# BMS3 Mechanics and the Black Hole Interior

Marc Geiller, Etera R. Livine, Francesco Sartini

(Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,
CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
)

###### Abstract

The spacetime in the interior of a black hole can be described by an
homogeneous line element, for which the Einstein–Hilbert action reduces to a
one-dimensional mechanical model. We have shown in [1] that this model
exhibits a symmetry under the $(2+1)$-dimensional Poincaré group. Here we
explain how this can be understood as a broken infinite-dimensional BMS3
symmetry. This is done by reinterpreting the action for the model as a
geometric action for BMS3, where the configuration space variables are
elements of the algebra $\mathfrak{bms}_{3}$ and the equations of motion
transform as coadjoint vectors. The Poincaré subgroup then arises as the
stabilizer of the vacuum orbit. This symmetry breaking is analogous to what
happens with the Schwarzian action in AdS2 JT gravity, although in the present
case there is no direct interpretation in terms of boundary symmetries. This
observation, together with the fact that other lower-dimensional gravitational
models (such as the BTZ black hole) possess the same broken BMS3 symmetries,
provides yet another illustration of the ubiquitous role played by this group.

###### Contents

  1. 1 Introduction
  2. 2 The black hole interior spacetime
    1. 2.1 Homogeneous metric, choice of time and Hamiltonian
    2. 2.2 Reduced action and classical solutions
  3. 3 Adjoint representation of BMS3
    1. 3.1 Symmetries and transformations of the action
      1. 3.1.1 Finite transformations
      2. 3.1.2 Infinitesimal transformations
    2. 3.2 Hamiltonian formulation and generators
  4. 4 Coadjoint representation of BMS3 and central charges
  5. 5 Discussion
  6. A The Virasoro and BMS3 groups
    1. A.1 The Virasoro group
      1. A.1.1 Tensor densities on $S^{1}$
      2. A.1.2 Central extension
      3. A.1.3 Coadjoint action
    2. A.2 The BMS3 group
  7. B Other gravitational systems
    1. B.1 4d black hole in Eddington–Finkelstein coordinates
    2. B.2 3d BTZ black hole interior
    3. B.3 2d dilatonic gravity
  8. C Cosmological constant and scalar field
  9. D 1d BMS3 invariant action
    1. D.1 Lagrangian analysis
    2. D.2 Hamiltonian analysis

##  Appendix 1 Introduction

Symmetries are an invaluable tool when studying classical theories and their
quantization. They help understand the structure of the classical field
equations and of their solutions, and representation theory of symmetry groups
organize the states and spectra at the quantum level. It is clear that the
quantization of gravity, if attainable, will require a profound understanding
of the role of symmetries in general relativity.

General relativity being a gauge theory, the role of symmetries therein is
particularly subtle. It has been known for a while that gauge symmetries
acquire a particular status in the presence of boundaries [2, 3, 4, 5, 6, 7,
8]. Boundaries indeed have the ability of turning gauge symmetries into
physical symmetries with a non-trivial charge. In the gravitational context,
this is related to the appearance of infinite-dimensional boundary symmetry
groups, whose first historical examples are the BMS4 enhancement of Poincaré
in 4-dimensional asymptotically flat spacetimes [9, 10, 11], and the double
Virasoro enhancement of SO(2,2) in AdS3 gravity [12]. In certain cases these
boundary symmetries can moreover be seen as the bulk symmetries of a lower-
dimensional theory describing the boundary dynamics [13, 14, 15, 16].

The particular case of the BMS4 group has received lots of attention in recent
years due to newly discovered connections with gravitational scattering and
memory effects [17, 18, 19, 20, 21, 22], and the prospect of constructing
holographic theories [23, 24, 25, 26, 27, 28, 29, 30]. This has motivated in
particular authors to propose various extensions of the 4-dimensional BMS
group [25, 31, 32, 33, 34, 35, 36]. The 3-dimensional group BMS3 and its
extensions have also been studied at length [37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48]. In all these works the BMS group is always associated with
the symmetries of some boundary structure, whether at infinity or on a black
hole horizon. In the present paper we study a setup where the group BMS3 plays
a role, although as far as we can tell there are no boundaries or asymptotic
symmetries involved.

To set the stage, let us recall that in a recent paper [1] we have studied the
dynamics of the black hole interior spacetime. This is described by an
homogeneous line element, the Kantowski–Sachs metric, for which the
Einstein–Hilbert action reduces to a 1-dimensional mechanical model, with time
reparametrization symmetry inherited from the 4-dimensional diffeomorphism
invariance. We have shown that the resulting action (and the Hamiltonian
theory), when gauge-fixed to proper time, exhibits a residual invariance under
the $(2+1)$-dimensional Poincaré group ISO$(2,1)$. This extends the results
obtained for FLRW spacetimes in [49, 50, 51, 52, 53].

The aim of the present work is to study how this Poincaré symmetry group in
the black hole interior can be embedded into its infinite-dimensional
enhancement BMS3, and how the results of [1] can be reinterpreted from this
point of view. As we will see, even if the theory is not invariant under the
full group but only under its global Poincaré subgroup, it is possible to
associate integrable generators to all the BMS3 transformations on the phase
space. We then show that the classical trajectories can be interpreted in
terms of coadjoint orbits of BMS3. More precisely, the equations of motion
themselves transform as coadjoint vectors. Motivated by this observation, we
show that an equivalent action for the black hole interior can be obtained in
the form of a geometric action for BMS3, i.e. as a pairing between coadjoint
vectors (the equations of motion) and elements of the Lie algebra
$\mathfrak{bms}_{3}$ (the configuration variables). With this interpretation
the theory picks up the gravitational central charge $c_{2}$ of BMS3. The new
action resembles closely the charges of the 2-dimensional BMS–Liouville field
theory introduced in [16, 15], although as we have mentioned the 1-dimensional
black hole interior action is not BMS3 invariant and is not obtained as a
boundary theory. Instead, there is a symmetry breaking and only the Poincaré
subgroup survives as the symmetry group of the theory.

It is known however that there are setups in which the boundary, instead of
enlarging the symmetries (to e.g. BMS), partially breaks the symmetry to a
subgroup. This is for example the case of the Schwarzian theory living at the
AdS boundary of JT gravity [54, 55, 56, 57, 58], where the original Virasoro
group is broken to its global SL$(2,{\mathbb{R}})$ part. All these
observations suggest that we are actually dealing with a symmetry breaking
inherited from a higher-dimensional theory, potentially related to the
boundary geometric actions appearing in 3-dimensional gravity, but the clear
relationship is still missing. What is also surprising is that although this
residual Poincaré invariance (and the related BMS interpretation) has been
initially found in the 4-dimensional black hole interior, it turns out that
there exist a broad class of other minisuperspace models in which the same
1-dimensional action appears. This is for example the null slicing of the
Schwarzschild solution, the BTZ black hole interior, and a class of
2-dimensional dilatonic gravity theories. This strongly suggests that there is
a universal yet still elusive mechanism at play and involving the
aforementioned symmetry groups.

The paper is organised as follows. In section 2 we review the classical setup
for the study of the homogeneous black hole interior spacetime. We explain in
particular the choice of lapse, the gauge fixing to proper time, and show how
the solutions to the equations of motion reconstruct the black hole interior
geometry. Then we move on in the next sections to the study of the symmetries
of the model. In 3 we define the adjoint representation of BMS, and use this
to act with finite and infinitesimal transformations on the action and to
derive the integrable Hamiltonian generators. This shows that the Poincaré
subgroup indeed defines symmetries, and also shows how a fully BMS3 invariant
theory can be obtained via a Stueckelberg mechanism. This fully invariant
theory is further studied in appendix D. In section 4 we then turn to the
coadjoint representation of BMS3, and explain how the equations of motion and
the broken Poincaré symmetry can be understood in terms of coadjoint orbits.
The main result is the new form (4.6) of the action, which is a pairing
between coadjoint vectors and the configuration space variables which
transform as elements of the Lie algebra $\mathfrak{bms}_{3}$. Appendix A
contains various definitions about the centrally-extended Virasoro and BMS3
groups. In appendix B we present other systems in 4, 3, and 2-dimensional
gravity which are governed by the same action as the 4-dimensional black hole
interior, and therefore admit the same symmetry structure.

##  Appendix 2 The black hole interior spacetime

In this first section we present the classical model underlying our
construction, namely the homogeneous black hole interior spacetime. We start
by explaining the choice of lapse and clock, and analyse the resulting
Hamiltonian. We then present the symmetry reduced action in proper time and
recall how the resulting equations of motion reconstruct the black hole
interior spacetime.

###  Homogeneous metric, choice of time and Hamiltonian

The region inside the horizon of a Schwarzschild black hole of mass $M$ is
described by the Kantowski–Sachs metric, with the homogeneous though
anisotropic line element

$\displaystyle\mathrm{d}s^{2}=-\left(\frac{2M}{T}-1\right)^{-1}\mathrm{d}T^{2}+\left(\frac{2M}{T}-1\right)\mathrm{d}r^{2}+T^{2}\mathrm{d}\Omega^{2}\,,$
(2.1)

where $\mathrm{d}\Omega^{2}$ is the metric on the unit 2-sphere at constant
$r$ and $T$. In these coordinates, the singularity is located at $T=0$ and the
horizon is at $T=2M$. In order to study the dynamics of this system, its
symmetries, perturbations around this classical solution and their possible
quantization, we introduce dynamical fields and consider the more general
homogeneous ansatz for the line element:

$\displaystyle\mathrm{d}s^{2}=-N(t)^{2}\mathrm{d}t^{2}+\frac{8V_{2}(t)}{V_{1}(t)}\mathrm{d}x^{2}+V_{1}(t)\mathrm{d}\Omega^{2}\,,$
(2.2)

where the spatial coordinate runs over the whole real line,
$x\in{\mathbb{R}}$. The spatial slices have the topology ${\mathbb{R}}\times
S^{2}$. Because of the spatial slices are non-compact, the evaluation of the
Einstein–Hilbert action on the line element (2.2) diverges, due to the
constant integration over $x$. This can be regularized with the introduction
of a fiducial length $L_{0}$ playing the role of an IR cutoff. Restricting the
integration range to $x\in[0,L_{0}]$, the vacuum Einstein–Hilbert action
evaluated on (2.2) becomes

$\displaystyle{\mathcal{S}}_{\text{EH}}^{(t)}[N,V_{i}]$
$\displaystyle=\frac{1}{16\pi
G}\int_{{\mathcal{M}}}\mathrm{d}^{4}x\,\sqrt{-g}\,R$ (2.3)
$\displaystyle=\frac{1}{G}\int_{{\mathbb{R}}}\mathrm{d}t\,L_{0}\left[\frac{V_{2}(4N^{2}V_{1}+V_{1}^{\prime
2})-2V_{1}V_{1}^{\prime}V_{2}^{\prime}}{2\sqrt{2}N(V_{1}^{3}V_{2})^{1/2}}+\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\sqrt{2}N}\big{(}\sqrt{V_{1}V_{2}}\big{)}^{\prime}\right)\right]\,,$
(2.4)

where the prime denotes the derivative with respect to coordinate time $t$.
The total derivative is the Gibbons–Hawking–York term associated with slices
at constant $t$. We will therefore drop this term since it does not play a
role in the study of the equations of motion and does not affect the
discussion of the symmetries.

In a previous work [1] we have shown that the analysis of the classical
symmetries of this 1-dimensional mechanical action is more transparent when
redefining the lapse as

$\displaystyle
N=L_{0}N_{\text{p}}\sqrt{\frac{V_{1}}{2V_{2}}}\,,\qquad\qquad\mathrm{d}s^{2}=-L_{0}^{2}N_{\text{p}}(t)^{2}\frac{V_{1}(t)}{2V_{2}(t)}\mathrm{d}t^{2}+\frac{8V_{2}(t)}{V_{1}(t)}\mathrm{d}x^{2}+V_{1}(t)\mathrm{d}\Omega^{2}\,.$
(2.5)

We emphasize that this redefinition actually corresponds to a field-dependent
and $L_{0}$-dependent time diffeomorphism111This is possible because we do not
impose any boundary condition on the lapse, but it nevertheless begs the
question of the equivalence between the two gauges, especially at quantum
level.. The action now reads

$\displaystyle{\mathcal{S}}_{\text{EH}}^{(t)}[N_{\text{p}},V_{i}]$
$\displaystyle=\frac{1}{G}\int\mathrm{d}t\,\left[N_{\text{p}}L_{0}^{2}+\frac{V^{\prime}_{1}(V_{2}V^{\prime}_{1}-2V_{1}V^{\prime}_{2})}{2N_{\text{p}}V_{1}^{2}}\right]\,.$
(2.6)

Then, we introduce the proper time gauge by working with
$\mathrm{d}\tau=N_{\text{p}}\mathrm{d}t$. With this redefinition of the lapse
and the choice of proper time, the lapse disappears from the action, which
becomes

$\displaystyle{\mathcal{S}}_{\text{EH}}^{(\tau)}[N_{\text{p}},V_{i}]=\frac{1}{G}\int\mathrm{d}\tau\,\left[L_{0}^{2}+\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}\right]\,,$
(2.7)

where the dot denotes the derivative with respect to proper time $\tau$. The
contribution of $L_{0}$ is now a boundary term. As such, it cannot contribute
to the equations of motion and should simply be discarded from the action. We
therefore redefine the action in proper time as

$\displaystyle{\mathcal{S}}_{0}[V_{i}]\equiv\frac{1}{G}\int\mathrm{d}\tau\,{\mathcal{L}}_{0}=\frac{1}{G}\int\mathrm{d}\tau\,\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}\,.$
(2.8)

On the one hand, one might think that this is a great feature of the choice of
time $\tau$. Indeed, this gauge fixing of the time reparametrization symmetry
to proper time not only removes the lapse but also gets rid of the IR cut-off
$L_{0}$. On the other hand, this IR scale is simply a pre-factor in front of
the whole action ${\mathcal{S}}_{\text{EH}}^{(t)}[N,V_{i}]$, so how can it
disappear?

This intriguing property is deeply intertwined with the role of the lapse.
Indeed, even if the lapse has formally disappeared from the action
${\mathcal{S}}_{\text{EH}}^{(\tau)}$, it still enters implicitly in the
definition of $\tau$. In fact, the equation of motion resulting from the
stationarity of the Lagrangian in ${\mathcal{S}}_{\text{EH}}^{(t)}$ with
respect to variations of the lapse translates into the fact that the
Hamiltonian of ${\mathcal{S}}_{\text{EH}}^{(\tau)}$ dictating the evolution in
proper time $\tau$ is a constant of motion and is equal to $L_{0}^{2}$. Thus
the Hamiltonian of the action in proper time does not vanish and the term
$L_{0}^{2}$ is now interpreted as a non-zero level for this Hamiltonian which
generates the evolution with respect to $\tau$ [1].

Let us check this in more details. Differentiating the action
${\mathcal{S}}_{\text{EH}}^{(t)}[N_{\text{p}},V_{i}]$ with respect to the
redefined lapse $N_{\text{p}}$ gives the following equation of motion:

$\displaystyle
L_{0}^{2}=\frac{1}{N_{\text{p}}^{2}}\left[\frac{V_{2}{V^{\prime}_{1}}^{2}}{2V_{1}^{2}}-\frac{V^{\prime}_{1}V^{\prime}_{2}}{V_{1}}\right]=\frac{V_{2}\dot{V}_{1}^{2}}{2V_{1}^{2}}-\frac{\dot{V}_{1}\dot{V}_{2}}{V_{1}}\,.$
(2.9)

Now, performing the canonical analysis of the Lagrangian ${\mathcal{L}}_{0}$
gives the following conjugate momenta:222Note that for convenience we perform
the Legendre transform of ${\mathcal{L}}_{0}$ alone and leave out the
prefactor $1/G$.

$\displaystyle
P_{1}=\frac{\partial{\mathcal{L}}_{0}}{\partial\dot{V}_{1}}=\frac{V_{2}\dot{V}_{1}-V_{1}\dot{V}_{2}}{V_{1}^{2}}\,,\qquad\qquad
P_{2}=\frac{\partial{\mathcal{L}}_{0}}{\partial\dot{V}_{2}}=-\frac{\dot{V}_{1}}{V_{1}}\,,$
(2.10)

and the following Hamiltonian:

$\displaystyle
H=P_{1}\dot{V}_{1}+P_{2}\dot{V}_{2}-{\mathcal{L}}_{0}=\frac{V_{2}\dot{V}_{1}^{2}}{2V_{1}^{2}}-\frac{\dot{V}_{1}\dot{V}_{2}}{V_{1}}=-P_{2}\left(V_{1}P_{1}+\frac{1}{2}V_{2}P_{2}\right)\,.$
(2.11)

It is then obvious that the equation of motion for the lapse
$N_{\text{p}}\propto N$ is exactly equivalent to fixing the value of the
energy for the evolution in proper time $\tau$, i.e.

$\displaystyle\frac{\delta{\mathcal{S}}_{\text{EH}}^{(t)}}{\delta
N_{\text{p}}}=0\quad\Leftrightarrow\quad H=L_{0}^{2}\,.$ (2.12)

This can at first seem like a surprising role for the IR cut-off as the
Hamiltonian level for the evolution in proper time, but we have shown that
this role is actually very natural.

Thus, at the end of the day, we can work in proper time and forget about the
lapse $N$ and the IR cut-off $L_{0}$, as long as we remember to fix the value
of the Hamiltonian to $L_{0}^{2}$ when inserting the on-shell fields in the
line element.

###  Reduced action and classical solutions

The starting point of the present work is the 1-dimensional action (2.8)
written in proper time $\tau$. The variation of this action can be written in
the compact form

$\displaystyle\delta{\mathcal{S}}_{0}=\frac{1}{G}\int\mathrm{d}\tau\,\big{[}J\delta
V_{1}+P\delta V_{2}+\mathrm{d}_{\tau}\theta\big{]}\,,$ (2.13)

with

$\displaystyle J$
$\displaystyle\coloneqq\frac{V_{2}\dot{V}_{1}^{2}}{V_{1}^{3}}-\frac{\dot{V}_{1}\dot{V}_{2}}{V_{1}^{2}}-\frac{V_{2}\ddot{V}_{1}}{V_{1}^{2}}+\frac{\ddot{V}_{2}}{V_{1}},$
(2.14a) $\displaystyle P$
$\displaystyle\coloneqq\frac{\ddot{V}_{1}}{V_{1}}-\frac{\dot{V}_{1}^{2}}{2V_{1}^{2}},$
(2.14b) $\displaystyle\theta$
$\displaystyle\coloneqq\frac{\dot{V}_{1}V_{2}-\dot{V}_{2}V_{1}}{V_{1}^{2}}\delta
V_{1}-\frac{\dot{V}_{1}}{V_{1}}\delta V_{2}.$ (2.14c)

The equation of motion are thus given by $J=0$ and $P=0$, and $\theta$ is the
pre-symplectic potential. As we will see in section 4, the names $J$ and $P$
have suggestively been chosen because of the relationship between the
equations of motion and the coadjoint representation of BMS3. One can notice
that by removing the boundary term in (2.3) (which is equivalent to adding the
GHY term), we have eliminated the second order time derivatives from the
action and obtained a well-defined variational principle for fixed fields
$V_{i}$ at initial and final time (i.e. a fixed metric).

We can now look at the solutions of the equations of motion $J=0=P$. They are
given by

$\displaystyle V_{1}$ $\displaystyle=\frac{A}{2}(\tau-\tau_{0})^{2}\,,$
(2.15a) $\displaystyle V_{2}$
$\displaystyle=B(\tau-\tau_{0})-\frac{L_{0}^{2}}{2}(\tau-\tau_{0})^{2}\,,$
(2.15b)

where $A$, $B$ and $\tau_{0}$ are constants of integration. In these solutions
the constant $\tau_{0}$ reflects the freedom in choosing the origin for the
time variable, and can be set to zero without loss of generality. The
constants $A$ and $B$ are integrals of motion, which can be identified as:

$\displaystyle A=\frac{\dot{V}_{1}^{2}}{2V_{1}}\,,\qquad
B=\frac{V_{2}\dot{V}_{1}-V_{1}\dot{V}_{2}}{V_{1}}\,.$ (2.16)

Finally, $L_{0}$ appears because we have used the constraint initially
enforced by the lapse $N_{\text{p}}$ and fixing the energy level of the
Hamiltonian, i.e.

$\displaystyle\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}=L_{0}^{2}\,.$
(2.17)

Inserting these solution into the line element (2.5), and changing the
variables as

$\displaystyle\tau-\tau_{0}=\sqrt{\frac{2}{A}}T\,,\qquad\qquad
x=\frac{1}{2L_{0}}\sqrt{\frac{A}{2}}r\,,$ (2.18)

we recover the Schwarzschild interior solution (2.1) with mass

$\displaystyle M=\frac{B\sqrt{A}}{\sqrt{2}L_{0}^{2}}\,.$ (2.19)

This provides the physical interpretation for the constants of motion $A$ and
$B$.

Before moving on to the study of the BMS structure, we would like to point out
that the symmetry-reduced action ${\mathcal{S}}_{0}$ is not specific to the
case of the 4-dimensional black hole interior, but emerges also in various
other mini-superspace gravitational systems. Interestingly, some of these
systems are not 4-dimensional. For instance, we find ${\mathcal{S}}_{0}$ for a
null slicing of the 4d black hole spacetime, for an homogeneous slice of the
3d BTZ black hole interior, and for a large class of 2d dilatonic gravity
models. These examples are briefly presented in appendix B. This strongly
hints at the presence of some universal and dimension-independent structure,
although its precise characterization remains elusive as for now.

Moreover, despite the fact that ${\mathcal{S}}_{0}$ has been found in
relationship with the black hole interior, and should be in principle limited
to the range in which $\tau$ is timelike, the fields $V_{i}$ are regular on
the whole real line. The classical evolution smoothly cross not only the
horizon, but also the singularity.

##  Appendix 3 Adjoint representation of BMS3

We now set out to explain the relationship between the action (2.8) and BMS3.
In our previous work [1] we have shown that the dynamical system describing
the black hole interior via (2.8) admits a set of symmetries whose finite form
is isomorphic to the 3-dimensional Poincaré group ISO$(2,1)$. Moreover, we
have shown that these symmetries can be realized by the embedding of the $2+1$
Poincaré group into the larger333Later on we will use the centrally-extended
group BMS3. The central extensions are defined in appendix A, where we use the
notation $\widehat{G}$ for the centrally-extended group $G$. In the core of
the text however we drop this hat in order to deal with lighter notations.
$\text{BMS}_{3}=\text{Diff}(S^{1})\ltimes\text{Vect}(S^{1})_{\text{ab}}$
group, where $\text{Diff}(S^{1})$ is the group of diffeomorphism of the
(compactified) time axis, while $\text{Vect}(S^{1})_{\text{ab}}$ is its Lie
algebra seen as an Abelian vector group. The natural question is now to
understand the reformulation of the Kantowski–Sachs mini-superspace model in
terms of the (centrally-extended) BMS group, and how the BMS symmetry is
broken down to Poincaré. For this we will first discuss in details the
symmetries of the action (2.8) using the adjoint representation, and then
explain in the next section the relationship with coadjoint representations.
We refer the reader to appendix A for more details on the properties of the
(centrally-extended) BMS3 group and its representations.

###  Symmetries and transformations of the action

The previous work [1] started from the Poincaré symmetry of the
Kantowski–Sachs reduced action and, trying to extend it to a BMS symmetry,
realized that the metric components $V_{1}$ and $V_{2}$ transform as elements
of the Lie algebra $\mathfrak{bms}_{3}$ in the adjoint representation of the
BMS group. Let us recall how this comes about.

We consider transformations $D_{f}\in\text{Diff}(S^{1})$ and
$T_{g}\in\text{Vect}(S^{1})_{\text{ab}}$, and define their action on the
configuration variables $V_{i}=(V_{1},V_{2})$ as

$\displaystyle D_{f}:\,$
$\displaystyle\left|\begin{array}[]{lcl}V_{i}&\mapsto&\widetilde{V}_{i}=(\dot{f}\,V_{i})\circ
f^{-1}\,,\end{array}\right.$ (3.2) $\displaystyle T_{g}:\,$
$\displaystyle\left|\begin{array}[]{lcl}V_{1}&\mapsto&\widetilde{V}_{1}=V_{1}\,,\\\
V_{2}&\mapsto&\widetilde{V}_{2}=V_{2}+g\dot{V}_{1}-\dot{g}V_{1}\,,\end{array}\right.$
(3.5)

where we recall that all these quantities depend on $\tau$. These are the
transformation laws derived in [1]. They are given here in a form acting only
on the dynamical fields $V_{i}$. To compute the transformation of the action
it will be convenient to act instead with $D_{f}$ equivalently rewritten as

$\displaystyle
D_{f}:\,\,\left|\begin{array}[]{lcl}\tau&\mapsto&\tilde{\tau}=f(\tau)\,,\vspace*{1mm}\\\
V_{i}&\mapsto&\widetilde{V}_{i}(\tilde{\tau})=\dot{f}(\tau)\,V_{i}(\tau)\,,\end{array}\right.\qquad\textrm{so}\qquad\left|\begin{array}[]{lcl}\mathrm{d}\tau&\mapsto&\mathrm{d}\tilde{\tau}=\dot{f}\mathrm{d}\tau\,,\vspace*{1mm}\\\
{\mathrm{d}_{\tau}V_{i}}&\mapsto&\mathrm{d}_{\tilde{\tau}}\widetilde{V}_{i}={\mathrm{d}_{\tau}V_{i}}+V_{i}\,\mathrm{d}_{\tau}\ln\dot{f}\,.\end{array}\right.$
(3.10)

Following the terminology introduced for asymptotic symmetries, we will
(abusively) refer to the transformations $D_{f}$ as superrotations and to
$T_{g}$ as supertranslations. In order to justify this name, we now show that
the composition

$\displaystyle\text{Ad}_{f,g}\coloneqq T_{g}\circ D_{f}$ (3.11)

does indeed define a representation of BMS3. For this, we use the properties

$\displaystyle D_{f^{-1}}\circ T_{g}\circ D_{f}=T_{(g\circ
f)/\dot{f}},\qquad\qquad D_{f_{1}}\circ D_{f_{2}}=D_{f_{1}\circ
f_{2}}\,,\qquad\qquad T_{g_{1}}\circ T_{g_{2}}=T_{g_{1}+g_{2}}\,,$ (3.12)

which allow us to show that the composition law for (3.11) is given by

$\displaystyle\text{Ad}_{f_{1},g_{1}}\circ\text{Ad}_{f_{2},g_{2}}$
$\displaystyle=T_{g_{1}}\circ D_{f_{1}}\circ T_{g_{2}}\circ D_{f_{2}}$ (3.13)
$\displaystyle=T_{g_{1}}\circ(D_{f_{1}}\circ T_{g_{2}}\circ
D_{f_{1}}^{-1})\circ(D_{f_{1}}\circ D_{f_{2}})$ (3.14)
$\displaystyle=T_{g_{1}+\dot{f}_{1}\,g_{2}\circ f_{1}^{-1}}\circ D_{f_{1}\circ
f_{2}}.$ (3.15)

The inverse composition is

$\displaystyle(T_{g}\circ D_{f})^{{-1}}=D_{f^{-1}}\circ T_{-g}=T_{-(g\circ
f)/\dot{f}}\circ D_{f^{-1}}\,.$ (3.16)

As desired, this is the group multiplication and its inverse for the Lie group
defined as the semi-direct product
$\text{Diff}(S^{1})\ltimes\text{Vect}(S^{1})_{\text{ab}}$.

The interpretation is then that the gravitational variables $V_{1}$ and
$V_{2}$ belong to the Lie algebra

$\displaystyle\mathfrak{bms}_{3}=\text{Vect}(S^{1})\oplus_{\text{ad}}\text{Vect}(S^{1})_{\text{ab}}\ni(V_{1},V_{2})\,,$
(3.17)

and are acted on by the group via the adjoint action (A.20).

####  3.1.1 Finite transformations

We now study how the action (2.8) transforms under finite BMS transformations.
It turns out to be easier to compute this using the inverse transformations to
(3.2). Acting jointly with superrotations and supertranslations we find

$\displaystyle\text{Ad}_{(f^{-1},g)}{\mathcal{S}}_{0}[V_{i},\tau]$
$\displaystyle={\mathcal{S}}_{0}[V_{i},\tilde{\tau}]-\frac{1}{G}\int\mathrm{d}\tilde{\tau}\left[\vphantom{\frac{1}{G}}(V_{2}-\dot{g}V_{1}+g\dot{V}_{1})\,\text{Sch}[f^{-1}]-V_{1}g^{(3)}\right.$
(3.18)
$\displaystyle\phantom{={\mathcal{S}}_{0}[V_{i},\tilde{\tau}]+\frac{1}{G}\int\mathrm{d}\tilde{\tau}\left[\right.~{}}+\left.\frac{\mathrm{d}}{\mathrm{d}\tilde{\tau}}\left(\frac{\ddot{f}}{\dot{f}}(V_{2}-\dot{g}V_{1}+g\dot{V}_{1})+\ddot{g}V_{1}-\frac{g\dot{V}_{1}^{2}}{2V_{1}}\right)\right]\,,$

where $\text{Sch}[\,\cdot\,]$ denotes the Schwarzian, whose properties are
recalled in appendix A. As expected, this expression has cross terms involving
both $f$ and $g$ since we have acted jointly with $\text{Ad}_{f,g}$. Focusing
on the sector $f$, it is immediate to see on this expression that the subgroup
of transformations with vanishing Schwarzian derivative are symmetries of the
reduced gravitational action. These are given by Möbius transformations,
isomorphic to the group SL$(2,{\mathbb{R}})$. Similarly, in the $g$ sector we
obtain a symmetry if $g$ has a vanishing third derivative, meaning that it is
a second degree polynomial. This shows that the subgroup of (3.2) which
describes the symmetries of (2.8) is given by
$\text{SL}(2,{\mathbb{R}})\ltimes{\mathbb{R}}^{3}$, which is the $2+1$
Poincaré group [1].

When the transformation $\text{Ad}_{f,g}$ does not belong to the Poincaré
subgroup, we see that the general BMS transformation produces terms
proportional to the dynamical variables $V_{1}$ and $V_{2}$ in the action. As
explained in appendix C, terms in $V_{1}$ and $V_{2}$ also appear when
introducing a cosmological constant or a scalar field in the model. While it
is tempting to interpret the BMS transformation as producing a cosmological
constant term or a scalar field, this interpretation does not strictly hold
since, as can be seen on (C.2) a scalar field comes with specific time
derivatives which cannot be obtained from $f$ and $g$ in (3.18). However, in
the case of pure supertranslations $g$ (i.e. with $f=\mathbb{I})$ one can
really interpret the term $g^{(3)}V_{1}$ as the contribution of a cosmological
constant. The action of a further finite supertranslation then shifts the
value of this cosmological constant.

As an alternative to this interpretation in terms of a cosmological constant
and a scalar field, we can also create a fully BMS invariant action by
including extra fields in the theory. These have to transform in a way which
compensates the Schwarzian and the $g^{(3)}$ produces by a finite BMS
transformation. This is achieved by the invariant action

$\displaystyle{\mathcal{S}}_{\text{inv}}[V_{i},\Phi,\Psi]\coloneqq{\mathcal{S}}_{0}[V_{i}]+\frac{1}{G}\int\mathrm{d}\tau\left[(V_{2}+\dot{\Psi}V_{1}-\Psi\dot{V}_{1})\text{Sch}[\Phi]+V_{1}\Psi^{(3)}\right],$
(3.19)

provided the new fields $\Phi$ and $\Psi$ (which can be seen as Stückelberg
fields for the BMS symmetry) transform as

$\displaystyle\text{Ad}_{f,g}:\,$
$\displaystyle\left|\begin{array}[]{lcl}\Phi&\mapsto&\widetilde{\Phi}=\Phi\circ
f^{-1}\,,\\\ \Psi&\mapsto&\widetilde{\Psi}=g+(\dot{f}\Psi)\circ f^{-1}\,.\\\
\end{array}\right.$ (3.22)

This BMS invariant action can be seen as the extension from the Virasoro group
to the BMS group of the conformally-invariant action for FLRW cosmology that
was introduced in [53]. The cosmological model can be understood as the
$V_{1}=0$ regime of the theory with $V_{2}$ playing the role of the
cosmological volume. In that case, there are no supertranslations but only
superrotations and the conformally-invariant cosmological action was defined
by introducing solely an extra $V_{2}\,\text{Sch}[\Phi]$ term.

In terms of group representation, this means that the fields are embedded in
the BMS3 group itself:

$\displaystyle(\Phi,\Psi)\in\text{BMS}_{3}=\text{Diff}(S^{1})\ltimes_{\text{Ad}}\text{Vect}(S^{1})_{\text{ab}}.$
(3.23)

This will inevitably modify the equations of motion for the initial fields
$V_{i}$, and we postpone the analysis of this fully invariant theory to
appendix D.

Note that here we have simply introduced the BMS invariant action as an
example of a 1-dimensional system with full BMS symmetry. This is to be
contrasted with the 2-dimensional BMS–Liouville theory (3.41) which will also
have a close relationship with the action ${\mathcal{S}}_{0}$ we started from.

####  3.1.2 Infinitesimal transformations

We now move on to the study of the infinitesimal transformations. Since the
supertranslations are Abelian, their infinitesimal generator, which we will
denote $\alpha$, acts in the same way as the finite generator $g$. For the
superrotations, we parametrize the transformation infinitesimally as
$f(\tau)=\tau+\epsilon X(\tau)$. Then the infinitesimal version of (3.2)
reproduces the $\mathfrak{bms}_{3}$ algebra (A.19). We then obtain

$\displaystyle\delta_{X}V_{i}\coloneqq\frac{\text{Ad}_{f,0}V_{i}-V_{i}}{\epsilon}=\dot{X}V_{i}-X\dot{V}_{i}=[V_{i},X],$
(3.24)

and

$\displaystyle\delta_{\alpha}V_{1}$ $\displaystyle=0,$ (3.25a)
$\displaystyle\delta_{\alpha}V_{2}$
$\displaystyle=\alpha\dot{V}_{1}-\dot{\alpha}V_{1}=-[V_{1},\alpha].$ (3.25b)

With this, the infinitesimal variation of the action is

$\displaystyle\delta_{X,\alpha}{\mathcal{S}}_{0}=\frac{1}{G}\int\mathrm{d}\tau\left[V_{2}X^{(3)}-V_{1}\alpha^{(3)}\right]+\mathrm{d}\tau\frac{\mathrm{d}}{\mathrm{d}\tau}\left[\frac{\dot{V}_{1}}{2V_{1}^{2}}\Big{(}2XV_{1}\dot{V}_{2}-\dot{V}_{1}(\alpha
V_{1}+XV_{2})\Big{)}+\ddot{\alpha}V_{1}-\ddot{X}V_{2}\right].$ (3.26)

Once again, we see that the Poincaré subgroup, which at the infinitesimal
level has generators such that $X^{(3)}=0=\alpha^{(3)}$, generates symmetries
of the system since this variation then reduces to a total derivative.

###  Hamiltonian formulation and generators

Having studied the action of finite and infinitesimal BMS transformations on
our system ${\mathcal{S}}_{0}$, we can now study the generators in the
(covariant) Hamiltonian formulation. From the general variational expression
we can read the symplectic potential $\theta$. Its expression can be
simplified if we change variables and introduce

$\displaystyle\varphi=-\log V_{1}\,,\qquad\qquad\xi=\frac{V_{2}}{V_{1}}\,.$
(3.27)

In terms of these variables $J$ and $P$ in (2.14) become

$\displaystyle
P=\frac{1}{2}\left(\dot{\varphi}^{2}-2\ddot{\varphi}\right)\,,\qquad\qquad J$
$\displaystyle=\ddot{\xi}-\dot{\xi}\dot{\varphi}\,,$ (3.28)

and the equations of motion are $J=0=P$. One can note that in terms of $W$
defined by $\dot{W}=e^{\varphi}$ we have $P=-\text{Sch}[W]$. The
transformations (3.2) become

$\displaystyle\text{Ad}_{f,g}^{-1}:\,$
$\displaystyle\left|\begin{array}[]{lcl}\varphi&\mapsto&\widetilde{\varphi}=\varphi\circ
f+\log\dot{f}\,,\\\
\xi&\mapsto&\widetilde{\xi}=\left(\xi+g\dot{\varphi}+\dot{g}\right)\circ
f\,,\\\ \end{array}\right.$ (3.31)

and we see in particular that under $D_{f}$ the variables $\varphi$ and $\xi$
transform respectively as a connection and a scalar. The infinitesimal
transformations (3.24) and (3.25) take the form

$\displaystyle\delta_{X,\alpha}\,\varphi$
$\displaystyle=-X\dot{\varphi}-\dot{X}\,,$ (3.32a)
$\displaystyle\delta_{X,\alpha}\,\xi$
$\displaystyle=-X\dot{\xi}-\alpha\dot{\varphi}-\dot{\alpha}\,.$ (3.32b)

Rewriting (2.14c) in terms of $\varphi$ and $\xi$, we find the symplectic
structure

$\displaystyle\omega=\delta\theta=-\delta\dot{\xi}\,\delta(e^{-\varphi})+\delta\dot{\varphi}\,\delta(\xi
e^{-\varphi})\,.$ (3.33)

From this we can rewrite the initial momenta as

$\displaystyle
P_{1}=-\dot{\xi}=\frac{\dot{V}_{1}V_{2}-\dot{V}_{2}V_{1}}{V_{1}^{2}}\,\qquad\qquad
P_{2}=\dot{\varphi}=-\frac{\dot{V}_{1}}{V_{1}}\,,\qquad\qquad\\{V_{i},P_{j}\\}=\delta_{ij}\,.$
(3.34)

With the symplectic structure at our disposal, we can ask whether there are
integrable generators associated with the infinitesimal transformations
(3.32). This is found by contracting $\omega$ with the variations
$\delta_{X,\alpha}$, and then using the equations of motion to get rid of the
second order derivatives. Quite surprisingly, even if the general BMS
transformations do not describe symmetries of the action, they have an
integrable generator. Separating the superrotations and supertranslations, we
find

$\displaystyle\delta_{X}\lrcorner\,\\!\\!\\!\lrcorner\,\omega$
$\displaystyle=-\delta_{X}\dot{\xi}\,\delta(e^{-\varphi})+\delta_{X}\dot{\varphi}\,\delta(\xi
e^{-\varphi})+\delta\dot{\xi}\,\delta_{X}(e^{-\varphi})-\delta\dot{\varphi}\,\delta_{X}(\xi
e^{-\varphi})$ (3.35a)
$\displaystyle=-\delta\left(e^{-\varphi}\left(\frac{\xi}{2}\dot{\varphi}^{2}-\dot{\varphi}\dot{\xi}\right)X+e^{-\varphi}(\xi\dot{\varphi}-\dot{\xi})\dot{X}+e^{-\varphi}\xi\ddot{X}\right)$
(3.35b)
$\displaystyle=-\delta\left(P_{2}\left(P_{1}V_{1}+\frac{1}{2}P_{2}V_{2}\right)X+(V_{2}P_{2}+V_{1}P_{1})\dot{X}+V_{2}\ddot{X}\right)$
(3.35c) $\displaystyle\coloneqq-\delta{\mathcal{D}}_{X}\,,$ (3.35d)
$\displaystyle\delta_{\alpha}\lrcorner\,\\!\\!\\!\lrcorner\,\omega$
$\displaystyle=-\delta_{\alpha}\dot{\xi}\,\delta(e^{-\varphi})+\delta_{\alpha}\dot{\varphi}\,\delta(\xi
e^{-\varphi})+\delta\dot{\xi}\,\delta_{\alpha}(e^{-\varphi})-\delta\dot{\varphi}\,\delta_{\alpha}(\xi
e^{-\varphi})$ (3.35e)
$\displaystyle=\delta\left(\frac{1}{2}e^{-\varphi}\dot{\varphi}^{2}\alpha+e^{-\varphi}\dot{\varphi}\dot{\alpha}+e^{-\varphi}\ddot{\alpha}\right)$
(3.35f)
$\displaystyle=\delta\left(\frac{1}{2}{P_{2}}^{2}V_{1}\alpha+P_{2}V_{1}\dot{\alpha}+V_{1}\ddot{\alpha}\right)$
(3.35g) $\displaystyle\coloneqq-\delta{\mathcal{T}}_{\alpha}\,.$ (3.35h)

It is now interesting to compute the Poisson brackets between these
generators. This can be one either by contracting twice the symplectic
structure with a variation $\delta_{X,\alpha}$, or using the phase space
expression for the generators and the usual definition of Poisson brackets.
For example we have

$\displaystyle\\{{\mathcal{D}}_{X},{\mathcal{D}}_{Y}\\}=-\delta_{X}\lrcorner\,\\!\\!\\!\lrcorner\,\delta_{Y}\lrcorner\,\\!\\!\\!\lrcorner\,\omega=\frac{\mathrm{d}{\mathcal{D}}_{X}}{\mathrm{d}V_{i}}\frac{\mathrm{d}{\mathcal{D}}_{Y}}{\mathrm{d}P_{i}}-\frac{\mathrm{d}{\mathcal{D}}_{X}}{\mathrm{d}P_{i}}\frac{\mathrm{d}{\mathcal{D}}_{Y}}{\mathrm{d}V_{i}}\,.$
(3.36)

Explicitly, we find

$\displaystyle\\{{\mathcal{D}}_{X},{\mathcal{D}}_{Y}\\}$
$\displaystyle=-{\mathcal{D}}_{[X,Y]}+\big{(}XY^{(3)}-YX^{(3)}\big{)}V_{2}\,,$
(3.37a) $\displaystyle\\{{\mathcal{T}}_{\alpha},{\mathcal{T}}_{\beta}\\}$
$\displaystyle=0\,,$ (3.37b)
$\displaystyle\\{{\mathcal{D}}_{X},{\mathcal{T}}_{\alpha}\\}$
$\displaystyle=-{\mathcal{T}}_{[X,\alpha]}+\big{(}\alpha
X^{(3)}-X\alpha^{(3)}\big{)}V_{1}\,,$ (3.37c)

where $[X,Y]=X\dot{Y}-Y\dot{X}$ and similarly for $[X,\alpha]$.

These brackets are consistent with the equations of motion. For example, one
can study the superrotation associated with a constant shift in time. This
actually corresponds to the Hamiltonian of the system, i.e.
${\mathcal{D}}_{X(\tau)=1}=H$, as one can also check using a Legendre
transform of the Lagrangian. Note that this is slightly peculiar since in the
usual spacetime picture the Hamiltonian belongs to the Abelian sector of the
symmetry algebra (i.e. Poincaré time translations), whereas here it belongs to
the superrotations. For a general (time-dependent) phase space function
${\mathcal{O}}(\tau)$, the time evolution is given by

$\displaystyle\dot{\mathcal{O}}\eqqcolon\mathrm{d}_{\tau}{\mathcal{O}}=\partial_{\tau}{\mathcal{O}}+\\{{\mathcal{O}},H\\}\,.$
(3.38)

Using the commutation relations (3.37) applied to ${\mathcal{D}}_{1}$, and the
fact that e.g. $\partial_{\tau}{\mathcal{D}}_{X}={\mathcal{D}}_{\dot{X}}$, we
then find

$\displaystyle\mathrm{d}_{\tau}{\mathcal{D}}_{X}$
$\displaystyle=\partial_{\tau}{\mathcal{D}}_{X}-\\{H,{\mathcal{D}}_{X}\\}=V_{2}X^{(3)},$
(3.39a) $\displaystyle\mathrm{d}_{\tau}{\mathcal{T}}_{\alpha}$
$\displaystyle=\partial_{\tau}{\mathcal{T}}_{\alpha}-\\{H,{\mathcal{T}}_{\alpha}\\}=-V_{1}\alpha^{(3)},$
(3.39b)

which is consistent with the equations of motion.

In spite of this consistency, we see of course that the bracket (3.37) between
the generators ${\mathcal{D}}$ and ${\mathcal{T}}$ do not close because of a
remaining field-dependency on the right-hand side (although up to the
$V_{i}$’s this would have been the centrally-extended BMS3 algebra). This is
to be expected since ${\mathcal{D}}$ and ${\mathcal{T}}$, although they are
integrable, do not generate symmetries of the theory. Consistently, we recover
a closed algebra for the Poincaré generators since these have a vanishing
third derivative.

One way to obtain charges closing on the BMS3 algebra is of course to consider
a BMS invariant field theory. Although we have proposed a 1-dimensional such
theory in (3.19), it turns out that a simpler field theory which has an
interesting relationship with the black hole interior action (2.8) is the
2-dimensional BMS–Liouville theory of [59, 44, 16]. In terms of the scalar
$\xi$ and the connection $\varphi$ its action is444The fields now depends on
two variables, but the symmetries are still in the form (3.31), with
$\displaystyle
X(u,\tau)=Y(\tau)\,,\qquad\qquad\alpha(u,\tau)=T(\tau)+u\,\frac{\mathrm{d}Y(\tau)}{\mathrm{d}\tau}\,.$
(3.40) Usually the $\tau$ coordinate is compactified and represented by an
angle $\phi$, but here we keep the name $\tau$ for a better comparison with
our case.

$\displaystyle{\mathcal{S}}_{\mathrm{BMS}}=\iint\mathrm{d}u\,\mathrm{d}\tau\big{[}\partial_{\tau}\xi\,\partial_{u}\varphi-(\partial_{\tau}\varphi)^{2}\big{]}\,.$
(3.41)

Performing the Hamiltonian analysis of this action with respect to the new
variable $u$, we find that the conserved charges generating the BMS3
symmetries are given by

$\displaystyle{\mathcal{Q}}=\int\mathrm{d}\tau\big{[}JX+P\alpha\big{]},$
(3.42)

where $J$ and $P$ are precisely the quantities (3.28), whose vanishing is
equivalent to the equations of motion. This can also be seen by computing the
charges as the contraction of the transformations (3.31) with the symplectic
structure $\omega=\delta(2\varphi-\xi)\delta\dot{\varphi}$ coming from (3.41).
As usual, one interprets this expression for the charges as a pairing between
elements $(X,\alpha)$ in the adjoint representation and elements $(J,P)$ in
the coadjoint representation. This particular relationship between the charges
of the 2d BMS3 invariant theory and the equations of motion of the 1d model
(2.8) becomes clearer if we look at the possibility of embedding our system in
the coadjoint representation of BMS3. This is the subject of the next section.

##  Appendix 4 Coadjoint representation of BMS3 and central charges

The understanding of coadjoint orbits of the symmetry groups is crucial in
order to go towards the quantum theory. In particular the classification of
the orbits of semi-direct product groups by their little groups will naturally
leads to irreducible representation of the full group. In the case of the BMS3
group, a detailed discussion is presented in [40]. In appendix A we report the
key ingredients about the coadjoint orbits which we need in the present work.

The main goal of this section is to explain how the identification of the
gravitational fields with the $\mathfrak{bms}_{3}$ Lie algebra elements
naturally leads to the introduction of 2-forms on the circle that transform as
covectors under the group action. Moreover, we show here that the action
itself can be seen as a bilinear form between vectors and covectors. For this,
let us first consider the equations of motion for the configuration fields
$V_{i}$, written in the form

$\displaystyle P$
$\displaystyle=\frac{\ddot{V}_{1}}{V_{1}}-\frac{\dot{V}_{1}^{2}}{2V_{1}^{2}}=\frac{1}{2}\left(\dot{\varphi}^{2}-2\ddot{\varphi}\right)\,,$
(4.1a) $\displaystyle J$
$\displaystyle=-\frac{\dot{V}_{2}\dot{V}_{1}}{V_{1}^{2}}+\frac{\ddot{V_{2}}}{V_{1}}+\frac{V_{2}}{V_{1}}\left(\frac{\dot{V}_{1}^{2}}{V_{1}^{2}}-\frac{\ddot{V}_{1}}{V_{1}}\right)=\ddot{\xi}-\dot{\xi}\dot{\varphi}\,.$
(4.1b)

A straightforward calculation reveals that $J$ and $P$ transform exactly as in
the coadjoint representation of the centrally-extended group $\text{BMS}_{3}$,
i.e. as (A.23), with central charges given by555The central charge $c_{2}$
appears here as a dimensionless quantity, while in 3-dimensional gravity it is
usually normalized to $c_{2}=3/G$. Nevertheless, by a constant and
dimensionfull rescaling of $P$ in (4.1a) we can arbitrarily change the value
of $c_{2}$ (and its dimension). The normalization in 3-dimensional gravity
comes from the fact that the zero mode of the supermomentum represents the
Bondi mass aspect, while here the on-shell value of $P$ is zero, so its
dimension is not important. We keep the choice leading to ligher notations.

$\displaystyle c_{1}=0\qquad\qquad c_{2}=1.$ (4.2)

More precisely, defining the coadjoint action as

$\displaystyle\text{Ad}^{*}_{f^{-1},g}(J,P)\coloneqq\left(\text{D}_{f^{-1}}\circ\text{T}_{g}\right)(J,P)\eqqcolon(\tilde{J},\tilde{P})\,,$
(4.3)

we find the transformation laws

$\displaystyle\tilde{P}=\dot{f}^{2}(P\circ
f)-\,\text{Sch}[f]\,,\qquad\qquad\tilde{J}=\dot{f}^{2}\left(J+g\dot{P}+2\dot{g}P-g^{(3)}\right)\circ
f\,.$ (4.4)

As a remark, one can note that the other central charge can be switched on by
the shift $J\to J+cP$, which leads to $c_{1}=c$. This shift can be obtained by
adding a term $cPV_{1}$ to the action (2.8). Remarkably, this shift does not
affect the equations of motion and only modifies the symplectic potential.
This comes from the fact that

$\displaystyle\delta(PV_{1})=P\delta
V_{1}+\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\delta\dot{V}_{1}-\frac{\delta
V_{1}\dot{V}_{1}}{V_{1}}\right),$ (4.5)

so that the variation of the new Lagrangian is
$\delta({\mathcal{L}}_{0}+cPV_{1})=(J+cP)\delta V_{1}+P\delta
V_{2}+\mathrm{d}_{\tau}\bar{\theta}$, where $\bar{\theta}$ is the sum of the
boundary term (4.5) and the potential (2.14c) coming from ${\mathcal{L}}_{0}$.
The equations of motion therefore combine to give once again $J=0$ and $P=0$.
It is very intriguing that this mechanism is the same as in 3-dimensional
gravity, where the BMS3 central charge $c_{1}$ (or a chiral mismatch
$c_{+}\neq c_{-}$ between the two Brown–Henneaux central charges in the AdS3
case) can be switched on by adding a Chern–Simons and a torsion term (together
these constitute Witten’s exotic Lagrangian [60]) to the first order
Lagrangian, without however modifying the equations of motion [61, 62, 63].

By adding an appropriate boundary term to the initial action (2.8), it is also
possible to write a new action as the bilinear form between vectors and
covectors. This rewriting is done by defining

$\displaystyle{\mathcal{S}}[V_{i}]$
$\displaystyle\coloneqq{\mathcal{S}}_{0}[V_{i}]-\frac{1}{G}\int\mathrm{d}\tau\frac{\mathrm{d}}{\mathrm{d}\tau}\left(\frac{\dot{V}_{1}V_{2}}{V_{1}}\right)$
(4.6)
$\displaystyle=\frac{1}{G}\int\mathrm{d}\tau\left[\frac{V_{2}\dot{V}_{1}^{2}}{2V_{1}^{2}}-\frac{\dot{V}_{1}\dot{V}_{2}}{V_{1}}+\ddot{V}_{2}\right]$
(4.7)
$\displaystyle=\frac{1}{G}\int\mathrm{d}\tau\big{[}JV_{1}+PV_{2}\big{]}.$
(4.8)

By construction this action leads to the same equations of motion as
${\mathcal{S}}_{0}$. It is however remarkable that the explicit variation is
again of the form

$\displaystyle\delta{\mathcal{S}}=\frac{1}{G}\int\mathrm{d}\tau\,\big{[}J\delta
V_{1}+P\delta V_{2}+\mathrm{d}_{\tau}\tilde{\theta}\,\big{]}\,,$ (4.9)

even though $J$ and $P$ (expressed in terms of the $V_{i}$’s as in (4.1)) have
of course been varied in (4.6) to obtain this expression! Consistently, the
only difference with (2.13) is the form of the symplectic potential (which we
have not displayed here because it won’t play a role in what follows).

The result is the 1-dimensional action (4.6) for the black hole interior in
proper time gauge. This action, which is not BMS3 invariant, is however
equivalent to the charge expression (3.42) for the 2-dimensional BMS3
invariant theory (3.41), provided we identify the infinitesimal parameters
$(X,\alpha)$ in the charge with the algebra element $(V_{1},V_{2})$. We note
that this new action transforms under BMS3 as

$\displaystyle\text{Ad}^{*}_{(f^{-1},g)}{\mathcal{S}}[V_{i},\tau]={\mathcal{S}}[V_{i},\tilde{\tau}]-\frac{1}{G}\int\mathrm{d}\tilde{\tau}\left[(V_{2}-\dot{g}V_{1}+g\dot{V}_{1})\,\text{Sch}[f^{-1}]-V_{1}g^{(3)}-\frac{\mathrm{d}}{\mathrm{d}\tilde{\tau}}(gV_{1}P)\right],$
(4.10)

which as expected differs from (3.18) only by a boundary term.

We can now give a nice interpretation of the Poincaré symmetry subgroup as an
orbit stabilizer. So far we have embedded the fields $V_{1}$ and $V_{2}$
within the ${\mathfrak{bms}}_{3}$ Lie algebra. We can build corresponding
group elements by considering the functions

$\displaystyle
f(\tau)\coloneqq\int^{\tau}\frac{\mathrm{d}s}{V_{1}(s)}\,,\qquad\qquad
g(\tau)\coloneqq\int^{\tau}\mathrm{d}s\,\left(\frac{V_{2}}{V_{1}}\right)\circ
f^{(-1)}(s)\,.$ (4.11)

Transforming these quantities under BMS3 reveals that they are indeed group
elements. If we now insert these functions into the transformation law (4.4)
for the covectors, starting from the point $(P=0,J=0)$ we find that the
transformed pair $(\tilde{J},\tilde{P})$ is precisely equal to the right-hand
side of (4.1). In other words, for the two functions $(f,g)$ given above we
have

$\displaystyle-\text{Sch}[f]=\frac{\ddot{V}_{1}}{V_{1}}-\frac{\dot{V}_{1}^{2}}{2V_{1}^{2}}=P\,,\qquad\qquad-\dot{f}^{2}\,g^{(3)}\circ
f=-\frac{\dot{V}_{2}\dot{V}_{1}}{V_{1}^{2}}+\frac{\ddot{V_{2}}}{V_{1}}+\frac{V_{2}}{V_{1}}\left(\frac{\dot{V}_{1}^{2}}{V_{1}^{2}}-\frac{\ddot{V}_{1}}{V_{1}}\right)=J.$
(4.12)

This shows that demanding that the fields satisfy the equations of motion is
equivalent to the condition that ($P=0,J=0$) be preserved under (4.4), which
means in turn that $(f,g)$ belongs to the stabilizer of the orbit. In general
the coadjoint orbit of BMS3 is defined by the Virasoro orbit spanned by the
supermomentum $P$. These orbits are usually classified for the diffeomorphism
group of the circle, while here we are dealing with functions on the real line
since $\tau\in{\mathbb{R}}$. We can however compactify the proper time with
e.g. the reparametrization $\tau\to\arctan\theta/2$. In particular, this maps
the point $(P=0,J=0)$ to $(P=-1/2,J=0)$, i.e. the constant representative of
the so called vacuum orbit [40]. The stabilizer of this orbit is known to be
ISO(2,1), which of course is precisely the symmetry group of our theory.

##  Appendix 5 Discussion

In [1] we have studied the ISO(2,1) (i.e. $(2+1)$-dimensional Poincaré)
symmetry of the homogeneous action (2.8) describing the black hole interior
spacetime, that is

$\displaystyle{\mathcal{S}}_{0}=\frac{1}{G}\int\mathrm{d}\tau\,\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}\,,$
(2.8)

where $V_{1}$ and $V_{2}$ are components of the 4-metric. This previous work
has hinted at the fact that the infinite-dimensional extension BMS3 of the
Poincaré group could also play a role in understanding the origin and the role
of this symmetry. In the present paper, we have pushed along this direction
and studied the action (2.8) for the geometry inside the black hole from the
point of view of BMS3.

For this, we have shown in section 3 how finite and infinitesimal BMS3
transformations act on (2.8). These transformations do not leave the action
invariant, and only the subgroup of transformations corresponding to Poincaré
does. Nevertheless, in the case of the supertranslations, one can interpret
the non-invariance of the action as creating a term corresponding to a
cosmological constant. Acting with a further supertranslation then preserves
the form of the Lagrangian while however changing the value of the
cosmological “constant”. We have shown that even if the BMS transformations
are not strictly speaking symmetries of the action, they nevertheless admit
integrable generators on the phase space of the theory. Their charge algebra,
given by (3.37), does however fail to reproduce the centrally-extended
$\mathfrak{bms}_{3}$. This is to be expected since these are indeed not
symmetries of the theory. We have explained how a fully BMS-invariant action
(3.19) can be obtained by promoting the BMS3 group elements to dynamical
variables. This higher order action was then studied in appendix D. Aside from
this 1-dimensional invariant action, there are 2-dimensional geometrical
actions typically arising in studies of the boundary dynamics of 3-dimensional
gravity [16, 15, 59]. This is for example the BMS3 invariant action (3.41).
Intriguingly, the charges of this 2-dimensional action are written in terms of
coadjoint vectors which turn out to be precisely the equations of motion of
our starting point action (2.8). This has motivated the study of the coadjoint
representation of BMS3 in section 4. Note that for the action (3.19) the BMS
symmetries are gauge symmetries, while in the case of (3.41) they are physical
symmetries.

We have then shown that the action (2.8) for the black hole interior can be
rewritten (up to a boundary term innocent for the equations of motion) as the
compact geometric action

${\mathcal{S}}=\frac{1}{G}\int\mathrm{d}\tau\left[\frac{V_{2}\dot{V}_{1}^{2}}{2V_{1}^{2}}-\frac{\dot{V}_{1}\dot{V}_{2}}{V_{1}}+\ddot{V}_{2}\right]=\frac{1}{G}\int\mathrm{d}\tau\big{[}JV_{1}+PV_{2}\big{]}\,.$
(4.6)

This is made possible by the fact that $J$ and $P$ (which are defined in terms
of $V_{1},V_{2}$ and their time derivatives) are coadjoint vectors under BMS3
with central charge $c_{2}=1$, while the configuration variables $V_{1}$ and
$V_{2}$ are elements of the Lie algebra $\mathfrak{bms}_{3}$. Remarkably, the
variation of this action (4.6) leads to the field equations $J=P=0$, which are
exactly equivalent to the original equations of motion. This property of the
geometric action also enables to turn on the other central charge $c_{1}$ by
adding a term to the Lagrangian without however modifying the equations of
motion. In this construction, the Poincaré subgroup corresponds to the
stabilizer of the vacuum orbit of BMS3.

These results show that even if the action (2.8) is only invariant under
Poincaré symmetries, there is a meaningful way in which one can understand
this invariance as a broken BMS3 symmetry. This confirms the intuition that
this latter symmetry group does indeed plays a foundational role in the
physics of the black hole interior, although it does not appear here in the
more usual way as an asymptotic or horizon boundary symmetries. It is also
intriguing, as shown in appendix B, that other gravitational systems in 4, 3,
and 2 spacetime dimensions involve the same action (2.8) and therefore admit
the same symmetry structure.

In order to investigate further the origin and the physical role of these
symmetries (both the Poincaré and the extended BMS one), there are several
directions to be developed:

  * •

One should investigate to which physical systems correspond the other BMS
coadjoint orbits. A related question is how the phase space structure of these
coadjoint orbits can be used to define a quantization of the system. This can
already be investigated in terms of Poincaré representations [64], but the
embedding into BMS could allow to shift the Casimirs and to describe new
physical processes related to the dynamics of the black hole interior.

  * •

An intriguing question remains that of the relationship between the BMS group
appearing in the present context and that appearing for asymptotic or near-
horizon symmetries. It is particularly interesting how the Poincaré symmetry
of (2.8) can be seen as a broken BMS symmetry as in the case of
SL$(2,\mathbb{R})$ and the Schwarzian action in JT gravity.

  * •

A related question is that of understanding BMS-invariant actions in mechanics
and field theory. It could be that (2.8) descends from a higher-dimensional
BMS-invariant action with a gauge-fixing and a dimensional reduction.

  * •

We have seen that the BMS transformations do not leave the action invariant,
but that in a sense (which is clear for supertranslations) they generate new
terms which can be interpreted e.g. as a cosmological constant. Then the BMS
transformations generate a flow in a space of theories, and preserve the form
of the action while changing its couplings. It would be very interesting to
investigate the extension of Noether’s theorem(s) to this type of
transformations (which are very similar to renormalization flow
transformations).

  * •

Finally, another possible generalization of this work, motivated in parts by
the symmetry structure of 3-dimensional gravity, is to study an extension of
(2.8) which admits as a symmetry group
$\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})$ instead of Poincaré’s
$\text{SL}(2,\mathbb{R})\ltimes{\mathbb{R}}^{3}$, and therefore involves a
double Virasoro algebra instead of BMS3. In the case of 3-dimensional gravity,
this is done by turning on the cosmological constant, while in the case of the
black hole interior, this is evidently not so. There are however other simple
terms to be added to the action (2.8) in order to obtain this deformation to
double Virasoro. The gravitational interpretation of these terms in the
present context is however not yet clear, but is the subject of ongoing work.

##  Appendix A The Virasoro and BMS3 groups

In this appendix we recall basic facts about the Virasoro and the BMS3 groups,
as well as their centrally-extended version. We give in particular the form of
the coadjoint action used in the main text.

###  The Virasoro group

We recall here some basic properties of the Virasoro group as the central
extension of the group of diffeomorphisms of the circle $S^{1}$. We denote by
$\text{Vect}(S^{1})$ the space of vector fields on the circle endowed with the
bracket

$\displaystyle\big{[}f(\theta)\,\partial_{\theta},\,g(\theta)\,\partial_{\theta}\big{]}=(fg^{\prime}-gf^{\prime})\partial_{\theta}\qquad\qquad
f\,\partial_{\theta},\,g\,\partial_{\theta}\in\text{Vect}(S^{1})\,,$ (A.1)

with $f$ and $g$ periodic functions of $\theta$ with period $2\pi$. A standard
basis is given by the Fourier modes $\ell_{n}\coloneqq
ie^{in\theta}\partial_{\theta}$, with the commutation relations
$[\ell_{n},\ell_{m}]=-i(n-m)\ell_{n+m}$. This is the Witt algebra. Moreover,
the vector space $\text{Vect}(S^{1})$ can been seen as the space of generators
of the (orientation preserving) diffeomorphisms on the circle, denoted by
$\text{Diff}^{+}(S^{1})$. Conversely, the diffeomorphisms can be endowed with
a Lie group structure, whose algebra is $\text{Vect}(S^{1})$.

####  A.1.1 Tensor densities on $S^{1}$

We denote by ${\mathcal{F}}_{\lambda}$ the space of tensor densities of degree
$\lambda\in{\mathbb{R}}$. It consists of elements of the form
$\alpha=\alpha(\theta)\mathrm{d}\theta^{\lambda}$. Diffeomorphisms act by the
adjoint action as

$\displaystyle\alpha(\theta)\xrightarrow{\varphi\,\in\,\text{Diff}^{+}(S^{1})}\text{Ad}^{*}_{\varphi^{-1}}\alpha(\theta)=\left(\varphi^{\prime}(\theta)\right)^{\lambda}\alpha\left(\varphi(\theta)\right).$
(A.2)

The infinitesimal version gives the adjoint action of $\text{Vect}(S^{1})$ on
densities, i.e. the Lie derivative of the differential along a vector field.
Taking $\varphi=\mathbb{I}+\epsilon X$, with $\epsilon\to 0$ and
$X=X(\theta)\partial_{\theta}$, we get

$\displaystyle
L^{(\lambda)}_{X}\alpha\coloneqq\text{ad}^{*}_{X}\alpha=X\alpha^{\prime}+\lambda
X^{\prime}\alpha.$ (A.3)

We can naturally define a bilinear form on
${\mathcal{F}}_{\lambda}\times{\mathcal{F}}_{1-\lambda}$ which is invariant
under Lie derivative as

$\displaystyle\braket{\alpha,\beta}\coloneqq\int_{S^{1}}\alpha\otimes\beta\
\qquad\qquad\forall\ \alpha\in{\mathcal{F}}_{\lambda}\,,\
\beta\in{\mathcal{F}}_{1-\lambda}\,.$ (A.4)

We then identify the dual of $\text{Vect}(S^{1})$ with the space of quadratic
differentials $p=p(\theta)\mathrm{d}\theta^{2}\in{\mathcal{F}}_{2}$, and we
have

$\displaystyle\braket{p,v}\coloneqq\int_{0}^{2\pi}\mathrm{d}\theta\,p(\theta)v(\theta)\qquad\qquad\forall\
v\in{\mathcal{F}}_{-1}\,.$ (A.5)

This means that the action of the vector fields on ${\mathcal{F}}_{2}$
coincides with the co-adjoint action, i.e.

$\displaystyle\braket{\text{ad}^{*}_{X}(p),Y}=-\braket{p,[X,Y]}\qquad\forall\
X,Y\in{\mathcal{F}}_{-1}\,,\ p\in{\mathcal{F}}_{2}\,.$ (A.6)

####  A.1.2 Central extension

We recall that the only non-trivial cocycle on $\text{Vect}(S^{1})$ is given
by the Gelfand–Fuchs 2-cocycle

$\displaystyle\omega:\text{Vect}(S^{1})\times\text{Vect}(S^{1})$
$\displaystyle\to$ $\displaystyle{\mathbb{R}}$ (A.7) $\displaystyle(X,Y)$
$\displaystyle\mapsto$
$\displaystyle\int\mathrm{d}\theta\,X^{\prime}Y^{\prime\prime}=\frac{1}{2}\int\mathrm{d}\theta(X^{\prime}Y^{\prime\prime}-X^{\prime\prime}Y^{\prime})\,.$

The Virasoro algebra is the central extension of the algebra of vector fields,
defined on the vector space
$\widehat{\text{Vect}(S^{1})}\coloneqq\text{Vect}(S^{1})\oplus{\mathbb{R}}$,
with the bracket

$\displaystyle\big{[}(X,a),(Y,b)\big{]}\coloneqq\big{(}[X,Y],\omega(X,Y)\big{)}\qquad\qquad\forall\
a,b\in{\mathbb{R}}\,,\ X,Y\in{\mathcal{F}}_{-1}\,.$ (A.8)

The corresponding group is introduced via the Bott–Thurston 2-cocycle in
$\mathrm{Diff}^{+}(S^{1})$, which is

$\displaystyle B:\text{Diff}^{+}(S^{1})\times\text{Diff}^{+}(S^{1})$
$\displaystyle\to$ $\displaystyle{\mathbb{R}}$ (A.9)
$\displaystyle(\varphi,\psi)$ $\displaystyle\mapsto$
$\displaystyle\frac{1}{2}\int\mathrm{d}\theta\log(\varphi^{\prime}\circ\psi)\left(\log(\psi^{\prime})\right)^{\prime}\,.$

We then define the Virasoro group as
$\widehat{\text{Diff}^{+}}(S^{1})=\text{Diff}^{+}(S^{1})\times{\mathbb{R}}$
equipped with the group law

$\displaystyle(\varphi,a)\circ(\psi,b)=\big{(}\varphi\circ\psi,b+B(\varphi,\psi)\big{)}\,.$
(A.10)

Consistently, we recover the Virasoro algebra by the observation that the
infinitesimal limit of $B$ is $\omega$. Indeed, defining the flows
$\varphi_{t}$ and $\psi_{s}$ corresponding respectively to the vector fields
$X(\theta)\partial_{\theta}$ and $Y(\theta)\partial_{\theta}$, we have

$\displaystyle\omega(X,Y)=\frac{\mathrm{d}^{2}}{\mathrm{d}s\,\mathrm{d}t}\big{(}B(\varphi_{t},\psi_{s})-B(\psi_{s},\varphi_{t})\big{)}\Big{|}_{t=s=0}\,.$
(A.11)

####  A.1.3 Coadjoint action

With the extended Lie bracket and the bilinear form at our disposal, we can
compute the coadjoint action of the Virasoro algebra on quadratic forms.
Taking the adjoint action of the algebra on itself, we define its coadjoint
action as in (A.6). A straightforward calculation leads to

$\displaystyle\text{ad}^{*}_{(X,a)}(p,c)=\big{(}p^{\prime}X+2X^{\prime}p-cX^{(3)},0\big{)}\qquad\qquad\forall\
X\in{\mathcal{F}}_{-1}\,,\ p\in{\mathcal{F}}_{2}\,,\ a,c\in{\mathbb{R}}\,.$
(A.12)

Its exponential gives the group coadjoint action

$\displaystyle\text{Ad}^{*}_{(f^{-1},a)}(p,c)=\big{(}p\bullet
f-c\,{\mathcal{S}}[f],\,c\big{)}\qquad\qquad\forall\
f\in\mathrm{Diff}^{+}(S^{1})\,,\ p\in{\mathcal{F}}_{2}\,,\
a,c\in{\mathbb{R}}\,,$ (A.13)

where

$\displaystyle p\bullet f=f^{\prime}(\theta)^{2}(p\circ
f)\,\mathrm{d}\theta^{2}\,,\qquad\qquad{\mathcal{S}}[f]=\text{Sch}[f]\mathrm{d}\theta^{2}=\left[\frac{f^{(3)}}{f^{\prime}}-\frac{3}{2}\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{2}\right]\,\mathrm{d}\theta^{2}\,.$
(A.14)

Finally, introducing $h=f^{\prime}$, we recall that the Schwarzian derivative
satisfies

$\displaystyle\text{Sch}[f]$
$\displaystyle=\frac{h^{\prime\prime}}{h}-\frac{3}{2}\left(\frac{h^{\prime}}{h}\right)^{2}=(\log
h)^{\prime\prime}-\frac{1}{2}\big{(}(\log h)^{\prime}\big{)}^{2}\,,$ (A.15a)
$\displaystyle\text{Sch}[f\circ g]$
$\displaystyle=\text{Sch}[g]+(g^{\prime})^{2}\,\text{Sch}[f]\circ g\,,$
(A.15b) $\displaystyle\text{Sch}[f^{-1}]$
$\displaystyle=-\frac{\text{Sch}[f]}{h^{2}}\,.$ (A.15c)

###  The BMS3 group

The centrally-extended BMS3 group is the semidirect product, under the adjoint
action, of the Virasoro group and its algebra seen as an Abelian vector group.
This is

$\displaystyle\widehat{\text{BMS}_{3}}=\widehat{\text{Diff}^{+}}(S^{1})\ltimes_{\text{Ad}}\widehat{\text{Vect}}(S^{1})_{\text{ab}}\,.$
(A.16)

We will (abusively) refer to the first factor as superrotations, and to the
second one as supertranslations. Its elements are quadruples $(f,a;g,b)$ with
$f\in\text{Diff}^{+}(S^{1})$, $g\in\text{Vect}(S^{1})_{\text{ab}}$, and
$a,b\in{\mathbb{R}}$. The adjoint action of the Virasoro group reads

$\displaystyle\text{Ad}_{f}(g,b)=\left((gf^{\prime})\circ
f^{-1},b-\int\mathrm{d}\theta\,\text{Sch}[f]g\right)\,.$ (A.17)

The corresponding algebra is

$\displaystyle\widehat{\mathfrak{bms}_{3}}=\widehat{\text{Vect}}(S^{1})\oplus_{\text{ad}}\widehat{\text{Vect}}(S^{1})_{\text{ab}}\,,$
(A.18)

whose elements are again quadruples $(X,a;\alpha,b)$ with
$X\in\text{Vect}(S^{1})$, $\alpha\in\text{Vect}(S^{1})_{\text{ab}}$, and
$a,b\in{\mathbb{R}}$. The commutation relations are

$\displaystyle\big{[}(X,a;\alpha,b),(Y,r;\beta,s)\big{]}=\big{(}[X,Y],\omega(X,Y);[X,\beta]-[Y,\alpha],\omega(X,\beta)-\omega(Y,\alpha)\big{)}\,.$
(A.19)

As always, we can define an adjoint action of the group on its algebra by
conjugation, which simply consists in exponentiating the commutation relation
above. In the specific case studied here this gives

$\displaystyle\text{Ad}_{(f;g)}(X;\alpha)=\left(\text{Ad}_{f}X;\text{Ad}_{f}\alpha+[\text{Ad}_{f}X,g]\right)\,.$
(A.20)

We are now interested in the coadjoint representation. The dual of the algebra
$\widehat{\mathfrak{bms}_{3}}$ is the space
$\widehat{\text{Vect}}(S^{1})^{*}\oplus\widehat{\text{Vect}}(S^{1})_{\text{ab}}^{*}$,
paired with the algebra element via the bilinear form

$\displaystyle\big{\\{}(J,c_{1};P,c_{2}),(X,a;\alpha,b)\big{\\}}\coloneqq\int_{0}^{2\pi}\left(JX+P\alpha\right)\mathrm{d}\theta+c_{1}a+c_{2}b\,.$
(A.21)

This leads to the coadjoint representations of $\widehat{\text{BMS}_{3}}$,
which is

$\displaystyle\text{Ad}^{*}_{f^{-1},g}(J,c_{1};P,c_{2})=\left(\tilde{J},c_{1};\tilde{P},c_{2}\right)$
(A.22)

with

$\displaystyle\tilde{P}=f^{\prime 2}(P\circ
f)-c_{2}\,\text{Sch}[f]\,,\qquad\qquad\tilde{J}=f^{\prime
2}\left(J+gP^{\prime}+2g^{\prime}P-c_{2}g^{(3)}\right)\circ
f-c_{1}\,\text{Sch}[f]\,.$ (A.23)

This is what we use in the main text, for example in (4.4), where there
however the prime ′ has become dot $\cdot$ since the variable is the proper
time $\tau$ instead of the angle $\theta$.

##  Appendix B Other gravitational systems

In this section we briefly present some other gravitational systems which
exhibit the same algebraic properties (encoded in the Poincaré symmetry and
its embedding into BMS3) as the action ${\mathcal{S}}_{0}$ which was our
starting point.

###  4d black hole in Eddington–Finkelstein coordinates

The first example is the 4-dimensional black hole spacetime written in
Eddington–Finkelstein null coordinates. This is described by the line element

$\displaystyle\mathrm{d}s^{2}_{\text{EF}}=\frac{8V_{2}(r)}{V_{1}(r)}\mathrm{d}u^{2}+4L_{0}\,\mathrm{d}u\,\mathrm{d}r+V_{1}(r)\mathrm{d}\Omega^{2}\,.$
(B.1)

Up to a boundary term, with $\tau$ replaced by $r$, and with the non-compact
coordinate $u$ regularized to $u\in[0,L_{0}]$, the Einstein–Hilbert action
evaluated on this line element gives ${\mathcal{S}}_{0}$, i.e.

$\displaystyle{\mathcal{S}}_{\text{EH}}^{(\text{EF})}=\frac{1}{G}\int\mathrm{d}r\,\left[L_{0}^{2}+\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}+\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{V_{2}\dot{V}_{1}+V_{1}\dot{V}_{2}}{V_{1}}\right)\right]\,,$
(B.2)

where the dot denotes derivative with respect to $r$.

###  3d BTZ black hole interior

Going lower in dimensions, it is interesting that the 3-dimensional BTZ black
hole, which is defined in AdS spacetime, also leads to the action
${\mathcal{S}}_{0}$. To see this, we consider the 3-dimensional line element

$\displaystyle\mathrm{d}s^{2}_{\text{BTZ}}$
$\displaystyle=-N^{2}\mathrm{d}t^{2}+\frac{2V_{2}}{\sqrt{V_{1}}}\mathrm{d}r^{2}+\sqrt{V_{1}}(\mathrm{d}\phi+N_{\phi}\mathrm{d}r)^{2},$
(B.3)
$\displaystyle=-N^{2}\mathrm{d}t^{2}+\left(\frac{2V_{2}}{\sqrt{V_{1}}}+N_{\phi}^{2}\sqrt{V_{1}}\right)\mathrm{d}r^{2}+2N_{\phi}\sqrt{V_{1}}\mathrm{d}r\,\mathrm{d}\phi+\sqrt{V_{1}}\mathrm{d}\phi^{2},$
(B.4)

with coefficients depending only on $t$, and where we have included angular
momentum $N_{\phi}$. With this line element we find $\sqrt{-g}=N\sqrt{2V_{2}}$
and

$\displaystyle\sqrt{-g}\,R=\frac{V^{\prime}_{1}(V_{2}V^{\prime}_{1}-2V_{1}V^{\prime}_{2})+2V_{1}(V_{1}N_{\phi}^{\prime})^{2}}{4\sqrt{2}NV_{1}^{2}\sqrt{V_{2}}}+\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\sqrt{2}V^{\prime}_{2}}{N\sqrt{V_{2}}}\right).$
(B.5)

We remark that the 3-dimensional Lagrangian has only a kinetic term and no
potential. This is feature of 3-dimensional gravity, where the potential term
is actually provided by the cosmological constant volume term. Dropping the
boundary term above, and integrating over $r\in[0,L_{0}]$ and
$\phi\in[0,2\pi]$ we find

$\displaystyle{\mathcal{S}}_{\text{3d}}=\frac{1}{2\pi}\int_{M}\mathrm{d}^{3}x\,\sqrt{-g}\,(R+2\Lambda)=\int_{\mathbb{R}}\mathrm{d}t\,L_{0}\left[\frac{V^{\prime}_{1}(V_{2}V^{\prime}_{1}-2V_{1}V^{\prime}_{2})+2V_{1}(V_{1}N_{\phi}^{\prime})^{2}}{4\sqrt{2}NV_{1}^{2}\sqrt{V_{2}}}+\Lambda
N\sqrt{8V_{2}}\right].$ (B.6)

Gauge fixing the lapse to introduce proper time
$\mathrm{d}\tau=\frac{N\sqrt{8V_{2}}}{L_{0}}\mathrm{d}t$, we find once again
the action ${\mathcal{S}}_{0}$ up to some differences:

  * •

The constant term $L_{0}^{2}$ is now multiplied by the cosmological constant
$\Lambda$.

  * •

There is an additional kinetic term depending on the angular momentum. This
term is actually covariant under the BMS transformations, provided that
$N_{\phi}$ is a scalar under superrotations and does not transform under
supertranslations, i.e.

$\displaystyle\mathrm{Ad}_{f,g}N_{\phi}(\tau)=N_{\phi}\circ f^{-1}.$ (B.7)

###  2d dilatonic gravity

Finally, another example leading to ${\mathcal{S}}_{0}$ can be found in
2-dimensional dilatonic gravity. In this context, we consider the 2-parameter
family of actions

$\displaystyle{\mathcal{S}}_{\text{2d}}=\int\mathrm{d}x^{2}\,\sqrt{-g}\,\Big{(}\Phi^{2}R+\alpha\Phi^{\alpha-2}+2\beta(\nabla\Phi)^{2}\Big{)}.$
(B.8)

This includes the spherically-symmetric section of 4d gravity
$(\alpha=2,\beta=1)$, the CGHS model $(\alpha=4,\beta=2)$, and JT gravity
$(\alpha=4,\beta=0)$. Let us then consider a gauge where the dilaton depends
only on $r$, and with 2-dimensional metrics of the form

$\displaystyle\mathrm{d}s^{2}=A(r)\mathrm{d}u^{2}+2B(r)\mathrm{d}u\,\mathrm{d}r\,.$
(B.9)

If we now choose the fields as

$\displaystyle A(r)$
$\displaystyle=\frac{\alpha-\beta}{\alpha}V_{1}^{\frac{\alpha}{2(\alpha-\beta)}}V_{2}^{-\frac{\beta}{\alpha-\beta}}\,,$
(B.10a) $\displaystyle B(r)$
$\displaystyle=\frac{L_{0}}{\alpha}V_{1}^{\frac{\alpha-2}{2(\alpha-\beta)}}V_{2}^{\frac{2-\alpha}{\alpha-\beta}}\,,$
(B.10b) $\displaystyle\Phi(r)$
$\displaystyle=V_{1}^{-\frac{1}{2\alpha-2\beta}}V_{2}^{\frac{1}{\alpha-\beta}}\,,$
(B.10c)

the dilatonic action reduces again to ${\mathcal{S}}_{0}$ up to a boundary
term if we choose $u\in[0,L_{0}]$, and replace $\tau$ with $r$. More
precisely, we find

$\displaystyle{\mathcal{S}}_{\text{2d}}=\int\mathrm{d}r\,\left[L_{0}^{2}+\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}+\frac{\mathrm{d}}{\mathrm{d}r}\left(\alpha\frac{V_{2}\dot{V}_{1}}{2V_{1}}-\beta\dot{V}_{2}\right)\right]\,,$
(B.11)

where dot now denotes derivation with respect to $r$.

##  Appendix C Cosmological constant and scalar field

In this appendix we briefly explain how the action (2.8) is modified in the
presence of a cosmological constant and an homogeneous scalar field. This
shows that the contribution of these matter sources is proportional to $V_{1}$
and $V_{2}$ in the action, just like the extra terms which have been produced
by the action of a finite BMS transformation in (3.18).

Using the metric with redefined lapse $N_{\text{p}}$ as in (2.5), working with
proper time $\tau$ and introducing the IR cutoff $L_{0}$ to regulate the
integration over $x$, the inclusion of a cosmological constant is done by
adding to the Einstein–Hilbert action the volume term

$\displaystyle{\mathcal{S}}_{\Lambda}=\frac{1}{8\pi
G}\int\mathrm{d}^{4}x\,\sqrt{-g}\,\Lambda=\int\mathrm{d}\tau\,L_{0}V_{1}\Lambda\,.$
(C.1)

Similarly, the contribution of a minimally-coupled scalar field $\phi$ in the
spherically-symmetric and homogeneous spacetime (2.5) is described by the
action

$\displaystyle{\mathcal{S}}_{\phi}=-\frac{1}{16\pi
G}\int\mathrm{d}^{4}x\,\sqrt{-g}\left(\frac{1}{2}(\nabla\phi)^{2}-{\mathcal{V}}(\phi)\right)=\frac{1}{2G}\int\mathrm{d}\tau\left(\dot{\phi}^{2}V_{2}+L_{0}V_{1}{\mathcal{V}}(\phi)\right)\,.$
(C.2)

Comparing these two actions with the transformation (3.18), one can see that
for pure supertranslations (i.e. with $f=\mathbb{I}$) we can interpret the
third derivative of $g$ as an effective cosmological constant, while for pure
superrotations (i.e. with $g=0$) we can interpret the term created by the
superrotation as the kinetic term of a scalar field (up to a redefinition
$\text{Sch}[f^{-1}]\mapsto\dot{\phi}^{2}$).

##  Appendix D 1d BMS3 invariant action

This appendix is devoted to the study of some properties of the full BMS
invariant theory (3.19), which is

$\displaystyle{\mathcal{S}}_{\text{inv}}[V_{i},\Phi,\Psi]=\frac{1}{G}\int\mathrm{d}\tau\left[\frac{\dot{V}_{1}(V_{2}\dot{V}_{1}-2V_{1}\dot{V}_{2})}{2V_{1}^{2}}+(V_{2}+\dot{\Psi}V_{1}-\Psi\dot{V}_{1})\text{Sch}[\Phi]+V_{1}\Psi^{(3)}\right].$
(D.1)

We first study the Lagrangian equations of motion and symmetries, and then
perform the Hamiltonian analysis.

###  Lagrangian analysis

First of all we need to remark that the field $\Phi$ does not appear in the
Lagrangian without derivatives, implying the existence of a ghost zero mode
for $\Phi$. On the other hand, both $\Psi$ and $\Phi$ appear with third
derivatives. Moreover, the action exhibits an infinite-dimensional gauge
symmetry (under BMS3), which will be reflected in the freedom to arbitrarily
choose two of the fields in the solutions.

In spite of these peculiarities, the equations of motion takes nevertheless a
remarkably simple form in terms of the coadjoint vectors defined in (2.14a)
and (2.14b). They are given by

$\displaystyle 0\equiv{\mathcal{E}}_{V_{1}}$
$\displaystyle=J+\Psi\,\frac{\mathrm{d}\,\text{Sch}[\Phi]}{\mathrm{d}\tau}+2\dot{\Psi}\,\text{Sch}[\Phi]+\Psi^{(3)}\,,$
(D.2a) $\displaystyle 0\equiv{\mathcal{E}}_{V_{2}}$
$\displaystyle=P+\text{Sch}[\Phi]\,.$ (D.2b)

The other two equations of motion arise as a consequence of these first two.
Indeed, one an check that we have

$\displaystyle 0\equiv{\mathcal{E}}_{\Psi}$
$\displaystyle=-\frac{1}{V_{1}}\frac{\mathrm{d}}{\mathrm{d}\tau}\left({\mathcal{E}}_{V_{2}}V_{1}^{2}\right)\,,$
(D.3a) $\displaystyle 0\equiv\dot{\Phi}{\mathcal{E}}_{\Phi}$
$\displaystyle=-\frac{1}{V_{1}}\frac{\mathrm{d}}{\mathrm{d}\tau}\left({\mathcal{E}}_{V_{1}}V_{1}^{2}\right)-\frac{1}{V_{2}}\frac{\mathrm{d}}{\mathrm{d}\tau}\left({\mathcal{E}}_{V_{2}}V_{2}^{2}\right)+\frac{1}{\Psi}\frac{\mathrm{d}}{\mathrm{d}\tau}\left[\frac{\Psi^{2}}{V_{1}}\frac{\mathrm{d}}{\mathrm{d}\tau}\left({\mathcal{E}}_{V_{2}}V_{1}^{2}\right)\right]\,.$
(D.3b)

This makes manifest that the theory posses gauge symmetries, provided by the
BMS3 group. The equations of motion are solved by the 2-parameter family of
solutions

$\displaystyle V_{1}=\frac{1}{\dot{\Phi}}\,,\qquad\qquad
V_{2}=\frac{\mathfrak{c_{1}}+\mathfrak{c_{2}}\Phi}{\dot{\Phi}}-\frac{1}{\dot{\Phi}}\frac{\mathrm{d}}{\mathrm{d}\tau}\big{(}\Psi\dot{\Phi}\big{)}\,,$
(D.4)

for some constant $\mathfrak{c}_{i}$’s. Moreover, we have that different
solutions can be mapped into each other according to the monodromy condition
satisfied by the supermomentum $P$ (one can compare the equations of motion
with (4.4) in order to see this), meaning that choosing a particular solution
is equivalent to picking up a point on a coadjoint orbit of BMS3 entirely
determined by the constants of integration $\mathfrak{c}_{1}$ and
$\mathfrak{c}_{2}$.

We can also compute the Noether charges associated to the BMS3 symmetry. They
are given by

$\displaystyle{\mathcal{D}}_{X}$
$\displaystyle=-X\left(V_{1}\big{(}J-\Psi\,\dot{P}+2\dot{\Psi}\,\text{Sch}[\Phi]+\Psi^{(3)}\big{)}+(V_{2}-2\Psi\dot{V}_{1})\big{(}P+\text{Sch}[\Phi]\big{)}\right)\,,$
(D.5a) $\displaystyle{\mathcal{T}}_{\alpha}$
$\displaystyle=\alpha\,V_{1}\big{(}P+\text{Sch}[\Phi]\big{)}\,,$ (D.5b)

that trivially vanishes on-shell, as expected for a gauge symmetry charges.

###  Hamiltonian analysis

Since the Lagrangian in (3.19) is higher-order, in order to perform the
Hamiltonian analysis we need to follow the Ostrogradski algorithm and
introduce auxiliary fields with appropriate constraints. First, we therefore
define

$\displaystyle\Phi_{1}=\dot{\Phi}\,,\qquad\qquad\Phi_{2}=\ddot{\Phi}\,,\qquad\qquad\Psi_{1}=\dot{\Psi}\,,\qquad\qquad\Psi_{2}=\ddot{\Psi}\,.$
(D.6)

With this, the action (3.19) can then be rewritten with at most first order
derivatives as

$\displaystyle{\mathcal{S}}_{\text{inv}}$
$\displaystyle={\mathcal{S}}_{0}+\frac{1}{G}\int\mathrm{d}\tau\left[(V_{2}+\Psi_{1}V_{1}-\Psi\dot{V}_{1})\left(\frac{\dot{\Phi}_{2}}{\Phi_{1}}-\frac{3}{2}\left(\frac{\Phi_{2}}{\Phi_{1}}\right)^{2}\right)+V_{1}\dot{\Psi}_{2}\right.$
(D.7)
$\displaystyle\phantom{={\mathcal{S}}_{0}+\frac{1}{G}\int\mathrm{d}\tau\left[\right.}+\Pi(\dot{\Phi}-\Phi_{1})+\Pi_{1}(\dot{\Phi}_{1}-\Phi_{2})+\Xi(\dot{\Psi}-\Psi_{1})+\Xi_{1}(\dot{\Psi}_{1}-\Psi_{2})\Bigg{]}\,.$
(D.8)

Here the second line is the set of constraints, imposed with Lagrange
multiplier $(\Pi,\Pi_{1},\Xi,\Xi_{1})$, ensuring that the auxiliary fields can
be eliminated in order to go back to the initial expression (3.19). As
expected, because this theory has gauge symmetries it is a constrained system.
In order to see this in the Hamiltonian formulation, we perform the Legendre
transform and introduce a 16-dimensional phase space with canonical pairs

$\displaystyle\big{\\{}(\Phi,\Pi),(\Phi_{1},\Pi_{1}),(\Phi_{2},\Pi_{2}),(\Psi,\Xi),(\Psi_{1},\Xi_{1}),(\Psi_{2},\Xi_{2}),(V_{1},P_{1}),(V_{2},P_{2})\big{\\}}\,,$
(D.9)

with

$\displaystyle\big{\\{}\Phi_{i},\Pi_{i}\big{\\}}=\big{\\{}\Psi_{i},\Xi_{i}\big{\\}}=\big{\\{}V_{i},P_{i}\big{\\}}=1\,.$
(D.10)

The extended Hamiltonian is then found to be

$\displaystyle H$
$\displaystyle=H_{0}+\dot{\Phi}_{2}\chi_{1}+\dot{\Psi}_{2}\chi_{2}\,,$ (D.11)
$\displaystyle H_{0}$
$\displaystyle=-P_{2}\left(V_{1}P_{1}+\frac{1}{2}V_{2}P_{2}\right)+(V_{2}+\Psi_{1}V_{1}+\Psi
V_{1}P_{2})\frac{3}{2}\left(\frac{\Phi_{2}}{\Phi_{1}}\right)^{2}+\Pi\Phi_{1}+\Pi_{1}\Phi_{2}+\Xi\Psi_{1}+\Xi_{1}\Psi_{2}\,,$

where $\dot{\Phi}_{2}$ and $\dot{\Psi}_{2}$ are Lagrange multipliers imposing
the primary constraints

$\displaystyle\chi_{1}$
$\displaystyle=\Pi_{2}-\frac{1}{\Phi_{1}}(V_{2}+\Psi_{1}V_{1}+\Psi
V_{1}P_{2})\,,$ (D.12a) $\displaystyle\chi_{2}$
$\displaystyle=V_{1}-\Xi_{2}\,.$ (D.12b)

We can now apply Dirac’s algorithm and evolve the primary constraints with
respect to the extended Hamiltonian. This evolution generates two secondary
constraints given by

$\displaystyle\chi_{3}$
$\displaystyle=\big{\\{}\chi_{1},H_{\text{tot}}\big{\\}}=\frac{1}{\Phi_{1}}\left(V_{1}P_{1}+V_{2}P_{2}-V_{1}\Psi_{2}+\frac{1}{2}V_{1}P_{2}^{2}\Psi\right)-\frac{2\Phi_{2}}{\Phi_{1}^{2}}(V_{2}+\Psi_{1}V_{1}+\Psi
V_{1}P_{2})-\Pi_{1}\,,$ (D.13a) $\displaystyle\chi_{4}$
$\displaystyle=\\{\chi_{2},H_{\text{tot}}\\}=-P_{2}V_{1}+\Xi_{1}\,.$ (D.13b)

The evolution of these secondary constraints then leads to two tertiary
constraints given by

$\displaystyle\chi_{5}=\\{\chi_{3},H_{\text{tot}}\\}$
$\displaystyle=\Pi-\frac{1}{2\Phi_{1}}P_{2}\big{(}P_{2}V_{2}-V_{1}(P_{2}\Psi_{1}-2P_{1}+2\Psi_{2})\big{)}$
(D.14a) $\displaystyle\phantom{=\
}+\frac{\Phi_{2}}{2\Phi_{1}^{2}}\big{(}2P_{2}V_{2}+V_{1}(2P_{1}+P_{2}^{2}\Psi-2\Psi_{2})\big{)}-\frac{\Phi_{2}^{2}}{2\Phi_{1}^{3}}\big{(}V_{2}+V_{1}(P_{2}\Psi+\Psi_{1})\big{)}\,,$
(D.14b) $\displaystyle\chi_{6}=\\{\chi_{4},H_{\text{tot}}\\}$
$\displaystyle=\frac{1}{2}P_{2}^{2}V_{1}-\Xi\,,$ (D.14c) $\displaystyle
H_{\text{tot}}$ $\displaystyle=H_{0}+\mu_{i}\chi_{i}\,.$ (D.14d)

The algorithm then stops without giving any conditions on the Lagrange
multipliers or on the fields. All the constraints are first class: their
brackets weakly vanish, and for $\chi_{5}$ and $\chi_{6}$ they actually
strongly vanish.

Now that we know the first class constraints, we need to find which
combination generates the infinitesimal BMS3 gauge transformations. A
straightforward calculation shows that these Hamiltonian generators are given
by

$\displaystyle{\mathcal{D}}_{X}$
$\displaystyle=X(-\Phi_{1}\chi_{5}+\chi_{2}\dot{\Psi}_{2}-\dot{\Phi}_{2}\chi_{1}+\Phi_{2}\chi_{3}+\Psi_{1}\chi_{6}-\Psi_{2}\chi_{4})+$
(D.15a) $\displaystyle\phantom{=\
}+\dot{X}(\Phi_{1}\chi_{3}-2\Phi_{2}\chi_{1}+\Psi_{2}\chi_{2}-\Psi\chi_{6})+\ddot{X}(\Psi\Phi_{4}-\Psi_{1}\chi_{2}-\Phi_{1}\chi_{1})-\Psi\chi_{2}X^{(3)}\,,$
$\displaystyle{\mathcal{T}}_{\alpha}$
$\displaystyle=-\alpha\chi_{6}+\dot{\alpha}\chi_{4}-\ddot{\alpha}\chi_{2}\,.$
(D.15b)

Finally, one can note that this is consistent with the Hamiltonian analysis of
the initial action ${\mathcal{S}}_{0}$, and that we obtain the
$\text{ISO}(2,1)$ generators of the symmetry-broken theory in the limit

$\displaystyle(\Psi,\Pi_{i},\Xi_{i})\to 0\,,\qquad\qquad\Phi(\tau)\to\tau\,.$
(D.16)

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