# All symmetries of near-horizon scattering

Ankit Aggarwal⋆,‡ and Nava Gaddam†

###### Abstract

Asymptotic symmetries are known to constrain the infrared behaviour of
scattering processes in asymptotically flat spacetimes. By the same token, one
expects symmetries of the black hole horizon to constrain near-horizon
gravitational scattering. In this paper, we make an important advance towards
establishing this connection. We find all near-horizon symmetries relevant for
gravitational scattering near the horizon of the Schwarzschild black hole. We
study large diffeomorphisms of linearised perturbations of the Schwarzschild
black hole in a partial wave basis and in gauge that allows for gravitational
radiation crossing the event horizon. This setup is ideally suited to study
processes involving near-horizon gravitons like scattering and black hole
evaporation. We find the most general near-horizon symmetries that are
consistent with the perturbations being finite at the horizon. With no further
restriction on the boundary conditions than regularity on the horizon, we find
the associated covariant charges to be finite and non-zero. These symmetries
are therefore physical. The complete symmetry algebra, however, does not
close. The maximal subset of symmetries that forms a closed algebra turns out
to be ${\rm Diff}(S^{2})$ in a semi-direct sum with two supertranslations. Our
boundary conditions in fact allow us to extend the asymptotic Killing vectors
everywhere outside the horizon and we show that a sub-algebra closes to all
orders in the near-horizon expansion parameter. Interestingly, for a large
black hole, the dominant symmetries are just two copies of $u(1)$, one of
which is not present in the maximal closed sub-algebra.

##  1 Introduction

Classical analyses of perturbations of the Schwarzschild black hole have a
rich history. Seeking an understanding of the classical stability of the
background, influential work on the subject dates back to several decades ago
[1, 2, 3]. These lead to further important developments due to Chandrasekhar
[4]. In addition to various applications [5, 6, 7, 8], an effort to cast the
perturbations in gauge invariant form has culminated in a rather practical and
covariant avatar [9, 10, 11, 12, 13, 14] which also allowed for a study of
classical gravitational radiation both crossing the horizon and reaching
infinity.

Owing to Hawking’s seminal work [15, 16], quantum aspects of black holes have
an equally captivating and yet, as is widely acknowledged, an incomplete
story. While the problem of information loss has been widely debated, quantum
aspects of the most general perturbations of the Schwarzschild black hole have
received considerably little attention. Only recently has a formalism to study
scattering processes in the near-horizon region been developed [17, 18, 19,
20, 21].111For previous work that led to these developments, see [22, 23, 24,
25]. These processes naturally include metric perturbations resulting in a new
regime of quantum gravity where elastic $2-2$ amplitudes eikonalise. This
regime is where centre of mass energies of scattering processes satisfy $EM\gg
M^{2}_{Pl}$ [17, 18], where $M$ is the mass of the black hole. The formalism
also allows for calculations of S-matrix elements away from the black hole
eikonal regime. Inelastic processes relevant for black hole evolution can also
be explicitly calculated [20]. A new soft limit is also expected to emerge in
the near-horizon limit where graviton momenta scale inversely with the
Schwarzschild radius in the large black hole limit [26]. These developments
are inherently based on the covariant avatar of the black hole perturbations
alluded to earlier.

Black holes notwithstanding, it has become increasingly apparent that the
infrared structure of gravity is far richer than previously thought even
perturbatively about flat space. Gravitational radiation reaching null
infinity is known to be intricately tied to infinite dimensional symmetries
arising from large gauge transformations that survive at future and past null
infinities [27, 28, 29]. In turn, a diagonal subgroup of these future and past
Bondi-Metzner-Sachs (BMS) groups has been argued to be a symmetry of the
asymptotically flat quantum gravity S-matrix [30]. The Ward identities
associated with these symmetries have been shown to be the same as Weinberg’s
soft graviton theorem in flat space [31, 32, 33].

It is then natural to ask if the radiation crossing the horizon has an
analogously rich relationship with asymptotic symmetries of the horizon and if
these are tied to the near-horizon scattering processes. Infinite dimensional
symmetries have in fact been shown to arise near the horizon of non-extremal
black holes [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49].
Their relevance to the black hole information problem has also been debated
[50, 51]. These symmetries can be thought as being emergent near the horizon,
once the collapse process of a large black hole has settled. They are valid
for as long as the black hole is semi-classically large ($M\gg M_{Pl}$). It is
not known if these symmetries in the literature are the complete set of all
symmetries of the horizon. Furthermore, any potential relationship between
these symmetries and scattering processes near the horizon is also a glaringly
open problem.

##### Results:

In this article, we take an important step towards addressing the questions
raised in the previous paragraph. Within the covariant approach to black hole
perturbations, we derive the set of all near- horizon symmetries assuming only
that the perturbations remain finite on the horizon. Remarkably, with no
further boundary conditions on the perturbations than their finiteness at the
horizon, we find the corresponding asymptotic charges to be finite. The
asymptotic Killing vectors thus found, form a closed abelian algebra at the
linearised level but there are non-linear obstructions to closure. The maximal
closed sub-algebra without such obstructions turns out to arise from a
restricted boundary condition that sets certain non-radiative data to zero.
The resulting sub-algebra contains $\text{Diff}\left(S^{2}\right)$ in semi-
direct sum with two supertranslations. In fact, we find the Killing vectors to
all orders in the near-horizon expansion parameter and see that their algebra
does not close to all orders. But a restriction of it (which now includes
arbitrary functions of the sphere and the near-horizon parameter) does in fact
close. Moreover, in the large black hole limit, the dominant symmetries form
two copies of $u(1)$, one of which is not contained in the maximal sub-
algebra. Despite some open questions which we discuss at the end, in this
paper, we report the set of all near-horizon symmetries relevant for near-
horizon scattering.

##### Organisation of this paper:

In Section 2, we set up the perturbations of the Schwarzschild black hole in a
partial wave basis and explain our choice of gauge that allows for
gravitational radiation crossing the horizon. In Section 3, we derive the
residual diffeomorphisms that preserve the said gauge choice. We also impose
regular boundary conditions for the gravitons on the horizon to find the near-
horizon symmetries, and work out the associated charges using covariant phase
space formalism. In Section 4, we turn to the algebra of these near-horizon
Killing vectors and identify the sub-algebras that close. We also study the
symmetries relevant for large black holes. Finally, we consider the extension
of our near-horizon symmetries to all orders in the distance away from the
horizon. We also find a subset of symmetries that forms a closed algebra to
all orders. We conclude with a brief summary of results and some open
questions in Section 5.

##  2 Black hole perturbations in partial waves

We are interested in metric perturbations, $h_{\mu\nu}$, defined via
${g}_{\mu\nu}\leavevmode\nobreak\ =\leavevmode\nobreak\
\bar{g}_{\mu\nu}+\kappa h_{\mu\nu}$ with $\kappa^{2}=8\pi G$. The background
is the Schwarzschild black hole denoted by $\bar{g}_{\mu\nu}$ that is
specified by

$\mathrm{d}s^{2}\leavevmode\nobreak\ =\leavevmode\nobreak\
A\left(u,v\right)\mathrm{d}u\mathrm{d}v+r\left(u,v\right)^{2}\mathrm{d}\Omega^{2}\quad\text{where}\quad
A\left(u,v\right)=\dfrac{R}{r}\exp\left(1-\dfrac{r}{R}\right)\,,$ (2.1)

where $R$ is the Schwarzschild radius and $r\left(u,v\right)$ is implicitly
determined from the relation:

$uv\leavevmode\nobreak\ =\leavevmode\nobreak\
2R^{2}\left(1-\dfrac{r}{R}\right)\exp\left(\dfrac{r}{R}-1\right)\,.$ (2.2)

The horizon is located at $r=R$ implying that either $u=0$ or $v=0$. We call
$u=0$ and $v=0$ as the future and past horizon, respectively. While we
restrict our attention to the past horizon (wherefrom outgoing radiation
emanates) in this article, analogous considerations directly apply to the
future horizon.

Perturbations of the Schwarzschild black hole are most naturally studied in
the partial wave decomposition222In this paper, following [17, 18], we use the
real representation of the spherical harmonics which satisfy the same
orthogonality relations as the complex ones. of Regge and Wheeler [1]. In this
basis, metric perturbations are split into the so-called even and odd modes,
respectively, as follows:

$\displaystyle h^{+}_{\mu\nu}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \sum_{\ell,m}\leavevmode\nobreak\
\begin{pmatrix}H_{ab}&-h^{+}_{a}D_{A}\\\ -h^{+}_{a}D_{A}&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
r^{2}\left(K+\frac{\ell(\ell+1)}{2}G\right)\gamma_{AB}+r^{2}GD_{A}D_{B}\end{pmatrix}Y_{\ell
m}\,,$ (2.3a) $\displaystyle h^{-}_{\mu\nu}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \sum_{\ell,m}\leavevmode\nobreak\
\begin{pmatrix}0&-h^{-}_{a}{\epsilon_{A}}^{B}D_{B}\\\
-h^{-}_{a}{\epsilon_{A}}^{B}D_{B}&\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-h_{\Omega}{\epsilon_{(A}}^{C}D_{B)}D_{C}\end{pmatrix}Y_{\ell m}\,.$ (2.3b)

All metric components in this decomposition naturally carry partial wave
indices which we have suppressed to avoid a clutter of notation. We will
continue to leave the partial wave indices implicit except when we choose to
remind ourselves of their presence. The longitudinal Kruskal coordinates
$(u,\,v)$ are labelled by lowercase Latin indices, while the corresponding
uppercase ones stand for the angular coordinates. Moreover, $D_{A}$ stands for
the covariant derivative on the unit two-sphere parametrised by the round
metric $\gamma_{AB}$. Finally, ${\epsilon_{A}}^{B}$ is the completely
antisymmetric tensor on the sphere with
${\epsilon_{\theta}}^{\phi}=\sin\theta$.

###  2.1 Radiation gauge

Consider a generic vector field decomposed into vector spherical harmonics as

$\displaystyle\bar{\chi}_{a}\leavevmode\nobreak\ =\leavevmode\nobreak\
\kappa\sum_{\ell m}\chi^{\ell m}_{a}Y_{\ell
m}\quad\text{and}\quad\bar{\chi}_{A}\leavevmode\nobreak\ =\leavevmode\nobreak\
\kappa\sum_{\ell m}\left(\chi^{+}_{\ell m}\partial_{A}+\chi^{-}_{\ell
m}{\epsilon_{A}}^{B}\partial_{B}\right)Y_{\ell m}\,.$ (2.4)

It is evident that three of the components of the vector field are even modes
while $\chi^{-}_{\ell m}$ is an odd mode. As discussed in [17, 18, 52, 53,
20], a convenient choice of gauge (the “Regge-Wheeler gauge”) that leaves no
residual gauge symmetry is one where the vector fields are chosen such that
$G=0$, $h_{\Omega}=0$, $h^{+}_{a}=0$. This ensures that the even graviton mode
is block diagonal whereas the odd graviton is entirely off-diagonal. However,
this gauge does not allow for any radiative data.

We will use a gauge that allows for gravitational radiation crossing the past
horizon, called the “radiation gauge”, first proposed in [14]:

$H_{uv}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0\,,\quad
H_{uu}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0\,,\quad\text{and}\quad
h_{u}^{\pm}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0\,.$ (2.5)

A similar condition can be imposed to study radiation on the future horizon
and analogous results can be obtained. Near the past (future) horizon,
$v=0\leavevmode\nobreak\ (u=0)$, it was argued in [14] that the free radiative
data can be chosen to be the leading component of $G$ when expanded in a small
$v$ series ($u$ series). This component is at $\mathcal{O}(v^{0})$. Moreover,
as pointed out in [14, 17, 18] and further exploited in [52, 53, 20], it may
be worth noting that there are no propagating degrees of freedom in the
monopole ($\ell=0$) and dipole ($\ell=1$) sectors. Only the multipole modes
with $\ell>1$ are physical and propagating.

##  3 Symmetries and covariant charges

In this section, we find the residual symmetries of the radiation gauge,
demand that the perturbations be regular at the horizon as boundary
conditions, and compute the near-horizon charge using the covariant phase
space method.

###  3.1 Residual gauge symmetry

With the parameterisation (2.4) of a generic diffeomorphism decomposed in
partial waves, the metric components in (2.3) transform as

$\displaystyle\delta H^{\ell m}_{ab}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \tilde{\nabla}_{a}\chi^{\ell
m}_{b}+\tilde{\nabla}_{b}\chi^{\ell m}_{a}\leavevmode\nobreak\ ,$ (3.1a)
$\displaystyle\delta h_{a,\ell m}^{+}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \chi^{\ell
m}_{a}+\tilde{\nabla}_{a}\chi^{+}_{\ell
m}-\dfrac{2}{r}\left(\partial_{a}r\right)\,\chi^{+}_{\ell
m}\leavevmode\nobreak\ ,$ (3.1b) $\displaystyle\delta K_{\ell
m}\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\dfrac{2}{r}g^{ab}\left(\partial_{a}r\right)\chi^{\ell
m}_{b}-\dfrac{\ell\left(\ell+1\right)}{r^{2}}\chi^{+}_{\ell m}\,,$ (3.1c)
$\displaystyle\delta G_{\ell m}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \dfrac{2}{r^{2}}\chi^{+}_{\ell
m}\leavevmode\nobreak\ ,$ (3.1d) $\displaystyle\delta h_{a,\ell
m}^{-}\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\nabla_{a}\chi^{-}_{\ell
m}-\dfrac{2}{r}\left(\partial_{a}r\right)\,\chi^{-}_{\ell
m}\leavevmode\nobreak\ ,$ (3.1e) $\displaystyle\delta h^{\ell
m}_{\Omega}\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
2\chi^{-}_{\ell m}\leavevmode\nobreak\ .$ (3.1f)

Here all the covariant derivatives are with respect to the background metric
and $\tilde{\nabla}$ refers to the covariant derivative restricted to the
longitudinal coordinates. We are interested in finding those gauge
transformations that leave the gravitons in the radiation gauge (2.5). The
radiation gauge imposes constraints on the diffeomorphisms in (2.4). Demanding
$H_{uu}^{\ell m}=0$ leads to

$\displaystyle 0\leavevmode\nobreak\ =\leavevmode\nobreak\
\tilde{\nabla}_{u}\chi^{\ell m}_{u}=\leavevmode\nobreak\
\partial_{u}\chi^{\ell
m}_{u}-\dfrac{\partial_{u}A\left(r\right)}{A\left(r\right)}\chi^{\ell
m}_{u}\,.$ (3.2)

Therefore, we find that the most general solution for $\chi^{\ell m}_{u}$ in
terms of an arbitrary integration constant $f_{1}^{\ell m}\left(v\right)$ is
of the form

$\chi^{\ell m}_{u}\leavevmode\nobreak\ =\leavevmode\nobreak\
A\left(r\right)f_{1}^{\ell m}\left(v\right)\,.$ (3.3)

Next, we have that $h^{+}_{a,\ell m}=0$ implies

$\displaystyle 0\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\left(\partial_{u}-\dfrac{2}{r}\partial_{u}r\right)\chi^{+}_{\ell
m}+\chi^{\ell m}_{u}\,.$ (3.4)

resulting in the solution

$\displaystyle\chi^{+}_{\ell m}\leavevmode\nobreak\ =\leavevmode\nobreak\
r^{2}\left(f_{2}^{\ell m}\left(v\right)-f_{1}^{\ell
m}\left(v\right)\int\mathrm{d}u\,\dfrac{A\left(r\right)}{r^{2}}\right)\,,$
(3.5)

where we introduced a new integration constant $f_{2}\left(v\right)$. The last
gauge condition, $H_{uv}^{\ell m}=0$, implies that

$\displaystyle 0\leavevmode\nobreak\ =\leavevmode\nobreak\
\partial_{u}\chi^{\ell m}_{v}+\partial_{v}\chi^{\ell m}_{u}\,,$ (3.6)

wherewe used that, in the background (2.1), $g^{ab}\Gamma^{c}_{ab}=0$.
Therefore, we find the following solution333Alternatively, the solution to
this equation may also be written in terms of an auxillary function $q^{\ell
m}\left(u,v\right)$ such that $\chi^{\ell
m}_{a}=g^{bc}\epsilon_{ac}\partial_{b}q^{\ell m}$ which implies $\chi^{\ell
m}_{v}=-\frac{1}{A\left(r\right)}\partial_{v}q^{\ell m}$. in terms of yet
another integration constant $f_{3}\left(v\right)$:

$\displaystyle\chi^{\ell m}_{v}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ f_{3}^{\ell
m}\left(v\right)-\partial_{v}f_{1}^{\ell
m}\left(v\right)\int\mathrm{d}u\,A\left(r\right)-f_{1}^{\ell
m}\left(v\right)\int\mathrm{d}u\,\partial_{v}A\left(r\right)\,.$ (3.7)

Finally, requiring $h^{-}_{u,\ell m}=0$ yields the equation

$\displaystyle 0\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\partial_{u}\left(\dfrac{\chi^{-}_{\ell m}}{r^{2}}\right)\quad\text{which has
the solution}\quad\chi^{-}_{\ell m}\leavevmode\nobreak\ =\leavevmode\nobreak\
r^{2}f^{\ell m}_{4}\left(v\right)\,.$ (3.8)

Collecting these components, we find the residual Killing vectors of the
radiation gauge to be

$\displaystyle\chi^{\ell m}_{u}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
A\left(r\right)f_{1}\left(v\right)\,,\quad\chi^{\ell
m}_{v}\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
f_{3}\left(v\right)-\partial_{v}f_{1}\left(v\right)\int\mathrm{d}u\,A\left(r\right)-f_{1}\left(v\right)\int\mathrm{d}u\,\partial_{v}A\left(r\right)$
(3.9a) $\displaystyle\chi^{-}_{\ell m}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
r^{2}f_{4}\left(v\right)\,,\qquad\chi^{+}_{\ell m}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
r^{2}\left(f_{2}\left(v\right)-f_{1}\left(v\right)\int\mathrm{d}u\,\dfrac{A\left(r\right)}{r^{2}}\right)\,.$
(3.9b)

The derivation of these equations did not require any near-horizon expansion
and they are valid to all orders in $v$.

###  3.2 Boundary conditions and equations of motion near the horizon

The only boundary conditions we demand are that the gravitons do not diverge
at the horizon. As we will see, this rather mild constraint leads to finite
near-horizon charges. The requirement that the perturbations remain finite
allows us to expand all the six remaining off-shell fields
$\Phi\in\left\\{G,\,K,\,h^{\pm}_{v},\,H_{vv},\,h_{\Omega}\right\\}$ in a
Madhavan-Taylor expansion around $v=0$:

$\Phi\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{n=0}^{\infty}\Phi^{(n)}v^{n}\leavevmode\nobreak\ .$ (3.10)

It may be checked that the $uu$-component of the leading order linearised
vacuum equations of motion imply that $K^{(0)}$ is non-radiative free data
determined by two arbitrary constants 444In [14], $K^{(0)}$ was set to 0 which
turns out to be very constraining for the near-horizon symmetries.
$c_{K^{(0)}}$ and $d_{K^{(0)}}$:

$\partial_{u}^{2}K^{(0)}\leavevmode\nobreak\ =\leavevmode\nobreak\
0\quad\text{which implies that}\quad K^{(0)}\leavevmode\nobreak\
=\leavevmode\nobreak\ c_{K^{(0)}}u+d_{K^{(0)}}\,.$ (3.11)

All the other even graviton fields can be expressed in terms of $G^{(0)}$
together with other non-radiative free data

$\displaystyle\partial_{u}G^{(1)}\leavevmode\nobreak$
$\displaystyle=\leavevmode\nobreak\
\dfrac{1}{2R^{2}}\left(u\partial_{u}G^{(0)}+2h_{v}^{(0)}\right)$ (3.12)
$\displaystyle\partial_{u}K^{(1)}\leavevmode\nobreak$
$\displaystyle=\leavevmode\nobreak\
-\dfrac{1}{2R^{2}}\left(\ell\left(\ell+1\right)\partial_{u}h_{v}^{(0)}-u\partial_{u}K^{(0)}+R^{2}\partial_{u}^{2}H_{vv}^{(0)}\right)$
$\displaystyle\partial_{u}^{2}h_{v}^{+(0)}\leavevmode\nobreak$
$\displaystyle=\leavevmode\nobreak\
-\dfrac{1}{2}\left(\ell-1\right)\left(\ell+2\right)\partial_{u}G^{(0)}-\partial_{u}K^{(0)}$
$\displaystyle\partial_{u}^{2}H_{vv}^{(0)}\leavevmode\nobreak$
$\displaystyle=\leavevmode\nobreak\
\dfrac{1}{2R^{2}}\left(\left(\ell-1\right)\ell\left(\ell+1\right)\left(\ell+2\right)G^{(0)}+2\left(\left(\ell+2\right)\left(\ell-1\right)\right)K^{(0)}-2u\partial_{u}K^{(0)}\right)\,,$

The first of these equations arises from the $\theta\phi$ component of the
equations of motion, the second from the $\phi\phi$ component, and the third
from the $u\phi$ component. Therefore, it is evident that near the past
(future) horizon, $v=0\leavevmode\nobreak\ (u=0)$, the free radiative data for
the even sector may be chosen to be the leading $\mathcal{O}\left(1\right)$
component of $G$ when expanded in a small $v$ series ($u$ series). Of course,
the leading components of $h^{+}_{v}$ or $H_{vv}$ are equally valid choices.
The above equations can be solved to yield

$\displaystyle K^{(1)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\dfrac{1}{2R^{2}}\left[2c_{K^{(0)}}u^{2}-\left\\{\ell\left(\ell+1\right)c_{h}+\left(\ell+2\right)\left(\ell-1\right)d_{K^{(0)}}\right\\}u\right]+d_{K^{(1)}}\leavevmode\nobreak\
,$ $\displaystyle h_{v}^{+(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\dfrac{1}{2}\left(\ell-1\right)\left(\ell+2\right)\left(\int\mathrm{d}uG^{(0)}\right)-\dfrac{u^{2}}{2}c_{K^{(0)}}+c_{h}u+d_{h}\leavevmode\nobreak\
,$ $\displaystyle\partial_{u}H_{vv}^{(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\dfrac{1}{2R^{2}}\big{[}\left(\ell-1\right)\ell\left(\ell+1\right)\left(\ell+2\right)\left(\int\mathrm{d}u\leavevmode\nobreak\
G^{(0)}\right)+\left(\ell^{2}+\ell-3\right)c_{K^{(0)}}{u^{2}}$
$\displaystyle\qquad+\leavevmode\nobreak\
2\left(\ell+2\right)\left(\ell-1\right)d_{K^{(0)}}u+d_{H}\big{]}\leavevmode\nobreak\
.$ (3.13)

Here we introduced additional constants of integration: $c_{h}$,
$d_{K^{(1)}}$, $d_{h}$, and $d_{H}$. In the odd sector, on the other hand, the
leading order linearised vacuum equations of motion imply

$\partial_{u}h_{\Omega}^{(0)}\leavevmode\nobreak\ =\leavevmode\nobreak\
-\dfrac{2R^{2}}{\left(\ell+2\right)\left(\ell-1\right)}\partial_{u}^{2}h_{v}^{-(0)}\leavevmode\nobreak\
.$ (3.14)

All the subleading fields can be expressed in terms of $h_{v}^{-(0)}$ or
$h_{\Omega}^{(0)}$ together with some non-radiative data. Thus, near the past
(future) horizon, $v=0\leavevmode\nobreak\ (u=0)$, the free radiative data for
the odd sector may be chosen to be the leading component of $h_{\Omega}$ when
expanded in a small $v$ series.

###  3.3 Asymptotic Killing vectors

The near-horizon expansions (3.10) together with (3.1) lead to the following
expansions for the Killing vector near the past horizon

$\chi^{\ell m}_{a}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{n=0}^{\infty}\chi_{a}^{(n)}v^{n}\leavevmode\nobreak\
,\quad\chi^{+}_{\ell m}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{n=0}^{\infty}\chi^{+(n)}v^{n}\leavevmode\nobreak\ ,\quad\chi^{-}_{\ell
m}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{n=0}^{\infty}\chi^{-(n)}v^{n}\leavevmode\nobreak\ ,$ (3.15)

where the leading order components can be found to be555We introduced
additional factors of $R$ in some coefficients to ensure that all constants
are dimensionless. These may be fixed by noting the dimensions of the metric
perturbations in (2.3) and deducing the dimensions of the Killing vectors from
(3.1). Based on this analysis, it can be checked that
$\left[\chi_{a}\right]=L^{0}$, $\left[\chi^{+}\right]=L^{1}$, and
$\left[\chi^{-}\right]=L$.

$\displaystyle\chi_{v}^{(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\dfrac{\alpha_{1}u^{2}}{2R^{2}}+\dfrac{\alpha_{2}u}{R}+\alpha_{3}\,,$ (3.16a)
$\displaystyle\chi_{u}^{(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \alpha_{1}\,,$ (3.16b)
$\displaystyle\chi^{+(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\alpha_{1}u+R\beta_{1}\leavevmode\nobreak\ ,$ (3.16c)
$\displaystyle\chi^{-(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ R\gamma_{1}$ (3.16d)

where the partial-wave dependent constants $\alpha_{i}$, $\beta_{i}$, and
$\gamma_{i}$ are determined in terms of the integration constants in (3.9).
Writing all integration constants appearing in (3.9) also in a power series as
$f_{i}(v)=\sum_{n=0}^{\infty}f^{(n)}_{i}v^{n}$, we have that
$\alpha_{1}=f^{(0)}_{1}$, $\alpha_{2}=-Rf^{(1)}_{1}$,
$\alpha_{3}=f_{3}^{(0)}$, $\beta_{1}=Rf^{(0)}_{2}$, and
$\gamma_{1}=Rf^{(0)}_{4}$. These solutions follow from (3.9) where we used
that, in the Schwarzschild background,
$\partial_{a}A\left(r\right)=\partial_{a}r\partial_{r}A\left(r\right)$ and
that

$r\left(u,v\right)=R-\frac{uv}{2R}-\frac{u^{2}v^{2}}{4R^{3}}+\mathcal{O}\left(v^{3}\right)\leavevmode\nobreak\
,\quad
A(u,v)=1+\frac{uv}{R^{2}}+\frac{9u^{2}v^{2}}{8R^{2}}+\mathcal{O}\left(v^{3}\right)\leavevmode\nobreak\
$ (3.17)

near the horizon. The subleading terms can similarly be determined to be

$\displaystyle\chi_{v}^{(1)}(u)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\dfrac{3\alpha_{1}}{4R^{4}}u^{3}+\dfrac{\alpha_{2}}{R^{3}}u^{2}+\dfrac{\alpha_{4}u}{R^{2}}+\dfrac{\alpha_{5}}{R}\,,$
(3.18a) $\displaystyle\chi_{u}^{(1)}(u)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\dfrac{\alpha_{1}u}{R^{2}}-\dfrac{\alpha_{2}}{R}\leavevmode\nobreak\ ,$
(3.18b) $\displaystyle\chi^{+(1)}(u)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\left(\dfrac{\alpha_{2}}{R}-\dfrac{\beta_{1}}{R}\right)u+\beta_{2}\leavevmode\nobreak\
,$ (3.18c) $\displaystyle\chi^{-(1)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\gamma_{2}-\frac{\gamma_{1}u}{R}\leavevmode\nobreak\ .$ (3.18d)

The new coefficients appearing in this expression are determined as
$\alpha_{4}=-2R^{2}f_{1}^{(2)}$, $\alpha_{5}=Rf_{3}^{(1)}$,
$\beta_{2}=R^{2}f_{2}^{(1)}$, and $\gamma_{2}=R^{2}f_{4}^{(1)}$. Plugging
these solutions into the variations (3.1), we find the following leading order
variations of the metric perturbations

$\displaystyle\delta\partial_{u}H^{(0)}_{vv}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\dfrac{3\alpha_{1}u^{2}}{2R^{4}}+2\dfrac{\alpha_{4}-\alpha_{3}}{R^{2}}\leavevmode\nobreak\
,$ (3.19a) $\displaystyle\delta h_{v}^{+(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\dfrac{3\alpha_{1}u^{2}}{2R^{2}}+\frac{2\alpha_{2}u}{R}+\alpha_{3}+\beta_{2}\leavevmode\nobreak\
,$ (3.19b) $\displaystyle\delta K^{(1)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\dfrac{\left(\ell^{2}+\ell+1\right)\alpha_{1}u^{2}}{R^{4}}-\dfrac{\ell\left(\ell+1\right)\alpha_{2}u}{R^{3}}+\dfrac{\alpha_{3}-\ell\left(\ell+1\right)\beta_{2}}{R^{2}}\,,$
(3.19c) $\displaystyle\delta G^{(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
-\dfrac{2\alpha_{1}u}{R^{2}}+\dfrac{2\beta_{1}}{R}\leavevmode\nobreak\ ,$
(3.19d) $\displaystyle\delta K^{(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\dfrac{\left(\ell^{2}+\ell+1\right)\alpha_{1}u}{R^{2}}-\dfrac{\ell\left(\ell+1\right)\beta_{1}}{R}$
(3.19e) $\displaystyle\delta h_{v}^{-(0)}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \gamma_{2}\leavevmode\nobreak\ ,$ (3.19f)
$\displaystyle\delta h^{(0)}_{\Omega}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ 2R\gamma_{1}\leavevmode\nobreak\ .$
(3.19g)

It can be verified that these variations are consistent with the equations of
motion we found in (3.2) and (3.14).

###  3.4 Covariant charge

The gravitational charges associated with the asymptotic Killing vectors (4.1)
can be found using the covariant phase space formalism [54, 55] (also see [56]
for a review):

$\displaystyle\cancel{\delta}Q_{\bar{\chi}}\left[\bar{g}_{\rho\sigma};h_{\rho\sigma}\right]\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\ \dfrac{\kappa}{16\pi
G}\int\left(\mathrm{d}^{2}x\right)_{\mu\nu}\sqrt{-\bar{g}}\left[\bar{\chi}^{\nu}\nabla^{\mu}h-\bar{\chi}^{\mu}\nabla_{\sigma}h^{\mu\sigma}+\bar{\chi}_{\sigma}\nabla^{\nu}h^{\mu\sigma}+\dfrac{1}{2}h\nabla^{\nu}\bar{\chi}^{\mu}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+\dfrac{1}{2}h^{\nu\sigma}\left(\nabla^{\mu}\bar{\chi}_{\sigma}-\nabla_{\sigma}\bar{\chi}^{\mu}\right)-\left(\mu\leftrightarrow\nu\right)\right]\,,$
(3.20)

where $\bar{\chi}$ was defined in (2.4), $\cancel{\delta}$ indicates that the
charges are not integrable in general and
$\left(\mathrm{d}^{2}x\right)_{\mu\nu}=\frac{1}{4}\epsilon_{\mu\nu\rho\sigma}\mathrm{d}x^{\rho}\wedge\mathrm{d}x^{\sigma}$.
Plugging in our solutions for the asymptotic Killing vectors from Section 3.3
and using the decomposition of the graviton into partial waves as before, we
find the charge in the even sector to be

$\displaystyle Q\left[\chi^{(0)\ell m}_{a},\chi^{+(0)}_{\ell
m}\right]\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\dfrac{1}{2}\bigg{[}\ell(\ell+1)\left(\partial_{u}{h_{v}^{(0)}}\left(R\beta_{1}-\alpha_{1}u\right)+2\alpha_{1}{h_{v}^{(0)}}\right)$
$\displaystyle\qquad\qquad\qquad+2R^{2}\alpha_{1}K^{(1)}+\left(2R\alpha_{2}-3u\alpha_{1}\right)K^{(0)}$
$\displaystyle\qquad\qquad\qquad+\left(u^{2}\alpha_{1}-2Ru\alpha_{2}-2R\alpha_{2}\right)\partial_{u}K^{(0)}\bigg{]}\leavevmode\nobreak\
.$ (3.21)

Here, we used the following orthogonality relation for spherical harmonics to
integrate over the two-sphere

$\int\mathrm{d}\Omega D^{A}Y(\Omega)_{\ell
m}D_{A}Y_{\ell^{\prime}m^{\prime}}(\Omega)\leavevmode\nobreak\
=\leavevmode\nobreak\
\ell(\ell+1)\delta_{\ell\ell^{\prime}}\delta_{m,m^{\prime}}.$ (3.22)

The $\ell,m$ labels are again implicit in the charges. Upon imposing equations
of motion (3.12), this charge reduces to

$\displaystyle Q\left[\chi^{\ell m}_{a},\chi^{+}_{\ell
m}\right]\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\dfrac{1}{2}\bigg{[}\ell\left(\ell+1\right)\left(\partial_{u}{h_{v}^{(0)}}\left(R\beta_{1}-\alpha_{1}u\right)+2\alpha_{1}{h_{v}^{(0)}}\right)$
$\displaystyle\qquad\qquad\qquad-u\left(\ell\left(\ell+1\right){c_{h}}+d_{K^{(0)}}\right)\alpha_{1}-\ell\left(\ell+1\right)\alpha_{1}c_{K^{(0)}}$
$\displaystyle\qquad\qquad\qquad+2R\left(\alpha_{2}d_{K_{(0)}}-R\alpha_{3}c_{K^{(0)}}\right)$
$\displaystyle\qquad\qquad\qquad+2\ell(\ell+1)\alpha_{1}d_{h}+2R^{2}\alpha_{1}d_{K^{(1)}}\bigg{]}\,.$
(3.23)

The first line of this expression contains free radiative data (which we chose
to be $h^{(0)}_{v}$), whereas the terms in the second line and below only
contain non-radiative data. Such terms may be eliminated by an appropriate
choice of boundary conditions that fix the integration constants appearing in
the solutions to equations of motion in (3.2).666Similar conditions were used
in the study of higher dimensional supertranslations in [57]. Similarly, in
the odd sector, we find the following linearised charge

$\displaystyle Q\left[\chi^{-}_{\ell m}\right]\leavevmode\nobreak\
=\leavevmode\nobreak\
\dfrac{R}{2}\left[\ell\left(\ell+1\right)\gamma_{1}^{\ell,m}\partial_{u}h^{-(0)}_{v}\right]\,,$
(3.24)

where we used the following orthogonality relation to integrate over the two-
sphere

$\int\mathrm{d}\Omega\leavevmode\nobreak\ \epsilon^{AB}D_{B}Y_{\ell
m}(\Omega){\epsilon_{A}}^{C}D_{C}Y_{\ell^{\prime}m^{\prime}}(\Omega)\leavevmode\nobreak\
=\leavevmode\nobreak\
\ell(\ell+1)\delta_{\ell\ell^{\prime}}\delta_{m,m^{\prime}}.$ (3.25)

We note that the Iyer-Wald [54] and Barnich-Brandt [55] charges coincide.
Furthermore, there are no central extensions in the charge algebra.

###  3.5 More restrictive boundary conditions

Some of the non-radiative data may be fixed by a slightly stronger boundary
condition

$\partial_{u}K^{(0)}\leavevmode\nobreak\ =\leavevmode\nobreak\
c_{K^{(0)}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0\,.$ (3.26)

This condition implies that $\alpha_{1}=0$ in the asymptotic Killing vector as
is evident from (3.19c). As we will see in Section 4.1, the most general
asymptotic Killing vectors do not form a closed algebra. However, setting
$\alpha_{1}=0$ indeed leads to a closed algebra. Therefore, these boundary
conditions are more natural in this sense. The charge in the even sector then
simplifies to

$\displaystyle Q\left[\chi^{\ell m}_{a},\chi^{+}_{\ell
m}\right]\Bigg{|}_{c_{K}^{(0)}=0,\alpha_{1}=0}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\dfrac{R}{2}\bigg{[}\ell\left(\ell+1\right)\left(\partial_{u}{h_{v}^{(0)}}\beta_{1}\right)+2\alpha_{2}d_{K_{(0)}}\bigg{]}\,.$
(3.27)

##  4 Symmetry algebra

In this section, we study the algebra of the near-horizon symmetries we found
in Section 3.3 and its closed sub-algebras, and the algebra of the symmetries
(in (3.9)) to all orders in $v$. We examine the closure of these algebras
under the usual Lie bracket and determine necessary conditions for closure. Of
the two resulting closed sub-algebras of the near-horizon asymptotic Killing
symmetries, one corresponds to a maximal sub-algebra while the other is the
dominant one for large black holes. It must be noted that in the linearised
theory, the algebra is abelian and therefore always closes. This can be seen
from (2.4) by noting that $\bar{\chi}$ is $\mathcal{O}(\kappa)$. Therefore,
the Lie bracket is $\mathcal{O}(\kappa^{2})$. So, all the statements we make
below about the closure (or otherwise) go beyond the linear approximation.

###  4.1 Near-horizon symmetry algebra

From the solutions for the asymptotic Killing vectors in (3.16) written in
terms of arbitrary partial wave dependent constants, we notice that the
complete asymptotic Killing vector can be parameterised in terms of arbitrary
functions $F\left(\Omega\right)$, $M\left(\Omega\right)$,
$P\left(\Omega\right)$ and $Z^{A}\left(\Omega\right)$ on the unit sphere as
follows:

$\displaystyle{\bar{\chi}}\left[F,M,P,Z^{A}\right]\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\kappa\bigg{[}\left(F\left(\Omega\right)\dfrac{u^{2}}{2R^{2}}+M\left(\Omega\right)\dfrac{u}{R}+P\left(\Omega\right)\right)\partial_{u}-F\left(\Omega\right)\partial_{v}-\frac{u}{R^{2}}\gamma^{AC}\partial_{C}F\left(\Omega\right)\partial_{A}$
$\displaystyle\qquad+\frac{Z^{A}\left(\Omega\right)}{R}\partial_{A}+\leavevmode\nobreak\
\cdots\bigg{]}\leavevmode\nobreak\ ,$ (4.1)

with the dots representing sub-leading terms and $\Omega$ a point on the unit
$2$-sphere. Here,

$\displaystyle F(\Omega)$ $\displaystyle=$
$\displaystyle\sum\limits_{\ell,m}\alpha_{1}^{\ell m}Y_{\ell
m}(\Omega)\leavevmode\nobreak\ ,$ (4.2) $\displaystyle M(\Omega)$
$\displaystyle=$ $\displaystyle-\sum\limits_{\ell,m}\alpha_{2}^{\ell m}Y_{\ell
m}(\Omega)\leavevmode\nobreak\ ,$ (4.3) $\displaystyle P(\Omega)$
$\displaystyle=$ $\displaystyle-\sum\limits_{\ell,m}\alpha_{3}^{\ell m}Y_{\ell
m}(\Omega)\leavevmode\nobreak\ ,$ (4.4) $\displaystyle Z^{A}(\Omega)$
$\displaystyle=$ $\displaystyle\sum\limits_{\ell,m}\beta_{1}^{\ell
m}\gamma^{AC}D_{C}Y_{\ell m}(\Omega)+\gamma_{1}^{\ell
m}\epsilon^{AB}D_{B}Y_{\ell m}(\Omega)\leavevmode\nobreak\ .$ (4.5)

Note that the spacetime index of the asymptotic Killing vector (3.16) must be
raised with the background metric ($\bar{g}^{\mu\nu}\bar{\chi}_{\nu}$) to
arrive at this expression. The even and odd modes on the sphere result in
independent even and odd functions on the sphere which have been combined into
two arbitrary functions $Z^{A}\left(\Omega\right)$ on the sphere. It can be
checked that the lie bracket of (4.1) does not close,

$\displaystyle[\bar{\chi}(F_{1},M_{1},P_{1},Z^{A}_{1}),\leavevmode\nobreak\
\bar{\chi}(F_{2},M_{2},P_{2},Z^{A}_{2})]\leavevmode\nobreak\ $
$\displaystyle\neq\leavevmode\nobreak\
\bar{\chi}\left(F_{12},M_{12},P_{12},Z^{A}_{12}\right)\leavevmode\nobreak\ ,$
(4.6)

for any arbitrary functions $F_{12},\leavevmode\nobreak\
M_{12},\leavevmode\nobreak\ P_{12},\leavevmode\nobreak\ Z_{12}$. However, this
is a non-linear statement since the right hand side is
$\mathcal{O}(\kappa^{2})$.

It is conceivable that the non-linear closure of this algebra requires the
addition of more terms to the asymptotic Killing vector (4.1) that may
resemble [43, 48, 49] where diffeomorphisms along the horizon (in addition to
${\rm Diff}(S^{2})$) were found. Indeed, it can be checked that the Lie
bracket (4.6) generates terms of the type ${F}_{n}(\Omega)u^{n}\partial_{u}$
for some arbitrary function on the sphere $F_{n}(\Omega)$ and an arbitrary
integer $n$. We defer further discussion to Section 5.

####  4.1.1 Maximal closed sub-algebra

The maximal sub-algebra that closes is that of vector fields with
$F\left(\Omega\right)=0$.

$\displaystyle[\bar{\chi}(M_{1},P_{1},Z^{A}_{1}),\leavevmode\nobreak\
\bar{\chi}(M_{2},P_{2},Z^{A}_{2})]=\bar{\chi}\left(M_{12},P_{12},Z^{A}_{12}\right)$
(4.7)

with

$\displaystyle M_{12}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\kappa\left[\dfrac{Z^{A}_{1}\partial_{A}M_{2}-Z^{A}_{2}\partial_{A}M_{1}}{R}\right]\leavevmode\nobreak\
,$ (4.8a) $\displaystyle P_{12}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\kappa\bigg{[}M_{1}P_{2}-M_{2}P_{1}+\frac{Z^{A}_{1}\partial_{A}P_{2}-Z^{A}_{2}\partial_{A}P_{1}}{R}\bigg{]}\leavevmode\nobreak\
,$ (4.8b) $\displaystyle Z^{A}_{12}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\kappa\left[\frac{Z^{B}_{1}\partial_{B}Z^{A}_{2}-Z^{B}_{2}\partial_{B}Z^{A}_{1}}{R}\right]\,.$
(4.8c)

As pointed out in Section 3.5, this sub-algebra naturally arises from a
stronger boundary condition (3.26) than finiteness of metric perturbations on
the horizon. It can be verified that it consists of diffeomorphisms of the
two-sphere, $\text{Diff}\left(S^{2}\right)$ generated by
$Z^{A}\left(\Omega\right)$, in semi-direct sum with two supertranslations,
generated by $M\left(\Omega\right)$ and $P\left(\Omega\right)$ on the horizon.
This is in agreement with the results of [40] and also the results found in
[36] where two copies of the Virasoro algebra were found and the complete set
of $\text{Diff}\left(S^{2}\right)$ was anticipated.

It is interesting to note that in our setup, in contrast to the analysis of
[40, 36], we see that one of the supertranslations (associated with
$M\left(\Omega\right)$) and ${\rm Diff}(S^{2})$ are suppressed in powers of
$R$ for a large black hole, $R\gg\kappa$.

####  4.1.2 Symmetries of large black holes

While we saw that $F(\Omega)=0$ was necessary for the maximal closed algebra
defined by the conventional Lie bracket, it is evident that for very large
black holes, the Killing vector (4.1) reduces to

$\displaystyle\bar{\chi}\left[F,P\right]\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\kappa\bigg{[}P\left(\Omega\right)\partial_{u}-F\left(\Omega\right)\partial_{v}+\mathcal{O}\left(\dfrac{1}{R}\right)\bigg{]}\,.$
(4.9)

The algebra generated by this asymptotic Killing vector clearly closes with a
trivial commutator leading to an algebra of two copies of $u(1)$. In the
maximal sub-algebra discussed in the previous subsection, the near-horizon
symmetry associated with $F\left(\Omega\right)$ did not contribute. However,
we see that for a large black hole this symmetry dominates and without any
obstructions to the algebra closure. Thus, it has physical significance for
scattering processes near large black holes which are of semi-classical
interest.

###  4.2 Symmetry algebra to all orders in $v$

Our boundary conditions allow us to write the asymptotic Killing vector fields
(4.1) to all order in $v$ using (3.9):

$\displaystyle{\bar{\chi}}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \kappa\bigg{[}\left(-\dfrac{\hat{\bf
F}\left(v,\Omega\right)}{A\left(r\right)}+\dfrac{\hat{A}(u,v)}{A\left(r\right)}\partial_{v}{\bf
F}\left(v,\Omega\right)+{\bf
F}\left(v,\Omega\right)\dfrac{\partial_{v}\hat{A}(u,v)}{A\left(r\right)}\right)\partial_{u}-{\bf
F}\left(v,\Omega\right)\partial_{v}$
$\displaystyle\qquad\qquad\qquad-\tilde{A}(u,v)\leavevmode\nobreak\
\gamma^{AC}{\partial_{C}{\bf F}\left(v,\Omega\right)}\partial_{A}+{{\bf
Z}^{A}\left(v,\Omega\right)}\partial_{A}\bigg{]}\leavevmode\nobreak\ \,$
(4.10)

where we defined the background functions

$\displaystyle\hat{A}(u,v)\leavevmode\nobreak\ \coloneqq\leavevmode\nobreak\
\int\mathrm{d}u\,A\left(r\right)\,,\quad\text{and}\quad\tilde{A}(u,v)\leavevmode\nobreak\
\coloneqq\leavevmode\nobreak\
\int\mathrm{d}u\,\dfrac{A\left(r\right)}{r^{2}}\,.$ (4.11)

Here,

$\displaystyle{\bf F}(v,\Omega)$ $\displaystyle=$
$\displaystyle\sum\limits_{\ell,m}f_{1}(v)^{\ell m}Y_{\ell
m}(\Omega)\leavevmode\nobreak\ ,$ (4.12) $\displaystyle\hat{\bf F}(v,\Omega)$
$\displaystyle=$ $\displaystyle\sum\limits_{\ell,m}f_{3}(v)^{\ell m}Y_{\ell
m}(\Omega)\leavevmode\nobreak\ ,$ (4.13) $\displaystyle{\bf Z}^{A}(v,\Omega)$
$\displaystyle=$ $\displaystyle\sum\limits_{\ell,m}f_{2}(v)^{\ell
m}\gamma^{AC}D_{C}Y_{\ell m}(\Omega)+f_{4}(v)^{\ell
m}\epsilon^{AB}D_{B}Y_{\ell m}(\Omega)\leavevmode\nobreak\ .$ (4.14)

The algebra of these vector fields does not close in general (since the
asymptotic algebra did not close (4.6))

$\displaystyle[\bar{\chi}(\hat{\bf F}_{1},{\bf F}_{1},,{\bf
Z}^{A}_{1}),\leavevmode\nobreak\ \bar{\chi}(\hat{\bf F}_{2},{\bf F}_{2},{\bf
Z}^{A}_{2})]\leavevmode\nobreak\ $ $\displaystyle\neq\leavevmode\nobreak\
\bar{\chi}\left(\hat{\bf F}_{12},{\bf F}_{12},{\bf
Z}^{A}_{12}\right)\leavevmode\nobreak\ .$ (4.15)

However, it can easily be checked that the sub-algebra formed by vector fields
with ${\bf F}\left(v,\Omega\right)=0$ does indeed close with

$\displaystyle\hat{\bf F}_{12}\leavevmode\nobreak\ =\leavevmode\nobreak\
\kappa\left[{{\bf Z}^{A}_{1}\partial_{A}\hat{\bf F}_{2}-{\bf
Z}^{A}_{2}\partial_{A}\hat{\bf F}_{1}}\right]\quad\text{and}\quad{\bf
Z}^{A}_{12}\leavevmode\nobreak\ =\leavevmode\nobreak\ \kappa\left[{\bf
Z}^{B}_{1}\partial_{B}{\bf Z}^{A}_{2}-{\bf Z}^{B}_{2}\partial_{B}{\bf
Z}^{A}_{1}\right]\,.$ (4.16)

Note that this sub-algebra coincides with the one formed by (4.1) for $F=M=0$,
in the near-horizon region. Therefore, it consists of only one
supertranslation instead of two. Notice that the Killing vector now contains
arbitrary functions of $v$ and a dependence on $u$ via the background function
$A\left(r\right)$. It would be interesting to find less stringent restrictions
on the function ${\bf F}\left(v,\Omega\right)$ that may still lead to a closed
algebra. In general, it is also of interest to find the general role and
meaning of these symmetries that do not lead to a closed algebra, both on the
horizon to leading order and also possibly to all orders in the distance to
the horizon, parametrised by $v$.

##  5 Conclusions and outlook

In this article, we studied the perturbations of the Schwarzschild black hole
in a partial wave basis in a gauge that allows for radiation crossing the
horizon. This setup is ideally suited for studying near-horizon scattering. We
found the residual symmetries that preserve the aforementioned “radiation
gauge”. By requiring only that the perturbations be finite on the horizon, we
found the Killing vectors not only near the horizon but also to all orders in
$v$ in Section 4.2. Remarkably, with no further restrictions we found the
charges on the horizon corresponding to these symmetries to be finite and non-
vanishing. The physical significance of the sub-leading terms is not clear as
they do not contribute to the charges. Nevertheless, there is some evidence
that such sub-leading terms can be related to sub-leading soft theorems for
scattering in flat spacetime [58, 59]. Our leading order vector fields
correspond to near-horizon symmetries that were not previously known. In the
linearised theory, the algebra is abelian and therefore closes to leading
order. However, it is curious that the complete algebra on the horizon does
not close with the conventional Lie bracket owing to sub-leading non-linear
obstructions. We identified two important sub-cases that lead to a closed
algebra. The first is obtained by setting $F(\Omega)=0$ and leads to an
algebra of two supertranslations in semi-direct sum with all diffeomorphisms
of the two sphere, $\text{Diff}\left(S^{2}\right)$. We found a specific
restriction on the boundary conditions that naturally leads to this maximal
sub-algebra. The second sub-algebra emerges in the large black hole limit and
is considerably smaller containing only two copies of $u(1)$. It would be
interesting to check if a modified bracket, like the one due to Barnich and
Troessaert [60, 61], closes the complete algebra with no restrictions.777AA
would like to thank Glenn Barnich for insightful discussions related to the
modified bracket.

##### Comparison to previous work:

Several authors have studied the symmetries of the black hole horizon in
recent years as mentioned in the introduction. Here we list a comparison
between some of these works and our results. In [36], the gauge that was used
is different from the radiation gauge that we employ. Consequently, we find
different near-horizon symmetries. In [37], the choice of gauge, fixed the
size of the horizon. In our paper, we found that some symmetries do change the
size of the black hole, while others affect its shape. In [42, 40, 49],
further restrictions on the symmetries were imposed in order to, for instance,
preserve the location of the (bifurcate) horizon. In this work, we allow for
symmetries that may change the location of the horizon. Indeed, scattering
processes near the black hole horizon may change the location of the horizon
in general. In the linearised theory, the perturbations (including but not
limited to changes in the location of the horizon) are small when $\kappa/R\ll
1$. However, we may improve our results order by order in $\kappa$, thereby
generating non-linear effects, all the while preserving our relaxed boundary
conditions. It would be interesting to compare the results of such non-linear
effects with the recent work of [48, 49]. In particular, it is of interest to
compare the resulting (potentially) closed algebra (by adding more terms to
our vector fields) with what is found there. The maximal sub-algebra that does
close is in line with the previous literature [36, 40]. However, we found an
additional symmetry which finds its place in a smaller commuting sub-algebra
that survives in the large black hole limit. Finally, in contrast to all the
above works, we find a hierarchy between the different supertranslations in
the large black hole limit.

The primary motivation for studying linear perturbations in the partial wave
basis, as opposed to the non-linear theory, is that scattering processes in
the near-horizon region arise naturally in this basis [17, 18, 19, 20, 21].
The results of this article can therefore to be seen as capturing all the
symmetries of near-horizon scattering. The next natural step is to study the
implications of these symmetries for near-horizon scattering. This warrants an
understanding of the Ward-Takahashi identities of these near-horizon
symmetries. The relation between these identities and emergent soft graviton
theorems near the horizon is an important question that we will report on in
an upcoming work [26].

In a dynamical problem of collapse, boundary conditions imposed on past null
infinity would in-principle dictate boundary conditions to be imposed on the
horizon. Similarly, if the black hole is to evaporate in its entirety,
boundary conditions on future null infinity would also determine those on the
horizon. However, determining these boundary conditions on the horizon in
practice is a difficult task. One may approach this by setting up a WKB-like
analysis where equations of motion are solved order by order both near the
horizon and near infinity. An appropriate matching condition in the
intermediate region will then determine boundary conditions on the horizon in
terms of the choices at infinity.888NG would like to thank Godwin Martin for
discussions regarding this idea. This might be possible in our formalism
because the symmetries are known to all orders in $v$ and therefore at any
point in spacetime.

Recently, in [62], a scattering algebra of ’t Hooft’s associated with
shockwaves was shown to be related to the soft algebra near infinity. This can
be exploited to provide a physical interpretation of the familiar antipodal
matching condition at spatial infinity from the perspective of scattering in
the bulk [63]. Our results in this paper, especially given their direct
relevance for near-horizon scattering and an analogous shockwave algebra in
black hole backgrounds, may potentially imply a similar antipodal matching
condition on the bifurcation sphere.999A primitive, and possibly inaccurate,
version of such an antipodal identification was anticipated and discussed in
[24, 64, 65, 66].

While we have entirely focussed on graviton perturbations in this paper, it is
also possible to study charged particle scattering near the black hole horizon
[67]. Our analysis can be extended in a straightforward manner101010See also
[68]. to study emergent and potentially new symmetries of QED near the black
hole horizon and ask about their relationship with an emergent soft-photon
theorem near the horizon.

Asymptotic symmetries near null infinity are subtle in different dimensions
[69, 70, 71, 72, 57, 73, 74, 44, 75, 76, 77, 78, 79, 80, 81, 82] because the
desirable asymptotic charges seem to diverge in a naive $r\rightarrow\infty$
limit. Such divergences do not arise in the case of higher dimensional black
holes where the near-horizon charges are computed in the $r\rightarrow R$
limit. Therefore, our formalism is easily adapted to the case of higher
dimensional black holes.111111A similar observation was made in [83]. An
appropriate choice of spherical harmonics would imply that in the final
expressions, only the eigenvalues would have to be adapted in comparison to
our results. Further potential subtleties, if any, may be interesting to
understand.

Finally, it is conceivable that a version of the memory effect in the near
horizon region owing to the symmetries near the horizon may have observable
consequences. For instance, it would be of great importance to understand
whether the radiation emerging from the horizon can leave an imprint on the
luminosity fluctuations of the spectra of stellar oscillations (of the S-stars
orbiting the Saggittarius A* for example). While there are other (potentially
stronger) sources for the said luminosity fluctuations like those due to
internal pressure and density dynamics of the orbiting stars, those owed to
near-horizon radiation considered in this paper are likely to occur on a much
shorter time-scale (inverse of the speed-of-light as opposed to the inverse of
the speed-of-sound). This separation of scales may make these signals
potentially observable in the near future. However, such measurements depend
greatly on an asteroseismological understanding of the spectral type, or any
other signature, that captures the average and/or dominant luminosity
fluctuations of the stellar oscillations of interest.

## Acknowledgements

It is a pleasure to thank Anupam A.H., Uddipan Banik, Glenn Barnich,
Chandramouli Chowdhury, Fabiano Feleppa, Nico Groenenboom, and Alok Laddha for
various helpful discussions. We are also grateful to the Cargese Summer School
2021 where this collaboration bore fruit, and Perimeter Institute, where a
part of this work was carried out, for hosting both authors.

AA is a Research Fellow of the Fonds de la Recherche Scientifique F.R.S.-FNRS
(Belgium). AA is partially supported by IISN – Belgium (convention 4.4503.15)
and by the Delta ITP consortium, a program of the NWO that is funded by the
Dutch Ministry of Education, Culture and Science (OCW).

NG was supported by the Delta-Institute for Theoretical Physics (D-ITP) that
is funded by the Dutch Ministry of Education, Culture and Science (OCW) during
the initial stages of this work and is currently supported by project RTI4001
of the Department of Atomic Energy, Govt. of India.

## References

  * [1] T. Regge and J. A. Wheeler, “Stability of a Schwarzschild singularity,” Phys. Rev. 108 (1957) 1063–1069.
  * [2] C. V. Vishveshwara, “Stability of the schwarzschild metric,” Phys. Rev. D 1 (1970) 2870–2879.
  * [3] F. J. Zerilli, “Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics,” Phys. Rev. D 2 (1970) 2141–2160.
  * [4] S. Chandrasekhar, The mathematical theory of black holes.  1985\.
  * [5] V. P. Frolov and I. D. Novikov, eds., Black hole physics: Basic concepts and new developments.  1998\.
  * [6] R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, “Gravitational radiation from Schwarzschild black holes: The Second order perturbation formalism,” Phys. Rept. 325 (2000) 41–81, arXiv:gr-qc/9807077.
  * [7] M. Sasaki and H. Tagoshi, “Analytic black hole perturbation approach to gravitational radiation,” Living Rev. Rel. 6 (2003) 6, arXiv:gr-qc/0306120.
  * [8] K. Martel, “Gravitational wave forms from a point particle orbiting a Schwarzschild black hole,” Phys. Rev. D 69 (2004) 044025, arXiv:gr-qc/0311017.
  * [9] V. Moncrief, “Gravitational perturbations of spherically symmetric systems. I. The exterior problem.,” Annals Phys. 88 (1974) 323–342.
  * [10] U. H. Gerlach and U. K. Sengupta, “GAUGE INVARIANT PERTURBATIONS ON MOST GENERAL SPHERICALLY SYMMETRIC SPACE-TIMES,” Phys. Rev. D 19 (1979) 2268–2272.
  * [11] O. Sarbach and M. Tiglio, “Gauge invariant perturbations of Schwarzschild black holes in horizon penetrating coordinates,” Phys. Rev. D 64 (2001) 084016, arXiv:gr-qc/0104061.
  * [12] C. T. Cunningham, R. H. Price, and V. Moncrief, “Radiation from collapsing relativistic stars. I - Linearized odd-parity radiation,” Astrophys. J. 224 (1978) 643.
  * [13] S. Jhingan and T. Tanaka, “Improvement on the metric reconstruction scheme in Regge-Wheeler-Zerilli formalism,” Phys. Rev. D 67 (2003) 104018, arXiv:gr-qc/0211060.
  * [14] K. Martel and E. Poisson, “Gravitational perturbations of the Schwarzschild spacetime: A Practical covariant and gauge-invariant formalism,” Phys. Rev. D 71 (2005) 104003, arXiv:gr-qc/0502028.
  * [15] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975) 199–220. [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * [16] S. W. Hawking, “Breakdown of Predictability in Gravitational Collapse,” Phys. Rev. D 14 (1976) 2460–2473.
  * [17] N. Gaddam, N. Groenenboom, and G. ’t Hooft, “Quantum gravity on the black hole horizon,” JHEP 01 (2022) 023, arXiv:2012.02357 [hep-th].
  * [18] N. Gaddam and N. Groenenboom, “Soft graviton exchange and the information paradox,” arXiv:2012.02355 [hep-th].
  * [19] P. Betzios, N. Gaddam, and O. Papadoulaki, “Black hole S-matrix for a scalar field,” JHEP 07 (2021) 017, arXiv:2012.09834 [hep-th].
  * [20] N. Gaddam and N. Groenenboom, “2 → 2N scattering: Eikonalisation and the Page curve,” JHEP 01 (2022) 146, arXiv:2110.14673 [hep-th].
  * [21] N. Gaddam and N. Groenenboom, “A toolbox for black hole scattering,” arXiv:2207.11277 [hep-th].
  * [22] G. ’t Hooft, “The Scattering matrix approach for the quantum black hole: An Overview,” Int. J. Mod. Phys. A 11 (1996) 4623–4688, arXiv:gr-qc/9607022.
  * [23] G. ’t Hooft, “Diagonalizing the Black Hole Information Retrieval Process,” arXiv:1509.01695 [gr-qc].
  * [24] G. ’t Hooft, “Black hole unitarity and antipodal entanglement,” Found. Phys. 46 no. 9, (2016) 1185–1198, arXiv:1601.03447 [gr-qc].
  * [25] P. Betzios, N. Gaddam, and O. Papadoulaki, “The Black Hole S-Matrix from Quantum Mechanics,” JHEP 11 (2016) 131, arXiv:1607.07885 [hep-th].
  * [26] A. Aggarwal and N. Gaddam, “In preparation,” (2023) .
  * [27] R. K. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128 (1962) 2851–2864.
  * [28] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-time,” Proc. Roy. Soc. Lond. A 270 (1962) 103.
  * [29] H. Bondi, M. G. van der Burg, and A. W. Metzner, “Gravitational waves in general relativity. 7. Waves from axi-symmetric isolated systems,” Proc. Roy. Soc. Lond. A 269 (1962) 21.
  * [30] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 (2014) 152, arXiv:1312.2229 [hep-th].
  * [31] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140 (1965) B516–B524.
  * [32] T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th].
  * [33] A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” arXiv:1703.05448 [hep-th].
  * [34] L. Donnay, G. Giribet, H. A. Gonzalez, and M. Pino, “Supertranslations and Superrotations at the Black Hole Horizon,” Phys. Rev. Lett. 116 no. 9, (2016) 091101, arXiv:1511.08687 [hep-th].
  * [35] S. W. Hawking, M. J. Perry, and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. 116 no. 23, (2016) 231301, arXiv:1601.00921 [hep-th].
  * [36] L. Donnay, G. Giribet, H. A. González, and M. Pino, “Extended Symmetries at the Black Hole Horizon,” JHEP 09 (2016) 100, arXiv:1607.05703 [hep-th].
  * [37] S. W. Hawking, M. J. Perry, and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes,” JHEP 05 (2017) 161, arXiv:1611.09175 [hep-th].
  * [38] S. Haco, S. W. Hawking, M. J. Perry, and A. Strominger, “Black Hole Entropy and Soft Hair,” JHEP 12 (2018) 098, arXiv:1810.01847 [hep-th].
  * [39] A. Aggarwal, A. Castro, and S. Detournay, “Warped Symmetries of the Kerr Black Hole,” JHEP 01 (2020) 016, arXiv:1909.03137 [hep-th].
  * [40] V. Chandrasekaran, E. E. Flanagan, and K. Prabhu, “Symmetries and charges of general relativity at null boundaries,” JHEP 11 (2018) 125, arXiv:1807.11499 [hep-th]. [Erratum: JHEP 07, 224 (2023)].
  * [41] 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 no. 4, (2020) 041601, arXiv:1908.09833 [hep-th].
  * [42] H. Adami, D. Grumiller, S. Sadeghian, M. M. Sheikh-Jabbari, and C. Zwikel, “T-Witts from the horizon,” JHEP 04 (2020) 128, arXiv:2002.08346 [hep-th].
  * [43] H. Adami, D. Grumiller, M. M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo, and C. Zwikel, “Null boundary phase space: slicings, news & memory,” JHEP 11 (2021) 155, arXiv:2110.04218 [hep-th].
  * [44] V. Chandrasekaran, E. E. Flanagan, I. Shehzad, and A. J. Speranza, “A general framework for gravitational charges and holographic renormalization,” Int. J. Mod. Phys. A 37 no. 17, (2022) 2250105, arXiv:2111.11974 [gr-qc].
  * [45] L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, “Extended corner symmetry, charge bracket and Einstein’s equations,” JHEP 09 (2021) 083, arXiv:2104.12881 [hep-th].
  * [46] L. Ciambelli and R. G. Leigh, “Isolated surfaces and symmetries of gravity,” Phys. Rev. D 104 no. 4, (2021) 046005, arXiv:2104.07643 [hep-th].
  * [47] M. M. Sheikh-Jabbari, “On symplectic form for null boundary phase space,” Gen. Rel. Grav. 54 no. 11, (2022) 140, arXiv:2209.05043 [gr-qc].
  * [48] V. Chandrasekaran and E. E. Flanagan, “The gravitational phase space of horizons in general relativity,” arXiv:2309.03871 [gr-qc].
  * [49] G. Odak, A. Rignon-Bret, and S. Speziale, “General gravitational charges on null hypersurfaces,” arXiv:2309.03854 [gr-qc].
  * [50] M. Mirbabayi and M. Porrati, “Dressed Hard States and Black Hole Soft Hair,” Phys. Rev. Lett. 117 no. 21, (2016) 211301, arXiv:1607.03120 [hep-th].
  * [51] A. Strominger, Black Hole Information Revisited, pp. 109–117.  2020\.  arXiv:1706.07143 [hep-th].
  * [52] R. Kallosh and A. A. Rahman, “Quantization of gravity in the black hole background,” Phys. Rev. D 104 no. 8, (2021) 086008, arXiv:2106.01966 [hep-th].
  * [53] R. Kallosh, “Quantization of gravity in spherical harmonic basis,” Phys. Rev. D 104 no. 8, (2021) 086023, arXiv:2107.02099 [hep-th].
  * [54] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D50 (1994) 846–864, gr-qc/9403028.
  * [55] G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conservation laws and central charges,” Nucl. Phys. B 633 (2002) 3–82, arXiv:hep-th/0111246.
  * [56] A. Fiorucci and G. Compère, Advanced Lectures in General Relativity.  PhD thesis, Brussels U., PTM, 2018.  arXiv:1801.07064 [hep-th].  http://inspirehep.net/record/1649253/files/arXiv:1801.07064.pdf.
  * [57] A. Aggarwal, “Supertranslations in Higher Dimensions Revisited,” Phys. Rev. D 99 no. 2, (2019) 026015, arXiv:1811.00093 [hep-th].
  * [58] E. Conde and P. Mao, “Remarks on asymptotic symmetries and the subleading soft photon theorem,” Phys. Rev. D 95 no. 2, (2017) 021701, arXiv:1605.09731 [hep-th].
  * [59] E. Conde and P. Mao, “BMS Supertranslations and Not So Soft Gravitons,” JHEP 05 (2017) 060, arXiv:1612.08294 [hep-th].
  * [60] G. Barnich and C. Troessaert, “BMS charge algebra,” JHEP 12 (2011) 105, arXiv:1106.0213 [hep-th].
  * [61] G. Barnich and P.-H. Lambert, “Einstein-Yang-Mills theory: Asymptotic symmetries,” Phys. Rev. D 88 (2013) 103006, arXiv:1310.2698 [hep-th].
  * [62] T. He, A.-M. Raclariu, and K. M. Zurek, “From Shockwaves to the Gravitational Memory Effect,” arXiv:2305.14411 [hep-th].
  * [63] N. Gaddam and T. He, “In preparation,” (2023) .
  * [64] P. Betzios, N. Gaddam, and O. Papadoulaki, “Antipodal correlation on the meron wormhole and a bang-crunch universe,” Phys. Rev. D 97 no. 12, (2018) 126006, arXiv:1711.03469 [hep-th].
  * [65] P. Betzios, N. Gaddam, and O. Papadoulaki, “Black holes, quantum chaos, and the Riemann hypothesis,” arXiv:2004.09523 [hep-th].
  * [66] G. ’t Hooft, “Quantum Clones inside Black Holes,” Universe 8 no. 10, (2022) 537.
  * [67] F. Feleppa, N. Gaddam, and N. Groenenboom, “Charged particle scattering near the horizon,” arXiv:2309.05791 [hep-th].
  * [68] U. H. Gerlach and U. K. Sengupta, “GAUGE INVARIANT COUPLED GRAVITATIONAL, ACOUSTICAL, AND ELECTROMAGNETIC MODES ON MOST GENERAL SPHERICAL SPACE-TIMES,” Phys. Rev. D 22 (1980) 1300–1312.
  * [69] S. Hollands, A. Ishibashi, and R. M. Wald, “BMS Supertranslations and Memory in Four and Higher Dimensions,” Class. Quant. Grav. 34 no. 15, (2017) 155005, arXiv:1612.03290 [gr-qc].
  * [70] S. Hollands and R. M. Wald, “Conformal null infinity does not exist for radiating solutions in odd spacetime dimensions,” Class. Quant. Grav. 21 (2004) 5139–5146, arXiv:gr-qc/0407014.
  * [71] D. Kapec, V. Lysov, S. Pasterski, and A. Strominger, “Higher-dimensional supertranslations and Weinberg’s soft graviton theorem,” Ann. Math. Sci. Appl. 02 (2017) 69–94, arXiv:1502.07644 [gr-qc].
  * [72] M. Pate, A.-M. Raclariu, and A. Strominger, “Gravitational Memory in Higher Dimensions,” JHEP 06 (2018) 138, arXiv:1712.01204 [hep-th].
  * [73] D. Colferai and S. Lionetti, “Asymptotic symmetries and the subleading soft graviton theorem in higher dimensions,” Phys. Rev. D 104 no. 6, (2021) 064010, arXiv:2005.03439 [hep-th].
  * [74] A. Campoleoni, D. Francia, and C. Heissenberg, “On asymptotic symmetries in higher dimensions for any spin,” JHEP 12 (2020) 129, arXiv:2011.04420 [hep-th].
  * [75] C. Chowdhury, A. A. H., and A. Kundu, “Generalized BMS algebra in higher even dimensions,” Phys. Rev. D 106 no. 12, (2022) 126025, arXiv:2209.06839 [hep-th].
  * [76] C. Chowdhury, R. Mishra, and S. G. Prabhu, “The Asymptotic Structure of Gravity in Higher Even Dimensions,” arXiv:2201.07813 [hep-th].
  * [77] F. Capone, Aspects of Holography beyond AdS BMS superrotations in higher dimensions.  PhD thesis, Southampton U., University of Southampton, U. Southampton, 2021.
  * [78] F. Capone, “General null asymptotics and superrotation-compatible configuration spaces in $d\geq 4$,” JHEP 10 (2021) 158, arXiv:2108.01203 [hep-th]. [Erratum: JHEP 02, 113 (2022)].
  * [79] O. Fuentealba, M. Henneaux, J. Matulich, and C. Troessaert, “Bondi-Metzner-Sachs Group in Five Spacetime Dimensions,” Phys. Rev. Lett. 128 no. 5, (2022) 051103, arXiv:2111.09664 [hep-th].
  * [80] O. Fuentealba, M. Henneaux, J. Matulich, and C. Troessaert, “Asymptotic structure of the gravitational field in five spacetime dimensions: Hamiltonian analysis,” JHEP 07 (2022) 149, arXiv:2206.04972 [hep-th].
  * [81] S. Lionetti, “Asymptotic symmetries and soft theorems in higher-dimensional gravity,” EPJ Web Conf. 270 (2022) 00034, arXiv:2209.10889 [hep-th].
  * [82] F. Capone, P. Mitra, A. Poole, and B. Tomova, “Phase Space Renormalization and Finite BMS Charges in Six Dimensions,” arXiv:2304.09330 [hep-th].
  * [83] E. T. Akhmedov and M. Godazgar, “Symmetries at the black hole horizon,” Phys. Rev. D 96 no. 10, (2017) 104025, arXiv:1707.05517 [hep-th].