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\begin{center}
{\LARGE {\bf Evaporating black holes and long range scaling}}\\

\vskip 1.5cm

{\large {\bf Hadi Salehi\footnote { Electronic address:
salehi@netware2.ipm.ac.ir}}}
\vskip 0.5 cm
 Department of Physics, Shahid Beheshti University, Evin,
Tehran 19834,  Iran.\\
\end{center}

 \vspace{2.5 cm}

\begin{abstract}

For an effective treatment of the evaporation process of a large
black hole the problem concerning the role played by the
fluctuations of the (vacuum) stress tensor close to the horizon
is addressed. We present arguments which establish a principal
relationship between the outwards fluctuations of the stress
tensor close to the horizon and quantities describing the onset of
the evaporation process. This suggest that the evaporation
process may be described by a fluctuation-dissipation theorem
relating the noise of the horizon to the black hole evaporation
rate.

\end{abstract}

\vspace{1.5 cm}
\section{Introduction}

One of the central questions in the theory of black hole
evaporation concerns the detailed understanding of the
characteristics scales involved in the treatment of the onset of
evaporation process. In the usual treatment the characteristic
scale of length for a black hole of mass $M$ is identified with
the Schwartzschild radius\footnote{We use units in which G=c=1.}
$2M$ which for a large black hole is a macroscopic length.
Therefore, for a large black hole one may be inclined to persist
on the paradigm of an effective description requiring a low
energy treatment of black hole evaporation involving only the
characteristic energy scale $\sim\frac{\hbar}{2M}$.

The point,
however, is that due to the infinite gravitational red-shift on
the horizon, the long time (and long distance) observations in
the outside region of a black hole exhibit correlations with the
physical situations in a high energy regime in the vicinity of
the horizon where the fluctuations of the (vacuum) stress tensor
due to the high energy gravitational effects can no longer be
neglected [1][2][3]. This remark demonstrates that a low energy
description involving only the characteristic scale
$\frac{\hbar}{2M}$ may not encompass the essential features of
black hole evaporation. In fact it seems that high energy
gravitational effects may profoundly affects the correct form of the
stress-tensor fluctuations near the horizon in such a way that
applicability of semiclassical methods for the treatment of outside
observations becomes questionable.

A promising idea towards clarifying this issue comes from the
application of the black hole complementarity principle [4] [5],
which emphasizes that the role played by the high energy effects
near the horizon depends essentially on whether we look at
physical states from the outside or the inside of the horizon.
For example, in the Hilbert space used in the description
of an outside static observer, the principle demands that the
high energy gravitational effects decouple themselves in form of a materialized
stretched horizon where the incoming information are transferred
into the outgoing thermal radiation. Thus as long as physical
states are strictly localized outside the (stretched) horizon one may assume
that the high energy gravitational
effects are suppressed in such a way that the correct
form of the stress-tensor fluctuations near the horizon can be
taken to posses only the appropriate low energy characteristics of a
small disturbance of black hole causal structure, so that
semiclassical methods may be applied to a good approximation.
The conditions of this effective Hilbert space outside the horizon
however can not be indicative of the observations made by an
infalling observer crossing the horizon because the high energy
effects never seem to decouple from the inside of the horizon so
that the Hilbert space becomes the wrong Hilbert space for the
fundamental low energy description of the physical state of an
infalling observer. To avoid any physical inconsistency one
requires that there is basically no way to combine the
description of the outside observations with the description of
observations made by infalling observers crossing the horizon,
the two descriptions are complementary descriptions. In this way
complementarity is claimed to reflect an important feature of
black hole evaporation.

For a detailed understanding of the principle of black hole
complementarity it is important to realize that the principle
essentially implies two things. Firstly it implies an assertion
about the impossibility of realizing the physical state of an
outside static observer and the physical state of an infalling
observer crossing the horizon in the same Hilbert space. In fact,
the two states are related to basically different sets of
boundary conditions inside the horizon. This is a statement about
the complementary properties of basically different Hilbert spaces
used by observers separated by mutually exclusive behavior of
their coordinates on the horizon. The second implication is that
for the effective description of observations outside the horizon
the principle requires a systematic link between a small
disturbance of black hole causal structure and the evaporation
process through the choice of a physical low energy state. In
order to establish such a link we shall study a model in which
the stress-tensor fluctuations near the horizon, expressed in
coordinates of an outside static observer, are taken to posses
the appropriate low energy characteristics of random fluctuations
arising from the noise of gravitational effects inside the
horizon (the horizon noise). According to the principle of black
hole complementarity this effective ansatz, which is studied in
this paper, is correct as long as the physical state of an static
observer is strictly localized outside the horizon. Inside the
horizon the methods of this effective ansatz are generally felt
to break down, because the correct form of the physical state of
an static observer inside the horizon never seem to posses the
low energy characteristics required by that effective ansatz.

This effective ansatz is very useful for posing a general
question concerning the dissipative effects of quantized fields
in the presence of a black hole. We would expect, namely, that
the random fluctuations of the stress tensor near the horizon,
expressed in coordinates of an outside static observer, to posses
a dissipative character. This kind of behavior is suggested in a
very general way by fluctuation-dissipation theorems which
systematically link the random fluctuations of a system to a
systematic effect, namely the dissipative behavior of the same
system over long time intervals. In the present
context the dissipation is represented by the black hole evaporation process.
Therefore it is important to determine how the random fluctuations
of the stress tensor near the horizon can be related by the
conditions of a semiclassical theory to the evaporation rate.
In this paper we shall study this relationship using a
two-dimensional Schwarzschild black hole model. The significance
of such a lower dimensional model lies in the fact that it may be
considered as a model arising from the geometric optics
approximation of a physical spherically symmetric black hole
model. Such a restriction to geometric optics approximations and
to the corresponding lower dimensional methods simplifies
considerably the analysis, and it is generally believed that the
qualitative features of the evaporation process will not alter
too much by this restriction. The organization of the paper is as
follows: In the subsequent two chapters we present the model and
discuss phenomenological arguments leading to  a long range
scaling law which controls the outwards
stress-tensor fluctuations near the horizon, expressed in the
coordinates of an outside static observer, in terms of a large
correlations length. In this model we deal
with the mean value of these fluctuations which is taken to be systematically
determined by the outwards component of the renormalized expectation value
of the stress tensor of a quantum field
taken in some appropriatly chosen quantum state.
Therefore the scaling law should basically understood
as a condition imposed
on this state.
In chapter 4 we present a dynamical
derivation of the scaling law on the basis of the backreaction
effect using a Planckian cutoff condition in the frame of an
observer who uses finite coordinates at the horizon. In chapter 5
we show that the scaling law can be represented in form of a
fluctuation-dissipation theorem which relates the mean value of
the outwards fluctuations of the stress tensor near the horizon
to the black hole evaporation rate.
Some arguments for
deriving corrections to the radiation temperature are then
presented in chapter 6. The paper ends with some concluding
remarks.

\section{The Model}

We  consider
a two-dimensional analog of the Schwarzschild
black hole of mass $M$, described in coordinates which are
indicative of the outside observations (Schwarzschild coordinates)
by the metric
\begin{equation}
ds^{2}=-\Omega(r)dt^{2}+\Omega^{-1}(r)dr^{2},~~~\Omega(r)=1-2M/r,
\label{1}\end{equation} to which a massless scalar quantum field
$\phi$ is taken to be minimally coupled. Let the stress tensor of
$\phi$ in the (t,r) coordinates be denoted by $T_{\mu\nu}$. We
are primarily interested in the form of the low energy
fluctuations of $T_{\mu\nu}$ near the horizon, i.e. in the
effective horizon limit $r\rightarrow 2M$. In a semiclassical
treatment these fluctuations must be considered as random
fluctuations due to the horizon noise. Denoting
their mean value by $\delta T_{\mu\nu}$, the first fundamental
task is how to control the typical magnitude of $\delta
T_{\mu\nu}(r\rightarrow 2M)$ in terms of quantities accessible to
a semiclassical treatment.

In a semicalssical theory the operator $T_{\mu\nu}$ arises as a
singular operator because it involves the product of the field
operator at a single point. Therefore one can generally assume
that the typical magnitude of $\delta T_{\mu\nu}(r\rightarrow
2M)$ may be related to the effective horizon limit $r\rightarrow
2M$ of the late-time configuration of the renormalized
expectation value $<T_{\mu\nu}>^{ren.}_{\omega}$ taken in some
appropriately chosen quantum state $\omega$, namely
\begin{equation}
\delta T_{\mu\nu}(r\rightarrow 2M)\sim <T_{\mu\nu}(r\rightarrow
2M)>^{ren.}_{\omega}. \label{1-a}\end{equation}
This relation
links two kinds of effects. The right hand side of (\ref{1-a})
is the systematic magnitude of the stress tensor near the horizon
which is accessible to a semiclassical treatment through the
choice of the quantum state $\omega$, whereas the left hand side
is the random magnitude due to random fluctuations. The relation
(\ref{1-a}) requires a systematic link between both
magnitudes. The quantum state $\omega$ is, therefore, assumed to
link the random magnitude of the stress tensor $T_{\mu\nu}$ near
the horizon with its systematic counterpart, namely the renormalized
expectation value of $<T_{\mu\nu}(r\rightarrow
2M)>^{ren.}_{\omega}$. This has an essential consequence for the
characterization of the state $\omega$.

\section{Long range scaling}

For the characterization of the state $\omega$ the determination
of the outwards component of $<T_{\mu\nu}>^{ren.}_{\omega}$ near
the horizon is very important because this component is the
indicative
quantity of the long-time observations in the outside
region. Let $<T_{uu}>^{ren.}_{\omega}$ denotes the outwards
component of $<T_{\mu\nu}>^{ren.}_{\omega}$ defined with respect
to the standard outward (retarded) time $u$ of the metric
(\ref{1}), namely
\begin{equation}
u=t-\stackrel{*}{r},~~~~\stackrel{*}{r}=r+2M\ln|\frac{r}{2M}-1|.
\label{2}\end{equation} The problem is how to determine the limit
$<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}$. The relation
(\ref{1-a}) tells us that $<T_{uu}(r\rightarrow
2M)>^{ren.}_{\omega}$ is related to $\delta T_{uu}(r\rightarrow
2M)$ which is the mean value of the outwards fluctuations of the
stress tensor near the horizon. There is a phenomenological
argument suggesting that these fluctuations shall be correlated
over all scales of lengths. In fact, null rays which are
equispaced along the future null infinity over long distances
crowd up near the horizon over small distances, indicating that
the outwards fluctuations of the stress tensor of a massless
field near the horizon shall be correlated over almost all scales
of lengths. This behavior implies that the typical scale
of the outward component of the renormalized stress tensor near
the horizon is set by a large correlation length $\xi$ which
could have in principle its value many orders of magnitudes away
from the Schwartzschid radius, giving us the scale hierarchy
\begin{equation}
\xi>>2M. \label{3}\end{equation}
In essence, such a scale
hierarchy indicates that long-range fluctuations dominate at
late-times. Therefore we can use dimensional arguments to arrive
at the long range scaling law
\begin{equation}
<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}\sim \xi^{-2}.
\label{4}\end{equation}
This scaling law implies that the characteristic order of
the magnitude of the expectation value $<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}$ is set
by the correlation length
of the outwards fluctuations of the stress tensor at late times, which is distinctly
separated by the scale hierarchy (\ref{3})
from the characteristic macroscopic length of the system, namely $2M$.

This feature implies, in particular, that the quantity of
$<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}$ should basically
decouple from the dynamical constraints of the renormalization
theory describing the effective change of the late-time
configuration of the renormalized expectation value
$<T_{\mu\nu}>^{ren.}_{\omega}$, because this change can
generically be expected to occur on those typical macroscopic
length scales which are distinctly much smaller than $\xi$. That
this decoupling actually happens  to be the case may be seen from
the following
consideration:\\
The dynamical constraints of the renormalization theory can be expressed
in form of a hydrodynamic constraint, namely the conservation law
\begin{equation}
\nabla^{\mu}<T_{\mu\nu}>^{ren.}_{\omega}=0.
\label{5}\end{equation}
This law can be used to determine the
static form of $<T_{\mu\nu}>^{ren.}_{\omega}$. In doing this we
ignore effects related to a preassigned time-dependence of the
expectation value $<T_{\mu\nu}>^{ren.}_{\omega}$. Naturally, we
assume that any time-dependence of $<T_{\mu\nu}>^{ren.}_{\omega}$
should be suppressed in the late-time limit. Therefore we look
for the static configuration of the expectation value
$<T_{\mu\nu}>^{ren.}_{\omega}$ which can be found to be [6][7]
\begin{equation}
<T_{\mu}^{\nu}>^{ren.}_{\omega}=T_{\mu}^{(1)\nu}+T_{\mu}^{(2)\nu}+T_{\mu}^{(3)\nu}
\label{6}\end{equation}
where in $(t,\stackrel{*}{r})$ coordinates
\begin{equation}
T_{\mu}^{(1)\nu}=
\left(\matrix{<T^{\alpha}_{\alpha}(r)>_{\omega}-\Omega^{-1}(r)H(r)&0\cr
0&\Omega^{-1}(r)H(r)~\cr}\right)
\label{6-a}\end{equation}
\begin{equation}
T_{\mu}^{(2)\nu}=
\Omega^{-1}(r)\frac{K}{M^{2}}\left(\matrix{1&-1\cr
1&-1\cr}\right)
\label{6b}\end{equation}
\begin{equation}
T_{\mu}^{(3)\nu}=
\Omega^{-1}(r)\frac{Q}{M^{2}}\left(\matrix{-2&0\cr
0&2\cr}\right).
\label{6c}\end{equation}
Here $K$ and $Q$ are arbitrary constants, and
\begin{equation}
H(r)=\frac{1}{2}\int^r_{2M}
\Big({{d\over{dr^\prime}}
\Omega(r^\prime)}\Big)<T^\alpha_\alpha(r^\prime)>_{\omega} dr^\prime.
\label{6d}\end{equation}
We can now determine the outwards component of this solution. We find
\begin{equation}
<T_{uu}>^{ren.}_{\omega}=\frac{1}{2}(H(r)+2Q/M^{2})-\frac{1}{4}\Omega(r)
<T^{\alpha}_{\alpha}(r)>_{\omega}
\label{7}\end{equation}
which near the horizon yields the scaling law
\begin{equation}
<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}\rightarrow Q/M^{2}.
\label{8}\end{equation}
Thus, if the
conservation law is applied, it is possible to describe the
outwards component of $<T_{\mu\nu}>^{ren.}_{\omega}$ near the
horizon in terms of an integration constant which is not
dependent upon the particular dynamical coupling of $\phi$ to the
metric (\ref{1}). This implies that the specification of that
component must reflect a model independent general characteristic
of the vacuum state as observed by those observers which are
strictly localized  outside the black hole. This
observation establishes the decoupling of
$<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}$ from
the dynamical constraint of the renormalization theory.

We should emphasize that the scaling law (\ref{4}) as it stands
has a highly phenomenological character, and thus it is desirable
to derive it on a dynamical basis.

\section{The dynamical derivation of the scaling law}

The phenomenological arguments that led to the scaling law
(\ref{4}) can find a dynamical justification by combining a
cutoff condition near the horizon with the backreaction effect of
black hole radiation. To this aim we first note that the
unrenormalized expectation value $<T_{uu}(r\rightarrow
2M)>_{\omega}$ is mathematically a singular quantity. The
corresponding renormalized value can be obtained firstly by
introducing a shortdistance cutoff length which is taken to be
the Planck length $l_P$, and secondly by specifying the reference
frame to which the Planckian cutoff is applied. The most natural
reference frame for the imposition of a Planckian cutoff is a
reference frame of an infalling observer crossing the horizon. The
coordinate system which is indicative of such an observer is a
coordinate system which is finite at the horizon, such as the
inwards and the outwards Kruskal time-coordinates, defined
respectively as
\begin{equation}
V=4M~ exp(v/4M),~~~ U=-4M~ exp(-u/4M)
\label{9}\end{equation}
where $v=t+\stackrel{*}{r}$ and $u$ as given by (\ref{2}). In this
coordinate system the imposition of a Planckian cutoff
on the expectation value $<T_{uu}(r\rightarrow 2M)>_{\omega}$
is taken to correspond to the requirement
that the stress-tensor fluctuations near the horizon
shall be correlated over a cutoff domain of Planckian size $\sim l_p$.
The
correlations length of these fluctuations should therefore be
taken as $l_P$. The length $l_P$ sets the typical scale of length for
the determination of $<T_{UU}(r\rightarrow 2M)>^{ren.}_{\omega}$.
Therefore on dimensional grounds we arrive at
the relation
\begin{equation}
<T_{UU}(r\rightarrow 2M)>^{ren.}_{\omega}\sim l^{-2}_p
\label{10}\end{equation}
which indicates that the state $\omega$ exhibits
large stress-tensor fluctuations in the frame of an infalling observer
crossing the horizon.
It is important to note that this feature which arises from the
cutoff condition
reflects the characteristic
feature of the black hole complementarity principle because it
indicates that strong gravitational effects may not decouple from
the inside of the horizon, so that
the Hilbert space of the state $\omega$ becomes the wrong Hilbert space
for the low energy description of the physical state of an infalling observer.

To derive the scaling law (\ref{4}) from the relation (\ref{10})
we first relate the
expectation value $<T_{UU}>^{ren.}_{\omega}$ to $<T_{uu}>^{ren.}_{\omega}$
using the coordinate transformation (\ref{9}) to find
\begin{equation}
<T_{UU}>^{ren.}_{\omega}=\frac{1}{4} ~exp(-r/M) V^{2} (r-2M)^{-2}
<T_{uu}>^{ren.}_{\omega}. \label{11}\end{equation}
We then try to
use this relation for the derivation of the scaling behavior of
$<T_{uu}>^{ren.}_{\omega}$ in the limit $r\rightarrow 2M$. The
first observation is that (\ref{11}) together with (\ref{10})
implies that $<T_{uu}>^{ren.}_{\omega}$ vanishes in the limit
$r\rightarrow 2M$. However this feature is an idealization of
neglecting the backreaction  due the Hawking effect. The
consideration of the backreaction implies that the effective
horizon limit $r\rightarrow 2M$ should be carried out with
respect to a mass scale which is slightly smaller than the mass
$M$, namely
\begin{equation}
r\rightarrow 2(M-\delta M)
\end{equation}
where $\delta M $ is of the order of the mass evaporated away
during the times just prior to the formation of the horizon. For
a sufficiently large black hole one can generally expect that
\begin{equation}
\delta M<< M
\end{equation}
holds. But it can be shown that $\delta M$ is even much smaller
than $l_P$. To show this we consider $M$ as a
function of the advanced time $M(v)$, and let $v_0$ be the value
of the advanced time at which the horizon would form if we
neglect the backreaction effect. The mass $\delta M$ can be
estimated by
\begin{equation}
\delta M\sim |\frac{dM}{dv}(\tilde{v})| (v_0-\tilde{v})
\label{12}\end{equation}
where the time $\tilde{v}$ is taken to characterize the onset of the
evaporation process, so it must be very close to the horizon formation time
$v_0$. In general the time difference $\delta v=v_0-\tilde{v}$ must be
taken as much smaller than the characteristic time-scale of the system which is
set by the black hole mass. Therefore one should have
\begin{equation}
\delta v<< 2M.\label{13}\end{equation}
Although the validity of
this relation seems to be apparent from the context, but it can
be justified also by noting that, for a value $\tilde{v}$ of the
advanced  time characterized by (\ref{13}), a null-geodesic after
its propagation through a collapsing objects will be characterized
by a retarded time $\tilde{u}\sim -4M\ln{(v_0-\tilde{v})/2M}$;
which
is characteristic to the onset of the evaporation process [8][3].
The correct order of magnitude of $\delta M$ can now be
determined if we estimate $\frac{dM}{dv}(\tilde{v})$ by the
Hawking law
\begin{equation}
-\frac{dM}{dv}(\tilde{v)}\sim (l_p)^2 \frac{1}{M^2}.
\label{14}\end{equation}
Using this law in (\ref{12}) we obtain
\begin{equation}
\delta M\sim l_p \frac{l_p}{M} \frac{\delta v}{M}
\label{15}\end{equation} which in conjunction with (\ref{13})
yields
\begin{equation}
\delta M<< l_p. \label{16}\end{equation}

This relation can be
used to estimate the renormalized expectation value
$<T_{uu}>^{ren.}_{\omega}$ near the horizon using the effective
horizon limit $r\rightarrow 2M-\delta M$ of the relation
(\ref{11}) together with (\ref{10}). We obtain
\begin{equation}
<T_{uu}(r\rightarrow 2M)>^{ren}_{\omega}\sim \xi^{-2}
\label{18}\end{equation}
where
\begin{equation}
\xi\sim\frac{l_P}{\delta M} 2M>>2M.
\label{19}\end{equation}
We arrive therefore at the scaling law (\ref{4}).


\section{A fluctuation-dissipation theorem}


The scaling law (\ref{4})  links  the scaling behavior of the
renormalized expectation value $<T_{uu}>^{ren.}_{\omega}$ near
the horizon with the large correlations length $\xi$ of the
stress-tensor fluctuations near the horizon through the choice of
the quantum state $\omega$ in the outside region of the black
hole. One can therefore expect this state $\omega$ to posses a
dissipative character related to these fluctuations. That this is
indeed the case can be seen directly from (\ref{4}). In fact if
we combine (\ref{4}) with the Hawking law (\ref{14}) we may write
(\ref{4}) as
\begin{equation}
<T_{uu}(r\rightarrow 2M)>^{ren.}_{\omega}=\gamma l^{-2}_p
\frac{dM}{dv} (v\rightarrow v_0),~~~ \gamma\sim(2M/\xi)^2<<1.
\label{20}\end{equation}
This formulation of the scaling law
(\ref{4}) is instructive because it shows that the quantum state
$\omega$ links the outwards fluctuations of the stress tensor
near the horizon to a long-time dissipative behavior, namely the
evaporation rate of the black hole. Therefore, the scaling law if
combined with the Hawking law may be brought into a form
suggesting a fluctuation-dissipation theorem. One can
alternatively consider the arguments presented in the previous
chapter as demonstrating as to how such a theorem can be derived
on dynamical basis from a cutoff condition near the horizon.

It is important to add the following remarks concerning
the correct physical interpretation of (\ref{20}).
The fluctuations that contribute to the left hand side of (\ref{20})
cause a small disturbance of the black hole
metric and the corresponding causal structure.
The relation (\ref{20}) describes how
this disturbance is related  to
the evaporation rate via the choice of
the quantum state $\omega$
for the undisturbated system, i.e, the black hole thermal state.
This is is very much in the sprit of the general framework of
the response theory which relates the response of a sytem
to a small disturbanse to the equilibrium characteristics of
the undisturbated system.



\section{Corrections to the Hawking effect}

It may be of interest to examine the effect of the scale-separation
$\xi>>2M$ on the Hawking effect. Generally, one expects to find
a deviation of the
black hole temperature from the Hawking temperature by a term
of the relative order $(2M/\xi)^\alpha$
where $\alpha$ is a characteristic exponent. To determine this exponent
we proceed as follows:
In two dimensions an outwards flux of thermal radiation
can be characterized at large $r$ by the energy momentum tensor
\begin{equation}
{\pi\over{12}}T^2\left(\matrix{-1&-1\cr
1&1\cr}\right)
\label{21}\end{equation}
in which $T$ is the temperature.
From (\ref{21}) one
infers
that a static, spherically symmetric
configuration of matter which describes the Hawking
radiation at large $r$ must have a stress tensor satisfying
the condition
\begin{equation}
T_{t}^{t}(r)\rightarrow T_{t}^{r}(r),~~~as ~r\rightarrow\infty
\label{22}\end{equation}
which means that the energy density and the flux are asymptotically equal.
If this condition is applied to the general solution (\ref{6}) one gets
\begin{equation}
K-\frac{1}{2}M^{2}[H(\infty)-<T_{\alpha}^{\alpha}(\infty)>_{\omega}+2Q]=0.
\label{23}\end{equation}

To derive the Hawking radiation from
such a relation one usually assumes two additional requirements.
The first one corresponds to the consistency of the trace anomaly
with respect to the two dimensional metric (\ref{1}), namely [6]
\begin{equation}
<T_{\alpha}^{\alpha}(r)>^{ren.}_{\omega}=\frac{M}{6\pi r^{3}}.
\label{24}\end{equation}
The second requirement concerns the
finiteness of the energy momentum tensor at the horizon with
respect to a coordinate system which is finite there. In order
to implement the second assumption the standard derivation takes
the value $Q=0$ which arises as a pure effect of the
transformation law (\ref{11}) in the limit $r\rightarrow 2M$.
There is however some objections for considering the value $Q=0$
as the correct one, because the finiteness condition of a quantum
stress tensor at the horizon requires us to investigate a cutoff
condition in the frame of an observer who uses finite coordinates
at the horizon, and we have seen that this leads to the scaling
law (\ref{4}) which together with (\ref{8}) predicts a
non-vanishing value $(\sim 2M/\xi)^2$ for $Q$. Thus for the
derivation of the Hawking radiation we may take the consistency
of the trace anomaly together with this value of $Q$. At large $r$
the latter condition predicts via the last term in (\ref{6}),
namely the tensor $T_{\mu}^{(3)\nu}$, a deviation of the thermal
radiation from the Hawking temperature of the relative order
$M^2/\xi^2$, leading to the characteristic exponent $\alpha=2$
which coincides in the present case with the dimensionality of space-time.

We should also remark that
the non-vanishing value $Q\sim M^2/\xi^2$ predicts that
the expectation value $<T_{\mu\nu}>^{ren.}_{\omega}$ has at large $r$
a term corresponding to a
background heat bath with the temperature $\sim 1/\xi$.
This follows if one compares the tensor $T_{\mu}^{(3)\nu}$ at large $r$
with the
stress tensor of an equilibrium gas, namely
\begin{equation}
{\pi\over{12}}(kT)^2\left(\matrix{-2&0\cr
0&2\cr}\right).
\label{15}\end{equation}
Such a model
may have some power at the cosmological level, in that
the background
heat bath may act as a  model for the thermal
equilibrium gas of an associated
cosmological horizon. In this way the cutoff condition on the horizon
may be linked with a small cosmological constant.


\section{Concluding remarks}

The paper has examined a new method for introducing a quantum
state $\omega$ for the outside region of a black hole. The
distinct feature of this method as compared with the standard
methods for introducing black hole states, such as those
discussed in [6] [7], is that it links via the scaling law
(\ref{4}) the choice of the quantum state $\omega$ outside the
horizon with the noise of gravitational effects inside
the horizon, and in this respect it emphasizes a general
relationship between a small disturbance of the black hole causal
structure and the choice of an external quantum state in the
absence of this disturbance. One may expect that the implications
that arise from this viewpoint may improve our uderstanding about
the nature of black hole evaporation.\\\\

{\bf Acknowledgement}\\
The author thanks the office of scientific research
of Sh.Beheshti University for financial support.\\\\


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