# Phase spaces and symmetries of Vaidya superspace

Salvatore Ribisi  Aix Marseille Univ, Université de Toulon, CNRS, CPT,
Marseille, France  Francesco Sartini  Okinawa Institute of Science and
Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495,
Japan

###### Abstract

We investigate the classical symmetries of the dynamics of the null-dust
spherically symmetric Vaidya spacetime. Einstein’s equations for this model
can be obtained as equations of motion of a two-dimensional field theory. We
discuss the transformations leaving invariant such equations of motion. These
are given by two distinct sets, the residual diffeomorphisms coming from
general relativity and the generalisation of the Schrödinger symmetry,
recently found for the static Schwarzschild black holes. Surprisingly, these
two sets represent the symmetries of two different action functionals, leading
to the same equations of motion, but with different phase spaces.

General relativity rests strongly on a symmetry principle, the invariance
under diffeomorphisms, corresponding to coordinate changes on spacetime.
However, these symmetries appear as gauge symmetries, redundancy of the
physical description. Hence, they are not associated with any physical
content. The Noetherian duality between symmetries and conserved quantities,
or charges, would give zero charge for gauge symmetries, leading to the
cumbersome task of defining observables in gravity. The situation changes
dramatically in the presence of boundaries, may they be asymptotic or at
finite distance. They can promote some gauge symmetries to have a non-zero
charge living on codimension-two corners of spacetime [1, 2, 3, 4, 5, 6, 7, 8]

Thinking of boundaries as bridges between different regions makes the corner
symmetry algebra very relevant for the study of entanglement entropy between
subregions. In this spirit, we can aim to use the representation theory of the
corner algebra as non-perturbative handles on quantum gravity [8]; but it also
allows us to, more conjecturally, make spacetime and its topology emerge from
quantum entanglement between subregions [9, 10].

However, even letting aside the boundaries, bulk symmetries might have their
relevance. Some transformations link different sets of bulk solutions in
general relativity. For example, the Newman-Janis algorithm allows obtaining
rotating black hole solutions, out of the Schwarzschild one, through a complex
coordinate transformation [11, 12]. In addition, we have regularities in the
tower of quasi-normal modes or responses to perturbation that come from
approximate near-horizon symmetries [13, 14]. The latter is also related to
boundary structure and black hole entropy [15, 16, 17]

Recently a very peculiar class of symmetries [18, 19, 20, 21], for some very
regular solutions of general relativity, has drawn some attention. These
highly-symmetric spacetimes, or minisuperspaces, can be described as
mechanical models, focusing on the evolution in just one spacetime direction
and freezing the other ones. This is analogous to selecting the zero modes of
geometry, but despite seeming very simple at first glance, they are relevant
for cosmological or near-singularity applications.

These symmetries fully encode the evolution of the physical spacetime, and
have an elegant interpretation in terms of geometrization of the dynamical
space. The configuration field space is endowed with a metric, constructed out
of the kinetic term of the reduced action [20, 19].

Originally discovered for the isotropic cosmological setup [22, 23], these
minisuperspace symmetries have also been uncovered for black hole models [18,
19, 21] and anisotropic cosmologies [20, 24]. A review of a systematic
approach to the exploration of homogeneous models can be found in [20, 24] or
see [19] for an equivalent technique, known as Eisenhart–Duval lift, based on
an extended phase space [25].

The interest in minisuperspaces goes beyond the elegant relationship between
symmetries, dynamics and geometrical structure of the field space. Recent
works have pointed out that astrophysically relevant models possess a symmetry
group equivalent to the Schrödinger group. This conformal group has a key role
in non-relativistic hydrodynamics and for some Bose-Einstein condensates. It
suggests an intriguing correspondence between the response to perturbation of
these gravitational systems and fluid analogues.

Moreover, there seems to be an intriguing feature of the minisuperspace
symmetries in relationship with the boundaries of spacetime. In all these
models we need indeed a regulator to deal with an infinite homogeneous slice.
This turns out to interplay with the symmetries, being modified by them [18,
24]. However, to better understand these structures we should go beyond the
simple homogeneous setup, by including inhomogeneity in cosmologies or non-
stationary processes for black holes.

In this paper, we will discuss the extension of the Schrödinger symmetry to
the simplest non-static generalisation of the Schwarzschild solution, known as
Vaidya spacetime. This will force at least one field to evolve in two
directions. On top of the radial dependence of the system, already considered
for the stationary minisuperspace, we will add the dependence on a null
coordinate. The richer spacetime structure makes the residual diffeomorphism
gauge freedom less trivial than the static case. We will then discuss how this
gauge symmetry interplays with the Schrödinger transformations.

The paper is organised as follows. We started in section I by introducing the
Vaidya superspace as a two-dimensional field theory, coming from the spherical
symmetric general relativity in a particular gauge. After verifying that the
solutions of the equations of motion are consistently given by the Vaidya
solutions alone, we will move to the discussion about their symmetries. In
section II we introduce these symmetries simply as the transformations mapping
solutions into solutions. However, to interpret them in a Noetherian sense, we
shall give a notion of phase space. In III we show that the two sets of
transformations, Schrödinger and gauge symmetries, are integrable on two
different phase spaces, coming from two different theories, leading to the
same classical equations.

Indeed the plural phase spaces in the title of this article is not a typo, we
can obtain the same classical spacetime from two inequivalent phase spaces
coming from different action functionals.

##  Action and equations of motion of Vaidya superspace

The Vaidya metric represents the simplest radiating solution for black holes
and provides a natural testbed to address questions related to black hole
evaporation [26, 27, 28, 29, 30]. The presence of hydrodynamical symmetries in
such a model might give new insights into the problem.

The metric is usually presented in the Eddington–Finkelstein gauge,
generalising the spherically symmetric ansatz by breaking the stationarity or
equivalently partially breaking the diffeomorphism invariance on the radial-
temporal plane. More in detail, we take a spherically symmetric ansatz in four
spacetime dimensions, which means separating the angular directions with
respect to the other two coordinates. The latter represents the radial-
temporal plane, whose compactification is the Penrose diagram. On top of this,
we force one of the coordinates to be null, imposing the gauge condition
$g_{rr}=0$. The ansatz that we take is thus

$\displaystyle\mathrm{d}s^{2}=\frac{B(v,r)}{X(r,v)}\mathrm{d}v^{2}+2N(r,v)\mathrm{d}v\mathrm{d}r+X(r,v)^{2}\mathrm{d}S_{(2)}^{2}\,,$
(1)

where the term $\mathrm{d}S_{(2)}^{2}$ represents the usual two-sphere metric.
The choice of parametrization for the $g_{vv}$ term is chosen in this way to
simplify the notation below. We insist again on the fact that this ansatz
partially breaks the covariance along the mixed $r$-$v$ direction, because it
does not contain the $rr$ term, and constrains $v$ to be a null coordinate. In
other words, the diffeomorphisms that preserve the ansatz (1) are the ones
generated by the vector fields

$\displaystyle\xi=\xi^{r}(r,v)\partial_{r}+\xi^{v}(v)\partial_{v}+\sigma[S_{(2)}]\,,$
(2)

with $\sigma$ generating the celestial sphere’s global $\mathrm{SO}(3)$
rotations. It has a non-null component only along the angular direction,
trivially commuting with the null and radial diffeomorphisms. We will later
show that the partial breaking of the full two-dimensional diffeomorphism
group on the $r$-$v$ plane will have the consequence of losing one Einstein
equation, corresponding to the mass conservation. On the other hand, this
allows us to obtain the non-static Vaidya metric as a solution. We shall
remark that in principle our ansatz contains both the emitting and absorbing
Vaidya pure radiation fields111With the notation chosen here we have an
absorbing field for positive $N$, for which we usually use the ingoing null
coordinate $v$. On the contrary, the outgoing null coordinate is typically
denoted by the letter $u$. We choose here to keep $v$ for the null coordinate,
also in the ingoing case. We use the sign of the field $N$ to flip between
absorbing and emitting cases, depending on the sign of $N$.

We would like to obtain Einstein’s equation from the variational principle of
some action functional. The most obvious being the reduced Einstein–Hilbert
action

$\displaystyle{\mathcal{S}}_{\text{EH}}$
$\displaystyle=\frac{1}{16\pi\ell_{\text{Pl}}^{2}}\int\mathrm{d}^{4}x\sqrt{|g|}{\mathcal{R}}$
$\displaystyle=\frac{1}{2\ell_{\text{Pl}}^{2}}\int\mathrm{d}v\mathrm{d}r\left[N+\frac{X^{\prime}(2\partial_{v}(NX)-BX^{\prime})}{N}\right]+\left.\frac{1}{4\ell_{\text{Pl}}^{2}}\int\mathrm{d}v\left[\frac{\partial_{r}(BX^{3})-2X^{2}\dot{N}}{NX^{2}}\right]\right|_{{r_{\text{i}}}}^{{r_{\text{f}}}}-\left.\frac{X^{2}}{2\ell_{\text{Pl}}^{2}}\right|_{{r_{\text{i}}},{v_{\text{i}}}}^{{r_{\text{f}}},{v_{\text{f}}}}\,,$
(3)

where the dot represents the derivative with respect to $v$ and the prime with
respect to the radial direction. The four-dimensional Ricci scalar is denoted
with ${\mathcal{R}}$, and we choose units such that the four-dimensional
Newton constant is $G=\ell_{\text{Pl}}^{2}$, with the Plank length
$\ell_{\text{Pl}}$. The action must be thought to describe the variational
problem in the region between two slices at a constant radius and two null
surfaces at constant $v$. However, for the moment let us neglect the
discussion about boundary conditions and just focus on the bulk equation of
motion.

The first part of the action (3) contains the bulk Lagrangian and the second
term is a boundary term, acting on the polarization of the phase space (when
evaluating the evolution in the $r$ direction). The last one is a corner term
for the symplectic potential and might play a role in the determination of the
charges and algebra. However, for the study of the classical solutions and
bulk symmetries, the only relevant part is given by the bulk term. We can thus
evaluate the variational principle of the two-dimensional field theory action

$\displaystyle{\mathcal{S}}_{0}=\frac{1}{2\ell_{\text{Pl}}^{2}}\int\mathrm{d}v\mathrm{d}r\left[N-\frac{X^{\prime}(B^{\prime}-2\partial_{v}(NX))}{N}\right].$
(4)

The first property that we shall verify is the consistency with general
relativity. The fact that the Euler–Lagrange equations for the Lagrangian (4)
give Einstein’s equations for the metric (1) is a highly non-trivial
statement. Indeed we will show that one equation is missing and the mass will
be allowed to evolve in the null direction. A straightforward computation
gives us the equations of motion for the field theory (4):

$\displaystyle 0$
$\displaystyle\approx\partial_{r}\left(X^{\prime}/N\right)\,,$ (5a)
$\displaystyle 0$ $\displaystyle\approx
N(BX^{\prime\prime}-4N\dot{X}^{\prime})-BX^{\prime}N^{\prime}-2X(N\dot{N}^{\prime}-\dot{N}N^{\prime})\,,$
(5b) $\displaystyle 0$ $\displaystyle\approx
N^{2}+X^{\prime}BX^{\prime}-N\partial_{v}\partial_{r}X^{2}\,.$ (5c)

These are not all independent, because of the residual gauge freedom generated
by (2). We can analytically solve the equations of motion for any function
$X$, that will be later identified as a dilatonic field from a two-dimensional
perspective [31, 32, 33, 34]. We can deparametrize the evolution with respect
to this field, and add two free functions depending only on $v$ as initial
conditions. The general solutions of (5) are222We use $\approx$ to denote on-
shell equalities

$\displaystyle B\approx B_{0}(v)-X\,n(v)^{2}+2n(v)X\dot{X}\,,\quad\quad
N\approx n(v)X^{\prime}\,.$ (6)

As expected, the solution space is not completely invariant under spacetime
reparametrization, but only under the action of the residual diffeomorphisms
(2). Among them, only the radial and null directions act non trivially on the
solution space, while the celestial sphere angular directions are gauged out
of the model. The coordinates $r$ and $v$ play two different roles, and our
ansatz (1) is invariant only under diffeomorphisms that leave $v$ as a null
coordinate. As already announced, from the variation of the action we miss the
mass conservation, leading to an on-shell Vaidya metric. Replacing the
solutions (6) into the ansatz (1) we explicitly get

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\left(n^{2}-\frac{B_{0}}{X}-2n\dot{X}\right)\mathrm{d}v^{2}+2nX^{\prime}\,\mathrm{d}v\mathrm{d}r+X(r,v)^{2}\mathrm{d}S_{(2)}^{2}$
$\displaystyle=-\left(1-\frac{2GM(v)}{X}\right)n^{2}\mathrm{d}v^{2}+2n\mathrm{d}v\,\mathrm{d}X+X^{2}\,\mathrm{d}S_{(2)}^{2}\,,\quad\quad
2\ell_{\text{Pl}}^{2}M(v)=B_{0}/n^{2}\,.$ (7)

The usual null coordinate of Eddington–Finkelstein parametrization of Vaidya
is given by $\pm\mathrm{d}V=n(v)\mathrm{d}v$, while $X$ represents the radial
coordinate 333The variational principle of the action (4) allows for both
positive and negative $N$, representing emitting and absorbing Vaidya
spacetimes. Moreover, the residual diffeomorphisms allow to flip the sign of
$N$ by exponentiation of a $\xi$ pointing backwards in the $v$ direction.

The missing equation, which should come from the variation of the action with
respect to the $g^{vv}$ term444The vanishing of the $g_{rr}$ term is, of
course, equivalent to the vanishing of $g^{vv}$, is the mass conservation.
Indeed the on-shell Einstein tensor for our ansatz has a non-vanishing
component

$\displaystyle
G_{vv}\approx\frac{n\dot{B}_{0}-2B_{0}\dot{n}}{X^{2}n^{2}}=\frac{2\ell_{\text{Pl}}^{2}\dot{M}}{X^{2}}\,.$
(8)

This means that the variational principle of the action (4) gives non-vacuum
Einstein’s equations for the bulk ansatz (1), effectively coupled to a pure
radiation field with stress-energy tensor satisfying

$\displaystyle
T_{\mu\nu}\approx\frac{1}{4\pi}\frac{\dot{M}}{X^{2}}l_{\mu}l_{\nu}\,,$ (9)

with the null form $l_{\mu}\mathrm{d}x^{\mu}=-\mathrm{d}v$. The full
diffeomorphism invariance on the $r-v$ plane can be restored by adding such
null dust to the Lagrangian [26, 27]. This will give the Vaidya solution for
an ansatz in which $v$ is not necessarily null, adding a $g_{rr}$ term, whose
variation will impose the total mass conservation. We chose here to hide the
matter contribution in the partial gauge fixing, to make the comparison with
the static case easier to handle.

###  I.1 Relationship with two-dimensional dilatonic gravity

Before moving to the discussion about the classical symmetries of this model,
let us open a small parenthesis on an interesting relationship between this
model and a general two-dimensional dilatonic theory. A general class of such
theories, whose dynamical content is given by a two-dimensional metric
$g_{\mu\nu}^{(2)}$ and a scalar field $\Phi$, is given by [33, 34]

$\displaystyle{\mathcal{S}}_{\text{DGT}}=\frac{1}{16\pi
G_{2D}}\int\mathrm{d}^{2}x\sqrt{|g^{(2)}|}\left(\Phi
R-U(\Phi)(\nabla\Phi)^{2}-2V(\Phi)\right)\,,$ (10)

where $U$ and $V$ are functions of the dilaton field $\Phi$, and $R$ is the
Ricci scalar of the two-dimensional metric. Within this class, we have models
such as JT gravity ($U=0$, $V=\Lambda\Phi$) or the CGHS model ($U=0$,
$V=\lambda$). The four-dimensional spherically symmetric gravity also belongs
to this class. Indeed we can identify the two-dimensional metric as the one
describing the radial and temporal direction (i.e. the Penrose diagram), while
the scalar field determines the measure of the celestial sphere (points on the
Penrose diagram). Let us consider a general four-dimensional spherically
symmetric ansatz

$\displaystyle\mathrm{d}s^{2}_{4D}=e^{2\Omega(\Phi)}g^{(2)}_{\mu\nu}\,\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\Phi\,\mathrm{d}S\,,\quad\mu,\nu\in\\{0,1\\}\,.$
(11)

The two-dimensional metric can be identified up to a conformal factor
$\Omega$, taken to be a function of the scalar dilaton. The conformal
rescaling interplays with the potential $U$, $V$ and we can use this fact to
set one of the potentials to zero. The Einstein–Hilbert action of the
spherically symmetric line-element (11) is

$\displaystyle S_{\text{4D}}$
$\displaystyle=\frac{1}{16\pi\ell_{\text{Pl}}^{2}}\int\mathrm{d}^{4}x\sqrt{|g^{(4D)}|}{\mathcal{R}}$
$\displaystyle=\frac{1}{4\ell_{\text{Pl}}^{2}}\int\mathrm{d}^{2}x\sqrt{|g^{(2)}|}\left(\Phi
R+2e^{2\Omega}+\frac{1}{2\Phi}(\nabla\Phi)^{2}-2\Phi\nabla^{2}\Omega\right)+\text{“boundary
terms"}\,.$ (12)

We can eliminate the kinetic term for the dilaton (up to a boundary term) by
choosing

$\displaystyle e^{2\Omega}=\frac{1}{\sqrt{\Phi}}\,.$ (13)

This identifies the reduced four-dimensional action as the one in the class
(10) with $U=0$, $V=-1/\sqrt{\Phi}$. At this point, this model still possesses
the whole invariance under two-dimensional diffeomorphism, but we can
immediately recognise that our ansatz (1) corresponds to the Bondi gauge of
the two-dimensional metric, and the field $X$ is the square root of the
dilaton. Taking the two-dimensional metric $g_{\mu\nu}^{(2)}$ in the Bondi
gauge

$\displaystyle\mathrm{d}s_{2D}^{2}=B(v,r)\mathrm{d}v^{2}+2X(r,v)N(r,v)\mathrm{d}v\mathrm{d}r\,$
(14)

and plugging this ansatz in the dilatonic action, with the choices $U=0$,
$V=-1/X$, $\Phi=X^{2}$, gives the “Vaidya action” (3), up to a boundary term,
meaning that

$\displaystyle
S_{\text{EH}}=\frac{1}{4\ell_{\text{Pl}}^{2}}\int\mathrm{d}^{2}x\sqrt{|g_{2D}|}\left(X^{2}R^{(2D)}+\frac{2}{X}\right)+\frac{1}{4\ell_{\text{Pl}}^{2}}\int\mathrm{d}v\frac{BX^{\prime}}{N}-\left.\frac{3X^{2}}{4\ell_{\text{Pl}}^{2}}\right|_{{r_{\text{i}}},{v_{\text{i}}}}^{{r_{\text{f}}},{v_{\text{f}}}}\,.$
(15)

As a consequence, this observation implies that one must be extremely careful
when plugging the Bondi gauge in off-shell quantities. As shown here, if we do
so in the two-dimensional case at the action level and then we evaluate the
Euler–Lagrange equation for the gauge fixed action, we will lose the mass
conservation equation. Although here we want precisely to use this fact to
obtain the Vaidya solution from the “wrong” gauge fixing without specifying
the matter content of the theory, this has important consequences for the
study of asymptotic symmetries.

###  I.2 Linear radial gauge

Inspired by the previous works on Schwartzschild black holes [19, 18, 24], we
do a further gauge choice by picking a linear square root of the dilaton, i.e.
imposing

$\displaystyle X(r,v)=A_{0}(v)\left(r-\phi_{0}(v)\right)\,.$ (16)

By doing so, we obtain a solution space spanned by four free functions
depending on the null coordinate. The same happens in [33], where they study
the integrability of asymptotic large diffeomorphisms in two-dimensional
gravity, even if therein the linearity condition is on $\Phi=X^{2}$. The
choice here is taken as a natural generalisation for the previous works in
black hole minisuperspaces and it is implemented by the condition
$N^{\prime}=0$, which is the same as the one chosen in [18, 19], while the one
in [33] is implemented by $\partial_{r}(XN)=0$.

The equations of motion get rewritten in this gauge as

$\displaystyle 0$ $\displaystyle\approx X^{\prime\prime}\,,$ (17a)
$\displaystyle 0$ $\displaystyle\approx
B^{\prime\prime}-4N\dot{X}^{\prime}\,,$ (17b) $\displaystyle 0$
$\displaystyle\approx
N^{2}+X^{\prime}B^{\prime}-N\partial_{v}\partial_{r}X^{2}\,,$ (17c)

and the solution space is now spanned by four functions of the null coordinate
alone $A_{0}$, $B_{0}$, $\phi_{0}$ and $n$,

$\displaystyle X$ $\displaystyle\approx A_{0}(v)\left(r-\phi_{0}(v)\right)\,,$
$\displaystyle B$ $\displaystyle\approx
B_{0}(v)-A_{0}\,n\left(r-\phi_{0}\right)(n-2(r-\phi_{0})\dot{A}_{0}+2A_{0}\dot{\phi}_{0})\,,$
(18) $\displaystyle N$ $\displaystyle\approx A_{0}(v)\,n(v)\,.$

##  Moebius symmetry and residual diffeomorphisms

With the solution space at hand, we can now turn to the study of the
symmetries of this model. In the first place, we will search for sets of
transformations that preserve the equations of motion, in the sense that they
map solutions of (17) into solutions. We also work within the linear gauge
$N^{\prime}=0$.

This section aims to show how it is possible to generalise the minisuperspace
Schrödinger symmetry, originally found for the static black holes to the
Vaidya model. For this let us recall that the two-dimensional Schrödinger
group splits into the semi-direct product

$\displaystyle\rm{Sh}(2)=\left(\mathrm{SL}(2,{\mathbb{R}})\times\mathrm{SO}(2)\right)\ltimes\left({\mathbb{R}}^{2}\times{\mathbb{R}}^{2}\right)\,,$
(19)

where the algebra generating the abelian part
$({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$ contains a central extension. In
quantum mechanics, where the group has been introduced in the first place, the
central charge is the Plank constant. Classically, the abelian symmetry group
corresponds to constant shifts of position and momenta of a free particle. The
$\mathrm{SO}(2)$ part represents the rotations of the two-dimensional plane,
and $\mathrm{SL}_{2}$ generates conformal symmetries of the Schrödinger
equation.

In particular, the realization of the
$\mathrm{SL}(2,{\mathbb{R}})\times\mathrm{SO}(2)$ subgroup on the black hole
superspace translates into a Moebius transformation on the radial coordinate,
while the metric coefficients transform as conformal fields of different
weights. At the level of spacetime, this corresponds to an anisotropic Weyl
rescaling of the geometry and is not a residual diffeomorphism [20, 24]. To
begin, we will focus on this subgroup to try to extend the symmetry to the
Vaidya superspace.

In the minsuperspace setup, the Schrödinger symmetry emerges naturally from a
second geometrization procedure, mapping the spacetime dynamics to a point
particle geodesic motion on the field space. From this perspective, the
symmetries are associated with conformal properties of the supermetric [19,
20]. In the case we study here, we lack such a point particle interpretation,
because of the presence of an infinite dimensional field space. However, we
can rewrite the equations of motion in a similar way to the finite-dimensional
space. We can achieve this by redefining the fields to get rid of the null
direction derivatives in the equations of motion. While this is already the
case for $X$ in (17a), it is convenient to introduce the quantity

$\displaystyle{\mathcal{B}}:=B-2rN\dot{X}\,.$ (20)

It’s easy to convince ourselves that the equation of motion (17b) for $B$ is
rewritten in a very simple way for the new field, namely
${\mathcal{B}}^{\prime\prime}\approx 0$. Although this field redefinition
makes the equations of motion have a nicer form, its geometrical
interpretation is not straightforward.

It also turns out to be useful to redefine the field $N$ in a way that makes
the last equation more compact. Let us consider the combination

$\displaystyle{\mathcal{N}}^{2}:=N^{2}-2N\dot{X}^{\prime}(X-rX^{\prime})\,.$
(21)

Combining the gauge condition $N^{\prime}=0$ and the equation of motion
$X^{\prime\prime}=0$, we can show that also ${\mathcal{N}}$ is independent of
the radial coordinate. Equivalently, on our solution space,
${\mathcal{N}}^{\prime}\approx 0$. This brings us to make the $v$ derivatives
completely disappear from the equation of motions, turning (17) into

$\displaystyle 0$ $\displaystyle\approx X^{\prime\prime}\,,$ (22a)
$\displaystyle 0$ $\displaystyle\approx{\mathcal{B}}^{\prime\prime}\,,$ (22b)
$\displaystyle 0$
$\displaystyle\approx{\mathcal{N}}^{2}+X^{\prime}{\mathcal{B}}^{\prime}\,.$
(22c)

In particular, the first two equations decouple the evolution of $X$ and
${\mathcal{B}}$ and are both in a form which is invariant under Moebius
reparametrization of the radial coordinate555The same form indeed appears in
the minisuperspace setup, once we write the field space in the appropriate
null variables [20], and it is related to the conformal invariance of the free
particle mechanics. Let us define the transformation

$\displaystyle r\ $ $\displaystyle\rightarrow\ \tilde{r}=h(r):=\frac{\alpha\
r+\beta}{\gamma\ r+\delta},\qquad\alpha\delta-\beta\gamma=1\,,\quad
h^{\prime}(r)=\frac{1}{(\gamma r+\delta)^{2}}\,,$ (23a) $\displaystyle X(r,v)\
$ $\displaystyle\rightarrow\
\widetilde{X}(\tilde{r},v)=\lambda\sqrt{h^{\prime}(r)}\
X(r,v)\,,\quad\quad\lambda=const\,,$ (23b) $\displaystyle{\mathcal{B}}(r,v)\ $
$\displaystyle\rightarrow\
\widetilde{{\mathcal{B}}}(\tilde{r},v)=\lambda^{-1}\sqrt{h^{\prime}(r)}\
{\mathcal{B}}(r,v)\,,$ (23c)

This is easily shown to leave the first two equations of motion (22a) and
(22b) invariant, we have indeed e.g. the first one

$\displaystyle
X^{\prime\prime}\mapsto\partial_{\tilde{r}}^{2}\widetilde{X}=\lambda\frac{h^{\prime
2}X^{\prime\prime}+X(2h^{\prime}h^{\prime\prime\prime}-3h^{\prime\prime
2})/4}{\sqrt{h^{\prime}}}=\lambda h^{\prime 3/2}\,X^{\prime\prime}\,,$ (24)

recognising the Schwarzian derivative, that vanishes for the Moebius
transformation above,

$\displaystyle\text{Sch}[h]:=\frac{h^{\prime\prime\prime}}{h^{\prime}}-\frac{3}{2}\left(\frac{h^{\prime\prime}}{h^{\prime}}\right)^{2}=0\quad\Leftrightarrow\quad
h=\frac{\alpha\ r+\beta}{\gamma\ r+\delta}\,.$ (25)

The same happens for the equation (22b) concerning the evolution of
${\mathcal{B}}$.

We now need to discuss the last equation of motion and the transformation for
${\mathcal{N}}$. The most naive way of defining how ${\mathcal{N}}$ transform,
is precisely through the equation of motion (22c). Indeed, for the
transformations to be symmetries, we must have

$\displaystyle\widetilde{{\mathcal{N}}}\approx\left(-\partial_{\tilde{r}}\widetilde{X}\partial_{\tilde{r}}\widetilde{\mathcal{B}}\right)^{1/2}=\frac{1}{\sqrt{h^{\prime}}}\left(-X^{\prime}{\mathcal{B}}^{\prime}-\frac{1}{2}\partial_{r}\left(\frac{h^{\prime\prime}}{h^{\prime}}X{\mathcal{B}}\right)\right)^{1/2}\approx\frac{1}{\sqrt{h^{\prime}}}\left({\mathcal{N}}^{2}-\frac{1}{2}\partial_{r}\left(\frac{h^{\prime\prime}}{h^{\prime}}X{\mathcal{B}}\right)\right)^{1/2}\,.$
(26)

We can use the last expression to define $\widetilde{\mathcal{N}}$. This
expression maps ${\mathcal{N}}$ to some $r$ dependent field, seeming to break
the gauge condition. However, once it is projected onto the solution space, it
happens to be consistent with the gauge choice. If the equations (22) hold,
and ${\mathcal{N}}^{\prime}=0$, then we also have
$\widetilde{\mathcal{N}}^{\prime}=0$

We can also work with on-shell quantities and find a closed expression for the
symmetry flow on the solution space (see also appendix A). This takes a very
compact form in terms of a particular combination of initial conditions. Let
us define the quantities

$\displaystyle\psi_{0}$
$\displaystyle:=\frac{n\partial_{v}(\phi_{0}A_{0})^{2}+n^{2}\phi_{0}A_{0}+B_{0}}{A_{0}n(2\phi_{0}\dot{A}_{0}+n)}\,,$
(27a) $\displaystyle Q_{0}$ $\displaystyle:=nA_{0}(2\phi_{0}\dot{A}_{0}+n)\,,$
(27b)

corresponding respectively to the zero and (minus) the first derivative of the
on-shell field ${\mathcal{B}}$. The
$\mathrm{SL}(2,{\mathbb{R}})\times\mathrm{SO}(2)$ group maps solutions for the
fields $X$ and ${\mathcal{B}}$ into solutions, and it acts non-trivially on
the solution space as

$\displaystyle A_{0}$ $\displaystyle\to\frac{\lambda
A_{0}}{\sqrt{h^{\prime}(\phi_{0})}}\,,$ (28a) $\displaystyle\phi_{0}$
$\displaystyle\to h(\phi_{0})\,,$ (28b) $\displaystyle Q_{0}$
$\displaystyle\to\frac{Q_{0}}{\lambda\sqrt{h^{\prime}(\psi_{0})}}\,,$ (28c)
$\displaystyle\psi_{0}$ $\displaystyle\to h(\psi_{0})\,,$ (28d)

for the Moebius function $h$ defined in (23). It is also useful to write the
corresponding infinitesimal transformations, both on the field and solution
spaces. We evaluate them at the same space-time point, meaning that we define
the variation of a field $\chi(r,v)$, or of a solution space parameter
$\psi(v)$ as

$\displaystyle\delta\chi$
$\displaystyle:=\tilde{\chi}(r,v)-\chi(r,v)=\tilde{\chi}(\tilde{r},\tilde{v})-\chi(r,v)-\delta
r\,\chi^{\prime}-\delta v\,\dot{\chi}\,,$ (29a) $\displaystyle\delta\psi$
$\displaystyle:=\tilde{\psi}(v)-\psi(v)=\tilde{\psi}(\tilde{v})-\psi(v)-\delta
v\,\dot{\psi}\,.$ (29b)

Let us remark that, for both fields $X$ and ${\mathcal{B}}$, we have assumed
that the finite conformal transformations (23) leave the null coordinate
invariant. We set then $\delta v=0$, and we will discuss later the residual
null reparametrization, coming from the reduction of gauge diffeomorphisms
(2). The infinitesimal generator of the Moebius transformation is given by a
second-degree polynomial [18], while $\lambda$ differs from the identity by a
small constant,

$\displaystyle h(r)$ $\displaystyle\sim
r+\epsilon(r)\,,\quad\quad\epsilon^{\prime\prime\prime}=0\,,$
$\displaystyle\lambda$ $\displaystyle\sim 1+\eta\,,\quad\quad\eta=const\,.$
(30)

This gives infinitesimal variations on the field space

$\displaystyle\delta
X=X\left(\frac{\epsilon^{\prime}}{2}+\eta\right)-\epsilon\,X^{\prime}\,,\quad\delta{\mathcal{B}}={\mathcal{B}}\left(\frac{\epsilon^{\prime}}{2}-\eta\right)-\epsilon\,{\mathcal{B}}^{\prime}\,,$
(31)

and on the solution space

$\displaystyle\delta A_{0}$
$\displaystyle=A_{0}\left(\xi-\frac{\epsilon^{\prime}(\phi_{0})}{2}\right)\,,$
(32a) $\displaystyle\delta\phi_{0}$ $\displaystyle=\epsilon(\phi_{0})\,,$
(32b) $\displaystyle\delta Q_{0}$
$\displaystyle=-Q_{0}\left(\xi+\frac{\epsilon^{\prime}(\psi_{0})}{2}\right)\,,$
(32c) $\displaystyle\delta\psi_{0}$ $\displaystyle=\epsilon(\psi_{0})\,.$
(32d)

We refer to appendix A for the transformation laws for $B_{0}$ and $n$.

On top of the conformal symmetries (23) there is also another, already known,
set of transformations that leave invariant the linear Bondi gauge in two-
dimensional dilatonic gravity. This is given by the residual diffeomorphisms
(2), after imposing the linear radial gauge. A three-parameter family vector
field generates them [33],

$\displaystyle\Xi={\mathcal{E}}(v)\partial_{v}+\left(X(v)r+H(v)\right)\partial_{r}\,.$
(33)

We use capital letters to distinguish these transformations from the previous
ones. The action of these diffeomorphisms can be equivalently thought of as
acting on the four-dimensional ansatz (1), or on the two-dimensional plane
(14), with $X$ as a scalar field. Under these, the solution space transforms
as

$\displaystyle\Delta A_{0}$ $\displaystyle=XA_{0}+{\mathcal{E}}\dot{A}_{0}\,,$
(34a) $\displaystyle\Delta\phi_{0}$
$\displaystyle=-X\phi_{0}+{\mathcal{E}}\dot{\phi}_{0}-H\,,$ (34b)
$\displaystyle\Delta B_{0}$
$\displaystyle=2B_{0}\dot{\mathcal{E}}+{\mathcal{E}}\dot{B}_{0}\,,$ (34c)
$\displaystyle\Delta n$
$\displaystyle=n\dot{\mathcal{E}}+{\mathcal{E}}\dot{n}\,.$ (34d)

In this case, the expressions take a more compact way in terms of $B_{0}$,
$n$, with respect to $Q_{0}$ and $\psi_{0}$. We see immediately that for this
transformation the mass is conserved, transforming as a scalar $\delta
M={\mathcal{E}}\dot{M}$. This is not the case for the Moebius transformation
(68). It is possible to show that the only field-independent transformation
that belongs to both classes is the constant translation in space, given by
the condition $\epsilon(r)=-H(v)=const$.

##  Phase space and conserved charges

In order to discuss the structure of the phase space and the charges
associated with the symmetries, we will use the covariant phase space
formalism. This has been developed to deal with the role of boundary
conditions in gauge theories, highlighting the integrability of charges and
their relationship with edge modes [35, 36, 37, 38, 39, 40, 41, 42]. At the
same time, it allows us to deal with the definition of a Poisson structure,
through a symplectic structure, for theories with gauge symmetries. For an
action ${\mathcal{S}}$, functional of the fields $\chi$, the variation is

$\displaystyle\delta{\mathcal{S}}=\int_{\mathcal{M}}\rm{EoM}\,\delta\chi+\int_{\partial{\mathcal{M}}}\theta\left(\chi,\delta\chi\right)\,,$
(35)

where $\theta$ is the presymplectic potential. It vanishes if we hold fixed
some boundary condition on the hypersurface $\partial{\mathcal{M}}$, to have a
well-defined variational principle. In principle, it might also contain a term
of co-dimension two, representing the so-called edge, or corner, modes [6, 5,
4, 8, 43, 44, 45, 7, 9, 10]. This contributes to the ambiguity of the
definition of the symplectic potential, together with a possible total
variation, coming from a boundary Lagrangian and changing the polarisation of
the phase space.

Let us start with the first order action (4). We recall that our field theory
is defined in a null stripe between ${v_{\text{i}}}$ and ${v_{\text{f}}}$
bounded by two surfaces at constant radius ${r_{\text{i}}}$ and
${r_{\text{f}}}$, which, in principle, might be either inside or outside the
black hole horizon. The gauge choice allows us to solve the equations of
motion explicitly in the radial direction, which makes it the natural
evolution parameter (in a Hamiltonian sense) for our theory. This makes us
identify the slice at ${r_{\text{i}}}$ as the Cauchy slice on which we shall
set the initial conditions for the variational problem, which then evolves
through the other slices $\Sigma(r)$ at a constant radius. The null
disconnected boundaries at ${v_{\text{i}}}$ and ${v_{\text{f}}}$ are then
collectively denoted by $\Gamma$ (see Fig. 1). We can decompose the boundary
of the support for the field theory as
$\partial{\mathcal{M}}=\Sigma({r_{\text{i}}})\cup\Sigma({r_{\text{f}}})\cup\Gamma$.

Figure 1: Schematic representation of the Penrose diagram of Vaidya spacetime.
We allow here for simplicity $M$ to vary only in the region between
${v_{\text{i}}}$ and ${v_{\text{f}}}$, starting from the Minkowski vacuum
$M({v_{\text{i}}})=0$, and settling down to a static black hole after
${v_{\text{f}}}$. The lines at constant radius are drawn in red. While we have
the apparent horizon at $X=2M$, that coincides with the event horizon only in
the static patch. The null boundary $\Gamma$ is then represented by the two
surfaces at ${v_{\text{i}}}$ and ${v_{\text{f}}}$, while the Cauchy slices are
at constant $r$. Let us remark that this is just a pictorial representation
because in principle the slices at constant $r$ do not coincide with the ones
at constant $X$, unless $A_{0}$ and $\phi_{0}$ are constants. Our model is
well defined in the bulk regardless of the signature of the slices $\Sigma$.

The presymplectic potential has two different components, coming from the
variation along the radial and null coordinates, these are respectively:

$\displaystyle\theta^{r}$ $\displaystyle=\frac{\delta BX^{\prime}+\delta
X(B^{\prime}+2\partial_{v}(NX))}{2N\ell_{\text{Pl}}^{2}}\,,$ (36a)
$\displaystyle\theta^{v}$ $\displaystyle=\frac{(X\delta N+N\delta
X)X^{\prime}}{N\ell_{\text{Pl}}^{2}}=\frac{X^{\prime}}{N\ell_{\text{Pl}}^{2}}\delta(XN)\,.$
(36b)

The first is integrated over the slices $\Sigma$, while the second one lives
on the boundary $\Gamma$. Once going on-shell of the bulk equations of motion,
we can explicitly carry the integration over $r$ and project the whole
symplectic potential on a slice $\Sigma$

$\displaystyle\Theta$
$\displaystyle:=\int\mathrm{d}v\left[\theta^{r}+\int\mathrm{d}r\,\partial_{v}\theta^{v}\right]$
$\displaystyle\approx\frac{1}{2\ell_{\text{Pl}}^{2}}\int\mathrm{d}v\left[-\frac{\delta
B_{0}}{n}+\frac{\dot{n}\delta(A_{0}^{2}\phi_{0}^{2})}{n}-\frac{\partial_{v}(A_{0}^{2}\phi_{0}^{2})\delta
n}{n}+\delta Y\right]\,,$ (37) $\displaystyle\text{with}\;\;Y$
$\displaystyle=r^{2}\frac{A_{0}^{2}\dot{n}}{n}+2rA_{0}\left(n-\frac{A_{0}\phi_{0}\dot{n}}{n}\right)-2A_{0}n\phi_{0}-\partial_{v}\left(A_{0}^{2}\phi_{0}^{2}\right)\,.$

We immediately see that we can renormalize the presymplectic potential, to
make it independent of the radius, by eliminating the total variation $Y$,
which in any case does not play a role for the symplectic structure
$\Omega=\delta\Theta$. The latter turns out to be always conserved along the
radial direction, without the need to impose extra boundary conditions on
$\Gamma$. By eliminating $Y$, we also recover a well-defined variational
principle for the boundary condition (on the Cauchy slice) $\delta B_{0}=0$,
$\delta n=0$, $\delta(A_{0}\phi_{0})=0$. The renormalized symplectic form is
at the end of the day

$\displaystyle\Omega_{0}$
$\displaystyle\approx\int_{v_{\text{i}}}^{v_{\text{f}}}\mathrm{d}v\left[\frac{\delta
B_{0}\curlywedge\delta
n}{2\ell_{\text{Pl}}^{2}n^{2}}+\partial_{v}\left(\frac{\delta(A_{0}^{2}\phi_{0}^{2})\curlywedge\delta
n}{2\ell_{\text{Pl}}^{2}n}\right)\right]$ (38)
$\displaystyle=\left.\frac{\delta(A_{0}^{2}\phi_{0}^{2})\curlywedge\delta
n}{2\ell_{\text{Pl}}^{2}n}\right|_{v_{\text{i}}}^{v_{\text{f}}}+\int_{v_{\text{i}}}^{v_{\text{f}}}\mathrm{d}v\,\delta
M\curlywedge\delta n\,,$

with the field space wedge product $\curlywedge$. We recognise the first part
to be a corner term, evaluated on two points in the two-dimensional picture or
two opposite homogenous celestial spheres in the four-dimensional point of
view. To lighten the notation we will drop the ${v_{\text{i}}}$ and
${v_{\text{f}}}$ from the formulas. The subscript 0 refers to the fact that we
have started from the action ${\mathcal{S}}_{0}$, adding boundary a Lagrangian
can change the corner term. We see that, as should be expected, the only bulk
degrees of freedom are the mass and its conjugate momentum which is the null
coordinate in the Eddington–Finkelstein gauge (see equation (7)). To make this
clearer, let us consider the bulk symplectic potential

$\displaystyle\Theta_{0}:=\int(n\delta M)\mathrm{d}v=\int(\delta
M)\mathrm{d}V\,,\quad\quad\mathrm{d}V=n\mathrm{d}v$ (39)

and assume an infalling thin shell, corresponding to a step function mass
$M(V)=M\Theta_{H}(V-V_{0})$, for some insertion time of the shell $V_{0}$.
Then we can explicitly integrate over the Cauchy slice and get

$\displaystyle\Theta_{0}:=\int\mathrm{d}V\left[\Theta_{H}(V-V_{0})\delta
M-\delta(V-V_{0})M\delta V_{0}\right]=V_{\text{f}}\,\delta
M-\delta(V_{0}M)\,,$ (40)

assuming $V_{\text{i}}<V_{0}<V_{\text{f}}$. Discarding the total variation
$\delta(V_{0}M)$, we see that the conjugate variable to the mass is the null
coordinate of the boundary $V_{\text{f}}$.

As already pointed out, the corner term in the symplectic current depends on
the boundary Lagrangian that we choose. For example, if we consider the
Einstein–Hilbert action (3) including the boundary terms, we get

$\displaystyle\Omega_{\text{EH}}\approx\frac{\delta(A_{0}^{2}\phi_{0}^{2})\curlywedge\delta
n}{4\ell_{\text{Pl}}^{2}n}+\frac{A_{0}\phi_{0}\,\delta
A_{0}\curlywedge\delta\phi_{0}}{2\ell_{\text{Pl}}^{2}}+\int\mathrm{d}v\,\delta
M\curlywedge\delta n\,.$ (41)

Let us assume the general case (that might correspond to different boundary
conditions on the Cauchy slice, e.g Dirichlet, Neumann or mixed)

$\displaystyle\Omega\approx\kappa_{1}\frac{\delta(A_{0}^{2}\phi_{0}^{2})\curlywedge\delta
n}{4\ell_{\text{Pl}}^{2}n}+\kappa_{2}\frac{A_{0}\phi_{0}\,\delta
A_{0}\curlywedge\delta\phi_{0}}{2\ell_{\text{Pl}}^{2}}+\int\mathrm{d}v\,\delta
M\curlywedge\delta n:=\omega_{\text{c}}+\int\mathrm{d}v\,\omega_{0}\,,$ (42)

with the codimension-2 term $\omega_{\text{c}}$, relevant to discuss the
integrability of large diffeomorphisms, and the codimension-1 term
$\omega_{0}$, capturing the bulk physical degrees of freedom. The presence of
local degrees of freedom, represented by the bulk term is a consequence of the
bad gauge fixing provided by the ansatz (1). The missing mass conservation in
the equations of motion, hiding some matter contribution, is translated on the
phase space as the seeming emergence of local degrees of freedom from the
gravitational action alone.

With the phase space at hand, we can discuss the realisation of the symmetries
(23) and (34) on it, the corresponding integrability of charges and their
algebra. We will begin with the study of the residual diffeomorphisms.

###  III.1 Integrability of the residual diffeomorphisms

Along the lines of [33], we will discuss the integrability of the residual
diffeomorphisms when they act non-trivially on the boundary $\Gamma$, or
equivalently on the corner part of the symplectic current. For this, we shall
contract the symplectic form $\Omega$ with the residual spacetime
diffeomorphisms (34).

From the codimension-1 term, we immediately see that we can hope to make them
integrable only on the solutions with constant mass (non-radiative). Indeed we
have

$\displaystyle\Delta_{\Xi}\cdot\omega_{0}={\mathcal{E}}\dot{M}\delta
n-\partial_{v}({\mathcal{E}}n)\delta M=\not{\delta}Q\,.$ (43)

As we stressed before, this is a consequence of ignoring the matter degrees of
freedom responsible for the collapse. We leave the problem of including them
in the analysis for future works.

Setting $\dot{M}=0$, we can search for a change of slicing on the phase space
that makes the charges integrable. This is also called the Pfaff problem [46,
47, 48, 49, 50, 51, 33] and amounts to finding a field-dependent choice for
the diffeomorphisms parameter (34), such that the variation (43) is exact. In
general, it is expected that for non-radiative phase spaces (without local
degrees of freedom passing through the boundary) such a problem has an
infinite number of solutions.

For this, let us take666Imposing this condition will be denoted by the symbol
$\hat{=}$, as it can be thought of as a non-radiative boundary condition on
$\Gamma$ $\partial_{v}M\,\hat{=}\,0$. We can easily find a field-dependent
parameter that makes the bulk piece integrable. Let us take
${\mathcal{E}}=\widetilde{\mathcal{E}}/n$

$\displaystyle\Delta_{\widetilde{\Xi}}\cdot\omega_{0}\,\hat{=}\,-\partial_{v}\widetilde{\mathcal{E}}\delta
M\,,$ (44)

that gives the mass aspect as the charge associated with reparametrization of
the null direction. As a side effect we also see that for constant mass
solutions, the last expression in the equation above is turned to a corner
term.

Using the redefinition of ${\mathcal{E}}$ to study the corner part of the
symplectic current, we get

$\displaystyle\Delta_{\widetilde{\Xi}}\cdot\omega_{\text{c}}=$
$\displaystyle\,\widetilde{\mathcal{E}}\left(\frac{\kappa_{1}}{4n^{2}\ell_{\text{Pl}}^{2}}\partial_{v}(A_{0}^{2}\phi_{0}^{2})\delta
n-\kappa_{2}\frac{\partial_{v}\phi_{0}^{2}\delta
A_{0}^{2}-\partial_{v}A_{0}^{2}\delta\phi_{0}^{2}}{8n\ell_{\text{Pl}}^{2}}\right)+\frac{\delta(A_{0}^{2}\phi_{0}^{2})}{4\ell_{\text{Pl}}^{2}}\left(\kappa_{2}X-\frac{\kappa_{1}}{n}\dot{\widetilde{\mathcal{E}}}\right)$
$\displaystyle-H\left(\frac{\kappa_{1}}{2\ell_{\text{Pl}}^{2}n}A_{0}^{2}\phi_{0}\delta
n-\frac{\kappa_{2}}{4\ell_{\text{Pl}}^{2}}\phi_{0}\delta A_{0}^{2}\right)\,.$
(45)

We can make this a $\delta$-exact form, by taking the field-dependent
transformations

$\displaystyle{\mathcal{E}}$
$\displaystyle=\frac{\widetilde{\mathcal{E}}}{n}\,,$ (46a) $\displaystyle X$
$\displaystyle=\widetilde{X}\frac{A_{0}^{2}}{\phi_{0}^{2}}\left(\frac{A_{0}^{2}\phi_{0}}{n^{\kappa_{1}/\kappa_{2}}}\right)^{\alpha}+\widetilde{H}\frac{A_{0}^{2}}{\phi_{0}^{2}}\left(\frac{A_{0}^{2}\phi_{0}}{n^{\kappa_{1}/\kappa_{2}}}\right)^{\beta}-\frac{\widetilde{\mathcal{E}}}{nA_{0}}\dot{A}_{0}+\frac{\kappa_{1}}{\kappa_{2}n}\dot{\widetilde{\mathcal{E}}}\,,$
(46b) $\displaystyle H$
$\displaystyle=\widetilde{H}\frac{A_{0}^{2}}{\phi_{0}}\left(\frac{A_{0}^{2}\phi_{0}}{n^{\kappa_{1}/\kappa_{2}}}\right)^{\beta}+\widetilde{X}\frac{A_{0}^{2}}{\phi_{0}}\left(\frac{A_{0}^{2}\phi_{0}}{n^{\kappa_{1}/\kappa_{2}}}\right)^{\alpha}+\frac{\widetilde{\mathcal{E}}}{nA_{0}}\partial_{v}(A_{0}\phi_{0})\,,$
(46c)

for some real numbers $\alpha$, $\beta$. With this choice, on constant mass
solutions, we get the codimension-2 charges

$\displaystyle\Delta_{\widetilde{\Xi}}\cdot\Omega\,\hat{=}\,\delta\left(\frac{\kappa_{2}}{2\ell_{\text{Pl}}^{2}}\frac{\widetilde{X}}{\alpha}\left(\frac{A_{0}^{2}\phi_{0}}{n^{\kappa_{1}/\kappa_{2}}}\right)^{\alpha}+\frac{\kappa_{2}}{2\ell_{\text{Pl}}^{2}}\frac{\widetilde{H}}{\beta}\left(\frac{A_{0}^{2}\phi_{0}}{n^{\kappa_{1}/\kappa_{2}}}\right)^{\beta}-\widetilde{\mathcal{E}}M\right)\,.$
(47)

The value of the charges seems to depend both on the choice of boundary
condition (i.e. the $\kappa$’s) and the particular solution of the Pfaff
problem that we choose (i.e. $\alpha$ and $\beta$). However, the charge
algebra turns out to be independent of these choices and it is the abelian
algebra

$\displaystyle\\{Q[\widetilde{\Xi}_{1}],Q[\widetilde{\Xi}_{2}]\\}=\Delta_{\widetilde{\Xi}_{1}}\cdot\Delta_{\widetilde{\Xi}_{2}}\cdot\Omega=0\,.$
(48)

This is consistent with the results in [33], except for the missing central
charge in our case. This difference can have its origin in the different gauge
fixing choices. We recall that while here we have fixed $\partial_{r}N=0$, the
choice in [33] and the usual literature about 2d gravity is instead
$\partial_{r}XN)=0$. The different gauge fixing can be interpreted as two
different reference frames, and thus two different observers [9, 10]. Thus, it
is not surprising that the algebras are different as different observers are
measuring different physical quantities. A more refined analysis taking into
account different gauge fixing and the mapping between them as a change of
reference frame is needed to further comment on the comparison with previous
results. This is beyond the scope of the present work and we postpone such
questions to future works. However, we would like to stress the independence
of the abelian algebra on the particular choice of boundary conditions. The
latter corresponds to different presymplectic boundary potentials, it is known
that the numerical value of the charge can depend on the choice of boundary
conditions [52, 53, 54], without affecting their algebra. On the quantum
level, this is mapped to the choice of different irreducible representations
of the same algebra.

We would like to end this section with a small remark concerning the edge
modes literature. An alternative way of making the charges integrable is to
add some edge modes fields, living on the boundary $\Gamma$, that can be
interpreted as the image of the residual degrees of freedom leaving in the
complementary region outside $\Gamma$. We chose here the approach in [33] of
slice changing, i.e. considering field-dependent diffeomorphisms, avoiding
going too deeply into the construction of reference frames or edge modes [9,
10, 55]. We expect however the two approaches to be related, as we can usually
interpret a field-dependent diffeomorphism as a change of reference frame [9,
10].

###  III.2 Conformal transformation and alternative action

For the Moebius transformation (23), we lack such freedom of redefining the
transformation parameter in a field-dependent way. The finite-dimensional
group $\mathrm{SL}(2,{\mathbb{R}})\times\mathrm{SO}(2)$ does not allow to take
the coefficients of $\epsilon$ and $\xi$ in (32) to vary along the null
direction. For this, we cannot aim for a change of slicing to make the charges
integrable and unfortunately, contracting the infinitesimal transformations
(32) into the symplectic current $\omega$, we find non-integrable quantities.
We refer to the appendix A for the full (lengthy) expression of
$\not{\delta}Q$, from which the takeaway message is the non-integrability of
the transformation (23).

This, however, is not too surprising. At some heuristic level, we can see that
the Moebius transformations (32) and the residual diffeomorphisms (34) look
very different. The former is more easily described by the pair of initial
conditions $Q_{0}$ and $\psi_{0}$, whose mechanical interpretation is
straightforward in terms of initial value and velocity of the field
${\mathcal{B}}$, while their spacetime interpretation is more vague.
Conversely, $B_{0}$ and $n$ are nicer geometric quantities, related to the
mass and the shell insertion, but their expression in terms of dynamical
quantities is more involved.

More rigorously, this contrast between the two sets of transformations is
manifest in the non-covariance of the Moebius transformation, meaning that
$\delta_{\\{\epsilon,\xi\\}}$ and $\Delta_{\Xi}$ do not form a field
independent closed Lie algebra. Although the Moebius transformation (23) is a
symmetry of the gauge fixed equations of motion
$\delta_{\\{\epsilon,\xi\\}}\text{EOM}\approx 0$, it is not a symmetry of the
reduced Lagrangian in the Noether sense. This is not something completely
uncommon in physics, even for the very simple model of a free particle, we
know that, in general, the conformal rescaling of the position is a symmetry
of the equation of motion, but it corresponds to a rescaling of the
Lagrangian, not to a total derivative, as required by the Noether theorem.

Nonetheless, we can still associate with the Moebius transformation some
conserved quantities along the radial direction. For this, let us consider, on
the gauge fixed field space, the following functional

$\displaystyle{\mathcal{S}}_{\text{mob}}=\int\mathrm{d}r\mathrm{d}v\left[{\mathcal{B}}^{\prime}X^{\prime}\right]\,.$
(49)

The corresponding Euler–Lagrange equations are still (17a) and (17b), the same
as for the reduced action coming from general relativity. But, in this case,
we lose the constraint (17c), which however can be obtained from the other
two. Indeed assuming ${\mathcal{B}}^{\prime\prime}\approx 0\approx
X^{\prime\prime}$ we trivially get the conservation along the radial direction
of the quantity ${\mathcal{B}}^{\prime}X^{\prime}$, that we can then define as
${\mathcal{N}}(v)^{2}$, mimicking the last equation of (17). From the point of
view of the new action functional, there is no gauge freedom, or redundancy in
the equation of motion. Equivalently, we can see this by the non-covariance of
the functional (49) under the residual diffeomorphisms generated by (33).

From the variational principle of the action (49) we get

$\displaystyle\delta{\mathcal{S}}_{\text{mob}}=-\int\mathrm{d}r\mathrm{d}v\left[{\mathcal{B}}^{\prime\prime}\delta
X+X^{\prime\prime}\delta{\mathcal{B}}\right]+\int\mathrm{d}v\left[{\mathcal{B}}^{\prime}\delta
X+X^{\prime}\delta{\mathcal{B}}\right]\,,$ (50)

and thus the symplectic form

$\displaystyle\Omega_{\text{mob}}$
$\displaystyle=\int\mathrm{d}v\left[\delta{\mathcal{B}}^{\prime}\curlywedge\delta
X+\delta X^{\prime}\curlywedge\delta{\mathcal{B}}\right]$ (51)
$\displaystyle\approx\int\mathrm{d}v\left[\delta
Q_{0}\curlywedge\delta(\phi_{0}A_{0})+\delta
A_{0}\curlywedge(\psi_{0}Q_{0})\right]\,.$ (52)

Contracting this with the Moebius transformations (32) we get the integrable
charges

$\displaystyle\delta_{\\{\epsilon,\xi\\}}\cdot\Omega_{\text{mob}}$
$\displaystyle\approx\int\mathrm{d}v\,\delta\left[\xi
Q_{0}A_{0}(\psi_{0}-\phi_{0})-A_{0}Q_{0}\left(\epsilon_{0}+\frac{\epsilon_{1}}{2}(\phi_{0}+\psi_{0})+\epsilon_{2}\psi_{0}\phi_{0}\right)\right]$
(53) $\displaystyle=\int\mathrm{d}v\,\delta\left[\xi
Q_{0}A_{0}(\psi_{0}-\phi_{0})-\frac{A_{0}Q_{0}}{2}\left(\epsilon(\phi_{0})+\epsilon(\psi_{0})-(\psi_{0}-\phi_{0})^{2}\frac{\epsilon^{\prime\prime}}{2}\right)\right]:=\delta
Q[\epsilon,\xi]\,,$

for $\epsilon(r):=\epsilon_{0}+\epsilon_{1}r+\epsilon_{2}r^{2}$. Their algebra
reproduces the Lie algebra of infinitesimal transformations

$\displaystyle\\{Q[\epsilon_{1},\xi_{1}],Q[\epsilon_{2},\xi_{2}]\\}=Q[\epsilon_{1}\epsilon_{2}^{\prime}-\epsilon_{2}\epsilon_{1}^{\prime},\,\xi=0]\,.$
(54)

Contrary to the covariant description, in this case, we don’t have any edge
modes or corner charges, the charges living on the codimension-1 Cauchy slice
$\Sigma(r)$. The action (49) looks like a mechanical action, with a kinetic
term quadratic in the radial derivatives, and there is no presence of the null
coordinate, except for the integration interval. In other words, the
redefinition of the fields, introducing ${\mathcal{B}}$, formally maps the
Vaidya superspace into an infinite set of decoupled mechanical models
isomorphic to the static case, labelled by the null coordinate $v$. The side
effect of this construction is the loss of manifest covariance, which can be
seen either as the non-covariance of the action (49) under residual
diffeomorphisms (33) or as the impossibility of making the gauge
transformations (34) integrable on the symplectic structure
$\Omega_{\text{mob}}$.

Regardless of the non-covariance, we can associate conserved quantities with
the Moebius symmetry, in the usual Noether sense. For this, we should work
with the presymplectic potential and infinitesimal transformations on the
fields ${\mathcal{B}}$ and $X$ as in (31)

$\displaystyle\delta_{\\{\epsilon,\xi\\}}\Omega_{\text{mob}}$
$\displaystyle\approx\int\mathrm{d}v\,\delta\left[\eta(X^{\prime}{\mathcal{B}}-{\mathcal{B}}^{\prime}X)+\epsilon(X^{\prime}{\mathcal{B}}^{\prime})+\epsilon^{\prime}(X{\mathcal{B}})^{\prime}+\frac{\epsilon^{\prime\prime}}{2}X{\mathcal{B}}\right]$
$\displaystyle=\int\mathrm{d}v\,\delta\left[\eta(X^{\prime}{\mathcal{B}}-{\mathcal{B}}^{\prime}X)+\frac{1}{2}(\epsilon
X{\mathcal{B}})^{\prime\prime}-\frac{3}{2}\epsilon^{\prime}(X{\mathcal{B}})^{\prime}\right]:=\delta
Q\,.$ (55)

These are conserved along the radial direction and agree with the on-shell
charges given in (53). Moreover, the interpretation as an infinite set of
mechanical models for each null cut is consistent with the fact that the
current inside the integral is conserved along the radial direction, even
without integrating along the whole Cauchy slice. For each point of
$\Sigma(r)$ we can define

$\displaystyle
j:=\eta(X^{\prime}{\mathcal{B}}-{\mathcal{B}}^{\prime}X)+\frac{1}{2}(\epsilon
X{\mathcal{B}})^{\prime\prime}-\frac{3}{2}\epsilon^{\prime}(X{\mathcal{B}})^{\prime}\,.$
(56)

and we have $j^{\prime}\approx 0$, corresponding to the Noether charge
associated with the Moebius symmetry as in the mechanical setup [20, 19].

###  III.3 Heisenberg extension and full Schrödinger symmetry

We can extend the Moebius transformation, mimicking the construction in [20,
19]. The linearity of the solutions for ${\mathcal{B}}$ and $X$, makes them
trivially invariant under the transformations

$\displaystyle X(r,v)\ $ $\displaystyle\rightarrow\
\widetilde{X}(r,v)=X(r,v)+p_{2}(v)r+q_{1}(v)\,,$ (57a)
$\displaystyle{\mathcal{B}}(r,v)\ $ $\displaystyle\rightarrow\
\widetilde{{\mathcal{B}}}(r,v)={\mathcal{B}}(r,v)+p_{1}(v)r+q_{2}(v)\,,$ (57b)

They correspond to an abelian symmetry of the equations of motion, for which
the infinitesimal and finite transformations coincide. The corresponding
action on the initial conditions is

$\displaystyle\delta A_{0}$ $\displaystyle=p_{2}\,,$ (58a)
$\displaystyle\delta\phi_{0}$
$\displaystyle=-\frac{q_{1}+\phi_{0}p_{0}}{A_{0}}\,,$ (58b)
$\displaystyle\delta Q_{0}$ $\displaystyle=-p_{1}\,,$ (58c)
$\displaystyle\delta\psi_{0}$
$\displaystyle=\frac{q_{2}+\phi_{0}p_{2}}{Q_{0}}\,.$ (58d)

Using this with the symplectic structure $\Omega_{\text{mob}}$ also gives an
infinite tower of conserved quantities

$\displaystyle\delta_{\\{\epsilon,\xi\\}}\cdot\Omega_{\text{mob}}$
$\displaystyle\approx\int\mathrm{d}v\,\delta\left[-q_{1}{\mathcal{B}}^{\prime}-q_{2}X^{\prime}+p_{1}(X-rX^{\prime})+p_{2}({\mathcal{B}}-r{\mathcal{B}}^{\prime})\right]$
(59)
$\displaystyle\approx\int\mathrm{d}v\,\delta\left[p_{2}\psi_{0}Q_{0}-p_{1}A_{0}\phi_{0}+q_{1}Q_{0}-q_{2}A_{0}\right]:=\delta
Q[p_{i},q_{i}]\,,$

providing a centrally extended algebra

$\displaystyle\\{Q[p_{i}^{(1)},q_{i}^{(1)}],Q[p_{i}^{(2)},q_{i}^{(2)}]\\}=\int
p_{i}^{(I)}q_{i}^{(J)}\epsilon_{IJ}\,\mathrm{d}v\,,$ (60)

with the totally antisymmetric symbol $\epsilon_{IJ}$. As for the Moebius
symmetry, we can define here some conserved current on each slice at constant
$v$. This means that the centrally extended part in the Schrödinger algebra,
which is finite-dimensional, is promoted to an infinite dimensional set in the
Vaidya model. This can be understood from the mechanical point of view because
we can interpret the action (49)as an infinite set of decoupled mechanical
models, one at each null cut $v=const$. We can arbitrarily deform the initial
conditions on the slice $\Sigma$ at any point $v$, and the charges
$Q[p_{i},q_{i}]$, measures precisely these initial conditions, namely the
initial value of the fields ${\mathcal{B}}$, $X$ and their velocity. The full
algebra is thus

$\displaystyle\left({\mathfrak{sl}}(2,{\mathbb{R}})\oplus{\mathfrak{so}}(2)\right)\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}}\left(C^{\infty}({\mathbb{R}}^{2})\oplus_{c}C^{\infty}({\mathbb{R}}^{2})\right)\,.$
(61)

As in the static case, this is an overcomplete set of charges on the phase
space. Although the initial conditions are an infinite set of numbers, they
can be represented as four (continuous) functions of the null direction. So
the charges $Q[p_{i},q_{i}]$ are sufficient to specify the initial value
problem and integrate the motion along the radial direction, by exponentiating
the charge corresponding to $\epsilon(r)=1$, generator of constant radial
translations. It is indeed possible to show that, as in the static setup, here
we can obtain the Moebius charges from quadratic combinations of the linear
charges $Q[p_{i},q_{i}]$, for example, we have

$\displaystyle
Q[\epsilon=1]=Q[q_{1}=q(v),q_{2}=0,p_{i}=0]\,Q[q_{1}=0,q_{2}=1/q(v),p_{i}=0]\,,$
(62)

and similar for $\epsilon=r$, $\epsilon=r^{2}$.

We would like to stress that all along these sections, we have completely
neglected the boundary conditions at the null boundary $\Gamma$. However,
restricting to some specific conditions on $\Gamma$ can only affect the fall-
off conditions of the initial conditions. Classically, because of the freedom
of the dynamics in the null direction, in the bulk, we are always free to set
$\\{A_{0},Q_{0},\psi_{0},\phi_{0}\\}$ as we desire. The only way of
constraining the bulk dynamics is to specify some profile for the infalling
null dust, that goes beyond the scope of this work. However, we should remark
that once we want to quantize the theory, the fall-off conditions become very
relevant, as they affect the structure of the Hilbert space, and consequently
the spectrum of some operators. We can imagine that different representations
of the symmetry group might correspond to different choices of boundary
conditions on $\Gamma$, exploring this direction can give an interesting
further development of this work.

## Discussion

In this article, we have discussed the symmetries of the Vaidya superspace. We
have shown that the evolution of a general Vaidya spacetime in the radial
direction can be obtained from two different action principles. The first one
we have discussed (4) possesses manifestly some residual gauge invariance,
leading to codimension-two charges on the corner of the Cauchy slice $\Sigma$.
The phase space for this action contains just the mass and the insertion time
for a null shell as conjugate variables. All the other degrees of freedom are
relevant only at the boundary, playing the role of edge modes fields. We have
shown that in order to make the corresponding charges integrable, we need to
consider field-dependent gauge transformation. Such a choice is not unique,
but it always leads to the same abelian charge algebra (48).

On top of the residual gauge freedom, we have shown that the partial gauge
fixed equations of motion (22) are invariant under a conformal
reparametrization of the radius (23), and an infinite set of linear
transformations (57). These correspond to non-integrable charges on the
gravitational reduced phase space, due to the intrinsic non-covariance nature
of the transformations. However, we have shown the existence of an alternative
action functional (49) leading to a different phase space, where the extended
Schrödinger symmetries are integrable. The price to pay is unfortunately the
loss of covariance of the theory, which now looks like an infinite set of
decoupled mechanical models.

Although both sets of transformations map solutions of the equations of motion
in the linear gauge into solutions, they are associated with two different
phase spaces. The difference between the two sets is manifest also in the fact
that their infinitesimal version separately provides Lie algebras, but
together they do not close into a bigger algebra. The commutator of a
diffeomorphism and a Moebius transformation gives a new transformation, or
equivalently we can say that the structure constants of the two algebras
together are field-dependent.

The difference in the phase space is almost irrelevant on the classical level,
because the classical equations of motion are the same for the two actions
${\mathcal{S}}_{\text{EH}}$ and ${\mathcal{S}}_{\text{mob}}$, but it becomes
crucial once we quantize the theory. A measurement involving quantum processes
could then in principle distinguish between the two models.

For the Moebius action, the physical degrees of freedom are more numerous than
in the general relativity phase space, which is limited to the mass and its
conjugated time. The rest of the initial conditions get a physical meaning
only on the boundary, as edge modes for the gauge transformation. Conversely
in the mechanical setup, they are all already physical in the bulk.

We would like to remark that this discussion does not represent a no-go
statement for a generalisation of these conformal symmetries in the full
theory, but points out a crucial difference with the boundary large
diffeomorphisms, on the contrary to what has been conjectured in previous
works [18, 24]. A more refined analysis including reference frames and
(matter) observers could shed light on the origin of the two actions and the
physical process underlying this difference.

We stress again that, despite not being integrable on the covariant phase
space, the Schrödinger symmetries are truly symmetries of the dynamics of
Vaidya superspace, mapping solution into solutions and being associated with
conserved quantities along the radial direction. As such, they play a role in
perturbation theory around the Vaidya background. In the static framework, the
conformal reparametrization of the radius is related to the vanishing of Love
numbers [56], i.e. to the response of black holes to tidal perturbations. The
presence of such symmetry in the Vaidya model opens interesting perspectives
on the study of its perturbations and the radiative/absorption processes of
black holes.

###### Acknowledgements.

FS is grateful for the hospitality of the association Esprit des Lieux in
Saint-Léger du Ventoux, France, where the initial stages of this work were
carried out. The authors would like to thank Marc Geiller, Alejandro Perez and
Jibril Ben Achour for the precious discussions and comments. This project was
made possible through the support of the ID# 62312 grant from the John
Templeton Foundation, as part of the ‘The Quantum Information Structure of
Spacetime’ Project (QISS). The opinions expressed in this project/publication
are those of the author(s) and do not necessarily reflect the views of the
John Templeton Foundation.

##  Appendix A Action of the symmetry on the solution space

Let us consider the symmetry transformations as defined in the main text in
(23). These map solutions of the gauge fixed equations (22a) and (22b) into
solutions. We also recall that the on-shell expressions for the corresponding
dynamical fields are of course linear,

$\displaystyle X\approx A_{0}(r-\phi_{0})\,,\quad{\mathcal{B}}\approx-
Q_{0}(r-\psi_{0})\,.$ (63)

Combining this with the symmetry transformation we get the on-shell version of
the transformed field, let’s take e.g. $X$,

$\displaystyle\widetilde{X}\approx\lambda
A_{0}\frac{r-\phi_{0}}{\gamma\,r+\delta}=\frac{\lambda
A_{0}}{\gamma\phi_{0}+\delta}\left(\frac{\alpha\,r+\beta}{\gamma\,r+\delta}-\frac{\alpha\,\phi_{0}+\beta}{\gamma\,\phi_{0}+\delta}\right)=\lambda\frac{A_{0}}{\sqrt{h^{\prime}(\phi_{0})}}(h(r)-h(\phi_{0}))\,,$
(64)

from which we easily read the transformation law for $A_{0}$ and $\phi_{0}$
presented in the main text’s equation (28). The exact same calculation for
${\mathcal{B}}$ gives us the transformation law for the other two initial
conditions $Q_{0}$ and $\psi_{0}$.

Concerning the off-shell transformation law for $N$, this must be such that
the constraint (17c) is preserved. As shown in the main text, working with the
rescaled field ${\mathcal{N}}$ is easier. Its value on-shell is given by

$\displaystyle{\mathcal{N}}^{2}\approx-X^{\prime}{\mathcal{B}}^{\prime}\approx
nA_{0}^{2}(n+2\phi_{0}\dot{A}_{0})=Q_{0}A_{0}\,,$ (65)

Now, the constraint (17c) transform as in (26)

$\displaystyle 0$
$\displaystyle\approx\widetilde{\mathcal{N}}^{2}+\partial_{\tilde{r}}\widetilde{X}\partial_{\tilde{r}}\widetilde{\mathcal{B}}$
(66)
$\displaystyle=\widetilde{\mathcal{N}}^{2}+\frac{1}{h^{\prime}}\left(X^{\prime}{\mathcal{B}}^{\prime}+\frac{1}{2}\partial_{r}\left(\frac{h^{\prime\prime}}{h^{\prime}}X{\mathcal{B}}\right)\right)$
$\displaystyle\approx\widetilde{\mathcal{N}}^{2}-\frac{A_{0}\left(B_{0}h^{\prime\prime}(\phi_{0})+2n^{2}A_{0}h^{\prime}(\phi_{0})+2n\phi_{0}\partial_{v}(A_{0}^{2}h^{\prime}(\phi_{0}))\right)}{2h^{\prime}(\phi_{0})^{2}}$
$\displaystyle=\widetilde{\mathcal{N}}^{2}-\frac{Q_{0}A_{0}}{\sqrt{h^{\prime}(\psi_{0})h^{\prime}(\phi_{0})}}=\widetilde{\mathcal{N}}^{2}-\widetilde{Q}_{0}\widetilde{A}_{0}\,,$

where in the last line we see that the transformations are indeed symmetries
of the constraint (65).

Inverting the definition of $Q_{0}$ and $\psi_{0}$, and using the variations
(32), we can also write the infinitesimal transformation on the alternative
version of the initial conditions $B_{0}$ and $n$:

$\displaystyle\delta B_{0}$
$\displaystyle=\frac{B_{0}\left(\phi_{0}\left(\dot{A}_{0}\left(\epsilon_{0}^{\prime}-2\xi\right)+A_{0}\epsilon_{0}^{\prime\prime}\dot{\phi}_{0}\right)+n\left(\epsilon_{0}^{\prime}-2\xi\right)\right)+2A_{0}^{2}n^{2}\dot{\phi}_{0}\left(\phi_{0}\left(\epsilon_{0}^{\prime}-4\xi\right)-2\epsilon_{0}\right)}{2(\phi_{0}\dot{A}_{0}+n)}\,,$
(67a) $\displaystyle\delta n$
$\displaystyle=-\frac{B_{0}\epsilon_{0}^{\prime\prime}+2nA_{0}\left(2\xi
n+\dot{A}_{0}\left(2\epsilon_{0}+\phi_{0}(6\xi-\epsilon_{0}^{\prime})\right)\right)}{4A_{0}(n+\phi_{0}\dot{A}_{0})}\,,$
(67b)

with the short hand notation $\epsilon_{0}=\epsilon(\phi_{0})$. Finally, let
us remark that this is not a spacetime diffeomorphism, as the mass content of
the spacetime is changed, i.e. it does not transform as a scalar field under
reparametrization of the null coordinates. We explicitly have

$\displaystyle\delta M=$
$\displaystyle\frac{A_{0}\left(\phi_{0}\left(A_{0}\dot{\phi}_{0}\left(A_{0}\left(\epsilon_{0}^{\prime}-4\xi\right)+\ell_{\text{Pl}}^{2}M\epsilon_{0}^{\prime\prime}\right)+\ell_{\text{Pl}}^{2}M\dot{A}_{0}\left(10\xi-\epsilon_{0}^{\prime}\right)\right)+\epsilon_{0}\left(4\ell_{\text{Pl}}^{2}M\dot{A}_{0}-2A_{0}^{2}\dot{\phi}_{0}\right)\right)}{2\ell_{\text{Pl}}^{2}A_{0}\left(\phi_{0}\dot{A}_{0}+n)\right)}$
$\displaystyle+\frac{Mn\left(A_{0}\left(\epsilon_{0}^{\prime}+2\xi\right)+2\ell_{\text{Pl}}^{2}M\epsilon_{0}^{\prime\prime}\right)}{2A_{0}\left(\phi_{0}\dot{A}_{0}+n)\right)}.$
(68)

Let us finally give the explicit expression for the non-integrability of the
Moebius transformation on the symplectic form $\Omega$, separating the
codimension-1 and codimension-2 terms we have

$\displaystyle\delta_{\\{\epsilon,\xi\\}}\cdot\omega_{0}=$
$\displaystyle\frac{4A_{0}n^{2}\left(A_{0}{}^{2}\delta
n\dot{\phi}_{0}\left(\phi_{0}\left(\epsilon_{0}^{\prime}-4\xi\right)-2\epsilon_{0}\right)+\xi\delta
B_{0}\right)+2A_{0}n\left(\dot{A}_{0}\delta
B_{0}\left(\phi_{0}\left(6\xi-\epsilon_{0}^{\prime}\right)+2\epsilon_{0}\right)B_{0}\delta
n\left(\epsilon_{0}^{\prime}-2\xi\right)\right)}{8\ell_{\text{Pl}}^{2}A_{0}n^{2}\left(\dot{A}_{0}\phi_{0}+n\right)}$
$\displaystyle+\frac{B_{0}\left(\epsilon_{0}^{\prime\prime}\left(2A_{0}{}^{2}\delta
n\phi_{0}\dot{\phi}_{0}+\delta B_{0}\right)+2A_{0}\dot{A}_{0}\delta
n\phi_{0}\left(\epsilon_{0}^{\prime}-2\xi\right)\right)}{8\ell_{\text{Pl}}^{2}A_{0}n^{2}\left(\dot{A}_{0}\phi_{0}+n\right)},$
(69a) $\displaystyle\delta_{\\{\epsilon,\xi\\}}\cdot\omega_{\text{c}}=$
$\displaystyle\frac{\phi_{0}\kappa_{1}}{8\ell_{\text{Pl}}^{2}n\left(\dot{A}_{0}\phi_{0}+n\right)}\left(B_{0}\delta(A_{0}\phi_{0})\epsilon_{0}^{\prime\prime}+2\dot{A}_{0}A_{0}{}^{2}\delta
n\,\phi_{0}\left(\phi_{0}\left(2\xi-\epsilon_{0}^{\prime}\right)+2\epsilon_{0}\right)\right)$
$\displaystyle+\frac{A_{0}n\phi_{0}}{4\ell_{\text{Pl}}^{2}\left(\dot{A}_{0}\phi_{0}+n\right)}\left(A_{0}\delta\phi_{0}\left(2\xi(\kappa_{1}+\kappa_{2})-\kappa_{2}\epsilon_{0}^{\prime}\right)-2\kappa_{2}\delta
A_{0}\epsilon_{0}+2\kappa_{1}\xi\delta A_{0}\phi_{0}\right)$
$\displaystyle+\frac{2A_{0}\phi_{0}^{2}}{4\ell_{\text{Pl}}^{2}\left(\dot{A}_{0}\phi_{0}+n\right)}\left(\kappa_{1}\delta
A_{0}\dot{A}_{0}\phi_{0}\left(6\xi-\epsilon_{0}^{\prime}\right)+A_{0}\left(\dot{A}_{0}\delta\phi_{0}\left(2\xi(3\kappa_{1}+\kappa_{2})-(\kappa_{1}+\kappa_{2})\epsilon_{0}^{\prime}\right)+\kappa_{1}\delta
n\left(2\xi-\epsilon_{0}^{\prime}\right)\right)\right)$
$\displaystyle+\frac{\epsilon_{0}A_{0}\phi_{0}}{2\ell_{\text{Pl}}^{2}\left(\dot{A}_{0}\phi_{0}+n\right)}\left((\kappa_{1}-\kappa_{2})\delta
A_{0}\dot{A}_{0}\phi_{0}+\kappa_{1}A_{0}\left(\dot{A}_{0}\delta\phi_{0}+\delta
n\right)\right).$ (69b)

##  Appendix B Spherically symmetric reduction and useful formulas

Let us write the spherically symmetric metric as

$\displaystyle\mathrm{d}s^{2}=g^{(4)}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=g_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}+\frac{\Phi(t,r)^{2}}{\lambda^{2}}\mathrm{d}\Omega^{2},$
(70)

where $g_{ab}$ is the 2-metric in the $(t,r)$ plane and
$[\lambda]=\text{length}^{-1}$. With this metric we get

$\displaystyle\sqrt{-g^{(4)}}=\sqrt{-g}\,\frac{\Phi^{2}}{\lambda^{2}}\sin\theta,\quad\quad
R^{(4)}=R+2\lambda^{2}\Phi^{-2}-2\Phi^{-2}(\nabla\Phi)^{2}-4\Phi^{-1}\nabla^{2}\Phi,$
(71)

and therefore the Einstein–Hilbert action becomes

$\displaystyle S$
$\displaystyle=\frac{1}{\kappa}\int_{M}\mathrm{d}^{4}x\,\sqrt{-g^{(4)}}\,R^{(4)}$
(72)
$\displaystyle=\frac{1}{\kappa\lambda^{2}}\int\mathrm{d}^{2}x\int_{0}^{\pi}\mathrm{d}\theta\,\sin\theta\int_{0}^{2\pi}\mathrm{d}\varphi\,\sqrt{-g}\,\Phi^{2}R^{(4)}$
(73)
$\displaystyle=\frac{4\pi}{\kappa\lambda^{2}}\int\mathrm{d}^{2}x\,\sqrt{-g}\,\Phi^{2}\Big{(}R+2\lambda^{2}\Phi^{-2}-2\Phi^{-2}(\nabla\Phi)^{2}-4\Phi^{-1}\nabla^{2}\Phi\Big{)}$
(74)
$\displaystyle=\frac{4\pi}{\kappa\lambda^{2}}\int\mathrm{d}^{2}x\,\sqrt{-g}\,\Big{(}\Phi^{2}R+2\lambda^{2}-2(\nabla\Phi)^{2}-4\Phi\nabla^{2}\Phi\Big{)}$
(75)
$\displaystyle=\frac{4\pi}{\kappa\lambda^{2}}\int\mathrm{d}^{2}x\,\sqrt{-g}\,\Big{(}\Phi^{2}R+2\lambda^{2}+2(\nabla\Phi)^{2}-4\nabla_{a}(\Phi\nabla^{a}\Phi)\Big{)},$
(76)

where later on we will set $\kappa=16\pi$. The action for a minimally coupled
massless scalar field $f$ is

$\displaystyle S_{\text{m}}=-\int_{M}\mathrm{d}^{4}x\,\sqrt{-g^{(4)}}\,(\nabla
f)^{2}=-\frac{2\pi}{\lambda^{2}}\int\mathrm{d}^{2}x\,\sqrt{-g}\,\Phi^{2}(\nabla
f)^{2}.$ (77)

Under variations we have

$\displaystyle\delta\Gamma^{\rho}_{\mu\nu}$
$\displaystyle=\frac{1}{2}g^{\rho\sigma}(\nabla_{\mu}\delta
g_{\sigma\nu}+\nabla_{\nu}\delta g_{\sigma\mu}-\nabla_{\sigma}\delta
g_{\mu\nu}),$ (78a) $\displaystyle\delta R_{\mu\nu}$
$\displaystyle=\nabla_{\rho}\delta\Gamma^{\rho}_{\mu\nu}-\nabla_{\nu}\delta\Gamma^{\rho}_{\mu\rho}=\frac{1}{2}(\nabla^{\rho}\nabla_{\mu}\delta
g_{\rho\nu}+\nabla^{\rho}\nabla_{\nu}\delta
g_{\mu\rho}-g^{\rho\sigma}\nabla_{\mu}\nabla_{\nu}\delta
g_{\rho\sigma}-\nabla^{2}\delta g_{\mu\nu}),$ (78b) $\displaystyle\delta R$
$\displaystyle=\delta g^{\mu\nu}R_{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}$ (78c)
$\displaystyle=\delta
g^{\mu\nu}R_{\mu\nu}+\nabla_{\mu}(g^{\rho\sigma}\delta\Gamma^{\mu}_{\rho\sigma}-g^{\rho\mu}\delta\Gamma^{\nu}_{\rho\nu})$
(78d) $\displaystyle=\delta
g^{\mu\nu}R_{\mu\nu}+\nabla^{\mu}\nabla^{\nu}(\delta
g_{\mu\nu}-g_{\mu\nu}g^{\rho\sigma}\delta g_{\rho\sigma})$ (78e)
$\displaystyle=\delta
g^{\mu\nu}R_{\mu\nu}+(\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{2})\delta
g_{\mu\nu}$ (78f) $\displaystyle=\delta
g^{\mu\nu}R_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}(g^{\mu\nu}g_{\rho\sigma}\delta
g^{\rho\sigma}-\delta g^{\mu\nu}),$ (78g) $\displaystyle\delta\sqrt{-g}$
$\displaystyle=\frac{1}{2}\sqrt{-g}\,g^{\mu\nu}\delta g_{\mu\nu}.$ (78h)

Under a conformal rescaling
$g_{\mu\nu}=\Omega^{2}\tilde{g}_{\mu\nu}=e^{2\sigma}\tilde{g}_{\mu\nu}$ we
have

$\displaystyle\sqrt{-g}$ $\displaystyle=\Omega^{2}\sqrt{-\tilde{g}},$ (79a)
$\displaystyle R$
$\displaystyle=\Omega^{-2}\big{(}\tilde{R}-2\Omega^{-1}\tilde{\nabla}^{2}\Omega+2\Omega^{-2}(\tilde{\nabla}\Omega)^{2}\big{)}=e^{-2\sigma}\big{(}\tilde{R}-2\tilde{\nabla}^{2}\sigma\big{)},$
(79b) $\displaystyle\nabla_{\mu}\phi$
$\displaystyle=\tilde{\nabla}_{\mu}\phi,$ (79c)
$\displaystyle(\nabla\phi)^{2}$
$\displaystyle=\Omega^{-2}(\tilde{\nabla}\phi)^{2},$ (79d)
$\displaystyle\nabla^{2}\phi$
$\displaystyle=\Omega^{-2}\tilde{\nabla}^{2}\phi,$ (79e)
$\displaystyle\nabla_{\mu}\nabla_{\nu}\phi$
$\displaystyle=\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\phi-\tilde{\nabla}_{\mu}\sigma\tilde{\nabla}_{\nu}\phi-\tilde{\nabla}_{\mu}\phi\tilde{\nabla}_{\nu}\sigma+\tilde{g}_{\mu\nu}\tilde{\nabla}^{\alpha}\sigma\tilde{\nabla}_{\alpha}\phi.$
(79f)

With this, for a Lagrangian of the form

$\displaystyle
L=\sqrt{-g}\,\Big{(}V(\Phi)R+U(\Phi)+W(\Phi)(\nabla\Phi)^{2}\Big{)},$ (80)

we can always remove the kinetic term for $\Phi$ by using

$\displaystyle\Omega(\Phi)=\exp\left(-\int_{\Phi_{0}}^{\Phi}\frac{W(z)}{2V^{\prime}(z)}\mathrm{d}z\right).$
(81)

## References

  * [1] T. Regge and C. Teitelboim, _Role of Surface Integrals in the Hamiltonian Formulation of General Relativity_ , _Annals Phys._ 88 (1974) 286.
  * [2] S. Carlip, _The Statistical Mechanics of the (2+1)-Dimensional Black Hole_ , _Phys. Rev. D_ 51 (1995) 632–637.
  * [3] A. P. Balachandran, L. Chandar and A. Momen, _Edge States in Canonical Gravity_ , .
  * [4] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity – I: Corner potentials and charges_ , _JHEP_ 11 (Nov., 2020) 026.
  * [5] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity – II: Corner metric and Lorentz charges_ , _JHEP_ 11 (Nov., 2020) 027.
  * [6] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity. Part III. Corner simplicity constraints_ , _JHEP_ 01 (Jan., 2021) 100.
  * [7] W. Donnelly, L. Freidel, S. F. Moosavian and A. J. Speranza, _Gravitational edge modes, coadjoint orbits, and hydrodynamics_ , _JHEP_ 09 (Sept., 2021) 008.
  * [8] L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, _Extended corner symmetry, charge bracket and Einstein’s equations_ , _JHEP_ 09 (Sept., 2021) 083.
  * [9] S. Carrozza and P. A. Hoehn, _Edge modes as reference frames and boundary actions from post-selection_ , _JHEP_ 02 (Feb., 2022) 172.
  * [10] S. Carrozza, S. Eccles and P. A. Hoehn, _Edge modes as dynamical frames: charges from post-selection in generally covariant theories_ , .
  * [11] E. T. Newman and A. I. Janis, _Note on the Kerr Spinning-Particle Metric_ , _J. Math. Phys._ 6 (1965) 915–917.
  * [12] S. P. D. P. Szekeres, _An explanation of the Newman-Janis Algorithm_ , _Gen. Rel. Grav._ 32 (2000) 445–458.
  * [13] B. Chen and J. Long, _Hidden Conformal Symmetry and Quasi-normal Modes_ , _Phys. Rev. D_ 82 (2010) 126013.
  * [14] Y.-W. Kim, Y. S. Myung and Y.-J. Park, _Quasinormal modes and hidden conformal symmetry in the Reissner-Nordstrom black hole_ , _Eur. Phys. J. C_ 73 (2013) 2440.
  * [15] D. Birmingham, I. Sachs and S. Sen, _Entropy of Three-Dimensional Black Holes in String Theory_ , _Phys. Lett. B_ 424 (1998) 275–280.
  * [16] S. Carlip, _Black Hole Entropy from Conformal Field Theory in Any Dimension_ , _Phys. Rev. Lett._ 82 (1999) 2828–2831.
  * [17] S. Carlip, _Black Hole Entropy from BMS Symmetry at the Horizon_ , _Phys. Rev. Lett._ 120 (Mar., 2018) 101301.
  * [18] M. Geiller, E. R. Livine and F. Sartini, _Symmetries of the black hole interior and singularity regularization_ , _SciPost Phys._ 10 (Jan., 2021) 022.
  * [19] J. Ben Achour, E. R. Livine, D. Oriti and G. Piani, _Schrödinger Symmetry in Gravitational Mini-Superspaces_ , .
  * [20] M. Geiller, E. R. Livine and F. Sartini, _Dynamical symmetries of homogeneous minisuperspace models_ , _Phys. Rev. D_ 106 (Sept., 2022) 064013.
  * [21] J. Ben Achour, E. R. Livine and D. Oriti, _Schrödinger symmetry of Schwarzschild-(A)dS black hole mechanics_ , .
  * [22] J. Ben Achour and E. R. Livine, _Thiemann complexifier in classical and quantum FLRW cosmology_ , _Phys. Rev. D_ 96 (Sept., 2017) 066025.
  * [23] J. Ben Achour and E. R. Livine, _Protected SL(2,R) Symmetry in Quantum Cosmology_ , _JCAP_ 09 (Sept., 2019) 012.
  * [24] F. Sartini, _Hidden Symmetries in Gravity : Black holes and other minisuperspaces_.  PhD Thesis, Laboratoire de Physique de l’ENS Lyon, France, ENS, Lyon, Lab. Phys., July, 2022.
  * [25] M. Cariglia, C. Duval, G. W. Gibbons and P. A. Horvathy, _Eisenhart lifts and symmetries of time-dependent systems_ , _Annals Phys._ 373 (Oct., 2016) 631–654.
  * [26] J. Bicak and K. Kuchar, _Null dust in canonical gravity_ , _Phys. Rev. D_ 56 (1997) 4878–4895.
  * [27] J. Louko, B. F. Whiting and J. L. Friedman, _Hamiltonian spacetime dynamics with a spherical null-dust shell_ , _Phys. Rev. D_ 57 (1998) 2279–2298.
  * [28] M. Campiglia, R. Gambini, J. Olmedo and J. Pullin, _Quantum self-gravitating collapsing matter in a quantum geometry_ , _Class. Quant. Grav._ 33 (Aug., 2016) 18LT01.
  * [29] P. Hajicek, _Quantum theory of gravitational collapse (lecture notes on quantum conchology)_ , in _Lect. Notes Phys._ , vol. 631, pp. 255–299.  2003\.  DOI.
  * [30] R. Eyheralde, M. Campiglia, R. Gambini and J. Pullin, _Quantum fluctuating geometries and the information paradox_ , _Class. Quant. Grav._ 34 (Nov., 2017) 235015.
  * [31] D. Grumiller, W. Kummer and D. V. Vassilevich, _Dilaton Gravity in Two Dimensions_ , _Phys. Rept._ 369 (2002) 327–430.
  * [32] H. Afshar, H. A. González, D. Grumiller and D. Vassilevich, _Flat space holography and the complex Sachdev-Ye-Kitaev model_ , _Phys. Rev. D_ 101 (Apr., 2020) 086024.
  * [33] R. Ruzziconi and C. Zwikel, _Conservation and Integrability in Lower-Dimensional Gravity_ , _JHEP_ 04 (Apr., 2021) 034.
  * [34] D. Grumiller, R. Ruzziconi and C. Zwikel, _Generalized dilaton gravity in 2d_ , _SciPost Phys._ 12 (Jan., 2022) 032.
  * [35] C. Crnkovic and E. Witten, _COVARIANT DESCRIPTION OF CANONICAL FORMALISM IN GEOMETRICAL THEORIES_ , .
  * [36] J. Lee and R. M. Wald, _Local symmetries and constraints_ , _J. Math. Phys._ 31 (1990) 725–743.
  * [37] A. Ashtekar, L. Bombelli and O. Reula, _THE COVARIANT PHASE SPACE OF ASYMPTOTICALLY FLAT GRAVITATIONAL FIELDS_ , .
  * [38] R. M. Wald, _Black Hole Entropy is Noether Charge_ , _Phys. Rev. D_ 48 (Oct., 1993) R3427–R3431.
  * [39] V. Iyer and R. M. Wald, _Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy_ , _Phys. Rev. D_ 50 (1994) 846–864.
  * [40] V. Iyer and R. M. Wald, _A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes_ , _Phys. Rev. D_ 52 (1995) 4430–4439.
  * [41] T. Jacobson, G. Kang and R. C. Myers, _On Black Hole Entropy_ , _Phys. Rev. D_ 49 (1994) 6587–6598.
  * [42] R. M. Wald and A. Zoupas, _A General Definition of ”Conserved Quantities” in General Relativity and Other Theories of Gravity_ , _Phys. Rev. D_ 61 (2000) 084027.
  * [43] M. Geiller and P. Jai-akson, _Extended actions, dynamics of edge modes, and entanglement entropy_ , _JHEP_ 09 (Sept., 2020) 134.
  * [44] A. J. Speranza, _Local phase space and edge modes for diffeomorphism-invariant theories_ , _JHEP_ 02 (Feb., 2018) 021.
  * [45] W. Donnelly and L. Freidel, _Local subsystems in gauge theory and gravity_ , _JHEP_ 09 (Sept., 2016) 102.
  * [46] G. Barnich and G. Compere, _Surface charge algebra in gauge theories and thermodynamic integrability_ , _J. Math. Phys._ 49 (2008) 042901.
  * [47] H. Adami, M. M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo and C. Zwikel, _Symmetries at Null Boundaries: Two and Three Dimensional Gravity Cases_ , _JHEP_ 10 (Oct., 2020) 107.
  * [48] F. Alessio, G. Barnich, L. Ciambelli, P. Mao and R. Ruzziconi, _Weyl Charges in Asymptotically Locally AdS$_3$ Spacetimes_ , _Phys. Rev. D_ 103 (Feb., 2021) 046003.
  * [49] G. Compère and A. Fiorucci, _Asymptotically flat spacetimes with BMS$_3$ symmetry_ , _Class. Quant. Grav._ 34 (Sept., 2017) 204002.
  * [50] D. Grumiller, A. Pérez, M. M. Sheikh-Jabbari, R. Troncoso and C. Zwikel, _Spacetime structure near generic horizons and soft hair_ , _Phys. Rev. Lett._ 124 (Jan., 2020) 041601.
  * [51] L. Ciambelli, S. Detournay and A. Somerhausen, _New Chiral Gravity_ , _Phys. Rev. D_ 102 (Nov., 2020) 106017.
  * [52] G. Odak and S. Speziale, _Brown-York charges with mixed boundary conditions_ , _JHEP_ 11 (Nov., 2021) 224.
  * [53] G. Odak, A. Rignon-Bret and S. Speziale, _Wald-Zoupas prescription with (soft) anomalies_ , _Phys. Rev. D_ 107 (Apr., 2023) 084028.
  * [54] G. Odak, A. Rignon-Bret and S. Speziale, _General gravitational charges on null hypersurfaces_ , .
  * [55] C. Goeller, P. A. Hoehn and J. Kirklin, _Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance_ , .
  * [56] J. B. Achour, E. R. Livine, S. Mukohyama and J.-P. Uzan, _Hidden symmetry of the static response of black holes: Applications to Love numbers_ , .