric is Minkowski in
the past and we consider again the simplified case of a meromorphic superboost
transition, assuming that the detector singled out in the timelike congruence
is away from the singularities on the celestial sphere. In other words, we
assume $\Phi_{-}=-\ln\sqrt{\mathring{q}}$
($q^{vac}_{AB}[\Phi_{-}]=\mathring{q}_{AB}$), $C_{-}=0$,
$\Phi_{+}=\phi(z)+\bar{\phi}(\bar{z})-\ln\sqrt{\mathring{q}}$ and
$C_{+}=C_{+}(x^{A})$ is arbitrary. The 4-velocity is now
$v^{\mu}\partial_{\mu}=\sqrt{\frac{2}{R[q]}}\partial_{u}+\mathcal{O}(\rho^{-1})$
and $R_{uAuB}=-\frac{\rho}{2}\partial^{2}_{u}C_{AB}+\mathcal{O}(\rho^{0})$ is
not modified. Following the same procedure, we provide the power expansion
$s^{A}=s^{A}_{0}+\frac{1}{\rho}\Delta s^{A}(u,x^{A})+\mathcal{O}(\rho^{-2})$
and away from the singular points on the sphere, we get

$\begin{split}\Delta s^{A}&=\frac{1}{2}q^{AB}C_{BC}s^{C}_{0}\\\
&=\frac{1}{2}\Theta(u)\mathring{q}^{AB}C_{BC}^{vac}s^{C}_{0}\\\
&=\frac{1}{2}\mathring{q}^{AB}\left(u\Theta(u)N_{BC}^{vac}+\Theta(u)C^{(0)}_{BC}+\Theta(u)CN_{BC}^{vac}\right)s^{C}_{0}.\end{split}$
(5.2.18)

The first term $\propto\,u\Theta(u)$ leads to the velocity kick memory effect.
The second term $\propto\,\Theta(u)C^{(0)}_{BC}$ leads to the displacement
memory effect due to a change of supertranslation field $C$ between the final
and initial states [98]. The third and last term
$\propto\,\Theta(u)CN^{vac}_{BC}$ is a new type of non-linear displacement
memory effect due to change of both the superboost field $\Phi$ and the
supertranslation field $C$. The four lowest spherical harmonics
$\texttt{{l}}=0,1$ of $C$, interpretated as the center-of-mass, do not
contribute to the standard displacement memory effect because they are zero
modes of the differential operator $C_{AB}^{(0)}$ in (5.1.13). Here, they do
contribute to the non-linear displacement memory effect. The modification of
the supertranslation field, in particular of the center-of-mass, is determined
by (5.2.12), as discussed earlier. Note crucially that these observations are
only relevant in the presence of superboost transitions driven by cosmic
events that are able to modify the leading structure of the Bondi metric at
null infinity. It is therefore not expected to be relevant for localized
events such as compact binary mergers, the latter being perfectly understood
in the reduced phase space built up from
$\mathring{\mathcal{S}}_{0}^{\text{Mink}}$. We finally close this discussion
of memory effects associated with generalized vacuum transitions in the
presence of both supertranslation and superboost symmetries and are on track
to address the last point of this thesis.

###  5.3 Generalized BMS4 finite charges

In section 5.1, we derived a closed-form expression of the orbit of Minkowski
spacetime under arbitrary super-Lorentz transformations. Thanks to this
knowledge, we were able to describe the gravitational memory effects related
to these leading symmetries in the previous section, 5.2. In this last
section, we describe the construction of finite integrable charges for the
Generalized BMS4 asymptotic symmetries of flat spacetime. We also compute the
associated fluxes of charges and match them with quantum results at the linear
level of the theory.

#### 5.3.1 Initial and late data in general super-Lorentz frames

For the purpose of this analysis, it is primordial to study the behavior of
the field at $\mathscr{I}^{+}_{-}$ and $\mathscr{I}^{+}_{+}$, which are
respectively the past and future boundaries of $\mathscr{I}^{+}$ for
$u\to-\infty$ and $u\to+\infty$. The asymptotic sphere $\mathscr{I}^{+}_{-}$
is bordering spatial infinity while its future counterpart
$\mathscr{I}^{+}_{+}$ touches future timelike infinity. Prescribing some fall-
off conditions in $u$ and asymptotic values when $|u|\to+\infty$ for the
various fields parametrizing the phase space amounts to defining a set of
initial and late data for any solution in the phase space which consists of an
additional physically motivated input of boundary conditions. Here we focus on
physically relevant solutions that start from a stationary configuration
without radiation in the past and revert back to stationarity in the future.
Note that, a priori, this assumption rules out black hole formation.

Let us describe the consequences of assuming early and late stationarity in
the restricted context of the asymptotically Minkowskian solution space
$\mathring{\mathcal{S}}_{0}^{\text{Mink}}$ (2.1.41). There is no allowed pure
super-Lorentz transformation, hence the class of spacetimes under
consideration is assumed to start from a stationary configuration in the far
past $u\to-\infty$, solely labeled by a supertranslation field $C_{-}(x^{A})$
and some zeros modes (ADM charges) and decay to another stationary
configuration in the far future $u\to+\infty$, labeled by another
supertranslation field $C_{+}(x^{A})$ and another bunch of zero modes
(frequently assumed to be zero). We have $C_{AB}\to C^{(0)}_{AB}[C_{\pm}]$ for
$u\to\pm\infty$, given by (5.1.13). In particular, $N_{AB}\to 0$ in this limit
and $C_{AB}$ is purely electric [112], i.e.

$\left.(D_{B}D^{C}C_{AC}-D_{A}D^{C}C_{BC})\right|_{\mathscr{I}^{+}_{\pm}}=0.$
(5.3.1)

This is a differential statement equivalent to say that $C_{AB}$ is pure-gauge
since it is only built from the supertranslation field $C_{-}$ or $C_{+}$.
Since we have stationary configurations as temporal boundary conditions, we
can equivalently say that the radiative phase is assumed to have compact
support on $\mathscr{I}^{+}$, because the Weyl tensor is required to fall off
to zero at large retarded times. Such radiative solutions have been defined in
a rigorous way by Christodoulou and Klainerman [269]. They showed that there
exists a class of Cauchy data which decay sufficiently fast at spatial
infinity such that the Cauchy problem leads to a smooth geodesically complete
solution. In fact, they prove that the non-linear stability of Minkowski
spacetime (i.e. the hypothesis to be in a – possibly supertranslated – vacuum
for $|u|\to+\infty$) requires that the Bondi news falls off as

$N_{AB}=\mathcal{O}\left(|u|^{-(1+\varepsilon)}\right)$ (5.3.2)

when $u\rightarrow\pm\infty$, for any $\varepsilon>0$ arbitrarily fixed, while
the Bondi functions $M$ and $N_{A}$ remain finite in the two limits and we can
endow the asymptotic vacua by these zero modes. Now, even if black holes form
in the spacetime, we do not expect that these quantities will behave
differently since they do not emit any radiation at early or late retarded
times. We can thus safely assume that all physical asymptotically flat
spacetimes worth of interest in $\mathring{\mathcal{S}}_{0}^{\text{Mink}}$
obey these additional boundary conditions. Using (2.1.44) and (2.1.45)
restricted on this solution space, we can check explicitly that $M$ and
$N_{A}$ become constant while approaching spatial infinity, forming the total
ADM mass and angular momentum of the spacetime after integration on
$\mathscr{I}^{+}_{-}$. The constraint (5.3.1), obeyed by the boundary tensor
$C_{AB}^{(0)}[C_{-}]$, has to be used explicitly in (2.1.45) to show that
$\partial_{u}N_{A}\to 0$ at spatial infinity.

Let us now see how these considerations can be extrapolated (or not) in the
Generalized BMS4 solution space $\mathring{\mathcal{S}}_{0}$ (2.1.40). Again,
we demand that any spacetime under consideration decays to stationary
configurations when $|u|\to+\infty$. By consistency, the spacetime in these
stationary regions is one of the Generalized BMS4 vacua possibly endowed with
zeros modes. Hence, the boundary conditions are described by a triplet of
boundary fields $(C_{-},\Phi_{-},\widetilde{\Phi}_{-})$ in the past and by
another triplet of boundary fields $(C_{+},\Phi_{+},\widetilde{\Phi}_{+})$ in
the future, according to the results of section 5.1.2. We write

$\displaystyle q_{AB}^{\pm}\equiv\lim_{u\rightarrow\pm\infty}q_{AB}$
$\displaystyle=q^{vac}_{AB}[\gamma_{\underline{AB}},\sqrt{\mathring{q}},\Phi_{\pm},\widetilde{\Phi}_{\pm}]+o(u^{0}),$
(5.3.3) $\displaystyle\lim_{u\rightarrow\pm\infty}C_{AB}$
$\displaystyle=C^{vac}_{AB}[q_{AB}^{\pm},\Phi_{\pm},C_{\pm}]+o(u^{0}),$
(5.3.4)

where $q_{AB}^{vac}$ and $C^{vac}_{AB}$ are defined in (5.1.11) and (5.1.13)
respectively. It follows that

$\lim_{u\rightarrow\pm\infty}N_{AB}=N^{vac}_{AB}[q^{\pm}_{AB},\Phi_{\pm}]+o(u^{-1})$
(5.3.5)

where $N^{vac}_{AB}$ is defined in (5.1.13). When $u$ evolves, the initial
fields $(C_{-},\Phi_{-},\Psi_{-})$ can change with hard processes involving
null radiation reaching $\mathscr{I}^{+}$. Even though this null radiation can
originate from matter or from gravity itself, here we want to consider only
gravity. This implies in particular that (2.1.42) holds, hence

$q_{AB}\equiv
q_{AB}^{-}=q_{AB}^{+}\,,\quad\Phi\equiv\Phi_{+}=\Phi_{-}\,,\quad\widetilde{\Phi}\equiv\widetilde{\Phi}_{+}=\widetilde{\Phi}_{-}.$
(5.3.6)

These conditions forbid transitions between the initial and final superboost
and superrotation fields. The class of spacetimes that we are considering is
therefore more general than those considered in [59, 269] but lacks to include
transitions by gravitational radiation in the boundary metric, since the
latter can only be driven by a dynamical Weyl rescaling $q_{AB}\to
e^{2\varphi(u,x^{C})}q_{AB}$ [46, 32], as indicated by (2.1.34). Let us
mention at this point that the Weyl rescaling can always be gauge fixed by a
Weyl transformation allowed within the Bondi gauge to get (2.1.39). This
repackages the arbitrary time dependence in $\varphi(u,x^{A})$ into the news
tensor whose fall-offs in $u$ can thus be arbitrary. Here, since we assume the
precise asymptotic behavior (5.3.5) for $N_{AB}$, we rule out solutions with
arbitrary time-dependence in the Weyl factor. For instance, general Robinson-
Trautman solutions [195, 286], whose only degree of freedom is precisely such
a Weyl factor with an arbitrary dependency in $u$, are not present in the
phase space. Recently, a proposal for more general phase spaces including
Weyl-sourced Bondi news tensors has been given in [92, 91] but without
allowing the Weyl factor to be time-dependent. It should be interesting to
extend further these analyses, then gauge-fix the Weyl factor as (2.1.39) for
the purpose to obtain a suitable generalization of (5.3.5) in that context.

In an arbitrary superboost frame, (5.3.5) reads as $N_{AB}\to
N^{vac}_{AB}[q_{AB},\Phi]+o(u^{-1})$ which, of course, does not obey the
Christodoulou-Klainermann fall off condition (5.3.2). This is fundamentally
due to the fact that the Bondi news tensor transforms inhomogenenously under
superboost, as stated by (2.1.73). Indeed, the term
$(D_{A}D_{B}D_{C}Y^{C})^{TF}$, shown to be zero for any global boost
(annihilated by three derivatives because of the conformal Killing equation
(2.1.56)), is not vanishing when extending the symmetries to superboosts.
Starting from the Minkowski global spacetime (for which the Weyl tensor is
identically zero) where $N_{AB}=0$, any superboost generates a non-vanishing
news tensor $N_{AB}^{vac}$ in the orbit of Minkowski while keeping a vanishing
Weyl tensor. Hence, $N_{AB}$ does not formalize, strictly speaking, the notion
we have of a “physical” Bondi news tensor that should be zero in the absence
of gravitational radiation. We define the shifted tensor

$\hat{N}_{AB}\equiv N_{AB}-N_{AB}^{vac}[\Phi]$ (5.3.7)

and we call it the physical news tensor for the following reasons. First of
all, from (5.3.5) and (5.3.6), we observe that $\hat{N}_{AB}$ obeys the fall-
off requirement (5.3.2) that is suitable to describe well-behaved
gravitational radiation around spatial infinity. Moreover, owing to (2.1.73)
and (5.1.28), we observe that

$\delta_{T,Y}\hat{N}_{AB}=(T\partial_{u}+\mathcal{L}_{Y}+\frac{u}{2}D_{C}Y^{C})\hat{N}_{AB},$
(5.3.8)

hence $\hat{N}_{AB}$ transforms homogeneously. Any vacuum configuration, with
a vanishing Weyl tensor, will verify, in particular, the suitable condition
$\hat{N}_{AB}=0$. This refines the notion of stationarity in a covariant
manner under the full Generalized BMS4 group. Inspired by the shift (5.3.7) we
also define

$\hat{C}_{AB}\equiv C_{AB}-uN_{AB}^{vac}[\Phi]$ (5.3.9)

whose transformation law is

$\delta_{T,Y}\hat{C}_{AB}=[(T+\frac{u}{2}D_{C}Y^{C})\partial_{u}+\mathcal{L}_{Y}-\frac{1}{2}D_{C}Y^{C}]\hat{C}_{AB}-2(D_{A}D_{B}T)^{TF}+TN_{AB}^{vac},$
(5.3.10)

and $\hat{N}_{AB}=\partial_{u}\hat{C}_{AB}$ is at most finite in $u$. In the
vacua, by virtue of (5.1.13), we have

$\lim_{u\to\pm\infty}\hat{C}_{AB}=C_{\pm}N_{AB}^{vac}[\Phi]+C_{AB}^{(0)}[C_{\pm}]=C_{\pm}N_{AB}^{vac}[\Phi]-(2D_{A}D_{B}C_{\pm})^{TF}$
(5.3.11)

which the generalization of a pure-supertranslation shear tensor (see (2.1.63)
and (2.1.64)) in an arbitrary superboosted frame. It is also purely electric
in the sense that

$\left.\left[(D_{B}D_{C}-\frac{1}{2}N_{BC}^{vac})\hat{C}{{}_{A}}^{C}-(D_{A}D_{C}-\frac{1}{2}N_{AC}^{vac})\hat{C}{{}_{B}}^{C}\right]\right|_{\mathscr{I}^{+}_{\pm}}=0.$
(5.3.12)

This generalizes (5.3.1) for any non-trivial superboost field $\Phi$ and can
be proved thanks to (5.1.15) and (5.1.19). Using (5.1.15), one can prove that
$N_{AB}^{vac}$ trivially satisfies (5.3.12), hence we can trade $\hat{C}_{AB}$
for $C_{AB}$ in this condition. Hence, to say that $C_{AB}$ is pure-gauge or
purely electric as (5.3.12) are equivalent statements. The incorporation of
ADM zero modes at spatial infinity does not change any of these conclusions,
so we can demand that the spacetime decays to vacuum configurations for
$|u|\to+\infty$ modulo these zero modes which however produce non-trivial
contributions in the Weyl tensor (the Newtonian potential for instance). This
ends our discussion on the behavior of the fields at $\mathscr{I}^{+}_{\pm}$.
Let us explain now how to use these observations to build finite and well-
behaved charges for the Generalized BMS4 symmetries.

#### 5.3.2 Finite Hamiltonians from the Generalized BMS4 charges

As we explained in section 2.2.4.2, the Generalized BMS4 infinitesimal surface
charges (2.2.126) are not integrable because the radiative flat spacetimes
emit some flux through future null infinity $\mathscr{I}^{+}$. In that
context, the definition of an Hamiltonian (i.e. an integrable part) associated
with any asymptotic symmetry $\xi$ is ambiguous because there is no preferred
split between integrable and non-integrable parts in the infinitesimal surface
charge in the absence of further boundary conditions allowing to fix this
ambiguity. In other words, the covariant phase space methods only fix the
infinitesimal charge variation, not the finite charge variation due to the
lack of integrability: some prescription is therefore required to define the
finite charge related to the asymptotic symmetries.

To that end, we introduce four requirements that are physically motivated and
provide a set of minimal conditions to be obeyed by the suitable finite
Hamiltonian $\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$. We denote by $\phi$ the
whole collection of metric parts as well as foliations and boundary data we
need, including the vacuum fields $\Phi,C_{\pm}$ previously defined. Here are
the requirements:

  1. I.

$\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ is finite in the non-compact Bondi
coordinates $r$ and $u$.

  2. II.

The flux of $\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ vanishes identically for
stationary configurations $\hat{N}_{AB}=0$.

  3. III.

$\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ is zero in any vacuum configuration
for which the Weyl tensor vanishes.

  4. IV.

The algebra of charges $\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ closes under
the Poisson bracket (2.3.3) at $\mathscr{I}^{+}_{\pm}$.

The requirement I is very natural since we want to manipulate well-defined
Hamiltonians on each point of $\mathscr{I}^{+}=\\{r\to+\infty\\}$ and consider
the limits to spatial and (future) timelike infinities. Thanks to the
regularization procedure developed in section 2.2.4 and confirmed by the
robust construction explained in section 4.3.3, the finiteness in $r$ is
already implemented for any Hamiltonian deducible from (2.2.126), including
the minimal choice $H_{\xi}^{\text{GBMS}_{4}}[\phi]$ given by (2.2.127). But
using (5.3.3)–(5.3.5), one can show that the latter is not finite in $u$,
hence it has to be denied as a good choice for the desired finite Hamiltonian.
Recalling (2.2.77), we define

$\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]\equiv
H_{\xi}^{\text{GBMS}_{4}}[\phi]+\Delta
H_{\xi}^{\text{GBMS}_{4}}[\phi]=\mathcal{O}(u^{0}).$ (5.3.13)

Among the prospective finite Hamiltonians, Wald and Zoupas [31] proposed to
adjust the definition of $\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ such that the
local flux $\partial_{u}\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]=0$ for
stationary configurations. In the standard BMS4 phase space, a stationary
configuration verifies $N_{AB}=0$. The latter is not a statement that is
invariant under super-Lorentz transformations, see (2.1.73) or the discussion
above (5.3.7). Instead we can impose that the physical news tensor
$\hat{N}_{AB}$ vanishes in any super-Lorentz frame since it transforms
homogeneously under all asymptotic symmetries, according to (5.3.8). We are
led to devise a generalized notion of stationarity demanding the vanishing of
$\hat{N}_{AB}$ instead of $N_{AB}$. Hence the requirement II is the extension
of Wald-Zoupas’ prescription to arbitrary super-Lorentz frames [45].

Particular stationary configurations are the vacuum configurations which are
locally flat (i.e. the Weyl tensor vanishes). The vacua of gravitational
fields are not expect to carry any mass or angular momentum that cannot be
created by diffeomorphism acting on Minkowski spacetime. From this point of
view, the suggestion III is the reasonable requirement that the Riemann-flat
vacuum is the reference that fixes the “offset” of the Hamiltonian
$\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$. The curvature being a tensor field, a
good choice of charges will give zero for this vacuum written in any
Generalized BMS4 frame.

Finally recalling the theorem (2.3.13) and its modification under some shift
of the Hamiltonian as (2.3.21), we deduce that (5.3.13) satisfies the
following algebra under the standard Poisson bracket [37]

$\left\\{\bar{H}_{\xi_{1}}^{\text{GBMS}_{4}}[\phi],\bar{H}_{\xi_{2}}^{\text{GBMS}_{4}}[\phi]\right\\}=\bar{H}_{[\xi_{1},\xi_{2}]_{\star}}^{\text{GBMS}_{4}}[\phi]+R_{\xi_{1},\xi_{2}}[\phi],$
(5.3.14)

where the residue

$R_{\xi_{1},\xi_{2}}[\phi]=K^{\text{GBMS}_{4}}_{\xi_{1},\xi_{2}}[\phi]-\Delta
H_{[\xi_{1},\xi_{2}]_{\star}}^{\text{GBMS}_{4}}[\phi]+\delta_{\xi_{2}}\Delta
H_{\xi_{1}}^{\text{GBMS}_{4}}[\phi]-\Xi_{\xi_{2}}^{\text{GBMS}_{4}}[\phi;\delta_{\xi_{1}}\phi]$
(5.3.15)

is defined from (2.3.24) and (2.2.128). Written in that way, it is not
manifestly antisymmetric. At early and late times, the absence of radiation
and physical flux suggests that the charge algebra should close under the
standard Poisson bracket which does not involve any non-integrable
contribution, since the latter would be intimately related to some non-
equilibrium physics at the boundary, nonexistent by hypothesis at the corners
$\mathscr{I}^{+}_{\pm}$. Hence demanding that IV is respected is equivalent to
affirm that there exists a shift (5.3.13) such that
$R_{\xi_{1},\xi_{2}}[\phi]\to 0$ when $|u|\to+\infty$ for any couple of
Generalized BMS4 generators $\xi_{1},\xi_{2}$. Considering (5.3.15), this
brings an involved constraint on the shift $\Delta
H_{\xi}^{\text{GBMS}_{4}}[\phi]$. Unlike the three constraints I–III that
allow for a considerable ambiguity on the finite Hamiltonian (i.e. any finite
term vanishing in the vacuum and whose time derivative is zero when
$\hat{N}_{AB}$ can be added without ruining any constraint), the last
requirement IV sharply restricts the set of admissible terms to be added to
the Hamiltonian. We do not attempt here to find the general solution of
(5.3.15) in terms of $\Delta H_{\xi}^{\text{GBMS}_{4}}[\phi]$ which is a
highly non-trivial task. But we can even restrict the possible ambiguity by
demanding that the shift (5.3.13) is a local function of $C_{AB}$, the initial
data $q_{AB}$, $N_{AB}^{vac}$ and the gauge parameters $f,Y^{A}$ only. The
dependency in the gauge parameters is assumed to be linear while the
dependency in $C_{AB}$ or $N_{AB}^{vac}$ is at most quadratic.

With these clarifications in mind, we claim that the finite Hamiltonians
satisfying the four prescriptions I–IV are given by

$\boxed{\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]=\oint_{S_{\infty}}\frac{\text{d}^{2}\Omega}{16\pi
G}\left[4\,T\,\bar{M}+2\,Y^{A}\,\bar{N}_{A}\right]}$ (5.3.16)

for

$\displaystyle\bar{M}$ $\displaystyle=M+\frac{1}{8}C_{AB}N^{AB}_{vac},$
(5.3.17) $\displaystyle\bar{N}_{A}$
$\displaystyle=N_{A}-u\partial_{A}\bar{M}+\frac{1}{4}C_{AB}D_{C}C^{BC}+\frac{3}{32}\partial_{A}(C_{BC}C^{BC})$
$\displaystyle\quad\,+\frac{u}{4}D^{B}(D_{B}D_{C}-\frac{1}{2}N_{BC}^{vac}){C_{A}}^{C}-\frac{u}{4}D^{B}(D_{A}D_{C}-\frac{1}{2}N_{AC}^{vac}){C_{B}}^{C}.$
(5.3.18)

This is a new prescription for the Generalized BMS4 Hamiltonians that reunites
the expressions of [45] and [37], the first paper focusing on both
requirements I and II and the second one further discussing both requirements
III and IV in the last section. We managed to reduce the amount of parentheses
in (5.3.18) but the $\mathcal{O}(u)$ should be read as $D^{B}S_{[AB]}$ for
$S_{AB}=(D_{A}D_{C}-\frac{1}{2}N_{AC}^{vac}){C_{B}}^{C}$. It is worth noticing
for the discussions below that the differential operator defining $S_{[AB]}$
first appeared in the generalized electricity condition (5.3.12).

We can immediately check by a straightforward computation that the charges
(5.3.16) are finite in $u$ at the corners $\mathscr{I}^{+}_{\pm}$. Indeed, we
can identify the leading terms in (2.1.44) and (2.1.45) when $|u|\to+\infty$,
providing the divergences in $\mathcal{O}(u)$ in $M$ and $N_{A}$ after
integration in $u$. This information can be used to prove that (5.3.17) and
(5.3.18) go to constants at early and late times, because
$\partial_{u}\bar{M}\to 0$ and $\partial_{u}\bar{N}_{A}\to 0$. We do not give
the detailed demonstration here.

The explicit shift that must be introduced in (5.3.13) is

$\boxed{\begin{aligned} \Delta
H_{\xi}^{\text{GBMS}_{4}}[\phi]=&\oint_{S_{\infty}}\frac{\text{d}^{2}\Omega}{16\pi
G}\left[\frac{1}{2}fC_{AB}N^{AB}_{vac}+\frac{1}{2}Y^{A}C_{AB}D_{C}C^{BC}+\frac{1}{8}Y^{A}\partial_{A}(C^{BC}C_{BC})\right.\\\
&+\frac{u}{2}\left.Y^{A}D^{B}(D_{B}D_{C}-\frac{1}{2}N_{BC}^{vac}){C_{A}}^{C}-\frac{u}{2}\,Y^{A}D^{B}(D_{A}D_{C}-\frac{1}{2}N_{AC}^{vac}){C_{B}}^{C}\right].\end{aligned}}$
(5.3.19)

The first term aims at subtracting the vacuum contribution to the Bondi mass
aspect (5.1.30). The second and third terms operate similarly on the vacuum
Bondi angular momentum aspect, which demonstrates that (5.3.16) is identically
zero if the Weyl curvature is zero and thus obeys the prescription III. The
last terms, linear in $C_{AB}$, are vanishing in the vacua for which
$\hat{C}_{AB}$ is purely electric, see (5.3.12) and do not contribute in the
limit $|u|\to+\infty$ for our choice of boundary conditions in $u$. Let us
stress again that $C_{AB}$ can be traded for $\hat{C}_{AB}$ without cost
thanks to (5.1.15). These terms were not included in the charge expression
given in [37] but do not change any conclusion of this paper since they vanish
identically in the vacuum as well as at $\mathscr{I}^{+}_{\pm}$. Nevertheless,
they are necessary to obey the prescription II in general at any retarded time
$u$, as shown in [45]. Indeed, nothing prevents $\hat{C}_{AB}$ from getting a
non-trivial magnetic part in the non-radiative phases for $|u|<+\infty$ when
$\hat{N}_{AB}=0$ and the latter can contribute to the flux, which should
contradict our hypothesis II. We claim that the shift (5.3.19) is the unique
local function of $q_{AB}$, $C_{AB}$ and $N_{AB}^{vac}$ only such that
(5.3.16) fulfills the requirements I–IV.

In a Bondi frame where $q_{AB}=\mathring{q}_{AB}$ and $N_{AB}^{vac}=0$, we
have

$\displaystyle\bar{M}$ $\displaystyle=M,$ (5.3.20) $\displaystyle\bar{N}_{A}$
$\displaystyle=N_{A}-u\partial_{A}M+\frac{1}{4}C_{AB}D_{C}C^{BC}+\frac{3}{32}\partial_{A}(C_{BC}C^{BC})$
$\displaystyle\quad\,+\frac{u}{4}D^{B}D_{B}D_{C}{C_{A}}^{C}-\frac{u}{4}D^{B}D_{A}D_{C}{C_{B}}^{C}.$
(5.3.21)

The “soft” $\mathcal{O}(u)$ terms still appear in (5.3.21) and only depend on
the magnetic part of the shear. Performing integrations by parts on the
sphere, we observe that these terms contribute as $\sim
uC_{zz}D_{z}^{3}Y^{z}+\text{c.c.}$ in the charges when working in
stereographic coordinates and therefore vanish identically for global Lorentz
symmetries. They are shown to be mandatory in the discussion of the subleading
soft theorem [270, 45], see section 5.3.4.2. But if we restrict only to
Lorentz generators $Y$, we have

$\boxed{\mathscr{L}_{Y}\equiv
H_{\xi(0,Y)}^{\text{GBMS}_{4}}[\phi]=\oint_{S_{\infty}}\frac{\text{d}^{2}\Omega}{16\pi
G}Y^{A}\Big{[}N_{A}-u\partial_{A}M+\frac{1}{4}C_{AB}D_{C}C^{BC}+\frac{3}{32}\partial_{A}(C_{BC}C^{BC})\Big{]}.}$
(5.3.22)

Our proposal for the angular momentum thus coincides with [98, 100], see the
matching of conventions regarding the Bondi angular momentum aspect below
(2.1.32).

#### 5.3.3 Flux algebra and realization at spatial infinity

Inserting the shift (5.3.19) into (5.3.15) while evaluating (2.2.128) and
(2.3.24), we can show after a quite long computation that
$R_{\xi_{1},\xi_{2}}[\phi]=0$ at the corners $\mathscr{I}^{+}_{-}$ and
$\mathscr{I}^{+}_{+}$. This is precisely what we wanted by imposing the
requirement IV and the Generalized BMS4 surface charge algebra closes under
the standard Poisson bracket at $\mathscr{I}^{+}_{-}$ and
$\mathscr{I}^{+}_{+}$ without any central extension, i.e.

$\boxed{\\{\bar{H}_{\xi_{1}}^{\text{GBMS}_{4}}[\phi],\bar{H}_{\xi_{2}}^{\text{GBMS}_{4}}[\phi]\\}\Big{|}_{\mathscr{I}^{+}_{\pm}}=\bar{H}_{[\xi_{1},\xi_{2}]_{\star}}^{\text{GBMS}_{4}}[\phi]\Big{|}_{\mathscr{I}^{+}_{\pm}}.}$
(5.3.23)

This is a highly non-trivial result. For vacuum configurations, this is direct
because of the hypothesis III since all Generalized BMS4 charges
$\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ are zero. But the result obtained here
is stronger and means that this algebra is valid also for non-vacuum
configurations with Poincaré charges such as mass and angular momentum, in the
presence of displacement memory and in arbitrary Lorentz and super-Lorentz
frames. It also applies at the corners of the past null boundary
$\mathscr{I}^{-}$ under similar boundary conditions. In fact, since there is
no flux at spatial infinity, all Generalized BMS4 Hamiltonians
$\bar{H}_{\xi}^{\text{GBMS}_{4}}[\phi]$ are conserved at spatial infinity and
the Generalized BMS4 charge algebra (5.3.23) is therefore realized also at
spatial infinity, including the super-Lorentz transformations. Since the
Generalized BMS4 Hamiltonians are all generically non-vanishing from their
explicit expressions (5.1) and (5.9) of [317], we have proven the following
theorem:  The asymptotic symmetry algebra of asymptotically locally flat
spacetimes in $\mathring{\mathcal{S}}_{0}$ with non-radiative boundary
conditions at early and late times is the Generalized BMS4 charge algebra
(5.3.23) without central extension. This means that the Generalized BMS4
asymptotic symmetry group acts on the whole boundary of flat spacetime,
including spatial infinity. This extends the result derived in [35, 33] in the
Hamiltonian formalism and includes super-Lorentz asymptotic symmetries as
well.

Now let us consider the local fluxes of the charges. They are given at any $u$
by

$\boxed{\begin{aligned} \partial_{u}&H^{\text{GBMS}_{4}}_{\xi(T,Y)}[\phi]\\\
&=-\frac{1}{32\pi
G}\oint_{S}\left[f\left(N^{AB}\hat{N}_{AB}-2D^{A}D^{B}\hat{N}_{AB}\right)-2Y^{A}C_{AB}D_{C}\hat{N}^{BC}-Y^{A}D_{B}(C_{AC}\hat{N}^{BC})\right.\\\
&\phantom{-\frac{1}{32\pi
G}Y^{A}\Big{[}}\quad\left.-uY^{A}D_{B}(D^{B}D^{C}-\frac{1}{2}N^{BC}_{vac})\hat{N}_{AC}+uY^{A}D_{B}(D_{A}D_{C}-\frac{1}{2}N_{AC}^{vac})\hat{N}^{BC}\right]\end{aligned}}$
(5.3.24)

where the parameter $f=T+\frac{u}{2}D_{A}Y^{A}$ has been retained against $T$
for the sake of conciseness. To obtain these expressions, several
manipulations are necessary. They involve the equations of motion (2.1.44) and
(2.1.45), the boundary conditions (5.3.3)–(5.3.5), the redefinitions (5.3.7)
and (5.3.9) as well as the identities (E.11) and (E.12). Thanks to the
“Christodoulou-Klainermann fall-off condition” $\hat{N}_{AB}=o(u^{-1})$, it is
easy to show that the local fluxes (5.3.24) are vanishing in the limit
$|u|\to+\infty$, in accordance with the requirements I–III. Hence the
Hamiltonians (5.3.16) go to constants at the corners, are finite in $u$ and
their fluxes go to zero when the spacetime reverse back to stationary
configurations at early and late times. We also observe that these fluxes
vanish identically on any interval $\mathscr{U}\subset\mathscr{I}^{+}$ if and
only if $\hat{N}_{AB}=0$ for all $u\in\mathscr{U}$. They are thus zero for
stationary configurations and obey the requirement III! Note crucially that
the two last terms in (5.3.13) are necessary to have the corresponding
differential operator $D_{A}D_{B}-\frac{1}{2}N_{AB}^{vac}$ acting on
$\hat{N}_{AB}$ in (5.3.24) instead of the whole $C_{AB}$ which may contains a
magnetic part contributing to the flux in generic radiative situations.

We define the total integrated fluxes at $\mathscr{I}^{+}$ as

$\bar{F}_{\xi}[\phi]=\bar{H}^{\text{GBMS}_{4}}_{\xi}[\phi]\Big{|}_{\mathscr{I}_{+}^{+}}-\bar{H}^{\text{GBMS}_{4}}_{\xi}[\phi]\Big{|}_{\mathscr{I}_{-}^{+}}=\int_{-\infty}^{+\infty}\text{d}u\,\partial_{u}\bar{H}^{\text{GBMS}_{4}}_{\xi}[\phi].$
(5.3.25)

We denote the supermomenta fluxes and super-Lorentz fluxes as
$\bar{P}_{T}[\phi]\equiv\bar{F}_{\xi(T,0)}[\phi]$ and
$\bar{J}_{Y}[\phi]\equiv\bar{F}_{\xi(0,Y)}[\phi]$, respectively. An immediate
consequence of (5.3.23) is the following algebra of Generalized BMS4 fluxes,

$\boxed{\\{\bar{P}_{T_{1}},\bar{P}_{T_{2}}\\}=0,\quad\\{\bar{J}_{Y_{1}},\bar{P}_{T_{2}}\\}=\bar{P}_{Y_{1}(T_{2})},\quad\\{\bar{J}_{Y_{1}},\bar{J}_{Y_{2}}\\}=\bar{J}_{[Y_{1},Y_{2}]}}$
(5.3.26)

where
$Y_{1}(T_{2})\equiv(Y_{1}^{A}\partial_{A}-\frac{1}{2}D_{A}Y^{A}_{1})T_{2}$.
Therefore, the algebra of Generalized BMS4 fluxes represents the Generalized
BMS4 algebra of asymptotic symmetries (2.1.57) without central extension. This
confirms the construction of [194], which can now be deduced from covariant
phase space methods where renormalization is provided by the flat limit of the
$\Lambda\neq 0$ holographic renormalization scheme and our new treatment of
corner terms, presented in section 4.3.3.

#### 5.3.4 Infrared structures and soft theorems

In this section, we give important comments on the various relationships
existing in the low energy sector of the asymptotically flat theory of
gravity, between physical observable predictions (gravitational memory
effects), mathematical features of the radiative phase spaces (the Generalized
BMS4 symmetries) and infrared factorization properties of the scattering
quantum amplitudes (the soft theorems) [57, 98]. We review the junction
conditions that must be imposed around spatial infinity in order to be
provided with a well-defined gravitational scattering problem [112], implying
asymptotic conservation laws for between past and future null infinities for
the finite Hamiltonian derived in section 5.3.2. Under these additional
boundary constraints, we show that the flux of charges computed in section
5.3.3 reproduces the flux-balance laws needed for the matching with the
leading [114, 113] and subleading [117, 58] soft theorems. Finally, we review
Strominger’s leading infrared triangle of relationships [57], involving the
supertranslation symmetry and contrasts it with an hybrid
overleading/subleading incomplete square of relationships, this time involving
the super-Lorentz symmetries.

##### 5.3.4.1 Gravitational scattering problem and junction conditions

The null motion in asymptotically flat spacetimes at null infinity is governed
by hyperbolic equations of motion for which initial conditions are given at
past null infinity $\mathscr{I}^{-}$ and the evolution of null rays propagates
this information to future null infinity $\mathscr{I}^{+}$. Both limiting
hypersurfaces come with the same structure, namely a null normal $\bm{T}$ and
a boundary volume $\sqrt{\mathring{q}}$. These structures are preserved by two
copies of the Generalized BMS4 group, one living on $\mathscr{I}^{+}$ and
denoted as G-BMS${}_{4}^{+}$ and an homologous one, denoted as
G-BMS${}_{4}^{-}$, living on $\mathscr{I}^{-}$ where the whole analysis
performed so far can be repeated up to considering advanced Bondi coordinates
$(v,r,x^{A})$ instead of retarded Bondi coordinates $(u,r,x^{A})$. The
evolution of null matter as well as gravitational waves from data imposed
around $\mathscr{I}^{-}$, crossing the bulk of spacetime and reaching
$\mathscr{I}^{+}$ defines the (null) scattering problem in asymptotically flat
spacetimes. It gives the fundamental element to build the S-matrix for
gravity, at least in the low-energy (infrared) regime, for which the “in”
states are incoming at $\mathscr{I}^{-}$ and the “out” states are outgoing at
$\mathscr{I}^{+}$. As argued by Strominger in [112], a prescription should be
given to relate the “in” and the “out” states in order to ensure the well-
definiteness of the scattering problem. In particular, without such matching
conditions, the two copies of Generalized BMS4 are independent, which is not
suitable: indeed, the action of a G-BMS${}_{4}^{-}$ transformation on the
initial data should be reflected at some point in the late data arriving at
$\mathscr{I}^{+}$. In other words, the two asymptotic groups cannot be
independent symmetries acting on both the initial and final states. Moreover,
there is one and only one copy of the Generalized BMS4 group acting at spatial
infinity, which also suggests that a diagonal subgroup of G-BMS${}_{4}^{-}$
$\times$ G-BMS${}_{4}^{+}$ has to be selected and manifests its presence at
spatial infinity [36, 35, 37, 33, 34].

The remaining question is: which diagonal subgroup is selected? Or in other
words, how to identify the Generalized BMS${}_{4}^{+}$ generators at
$\mathscr{I}^{+}$ with the ones at $\mathscr{I}^{-}$? The crucial clue is that
propagating null fields in the Minkowski vacuum obey universal antipodal
matching conditions at spatial infinity. For each bulk field $\phi^{i}$, one
can define its limit at $\mathscr{I}^{+}$ and then take $u\rightarrow-\infty$
which defines the field at $\mathscr{I}^{+}_{-}$. The field at
$\mathscr{I}^{-}_{+}$ is obtained in a similar way up to the standard trading
of $u$ for $v$. An antipodal matching condition would mean that

$\phi^{i}(\theta,\varphi)\Big{|}_{\mathscr{I}^{+}_{-}}=\phi^{i}(\pi-\theta,\varphi+\pi)\Big{|}_{\mathscr{I}^{-}_{+}}$
(5.3.27)

where $(\theta,\varphi)$ are the colatitude and azimuth angles on the sphere
at $|u|\to+\infty$. For instance, it turns out that the electromagnetic
Liénard-Wiechert field describing the retarded electromagnetic field of a
uniformly moving source obeys these antipodal matching conditions [318, 319].
The metric of the boosted Kerr black hole [320] is also expected to obey these
conditions. Moreover, these conditions are $CPT$ invariant and Lorentz
invariant. Since the seminal work [112], it is admitted that the metric field
in asymptotically flat spacetimes obeys the antipodal matching conditions.
Yet, it has not been derived whether or not the antipodal map is generally
valid for any subleading field in the asymptotic expansion close to null
infinity and for other types of causal structures, such as spacetimes
containing a black hole formed from stellar collapse. For the leading order
fields, consistent boundary conditions, which admit antipodal matching
boundary conditions, were found both in $3d$ [36] and $4d$ Einstein gravity
[35, 33, 34].

From the perspective of perturbative quantum gravity, the antipodal matching
conditions imply the BMS invariance of the gravitational S-matrix in the sense
that acting with the Generalized BMS4 fluxes $\bar{P}_{T}$, $\bar{J}_{Y}$ on
the “in” states or “out” states are two commuting operations:

$\bar{P}_{T}^{+}\text{S}=\text{S}\bar{P}_{T}^{-},\qquad\bar{J}_{Y}^{+}\text{S}=\text{S}\bar{J}_{Y}^{-},$
(5.3.28)

where the superscript $+$ (resp. $-$) denotes the evaluation of the fluxes
through $\mathscr{I}^{+}$ (resp. $\mathscr{I}^{-}$). The study for global BMS4
has been developed in [112, 113] and extended to Diff($S^{2}$) super-Lorentz
symmetries in [58, 59]. The invariance of the gravitational S-matrix under the
asymptotic symmetry groups leads to Ward identities for supertranslations and
Lorentz transformations or their extension. The interesting – although not so
suprising – thing is that the supertranslation Ward identities are identical,
after a change of notation, to the soft graviton theorem [113], derived by
Weinberg in 1965 [114] for excitations around the Minkowski vacuum. At
subleading order, it was shown in [58] that the Diff$(S^{2})$ superrotation
Ward identities are identical, after rewriting, to the newly found Cachazo-
Strominger subleading soft graviton theorem [117]. The antipodal matching
conditions are therefore compatible with the soft theorems, which validates
their range of applicability around Minkowski spacetime. Indeed, sandwiching
(5.3.28) between some “in” and “out” quantum states at $\mathscr{I}^{-}$ and
$\mathscr{I}^{+}$ in order to compute the scattering amplitude
$\mathcal{M}_{n}(p_{1},\dots,p_{n})$ for a $n$-particle process, one gets

$\mathcal{M}_{n+1}(p_{0},p_{1},\dots,p_{n})=\left[\frac{1}{p_{0}}S^{(0)}+S^{(1)}+\mathcal{O}(p_{0})\right]\mathcal{M}_{n}(p_{1},\dots,p_{n})$
(5.3.29)

in the limit $p_{0}\to 0$. This states that in any scattering process
involving $n+1$ particules among which one finds a massless particle with very
weak momentum $p_{0}$ (a “soft particle”), the contribution of the latter
factorizes out in the scattering amplitude in the limit $p_{0}\to 0$. The soft
factors $S^{(0)}$ and $S^{(1)}$ are universal and only depend on the nature of
the soft particle. The presence of $S^{(0)}$ is due to the supertranslation
symmetry while the second factor $S^{(1)}$ is tightly related to super-Lorentz
symmetries. Proposals to continue the Taylor series in $p_{0}$ in the soft
limit have also been discussed e.g. in [321, 322] although the underlying
tower of symmetry principles remains to be found.

Assuming that the antipodal matching conditions hold in generality directly
leads to conservation laws. Indeed, physical quantities depend upon the fields
and if all relevant fields (including the symmetry parameters) are antipodally
matched, the finite Hamiltonian charges $\bar{H}_{\xi}^{\text{GBMS}_{4}}$ at
$\mathscr{I}^{+}_{-}$ and $\mathscr{I}^{-}_{+}$ (either supertranslations or
super-Lorentz) are related by the antipodal map symbolically denoted by
$\text{AntiPodMap}(\circ)$,

$\bar{H}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{+}_{-}}=\text{AntiPodMap}(\bar{H}_{\xi}^{\text{GBMS}_{4}})\Big{|}_{\mathscr{I}^{-}_{+}}$
(5.3.30)

Using the flux-balance laws of these charges, we deduce by integration along
$u$ and $v$ the conservation laws

$\begin{gathered}\int_{\mathscr{I}^{+}}\text{d}u\>\partial_{u}\bar{H}_{\xi}^{\text{GBMS}_{4}}+\bar{H}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{+}_{+}}=\text{AntiPodMap}\Big{(}\int_{\mathscr{I}^{-}}\text{d}v\>\partial_{v}\bar{H}_{\xi}^{\text{GBMS}_{4}}+\bar{H}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{-}_{-}}\Big{)}\\\
\Longleftrightarrow\\\
\bar{F}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{+}}+\bar{H}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{+}_{+}}=\text{AntiPodMap}\Big{(}\bar{F}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{-}}+\bar{H}_{\xi}^{\text{GBMS}_{4}}\Big{|}_{\mathscr{I}^{-}_{-}}\Big{)}\end{gathered}$
(5.3.31)

including the eventual contribution of matter originating at past timelike
infinity $\mathscr{I}^{-}_{-}$ and terminating at future timelike infinity
$\mathscr{I}^{+}_{+}$. These supertranslation and super-Lorentz conservation
laws are the conservation of energy and angular momentum at each angle on the
celestial sphere [112].

##### 5.3.4.2 Leading and subleading soft theorems

Let us show how to match our expressions for the fluxes with the expressions
of the literature used in the Ward identities that are equivalent to the
leading [114] and subleading [117] soft graviton theorems. The final flux
(5.3.24) can be decomposed in soft (i.e. linear in $\hat{C}_{AB}$,
$\hat{N}_{AB}$) and hard parts (i.e. quadratic in $\hat{C}_{AB}$,
$\hat{N}_{AB}$). Let us first consider the flux associated with
supertranslations $\xi(T,0)$. We have

$\bar{F}_{\xi(T,0)}^{\text{GBMS}_{4}}[\phi]=\int_{\mathscr{I}^{+}}\text{d}u\,\partial_{u}\bar{H}^{\text{GBMS}_{4}}_{\xi(T,0)}[\phi]=F_{S}[T]+F_{H}[T],$
(5.3.32)

that is, after integrations by parts on the sphere,

$F_{S}[T]=\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\;\hat{N}^{AB}D_{A}D_{B}T=-\frac{1}{32\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\;\hat{N}^{AB}\delta_{T}^{I}\hat{C}_{AB}$
(5.3.33)

and

$F_{H}[T]=-\frac{1}{32\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\;T\,\hat{N}^{AB}N_{AB}=-\frac{1}{32\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\;\hat{N}^{AB}\delta_{T}^{H}\hat{C}_{AB}.$
(5.3.34)

Here $\delta_{T}^{H}\hat{C}_{AB}$ and $\delta_{T}^{I}\hat{C}_{AB}$ represents
the homogeneous and inhomogeneous parts of the transformation under
$\xi=\xi(T,0)$ of $\hat{C}_{AB}$ defined as (5.3.9), which can be
straightforwardly deduced from (5.3.10). In the standard case where
$N_{AB}^{vac}=0$, the flux of supermomentum (5.3.33)-(5.3.34) reproduces
(2.11) of [113] up to a conventional overall sign, which itself agrees with
previous results [218]. After imposing the antipodal matching condition on the
renormalized Bondi mass aspect $\bar{M}$ (5.3.17) at spatial infinity, one can
equate the flux on $\mathscr{I}^{+}$ with the antipodally related flux on
$\mathscr{I}^{-}$. The result of [113] is precisely that the quantum version
of this identity is the Ward identity of the leading soft graviton theorem. We
have now obtained a generalization in the presence of a superboost background
field.

The treatment of super-Lorentz fluxes requires a bit more care. Starting again
from (5.3.24), one can isolate the terms associated with vectors $\xi(0,Y)$.
After a non-trivial computation involving the identities (E.11) and (E.12), we
can decompose the flux as

$\bar{F}_{\xi(0,Y)}^{\text{GBMS}_{4}}[\phi]=\int_{\mathscr{I}^{+}}\text{d}u\,\partial_{u}\bar{H}^{\text{GBMS}_{4}}_{\xi(0,Y)}[\phi]=F_{S}[Y]+F_{H}[Y]+Q_{\mathscr{I}^{+}_{\pm}}[Y].$
(5.3.35)

The hard term is completely analogous to (5.3.34) and reads as

$\begin{split}F_{H}[Y]&=-\frac{1}{32\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\;\hat{N}^{AB}\left(\mathcal{L}_{Y}\hat{C}_{AB}+\frac{u}{2}D_{C}Y^{C}\hat{N}_{AB}-\frac{1}{2}D_{C}Y^{C}\hat{C}_{AB}\right)\\\
&=-\frac{1}{32\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\;\hat{N}^{AB}\delta^{H}_{Y}\hat{C}_{AB}\end{split}$
(5.3.36)

where $\delta^{H}_{Y}$ is the homogeneous part of the transformation of
$\hat{C}_{AB}$. The expression matches (up to the overall conventional sign)
with equation (40) of [59] (see also (4.2) of [194]). The soft part simplifies
to

$F_{S}[Y]=\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\text{d}^{2}\Omega\,u\,\hat{N}^{AB}\Sigma_{AB}$
(5.3.37)

for a symmetric traceless tensor $\Sigma_{AB}$ given by

$\begin{split}\Sigma_{AB}&=\left[D_{A}D_{B}D_{C}Y^{C}-\mathcal{L}_{Y}N_{AB}^{vac}+\frac{1}{2}R[q]D_{(A}Y_{B)}-\frac{1}{2}D_{(A}(D^{C}D_{C}+\frac{1}{2}R[q])Y_{B)}\right]^{TF}\\\
&=\left[-\delta_{Y}N_{AB}^{vac}+\frac{1}{4}R[q]\delta_{Y}q_{AB}-\frac{1}{2}D_{(A}D^{C}\delta_{Y}q_{B)C}\right]^{TF}\end{split}$
(5.3.38)

using (5.1.28) and (2.1.71) in the form
$D^{C}\delta_{Y}q_{AC}=D_{C}D^{C}Y_{A}+\frac{1}{2}R[q]Y_{A}$. The tensor
(5.3.38) is recognized as the generalization of (47) of [59] in the presence
of a non-trivial boundary curvature and $N_{AB}^{vac}\neq 0$ (see also (4.14)
of [194]). Since $N_{AB}^{vac}$ and $q_{AB}$ are invariant under Lorentz
transformations, $\Sigma_{AB}=0$ for any boost or rotation. The last piece is
the boundary term

$Q_{\mathscr{I}^{+}_{\pm}}[Y]=\frac{1}{32\pi
G}\int_{\mathscr{I}^{+}_{\pm}}\text{d}^{2}\Omega\left(Y^{A}\hat{C}_{AB}D_{C}\hat{C}^{BC}+\frac{1}{4}Y^{A}\partial_{A}(\hat{C}_{BC}\hat{C}^{BC})\right)$
(5.3.39)

where the notation $\int_{\mathscr{I}^{+}_{\pm}}$ stands for the difference of
the sphere integrals between $\mathscr{I}^{+}_{+}$ and $\mathscr{I}^{+}_{-}$
(also proposed in [194] in a different form). This is precisely the flux
contribution of the shift needed to subtract the non-trivial super-Lorentz
charges of the vacua and vanishes when $Y^{A}$ generates a Lorentz symmetry
[99, 45]. Hence the flux of angular momentum is only (5.3.36) in agreement
with [14]. Furthermore, as a boundary term in the flux, it cancels out in the
Ward identity and does not affect the soft theorems [194].

Now that we identified our expressions with the ones of [59, 194], we can use
their results. After imposing the antipodal matching condition on
$\bar{N}_{A}$ at spatial infinity, one can equate the flux of super-Lorentz
charge on $\mathscr{I}^{+}$ with the antipodally related flux on
$\mathscr{I}^{-}$ as originally proposed in [44] (but where the expression for
$\bar{N}_{A}$ should be modified to (5.3.18)). The result of [59] is precisely
that this identity is the Ward identity of the subleading soft graviton
theorem [117]. The same result holds here in an arbitrary super-Lorentz frame.

Finally, considering only the background Minkowski spacetime
($q_{AB}=\mathring{q}_{AB}$ and $N_{AB}^{vac}=0$), one can check that in
stereographic coordinates one has
$\Sigma_{zz}=\partial_{z}^{3}Y^{z}=D_{z}^{3}Y^{z}$. The soft piece of the
super-Lorentz flux then reads as

$F_{S}[Y]=\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}z\,\sqrt{\mathring{q}}\,(u\hat{N}^{zz}D_{{z}}^{3}Y^{{z}}+u\hat{N}^{\bar{z}\bar{z}}D_{\bar{z}}^{3}Y^{\bar{z}})$
(5.3.40)

where we keep
$Y^{A}\partial_{A}=Y^{z}(z,\bar{z})\partial_{z}+Y^{\bar{z}}(z,\bar{z})\partial_{\bar{z}}$
arbitrary. In the case of meromorphic super-Lorentz tranformations, this
reproduces equation (5.3.17) of [57] (up to the conventional global sign).
This concludes our checks with the literature concerning the soft theorems. It
shows that the Ward identities of supertranslations and super-Lorentz
transformations are equivalent to the leading and subleading soft graviton
theorems following the arguments of [113, 58].

##### 5.3.4.3 Infrared relationships

We close this chapter by summarizing the results discussed so far while giving
some diagrammatic relationships between them.

Since the works of Strominger and collaborators in the previous decade, we
know that the BMS4 supertranslation symmetry, the leading soft graviton
theorem and the displacement memory effect form three corners of a triangle
describing the leading infrared structure of asymptotically flat spacetimes at
null infinity [57]. The three edges of the triangle can be described in the
language of vacuum transitions, Ward identities and Fourier transforms, see
Figure 5.6, in the following sense:

  * $\rhd$

A supertranslation $T$ induces a vacuum transition mapping the Minkowski
vacuum $\eta_{\mu\nu}$ on a vacuum metric $g_{\mu\nu}[C]$ now labeled by a
time-independent field whose non-trivial value is generated by (5.1.26).
Although both vacuum carry zero charges, it is possible to probe the vacuum
transition by the mean of a Bondi detector described in section 5.2.2 built up
from two geodesic observers far away from the sources of radiation near
$\mathscr{I}^{+}$. The permanent shift they will experience after the vacuum
transition is the linear displacement memory effect that we reviewed in
section 5.2.2 and is precisely controlled by the (angle-dependent) amplitude
of the supertranslation involved in the process. That gives the left arrow of
the triangle.

  * $\rhd$

Any symmetry is associated with a finite Hamiltonian as defined in section
5.3.2 which obeys conservation laws for the scattering problem around spatial
infinity assuming some matching conditions between past and future null
infinities. We gave the motivations for the choice of antipodal matching
conditions [112] in section 5.3.4.1. Under such conditions, the conservation
laws can be sandwiched by “in” and “out” quantum states defined respectively
at past and future null infinity and lead to a Ward identity for the
supertranslation symmetry at quantum level [98, 113]. The Ward identity
implies the factorization in quantum scattering amplitudes of the low-energy
(soft) contributions, which is nothing but Weinberg’s soft graviton theorem
[114]. That gives the right arrow of the triangle.

  * $\rhd$

Finally, as shown in [98], if we refine the soft factor as
$p_{0}^{-1}S^{(0)}\equiv\pi^{\mu\nu}S_{\mu\nu}$ where $\pi_{\mu\nu}$ is the
polarization tensor of the soft graviton, the symmetric transverse traceless
tensor $S_{\mu\nu}$ can be recast as the leading stationary phase
approximation of the Fourier transform between time and momentum variables $u$
and $p_{0}$ of the metric shift occasioned by the displacement memory effect.
The latter shall be expressed as the difference in the linear perturbation
metric across the vacuum transition, in transverse traceless de Donder gauge
[95]. In short we can say that the Weinberg’s soft graviton theorem is the
Fourier transform of the kinematic equation governing the memory effect,
written as (5.2.3) in the Bondi gauge, which closes the triangle by the bottom
arrow.

Displacement memory effect Leading soft graviton theorem Supertranslation
Transformation Charge Vacuum transitionWard identityFourier transform Leading
BMS triangle Figure 5.6: Leading infrared triangular relationship.

The exceptional feature of the leading infrared triangle is that the diagram
closes and can be looped in any direction. This is due to the fact that the
supertranslation transformation as well as the associated charge describe
objects at the same order in $1/r$. Indeed, the supertranslation symmetry does
not modify the metric field at leading order on $\mathscr{I}^{+}$, i.e.
$\delta_{T}q_{AB}=0$. It impacts only the subleading field $C_{AB}$ and the
more suppressed orders in $1/r$ in the power series of $g_{AB}$. On the other
hand, the supertranslation charge is related to the subleading structure of
the gravitational field, i.e. the Bondi mass aspect, which appears also as a
$1/r$ field in the metric tensor. Therefore, there is no bifurcation at the
vertices of the diagram.

The situation is radically different in the case of super-Lorentz symmetry.
Indeed, for this complementary part of the Generalized BMS group, one needs to
distinguish the symmetry – which is “overleading” in the sense that it
modifies the leading order metric at future null infinity – and the charges
that still describe the subleading structure of gravitational fields, i.e. the
angular momentum aspect. If we attempt to build a triangle as in Figure 5.6
with the super-Lorentz symmetry sitting at the apex, this will be the locus of
a bifurcation. The branch induced to the left from the “transformation” node
will be a vacuum transition responsible for a shift in $q_{AB}$ at overleading
order with respect to the shift induced by supertranslations, while the other
branch running to the right from the “charge” node will lead to a subleading
Ward identity which is definitely not in the same footing that the left
branch. Hence we expect to get an open diagram which cannot be a cycle
anymore. Let us find out what are the edges of this novel diagram.

Refraction memory effect Subleading soft graviton theorem Super-Lorentz
Transformation Charge Spin memory effect Superboost transition Ward identity
Fourier transform Overleading/Subleading BMS square Figure 5.7:
Overleading/Subleading infrared relationship.

Two edges have been previously drawn relating super-Lorentz charges to the
subleading soft graviton theorem by Ward identities [56, 58, 59] and relating
the subleading soft graviton theorem to the spin memory effect by a Fourier
transformation [100] in a perfect analogy with the leading set of
relatioships. The spin memory effect is not described by a local memory field
in the Bondi gauge, which is the reason of its absence in our description of
gravitational memories. The spin memory effect is related to the time integral
of the magnetic part of $C_{AB}$ during radiative phases, see [211] for more
explanations. It was also shown that a local memory field leading to the spin
memory effect can be found in retarded harmonic gauge among the radiative
components of the Weyl tensor [118]. This completes the right chain of arrows
in the Figure 5.7 connecting nodes which involve consistent objects at the
same order in $1/r$ (namely $N_{A}$ and the magnetic part of $C_{AB}$). In
section 5.2.3, we clarified how the superboost transitions lead to the
refraction or velocity kick memory effect at null infinity. The latter are
different from the spin memory effect and are encoded with local functions in
the Bondi gauge. This gives the left arrow in the Figure 5.7 involving now
exclusively overleading quantities with respect to supertranslations. This
definitively suggests to describe the infrared structure related to super-
Lorentz transformations by an hybrid overleading/subleading structure which
looks like an (incomplete) square instead of a triangle.

It could be expected that a similar overleading/subleading square structure
will also appear in the description of other gauge and gravity theories when
the symmetry transformations and the associated charges do not probe and
involve the fields at the same order in $1/r$. This is the last comment that
we shall make to close this dissertation on infrared structures of
asymptotically flat spacetimes. $\blacksquare$

##  Chapter 6 Conclusion and perspectives

In this final chapter, we summarize the main points from the various results
discussed throughout the thesis, and then offer some ideas for further
examination that we propose to follow in the future.

###  6.1 Summary of the main results

We begin with a summary of the ideas and results defended in this manuscript,
providing an overview of what has been discussed throughout the whole
dissertation, which could be used as a memory refresh after reading.

###### Gravity with leaky boundary conditions

We began with a theoretical presentation of covariant phase space methods in
gravity, mixing a review of well-known results and some brand new
improvements.

  * $\rhd$

We introduced the dichotomic terminology to distinguish between conservative
boundary conditions leading to stationary variational principles and leaky
boundary conditions designed to model non-stationary situations for which some
gauged degrees of freedom are fluxing through the boundary under
consideration. The second class is the mandatory mathematical framework to
formulate radiative solutions in General Relativity which lies at the core of
our discussion and interests.

  * $\rhd$

The permeability of the boundary yields non-equilibrium physics giving rise to
non-conserved and non-integrable charges. The algebraic properties of non-
integrable charges are encapsulated by the Barnich-Troessaert bracket, for
which the representation theorem has been extended and proven using few
assumptions, namely the existence of a phase space provided with a
differential structure.

This long technical introduction was required to understand the physical
results obtained further in two (not so) distinct contexts, the Al(A)dS and
the asymptotically flat spacetimes.

###### Asymptotically locally (A)dS spacetimes

First, we focused on the class of radiative Al(A)dS spacetimes in $d+1$
dimensions.

  * $\rhd$

In the absence of further boundary conditions beyond the minimal fall-offs
allowing for conformal compactification, Al(A)dSd+1 spacetimes admit a
permeable boundary similar to the null boundary of asymptotically flat
spacetimes. Arbitrary boundary diffeomorphisms are associated with finite
surface Hamiltonian charges that obey a flux-balance law while Weyl rescalings
admit a non-vanishing Hamiltonian charge only for even $d$. For these
particular cases, this peculiar feature is seen as the manifestation of the
Weyl anomaly in the dual conformal field theory.

  * $\rhd$

We proved that the Hamiltonians for the class of boundary diffeomorphisms
represent the diffeomorphism algebra under the Barnich-Troessaert bracket
without non-trivial central extensions for odd $d$. In even $d$ the non-
vanishing contribution of the Weyl symmetry to the charges brings a non-
trivial field-dependent 2-cocycle at the level of the algebra, reducing to the
Brown-Henneaux central charge when $d=2$ with Dirichlet boundary conditions.

  * $\rhd$

In order to distinguish the radiative boundary degrees of freedom among the
components of the boundary metric and holographic stress-tensor, we proposed a
boundary gauge fixing which can be reached locally without constraining the
initial value problem. It consists in fixing a boundary foliation and measure,
which reduces the boundary diffeomorphism algebra to the $\Lambda$-BMSd+1
algebroid.

  * $\rhd$

The boundary gauge fixing is a weaker condition than Dirichlet boundary
conditions in bulk dimensions higher than three, since for $d=2$, they reduce
to the Dirichlet boundary conditions and the asymptotic symmetry group is
therefore nothing else than the standard two copies of the De Witt algebra.
Instead, in $d+1>3$ dimensions, the codimension two boundary metric $q_{AB}$
orthogonal to the foliation still admits $d-1$ arbitrary functions (since its
determinant is fixed). These free functions determine the structure constants
of the $\Lambda$-BMSd+1 algebroid.

  * $\rhd$

The flat limit of the $\Lambda$-BMSd+1 algebroid gives the Generalized BMS
algebra in $d+1$ dimensional asymptotically flat spacetimes, consisting of an
abelian ideal of smooth supertranslations in semi-direct product with super-
Lorentz transformations, both being field-independent.

We then particularized this analysis to the physical case of four spacetime
dimensions.

  * $\rhd$

We obtained the most general solution space in the Bondi gauge with a non-
vanishing cosmological constant, written within the convenient $\Lambda$-BMS
boundary gauge fixing. Thanks to a diffeomorphism between the SFG and Bondi
gauges, we matched the radiative data naturally defined in the Bondi gauge
with the holographic fields provided by the SFG expansion as well as the
asymptotic symmetry parameters.

  * $\rhd$

In particular, we found that the role of the couple $(N_{AB},C^{AB})$ formed
by the asymptotic shear and the Bondi news tensors when $\Lambda=0$ is played,
in the presence of $\Lambda\neq 0$, by the symplectic pairing
$(T_{AB}^{TF},g^{AB}_{(0)})$ in SFG coordinates or $(J_{AB},q^{AB})$ in Bondi
coordinates. We have deduced flux-balance laws for the fluxes of
$\Lambda$-BMS4 charges through conformal infinity driven by these radiative
data and checked their consistency with the conservation equation of the
holographic stress-tensor.

  * $\rhd$

In the Bondi gauge, we performed the consistency check to map the Al(A)dS4
solution space on the asymptotically locally flat solution space invariant
under Generalized BMS4 transformations through a flat limit process. In order
to promote the latter at the level of the action, the symplectic structure and
the Hamiltonian charges, we found that corner terms must supply the action
principle in addition to the standard holographic counterterms. More
precisely, we introduced a codimension two kinetic Lagrangian for $q_{AB}$ and
prescribed the boundary terms to be added to the symplectic structure, which
completes the standard prescription (3.2.32) to bring effects of boundary-
covariant renormalization schemes, such as holographic renormalization in the
SFG gauge, at the level of the presymplectic structure. We expect that our
procedure for treating corner boundaries in the presence of fluxes could also
be useful in the context of finite boundaries (see e.g. [253, 254, 255, 256,
323, 252, 257]).

###### Asymptotically locally flat spacetimes

The second axis of research was devoted to addressing various issues and
questions induced by the extension of the BMS4 group into the Generalized BMS4
group including infinite-dimensional super-Lorentz transformations.

  * $\rhd$

Because the super-Lorentz transformations modify the boundary structure at
null infinity, the well-definiteness of the Generalized BMS4 surface charge
algebra requires a renormalization procedure [45], which cannot be derived
from a covariant prescription in terms of the bulk metric alone [60]. We
circumvented this difficulty by obtaining the Generalized BMS4 surface charge
algebra as a contraction of the $\Lambda$-BMS4 charge algebroid induced by a
flat limit $\Lambda\to 0$ at the level of the phase space. The renormalization
of the surface charges follows from holographic renormalization of the
embedding Al(A)dS4 spacetimes combined with a new prescription for treating
corner terms in the presence of asymptotic fluxes.

  * $\rhd$

Such a renormalization is based on the introduction of a background structure
that consists of an asymptotic bulk foliation by codimension one
hypersurfaces, a boundary foliation by codimension two hypersurfaces and a
boundary measure. It therefore provides a geometrical and covariant framework
for the asymptotically flat renormalization procedure introduced in components
in [45], which gives another motivation to consider the $\Lambda$-BMS4
extension of the standard finite-dimensional asymptotic groups for
$\Lambda\neq 0$.

  * $\rhd$

We derived the asymptotic symmetry algebra at the corners
$\mathscr{I}^{+}_{+},\mathscr{I}^{+}_{-},\mathscr{I}^{-}_{+},\mathscr{I}^{-}_{-}$
of future and past null infinity of asymptotically flat spacetimes that are
non-radiative at early and late times. We showed that the Generalized BMS4
algebra including super-Lorentz transformations is realized there as
asymptotic symmetry algebra without non-trivial central extension. Moreover,
we gave the prescription for fixing the ambiguity in the definition of the
BMS4 surface charges such that these charges identically vanish for vacuum
configurations and such that the central extension explicitly vanishes.
Conservation of the Hamiltonian charges at spatial infinity implies that the
asymptotic symmetry group is realized at spatial infinity as well. As a
corollary of the representation theorem, the fluxes at null infinity obey the
BMS4 algebra without central extension.

  * $\rhd$

Finally, we computed the class of vacuum solutions invariant under the
Generalized BMS4 group. It is parametrized by three fields on the sphere,
transforming independently under superrotations, superboosts and
supertranslations. We then analyzed the kinematics and the dynamics of
gravitational memory effects observed for transitions among these vacua and
provided physical processes inducing these transitions. Like supertranslations
that give a symmetry principle underlying the usual displacement memory
effect, superboost transitions yield the refraction and velocity kick effects
on asymptotic Bondi detectors.

###  6.2 Some future perspectives

We conclude our journey by examining some perspectives and research
opportunities left ahead of us by our research program.

######  $\Lambda$-BMS phase space in arbitrary dimensions

In this thesis, we demonstrated how to gain control on radiative data as well
as the associated complete set of non-trivial symmetries thanks to a
diffeomorphism between the SFG coordinates in which Al(A)dS gravity has been
built and understood for decades and the Bondi gauge well-suited to tackle
radiative problems (see section 4.2.2). Moreover, the flat limit, impossible
in the SFG gauge, works almost directly in Bondi coordinates because the
latter are defined irrespectively of the cosmological constant (see sections
4.3 and 4.3.3). One might ask if the procedure is exportable to higher
dimensional cases.

In the presence of a cosmological constant, the structure lifts easily and
includes the $\Lambda$-BMS algebroid as well as the same pattern of radiative
fields $(T_{AB}^{TF},g^{AB}_{(0)})$. The asymptotically flat phase space,
however, does not call for an infinite-dimensional extension of Poincaré like
BMS to include interesting radiative solutions [324, 325, 326, 327, 328, 329,
330, 331, 48], in contrast with the $d=3$ case, where at least the presence of
proper supertranslations is required to allow for an arbitrary $C_{AB}$. The
appearance of radiative data at subleading orders in the Bondi gauge could be
explained by the fact that the holographic stress-tensor also appears at
deeper orders in the SFG expansion for $d>3$, modulo the generalization of the
diffeomorphism to the Bondi gauge and the flat limit process. However, the
reduction of the asymptotic symmetry group is less clear. Other crucial
questions should be addressed, such as the fact that the Weyl charges are non-
vanishing in the flat case [32, 91, 92] although our general analysis in
Al(A)dS4 without boundary conditions [37] tells us that they are zero before
taking the flat limit. In higher odd bulk dimensions, it would also be
interesting to study the impact of the Weyl anomalies in (A)dS in the flat
phase space obtained by a suitable flat limit process and to compute the Weyl
charges and charge algebra in that context.

###### Content of the $\Lambda$-BMS phase space and Bondi mass loss

Another perspective is the analysis of particular solutions on which one could
perform a stress-test of our formalism and results, in order to effectively
observe the practical utility of our leaky boundary conditions. In particular,
if the boundary gauge fixing presented in section 4.1.2 does not rule any
radiative solution, it is rare to find such solutions already written in these
adequate coordinates. One could think, for instance, of Robinson-Trautman
solutions [195] in AdS [285, 197, 198], for which two approaches are possible.
The first approach is to keep in force the Bondi gauge without assuming the
boundary gauge fixing, in particular allowing the boundary volume form to be
variable and arbitrarily time-dependent. This approach would require a further
extension of the phase space, in the spirit of [46, 91, 92], to include Weyl
symmetries on transverse spheres as part of the asymptotic group in (A)dS. The
second one takes the more direct route to perform a diffeomorphism within the
Bondi gauge to put the Robinson-Trautman waves in $\Lambda$-BMS boundary
coordinates. A sketchy analysis confirms that maintaining the boundary
foliation while fixing the boundary measure is possible at the price to
transfer the time dependence in the remaining degrees of freedom of $q_{AB}$.
This would be an interesting arena to investigate, for example, the question
of the positivity of the mass loss in (A)dS and the implications for non-
equilibrium holography.

###### Infrared structure of gravity with $\Lambda\neq 0$

In the context of asymptotically flat spacetimes, we have seen that memory
effects observed by inertial detectors traveling near null infinity are
related to vacuum transitions driven by non-stationary phenomena. Consistency
of the phase space under the action of Generalized BMS4 implies that the
kinematics of these memory effects is encoded into the asymptotic symmetry
group (i.e. proper supertranslations or super-Lorentz transformations)
relating both vacua before and after the non-stationary phase. In the presence
of a cosmological constant, the same kind of memory effects are expected,
eventually enhanced by correcting terms in $\Lambda$ (see e.g. [332, 333, 154,
334, 335, 336, 337, 338]). It would be interesting to see how these effects
are related to the $\Lambda$-BMS symmetries derived in section 4.1.2. Although
we treated AdS and dS asymptotics on the same footing because of the
analytical similarities between both cases, any physical discussion, such as
the derivation of a memory effect, will be radically different and should
exhibit all the complexity of physics in dS. On one hand, the description of
the memory effect as perceived by a pair of inertial timelike observers near
the conformal boundary of AdS should not pose any conceptual issue and should
be very reminiscent of its homologous effect in the flat limit. On the other
hand, in dS, physical free-falling observers travel towards the timelike
boundary and the setup is radically different. Thus, it will also be
interesting to consider memory effects in higher dimensions, as has been done
in the flat context [105].

To complete a tentative infrared triangle, it would be worth investigating how
suitable past boundary conditions could yield a well-defined scattering
problem in (A)dS as is the case for $\Lambda=0$ thanks to the remarkable work
of Strominger and collaborators in the field. The Ward identities that would
stand for the quantum version of the obtained conservation laws for
$\Lambda$-BMS charges could provide symmetry principles underlying the
consistency conditions for correlation functions [339, 340, 341, 342, 343,
344, 125, 345, 346], which are the closest analog for $\Lambda\neq 0$ of
Weinberg’s factorization properties derived for $\Lambda=0$.

###### Looking towards flat holography

The holographic principle experienced its most brilliant realization through
the AdS/CFT correspondence. Even at the classical level, we observe that all
of the ingredients are present to ensure its good functioning, namely the
conformal compactness of the bulk spacetime bounded by a codimension one
spacelike surface on which a clear notion of chronology can be defined
everywhere. Otherwise, in the flat case, an analogous holographic
correspondence is much less known and mastered, mainly because of the null
nature of the conformal boundary. Even if the conformal compactification
process can be performed, there is no more SFG expansion allowing to
parametrize the gravitational phase space uniquely by codimension one boundary
data in a transparent fashion. This thesis reviewed the usual method to
approach null infinity, which consists in following the directions of null
cones in the bulk that precisely define the Bondi gauge null foliation. In
(A)dS, we provided all technicities to translate the boundary data commonly
used in holography to Bondi radiative fields, at least in four dimensions. We
also proposed a well-defined flat limit process within the Bondi gauge at the
level of the phase space, which is enough to obtain all dynamical information
about the bulk gravity that we need to build a more robust notion of flat
holography. We have shown that the renormalization of the asymptotically flat
spacetime can be understood as a relic of the holographic renormalization
procedure in (A)dS coupled to a careful management of the corner terms before
pushing the cosmological constant to zero. We believe that all of the
technical elements to build a consistent notion flat holography on the
shoulders of the famous AdS/CFT correspondence have been derived through to
the $\Lambda$-BMS program.

We plan to pursue this research direction in the future. Crucial progress
toward this endeavor has been made recently, namely the study of the coadjoint
representation of BMS4 [347] under which the non-radiative solution space in
the Bondi gauge transforms. In three-dimensional gravity genuinely exempt of
radiation, the coadjoint representation of BMS3 is necessary and sufficient to
encapsulate the whole solution space [39], and the theory has been shown to be
perfectly described by a geometric action [348] for a theory living on the
boundary. The inclusion of radiation in four-dimensional gravity would amount
to couple the dual geometric action defined on the coadjoint orbits of BMS4
spanning the stationary subsector of the theory to external sources encoded in
the asymptotic shear and Bondi news tensors. This would lead to a holographic
description of asymptotically flat gravity at null infinity.

###### Gauge theories with external sources

Beyond its concrete contributions to a better understanding of the asymptotic
symmetries of General Relativity, this thesis was an opportunity to gather
knowledge about the leaky phase spaces that can be constructed for this theory
and to emphasize their importance in the theoretical discussion of the
phenomena involving gravitational radiation. Although this is a very old
research topic first initiated by Bondi, Metzner, Sachs and van der Burg
almost 60 years ago, several technical results are still missing or
incompletely understood in the literature. For instance, gauge theories with
non-conserved and non-integrable charges need more than ever renewed attention
from a purely mathematical point of view. The algebraic properties of the
surface charges as postulated by Barnich and Troessaert for the very
restricted case of the BMS4 algebra [30] were re-analyzed very recently
considering radiative gravity as a Hamiltonian open system (see e.g. [266, 90,
88]) with more care given to the treatment of sources of non-conservation and
non-integrability. New prescriptions for the charge bracket in the presence of
anomalies due to background structures have also been studied [91, 92]. In the
future, we would like to extend the analysis to any gauge theory coupled with
external sources, with a framework that clearly separates the dynamical
degrees of freedom that contribute to the physical flux and are responsible
for non-integrability as well as non-conservation of the charges among all
gauged degrees of freedom. These investigations could help in proving the
conjecture of [259], discussed in section 2.2.3.4 for gravity. Moreover,
considering the low number of hypotheses and the generality of the
developments presented in section 2.3.2, we would also like to address the
relevance of the Barnich-Troessaert bracket for theorizing the dynamics of
open systems in classical Hamiltonian and Lagrangian mechanics, where
dissipative phenomena and exchanges with the environment in the grand-
canonical ensemble are often incorporated by hand in the equations of motion
without completely controlling the modified phase space features and
symmetries disturbed by the presence of the external sources. This ambitious
research endeavor has already been started [94], encouraged by the current
stimulating interest around these fundamental problematics.

###### A few last words

This concludes our exploration of leaky covariant phase spaces in General
Relativity as well as four years of intense personal and professional
enrichment. The exercise of completing a Ph.D. thesis is not a series of
solitary computations, but a living adventure during which one acquires a more
solid technical mastery while sharing ideas and projects with wonderful
people. I hope that all doctoral students will have as fruitful and pleasant
an experience as I had. Although what lies ahead contains that bundle of
unexpectedness and hope that makes it so appealing, it should never be
forgotten that it is built on the beautiful experiences and fond memories of
the past. In writing these last lines I now take the step to forever put my
studies behind me, with a firm desire to do justice to what I have been taught
and to continue to contribute honorably to a quest definitely promised a
bright future! $\blacksquare$

## Appendices

###  C Killing vectors of (A)dS4 in Bondi gauge

In this appendix, we review the expression of the Killing vectors for the
global vacua of 4$d$ gravity (Minkowksi, (A)dS4) in the Bondi gauge. Once the
results have been presented for AdS4 (with radius $\ell$), one can readily
extrapolate to the dS4 case thanks to the analytical continuation $\ell\to
i\ell$. The isometries of flat space are finally obtained by taking the flat
limit $\ell\to+\infty$.

For the purpose to compute the symmetries of AdS4, it is convenient to see
this spacetime isometrically immersed into $\mathbb{R}^{(2,3)}$ as the
hypersurface

$\\{X^{\mu}\in\mathbb{R}^{(2,3)}|-X_{0}^{2}-X_{\underline{0}}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}=-\ell^{2}\\},$
(C.1)

see section 3.1.1 for a review. In this point of view, the exact symmetry
group of AdS4 is the homogeneous part of the isometry group of
$\mathbb{R}^{(2,3)}$, $ISO(2,3)$, which is $SO(2,3)$. The generators of
$SO(2,3)$ algebra

$\mathcal{J}_{ab}=\mathcal{J}_{[ab]}=X_{b}\partial_{a}-X_{a}\partial_{b},$
(C.2)

$a,b\in\\{0,\underline{0},1,2,3\\}$, directly lead to the Killing vectors of
AdS4 after pullback on (C.1).

In retarded Bondi coordinates $(u,r,x^{A})$, the AdS4 line element takes the
form (4.2.46) where we take $x^{A}=(z,\bar{z})$ as the stereographic
coordinates on the 2-sphere with unit-round metric $\mathring{q}_{AB}$. The
Minkowski metric in retarded coordinates (5.1.1) is recovered in the flat
limit $\ell\rightarrow\infty$. In these coordinates, the Killing vectors are
given by $\mathcal{J}=\mathcal{J}^{\mu}\partial_{\mu}$ with

$\mathcal{J}_{ab}^{u}=f,\qquad\mathcal{J}_{ab}^{r}=-\frac{r}{2}\mathring{D}_{A}\mathcal{J}_{ab}^{A},\qquad\mathcal{J}_{ab}^{A}=Y^{A}-\frac{1}{r}\partial^{A}f$
(C.3)

where $\mathring{D}_{A}$ is the covariant derivative on the sphere and the
functions $f,Y^{A}$ obey [222]

$\partial_{u}f=\frac{1}{2}\mathring{D}_{A}Y^{A}\,,\qquad\partial_{u}Y^{A}=\frac{1}{\ell^{2}}\partial^{A}f\,,\qquad
2\mathring{D}_{(A}Y_{B)}-\mathring{q}_{AB}\mathring{D}_{C}Y^{C}=0.$ (C.4)

The last equation fragments the set of Killing vectors of AdS4 in two
categories, namely the 4 vectors associated with exact isometries of the
boundary cylinder (with $\mathring{D}_{A}Y^{A}=0$):

  1. 1.

Rotations:

$\begin{split}\mathcal{J}_{12}&=[0,0,-iz,-i\bar{z}]=\partial_{\phi},\\\
\mathcal{J}_{13}&=\Big{[}0,0,\frac{1}{2}(-1-z^{2}),\frac{1}{2}(-1-\bar{z}^{2})\Big{]},\\\
\mathcal{J}_{23}&=\Big{[}0,0,\frac{1}{2}i(-1+z^{2}),\frac{1}{2}i(-1+\bar{z}^{2})\Big{]};\end{split}$
(C.5)

  2. 2.

Time translation:

$\mathcal{J}_{0\underline{0}}=\partial_{u};$ (C.6)

and the 6 vectors associated with the conformal isometries of the boundary
cylinder, for which $\mathring{D}_{A}Y^{A}\neq 0$:

  1. 3.

Boosts on the first timelike coordinate: with the shorthand notation
$\mu(u/\ell,r)$ $\equiv$ $r$ $\cos(u/\ell)$ $+$ $\ell\sin(u/\ell)$, we have

$\begin{split}\mathcal{J}_{01}&=\Big{[}\frac{(z+\bar{z})\ell\sin(\frac{u}{\ell})}{1+z\bar{z}},-\frac{(z+\bar{z})\mu(\frac{u}{\ell},r)}{(1+z\bar{z})},\frac{(-1+z^{2})\mu(\frac{u}{\ell},r)}{2r},-\frac{(-1+\bar{z}^{2})\mu(\frac{u}{\ell},r)}{2r}\Big{]},\\\
\mathcal{J}_{02}&=\Big{[}-\frac{(z-\bar{z})\ell\sin(\frac{u}{\ell})}{1+z\bar{z}},\frac{i(z-\bar{z})\mu(\frac{u}{\ell},r)}{(1+z\bar{z})},-\frac{i(1+z^{2})\mu(\frac{u}{\ell},r)}{2r},\frac{i(1+\bar{z}^{2})\mu(\frac{u}{\ell},r)}{2r}\Big{]},\\\
\mathcal{J}_{03}&=\Big{[}\frac{(-1+z\bar{z})\ell\sin(\frac{u}{\ell})}{1+z\bar{z}},\frac{(-1+z\bar{z})\mu(\frac{u}{\ell},r)}{(1+z\bar{z})},-\frac{z\mu(\frac{u}{\ell},r)}{r},-\frac{\bar{z}\mu(\frac{u}{\ell},r)}{r}\Big{]};\\\
\end{split}$ (C.7)

  2. 4.

Boosts on the second timelike coordinate: by virtue of the $SO(2,3)$ algebra,

$\mathcal{J}_{\underline{0}i}=-[\mathcal{J}_{0\underline{0}},\mathcal{J}_{0i}]=-\partial_{u}\mathcal{J}_{0i}.$
(C.8)

The Lorentz algebra can be explicitly checked for the rotations
$V_{i}={\epsilon_{i}}^{jk}\mathcal{J}_{jk}$ together with the boosts along $0$
time direction, $K_{i}=\mathcal{J}_{0i}$, or the boosts along $\underline{0}$
time direction, $K_{i}=\ell\mathcal{J}_{\underline{0}i}$:

$[V_{i},V_{j}]=\epsilon_{ijk}V_{k}\,,\quad[K_{i},K_{j}]=-\epsilon_{ijk}V_{k}\,,\quad[V_{i},K_{j}]=\epsilon_{ijk}K_{k}.$
(C.9)

The time translation and the three rotations have a trivial flat limit. When
$\ell\to+\infty$ the boosts $\mathcal{J}_{0i}$ become the Lorentz boosts

$\begin{split}\mathcal{J}_{01}&\rightarrow\Big{[}\frac{u(z+\bar{z})}{1+z\bar{z}},-\frac{(r+u)(z+\bar{z})}{1+z\bar{z}},\frac{(r+u)(-1+z^{2})}{2r},\frac{(r+u)(-1+\bar{z}^{2})}{2r}\Big{]},\\\
\mathcal{J}_{02}&\rightarrow\Big{[}-i\frac{u(z-\bar{z})}{1+z\bar{z}},i\frac{(r+u)(z-\bar{z})}{1+z\bar{z}},-i\frac{(r+u)(1+z^{2})}{2r},i\frac{(r+u)(1+\bar{z}^{2})}{2r}\Big{]},\\\
\mathcal{J}_{03}&\rightarrow\Big{[}\frac{u(-1+z\bar{z})}{1+z\bar{z}},-\frac{(r+u)(-1+z\bar{z})}{1+z\bar{z}},-\frac{(r+u)z}{r},-\frac{(r+u)\bar{z}}{r}\Big{]},\end{split}$
(C.10)

while the boosts $\mathcal{J}_{\underline{0}i}$ become the spatial
translations

$\begin{split}\mathcal{J}_{\underline{0}1}&\rightarrow\Big{[}\frac{z+\bar{z}}{1+z\bar{z}},-\frac{z+\bar{z}}{1+z\bar{z}},\frac{-1+z^{2}}{2r},\frac{-1+\bar{z}^{2}}{2r}\Big{]},\\\
\mathcal{J}_{\underline{0}2}&\rightarrow\Big{[}-i\frac{z-\bar{z}}{1+z\bar{z}},i\frac{z-\bar{z}}{1+z\bar{z}},-i\frac{1+z^{2}}{2r},i\frac{1+\bar{z}^{2}}{2r}\Big{]},\\\
\mathcal{J}_{\underline{0}3}&\rightarrow\Big{[}-\frac{1-z\bar{z}}{1+z\bar{z}},\frac{1-z\bar{z}}{1+z\bar{z}},-\frac{z}{r},-\frac{\bar{z}}{r}\Big{]}.\end{split}$
(C.11)

One can check explicitly that they verify the constraints (C.4) in the limit
$\ell\to+\infty$.

###  D Map from Bondi to Starobinsky/Fefferman-Graham gauge

This appendix contains the detailed material complementing the section 4.2.2.
It is devoted to discuss the change of coordinates that maps a general vacuum
asymptotically locally (A)dS4 spacetime ($\Lambda\neq 0$) in the Bondi gauge
to the Starobinsky/Fefferman-Graham gauge. In particular, we give here the
explicit forms of the various coefficients appearing in the perturbative
change of coordinate (4.2.49) in terms of the metric fields.

We use the following shorthand notations for subleading fields in the Bondi
gauge:

$\begin{split}\frac{V}{r}&=\frac{\Lambda}{3}e^{2\beta_{0}}r^{2}+r\
V_{(1)}(t,x^{A})+V_{(0)}(t,x^{A})+\frac{2M}{r}+\mathcal{O}(r^{-2}),\\\
U^{A}&=U_{0}^{A}(t,x^{B})+\frac{1}{r}U_{(1)}^{A}(t,x^{B})+\frac{1}{r^{2}}U_{(2)}^{A}(t,x^{B})+\frac{1}{r^{3}}U_{(3)}^{A}(t,x^{B})+\mathcal{O}(r^{-4}),\\\
\beta&=\beta_{0}(t,x^{A})+\frac{1}{r^{2}}\beta_{(2)}(t,x^{A})+\mathcal{O}(r^{-4}).\end{split}$
(D.1)

whose explicit on-shell values can be read off in (4.2.23) and (4.2.20). That
allows to write the equations in a more compact way. All fields are now
evaluated on $(t,x^{A})$ since the boundary time coordinate can be defined as
$t$ as well as $u$. We also define some recurrent structures appearing in the
diffeomorphism as differential operators on boundary scalar fields
$\varphi(t,x^{A})$, $\chi(t,x^{A})$:

$\begin{split}P[\varphi]&=\frac{1}{2}e^{-4\beta_{0}}(\partial_{t}\varphi+U_{0}^{A}\partial_{A}\varphi),\\\
Q[\varphi;\chi]&=P[\varphi]-2P[\chi]\varphi,\\\
B_{A}[\varphi]&=\frac{1}{2}e^{-2\beta_{0}}(\partial_{A}-2\partial_{A}\beta_{0})\varphi.\end{split}$
(D.2)

Below, $P^{n}[\varphi]$ denotes $n$ applications of $P$ on $\varphi$, for
example $P^{2}[\varphi]\equiv P[P[\varphi]]$. Now we can write down the
perturbative change of coordinate to Starobinsky/Fefferman-Graham gauge:

$\displaystyle R_{1}(t,x^{A})$ $\displaystyle=-\frac{3}{\Lambda},$
$\displaystyle R_{2}(t,x^{A})$
$\displaystyle=\frac{9}{2\Lambda^{2}}e^{-2\beta_{0}}V_{(1)},$ $\displaystyle
R_{3}(t,x^{A})$
$\displaystyle=\frac{3}{2\Lambda}\beta_{(2)}-\frac{3}{\Lambda^{2}}\Big{(}1+\frac{3}{4}e^{-2\beta_{0}}V_{(0)}\Big{)}+\frac{27}{2\Lambda^{3}}\Big{(}Q[V_{(1)};\beta_{0}]-\frac{3}{8}e^{-4\beta_{0}}V_{(1)}^{2}\Big{)},$
$\displaystyle R_{4}(t,x^{A})$
$\displaystyle=\frac{3}{\Lambda^{2}}e^{-2\beta_{0}}\Big{(}M+2e^{4\beta_{0}}P[\beta_{(2)}]-\frac{5}{2}V_{(1)}\beta_{(2)}\Big{)}$
$\displaystyle\quad-\frac{9}{\Lambda^{3}}\Big{\\{}Q[V_{(0)};\beta_{0}]+\frac{1}{4}e^{-4\beta_{0}}\Big{[}U^{A}_{(1)}\partial_{A}V_{(1)}-2V_{(1)}U^{A}_{(1)}\partial_{A}\beta_{0}-3V_{(1)}(2e^{2\beta_{0}}+V_{(0)})\Big{]}\Big{\\}}$
$\displaystyle\quad+\frac{27}{\Lambda^{4}}e^{2\beta_{0}}\Big{[}P^{2}[V_{(1)}]-2V_{(1)}\Big{(}P^{2}[\beta_{0}]+\frac{1}{2}e^{-4\beta_{0}}Q[V_{(1)};\beta_{0}]-\frac{3}{32}e^{-8\beta_{0}}V_{(1)}^{2}\Big{)}-2P[\beta_{0}]P[V_{(1)}]\Big{]},$
$\displaystyle T_{1}(t,x^{A})$
$\displaystyle=(1-e^{-2\beta_{0}})R_{1}(t,x^{A}),$ $\displaystyle
T_{2}(t,x^{A})$
$\displaystyle=(1-e^{-2\beta_{0}})R_{2}(t,x^{A})-\frac{18}{\Lambda^{2}}\Big{(}P[\beta_{0}]-\frac{1}{4}e^{-4\beta_{0}}V_{(1)}\Big{)},$
$\displaystyle T_{3}(t,x^{A})$
$\displaystyle=(1-e^{-2\beta_{0}})R_{3}(t,x^{A})-\frac{3}{\Lambda^{2}}e^{-2\beta_{0}}(1+e^{-2\beta_{0}}V_{(0)}-2\partial^{A}\beta_{0}\partial_{A}\beta_{0})$
$\displaystyle\quad+\frac{9}{\Lambda^{3}}e^{-2\beta_{0}}\Big{(}Q[V_{(1)};\beta_{0}]-4e^{4\beta_{0}}P^{2}[\beta_{0}]-\frac{1}{2}e^{-4\beta_{0}}V_{(1)}^{2}\Big{)},$
$\displaystyle T_{4}(t,x^{A})$
$\displaystyle=(1-e^{-2\beta_{0}})R_{4}(t,x^{A})$
$\displaystyle\quad+\frac{9}{2\Lambda^{2}}\Big{[}e^{-4\beta_{0}}\Big{(}M-\beta_{(2)}V_{(1)}-\frac{1}{3}U_{(2)}^{A}\partial_{A}\beta_{0}\Big{)}-\frac{1}{2}(P[\beta_{(2)}]-8\beta_{(2)}P[\beta_{0}])\Big{]}$
$\displaystyle\quad-\frac{27}{\Lambda^{3}}\Big{\\{}\frac{1}{8}e^{-2\beta_{0}}\Big{(}3Q[V_{(0)};\beta_{0}]-\frac{8}{3}P[\beta_{0}]V_{(0)}-2e^{-4\beta_{0}}V_{(1)}V_{(0)}\Big{)}$
$\displaystyle\qquad\qquad+\frac{1}{3}e^{-2\beta_{0}}\Big{(}P[U_{(1)}^{A}]\partial_{A}\beta_{0}+\frac{3}{2}U_{(1)}^{A}\partial_{A}P[\beta_{0}]\Big{)}$
$\displaystyle\qquad\qquad-\frac{1}{12}e^{-4\beta_{0}}\Big{[}U_{(1)}^{A}B_{A}[V_{(1)}]+6V_{(1)}-2(V_{(1)}\partial_{A}\beta_{0}+2\partial_{B}\beta_{0}\partial_{A}U_{0}^{B})\partial^{A}\beta_{0}\Big{]}\Big{\\}}$
$\displaystyle\quad+\frac{81}{\Lambda^{4}}\Big{\\{}-\frac{2}{3}e^{4\beta_{0}}\Big{(}P^{3}[\beta_{0}]+2P[\beta_{0}]P^{2}[\beta_{0}]\Big{)}+\frac{1}{4}\Big{(}P^{2}[V_{(1)}]-2V_{(1)}P^{2}[\beta_{0}]\Big{)}$
$\displaystyle\qquad\qquad+\frac{1}{6}P[\beta_{0}]\Big{[}\Big{(}\frac{13}{4}e^{-4\beta_{0}}V_{(1)}-8P[\beta_{0}]\Big{)}V_{(1)}+P[V_{(1)}]\Big{]}$
$\displaystyle\qquad\qquad-\frac{1}{16}e^{-4\beta_{0}}V_{(1)}\Big{(}5P[V_{(1)}]-e^{-4\beta_{0}}V_{(1)}^{2}\Big{)}\Big{\\}},$
$\displaystyle X_{1}^{A}(t,x^{B})$ $\displaystyle=(T_{1}-R_{1})U_{0}^{A},$
$\displaystyle X_{2}^{A}(t,x^{B})$
$\displaystyle=(T_{2}-R_{2})U_{0}^{A}-\frac{3}{2\Lambda}e^{-2\beta_{0}}U_{(1)}^{A}+\frac{9}{\Lambda^{2}}P[U_{0}^{A}],$
$\displaystyle X_{3}^{A}(t,x^{B})$
$\displaystyle=(T_{3}-R_{3})U_{0}^{A}+\frac{1}{\Lambda}e^{-2\beta_{0}}U_{(2)}^{A}$
$\displaystyle\quad-\frac{6}{\Lambda^{2}}\Big{[}Q[U_{(1)}^{A};\beta_{0}]+\frac{1}{2}B^{A}[V_{(1)}]+\frac{1}{4}e^{-4\beta_{0}}(U_{(1)}^{B}\partial_{B}U^{A}_{0}-V_{(1)}U_{(1)}^{A})\Big{]}$
$\displaystyle\quad+\frac{18}{\Lambda^{3}}e^{2\beta_{0}}Q[P[U_{0}^{A}];\beta_{0}],$
$\displaystyle X_{4}^{A}(t,x^{B})$
$\displaystyle=(T_{4}-R_{4})U_{0}^{A}-\frac{3}{4\Lambda}e^{-2\beta_{0}}\Big{[}U_{(3)}^{A}+\frac{1}{2}e^{2\beta_{0}}(\partial^{A}\beta_{(2)}-8\beta_{(2)}\partial^{A}\beta_{0})\Big{]}$
$\displaystyle+\frac{9}{2\Lambda^{2}}\Big{[}Q[U_{(2)}^{A};\beta_{0}]-\frac{1}{2}e^{-4\beta_{0}}\Big{(}V_{(1)}U_{(2)}^{A}-\frac{1}{3}U_{(2)}^{B}\partial_{B}U_{(0)}^{A}\Big{)}-2\beta_{(2)}P[U_{0}^{A}]$
$\displaystyle\qquad\qquad+\frac{1}{4}B^{A}[V_{(0)}]+\frac{1}{2}C^{AC}B_{C}[V_{(1)}]+\frac{1}{2}e^{-2\beta_{0}}U_{(1)}^{C}B_{C}[U^{A}_{(1)}]\Big{]}$
$\displaystyle-\frac{27}{\Lambda^{3}}\Big{\\{}e^{2\beta_{0}}\Big{(}P[Q[U_{(1)}^{A};\beta_{0}]]+P[B^{A}[V_{(1)}]]-\frac{1}{2}q^{AC}P[B_{C}[V_{(1)}]]\Big{)}$
$\displaystyle\qquad\qquad-\frac{1}{2}e^{-2\beta_{0}}\Big{[}V_{(1)}P[U_{(1)}^{A}]-\frac{2}{3}\Big{(}P[U_{(1)}^{C}]+\frac{1}{2}B^{C}[V_{(1)}]-5P[\beta_{0}]U_{(1)}^{C}\Big{)}\partial_{C}U_{0}^{A}$
$\displaystyle\qquad\qquad+\frac{1}{2}P[V_{(1)}]U_{(1)}^{A}-\frac{2}{3}(V_{(0)}-8e^{2\beta_{0}}\partial^{B}\beta_{0}\partial_{B}\beta_{0})P[U_{0}^{A}]-U_{(1)}^{B}P[\partial_{B}U_{0}^{A}]$
$\displaystyle\qquad\qquad+\frac{1}{2}(\partial^{A}U_{0}^{C})B_{C}[V_{(1)}]\Big{]}+3P[\beta_{0}]V_{(1)}\partial^{A}\beta_{0}$
$\displaystyle\qquad\qquad-e^{-4\beta_{0}}\Big{[}\frac{3}{32}\Big{(}\partial^{A}(V_{(1)}^{2})-\frac{20}{3}\partial^{A}\beta_{0}V_{(1)}^{2}\Big{)}+\frac{1}{6}(V_{(1)}\partial^{B}\beta_{0}-\partial^{C}\beta_{0}\partial_{C}U_{0}^{B})\partial_{B}U_{0}^{A}\Big{]}\Big{\\}}$
$\displaystyle+\frac{81}{\Lambda^{4}}\Big{\\{}\frac{1}{3}e^{4\beta_{0}}P^{3}[U_{0}^{A}]+\Big{[}\frac{1}{4}e^{-4\beta_{0}}V_{(1)}^{2}-\frac{1}{3}Q[V_{(1)};\beta_{0}]-\frac{4}{3}e^{4\beta_{0}}(P^{2}[\beta_{0}]+P[\beta_{0}]^{2})\Big{]}P[U_{0}^{A}]\Big{\\}}.$

The function $R_{1}$ is not constrained by the gauge conditions (4.2.48) but
is simply fixed by requiring that the Weyl rescalings of the transverse metric
on the sphere agree between both gauges, i.e. $g^{(0)}_{AB}=q_{AB}$. Several
consistency checks can be performed at each stage of the computation. The
boundary metric in the SFG gauge must be equivalent to the pulled-back metric
on the hypersurface $\\{r\to\infty\\}$ in the Bondi gauge, up to the usual
replacement $u\to t$. This is the case because of (4.2.50). At subleading
orders, $g^{(1)}_{ab}$ and $g^{(2)}_{ab}$ must be algebraically determined by
$g_{ab}^{(0)}$ and its first and second derivatives, what turns out to be the
case. The constraint (4.2.24) enforces $g^{(1)}_{ab}=0$ while the annulation
of $\mathcal{D}_{AB}(t,x^{C})$ (4.2.25) results in

$g^{(2)}_{ab}=\frac{3}{\Lambda}\Big{[}R^{(0)}_{ab}-\frac{1}{4}R_{(0)}g^{(0)}_{ab}\Big{]}.$
(D.3)

This agrees with the $d=3$ version of the equation of motion (3.1.33). We will
not give the full general form of $g^{(3)}_{ab}$, but it can be proven that
this tensor is traceless with respect to $g^{(0)}_{ab}$ and that the equations
of motion in the Bondi gauge are necessary and sufficient to show its
conservation $D^{(0)}_{a}g_{(3)}^{ab}=0$, as we argued in the main text.

After partial boundary gauge fixing $\beta_{0}=0$, $U_{0}^{A}=0$ (Gaussian
normal coordinates on the boundary), but leaving $\sqrt{q}$ arbitrary, the
expressions of each coefficient in the diffeomorphism simplifies drastically:

$\displaystyle R_{1}(t,x^{A})$ $\displaystyle=-\frac{3}{\Lambda},$
$\displaystyle R_{2}(t,x^{A})$ $\displaystyle=\frac{9}{2\Lambda^{2}}V_{(1)},$
$\displaystyle R_{3}(t,x^{A})$
$\displaystyle=\frac{3}{2\Lambda}\beta_{(2)}-\frac{3}{\Lambda^{2}}\Big{(}1+\frac{3}{4}V_{(0)}\Big{)}+\frac{27}{2\Lambda^{3}}\Big{(}\frac{1}{2}\partial_{t}V_{(1)}-\frac{3}{8}V_{(1)}^{2}\Big{)},$
$\displaystyle R_{4}(t,x^{A})$
$\displaystyle=\frac{3}{\Lambda^{2}}\Big{(}M+\partial_{t}\beta_{(2)}-\frac{5}{2}V_{(1)}\beta_{(2)}\Big{)}-\frac{9}{\Lambda^{3}}\Big{[}\frac{1}{2}\partial_{t}V_{(0)}-\frac{3}{4}V_{(1)}(2+V_{(0)})\Big{]}$
$\displaystyle\quad+\frac{27}{\Lambda^{4}}\Big{(}\frac{1}{4}\partial_{t}^{2}[V_{(1)}]-\frac{1}{4}\partial_{t}V_{(1)}^{2}+\frac{6}{32}V_{(1)}^{3}\Big{)}.$
$\displaystyle T_{1}(t,x^{A})$ $\displaystyle=0,$ $\displaystyle
T_{2}(t,x^{A})$ $\displaystyle=\frac{9}{2\Lambda^{2}}V_{(1)},$ $\displaystyle
T_{3}(t,x^{A})$
$\displaystyle=-\frac{3}{\Lambda^{2}}(1+V_{(0)})+\frac{9}{2\Lambda^{3}}\Big{(}\partial_{t}V_{(1)}-V_{(1)}^{2}\Big{)},$
$\displaystyle T_{4}(t,x^{A})$
$\displaystyle=\frac{9}{2\Lambda^{2}}\Big{[}M-\frac{1}{4}(\partial_{t}\beta_{(2)}+4V_{(1)}\beta_{(2)})\Big{]}+\frac{9}{\Lambda^{3}}\Big{[}-\frac{9}{16}\partial_{t}V_{(0)}+\frac{3}{2}V_{(1)}(1+\frac{1}{2}V_{(0)})\Big{]}$
$\displaystyle\quad+\frac{81}{\Lambda^{4}}\Big{(}\frac{1}{16}\partial_{t}^{2}V_{(1)}-\frac{5}{64}\partial_{t}V_{(1)}^{2}+\frac{1}{16}V_{(1)}^{3}\Big{)}.$
$\displaystyle X_{1}^{A}(t,x^{B})$ $\displaystyle=X_{2}^{A}(t,x^{B})=0,$
$\displaystyle X_{3}^{A}(t,x^{B})$
$\displaystyle=\frac{1}{\Lambda}U_{(2)}^{A}-\frac{3}{2\Lambda^{2}}\partial^{A}V_{(1)},$
$\displaystyle X_{4}^{A}(t,x^{B})$
$\displaystyle=-\frac{3}{4\Lambda}\Big{(}U_{(3)}^{A}+\frac{1}{2}\partial^{A}\beta_{(2)}\Big{)}+\frac{9}{2\Lambda^{2}}\Big{(}\frac{1}{2}\partial_{t}U_{(2)}^{A}-\frac{1}{2}V_{(1)}U_{(2)}^{A}+\frac{1}{8}\partial^{A}V_{(0)}\Big{)}$
$\displaystyle\quad-\frac{27}{16\Lambda^{3}}\
q^{AB}\Big{(}\partial_{t}\partial_{B}V_{(1)}+\frac{1}{2}V_{(1)}\partial_{B}V_{(1)}\Big{)}.$

In particular, $t=u+\mathcal{O}(\rho)$ and both time coordinates can be
identified on the boundary.

###### Note.

The expressions of the diffeomorphism and the metric elements $g_{ab}^{(0)}$,
$g_{ab}^{(2)}$ and $g_{ab}^{(3)}$ (the latter within the additional boundary
gauge fixing) can be found in the Mathematica file BMS_TO_FG.nb attached to
[271] on the arXiv publication. The package RGtensors [349] is needed to run
this notebook.

###  E Useful relations for computations in the Bondi gauge

In this appendix, we give some useful relations in the Bondi gauge that are
widely used in sections 2.2.4, 2.3.3 and 4.3.3. The metric on the celestial
sphere is written as $q_{AB}$. From $\delta q^{AB}=-q^{AC}q^{BD}\delta
q_{CD}$, we have

$T_{AB}\delta q^{AB}=-T^{AB}\delta q_{AB}.$ (E.1)

for any symmetric tensor $T_{AB}$. For any $C_{AB}$ symmetric traceless tensor
(i.e. $C_{AB}=C_{(AB)}$, $q^{AB}C_{AB}=0$), it follows that

$q_{AB}\delta C^{AB}=C_{AB}\delta q^{AB}.$ (E.2)

Requiring further that $\delta\sqrt{q}=0$ one finds

$\displaystyle\delta(C_{AB}C^{AB})$ $\displaystyle=2C_{AB}\delta C^{AB},$
(E.3) $\displaystyle\delta(C_{AB}C^{AB})$
$\displaystyle=q_{AB}\delta(C^{AC}C_{C}^{B}),$ (E.4) $\displaystyle
M_{AB}\delta C^{AB}$ $\displaystyle=M^{AB}\delta C_{AB},$ (E.5)

where $M_{AB}$ is also an arbitrary symmetric traceless tensor. Considering
two variations $\delta,\delta^{\prime}$ one can prove the following identity,
again with $\delta\sqrt{q}=0$ :

$T_{AB}\delta\delta^{\prime}q^{AB}=-T^{AB}\delta\delta^{\prime}q_{AB}-T\delta
q_{AB}\delta^{\prime}q^{AB},$ (E.6)

where $T\equiv q^{AB}T_{AB}$. It follows from (E.1) that

$\delta T_{AB}\delta^{\prime}q^{AB}=-\delta
T^{AB}\delta^{\prime}q_{AB}+T\delta q_{AB}\delta^{\prime}q^{AB}.$ (E.7)

For a traceless tensor such as $M_{AB}$, we have

$\delta M_{AB}\delta^{\prime}q^{AB}=-\delta M^{AB}\delta^{\prime}q_{AB}.$
(E.8)

For the metric $q_{AB}$ itself, $T=2$ and

$\delta q_{AB}\delta^{\prime}q^{AB}=\delta q^{AB}\delta^{\prime}q_{AB}.$ (E.9)

Finally from (E.5) we get

$\delta M_{AB}\delta^{\prime}C^{AB}=\delta M^{AB}\delta^{\prime}C_{AB}.$
(E.10)

While manipulating rank 2 symmetric traceless tensors $C_{AC}$, $M_{AB}$ on
the sphere, the following identities turn out to be very useful:

$\displaystyle\frac{1}{2}(C_{AB}M^{BC}+M_{AB}C^{BC})=\frac{1}{2}{\delta_{A}}^{B}(C_{CD}M^{CD}),$
(E.11) $\displaystyle D_{A}C_{BC}M^{BC}=D_{B}C_{AC}M^{BC}+M_{AB}D_{C}C^{BC}.$
(E.12)

###  F Technical material from quadratic curvature gravity

In this appendix, we review some material about the quadratic curvature
gravity theory [282, 281] (also coined as new massive gravity theory [350]).
Indeed, the patterns of this theory appear naturally when considering the Weyl
charges in the $d=4$ case.

The quadratic gravity Lagrangian in $d$ dimensions contains generally three
pieces

$\bm{L}_{QCG}[g]=\bm{L}_{EH}[g]+\beta_{1}\,\bm{L}_{QCG(1)}[g]+\beta_{2}\,\bm{L}_{QCG(2)}[g]$
(F.1)

where $g$ denotes an arbitrary metric tensor in $d$ dimensions,
$\beta_{1},\beta_{2}$ are real constants and

$\begin{split}\bm{L}_{EH}[g]&=\frac{\sqrt{|g|}}{16\pi
G}\left(R+\eta\frac{d(d-1)}{\ell^{2}}\right)\leavevmode\nobreak\
\text{d}^{d}x,\\\ \bm{L}_{QCG(1)}[g]&=\frac{\sqrt{|g|}}{16\pi
G}R_{ab}R^{ab}\leavevmode\nobreak\
\text{d}^{d}x,\qquad\bm{L}_{QCG(2)}[g]=\frac{\sqrt{|g|}}{16\pi
G}R^{2}\leavevmode\nobreak\ \text{d}^{d}x.\end{split}$ (F.2)

Taking the variation of these Lagrangians, we have

$\begin{split}\delta\bm{L}_{EH}[g]&=\frac{\delta\bm{L}_{EH}[g]}{\delta
g^{ab}}\delta g^{ab}+\text{d}\bm{\Theta}_{EH}[g;\delta g],\\\
\delta\bm{L}_{QCG(1)}[g]&=\frac{\delta\bm{L}_{QCG(1)}[g]}{\delta g^{ab}}\delta
g^{ab}+\text{d}\bm{\Theta}_{QCG(1)}[g;\delta g],\\\
\delta\bm{L}_{QCG(2)}[g]&=\frac{\delta\bm{L}_{QCG(2)}[g]}{\delta g^{ab}}\delta
g^{ab}+\text{d}\bm{\Theta}_{QCG(2)}[g;\delta g]\end{split}$ (F.3)

where

$\displaystyle\frac{\delta\bm{L}_{EH}[g]}{\delta g^{ab}}$
$\displaystyle=\frac{\sqrt{|g|}}{16\pi
G}\left[R_{ab}-\frac{1}{2}g_{ab}-\eta\frac{d(d-1)}{2\ell^{2}}g_{ab}\right]\leavevmode\nobreak\
\text{d}^{d}x,$ $\displaystyle\frac{\delta\bm{L}_{QCG(1)}[g]}{\delta g^{ab}}$
$\displaystyle=\frac{\sqrt{|g|}}{16\pi
G}\left[2R_{acbd}R^{cd}-D_{a}D_{b}R+D^{c}D_{c}R_{ab}+\frac{1}{2}g_{ab}\left(D^{c}D_{c}R-R_{cd}R^{cd}\right)\right]\leavevmode\nobreak\
\text{d}^{d}x,$ $\displaystyle\frac{\delta\bm{L}_{QCG(2)}[g]}{\delta g^{ab}}$
$\displaystyle=\frac{\sqrt{|g|}}{16\pi
G}\left[2RR_{ab}-2D_{a}D_{b}R+g_{ab}\left(2D^{c}D_{c}R-\frac{1}{2}R^{2}\right)\right]\leavevmode\nobreak\
\text{d}^{d}x$ (F.4)

and $\bm{\Theta}_{EH}[g;\delta g]=\Theta_{EH}^{a}[g;\delta
g]\leavevmode\nobreak\ (\text{d}^{d-1}x)_{a}$ (idem for
$\bm{\Theta}_{QCG(1)}[g;\delta g],\bm{\Theta}_{QCG(2)}[g;\delta g]$) with

$\displaystyle\Theta^{a}_{EH}[g;\delta g]$
$\displaystyle=\frac{\sqrt{|g|}}{16\pi G}\left[D_{b}(\delta
g)^{ab}-D^{a}(\delta g)^{c}_{c}\right],$