TUW–20–02

††institutetext: a School of Physics, Institute for Research in Fundamental
Sciences (IPM),
P.O.Box 19395-5531, Tehran, Iran††institutetext: b Institute for Theoretical
Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna,
Austria††institutetext: c Wigner Research Centre for Physics, Konkoly-Thege
Miklos u. 29-33, 1121 Budapest, Hungary††institutetext: d The Abdus Salam
ICTP, Strada Costiera 11, Trieste, Italy

#  T-Witts from the horizon

H. Adami b , D. Grumiller a,c , S. Sadeghian a,d , M.M. Sheikh-Jabbari b and
C. Zwikel<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Expanding around null hypersurfaces, such as generic Kerr black hole horizons,
using co-rotating Kruskal–Israel-like coordinates we study the associated
surface charges, their symmetries and the corresponding phase space within
Einstein gravity. Our surface charges are not integrable in general. Their
integrable part generates an algebra including superrotations and a BMS3-type
algebra that we dub “T-Witt algebra”. The non-integrable part accounts for the
flux passing through the null hypersurface. We put our results in the context
of earlier constructions of near horizon symmetries, soft hair and of the
program to semi-classically identify Kerr black hole microstates.

††preprint: IPM/P-2019/041

##  1 Introduction

In this paper we study non-extremal Kerr black holes Kerr:1963ud  from a near
horizon perspective within Einstein gravity Einstein:1915ca . This work is
organized as follows:

  * •

In section 2 we introduce co-rotating Kruskal–Israel like coordinates for Kerr
black holes.

  * •

In section 3 we provide fall-off conditions suitable to describe the region
around the bifurcation surface, discuss the near bifurcation Killing vector
algebra and derive the associated surface charges, which turn out to be
finite, integrable and conserved.

  * •

In section 4 we consider more generic boundary conditions that only assume the
existence of a null hypersurface as (fiducial or actual) boundary as well as
additional technical assumptions (co-rotation and Taylor-expandability). We
derive the near null hypersurface Killing vectors that preserve these fall-off
conditions and deduce the associated null hypersurface symmetry algebra, which
contains a T-Witt subalgebra.

  * •

In section 5 we construct the surface charge variations associated with the
near horizon symmetry generators of section 4. They are neither integrable nor
conserved, in general. We employ the Barnich–Troessaert modified bracket
method to separate the integrable part of the charge and establish a
generalized charge conservation equation that relates its time-derivative to
the non-integrable part, which is interpreted as flux through the null
hypersurface. We also discuss ambiguities of the split into integrable and
flux-part and fix them such that the modified bracket algebra closes without
central extension, thereby recovering precisely the algebra discussed in
section 4.

  * •

In section 6 we examine the on-shell phase space and possible redundancies
that can be gauge fixed. We verify for generic and special cases (non-
expanding or stationary null hypersurfaces) that there is always a precise
match between the independent functions on the boundary phase space and the
independent functions in the symmetry generators.

  * •

In section 7 we impose physically well-motivated conditions on the allowed
variations and recover earlier constructions of near horizon symmetries as
special cases of our analysis.

  * •

In section 8 we conclude with a summary and give an outlook to current and
future research directions.

##  2 Kerr geometry and metric near its bifurcate horizon

The Kerr metric in Boyer–Lindquist coordinates

$\begin{split}\operatorname{d}\\!{s}^{2}=-\frac{\Delta}{\rho^{2}}\big{(}\operatorname{d}\\!{t}-a\sin^{2}\theta\operatorname{d}\\!{\varphi}\big{)}^{2}+\frac{\rho^{2}}{\Delta}\operatorname{d}\\!{r}^{2}+\rho^{2}\operatorname{d}\\!{\theta}^{2}+\frac{(r^{2}+a^{2})^{2}\sin^{2}\theta}{\rho^{2}}\big{(}\operatorname{d}\\!{\varphi}-\frac{a}{r^{2}+a^{2}}\operatorname{d}\\!{t}\big{)}^{2}\end{split}$
(2.1)

with

$\Delta=r^{2}-2Mr+a^{2}\qquad\qquad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$ (2.2)

depends parametrically on mass $M$ and angular momentum $J=Ma$. The angular
coordinates (also called “angular part”) are the usual polar angle
$\theta\in[0,\pi]$ and the azimuthal angle $\varphi\sim\varphi+2\pi$. The time
and radial coordinates (also referred to as “spacetime part”) are non-compact,
$t\in(-\infty,\,\infty)$ and $r\in[0,\,\infty)$.

The inner and outer horizon radii $r_{\pm}$ are respectively the smaller and
bigger roots of $\Delta=0$,

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}\,.$ (2.3)

The loci $r=r_{\pm}$ are bifurcate Killing horizons with a bifurcation
2-sphere $\cal B$. The outer one is simultaneously a black hole event horizon
and will be our main region of interest. Surface gravity at the black hole
horizon is given by

$\kappa=\frac{r_{+}-r_{-}}{2(r_{+}+r_{-})r_{+}}\,.$ (2.4)

Since Boyer–Lindquist coordinates are not well-adapted to near horizon
expansions we introduce new coordinates below.

###  2.1 Near horizon coordinates

For near horizon analyses various coordinate choices have been used in the
literature, e.g. Eddington–Finkelstein (EF) types of coordinates
Donnay:2015abr ; Donnay:2016ejv ; Chandrasekaran:2018aop , conformal types of
coordinates (HHPS) Haco:2018ske  and Rindler coordinates Grumiller:2019fmp .
Of course, there is also the venerable set of Kruskal coordinates
Kruskal:1959vx  or simplifications thereof Israel:1966 . Each of these have
their own merits and drawbacks. For our purpose the most striking difference
between various choices is the way the horizon is approached and whether or
not the bifurcation 2-sphere can be covered. We found none of these coordinate
systems suitable for our purposes.

Thus, we introduce a Kruskal–Israel-inspired coordinate system that is well-
adapted to study not only the black hole horizon, but specifically the region
near the bifurcation 2-sphere. The main features of our coordinates are

  * •

co-rotation with the horizon (as opposed to Kruskal- or Israel-coordinates)

  * •

no mixing of angular and spacetime coordinates (as opposed to HHPS)

  * •

cover open region around the bifurcation 2-sphere (as opposed to EF or
Rindler)

We describe now explicitly the new coordinates, starting with the angular
part.

Since we are interested in an expansion around the outer horizon, $r\sim
r_{+}$, it is useful to transform the azimuthal angle so that our coordinate
frame is co-rotating with the outer horizon

$\phi={\varphi}-\Omega_{\textrm{\tiny H}}\,t\qquad\qquad\Omega_{\textrm{\tiny
H}}=\frac{a}{r_{+}^{2}+a^{2}}\,.$ (2.5)

As we are content with the polar angle $\theta$ we do not transform it.

We address now the spacetime part. We introduce Kruskal–Israel-like
coordinates that we denote by $x^{\pm}$, defined by

$x^{+}=\pm\sqrt{\left|\frac{r-r_{+}}{r_{+}-r_{-}}\right|}\,e^{\kappa
t}\qquad\qquad
x^{-}=\pm\sqrt{\left|\frac{r-r_{+}}{r_{+}-r_{-}}\right|}\,e^{-\kappa t}$ (2.6)

the inverse of which is

$t=\frac{1}{2\kappa}\,\ln{\left|\frac{x^{+}}{x^{-}}\right|}\qquad\qquad
r=r_{+}-(r_{+}-r_{-})\,x^{+}x^{-}$ (2.7)

and hence

$\Delta=(r_{+}-r_{-})^{2}x^{+}x^{-}(x^{+}x^{-}-1)\qquad\qquad\rho^{2}=(r_{+}-(r_{+}-r_{-})x^{+}x^{-})^{2}+r_{+}r_{-}\cos^{2}\theta\,.$
(2.8)

In this coordinate system $x^{\pm}\in\mathbb{R}$. The signs in (2.6) are fixed
as follows. As in the usual Kruskal coordinates, the outside region
corresponds to $x^{+}>0,x^{-}<0$, the black hole region to $x^{\pm}>0$, the
white hole region to $x^{\pm}<0$ and the second outside region to
$x^{+}<0,x^{-}>0$. The locus $x^{+}x^{-}=0$ describes the bifurcate black hole
horizon and $x^{+}=x^{-}=0$ the bifurcation 2-sphere $\cal B$. The inner
(Cauchy) horizon corresponds to $x^{+}x^{-}=1$ and our coordinates break down
on it. Thus, the only restriction on the range of the coordinates $x^{\pm}$ is
that their product is smaller than unity, $x^{+}x^{-}<1$.

In these new coordinates the Kerr black hole is described by metrics of the
form

$\operatorname{d}\\!s^{2}=\rho^{2}\operatorname{d}\\!\theta^{2}-\rho^{2}\frac{(x^{-}\operatorname{d}\\!x^{+}+x^{+}\operatorname{d}\\!x^{-})^{2}}{x^{+}x^{-}(1-x^{+}x^{-})}+\frac{\sin^{2}\theta}{\rho^{2}}\Big{(}a\big{(}x^{+}\operatorname{d}\\!x^{-}-x^{-}\operatorname{d}\\!x^{+}\big{)}\big{(}2r_{+}-(r_{+}-r_{-})x^{+}x^{-}\big{)}\\\
+\big{(}r_{+}^{2}(1-x^{+}x^{-})^{2}+r_{-}^{2}(x^{+}x^{-})^{2}+r_{+}r_{-}(1+2x^{+}x^{-}(1-x^{+}x^{-}))\big{)}\,\operatorname{d}\\!\phi\Big{)}^{2}\\\
+\frac{1-x^{+}x^{-}}{\rho^{2}\,x^{+}x^{-}}\,\Big{(}\rho_{+}^{2}\,\big{(}x^{-}\operatorname{d}\\!x^{+}-x^{+}\operatorname{d}\\!x^{-}\big{)}-a\sin^{2}\theta\,(r_{+}-r_{-})\,x^{+}x^{-}\operatorname{d}\\!\phi\Big{)}^{2}$
(2.9)

with $a=\sqrt{r_{+}r_{-}}$ and
$\rho_{+}^{2}=r_{+}(r_{+}+r_{-}\cos^{2}\theta)$. In the extremal case,
$r_{+}=r_{-}>0$, the metric (2.9) simplifies to the near horizon extremal Kerr
(NHEK) geometry Bardeen:1999px

$\operatorname{d}\\!s^{2}_{\textrm{\tiny{NHEK}}}=r_{+}^{2}\,(1+\cos^{2}\theta)\,\Big{(}-\frac{(x^{-}\operatorname{d}\\!x^{+}+x^{+}\operatorname{d}\\!x^{-})^{2}}{x^{+}x^{-}(1-x^{+}x^{-})}+\frac{(x^{-}\operatorname{d}\\!x^{+}-x^{+}\operatorname{d}\\!x^{-})^{2}(1-x^{+}x^{-})}{x^{+}x^{-}}\\\
+\operatorname{d}\\!\theta^{2}+\frac{4\sin^{2}\theta}{(1+\cos^{2}\theta)^{2}}\,\big{(}\operatorname{d}\\!\phi-x^{-}\operatorname{d}\\!x^{+}+x^{+}\operatorname{d}\\!x^{-}\big{)}^{2}\Big{)}$
(2.10)

The reason why we obtain NHEK rather than extremal Kerr is because the
coordinate transformation (2.7) is singular in the extremal case and zooms
into the region $r=r_{+}$ for any finite values of $x^{\pm}$. So, our
coordinate system captures generic and near extremal cases. For convenience we
collect the ranges of the coordinates: $x^{\pm}\in\mathbb{R}$, $x^{+}x^{-}<1$,
$\theta\in[0,\pi]$ and $\phi\sim\phi+2\pi$.

The Killing vectors of the Kerr metric in the Boyer–Lindquist coordinates
(2.1) are $\partial_{t}$ and $\partial_{\varphi}$, associated with
stationarity and axi-symmetry, respectively. The combination
$\zeta_{\textrm{\tiny H}}=\partial_{t}+\Omega_{\textrm{\tiny
H}}\partial_{\varphi}$ is null, tangential and normal at the black hole
horizon $r=r_{+}$, so we use it instead of $\partial_{t}$. In our new
coordinates (2.9) the Killing vectors read

$\zeta_{\textrm{\tiny
H}}=\kappa\,\big{(}x^{+}\partial_{+}-x^{-}\partial_{-}\big{)}\qquad\qquad\zeta_{\phi}=\partial_{\phi}\,.$
(2.11)

Consistently, $\zeta_{\textrm{\tiny H}}$ is null at the bifurcate horizon
$x^{+}x^{-}=0$ and vanishes at the bifurcation 2-sphere $\cal B$, whose
properties we discuss next.

###  2.2 Bifurcation 2-sphere

The bifurcation surface ${\cal B}$ is a compact surface of $S^{2}$ topology
with metric

$\operatorname{d}\\!{\sigma}^{2}_{{}_{\cal
B}}=\frac{(2Mr_{+})^{2}\sin^{2}\theta}{\rho_{+}^{2}}\operatorname{d}\\!{\phi}^{2}+\rho_{+}^{2}\operatorname{d}\\!{\theta}^{2}$
(2.12)

where

$\rho_{+}^{2}=r_{+}(r_{+}+r_{-}\cos^{2}\theta)\,.$ (2.13)

The area of the horizon ${\cal B}$ is

$A=\int_{{\cal B}}\
\operatorname{d}\\!{\theta}\operatorname{d}\\!{\phi}\,(2Mr_{+})\sin\theta=8\pi
Mr_{+}\,.$ (2.14)

The metric (2.12) can be written as conformal to the metric of round 2-sphere

$\operatorname{d}\\!{\sigma}^{2}_{{}_{\cal
B}}=\Omega_{\text{\tiny{Kerr}}}\,\frac{4\operatorname{d}\\!{z}\operatorname{d}\\!{\bar{z}}}{(1+z\bar{z})^{2}}\,.$
(2.15)

by a Weyl factor

$\Omega_{\text{\tiny{Kerr}}}=\frac{(2Mr_{+})^{2}\sin^{2}\theta}{4\rho_{+}^{2}}\Big{(}\mu(\theta)+\frac{1}{\mu(\theta)}\Big{)}^{2}$
(2.16)

with coordinates defined by

$z=e^{i\phi}\mu(\theta)\qquad\qquad\bar{z}=e^{-i\phi}\mu(\theta)\qquad\qquad\mu(\theta)=e^{-\frac{r_{-}}{2M}\cos{\theta}}\cot{\frac{\theta}{2}}\,.$
(2.17)

##  3 Near horizon symmetries at the bifurcation surface

In this section we analyse the horizon symmetries and associated charges
around the bifurcation surface ${\cal B}$, which we refer to as “near
bifurcation symmetries”. In section 3.1 we discuss the most general fall-off
behavior for metric fluctuations preserving the bifurcation surface. In
section 3.2 we derive the near bifurcation Killing vectors compatible with the
fall-off conditions. In section 3.3 we study the ensuing near bifurcation
Killing vector algebra. In section 3.4 we determine the charges associated
with the near bifurcation Killing vectors, investigate their integrability and
present the algebra generated by them.

###  3.1 Fall-off conditions

To motivate our fall-off conditions, we expand the Kerr metric (2.9) near the
bifurcation surface ${\cal B}$ at $x^{\pm}=0$,

$\operatorname{d}\\!{s}^{2}=-4\rho_{+}^{2}\operatorname{d}\\!{x}^{+}\operatorname{d}\\!{x}^{-}-{8Ma\left(\frac{r_{+}^{2}}{{\rho_{+}^{2}}}+\kappa
r_{+}\right)\sin^{2}\theta}\,\big{(}x^{-}\operatorname{d}\\!{x}^{+}-x^{+}\operatorname{d}\\!{x}^{-}\big{)}\operatorname{d}\\!{\phi}+\operatorname{d}\\!{\sigma}^{2}_{{}_{\cal
B}}+\cdots$ (3.1)

where $\operatorname{d}\\!{\sigma}^{2}_{{}_{\cal B}}$ is given in (2.12) and
the ellipsis denotes higher order terms in $x^{\pm}$. In this expansion we are
assuming $x^{+}$ and $x^{-}$ to be small and of the same order.

As suggested by the expansion (3.1) we postulate the near bifurcation fall-off
conditions

$\displaystyle g_{\pm\pm}$ $\displaystyle=\mathcal{O}(x^{2})$ $\displaystyle
g_{\pm A}$ $\displaystyle=x^{\mp}C^{\pm}_{A}(x^{B})+\mathcal{O}(x^{3})$ (3.2a)
$\displaystyle g_{+-}$ $\displaystyle=\eta(x^{B})+\mathcal{O}(x^{2})$
$\displaystyle g_{AB}$ $\displaystyle=\Omega_{AB}(x^{C})+\mathcal{O}(x^{2})$
(3.2b)

where $x^{A}=(\theta,\phi)$ denote the coordinates on ${\cal B}$. To avoid
clutter we use $\mathcal{O}(x^{n})$ as a shorthand for
$\mathcal{O}((x^{\pm})^{n})$. The near bifurcation expansion functions,
$\Omega_{AB},C_{A},\ \eta$, are not constrained by the Einstein field
equations to leading order. The fall-off conditions (3.2) are the most general
ones (subject to Taylor-expandability) preserving the bifurcation 2-sphere at
$x^{\pm}=0$.

###  3.2 Near bifurcation Killing vectors

The diffeomorphisms that keep the near bifurcation expansion (3.2) intact are
generated by “near bifurcation Killing vectors” given by

$\xi^{\pm}=\pm
x^{\pm}T^{\pm}(x^{A})+\mathcal{O}(x^{3})\qquad\qquad\xi^{A}=Y^{A}(x^{B})+\mathcal{O}(x^{2})\,.$
(3.3)

Under a transformation generated by near bifurcation Killing vector fields
(3.3), the leading order metric functions transform as

$\displaystyle\delta_{\xi}\eta$
$\displaystyle=Y^{A}\partial_{A}\eta+\left(T^{+}-T^{-}\right)\eta$ (3.4a)
$\displaystyle\delta_{\xi}C^{\pm}_{A}$
$\displaystyle=Y^{B}\partial_{B}C^{\pm}_{A}+C^{\pm}_{B}\partial_{A}Y^{B}+\left(T^{+}-T^{-}\right)C^{\pm}_{A}\mp\eta\partial_{A}T^{\mp}$
(3.4b) $\displaystyle\delta_{\xi}\Omega_{AB}$
$\displaystyle=Y^{C}\partial_{C}\Omega_{AB}+\Omega_{AC}\partial_{B}Y^{C}+\Omega_{BC}\partial_{A}Y^{C}$
(3.4c)

In the case above $\eta$, $C_{A}^{\pm}$ and $\Omega_{AB}$ are tensors (scalar,
1-forms and 2-tensor, respectively) on ${\cal B}$, while in later sections we
shall also encounter tensor-densities. The $\xi^{\pm}$ in (3.3) generate
$x^{A}$-dependent scalings in $x^{\pm}$ directions. As required, these
transformations map $x^{\pm}=0$ to itself and hence keep ${\cal B}$ intact.

###  3.3 Near bifurcation Killing vector algebra

Under the usual Lie bracket, the near bifurcation Killing vector fields (3.3)
satisfy the algebra

$\left[\xi(T^{+}_{1},T^{-}_{1},Y^{A}_{1}),\,\xi(T^{+}_{2},T^{-}_{2},Y^{A}_{2})\right]=\xi(T^{+}_{12},T^{-}_{12},Y^{A}_{12})$
(3.5)

where

$T^{\pm}_{12}=Y^{A}_{1}\partial_{A}T^{\pm}_{2}-Y^{A}_{2}\partial_{A}T^{\pm}_{1}\qquad\qquad
Y^{A}_{12}=Y^{B}_{1}\partial_{B}Y^{A}_{2}-Y^{B}_{2}\partial_{B}Y^{A}_{1}\,.$
(3.6)

The generators $T^{\pm},Y^{A}$ are functions on ${\cal B}$. As (3.6) shows,
$Y^{A}$ generate general 2d diffeomrphisms on ${\cal B}$. The $T^{\pm}$
transform as scalars under 2d diffeomorphisms, while commuting among
themselves and with each other. Thus, the $T^{\pm}$ generate so-called
supertranslations.

Analogously to Barnich:2009se ; Barnich:2010eb  for simplicity we impose a
further restriction on the metric, namely conformality to the round $S^{2}$ of
the co-dimension two metric $\Omega_{AB}$,

$\Omega_{AB}\operatorname{d}\\!{x}^{A}\,\operatorname{d}\\!{x}^{B}=\Omega\,\gamma_{AB}\,\operatorname{d}\\!{x}^{A}\,\operatorname{d}\\!{x}^{B}=\Omega\,\frac{4\,\operatorname{d}\\!{z}\,\operatorname{d}\\!{\bar{z}}}{(1+z\,\bar{z})^{2}}\,.$
(3.7)

The assumption (3.7) allows us to work with a single function $\Omega$ instead
of $\Omega_{AB}$. That is, $\eta,\Omega,C^{\pm}_{A}$ are now the functions
parametrizing our phase space of this section. At the level of diffeos, this
amounts to working with a subclass of the $Y^{A}$ generating Weyl rescalings
of ${\cal B}$, i.e., we are restricting to superrotations rather than generic
2d diffeomorphisms.

In the coordinates $z,\bar{z}$ defined in (3.7) the generators expand as

$\displaystyle\mathcal{T}^{\pm}_{n,m}:=\pm
z^{n}\bar{z}^{m}x^{\pm}\partial_{\pm}$ $\displaystyle\xi^{\pm}\partial_{\pm}$
$\displaystyle=\sum_{n,m\in\mathbb{Z}}\tau^{\pm}_{nm}\mathcal{T}^{\pm}_{n,m}$
(3.8a) $\displaystyle\mathcal{L}_{n}:=-z^{n+1}\partial_{z}\
,\quad{\bar{\mathcal{L}}}_{n}=-\bar{z}^{n+1}\partial_{\bar{z}}$ $\displaystyle
Y^{A}(x^{B})\partial_{A}$ $\displaystyle=\sum_{n\in\mathbb{Z}}\big{(}{\cal
Y}_{n}\mathcal{L}_{n}+\bar{\cal Y}_{n}{\bar{\mathcal{L}}}_{n}\big{)}$ (3.8b)

where $\tau^{\pm}_{nm}$, ${\cal Y}_{n}$ and $\bar{\cal Y}_{n}$ are arbitrary
numbers.

In the basis spanned by ${\cal L}_{n},{\cal{\bar{L}}}_{n},{\cal
T}^{\pm}_{n,m}$ the algebra (3.5) takes the form

$\displaystyle[{\mathcal{L}}_{n},{\mathcal{L}}_{m}]$
$\displaystyle=(n-m)\mathcal{L}_{n+m}$
$\displaystyle[{\bar{\mathcal{L}}}_{n},{\bar{\mathcal{L}}}_{m}]$
$\displaystyle=(n-m){\bar{\mathcal{L}}}_{n+m}$ (3.9a)
$\displaystyle[\mathcal{L}_{k},\mathcal{T}^{\pm}_{n,m}]$ $\displaystyle=-n\
\mathcal{T}^{\pm}_{n+k,m}$
$\displaystyle[{\bar{\mathcal{L}}}_{k},\mathcal{T}^{\pm}_{n,m}]$
$\displaystyle=-m\ \mathcal{T}^{\pm}_{n,m+k}$ (3.9b)
$\displaystyle[\mathcal{L}_{n},{\bar{\mathcal{L}}}_{m}]$ $\displaystyle=0$
$\displaystyle[\mathcal{T}^{\pm}_{n,m},\mathcal{T}^{\pm}_{k,l}]$
$\displaystyle=0=[\mathcal{T}^{+}_{n,m},\mathcal{T}^{-}_{k,l}]\,.$ (3.9c)

The algebra (3.9a) consists of a Witt $\oplus$ Witt algebra (the
“superrotation part”), generated by $\mathcal{L}_{n},{\bar{\mathcal{L}}}_{m}$,
and two towers of supertranslations generated by $\mathcal{T}^{\pm}_{n,m}$.
This algebra closely resembles the DGGP algebra Donnay:2015abr  but we have
two copies of supertranslations, instead of one copy there. While the second
copy will not play any role in this section, it will become important from
section 4 onwards.

###  3.4 Conserved charges and their algebra

There are different methods to compute the charges and their algebras,
including the Hamiltonian formulation Regge:1974zd ; Brown:1986nw ;
Henneaux:2018cst , action based Lee:1990nz ; Iyer:1994ys  or field-equation
based Barnich:2001jy ; Barnich:2006av  formulations. These methods provide a
formula for computing surface charge variations associated with the near
bifurcation Killing vectors, which generically may be written as

$\not{\delta}Q_{\xi}[g;\,\delta g]=\frac{1}{8\pi G}\int_{\cal B}{\cal
Q}^{\mu\nu}[g,\nabla_{\mu};\delta g,\xi]\,\operatorname{d}\\!{x}_{\mu\nu}$
(3.10)

where $g$ is the metric of the background, $\delta g$ the allowed metric
variations around this background and $\xi$ is any near bifurcation symmetry
generator. The two-form components ${\cal Q}^{\mu\nu}[g,\nabla_{\mu};\delta
g,\xi]$ are linear in both $\delta g$ and in $\xi$. Index manipulations and
covariant derivative $\nabla_{\mu}$ are both defined with respect to the
background metric $g$. Finally, $\operatorname{d}\\!\sigma_{\mu\nu}$ denotes
the induced co-dimension two volume form on ${\cal B}$.

The explicit form of ${\cal Q}$ is determined by the theory (and somewhat by
the method used as each of the charge-variation computation methods comes with
some different ambiguities to be dealt with). The $\not{\delta}$ on the left-
hand-side of (3.10) is used to stress that $\not{\delta}Q_{\xi}[g;\delta g]$
is not necessarily integrable in phase space. That is, there may or may not be
a charge $Q_{\xi}[g]$ as a function over the phase space of metrics, such that
$\delta Q_{\xi}[g]=\not{\delta}Q_{\xi}[g;\delta g]$. If this equality holds we
call $\not{\delta}Q_{\xi}[g;\delta g]$ integrable and replace
$\not{\delta}\to\delta$. If the charge is integrable then the fundamental
theorem of the covariant phase space method (see Compere:2018aar  and refs.
therein) states

$\delta_{\xi_{2}}Q_{\xi_{1}}=\\{Q_{\xi_{1}},Q_{\xi_{2}}\\}=Q_{[\xi_{1},\xi_{2}]}+K(\xi_{1},\xi_{2})$
(3.11)

where the bracket is defined by the first equality and $K(\xi_{1},\xi_{2})$ is
a possible central extension.

In the present case the metric $g+\delta g$ is given by the near bifurcation
fall-off (3.2), the symmetry generators by the near bifurcation Killing
vectors (3.3) and the theory is general relativity. The action based covariant
phase space method Lee:1990nz ; Iyer:1994ys  yields

$\not{\delta}Q_{\xi}=\frac{1}{8\pi G}\oint_{{\cal
B}}\text{d}^{2}x_{\mu\nu}\sqrt{-g}\,\Big{(}h^{\lambda[\mu}\nabla_{\lambda}\xi^{\nu]}-\xi^{\lambda}\nabla^{[\mu}h^{\nu]}_{\lambda}-\frac{1}{2}h\nabla^{[\mu}\xi^{\nu]}+\xi^{[\mu}\nabla_{\lambda}h^{\nu]\lambda}-\xi^{[\mu}\nabla^{\nu]}h\Big{)}$
(3.12)

where $h_{\mu\nu}=\delta g_{\mu\nu}$ is a metric variation allowed by the near
bifurcation fall-off (3.2).

Assuming that the generators $T^{\pm}$, $Y^{A}$ are field-independent the
charges are integrable and independent from the difference $T^{+}-T^{-}$,

$Q(T,\,Y^{A})\equiv\int_{\cal B}\operatorname{d}\\!^{2}x\,\big{(}T\,{\cal
P}+Y^{A}\,{\cal J}_{A}\big{)}$ (3.13)

with $T=(T^{+}+T^{-})/2$ and the charge densities

${\cal P}=\frac{\Omega}{8\pi G}\qquad\qquad{\cal J}_{A}=-\frac{\Omega}{16\pi
G}\frac{C^{+}_{A}-C^{-}_{A}}{\eta}\,.$ (3.14)

Since there are fewer charges in (3.14) than functions parametrizing our phase
space, $\eta$, $\Omega$, $C_{A}^{\pm}$, there is a redundancy in our phase
space. We discuss its physical meaning and how to fix it in section 6.1.

As an example we consider Kerr black holes (2.9), the charges of which are
given by

${\cal P}^{\textrm{\tiny Kerr}}=\frac{r_{+}(r_{+}+r_{-})}{8\pi
G}\,\sin\theta\qquad\qquad{\cal J}^{\textrm{\tiny Kerr}}_{A}={\cal
P}^{\textrm{\tiny Kerr}}\,\varepsilon_{A}{}^{B}\partial_{B}\psi^{\textrm{\tiny
Kerr}}$ (3.15)

with
$\varepsilon_{\phi}{}^{\theta}=-g^{\theta\theta}\,\sqrt{\det\Omega_{AB}}=-(r_{+}+r_{-})\sin\theta/(r_{+}+r_{-}\cos^{2}\theta)$,
and

$\psi^{\textrm{\tiny
Kerr}}=4\arctan\big{(}\sqrt{r_{-}/r_{+}}\cos\theta\big{)}+2\,\frac{r_{+}-r_{-}}{r_{+}+r_{-}}\,\sqrt{r_{-}/r_{+}}\cos\theta$
(3.16)

which essentially coincides with Eq. (VII.1) in Grumiller:2019fmp .

Let us now come back to the general result for the charges (3.13) and derive
the algebra generated by them using (3.11). The transformation laws (3.4)
yield

$\displaystyle\delta_{\xi}\mathcal{P}$
$\displaystyle=Y^{A}\partial_{A}\mathcal{P}+\mathcal{P}\partial_{A}Y^{A}$
(3.17a) $\displaystyle\delta_{\xi}\mathcal{J}_{A}$
$\displaystyle=Y^{B}\partial_{B}\mathcal{J}_{A}+\mathcal{J}_{B}\partial_{A}Y^{B}+\mathcal{J}_{A}\partial_{B}Y^{B}+\mathcal{P}\partial_{A}T$
(3.17b)

which is the usual transformation behavior of scalar- and vector-densities of
weight one under 2d diffeos generated by $Y^{A}$. The algebra above coincides
with the DGGP algebra Donnay:2015abr , see also Grumiller:2019fmp , which
notably features no central extension, i.e., the quantity
$K(\xi_{1},\,\xi_{2})$ in (3.11) vanishes. Thus, the near bifurcation boundary
conditions recover known results.

In the next section we generalize these boundary conditions by dropping the
assumption of a preserved bifurcation 2-sphere. This is physically well-
motivated since real black holes that formed through some collapse of matter
do not have bifurcate horizons.

##  4 T-Witt symmetries on null hypersurfaces

In this section we study near horizon symmetries associated with generic
points on the future horizon. For a black hole like Kerr, the future horizon
is not just an arbitrary null surface but has more properties and structure,
e.g. it is a Killing and an event horizon. However, these extra properties are
not generically respected by physically important processes like objects
falling into the black hole or Hawking radiation. To allow for such processes
we relax some of these properties and impose merely the existence of a (co-
rotating) null hypersurface with suitable fall-off conditions that we refer to
as “null hypersurface conditions”; the associated symmetries are captured by
the null hypersurface symmetry algebra.

In section 4.1 we introduce near null hypersurface fall-off behavior, taking
into account on-shell constraints. In section 4.2 we derive the near null
hypersurface Killing vectors. In section 4.3 we introduce the null
hypersurface symmetry algebra associated with the near null hypersurface
Killing vectors, and its particularly important subalgebra, which we call
“T-Witt algebra”.

###  4.1 Near null hypersurface expansion and on-shell conditions

Consider a null hypersurfaces which with no loss of generality we assume to be
located at $x^{-}=0$. The 1-form normal to the surface,
$l_{\mu}\propto\delta^{-}_{\mu}$ at $x^{-}=0$, by definition has vanishing
norm

$g^{\mu\nu}l_{\mu}l_{\nu}\big{|}_{x^{-}=0}=0\qquad\Longrightarrow\qquad
g^{--}|_{x^{-}=0}=0\,.$ (4.1)

The hypersurface is parametrized by $x^{+}$ and $x^{A}$, where $x^{+}$ is the
‘time’ along the null hypersurface and $x^{A}$ denotes the angular part. As
before we assume hypersurface-orthogonality (which in previous sections
implied co-rotation) with respect to the null hypersurface

$l^{\mu}\partial_{\mu}x^{A}|_{x^{-}=0}=0\qquad\Longrightarrow\qquad
g^{-A}\big{|}_{x^{-}=0}=0\,.$ (4.2)

A physical consequence of this geometric assumption is that the normal vector
field $l^{\mu}\partial_{\mu}\propto\partial_{+}$ has no angular components and
hence can be identified with the Hamiltonian (for a suitable choice of the
proportionality factor). This justifies to label $x^{+}$ as ‘time’.

The conditions (4.1) and (4.2) together with regularity of the metric at
$x^{-}=0$ yields the following general fall-off behavior near the null
hypersurface

$\displaystyle g_{++}$ $\displaystyle={x^{-}\eta\,{\cal
G}}+\mathcal{O}\left([x^{-}]^{2}\right)$ $\displaystyle g_{--}$
$\displaystyle=\mathcal{F}+\mathcal{O}\left(x^{-}\right)$ (4.3a)
$\displaystyle g_{+A}$
$\displaystyle=x^{-}\mathcal{V}_{A}+\mathcal{O}\left([x^{-}]^{2}\right)$
$\displaystyle g_{-A}$
$\displaystyle=\mathcal{U}_{A}+\mathcal{O}\left(x^{-}\right)$ (4.3b)
$\displaystyle g_{AB}$
$\displaystyle=\Omega_{AB}+\mathcal{O}\left(x^{-}\right)$ $\displaystyle
g_{+-}$ $\displaystyle=\eta+\mathcal{O}\left(x^{-}\right)$ (4.3c)

where all fields appearing in (4.3) are functions of time $x^{+}$ and angles
$x^{A}$. To have non-vanishing determinant of the metric we require $\eta\neq
0$ and $\det\Omega_{AB}\neq 0$. At this stage it is not clear which
(combinations of the) fields correspond to physical boundary degrees of
freedom and which of them are pure gauge. We shall clarify this after
computing the charges. As an example the metric expansion of the Kerr black
hole metric (2.9) around the future null horizon $x^{-}=0$ is displayed in
appendix A.

We consider now on-shell conditions. Unlike the near bifurcation boundary
conditions in section 3.1, the dynamics of the metric functions in (4.3) are
partly determined by the near null hypersurface Einstein equations,
corresponding to the leading orders in $x^{-}$ of the full field equations
$R_{\mu\nu}=0$. The $\mathcal{O}(1)$ of $R_{++}=0$ in general imposes a
relation between time derivatives of $\det\Omega_{AB}$ and
$\det(\partial_{+}\Omega_{AB})$. For our choice (3.7),111The physical
motivation for the restriction (3.7) is that it eliminates part of the
gravitational wave excitations characterized by the traceless part of
$\Omega_{AB}$ and thus allows us to focus on the boundary excitations, the
main interest of our present work. To be more precise, we are eliminating all
outgoing gravitational waves propagating along the null hypersurface and all
ingoing gravitational waves that would yield a memory effect, i.e., an imprint
on the null hypersurface. The remaining gravitational waves allowed by the
restriction (3.7) are ingoing gravitational waves with a node on the null
hypersurface.  where $\sqrt{\det\Omega_{AB}}=\gamma\Omega$ in which $\gamma$
is a fixed given function of $z,\bar{z}$, this equation is

$\partial^{2}_{+}\Omega+\frac{1}{2}\partial_{+}\Omega\,\Big{(}\Gamma-\frac{\partial_{+}\eta}{\eta}-\frac{\partial_{+}\Omega}{\Omega}\Big{)}=0$
(4.4)

where

$\Gamma:={\cal G}-\frac{\partial_{+}\eta}{\eta}\,.$ (4.5)

The $\mathcal{O}(1)$ of $R_{+A}=0$ gives

$\partial_{+}\Upsilon_{A}+{\Omega\partial_{A}\Gamma}-\Omega\partial_{+}\partial_{A}\ln\Omega+\partial_{+}\Omega\,\frac{\partial_{A}\eta}{\eta}=0$
(4.6)

where

$\Upsilon_{A}\,[\mathcal{V}_{A},\mathcal{U}_{A},\eta,\Omega]:=-\frac{\Omega}{\eta}\,\Big{(}\mathcal{V}_{A}-\partial_{+}\mathcal{U}_{A}+\frac{\partial_{+}\Omega}{\Omega}\,\mathcal{U}_{A}\Big{)}\,.$
(4.7)

The other components of the field equations do not yield further constraints
for the leading order functions.

###  4.2 Near null hypersurface Killing vectors and field variations

The fall-off conditions (4.3), the null hypersurface at $x^{-}=0$ and the
direction of the 1-form normal $l_{\mu}$ are preserved by the near null
hypersurface Killing vector fields

$\displaystyle\xi^{+}$
$\displaystyle=T^{+}(x^{+},x^{B}){+x^{-}\xi_{1}^{+}(x^{+},x^{B})}+\mathcal{O}\left([x^{-}]^{2}\right),$
(4.8a) $\displaystyle\xi^{-}$
$\displaystyle=-x^{-}T^{-}({x^{+}},x^{B})+\mathcal{O}\left([x^{-}]^{2}\right)$
(4.8b) $\displaystyle\xi^{A}$
$\displaystyle=Y^{A}(x^{B})+x^{-}\,\xi_{1}^{A}(x^{+},x^{B})+\mathcal{O}\left([x^{-}]^{2}\right)$
(4.8c)

where $Y^{A}$ satisfy the conformal Killing equation for the metric
$\mathcal{L}_{Y}\gamma_{AB}=(\bar{\nabla}_{C}Y^{C})\gamma_{AB}$ with
$\bar{\nabla}_{A}$ being the covariant derivative with respect to
$\gamma_{AB}$, defined in (3.7).

The remarkable feature of these symmetry generators is the appearance of two
time-dependent functions $T^{\pm}(x^{+},x^{A})$. As discussed below, this time
dependence brings novel features and new physics.

Under the transformations generated by the near null hypersurface Killing
vector fields (4.8) the functions in the near null hypersurface metric (4.3)
transform as

$\displaystyle\delta_{\xi}\Omega_{AB}$
$\displaystyle=T^{+}\partial_{+}\Omega_{AB}+Y^{C}\partial_{C}\Omega_{AB}+\Omega_{AC}\partial_{B}Y^{C}+\Omega_{BC}\partial_{A}Y^{C}$
(4.9a) $\displaystyle\delta_{\xi}\eta$
$\displaystyle=\partial_{+}\big{(}T^{+}\eta\big{)}-T^{-}\eta+Y^{A}\partial_{A}\eta$
(4.9b) $\displaystyle\delta_{\xi}\mathcal{G}$
$\displaystyle=-2\partial_{+}T^{-}+\partial_{+}(\mathcal{G}\,T^{+})+Y^{A}\partial_{A}\mathcal{G}\,.$
(4.9c)

Using (3.7) and that $\gamma_{AB}$ is fixed under the variations, i.e.
$\delta_{\xi}\gamma_{AB}=0$, yields

$\delta_{\xi}\Omega=T^{+}\partial_{+}\Omega+Y^{A}\partial_{A}\Omega+\Omega\,\bar{\nabla}_{A}Y^{A}$
(4.10)

and

$\displaystyle\delta_{\xi}\mathcal{F}$
$\displaystyle=T^{+}\partial_{+}\mathcal{F}-2T^{-}\mathcal{F}+Y^{A}\partial_{A}\mathcal{F}+2\,\eta\,\xi^{+}_{1}+2\,\mathcal{U}_{A}\xi^{A}_{1}$
(4.11a) $\displaystyle\delta_{\xi}\mathcal{U}_{A}$
$\displaystyle=T^{+}\partial_{+}\mathcal{U}_{A}-T^{-}\mathcal{U}_{A}+\eta\partial_{A}T^{+}+\Omega_{AB}\xi^{B}_{1}+Y^{B}\partial_{B}\mathcal{U}_{A}+\mathcal{U}_{B}\partial_{A}Y^{B}$
(4.11b) $\displaystyle\delta_{\xi}\mathcal{V}_{A}$
$\displaystyle=\partial_{+}\left(T^{+}\mathcal{V}_{A}\right)-T^{-}\mathcal{V}_{A}-(\eta\partial_{A}+\mathcal{U}_{A}\partial_{+})T^{-}+\eta\,\mathcal{G}\,\partial_{A}T^{+}+\Omega_{AB}\partial_{+}\xi^{B}_{1}$
$\displaystyle\quad+Y^{B}\partial_{B}\mathcal{V}_{A}+\mathcal{V}_{B}\partial_{A}Y^{B}\,.$
(4.11c)

The composite fields which appear in the field equations (4.4) and (4.6),
$\Gamma$ and $\Upsilon_{A}$ defined in (4.5) and (4.7) vary as

$\displaystyle\\!\\!\delta_{\xi}\Upsilon_{A}$
$\displaystyle=T^{+}\partial_{+}\Upsilon_{A}-\Omega\,\Gamma\,\partial_{A}T^{+}-\partial_{+}\Omega\,\partial_{A}T^{+}+2\Omega\,\partial_{A}\tilde{T}^{-}$
$\displaystyle\quad+Y^{B}\partial_{B}\Upsilon_{A}+\Upsilon_{B}\partial_{A}Y^{B}+\Upsilon_{A}\partial_{B}Y^{B}$
(4.12) $\displaystyle\\!\\!\delta_{\xi}\Gamma$
$\displaystyle=T^{+}\partial_{+}\Gamma+\Gamma\,\partial_{+}T^{+}-2\partial_{+}\tilde{T}^{-}+Y^{A}\partial_{A}\Gamma$
(4.13)

with $\tilde{T}^{-}:=\frac{1}{2}\,(T^{-}+\partial_{+}T^{+})$. The
transformation laws above show that $\Upsilon_{A}$ is a 1-form density of
weight one with respect to $Y$-diffeos and mixes with $\Gamma$ under
$T^{+}$-diffeos. The quantity $\Gamma$ is a scalar with respect to $Y$-diffeos
and a 1-form with respect to $T^{+}$-diffeos.

###  4.3 T-Witt and null hypersurface symmetry algebras

As may be anticipated on general grounds (and as shown explicitly in the next
section) the charges associated with subleading terms of the near null
hypersurface Killing vectors, parameterized by $\xi_{1}^{+},\xi_{1}^{A}$ in
(4.8), vanish. Therefore, here we only consider non-zero $T^{\pm},Y^{A}$ in
the near null hypersurface Killing vectors (4.8). Their Lie bracket is

$\left[\xi(T^{+}_{1},T^{-}_{1},Y^{A}_{1}),\xi(T^{+}_{2},T^{-}_{2},Y^{A}_{2})\right]=\xi(T^{+}_{12},T^{-}_{12},Y^{A}_{12})$
(4.14)

where

$\displaystyle T^{+}_{12}$
$\displaystyle=T_{1}^{+}\partial_{+}T_{2}^{+}-T_{2}^{+}\partial_{+}T_{1}^{+}+Y^{A}_{1}\partial_{A}T^{+}_{2}-Y^{A}_{2}\partial_{A}T^{+}_{1}$
(4.15a) $\displaystyle T^{-}_{12}$
$\displaystyle=T_{1}^{+}\partial_{+}T_{2}^{-}-T_{2}^{+}\partial_{+}T_{1}^{-}+Y^{A}_{1}\partial_{A}T^{-}_{2}-Y^{A}_{2}\partial_{A}T^{-}_{1}$
(4.15b) $\displaystyle Y^{A}_{12}$
$\displaystyle=Y^{B}_{1}\partial_{B}Y^{A}_{2}-Y^{B}_{2}\partial_{B}Y^{A}_{1}\,.$
(4.15c)

With the Taylor–Maclaurin expansions in $x^{+}$ and the Laurent expansions in
$z$ and $\bar{z}$

$\displaystyle T^{+}(x^{+};z,\bar{z})\,\partial_{+}$
$\displaystyle=\sum_{n=-1}^{\infty}\sum_{p,q=-\infty}^{\infty}{\tau}^{n}_{pq}\,{\cal
R}_{n}^{p,q}$ $\displaystyle{\cal R}_{n}^{p,q}$
$\displaystyle:=-(x^{+})^{n+1}z^{p}\bar{z}^{q}\,\partial_{+}$ (4.16a)
$\displaystyle T^{-}(x^{+};z,\bar{z})\,\partial_{-}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{p,q=-\infty}^{\infty}{\rho}^{n}_{pq}\,{\cal
T}_{n}^{p,q}$ $\displaystyle{\cal T}_{n}^{p,q}$
$\displaystyle:=-(x^{+})^{n}z^{p}\bar{z}^{q}\,\partial_{-}$ (4.16b)
$\displaystyle Y^{A}(z,\bar{z})\,\partial_{A}$
$\displaystyle=\sum_{p=-\infty}^{\infty}\left({\cal Y}_{p}\,{\cal
L}^{p}+{\bar{\cal Y}}_{p}\,{\cal{\bar{L}}}^{p}\right)$ $\displaystyle{\cal
L}^{p}$
$\displaystyle:=-z^{p+1}\,\partial_{z}\,,\quad{\cal{\bar{L}}}^{p}:=-{\bar{z}}^{p+1}\,\partial_{\bar{z}}$
(4.16c)

where ${\tau}^{n}_{pq},{\rho}^{n}_{pq},{\cal Y}_{n}$ and ${\bar{\cal Y}}_{n}$
are some arbitrary numbers, the algebra (4.14) reads

$\displaystyle[{\cal R}_{n}^{p,q},{\cal R}_{m}^{r,s}]$
$\displaystyle=(n-m)\,{\cal R}_{n+m}^{p+r,q+s}$ $\displaystyle[{\cal
R}_{n}^{p,q},{\cal T}_{m}^{r,s}]$ $\displaystyle=-m\,{\cal T}_{n+m}^{p+r,q+s}$
$\displaystyle[{\cal T}_{n}^{p,q},{\cal T}_{m}^{r,s}]$ $\displaystyle=0$
(4.17a) $\displaystyle[{\cal L}^{p},{\cal L}^{q}]$ $\displaystyle=(p-q)\,{\cal
L}^{p+q}$ $\displaystyle[{\bar{\cal L}}^{p},{\cal{\bar{L}}}^{q}]$
$\displaystyle=(p-q)\,{\cal{\bar{L}}}^{p+q}$ $\displaystyle[{{\cal
L}}^{p},{\cal{\bar{L}}}^{q}]$ $\displaystyle=0$ (4.17b) $\displaystyle[{\cal
L}^{r},{\cal R}_{m}^{p,q}]$ $\displaystyle=-p\,{\cal R}_{m}^{p+r,q}$
$\displaystyle[{\cal{\bar{L}}}^{r},{\cal R}_{m}^{p,q}]$
$\displaystyle=-q\,{\cal R}_{m}^{p,q+r}$ (4.17c) $\displaystyle[{\cal
L}^{r},{\cal T}_{m}^{p,q}]$ $\displaystyle=-p\,{\cal T}_{m}^{p+r,q}$
$\displaystyle[{\cal{\bar{L}}}^{r},{\cal T}_{m}^{p,q}]$
$\displaystyle=-q\,{\cal T}_{m}^{p,q+r}$ (4.17d)

The Taylor–Maclaurin indices are such that they guarantee smoothness in the
$x^{+}\to 0$ limit, that is $n\geq-1$ for ${\cal R}_{n}^{p,q}$ and $n\geq 0$
for ${\cal T}_{n}^{p,q}$.222We address generalizations of this smoothness
condition in the concluding section 8. The Laurent indices $p,q,r,s$ are
integers.

We refer to the full algebra (4.17) as “null hypersurface symmetry algebra”
(NHS); it has several interesting subalgebras (see appendix B). We call the
one generated by ${\cal R}_{n}^{p,q}$ and ${\cal T}_{n}^{p,q}$ “T-Witt
algebra” (4.17a), and the one generated only by ${\cal R}_{n}^{p,q}$ “S-Witt
algebra”. At a given point on the bifurcation sphere, the S-Witt algebra
reduces to a Witt algebra and the T-Witt algebra reduces to an algebra closely
related to the BMS3 algebra Ashtekar:1996cd ; Barnich:2006av . It differs from
the BMS3 algebra only in the bracket $[{\cal R}_{n},{\cal T}_{m}]$ that is
proportional to $-m$ instead of $(n-m)$. This algebra was called
BMS${}_{3}^{(0)}$ in Grumiller:2019fmp  and $W(0,0)$ in Parsa:2018kys . The
subalgebra (4.17b) generated by ${\cal L}^{p}$ and ${\cal L}^{q}$ yields
superrotations in the sense of Barnich:2009se ; Barnich:2011ct . We discuss
the relation to previous near horizon and null hypersurface symmetry algebras
in section 7, but point out already here that the T-Witt algebra was missed in
all previous constructions.

##  5 Charges, fluxes and generalized charge conservation

We first compute the variation of the charges associated to the symmetries
(4.8) in section 5.1. We find out that they are not integrable. One can
separate the integrable and non-integrable parts of the charge variation.
There is, however, an ambiguity in doing so. In section 5.2 we use the
Barnich–Troessaert modified bracket method Barnich:2011mi  to fix this
ambiguity by demanding the absence of a central extension in the null
hypersurface symmetry algebra. Section 5.3 establishes a generalized charge
conservation equation relating the non-integrable part to the time-derivative
of the integrable part of the charge.

###  5.1 Charge variations and non-integrability of the charge

Inserting the near null hypersurface expansion (4.3) together with the near
null hypersurface Killing vectors (4.8) into the surface charge variations
(3.12) yields

$\not{\delta}Q_{\xi}=\int_{\gamma}\,\delta\big{(}\Omega\partial_{+}\
T^{+}-2T^{+}\partial_{+}\Omega+\Omega\,T^{-}+Y^{A}\Upsilon_{A}\big{)}\\\
+\int_{\gamma}\,T^{+}\left[\left(-\Gamma+\frac{\partial_{+}\Omega}{\Omega}\right)\delta\Omega+\frac{\partial_{+}\Omega}{\eta}\delta\eta\right]$
(5.1)

where we used the conformal form of the angular part, so that the integrals
are over a 2d space with reference metric $\gamma_{AB}$ defined in (3.7). For
convenience we absorb an overall factor containing Newton’s constant into the
definition of the integral.

$\int_{\gamma}\ \bullet\,\equiv\,\frac{1}{16\pi G}\int\
\operatorname{d}\\!{}^{2}x\sqrt{\gamma}\ \bullet$ (5.2)

For later purposes we highlight that the charge variation (5.1) is independent
from $\xi_{1}^{+}$ and from $\xi_{1}^{A}$.

While the first line in (5.1) is manifestly the variation of a function over
the phase space, the second line is not. Therefore, we split (5.1)

$\not{\delta}Q_{\xi}=\int_{\gamma}\left(\delta\mathcal{Q}^{\text{I}}_{\xi}+\mathcal{F}_{\xi}(\delta
g)\right)$ (5.3)

into integrable part

$\mathcal{Q}^{\text{I}}_{\xi}=\Omega\partial_{+}T^{+}-2T^{+}\partial_{+}\Omega+\Omega\,T^{-}+Y^{A}\Upsilon_{A}$
(5.4)

and non-integrable part

$\mathcal{F}_{\xi}(\delta
g)=T^{+}\,\bigg{[}\bigg{(}-\Gamma+\frac{\partial_{+}\Omega}{\Omega}\bigg{)}\delta\Omega+\frac{\partial_{+}\Omega}{\eta}\delta\eta\bigg{]}\,.$
(5.5)

This split is not unique due to the ambiguity

$\mathcal{Q^{\text{I}}}_{\xi}\to\mathcal{\tilde{Q}}^{\text{I}}_{\xi}=\mathcal{Q^{\text{I}}}_{\xi}+\mathcal{A}_{\xi}(g)\qquad\qquad{\mathcal{F}}_{\xi}(\delta
g)\to{\cal{\tilde{F}}}_{\xi}(\delta g)={\mathcal{F}}_{\xi}(\delta
g)-\delta\mathcal{A}_{\xi}(g)$ (5.6)

for any arbitrary functional $\mathcal{A}_{\xi}(g)$. We shall discuss below
how this ambiguity can be fixed.

###  5.2 Barnich–Troessaert method and modified bracket algebra

As discussed in the first two paragraphs of section 3.4, for the integrable
charges the fundamental theorem of the covariant phase space method implies
that the algebra of charges is the same as the algebra of asymptotic (near
bifurcation surface/near horizon/near null hypersurface) Killing vectors (up
to a possible central extension). One may use a similar criterion for fixing
the ambiguity in separation of the integrable and non-integrable parts when
charges are not integrable, as in the case we are dealing with. This method
was developed by Barnich and Troessaert in Barnich:2011mi . They proposed a
“modified bracket” of the integrable part of charges,
$Q_{\xi}^{\text{\tiny{BT}}}$, such that333One should note the difference
between “adjusted brackets” which arise due to field dependence of the
generators, e.g. see Compere:2015knw , and the “modified bracket”
Barnich:2011mi  which arise in the non-integrability context.

$\\{Q_{\xi_{1}}^{\text{\tiny{BT}}},Q_{\xi_{2}}^{\text{\tiny{BT}}}\\}^{*}:=\delta_{\xi_{2}}Q^{\text{\tiny{BT}}}_{\xi_{1}}+F^{\text{\tiny{BT}}}_{\xi_{2}}(\delta_{\xi_{1}}g)$
(5.7)

where
$Q_{\xi_{1}}^{\text{\tiny{BT}}}=\int_{\gamma}\mathcal{Q}_{\xi}^{\text{\tiny{BT}}}$
and
$F^{\text{\tiny{BT}}}_{\xi}=\int_{\gamma}\mathcal{F}^{\text{\tiny{BT}}}_{\xi}$.
On-shell the modified bracket satisfies

$\\{Q_{\xi_{1}}^{\text{\tiny{BT}}},Q_{\xi_{2}}^{\text{\tiny{BT}}}\\}^{*}=Q^{\text{\tiny{BT}}}_{[\xi_{1},\xi_{2}]}+K_{\xi_{1},\xi_{2}}$
(5.8)

where $K_{\xi_{1},\xi_{2}}$ is a central extension term. In our case on-shell
the latter is given by

$K_{\xi_{1},\xi_{2}}=-\int_{\gamma}\Omega\left[\left(T_{1}^{+}\partial_{+}^{2}T_{2}^{+}-T_{2}^{+}\partial_{+}^{2}T_{1}^{+}\right)+\left(T_{1}^{+}\partial_{+}T_{2}^{-}-T_{2}^{+}\partial_{+}T_{1}^{-}\right)\right]$
(5.9)

The central extension (5.9) is in general non-vanishing.

Under the ambiguity (5.6) the form of the modified bracket (5.8) is preserved
but the central extension term shifts,

$\tilde{K}_{\xi_{1},\xi_{2}}=K_{\xi_{1},\xi_{2}}+\delta_{\xi_{2}}A_{\xi_{1}}(g)-\delta_{\xi_{1}}A_{\xi_{2}}(g)-A_{[\xi_{1},\xi_{2}]}(g)$
(5.10)

where $A_{\xi}(g)=\int_{\gamma}\mathcal{A}_{\xi}(g)$. This begs the question
whether or not there is a choice for the ambiguity such that the central
extension vanishes. We show now that the answer is yes. To be minimalistic we
consider only ambiguities that are linear combinations of terms resembling the
ones already present in the non-integrable part (5.5), namely
$T^{+}(\partial_{+}\Omega)\log\Omega$, $T^{+}(\partial_{+}\Omega)\log\eta$ and
$T^{+}\,\Gamma$. We find that the choice

$\mathcal{A}_{\xi}(g)=-T^{+}\,\Omega\,\Gamma$ (5.11)

for the ambiguity leads to vanishing central extension,
$\tilde{K}_{\xi_{1},\xi_{2}}=0$. (The choice (5.11) is not unique; for
instance, adding to $\mathcal{A}_{\xi}(g)$ a term proportional to
$T^{+}\partial_{+}\Omega$ would not change our conclusions; however, the
choice above is minimalistic.)

Thus, we fix the ambiguity by demanding the absence of a central extension.
Using (5.11) the integrable and non-integrable parts of the charge become

$\displaystyle Q^{\text{\tiny{BT}}}_{\xi}$
$\displaystyle=\int_{\gamma}\Big{[}\big{(}\partial_{+}T^{+}+T^{-}\big{)}\
\Omega-T^{+}\big{(}2\partial_{+}\Omega+\Omega\,\Gamma\big{)}+Y^{A}\Upsilon_{A}\Big{]}$
(5.12) $\displaystyle F^{\text{\tiny{BT}}}_{\xi}(\delta g)$
$\displaystyle=\int_{\gamma}T^{+}\,\bigg{[}\frac{\partial_{+}\Omega}{\Omega}\delta\Omega+\frac{\partial_{+}\Omega}{\eta}\delta\eta+\Omega\delta\Gamma\bigg{]}\,.$
(5.13)

Introducing modes for the charges analogous to (4.16),

$\displaystyle R_{n}^{p,q}$ $\displaystyle:=Q^{\text{\tiny{BT}}}_{\xi}$
$\displaystyle\textrm{for}\qquad\xi=-(x^{+})^{n+1}z^{p}\bar{z}^{q}\partial_{+}$
(5.14a) $\displaystyle T_{n}^{p,q}$
$\displaystyle:=Q^{\text{\tiny{BT}}}_{\xi}$
$\displaystyle\textrm{for}\qquad\xi=-x^{-}(x^{+})^{n}z^{p}\bar{z}^{q}\partial_{-}$
(5.14b) $\displaystyle L^{p}$ $\displaystyle:=Q^{\text{\tiny{BT}}}_{\xi}$
$\displaystyle\textrm{for}\qquad\xi=-z^{p+1}\partial_{z}$ (5.14c)
$\displaystyle\bar{L}^{p}$ $\displaystyle:=Q^{\text{\tiny{BT}}}_{\xi}$
$\displaystyle\textrm{for}\qquad\xi=-\bar{z}^{p+1}\partial_{\bar{z}}$ (5.14d)

their modified bracket algebra

$\displaystyle\\{R_{n}^{p,q},\,R_{m}^{r,s}\\}^{\ast}$
$\displaystyle=(n-m)\,R_{n+m}^{p+r,\,q+s}$
$\displaystyle\\{R_{n}^{p,q},\,T_{m}^{r,s}\\}^{\ast}$
$\displaystyle=-m\,T_{n+m}^{p+r,\,q+s}$ (5.15a)
$\displaystyle\\{T_{n}^{p,q},T_{m}^{r,s}\\}^{\ast}$ $\displaystyle=0$
$\displaystyle\\{L^{p},\,{\bar{L}}^{q}\\}^{\ast}$ $\displaystyle=0$ (5.15b)
$\displaystyle\\{L^{p},\,L^{q}\\}^{\ast}$ $\displaystyle=(p-q)\,L^{p+q}$
$\displaystyle\\{{\bar{L}}^{p},\,{\bar{L}}^{q}\\}^{\ast}$
$\displaystyle=(p-q)\,{\bar{L}}^{p+q}$ (5.15c)
$\displaystyle\\{L^{r},\,R_{m}^{p,q}\\}^{\ast}$
$\displaystyle=-p\,R_{m}^{p+r,\,q}$
$\displaystyle\\{{\bar{L}}^{r},\,R_{m}^{p,q}\\}^{\ast}$
$\displaystyle=-q\,R_{m}^{p,q+r}$ (5.15d)
$\displaystyle\\{L^{r},\,T_{m}^{p,q}\\}^{\ast}$
$\displaystyle=-p\,T_{m}^{p+r,\,q}$
$\displaystyle\\{{\bar{L}}^{r},\,T_{m}^{p,q}\\}^{\ast}$
$\displaystyle=-q\,T_{m}^{p,\,q+r}$ (5.15e)

coincides with the null hypersurface symmetry algebra (4.17) without central
extension. This is one of the key results of our paper.

###  5.3 Generalized charge conservation, Hamiltonian and entropy

We study now the time evolution of the charge $Q^{\text{\tiny{BT}}}_{\xi}$
(5.12). Depending on the observer, there are different interesting cases for
the choice of time coordinate. Below, we discuss two such cases.

#### Null hypersurface observer ($\boldsymbol{x^{+}}$ as time).

The natural time coordinate for an observer temporarily residing at the
$x^{-}=0$ null hypersurface is $x^{+}$. The corresponding symmetry generator
is translation along $x^{+}$, ${\cal R}^{0,0}_{-1}=-\partial_{+}$ and the
corresponding charge (5.12), the Hamiltonian, is
$Q^{\text{\tiny{BT}}}_{\partial_{+}}=R^{0,0}_{-1}=\int_{\gamma}(2\partial_{+}\Omega+\Omega\Gamma)$.
The charge algebra (5.15) implies that this Hamiltonian commutes with
superrotations $L^{r},{\bar{L}}^{s}$ but does not commute with the T-Witt
generators $R_{n}^{p,q},T_{n}^{p,q}$. In the stationary limit, where
$\partial_{+}\Omega=\Gamma=0$, the Hamiltonian
$Q^{\text{\tiny{BT}}}_{\partial_{+}}$ vanishes and is thus not a useful
observable. So for (almost) stationary situations one should use the time
coordinate below instead.

Nevertheless, if one considers the charge associated with the Hamiltonian
$Q^{\text{\tiny{BT}}}_{\partial_{+}}$ and evaluates its bracket with another
charge $Q^{\text{\tiny{BT}}}_{\xi}$ for generic $\xi$ then (5.8) yields

$\\{Q^{\text{\tiny{BT}}}_{\partial_{+}},{Q}_{\xi}^{\text{\tiny{BT}}}\\}^{*}={Q}_{[\partial_{+},\xi]}^{\text{\tiny{BT}}}={Q}_{\partial_{+}\xi}^{\text{\tiny{BT}}}$
(5.16)

while (5.7) yields

$\\{Q_{\partial_{+}}^{\text{\tiny{BT}}},{Q}_{\xi}^{\text{\tiny{BT}}}\\}^{*}=-\delta_{\partial_{+}}Q_{\xi}^{\text{\tiny{BT}}}-{F}^{\text{\tiny{BT}}}_{\partial_{+}}(\delta_{\xi}g)\,.$
(5.17)

As implied by (4.9) and (4.10), we have
$Q_{\partial_{+}\xi}^{\text{\tiny{BT}}}+\delta_{\partial_{+}}Q_{\xi}^{\text{\tiny{BT}}}={\partial_{+}}Q_{\xi}^{\text{\tiny{BT}}}$
and therefore arrive at the generalized charge conservation equation (GCCE)

${\partial_{+}}Q_{\xi}^{\text{\tiny{BT}}}=-{F}^{\text{\tiny{BT}}}_{\partial_{+}}(\delta_{\xi}g)\,.$
(5.18)

The GCCE means that the non-integrable part of the charge is the source of the
non-conservation of the integrable part and therefore represents the flux
passing through the null hypersurface. The GCCE (5.18) provides a consistency
check of our boundary conditions and is expected to hold on general grounds
Barnich:2011mi , see also Bunster:2018yjr  for the canonical viewpoint and
Donnay:2016ejv  for similar discussions and results in a special case
addressed in section 7.2.

#### Stretched horizon observer
($\boldsymbol{t=\frac{1}{2\kappa}\ln(-x^{+}/x^{-})}$ as time).

For an observer just outside the horizon in the stationary case the most
natural time-translation generator is the one associated with the Killing
vector $\zeta_{\textrm{\tiny{H}}}$ given in (2.11). The corresponding time
coordinate is proportional to $\ln(-x^{+}/x^{-})$, see (2.7). Even for non-
stationary situations we can consider this time-coordinate. Following
Grumiller:2019fmp  we define the near horizon Hamiltonian $H_{\text{\tiny
NH}}$ as

$H_{\text{\tiny
NH}}:=Q_{\partial_{t}}^{\text{\tiny{BT}}}=\kappa\,\big{(}T_{0}^{0,0}-R_{0}^{0,0}\big{)}\,.$
(5.19)

The Hamiltonian $H_{\text{\tiny NH}}$ commutes with superrotations and all
$T^{p,q}_{n}$. In the stationary case it is conserved and commutes with all
other charges, and hence the associated physical excitations are soft hair, as
explained in Grumiller:2019fmp . For the Kerr background (see section 2), the
near horizon Hamiltonian is given by the Wald entropy times Hawking
temperature, thereby recovering the near horizon first law Donnay:2015abr .

In full generality, however, $H_{\text{\tiny NH}}$ is not conserved since
$H_{\text{\tiny NH}}$ does not commute with $R^{p,q}_{n}$. This non-
conservation in particular, and time variation of the charges in general, is
captured by a GCCE similar to (5.18):

${\partial_{t}}Q_{\xi}^{\text{\tiny{BT}}}=-{F}^{\text{\tiny{BT}}}_{\partial_{t}}(\delta_{\xi}g)\,.$
(5.20)

In the adiabatic limit, the left hand side of the above equation for
$\xi=\zeta_{\textrm{\tiny{H}}}$ (2.11) is the time variation of the entropy,
which equals to the flux through the null hypersurface. We shall return to
this point in section 8.

##  6 Redundancies and on-shell phase space

The phase spaces defined in the previous three sections describe boundary
excitations [since we switched off the bulk excitations through our assumption
(3.7)]. These boundary excitations correspond to edge states and are physical,
modulo residual pure gauge redundancies, so that the number of functions
parametrizing these phase spaces are typically larger than the number of
surface charges. We have seen this effect already in section 3: the near
bifurcation Killing vectors involve $T^{\pm}$, while the charges (3.13) depend
only on their sum. So there is a pure gauge degeneracy in the near bifurcation
Killing vectors. This particular degeneracy is lifted in sections 4-5.
However, even in the phase space defined through the boundary conditions in
section 4 and the charge analysis in section 5 there are still gauge
redundancies, which we address in this section.

In section 6.1 we perform residual gauge fixing to remove gauge redundancies
from the boundary phase space. In section 6.2 we discuss the on-shell phase
space to verify that the number of charges matches with the number of
functions parametrizing our boundary phase space. Two special cases require
separate discussion, namely non-expanding and sationary null hypersurfaces,
which we deal with in section 6.3.

###  6.1 Fixing residual gauge redundancies

The first type of pure gauge redundancy arises when certain symmetry
generators do not contribute to the charges at all. In the present case the
functions $\xi_{1}^{+}$ and $\xi^{A}_{1}$ appearing in the near null
hypersurface Killing vectors (4.8) do not contribute to the charge variation
(5.1). Therefore, the diffeomorphisms generated by near null hypersurface
Killing vectors that have only these functions switched on are proper
diffeomorphisms, even at the null hypersurface boundary. Anticipating this
result we considered only the remaining free functions, $T^{\pm}$ and $Y^{A}$,
when deriving the near null hypersurface Killing vector algebra (4.17).

There is a second type of pure gauge redundancies: it could be that not all
linear combinations of free functions appearing in the near null hypersurface
expansion of the metric appear in the charges. Linear combinations that do not
appear thus do not label any physical states and can be gauge fixed to a
convenient form. In the present case there are four scalar fields, $\eta,{\cal
F},{\cal G}$ and $\Omega$, and two vector fields, ${\cal V}_{A}$ and ${\cal
U}_{A}$, appearing in our parametrization of the near null hypersurface metric
(4.3). However, the expressions for the charges (5.12) and fluxes (5.13)
depend only on three of the scalar fields — there is no dependence on $\cal F$
— and on one combination of the vector fields, namely $\Upsilon_{A}$ defined
in (4.7). This means that ${\cal F}$ and one combination of ${\cal V}_{A}$ and
${\cal U}_{A}$ that is linearly independent from $\Upsilon_{A}$ can be gauge-
fixed conveniently.

Recalling (4.11), we eliminate now $\cal F$ and ${\cal U}_{A}$ by a
diffeomorphism generated by a near null hypersurface Killing vector (4.8) with
non-zero functions444To eliminate ${\cal F}$ and ${\cal U}_{A}$ we need to
consider the finite form of diffeomorphisms generated infinitesimally by
$\xi_{1}$. This means in particular that we need to keep the terms to second
order in $\xi_{1}$ in our transformations, whereas (4.11) are only showing the
first order ones. The transformation for ${\cal F}$ is then modified to
$\delta_{\xi}\mathcal{F}+\Omega_{AB}\xi_{1}^{A}\xi_{1}^{B},$ where
$\delta_{\xi}{\cal F}$ is given in (4.11). The expression for
$\delta_{\xi}{\cal U}_{A}$ does not receive second order corrections.

${\xi_{1}^{+}}=-\frac{1}{2\eta}({\cal F}-\Omega^{AB}{\cal U}_{A}{\cal
U}_{B})\qquad\qquad\xi_{1}^{A}=-\Omega^{AB}{\cal U}_{B}\,.$ (6.1)

Removing the gauge freedom from our phase space reduces the near null
hypersurface expansion (4.3) of the metric

$\displaystyle g_{++}$ $\displaystyle=x^{-}\eta\,{\cal
G}+\mathcal{O}\left([x^{-}]^{2}\right)$ $\displaystyle g_{--}$
$\displaystyle=\mathcal{O}\left(x^{-}\right)$ (6.2a) $\displaystyle g_{+A}$
$\displaystyle=x^{-}\theta_{A}+\mathcal{O}\left([x^{-}]^{2}\right)$
$\displaystyle g_{-A}$ $\displaystyle=\mathcal{O}\left(x^{-}\right)$ (6.2b)
$\displaystyle g_{AB}$
$\displaystyle=\Omega_{AB}+\mathcal{O}\left(x^{-}\right)$ $\displaystyle
g_{+-}$ $\displaystyle=\eta+\mathcal{O}\left(x^{-}\right)$ (6.2c)

where $\theta_{A}:=-\frac{\eta}{\Omega}\Upsilon_{A}$. The dynamical fields
${\cal G}$, $\eta$, $\Omega_{AB}=\gamma_{AB}\,\Omega$ [see (3.7)] and
${\theta}_{A}$ are functions of time $x^{+}$ and angles $x^{A}$ and
parameterize our phase space. Terms not displayed are subleading and do not
contribute to the charges or fluxes.

Higher powers of $x^{-}$ in the near null hypersurface Killing vectors (4.8)
do not contribute to the charges and thus generate gauge redundancies that can
be used to remove the higher powers in the $g_{-\mu}$ components of the
metric. The fully gauge-fixed version of the metric (6.2)

$\displaystyle g_{++}$ $\displaystyle=x^{-}\eta\,{\cal
G}+\mathcal{O}\left([x^{-}]^{2}\right)$ $\displaystyle g_{--}$
$\displaystyle=0$ (6.3a) $\displaystyle g_{+A}$
$\displaystyle=x^{-}\theta_{A}+\mathcal{O}\left([x^{-}]^{2}\right)$
$\displaystyle g_{-A}$ $\displaystyle=0$ (6.3b) $\displaystyle g_{AB}$
$\displaystyle=\gamma_{AB}\,\Omega+\lambda_{AB}\,x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
$\displaystyle g_{+-}$ $\displaystyle=\eta\,.$ (6.3c)

is preserved by near null hypersurface Killing vectors

$\xi=T^{+}(x^{+},x^{B})\,\partial_{+}-x^{-}T^{-}(x^{+},x^{B})\,\partial_{-}+Y^{A}(x^{B})\,\partial_{A}+\xi^{\textrm{\tiny
fix}}\,.$ (6.4)

The subleading term $\xi^{\textrm{\tiny
fix}}=x^{-}\hat{\xi}^{A}\,\partial_{A}+x^{-\,2}\hat{\xi}^{-}\,\partial_{-}\hat{\xi}^{A}$
is completely determined by the metric and the leading order functions,
$\hat{\xi}^{A}=-\frac{\eta}{\Omega}\gamma^{AB}\partial_{B}T^{+}+{\cal
O}(x^{-})$ and
$\hat{\xi}^{-}=-\frac{1}{2\eta}\,\hat{\xi}^{A}\big{(}\theta_{A}+\partial_{A}\eta\big{)}+{\cal
O}(x^{-})$. This shows explicitly that all residual gauge redundancies present
in the near null hypersurface Killing vectors (4.8) are removed and the
remaining free functions, $T^{\pm}$ and $Y^{A}$, generate null hypersurface
symmetries that are not gauge redundancies in general.

After removing all residual gauge redundancies our boundary phase space is
therefore characterized by three scalar functions, $\eta,\Gamma,\Omega$ and a
vector $\Upsilon_{A}$; the quantity $\lambda_{AB}$ does not appear in the
charges and captures (ingoing and memoryless) gravitational wave degrees of
freedom that are part of the bulk phase space.

Before moving on we mention a similar redundancy that is also present in the
near bifurcate horizon analysis of section 3, since the charges (3.13) depend
only on the function $\Omega$ and the combination
$(C_{A}^{+}-C_{A}^{-})/\eta$, as evident from (3.14). The phase space is
parametrized by four functions $\eta$, $\Omega$ and $C^{\pm}_{A}$, and
therefore there are two redundant functions. The redundancy is eliminated by
gauge fixing, e.g. $\eta=1$ and $C_{A}^{-}=0$.555Note that, given
$\xi_{\tiny{{\cal
P}}}:=\frac{1}{2}T(x^{A})(x^{+}\partial_{+}-x^{-}\partial_{-})$ and
$\xi_{\tiny{{\cal
D}}}:=\tilde{T}(x^{A})(x^{+}\partial_{+}+x^{-}\partial_{-})$, ${\cal P}$ is
the charge associated with $\xi_{\tiny{{\cal P}}}$ and the charge for
$\xi_{\tiny{{\cal D}}}$ vanishes on our phase space. That is,
$\xi_{\tiny{{\cal D}}}$ generates trivial transformations which can be used to
gauge fix $\eta$ to 1.

###  6.2 On-shell phase space for generic null hypersurfaces

At first sight there seems to be a mismatch between the degrees of freedom of
the boundary phase space and the number of surface charges since there are
more functions on the phase space, $\eta,\Gamma,\Omega,\Upsilon_{A}$, than
null hypersurface symmetry generators, $T^{\pm},Y^{A}$. The reason for this is
that we did not consider all on-shell conditions yet. Therefore, we analyze
now the equations of motion (eom) (4.4) and (4.6) and evaluate the charge and
the flux (5.12) and (5.13) on-shell.

The eom yield a dichotomy: generic solutions have $\partial_{+}\Omega\neq 0$,
while special solutions have a non-expanding null hypersurface,
$\partial_{+}\Omega=0$. We postpone a discussion of the latter case to section
6.3 and focus here on generic solutions.

In the generic case the eom (4.4) can be solved for $\Gamma$,

$\Gamma=\Gamma(\eta,\Omega)=\partial_{+}\left(\ln{\frac{\eta\,\Omega}{(\partial_{+}\Omega)^{2}}}\right)\,.$
(6.5)

Then the eom (4.6) yields

$\partial_{+}\Upsilon_{A}=2\partial_{+}\Big{(}\Omega\
\partial_{A}\Big{(}\ln{\frac{\partial_{+}\Omega}{\sqrt{\eta}}}\Big{)}-\partial_{A}\Omega\Big{)}\,.$
(6.6)

So $\Upsilon_{A}$ is fully determined by $\eta,\Omega$ and the initial value
$\Upsilon_{A}^{0}:=\Upsilon_{A}(x^{+}=0)$.

Thus, generic points in the phase space (modulo possible ‘accidental gauge
redundancies’ which we discuss below) are uniquely specified on-shell by two
arbitrary functions of time $x^{+}$, namely $\Omega$ and $\eta$, together with
the initial value $\Upsilon_{A}^{0}(x^{B})$. These on-shell degrees of freedom
match the symmetry generating functions appearing in the near horizon Killing
vectors, $T^{\pm}(x^{+},x^{A})$ and $Y^{A}(x^{B})$, and the associated charges

$\begin{split}{\cal Q}_{\xi}^{\textrm{\tiny{eom}}}=&{\cal
Q}_{\xi}^{\textrm{\tiny{eom}}}[\Omega,\eta,\Upsilon_{A}^{0};T^{\pm},Y^{A}]\\\
:=&(16\pi G)\left[\tilde{T}^{-}{\cal P}_{-}+Y^{A}{\cal J}_{A}+T^{+}{\cal
P}_{+}\right]\end{split}$ (6.7)

where $\tilde{T}^{-}:=\frac{1}{2}(T^{-}+\partial_{+}T^{+})$. The charge
densities ${\cal P}_{\pm},{\cal J}_{A}$ read

$\displaystyle{\cal J}_{A}$ $\displaystyle=\frac{\Upsilon_{A}}{16\pi G}$
(6.8a) $\displaystyle{\mathbb{\cal P}}_{-}$ $\displaystyle=\frac{\Omega}{8\pi
G}$ (6.8b) $\displaystyle{\cal P}_{+}$ $\displaystyle=\frac{\Omega}{8\pi
G}\partial_{+}\left[\ln{\frac{\partial_{+}\Omega}{\Omega}}-\ln{\sqrt{\eta\Omega}}\right]$
(6.8c)

and their variations yield 666Note that ${\cal P}_{\pm}$ are scalar densities
and ${\cal J}_{A}$ is a vector density. Therefore, $Y^{A}\partial_{A}{\cal
P}_{\pm}+{\cal
P}_{\pm}\bar{\nabla}_{A}Y^{A}=\bar{\nabla}_{A}\left(Y^{A}\,{\cal
P}_{\pm}\right)$, where $\bar{\nabla}_{A}$ is the covariant derivative with
respect to metric $\gamma_{AB}$.

$\displaystyle\delta_{\xi}{\cal J}_{A}$ $\displaystyle={\cal
P}_{-}\partial_{A}\tilde{T}^{-}+T^{+}\partial_{+}{\cal J}_{A}+{\cal
P}_{+}\partial_{A}T^{+}+\frac{1}{2}{\partial_{+}}{\cal
P}_{-}\partial_{A}T^{+}$
$\displaystyle\quad+Y^{B}\partial_{B}\mathcal{J}_{A}+\mathcal{J}_{B}\partial_{A}Y^{B}+\mathcal{J}_{A}\partial_{B}Y^{B}$
(6.9a) $\displaystyle\delta_{\xi}{{\cal P}}_{-}$
$\displaystyle=T^{+}\partial_{+}{\cal P}_{-}+Y^{A}\partial_{A}{\cal
P}_{-}+{\cal P}_{-}\bar{\nabla}_{A}Y^{A}$ (6.9b)
$\displaystyle\delta_{\xi}{\cal P}_{+}$
$\displaystyle=\partial_{+}\tilde{T}^{-}{\cal
P}_{-}+\partial_{+}\left(T^{+}{\cal P}_{+}\right)+Y^{A}\partial_{A}{\cal
P}_{+}+{\cal P}_{+}\bar{\nabla}_{A}Y^{A}\,.$ (6.9c)

Thus, there are exactly as many charges, ${\cal P}_{\pm},{\cal J}_{A}$, as
there are independent functions on the boundary phase space,
$\Omega,\eta,\Upsilon^{0}_{A}$. As (6.9) show, the symmetry algebra of the on-
shell charges is isomorphic to the symmetry algebra of the off-shell charges
(5.15).

While the charges ${\cal J}_{A},{\cal P}_{-}$ are non-zero even in the limit
of vanishing expansion, this is not true for ${\cal P}_{+}$. A reason for this
significant difference is that the two former are integrable and conserved
while the latter is not. This may be seen from the fact that the flux (5.13)
is proportional to $T^{+}$ and is independent of $T^{-},Y^{A}$. Explicitly,
the on-shell value of the flux is

${\cal F}_{\xi}^{\textrm{\tiny{eom}}}={\cal
F}_{\xi}^{\textrm{\tiny{eom}}}[\Omega,\eta;T^{+}]=2T^{+}\Big{[}\partial_{+}\big{(}\Omega\delta\ln\sqrt{\eta\Omega}\big{)}-\Omega\partial_{+}\big{(}\delta\ln(\partial_{+}\Omega)\big{)}\Big{]}\,.$
(6.10)

The expression for the on-shell flux (6.10) is a function over the same on-
shell phase space as the charges, and it is related to the latter through our
GCCE (5.18).

###  6.3 On-shell phase space for non-expanding and stationary null
hypersurfaces

We consider now non-expanding null hypersurfaces, $\partial_{+}\Omega=0$. This
is a consistent truncation of the phase space since the condition
$\partial_{+}\Omega=0$ is preserved under the orbit of the near null
hypersurface Killing vectors.

In this case $\Gamma$ may not be solved for since the eom (4.4) is satisfied
identically. The eom (4.6) and the expressions for charge (5.12) and flux
(5.13) simplify to

$\displaystyle\partial_{+}{\Upsilon_{A}}+$
$\displaystyle\,\Omega\,\partial_{A}\Gamma=0$ (6.11a)
$\displaystyle{\mathcal{Q}}^{\text{\tiny{non-ex}}}_{\xi}=$
$\displaystyle\,2\,\tilde{T}^{-}\,\Omega+Y^{A}\Upsilon_{A}-T^{+}\,\Omega\,\Gamma$
(6.11b) $\displaystyle{\cal{F}}^{\text{\tiny{non-ex}}}_{\xi}(\delta g)=$
$\displaystyle\,T^{+}\Omega\,\delta\Gamma\,.$ (6.11c)

Within the special case of vanishing expansion we still encounter a dichotomy:
generically, $\delta\Gamma\neq 0$, so despite of vanishing expansion there is
a non-trivial flux, but for stationary solutions $\delta\Gamma=0$ and the flux
vanishes. We discuss first the generic case.

#### Generic non-expanding case.

Our goal is to compare the number of free functions on the on-shell phase
space with the number of free functions appearing in the surface charges.
Since the functions ${\cal G}$ and $\eta$ always appear through the
combination $\Gamma$ there is an accidental gauge symmetry, namely any
transformation that changes $\eta,{\cal G}$ (4.9b), (4.9c) but keeps $\Gamma$
invariant. With no loss of generality we exploit this gauge symmetry to fix
${\cal G}=0$, implying

$\Gamma=-\partial_{+}\ln\eta\qquad\qquad\partial_{+}\Big{(}\Upsilon_{A}-\Omega\,\frac{\partial_{A}\eta}{\eta}\Big{)}=0$
(6.12)

where in the right equation we used (6.11b). For consistency also its
variation (4.9c) must vanish, yielding $\partial_{+}T^{-}=0$ so that
$T^{-}=T^{-}(x^{A})$, where we take $T^{-}$ to be field independent. The
remaining eom (6.11a) implies that the on-shell degrees of freedom are
$\Gamma(x^{+},x^{A})$, $\Omega(x^{A})$ and $\Upsilon_{A}^{0}(x^{B})$. This
matches precisely with the freedom contained in the symmetry generators
$T^{+}(x^{+},x^{A})$, $T^{-}(x^{A})$ and $Y^{A}(x^{B})$.

For completeness, we also present the on-shell variation of the remaining
fields in this gauge from which one can read the algebra of charges. These
variations may be obtained by setting $\partial_{+}\Omega=0$ and using
$\partial_{+}T^{-}=0$ in (4.10), (4.12) and (4.13),

$\displaystyle\delta_{\xi}\Omega$
$\displaystyle=Y^{A}\partial_{A}\Omega+\Omega\,\bar{\nabla}_{A}Y^{A}$ (6.13a)
$\displaystyle\delta_{\xi}\Upsilon_{A}$
$\displaystyle=\Omega\,\partial_{A}{T}^{-}+\Omega\partial_{A}(\partial_{+}T^{+}-T^{+}\Gamma)+Y^{B}\partial_{B}\Upsilon_{A}+\Upsilon_{B}\partial_{A}Y^{B}+\Upsilon_{A}\partial_{B}Y^{B}$
(6.13b) $\displaystyle\delta_{\xi}\Gamma$
$\displaystyle=\partial_{+}(T^{+}\Gamma)-\partial^{2}_{+}T^{+}+Y^{A}\partial_{A}\Gamma\,.$
(6.13c)

In $\delta_{\xi}\Upsilon_{A}$ we dropped the term
$T^{+}\left(\partial_{+}\Upsilon_{A}+\Omega\partial_{A}\ \Gamma\right)$, which
vanishes on-shell (6.11a). In the ${\cal G}=0$ gauge the variation (6.13c) is
equivalent to (4.9b) for $\partial_{+}T^{-}=0$. For this sector we have the
charges $R^{p,q}_{n}$, $T^{p,q}_{0}$, $L^{p}$ and ${\bar{L}}^{r}$. The T-Witt
part of the algebra hence reduces to a Witt algebra among $R^{p,q}_{n}$ and
$T^{p,q}_{0}$ generators which commute with all the $R^{r,s}_{n}$. The
generators $T^{p,q}_{0}$, $L^{p}$ and ${\bar{L}}^{r}$ form an algebra obtained
in Donnay:2015abr  for the near horizon Kerr geometry.

#### Stationary case.

Finally, for the special case of stationary non-expanding null hypersurfaces,
$\delta\Gamma=0$, the flux (6.11c) vanishes and the charge (6.11b) is
integrable

${{Q}}^{\text{\tiny{stat}}}_{\xi}=\int_{\gamma}\big{(}\hat{T}\,\Omega+Y^{A}\,\Upsilon_{A}\big{)}\qquad\qquad\mathcal{F}_{\xi}=0$
(6.14)

where $\hat{T}:=T^{-}+\partial_{+}T^{+}-\Gamma\,T^{+}$. The value of the
charge and the charge variation has no $\eta$-dependence (or equivalently
$\Gamma$-dependence). It implies that there is a new accidental gauge symmetry
in the stationary sector. One can use this to gauge-fix $\eta$ to a constant
and thus $\Gamma=0$. In this gauge $\hat{T}=T^{-}+\partial_{+}T^{+}$ and
(6.13c) implies $\partial_{+}^{2}T^{+}=0$. Therefore,
$\partial_{+}\hat{T}=\partial_{+}(T^{-}+\partial_{+}T^{+})=0$. Moreover, as a
consequence of the eom (6.12), $\partial_{+}\Upsilon_{A}=0$. The charge (6.14)
is time independent and thus conserved, consistently with the GCCE (5.18).

Thus, also the stationary subsector of the boundary phase space shows a
matching between charges, $\Omega(x^{A})$ and $\Upsilon_{A}(x^{B})$, and
symmetry generators, $\hat{T}(x^{A})$ and $Y^{A}(x^{B})$. For the stationary
phase space, $\Gamma=\partial_{+}\Omega=0$, the charge variation simplifies to

$\displaystyle\delta_{\xi}\Omega$
$\displaystyle=Y^{A}\partial_{A}\Omega+\Omega\,\bar{\nabla}_{A}Y^{A}$ (6.15a)
$\displaystyle\delta_{\xi}\Upsilon_{A}$
$\displaystyle=2\,\Omega\,\partial_{A}{\hat{T}}+Y^{B}\partial_{B}\Upsilon_{A}+\Upsilon_{B}\partial_{A}Y^{B}+\Upsilon_{A}\partial_{B}Y^{B}\,.$
(6.15b)

The above is precisely what we obtained in section 3 as the symmetry algebra
near the bifurcation point as well as the algebra obtained in
Grumiller:2019fmp . See also section 7.3 for further discussions.

We come back to bulk aspects of the phase space and gravitational wave
excitations in the concluding section 8.

##  7 Recovering other near horizon symmetries

In this section we curtail the analysis of sections 4, 5 and 6 by imposing
physically well-motivated conditions on the general variations around the null
hypersurface $x^{-}=0$. Thereby, we recover near horizon symmetries discussed
earlier as special cases.

In section 7.1 we restrict the null hypersurface generator to have a constant
non-affinity parameter over spacetime and phase space. In section 7.2 we
discuss the Gaussian null coordinates. In section 7.3 we require the null
hypersurface to be a Killing horizon, which in a special case recovers the
phase space established in section 3.

###  7.1 Null hypersurfaces with constant non-affinity parameter

The generator of the $x^{-}=0$ null hypersurface in the metric (6.2) to
leading order in $x^{-}$ is

$l=\frac{N}{\eta}\,\partial_{+}+\mathcal{O}(x^{-})$ (7.1)

where $N$ is an arbitrary function of $x^{+}$ and $x^{A}$. Its non-affinity
parameter $\kappa_{l}$ defined through $l\cdot\nabla
l^{\mu}=\kappa_{l}l^{\mu}$ on the surface $x^{-}=0$ is

$\kappa_{l}=\frac{1}{\eta}\left(\partial_{+}N-\frac{\mathcal{G}N}{2}\right)\,.$
(7.2)

The subscript $l$ is a reminder that the non-affinity parameter $\kappa_{l}$
in general is not associated with surface gravity $\kappa$, since we are
currently not assuming to have a Killing horizon; for the special case of a
Killing horizon both quantities coincide with each other.

Requiring $\kappa_{l}$ to be constant on both spacetime and phase space, i.e.,
$\partial_{\mu}\kappa_{l}=0$ and $\delta\kappa_{l}=0$, fixes $N$ as a function
of ${\cal G}$ and $\eta$. This does not impose any further restriction on our
phase space. However, physically motivated choices can lead to further
restrictions. Here we focus on the case $N\propto x^{+}\eta$. Constancy of
$\kappa_{l}$ in this case leads to

$\mathcal{G}=2\frac{\partial_{+}\eta}{\eta}\,.$ (7.3)

The above condition constitutes a special case for which the $\eta,{\cal
G}$-dependence drops out of the field equation (4.4), yielding

$\Omega=\Omega_{0}+2\sqrt{\Omega_{0}\Omega_{1}}\ x^{+}+\Omega_{1}\
(x^{+})^{2}\qquad\qquad\Omega_{0}=\Omega_{0}(x^{A}),\
\Omega_{1}=\Omega_{1}(x^{A})\,.$ (7.4)

Recalling the variations (4.9), to preserve the condition (7.3) on the phase
space the transformation parameter $T^{+}$ has to obey

$\partial^{2}_{+}T^{+}=0\qquad\Longrightarrow\qquad T^{+}=T_{0}+x^{+}T_{1}$
(7.5)

where $T_{0},T_{1}$ are arbitrary functions of $x^{A}$, while $T^{-}$ and
$Y^{A}$ are unrestricted. Therefore, the on-shell phase space in general is
determined by
$\eta(x^{+},x^{A}),\Omega_{0}(x^{A}),\Omega_{1}(x^{A}),\Upsilon_{A}^{0}(x^{B})$
and the symmetry generators are
$T^{-}(x^{+},x^{A}),T_{0}(x^{A}),T_{1}(x^{A}),Y^{A}(x^{B})$.

###  7.2 Null hypersurfaces in Gaussian null coordinates

Many previous near horizon studies invoked Gaussian null coordinates, see e.g.
Donnay:2015abr ; Donnay:2016ejv ; Chandrasekaran:2018aop ;
Chandrasekaran:2019ewn . It is one of the purposes of this subsection to show
that there is loss of generality in assuming Gaussian null coordinates,
because the coordinate transformations required to transform to these
coordinates are not proper ones within our more general set of boundary
conditions (6.3). Technically, it turns out that Gaussian null coordinates
require to fix $\eta=1$ and $\delta\eta=0$ on the phase space of metrics
(6.3). This leads to a reduced phase space. Hence there is loss of generality
in making these assumptions. In the remainder of this subsection we make
explicit the consequences of the choice $\eta=1$ (and $\delta\eta=0$), in
particular for the metric and for the symmetry generators.

The condition $\eta=1$ is achieved as follows. Consider the coordinate system
in section 6.1, in particular the metric (6.3). Next, perform the coordinate
transformation

$x^{-}\to x^{-}\mathcal{C}$ (7.6)

where $\mathcal{C}$ is an arbitrary function of $x^{+}$ and $x^{A}$. The
functions in the metric (6.3) then transform as

$\eta\to{\cal C}\eta\qquad\qquad{{\cal G}}\to{{\cal
G}}+2\,\frac{\partial_{+}{\cal C}}{{\cal
C}}\qquad\qquad\frac{1}{\eta}\theta_{A}\to\frac{1}{\eta}\theta_{A}+\frac{\partial_{A}{\cal
C}}{{\cal C}}\,.$ (7.7)

The coordinate transformation (7.6) is generated by $T^{-}$ in (4.8). The
transformations of the metric functions (7.7) leave invariant $\Omega$ and
${\cal G}-2\partial_{+}\eta/\eta$, while ${\cal G}$, $\eta$, and
$\Upsilon_{A}$ transform. Therefore, the coordinate transformation (7.6) is a
nontrivial diffeomorphism since there is a non-vanishing surface charge
associated with it. The choice ${\cal C}=1/\eta$ yields the desired reduction
of the phase space to a sector in which $\eta=1$. The metric (6.3) is then
manifestly in the Gaussian null coordinates,

$\operatorname{d}\\!{s}^{2}=-2x^{-}{\cal K}\
(\operatorname{d}\\!{x}^{+}){}^{2}+2\operatorname{d}\\!{x}^{-}\operatorname{d}\\!{x}^{+}+2x^{-}\tilde{\theta}_{A}\operatorname{d}\\!{x}^{+}\operatorname{d}\\!{x}^{A}+\Omega_{AB}\operatorname{d}\\!{x}^{A}\operatorname{d}\\!{x}^{B}+\cdots$
(7.8)

where

${\cal
K}=-\frac{\mathcal{G}}{2}+\frac{\partial_{+}\eta}{\eta}\qquad\qquad\tilde{\theta}_{A}=\frac{1}{\eta}(\theta_{A}-\partial_{A}\eta)\,.$
(7.9)

Due to the form of the near null hypersurface Killing vectors (4.8) a generic
transformation moves us away from $\eta=1$. To keep this condition we require
$\delta_{\xi}\eta=0$, which by virtue of (4.9b) yields

$T^{-}=\partial_{+}T^{+}\,.$ (7.10)

One may use (4.4) to solve ${\cal G}$ in terms of $\Omega$ and then (4.6) to
solve $\Upsilon_{A}$ in terms of $\Omega$ and an initial value function
$\Upsilon_{A}^{0}$. So in this sector our on-shell phase space is described by
$\Omega(x^{+},x^{A}),\Upsilon_{A}^{0}(x^{B})$ and the independent symmetry
generators are $T^{+}(x^{+},x^{A}),Y^{A}(x^{B})$. The T-Witt algebra (4.17a)
is then reduced to the S-Witt algebra, generated by ${\cal R}_{n}^{p,q}$.

In addition to the case $\eta=1$ considered here, one can demand the generator
$l^{\mu}\partial_{\mu}=N\,\partial_{+}+\mathcal{O}(x^{-})$ to have a constant
non-affinity parameter, as it was done in section 7.1. The constant non-
affinity parameter case with $N\propto x^{+}\eta$ yields (7.3) and does not
constrain $T^{-}$. It is hence compatible with the $\eta=1$ constraint and the
two restrictions may be combined.

Requiring constancy of the non-affinity parameter in (7.3) together with
$\eta=1$ implies ${\cal G}=0$. Preserving this condition yields

$\partial^{2}_{+}T^{+}=0,\ T^{-}=\partial_{+}T^{+}\qquad\Longrightarrow\qquad
T^{+}=T_{0}+x^{+}T_{1}\,,\ T^{-}=T_{1}$ (7.11)

where $T_{0},T_{1}$ are arbitrary functions of $x^{A}$. The field equations
for $\Omega$ yield (7.4) and for $\Upsilon_{A}$ (4.6) reduce to
$\partial_{+}\Upsilon_{A}=\Omega\partial_{+}\partial_{A}\ln\Omega$. Therefore,
time dependence of $\Upsilon_{A}$ is completely determined in terms of
$\Omega$, the on-shell phase space in general is specified by
$\Omega_{0}(x^{A}),\Omega_{1}(x^{A}),\Upsilon_{A}^{0}(x^{B})$, and the
symmetry generators are $T_{0}(x^{A}),T_{1}(x^{A}),Y^{A}(x^{B})$. The symmetry
generators, charge and flux read

$\displaystyle\xi$
$\displaystyle=T_{0}\partial_{+}+T_{1}(x^{+}\partial_{+}-x^{-}\partial_{-})+Y^{A}\partial_{A}$
(7.12a) $\displaystyle{\cal Q}_{\xi}$
$\displaystyle=2\Omega\partial_{+}T^{+}-2T^{+}\partial_{+}\Omega+Y^{A}\Upsilon_{A}$
(7.12b) $\displaystyle{\cal F}_{\xi}$
$\displaystyle={4\,}T^{+}\sqrt{\Omega_{1}}\ \delta(\sqrt{\Omega})\,.$ (7.12c)

The charge becomes integrable when the flux vanishes, i.e., for
$\Omega_{1}=0$. This is the stationary case with $\partial_{+}\Omega=0$,
discussed in section 6.3.

We finally note that upon the coordinate transformation

$x^{+}=\ e^{{\kappa}v}+\mathcal{O}(\rho^{2})\qquad\qquad x^{-}=\
\frac{e^{-{\kappa}v}}{\kappa}\rho+\mathcal{O}(\rho^{2})\qquad\qquad x^{A}=\
y^{A}+\mathcal{O}(\rho^{2})\,,$ (7.13)

with $\kappa:=\kappa_{l}$, the line-element (6.3) (for $\eta=1,{\cal G}=0$)
takes the standard Gaussian null form

$\operatorname{d}\\!{s}^{2}=-2\,{\kappa}\,\rho\,\operatorname{d}\\!{v}^{2}+2\,\operatorname{d}\\!{v}\,\operatorname{d}\\!{\rho}+2\,\rho\,{\theta}_{A}\,\operatorname{d}\\!{v}\,\operatorname{d}\\!{y}^{A}+\left({\Omega}_{AB}+\rho\,{\lambda}_{AB}\right)\,\operatorname{d}\\!{y}^{A}\,\operatorname{d}\\!{y}^{B}+\mathcal{O}\left(\rho^{2}\right)\,.$
(7.14)

The null hypersurface $x^{-}=0$ is mapped to the surface $\rho=0$ in (7.14)
and is generated by $\zeta_{\textrm{\tiny
H}}=\kappa\,(x^{+}\partial_{+}-x^{-}\partial_{-})=\partial_{v}$ whose the non-
affinity parameter $\kappa$ is constant. Note that the coordinate
transformation (7.13) is a proper gauge transformation over the restricted
phase space we are considering in this section.

The near horizon Killing vector (7.12) in Gaussian null coordinates

$\xi=f\partial_{v}-\partial_{v}f\rho\partial_{\rho}+\left(Y^{A}-\rho\,{\Omega}^{AB}\partial_{B}f\right)\partial_{A}+\mathcal{O}(\rho^{2})\quad\textrm{with}\quad
f=\frac{1}{\kappa}\,\big{(}T_{1}+e^{-\kappa v}T_{0}\big{)}$ (7.15)

is exactly the near-horizon symmetry generator for isolated horizons presented
in Donnay:2016ejv ; Chandrasekaran:2019ewn . The near horizon symmetry algebra
in this case is given by (4.17), but with the T-Witt sector restricted to
${\cal R}^{p,q}_{0}\simeq{\cal R}^{p,q}_{0}+\mathcal{T}^{p,q}_{0}$ (which in
terms of differential operators is $-\zeta_{\textrm{\tiny H}}/\kappa$) and
${\cal R}^{p,q}_{-1}$.

###  7.3 Killing horizons

In this subsection we restrict the phase space to describe Killing horizons.
It means we require the existence of a vector field $\zeta_{\text{\tiny H}}$
normal to the null hypersurface $x^{-}=0$ such that ${\cal
L}_{\zeta_{\text{\tiny H}}}g_{\mu\nu}=0$ and $\zeta_{\text{\tiny H}}^{2}=0$ at
$x^{-}=0$. The geometry (6.3) admits such a Killing vector field,

$\zeta_{\text{\tiny
H}}=\frac{N}{\eta}\partial_{+}+l_{(1)}^{\mu}\,x^{-}\,\partial_{\mu}+\mathcal{O}(x^{-})^{2}$
(7.16)

where $N$ is a possibly field-dependent function over the phase space, if

$\displaystyle\partial_{+}\Omega$ $\displaystyle=0$ $\displaystyle\Gamma$
$\displaystyle=\partial_{+}\ln\frac{N^{2}}{\eta}-2\frac{\eta}{N}\kappa$
(7.17a) $\displaystyle\Omega_{AB}l_{(1)}^{B}$
$\displaystyle=-N\partial_{A}\ln\frac{N}{\eta}$ $\displaystyle l_{(1)}^{-}$
$\displaystyle=-\frac{\partial_{+}N}{\eta}$ $\displaystyle l_{(1)}^{+}$
$\displaystyle=0$ (7.17b)
$\displaystyle\partial_{+}{\Upsilon_{A}}+\Omega\,\partial_{A}\Gamma$
$\displaystyle=\frac{2\eta}{N}\partial_{A}\kappa$ (7.17c)

where $\kappa(x^{A})$ is the non affinity parameter of (7.16). Moreover the
eom $\Upsilon_{A}$ (6.11a) implies $\partial_{A}\kappa=0$. This is the zeroth
law of black hole thermodynamics stating that the surface gravity $\kappa$ has
to be constant over the Killing horizon.

The Killing horizon case is therefore a special case of the non-expanding case
of section 6.3 which has constant surface gravity. The charges and their
algebra are hence the same as the one discussed below (6.13). As in section
6.3, one may fix ${\cal G}=0$ gauge in which $\partial_{+}N=\kappa\eta$, and
thus $N=\kappa\,\int^{x^{+}}\eta+U(x^{A})$. However the integration function
$U(x^{A})$ has to vanish in order to consistently describe the bifurcation
point. The variation of $N$ is

$\delta_{\xi}{N}=T^{+}\partial_{+}N-T^{-}N+Y^{A}\partial_{A}{N}$ (7.18)

where we used (4.9b) and the fact that $T^{-}=T^{-}(x^{A})$.

As in the previous subsection, one may further restrict $N$. For example, for
the case of $N=\kappa\eta x^{+}$, $\partial_{+}N=\kappa\eta$ yields
$\partial_{+}\eta=0$ and (7.17b) yields $l_{(1)}^{A}=0$. Therefore,

$\zeta_{\text{\tiny
H}}=\kappa\left(x^{+}\partial_{+}-x^{-}\partial_{-}\right)+\mathcal{O}(x^{-})^{2}\,.$
(7.19)

Moreover, in this case $\Gamma=0,\delta\Gamma=0$ and hence we recover the
stationary case of section 6.3. That is, after fixing the extra accidental
symmetries we find in this case, the on-shell physical phase space is
described by $\Omega(x^{A}),\Upsilon_{A}^{0}(x^{B})$ and the symmetry algebra
reduces to that of section 3.

##  8 Concluding remarks

We conclude with a brief summary of the main results and a perspective on
future research directions.

Motivated by the desire to understand non-extremal Kerr black holes from a
near horizon perspective we introduced new coordinates (2.9) that are co-
rotating with the event horizon but otherwise do not mix angular and spacetime
coordinates and allow to cover the whole Kerr manifold up to the Cauchy
horizon. Expanding around the bifurcation 2-sphere of the Kerr geometry led us
to propose bifurcation fall-off conditions (3.2). The associated bifurcation
Killing vector algebra (3.9a) consisted of superrotations and two towers of
supertranslations. However, only one such copy entered the surface charges
(3.13), while as we discussed in the end of section 6.1, the other one turned
out to be pure gauge.

We generalized the fall-off conditions to generic (but co-rotating) null
hypersurfaces (regardless of whether they describe bifurcate Killing horizons,
isolated horizons or no horizons), see (4.3). The near null hypersurface
Killing vectors (4.8) generated an algebra (4.17) that contained again
superrotations and two towers of supertranslations, which among themselves
formed a subalgebra (4.17a) that we dubbed ‘T-Witt’ algebra. In contrast to
the previous case both towers of supertranslations turned out to be relevant
for the charges (5.1), which no longer were integrable. Their split into
integrable and flux parts was ambiguous, as expected on general grounds, but
the ambiguity could be largely fixed by demanding the absence of a central
extension. This led us to the integrable part (5.12) and the flux part (5.13),
which together obey a generalized charge conservation equation (5.18) [or
(5.20)] that justifies the label ‘flux’. We verified in section 6 that there
are always equally many functions parametrizing the on-shell phase space and
functions appearing in the symmetry generators, which is why we referred to
the phase space as ‘boundary phase space’. In section 7 we recovered special
cases considered in earlier literature on near horizon symmetries.

For readers exclusively interested in results pertaining to Kerr black holes
(2.9) we summarize them briefly in this paragraph. The near bifurcation
charges and their variations are displayed in (3.15)-(3.17). The more generic
near horizon expansion of Kerr is displayed in appendix A.

generic: T-Witt ${\cal R}_{n}^{p,q}$, ${\cal T}_{n}^{p,q}$ Eq. (4.17a) non-
expanding ${\cal R}_{n}^{p,q}$, ${\cal T}_{0}^{p,q}$ Sec. 6.3 GNC: S-Witt
${\cal R}_{n}^{p,q}$ Sec. 7.2 const. non-affinity ${\cal R}_{-1}^{p,q}$,
${\cal R}_{0}^{p,q}$, ${\cal T}_{n}^{p,q}$ Sec. 7.1 GNC & const. $\kappa$
${\cal R}_{-1}^{p,q}$, ${\cal R}_{0}^{p,q}$ Sec. 7.2 stationary ${\cal
R}_{0}^{p,q}\simeq{\cal R}_{0}^{p,q}+{\cal T}_{0}^{p,q}$ Sec. 3 & 6.3 & 7.3
Figure 1: Selected infinite subalgebras of T-Witt algebra. The null
hypersurface symmetry algebra consists of superrotations and the T-Witt. Some
physically motivated restrictions, such as vanishing expansion, leave intact
the superrotations but reduce T-Witt to one of its subalgebras discussed in
appendix B. Particularly the assumption of Gaussian null coordinates reduces
T-Witt to S-Witt. Assuming additionally constant surface gravity and/or
stationarity yields further reductions, recovering near horizon symmetries
also found in previous literature.

In this work we considered some subalgebras of the T-Witt algebra motivated by
geometric and/or physical restrictions of the boundary conditions, see fig. 1
and appendix B. There are numerous further algebraic aspects that deserve
further study. We list here some of them. It could be interesting to fully
classify all maximal finite subalgebras and the infinite subalgebras of the
T-Witt algebra, as well as all their central extensions. To obtain the latter
we would need to relax the regularity assumption stated below (4.17) (e.g. by
compactifying the null hypersurface or by allowing a singular point $x^{+}=0$)
and allow for positive and negative mode numbers. Besides these purely
algebraic aspects, it would be physically interesting to find a geometric
realizations of boundary conditions that switch on some of the algebraically
allowed central extensions. Also the superrotation part of the algebra may
deserve further study, see e.g. Campiglia:2020qvc  and refs. therein for a
recent discussion in the context of asymptotically flat spacetimes.

Some current and future research directions that have strong overlap with the
results of our work are listed below.

  * •

Bulk and boundary. The discussions in our work also reflect on purely
theoretical considerations relevant for field theories in the presence of
boundaries, see e.g. Compere:2018aar ; Strominger:2017zoo ; Harlow:2019yfa ;
Geiller:2019bti ; Barnich:2019xhd  and refs. therein. On general grounds, the
phase space splits into a bulk part, capturing the local physical excitations
(in our case two helicities of massless gravitational waves), and a boundary
part, capturing physical boundary degrees of freedom. While there seems
universal agreement on the bulk, there are different perspectives in the
literature on how to interpret the boundary degrees of freedom, sometimes even
to the extent that it is questioned whether they are physical. A common
perspective is to view the surface charges as a necessary boundary
contribution to the canonical gauge generators to render them functionally
differentiable. Following this line of thought to its conclusion means that
the surface charges parametrize those boundary condition preserving
transformations that are pure gauge in the bulk, but fail to be pure gauge at
the boundary. Hence, there are fewer gauge degrees of freedom at the boundary,
which is the essence of why physical boundary excitations emerge. An
alternative perspective is to explicitly add boundary degrees of freedom with
their own boundary equations of motion to restore gauge invariance and
factorizability of the Hilbert space under splitting of spacetime into
subregions, see Geiller:2019bti  and refs. therein. Either way, the total
physical phase space consists of two parts, the bulk phase space (describing
gravitational wave excitations in Einstein gravity) and the boundary phase
space (described by surface charges and governed by the symmetries they
generate). In our work we focused almost exclusively on the latter. It would
be interesting to generalize our discussion to address also the bulk phase
space (the gravitational waves) and its interaction with boundary degrees of
freedom, for instance to describe physical processes such as formation of a
black hole, things falling into the hole or black hole evaporation and their
impact on the surface charges.

  * •

Gravitational waves. As a first step to generalize our work to include a
discussion of bulk degrees of freedom we can identify where gravitational wave
excitations are hidden in our construction. In the coordinates adapted to null
hypersurfaces we have an ingoing direction $x^{-}$, an outgoing direction
$x^{+}$ and transversal directions $x^{A}$, see for instance the fully gauge-
fixed version of metrics (6.3). Ingoing gravitational waves can be Fourier-
decomposed with factors $e^{ik_{-}x^{-}}$, and thus are captured by metric
functions such as $\lambda_{AB}$ (and subleading terms) in (6.3). Outgoing
gravitational waves with Fourier factors $e^{ik_{+}x^{+}}$ are switched off by
the conformality assumption (3.7), so by relaxing this assumption the
information about such waves would be contained in $\Omega_{AB}$. Thus, we see
that three quarters of all gravitational wave modes require to drop the
assumption (3.7); the remaining quarter, which is already included in our
current work, are ingoing gravitational waves with nodes on the null
hypersurface. The latter do not influence the surface charges, while the
remaining three quarters will generically induce non-trivial changes of the
surface charges and fluxes. It should be rewarding to consider some elementary
processes — e.g. capture or emission of a single plane graviational wave — to
unravel the possible interactions between surface charges and gravitational
waves, and more generally between boundary and bulk degrees of freedom.

  * •

Charges and fluxes. The charges associated with boundary conditions in the
present work are not generically integrable. The Barnich–Troessaert method of
using a modified bracket almost uniquely allows to separate the integrable
from the non-integrable part (there remains an inessential ambiguity, up to
algebra 2-cocycles). The generalized charge conservation equation (5.18)
relates the non-integrability of the charge to the non-conservation of the
integrable part. Instead of using the modified bracket one could elevate the
generalized charge conservation equation to a postulate. This is essentially
the idea behind the Wald–Zoupas method Wald:1999wa : if there is a region of
stationary configurations in the phase space one expects there the charges to
be conserved (and hence integrable), and the fluxes to vanish. This region may
then be used as reference point for charge-flux separation. In all examples
that both the Barnich–Troessaert and Wald–Zoupas methods can be applied to
they yield the same result. In our case, however, as the discussions in
section 6 reveal, in general we do not have a reference point of stationary
configurations. Hence, one may not unambiguously employ the Wald–Zoupas
method. It could be interesting to explore if the Wald–Zoupas method can be
generalized to cover cases like ours.

  * •

Soft hair. Numerous previous papers considered near horizon aspects of black
holes, see e.g. tHooft:1990fkf ; tHooft:1991uqr ; Susskind:1993if ;
Hayward:1993wb ; Carlip:1994gy ; Balachandran:1994up ; Carlip:1995cd ;
Strominger:1997eq ; Ashtekar:1997yu ; Carlip:1998wz ; Hotta:2000gx ;
Ashtekar:2000sz ; Ashtekar:2002ag ; tHooft:2006xjp ; Majhi:2012tf ;
Penna:2015gza ; Afshar:2015wjm  and refs. therein. A new aspect addressed
during the past five years is the emergence of soft hair excitations. The name
‘soft hair’ was coined by Hawking, Perry and Strominger Hawking:2016msc ,
where ‘soft’ refers to zero energy excitations and ‘hair’ to these excitations
being physical. This line of research has engendered a lot of recent
activities, see for instance Hooft:2016itl ; Averin:2016ybl ; Compere:2016hzt
; Afshar:2016wfy ; Eling:2016xlx ; Afshar:2016uax ; Mirbabayi:2016axw ;
Donnay:2016ejv ; Hopfmuller:2016scf ; Hawking:2016sgy ; Afshar:2016kjj ;
Wieland:2017zkf ; Bousso:2017dny ; Strominger:2017aeh ; Akhmedov:2017ftb ;
Lust:2017gez ; Blommaert:2018rsf ; Hooft:2018gtw ; Donnelly:2018nbv ;
Chandrasekaran:2018aop ; Donnay:2018ckb ; Haco:2018ske ; Penna:2018gfx ;
Donnay:2019jiz ; Wieland:2019hkz ; Grumiller:2019tyl ; Chandrasekaran:2019ewn
; Grumiller:2019fmp ; Grumiller:2019ygj ; Bagchi:2019clu ; Ashtekar:2020ifw .
The first papers where soft hair was constructed as near horizon excitations
were based on boundary conditions that kept fixed surface gravity and had
integrable surface charges Donnay:2015abr ; Afshar:2016wfy , unlike the
present work. A subtlety in these constructions is the precise state-
dependence of the near horizon Killing vectors: depending on the assumptions
one can construct different near horizon symmetries, with or without central
charges, including BMS symmetries with supertranslation generators of
arbitrary spin and Heisenberg symmetries Grumiller:2019fmp . It is possible
(and could be useful for applications) to generalize the construction in our
present work to allow for suitable state-dependence of the near null
hypersurface Killing vectors (4.8), i.e., to allow certain field-variations of
the functions $T^{\pm}$ and $Y^{A}$ appearing therein. Like in
Grumiller:2019fmp  this would model the physical properties of the black hole
and its interactions with some fiducial thermal bath, much like macroscopic
electrodynamics models certain materials through suitable boundary conditions
without having to deal with details of the microphysics. The novel aspect
provided by our results in the context of this program compared to the
discussions in Grumiller:2019fmp  is the possibility to allow for non-
integrable charges and hence fluxes through the horizon.

  * •

Thermodynamical aspects. In the stationary limit there is a simple near
horizon first law relating variations of the near horizon energy to
temperature times the variation of the black hole entropy Donnay:2015abr , as
expected from the Iyer–Wald derivation Iyer:1994ys  of the first law of black
hole mechanics (see also Hajian:2015xlp  for generalizations of the Iyer–Wald
analysis). For time-dependent situations considered in the present work it is
plausible that at least in the adiabatic limit there could be a generalized
first law relating time variations of the near horizon energy to time
variations of entropy and fluxes across the horizon. It will be interesting to
establish such a generalized first law, based on the generalized charge
conservation equation (5.20).

  * •

Kerr entropy. One of the outstanding goals is to provide a microscopic picture
of the entropy of generic Kerr black holes. While for the extremal case one
may invoke the Kerr/CFT correspondence Guica:2008mu ; Compere:2012jk  to count
the black hole microstates, it remains unclear what these microstates are and
how to generalize this construction to generic Kerr black holes. Haco,
Hawking, Perry and Strominger Haco:2018ske  proposed the two-dimensional
conformal algebra (with central charges given by the black hole angular
momentum times 12) near the horizon of generic Kerr black holes as responsible
for the microstates and verified that applying the Cardy formula to the
conformal algebra reproduces the correct Bekenstein–Hawking entropy. A
discussion by Aggarwal, Castro and Detournay Aggarwal:2019iay  suggests
alternatively at the same level of rigor centrally extended Virasoro–Kac–Moody
symmetries. In the present work as in many previous works, e.g. Donnay:2015abr
; Donnay:2016ejv ; Chandrasekaran:2018aop ; Chandrasekaran:2019ewn ;
Grumiller:2019fmp ; Flanagan:2019vbl , the near horizon symmetry algebras do
not have central charges. Absence of central terms in the canonical
realization of the near horizon symmetry algebra bars simple ‘Cardyology’ but
does not mean that these symmetry algebras cannot be used for microstate
counting, as long as there is a gap between the vacuum and the black holes
state. While we do not know if the T-Witt symmetries of our present work will
ultimately be useful for a generic Kerr black hole microstate counting, it is
an exciting possibility to ponder about in the future.

  * •

Semi-classical microstate construction. Even if the microstate counting based
on symmetries did work, it does not tell us precisely what these microstates
are. A more ambitious approach is to explicitly construct these microstates,
at least in some semi-classical limit (large black holes, sufficiently far
away from extremality). We have no idea if this ever will work for generic
Kerr black holes, but we offered a concrete “fluff” proposal for three-
dimensional black holes in anti-de Sitter space Afshar:2016uax ;
Afshar:2017okz  that was based on near horizon soft hair. This construction
gives at least a glimpse of hope that near horizon symmetries, such as the
ones discussed in the present work, will be a key aspect in semi-classical
microstate constructions of generic Kerr black holes.

  * •

Information loss. Charge non-conservation/non-integrability and generalized
conservation equations like (5.18) and (5.20) may also be useful to move
beyond stationary questions (like microstates) and address dynamical ones,
e.g. Hawking radiation and the information loss problem. It would be rewarding
to explore this further.

Finally, the effect that assuming some a priori gauge like Gaussian null
coordinates can reduce the boundary phase space is independent from whether
one expands near the horizon or in the asymptotic region. Indeed, a similar
effect was observed already in three-dimensional gravity where the assumption
of Fefferman–Graham gauge reduces the physical phase space Grumiller:2016pqb ;
Grumiller:2017sjh . Thus, it seems plausible that also in the context of four-
dimensional asymptotically flat gravity standard assumptions like Bondi-gauge
can reduce the boundary phase space. In conclusion, it would be interesting to
reconsider yet-another-time asymptotic symmetries of asymptotically flat
spacetimes to verify whether or not the same effect arises there. It cannot be
excluded that an enlarged set of asymptotic symmetries will allow to make
novel statements about the infrared triangle asymptotic symmetries/memory
effects/soft theorems Strominger:2013jfa ; Strominger:2017zoo ;
Pasterski:2019msg .

## Acknowledgement

We are grateful to Glenn Barnich, Laura Donnay, Laurent Freidel, Marc Geiller,
Gaston Giribet, Mehdi Hakami Shalamzari, Marc Henneaux, Malcolm Perry, Kartik
Prabhu, Andrea Puhm, Romain Ruzziconi and Wolfgang Wieland for useful
discussions. DG and MMShJ thank Hamid Afshar, Martin Ammon, Stephane
Detournay, Hernán González, Philip Hacker, Wout Merbis, Blagoje Oblak, Alfredo
Pérez, Stefan Prohazka, David Tempo, Ricardo Troncoso, Raphaela Wutte and
Hossein Yavartanoo for previous collaboration on near horizon symmetries.

HA, SS and MMShJ acknowledge the support by INSF grant No. 950124 and
Saramadan grant No. ISEF/M/98204. DG was supported by the Austrian Science
Fund (FWF), projects P 30822 and P 32581. CZ was supported by the Austrian
Science Fund (FWF), projects P 30822 and M 2665. DG, SS and MMShJ acknowledge
the Iran-Austria IMPULSE project grant, supported and run by Khawrizmi
University and OeAD. MMShJ would like to thank the hospitality of ICTP HECAP
and ICTP EAIFR where a part of this work carried out.

##  Appendix A Near horizon expansion for Kerr

Here we present the expansion of the Kerr geometry around an arbitrary point
on the future horizon, i.e., we start from (2.9) and expand around $x^{-}=0$
for arbitrary (but finite) $x^{+}$. We display only non-zero metric components

$\displaystyle g_{++}$ $\displaystyle=\mathcal{O}\left([x^{-}]^{2}\right)$
(A.1) $\displaystyle g_{+-}$
$\displaystyle=-2r_{+}(r_{+}+r_{-}\cos^{2}\\!\theta)-\frac{4r_{+}^{2}r_{-}\sin^{2}\\!\theta}{r_{+}+r_{-}\cos^{2}\\!\theta}\,x^{+}x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
(A.2) $\displaystyle g_{--}$
$\displaystyle=\frac{2r_{+}(r_{+}^{2}-2r_{-}(r_{+}+r_{-})\cos^{2}\\!\theta-
r_{-}^{2}\cos^{4}\theta)}{r_{+}+r_{-}\cos^{2}\\!\theta}\,x^{+\,2}+F(\theta)\,x^{+\,3}x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
(A.3) $\displaystyle g_{+\phi}$
$\displaystyle=-\sin^{2}\\!\theta\,H(\theta)\,x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
(A.4) $\displaystyle g_{-\phi}$
$\displaystyle=\sin^{2}\\!\theta\,H(\theta)\,x^{+}-(r_{+}-r_{-})\,\sin^{2}\\!\theta\,K(\theta)\,x^{+\,2}x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
(A.5) $\displaystyle g_{\phi\phi}$
$\displaystyle=\frac{r_{+}(r_{+}+r_{-})^{2}\sin^{2}\\!\theta}{r_{+}+r_{-}\cos^{2}\\!\theta}-(r_{+}-r_{-})\sin^{2}\\!\theta\,G(\theta)\,x^{+}x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
(A.6) $\displaystyle g_{\theta\theta}$
$\displaystyle=r_{+}(r_{+}+r_{-}\cos^{2}\\!\theta)-2r_{+}(r_{+}-r_{-})x^{+}x^{-}+\mathcal{O}\left([x^{-}]^{2}\right)$
(A.7)

with the definitions

$\displaystyle F(\theta)$
$\displaystyle=r_{+}^{2}-2r_{-}^{2}-r_{+}r_{-}\cos^{2}\\!\theta-8r_{+}r_{-}\cos^{2}\\!\theta\,\frac{r_{+}^{2}-r_{-}^{2}}{(r_{+}+r_{-}\cos^{2}\\!\theta)^{2}}$
(A.8) $\displaystyle G(\theta)$
$\displaystyle=r_{+}-r_{-}+\frac{r_{+}(r_{+}^{2}-r_{-}^{2})+r_{-}(3r_{+}^{2}+4r_{+}r_{-}+r_{-}^{2})\cos^{2}\\!\theta}{(r_{+}+r_{-}\cos^{2}\\!\theta)^{2}}$
(A.9) $\displaystyle H(\theta)$
$\displaystyle=\sqrt{r_{+}r_{-}}\,(r_{+}-r_{-})+2\sqrt{r_{+}r_{-}}\,\frac{r_{+}\,\left(r_{+}+r_{-}\right)}{r_{+}+r_{-}\cos^{2}\\!\theta}$
(A.10) $\displaystyle K(\theta)$
$\displaystyle=\sqrt{r_{+}r_{-}}-(r_{+}+r_{-})\,\sqrt{r_{+}r_{-}}\,\frac{r_{+}-3r_{-}\cos^{2}\\!\theta}{(r_{+}+r_{-}\cos^{2}\\!\theta)^{2}}\,.$
(A.11)

Comparing the results above with the fall-off conditions (4.3) yields

$\displaystyle{\cal G}$ $\displaystyle={\cal U}_{\theta}={\cal
V}_{\theta}=\Omega_{\theta\phi}=0$ $\displaystyle\frac{{\cal F}}{x^{+\,2}}$
$\displaystyle=\frac{4r_{+}^{2}(r_{+}+r_{-})}{r_{+}+r_{-}\cos^{2}\\!\theta}-2r_{+}\big{(}r_{+}+r_{-}(2+\cos^{2}\\!\theta)\big{)}$
(A.12a) $\displaystyle\eta$
$\displaystyle=-2r_{+}(r_{+}+r_{-}\cos^{2}\\!\theta)$
$\displaystyle\frac{{\cal U}_{\phi}}{x^{+}}$ $\displaystyle=-{\cal V}_{\phi}\
=\sin^{2}\\!\theta\,H(\theta)$ (A.12b) $\displaystyle\Omega_{\theta\theta}$
$\displaystyle=r_{+}(r_{+}+r_{-}\cos^{2}\\!\theta)$
$\displaystyle\Omega_{\phi\phi}$
$\displaystyle=\frac{r_{+}(r_{+}+r_{-})^{2}\sin^{2}\\!\theta}{r_{+}+r_{-}\cos^{2}\\!\theta}\,.$
(A.12c)

##  Appendix B Some subalgebras of the null hypersurface symmetry algebra

The null hypersurface symmetry algebra (4.17) has different subalgebras which
on their own may arise as near horizon symmetry algebras for more restrictive
sets of boundary conditions (like in section 7). In this appendix we list some
of these subalgebras.

###  B.1 Finite subalgebras

Finite subalgebras of the near horizon/asymptotic symmetry algebras are
typically associated with the global symmetries (isometries) of the
corresponding background or their conformal extensions. Finite maximal
subalgebras are subalgebras where no additional generator can be added without
producing an infinite algebra.

Rather than being encyclopedic we focus on the physically most interesting
finite maximal subalgebra, where the superrotations are restricted to ${\cal
L}^{a}$, ${\cal\bar{L}}^{a}$, $a=\pm 1,0$, which form an
sl$(2,\mathbb{R})\oplus\,$sl$(2,\mathbb{R})\simeq\,$so$(2,2)$ algebra.
Geometrically, this algebra is the global part of the conformal algebra acting
on the bifurcation surface ${\cal B}$. The supertranslations are restricted to
${\cal R}^{0,0}_{a},\ a=\pm 1,0$ and ${\cal T}^{0,0}_{0}$. These generators
commute with the conformal algebra and form an $sl(2,\mathbb{R})\oplus u(1)$
subalgebra. Geometrically, ${\cal R}_{-1}^{0,0}$ generates rigid timeshifts,
$x^{+}\to x^{+}+x^{+}_{0}$, while ${\cal R}^{0,0}_{0}$ and ${\cal
T}^{0,0}_{0}$ generate rigid scalings (also known as conformal boosts) in
$x^{+}$ and $x^{-}$ directions, respectively, i.e., $x^{+}\to\lambda_{+}x^{+}$
and $x^{-}\to\lambda_{-}x^{-}$. Therefore, ${\cal R}^{0,0}_{0}-{\cal
T}^{0,0}_{0}$ generates boosts and ${\cal R}^{0,0}_{0}+{\cal T}^{0,0}_{0}$
dilatations on the $x^{\pm}$ plane. The remaining generator ${\cal
R}_{1}^{0,0}$ generates special conformal translations along $x^{+}$. Notably,
we do not have translations (nor special conformal translations) along $x^{-}$
and hence the 2d Poincaré algebra is not contained in our subalgebra.
Physically, this is expected since we fix the locus of the null hypersurface
to be $x^{-}=0$, a condition which manifestly breaks translation invariance
along $x^{-}$.

In conclusion, an interesting finite maximal subalgebra of the null
hypersurface symmetry algebra (4.17) is
so$(2,2)\oplus\,$sl$(2,\mathbb{R})\oplus\,$u$(1)$, with the first part
corresponding to global conformal transformations of the bifurcation surface,
the second part to a chiral half of global conformal transformations in the
$x^{\pm}$ plane and the abelian factor corresponding to boosts along $x^{-}$.
As differential operators the generators of this algebra are given by
$\partial_{z}$, $z\partial_{z}$, $z^{2}\partial_{z}$, $\partial_{\bar{z}}$,
$\bar{z}\partial_{\bar{z}}$, $\bar{z}^{2}\partial_{\bar{z}}$, $\partial_{+}$,
$x^{+}\partial_{+}$, $(x^{+})^{2}\partial_{+}$ and $x^{-}\partial_{-}$.

###  B.2 Infinite subalgebras

One can identify many different infinite dimensional subalgebras of the null
hypersurface symmetry algebra (4.17). Here we list some of the most important
ones.

  1. 1.

One can switch off everything except for the T-Witt algebra generated by
${\cal R}_{n}^{p,q}$ and ${\cal T}_{n}^{p,q}$, keeping rigid the bifurcation
surface $\cal B$. The T-Witt algebra itself also has some interesting
subalgebras, for instance the BMS${}_{3}^{(s=0)}$ algebra (also known as
‘warped Witt algebra’) generated by ${\cal R}_{n}:={\cal R}_{n}^{0,0}$ as Witt
generators and ${\cal T}_{n}:={\cal T}_{n}^{0,0}$ as lower-spin
supertranslations, or the sl$(2,\mathbb{R})\oplus u(1)$ extended Witt algebra
generated by ${\cal R}_{a}^{p,q},a=\pm 1,0$ and ${\cal T}_{0}^{p,q}$.

  2. 2.

Alternatively, one can switch off the T-Witt part of the algebra. The
remainder is a Witt$\,\oplus\,$Witt subalgebra, the conformal algebra over
${\cal B}$, generated by ${\cal L}^{p}$, ${\cal{\bar{L}}}^{p}$.

  3. 3.

Physically more relevant are combined subalgebras that contain part of the
T-Witt algebra and the full conformal algebra over ${\cal B}$. For example,
one can keep the T-Witt generators ${\cal R}_{n}^{0,0}$, ${\cal T}_{n}^{0,0}$
and all Witt generators ${\cal L}^{p}$, ${\cal{\bar{L}}}^{q}$, yielding the
subalgebra Witt$\,\oplus\,$Witt$\,\oplus$ BMS${}_{3}^{(s=0)}$. This subalgebra
is an infinite-dimensional lift of the maximal finite subalgebra discussed
above. In the remaining cases below we single out some of these algebras that
either appeared in earlier literature or in the present work by imposing
additional restrictions on the boundary conditions. We start with the former.

  4. 4.

There are infinitely many DGGP type Donnay:2015abr  subalgebras, generated by
${\cal L}^{p}$, ${\cal{\bar{L}}}^{p}$, ${\cal R}_{n}^{p,q}$ or ${\cal L}^{p}$,
${\cal{\bar{L}}}^{p}$, ${\cal T}_{n}^{p,q}$ for any fixed $n$. These algebras
are BMS${}_{4}^{(s=0)}$ in the notation of Grumiller:2019fmp , or ${\cal
W}(0,0;0,0)$ in the notation of Safari:2019zmc . The only difference to the
original BMS4 algebra Bondi:1962 ; Sachs:1962 ; Barnich:2011ct  is the lower
spin of the supertranslations, see the discussion in the appendix of
Grumiller:2019fmp .

  5. 5.

The algebra discussed in Donnay:2016ejv ; Chandrasekaran:2018aop ;
Chandrasekaran:2019ewn  is a subalgebra generated by ${\cal L}^{p}$,
${\cal{\bar{L}}}^{p}$, ${\cal R}_{-1}^{p,q}$ and ${\cal R}_{0}^{p,q}$. Upon
changing the basis777More generally, the algebra automorphism ${\cal
R}_{n}^{p,q}\to{\cal R}_{n}^{p,q}+\alpha(n+1){\cal T}_{n}^{p,q}$ (with some
constant $\alpha$) amounts to shifting the null hypersurface symmetry
generator $T^{-}\to T^{-}+\alpha\partial_{+}T^{+}$.  ${\cal
R}_{0}^{p,q}\to{\cal R}_{0}^{p,q}+{\cal T}_{0}^{p,q}$ this subalgebra arises
as a special case discussed in section 7 where surface gravity is constant and
Gaussian null coordinates are used (with loss of generality), see the next
item.

  6. 6.

Finally, we address subalgebras that emerged as special cases in sections 6 or
7. The non-expanding case discussed in section 6.3 yields a subalgebra
generated by ${\cal L}^{p}$, $\bar{\cal L}^{p}$, ${\cal T}^{p,q}_{0}$ and
${\cal R}_{n}^{p,q}$, see the statements after (6.13). Constant surface
gravity with (7.3) leads to a subalgebra generated by ${\cal L}^{p}$,
$\bar{\cal L}^{p}$, ${\cal T}^{p,q}_{n}$, ${\cal R}_{-1}^{p,q}$ and ${\cal
R}_{0}^{p,q}$. Gaussian null coordinates, i.e., constant $\eta$, lead to a
subalgebra generated by ${\cal L}^{p}$, $\bar{\cal L}^{p}$ and ${\cal
R}^{p,q}_{n}$, see the statements below (7.10). Combining both cases, i.e.,
assuming constant $\eta$ and constant surface gravity, leads to a smaller
subalgebra, generated by ${\cal L}^{p}$, $\bar{\cal L}^{p}$, ${\cal
R}_{-1}^{p,q}$ and ${\cal R}_{0}^{p,q}+{\cal T}_{0}^{p,q}$, see the statements
below (7.15). Killing horizons in general also yield the constant surface
gravity subalgebra, but when restricted to near bifurcation boundary
conditions recover the smaller algebra (3.9a) found in section 3, which is a
subalgebra generated by ${\cal L}^{p}$, $\bar{\cal L}^{p}$, ${\cal
T}_{0}^{p,q}$ and ${\cal R}_{0}^{p,q}$. [Notational alert: in (3.9a) all mode
indices appear as lower indices and are labelled by letters $n,m,l,k$, as
opposed to (4.17).] Since only the linear combination ${\cal
R}_{0}^{p,q}+{\cal T}_{0}^{p,q}$ enters the charges this subalgebra further
reduces to the DGGP algebra Donnay:2015abr , generated by ${\cal L}^{p}$,
$\bar{\cal L}^{p}$ and supertranslations ${\cal R}_{0}^{p,q}+{\cal
T}_{0}^{p,q}$.

## References

  * (1) R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,” Phys. Rev. Lett. 11 (1963) 237–238.
  * (2) A. Einstein, “Die Feldgleichungen der Gravitation,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1915 (1915) 844–847.
  * (3) L. Donnay, G. Giribet, H. A. González, and M. Pino, “Supertranslations and Superrotations at the Black Hole Horizon,” Phys. Rev. Lett. 116 (2016), no. 9, 091101, 1511.08687.
  * (4) L. Donnay, G. Giribet, H. A. González, and M. Pino, “Extended Symmetries at the Black Hole Horizon,” JHEP 09 (2016) 100, 1607.05703.
  * (5) V. Chandrasekaran, É. É. Flanagan, and K. Prabhu, “Symmetries and charges of general relativity at null boundaries,” JHEP 11 (2018) 125, 1807.11499.
  * (6) S. Haco, S. W. Hawking, M. J. Perry, and A. Strominger, “Black Hole Entropy and Soft Hair,” JHEP 12 (2018) 098, 1810.01847.
  * (7) 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 (2020) 041601, 1908.09833.
  * (8) M. D. Kruskal, “Maximal extension of Schwarzschild metric,” Phys. Rev. 119 (1960) 1743–1745.
  * (9) W. Israel, “New interpretation of the extended Schwarzschild manifold,” Phys.Rev. 143 (1966) 1016.
  * (10) J. M. Bardeen and G. T. Horowitz, “The extreme Kerr throat geometry: A vacuum analog of AdS(2) x S(2),” Phys. Rev. D60 (1999) 104030, hep-th/9905099.
  * (11) G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105 (2010) 111103, 0909.2617.
  * (12) G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP 1005 (2010) 062, 1001.1541.
  * (13) T. Regge and C. Teitelboim, “Role of surface integrals in the Hamiltonian formulation of general relativity,” Ann. Phys. 88 (1974) 286\.
  * (14) J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. 104 (1986) 207–226.
  * (15) M. Henneaux and C. Troessaert, “BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach,” JHEP 03 (2018) 147, 1801.03718.
  * (16) J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys. 31 (1990) 725–743.
  * (17) V. Iyer and R. M. Wald, “Some properties of Nöther charge and a proposal for dynamical black hole entropy,” Phys. Rev. D50 (1994) 846–864, gr-qc/9403028.
  * (18) G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conservation laws and central charges,” Nucl. Phys. B633 (2002) 3–82, hep-th/0111246.
  * (19) G. Barnich and G. Compere, “Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions,” Class.Quant.Grav. 24 (2007) F15–F23, gr-qc/0610130.
  * (20) G. Compère and A. Fiorucci, “Advanced Lectures on General Relativity,” Lect. Notes Phys. 952 (2019) 150, 1801.07064.
  * (21) A. Ashtekar, J. Bicak, and B. G. Schmidt, “Asymptotic structure of symmetry reduced general relativity,” Phys.Rev. D55 (1997) 669–686, gr-qc/9608042.
  * (22) A. Farahmand Parsa, H. R. Safari, and M. M. Sheikh-Jabbari, “On Rigidity of 3d Asymptotic Symmetry Algebras,” JHEP 03 (2019) 143, 1809.08209.
  * (23) G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS CNCFG2010 (2010) 010, 1102.4632. [Ann. U. Craiova Phys.21,S11(2011)].
  * (24) G. Barnich and C. Troessaert, “BMS charge algebra,” JHEP 1112 (2011) 105, 1106.0213.
  * (25) G. Compère, P.-J. Mao, A. Seraj, and M. M. Sheikh-Jabbari, “Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons,” JHEP 01 (2016) 080, 1511.06079.
  * (26) C. Bunster, A. Gomberoff and A. Pérez, “Regge–Teitelboim analysis of the symmetries of electromagnetic and gravitational fields on asymptotically null spacelike surfaces,” 1805.03728.
  * (27) V. Chandrasekaran and K. Prabhu, “Symmetries, charges and conservation laws at causal diamonds in general relativity,” JHEP 10 (2019) 229, 1908.00017.
  * (28) M. Campiglia and J. Peraza, “Generalized BMS charge algebra,” 2002.06691.
  * (29) A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” 1703.05448.
  * (30) D. Harlow and J.-Q. Wu, “Covariant phase space with boundaries,” 1906.08616.
  * (31) M. Geiller and P. Jai-akson, “Extended actions, dynamics of edge modes, and entanglement entropy,” 1912.06025.
  * (32) G. Barnich, “Black hole entropy from nonproper gauge degrees of freedom: The charged vacuum capacitor,” Phys. Rev. D99 (2019), no. 2, 026007, 1806.00549.
  * (33) R. M. Wald and A. Zoupas, “A General definition of ’conserved quantities’ in general relativity and other theories of gravity,” Phys.Rev. D61 (2000) 084027, gr-qc/9911095.
  * (34) G. ’t Hooft, “The black hole interpretation of string theory,” Nucl. Phys. B335 (1990) 138–154.
  * (35) G. ’t Hooft, “The Black hole horizon as a quantum surface,” Phys. Scripta T36 (1991) 247–252.
  * (36) L. Susskind, L. Thorlacius, and J. Uglum, “The Stretched horizon and black hole complementarity,” Phys.Rev. D48 (1993) 3743–3761, hep-th/9306069.
  * (37) S. A. Hayward, “General laws of black hole dynamics,” Phys. Rev. D49 (1994) 6467–6474.
  * (38) S. Carlip, “The statistical mechanics of the (2+1)-dimensional black hole,” Phys. Rev. D51 (1995) 632–637, gr-qc/9409052.
  * (39) A. P. Balachandran, L. Chandar, and A. Momen, “Edge states in gravity and black hole physics,” Nucl. Phys. B461 (1996) 581–596, gr-qc/9412019.
  * (40) S. Carlip, “Statistical mechanics and black hole entropy,” in Field Theory, Integrable Systems and Symmetries.  1997\.  gr-qc/9509024.
  * (41) A. Strominger, “Black hole entropy from near-horizon microstates,” JHEP 02 (1998) 009, hep-th/9712251.
  * (42) A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, “Quantum geometry and black hole entropy,” Phys. Rev. Lett. 80 (1998) 904–907, gr-qc/9710007.
  * (43) S. Carlip, “Black hole entropy from conformal field theory in any dimension,” Phys. Rev. Lett. 82 (1999) 2828–2831, hep-th/9812013.
  * (44) M. Hotta, K. Sasaki, and T. Sasaki, “Diffeomorphism on horizon as an asymptotic isometry of Schwarzschild black hole,” Class. Quant. Grav. 18 (2001) 1823–1834, gr-qc/0011043.
  * (45) A. Ashtekar et al., “Isolated horizons and their applications,” Phys. Rev. Lett. 85 (2000) 3564–3567, gr-qc/0006006.
  * (46) A. Ashtekar and B. Krishnan, “Dynamical horizons: Energy, angular momentum, fluxes and balance laws,” Phys. Rev. Lett. 89 (2002) 261101, gr-qc/0207080.
  * (47) G. ’t Hooft, “The Black hole horizon as a dynamical system,” Int. J. Mod. Phys. D15 (2006) 1587–1602, gr-qc/0606026.
  * (48) B. R. Majhi and T. Padmanabhan, “Noether current from the surface term of gravitational action, Virasoro algebra and horizon entropy,” Phys. Rev. D86 (2012) 101501, 1204.1422.
  * (49) R. F. Penna, “BMS invariance and the membrane paradigm,” JHEP 03 (2016) 023, 1508.06577.
  * (50) H. Afshar, S. Detournay, D. Grumiller, and B. Oblak, “Near-Horizon Geometry and Warped Conformal Symmetry,” JHEP 03 (2016) 187, 1512.08233.
  * (51) S. W. Hawking, M. J. Perry, and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. 116 (2016), no. 23, 231301, 1601.00921.
  * (52) G. ’t Hooft, “Black hole unitarity and antipodal entanglement,” Found. Phys. 46 (2016), no. 9, 1185–1198, 1601.03447.
  * (53) A. Averin, G. Dvali, C. Gomez, and D. Lust, “Gravitational Black Hole Hair from Event Horizon Supertranslations,” JHEP 06 (2016) 088, 1601.03725.
  * (54) G. Compère and J. Long, “Classical static final state of collapse with supertranslation memory,” Class. Quant. Grav. 33 (2016), no. 19, 195001, 1602.05197.
  * (55) H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo, and R. Troncoso, “Soft Heisenberg hair on black holes in three dimensions,” Phys. Rev. D93 (2016), no. 10, 101503, 1603.04824.
  * (56) C. Eling and Y. Oz, “On the Membrane Paradigm and Spontaneous Breaking of Horizon BMS Symmetries,” JHEP 07 (2016) 065, 1605.00183.
  * (57) H. Afshar, D. Grumiller, and M. M. Sheikh-Jabbari, “Near horizon soft hair as microstates of three dimensional black holes,” Phys. Rev. D96 (2017), no. 8, 084032, 1607.00009.
  * (58) M. Mirbabayi and M. Porrati, “Dressed Hard States and Black Hole Soft Hair,” Phys. Rev. Lett. 117 (2016), no. 21, 211301, 1607.03120.
  * (59) F. Hopfmüller and L. Freidel, “Gravity Degrees of Freedom on a Null Surface,” Phys. Rev. D95 (2017), no. 10, 104006, 1611.03096.
  * (60) S. W. Hawking, M. J. Perry, and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes,” JHEP 05 (2017) 161, 1611.09175.
  * (61) H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo, and R. Troncoso, “Soft hairy horizons in three spacetime dimensions,” Phys. Rev. D95 (2017), no. 10, 106005, 1611.09783.
  * (62) W. Wieland, “New boundary variables for classical and quantum gravity on a null surface,” Class. Quant. Grav. 34 (2017), no. 21, 215008, 1704.07391.
  * (63) R. Bousso and M. Porrati, “Soft Hair as a Soft Wig,” Class. Quant. Grav. 34 (2017), no. 20, 204001, 1706.00436.
  * (64) A. Strominger, “Black Hole Information Revisited,” pp. 109–117.  2020\.  1706.07143.
  * (65) E. T. Akhmedov and M. Godazgar, “Symmetries at the black hole horizon,” Phys. Rev. D96 (2017), no. 10, 104025, 1707.05517.
  * (66) D. Lust, “Supertranslations and Holography near the Horizon of Schwarzschild Black Holes,” Fortsch. Phys. 66 (2018), no. 2, 1800001, 1711.04582.
  * (67) A. Blommaert, T. G. Mertens, H. Verschelde, and V. I. Zakharov, “Edge State Quantization: Vector Fields in Rindler,” JHEP 08 (2018) 196, 1801.09910.
  * (68) G. ’t Hooft, “What happens in a black hole when a particle meets its antipode,” 1804.05744.
  * (69) W. Donnelly and S. B. Giddings, “Gravitational splitting at first order: Quantum information localization in gravity,” Phys. Rev. D98 (2018), no. 8, 086006, 1805.11095.
  * (70) L. Donnay, G. Giribet, H. A. González, and A. Puhm, “Black hole memory effect,” Phys. Rev. D98 (2018), no. 12, 124016, 1809.07266.
  * (71) R. F. Penna, “Near-horizon Carroll symmetry and black hole Love numbers,” 1812.05643.
  * (72) L. Donnay and C. Marteau, “Carrollian Physics at the Black Hole Horizon,” 1903.09654.
  * (73) W. Wieland, “Generating functional for gravitational null initial data,” Class. Quant. Grav. 36 (2019), no. 23, 235007, 1905.06357.
  * (74) D. Grumiller and W. Merbis, “Near horizon dynamics of three dimensional black holes,” SciPost Phys. 8 (2020) 010, 1906.10694.
  * (75) D. Grumiller, M. M. Sheikh-Jabbari, C. Troessaert, and R. Wutte, “Interpolating Between Asymptotic and Near Horizon Symmetries,” 1911.04503.
  * (76) A. Bagchi, R. Basu, A. Mehra, and P. Nandi, “Field Theories on Null Manifolds,” 1912.09388.
  * (77) A. Ashtekar, “Black Hole evaporation: A Perspective from Loop Quantum Gravity,” Universe 6 (2020), no. 2, 21, 2001.08833.
  * (78) K. Hajian and M. M. Sheikh-Jabbari, “Solution Phase Space and Conserved Charges: A General Formulation for Charges Associated with Exact Symmetries,” Phys. Rev. D93 (2016), no. 4, 044074, 1512.05584.
  * (79) M. Guica, T. Hartman, W. Song, and A. Strominger, “The Kerr/CFT Correspondence,” Phys. Rev. D80 (2009) 124008, 0809.4266.
  * (80) G. Compere, “The Kerr/CFT correspondence and its extensions: a comprehensive review,” Living Rev.Rel. 15 (2012) 11, 1203.3561.
  * (81) A. Aggarwal, A. Castro and S. Detournay, “Warped Symmetries of the Kerr Black Hole,” JHEP 01 (2020) 016, 1909.03137.
  * (82) E. E. Flanagan, K. Prabhu, and I. Shehzad, “Extensions of the asymptotic symmetry algebra of general relativity,” JHEP 01 (2020) 002, 1910.04557.
  * (83) H. Afshar, D. Grumiller, M. M. Sheikh-Jabbari, and H. Yavartanoo, “Horizon fluff, semi-classical black hole microstates — Log-corrections to BTZ entropy and black hole/particle correspondence,” JHEP 08 (2017) 087, 1705.06257.
  * (84) D. Grumiller and M. Riegler, “Most general AdS3 boundary conditions,” JHEP 10 (2016) 023, 1608.01308.
  * (85) D. Grumiller, W. Merbis, and M. Riegler, “Most general flat space boundary conditions in three-dimensional Einstein gravity,” Class. Quant. Grav. 34 (2017), no. 18, 184001, 1704.07419.
  * (86) A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 (2014) 152, 1312.2229.
  * (87) S. Pasterski, “Implications of Superrotations,” Phys. Rept. 829 (2019) 1–35, 1905.10052.
  * (88) H. R. Safari and M. M. Sheikh-Jabbari, “BMS4 algebra, its stability and deformations,” JHEP 04 (2019) 068, 1902.03260.
  * (89) H. Bondi, M. van der Burg, and A. Metzner, “Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems,” Proc. Roy. Soc. London A269 (1962) 21–51.
  * (90) R. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128 (1962) 2851–2864.