BMS symmetries of
gravitational scattering

Xavier Kervyn
<EMAIL_ADDRESS>/<EMAIL_ADDRESS>
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Centre for Mathematical Sciences,
Wilberforce Rd, Cambridge CB3 0WA, United Kingdom

August 2023

After motivating the relevance of the Bondi-Metzner-Sachs (BMS) group over the
last decades, we review how concepts such as Penrose diagrams and the
covariant phase space formalism can be used to understand the asymptotic
structure of asymptotically flat spacetimes (AFS). We then explicitly
construct the asymptotic symmetry group of AFS in $3+1$ dimensions, the BMS
group. Next, we apply this knowledge to the usual far-field scattering problem
in general relativity, which leads to the unravelling of the intrinsic
features of gravity in the infrared. In particular, we work out the
connections between asymptotic symmetries, soft theorems in quantum field
theories and gravitational memory effects. We restrict to the study of this
infrared triangle through the lens of supertranslations here, but the
analogous features that can be found in the case of superrotations or for
other gauge theories are also motivated at the end of our discussion. We
conclude with an overview of the implications of the infrared triangle of
gravity for the formulation of an approach to quantum gravity through
holography, as well as a brief discussion of its potential in tackling the
black hole information paradox.

This review article arose from an essay submitted for the partial fulfillment
of the requirements for the degree of Master of Advanced Study in Applied
Mathematics (Part III of the Mathematical Tripos) at the University of
Cambridge, set by Dr. Prahar Mitra and submitted in May 2023. It is aimed at
advanced undergraduate students or early postgraduate students willing to
learn about the role of asymptotic symmetries in the context of flat space
holography, with only basic knowledge of quantum field theory and general
relativity assumed.

###### Contents

  1. 1 Introduction
  2. 2 Asymptotic structure of Minkowski spacetime
    1. 2.1 Conformal compactification and scattering in Minkowski
    2. 2.2 Gravitational scattering at null infinity: coordinate conventions
    3. 2.3 Covariance, asymptotic symmetries and Covariant Phase Space formalism
    4. 2.4 Symmetries of flat space and Poincaré group
  3. 3 Asymptotic symmetries of asymptotically flat spacetimes
    1. 3.1 Asymptotically flat spacetimes
    2. 3.2 Bondi-Sachs metric
    3. 3.3 Bondi-Metzner-Sachs group in General Relativity
      1. 3.3.1 $\mathfrak{bms}_{4}$ algebra
      2. 3.3.2 Supertranslations
      3. 3.3.3 Lorentz transformations and superrotations
      4. 3.3.4 Structure of the BMS group
  4. 4 Infrared structure of gravity through supertranslations
    1. 4.1 Christodoulou-Klainerman spaces
    2. 4.2 BMS supertranslations and leading soft theorem
      1. 4.2.1 Supertranslation charges
      2. 4.2.2 Supertranslation invariance, soft graviton and Ward identity
      3. 4.2.3 Leading soft theorem
    3. 4.3 Gravitational memory effects
      1. 4.3.1 Displacement memory effect
      2. 4.3.2 Vacuum transitions
      3. 4.3.3 Memory and soft theorems
  5. 5 Conclusion and Outlook
    1. 5.1 The infrared triangle for BMS supertranslations
    2. 5.2 Outlook
      1. 5.2.1 Other copies of the infrared triangle
      2. 5.2.2 Towards flat space holography
      3. 5.2.3 Black hole information and soft hairs
  6. Acknowledgements
  7. A Appendix: detailed calculations
    1. A.1 BMS4 equations of motion and constraint equations for $m$ and $N^{A}$
    2. A.2 Determination of the BMS+ generators
    3. A.3 Effect of a supertranslation

##  1 Introduction

The current theoretical exploration in the search for further unification of
the forces of Nature is very often based on the search for new symmetries of
the theory. As Weinberg said,

> “Nothing in physics seems so hopeful to me as the idea that it is possible
> for a theory to have a high degree of symmetry was hidden from us in
> everyday life. The physicist’s task is to find this deeper symmetry.” – The
> Forces of Nature. (1976)

In mathematics, this notion ties beautifully to the concepts of group theory,
which in turn provides a general framework for the study of the laws of
physics. Gravity is no exception. In this article, we review the insights
provided by the study of asymptotic symmetries and their associated asymptotic
symmetry group on general relativity, the BMS group, restricting ourselves to
$3+1$ dimensions.

### Historical overview

Since Einstein’s publication of the theory of general relativity in 1915,
group-theoretical methods have been instrumental not only in deriving new
solutions of the associated Einstein equations and understanding their
properties [1, 2], but also in probing the asymptotic and conformal structure
of spacetime, notably through the work of Penrose and collaborators in the
1960s [3, 4, 5, 6]. At the same time, Bondi, Metzner and Sachs [7, 8, 9] first
formulated a set of symmetries that describe the behavior of gravitational
waves in asymptotically flat spacetimes. These were shown to be the building
blocks of a so-called asymptotic symmetry group, the BMS group, of which
McCarthy and collaborators investigated the representations and a number other
properties in the 1970s [10, 11, 12, 13]. However, while this work provided a
foundational understanding, it was not immediately clear how to incorporate
the BMS group into a broader theory of gravity.

In the subsequent decades, research in theoretical physics shifted towards
other areas, such as quantum field theory, string theory, and the study of
black holes. These fields garnered significant attention and resources,
leading to a temporary diversion of focus away from BMS-related work. In 1993,
Christodoulou and Klainerman (CK) [14] proved the nonlinear gravitational
stability of Minkowski space-time. In doing so, they also provided a
prescription for matching data incoming from future and past null infinity,
which is necessary to study scattering processes. At the time though, string
theory was still in full bloom, retaining much of the attention of the
scientific community, while the result did not seem of big importance to the
general relativity community. It was only in the early 2010s that Barnich and
Trossaert reignited interest in the BMS group [15], opening new avenues of
research along the lines of string theory and holography, where symmetries are
key ingredients of the duality. Combining their results with the work of CK,
Strominger, along with collaborators, then made significant contributions to
our understanding of the BMS group [16, 17, 18, 19, 20], see [21] and
references therein. Among others, their work connected the BMS symmetries to
the holographic principle, which relates gravitational theories in higher
dimensions to lower-dimensional quantum field theories, as well as to the
black hole information paradox.

Interestingly, these developments were concomitant with the successful
detection of gravitational waves by the Laser Interferometer Gravitational-
Wave Observatory (LIGO) in 2015, about 50 years after the publication of BMS’s
seminal papers. While no experimental data was needed for the aforementioned
theoretical progress, the experimental breakthroughs highlighted the relevance
of the BMS group and its potential implications for the fundamental nature of
spacetime. We briefly mention some the key insights provided so far by this
prism of study of the BMS group in the next paragraphs, in hindsight of and
with references to our discussion in the rest of this review article.

### Key insights from the BMS group

##### Infrared structure of gauge theories

BMS were originally expecting gravity in asymptotically flat spacetimes to
resemble special relativity as one recedes from an isolated gravitational
system. Solving the Killing equation to obtain the vector fields preserving
the form of their metric, they instead found that the BMS group is larger than
the Poincaré group. We follow their steps and construct the generators of the
BMS group in section 3. Strominger shed a new light on the implications of
this feature by showcasing a triangular equivalence relation between
asymptotic symmetries, soft theorems and memory effects as pictured in Fig. 1
[16, 22, 18].

Figure 1: The infrared triangle

In this diagram, asymptotic symmetries – such as the BMS symmetries – are only
one of the three equivalent entries to a much richer dictionary, allowing for
a better understanding of of each of the corners. The explicit derivation of
this triangle in the case of BMS supertranslations will be the backbone of
this review. We shall see that it has profound consequences for gravity,
showing for instance that the vacuum in general relativity is not unique.
Copies of the same triangle were subsequently constructed for electromagnetism
[23, 24], BMS superrotations [25], and other gauge theories [26, 27, 28], each
involving various soft theorems and unravelling new symmetries or memory
effects (work in progress).

##### Gravitational wave astronomy

The upper corner of Fig. 1, memory effects, refers to long-lasting changes in
the spacetime geometry caused by the passage of gravitational waves. These
effects can lead to a permanent displacement of test masses or the acquisition
of a new state of motion. Since BMS symmetries govern the asymptotic behavior
of gravitational waves in asymptotically flat spacetimes, they play a crucial
role in their propagation. Understanding them is thus essential for accurately
analyzing and interpreting the signals received from gravitational wave
detectors. As shall be made explicit in section 4.3 of this review, the BMS
group provides insights into the radiation patterns, conservation laws, and
energy-momentum properties of gravitational waves.

##### Quantum gravity and holography

The holographic conjecture is a major advance in theoretical physics which
relates gravitational theories in higher dimensions to lower-dimensional
quantum field theories; the prime example of it being the AdS/CFT
correspondence [29, 30]. This is still an active area of research and many
results remain unknown, especially for (asymptotically) flat spacetimes – in
fact in any but asymptotically negatively curved backgrounds. Recent research
suggested that the BMS group could be used to address this problem in
asymptotically flat spacetimes, by allowing to recast scattering amplitudes of
any four-dimensional theory with non-abelian gauge group as two-dimensional
correlation functions on the asymptotic two-sphere at null infinity [31]. This
led to the proposal that gravity in four-dimensional asymptotically flat
spacetimes may be dual to a theory living on the “celestial sphere” at
infinity [32], a program known as celestial holography [33]. The BMS
symmetries have been linked to the symmetries of the boundary theory in the
holographic dual, offering potential insights into the nature of spacetime,
information, and quantum gravity [34]. Many complexities and challenges
however still need to be carefully addressed.

##### Black hole physics

BMS symmetries also led to crucial insights into the conservation laws of
black holes. They govern the behavior of the waves they emit and thus provide
a framework to compute conserved quantities such as energy, momentum, and
angular momentum which are needed to characterize the dynamics and evolution
of such objects. Second, the BMS group also has connections to the black hole
information paradox [35], which arises due to the conflict between the
principles of quantum mechanics and classical general relativity, suggesting
that information can be lost in the process of black hole evaporation. In
particular, BMS supertranslations within the BMS group have been linked to
changes in the geometry and dynamics of the event horizon through soft hairs
[20, 36, 19]. The links of the BMS group to quantum gravity and the
holographic principle could help to understand the holographic nature of black
holes and the role of the event horizon in encoding information about the
interior [37], see also [21] and references therein. This research direction
thus has the potential to shed light on the fundamental nature of black holes
and their relation to quantum gravity, though many questions remain to be
addressed.

### Plan of the article

This review article is structured as follows. In section 2, we introduce key
concepts (Penrose diagram, asymptotic symmetries, covariant phase space
formalism, Poincaré group) as well as coordinate conventions to be used
throughout this article. Section 3 is then devoted to the construction of the
BMS group. In section 4 we eventually build upon these newly constructed
asymptotic symmetries of gravitational scattering to understand the intricate
structure of gravity. We focus on the role of supertranslations and their
connections to soft theorems and memory effects. Section 5 wraps up our
discussion by presenting the infrared triangle of gravity, its analogue for
BMS superrotations or other gauge theories, and then mentioning the role of
the latter in formulating an approach to quantum gravity through flat space
holography or tackling the black hole information paradox.

##  2 Asymptotic structure of Minkowski spacetime

In this section, we carry out the conformal compactification of Minkowski and
introduce the different regions of interest for the scattering problem in
gravity. Next, we explicit the scattering problem and set up the coordinate
system that will predominantly be used throughout this article. Finally, we
introduce the covariant phase space formalism as a set of methods allowing for
the construction of a symplectic form on the phase space of a covariant field
theory. We end with a lightning overview of the generators of the Poincaré
group, the group of isometries of flat space in $3+1$ dimensions.

###  2.1 Conformal compactification and scattering in Minkowski

The goal of conformal compactification is to extend the notion of infinity in
a geometric space, such as a manifold or a metric space, in a way that makes
it possible to study the properties of the space near infinity [38]. In
practice, it consists of a succession of clever coordinate transformations
that allow to bring infinity "at a finite coordinate distance away".

Let us carry out this procedure for Minkowski, $\mathbb{R}^{1,3}$. The latter
is usually described by Cartesian coordinates
$(x^{0},x^{1},x^{2},x^{3})\in\mathbb{R}^{4}$ and the flat metric
$\eta=\text{diag}(-1,+1,+1,+1)$, such that the invariant line element writes
$\mathrm{d}s^{2}=\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$. Our first
action will be to transform to spherical coordinates
$(x^{1},x^{2},x^{3})\to(r,\theta,\phi)$, with $x^{0}\equiv t$. In these
coordinates, the line element writes

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},\quad\text{where
}\mathrm{d}\Omega^{2}=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}.$

We then switch to the retarded and advanced null coordinates

$u=t-r,\quad v=t+r,\qquad-\infty<u\leq v<\infty,$ (2.1)

in terms of which we now have
$\mathrm{d}s^{2}=-\mathrm{d}u\mathrm{d}v+\frac{1}{4}(v-u)^{2}\mathrm{d}\Omega^{2}$.
Defining $U=\arctan u$ and $V=\arctan v$, we get

$\mathrm{d}s^{2}=\frac{1}{4\cos^{2}U\cos^{2}V}[-4\mathrm{d}U\mathrm{d}V+\sin^{2}(V-U)\mathrm{d}\Omega^{2}],\quad-\pi/2<U\leq
V<\pi/2.$

Transforming back to timelike coordinate $T$ and radial coordinate $R$ via
$T=V+U$, $R=V-U$, in the respective ranges $0\leq R<\pi$,
$\absolutevalue{T}+R<\pi$, we arrive at

$\mathrm{d}s^{2}=\frac{1}{\omega^{2}(T,R)}\underbrace{\left(-\mathrm{d}T\otimes\mathrm{d}T+\mathrm{d}R\otimes\mathrm{d}R+\sin^{2}R\mathrm{d}\Omega^{2}\right)}_{=:\mathrm{d}\tilde{s}^{2}},\quad\omega=2\cos
U\cos V=\cos T+\cos R.$

The original line element $\mathrm{d}s^{2}$ is conformally equivalent to the
unphysical line element $\mathrm{d}\tilde{s}^{2}$. The latter describes the
manifold $\mathbb{R}\times S^{3}$, which is no other than a region of Einstein
static universe, pictured in Fig. 2(a).

(a)

(b)

Figure 2: Conformal compactification of Minkowski space [21]

The shaded region can be unrolled to portray Minkowski space as a triangle.
This is our conformal diagram. Note that our manipulations did not affect the
compact coordinates $\theta,\phi$. Hence, each point on the conformal diagram
should be thought of a 2-sphere. It is convenient to mirror that triangle to
obtain a symmetric picture of the conformal diagram as in Fig. 2(b). On this
picture, each two-sphere of constant $(R>0,T)$ is represented by two points,
one on the left and one on the right, which are exchanged by the antipodal
map.

##### Asymptotic regions

The causality of Minkowski translates into several regions of interest in the
conformal diagram, also pictured in Figure 2(b): first, (future/past) timelike
infinity $i^{\pm}$ is such that all timelike geodesics begin at $i^{-}$ and
end at $i^{+}$. In the unphysical coordinates, $i^{+}=(T=\pi,R=0)$,
$i^{-}=(T=-\pi,R=0)$. Next, (future/past) null infinity $\cal I^{\pm}$ are
such that all null geodesics begin at $\mathcal{I}^{-}$ and end at $\cal
I^{+}$. In the unphysical coordinates, $\mathcal{I}^{-}=(T=-\pi+R,0<R<\pi)$,
$\mathcal{I}^{+}=(T=\pi-R,0<R<\pi)$. These are null surfaces with topology of
$\mathbb{R}\times\mathbb{S}^{2}$. Last but not least, spatial infinity $i^{0}$
is defined such that spacelike geodesics both begin and end at $i^{0}$
($i^{0}=(T=0,R=\pi)$). We now build upon these concepts to define our
scattering problem.

###  2.2 Gravitational scattering at null infinity: coordinate conventions

When studying scattering processes, one is usually interested in knowing how a
given initial state of a system transforms into a final state. In quantum
mechanics, this is associated to the amplitude

$\mathcal{A}_{\text{out,in}}=\langle out|in\rangle,$

where the scattering matrix ($\cal S$-matrix) relates the ingoing and outgoing
states, $|out\rangle=\mathcal{S}|in\rangle$. In this work, we ultimately want
to look at the scattering of massless particles (or wavepackets) in
asymptotically flat spacetimes, such as gravitational waves propagating. To
gain insight into this problem, we will need to specify $|in\rangle$, i.e.
provide initial data at $\cal I^{-}$ (since we do not consider any stable
massive particles, hence need not provide initial data at $i^{-}$).
Furthermore, one usually assumes that the theory is weakly interacting in the
far past and future. Incoming wavepackets then evolve towards each other,
interact and come out on $\cal I^{+}$ (again, we do not consider stable
massive particles here, so nothing happens at $i^{+}$). From Fig. 2(b), we
immediately see that what goes on near $i^{0}$ will be important. In
particular, we will need to understand how to relate fields at $\cal
I^{-}_{+}$ to fields at $\cal I^{+}_{-}$ by means of suitably defined matching
conditions. Both these points will be addressed in section 4. We now introduce
a convenient set of coordinates for describing null infinity.

##### Retarded and advanced coordinates

Retarded and advanced null coordinates (2.1) naturally parametrize $\cal
I^{\pm}$ respectively. As a matter of fact, the Minkowski metric in these
coordinates writes

$\mathrm{d}s^{2}=-\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}r+r^{2}\gamma_{AB}(x)\mathrm{d}x^{A}\mathrm{d}x^{B}=-\mathrm{d}v^{2}+2\mathrm{d}v\mathrm{d}r+r^{2}\gamma_{AB}(\tilde{x})\mathrm{d}\tilde{x}^{A}\mathrm{d}\tilde{x}^{B}.$
(2.2)

Here, $x^{A}$ and $\tilde{x}^{A}$ ($A\in\\{2,3\\}$) are antipodal coordinate
systems on the asymptotic 2-sphere, while $\gamma_{AB}$ is the round metric on
the 2-sphere. Future null infinity $\cal I^{+}$ is then best described in
retarded coordinates $(u,r,x^{A})$ as the boundary located at $r=\infty$,
keeping $(u,x^{A})$ fixed, whereas $\cal I^{-}$ is best described in advanced
coordinates $(v,r,\tilde{x}^{A})$ as the limit $r\to\infty$, keeping
$(v,\tilde{x}^{A})$ fixed. A specific choice for $x^{A}$ is preferred in the
literature: stereographic coordinates.

##### Stereographic coordinates

It is useful to think of $\mathbb{S}^{2}$ as the Riemann sphere
$\mathbb{C}\cup\infty$. Starting with standard spherical coordinates
$(\theta,\phi)\in[0,\pi[\times[0,2\pi[$, we use the stereographic projection
St to describe the sphere $\mathbb{S}^{2}$ as a complex 1-dimensional manifold
[39]:

$\text{St}:\mathbb{S}^{2}\backslash\\{(0,0,1)\\}\to\mathbb{C}\cong\mathbb{R}^{2},\quad\text{St}(\theta,\phi)\mapsto
z:=e^{i\phi}\tan\frac{\theta}{2},$ (2.3)

so that $\bar{z}=e^{-i\phi}\tan(\theta/2)$. This projection can be extended to
a diffeomorphism $\kappa$ such that

$\kappa:\mathbb{S}^{2}\to\mathbb{C}\cup\\{\infty\\},\quad\kappa(\mathbf{z})=\left\\{\begin{tabular}[]{ll}\text{St}($\mathbf{z}$)&$\mathbf{z}\in\mathbb{S}^{2}\backslash\\{(0,0,1)\\}$\\\
$\infty$&$\text{else}$\end{tabular}\right.$

where we denoted the resulting complex coordinates by
$\mathbf{z}=(z,\bar{z})$. The sets of coordinates $(u,r,z,\bar{z})$ and
$(v,r,z,\bar{z})$ are pictured in Figure 3(a) and 3(b) respectively.

(a)

(b)

Figure 3: Retarded and advanced coordinates on Minkowski space [21]

The former is related to standard Cartesian coordinates by

$u=t-\sqrt{x_{i}x^{i}},\quad r=\sqrt{x_{i}x^{i}},\quad
z=\frac{x^{1}+ix^{2}}{x^{3}+\sqrt{x_{i}x^{i}}},\quad\bar{z}=\frac{x^{1}-ix^{2}}{x^{3}+\sqrt{x_{i}x^{i}}},$
(2.4)

and inversely

$t=u+r,\quad x^{1}+ix^{2}=\frac{2rz}{1+z\bar{z}},\quad
x^{3}=\frac{r(1-z\bar{z})}{1+z\bar{z}}.$ (2.5)

In these coordinates, the $\mathbb{S}^{2}$ line element takes the form
$\mathrm{d}\Omega^{2}=2\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}$, with
$\gamma_{z\bar{z}}=2/(1+z\bar{z})^{2}$ and
$\gamma_{zz}=\gamma_{\bar{z}\bar{z}}=0$. The Minkowski line element (2.2)
writes

$\mathrm{d}s^{2}=-\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}r+2r^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}=-\mathrm{d}v^{2}+2\mathrm{d}v\mathrm{d}r+2r^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}.$
(2.6)

From (2.3) wee see that $z$ runs over entire complex plane, $z=0$ is the north
pole, $z=\infty$ is the south pole, $z\bar{z}=1$ is the equator and the
antipodal stereographic coordinates $(\tilde{z},\tilde{\bar{z}})$ are related
to $(z,\bar{z})$ as $\tilde{z}=-1/\bar{z}$, $\tilde{\bar{z}}=-1/z$. The minus
signs ensure that the $z$ in the advanced coordinates denotes the antipodal
point on the sphere to the $z$ in the retarded coordinates.111Hence, for a
light ray that crosses Minkowski space, the initial value of $z$ in advanced
coordinates is equal to the final value of $z$ in retarded coordinates.

###  2.3 Covariance, asymptotic symmetries and Covariant Phase Space formalism

In general relativity, the principle of general covariance establishes the
equivalence of all observers. This translates in gravity being a gauge theory
under the gauge group of diffeomorphic automorphisms over the spacetime
manifold. However, when analyzing the asymptotic behavior of the gravitational
field around a boundary – be it repelled to infinity – the situation
drastically changes.

##### Asymptotic symmetries

The presence of a boundary explicitly breaks general covariance by imposing a
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 under consideration [40, 41]. While a majority of transformations still
describe a pure redundancy of the theory with zero charge, some of the
residual gauge transformations preserving the structure around the boundary
are promoted to physical symmetries of the theory.

One defines residual symmetries as the symplectomorphisms preserving the
dynamics in the bulk and the boundary conditions. Among these, trivial
symmetries still have vanishing Noether charges (and therefore are true
redundancies of the system), while asymptotic symmetries222The name is rather
misleading, since these are not approximate but exact symmetries of the theory
in the asymptotic region (e.g. infinity) of spacetime. now acquire non-
vanishing Noether charges. They are, therefore, physical transformations
acting non-trivially on the field space, mapping the system into an
inequivalent configuration. From the perspective of the scattering problem
mentioned in the previous section, it will be useful to study the action of
the asymptotic symmetry group on the $\cal S$-matrix,

$\text{asymptotic symmetry group}=\frac{\text{residual gauge
symmetries}}{\text{trivial gauge symmetries}}.$

In fact, the restriction to asymptotic symmetries is needed if one wishes to
deal with a well-defined Poisson bracket of physical charges – see the notes
of Ciambelli et al. for a pedagogical review [41].

In the following, looking at radiative gravity in four dimensions will lead us
to work in asymptotically flat spacetimes (AFS), for which boundary conditions
are needed (section 3). There is a large freedom in the choice of falloffs and
gauge conditions one can pick, provided these are weak enough to allow for all
physically reasonable solutions, but strong enough to permit the construction
of the charges of asymptotic symmetries. We will follow the analysis of Bondi,
van der Burg and Metzner [7] to derive the asymptotic symmetry group of
asymptotically flat spacetimes at null infinity: the BMS group. Most of
section 4 will then be devoted to investigating the effect of the relevant
charges on the $\cal S$-matrix and their implications on the infrared
structure of gravity. Before doing so, we now formulate a prescription for
deriving the charges associated to asymptotic symmetries.

We have just fleshed out the implications of non-trivial conserved charges for
gravitational scattering. While the scattering problem is usually most
naturally discussed within the Hamiltonian formalism, in relativistic theories
however it is difficult to use since the choice of a preferred set of time
slices inevitably destroys manifest covariance. This problem is common to all
relativistic covariant field theories, and usually avoided by restricting to
Lagrangians (classical) and path integrals (quantum) approaches, but there
remain some applications, such as the initial value problem, for which the
tools inherited from the Hamiltonian formalism are too convenient to dispense
with.333For example it is only in the Hamiltonian formalism that one can do a
proper accounting of the degrees of freedom in a system and define
thermodynamic quantities such as energy and entropy [42].

##### CPS procedure

Crnkovic, Witten, Iyer, Lee, Wald and Zoupas [43, 44, 45, 46, 47] developed a
formalism which incorporates the powerful features of phase space analyses
without abandoning covariance: the covariant phase space (CPS) formalism. It
allows to build the phase space444What physicists refer to as a phase space is
also known as symplectic manifold $(\Gamma,\Omega)$ by mathematicians, that is
a smooth manifold $\Gamma$ equipped with a closed, degenerate two-form
$\Omega$. Recall a $q$-form $\Omega\in\Omega^{q}(\Gamma)$ is an antisymmetric
$(0,q)$ tensor field on $\Gamma$, which is said to be closed if:
$\mathrm{d}\Omega=0$, and non-degenerate if: $i_{X}\Omega=0\Rightarrow X=0$,
$\forall X\in T\Gamma$. of a covariant theory starting from its Lagrangian,
without having to explicitly refer to the Hamiltonian. The construction of
surface charges at the boundary then builds upon the variational
principle555Another definition for the surface charges was developed in
parallel by Barnich and Brandt [48], but relying on the equations of motion
rather than the variational principle [40]. as well as symplectic methods. At
the end of the day, the CPS formalism boils down to an algorithm involving six
essential steps, schematically pictured in Fig. 4 and summarized in what
follows.

  1. 1.

Starting from a four-dimensional spacetime $(\mathcal{M},g)$, the
configuration space $\mathfrak{F}$ is first obtained as the space of all
allowed field configurations, defined by imposing boundary conditions on the
field of the generally covariant theory. The Lagrangian density $\cal L$
governing the dynamics of the system can then be constructed. It generally
depends both on the metric $g_{ab}$, the matter fields $\psi$ and a finite
number of their derivatives, which we combine under the collective variable
$\phi\equiv(g_{ab},\psi)$ in the following. Under the variation
$\phi\to\phi+\delta\phi$, $\cal L$ changes as
$\mathcal{L}\to\mathcal{L}+\delta\mathcal{L}$, with

$\delta\mathcal{L}=\delta\phi^{i}\frac{\partial\mathcal{L}}{\partial\phi^{i}}+\partial_{\mu}\delta\phi^{i}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi^{i})}+...\equiv\delta\phi^{i}\mathcal{E}_{i}+\partial_{\mu}\theta^{\mu}[\phi,\delta\phi],$
(2.7)

where

$\mathcal{E}_{i}\equiv\frac{\partial\mathcal{L}}{\partial\phi^{i}}-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi^{i})}\right)+\partial_{\mu}\partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\partial_{\nu}\phi^{i})}\right)+...$

are the Euler-Lagrange terms and $\theta$ is the symplectic potential current
density obtained by successive application of the Leibniz rule to get rid of
the terms of the form $\partial_{\mu}\delta\phi^{i}$,
$\partial_{\mu}\partial_{\nu}\delta\phi^{i}$, etc. and factor out the
variation $\delta\phi^{i}$ in front of $\mathcal{E}_{i}$. From the perspective
of the variational bicomplex, the Lagrangian form $L\equiv\epsilon\mathcal{L}$
is a $4$-form on $\cal M$ and a function on $\mathfrak{F}$ [49]. In the
language of differential forms, (2.7) writes

$\delta L=\delta\phi^{i}\frac{\delta
L}{\delta\phi^{i}}+\mathrm{d}\theta[\phi,\delta\phi].$ (2.8)

$\theta$ is now the $3$-form on $\mathcal{M}$ associated with the symplectic
current potential. By construction, it is defined only up to a closed form in
spacetime $\kappa\in\Omega^{2}(\mathcal{M})$,
$\theta\to\theta+\mathrm{d}\kappa$.

  2. 2.

Requiring that the variational principle hold amounts to the Euler-Lagrange
equations $\mathcal{E}_{i}=0$. This takes us to the solution space
$\mathfrak{S}\subset\mathfrak{F}$. Configurations in $\mathfrak{S}$ are said
to be on-shell.

  3. 3.

Next, we define the symplectic current density $\mathbf{\omega}$ as the
exterior derivative of $\theta$ on $\mathfrak{S}$,

$\omega=\mathrm{d}\theta\in\Omega^{3}(\mathcal{M})\times\Omega^{2}(\mathfrak{S}).$

By construction, $\omega$ is closed in $\mathfrak{S}$ and $\cal M$ [49]. For
two field variations $\delta_{1}\phi$ and $\delta_{2}\phi$, we have

$\omega[\phi,\delta_{1}\phi,\delta_{2}\phi]=\delta_{1}\theta[\phi,\delta_{2}\phi]-\delta_{2}\theta[\phi,\delta_{1}\phi].$
(2.9)

  4. 4.

Our next task is to integrate $\omega$ and $\theta$ over a Cauchy slice666A
Cauchy surface is a subset of the manifold which is intersected by every
maximal causal curve exactly once. Once the initial data is fixed on such a
codimension 1 surface, the field equations lead to the evolution of the system
in the entire spacetime [50]. $\Sigma$ of $\mathcal{M}$. This yields the
presymplectic form

$\tilde{\Omega}_{\Sigma}[\phi,\delta_{1}\phi,\delta_{2}\phi]\equiv\int_{\Sigma}\omega[\phi,\delta_{1}\phi,\delta_{2}\phi]$

and presymplectic potential $\tilde{\Theta}_{\Sigma}$ such that
$\tilde{\Omega}_{\Sigma}=\mathrm{d}\tilde{\Theta}_{\Sigma}$. By construction
the former is closed and thus a good candidate for the symplectic form of the
phase space of our theory, but nothing guarantees that it is indeed non-
degenerate (hence the name "presymplectic"). In particular, the freedom in
defining $\theta$ up to a closed 2-form in (2.8) implies that
$\tilde{\Omega}_{\Sigma}$ is not uniquely defined.

  5. 5.

We thus need to construct the equivalence classes [$\sim$] of degeneracy
subspaces of $\tilde{\Omega}_{\Sigma}$, and then quotient $\mathfrak{S}$ by
the latter. This fixes $\Theta_{\Sigma}=\tilde{\Theta}_{\Sigma}|_{\Gamma}$ and
$\Omega_{\Sigma}=\tilde{\Omega}_{\Sigma}|_{\Gamma}$. The latter is now
nondegenerate and independent of $\Sigma$ provided the variations
$\delta_{1}\phi$ and $\delta_{2}\phi$ obey the equations of motion [51].

  6. 6.

The phase space of our theory is $\Gamma=\mathfrak{S}/\sim$, with associated
symplectic form $\Omega_{\Sigma}$ and symplectic potential $\Theta_{\Sigma}$.

Spacetime $(\mathcal{M},g)$ Configuration space $\mathfrak{F}$ Lagrangian form
$L=\epsilon\mathcal{L}$ $\delta L$, equations of motion Solution space
$\mathfrak{S}\subset\mathfrak{F}$ define $\omega$ obtain
$\tilde{\Omega}_{\Sigma}$ and $\tilde{\Theta}_{\Sigma}$ identify equivalence
class of degeneracy subspaces Phase space $\Gamma\equiv\mathfrak{S}/\sim$
Canonical transformations & Poisson brackets 123456 symplectic geometry Figure
4: Covariant phase space formalism

##### Canonical transformations

We sketched how one constructs the symplectic form for a theory on a given
Cauchy surface. Let us now see how the conserved charges follow. Omitting the
reference to $\Sigma$ for now, we have [49]

$\Omega=\mathrm{d}\Theta\text{ (up to closed one-form)
}\Rightarrow\Omega(X,Y)=X(\Theta(Y))-Y(\Theta(X))-\Theta([X,Y]).$

In particular, $\Omega\in\Omega^{2}(\Gamma)$ but can also be thought as
$\Omega\in\Omega^{1}(\Gamma)$ by leaving one slot empty,

$\Omega:T\Gamma\to\Omega^{1}(\Gamma),\quad\Omega(X)\equiv-\iota_{X}\Omega\quad\text{such
that }\Omega(X)(Y)=-\Omega(Y)(X)=-\Omega(X,Y),$ (2.10)

with $\iota_{X}$ denoting the interior derivative777The interior product
$\iota_{X}\Omega$ is the contraction of the differential form $\Omega$ with
the vector field $X$, $\iota_{X}:\Omega^{p}(M)\to\Omega^{p-1}(M)$ with
$(\iota_{X}\Omega)(X_{1},...,X_{p-1})=\Omega(X,X_{1},...,X_{p-1})$. on
$\Gamma$ and $T\Gamma$ the tangent space to the symplectic manifold $\Gamma$.
$\Omega$ being non-degenerate, one can find an inverse map
$\Omega^{-1}:\Omega^{1}(\Gamma)\to T\Gamma$ that also acts as antisymmetric
bilinear map acting on $\Omega^{1}(\Gamma)\times\Omega^{1}(\Gamma)$.
Diffeomorphisms on $\Gamma$ that preserve the symplectic form $\Omega$ are
called canonical transformations. They are generated by Hamiltonian vector
fields $X_{f}$ obeying

$\mathcal{L}_{X_{f}}\Omega=0,$ (2.11)

with $\cal L$ the Lie derivative with respect to the metric on our spacetime.
Using Cartan’s homotopy formula
$\mathcal{L}_{X}=\mathrm{d}\iota_{X}+\iota_{X}\mathrm{d}$ and the closedness
of $\Omega$, one obtains

$\mathcal{L}_{X_{f}}\Omega=\mathrm{d}\iota_{X_{f}}\Omega+\iota_{X_{f}}\cancel{\mathrm{d}\Omega}\overset{!}{=}0\Rightarrow\iota_{X_{f}}\Omega=-\mathrm{d}f\in\Omega^{1}(\Gamma),$

for $f\in\mathcal{F}(\Gamma)$ the Hamiltonian charge. $\mathrm{d}f$ can here
be seen as the infinitesimal charge associated with the symmetry generated by
the Hamiltonian vector field $X_{f}$. Using (2.10), we can write an explicit
relation between this infinitesimal charge and the symplectic (one-)form:

$\Omega(X_{f})=\mathrm{d}f\Leftrightarrow X_{f}=\Omega^{-1}(\mathrm{d}f).$
(2.12)

Restoring the reference to the Cauchy slice of interest, we write

$\cancel{\delta}Q_{\xi}[\phi,\delta\phi]=\int_{\Sigma}\omega[\phi,\delta\phi,\delta_{\xi}\phi]=\Omega_{\Sigma}[\phi,\delta\phi,\delta_{\xi}\phi]$
(2.13)

for the infinitesimal charge associated to a symmetry generated by the vector
field $\xi$. Here, we introduced the notation $\cancel{\delta}$ used by
Alessio and Arzano [51] to emphasize that (2.13) might not be an exact
differential, or integrable. It is straightforward to build upon this
symplectic structure and define the Hamiltonian charges associated to
canonical transformations generated by the Lie bracket of Hamiltonian vector
fields, which we call Poisson bracket. Let $X_{f}$ and $X_{g}$ be two such
vector fields corresponding to two Hamiltonian charges $f,g$ on phase space
respectively. Then (2.11) $\Rightarrow$ (2.12) can be rewritten as

$\mathcal{L}_{[X_{f},X_{g}]}\Omega=[\mathcal{L}_{X_{f}},\mathcal{L}_{X_{g}}]\Omega=0\Rightarrow\mathrm{d}h=\Omega(X_{f},X_{g})\equiv-\\{f,g\\}\quad\text{(Poisson
bracket).}$

With this definition, we recover the usual properties of the Poisson brackets
in analytical mechanics.

###  2.4 Symmetries of flat space and Poincaré group

Before looking at gravity in asymptotically flat spacetimes, let us briefly
identify the symmetries of flat space. In $3+1$ dimensional Minkowski, there
are ten isometries which form the well-known Poincaré group. The generators
$\zeta$ of the latter solve the Killing equation

$\mathcal{L}_{\zeta}g_{ab}=0\Leftrightarrow\nabla_{a}\zeta_{b}+\nabla_{b}\zeta_{a}=0.$

This gives rise to three rotation generators

$\xi_{12}=x^{1}\partial_{2}-x^{2}\partial_{1},\quad\xi_{13}=-x^{1}\partial_{3}+x^{3}\partial_{1},\quad\xi_{32}=x^{2}\partial_{3}-x^{3}\partial_{2}$
(2.14)

and three boost generators

$\xi_{01}=x^{1}\partial_{0}+x^{0}\partial_{1},\quad\xi_{02}=x^{2}\partial_{0}+x^{0}\partial_{2},\quad\xi_{03}=x^{3}\partial_{0}+x^{0}\partial_{3}$
(2.15)

which together form the Lorentz group. On top of this, one also gets an
Abelian normal subgroup of four spacetime translations

$\xi_{0}=\partial_{0},\quad\xi_{1}=\partial_{1},\quad\xi_{2}=\partial_{2},\quad\xi_{3}=\partial_{3}.$
(2.16)

As a result, we have the group structure:
$\text{Poincar\'{e}}=\text{Lorentz}\ltimes\text{Translations}.$ We do not
prove these standard results here and refer the interested reader to e.g. the
work of Hirata [52] or Compère and Fiorucci [50] for the construction of these
generators and their associated charges. We now turn to the asymptotic
symmetries of general relativity in asymptotically flat spacetimes. In
particular, we will see that the asymptotic symmetry group of such spaces is
larger than the Poincaré group, meaning that general relativity does not
reduce to special relativity at large distances!

##  3 Asymptotic symmetries of asymptotically flat spacetimes

In section 2.3, we explained how boundary conditions give rise to asymptotic
symmetries. We now apply these ideas to look at asymptotically flat
spacetimes. We first define the latter and then introduce a metric treatment
that will be convenient of the study of gravitational scattering: the Bondi-
Sachs formalism. We later on make good use of this formalism to derive the
asymptotic symmetry group of asymptotically flat spacetimes, the BMS group.

###  3.1 Asymptotically flat spacetimes

In general relativity, an asymptotically flat spacetime (AFS) corresponds to
the intuitive notion of an isolated system. The formal definition of the
latter is however not so straightforward, notably because the metric acts both
as physical field and background. Consider a system alone in the universe,
described by a spacetime $\cal M$. As one recedes from the system, we expect
its influence to decrease, so we expect $\cal M$ to resemble flat Minkowski
spacetime $\mathbb{R}^{1,3}$, with this approximation becoming even better the
farther away we go. This is the intuitive picture. The formal definition of an
AFS involves the concept of asymptotically simple spacetime [53, 54]. We omit
the formal definition here for conciseness, only stating that an
asymptotically simple spacetime provides all the necessary conditions to
perform a conformal compactification in the spirit of what we did in section
2.1. To arrive at an AFS, one must make additional assumptions on behaviour of
the curvature near $\cal I$.

##### Asymptotic flatness

With an asymptotically simple spacetime at hand, there are two ways to define
asymptotic flatness according to Compère and Fiorucci [50], either

  1. 1.

using covariant objects but involving unphysical fields such as a conformal
factor used to do a Penrose compactification of spacetime (one would then
expect the causal structure of the AFS to resemble the one of Minkowski seen
in section 2.1); or

  2. 2.

using an adapted coordinate system and specifying fall-off conditions.

We follow the second route in this article, which allows for an easier
analysis of the details the asymptotic structure, even though it involves a
choice of coordinates which makes unclear whether the definition is still
covariant. Both approaches are of course expected to be equivalent, and the
choices made by BMS can indeed be justified in terms of Penrose
compactification of $\mathcal{I}^{+}$ [55, 56].

###  3.2 Bondi-Sachs metric

The description of an AFS by means of suitable coordinates and falloffs is due
to BMS [7], who described the propagation of gravitational waves in four-
dimensional AFS endowed with an additional axial and reflexion symmetry
($\phi\to-\phi$). To do so, they introduced a convenient choice of metric
satisfying the so-called Bondi gauge.

##### Bondi gauge

As motivated in section 2.2, it is convenient to use set of coordinates
$(u,r,\theta,\phi)$, where $u$ is the retarded time encountered previously in
(2.1), and $\theta,\phi$ the usual spherical coordinates. One can foliate the
original spacetime in a family of null hypersurfaces
$\Gamma_{c}=\\{(u,r,x^{A}):u=c\\}$ and define $r$ to be the future-pointing
null radial coordinate along these

Figure 5: Bondi coordinates

hypersurfaces, while $x^{A}$ ($A=2,3$) are two compact angular coordinates.
This setup is schematically depicted in Figure 5. Null hypersurfaces have null
normal vector $k_{\mu}=-\partial_{\mu}u$, which is also tangent to the surface
[57]. In particular,

$k^{2}=k^{\mu}k_{\mu}=g^{\mu\nu}(\partial_{\mu}u)(\partial_{\nu}u)=0\quad\Rightarrow\quad
g^{uu}=0.$

Since $x^{A}$ and $u$ are orthogonal directions we also have

$k^{\mu}\partial_{\mu}x^{A}=-g^{\mu\nu}(\partial_{\mu}u)(\partial_{\nu}x^{A})=0\quad\Rightarrow\quad
g^{uA}=0.$ (3.1)

Hence for $g^{\mu\nu}$ to be non-degenerate we need $g^{ur}\neq 0$. Actually,
we can be more precise. For $X$ and $Y$ two causal vectors belonging to the
same connected component of the light cone, one must have that
$g_{ab}X^{a}Y^{b}\leq 0$ [58] so in our case $g^{ur}<0$. Schematically,

$g^{\mu\nu}=\begin{pmatrix}0&g^{ur}&0&0\\\
g^{ur}&g^{rr}&g^{r\theta}&g^{r\phi}\\\ 0&g^{r\theta}&g^{22}&g^{33}\\\
0&g^{r\phi}&g^{23}&g^{33}\end{pmatrix}^{\mu\nu}$

where we introduced the inverse $2\times 2$ metric $g^{AB}$ for the angular
components. Inverting $g^{\mu\nu}$, one can easily check that we get
$g_{rr}=g_{rA}=0$. Meanwhile, the radial coordinate $r$ is still unspecified.
The choice of BMS was to define it as the radial luminosity distance along a
null geodesic and impose

$\partial_{r}\text{det}(r^{-2}g_{AB})=0.$ (3.2)

We have arrived at the Bondi gauge

$g_{rr}=g_{rA}=0\quad\text{and}\quad\partial_{r}\text{det}\left(\frac{g_{AB}}{r^{2}}\right)=0.$
(3.3)

It is worth emphasizing that our construction is possible for any metric,
since in $D$ dimensions there are $D$ coordinate choices to be made, and the
Bondi gauge amounts precisely to $1+(D-2)+1=D$ conditions. Let us now see how
to parametrize the metric in this gauge.

##### Bondi-Sachs metric

We now construct the metric in the Bondi gauge. We saw that $g^{ur}<0$. A
natural parametrization is thus $g^{ur}=-e^{2\beta}$ with
$\beta\equiv\beta(u,r,x^{A})$. We can then define $g^{rr}\equiv e^{-4\beta}U$
and $g^{rA}\equiv\frac{1}{2}e^{-2\beta}U^{A}$ with $U,U^{A}$ all functions of
$(u,r,x^{A})$ to get

$g^{\mu\nu}=\begin{pmatrix}0&-e^{-2\beta}&0&0\\\
-e^{-2\beta}&e^{-4\beta}U&\frac{1}{2}e^{-2\beta}U^{2}&\frac{1}{2}e^{-2\beta}U^{3}\\\
0&\frac{1}{2}e^{-2\beta}U^{2}&g^{22}&g^{23}\\\
0&\frac{1}{2}e^{-2\beta}U^{3}&g^{23}&g^{33}\end{pmatrix}^{\mu\nu}.$ (3.4)

The form of these coefficients is motivated by the work of BMS [7], albeit
slightly different in order to match the notation of Strominger [21] and
simplify calculations later on; the key point being that $g^{ur}<0$. Inverting
(3.4), one finds for the covariant components

$g_{\mu\nu}=\begin{pmatrix}-U+\frac{1}{4}g_{AB}U^{A}U^{B}&-e^{2\beta}&\frac{1}{2}g_{2A}U^{A}&\frac{1}{2}g_{3A}U^{A}\\\
-e^{2\beta}&0&0&0\\\ \frac{1}{2}g_{2A}U^{A}&0&g_{22}&g_{23}\\\
\frac{1}{2}g_{3A}U^{A}&0&g_{23}&g_{33}\end{pmatrix}_{\mu\nu}.$ (3.5)

Such a metric corresponds to the line element

$\mathrm{d}s^{2}=-U\mathrm{d}u^{2}-2e^{2\beta}\mathrm{d}u\mathrm{d}r+g_{AB}\left(\mathrm{d}x^{A}+\frac{1}{2}U^{A}\mathrm{d}u\right)\left(\mathrm{d}x^{B}+\frac{1}{2}U^{B}\mathrm{d}u\right).$
(3.6)

This is the retarded Bondi-Sachs metric. Choosing the advanced time $v$
instead of $u$ in our parametrization similarly yields the advanced Bondi-
Sachs metric

$\mathrm{d}s^{2}=-V\mathrm{d}v^{2}+2e^{2\beta^{-}}\mathrm{d}v\mathrm{d}r+g_{AB}^{-}\left(\mathrm{d}x^{A}-\frac{1}{2}V^{A}\mathrm{d}v\right)\left(\mathrm{d}x^{B}-\frac{1}{2}V^{B}\mathrm{d}v\right),$

where we introduced $\beta^{-}$, $g_{AB}^{-}$, $V$ and $V^{A}$. Switching
$x^{A}=(\theta,\phi)\to(z,\bar{z})$ as in (2.3), we have

$\mathrm{d}s^{2}=-U\mathrm{d}u^{2}-2e^{2\beta}\mathrm{d}u\mathrm{d}r+\left[g_{zz}\left(\mathrm{d}z+\frac{1}{2}U^{z}\mathrm{d}u\right)^{2}+g_{z\bar{z}}\left(\mathrm{d}z+\frac{1}{2}U^{z}\mathrm{d}u\right)\left(\mathrm{d}\bar{z}+\frac{1}{2}U^{\bar{z}}\mathrm{d}u\right)+\text{c.c.}\right]\\\
=-U\mathrm{d}u^{2}-2e^{2\beta}\mathrm{d}u\mathrm{d}r+\left[g_{zz}\left(\mathrm{d}z^{2}+U^{z}\mathrm{d}u\mathrm{d}z+\frac{1}{4}U^{z}U^{z}\mathrm{d}u^{2}\right)\right.\\\
\left.+g_{z\bar{z}}\left(\mathrm{d}z\mathrm{d}\bar{z}+\frac{1}{2}U^{\bar{z}}\mathrm{d}u\mathrm{d}z+\frac{1}{2}U^{z}\mathrm{d}u\mathrm{d}\bar{z}+\frac{1}{4}U^{z}U^{\bar{z}}\mathrm{d}u^{2}\right)+\text{c.c.}\right],$
(3.7)

while for advanced coordinates one gets

$\mathrm{d}s^{2}=-V\mathrm{d}v^{2}-2e^{2\beta^{-}}\mathrm{d}v\mathrm{d}r+\left[g^{-}_{zz}\left(\mathrm{d}z^{2}-V^{z}\mathrm{d}v\mathrm{d}z+\frac{1}{4}V^{z}V^{z}\mathrm{d}v^{2}\right)\right.\\\
\left.+g^{-}_{z\bar{z}}\left(\mathrm{d}z\mathrm{d}\bar{z}-\frac{1}{2}V^{\bar{z}}\mathrm{d}v\mathrm{d}z-\frac{1}{2}V^{z}\mathrm{d}v\mathrm{d}\bar{z}+\frac{1}{4}V^{z}V^{\bar{z}}\mathrm{d}v^{2}\right)+\text{c.c.}\right].$
(3.8)

Again, we stress that this construction is general, meaning that any metric
can be put in this form.

##### Asymptotics at infinity

Now, if we want to focus on AFS, we have to impose falloff conditions on the
components of the metric. At $\cal I^{+}$, this amounts to performing an
analytic expansion888It is assumed [7] that for any choice of $u$ one can take
the limit $r\to\infty$ along each ray. Newman and Unti [59] replaced this
condition with a weaker statement. As a matter of fact, Penrose [3, 60] showed
that the peeling theorem is violated in four dimensions, and we cannot expect
the expansion to be analytic in general. However, in turns out that the
analycity assumption is sufficient in the case of BMS supertranslations, which
is precisely within the scope of our considerations in this article. of the
coefficients $U,\beta$ and $U^{A}$ in powers of $1/r$ at large distances.
There is no a priori preferred method to determine these falloff conditions on
the metric components. Following [7, 9, 61] we write:

$\displaystyle U(u,r,x^{A})$
$\displaystyle=1-\frac{2m(u,x^{A})}{r}+\frac{U_{2}(u,x^{A})}{r^{2}}+\mathcal{O}(r^{-3}),$
(3.9) $\displaystyle\beta(u,r,x^{A})$
$\displaystyle=\frac{\beta_{1}(u,x^{A})}{r}+\frac{\beta_{2}(u,x^{A})}{r^{2}}+\frac{\beta_{3}(u,x^{A})}{r^{3}}+\mathcal{O}(r^{-4}),$
$\displaystyle U^{A}(u,r,x^{B})$
$\displaystyle=\frac{U^{A}_{2}(u,x^{B})}{r^{2}}+\frac{U^{A}_{3}(u,x^{B})}{r^{3}}+\frac{U^{A}_{4}(u,x^{B})}{r^{4}}+\mathcal{O}(r^{-5}),$
$\displaystyle g_{AB}(u,r,x^{C})$
$\displaystyle=r^{2}\gamma_{AB}(x^{C})+rC_{AB}(u,x^{C})+D_{AB}(u,x^{C})+\mathcal{O}(r^{-1}).$

We adopt the convention that capital Roman indices $A,B$ etc. are raised and
lowered with the round metric on the 2-sphere $\gamma_{AB}$ and its inverse
$\gamma^{AB}$, and we denote by $D_{A}$ the covariant derivative associated
with $\gamma_{AB}$. In particular, we have that the trace of $C_{AB}$ vanishes
by (3.2), since

$\text{det}\left(\frac{g_{AB}}{r^{2}}\right)=\text{det}\left(\gamma_{AB}+\frac{C_{AB}}{r}+\mathcal{O}(r^{-2})\right)=\text{det}(\gamma)\left(1+\frac{C^{A}\,_{A}}{r}+\mathcal{O}(r^{-2})\right)$

and thus $\partial_{r}\text{det}(g_{AB}/r^{2})=0$ requires
$\gamma^{AB}C_{AB}=0$ to $\mathcal{O}(r^{-1})$. For $x^{A}=(z,\bar{z})$ the
stereographic coordinates (2.3),

$\gamma^{AB}C_{AB}=0\Leftrightarrow
2\frac{(1+z\bar{z})^{2}}{2}C_{z\bar{z}}=0\Rightarrow C_{z\bar{z}}=0,$

while $C_{zz}$ and $C_{\bar{z}\bar{z}}$ are left unspecified. From (3.7) we
can read off the metric coefficients

$\displaystyle
g_{uu}=-U+\frac{1}{4}g_{zz}U^{z}U^{z}+\frac{1}{4}g_{\bar{z}\bar{z}}U^{\bar{z}}U^{\bar{z}}+\frac{1}{2}g_{z\bar{z}}U^{z}U^{\bar{z}},\quad\text{and}\quad
g_{ur}=-e^{2\beta}$ (3.10) $\displaystyle
g_{uz}=\frac{1}{2}g_{zz}U^{z}+\frac{1}{2}g_{z\bar{z}}U^{\bar{z}},\quad\text{and}\quad
g_{u\bar{z}}=\frac{1}{2}g_{\bar{z}\bar{z}}U^{\bar{z}}+\frac{1}{2}g_{z\bar{z}}U^{z}.$
(3.11)

The angular components of the metric write to $\mathcal{O}(r)$

$g_{zz}=r^{2}\cancel{\gamma_{zz}}+rC_{zz},\quad
g_{\bar{z}\bar{z}}=r^{2}\cancel{\gamma_{\bar{z}\bar{z}}}+rC_{\bar{z}\bar{z}},\quad\text{and}\quad
g_{z\bar{z}}=r^{2}\gamma_{z\bar{z}}+r\cancel{C_{z\bar{z}}}.$

These together with the falloffs (3.9) for $U$, $\beta$ and $U^{A}$ help
rewrite

$\displaystyle g_{uu}$
$\displaystyle=-\left(1-\frac{2m}{r}\right)+\left[\frac{rC_{zz}}{4}\left(\frac{U^{z}_{2}}{r^{2}}+\frac{U^{z}_{3}}{r^{3}}\right)^{2}+\text{c.c.}\right]+\frac{r^{2}\gamma_{z\bar{z}}}{2}\left(\frac{U^{z}_{2}}{r^{2}}+\frac{U^{z}_{3}}{r^{3}}\right)\left(\frac{U^{\bar{z}}_{2}}{r^{2}}+\frac{U^{\bar{z}}_{3}}{r^{3}}\right)$
(3.12) $\displaystyle=-1+\frac{2m}{r}+\mathcal{O}(r^{-2}).$

Assuming $\beta_{1}/r\ll 1$, we also get

$g_{ur}=-\exp\left(\frac{2\beta_{1}}{r}+\mathcal{O}(r^{-2})\right)=-1-\frac{2\beta_{1}}{r}+\mathcal{O}(r^{-2}),$

and finally

$\displaystyle g_{uz}$
$\displaystyle=\frac{rC_{zz}}{2}\left(\frac{U^{z}_{2}}{r^{2}}+\frac{U^{z}_{3}}{r^{3}}\right)+\frac{r^{2}\gamma_{z\bar{z}}}{2}\left(\frac{U^{\bar{z}}_{2}}{r^{2}}+\frac{U^{\bar{z}}_{3}}{r^{3}}\right)=\frac{1}{2}U_{2z}+\frac{1}{r}\left(\frac{1}{2}C_{zz}U^{z}_{2}+\frac{1}{2}U_{3z}\right)+\mathcal{O}(r^{-2})$
$\displaystyle\text{and}\quad g_{u\bar{z}}$
$\displaystyle=\frac{1}{2}U_{2\bar{z}}+\frac{1}{r}\left(\frac{1}{2}C_{\bar{z}\bar{z}}U^{\bar{z}}_{2}+\frac{1}{2}U_{3\bar{z}}\right)+\mathcal{O}(r^{-2})\qquad\Rightarrow
g_{uA}=\frac{1}{2}\gamma_{AB}U^{B}_{2}+\mathcal{O}(r^{-1}).$

These coefficients correspond to a metric of the form

$g_{\mu\nu}=\begin{pmatrix}-\left(1-\frac{2m}{r}\right)&-1-\frac{2\beta_{1}}{r}&g_{uz}&g_{u\bar{z}}\\\
-1-\frac{2\beta_{1}}{r}&0&0&0\\\ g_{uz}&0&rC_{zz}&r^{2}\gamma_{z\bar{z}}\\\
g_{u\bar{z}}&0&r^{2}\gamma_{z\bar{z}}&rC_{\bar{z}\bar{z}}\end{pmatrix}_{\mu\nu}+\mathcal{O}(r^{-2}).$
(3.13)

To determine the various coefficients in the expansion (3.9), we require that
(3.5) satisfy the Einstein equations

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}.$ (3.14)

We now need to specify the behaviour of $T_{\mu\nu}$ as $r\to\infty$ to make
progress [61]

$\displaystyle
T_{uu}=\frac{1}{r^{2}}\hat{T}_{uu}(u,x^{A})+\mathcal{O}(r^{-3}),\qquad
T_{rr}=\frac{1}{r^{4}}\hat{T}_{rr}(u,x^{A})+\frac{1}{r^{5}}\tilde{T}_{rr}(u,x^{A})+\mathcal{O}(r^{-6}),$
$\displaystyle
T_{uA}=\frac{1}{r^{2}}\hat{T}_{uA}(u,x^{A})+\mathcal{O}(r^{-3}),\qquad
T_{rA}=\frac{1}{r^{3}}\hat{T}_{rA}(u,x^{A})+\mathcal{O}(r^{-4}),$
$\displaystyle
T_{AB}=\frac{1}{r}\hat{T}(u,x^{A})\gamma_{AB}+\mathcal{O}(r^{-2}),\qquad
T_{ur}=\mathcal{O}(r^{-4}).$

Using stress-energy conservation $\nabla^{\beta}T_{\alpha\beta}=0$, one is
then able to simplify $\hat{T}_{rA}$ and ultimately obtain and expression for
the coefficient of our expansion by enforcing (3.14). For simplicity, we will
later on restrict to the case where $\hat{T}=0$, as done by Strominger [16],
Kapec et al. [17], He et al. [22] or Alessio and Arzano [51]. The detailed
calculations of the expansion coefficients was carried out with the help of
`Mathematica` (Appendix A.1). We summarize here the results. The gauge
condition (3.2) gives

$D_{z\bar{z}}=\frac{1}{2}\gamma^{z\bar{z}}C_{\bar{z}\bar{z}}C_{zz}.$

From the $rA$ piece of (3.14), we get

$U_{2A}=D^{B}C_{BA}\quad\Rightarrow\quad
g_{uA}=\frac{1}{2}D^{B}C_{BA}+\mathcal{O}(r^{-1}).$ (3.15)

Finding an expression for $U_{3A}$ is more involving; one needs to keep
additional terms in the expansion. It is convenient to write it as

$U_{3z}=\frac{4}{3}N_{z}+\frac{4}{3}uD_{z}m-\frac{1}{4}D_{z}C_{zz}C^{zz}-C_{zz}\gamma^{z\bar{z}}D^{\bar{z}}C_{\bar{z}\bar{z}},$

implying

$g_{uz}=\frac{1}{2}D^{z}C_{zz}+\frac{1}{6r}C_{zz}D_{z}C^{zz}+\frac{2}{3r}N_{z}+\mathcal{O}(r^{-2}).$

Meanwhile, the $rr$ piece of (3.14) yields

$\beta_{1}=0\text{ and
}\beta_{2}=-\frac{1}{32}C_{AB}C^{AB}\quad\Rightarrow\quad
g_{ur}=-1+\mathcal{O}(r^{-2}).$ (3.16)

The other terms in the expansion will not play a role in what we consider
later on. We have arrived at our desired result: the class of allowed metrics
in Bondi gauge whose falloffs ensure a description of an AFS at large radial
distances, with line element given by

$\mathrm{d}s^{2}=-\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}r+2r^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}+\frac{2m}{r}\mathrm{d}u^{2}+rC_{zz}\mathrm{d}z^{2}+rC_{\bar{z}\bar{z}}\mathrm{d}\bar{z}^{2}\\\
+D^{z}C_{zz}\mathrm{d}u\mathrm{d}z+D^{\bar{z}}C_{\bar{z}\bar{z}}\mathrm{d}u\mathrm{d}\bar{z}+\frac{1}{r}\left(\frac{4}{3}(N_{z}+uD_{z}m)-\frac{1}{4}D_{z}(C_{zz}C^{zz})\right)\mathrm{d}u\mathrm{d}z+\text{c.c.}+...$
(3.17)

If we readily identify the flat Minkowski metric (2.6) in the first line, it
is not obvious why the remainder of this expression takes this particular
form, with factors of $4/3$ and all that. It is however common practice in the
literature to write such expressions in hindsight of the calculations to come,
which often helps simplifying expressions or making explicit the physical
interpretation of some of the coefficients. Speaking of physical significance,
let us thus pause for a moment to explicit the several physically relevant
fields we encountered in our expansion of the metric coefficients:

  * •

the Bondi mass aspect $m(u,z,\bar{z})$ gives the angular density of energy of
the spacetime as measured from a point at $\cal I^{+}$ labeled by $u$ and in
the direction pointed out by the angles $z,\bar{z}$. The Bondi mass is
obtained after performing an integration of $m$ on the sphere:

$M_{B}=\oint_{\mathbb{S}^{2}_{\infty}}\mathrm{d}^{2}\Omega\,m(u,z,\bar{z}).$

One can show that $\partial_{u}M_{B}(u)\leq 0$ for pure gravity or gravity
coupled to matter obeying the null energy condition. Physically, the radiation
carried by gravitational waves or null matter such as electromagnetic fields
escapes through $\cal I^{+}$ and lowers the energy of spacetime when the
retarded time $u$ evolves: this is the mass loss [7]. At $u\to\infty$, the
Bondi mass equates the ADM energy, or total energy of a Cauchy slice of
spacetime [50].

  * •

the traceless and symmetric field $C_{AB}$. $C_{zz}$ and $C_{\bar{z}\bar{z}}$
can be seen as two polarization modes for the gravitational waves. Later on,
we will see that they also encode the helicity modes of gravitons in the
context of soft theorems. The (retarded) Bondi news tensor is defined as

$N_{AB}=\partial_{u}C_{AB}.$ (3.18)

Its square is proportional to the energy flux across $\cal I^{+}$.

  * •

finally, we also introduced the angular momentum aspect $N^{A}(u,z,\bar{z})$.
It is closely related to the angular density of angular momentum with respect
to the origin ($r=0$).

Great. We now have an expression for the asymptotic behaviour of the metric of
an AFS at $\cal I^{+}$. However, we have not yet imposed all of Einstein’s
equations. Looking at the $uu$ and $uA$ parts of (3.14) gives us additional
constraints (Appendix A.1),

$\displaystyle\partial_{u}m$ $\displaystyle=-4\pi
G\hat{T}_{uu}-\frac{1}{4}N_{zz}N^{zz}+\frac{1}{4}[D^{z2}N_{zz}+D^{\bar{z}2}N_{\bar{z}\bar{z}}]$
(3.19)
$\displaystyle=-T_{uu}+\frac{1}{4}[D^{z2}N_{zz}+D^{\bar{z}2}N_{\bar{z}\bar{z}}]=-T_{uu}+\frac{1}{4}\partial_{u}[D^{z}U_{2z}+D^{\bar{z}}U_{2\bar{z}}],$

and

$\partial_{u}N_{z}=\frac{1}{4}D_{z}[D_{z}^{2}C^{zz}-D_{\bar{z}}^{2}C^{\bar{z}\bar{z}}]\
-uD_{z}\partial_{u}m+\frac{1}{4}D_{z}(C_{zz}N^{zz})+\frac{1}{2}C_{zz}D_{z}N^{zz}-8\pi
G\hat{T}_{uz}.$ (3.20)

These are the constraint equations, which tell us the time evolution of $m$
and $N_{z}$. We learn that to first and second subleading order in the $1/r$
expansion, $C_{zz}$ and $C_{\bar{z}\bar{z}}$ is the only "free data" that we
need to assign, since all the other components of the metric are determined
through (3.19) and (3.20) once initial conditions for $m$, $N^{A}$ and
$C_{AB}$ have been provided.

We can repeat the same steps when looking at the behaviour of the metric at
$\cal I^{-}$. Starting from (3.8), we find that the metric in advanced Bondi
coordinates $(v,r,z,\bar{z})$ has the large $r$ expansion

$\mathrm{d}s^{2}=-\mathrm{d}v^{2}+2\mathrm{d}v\mathrm{d}r+2r^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}+\frac{2m^{-}}{r}\mathrm{d}v^{2}+rD_{zz}\mathrm{d}z^{2}+rD_{\bar{z}\bar{z}}\mathrm{d}\bar{z}^{2}+V_{2z}\mathrm{d}v\mathrm{d}z+V_{2\bar{z}}\mathrm{d}v\mathrm{d}\bar{z}+...$

with $m^{-}$ the advanced Bondi mass, $V_{2z}=-D^{z}D_{zz}$,
$V_{2\bar{z}}=-D^{\bar{z}}D_{\bar{z}\bar{z}}$ and the constraint

$\partial_{v}m^{-}=-T_{vv}-\frac{1}{4}\partial_{v}[D^{z}V_{2z}+D^{\bar{z}}V_{2\bar{z}}],$
(3.21)

where $T_{vv}$ is the total incoming radiation flux at $\cal I^{-}$. We assume
$g_{AB}^{-}=r^{2}\gamma_{AB}+rD_{AB}+\mathcal{O}(1)$ and also denote by
$M_{AB}=\partial_{v}D_{AB}$ the advanced Bondi news tensor. In the following,
we will mostly focus on what happens at $\cal I^{+}$. The analogous statements
for $\cal I^{-}$ can be read off from $U^{A}\to-V^{A}$.

###  3.3 Bondi-Metzner-Sachs group in General Relativity

We now have a description of the behaviour of the Bondi metric of an AFS at
our disposal. Our next task will be to find the form of the most general
diffeomorphism $\xi$ which preserves the Bondi gauge conditions and the
falloffs we required for an AFS: the generator of asymptotic symmetries. This
will lead us to constructing the BMS algebra in four dimensions,
$\mathfrak{bms}_{4}$.

####  3.3.1 $\mathfrak{bms}_{4}$ algebra

Recall that the variation $g\to g+\delta g$ of the metric under a
diffeomorphism generated by the vector field $\xi$ is given in a coordinate
basis by [58]

$\delta
g_{\mu\nu}\equiv\mathcal{L}_{\xi}g_{\mu\nu}=\xi^{\rho}\partial_{\rho}g_{\mu\nu}+g_{\mu\rho}\partial_{\nu}\xi^{\rho}+g_{\nu\rho}\partial_{\mu}\xi^{\rho}.$
(3.22)

Hence, preservation of the Bondi gauge conditions (3.3) under $\xi$ writes

$\boxed{\mathcal{L}_{\xi}g_{rr}=0,\quad\mathcal{L}_{\xi}g_{rA}=0,\quad\text{and}\quad\mathcal{L}_{\xi}\partial_{r}\text{det}\frac{g_{AB}}{r^{2}}=0\Rightarrow
g^{AB}\mathcal{L}_{\xi}g_{AB}=0,}$ (3.23)

where the last implication stems from the fact that

$\displaystyle\mathcal{L}_{\xi}(\text{det}g_{AB})$
$\displaystyle=\lim_{\epsilon\to
0}\frac{\text{det}(g^{\prime}_{AB})-\text{det}(g_{AB})}{\epsilon}=\lim_{\epsilon\to
0}\frac{\exp\text{Tr}[\log(g_{AB}+\mathcal{L}_{\xi}g_{AB}\epsilon)]-\text{det}(g_{AB})}{\epsilon}.$

Here we expanded
$g_{AB}^{\prime}=g_{AB}+\mathcal{L}_{\xi}g_{AB}\epsilon+\mathcal{O}(\epsilon^{2})$
and made use of the relation $\text{det}(A)=\exp\text{Tr}(\ln A)$ for any
square matrix $A$. Then a Taylor series expansion of $\exp$ and $\log$ gives

$\mathcal{L}_{\xi}(\text{det}g_{AB})\approx\lim_{\epsilon\to
0}\frac{\text{det}(g_{AB})+g^{AB}\mathcal{L}_{\xi}g_{AB}\epsilon+...-\det(g_{AB})}{\epsilon}=g^{AB}\mathcal{L}_{\xi}g_{AB}\overset{!}{=}0.$

This last constraint amounts to asking that the angular metric $g_{AB}$ does
not undergo any conformal rescaling under the transformation [62]. Requiring
that the asymptotic behaviour of (3.17) hold true999Horn [63] investigated
asymptotic symmetries which preserve the Bondi gauge condition but do not
preserve the asymptotic falloff conditions for the metric near null boundary.
under $\xi$ amounts to enforcing:

$\boxed{\mathcal{L}_{\xi}g_{uu}=\mathcal{O}(r^{-1}),\quad\mathcal{L}_{\xi}g_{ur}=\mathcal{O}(r^{-2}),\quad\mathcal{L}_{\xi}g_{uA}=\mathcal{O}(1),\quad\mathcal{L}_{\xi}g_{AB}=\mathcal{O}(r).}$
(3.24)

The procedure for determining $\xi$ is as follows: one first solves the
constraints (3.23) exactly, which allows to express the 4 components of
$\xi^{\mu}$ in terms of 4 functions of $u,x^{A}$. The falloff preservation
condition (3.24) can then be solved to reduce these 4 functions to only 3
functions on the 2-sphere, namely $T(x^{A})$ and $R^{B}(x^{A})$. These
calculations are carried out in detail in Appendix A.2 and yield

$\left.\xi_{T,R}\right|_{\mathcal{I}^{+}}=\left[T(x^{C})+\frac{u}{2}D_{A}R^{A}(x^{C})\right]\partial_{u}+\left[R^{A}(x^{C})\right]\partial_{A}.$
(3.25)

They correspond to the exact Killing vector fields at $\cal I^{+}$. In this
expression $T(x^{A})$ is unconstrained, while $R^{A}(x^{B})$ obey the
conformal Killing equation on the 2-sphere (Appendix A.2),

$D_{A}R_{B}+D_{B}R_{A}=\gamma_{AB}D_{C}R^{C}.$ (3.26)

In stereographic coordinates $(z,\bar{z})$, (3.26) implies that $R^{z}$ is
holomorphic, $R^{z}\equiv R^{z}(z)$ and $R^{\bar{z}}$ antiholomorphic,
$R^{\bar{z}}\equiv R^{\bar{z}}(\bar{z})$; more on this in section 3.3.3.
Extending (3.25) from future null infinity into the interior of the spacetime
while still asking that (3.23) and (3.24) hold, we get asymptotic Killing
vectors (Appendix A.2)

$\xi_{T,R}=f\partial_{u}+\left[R^{A}-\frac{D^{A}f}{r}+\frac{(D_{B}f)C^{AB}}{2r^{2}}+\mathcal{O}(r^{-3})\right]\partial_{A}\\\
+\left[-\frac{rD_{A}R^{A}}{2}+\frac{D^{2}f}{2r}-\frac{2(D_{A}C^{AB})D_{B}f+C^{AB}D_{A}D_{B}f}{4r}+\mathcal{O}(r^{-2})\right]\partial_{r}$
(3.27)

with

$f(u,x^{A})=T(x^{A})+\frac{u}{2}D_{B}R^{B}(x^{A}).$ (3.28)

The vectors (3.27) are known as the (asymptotic) BMS generators. The most
general diffeomorphisms generating a variation of our metric compatible with
the Bondi gauge and the required falloffs for an AFS are obtained from
$\xi_{T,R}$. They generate the asymptotic $\mathfrak{bms}_{4}$ algebra, which
appears to be larger than the Poincaré algebra. This is further studied in the
remainder of this section.

#####  $\mathfrak{bms}_{4}$ brackets

Our notation involving $T$ and $R$ suggests that there are two types of
transformation generators of $\mathfrak{bms}_{4}$. Let us make this more
explicit by computing the brackets of this algebra. First and foremost, note
that there are two equivalent ways to go around this. One can start with
(3.25) and compute the usual Lie bracket
$[\xi_{1},\xi_{2}]\equiv[\xi_{T_{1},R_{1}},\xi_{T_{2},R_{2}}]$ (omitting the
subscript $\cal I^{+}$). The vectors (3.25) are however only defined at $\cal
I^{+}$ so one then has to check using (3.27) that the commutation relations
hold even away from the boundary (see [21] for an explicit computation). The
alternative is to use a modified Lie bracket

$[\xi_{1},\xi_{2}]_{M}=[\xi_{2},\xi_{2}]-\delta^{g}_{\xi_{1}}\xi_{2}+\delta^{g}_{\xi_{2}}\xi_{1},$

where $\delta^{g}_{\xi_{1}}\xi_{2}$ denotes the variation in $\xi_{2}$ under
the variation of the metric induced by $\xi_{1}$. This accounts for the
dependence of the asymptotic vector fields on the background metric, notably
through $C_{AB}$ in (3.27). Barnich and Troessaert [15] showed that the latter
vectors provide a faithful representation of $\mathfrak{bms}_{4}$ when
equipped with $[\cdot,\cdot]_{M}$ and for $R^{A}$ a conformal Killing vector
of the 2-sphere. In either way, denoting $\xi_{T,0}\equiv\xi_{T}$ and
$\xi_{0,R}\equiv\xi_{R}$, we get

$\boxed{[\xi_{T_{1}},\xi_{T_{2}}]=0,\quad\text{and}\quad[\xi_{R_{1}},\xi_{R_{2}}]=\hat{R}^{A}\partial_{A}+\frac{u}{2}D_{A}\hat{R}^{A}\partial_{u}\equiv\xi_{\hat{R}}}$
(3.29)

with
$\hat{R}^{A}=R_{1}^{C}\partial_{C}R_{2}^{A}-R_{2}^{C}\partial_{C}R_{1}^{A}$,
as well as

$\boxed{[\xi_{R},\xi_{T}]=\left(R^{A}\partial_{A}T-\frac{T}{2}D_{B}R^{B}\right)\partial_{u}\equiv\hat{T}\partial_{u}\equiv\xi_{\hat{T}}.}$
(3.30)

These relations define the $\mathfrak{bms}_{4}$ algebra at $\cal I^{+}$, of
which trivial boundary diffeomorphisms with $T=R^{A}\equiv 0$ form an ideal.
Taking the quotient by this ideal, we are left with the asymptotic algebra of
asymptotically flat spacetimes compatible with the Bondi-Sachs boundary
conditions at $\cal I^{+}$. We will see later on that this algebra
exponentiates to the group called BMS+, and that replicating the exact same
analysis for $\cal I^{-}$ starting this time from (3.8) yields a second copy
of the BMS group, BMS-, acting on ingoing data at past null infinity. Before
doing so, let us investigate the action of $\xi_{T}$ and $\xi_{R}$. In the
next section, we show that the former vectors generate the so-called
supertranslations, while the latter generate the Lorentz transformations
encountered in section 2.

####  3.3.2 Supertranslations

From the first of (3.29), we learn that the generators of supertranslations
$\xi_{T}=T(x^{A})\partial_{u}$ at $\cal I^{+}$ form an Abelian ideal of the
$\mathfrak{bms}_{4}$ algebra. Inside the spacetime we have for the asymptotic
Killing vector (3.27), working in the stereographic $x^{A}=(z,\bar{z})$
coordinates

$\xi_{\text{T}}^{+}\approx
f\partial_{u}+\left[-\frac{1}{r}(D^{z}f\partial_{z}+D^{\bar{z}}f\partial_{\bar{z}})+\frac{1}{2r^{2}}(C^{zz}D_{z}f\partial_{z}+C^{\bar{z}\bar{z}}D_{z}f\partial_{\bar{z}})\right]+D^{z}D_{z}f\partial_{r}.$
(3.31)

Here, we have renamed $T(x^{A})\equiv f(x^{A})$ since $R^{B}\equiv 0$ in
(3.28) and used $\frac{1}{2}D^{2}f=D^{z}D_{z}f$ to match the notation of
Strominger [21]. Similarly, one obtains in advanced Bondi coordinates

$\xi_{\text{T}}^{-}\approx
f^{-}\partial_{v}+\left[\frac{1}{r}(D^{\bar{z}}f^{-}\partial_{\bar{z}}+D^{z}f^{-}\partial_{z})+\frac{1}{2r^{2}}(C^{zz}D_{z}f^{-}\partial_{z}+C^{\bar{z}\bar{z}}D_{\bar{z}}f^{-}\partial_{\bar{z}})\right]-D^{z}D_{z}f^{-}\partial
r.$ (3.32)

Since $f$ and $f^{-}\equiv f^{-}(z,\bar{z})$ are arbitrary scalar fields on
$\mathbb{S}^{2}$, the exponentiation of these vectors gives rise to an Abelian
subgroup $S$ of BMS4, which is infinite-dimensional. In fact, $S$ admits one
unique normal finite subgroup that reproduces the Poincaré translations. This
is discussed in what follows.

##### Spacetime translations

We said that supertranslations are parametrized by an arbitrary function
$f(z,\bar{z})$ on $\mathbb{S}^{2}$. As such, it is natural to think of the
latter as a linear superposition of spherical harmonics $Y^{l}_{m}$,

$f(z,\bar{z})=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}f_{l,m}Y^{l}_{m}(z,\bar{z}),\quad
f_{l,-m}=(-1)^{m}\bar{f}_{l,m}.$ (3.33)

For $l=0$ and $l=1$, we have for $\xi_{T}^{+}(f)$ given by (3.31)

$\xi_{T}^{+}(Y^{0}_{0})=Y^{0}_{0}\partial_{u},\quad\text{and}\quad\xi_{T}^{+}(Y_{1}^{m})=Y_{1}^{m}\partial_{u}-\frac{\gamma^{AB}\partial_{B}Y_{1}^{m}}{r}\partial_{A}+\frac{1}{2}D^{2}Y_{1}^{m}\partial_{r}.$
(3.34)

Using (2.3) to write

$\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}=-\sin\phi,\quad\frac{z\bar{z}-1}{1+z\bar{z}}=-\cos\theta\quad\text{and}\quad\frac{2\sqrt{z\bar{z}}}{1+z\bar{z}}=\sin\theta,$

the first few spherical harmonics can be recast in the $(u,r,z,\bar{z})$
system as

$Y^{0}_{0}=1,\quad Y^{0}_{1}=\frac{1-z\bar{z}}{1+z\bar{z}},\quad
Y_{1}^{1}=\frac{z}{1+z\bar{z}},\quad Y^{-1}_{1}=\frac{\bar{z}}{1+z\bar{z}}.$

Inserting this in (3.34) gives

$\displaystyle\xi_{T}^{+}(Y^{0}_{0})=\partial_{u},\quad\xi_{T}^{+}(Y^{0}_{1})=\frac{1-z\bar{z}}{1+z\bar{z}}(\partial_{u}-\partial_{r})+\frac{z}{r}\partial_{z}+\frac{\bar{z}}{r}\partial_{\bar{z}}$
$\displaystyle\xi_{T}^{+}(Y^{1}_{1})=\frac{z}{1+z\bar{z}}(\partial_{u}-\partial_{r})+\frac{z^{2}}{2r}\partial_{z}-\frac{1}{2r}\partial_{\bar{z}},\quad\xi_{T}^{+}(Y^{-1}_{1})=\frac{\bar{z}}{1+z\bar{z}}(\partial_{u}-\partial_{r})-\frac{1}{2r}\partial_{z}+\frac{\bar{z}^{2}}{2r}\partial_{\bar{z}},$

where we used that
$\frac{1}{2}D^{2}f(z,\bar{z})=\frac{1}{2}(D^{z}D_{z}+D^{\bar{z}}D_{\bar{z}})f(z,\bar{z})=\gamma^{z\bar{z}}\partial_{z}\partial_{\bar{z}}f(z,\bar{z}).$
Besides, global spacetime translations (2.16) write in Bondi coordinates [21]

$\displaystyle\xi_{0}$ $\displaystyle=\partial_{u}$ $\displaystyle\xi_{1}$
$\displaystyle=-\frac{z+\bar{z}}{1+z\bar{z}}\partial_{u}+\frac{z+\bar{z}}{1+z\bar{z}}\partial_{r}+\frac{1-z^{2}}{2r}\partial_{z}+\frac{1-\bar{z}^{2}}{2r}\partial_{\bar{z}},$
$\displaystyle\xi_{2}$
$\displaystyle=\frac{i(z-\bar{z})}{1+z\bar{z}}\partial_{u}-\frac{i(z-\bar{z})}{1+z\bar{z}}\partial_{r}+\frac{i(1+z^{2})}{2r}\partial_{z}-\frac{i(1+\bar{z}^{2})}{2r}\partial_{\bar{z}},$
$\displaystyle\xi_{3}$
$\displaystyle=-\frac{1-z\bar{z}}{1+z\bar{z}}\partial_{u}+\frac{1-z\bar{z}}{1+z\bar{z}}\partial_{r}-\frac{z}{r}\partial_{z}-\frac{\bar{z}}{r}\partial_{\bar{z}}.$

Therefore, we identify

$\displaystyle\xi_{T}^{+}(Y_{0}^{0})=\xi_{0},\quad\xi_{T}^{+}(Y_{1}^{0})=-\xi_{3},\quad\xi_{T}^{+}(Y_{1}^{1})=-\frac{\xi_{1}+i\xi_{2}}{2},\quad\xi_{T}^{+}(Y^{-1}_{1})=-\frac{\xi_{1}-i\xi_{2}}{2}.$

Thus the usual spacetime translations are contained in the BMS
supertranslations. Time translation $\xi_{0}$ implies energy conservation,
while the three spatial translations $\xi_{i}$ imply ADM momentum conservation
[64]. We have shown so far that $S$ is Abelian and contains the usual
translations of Minkowski. Showing that it is in fact a normal subgroup of BMS
requires a more careful study of the structure of BMS transformations and
their interplay [65]. In the next paragraph, we look at the action of
supertranslations on the data we specified for our metric (3.17).

##### Effect of a supertranslation

The supertranslations at $\cal I^{+}$ act to shift individual light rays of
null infinity forwards or backwards in retarded time. In practice, writing
$\mathcal{L}_{f}\equiv\mathcal{L}_{\xi_{f}^{+}}$ with $f\equiv T$ for
notational simplicity, we obtain at $\cal I^{+}$ (Appendix A.3)

$\displaystyle\mathcal{L}_{f}C_{AB}$
$\displaystyle=f\partial_{u}C_{AB}+\gamma_{AB}D^{2}f-2D_{A}D_{B}f,$ (3.35)
$\displaystyle\mathcal{L}_{f}N_{AB}$ $\displaystyle=f\partial_{u}N_{AB},$
(3.36) $\displaystyle\mathcal{L}_{f}m$
$\displaystyle=f\partial_{u}m+\frac{1}{4}\left(N^{AB}D_{A}D_{B}f+2D_{A}fD_{B}N^{AB}\right),$
(3.37) $\displaystyle\mathcal{L}_{f}U_{2z}$
$\displaystyle=f\partial_{u}U_{2z}+D^{z}f\partial_{u}C_{zz}-2D^{z}D^{2}_{z}f$
(3.38)

while at $\cal I^{-}$ we have, writing $f\equiv f^{-}$ again to simplify
notation

$\displaystyle\mathcal{L}_{f}D_{zz}$
$\displaystyle=f\partial_{v}D_{zz}+2D^{2}_{z}f,$ (3.39)
$\displaystyle\mathcal{L}_{f}M_{zz}$ $\displaystyle=f\partial_{v}M_{zz}.$
(3.40) $\displaystyle\mathcal{L}_{f}V_{2z}$
$\displaystyle=-D^{z}f\partial_{v}D_{zz}+f\partial_{v}V_{2z}-2D^{z}D^{2}_{z}f.$
(3.41)

####  3.3.3 Lorentz transformations and superrotations

We now focus on the second set of generators of $\mathfrak{bms}_{4}$ algebra,
$\xi_{0,R}\equiv\xi_{R}$. Note that $f$ is again given here by (3.28) with
$T=0$, and thus a function of $u$ and $x^{A}$. The vectors $\xi_{R}$ from
(3.27) are the generators of superrotations and write

$\xi_{R}=\frac{u}{2}D_{A}R^{A}\partial_{u}+\left[R^{A}-\frac{u}{2r}D^{A}D_{B}R^{B}+\mathcal{O}(r^{-2})\right]\partial_{A}+\left[-\frac{r}{2}D_{A}R^{A}+\frac{u}{4}D^{2}D_{B}R^{B}+\mathcal{O}(r^{-1})\right]\partial_{r}$

with $R^{A}(x^{A})$ satisfying the conformal Killing equation (3.26). In
stereographic coordinates

$\displaystyle\xi^{+u}_{R}$
$\displaystyle\approx\frac{u}{2}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}}),$
$\displaystyle\xi^{+z}_{R}$ $\displaystyle\approx
R^{z}-\frac{u}{2r}D^{z}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}})=R^{z}+\frac{u}{2r}(R^{z}-D^{z}D_{\bar{z}}R^{\bar{z}})$
$\displaystyle\xi^{+\bar{z}}_{R}$ $\displaystyle\approx
R^{\bar{z}}-\frac{u}{2r}D^{\bar{z}}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}})=R^{\bar{z}}+\frac{u}{2r}(R^{\bar{z}}-D^{\bar{z}}D_{z}R^{z})$

having used that $D^{z}D_{z}R^{z}=-R^{z}$ in the last two lines, and

$\displaystyle\xi_{R}^{+r}$
$\displaystyle\approx-\frac{r}{2}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}})+\frac{u}{4}(D^{z}D_{z}+D^{\bar{z}}D_{\bar{z}})(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}})=-\frac{r+u}{2}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}}).$

Therefore, in the $(u,r,z,\bar{z})$ system, our asymptotic Killing vector
$\xi_{R}^{+}$ writes

$\xi_{R}^{+}\approx\frac{u}{2}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}})\partial_{u}-\frac{r+u}{2}(D_{z}R^{z}+D_{\bar{z}}R^{\bar{z}})\partial_{r}\\\
+\left[R^{z}+\frac{u}{2r}(R^{z}-D^{z}D_{\bar{z}}R^{\bar{z}})\right]\partial_{z}+\left[R^{\bar{z}}+\frac{u}{2r}(R^{\bar{z}}-D^{\bar{z}}D_{z}R^{z})\right]\partial_{\bar{z}}.$
(3.42)

##### Lorentz transformations

So far, we have not imposed any restrictions on $R^{A}$ apart from the fact
that they are CKVs on the unit two-sphere. We have also mentioned that (3.26)
implies that $R^{z}$ and $R^{\bar{z}}$ are repectively holomorphic and
antiholomorphic, $\partial_{z}R^{\bar{z}}=\partial_{\bar{z}}R^{z}=0$. Indeed,
from (3.26) we learn that $D_{z}R^{\bar{z}}=D_{\bar{z}}R^{z}=0$, which, using
the Christoffel symbols for $\gamma$ in stereographic coordinates, imply

$D_{z}R^{\bar{z}}=\partial_{z}R^{\bar{z}}+\cancel{\Gamma^{\bar{z}}_{zz}}R^{z}+\cancel{\Gamma^{\bar{z}}_{\bar{z}z}}R^{\bar{z}}\overset{!}{=}0\Rightarrow\partial_{z}R^{\bar{z}}=0,$

and similarly
$D_{\bar{z}}R^{z}\overset{!}{=}0\Rightarrow\partial_{\bar{z}}R^{z}=0$. As
such, both $R^{z}$ and $R^{\bar{z}}$ appear as a sum of monomial terms
$R^{z}=z^{k}$, $k\in\mathbb{Z}$ when expanded in Laurent series. This
restricts the possible choices for $R^{z}$. Indeed, considering
$v_{k}=z^{k}\partial_{z}$, we have that when $k<0$, $z^{k}$ is singular at the
origin $z=0$ of the Riemann sphere, while for $k>2$ $z^{k}$ is
singular101010To see this, it is helpful to consider the transformation $z\to
w=z^{-1}$, such that $v^{k}\to-w^{2-k}\partial_{w}$. at the point at infinity
$z=\infty$. Therefore $R^{z}$ is only well-defined for $k=0,1,2$, and
similarly for $R^{\bar{z}}$, giving six valid asymptotic Killing vectors
$\xi_{R}^{+}$ in total. These are parametrized by

$R^{z}\in\\{1,z,z^{2},i,iz,iz^{2}\\},\quad R^{\bar{z}}=\overline{R^{z}}.$

Here comes the punchline: these six vector fields generate exactly the Lorentz
transformations we know and love! Put differently, $R^{z}$ and $R^{\bar{z}}$
are well-defined globally on $\mathbb{S}^{2}$ if and only if the
superrotations are in fact Lorentz transformations (LT). To see this, note
that the LT Killing vectors (2.14) and (2.15) write in the $(u,r,z,\bar{z})$
coordinates [21]

$\displaystyle\xi_{01}$
$\displaystyle=-\frac{u(z+\bar{z})}{1+z\bar{z}}\partial_{u}+\frac{(r+u)(z+\bar{z})}{1+z\bar{z}}\partial_{r}+\frac{(1-z^{2})(r+u)}{2r}\partial_{z}+\frac{(1-\bar{z}^{2})(r+u)}{2r}\partial_{\bar{z}}$
$\displaystyle\xi_{02}$
$\displaystyle=\frac{iu(z-\bar{z})}{1+z\bar{z}}\partial_{u}-\frac{i(r+u)(z-\bar{z})}{1+z\bar{z}}\partial_{r}+\frac{i(1+z^{2})(r+u)}{2r}\partial_{z}-\frac{i(1+\bar{z}^{2})(r+u)}{2r}\partial_{\bar{z}}$
$\displaystyle\xi_{03}$
$\displaystyle=-\frac{u(1-z\bar{z})}{1+z\bar{z}}\partial_{u}+\frac{(r+u)(1-z\bar{z})}{1+z\bar{z}}\partial_{r}-\frac{z(r+u)}{r}\partial_{z}-\frac{\bar{z}(r+u)}{r}\partial_{\bar{z}}$

for the boosts and

$\xi_{21}=iz\partial_{z}-i\bar{z}\partial_{\bar{z}},\quad\xi_{32}=\frac{i(z^{2}-1)}{2}\partial_{z}-\frac{i(\bar{z}^{2}-1)}{2}\partial_{\bar{z}},\quad\xi_{13}=\frac{(z^{2}+1)}{2}\partial_{z}+\frac{(\bar{z}^{2}+1)}{2}\partial_{\bar{z}}.$

for the spacetime rotations. Following [21], we write
$\xi_{R^{z}}\equiv\zeta(a,b,c)$ for $a,b,c\in\mathbb{C}$

$\displaystyle\zeta^{u}$
$\displaystyle=\frac{u}{2}\frac{2z(c-a^{\ast})+2\bar{z}(c^{\ast}-a)+(b+b^{\ast})(1-z\bar{z})}{1+z\bar{z}},$
$\displaystyle\zeta^{r}$
$\displaystyle=-\frac{u+r}{2}\frac{2z(c-a^{\ast})+2\bar{z}(c^{\ast}-a)+(b+b^{\ast})(1-z\bar{z})}{1+z\bar{z}},$
$\displaystyle\zeta^{z}$
$\displaystyle=a+bz+cz^{2}+\frac{u}{2r}((a-c^{\ast})+(b+b^{\ast})z+(c-a^{\ast})z^{2}),\quad\zeta^{\bar{z}}=(\zeta^{z})^{\ast}$

and thus identify

$\displaystyle\xi_{12}=\zeta(0,i,0),\quad\xi_{23}=\zeta(-i/2,0,i/2),\quad\xi_{31}=\zeta(1/2,0,1/2)$
$\displaystyle\xi_{01}=\zeta(1/2,0,-1/2),\quad\xi_{02}=\zeta(i/2,0,i/2),\quad\xi_{03}=\zeta(0,-1,0).$

Here, we used

$D_{z}R^{z}=\partial_{z}R^{z}-\frac{2\bar{z}}{1+z\bar{z}}R^{z}\quad\text{and}\quad
D_{\bar{z}}R^{\bar{z}}=\partial_{\bar{z}}R^{\bar{z}}-\frac{2z}{1+z\bar{z}}R^{\bar{z}}.$

This shows that BMS vector fields obtained from a linear combination of
$\\{1,z,z^{2},i,iz,iz^{2}\\}$ indeed correspond to the usual Lorentz
generators, so the globally defined $\xi_{R}^{+}$ are simply the asymptotic
Lorentz transformations. This is the historic form of the BMS group, sometimes
also called global BMS. Allowing for more general (and hence singular)
superrotations gives rise to extensions of the BMS group, some of which will
be introduced in the next section.

##### Effect of a BMS superrotation

The determination of the action of a superrotation on the data at $\cal I^{+}$
follows the exact same procedure as for supertranslations. One computes the
change in the metric components under the diffeomorphism using (A.1), and then
looks at the relevant order in $1/r$ to identify e.g. $\mathcal{L}_{\xi}m$,
etc. We do not reproduce these calculations here for the sake of brevity, and
instead state the results. We have at $\cal I^{+}$ [51]

$\displaystyle\mathcal{L}_{\xi}C_{zz}$ $\displaystyle=\frac{u}{2}D\cdot
RN_{zz}+\mathcal{L}_{R}C_{zz}-\frac{1}{2}D\cdot RC_{zz}-uD^{3}_{z}R^{z},$
(3.43) $\displaystyle\mathcal{L}_{\xi}N_{zz}$ $\displaystyle=\frac{u}{2}D\cdot
R\partial_{u}N_{zz}+\mathcal{L}_{R}N_{zz}-D^{3}_{z}R^{z}$ (3.44)

where $\mathcal{L}_{R}C_{zz}=R\cdot DC_{zz}+2D_{z}R^{z}C_{zz}$ and similarly
on $N_{zz}$.

####  3.3.4 Structure of the BMS group

Our previous study of the BMS generators taught us that the standard BMS group
is composed of an infinite-dimensional subgroup of supertranslations, $S$,
containing the usual spacetime translations of flat space, as well as
superrotations. If we discard the singular generators for the latter, we fall
back on the well-known Lorentz group. Hence, we eventually get after
exponentiation111111The problem of the existence of exponentials for the BMS
group is discussed by Prinz and Schmeding [39].

$\text{BMS}_{4}=\text{SO}(3,1)\ltimes
S=\text{Lorentz}\ltimes\text{Supertranslations}$

This is the historical form of the BMS group, which reproduces the semi-direct
structure of the Poincaré group: the Lorentz group acts non-trivially on $S$
as it does on the usual global translations, which could be seen from the
bracket (3.30). Since Translations $\subset$ Supertranslations, we have

$\text{Poincar\'{e}}=\text{Lorentz}\ltimes\text{Translations}\subset\text{BMS}_{4}.$

The only difference between Poincaré and global BMS is thus that the
translational part is enhanced in the latter. We shall see that this implies
the degeneracy of the gravitational Poincaré vacua [66], a phenomenon at which
we will look from the viewpoint of memory effects in section 4.3. General
relativity does indeed not reduce to special relativity at large distances.

##### Variants of the BMS group

The restriction of $R^{z}(z)$ to be a globally well-defined CKV on the two-
sphere has been debated in the literature.121212As a motivation for relaxing
the well-definedness of $R$, note that crucial insights have been obtained in
conformal field theories from singular conformal transformations on the
sphere. Two variants of the BMS group arise from the original BMS group
discussed above by replacing the Lorentz group by a larger symmetry group:

  * •

the extended BMS group eBMS, obtained by using two copies of the Bott-Virasoro
group as superrotations [15, 67, 68]. This group emerged from the context of
flat space holography, with the AdS/CFT correspondence in mind [64].

  * •

the generalized BMS group gBMS, obtained by allowing the index $k$ in the
Laurent series of $R^{z}$ to run over all of $\mathbb{Z}$. As a result,
superrotations comprise the entire group of diffeomorphisms on the 2-sphere,
Diff$(\mathbb{S}^{2})$. This extension first stem from the study of
gravitational scattering and the equivalence of Ward identities with soft
theorems [24, 69]. It is nowadays believed that gBMS is the correct asymptotic
symmetry group of the theory as it is the only one to admit a generalization
to higher spacetime dimensions.

This concludes our construction of the asymptotic symmetry group of four-
dimensional AFS. Next, we study the rich implications of this structure for
the gravitational scattering problem.

##  4 Infrared structure of gravity through supertranslations

We are now able to exploit the BMS symmetries constructed above to probe the
behaviour of gravitational scattering in the infrared (i.e. in AFS, at large
distances from an isolated system). In particular, we derive equivalence
relations between the conservation laws obtained and so-called soft theorems.

Note beforehand that two types of conservation laws can be singled out from
the BMS group [61]:

  1. 1.

those that relate quantities at one cross-section of $\cal I^{+}$ to another,
and

  2. 2.

those that relate quantities at $\cal I^{-}$ to quantities at $\cal I^{+}$.

In the context of gravitational scattering where one seeks a
$\mathcal{S}$-matrix relating $|out\rangle=\mathcal{S}|in\rangle$, it is
natural to focus on the second type and ask whether we can find a relation of
the form

$B^{+}\mathcal{S}-\mathcal{S}B^{-}=0,$ (4.1)

for $B^{\pm}$ the infinitesimal generators (charges) of BMS±. However, there
is a subtlety we need to address before we further develop this approach:
(4.1) implicitly relates what happens at past- and future null infinity since
we scatter massless particles. This means that what goes on near spatial
infinity $i^{0}$ will be important. Specifically, we need to understand how to
match on final data at $\mathcal{I}^{+}_{-}$ given initial data at
$\mathcal{I}^{-}_{+}$ for it to make sense. Such a prescription was provided
by Christodoulou and Klainerman [14] and is precisely the topic of the next
subsection.

###  4.1 Christodoulou-Klainerman spaces

The particles considered in our scattering process are assumed to be weakly
interacting in the far past and future. One could thus be tempted to naively
equate the data at $\cal I^{-}$ with the data at $\cal I^{+}$ at spacelike
infinity $i^{0}$. Things are however not so straightforward… Indeed, $i^{0}$
is a singular point in the conformal compactification of general AFS,
preventing a fully general canonical identification between $\cal I^{+}$ and
$\cal I^{-}$. There is a way out though, provided the metric under study lies
in a suitable neighbourhood of the Minkowski metric. As argued by Strominger
[16], we can assume that the configurations we analyze correspond to weakly
interacting Christodoulou-Klainerman (CK) geometries. CK showed [14] that
there exists a class of initial data which decays sufficiently fast as spatial
infinity, such that the "mapping" from $\cal I^{-}$ to $\cal I^{+}$
corresponds to a smooth geodesically complete solution. In such
configurations, the Bondi news tensor falls off as

$N_{zz}(u)\to|u|^{-3/2}\quad\text{or faster as }u\to\pm\infty,$ (4.2)

while $m$ and $N_{A}$ remain finite in the two limits. This will help us
impose matching conditions for initial and late data. Recall that we seek
asymptotically flat solutions to the Einstein equations which revert to the
vacuum in the far past and future. Taking care of the Bondi news trivially
determines $C_{zz}$, $C_{\bar{z}\bar{z}}$ up to an integration function.
Meanwhile, from (3.35) we have

$\mathcal{L}_{f}C_{zz}=f\partial_{u}C_{zz}-2D^{2}_{z}f=fN_{zz}-2D^{2}_{z}f.$

Hence vanishing of $N_{zz}$ as $u\to\pm\infty$ in CK spaces implies

$\lim_{u\to\pm\infty}\mathcal{L}_{f}C_{zz}=-2D^{2}_{z}f|_{\mathcal{I}^{+}_{\pm}}\Leftrightarrow\left.C_{zz}\right|_{\mathcal{I^{+}_{\pm}}}=\left.-2D^{2}_{z}C\right|_{\mathcal{I}^{+}_{\pm}},\quad\text{with
$C(z,\bar{z})$ such that }\mathcal{L}_{f}C=f,$ (4.3)

having used that
$C(z,\bar{z})|_{\mathcal{I}^{+}_{\pm}}=C(z,\bar{z})|_{\mathcal{I}^{-}_{\mp}}$
from the antipodal map relating stereographic coordinates. Similarly, we
deduce at past null infinity $\cal I^{-}$ from (3.39) that (it is also assumed
that the advanced Bondi news $M_{zz}$ vanishes as $v\to\pm\infty$ in CK
spaces)

$D_{zz}|_{\mathcal{I}^{-}_{\pm}}=+2D^{2}_{z}D|_{\mathcal{I}^{-}_{\pm}},\quad\text{for
}D(z,\bar{z})\text{ such that }\mathcal{L}_{f^{-}}D=f^{-}.$

We now have a prescription on how to relate initial and late data, given by
the continuity condition

$\boxed{C_{zz}|_{\mathcal{I}^{+}_{-}}=-D_{zz}|_{\mathcal{I}^{-}_{+}}\Leftrightarrow
C(z,\bar{z})=-D(z,\bar{z}).}$ (4.4)

Let us turn to the study of the effect of BMS supertranslation generators
(charges) on the $\cal S$-matrix.

###  4.2 BMS supertranslations and leading soft theorem

In the following, we make use of the covariant phase space formalism
introduced in section 2.3 to discuss the construction of the conserved charges
associated to supertranslations. We then show that these charges yield a
symmetry of the $\cal S$-matrix, implying energy conservation and a Ward
identity. To conclude, we relate this identity to Weinberg’s soft graviton
theorem.

####  4.2.1 Supertranslation charges

To get the conserved charges, we first need to construct the symplectic form
of the phase space of our theory. This is where the algorithm we fleshed out
in section 2.3 comes in handy. The Lagrangian form considered here is no other
than the Einstein-Hilbert Lagrangian

$L=\frac{1}{16\pi G}\sqrt{-g}g^{\mu\nu}R_{\mu\nu}(\mathrm{d}^{4}x).$ (4.5)

Steps 1 to 4 of Figure 4 were carried out by Alessio and Arzano [51], who
assumed that

$\lim_{u\to\infty}C_{zz}(u,z,\bar{z})=\phi^{+}_{zz}(z,\bar{z})=D^{2}_{z}C^{+},\quad\lim_{u\to-\infty}C_{zz}(u,z,\bar{z})=\phi^{-}_{zz}(z,\bar{z})=D^{2}_{z}C^{-}$

where $\phi^{\pm}_{zz}(z,\bar{z})$ and
$C^{\pm}(z,\bar{z})=-2C|_{\mathcal{I}^{+}_{\pm}}$ are smooth, non-vanishing
functions on $\mathbb{S}^{2}$. Integrating the Bondi news tensor (3.18) along
retarded time, one gets

$C_{zz}(u,z,\bar{z})-\phi^{-}_{zz}(z,\bar{z})=\int^{u}_{-\infty}\mathrm{d}u^{\prime}N_{zz}(u^{\prime},z,\bar{z}),\quad\phi^{+}_{zz}(z,\bar{z})-C_{zz}(u,z,\bar{z})=\int^{\infty}_{u}\mathrm{d}u^{\prime}N_{zz}(u^{\prime},z,\bar{z}).$

Adding the two,

$\int_{-\infty}^{\infty}\mathrm{d}u^{\prime}N_{zz}(u^{\prime},z,\bar{z})=\phi^{+}_{zz}-\phi^{-}_{zz}\equiv\Delta\phi_{zz}=D^{2}_{z}N,$
(4.6)

for $N(z,\bar{z})$ a real boundary field. Meanwhile, substracting yields

$C_{zz}(u,z,\bar{z})\equiv\frac{1}{2}\Delta\phi_{zz}(z,\bar{z})+\phi^{-}_{zz}(z,\bar{z})+\hat{C}_{zz}(u,z,\bar{z}).$

where

$\hat{C}_{zz}(u,z,\bar{z})\equiv\frac{1}{2}\left[\int^{u}_{-\infty}\mathrm{d}u^{\prime}N_{zz}(u^{\prime},z,\bar{z})-\int_{u}^{\infty}\mathrm{d}u^{\prime}N_{zz}(u^{\prime},z,\bar{z})\right],$

is the bulk contribution and the remaining terms are boundary terms. In
particular, this means

$N_{zz}=\partial_{u}C_{zz}=\partial_{u}\hat{C}_{zz}.$

Choosing $\Sigma=\cal I^{+}$ as the Cauchy slice, one arrives at the
presymplectic form

$\tilde{\Omega}_{\mathcal{I}^{+}}=\frac{1}{16\pi
G}\int_{\mathcal{I}^{+}}\gamma_{z\bar{z}}\mathrm{d}^{2}z\mathrm{d}u\delta\hat{C}_{zz}\wedge\delta
N^{zz}+\frac{1}{32\pi
G}\int\gamma_{z\bar{z}}\mathrm{d}^{2}z(\delta\phi^{-}_{zz}\wedge\delta\Delta\phi^{zz}+\delta\phi^{-}_{\bar{z}\bar{z}}\wedge\delta\Delta\phi^{\bar{z}\bar{z}}).$
(4.7)

To get the symplectic form, we need to carry out steps 5 and 6 of Fig. 4 and
define our phase space.

##### Phase space

As mentioned in Section 4.1, we focus on CK spaces where the falloff (4.2) of
the Bondi news as $u\to\pm\infty$ guarantees that the first integral in (4.7)
converges. We argued that in such spaces, the only data we need specify is
$C_{zz}$, which is further constrained by the condition (4.4). As a result, we
choose for our phase space $\Gamma^{+}$ the set of $C_{zz}$ such that these
hold,

$\Gamma^{+}:=\left\\{C_{zz}:C_{zz}|_{\mathcal{I}^{+}_{\pm}}=D^{2}_{z}C^{\pm}+\mathcal{O}(u^{-\epsilon}),\epsilon>0\right\\},$

and similarly for $\Gamma^{-}$ at $\cal I^{-}$

$\Gamma^{-}:=\left\\{D_{zz}:D_{zz}|_{\mathcal{I}^{-}_{\pm}}=D^{2}_{z}D^{\pm}+\mathcal{O}(u^{-\epsilon}),\epsilon>0\right\\}.$

We also demand the variations $\delta C_{zz}$ and $\delta N_{zz}$ be also CK,
that is to say

$\displaystyle\delta
C_{zz}|_{\mathcal{I}^{+}_{\pm}}=C^{\prime\pm}_{zz}|_{\mathcal{I}^{+}_{-}}-C^{\pm}_{zz}|_{\mathcal{I}^{+}_{-}}=D^{2}_{z}C^{\prime\pm}-D_{z}^{2}C^{\pm}\overset{!}{=}D_{z}^{2}\delta
C^{\pm}$ and $\displaystyle\delta
N_{zz}\overset{u\to\pm\infty}{\longrightarrow}0\text{ as
}u^{1-\epsilon},\quad\epsilon>0,\quad\text{and similarly at $\cal I^{-}$}.$

##### Symplectic form

These considerations allow to go from (4.7) to the symplectic form [51]

$\Omega_{\mathcal{I}^{+}}=\frac{1}{16\pi
G}\int_{\mathcal{I}^{+}}\gamma_{z\bar{z}}\mathrm{d}^{2}z\mathrm{d}u\delta\hat{C}_{zz}\wedge\delta
N^{zz}+\frac{1}{16\pi G}\int\gamma_{z\bar{z}}\mathrm{d}^{2}zD^{2}_{z}\delta
C\wedge D^{2z}\delta N,$ (4.8)

where one denotes $C^{-}\equiv C$ for notational simplicity. This now
converges on $\Gamma$ and we are in a position to apply the tools of
symplectic geometry and derive Poisson brackets.

##### Charges

From (4.8), we can read off the non-vanishing Poisson brackets:

bulk-bulk:
$\displaystyle\quad\\{N_{\bar{z}\bar{z}}(u,z,\bar{z}),\hat{C}_{ww}(u^{\prime},w,\bar{w})\\}=16\pi
G\delta^{2}(z-w)\delta(u-u^{\prime})\gamma_{z\bar{z}}$ boundary-boundary:
$\displaystyle\quad\\{D^{2}_{\bar{z}}N(z,\bar{z}),D^{2}_{w}C(w,\bar{w})\\}=16\pi
G\delta^{2}(z-w)\gamma_{z\bar{z}}.$

Now, recall that in symplectic geometry, the symplectic form
$\Omega_{\mathcal{I}^{+}}$ and the infinitesimal charge
$\cancel{\delta}Q_{\xi}$ associated with the symmetry generated by a given
vector field $\xi$ are related by (2.13). The finite charge can then be
obtained by integrating $\cancel{\delta}Q_{\xi}$ along a path in the field
space.131313Such charge is said to be integrable if the integral does not
depend on the particular path chosen, i.e. if there exists a functional
$Q_{\xi}$ such that $\cancel{\delta}Q_{\xi}=\delta(Q_{\xi})$. $Q_{\xi}$ is
conserved on shell [51]. Eq.(4.8) yields $\cancel{\delta}Q_{f}=\delta Q_{f}$
where [51]

$Q_{f}=-\frac{1}{16\pi
G}\int_{\mathcal{I}^{+}}\gamma_{z\bar{z}}\mathrm{d}^{2}z\mathrm{d}ufN_{zz}N^{zz}+\frac{1}{8\pi
G}\int\gamma_{z\bar{z}}\mathrm{d}^{2}zD^{2}_{z}fD^{2z}N\equiv
Q^{\mathcal{H}}_{f}+Q^{\mathcal{S}}_{f}$ (4.9)

is our conserved charge under supertranslations. Note that it naturally
decomposes into a soft (linear in the fields) and hard (quadratic in the
fields) part, respectively denoted $Q^{\mathcal{S}}_{f}$ and
$Q^{\mathcal{H}}_{f}$. We also introduced the boundary field $N(z,\bar{z})$
[51] such that

$D^{2}_{z}N(z,\bar{z})=\int\mathrm{d}u^{\prime}N_{zz}(u^{\prime},z,\bar{z}).$

Denoting $Q_{f}\equiv T^{+}$ to match the results of [16] we have

$T^{+}(f)=-\frac{1}{16\pi
G}\int_{\mathcal{I}^{+}}\gamma_{z\bar{z}}\mathrm{d}^{2}z\mathrm{d}ufN_{zz}N^{zz}+\frac{1}{8\pi
G}\int\gamma_{z\bar{z}}\mathrm{d}^{2}zD^{2}_{z}fD^{2z}N=\frac{1}{4\pi
G}\int_{\mathcal{I}^{+}_{-}}\mathrm{d}^{2}z\gamma_{z\bar{z}}fm.$ (4.10)

Here, we performed an integration by parts on the last term in the first
equality and used the constraint equation for the Bondi mass (3.19). From the
brackets, one finds [16, 51]

$\displaystyle\\{T^{+}(f),N_{zz}\\}=f\partial_{u}N_{zz},\quad\\{T^{+}(f),C_{zz}\\}=f\partial_{u}C_{zz}-2D_{z}^{2}f,$
$\displaystyle\\{T^{+}(f),N\\}=0,\quad\text{and}\quad\\{T^{+}(f),C\\}=-2f.$

Comparing this with (3.35) and (3.36) confirms that $T^{+}(f)$ is the
generator of supertranslations at $\cal I^{+}$. Carrying out the same analysis
at $\cal I^{-}$, one arrives at the supertranslation charge

$\displaystyle T^{-}(f^{-})$ $\displaystyle=-\frac{1}{16\pi
G}\int\mathrm{d}v\mathrm{d}^{2}zf^{-}\gamma_{z\bar{z}}M_{zz}M^{zz}-\frac{1}{8\pi
G}\int\mathrm{d}^{2}z\gamma_{z\bar{z}}f^{-}D_{z}^{2}D_{\bar{z}}^{2}M,$ (4.11)
$\displaystyle=\frac{1}{4\pi
G}\int_{\mathcal{I}^{-}_{+}}\mathrm{d}^{2}z\gamma_{z\bar{z}}f^{-}m^{-}\quad\text{having
used that }D_{z}^{2}M=\int\mathrm{d}u^{\prime}M_{zz}(u^{\prime},z,\bar{z}).$

This charge has brackets

$\displaystyle\\{T^{-}(f^{-}),M_{zz}\\}=f^{-}\partial_{v}M_{zz},\quad\\{T^{-}(f^{-}),D_{zz}\\}=f^{-}\partial_{v}D_{zz}+2D_{z}^{2}f^{-},$
$\displaystyle\\{T^{-}(f^{-}),M\\}=0,\quad\text{and}\quad\\{T^{-}(f^{-}),D\\}=2f^{-}.$

Inspecting (3.39) and (3.40) shows again that $T^{-}(f^{-})$ indeed generates
supertranslations at $\cal I^{-}$.

####  4.2.2 Supertranslation invariance, soft graviton and Ward identity

Having obtained the supertranslation charges, we want to show that some
supertranslations leave the $\cal S$-matrix invariant, i.e. that $B^{\pm}$
commute with $\cal S$ for the generators $\xi_{T}$ in a subgroup BMS0 of
$\text{BMS}^{+}\times\text{BMS}^{-}$. Let us determine this subgroup. From
(3.38) and (3.41), we learn that the supertranslations that are compatible
with (4.4) are those satisfying

$V_{2z}|_{\mathcal{I}^{-}_{+}}=U_{2z}|_{\mathcal{I}^{+}_{-}}\Rightarrow
f^{-}(z,\bar{z})=f(z,\bar{z}).$ (4.12)

To match the charges (4.10) and (4.11), we also need to impose the matching of
the Bondi mass aspect

$m(z,\bar{z})|_{\mathcal{I}^{+}_{-}}=m^{-}(z,\bar{z})|_{\mathcal{I}^{-}_{+}}.$
(4.13)

Note that in these expressions, the coordinates $(z,\bar{z})$ on both sides of
the equality are related by the antipodal map. The vector fields satisfying
these two conditions constitute the supertranslations of BMS0, which is
usually referred to as the diagonal BMS group.141414The superrotations of BMS0
will be discussed in section 5.2.1.

We have not shown that the $\cal S$-matrix is invariant under
$\xi_{T}\in\text{BMS}^{0}$ yet. To see this, note from (4.10) that
$T^{+}(1)=M$. Put differently, $T^{+}(1)$ is the ADM Hamiltonian (times $G$).
Moreover, the charges at $\cal I^{+}$ obey [16]

$\\{T^{+}(f),T^{+}(f^{\prime})\\}=0.$ (4.14)

From our above matching conditions (4.12) and (4.13), we can finally write for
$\xi_{T}\in\text{BMS}^{0}$

$T(f)\equiv T^{+}(f)=T^{-}(f).$ (4.15)

Since $T(1)$ is the Hamiltonian, all $T(f)$ Poisson-commute with $T(1)$ (4.14)
and $\cal S$ is constructed from exponentials of the Hamiltonian, we have
$[T(f),\mathcal{S}]=0$. This, combined with (4.15), is exactly what we needed
to argue the supertranslation invariance of the $\cal S$-matrix under BMS0,

$\boxed{T^{+}(f)\mathcal{S}-\mathcal{S}T^{-}(f)=0\quad\text{for $f$ and $m$
satisfying \eqref{eq: BMS0 supertranslation condition} and \eqref{eq: bondi
mass matching condition}}.}$ (4.16)

So far, all our considerations were purely classical. By writing (4.16) we
have now just quantized our theory by promoting the charges to operators
acting on Hilbert spaces and on the $\cal S$-matrix.

##### Energy conservation

There is another nice interpretation to the statement (4.16). Using the
constraints (3.19) and (3.21) (assuming non vanishing stress tensor this
time), we can rewrite the supertranslation charges at $\cal I^{+}$ (4.10) and
$\cal I^{-}$ (4.11) as

$T^{+}(f)=\frac{1}{4\pi
G}\int\mathrm{d}u\mathrm{d}^{2}zf\gamma_{z\bar{z}}\left[-T_{uu}+\frac{1}{4}(D^{2}_{z}N^{zz}+D^{2}_{\bar{z}}N^{\bar{z}\bar{z}})\right]$

and

$T^{-}(f)=\frac{1}{4\pi
G}\int\mathrm{d}v\mathrm{d}^{2}zf\gamma_{z\bar{z}}\left[-T_{vv}-\frac{1}{4}(D^{2}_{z}M^{zz}+D^{2}_{\bar{z}}M^{\bar{z}\bar{z}})\right].$

We see that the local energy at a point includes not only a contribution from
the stress tensor but also a term which is linear in the Bondi news and a
total $u$ derivative. Then in particular for $f=\delta^{2}(z-w)$ our equality
(4.15) writes

$\int_{\mathcal{I}^{+}}\mathrm{d}u\gamma_{z\bar{z}}\left[-T_{uu}+\frac{1}{4}(D^{2}_{z}N^{zz}+D^{2}_{\bar{z}}N^{\bar{z}\bar{z}})\right]=\int_{\mathcal{I}^{-}}\mathrm{d}v\gamma_{z\bar{z}}\left[-T_{vv}-\frac{1}{4}(D^{2}_{z}M^{zz}+D^{2}_{\bar{z}}M^{\bar{z}\bar{z}})\right].$
(4.17)

This has a nice consequence: the energy flux at a point $w$ on $\cal I^{+}$ is
equal to the integrated energy flux at the antipodal point $w$ on $\cal
I^{-}$. In other words, the total accumulated energy incoming from every angle
$(z,\bar{z})$ on $\cal I^{-}$ equals the total accumulated energy emerging at
the angle $(z,\bar{z})$ on $\cal I^{+}$ [16].

##### Soft graviton

We now seek to relate the supertranslation invariance of the $\cal S$-matrix
to a quantum Ward identity involving so called soft gravitons. In quantum
field theory, Ward identities are fundamental relationships expressing the
symmetries of a physical system by relating correlation functions of fields to
each other. In the same way as fluctuations of quantum fields give rise to
particles in quantum field theory, gravitons denote the particles
corresponding to small fluctuations of the spacetime metric around a flat
background (Minkowski metric)

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}.$

The tensor $h_{\mu\nu}$ is then referred to as the graviton field. Looking
back at the form (3.17) of the asymptotic metric for an AFS in retarded
coordinates,

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}r+2r^{2}\gamma_{z\bar{z}}\mathrm{d}z\mathrm{d}\bar{z}+\frac{2m}{r}\mathrm{d}u^{2}+rC_{zz}\mathrm{d}z^{2}+rC_{\bar{z}\bar{z}}\mathrm{d}\bar{z}^{2}+...$
$\displaystyle=\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+rC_{zz}\mathrm{d}z^{2}+rC_{\bar{z}\bar{z}}\mathrm{d}\bar{z}^{2}+...$

we deduce that the graviton field in the $(u,r,z,\bar{z})$ system is encoded
through $C_{AB}$ since we can easily read off (up to normalization) [22]

$C_{zz}(u,z,\bar{z})=\kappa\lim_{r\to\infty}\frac{1}{r}h^{out}_{zz}(r,u,z,\bar{z}),\quad\kappa^{2}=32\pi
G.$ (4.18)

Thus $C_{zz}$ and $C_{\bar{z}\bar{z}}$ can indeed be seen as the two helicity
modes of the particle, as was already pointed out around (3.18). Since we are
dealing with outgoing radiation at $\cal I^{+}$, $h_{zz}^{out}$ in fact
denotes the free graviton field. It is common practice in QFT to expand the
fields in momentum space

$h_{\mu\nu}^{out}(x)=\sum_{\alpha=\pm}\int\frac{\mathrm{d}^{3}q}{(2\pi)^{3}}\frac{1}{2\omega_{q}}\left[\varepsilon^{\alpha\ast}_{\mu\nu}(\vec{q})a_{\alpha}^{out}(\vec{q})e^{iq\cdot
x}+\varepsilon^{\alpha}_{\mu\nu}(\vec{q})a_{\alpha}^{out}(\vec{q})^{\dagger}e^{-iq\cdot
x}\right].$ (4.19)

Here $q^{0}\equiv\omega_{q}=\absolutevalue{\vec{q}}$ since the graviton is
massless, $\alpha=\pm$ are the two helicities of the particle and
$\varepsilon_{\mu\nu}^{\alpha}$ is its polarization tensor. The quantization
comes with a set of commutation relations for the graviton creation and
annihilation operators $a_{\alpha}^{out\dagger}$ and $a_{\alpha}^{out}$

$[a_{\alpha}^{out}(\vec{q}),a_{\beta}^{out}(\vec{q}^{\prime})^{\dagger}]=\delta_{\alpha\beta}(2\omega_{q})(2\pi)^{3}\delta^{(3)}(\vec{q}-\vec{q}^{\prime}),\quad\text{all
the others vanish.}$

To proceed, it is useful to introduce a careful parametrization151515This is
motivated by the fact that for $t,r\to\infty$, the wave packet for a massless
particle with spatial momentum centered around $\vec{p}$ localizes on the
conformal sphere near
$\vec{p}=\omega\hat{x}\equiv\omega\frac{\vec{x}}{r}=\frac{\omega}{1+z\bar{z}}(z+\bar{z},-i(z-\bar{z}),1-z\bar{z}),\quad\text{where
$\vec{p}\cdot\vec{p}=\omega^{2}$
\cite[cite]{[\@@bibref{Number}{he_bms_2015}{}{}]}.}$ of the graviton four-
momentum $q^{\mu}$,

$q^{\mu}=\frac{\omega_{q}}{1+w\bar{w}}(1+w\bar{w},w+\bar{w},-i(w-\bar{w}),1-w\bar{w}),$

as well as of the polarization tensor
$\varepsilon^{\pm\mu\nu}=\varepsilon^{\pm\mu}\varepsilon^{\pm\nu}$,

$\varepsilon^{+\mu}(\vec{q})=\frac{1}{\sqrt{2}}(\bar{w},1,-i,-\bar{w}),\quad\varepsilon^{-\mu}(\vec{q})=\frac{1}{\sqrt{2}}(w,1,i,-w),\quad\text{such
that }\varepsilon^{\pm\mu\nu}q_{\nu}=\varepsilon^{\pm\mu}\,_{\mu}=0.$

Now applying the chain rule
$h_{zz}=\partial_{z}x^{\mu}\partial_{z}x^{\nu}h_{\mu\nu}$ to our mode
expansion (4.19) for the graviton field

$h_{zz}=\partial_{z}x^{\mu}\partial_{z}x^{\nu}\sum_{\alpha=\pm}\int\frac{\mathrm{d}^{3}q}{(2\pi)^{3}}\frac{1}{2\omega_{q}}\left[\varepsilon^{\alpha\ast}_{\mu\nu}(\vec{q})a_{\alpha}^{out}(\vec{q})e^{iq\cdot
x}+\varepsilon^{\alpha}_{\mu\nu}(\vec{q})a_{\alpha}^{out}(\vec{q})^{\dagger}e^{-iq\cdot
x}\right]$

and using the fact that with the above parametrization choices

$\partial_{z}x^{\mu}\varepsilon^{+}_{\mu}(\vec{q})=\frac{\sqrt{2}r\bar{z}(\bar{w}-\bar{z})}{(1+z\bar{z})^{2}},\quad\partial_{z}x^{\mu}\varepsilon^{-}_{\mu}(\vec{q})=\frac{\sqrt{2}r(1+w\bar{z})}{(1+z\bar{z})^{2}}$

along with (4.18), we can recast $C_{zz}$ directly in terms of the graviton
mode operators [22]

$C_{zz}=\kappa\lim_{r\to\infty}\frac{1}{r}\partial_{z}x^{\mu}\partial_{z}x^{\nu}\sum_{\alpha=\pm}\int\frac{\mathrm{d}^{3}q}{(2\pi)^{3}}\frac{1}{\omega_{q}}\left[\varepsilon^{\alpha\ast}_{\mu\nu}(\vec{q})a_{\alpha}^{out}(\vec{q})e^{-i\omega_{q}u-i\omega_{q}r(1-\cos\theta)}+\text{h.c.}\right].$

Here $\theta$ is the angle between $\vec{x}$ and $\vec{q}$ and we used the
expressions (2.5) for $x^{\mu}$ in terms of $(u,r,z,\bar{z})$. Taking
$r\to\infty$, this integral is can be evaluated by resorting to the stationary
phase approximation, i.e. we approximate the oscillatory integrand by its
dominant contributions, which occur when the phase of the integrand is
stationary ($\theta=0,\pi$). The contributions from other regions of the
integrand that rapidly oscillate will cancel out or be negligible due to their
oscillatory nature. As it turns out, the contribution from $\theta=\pi$ also
vanishes as $r\to\infty$, leaving us with

$C_{zz}=-\frac{i\kappa}{4\pi^{2}(1+z\bar{z})^{2}}\int_{0}^{\infty}\mathrm{d}\omega_{q}\left[a_{+}^{out}(\omega_{q}\hat{x})e^{-i\omega_{q}u}-a_{-}^{out}(\omega_{q}\hat{x})^{\dagger}e^{i\omega_{q}u}\right].$
(4.20)

Hence $C_{zz}$ is nothing but a Fourier transform of the momentum-space
creation and annihilation operators. Acting with the inverse transform and
defining the Fourier modes of the Bondi news

$N^{\omega}_{zz}(z,\bar{z})\equiv\int_{-\infty}^{\infty}\mathrm{d}ue^{i\omega
u}\partial_{u}C_{zz},$ (4.21)

our expression (4.20) becomes

$N_{zz}^{\omega}(z,\bar{z})=-\frac{\kappa}{2\pi(1+z\bar{z})^{2}}\int_{0}^{\infty}\mathrm{d}\omega_{q}\omega_{q}\left[a_{+}^{out}(\omega_{q}\hat{x})\delta(\omega_{q}-\omega)+a_{-}^{out}(\omega_{q}\hat{x})^{\dagger}\delta(\omega_{q}+\omega)\right].$

The sign of $\omega$ will affect which of the terms in the integrand
contributes. For $\omega=0$, we average on both contributions and define the
(hermitian) zero-mode $N^{0}_{zz}$ of the Bondi news as

$N^{0}_{zz}(z,\bar{z})\equiv\lim_{\omega\to
0^{+}}\frac{1}{2}(N_{zz}^{\omega}+N_{zz}^{-\omega})=-\frac{\kappa}{4\pi(1+z\bar{z})^{2}}\lim_{\omega\to
0^{+}}\left[\omega a_{+}^{out}(\omega\hat{x})+\omega
a_{-}^{out}(\omega\hat{x})^{\dagger}\right].$ (4.22)

Carrying out the same steps at $\cal I^{-}$ yields [22]

$M^{0}_{zz}=-\frac{\kappa}{4\pi(1+z\bar{z})^{2}}\lim_{\omega\to
0^{+}}\left[\omega a_{+}^{in}(\omega\hat{x})+\omega
a_{-}^{in}(\omega\hat{x})^{\dagger}\right].$ (4.23)

This achieves to show how $C_{zz}$ relates to gravitons. Both formuli will
play a key role later on.

Now, what do we mean by soft gravitons? In scattering amplitudes, soft
contributions refer to the contributions from low-energy, low-momentum
particles. In quantum field theory, we see that these contributions are linear
in the fields because they correspond to the first-order terms in a
perturbative expansion, where each order is proportional to a power of the
coupling constant. In contrast, hard contributions correspond to high-energy,
high-momentum particles or radiation which cannot be treated as small
fluctuations and thus require a non-perturbative treatment. The non-linear
terms in the perturbative expansion become important at higher orders, where
they contribute to the hard contributions to the scattering amplitudes. This
motivates our splitting of the charge (4.9) since we now see that we only need
to consider the soft parts of the charges to first order in perturbation
theory. Incidentally, these are also related to the zero-modes of the advanced
and retarded Bondi news we just derived through $C_{zz}$, etc. Great! We now
have all the tools we need to reformulate (4.16) as a Ward identity, viewing
it first as a statement for a soft graviton current.

##### Soft graviton current

Time has finally come to look at scattering. Let us denote a in-state with
energies $E_{k}^{in}$ incoming at points $z_{k}^{in}$ on the conformal
$\mathbb{S}^{2}$ by $|z_{1}^{in},z_{2}^{in},...\rangle$. Then the
supertranslation generator $T^{-}$ acts as [16]

$T^{-}(f)|z_{1}^{in},z_{2}^{in},...\rangle=F^{-}|z_{1}^{in},z_{2}^{in},...\rangle+\sum_{k}E_{k}^{in}f(z_{k}^{in})|z_{1}^{in},z_{2}^{in},...\rangle,$
(4.24)

where $F^{-}$ denotes the soft part of (4.11)

$F^{-}=-\frac{1}{8\pi
G}\int_{\mathcal{I}^{-}}\mathrm{d}v\mathrm{d}^{2}zD^{2}_{\bar{z}}fM^{\bar{z}}\,_{z},\quad
M_{zz}=\partial_{v}D_{zz}.$

Labelling an outgoing state with energies $E_{k}^{out}$ by
$|z_{1}^{out},z_{2}^{out},...\rangle$, we see that $T^{+}$ similarly acts as

$\langle z_{1}^{out},z_{2}^{out},...|T^{+}(f)=\langle
z_{1}^{out},z_{2}^{out},...|F^{+}+\sum_{k}E_{k}^{out}f(z_{k}^{out})\langle
z_{1}^{out},z_{2}^{out},...|,$ (4.25)

with $F^{+}$ the soft part of (4.10)

$F^{+}=\frac{1}{8\pi
G}\int_{\mathcal{I}^{+}}\mathrm{d}u\mathrm{d}^{2}zD_{\bar{z}}^{2}fN^{\bar{z}}\,_{z}.$

Let us also define

$F\equiv F^{+}-F^{-}=\frac{1}{8\pi
G}\int\mathrm{d}^{2}z\gamma^{z\bar{z}}D_{\bar{z}}^{2}f\left[\int_{\mathcal{I}^{-}}\mathrm{d}vM_{zz}+\int_{\mathcal{I}^{+}}\mathrm{d}uN_{zz}\right].$
(4.26)

The supertranslation Ward identity for the time ordered product
$:F\mathcal{S}:=F^{+}\mathcal{S}-\mathcal{S}F^{-}$ is then straightforwardly
deduced from (4.16), (4.24) and (4.25)

$\langle
z^{out}_{1},z_{2}^{out},...|:F\mathcal{S}:|z^{in}_{1},z_{2}^{in},...\rangle=\langle
z^{out}_{1},z_{2}^{out},...|F^{+}\mathcal{S}-\mathcal{S}F^{-}|z^{in}_{1},z_{2}^{in},...\rangle\\\
=\sum_{k}(E_{k}^{in}f(z_{k}^{in})-E_{k}^{out}f(z_{k}^{out}))\langle
z^{out}_{1},z_{2}^{out},...|\mathcal{S}|z^{in}_{1},z_{2}^{in},...\rangle.$
(4.27)

This relates $\cal S$-matrix elements with and without the insertion of $F$.
In light of our expressions (4.22) and (4.23) for the modes of $N_{zz}$ and
$M_{zz}$ in terms of graviton annihilation and creation operators, it is
natural to think of $F$ as the operator corresponding to the insertion of a
soft graviton to the scattering process. If there are $n$ incoming and $m$
outgoing particles, then total energy conservation requires

$\sum_{k=1}^{m}E_{k}^{out}=\sum_{k=1}^{n}E_{k}^{in}.$

Choosing $f(w,\bar{w})=(z-w)^{-1}$ in (4.26), we rewrite $F$ as the soft
graviton current $P_{z}$

$F\left(\frac{1}{z-w}\right)\equiv
P_{z}=\frac{1}{2G}\left(\left.U_{2z}\right|_{\mathcal{I}^{+}_{-}}^{\mathcal{I}^{+}_{+}}-\left.V_{2z}\right|_{\mathcal{I}^{-}_{-}}^{\mathcal{I}^{-}_{+}}\right)=\frac{1}{2G}\left(\int_{-\infty}^{\infty}\mathrm{d}v\partial_{v}V_{2z}-\int_{-\infty}^{\infty}\mathrm{d}u\partial_{u}U_{2z}\right).$
(4.28)

Using (4.12), the matching (4.17) of the charges at $\cal I^{\pm}$ can be
rewritten as

$\displaystyle\int_{\mathcal{I}^{+}}\mathrm{d}u\left[-\gamma_{z\bar{z}}T_{uu}+\frac{1}{4}\partial_{u}(\partial_{z}U_{2\bar{z}}+\partial_{\bar{z}}U_{2z})\right]$
$\displaystyle=\int_{\mathcal{I}^{-}}\mathrm{d}v\left[-\gamma_{z\bar{z}}T_{vv}+\frac{1}{4}\mathrm{d}v(\partial_{z}V_{2\bar{z}}+\partial_{\bar{z}}V_{2z})\right]$
$\displaystyle\Leftrightarrow\gamma_{z\bar{z}}\left(\int_{\mathcal{I}^{+}}\mathrm{d}uT_{uu}-\int_{\mathcal{I}^{-}}\mathrm{d}vT_{vv}\right)$
$\displaystyle=\partial_{\bar{z}}\left[\left.U_{2z}\right|_{\mathcal{I}^{+}_{-}}^{\mathcal{I}^{+}_{+}}-\left.V_{2z}\right|_{\mathcal{I}^{-}_{-}}^{\mathcal{I}^{-}_{+}}\right]\overset{\eqref{eq:
def soft graviton current Pz}}{=}4G\partial_{\bar{z}}P_{z}.$

We can solve for $P_{z}$ in this special case by means of a Green function for
$\partial_{\bar{z}}$,

$P_{z}\equiv\frac{1}{4\pi
G}\int\mathrm{d}^{2}w\frac{\gamma_{w\bar{w}}}{z-w}\left(\int\mathrm{d}uT_{uu}-\int\mathrm{d}vT_{vv}\right).$
(4.29)

##### Ward identity

We now construct the Ward identity for the soft graviton current. Consider the
simple case where

$G\sum_{k}E_{k}^{out}=G\sum_{k}E_{k}^{in}$

and the incoming and outgoing particles are localized at $(v_{k},z_{k}^{in})$
and $(u_{k},z_{k}^{out})$ at $\cal I^{-}$ and $\cal I^{+}$ respectively. Then
$T_{uu}$ and $T_{vv}$ take the simple form

$T_{uu}=4\pi
G\sum_{k}E_{k}^{out}\delta(u-u_{k})\frac{\delta^{2}(z-z_{k}^{out})}{\gamma_{z\bar{z}}}\quad\text{and}\quad
T_{vv}=4\pi
G\sum_{k}E_{k}^{in}\delta(v-v_{k})\frac{\delta^{2}(z-z_{k}^{in})}{\gamma_{z\bar{z}}}$

while (4.29) simplifies to

$P_{z}=\sum_{k=1}^{m}\frac{E_{k}^{out}}{z-z_{k}^{out}}-\sum_{k=1}^{n}\frac{E_{k}^{in}}{z-z_{k}^{in}},$

such that the matrix element (4.27) reads

$\boxed{\langle
z^{out}_{1},z_{2}^{out},...|:P_{z}\mathcal{S}:|z^{in}_{1},z_{2}^{in},...\rangle=\left[\sum_{k=1}^{m}\frac{E_{k}^{out}}{z-z_{k}^{out}}-\sum_{k=1}^{n}\frac{E_{k}^{in}}{z-z_{k}^{in}}\right]\langle
z^{out}_{1},z_{2}^{out},...|\mathcal{S}|z^{in}_{1},z_{2}^{in},...\rangle.}$
(4.30)

Thus (4.30) relates $\cal S$-matrix elements with and without insertions of
the soft graviton current $P_{z}$. We now show that this is equivalent to
Weinberg’s leading soft graviton theorem.

####  4.2.3 Leading soft theorem

As mentioned before, there are certain limits in which the scattering matrix
of a particular process simplifies. In particular, if the energy of one or
more of the massless particles involved in the collision is taken to be small
in comparison to the energy or masses of the other particles in the process,
universal properties of Feynman diagrams and scattering amplitudes emerge.
This is the soft limit, which gives rise to soft theorems. In what follows, we
first state the results of Weinberg [70], who formulated these concepts for
gravity. We then show how his results can be related to the Ward identity
(4.30) derived previously from the BMS supertranslation invariance of the
$\cal S$-matrix.

##### Weinberg’s soft graviton theorem

We focus here on the case of a free massless scalar field, as done by He et
al. [22]. We are interested in computing the on-shell amplitude
$\mathcal{M}(p_{1}^{\prime},...,p_{m}^{\prime},p_{1},...,p_{n})$ involving $n$
incoming (momenta $p_{1},...,p_{n}$) and $m$ outgoing (momenta
$p_{1}^{\prime},...,p_{m}^{\prime}$) massless scalars. Now, consider the exact
same amplitude but with an additional outgoing soft graviton of momentum $q$
and polarization $\epsilon_{\mu\nu}(q)$ satisfying the gauge condition
$q^{\mu}\epsilon_{\mu\nu}=\frac{1}{2}q_{\nu}\epsilon^{\mu}\,_{\mu}$,
$\mathcal{M}_{\mu\nu}(q,p_{1}^{\prime},...,p^{\prime}_{m},p_{1},...,p_{n})$.
Weinberg’s soft graviton theorem relates the two as

$\mathcal{M}_{\mu\nu}(q,p_{1}^{\prime},...,p^{\prime}_{m},p_{1},...,p_{n})=\frac{\kappa}{2}\left[\sum_{k=1}^{m}\frac{p^{\prime}_{k\mu}p^{\prime}_{k\nu}}{p^{\prime}_{k}\cdot
q}-\sum_{k=1}^{n}\frac{p_{k\mu}p_{k\nu}}{p_{k}\cdot
q}\right]\mathcal{M}(p_{1}^{\prime},...,p_{m}^{\prime},p_{1},...,p_{n}),$
(4.31)

with $\kappa^{2}=32\pi G$ as before and where the terms in brackets is the
soft factor, $S_{\mu\nu}^{m-n}$. The latter is universal and gauge invariant
[22], meaning that the formula does not depend on any of the quantum numbers
of the asymptotic particles involved in the $\mathcal{S}$-matrix element.

##### Equivalence

We now want to show that (4.30) and (4.31) are equivalent statements. To this
end, we need to recast the soft graviton current (4.28) in terms of standard
momentum space creation and annihilation operators. From (4.21) and (4.6), we
see that

$N^{0}_{zz}(z,\bar{z})=\int_{-\infty}^{\infty}\mathrm{d}u\partial_{u}C_{zz}=D_{z}^{2}N.$

Similar considerations at $\cal I^{-}$ yield
$M^{0}_{zz}(z,\bar{z})=D_{z}^{2}M$, for $M$ also a boundary field. Then
(4.28), together with $U_{2z}=D^{z}C_{zz}$ as in (3.15) and
$V_{2z}=-D^{z}D_{zz}$ means that we can write

$P_{z}=-\frac{1}{2G}\left(\int_{-\infty}^{\infty}\mathrm{d}v\partial_{v}D^{z}D_{zz}+\int_{-\infty}^{\infty}\mathrm{d}u\partial_{u}D^{z}C_{zz}\right)=\frac{1}{2G}D^{z}\left(M^{0}_{zz}+N^{0}_{zz}\right)=\frac{1}{4G}\gamma^{z\bar{z}}\partial_{\bar{z}}\mathcal{O}_{zz}$
(4.32)

with $\mathcal{O}_{zz}=N_{zz}^{0}+M_{zz}^{0}$ the sum of the zero Fourier
modes of the Bondi news (4.22) and (4.23). Recall that in these formuli,
$a^{in}_{\pm}$ ($a^{out}_{\pm}$) and $a_{\pm}^{in\dagger}$
($a_{\pm}^{out\dagger}$) annihilate and create incoming (outgoing) gravitons
on $\cal I^{-}$ ($\cal I^{+}$) with positive ($+$) or negative ($-$) helicity.
Now, consider the matrix element

$\langle z_{1}^{out},...|:\mathcal{O}_{zz}\mathcal{S}:|z_{1}^{in},...\rangle.$

On the one hand, using (4.22) and (4.23) we have

$\displaystyle\langle
z_{1}^{out},...|:\mathcal{O}_{zz}\mathcal{S}:|z_{1}^{in},...\rangle$
$\displaystyle=-\frac{\kappa}{4\pi(1+z\bar{z})^{2}}\lim_{\omega\to
0^{+}}\langle z_{1}^{out},...|\left[\omega
a_{+}^{out}(\omega\hat{x})+\omega\cancel{a^{out}_{-}(\omega\hat{x})^{\dagger}}\right]\mathcal{S}|z_{1}^{in},...\rangle$
$\displaystyle\qquad-\frac{\kappa}{4\pi(1+z\bar{z})^{2}}\lim_{\omega\to
0^{+}}\langle
z_{1}^{out},...|\mathcal{S}\left[\omega\cancel{a_{+}^{in}(\omega\hat{x})}+\omega
a_{-}^{in}(\omega\hat{x})^{\dagger}\right]|z_{1}^{in},...\rangle$
$\displaystyle=-\frac{\kappa}{2\pi(1+z\bar{z})^{2}}\lim_{\omega\to
0^{+}}\omega\langle
z_{1}^{out},...|a_{+}^{out}(\omega\hat{x})\mathcal{S}|z_{1}^{in},...\rangle.$

Here we used that $\langle
z_{1}^{out},...|a_{+}^{out}(\omega\hat{x})\mathcal{S}|z_{1}^{in},...\rangle=\langle
z_{1}^{out},...|\mathcal{S}a_{-}^{in}(\omega\hat{x})^{\dagger}|z_{1}^{in},...\rangle$
in the last line. On the other hand, (4.31) with a positive helicity outgoing
graviton ($a_{+}^{out}(\vec{q})$) allows to write [22]

$\langle
z_{1}^{out},...|:\mathcal{O}_{zz}\mathcal{S}:|z_{1}^{in},...\rangle=\frac{8G}{(1+z\bar{z})}\langle
z_{1}^{out},...|\mathcal{S}|z_{1}^{in},...\rangle\\\
\times\left[\sum_{k=1}^{m}\frac{E_{k}^{out}(\bar{z}-\bar{z}_{k}^{out})}{(z-z_{k}^{out})(1+z_{k}^{out}\bar{z}_{k}^{out})}-\sum_{k=1}^{n}\frac{E_{k}^{in}(\bar{z}-\bar{z}_{k}^{in})}{(z-z_{k}^{in})(1+z_{k}^{in}\bar{z}_{k}^{in})}\right].$

Then (4.32) gives us

$\displaystyle\langle
z_{1}^{out},...|:P_{z}\mathcal{S}:|z_{1}^{in},...\rangle$
$\displaystyle=\frac{1}{4G}\gamma^{z\bar{z}}\partial_{\bar{z}}\langle
z_{1}^{out},...|:\mathcal{O}_{zz}\mathcal{S}:|z_{1}^{in},...\rangle$
$\displaystyle=\langle
z_{1}^{out},...|\mathcal{S}|z_{1}^{in},...\rangle\left[\sum_{k=1}^{m}\frac{E_{k}^{out}}{z-z_{k}^{out}}-\sum_{k=1}^{n}\frac{E^{in}_{k}}{z-z_{k}^{in}}\right]$
$\displaystyle\qquad+\langle
z_{1}^{out},...|\mathcal{S}|z_{1}^{in},...\rangle\cancel{\left[\sum_{k=1}^{m}\frac{E_{k}^{out}\bar{z}_{k}^{out}}{1+z_{k}^{out}\bar{z}_{k}^{out}}-\sum_{k=1}^{n}\frac{E^{in}_{k}\bar{z}_{k}^{in}}{1+z_{k}^{in}\bar{z}_{k}^{in}}\right]}.$

The last term vanishes because of total momentum conservation, meaning that we
recover exactly (4.30); the Ward identity for supertranslation invariance is
equivalent to Weinberg’s leading soft graviton theorem for the soft graviton
current. Note that (4.31) naturally diverges as $q\to 0$. It is only thanks to
the factor of $\omega$ in (4.22) and (4.23) that we are able to cancel this
divergence in the soft limit and thereby pick up the residue of the soft
factor.

We now turn to the study of another aspect of the infrared structure of
gravitational scattering that is highlighted by supertranslations: memory
effects.

###  4.3 Gravitational memory effects

In this subsection we relate the BMS supertranslations to physical phenomena
known as gravitational memory effects. We first describe how the latter arise
in general relativity, prior to describing how they relate to BMS
supertranslations via so-called vacuum transitions. In a second time, we show
that memory effects can also be linked to soft theorems, in the spirit of what
we saw in section 4.2.

####  4.3.1 Displacement memory effect

Let us introduce memory effects in gravity. Consider a pair of inertial
observers ("detectors") travelling near future infinity, and suppose that a
gravitational wave travels between the two during the retarded time interval
$\Delta u\equiv u_{f}-u_{i}\ll r$. We assume that for any
$u\notin[u_{i},u_{f}]$, $N_{AB}=T_{\mu\nu}=0$. This situation is depicted in
Figure 6.

Figure 6: Penrose diagram for the displacement memory effect [50]

It has been known since the work of Zeldovich [71], Christodoulou [72],
Braginsky and Thorne [73] and others that the passage of gravitational
radiation in such a setup induces a permanent shift in the relative separation
$s^{\mu}$ of the detectors.

To see this, assume that the two inertial observers move along geodesics with
4-velocity $v^{\mu}$. Since both are located in the vicinity of $\cal I^{+}$,
we can posit that $v^{\mu}\partial_{\mu}=\partial_{u}$ to leading order. We
want to see how the passage of a gravitational wave affects our observers. We
thus resort to the equation of geodesic deviation, which quantifies the effect
of tidal gravitational forces on neighboring free-falling objects. This writes
[21, 50, 58]

$\nabla_{v}\nabla_{v}s^{\mu}=R^{\mu}\,_{\alpha\beta\gamma}v^{\alpha}v^{\beta}s^{\gamma},$

where $\nabla_{v}=v^{\mu}\nabla_{\mu}$ is the directional derivative along
$v^{\mu}$. Both detectors being located on the same celestial sphere (i.e. at
the same value of $r$), we have $s^{r}=0$. Expanding, one gets

$r^{2}\gamma_{AB}\partial_{u}^{2}s^{B}=R_{uAuB}s^{B}.$

Noting that in Bondi gauge
$R_{uAuB}=-\frac{r}{2}\partial_{u}^{2}C_{AB}+\mathcal{O}(1)$ [50], this
simplifies to

$\gamma_{AB}\partial_{u}^{2}s^{B}=\frac{1}{2r}\partial_{u}^{2}C_{AB}s^{B}.$

To proceed, let us expand $s^{B}=s^{B}_{i}+s^{B}_{sub}/r$ and integrate the
above equation over $u$ to find

$\gamma_{AB}\Delta s_{sub}^{B}=\frac{1}{2r}\Delta
C_{AB}s^{B}_{i}+\mathcal{O}(r^{-2}),\quad\Delta f(u)\equiv f(u_{f})-f(u_{i}).$

The case $\Delta s_{sub}^{B}\neq 0$ leads to the so-called displacement
effect. To find the causes of this phenomenon, one must look at what could
possibly lead to a change in $C_{AB}$. In their lecture notes, Compère and
Fiorucci [50] showed, using the constraint (3.19) on the Bondi mass aspect,
that this either happens when $m$ varies between $u_{i}$ and $u_{f}$, when
null matter reaches $\cal I^{+}$ between $u_{i}$ and $u_{f}$ (ex.
electromagnetic radiation), or when gravitational waves pass through $\cal
I^{+}$ in the same interval. We have indeed argued above that the difference
$\Delta C_{zz}$ need not vanish in our case since $C_{zz}$ (and
$C_{\bar{z}\bar{z}}$) is left unconstrained in AFS, meaning that a shift in
$s^{\mu}$ is possible: this is sometimes referred to as the Christodoulou
effect. See Fig. 7 for an illustration of the phenomenon, where the passage of
the gravitational wave induces a permanent shift in the metric deformation.

Figure 7: Sketch of the metric perturbation over time [41]

We now show that this can be seen as a consequence of BMS symmetry.

####  4.3.2 Vacuum transitions

Recall how supertranslations act on $C_{zz}$ (3.35), $N_{zz}$ (3.36) and $m$
(3.37). Therefore, starting with $C_{zz}=0$, we have that after a pure161616By
”pure”, we mean one that is not an ordinary spacetime translation so that
$D_{z}^{2}f\neq 0$. supertranslation
$C_{zz}^{\prime}=C_{zz}+\mathcal{L}_{f}C_{zz}\neq 0$. However, as $C_{zz}$
characterizes the vacuum ($N_{zz}=0$), it must be that $C_{zz}^{\prime}$ and
$C_{zz}$ effectively describe physically inequivalent configurations. This
shows that BMS supertranslations are the source of an infinite-dimensional
degeneracy of vacua.

Let us show that a vacuum transition under a supertranslation is equivalent to
the displacement memory effect discussed above. Starting with the same setup
as Fig. 6, we consider spacetimes for which

$\displaystyle m=M_{i}=\text{constant, and }C_{zz}=0,N_{zz}=0\quad\text{for
}u<u_{i},$ $\displaystyle m=M_{f}=\text{constant, and }C_{zz}\neq
0,N_{zz}=0\quad\text{for }u>u_{f},$

while for $u_{i}\leq u\leq u_{f}$ $N_{zz}$ and/or $T_{uu}$ are nonzero on
$\cal I^{+}$. We saw in (4.3) that in CK spaces we can find $C(z,\bar{z})$
such that $C_{zz}=-2D^{2}_{z}C(z,\bar{z})$ with $\mathcal{L}_{f}C=f$. In light
of our discussion of vacuum transitions, we can now reinterpret this result as
the statement that different vacua are related by supertranslations under
which $C\to C+f$. Let us construct the supertranslation associated with a
displacement memory effect. Integrating once again the constraint (3.19) on
$m$ between $u_{i}$ and $u_{f}$ we find

$\displaystyle\partial_{u}m$
$\displaystyle=-T_{uu}+\frac{1}{4}\left[D^{2z}N_{zz}+D^{\bar{z}2}N_{\bar{z}\bar{z}}\right]=-T_{uu}+\frac{1}{2}D_{z}^{2}\partial_{u}C^{zz}$
$\displaystyle\Leftrightarrow\quad D^{2}_{z}\Delta C^{zz}$
$\displaystyle=2\int^{u_{f}}_{u_{i}}\mathrm{d}uT_{uu}+2\Delta m.$

The supertranslation $\Delta C\equiv f$ corresponding to such a change can be
found by means of the Green function for $D_{z}^{2}D_{\bar{z}}^{2}$:

$G(z,\bar{z};z^{\prime},\bar{z}^{\prime})=-\frac{1}{\pi}\sin^{2}\frac{\Theta}{2}\log\sin^{2}\frac{\Theta}{2},\quad\sin^{2}\frac{\Theta(z,z^{\prime})}{2}\equiv\frac{\absolutevalue{z-z^{\prime}}^{2}}{(1+z^{\prime}\bar{z}^{\prime})(1+z\bar{z})}$

such that
$D^{2}_{z}D^{2}_{\bar{z}}G(z,z^{\prime};z^{\prime},\bar{z}^{\prime})=-\gamma_{z\bar{z}}\delta^{2}(z-z^{\prime})+...$,
which yields

$\Delta
C(z,\bar{z})=2\int\mathrm{d}^{2}z^{\prime}\gamma_{z^{\prime}\bar{z}^{\prime}}G(z,\bar{z};z^{\prime},\bar{z}^{\prime})\left(\int_{u_{i}}^{u_{f}}\mathrm{d}uT_{uu}(z^{\prime},\bar{z}^{\prime})+\Delta
m\right).$ (4.33)

Indeed, a tedious calculation [18] shows that

$\Delta
C(z,\bar{z})=-\frac{2}{\pi}\int\mathrm{d}^{2}z^{\prime}\gamma_{z^{\prime}\bar{z}^{\prime}}\frac{\absolutevalue{z-z^{\prime}}^{2}}{(1+z^{\prime}\bar{z}^{\prime})(1+z\bar{z})}\log\frac{\absolutevalue{z-z^{\prime}}^{2}}{(1+z^{\prime}\bar{z}^{\prime})(1+z\bar{z})}\left(\int_{u_{i}}^{u_{f}}\mathrm{d}uT_{uu}(z^{\prime},\bar{z}^{\prime})+\Delta
m\right)$

is such that

$D_{z}^{2}D_{\bar{z}}^{2}\Delta
C=2\left(\int_{u_{i}}^{u_{f}}\mathrm{d}uT_{uu}(z^{\prime},\bar{z}^{\prime})+\Delta
m\right)$

and $C_{zz}=-2D^{2}_{z}C(z,\bar{z})$ then gives us our result (4.33). This is
an explicit expression for the supertranslation induced by waves passing
through $\cal I^{+}$, showing that the displacement effect is indeed
equivalent to BMS supertranslations through vacuum transitions. Let us now
wrap up this discussion by looking at how memory effects are related to the
soft theorems of section 4.2.

####  4.3.3 Memory and soft theorems

Finding a connection between soft theorems and memory effects is relatively
straightforward in the case where both are associated to BMS
supertranslations. On the one hand, Weinberg’s soft graviton theorem (4.31)
can be rewritten as

$\displaystyle\mathcal{M}_{\mu\nu}(q,p_{1}^{\prime},...,p^{\prime}_{m},p_{1},...,p_{n})\epsilon^{\mu\nu}$
$\displaystyle=\frac{\kappa}{2}\left[\sum_{k=1}^{m}\frac{p^{\prime}_{k\mu}p^{\prime}_{k\nu}}{p^{\prime}_{k}\cdot
q}-\sum_{k=1}^{n}\frac{p_{k\mu}p_{k\nu}}{p_{k}\cdot
q}\right]\epsilon^{\mu\nu}\mathcal{M}(p_{1}^{\prime},...,p_{m}^{\prime},p_{1},...,p_{n})$
$\displaystyle=\frac{\kappa}{2}\left[\sum_{k=1}^{m}\frac{p^{\prime}_{k\mu}p^{\prime}_{k\nu}}{p^{\prime}_{k}\cdot\omega
k}-\sum_{k=1}^{n}\frac{p_{k\mu}p_{k\nu}}{p_{k}\cdot\omega
k}\right]^{TT}\epsilon^{\mu\nu}\mathcal{M}(p_{1}^{\prime},...,p_{m}^{\prime},p_{1},...,p_{n})$

by making an explicit reference to the transverse polarization
$\epsilon_{\mu\nu}$ and the graviton $4$-momentum by
$q=(\omega,\omega\vec{k})$ ($\omega\to 0$). Here, the superscript $TT$ denotes
the transverse traceless part of the brackets. To go from the first to second
line, we used that the graviton always couples to transverse and traceless
polarizations, which is singled out from contraction of the terms in square
brackets with $\epsilon^{\mu\nu}$. Meanwhile, Braginski and Thorne [73] showed
in previous work that the shift in this $TT$ part of the asymptotic metric at
$\cal I^{+}$ resulting from the collision of large massive objects is [18]

$\Delta
h_{\mu\nu}^{TT}(\omega,\vec{k})=\frac{1}{r_{0}}\sqrt{\frac{G}{2\pi}}\left(\sum_{j=1}^{n}\frac{p^{\prime}_{j\mu}p^{\prime}_{j\nu}}{\omega
k\cdot p^{\prime}_{j}}-\sum_{j=1}^{m}\frac{p_{j\mu}p_{j\nu}}{\omega k\cdot
p_{j}}\right)^{TT}$

for $n$ ($m$) incoming (outgoing) massive objects with momenta $p_{j\mu}$
($p^{\prime}_{j\mu}$) and where $\vec{k}$ is the null vector pointing from the
collision region to infinity. To gain insight, it is again helpful to consider
the graviton field in Fourier space, assuming that the stationary
approximation holds as at large $r$,

$h_{\mu\nu}^{TT}(\omega,\vec{k})=4\pi
i\lim_{r\to\infty}r\int\mathrm{d}ue^{i\omega u}h_{\mu\nu}^{TT}(u,r\vec{k}).$

The field $h^{TT}_{\mu\nu}(u,r\vec{k})$ is expected to decay at and approach
different finite values as $u\to\pm\infty$ (with $r=r_{0}$ large). Then it
must be that

$\Delta h_{\mu\nu}^{TT}(\omega,\vec{k})=\frac{1}{4\pi r_{0}}\lim_{\omega\to
0}\left(-i\omega h_{\mu\nu}^{TT}(\omega,\vec{k})\right),$

Finally, we find an expression for $\omega h_{\mu\nu}^{TT}$ noting that [18]

$\displaystyle\lim_{\omega\to 0}\omega
h_{\mu\nu}^{TT}(\omega,k)\epsilon^{\mu\nu}$ $\displaystyle=\lim_{\omega\to
0}\frac{\omega\mathcal{A}_{m+n+1}\left(p_{1},\ldots
p_{n};p_{1}^{\prime},\ldots p_{m}^{\prime},\left(\omega
k,\epsilon_{\mu\nu}\right)\right)}{\mathcal{A}_{m+n}\left(p_{1},\ldots
p_{n};p_{1}^{\prime},\ldots p_{m}^{\prime}\right)}$ $\displaystyle=\sqrt{8\pi
G}\epsilon^{\mu\nu}\lim_{\omega\to 0}\omega S_{\mu\nu}(\omega k)=\sqrt{8\pi
G}\epsilon^{\mu\nu}\left(\sum_{j=1}^{m}\frac{p_{j\mu}p_{j\nu}}{k\cdot
p_{j}}-\sum_{j=1}^{n}\frac{p_{j\mu}^{\prime}p_{j\nu}^{\prime}}{k\cdot
p_{j}^{\prime}}\right)^{TT},$

where $\mathcal{A}_{m+n}$ denotes a collision involving $n$ incoming objects
with $m$ other massive objects. In short, acting with a Fourier transform on
the momentum space formula allows to get to the expression we obtained from
Weinberg’s soft graviton theorem. This concludes our study of
supertranslations and their implications for the infrared structure of
gravity.

##  5 Conclusion and Outlook

Let us now summarize the review. Starting from simple considerations of
gravitational scattering processes between $\cal I^{-}$ and $\cal I^{+}$, we
constructed the asymptotic symmetry group of asymptotically flat spacetimes,
the (global) BMS group. This group turned out to be larger than the Poincaré
group of isometries of Minkowski space, showing that general relativity does
not reduce to its special counterpart at large distances from an isolated
system. Recently, Compère et al. [74] extended these concepts to $i^{\pm}$ and
$i^{0}$, showing how individual ingoing and outgoing massive bodies may be
ascribed initial or final BMS charges and deriving associated global
conservation laws. The BMS analysis has also been carried out in three
spacetime dimensions, see [75] for details. The framework was extended to
higher dimensions as well as of late by Capone and collaborators [76].

###  5.1 The infrared triangle for BMS supertranslations

Building upon our knowledge of the Killing vectors of BMS4, we were able to
derive a series of relationships between asymptotic symmetries, soft theorems
and memory effects. Figure 8 summarizes what we learned from this in the case
of supertranslations.

Figure 8: The infrared triangle of gravity for BMS supertranslations

After having obtained the form of the generators of supertranslations, we were
able to put the covariant phase space formalism to use and compute the
conserved charges associated with this symmetry. Considering the diagonal