5pt

Université Libre de Bruxelles and International Solvay Institutes CP 231,
B-1050 Brussels, Belgium

Thesis submitted in fulfilment of the requirements of the _PhD Degree in
Sciences_. Academic year 2020-2021.

Leaky covariant phase spaces
Theory and application to $\bm{\Lambda}$-BMS symmetry

Adrien Fiorucci

Supervisors:

Prof. Geoffrey Compère & Prof. Glenn Barnich.

Thesis jury:
Prof. Frank Ferrari (Université Libre de Bruxelles),

Prof. Andrès Collinucci (Université Libre de Bruxelles),

Prof. Daniel Grumiller (Technische Universität Wien),

Prof. Simone Speziale (Centre de Physique Théorique Marseille).

— Abstract —

The present thesis aims at providing a unified description of radiative phase
spaces in General Relativity for any value of the cosmological constant using
covariant phase space methods. We start by considering generic asymptotically
locally (A)dS spacetimes with leaky boundary conditions in the
Starobinsky/Fefferman-Graham gauge and in arbitrary dimensions. The boundary
structure is allowed to fluctuate and plays the role of source yielding some
flux of gravitational radiation at the boundary. The holographic
renormalization procedure is employed to remove divergences from the
presymplectic structure, which leads to finite surface charges for the whole
class of boundary diffeomorphisms and Weyl rescalings. The charge algebra
represents this asymptotic symmetry algebra under the Barnich-Troessaert
bracket, up to a field-dependent 2-cocycle in odd spacetime dimensions. We
then propose a boundary gauge fixing isolating the radiative components among
the boundary degrees of freedom without constraining the Cauchy problem in
asymptotically de Sitter spacetimes. This additional gauge fixing reduces the
set of allowed boundary diffeomorphisms to the infinite-dimensional
$\Lambda$-BMS algebroid, which is the counterpart to the Generalized BMS
algebra of smooth supertranslations and super-Lorentz transformations in the
flat limit.

In a second round, the analysis is repeated in the Bondi gauge, which is
better suited to discuss radiative phenomena as well as construct a flat limit
process at the level of the solution space. Thanks to a diffeomorphism we
translate the results previously obtained in the Starobinsky/Fefferman-Graham
coordinates and identify the analogues of the Bondi news, mass and angular
momentum aspects in the presence of a non-vanishing cosmological constant. We
give a prescription to perform the flat limit at the level of the phase space
and demonstrate how to use this connection to renormalize the corresponding
phase space of asymptotically locally flat spacetimes at null infinity. The
latter is made necessary as soon as the boundary structure of the
gravitational field is allowed to vary under arbitrary super-Lorentz
transformations.

The last part of the manuscript is devoted to discussing the various
implications of these super-Lorentz transformations as genuine asymptotic
symmetries of asymptotically flat Einstein’s gravity. In particular, we derive
a closed-form expression of the orbit of gravitational vacua under the
Generalized BMS symmetries. Transitions among these vacua are related to the
refraction/velocity kick memory effect and the displacement memory effect.
Finally, we give a physical prescription to define finite Hamiltonian
generators canonically conjugated to Generalized BMS transformations on the
subclass of physical solutions that are stationary at early and late times and
comment on the enhancement of the infrared structure of gravity in the
presence of super-Lorentz transformations.

The research compiled in this doctoral thesis was supported by the
FRS-F.N.R.S. Belgium. It was carried out at the Université de Bruxelles
(Belgium) from the 1st of October 2017 to the 1st of May 2021.

The present document was typeset in LaTeX report class and written using
_Texmaker_ (v.5.0.4) running on _Microsoft Windows 10 Pro_. The figures are
vector graphics drawn with the creation tools PGF/TikZ. Some symbolic
computations were performed by means of the Wolfram Mathematica software
(v.12.2).

[ arXiv version ]

© Adrien Fiorucci, 2021.

## Acknowledgements

Completing a doctoral thesis is as much a personal scientific challenge as a
shared human adventure. Therefore, it seems fair to me to begin this
manuscript by paying tribute to the various people who, from near or far, have
contributed to making this first research experience truly magical.

First, I wish to show all my gratitude to Geoffrey Compère for his exemplary
supervision. Since early in my master studies, Geoffrey has been a passionate
guide in the discovery of my future profession. Gifted with an impressive
intuition, an extensive knowledge on both theoretical and experimental aspects
of modern gravitational physics and a formidable mathematical ease, he
represents the ideal of a brilliant researcher towards which I will strive.
His unquenchable thirst for new scientific challenges, his constant enthusiasm
and his attentive support always acted as boosters that made me progress and
surpass myself. For his constant availability to discuss deep conceptual
issues and provide concrete help in concluding a computation, for his
patience, for his pedagogical qualities and finally for providing me with many
project opportunities and stimulating ideas during five years of
collaboration, I give him my heartfelt thanks.

Furthermore, I am grateful to my co-supervisor, Glenn Barnich, for the
numerous discussions we had, during which he shared to me his prodigious
knowledge on asymptotic symmetries in gauge theories among many other areas of
physics. Working with Glenn is a true immersion into the finest subtleties in
the field from which I always came back enriched. His ability to directly
detect the core of the problem, formalize it in a mathematically robust and
unequivocal way and never relax concentration before fully understanding the
outcome are all rare qualities to be treasured in science. His powerful,
rigorous and independent vision of scientific research will definitively
inspire me in my future career.

The third main actor who participated in the success of my early scientific
enterprise was my colleague and friend Romain Ruzziconi, in whom I discovered
the most wonderful collaborator. I could spend several pages detailing the
importance of his contribution to the numerous ideas developed in this
manuscript, or even remembering the passionate and stimulating discussions we
had. I thank him for the fantastic human he is, for the scientific journey we
made together and for the future of our fruitful collaboration, which already
promises to be rich in exciting projects.

In parallel to this research, I was able to free up a few hours to devote
myself to another of my greatest passions: teaching. Therefore, I would also
like to thank Glenn Barnich, Nicolas Boulanger, Frank Ferrari and Serge Massar
for giving me the opportunity to teach alongside them during the past three
years. I am delighted to have benefited from their advice and teaching
experience to improve my pedagogical methods.

I gratefully acknowledge the support of Riccardo Argurio and Andrès
Collinucci, who formed my accompaniment committee with Geoffrey and Glenn.
Moreover, I would like to express my sincere thanks to Andrès Collinucci,
Daniel Grumiller, Frank Ferrari and Simone Speziale for agreeing to be my
examiners for the final step of this Ph.D. adventure, and for the precious
time they will dedicate to reading this document.

On the scientific level, I thank Abhay Ashtekar, Luca Ciambelli, Laura Donnay,
Laurent Freidel, Marc Geiller, Marc Henneaux, Yannick Herfray, Stefan
Hollands, Yegor Korovin, Charles Marteau, Prahar Mitra, Kévin Nguyen, Blagoje
Oblak, Roberto Oliveri, Sabrina Pasterski, Daniele Pranzetti, Simone Speziale,
Cédric Troessaert, Amitabh Virmani and Céline Zwikel for useful discussions
and correspondence during my Ph.D. (apologies if I unintentionally forgot
someone). I would also like to thank all the members of my research group for
the stimulating dynamic they created around me.

Looking towards the past, I would also like to pay tribute to Didier Salmon,
my first physics teacher (Athénée Royal de Gembloux, Belgium), who shared with
me a passion for mathematical rigor as well as the wonder of scientific
discovery. I also thank my teachers at the University of Namur (Belgium) for
providing me the necessary push towards my final choice when I was undecided
between an artistic and a scientific career. I feel especially indebted to
André Füzfa for having ignited in me the spark of love for General Relativity.

On the private level, I would really like to thank my friends for their
unwavering support and for making my life more exciting. My thanks goes
particularly to Thomas Evrard and Marvyn Gulina for our unbreakable friendship
which has lasted for 10 years now and to Antoine Pasternak for countless
fantastic evenings spent between science and entertainment, in Brussels or
abroad, that will remain among the best memories of my youth.

Finally, I would like to express my infinite gratitude to my parents, who are
undoubtedly the determining agents of my success. From the beginning, with
love, attention and benevolence, they made every effort to provide a peaceful
and comfortable work space for me to evolve and progress. They also
demonstrated a boundless patience facing my mood swings, my passionate
outbursts and sometimes difficult moments. They served as solid examples of
upright, courageous and respectable people, instilling in my mind
determination and refusal of failure that characterizes quite well my
personality. That marks a path that I am committed to follow during my whole
life.

My work was supported by the FRS-F.N.R.S. Belgium (2017-2021), to which I
address my earnest thanks for giving me the possibility to perform this
exciting research. $\blacksquare$

## Conventions and notations

###### Geometry and units

We choose units such that the speed of light $c=1$ but we keep the Newton-
Cavendish constant $G$ explicit. Time and lengths are therefore measured in SI
units of meters. The spacetime manifold is denoted by the couple
$(\mathscr{M},g_{\mu\nu})$. The signature of the Lorentzian metric
$g_{\mu\nu}$ obeys the mostly plus convention $(-,+,+,\dots)$. Bulk
hypersurfaces and boundaries in $\mathscr{M}$ are denoted by curly letters
$\mathscr{B},\mathscr{I},\mathscr{U}$ etc. The spacetime dimension is
$n\in\mathbb{N}_{0}$ ($n>1$). When a natural foliation is defined in the bulk,
we write explicitly $n=d+1$ where $d$ represents the number of transverse
dimensions with respect to the foliation.

###### Indices

The notation of spacetime coordinates is as follows. Greek indices
$\mu,\nu,\dots$ span the full dimension of spacetime $n=d+1$, so
$\mu\in\\{0,\dots,d\\}$. The index $0$ represents the coordinate along the
direction of foliation if present. Latin indices label the transverse
coordinates $x^{a}$ with $a\in\\{1,\dots,d\\}$. Capital Latin letters are used
to denote compact (angular) coordinates $x^{A}$ among the transverse
coordinates. We use the Einstein convention for summation: repeated indices
are supposed to be summed over their full range.

###### Affine connection and curvature

Concerning the definition of geometrical objects used in General Relativity,
we follow the conventions adopted in the book Gravitation [1] by Wheeler,
Thorne and Misner. In particular, the Riemann-Christoffel tensor is determined
in terms of the Christoffel symbols ${\Gamma^{\mu}}_{\nu\rho}$ as

$R^{\mu}_{\phantom{\mu}\nu\alpha\beta}=\partial_{\alpha}\Gamma^{\mu}_{\phantom{\mu}\nu\beta}-\partial_{\beta}\Gamma^{\mu}_{\phantom{\mu}\nu\alpha}+\Gamma^{\mu}_{\phantom{\mu}\kappa\alpha}\Gamma^{\kappa}_{\phantom{\kappa}\nu\beta}-\Gamma^{\mu}_{\phantom{\mu}\kappa\beta}\Gamma^{\kappa}_{\phantom{\kappa}\nu\alpha}.$

In this convention, the $n$-sphere has positive Ricci scalar curvature,
$R=R^{\alpha}_{\phantom{\alpha}\alpha}$, where the Ricci tensor is
$R^{\alpha}_{\phantom{\alpha}\mu\alpha\nu}$. Accordingly, the Ricci curvature
for vacuum configurations has the same sign as the cosmological constant, i.e.
$R>0$ for the de Sitter spacetime and $R<0$ for the anti-de Sitter spacetime.
The Levi-Civita connection with respect to the bulk geometry on $\mathscr{M}$
is denoted by $\nabla_{\mu}$, while the induced connection on codimension 1
hypersurfaces is written $\mathcal{D}_{a}$. Additionally, the Levi-Civita
connection with respect to the boundary geometry and its restriction to
(codimension 2) angular coordinates will be denoted by $D_{a}$ and $D_{A}$
respectively.

###### Differential forms

In some coordinate patch $\\{x^{\mu}\\}$, we define the basis of $(n-k)$
differential forms (also codimension $k$ forms) as

$(\text{d}^{n-k}x)_{\mu_{1}\dots\mu_{k}}=\frac{1}{k!(n-k)!}\varepsilon_{\mu_{1}\dots\mu_{k}\nu_{1}\dots\nu_{n-k}}\text{d}x^{\nu_{1}}\wedge\cdots\wedge\text{d}x^{\nu_{n-k}},$

where $\varepsilon_{\mu_{1}\dots\mu_{n}}$ denotes the (numerically invariant)
Levi-Civita symbol in $n$ dimensions. The $(n-k)$ differential forms on
$\mathscr{M}$ are denoted by boldface symbols

$\bm{A}=A^{\mu_{1}\dots\mu_{k}}(\text{d}^{n-k}x)_{\mu_{1}\dots\mu_{k}}\in\Omega^{n-k}(\mathscr{M}),$

where $\Omega^{p}(\mathscr{M})$ represents the space of $p$-forms on
$\mathscr{M}$. The convention for exterior derivative is
$\text{d}=\text{d}x^{\sigma}\partial_{\sigma}$, which gives

$\text{d}\bm{A}=\partial_{\sigma}A^{\mu_{1}\dots\mu_{k-1}\sigma}(\text{d}^{n-k+1}x)_{\mu_{1}\dots\mu_{k}-1}.$

The volume form is $\sqrt{|g|}(\text{d}^{n}x)$. If $r,t$ are two coordinates
on $\mathscr{M}$, the integration measure on constant $r$ and constant $r,t$
hypersurfaces are given respectively by
$(\text{d}^{n-1}x)_{r}\equiv(\text{d}^{n-1}x)$ and
$(\text{d}^{n-2}x)_{rt}\equiv(\text{d}^{n-2}x)$ where the subscripts are
dropped for the sake of conciseness if the submanifold on which we integrate
is clearly specified.

###### Covariant phase space formalism

We use the conventions of [2] regarding the formal tools of the covariant
phase space formalism. Unless otherwise stated, $\phi=(\phi^{i})$ represents a
collection of fields and background structures needed to define the theory
under consideration (with arbitrary index structure summarized by the compact
index $i$ including tensorial index structure as well as labels of the various
fields). $S[\phi]$ denotes the action integral of the theory, built from the
Lagrangian top-form $\bm{L}[\phi]$. The variational operator on the phase
space is denoted as $\delta$. Assuming general covariance, a field $\phi^{i}$
is modified under an infinitesimal diffeomorphism generated by the vector
$\xi=\xi^{\mu}\partial_{\mu}$ by the Lie derivative
$\delta_{\xi}\phi^{i}=+\mathcal{L}_{\xi}\phi^{i}$. Finally, the Poisson
bracket is defined as
$\\{H_{\xi_{1}},H_{\xi_{2}}\\}=\delta_{\xi_{2}}H_{\xi_{1}}$ in order to
represent the algebra of symmetry generators with correct signs.
$\blacksquare$

###### Table of contents

  1. Acknowledgements
  2. Conventions and notations
  3. 1 Introduction
    1. 1.1 A brief history of gravitational waves
    2. 1.2 Asymptotic symmetries in General Relativity
      1. 1.2.1 Gauge theories with boundaries
      2. 1.2.2 The BMS group and its extensions
      3. 1.2.3 Formal tools and covariant phase spaces
      4. 1.2.4 Memory effects
      5. 1.2.5 Quantum aspects
    3. 1.3 Exploring other types of asymptotics
      1. 1.3.1 Positive cosmological constant and universal expansion
      2. 1.3.2 Negative cosmological constant and holography
    4. 1.4 Original contributions
      1. 1.4.1 Super-Lorentz extension of the BMS group
      2. 1.4.2 Radiative phase spaces with cosmological constant
    5. 1.5 Reading guide and plan of the thesis
  4. 2 Covariant phase spaces in General Relativity
    1. 2.1 Notion of asymptotic symmetries
      1. 2.1.1 Preliminary definitions
        1. 2.1.1.1 Gauge symmetries
        2. 2.1.1.2 Gauge fixing conditions
        3. 2.1.1.3 Residual gauge transformations
      2. 2.1.2 Solution space and boundary conditions
        1. 2.1.2.1 The art of fixing boundary conditions
        2. 2.1.2.2 Solution space and equations of motion
      3. 2.1.3 Variational principle
        1. 2.1.3.1 Conservative boundary conditions
        2. 2.1.3.2 Leaky boundary conditions and open action principles
      4. 2.1.4 Asymptotic symmetries
        1. 2.1.4.1 General definition
        2. 2.1.4.2 Asymptotic symmetry algebra
        3. 2.1.4.3 Representation on the solution space
      5. 2.1.5 Application to asymptotically locally flat spacetimes at null infinity
        1. 2.1.5.1 Boundary conditions
        2. 2.1.5.2 Solution space in the Bondi gauge
        3. 2.1.5.3 Asymptotic symmetries and the BMS4 group
    2. 2.2 Surface charges
      1. 2.2.1 The puzzle to define canonical charges for gauge theories
        1. 2.2.1.1 Noether’s first theorem and trivial currents
        2. 2.2.1.2 Lower degree conservation laws
        3. 2.2.1.3 Surface charges in generally covariant theories
      2. 2.2.2 Covariant phase space formalism
        1. 2.2.2.1 Hamiltonian mechanics
        2. 2.2.2.2 Field fibration and jet bundle
        3. 2.2.2.3 Presymplectic structure
        4. 2.2.2.4 Noether’s second theorem: an important lemma
        5. 2.2.2.5 Fundamental theorem of covariant phase space formalism
        6. 2.2.2.6 Some residual ambiguities
      3. 2.2.3 Iyer-Wald surface charges
        1. 2.2.3.1 Definition of the charges
        2. 2.2.3.2 Conservation of the charges
        3. 2.2.3.3 Integrability of the charges
        4. 2.2.3.4 Physical content of the presymplectic flux
        5. 2.2.3.5 Expressions in General Relativity
        6. 2.2.3.6 Link with the Barnich-Brandt formalism
      4. 2.2.4 Asymptotically locally flat radiative phase spaces
        1. 2.2.4.1 Variational principle and renormalization
        2. 2.2.4.2 Symplectic structure and charges
    3. 2.3 Surface charge algebra
      1. 2.3.1 Integrable case: the Poisson bracket
      2. 2.3.2 Non-integrable case: the Barnich-Troessaert bracket
        1. 2.3.2.1 Generalized representation theorem
        2. 2.3.2.2 Field-dependent 2-cocycle
      3. 2.3.3 The Generalized BMS4 charge algebra
  5. 3 Asymptotically locally (A)dS spacetimes
    1. 3.1 Gravity in Starobinsky/Fefferman-Graham gauge
      1. 3.1.1 Asymptotic properties of (A)dS spacetimes
        1. 3.1.1.1 The de Sitter solution
        2. 3.1.1.2 The anti-de Sitter solution
      2. 3.1.2 Conformally compact manifolds
      3. 3.1.3 Starobinsky/Fefferman-Graham gauge
      4. 3.1.4 Solution space
        1. 3.1.4.1 Organization of the Einstein equations
        2. 3.1.4.2 Trace of metric coefficients
        3. 3.1.4.3 Divergence of metric coefficients
        4. 3.1.4.4 Explicit expressions of metric coefficients
      5. 3.1.5 Residual gauge diffeomorphisms
      6. 3.1.6 Variations of the solution space
    2. 3.2 Holographic renormalization
      1. 3.2.1 Variational principle
        1. 3.2.1.1 Regulated variational principle
        2. 3.2.1.2 Holographic stress-tensor
      2. 3.2.2 Renormalized presymplectic structure
        1. 3.2.2.1 Dynamical fields and background structures
        2. 3.2.2.2 Renormalization of the presymplectic potential
        3. 3.2.2.3 Boundary conditions and the Cauchy problem
    3. 3.3 Charge algebra in asymptotically locally (A)dS spacetimes
      1. 3.3.1 Canonical surface charges
      2. 3.3.2 Charge algebra
        1. 3.3.2.1 Computation of the current algebra
        2. 3.3.2.2 Properties of the charge algebra
    4. 3.4 Application to more restrictive boundary conditions
      1. 3.4.1 Dirichlet boundary conditions
        1. 3.4.1.1 Asymptotic symmetry algebra
        2. 3.4.1.2 Charge algebra
      2. 3.4.2 Neumann boundary conditions
        1. 3.4.2.1 Residual diffeomorphisms
        2. 3.4.2.2 Charge algebra
  6. 4 $\Lambda$-BMS and the flat limit
    1. 4.1 Leaky boundary conditions and $\Lambda$-BMSd+1
      1. 4.1.1 Boundary conditions and dynamical degrees of freedom
      2. 4.1.2 Residual gauge symmetries: the $\Lambda$-BMSd+1 algebroid
      3. 4.1.3 $\Lambda$-BMSd+1 charge algebra
    2. 4.2 Aspects of the $\Lambda$-BMS4 phase space
      1. 4.2.1 Solution space in Bondi gauge
        1. 4.2.1.1 Minimal fall-off requirements
        2. 4.2.1.2 Organization of Einstein’s equations
        3. 4.2.1.3 Solution to the algebraic equations
        4. 4.2.1.4 $\Lambda$-BMS4 boundary gauge fixing
        5. 4.2.1.5 Constraint equations as Bondi evolution equations
      2. 4.2.2 Dictionary between Starobinsky/Fefferman-Graham and Bondi gauges
      3. 4.2.3 Bondi news and Bondi mass in (A)dS4
      4. 4.2.4 $\Lambda$-BMS4 algebroid in Bondi gauge
        1. 4.2.4.1 Bondi gauge parameters and constraints
        2. 4.2.4.2 Explicit solutions and flat limit
        3. 4.2.4.3 $\Lambda$-BMS4 charges and algebra
      5. 4.2.5 Conservative subsectors and stationary solutions
        1. 4.2.5.1 Mixed Neumann conservative boundary conditions
        2. 4.2.5.2 Asymptotic symmetry algebra
        3. 4.2.5.3 Stationary solutions
    3. 4.3 Flat limit of the $\Lambda$-BMS4 phase space
      1. 4.3.1 The philosophy of the flat limit
      2. 4.3.2 Flat limit of the solution space
      3. 4.3.3 Flat limit of the action and corner terms
        1. 4.3.3.1 Corner terms and prescription for renormalization
        2. 4.3.3.2 Corner contributions to the charge algebra
        3. 4.3.3.3 Symplectic structure at flat limit and discussion
  7. 5 Super-Lorentz transformations and Gravitational Memories
    1. 5.1 Vacua of the gravitational field
      1. 5.1.1 Generation of the vacua
      2. 5.1.2 The superboost, superrotation and supertranslation fields
    2. 5.2 Gravitational memory effects
      1. 5.2.1 Axiomatic definition and relation to gauge symmetries
      2. 5.2.2 Linear displacement memory effect
      3. 5.2.3 Superboost transitions
        1. 5.2.3.1 Penrose’s gravitational shockwaves
        2. 5.2.3.2 General impulsive gravitational wave transitions
        3. 5.2.3.3 Evolution of the Bondi mass aspect and the center-of-mass
        4. 5.2.3.4 Refraction/Velocity kick memory
        5. 5.2.3.5 A new non-linear displacement memory
    3. 5.3 Generalized BMS4 finite charges
      1. 5.3.1 Initial and late data in general super-Lorentz frames
      2. 5.3.2 Finite Hamiltonians from the Generalized BMS4 charges
      3. 5.3.3 Flux algebra and realization at spatial infinity
      4. 5.3.4 Infrared structures and soft theorems
        1. 5.3.4.1 Gravitational scattering problem and junction conditions
        2. 5.3.4.2 Leading and subleading soft theorems
        3. 5.3.4.3 Infrared relationships
  8. 6 Conclusion and perspectives
    1. 6.1 Summary of the main results
    2. 6.2 Some future perspectives
  9. Appendices
    1. C Killing vectors of (A)dS4 in Bondi gauge
    2. D Map from Bondi to Starobinsky/Fefferman-Graham gauge
    3. E Useful relations for computations in the Bondi gauge
    4. F Technical material from quadratic curvature gravity

###### List of Figures

  1. 2.1 Asymptotic flatness at $\mathscr{I}^{+}$ in the Bondi gauge.
  2. 2.2 BMS supertranslations.
  3. 2.3 Structure of the jet bundle.
  4. 2.4 Kinematical _vs._ dynamical boundary degrees of freedom (BDoF).
  5. 2.5 Schematic contour for the variational principle.
  6. 3.1 Causal structure of dSd+1 spacetime.
  7. 3.2 Causal structure of AdSd+1 spacetime.
  8. 3.3 Schematic picture of the  Starobinsky/Fefferman-Graham foliation.
  9. 3.4 Conservative _vs._ leaky boundary conditions in Al(A)dS spacetimes.
  10. 3.5 Universal boundary structure  for Dirichlet boundary conditions.
  11. 4.1 Universal boundary structure  for $\Lambda$-BMSd+1 boundary conditions.
  12. 4.2 Energy flow in the bulk and flat limit of (A)dS asymptotics.
  13. 4.3 Geometry of Al(A)dS4 boundaries with background structure.
  14. 5.1 Model of detector for gravitational memories.
  15. 5.2 Displacement memory effect.
  16. 5.3 Penrose’s “scissor and paste” construction.
  17. 5.4 Gravitational shockwave induced by cosmic string snapping.
  18. 5.5 Velocity kick memory effect.
  19. 5.6 Leading infrared triangular relationship.
  20. 5.7 Overleading/Subleading infrared relationship.

##  Chapter 1 Introduction

###  1.1 A brief history of gravitational waves

Einstein’s theory of General Relativity [3] is often considered as one of the
most exceptional achievements of science. Based on a small number of natural
and robust postulates, among which one finds the equivalence principles or the
principle of general covariance, it has initiated the modern way of
geometrizing physics from first principles with the broad success we all
celebrate, for instance in the building of the standard model of particle
physics. Although refractory to quantification and technically very intricate,
Einstein’s paradigm of gravity is still capable of describing with astonishing
precision all observable gravitational phenomena, whether at astrophysical or
cosmological scales. This is true more than 100 years after the theory’s
development from the pure genius of a man guided by powerful intuition and the
observationally-driven Gedankenexperimenten he liked so much. There is no
surprise that General Relativity still inspires extensive research today and
that during the past century, its fantastic predictions have fed the
imagination of many authors and filmmakers, which has also contributed to the
notoriety of the theory among the public. The new century has opened new
routes of active research on General Relativity, as new technology,
experimental equipment and computer performance levels now allow for probing
the deepest abysses of the cosmos and ultimately testing the efficiency and
the limits of the theory. For this reason, the interest of cutting-edge
researchers in General Relativity is more justified than ever.

Among the stunning predictions of General Relativity, one finds the existence
of gravitational waves, which consist of small curvature oscillations around a
given steady solution propagating at the speed of light and are conceivable as
soon as the spacetime itself is endowed with a dynamical structure. Einstein
considered very early linearized perturbations around the Minkowski flat
vacuum spacetime [4]. He proved that gravitational radiation can emerge from
periodic motions of massive bodies, such as binaries orbiting around each
other and presenting at least a quadrupolar component. This is a major
difference with electromagnetic waves that can be excited by a dipole. The
second difference is that gravitational waves are ripples of the spacetime
itself and are thus not stopped by ambient matter, with an obvious crucial
interest in multi-source astronomy. The modes deform circular arrangements of
test masses by quadrupolar oscillations, with two independent polarizations
separated by an angle of 45∘. Einstein also derived the famous quadrupole
formula linking the sources of radiation with the energy flux they are losing
by gravitational emission. As interesting as it was, Einstein doubted his
discovery and the potential for its physical realization, changing several
times his point of view. His hesitation was the origin of a controversy
concerning the status of gravitational waves (see e.g. [5] for a review) :
were they an artifact of linear theory that could be eliminated by a change of
coordinates, or could we observe gravitational waves carrying energy over long
distances in the full nonlinear theory as well? The debate raged for 40 long
years mainly because of the innate difficulty of addressing this question in a
suitable framework to offer a meaningful answer. In particular, the isolation
of the propagating degrees of freedom among the 10 components of the metric
field was a long-standing problem in the absence of a canonical prescription
that General Relativity cannot provide.

It was not until the 60’s that Bondi, Metzner, van der Burg [6] and Sachs [7,
8] finally proposed a sturdy framework to address the question in an
unambiguous and elegant manner. They introduced a set of coordinates
exploiting the fact that gravitational radiation travels along light cones
together with a set of boundary conditions assuming that the sources of
emission are localized in the bulk of spacetime. The formalism provided a
parametrization of the metric field and a well-controlled expansion in the
asymptotic zone, far from the emitting sources, in which the separation
between kinematical and radiative degrees of freedom could be explicitly
achieved. In particular, they were capable to derive the Bondi mass loss
formula, proving the existence of gravitational waves in the fully non-linear
theory. They showed that some fields in their parametrization are responsible
for a net flux of energy propagating at very large distances and identified
them as encoded in what is broadly known as the Bondi news tensor. Fall-offs
of the components of the Weyl tensor in the presence of radiation (peeling)
were worked out by Sachs [7, 8], providing constraints to firmly and
completely describe the gravitational field as observed remotely from the
source. Shortly after, the construction was confirmed by Penrose, who
elaborated conformal techniques to model relativistic infinities in a fully
covariant manner [9], leading to the characterization of fundamental
properties of null asymptotes, the conformal compactness of the spacetime or
the asymptotic flatness at null infinity heuristically theorized by Bondi and
coworkers.

Although these arguments finally ended the controversy among theoretical
physicists, there was not yet any experimental evidence available. After
Weber’s unsuccessful attempts in 1969 [10], the first indirect experimental
clue in favor of the existence of gravitational waves came from the
observation of the binary pulsar PSR B1913+16 by Hulse and Taylor in 1974
[11]. After monitoring for several years radio pulses emitted by the system
and following the evolution of its orbital parameters, they could deduce the
loss of energy causing the inspiral motion of the system and established a
perfect match with the prediction of General Relativity. Their discovery was
awarded the Nobel Price of Physics in 1993. During the subsequent decades,
gravitational radiation physics in the asymptotically flat context has
benefited from strong interest. In the wake of Penrose, purely geometrical
tools were developed – namely by Geroch, Ashtekar and collaborators [12, 13,
14] – in order to discuss the asymptotic structure at null infinity and the
associated radiative data through coordinate independent objects. New
frameworks have been successfully proposed to complement the Bondi-Sachs
formalism and have proven to be more convenient for studying gravitational
wave generation as well as for applications to the data analysis of possible
experimental events. For instance, in order to find a connection between the
material sources generating the waves and the asymptotic structure of the
radiative gravitational field, Blanchet and Damour developed the multipolar
post-Minkowskian formalism [15, 16, 17, 18, 19] combining the multipole
expansion for the field in the near zone around the source, naturally defined
for the linearized theory, with an expansion of the gravitational field in
powers of the gravitational constant, perturbatively encapsulating the
intrinsic non-linearity of gravitational dynamics. At linear order, it reduces
to the solution presented by Thorne in 1980 [20] as the general solution of
the linearized Einstein equations in terms of gravitational multipoles, for
which Einstein’s quadrupolar result is the leading order approximation.
Despite constant efforts in the field and outstanding theoretical
improvements, the experimental evidence is long overdue.

The main issue encountered by experimenters is the notable weakness of the
gravitational interaction compared to the other three fundamental forces
(about 40 orders of magnitude weaker). As a result, the supposed
characteristic wave length of the gravitational radiation makes detection by
light interferometry impossible without the use of gigantic experimental
devices spanning over kilometers. In September 2015, nearly one century after
Einstein’s prediction, gravitational waves were finally declared to be
directly detected by LIGO (Laser Interferometer Gravitational-Wave
Observatory, U.S.A.) [21] while observing the inspiral and merging of two
black holes orbiting at 1,3 billion light-years from Earth. Officially
confirmed in February 2016, this spectacular observation earned Thorne, Weiss
and Barish the 2017 Nobel Prize for their previous works on gravitational
waves as well as their involvement in the LIGO project and also paved the way
for gravitational astronomy. The technological endeavor is now focused on
improving the accuracy of measurements. For instance, space probe
interferometers currently in project such as eLISA (Laser Interferometer Space
Antenna, Europa) [22], which is scheduled for 2032 and designed to span over
2,5 million kilometers in the interstellar vacuum, should give us access to
more precise acquisitions free of earthquake noise.

###  1.2 Asymptotic symmetries in General Relativity

#### 1.2.1 Gauge theories with boundaries

The study of dynamics at null infinity initiated by Bondi and coworkers has
raised many questions and mobilized constant efforts over decades, at the
junction of geometrical formalization and physical intuitions. One central
question regards the symmetries of General Relativity in the presence of
boundaries and the conserved charges associated with them, in the spirit of
Noether’s theorem [23]. From the beginning, Einstein’s theory was meant to be
invariant under arbitrary changes of coordinates (diffeomorphisms), in
accordance with the covariance principle establishing the equivalence of all
observers. Believed to hold for any theory of gravitational interaction, this
statement can be rephrased as gravity being a gauge theory under the gauge
group of diffeomorphic automorphisms over the spacetime manifold. Two
solutions for Einstein’s field equations that are related by diffeomorphism
are physically indistinguishable, as they are just two equivalent
interpretations of the same physical system described by two different
observers and the diffeomorphism shall carry no bulk charge. However, when
analyzing the asymptotic behavior of the gravitational field around a
boundary, be it repelled to infinity, the situation drastically changes.
Indeed, when a boundary is present, the general covariance is explicitly
broken by the choice of a particular class of observers that all agree with
the position of the boundary and the assorted set of boundary conditions for
the dynamical fields (gravity or matter) under consideration or even live on
it. For instance, when the arms of LIGO are measuring gravitational radiation,
they lay at future null infinity of a modelized asymptotically flat spacetime
where the sources of radiation emit from deep bulk regions. By the fixation of
a boundary and boundary conditions describing the behavior of the physical
fields when approaching the boundary, a class of gauge transformations play a
physical role as soon as they acquire a non-vanishing charge. While a majority
of transformations still describe a pure redundancy of the theory with zero
charge, the charged symmetries among the residual gauge transformations
preserving the structure around the boundary become physical symmetries of the
theory, and go under the name of asymptotic symmetries.

#### 1.2.2 The BMS group and its extensions

In the context of radiative Einstein’s gravity in four dimensions, the second
important outcome of the seminal work of Bondi, Metzner, Sachs and van der
Burg [6, 7, 8] was the derivation of the asymptotic symmetry group of
asymptotically flat spacetimes at null infinity. Far from the sources of
gravitation, as the spacetime geometry decays back to flat space, one could
naively expect to find only the exact isometries of Minkowski space – the
Poincaré group. Bondi and coworkers demonstrated that the structure of the
asymptotic group was richer than Poincaré, enhancing the four translations of
the latter into an infinite-dimensional normal subgroup of supertranslations
(i.e. angular-dependent translations) and earning the name of “BMS group.” In
fact, the inclusion of supertranslations was unavoidable to encompass the
interesting sector of radiative solutions. Indeed, any reduction to Poincaré
by freezing the pure supertranslations amounts to canceling the radiative
boundary degrees of freedom [14]. This result contrasts with previous analyses
by Arnowitt, Deser and Misner at spatial infinity [24], where Poincaré
invariances were necessary and sufficient to define total (ADM) mass and
angular momentum of spacetime. The BMS group enjoyed a constant interest in
the subsequent decades, unveiling its algebraic properties in a fully
covariant language [25, 26, 27, 28, 12] and discussing associated radiative
phase space and fluxes of charges (see e.g. [14, 29, 30, 31, 32]). Thereafter,
BMS symmetries proved their relevance in many other physical contexts,
distinct from future null infinity, such as at spatial infinity of
asymptotically flat spacetimes [33, 34, 35, 36, 37], in three-dimensional
gravity [38, 39, 40, 41] and at the black hole horizon [42, 43, 44].

As the content of the asymptotic symmetry group is obviously constrained by
the precise boundary conditions under consideration, it can be further
enlarged by weakening the definition of asymptotic flatness. The natural
extension of the BMS group lies in enhancing the Lorentz factor into an
infinite-dimensional bunch of super-Lorentz transformations (in the
terminology of [45]). Barnich and Troessaert recently introduced such an
extension into the infinite-dimensional group of conformal transformations on
the two dimensional sphere at infinity [46, 47, 30, 48], which they called
“superrotations”. Their primary motivation was an interest in two dimensional
conformal quantum field theories [49] for holographic perspectives like in
asymptotically AdS3 gravity [50] (see section 1.3.2) and resulted a bit later
in the celestial sphere holography program [51, 52, 53, 54, 55, 56, 57] in
asymptotically flat spacetimes. As generic holomorphic functions of the
complex stereographic coordinates on the sphere, Barnich-Troessaert’s
superrotations are singular beyond the Lorentz transformations which are the
six globally-defined Laurent modes. They introduce poles on the sphere that
destroy the common sense of asymptotic flatness on a countable set of points;
by consistency of the BMS algebra, the supertranslations become singular as
well. Discarding these singularities requires focusing only on local
properties of conformal isometries or trading the sphere for the cylinder as
topology of the null generators of null infinity.

Campiglia and Laddha later proposed another definition of super-Lorentz
extension by allowing the whole class of smooth diffeomorphisms on the
celestial sphere as part of the asymptotic symmetry group [58, 59, 45, 60],
leading to the Generalized BMS group. The motivation for this wark came from
infrared properties of quantum scattering amplitudes involving low-energy
gravitons (see section 1.2.5). Here the smooth super-Lorentz modify the
leading-order structure at null infinity, which requires a firm relaxing of
the definition of asymptotic flatness. The more general boundary conditions
have to include solutions that are no longer strictly asymptotically Minkowski
but with a fluctuating boundary metric, allowing for the choice of any
arbitrary frame on the celestial sphere. We comprehensively discuss this
extension later in the manuscript.

#### 1.2.3 Formal tools and covariant phase spaces

At the theoretical level, the determination of produced and flowing amounts of
energy in the presence of gravitational radiation poses an intricate problem.
As a diffeomorphism covariant theory, and in accordance with the equivalence
principle, there is no notion of local stress-energy for the gravitational
field, because the latter can always be locally suppressed by a change of
coordinates. For instance, the Hamiltonian associated with the gravitational
field is a pure boundary term and vanishes on-shell. The definition of
conserved charges by the usual Noetherian procedure is thus doomed to failure
in General Relativity, even in the presence of exact isometries, as we review
in the main text. Due to the gauged nature of the invariance under
diffeomorphisms, non-trivial gravitational charges are codimension two
(“surface”) objects living on the boundary of the spacetime volume whose
energy, linear and angular momenta, for instance, have to be evaluated. For
observers far away from the sources of gravitation, the useful boundaries are
repelled towards conformal infinity and the desired dynamical quantities are
canonically conjugated with asymptotic symmetries.

At spatial infinity, the Arnowitt-Deser-Misner (ADM) Hamiltonian analysis
allows for computing the total Poincaré charges of any asymptotically flat
spacetime [24]. With suitable asymptotically flat boundary conditions and
assuming a parity condition on the fields, Regge and Teitelboim prescribed the
boundary terms to supply to the gravitational Hamiltonian that reproduce the
Poincaré charges at spatial infinity [61]. These Hamiltonian charges are
integrable on the solution space and conserved in time because no radiation
can escape through spatial infinity. They form an algebra of phase space
functionals representing the diffeomorphism algebra of asymptotic symmetries
up to an eventual central extension, as shown by Brown and Henneaux in [50].
The result was immediately verified in three-dimensional [62] and higher-
dimensional [63, 64] Einstein gravity with negative cosmological constant.
Boundary structure at the spatial infinity of asymptotically flat spacetimes
was further discussed in [65, 66, 67, 68, 69, 13, 70, 71, 72], culminating
with the recent realization of enhancement of Poincaré to BMS symmetry also at
spatial infinity [33, 34, 35, 36].

Things change substantially when discussing an equivalent notion of “conserved
charges” from the perspective of null infinity, mainly because energy fluxes
cannot be discarded and the evolution parameter along the boundary is a null
coordinate. The system genuinely exchanges energy with its environment when
radiative modes are turned on, causing potential non-integrability, but surely
non-conservation, of the surface charges. As previously noted, the first
promising attempts to quantify the radiated flux of energy carried by
gravitational radiation crossing null infinity date back to Trautman [73] and
Bondi [6] analyses. The definitions of the so-called Bondi mass and momenta
ensure that they decay to the ADM quantities near spatial infinity and that
the associated fluxes provide a negative balance from the bulk point of view,
asserting that energy is escaping through future null infinity while the Bondi
mass decreases in (retarded) time, as it should according to elementary
physical intuition. Ashtekar and Steubel [14] later observed that the BMS
transformations act as canonical transformations on the exact radiative modes
in the full non-linear theory characterized covariantly and intrinsically at
null infinity [13, 74]. The generating Hamiltonians are the integrated fluxes
of energy on the whole boundary, which offered no information about local
charges defined on constant-time sections of null infinity until Dray and
Streubel [29] introduced a definition of BMS momenta and supermomenta (for
proper supertranslations) whose fluxes agree with [14].

With the aim of deducing the expression of local surface charges at null
infinity, covariant phase space methods – initiated by [75, 76, 68] – were
invoked and refined over the years. The formalism consists of a smart fusion
between Hamiltonian and Lagrangian methods, incorporating the powerful
features of phase space analyses into a flexible language that does not
abandon covariance. A considerable research effort has been pushed in this
direction by Wald and coworkers in [77, 78, 79]. The construction of surface
charges at the boundary of any Lagrangian covariant theory of fields was then
routed through an algorithm. The latter is inspired by the symplectic methods
for Hamiltonian classical mechanics, and provides as a first step the
symplectic structure to be equipped on the solution space from the variational
principle. The charges are then computed as a contraction of this symplectic
structure with a gauge transformation. Since the integrability of the charges
is no longer guaranteed in such a non-equilibrium physics experienced at null
infinity, the definition of a Hamiltonian is ambiguous until a clear
separation has been made between physical charges and physical fluxes. Wald
and Zoupas introduced a prescription to fix this ambiguity by demanding that
the flux of suitable charges is zero on stationary configurations in the
absence of radiation [31].

Another definition for the surface charges was developed in parallel by
Barnich and Brandt in [80, 81], but relying on the equations of motion rather
than the variational principle. BRST techniques are at the core of this second
definition, mainly motivated by cohomogical approaches of asymptotic
symmetries [82] discussed with Henneaux in [83, 84]. Free of the various
ambiguities in boundary terms arising from Wald’s definition, Barnich and
Brandt’s prescription fixes the local charges as soon as the dynamical
equations for the fields are given, and is applicable to a more general class
of gauge theories than the only generally covariant theories. The non-linear
theory for exact and asymptotic symmetries in this framework was developed in
[2, 85].

Echoing the representation theorem obtained in Hamiltonian formalism, a
natural question to ask was, are the BMS symmetries represented at the level
of the charges? Barnich and Troessaert [30] observed that the presence of non-
vanishing physical flux through null infinity represents an obstruction for
the BMS surface charges to close an algebra under the usual covariant Poisson
bracket [86]. They introduced a new definition of the charge bracket that
includes a supplementary term vanishing in the absence of Bondi news and
proved that this extension was sufficient to obtain an avatar of the
representation theorem for BMS charges, up to a field-dependent “central
extension.” Despite many checks in various contexts (see e.g. [32, 37, 87])
and theoretical progress (see e.g. [88, 89, 90, 91, 92, 93]), the right
framework into which a general proof of this generalized representation
theorem for open gauged systems could be formulated is still missing and
mobilizes some current research efforts [94].

As the title of the thesis suggests, we review and discuss the advances and
difficulties related to this problem later in the manuscript. The following
sub-sections return to physics.

#### 1.2.4 Memory effects

An important aspect of asymptotic symmetries is that they govern the
kinematics of physically measurable effects recast as transitions among
solutions in the phase space. In General Relativity for instance, the passage
of gravitational radiation causes a permanent shift in the relative angular
positions of a detector comprised of a pair of free-falling observers in the
asymptotic region. Since the geometry keeps track of the radiative phenomenon
after its extinction, this constitutes a gravitational memory effect. Before
and after the non-stationary phase, the gravitational field is supposed to be
in a steady vacuum configuration, but the radiation has driven a vacuum
transition from the initial vacuum to another inequivalent one. The closure of
the solution space under the action of the asymptotic symmetry group implies
that both vacua are diffeomorphic to each other, and the permanent shift
experienced by the inertial observers defined above is precisely controlled by
the amplitude of the involved asymptotic symmetry. In the context of
asymptotically flat spacetimes in four dimensions, the displacement memory
effect [15, 95, 96, 97, 16] induced by null matter or gravitational radiation
reaching null infinity is understood as a vacuum transition driven by a proper
supertranslation [98, 99]. At the dynamical level, the flux-balance law for
the BMS supertranslation charge offers the key to relate the amplitude of the
total displacement memory effect and the strain of radiation among other
observable parameters (such as the orbital parameters of the black hole merger
producing the waves).

Many other memory effects have been identified in four-dimensional General
Relativity, such as the spin memory effect [100], concerning the phase shift
experienced by photonic loops under the action of gravitational waves; the
center-of-mass memory effect [101], modifying the angular momentum of
spacetime in the center-of-mass frame; the velocity kick memory effect [102,
103], predicting a cumulative geodesic deviation and finally the black hole
memory effect [104], which is the dynamical response of a black hole horizon
when subjected to some incoming gravitational burst. In comparison to the
ripples recorded by LIGO for typical black hole mergers, these memory effects
appear as subleading deviations in the inspiral-merging curve profiles and are
currently beyond the experimental tolerance of detectors. The hope is that
these limitations will soon be lifted by the use of space probe
interferometers. The rich panel of gravitational memories, even for higher
dimensional configurations [105], has inspired a reinforced interest to
discuss similar effects in other gauge theories, namely in Maxwell [106, 107,
108, 109] and Yang-Mills theories [110, 111].

#### 1.2.5 Quantum aspects

The exploration of the structure of asymptotic symmetry groups in General
Relativity and the algebraic properties of the associated conserved charges is
also crucial to address the problem of building a quantum theory of gravity.
Without even evoking very precise motivations like in holography that we later
discuss (see sub-section 1.3.2), the existence of rich symmetry structures in
a classical theory and the control of their representations tells us a lot
about how the Hilbert space of quantum states should be organized in the
quantized theory. In particular, the charges become observables whose
canonical commutation with the fields generate the symmetry at the quantum
level. This is particularly significant for gravity, for which the quantum
theory is still beyond our reach. The key point is that asymptotic symmetries
provide a clear and robust anchorage for progress towards the quantification
while lacking the full non-linear quantum theory.

An important research program started by Strominger in 2013 set out to study
the quantum aspects of the low-energy regime of gravity. The program sought to
discuss the scattering problem of gravitational waves in asymptotically flat
spacetimes at both past and future null infinity, where two different BMS
groups seem to act independently in the naive picture. The scattering process
is well-defined if one selects the diagonal subgroup of BMS transformations
antipodally identified near spatial infinity [112]. This yields an infinite
tower of conservation laws for the supermomenta whose quantum versions state
nothing but the invariance of the gravitational S-matrix under the BMS
transformations. The Ward identity associated with supertranslations is then
equivalent to the Weinberg soft graviton theorem [113], which is an old result
of pure quantum field theory [114], according to which the contributions of
soft particles (i.e. massless particles with nearly zero momentum) are
universal and factorize in quantum scattering amplitudes. Interestingly,
performing a Fourier transform on the universal soft factor appearing in this
result reproduces the metric deviation involved in the kinematics of the
displacement memory effect [98].

The presence of the supertranslation symmetry at the quantum level in the
linearized theory sheds new light on the infrared (low-energy) structure of
gravity, organized as a triangle whose vertices bear the symmetry itself, the
associated Ward identity in the form of Weinberg’s theorem and the conjugated
dynamical process that is the displacement memory effect [57]. It is
remarkable that this tripartite structure has also been subsequently uncovered
for other gauge theories, such as electrodynamics [115, 116] and
chromodynamics [110]. In gravity, the enhancement of the BMS group has led to
subleading connections involving super-Lorentz transformations [46, 47, 48,
58, 59, 45] whose Ward identity was shown to be equivalent to a subleading
version of the soft gravitation theorem [117, 56, 118], and assorted with new
memory effects such as the spin memory effect [100] or the center-of-mass
memory effect [101]. Similar discussions for Maxwell’s theory were again quick
to follow [119, 120, 115].

In any physical process, one expects to observe a certain amount of emitted
soft gravitons. In particular this emission could occur during the quantum
evaporation of a black hole, as predicted by Hawking’s seminal computation
[121] leading to the famous black hole information paradox [122], which
results from the deterioration of the pure states inside the black hole into
thermal mixed outgoing Hawking radiation. The counting and matching of degrees
of freedom in the various attempts to solve this paradox must take into
account the information encoded in the emitted soft modes, and the infrared
triangular structure is very helpful in that task. Indeed, thanks to the
correspondence between soft gravitons and BMS invariances, black holes are
classically dressed with BMS charges (“soft hairs”), whose associated flux-
balance laws at the horizon may be of crucial importance in the flow of
quantum information [43, 44, 123] (see also [124, 125]).

###  1.3 Exploring other types of asymptotics

#### 1.3.1 Positive cosmological constant and universal expansion

The program started by the pioneer works of Bondi and coworkers focuses its
attention on asymptotically flat spacetimes. However, during the course of the
past century, we discovered strong evidence that our Universe experiences a
phase of accelerated expansion, understood in cosmology as the contribution of
a small, but non-vanishing positive cosmological constant $\Lambda$ in
Einstein’s equations [126, 127, 128]. $\Lambda$ is evaluated around
$10^{-52}m^{-2}$ according to the final results of the Planck probe mission
[129]. Again assuming the localization of the sources of gravity (stars,
planets, black holes, interstellar dust etc.) deep in the bulk, the geometry
of the Universe is shown to asymptote to the de Sitter spacetime [130, 131,
132], which is the equivalent of the Minkowski vacuum when $\Lambda>0$. The
class of radiative asymptotically de Sitter spacetimes is surely worth
studying as soon as it would be possible to perform experimental detections of
gravitational waves emitted by dynamical sources separated from Earth by
cosmological-scale distances.

Physics in de Sitter presents notable peculiarities in comparison of the well-
known flat case. Observers in the asymptotic region of de Sitter have no
access to the entire spacetime because of the constantly accelerated
cosmological expansion: for each observer, a (past) cosmological horizon
exists, limiting the description of the physics to a half-causal wedge. In the
flat space, in contrast, any asymptotic Bondi observer has access to the whole
history of the spacetime up to its current (retarded) time. Moreover, the
(future) conformal boundary where null rays come to an end is now a timelike
surface, drawn as a horizontal line segment in the conformal diagram. There is
no chronology provided on the boundary; therefore, the initial value problem
for boundary data as well as the discussion of the loss of energy carried out
by gravitational radiation crossing the boundary are more intricate. For that
reason, one cannot distinguish between translations and rotations among the
diffeomorphisms acting on the boundary as is possible for the BMS symmetries
in the flat case. This explains why the de Sitter case is less covered in the
literature and why any step forward is tricky and time-consuming.

A renewal of interest around de Sitter asymptotics has been observed during
the two last decades, mainly due to the success of holography in the presence
of negative cosmological constant (see sub-section 1.3.2). Several attempts
have been made to define boundary conditions at the future conformal boundary,
including [133, 134, 135]. However, as recently emphasized in [136, 137, 135],
that enterprise comes with a strong drawback: imposing future boundary
conditions amounts to restricting the initial data backwards in time and
therefore the bulk dynamics. Allowing generic initial data and, in particular,
generic bulk gravitational waves prevents the imposition of any boundary
conditions at the future boundary of de Sitter. An interesting question is
then how to extend the notions of the radiative data present in Bondi
formalism to asymptotically locally de Sitter spacetimes (see e.g. [138, 139,
140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152]). A
considerable step forward in the understanding of gravitational radiation in
the presence of a cosmological constant has been made very recently by
Ashtekar and coworkers in a series of papers discussing the extension of the
fully-geometrical asymptotic treatment to de Sitter [136], inventorying the
properties of linearized gravitational waves near the future conformal
boundary while precisely quantifying the information lost by imposing further
boundary conditions [137] and deriving a generalization of Einstein’s
quadrupole formula in this new context [153]. The cosmological version of the
displacement memory was also unveiled afterwards [154]. The Hamiltonian phase
space for gravity in de Sitter has also been worked out [155] in the spirit of
Regge-Teitelboim’s analysis in flat space [61] (see also [33, 34]). One of the
main goals of this thesis is to take advantage of this impetus to repeat the
Bondi analysis for asymptotically de Sitter spacetimes and to discuss the
relation with the well-known flat case through a flat limit process. The
construction of radiative phase spaces in the presence of a cosmological
constant is appealing and mainly motivated by uncovering the realization of a
BMS-like symmetry and defining the right notion of Bondi news in de Sitter.

#### 1.3.2 Negative cosmological constant and holography

In contrast to the de Sitter case, Einstein’s gravity with negative
cosmological constant has enjoyed stimulating research, mainly because of its
crucial role in the holographic principle [156, 157].

The paradigm of holography is a promising road to understand quantum gravity.
It is based on the hope that a theory of quantum gravity can be described in
terms of a lower-dimensional dual quantum field theory. Even if this statement
has not been proven yet in general, evidence in favor of holography has been
accumulated during the last 20 years in the context of the AdS/CFT
correspondence [158], according to which any asymptotically anti-de Sitter
solution of gravity is dual to a conformal field theory living on the
conformal codimension one boundary of the spacetime. The holographic
dictionary provides a translation guide between gravitation and fields living
in the bulk of spacetime and the dynamics of dual boundary quantum fields. At
the level of the symmetries, the asymptotic symmetries of the bulk theory are
dual to the global symmetries acting on the ultraviolet modes of the boundary
theory, and the study of asymptotic symmetries on the gravity side of the
duality has been shown to be extremely efficient in extracting some patterns
of the holographic correspondence.

One of the crucial properties of asymptotically anti-de Sitter solutions that
make them the natural breeding ground for holographic investigations is that
their conformal boundaries look like timelike infinite cylinders, on which a
notion of chronology can be defined. This feature contrasts with the de Sitter
case, where no such boundary time direction even exists, and the flat case for
which the asymptotic boundary is a null manifold. However, from the point of
view of the bulk theory, this peculiar feature allows the null rays to reach
the conformal boundary after a finite amount of coordinate time: a well-posed
initial value problem requires to supplement the data on any Cauchy slice from
which the null motion originates by a set of boundary conditions predicting
the fate of the null rays after they encountered the conformal boundary.
Satisfactory boundary conditions for holographic purposes are Dirichlet
boundary conditions that aim at reflecting the null rays back towards the
bulk. This causally links the interior of the spacetime with its boundary and
ensures the unitarity of the quantum dual theory. In bulk dimensions greater
than three, Dirichlet boundary conditions reduce the asymptotic symmetry group
to the lower-dimensional conformal group, which is the (finite-dimensional)
isometry group of vacuum anti-de Sitter solution [159, 160, 64, 63, 161]. In
three dimensions, the asymptotic symmetry algebra under Dirichlet boundary
conditions is infinite-dimensional: this is the conformal group in two
dimensions consisting of a double copy of the Virasoro algebra. The associated
charge algebra was shown to represent this double Virasoro symmetry with the
famous Brown-Henneaux central extension [62]. The latter provided a first
encouraging hint leading to the holographic conjecture. Indeed, this central
extension was later related to the conformal anomaly of the dual conformal
field theory [162]. Furthermore, it was used to reproduce the BTZ black hole
entropy using the Cardy formula [163]. Many other boundary conditions in
asymptotically anti-de Sitter spacetimes have been proposed in the literature
[164, 165, 166, 167, 168, 169, 170], leading to different asymptotic
symmetries and holographic dualities. As in the flat case, the analysis of
conserved charges associated with the asymptotic symmetries in AdS has
attracted considerable attention (see e.g. [160, 161, 70, 171, 172, 173, 174,
139, 64, 62]).

Nevertheless, the motivation to fix these various classes of boundary
conditions was the existence of a well-defined variational principle [175] in
the usual sense that the action is stationary on any solution of the equations
of motion. This condition was guaranteed by the absence of flux through the
conformal boundary, in complete contrast with what is observed for radiative
asymptotically flat or de Sitter spacetimes. This feature presents several
drawbacks if one pays attention to recent developments in the field. A first
example occurs when trying to address the black hole information paradox in
the language of the AdS/CFT correspondence. Indeed, reflexive boundary
conditions preventing leaks at the conformal boundary do not allow for
dynamical black hole evaporation [176] as the energy flow is permanently re-
injected into the bulk, invariably leading to eternal black holes [177]. In
particular, it was recently shown in the “islands” program [178, 179, 180]
that the usual boundary conditions prevent from deriving the Page curve [181,
182] under the assumption of unitarity from quantum gravity path integral
arguments. However, the Page curve is a central ingredient in the discussion,
as it is assumed to govern the time evolution of the total quantum entropy
during the evaporation (see e.g. section 5 of [180] for a review). In this
context, it has been useful to allow some radiation to escape the spacetime
boundary so that the black hole can evaporate in anti-de Sitter as it does in
flat space. This method was implemented in practice by gluing an
asymptotically flat region to the conformal boundary, providing a reservoir
for the outgoing radiation, and coupling the dual theory to a thermal bath.
Another example appears when considering brane worlds interacting with ambient
higher-dimensional spacetimes with negative cosmological constant in the
spirit of Randall and Sundrum [183, 184]. This picture naturally yields
holographic dualities with permeable boundaries, mathematically implemented by
a dynamical boundary metric, or equivalently said, quantum gravity on the
boundary [85]. These examples are certainly appealing to investigate a more
general class of boundary conditions that allow for some flux at infinity and
fluctuating boundary structure.

###  1.4 Original contributions

The purpose of this thesis is to provide a unified description of radiative
covariant phase spaces with leaky boundary conditions whatever the value of
the cosmological constant. The present manuscript – conceived to introduce the
original results achieved during these four last years of research,
contextualize them in the state of the art and describe our current
understanding about them – is based on the following papers:

  1. [P1]

G. Compère, A. Fiorucci, and R. Ruzziconi, Superboost transitions, refraction
memory and super-Lorentz charge algebra, JHEP 11 (2018) 200, 1810.00377.

  2. [P2]

G. Compère, A. Fiorucci, and R. Ruzziconi, The $\Lambda$-BMS4 group of dS4 and
new boundary conditions for AdS4, Class. Quant. Grav. 36 (2019), no. 19,
195017, 1905.00971.

  3. [P3]

G. Compère, A. Fiorucci, and R. Ruzziconi, The $\Lambda$-BMS4 charge algebra,
JHEP 10 (2020) 205, 2004.10769.

  4. [P4]

A. Fiorucci and R. Ruzziconi, Charge Algebra in Al(A)dSn Spacetimes, JHEP 05
(2021) 210, 2011.02002.

Some review parts of the manuscript also reproduce the following lecture
notes:

  1. [P5]

G. Compère and A. Fiorucci, Advanced Lectures on General Relativity,
1801.07064.

The following sub-sections summarize the content of these articles, focusing
on the most striking results, in order to familiarize the reader with the
original content of the thesis and provide a brief overview of the origin of
each argument and development detailed later in the main text.

#### 1.4.1 Super-Lorentz extension of the BMS group

###### Super-Lorentz symmetry and memory effects

Staged in the asymptotic flat land in four dimensions, the first paper [P1]
addresses the several issues encountered when the BMS4 group is augmented with
an infinite super-Lorentz extension as well as the new memory effects related
to these overleading symmetries. We started with the derivation of the orbit
of Minkowski spacetime under arbitrary Diff$(S^{2})$ super-Lorentz
transformations and supertranslations in the Bondi gauge, in the spirit of
[185, 99, 186]. We determined that such vacua are labelled by a set of three
fields on the celestial sphere, which we identified as the superboost,
superrotation and supertranslation fields. The terminology “superboost” and
“superrotation” is reserved for transformations that are driven by curl-free
and divergence-free vectors on the sphere, respectively. The knowledge gained
from this careful study of the vacuum structure of the theory was used to
identify the memory effects, which are understood as transitions between vacua
generated by non-stationary phases occurring when gravitational radiation or
more exotic cosmic events perturb the metric irreversibly. We showed how
impulsive transitions [187, 188, 189] among vacua are related to the velocity
kick/refraction memory effect [190, 191, 192] and the displacement memory
effect [97, 96, 193, 15, 16, 98].

###### Renormalized phase space

The second main goal of the paper was to construct, in a rigorous manner and
from first principles, a covariant phase space whose asymptotic symmetry group
consists of arbitrary Diff$(S^{2})$ super-Lorentz transformations and smooth
supertranslations. The motivations for doing so are numerous, but it is
important to note that besides its relevance for memory effects, the smooth
super-Lorentz extension is required to prove [58, 59, 194] that the Ward
identities for the transverse sector of the BMS4 transformations are
equivalent to the Cachazo-Strominger subleading soft graviton theorem [117].
Since these symmetries modify the leading structure of the gravitational field
at null infinity, their inclusion in the phase space introduces divergences in
the on-shell action as well as in the symplectic structure and finally the
surface charges. Therefore, a renormalization of the symplectic structure is
necessary. The renormalization procedure was performed heuristically using
non-covariant objects due to the background structure induced by the Bondi
gauge fixing as well as the boundary conditions. Nevertheless, the
renormalization scheme was consolidated by taking the flat limit of the
radiative asymptotically (A)dS phase spaces in [P3].

After obtaining radially finite charges, we defined a subset of solutions that
contain temporary non-stationary effects, coming from a steady gravitational
field in the past to relax back to another steady field in the future. This is
the natural frame to formalize transitions leading to memory effects as well
as realistic localized sources. For this class of solutions, we defined finite
Hamiltonian conjugated to the Generalized BMS4 diffeomorphisms by controlling
the time divergence at the corners of null infinity and requiring that their
canonical fluxes were zero in the absence of gravitational radiation,
extending the prescription of [31] to general super-Lorentz frames. Finally,
we showed that our flux expressions are consistent with the leading [114, 98,
113] and subleading [117] soft graviton theorems and computed the charge
algebra under the Barnich-Troessaert bracket [46]. We concluded by contrasting
the leading BMS triangle structure [57] with the mixed overleading/subleading
BMS square structure.

#### 1.4.2 Radiative phase spaces with cosmological constant

As previously mentioned, the radiative phase spaces with a non-vanishing
cosmological constant $\Lambda$ are significantly less studied in the
literature compared to the $\Lambda=0$ case. In the AdS ($\Lambda<0$) case,
some radiative configurations, such as Robinson-Trautman solutions [195], have
been considered [196, 197, 198] as concrete models for non-equilibrium
holography, but overall the conformal boundary of AdS has remained resolutely
impermeable, which is suitable for the sake of unitarity of the dual quantum
theory. The dS case ($\Lambda>0$), although relevant from cosmological
perspectives, has not received much interest and only recent analyses [136,
137, 153] address the questions of consistent boundary conditions at the
future conformal boundary of dS compatible with generic radiative sources in
the bulk (see sub-section 1.3.1).

The second line of research pursued in this thesis proposes to repeat the
Bondi analysis, which was very successful in the flat case, to identify the
fields encoding the radiation flux at conformal infinity as well as to have an
infinite-dimensional symmetry group mimicking the features of the
(Generalized) BMS group of flat space. Such a group shall encompass more
general diffeomorphisms than the conformal isometries of the boundary, and
thus act non-trivially on the boundary metric. However, as soon as the
symplectic flux in (A)dS is known to be sourced by fluctuations of the
boundary metric [85, 166], the search for a non-trivial infinite-dimensional
asymptotic symmetry group becomes equivalent to the search for leaky boundary
conditions in (A)dS. From this program, we can expect advances in two ways:
first, we can import techniques and results derived in the well-mastered
asymptotically flat context to (A)dS asymptotics, with some necessary
precautions about the physical differences between flat null infinity and
(A)dS conformal boundaries; second, if we build a mathematically robust flat
limit process allowing for linking the results obtained for $\Lambda\neq 0$,
where one has better control on holographic features, to the flat case, we can
expect to gain a better understanding of radiative spacetimes with fluctuating
boundary structures for $\Lambda=0$.

###### BMS-like symmetries in presence of cosmological constant

In a first paper [P2] inspired by the recent work [150], we derived the
general four-dimensional solution space with asymptotically locally (A)dS
boundary conditions in the Bondi gauge, and gave a holographic interpretation
to the Bondi fields thanks to a diffeomorphism to the Starobinsky/Fefferman-
Graham [199, 200] gauge, where all the holographic features of (A)dS
spacetimes are more transparent. It is worth emphasizing that diffeomorphisms
breaking the SFG gauge are not considered hereafter, although they might be
associated with further non-trivial charges as in three-dimensional Einstein
gravity (see e.g. [167, 201, 202]). Using this dictionary between the gauges,
we proved that the dynamical equations in the Bondi gauge admit a well-defined
flat limit $\Lambda\to 0$. More crucially, we also identified the analogues of
the Bondi news, Bondi mass and Bondi angular momentum aspects at the conformal
boundary.

This finding led us to separate the boundary degrees of freedom between
dynamical (or radiative) and kinematical pieces by the choice of a particular
universal boundary structure, consisting of a foliation and a measure on the
transverse spaces. We argued that this fixation is always reachable by
diffeomorphism and is thus merely a boundary gauge fixing that does not rule
out any solution for the phase space and does not constrain the initial value
problem, more particularly in dS. We introduced the $\Lambda$-BMS4 group as
the set of residual symmetries of the Bondi gauge after this additional
boundary gauge fixing. It consists of infinite-dimensional non-abelian
“supertranslations” and “superrotations” but it reduces in the asymptotically
flat limit to the Generalized BMS4 group [58, 59, 194, 45] . Given that the
constraints among the diffeomorphism parameters are field-dependent, one
should adopt a precise terminology stating that $\Lambda$-BMS4 is a Lie
groupoid, associated with a Lie algebroid of infinitesimal diffeomorphisms
closing with solution-dependent “structure constants” [203, 204, 205].

Finally, we presented new boundary conditions for asymptotically locally AdS4
spacetimes insuring a well-posed Cauchy problem, that admit $\mathbb{R}$ times
the group of area-preserving diffeomorphisms as the asymptotic symmetry group.
This conservative subset of the $\Lambda$-BMS4 boundary conditions amounts to
fixing two components of the holographic stress-tensor while allowing two
components of the boundary metric to fluctuate. They correspond to a
deformation of a holographic conformal field in three dimensions, which is
coupled to a fluctuating spatial metric of fixed area.

###### Radiative phase spaces with a cosmological constant and flat limit

We devoted a second paper [P3] to deriving the surface charge algebra
associated with these novel $\Lambda$-BMS4 symmetries. We started with a
derivation of the surface charge (and algebra) of generic asymptotically
locally (A)dS4 spacetimes without matter. The computations were performed
without assuming any further boundary conditions than the existence of a
conformal completion [9, 160]. Surface charges associated with boundary Weyl
rescalings [206, 207, 208] were found to be vanishing, while the boundary
diffeomorphism charge algebra was non-trivially represented without central
extension under the Barnich-Troessaert bracket [46]. We particularized the
result by specifying a boundary foliation and a boundary measure and obtained
the $\Lambda$-BMS4 charge algebra.

Pursuing the investigations of [P2], we considered the feasibility of a well-
defined flat limit at the level of the phase space. The procedure requires the
incorporation of corner terms in the action principle and symplectic structure
that are defined from the boundary foliation and measure to exempt them from
poles in $1/\Lambda$. The flat limit then reproduces the Generalized BMS4
phase space and charge algebra of supertranslations and super-Lorentz
transformations acting on the class of asymptotically locally flat spacetimes
studied in [P1]. In particular, the renormalization procedure needed in the
flat case when super-Lorentz diffeomorphisms act at null infinity is morally
equivalent to the completely covariant corner subtraction of $1/\Lambda$ poles
in the (A)dS case, confirming the result of [P1] in a somewhat indirect and
non-trivial way. Finally, we again added a tiny layer of refinement to the
results of [P1] and proposed a second prescription for the finite Hamiltonians
associated with Generalized BMS4 transformations (in agreement with [44,
194]), which enjoy the desirable property of representing the Generalized BMS4
algebra without central extension at the corners of null infinity under the
standard Poisson bracket. This feature implies that the BMS4 flux algebra
admits no non-trivial central extension.

###### Generalization to arbitrary dimensions

The series is completed by a third paper [P4] in which the analysis of [P3]
was extended to generic asymptotically locally (A)dS spacetimes in $n\geq 3$
dimensions, with the only exception that the computations were performed
within the Starobinsky/Fefferman-Graham gauge. Again, we obtained the
gravitational charge algebra for boundary diffeomorphisms and Weyl rescalings
without assuming any further boundary condition than the minimal falloffs
allowing for conformal compactification. In particular, the whole boundary
structure was free to fluctuate and play the role of source yielding some
symplectic flux at the boundary.

Using the holographic renormalization procedure [207, 209, 210], the
divergences in the holographic coordinate were removed from the symplectic
structure in any dimension, leading to finite expressions [85, 166]. The
charges associated with boundary diffeomorphisms were found to be generically
non-vanishing, non-integrable and not conserved, which was interpreted as the
manifestation of the leaks of gravitational radiation through the conformal
boundary, eventually collected by an outer reservoir [179, 178, 180]. However,
the charges associated with boundary Weyl rescalings were non-vanishing only
in odd dimensions, which we interpreted as the presence of Weyl anomalies in
the dual theory [162, 207, 166].

Due to the presence of leaks, the charge algebra was computed using the
Barnich-Troessaert bracket [46] and exhibited a field-dependent $2$-cocycle in
odd dimensions. When the general framework was restricted to three-dimensional
asymptotically AdS spacetimes with Dirichlet boundary conditions, the
$2$-cocycle reduced to the Brown-Henneaux central extension [62]. The analysis
was finally specified to leaky boundary conditions in asymptotically locally
(A)dS spacetimes that lead to the $\Lambda$-BMS asymptotic symmetry group,
generalizing the boundary conditions introduced in [P2]. In the flat limit,
the latter contracts into the Generalized BMS group in $n$ dimensions,
including smooth supertranslations and arbitrary smooth codimension 2 vectors
on the $(n-2)$-dimensional celestial sphere.

###  1.5 Reading guide and plan of the thesis

We structure our presentation as follows.

Chapter 2 consists of a pedagogical introduction to the notion of asymptotic
symmetries and their associated charges in gauge theories, with a main focus
on General Relativity, leaning on the first and third chapters of the lecture
notes [P5]. We provide the fundamental definitions and results to address
problems of gravitational physics in the presence of boundaries, including the
well-definiteness of the variational principle, the fixation of consistent
boundary conditions and the associated residual gauge symmetries among which
some gauge generators turn out to be canonically charged. We then discuss the
mathematical structure of the solution spaces promoted as covariant phase
spaces using a notion of presymplectic form that directly derives from the
variational principle. We show how to use the functionalities of this
particular kind of field space to compute canonical surface charges and
address the crucial questions of their conservation and integrability.
Finally, we review how the set of conserved integrable charges form an algebra
representing the asymptotic symmetry algebra, and explain how to generalize
this result when the charges are neither conserved nor integrable, features in
which we are mainly interested in this thesis.

Throughout the chapter, the concepts defined and articulated in various
theorems are exemplified on the class of radiative asymptotically flat
spacetimes with fluctuating boundary structure. Besides the obvious
pedagogical utility, this basis provides a rich ground on which to build the
ideas and methods developed in the subsequent chapters for other classes of
radiative phase spaces. The asymptotic symmetry group under consideration is
the Generalized BMS4 symmetry group, consisting of a double infinite-
dimensional extension of the Poincaré group into smooth supertranslations and
super-Lorentz transformations. We describe the various features in the Bondi
gauge, which are particularly well-adapted to treat radiative problems in
gravity. The central notions of the Bondi analysis are reviewed in our
notations and framework before presenting the renormalized phase space
discussed in [P1], including the overleading super-Lorentz symmetries as
genuine asymptotic symmetries.

In Chapter 3, after reviewing interesting facts about geometry and physics in
dS and AdS spacetimes, we study the most general solution space with
asymptotically locally (A)dS boundary conditions and the residual gauge
transformations in the Starobinsky/Fefferman-Graham gauge. We use the
holographic renormalization procedure to control and remove the divergences
from the presymplectic structure, leading to the definition of general
radiative phase spaces with a non-vanishing cosmological constant, without
assuming any further boundary condition than the minimal fall-offs ensuring
conformal compactness. Within this framework, we compute the surface charges
associated with the residual gauge symmetries as well as their algebra. We
finally apply the formalism to well-known sub-cases of conservative boundary
conditions at conformal infinity.

We extend the discussion in Chapter 4, where we identify a consistent boundary
gauge fixing that singles out a set of residual gauge diffeomorphisms whose
structure is very similar to the BMS group of flat space and reduces to it at
flat limit. It therefore receives the name of $\Lambda$-BMS algebroid, because
of the unavoidable field-dependence in the gauge parameters. We explain why
the boundary conditions imposing this further gauge fixing are suitable to
describe genuine radiative modes at conformal infinity, especially in
asymptotically dS spacetimes where the Cauchy problem is left unconstrained.
We then detail some aspects of the $\Lambda$-BMS phase space in four
dimensions, principally in the Bondi gauge. We take care to derive the most
general solution of Einstein’s equations in this coordinate system. Thanks to
a diffeomorphism between both gauges, we translate the dynamical quantities
and the symmetry parameters from Starobinsky/Fefferman-Graham to Bondi
coordinates, and identify the analog of the Bondi news tensor for radiative
phase spaces with a non-vanishing cosmological constant. After describing an
interesting sub-sector of conservative boundary conditions, we discuss the
flat limit of the $\Lambda$-BMS4 phase space and explain our method to
regulate poles in $\Lambda$ in the presymplectic structure by addition of
covariant corner terms at the boundaries of conformal infinity.

The flat limit process impressively gives the phase space of radiative
asymptotically locally flat spacetimes and confirms the renormalization
procedure previously discussed in Chapter 2. To close the loop opened at the
beginning of the thesis, Chapter 5 recovers the Einstein gravity with a
vanishing cosmological constant and aims at answering several questions raised
by the super-Lorentz extension of the BMS4 group. First we derive the class of
vacua obtained by exponentiating the Generalized BMS4 group around Minkowski
space, and identify the set of fields needed to parametrize this gauge orbit.
We make good use of this knowledge to study the memory effects observed for
vacuum transitions driven by non-stationary events localized in time and make
contact with well-known effects such as the displacement memory effect.
Finally, we give a prescription to define finite Generalized BMS4 Hamiltonians
for the important sub-class of asymptotically flat solutions without radiation
at early and late times, which is the natural arena to model standard
astrophysical configurations as well as the above-mentioned memory effects. We
show that the integrated fluxes of these Hamiltonians are compatible with the
soft theorems obtained at quantum level for the linearized theory and comment
on the enhancing of the infrared structure of gravity via super-Lorentz
symmetries.

The manuscript is concluded in Chapter 6, which gives an overview of the
results derived together with an outlook on the many interesting questions
raised by our research program. The thesis also contains Appendices C–F, which
gather additional technical material referenced in the main text.

A considerable effort was made to ensure that this thesis is self-contained,
whether at the conceptual level or at the level of notations. Only some
lengthy mathematical developments were not reproduced to provide a smoother
reading experience. In order to be accessible to a broader readership beyond
specialists, numerous well-known facts and results have been reviewed before
discussing the original contributions of this research. Although resulting in
the lengthening of the manuscript, we hope that this choice improves the
experience of the reader who does not necessarily have extensive base
knowledge, and thus would have the unpleasant task of interrupting reading for
incessant dives into the abundant literature in the field. Having outlined the
thesis plan, we now progress to the formulation of covariant phase spaces in
General Relativity. $\blacksquare$

##  Chapter 2 Covariant phase spaces in General Relativity

This second chapter is a self-contained and duly contextualized introduction
to the notion of covariant phase spaces in generally covariant theories. In
section 2.1, we rigorously define the notion of asymptotic symmetries in
general gauge theories within the gauge fixing approach. The problem of fixing
consistent boundary conditions and the formulation of the variational
principle are discussed. We then give the extensive example of asymptotically
flat Einstein’s gravity in four dimensions, which allows us to highlight
several results and features that are widely used in the following chapters
and to pedagogically illustrate the theoretical concepts introduced thus far.
We primarily consider the BMS4 asymptotic symmetry group and its various
super-Lorentz extensions. Section 2.2 subsequently discusses how to build
canonical surface charges associated with asymptotic symmetries in gravity
within the framework of the covariant phase space formalism, which is directly
inspired by the Hamiltonian formulation of classical mechanics that we also
briefly review. We address the crucial questions of the conservation and the
integrability of the charges and particularize again to the radiative
asymptotically flat spacetimes. Section 2.3 concludes with an analysis of the
algebraic properties of the surface charges and describes in which sense they
represent the asymptotic symmetry algebra in full generality, without assuming
conservation or integrability. Finally, we show that the surface charge
algebra in asymptotically flat spacetimes contains deep physical information
on the flux of gravitational radiation at null infinity and make several
comments on the physics of thermodynamically open gauge systems.

This chapter consists of an updated and augmented version of the lecture notes
[211] with additional elements and precisions gleaned in [204, 46, 212, 213,
214, 2, 77, 78, 80, 83, 215, 216, 217] in order to improve pedagogical
efficiency and to cover all of the features that are observed in the concrete
realizations explained further in the manuscript. Although it looks like an
oriented review of covariant phase space methods, this hybrid introductory
part also contains original results. Sections 2.1.5, 2.2.4 and 2.3.3 are
essentially reproduced from [45], while section 2.3.2 is based on future work
to appear [94]. In summary, the aim of this formal prolegomenon – which
comprises the longest chapter of this thesis – is to present the current
understanding about the central notion of asymptotic symmetries and their
surface charges in a fluid and pedagogical way and to offer our contribution
to the field providing the basis for the more evolved considerations that are
explored in the following chapters.

###  2.1 Notion of asymptotic symmetries

This first section is aimed at defining the notion of asymptotic symmetries in
a precise way, within the gauge fixing approach. The structure of the
presentation exactly reflects the rigorous algorithmic procedure used to
derive and study the asymptotic structure of any gauge theory, exemplified
throughout with the 4$d$ Einstein theory of gravity with (not so standard)
asymptotically flat boundary conditions. We successively discuss the
definition of the action integral for gauge theories, the fixation of a
suitable gauge in which boundary conditions can be formulated in a consistent
and meaningful manner, the derivation of the solution space in this gauge for
the considered boundary conditions and finally the uncovering of the residual
gauge symmetries which are tangent to this solution space, which we eventually
call asymptotic symmetries.

We stress that we restrain ourselves to the gauge fixing approach which was
the route chosen in the various works compiled in this thesis, but alternative
ways to study the asymptotic behavior of gauge theories also exist. The fully
geometric approach, mostly relevant for gravity [25, 9, 218, 14, 219, 220],
has the privilege to remain manifestly gauge invariant and does not rely on
any coordinate system in the discussion of asymptotic symmetries. The price to
pay for this elegance is some lack of flexibility in imposing boundary
conditions, implemented by assuming the existence of universal boundary
structures that must be preserved by the symmetries. The modification of these
boundary structures is often involved and consequently first worked out in the
gauge fixing approach and re-understood later in terms of covariant
structures. This holds also for the original content of this thesis, in which
we try to give a purely geometrical equivalent to the various boundary
conditions we are considering. Another successful approach that is worth
mentioning is the Hamiltonian formalism, for which a slicing of the spacetime
by constant time hypersurfaces is assumed to exist. This framework is
logically well-adapted to treat spacelike infinities in gravity (see e.g. [61,
62, 64, 34]) without requiring any further gauge fixing than the existence of
a time coordinate, but suffers from some weaknesses when treating other kinds
of boundaries, namely null boundaries. As a consequence, in the following, we
always work in particular systems of coordinates in order to gain in
pedagogical, interpretative and computational efficiency.

#### 2.1.1 Preliminary definitions

We begin by providing basic definitions and notations that are necessary to
formulate the notion of asymptotic structure in a mathematically robust way.
Although our main concern is invariably Einstein’s gravity, we keep the
concepts and the notations sufficiently general to include a large class of
Lagrangian theories with gauge invariances.

##### 2.1.1.1 Gauge symmetries

Consider a classical field theory defined on some $n$-dimensional spacetime
manifold $(\mathscr{M},g)$, with a certain amount of fields $\phi^{i}$
collectively denoted by $\phi=(\phi^{i})$. $\mathscr{M}$ is generically
covered by local coordinates $\\{x^{\mu}\\}$ (at least in a sufficiently large
open neighborhood of the events under consideration). The dynamics of the
theory is encoded into the action integral

$S[\phi]=\int_{\mathscr{M}}\bm{L}[\phi,\partial_{\mu}\phi,\partial_{\mu}\partial_{\nu}\phi,\dots]$
(2.1.1)

where $\bm{L}=L\,\text{d}^{n}x$ is the Lagrangian $n$-form. A general
transformation of the fields $\phi=(\phi^{i})$ is defined as

$\delta_{Q}\phi^{i}=Q^{i}.$ (2.1.2)

The characteristics $Q$ is generically a collection of local functions, which
means that each $Q^{i}$ depends on the spacetime coordinates as well as on the
fields and their derivatives. Such a transformation is a symmetry of the
theory if and only if it preserves the action integral (2.1.1), or in other
words

$\delta_{Q}\bm{L}=\text{d}\bm{B}_{Q}$ (2.1.3)

under the transformation (2.1.2), for some codimension 1 form $\bm{B}_{Q}$.
Among all the transformations acting on the system, we can find some of them
that depend on parameters $\lambda=(\lambda^{\alpha})$ that are arbitrary
spacetime functions: they are coined as gauge transformations. The number of
independent gauge parameters in $\lambda$ is denoted as
$n_{g}\in\mathbb{N}_{0}$. At infinitesimal level, we represent the action of a
gauge transformation on the fields by [213]

$\delta_{\lambda}\phi^{i}=R^{i}[\lambda]=\sum_{(\mu)}R^{i(\mu)}_{\alpha}\partial_{(\mu)}\lambda^{\alpha}$
(2.1.4)

where the characteristics $R^{i(\mu)}_{\alpha}$, $\forall i,\alpha,k$, are
also local functions of the spacetime coordinates, the fields and their
derivatives. $(\mu)$ is a multi-index notation representing sets of implicitly
symmetrized indices $(\mu)=(\mu_{1},\mu_{2},\dots,\mu_{k})$. The cardinal of
$(\mu)$ is denoted as $|\mu|=k$ and
$\partial_{(\mu)}=\partial_{\mu_{1}}\partial_{\mu_{2}}\dots\partial_{\mu_{k}}$
in that case. The writing (2.1.4) assumes that the characteristics of the
transformation $R[\lambda]$ depend linearly and homogeneously on the gauge
parameters $\lambda$. The gauge transformation defined by the set of paramters
$(\lambda^{\alpha})$ is a gauge symmetry of the theory if and only if they
obey the condition (2.1.3), i.e. there exists a boundary term
$\bm{B}_{\lambda}\in\Omega^{n-1}(\mathscr{M})$ such that
$\delta_{\lambda}\bm{L}=\text{d}\bm{B}_{\lambda}$ under the infinitesimal
transformation (2.1.4).

As a pedagogical example, we can consider free Maxwell theory, whose the only-
field is the 4-potential 1-form $\bm{A}=A_{\mu}\text{d}x^{\mu}$. The
Lagrangian is built up from the Faraday’s tensor
$F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ and reads as
$\bm{L}[\bm{A}]=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\text{d}^{n}x$. The
transformation $\delta_{\lambda}\bm{A}=\text{d}\lambda$ for any smooth
function $\lambda(x)$ is a gauge symmetry of the theory: it simply leaves the
Lagrangian invariant. Now let us consider Einstein’s gravity, for which the
dynamical field is $g$. The action integral is given by the Einstein-Hilbert
functional

$S_{EH}[g]=\frac{1}{16\pi
G}\int_{\mathscr{M}}(R-2\Lambda)\sqrt{-g}\,\text{d}^{n}x$ (2.1.5)

where $R$ is the Ricci scalar curvature, $\sqrt{-g}\text{d}^{n}x$ the volume
form on $\mathscr{M}$ and $\Lambda$ the cosmological constant. A
diffeomorphism $\xi$ is a gauge transformation which acts infinitesimally on
the metric by a Lie derivative,
$\delta_{\xi}g_{\mu\nu}=\mathcal{L}_{\xi}g_{\mu\nu}=\xi^{\rho}\partial_{\rho}g_{\mu\nu}+2g_{\rho(\mu}\partial_{\nu)}\xi^{\rho}$.
It is straightforward to show that the gauge transformation generated by $\xi$
is a symmetry of the theory, in accordance with the general covariance
principle. It is remarkable that for gravity, as well as for electromagnetism,
the functions $R^{i(\mu)}_{\alpha}$ are vanishing for any $|\mu|>1$.

##### 2.1.1.2 Gauge fixing conditions

In the presence of gauge symmetries, a physical theory has natural
redundancies along the gauge orbits. Since we have $n_{g}$ parameters at our
disposal, we can use them to perform a collective gauge transformation on the
fields in order to satisfy some set of gauge fixing conditions

$\mathcal{G}[\phi]=\left\\{c_{1}[\phi]=0,c_{2}[\phi]=0,\dots,c_{n_{g}}[\phi]=0\right\\}$
(2.1.6)

where the constraints $c_{i}[\phi]=0$ are assumed to be mutually independent.
They can be either algebraic or differentiable but must be reachable by a
gauge transformation included in the set of symmetries of the theory. For
example, the Weyl gauge fixing in electromagnetism is algebraic and amounts to
setting $A_{0}=0$ which is always reachable since
$A_{0}+\partial_{0}\lambda=0$ has always a solution for $\lambda$. The most
famous gauge fixing, the Lorenz gauge, is differential and requires that
$\partial_{\mu}A^{\mu}=0$, reachable by a transformation satisfying
$\partial_{\mu}A^{\mu}+\partial_{\mu}\partial^{\mu}\lambda=0$ which admits a
solution for any field $\bm{A}$.

In gravity, any gauge fixing consists in providing a set of coordinates to
parametrize the metric field. When interested in gravitational wave physics,
since the latter travel along light cones, one can use a coordinate system
well-adapted to treat null radiation, i.e. in which, concretely, null
congruences live on constant-coordinate slices. Let us consider a family of
null hypersurfaces labeled by constant $u$ coordinate in a $4$-dimensional
spacetime ($n=4$). The normal vector of these hypersurfaces
$n^{\mu}=g^{\mu\nu}\partial_{\nu}u$ is null by construction, so we fix
$g^{uu}=0$. We define angular coordinates $x^{A}=(\theta,\phi)$ such that the
directional derivative along the normal $n^{\mu}$ is zero,
$n^{\mu}\partial_{\mu}x^{A}=0\Rightarrow g^{uA}=0$. We finally select the
radial coordinate $r$ to be the luminosity distance, i.e. we fix
$\partial_{r}(\det g_{AB}/r^{4})=0$. The so defined coordinates
$x^{\mu}=(u,r,x^{A})$ are known in the literature as the Bondi-Sachs
coordinate system or Bondi gauge [6, 7, 8]. $u$ has the meaning of a retarded
time, hence the Bondi gauge we have just defined should be qualified as
retarded Bondi gauge since null rays at constant $u$ are outgoing in a sense
that will be precised after giving a notion of asymptotics and imposing
boundary conditions. After lowering the indices, we obtain the Bondi gauge
fixing conditions on $g$ reading as $\\{c_{i}[\phi]=0\\}_{i=1}^{4}$ with

$g_{rr}=0,\qquad g_{rA}=0,\qquad\partial_{r}\left(\frac{\det
g_{AB}}{r^{4}}\right)=0.$ (2.1.7)

Let us notice that the conditions $c_{1}=0$, $c_{2}=0$ and $c_{3}=0$ are
algebraic while $c_{4}=0$, coined as the Bondi determinant condition, is
differential. It is worth emphasizing that the condition $c_{4}$ is weaker
than the historical determinant condition demanding that $\det
g_{AB}=r^{2}\det\mathring{q}_{AB}$ [6, 7, 8]. As we will see later, this
slight extension allows to modify the boundary volume element by performing
boundary Weyl rescalings [46, 91]. In the Bondi parametrization, the line
element takes the form

$\text{d}s^{2}=e^{2\beta}\frac{V}{r}\text{d}u^{2}-2e^{2\beta}\text{d}u\text{d}r+g_{AB}(\text{d}x^{A}-U^{A}\text{d}u)(\text{d}x^{B}-U^{B}\text{d}u)$
(2.1.8)

where $\beta,U^{A}$ and $V$ are arbitrary functions of the coordinates. As a
gauge fixing always reachable by diffeomorphism [7, 8], any metric field can
be expressed in these coordinates. For example, the Minkowski line element can
be written as (2.1.8) with $\beta=0$, $U^{A}=0$, $\frac{V}{r}=-1$ and
$g_{AB}=\mathring{q}_{AB}$ the unit-round sphere metric on the 2-sphere,

$\mathring{q}_{AB}\text{d}x^{A}\text{d}x^{B}=\text{d}\theta^{2}+\sin^{2}\theta\text{d}\phi^{2}.$
(2.1.9)

Note crucially that fixing the gauge in gravity explicitly breaks general
covariance by selecting a preferred class of observers who will express the
physical quantities and perform calculations. In other words, the gauge fixing
can be seen as the adjunction of a background structure while defining the
dynamical fields of the theory. For Bondi gauge, this background structure is
represented by a set of fixed null hypersurfaces and a particular tangent
vector for their null generators.

Another choice for the radial coordinate is to take it as an affine parameter
$\rho$ along the null generators of the constant $u$ hypersurfaces. This
defines the Newmann-Unti gauge [221] with gauge conditions $g_{\rho\rho}=0$,
$g_{\rho A}=0$ and $g_{u\rho}=-1$, coming with the advantage to be completely
algebraic. This gauge fixing exists as an alternative to the Bondi gauge and
is particularly well-suited to make the link between the second order
formulation of gravity that we present here and the first order tetrad
formalism [27].

##### 2.1.1.3 Residual gauge transformations

When the gauge fixing conditions (2.1.6) are imposed, the set of allowed gauge
transformations is reduced, sometimes drastically, but not completely
destroyed. For example, fixing the Lorenz gauge in electromagnetism still
allows for gauge transformations generated by harmonic functions
($\partial_{\mu}\partial^{\mu}\lambda=0$). This is a very simple example of
residual gauge transformations defined as gauge transformations that preserve
the gauge fixing conditions (2.1.6), namely $\delta_{\lambda}c_{i}[\phi]=0$,
$i=1,\dots,n_{g}$. The generators $\lambda$ that solve this set of conditions
can be re-parametrized in terms of local functions of $(n-1)$ coordinates,
i.e. $\lambda=\lambda(s)$ for $s=\\{s_{1},s_{2},...\\}$.

Turning back to our favorite example of gravity in the Bondi gauge,
infinitesimal diffeomorphisms $\xi$ that preserve the gauge fixing conditions
(2.1.7) have to satisfy [6, 46, 45]

$\mathcal{L}_{\xi}g_{rr}=0,\quad\mathcal{L}_{\xi}g_{rA}=0,\quad
g^{AB}\mathcal{L}_{\xi}g_{AB}=4\omega(u,x^{C}).$ (2.1.10)

While the two first constraints are readily understandable, the third one
needs some work. The Bondi determinant condition can be integrated in
$\det(g_{AB})=r^{4}\chi(u,x^{C})$ where $\chi$ is an arbitrary function of
$(u,x^{C})$. So

$g^{AB}\mathcal{L}_{\xi}g_{AB}=\delta_{\xi}\ln[\det(g_{AB})]=\delta_{\xi}\ln
r^{4}+\delta_{\xi}\ln\chi=4\omega(u,x^{C})$ (2.1.11)

where $\omega(u,x^{C})$ is another arbitrary function of $(u,x^{C})$ and the
prefactor $4$ is introduced for convenience. Developing the Lie derivatives in
(2.1.10) and integrating the resulting partial differential equations yields

$\begin{split}\xi^{u}&=f,\\\ \xi^{A}&=Y^{A}+I^{A},\quad
I^{A}=-\partial_{B}f\int_{r}^{\infty}\text{d}r^{\prime}(e^{2\beta}g^{AB}),\\\
\xi^{r}&=-\frac{r}{2}(\mathcal{D}_{A}Y^{A}-2\omega+\mathcal{D}_{A}I^{A}-\partial_{B}fU^{B}+\frac{1}{2}fg^{-1}\partial_{u}g),\\\
\end{split}$ (2.1.12)

where $\partial_{r}f=0=\partial_{r}Y^{A}$ and $g=\det(g_{AB})$ [46, 222]. The
covariant derivative $\mathcal{D}_{A}$ is associated with the $2$-dimensional
metric $g_{AB}$. The residual gauge transformations are thus parametrized by 4
functions $\omega$, $f$ and $Y^{A}$ of $(u,x^{A})$. Let us remark that some
components of the metric field explicitly appear in (2.1.12)! This is a
generic result: since the gauge fixing conditions $c_{i}[\phi]=0$ involve the
fields, the preservation conditions $\delta_{\lambda}c_{i}[\phi]=0$ may have
field-dependent solutions. As we have seen, the most striking example of that
is gravity, because the spacetime itself has a dynamical structure and the
gauge field is the metric: simply raising some indices in one of these
conditions is sufficient to bring field-dependence into the game.

#### 2.1.2 Solution space and boundary conditions

For any local theory such as (2.1.1), the dynamical equations for the fields
$\phi$ are partial differential equations which need boundary conditions in
order to be solved. This is the unavoidable step to precise the kinematics of
the system. Like the gauge fixing, the choice of boundary conditions is
completely free but should be motivated by the class of physical phenomena one
wants to describe. This is the most “artistic” part in the path leading to the
formulation of a phase space, since there is neither mathematical nor physical
generic prescription to define the right boundary conditions for the right
purpose. Their deduction, or better, discovery, is often the achievement of a
quest summarized by the optimization problem that we describe now.

##### 2.1.2.1 The art of fixing boundary conditions

Who says “boundary conditions” says in particular “boundary,” a concept that
can appear under multiple avatars: a black hole horizon, an hypersurface
located somewhere in the spacetime, or also some notion of infinity at large
distances. Let us define a boundary $\mathscr{B}$ in $\mathscr{M}$ and
$\mathscr{U}$ an open neighborhood of $\mathscr{B}$. While discussing
asymptotics, $\mathscr{B}$ can be repelled at infinity in a sense that we will
precise in a few lines. Giving boundary conditions in the vicinity of
$\mathscr{B}$ consists in prescribing the behavior of the fields in the
neighborhood $\mathscr{U}$ (fall-off conditions) as well as their value on
$\mathscr{B}$ – for those that do not decay sufficiently fast while traveling
$\mathscr{U}$ towards $\mathscr{B}$. We can choose a coordinate system
covering $\mathscr{U}$ such that one of these coordinates, say $r$, tends to
infinity when approaching $\mathscr{B}$. Fall-off conditions dictate the
behavior in $r$ of the various fields when $r$ is large. The latter may be
singular on $\mathscr{B}$ but this issue can often be cured by a change of
coordinates on $\mathscr{U}$, or by taking another picture of the situation if
$\mathscr{B}$ stands for some notion of infinity (for example, through a
conformal compactification process [9]). A set of boundary conditions,
encompassing fall-off conditions in $r$ as well as codimension 1 conditions on
the fields pulled back to $\mathscr{B}$, is considered to be physically
relevant if the following requirements are met:

  * $\rhd$

The conditions are sufficiently weak to have a non-trivial associated set of
solutions while keeping a sufficient amount of symmetries acting on it. The
conserved quantities in the evolution of the system should be finite while $r$
runs to infinity and generically non-vanishing for a non-trivial subset of
solutions.

  * $\rhd$

The conditions are sufficiently strong to avoid dealing with unphysical
solutions (in a sense that obviously depends on the situation) and infinite
conserved quantities.

We can help ourselves by a careful study of the asymptotic structure of some
well-known solutions that we want to include in the solution space compatible
with the boundary conditions. Anyways, the best boundary conditions must
minimize the tension between the two requirements enunciated here above,
although the task to derive them is highly non-trivial. Let us finally
emphasize that any set of boundary conditions underlies a fixation of a
boundary structure including the boundary itself. The latter is said universal
because it is shared by all of the solutions and can take different forms: a
boundary foliation, a metric tensor fixed on $\mathscr{B}$, a codimension 1
source for the fields, etc. We will particularize this quite philosophical
discussion to concrete examples in sections 2.1.5 and 4.1.1.

##### 2.1.2.2 Solution space and equations of motion

Let us consider again our general theory (2.1.1) for which we have imposed the
gauge fixing (2.1.6) and given some boundary conditions. The elementary pieces
of kinematics are right in place to study the solution of the equations of
motion that prescribe the evolution of the fields $\phi$. These are the Euler-
Lagrange equations, written formally as

$\frac{\delta\bm{L}}{\delta\phi^{i}}=\sum_{(\mu)}(-1)^{|\mu|}\partial_{(\mu)}\left(\frac{\partial\bm{L}}{\partial\partial_{(\mu)}\phi^{i}}\right)=0.$
(2.1.13)

The solution space $\mathcal{S}$ of the theory is defined as the set of field
configurations that solve the Euler-Lagrange equations while satisfying the
gauge fixing conditions as well as the boundary conditions. After formal
resolution, any solution $\phi\in\mathcal{S}$ can be parametrized by arbitrary
functions of $(n-1)$ coordinates $p=\\{p_{1},p_{2},\dots\\}$, i.e.
$\phi=\phi(p)$. We will observe this feature on the solution space of
asymptotically locally flat spacetimes.

#### 2.1.3 Variational principle

Since the pioneer works of d’Alembert and Lagrange, we know that the dynamical
equations (2.1.13) are in fact a consequence of a variational principle which
is traditionally formulated as a optimization problem involving the action
integral (2.1.1) as a functional on the fields $\phi$. The variations
$\delta\phi$ are defined as infinitesimal perturbations of
$\phi\in\mathcal{S}$ such that $\phi+\delta\phi$ also belongs to
$\mathcal{S}$. This means in particular that $\delta\phi$ must obey the
boundary conditions imposed in the vicinity of $\mathscr{B}$. Considering an
arbitrary variation $\delta\phi$, we have [77, 78]

$\begin{split}\delta\bm{L}=\delta\phi^{i}\frac{\partial\bm{L}}{\partial\phi^{i}}+\partial_{\mu}\delta\phi^{i}\frac{\delta\bm{L}}{\delta\partial_{\mu}\phi^{i}}+\cdots=\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}+\text{d}\bm{\Theta}[\phi;\delta\phi].\end{split}$
(2.1.14)

The second equality is obtained by performing inverse Leibniz rules on the
spacetime derivatives $\partial_{\mu},\partial_{\mu}\partial_{\nu}\dots$ in
order to recognize (2.1.13) in front of the arbitrary variation $\delta\phi$.
The residue of these integrations by part is a boundary term appearing as the
codimension 1 form $\bm{\Theta}[\phi;\delta\phi]$ depending linearly on
$\delta\phi$, $\partial_{\mu}\delta\phi$,
$\partial_{\mu}\partial_{\nu}\delta\phi\dots$ It goes under the name of
presymplectic potential [77] for reasons that will be clarified in section
2.2.2. Integrating over $\mathscr{M}$, we get

$\delta
S[\phi]=\int_{\mathscr{M}}\delta\phi^{i}\frac{\delta\bm{L}}{\delta\phi^{i}}+\int_{\mathscr{B}}\bm{\Theta}[\phi;\delta\phi]$
(2.1.15)

thanks to the Stokes theorem. The cancellation of the bulk term for any set of
arbitrary variations leads to the equations of motion (2.1.13), so much so
that the on-shell variational principle, i.e. considered for variations
$\delta\phi$ around a point of $\mathcal{S}$, is given by [77]

$\delta S[\phi]=\int_{\mathscr{B}}\bm{\Theta}[\phi;\delta\phi].$ (2.1.16)

Contributions coming from the boundaries of $\mathscr{B}$ (“corners”) will be
ignored in the current discussion for the sake of simplicity. Although they
are crucial for the definition of the complete variational principle, we are
mainly interested in dynamical (radiative) parts of $\mathscr{B}$ and not the
corners that are often rigid boundaries without dynamics. The problem to
incorporate these corner contributions will be addressed later in this thesis
(see section 2.2.2.6 for details concerning codimension 2 ambiguity terms in
the presymplectic potential and sections 2.2.4.1 and 4.3.3 for concrete
examples).

The result (2.1.16) is an important observation: the pull-back of the
presymplectic potential on the boundary $\mathscr{B}$ controls the on-shell
variational principle. The right-hand side of this equation is called the
presymplectic flux through $\mathscr{B}$. The precise value of it depends on
the fixation of the boundary conditions around $\mathscr{B}$. That is where
physics comes into the setup! In many situations (as in standard classical
mechanics) we are interested in systems without presymplectic flux at the
boundary, but we will argue that sometimes allowing for some non-vanishing
presymplectic flux through the boundary is mandatory in order to preserve the
whole dynamics of the system. We are thus facing a dichotomous choice in the
boundary conditions, that can either forbid or allow for a non-vanishing
presymplectic flux.

##### 2.1.3.1 Conservative boundary conditions

We qualify as conservative boundary conditions a set of conditions imposed on
$\mathscr{B}$ such that the presymplectic flux vanishes for any variation. In
other words, the action $S[\phi]$ is stationary on-shell,

$\delta S[\phi]=0.$ (2.1.17)

This is the case, for instance, if $\bm{\Theta}[\phi;\delta\phi]$ decays
sufficiently fast when approaching $\mathscr{B}$, or if the variations
$\delta\phi$ are designed to have sufficiently strong fall-offs to cancel out
on the boundary. The requirement of vanishing symplectic flux is obviously
defined up to a redefinition of the action integral (2.1.1) by pure boundary
terms that do not affect the Euler-Lagrange equations (2.1.13). Therefore in
general, a set of boundary conditions is declared to be conservative if and
only if there exists a codimension 1 form $\bm{B}[\phi|\bar{\chi}]$ depending
on the fields $\phi$ as well as the background structures $\bar{\chi}$ needed
to fix the boundary conditions at $\mathscr{B}$, such that

$S\to S^{\prime}=S+\int_{\mathscr{B}}\bm{B}[\phi|\bar{\chi}],$ (2.1.18)

is stationary on-shell, $\delta S^{\prime}[\phi]=0$. Again using Stokes
theorem, we see that
$\bm{L}^{\prime}[\phi]=\bm{L}[\phi]+\text{d}\bm{B}[\phi|\bar{\chi}]$.
Performing a variation as in (2.1.14), we see that $\delta S^{\prime}[\phi]=0$
is ensured if and only if

$\bm{\Theta}[\phi;\delta\phi]\Big{|}_{\mathscr{B}}=-\delta\bm{B}[\phi|\bar{\chi}],$
(2.1.19)

(see e.g. [223, 217]). Conservative boundary conditions describe theories
without any dynamics through the boundary $\mathscr{B}$, since traveling
$\mathcal{S}$ from $\phi$ to $\phi+\delta\phi$ conserves the total action
integral of the system which contains all the information about it. When a
covariant phase space will be rigorously defined, we will see that this kind
of boundary conditions lead to conserved charges, which justifies a posteriori
the terminology.

##### 2.1.3.2 Leaky boundary conditions and open action principles

In contrast, boundary conditions allowing for some presymplectic flux through
the $\mathscr{B}$ will be called leaky boundary conditions. In these
configurations, the condition (2.1.19) must be negated: it implies that there
is no boundary term that can supplement the action integral in order to have a
refined action which is stationary on solutions. In that context, what we
consider to be a “well-defined variational principle” must be revised, since
the stationarity is no longer a criterion that can be imposed. We say that the
variational principle (2.1.16) is well-defined if the two following
assumptions are met[87, 224]:

  1. (A1)

The variation of the action $\delta S[\phi]$ on $\mathcal{S}$ is finite on-
shell, for the sake of definiteness and regularity of the presymplectic flux.

  2. (A2)

The stationarity of the action is restored for any sub-class of non-radiative
solutions, up to the incorporation of a finite counter-term like (2.1.18).

In other words, (A2) stipulates that a subset of conservative boundary
conditions must exist as a limiting case of leaky boundary conditions when
turning off the “leaking fields” that source the presymplectic flux and
demanding that $\bm{\Theta}[\phi;\delta\phi]$ is selected in such a way that
$\delta S[\phi]=0$ when these sources are absent. In general, the definition
of a renormalization scheme will be necessary in order to obey the assumption
(A1), while an adjustment of the finite part of the renormalized presymplectic
potential will easily ensure the second assumption (A2). In our sense, a well-
defined variational principle is therefore more general than (2.1.17): the on-
shell action is not supposed to be preserved when moving from $\phi$ to
$\phi+\delta\phi$: the discrepancy between the two values $S[\phi]$ and
$S[\phi]+\delta S[\phi]$ is precisely controlled by the presymplectic flux
through $\mathscr{B}$. There is an exchange between the system under
consideration, enclosed in $\mathscr{B}$, and the exterior environment outside
$\mathscr{B}$. Leaky boundary conditions are therefore unavoidable when
treating dissipative phenomena as well as open systems. In particular, the
various charges that are defined on the system (such as the mass, the energy,
the electric charge etc.) will be associated with some flux passing through
$\mathscr{B}$ and, therefore, will not be expected to be conserved. In this
thesis, we will give several examples of such open systems for Einstein’s
gravity in various dimensionalities and for different values of the
cosmological constant. We will motivate the importance of considering open
variational systems for which $S[\phi]$ is not stationary on-shell for a
fundamental study of the radiative sector of the theory.

At this point, we insist on the fact that the statement “$\delta S\neq 0$ on-
shell” is not at all a rejection of the principle of least action, the latter
being without any doubt one of the most robust pillars of any theoretical
approach of Physics. The non-stationarity is due to the fact that we formulate
the variational principle for the system with boundary $\mathscr{B}$ without
telling more about what is beyond $\mathscr{B}$ and what is collecting the
symplectic flux going through this boundary. In this point of view, (2.1.16)
is nothing but an avatar of some continuity equation through the interface
$\mathscr{B}$ and characterizes how far we are from an equilibrium state
without leaks. When the action principle is well-defined in the sense of the
hypotheses (A1) and (A2), we can also say that it is compatible with a global
stationary variational principle encompassing the dynamics of the system as
well as its environment. Indeed, when these two assumptions are obeyed, it is
possible to define a complete variational principle for the system and one
particular environment whose dynamics has to be precised. In particular, if
the system and the environment form together a closed system, the stationarity
of the complete on-shell action can be restored.

This temporarily closes our discussion on the variational principle. We will
address more precisely the questions on what is physical or not in the
presymplectic flux and what are the laws controlling the non-conservation of
the dynamical quantities of the system at the boundary $\mathscr{B}$ when
canonical charges associated with gauge symmetries have been defined in
section 2.2. This will need a bunch of formal tools that we do not want to
introduce now in order to get directly into the notion of asymptotic
symmetries.

#### 2.1.4 Asymptotic symmetries

We have enough material to define rigorously the notion of asymptotic
symmetries in our approach. Let us quickly recall the main steps and the
necessary ingredients. We started from a Lagrangian theory of fields (2.1.1)
with some gauge symmetries (2.1.4) parametrized by $\lambda$. In order to
eliminate a considerable part of trivial gauge transformations mapping
equivalent physical situations to each other, we imposed the consistent set of
gauge fixing conditions $\mathcal{G}[\phi]$ (see (2.1.6)) which are always
reachable by a gauge transformation and bring generically constraints among
the gauge parameters because $\delta_{\lambda}\mathcal{G}[\phi]=0$ by
definition. The transformations compatible with these constraints have been
coined as residual gauge transformations and are parametrized as
$\lambda=\lambda(s)$ where $s$ denote a collection of codimension 1 functions.
Next, we can impose boundary conditions and derive the associated class
$\mathcal{S}$ of solutions to Euler-Lagrange equations (2.1.13). The elements
of $\mathcal{S}$ are parametrized as $\phi=\phi(p)$ where $p$ denotes another
set of codimension 1 functions. Finally, the last question we want to address
is: what are the gauge symmetries that survive on the solution space after the
imposition of the boundary conditions? In other words, are all the parameters
in $s$ still pertinent to describe the residual gauge transformations, or do
some of them have been ruled out by the boundary conditions?

##### 2.1.4.1 General definition

A gauge transformation $\delta_{\lambda}\phi=R[\lambda]$ is tangent to the
solution space $\mathcal{S}$ if it preserves the gauge fixing conditions
(2.1.6) and the boundary conditions defining $\mathcal{S}$. The first
requirement implies that $\lambda$ has to be found among the residual gauge
transformations. The fact that $\delta_{\lambda}\phi$ is tangent to the
solution space is obviously an on-shell notion and gives some constraints
among the parameters $s$ which are generically dependent on the field
configuration $\phi$. The intuitive point of view is that such a gauge
transformation preserves the location of the limiting hypersurface
$\mathscr{B}$ defined in section (2.1.2) and the boundary structure defined on
$\mathscr{B}$ to implement the boundary conditions. Because this kind of
residual gauge transformation is designed to preserve the universal boundary
structure leading to the boundary conditions under consideration, it receives
the name of asymptotic symmetry. In gravity, when the boundary structure is a
fixed induced metric, a particular class of asymptotic symmetries, that
generalize the concept of exact isometries, are diffeomorphisms $\xi$ that
solve the Killing equation $\nabla_{(\mu}\xi_{\nu)}=0$ asymptotically. For
this reason they go under the name of asymptotic Killing vectors although this
nomenclature is not appropriate for more general sets of boundary conditions.
Even though our interest is regarding only to asymptotic boundaries, we
mention that numerous classes of asymptotic symmetries around boundaries
located at finite distance in the bulk of spacetime have been studied in the
literature, namely at the black hole horizon [44, 43, 123, 225, 226, 227, 228,
86, 229, 230, 231]. The study of asymptotic symmetries benefits also of a
popular research movement in other gauge theories (see e.g. [222, 232] for
Yang-Mills theories or [233, 234, 235] for electrodynamics and $k$-forms
theories).

##### 2.1.4.2 Asymptotic symmetry algebra

By definition, the asymptotic symmetries act internally on the solution space.
As usual, two symmetries can be performed successively on the physical system
described by the action integral (2.1.1) and the operation is equivalent to
another transformation performed in a single stroke. Each symmetry is meant to
be inverted and there exists a neutral transformation representing the absence
of transformation. In short, there is a notion of group of transformations
underlying the class of asymptotic symmetries. The composition of two gauge
transformations is translated at infinitesimal level by a commutation law for
the generators $\lambda$. Given a couple of generators
$(\lambda_{1},\lambda_{2})$, we define the modified bracket as [204, 46, 212]

$[\lambda_{1},\lambda_{2}]_{\star}\equiv[\lambda_{1},\lambda_{2}]-\delta_{\lambda_{1}}\lambda_{2}+\delta_{\lambda_{2}}\lambda_{1}.$
(2.1.20)

The first $[\lambda_{1},\lambda_{2}]$ represents the standard Lie bracket
between gauge parameters generically defined as [80]

$[\lambda_{1},\lambda_{2}]\equiv\sum_{(\mu),(\nu)}C_{\alpha\beta}^{(\mu)(\nu)}\partial_{(\mu)}\lambda_{1}^{\alpha}\partial_{(\nu)}\lambda_{2}^{\beta}$
(2.1.21)

where the structure functions $C_{\alpha\beta}^{(\mu)(\nu)}$ are anti-
symmetric in $(\alpha,\beta)$ and possibly field-dependent. For example, the
vector Lie bracket on the tangent bundle of $\mathscr{M}$,
$[\xi_{1},\xi_{2}]=(\xi_{1}^{\mu}\partial_{\mu}\xi_{2}^{\nu}-\xi_{2}^{\mu}\partial_{\mu}\xi_{1}^{\nu})\partial_{\nu}$,
falls into this definition. The additional terms
$-\delta_{\lambda_{1}}\lambda_{2}+\delta_{\lambda_{2}}\lambda_{1}$ in (2.1.20)
must be included as soon as the generators are field-dependent. They bring a
correction to extract the true action of one generator, say $\lambda_{1}$, on
the other, $\lambda_{2}$, regardless of the peculiar action of $\lambda_{1}$
on the field-dependence of $\lambda_{2}$. It can be shown that the so-defined
modified bracket is a Lie bracket (i.e. it acts as a antisymmetric bilinear
operator on the vector space of generators and satisfies the Jacobi identity)
since the standard bracket is itself a Lie bracket. This implies that the
asymptotic symmetry generators form an algebra, coined as the asymptotic
symmetry algebra.

##### 2.1.4.3 Representation on the solution space

The asymptotic symmetries act infinitesimally on the solution space
$\mathcal{S}$ as $\delta_{\lambda}\phi=\phi[p+\delta_{\lambda}p]-\phi[p]$
where the difference is evaluated by retaining only the linear order in
$\delta_{\lambda}p$. This is the statement that the formal structure of the
solution $\phi(p)$ is preserved in order to keep solving the equations of
motion for the prescribed the boundary conditions after that the
transformation has been performed. Only the parameters $p$ vary non-trivially
under the action of the generators $\lambda$ and this transformation is fixed
by the preservation requirement we have just mentioned. The algebra of
generators under the modified bracket (2.1.20) is therefore represented on
$\mathcal{S}$ because

$\delta_{\lambda_{1}}R[\lambda_{2}]-\delta_{\lambda_{2}}R[\lambda_{1}]=-R\left[[\lambda_{1},\lambda_{2}]_{\star}\right]$
(2.1.22)

holds on-shell, i.e. for any $\phi\in\mathcal{S}$ (see e.g. [213, 214, 2] for
a proof). The left-hand side defines a bracket on the variations as

$[\delta_{\lambda_{1}},\delta_{\lambda_{2}}]\phi\equiv\left[R[\lambda_{1}],R[\lambda_{2}]\right]$
(2.1.23)

while the right-hand side is nothing but the variation under the
transformation generated by the modified bracket of generators, i.e.
$\delta_{[\lambda_{1},\lambda_{2}]_{\star}}\phi=R[[\lambda_{1},\lambda_{2}]_{\star}]$.
Hence the representation result can be written as

$\boxed{[\delta_{\lambda_{1}},\delta_{\lambda_{2}}]\phi=-\delta_{[\lambda_{1},\lambda_{2}]_{\star}}\phi.}$
(2.1.24)

Because of the generic field-dependence of the asymptotic symmetries, there is
a tight link between them and the solution space from which they emerge. For
that reason, the natural structure appearing here is not properly a Lie
algebra, but rather a Lie algebroid [203, 204, 205] of asymptotic symmetries.

The axiomatic definition is as follows. Let $\mathscr{N}$ be a differentiable
manifold. A Lie algebroid over $\mathscr{N}$ consists of a vector bundle
$\mathscr{A}$ together with an anchor map $\rho_{\mathscr{A}}:\mathscr{A}\to
T\mathscr{N}$ from $\mathscr{A}$ onto the tangent bundle of $\mathscr{N}$ and
a Lie bracket $[\cdot,\cdot]_{\mathscr{A}}$ on the space of sections
$\Gamma(\mathscr{A})$, satisfying the Leibniz rule
$[\alpha,f\beta]_{\mathscr{A}}=f[\alpha,\beta]_{\mathscr{A}}+\beta\mathcal{L}_{\rho_{\mathscr{A}}(\alpha)}f$
for all $\alpha,\beta$ in the vector space $\Gamma(\mathscr{A})$ and all
smooth $f$ on $\mathscr{N}$. This formalizes the idea that a Lie algebroid is
a Lie algebra which depends on the point of $\mathscr{N}$ we want to consider
in the section $\Gamma(\mathscr{A})$. The most intuitive example of a Lie
algebroid is when $\mathscr{A}=T\mathscr{N}$ with the identity map as anchor
and $[\cdot,\cdot]_{\mathscr{A}}$ is the vector commutator. The tangent space
in a point $P\in\mathscr{N}$ is obviously depending on $P$ for general
geometrical configurations since it is a local notion defined from the tangent