text:
\section{Introduction

In Minkowski space
the vacuum
of quantum fields, the ground state of the Hamiltonian,
is globally
defined.
For instance, for a massless free field $\phi$ in two
dimensions,
it is the
state annihilated by the destruction operators $a_k$ associated with the
positive norm
modes $\phi_k = e^{-i\Om t +kz}/(4\pi \Om)^{1/2}$ where $\Om = |k| > 0$.
When probed locally, it determines
correlation functions, such as that of the stress-energy tensor.
Considering
$T_{UU} = (\partial_U \phi)^2$,
where $U$ ($V$) is the retarded (advanced) null
coordinate $U= t-z$, ($V= t+z$), the connected function is
\ba
\langle T_{UU}({U}) \,  T_{UU}({U_0})\rangle &=&
\Big(\partial_{{U}}\partial_{{U_0}}\,  \langle \phi({U}) \;  \phi({U_0})\rangle \Big)^2
\nonumber\\
&=&
\left(\frac{1}{4\pi}  \frac{1}{({U} - {U_0} - i \epsilon)^2}   \right)^2 .
\label{TUU}
\ea
The regulator $\epsilon \to 0^+$ specifies the nature
of the singularity in the
limit ${U} \to {U_0}$. It arises from the fact
that only positive frequency $\Om = i\partial_{{U}}$ contribute to \eqr{TUU}.

Even though
this
function
monotonically behaves
in $\Delta U^{-4}$,
it determines
subtle
effects.
This
can
be seen by considering
accelerated systems where
 both local and non-local phenomena
related to the Unruh
effect~\cite{Unruh:1976db,Unruh:1983ms,Carlitz:1986ng,Massar:1995im,Brout:1995rd,Obadia:2002qe,Massar:2006ev}
can be obtained from \eqr{TUU}.
Moreover
the correlations of $T_{UU}$ in the
(Unruh)
 vacuum
across the future horizon of a black hole behave as in \eqr{TUU}.
However since the black hole geometry is non uniform, as one probes the vacuum further away from the horizon,
there is a gradual change
which results, on one hand, in correlations typical of a thermal flux at
the Hawking temperature~\cite{Hawking:1974sw,Davies:1976ei,Carlitz:1986ng}
and, on the other,
in
correlations across the horizon between Hawking quanta and their partners~\cite{Massar:1996tx}.

The aim of the present paper is to analyze
the black hole
case in the light of the Minkowski case.
We show
that the gradual change in the correlations
comes through the observables used
to probe the state, and not from $\phi$
which is conformally invariant and thus insensitive to the
geometry. We relate this gradual change
to that obtained when considering
renormalized expectation values which also depend on
the conformal factor.
In the last part,
we consider dispersive field theories~\cite{Unruh:1994je},
or analogue black holes in condensed matter~\cite{Unruh:1980cg,Barcelo}.
First,
we show that
the preferred rest frame~\cite{Jacobson:1996zs} associated with dispersion defines
new scalars (observables)
which give an unambiguous meaning to the gradual change of
the
stress energy
correlations.
Second we study how dispersion affects the correlation pattern.\footnote{
While completing this work, we became aware of \cite{Schutzhold:2010ig} where
similar conclusions are
 reached.}
In addition to the robustness of the spectral properties of Hawking
radiation~\cite{Unruh:1994je,Brout:1995wp,Corley96},
that  of the long distance correlations across the horizon
was established
by constructing wave packets of $in$ modes~\cite{Brout:1995wp}.
We shall return to this observation and make contact with
the analysis of density-density correlations in Bose-Einstein
condensates~\cite{Balbinot:2007de,Iacopo08,Macher:2009nz}.

We have organized the paper as follows.
In Section II, we describe the behavior of the stress energy 2pt function in flat space,
and interpret
its properties by introducing external
(accelerated) systems which probe the field.
In Section III, we perform the same analysis in a black hole geometry.
Using affine coordinates,
we present an invariant
description of the transition from
its near horizon behavior, which is similar to \eqr{TUU}, to its long distance behavior.
Finally, in Section IV,
we consider the stress energy correlations in
dispersive theories,
and
investigate what are the modifications
induced by dispersion. We conclude with
remarks about the similarities between these dispersive effects and those
obtained with relativitic fields propagating in fluctuating metrics~\cite{Barrabes:2000fr
Parentani:2007mb}.

\section{Unruh effect and correlations across a Rindler horizon}

We introduce the null coordinates $a u= - \ln(-aU)$, $a v = \ln(aV) $,  defined
respectively for $U< 0$ and $V > 0$.
These are related by $\tau = (u+v)/2$
to $\tau$, the proper time of an accelerated system which follows
$z^2 - t^2 = 1/a^2$ in $R$, the right quadrant $z > |t|$. ($a$ is the constant acceleration.)
When ${U}$ and ${U_0}$ are
negative, i.e. with both points on the right of $U=0$, the future horizon
 of the accelerated system,
\eqr{TUU}
can be re-written using $u$.
Introducing $T^R_{uu} = (\partial_u \phi)^2 = T_{UU} \, e^{- 2 a u}$, \eqr{TUU} becomes
\ba
\langle T^R_{uu}({u}) \,  T^R_{uu}({u_0})\rangle &=& \left(\frac{1}{16 \pi}
\frac{a^2}{( \sinh \frac{a}{2}({u} - {u_0} - i\epsilon))^2}\right)^2 .
\label{Tuu}
\ea
Since $1/(\sinh y)^2= \Sigma_n\,  (y + i n \pi)^{-2}$,
where the sum over $n$
goes from $-\infty$ to $\infty$,
$\langle T^R_{uu}  T^R_{uu}\rangle $
 is {\it the} periodic function in Im$u$
of period $2\pi/a$ whose zeroth term, $n=0$,
behaves as in the vacuum, see \eqr{TUU}.
Hence \eqr{Tuu} is
the 2-point
function
in a thermal bath
at a temperature equal to $a/2\pi$.
Noticing that
along the accelerated trajectory, one has $\tau = u$ (because $v=u$)
\eqr{Tuu}
determines
the
correlation function of the
energy density
probed
by the accelerated system.
In this way we recover
the Unruh effect: accelerated systems perceive the vacuum as a thermal ensemble at that
temperature~\cite{Unruh:1976db}.

It is less usual but equally
interesting to consider \eqr{TUU} with one point on either side of the horizon.
Using $u$ and $T_{uu}$ on the right,
and their symmetrical counterpart on the left,
$a \bar u = \ln(aU)$
and $T^L_{u u} = (\partial_{\bar u} \phi)^2 = T_{UU} \, e^{2 a \bar u}$ for $U > 0$,
\eqr{TUU} becomes
\ba
\langle T^L_{u u}(\bar {u}) \,  T^R_{uu}({u_0})\rangle &=& \left(\frac{1}{16 \pi}
\frac{ a^2}{(\cosh \frac{a}{2}(\bar {u} + {u_0}))^2}\right)^2  .
\label{ldTuu}
\ea
Quite
surprizingly
$\langle T^L_{u u} T^R_{uu}\rangle$ possesses a
'bump'
which is centered at $\bar {u} + {u_0} = 0$.
In terms of the affine coordinate $U$,
it is located at ${U} = - {U_0}$,
on the opposite value from the horizon.
This
is unnexpected
because,
at {\it fixed} $\Delta V$, $\langle \phi(U, V) \phi(U_0,V_0) \rangle
 = \frac{-1}{ 4\pi} \ln \Delta U
 + Cst. $
 has no maximum
when expressed in terms of $\bar u$ and $u_0$.
In fact, the
maximum in \eqr{ldTuu}
results from the combined effect of the Jacobians $dU/du$
associated with the tensorial character of $T_{UU}$,
and the monotonic power law decay of \eqr{TUU}.\footnote{My attention was drawn to this point
in a discussion with W. Unruh and T. Jacobson that took place during
the workshop "Towards the observation of
Hawking radiation in condensed matter systems"
held at IFIC in Valencia in Feb. 2009.}
These
properties
apply to massless fields in any space-time dimension $d$. Because of the dimensionality of the $\phi$ field
($= (d-2)/2$)
the location of the maximum is shifted at
$\bar {u} + {u_0} = -\frac{2}{a} {\rm arctanh} \frac{d-2}{d+2}.$

So far however
\eqr{ldTuu} is simply a
re-expression of \eqr{TUU}:
no physics is gained when writing a tensor in a new coordinate system.
So, isn't the maximum of \eqr{ldTuu} just a coordinate artefact ?
No, \eqr{ldTuu}, as \eqr{Tuu}, governs physical effects.
For instance, when looking at the
state of two opposite
accelerated systems~\cite{Massar:2006ev},
the correlation matrix of this bi-partite system
exhibits a non-trivial entanglement  near
the maximum of \eqr{ldTuu}.
Similarly the scattering by two opposite
accelerated mirrors destructively interferes near that maximum~\cite{Obadia:2002qe}.
As another example~\cite{Massar:1995im},
expressions closely related to \eqr{ldTuu}
obtain when looking at the
value of $T^L_{uu}$ which is correlated
to the fact that an accelerated detector in $R$ has
made a transition.

In all cases the physics is the same:
measurements performed in  $R$
are statistically correlated to those
performed on the other side of the horizon.

To convey the idea that
the re-writing of \eqr{TUU} as Eqs. (\ref{Tuu},
\ref{ldTuu}) is not merely a
coordinate change
but has to do with the physics of accelerated systems,
we make two observations. First
it is conceptually useful
to
introduce the ($U$ part of the) {\it scalar}
energy density that such system in $R$ will measure:
$\rho^R = T_{\mu \nu} u_{R}^\mu  u_{R}^\mu $,
where $u_{R}^\mu= dx_{R}^\mu/d\tau$ is the unit vector field tangeant to
its trajectory. Then the l.h.s of \eqr{Tuu} is
$\langle \rho^R
(\tau) \, \rho^R
(\tau_0)
\rangle$.
Similarly the l.h.s of \eqr{ldTuu} is
$\langle \rho^L(\bar\tau) \, \rho^R
(\tau_0) \rangle$.
Hence both
\eqr{Tuu} and \eqr{ldTuu}
have a clear
interpretation
as energy-energy correlation functions of accelerated systems.
Secondly, we
notice that
\eqr{Tuu} only depends on ${u} - {u_0}$.
This
results from the invariance of the vacuum under
Lorentz transformations
$u \to u + b, U \to U e^{-ab}$.
In fact the
Minkowski
distance between two points situated along the accelerated trajectory $v=u$
is
\be
s^2 = - \Delta t ^2 + \Delta z ^2
= - \frac{4 \sinh^2(a (\tau- \tau_0)/2)}{a^2}.
\label{s1}
\ee
We therefore see that the stationarity
of \eqr{Tuu} in $u$ for {\it arbitrary values}  of $v$
are related to the properties of $s^2(\Delta\tau)$ evaluated along $v=u$.
In a similar fashion, \eqr{ldTuu} is related to
the
distance $s_{\rm op}^2$ bewteen two points situated along the
opposite accelerated trajectories defined by $z^2 - t^2 = 1/a^2$. Using the proper time $\tau_0$ ($\bar \tau$)
to localize the point on the right (left) trajectory, one gets
\be
s_{\rm op}^2 =
\frac{4 \cosh^2(a (\bar \tau + \tau_0)/2)}{a^2}.
\label{s2}
\ee
The maximum of correlation in \eqr{ldTuu} thus coincides with
the minimum (space-like) distance bewteen the two
trajectories.
 Because of Lorentz invariance no
preferred spatial distance $\Delta z ^2$ is selected though.

Yet, in preparation for the black hole problem,
one would like to know what is the origin of
the {\it mathematical} properties of \eqr{Tuu} and \eqr{ldTuu}.
In fact,  these
must be deeply rooted
since
\eqr{ldTuu} is the analytical continuation of \eqr{Tuu} obtained by
subtracting
half of the imaginary
period
that fixes the Unruh
temperature.
In addition, we wish to identify what could explain
the maximum of correlation in \eqr{ldTuu},
which is independent of
$\Delta V$, and thus
{not necessarily} refers to \eqr{s2} and
 accelerated systems in $L$.
As shown in the Appendices,
these
properties can be understood
when using
the
Unruh modes~\cite{Unruh:1976db}.
In brief, the maximum of \eqr{ldTuu} finds its origin in the
entanglement in Fock space between states with opposite Killing frequency $\om = i\partial_u$.
This entanglement translates into space-time correlations because, in each pair,
the partner mode lives on the other side of the Rindler horizon, {\it and} is weighted in \eqr{Um}
by a factor $z_\om$ which is real for all $\om$.
In addition the  maximum value of correlations is directly related to the mean occupation
number of Rindler quanta.

\section{Hawking effect and
correlations across the
 horizon}

We have a double aim. First, we wish to interpret the equivalent of \eqr{TUU}
and
Eqs. (\ref{Tuu},\ref{ldTuu}) in a black hole geometry.
Second, we aim to
analyze the differences
 between the black hole case and Minkowski.
In particular we are looking for a smooth and {\it coordinate invariant} interpolation from the near horizon region,
where \eqr{TUU} should make sense, to the asymptotic regions where Eqs. (\ref{Tuu},\ref{ldTuu})
should do.

To reach these ends,
we
need to identify what corresponds to
the Minkowski vacuum,
the affine coordinate $U$, and
the
$u$  coordinate
associated
with boosts
$u \to u+b, U \to U e^{ab}$.
For simplicity we shall work with a two dimensional conformally invariant field
 $\phi$.
Then the identification is rather easy and known~\cite{Unruh:1976db,Brout:1995rd}.
The novelty mainly consists in the attention paid
to the stress-energy correlation function.

\subsection{Prelude: conformal invariance}

We
work with a
conformally invariant field
because
the simplicity of the expressions
will oblige us to identify under which conditions
identities
valid in all space-times
acquire a
physical meaning related to the Unruh and/or to the Hawking effect.

In
two dimensions
the line element can always be written in double null coordinates as
\be
ds^2 = - C(U,V)\,  dU dV \, .
\ee
In these coordinates,
$\phi$ obeys
$\partial_U \partial_V \phi = 0$.
Therefore the conformal factor
$C$
drops out and the right and left moving sectors
remain
decoupled $\phi = \phi(U) + \phi(V)$.
As a result, for {any} $U$ coordinate, one can consider the $U$-vacuum state
defined by the positive frequency modes $\phi_\Om = e^{-i \Om U}/(4 \pi \Om)^{1/2}$.
In that state, the {\it connected}
correlation function of
$T_{UU} = (\partial_U \phi)^2$
\be
\langle T_{UU}({U}) \,  T_{UU}({U_0})\rangle_{c}
\equiv
 \langle T_{UU}({U}) \,  T_{UU}({U_0})\rangle-  \langle T_{UU}({U}) \rangle \,   \langle T_{UU}({U_0}) \rangle,
\label{TUUC}
\ee
obeys \eqr{TUU},
independently
of the regularization scheme used to compute $\langle T_{UU}({U}) \rangle$.\footnote{
The
connected 2pt function of $T_{\mu \nu}$
governs gravitational back-reaction effects
beyond those included in the semi-classical Einstein equations.~\cite{Parentani:2007mb}
This provides its dynamical relevance.}
Introducing the coordinates $au =\ln( -aU)$
and $a \bar u = \ln (aU)$ for respectively negative and positive values of $U$, \eqr{Tuu} and \eqr{ldTuu}
also follow, as
{\it mathematical identities} valid in all space-times and for any coordinate $U$.
Some physical input is therefore needed
to transform these identities into meaningful relations among observables.

To relate \eqr{Tuu}
to the Unruh effect is straightforward. It suffices to consider
a
particle detector following the orbit $u=v$. It will perceive the $U$-vacuum
as a thermal state at a temperature $a/2\pi$. (In general $a$ is no longer the proper acceleration,
consider e.g.
de Sitter space where the thermal bath is also perceived by inertial detectors.) Similarly, for \eqr{ldTuu},
when introducing another detector which follows
the mirror trajectory $\bar u = \bar v$,
the combined state of these two detectors will be entangled as that of accelerated detectors
in Minkowski~\cite{Massar:2006ev}, again as a direct consequence
of the conformal invariance of $\phi$.
This generalization of the Unruh effect
applies to any
field and in any dimension, but only approximatively provided the acceleration is much higher
that the space-time curvature.
(This
limit is also used
in the thermodynamic analysis of space-time~\cite{Jacobson:1995ab}.
However the fact that \eqr{ldTuu} always applies
indicates that the purity of the quantum state, and not only the mean fluxes, is
a key ingredient.)

To relate
\eqr{Tuu} and \eqr{ldTuu} to the Hawking effect is more elaborate.
As we shall see, the metric
plays a crucial role in
"selecting" the $U$-vacuum and the coordinate $u$.
This follows from
the fact that, unlike the connected
2pt function of \eqr{TUUC},
other
observables are {\it not} conformally invariant,
as for
instance the renormalized
expectation value of $T_{UU}$~\cite{Davies:1976ei}

\be
\langle T_{UU}\rangle^{\rm ren} = - \frac{1}{12 \pi}  C^{1/2} \, \partial_U^2 \frac{1}{C^{1/2}} .
\label{Tuuren}\ee
In Sec. III.D, we shall see that
the breakdown of the conformal invariance comes from the
renormalization scheme and not from the "bare" operator.

\subsection{The correspondence}

We work with one dimensional
 stationary  black hole metrics that we describe using
Eddington-Finkelstein (EF) coordinates $v,r$
in which the line element reads
\ba
ds^2 &=& -(1- w^2(r))\, dv^2 + 2 dv dr .
\label{EF}
\ea
We shall not consider specific functions for $w^2$,
as e.g. $2M/r$ which describes a
Schwarzschild black hole.
Hence the coordinate $r$ should not be
conceived as associated
with spherical symmetry. In fact, from a 2D point of view,
$r$ in \eqr{EF}
is an affine coordinate
along $v=cst.$ We also notice that $r$ is affine along $\tau = cst.$,
where $\tau$ is the proper time appearing in Painlev\'e-Gullstrand
(PG) coordinates~\cite{Barcelo} where $ds^2 = - d\tau^2 + (dr - w d\tau )^2$.
This guarantees that the forthcoming analysis can be made in PG coordinates.

When $w$ is
a constant, the geometry is
Minkowski.
Therefore the differences with
Sec. II will {\it all} stem from
the gradient of
$w(r)$.
We have adopted these coordinates
because they are well behaved
in the late time portion of the space-time
which is relevant for the Hawking effect and the correlations across the horizon, see e.g.  \cite{Brout:1995rd}.
In particular they stay regular
when $w^2$ crosses $1$, say at $r= r_h $.

This locus is the future horizon (for the static observers at fixed $r > r_h$)
and it is a null line $U= cst$.
The absence of the corresponding $V$ (past) horizon
induces a
disymmetry:
left moving modes
are regular across the horizon $U=cst.$ and
play
no role in the following.

Assuming that $w(r)\to w_\infty = cst.$ for $r \to \infty$,
the asymptotic observers at fixed $r$ are inertial, at rest with the black hole, and
their {proper} time is
$\tau_{\rm as} = v
(1 - w_\infty^2)^{1/2}$.
By rescaling
$v$ and $r$, one can
work in a gauge in which $w_\infty= 0$.
From now on we adopt it.
The positive norm modes that these observers will use
are
$ \phi_\om^{\rm as} = {e^{- i\om u } /
(4\pi \om)^{1/2}},
$
with $\om > 0$.
When $r \to \infty$,
 $u,v$ are related to
$\tau_{as}$ by $\tau_{as} = (u+v)/2$.
At finite $r$,
one has $u
= v - 2 \int dr/(1-w^2)$.
One sees that
$u$ might diverge when $w^2$ crosses $1$.

To settle this,
one needs to know how $w$ behaves across $r_h$. We assume a regular behavior
$w^2(r) \sim  1 - 2  \kappa (r - r_h)$
as this is
the case for regular collapses. ($\kappa > 0$ coincides with the
surface gravity since
$w_\infty = 0$~\cite{Jacobson:2007jx}. The extreme case $\kappa = 0$ requires a special
treatment~\cite{Balbinot:2007kr}.)
At fixed $v$,
one finds $u \sim - \kappa \ln(r- r_h)$.
Therefore,
when
$r \to r_h^+$ at fixed $v$,
$\phi^{\rm as}_\om$
behaves as
\be
\phi^{\rm as}_\om(u) \sim
{(r_v(u)-r_h)^{i\om/ \kappa} \over (4\pi \om)^{1/2}}
,
\label{bhm}
\ee
and can be taken to vanish for $r< r_h$. The function $r_v(u)$ obeys
\be
{dr_v} = -\frac{1}{2} \, C(u,v) du ,
\label{rv}
\ee
where $C(u,v)= (1- w^2)$ is the conformal factor in the $u,v$ coordinates.
It should be emphasized that $r_v(u)$ is an affine parameter for all $w(r)$,
and this property is governed by
$C$ which appears here for the second time.
(It was implicitly used before
when imposing that the coordinates $u, v$ are related to the asymptotic
proper time by $2\tau_{as} = u+v$.)

When replacing the acceleration $a$ by $ \kappa$ in \eqr{bhm}, the correspondance with
 \eqr{Um} is manifest, and
physically
meaningful
 because
$r_v$
is affine,
as  $U$ is
in Minkowski space.
However, this correspondence is confined near the horizon
since
$\phi^{\rm as}_\om$
behaves as $e^{i 2\om r}$ for $r\to \infty$ at fixed $v$.
Therefore, the gradual change of the mode $\phi_\om$
from
$\sim (r-r_h)^{i\om/\kappa}$ to
$\sim e^{i 2\om r}$ is what distinguishes
the back hole case from Minkowski. These
facts have been recognized in~\cite{Unruh:1976db,Damour:1976jd} and shall be further exploited
below.

The second part of the correspondence concerns the state of $\phi$.
While in Minkowski space the notion of vacuum is unambiguous,
in a black hole geometry
this
is lost. Nevertheless
the late time behavior~\cite{Hawking:1974sw}
is universal {\it and} stationary when described in terms of $u$ (or $\tau_{\rm as}$).
This can be
verified by considering different collapses and different initial states:
in each case there are transients,
but after a few e-folding $u$-times
$1/\kappa$, these fade out as
$\sim e^{-\kappa u}$
and the stationary values set in
(unless the collapse and/or the state is singular).
Therefore, as far as the description in terms of $u$
is concerned,
a {\it single} stationary state
is
selected.
As
noticed in~\cite{Unruh:1976db}, this state is most simply characterized
by Unruh
modes,  $\phi_\om$,
exactly as the Minkowski vacuum can also be, see the discussion after \eqr{Um}.
The particular combination of modes
weighted by $z_\om = e^{- \pi \om/\kappa}$ means that $\phi_\om$ only contains positive frequencies
$\Omega_K= i\partial_{U_K}$
where $U_K$ is {\it the} regular
coordinate
across $r_h$ which is related to $u$ by
$d U_K/du = e^{-\kappa u}$.
Indeed, stationarity requires an exponential relation bewteen $u$ and $U_K$,
whereas regularity across $r_h$ fixes the decay rate $d \ln U_K /du$
to be
$\kappa$.

The correspondence is now completed:
the regularity and the stationarity of the state
allow to characterize it
by the modes
 of
\eqr{Um}, with
 $U$ replaced by
$U_K
= -e^{-\kappa u}/\kappa$.
This exponential
is no longer related to a Lorentz transformation,
but is still related to an isometry whose
Killing field norm $\propto (1 - w^2)^{1/2}$
vanishes on the horizon.

\subsection{Correlation functions and asymptotic quanta}

In the Unruh vacuum,
the connected 2pt function  of  $T_{UU}= (\partial_{U_K} \phi)^2$, see \eqr{TUUC},
is {exactly}
given by \eqr{TUU}.
Then,
since
$\kappa u = -\ln (-\kappa U_K)$,
\eqr{Tuu} with $a \to \kappa$
also obtains.
However, as discussed in III. A, there is nothing special about these identities.
What
makes
\eqr{Tuu}
meaningfull here is
that both $U_K$ and $u$ are affine, respectively across the horizon and asymptotically.
Hence \eqr{Tuu} tells us that
when evaluated in the {regular}
state,
the 2pt function used by {inertial} asymptotic observers
 is that of
a thermal flux at the Hawking temperature $\kappa/2\pi$~\cite{Carlitz:1986ng}.

Let us now consider the equivalent of \eqr{ldTuu}. Introducing on the other side of the horizon
$\kappa \bar u = \ln \kappa U_K$,

 \eqr{ldTuu} with $a \to \kappa$ automatically obtains. But what does it mean~?
Unlike for the Unruh effect, we cannot refer to accelerated systems in the $L$ quadrant.
Nevertheless the procedure of \cite{Massar:2006ev} applies to black holes,
and one can study the $T_{\mu \nu}$ correlated to the detection of an asymptotic quantum.
As in Minkowski,
there is a reduction of the state:
 expectation values
should be computed with the reduced density matrix, see App.~C.
Then, because of the entanglement of \eqr{sqs},
there is a correlation between $T_{\mu \nu}$ evaluated inside the horizon and a detection
on $\cal I^+$~\cite{Massar:1996tx}.
Equivalently,
one can directly look for correlations in energy across the horizon and obtain
\eqr{ldTuu}.

There is however an important difference between \eqr{Tuu} and \eqr{ldTuu}. It
originates from the different status of the coordinates $u$ and $\bar u$.
\eqr{Tuu} has a clear interpretation because
$du $ at fixed $v$ is affine for $r\to \infty$.
If we can
 consider
the equivalent of \eqr{Tuu} with both points inside, and obtain the same answer when using $\bar u$,
we should ask under which conditions would $\langle T^L_{uu} T^L_{uu}\rangle$ posses
an intrinsic meaning (without any reference to the external region).
For this, it "suffices" that the inside region be also infinite
and that $w \to const.$ for $r \to - \infty$. Whereas it is unlikely that this be relevant for
black holes,
we assume it is the case
and consider the consequences.
(In a next Section we discusss analogue black holes
where this possibility can easily be realized.)
When $w \to const.$ for $r \to - \infty$,
the notion of asymptotic quanta equally applies to the negative frequency partners~\cite{Macher:2009tw}.
Then, $\langle T^L_{uu} T^L_{uu}\rangle$
has
the same meaning as $\langle T^R_{uu} T^R_{uu}\rangle$,
and
the long distance correlation $\langle T^L_{uu} T^R_{uu}\rangle$ of \eqr{ldTuu}
can be probed.

\subsection{Locality, covariance and renormalization}

To interprete \eqr{TUUC}
we have
so far used the asymptotic properties of the metric.
However we would like an {intrinsic} description of the
{\it gradual}
transition from the horizon
to the asymptotic regions.
To this end we consider the renormalization procedure.
As we shall see its
covariance
supplies the intrinsic description we are looking for, and this by breaking the conformal invariance.
(It should be clear
that different rules
on how to interpolate
will give different behaviors since the 2pt function of $T_{UU}$
is not a scalar).

We noticed
that both $u$ and $U_K$
are
affine
respectively for $r \to \infty$ and $r \sim r_h$.
We also noticed that $r_v$ of \eqr{rv} is affine
all the way through.
In what follows we exploit this
to relate the 2pt
function of $T_{UU}$ to its {renormalized} value of \eqr{Tuuren}.
To this end,
we study the coincidence point limit of \eqr{TUUC} in the Unruh vacuum
using
\be
T_{rr} = (\partial_r \phi)^2 =  \left(\frac{dU}{dr}\right)^2\,  T_{UU} ,
\label{Trr}
\ee which
is "coordinate invariant" since $r_v$ is,  up to a scale,
globally defined.
To present the concepts we first work at spatial infinity. There, in the limit $u_0 \to u$, one has
\ba
\langle T_{uu}(u) T_{uu}(u_0)\rangle_{K} &=&
\left( \frac{dU_K(u)}{du}  \frac{dU_K(u_0)}{du_0} \right)^2
\left(
\frac{1}{4\pi}\frac{1}{({U_K(u)} - {U_K(u_0)} - i \epsilon)^2}   \right)^2
\nonumber \\
&=&
\left(\frac{1}{4\pi}  \frac{1}{({u} - {u_0} - i \epsilon)^2}   \right)^2
+
\frac{ 2\, \langle T_{uu}(u)\rangle^{\rm ren}_{K} }{4\pi({u} - {u_0} - i \epsilon)^2}
+ O\left(\frac{1}{u - u_0}\right),
\label{TTrenas}
\ea
where $\langle T_{uu}(u)\rangle^{\rm ren}_{K}$ is the asymptotic
expectation value of $T_{uu}$
in the Unruh vacuum. Indeed, it is
defined~\cite{Davies:1976ei,Brout:1995rd} as
\ba
\langle T_{uu}(u)\rangle^{\rm ren}_{K} &=& \lim_{u_0 \to u} \partial_u \partial_{u_0}
 \Big(\langle \phi(u) \phi(u_0)\rangle_{K} - \langle \phi(u) \phi(u_0)\rangle_M \Big)
\nonumber \\
&=& \frac{1}{12 \pi}
\left( \frac{dU_K}{du} \right)^{1/2}  \partial_{u}^2
\left( \frac{dU_K}{du} \right)^{-1/2}
= \frac{\kappa^2}{48\pi},
\label{Tuuas}
\ea
where $\langle \phi(u) \phi(u_0)\rangle_M = \frac{-1}{4\pi}\ln ({u} - {u_0} - i \epsilon)$ is
evaluated in the asymptotic (Boulware) Minkowski-like vacuum.
The procedure of \eqr{TTrenas} is clear: by subtracting the asymptotic vacuum divergence,
one extracts the excess in the "noise" and thus identifies
the mean value of \eqr{Tuuas}.
Moreover, this procedure can be {\it univocally} covariantized and applied
at every space time point.
This is
achieved
by making use of $r_v(u)$:

\ba
\langle T_{rr}(u)\,  T_{rr}(u_0)\rangle_{K} &=&
\left(\frac{1}{4\pi}
 \frac{1}{(r_{v}{(u)} - r_v{(u_0)} + i \epsilon)^2}   \right)^2
 \nonumber \\
&& + 2 \, {\langle T_{rr}(u,v)\rangle^{\rm ren}_{K} }
\, \left(\frac{1}{4\pi}
 \frac{1}{(r_{v}{(u)} - r_v{(u_0)} + i \epsilon)^2}   \right) + ...
\label{loccol}
\ea
where $\langle T_{rr}(u,v)\rangle^{\rm ren}_{K}$ is the
renormalized value of $T_{rr}$
which is
defined as
\ba
\langle T_{rr}(u,v)\rangle^{\rm ren}_{K}
&=& \lim_{u_0 \to u} \partial_r \partial_{r_0}
 \Big(\langle \phi(u) \phi(u_0)\rangle_{K} - \langle \phi(u) \phi(u_0)\rangle_{\rm local}
\Big)\nonumber \\
&=&
\frac{1}{12 \pi}
\left( \frac{dU_K}{dr} \right)^{1/2}  \partial_{r}^2
\left( \frac{dU_K}{dr} \right)^{-1/2},
\label{Tuuloc}
\ea
where
\be
\langle \phi(u) \phi(u_0)\rangle_{\rm local} = - \frac{1}{4\pi}

\ln (r_v(u) - r_v(u_0) + i \epsilon).
\label{localG}
\ee
In virtue of the covariance, when using
the affine
coordinate $r_v$, the subtraction term
possesses this universal form.
This explains why \eqr{Tuuloc}
generalizes \eqr{Tuuas} at every point.

Several remarks should be made.
Firstly, using \eqr{rv}, one verifies that \eqr{Tuuloc} gives back \eqr{Tuuren}
in all space-times and all vacua. Hence \eqr{Tuuloc} can be seen as an alternative
expression for it.
Secondly, since
the vacuum
is
defined through the coordinate $U_K$, and the subtraction only refers to the metric through
\eqr{rv},
$\langle T_{rr}\rangle^{\rm ren}_{K}$ is
governed by $dU_K/dr$ and nothing else.
Thirdly,
even though $\langle T_{UU} T_{UU}\rangle_K$ is independent of
$v$,
the covariance of the divergent terms in \eqr{loccol}
unambiguously
defines the $v$-dependence
of
$\langle T^{\rm ren}_{rr}\rangle_K$ in \eqr{Tuuloc}.

\subsection{Covariant description of stress tensor correlations}

Following the same logic, we use
$T_{rr}$ of \eqr{Trr}
to characterize the
stress-energy correlations
in the Unruh vacuum at every point. Using \eqr{TUU}, \eqr{TUUC} gives

\ba
\langle T_{rr}(v,r)\,  T_{rr}(v_0 , r_0)\rangle_{K} &=&
\Big(  \partial_{r} \partial_{r_0}
\langle \phi(v,r)\, \phi(v_0,r_0)\rangle_{K} \Big)^2
\nonumber\\
&=&
\left( \frac{dU_K}{dr}  \frac{dU_K}{dr_0} \right)^2 \left(
\frac{1}{4\pi}\frac{1}{({U_K(v,r)} - {U_K(v_0,r_0)} - i \epsilon)^2 }  \right)^2 .
\label{TTrr}
\ea
This bi-tensor field
depends on $\Delta U_K^{-4}$ since the state is the Unruh vacuum, but unlike \eqr{TUU},
 it depends on the actual location of the two
points through
the $v$-dependence of
the Jacobians $dU_K/dr$.
Using $U_K(v,r) = e^{-\kappa v}\, {\cal{U}}_K(r)$ which follows from the stationarity of the metric,
we can extract this $v$-dependence and obtain
\be
\langle T_{rr}(v,r)\,  T_{rr}(v_0 , r_0)\rangle_{K} =
\left( \frac{d{\cal{U}}_K}{dr}  \frac{d{\cal{U}}_K}{dr_0} \right)^2
\left(
{4\pi}
{\left( e^{\frac{-\kappa (v-v_0)}{2}}\, {{\cal{U}}_K} -
e^{\frac{\kappa (v-v_0)}{2}}\, {{\cal{U}}_K^0} - i \epsilon\right)^2 }  \right)^{-2} .
\label{TTrr2}
\ee
There is yet another interesting way to write this correlator.
Using
\be
1 - w^2 = 2 \kappa \left({d\ln {\cal{U}}_K \over dx}\right)^{-1},
\label{relwU}
\ee
which follows from \eqr{rv}, we get
\be
\langle T_{rr}(v,r)\,  T_{rr}(v_0 , r_0)\rangle_{K} =
\left(\frac{\kappa^2}{\pi} \frac{{U_K} {U^0_K}}
{(1 - w^2)(1 - w_0^2) }\,
\frac{1}{\left( {U_K} - {U^0_K
- i \epsilon\right)^{2}} \right)^{2} .
\label{TTrr3}
\ee

From \eqr{TTrr2}, it is clear that the correlator
is a function of only 3 variables, $r, r_0$ and $v-v_0$, the state (Unruh vacuum) being stationary.
Because it still
depends on 3 variables, its behavior in different 2 dimensional sections
illustrates different aspects of the correlations associated with the Hawking effect.
When fixing  $r_0,v_0$
on ${\cal I}^+$,
\eqr{TTrr}, function of $r, v$,
describes the correlations associated with
a late detection on ${\cal I}^+$~\cite{Massar:1996tx}.
Instead, at equal time
 $v_0 = v $,
\eqr{TTrr}
describes the correlations in the $r,r_0$ plan
that have been accumulated in the past of that time~\cite{Iacopo08}.

Before studying these two cases, we need to be more precise about the
black hole
geometries we shall work with. As explained before,
we consider profiles
that become
constant for $r \to \pm \infty$.
To have a simple example at hand, we
choose directly ${\cal U}_K(r)$
since
 it is the only relevant function in \eqr{TTrr2}:
\be
\kappa \, {\cal{U}}_K(x) = -
\left( e^{2 \kappa x} - e^{- 2 \bar \kappa x}\right) ,
\label{wex}
\ee
where $x= r - r_h$. This
 is a kind of symmetrized version
of the Schwarzschild metric where ${\cal{U}}_K = -x \, e^{2\kappa x}$.
Using \eqr{relwU}, one gets
\be
1 - w^2 =
\frac{e^{2 \kappa x} - e^{- 2 \bar \kappa x}}{e^{2 \kappa x} + \frac{\bar \kappa}{\kappa} e^{- 2 \bar \kappa x}}.
\label{relwU2}
\ee
Near the horizon, for $\kappa x \ll 1$,
one has $w^2 = 1 - 2 \kappa x $ for all values of $\bar \kappa$,
and the asymptotic values are
$w_\infty = 0$ for $x \to \infty$, and
$w^2_{-\infty} = 1+ \bar \kappa/\kappa$ for $x \to -\infty$.

\subsubsection{Correlations to a late detection}

We fix $x_0,v_0$ on ${\cal I}^+_R$, on the future right infinity, and label it with $u_0$. We consider \eqr{TTrr}
\be
\bar T_{rr}(r,v)\vert_{ u_0, \, {\cal I}^+} \equiv
\langle T_{rr}(v,r)\,  T_{rr}(u_0,{\cal I}^+_R)\rangle_{K},
\label{Cinfty}
\ee
as
a 1pt function. As such $\bar T_{rr}(r,v)$ is a special case of the conditional valu
\be
\bar T_{rr}(r,v)\vert_{ {\Pi}_R} =  \langle T_{rr}(v,r)\,  \hat \Pi_{{\cal I}^+_R} \rangle_{K},
\label{cond}
\ee
where $\hat \Pi_{{\cal I}^+_R}$
is a projector that specifies
 a state on ${\cal I}^+_R$, see App. C for a brief account, and App. C of~\cite{Brout:1995rd}
for details.
The projector can be chosen at will.
If it is taken to be $a^{as\, \dagger}_\om a^{as}_\om$
it selects the subset of states (present in the Unruh)
which contains that asymptotic quantum
without specifying when it is detected. One can also consider the
other limit in which one only specifies the moment of detection $u_0$. In that
case, \eqr{cond} is identical to \eqr{Cinfty}, up to an overal constant factor,
as can be seen by taking $f(\tau) = \delta(\tau - u_0)$ in Eqs.(55, 56)
of \cite{Massar:1996tx}. From this we learn the physical meaning of \eqr{Cinfty}:
it gives the mean value of $T_{rr}$ when
the $in$
state is Unruh vacuum, {\it and} when
a particle is
detected on ${\cal I}^+$ at $u=u_0$.
\begin{figure}
\includegraphics[width=70mm]{Fig1v10.eps}
\label{fig}
\caption{{\it The stress-energy conditional to a late detection}. We represent $\bar T_{rr}$
of \eqr{Cinfty} in the $x,v$ plane ($v$ being vertical) for a detection at
$x_0 = 4,v_0=0$, outside the picture in the top right-hand region where $w(r)$ is constant.
The horizon is at $x=0$ and
$\kappa = \bar \kappa = 1$ in \eqr{relwU2}.
When $\vert x \vert \geq 1$,
on both sides, the pattern is translation invariant along the null direction because $w$ is constant.
Instead for  $\vert x \vert < 1$, one sees the endless focusing of the null lines for $v\to -\infty$.
The
"post-selected" partner propagates along $v + 2x = -8=v_0 - 2 x_0$, i.e. along the opposite trajectory
fixed by $x_0,v_0$.
These features were found in \cite{Massar:1996tx} considering \eqr{cond}
for a typical Hawking quantum.}
\end{figure}

When $(r,v)$ is also on ${\cal I}^+_R$ and coordinated by $u$, \eqr{Cinfty} is given
by \eqr{Tuu} (times $4^2$), as can be seen using \eqr{TTrr3}, and $\kappa u = -\ln(-\kappa U_K)$.
In this we recover the thermal correlations~\cite{Carlitz:1986ng} of the asymptotic radiation.
When $(r,v)$ is taken on the opposite null infinity ${\cal I}^+_L$,  or
{sufficiently far away} from the horizon so that $w$ is constant,
$\bar T_{rr}$ behaves as
\eqr{ldTuu} when using the mirror coordinate
$\kappa \bar u =\ln(\kappa U_K)$.
Moreover, when $(r,v)$ is near the horizon, $\kappa (r-r_h) \ll 1$,
and parameterized by $U_K$, $\bar T_{rr}$
behaves as in \eqr{TUU},
as can be seen using \eqr{TTrr} and $dU_K/dr \sim -2 $ for $r=r_h$.
In this we recover that when probed near the horizon, for
$\kappa x \ll 1$, Unruh vacuum behaves like
Minkowski vacuum.

In addition to these three asymptotic behaviors,
the non-trivial information contained in $\bar T_{rr}$  of \eqr{Cinfty}
is the
smooth
interpolation from one to the other, which is represented
 in Fig. 1. From this we clearly see the gradual emergence from $v=-8$
of the energy flux associated with the partner on the other side of the horizon.
What is non-trivial is the following.
In the past of that time, $\bar T_{rr}$ is essentially constant along the outgoing null lines $U_K= const$.
and, as could have been expected, behaves exactly as $\bar T_{UU}$ would in Minkowski.
On the contrary, near the horizon and in the future, $\bar T_{rr}$ behaves very differently
since the lines $\bar T_{rr}=const.$
cross the horizon. This peculiar behavior could not have be found had we studied
the 2pt function of \eqr{TUUC}
because the latter obeys \eqr{TUU} and depends only on $U$, even in a black hole geometry.
This establishes that the use of $T_{rr}$ of \eqr{Trr}
with $r$ affine
is truly necessary. It should be also noticed that the above behavior of $\bar T_{rr}$
{cannot} be found in Minkowski using affine coordinates either. In fact, it is characteristic
 of pair creation processes, as can be seen by comparing Fig.1 to Fig. 1.1 of~\cite{Brout:1995rd}
which describes pair creation in an electric field.
As noticed in~\cite{Massar:1996tx}, these properties of  $\bar T_{rr}$ provide a
clear answer to the long standing question: where is a Hawking quantum "born" ?~\cite{Unruh1977}.

\subsubsection{Equal time correlations}

We consider \eqr{TTrr} at equal EF time $v=v_0$. Since $U_K = e^{-\kappa v} {\cal U}_K$,
the correlator
\be
C_K(r,r_0) = \langle T_{rr}(r)\,  T_{rr}(r_0)\rangle_{K} = \Big(\,  \partial_{r} \partial_{r_0}
\langle \phi(r)\, \phi(r_0)\rangle_{K} \, \Big)^2,
\label{Crr}
\ee
is given by \eqr{TTrr3} with $U_K$ replaced by ${\cal U}_K$.
Using ${\cal U}_K$ of \eqr{wex},
$C_K(r,r_0)$
 diverges as $\sim(r -r_0)^{-4}$ when $r \to r_0$,
as expected since the Unruh vacuum is a Hadamard state.
Moreover,
when $r\to \infty$, since ${\cal U}_K \sim - e^{-2\kappa r}$, one has
\be
C_K (r,r_0)= \frac{\kappa^4}{16 \pi^2 \sinh^4(\kappa(r-r_0))}.
\label{CWC}
\ee
The thermal noise of \eqr{Tuu} associated with the asymptotic radiation
is properly encoded in $C_K$
since at fixed $v$, $du= -2 dr$.
Similarly, when $x$ and $x_0$ are on  opposite sides of the horizon,
since ${\cal U}_K \sim - e^{-2\bar \kappa \vert x \vert}$ for $x \to -\infty$,
one asymptotically gets
\be
C_K(r,r_0) = \frac{\kappa^2 \bar \kappa^2}{16 \pi^2 \cosh^4(\bar \kappa x + \kappa x_0))},
\label{BC}\ee
thereby recovering
\eqr{ldTuu}, and making contact with \cite{Balbinot:2007de,Iacopo08},
see Fig. 2 on the left.
\begin{figure}
\includegraphics[width=60mm]{fig2Ln.eps}\quad \quad  \includegraphics[width=60mm]{fig2Rn1.eps}\label{fig2}
\caption{{\it Equal time correlations.} On the left, we represent
\eqr{Crr}, and on the right,
 \eqr{Sr}, both in the metric of Fig.~1.
The horizon is at $x, x_0=0$.
 On the left, the signal diverges for $x \to x_0$
whereas the pattern along $x+x_0= 0$
represents \eqr{BC}. It is
translation invariant in $x-x_0$
once $w$ has reached a constant. On the right,  the subtracted
$S_K$
is everywhere finite and regular. It is dominated by the
correlations across the horizon.
The subdominant patterns centered along $x-x_0= 0$
are due to the fact that the un-subtracted
correlator in
\eqr{Sr}
decreases faster than the subtraction.
On the diagonal,
$S_K(x,x) = \langle T_{rr}(x)\rangle^{\rm ren}_K$,
the renormalized flux
 of \eqr{Tuuloc}.}
\end{figure}
To further investigate the gradual change from near horizon
configurations
to long distance correlations,
we consider
the subtracted 2pt function\footnote{For simplicity we worked with the square root of \eqr{Crr}
 rather than $C_K$ itself.
The reason is that
the subtraction needed to obtain a finite expression for $r \to r_0$ is more complicated, as it requires three terms.
The subtracted correlator of $\langle T_{rr} T_{rr}\rangle_{K}$ possesses the same behavior as $S_K$.}
\be
S_K (r,r_0)= \partial_{r} \partial_{r_0}\left[
\langle \phi(r)\, \phi(r_0)\rangle_{K} -
\frac{(-1)}{4\pi} \ln(r-r_0)\right].
\label{Sr}
\ee
The substraction term is the same as in \eqr{localG}, but considered here for all values
of $r-r_0$.
From the the right plot in Fig. 2, we see that the {\it residual} signal is free of
UV divergences and
describes the emergence of the pairs
on distances $\kappa (r-r_h)= \kappa x \sim 1$, i.e. characterized by the geometry.

One first notices that the correlations across the horizon in $S_K$
are negative (as in inflation~\cite{Campo:2003pa}). In fact the
correlator
$ \langle \partial_{r} \phi\,\partial_{r_0} \phi\rangle_{K} $
is negative "everywhere", as can be seen from \eqr{TTrr2} and \eqr{wex}. We have added quotation marks because this correlator is  a distribution.
In fact, the coincidence point limit is ruled by the $i \epsilon$, see \eqr{TUU}.
It specifies that
(the real part of) $ \langle \partial_{r} \phi\,\partial_{r_0} \phi\rangle_{K} $
diverges positively for $r \to r_0$ same $v$, and it ensures that  the integral
$\int_{-\infty}^\infty dU  \langle \partial_{U}\phi\,\partial_{U_0}\phi\rangle$
vanishes in the Minkowski vacuum. (This is reminiscent
of the behavior of $\langle T_{UU} \rangle_{FR}$,
the mean
flux evaluated in the Fulling-Rindler vacuum~\cite{Parentani:1993yz}.)
We also
notice that for close points, $S_K$ is
positive because the subtraction is larger than the "bare" term.
Moreover when evaluated at the same point,
$S_K(x,x)$ is equal to $\langle T_{rr}(x)\rangle^{\rm ren}_K$
 of \eqr{Tuuloc}. For $x \to \infty$  one
 verifies that $S_K(x,x) = (-1/3)\times S_K(x,-x)$.
Finally we
notice that Fig.2. is symmetrical under  $x \to -x$. This follows
from the symmetry
of $1- w^2$ in \eqr{relwU2} when $\kappa = \bar \kappa$.
When $\bar \kappa/\kappa \gg 1 $, this
is lost, and $1- w^2$ resembles more to Schwarzschild.
We hope to report on this case soon.

\subsection{Time-dependent growth of flux and correlations}

So far we considered the stationary correlation patterns
found in the Unruh vacuum.
We now consider the early transient effects.
As already said in Sec. III.B, they
are not universal. However
their late time behavior
is
when the state contains no high frequency excitations, i.e. when it is an Hadamard state.
There is a simple and efficient way to characterize this behavior.
It consists in assuming that the
initial state specified at $v=v_{\rm in}$ is the Minkowski vacuum. (This state can be found
in a gravitational
collapse when the infalling matter is a light-like thin shell~\cite{Massar:1996tx}.)
In this state, $v= v_{\rm in}$, the 2pt function at
 is
$ G^{\rm in}(r,r_0)
= \frac{-1}{4\pi} \ln(r-r_0 + i\epsilon)$, as in
\eqr{localG}.
In
the future, one has
\be
G^{\rm in}(r,v; r_0, v_0) = -\frac{1}{4\pi} \ln(X^{\rm in}(r,v)-X^{\rm in}(r_0,v_0) + i\epsilon),
\label{localG2} \ee
where $X^{\rm in}(r,v)$ gives the value of $x = r- r_h$ hit
by the outgoing null ray
issued from $r,v$ when it crosses $v_{\rm in}$. In this state, the mean flux of \eqr{Tuuloc}
becomes
\be
\langle T_{rr}(r,v)\rangle^{\rm ren}_{\rm in}
= \frac{1}{12 \pi}
\left( \frac{dX^{\rm in}(r,v)}{dr} \right)^{1/2}  \partial_{r}^2
\left( \frac{dX^{\rm in}(r,v)}{dr} \right)^{-1/2},
\label{Tuuloctd}
\ee
and the correlator of \eqr{TTrr} is
\be
\langle T_{rr}(v,r)\,  T_{rr}(v_0 , r_0)\rangle_{\rm in} =
\left( \frac{dX^{\rm in}}{dr}  \frac{dX^{\rm in}}{dr_0} \right)^2
\left( \frac{1}{4\pi}\frac{1}{({X^{\rm in}(v,r)} - {X^{\rm in}(v_0,r_0)} + i \epsilon)^2 }  \right)^2 .
\label{TTrrtd}
\ee
In both expressions one has simply replaced $U_K$, the Kruskal coordinate encoding the Unruh vacuum,
by $X^{\rm in}$ which encodes the time-dependent state which is vacuum at $v_{\rm in}$.
\begin{figure}
\includegraphics[width=55mm]{fig3L1n.eps}\quad \quad  \includegraphics[width=55mm]{fig3R1n1.eps}\\
\includegraphics[width=55mm]{fig3L2n.eps}\quad \quad  \includegraphics[width=55mm]{fig3R2n.eps}
\label{fig3}
\caption{{\it The growth of equal time correlations.}
We represent \eqr{TTrrtd} at equal time on the left,
 and  \eqr{Sr} on the right,
after a lapse of time $\kappa (v - v_{\rm in}) = 1$ in the upper plots, and $\kappa (v - v_{\rm in}) = 4$ in the lower ones.
On the left,
one observes the growth of the correlations across the horizon centered along $x+x_0 = 0$.
One also observes a narrowing
 of the correlations centered along $x = x_0 $.
On the right,
$S_{\rm in}(r,r_0)$ displays two distinct features. A strong signal associated with the
building up of the
correlations across the horizon, and
subdominant patterns on both sides of the horizon
due to the growth
of thermal correlations of \eqr{CWC}.
At late times, both patterns asymptote to the stationary ones of Fig. 2.}\end{figure}

In the metrics of \eqr{relwU2},
these expressions flow towards
 \eqr{Tuuloc} and \eqr{TTrr}, exponentially  in $v-v_{\rm in}$
in the near horizon region
where $\partial_r w = \kappa$,
and linearly when $\partial_r w \sim 0$.
To show this
we
make use of ${\cal U}_K(x)$,
 solution of \eqr{relwU}.
Calling $X({\cal U}_K)$ the inverse function,
and using $ U_K(x,v)= {\cal U}_K(x) e^{\kappa v}$, we obtain
\be
X^{\rm in}(r,v) = X\left[{\cal U}_K(x) \,  e^{-\kappa (v - v_{\rm in})}\right],
\ee
for all profiles $w^2(x)$.
To give a simple example, we use \eqr{wex} with $\bar \kappa = \kappa$, and we get
\ba
2 \kappa X^{\rm in}(r,v) &=&
{\rm arcsinh}\left( e^{-\kappa (v - v_{\rm in})}\sinh(2\kappa x)\right),
\nonumber \\
\frac{dX^{\rm in}}{dx} &=& \frac{\cosh(2\kappa x)}{\left[ e^{2\kappa (v - v_{\rm in})}
+ \sinh^2(2\kappa x)\right]^{1/2}}.
\label{exX}
\ea
In Fig. 3, on the left, we  represent
\eqr{TTrrtd} evaluated at equal time
for different values of $v-v_{\rm in}$.
In these plots, one
clearly sees the growth of the correlations across the horizon with a rate given by $\kappa$~\cite{Iacopo08}.
One also observes a narrowing of the spread of the dominant correlations centered along $x = x_0 $.
This is due to the
progressive replacement of vacuum correlations $\sim 1/(x-x_0)^{2}$ by the thermal ones
$\sim \kappa^2/\sinh^{2}\kappa(x-x_0)$ of \eqr{CWC}.
On the right plots, the subtracted function $S_{\rm in}(r,r_0)$,
the equivalent of \eqr{Sr} evaluated in the $in$ vacuum, displays
both the growth of the long distance correlations,
and the modifications of the local correlations.
 \begin{figure}
\includegraphics[width=59mm]{fig4Ln1.eps}\quad \quad \includegraphics[width=60mm]{fig4Rn.eps}\label{fig4}
\caption{{\it Time dependence of the mean flux and of the asymptotic correlations.}
On the left, we represent \eqr{Tuuloctd} with \eqr{exX}
from the onset of the vacuum at $v=0$
till $\kappa v = 8$, and for $\kappa x$ from $0$ to $4$ (It is symmetric under $x \to -x$).
The transients propagate on null lines $u=v-2x \sim 0$. After they passed,
 $\langle T_{rr}\rangle_{\rm in}$ is $v$-independent.
On the right plot, we represent $\bar T_{uu}(\bar u)\vert_{u_0}$ of \eqr{Cinfty} in the $u_0,\bar u$ plane
with $\bar u$ vertical, with $u_0$ and $\bar u$ defined on the future null infinity $v_0=v=\infty$.
For $u_0< 0$, before the transients, there is no correlations across the horizon. Instead for $u_0 > 3$, $\bar T_{uu}(\bar u)\vert_{u_0}$
only depends of $u_0+\bar u$, and is given by \eqr{ldTuu}.}
\end{figure}

It is worth analyzing these time dependent effects through two other perspectives.
In Fig. 4, on the left, we present
the mean flux of \eqr{Tuuloctd} in the $x,v$ plane from the onset of the vacuum at $v=v_{\rm in}=0$.
$\langle T_{rr}\rangle_{\rm in}$
contains transients which propagate along $u=v-2x \sim 0$.
It then reaches a constant $x$-dependent profile.
For $x\to \infty$ one recovers the standard value $\kappa^2/12 \pi$~\cite{Davies:1976ei}.
In the metric of \eqr{wex} with $\kappa= \bar \kappa$, $\langle T_{rr}\rangle_{\rm in}$
 crosses $0$ for $\sinh 2x = \sqrt{2}$, i.e. $x \sim \pm .57$, and on the horizon,
it is negative and equal to $ -2\kappa^2/12 \pi$. When including gravitational back-reaction effects,
this term participates to the evaporation of the black hole, see~\cite{Parentani:1994ij}
for a numerical analysis in similar settings.
On the right plot, we show
$\bar T_{\bar u \bar u}(\bar u)\vert_{u_0}$ of \eqr{Cinfty}, the
flux correlated to a late detection at $u_0$ on ${\cal I}^+_R$,
that we evaluate on ${\cal I}^+_L$ and parameterize with $\bar u$, the mirror coordinate $\kappa \bar u =
\ln \kappa U_K$. There is no correlations from ${\cal I}^+_R$ to ${\cal I}^+_L$ for $u_0< - 1$ in conformity
to the fact that no Hawking radiation as yet reached the null infinites. Then for positive $u_0$, the
correlations settle to a stationary pattern centered around $u_0 + \bar u = 0$ found in the Unruh vacuum.

In conclusion it is interesting to observe that, even though the transients
give rise to a higher value of the mean flux, they are {\it not} associated
with stronger correlations across the horizon, and this because, unlike the steady Hawking radiation,
the transients are not composed of entangled pairs of opposite frequency $\om$.
This can be checked
by
comparing
$\bar T_{uu}(\bar u)\vert_{u}$ to
the asymptotic flux $\langle T_{uu}\rangle_{\rm in}$ evaluated for $x,v \to \infty$,
with
$u= v - 2x $ fixed:
\ba
\langle T_{uu}(u)\rangle^{\rm ren}_{\rm in}
&=& \frac{\kappa^2}{48 \pi}
\frac{1 + e^{-2 \kappa u}}{\left( 1+ e^{-2 \kappa u} /4\right)^{2}},\nonumber \\
\bar T_{\bar u \bar u}(\bar u)\vert_{u_0}
&= &\left(\frac{\kappa^2}{\pi}\right)^2  \Big( \left({e^{2 \kappa u_0} +1/4}\right)
\left(
{e^{-2 \kappa \bar u} +1/4} \right)
{\left( {\rm arcsinh}\left( e^{-\kappa u_0}/2 \right) +  {\rm arcsinh}\left( e^{+\kappa\bar u}/2 \right)  \right)^4}
\Big)^{-1} .
\label{TTrrtd2}
\ea

\section{Dispersive theories and Analogue black holes}

In a non-homogeneous medium, linear density fluctuations
obey a relativistic d'Alembert equation in a curved
metric
when their wave lengths are  larger than the inter-atomic distance~\cite{Unruh:1980cg}.
Instead, for shorter wave lengths, the propagation becomes dispersive~\cite{Jacobson:1991gr}.
Assuming the speed of sound is constant and set to $1$, it can be
described by
\be
\Om^2 = (\om - w p)^2 = F^2(p^2),
\label{dr}
\ee
where  $\Om$ is the co-moving frequency measured w.r.t. to
the atoms, $w$ the velocity of the fluid, and $p,\om$ the wave vector
and the frequency measured in the lab.
The dispersion can be incorporated in
a modified
field equation~\cite{Unruh:1994je}  \be
\left(\partial_\tau
+ \partial_x w \right) \left( \partial_\tau
+ w \partial_x \right) \phi - \partial^2_x \phi \pm \frac{1}{\Lambda^2} \partial_x^4 \phi =0 ,
\label{modeeq}
\ee
which reduces to the relativistic one when sending the dispersive scale $\Lambda \to \infty$.

For simplicity, we chose
quartic, super (+) or subluminal (-) dispersions: $F^2 = p^2 \pm p^4/\Lambda^2$.

\eqr{modeeq} can then be used to study the impact of dispersion on Hawking radiation.
In ~\cite{Unruh:1994je}, the thermicity and the stationarity of the asymptotic
radiation have been shown to be robust,
i.e. hardly affected by dispersion when  $\kappa \ll \Lambda$.
This is
sufficient for recovering
\eqr{CWC}.
By constructing
wave packets of $in$ modes,
it was then shown~\cite{Brout:1995wp} that
at large distance from the horizon one also recovers the
correlations between Hawking quanta and their partners, and this, even though
the early propagation was radically affected by dispersion.
This second aspect
is sufficient for obtaining
\eqr{BC}, Fig. 2,
and the late time properties of
Fig. 1.

To clarify this,
we shall compare
the properties
of wave packets of $\phi^{in}_\om$, the $in$ modes of \eqr{modeeq},
\be
\bar \phi_{\bar \om}(\tau, x) = \int d\om  \,   e^{- i \om \tau} \phi^{in}_\om(x) \, \bar f_\om,
\label{wpbhx}
\ee
where $\bar f_\om$
 selects the wave packet,
with the
correlation function
\ba
G^{in}(\tau, x ;\tau_0,x_0)& \equiv&  \langle \phi(\tau, x) \, \phi(\tau_0,x_0) \rangle_{in} \nonumber
\\
&=& \int d\om   \,
e^{- i \om \tau} \phi^{in}_\om(x)\,  [ e^{- i \om \tau_0} \phi^{in}_\om(x_0)]^* ,
\label{Gbhx}
\ea
evaluated in the stationary $in$ vacuum.
When dispersion is weak, the similarity of
expressions
guarantees that
similar patterns will be found. However,
since dispersion grows as approaching the horizon, the way one probes the correlations,
i.e. by extracting some limited range of
 $\om$ through $\bar f_\om$ in \eqr{wpbhx},
or not as in \eqr{Gbhx}, can lead to different behaviors.
We now review the relevant
points for achieving this comparison following~\cite{Brout:1995wp,Jacobson:2007jx};
other treatments are mentioned in~\cite{Jacobson:1999zk}.

\subsection{Kinematics}

From a relativistic point of view,
the presence of dispersion
defines a preferred frame~\cite{Jacobson:1996zs} which allows to define new scalars.
This is best seen by "covariantizing" \eqr{dr},  i.e.
by introducing a unit time-like vector field  $u^\mu$,
and viewing the field $\phi$
as propagating on a manifold endowed with both the metric and $u^\mu$.
Then the energy in the preferred frame,
and the spatial momentum perpendicular to it are respectively
\be
\Om = u^\mu p_\mu, \quad p = s^\mu p_\mu,
\label{Om2}
\ee
where $- u^2 = 1 = s^2$ and $s^\mu u_\mu= 0$. The PG coordinates used in \eqr{modeeq}
can then be {invariantly}
defined by $\partial_x = s^\mu \partial_\mu$ and $d\tau = u_\mu dx^u$.
The field $u^\mu$ also defines the scalar density $\rho = u^\mu u^\nu T_{\mu\nu}$
which corresponds to the proper energy that observers following
$dx^\mu/d \tau = u^\mu$ would measure.
It is interesting to note that in the hydrodynamical limit one gets
\be
\rho =  T_{xx} = (\partial_x \phi)^2.
\label{rho}
\ee
 In other words,
$\rho$
coincides with $T_{rr}$ of \eqr{Trr} with
 $r$ defined by \eqr{rv}. This non-trivial correspondence follows from the affinity of $r$ at fixed
$v$ and $\tau$,
see the remarks after \eqr{EF}.

In the same spirit, we notice that the surface gravity measured with respect to the preferred frame is also
scalar. It is given by the expansion~\cite{Jacobson:2008cx}
$\theta = u^\mu_{;\mu} = \partial_x w$ evaluated on the horizon.
(The second expression is valid in PG coordinates).
The ambiguity of the scale of the surface gravity in covariant theories is thus removed
when using
$u^\mu$.

\subsection{The modified modes}

In linear field theories, the modifications of the stress energy correlations due to
dispersion will stem from the modifications of the modes, solutions of \eqr{modeeq}. When $\Lambda \gg\kappa$,
 these
are modifications localized near the horizon, for $\kappa(r - r_h)= \kappa x \ll 1$.
As noticed in \cite{Brout:1995wp} it is appropriate to work in the $p$-representation (with $x = i \partial_p$)
with $w(x)$ linearized: $w= -1 + \kappa x  = -1 + i \kappa \partial_p$.
 In this
representation,
\eqr{modeeq} becomes
\be
 \left(\omega - p w \right) \left( \omega -  w p \right)\tilde{\phi}_{\omega} =
 F^2\,  \tilde{\phi}_{\omega}.
\ee
Using $w= -1 + i \kappa \partial_p$,
the modified modes
have the form, for details see the Appendix of~\cite{Jacobson:2007jx},
\be
\tilde{\phi}_{\omega} = \tilde{\phi}^0_{\omega} \times e^{- i p/\kappa } \chi(p),
\ee
where $\tilde \phi^0_\om =
\vert p\vert ^{-i \om/\kappa - 1}$
is the standard
dispersion less mode,
and where $\chi$ obeys
\be
 - \kappa^2 \partial_p^2 \,  \chi = \frac{F^2}{p^2}\, \chi= H^2\, \chi .
\label{ceq}
\ee
 The $in$ state
which generalizes the notion of the Unruh vacuum, the "free falling vacuum",
is characterized by the positive norm modes which contain only positive $\Om$ of \eqr{Om2}.
These
$in$ modes
are related to the Unruh modes, see \eqr{Ump},
\be
\tilde \phi^U_\om = \theta(p) \, \frac{ p^{-i \om/\kappa - 1}}{(4 \pi \kappa)^{1/2}},
\label{bhUm}
\ee
by~\cite{Brout:1995wp}
\be
\tilde{\phi}^{in}_{\omega} = \tilde{\phi}^U_{\omega} \,\times\,  e^{- i p/\kappa }  \chi(p),
\label{bmpsm}
\ee
where $\chi(p)$ is the solution of \eqr{ceq} with a
Wronskian equal to $\chi^* \partial_p \chi -\chi \partial_p \chi^* = 2i /\kappa$.
The corresponding WKB solution is
\be
\chi(p) = \frac{1}{(H)^{1/2}} \,
 \exp\left({ i \int^p_{p_0} H(p')dp'/\kappa}\right) .
\label{Wchi}
\ee
It provides a reliable approximation
when $\Lambda\gg \kappa $.
In the limit $\Lambda \to \infty$ fixed $p$, $H \to 1$ and $e^{- i p/\kappa }\chi \to 1$,
thereby implying that $\tilde{\phi}^{in}_{\omega}$
smoothly gives back $\tilde \phi^U_\om$.\footnote{At fixed $x$
instead, the limit $\Lambda \to \infty$
can be singular as some roots $p(x,\om)$ of \eqr{dr} are sent to infinity.
The $p$-WKB approximation should not be confused with the usual one defined in $x$-space.
For the Airy function,
the modes in $p$ are exactly given by their $p$-WKB approximation.
Similarly here, the corrections to
\eqr{Wchi}
are negligible when $\Lambda/\kappa \gg 1$.
This has been confirmed by numerical analysis,
see~\cite{Macher:2009tw} for a detailed study.
From now on we neglect them and work in the adiabatic approximation with \eqr{Wchi}.}

To understand the impact of dispersion,
we now study the
characteristics of
\eqr{modeeq}
since the maximum of correlations will be localized along them.
Having already the
modes in $p$-space,
the simplest way to get them
is to consider
\eqr{wpbhx}  in $p$-space, and
 look for the stationary phase condition in $\om$.
Using \eqr{bmpsm}, since $\chi$ is independent of $\om$,
 one gets
\be
p(\tau) = p_0 \, e^{- \kappa \tau},
\label{pevol}
\ee
irrespectively of the dispersion relation $F$, and thus as in relativistic theories.

To get the modified characteristics in $x$, we use
 $\om - wp = F$, the root of \eqr{dr}
describing the right moving modes.
Using
$w\sim -1 + \kappa x$,
 one gets~\cite{Balbinot:2006ua}
\ba
\frac{\kappa}{2} \left[ x_\om(p) - x_{-\om}(p)\right] &=& \frac{ \om} {p} ,
\nonumber \\
\frac{\kappa}{2} \left[ x_\om(p) + x_{-\om}(p)\right]
 &=& 1- H(p).
\label{modchar}
\ea
The first equation is again
independent of $F$ and coincides what is found in
relativistic theories, namely,
when propagated backwards in time,
pairs of characteristics pill up
exponentially in PG time.
The second equation tells us that the "center of mass" of a pair which is centered on the relativistic
horizon for $1-H \ll 1$, i.e. $p \ll \Lambda$, gradually moves
away
as $p$ increases.
For sub (super) luminal
dispersion, $H< 1$ ($H> 1$), the pair is sent outwards (inwards).
For quartic dispersion
the momentum at the turning point is given by
$p_{t.p.}^3 = 2 \Lambda^2 \vert \om\vert $~\cite{Corley}.
Because of this movement away from the horizon,
the increase of $p$ and the focusing of
$x_\om - x_{-\om}$ will stop when the pair reaches $\vert \kappa x \vert \sim D$,
where
the gradient $\partial_x w$ drops down.
A straightforward calculation gives that the focusing stops for $p\sim \Lambda D^{1/2}$.

The end of the focusing and the movement away from the horizon are the
principal consequences of dispersion.
They imply that the early properties
of Fig. 1 are inevitably modified, as we shall see below.

\subsection{Correlations from wave packets}

To show how dispersion affects
the correlation pattern encoded in \eqr{wpbhx},
we need
\eqr{bmpsm} in $x$-space
\be
\phi^{in}_\om(x) = \int_0^\infty {dp \over (2\pi)^{1/2} }  e^{ipx} \tilde\phi^{in}_\om(p).
\label{phi_in}
\ee
Far away from the turning point, one can
evaluate this integral at the saddle point approximation (since it is reliable~\cite{Brout:1995wp,Corley}),
and decompose
\eqr{phi_in}  in terms of
outgoing modes defined for low momenta. Doing so one finds
\be
\phi^{in}_\om(x) = \alpha_\om \left[ \theta(x) \, \varphi_\om(x) + z_\om \, \theta(-x) \,(\varphi_{-\om}(x))^* \right],
\label{aBog}\ee
where $\varphi_\om(x)$ is the $x$-WKB mode of \eqr{modeeq} with unit norm.
Explicitely it is given by
\be
\varphi_\om(x) =  \sqrt{\frac{\partial p_\om(x)}{\partial \om}}\,
\frac{\exp \left(i \int^x_{x^0} dx' p_\om(x')\right)}{\sqrt{4 \pi  \Omega(p_\om(x))}},
\label{WKBmodes}
\ee
where $p_\om(x)$ is the corresponding low momentum root of \eqr{dr}.
One easily verifies that these $out$ modes are identical to the relativistic ones for $p_\om \ll \Lambda$.
Moreover, for $\kappa \ll \Lambda$, up to a phase, one finds $z_\om = e^{- \pi \om /\kappa}$.
Hence \eqr{aBog} gives  the equivalent of \eqr{Um}.~\footnote{This approximation
is valid provided $\om$ is sufficiently small. For quartic dispersion
there is
a critical frequency $\om_{\rm max}$,
related to both $\Lambda$ and the
asymptotic velocities $w(\pm \infty)$,
above which $z_\om$ identically vanishes~\cite{Macher:2009tw}.
 From the numerical results of that ref., a good fit
is
$\vert z_\om^F \vert  =  e^{- \pi \om /\kappa}\,
\left(1 - \om/\om_{\rm max}\right)^{1/4} $.\label{Macherz}}

In addition to the above two low momentum modes, there is a third saddle
--on the left (right) of the horizon for super (sub) luminal dispersion,
in conformity with \eqr{modchar}--
which gives a
high momentum mode.
Its WKB wave is also given by \eqr{WKBmodes}
with $p_\om$ being
 the unique large positive real root of \eqr{dr}. One
verifies that its overall coefficient
is unity in conformity with the fact that it describes the incoming
 mode that shall be scattered.

Therefore, considering
\eqr{wpbhx}
with $\bar f_\om$
centered
around
$0 < \bar \om \ll \om_{\rm max} $, we get two results.
First, at late times, using \eqr{aBog} and \eqr{WKBmodes},
one finds two low momentum packets following \eqr{modchar} with $\om= \pm \bar \om$, where the negative
frequency packet
has its amplitude reduced by $z_{\bar \om}$, as in \eqr{wp}.
Since $z_\om = e^{- \pi \om /\kappa}$, and since both $\varphi_\om$ of \eqr{aBog}
behave as relativistic modes once $p$ is small enough ($F-p \ll p$), at large distances,
the pattern is {\it indistinguishable}
from
the relativistic one obtained by replacing \eqr{bmpsm} by \eqr{bhUm}.

Second, at early times, only the incoming high momentum mode
constructively interferes. It has a mean positive frequency $\bar \om$,
 follows the second line of \eqr{modchar}
with $p \gg \bar \om$, and leaves the near horizon
region with $p \sim \Lambda$.
This is completely different from what is obtained in the relativistic case.
Indeed,
using \eqr{bhUm},
$p$ would keep increasing  for ever following \eqr{pevol},
and the spread in $x$ correspondingly decrease as $\sim 1/p$.

From the analysis of wave packets, we have thus reached two important results.
On one hand, the low momentum (late time) properties of the relativistic pattern of Fig. 1
is unaffected by dispersion. On the other hand, the early properties of this pattern will
be radically affected by dispersion since the peak of correlations will follow  \eqr{modchar},
as represented in Fig. 4 of~\cite{Brout:1995wp}.
To further investigate how dispersion affects the properties of Figs. 1 and 2, we now consider
the pattern encoded in \eqr{Gbhx} rather than in \eqr{wpbhx}.

\subsection{Correlations in energy density}

We start with
the correlations of a relativistic field expressed in the present language.
Since $\Om = p$,
the correlation function of $\rho$ of \eqr{rho} is
\be
\langle\rho(x,\tau) \rho(x_0,\tau_0)\rangle_{in} =  \left(
\partial_x \partial_{x_0}\int_{-\infty}^{\infty} d\om e^{-i \om (\tau - \tau_0)} G^{in}_\om(x,x_0)\right)^2,
\label{prho}
\ee
where $G^{in}_\om = \phi^{in}_\om(x) (\phi^{in}_\om(x_0))^*$ is the $\om$ component of
$G^{in}$
of \eqr{Gbhx}.
In the $p$-representation, using
\eqr{bhUm}, one gets
\ba
\tilde G_K(p,\tau;p_0,0) &=& \int_{-\infty}^{\infty} d\om e^{-i \om \tau} \tilde G^{in}_\om(p,p_0)
\nonumber \\
&=& \theta(p) \theta(p_0)\,  {1 \over 2 p p_0} \, \delta(\kappa \tau + \ln(p/p_0)).
\label{Gnodisp_p}
\ea
On one hand, we recover the classical evolution law of \eqr{pevol}. On the other we learn that
in the Unruh-vacuum, at $\tau = \tau_0$, only configurations with equal values of $p$ contribute.
There is no spread in $p$ in this state.
When inverse Fourier transform, one gets
\be
G_K(x,\tau;x_0,0) = - \frac{1}{ 4 \pi} \ln(x  - x_0\, e^{\kappa \tau}+ i \epsilon).
\label{Gnodisp_x}
\ee
We recover
the standard result, \eqr{2ptphi},
expressed in PG coordinates.
At equal PG time, we notice also that
the argument of the log is $x-x_0$.
We shall return to this point below.
When computing $\partial_x \partial_{x_0} G_K$
one obtains
\be
\partial_x \partial_{x_0} G_K(x,\tau; x_0,0) = - \frac{1}{ 4 \pi}
\frac{e^{\kappa \tau}}{(x - x_0\, e^{\kappa \tau} + i \epsilon)^2},
\label{ddcf}
\ee
which is the square root of \eqr{TTrr2} in the near horizon region where ${\cal U}_K \sim -x$.

When introducing dispersion,
$\rho$
receives corrections with respect to $(\partial_x \phi)^2$ due to the non-linearities of $F^2$.
This is hardly relevant for us, because in the near horizon region, the momenta $p$ are much smaller
than the UV scale $\Lambda$. They are of course modifications in the UV sector of the theory,
but these ultra local effects are the same as in Minkowski space.
Therefore the main modifications will come from the replacement of the Unruh modes
by the modified ones.
 Using \eqr{bmpsm}, \eqr{Gnodisp_p} is replaced by~\footnote{We proceed as in \cite{Schutzhold:2010ig}.
Nevertheless the forthcoming equations differ in several respects.}
\ba
\tilde G^{in}(p, \tau; p_0, 0) &=&
\tilde G_K(p, \tau; p_0, 0) \times
\frac{\exp  i \int^p_{p_0} [H(p')- 1] dp'/\kappa}{(H(p) H(p_0))^{1/2}} .
\label{Gdisp_p}
\ea
It should be noticed that
the frequency $\om_{\rm max}$ mentioned in footnote \ref{Macherz},
will cut out the integral in \eqr{prho}, below $-\om_{\rm max}$ for superluminal dispersion,
and above $\om_{\rm max}$ for sub luminal.
We ignored for this UV cutoff in computing \eqr{Gdisp_p} because \eqr{bmpsm} is no longer trustworthy
anyway when $\om \to \om_{\rm max}$. In fact, in the adiabatic approximation
of \eqr{Wchi}, there are cancelling
errors, in that the next equation can be shown to be exact.

\subsubsection{Equal time correlations}

At equal times, \eqr{Gdisp_p} gives
\ba
\tilde G^{in}(p,p_0, \delta \tau= 0) &=&  \theta(p) \, \frac{\delta(p -p_0)}{2 \Om(p)} ,
\label{etGdisp_p}
\ea
and in the $x$-representation,
one has
\ba
G^{in}(x,x_0, \delta \tau= 0) &=&  \int_0^\infty {dp \over 4\pi } {e^{ip(x-x_0) }\over \Om(p)}.
\label{Geqt}
\ea
The only effect of dispersion is to replace in the denominator
the relativistic law $\Om = p$ by $\Om = F(p)$.
Therefore \eqr{etGdisp_p} is {exactly} what one obtains in Minkowski
vacuum in the preferred frame.
The reason
is again that $x$ is affine at fixed $\tau$: $ds^2 = dx^2$.
In fact, as in the relativistic case, see Sec. III.F, a non-stationary vacuum can be defined
at a given time but for {all} values of $x$
by
plane waves $e^{ipx}$ with $p>0$.
Then,
the negligible character of the non-adiabatic corrections to \eqr{Wchi} in {the near horizon region}
gives
 \eqr{Gdisp_p} which implies that in {\it that} region but at {\it all} times
the vacuum
stays characterized by $p > 0$.\footnote{
\eqr{etGdisp_p} is also
 obtained in the adiabatic approximation,
in cosmological backgrounds when the preferred frame is aligned along the
cosmic frame. Moreover,
this correspondence becomes exact ({\it beyond} the adiabatic approximation)
 when considering de Sitter space when
the Hubble parameter $H=\kappa$
since
the linearized expression $w = -1 + \kappa x$
describes this space in PG coordinates when $-\infty < x < \infty$.}
This also implies that
the dispersive version of the time-dependent
\eqr{localG2}
will evolve towards the stationary \eqr{Gbhx}, as
\eqr{localG2} evolved into
the stationary
 function in
the Unruh vacuum.

From \eqr{Geqt} several consequences can be drawn.
If one probes the $in$ state for
$\kappa \vert x \vert \ll 1$,
the deviations w.r.t. to the relativistic case for $x-x_0 < 1/\Lambda$ ($p > \Lambda$)
are the same as in Minkowski,
and are therefore insensitive to
presence of the black hole.
If one probes
 the $in$ state
further away from the horizon and for momenta $p < \Lambda$, since $\Om \sim p$,
\eqr{Gbhx}
will behave as the
relativistic function, as it obeys the same equation, and possesses the same initial conditions.
Hence the whole analysis of Sec. III.E.2 applies.
In particular,
as soon as $w$ is constant, $\partial_x w \ll \kappa$,
\eqr{prho}
will obey
\eqr{CWC} when both points are on the same side of the horizon,
and
\eqr{BC} when one is on either side.~\footnote{In this we
recover
what has been found
in Bose Einstein
condensates when looking at the density-density correlation function~\cite{Balbinot:2007de,Iacopo08}.
In that case, in the hydrodynamical limit,
the atom density fluctuation
is given by $\partial_x \phi$, and
the correlation corresponds to
\eqr{ddcf}.}
Thus the properties of Fig. 2 (left) are not affected by dispersion when $\kappa \ll \om_{\rm max}$.
Those of Fig. 2 (right) are not either when subtracting the dispersive expression that replaces the
log in \eqr{Sr}, because $S_K$ varies on scales $~1/\kappa \gg 1/\Lambda$.

The insensitivity of Fig.2 against introducing dispersion
is quite surprising since, as discussed before, we expect that the
properties of Fig. 1 be affected by
the drift of \eqr{modchar}
which occurs for rather low momenta $\sim \Lambda^{2/3} \kappa^{1/3} \ll \Lambda$.
 The reason of the disappearance of the drift (at equal PG time)
is the following. In \eqr{prho},
because we are summing over $\om$, we erase the coherence in $x$-space
that exists in each $\om$ sector,
thereby
 recovering the
translation invariance of the $in$ state, as in \eqr{2ptphi}.
In other words, it is only when isolating some $\om$ content out of all vacuum
configurations that the {\it early} pattern characteristic of wave packets
emerges. This deserves further comments.

Given \eqr{Geqt}, what can be said about the entanglement entropy ?
The
regular behavior of the dispersive $in$ modes and the
entanglement in Fock space between states of opposite $\om$, see \eqr{sqs},

were exploited in \cite{Jacobson:2007jx}
to argue that the entanglement entropy of a black hole is finite (in 1+1 dimensions).
However, using \eqr{Geqt} one would conclude that upon tracing over inside configurations $x< 0$,
one would obtain the same (diverging) result as in Minkowski~\cite{Bombelli:1986rw}.
This
conflictual result
 indicates
 that
there is probably no unique notion of the entanglement entropy.
Therefore to get a well defined result, it is needed to specify what one exactly means by
"tracing over the inside configurations".

We saw that the 2pt
correlation function (at equal time)
does not display the characteristic pattern of wave packets with a given frequency content.
This is quite general.
It was discussed in \cite{Obadia:2002qe}
when studying the correlations amongst particles emitted by accelerator mirrors,
and in a inflationary context
in \cite{Campo:2003pa}.\footnote{These remarks raise the question of the choice of (the set of) observables
used to probe a  quantum state. To give a concrete ex.:
in inflationary cosmology, it is generally assumed that the
large amplification experienced by primordial fluctuations erase all
quantum properties and would give a state indistinguishable from a stochastic ensemble
of classical fluctuations. In \cite{Campo:2005sv} it was shown that irrespectively of the amplification
there exist observables exhibiting violations of Bell inequalities (for linearized modes).}

\subsubsection{Correlations at different times}

When $\tau \neq 0$, in the relativistic case one gets \eqr{Gnodisp_x}.
Instead, \eqr{Gdisp_p} gives
\ba
G^{in}(x,\tau; x_0, 0) &=&  \int_0^\infty {dp \over 4\pi } e^{i
p \delta
}
{e^{\kappa \tau/2}
\over  {[\Om(p)\,  \Om(p e^{ \kappa \tau} ) ]^{1/2}}}
\, \, \exp  i \int^{p}_{p e^{ \kappa \tau} } [H(p')- 1] dp'/\kappa,
\label{mG}
\ea
where
$\delta = x
- x_0 e^{ \kappa \tau}$.
The non-trivial modifications of the correlations due to dispersion are best seen
by evaluating this integral at the saddle point approximation. The value of the saddle $p_*$
answers the classical question: given that one starts at $x_0, \tau_0=0$
and ends at $x, \tau$, what is the momentum at that time ?
It is given by
\be
\kappa \delta =
\left[
e^{\kappa \tau} (H(p_* e^{\kappa \tau} ) - 1 )
- (H(p_*) - 1)
\right] .
\label{Gnoteqt}
\ee
To understand the implications of this expression, we consider three regimes.
First, if $\kappa \delta\ll 1$ and $\kappa \tau \sim 1$, the two points are almost connected by
a
null ray and the red-shifting effect is moderate.
In this case $p_*/ \Lambda \ll 1$ and one recovers the relativistic behavior of \eqr{Gnodisp_x}.

Second, we study the non-trivial correlation far away from the light cone (but still in the near horizon region).
For definiteness we restrict attention to
quartic laws $F^2 = p^2 \pm p^4/\Lambda^2$.
We expand
\eqr{Gnoteqt}
to first order in $1/\Lambda^2$, and using $H - 1 \sim \pm p^2/2 \Lambda^2$,
we get
\be
\kappa \delta = \pm {p_*^2 \over 2 \Lambda^2} \, (e^{3\kappa \tau} - 1 )
\label{kd}
\ee
When
$x = x_0 > 0$,
irrespectively of the sign of $\tau$,

there is no (real) saddle for the $+$ sign, i.e., superluminal dispersion,
in agreement with \eqr{modchar} which says that both
partners
are dragged inside the black hole horizon. Instead, for subluminal dispersion, since they are both
dragged outside, there must be a non trivial solution.
To confirm this,
we take
$\kappa \tau $  such that $e^{\kappa \tau}
\gg e^{-\kappa \tau}$. In this regime, \eqr{kd} reduces to $\kappa x_0   = p_*^2 e^{2\kappa \tau} /2\Lambda^2 $.
From this we can deduce $\om_*$ the mean value of the frequency corresponding to the trajectory
that goes from
$x_0$ back to it in a lapse equal to $\tau$.
It is approximatively given by
\be
\om_* = \sqrt{2}\Lambda \, (\kappa x_0)^{3/2} \,  e^{-\kappa \tau} .
\ee
This result can also be derived using \eqr{modchar} (and applied to superluminal dispersion for $x< 0$).
Thus, when studying $G(x,\tau;x_0,0)$ at sufficiently large $\kappa \tau$,
unlike what we found in \eqr{Geqt},
the correlations are now in agreement
with the locus of constructive interferences of wave packets
because only a limited range of frequencies centered about $\om_*$
significantly contributes. This confirms that near horizon behavior of
\eqr{prho}
 will completely differ from that of Fig. 1, and will be similar to those of Fig. 1. of \cite{Jacobson:2007jx}.
What remains to be clarified concerns
the profile of \eqr{prho} at early times. Namely, at fixed $x_0,\tau_0$,
what is the trajectory
of the maximum of \eqr{prho}, and what is its spread in $x$ as a function
of $ x_0, \tau-\tau_0$, and $ \Lambda$ ? We conjecture that both of these quantities are
ruled by $\om_{\rm max}$ of footnote \ref{Macherz}.

What can be studied~\cite{Schutzhold:2010ig} is the "off-shell"
limit of very large blue-shift $e^{\kappa \tau}\gg 1$
encoded in a backward propagation at fixed $x$ and fixed $\delta = x - x_0 e^{\kappa \tau}$.
This limit displays how dispersion tames the "trans-Planckian" behavior
found for the relativistic field.
In that case, \eqr{ddcf} gives
\be
\partial_x \partial_{x_0} G_K(x,0 ; x_0, -\tau) = - \frac{1}{ 4 \pi}
\frac{e^{\kappa \tau}}{(\delta+ i \epsilon)^2},
\label{ddcfd}
\ee
how ever large is $\kappa \tau > 0$, in agreement with Fig. 1.
In the dispersive case, using \eqr{mG} and $H(p) - 1 \sim \pm p^2/2 \Lambda^2$,
one has
\ba
\partial_x \partial_{x_0} G^{in}(x,0 ; x_0, -\tau) &=&
\int_0^\infty {dp \over 4\pi }
{p^2  e^{3 \kappa \tau/2}
\over  {[\Om(p)\,  \Om(p e^{ \kappa \tau} ) ]^{1/2}}}
\, \exp i \left(  p \delta \mp {p^3  e^{3\kappa \tau} \over 6 \Lambda^2 \kappa}\right).
\label{mGd}
\ea
When the blue shift is moderate, i.e.  $e^{\kappa \tau} < \kappa \delta \,  (\Lambda/\kappa)^{2/3}$,
\eqr{mGd} behaves as \eqr{ddcfd} plus corrections in $e^{3\kappa \tau} /\Lambda^2 \kappa \delta^3 \ll 1 $
that can be computed perturbatively, as can be seen by changing variable $p \to q = p\delta$.
Instead, when the blue shift is
large: $e^{\kappa \tau} >  \kappa \delta \,  (\Lambda/\kappa)^{2/3}$,
the integral becomes independent of $\delta$ as is seen by using $k = p e^{\kappa \tau}/ (\Lambda^2\kappa)^{1/3}$.
Explicitly one finds
\be
\partial_x \partial_{x_0} G^{in}(x,0 ; x_0, -\tau) \sim  (\Lambda^2 \kappa)^{2/3} e^{-\kappa \tau}
\times C_\pm(\kappa/\Lambda,\tau),
\label{feq}
\ee
where $C_\pm(\kappa/\Lambda,\tau)$ are slowly varying functions which stay bounded for $\tau \to \infty$.
This exponentially decreasing result can be seen as the contribution on the horizon
of the tail of the configurations with high $p$ which follow the second equation in \eqr{modchar}.

This smoothing out of the relativistic behavior is very reminiscent to what was found in~\cite{Barrabes:2000fr}
when studying the backwards evolution of $ G^{in}(x,0 ; x_0, -\tau)$ of a {\it relativistic} field
propagating in a stochastically fluctuating black hole metric.
In addition, for nearby points, $ G^{in}$ in a stochastic geometry also behaved as \eqr{Geqt},
as can be seen in Eq. (4.7).
Based on this similarity it was argued~\cite{Parentani:2007mb} that when taking into account
the gravitational radiative corrections,
the dressed Green functions should
effectively behave near a black hole horizon as
in \eqr{feq}, thereby
reinforcing the
idea
that the unbounded growth of \eqr{ddcfd} cannot "accommodate gravitational non-linearities".

\section{Conclusions}

We showed that the monotonic energy correlations found in the vacuum \eqr{TUU}
gives rise to a maximum of correlation across a Rindler horizon
when re-expressed in terms of coordinates associated with accelerated systems, see \eqr{ldTuu}.
This maximum is not a mere coordinate artefact as it affects the combined state
of co-accelerating systems.

In Sec. III, we transposed this analysis to stationary black hole geometries, and recalled
that the regularity of the state across the horizon and the inertial character of
asymptotic observers are essential to provide a physical meaning to the thermal
correlations of \eqr{Tuu}. When considering  black hole geometries which contain
asymptotic regions on both sides of the horizons, the correlations of
\eqr{ldTuu} are found at large distances when using inertial coordinates.
We then make use of the affine parameter $r_v$ of \eqr{rv} to obtain
an invariant description of the energy correlations in the entire space-time.
We compared the correlation pattern associated with a late detection, Fig. 1, to that
obtained at equal EF time, Fig. 2.
In both cases the gradual emergence of a maximum of correlations across the horizon
is clearly visible.
 By considering the subtracted
correlations of \eqr{Sr}, we saw that the remaining signal is dominated by the
long distance correlations across the horizon, and also contains a
sub-dominant local contribution associated with \eqr{Tuu}.
This analysis was generalized in III.F. by including the transients effects which
precede the stationary patterns found in the Unruh vacuum.

In Sec. IV. by studying both wave packets and correlation functions,
we studied
 how these patterns are modified by dispersion.
Far away from the horizon, the pattern is robust, i.e. hardly affected by dispersion.
Close to the horizon we saw that dispersive effects show up differently depending on how
one probes the state. When probed at equal PG time, the correlation function is translation
invariant, and as in Minkowski, see \eqr{Geqt}. Instead wave packets of $in$ modes
centered around a given frequency $\om$
display a characteristic pattern which follows the modified characteristics of \eqr{modchar}.
When the momentum has sufficiently increased (in a backward in time propagation)
the wave packets are dragged away from the horizon,
and, the blue shift effect saturates.
This behavior is recovered from the correlation function when considered at different times
and appears through a non-trivial saddle point in
\eqr{mG}.
When considering the correlation function for two points separated by a very large PG time,
the drag w.r.t. the relativistic horizon results in an exponentially suppressed amplitude
in the place of the exponentially growing result found in relativistic theories, compare
\eqr{ddcfd} with \eqr{mGd}. These properties are reminiscent to what was
found
when considering field propagation in a
fluctuating black hole metric, and could possibly be found when taking into
account gravitational interactions at the quantum level.

\begin{acknowledgments}
We would like to thank T. Jacobson for discussions concerning the affinity of $r$,
and R. Balbinot, S. Finazzi and N. Obadia for useful remarks.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
The HamburgAmbassadors are the representatives of Hamburg Metropolitan Region all over the world. What unites all of them is a strong commitment to Hamburg. Either they were born in the Metropolitan Region, or they have lived, worked or studied in Hamburg, which is why they feel a strong connection towards the city.
The program started in 2005 and has 36 ambassadors all over the world, contributing to Hamburg's strong network. The ambassadors engage themselves in different contexts to promote Hamburg. This is done via events, lectures, talks or by organizing the so-called "Hamburg Nights". Additionally, HamburgAmbassadors also support the organisation of trade fairs, meetings and conferences. In doing so, they strengthen existing contacts and develop new connections for Hamburg.
Once a year, all HamburgAmbassadors come together in Hamburg to get to know each other and partners within the city. They obtain the latest information, developments and trends within Hamburg.
For further information, please follow this link.
HamburgAmbassadors in India
Since May 2012, Amita Desai is a HamburgAmbassador based in Hyderabad. She completed her M.A. as well as her M.Ed. at the University of Hamburg in Education, English Language and American Literature. Before becoming the Executive Director of the Goethe-Institut in Hyderabad in 2003, she worked at the Indo-German Chamber of Commerce and afterwards in the Goethe-Institut Max Mueller Bhavanin Mumbai.

redpajama_set_name: RedPajamaCommonCrawl


text:
HomeReviewsAlbum ReviewsE.G. Phillips "At Home At Sea" (Independent, 2019)
E.G. Phillips "At Home At Sea" (Independent, 2019)
August 6, 2019 Rick Bayles Album Reviews 0
As a singer/songwriter it's always considered an advantage to have a distinctive voice – but can a voice be a little too "distinctive"? You can argue that a very distinctive voice never harmed the sales of the likes of Tiny Tim, Tom Waits, Neil Young and many more – but it will often polarise opinion; for every fan of Tom Waits' signature growl there will be someone who is less than enthusiastic.
E.G. Phillips definitely has a distinctive voice and it's likely to be another one that polarises opinion. It's an odd, loping singing style that has been described as "lounge crooner meets street busker", which isn't a bad description.
On listening to his latest release, "At Home At Sea", this reviewer was reminded a lot of the late British singer, songwriter and poet, Jake Thackray. Phillips has the same stylised delivery and often over precise pronunciation and it's a style that can tend to dominate a song to the point where you only hear the voice and not the song itself. Which is a shame because there are some good songs here, especially once you get past novelty opener 'The Albatross Song'. 'An Alternate Route to Your Heart' is a gentle, piano driven song of unrequited love that is clever and poignant – "How do you reach into my being so effortlessly? Ever so distant, no matter how near we are". There's also a nice, laid back vibe to 'All I Can Share is Photos' with its slightly discordant Bossa nova tempo and almost spoken delivery.
Perhaps the most striking cut on the album is penultimate track 'You Will Sail With Me', which has an eerie quality and a more serious feel to it than many of his other songs, with the repeated line "I know it's just a dreamboat, but I like it anyway" and some fine instrumental work at the tail end of the track with sweeping guitar chords and tinkling percussion; it's a track that really does bear repeated listening and becomes a bit of an earworm quite quickly.
Unfortunately, the rest of the album is too whimsical and too busy trying to be lyrically clever. Song titles like 'A Finnish Midsummer Midnight', 'The Comet and the Wandering Moon', 'Your Inexorable Pull', all betray a writer who is too wordy, often trying to cram lines that don't scan into too short a musical phrase.
Phillips has said that this album was about showing off the influence of Jazz music on his writing and you can detect a strong element of jazz phrasing in his music, along with elements of his other cited influences – Tom Lehrer and Bob Dylan, among others. This has also, apparently, been an exercise in resurrecting and re-writing older material and this seems to add to the disjointed nature of the album. I suspect this is a deliberate decision on Phillips' part and it does have a definite effect, though not always a good one. The album is well produced by Ben Osheroff who also contributes some very tasty guitar work but it's hard to escape the feeling that this album is quirky for the sake of it, not because it serves the music.
If you like intricate lyrics and the sort of musical jokes delivered by the likes of They Might Be Giants then this might well be one to check out but it's hard to see this album having wide appeal.
At Home At Sea fails to float the boat
E.G. Phillips
Video: Angela Perley "Let Go"
The Rails, The Lexington, London, 31 July 2019


text:
Tully's Tales
May 13, 2010 by Marion Maneker
The Master, Judd Tully, has some blow-by-blow details from Sotheby's Contemporary success last night:
Another rare-to-market Ab Ex work, Jackson Pollock's enamel and gesso ground on paper Number 12A, 1948: Yellow, Gray, Black (1948), ignited interest and sold to Gabriel Catone of Manhattan art advisors Ruth Catone for $8,762,500 against a $4-6 million estimate. The dynamic work on paper, once owned by avant-garde dealer Betty Parsons, last sold at auction at the same house in November 1987 for $396,000.
auction newcomer Jon Colby, who was identified by the New York Times as chasing Silver Liz (1963) at Christie's, had better luck tonight, snaring Joan Mitchell's majestic abstraction Vera Cruz (1960-62) for $4,002,500 (est. $3.5-4.5 million) and Anish Kapoor's untitled 2005 stainless-steel mirrored disc — an artist's proof from an edition of three (est. $600-900,000) — for $1,082,500. The Miami collector prefers to stand and call out his bids, commanding lots of attention along the way. The Mitchell last sold at auction at Christie's New York in November 1995 for $288,500.
the penultimate lot took ages to complete as a bidding war for Jean-Michel Basquiat's jazz-infused 1983 Untitled (Stardust) (est. $1.8-2.5 million) unfolded between Larry Gagosian and an elegantly attired woman seated across the aisle from him. Gagosian joined the fray at around the $2.9 million mark, after Jose Mugrabi dropped out, and the price climbed in $100,000 increments. Gagosian didn't hesitate in casting his bids, simply lifting his paddle again and again, as his rival consulted with her client via cell phone and then demurely flicked her paddle in the direction of auctioneer Tobias Meyer. Surprisingly — since this was, after all, Larry Gagosian — the anonymous rival succeeded in buying the painting, freshly consigned directly from the estate of the artist, for a thundering $7,250,500.
Warhol Edges Out Rothko (and Johns) at a Buoyant Sotheby's Sale (ArtInfo.com)
Buying Abstract Art in Bulk
Pink Panther in Perspective
Sotheby's Mid-Season Cont. Data Sheet
Sotheby's NY Cont Eve Sale = $266.6m
Contemporary Art Day-Sale Results
Sotheby's Has Strong Sales; Ovitz Decries Crass Commericalism
Filed Under: Auction Results Tagged With: Contemporary, New York, Sotheby's


Startups. Alternative Finance. Borderless Living.
Hi, I'm Nathan. I'm an author & digital marketer with a passion for technology, minimalism and freedom.
You can read more about me and get in touch here.
Spent time in over 30 countries. I hold legal residency in three of them.
Started my own agency, operating entirely location-independently.
I believe the early 21st century is the most exciting time to be alive in all of history. Technological changes have enabled people to hire remotely, do business internationally, and live wherever they choose. They are Citizens of the World – and not in some vague, cultural sense – but in a very real, practical sense. This website is dedicated to helping you make the most of it.
I write about startups, alternative finance, and borderless living. Get my best resources for living as a Citizen of the World, and sample chapters of my books.
Minimalism – Is It At Odds With Digital Marketing?
The System Is Dead. Long Live The System.
Hacking The System – Is It Being Smart, Or Cheating?


text:
DC's Legends of Tomorrow: The Complete Second Season (Blu-ray)(Region Free)
DESCRIPTION REGION CODING / SHIPPING TRAILERS
DC's Legends of Tomorrow: The Complete Second Season Blu-ray
Warner Bros. | 2016 | Season 1 | 681 min | Not rated | Aug 15, 2017
Codec: MPEG-4 AVC (11.92 Mbps)
English SDH, French, Portuguese, Spanish, Danish, Dutch, Finnish, Norwegian, Swedish
Three-disc set (3 BD-50)
DC's Legends of Tomorrow: The Complete Second Season (TV) (2016) Blu-ray
Time Master Rip Hunter travels back in time to the present day where he brings together a team of heroes and villains in an attempt to prevent Vandal Savage from destroying the world and time itself.
Directors: Dermott Downs, Rachel Talalay, Glen Winter
Writers: Phil Klemmer, Greg Berlanti, Marc Guggenheim, Andrew Kreisberg
Starring: Victor Garber, Brandon Routh, Caity Lotz, Arthur Darvill, Ciara Renée, Franz Drameh
- DC's Legends of Tomorrow: 2016 Comic-Con Panel
- Allied: The Invasion Complex (DC's Legends of Tomorrow)
- Gag-Reel
DISC & SHIPPING:
- Brand New Factory Sealed.
- Free Shipping Worldwide.
- Ships from the USA.
BLU-RAY, DVD & DIGITAL COPY:
- The Blu-ray Disc is Region Free and will play on all Blu-ray Players.
- If the product comes with a DVD, the DVD is Region A and will only play on NTSC/Region Free Players.
- If the product comes with a Digital Copy, the Digital Copy may not play in your region.
SLIPCOVER:
- We cannot guarantee that this item will come with a slipcover if applicable.


If there's one component of your vehicle that you should really check every so often – it's your car brakes. Gaining speed and being able to stop when needed are equally important. Faulty brakes are dangerous because more than just car damage, it can actually cost you your life, or even someone else's. So keep it in check at all times!
Visual inspection – This is the easiest and simplest way to check your brakes. Have a look at the thickness of your brake pads and it should be at least a quarter of an inch thick. If the pads look really thin, then it means they're worn out and it's time to get new ones.
Screeching sounds – When brakes wear down, there's a tiny piece of metal that touches the brake rotor thus making quite the unpleasant sound. This is actually intentional so you know right away if there's a brake problem.
Less braking power – Just like anything else, the newer the item, the better it performs. This is the same with brakes. The more you use it, the more it gets worn out and becomes not as effective when it was brand new. You'll feel it when you need a longer braking distance before your car fully stops. So before you get into any accidents, have those brakes checked and replaced.
Driving distance – In general, brake pads should be replaced when you've travelled around 50,000 miles but this may vary depending on your usual driving conditions. If you're usually driving with a full load or on steep roads, your brakes may get more worn out faster than normal.
Warning lights – If the brake light turns up on your dashboard, then you have to immediately check your brakes. The light is a definite sign that something's going on and it doesn't just turn on without any reason.
Burning smell – If something starts smelling weird when you're stepping on those brakes, then you may want to consider checking them out. It can mean your brakes or your clutch is overheating.
These are just some signs that can tell you about the condition of your brakes but did you know that there's a more accurate way to check them?
Safe T Stop is an advanced brake and wheel alignment test equipment. It can provide accurate brake performance and wheel alignment test results in less than 10 seconds!
And yes, we now have it here at Ipswich City Mechanical and Auto Electrical! We have recently installed Safe T Stop lanes so we can provide you with better wheel and brake repair services! And the best part? It is now part of all the services we offer! Click here to see it in action!
So if you feel that your breaks are starting to wear out, just bring your car to our shop and we'll be more than happy to check it out. We take pride in providing personalised car maintenance so no job is too small or too big for us.


text:
Q: Flowtypes, how to reduce duplication in Model-like class definitions We're currently adopting flow, and I'm running into an interesting problem. In one of our code-bases, we use a pattern roughly as follows.
We have 'models', which represent a 'Value object' or 'Data transfer object'. So our 'models' hold a single record. This is one of the simplest possible models:
class UserModel extends BaseModel {

}

The BaseModel defines a constructor, which roughly looks like this:
class BaseModel {

  constructor(properties: Object) {
    for (let key in properties) {
      this[key] = properties[key];
    }
  }
}

This allows you to do stuff like this:
const user = new UserModel({
  firstName: 'Evert',
  lastName: 'Pot'
});

I'm looking for an elegant way to add type-safety to this. One of the issues I'm facing is that I'd like both typesafety in places where an instance of UserModel is passed around, as well as during the construction of UserModel. Ideally I'd like UserModel to not be allowed to be instantiated in incomplete/invalid forms.
My first pass on this (which meets this goal) is this:
/* @flow */

interface UserInterface {

  firstName: string;
  lastName: string;

}

class UserModel extends BaseModel implements UserInterface {

   firstName: string;
   lastName: string;

   constructor(props: UserInterface) {
      this.firstName = props.firstName;
      this.lastName = props.lastName;
   }

}

const user = new User({
   firstName: 'Evert',
   lastName: 'Pot'
});

I'm re-using UserInterface both as a type for the constructor as well as the class itself.
This works, but has the drawback that I'm repeating every property name 4 times. When defining the interface, when defining the class and when setting the property in the constructor.
Being fairly new to flow, I wonder if there's a way to reduce this. Can I remove the duplication of property names?

A: You could change your BaseModel to have an internal dictionary for properties like:
class BaseModel {
  props: Object = {}

  constructor(props: Object) {
    this.props = props
  }
}

You could even use a proxy to set up dynamic getters/setters like in this answer or iterate over your properties and use defineProperty (similar to this fiddle I whipped up. You could alternatively just add a get method or something that accesses those values for you (similar to Backbone).
Then you can just have your UserInterface as a type if you want and do the following:
type UserInterface = {
  firstName: string;
  lastName: string;
}

class UserModel extends BaseModel {
  // The constructor is only necessary for flow to limit the values allowed
  constructor(props: UserInterface) {
    super(props)
  }
}

redpajama_set_name: RedPajamaStackExchange


One of the biggest issues in eCommerce is the difficulty of verifying a customer's identity. Credit card fraud causes huge problems for both shoppers and merchants alike, what with chargebacks, lost merchandise, lost profits, and the overall damage to the reputation of eCommerce. Thankfully, we now have more thorough ways to authenticate transactions, including Verified by Visa, based on the 3D Secure 2.0 protocol. To help verify that a purchase is being made by the actual cardholder, a large amount of information is passed between the merchant and financial institution to ensure that the purchase meets the customer's established buying patterns, comes from a recognized device and location, and more.
Verified by Visa is available for both merchants and customers. Customers can enroll any Visa-branded card to require the entry of an extra, personal password to authenticate their online purchases. Merchants who set up Verified by Visa on their store will be protected from high-risk transactions (transactions that do not match the data for the specific customer) by requiring an additional step of identity verification.
The additional verification takes the form of a password known only to the cardholder. The password isn't shared with the merchant, but goes straight to the financial institution by secure connection. Once the authentication goes through, the cardholder completes their purchase as normal.
Over half of consumers are still hesitant to shop online, but with the additional security provided by Verified by Visa and other 3D Secure 2.0-based services, eCommerce is becoming safer than ever. Even if a credit card is lost or stolen, a thief won't be able to make online purchases without the password, making the card useless to anyone but the cardholder.
As an eCommerce merchant, you'll benefit from this too. Less potential for fraud means fewer chargebacks, and less risk of losing merchandise to fraudulent sales. Your bottom line is more solid and you gain the trust of your customers. It works out great for all involved, except the thieves!


Either street entrance to the church building, off Plyers Mill Road or Windham Lane, will bring you into the main church parking lot. From the lot, you can approach the church from either the front side along Georgia Avenue, or from the back through the courtyard area.
From the front side, the sidewalk will take you past three sets of double doors. The second set of doors provides access directly ahead to the chapel, where the El Buen Samaritano (EBS) conducts Spanish language services. The third set of doors is the entrance to the main sanctuary. There is a ramp for handicapped entry just before the stairs.
Back-side access to the church for those without special needs is through the azalea garden, up the stairs straight across from where you enter it (notice the large directional sign on the wall of the sanctuary). This entrance will bring you to the fellowship foyer. Go up the stairs and either turn right to access the main sanctuary, or turn left and proceed down the hallway to the chapel and EBS services. For other facilities, go down the stairs to your left and immediately turn right to access rest rooms, the music area, church offices and meeting rooms.
Alternatively, from the azalea garden you can take the set of descending stairs on the right to access the lower level, including the church office, staff offices, choir area and meeting rooms, all directly below the main sanctuary.
During the week all doors on the Georgia Avenue side of the facility are locked, so access is usually through the rear courtyard area of the facility. Church office hours are ordinarily from 9 a.m. to 3 p.m. Monday-Friday. The church offices, staff offices and meeting rooms are accessed from the descending stairs off the azalea garden to the right. Ring the buzzer during office hours to be admitted. The entrance to Evergreen School is through the door in the education wing just to the left of the azalea garden. Again, you will need to buzz at the entrance to be admitted.
One other set of doors may be open at times during the week when there are activities in the gym or the social hall. These facilities are in the section of the building closest to the main parking area and can be accessed by the black wrought iron stairs adjacent to the parking lot. Go up the stairs to the gym or down the stairs to the social hall. The rest room on the gym level is through the double doors at the front of the gym (toward Georgia Avenue), then to the right. The rest rooms on the lower level are through the double doors at the end of the social hall opposite the kitchen (again, toward Georgia Avenue).