# New horizon symmetries, hydrodynamics, and quantum chaos

Maria Knysh<EMAIL_ADDRESS>Theoretische Natuurkunde, Vrije Universiteit
Brussel (VUB) and The International Solvay Institutes, Pleinlaan 2, B-1050
Brussels, Belgium  Hong Liu<EMAIL_ADDRESS>Center for Theoretical
Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave.,
Cambridge, MA 02139  Natalia Pinzani-Fokeeva<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia, Universitá di Firenze, Via G. Sansone 1,
I-50019, Sesto Fiorentino, Firenze, Italy  INFN, Sezione di Firenze; Via G.
Sansone 1, I-50019, Sesto Fiorentino, Firenze, Italy

###### Abstract

We generalize the formulation of horizon symmetries presented in previous
literature to include diffeomorphisms that can shift the location of the
horizon. In the context of the AdS/CFT duality, we show that horizon
symmetries can be interpreted on the boundary as emergent low-energy gauge
symmetries. In particular, we identify a new class of horizon symmetries that
extend the so-called shift symmetry, which was previously postulated for
effective field theories of maximally chaotic systems. Additionally, we
comment on the connections of horizon symmetries with bulk calculations of
out-of-time-ordered correlation functions and the phenomenon of pole-skipping.

Keywords: Asymptotic symmetries, horizon symmetries, black holes,
hydrodynamics, chaos

###### Contents

  1. I Introduction
  2. II Formulation of horizon symmetries
    1. II.1 Setup
    2. II.2 Review
    3. II.3 New horizon symmetries
    4. II.4 $\chi^{A}$ along the horizon
    5. II.5 General case
    6. II.6 Simplifications of the horizon symmetry equations
    7. II.7 An explicit example
    8. II.8 Passive perspective
  3. III Hydrodynamical interpretation
    1. III.1 Review
    2. III.2 Horizon symmetries as gauge symmetries of quantum hydrodynamics
    3. III.3 An explicit example
  4. IV Implications for quantum chaos
    1. IV.1 Connection with the shock wave geometry
    2. IV.2 Connections with pole skipping
  5. V Conclusions and discussions
  6. A Freedom in choosing $n^{A}$
    1. A.1 Independence of $n^{A}$
  7. B Details of some derivations
  8. C Different descriptions of diffeomorphism transformations

##  I Introduction

Asymptotic symmetries are local symmetries that do not vanish at infinity and
preserve the boundary conditions. They are believed to represent physical
symmetries of the system. For instance, in the context of the AdS/CFT duality,
asymptotic symmetries in an asymptotic AdS spacetime correspond to global
symmetries of the boundary system.

For a black hole geometry, the focus is often on the physics outside the
horizon. In this case, it is convenient to treat the event horizon as a
“boundary” in an effective sense as it has been done, for example, in the so-
called membrane paradigm [1]. It is natural to extend the discussion of
asymptotic symmetries to an event horizon and consider diffeomorphisms that
preserve the horizon of a black hole geometry [2, 3, 4, 5, 6], as well as
their physical implications. See e.g., [7, 8, 9, 10, 11, 12, 13, 14] for
further discussions on this topic and [15, 16, 17, 18, 19, 20, 21] for
applications to the black hole microstate counting. Since a horizon is not a
genuine physical boundary, such horizon-preserving diffeomorphisms are not
true asymptotic symmetries, and their precise physical interpretations remain
unclear. Below we will follow the standard terminology and refer to them as
“horizon symmetries”.

To probe physical interpretations and implications of horizon symmetries, it
is helpful to consider them in a familiar context. In this paper, we consider
horizon symmetries in an eternal black hole in AdS and seek a precise physical
interpretation in terms of the boundary CFT. We will first generalize the
formulation of horizon symmetries given in [5, 6, 12], then show that they can
be interpreted in the boundary theory as emergent low-energy gauge symmetries,
and finally discuss their connections with hydrodynamics and maximal chaos.

The main results of this paper can be summarized as follows:

  1. 1.

We define horizon symmetries as bulk diffeomorphisms that preserve both the
null vector field and its non-affine parameter on the horizon. These include
those that can move the location of the horizon, i.e. without restricting to
diffeomorphisms that lie along the horizon as in [22].

For a stationary black hole in a suitable coordinate system, the horizon
symmetries are

$\displaystyle\sigma^{0}\rightarrow\sigma^{0}+\lambda(\vec{\sigma})+\alpha(\vec{\sigma})e^{-\kappa_{0}\sigma^{0}}+\tilde{\gamma}(\vec{\sigma})e^{\kappa_{0}\sigma^{0}}\,,$
(1)
$\displaystyle\sigma^{i}\rightarrow\sigma^{i}+\zeta^{i}(\vec{\sigma})+a\,\partial_{i}\tilde{\gamma}(\vec{\sigma})e^{\kappa_{0}\sigma^{0}}\,,$
(2)

where $(\sigma^{0},\vec{\sigma})$ are horizon coordinates, $\lambda$,
$\alpha$, $\tilde{\gamma}$, and $\zeta^{i}$ are generic functions of the
spatial coordinates $\vec{\sigma}$, $a$ is a constant that depends on the
specific black hole metric, and $\kappa_{0}$ is the non-affinity parameter.
The novel symmetry is parameterized by $\tilde{\gamma}(\vec{\sigma})$ and is
proportional to an exponentially growing term $e^{\kappa_{0}\sigma^{0}}$.

  2. 2.

We show that the horizon symmetries (1) and (2) in fact correspond to gauge
symmetries of the dual boundary hydrodynamic and maximally chaotic effective
field theory. In particular, the symmetries parametrized by $\lambda$ and
$\zeta^{i}$ correspond to certain gauge symmetries of any hydrodynamic
effective field theory defined in [23, 24, 25, 26, 27]. The symmetries
parameterized by $\alpha$ and $\tilde{\gamma}$ in (1), dubbed as shift
symmetries, correspond to symmetries of maximally chaotic systems as proposed
by [28, 29].

This claim relies on the fact that both the hydrodynamic [23, 24, 25, 26, 27]
and maximally chaotic effective field theories [28, 29] are based on the
definition of appropriate hydrodynamical variables. Their gravitational
counterpart has been discussed in [30, 31, 32] which identify the dual field
theory fluid variables with the relative embedding between the boundary and
the horizon. Horizon symmetry transformations, like (1) and (2), then lead
directly to symmetry transformations of the hydrodynamical variables. Given
that horizon symmetries are bulk diffeomorphisms, the corresponding symmetry
transformations of the hydrodynamical variables are then identified as gauge
symmetries of the dual boundary hydrodynamic and chaotic effective field
theory.

  3. 3.

Two special features of maximally chaotic systems as the ones arising in the
context of holography [33, 34, 35] and SYK models [36, 37, 38, 39] are the
exponential growth of Out-of-Time Ordered Correlators (OTOCs) and the pole-
skipping phenomenon [40, 28, 41]. We show that the horizon symmetries
parameterized by $\tilde{\gamma}$ can be used to create shock wave geometries,
analogous to the ones analyzed in [33, 34, 35] that lead to the exponential
growth of OTOCs in holographic systems. Moreover, we will comment on the
connection between horizon symmetries and the pole-skipping phenomenon in the
energy density retarded two-point function.

This paper is organized as follows. In Section II we start by illustrating our
setup and defining the generators that preserve the horizon structure. In
particular, we will show how symmetries like (1) and (2) arise as symmetries
of black hole horizons. Then, we construct the holographic dual degrees of
freedom for boundary hydrodynamics in Section III. We show that these objects
inherit the horizon symmetries (1) and (2), establishing the correspondence
and the interpretation of the horizon symmetries as symmetries of the dual
fluid dynamical variables. In Section IV we use the horizon symmetries to
generate shock wave geometries and elaborate on their connection to the
chaotic features of OTOCs as well as the pole-skipping phenomenon. We conclude
with a summary and discussion in Section V. Details of some of the derivations
are deferred to Appendices A-C.

##  II Formulation of horizon symmetries

###  II.1 Setup

Consider a $(d+1)$-dimensional black hole spacetime ${\cal M}$ with a
horizon111We focus on black hole horizons despite much of this discussion can
be applied to any null surface. denoted as ${\cal H}$. The spacetime is
described by a metric $G_{AB}$ with coordinates $x^{A}$. ${{\mathcal{H}}}$ can
be specified by a set of embedding functions $X^{A}(\sigma^{a})$ in
${{\mathcal{M}}}$, where
$\sigma^{a}=(\sigma^{0},\sigma^{i})=(\sigma^{0},\vec{\sigma})$ denote
intrinsic coordinates on ${{\mathcal{H}}}$. We will denote bulk quantities
with capital latin indices $A,B,\dots$ and horizon quantities with
$a,b,\dots$.

The intrinsic metric on ${{\mathcal{H}}}$ can be written as

$\displaystyle ds^{2}_{\cal H}=h_{ab}d\sigma^{a}d\sigma^{b},\qquad
h_{ab}{\;\,\triangleq\;\,}G_{AB}(X)\partial_{a}X^{A}\partial_{b}X^{B}\ ,$ (3)

where $"{\;\,\triangleq\;\,}"$ is an equality evaluated at the horizon. By
definition, $h_{ab}$ is degenerate, and has a single eigenvector
$\hat{\ell}^{a}$ with zero eigenvalue,

$h_{ab}\hat{\ell}^{b}=0\,.$ (4)

Through $X^{A}(\sigma^{a})$, we can push forward $\hat{\ell}^{a}$ to a
spacetime vector field $\ell^{A}$,

$\ell^{A}{\;\,\triangleq\;\,}\hat{\ell}^{a}\partial_{a}X^{A}\,.$ (5)

The vector $\ell^{A}$ will be assumed to have a smooth extension away from
${{\mathcal{H}}}$, and our discussion below will not depend on its extension.
By definition $\ell^{A}$ is null on ${{\mathcal{H}}}$, that is

$G_{AB}\ell^{A}\ell^{B}{\;\,\triangleq\;\,}0\,,$ (6)

and it is normal to ${\cal H}$ as (4) implies that

$\ell_{A}\partial_{a}X^{A}{\;\,\triangleq\;\,}0\,,\quad\ell_{A}=G_{AB}\ell^{B}\,.$
(7)

Moreover, $\ell^{A}$ can be shown to satisfy the geodesics equation

$\ell^{B}\nabla_{B}\ell^{A}{\;\,\triangleq\;\,}\kappa\ell^{A}\,,$ (8)

with some scalar $\kappa$ (non-affine parameter).

There is no intrinsic way to normalize $\hat{\ell}^{a}$, and under a
rescaling,

$\displaystyle\hat{\ell}^{a}\to\hat{\ell}^{\prime
a}=e^{\rho}\hat{\ell}^{a}\,,\quad\kappa\to
e^{\rho}(\kappa+{{\mathcal{L}}}_{\ell}\rho){\;\,\triangleq\;\,}e^{\rho}(\kappa+{{\mathcal{L}}}_{\hat{\ell}}\rho)\,,$
(9)

where $\rho$ is an arbitrary scalar on the horizon, and
${{\mathcal{L}}}_{\ell}$ and ${{\mathcal{L}}}_{\hat{\ell}}$ denote the Lie
derivative along $\ell^{A}$ and $\hat{\ell}^{a}$ respectively. Equations (9)
define an equivalence relation: the pairs $(\hat{\ell}^{a},\kappa)$ related by
such rescalings should be viewed as being equivalent. The equivalence class of
the pair $(\hat{\ell}^{a},\kappa)$ under (9) is referred to as the horizon
structure, see, e.g., [22]. Later, we will often use the freedom (9) to fix

$\kappa=\kappa_{0}={\rm const}\ .$ (10)

It is convenient to introduce another spacetime 1-form $n_{A}$ that is null on
${{\mathcal{H}}}$, satisfying

$\ell^{A}n_{A}{\;\,\triangleq\;\,}-1,\quad
G_{AB}n^{A}n^{B}{\;\,\triangleq\;\,}0\,,\quad n^{A}=G^{AB}n_{B}\ .$ (11)

We can pull back $n_{A}$ to ${{\mathcal{H}}}$ to define a 1-form
$\hat{n}_{a}$,

$\hat{n}_{a}{\;\,\triangleq\;\,}\partial_{a}X^{A}n_{A}\,,\quad\hat{n}_{a}\hat{\ell}^{a}=-1\
.$ (12)

Using (11), we can also define a projector

$\Pi_{A}{{}^{B}}=\delta_{A}^{B}+\ell_{A}n^{B}\,,\quad\Pi_{A}{{}^{B}}\ell_{B}{\;\,\triangleq\;\,}0\,,\quad
n^{A}\Pi_{A}{{}^{B}}{\;\,\triangleq\;\,}0\ .$ (13)

The quantity $\Pi_{A}{{}^{B}}$ is asymmetric; acting with it on a vector from
the left projects to the space orthogonal to $n^{A}$, while acting from the
right projects to the space orthogonal to $\ell_{A}$. We can also define a
symmetric, codimension-two, transverse projector

$q_{A}{{}^{B}}={{\delta}}_{A}^{B}+\ell_{A}n^{B}+n_{A}\ell^{B}\,,\quad
q_{A}{{}^{B}}\ell_{B}{\;\,\triangleq\;\,}q_{A}{{}^{B}}n_{B}{\;\,\triangleq\;\,}0\,,\quad\ell^{A}q_{A}{{}^{B}}{\;\,\triangleq\;\,}n^{A}q_{A}{{}^{B}}{\;\,\triangleq\;\,}0\,,$
(14)

which acts as a projector onto the base space of the horizon ${\cal H}$. That
is, if we assume the horizon to have a topology ${\cal H}={\cal R}\times{\cal
Z}$, where ${\cal R}$ is the null direction, $q_{A}{}^{B}$ projects onto the
space ${\cal Z}$.

Notice that the choice of $n_{A}$ is not unique. It is always possible to
shift its value as long as (11) is satisfied as we discuss in detail in
Appendix A. Our discussion will not depend on a specific choice of $n_{A}$.

The most general horizon metric can be written in the form

$ds^{2}_{\cal
H}=\gamma_{ij}(d\sigma^{i}-a^{i}d\sigma^{0})(d\sigma^{j}-a^{j}d\sigma^{0})\,,$
(15)

where $\gamma_{ij}$ is a non-degenerate spatial metric. We then have

$\displaystyle\hat{\ell}^{a}={\cal
C}(1,a^{i})\quad\text{i.e.}\quad\hat{\ell}^{a}={\cal
C}\left(\left({\partial\over\partial\sigma^{0}}\right)^{a}+a^{i}\left({\partial\over\partial\sigma^{i}}\right)^{a}\right),$
(16) $\displaystyle\ell^{A}{\;\,\triangleq\;\,}{\cal C}\left({\partial
X^{A}\over\partial\sigma^{0}}+a^{i}{\partial
X^{A}\over\partial\sigma^{i}}\right)\,,$ (17)

where ${\cal C}$ is an arbitrary normalization. Suppose we choose the horizon
coordinates $\sigma^{a}$ such that

$\hat{\ell}^{a}=\left({\partial\over\partial\sigma^{0}}\right)^{a},$ (18)

then $a^{i}=0$, and

$ds^{2}_{\cal H}=\gamma_{ij}d\sigma^{i}d\sigma^{j}\,.$ (19)

###  II.2 Review

In this subsection we review the formulation of horizon symmetries given in
[22]. Consider an infinitesimal diffeomorphism on ${{\mathcal{H}}}$ generated
by a vector field $\xi^{a}(\sigma^{b})$ under which

$\hat{\ell}^{a}\to\hat{\ell}^{\prime
a}=\hat{\ell}^{a}+{{\mathcal{L}}}_{\xi}\hat{\ell}^{a}=\hat{\ell}^{a}+[\xi,\hat{\ell}]^{a}\,,\quad\kappa\to\kappa^{\prime}=\kappa+{{\mathcal{L}}}_{\xi}\kappa=\kappa+\xi^{a}\partial_{a}\kappa\
.$ (20)

A general $\xi^{a}$ does not preserve the horizon structure specified by the
equivalence class of the pair $(\hat{\ell}^{a},\kappa)$. The horizon
symmetries of [22] are defined by those $\xi^{a}$ that preserve the horizon
structure. That is, the corresponding
$(\hat{\ell}^{a^{\prime}},\kappa^{\prime})$ lies in the equivalence class of
$(\hat{\ell}^{a},\kappa)$. This leads to the requirements

$\displaystyle{{\mathcal{L}}}_{\xi}\hat{\ell}^{a}=b\hat{\ell}^{a}\,,$ (21)
$\displaystyle{{\mathcal{L}}}_{\xi}{\kappa}=b\kappa+{{\mathcal{L}}}_{\hat{\ell}}b\,,$
(22)

for some infinitesimal scalar function $b$ on ${{\mathcal{H}}}$.

To see the implications of the conditions (21) and (22), it is convenient to
decompose $\xi^{a}$ into a part that is along the null direction
$\hat{\ell}^{a}$ and a part that is orthogonal to it,

$\xi^{a}=f\hat{\ell}^{a}+\hat{Y}^{a}\,,\qquad\hat{Y}^{a}\hat{n}_{a}=0\,,$ (23)

where $f$ and $\hat{Y}^{a}$ are functions of the horizon coordinates. These
functions have been referred to as supertranslations and superrotations
respectively, in analogy with the asymptotic symmetry structure and
terminology of flat spacetime. It can be shown that, with the decomposition
(23), equations (21) and (22) become the following conditions on $f$ and
$\hat{Y}^{a}$,

$\displaystyle{{\mathcal{L}}}_{\hat{\ell}}\hat{Y}^{a}\propto\hat{\ell}^{a}\,,$
(24)
$\displaystyle{{\mathcal{L}}}_{\hat{\ell}}({{\mathcal{L}}}_{\hat{\ell}}+\kappa)f+\hat{Y}^{a}{{\mathcal{L}}}_{\hat{\ell}}({{\mathcal{L}}}_{\hat{\ell}}+\kappa)\hat{n}_{a}+\hat{Y}^{a}\partial_{a}\kappa=0\,,$
(25)

see, e.g., [22, 14]. With (24) written as

${{\mathcal{L}}}_{\hat{\ell}}\hat{Y}^{a}=a\hat{\ell}^{a}\,,\quad
a=-\hat{n}_{a}{{\mathcal{L}}}_{\hat{\ell}}\hat{Y}^{a}\,,$ (26)

equation (25) can be written alternatively as

${{\mathcal{L}}}_{\hat{\ell}}({{\mathcal{L}}}_{\hat{\ell}}+\kappa)f+({{\mathcal{L}}}_{\hat{\ell}}+\kappa)a+\hat{Y}^{a}\partial_{a}\kappa=0\,.$
(27)

While in (25) there is an apparent dependence on $\hat{n}_{a}$ (and thus the
choice of $n_{A}$), it can be explicitly verified that there is, in fact,
none. See Appendix A for details.

To explicitly see the form of the horizon symmetry implied by (26) and (27),
we can choose the normalization of $\hat{\ell}^{a}$ and the intrinsic
coordinates $\sigma^{a}$ such that (10) and (18) hold. Then, with the simple
choice of $\hat{n}_{a}=(-1,0)$, equation (23) has the form

$\xi^{a}\partial_{a}=f\partial_{0}+\hat{Y}^{i}\partial_{i}\ .$ (28)

In this case, (26) becomes

$\partial_{0}\hat{Y}^{i}=0,\quad a=0,$ (29)

and equation (27) reduces to

$\displaystyle\partial_{0}^{2}f+\kappa_{0}\partial_{0}f=0\ .$ (30)

Equations (29)–(30) are solved by

$\displaystyle\hat{Y}^{i}=\zeta^{i}(\vec{\sigma})\,,$ (31) $\displaystyle
f=\lambda(\vec{\sigma})+\alpha(\vec{\sigma})e^{-\kappa_{0}\sigma^{0}}\,,$ (32)

where $\zeta^{i}$, $\lambda$ and $\alpha$ are arbitrary functions of the
spatial horizon coordinates $\vec{\sigma}$.

The explicit form of the horizon symmetries (31) and (32) has also been
derived before in [5, 6], where the authors considered black hole horizons
expressed in Gaussian null coordinates and defined the horizon symmetries to
be those bulk diffeomorphisms that preserve these coordinates. Other proposals
for horizon symmetries include [42, 19, 12].

###  II.3 New horizon symmetries

In this subsection, we present our formulation of horizon symmetries that
leads to additional symmetries beyond those given in (31) and (32). Consider
an infinitesimal variation of the metric $G_{AB}\to
G_{AB}^{\prime}=G_{AB}+f_{AB}$, which induces a variation of the metric on the
horizon hypersurface ${{\mathcal{H}}}$,

${{\delta}}h_{ab}{\;\,\triangleq\;\,}f_{AB}\partial_{a}X^{A}\partial_{b}X^{B}\
.$ (33)

We say that the variation $f_{AB}$ preserves the horizon structure if the
following two conditions are satisfied:

  1. 1.

The vector $\hat{\ell}^{a}$ on ${{\mathcal{H}}}$ has still zero eigenvalue
under the new horizon metric, i.e.,

${{\delta}}h_{ab}\hat{\ell}^{b}{\;\,\triangleq\;\,}f_{AB}\partial_{a}X^{A}\partial_{b}X^{B}\hat{\ell}^{b}{\;\,\triangleq\;\,}f_{AB}\partial_{a}X^{A}\ell^{B}{\;\,\triangleq\;\,}0,$
(34)

which is equivalent to

$\displaystyle f_{AB}\ell^{B}{\;\,\triangleq\;\,}-c\ell_{A},\quad
c{\;\,\triangleq\;\,}f_{AB}\ell^{B}n^{A}\,,$ (35)

for some infinitesimal function $c$ defined on ${{\mathcal{H}}}$.

  2. 2.

We require $\kappa$ to be unchanged under the new metric, i.e., with
${\nabla}_{A}^{\prime}$ denoting the covariant derivative associated with
$G_{AB}^{\prime}$,

$\ell^{B}{\nabla}_{B}^{\prime}\ell^{A}{\;\,\triangleq\;\,}\kappa\ell^{A}\,,$
(36)

which can also be written as

$\ell^{B}{{\delta}}{\nabla}_{B}\ell^{A}{\;\,\triangleq\;\,}0,\quad{\nabla}_{B}^{\prime}={\nabla}_{B}+{{\delta}}{\nabla}_{B}\
.$ (37)

Now suppose the metric variation is generated by a diffeomorphism, i.e.,

$f_{AB}={{\mathcal{L}}}_{\chi}G_{AB}\,,$ (38)

for some infinitesimal vector field $\chi^{A}$. We say that $\chi^{A}$
generates a horizon symmetry if (38) preserves the horizon structure as
defined above. This requirement translates into the following two conditions

$\displaystyle{{\mathcal{L}}}_{\chi}\ell^{A}-G^{AB}{{\mathcal{L}}}_{\chi}\ell_{B}{\;\,\triangleq\;\,}c\,\ell^{A}\,,$
(39)
$\displaystyle{{\frac{1}{2}}}\ell^{B}\ell^{C}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{\chi}G_{BC}+c\kappa{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\ell}c\
.$ (40)

To derive equation (39), we use that

$\ell^{A}{{\mathcal{L}}}_{\chi}G_{AB}={{\mathcal{L}}}_{\chi}\ell_{B}-G_{AB}{{\mathcal{L}}}_{\chi}\ell^{A}\,.$
(41)

Combined with (35), this expression leads to

${{\mathcal{L}}}_{\chi}\ell_{B}-G_{AB}{{\mathcal{L}}}_{\chi}\ell^{A}{\;\,\triangleq\;\,}-c\,\ell_{B}\,,$
(42)

which readily gives (39) with $c$ given by

$c{\;\,\triangleq\;\,}\ell^{A}n^{B}{{\mathcal{L}}}_{\chi}G_{AB}{\;\,\triangleq\;\,}n^{B}{{\mathcal{L}}}_{\chi}\ell_{B}-n_{A}{{\mathcal{L}}}_{\chi}\ell^{A}\
.$ (43)

The derivation of (40) is contained in Appendix B. Using
${{\mathcal{L}}}_{n}{{\mathcal{L}}}_{\chi}-{{\mathcal{L}}}_{\chi}{{\mathcal{L}}}_{n}={{\mathcal{L}}}_{[n,\chi]}$,
we can further rewrite the first term of (40) as

$\displaystyle\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{\chi}G_{AB}$
$\displaystyle=$
$\displaystyle\ell^{A}\ell^{B}{{\mathcal{L}}}_{\chi}{{\mathcal{L}}}_{n}G_{AB}+\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\chi]}G_{AB}$
(44) $\displaystyle=$
$\displaystyle{{\mathcal{L}}}_{\chi}(\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}G_{AB})-2\ell^{B}({{\mathcal{L}}}_{\chi}\ell^{A})({{\mathcal{L}}}_{n}G_{AB})+\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\chi]}G_{AB}\,,$

which will be convenient below.

###  II.4 $\chi^{A}$ along the horizon

To see the implications of (39) and (40), we consider first $\chi^{A}$ along
the horizon, i.e.,

$\chi_{A}\ell^{A}{\;\,\triangleq\;\,}0\,.$ (45)

In this case, the diffeomorphisms at the horizon can be thought of as being
intrinsic to the hypersurface and, as we will show below, our formulation of
the horizon symmetries is equivalent to that of Sec. II.2. In the next
subsection, we will consider the general case with $\chi_{A}\ell^{A}\neq 0$,
allowing us to derive new horizon symmetries.

For $\chi^{A}$ satisfying (45), we can “pull it back” to ${{\mathcal{H}}}$ as
follows

$\xi^{a}{\;\,\triangleq\;\,}\chi^{A}P_{A}{{}^{a}}\,,\quad\chi^{A}{\;\,\triangleq\;\,}\partial_{a}X^{A}\xi^{a}\,,$
(46)

where $P_{A}{{}^{a}}$ is the inverse of $\partial_{a}X^{A}$ in the subspace
orthogonal to $\ell_{A}$ (recall that
$\partial_{a}X^{A}\ell_{A}{\;\,\triangleq\;\,}0$)222We may say that the
subspace spanned by $\partial_{a}X^{A}$ does not contain $n^{A}$ while that
spanned by $P_{A}{{}^{a}}$ does not contain $\ell_{A}$., i.e., it is the
matrix satisfying

$n^{A}P_{A}{{}^{a}}{\;\,\triangleq\;\,}0\,,\quad
P_{A}{{}^{a}}\,\partial_{a}X^{B}{\;\,\triangleq\;\,}\Pi_{A}{{}^{B}}\,,\quad\partial_{a}X^{A}P_{A}{{}^{b}}{\;\,\triangleq\;\,}{{\delta}}_{a}^{b}\
.$ (47)

With (45), (34) can be written as

$\displaystyle
0{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\chi}G_{AB}\,\partial_{a}X^{A}\ell^{B}{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\chi}(h_{ab}\hat{\ell}^{b})-G_{AB}\ell^{B}{{\mathcal{L}}}_{\chi}(\partial_{a}X^{A})-G_{AB}\partial_{a}X^{A}{{\mathcal{L}}}_{\chi}\ell^{B}$
(48)
$\displaystyle{\;\,\triangleq\;\,}0+\ell_{A}\partial_{b}X^{A}\partial_{a}\xi^{b}-h_{ab}{{\mathcal{L}}}_{\xi}\hat{\ell}^{b}{\;\,\triangleq\;\,}-h_{ab}{{\mathcal{L}}}_{\xi}\hat{\ell}^{b}\,,$
(49)

where we have used that for any vector of the form
$V^{A}=v^{a}\partial_{a}X^{A}$,

$\displaystyle{{\mathcal{L}}}_{\chi}V^{A}=\chi^{B}\partial_{B}V^{A}-V^{B}\partial_{B}\chi^{A}{\;\,\triangleq\;\,}\xi^{b}\partial_{b}(\partial_{a}X^{A}v^{a})-v^{b}\partial_{b}(\partial_{a}X^{A}\xi^{a}){\;\,\triangleq\;\,}\partial_{a}X^{A}{{\mathcal{L}}}_{\xi}v^{a},$
(50)
$\displaystyle{{\mathcal{L}}}_{\chi}\partial_{a}X^{A}=\chi^{B}\partial_{B}\partial_{a}X^{A}-\partial_{a}X^{B}\partial_{B}\chi^{A}{\;\,\triangleq\;\,}\xi^{b}\partial_{b}\partial_{a}X^{A}-\partial_{a}(\partial_{b}X^{A}\xi^{b}){\;\,\triangleq\;\,}-\partial_{b}X^{A}\partial_{a}\xi^{b}\
.$ (51)

From (49) we conclude that (34) (and therefore (39)) is equivalent to

${{\mathcal{L}}}_{\xi}\hat{\ell}^{a}=b\hat{\ell}^{a}\,,$ (52)

for some infinitesimal function $b$, which recovers (21).

We now prove the equivalence of (37) (and therefore (40)) with (22).
Multiplying $\partial_{a}X^{A}$ on both sides of (52) gives

$\partial_{a}X^{A}{{\mathcal{L}}}_{\xi}\hat{\ell}^{a}{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\chi}\ell^{A}{\;\,\triangleq\;\,}b\ell^{A}\quad\to\quad
b{\;\,\triangleq\;\,}-n_{A}{{\mathcal{L}}}_{\chi}\ell^{A}\,.$ (53)

Moreover, for $\chi$ given by (46), we have automatically

${{\mathcal{L}}}_{\chi}\ell_{B}{\;\,\triangleq\;\,}\tilde{b}\ell_{B}\quad\to\quad\tilde{b}{\;\,\triangleq\;\,}-n^{A}{{\mathcal{L}}}_{\chi}\ell_{A},$
(54)

which can be seen from

$\partial_{a}X^{B}{{\mathcal{L}}}_{\chi}\ell_{B}={{\mathcal{L}}}_{\chi}(\partial_{a}X^{B}\ell_{B})-\ell_{B}{{\mathcal{L}}}_{\chi}\partial_{a}X^{B}{\;\,\triangleq\;\,}0\
.$ (55)

From equations (43), (53) and (54), we find

$c=b-\tilde{b}\ .$ (56)

Now, consider (40) for $\chi^{A}$ given by (46). Various terms in (44) can be
evaluated as follows,

$\displaystyle{{\mathcal{L}}}_{\chi}(\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}G_{AB})=2{{\mathcal{L}}}_{\chi}(\ell_{A}\ell^{B}\nabla_{B}n^{A}){\;\,\triangleq\;\,}-2{{\mathcal{L}}}_{\chi}(n^{A}\ell^{B}\nabla_{B}\ell_{A}){\;\,\triangleq\;\,}2{{\mathcal{L}}}_{\chi}\kappa,$
(57)
$\displaystyle\ell^{B}({{\mathcal{L}}}_{\chi}\ell^{A})({{\mathcal{L}}}_{n}G_{AB}){\;\,\triangleq\;\,}b\ell^{B}\ell^{A}{{\mathcal{L}}}_{n}G_{AB}{\;\,\triangleq\;\,}2b\kappa,$
(58)
$\displaystyle\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\chi]}G_{AB}=2\ell_{A}\ell^{B}\nabla_{B}{{\mathcal{L}}}_{n}\chi^{A}{\;\,\triangleq\;\,}-2{{\mathcal{L}}}_{\ell}\tilde{b}-2{{\mathcal{L}}}_{n}\chi^{A}\ell^{B}\nabla_{B}\ell_{A}{\;\,\triangleq\;\,}-2{{\mathcal{L}}}_{\ell}\tilde{b}+2\tilde{b}\kappa,$
(59)

where in the second line we have used (53) and in the third line (54). Using
them in (40), we then find that

${{\mathcal{L}}}_{\chi}\kappa-{{\mathcal{L}}}_{\ell}\tilde{b}-b\kappa{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\ell}c\quad\to\quad{{\mathcal{L}}}_{\chi}\kappa{\;\,\triangleq\;\,}b\kappa+{{\mathcal{L}}}_{\ell}b,$
(60)

which is (22) given that $\kappa$ and $b$ are functions on the horizon ${\cal
H}$ and therefore

${{\mathcal{L}}}_{\chi}\kappa{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\xi}\kappa\,,\quad{{\mathcal{L}}}_{\ell}b{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\hat{\ell}}b\,.$
(61)

###  II.5 General case

We now consider a general vector field $\chi^{A}$ which we parameterize as
follows

$\chi^{A}{\;\,\triangleq\;\,}f\ell^{A}+Y^{A}+Zn^{A}=\tilde{\chi}^{A}+Zn^{A}\,,\quad
Y^{A}n_{A}{\;\,\triangleq\;\,}0\,,\quad Y^{A}\ell_{A}{\;\,\triangleq\;\,}0\,,$
(62)

where $f$, $Y^{A}$, and $Z$ are functions of the horizon coordinates,
$Z{\;\,\triangleq\;\,}-\chi^{A}\ell_{A}$, and
$\tilde{\chi}^{A}\ell_{A}{\;\,\triangleq\;\,}0$ lies along ${{\mathcal{H}}}$.
The horizon symmetry conditions on $\chi^{A}$ are (39) and (40). It can be
shown that (see Appendix B for detailed derivations) equation (39) reduces to

$\displaystyle{{\mathcal{L}}}_{\ell}Z-\kappa Z{\;\,\triangleq\;\,}0\,,$ (63)
$\displaystyle{{\mathcal{L}}}_{\ell}Y^{A}-q^{AB}{\nabla}_{B}Z+Z\eta^{A}_{\perp}{\;\,\triangleq\;\,}a\ell^{A}\,,$
(64)

for some function $a$ which can be written as

$a{\;\,\triangleq\;\,}-n_{A}{{\mathcal{L}}}_{\ell}Y^{A}\,,$ (65)

while equation (40) reduces to

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+({{\mathcal{L}}}_{\ell}+\kappa)a+{{\mathcal{L}}}_{Y}\kappa+\eta^{A}_{\perp}{\nabla}_{A}Z+Z{\mathcal{F}}{\;\,\triangleq\;\,}0\,,$
(66)
$\displaystyle{\mathcal{F}}{\;\,\triangleq\;\,}{{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}-\eta^{2}_{\perp}-({{\mathcal{L}}}_{\ell}-\kappa)\lambda\,,$
(67)

with

$\displaystyle\eta_{A}\equiv\ell^{B}({{\mathcal{L}}}_{n}G_{AB})={{\mathcal{L}}}_{n}\ell_{A}-G_{AB}{{\mathcal{L}}}_{n}\ell^{B}\,,\quad\lambda\equiv
n^{A}\eta_{A}{\;\,\triangleq\;\,}\ell_{B}n^{A}{\nabla}_{A}n^{B}\,,\quad\eta_{\perp}^{A}=q^{A}{{}_{B}}\eta^{B}\
.$ (68)

We can also rewrite (66) (by using (65) in (66)) as

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+Y^{A}({{\mathcal{L}}}_{\ell}+\kappa){{\mathcal{L}}}_{\ell}n_{A}+{{\mathcal{L}}}_{Y}\kappa+2q^{AB}{\nabla}_{A}Z\ell^{C}{\nabla}_{C}n_{B}+Z{\tilde{\mathcal{F}}}{\;\,\triangleq\;\,}0\,,$
(69)
$\displaystyle{\tilde{\mathcal{F}}}{\;\,\triangleq\;\,}{{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}+2n_{A}\ell^{C}{\nabla}_{C}\eta^{A}_{\perp}-({{\mathcal{L}}}_{\ell}-\kappa)\lambda\
.$ (70)

Equations (63), (64), and (66) constitute a central result of our work. They
are constraint equations that select a specific class of diffeomorphisms which
we call horizon symmetries. They are parameterized by the supertranslations
$f$, the superrotations $Y^{A}$, and the radial displacement $Z$. The part
dependent on $Z$ leads to the new horizon symmetry and will play a central
role in characterizing the chaotic regime of a black hole as we will see
explicitly later on.

The authors of [22] always worked in a regime where
$\chi_{A}\ell^{A}{\;\,\triangleq\;\,}-Z=0$ which allowed them to derive the
constraint equations (24) and (25) that lead to (31) and (32). Moreover, they
always imposed $c=0$ by requiring $b=\tilde{b}$. This latter condition is
equivalent to a constraint on terms in $\chi^{A}$ away from the horizon,
without affecting the form of $f$ and $Y^{A}$. The authors of [6], while
allowing a general radial displacement $Z$ and deriving similar intermediate
equations to ours in Gaussian null coordinates, always effectively imposed
$Z=0$ by requiring independence of their horizon symmetries from the fields
parameterizing the horizon metric (apart from $\kappa$). After additionally
requiring that $\kappa=const.$, they derived equations (31) and (32). Their
proposal of horizon symmetries also constrained some components of $\chi^{A}$
away from the horizon.

In the following two subsections, we derive simple solutions to the constraint
equations (63), (64), and (66) by using a specific parameterization of the
horizon coordinates and then focusing on an explicit example.

###  II.6 Simplifications of the horizon symmetry equations

We now further simplify (63)–(64) and (66) by choosing a convenient set of
horizon coordinates. Recall that

$\ell_{A}Y^{A}{\;\,\triangleq\;\,}n_{A}Y^{A}{\;\,\triangleq\;\,}0\ .$ (71)

Therefore, introducing

$\hat{Y}^{a}{\;\,\triangleq\;\,}Y^{A}P_{A}{{}^{a}},\quad
Y^{A}{\;\,\triangleq\;\,}\hat{Y}^{a}\partial_{a}X^{A}\,,$ (72)

we have

$\hat{Y}^{a}\hat{n}_{a}=0\ .$ (73)

We should view quantities $f,\hat{Y}^{a},Z$ all as functions defined on
${{\mathcal{H}}}$, i.e., they are functions of the horizon coordinates
$\sigma^{a}$. The quantities introduced in (68) should be evaluated on the
horizon and thus are also considered as functions of $\sigma^{a}$.

We will again choose the normalization of $\hat{\ell}^{a}$ and the intrinsic
coordinates $\sigma^{a}$ such that (10) and (18) hold. We will choose $n^{A}$
such that $\hat{n}_{a}=(-1,0)$, and thus
$\hat{Y}^{a}=\hat{Y}^{i}{{\delta}}_{i}^{a}$ where $\hat{Y}^{i}$ is a spatial
vector on the horizon. Similarly, we have

$\displaystyle
q^{AB}{\;\,\triangleq\;\,}\gamma^{ij}\partial_{i}X^{A}\partial_{j}X^{B},\quad\eta^{A}_{\perp}{\;\,\triangleq\;\,}\hat{\eta}^{a}\partial_{a}X^{A},\quad\hat{\eta}^{a}{\;\,\triangleq\;\,}\hat{\eta}^{i}{{\delta}}_{i}^{a}\
.$ (74)

where $\hat{\eta}^{i}$ is also a spatial vector on the horizon such that
$\hat{\eta}^{a}\hat{n}_{a}=0$.

Equation (63) then becomes,

$\ell^{A}\partial_{A}Z{\;\,\triangleq\;\,}{\partial\over\partial\sigma^{0}}Z=\kappa_{0}Z\quad\to\quad
Z={{\gamma}}(\vec{\sigma})e^{\kappa_{0}\sigma^{0}}\ .$ (75)

where $\gamma$ is a generic function of the spatial horizon coordinates
$\vec{\sigma}$. This exponentially growing solution is the new horizon
symmetry, key to our later discussion in connection to many-body quantum chaos
in Sec. IV.

Equation (64) can be written as

$\displaystyle\partial_{0}(\hat{Y}^{i}\partial_{i}X^{A})-\hat{Y}^{i}\partial_{i}(\partial_{0}X^{A})-\gamma^{ij}\partial_{i}X^{A}\partial_{j}Z+Z\hat{\eta}^{i}\partial_{i}X^{A}=a\partial_{0}X^{A}$
(76)
$\displaystyle\to\quad(\partial_{0}\hat{Y}^{i}-\gamma^{ij}\partial_{j}Z+Z\hat{\eta}^{i})\partial_{i}X^{A}=0,\quad
a=0\,,$ (77)

which implies that

$\partial_{0}\hat{Y}^{i}-\gamma^{ij}\partial_{j}Z+Z\hat{\eta}^{i}=0\ .$ (78)

Finally, equation (66) becomes

$\displaystyle\partial_{0}(\partial_{0}+\kappa_{0})f+\hat{\eta}^{i}\partial_{i}Z+Z{{\mathcal{F}}}=0\,,$
(79)
$\displaystyle{{\mathcal{F}}}{\;\,\triangleq\;\,}{{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}-\gamma_{ij}\hat{\eta}^{i}\hat{\eta}^{j}-(\partial_{0}-\kappa_{0})\lambda\
.$ (80)

It is not possible to solve equations (78) and (79) in full generality as they
still depend on the details of the horizon metric $\gamma^{ij}$, on
$\hat{\eta}^{i}$ and $\lambda$. In the next subsection, we will choose a
specific example that will allow us to obtain an explicit form of the horizon
symmetries.

###  II.7 An explicit example

Consider the boosted black brane solution in AdSd+1. Its metric in Eddington-
Finkelstein coordinates $(r,x^{\mu})$ has the form

$ds^{2}=G_{AB}dx^{A}dx^{B}=-2u_{\mu}dx^{\mu}dr+\chi_{\mu\nu}dx^{\mu}dx^{\nu},$
(81)

where $u_{\mu}$ is a constant vector satisfying $u^{\mu}u_{\mu}=-1$, and the
indices $\mu,\nu,\dots$ are raised by the Minkowski metric $\eta^{\mu\nu}$,
and

$\displaystyle\chi_{\mu\nu}=-F(r)u_{\mu}u_{\nu}+g(r){{\Delta}}_{\mu\nu},\quad{{\Delta}}_{\mu\nu}=\eta_{\mu\nu}+u_{\mu}u_{\nu}\,,$
(82) $\displaystyle F(r)={r^{2}\over R^{2}}\left(1-{r_{0}^{d}\over
r^{d}}\right),\quad g(r)={r^{2}\over R^{2}}\ .$ (83)

Here, a constant $r_{0}$ is the radial location of the horizon, $R$ is the AdS
scale, and $\Delta_{\mu\nu}$ is a projector in the directions that are
orthogonal to $u_{\mu}$, that is $\Delta_{\mu\nu}u^{\mu}=0$.

With the embedding function $X^{A}(x^{\mu})=(r_{0},x^{\mu})$ where the horizon
coordinates $x^{\mu}$ are taken to be the same as the spacetime coordinates
$x^{\mu}=(x^{0},\vec{x})$, the induced metric on the horizon is

$ds_{\cal
H}^{2}=g(r_{0}){{\Delta}}_{\mu\nu}dx^{\mu}dx^{\nu}=g(r_{0})(\eta_{\mu\nu}+u_{\mu}u_{\nu})dx^{\mu}dx^{\nu}\,,$
(84)

which gives

$\hat{\ell}^{\mu}={\cal
C}u^{\mu}\quad\to\quad\ell^{A}{\;\,\triangleq\;\,}{\cal
C}(0,u^{\mu})\,,\quad\ell_{A}={\cal C}(1,Fu_{\mu}){\;\,\triangleq\;\,}{\cal
C}(1,0)\,,$ (85)

with an arbitrary normalization factor ${\cal C}$. We can also readily
calculate

$\displaystyle\ell^{B}{\nabla}_{B}\ell^{A}{\;\,\triangleq\;\,}\ell^{A}\left(u^{\mu}\partial_{\mu}{\cal
C}+{{\cal C}\over
2}\partial_{r}F\right)\quad\to\quad\kappa=u^{\mu}\partial_{\mu}{\cal C}+{\cal
C}\hat{\kappa}\,,$ (86)

where

$\hat{\kappa}={1\over 2}\partial_{r}F\bigg{|}_{r_{0}}={2\pi\over{\beta}},$
(87)

is the non-affine parameter for the Killing vector
$\left({\partial\over\partial x^{0}}\right)^{A}$, and ${\beta}=1/T$ is the
(constant) inverse temperature of the black brane. For the condition (10) to
hold, we choose a constant ${\cal C}$ that satisfies

${\cal C}={\kappa_{0}\over\hat{\kappa}}={\kappa_{0}\over 2\pi}{\beta}\ .$ (88)

The horizon coordinates $\sigma^{a}$ satisfying the conditions (18) are
obtained from $x^{\mu}$ via the Lorentz boost that takes $u^{\mu}$ to
$(1,\vec{0})$,

$\displaystyle\sigma^{0}=-{1\over{\cal
C}}u_{\mu}x^{\mu},\quad\sigma^{i}=x^{i}-u^{i}x^{0}+{u^{i}\over
1+u^{0}}u_{j}x^{j}\,.$ (89)

The inverse transformation is

$\displaystyle x^{0}=-{\cal C}u_{0}\sigma^{0}+u_{i}\sigma^{i},\quad
x^{i}=\sigma^{i}+{\cal C}u^{i}\sigma^{0}+{u^{i}\over
1+u^{0}}u_{j}\sigma^{j}\,,$ (90)

which gives the embedding functions
$X^{A}(\sigma^{a})=(r_{0},x^{\mu}(\sigma^{a}))$. In terms of $\sigma^{a}$, the
horizon metric has the form

$ds_{\cal H}^{2}=g(r_{0})d\vec{\sigma}^{2}\ ,$ (91)

with $\gamma_{ij}=g(r_{0})\delta_{ij}$. We also choose

$\displaystyle n^{A}{\;\,\triangleq\;\,}-{1\over{\cal
C}}\left({\partial\over\partial r}\right)^{A}=-{{\cal C}^{-1}}(1,0),\quad
n_{A}{\;\,\triangleq\;\,}{1\over{\cal C}}\left(0,u_{\mu}\right)\,,$ (92)

which leads to the projector

$\displaystyle
q^{A}{{}_{B}}{\;\,\triangleq\;\,}{{\delta}}^{A}_{B}+{{\delta}}^{A}_{\mu}{{\delta}}_{B}^{\nu}u^{\mu}u_{\nu}-{{\delta}}^{A}_{r}{{\delta}}_{B}^{r}={{\delta}}^{A}_{\mu}{{\delta}}_{B}^{\nu}{{\Delta}}^{\mu}{{}_{\nu}}\,.$
(93)

The infinitesimal diffeomorphism vector $\chi^{A}$ can then be decomposed
using (62) as

$\displaystyle\chi^{A}{\;\,\triangleq\;\,}{\cal
C}fu^{\mu}\left({\partial\over\partial x^{\mu}}\right)^{A}+Y^{A}-{Z\over{\cal
C}}\left({\partial\over\partial r}\right)^{A}\,,$ (94) $\displaystyle
Y^{A}{\;\,\triangleq\;\,}\hat{Y}^{i}\Lambda_{i}^{A},\quad\Lambda_{i}^{A}{\;\,\triangleq\;\,}{{\delta}}^{A}_{\mu}\lambda_{i}{{}^{\mu}}\,,$
(95)

where we have introduced

$\displaystyle\Lambda_{i}^{A}=\frac{\partial
X^{A}}{\partial\sigma^{i}}\,,\quad\lambda_{i}{{}^{\mu}}\equiv{\partial
x^{\mu}\over\partial\sigma^{i}}=\left(u_{i},{{\delta}}_{i}^{j}+{u_{i}u^{j}\over
1+u^{0}}\right),\quad u_{\mu}\lambda_{i}{{}^{\mu}}=0,$ (96)
$\displaystyle\eta_{\mu\nu}\lambda_{i}{{}^{\mu}}\lambda_{j}{{}^{\nu}}={{\delta}}_{ij},\quad\delta^{ij}\lambda_{i\mu}\lambda_{j\nu}={{\Delta}}_{\mu\nu}\
,\quad\lambda_{i\mu}=\eta_{\mu\nu}\lambda_{i}^{\nu}\,.$ (97)

Moreover, we have

$\displaystyle{{\mathcal{L}}}_{n}G_{AB}{\;\,\triangleq\;\,}-{1\over{\cal
C}}\partial_{r}G_{AB},\quad\eta_{A}=\ell^{B}{{\mathcal{L}}}_{n}G_{AB}{\;\,\triangleq\;\,}-u^{\mu}\partial_{r}G_{\mu
A},\quad\lambda{\;\,\triangleq\;\,}\frac{1}{{\cal
C}}G^{Ar}\partial_{r}G_{Ar}{\;\,\triangleq\;\,}\frac{1}{\cal
C}u^{\mu}\partial_{r}G_{\mu r}\,,$ (98)
$\displaystyle\quad\eta^{A}_{\perp}=q^{A}{{}_{C}}G^{BC}\eta_{B}{\;\,\triangleq\;\,}\hat{\eta}^{i}\Lambda_{i}^{A},\quad\hat{\eta}^{i}={{\gamma}}^{ij}\Lambda_{j}^{B}\eta_{B},\quad{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}{\;\,\triangleq\;\,}{1\over{{\cal
C}^{2}}}\partial_{r}^{2}G_{AB}\,,$ (99)

which gives

$\displaystyle\hat{\eta}^{i}=-{1\over
g(r_{0})}\delta^{ij}\lambda_{j}^{\mu}u^{\nu}\partial_{r}G_{\mu\nu}=0,\quad\lambda={2\pi\over(\kappa_{0}{\beta})}u^{\mu}\partial_{r}G_{\mu
r}=0\,,$ (100)
$\displaystyle{{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}{\;\,\triangleq\;\,}{{\frac{1}{2}}}u^{\mu}u^{\nu}\partial_{r}^{2}G_{\mu\nu}=-{{\frac{1}{2}}}F^{\prime\prime}(r_{0})={d(d-3)\over
2R^{2}}\ .$ (101)

Equation (75) can now be written as

$Z={{\gamma}}(\vec{\sigma})e^{\kappa_{0}\sigma^{0}}={{\gamma}}(\vec{\sigma})e^{-{2\pi\over{\beta}}u_{\mu}x^{\mu}}\,,$
(102)

where $\vec{\sigma}$ is given by (89). Equation (78) becomes

$\displaystyle\partial_{0}\hat{Y}^{i}=\frac{1}{g(r_{0})}\partial_{i}Z-Z\hat{\eta}^{i}=\frac{1}{g(r_{0})}\partial_{i}{{\gamma}}(\vec{\sigma})e^{\kappa_{0}\sigma^{0}}\,,$
(103)
$\displaystyle{\Longrightarrow}\quad\hat{Y}^{i}=\zeta^{i}(\vec{\sigma})+{1\over\kappa_{0}g(r_{0})}\partial_{i}{{\gamma}}(\vec{\sigma})e^{-{2\pi\over{\beta}}u_{\mu}x^{\mu}}\,,$
(104)

and equation (79) becomes

$\displaystyle\partial_{0}(\partial_{0}+\kappa_{0})f=-Z{\cal F},\quad{\cal
F}=-{{\frac{1}{2}}}F^{\prime\prime}(r_{0}),$ (105)
$\displaystyle{\Longrightarrow}\quad
f=\lambda(\vec{\sigma})+{\alpha}(\vec{\sigma})e^{{2\pi\over{\beta}}u_{\mu}x^{\mu}}+{1\over
4\kappa_{0}^{2}}F^{\prime\prime}(r_{0}){{\gamma}}(\vec{\sigma})e^{-{2\pi\over{\beta}}u_{\mu}x^{\mu}}\
.$ (106)

A special case is the black brane with $u_{\mu}=(-1,0)$, for which we have

$\displaystyle Z={{\gamma}}({\vec{x}})e^{{2\pi\over{\beta}}x^{0}}\,,$ (107)
$\displaystyle
Y^{i}=\zeta^{i}({\vec{x}})+{1\over\kappa_{0}g(r_{0})}\partial_{i}{{\gamma}}({\vec{x}})e^{{2\pi\over{\beta}}x^{0}}\,,$
(108) $\displaystyle
f=\lambda({\vec{x}})+{\alpha}({\vec{x}})e^{-{2\pi\over{\beta}}x^{0}}+{1\over
4\kappa_{0}^{2}}F^{\prime\prime}(r_{0}){{\gamma}}({\vec{x}})e^{{2\pi\over{\beta}}x^{0}}\
.$ (109)

In the context of two-dimensional gravities, the exponentially growing and
exponentially decaying behavior have also been discussed for the
reparameterization of the time coordinate at the horizon in [8, 43].

###  II.8 Passive perspective

Before concluding this section, we comment on the passive perspective of the
horizon symmetries. Instead of varying the metric with the embedding fixed as
we have done in Section II.3 with $G_{AB}\rightarrow
G_{AB}^{\prime}=G_{AB}+{{\mathcal{L}}}_{\chi}G_{AB}$, we can alternatively fix
the metric and change the embedding of the horizon. In the new coordinates,
the embedding functions become

$X^{\prime A}(\sigma)=X^{A}(\sigma)+\chi^{A}(X(\sigma))\,,$ (110)

where $\chi^{A}$ is the vector parameterizing linearized diffeomorphisms and
should be evaluated at the horizon.

The first condition for the horizon structure to be preserved is the same as
(34), i.e.,

$h^{\prime}_{ab}\hat{\ell}^{a}=0\,,$ (111)

with now

$h^{\prime}_{ab}{\;\,\triangleq\;\,}G_{AB}(X^{\prime})\partial_{a}X^{\prime
A}\partial_{b}X^{\prime B}\ .$ (112)

With $\hat{\ell}^{a}$ unchanged, we have

$\ell^{\prime A}{\;\,\triangleq\;\,}\partial_{a}X^{\prime A}\hat{\ell}^{a}\,,$
(113)

and the second condition (36) becomes

$\ell^{\prime A}{\nabla}_{A}\ell^{\prime
B}{\;\,\triangleq\;\,}\kappa\ell^{\prime B}\ .$ (114)

The two descriptions are effectively equivalent as it is shown generically in
Appendix C.

##  III Hydrodynamical interpretation

We now consider a gravity system in asymptotically AdSd+1 spacetime with a
strongly coupled conformal field theory (CFT) dual. In particular, we will
consider a black hole in AdSd+1, dual to a boundary system at a finite
temperature. For definiteness we take the boundary to be
${\mathbb{{R}}}^{1,d-1}$ with Minkowski metric $\eta_{\mu\nu}$.333The
discussion can be generalized straightforwardly to a curved boundary metric.
In what follows, we will argue that the horizon symmetries can be interpreted
as emergent gauge symmetries of an effective field theory (quantum
hydrodynamics) of the boundary system.

###  III.1 Review

We start by reviewing some necessary ingredients for a theory of (quantum)
hydrodynamics. In the effective field theory approach to hydrodynamics based
on an action principle developed in [24, 27, 44] (see also [23, 25, 26]), the
dynamical variables are maps ${\cal Y}^{\mu}(\sigma^{a})$ between the fluid
spacetime labeled by the coordinates $\sigma^{a}=(\sigma^{0},\sigma^{i})$ and
the physical spacetime with coordinates $y^{\mu}=(t,y^{i})$. The spatial
coordinates $\sigma^{i}$ in the fluid spacetime label fluid elements while
$\sigma^{0}$ can be considered as their “internal time”. For a fixed
$\sigma^{i}$, ${\cal Y}^{\mu}(\sigma^{0},\sigma^{i})$ gives the spacetime
trajectory of a fluid element with label $\sigma^{i}$, parameterized by
$\sigma^{0}$. We denote the inverse map of ${\cal Y}^{\mu}(\sigma^{a})$ as
$\Sigma^{a}(y^{\mu})$, which describes the fluid elements passing through the
point $\vec{y}$ at time $t$. The dynamical variables such as the velocity
field $u^{\mu}$ and temperature $T$ of the more conventional approach can be
expressed in terms of ${\cal Y}^{\mu}(\sigma^{a})$ as

$\beta^{\mu}\equiv\beta u^{\mu}=\frac{\partial{\cal
Y}^{\mu}}{\partial\sigma^{0}},\quad\eta_{\mu\nu}u^{\mu}u^{\nu}=-1\ .$ (115)

See, e.g., [44] for more details and references.

An important element in the formulation of the effective action of
hydrodynamics is that the action should be invariant under the “gauge
symmetries”

$\sigma^{0}\rightarrow\sigma^{\prime\,0}=\sigma^{0}+\lambda(\vec{\sigma})\,,\quad\sigma^{i}\rightarrow\sigma^{\prime\,i}=\sigma^{i}+\zeta^{i}(\vec{\sigma})\,,$
(116)

where $\lambda$ and $\zeta^{i}$ are arbitrary functions of the spatial
variables $\vec{\sigma}$. The symmetries (116) can be interpreted as a
reparameterization freedom of the fluid variables and of their internal time.
They played a crucial role in constraining the form of the hydrodynamic
effective action and the resulting correlation functions.

For a strongly coupled CFT at finite temperature $T=1/{\beta}$, physics at
distances and time scales $L$ much greater than ${\beta}$ is described by
hydrodynamics, usually formulated in a derivative expansion with an effective
expansion parameter ${{\beta}\over L}$. Nevertheless, the recent reformulation
of hydrodynamics indicates that the theory can in fact be extended to scales
$L\sim O({\beta})$, well beyond the usual regime of validity of hydrodynamics.
Such a theory, called quantum hydrodynamics, is nonlocal and yet may have
predictive power.

In fact, in [28, 29], quantum hydrodynamics was used to postulate an effective
field theory for maximally chaotic systems.444The effective field theory of
[28, 29] was constructed for systems with only energy conservation (no
momentum conservation). A full effective field theory for translationally
invariant maximally chaotic systems has not yet been constructed. In addition
to (116), it was assumed that such a theory also possesses a shift symmetry of
the form

$\sigma^{0}\rightarrow\sigma^{\prime\,0}=\sigma^{0}+{\alpha}(\vec{\sigma})e^{-\kappa_{0}\sigma_{0}}+\tilde{\gamma}(\vec{\sigma})e^{\kappa_{0}\sigma_{0}}\,,\quad\sigma^{i}\rightarrow\sigma^{\prime\,i}=\sigma^{i}\,,$
(117)

that leads to a transformation of the velocity field as follows

$\displaystyle\beta^{0}\rightarrow\beta^{\prime\,0}=\beta\left(1-\kappa_{0}{\alpha}(\vec{y})e^{-{\kappa_{0}\over\beta}t}+\kappa_{0}\tilde{\gamma}(\vec{y})e^{{\kappa_{0}\over\beta}t}\right)\,,$
(118) $\displaystyle\beta^{i}\rightarrow\beta^{\prime\,i}=\beta^{i}\,,$ (119)

where ${\alpha}$ and $\tilde{\gamma}$ are generic functions of the boundary
spatial coordinates $\vec{y}$.555In writing this equation, we imposed
$\lambda$ of [28] to be $\lambda=\kappa_{0}$ to match our notation. The
difference is in the choice of the equilibrium configuration. As we will see
shortly, we impose $\Sigma^{0}=\frac{1}{\beta}t$ while [28] have
$\Sigma^{0}=t$.

###  III.2 Horizon symmetries as gauge symmetries of quantum hydrodynamics

Here we show that the symmetries used to formulate hydrodynamics (116), as
well as the shift symmetry (117) postulated for maximally chaotic systems, can
be understood on the gravity side as horizon symmetries in the context of the
AdS/CFT correspondence.

For a holographic system in a local equilibrium state, hydrodynamical
variables $\Sigma^{a}(y^{\mu})$ (or equivalently the inverse maps ${\cal
Y}^{\mu}(\sigma^{a})$) can be obtained from the relative embedding between the
boundary (with coordinates $y^{\mu}$) and the horizon (with coordinates
$\sigma^{a}$) as developed in [30, 31, 32]. More explicitly, given a bulk
metric $G_{AB}$, consider shooting a null geodesic with tangent vector
$T^{A}(y^{\mu})$ from the boundary to the horizon. The geodesic leaves the
boundary at $y^{\mu}$, and reaches the horizon ${{\mathcal{H}}}$ at point
$\sigma^{a}$, establishing the map $\sigma^{a}=\Sigma^{a}(y^{\mu})$ and its
inverse $y^{\mu}={\cal Y}^{\mu}(\sigma^{a})$.

The velocity field and temperature can then be obtained from the definition
(115) in the effective field theory approach. Using the map ${\cal
Y}^{\mu}(\sigma^{a})$, we can push forward the null vector $\hat{\ell}^{a}$
defined on the horizon ${{\mathcal{H}}}$ to a vector ${\beta}^{\mu}$ on the
boundary,

${\partial{\cal Y}^{\mu}\over\partial\sigma^{0}}={\partial{\cal
Y}^{\mu}\over\partial\sigma^{a}}\hat{\ell}^{a}={\beta}^{\mu}\,,$ (120)

where we have used our choice of the horizon coordinates (18). Below, we will
see explicitly that by choosing $\kappa_{0}$ in (10) to be

$\kappa_{0}=2\pi$ (121)

for an equilibrium configuration, ${\beta}$ obtained from (120) coincides with
the boundary inverse temperature. Thus, we can interpret ${\beta}$ and
$u^{\mu}$ defined by (120)–(121) respectively as the local fluid inverse
temperature and velocity for a general non-equilibrium case.

Notice that the given definition of the map ${\cal Y}^{\mu}(\sigma^{a})$ in
gravity is not unique. Using the Fefferman-Graham coordinates near the
boundary, we can parameterize $T^{A}$ in terms of a time-like vector $a^{\mu}$
as follows

$T^{A}(y^{\mu})=(1,a^{\mu}(y^{\nu})),\quad\eta_{\mu\nu}a^{\mu}a^{\nu}=-1\ .$
(122)

The choice of $a^{\mu}$ corresponds to a choice of the local Lorentz frame
characterizing the boundary fluid.

We now consider more explicitly how horizon symmetries act on the
hydrodynamical variables $\Sigma^{a}(y^{\mu})$ or ${\cal
Y}^{\mu}(\sigma^{a})$. For this purpose, it is convenient to write the bulk
metric in terms of the Eddington-Finkelstein coordinates as follows

$ds^{2}=-2a_{\mu}(x)e^{\eta}dx^{\mu}dr+\chi_{\mu\nu}{(r,x)}dx^{\mu}dx^{\nu},$
(123)

in which null geodesics represented by lines of $x^{\mu}={\rm const}$ are
precisely those with the tangent vector at the boundary specified by (122),
see [45]. We can also choose the radial coordinate $r$ so that
${{\mathcal{H}}}$ lies at $r=r_{0}={\rm const}$. The induced metric at the
horizon can then be written as

$ds_{\cal
H}^{2}=\lambda_{\mu\nu}(x)dx^{\mu}dx^{\nu},\quad\lambda_{\mu\nu}=\chi_{\mu\nu}(r=r_{0})\
.$ (124)

With the horizon coordinates $\sigma^{a}$ being specified by (10) and (18),
the horizon embedding functions $X^{A}(\sigma^{a})$ are given by $r=r_{0}$ and
$X^{\mu}(\sigma^{a})=x^{\mu}(\sigma^{a})$.

In (123), the radial geodesic starting from $y^{\mu}$ with tangent vector
(122) is simply given by $x^{\mu}(s)=y^{\mu}$ with $s$ a parameter along the
geodesic. That is, starting from $y^{\mu}$ on the boundary, the horizon is
reached at $(r_{0},y^{\mu})$ and the map ${\cal Y}^{\mu}(\sigma^{a})$ is then
given by

${\cal Y}^{\mu}(\sigma^{a})=X^{\mu}(\sigma^{a})=x^{\mu}(\sigma^{a})\ .$ (125)

Thus the boundary local temperature and velocity field are given by

${\beta}^{\mu}={\partial X^{\mu}\over\partial\sigma^{0}}={\partial
x^{\mu}\over\partial\sigma^{0}}\ .$ (126)

Notice that each gravity solution $G_{AB}$ determines a solution of
$\Sigma^{a}(y^{\mu})$ and ${\cal Y}^{\mu}(\sigma^{a})$, which can be defined
without performing any derivative expansion. Thus the effective field theory
of $\Sigma^{a}(y^{\mu})$ may be considered as “quantum hydrodynamics” which is
valid for spacetime variations of order ${\beta}$ and goes beyond the usual
effective theory arising in the context of the fluid/gravity correspondence.

Now consider making a horizon symmetry transformation generated by some
infinitesimal vector field $\chi^{A}$ parameterized by (62). As discussed in
Sec. II.8, we can equivalently describe the transformation in terms of a
linearized shift in the embedding functions

$X^{A}\to X^{\prime A}(\sigma^{a})=X^{A}(\sigma^{a})+\chi^{A}(\sigma^{a})$
(127)

with $G_{AB}$ unchanged. More explicitly, we have

$r(\sigma^{a})=r_{0}\to r^{\prime}(\sigma)=r_{0}+\chi^{r}(\sigma^{a}),\quad
x^{\mu}(\sigma^{a})\to
x^{\prime\mu}(\sigma^{a})=x^{\mu}(\sigma^{a})+\chi^{\mu}(\sigma^{a})\ .$ (128)

Since the metric does not change, the geodesic starting at $y^{\mu}$ now hits
the horizon at $(r_{0}^{\prime}(y^{\mu}),y^{\mu})$ where

$r^{\prime}_{0}(y^{\mu})=r_{0}+\chi^{r}(\sigma^{a}(y^{\mu}))\,,$ (129)

and $\sigma^{a}(y^{\mu})$ is the inverse of $x^{\mu}(\sigma^{a})$ evaluated at
$x^{\mu}=y^{\mu}$. The new boundary to horizon map is given by

${\cal Y}^{\prime\,\mu}(\sigma^{a})=x^{\prime\,\mu}(\sigma^{a})={\cal
Y}^{\mu}(\sigma^{a})+\chi^{\mu}(\sigma^{a})\,.$ (130)

The new velocity field is then

${\beta}^{\,\mu}=\partial_{a}{\cal
Y}^{\prime\,\mu}\hat{\ell}^{a}={\beta}^{\mu}+\partial_{0}\chi^{\mu}\ .$ (131)

Since horizon symmetry transformations are bulk diffeomorphisms, they take a
gravity solution to another. Moreover, the two solutions should be physically
equivalent. This implies that the corresponding transformations (130)–(131)
should again be viewed as “gauge symmetries” of the dual effective field
theory of quantum hydrodynamics. In particular, such transformations could in
principle change the local boundary inverse temperature and velocity, and lead
to nontrivial constraints on the hydrodynamic equations. Below we will look at
an explicit example.

###  III.3 An explicit example

As an illustration of the above abstract discussion, we consider the boundary
system on ${\mathbb{{R}}}^{1,d-1}$ in a thermal equilibrium at inverse
temperature ${\beta}$, which is described in the bulk by the solution
(81)–(83) with $u_{\mu}=(-1,\vec{0})$, i.e. with $x^{0}=v$,

$\displaystyle ds^{2}=G_{AB}dx^{A}dx^{B}=2dvdr-F(r)dv^{2}+g(r)d{\vec{x}}^{2},$
(132) $\displaystyle F(r)={r^{2}\over R^{2}}\left(1-{r_{0}^{d}\over
r^{d}}\right),\quad g(r)={r^{2}\over R^{2}}\ .$ (133)

The horizon ${{\mathcal{H}}}$ is located at the constant value $r=r_{0}$ and
the induced metric on ${{\mathcal{H}}}$ is given by

$ds^{2}_{\cal H}=g(r_{0})d{\vec{x}}^{2}\ .$ (134)

The boundary inverse temperature is given by

${\beta}={4\pi\over F^{\prime}(r_{0})}={2\pi\over\hat{\kappa}}\ .$ (135)

Using the already derived relations (89), the horizon coordinates $\sigma^{a}$
satisfying the conditions (10), (18) (88), and (121) are

$\sigma^{0}={1\over{\beta}}v,\quad\sigma^{i}=x^{i}\ .$ (136)

Given the metric (132), we choose $a^{\mu}=(1,\vec{0})$ for the radial null
geodesics defined in (122). In this way, a point $y^{\mu}$ on the boundary is
simply mapped to $x^{\mu}(s)=y^{\mu}$. We then have

$\Sigma^{0}(y^{\mu})={1\over{\beta}}t,\quad\Sigma^{i}=y^{i},\quad{\cal
Y}^{0}={\beta}\sigma^{0},\quad{\cal Y}^{i}=\sigma^{i},$ (137)

and thus the velocity field is given by

${\beta}^{\mu}={\beta}(1,\vec{0})$ (138)

with the inverse temperature $\beta$ given by (135) as anticipated around
(121).

The set of horizon symmetry transformations parameterized by the vector field
$\chi^{A}$ was given in (94) and (107)–(109), which we copy here for
convenience

$\displaystyle\chi^{A}=f{\beta}\left({\partial\over\partial
v}\right)^{A}+\hat{Y}^{i}\left({\partial\over\partial
x^{i}}\right)^{A}-{Z\over{\beta}}\left({\partial\over\partial r}\right)^{A},$
(139) $\displaystyle Z={{\gamma}}(\vec{\sigma})e^{2\pi\sigma^{0}}\,,$ (140)
$\displaystyle\hat{Y}^{i}=\zeta^{i}(\vec{\sigma})+{1\over 2\pi
g(r_{0})}\partial_{i}{{\gamma}}(\vec{\sigma})e^{2\pi\sigma^{0}},$ (141)
$\displaystyle
f=\lambda(\vec{\sigma})+{\alpha}(\vec{\sigma})e^{-2\pi\sigma^{0}}+{1\over
16\pi^{2}}F^{\prime\prime}(r_{0}){{\gamma}}(\vec{\sigma})e^{2\pi\sigma^{0}}\
.$ (142)

From (131), we then find the transformation of the boundary inverse
temperature and velocity field are given by

$\displaystyle{\beta}^{{}^{\prime}\,0}(y^{\mu})={\beta}\left(1-2\pi{\alpha}(\vec{y})e^{-{2\pi\over{\beta}}t}+{1\over
8\pi}F^{\prime\prime}(r_{0}){{\gamma}}(\vec{y})e^{{2\pi\over{\beta}}t}\right),$
(143) $\displaystyle{\beta}^{{}^{\prime}\,i}(y^{\mu})={1\over
g(r_{0})}\partial_{i}{{\gamma}}(\vec{y})e^{{2\pi\over{\beta}}t}\,.$ (144)

We note that $\zeta^{i}(\vec{\sigma})$ and $\lambda(\vec{\sigma})$ do not lead
to any change in ${\beta}^{\mu}$. They lead to a diffeomorphism on the horizon
of the form

$\sigma^{0}\rightarrow\sigma^{\prime\,0}=\sigma^{0}+\lambda(\vec{\sigma})\,,\qquad\sigma^{\prime\,i}=\sigma^{i}\rightarrow\sigma^{i}+\zeta^{i}(\vec{\sigma})\,,$
(145)

which matches the field theory symmetry of hydrodynamics (116). The velocity
field transformation (143) matches the previous postulate (118) with our
choice of $\kappa_{0}$ given in (121) and
$\tilde{\gamma}(\vec{\sigma})={1\over
16\pi^{2}}F^{\prime\prime}(r_{0}){{\gamma}}(\vec{\sigma})$. Equation (144) is
instead a new prediction with respect to (119).

##  IV Implications for quantum chaos

In Section III, we discussed how horizon symmetries formulated in Section II
lead to low-energy gauge symmetries in the effective field theory (quantum
hydrodynamics) of the boundary theory in the context of the AdS/CFT
correspondence. The symmetry transformations associated with the parameters
$\lambda(\vec{\sigma})$ and $\zeta^{i}(\vec{\sigma})$ are related to
symmetries of the dual hydrodynamic theory [27]. The symmetry transformations
associated with parameters ${\alpha}(\vec{\sigma})$ and
${{\gamma}}(\vec{\sigma})$ are related to the shift symmetries (117)
postulated in [28, 29] for maximally chaotic quantum many-body systems as we
have seen explicitly towards the end of the previous section. We now comment
in more detail on the implications of these symmetries on quantum chaos by
considering two probes, the OTOCs and the phenomenon of pole-skipping.

In the effective field theory formulation of [28, 29], the shift symmetry
associated with ${{\gamma}}(\vec{\sigma})$ (i.e. the exponentially growing
part) implies that the linearized hydrodynamic field $\epsilon$, defined as
${\cal Y}^{0}\sim\beta\sigma^{0}+\epsilon$, has an exponentially growing gauge
symmetry transformation which in turn implies an exponentially growing
propagator $G_{\epsilon\epsilon}\propto e^{{2\pi\over{\beta}}t}$. This feature
leads to the exponential growth of finite temperature OTOCs,
${\left\langle{A(t)B(0)A(t)B(0)}\right\rangle}_{\beta}$, for two general
scalar operators $A,B$. Furthermore, invariance of the system under the shift
symmetry associated with ${\alpha}(\vec{\sigma})$ (i.e. the exponentially
decreasing part) is crucial for the absence of the exponentially growing
behavior in the time-ordered correlators (TOCs),
${\left\langle{B(0)A(t)A(t)B(0)}\right\rangle}_{\beta}$. Given our
identification of horizon symmetries with the symmetries of the dual quantum
hydrodynamic theory, we thus see that the exponential growth of OTOCs (and the
absence of the exponential growth of TOCs) for theories with a holographic
dual has its gravitational origin in horizon symmetries.666Horizon symmetries
in connection with quantum chaos have also been discussed in [43, 13].

In [28, 29], the stress-energy tensor is seen as a composite operator of the
hydrodynamic fields. Therefore, the presence of the exponentially growing
behavior in the two-point function of $\epsilon$ may cause concerns as it may
lead to the exponential growth in correlation functions of the stress tensor
too. This would imply instabilities that are generically not allowed.
Nevertheless, in [28] it was further argued that the shift symmetries ensure
that such an exponentially growing behavior does not appear in the correlation
functions of the stress tensor through the phenomenon called pole-skipping
[40, 28, 41]. We thus conclude that the pole-skipping phenomenon in
holographic theories is also intimately related to horizon symmetries.777As we
will elaborate more explicitly below, only pole-skipping in a particular
channel of the stress tensor, the energy conservation channel, should be due
to the horizon symmetries.

That the horizon plays a crucial role in both the exponential growth of OTOCs
and the pole-skipping phenomenon has of course been well understood. In the
original calculation of OTOCs from holography [33], the energy of small
perturbations to a thermofield double state long in the past is exponentially
blueshifted from the horizon, which leads to the exponential growth
$e^{{2\pi\over{\beta}}t}$ of the OTOCs. The pole-skipping phenomenon was
discovered on the gravity side from an analysis of the Einstein equations near
the horizon in [40] and elucidated in [41].

In this section, we elaborate a bit further on these gravity calculations from
the perspective of horizon symmetries.

###  IV.1 Connection with the shock wave geometry

For a holographic system, an OTOC can be obtained from the dual bulk gravity
system by calculating two-point functions of $B$ in a shock wave geometry
generated by the operator $A$ inserted in the infinite past of a black hole
geometry, see, e.g., [33, 46, 35]. We may view the shock wave as an effective
description of multiple graviton exchanges between $A$ and $B$, which can in
turn be interpreted in the boundary system in terms of exchanges of the
hydrodynamic modes corresponding to the stress tensor.

Figure 1: A geometry with a shock wave inserted at $V=0$ that leads to a
discontinuity of the $U=0$ surface.

In this subsection, we show that the shock wave geometry of [47, 33, 46, 35]
(see, e.g., Fig. 1) can be interpreted as being generated by a horizon
symmetry transformation. More explicitly, we will show that near the horizon,
the shock wave generated by inserting $A$ at early times may be viewed as
patching together two spacetimes with metrics $G$ and $G^{\prime}$ along the
trajectory of the quantum created by $A$. Here $G$ is the black brane
spacetime describing the equilibrium state, while $G^{\prime}$ is related to
$G$ by a horizon symmetry transformation. In other words, we may say that an
$A$ insertion generates a horizon symmetry transformation. We will now
describe this connection in more detail using the example of (132).

For this purpose it is convenient to use the Kruzkal-Skezeres (KS) coordinates
in terms of which the black brane metric (132) has the form

$ds^{2}=Q(r)dUdV+g(r)d\vec{x}^{2}\,,$ (146)

where the coordinates $U$ and $V$ are given as follows

$U=-e^{-\hat{\kappa}\left(v-2r_{*}(r)\right)}\,,\quad
V=e^{\hat{\kappa}v}\,,\quad
UV=-e^{2\hat{\kappa}r_{*}(r)}\,\quad\partial_{r}r_{*}(r)=F(r)^{-1}\,,$ (147)

and

$Q(r)=\frac{F(r)}{\hat{\kappa}^{2}UV}.$ (148)

In KS coordinates, the black hole’s future horizon $\cal{H}^{+}$ is located at
$U=0$, while the past horizon $\cal{H}^{-}$ is at $V=0$.

The shock wave geometry $G_{\rm shock}$ generated by a high energy particle
with trajectory $V=0$ similar to that used in [33, 46, 35, 47], is given by

$\displaystyle G_{\rm shock}=G+\Theta(V){\cal L}_{\xi}G$ (149)

where $\Theta(V)$ is the Heaviside stepfunction. It is described by an
unperturbed black brane with a metric given in equation (146) for $V<0$, while
for $V\geq 0$ the coordinates are shifted by $x^{A}\rightarrow x^{A}+\xi^{A}$
where $\xi^{A}$ is given by

$\displaystyle\xi^{V}=0\,,\quad\xi^{i}=0\,,\quad\xi^{U}=q\left(\vec{x}\right)\,.$
(150)

The function $q\left({\vec{x}}\right)$ is determined by the Einstein’s
equations

$\left(\nabla^{2}-d\pi
Tg^{\prime}(r_{0})\right)q(\vec{x}^{\prime})=a_{0}Ee^{\hat{\kappa}t}\delta(\vec{x}^{\prime})\,.$
(151)

where $a_{0}$ is some constant, $E$ is the energy of the particle, and the
exponential factor $e^{\hat{\kappa}t}$ comes from a boost near the horizon.

Consider now the effect of the horizon symmetry transformation (140)-(142)
parameterized by $\gamma$. Near the future horizon, this linearized888 Notice
that we work at the linearized level in $\gamma(\vec{x})V$. With this
approximation, one can probe boundary times for which the backreaction due to
$A$ is substantial such that we can see the exponential growth similar to
[33]. This aligns with expectations from [28, 29] for maximally chaotic
quantum many-body systems. transformation can be expressed in terms of the KS
coordinates as

$\displaystyle\chi^{V}={1\over
8\pi}F^{\prime\prime}(r_{0}){{\gamma}}(\vec{x})V^{2}+{\cal O}(U)\,,$ (152)
$\displaystyle\chi^{i}={1\over 2\pi
g(r_{0})}\partial_{i}{{\gamma}}(\vec{x})V+\mathcal{O}\left(U\right)\,,$ (153)
$\displaystyle\chi^{U}=-\frac{\gamma(\vec{x})}{dr_{0}}+\mathcal{O}\left(U\right)\,.$
(154)

Notice that at $V=0$, the transformation in (152)-(154) reduce to that in
(150) upon the identification $q(\vec{x})=-\frac{\gamma({\vec{x}})}{dr_{0}}$
near the future horizon.

One can use these transformations to construct a geometry with metric $G_{\rm
shock}^{\prime}$,

$\displaystyle G^{\prime}_{\rm shock}=G+\Theta(V){{\mathcal{L}}}_{\chi}G\,,$
(155)

described by the unperturbed black hole with metric (146) for $V<0$, and by
the metric (146) with the coordinate transformation in (152)-(154) for $V\geq
0$. Crucially, the geometry described by $G_{\rm shock}$ and the one described
by $G^{\prime}_{\rm shock}$ are related by a coordinate transformation near
the future horizon as follows

$\displaystyle G_{\rm shock}=G^{\prime}_{\rm
shock}+{{\mathcal{L}}}_{\rho}G^{\prime}_{\rm shock}\,,$ (156)

with

$\rho=-\Theta(V)\left(\chi-\xi\right)$ (157)

and $q(\vec{x})=-\frac{\gamma({\vec{x}})}{dr_{0}}$ upon linearization. In
deriving (157) we have used the fact that the delta functions arising from
derivatives on $\Theta(V)$ do not contribute as they are always multiplied by
factors of $V$ near the future horizon.

Hence, near the future horizon, the two seemingly different shock wave metrics
$G_{\rm shock}$ and $G_{\rm shock}^{\prime}$ are related by a diffeomorphism
and are thus physically equivalent. Notice that to extend this argument beyond
the near-horizon limit, one needs to know how the horizon symmetry
transformation (140)-(142) acts away from the horizon. An interesting set of
transformations are those for which $\rho^{A}\big{|}_{V=0}=0$. In this case,
$G_{\rm shock}$ and $G^{\prime}_{\rm shock}$ are related by the coordinate
transformation (157) even away from the future horizon. Consequently, gluing
of $G$ to $G+\Theta(V){{\mathcal{L}}}_{\chi}G$ using the horizon symmetry
$\gamma$ serves as a solution-generating technique for the shock wave geometry
$G_{\rm shock}^{\prime}$ created by a highly energetic particle released from
the right boundary long in the past.

The procedure outlined above can also be used to construct the shock wave
geometry created by a particle released from the left boundary that follows a
null trajectory close to the future horizon located at $U=0$. This geometry
corresponds to gluing two halves of the unperturbed black hole along the $U=0$
slice with a shift in the $V$ direction $V\to V+q\left(\vec{x}\right)$. To
generate this kind of geometry using horizon symmetries, it is necessary to
consider the symmetries of the past horizon using outgoing EF coordinates and
perform the gluing similar to the construction above.

Finally, we want to comment on the special case of $d=3$. In this case, the
coordinates $v$ and $V$ will not transform due to the vanishing of
$F^{\prime\prime}(r_{0})$ as can be seen from equation (106). Nevertheless,
the coordinate $U$ will transform, and thus a shockwave geometry can be
constructed. Therefore, we do expect that also for $d=3$ the OTOCs will grow
exponentially due to the symmetry transformation related to $\gamma$.

###  IV.2 Connections with pole skipping

We now elaborate on the connection between the horizon symmetries and the
phenomenon of pole skipping.

First, we review the argument of pole-skipping contained in [28]. There it was
shown that the exponentially growing part of the symmetry in (117)
parameterized by $\gamma(\vec{\sigma})$ implies that the momentum space
retarded two-point function $G_{\epsilon\epsilon}(\omega,{\vec{k}})$ of the
linearized hydrodynamic mode $\epsilon$ has a pole in the upper half complex
$\omega$-plane at

$\omega=i{2\pi\over{\beta}}\,,\quad{\vec{k}}^{2}=-k_{c}^{2},\quad
k_{c}={2\pi\over{\beta}}{1\over v_{B}}\,,$ (158)

which gives rise to the coordinate space behavior

$G_{\epsilon\epsilon}(t,\vec{x})\sim
e^{\frac{2\pi}{\beta}\left(t-\frac{|\vec{x}|}{v_{B}}\right)}\,,$ (159)

where $\frac{2\pi}{\beta}$ is the maximal Lyapunov exponent and $v_{B}$ is the
butterfly velocity.

The stress tensor $T^{\mu\nu}$ in the hydrodynamic effective field theory
approach is expressed as a composite operator in terms of the hydrodynamic
fields $\epsilon^{\mu}$ where ${\cal Y}^{\mu}\sim x^{\mu}+\epsilon^{\mu}$.
Therefore its correlation functions are determined by those of
$\epsilon^{\mu}$. However, the behavior (159), and thus the pole (158), cannot
appear in the two-point functions of $T^{\mu\nu}$, and thus must be canceled
through how $T^{\mu\nu}$ depends on $\epsilon^{\mu}$. Since $T^{\mu\nu}$ is a
physical observable, its dependence on $\epsilon^{\mu}$ must be invariant
under the shift symmetries parameterized by $\alpha$ and $\gamma$ contained in
(142), which are gauge symmetries. Working in the regime of no momentum
conservation, it has been argued in [28] that the gauge invariance of
$T^{\mu\nu}$ in fact warrants that the pole (158) is always canceled in the
retarded correlator of the energy density operator $T^{00}$ by the appearance
of a zero at the same location. This is the phenomenon of pole-skipping.

Notice that while the pole in $\omega$ given in (158) is universal, the values
of $k_{c}$ and the butterfly velocity $v_{B}$ depend on the specific theory
and cannot be determined from symmetries. On the gravity side, their values
depend on the specific background via the bulk equations of motion, see, e.g.,
[40, 41]. This can also be seen from the discussion on the shock wave
geometries of the previous subsection. While the horizon symmetries allow any
choice of ${{\gamma}}(\vec{\sigma})$, the explicit profile
$\gamma(\vec{\sigma})$ is determined through the shock wave equation (151)
which in turn comes from the specific form of the Einstein equations. In fact,
the solution to the shock wave equation determines the spatial dependence in
(159), and thus the location of the skipped pole in (158). Indeed it was shown
explicitly in [41] from an analysis of the Einstein equations near the horizon
that precisely the shock wave equation (151) determines the location of
$k_{c}$ in the pole-skipping phenomenon. Thus horizon symmetries lead to the
phenomenon of pole-skipping and specify the frequency in (158), but more
information is needed beyond horizon symmetries to determine $k_{c}$ and thus
the full location of the pole.

We emphasize that in [28] the authors worked within the energy density
conservation channel were the momentum dependence was explicitly turned off,
that is $\epsilon^{i}=0$. Thus, the Green’s function for $T^{00}$ was solely
given in terms of its dependence on $\epsilon$ and there was no symmetry
associated to $\sigma^{i}$ coordinates in (117) and (119). Our prediction
arising from horizon symmetries given in (144) could play a role in the
construction of a complete effective field theory for maximally chaotic
systems. In particular, including momentum dependence would lead to a
contribution of the propagator $G_{\epsilon^{i}\epsilon^{i}}$ even to $T^{00}$
and thus needs to be considered. The constraints on
$G_{\epsilon^{i}\epsilon^{i}}$ arising from (144) might be crucial to
understand the pole-skipping phenomenon in a full effective field theory for
maximally chaotic systems.

This expectation seems even more true in light of the pole-skipping phenomenon
for holographic systems obtained directly from gravity calculations in [40,
41]. There, it was shown from a near horizon analysis of the Einstein’s
equations of motion, that the pole-skipping phenomenon arises for every
dimensionality of the bulk spacetime. Nevertheless, from our horizon symmetry
analysis and from the arguments outlined in [28], it would seem that in $d=3$
dimensions there would be no pole-skipping phenomenon due to the absence of an
exponentially growing symmetry transformation of $\epsilon$ as can be seen in
(142) with $F$ given by (106). Thus, the prediction given in (144) could be
pivotal in proving the pole-skipping phenomenon for holographic systems from
an effective field theory approach. We reserve an explicit analysis of this
point for future work.

Since the initial discussions of [40, 41] of the pole-skipping phenomenon in
the sound channel (or energy diffusion channel in the absence of momentum
conservation) associated to chaos, an infinite tower of pole-skipping points
has been found in the lower half of the complex $\omega$ plane for all
channels, and for many other types of fields, see, e.g., [48]. From the
perspective of horizon symmetries considered in this paper, only the pole-
skipping phenomenon associated with the energy density operator for (158) is
implied by the horizon symmetries. The pole-skipping phenomenon in the other
channels/fields, while clearly also having to do with the horizon dynamics,
appears to be unrelated to the horizon symmetries identified in this work
which should only affect hydrodynamic modes. Thus we would conclude the pole-
skipping phenomenon in other fields and other gravitational channels likely
arises from different physics.

##  V Conclusions and discussions

In this work, we generalized the discussion of [22, 5, 6] of horizon
symmetries to a larger class and gave a boundary interpretation of the
symmetries as the gauge symmetries of the dual quantum hydrodynamic theory in
the context of the AdS/CFT correspondence. We worked out the symmetries in
some simple examples and showed the existence of a new exponentially growing
symmetry that has close connections with the behavior of OTOCs and the pole-
skipping phenomenon.

In particular, we provided a new prediction for the gauge symmetries of any
maximally chaotic effective field theory that has a gravitational dual. We
showed that horizon symmetry transformations can be used as a solution
generating technique to build up shock wave geometries as those arising in the
holographic computations of OTOCs [33, 35]. Finally, we showed that the pole
in $\omega$ in the pole-skipping phenomenon is universal and implied by the
horizon symmetries, while the argument for the existence of a pole in
$\vec{k}$ is model-dependent and necessitates additional bulk information.

We now point out some interesting future directions. Among the most
straightforward generalizations to consider are more general black holes, such
as charged black holes and those corresponding to far-from-equilibrium states.
It would also be useful to understand what happens to the horizon symmetries
when including higher derivative or stringy corrections. The latter correspond
to instances of non-maximal chaos which is subject of current ongoing
research.

Another avenue is to consider spacetimes that are not asymptotically AdS since
much of our discussion can be generalized to the case of other signs of the
cosmological constant. In fact, the hydrodynamic degrees of freedom can also
be defined as maps between the horizon and some timelike boundary that is
embedded in the interior of the bulk spacetime, not necessarely at asymptotic
infinity. In this way, it is possible to analyse the hydrodynamic and chaotic
regime (quantum hydrodynamic) of a putative dual theory that lives on that
timelike boundary without knowing the precise details of that theory and the
bulk asymptotics. See, e.g., [49, 50, 51, 52, 53, 54] for the fluid/gravity
correspondence of these so called Rindler fluids.

To have additional support to the findings of this work, it would be useful to
extend the effective field theory for maximally chaotic systems formulated in
[28, 29] to also include momentum conservation and derive the implications of
the transformation of the spatial coordinates due to $\gamma(\vec{\sigma})$ in
(140) on the OTOCs and the phenomenon of pole-skipping. In particular, this
could possibly help the issue arising in $d=3$ we highlighted at the end of
Section IV.2. A more challenging task is to have an explicit derivation of the
chaos effective action formulated in [28, 29] from holography using the
methods developed in [31, 32], perhaps also generalizing the findings of [8].

In this paper, we have identified the gravitational dual to the gauge shift
symmetries and the hydrodynamic symmetries. It would also be desirable to
identify the gravitational dual and its relation to horizon symmetries of
another symmetry appearing in the context of effective field theories of
hydrodynamics: the one responsible for the entropy production [55]. See, e.g.,
[14, 56] for results in this direction. At the same time, one might wonder
whether it is possible to identify gravitational and effective field theory
symmetries related to the additional, infinite set of skipped poles found in
[48] in various channels and for various fields.

It has been argued that horizon symmetries lead to local charges upon using
the Wald-Zupas [57] or the Barnich and Brandt formalism [58] (see, e.g., [5,
6, 22]). It would be interesting to explore whether our horizon symmetries
lead to local charges and what are their implications. In particular, local
charges have to satisfy an integrability condition, that is they should be
independent of the path taken in the so-called phase space, see, e.g., [59],
for details. This problem has been circumvented in [18] by introducing edge
modes: extra dynamical fields that ensure the resulting charges always respect
the integrability condition. It would be interesting to explore possible
relations between the gravitational edge modes of [18] and the fluid dynamic
degrees of freedom described in these notes.

## Acknowledgments

We would like to thank Ben Craps, Juan Hernandez, and Mikhail Khramtsov for
useful discussions. NPF is supported by the European Commission through the
Marie Sklodowska-Curie Action UniCHydro (grant agreement ID: 886540). NPF
would also like to acknowledge support from the Center for Theoretical Physics
and the Department of Physics at the Massachusetts Institute of Technology
where most of the work was carried out. NPF was also supported in part by the
National Science Foundation under the Grant No. PHY-1748958 to the Kavli
Institute for Theoretical Physics (KITP) during the workshop refluids23 where
part of the work was done. MK is supported by FWO-Vlaanderen project G012222N
and by the VUB Research Council through the Strategic Research Program High-
Energy Physics. HL is supported by the Office of High Energy Physics of U.S.
Department of Energy under grant Contract Number DE-SC0012567 and DE-SC0020360
(MIT contract # 578218).

##  Appendix A Freedom in choosing $n^{A}$

Having defined $\hat{\ell}^{a}$ as the unique vector that satisfies
$h_{ab}\hat{\ell}^{a}=0$ on the horizon ${\cal H}$ up to rescalings (9), there
is yet a freedom in choosing the vector $n_{A}$. This can be shown as follows.
Consider a basis of tangent vectors to ${{\mathcal{M}}}$ consisting of
$(\ell^{A},n^{A},V^{A}_{i})$ satisfying on ${{\mathcal{H}}}$

$G_{AB}\ell^{A}V_{i}^{B}{\;\,\triangleq\;\,}0,\quad
G_{AB}n^{A}V_{i}^{B}{\;\,\triangleq\;\,}0,\quad
G_{AB}V^{A}_{i}V_{j}^{B}{\;\,\triangleq\;\,}{{\delta}}_{ij}\,,$ (160)

as well as the usual relations

$G_{AB}n^{A}n^{B}{\;\,\triangleq\;\,}0\,,\qquad
G_{AB}\ell^{A}n^{B}{\;\,\triangleq\;\,}-1\,.$ (161)

We can always reparameterize $n^{A}$ and therefore $V_{i}^{A}$ on ${\cal H}$
at will, so long as the above conditions are left invariant.

The most general transformation of this kind $n^{A}\rightarrow\tilde{n}^{A}$
and $V_{i}^{A}\rightarrow\tilde{V}_{i}^{A}$ can be parameterized as follows

$\tilde{n}^{A}{\;\,\triangleq\;\,}n^{A}+a\ell^{A}+c_{i}V^{A}_{i},\quad\tilde{V}^{A}_{i}{\;\,\triangleq\;\,}d_{ij}V^{A}_{j}+b_{i}\ell^{A},$
(162)

then

$\displaystyle\ell^{A}\tilde{V}_{A\,i}{\;\,\triangleq\;\,}0\,,\quad\ell^{A}\tilde{n}_{A}{\;\,\triangleq\;\,}-1,\quad\tilde{n}^{A}\tilde{n}_{A}{\;\,\triangleq\;\,}-2a+c_{i}^{2},\quad\tilde{n}_{A}\tilde{V}^{A}_{i}{\;\,\triangleq\;\,}-b_{i}+d_{ij}c_{j}\,,$
(163)
$\displaystyle\tilde{V}^{A}_{i}\tilde{V}_{Aj}{\;\,\triangleq\;\,}d_{ik}d_{jk}\
.$ (164)

We thus need $d_{ij}$ to be an orthogonal matrix, which we can take to be the
identity matrix. The other quantities are then set as

$d_{ij}{\;\,\triangleq\;\,}\delta_{ij}\,,\quad
a{\;\,\triangleq\;\,}{{\frac{1}{2}}}W^{2},\quad
c_{i}=b_{i}{\;\,\triangleq\;\,}W_{A}V^{A}_{i},\quad
W^{A}\ell_{A}{\;\,\triangleq\;\,}W^{A}n_{A}{\;\,\triangleq\;\,}0\,.$ (165)

Thus, to summarize, we may always redefine $n_{A}$ on the horizon as follows

$\displaystyle n_{A}\rightarrow\tilde{n}_{A}=n_{A}+\Phi_{A}\,,$ (166)
$\displaystyle\Phi^{A}{\;\,\triangleq\;\,}\hat{\Phi}^{A}{\;\,\triangleq\;\,}W^{A}+\frac{1}{2}\ell^{A}W^{2}\,,\quad
W^{A}\ell_{A}{\;\,\triangleq\;\,}0\,,\quad W^{A}n_{A}{\;\,\triangleq\;\,}0\,,$
(167)

where $\Phi^{A}$ is a generic vector while $\hat{\Phi}^{A}$ is its value
evaluated at the horizon ${\cal H}$ which can be decomposed into a part that
is parallel to $\ell^{A}$ and a part that is orthogonal to it. The vector
$W^{A}$ has to be taken as a function of the horizon coordinates. The
redefinition (166) implies a redefinition of the projector $q_{A}{}^{B}$ too,

$\displaystyle
q_{A}{}^{B}\rightarrow\tilde{q}_{A}{}^{B}=q_{A}{}^{B}+\Phi_{A}\ell^{B}+\ell_{A}\Phi^{B}\,,$
(168)

which will be useful later on.

The reparameterization freedom (166) and (168) is akin to the Carrollian boost
transformation which arises in the context of ultra relativistic field
theories. It has been shown that it also a symmetry of null surfaces when they
are embedded in a Lorentzian spacetime of one dimension higher, see, e.g.,
[60].

###  A.1 Independence of $n^{A}$

The arbitrariness in choosing $n^{A}$ seems to naively affect the horizon
constraint equations derived in (24), (25) as well as in (63), (64), and (66).
However, we will now show that this is not the case.

$\bullet$ $\chi^{A}\ell_{A}{\;\,\triangleq\;\,}0$

We start with the case where $\chi^{A}$ is tangent to the horizon, that is
$\chi^{A}\ell_{A}{\;\,\triangleq\;\,}0$. In this case,
$\chi^{A}{\;\,\triangleq\;\,}\partial_{a}X^{A}\xi^{a}$ and $\xi^{a}$ is
decomposed as in (23). Given that
$\hat{n}_{a}{\;\,\triangleq\;\,}\partial_{a}X^{A}n_{A}$, the
reparameterization freedom of $n_{A}$ obtained in (166) implies a
reparameterization freedom of $\hat{n}_{a}$ and $\hat{q}_{a}{}^{b}$ as follows

$\hat{n}_{a}\rightarrow\tilde{\hat{n}}_{a}=\hat{n}_{a}+\hat{\Phi}_{a}\,,\quad\hat{q}_{a}{}^{b}\rightarrow\tilde{\hat{q}}_{a}{}^{b}=\hat{q}_{a}{}^{b}+\hat{n}_{a}\hat{\ell}^{b}\,,$
(169)

with

$\hat{\Phi}_{a}{\;\,\triangleq\;\,}\partial_{a}X^{A}\Phi_{A}\,,\quad\hat{\Phi}_{a}\hat{\ell}^{a}=0\,,\quad\hat{q}_{a}{}^{b}=\delta_{a}{}^{b}+\hat{n}_{a}\hat{\ell}^{b}\,.$
(170)

Given that $\xi^{a}$ is a generic diffeomorphism on the horizon and should be
independent of the chosen $\hat{n}_{a}$, this arbitrariness affects the
parameterization (23) via the following redefinitions

$\tilde{f}=-\xi^{a}\tilde{\hat{n}}_{a}=f-h,\quad
Y^{a}\to\tilde{Y}^{a}=\xi^{b}\tilde{\hat{q}}_{b}{}^{a}=Y^{a}+h\hat{\ell}^{a},\quad
a\to\tilde{a}=a+{{\mathcal{L}}}_{\hat{\ell}}h\,,$ (171)

with

$h=Y^{a}\hat{\Phi}_{a}\ .$ (172)

It can be readily checked that equation (27) is equivalent to that with
$f,a,Y^{a}$ replaced by tilded quantities. In this way, it is clear that
equations (24) and (25) (as well as (27)) do not depend on the specific choice
of $\hat{n}_{a}$ as expected given the equivalence to (21)–(22) that are
explicitly independent of it.

$\bullet$ $\chi^{A}\ell_{A}\neq 0$

Consider now the case with $\chi^{A}\ell_{A}\neq 0$. The most general
parameterization of $\chi^{A}$ is given in (62), and under the redefinition of
$n_{A}$ and $q_{A}{}^{B}$ given in (166) and (168), we have

$\displaystyle
f\rightarrow\tilde{f}{\;\,\triangleq\;\,}-\chi^{A}\tilde{n}_{A}{\;\,\triangleq\;\,}f-H\,,$
(173) $\displaystyle
Y^{A}\rightarrow\tilde{Y}^{A}{\;\,\triangleq\;\,}\tilde{q}^{A}{}_{B}\chi^{B}{\;\,\triangleq\;\,}Y^{A}+H\ell^{A}-Z\Phi^{A}\,,$
(174) $\displaystyle
Z\rightarrow\tilde{Z}{\;\,\triangleq\;\,}-\chi^{A}\ell_{A}{\;\,\triangleq\;\,}Z\,,$
(175)

with

$\displaystyle\Phi_{A}{\;\,\triangleq\;\,}\hat{\Phi}_{A}{\;\,\triangleq\;\,}W_{A}+\frac{1}{2}W^{2}\ell_{A}\,,\quad
W^{A}\ell_{A}{\;\,\triangleq\;\,}0\,,\quad W^{A}n_{A}{\;\,\triangleq\;\,}0\,,$
(176) $\displaystyle H=h-\frac{1}{2}W^{2}Z\,,\quad h=Y^{A}W_{A}\,.$ (177)

The following expressions will be useful

$\Phi_{A}\ell^{A}{\;\,\triangleq\;\,}0\,,\quad\Phi_{A}n^{A}{\;\,\triangleq\;\,}-\frac{1}{2}W^{2}\,,\quad\Phi^{2}{\;\,\triangleq\;\,}W^{2}\,.$
(178)

The symmetry constraint equation (63) is clearly invariant under (166) and
(175). We now want to show that the expression (64) is also covariant under
the above transformations, that is it has the same form once all the
quantities have been replaced by tilded ones. To do so, let us perform the
intermediate transformations evaluated at the horizon

$\displaystyle{{\mathcal{L}}}_{\ell}Y^{A}\rightarrow{{\mathcal{L}}}_{\ell}Y^{A}+\ell^{A}{{\mathcal{L}}}_{\ell}H-\kappa
Z\hat{\Phi}^{A}-Z{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}\,,$ (179)
$\displaystyle-q^{AB}\nabla_{B}Z\rightarrow-q^{AB}\nabla_{B}Z-\kappa
Z\hat{\Phi}^{A}-\ell^{A}{{\mathcal{L}}}_{\hat{\Phi}}Z\,,$ (180) $\displaystyle
Z\eta^{A}_{\perp}\rightarrow Z\eta^{A}_{\perp}+2\kappa
Z\hat{\Phi}^{A}+Z\ell^{A}\eta^{B}_{\perp}W_{B}+\kappa
ZW^{2}\ell^{A}+Z{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}+Zn_{B}\ell^{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{B}+Z\ell^{A}\hat{\Phi}_{B}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{B}\,,$
(181) $\displaystyle a\rightarrow
a+{{\mathcal{L}}}_{\ell}H+Zn_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}-{{\mathcal{L}}}_{\hat{\Phi}}Z+Z\eta^{A}_{\perp}W_{A}+\kappa
ZW^{2}+Z\hat{\Phi}_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}\,,$ (182)

where we used (63) and (64) to simplify the expressions, the fact that
$\ell^{A}\nabla_{A}$ is a derivative along the horizon, as well as
$\eta^{A}\ell_{A}{\;\,\triangleq\;\,}2\kappa$,
$\eta^{A}\Phi_{A}{\;\,\triangleq\;\,}\eta^{A}_{\perp}W_{A}+\kappa W^{2}$,
${{\mathcal{L}}}_{\ell}\Phi^{A}{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}$,
and

$q^{AB}{{\mathcal{L}}}_{\Phi}\ell_{A}{\;\,\triangleq\;\,}0\,,\quad\ell^{A}{{\mathcal{L}}}_{\Phi}\ell_{A}{\;\,\triangleq\;\,}0\,,\quad\Phi^{A}{{\mathcal{L}}}_{\Phi}\ell_{A}{\;\,\triangleq\;\,}0\,,\quad\ell_{A}{{\mathcal{L}}}_{\Phi}\ell^{A}{\;\,\triangleq\;\,}0\,.$
(183)

These latter equations can be easily derived using the fact that
$\ell^{2}{\;\,\triangleq\;\,}0$, $q^{AB}\nabla_{B}$ and $\ell^{B}\nabla_{B}$
are derivatives along the horizon, and the fact that $\ell_{A}$ is
hypersurface orthogonal, that is

$q^{AB}{{\mathcal{L}}}_{\Phi}\ell_{A}=q^{AB}(\Phi^{C}\nabla_{C}\ell_{A}+\ell_{C}\nabla_{A}\Phi^{C}){\;\,\triangleq\;\,}q^{AB}W^{C}(\nabla_{C}\ell_{A}-\nabla_{A}\ell_{C}){\;\,\triangleq\;\,}0\,.$
(184)

Combining the transformations (179)-(182) we find that (64) is equivalent to
the same equation with tilded variables.

Let us now consider (66) which we copy here for convenience

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+({{\mathcal{L}}}_{\ell}+\kappa)a+{{\mathcal{L}}}_{Y}\kappa+\eta^{\perp\,A}\nabla_{A}Z+Z{\cal
F}{\;\,\triangleq\;\,}0\,,$ $\displaystyle{\cal
F}=\frac{1}{2}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}-\eta^{2}-{{\mathcal{L}}}_{\ell}\lambda-3\kappa\lambda\,.$
(185)

and we have performed a slight rewriting of ${\cal F}$ using

$\eta^{2}_{\perp}{\;\,\triangleq\;\,}\eta^{2}-4\kappa\lambda\,.$ (186)

Under the redefinition (166), the various terms, when evaluated on the horizon
${\cal H}$, transform as follows

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f\rightarrow{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f-{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)H\,,$
(187)
$\displaystyle({{\mathcal{L}}}_{\ell}+\kappa)a\rightarrow({{\mathcal{L}}}_{\ell}+\kappa)a+{{\mathcal{L}}}_{\ell}{{\mathcal{L}}}_{\ell}H+\kappa{{\mathcal{L}}}_{\ell}H+2\kappa
Zn_{A}{{\mathcal{L}}}_{\ell}{\hat{\Phi}}^{A}+Z{{\mathcal{L}}}_{\ell}(n_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A})+$
$\displaystyle-Z{{\mathcal{L}}}_{\hat{\Phi}}\kappa-2\kappa{{\mathcal{L}}}_{\hat{\Phi}}Z+2\kappa
Z\eta^{\perp A}W_{A}+Z{{\mathcal{L}}}_{\ell}(\eta^{\perp A}W_{A})+2\kappa
Z\hat{\Phi}_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}+Z{{\mathcal{L}}}_{\ell}(\hat{\Phi}_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A})+$
$\displaystyle-{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}\nabla_{A}Z+Z{{\mathcal{L}}}_{\ell}(\kappa
W^{2})+2\kappa^{2}W^{2}Z\,,$ (188)
$\displaystyle{{\mathcal{L}}}_{Y}\kappa\rightarrow{{\mathcal{L}}}_{Y}\kappa+H{{\mathcal{L}}}_{\ell}\kappa-Z{{\mathcal{L}}}_{\hat{\Phi}}\kappa\,,$
(189)
$\displaystyle\eta^{A}\rightarrow\eta^{A}+G^{AB}{{\mathcal{L}}}_{\Phi}\ell_{B}-{{\mathcal{L}}}_{\Phi}\ell^{A}\,,$
(190)
$\displaystyle\eta^{\perp\,A}\nabla_{A}Z\rightarrow\eta^{\perp\,A}\nabla_{A}Z+\kappa
Z\eta^{\perp\,B}W_{B}+2\kappa{{\mathcal{L}}}_{\hat{\Phi}}Z+\kappa^{2}W^{2}Z+{{\mathcal{L}}}_{\ell}\hat{\Phi}^{B}\nabla_{B}Z+$
$\displaystyle+\kappa Zn_{B}{{\mathcal{L}}}_{\ell}{\hat{\Phi}}^{B}+\kappa
Z\hat{\Phi}_{B}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{B}\,,$ (191)
$\displaystyle\frac{1}{2}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}\rightarrow\frac{1}{2}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}+2{{\mathcal{L}}}_{\hat{\Phi}}\kappa-2\eta_{A}{{\mathcal{L}}}_{\hat{\Phi}}\ell^{A}-\delta\eta_{A}{{\mathcal{L}}}_{\hat{\Phi}}\ell^{A}-{{\mathcal{L}}}_{\ell}{\cal
A}+\kappa{\cal A}\,,$ (192)
$\displaystyle\eta^{2}\rightarrow\eta^{2}+4\kappa{\cal
A}-2\eta_{A}{{\mathcal{L}}}_{\hat{\Phi}}\ell^{A}-\delta\eta_{A}{{\mathcal{L}}}_{\hat{\Phi}}\ell^{A}\,,$
(193) $\displaystyle\lambda\rightarrow\lambda-{\cal
A}+n_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}+\eta^{A}_{\perp}W_{A}+\kappa
W^{2}+\hat{\Phi}_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A}\,,$ (194)
$\displaystyle{{\mathcal{L}}}_{\ell}\lambda\rightarrow{{\mathcal{L}}}_{\ell}\lambda-{{\mathcal{L}}}_{\ell}{\cal
A}+{{\mathcal{L}}}_{\ell}(n_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A})+{{\mathcal{L}}}_{\ell}(\eta^{A}_{\perp}W_{A})+{{\mathcal{L}}}_{\ell}(\kappa
W^{2})+{{\mathcal{L}}}_{\ell}(\hat{\Phi}_{A}{{\mathcal{L}}}_{\ell}\hat{\Phi}^{A})\,,$
(195)

where we used

$\displaystyle{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{\Phi}G_{AB}={{\mathcal{L}}}_{\Phi}{{\mathcal{L}}}_{n}G_{AB}+{{\mathcal{L}}}_{[n,\Phi]}G_{AB}={{\mathcal{L}}}_{\Phi}{{\mathcal{L}}}_{n}G_{AB}+2\nabla_{A}(G_{BC}{{\mathcal{L}}}_{n}\Phi^{C})\,,$
(196)
$\displaystyle\ell_{C}{{\mathcal{L}}}_{n}\Phi^{C}=-\ell_{C}{{\mathcal{L}}}_{\Phi}n^{C}{\;\,\triangleq\;\,}n^{C}{{\mathcal{L}}}_{\Phi}\ell_{C}{\;\,\triangleq\;\,}-{\cal
A}\,,$ (197)

and we have defined

${{\mathcal{L}}}_{\Phi}\ell_{C}{\;\,\triangleq\;\,}{\cal
A}\ell_{C}\,,\quad{\cal
A}{\;\,\triangleq\;\,}-n^{C}{{\mathcal{L}}}_{\Phi}\ell_{C}\,,$ (198)

using (183). In all the steps above we repeadetely used
$\ell^{A}n_{A}{\;\,\triangleq\;\,}-1$, $\ell^{2}{\;\,\triangleq\;\,}0$,
$\ell^{A}\hat{\Phi}_{A}{\;\,\triangleq\;\,}0$,
$\delta\eta_{A}={{\mathcal{L}}}_{\Phi}\ell_{A}-G_{AB}{{\mathcal{L}}}_{\Phi}\ell^{B}$,
$\eta^{A}\ell_{A}{\;\,\triangleq\;\,}2\kappa$,
$\eta^{A}n_{A}{\;\,\triangleq\;\,}\lambda$,
$\delta\eta^{B}\ell_{B}{\;\,\triangleq\;\,}0$, and the fact that various
derivatives are taken to be along the horizon.

Combining all the results above, it is straightforward to show that equation
(66) is equivalent to the one with the tilded variables as we wanted to
demonstrate.

##  Appendix B Details of some derivations

$\bullet$ Proof of (40):

To derive equation (40), first we note that

$\displaystyle\ell^{B}{{\delta}}{\nabla}_{B}\ell^{A}=\ell^{B}{{\delta}}{{\Gamma}}^{A}_{BC}\ell^{C}=\frac{1}{2}\ell^{B}\ell^{C}(\nabla_{B}f^{A}_{C}+\nabla_{C}f^{A}_{B}-\nabla^{A}f_{BC})$
$\displaystyle=\ell^{B}\ell^{C}\nabla_{B}f^{A}_{C}-{{\frac{1}{2}}}\ell^{B}\ell^{C}\nabla^{A}f_{BC}\,.$
(199)

Assuming (35), we have

$\displaystyle\ell^{B}\ell^{C}\nabla_{B}f^{A}_{C}=\ell^{B}\nabla_{B}(f^{A}_{C}\ell^{C})-\ell^{B}f^{A}_{C}\nabla_{B}\ell^{C}{\;\,\triangleq\;\,}-\ell^{A}{{\mathcal{L}}}_{\ell}c,$
(200)
$\displaystyle\ell^{B}\ell^{C}\nabla_{A}f_{BC}{\;\,\triangleq\;\,}\nabla_{A}(f_{BC}\ell^{B}\ell^{C})+2c\,\ell_{C}\nabla_{A}\ell^{C}{\;\,\triangleq\;\,}\partial_{A}(f_{BC}\ell^{B}\ell^{C}+c\,\ell_{C}\ell^{C})\
.$ (201)

where we used the fact that $\ell^{2}=\ell_{C}\ell^{C}{\;\,\triangleq\;\,}0$.
Contracting these expressions with $\ell_{A}$ or $q_{A}{}^{D}$ leads to a
vanishing result given that $f_{BC}\ell^{B}\ell^{C}{\;\,\triangleq\;\,}0$ (due
to (35)), $\ell^{2}{\;\,\triangleq\;\,}0$, and derivatives along
$\ell^{A}\nabla_{A}$ and $q^{AD}\nabla_{A}$ are along the horizon surface and
therefore vanish if their argument is vanishing along the horizon.

Contracting (200) and (201) along $n_{A}$ instead leads to a nontrivial
result. We get

$\displaystyle
n_{A}\ell^{B}\ell^{C}\nabla_{B}f^{A}_{C}{\;\,\triangleq\;\,}{{\mathcal{L}}}_{\ell}c\,,$
(202) $\displaystyle
n^{A}\ell^{B}\ell^{C}\nabla_{A}f_{BC}{\;\,\triangleq\;\,}{{\mathcal{L}}}_{n}(f_{BC}\ell^{B}\ell^{C}+c\,\ell_{C}\ell^{C})$
(203)
$\displaystyle=\ell^{B}\ell^{C}{{\mathcal{L}}}_{n}f_{BC}-c\ell_{A}{{\mathcal{L}}}_{n}\ell^{A}+c\ell^{A}{{\mathcal{L}}}_{n}\ell_{A}=\ell^{B}\ell^{C}{{\mathcal{L}}}_{n}f_{BC}+2\kappa
c$ (204)

where we have used that

$\ell^{A}{{\mathcal{L}}}_{n}\ell_{A}-\ell_{A}{{\mathcal{L}}}_{n}\ell^{A}=2\ell^{A}\ell_{B}{\nabla}_{A}n^{B}{\;\,\triangleq\;\,}2\kappa\
.$ (205)

as well as $n_{A}\ell^{A}{\;\,\triangleq\;\,}-1$ and the fact that
$\ell^{A}\nabla_{A}$ is a derivative along the horizon hypersurface. Combining
these two expressions into (199) we get (40) as desired.

$\bullet$ Proof of (63), (64), (66), and (69):

Here we give details of the derivations of (63)–(64), (66), and (69). For this
purpose, we copy equations (39)–(43) and (40) with the decomposition (44) here
for convenience.

$\displaystyle\ell^{A}{{\mathcal{L}}}_{\chi}G_{AB}={{\mathcal{L}}}_{\chi}\ell_{B}-G_{AB}{{\mathcal{L}}}_{\chi}\ell^{A}={{\mathcal{L}}}_{\chi}\ell_{B}+G_{AB}{{\mathcal{L}}}_{\ell}\chi^{A}\propto\ell_{B}\,,$
(206)
$\displaystyle\frac{1}{2}{{\mathcal{L}}}_{\chi}(\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}G_{AB})-\ell^{B}({{\mathcal{L}}}_{\chi}\ell^{A})({{\mathcal{L}}}_{n}G_{AB})+\frac{1}{2}\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\chi]}G_{AB}+c\kappa-{{\mathcal{L}}}_{\ell}c=0\,,$
(207) $\displaystyle
c\equiv\ell^{A}n^{B}{{\mathcal{L}}}_{\chi}G_{AB}=n^{B}{{\mathcal{L}}}_{\chi}\ell_{B}-n_{A}{{\mathcal{L}}}_{\chi}\ell^{A}\
.$ (208)

Using the decomposition of the diffeomorphisms vector $\chi^{A}$ given in
(62), we can separate the equations (206)–(208) into the parts involving
$\tilde{\chi}^{A}$ and the parts involving $Zn^{A}$. Consider first (206). It
can be written as

$\displaystyle
Z{{\mathcal{L}}}_{n}\ell_{B}-{\nabla}_{B}Z+G_{AB}{{\mathcal{L}}}_{\ell}Y^{A}-ZG_{AB}{{\mathcal{L}}}_{n}\ell^{A}+n_{B}{{\mathcal{L}}}_{\ell}Z+{{\mathcal{L}}}_{\tilde{\chi}}\ell_{B}\propto\ell_{B}\
,$ (209)

and further simplified as

$\displaystyle
Z\eta_{B}+2n_{B}{{\mathcal{L}}}_{\ell}Z-q_{B}^{A}{\nabla}_{A}Z+G_{AB}{{\mathcal{L}}}_{\ell}Y^{A}\propto\ell_{B}\,,$
(210)

where we have used the definition (68) and
${{\mathcal{L}}}_{\tilde{\chi}}\ell_{B}\propto\ell_{B}$ as discussed in (54).

Multiplying both sides of (210) by $\ell^{B}$ and using

$\displaystyle\ell^{A}\eta_{A}=\ell^{A}\ell^{B}({{\mathcal{L}}}_{n}G_{AB})=2\ell^{A}\ell^{B}{\nabla}_{A}n_{B}{\;\,\triangleq\;\,}-2\ell^{A}n_{B}{\nabla}_{A}\ell^{B}{\;\,\triangleq\;\,}2\kappa$
(211)
$\displaystyle\ell_{A}{{\mathcal{L}}}_{\ell}Y^{A}=-Y^{A}{{\mathcal{L}}}_{\ell}\ell_{A}{\;\,\triangleq\;\,}0\,,$
(212)

where again we used $Y^{A}\ell_{A}{\;\,\triangleq\;\,}0$ and the fact that
$\ell^{A}\nabla_{A}$ is a derivative along the horizon, we find that

${{\mathcal{L}}}_{\ell}Z-\kappa Z{\;\,\triangleq\;\,}0\ ,$ (213)

which is (63).

Using (213) in (210) then gives

$\displaystyle{{\mathcal{L}}}_{\ell}Y^{A}-q^{AB}{\nabla}_{B}Z+Z\eta^{A}+2\kappa
Zn^{A}{\;\,\triangleq\;\,}\propto\ell^{A},$ (214)

which, using (68) and the fact that
$\eta^{A}\ell_{A}{\;\,\triangleq\;\,}2\kappa$, can be further simplified to

${{\mathcal{L}}}_{\ell}Y^{A}-q^{AB}{\nabla}_{B}Z+Z\eta^{A}_{\perp}{\;\,\triangleq\;\,}a\ell^{A}\,,$
(215)

which is the desired equation (64), where $a$ can be written as

$a=-n_{A}{{\mathcal{L}}}_{\ell}Y^{A}\ .$ (216)

Finally, to prove (66), consider (207), whose LHS can be written as

${\rm LHS}={\rm LHS}_{1}+{\rm LHS}_{2}\,,$ (217)

where ${\rm LHS}_{1}$ denotes the part involving $\tilde{\chi}^{A}$ and ${\rm
LHS}_{2}$ denotes the part involving $Zn^{A}$. First note that

$\displaystyle
c=n^{B}{{\mathcal{L}}}_{\tilde{\chi}}\ell_{B}+n^{B}{{\mathcal{L}}}_{Zn}\ell_{B}+n_{A}{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A}+n_{A}{{\mathcal{L}}}_{\ell}(Zn^{A})$
(218)
$\displaystyle{\;\,\triangleq\;\,}\tilde{c}+Z({{\mathcal{L}}}_{n}(n^{B}\ell_{B})+n_{A}{{\mathcal{L}}}_{\ell}n^{A})-{{\mathcal{L}}}_{n}Z{\;\,\triangleq\;\,}\tilde{c}+Z\lambda-{{\mathcal{L}}}_{n}Z,$
(219) $\displaystyle\tilde{c}\equiv
n^{B}{{\mathcal{L}}}_{\tilde{\chi}}\ell_{B}+n_{A}{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A},$
(220)
$\displaystyle-{{\mathcal{L}}}_{\ell}c{\;\,\triangleq\;\,}-{{\mathcal{L}}}_{\ell}\tilde{c}-{{\mathcal{L}}}_{\ell}(Z\lambda-{{\mathcal{L}}}_{n}Z){\;\,\triangleq\;\,}-{{\mathcal{L}}}_{\ell}\tilde{c}-Z({{\mathcal{L}}}_{\ell}+\kappa)\lambda+{{\mathcal{L}}}_{\ell}{{\mathcal{L}}}_{n}Z\,,$
(221)

where we used the fact that $n^{2}=n_{A}n^{A}{\;\,\triangleq\;\,}0$, that
$\ell^{A}\nabla_{A}$ is a derivative along the horizon, and equation (213).
The parts of the terms in (207) that involve $Zn^{A}$ have the form

$\displaystyle{{\frac{1}{2}}}{{\mathcal{L}}}_{\chi}(\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}G_{AB})=Zn^{C}{\nabla}_{C}(\ell^{A}\ell^{B}{\nabla}_{A}n_{B})+\cdots={{\frac{1}{2}}}Z{{\mathcal{L}}}_{n}(\ell^{A}\eta_{A})+\cdots\,,$
(222)
$\displaystyle-\ell^{B}({{\mathcal{L}}}_{\chi}\ell^{A})({{\mathcal{L}}}_{n}G_{AB}){\;\,\triangleq\;\,}+Z\eta_{A}({{\mathcal{L}}}_{\ell}n^{A}+\kappa
n^{A})+\cdots=+Z\eta_{A}{{\mathcal{L}}}_{\ell}n^{A}+\kappa Z\lambda+\cdots\,,$
(223)
$\displaystyle{{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\chi]}G_{AB}=\ell^{A}\ell^{B}{\nabla}_{A}[n,Zn]_{B}+\cdots\,,$
(224)
$\displaystyle=\ell^{A}\ell^{B}{\nabla}_{A}(n_{B}{{\mathcal{L}}}_{n}Z)+\cdots{\;\,\triangleq\;\,}-{{\mathcal{L}}}_{\ell}{{\mathcal{L}}}_{n}Z+\kappa{{\mathcal{L}}}_{n}Z+\cdots\,,$
(225)

where $\cdots$ denote the parts involving $\tilde{\chi}^{A}$ and we used
repeadetely $\ell^{A}n_{A}{\;\,\triangleq\;\,}-1$,
$n^{2}{\;\,\triangleq\;\,}0$, and the fact that $\ell^{A}\nabla_{A}$ is a
derivative along the horizon. From the above equations we then find

$\displaystyle{\rm
LHS}_{2}{\;\,\triangleq\;\,}{{\frac{1}{2}}}Z{{\mathcal{L}}}_{n}(\ell^{A}\eta_{A})+Z\eta_{A}{{\mathcal{L}}}_{\ell}n^{A}+\kappa
Z\lambda-{{\mathcal{L}}}_{\ell}{{\mathcal{L}}}_{n}Z+\kappa{{\mathcal{L}}}_{n}Z$
(226)
$\displaystyle+\kappa(Z\lambda-{{\mathcal{L}}}_{n}Z)-Z({{\mathcal{L}}}_{\ell}+\kappa)\lambda+{{\mathcal{L}}}_{\ell}{{\mathcal{L}}}_{n}Z$
(227)
$\displaystyle={{\frac{1}{2}}}Z{{\mathcal{L}}}_{n}(\ell^{A}\eta_{A})+Z\eta_{A}{{\mathcal{L}}}_{\ell}n^{A}-Z({{\mathcal{L}}}_{\ell}-\kappa)\lambda$
(228)
$\displaystyle={{\frac{1}{2}}}Z\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}-Z({{\mathcal{L}}}_{\ell}-\kappa)\lambda\
.$ (229)

For the $\tilde{\chi}$ part of (207), we have

$\displaystyle{\rm
LHS}_{1}=\frac{1}{2}{{\mathcal{L}}}_{\tilde{\chi}}(\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}G_{AB})-\ell^{B}({{\mathcal{L}}}_{\tilde{\chi}}\ell^{A})({{\mathcal{L}}}_{n}G_{AB})+\frac{1}{2}\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\tilde{\chi}]}G_{AB}+\tilde{c}\kappa-{{\mathcal{L}}}_{\ell}\tilde{c}\,,$
(230)

where

$\displaystyle\tilde{c}\equiv
n^{B}{{\mathcal{L}}}_{\tilde{\chi}}\ell_{B}+n_{A}{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A}=-\tilde{b}+n_{A}{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A},$
(231)
$\displaystyle{{\mathcal{L}}}_{\tilde{\chi}}\ell_{B}{\;\,\triangleq\;\,}\tilde{b}\ell_{B},\quad\tilde{b}{\;\,\triangleq\;\,}-n^{A}{{\mathcal{L}}}_{\tilde{\chi}}\ell_{A}=\ell_{A}{{\mathcal{L}}}_{\tilde{\chi}}n^{A}=-\ell_{A}{{\mathcal{L}}}_{n}\tilde{\chi}^{A}\
.$ (232)

From (64) we get

$\displaystyle{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A}{\;\,\triangleq\;\,}\ell^{A}{{\mathcal{L}}}_{\ell}f+{{\mathcal{L}}}_{\ell}Y^{A}{\;\,\triangleq\;\,}({{\mathcal{L}}}_{\ell}f+a)\ell^{A}+q^{AB}{\nabla}_{B}Z-Z\eta^{A}_{\perp}$
(233) $\displaystyle
n_{A}{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A}=-({{\mathcal{L}}}_{\ell}f+a)\,,\quad{\Longrightarrow}\quad\tilde{c}=-(\tilde{b}+{{\mathcal{L}}}_{\ell}f+a)\
.$ (234)

Now note that

$\displaystyle{{\frac{1}{2}}}{{\mathcal{L}}}_{\tilde{\chi}}(\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}G_{AB}){\;\,\triangleq\;\,}{{\mathcal{L}}}_{\tilde{\chi}}\kappa\,,$
(235)
$\displaystyle{{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{[n,\tilde{\chi}]}G_{AB}=\ell_{A}\ell^{B}\nabla_{B}{{\mathcal{L}}}_{n}\tilde{\chi}^{A}{\;\,\triangleq\;\,}-{{\mathcal{L}}}_{\ell}\tilde{b}-{{\mathcal{L}}}_{n}\tilde{\chi}^{A}\ell^{B}\nabla_{B}\ell_{A}{\;\,\triangleq\;\,}-{{\mathcal{L}}}_{\ell}\tilde{b}+\tilde{b}\kappa\,,$
(236)

and from (233) we have

$\displaystyle-\ell^{B}({{\mathcal{L}}}_{\tilde{\chi}}\ell^{A})({{\mathcal{L}}}_{n}G_{AB})=+\eta_{A}{{\mathcal{L}}}_{\ell}\tilde{\chi}^{A}{\;\,\triangleq\;\,}+\eta_{A}(({{\mathcal{L}}}_{\ell}f+a)\ell^{A}+q^{AB}{\nabla}_{B}Z-Z\eta^{A}_{\perp})$
(237)
$\displaystyle{\;\,\triangleq\;\,}+2({{\mathcal{L}}}_{\ell}f+a)\kappa+\eta^{A}_{\perp}{\nabla}_{A}Z-Z\eta^{2}_{\perp},\quad$
(238)

In deriving these steps we used several times the fact that
$n_{A}\ell^{A}{\;\,\triangleq\;\,}-1$ and that $\tilde{\chi}^{A}\nabla_{A}$ is
a derivative along the horizon. We thus find that equation (230) becomes

$\displaystyle{\rm
LHS}_{1}={{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+({{\mathcal{L}}}_{\ell}+\kappa)a+{{\mathcal{L}}}_{Y}\kappa+\eta^{A}_{\perp}{\nabla}_{A}Z-Z\eta^{2}_{\perp}\
.$ (239)

Combining with (229) we find

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+({{\mathcal{L}}}_{\ell}+\kappa)a+{{\mathcal{L}}}_{Y}\kappa+\eta_{A}^{\perp}{\nabla}_{C}Z+Z{\cal
F}=0,$ (240) $\displaystyle{\cal
F}={{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}-\eta^{2}_{\perp}-({{\mathcal{L}}}_{\ell}-\kappa)\lambda\,$
(241)

which is the desired equation (66).

Alternatively, using (216) we can write (240) as

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+Y^{A}({{\mathcal{L}}}_{\ell}+\kappa){{\mathcal{L}}}_{\ell}n_{A}+{{\mathcal{L}}}_{Y}\kappa+q^{AC}{\nabla}_{C}Z(\eta_{\perp
A}+{{\mathcal{L}}}_{\ell}n_{A})+Z{\tilde{\cal F}}=0,$ (242)
$\displaystyle{\tilde{\cal F}}={{\cal
F}}-\eta_{\perp}^{A}{{\mathcal{L}}}_{\ell}n_{A}={{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}-\eta^{A}_{\perp}(\eta_{\perp
A}+{{\mathcal{L}}}_{\ell}n_{A})-({{\mathcal{L}}}_{\ell}-\kappa)\lambda\ ,$
(243)

where we used (64), the identity
$\ell^{A}{{\mathcal{L}}}_{\ell}n_{A}{\;\,\triangleq\;\,}0$, the fact that
$Y^{A}n_{A}{\;\,\triangleq\;\,}0$, and that $\ell^{A}\nabla_{A}$ is a
derivative along the horizon. Note that

$\displaystyle q^{AC}\left(\eta_{\perp
C}+{{\mathcal{L}}}_{\ell}n_{C}\right)=q^{AB}\ell^{C}({\nabla}_{B}n_{C}+{\nabla}_{C}n_{B})+q^{AC}(\ell^{B}{\nabla}_{B}n_{C}+n_{B}{\nabla}_{C}\ell^{B})$
(244) $\displaystyle=2q^{AB}\ell^{C}{\nabla}_{C}n_{B}\ .$ (245)

So, equations (242)–(243) can be further simplified to

$\displaystyle{{\mathcal{L}}}_{\ell}({{\mathcal{L}}}_{\ell}+\kappa)f+Y^{A}({{\mathcal{L}}}_{\ell}+\kappa){{\mathcal{L}}}_{\ell}n_{A}+{{\mathcal{L}}}_{Y}\kappa+2q^{AB}{\nabla}_{A}Z\ell^{C}{\nabla}_{C}n_{B}+Z{\tilde{\cal
F}}=0,$ (246) $\displaystyle{\tilde{\cal
F}}={{\frac{1}{2}}}\ell^{A}\ell^{B}{{\mathcal{L}}}_{n}{{\mathcal{L}}}_{n}G_{AB}+2n_{A}\ell^{C}{\nabla}_{C}\eta^{A}_{\perp}-({{\mathcal{L}}}_{\ell}-\kappa)\lambda\
,$ (247)

which is (69), where again we used
$\eta^{A}_{\perp}n_{A}{\;\,\triangleq\;\,}0$ and the fact that
$\ell^{A}\nabla_{A}$ is a derivative along the horizon.

##  Appendix C Different descriptions of diffeomorphism transformations

Consider a submanifold $N$ embedded in $M$ described by $X^{A}(\xi^{a})$ where
$x^{A}$ denote coordinates of $M$ and $\xi^{a}$ denote coordinates of $N$.
Consider

$G_{AB}\to G_{AB}^{\prime}=G_{AB}+{{\mathcal{L}}}_{\chi}G_{AB}$ (248)

which gives

$h^{\prime}_{ab}=h_{ab}+({\nabla}_{A}\chi_{B}+{\nabla}_{B}\chi_{A})\partial_{a}X^{A}\partial_{b}X^{B}\
.$ (249)

Alternatively, we can take $G_{AB}$ to be fixed while changing the embedding
$X^{A}\to X^{\prime A}=X^{A}+\chi^{A}$ under which

$\displaystyle
h^{\prime}_{ab}=h_{ab}+G_{AB}(\partial_{a}\chi^{A}\partial_{b}X^{B}+\partial_{a}X^{A}\partial_{b}\chi^{B})+\partial_{C}G_{AB}\chi^{C}\partial_{a}X^{\prime
A}\partial_{b}X^{\prime B}\,,$ (250)

which is equivalent to (249).

Now consider a deformation of $N$ by taking $X^{A}(\xi)\to X^{\prime
A}=X^{A}+{{\delta}}X^{A}$. The restriction $\phi(\xi)$ of a scalar function
$\Phi(x)$ to $N$ now changes to

$\phi^{\prime}(\xi)=\Phi(X^{\prime A}(\xi))\,,$ (251)

and the induced metric on $N$ now has the form

$h^{\prime}_{ab}d\xi^{a}d\xi^{b}=G_{AB}(X^{\prime})\partial_{a}X^{\prime
A}\partial_{b}X^{\prime B}\ .$ (252)

Alternatively, we can keep the embedding $X^{A}$ fixed and consider a
diffeomorphism transformation on $\Phi$ and $G_{AB}$

$\displaystyle\phi^{\prime}(\xi)=\Phi^{\prime}(X(\xi)),\quad
h^{\prime}_{ab}d\xi^{a}d\xi^{b}=G^{\prime}_{\mu\nu}(X)\partial_{a}X^{A}\partial_{b}X^{B},$
(253) $\displaystyle\Phi^{\prime}(x)=\Phi(x^{\prime}),\quad
G^{\prime}_{AB}(x)=G_{CD}(x^{\prime})\partial_{A}x^{\prime
C}\partial_{B}x^{\prime D}\ .$ (254)

The two descriptions are completely equivalent.

## References

  * Thorne  _et al._ [1986] K. Thorne, K. Thorne, R. Price, and D. MacDonald,  _Black Holes: The Membrane Paradigm_, The Silliman Memorial Lectures Series (Yale University Press, 1986).
  * Koga [2001] J.-i. Koga, Asymptotic symmetries on Killing horizons, Phys. Rev. D 64, 124012 (2001), arXiv:0107096 [gr-qc] .
  * Hotta  _et al._ [2001] M. Hotta, K. Sasaki, and T. Sasaki, Diffeomorphism on horizon as an asymptotic isometry of Schwarzschild black hole, Class. Quant. Grav. 18, 1823 (2001), arXiv:gr-qc/0011043 .
  * Hotta [2002] M. Hotta, Holographic charge excitations on horizontal boundary, Phys. Rev. D 66, 124021 (2002), arXiv:hep-th/0206222 .
  * Donnay  _et al._ [2016a] L. Donnay, G. Giribet, H. A. Gonzalez, and M. Pino, Supertranslations and Superrotations at the Black Hole Horizon, Phys. Rev. Lett. 116, 091101 (2016a), arXiv:1511.08687 [hep-th] .
  * Donnay  _et al._ [2016b] L. Donnay, G. Giribet, H. A. González, and M. Pino, Extended Symmetries at the Black Hole Horizon, JHEP 09, 100, arXiv:1607.05703 [hep-th] .
  * Eling and Oz [2016] C. Eling and Y. Oz, On the Membrane Paradigm and Spontaneous Breaking of Horizon BMS Symmetries, JHEP 07, 065, arXiv:1605.00183 [hep-th] .
  * Eling [2017] C. Eling, Spontaneously Broken Asymptotic Symmetries and an Effective Action for Horizon Dynamics, JHEP 02, 052, arXiv:1611.10214 [hep-th] .
  * Penna [2017] R. F. Penna, Near-horizon BMS symmetries as fluid symmetries, JHEP 10, 049, arXiv:1703.07382 [hep-th] .
  * Donnay and Marteau [2019] L. Donnay and C. Marteau, Carrollian Physics at the Black Hole Horizon, Class. Quant. Grav. 36, 165002 (2019), arXiv:1903.09654 [hep-th] .
  * Donnelly  _et al._ [2021] W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, Gravitational edge modes, coadjoint orbits, and hydrodynamics, JHEP 09, 008, arXiv:2012.10367 [hep-th] .
  * Chandrasekaran and Speranza [2021] V. Chandrasekaran and A. J. Speranza, Anomalies in gravitational charge algebras of null boundaries and black hole entropy, JHEP 01, 137, arXiv:2009.10739 [hep-th] .
  * Pasterski and Verlinde [2021] S. Pasterski and H. Verlinde, HPS meets AMPS: how soft hair dissolves the firewall, JHEP 09, 099, arXiv:2012.03850 [hep-th] .
  * Marjieh  _et al._ [2022] R. Marjieh, N. Pinzani-Fokeeva, B. Tavor, and A. Yarom, Black Hole Supertranslations and Hydrodynamic Enstrophy, Phys. Rev. Lett. 128, 241602 (2022), arXiv:2111.00544 [hep-th] .
  * Carlip [1999] S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82, 2828 (1999), arXiv:hep-th/9812013 .
  * Bagchi  _et al._ [2013] A. Bagchi, S. Detournay, R. Fareghbal, and J. Simón, Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett. 110, 141302 (2013), arXiv:1208.4372 [hep-th] .
  * Hawking  _et al._ [2016] S. W. Hawking, M. J. Perry, and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116, 231301 (2016), arXiv:1601.00921 [hep-th] .
  * Donnelly and Freidel [2016] W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09, 102, arXiv:1601.04744 [hep-th] .
  * Afshar  _et al._ [2016a] 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. D 93, 101503 (2016a), arXiv:1603.04824 [hep-th] .
  * Carlip [2018] S. Carlip, Black Hole Entropy from Bondi-Metzner-Sachs Symmetry at the Horizon, Phys. Rev. Lett. 120, 101301 (2018), arXiv:1702.04439 [gr-qc] .
  * Haco  _et al._ [2018] S. Haco, S. W. Hawking, M. J. Perry, and A. Strominger, Black Hole Entropy and Soft Hair, JHEP 12, 098, arXiv:1810.01847 [hep-th] .
  * Chandrasekaran  _et al._ [2018] V. Chandrasekaran, E. E. Flanagan, and K. Prabhu, Symmetries and charges of general relativity at null boundaries, JHEP 11, 125, arXiv:1807.11499 [hep-th] .
  * Haehl  _et al._ [2016a] F. M. Haehl, R. Loganayagam, and M. Rangamani, The Fluid Manifesto: Emergent symmetries, hydrodynamics, and black holes, JHEP 01, 184, arXiv:1510.02494 [hep-th] .
  * Crossley  _et al._ [2017] M. Crossley, P. Glorioso, and H. Liu, Effective field theory of dissipative fluids, JHEP 09, 095, arXiv:1511.03646 [hep-th] .
  * Haehl  _et al._ [2016b] F. M. Haehl, R. Loganayagam, and M. Rangamani, Topological sigma models & dissipative hydrodynamics, JHEP 04, 039, arXiv:1511.07809 [hep-th] .
  * Jensen  _et al._ [2018] K. Jensen, N. Pinzani-Fokeeva, and A. Yarom, Dissipative hydrodynamics in superspace, JHEP 09, 127, arXiv:1701.07436 [hep-th] .
  * Glorioso  _et al._ [2017] P. Glorioso, M. Crossley, and H. Liu, Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current, JHEP 09, 096, arXiv:1701.07817 [hep-th] .
  * Blake  _et al._ [2018a] M. Blake, H. Lee, and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10, 127, arXiv:1801.00010 [hep-th] .
  * Blake and Liu [2021] M. Blake and H. Liu, On systems of maximal quantum chaos, JHEP 05, 229, arXiv:2102.11294 [hep-th] .
  * Nickel and Son [2011] D. Nickel and D. T. Son, Deconstructing holographic liquids, New J. Phys. 13, 075010 (2011), arXiv:1009.3094 [hep-th] .
  * Crossley  _et al._ [2016] M. Crossley, P. Glorioso, H. Liu, and Y. Wang, Off-shell hydrodynamics from holography, JHEP 02, 124, arXiv:1504.07611 [hep-th] .
  * de Boer  _et al._ [2015] J. de Boer, M. P. Heller, and N. Pinzani-Fokeeva, Effective actions for relativistic fluids from holography, JHEP 08, 086, arXiv:1504.07616 [hep-th] .
  * Shenker and Stanford [2014a] S. H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03, 067, arXiv:1306.0622 [hep-th] .
  * Roberts  _et al._ [2015] D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, JHEP 03, 051, arXiv:1409.8180 [hep-th] .
  * Shenker and Stanford [2015] S. H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05, 132, arXiv:1412.6087 [hep-th] .
  * Kitaev [2015] A. Kitaev, Talk at KITP (2015), http://online.kitp.ucsb.edu/online/entangled15/kitaev/ .
  * Polchinski and Rosenhaus [2016] J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04, 001, arXiv:1601.06768 [hep-th] .
  * Maldacena and Stanford [2016] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016), arXiv:1604.07818 [hep-th] .
  * Kitaev and Suh [2018] A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05, 183, arXiv:1711.08467 [hep-th] .
  * Grozdanov  _et al._ [2018] S. Grozdanov, K. Schalm, and V. Scopelliti, Black hole scrambling from hydrodynamics, Phys. Rev. Lett. 120, 231601 (2018), arXiv:1710.00921 [hep-th] .
  * Blake  _et al._ [2018b] M. Blake, R. A. Davison, S. Grozdanov, and H. Liu, Many-body chaos and energy dynamics in holography, JHEP 10, 035, arXiv:1809.01169 [hep-th] .
  * Afshar  _et al._ [2016b] H. Afshar, S. Detournay, D. Grumiller, and B. Oblak, Near-Horizon Geometry and Warped Conformal Symmetry, JHEP 03, 187, arXiv:1512.08233 [hep-th] .
  * Lin  _et al._ [2019] H. W. Lin, J. Maldacena, and Y. Zhao, Symmetries Near the Horizon, JHEP 08, 049, arXiv:1904.12820 [hep-th] .
  * Liu and Glorioso [2018] H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoS TASI2017, 008 (2018), arXiv:1805.09331 [hep-th] .
  * Bhattacharyya  _et al._ [2008] S. Bhattacharyya, V. E. Hubeny, R. Loganayagam, G. Mandal, S. Minwalla, T. Morita, M. Rangamani, and H. S. Reall, Local Fluid Dynamical Entropy from Gravity, JHEP 06, 055, arXiv:0803.2526 [hep-th] .
  * Shenker and Stanford [2014b] S. H. Shenker and D. Stanford, Multiple shocks, JHEP 2014 (12), arXiv:1312.3296 [hep-th] .
  * Sfetsos [1995] K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436, 721 (1995), arXiv:hep-th/9408169 .
  * Blake  _et al._ [2020] M. Blake, R. A. Davison, and D. Vegh, Horizon constraints on holographic Green’s functions, JHEP 01, 077, arXiv:1904.12883 [hep-th] .
  * Bredberg  _et al._ [2011] I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, Wilsonian Approach to Fluid/Gravity Duality, JHEP 03, 141, arXiv:1006.1902 [hep-th] .
  * Bredberg  _et al._ [2012] I. Bredberg, C. Keeler, V. Lysov, and A. Strominger, From Navier-Stokes To Einstein, JHEP 07, 146, arXiv:1101.2451 [hep-th] .
  * Compere  _et al._ [2011] G. Compere, P. McFadden, K. Skenderis, and M. Taylor, The Holographic fluid dual to vacuum Einstein gravity, JHEP 07, 050, arXiv:1103.3022 [hep-th] .
  * Eling  _et al._ [2012] C. Eling, G. Chirco, and S. Liberati, Hydrodynamics and viscosity in the Rindler spacetime, AIP Conf. Proc. 1458, 69 (2012).
  * Compere  _et al._ [2012] G. Compere, P. McFadden, K. Skenderis, and M. Taylor, The relativistic fluid dual to vacuum Einstein gravity, JHEP 03, 076, arXiv:1201.2678 [hep-th] .
  * Pinzani-Fokeeva and Taylor [2015] N. Pinzani-Fokeeva and M. Taylor, Towards a general fluid/gravity correspondence, Phys. Rev. D 91, 044001 (2015), arXiv:1401.5975 [hep-th] .
  * Glorioso and Liu [2016] P. Glorioso and H. Liu, The second law of thermodynamics from symmetry and unitarity (2016), arXiv:1612.07705 [hep-th] .
  * Bu  _et al._ [2022] Y. Bu, X. Sun, and B. Zhang, Holographic Schwinger-Keldysh field theory of SU(2) diffusion, JHEP 08, 223, arXiv:2205.00195 [hep-th] .
  * Wald and Zoupas [2000] R. M. Wald and A. Zoupas, A General definition of ’conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61, 084027 (2000), arXiv:gr-qc/9911095 .
  * Barnich and Brandt [2002] G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633, 3 (2002), arXiv:hep-th/0111246 .
  * Ciambelli [2023] L. Ciambelli, From Asymptotic Symmetries to the Corner Proposal, PoS Modave2022, 002 (2023), arXiv:2212.13644 [hep-th] .
  * Hartong [2015] J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08, 069, arXiv:1505.05011 [hep-th] .