††institutetext: School of Mathematical Sciences,
Highfield, University of Southampton,
SO17 1BJ Southampton, UK

# Holographic reconstruction of flat spacetime

Zezhuang Hao<EMAIL_ADDRESS>
###### Abstract

The flat/CFT dictionary between the bulk gravitational theory and boundary
conformal field theory is systematically developed in this paper.
Asymptotically flat spacetime is built up by asymptotically AdS hyperboloid
slices in terms of Fefferman Graham coordinates together with soft modes
propagating between different slices near the null boundary. Then we construct
the flat holography dictionary based on studying Einstein equation at zero and
first order and it turns out that these correspond to the description of hard
and soft sector for the field theory from the boundary point of view. The
explicit expression for energy-stress tensor is also determined by performing
holographic renormalisation on the Einstein Hilbert action. By studying the
anomalies of the energy-stress tensor, we obtain the leading and subleading
contribution to the central charge. Einstein equations in the bulk are related
to the Ward identities of the boundary theory and we find that the boundary
CFT energy-stress tensor is not conserved due to the existence of radiative
soft modes which will generate the energy flow through the null boundary.

##  1 Introduction

The AdS/CFT correspondence Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc
has been brought up for long time while there are lots of attempts
deHaro:2000wj ; deBoer:2003vf ; Mann:2005yr ; Costa:2012fm ;
Strominger:2017zoo ; Ball:2019atb ; raclariu2021lectures ; Pasterski:2021rjz ;
Nguyen:2021ydb  aiming to construct the duality relation between the quantum
gravity theory with (asymptotically) flat background and the boundary
conformal field theories. It is not clear whether the flat/CFT could work in
the same way as AdS/CFT or not until recently the construction of flat/CFT
dictionary from the bottom-up point of view Hao:2023wln . Basic principles and
underlying physical interpretations for flat holography dictionary are
discussed using the scalar field as an example. Here, following the same
spirit, we will establish the dictionary between the $d+2$ dimensional bulk
theories including gravity and the $d$ dimensional boundary conformal field
theories on the celestial sphere thus make the construction of flat/CFT
duality more complete and concrete.

To study the gravitational theory on asymptotically flat background in a
formal way, the first step is to specify the definition of asymptotic
flatness. The general guideline for defining asymptotic flatness is that it
must make the spacetime close enough to flat case while the deformations
should at the same time contain enough non trivial physical contents for us to
investigate. For example, the well studied Bondi gauge Bondi:1962px ;
Sachs:1962wk ; Sachs:1962zza  in gravity literature. To establish the flat/CFT
dictionary, one also needs to construct the map of data between the bulk and
boundary but there is no clue on how to decompose the bulk data in Bondi
gauge. Noting that Minkowski spacetime is foliated by AdS hyperboloids and the
specific map of data for AdS/CFT is clearly studied in terms of Fefferman
Graham coordinates AST_1985__S131__95_0 ; Graham:1999jg ; Fefferman:2007rka ;
Henningson:1998gx ; deHaro:2000vlm ; Balasubramanian:1999re , therefore in
this paper we choose to define the asymptotically flat spacetime in terms of
Fefferman Graham like coordinates written as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-d\tau^{2}+\tau^{2}\left(\frac{d^{2}\rho}{4\rho^{2}}+\rho
g_{ij}(\rho,x)dx^{i}dx^{j}\right)$ (1)
$\displaystyle+\frac{\tau}{\rho^{2}}m(\rho,x)d^{2}\rho+\tau\rho\;\sigma_{ij}(\rho,x)dx^{i}dx^{j}+\tau
A_{i}(\rho,x)d\rho dx^{i}+\cdots,$

where the first line is the leading contribution to the spacetime coming from
the asymptotic AdS slices and the second line is the subleading contribution
for large $\tau$. We will see that most of the non trivial physical results in
addition to the AdS/CFT duality would come from the existence of such
subleading sector. In section 2, we will explore such gauge in a careful way
by determining the asymptotic symmetries and solving Einstein equation at
different order of $\tau$ and $\rho$. The strategy here is that we choose to
expand the parameters
$g_{ij}(\rho,x),m(\rho,x),\sigma_{ij}(\rho,x),A_{i}(\rho,x)$ in terms of
$1/\rho$ and determine the constraints between the coefficients
$g^{(2k)}(x),m^{(2k)}(x),\sigma^{(2k)}(x),A_{i}^{(2k)}(x)$ by solving the
Einstein equations $R_{\mu\nu}=0$ at the zero and first order of $1/\tau$.

After the study of asymptotically flat spacetime, for $d=2$, we propose the
flat/CFT dictionary as

${\rm exp}\;\big{(}iS_{{\rm gr,ren}}[G]\big{)}=\Big{\langle}{\rm
exp}\;\frac{1}{2}\int_{S^{2}}d^{2}x\sqrt{\bar{G}}\;\bar{G}^{ij}\;T_{ij}\Big{\rangle}$
(2)

where $G$ and $\bar{G}$ are the bulk metric and the background metric for the
CFT, respectively. $T_{ij}$ is the energy-stress tensor of the boundary CFT.
To make such dictionary well defined and work the same way as the AdS/CFT
dictionary, one further needs to perform proper renormalisation procedure on
the bulk gravitational action $S_{\rm gr}[G]$ making it finite and to specify
the exact map of data between two sides. Here the exact map means that given
the bulk metric we should be able determine boundary data $\bar{G}$ and
$T_{ij}$ or vice versa. These are two main obstacles during the development of
flat holography dictionary and we will discuss them in section 3.

In the context of AdS/CFT, one needs to perform holographic renormalisation in
order to obtain the finite renormalised action $S_{\rm gr,ren}[G]$. The
infinity coming from the integral over the whole spacetime is treated as the
IR divergence and is regulated by choosing the AdS spatial radius $\rho$ as
the IR cut off. Here for flat spacetime, we have one more timelike non-compact
direction labeled by $\tau$ and we simply choose to impose a bound $L$ on it,
i.e $\tau\in[0,L]$. After performing the holographic renormalisation in the
given interval $[0,L]$, we obtain the renormalised gravitational action and
then further propose the map of data between bulk and boundary as

$\bar{G}_{ij}=g^{(0)}_{ij}+\frac{1}{L}\sigma_{ij}^{(0)}+\cdots,$ (3)

where $\bar{G}_{ij}=g^{(0)}_{ij}$ is originally the AdS/CFT map and now we are
taking the soft sector into consideration treating $L$ as the energy cut-off
of the boundary CFT. Using the flat/CFT dictionary, we obtain a specific
expression for the energy stress tensor $\langle T_{ij}\rangle$, given by

$\displaystyle\langle T_{ij}\rangle=\frac{iL^{2}}{16\pi
G_{N}}(g_{(2)ij}-g_{(0)ij}{\rm Tr}g_{(2)})-\frac{iL}{8\pi G_{N}}\Big{(}({\rm
Tr}\;\sigma^{(2)}-\sigma^{(0)}_{kl}\;g_{(2)}^{kl})\;g^{(0)}_{ij}$ (4)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+{\rm
Tr}g^{(2)}\sigma_{ij}^{(0)}-\sigma^{(2)}_{ij}+2m^{(0)}(g^{(2)}_{ij}-g^{(0)}_{ij}{\rm
Tr}g^{(2)})\Big{)}+\cdots.$

Moreover, by considering the anomalies of the energy-stress tensor $\langle
T^{i}_{i}\rangle$, the central charge is then determined to be

$c=\frac{i3L^{2}}{4G_{N}}+\frac{iML3(\alpha-2)}{2G_{N}}$ (5)

where $M=m^{(0)}$ and $\alpha$ are constants characterising the behaviour of
asymptotically flat spacetime. The behaviour of the leading term is already
argued in the work Cheung:2016iub ; Pasterski:2022lsl ; Ogawa:2022fhy  and
here we determine its value in a precise way. With the help of flat/CFT
dictionary, we can in principle determine all the subleading contributions and
in this paper we just present the first order term.

The Einstein equations will impose constraints on the parameters in the metric
and those constrains on the boundary will correspond to the Ward identity of
the stress tensor. The story for AdS/CFT or for the hard sector is that the
bulk equation of motion implies that the conservation of energy-stress tensor
$\nabla^{j}\langle T_{ij}\rangle=0$. By checking the Einstein equations at the
first order, we find that it will not give us the conservation of the soft
energy-stress tensor while we have

$\bar{\nabla}^{j}\langle T_{ij}\rangle=\frac{iL}{16\pi
G_{N}}(8\nabla_{i}m^{(2)}+4M\nabla_{i}{\rm
Tr}g^{(2)}-\sigma^{mk}_{(0)}\nabla_{m}g^{(2)}_{ki}),$ (6)

where ${\bar{\nabla}}_{i},\nabla_{i}$ are the covariant derivative with
respect to $\bar{G}_{ij},g^{(0)}_{ij}$, respectively. Thus one can interpret
the soft modes as the radiation modes which generate flow of energy at the
null boundary leading to a non conserved energy-stress tensor from the
boundary point of view.

##  2 Fefferman Graham Gauge

In this section, we will start from the coordinate for asymptotically flat
spacetime then recast it to the asymptotically AdS form. For the spacetime of
dimension $d+2$, we first introduce the standard coordinate $X^{\mu}$ for
$\mu=0,1,\cdots,d+1$ in which the flat metric takes the form $\eta_{00}=-1$,
$\eta_{11}=\cdots\eta_{d+1,d+1}=1$. The Euclidean AdS of dimension $d+1$ can
be regarded as the hyperboloid embedded in the $d+2$ flat spacetime given by
the relation 111Such patch is called Milne wedge for Minkowski while one can
study Rindler wedge by the analytic continuation of the radius
$\tau\rightarrow i\tau$. The Rindler wedge is sliced by dS hypersurfaces.

$-(X^{0})^{2}+(X^{1})^{2}+\cdots+(X^{d+1})^{2}=-\tau^{2},$ (7)

in which $\tau\geq 0$ is the radius of the AdS surface.

Motivated by such foliation, we now choose to write the asymptotically flat
spacetime given by the metric $G(X)$ into the form

$ds^{2}=G_{\mu\nu}(X)dX^{\mu}dX^{\nu}=-d\tau^{2}+\tau^{2}\hat{G}_{ab}(\tau,y)dy^{a}dy^{b},$

where $a,b=1,\cdots,d+1$ and we have fixed the gauge in order to make the
$d\tau dy^{a}$ term vanish. One can always manage to find such gauge by
performing local diffeomorphism transformation. For flat spacetime, the metric
$\hat{G}_{ab}$ will be independent of $\tau$ and reduced to the metric for
$AdS_{d+1}$ described by $y$ while it will become asymptotically AdS when the
spacetime is near flat. At the null boundary, we can expand the metric
$\hat{G}_{ab}(\tau,y)$ around $1/\tau=0$ written as

$\hat{G}_{ab}(\tau,y)=\hat{G}^{0}_{ab}(y)+\hat{G}^{1}_{ab}(y)\frac{1}{\tau}+\cdots=\hat{G}^{0}_{ab}+h_{ab}(\tau,y),$
(8)

in which $\hat{G}^{0}_{ab}(y)$ is the metric for $d+1$ dimensional
asymptotically AdS while $\hat{G}^{n}_{ab}(y)$ are higher order corrections
and they together form a complete description of asymptotically flat
spacetime. We put all the higher order corrections into the term $h_{ab}$. In
this article, to study asymptotically AdS spacetime $G^{0}_{ab}$, we choose to
use Fefferman and Graham coordinates written as

$\displaystyle\hat{G}^{0}_{ab}dy^{a}dy^{b}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}(dr^{2}+g_{ij}(r,x)dx^{i}dx^{j})$ (9)
$\displaystyle=$
$\displaystyle\frac{d\rho^{2}}{4\rho^{2}}+\rho\;g_{ij}(x,\rho)dx^{i}dx^{j},$
(10)

in which $y=(r,x^{i})$ for $i=1,\cdots d$ are coordinates on the AdS
hyperboloid and $\rho=1/r^{2}$. The $d$ dimensional metric $g_{ij}(x,\rho)$
has been extensively studied in the AdS/CFT literature and the main method is
that one can organise the data by doing expansion of the order $\rho$ for
$\rho\rightarrow\infty$. The leading term will contribute to the AdS spacetime
while lower order terms are asymptotically AdS corrections.

###  2.1 Asymptotic Symmetries

To illustrate the spacetime structure introduced above in a more precise way,
for simplicity, we take $d=2$ as an example therefore the spacetime becomes
asymptotically Minkowski. In this case, the metric is then given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-d\tau^{2}+\tau^{2}\left(\frac{d^{2}\rho}{4\rho^{2}}+\rho\;dzd\bar{z}+\cdots\right)$
(11)
$\displaystyle+\frac{\tau}{\rho^{2}}m(\rho,z,\bar{z})d^{2}\rho+\tau\rho\;\sigma_{z\bar{z}}(\rho,z,\bar{z})dzd\bar{z}+2\tau
A_{z}(\rho,z,\bar{z})d\rho dz+\tau\rho\;\sigma_{zz}(\rho,z,\bar{z})dzdz$
$\displaystyle+\;{\rm c.c}\;+\;\cdots,$

in which the Minkowski space is written in terms of Milne coordinates
$(\tau,\rho,z,\bar{z})$ at the first line while the first order deviations
with respect to $\tau$ are described by the functions $m(\rho,z,\bar{z})$,
$A_{z}(\rho,z,\bar{z})$, $\sigma_{z\bar{z}}(\rho,z,\bar{z})$,
$\sigma_{zz}(\rho,z,\bar{z})$ together with the corresponding complex
conjugates. The dots in the first line represent the asymptotically AdS
deformation while the dots in the third line represent higher order
contributions according to $1/\tau$. Written in the form of $G_{\mu\nu}$, the
metric are

$G_{\rho\rho}=\frac{\tau^{2}}{4\rho^{2}}+\frac{\tau}{\rho^{2}}m+\mathcal{O}(\tau^{0})$
(12)

for $\rho\rho$ component and for $z\bar{z}$ component we have

$G_{z\bar{z}}=\frac{\tau^{2}\rho}{2}+\tau\rho\;\sigma_{z\bar{z}}+\mathcal{O}(\tau^{0}).$
(13)

At leading order, $G_{\rho z}$ $G_{zz}$ will vanish while for asymptotic flat
spacetime there could be subleading contribution like

$G_{\rho z}=\tau A_{z}+\mathcal{O}(\tau^{0})\qquad G_{\rho\bar{z}}=\tau
A_{\bar{z}}+\mathcal{O}(\tau^{0})$ (14)

and

$G_{zz}=\tau\rho\;\sigma_{zz}+\mathcal{O}(\tau^{0})\qquad
G_{\bar{z}\bar{z}}=\tau\rho\;\sigma_{\bar{z}\bar{z}}+\mathcal{O}(\tau^{0}).$
(15)

Now, we will study asymptotic symmetry which is the diffeomorphism that
preserves the metric of the form shown in (11). Given the Killing vector
$\xi(\tau,\rho,z,\bar{z})$, the variation of the metric $\delta G_{\tau\rho}$
is then deduced to be

$\displaystyle\delta G_{\tau\rho}$ $\displaystyle=$
$\displaystyle-\partial_{\rho}\xi^{\tau}+\frac{\tau^{2}}{4\rho^{2}}\partial_{\tau}\xi^{\rho}$
(16) $\displaystyle+\frac{\tau}{\rho^{2}}m\partial_{\tau}\xi^{\rho}+\tau
A_{z}\partial_{\tau}\xi^{z}+\tau
A_{\bar{z}}\partial_{\tau}\xi^{\bar{z}}+\mathcal{O}(\tau^{0})$

and for $\delta G_{\tau z}$ we have

$\displaystyle\delta G_{\tau z}$ $\displaystyle=$
$\displaystyle-\partial_{z}\xi^{\tau}+\frac{\tau^{2}\rho}{2}\partial_{\tau}\xi^{\bar{z}}$
(17) $\displaystyle+\tau
A_{z}\partial_{\tau}\xi^{\rho}+\tau\rho\;\sigma_{zz}\partial_{\tau}\xi^{z}+\tau\rho\;\sigma_{z\bar{z}}\partial_{\tau}\xi^{\bar{z}}+\mathcal{O}(\tau^{0}),$

in which we have shown the variation up to $\mathcal{O}(\frac{1}{\tau})$ and
higher order terms are omitted. For other terms we have

$\delta
G_{z\bar{z}}=\frac{\tau^{2}}{2}\xi^{\rho}+\frac{\tau^{2}\rho}{2}(\partial_{z}\xi^{z}+\partial_{\bar{z}}\xi^{\bar{z}})+\mathcal{O}(\tau),$
(18) $\delta
G_{zz}=\tau^{2}\rho\;\;\partial_{z}\xi^{\bar{z}}+\mathcal{O}(\tau),$ (19)
$\delta G_{\rho
z}=\frac{\tau^{2}}{4\rho^{2}}\partial_{z}\xi^{\rho}+\frac{\tau^{2}\rho}{2}\partial_{\rho}\xi^{\bar{z}}+\mathcal{O}(\tau),$
(20) $\delta
G_{\rho\rho}=\frac{\tau^{2}}{2\rho^{3}}(\rho\partial_{\rho}\xi^{\rho}-\xi^{\rho})+\mathcal{O}(\tau).$
(21)

To find out the asymptotic symmetries, one need to determine the Killing
vector $\xi$ which can be expanded as

$\xi^{\mu}(\tau,\rho,z,\bar{z})=\xi^{\mu}_{0}(\rho,z,\bar{z})+\frac{1}{\tau}\xi^{\mu}_{1}(\rho,z,\bar{z})+\cdots$
(22)

where $\xi^{\mu}_{k}$ for $k\in N$ are coefficients associated to the term
$1/\tau^{k}$. For $\delta G_{\tau\rho}$ and $\delta G_{\tau z}$ they have to
vanish since we are working in the gauge $G_{\tau\rho}=G_{\tau z}=0$. From the
expression (16) and (17), at the order of $\mathcal{O}(1)$ we obtain

$\partial_{z}\xi^{\tau}_{0}+\frac{\rho}{2}\xi^{\bar{z}}_{1}=0\qquad\partial_{\rho}\xi^{\tau}_{0}+\frac{1}{4\rho^{2}}\xi^{\rho}_{1}=0.$
(23)

Moreover, we have

$A_{z}\xi^{\rho}_{1}+\rho\;\sigma_{zz}\xi^{z}_{1}+\rho\;\sigma_{\bar{z}\bar{z}}\xi^{\bar{z}}_{1}=0$
(24)

and

$m\;\xi^{\rho}_{1}+\rho^{2}A_{z}\xi^{z}_{1}+\rho^{2}A_{\bar{z}}\xi^{\bar{z}}_{1}=0$
(25)

when considering the $\delta G_{\tau\rho}=\delta G_{\tau z}=0$ of the order
$\tau^{2}$. For $\delta G_{zz}$ and $\delta G_{\bar{z}\bar{z}}$, the
contribution at the order $1/\tau$ should vanish in order to preserve the
condition of asymptotic flatness therefore we have

$\partial_{z}\xi^{\bar{z}}_{0}=\partial_{\bar{z}}\xi^{z}_{0}=0.$ (26)

For the same reason we have

$\rho\partial_{\rho}\xi^{\rho}_{0}-\xi^{\rho}_{0}=0$ (27)

when considering the component $G_{\rho\rho}$ in the (21). Then the killing
vector can be written as

$\displaystyle\xi^{\tau}=\chi(z,\bar{z}),$ (28) $\displaystyle\xi^{\rho}=0,$
(29) $\displaystyle\xi^{z}=Y^{z}(z,\bar{z}),$ (30)
$\displaystyle\xi^{\bar{z}}=Y^{\bar{z}}(z,\bar{z}),$ (31)

where $\chi$ is an arbitrary function on $z,\bar{z}$ and we have
$\partial_{z}Y^{\bar{z}}=\partial_{\bar{z}}Y^{z}=0$ therefore the BMS group
Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza  is recovered at leading order.
The transformation of the spacetime metric under such symmetry group is then
given by

$\displaystyle\delta g_{z\bar{z}}$ $\displaystyle=$ $\displaystyle
g_{z\bar{z}}(\partial_{z}Y^{z}+\partial_{\bar{z}}Y^{\bar{z}})+Y^{z}\partial_{z}g_{z\bar{z}}+Y^{\bar{z}}\partial_{\bar{z}}g_{z\bar{z}}$
(32) $\displaystyle\delta g_{zz}$ $\displaystyle=$ $\displaystyle
2g_{zz}\;\partial_{z}Y^{z}+Y^{z}\partial_{z}g_{zz}+Y^{\bar{z}}\partial_{\bar{z}}g_{zz}$
(33) $\displaystyle\delta A_{z}$ $\displaystyle=$ $\displaystyle
Y^{z}\partial_{z}A_{z}+Y^{\bar{z}}\partial_{\bar{z}}A_{z}+A_{z}\partial_{z}Y^{z}$
(34) $\displaystyle\delta\sigma_{zz}$ $\displaystyle=$ $\displaystyle
Y^{z}\partial_{z}\sigma_{zz}+Y^{\bar{z}}\partial_{\bar{z}}\sigma_{zz}+2\sigma_{zz}\partial_{z}Y^{z}+2\chi
g_{zz}$ (35) $\displaystyle\delta\sigma_{z\bar{z}}$ $\displaystyle=$
$\displaystyle
Y^{z}\partial_{z}\sigma_{z\bar{z}}+Y^{\bar{z}}\partial_{\bar{z}}\sigma_{z\bar{z}}+(\partial_{z}Y^{z}+\partial_{\bar{z}}Y^{\bar{z}})\sigma_{z\bar{z}}+2\chi
g_{z\bar{z}}$ (36)

where we have used the complex metric $g_{z\bar{z}}$ for the hard sector. From
above translation rules, one can see that the superrotation part described by
$Y^{z}$, $Y^{\bar{z}}$ will act on the leading and subleading part of the
metric while the supertranslation part described by $\chi$ will only act on
the subleading part.

###  2.2 Equation of Motion

Now we turn to study the dynamics of the gravitational system. The general
$d+2$ dimensional asymptotic flat spacetime is described by the coordinate

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-d\tau^{2}+\tau^{2}\left(\frac{d^{2}\rho}{4\rho^{2}}+\rho
g_{ij}(\rho,x)dx^{i}dx^{j}\right)$ (37)
$\displaystyle+\frac{\tau}{\rho^{2}}m(\rho,x)d^{2}\rho+\tau\rho\;\sigma_{ij}(\rho,x)dx^{i}dx^{j}+\tau
A_{i}(\rho,x)d\rho dx^{i}+\cdots$

in terms of the real coordinates $(\tau,\rho,x^{i})$ for $i=1,\dots,d$. In the
first line, the metric is mainly built up from asymptotically AdS
hyperboloids. It describes the hard sector of the gravitational theory and
manifests the superrotation symmetry. The second line is the subleading
contribution to the asymptotically flat spacetime according to $1/\tau$. It
describes the soft modes coming from the radiation and manifests the
superrotation and supertranslation symmetry.

In this section, we are going to determine the constraints on the metric
$g_{ij}$ and the soft parameters by checking the Einstein equations
$R_{\mu\nu}=0$ at different orders. Starting with the connections, they are
given by

$\Gamma^{\tau}_{ab}=\tau\hat{G}_{ab}+\frac{1}{2}\tau^{2}\partial_{\tau}\hat{G}_{ab}$
(38)

and

$\displaystyle\Gamma^{a}_{b\tau}=\frac{\delta^{a}_{b}}{\tau}+\frac{1}{2}\hat{G}^{ac}\partial_{\tau}\hat{G}_{cb}$
(39)

while we have $\Gamma^{\tau}_{\tau\tau}=\Gamma^{\tau}_{\tau
a}=\Gamma^{a}_{\tau\tau}$=0 and
$\Gamma_{bc}^{a}=\hat{\Gamma}_{bc}^{a}[\hat{G}]$. In our definition the Ricci
tensor is now given by

$R_{ab}[G]=R^{d+1}_{ab}[\hat{G}]-\Gamma_{ab}^{\tau}\Gamma^{c}_{c\tau}+2\Gamma^{c}_{a\tau}\Gamma^{\tau}_{bc}-\partial_{\tau}\Gamma^{\tau}_{ab},$
(40)

which can be further decomposed into

$\displaystyle R_{ab}[G]$ $\displaystyle=$ $\displaystyle
R^{d+1}_{ab}[\hat{G}]-d\hat{G}_{ab}$ (43)
$\displaystyle-\frac{d+1}{2}\tau\partial_{\tau}\hat{G}_{ab}-\frac{1}{2}\tau^{2}\partial^{2}_{\tau}\hat{G}_{ab}-\frac{\tau}{2}\hat{G}_{ab}\hat{G}^{cd}\partial_{\tau}\hat{G}_{cd}$
$\displaystyle+\frac{1}{2}\tau^{2}\partial_{\tau}\hat{G}_{cb}\partial_{\tau}\hat{G}_{da}\hat{G}^{cd}.$

For $\tau\tau$ component we have

$R_{\tau\tau}[G]=\frac{1}{\tau}\hat{G}^{ab}\partial_{\tau}\hat{G}_{ab}+\frac{1}{2}\partial_{\tau}(\hat{G}^{ab}\partial_{\tau}\hat{G}_{ab})+\frac{1}{4}\hat{G}^{ac}\hat{G}^{bd}\partial\tau\hat{G}_{ab}\partial_{\tau}\hat{G}_{cd}$
(44)

while for $\tau a$ one can deduce that

$R_{a\tau}[G]=\frac{1}{2}\hat{\nabla}_{a}(\hat{G}^{bc}\partial_{\tau}\hat{G}_{bc})-\frac{1}{2}\hat{\nabla}_{b}(\hat{G}^{bc}\partial_{\tau}\hat{G}_{ca}),$
(45)

where the covariant derivative is with respect to the metric $\hat{G}_{ab}$.
In practice, to study the equation of motion at different orders, we choose to
write the Ricci curvature perturbatively according to the expansion (8), which
means we have

$R_{\mu\nu}[G^{0}+h]=R^{0}_{\mu\nu}[G^{0}]+\frac{1}{\tau}R^{1}_{\mu\nu}+\cdots$
(46)

where the zero order mainly comes from the hard sector of the metric described
by $G^{0}$ and the soft modes will go into the first or higher order terms.

####  2.2.1 Zero Order

For convenience, we will denote $\hat{G}^{0}_{ab}(y)$ as $\hat{G}_{ab}(y)$ in
this subsection and consider the equation of motion at leading order of
$1/\tau$. In such case, connections involving $\tau$ component are given by
$\Gamma^{\tau}_{ab}=\tau\hat{G}_{ab},\Gamma^{a}_{b\tau}=\frac{1}{\tau}\delta^{a}_{b}$
and $\Gamma^{\tau}_{\tau\tau}=\Gamma^{\tau}_{\tau a}=\Gamma^{a}_{\tau\tau}=0$.
For the connections not involving $\tau$ component denoted as
$\Gamma^{a}_{bc}$, they are given by the direct $d+1$ dimensional calculation
using the AdS metric $\tau^{2}\hat{G}_{ab}(y)$. Now, to solve the vacuum
Einstein equation with zero cosmology constant

$R_{\mu\nu}[G]=0,$ (47)

we should deduce the Ricci curvature $R$ written as
$R[G]=R^{a}_{a}[G]+R^{\tau}_{\tau}[G]$. One can easily verify that
$R_{\tau\mu}=0$, while for $R_{ab}$ one has

$R_{ab}[G]=R^{(d+1)}_{ab}[\hat{G}]-d\hat{G}_{ab},$ (48)

in which we have introduced the notion $R^{(d+1)}[\hat{G}]$ to denote that the
Ricci curvature induced on the $d+1$ dimensional AdS hyperboloid. Therefore,
the Ricci curvature for near flat spacetime is then deduced to be

$R[G]=R^{(d+1)}[\hat{G}]-{d(d+1)}$ (49)

and the Einstein equation in (47) is equivalent to

$R^{(d+1)}_{ab}[\hat{G}]-\frac{1}{2}R^{(d+1)}[\hat{G}]\;\hat{G}_{ab}=\Lambda\hat{G}_{ab}$
(50)

for

$\Lambda=-\frac{d(d-1)}{2}.$ (51)

We can treat $\Lambda$ as the effective cosmology constant and $\tau$ as the
effective AdS radius since we are recasting the curvature $R[G]$ for $d+2$
dimensional near flat metric $\hat{G}_{\mu\nu}$ into the induced curvature
$R^{(d+1)}[\hat{G}]$ for $d+1$ dimensional asymptotically AdS metric
$\hat{G}_{ab}(y)$.

In terms of Fefferman Graham gauge, the study of equation of motion at zero
order is equivalent to the study of the differential equation for
$g_{ij}(x,\rho)$. The function $g_{ij}(x,\rho)$ are determined by
Graham:1999jg ; Fefferman:2007rka ; Henningson:1998gx ; deHaro:2000vlm

$\displaystyle\rho^{2}(2\rho g^{\prime\prime}_{ij}+4g^{\prime}_{ij}-2\rho
g^{lm}g^{\prime}_{mj}g^{\prime}_{li}+\rho
g^{lm}g^{\prime}_{lm}\;g^{\prime}_{ij})+R_{ij}[g]+(d-2)\rho^{2}g^{\prime}_{ij}+\rho^{2}g^{lm}g^{\prime}_{lm}\;g_{ij}=0$
$\displaystyle\nabla_{i}(\;g^{lm}g^{\prime}_{lm})-\nabla^{j}g^{\prime}_{ij}=0$
(52) $\displaystyle g^{ij}(\rho
g^{\prime\prime}_{ij}+2g^{\prime}_{ij})-\frac{1}{2}\rho
g^{ik}g^{jm}g^{\prime}_{ij}g^{\prime}_{km}=0$

according to the equation (47) or (50) and the covariant derivative is with
respect to the metric $g_{ij}$. A brief study of such equation is shown in
appendix B and here we have $g^{\prime}_{ij}=\partial_{\rho}g_{ij}$. Moreover,
for even $d$, one has the expansion

$g_{ij}(x,\rho)=g^{(0)}_{ij}+\rho^{-1}g^{(2)}_{ij}+\cdots+\rho^{-d/2}g^{(d)}_{ij}+c_{ij}\rho^{-d/2}{\rm
log}\rho+\cdots$ (53)

when $\rho$ goes to infinity. Here, following the convention from previous
literature, the superscript $2k$ in the coefficients $g^{(2k)}_{ij}$ are used
to keep track of the order of $r$. Or equivalently, $k$ is used to keep track
of the order of $\rho$. Coefficients $g^{(2k)}_{ij}$ are uniquely determined
by the lower order terms via checking the equation (52) at the order of
$\rho^{k}$ while such procedure will fail until $2k=d$. In such case, the
equation of motion will only allow us to determine ${\rm Tr}g^{(d)}$ and it
also leaves us freedom to introduce the traceless algorithm term parameterised
by $c_{ij}$. Fox example, by checking the first two equations at leading
order, one can obtain the relation

$R_{ij}[g^{(0)}]=(d-2)g^{(2)}_{ij}+g^{(0)}_{ij}{\rm Tr}g^{(2)}$ (54)

together with

$\nabla_{i}({\rm Tr}g^{(2)})-\nabla^{j}g^{(2)}_{ji}=0$ (55)

where the covariant derivative $\nabla_{i}$ and the trace ${\rm Tr}$ now are
with respective to the metric $g^{(0)}_{ij}$ while we will keep such
convention in the following part of this paper. One can see that
$g^{(2)}_{ij}$ is fully determined as the function of $g^{(0)}_{ij}$ for
$d\neq 2$ while only the trace part is fixed when $d=2$.

####  2.2.2 First Order

We have studied $R^{0}_{\mu\nu}$ in the previous section while here we are
going to deal with $R^{1}_{\mu\nu}$ therefore determine equation of motion at
first order. It is easier to calculate the $R_{\tau\tau}$ and $R_{\tau a}$
components by checking the formula (44) and (45). The $R_{\tau\tau}=0$ will be
trivial at first order while for $R^{1}_{\tau a}$, we have

$R^{1}_{\tau
a}=\frac{1}{2}\hat{\nabla}_{a}(\hat{G}_{0}^{bc}\hat{G}^{1}_{bc})-\frac{1}{2}\hat{\nabla}^{b}\hat{G}_{ab}^{1}$
(56)

where the covariant derivative $\hat{\nabla}_{a}$ is with respect to the
metric $\hat{G}^{0}_{ab}.$ For the $R_{ab}$ components, we have

$R^{1}_{ab}=R^{d+1,1}_{ab}-\frac{d+1}{2}\hat{G}_{ab}^{1}+\frac{1}{2}\hat{G}_{ab}^{1}G^{cd}_{0}G_{cd}^{1}$
(57)

from which one can see that the first order contribution $R^{d+1,1}_{ab}$ of
$R^{d+1}_{ab}$ will also contribute to the first order $R^{1}_{ab}$ therefore
it will make the results more complicated. The strategy here is to determine
the first order term in $R^{d+1}_{ab}$ then add the other terms in (57) which
also contribute at the first order.

Such soft sector is described by the first order term $h_{\mu\nu}$. More
precisely, it is determined by the parameter $m$, $\sigma_{ij}$ and $A_{i}$
once the gauge is fixed. To simplify the calculation further, now we consider
the expansion of parameter $m$, $\sigma_{ij}$ and $A_{i}$ by the order
$1/\rho$. Taking the parameter $m(\rho,x)$ for example, we have

$m(\rho,x)=m^{(0)}(x)+\frac{1}{\rho}m^{(2)}(x)+\cdots,$ (58)

in which $m^{(0)}(x)$ is the leading term while $m^{(2)}(x)$ is the subleading
contribution. $m^{(0)}$, $m^{(2)}$ describe the zero order and first order
contribution to the soft sector according to the spatial radius of AdS
hyperboloid $\rho$. For the parameter $\sigma_{ij}$, $A_{i}$ we adopt similar
convention and the corresponding coefficients are denoted as $A_{i}^{(2k)}$
and $\sigma^{(2k)}_{ij}$.

Therefore following the equation of motion given by the Einstein equation
explicitly showed in the appendix C, we obtain the constraints

${\rm Tr}\;\sigma^{(0)}=4dm^{(0)}$ (59)

by considering $R_{\rho\rho}=R_{\rho\tau}=0$. Moreover we have

$A^{(0)}_{i}=0$ (60)

by checking the equation of motion $R_{i\tau}=0$ at the zero order of
$1/\rho$. For the equation $R_{ij}=0$, again at leading order, we obtain

$\frac{d-1}{2}\sigma^{(0)}_{ij}+\frac{1}{2}g_{ij}^{(0)}\;{\rm
Tr}\;\sigma^{(0)}-2(2d-1)m^{(0)}g_{ij}^{(0)}=0,$ (61)

which is compatible with the constraint (59) after taking the trace on both
sides. One can also obtain the relation

$\nabla^{j}\sigma^{(0)}_{ji}=4\nabla_{i}m^{(0)}$ (62)

after acting the covariant derivative $\nabla^{j}$ on both sides.

Now we consider the Einstein equations at first order of $1/\rho$ in order to
determine $m^{(2)}$, $A^{(2)}_{i}$ and $\sigma^{(2)}_{ij}$. By checking the
equation of motion $R_{i\tau}=0$ at the first order, we have

$2\nabla_{i}m^{(0)}=A^{(2)}_{i}$ (63)

while for $R_{\rho i}=0$ we obtain the relation

$\frac{3d-1}{2}A^{(2)}_{i}-d\nabla_{i}m^{(0)}+\frac{1}{4}\nabla^{j}\sigma^{(0)}_{ij}-\frac{1}{4}\nabla_{i}{\rm
Tr}\sigma^{(0)}=0.$ (64)

Given the above equation, we can deduce that $A^{(2)}_{i}=0$ after using the
relation (59),(62) and (63). This tells us that the parameter $m^{(0)}$ should
be a constant and the zero order coefficient $\sigma_{ij}^{(0)}$ is conserved
with respect to the metric $g^{(0)}_{ij}$, written as

${\rm Tr}\sigma^{(0)}=4dM\qquad\nabla^{j}\sigma^{(0)}_{ij}=0$ (65)

where we have denoted the parameter $m^{(0)}$ as constant $M$. For the
equation of motion $R_{\rho\tau}=0$ and $R_{\rho\rho}=0$, they will give us
the relations

$dm^{(2)}+\frac{1}{4}{\rm Tr}\;\sigma^{(2)}-{\rm Tr}g_{(2)}m^{(0)}=0$ (66)

and

$\frac{d}{2}m^{(2)}+\frac{1}{8}{\rm
Tr}\;{\sigma^{(2)}}-\frac{1}{8}g^{ij}_{(2)}\sigma^{(0)}_{ij}=0.$ (67)

From these two equations one can see that, in order to make $m^{(2)}$ and
$\sigma_{ij}^{(2)}$ solvable, one should further impose constraint on
$g^{ij}_{(2)}\sigma^{(0)}_{ij}$ thus we have

$\displaystyle g^{ij}_{(2)}\sigma^{(0)}_{ij}=4{\rm Tr}g^{(2)}m^{(0)}.$ (68)

To obtain $\sigma^{(2)}_{ij}$, one needs to check $R_{ij}=0$ explicitly. The
equation is more involved and here we just present the result

$\displaystyle\frac{3-d}{2}\sigma^{(2)}_{ij}+(6-4d)m^{(2)}g^{(0)}_{ij}-6g^{(2)}_{ij}m^{(0)}-\frac{1}{2}({\rm
Tr}\sigma^{(2)}-g^{lk}_{(2)}\sigma^{0}_{lk})g^{(0)}_{ij}$ (69)
$\displaystyle\qquad\qquad\qquad\qquad+4{\rm
Tr}g^{(2)}m^{(0)}g^{(0)}_{ij}-{\rm Tr}g^{(2)}\sigma^{(0)}_{ij}+\frac{1}{2}{\rm
Tr}\sigma^{(0)}g^{(2)}_{ij}+\delta R_{ij}=0,$

where $R_{ij}[g+\frac{1}{\tau}\sigma]=R_{ij}[g]+\delta R_{ij}$ 222More
precisely, we have $\delta
R_{ij}=\frac{1}{2}(\nabla^{m}\nabla_{m}\sigma_{ij}+\nabla_{i}\nabla_{j}\sigma^{m}_{m}-\nabla_{k}\nabla_{i}\sigma^{k}_{j}-\nabla_{k}\nabla_{j}\sigma^{k}_{i}).$
and we can see that, given the value of $m^{(2)}$, $\sigma^{(2)}_{ij}$ is
determined by solving the equation. After taking the trace on both sides, we
have

$\displaystyle\frac{3-2d}{2}{\rm Tr}\sigma^{(2)}+(6-4d)dm^{(2)}-6{\rm
Tr}g^{(2)}\;m^{(0)}+\frac{d}{2}g^{lm}_{(2)}\sigma^{(0)}_{lm}+2dm^{0}{\rm
Tr}g^{(2)}=0,$ (70)

from which we will obtain the same relation as (68) after the substitution of
(66) or (67).

From above calculation, we see that the value of $m^{(2k)}$ are related to the
trace of soft metric coefficients ${\rm Tr}\sigma^{(2k)}$. More precisely, by
considering the constraint (59), (66) and (68), one can find that they obey
the more compact relation

$\displaystyle{\rm Tr}^{(2k)}(\sigma-4mg)=0$ (71)

where ${\rm Tr}^{(2k)}$ is defined as the trace over the metric
$g^{(2k)}_{ij}$ and we denote ${\rm Tr}^{(0)}={\rm Tr}$. The relation between
(71) and the Einstein equations is not clear but we expect this is true when
going to the higher order.

Further more, by checking the equation of motion $R_{i\tau}=0$ at the order of
$1/\rho^{2}$, we obtain the relations involving $\nabla^{j}\sigma_{ij}^{(2)}$
written as

$\displaystyle(2-d)A_{i}^{(4)}+2\nabla_{i}m^{(2)}+\frac{1}{2}\nabla_{i}({\rm
Tr}\sigma^{(2)}-g^{kl}_{(2)}\sigma^{(0)}_{kl})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{1}{2}\nabla^{j}\sigma^{(2)}_{ij}+\frac{1}{4}\nabla_{k}g^{mk}_{(2)}\sigma^{(0)}_{mi}+\frac{1}{4}\nabla_{i}g^{mn}_{(2)}\sigma^{(0)}_{mn}=0$
(72)

where we have used the relation (55) and the last two terms come from the
variation of the connection $\delta\Gamma^{i}_{jk}$. It turns out that,
although a little bit tedious, equation (72) will be useful for us to study
Ward identities of the boundary conformal field theory with the help of
flat/CFT dictionary. Moreover, by studying the equation of motion
$R_{i\rho}=0$ at second order of $1/\rho^{2}$ one should be able to determine
$A^{(4)}_{i}$ once $m^{(2)}$ or equivalently ${\rm Tr}\;\sigma^{(2)}$ is
fixed.

##  3 The flat/CFT Dictionary

The Einstein-Hilbert action for the gravitational theory on a four dimensional
asymptotically flat manifold $M$ with boundary $\partial M$ is given by
Gibbons:1976ue

$S_{{\rm gr}}[G]=\frac{1}{16\pi
G_{N}}\left(\int_{M}d^{4}X\sqrt{-G}\;R[G]+\int_{\partial
M}d^{3}X\sqrt{-\gamma}\;2K\right),$ (73)

in which $K$ is the trace of the second fundamental form and $\gamma$ is the
induced metric on the boundary. To evaluate the action, we first choose to use
the equation of motion (47) and set the boundary as the surface with constant
AdS spatial radius $\rho=1/\epsilon$. Then the regulated action is given by

$S_{{\rm gr,reg}}=\frac{1}{16\pi
G_{N}}\int_{0}^{\infty}d\tau\int_{S^{2}}d^{2}x\sqrt{-\gamma}\;2K\Big{|}_{\rho=1/\epsilon}$
(74)

where we have $\gamma_{\tau\tau}=-1,\gamma_{ij}=\tau^{2}\hat{G}_{ij}$.333In
fact, there are three components that belong to the boundary $\partial M$. One
is at $\rho=1/\epsilon$ while the other two are at $\tau=0$ and $\tau=\infty$.
In this paper we focus on the renormalisation of the divergence at
$\rho=1/\epsilon$. For the integral along the surface of constant $\tau$, we
treat them as the assignment of initial and final data. The treatment of the
integral at the constant time surface is equivalent to the procedure that we
fix the initial modes by hand or inserting proper $i\varepsilon$ description
in the path integral. More rigorously, like the treatment for real time
holography Skenderis:2008dg ; Skenderis:2008dh ; Hao:2023ijx , one can choose
to glue an Euclidean cap at $\tau=0$ surface and make divergence cancelled.

Moreover, to calculate the integral over $K$, one should notice the relation
$K_{ij}=\hat{\nabla}_{i}n_{j}$ thus we obtain 444We are abusing the notion
here and $\hat{\nabla}$ means the covariant derivative with respect to the
metric $\hat{G}_{ij}$.

$\int_{0}^{\infty}d\tau\int
d^{2}x\sqrt{-\gamma}\;2K\Big{|}_{\rho=1/\epsilon}=\int_{0}^{\infty}\tau
d\tau\int_{S^{2}}d^{2}x\sqrt{\hat{G}}\frac{\hat{G}^{\rho\mu}}{\sqrt{\hat{G}^{\rho\rho}}}(2\hat{G}^{ij}\partial_{i}\hat{G}_{j\mu}-\hat{G}^{ij}\partial_{\mu}\hat{G}_{ij})\Big{|}_{\rho=1/\epsilon}$
(75)

in which $n^{\mu}$ is the outward unit normal for the boundary $\partial M$.
In our case, for the boundary $\rho=1/\epsilon$, the only non-zero component
is $n^{\rho}=\sqrt{G^{\rho\rho}}$. To see the divergent part of the regulated
action in a more precise way, for even $d$, one can use the expansion for $g$
shown in (53) and extract the infinite part written as

$S_{{\rm gr,reg}}=\frac{1}{16\pi
G_{N}}\int_{0}^{\infty}d\tau\tau\int_{S^{2}}d^{2}x\;\sqrt{g_{(0)}}\left(\epsilon^{-1}a_{1}(\tau,x)+a_{0}(\tau,x)+{\rm
log}\epsilon\;b(\tau,x)+\mathcal{O}(\epsilon^{0})\right),$ (76)

where $a_{i}$ and $b$ are the corresponding coefficients. To get the
renormalised action $S_{{\rm gr,ren}}$, one should introduce the local and
covariant counterterm $S_{{\rm gr,ct}}$ to eliminate the divergence, which
takes the form

$S_{{\rm
gr,ct}}=\int_{0}^{\infty}d\tau\int_{S^{2}}d^{2}x\;f(\tau,z)\sqrt{-\gamma}+\int_{0}^{\infty}d\tau\int_{S^{2}}d^{2}x\;g(\tau,z)\sqrt{-\gamma}\;R\;[\gamma]+\cdots$
(77)

where $f,g$ are scalar functions of $\tau$, $x$ and they are determined by the
coefficients $a_{i}$, $b$ in (76). Given the renormalised action $S_{{\rm
gr,ren}}=S_{{\rm gr,reg}}+S_{{\rm gr,ct}}$ together with the dictionary

${\rm exp}\;\big{(}iS_{{\rm gr,ren}}[G]\big{)}=\Big{\langle}{\rm
exp}\;\frac{1}{2}\int_{S^{2}}d^{2}x\sqrt{\bar{G}}\;\bar{G}^{ij}\;T_{ij}\Big{\rangle}$
(78)

the CFT stress tensor $T_{ij}$ is then deduced to be

$\langle T_{ij}\rangle=\lim_{\epsilon\rightarrow
0}\frac{2i}{\sqrt{\bar{G}}}\frac{\delta S_{{\rm
gr,ren}}}{\;\delta\bar{G}^{ij}},$ (79)

where $\bar{G}_{ij}$ is the background metric of the boundary CFT. Before
going into the detail of holographic renormalisation, here we briefly discuss
the structure of renormalised gravity effective action. Through the
renormalisation procedure, the IR divergence is regulated by choosing the AdS
spatial radius $\rho=1/\epsilon$ as the low energy cutoff. However, for the
flat spacetime, we also have timelike direction labelled by $\tau$ and here we
treat it as the UV cutoff from boundary point of view by specifying the range
of integral $0\leq\tau\leq L$. Therefore, organized by the powers of $L$, the
action takes the form

$S_{\rm gr,reg}=S^{0}_{\rm gr,reg}+S^{1}_{\rm gr,reg}+\cdots.$ (80)

where $S^{0}_{\rm gr,ren}$ is the leading contribution while $S^{0}_{\rm
gt,ren}$ is subleading. After performing the integral of $\tau$ during the
holographic renormalisation, we have

$S_{\rm gr,ren}=L^{2}S^{0}_{\rm gr,ren}+LS^{1}_{\rm
gr,ren}+\mathcal{O}(\log{L}),$ (81)

in which $S^{0}_{\rm gr,ren}$ is the contribution to the renormalised action
of the order $L^{2}$ and $S^{1}_{{\rm gr,ren}}$ is the lower order term. We
identify $S^{0}_{\rm gr,ren}$ as the hard sector since it comes from the AdS
hyperboloid while the soft sector is identified as $S^{1}_{\rm gr,ren}$ coming
from soft modes in the metric.

Given the dictionary (78), to perform the calculation and make it work the
same as the AdS/CFT dictionary, we need the specific map between the boundary
and bulk data and here we propose the relation to be

$\bar{G}_{ij}=g^{(0)}_{ij}+\frac{1}{L}\sigma_{ij}^{(0)}+\cdots$ (82)

where the boundary background metric is expanded by the order of energy cut
off $L$ given by the bulk data $g^{(0)}_{ij}$ and $\sigma^{(0)}_{ij}$.

###  3.1 Hard Sector

In this section, we choose to perform the holographic renormalisation for the
hard sector ignoring the soft contribution from $h_{ij}$. It turns out the
treatment of the hard sector is equivalent to the linear summation over all
the AdS hyperboloid contribution and one can regard this part as the review of
AdS/CFT holographic renormalisation. At zero order, the onshell action takes
the form

$S_{{\rm gr,reg}}=\frac{-1}{16\pi
G_{N}}\int_{0}^{L}d\tau\tau\int_{S^{2}}d^{2}x\;\sqrt{g}\;2\rho^{2}\left(\frac{2}{\rho}+g^{ij}\partial_{\rho}g_{ij}\right)\Big{|}_{\rho=1/\epsilon}$
(83)

and the counterterm is given by

$S_{{\rm gr},{\rm ct}}=-\frac{d-1}{8\pi
G_{N}}\int_{0}^{L}d\tau\int_{S^{2}}d^{2}x\;\frac{1}{\tau}\sqrt{-\gamma}.$ (84)

Now, following the above discussion, we will show the result for $d=2$. In
such case, $g_{(0)ij}$ is the metric on the sphere and the regulated stress
tensor $T^{{\rm reg}}_{ij}$ on the celestial sphere is therefore given by

$T^{{\rm reg}}_{ij}=\frac{iL^{2}}{16\pi
G_{N}}(K_{ij}-\gamma_{ij}K)=\frac{iL^{2}}{16\pi
G_{N}}\rho\left((d-1)g_{ij}+\rho
g_{ij}g^{lk}\partial_{\rho}g_{lk}-\rho\partial_{\rho}g_{ij}\right)\Big{|}_{\rho=1/\epsilon},$
(85)

in which $L$ is the upper bound of the integral over $\tau$ and we have set
$\rho=1/\epsilon$. After the subtraction of the counterterm

$T^{{\rm ct}}_{ij}=-\frac{iL^{2}}{16\pi
G_{N}}\;\frac{(d-1)g_{ij}}{\epsilon}+\cdots,$ (86)

one then obtain the stress tensor

$\langle T_{ij}\rangle=\frac{iL^{2}}{16\pi G_{N}}(g_{(2)ij}-g_{(0)ij}{\rm
Tr}g_{(2)})$ (87)

by taking the limit $\epsilon\rightarrow 0$. Moreover, with the help of the
relation

$g_{(2)ij}=\frac{1}{2}(Rg_{(0)ij}+t_{ij}),\qquad{\rm Tr}\;t=-R,$ (88)

where $R=R\;[g_{(0)}]$ and $t_{ij}$ is a conserved symmetic tensor
$\nabla^{i}t_{ij}=0$, we have

$\langle T_{ij}\rangle=\frac{iL^{2}}{32\pi G_{N}}\;t_{ij}.$ (89)

Therefore, after taking the trace, the Weyl anomaly is then deduced to be

$\langle T^{i}_{i}\rangle=-\frac{c}{24\pi}R,$ (90)

in which $c$ is the central charge on the celestial sphere

$c=\frac{i3L^{2}}{4G_{N}}.$ (91)

One can see the central charge will approach $i\infty$ as argued in
Cheung:2016iub ; Pasterski:2022lsl ; Ogawa:2022fhy  if one treats $L$ as the
scale of energy.

The infinite behaviour of the central charge can be easily understood given
the detail of the flat/CFT dictionary, which has been extensively studied in
the case for scalar fields Hao:2023wln . The onshell scalar field is studied
by the separation of variables which splits the $\tau$ direction and other
coordinates on the AdS hypersurfaces and it turns out that one could decompose
a single scalar field into infinite number of modes labelled by the complex
number $k$ making the scale dimension of the dual operator live on the
principal series. Here we have the metric $G_{\mu\nu}$ while it is hard to
apply the variable separation method to split the nonlinear Einstein equation
into the $\tau$ dependent part and the other part which describes the equation
of motion on the AdS hyperboloid therefore decompose the metric into the form
of

$G_{\mu\nu}(\tau,\rho,x)\longrightarrow G_{\mu\nu}(\rho,x;k)$ (92)

labelled by the parameter $k$. But here it is still reasonable to assume that
the bulk metric is dual to infinite number of operators on the boundary
described by stress-tensor modes denoted as $T_{ij}(x;k)$ and the energy-
stress tensor calculated here are in fact the summation of all these modes.
Each mode will contribute to the central charge in a finite way while the
total effect will become infinite after summing over all the modes labelled by
$k$ treated as the frequency space dual to the $\tau$ direction.

###  3.2 Soft Sector

Now, based on our study of hard sector, we move on to the study of soft
sector. In order to obtain the next leading order correction $S^{1}_{\rm
gr,ren}$, one needs to consider the higher order terms in $1/\tau$ of the
onshell action (75).

$\displaystyle S^{0}_{\rm gr,reg}+S^{1}_{\rm gr,reg}=\frac{-1}{16\pi
G_{N}}\int^{L}_{0}d\tau\tau\int_{S^{2}}d^{2}x\sqrt{\hat{G}}\frac{4\rho}{\sqrt{{\hat{G}}^{\rho\rho}}}\big{(}(d+\rho
g^{ij}\partial_{\rho}g_{ij})$ (93)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{1}{\tau}(2\nabla^{i}A_{i}-g^{ij}\rho\partial_{\rho}\sigma_{ij}+\sigma^{ij}\rho\partial_{\rho}g_{ij})\big{)},$

therefore regulated action at the first order now becomes

$\displaystyle S_{{\rm gr,reg}}^{1}=\frac{L}{8\pi
G_{N}}\int_{S^{2}}d^{2}z\;\sqrt{g_{(0)}}\;\rho\;\Big{(}2\nabla^{i}A_{i}-\rho
g^{ij}\partial_{\rho}\sigma_{ij}+\sigma^{ij}\rho\partial_{\rho}g_{ij}$ (94)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-(2m+2dM)(d+\rho
g^{ij}\partial_{\rho}g_{ij})\Big{)}\Big{|}_{\rho=1/\epsilon}$

where the integral over $\tau$ has already been performed and the contribution
in the second line comes from the determinant $\sqrt{\hat{G}}$ and the norm
vector factor $\sqrt{G_{\rho\rho}}$. Together with the expression of the
extrinsic curvature

$K_{ij}=-\rho(\rho\partial_{\rho}g_{ij}+g_{ij})+\frac{\rho}{\tau}\Big{(}\nabla_{i}A_{j}+\nabla_{j}A_{j}-\rho\partial_{\rho}\sigma_{ij}-\sigma_{ij}+2m(\rho\partial_{\rho}g_{ij}+g_{ij})\Big{)},$
(95)

one can obtain the soft stress tensor $T_{ij}^{1,{\rm reg}}$

$\displaystyle T^{1,{\rm reg}}_{ij}=\frac{iL}{8\pi
G_{N}}\rho\Big{(}(\nabla_{i}A_{j}+\nabla_{j}A_{i}-\rho\partial_{\rho}\sigma_{ij}-\sigma_{ij})+(\rho
g^{lk}\partial_{\rho}g_{lk}+d)\sigma_{ij}$ (96)
$\displaystyle-(2\nabla_{i}A^{i}-\rho
g^{lk}\partial_{\rho}\sigma_{lk}+\rho\sigma^{lk}\partial_{\rho}g_{lk})g_{ij}+2m(\rho\partial_{\rho}g_{ij}+g_{ij})-2m(\rho
g^{lk}\partial_{\rho}g_{lk}+d)g_{ij}\Big{)}$

by checking the first order of $K_{ij}-\gamma_{ij}K$ then performing the
integral over $\tau$ like we have done for the hard sector. From the above
expression one can see the stress tensor will go to infinity at large $\rho$.
One of the divergent term comes from $\nabla_{i}A_{j}$ while the other term
comes from the first order metric on the sphere $\sigma_{ij}$. However, taking
the constraint $A^{(0)}_{i}=A^{(2)}_{i}=0$ and the counterterm

$S_{{\rm gr},{\rm ct}}=-\frac{d-1}{8\pi
G_{N}}\int_{0}^{L}d\tau\int_{S^{2}}d^{2}x\;\tau\sqrt{\hat{G}}\;\big{(}4\rho^{2}\hat{G}_{\rho\rho}\big{)}^{-\frac{1}{2}}$
(97)

into consideration, we have

$T_{ij}^{\rm ct}=\frac{iL^{2}}{16\pi
G_{N}}(1-d)\rho\big{(}g_{ij}+\frac{2}{L}(\sigma_{ij}-2mg_{ij})\big{)}+\cdots$
(98)

therefore the corresponding finite renormalised stress tensor at first order
becomes

$\displaystyle\langle T_{ij}^{1}\rangle=-\frac{iL}{8\pi G_{N}}\Big{(}({\rm
Tr}\;\sigma^{(2)}-\sigma^{(0)}_{kl}\;g_{(2)}^{kl})\;g^{(0)}_{ij}+{\rm
Tr}g^{(2)}\sigma_{ij}^{(0)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\sigma^{(2)}_{ij}+2m^{(0)}(g^{(2)}_{ij}-g^{(0)}_{ij}{\rm
Tr}g^{(2)})\Big{)}.$ (99)

###  3.3 Ward Identities

Given the flat/CFT dictionary and the specific expression of the energy-stress
tensor, now we turn to study Ward identities concerning $\langle
T_{ij}\rangle$ with the help of constraints on the gravity metric studied
before. For Weyl anomaly, we will perform the calculation from the boundary
point of view which means that the indices now are raised and lowered by the
metric $\bar{G}_{ij}$. After taking the trace, for the soft stress tensor, we
have555Here $\langle T^{i}_{i}\rangle^{1}$ represents the first order of
$\bar{G}^{ij}\langle T_{ij}\rangle$, i.e $\langle T^{i}_{i}\rangle^{1}=\langle
T_{ij}^{1}\rangle g_{(2)}^{ij}-\langle T^{0}_{ij}\rangle\sigma_{(0)}^{ij}$.

$\displaystyle\langle T_{i}^{i}\rangle^{1}=-\frac{iL}{16\pi
G_{N}}\Big{(}2(d-1)\big{(}{\rm
Tr}\sigma^{(2)}-\sigma^{(0)}_{ij}g_{(2)}^{ij}-2m^{(0)}{\rm Tr}g^{(2)}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+{\rm Tr}g^{(2)}{\rm
Tr}\sigma^{(0)}-\sigma^{(0)}_{ij}g_{(2)}^{ij}\Big{)},$ (100)

which is equivalent to

$\langle T^{i}_{i}\rangle^{1}=\frac{iL}{\pi G_{N}}m^{(2)}$ (101)

after using the relation (66), (68) and setting $d=2$. As we have studied,
$m^{(2)}$ could be an arbitrary scalar function therefore it will contribute
to the anomaly at the subleading order in an arbitrary way. At the same time,
one should note that the form of anomaly are highly constrained in two
dimensional conformal field theory Deser:1993yx ; Bonora:1985cq , the anomaly
should be proportional to the Euler density, which means one should consider a
special class of asymptotically flat spacetime in order to make the boundary
field theory conformal. Here, we choose to consider a set of solutions of
$m^{(2)}$ written as

${\rm Tr}\sigma^{(2)}=\alpha M{\rm Tr}g^{(2)}\qquad
dm^{(2)}=(1-\frac{\alpha}{4})M{\rm Tr}g^{(2)}$ (102)

satisfying the constraint (66) and (67) for a real parameter $\alpha$. Such
choice could be treated as the fix of gauge for soft sector like the gauge of
asymptotic AdS hyperboloids are fixed in terms of Fefferman Graham coordinates
where $G_{\rho\rho}^{0}=1/4\rho^{2}$. Therefore, we will have

$\langle T^{i}_{i}\rangle^{1}=-\frac{iL}{16\pi G_{N}}(2\alpha-8){\rm
Tr}g^{(2)}m^{(0)}.$ (103)

In such case, we can treat the contribution from $m^{(2)}$ as part of the
central charge at subleading order. To determine the central charge at the
order of $1/L$, we can use the relation

$\langle T^{i}_{i}\rangle=-\frac{c}{24\pi}R\;[\bar{G}].$ (104)

where $R[\bar{G}]=R[g_{0}]+\frac{1}{L}(g^{ij}_{(0)}\delta
R_{ij}-\sigma^{ij}_{(0)}R_{ij}[g_{(0)}])$. By checking the formula (103) and
(104) specifically, we have

$c=\frac{i3L^{2}}{4G_{N}}+\frac{iML3(\alpha-2)}{2G_{N}}$ (105)

from which we can see the central charge will have first order correction that
depends on the geometry of spacetime characterised by the parameter $M$ and
$\alpha$ while we leave higher order correction for further investigation.

Before going to study the conservation laws of energy-stress tensor, we first
recall some lessons learnt from the AdS/CFT correspondence. For the
asymptotically AdS case, the spacetime behaves like a box and no particle
could finally reach the infinity while this fits the calculation that the dual
CFT energy-stress tensor is conserved. Following our definition of asymptotic
flatness, the hard sector is built up by the AdS hyperboloid therefore the
dual energy-stress tensor on the celestial sphere is expected to be conserved
at leading order, written as

$\bar{\nabla}^{j}\langle T_{ij}\rangle^{0}=0,$ (106)

which can be deduced using (68) and (87). Comparing with the AdS spacetime,
one of the main feature for asymptotically flat spacetime is that there could
be gravitational radiation at the boundary and thus the system is not strictly
closed i.e, energy could be carried in or away by particles passing through
the null boundary. Now, for the stress tensor at subleading order, we have

$\displaystyle\bar{\nabla}^{j}\langle T_{ij}\rangle^{1}=-\frac{iL}{16\pi
G_{N}}\big{(}2\nabla_{i}({\rm
Tr}\sigma^{(2)}-\sigma^{(0)}_{kl}g^{kl}_{(2)})-2\nabla^{j}\sigma^{(2)}_{ji}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\sigma_{(0)}^{mk}\nabla_{m}g^{(2)}_{ki}+\nabla^{j}{\rm
Tr}g_{(2)}\sigma^{(0)}_{ij}\big{)}$ (107)

where we have used the the relation (55) and the covariant derivative
$\bar{\nabla}_{i}$ is with respect to the background metric $\bar{G}_{ij}$ of
the boundary CFT. Moreover, taking the constraint (72) into consideration, we
have

$\displaystyle\bar{\nabla}^{j}\langle T_{ij}\rangle=\frac{iL}{16\pi
G_{N}}(8\nabla_{i}m^{(2)}+4M\nabla_{i}{\rm
Tr}g^{(2)}-\sigma^{mk}_{(0)}\nabla_{m}g^{(2)}_{ki}),$ (108)

from which we can see that the stress tensor is not conserved at the
subleading order due to the existence of soft modes therefore we can interpret
such soft modes as the radiation modes which generate the flow of energy
through the null boundary.

##  4 Conclusion

In this paper, we have developed the holographic renormalisation procedure for
the gravitational action on the asymptotically flat background then obtain the
flat/CFT dictionary between the $d+2$ dimensional theory in the bulk and the
$d$ dimensional CFT on the celestial sphere. Based on the construction of
flat/CFT dictionary, we then obtain a precise map between the asymptotic bulk
data and the conformal energy stress tensor. By considering the conformal
anomalies, we deduce the value of the central charge up to the first
subleading order, which comes from the soft sector of the energy stress
tensor. It turns out that the central charge takes complex value approaching
infinity. Such behaviour has already been argued for long while here, given
the flat/CFT duality relation, we may ask if the complex central charge
implies that the boundary CFT is not unitary and we will leave this to further
investigation. Moreover, we can see the value of central charge is expressed
as power expansion according to the energy cut off $L$ and the physical
implication of such expansion is also not clear from either bulk or boundary
point of view.

The introduction of the cuff off $L$ can be treated as the extra input when
performing the renormalisation for the gravitational action. For scalar case
Hao:2023wln , we also have the integral over $\tau$ from zero to infinity
while there is no need to introduce the cut off to make the action finite.
That is because the integral over $\tau$ together with the weight function and
$\tau$ modes will produce orthogonal relations telling us the coupling between
different modes labelled by $k$. We have seen such orthogonal relation by
solving Klein-Gordon equation explicitly while here the treatment for Einstein
equation will be much harder as we have briefly discussed in section 3. In
this paper, we have not studied each graviton mode in a microscopic way while
we choose to study those infinite number of modes macroscopically by
introducing the cut off $L$.

As we have seen, for the flat/CFT duality, most of the nontrivial results come
from the contribution of the soft sector. It will lead to a non-conserved
stress-energy tensor from boundary point of view. Such stress tensor makes the
behaviour of boundary CFT more complicated while it enables us to investigate
the gravitational bulk radiation from boundary point of view. Therefore the
interpretation of non-conserved part of the energy-stress tensor is more like
the introduction of heat bath or matter fields studied in the AdS/CFT
correspondence.

The definition of asymptotic flatness is clarified in the whole paper as (37)
while one may ask if we could consider the asymptotically flat spacetime in a
more general sense. It is interesting to explore how the renormalisation works
if Fefferman Graham gauge is broken. For example, one can consider the case
that $G_{\rho\rho}^{0}$ takes arbitrary form or $G_{\rho i}^{0}\neq 0$. For
the spacetime in (37), the choice of spatial radius $\rho$ on the AdS
hyperboloid as the IR regulator is straight forward since it will not break
the asymptotic symmetry while the development of holographic renormalisation
will become more complicated if one wants to deal with more general metric.

For the soft sector, we also meet the similar problem like the gauge fixing of
the AdS hyperboloid and this comes from the freedom of the choice of ${\rm
Tr}\sigma^{(2)}$ or equivalently $m^{(2)}$. As we have seen, the trace part
${\rm Tr}\sigma^{(2)}$ tends to contribute to the subleading part of the
anomalies of the stress tensor $\langle T^{i}_{i}\rangle$ in a arbitrary way
while the form of Weyl anomaly is highly constrained from the CFT point of
view. After the holographic renormalisation, to make the field theory coming
from the bulk gravitational theory conformal, we have to further fix the gauge
of ${\rm Tr}\sigma^{(2)}$ as shown in (102). Here we have the freedom to do so
but this leads to the problem that if all the asymptotically flat
gravitational theory is dual to the CFT on the boundary. In fact, the
definition of asymptotically flat spacetime is a vague concept in gravity as
we have discussed in the introduction. In addition to the Ricci flat condition
$R_{\mu\nu}=0$, one should also make the spacetime approach to flat at
infinity so that recover enough flat space results and properties. On the
other hand, the CFT is well studied thus such mismatching makes the
construction of flat/CFT duality challenging.

At the end, we illustrate some connections to the celestial holography. The
duality between the bulk metric and the boundary stress-energy tensor has also
been studied in the work Kapec:2016jld ; Cheung:2016iub  in order to manifest
the duality between bulk scattering soft theorems and boundary Ward
identities. Here we derived the expression for the boundary energy-stress
tensor with the guidance of the flat/CFT dictionary (78) while this enables us
to determine both of the hard and soft sector of the energy-stress tensor in
terms of expansions from bulk metric in a local way therefore the boundary
Ward identities will naturally correspond to the Einstein equation in the
bulk. It is interesting to quantize those gravitational modes and find the
relation between these two kinds of bulk gravity and boundary energy-stress
tensor duality. We will leave this to further investigation.

###### Acknowledgements.

I would like to thank my father Qinghe Hao and mother Xiulan Xu for providing
funding for the tuition and accommodation fees when studying PhD at the
University of Southampton (Fourth year tuition fee is covered by university
funding for teaching). I also wish to thank Enrico Parisini for various
insightful discussions on the structure of asymptotcially flat spacetime. I am
grateful to Kostas Skenderis and Marika Taylor for valuable comments on the
manuscript.

##  Appendix A Asymptotic symmetries

Given the Killing vector $\xi$, we can then further write down the variation
of the metric as

$\mathcal{L}_{\xi}G_{\mu\nu}=\xi^{\sigma}\partial_{\sigma}G_{\mu\nu}+G_{\mu\sigma}\;\partial_{\nu}\xi^{\sigma}+G_{\nu\sigma}\;\partial_{\mu}\xi^{\sigma},$
(109)

in which the expression is true for generic metric. In this article we will
work in the Fefferman Graham gauge which means we have $G_{\tau\tau}=-1$ and
$G_{\tau a}=0$. In such gauge, for $\tau\tau$ component, we have

$\mathcal{L}_{\xi}G_{\tau\tau}=\xi^{\sigma}\partial_{\sigma}G_{\tau\tau}+2G_{\tau\sigma}\;\partial_{\tau}\xi^{\sigma}=-2\partial_{\tau}\xi^{\tau}$
(110)

while for the $\tau\rho$ and $\tau z$ component we have

$\displaystyle\mathcal{L}_{\xi}G_{\tau\rho}$ $\displaystyle=$ $\displaystyle
G_{\tau\sigma}\;\partial_{\rho}\xi^{\sigma}+G_{\rho\sigma}\;\partial_{\tau}\xi^{\sigma}$
(111) $\displaystyle=$
$\displaystyle-\partial_{\rho}\xi^{\tau}+G_{\rho\rho}\partial_{\tau}\xi^{\rho}+G_{\rho
z}\partial_{\tau}\xi^{z}+G_{\rho\bar{z}}\partial_{\tau}\xi^{\bar{z}}$ (112)

and

$\displaystyle\mathcal{L}_{\xi}G_{\tau z}$ $\displaystyle=$ $\displaystyle
G_{\tau\sigma}\;\partial_{z}\xi^{\sigma}+G_{z\sigma}\;\partial_{\tau}\xi^{\sigma}$
(113) $\displaystyle=$ $\displaystyle-\partial_{z}\xi^{\tau}+G_{\rho
z}\partial_{\tau}\xi^{\rho}+G_{zz}\partial_{\tau}\xi^{z}+G_{z\bar{z}}\partial_{\tau}\xi^{\bar{z}}.$
(114)

For the spatial $\rho z\bar{z}$ part, we have

$\mathcal{L}_{\xi}G_{z\bar{z}}=\xi^{\sigma}\partial_{\sigma}G_{z\bar{z}}+G_{z\sigma}\partial_{\bar{z}}\xi^{\sigma}+G_{\bar{z}\sigma}\partial_{z}\xi^{\sigma}$
(115)
$\mathcal{L}_{\xi}G_{zz}=\xi^{\sigma}\partial_{\sigma}G_{zz}+2G_{z\sigma}\partial_{z}\xi^{\sigma}$
(116)
$\mathcal{L}_{\xi}G_{\rho\rho}=\xi^{\sigma}\partial_{\sigma}G_{\rho\rho}+2G_{\rho\sigma}\partial_{\rho}\xi^{\sigma}$
(117) $\mathcal{L}_{\xi}G_{\rho z}=\xi^{\sigma}\partial_{\sigma}G_{\rho
z}+G_{\rho\sigma}\partial_{z}\xi^{\sigma}+G_{z\sigma}\partial_{\rho}\xi^{\sigma}.$
(118)

##  Appendix B Fefferman and Graham Coordinates

In terms of Fefferman and Graham coordinates $(r,x_{i})$. The metric for
asymptotic AdS spacetime takes the form

$\hat{G}_{rr}=\frac{1}{r^{2}},\qquad\hat{G}_{ij}=\frac{g_{ij}}{r^{2}},\qquad\hat{G}_{ri}=0,$
(119)

in which the asymptotic behaviour is described by the function
$g_{ij}(r,x_{i})$. In such coordinates, the connection is then given by

$\Gamma_{rr}^{r}=-\frac{1}{r},\qquad\Gamma^{r}_{ri}=\Gamma^{i}_{rr}=0.$ (120)

For the other components that involve $r$ component, in terms of the function
$g_{ij}$, the connections can be written as

$\Gamma^{r}_{ij}=-\frac{r^{2}}{2}\;\partial_{r}\hat{G}_{ij}=\frac{g_{ij}}{r}-\frac{1}{2}\partial_{r}g_{ij}$
(121)

and

$\Gamma^{i}_{rj}=\frac{1}{2}\hat{G}^{ik}\partial_{r}\hat{G}_{kj}=-\frac{\delta^{i}_{j}}{r}+\frac{1}{2}g^{ik}\partial_{r}g_{kj}$
(122)

For the connections that do not involve the $r$ component, they are determined
by the function $g_{ij}$ and one can treat them as the connection of $g_{ij}$,
i.e, $\Gamma^{i}_{jk}=\hat{\Gamma}^{i}_{jk}[g]$.

Given the connections, we can use them to calculate the Ricci tensor following
the definition (128), the $rr$ component is given by

$\displaystyle R_{rr}^{d+1}[\hat{G}]$ $\displaystyle=$
$\displaystyle\partial_{r}\Gamma^{k}_{rk}+\Gamma^{l}_{rk}\Gamma^{k}_{rl}-\Gamma^{r}_{rr}\Gamma^{k}_{kr}$
(123) $\displaystyle=$
$\displaystyle\frac{1}{2}\;g^{ij}\;\partial^{2}_{r}\;g_{ij}-\frac{1}{2r}g^{ij}\partial_{r}g_{ij}-\frac{1}{4}g^{ij}\;g^{lm}\;\partial_{r}g_{il}\;\partial_{r}g_{jm}+\frac{d}{r^{2}}$

and the $ir$ components is determined to be

$\displaystyle R_{ir}^{d+1}[\hat{G}]$ $\displaystyle=$
$\displaystyle\partial_{i}\Gamma_{rk}^{k}-\partial_{k}\Gamma^{k}_{ri}+\Gamma^{k}_{rl}\Gamma^{l}_{ik}-\Gamma^{m}_{ri}\Gamma^{k}_{km}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{i}(\;g^{lm}\;\partial_{r}g_{lm})-\frac{1}{2}\partial_{k}(\;g^{kl}\;\partial_{r}g_{il})+\frac{1}{2}g^{km}\;\Gamma^{l}_{ik}\;\partial_{r}g_{lm}-\frac{1}{2}g^{mk}\;\Gamma^{l}_{lm}\;\partial_{r}g_{ik}.$

Moreover, in terms of covariant derivative $\nabla_{i}$ with respective to the
metric $g_{ij}$, $R_{ir}$ can be simplified to

$R_{ir}^{d+1}[\hat{G}]=\frac{1}{2}\nabla_{i}(g^{lm}\;\partial_{r}g_{lm})-\frac{1}{2}\nabla^{j}\partial_{r}g_{ji}.$
(125)

The $ij$ component is given by

$\displaystyle
R_{ij}^{d+1}[\hat{G}]=R_{ij}[g]-\partial_{r}\Gamma^{r}_{ij}+\Gamma^{k}_{ir}\Gamma^{r}_{jk}+\Gamma^{r}_{il}\Gamma^{l}_{jr}-\Gamma^{r}_{ij}\Gamma^{l}_{lr}-\Gamma^{r}_{ij}\Gamma^{r}_{rr}$
(126)
$\displaystyle\qquad\quad=R_{ij}[g]+\frac{1}{2}\partial^{2}_{r}g_{ij}+\frac{d}{r^{2}}g_{ij}+\frac{1-d}{2r}\partial_{r}g_{ij}-\frac{1}{2}g^{km}\partial_{r}g_{ki}\partial_{r}g_{mj}$
$\displaystyle\qquad\qquad+\frac{1}{4}\partial_{r}g_{ij}g^{lm}\partial_{r}g_{lm}-\frac{1}{2r}g_{ij}g^{lm}\partial_{r}g_{lm},$

in which the induced Ricci tensor of $g_{ij}$ is denoted as $R_{ij}[g]$.

##  Appendix C Equation of Motion

To calculate the Ricci tensor, we will use the convention

$R_{\mu\nu\rho}^{\;\;\;\;\;\;\sigma}=\partial_{\mu}\Gamma_{\nu\rho}^{\sigma}-\partial_{\nu}\Gamma_{\mu\rho}^{\sigma}+\Gamma_{\mu\lambda}^{\sigma}\Gamma^{\lambda}_{\nu\rho}-\Gamma_{\nu\lambda}^{\sigma}\Gamma^{\lambda}_{\mu\rho}$
(127)

so that the tensor is given by

$R_{\mu\nu}=R_{\mu\rho\nu}^{\;\;\;\;\;\;\rho}=\partial_{\mu}\Gamma_{\nu\rho}^{\rho}-\partial_{\rho}\Gamma_{\mu\nu}^{\rho}+\Gamma_{\mu\lambda}^{\rho}\Gamma^{\lambda}_{\nu\rho}-\Gamma_{\rho\lambda}^{\rho}\Gamma^{\lambda}_{\mu\nu}.$
(128)

For $d$ dimensional spacetime, in terms of the Milne coordinates, the Einstein
equation at linear level could be written as

$R_{\tau\tau}[G]=\frac{4\rho^{2}}{\tau}\partial_{\tau}h_{\rho\rho}+2\rho^{2}\partial^{2}_{\tau}h_{\rho\rho}+\frac{1}{2}g^{ij}\partial^{2}_{\tau}h_{ij}+\frac{1}{\tau}g^{ij}\partial_{\tau}h_{ij}$
(129) $\displaystyle R_{\tau
i}[G]=2\rho^{2}\nabla_{i}\partial_{\tau}h_{\rho\rho}+\frac{1}{2\rho}\nabla_{i}(\;g^{jm}\partial_{\tau}h_{jm})-\frac{1}{2\rho}\partial_{\tau}\nabla^{k}h_{ki}-2\rho^{2}\partial_{\rho}\partial_{\tau}h_{i\rho}$
$\displaystyle\qquad\qquad-4\rho\partial_{\tau}h_{i\rho}-2\rho^{2}\partial_{\tau}h_{\rho
i}\Gamma^{a}_{a\rho}$ (130)

in which we have

$\Gamma^{a}_{a\rho}=\frac{d-2}{2}\frac{1}{\rho}+\frac{1}{2}g^{lm}\partial_{\rho}g_{lm}$
(131)

and $R_{\tau\rho}$ is deduced to be

$\displaystyle
R_{\tau\rho}[G]=-\frac{1}{2\rho^{2}}g^{ij}\partial_{\tau}h_{ij}+\frac{1}{2\rho}\partial_{\rho}(g^{ij}\partial_{\tau}h_{ij})-2\rho\partial_{\tau}h_{\rho\rho}-\frac{1}{2\rho}\partial_{\tau}\nabla^{i}h_{i\rho}$
$\displaystyle\qquad\qquad\qquad-2\rho^{2}\partial_{\tau}h_{\rho\rho}\Gamma^{a}_{a\rho}+\frac{1}{2\rho}g^{ij}\partial_{\tau}h_{jk}\Gamma^{k}_{i\rho}.$
(132)

For other components, with the help of the expansion of the Ricci curvature,

$R_{ab}^{d+1}[\hat{G}^{(0)}+h]=R_{ab}^{d+1}[\hat{G}^{(0)}]+\frac{1}{2}(\hat{\nabla}^{2}h_{ab}+\hat{\nabla}_{a}\hat{\nabla}_{b}h-\hat{\nabla}_{c}\hat{\nabla}_{a}h^{c}_{b}-\hat{\nabla}_{c}\hat{\nabla}_{b}h^{c}_{a})+\mathcal{O}(h^{2})$
(133)

where $\hat{\nabla}_{a}$ is the covariant derivative with respect to the
metric $\hat{G}^{(0)}_{ab}$. Then one can obtain

$\displaystyle
R_{\rho\rho}[G]=-dh_{\rho\rho}-\frac{d+2}{2}\tau\partial_{\tau}h_{\rho\rho}-\frac{1}{2}\tau^{2}\partial^{2}_{\tau}h_{\rho\rho}-\frac{\tau}{8\rho^{3}}g^{ij}\partial_{\tau}h_{ij}+(4\rho^{2}\Gamma^{i}_{\rho
j}\Gamma^{j}_{i\rho}-\frac{g^{ij}}{\rho^{2}}\Gamma^{\rho}_{ij}$
$\displaystyle\qquad\qquad+\frac{g^{ij}}{\rho}\Gamma^{k}_{i\rho}\Gamma^{\rho}_{jk}-4(d+\rho
g^{ij}\partial_{\rho}g_{ij}))h_{\rho\rho}-(2\rho(d+\rho
g^{ij}\partial_{\rho}g_{ij})+\frac{g^{ij}}{2\rho}\Gamma^{\rho}_{ij})\partial_{\rho}h_{\rho\rho}$
$\displaystyle\qquad\qquad+\frac{1}{2\rho}\nabla^{i}\nabla_{i}h_{\rho\rho}-\frac{1}{\rho}\nabla_{i}\partial_{\rho}h_{\rho}^{i}-\frac{2}{\rho}\Gamma^{k}_{i\rho}\nabla^{i}h_{k\rho}-\frac{1}{\rho^{2}}\nabla_{i}h^{i}_{\rho}+\frac{1}{2\rho}\partial^{2}_{\rho}h^{i}_{i}-\frac{1}{2\rho^{2}}\partial_{\rho}h^{i}_{i}$
$\displaystyle\qquad\qquad+\frac{1}{\rho}\Gamma^{i}_{\rho
j}\partial_{\rho}h^{j}_{i}+\frac{1}{2\rho^{3}}h^{i}_{i}+\frac{1}{\rho}g^{ij}\Gamma^{k}_{i\rho}\Gamma^{l}_{j\rho}h_{kl}-\frac{1}{\rho^{2}}\Gamma^{i}_{\rho
j}h^{j}_{i}-\frac{1}{\rho}\Gamma^{i}_{\rho j}\Gamma^{k}_{\rho i}h^{j}_{k}$
(134)

where we have $h^{i}_{j}=g^{ik}h_{kj}$ and $h^{i}_{\rho}=g^{ik}h_{k\rho}$. The
connections used here are given by $\Gamma^{\rho}_{\rho\rho}=-\frac{1}{\rho}$,
$\Gamma^{i}_{j\rho}=\frac{\delta^{i}_{j}}{2\rho}+\frac{1}{2}g^{ik}\partial_{\rho}g_{kj}$
and $\Gamma^{\rho}_{ij}=-2\rho^{2}g_{ij}-2\rho^{3}\partial_{\rho}g_{ij}$. For
the components of $R_{i\rho}$, $R_{ij}$ we have

$\displaystyle
R_{i\rho}[G]=-\frac{d+1}{2}\tau\partial_{\tau}h_{i\rho}-\frac{1}{2}\tau^{2}\partial^{2}_{\tau}h_{i\rho}-dh_{i\rho}-2\rho\nabla_{i}h_{\rho\rho}-2\rho^{2}\nabla_{k}(\Gamma^{k}_{i\rho}h_{\rho\rho})-\frac{1}{2\rho}\nabla^{m}(\Gamma^{\rho}_{mi}h_{\rho\rho})$
$\displaystyle\qquad\qquad-2\rho^{2}\Gamma^{a}_{a\rho}\nabla_{i}h_{\rho\rho}+\frac{1}{2\rho}(\nabla^{m}\nabla_{m}h_{i\rho}-\nabla_{k}\nabla_{i}h^{k}_{\rho})+2\rho^{2}\partial_{\rho}(\Gamma^{j}_{\rho
i}h_{\rho j})-\frac{1}{2\rho}\partial_{\rho}(\Gamma^{\rho}_{ij}h^{j}_{\rho})$
$\displaystyle\qquad\qquad+\frac{3}{2}\Gamma^{\rho}_{mi}\Gamma^{k}_{n\rho}g^{mn}h_{k\rho}+\frac{g^{mn}}{2\rho}\Gamma^{\rho}_{mn}\Gamma^{j}_{\rho
i}h_{j\rho}+\frac{1}{2\rho}g^{mn}\Gamma^{k}_{m\rho}\Gamma^{\rho}_{nk}h_{i\rho}+\frac{1}{2\rho}\Gamma^{j}_{k\rho}\Gamma^{\rho}_{ij}h^{k}_{\rho}$
$\displaystyle\qquad\qquad-\frac{1}{2\rho^{2}}g^{mn}\Gamma^{\rho}_{mn}h_{i\rho}+2\rho^{2}\Gamma^{j}_{k\rho}\Gamma^{k}_{j\rho}h_{\rho
i}+\Gamma^{a}_{a\rho}(-2\rho h_{\rho i}+2\rho^{2}\Gamma^{j}_{\rho i}h_{\rho
j}+2\rho^{2}\Gamma^{k}_{i\rho}h_{\rho
k}-\frac{1}{2\rho}\Gamma^{\rho}_{ij}h^{j}_{\rho})$
$\displaystyle\qquad\qquad+8\rho\Gamma^{j}_{i\rho}h_{\rho
j}-2h_{i\rho}+\frac{1}{\rho}\Gamma^{\rho}_{ij}\partial_{\rho}h^{j}_{\rho}-2\rho\partial_{\rho}h_{i\rho}-2\rho^{2}\Gamma^{a}_{a\rho}\partial_{\rho}h_{\rho
i}+\frac{1}{\rho}\nabla_{k}(\Gamma^{l}_{i\rho}h^{k}_{l})$
$\displaystyle\qquad\qquad-\frac{1}{2\rho}\Gamma^{k}_{m\rho}\nabla^{m}h_{ik}+\frac{1}{2\rho}\Gamma^{k}_{j\rho}\nabla_{i}h^{j}_{k}-\frac{1}{2\rho}\nabla^{j}(\Gamma^{k}_{j\rho}h_{ik})-\frac{1}{2\rho}\nabla_{j}(\Gamma^{j}_{\rho
k}h^{k}_{i})+\frac{1}{2\rho}\Gamma^{j}_{k\rho}\nabla_{j}h^{k}_{i}$
$\displaystyle\qquad\qquad+\frac{1}{2\rho^{2}}\nabla_{k}h^{k}_{i}-\frac{1}{2\rho^{2}}\nabla_{i}h-\frac{1}{2\rho}\Gamma^{k}_{i\rho}\nabla_{k}h+\frac{1}{2\rho}\nabla_{i}\partial_{\rho}h-\frac{1}{2\rho}\nabla_{j}\partial_{\rho}h^{j}_{i}$
(135)

and

$\displaystyle
R_{ij}[G]=-dh_{ij}-\frac{d+1}{2}\tau\partial_{\tau}h_{ij}-\frac{1}{2}\tau^{2}\partial_{\tau}^{2}h_{ij}-\frac{\tau}{2}g_{ij}(g^{mn}\partial_{\tau}h_{mn}+4\rho^{3}\partial_{\tau}h_{\rho\rho})$
$\displaystyle\qquad\qquad+\partial_{\rho}(4\rho^{2}\Gamma^{\rho}_{ij}h_{\rho\rho})-2\rho^{2}(\Gamma^{k}_{\rho
i}\Gamma^{\rho}_{kj}h_{\rho\rho}+\Gamma^{k}_{\rho
j}\Gamma^{\rho}_{ki}h_{\rho\rho})+\frac{1}{\rho}\Gamma^{\rho}_{ni}\Gamma^{\rho}_{nj}g^{mn}h_{\rho\rho}+2\rho^{2}\nabla_{i}\nabla_{j}h_{\rho\rho}$
$\displaystyle\qquad\qquad+4\rho^{2}\Gamma^{a}_{a\rho}\Gamma^{\rho}_{ij}h_{\rho\rho}-\Gamma^{\rho}_{ij}\partial_{\rho}(2\rho^{2}h_{\rho\rho})-2\partial_{\rho}(\rho^{2}\nabla_{i}h_{\rho
j})-2\partial_{\rho}(\rho^{2}\nabla_{j}h_{\rho i})$
$\displaystyle\qquad\qquad-2\rho^{2}\nabla_{k}(\Gamma^{k}_{i\rho}h_{j\rho})-2\rho^{2}\nabla_{k}(\Gamma^{k}_{j\rho}h_{i\rho})+\frac{1}{2\rho}\Gamma^{\rho}_{kj}\nabla_{i}h^{k}_{\rho}+\frac{1}{2\rho}\Gamma^{\rho}_{ki}\nabla_{j}h^{k}_{\rho}+\frac{1}{\rho}\nabla_{k}(\Gamma^{\rho}_{ij}h^{k}_{\rho})$
$\displaystyle\qquad\qquad-2\rho^{2}\Gamma^{a}_{a\rho}(\nabla_{i}h_{j\rho}+\nabla_{j}h_{i\rho})-2\rho^{2}\Gamma^{k}_{\rho
i}h_{\rho j}-2\rho^{2}\Gamma^{k}_{\rho j}h_{\rho i}-2\rho^{2}\Gamma^{k}_{\rho
j}\nabla_{i}h_{k\rho}-2\rho^{2}\Gamma^{k}_{\rho i}\nabla_{j}h_{k\rho}$
$\displaystyle\qquad\qquad-\frac{1}{2\rho}\nabla^{n}(\Gamma^{\rho}_{ni}h_{\rho
j})-\frac{1}{2\rho}\nabla^{n}(\Gamma^{\rho}_{nj}h_{i\rho})-\frac{1}{2\rho}\Gamma^{\rho}_{mi}\nabla^{m}h_{\rho
j}-\frac{1}{2\rho}\Gamma^{\rho}_{mj}\nabla^{m}h_{\rho i}$
$\displaystyle\qquad\qquad-\frac{1}{2\rho}(\nabla_{k}\nabla_{i}h^{k}_{j}+\nabla_{k}\nabla_{j}h^{k}_{i})+\frac{1}{2\rho}\nabla_{i}\nabla_{j}h+\frac{1}{2\rho}\nabla_{m}\nabla^{m}h_{ij}+2\rho^{2}\partial^{2}_{\rho}h_{ij}+2\rho\partial_{\rho}h_{ij}$
$\displaystyle\qquad\qquad-2\rho^{2}\big{(}\partial_{\rho}(\Gamma^{k}_{\rho
i}h_{kj})+\partial_{\rho}(\Gamma^{k}_{\rho j}h_{ki})+\Gamma^{k}_{\rho
i}\partial_{\rho}h_{kj}+\Gamma^{k}_{\rho
j}\partial_{\rho}h_{ki}\big{)}+\frac{1}{2\rho^{2}}\Gamma^{\rho}_{ij}h-\frac{1}{2\rho}\Gamma^{\rho}_{ij}\partial_{\rho}h$
$\displaystyle\qquad\qquad+\frac{1}{2}\Gamma^{\rho}_{ki}\partial_{\rho}(\frac{1}{\rho}h^{k}_{j})+\frac{1}{2}\Gamma^{\rho}_{kj}\partial_{\rho}(\frac{1}{\rho}h^{k}_{i})-\frac{1}{2}\partial_{\rho}(\frac{1}{\rho}\Gamma^{\rho}_{ki}h^{k}_{j})-\frac{1}{2}\partial_{\rho}(\frac{1}{\rho}\Gamma^{\rho}_{kj}h^{k}_{i})$
$\displaystyle\qquad\qquad+2\rho^{2}(\Gamma^{k}_{\rho i}\Gamma^{m}_{\rho
k}h_{mj}+\Gamma^{k}_{\rho j}\Gamma^{m}_{\rho k}h_{mi}+2\Gamma^{k}_{\rho
i}\Gamma^{m}_{\rho
j}h_{mk})-\frac{1}{2\rho}(\Gamma^{\rho}_{ki}\Gamma^{m}_{\rho
j}+\Gamma^{\rho}_{kj}\Gamma^{m}_{\rho i})h^{k}_{m}$
$\displaystyle\qquad\qquad+\frac{1}{2\rho}(\Gamma^{\rho}_{ki}\Gamma^{k}_{\rho
m}h^{m}_{j}+\Gamma^{\rho}_{kj}\Gamma^{k}_{\rho m}h^{m}_{i}+\Gamma^{k}_{\rho
i}\Gamma^{\rho}_{km}h^{m}_{j}+\Gamma^{k}_{\rho
j}\Gamma^{\rho}_{km}h^{m}_{i})-2\rho\Gamma^{k}_{\rho
i}h_{kj}-2\rho\Gamma^{k}_{\rho j}h_{ki}$
$\displaystyle\qquad\qquad-\frac{1}{2\rho}\Gamma^{a}_{a\rho}(\Gamma^{\rho}_{im}h^{m}_{j}+\Gamma^{\rho}_{jm}h^{m}_{i})+\frac{g^{mn}}{2\rho}\Big{(}\Gamma^{\rho}_{mn}(\Gamma^{k}_{i\rho}h_{kj}+\Gamma^{k}_{\rho
j}h_{ki})+\Gamma^{k}_{n\rho}(\Gamma^{\rho}_{mj}h_{ki}$
$\displaystyle\qquad\qquad+\Gamma^{\rho}_{mi}h_{kj})-\Gamma^{\rho}_{mn}\partial_{\rho}h_{ij}\Big{)}.$
(136)

Given the Einstein tensor, then one should be able to deduce $h_{ab}$ exactly
by solving the Einstein equation $R_{\mu\nu}=0$. For simplicity, we consider
the equation at the leading order which means for $h_{ab}$ we have

$h_{\rho\rho}=\frac{1}{\rho^{2}\tau}m(\rho,z,\bar{z}),\qquad
h_{ij}=\frac{\rho}{\tau}\sigma_{ij}(\rho,z,\bar{z}),\qquad h_{\rho
i}=\frac{1}{\tau}A_{i}(\rho,z,\bar{z})$ (137)

where higher order terms of $1/\tau$ are omitted.

## References

  * (1) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (2) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (3) S.S. Gubser, I.R. Klebanov and A.M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105 [hep-th/9802109].
  * (4) S. de Haro, K. Skenderis and S.N. Solodukhin, _Gravity in warped compactifications and the holographic stress tensor_ , _Class. Quant. Grav._ 18 (2001) 3171 [hep-th/0011230].
  * (5) J. de Boer and S.N. Solodukhin, _A Holographic reduction of Minkowski space-time_ , _Nucl. Phys. B_ 665 (2003) 545 [hep-th/0303006].
  * (6) R.B. Mann and D. Marolf, _Holographic renormalization of asymptotically flat spacetimes_ , _Class. Quant. Grav._ 23 (2006) 2927 [hep-th/0511096].
  * (7) R.N.C. Costa, _Holographic Reconstruction and Renormalization in Asymptotically Ricci-flat Spacetimes_ , _JHEP_ 11 (2012) 046 [1206.3142].
  * (8) A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ (3, 2017), [1703.05448].
  * (9) A. Ball, E. Himwich, S.A. Narayanan, S. Pasterski and A. Strominger, _Uplifting AdS 3/CFT2 to flat space holography_, _JHEP_ 08 (2019) 168 [1905.09809].
  * (10) A.-M. Raclariu, _Lectures on celestial holography_ , _arXiv preprint arXiv:2107.02075_ (2021) .
  * (11) S. Pasterski, _Lectures on celestial amplitudes_ , _Eur. Phys. J. C_ 81 (2021) 1062 [2108.04801].
  * (12) K. Nguyen and J. Salzer, _Celestial IR divergences and the effective action of supertranslation modes_ , _JHEP_ 09 (2021) 144 [2105.10526].
  * (13) Z. Hao and M. Taylor, _Flat holography and celestial shockwaves_ , _JHEP_ 02 (2024) 090 [2309.04307].
  * (14) H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, _Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems_ , _Proc. Roy. Soc. Lond. A_ 269 (1962) 21.
  * (15) R.K. Sachs, _Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times_ , _Proc. Roy. Soc. Lond. A_ 270 (1962) 103.
  * (16) R. Sachs, _Asymptotic symmetries in gravitational theory_ , _Phys. Rev._ 128 (1962) 2851.
  * (17) C. Fefferman and C.R. Graham, _Conformal invariants_ , in _Élie Cartan et les mathématiques d’aujourd’hui - Lyon, 25-29 juin 1984_ , no. S131 in Astérisque, Société mathématique de France (1985).
  * (18) C.R. Graham, _Volume and area renormalizations for conformally compact Einstein metrics_ , _Rend. Circ. Mat. Palermo S_ 63 (2000) 31 [math/9909042].
  * (19) C. Fefferman and C.R. Graham, _The ambient metric_ , _Ann. Math. Stud._ 178 (2011) 1 [0710.0919].
  * (20) M. Henningson and K. Skenderis, _The Holographic Weyl anomaly_ , _JHEP_ 07 (1998) 023 [hep-th/9806087].
  * (21) S. de Haro, S.N. Solodukhin and K. Skenderis, _Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence_ , _Commun. Math. Phys._ 217 (2001) 595 [hep-th/0002230].
  * (22) V. Balasubramanian and P. Kraus, _A Stress tensor for Anti-de Sitter gravity_ , _Commun. Math. Phys._ 208 (1999) 413 [hep-th/9902121].
  * (23) C. Cheung, A. de la Fuente and R. Sundrum, _4D scattering amplitudes and asymptotic symmetries from 2D CFT_ , _JHEP_ 01 (2017) 112 [1609.00732].
  * (24) S. Pasterski and H. Verlinde, _Chaos in celestial CFT_ , _JHEP_ 08 (2022) 106 [2201.01630].
  * (25) N. Ogawa, T. Takayanagi, T. Tsuda and T. Waki, _Wedge holography in flat space and celestial holography_ , _Phys. Rev. D_ 107 (2023) 026001 [2207.06735].
  * (26) G.W. Gibbons and S.W. Hawking, _Action Integrals and Partition Functions in Quantum Gravity_ , _Phys. Rev. D_ 15 (1977) 2752.
  * (27) K. Skenderis and B.C. van Rees, _Real-time gauge/gravity duality: Prescription, Renormalization and Examples_ , _JHEP_ 05 (2009) 085 [0812.2909].
  * (28) K. Skenderis and B.C. van Rees, _Real-time gauge/gravity duality_ , _Phys. Rev. Lett._ 101 (2008) 081601 [0805.0150].
  * (29) Z. Hao, _Real time holography for higher spin theories_ , 2311.05382.
  * (30) S. Deser and A. Schwimmer, _Geometric classification of conformal anomalies in arbitrary dimensions_ , _Phys. Lett. B_ 309 (1993) 279 [hep-th/9302047].
  * (31) L. Bonora, P. Pasti and M. Bregola, _WEYL COCYCLES_ , _Class. Quant. Grav._ 3 (1986) 635.
  * (32) D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, _2D Stress Tensor for 4D Gravity_ , _Phys. Rev. Lett._ 119 (2017) 121601 [1609.00282].