# Correlation functions of the Bjorken flow in the holographic Schwinger-
Keldysh approach

Toshali Mitra<EMAIL_ADDRESS>Asia Pacific Center for Theoretical
Physics, Pohang 37673, Korea  The Institute of Mathematical Sciences, Chennai
600113, India  Homi Bhabha National Institute, Training School Complex,
AnushaktiNagar, Mumbai 400094, India  Avik Banerjee<EMAIL_ADDRESS>Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France  Center
for Strings, Gravitation and Cosmology, Department of Physics, Indian
Institute of Technology Madras, Chennai 600036, India  Ayan Mukhopadhyay
<EMAIL_ADDRESS>Instituto de Física, Pontificia Universidad Católica
de Valparaíso, Avenida Universidad 330, Valparaíso, Chile  Center for Strings,
Gravitation and Cosmology, Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  [

###### Abstract

One of the outstanding problems in the holographic approach to many-body
physics is the explicit computation of correlation functions in nonequilibrium
states. We provide a new and simple proof that the horizon cap prescription of
Crossley-Glorioso-Liu for implementing the thermal Schwinger-Keldysh contour
in the bulk is consistent with the Kubo-Martin-Schwinger periodicity and the
ingoing boundary condition for the retarded propagator at any arbitrary
frequency and momentum. The generalization to the hydrodynamic Bjorken flow is
achieved by a Weyl rescaling in which the dual black hole’s event horizon
attains a constant surface gravity and area at late time although the
directions longitudinal and transverse to the flow expands and contract
respectively. The dual state’s temperature and entropy density thus become
constants (instead of the perfect fluid expansion) although no time-
translation symmetry emerges at late time. Undoing the Weyl rescaling, the
correlation functions can be computed systematically in a large proper time
expansion in inverse powers of the average of the two reparametrized proper
time arguments. The horizon cap has to be pinned to the nonequilibrium event
horizon so that regularity and consistency conditions are satisfied.
Consequently, in the limit of perfect fluid expansion, the Schwinger-Keldysh
correlation functions with space-time reparametrized arguments are simply
thermal at an appropriate temperature. A generalized bilocal thermal structure
holds to all orders. We argue that the Stokes data (which are functions rather
than constants) for the hydrodynamic correlation functions can decode the
quantum fluctuations behind the horizon cap pinned to the evolving event
horizon, and thus the initial data.

###### Contents

  1. I Introduction
    1. I.1 Motivation and aims
    2. I.2 A brief historical review
    3. I.3 Summary of the results
    4. I.4 Plan
  2. II The CGL horizon cap of the thermal black brane
  3. III Bjorken flow, Weyl rescaling and the holographic dual
    1. III.1 Bjorken flow and its Weyl rescaling
    2. III.2 Gravitational setup
  4. IV The bulk scalar field and the horizon cap of the Bjorken flow
  5. V The real time out-of-equilibrium correlation functions
    1. V.1 Some useful relations and their consequences
    2. V.2 General structure of the hydrodynamic correlation functions
    3. V.3 In the limit of the perfect fluid expansion
    4. V.4 First and higher orders in the late proper time expansion
    5. V.5 Consistency checks
  6. VI On initial conditions and seeing behind the event horizon
  7. VII Conclusions and outlook
  8. A Details of the five dimensional metric dual to the Bjorken flow upto second order in the late-time expansion
  9. B The event horizon and the apparent horizon of the Bjorken flow
    1. B.1 Event horizon:
    2. B.2 Apparent horizon:
  10. C More details of the bulk scalar field in the $AdS_{5}$ Bjorken flow background
  11. D Derivation of Eqs. (131)
  12. E Transients and $\gamma_{0}$

##  I Introduction

###  I.1 Motivation and aims

One of the outstanding issues in the holographic duality, that maps strongly
interacting quantum systems to semi-classical gravity in one higher dimension,
is to understand the dictionary in real time. The applications of the
holographic approach to many-body physics are especially limited without
explicit and implementable prescriptions for computing out-of-equilibrium
correlation functions. Even in the weak-coupling limit, these correlation
functions are fundamental tools for studying decoherence and thermalization,
e.g. to understand how the commutator and the anti-commutator evolve to
satisfy the fluctuation-dissipation relation leading to the emergence of the
Kubo-Martin-Schwinger periodicity at an appropriate temperature, and how the
occupation numbers of quasi-particles equilibrate or evolve to new fixed
points Berges (2004); Sieberer  _et al._ (2016); Heyl (2018); Mori  _et al._
(2018). These correlation functions are actually indispensable for
understanding dynamics far from equilibrium in the strong coupling limit where
the system cannot be described by quasi-particles. Although one can compute
the one-point functions such as the energy-momentum tensor in real time using
the correspondence between time-dependent geometries with regular horizons and
states in the dual theory, with the remarkable fluid-gravity correspondence
Policastro  _et al._ (2001); Bhattacharyya  _et al._ (2008); Baier  _et al._
(2008); Rangamani (2009) providing a primary example, and numerical relativity
Chesler and Yaffe (2014) providing a powerful tool, an explicit computation of
a generating functional for hydrodynamic Schwinger-Keldysh correlation
functions has not been achieved yet.111A limited number of observables can
still be computed analytically or numerically. The equal time two-point
functions can be computed via the geodesic approximation when the operator has
a large scaling dimension, even out of equilibrium. The out-of-equilibrium
retarded correlation function can also be computed by implementing linear
causal response appropriately – see Banerjee  _et al._ (2016) for a general
prescription. Furthermore, equal-time Green’s function can be computed for
operators with large anomalous scaling dimensions in the geodesic
approximation and has been used to understand thermalization Balasubramanian
_et al._ (2011).

The object of interest is the generating functional

$\displaystyle\exp(iW[J_{1},J_{2}])={\rm
Tr}\left(\hat{\rho}_{0}~{}T_{c}\exp\left(-i\oint{\rm d}t\int{\rm
d}^{d-1}x\,\,J(t,\mathbf{x})\hat{O}(t,\mathbf{x})\right)\right)$ (1)

in the dual field theory, where $\hat{\rho}_{0}$ denotes the initial density
matrix, $\oint$ the closed time Schwinger-Keldysh contour composed of the
forward and backward arms —
$\oint=\int_{-\infty}^{\infty}+\int^{-\infty}_{\infty}$, where the source
$J(t,\mathbf{x})$ is specified such that it is $J_{1}(t,\mathbf{x})$ and
$J_{2}(t,\mathbf{x})$ on the forward and backward arms of the contour,
respectively, and $T_{c}$ denotes contour ordering. Formally, we can rewrite
the density-matrix as

$\hat{\rho}_{0}=\int[\mathcal{D}\phi_{1}][\mathcal{D}\phi_{2}]\,\,\rho_{0}(\phi_{1},\phi_{2})\,|\phi_{1}\rangle\langle\phi_{2}|$
(2)

in terms of a basis $|\phi\rangle$ of field configurations, and construct the
kernel

$\mathcal{K}(\phi_{1},\phi_{2};J_{1},J_{2})=\langle\phi_{2}|T_{c}\exp\left(-i\oint{\rm
d}t\int{\rm
d}^{d-1}x\,\,J(t,\mathbf{x})\hat{O}(t,\mathbf{x})\right)|\phi_{1}\rangle.$ (3)

Then the functional $W[J_{1},J_{1}]$ can be obtained from

$\exp(iW[J_{1},J_{2}])=\int[\mathcal{D}\phi_{1}][\mathcal{D}\phi_{2}]\,\,\rho_{0}(\phi_{1},\phi_{2})\mathcal{K}(\phi_{1},\phi_{2};J_{1},J_{2}).$
(4)

When $\hat{\rho}_{0}$ is the thermal density matrix, this computation
simplifies drastically. One needs to just add an appendage of length $T^{-1}$
at the end of the closed real-time contour parallel to the negative imaginary
axis, and impose periodic boundary conditions on the full contour implementing
Kubo-Martin-Schwinger (KMS) periodicity Le Bellac (1996).

Figure 1: Left: The general Schwinger-Keldysh contour for an arbitrary initial
density matrix $\hat{\rho}_{0}$. Field configurations need to be specified at
the two ends of the closed time contour which are represented in bold. Right:
In thermal equilibrium this simplifies. One needs an appendage to the contour
extending along the negative imaginary axis by $\beta=T^{-1}$ and then impose
periodic boundary conditions for the full contour.

One can also expect a similar simplification for the Bjorken flow which
provides the simplest example of the evolution of an expanding system on the
forward light cone (see Sec III.1 for details). The state is assumed to have
boost invariance, and also translational and rotational invariance along the
transverse plane so that the energy-momentum tensor can be expressed only in
terms of the energy density $T_{\tau\tau}=\epsilon(\tau)$ via Ward identities,
where $\tau=\sqrt{t^{2}-z^{2}}$ is the proper time of an observer co-moving
with the flow and $z$ being the longitudinal coordinate along which the
expansion happens. (Here we use units where $c=1$.) At late time,
$\epsilon(\tau)$ is described by hydrodynamics and thus it reaches a perfect
fluid expansion, so that

$\epsilon(\tau)\approx\epsilon_{0}\left(\frac{\tau_{0}}{\tau}\right)^{\frac{d}{d-1}}.$
(5)

In the hydrodynamic regime, it can be described by a single constant
parameter, namely

$\mu:=\epsilon_{0}\tau_{0}^{\frac{d}{d-1}}.$ (6)

The full hydrodynamic series for $\epsilon(\tau)$ in powers of
$\tau^{-\frac{d-2}{d-1}}$ (essentially a derivative expansion) is given in
terms of $\mu$ (which is determined by initial conditions) and the transport
coefficients which are determined by the fundamental microscopic theory. In
this case, it is natural to ask whether there can be a simpler computation of
the Schwinger-Keldysh partition function of $W[J_{1},J_{2}]$ in the
hydrodynamic limit, since just like the thermal case, the state can be
essentially captured by a single parameter.

More generally, we would expect that general methods for computing
$W[J_{1},J_{2}]$ would exist in the hydrodynamic regime where the energy-
momentum tensor and conserved currents are described only by the hydrodynamic
variables, namely the four-velocity $u^{\mu}(t,\mathbf{x})$, the energy
density $\epsilon(t,\mathbf{x})$ (or equivalently the temperature
$T(t,\mathbf{x})$), etc., and we would not require the knowledge of the
detailed (off-diagonal) matrix elements $\rho(\phi_{1},\phi_{2})$ of the state
or the kernel $\mathcal{K}(\phi_{1},\phi_{2};J_{1},J_{2})$ explicitly. In
fact, $W[J_{1},J_{2}]$ is related to the generalization of the thermodynamic
free energy to hydrodynamics via Legendre transform, and the latter especially
in the context of macroscopic space-time configurations of conserved currents
is also known as the large deviation functional Touchette (2009) which can be
computed in many models studied in classical nonequilibrium statistical
mechanics.222The large deviation functional gives the probability for a
macroscopic space-time profile of a conserved charge or current density which
does not necessarily satisfy the hydrodynamic equations. In a quantum system,
the off-diagonal matrix elements in the basis of macroscopic field
configurations for the conserved charges and currents eventually decohere, but
the decoherence would be of interest. See Bernard (2021) for a recent
discussion on the possibility of a quantum generalization of large deviation
functional methods.

The primary aim of this work is to show how the explicit computation of
$W[J_{1},J_{2}]$ can be achieved by holographic methods in the hydrodynamic
limit of the Bjorken flow. We will also present concrete steps for
understanding how to go beyond the hydrodynamic limit and recover the initial
state.

###  I.2 A brief historical review

The first major advance in understanding thermal real-time correlation
functions in holography was the Son-Starinets prescription for computing the
retarded correlation function, according to which the ingoing boundary
condition at the horizon implements the causal linear response in the
classical gravity (large $N$ and infinite strong coupling) approximation Son
and Starinets (2002). Using the Chesler and Yaffe method for causal time
evolution in the bulk Chesler and Yaffe (2014), this approach was suitably
generalized to compute the out-of-equilibrium retarded correlation function in
holography Banerjee  _et al._ (2016). The first concrete implementation of the
Schwinger-Keldysh contour in holography is due to Son and Herzog utilizing the
eternal black hole geometry Herzog and Son (2003). The two boundaries of the
eternal black hole were shown to provide the forward and backward arms of the
Schwinger-Keldysh closed time contour with the backward arm displaced by
$-i\beta/2$ (note $\beta=T^{-1}$) along the imaginary axis.

The most concrete prescription for real-time gauge-gravity duality for general
initial states is due to Skenderis and van Rees Skenderis and van Rees (2008,
2009). This however requires detailed understanding of the state in terms of
semiclassical field configurations of dual gravity, and is best defined for
states which can be constructed using Euclidean path integrals. In this
prescription, one explicitly constructs the bulk geometry corresponding to the
boundary Schwinger-Keldysh contour with specified sources, and extends data on
the field theory contour into the bulk in an appropriate manner. It is,
however, not easy to apply this approach to realistic computations for generic
initial states. Furthermore, as mentioned above, we would expect a simpler
approach in the hydrodynamic regime. We will compare this approach with ours
in Sec VI.

Our method is based on generalizing the recently proposed horizon cap
prescription due to Crossley, Glorioso and Liu (CGL) Glorioso  _et al._ (2018)
for the static black brane dual to the thermal state. Here the Schwinger-
Keldysh contour is realized by a horizon cap, in which the ingoing Eddington-
Finkelstein radial coordinate goes around the horizon in the complex plane in
a little circle of radius $\epsilon$ (not to be confused with the energy
density) before going back to the real axis and reaching the second boundary.
Thus the two arms of the Schwinger-Keldysh contour at the two boundaries are
connected continuously in the bulk through the bulk radial contour. The
horizon cap implements the appropriate analytic continuation of bulk fields
from one arm of the bulk geometry to the other with the sources $J_{1}$ and
$J_{2}$ specified independently at the two boundaries.

###  I.3 Summary of the results

Our first result is the demonstration that the CGL horizon cap prescription
reproduces both the KMS periodicity and the ingoing boundary condition
prescription of Son and Starinets (needed to obtain the retarded correlation
function from causal response) in thermal equilibrium. Earlier in Glorioso
_et al._ (2018), both of these were demonstrated only up to quadratic orders
in the frequency. Our proof relies on a simple and novel matrix factorization
of thermal correlation functions which is reproduced by the holographic method
for extracting the correlation function of a scalar operator in the field
theory via the on-shell action of the dual bulk scalar field whose mass is
determined by the scaling dimension of the operator.333The most important
ingredient for realization of the matrix factorization is that only the terms
which are obtained from the product of an ingoing mode with an outgoing mode
can contribute to the quadratic on-shell action of the free bulk scalar field
minimally coupled to gravity.

Our primary tool for extending the CGL prescription into the (conformal)
hydrodynamic Bjorken regime is Weyl rescaling that maps the asymptotic perfect
fluid expansion to a flow in which the temperature and entropy density become
constant at late time in a non-trivial background metric without any time-like
Killing vector. This Weyl transformation can be lifted to a bulk
diffeomorphism Henningson and Skenderis (1998); de Haro  _et al._ (2001) in
which the dual black hole’s event and apparent horizon coincide at a fixed
radial location at a very late proper time. The area and surface gravity of
the horizons, and therefore the entropy density and the temperature of the
dual fluid, remain constant at late time, but the horizons shrink in the
directions transverse to the flow and expand in the longitudinal direction. In
the absence of any time-like Killing vector, the late-time behavior of the
Weyl transformed Bjorken flow is not thermal although the temperature and
energy density become constant.

The requirement that the dual geometry has a regular future black hole horizon
necessitates viscous corrections to the perfect fluid Bjorken flow, which
implies the same for the Weyl transformed version mentioned above since the
bulk dual of the latter is obtained simply via bulk diffeomorphism.
Furthermore, bulk regularity determines the precise values of all transport
coefficients of the holographic theory order by order in the derivative
expansion (equivalent to the large proper time expansion) with the Bjorken
flow giving a special case of the fluid-gravity correspondence Bhattacharyya
_et al._ (2008). The large proper time expansion of the holographic Bjorken
flow has been worked out to very high orders Heller  _et al._ (2013). We
discuss the bulk dual of the Bjorken flow and its Weyl transformed version in
Section III in detail.

We want to emphasize that the Weyl rescaling (and hence the dual bulk
diffeomorphism) is determined purely by the late perfect fluid flow regime
that is mapped to that of a constant temperature in a non-trivial background
metric in the field theory (which is also the boundary metric of the dual
black brane geometry after the bulk diffeomorphism) as mentioned above. The
Weyl transformation and consequently the Weyl rescaled background metric of
the field theory do not receive any correction at first and higher orders in
the derivative (i.e. large proper time) expansion.

As a result of the bulk diffeomorphism which implements the Weyl
transformation of the dual Bjorken flow, the Klein-Gordon equation for a bulk
scalar field with arbitrary mass can be mapped to that of a static black brane
at late proper time. In this map, the frequency needs to be appropriately
scaled, and the momenta in the static black brane geometry are identified with
co-moving longitudinal and transverse momenta.444Note that the map to the
static black brane holds only at the leading order in the large proper time
expansion. The co-moving momenta however are also defined at the leading order
itself via the Weyl transformed background metric of the field theory
(boundary metric of the bulk geometry). For vanishing momenta, this reproduces
the result of Janik and Peschanski Janik and Peschanski (2006a).

The utility of this map to the static black brane is to establish the horizon
cap prescription in the asymptotic perfect fluid limit. It follows that the
Schwinger-Keldysh correlation functions of the operator dual to the bulk
scalar field can be mapped to thermal correlation functions after suitable
space-time reparametrizations in the asymptotic perfect fluid limit of the
Bjorken flow (i.e. when both the proper time arguments of the two-point
correlation functions are sufficiently large). The non-thermal nature of the
Schwinger-Keldysh correlation functions can thus be absorbed into these space-
time reparametrizations up to overall (proper time-dependent) Weyl factors.

We also show that the horizon cap prescription extends to all orders in the
late proper time expansion. Firstly, at first and higher orders, the Klein-
Gordon equation of the bulk scalar takes the same form as at the leading order
but with source terms, and can be systematically solved at each order in the
large proper time expansion such that the leading behavior of the ingoing and
outgoing modes are exactly the same as at the zeroth order (that can be mapped
to the static black brane) implying that the standard analytic continuation at
the horizon cap can be performed at each order. Remarkably, this is possible
only if the horizon cap is pinned to the time-dependent event horizon (and not
the apparent or other dynamical horizons which do not coincide with the event
horizon beyond the zeroth order).

Second, the requirement of the near-horizon behavior of the ingoing and
outgoing modes at the time-dependent event horizon does not completely
determine all the order-by-order corrections. The undetermined coefficients
are those that give the leading behavior of the ingoing modes at the horizon
at first and higher orders in the proper time expansion. However, when the on-
shell action of the scalar field is used to extract the Schwinger-Keldysh
correlation functions, we find that they are consistent with all field theory
identities provided one of these identities which hold for arbitrary
nonequilibrium states provided one of these identities is used to determine
the unfixed coefficients that give the leading near-horizon behavior of the
ingoing modes.

Therefore, we establish the horizon cap prescription can be unambiguously
extended to determine the Schwinger-Keldysh correlation functions of the
entire hydrodynamic tail of the Bjorken flow at any arbitrary order in the
large proper time expansion. Aside from satisfying all field theory
identities, we find that the retarded propagator given by the ingoing modes
reproduces the normalizable bulk solutions at complex frequencies (which map
to quasi-normal modes of the static black brane Janik and Peschanski (2006a))
with vanishing sources to all orders although the relevant phase factors at
first and higher orders are determined as functions of the frequency via the
near-horizon behavior of the outgoing modes at real frequencies. The latter
implies non-trivially that one can obtain the normalizable bulk solutions dual
to the transients (nonequilibrium generalization of the collective excitations
of the system) to all orders in the large proper time expansion from the
retarded correlation function generalizing how quasinormal modes are obtained
from the poles of the retarded propagator in thermal equilibrium. To all
orders, the Schwinger-Keldysh correlation functions satisfy a matrix
factorized form implying a bilocal thermal structure.

Let us briefly provide some explicit expressions. The $d$-dimensional Bjorken
flow in the field theory is conveniently expressed in the Milne coordinates,
$\tau=\sqrt{t^{2}-z^{2}}$ (the proper time with $z$ being the longitudinal
Minkowski coordinate), the rapidity $\zeta={\rm arctanh}(z/t)$ and the
transverse coordinates $\vec{x}_{\perp}$.

It is convenient to define

$\sigma=\tau^{\frac{d-2}{d-1}}\tau_{0}^{\frac{1}{d-1}},\quad\hat{\zeta}=\zeta\tau_{0}.$
(7)

with $\tau_{0}$ being an arbitrarily chosen fixed proper time. Furthermore,
for the arguments $(\sigma_{1},\zeta_{1},\vec{x}_{\perp 1})$ and
$(\sigma_{2},\zeta_{2},\vec{x}_{\perp 2})$ of the two-point correlation
function let us define

$\displaystyle\overline{\sigma}=\frac{1}{2}(\sigma_{1}+\sigma_{2}),\quad\sigma_{r}=\sigma_{1}-\sigma_{2},\quad\widetilde{\zeta}_{r}=(\hat{\zeta}_{1}-\hat{\zeta}_{2})\frac{\overline{\sigma}}{\tau_{0}},\quad\widetilde{x_{\perp}}_{r}=|\vec{x}_{\perp
1}-\vec{x}_{\perp
2}|\left(\frac{\tau_{0}}{\overline{\sigma}}\right)^{\frac{1}{d-2}}.$ (8)

As discussed above, at late proper time, the Schwinger-Keldysh correlation
functions
$G(\sigma_{1},\sigma_{2},\hat{\zeta}_{1}-\hat{\zeta}_{2},|\vec{x}_{\perp
1}-\vec{x}_{\perp 2}|)$ of the Bjorken flow can be mapped to thermal
correlation functions $G_{\beta}$ at a specific temperature $T=\beta^{-1}$ as
follows

$G(\sigma_{1},\sigma_{2},\hat{\zeta}_{1}-\hat{\zeta}_{2},|\vec{x}_{\perp
1}-\vec{x}_{\perp
2}|)\rightarrow\left(\frac{\tau_{0}}{\overline{\sigma}}\right)^{\frac{2\Delta_{O}}{d-2}}{G}_{\beta}\left(\frac{d-1}{d-2}\sigma_{r},\sqrt{\widetilde{\zeta}_{r}^{2}+\widetilde{x_{\perp}}_{r}^{2}}\right).$
(9)

This limit implies that both $\sigma_{1}$ and $\sigma_{2}$ are large. Above
$\Delta_{O}$ is the scaling dimension of the operator (which is determined by
the mass of the dual bulk scalar field). Also both $G$ and $G_{\beta}$ are
$2\times 2$ matrices with indices determining whether the arguments
$(\sigma_{1},\zeta_{1},\vec{x}_{\perp 1})$ and
$(\sigma_{2},\zeta_{2},\vec{x}_{\perp 2})$ are in the forward or backward legs
of the Schwinger-Keldysh time-contour. Finally, the temperature $T$ of the
thermal is determined by the (only free) parameter $\epsilon_{0}$ of the
Bjorken flow (see Eq. (5)) as follows

$\beta=T^{-1}=\frac{4\pi\varepsilon_{0}^{1/d}}{d},\quad\varepsilon_{0}=\frac{16\pi
G_{N}}{d-1}\epsilon_{0},$ (10)

where the $d+1$-dimensional gravitational constant $G_{N}$ is given by the
rank of the gauge group of the dual theory (as for instance, in
$\mathcal{N}=4$ super Yang-Mills theory with $SU(N)$ gauge group in $4$
dimensions, $G_{N}^{-1}=2N^{2}/\pi$).555Apparently, the thermal form on the
right hand side of Eq. 10 depends on the choice of $\tau_{0}$. However, a
shift in $\tau_{0}$ also changes the parameter $\epsilon_{0}$ appropriately so
that the thermal form in Eq. 10 is unambiguous. This is explained in Section
V.3 explicitly. Remarkably, the map to the thermal form involving space-time
reparametrization implies the emergence of $SO(d-1)$ rotational symmetry, and
also time translation symmetry although these are absent in the original
coordinates $\tau$ and $\zeta$.

As mentioned above, we can systematically include viscous and higher-order
corrections to the correlation functions and obtain

$G(\sigma_{1},\sigma_{2},\hat{\zeta}_{1}-\hat{\zeta}_{2},|\vec{x}_{\perp
1}-\vec{x}_{\perp
2}|)=\left(\frac{\tau_{0}}{\overline{\sigma}}\right)^{\frac{2\Delta_{O}}{d-2}}\sum_{n=0}^{\infty}\frac{1}{\overline{\sigma}^{n}\varepsilon_{0}^{n/d}}G_{n}\left(T_{0}\sigma_{r},T_{0}\widetilde{\zeta}_{r},T_{0}\widetilde{x_{\perp}}_{r}\right)$
(11)

where $G_{0}$ coincides with the thermal correlation function $G_{\beta}$
given by (9).

We also note that our result that the horizon cap for the holographic
hydrodynamic Bjorken flow should be pinned to the nonequilibrium event horizon
captures the causal nature of the Schwinger-Dyson equations for the real-time
(out-of-equilibrium) correlation functions in the dual field theory.666The
latter is manifest when written in terms of the coupled evolution of the
statistical and spectral functions Berges (2004).

The series in Eq. (11) is not expected to be convergent, and therefore
requires a trans-series completion with appropriate Stokes data (distinct from
the Stokes data for the expectation value of the energy density which defines
the bulk geometry) which should be actually functions of $\sigma_{r}$,
$\widetilde{\zeta}_{r}$ and $\widetilde{x_{\perp}}_{r}$. We discuss their
physical role in deciphering the information of the initial state which is
lost in hydrodynamization, and also how they can be used to decode the
interior of the event horizon.

###  I.4 Plan

The paper is organized as follows. In Section II, we introduce the Crossley-
Glorioiso-Liu (CGL) horizon cap prescription for the thermal Schwinger-Keldysh
correlation functions in holography. We prove that the prescription indeed
reproduces the KMS periodicity so that they are given just in terms of the
retarded correlation function, and that the latter is exactly what we obtain
from the Son-Starinets prescription. As mentioned, we use a new matrix
factorization of thermal correlation functions. In Section III, we review the
Bjorken flow and its holographic dual. We also introduce the Weyl rescaling of
the Bjorken flow along with the dual bulk diffeomorphism such that the final
state has a fixed temperature and entropy density. As mentioned, although the
event horizon has a constant surface gravity and area at late time, it
stretches and expands in the directions longitudinal and transverse to the
flow respectively. Additionally, we discuss the proper residual gauge
transformation corresponding to radial reparametrization.

In Section IV, we study the probe bulk scalar field in the gravitational
background dual to the hydrodynamic Bjorken flow and show how we can preserve
the analytic structure of the horizon cap to all orders in the proper time
expansion. Crucially, we find that it requires the horizon cap to be pinned to
the nonequilibrium event horizon. In Section V, we use these results to
extract the real-time correlation functions of the hydrodynamic Bjorken flow.
After presenting the result for the perfect fluid limit in terms of a thermal
propagator with space-time reparametrizations, we show how we systematically
obtain the corrections in a proper time expansion. We also discuss many non-
trivial consistency checks of our results. In Section VI, we present a
discussion on how a trans-series completion of this expansion can lead to
seeing the quantum fluctuations behind the nonequilibrium event horizon, and
matching with initial data lost during hydrodynamization.

Finally, we conclude in Section VII with an outlook.

##  II The CGL horizon cap of the thermal black brane

The Crossley-Glorioso-Liu (CGL) horizon cap prescription is a simple proposal
for the holographic realization of the Schwinger-Keldysh contour at thermal
equilibrium Glorioso  _et al._ (2018). The thermal nature of the correlation
functions obtained from this prescription, including their consistency with
the Kubo-Martin-Schwinger (KMS) periodicity, has been explicitly verified up
to quadratic order in the small frequency expansion in Glorioso  _et al._
(2018). In Glorioso  _et al._ (2018), it has also been verified that the
retarded correlation function is implied by the ingoing boundary condition, as
demanded by the Son-Starinets prescription Son and Starinets (2002) up to the
quadratic order in frequency. These were sufficient to obtain a rudimentary
effective theory of diffusion and dissipative hydrodynamics from holography
Ghosh  _et al._ (2021); Crossley  _et al._ (2017); Glorioso  _et al._ (2017,
2019); Jensen  _et al._ (2018); Liu and Glorioso (2018); Glorioso  _et al._
(2018); de Boer  _et al._ (2019); Chakrabarty  _et al._ (2020); He  _et al._
(2022a); Jana  _et al._ (2020); He  _et al._ (2022b). Here, we present an
elegant proof that the CGL horizon cap indeed gives thermal correlation
functions satisfying KMS periodicity, and that it also implies the Son-
Starinets prescription for the retarded correlation function, at any arbitrary
frequency and momenta.777The real-time prescription Skenderis and van Rees
(2009, 2008) of van Rees and Skenderis leads to the ingoing boundary condition
as shown in van Rees (2009) in thermal equilibrium. Our methods discussed here
discuss a natural generalization away from equilibrium for out-of-equilibrium,
especially hydrodynamic states. For other approaches, see Jana  _et al._
(2020).

The generating functional for the thermal Schwinger-Keldysh correlation
functions in a quantum field theory is888Unless specified, we will always put
the backward arm of the Schwinger-Keldysh time contour infinitesimally below
the real axis. Also, we will often omit explicit mention of the appendage of
the contour along the imaginary axis which creates the thermal state in the
infinite past.

$e^{W[J_{1},J_{2}]}={\rm Tr}\left[\hat{\rho}_{\beta}T_{c}e^{i\int{\rm
d}t\int{\rm
d}^{d-1}x\,(\hat{O}_{1}(\mathbf{x},t)J_{1}(\mathbf{x},t)-\hat{O}_{2}(\mathbf{x},t)J_{2}(\mathbf{x},t))}\right]$
(12)

with $\rho_{\beta}$ denoting the thermal density matrix, $1$ and $2$ denoting
the forward and backward arms of the contour, and $T_{c}$ denoting (time)
contour ordering. The contour ordering implies that (with
$\langle\cdot\rangle\equiv{\rm Tr}(\hat{\rho}_{\beta}\cdot)$)

$\displaystyle
G_{11}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})=\frac{\delta^{2}W}{\delta
J_{1}(t,\mathbf{x})\delta J_{1}(t^{\prime},\mathbf{x}^{\prime})}=-i\langle
T(\hat{O}(t,\mathbf{x})\hat{O}(t^{\prime},\mathbf{x}^{\prime}))\rangle,$ (13)
$\displaystyle-
G_{12}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})=\frac{\delta^{2}W}{\delta
J_{1}(t,\mathbf{x})\delta
J_{2}(t^{\prime},\mathbf{x}^{\prime})}=i\langle\hat{O}(t^{\prime},\mathbf{x}^{\prime})\hat{O}(t,\mathbf{x}))\rangle,$
$\displaystyle-
G_{21}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})=\frac{\delta^{2}W}{\delta
J_{2}(t,\mathbf{x})\delta
J_{1}(t^{\prime},\mathbf{x}^{\prime})}=i\langle\hat{O}(t,\mathbf{x})\hat{O}(t^{\prime},\mathbf{x}^{\prime}))\rangle,$
$\displaystyle
G_{22}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})=\frac{\delta^{2}W}{\delta
J_{2}(t,\mathbf{x})\delta
J_{2}(t^{\prime},\mathbf{x}^{\prime})}=-i\langle\overline{T}(\hat{O}(t,\mathbf{x})\hat{O}(t^{\prime},\mathbf{x}^{\prime}))\rangle.$

Succinctly we can write the above as

$\displaystyle G_{ij}(x_{1},x_{2})=\frac{\delta^{2}W}{\delta
J_{i}(x_{1})\delta J_{j}(x_{2})}(-)^{i+j}$ (14)

with $(i,j)=(1,2)$. The above holds even out of equilibrium with
$\rho_{\beta}$ in (12) replaced by an arbitrary initial state $\rho_{0}$.

It can readily be shown that the KMS periodicity (arising from the ${\rm Tr}$
in (12) after extending the contour along the negative imaginary axis by
$\beta$ as shown in Fig. 1) implies that the thermal correlation functions in
Fourier space (with $i$ and $j$ standing for the $1$ (forward) or $2$
(backward) arms of the contour) defined as

$G_{ij}(\omega,\mathbf{k})=\int{\rm d}t~{}{\rm
d^{d-1}}x\,e^{i\omega(t-t^{\prime})}e^{-i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x}^{\prime})}G_{ij}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})$
(15)

assume the form

$\displaystyle G_{ij}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}G_{R}(\omega,\mathbf{k})(1+n(\omega))-G_{A}(\omega,\mathbf{k})n(\omega)\,&\,-(G_{R}(\omega,\mathbf{k})-G_{A}(\omega,\mathbf{k})n(\omega)\\\
-(G_{R}(\omega,\mathbf{k})-G_{A}(\omega,\mathbf{k})(1+n(\omega))\,&\,G_{R}(\omega,\mathbf{k})n(\omega)-G_{A}(\omega,\mathbf{k})(1+n(\omega))\end{pmatrix}$

where

$G_{R}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})=-i\theta(t-t^{\prime})\langle[\hat{O}(t,\mathbf{x}),\hat{O}(t^{\prime},\mathbf{x}^{\prime})]\rangle$
(17)

is the retarded propagator,

$G_{A}(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime})=-i\theta(t^{\prime}-t)\langle[\hat{O}(t^{\prime},\mathbf{x}^{\prime}),\hat{O}(t,\mathbf{x})]\rangle$
(18)

is the advanced propagator, and $n(\omega)=1/(e^{\beta\omega}-1)$ is the Bose-
Einstein distribution function. It is easy to see from these definitions that

$G_{A}^{*}(\omega,\mathbf{k})=G_{R}(\omega,\mathbf{k})~{}.$ (19)

The crucial element of the proof of why the CGL prescription works is a simple
and general factorization property of thermal correlation functions in field
theory (irrespective of whether the theory is holographic or not). The
Schwinger-Keldysh thermal correlation functions (II) obtained by
differentiating the real-time partition function at a temperature
$T=\beta^{-1}$ can be factorized as shown below

$\displaystyle
G_{ij}(\omega,\mathbf{k})=\sigma_{3}\cdot\begin{pmatrix}A(\omega,\mathbf{k})\,&\,B(\omega,\mathbf{k})\\\
A(\omega,\mathbf{k})\,&\,B(\omega,\mathbf{k})e^{\beta\omega}\end{pmatrix}\cdot\begin{pmatrix}a(\omega,\mathbf{k})\,&\,b(\omega,\mathbf{k})\\\
a(\omega,\mathbf{k})\,&\,b(\omega,\mathbf{k})e^{\beta\omega}\end{pmatrix}^{-1},$
(20)

in which $\sigma_{3}={\rm diag}(1,-1)$ is the third Pauli matrix, and

$G_{R}(\omega,\mathbf{k})=\frac{A(\omega,\mathbf{k})}{a(\omega,\mathbf{k})}=\frac{B^{*}(\omega,\mathbf{k})}{b^{*}(\omega,\mathbf{k})}=G_{A}^{*}(\omega,\mathbf{k}).$
(21)

Clearly $A\rightarrow\lambda A$, $B\rightarrow\tilde{\lambda}B$,
$a\rightarrow\lambda a$ and ${b}\rightarrow\tilde{\lambda}b$ gives the same
thermal matrix, so the factorization is unique up to the multiplicative
complex functions $\lambda(\omega,\mathbf{k})$ and
$\tilde{\lambda}(\omega,\mathbf{k})$.

The CGL horizon cap glues two copies of the black brane geometry, whose
boundaries represent the forward and backward arms of the Schwinger-Keldysh
time contour respectively, at the horizon as shown in Fig. 2. For reasons to
become clear later, this prescription is easily implemented in the ingoing
Eddington-Finkelstein (EF) coordinates. The ingoing EF radial coordinates of
the two geometries, representing the forward ($1$) and backward ($2$) arms of
the time contour respectively, are displaced along the imaginary axis by
$\mp\epsilon$ (i.e. $r_{1}\rightarrow r_{1}-i\epsilon$ and $r_{2}\rightarrow
r_{2}+i\epsilon$). The smooth gluing is achieved by the encircling of the
complexified radial coordinate around the horizon $r=r_{h}$ clockwise along a
circle of radius $\mathcal{O}(\epsilon)$ as it is analytically continued from
the (first) copy dual to the forward contour to the (second) copy dual to the
backward contour. The direction of time in the second copy has to be reversed
so that full complexified bulk geometry has a single orientation. Therefore,
the analytic continuation of the radial coordinate automatically necessitates
the closed Schwinger-Keldysh time contour.

Figure 2: Top: The radial contour in the complexified two-sheeted black brane
geometry on a constant time hypersurface. The radial coordinate goes around
the horizon $r=r_{h}$ in the complex plane forming the horizon cap, and
connects the two arms of the Schwinger-Keldysh contour at the two boundaries
at a constant Lorentzian time. The analytic continuation along the horizon cap
gives a well-defined Dirichlet value problem for the bulk fields in terms of
their sources specified at the two boundaries. Bottom: Shows the time
evolution of the radial contour (shown in blue lines) in the full complexified
space-time ending at the two Lorentzian arms at the two boundaries. The
reversal of the direction of time in the second copy implies that the full
complexified space-time has a single orientation after the two copies are
smoothly glued with the horizon cap.

Explicitly, the $AdS_{d+1}$ static black brane geometry in the ingoing
Eddington-Finkelstein coordinates is

$\displaystyle{\rm d}s^{2}=-\frac{2}{r^{2}}{\rm d}r{\rm
d}t-\frac{1}{r^{2}}\left(1-r^{d}\varepsilon_{0}\right){\rm
d}t^{2}+\frac{1}{r^{2}}({\rm d}x_{1}^{2}+\cdots+{\rm d}x_{d-1}^{2}).$ (22)

where $r$ is the bulk radial coordinate, $t$ is the Eddington-Finkelstein time
and the horizon is at $r=r_{h}=\varepsilon_{0}^{-1/d}$. The on-shell action
for bulk fields in this geometry is identified with the generating functional
of connected real-time correlation functions of the dual operators at the
boundary. A bulk scalar field configuration can be written in the form

$\displaystyle\Phi(r,t,\mathbf{x})=\int\frac{{\rm d}\omega}{2\pi}\frac{{\rm
d}^{d-1}k}{(2\pi)^{d-1}}\,e^{-i\omega
t}e^{i\mathbf{k}\cdot\mathbf{x}}\Phi(r,\omega,\mathbf{k}).$ (23)

On-shell, $\Phi(r,\omega,\mathbf{k})$ is a sum of two linearly independent
solutions $\phi_{\rm in}(r,\omega,\mathbf{k})$ and $\phi_{\rm
out}(r,\omega,\mathbf{k})$ which are ingoing and outgoing at the horizon
respectively. Therefore,

$\displaystyle\Phi(r,\omega,\mathbf{k})=\phi_{\rm
in}(r,\omega,\mathbf{k})p(\omega,\mathbf{k})+\phi_{\rm
out}(r,\omega,\mathbf{k})q(\omega,\mathbf{k})$ (24)

generally with $p(\omega,\mathbf{k})$ and $q(\omega,\mathbf{k})$ representing
the arbitrary Fourier coefficients of the solutions which are ingoing and
outgoing at the horizon respectively. The latter thus provide a basis of
solutions for given $\omega$ and $\mathbf{k}$, and can be uniquely defined via
the following conditions

$\phi_{\rm in}(r_{h},\omega,\mathbf{k})=1,\qquad\lim_{r\rightarrow
r_{h}}\phi_{\rm
out}(r,\omega,\mathbf{k})(r_{h}-r)^{-\frac{i\beta\omega}{2\pi}}=1,$ (25)

where

$\beta=T^{-1}=\frac{4\pi r_{h}}{d}$ (26)

is the inverse Hawking temperature of the black brane, and
$r_{h}=\varepsilon_{0}^{-1/d}$ is the radial location of the horizon. Indeed,
near the horizon ($r\approx r_{h}$),

$\Phi(r,\omega,\mathbf{k})\approx
p(\omega,\mathbf{k})+(r_{h}-r)^{\frac{i\beta\omega}{2\pi}}q(\omega,\mathbf{k})$
(27)

as should follow from the universal validity of the geometrical optics
approximation at the horizon. The CGL horizon cap prescription for the
analytic continuation of the radial coordinate from one copy of the bulk
space-time to another then implies that the Fourier coefficients of the on-
shell solutions in the two copies are related by

$p_{2}(\omega,\mathbf{k})=p_{1}(\omega,\mathbf{k}),\qquad
q_{2}(\omega,\mathbf{k})=e^{\beta\omega}q_{1}(\omega,\mathbf{k}),$ (28)

with $1$ and $2$ denoting the copies ending on the forward and backward arms
of the time contour respectively at their boundaries. The on-shell solution in
the full geometry can therefore be written in the following matrix form:

$\displaystyle\begin{pmatrix}\Phi_{1}(r,t,\mathbf{x})\\\
\Phi_{2}(r,t,\mathbf{x})\end{pmatrix}=\int\frac{{\rm d}\omega}{2\pi}\frac{{\rm
d}^{d-1}k}{(2\pi)^{d-1}}\,e^{-i\omega
t}e^{i\mathbf{k}\cdot\mathbf{x}}\mathcal{M}(r,\omega,\mathbf{k})\cdot\begin{pmatrix}p(\omega,\mathbf{k})\\\
q(\omega,\mathbf{k})\end{pmatrix}$ (29)

with the matrix

$\mathcal{M}(r,\omega,\mathbf{k})=\begin{pmatrix}\phi_{\rm
in}(r,\omega,\mathbf{k})~{}~{}~{}~{}~{}\phi_{\rm out}(r,\omega,\mathbf{k})\\\
\phi_{\rm in}(r,\omega,\mathbf{k})~{}~{}~{}~{}~{}e^{\beta\omegaup}\phi_{\rm
out}(r,\omega,\mathbf{k})\end{pmatrix}$ (30)

providing a basis of solutions for the entire complexified space-time
comprising of the two copies smoothly glued at the horizon. The sources
$J_{1}(\omega,\mathbf{k})$ and $J_{2}(\omega,\mathbf{k})$ specified at the two
boundaries (see below) implement the Dirichlet boundary conditions that
determine $p(\omega,\mathbf{k})$ and $q(\omega,\mathbf{k})$ uniquely for real
frequencies and momenta, and thus yielding a unique bulk field configuration
in the full complexified space-time.

According to the holographic dictionary, the generating functional for the
connected correlation functions is identified with the on-shell action for the
scalar field dual to the operator $\hat{O}$, on the full complexified space-
time, i.e.

$W[J_{1},J_{2}]=iS_{\rm on-shell}.$ (31)

Assuming minimal coupling to gravity, the on-shell quadratic action for the
bulk scalar field $\Phi$ dual to a scalar operator takes the form

$\displaystyle S_{on-shell}=S_{in-in}+S_{in-out}+S_{out-out},$ (32)

The first piece $S_{in-in}$ is quadratic in the ingoing mode. Since the
ingoing mode is analytic at the horizon, the contributions from the two arms
cancel each other out (as the solutions are the same on the two arms) while
the circle around the horizon does not contribute as well. Therefore, $S_{in-
in}=0$. Note if we keep the ingoing mode alone, then $J_{1}=J_{2}$. In the
field theory, $W[J_{1}=J_{2}]=0$ because the partition function $e^{W}$ with
the same unitary evolution forward and backward in time, equals unity.
Therefore, $S_{in-in}=0$ is consistent with field theory.

The second piece $S_{in-out}$, which is the sum of cross-terms between the in
and outgoing modes, has a branch point at the horizon. Integrating over the
two arms amounts to integrating around a branch cut, and results in the two
boundary terms on-shell, i.e.

$\displaystyle S_{in-out}=S_{bdy1}+S_{bdy2}.$ (33)

The third piece $S_{out-out}$ has a possibility of a pole at the horizon, i.e.
$(r_{h}-r)^{-1}$ terms in the Lagrangian density arising from the radial
derivative acting on the non-analytic piece
$(r_{h}-r)^{\frac{i\beta\omega}{2\pi}}$, which we denote collectively as
$S_{\epsilon}$. Essentially $S_{\epsilon}$ gets contributions from the
following two terms:

$\displaystyle S_{\epsilon}\propto\int{\rm d}\omegaup\int{\rm
d}^{d}k\oint_{\epsilon}\,{\rm
d}r\sqrt{-G}\Bigg{(}G^{rr}\partial_{r}\phi_{out}^{*}\partial_{r}\phi_{out}+G^{rt}\left(\partial_{r}\phi_{out}^{*}\partial_{t}\phi_{out}+\partial_{r}\phi_{out}\partial_{t}\phi_{out}^{*}\right)\Bigg{)}.$
(34)

Remarkably, the poles originating from these two terms cancel each other out
(note $\beta$ is given by (26)) resulting in $S_{\epsilon}=0$. The remaining
terms quadratic in the outgoing mode are analytic, so that $S_{out-out}$ is
also the sum of two boundary terms. These two boundary terms cancel each other
out as in the ingoing case. On the gravitational side, the easy way to see
this is by first writing the contributions from the forward and backward arms
of the contour separately. The boundary contributions from one arm in the
integrand would be proportional to

$\phi_{b}(-\omega,k)(\partial_{r}\phi)_{b}(\omega,k)+\cdots$

with $\phi_{b}$ and $(\partial_{r}\phi)_{b}$ denoting the boundary values of
$\phi$ and its radial derivative respectively, and $\cdots$ include counter-
terms too. The contributions from the forward and backward parts of the
contour come with opposite signs. If we consider the terms quadratic in the
outgoing mode, then $\phi_{b}(\omega,k)$ picks up a factor of
$e^{\beta\omega}$ while $\phi_{b}(-\omega,k)$ picks up a factor of
$e^{-\beta\omega}$ via analytic continuation through the horizon cap, and the
product of these factors is unity. Therefore, the contributions from the
forward and backward contours cancel out leading to $S_{out-out}=0$.

It is useful to see this also from the field theory perspective. If we keep
the outgoing modes only, then
$J_{2}(\omega,\mathbf{k})=J_{1}(\omega,\mathbf{k})e^{\beta\omega}$. In any
field theory Liu and Glorioso (2018)

$W[J_{1}(t,\mathbf{x}),J_{2}(t,\mathbf{x})]=W_{T}[J_{2}(t-i\beta,\mathbf{x}),J_{1}(t,\mathbf{x})],\,\,\,{\rm
i.e.}\,\,\,W[J_{1}(t,\mathbf{x}),J_{2}(t+i\beta,\mathbf{x})]=W_{T}[J_{2}(t,\mathbf{x}),J_{1}(t,\mathbf{x})],$

where $W_{T}$ stands for the time-reversed process in which we specify with
the same density matrix in the future instead of the initial
time.999Succinctly, $W[J_{1},J_{2}]={\rm
Tr}(\rho_{0}\tilde{U}[J_{2}]U[J_{1}])$ where $U[J_{1}]$ is forward evolution
with source $J_{1}$ and $\tilde{U}[J_{2}]$ is backward evolution with source
$J_{2}$. Similarly, $W_{T}[J_{1},J_{2}]={\rm
Tr}(\tilde{U}[J_{2}]\rho_{0}U[J_{1}])={\rm
Tr}(\rho_{0}U[J_{1}]\tilde{U}[J_{2}]).$ For
$J_{1}(t,\mathbf{x})=J_{2}(t,\mathbf{x})=J(\omega,\mathbf{k})$, this amounts
to

$W_{T}[J(\omega,\mathbf{k}),J(\omega,\mathbf{k}))]=W[J(\omega,\mathbf{k}),J(\omega,\mathbf{k})e^{\beta\omega}].$

The LHS of the above equation vanishes because once again the forward and
backward evolution with the same source are inverses of each other (there is
no operator insertion in the past now although the state is specified in the
future). Therefore, the RHS of the above equation should vanish too, implying
that

$W[J_{2}(\omega,\mathbf{k})=J_{1}(\omega,\mathbf{k})e^{\beta\omega}]=0.$ (35)

Thus, $S_{out-out}=0$ is consistent with field theory. See also footnote 12
for a more straightforward verification that the thermal correlators in the
dual theory originate from $S_{in-out}$ alone.

The upshot is that we obtain only two boundary contributions from the cross-
term between the ingoing and outgoing modes, so that

$\displaystyle S_{on-shell}\equiv S_{in-out}=S_{bdy1}+S_{bdy2}$ (36)

where $S_{bdy1}$ and $S_{bdy2}$ are the contributions from the two boundaries
after taking into account counter-terms necessary for holographic
renormalization Skenderis (2002). This implies that

$S_{\rm on-shell}[J_{1},J_{2}]=\int{\rm d}t\int{\rm d}^{d-1}x\,(\langle
O_{1}(t,\mathbf{x})\rangle J_{1}(t,\mathbf{x})-\langle
O_{2}(t,\mathbf{x})\rangle J_{2}(t,\mathbf{x}))$ (37)

with

$\displaystyle\begin{pmatrix}J_{1}(\omega,\mathbf{k})\\\
J_{2}(\omega,\mathbf{k})\end{pmatrix}$ $\displaystyle=$
$\displaystyle\mathcal{S}(\omega,\mathbf{k})\cdot\begin{pmatrix}p(\omega,\mathbf{k})\\\
q(\omega,\mathbf{k})\end{pmatrix},$ $\displaystyle\begin{pmatrix}\langle
O_{1}(\omega,\mathbf{k})\rangle\\\ \langle
O_{2}(\omega,\mathbf{k})\rangle\end{pmatrix}$ $\displaystyle=$
$\displaystyle\mathcal{R}(\omega,\mathbf{k})\cdot\begin{pmatrix}p(\omega,\mathbf{k})\\\
q(\omega,\mathbf{k})\end{pmatrix}$ (38) $\displaystyle=$
$\displaystyle(\mathcal{R}\cdot\mathcal{S}^{-1})(\omega,\mathbf{k})\cdot\begin{pmatrix}J_{1}(\omega,\mathbf{k})\\\
J_{2}(\omega,\mathbf{k})\end{pmatrix},$

The matrices $\mathcal{R}$ and $\mathcal{S}$ are defined as follows. Let the
asymptotic ($r\approx 0$) expansions of the ingoing and outgoing modes
be101010The scaling dimension $\Delta$ is related to the mass via
$\Delta=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m^{2}L^{2}}$ with $L$ being the AdS
radius.

$\displaystyle\phi_{\rm in}(r,\omega,\mathbf{k})$ $\displaystyle=$
$\displaystyle
r^{d-\Delta}(a_{0}(\omega,\mathbf{k})+\cdots)+r^{\Delta}(A_{0}(\omega,\mathbf{k})+\cdots),$
$\displaystyle\phi_{\rm out}(r,\omegaup,\mathbf{k})$ $\displaystyle=$
$\displaystyle
r^{d-\Delta}(b_{0}(\omega,\mathbf{k})+\cdots)+r^{\Delta}(B_{0}(\omega,\mathbf{k})+\cdots).$
(39)

Then

$\mathcal{S}(\omega,\mathbf{k})=\begin{pmatrix}a_{0}(\omega,\mathbf{k})~{}~{}~{}~{}b_{0}(\omega,\mathbf{k})\\\
a_{0}(\omega,\mathbf{k})~{}~{}~{}~{}e^{\beta\omega}b_{0}(\omega,\mathbf{k})\end{pmatrix}=\lim_{r\rightarrow
0}r^{\Delta-d}\mathcal{M}(r,\omega,\mathbf{k})$ (40)

and111111Note that, asymptotically,
$\mathcal{M}(r,\omega,\mathbf{k})=r^{d-\Delta}(\mathcal{S}(\omega,\mathbf{k})+\cdots)+r^{\Delta}(\mathcal{R}(\omega,\mathbf{k})+\cdots)$.

$\mathcal{R}(\omega,\mathbf{k})=(2\Delta-d)\begin{pmatrix}A_{0}(\omega,\mathbf{k})~{}~{}~{}~{}B_{0}(\omega,\mathbf{k})\\\
A_{0}(\omega,\mathbf{k})~{}~{}~{}~{}e^{\beta\omega}B_{0}(\omega,\mathbf{k})\end{pmatrix}+\cdots.$
(41)

The $\cdots$ above stands for (state-independent) contact terms which we
ignore. Denoting

$\widehat{G}(\omega,\mathbf{k})=(\sigma_{3}\cdot\mathcal{R}\cdot\mathcal{S}^{-1})(\omega,\mathbf{k})$
(42)

(with $\sigma_{3}={\rm diag}(1,-1)$) we find from (37), (II), (40), (41) and
(42) that121212$J_{i}(\omega,\mathbf{k})=\int{\rm d}t\,{\rm
d}^{d-1}x\,e^{i\omega t}e^{-i\mathbf{k}\cdot\mathbf{x}}J_{i}(t,\mathbf{x})$.

$\displaystyle S_{\rm on-shell}[J_{1},J_{2}]$ $\displaystyle=$
$\displaystyle\int\frac{{\rm d}\omega}{2\pi}\frac{{\rm
d}^{d-1}k}{(2\pi)^{d-1}}(J_{1}(-\omega,-\mathbf{k})\langle
O_{1}(\omega,\mathbf{k})\rangle-J_{2}(-\omega,-\mathbf{k})\langle
O_{2}(\omega,\mathbf{k})\rangle)$ (43) $\displaystyle=$
$\displaystyle\int\frac{{\rm d}\omega}{2\pi}\frac{{\rm
d}^{d-1}k}{(2\pi)^{d-1}}(J_{1}(-\omega,-\mathbf{k})\,\,J_{2}(-\omega,-\mathbf{k}))\cdot\widehat{G}(\omega,\mathbf{k})\cdot\begin{pmatrix}J_{1}(\omega,\mathbf{k})\\\
J_{2}(\omega,\mathbf{k})\end{pmatrix}.$

Therefore, the identification (31) together with (13) implies that,131313The
reader can check that substituting
$p(\omega,\mathbf{k})+q(\omega,\mathbf{k})=J_{1}(\omega,\mathbf{k})$ and
$p(\omega,\mathbf{k})+q(\omega,\mathbf{k})e^{\beta\omegaup}=J_{2}(\omega,\mathbf{k})$
in the on-shell action (43), and using the thermal form of the propagators
below, that indeed only cross-terms between $p$ and $q$, i.e. the in and
outgoing modes appear. There are no contributions from terms quadratic in $p$
or in $q$, implying that $S_{in-in}=S_{out-out}=0$ as claimed above, and also
$S_{on-shell}=S_{in-out}$.

$\displaystyle G_{ij}(\omega,\mathbf{k})=\frac{\partial^{2}S_{\rm on-
shell}}{\partial J_{i}(-\omega,-\mathbf{k})\partial
J_{j}(\omega,\mathbf{k})}(2\pi)^{d}=\widehat{G}(\omega,\mathbf{k}),$ (44)

From the matrix factorization of thermal correlation functions given by (20),
we readily find from (40), (41) and (42) that the correlation functions
obtained by differentiating the on-shell gravitational action are thermal,
i.e. assume the form (II) provided141414It is obvious that to map to the
factorization in (20), we have to set $A=(2\Delta-d)A_{0}$,
$B=(2\Delta-d)B_{0}$, $a=a_{0}$ and $b=b_{0}$.

$G_{R}(\omegaup,\mathbf{k})=(2\Delta-d)\frac{A_{0}(\omegaup,\mathbf{k})}{a_{0}(\omegaup,\mathbf{k})},\qquad
G_{A}(\omegaup,\mathbf{k})=(2\Delta-d)\frac{B_{0}(\omegaup,\mathbf{k})}{b_{0}(\omegaup,\mathbf{k})}.$
(45)

Remarkably, the above are exactly the Son-Starinets prescriptions Son and
Starinets (2002) for the retarded and advanced propagators according to which
they are obtained from the ingoing and outgoing boundary conditions at the
horizon respectively. Furthermore, since the outgoing mode is time reverse of
the ingoing mode (which is not manifest in the Eddington-Finkelstein gauge but
can be evident from transforming to Schwarzchild-like coordinates),151515Note
that the notion of in/outgoing modes are gauge-invariant up to overall
multiplicative factors, but these cancel out in the ratio of the normalizable
to the non-normalizable modes. This is why the Son-Starinets prescription is
also gauge-invariant. we should have

$\frac{B_{0}^{*}(\omegaup,\mathbf{k})}{b_{0}^{*}(\omegaup,\mathbf{k})}=\frac{A_{0}(\omegaup,\mathbf{k})}{a_{0}(\omegaup,\mathbf{k})},\quad{\rm
i.e.}\quad G_{A}^{*}(\omegaup,\mathbf{k})=G_{R}(\omegaup,\mathbf{k})\quad{\rm
holds}.$ (46)

We therefore conclude that the CGL horizon cap prescription reproduces the
Son-Starinets prescription for the retarded propagator together with KMS
periodicity and the thermal structure of the correlation functions at any
frequency and momentum. A similar approach was adopted earlier by Son and
Herzog by identifying the two sides of the eternal black hole with the forward
and backward arms of the Schwinger-Keldysh contour Son and Teaney (2009).
However, in this case, the backward part of the time contour needs to be
shifted by $\beta/2$ along the negative imaginary axis. The main advantage of
the CGL prescription is that we do not need an eternal black hole geometry for
its implementation suggesting that its nonequilibrium generalization would be
generically more feasible. Furthermore, it is also not clear if out-of-
equilibrium correlation functions can be analytically continued in their time
arguments as required by the Son and Herzog implementation of the Schwinger-
Keldysh contour. Also, it should be possible to define integration over bulk
vertices and bulk quantum loops in the CGL prescription as well via the
analytic structure of the complexified space-time with the horizon cap.
However, this is outside the scope of the present work, and therefore we do
not further discuss about this issue. Finally, we note that the arguments
presented here are simpler compared with Glorioso  _et al._ (2018) since we do
not employ an expansion about $\omega=0$ which obscures the analytic
continuation at the horizon cap by producing $(\log\omega)^{n}$ terms.

##  III Bjorken flow, Weyl rescaling and the holographic dual

###  III.1 Bjorken flow and its Weyl rescaling

Figure 3: The schematic diagram of the Bjorken flow illustrates the evolution
of an expanding system on the forward light cone. Here $z$ is the longitudinal
direction along which the system expands. The transverse directions have been
suppressed. Initial data is specified on a constant $\tau$ hyperboloid.

Bjorken flow Bjorken (1983) is a simple model describing the expanding plasma
produced by heavy ion collisions. This model is based on the assumptions of
boost invariance, and translational and rotational symmetries in the
transverse directions of an expanding system. The evolution occurs inside a
forward light cone as shown in Fig. 3. It is convenient to describe the
Bjorken flow in the Milne proper time $\tau$ and the rapidity $\zeta$ which
are related to the Minkowski (lab frame) time $t$ and the longitudinal
coordinate $z$ (along which the system is expanding) as

$t=\tau\cosh{\zeta}\hskip 28.45274pt\text{and}\hskip
28.45274ptz=\tau\sinh{\zeta}.$

The transverse coordinates $x_{\perp}$ are the same in both Milne and
Minkowski coordinate systems. In the Milne coordinates, the Minkowski metric
takes the form

${\rm d}s^{2}=-{\rm d}\tau^{2}+\tau^{2}{\rm d}\zeta^{2}+{\rm
d}s_{\perp}^{2}~{},$ (47)

where ${\rm d}s_{\perp}$ is the line element in the transverse plane.

The symmetries of the Bjorken flow imply that the expectation value of any
operator depends only on the proper time $\tau$. Thus an ansatz for the
expectation value of the energy-momentum tensor in a $d$-dimensional theory,
which is also consistent with the transverse translational and rotational
symmetries of the Bjorken flow, can take the form

$\langle T_{\mu\nu}\rangle={\rm
diag}(\epsilon(\tau),\,\tau^{2}p_{L}(\tau),\,\underbrace{p_{T}(\tau),\,\cdots,\,p_{T}(\tau)}_{(d-2)\,\,{\rm
times}}),$ (48)

in the Milne coordinates for $d>2$. Clearly, $\epsilon$, $p_{L}$ and $p_{T}$
denote the energy density, longitudinal and transverse pressures respectively.
The local conservation of energy and momentum $\nabla_{\mu}T^{\mu\nu}=0$
implies that

$p_{L}(\tau)=-\epsilon(\tau)-\tau\partial_{\tau}\epsilon(\tau),$ (49)

and the conformal Ward identity $T^{\mu}_{\,\,\mu}=0$ imposes

$p_{T}(\tau)=\frac{2}{d-2}\left(\epsilon(\tau)+\frac{1}{2}\tau\partial_{\tau}\epsilon(\tau)\right).$
(50)

Consequently, the evolution of the energy-momentum tensor is determined by
$\epsilon(\tau)$ in a conformal field theory.

At large proper time $\tau$, the Bjorken flow admits a hydrodynamic
description Jeon and Heinz (2015). To explicitly map the energy-momentum
tensor (48) to that of a fluid, we need to set the flow velocity as

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

i.e. ${\rm d}\tau$ is co-moving with the flow in the Milne coordinates. In a
conformal system, the large proper time expansion of $\epsilon(\tau)$ is given
by a single parameter, namely

$\mu:=\epsilon_{0}\tau_{0}^{\frac{d}{d-1}},$ (51)

which is determined by the initial conditions — $\epsilon_{0}$ is a constant
energy density, and $\tau_{0}$ can be chosen to be the value of $\tau$ where
we intialize. The large proper time expansion of $\epsilon(\tau)$ takes the
form

$T^{\tau}_{\,\,\tau}=-\epsilon(\tau)=-\mu\tau^{-\frac{d}{d-1}}\left(1+\sum_{n=1}^{\infty}\lambda_{n}\,\mu^{-\frac{n}{d}}\tau^{-n\frac{d-2}{d-1}}\right),$
(52)

where $\lambda_{n}$ are (state-independent) constants that are determined by
the transport coefficients of the microscopic theory. As for instance,
$\lambda_{1}$ is related to the shear viscosity $\eta$ as

$\lambda_{1}=-\frac{\eta(\epsilon)}{\epsilon^{\frac{d-1}{d}}},$ (53)

which should indeed be a constant in a conformal theory. The leading term of
the expansion $\propto\tau^{-d/(d-1)}$ gives an exact solution of the Euler
equations, and thus represents the expansion of a conformal perfect fluid.

In what follows, we will need a Weyl transformation of the Bjorken flow. In a
conformal theory, the hydrodynamic equations are Weyl covariant Rangamani
(2009). We are ignoring the Weyl anomaly for the moment, but we will
explicitly mention it later. Under a Weyl transformation which transforms the
metric and the energy-momentum tensor as

${\rm d}s^{2}\rightarrow\widetilde{{\rm d}s}^{2}=\Omega(x)^{2}{\rm
d}s^{2},\quad
T_{\mu\nu}\rightarrow\widetilde{T}_{\mu\nu}=\Omega(x)^{-d+2}T_{\mu\nu},$ (54)

the new solutions of the hydrodynamic equations are given by

$u^{\mu}\rightarrow\tilde{u}^{\mu}=\Omega(x)^{-1}u^{\mu},\quad\epsilon\rightarrow\tilde{\epsilon}=\Omega(x)^{-d}\epsilon,$
(55)

in any conformal theory. Consider the combined operation of the time
reparametrization

$\sigma=\tau^{\frac{d-2}{d-1}}\tau_{0}^{\frac{1}{d-1}}$ (56)

and the Weyl scaling with

$\Omega(\sigma)=\left(\frac{\tau_{0}}{\sigma}\right)^{\frac{1}{d-2}},$ (57)

under which the Milne metric (47) transforms to (with
$\hat{\zeta}=\zeta\tau_{0}$)

$\widetilde{{\rm d}s}^{2}=-\frac{(d-1)^{2}}{(d-2)^{2}}{\rm
d}\sigma^{2}+\frac{\sigma^{2}}{\tau_{0}^{2}}{\rm
d}{\hat{\zeta}}^{2}+\left(\frac{\sigma}{\tau_{0}}\right)^{-\frac{2}{d-2}}{\rm
d}s_{\perp}^{2},$ (58)

and the energy-momentum tensor given by (48), (49) and (50) transforms to

$\displaystyle\widetilde{T}_{\sigma\sigma}$ $\displaystyle=$
$\displaystyle\Omega^{-d+2}\tau^{\prime}(\sigma)^{2}\epsilon(\tau(\sigma))$
$\displaystyle=$
$\displaystyle\frac{(d-1)^{2}}{(d-2)^{2}}\left(\frac{\sigma}{\tau_{0}}\right)^{\frac{d}{d-2}}\epsilon(\tau(\sigma))$
$\displaystyle=$
$\displaystyle\frac{(d-1)^{2}}{(d-2)^{2}}\widetilde{\epsilon}(\sigma),\quad\left({\rm
Note}\,\,\widetilde{\epsilon}(\sigma):=\left(\frac{\sigma}{\tau_{0}}\right)^{\frac{d}{d-2}}\epsilon(\tau(\sigma))\right)$
$\displaystyle\widetilde{T}_{\hat{\zeta}\hat{\zeta}}$ $\displaystyle=$
$\displaystyle\Omega^{-d+2}\left(\frac{\tau(\sigma)}{\tau_{0}}\right)^{2}p_{L}(\tau(\sigma))$
$\displaystyle=$
$\displaystyle-\left(\frac{\sigma}{\tau_{0}}\right)^{\frac{3d-4}{d-2}}\Big{(}\epsilon(\tau(\sigma))+\tau(\sigma)\epsilon^{\prime}(\tau(\sigma))\Big{)}$
$\displaystyle=$
$\displaystyle\left(\frac{\sigma}{\tau_{0}}\right)^{2}\frac{1}{d-1}\Big{(}\widetilde{\epsilon}(\sigma)-(d-2)\sigma\tilde{\epsilon}^{\prime}(\sigma)\Big{)},$
$\displaystyle\widetilde{T}_{ii}$ $\displaystyle=$
$\displaystyle\Omega^{-d+2}P_{T}(\tau(\sigma))$ (59) $\displaystyle=$
$\displaystyle\frac{2}{d-2}\frac{\sigma}{\tau_{0}}\Big{(}\epsilon(\tau(\sigma))+\frac{1}{2}\tau(\sigma)\epsilon^{\prime}(\tau(\sigma))\Big{)}$
$\displaystyle=$
$\displaystyle\left(\frac{\sigma}{\tau_{0}}\right)^{-\frac{2}{d-2}}\frac{1}{d-1}\Big{(}\widetilde{\epsilon}(\sigma)+\sigma\tilde{\epsilon}^{\prime}(\sigma)\Big{)},$

with $\widetilde{T}_{ii}$ denoting the diagonal transverse components and ′
denoting the derivative w.r.t. the argument of the corresponding function. It
follows that in the hydrodynamic limit, the Bjorken expansion (52) takes the
resultant form

$\widetilde{T}^{\sigma}_{\,\,\sigma}\equiv-\widetilde{\epsilon}(\sigma)=-\left(\frac{\sigma}{\tau_{0}}\right)^{\frac{d}{d-2}}\epsilon(\tau(\sigma))=-\epsilon_{0}\left(1+\sum_{n=1}^{\infty}\lambda_{n}\,\epsilon_{0}^{-\frac{n}{d}}\sigma^{-n}\right).$
(60)

The Weyl scaled metric (58) has the property that

$\sqrt{-\widetilde{g}}=\frac{d-1}{d-2}~{},$ (61)

is a constant, and the spatial volume factor is unity, same as in the
Minkowski coordinates. However, the longitudinal volume expands, while the
transverse volume contracts with the evolution. Also note that for the Weyl
scaled Bjorken flow (60), we have

$\lim_{\sigma\rightarrow\infty}\widetilde{\epsilon}(\sigma)=\epsilon_{0}.$
(62)

Therefore, instead of a perfect fluid expansion, the flow attains a constant
temperature, energy and entropy densities at late time although no time-like
Killing vector exists in the background metric. The latter feature leads to
viscous and higher-order corrections. The large (reparametrised) proper time
expansion is determined by $\epsilon_{0}$, the final thermal value of the
energy density, while $\tau_{0}$ appears in the Weyl scaling factor $\Omega$
as explicit in Eq. (57).

The Weyl scaling depends explicitly on $\tau_{0}$. However, note that for
$\tau_{0}\rightarrow\xi\tau_{0}$, we obtain from (56) that
$\sigma\rightarrow\xi^{1/(d-1)}\sigma$. Thus the Weyl factor given by (57)
scales as $\Omega\rightarrow\xi^{1/(d-1)}\Omega$, implying that
$\widetilde{{\rm d}s}^{2}\rightarrow\xi^{2/(d-1)}\widetilde{{\rm d}s}^{2}$ and
$\epsilon_{0}\rightarrow\xi^{-1/(d-1)}\epsilon_{0}.$ Therefore, the
dimensionless variables $\epsilon_{0}^{-1/d}\sigma^{-1}$ (which provides the
proper time expansion parameter) and
$\sigma^{n}\widetilde{T}^{\sigma}_{\,\,\sigma}$ are invariant under
$\tau_{0}\rightarrow\xi\tau_{0}$, and are thus independent of $\tau_{0}$.

We also note that the Weyl transformation studied here is itself not corrected
at first and higher orders in the large and proper time expansion.

###  III.2 Gravitational setup

The gravitational dual of the Bjorken flow Shuryak  _et al._ (2007); Janik and
Peschanski (2006b); Janik (2007); Kinoshita  _et al._ (2009); Beuf  _et al._
(2009); Heller  _et al._ (2013) has been extensively studied in the literature
with the late time evolution providing a primary example of the fluid/gravity
correspondence Policastro  _et al._ (2001); Bhattacharyya  _et al._ (2008);
Baier  _et al._ (2008); Rangamani (2009) where large order resummation of the
hydrodynamic series Heller  _et al._ (2013) has been explicitly carried out
revealing the hydrodynamization Heller and Spalinski (2015) of a far-from-
equilibrium state. When a state hydrodynamizes, the energy-momentum tensor can
be described as an optimally truncated (divergent and asymptotic) hydrodynamic
series even when it is far from equilibrium Heller  _et al._ (2013); Heller
and Spalinski (2015). In the context of the Bjorken flow, the evolution of the
energy density approaches a hydrodynamic attractor Heller  _et al._ (2013);
Heller and Spalinski (2015). This is a generic property of a many-body
relativistic system irrespective of whether its degrees of freedom interact
weakly or strongly (see Soloviev (2022) for a recent review).

Here we will review the gravitational dual of the Bjorken flow in the
hydrodynamic limit and then describe its Weyl transformation in detail. This
Weyl transformation is what has been described in the previous subsection. In
the bulk it is implemented by an appropriate diffeomorphism. As a result of
this transformation, the state reaches a constant temperature and entropy
density at late proper time instead of attaining perfect fluid expansion. The
dual black hole also attains a horizon with constant surface gravity and area.
However, even at late proper time there is no time-like Killing vector – the
directions longitudinal and transverse to the flow expand and contract
respectively such that the horizon area remains constant at late proper time.
Along with the Weyl transformation of the metric and the energy-momentum
tensor described in the previous subsection, the holographic dual also
produces the Weyl anomaly. The Weyl transformation will be an important tool
in implementing the horizon cap prescription out of equilibrium.

Additionally, we will focus on the residual gauge freedom which allows us to
fix the nonequilibrium event or apparent horizon at a fixed radial location.
We will see that it is crucial to pin the horizon cap at the nonequilibrium
event horizon for regularity, and therefore this gauge freedom will play an
important role. We will explicitly show that this gauge freedom does not
affect the dual metric or the dual energy-momentum tensor (and is thus a
proper gauge transformation).

The holographic dual of the Bjorken flow in a $d$-dimensional conformal theory
is a $(d+1)$-dimensional geometry which satisfies the Einstein’s equations
with a negative cosmological constant:

$R_{MN}-\frac{1}{2}RG_{MN}-\frac{d(d-1)}{2L^{2}}G_{MN}=0.$ (63)

In what follows, we will set $L=1$ for convenience. In addition to the field
theory coordinates, we need an extra radial coordinate to describe the dual
geometry. The state of the conformal theory dual to a specific solution of
(63), lives at the boundary ($\rho=0$) in the boundary metric, which is
defined as

$g^{\rm b}_{\mu\nu}=\lim_{\rho\rightarrow 0}\rho^{2}G_{\mu\nu},$ (64)

where $a$ and $b$ stand for the field theory indices. Since we are considering
the Bjorken flow in the Milne metric (47), $g^{\rm b}_{\mu\nu}$ should
coincide with it. Similarly, if we consider the Weyl scaled version of the
Bjorken flow, the boundary metric should coincide with (58).

Before considering the Bjorken flow, it is useful to first understand the
vacuum solution, which is pure (maximally symmetric) $AdS_{d+1}$ space-time
with the desired boundary metric. In the ingoing Eddington-Finkelstein gauge,
the vacuum state in the Milne metric (47) is thus dual to

$\displaystyle{\rm d}s^{2}=-\frac{2}{r^{2}}{\rm d}r{\rm
d}\tau-\frac{1}{r^{2}}{\rm d}\tau^{2}+\left(1+\frac{\tau}{r}\right)^{2}{\rm
d}\zeta^{2}+\frac{1}{r^{2}}{\rm d}s_{\perp}^{2}~{},$ (65)

where $r$ is the radial coordinate. Similarly, the vacuum in Weyl scaled
metric (58) is dual to

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-\frac{2}{v^{2}}\frac{d-1}{d-2}{\rm d}v{\rm
d}\sigma-\frac{1}{v^{2}}\left(\frac{(d-1)^{2}}{(d-2)^{2}}+\frac{2(d-1)v}{(d-2)^{2}\sigma}\right){\rm
d}\sigma^{2}+\frac{1}{\tau_{0}^{2}}\left(1+\frac{\sigma}{v}\right)^{2}{\rm
d}{\hat{\zeta}}^{2}$ (66)
$\displaystyle+\frac{1}{v^{2}}\left(\frac{\sigma}{\tau_{0}}\right)^{-\frac{2}{d-2}}{\rm
d}s_{\perp}^{2},$

where $v$ is the radial coordinate. These bulk metrics (65) and (66) are
related by the diffeomorphism

$\tau=\tau_{0}^{-\frac{1}{d-2}}\sigma^{\frac{d-1}{d-2}},\quad
r=v\left(\frac{\sigma}{\tau_{0}}\right)^{\frac{1}{d-2}}.$ (67)

For both cases, (65) and (66), we obtain the boundary metrics (47) and (58)
from (64), after replacing $\rho$ with $r$ and $v$, respectively. Any Weyl
transformation at the boundary is dual to a bulk diffeomorphism. Since the
boundary metrics (47) and (58) are related by a Weyl transformation, (67) is
simply a specific instance of this general feature of holographic duality.
Note that $\tau$ and $\sigma$ are related exactly by the time
reparametrization (56) at the boundary.161616Diffeomorphisms such as (67)
which implement global transformations on the dual state are called improper
diffeomorphisms which are always part of residual gauge freedom after gauge
fixing in the bulk. The latter can also have additional proper diffeomorphisms
which do not affect the dual physical quantities.

Holographic renormalization Henningson and Skenderis (1998); Balasubramanian
and Kraus (1999); de Haro  _et al._ (2001); Skenderis (2002) provides the
framework for extracting the $\langle T_{\mu\nu}\rangle$ corresponding to the
state in the field theory dual to a specific asymptotically $AdS_{d+1}$ bulk
geometry. The procedure essentially amounts to covariantly regularizing the
Brown-York tensor on a cut-off hypersurface with local counterterms built out
of the induced metric, and then taking this surface to the boundary. For the
bulk geometry (65), $\langle T_{\mu\nu}\rangle=0$ in the dual vacuum state
living in the flat Milne metric (47) at the boundary. On the other hand, for
the vacuum state living on the Weyl transformed Milne metric (58) which is
dual to the bulk geometry (66), $\langle T_{\mu\nu}\rangle=0$ only if $d$ is
odd. For even $d$, holographic renormalization reproduces the Weyl anomaly of
the dual field theory. In the case of $d=4$, we obtain (using minimal
subtraction scheme)

$\displaystyle\langle\widetilde{T}_{\mu\nu}\rangle=\frac{1}{8\pi
G_{N}}\mathcal{A}_{\mu\nu}$ (68)

with

$\displaystyle\mathcal{A}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left(\frac{4}{3}\widetilde{R}_{\mu\nu}\widetilde{R}-2\widetilde{R}_{\mu\rho}\widetilde{R}^{\rho}_{\,\,\nu}-\tilde{g}_{\mu\nu}\left(\frac{1}{2}\widetilde{R}^{2}-\widetilde{R}_{\rho\sigma}\widetilde{R}^{\rho\sigma}\right)\right),$
(69) $\displaystyle=$ $\displaystyle{\rm
diag}\left(-\frac{1}{32\sigma^{4}},-\frac{11}{648\sigma^{2}\tau_{0}^{2}},\frac{25\tau_{0}}{648\sigma^{5}},\frac{25\tau_{0}}{648\sigma^{5}}\right)~{},$

where $\tilde{g}$ denotes the Weyl rescaled background metric (58),
$\widetilde{R}_{\mu\nu}$ is the Ricci tensor built out of it, etc. It is easy
to verify that

$\widetilde{\nabla}_{\mu}\widetilde{T}^{\mu}_{\,\,\nu}=0,$ (70)

i.e. energy and momentum are conserved in the Weyl rescaled background metric
(58) (with $\widetilde{\nabla}$ being the covariant derivative built out of
it), and

$\displaystyle\widetilde{T}^{\mu}_{\,\,\mu}=-\frac{1}{8\pi
G_{N}}\left(\frac{1}{24}\widetilde{R}^{2}-\frac{1}{8}\widetilde{R}_{\rho\sigma}\widetilde{R}^{\rho\sigma}\right)=\frac{1}{8\pi
G_{N}}\frac{2}{27\sigma^{4}}.$ (71)

Using

$\frac{1}{G_{N}}=\frac{2N^{2}}{\pi},$ (72)

we can readily find that (71) reproduces the Weyl anomaly of $SU(N)$
$\mathcal{N}=4$ super-symmetric Yang-Mills theory Henningson and Skenderis
(1998); Balasubramanian and Kraus (1999).

An asymptotically $AdS_{d+1}$ metric dual to a Bjorken flow on the flat Milne
metric (47) at the boundary, is a solution to the vacuum Einstein’s equations
(63) which takes the form

$\displaystyle ds^{2}=-\frac{2}{r^{2}}{\rm d}r{\rm
d}\tau-\frac{A(r,\tau)}{r^{2}}{\rm
d}\tau^{2}+\left(1+\frac{\tau}{r}\right)^{2}e^{L(r,\tau)}{\rm
d}\zeta^{2}+\frac{e^{K(r,\tau)}}{r^{2}}{\rm d}s_{\perp}^{2}$ (73)

with the following Dirichlet asymptotic boundary conditions

$A(r,\tau)\rightarrow 1,\hskip 17.07182ptK(r,\tau)\rightarrow 0\hskip
17.07182pt\text{and}\hskip 17.07182ptL(r,\tau)\rightarrow 0\hskip
17.07182pt\text{as}\quad r\rightarrow 0.$ (74)

These boundary conditions ensure that the boundary metric (64) (with $r$ being
the radial coordinate in place of the generic $\rho$) coincides with the Milne
metric (47).

The Einstein equations (63) can be readily solved in the late-time expansion,
as functions of the scaling variable

$s=r\left(\frac{\tau_{0}}{\tau}\right)^{\frac{1}{d-1}},$ (75)

and with the expansion parameter being

$\left(\tilde{\mu}^{1/d}\tau^{\frac{d-2}{d-1}}\right)^{-1}$ (76)

where $\tilde{\mu}:=\varepsilon_{0}\tau_{0}^{\frac{d}{d-1}}$ is a constant
which will be related to the single parameter $\mu$ of the Bjorken flow
defined in (51) (or equivalently to $\epsilon_{0}$) below. Explicitly,

$\displaystyle A(r,\tau)$ $\displaystyle=$ $\displaystyle
1-\varepsilon_{0}s^{d}+\sum_{i=1}^{\infty}\left(\tilde{\mu}^{1/d}\tau^{\frac{d-2}{d-1}}\right)^{-i}a_{(i)}(\varepsilon_{0}^{\frac{1}{d}}s),$
$\displaystyle L(r,\tau)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\left(\tilde{\mu}^{1/d}\tau^{\frac{d-2}{d-1}}\right)^{-i}l_{(i)}(\varepsilon_{0}^{\frac{1}{d}}s),$
$\displaystyle K(r,\tau)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\left(\tilde{\mu}^{1/d}\tau^{\frac{d-2}{d-1}}\right)^{-i}k_{(i)}(\varepsilon_{0}^{\frac{1}{d}}s).$
(77)

The functions $a_{(i)}$, $l_{(i)}$ and $k_{(i)}$ satisfy ordinary differential
equations with source terms at each order. We require that these functions do
not blow up at the perturbative horizon which is at
$s_{h}=\varepsilon_{0}^{-1/d}$, i.e. at

$r_{h}=\varepsilon_{0}^{-\frac{1}{d}}\left(\frac{\tau}{\tau_{0}}\right)^{\frac{1}{d-1}}$
(78)

Together with the Dirichlet boundary conditions (74), these finiteness
conditions ensure that we obtain solutions which are free of naked
singularities in the perturbative expansion and which are unique up to terms
which are determined by a single coefficient Heller  _et al._ (2013). This
coefficient captures the residual gauge freedom of the ingoing Eddington-
Finkelstein coordinates which is the reparametrization of the radial
coordinate $r$ (without spoiling the manifest translational and rotational
symmetries along the transverse directions). Usually, this residual gauge
freedom is fixed by setting the radial location of the apparent or event
horizon at (78) to all orders in the late proper time expansion Chesler and
Yaffe (2014). However, in what follows, we will show that the residual gauge
freedom should actually be fixed by the regularity of the horizon cap. Note
that although the requirement that the horizon cap should be pinned to the
evolving event horizon for regularity is a gauge-invariant statement, the
residual gauge freedom will be crucial to implement the prescription in the
Eddington-Finekelstein coordinates. We therefore keep this gauge freedom
unfixed here and later show how it is fixed by the regularity of the horizon
cap such that the latter is pinned to the evolving event horizon (which lies
at a fixed radial location after the residual gauge fixing).

It is also crucial to emphasize that the residual gauge freedom involving the
reparametrization of the radial coordinate is a proper diffeomorphism, i.e. it
leaves both the boundary metric (which is the flat Milne background (47)) and
also the $\langle T_{\mu\nu}\rangle$ of the dual Bjorken flow extracted from
holographic renormalization is invariant. It’s useful to see this explicitly.
For illustration, let’s consider the $AdS_{5}$ case. The asymptotic expansions
take the form:

$\displaystyle A(r,\tau)$ $\displaystyle=$ $\displaystyle
1+ra_{1}(\tau)+r^{2}\left(\frac{a_{1}(\tau)^{2}}{4}-a_{1}^{\prime}(\tau)\right)+r^{4}a_{4}(\tau)+\cdots,$
$\displaystyle K(r,\tau)$ $\displaystyle=$ $\displaystyle
ra_{1}(\tau)-r^{2}\frac{a_{1}(\tau)^{2}}{4}+r^{3}\frac{a_{1}(\tau)^{3}}{12}+r^{4}k_{4}(\tau)+\cdots,$
$\displaystyle L(r,\tau)$ $\displaystyle=$ $\displaystyle
ra_{1}(\tau)-r^{2}\left(\frac{a_{1}(\tau)^{2}}{4}+\frac{a_{1}(\tau)}{\tau}\right)$
(79)
$\displaystyle+r^{3}\left(\frac{a_{1}(\tau)^{3}}{12}+\frac{a_{1}(\tau)^{2}}{2\tau}+\frac{a_{1}(\tau)}{\tau^{2}}\right)+r^{4}l_{4}(\tau)+\cdots.$

Above, the function $a_{1}(\tau)$ is related to the residual gauge freedom,
and can be chosen arbitrarily. Furthermore, the constraints of Einstein’s
equations (63) impose

$\displaystyle a_{4}^{\prime}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{a_{1}(\tau)^{4}}{12\tau}-\frac{4}{3\tau}a_{4}(\tau)-\frac{8}{3\tau}k_{4}(\tau),$
$\displaystyle l_{4}(\tau)+2k_{4}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{a_{1}(\tau)}{\tau^{3}}-\frac{3}{4}\frac{a_{1}(\tau)^{2}}{\tau^{2}}-\frac{1}{4}\frac{a_{1}(\tau)^{3}}{\tau}-\frac{3}{32}a_{1}(\tau)^{4}.$
(80)

Using these constraints, one can find via holographic renormalization that

$\displaystyle\langle T_{\mu\nu}\rangle$ $\displaystyle=$
$\displaystyle\frac{3}{16\pi G_{N}}\times$
$\displaystyle\begin{pmatrix}-a_{4}(\tau)~{}&0~{}&0~{}&0\\\
0~{}&\tau^{2}\left(a_{4}(\tau)+\tau a_{4}^{\prime}(\tau)\right)~{}&0~{}&0\\\
0~{}&0~{}&-a_{4}(\tau)-\frac{1}{2}\tau a_{4}^{\prime}(\tau)~{}&0\\\
0~{}&0~{}&0~{}&-a_{4}(\tau)-\frac{1}{2}\tau
a_{4}^{\prime}(\tau)\end{pmatrix}.$

Firstly, the above result is exact to all orders in the late proper time
expansion. Second, we readily find that $\langle T_{\mu\nu}(\tau)\rangle$ is
determined by $a_{4}(\tau)$ alone and is independent of the arbitrary function
$a_{1}(\tau)$ capturing the residual gauge freedom in the asymptotic expansion
after utilizing the constraints (III.2) in the renormalized Brown-York stress
tensor. Thus $\langle T_{\mu\nu}(\tau)\rangle$ is invariant under the residual
gauge transformation. Furthermore, comparing (III.2) with (48), (49) and (50)
(for $d=4$) we find that $\langle T_{\mu\nu}(\tau)\rangle$ takes the general
form of the energy-momentum tensor of Bjorken flow with the identification

$\epsilon(\tau)=-\frac{3}{16\pi G_{N}}a_{4}(\tau).$ (82)

One can repeat the same exercise in arbitrary dimensions ($d>2$) and show that
$\langle T_{\mu\nu}(\tau)\rangle$ obtained from holographic renormalization
takes the general form given by (48), (49) and (50) with the identification

$\epsilon(\tau)=-\frac{d-1}{16\pi G_{N}}a_{d}(\tau),$ (83)

where $a_{d}(\tau)$ is the coefficient of $r^{d}$ in the asymptotic expansions
of $A(r,\tau)$.

Furthermore, extracting $a_{d}(\tau)$ from (III.2) we obtain that at the
leading order in the late proper time expansion

$\epsilon(\tau)\approx\frac{d-1}{16\pi
G_{N}}\varepsilon_{0}\left(\frac{\tau_{0}}{\tau}\right)^{\frac{d}{d-1}}=\frac{d-1}{16\pi
G_{N}}r_{h}(\tau)^{-d}=\frac{d-1}{16\pi G_{N}}\left(\frac{4\pi
T(\tau)}{d}\right)^{d},$ (84)

where we have used (78), and also (26) to define an instantaneous Hawking
temperature $T(\tau)$ given by

$T(\tau)=\frac{d}{4\pi}\varepsilon_{0}^{\frac{1}{d}}\left(\frac{\tau_{0}}{\tau}\right)^{\frac{1}{d-1}}.$
(85)

Once again comparing with the general (hydrodynamic) late proper time
expansion (52), we find that

$\epsilon_{0}=\frac{d-1}{16\pi G_{N}}\varepsilon_{0}.$ (86)

For the case of $AdS_{5}$, the identification (72) implies that at late proper
time

$T(\tau)\approx\frac{1}{\pi}\varepsilon_{0}^{1/4}\left(\frac{\tau_{0}}{\tau}\right)^{\frac{1}{3}},\qquad\quad\epsilon(\tau)\approx\frac{3}{8}\pi^{2}N^{2}T(\tau)^{4}=\frac{3}{8\pi^{2}}N^{2}\varepsilon_{0}\left(\frac{\tau_{0}}{\tau}\right)^{\frac{4}{3}}.$
(87)

For any $d$, $\epsilon(\tau)$ is given by a perfect fluid expansion given by
(84) at late time with

$p_{L}(\tau)\approx
p_{T}(\tau)\approx\frac{1}{d-1}\epsilon(\tau)\approx\frac{1}{16\pi
G_{N}}r_{h}(\tau)^{-d}.$ (88)

Thus, at late proper time, the energy density $\epsilon(\tau)$ and the
pressures $p_{L}(\tau)$ and $p_{T}(\tau)$ are thus given by the thermal
equation of state obtained from a static black brane geometry, but with a
time-dependent temperature (85) which satisfy the Euler equations.

To construct a regular horizon cap, it is useful to change coordinates from
$r$ and $\tau$ to $v$ and $\sigma$ following (67) as in the case of the space-
time dual to the vacuum state. Note, it follows from (75) that $v$ and $s$ are
the same. In these new coordinates, the metric (73) takes the form:

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-\frac{2}{v^{2}}\frac{d-1}{d-2}{\rm d}v{\rm
d}\sigma-\frac{1}{v^{2}}\left(\frac{(d-1)^{2}}{(d-2)^{2}}A(v,\sigma)+\frac{2(d-1)v}{(d-2)^{2}\sigma}\right){\rm
d}\sigma^{2}$ (89)
$\displaystyle+\frac{1}{\tau_{0}^{2}}\left(1+\frac{\sigma}{v}\right)^{2}e^{L(v,\sigma)}{\rm
d}{\hat{\zeta}}^{2}+\frac{1}{v^{2}}\left(\frac{\sigma}{\tau_{0}}\right)^{-\frac{2}{d-2}}e^{K(v,\sigma)}{\rm
d}s_{\perp}^{2},$

and is dual to Bjorken flow on the Weyl rescaled background metric (58) at the
boundary. We want to emphasize that just like the Weyl transformation, the
dual bulk diffeomorphism which achieves the above form of the metric is given
by (67) exactly, and is therefore not corrected at first and higher orders in
the large proper time expansion.

Holographic renormalization and the constraints of the Einstein’s equations
(63) imply that the energy-momentum tensor of the dual Bjorken flow takes the
form

$\langle\widetilde{T}_{\mu\nu}\rangle=T^{W}_{\mu\nu}+\frac{1}{8\pi
G_{N}}\mathcal{A}_{\mu\nu},$ (90)

where $T^{W}_{\mu\nu}$ takes the general Bjorken form with non-vanishing
components given by (III.1) in which

$\tilde{\epsilon}(\sigma)=-\frac{d-1}{16\pi
G_{N}}\frac{(d-1)^{2}}{(d-2)^{2}}a_{d}(\sigma),$ (91)

and $\mathcal{A}_{\mu\nu}$ is the Weyl anomaly appearing for even $d$.
Comparing with (83), we indeed verify that the bulk diffeomorphism (67)
implements the Weyl transformation and time reparametrization in the dual
theory (with the Weyl factor given by (57)) and also reproduces its Weyl
anomaly. Particularly, for $d=4$, we recall that $\mathcal{A}_{\mu\nu}$ is
simply given by (69). The anomalous term is state-independent (and is always
the same as in the Weyl transformed vacuum state). We again note that
$\langle\widetilde{T}_{\mu\nu}\rangle$ is invariant under the residual gauge
symmetry since it is independent of $a_{1}(\tau)$ after we implement the
gravitational constraints following the previous discussion.

Obviously, the late proper time expansion (III.2) takes the form

$\displaystyle A(v,\sigma)$ $\displaystyle=$ $\displaystyle
1-\varepsilon_{0}v^{d}+\sum_{i=1}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-i}a_{(i)}(\varepsilon_{0}^{1/d}v),$
$\displaystyle L(v,\sigma)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-i}l_{(i)}(\varepsilon_{0}^{1/d}v),$
$\displaystyle K(v,\sigma)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-i}k_{(i)}(\varepsilon_{0}^{1/d}v).$
(92)

Explicitly, for $d=4$,

$\displaystyle a_{(1)}(x)$ $\displaystyle=$
$\displaystyle\alpha_{1}\frac{x(1+x^{4})}{3}+\frac{2x^{4}(1+x)}{3},$
$\displaystyle k_{(1)}(x)$ $\displaystyle=$
$\displaystyle\alpha_{1}\frac{x}{3}+\frac{1}{2}g(x),$ $\displaystyle
l_{(1)}(x)$ $\displaystyle=$ $\displaystyle\alpha_{1}\frac{x}{3}-g(x),$ (93)

where

$g(x)=\frac{4}{3}x-\frac{1}{3}\ln(x^{2}+1)-\frac{2}{3}\ln(x+1)-\frac{2}{3}\arctan
x.$ (94)

Above $\alpha_{1}$ is the dimensionless parameter associated with the residual
gauge freedom. At any order in the late proper time expansion, the terms
multiplying $\alpha_{1}$ in (III.2) remain the same, however, we should
replace $\alpha_{1}$ by $\alpha_{n}$ at the $n$-th order. It is also easy to
see that $g(x)$ is finite at $x=1$ implying that the metric is regular (with
no naked singularity) at the perturbative horizon

$v_{h}=\varepsilon_{0}^{-1/d}+\mathcal{O}(\sigma^{-1}).$ (95)

At late proper time the dual black brane has a constant surface gravity and
area although the directions longitudinal to the flow keep expanding and those
transverse to the flow keep contracting.

The above metric reproduces the late proper time expansion of
$\widetilde{\epsilon}(\sigma)$ which takes the form (60) with $\epsilon_{0}$
given by (83) and $\lambda_{n}$ taking specific values for a given $d$.
Particularly, for any $d$, we obtain

$\lambda_{1}=-\frac{1}{(d-1)^{\frac{d-1}{d}}(16\pi G_{N})^{1/d}}.$ (96)

It is easy to verify from (53) and the equation of state (see (84))

$T=\beta^{-1}=\frac{d}{4\pi}\varepsilon_{0}^{1/d},\quad\epsilon=\frac{d-1}{16\pi
G_{N}}\varepsilon_{0}=\frac{d-1}{16\pi G_{N}}\left(\frac{4\pi
T}{d}\right)^{d}$ (97)

that (96) implies via (53) that

$\frac{\eta}{s}=\frac{1}{4\pi}~{},$ (98)

for any $d>2$. More details of the perturbative expansion are in Appendix A.

##  IV The bulk scalar field and the horizon cap of the Bjorken flow

The key to obtaining the real-time correlation functions is solving the
dynamics of the scalar field in the gravitational background dual to the
Bjorken flow. The starting point, however, is to construct the analog of the
bulk Schwinger-Keldysh contour with the horizon cap for the gravitational
background itself. This is straightforward. The metric dual to the Weyl scaled
Bjorken flow (given by Eqs. (89) and (III.2)) reaches a constant horizon
temperature at late time although the boundary metric has time-dependent
spatial components. Since we would be working perturbatively in the late
proper time expansion, we will fix the horizon cap at the constant late-time
value $v=\varepsilon_{0}^{-1/d}$ to all orders in the perturbative late proper
time expansion while keeping the residual gauge freedom of radial
reparametrization unfixed as mentioned above. The metric is analytic to all
orders at the horizon cap. Therefore, there is no modification to the metric
on the other arm of the bulk space-time as it is reached via the complexified
$v$ contour encircling the horizon as shown in Fig. 2 (as emphasized earlier,
there is no analytic continuation in $\sigma$ and other coordinates). Exactly
the same gravitational background is valid on both arms of the complex $v$
contour. It is also easy to see that the on-shell Einstein-Hilbert action on
the two arms cancel each other out implying that the dual (non-)equilibrium
partition function in the absence of additional sources (and with the same
boundary metric on the two arms) is exactly zero, as should be the
case.171717In the field theory, this is equivalent to the statement that
$W[J_{1}=J_{2}=0]=0$.

There is a crucial subtlety to this rather simple construction. We should
worry about the residual gauge symmetries at first and higher orders in the
late proper time expansion. In what follows, we will show that the analytic
behavior of the sourced scalar field retains its equilibrium nature to all
orders in the late proper time expansion at the horizon cap, provided the
residual gauge symmetry is fixed in a unique way at each order. This addresses
an issue which would have arisen if we had fixed the residual gauge symmetries
to keep the apparent or the event horizon at $v=\varepsilon_{0}^{-1/d}$ to all
orders in the late proper time expansion. However, the location of these two
horizons differ at second and higher orders in the proper time expansion. We
find that the gauge fixing which implements the regularity of the horizon cap
is exactly the same which fixes the event (but not the apparent) horizon at
$v_{h}=\varepsilon_{0}^{1/d}$ up to third order in the proper time expansion
in the case of $AdS_{4}$ and $AdS_{5}$. Although we do not have an analytic
proof that this feature will continue to hold at higher orders, we expect it
to be the case as we explicitly find that the gauge fixing is independent of
the mass of the bulk scalar field, and it should also hold for fermion, vector
and higher rank tensor fields.

At the outset, we repeat our emphasis that the Weyl rescaling (and hence the
dual bulk diffeomorphism) is determined purely by the late proper time perfect
flow regime that is mapped to that of a constant temperature in a non-trivial
background metric in the field theory (which is also the boundary metric of
the dual black brane geometry after the bulk diffeomorphism). This should be
evident from our construction of the bulk dual of the Weyl-transformed Bjorkan
flow (in the $v$ and $\sigma$ coordinates) in the previous section. Both these
Weyl transformation and the background metric of the field theory do not
receive any correction at first and higher orders in the derivative (i.e.
large proper time) expansion. Nevertheless, with such a Weyl transformation
(bulk diffeomorphism) we will be able to ensure that the Klein-Gordon equation
of the bulk scalar takes the same form at first and higher orders as that at
the leading order but with source terms, and that the leading near-horizon
behavior is exactly the same as that in the static black brane geometry (in
the $v$ and $\sigma$ coordinates) to all orders in the large proper time
expansion when the horizon cap is pinned to the evolving event horizon. This
will be instrumental in establishing the horizon cap prescription in the
hydrodynamic tail of the Bjorken flow.

To see the main advantage of working in the $v$ and $\sigma$ coordinates, note
that the explicit form of the leading order metric (as evident from Eqs. (89)
and (III.2)) is:

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-\frac{2}{v^{2}}\frac{d-1}{d-2}{\rm d}v{\rm
d}\sigma-\frac{1}{v^{2}}\left(\frac{(d-1)^{2}}{(d-2)^{2}}(1-v^{d}\varepsilon_{0}+\mathcal{O}(\sigma^{-1}))+\frac{2(d-1)v}{(d-2)^{2}\sigma}\right){\rm
d}\sigma^{2}$ (99)
$\displaystyle+\frac{1}{\tau_{0}^{2}}\left(1+\frac{\sigma}{v}\right)^{2}(1+\mathcal{O}(\sigma^{-1})){\rm
d}{\hat{\zeta}}^{2}+\frac{1}{v^{2}}\left(\frac{\sigma}{\tau_{0}}\right)^{-\frac{2}{d-2}}(1+\mathcal{O}(\sigma^{-1})){\rm
d}s_{\perp}^{2}.$

It is useful to define the comoving momenta which depend on $\sigma$

$\kappa_{L}=k_{L}\frac{\tau_{0}}{\sigma},\quad\vec{\kappa}_{T}=\vec{k}_{T}\left(\frac{\tau_{0}}{\sigma}\right)^{-\frac{1}{d-2}}.$
(100)

such that it takes care of the longitudinal expansion and transverse
contraction of the boundary metric. A natural ansatz for the scalar field
consistent with boost invariance of the background geometry is:

$\Phi(v,\sigma,\hat{\zeta},\vec{x}_{\perp})\approx
e^{i\left(-\frac{d-1}{d-2}\omegaup\sigma+k_{L}\hat{\zeta}+\vec{k}_{T}\cdot\vec{x}_{\perp}\right)}f(v,\omegaup,\kappa_{L},\vec{\kappa}_{T}).$
(101)

where remarkably there is no explicit $\sigma$ dependence in $f$ (only
implicitly through $\kappa_{L}$ and $\vec{\kappa}_{T}$) while in the phase we
have the usual fixed conjugate momenta $k_{L}$ and $\vec{k}_{T}$. This is
exact at leading order and can be further corrected to obtain a systematic
expansion of the equations of motion in powers of $\sigma^{-1}$ as shown
below. Particularly, the Klein Gordon equation $(\Box-m^{2})\Phi=0$ for the
bulk scalar field at the leading order $\sigma^{0}$ is then181818To obtain
this we should substitute $k_{L}$ and $\vec{k}_{T}$ in (101) by the redefined
momenta (100) and remember to differentiate $f$ w.r.t. $\sigma$ as the
redefined momenta (100) depend on $\sigma$.

$\displaystyle\mathcal{D}f=\mathcal{O}(\sigma^{-1})$ (102)

where

$\mathcal{D}=v^{2}(1-v^{d}\varepsilon_{0})\partial_{v}^{2}-v(d-1-2iv\omegaup+v^{d}\varepsilon_{0})\partial_{v}-(m^{2}+v(v(\kappa_{L}^{2}+{\kappa}_{T}^{2})+i(d-1)\omegaup)).$
(103)

Remarkably, the left hand side of (102) is just the Klein Gordon equation for
the massive bulk scalar in the $AdS_{d+1}$ static black brane geometry (22) at
temperature $\beta^{-1}$ given by (97), with $v$ substituting for $r$, and
$\omegaup$, $\kappa_{L}$ and $\vec{\kappa}_{T}$ identified with the canonical
frequency and momenta. The implication is that $f$ will have exactly the same
solutions at the leading order as in the static black brane geometry and
therefore the same analytic structure at the horizon cap at the leading order.
For the homogeneous ($k_{L}=\vec{k}_{T}=0$) and massless case, this result
reduces to the observation made by Janik and Peschanski Janik and Peschanski
(2006a) in the context of the transients of the Bjorken flow.

Note that the dependence of $f$ in Eq. (101) on the co-moving momenta implies
that we do not have separation of variables, but this should not be expected
as the background at late time corresponds to an expanding boost-invariant
perfect fluid, which has no time-like Killing vector. Nevertheless, the map to
the Laplacian of a static black hole geometry is possible at leading order
because the boost invariant perfect fluid is given only by a time-dependent
temperature.

The ansatz (101) can be corrected to incorporate the viscous and all higher
order corrections to the gravitational Bjorken flow background systematically
while ensuring the regularity of the horizon cap after fixing the residual
gauge freedom perturbatively in late proper time expansion. This modified
ansatz involves an expansion in $\sigma^{-1}$ with coefficients which are
functions of $v$, $\omegaup$ and the co-moving momenta, such that the
equations of motion can be solved systematically in $\sigma^{-1}$ expansion as
well. Obviously, this implies that we also take into account the implicit
dependence of the co-moving momenta on $\sigma$ while obtaining the equations
of motion order by order in $\sigma^{-1}$.

Explicitly, the ansatz for the bulk scalar field in the full complexified
space-time with $1$ and $2$ labeling the sheets of the bulk space-time ending
at the forward and backward arms of the Schwinger-Keldysh contour respectively
at their boundaries is

$\displaystyle\begin{pmatrix}\Phi_{1}(v,\sigma,\hat{\zeta},\vec{x}_{\perp})\\\
\Phi_{2}(v,\sigma,\hat{\zeta},\vec{x}_{\perp})\end{pmatrix}$ $\displaystyle=$
$\displaystyle\int{\rm d}\omegaup{\rm d}k_{L}{\rm
d}^{d-2}k_{T}\,\,e^{i\left(k_{L}\hat{\zeta}+\vec{k}_{T}\cdot\vec{x}_{\perp}\right)}e^{-i\frac{d-1}{d-2}\omegaup\sigma}\left(\frac{\sigma}{\tau_{0}}\right)^{i\gamma_{0}(\omegaup/\varepsilon_{0}^{1/d})}$
(104)
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\mathcal{M}_{n}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})\cdot\begin{pmatrix}p(\omegaup,k_{L},k_{T})\\\
q(\omegaup,k_{L},k_{T})\end{pmatrix},$

where

$\mathcal{M}_{n}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})=\begin{pmatrix}\phi_{n,in}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})~{}&\phi_{n,out}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})\\\
\phi_{n,in}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})~{}&e^{\beta\omegaup}\phi_{n,out}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})\end{pmatrix}.$
(105)

Above, in (104), a non-analytic term proportional to $\sigma^{i\gamma_{0}}$
has been introduced with $\gamma_{0}$ being a new dimensionless function of
only $\omegaup/\varepsilon_{0}^{1/d}$. This non-analytic factor is crucial to
have a regular horizon cap as shown below. However, it does not affect the
zeroth order equation of motion which takes the form of the Laplacian on a
static black brane as discussed above. For any $\omegaup$, (104) has the
appearance of a trans-series in $\sigma^{-1}$ with $\omegaup$ characterizing
the continuous instanton exponent. Note $p$ and $q$ are functions of
$\omegaup$, $k_{L}$ and $k_{T}$ (and not the redefined momenta $\kappa_{L}$
and $\vec{\kappa}_{T}$), since we are integrating over $\omegaup$, $k_{L}$ and
$k_{T}$, and utilizing the superposition principle to obtain the general
solution with right behavior at the horizon cap. However $\mathcal{M}_{n}$
depend on $\kappa_{L}$ and $\vec{\kappa}_{T}$, and therefore they have non-
trivial derivatives w.r.t. $\sigma$. The coefficients $p$ and $q$, which are
functions of the ordinary momenta ($k_{L}$ and $\vec{k}_{T}$), and
coefficients of the ingoing and outgoing modes respectively (see more below),
are determined by imposing Dirichlet boundary conditions at the two
boundaries.

The structure of $\mathcal{M}_{n}$ in (105) is determined as follows. Note
that at the zeroth order, i.e. at $n=0$, we obtain exactly the solutions of
the static black brane as noted above. We can choose the basis of solutions
which are ingoing and outgoing at the horizon, and satisfying the
normalization conditions given by (25). Explicitly, near the perturbative
horizon $v=\varepsilon_{0}^{1/d}$ where the horizon cap is located, they take
the forms191919$\phi_{0,in}\rightarrow 1$ and
$\left(\varepsilon_{0}^{-1/d}-v\right)^{-i\frac{2\omegaup}{d\varepsilon_{0}^{1/d}}}\phi_{0,out}\rightarrow
1$ as $v\rightarrow\varepsilon_{0}^{-1/d}$ are just normalizations.

$\displaystyle\phi_{0,in}$ $\displaystyle=$ $\displaystyle
1+\sum_{k=1}^{\infty}p_{0,k}\left(\varepsilon_{0}^{-1/d}-v\right)^{k},$
$\displaystyle\phi_{0,out}$ $\displaystyle=$
$\displaystyle\left(\varepsilon_{0}^{-1/d}-v\right)^{i\frac{2\omegaup}{d\varepsilon_{0}^{1/d}}}\left(1+\sum_{k=1}^{\infty}q_{0,k}\left(\varepsilon_{0}^{-1/d}-v\right)^{k}\right),$
(106)

respectively. Note we have used (97) to set
$\beta\omegaup/(2\pi)=2\omegaup/(d\varepsilon_{0}^{1/d})$. For $n\geq 1$,
$\phi_{n,in}$ and $\phi_{n,out}$ represent corrections to these zeroth order
solutions as discussed below. In (105), we have assumed that $\phi_{n,in}$ and
$\phi_{n,out}$ have the same behavior at the horizon cap
$v=v_{h}=\varepsilon_{0}^{-1/d}$ for $n\geq 1$as in the case of the
equilibrium (or equivalently at the zeroth order). This indeed turns out to be
the case with appropriate residual gauge fixing as mentioned before and
explicitly shown below.202020In the following section, we show that preserving
the near-horizon behavior to all orders is required for satisfying many
consistency conditions, such as ensuring that the retarded correlation
function is given by causal response.

At higher orders, the equations determining $\phi_{n,in}$ and $\phi_{n,out}$
can be obtained from expanding $(\Box-m^{2})\Phi=0$ in the late proper time
expansion after substituting $k_{L}$ and $\vec{k}_{T}$ in (104) by the
redefined momenta (100), and isolating the $\sigma^{-n}$ term. We obtain that
$\phi_{n,in}$ and $\phi_{n,out}$ satisfies the linear inhomogeneous ordinary
differential equations

$\mathcal{D}\phi_{n,in}=\mathcal{S}_{n,in},\quad\mathcal{D}\phi_{n,out}=\mathcal{S}_{n,out}$
(107)

with $\mathcal{D}$ being the same operator (103), which is simply that
corresponding to the Klein-Gordon equation for the static black brane at
temperature $\beta^{-1}$ given by (97), at all orders. The sources
$\mathcal{S}_{n,in}$ and $\mathcal{S}_{n,out}$ are functions of $v$,
$\omegaup$, $\kappa_{L}$ and $\vec{\kappa}_{T}$, and depend on $\phi_{m,in}$
and $\phi_{m,out}$ respectively for $m<n$. Both of these sources also depend
on the functions $a_{(i)}$, $k_{(i)}$ and $l_{(i)}$, which appear in the late
proper time expansion (III.2) of the background metric, with $i\leq n$. Note
that the source is linear in the bulk field $\Phi$, and therefore splits into
$\mathcal{S}_{n,in}$ and $\mathcal{S}_{n,out}$ at each order in the
$\sigma^{-1}$ expansion for $n\geq 1$. Since $\phi_{n,in}$ for $n\geq 1$
corrects $\phi_{0,in}$, we include the particular solution with only
$\phi_{m,in}$ (and $m<n$) appearing in the source term $\mathcal{S}_{n,in}$ in
it. The particular solutions sourced by $\phi_{m,out}$ with $0\leq m<n$ which
appear in $\mathcal{S}_{n,out}$ are added by definition to $\phi_{n,out}$
similarly.

The regularity of the horizon cap implies that at first and higher orders in
the proper time expansion, we should have

$\displaystyle\lim_{v\rightarrow\varepsilon_{0}^{-1/d}}\left(\varepsilon_{0}^{-1/d}-v\right)^{-i\frac{2\omegaup}{d\varepsilon_{0}^{1/d}}}\phi_{n,out}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle-i\gamma_{n,out}(\omegaup,\kappa_{L},\vec{\kappa}_{T}),$
$\displaystyle\lim_{v\rightarrow\varepsilon_{0}^{-1/d}}\phi_{n,in}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle-i\gamma_{n,in}(\omegaup,\kappa_{L},\vec{\kappa}_{T}),$ (108)

with $\gamma_{n,out}$s and $\gamma_{n,in}$s as new functions of $\omegaup$,
$\kappa_{L}$ and $\vec{\kappa}_{T}$ for $n\geq 1$. This will ensure that the
analytical dependence of the field on $v$ at the horizon cap
$v=v_{h}=\varepsilon_{0}^{-1/d}$ is the same to all orders in the proper time
expansion. Note that at the zeroth order, the analogous conditions for
$\phi_{0,in}$ and $\phi_{0,out}$ are simply given by $1$ on the RHS in both
cases by choice (see (IV)). We will show that the $\gamma_{n,out}$s and
$\gamma_{n,in}$s can be determined uniquely for $n\geq 1$ via horizon cap
regularity and field theory identities.

Firstly, note that (IV) implies that $\phi_{n,in}$ and $\phi_{n,out}$ can be
written in the form

$\displaystyle\phi_{n,in}=\phi_{n(p),in}-i\gamma_{n,in}\,\,\phi_{0,in},\quad\phi_{n,out}=\phi_{n(p),out}-i\gamma_{n,out}\,\,\phi_{0,out}$
(109)

for $n\geq 1$ such that $\phi_{n(p),in}$ and $\phi_{n(p),out}$ are the
particular solutions of the inhomogeneous ordinary differential equations
(107). Both $\phi_{n(p),in}$ and $\phi_{n(p),out}$ are determined by the
sources $\mathcal{S}_{n,in}$ and $\mathcal{S}_{n,out}$, and are proportional
to the coefficients $p$ and $q$, respectively (i.e. they vanish when $p=q=0$).
Near the horizon cap $v\approx\varepsilon_{0}^{1/d}$, we explicitly find that
$\phi_{n(p),in}$ and $\phi_{n(p),out}$ behave as

$\displaystyle\lim_{v\rightarrow\varepsilon_{0}^{-1/d}}\left(\varepsilon_{0}^{-1/d}-v\right)^{-i\frac{2\omegaup}{d\varepsilon_{0}^{1/d}}}\phi_{n(p),out}$
$\displaystyle=$
$\displaystyle\mathcal{O}\left(\varepsilon_{0}^{-1/d}-v\right),$
$\displaystyle\lim_{v\rightarrow\varepsilon_{0}^{-1/d}}\phi_{n(p),in}$
$\displaystyle=$
$\displaystyle\mathcal{O}\left(\varepsilon_{0}^{-1/d}-v\right),$ (110)

respectively. Equivalently,

$\displaystyle\phi_{n(p),out}$ $\displaystyle=$
$\displaystyle\left(\varepsilon_{0}^{-1/d}-v\right)^{i\frac{2\omegaup}{d\varepsilon_{0}^{1/d}}}\sum_{k=1}^{\infty}q_{n,k}\left(\varepsilon_{0}^{-1/d}-v\right)^{k},$
$\displaystyle\phi_{n(p),in}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}p_{n,k}\left(\varepsilon_{0}^{-1/d}-v\right)^{k},$
(111)

where the coefficients should be determined by the equations of motion. For
the outgoing solution, we find that this behavior is possible only when the
residual gauge parameter $\alpha_{n}$ appearing in the background metric and
$\gamma_{n,out}$ are chosen appropriately to cancel double and single poles
appearing in the equation of motion (107) at the horizon cap for each $n\geq
1$. Furthermore, $\alpha_{n}$s are simply numerical constants (as they are
defined to be), and $\gamma_{n,out}$s are linear functions of
$\omegaup/\varepsilon_{0}^{1/d}$ only. Thus the outgoing solutions appearing
in $\mathcal{M}_{n}$ in our ansatz (104) are determined uniquely. As discussed
before, and will be explicitly shown again in the next section this completely
determines the advanced propagator of the Bjorken flow. The nonequilibrium
retarded propagator then is also determined uniquely, since even out of
equilibrium, the advanced and retarded propagators are related by the exchange
of the spatial and temporal arguments. Utilizing this, we can determine
$\gamma_{n,in}$s uniquely as well for $n\geq 1$ as will be shown in the next
section. In this section, we will focus mainly on the outgoing mode.

It is easy to see that (IV) (equivalently (IV)) implies that $\gamma_{n,in}$
and $\gamma_{n,out}$ are simply the coefficients of the homogeneous solutions
of the equations of motion (107) for $n\geq 1$. Therefore $\gamma_{n,in}$ and
$\gamma_{n,out}$ appear in $\mathcal{S}_{m,in}$ and $\mathcal{S}_{m,out}$
respectively for $m>n$. We will illustrate by example how requiring the
regularity condition (IV) (equivalently (IV)) at the $n$-th order determines
$\gamma_{n-1,out}$ along with the gauge parameter $\alpha_{n}$ (which is a
constant) recursively.

Unlike the case of the outgoing mode, the ingoing mode is always analytic at
the horizon cap. Therefore, we need to use consistency conditions for the
Schwinger-Keldysh correlation functions to determine $\gamma_{n,in}$s for
$n\geq 1$. However, $\gamma_{0}$ in the ansatz (104) appears in both the
ingoing and outgoing modes. Our construction passes a significant consistency
test that the same function $\gamma_{0}(\omegaup/\varepsilon_{0}^{1/d})$
determines the homogeneous transients (sourceless solutions which are ingoing
at the horizon) with the argument $\omegaup/\varepsilon_{0}^{1/d}$ taking
values corresponding to the appropriately rescaled quasi-normal mode
frequencies of the static black hole, as will be discussed in Section V.5.
This is remarkable as we determine the function $\gamma_{0}$ analytically by
imposing the regularity condition (IV) on the outgoing mode at the horizon
cap.

In what follows, we illustrate how we determine the outgoing mode and the
residual gauge fixing uniquely in the case of $AdS_{5}$. Let us first see how
we determine $\gamma_{0}$ and $\alpha_{1}$ at the first order in the proper
time expansion. This requires the first-order correction to the background
metric given by (III.2) and (94). We find that the equation of motion (107)
for $\phi_{1,out}$ explicitly takes the following form near the horizon cap
$v\approx\varepsilon_{0}^{1/4}$ up to overall proportionality factors:

$\displaystyle\frac{1}{(\varepsilon_{0}^{1/4}v-1)^{2}}\frac{\omegaup}{4\varepsilon_{0}^{1/4}}(\alpha_{1}+3)(\omegaup\varepsilon_{0}^{-1/4}+2i)$
$\displaystyle+\frac{1}{(\varepsilon_{0}^{1/4}v-1)}\frac{i\omegaup}{\varepsilon_{0}^{1/4}}\Bigg{(}\frac{2\kappa_{L}^{2}+2\kappa_{T}^{2}-m^{2}-3\omegaup^{2}+240\omegaup\varepsilon_{0}^{1/4}}{24\varepsilon_{0}^{1/2}}(\alpha_{1}+3)$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{(16+5\alpha_{1})\omegaup^{2}-4\gamma_{0}\omegaup\varepsilon_{0}^{1/4}}{\varepsilon_{0}^{1/2}}\Bigg{)}+\cdots,$
(112)

with $\cdots$ denoting terms which are regular at $v=\varepsilon_{0}^{1/4}$ if
$\phi_{1,out}$ is of the form (IV). See Appendix C for more details. We
readily see that To have a solution of the desired form (IV) we must impose

$\alpha_{1}=-3,\quad\gamma_{0}=\frac{\omegaup}{4\varepsilon_{0}^{1/4}}$ (113)

so that the double and single pole terms of the equation of motion appearing
at the horizon cap vanish. The double pole term determines $\alpha_{1}$ and
the single pole term determines $\gamma_{0}$. As claimed before, we find that
$\gamma_{0}$ is indeed a simple linear function of
$\omegaup/\varepsilon_{0}^{1/4}$ (rather just proportional to it) while the
gauge parameter $\alpha_{1}$ is a numerical constant as it should be.

At the second order in the proper time expansion, similarly $\alpha_{2}$ and
$\gamma_{1}$ are determined by the vanishing of the double and single pole
terms in the equation of motion for $\phi_{2,out}$ respectively. Here we have
to utilize the explicit second-order correction to the background metric given
in Appendix A. Explicitly,

$\alpha_{2}=\frac{1}{72}\left(20+9\pi-12\ln
2\right),\quad\gamma_{1}=\frac{1}{16}\left(4i-\frac{9\pi+4-24\ln
2}{9}\frac{\omegaup}{\varepsilon_{0}^{1/4}}\right).$ (114)

We find once again $\alpha_{2}$ is just a numerical constant as it should be
and $\gamma_{1}$ is a linear function of $\omegaup/\varepsilon_{0}^{1/4}$.

It is indeed crucial that the $\gamma_{n,out}$s for $n\geq 0$ are functions of
$\omegaup$ only and are independent of $\kappa_{L}$ and $\vec{\kappa}_{T}$
which depend on $\sigma$. Otherwise, the central assumption (IV) (and thus
(IV)) are not valid for the outgoing mode at the horizon cap, and should be
corrected by log terms. The latter would have implied that the behavior near
the horizon cap at first and higher orders is different from the zeroth order
which is the same as in thermal equilibrium. Note that the ansatz for the
ingoing mode in (IV) remains valid to all orders even if $\gamma_{n,in}$s
depend on $\kappa_{L}$ and $\vec{\kappa}_{T}$. The coefficients $p_{n,k}$ in
(IV) also involve derivatives of $\gamma_{m,in}$s w.r.t. $\kappa_{L}$ and
$\vec{\kappa}_{T}$ with $m<n$. See Appendix C for more details.

Finally, we have explicitly verified that the values of the residual gauge
parameters $\alpha_{1}$ and $\alpha_{2}$ are such that the event horizon is
pinned to the horizon cap $v=v_{h}=\varepsilon_{0}^{-1/d}$ at the first and
second orders respectively for both $AdS_{4}$ and $AdS_{5}$. Note that the
apparent horizon differs from the event horizon from second order onwards, so
the evolving apparent horizon is behind the horizon cap. See Appendix B for
details. We expect that this feature persists to all orders so that although
the interior of the event horizon is excised, the full double-sheeted geometry
with the horizon cap still covers the entire bulk regions which can send
signals to the boundary.212121Although we do not have a rigorous proof to all
orders, we believe that this follows from the general result that the event
horizon is generated by null geodesics which also determine the singularities
arising in the equation of motion at the horizon cap. This feature mirrors the
causal nature Berges (2004) of the Schwinger-Dyson equations for the
correlation functions in the field theory. We will discuss more about the
consistency of this result in Sec V.5.

The quadratic on-shell action for the bulk scalar field is the sum of three
pieces, namely $S_{in-in}$ and $S_{out-out}$ which are quadratic in the
ingoing and outgoing modes respectively, and the cross-term $S_{in-out}$. As
in the thermal case discussed in Section II, $S_{in-in}=0$. The ingoing mode
is analytic at the horizon and the contributions from the forward and backward
arms of the radial contour cancel out. Once again this is required for
consistency, as if we keep only the ingoing mode by setting $q=0$ in (104),
then $J_{1}=J_{2}$ and $W[J_{1}=J_{2}]=0$ for an arbitrary initial (non-
thermal) state. (Recall $W$ is identified with $iS_{on-shell}$.) The cross
term $S_{in-out}$ has a branch point at the horizon cap and the integration
over the radial contour results in the two boundary terms like in the thermal
case discussed in Section II. $S_{out-out}$ potentially has a single pole
$(v_{h}-v)^{-1}$ divergence which we denote as $S_{\epsilon}$. Explicitly,

$\displaystyle S_{\epsilon}$ $\displaystyle\propto$ $\displaystyle\int{\rm
d}\omegaup\int{\rm d}k_{L}\int{\rm d}^{d-2}k_{T}\oint_{\epsilon}\,{\rm
d}v\sqrt{-G}\Bigg{(}G^{vv}\partial_{v}\phi_{n,out}^{*}\partial_{v}\phi_{n,out}\qquad\qquad$
(115)
$\displaystyle+G^{v\sigma}\left(\partial_{v}\phi_{n,out}^{*}\partial_{\sigma}\phi_{n,out}+\partial_{v}\phi_{n,out}\partial_{\sigma}\phi_{n,out}^{*}\right)+\cdots\Bigg{)}$

at the $n$-th order in the late proper time expansion. We have verified that
$S_{\epsilon}=0$ for the solution with the regular behavior at the horizon cap
given by (IV), obtained for the appropriate choices of $\alpha_{n}$ and
$\gamma_{n-1,out}$ as discussed above. One can also check that terms like
$\phi_{k,out}^{*}~{}\phi_{l,out}$ with $k+l\leq 2n$ also do not contribute to
$S_{\epsilon}$. However, our arguments for $S_{out-out}=0$ for the thermal
case in Section II do not go through here since they rely on the KMS boundary
condition. Nevertheless, since the pole at the horizon vanishes, $S_{out-out}$
is the sum of two boundary terms as well.

Since we preserve the horizon cap regularity at each order, the analytic
continuation of the outgoing mode across the horizon cap at each order works
exactly in the same way as in the case of thermal equilibrium, i.e.
$\phi_{n,out}(\omegaup,v,\kappa_{L},\kappa_{T})$ picks up a factor of
$e^{\beta\omegaup}$ as evident from (109) and (IV). Repeating the argument in
Section II, each boundary term in $S_{out-out}$ involves one
$\phi_{b}(\omega,\kappa_{L},\kappa_{T})$ and another
$\phi_{b}(-\omega,\kappa_{L},\kappa_{T})$ (with $b$ standing for the boundary
value) or the corresponding boundary values of the radial derivatives. Also,
the contribution from the forward and backward arms come with opposite signs.
While $\phi_{b}(\omega,\kappa_{L},\kappa_{T})$ picks up a $e^{\beta\omega}$
factor via analytic continuation across the horizon cap,
$\phi_{b}(-\omega,\kappa_{L},\kappa_{T})$ picks up a $e^{-\beta\omega}$
factor, and these multiply to unity. Therefore, the boundary term
contributions from the two arms cancel out resulting in $S_{out-out}=0$.

Finally, as in the thermal case, the on-shell action is simply the sum of two
boundary terms (including the counter-terms for holographic renormalization)
obtained from $S_{in-out}$. We can then readily differentiate this on-shell
action to obtain the Schwinger-Keldysh correlation functions of the Bjorken
flow.

As shown in the following section, the boundary correlation functions obtained
solely from $S_{in-out}$ ensure that the (nonequilibrium) retarded correlation
function is given always by the linear causal response. Furthermore, a major
consistency check is that we reproduce the homogeneous transients as poles of
the retarded Green’s function in complexified $\omegaup$ not only at the
leading order as computed in Janik and Peschanski (2006a), but also at the
subleading orders as shown in Appendix E and discussed further below. We will
discuss more consistency tests in Section V.5.

##  V The real time out-of-equilibrium correlation functions

In the previous section, we have shown that via the bulk diffeomorphism dual
to the Weyl transformation we can obtain the solution of the bulk scalar field
which has the same analytic thermal nature of the near-horizon modes to all
orders in the (hydrodynamic) large proper time expansion. Furthermore, this
solution is unique up to the coefficients that give the leading near-horizon
behavior of the ingoing modes, and the on-shell action to all orders is the
product of the ingoing and outgoing modes, as the terms which are quadratic in
the ingoing modes or the outgoing modes cancel exactly between the radial arms
corresponding to the forward and backward parts of the time contour due to the
thermal nature of their leading near-horizon behavior. Here we will compute
the Schwinger-Keldysh correlation functions from the on-shell action of the
bulk scalar field in an appropriate late-time expansion. We will show that the
correlation functions are consistent with the field theory identities which
hold even for out-of-equilibrium states and one of them can be used to fix the
coefficients that give the leading near-horizon behavior of the ingoing modes
and thus determine the bulk scalar field solution uniquely. Furthermore, we
will show that the all-order hydrodynamic correlation function has a matrix
factorized form giving a bilocal generalization of the corresponding form of
the thermal correlation functions.

In Sec. V.1, we will discuss the general identities satisfied by the
Schwinger-Keldysh correlation functions and discuss how they transform under
Weyl transformations. In Sec. V.2, we will obtain the Schwinger-Keldysh
correlation functions from the on-shell action and establish its matrix
factorized form which is consistent with the field theory identities. We will
also show that one of the latter identities can determine the bulk solution of
the dual scalar field uniquely and therefore the Schwinger-Keldysh correlation
functions. In Sec. V.3, we will obtain the correlation functions in the
perfect fluid limit and show that it is thermal after space-time
reparametrizations up to overall Weyl factors. In Sec. V.4, we will establish
the systematic late-time expansion of the correlation functions. Finally, in
Sec. V.5, we will discuss consistency checks including the feature find that
the retarded propagator given by the ingoing modes reproduces the normalizable
bulk solutions at complex frequencies (which map to quasi-normal modes of the
static black brane Janik and Peschanski (2006a)) with vanishing sources to all
orders although the relevant phase factors at first and higher orders are
determined as functions of the frequency via the near-horizon behavior of the
outgoing modes at real frequencies. The latter implies non-trivially that one
can indeed obtain the normalizable bulk solutions dual to the nonequilibrium
transients of the dual field theory to all orders in the large proper time
expansion from the retarded correlation function generalizing how we can
obtain the quasinormal modes (thermal transients) as poles of the thermal
retarded correlation function.

###  V.1 Some useful relations and their consequences

Some crucial identities are valid for the Schwinger-Keldysh correlation
functions even in out-of-equilibrium states. The first such identity of
interest is

$\displaystyle G_{R}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-i\theta(x_{1}^{0}-x_{2}^{0}){\rm
Tr}(\rho[\hat{O}(x_{1}),\hat{O}(x_{2})])$ (116) $\displaystyle=$
$\displaystyle-i\left({\rm Tr}(\rho\,{\rm
T}(\hat{O}(x_{1})\hat{O}(x_{2})))-{\rm
Tr}(\rho\hat{O}(x_{2})\hat{O}(x_{1}))\right)$ $\displaystyle=$ $\displaystyle
G_{11}(x_{1},x_{2})-G_{12}(x_{1},x_{2}).$

Similarly, we can arrive at the identity

$\displaystyle G_{A}(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle
i\theta(x_{2}^{0}-x_{1}^{0}){\rm Tr}(\rho[\hat{O}(x_{1}),\hat{O}(x_{2})])$
(117) $\displaystyle=$ $\displaystyle-i\left({\rm Tr}(\rho\,{\rm
T}(\hat{O}(x_{1})\hat{O}(x_{2})))-{\rm
Tr}(\rho\hat{O}(x_{1})\hat{O}(x_{2}))\right)$ $\displaystyle=$ $\displaystyle
G_{11}(x_{1},x_{2})-G_{21}(x_{1},x_{2}).$

These identities give the retarded and advanced correlation functions in the
usual Schwinger-Keldysh basis. The definitions of the correlation functions
also imply that in any arbitrary state

$G_{11}(x_{1},x_{2})=G_{11}(x_{2},x_{1}),\quad
G_{12}(x_{1},x_{2})=G_{21}(x_{2},x_{1}),\quad
G_{22}(x_{1},x_{2})=G_{22}(x_{2},x_{1}).$ (118)

Finally, it is obvious also that in any arbitrary (nonequilibrium) state

$G_{R}(x_{1},x_{2})=G_{A}(x_{2},x_{1}).$ (119)

We will show that these identities imply the following for the horizon cap of
the Bjorken flow. Firstly, will use (116) to show that the retarded
correlation function is always obtained from the ingoing mode, and (117) to
show that the advanced correlation function is always obtained from the
outgoing mode. Second, due to (119), we can uniquely determine $\gamma_{n,in}$
for $n\geq 1$. These coefficients cannot be determined by the regularity
condition (IV) of the horizon cap unlike $\gamma_{n,out}$ as discussed
previously, but together with $\gamma_{n,out}$ they provide unique solution of
the bulk scalar field for specified sources at the two boundaries via our
ansatz (104). We will also see that by satisfying (119) we can also ensure the
validity of (118).

Another important issue is the Weyl transformation. Consider a background
metric $g_{\mu\nu}$ and its Weyl rescaled version $\Omega^{2}(x)g_{\mu\nu}$
such that $\Omega(x)$ is a non-vanishing function. Disregarding Weyl anomaly,
the correlations functions of a scalar primary operator $\hat{O}$ of conformal
dimension $\Delta_{O}$ in a conformal field theory living in these two
background metrics would be related by

${\rm
Tr}(\tilde{\rho}\hat{O}(x_{1})\cdots\hat{O}(x_{n}))_{\Omega^{2}g_{\mu\nu}}=\Omega^{-\Delta_{O}}(x_{1})\cdots\Omega^{-\Delta_{O}}(x_{n}){\rm
Tr}(\rho\hat{O}(x_{1})\cdots\hat{O}(x_{n}))_{g_{\mu\nu}}$ (120)

where $\tilde{\rho}=U^{\dagger}\rho U$ with $U$ denoting the unitary operator
implementing the Weyl transformation. In the context of the holographic
Bjorken flow, the states $\rho$ and $\tilde{\rho}$ would be described by the
holographic geometries whose boundary metrics are the Milne metric and its
Weyl rescaled version respectively, and the respective energy-momentum tensors
are also appropriately Weyl transformed including the correct holographic Weyl
anomaly. This has been described in detail already in Section III.2.

We have seen that it is easier to implement the out-of-equilibrium horizon cap
prescription in the Weyl transformed geometry (state) in which the temperature
and entropy density become a constant at late proper time, and the Klein-
Gordon equation in the bulk assumes the form of that in a static black brane
at the leading order. It is convenient to compute the correlation functions in
this background first, and then go back to the usual Bjorken flow background
by Weyl transformation. In this case, we should use

$\Omega^{\Delta_{O}}(\tau_{1})\Omega^{\Delta_{O}}(\tau_{2})\widetilde{G}(x_{1},x_{2})$

with $\widetilde{G}(x_{1},x_{2})$ being the correlation functions computed
after Weyl rescaling to obtain the correlation functions in the usual Bjorken
flow background with $\Omega$ given by (56) and (57). We disregard Weyl
anomalies since they are not state-dependent (same as in vacuum).

Note that the full correlation functions to all orders in the late proper time
expansion should depend on $\hat{\zeta}_{1}$ and $\hat{\zeta}_{2}$ only
through the relative separation $\hat{\zeta}_{1}-\hat{\zeta}_{2}$ in rapidity
due to the boost invariance of the background, and similarly only on
$|\vec{x}_{\perp 1}-\vec{x}_{\perp 2}|$ due to translation and rotation
symmetries of the background in the transverse spatial plane.

We note that we can practically use the Weyl transformation to compute the
correlation functions is because we are in the boost invariant hydrodynamic
regime where the state is characterized only by a proper time-dependent
temperature. A Weyl rescaling makes the temperature constant at late time
leading us to set-up a perturbative derivative expansion controlled by a fixed
dimensionful parameter. However, there will be contributions to the
correlation functions in the form of a generalized trans-series which depends
on the initial conditions, and are not constructed from hydrodynamic data.
Such contributions which represent transients leading to hydrodynamization of
correlation functions are discussed in Section VI. In this section, we discuss
the Schwinger-Keldysh correlation functions in the hydrodynamic regime using a
well-defined late proper time expansion.

###  V.2 General structure of the hydrodynamic correlation functions

It is useful to define a new variable

$s(\sigma)=\frac{d-1}{d-2}\sigma-\tilde{\gamma}_{0}\varepsilon_{0}^{-1/d}\ln\left(\sigma/\tau_{0}\right)$
(121)

with $\tilde{\gamma}_{0}=\gamma_{0}\varepsilon_{0}^{1/d}/\omegaup$ being a
numerical constant (independent of $\omegaup$). We have already seen that the
smoothness of the horizon cap requires that in the case of $AdS_{5}$,
$\tilde{\gamma}_{0}=1/4$ (see (113)).222222In the case of $AdS_{4}$,
$\tilde{\gamma}_{0}=2/3$. In terms of this variable, we can readily see from
(104) that the non-normalizable mode of the scalar field in the gravitational
background dual to the Bjorken flow takes the form:

$\displaystyle\begin{pmatrix}J_{1}(\sigma,\hat{\zeta},\vec{x}_{\perp})\\\
J_{2}(\sigma,\hat{\zeta},\vec{x}_{\perp})\end{pmatrix}$ $\displaystyle=$
$\displaystyle\int{\rm d}\omegaup{\rm d}k_{L}{\rm
d}^{d-2}k_{T}\,\,e^{-i\omegaup
s(\sigma)}e^{i\left(k_{L}\hat{\zeta}+\vec{k}_{T}\cdot\vec{x}_{\perp}\right)}$
(122)
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\mathcal{S}_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})\cdot\begin{pmatrix}p(\omegaup,k_{L},k_{T})\\\
q(\omegaup,k_{L},k_{T})\end{pmatrix}$

and similarly, the expectation value of the dual operator (up to state-
independent contact terms) is

$\displaystyle\begin{pmatrix}\langle
O_{1}(\sigma,\hat{\zeta},\vec{x}_{\perp})\rangle\\\ \langle
O_{2}\rangle(\sigma,\hat{\zeta},\vec{x}_{\perp})\end{pmatrix}$
$\displaystyle=$ $\displaystyle(2\Delta_{O}-d)\int{\rm d}\omegaup{\rm
d}k_{L}{\rm d}^{d-2}k_{T}\,\,e^{-i\omegaup
s(\sigma)}e^{i\left(k_{L}\hat{\zeta}+\vec{k}_{T}\cdot\vec{x}_{\perp}\right)}$
(123)
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\mathcal{R}_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})\cdot\begin{pmatrix}p(\omegaup,k_{L},k_{T})\\\
q(\omegaup,k_{L},k_{T})\end{pmatrix}$

where $\mathcal{S}_{n}$ and $\mathcal{R}_{n}$ can be defined from the
asymptotic expansion of $\mathcal{M}_{n}$:

$\mathcal{M}_{n}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})=v^{d-\Delta_{O}}\left(\mathcal{S}_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right)+v^{\Delta_{O}}\left(\mathcal{R}_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right),$
(124)

and the labels $1$ and $2$ stand for the sheets of the bulk space-time ending
on the forward and backward arms of the Schwinger-Keldysh contour
respectively.

It is also useful to define

$\displaystyle\mathcal{R}(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\mathcal{R}_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T}),$
$\displaystyle\mathcal{S}(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\mathcal{S}_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T}).$
(125)

Clearly (105) leads to

$\displaystyle\mathcal{S}(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\begin{pmatrix}a(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\,&\,b(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\\\
a(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\,&\,e^{\beta\omegaup}b(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\end{pmatrix},$
$\displaystyle\mathcal{R}(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\begin{pmatrix}A(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\,&\,B(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\\\
A(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\,&\,e^{\beta\omegaup}B(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})\end{pmatrix},$
(126)

where

$\displaystyle\phi_{in}(v,\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\phi_{n,in}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T}),$
$\displaystyle\phi_{out}(v,\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}\phi_{n,out}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T}),$
(127)

gives us $a(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$,
$A(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$,
$b(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$ and
$B(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$ from their following near
boundary expansions

$\displaystyle\phi_{in}(v,\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$ $\displaystyle
v^{d-\Delta_{O}}\left(a(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right)+v^{\Delta_{O}}\left(A(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right),$
$\displaystyle\phi_{out}(v,\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$ $\displaystyle
v^{d-\Delta_{O}}\left(b(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right)+v^{\Delta_{O}}\left(B(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right).$
(128)

It should also be obvious that if we define $a_{n}$, $A_{n}$, $b_{n}$ and
$B_{n}$ via

$\displaystyle\phi_{n,in}(v,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$ $\displaystyle
v^{d-\Delta_{O}}\left(a_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right)+v^{\Delta_{O}}\left(A_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right),$
$\displaystyle\phi_{n,out}(v,\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$ $\displaystyle
v^{d-\Delta_{O}}\left(b_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right)+v^{\Delta_{O}}\left(B_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T})+\cdots\right).$
(129)

then

$\displaystyle a(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}a_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T}),$
$\displaystyle A(\sigma,\omegaup,\kappa_{L},\vec{\kappa}_{T})$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\varepsilon_{0}^{1/d}\sigma\right)^{-n}A_{n}(\omegaup,\kappa_{L},\vec{\kappa}_{T}),$