Accelerating Observers, Area and Entropy
Jarmo M\"akel\"a
We consider an explicit example of a process, %%@ where the entropy carried by radiation through an %%@ accelerating spacelike two-plane is proportional %%@ to the decrease in the area of that two-plane %%@ even when the two-plane is not a part of any %%@ horizon of spacetime. Our results seem to support %%@ the view that entropy proportional to area is %%@ possessed not only by horizons but by all %%@ spacelike two-surfaces of spacetime.
gr-qc0506087.tex
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vtex4}
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\begin{document}
		  
\title{Accelerating Observers, Area and Entropy}

\author{Jarmo M\"akel\"a} 
\email[Electronic address: %%@
]{jarmo.makela@phys.jyu.fi}  
\affiliation{Department of Physics, University of %%@
Jyv\"askyl\"a, PB 35 (YFL), FIN-40351 %%@
Jyv\"askyl\"a, Finland}


\begin{abstract}
We consider an explicit example of a process, %%@
where the entropy carried by radiation through an %%@
accelerating spacelike two-plane is proportional %%@
to the decrease in the area of that two-plane %%@
even when the two-plane is not a part of any %%@
horizon of spacetime. Our results seem to support %%@
the view that entropy proportional to area is %%@
possessed not only by horizons but by all %%@
spacelike two-surfaces of spacetime.  
\end{abstract}

\pacs{04.70.Dy}
\keywords{accelerating observers, area change, %%@
entropy}

		  
\maketitle
		  
\section{Introduction}
		  
		  A very interesting discovery was %%@
published by
                  Jacobson in 1995 \cite{eka}. He %%@
had found that, in a certain sense, Einstein's %%@
field equation may be viewed as a thermodynamic %%@
equation of state of spacetime and matter fields. %%@
This discovery was based on the fact that when %%@
the Rindler horizon of an accelerating observer %%@
emits the so called Unruh radiation, the area of %%@
the considered part of the Rindler horizon %%@
shrinks. Jacobson observed that if one {\it %%@
assumes} that the entropy carried away from the %%@
considered part of the Rindler horizon is, in %%@
natural units, exactly one quarter of the %%@
decrease in the area of that part, then %%@
Einstein's field equation follows from the first %%@
law of thermodynamics. In other words, it turned %%@
out that Einstein's field equation is just a %%@
consequence from the first law of thermodynamics, %%@
and an assumption that entropy equal to one %%@
quarter of the horizon area is possessed not only %%@
by black hole and cosmological horizons, but also %%@
by the Rindler horizon. This conclusion supports %%@
the view that classical general relativity %%@
actually describes
the thermodynamic properties of spacetime %%@
\cite{toka}. If one wants to construct a %%@
microscopic theory of gravitation, the correct %%@
way to do this may not be to quantize general %%@
relativity but to postulate an existence of %%@
certain microscopic constituents of spacetime %%@
obeying laws which, in the thermodynamic limit, %%@
reproduce Einstein's field equations. 

        An essential feature in Jacobson's %%@
analysis was that a {\it horizon} was looked at: %%@
When radiation is emitted by a horizon, the %%@
horizon shrinks, and the entropy of the horizon %%@
is converted to the entropy of the radiation. It %%@
is interesting to investigate, whether entropy %%@
would be possessed not only by horizons but by %%@
{\it all} spacelike two-surfaces of spacetime. %%@
Arguments supporting this claim have been given, %%@
for instance, in \cite{kolmas}, where it was %%@
proposed that every spacelike two-surface of %%@
spacetime possesses entropy which, in natural %%@
units, is one quarter of the area of that %%@
two-surface. If true, this proposal brings great %%@
clarity to the concept of gravitational entropy: %%@
Entropy is a property of spacetime, and the %%@
entropy of a spacelike two-surface is always the %%@
same, no matter whether that two-surface is a %%@
part of a horizon or not.

      To support the view that entropy is %%@
possessed not only by the horizons, but by all %%@
spacelike two-surfaces of spacetime we consider, %%@
in this paper, a slight modification of %%@
Jacobson's analysis. More precisely, we consider %%@
a plane parallel to the $xy$-plane, which is %%@
accelerating with a constant proper acceleration %%@
along the positive $z$-axis. We assume that, from %%@
the point of view an observer at rest with %%@
respect to the flat Minkowski coordinates $x$, %%@
$y$ and $z$, spacetime is filled with %%@
electromagnetic radiation in a thermal %%@
equilibrium, from which it follows that there is %%@
a certain net flow of radiation through the %%@
accelerating plane to the negative $z$-direction. %%@
We investigate the properties of this radiation, %%@
and it turns out that when radiation propagates %%@
through the accelerating plane, the plane %%@
shrinks. Using the first law of thermodynamics we %%@
shall arrive at an interesting conclusion that %%@
the entropy carried by Unruh radiation through %%@
the accelerating plane is exactly one half of the %%@
decrease in the area of that plane. In other %%@
words, it seems that we a have found a process %%@
where the entropy of a two-plane is converted to %%@
the entropy of radiation even when that two-plane %%@
is not a part of any horizon of spacetime. This %%@
result strongly supports the proposal that not %%@
only horizons, but all spacelike two-surfaces of %%@
spacetime possess a certain amount of entropy.

    This paper is organized as follows: In %%@
Section 2 we consider the energy flow and %%@
intensity of radiation through an accelerating %%@
plane. In Section 3 we investigate the change in %%@
the area of the accelerating two-plane when %%@
radiation flows through that plane. Finally, in %%@
Section 4 we bring to the stage the first law of %%@
thermodynamics. The first law of thermodynamics, %%@
together with the results of Sections 2 and 3, %%@
implies the desired relationship between area and %%@
entropy. We end our discussion with concluding %%@
remarks. Unless otherwise stated, we shall always %%@
use the natural units, where $\hbar=c=G=k_B=1$. 

\maketitle


\section{Flow of Energy}

      When flat Minkowski spacetime is filled %%@
with electromagnetic radiation in thermal %%@
equilibrium, there is no net flow of energy in %%@
any direction from the point of view of an %%@
observer at rest with respect to the flat %%@
Minkowski coordinates $t$, $x$, $y$, and $z$. The %%@
only non-zero components of the energy momentum %%@
stress tensor $T^{\mu\nu}$ of radiation are:
\begin{equation}
T^{00}=\rho,
\end{equation}
where $\rho$ is the energy density of radiation, %%@
and
\begin{equation}
T^{11}=T^{22}=T^{33}=p,
\end{equation}
where $p$ is the pressure of the radiation. %%@
Because for electromagnetic radiation
\begin{equation}
p=\frac{1}{3}\rho,
\end{equation}
we find that
\begin{equation}
T:=T^0_0+T^1_1+T^2_2+T^3_3=0.
\end{equation}
In other words, the tensor $T^{\mu\nu}$ is %%@
trace-free. This result will turn out very useful %%@
in our investigations.

       Although there is no net flow of energy %%@
from the point of view of an observer at rest %%@
with respect to the coordinates $x$, $y$ and $z$, %%@
there is, however, a net flow of energy to the %%@
negative $z$-direction from the point of view of %%@
an observer in a uniformly accelerating motion %%@
along the positive $z$-axis. The world line of %%@
such an observer satisfies the equation 
\begin{equation}
z^2-t^2=\frac{1}{a^2},
\end{equation}
where $a$ is the proper acceleration of the %%@
observer. This world line may be parametrized by %%@
the observer's proper time $\tau$ such that 
\begin{subequations}
\begin{eqnarray}
t&=&\frac{1}{a}\sinh(a\tau),  \\
z&=&\frac{1}{a}\cosh(a\tau). 
\end{eqnarray}
\end{subequations}
The time- and the $z$-coordinates of the %%@
accelerating observer are $t'$ and $z'$, %%@
respectively, such that for infinitesimal changes %%@
of $t'$ and $z'$:
\begin{subequations}
\begin{eqnarray}
dt'=d\tau &=&\frac{dt-v\,dz}{\sqrt{1-v^2}}, \\
dz' &=& \frac{dz-v\,dt}{\sqrt{1-v^2}}, 
\end{eqnarray}
\end{subequations}
where
\begin{equation}
v:=\tanh(a\tau).
\end{equation}
is the speed of the accelerating observer with %%@
respect to the observer at rest. The flow of %%@
energy, or intensity of radiation to the negative %%@
$z$-direction through the $xy$-plane at rest with %%@
respect to the accelerating observer is
\begin{equation}
I=-T'^{03}=-\frac{\partial x'^0}{\partial %%@
x^\mu}\frac{\partial x'^3}{\partial %%@
x^\nu}T^{\mu\nu}=\frac{1}{2}\sinh(2a\tau)(T^{00}+%%@
T^{33}),
\end{equation}
and it follows from Eqs. (2.1)-(2.3) that
\begin{equation}
I=\frac{2}{3}\sinh(2a\tau)\rho.
\end{equation}
Hence we see that if we pick up a region with %%@
area $A$ from the $xy$-plane, the energy carried %%@
by radiation through that region during an %%@
infinitesimal proper time interval $d\tau$ is 
\begin{equation}
\delta Q = \frac{2}{3}A\rho\sinh(2a\tau)\,d\tau.
\end{equation}

\section{Change of Area}

In the previous Section we assumed that spacetime %%@
is flat, and filled with electromagnetic %%@
radiation in thermal equlibrium. Interaction %%@
between radiation and the geometry of spacetime, %%@
however, makes spacetime curved, and as a result %%@
there is a change in the area of the accelerating %%@
plane we considered in the previous Section. When %%@
we evaluate the change in the area of an %%@
accelerating plane in the course of time, we %%@
might use, of course, the so called Raychaudhuri %%@
equation \cite{neljas}, and that was indeed the equation on %%@
which Jacobson's investigations were based. %%@
Raychaudhuri equation, however, describes the %%@
behavior of {\it geodesics} of spacetime, and %%@
because an accelerating observer does not move %%@
along a spacetime geodesic, it is not quite clear %%@
how to apply that equation. Because of the %%@
conceptual problems involved in the Raychaudhuri %%@
equation when studying the geometric properties %%@
of accelerating planes, we shall instead apply %%@
the linear field approximation of Einstein's %%@
field equation. 

    The starting point of the linear field %%@
approximation is to write the spacetime metric %%@
as:
\begin{equation}
ds^2 = (\eta_{\mu\nu} + %%@
h_{\mu\nu})\,dx^\mu\,dx^\nu.
\end{equation}
In this equation, $x^\mu$'s are the flat %%@
Minkowski coordinates, and %%@
$\eta_{\mu\nu}:=diag(1,-1,-1,-1)$ is the flat %%@
Minkowski metric. The field $h_{\mu\nu}$ %%@
represents a small deviation from the flat %%@
spacetime geometry. The linear field %%@
approximation is therefore particularly useful %%@
for the investigation of the small changes in the %%@
spacetime geometry. In what follows, we shall %%@
assume that at the spacetime point %%@
$P=(t,x,y,z)=(0,0,0,1/a)$, where our accelerating %%@
observer is at rest with respect to the %%@
radiation, the field $h_{\mu\nu}$ and its first %%@
derivatives vanish. In other words, we shall %%@
assume that at the point $P$ spacetime metric is %%@
that of flat spacetime. When we move away from %%@
the point $P$, spacetime metric deviates slightly %%@
from its flat spacetime form. We shall also %%@
assume that the spacetime metric (3.1) remains %%@
unchanged when we interchange the spatial %%@
coordinates representing deviations from the %%@
point P. In other words, we shall assume that, %%@
when looked from the point $P$, spacetime %%@
geometry looks the same in all spatial %%@
directions. This requirement is motivated by the %%@
fact that the components of the energy momentum %%@
stress tensor $T^{\mu\nu}$ remain unchanged %%@
under the interchanges of the spatial coordinates %%@
of spacetime. It is reasonable to expect that the %%@
same invariance property is possessed also by the %%@
spacetime metric itself.

   When $h_{\mu\nu}$ satisfies the so called %%@
Hilbert gauge condition
\begin{equation}
\partial_\mu(h^{\mu\nu} - %%@
\frac{1}{2}\eta^{\mu\nu}h) = 0,
\end{equation}
the linear field approximation of Einstein's %%@
field equation may be written as:
\begin{equation}
\partial_\lambda\partial^\lambda(h^{\mu\nu} - %%@
\frac{1}{2}\eta^{\mu\nu}h) = -16\pi T^{\mu\nu}.
\end{equation}
In these equations, $h:=\eta^{\mu\nu}h_{\mu\nu}$, %%@
and the components of $T^{\mu\nu}$ are given by %%@
Eqs.(2.1)-(2.3). Since the electromagnetic %%@
radiation which fills spacetime is assumed to be in thermal %%@
equilibrium, and therefore homogeneous, we shall %%@
look for a solution, where the diagonal elements %%@
of the metric depend on the time $t$ only. It is %%@
easy to see that there is just one solution to %%@
Eqs.(3.2) and (3.3), which satisfies this %%@
requirement, together with the other two %%@
requirements given before. The solution in %%@
question is the one, where the spacetime metric %%@
(3.1) takes the form:
\begin{eqnarray}
ds^2 &=& (1 - 8\pi\rho t^2)\,dt^2 - %%@
\frac{32}{3}\pi\rho t(x\,dx + y\,dy + %%@
\tilde{z}\,d\tilde{z})\,dt \nonumber \\ & &-(1 + %%@
\frac{8}{3}\pi\rho t^2)(dx^2 + dy^2 + %%@
d\tilde{z}^2) - \frac{16}{3}\pi\rho(xy\,dx\,dy + %%@
x\tilde{z}\,dx\,d\tilde{z} + %%@
y\tilde{z}\,dy\,d\tilde{z}),
\end{eqnarray}
where we have defined:
\begin{equation}
\tilde{z} := z - \frac{1}{a}.
\end{equation}

    We are now prepared to calculate the change %%@
in the area of a considered region of the %%@
accelerating $xy$-plane, from the point of view %%@
of an observer at rest with respect to that %%@
plane. In general, the area of a region on the %%@
$xy$-plane is
\begin{equation}
A = \int_{\mathcal{A}}\sqrt{q}\,dx\,dy,
\end{equation}
where $q$ is the determinant of the metric %%@
induced on that plane, and $\mathcal{A}$ is the %%@
domain of integration. Change in the area during %%@
the course of time is therefore a result from %%@
changes both in the metric and the domain of %%@
integration. When a plane is in accelerating %%@
motion in curved spacetime, the points of the %%@
region under consideration move on the plane with %%@
respect to the coordinates on that plane (or, %%@
rather, the coordinates move with respect to the %%@
points), and the geometry of the plane itself %%@
changes. Both of these contributions must be %%@
taken into account, when we calculate the area %%@
change. 

   To begin with, we investigate the movements of %%@
the points on the plane. Consider the point %%@
$Q=(x,y)$ on the $xy$-plane, which has the point
\begin{equation}
\mathcal{O} := (\frac{1}{a}\sinh(a\tau),0,0, %%@
\frac{1}{a}\cosh(a\tau))
\end{equation}
of spacetime as its origin. Our aim is to %%@
calculate the coordinates $(x',y')$ of $Q$ after %%@
the point $\mathcal{O}$ has been transported to the point %%@
$\mathcal{O}'$, where $\tau$ has been replaced by %%@
$\tau + d\tau$. We shall assume that when %%@
$\tau=0$, the point $Q$ is at rest with respect %%@
to the coordinates $x$, $y$ and $z$. In other %%@
words, we shall assume that when $\tau=0$, the %%@
only non-zero component of the four-velocity %%@
$u^\mu$ of the point $Q$ is
\begin{equation}
u^0 = 1.
\end{equation}
To calculate the four-velocity of $Q$, when $\tau %%@
\neq 0$, we parallel transport the vector $u^\mu$ %%@
from the point $(0,x,y,1/a)$ to the point %%@
$(\frac{1}{a}\sinh(a\tau),x,y, %%@
\frac{1}{a}\cosh(a\tau))$. In infinitesimal %%@
parallel transport from point $x^\mu$ to the %%@
point $x^\mu + dx^\mu$ the change experienced by %%@
$u^\mu$ is, in general,
\begin{equation}
\delta u^\mu = %%@
-\Gamma^\mu_{\alpha\beta}u^\alpha\,dx^\beta,
\end{equation}
where $\Gamma^\mu_{\alpha\beta}$ is the %%@
Christoffel symbol. In the linear field %%@
approximation $\Gamma^\mu_{\alpha\beta}$ takes %%@
the form:
\begin{equation}
\Gamma^\mu_{\alpha\beta} = %%@
\frac{1}{2}\eta^{\mu\sigma}(\partial_\beta %%@
h_{\sigma\alpha} + \partial_\alpha %%@
h_{\beta\sigma} - \partial_\sigma %%@
h_{\alpha\beta}).
\end{equation}
Using Eqs.(3.4) and (3.9) we find that at the %%@
point $(\frac{1}{a}\sinh(a\tau),x,y,
\frac{1}{a}\cosh(a\tau))$ we have:
\begin{subequations}
\begin{eqnarray}
u^1 &=& -\frac{16}{3}\pi\rho x\tau,\\
u^2 &=& -\frac{16}{3}\pi\rho y\tau,
\end{eqnarray}
\end{subequations}
where we have kept the terms linear in $\tau$ %%@
only. When $\tau$ is changed to $\tau + d\tau$, %%@
the coordinates $(x,y)$ of the point $Q$ on the %%@
$xy$-plane transform to
\begin{subequations}
\begin{eqnarray}
x' &=& x + u^1\,d\tau = (1 - %%@
\frac{16}{3}\pi\rho\tau\,d\tau)x,\\
y' &=& y + u^2\,d\tau = (1 - %%@
\frac{16}{3}\pi\rho\tau\,d\tau)y.
\end{eqnarray}
\end{subequations}
In other words, we have just re-scaled the %%@
coordinates $x$ and $y$ by the factor $(1 - %%@
\frac{16}{3}\pi\rho\tau\,d\tau)$.

     It only remains to calculate the change in %%@
the metric. The metric induced on the $xy$-plane %%@
is
\begin{equation}
dL^2 = (1 + \frac{8}{3}\pi\rho t^2)(dx^2 + dy^2) %%@
- \frac{16}{3}\pi\rho xy\,dx\,dy.
\end{equation}
The change experienced by $\sqrt{q}$ when $\tau$ %%@
is changed to $\tau + d\tau$ is therefore:
\begin{equation}
\delta\sqrt{q} = \Big(\frac{\partial}{\partial %%@
t}\sqrt{q}\Big)\frac{dt}{d\tau}\,d\tau = %%@
\frac{16}{3}\pi\rho\tau\, d\tau,
\end{equation}
where we have, again, kept the terms linear in %%@
$\tau$ only. We have also neglected the terms %%@
non-linear in $h_{\mu\nu}$. The relationship %%@
between $t$ and $\tau$ is given by Eq.(2.6a). %%@
Combining Eqs.(3.6), (3.12) and (3.14) we find %%@
that during the proper time interval $d\tau$ the %%@
area $A$ of the region under consideration %%@
becomes to
\begin{equation}
A' = (1 + \frac{16}{3}\pi\rho\tau\,d\tau)(1 - %%@
\frac{16}{3}\pi\rho\tau\,d\tau)^2A.
\end{equation}
Because $d\tau$ is infinitesimal, the change %%@
experienced by $A$ is therefore:
\begin{equation}
dA = -\frac{16}{3}\pi A\rho\tau\,d\tau.
\end{equation}
As one can see, the plane shrinks, when radiation %%@
flows through the plane to the negative %%@
$z$-direction.




\section{Area and Entropy}

        We denoted in Eq.(2.11) the amount of %%@
energy carried by radiation through the %%@
$xy$-plane by $\delta Q$ for a very good reason: %%@
Energy comes through the $xy$-plane as {\it %%@
heat}. According to the first law of %%@
thermodynamics the change in the heat of a system %%@
may be written as:
\begin{equation}
\delta Q = dE + p\,dV,
\end{equation}
where $dE$ is the (infinitesimal) change in the %%@
total energy, and $dV$ in the volume of the %%@
system. It is easy to see that both terms on the %%@
right hand side of Eq.(4.1) are present on the %%@
right hand side of Eq.(2.11): The increase in the %%@
total energy in the spatial region "behind" the %%@
accelerating plane from the point of view of the %%@
accelerating observer during an infinitesimal %%@
proper time interval $d\tau$ is
\begin{equation}
dE = \frac{1}{2}A\rho\sinh(2a\tau)\,d\tau,
\end{equation}
and because the increase in the three-volume $V$ %%@
of space "behind" the accelerating plane is, from %%@
the point of view of the accelerating observer, 
\begin{equation}
dV = A\sinh(a\tau)\,d\tau,
\end{equation}
we find, using Eq.(2.3), that the "work term" is
\begin{equation}
p\,dV = \frac{1}{6}A\rho\sinh(2a\tau)\,d\tau.
\end{equation}
When put together, the terms $dE$ and $p\,dV$ %%@
give the right hand side of Eq.(2.11).

   After having convinced ourselves that the %%@
right hand side of Eq.(2.11) really gives the %%@
heat transported through the accelerating plane %%@
from the point of view of the accelerating %%@
observer, we may now turn to the relationship %%@
between area and entropy. Because between the %%@
infinitesimal changes in the heat $Q$ and the %%@
entropy $S$ of a system there is a relationship:
\begin{equation}
\delta Q = T\,dS,
\end{equation}
where $T$ is the absolute temperature of the %%@
system, and because it follows from Eq.(2.11) %%@
that, for very small $\tau$:
\begin{equation}
\delta Q = \frac{4}{3}A\rho a\tau\,d\tau,
\end{equation}
we find, using Eq.(3.16), that the entropy $dS$ %%@
carried by radiation through the accelerating %%@
plane is related to the change $dA$ in the area %%@
of that plane such that:
\begin{equation}
T\,dS = -\frac{a}{4\pi}\,dA.
\end{equation}

      It only remains to fix the absolute %%@
temperature $T$ of the electromagnetic radiation. %%@
At this point we turn to the Unruh effect. %%@
According to that effect an accelerating observer %%@
experiences himself to be immersed in a heat bath of %%@
thermal particles. The temperature of this heat %%@
bath is the Unruh temperature \cite{viides}
\begin{equation}
T_U := \frac{a}{2\pi}.
\end{equation}
Suppose that when $\tau=0$, spacetime is, from %%@
the point of view of the accelerating observer, %%@
filled with electromagnetic radiation in thermal %%@
equilibrium at the Unruh temperature $T_U$. When %%@
$\tau=0$, there is no net flow of heat in any %%@
direction in the accelerating observer's frame of %%@
reference. After a very short elapsed proper time %%@
$\tau$, however, there is a net heat flow to the %%@
negative $z$-direction in the observer's frame of %%@
reference, and the amount of heat transported %%@
through the $xy$-plane during a proper time %%@
interval $d\tau$ is given by Eq.(4.6). The %%@
temperature of this heat flow is still the Unruh %%@
temperature $T_U$, and we may substitute $T_U$ %%@
for $T$ in Eq.(4.7). When we perform this %%@
substitution, we get:
\begin{equation}
dS = -\frac{1}{2}\,dA.
\end{equation}
In other words, the entropy carried by radiation %%@
through the accelerating plane is, in natural %%@
units, exactly one half of the decrease in the %%@
area of that plane. This is the final result of %%@
this paper, and it holds whenever the temperature %%@
of the radiation is equal to the Unruh %%@
temperature of an observer at rest with respect %%@
to the accelerating plane. It is remarkable that %%@
we have obtained a simple linear dependence %%@
between area and entropy even when our %%@
accelerating two-plane is not a part of any %%@
horizon of spacetime.


    \section{Concluding Remarks}

    In this paper we have considered an explicit %%@
example of a process, where radiation flows %%@
through a spacelike two-surface in such a way %%@
that the entropy carried through that two-surface %%@
is proportional to the decrease  in its area even %%@
when that two-surface is not a part of any %%@
horizon of spacetime. More precisely, we %%@
considered the flow of electromagnetic radiation %%@
through an accelerating two-plane, and we found %%@
that if the temperature of the radiation is the %%@
Unruh temperature of an observer at rest with %%@
respect to the accelerating plane, the entropy %%@
carried by radiation through that plane is, in %%@
natural units, exactly one half of the decrease %%@
in the area of that plane. This result bears a %%@
close resemblance to the well-known results %%@
concerning the radiation emitted by the black %%@
hole, cosmological, and Rindler horizons of %%@
spacetime. According to those results the entropy %%@
carried by radiation out from a horizon is, in %%@
natural units, exactly one quarter of the %%@
decrease in its area, whereas we found the same %%@
relationship -with an important exception that %%@
the constant of proportionality is not one %%@
quarter but one half- between the entropy flow %%@
and the area decrease when the two-surface is not %%@
a horizon but an arbitrary accelerating %%@
two-plane.

   The derivation of our result should be rather %%@
uncontroversial. Indeed, we have used just the %%@
fundamental results of thermodynamics and %%@
classical general relativity, together with the %%@
basic properties of Unruh radiation. We first %%@
assumed that spacetime is filled with %%@
electromagnetic radiation in thermal equilibrium, %%@
and then calculated the heat flow through a %%@
uniformly accelerating two-plane to the direction %%@
opposite to the direction of the motion of that %%@
two-plane. When radiation flows through the %%@
two-plane, the plane shrinks, and the decrease in %%@
its area may be calculated by means of the %%@
linear field approximation. Using the first law of %%@
thermodynamics one obtains the relationship %%@
between the flow of entropy and the decrease in %%@
area. Finally, if one substitutes for the %%@
temperature the Unruh temperature measured by an %%@
observer at rest with respect to the accelerating %%@
two-plane, one obtains the core result of this %%@
paper.

   Our result seems to support the view that %%@
entropy is possessed not only by horizons but, in %%@
addition, by {\it all} accelerating, spacelike %%@
two-planes of spacetime. When radiation flows %%@
through a two-plane, the plane shrinks, and the %%@
entropy of  the two-plane is converted to the %%@
entropy of radiation. If one accepts the view %%@
that entropy proportional to area is possessed by %%@
all accelerating two-planes, there is only a %%@
small step left to the idea that entropy is %%@
possessed by {\it all} spacelike two-surfaces of %%@
spacetime, no matter whether the two-surface %%@
under consideration is a part of any horizon or %%@
not.

   An enigmatic feature of our analysis is that %%@
it suggests that the entropy of an accelerating %%@
two-plane is, in natural units, not one quarter %%@
as one might expect, but {\it one half} of its %%@
area. An explanation of this curious result may %%@
lie in the fact that the radiation process of a %%@
horizon, and the radiation process considered in %%@
this paper are completely different: When a %%@
horizon radiates, its geometry interacts with the %%@
quanta of its radiation by its {\it one side} %%@
only. When radiation flows through a two-plane, %%@
however, the quanta of radiation interact with %%@
the geometry of spacetime on the {\it both sides} %%@
of the two-plane. So it is possible that %%@
radiation picks up entropy from the both sides of %%@
the two-plane. This may explain why the numerical %%@
value given by our analysis for the entropy of an %%@
accelerating two-plane is exactly twice the %%@
entropy of a spacetime horizon with the same %%@
area.

\begin{acknowledgements}

I thank Jorma Louko and Ari Peltola for their constructive criticism
during the preparation of this paper.

\end{acknowledgements}

\begin{thebibliography}{99}

\bibitem{eka} T. Jacobson, Phys. Rev. Lett. %%@
\textbf{75} (1995) 1260. See also, T. Jacobson, %%@
Found. Phys. \textbf{33} (2003) 323.

\bibitem{toka} The idea of a close connection %%@
between general relativity and thermodynamics has %%@
been extensively studied, among others, by T. %%@
Padmanabhan, Gen. Rel. Grav. \textbf{34} (2002) %%@
2029, Gen. Rel. Grav. \textbf{35} (2003) 2097, %%@
Mod. Phys. Lett. \textbf{A17} (2002) 1147, %%@
Astrophys. Space Sci. \textbf{285} (2003) 407, %%@
Phys. Rep. \textbf{406} (2005) 49, gr-qc/0411044, %%@
gr-qc/0412068.

\bibitem{kolmas} J. M\"akel\"a and A. Peltola, %%@
gr-qc/0406032.

\bibitem{neljas} See, for example, R. M. Wald, {\it General
    Relativity} (University of Chicago Press, Chicago 1984)

\bibitem{viides} W. G. Unruh, Phys. Rev. %%@
D\textbf{14} (1976) 870.

\end{thebibliography}
  		  
  \end{document}
  

gr-qc0506087.extracted_bib
\begin{thebibliography}{99}

\bibitem{eka} T. Jacobson, Phys. Rev. Lett. %%@
\textbf{75} (1995) 1260. See also, T. Jacobson, %%@
Found. Phys. \textbf{33} (2003) 323.

\bibitem{toka} The idea of a close connection %%@
between general relativity and thermodynamics has %%@
been extensively studied, among others, by T. %%@
Padmanabhan, Gen. Rel. Grav. \textbf{34} (2002) %%@
2029, Gen. Rel. Grav. \textbf{35} (2003) 2097, %%@
Mod. Phys. Lett. \textbf{A17} (2002) 1147, %%@
Astrophys. Space Sci. \textbf{285} (2003) 407, %%@
Phys. Rep. \textbf{406} (2005) 49, gr-qc/0411044, %%@
gr-qc/0412068.

\bibitem{kolmas} J. M\"akel\"a and A. Peltola, %%@
gr-qc/0406032.

\bibitem{neljas} See, for example, R. M. Wald, {\it General
    Relativity} (University of Chicago Press, Chicago 1984)

\bibitem{viides} W. G. Unruh, Phys. Rev. %%@
D\textbf{14} (1976) 870.

\end{thebibliography}