boundary conditions. In the first case, the flux of charges is positive in
(2.2.72) while in the second it is negative. For example, we will see below
that the presence of gravitational waves in the bulk of any asymptotically
flat spacetime is responsible for some outgoing flux at future null infinity.
If the gravitational system is sourced by another and can be approximated by
an asymptotically flat spacetime, one can also imagine that some gravitational
waves are free to enter through past null infinity, describing a toy-model of
ingoing radiative conditions.

$\leftarrow$ Dynamical BDoF (sources)Radiative open system Outgoing radiation
Kinematical BDoF $\downarrow$ $\mathscr{B}$$\delta S_{sys}\neq 0$ Environment
$\delta S_{env}\neq 0$ $\delta S_{tot}=0$ Figure 2.4: Kinematical _vs._
dynamical boundary degrees of freedom (BDoF).

###### Boundary degrees of freedom and physical flux

We will show below, by means of concrete examples, that it is convenient to
interpret the primary causes of the leaks as external sources or “boundary
degrees of freedom.” From an asymptotic analysis considering only the physical
system with action principle defined by $S$ and a set of boundary conditions
at $\mathscr{B}$, the dynamics of these boundary degrees of freedom is
unconstrained. However, if one would include the environment into the
analysis, then they would acquire some constrained dynamics. The latter would
be dictated by the precise nature of the environment under consideration. As a
result, the concatenation between both the leaky physical system and its
environment would restore a well-defined variational principle. In the
statement (2.1.16), there is merely an apparent violation of this because we
are tracing on the states of the environment in the analysis of asymptotics.
This is a local statement that reflects our ignorance about the outer region
which collects the outcoming flux of charges or produces some ingoing sources
for the charges. The most beautiful thing is that the framework we are
discussing in this chapter has enough strength to give the possibility to
perform the complete analysis of the asymptotics without knowing the precise
nature of the environment and the dynamics of the external sources. For
instance, in gravitational astronomy, the experimenters and their experimental
devices act as observers sitting at the future null infinity of a spacetime
where the phenomenon under interest has occurred. In order to provide them
with predictions, we just need to get some information about the dynamics at
$\mathscr{B}$, but what lies beyond is out of our scope and our interest. This
is the point of view we have adopted in the present thesis.

###### Radiative and kinematical degrees of freedom

There is however a subtlety to be mentioned here regarding the physical
interpretation of the symplectic flux. A non-vanishing symplectic current on
$\mathscr{B}$ does not necessarily mean that physical dynamical degrees of
freedom contribute to the genuine flux through the boundary. For example, one
can have a non-radiative gravitational system enclosed by $\mathscr{B}$ with a
fluctuating boundary structure producing non-trivial contributions in
$\bm{\omega}[\phi;\delta\phi,\delta\phi]|_{\mathscr{B}}$. This feature was
observed e.g. in [224] for 3$d$ gravity where there is no gravitational
propagating degree of freedom (the Weyl tensor is zero), but a non-vanishing
presymplectic flux at infinity is observed due to fluctuations of the metric
field at leading order. Furthermore, the charges can be made integrable by
changing the slicing of the phase space. This leads us to a natural
distinction between the dynamical part of
$\bm{\omega}[\phi;\delta\phi,\delta\phi]|_{\mathscr{B}}$ which encodes genuine
radiation flowing through $\mathscr{B}$ and the kinematical part of
$\bm{\omega}[\phi;\delta\phi,\delta\phi]|_{\mathscr{B}}$ which is related to
intrinsic boundary degrees of freedom that are not sourced by the radiative
modes (see figure 2.4). To express this with a tensorial object, it suffices
to compute the Weyl curvature and check that its asymptotic value only
involves the dynamical boundary degrees of freedom.

The natural conjecture that can be derived from this discussion is the
following: “when the dynamical part of the presymplectic flux at $\mathscr{B}$
vanishes, the surface charges are integrable” [259, 224]. Up to our knowledge,
there is no counterexample that invalidates this statement and the various
phase spaces studied in this thesis are also consistent with this. Since only
the dynamical part of the symplectic flux is assumed to vanish, the
conservation of the charges is not guaranteed in general. The example of 3$d$
gravity is again very instructive, because the dynamical symplectic flux is
genuinely zero. In [170], the boundary structure contains kinematical degrees
of freedom, namely the Weyl rescalings on the circle. These are responsible
for a non-vanishing presymplectic flux leading to some breaking of the
conservation laws, yet without any gravitational radiation reaching the
boundary. This is nicely interpreted in our framework: the variation of the
bulk action $S$ under a Weyl rescaling $\sigma$ reads as
$\delta_{\sigma}S=\int_{\mathscr{B}}\mathcal{A}\sigma$ on-shell, which gives
rise to the Weyl anomaly $\mathcal{A}$ in three dimensions. The latter
generates some kinematical symplectic flux at infinity although gravitational
radiation is absent. Consequently, the presymplectic current is sourced by
$\mathcal{A}$ and the charges cannot be conserved when $\mathcal{A}$ is non-
vanishing.

The kinematical boundary degrees of freedom can be recognized among others
because we can get rid of them by using the residual gauge freedom. For
instance, in asymptotically flat gravity, the boundary area $\sqrt{q}$ can be
fixed for free by using the diffeomorphism freedom among the residual gauge
diffeomorphism in the Bondi gauge without ruling out any solution [45, 37]. It
is not the same for $C_{AB}$ which enters in the symplectic flux as a
dynamical degree of freedom, as we will show in section 2.2.4. Apart from a
pure-gauge part (2.1.65) which can be brought to zero by a suitable BMS4
supertranslation, the two functions forming the rank 2 symmetric traceless
tensor $C_{AB}$ cannot be gauged away by a choice of coordinates in the
presence of gravitational radiation [14, 136, 241]. The latter, escaping from
$\mathscr{M}$ through future null infinity, sets the open system in a non-
equilibrium state, for which the integrability of the charges can only be
restored as soon as $N_{AB}=0$, or $\partial_{u}C_{AB}=0$ [31, 30].

Let us conclude this quite philosophical section by a crucial comment. The
fact that a kinematical boundary degree of freedom can be fixed does not mean
that it is not related to interesting asymptotic symmetries. This is the case,
for instance, for the Weyl rescalings that will be fixed on the boundary
throughout this thesis. The Weyl symmetry is always present as soon as the
manifold under consideration is conformally compact [9] (see 3.1.2 for a
rigorous definition) because it emerges from the ambiguity on the choice of
the conformal factor in the treatment of asymptotics through the conformal
compactification process. Generically conjugated with non-trivial charges (see
[170, 87, 92, 32]), it is definitely a non-trivial asymptotic symmetry.
Freezing it thanks to a boundary gauge fixing (namely demanding that boundary
transformations preserve the area of the celestial sphere) will make these
charges disappear but no solution will be rejected from the phase space. Hence
the choice to maintain arbitrary Weyl rescalings on the celestial sphere or
ruling them out is a matter of how general we want to be. In this perspective,
which is the status of the remaining degrees of freedom in $q_{AB}$ after the
fixation of the area and why not also fix them? For vacuum solutions, it is
true that the boundary condition (2.1.39) can always be reached by a gauge
transformation starting from any time-dependent area element $\sqrt{q}$,
because the Weyl parameter $\omega$ has also an arbitrary time dependence, see
(2.1.10). In this case, (2.1.34) implies that $\partial_{u}q_{AB}=0$ and the
remaining allowed transformations on the celestial sphere are smooth
diffeomorphisms $Y^{A}(x^{B})$ by virtue of (2.1.47). We can thus select some
$Y^{A}$ to send the arbitrary $q_{AB}(x^{C})$ to $\mathring{q}_{AB}(x^{C})$
and get back to asymptotically Minkowskian conditions. However, we will see in
chapter 5 that the coupling with matter can induce transitions in $q_{AB}$ and
subsequent interesting gravitational memory effects. While it is always
possible to fix the area element to $\sqrt{\mathring{q}}$ thanks a time-
dependent Weyl rescaling on the whole boundary, the super-Lorentz
transformations $Y^{A}(x^{B})$ are no longer sufficient to trivialize the
boundary metric everywhere since $\partial_{u}Y^{A}=0$. It shows that the two
boundary degrees of freedom in $q_{AB}$ which survive to the Weyl fixing are
dynamical in general and have to be included explicitly in the flux. We close
here our theoretical discussion on leaky boundary conditions: more precise
considerations will be given during the study of concrete examples of
radiative phase spaces.

##### 2.2.3.5 Expressions in General Relativity

Let us devote this little subsection to particularize the fundamental
quantities developed in the covariant phase space formalism for Einstein’s
theory without matter, whose action integral is (2.1.5). Under a linearized
variation $\delta g_{\mu\nu}$ around $g_{\mu\nu}$, this action transforms as

$\delta
S_{EH}[g]=\int_{\mathscr{M}}\text{d}^{n}x\,\left[\frac{\sqrt{-g}}{16\pi
G}(G_{{\mu\nu}}+\Lambda g_{\mu\nu})\delta
g^{\mu\nu}+\partial_{\mu}\Theta^{\mu}_{EH}[g;\delta g]\right]$ (2.2.80)

for the presymplectic potential [78]

$\Theta^{\mu}_{EH}[g;\delta g]=\frac{\sqrt{-g}}{16\pi
G}\left(\nabla_{\nu}(\delta g)^{\mu\nu}-\nabla^{\mu}{(\delta
g)^{\nu}}_{\nu}\right).$ (2.2.81)

Note that $(\delta g)^{\mu\nu}$ is obtained by varying the metric $g_{\mu\nu}$
before raising the indices with $g^{\mu\nu}$, hence $(\delta
g)^{\mu\nu}=-\delta g^{\mu\nu}=-g^{\mu\alpha}g^{\nu\beta}\delta
g_{\alpha\beta}$. If the variation is contracted with the action of a
diffeomorphism
$\delta_{\xi}g_{\mu\nu}=\mathcal{L}_{\xi}g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$,
we get

$\Theta^{\mu}_{EH}[g;\delta_{\xi}g]=\frac{\sqrt{-g}}{8\pi
G}\left(\nabla_{\nu}\nabla^{(\mu}\xi^{\nu)}-\nabla^{\mu}\nabla_{\nu}\xi^{\nu}\right).$
(2.2.82)

Using Einstein’s field equations, we get
$\nabla^{\mu}\nabla_{\nu}\xi^{\nu}=\nabla_{\nu}\nabla^{\mu}\xi^{\nu}+{{{R^{\nu}}_{\alpha}}^{\mu}}_{\nu}\xi^{\alpha}=\nabla_{\nu}\nabla^{\mu}\xi^{\nu}+\frac{2\Lambda}{n-2}\xi^{\mu}$,
so

$\Theta^{\mu}_{EH}[g;\delta_{\xi}g]=\frac{\sqrt{-g}}{16\pi
G}\left[\nabla_{\nu}\left(\nabla^{\nu}\xi^{\mu}-\nabla^{\mu}\xi^{\nu}\right)-\frac{4\Lambda}{n-2}\xi^{\mu}\right].$
(2.2.83)

This presymplectic potential gives us access to the Noether-Wald charge
(2.2.56) after the application of the homotopy operator (2.2.55). The latter
only hits terms with at least one derivative on $\xi^{\mu}$, so the last term
involving the cosmological constant disappears. A little calculation gives
[78]

$\bm{Q}^{EH}_{\xi}[g]=\frac{\sqrt{-g}}{16\pi
G}(\nabla^{\mu}\xi^{\nu}-\nabla^{\nu}\xi^{\mu})\,(\text{d}^{n-2}x)_{\mu\nu}.$
(2.2.84)

The contribution (2.2.84) in the Iyer-Wald codimension 2 form (2.2.58) is
often called the Komar term, in reference to the Komar integrals [258] that
give the mass and angular momentum of simple spacetimes when
$\frac{\sqrt{-g}}{8\pi G}\nabla^{[\mu}\xi^{\nu]}$ for an isometry $\xi$ is
evaluated on an asymptotic 2-sphere. The last ingredient we need to build the
local charge is

$\iota_{\xi}\bm{\Theta}_{EH}=\xi^{\mu}\frac{\partial}{\partial\text{d}x^{\mu}}\Theta_{EH}^{\nu}(\text{d}^{n-1}x)_{\nu}=(\xi^{\mu}\Theta_{EH}^{\nu}-\xi^{\nu}\Theta_{EH}^{\mu})\,(\text{d}^{n-2}x)_{\mu\nu}.$
(2.2.85)

Denoting $h_{\mu\nu}=\delta g_{\mu\nu}$ and $h\equiv{h^{\mu}}_{\mu}$, we
obtain [78] (see also [217, 211])

$\bm{k}_{\xi}^{EH}[g;h]=\frac{\sqrt{-g}}{8\pi
G}\left[\xi^{\mu}\nabla_{\sigma}h^{\nu\sigma}-\xi^{\mu}\nabla^{\nu}h+\xi_{\sigma}\nabla^{\nu}h^{\mu\sigma}+\frac{h}{2}\nabla^{\nu}\xi^{\mu}-h^{\rho\nu}\nabla_{\rho}\xi^{\nu}\right]\,(\text{d}^{n-2}x)_{\mu\nu}$
(2.2.86)

after some tensorial algebra. This gives the explicit expression of the Iyer-
Wald codimension 2 form in General Relativity. Notice that the terms in
$\delta\xi$ appearing in $\delta\bm{Q}_{\xi}[g]$ have been canceled by
$\bm{Q}_{\delta\xi}[g]$ as expected. At the pedagogical level, it is very
illuminating to take a concrete although pretty simple example. For instance,
let us evaluate (2.2.86) for the time translation symmetry of the
Schwarzschild black hole. In spherical coordinates $(t,r,\theta,\phi)$, the
region outside the horizon is described by the line element

$g_{\mu\nu}[M]=-\left(1-\frac{2M}{r}\right)\text{d}t^{2}+\left(1-\frac{2M}{r}\right)^{-1}\text{d}r^{2}+r^{2}(\text{d}\theta^{2}+\sin^{2}\theta\text{d}\phi^{2}).$
(2.2.87)

The only parameter that labels the metric is the radius $M$, so
$h_{\mu\nu}[M;\delta M]=\frac{\partial g_{\mu\nu}}{\partial M}\delta M$ at
first order. Picking $\xi=\partial_{t}$, a little massage of (2.2.86) shows
that $\bm{k}_{\xi}[M;\delta M]=\frac{\delta M}{4\pi
G}\delta_{t}^{\mu}\delta_{r}^{\nu}(\text{d}^{n-2}x)_{\mu\nu}$. Integrating on
a sphere $S$ of constant $t,r$ gives

$\delta H_{\xi}=\oint_{S}(\text{d}^{n-2}x)_{tr}\frac{\delta M}{4\pi
G}=\int_{0}^{2\pi}\text{d}\phi\>\int_{0}^{\pi}\text{d}\theta\>\sin\theta\frac{\delta
M}{4\pi G}=\frac{\delta M}{G}=\delta\mathcal{M}$ (2.2.88)

where $\mathcal{M}=M/G$ is the total mass of spacetime. The charge is
trivially integrable and, after a simple path integration between the
Minkowski metric ($M=0$) and a target metric with given $M>0$, we get the
right result according to which $\mathcal{M}$ is the total energy of the
Schwarzschild black hole. Note that in this simple case, (2.2.88) contains
only the local expression of the Komar charge [258].

##### 2.2.3.6 Link with the Barnich-Brandt formalism

We conclude the presentation of the formalism for the surface charge by
quickly reviewing the alternative formulation provided by Barnich and Brandt
in [80, 81] and making the link with the Iyer-Wald definition.

The great advantage of the Barnich-Brandt definition for the local codimension
2 forms $\bm{k}_{\xi}^{BB}[\phi;\delta\phi]$ consists in not requiring that
the theory under consideration is generally covariant. In contrast with the
Iyer-Wald definition, it is applicable for any Lagrangian gauge theory and
leads to a particular prescription to fix the boundary ambiguity in the
presymplectic form. Moreover, the definition of the presymplectic current
relies only on the datum of some equations of motion, regardless of whether
the gauge theory admits a Lagrangian formulation or not! Up to this major
difference, the Barnich-Brandt method functions in a way similar to Iyer-
Wald’s procedure and also relies on the link between the symplectic structure
on $\mathscr{J}$ and lower-degree conserved currents.

We first introduce a more formal procedure for performing integration by parts
on expressions depending on the fields $\phi^{i}$ but not necessarily on the
gauge parameters. It involves Anderson’s homotopy operator
$I^{p}_{\delta\phi}$ [261, 250], which bears some resemblance with the
operator $I^{p}_{\xi}$ constructed above. Using the Grassmann odd convention
for $\delta$, it can be defined as follows :

$\displaystyle-\text{d}I^{n}_{\delta\phi}+\delta\phi^{i}\frac{\delta}{\delta\phi^{i}}$
$\displaystyle=\delta\quad\text{ when acting on }n\text{-forms ;}$ (2.2.89)
$\displaystyle-\text{d}I^{p}_{\delta\phi}+I^{p+1}_{\delta\phi}\text{d}$
$\displaystyle=\delta\quad\text{ when acting on }p\text{-forms }(p<n).$
(2.2.90)

The cases $p=n$ and $p=n-1$ can be worked out explicitly as

$\displaystyle I^{n}_{\delta\phi}$
$\displaystyle=\left[\delta\phi^{i}\frac{\partial}{\partial\partial_{\mu}\phi^{i}}-\delta\phi^{i}\partial_{\nu}\frac{\partial}{\partial\partial_{\mu}\partial_{\nu}\phi^{i}}+\partial_{\nu}\delta\phi^{i}\frac{\partial}{\partial\partial_{\mu}\partial_{\nu}\phi^{i}}+\cdots\right]\frac{\partial}{\partial\text{d}x^{\mu}},$
(2.2.91) $\displaystyle I^{n-1}_{\delta\phi}$
$\displaystyle=\left[\frac{1}{2}\delta\phi^{i}\frac{\partial}{\partial\partial_{\mu}\phi^{i}}-\frac{1}{3}\delta\phi^{i}\partial_{\nu}\frac{\partial}{\partial\partial_{\mu}\partial_{\nu}\phi^{i}}+\frac{2}{3}\partial_{\nu}\delta\phi^{i}\frac{\partial}{\partial\partial_{\mu}\partial_{\nu}\phi^{i}}+\cdots\right]\frac{\partial}{\partial\text{d}x^{\mu}}.$
(2.2.92)

Applying (2.2.89) on the Lagrangian $\bm{L}$, we get

$\delta\bm{L}=\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}-\text{d}I^{n}_{\delta\phi}\bm{L}$
(2.2.93)

implying that $\bm{\Theta}=I^{n}_{\delta\phi}\bm{L}$ thanks to (2.2.32). Using
(2.2.90), we can apply $I_{\delta\phi}^{n}$ on both sides of (2.2.93) to get :

$\displaystyle
I^{n}_{\delta\phi}\delta\bm{L}=I^{n}_{\delta\phi}\left(\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}\right)-I_{\delta\phi}^{n}\text{d}I_{\delta\phi}^{n}\bm{L}=I^{n}_{\delta\phi}\left(\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}\right)-\delta
I_{\delta\phi}^{n}\bm{L}-\text{d}I^{n-1}_{\delta\phi}I^{n}_{\delta\phi}\bm{L}$
(2.2.94) $\displaystyle\Rightarrow I^{n}_{\delta\phi}\delta\bm{L}+\delta
I_{\delta\phi}^{n}\bm{L}=I^{n}_{\delta\phi}\left(\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}\right)-\text{d}I^{n-1}_{\delta\phi}I^{n}_{\delta\phi}\bm{L}.$
(2.2.95)

Since $[\delta,I^{n}_{\delta\phi}]=0$ because $\delta^{2}=0$, the left-hand
side is nothing but $2\delta
I^{n}_{\delta\phi}\bm{L}=2\bm{\omega}[\delta\phi,\delta\phi,\phi]$ by
definition, (2.2.34). We conclude that [2]

$\bm{\omega}[\phi;\delta\phi,\delta\phi]=\bm{W}[\phi;\delta\phi,\delta\phi]+\text{d}\bm{E}[\phi;\delta\phi,\delta\phi],$
(2.2.96)

where we have isolated the invariant presymplectic current

$\bm{W}[\phi;\delta\phi,\delta\phi]=\frac{1}{2}I^{n}_{\delta\phi}\left(\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}\right).$
(2.2.97)

It differs from Iyer-Wald’s presymplectic current by a boundary term that
reads as

$\bm{E}[\phi;\delta\phi,\delta\phi]=\frac{1}{2}I^{n-1}_{\delta\phi}I^{n}_{\delta\phi}\bm{L}.$
(2.2.98)

Barnich and Brandt proposed to choose $\bm{W}$ instead of $\bm{\omega}$ as
symplectic form to build conserved surface charges [80]. $\bm{W}$ is called
“invariant” because it is defined in terms of the equations of motion and does
not depend upon the boundary terms added to the action. No ambiguity appears
in this formulation, which can be thought as a force as well as a weakness.
Indeed, we will show later in this thesis, on (at least) two concrete
examples, that the Iyer-Wald ambiguity remains nonetheless very useful when
renormalization of the presymplectic structure is mandatory. The offer of this
freedom allows to bring easily the results of the renormalization scheme
defined for the on-shell variational principle at the level of the
presymplectic structure and consequently to the local surface charges.

Let us consider a diffeomorphic transformation $\xi$ existing for a covariant
field theory $\bm{L}[\phi^{i}]$.
$\delta_{\xi}\phi^{i}=\mathcal{L}_{\xi}\phi^{i}$ by hypothesis. Restarting
again from Noether’s second theorem

$\text{d}\bm{S}_{\xi}=\frac{\delta\bm{L}}{\delta\phi^{i}}\mathcal{L}_{\xi}\phi^{i},$
(2.2.99)

we can apply the homotopy operator $I^{n}_{\delta\phi}$ to obtain

$\displaystyle I^{n}_{\delta\phi}\text{d}\bm{S}_{\xi}$
$\displaystyle=I^{n}_{\delta\phi}\left(\frac{\delta\bm{L}}{\delta\phi^{i}}\mathcal{L}_{\xi}\phi^{i}\right)=\delta\bm{S}_{\xi}+dI^{n-1}_{\delta\phi}\bm{S}_{\xi}.$
(2.2.100)

After some non-trivial steps, it can be proved that the contraction of
$\bm{W}$ on-shell with the transformation $\delta_{\xi}\phi$ reads as

$\bm{W}[\phi;\mathcal{L}_{\xi}\phi,\delta\phi]=i_{\mathcal{L}_{\xi}\phi}\bm{W}[\phi;\delta\phi,\delta\phi]=I^{n}_{\mathcal{L}_{\xi}\phi}\left(\frac{\delta\bm{L}}{\delta\phi^{i}}\delta\phi^{i}\right)=-I^{n}_{\delta\phi}\left(\frac{\delta\bm{L}}{\delta\phi}\mathcal{L}_{\xi}\phi\right),$
(2.2.101)

see property 13 of [2]. Compiling both pieces of information and recalling
that $\delta\bm{S}_{\xi}$ vanishes as soon as $\phi^{i}$ and $\delta\phi^{i}$
are on-shell, we get Barnich-Brandt’s version of (2.2.57):

$\boxed{\bm{W}[\phi;\delta_{\xi}\phi,\delta\phi]=\text{d}\bm{k}^{BB}_{\xi}[\phi;\delta\phi],}$
(2.2.102)

which holds on-shell and involves the _invariant_ or Barnich-Brandt
codimension 2 form [80]

$\boxed{\bm{k}^{BB}_{\xi}[\phi;\delta\phi]=-I^{n-1}_{\delta\phi}\bm{S}_{\xi}\left[\frac{\delta\bm{L}}{\delta\phi},\phi\right].}$
(2.2.103)

The surface charges are obtained by integration on a codimension 2 surface and
on the phase space, as before. The computation of the Barnich-Brandt charge
for General Relativity can be performed thanks to this formula and with the
mere knowledge of $\bm{S}_{\xi}$ reading as (2.2.48). Here we can take a
shortcut by gathering the relations (2.2.57), (2.2.102) and (2.2.96) to get

$\bm{k}^{IW}_{\xi}[\phi;\delta\phi]=\bm{k}^{BB}_{\xi}[\phi;\delta\phi]+\bm{E}[\phi;\delta_{\xi}\phi,\delta\phi]+\text{d}(\cdot),$
(2.2.104)

again dropping the irrelevant total derivatives. Using the definition of the
ambiguity $\bm{E}$ in terms of $\bm{L}$ and assuming that the latter is the
Einstein-Hilbert Lagrangian, we have

$\bm{E}[\delta g,\delta g;g]=\frac{1}{32\pi G}(\delta
g)^{\mu}_{\phantom{\mu}\alpha}\wedge(\delta
g)^{\alpha\nu}(\text{d}^{n-2}x)_{\mu\nu}.$ (2.2.105)

Denoting again $\phi=g_{\mu\nu}$ and $\delta\phi=h_{\mu\nu}$, we get

$\begin{split}\bm{k}^{\mu\nu}_{\xi}[g;h]=\frac{\sqrt{-g}}{8\pi
G}\Big{[}&\xi^{\mu}\nabla^{\sigma}h^{\nu}_{\sigma}-\xi^{\mu}\nabla^{\nu}h+\xi^{\sigma}\nabla^{\nu}h^{\mu}_{\sigma}\\\
&+\frac{1}{2}h\nabla^{\nu}\xi^{\mu}+\frac{1}{2}h^{\nu}_{\sigma}(\nabla^{\mu}\xi^{\sigma}-\nabla^{\sigma}\xi^{\mu})\Big{]}(\text{d}^{n-2}x)_{\mu\nu}.\end{split}$
(2.2.106)

This formula was also obtained by Abbott and Deser by a similar procedure
involving integrations by parts [139, 262, 263], without using the formal
operators (2.2.89)-(2.2.90). To conclude on very good news at practical level,
one can check that the boundary term (2.2.105) is often zero when contracted
with the variation $\delta_{\xi}g_{\mu\nu}$. This is true for exact
(respectively asymptotic) Killing vectors, for which
$\delta_{\xi}g_{\mu\nu}=0$ everywhere (respectively in the asymptotic region).
Moreover, in many coordinate systems (and particularly in the Bondi, Newman-
Unti and Starobinsky/Fefferman-Graham gauges used in this thesis),
$\bm{E}[g;\delta_{\xi}g,\delta g]=0$ for any residual diffeomorphism $\xi$.
This last observation allows us to trade one formalism for the other without
worrying so much about the slight difference, irrelevant in the coordinate
systems we will work in practice. We close this short digression to return
back to concrete considerations regarding asymptotically locally flat
radiative phase spaces.

#### 2.2.4 Asymptotically locally flat radiative phase spaces

In section 2.1.5, we have discussed the boundary conditions and the solution
space $\mathring{\mathcal{S}}_{0}$ for Einstein’s gravity with asymptotically
locally flat behavior at future null infinity, as well as the group of
asymptotic symmetries coined as the Generalized BMS4 group. We connected this
extended solution space to the historical set of asymptotically Minkowskian
solutions $\mathring{\mathcal{S}}^{\text{Mink}}_{0}$, whose asymptotic group
is the global BMS4 group. Here, we promote these solution spaces into phase
spaces thanks to the covariant phase space formalism, compute asymptotic
surface charges and discuss their properties. Note that, sometimes, we will
reserve the full discussion including super-Lorentz transformations for the
more involved investigations summarized in chapter 5. At this point, besides
providing new sources for leaks at infinity and increasing the complexity of
the analysis, they are not needed to tackle the core of the discussion, since
all of the interesting features of the surface charges (namely non-
conservation and non-integrability) are yet observable for solutions in
$\mathring{\mathcal{S}}^{\text{Mink}}_{0}$.

##### 2.2.4.1 Variational principle and renormalization

Let us consider $g_{\mu\nu}\in\mathring{\mathcal{S}}_{0}$. The Einstein-
Hilbert action $S_{EH}[g]$ is given as (2.1.5) with $\Lambda=0$. In order to
prevent some eventual divergences in the variational principle, we evaluate
the action integral on a volume $\mathscr{V}$ of $\mathscr{M}$ bounded by the
three hypersurfaces $\mathscr{U}^{-}=\\{u=u^{-},r\leq r^{+}\\}$,
$\mathscr{I}^{+}_{r}=\\{u^{-}\leq u\leq u^{+},r=r^{+}\\}$,
$\mathscr{U}^{+}=\\{u=u^{+},r\leq r^{+}\\}$ for some radial cut-off
$r^{+}<+\infty$ and two time cut-offs $-\infty<u^{-}<u^{+}<+\infty$, as
represented in Figure 2.5. We will consider the limit
$r^{+}\rightarrow+\infty$, $u^{\pm}\rightarrow\pm\infty$ afterwards.

$\mathscr{I}^{+}_{+}$$\mathscr{I}^{+}_{-}$$\mathscr{I}^{+}$$r=r^{+}$$u=u^{+}$$u=u^{-}$Radiation
Figure 2.5: Schematic contour for the variational principle.

Using (2.2.80) and (2.2.81), the on-shell variational principle is evaluated
to be

$\delta S_{EH}=\int_{\partial\mathscr{V}}\bm{\Theta}_{EH}[g,\delta
g]=\int_{\mathscr{U}^{-}}\text{d}r\,\text{d}^{2}x\,\Theta_{EH}^{u}+\int_{\mathscr{I}^{+}_{r}}\text{d}u\,\text{d}^{2}x\,\Theta_{EH}^{r}+\int_{\mathscr{U}^{+}}\text{d}r\,\text{d}^{2}x\,\Theta_{EH}^{u}$
(2.2.107)

with

$\displaystyle\Theta^{u}_{EH}$
$\displaystyle=r\,\Theta^{u}_{(div)}+\Theta^{u}_{(fin)}+\mathcal{O}(r^{-2}),$
(2.2.108) $\displaystyle\Theta^{r}_{EH}$
$\displaystyle=r\,\Theta^{r}_{(div)}+\Theta^{r}_{(fin)}+\mathcal{O}(r^{-1}).$
(2.2.109)

in the large $r$ limit. We have $\Theta^{u}_{(div)}\propto\delta\sqrt{q}$,
therefore vanishing because of the boundary condition (2.1.27). The other
components are

$\displaystyle\Theta^{u}_{(fin)}$ $\displaystyle=\frac{\sqrt{q}}{16\pi
G}\frac{1}{2}C_{AB}\delta q^{AB},$ (2.2.110) $\displaystyle\Theta^{r}_{(div)}$
$\displaystyle=-\frac{\sqrt{q}}{16\pi G}\delta
R[q]-\frac{1}{2}\frac{\sqrt{q}}{16\pi G}N_{AB}\delta q^{AB},$ (2.2.111)
$\displaystyle\Theta^{r}_{(fin)}$ $\displaystyle=\frac{\sqrt{q}}{16\pi
G}\,\delta\left[-\frac{1}{8}C^{AB}N_{AB}+2M+D_{A}\mathcal{U}^{A}\right]$
$\displaystyle+\bar{\Theta}_{flux}-\frac{\sqrt{q}}{16\pi
G}D_{A}(\mathcal{U}_{B}\delta q^{AB}),$ (2.2.112)

where $\mathcal{U}^{A}\equiv-\frac{1}{2}D_{B}C^{AB}$ and we define with
hindsight the important quantity

$\bar{\Theta}_{flux}[q_{AB},C_{AB};\delta q^{AB},\delta
C^{AB}]\equiv\frac{\sqrt{q}}{16\pi G}\Big{[}\frac{1}{2}N_{AB}\delta
C^{AB}-\frac{1}{4}R[q]C_{AB}\delta q^{AB}+\mathcal{U}_{B}D_{A}\delta
q^{AB}\Big{]}.$ (2.2.113)

Some useful identities for the computation of $\Theta^{u}_{EH}$ and
$\Theta^{r}_{EH}$, involving variations of symmetric traceless tensors as well
as a fluctuating boundary metric, can be found in the appendix E. The bare
action principle is divergent on-shell because of the pole in $\mathcal{O}(r)$
in the presymplectic flux when $r\to+\infty$. The latter is due to the
fluctuations of the metric on celestial sphere and disappear when we restrict
the analysis to $\mathring{\mathcal{S}}_{0}^{\text{Mink}}$. Allowing
variations of the boundary structure calls for a renormalization procedure of
the on-shell variational principle, at least for the radial divergences. At
this point, it is useful to remark that a total derivative and a total
variation can be isolated in the divergent pieces:

$\displaystyle\Theta^{u}_{(fin)}$ $\displaystyle=$
$\displaystyle-\partial_{r}Y^{ur},$ (2.2.114) $\displaystyle
r\,\Theta^{r}_{(div)}$ $\displaystyle=$
$\displaystyle-\partial_{u}Y^{ru}+\delta(-\sqrt{q}\mathring{R}\;r)=-\partial_{u}Y^{ru}-\partial_{A}Y^{rA}$
(2.2.115)

where

$Y^{ur}=-Y^{ru}=-r\frac{1}{2}\frac{\sqrt{q}}{16\pi G}C_{AB}\delta q^{AB},$
(2.2.116)

and

$Y^{rA}=r\,\frac{1}{16\pi G}\,\Theta^{A}_{2d}(\delta q;q)$ (2.2.117)

is $r$ times the presymplectic potential of the 2-dimensional Einstein-Hilbert
action, $\partial_{A}\Theta^{A}_{2d}=\delta(\sqrt{q}\mathring{R})$. Since the
boundary of a boundary is zero, the corner terms $\propto\,Y^{ur}$ in the
variational principle drop. After integration over the sphere, the total
derivatives $\propto\,\partial_{A}v^{A}$ also drop. The radially divergent
contribution to the action is therefore the integrable term

$\frac{1}{4}\,\delta\left[r\int_{u^{-}}^{u^{+}}\text{d}u\,\chi[q]\right],\qquad\chi\equiv\frac{1}{4\pi}\oint_{S_{\infty}}\sqrt{q}R[q]$
(2.2.118)

where $\chi[q]$ is the topological Euler number of $q_{AB}$. We have
$\chi[\mathring{q}]=2$ for the unit-round sphere metric, which is assumed to
be part of the phase space since $\sqrt{q}=\sqrt{\mathring{q}}$ on
$\mathring{\mathcal{S}}_{0}$. Since this condition is diffeomorphic and Weyl
invariant, it is consistent with the action of super-Lorentz transformations
and supertranslations. Under an infinitesimal smooth variation, a topological
number cannot change. The action of boundary diffeomorphisms as well as Weyl
transformations preserves $\chi[q]=2$, hence $\delta_{Y}\chi[q]=0$ under any
smooth super-Lorentz transformation. If we allow singular infinitesimal
changes, such as the ones generated by singular super-Lorentz transformations
that arise in the snapping of cosmic strings, the boundary topology changes
and one would require to add a boundary term in the action to cancel this
divergence. Here we can discard this divergent term by imposing the
supplementary boundary condition

$\delta\left[\frac{1}{4\pi}\oint_{S_{\infty}}\sqrt{q}R[q]\right]=0$ (2.2.119)

which is perfectly compatible with the action of super-Lorentz
transformations.

Let us now evaluate the symplectic flux. In (2.2.112), the total derivative
terms will drop after integration on the sphere, hence the radial symplectic
flux contains a integrable term which has no influence on the presymplectic
current and the charges. We can remove it if we supplement the Einstein-
Hilbert action by the following boundary term:

$S_{ren}=S_{EH}-\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}\Omega\,\Big{(}2M-\frac{1}{8}C^{AB}N_{AB}\Big{)}$
(2.2.120)

where
$\text{d}^{2}\Omega\equiv\sqrt{q}\,\text{d}^{2}x=\frac{1}{2}\sqrt{q}\,\epsilon_{AB}\text{d}x^{A}\text{d}x^{B}$.
Notice that the last term, $D_{A}\mathcal{U}^{A}$, has not been included since
it gives no contribution after integration on the sphere. We do not provide a
covariant formulation of this boundary term, or the boundary terms
$Y^{\mu\nu}$, which would require geometrical tools on boundary null surfaces
[264, 265, 266, 267] or a prescription from holographic renormalization [268,
65]. The renormalized variational principle in $\mathscr{V}$ reads thus as

$\delta
S_{ren}=\int_{\mathscr{U}^{-}}\text{d}r\,\text{d}^{2}x\,\Theta_{in}^{u}+\int_{\mathscr{I}^{+}_{r}}\text{d}u\,\text{d}^{2}x\,\bar{\Theta}_{flux}+\int_{\mathscr{U}^{+}}\text{d}r\,\text{d}^{2}x\,\Theta_{in}^{u}$
(2.2.121)

where $\Theta_{in}^{u}=\mathcal{O}(r^{-2})$. In the limit $|u|\to+\infty$, the
first term drops since we assume that there are no incoming gravitational
radiation at $\mathscr{I}^{-}$. Ignoring for the moment the corner divergences
in $u$ whose discussion is postponed to chapter 5, the variational principle
gives

$\delta
S_{ren}=\int_{\mathscr{I}^{+}}\text{d}u\,\text{d}^{2}x\,\bar{\Theta}_{flux}$
(2.2.122)

in the limit $r\to+\infty$. An asymptotically locally flat spacetime is thus
considered in general as an open system with physical flux leaking through the
boundary $\mathscr{I}^{+}$. This leak is indispensable for allowing
gravitational radiation in the asymptotic region. Observing the form of
(2.2.113), there are two sources for the flux: the asymptotic shear $C_{AB}$
which is related to standard gravitational radiation [6, 7, 8, 219, 46] and
$q_{AB}$ which can be varied when one includes cosmic events, whose action
modify the metric tensor at leading order at $\mathscr{I}^{+}$ [187, 45], in
the phase space. Notice that even in the simplest case of the asymptotically
Minkowskian phase space where $\delta q_{AB}=0$, the symplectic flux is non-
vanishing, hence the action $S_{ren}$ is not stationary on solutions! This
suggests that suitable boundary conditions allowing for modifications of the
asymptotic shear $C_{AB}$ are naturally leaky. We will verify that this is the
case and make the link with the presence of gravitational waves in the bulk of
$\mathscr{M}$.

##### 2.2.4.2 Symplectic structure and charges

In the previous paragraph, we have motivated that the divergences in the
Einstein-Hilbert presymplectic potential could be compensated by the
incorporation of corner terms $Y^{ur}$ and $Y^{rA}$, see (2.2.114)-(2.2.115).
We can make good use of the Iyer-Wald ambiguity
$\bm{\Theta}_{EH}\to\bm{\Theta}_{ren}=\bm{\Theta}_{EH}+\text{d}\bm{Y}$ to
bring the renormalization at the level of the presymplectic potential. The
components $Y^{ur}$ and $Y^{rA}$ of the codimension 2 form $\bm{Y}$ have
already been identified in (2.2.116)-(2.2.117). Since $Y^{rA}$ is a
$\delta$-exact term, it does not contribute to the presymplectic current.
After renormalization, the presymplectic potential reads as

$\Theta^{u}_{ren}=\mathcal{O}(r^{-2}),\quad\Theta^{r}_{ren}=\bar{\Theta}_{flux}+\mathcal{O}(r^{-1}),$
(2.2.123)

leading to the following presymplectic current evaluated on $\mathscr{I}^{+}$

$\begin{split}&\bm{\omega}_{ren}[\phi;\delta_{1}\phi,\delta_{2}\phi]\Big{|}_{\mathscr{I}^{+}}=\\\
&\qquad\boxed{\frac{\text{d}u\,\text{d}^{2}\Omega}{16\pi
G}\left[\frac{1}{2}\delta_{1}\left(N^{AB}+\frac{1}{2}R[q]q^{AB}\right)\wedge\delta_{2}C_{AB}-\delta_{1}(D_{(A}\mathcal{U}_{B)})\wedge\delta_{2}q^{AB}\right].}\end{split}$
(2.2.124)

Here, the notation
$\phi=\\{g_{\mu\nu},q_{AB},C_{AB},\bm{T},\sqrt{\mathring{q}}\\}$ encompasses
the sources for the presymplectic flux and the background structures. The
latter, like the renormalization procedure, are defined in the specific radial
foliation given in the Bondi gauge: the definitions of $q_{AB}$, $C_{AB}$ and
the boundary term $\bm{Y}$ are not meant to be bulk covariant because they
depend on the additional background structure close to $\mathscr{I}^{+}$ (such
as the null foliation $\bm{T}$ or the fixed codimension 2 volume form).
Although a gauge-invariant formulation of the renormalization process would be
beneficial, we do not plan to discuss that further here, since we are just
using the asymptotically flat case as an illustrative example for the
covariant phase space methods and a kind of prequel story for our discussion
in (A)dS in the following chapters 3 and 4. Moreover, we will not need to
detail these boundary structures in the following discussion and will just
start from now by assuming that the presymplectic current is given by
(2.2.124), which defines the following symplectic structure at
$\mathscr{I}^{+}$

$\displaystyle\mathcal{W}_{flux}[\delta_{1}\phi,\delta_{2}\phi]$
$\displaystyle\equiv\frac{1}{16\pi
G}\int_{\mathscr{I}^{+}}\text{d}u\text{d}^{2}\Omega\left[\frac{1}{2}\delta_{1}\left(N^{AB}+\frac{1}{2}R[q]q^{AB}\right)\wedge\delta_{2}C_{AB}-\delta_{1}(D_{(A}\mathcal{U}_{B)})\wedge\delta_{2}q^{AB}\right]$
$\displaystyle=\int_{\mathscr{I}^{+}}\text{d}u\text{d}^{2}\Omega\left(\delta_{1}\bar{\Theta}_{flux}[\phi;\delta_{2}\phi]-\delta_{2}\bar{\Theta}_{flux}[\phi;\delta_{1}\phi]\right).$
(2.2.125)

In the second equality, $\bar{\Theta}_{flux}$ is still defined as (2.2.113)
but a total derivative has been discarded.

By means of the fundamental theorem of the covariant phase space formalism,
the surface charges associated with the Generalized BMS4 symmetries $\xi(T,Y)$
are defined from the codimension 2 forms
$\bm{k}^{\text{GBMS}_{4}}_{\xi}[\phi;\delta\phi]$ obeying
$\text{d}\bm{k}_{\xi}[\phi;\delta\phi]=\bm{\omega}_{ren}[\phi;\delta_{\xi}\phi,\delta\phi]$
with $\bm{\omega}_{ren}$ given by (2.2.124). Since we need to compute charges
evaluated on spheres $S$ of $u$ at infinity, we are interested in the radial
component of the current i.e.
$\partial_{u}k^{ur}_{\xi}[\phi;\delta\phi]+\partial_{A}k^{Ar}_{\xi}[\phi;\delta\phi]=\omega^{r}_{ren}[\phi;\delta_{\xi}\phi,\delta\phi]$.
The contribution of the angular components $k^{Ar}_{\xi}$ disappears after the
integration on $S$, so we just need $k^{ur}_{\xi}[\phi;\delta\phi]$. The
computation of this quantity is outlined in [45] and we just present the final
result here. The surface charges are

$\delta/H_{\xi}^{\text{GBMS}_{4}}[\phi]=\oint_{S_{\infty}}\bm{k}_{\xi}^{\text{GBMS}_{4}}[\phi;\delta\phi]\equiv\delta
H^{\text{GBMS}_{4}}_{\xi}[\phi]+\Xi^{\text{GBMS}_{4}}_{\xi}[\phi;\delta\phi]$
(2.2.126)

for the natural split between the integrable part [45, 30] (see also [240,
44])

$H^{\text{GBMS}_{4}}_{\xi}[\phi]=\frac{1}{16\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\left[4fM+2Y^{A}N_{A}+\frac{1}{16}Y^{A}D_{A}(C_{BC}C^{BC})\right]$
(2.2.127)

and the non-integrable part [45]

$\begin{split}\Xi_{\xi}^{\text{GBMS}_{4}}[\phi,\delta\phi]&=\frac{1}{16\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\left[\frac{1}{2}f\left(N^{AB}+\frac{1}{2}q^{AB}R[q]\right)\delta
C_{AB}-2\partial_{(A}f\mathcal{U}_{B)}\delta q^{AB}\right.\\\
&\qquad\qquad\qquad\qquad\left.-fD_{(A}\mathcal{U}_{B)}\delta
q^{AB}-\frac{1}{4}D_{C}D^{C}fC_{AB}\delta q^{AB}\right],\end{split}$ (2.2.128)

keeping $f=T+\frac{1}{2}uD_{A}Y^{A}$ instead of $T$ for the sake of
readability. $M$ and $N_{A}$ appear as conjugated to $T$ (or equivalently $f$)
and $Y^{A}$ in the expression (2.2.127), which justify their denominations as
mass and angular momentum aspects. For the Kerr black hole for instance, the
surface charges associated with the Killing vectors $\partial_{u}$ and
$\partial_{\phi}$ are just given by these integrable pieces, providing the
total mass and angular momentum of the black hole as the integral on sphere of
$M$ and $N_{A}$ respectively. When $\delta q_{AB}=0$ and $Y^{A}$ is a
conformal Killing vector on $S_{\infty}$ (except at poles), we recover the
result of [30].

Clearly, the charges are non-integrable and we observe that the sources
$(\delta q^{AB},\delta C_{AB})$ for the symplectic flux are responsible for
this non-integrability. Therefore, the split (2.2.126) is ambiguous and the
integrable piece (2.2.127) is just provided here as the natural choice coming
out from the computation without giving more justification. The fixation of
the finite Hamiltonian canonically conjugated with $\xi(T,Y)$ needs some
additional physical inputs, such as regularity conditions at the corners
$|u|\to+\infty$ [269, 45] and conditions on the flux of charges [31, 45]. An
important choice of Hamiltonian enjoying numerous suitable properties will be
presented in section 5.3.

Moreover, since the contraction of (2.2.124) on a diffeomorphism parameter
$\xi(T,Y)$ is non-zero at null infinity, the charges are also not conserved in
(retarded) time:

$\frac{\text{d}}{\text{d}u}\delta/H^{\text{GBMS}_{4}}_{\xi}[\phi;\delta\phi]=\oint_{S_{\infty}}\omega^{r}_{ren}[\phi;\delta_{\xi}\phi,\delta\phi]\neq
0.$ (2.2.129)

The integrated flux of charges associated with the diffeomorphism $\xi$ is
defined as

$\begin{split}F^{\text{GBMS}_{4}}_{\xi}[\phi]&=\int_{\gamma}\int_{-\infty}^{+\infty}\text{d}u\,\partial_{u}\delta/H_{\xi}^{\text{GBMS}_{4}}[\phi;\delta\phi]\\\
&=\int_{\gamma}\int_{\mathscr{I}^{+}}\text{d}\bm{k}_{\xi}^{\text{GBMS}_{4}}[\phi;\delta\phi]=\int_{\gamma}\mathcal{W}_{flux}[\phi;\delta_{\xi}\phi,\delta\phi].\end{split}$
(2.2.130)

Hence the presence of the presymplectic flux $\bar{\Theta}_{flux}$ in
(2.2.122) is responsible for the non-conservation of the surface charges, i.e.
$F_{\xi}^{\text{GBMS}_{4}}[\phi]\neq 0$.

Let us briefly discuss one important particular case. The (retarded) time
translation $\xi(1,0)$ $=\partial_{u}$ being clearly part of the asymptotic
group, we consider the flux-balance law for the associated surface charge
$\delta/H_{\xi(1,0)}^{\text{GBMS}_{4}}$ that would play the role of the
Hamiltonian if $u$ was a timelike coordinate. We have explicitly

$\frac{\text{d}}{\text{d}u}\delta/H_{\xi(1,0)}^{\text{GBMS}_{4}}[\phi;\delta\phi]=\frac{1}{32\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\left[\partial_{u}N_{AB}\delta
C^{AB}-\frac{1}{2}\delta(N_{AB}N^{AB})\right]$ (2.2.131)

using the variations (2.1.71)–(2.1.75) with $\xi(1,0)$ and $2N_{AB}\delta
N^{AB}=\delta(N_{AB}N^{AB})$ when $\delta\sqrt{q}=0$. The left-hand side reads
as $\frac{1}{32\pi G}[8\delta\partial_{u}M+\partial_{u}N_{AB}\delta
C^{AB}+\frac{1}{2}\delta(N_{AB}N^{AB})]$ and the non-integrable pieces
involving $\partial_{u}N_{AB}$ cancel out. We get rid of $\delta$’s quite
easily to obtain

$\frac{\text{d}}{\text{d}u}H^{\text{GBMS}_{4}}_{\xi(1,0)}=\frac{\text{d}}{\text{d}u}\oint_{S_{\infty}}\text{d}^{2}\Omega\frac{1}{4\pi
G}M(u,x^{C})=-\frac{1}{32\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\,N_{AB}N^{AB}.$ (2.2.132)

Recalling the definition of the Bondi mass as
$\mathcal{M}=\oint_{S_{\infty}}\text{d}^{2}\Omega\,\frac{M}{4\pi G}$, we have
in fact re-derived the long-time celebrated Bondi mass loss formula [6]

$\boxed{\frac{\text{d}}{\text{d}u}\mathcal{M}=-\frac{1}{32\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\,N_{AB}N^{AB}.}$ (2.2.133)

As a check, we can observe that (2.2.133) is consistent with the time
evolution for the Bondi mass aspect (2.1.44). Historically, this flux-balance
formula for the Hamiltonian $H^{\text{GBMS}_{4}}_{\xi(1,0)}$ canonically
conjugated to the (retarded) time translation at $\mathscr{I}^{+}$ was used as
a pretty convincing argument that gravitational waves exist beyond the linear
approximation of the theory [6] and, therefore, are not an artifact of
linearization. The latter would precisely kill the quadratic term appearing in
(2.2.133). This formula allows also to give the physical meaning of the Bondi
news $N_{AB}$, as the analog of the Maxwell field strengh for gravitational
radiation, since its square is proportional to the energy flux carried out by
the gravitational waves across $\mathscr{I}^{+}$. This discussion clarifies a
bit the role of the source $C_{AB}$ for non-integrability and non-conservation
and establishes the boundary conditions leading to asymptotically flat phase
spaces as leaky in our terminology. In the asymptotically Minkowskian case,
$C_{AB}$ is the unique source to be turned on. Imposing a subset of
conservative boundary conditions in that context requires to restrict the
phase space to stationary spacetimes where $\partial_{u}C_{AB}=N_{AB}=0$,
which completely kills the dynamics of gravitational waves. When we generalize
to asymptotically locally flat spacetimes where a second set of sources is
turned on in the boundary metric, the notion of stationarity is a bit less
evident and we need more work to interpret the physical flux at
$\mathscr{I}^{+}$. We postpone these more evolved considerations to chapter 5.

###  2.3 Surface charge algebra

As we reviewed symplectic methods for Hamiltonian mechanics, we have seen that
Hamiltonian charges form an algebra of smooth functions on the phase space
under the Poisson bracket. Moreover, they represent the algebra of symmetries,
defined as Hamiltonian vectors on the phase space. Here we show how to import
these results in the framework of the covariant phase space formalism, first
for integrable charges. The task is not so difficult if we focus on the
charges (2.2.69) rather than the codimension 2 form
$\bm{k}_{\xi}[\phi;\delta\phi]$. In particular, the proof of the so-called
representation theorem [62, 50] (see also [86]) proceeds in the same way that
in classical mechanics and relies only on the structural properties
(2.2.28)–(2.2.31) of the jet bundle. Local considerations involving
$\bm{k}_{\xi}[\phi;\delta\phi]$ can be found among the profound discussions of
[2] but will not be part of the developments presented here below.

When the integrability of the charges (2.2.69) cannot be assumed, the
representation theorem looses its central hypothesis and must be partially
extended. The failure has to be ascribed to the Poisson bracket that is no
longer well-suited to compute the charge algebra in the presence of non-
integrability. In [30], Barnich and Troessaert heuristically introduced a new
definition for the charge bracket while attempting to compute the algebra
formed the BMS4 non-integrable charges in asymptotically flat spacetimes. They
showed that this algebra is closed under their modified bracket up to a field-
dependent “central extension.” Since then, similar computations were
reproduced in different contexts (see e.g. [45, 37, 88, 270, 93, 91]),
accumulating evidences in favor this generalized charge algebra. The good news
is that it is possible to give a completely general proof of the
representation theorem for non-integrable charges involving the Barnich-
Troessaert bracket. As before, it relies only on the existence of a
differential phase space with trivial topological features and the datum of an
infinitesimal charge split between integrable and non-integrable pieces. The
result will only be mentioned here and its proof will be discussed elsewhere
[94].

#### 2.3.1 Integrable case: the Poisson bracket

Recalling the definition (2.2.69) of the infinitesimal surface charges, the
hypothesis of integrability yields

$i_{\delta_{\xi}}\mathcal{W}[\phi;\delta\phi,\delta\phi]=\delta H_{\xi}[\phi]$
(2.3.1)

This states that $\delta_{\xi}\phi$ is a Hamiltonian vector field on the jet
bundle associated with the Hamiltonian function $H_{\xi}[\phi]$, in analogy
with (2.2.17) in classical mechanics. The dynamics of the system is completely
encoded into the transformations of the charges under gauge transformations:
it is sufficient to know $\delta_{\xi_{2}}H_{\xi_{1}}$ for any couple of
diffeomorphisms $\xi_{1},\xi_{2}$ to have complete control on the evolution of
the physical system under consideration. For instance, a rotation in the
azimuthal angle $\varphi$ translates as
$\partial_{\varphi}H_{\xi}+\delta_{\partial_{\varphi}}H_{\xi}$ at the level of
the charges $H_{\xi}$ for any $\xi$; the evolution of the system itself is
modelized by the collection of time derivatives
$\partial_{t}H_{\xi}+\delta_{\partial_{t}}H_{\xi}$ for a timelike coordinate
$t$. As in classical mechanics, the first term represents the explicit
coordinate dependence while the second term is the net variation under the
action of the symmetry. Using (2.2.29), (2.3.1) can be contracted a second
time to give

$\begin{split}\delta_{\xi_{2}}H_{\xi_{1}}[\phi]&=i_{\delta_{\xi_{2}}}\delta
H_{\xi_{1}}[\phi]=i_{\delta_{\xi_{2}}}i_{\delta_{\xi_{1}}}\mathcal{W}[\phi;\delta\phi,\delta\phi].\end{split}$
(2.3.2)

It is not hard to show that this expression defines a Lie bracket between
Hamiltonian functions associated with diffeomorphisms. This is the exact
equivalent to the classical Poisson bracket for field theories

$\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}\equiv
i_{\delta_{\xi_{2}}}i_{\delta_{\xi_{1}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]=\delta_{\xi_{2}}H_{\xi_{1}}[\phi],$
(2.3.3)

see [62, 50, 213] for the pure Hamiltonian version and [2, 86] for the
covariant phase space version. As announced, a strong link exists between the
Lie algebra of diffeomorphisms $\xi$ and the corresponding Lie algebra of
canonically conjugated Hamiltonian functions $H_{\xi}[\phi]$: this is the
concern of the following theorem [50, 2]

Charge representation theorem (integrable case). Assuming integrability, the
conserved charges associated with a Lie algebra of diffeomorphisms also form
an algebra under the Poisson braket
$\\{H_{\xi_{1}},H_{\xi_{2}}\\}=\delta_{\xi_{2}}H_{\xi_{1}}$ which is
isomorphic to the Lie algebra of diffeomorphisms up to a central extension
depending only on a reference solution $\bar{\phi}$:
$\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}=H_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+K_{\xi_{1},\xi_{2}}[\bar{\phi}].$
(2.3.4)

The proof is similar to the representation theorem for Hamiltonian charges in
classical mechanics and is developed as follows. We contract the presymplectic
form with the variation along the vector $[\xi_{1},\xi_{2}]_{\star}$:

$\begin{split}&i_{\delta_{[\xi_{1},\xi_{2}]_{\star}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]\\\
&=-i_{[\delta_{\xi_{1}},\delta_{\xi_{2}}]}\mathcal{W}[\phi;\delta\phi,\delta\phi]\\\
&=-i_{\delta_{\xi_{1}}}\delta_{\xi_{2}}\mathcal{W}[\phi;\delta\phi,\delta\phi]+\delta_{\xi_{2}}i_{\delta_{\xi_{1}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]\\\
&=-i_{\delta_{\xi_{1}}}i_{\delta_{\xi_{2}}}\delta\mathcal{W}[\phi;\delta\phi,\delta\phi]-i_{\delta_{\xi_{1}}}\delta
i_{\delta_{\xi_{2}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]+\delta_{\xi_{2}}i_{\delta_{\xi_{1}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]\\\
&=\delta_{\xi_{2}}\mathcal{W}[\phi;\delta_{\xi_{1}}\phi,\delta\phi]-i_{\delta_{\xi_{1}}}\delta\mathcal{W}[\phi;\delta_{\xi_{2}}\phi,\delta\phi]\end{split}$
(2.3.5)

The first equality uses (2.2.31), the second one is due to (2.2.30), the third
one uses Cartan’s magic formula (2.2.29) while the fourth one needs
$\delta\mathcal{W}=0$ because $\mathcal{W}$ is $\delta$-exact. We use now the
assumption of integrability (2.3.1). Using $\delta^{2}=0$ a couple of times,
(2.3.5) becomes

$i_{\delta_{[\xi_{1},\xi_{2}]_{\star}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]=\delta_{\xi_{2}}\delta
H_{\xi_{1}}[\phi]=\delta i_{\delta_{\xi_{2}}}\delta
H_{\xi_{1}}[\phi]=\delta(\delta_{\xi_{2}}H_{\xi_{1}}[\phi])$ (2.3.6)

again thanks to (2.2.29). Now injecting the definition of the Poisson bracket
(2.3.3), we have proven that
$i_{\delta_{[\xi_{1},\xi_{2}]_{\star}}}\mathcal{W}$ is integrable and

$\delta H_{[\xi_{1},\xi_{2}]_{\star}}[\phi]\equiv
i_{\delta_{[\xi_{1},\xi_{2}]_{\star}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]=\delta\left(\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}\right),$
(2.3.7)

which can be integrated on the phase space between a reference solution
$\bar{\phi}$ to a target solution $\phi$ as (2.3.4) up to some functional
$K_{\xi_{1},\xi_{2}}[\bar{\phi}]$ depending only on $\bar{\phi}$. Because
$\delta K_{\xi_{1},\xi_{2}}=0$, it commutes with any surface charge $H_{\xi}$
under the Poisson bracket (2.3.3), so it belongs to the center of the charge
algebra: we obtain thus a central extension when we consider the charge
algebra instead of the associated vector algebra. Evaluating (2.3.4) for the
reference solution $\bar{\phi}$, one gets

$K_{\xi_{1},\xi_{2}}[\bar{\phi}]=\delta_{\xi_{2}}N_{\xi_{1}}[\bar{\phi}]-N_{[\xi_{1},\xi_{2}]_{\star}}[\bar{\phi}].$
(2.3.8)

By consistency of the charge algebra, the central extension must be
antisymmetric under the exchange $\xi_{1}\leftrightarrow\xi_{2}$ and satisfy

$K_{[\xi_{1},\xi_{2}]_{\star},\xi_{3}}[\bar{\phi}]+\text{cyclic}(1,2,3)=0$
(2.3.9)

as a consequence of the Jacobi identity for the bracket (2.3.3). In other
words, $K_{\xi_{1},\xi_{2}}[\bar{\phi}]$ forms a 2-cocycle on the Lie algebra
of diffeomorphisms, which is said to be non-trivial if and only if it cannot
be absorbed by a normalization of the charges for the reference $\bar{\phi}$,
see (2.3.8).

#### 2.3.2 Non-integrable case: the Barnich-Troessaert bracket

Since the result (2.3.4) relies on the integrability of the charge, it cannot
be directly extended to general charges that do not satisfy the integrability
condition (2.2.73). Defining a split between integrable and non-integrable
parts as (2.2.76), we analyze here how to extend the result of the
representation theorem in this more general context.

##### 2.3.2.1 Generalized representation theorem

The generalization first requires a new definition for the bracket between
charges $\\{H_{\xi_{1}},H_{\xi_{2}}\\}_{\star}$, which will involve the
integrable parts of the charges only to conserve some similarities with
(2.3.4). Keeping in mind the definition of the Poisson bracket (2.3.3), let us
compute

$\begin{split}i_{\delta_{\xi_{2}}}i_{\delta_{\xi_{1}}}\mathcal{W}[\phi;\delta\phi,\delta\phi]&=i_{\delta_{\xi_{2}}}(\delta
H_{\xi_{1}}[\phi]+\Xi_{\xi_{1}}[\phi;\delta\phi])=\delta_{\xi_{2}}H_{\xi_{1}}[\phi]+\Xi_{\xi_{1}}[\phi;\delta_{\xi_{2}}\phi].\end{split}$
(2.3.10)

The double contraction of the presymplectic form $\mathcal{W}$ fails to give a
good prescription for the charge bracket because of the non-integrable piece.
But the aforementioned obstruction delivers some inspiration to define the
following modified bracket

$\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}_{\star}\equiv\mathcal{W}[\phi;\delta_{\xi_{1}}\phi,\delta_{\xi_{2}}\phi]-\Xi_{\xi_{1}}[\phi;\delta_{\xi_{2}}\phi]+\Xi_{\xi_{2}}[\phi;\delta_{\xi_{1}}\phi]$
(2.3.11)

which is manifestly antisymmetric under the exchange
$\xi_{1}\leftrightarrow\xi_{2}$ and for which the Jacobi identity can be
checked. Using (2.3.10), we get

$\boxed{\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}_{\star}=\delta_{\xi_{2}}H_{\xi_{1}}[\phi]+\Xi_{\xi_{2}}[\phi;\delta_{\xi_{1}}\phi].}$
(2.3.12)

The modification $\Xi_{\xi_{2}}[\phi;\delta_{\xi_{1}}\phi]$ to the “standard
Poisson bracket” $\delta_{\xi_{2}}H_{\xi_{1}}[\phi]$ was first introduced in
[30] in the context of asymptotically flat spacetimes and the bracket (2.3.12)
goes nowadays under the name of the Barnich-Troessaert bracket. Using the
techniques involved in the proof of (2.3.4), we can demonstrate the following
result [94]:  Charge representation theorem (non-integrable case). Let
$\delta/H_{\xi}=\mathcal{W}[\phi;\delta_{\xi}\phi,\delta\phi]$ be the (non-
necessarly integrable) infinitesimal surface charges associated with
diffeomorphisms $\xi$ evaluated for the solution $\phi$. Let also $\delta
H_{\xi}[\phi]+\Xi_{\xi}[\phi;\delta\phi]$ be an arbitrary split of
$\delta/H_{\xi}$ into integrable and non-integrable parts $H_{\xi}[\phi]$ and
$\Xi_{\xi}[\phi;\delta\phi]$. The phase space functions $H_{\xi}[\phi]$
associated with the Lie algebra of diffeomorphisms $\xi$ also form an algebra
under the Barnich-Troessaert bracket
$\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}_{\star}=\delta_{\xi_{2}}H_{\xi_{1}}[\phi]+\Xi_{\xi_{2}}[\phi;\delta_{\xi_{1}}\phi]$
which is isomorphic to the Lie algebra of diffeomorphisms up to a field-
dependent 2-cocycle:
$\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}_{\star}=H_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+K_{\xi_{1},\xi_{2}}[\phi].$
(2.3.13) We do not present the detailed proof in this manuscript. The main
difference with (2.3.4) is that the “central extension”
$K_{\xi_{1},\xi_{2}}[\phi]$ reads now as a field-dependent object whose main
properties are discussed in the next paragraph.

##### 2.3.2.2 Field-dependent 2-cocycle

The charge algebra under the Barnich-Troessaert bracket reveals a field-
dependent extension that is not central because $\delta
K_{\xi_{1},\xi_{2}}[\phi]\neq 0$ in general. This function on the phase space
is not completely arbitrary, because it must obey the following conditions, in
order to maintain the consistency of the algebra (2.3.13):

$\displaystyle K_{\xi_{1},\xi_{2}}[\phi]=-K_{\xi_{2},\xi_{1}}[\phi],$ (2.3.14)
$\displaystyle
K_{[\xi_{1},\xi_{2}]_{\star},\xi_{3}}[\phi]-\delta_{\xi_{3}}K_{\xi_{1},\xi_{2}}[\phi]+\text{cyclic}(1,2,3)=0.$
(2.3.15)

This couple of conditions define a Lie algebroid 2-cocycle on the Lie
algebroid formed by the diffeomorphisms. The second condition (2.3.15), and
necessary to ensure the Jacobi identity for the Barnich-Troessaert bracket
(2.3.12) [30], is the generalization of (2.3.9) taking the field-dependence of
$K_{\xi_{1},\xi_{2}}[\phi]$ into account. The algebraic properties are
completed by the following identities [30]

$\displaystyle\\{K_{\xi_{1},\xi_{2}}[\phi],H_{\xi_{3}}[\phi]\\}_{\star}=-\\{H_{\xi_{3}}[\phi],K_{\xi_{1},\xi_{2}}[\phi]\\}_{\star}=\delta_{\xi_{3}}K_{\xi_{1},\xi_{2}}[\phi],$
(2.3.16)
$\displaystyle\\{K_{\xi_{1},\xi_{2}}[\phi],K_{\xi_{3},\xi_{4}}[\phi]\\}_{\star}=0.$
(2.3.17)

The first one is quite logical since it amounts to say that the action of the
charge on any phase space function is meant to generate the symmetry. The
second one states that the extension is “central” in the sense that it does
not generate any symmetry.

This is perhaps the good moment to discuss the impact on the algebra of the
ambiguity (2.2.77) of the split (2.2.76) between integrable and non-integrable
parts. Let us set

$H^{\prime}_{\xi}[\phi]=H_{\xi}[\phi]-\Delta
H_{\xi}[\phi],\qquad\Xi^{\prime}_{\xi}[\phi;\delta\phi]=\Xi_{\xi}[\phi;\delta\phi]+\delta\Delta
H_{\xi}[\phi]$ (2.3.18)

such that $\mathcal{W}[\phi,\delta_{\xi}\phi,\delta\phi]=\delta
H_{\xi}[\phi]+\Xi_{\xi}[\phi;\delta\phi]=\delta
H^{\prime}_{\xi}[\phi]+\Xi^{\prime}_{\xi}[\phi;\delta\phi]$. Defining the
Barnich-Troessaert bracket (2.3.12) with primed objects

$\\{H^{\prime}_{\xi_{1}}[\phi],H^{\prime}_{\xi_{2}}[\phi]\\}_{\star}=\delta_{\xi_{2}}H^{\prime}_{\xi_{1}}[\phi]+\Xi^{\prime}_{\xi_{2}}[\phi;\delta_{\xi_{1}}\phi],$
(2.3.19)

we can show that (2.3.13) is preserved up to some modification in the
2-cocycle. Indeed,

$\displaystyle\\{H^{\prime}_{\xi_{1}}[\phi],H^{\prime}_{\xi_{2}}[\phi]\\}_{\star}$
$\displaystyle=\\{H_{\xi_{1}}[\phi],H_{\xi_{2}}[\phi]\\}_{\star}-\delta_{\xi_{2}}\Delta
H_{\xi_{1}}[\phi]+\delta_{\xi_{1}}\Delta H_{\xi_{2}}[\phi]$
$\displaystyle=H_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+K_{\xi_{1},\xi_{2}}[\phi]-\delta_{\xi_{2}}\Delta
H_{\xi_{1}}[\phi]+\delta_{\xi_{1}}\Delta H_{\xi_{2}}[\phi]$
$\displaystyle=H^{\prime}_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+\left(K_{\xi_{1},\xi_{2}}[\phi]-\delta_{\xi_{2}}\Delta
H_{\xi_{1}}[\phi]+\delta_{\xi_{1}}\Delta H_{\xi_{2}}[\phi]+\Delta
H_{[\xi_{1},\xi_{2}]_{\star}}[\phi]\right)$ $\displaystyle\equiv
H^{\prime}_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+K^{\prime}_{\xi_{1},\xi_{2}}[\phi]$
(2.3.20)

where the new 2-cocycle

$K^{\prime}_{\xi_{1},\xi_{2}}[\phi]\equiv
K_{\xi_{1},\xi_{2}}[\phi]-\delta_{\xi_{2}}\Delta
H_{\xi_{1}}[\phi]+\delta_{\xi_{1}}\Delta H_{\xi_{2}}[\phi]+\Delta
H_{[\xi_{1},\xi_{2}]_{\star}}[\phi]$ (2.3.21)

directly satisfies the cocycle condition (2.3.15) if and only if
$K_{\xi_{1},\xi_{2}}[\phi]$ is also a 2-cocycle. As a result, the structure of
the algebra is left intact by the change of prescription for the split
(2.2.77) while only the 2-cocycle is affected [30]. We say that
$K_{\xi_{1},\xi_{2}}[\phi]$ is trivial if there exists some function $\Delta
H_{\xi}[\phi]$ such that $\delta K^{\prime}_{\xi_{1},\xi_{2}}[\phi]=0$, i.e.
it is possible to absorb the field-dependence of the 2-cocycle by a smart
choice of the integrable part. Otherwise, it is non-trivial and consequently
non-absorbable.

#### 2.3.3 The Generalized BMS4 charge algebra

As we mentioned, the first occurrence of the generalized result (2.3.13) was
found in [30] in the context of asymptotically flat gravity with the
restricted boundary conditions $\delta q_{AB}$ but allowing for meromorphic
super-Lorentz transformations. It can be checked explicitly that the result
holds also with the inclusion of smooth Diff($S^{2}$) super-Lorentz symmetries
on the phase space and the proof, similar to the original derivation
extensively detailed in [30], is lead as follows. Starting from the integrable
part (2.2.127), we can compute $\delta_{\xi_{2}}H_{\xi_{1}}^{\text{GBMS}_{4}}$
for any generators $\xi_{1}$, $\xi_{2}$ of the Generalized BMS4 owing to the
transformations (2.1.71)–(2.1.75) on the asymptotically locally flat solution
space $\mathring{\mathcal{S}}_{0}$. During the computations, any total
derivative on the sphere can be ignored since we required that all fields, as
well as the transformations, are smooth functions of the angles $x^{A}$. These
variations can also be plugged into the non-integrable part (2.2.128) to
reconstitute the Barnich-Troessaert bracket

$\\{H_{\xi_{1}}^{\text{GBMS}_{4}}[\phi],H_{\xi_{2}}^{\text{GBMS}_{4}}[\phi]\\}_{\star}=\delta_{\xi_{2}}H^{\text{GBMS}_{4}}_{\xi_{1}}[\phi]+\Xi^{\text{GBMS}_{4}}_{\xi_{2}}[\phi;\delta_{\xi_{1}}\phi].$
(2.3.22)

Again, $\phi$ designates here the metric field $g$ with any background
structure needed to fix the gauge as well as the boundary conditions, or
equivalently the solution space parameters appearing in (2.1.40). Secondly, we
have to evaluate the charge (2.2.127) for $\xi=[\xi_{1},\xi_{2}]_{\star}$ with
parameters given as (2.1.49)–(2.1.50). After an involved computation, we get

$\\{H_{\xi_{1}}^{\text{GBMS}_{4}}[\phi],H_{\xi_{2}}^{\text{GBMS}_{4}}[\phi]\\}_{\star}=H^{\text{GBMS}_{4}}_{[\xi_{1},\xi_{2}]_{\star}}[\phi]+K^{\text{GBMS}_{4}}_{\xi_{1},\xi_{2}}[\phi]$
(2.3.23)

with the particular 2-cocycle

$\begin{split}{K}^{\text{GBMS}_{4}}_{\xi_{1},\xi_{2}}[\phi]=\frac{1}{16\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\,\left[\frac{1}{2}f_{2}D_{A}f_{1}D^{A}R[q]+\frac{1}{2}C^{BC}f_{2}D_{B}D_{C}D_{D}Y^{D}_{1}\right]-(1\leftrightarrow
2),\end{split}$ (2.3.24)

generalizing the expression of [30] to arbitrary metrics on the celestial
sphere. This is a particular realization of the result (2.3.13) and this is
not the last discussed in this manuscript. The reference solution $\bar{\phi}$
is taken to be the global Minkowski vacuum with identically vanishing charges.
The 2-cocycle (2.3.24) is believed to be non-trivial thanks to the following
arguments.

  1. 1.

Two supertranslations commute (see (2.1.49)) and therefore the 2-cocycle
$K^{\text{GBMS}_{4}}_{\xi_{1}(T_{1});\xi_{2}(T_{2})}\neq 0$,
$\xi(T)\equiv\xi(T,0)$, does not transform upon shifting the integrable charge
according to (2.2.77) with a supertranslation charge that depends on $q_{AB}$
and not $C_{AB}$, i.e.
$K^{{}^{\prime}\text{GBMS}_{4}}_{\xi_{1}(T_{1});\xi_{2}(T_{2})}$ $=$
$K^{\text{GBMS}_{4}}_{\xi_{1}(T_{1});\xi_{2}(T_{2})}$ using (2.3.21) and
(2.1.71) implying $\delta_{T}q_{AB}=0$. Since
$K^{\text{GBMS}_{4}}_{\xi_{1}(T_{1});\xi_{2}(T_{2})}\neq 0$ and only depends
upon $q_{AB}$ the first term in the 2-cocycle is non-trivial.

  2. 2.

A cohomological formulation of the second term of (2.3.24) can be found in
[205]. After semi-classical quantization, the second term of the 2-cocyle can
also be related to the non-commutativity of the double soft limit of gravitons
[270].

When restricting to the subspace (2.1.41) of asymptotically Minkowskian
solutions, ${K}^{\text{GBMS}_{4}}_{\xi_{1},\xi_{2}}[\phi]$ vanishes
identically for any parameters since $R[q]=2$ and $Y^{A}$ is solution of the
conformal Killing equation (2.1.56). Hence, the Barnich-Troessaert bracket
allows the BMS4 charge algebra to represent the BMS4 algebra without central
extension. Because of the split ambiguity (2.2.77), this statement is
obviously dependent on the particular representative for the integrable part
$H_{\xi}^{\text{GBMS}_{4}}[\phi]$, but this is also a motivation to choose
(2.2.127) as Hamiltonians among the infinite bunch of possibilities.
Nevertheless, we will argue later that it is not the best option from physical
perspectives in section 5.3 and will give additional prescriptions to fix the
expression of the Hamiltonians for the Generalized BMS4 group.

Now let us illustrate and confirm the intuitions given in the opening of
section 2.3.1 to motivate the study of the charge algebra, namely that the
algebra (2.3.23) completely controls the dynamics of the gravitational system
and contains useful pieces of information, even for practical purposes. For
instance, as observed in the seminal work [30], it controls the non-
conservation of the charges through the presence of the modification term
involving the non-integrable part. The reasoning is as follows. As in
Hamiltonian mechanics, the total evolution of the quantity
$H_{\xi}^{\text{GBMS}_{4}}[\phi]$ is given by

$\frac{\text{d}}{\text{d}u}H_{\xi}^{\text{GBMS}_{4}}[\phi]=\frac{\partial}{\partial
u}H_{\xi}^{\text{GBMS}_{4}}[\phi]+\delta_{\partial_{u}}H_{\xi}^{\text{GBMS}_{4}}[\phi],$
(2.3.25)

where the first term represents the explicit dependence on $u$ while the
second term gives the net evolution along the flow of the vector
$\partial_{u}$. The first contribution is easy to derive since only the
parameter $\xi$ has some explicit dependence in $u$. The fields involved in
the charge have a constrained time evolution on-shell which is considered as
an implicit dependence already included in the second term. For example, we
see from (2.1.74) that
$\delta_{\partial_{u}}M=\delta_{\xi(1,0)}M=\partial_{u}M$, given on-shell by
(2.1.35). So we have

$\frac{\partial}{\partial
u}H_{\xi}^{\text{GBMS}_{4}}[\phi]=H^{\text{GBMS}_{4}}_{\partial_{u}\xi}[\phi]=-H^{\text{GBMS}_{4}}_{[\xi,\partial_{u}]_{\star}}[\phi].$
(2.3.26)

We insert (2.3.26) and (2.3.23) in (2.3.25) to obtain

$\frac{\text{d}}{\text{d}u}H_{\xi}^{\text{GBMS}_{4}}[\phi]=-\Xi^{\text{GBMS}_{4}}_{\partial_{u}}[\phi;\delta_{\xi}\phi]-K^{\text{GBMS}_{4}}_{\partial_{u},\xi}[\phi]=-\Xi^{\text{GBMS}_{4}}_{\partial_{u}}[\phi;\delta_{\xi}\phi],$
(2.3.27)

since (2.3.24) is not present when $\xi_{1}$ is $\partial_{u}$. A quick
comparison between (2.2.113) and (2.2.128), at the price to perform some
integrations by parts on the celestial sphere $S$, finally gives [45, 37]

$\boxed{\frac{\text{d}}{\text{d}u}H_{\xi}^{\text{GBMS}_{4}}[\phi]=-\oint_{S_{\infty}}\iota_{\partial_{u}}{\bm{\Theta}}_{ren}[\phi;\delta_{\xi}\phi]=-\oint_{S_{\infty}}\text{d}^{2}\Omega\,\bar{\Theta}_{flux}[q_{AB},C_{AB};\delta_{\xi}q^{AB},\delta_{\xi}C^{AB}].}$
(2.3.28)

This generalizes the flux formula previously obtained in [31] assuming $\delta
q_{AB}=0$. The leading contribution of the presymplectic flux, sourced by the
traditional radiative degrees of freedom in $C_{AB}$ and the additional
kinematics encoded in $\delta q_{AB}$, is responsible for the non-conservation
of the surface charges associated with the diffeomorphisms. The key point is
that the flux-balance laws (2.3.28) are a direct consequence of the charge
algebra (2.3.23), hence hold independently of the representative of the
integrable part of the charge. Taking in particular $\xi=\partial_{u}$,
$H^{\text{GBMS}_{4}}_{\xi}$ is nothing but the Bondi mass and (2.3.28) implies
[30]

$\frac{\text{d}}{\text{d}u}\mathcal{M}=-\frac{1}{32\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\,(N_{AB}\delta_{\partial_{u}}C^{AB})=-\frac{1}{32\pi
G}\oint_{S_{\infty}}\text{d}^{2}\Omega\,(N_{AB}N^{AB})$ (2.3.29)

which is exactly the Bondi mass loss formula (2.2.133) stating that the time
derivative of the Bondi mass is negative definite and vanishes if and only if
there is no Bondi news.

This aesthetic discussion about the algebraic properties of the gravitational
surface charges marks the end point of this quite long chapter devoted to the
covariant phase space formalism. We leave for the moment the flat land to
explore more spectacular and fascinating asymptotics in the presence of a
cosmological constant… $\blacksquare$

##  Chapter 3 Asymptotically locally (A)dS spacetimes

In this chapter, we progress our exploration towards the asymptotic regions of
solutions of Einstein’s gravity in the presence of a non-vanishing
cosmological constant. We export there the various features, techniques and
nomenclatures discussed in the previous chapter and define genuine radiative
phase spaces for these crucial kinds of asymptotics, broadly highlighted in
holography as well as in modern cosmology.

First, we review the causal and asymptotic properties of the (A)dS spacetimes,
in order to get inspired for the definition of phase spaces with solutions
that share the same asymptotic properties. In the spirit of Penrose’s analysis
of relativistic infinities [9], we leverage the conformal compactness of these
solutions to write the covariant phase space in the very convenient
Starobinsky/Fefferman-Graham coordinate system [199, 200]. We perform the
computations and interpret the outcomes in any bulk dimension and without
assuming further hypotheses. In particular, the boundary structure is allowed
to fluctuate and plays the role of source yielding some symplectic flux at the
conformal boundary [166, 85]. The class of spacetimes under consideration is
thus generically radiative and the necessity to consider leaky boundary
conditions even when the cosmological constant is present is debated at the
same time. The most famous conservative sub-cases [62, 160, 64, 63, 161] are
reviewed within our formalism in order to anchor our results in the extensive
literature in the field and show the improvements that we have introduced thus
far.

Using the holographic renormalization procedure [207, 209, 210], we show how
to remove the divergences from the symplectic structure [166, 85]. The charges
associated with the whole class of residual gauge diffeomorphisms in the
Starobinsky/Fefferman-Graham gauge are computed. All non-trivial boundary
diffeomorphisms are shown to be canonically conjugated with non-vanishing,
non-integrable and not conserved surface charges, while those associated with
boundary Weyl rescalings are non-vanishing only in odd dimensions, due to the
presence of Weyl anomalies in the dual theory [162, 207, 166]. Finally, the
charge algebra under the Barnich-Troessaert bracket [30] is derived and its
main features are scrutinized. In particular, a field-dependent $2$-cocycle
manifests in odd dimensions. When the general framework is restricted to
three-dimensional asymptotically AdS spacetimes with Dirichlet boundary
conditions, this cocycle reduces to the well-known Brown-Henneaux central
extension [62].

Apart from the review sections 3.1.1 and 3.1.2, this chapter essentially
reproduces [87] assorted with some elements taken from [271, 37].

###  3.1 Gravity in Starobinsky/Fefferman-Graham gauge

In this introductive section, we review how to define the notion of (A)dS
asymptotics through the conformal compactification process. We start by
reviewing some key features of global de Sitter and anti-de Sitter spacetimes
and developing about their asymptotic structure. We continue by recalling some
general statements about conformally compact spacetimes, for which one can
perform a conformal transformation to add meaningfully the closure of the
unphysical spacetime at finite distance and consider this as a robust
idealization of a manifold at infinity. Finally, building on these
considerations, we review the notion of asymptotically locally (anti-) de
Sitter spacetime (Al(A)dS for short) within the common and powerful coordinate
system often used to explore the physical features of these kinds of
asymptotics. For the purpose of our dissertations, we restrict ourselves to
coordinate-dependent discussions and directly provide explicit formulas. As
for the flat case, completely covariant formalisms were developed in parallel
of the various progressions in the gauge-fixing approach (see e.g. [160, 161,
136]). They manage to confirm the results obtained by explicit methods thanks
to elegant reformulations using fundamental geometrical objects, but we will
not review them here. The mathematical developments of this section summarize
the seminal works [207, 162, 272, 209, 210] and allow us to install our
conventions while defining rigorously the general framework in which we will
construct the phase space.

#### 3.1.1 Asymptotic properties of (A)dS spacetimes

This opening section consists of a selective review of the interesting
properties of the de Sitter and anti-de Sitter spacetimes. These are the
fundamental solutions whose we want to study the asymptotic behavior in order
to define a phase space of solutions mimicking (at least locally) their
asymptotic structure at infinity. For the interested reader, more details can
be found in the reviews [273, 211, 274, 275, 276] and references therein.

##### 3.1.1.1 The de Sitter solution

The de Sitter spacetime dSd+1 (in $n=d+1$ dimensions, $d>1$) is the only
maximally symmetrical Lorentzian manifold of positive Ricci curvature [130,
131, 132]. In the global coordinate system $\\{\tau,\chi,x^{A}\\}$ covering
the whole spacetime ($\tau\in\mathbb{R},\chi\in[0,\pi]$ and $x^{A}$,
$A=2,\dots,d$, are coordinates on the $S^{d-1}$ sphere), its line element
reads as

$\text{d}s^{2}_{\text{dS}}=-\text{d}\tau^{2}+\ell^{2}\cosh^{2}(\tau/\ell)\left(\text{d}\chi^{2}+\sin^{2}\chi\,\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}\right).$
(3.1.1)

The real parameter $\ell>0$ controls the Ricci curvature as
$R=d(d+1)/\ell^{2}$ and $\mathring{q}_{AB}$ represents the unit-round metric
on the $S^{d-1}$ sphere. At the physical level, (3.1.1) describes an
Einsteinian manifold solving the vacuum Einstein field equations for a
positive value of the cosmological constant given by

$\Lambda=\frac{d(d-1)}{2\ell^{2}}.$ (3.1.2)

In that sense, this solution is the exact analog of the Minkowski
gravitational vacuum in the presence of a strictly positive cosmological
constant. By inspection of (3.1.1), we learn that the de Sitter spacetime has
the topology of $\mathbb{R}\times S^{d}$ and can be seen as a $d$-sphere first
contracting then expanding when the “evolution parameter” $\tau$ travels the
real line $\mathbb{R}$. Any fixation of the couple $(\tau,\chi)$ describes a
$(d-1)$-sphere of radius $\sin\chi$ at fixed $\tau$ except for the extremal
values $\chi=0$ and $\chi=\pi$ which represent actual points located at the
north and south poles of the $d$-sphere. Geometrically, the de Sitter space
can be thought of as an hyperboloid $\mathscr{H}_{\text{dS}}$ embedded in the
$(d+2)$-dimensional Minkowski spacetime, defined as the geometrical locus of
points in $\mathbb{R}^{d+1,1}$ at a fixed proper distance $\ell$ from the
origin. For that reason, $\ell$ goes under the name of de Sitter radius. The
isometry group of $\mathscr{H}_{\text{dS}}$ is then the subgroup of Minkowski
isometries in $d+2$ dimensions that stabilizes the origin, i.e. the
$(d+2)$-dimensional-Lorentz group $SO(d+1,1)$. This includes rotations on the
sections of $\mathscr{H}_{\text{dS}}$ (which are $S^{d}$ spheres at constant
$\tau$) as well as boosts along the revolution axis of the hyperboloid.

To investigate the causal structure of de Sitter and figure out what is its
asymptotic behavior, it is convenient to trade the global time coordinate
$\tau$ for the conformal time coordinate $T$ such that
$\text{d}s^{2}_{\text{dS}}=f(T)^{2}(-\text{d}T^{2}+\text{d}\chi^{2}+\sin^{2}\chi\text{d}\Omega_{d-1}^{2})$.
The defining function of the conformal rescaling reads as
$f(T)=\ell\cosh(\tau/\ell)$ where $\tau=\tau(T)$ and
$\text{d}\tau=f\text{d}T$. Integrating this condition yields
$T=\arctan\sinh(\tau/\ell)$ if $T(\tau=0)=0$. The real line
$\tau\in\mathbb{R}$ is mapped onto $T\in]-\pi/2,\pi/2[$. In the conformal
coordinates $\\{T,\chi,x^{A}\\}$ the metric is manifestly conformal to a
portion of the flat spacetime, which is smooth everywhere including the
boundary represented here by extremal values of the conformal coordinate $T$.
This portion is a square in the $(T,\chi)$ plane because both coordinates have
an equal finite range of $\pi$: the Penrose-Carter diagram is represented by
the Figure 3.1. Each point of the diagram is a $S^{d-1}$ except the lateral
edges which represent the poles of the $S^{d}$ slices of constant $T$ (or
constant $\tau$). These lines are thus collections of true points. Strictly
speaking, they are not part of the conformal boundary
$\mathscr{I}_{\text{dS}}\equiv\mathscr{I}_{\text{dS}}^{-}\dot{\cup}\mathscr{I}_{\text{dS}}^{+}$
which reunites the past conformal boundary at $T=-\pi/2$ and the future
conformal boundary at $T=+\pi/2$. These are spacelike hypersurfaces where any
null geodesic has its end-points, starting from a point of
$\mathscr{I}_{\text{dS}}^{-}$ and terminating at a point of
$\mathscr{I}_{\text{dS}}^{+}$. Therefore, $\mathscr{I}_{\text{dS}}^{+}$ is the
future null infinity of de Sitter, which will be of major interest in this
thesis. The radial light rays, which have no motion along the $S^{d-1}$
spheres, travel the square at 45 degrees: such a ray originating from the
north pole will exactly cross the diagram to reach the south pole. Other light
rays with non-trivial motion in the angular directions travel the diagram at
less than 45 degrees.

Southern causal diamondCausal past $\mathcal{O}^{-}$Causal future
$\mathcal{O}^{+}$South poleNorth
pole$\mathscr{I}^{+}_{\text{dS}}$$\mathscr{I}^{-}_{\text{dS}}$$(T,\chi)=(\frac{\pi}{2},0)$$(T,\chi)=(\frac{\pi}{2},\pi)$$(T,\chi)=(-\frac{\pi}{2},0)$$(T,\chi)=(-\frac{\pi}{2},\pi)$$\bm{\mathcal{O}}$
Figure 3.1: Causal structure of dSd+1 spacetime.

One important although peculiar feature of the de Sitter spacetime is the
presence of two frame-dependent horizons for any observer. It is a fact: an
isolated observer in de Sitter cannot have access to the whole story of the
spacetime events and cannot send a (light-driven) message to any future
region. This contrasts drastically with the asymptotically flat case where a
timelike observer can have access to the entire history of the spacetime if he
waits for a sufficiently long time, enough to allow its past light cone to
intercept all of the observable past events. In the Penrose-Carter diagram
3.1, we represented the accessible regions for an observer $\mathcal{O}$
sitting at south pole (for example). His causal past $\mathcal{O}^{-}$ is
determined by the union of all his past light cones, which are below the null
direction connecting $\mathscr{I}_{\text{dS}}^{-}$ at north pole to
$\mathscr{I}_{\text{dS}}^{+}$ at south pole. The latter thus defines the past
horizon associated with $\mathcal{O}$ that prevents him from getting some
information about the regions beyond. Similarly, the other diagonal null
direction, connecting $\mathscr{I}_{\text{dS}}^{-}$ at south pole to
$\mathscr{I}_{\text{dS}}^{+}$ at north pole, describes a future horizon for
$\mathcal{O}$ because it delimits the region $\mathcal{O}^{+}$ to which the
observer can send a message, identified as the union of all his future light
cones. The intersection $\mathcal{O}^{-}\cap\mathcal{O}^{+}$ is coined as the
southern causal diamond and represents the portion of de Sitter spacetime
fully accessible to $\mathcal{O}$ living at the south pole. These causal
properties are transparent in the coordinate chart $\\{t,r,x^{A}\\}$ adapted
to the southern observer $\mathcal{O}$, in which the de Sitter line element is
written as

$\text{d}s_{\text{dS}}^{2}=-\left(1-\frac{r^{2}}{\ell^{2}}\right)\text{d}t^{2}+\frac{\text{d}r^{2}}{\left(1-\frac{r^{2}}{\ell^{2}}\right)}+r^{2}\,\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}.$
(3.1.3)

One calls this set of coordinates the static coordinates of de Sitter because
$\partial_{t}$ is here a manifest Killing vector and (3.1.3) does not contain
any cross-term involving $\text{d}t$. It also allows to give another
interpretation to the parameter $\ell$. Indeed, choosing $r$ negligible to
$\ell$ transforms (3.1.3) into the Minkowski line element in spherical
coordinates. This means that at small distances the de Sitter spacetime looks
locally like the flat space. The cosmological effects become important when
the phenomena we want to consider occur for large radii, commensurable to
$\ell$. The static coordinates cover only one of the four causal diamonds
delimited by the horizons $r^{2}=\ell^{2}$ since (3.1.3) becomes singular for
these values, but it can be defined in each of them. For instance, the
superior diamond adapted for observers at infinity is defined for $r>\ell$. It
will particularly interest us later on in this thesis. To make contact with
our previous discussion, let us cover the southern causal diamond. The past
horizon is determined as $r=\ell$ while the future horizon is located at
$r=-\ell$. Between these values, the norm of the Killing vector $\partial_{t}$
is negative, hence its flow correctly describes the time evolution.
$\partial_{t}$ becomes null on the horizons and spacelike in the upper and
lower causal diamonds. The absence of a global timelike Killing vector in the
de Sitter vacuum has deep and unavoidable consequences for the evolution
problem in this spacetime. Without entering into details about the complexity
of defining the Hamiltonian in field theories living on de Sitter – which
would be desirable from the point of view of quantum theory, we will just
mention some aspects of this difficulty below in the text, even at classical
level while dealing with radiative asymptotically de Sitter spacetimes.

##### 3.1.1.2 The anti-de Sitter solution

Conversely, one defines the anti-de Sitter spacetime in $n$ dimensions as the
only maximally symmetrical Lorentzian manifold of negative Ricci curvature. In
the set of hyperbolic coordinates $\\{\tau,\chi,x^{A}\\}$
($\tau\in\mathbb{R}$, $\chi\in\mathbb{R}^{+}$ and $x^{A}$ again parametrize
the $S^{d-1}$ sphere), its line element reads as

$\text{d}s^{2}_{\text{AdS}}=-\cosh^{2}\chi\text{d}\tau^{2}+\ell^{2}\left(\text{d}\chi^{2}+\sinh^{2}\chi\,\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}\right)$
(3.1.4)

for the real parameter $\ell>0$ defined this time as the AdS radius. (3.1.4)
is manifestly static in this patch and solves the Einstein field equations for
a negative value of the cosmological constant given by

$\Lambda=-\frac{d(d-1)}{2\ell^{2}},$ (3.1.5)

or, equivalently, $R=-d(d+1)/\ell^{2}$. In contrast to the de Sitter solution,
the topology of the AdS spacetime is $\mathbb{R}\times H^{d}$ where $H^{d}$
denotes the upper layer of the $d$ dimensional hyperboloid whose line element
is
$\text{d}\chi^{2}+\sinh^{2}\chi\,\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}$.
The latter has the topology of $\mathbb{R}^{d}$, hence AdS has in fact the
topology of $\mathbb{R}^{d+1}$, which is non-compact in all directions. This
constitutes the major difference between dS and AdS global vacua and will
impact the conformal compactification which will only be partial in the second
case.

Geometrically, the AdS spacetime can also be seen as an hyperboloid
$\mathscr{H}_{\text{AdS}}$ of radius $\ell$, but with more peculiar features,
since we have to embed it in the Riemannian flat space $\mathbb{R}^{d,2}$
provided with two time directions $t_{1}$ and $t_{2}$. The isometries of
AdSd+1 form the homogeneous subgroup of isometries of $\mathbb{R}^{d,2}$, that
is, $SO(d,2)$. It encompasses the full rotation group in the hyperspace for
constant $t_{1},t_{2}$ directions as well as two copies of the proper
orthochronous Lorentz group $SO(d,1)$ mixing $t_{1}$ or $t_{2}$ and transverse
spacelike coordinates.

Another practical coordinate system, adapted to static observers in AdS
spacetime, is the global static patch $\\{t,r,x^{A}\\}$, $t\in\mathbb{R}$
being the global time and $r\in\mathbb{R}^{+}$ a radial coordinate, for which
the line element (3.1.4) becomes

$\text{d}s_{\text{AdS}}^{2}=-\left(1+\frac{r^{2}}{\ell^{2}}\right)\text{d}t^{2}+\frac{\text{d}r^{2}}{\left(1+\frac{r^{2}}{\ell^{2}}\right)}+r^{2}\,\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}.$
(3.1.6)

This is the analytical continuation of (3.1.3) for $\ell\to i\ell$. It shows
explicitly that AdS is also locally flat for distances $r\ll\ell$. In this
second coordinate system, it is easy to compactify the radial direction by
proposing the change of coordinate $R=\arctan(r/\ell)$. While $r$ runs on the
semi-line $[0,+\infty[$, $R$ is limited to $[0,\pi/2[$ and the line element is
$\text{d}s_{\text{AdS}}^{2}=\ell^{2}\cos^{-2}R(-\text{d}t^{2}/\ell^{2}+\text{d}R^{2}+\sin^{2}R\,\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B})$.
The AdS metric is thus conformal to the $\mathbb{R}\times S^{d}$ cylinder. The
conformal boundary is the spatial infinity $r\to\infty$ located at the double
pole $R=\pi/2$ of the metric. We denote it as $\mathscr{I}_{\text{AdS}}$,
whose induced geometry is a $d$ dimensional cylinder aligned on the time
direction with $S^{d-1}$ spheres as constant $t$ sections. It is noteworthy
that, since the range of the timelike coordinate $t$ remains infinite while
the range of the conformal radius $R$ is finite, there is no way to compress
further the AdSd+1 spactime into a finite range of coordinates for both time
and radius if one wants to preserve the condition that null rays evolve along
straight lines at 45 degrees. If one attempts to perform another conformal
transformation to reduce $t$ in a finite range, the spheres generating the
conformal boundary will be mapped to points, since the interval of the
conformal radius $R$ will be squeezed to a single point. The Penrose-Carter
diagram of AdSd+1, depicted in Figure 3.2, is thus an infinite cylinder, whose
upper and lower boundaries $i^{+}_{\text{AdS}}$ and $i^{-}_{\text{AdS}}$,
rejected to infinity, are respectively future and past timelike infinities.
These are not part of the conformal boundary: the AdSd+1 spacetime is only
partially conformally compact because the relativistic infinities cannot be
brought back to finite distance by a conformal transformation.

$\mathscr{I}_{\text{AdS}}$$S^{d-1}$$t$$i^{+}_{\text{AdS}}$$i^{-}_{\text{AdS}}$$R=0$$R=\frac{\pi}{2}$
Figure 3.2: Causal structure of AdSd+1 spacetime.

A static observer whose proper time is $t$, sitting at the coordinates
$(r,x^{A})$, has neither past nor future horizon and has thus access to the
whole spacetime history. This pleasant feature contrasting with the dS case is
however counter-balanced by the major oddness of the AdSd+1 spacetime, which
is that null rays (in blue on Figure 3.2) reach the conformal boundary after a
finite amount of coordinate time. The data given on a spacelike slice (or
Cauchy slice) in the past are not sufficient to describe causally what is
happening after the rays have encountered the conformal boundary. Hence the
Cauchy problem is ill-defined until boundary conditions on
$\mathscr{I}_{\text{AdS}}$ are imposed to decide of the fate of the light ray.
For the standard reflexive boundary conditions, the null geodesic bounces on
the boundary to dig back to the center of the spacetime, as represented in the
Figure 3.2. This property connecting the conformal boundary and the bulk of
the spacetime manifold is peculiar to AdS and spearheads the AdS/CFT
correspondence [158, 277] debated in string theory for decades now. AdS is
thus a prototypic “spacetime in a box” where any test body or radiation is
naturally aimed to be repelled by the boundary towards the bulk. This is the
case for massive particles whose worldlines are schematized in red on the
Penrose-Carter diagram 3.2: the Christoffel symbols contain a counter-
potential in $\mathcal{O}(r^{2})$ whose importance grows as one approaches
$\mathscr{I}_{\text{AdS}}$ and repulse timelike geodesics back to the center
of the spacetime. This observation concludes our review of the asymptotic
behavior of AdS et dS spacetimes. We thought it would be valuable to briefly
summarize their properties and contrast them with the asymptotic behavior of
flat space before discussing with more details the desirable boundary
conditions that we are lead to impose in practice.

#### 3.1.2 Conformally compact manifolds

Now that we have reviewed the asymptotic properties we want to focus on, let
us describe generically the class of spacetimes for which the conformal
compactification can be performed (at least partially, as for the AdSd+1
spacetime). This is Penrose’s powerful idea to treat relativistic infinities
by bringing them back to finite distances and describe asymptotics as
geometric properties of the boundary of the conformally compactified
unphysical spacetime [9]. In that context, a gauge-invariant and robust
definition of “asymptotically (A)dS” spacetime – which is merely a conformally
compact manifold whose conformal boundary is spacelike (for dS) or timelike
(for AdS) – can be obtained. This constitutes the last step before defining a
coordinate system appropriate to discuss the solution space of Al(A)dSd+1.

Let us consider a Lorentzian manifold $(\mathscr{M},g)$ with boundary
$\mathscr{I}$ on which the metric tensor $g$ is possibly singular (due to the
intrinsic nature of infinity, located “at large distances”) but with the
crucial hypothesis that this singularity on $\mathscr{I}$ is a double pole.
This latter holds for the Minkowski vacuum as well as for the (A)dS global
solutions. Hence, there exists a smooth function $F$ on
$\bar{\mathscr{M}}=\mathscr{M}\dot{\cup}\mathscr{I}$ such that (1) $F>0$ on
$\mathscr{M}$, $F|_{\mathscr{I}}=0$, $\text{d}F|_{\mathscr{I}}\neq 0$ and (2)
$\bar{g}=F^{2}g$ can be extended smoothly on the whole manifold
$\bar{\mathscr{M}}$. The evaluation $|_{\mathscr{I}}$ denotes the pull-back to
the boundary $\mathscr{I}$ and the unphysical metric $\bar{g}$ is supposed to
be continuously differentiable up to a sufficiently large order allowing us to
take as many derivatives as we need. If $(\mathscr{M},g)$ satisfies all of
these hypotheses, it is called a conformally compact manifold [9, 278]. The
first hypothesis (1) allows to choose $F$ as a coordinate on the unphysical
spacetime $(\bar{\mathscr{M}},\bar{g})$ while the second (2) locates the
boundary $\mathscr{I}$ at the poles of the physical metric $g$. We can thus
identify the spacetime infinity with the boundary manifold $\mathscr{I}$.
Obviously, the conformal factor $F$ is not uniquely fixed by the above
definition and can be rescaled by a non-vanishing smooth function on
$\mathscr{M}$, $F\to e^{f}F$ without withdrawing any of the previous
hypotheses. As a consequence, the unphysical metric is only defined up to the
choice of a conformal frame and the conformal compactification process
precisely associates a physical metric $g$ to a conformal class of unphysical
metrics $\bar{g}\sim e^{2f}\bar{g}$. We will see later how to treat this
ambiguity in practice: for instance, in a particular set of coordinates, one
can fix the boundary metric $\bar{g}|_{\mathscr{I}}$ without loosing this
conformal invariance, which will be translated into a residual gauge
transformation.

The property of being conformally compact is kinematical and does not bring
more constraints on the factor $F$. Now assuming that $(\mathscr{M},g)$ is a
solution of Einstein gravity, the physical metric tensor $g=F^{-2}\bar{g}$ has
to solve the dynamical Einstein’s field equations. This requirement fixes the
norm of the vector $N=(\bar{g}^{\mu\nu}\partial_{\nu}F)\partial_{\mu}$ normal
to $\mathscr{I}$, i.e. the quantity
$N^{2}=\bar{g}^{\mu\nu}\partial_{\mu}F\partial_{\nu}F$ which extends smoothly
to $\mathscr{I}$. Indeed, a straightforward evaluation of the curvature tensor
associated with $g$ yields (see e.g. [279])

$R_{\mu\nu\alpha\beta}=-N^{2}(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha})+\mathcal{O}(F^{-3}),$
(3.1.7)

which implies that a conformally compact manifold asymptotes (when $F\to 0$)
to a constant curvature manifold. The first term is $\mathcal{O}(F^{-4})$
since the divergence of $g$ at infinity is a double pole. Focusing on this
leading term and imposing the Einstein field equations $G_{\mu\nu}+\Lambda
g_{\mu\nu}=0$, it is not hard to show that

$N^{2}\Big{|}_{\mathscr{I}}=\frac{\eta}{\ell^{2}},$ (3.1.8)

where $\eta=-\text{sgn}(\Lambda)$ and the length parameter $\ell$ is related
to $\Lambda$ as (3.1.2) and (3.1.5), or

$\Lambda=-\eta\frac{d(d-1)}{2\ell^{2}}.$ (3.1.9)

The Lorentzian manifold $(\mathscr{M},g)$ is asymptotically locally (A)dSd+1
if it is conformally compact and satisfies (3.1.8) for $\eta=-1$ (dS case) or
$\eta=1$ (AdS case) [160, 161, 207, 166]. As an immediate consequences of the
definitions, we see that $\mathscr{I}$ is timelike when $\eta=1$ and spacelike
when $\eta=-1$. This matches perfectly with the asymptotic analysis for global
AdS and dS spacetimes. In particular, the global AdS et dS metrics (in $d+1$
dimensions) are maximally symmetric solutions of Einstein’s gravity, hence
constant curvature manifolds. The equation (3.1.7) is thus the statement, in a
covariant way, that the metric $g$ approaches the (A)dS metric near infinity.
But this asymptote is just local because (3.1.7) is a pointwise equation. It
does not impose such rigid notion of strict asymptotically (A)dS spacetimes
where the metric at infinity is exactly the (A)dS line element. It brings also
no restriction on the conformal structure or the topology of the boundary. For
instance, the boundary metric on $\mathscr{I}$ is completely free since
(3.1.7) is just the property of being conformally compact phrased by means of
the curvature tensor.

#### 3.1.3 Starobinsky/Fefferman-Graham gauge

The task now is to make good use of the features discussed in the previous
section to define a set of coordinates appropriate to the treatment of
Al(A)dSd+1 spacetimes with the aim of defining an associated phase space later
on. Historically, it was first discovered heuristically by Starobinsky [199]
while studying the asymptotic properties of non-isotropic cosmological
expansions tending towards isotropization, then established mathematically by
Fefferman and Graham [200] as the patch exploiting at best the conformal
compactness in the presence of a non-vanishing cosmological constant. We call
it the Starobinsky/Fefferman-Graham coordinate system (SFG for short) and this
is how it is built.

Let $\mathscr{U}$ be a finite neighborhood of $\mathscr{I}$ in the unphysical
manifold $\bar{\mathscr{M}}$ set in the conformal frame defined by $F$. The
latter being not fixed, one can choose another frame in the conformal class in
order to get a modified defining function $\rho\equiv e^{f}F$ satisfying
$|\text{d}\rho|^{2}=\eta/\ell^{2}$ as in (3.1.8). The most simple way to
achieve this construction is to freeze the original conformal frame and set
$f|_{\mathscr{I}}=0$. Recalling the first point of the definition of conformal
compactness, $\rho$ can be seen as a coordinate on $\mathscr{U}$. The patch is
completed by $d$ coordinates $\\{x^{a}\\}$ defined as orthogonal to $\rho$ in
$\mathscr{U}$ according to $\bar{g}$. The orthogonality can always be demanded
by gauge-fixing $d+1$ functions in $g$ (i.e. fixing the lapse and the shift
with respect to displacements along the $\rho$ coordinate), which amounts to
build asymptotically a set of Gaussian normal coordinates around infinity
$\mathscr{I}$, located at $\rho\to 0$ by definition. This is precisely what
the SFG coordinate system is and the line element of any Al(A)dSd+1 spacetime
can be written as

$\text{d}s^{2}=\eta\frac{\ell^{2}}{\rho^{2}}\text{d}\rho^{2}+\gamma_{ab}(\rho,x^{c})\text{d}x^{a}\text{d}x^{b}.$
(3.1.10)

In (3.1.10), $\gamma_{ab}(\rho,x^{c})$ is the induced metric on leaves of
constant $\rho$ and verifies $\gamma_{ab}=\mathcal{O}(\rho^{-2})$ by assuming
conformal compactness .The local geometry around $\mathscr{I}$ is schematized
in Figure 3.3. We call $\rho\geq 0$ the holographic coordinate because it
lifts the information intrinsically defined on the boundary $\mathscr{I}$
located at $\rho=0$ into the bulk $\rho>0$. We choose it to have units of
inverse of length and the transverse coordinates $x^{a}$ are taken to be
dimensionless. $\rho$ is timelike for dS asymptotics ($\eta<0$) and spacelike
for AdS asymptotics ($\eta>0$). As a consequence, the boundary $\mathscr{I}$
is spacelike when $\Lambda>0$ and timelike when $\Lambda<0$.

$\bm{N}$$\mathscr{I}$$\gamma_{ab}(\rho,x^{c})$$\rho$ Figure 3.3: Schematic
picture of the
Starobinsky/Fefferman-Graham foliation.

The induced metric $\gamma_{ab}(\rho,x^{c})$ represents the only dynamical
field that will be constrained by the Einstein field equations. Apart of its
asymptotic behavior, the construction above does not give more information on
$\gamma_{ab}$. In order to solve the equations of motion, one has to start by
assigning a power-series in $\rho$ to this field, see what occurs practically
for the various coefficients and check the consistency of the ansatz. One sets
[207, 162]

$\gamma_{ab}=\frac{1}{\rho^{2}}g_{ab}^{(0)}+\frac{1}{\rho}g_{ab}^{(1)}+g_{ab}^{(2)}+\dots+\rho^{d-2}g_{ab}^{(d)}+\rho^{d-2}\ln\rho^{2}\tilde{g}_{ab}^{[d]}+\mathcal{O}(\rho^{d-1}),$
(3.1.11)

where the logarithmic term appears only for even $d$. We will review how to
derive the solution space in great details in the next section, but at this
point, we can already get some flavor of what we have to expect. First of all,
since (3.1.7) is imposed by design, we do not expect to get more constraints
on the leading order coefficient $g^{(0)}_{ab}(x^{c})\equiv\lim_{\rho\to
0}\rho^{2}\gamma_{ab}(\rho,x^{c})$. We call this field the boundary metric: it
constitutes the first set of free data that we have at disposal. To be
precise, from the conformal compactification process, $g^{(0)}_{ab}$ is only
defined up to a conformal rescaling and we should call it a representative of
the conformal class $[g^{(0)}_{ab}]=\\{g^{(0)}_{ab}\sim
e^{2\tilde{f}}g^{(0)}_{ab}\\}$ where $\tilde{f}$ is a smooth boundary field.
The first term in (3.1.10) being invariant under $\rho\to e^{\tilde{f}}\rho$,
the conformal frame can be set on the boundary and the possibility to perform
conformal transformations will be translated into a residual gauge
diffeomorphism acting as a boundary Weyl rescaling, see section 2.1.1.3. So
there is nothing to worry about the fixation of the representative of the
boundary metric, which is thus a well-defined object.

Since the Einstein equations are second-order partial differential equations,
starting from $g^{(0)}_{ab}$, one gets, among other constraints, a second-
order recursion relation fixing the subleading coefficients $g^{(k)}_{ab}$,
$k<d$, as unique local functions of the data encoded in the boundary metric
(see e.g. [274, 207, 280]). In this recursion process, a particular behavior
is observed for the step aimed at constraining $g^{(d)}_{ab}$ for each
dimension $d$, because this coefficient does not intervene in the algebraic
constraint at this order. Although this algebraic relation is tautologic for
odd $d$, it brings an additional and unwelcome constraint on lower order
coefficients in even $d$. This motivates the introduction of a logarithmic
term in (3.1.11) for even $d$ only in order to maintain the consistency of the
ansatz after imposing the equations of motion. The new field
$\tilde{g}_{ab}^{[d]}$ that is introduced is also a local function of the
boundary metric. The fact that $g^{(d)}_{ab}$ is absent from the algebraic
constraint means that it provides a new set of free data. The latter does not
share the same nature as $g^{(0)}_{ab}$ because it appears further in the
expansion in $\rho$ and, therefore, does care about the bulk dynamics. In
other words, $g^{(d)}_{ab}$ is subjected to Hamiltonian and momentum
constraints on the leaves of constant $\rho$ and sees thus its trace and
covariant divergence fixed by the equations of motion. Beyond this subtlety,
the recursion process resumes and gives the higher order coefficients
$g^{(k)}_{ab}$ for $k>d$ as local functions of $g^{(0)}_{ab}$ and
$g^{(d)}_{ab}$ and the expansion (3.1.11) is completely determined on-shell by
specifying $g_{ab}^{(0)}$ and ${g}_{ab}^{(d)}$. For future purposes, we define
the holographic stress-energy tensor as

$\boxed{T_{ab}^{[d]}=\frac{d}{16\pi
G}\frac{\eta}{\ell}\left(g_{ab}^{(d)}+X^{[d]}_{ab}[g^{(0)}]\right),}$ (3.1.12)

where $X_{ab}^{[d]}[g^{(0)}]$ is a function which vanishes for $d$ odd [207,
272] and is aimed at ensuring that (3.1.12) is divergence free (see (3.1.30)
and (3.1.31) for expressions in $d=2$ and $d=4$).

Finally, let us see how the (A)dS global vacua are written in the SFG gauge
(3.1.10). The interesting point is that they are conformally flat (the Weyl
tensor vanishes identically). Consequently, it can be shown that
$g^{(d)}_{ab}=0$, which is related to the fact that $\mathscr{I}$ has no
extrinsic curvature when embedded into the unphysical spacetime
$\bar{\mathscr{M}}$. Within these hypotheses, the recursion process terminates
with a finite number of terms and
$\gamma_{ab}=\rho^{-2}g^{(0)}_{ab}+g^{(2)}_{ab}+\rho^{2}g^{(4)}_{ab}$ is the
exact expansion in $\rho$ for the induced metric. We have

$g^{(0)}_{ab}\text{d}x^{a}\text{d}x^{b}=-\frac{\eta}{\ell^{2}}\text{d}t^{2}+\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}$
(3.1.13)

in a set of boundary coordinates $(t,x^{A})$ foliating $\mathscr{I}$ by
constant $t$ $(d-1)$ unit-round spheres covered by the angles $x^{A}$. The
subleading pieces are determined in terms of the boundary metric as (3.1.33)
and $g^{(4)}_{ab}=\frac{1}{4}g^{(2)}_{ac}g_{(0)}^{cd}g^{(2)}_{bd}$.

#### 3.1.4 Solution space

Let us make this discussion more precise by entering into the explicit
resolution of the Einstein equations. We closely follow [162, 207] up to some
different choices of conventions and notations. In the SFG gauge
$g_{\rho\rho}=\eta\frac{\ell^{2}}{\rho^{2}}$, $g_{\rho a}=0$, the coefficients
of the Levi-Civita connection can be expressed in terms of the induced metric
$\gamma_{ab}(\rho,x^{c})$ on leaves of constant $\rho$ as

$\begin{split}&\Gamma^{\rho}_{\rho\rho}=-\frac{1}{\rho},\quad\Gamma^{\rho}_{\rho
a}=0,\quad\Gamma^{a}_{\rho\rho}=0,\\\
&\Gamma^{a}_{b\rho}=\frac{1}{2}\gamma^{ac}\partial_{\rho}\gamma_{bc},\quad\Gamma^{\rho}_{ab}=-\frac{1}{2}\eta\frac{\rho^{2}}{\ell^{2}}\partial_{\rho}\gamma_{ab},\quad\Gamma^{a}_{bc}=\Gamma^{a}_{bc}[\gamma].\end{split}$
(3.1.14)

For practical computations, it is useful to notice that $\Gamma^{a}_{\rho
a}=-\eta\frac{\ell^{2}}{\rho^{2}}\gamma^{ab}\Gamma^{\rho}_{ab}$. Denoting by
$\mathcal{D}_{a}$ the Levi-Civita connection associated with
$\gamma_{ab}(\rho,x^{c})$, the components of the Ricci tensor read as

$\begin{split}R_{ab}&=R_{ab}[\gamma]+(\partial_{\rho}+\Gamma^{\rho}_{\rho\rho})\Gamma^{\rho}_{ab}+\Gamma^{\rho}_{ab}\Gamma^{c}_{c\rho}-2\Gamma^{\rho}_{c(a}\Gamma^{c}_{b)\rho}\\\
&=R_{ab}[\gamma]-\frac{1}{2}\eta\frac{\rho^{2}}{\ell^{2}}\left[\Big{(}\partial_{\rho}+\frac{1}{\rho}\Big{)}\partial_{\rho}\gamma_{ab}-\partial_{\rho}\gamma_{c(a}\gamma^{cd}\partial_{\rho}\gamma_{b)d}+\frac{1}{2}(\gamma^{cd}\partial_{\rho}\gamma_{cd})\partial_{\rho}\gamma_{ab}\right],\\\
R_{\rho
a}&=-2\mathcal{D}_{[a}\Gamma^{b}_{b]\rho}=\frac{1}{2}\left[\mathcal{D}^{b}(\partial_{\rho}\gamma_{ab})-\mathcal{D}_{a}(\gamma^{bc}\partial_{\rho}\gamma_{bc})\right],\\\
R_{\rho\rho}&=-(\partial_{\rho}-\Gamma^{\rho}_{\rho\rho})\Gamma^{a}_{a\rho}-\Gamma^{a}_{b\rho}\Gamma^{b}_{a\rho}=-\frac{1}{2}\left(\partial_{\rho}+\frac{1}{\rho}\right)(\gamma^{ab}\partial_{\rho}\gamma_{ab})-\frac{1}{4}\gamma^{ac}\gamma^{bd}\partial_{\rho}\gamma_{ab}\partial_{\rho}\gamma_{cd}.\end{split}$
(3.1.15)

The Ricci curvature is

$R=\gamma^{ab}R_{ab}[\gamma]-\eta\frac{\rho^{2}}{\ell^{2}}\left[\gamma^{ab}\left(\partial_{\rho}+\frac{1}{\rho}\right)\partial_{\rho}\gamma_{ab}-\frac{3}{4}\gamma^{ac}\gamma^{bd}\partial_{\rho}\gamma_{ab}\partial_{\rho}\gamma_{cd}+\frac{1}{4}(\gamma^{ab}\partial_{\rho}\gamma_{ab})^{2}\right].$
(3.1.16)

##### 3.1.4.1 Organization of the Einstein equations

Due to the particular form of the bulk metric, vacuum Einstein’s equations
read as follows:

$\displaystyle R_{\rho\rho}$ $\displaystyle=-\frac{d}{\rho^{2}},$ (3.1.17)
$\displaystyle R_{\rho a}$ $\displaystyle=0,$ (3.1.18) $\displaystyle R_{ab}$
$\displaystyle=-\eta\frac{d}{\ell^{2}}\gamma_{ab}$ (3.1.19)

when the Ricci curvature is set to its on-shell value

$R=-\eta\frac{d(d+1)}{\ell^{2}}.$ (3.1.20)

The four equations (3.1.17)-(3.1.20) can be solved order by order in $\rho$ if
$\gamma_{ab}(\rho,x^{c})$ is expressed as (3.1.11). We can already eliminate
$g_{ab}^{(1)}$ since the leading order $\mathcal{O}(1/\rho)$ of (3.1.19)
imposes that $(d-1)g^{(1)}_{ab}=0$, hence this field is zero for any
interesting case $d\geq 2$. The ansatz (3.1.11) is designed to ensure
consistency of the Einstein equations at each order in $\rho$, including the
logarithmic branch about which we will give some comments below. Hence the
radial dependence does not have to be determined and is spoiled out directly
by the power series. As a consequence, in the following, all fields under
consideration are local functions of the boundary coordinates $\\{x^{a}\\}$
and indices are lowered and raised by the boundary metric $g^{(0)}_{ab}$ and
its inverse $g_{(0)}^{ab}$, e.g.
$g_{(2)}^{cd}=g^{ca}_{(0)}g^{db}_{(0)}g^{(2)}_{ab}$.

The purely radial constraint (3.1.17) at order $\mathcal{O}(\rho^{k-2})$ fixes
the trace of $g_{ab}^{(k)}$ with respect to $g^{(0)}_{ab}$ for $k\geq 3$. The
$\mathcal{O}(\rho^{0})$ of the equation is tautologic, hence the first non-
trivial trace, $g_{(0)}^{ab}g^{(2)}_{ab}$, is not actually fixed by (3.1.17)
and requires a particular treatment. In fact, using
$R[\gamma]=\rho^{2}R^{(0)}+\mathcal{O}(\rho^{3})$, that trace can be extracted
easily from (3.1.20) at leading order as

$g^{ab}_{(0)}g^{(2)}_{ab}=-\frac{\eta\,\ell^{2}}{2(d-1)}R^{(0)}.$ (3.1.21)

For any dimension $d$, the leading logarithmic term in (3.1.17) imposes that
$\tilde{g}^{[d]}_{ab}$ is trace-free.

Once the traces of the various coefficients have been fixed, the contribution
of the momentum constraint (3.1.18) at order $\mathcal{O}(\rho^{k-1})$ fixes
the covariant divergence of $g^{(k)}_{ab}$ with respect to $g^{(0)}_{ab}$ for
$k\geq 2$. In particular, $D^{b}g^{(k)}_{ab}=0$ is identically zero when $k$
is odd. For any dimension $d$, the leading logarithmic term in (3.1.18)
imposes that $\tilde{g}^{[d]}_{ab}$ is divergence-free.

Once the traces and divergences of the various coefficients have been fixed by
(3.1.17) and (3.1.18), the transverse components (3.1.19) are solved
algebraically to get the explicit expressions of the aforementioned
coefficients in terms of lower order coefficients. Starting from
$g^{(1)}_{ab}=0$, one can show inductively that $(d-k)g^{(k)}_{ab}=0$ for $k$
odd at order $\mathcal{O}(\rho^{k-2})$, hence only even powers of $\rho$
appear in (3.1.11) up to $\mathcal{O}(\rho^{2[\frac{d-1}{2}]})$ where $[x]$
indicates the integer part of $x$. For $k$ even, one gets generically
$(d-k)g^{(k)}_{ab}=\mathcal{C}^{(k)}[g^{(0)},...,g^{(k-1)}]$ for $k<d$,
${\mathcal{C}^{\prime}}^{(d)}[g^{(0)},...,g^{(d-1)},\tilde{g}^{[d]}]=0$ for
$k=d$ and
$(d-k)g^{(k)}_{ab}={\mathcal{C}^{\prime\prime}}^{(k)}[g^{(0)},...,g^{(d)},\tilde{g}^{[d]},...,g^{(k-1)}]$
for $k>d$. In any dimension $d$, we see that the expression $g_{ab}^{(d)}$ is
always left unfixed by the equations of motion, only its trace and divergence
can be computed from (3.1.17) and (3.1.18) respectively. We also remark that
the logarithmic piece $\rho^{(d-2)}\ln\rho^{2}\tilde{g}^{[d]}_{ab}$ is
necessary to ensure the consistency of Einstein’s equations when $d$ is even.
Indeed, if we do not introduce the field $\tilde{g}^{[d]}_{ab}$, the equation
$\tilde{\mathcal{C}}^{(d)}[g^{(0)},...,g^{(d-1)}]=0$ brings in general an
additional constraint on the coefficients $g^{(0)}_{ab},...,g^{(d-1)}_{ab}$
which are supposed to have been fixed at lower order in $\rho$. Rather than
that, ${\mathcal{C}^{\prime}}^{(d)}[g^{(0)},...,g^{(d-1)},\tilde{g}^{[d]}]=0$
gives in these cases the explicit solution for $\tilde{g}^{[d]}_{ab}$ in terms
of lower order coefficients. To summarize,

$\tilde{g}^{[2k+1]}_{ab}=0,\quad\tilde{g}^{[2k]}_{ab}\neq 0.$ (3.1.22)

A notable exception occurs for $d=2$ as we will see below in more details,
simply because ${\mathcal{C}^{\prime}}^{(2)}[g^{(0)}]=0$ on-shell without the
help of any additional field.

##### 3.1.4.2 Trace of metric coefficients

Apart of the trace of $g^{(2)}_{ab}$ set as (3.1.21), we extract the various
traces from the equation (3.1.17), starting at $\mathcal{O}(\rho)$ order:

$\displaystyle g^{ab}_{(0)}g^{(3)}_{ab}$ $\displaystyle=0,$ (3.1.23)
$\displaystyle g^{ab}_{(0)}g^{(4)}_{ab}$
$\displaystyle=\frac{1}{4}g^{ab}_{(2)}g^{(2)}_{ab},$ (3.1.24) $\displaystyle
g^{ab}_{(0)}g^{(5)}_{ab}$ $\displaystyle=0,$ (3.1.25) $\displaystyle
g^{ab}_{(0)}g^{(6)}_{ab}$
$\displaystyle=\frac{2}{3}g_{(2)}^{ab}g^{(4)}_{ab}-\frac{1}{6}(g_{(0)}^{ab}g^{(2)}_{ab})^{3},\
\dots$ (3.1.26)

These equations help to compute the trace of the holographic stress-tensor as

$\mathcal{T}^{[2k+1]}\equiv
g_{(0)}^{ab}T_{ab}^{[2k+1]}=0,\,\forall\,k\in\mathbb{N}_{0}.$ (3.1.27)

Furthermore, for $d=2$, we have

$\mathcal{T}^{[2]}\equiv
g_{(0)}^{ab}T_{ab}^{[2]}=\frac{c}{24\pi}R^{(0)}=\frac{\ell}{\sqrt{|g^{(0)}|}}L_{EH}[g^{(0)}]$
(3.1.28)

where $c=\frac{3\ell}{2G}$ is the Brown-Henneaux central charge [62] and
$L_{EH}$ is the Einstein-Hilbert Lagrangian density. For $d=4$, we have

$\begin{split}\mathcal{T}^{[4]}\equiv
g_{(0)}^{ab}T_{ab}^{[4]}=&-\frac{1}{16\pi
G}\frac{\eta}{\ell}\left[(g^{cd}_{(0)}g_{cd}^{(2)})^{2}-g_{(2)}^{cd}g^{(2)}_{cd}\right]=\frac{\eta\,\ell^{3}}{64\pi
G}\left(R_{ab}^{(0)}R^{ab}_{(0)}-\frac{R_{(0)}^{2}}{3}\right)\\\
=&\frac{\eta\,\ell^{3}}{4\sqrt{|g^{(0)}|}}\left(L_{QCG(1)}[g^{(0)}]-\frac{1}{3}L_{QCG(2)}[g^{(0)}]\right)\end{split}$
(3.1.29)

where we used (3.1.24) and (3.1.33) explicitly. We see that (3.1.28) and
(3.1.29) reproduce the conformal anomaly in three and five dimensions [162].
Interestingly, $\mathcal{T}^{[4]}$ involves the Lagrangian densities
$L_{QCG(1)}[g^{(0)}]$ and $L_{QCG(2)}[g^{(0)}]$ of the quadratic curvature
gravity on the boundary [281, 282]. The expressions of $L_{QCG(1)}$ and
$L_{QCG(2)}$ are included in appendix F.

##### 3.1.4.3 Divergence of metric coefficients

Developing the momentum constraint (3.1.18) leads to the generic form
$D^{b}g^{(k)}_{ab}=-D^{b}X^{[k]}_{ab}+V_{a}^{[k]}$ where $D_{a}$ is the Levi-
Civita connection with respect to $g^{(0)}_{ab}$. This equation is crucial
because it allows to check explicitly that the boundary stress-tensor
$T_{ab}^{[d]}$ is indeed divergence-free. For any odd $k$, we simply have
$X_{ab}^{[k]}=0$ and $V_{a}^{[k]}=0$. For even $k$, we can show that

$\displaystyle X^{[2]}_{ab}$
$\displaystyle=-(g^{cd}_{(0)}g_{cd}^{(2)})g_{ab}^{(0)},\quad V_{a}^{[2]}=0,$
(3.1.30) $\displaystyle X^{[4]}_{ab}$
$\displaystyle=-\frac{1}{8}g_{ab}^{(0)}\left[(g_{(0)}^{cd}g^{(2)}_{cd})^{2}-g_{(2)}^{cd}g^{(2)}_{cd}\right]-\frac{1}{2}g_{(2)}^{ac}g_{(0)}^{cd}g^{(2)}_{bd}+\frac{1}{4}(g_{(0)}^{cd}g^{(2)}_{cd})g^{(2)}_{ab},\quad
V_{a}^{[4]}=0,\ \dots$ (3.1.31)

These results show that the combination $g_{ab}^{(d)}+X^{[d]}_{ab}$ is
sufficient to get the conservation of the holographic stress-tensor for
dimensions $d\leq 4$,

$\boxed{D^{a}T_{ab}^{[d]}=0.}$ (3.1.32)

One can notice the presence of a non-vanishing $V_{a}$ covector for $d=6$ and
generically beyond. It makes the construction of candidates for $T_{ab}^{[d]}$
from the equations of motion slightly more involved. Moreover, one can also
check that $X_{ab}^{[6]}$ contains an anti-symmetric part, which was not the
case for lower dimensions. However one can perform a field redefinition from
$g_{ab}^{(6)}$ in order to get a covariantly conserved symmetrical tensor
[207].

##### 3.1.4.4 Explicit expressions of metric coefficients

Now that we have determined the trace and the divergence of the metric
coefficients we are interested in, the last equation (3.1.19) allows to derive
their explicit form in terms of lower order coefficients. At leading order
$\mathcal{O}(\rho^{0})$, we find

$g_{ab}^{(2)}=-\frac{\eta\ell^{2}}{d-2}\left(R_{ab}^{(0)}-\frac{R^{(0)}}{2(d-1)}g_{ab}^{(0)}\right),\
\forall d\geq 3.$ (3.1.33)

For $d=2$, the $\mathcal{O}(\rho^{0})$ contribution of (3.1.19) does not
constrain $g_{ab}^{(2)}$ but fixes the logarithmic term $\tilde{g}_{ab}^{[2]}$
in the expansion and states nothing but
$\tilde{g}^{[2]}_{ab}=\frac{1}{2}\eta\ell^{2}(R_{ab}^{(0)}-\frac{1}{2}R^{(0)}g_{ab}^{(0)})$.
This leads to

$\tilde{g}_{ab}^{[2]}=0,$ (3.1.34)

since the Einstein tensor vanishes identically in two dimensions. At next
order $\mathcal{O}(\rho)$, we find

$(d-3)g^{(3)}_{ab}=0,$ (3.1.35)

which means that $g^{(3)}_{ab}$ is identically zero except for $d=3$ where it
is unconstrained. Going further at order $\mathcal{O}(\rho^{2})$, we find for
$d\neq 4$

$(d-4)g^{(4)}_{ab}+g_{ac}^{(2)}g_{(0)}^{cd}g_{bd}^{(2)}-\frac{1}{4}g_{(2)}^{cd}g^{(2)}_{cd}g_{ab}^{(0)}+\frac{1}{2}\eta\ell^{2}R^{(2)}_{ab}[g^{(0)}]=0$
(3.1.36)

where we used the notation
$R_{ab}[\gamma]=R^{(0)}_{ab}[g^{(0)}]+\rho^{2}R_{ab}^{(2)}[g^{(0)}]+\mathcal{O}(\rho^{3})$.
In order to extract some information from this equation, we need to compute
the first subleading piece of the Ricci tensor. First of all, we can remark
that
${\Gamma^{a}}_{bc}[\gamma]={\Gamma^{a}}_{bc}[g^{(0)}]+\rho^{2}\Gamma^{a}_{(2)bc}[g^{(0)}]+\mathcal{O}(\rho^{3})$,
where

$\begin{split}\Gamma^{a}_{(2)bc}[g^{(0)}]&=\frac{1}{2}g_{(0)}^{ad}(\partial_{b}g_{dc}^{(2)}+\partial_{c}g_{bd}^{(2)}-\partial_{d}g^{(2)}_{bc})-\frac{1}{2}g_{(2)}^{ad}(\partial_{b}g_{dc}^{(0)}+\partial_{c}g_{bd}^{(0)}-\partial_{d}g^{(0)}_{bc})\\\
&=\frac{1}{2}g_{(0)}^{ad}(D_{b}g_{cd}^{(2)}+D_{c}g_{bc}^{(2)}-D_{d}g^{(2)}_{bc})\end{split}$
(3.1.37)

is a covariant $(1,2)$-tensor with respect to the boundary geometry and only
depends on $g^{(0)}_{ab}$ and its curvature because of (3.1.33). As a
consequence,

$\displaystyle
R_{ab}^{(2)}[g^{(0)}]=D_{c}\Gamma^{c}_{(2)ab}[g^{(0)}]-D_{b}\Gamma^{c}_{(2)ac}[g^{(0)}]$
$\displaystyle\quad=D^{c}D_{(a}g^{(2)}_{b)c}-\frac{1}{2}D^{2}g_{ab}^{(2)}-\frac{1}{2}D_{a}D_{b}(g^{cd}_{(0)}g^{(2)}_{cd})$
(3.1.38)
$\displaystyle\quad=-\frac{\eta\,\ell^{2}}{d-2}\left[R_{ca}^{(0)}R^{c}_{(0)b}-R_{acbd}^{(0)}R^{cd}_{(0)}+\frac{(d-2)}{4(d-1)}D_{a}D_{b}R^{(0)}-\frac{1}{2}D^{c}D_{c}R_{ab}^{(0)}+\frac{D^{c}D_{c}R^{(0)}}{4(d-1)}g^{(0)}_{ab}\right].$

We can deduce that

$R[\gamma]=\gamma^{ab}R_{ab}[\gamma]=\rho^{2}R^{(0)}+\rho^{4}\frac{\eta\,\ell^{2}}{(d-2)}\left[R^{ab}_{(0)}R_{ab}^{(0)}-\frac{1}{2(d-1)}(R^{(0)})^{2}\right]+\mathcal{O}(\rho^{5}).$
(3.1.39)

Finally computing

$\begin{split}g_{(2)a}^{c}g_{bd}^{(2)}&=\frac{\ell^{4}}{(d-2)^{2}}\left[R_{(0)a}^{c}R_{bd}^{(0)}-\frac{R^{(0)}}{d-1}R_{ab}^{(0)}+\frac{(R^{(0)})^{2}}{4(d-1)^{2}}g^{(0)}_{ab}\right],\\\
g_{(2)}^{cd}g^{(2)}_{cd}&=\frac{\ell^{4}}{(d-2)^{2}}\left[R^{cd}_{(0)}R_{cd}^{(0)}+\frac{(4-3d)}{4(d-1)^{2}}(R^{(0)})^{2}\right],\end{split}$
(3.1.40)

we can solve (3.1.36) for $g^{(4)}_{ab}$ if $d\neq 4$ to get

$\displaystyle g^{(4)}_{ab}=\frac{\ell^{4}}{(d-4)}$
$\displaystyle\left[\frac{1}{8(d-1)}D_{a}D_{b}R^{(0)}-\frac{1}{4(d-2)}D_{c}D^{c}R_{ab}^{(0)}+\frac{D_{c}D^{c}R^{(0)}}{8(d-1)(d-2)}g_{ab}^{(0)}\right.$
(3.1.41)
$\displaystyle\quad-\frac{1}{2(d-2)}R_{acbd}^{(0)}R^{cd}_{(0)}+\frac{(d-4)}{2(d-2)^{2}}R_{a}^{(0)c}R^{(0)}_{cb}+\frac{R^{(0)}}{(d-1)(d-2)^{2}}R_{ab}^{(0)}$
$\displaystyle\quad\left.+\,\frac{1}{4(d-2)^{2}}R^{cd}_{(0)}R_{cd}^{(0)}g^{(0)}_{ab}-\frac{3d}{16(d-1)^{2}(d-2)^{2}}(R^{(0)})^{2}g^{(0)}_{ab}\right].$

When $d=4$, (3.1.36) fixes the logarithmic term $\tilde{g}^{[4]}_{ab}$ in the
expansion. After some algebra we obtain

$\begin{split}\tilde{g}^{[4]}_{ab}&=\frac{1}{2}g_{ac}^{(2)}g^{cd}_{(0)}g^{(2)}_{bd}-\frac{1}{8}g_{(2)}^{cd}g^{(2)}_{cd}g^{(0)}_{ab}+\frac{\eta\ell^{2}}{4}R^{(2)}_{ab}\\\
&=\frac{1}{2}g_{ac}^{(2)}g^{cd}_{(0)}g^{(2)}_{bd}-\frac{1}{8}g_{(2)}^{cd}g^{(2)}_{cd}g^{(0)}_{ab}+\frac{\eta\ell^{2}}{8}\left[2D^{c}D_{(a}g^{(2)}_{b)c}-D^{2}g_{ab}^{(2)}-D_{a}D_{b}(g^{cd}_{(0)}g^{(2)}_{cd})\right],\end{split}$
(3.1.42)

or explicitly in terms of $g^{(0)}_{ab}$

$\displaystyle\tilde{g}^{[4]}_{ab}$
$\displaystyle=\frac{\ell^{4}}{8}R_{acbd}^{(0)}R^{cd}_{(0)}-\frac{\ell^{4}}{48}D_{a}D_{b}R^{(0)}+\frac{\ell^{4}}{16}D_{c}D^{c}R_{ab}^{(0)}-\frac{\ell^{4}}{24}R^{(0)}R^{(0)}_{ab}$
(3.1.43) $\displaystyle\quad\
+\frac{\ell^{4}}{96}\left[(R^{(0)})^{2}-D_{c}D^{c}R^{(0)}-3R^{cd}_{(0)}R^{(0)}_{cd}\right]g^{(0)}_{ab}.$

The algorithm can be pursued as follows. We can check that $(d-5)g^{(5)}_{ab}$
is linear in $g^{(3)}_{ab}$, so $g^{(5)}_{ab}=0$ except for $d=5$ as expected.
We have $(d-6)g^{(6)}_{ab}=\mathcal{C}^{(6)}[g^{(0)}]$ for $d<6$,
$(d-6)g^{(6)}_{ab}=\mathcal{C}^{(6)}[g^{(0)},\tilde{g}^{(6)}_{ab}]$ for $d>6$
and the same equation fixes $\tilde{g}^{[6]}_{ab}$ in terms of $g^{(0)}_{ab}$
when $g^{(2)}_{ab}$ and $g^{(4)}_{ab}$ hit their on-shell values. We will not
present the results in details here because we have already derived all that
we need. The interested reader can refer to [207] for the explicit expressions
in $d=5,6$.

#### 3.1.5 Residual gauge diffeomorphisms

The second step in the path towards the formulation of the Al(A)dSd+1 phase
space consists in determining the residual symmetries among the
diffeomorphisms on $\mathscr{M}$ that survive after the gauge fixing (3.1.10).
Such diffeomorphisms are generated at the infinitesimal level by vector fields
$\xi=\xi^{\rho}\partial_{\rho}+\xi^{a}\partial_{a}$ tangent to $\mathscr{M}$
such that the transformed metric $g_{\mu\nu}\to
g_{\mu\nu}+\mathcal{L}_{\xi}g_{\mu\nu}$, expressed in the same coordinates
$\\{\rho,x^{a}\\}$, still belongs to the SFG solution space, i.e. reads as
(3.1.10) for another dynamical field $\gamma^{\prime}_{ab}$. We have the
following $d+1$ conditions

$\mathcal{L}_{\xi}g_{\rho\rho}=0,\quad\mathcal{L}_{\xi}g_{\rho a}=0$ (3.1.44)

which are first order differential constraints on the gauge parameter $\xi$.
The first condition leads to the equation

$\partial_{\rho}\xi^{\rho}=\frac{1}{\rho}\xi^{\rho},$ (3.1.45)

which can be solved for $\xi^{\rho}$ as

$\xi^{\rho}=\sigma(x^{a})\rho,$ (3.1.46)

where $\sigma(x^{a})$ is a possibly field-dependent arbitrary function of the
boundary coordinates that parametrizes the Weyl rescalings on the boundary.
The second condition leads to the equation

$\rho^{2}\gamma_{ab}\partial_{\rho}\xi^{b}+\eta\ell^{2}\partial_{a}\xi^{\rho}=0,$
(3.1.47)

which can be solved for $\xi^{a}$ as

$\xi^{a}=\bar{\xi}^{a}(x^{b})-\eta\ell^{2}\partial_{b}\sigma\int_{0}^{\rho}\frac{\text{d}\rho^{\prime}}{\rho^{\prime}}\gamma^{ab}(\rho^{\prime},x^{c}),$
(3.1.48)

where $\bar{\xi}^{a}(x^{b})$ is a possibly field-dependent arbitrary
diffeomorphism on the spacetime boundary. The subset of residual gauge
diffeomorphisms parametrized by $\sigma$ are referred as the Penrose-Brown-
Henneaux (PBH) transformations [62, 278, 206]. General diffeomorphisms
$\chi_{1},\chi_{2}$ on $\mathscr{M}$ are known to form a Lie algebra for the
Lie bracket defined as

$[\chi_{1},\chi_{2}]=(\chi_{1}^{\mu}\partial_{\mu}\chi_{2}^{\nu}-\chi_{2}^{\mu}\partial_{\mu}\chi_{1}^{\nu})\partial_{\nu}.$
(3.1.49)

The subclass of residual gauge diffeomorphisms $\xi$ satisfying (3.1.46) and
(3.1.48) also form a Lie algebra but under the modified Lie bracket (2.1.20)
that takes into account the field-dependence of the vectors fields[2, 46,
208],

$[\xi_{1},\xi_{2}]_{\star}=[\xi_{1},\xi_{2}]-\delta_{\xi_{1}}\xi_{2}+\delta_{\xi_{2}}\xi_{1}.$
(3.1.50)

The modification has to be interpreted as follows. In gravity, the gauge
parameter $\xi$ in a whole is rarely field-independent precisely because the
gauge-preserving conditions involves the metric field explictly. Here a joint
solution of (3.1.46) and (3.1.48) is clearly depending on the dynamical field
$\gamma_{ab}$, hence the gauge parameters $\xi$ have to be considered as
functions of the coordinates as well as on the solution space. We can thus
give sense to the variation $\delta_{\xi_{1}}\xi_{2}$ for any couple of
residual gauge diffeomorphisms $\xi_{1},\xi_{2}$: it represents the result of
the action of $\xi_{1}$ on the dynamical field implicitly present in
$\xi_{2}$. This action ensures the preservation of the gauge when performing
$\xi_{1}$ and $\xi_{2}$ successively. When computing the commutator
$[\xi_{1},\xi_{2}]$, one includes this action unintentionally but then the
usual Lie bracket does not reflect the true commutation relation between the
vectors themselves. This explains why the bracket (3.1.50) subtracts these
contributions in order to correctly close the algebra of vectors: this
contextualizes the discussion of section 2.1.4.2.

When (3.1.46) and (3.1.48) are solved, the residual diffeomorphisms are
uniquely defined if a set of gauge parameters $(\sigma,\bar{\xi}^{a})$ is
given. We denote this as $\xi=\xi(\sigma,\bar{\xi}^{a})$. How is the algebra
(3.1.50) encoded at the level of the parameters $\sigma$ and $\bar{\xi}^{a}$?
Let $\xi_{1}\equiv\xi(\sigma_{1},\bar{\xi}_{1})$ and
$\xi_{2}\equiv\xi(\sigma_{2},\bar{\xi}_{2})$ be two residual gauge
diffeomorphisms of the SFG gauge. Both satisfy the condition (3.1.47), hence
the computation of $[\xi_{1},\xi_{2}]^{\rho}_{\star}$ is straightforward and
gives

$\frac{1}{\rho}[\xi_{1},\xi_{2}]^{\rho}_{\star}=\left(\xi^{a}_{1}\partial_{a}\sigma_{2}-\xi^{a}_{2}\partial_{a}\sigma_{1}\right)-\delta_{\xi_{1}}\sigma_{2}+\delta_{\xi_{2}}\sigma_{1}.$
(3.1.51)

Taking a derivative with respect to $\rho$ and using again (3.1.47), we get

$\partial_{\rho}\left(\frac{1}{\rho}[\xi_{1},\xi_{2}]^{\rho}_{\star}\right)=\partial_{\rho}\xi^{a}_{1}\partial_{a}\sigma_{2}-\partial_{\rho}\xi_{2}^{a}\partial_{a}\sigma_{1}=0,$
(3.1.52)

which shows that $[\xi_{1},\xi_{2}]^{\rho}_{\star}=\rho\hat{\sigma}(x^{a})$
for

$\hat{\sigma}=\frac{1}{\rho}[\xi_{1},\xi_{2}]^{\rho}_{\star}\Big{|}_{\rho=0}=\bar{\xi}_{1}^{a}\partial_{a}\sigma_{2}-\bar{\xi}_{2}\partial_{a}\sigma_{1}-\delta_{\xi_{1}}\sigma_{2}+\delta_{\xi_{2}}\sigma_{1}.$
(3.1.53)

Let us now consider the transverse components. By evaluating the commutator at
leading order, we derive that

$\hat{\bar{\xi}}^{a}=\lim_{\rho\to
0}[\xi_{1},\xi_{2}]^{a}_{\star}=[\bar{\xi}_{1},\bar{\xi}_{2}]^{a}-\delta_{\xi_{1}}\bar{\xi}_{2}^{a}+\delta_{\xi_{2}}\bar{\xi}_{1}^{a}.$
(3.1.54)

Recalling that
$\delta_{\xi}\gamma^{ab}=\mathcal{L}_{\xi}\gamma^{ab}=\rho\sigma_{\xi}\partial_{\rho}\gamma^{ab}+\xi^{c}\partial_{c}\gamma^{ab}-2\gamma^{c(a}\partial_{c}\xi^{b)}$
and explicitly using (3.1.47) to express $\partial_{\rho}\xi^{a}_{1}$ and
$\partial_{\rho}\xi^{b}_{2}$ in terms of $\sigma_{1}$ and $\sigma_{2}$,
respectively, a direct computation yields

$\partial_{\rho}\left([\xi_{1},\xi_{2}]^{a}_{\star}\right)=-\eta\ell^{2}\frac{1}{\rho}\gamma^{ab}\partial_{b}\hat{\sigma}.$
(3.1.55)

The reasoning above transposes a similar proof given in [46] for BMS
symmetries.

As a result, the residual gauge diffeomorphisms $\xi_{1}$ and $\xi_{2}$ of the
SFG gauge satisfy

$\boxed{\begin{gathered}{}[\xi(\sigma_{1},\bar{\xi}_{1}^{a}),\xi(\sigma_{2},\bar{\xi}_{2}^{a})]_{\star}=\xi(\hat{\sigma},\hat{\bar{\xi}}^{a}),\quad\text{where}\
\left\\{\begin{gathered}\,\,\hat{\sigma}=\bar{\xi}_{1}^{a}\partial_{a}\sigma_{2}-\delta_{\xi_{1}}\sigma_{2}-(1\leftrightarrow
2),\\\
\,\,\hat{\bar{\xi}}^{a}=\bar{\xi}^{b}_{1}\partial_{b}\bar{\xi}^{a}_{2}-\delta_{\xi_{1}}\bar{\xi}_{2}^{a}-(1\leftrightarrow
2).\end{gathered}\right.\end{gathered}}$ (3.1.56)

This extends the analysis of [208] where the bracket (3.1.53) was applied for
the subclass of PBH transformations. In this derivation, we do not assume that
the residual gauge diffeomorphism parameters $\sigma$ and $\bar{\xi}^{a}$ are
field-independent, although the majority of previous analyses in the field
made this assumption. This is due to the fact that usual boundary conditions
in gravity fix enough boundary degrees of freedom of the gravitational field
in order to forbid the residual gauge constraints to be field-dependent, hence
the solution for the residual gauge parameters is also field-independent.
However, we will see in section 3.4 below that it is crucial to remain the
most general possible in treating the actual dependence of the parameters if
we want to consider generalized sets of boundary conditions that are less
restrictive than the usual ones and allow for some leaks at infinity.
Therefore, the terms $\delta_{\xi}\sigma$ and $\delta_{\xi}\bar{\xi}$ have to
be kept in (3.1.56). Finally, if one assumes that the parameters are field-
independent at this stage ($\delta_{\xi}\sigma=0$ and
$\delta_{\xi}\bar{\xi}^{a}=0$), then the commutation relations (3.1.56) reduce
to those of the semi-direct sum
$\text{Diff}(\mathscr{I})\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}\mathbb{R}$ where
Diff($\mathscr{I}$) denotes the diffeomorphisms on the boundary $\mathscr{I}$,
parametrized by $\bar{\xi}^{a}$ and $\mathbb{R}$ denotes the abelian Weyl
rescalings on the boundary, parametrized by $\sigma$.

#### 3.1.6 Variations of the solution space

We conclude our presentation of the kinematics by specifying the action of the
residual gauge diffeomorphisms $\xi(\sigma,\bar{\xi})$ on the solution space
described in section 3.1.4. Since any Al(A)dS solution in the SFG gauge is
characterized by the couple of free data $(g^{(0)}_{ab},T_{ab}^{[d]})$, we
just have to understand how these tensors transform under the action of $\xi$.
Up to some conventions, the derivation is a rephrasing of the one provided in
[206, 207].

Under the full set of residual gauge diffeomorphisms
$\xi=\xi^{\rho}\partial_{\rho}+\xi^{a}\partial_{a}$, the metric varies as

$\delta_{\xi}\gamma_{ab}(\rho,x^{c})=\mathcal{L}_{\xi}\gamma_{ab}(\rho,x^{c})=\left(\sigma\rho\partial_{\rho}+\mathcal{L}_{\xi^{c}}\right)\gamma_{ab}(\rho,x^{c}).$
(3.1.57)

The condition (3.1.46) has been incorporated directly in the first term. Owing
to (3.1.48), we can obtain the transformation of the boundary metric by
extracting the leading order $\mathcal{O}(1/\rho^{2})$ of (3.1.57). We get
[206, 280]

$\boxed{\delta_{\xi}g^{(0)}_{ab}=\mathcal{L}_{\bar{\xi}}g^{(0)}_{ab}-2\sigma
g_{ab}^{(0)}.}$ (3.1.58)

This justifies a posteriori the codification adopted for the parameters
$(\sigma,\bar{\xi})$. The first term in (3.1.58) is the expected action of a
boundary diffeomorphism $\bar{\xi}$ on the induced metric at the boundary
$\mathscr{I}$. The second term is the infinitesimal implementation of a Weyl
rescaling on $\mathscr{I}$. Indeed, if $\xi=\xi(\sigma,\bar{\xi}=0)$, (3.1.46)
can be readily exponentiated and generates the transformation $g^{(0)}_{ab}\to
e^{-2\sigma}g^{(0)}_{ab}$ on $g^{(0)}_{ab}$, by looking at the leading
$\mathcal{O}(1/\rho^{2})$ of (3.1.10).

The transformation of the holographic stress-energy needs to go further into
the power series in $\rho$. Writing the solution (3.1.48) as
$\xi^{a}(\rho,x^{b})=\bar{\xi}^{a}(x^{b})+\sum_{k=1}^{+\infty}\xi^{a}_{(k)}(\rho,x^{b})\rho^{k}$,
we have

$\xi^{a}_{(1)}=0,\quad\xi^{a}_{(2)}=-\frac{1}{2}\eta\ell^{2}g^{ab}_{(0)}\partial_{b}\sigma,\quad\xi^{a}_{(3)}=0,\quad\xi^{a}_{(4)}=\frac{1}{4}\eta\ell^{2}g^{ab}_{(2)}\partial_{b}\sigma.$
(3.1.59)

Evaluating (3.1.57) order by order in $\rho$ yields the following tower of
variations:

$\displaystyle\delta_{\xi}g^{(2)}_{ab}$
$\displaystyle=\mathcal{L}_{\bar{\xi}}g^{(2)}_{ab}+2D_{(a}\xi^{(2)}_{b)},$
(3.1.60) $\displaystyle\delta_{\xi}g^{(3)}_{ab}$
$\displaystyle=\mathcal{L}_{\bar{\xi}}g^{(3)}_{ab}+\sigma g^{(3)}_{ab},$
(3.1.61) $\displaystyle\delta_{\xi}g^{(4)}_{ab}$
$\displaystyle=\mathcal{L}_{\bar{\xi}}g^{(4)}_{ab}+2\sigma(g^{(4)}_{ab}+\tilde{g}^{[4]}_{ab})+2D_{(a}\xi_{b)}^{(4)}+\xi_{(2)}^{c}D_{c}g^{(2)}_{ab}+2g^{(2)}_{c(a}D_{b)}\xi^{c}_{(2)},\dots$
(3.1.62)

As a consistency check, we can verify by some straightforward algebra that
(3.1.60)-(3.1.62) are compatible with their explicit solutions presented in
Section 3.1.4 when (3.1.58) holds. For each dimension $d$, one takes benefit
from these variations to compute case by case the transformation of the
holographic stress-energy tensor. It yields in general [207, 280]

$\boxed{\delta_{\xi}T_{ab}^{[d]}=\mathcal{L}_{\bar{\xi}}T_{ab}^{[d]}+(d-2)\sigma
T_{ab}^{[d]}+A_{ab}^{[d]}[\sigma]}$ (3.1.63)