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\begin{document}
\title{Modular Inclusion, the Hawking Temperature and Quantum Field Theory
in Curved Space-Time}
\author{Stephen J. Summers \thanks{Department of Mathematics, University of
Florida, Gainesville, Florida 32611, U.S.A.} \ \ and \ \ Rainer Verch
\thanks{II. Institut f\"ur Theoretische Physik, Universit\"at Hamburg,
22761 Hamburg, Germany}}
\date{\normalsize{April 1995}}
\maketitle
{\abstract
A recent result by Borchers connecting geometric modular
action, modular inclusion and the spectrum condition, is applied
in quantum field theory on spacetimes with a bifurcate
Killing horizon (these are generalizations of black hole
space-times, comprising the familiar black hole spacetime
models). Within this framework we give sufficient,
model-independent conditions ensuring that the temperature
of thermal equilibrium quantum states is the Hawking temperature. }
\\[24pt]
{\bf 1. Introduction}
\\[18pt]
\noindent
Since Hawking [11] suggested that quantum fields on
black hole space-times have a thermal distribution corresponding to a
certain temperature now come to be known as the Hawking temperature, there
has been a series of papers attempting to understand this in
a mathematically rigorous way (see e.g. [15], [10], [12], [9], [14]). In
this note we wish to contribute to this discussion by pointing out a rigorous,
model-independent and, we believe, clarifying connection between the
geometrical action of certain modular objects and the necessity of thermal
states having the Hawking temperature. We shall address this subject in the
context of algebraic quantum field theory on a class of space-times with a
bifurcate Killing horizon. Spacetimes with a bifurcate Killing horizon -- see
the next section for a definition -- may be viewed as generalizations of
black-hole spacetimes. Some important examples of such space-times are the
Minkowski, the Schwarzschild-Kruskal, the Schwarzschild-deSitter, and a
family of Kerr-Newman space-times (see [14]). A natural question of physical
interest is under which conditions thermal equilibrium states of quantum
fields propagating in such a background are forced to assume the Hawking
temperature. \par
We wish to emphasize that the Hawking temperature itself provides
information about the geometry of the background space-time, since it is in
one-one correspondence with the surface gravity of the bifurcate Killing
horizon of the underlying space-time, which itself is closely related to
the mass of the black hole [6]. This is mentioned because we are also
interested in the degree to which the modular objects associated with certain
states on (subalgebras of) elements of nets of $C^*$-algebras on a space-time
with a bifurcate Killing horizon contain information about the underlying
space-time geometry. In studying physically significant states from the point
of view of the geometric content of their modular objects, we are motivated by
a program outlined in [5] (see also [3], [4]) which hopes to
characterize such states by their geometric modular action. The tools we shall
be employing here are due to Borchers [1] and Wiesbrock [16], who established
the intimate interconnection of a weak form of geometric modular action, the
spectrum condition and the notion of modular inclusion (to be discussed below).
\par
In the next section we introduce the geometrical setting of globally
hyperbolic spacetimes with a bifurcate Killing horizon according to the
detailed exposition by Kay and Wald [14]. We define the central
notion of a symmetry-improving restriction (to the bifurcate Killing horizon)
of a net of local observable algebras over such a spacetime in
Section 3. In that same section we use this notion to prove
a theorem providing model-independent sufficient conditions
for a KMS (thermal equilibrium) state on a net of local observable algebras
over a spacetime with a bifurcate Killing horizon to have the
Hawking temperature.
We view these conditions as rather general -- in the final section we show
that the net of observable algebras associated with a linear Hermitean scalar
field on such a space-time admits the required symmetry-improving restriction;
moreover, there do exist states over these space-times which manifest the
geometric modular action required by our theorem (see Section 4).
\\[24pt]
%%%
{ \bf 2. Space-times With a Bifurcate Killing Horizon}
\\[18pt]
In this section we briefly summarize parts of the discussion in [14] on
spacetimes with a bifurcate Killing horizon. We refer to this reference for
further details and illustrations. \par
We recall the following notational conventions:
Let $( M,g)$ be a $C^{\infty}$-spacetime manifold $M$ with smooth Lorentzian
metric $g$, admitting a time-orientation. Then for ${\cal O} \subset M$,
$J^{\pm}(\cal O)$ denotes the set of all points in $M$ which can be reached
by future/past directed causal curves emanating from $\cal O$.
$I^{\pm}(\cal O)$ is defined analogously for timelike curves. In contrast to
[14], we do not use the abstract index notation for tensor fields in the
present Letter. \par
A space-time with a {\it bifurcate Killing horizon} is
characterized by a quintupel $(M,g,\tau_{t},\Sigma,{\bf h})$, where $(M,g)$
is a globally hyperbolic space-time, $\{\tau_{t}\}$ is a (nontrivial)
one-parameter group of isometries of $(M,g)$, $\Sigma$ is a two-dimensional
connected spacelike submanifold, contained in a spacelike Cauchy-surface of
$M$, which is left pointwise invariant under the action of $\{\tau_{t}\}$, i.e.
$\tau_{t}(p) = p$ for all $t \in {\bf R}$, $p \in \Sigma$, and {\bf h} is
the bifurcate Killing horizon, i.e. the union of two three-dimensional
$C^{\infty}$-manifolds in
$M$ formed by the lightlike geodesics emanating from $\Sigma$ with
$\Sigma$-orthogonal tangent vectors; it is
assumed that a choice of two continuous, linearly independent, lightlike,
future-directed, $\Sigma$-orthogonal
vector fields $\chi_{A}$, $\chi_{B}$ along $\Sigma$
can be made. \par
Let $\gamma_{Ap}$ and $\gamma_{Bp}$ be the maximal geodesics defined by
$\chi_{A}(p)$ and $\chi_{B}(p)$, respectively, for $p \in \Sigma$.
These are lightlike geodesics which are left invariant under the action of
$\{\tau_{t}\}$. This means, in particular, that the corresponding Killing field
$\xi$ is tangent to these geodesics, and geodesics starting at different
points $p \in \Sigma$ cannot cross. One defines the following pieces of
{\bf h}:
${\bf h}_{X}$ is the subset of {\bf h} generated by $\gamma_{Xp}$, $p \in
\Sigma$, and ${\bf h}_{X}^{\pm} \equiv {\bf h}_{X} \cap I^{\pm}(\Sigma)$ for
$X \in \{A,B\}$. Then we set
$$ {\bf h}_{A}^{R} \equiv {\bf h}_{A}^{+}\,,\ \ \
{\bf h}_{A}^{L} \equiv {\bf h}_{A}^{-} $$
and
$$ {\bf h}_{B}^{R} \equiv {\bf h}_{B}^{-}\,,\ \ \ {\bf h}_{B}^{L} \equiv
{\bf h}_{B}^{+}\, . $$
By convention, it will be assumed that $\xi$ is future oriented on
${\bf h}_{A}^{R}$. A point $q \in {\bf h}_{A}$ $({\bf h}_{B})$ can be
coordinatized by a pair $(U,p)$ (resp. $\,(V,p)\,$), where the point
$p \in \Sigma$ determines on which geodesic $q$ lies and the affine parameter
$U$ (resp. $V$) indicates where on the specified geodesic $q$ lies, so that we
have $\gamma_{Ap}(U) = q$ (resp. $\,\gamma_{Bp}(V) = q\,$). We assume that the
affine parameters are chosen such that $\gamma_{Ap}(U=0) = p$ (resp.
$\, \gamma_{Bp}(V=0) = p\,$). As we have already mentioned, $\xi$ is
tangent to the geodesics $\gamma_{Ap}$ and $\gamma_{Bp}$, whose tangent vector
fields will be denoted by $\chi_{Ap}$ and $\chi_{Bp}$, and it can be
shown that there exists a smooth function $f_{A}^{R}$, defined on
${\bf R}^{+} \times \Sigma$,\footnote{where ${\bf R}^+ = (0,\infty)$} positive
and strictly increasing with $U$, such that
\begin{equation}
\xi(U,p) \: = \: f_{A}^{R}(U,p)\,\chi_{Ap}(U) \, ,
\end{equation}
for points $(U,p) \in {\bf h}_{A}^{R} \equiv {\bf R}^{+} \times \Sigma$,
and the quantity
\begin{equation}
\kappa \: \equiv \: \xi\cdot{\rm ln}(f_{A}^{R}) \: > \: 0
\end{equation}
is a constant, i.e. independent of $U$ and $p$ (see [14]). $\kappa$ is
called the {\it surface gravity} of ${\bf h}_{A}$. Similar arguments
apply with functions $f_{A}^{L},f_{B}^{R}$ and $f_{B}^{L}$ for the other parts
of {\bf h}, yielding the same $\kappa$.%
\footnote{The $f^{L}_{...}$
are negative and so one has to take $-f^{L}_{...}$
in the argument of the logarithm in (2).}
This implies that the action of
$\{\tau_{t}\}$ on points of the bifurcate Killing horizon is of the
following form, which relates the action of the Killing flow on the horizon
to the action of the affine dilatations:\\\\
%%%
%%%
\noindent {\bf Lemma 1:} Under the stated assumptions, one has
$$ \tau_{t}(U,p) \: = \: (e^{\kappa t}U,p) \ \ \ \ \ {\rm and} \ \ \ \ \
\tau_{t}(V,p) \: = \: (e^{-\kappa t}V,p). $$
\nind {\it Proof:}
Choose arbitrary $U > 0$ and $p \in \Sigma$.
The lightlike geodesic $\gamma_{Ap}$ is left invariant
under the action of $\{ \tT \}$, thus by the properties
of the chosen coordinatization we have that
$\tT (U,p) = (\thT (U),p)$ with a group action
$\{ \thT \}$ on ${\bf R}^+$. Using this, one deduces from
(1) that
\begin{equation}
\frac{d}{dt} \thT (U) = f_A^R ( \thT(U),p) \,.
\end{equation}
>From (2) it follows that
$$ \kappa =
\frac{d}{dt} {\rm ln}(f_A^R (\thT (U),p)) =
\frac{d}{dt} \thT (U) \cdot
\frac{1}{f_A^R (\thT (U),p) }
\cdot \left .
\frac{\partial}{\partial x}
f_A^R (x,p) \right | _{ x = \thT (U) } \, .
$$
Hence it follows that
$f_A^R (\thT (U),p) = e^{\kappa t + C(U)}$,
and thus, with (3),
$$
\widehat{\tau}_{t}(U) \: = \: \frac{1}{\kappa}\,e^{\kappa t + C(U)} +
\tilde{C}(U) \, .
$$
By the group property $\widehat{\tau}_{t'}\,\widehat{\tau}_{t}(U) =
\widehat{\tau}_{t'+t}(U)$ for all $t',t \in {\bf R}$, one easily obtains
$\tilde{C}(U) = 0$. Then
$U = \widehat{\tau}_{0}(U) = \frac{1}{\kappa}e^{C(U)}$, implying
$\widehat{\tau}_{t}(U) = e^{\kappa t}U$. The argument for
${\bf h}_{A}^{L}$ $(U < 0)$ and for ${\bf h}_{B}$ (i.e.
$\tau_{t}(V,p) = (e^{-\kappa t}V,p)$) is similar. $\Box$ \\
Another natural group action on ${\bf h}_A$ is that of the affine
translations:
$$
\ell_a(U,p) = (U-a,p), \qquad \ell_a(V,p) = (V-a,p),
$$
which we shall see also play an important role in our considerations.
\\[24pt]
{\bf 3.
Symmetry-Improving Restrictions of Nets of von Neumann Algebras,
Modular Inclusion and the Hawking Temperature}
\\[18pt]
Let $\net$ be a net\footnote{strictly speaking,
an inclusion-preserving map will do} of
$C^*$-algebras over $M$ indexed by a basis $\Rs$ for the topology of $M$ and
contained in a $C^*$-algebra $\As$. The net $\net$ need not be covariant with
respect to any transformation group associated with the manifold $M$. We shall
denote by $\Ks$ a set of open subsets of ${\bf h}_A$ (resp. ${\bf h}_B$) which
is invariant under the Killing flow and the affine
translations. In other words, for every $\Gs \in \Ks$ and $t,a \in {\bf R}$,
there exist $\Gs_t,\Gs_a \in \Ks$ such that $\tau_t \Gs = \Gs_t$ and
$\ell_a \Gs = \Gs_a$. In addition, with $\Ks_X$ defined as the set
$\{{\bf h}_A^X \cap \Gs \mid \Gs \in \Ks \}$ (resp.
$\{{\bf h}_B^X \cap \Gs \mid \Gs \in \Ks \}$), we require that
$\Ks_X \subset \Ks$, for $X = L,R$. If $\nnet$ is a net of $C^*$-algebras
indexed by $\Ks$ and contained in the
$C^*$-algebra $\Ns$, we shall denote by $\Ns_X$ the sub-$C^*$-algebra of $\Ns$
generated by the algebras $\{\Ns(\Gs \cap {\bf h}_A^X) \mid \Gs \in \Ks \}$
(resp. $\{\Ns(\Gs \cap {\bf h}_B^X) \mid \Gs \in \Ks \}$), for $X = L,R$.
The central notion to be introduced here is that of a symmetry-improving
restriction. \\
\\
\noindent {\bf Definition:} Let $\nnet$ be a net of $C^*$-algebras
(contained in the $C^*$-algebra $\Ns$) over ${\bf h}_A$ (resp. ${\bf h}_B$)
indexed by a collection $\Ks$ as described above. Such a net $\nnet$ is said
to be a {\it symmetry-improving restriction} to ${\bf h}_A$ (resp. ${\bf h}_B$)
of the net $\net$ if: \par
(1) it is covariant with respect to the Killing flow and the affine
translations, i.e. if there exist (suitably) continuous\footnote{We shall
indicate later which continuity we require for our present purposes; however,
since we feel that this notion will have other uses besides the application
presented here, we shall not at this point further specify the continuity.}
group representations
$$
{\bf R} \ni t \mapsto \alpha_t \in \Aut\,\Ns \quad , \quad
{\bf R} \ni a \mapsto \lambda_a \in \Aut\,\Ns \quad ;
$$
such that for all $a,t \in {\bf R}$ and $\Gs \in \Ns$ one has
$$
\alpha_t(\Ns(\Gs)) = \Ns(\tau_t \Gs) \quad , \quad
\lambda_a(\Ns(\Gs)) = \Ns(\ell_a \Gs) \quad ;
$$
(2) one has the inclusion
$\Ks \subset \{ \Os \cap {\bf h}_A \mid \Os \in \Rs \}$ (resp.
\newline $\Ks \subset \{ \Os \cap {\bf h}_B \mid \Os \in \Rs \}$) and
for each $\Ks \ni \Gs = \Os \cap {\bf h}_A$ (resp.
$\Ks \ni \Gs = \Os \cap {\bf h}_B$), $\Os \in \Rs$, also
$\Ns(\Gs)$ is a subalgebra of $\As(\Os)$. \\
We shall show in the next section that there exist examples of
such symmetry-improving restrictions on general space-times with bifurcate
Killing horizons. The name symmetry-improving restriction is, of course,
motivated by the fact that the original net $\net$ need not be covariant
under the Killing flow and certainly not under the affine translations, which
only act on the horizons, whereas both conditions are required for $\nnet$.
One can well imagine that this concept could be
useful in any space-time which possesses a submanifold having a group of
isometries which are not the restriction of isometries of the larger manifold.
\\\\
{\bf Remark:} We note here that in the case where the spacetime $(M,g)$ is
Minkowski spacetime and the bifurcate Killing horizon {\bf h} is formed by the
horizons of adjacent and causally complementary wedge regions, each
Poincar\'e covariant net $\An$ (of von Neumann algebras) in the
vacuum representation possesses symmetry-improving (or rather,
symmetry-preserving, in this case) restrictions
to {\bf h}. A detailed investigation of such a situation has
been carried
out by Driessler [8].
\\\\
Beginning with [1] and continuing with [16], [17], [18], [5], [3], [2]
and [4], interesting connections between the spectrum condition and the
`geometric' action of modular objects have been established. We recall the
first result of this nature. \\
\noindent {\bf Theorem 2 [1]:} Let $\cal M$ be a von Neumann algebra acting
on some Hilbert space $\cal H$ and assume that $\Omega \in \cal H$ is cyclic
and separating for $\cal M$. Then let $\Delta$, J be the modular operator
and modular conjugation corresponding to $({\cal M} , \Omega )$. Let
${\sf U}(a)$, $a \in {\bf R}$, be a strongly continuous one-parameter group
with positive generator leaving $\Omega$ invariant. If, in addition,
${\sf U}(a){\cal M}{\sf U}(a)^* \subset \cal M$ for $a \geq 0$, then it
follows that
$$ \Delta^{it}{\sf U}(a)\Delta^{-it} \: = \: {\sf U}( e^{-2\pi t}a)
\,\,\, \rm{and} \,\,\, {\rm J}{\sf U}({\it a}){\rm J} \: = \: {\sf U}
(-{\it a}) \, ,$$
for all $t, a \in {\bf R}$; if, instead,
${\sf U}(a){\cal M}{\sf U}(a)^* \subset \cal M$ for $a \leq 0$, then it
follows that
$$ \Delta^{it}{\sf U}(a)\Delta^{-it} \: = \: {\sf U}( e^{2\pi t}a) \,\,\,
{\rm and} \,\,\, {\rm J}{\sf U}(a){\rm J} \: = \: {\sf U}(-a)\, ,$$
for all $t, a \in {\bf R}$.
\\[6pt]
\noindent (For the proof, see Theorem II.9 in [1].)\\
\\
\noindent {\bf Remark}: In [16] Wiesbrock has proven an interesting converse
to Borchers' result. He showed that if ${\sf U}(a)$ is a continuous unitary
group such that ${\sf U}(a){\cal M}{\sf U}(a)^* \subset \cal M$ for $a \geq 0$,
and if
$$ \Delta^{it}{\sf U}(a)\Delta^{-it} \: = \: {\sf U}( e^{-2\pi t}a)
\,\,\, \rm{and} \,\,\, J{\sf U}({\it a})J \: = \:
{\sf U}(-{\it a}) \, ,$$
for all $t, a \in {\bf R}$, then it follows that the generator of ${\sf U}(a)$
is positive. In [5] it was shown that in this converse it is not necessary to
assume the condition ${\rm J}{\sf U}(a){\rm J}
\: = \: {\sf U}(-a)$. \\
We shall use Theorem 2 to show that any state on a symmetry-improving
restriction $\nnet$ which is a ground state with respect to $\{\lambda_{a}\}$
\footnote{Here we use the term in the sense of algebraic quantum
field theory;
hence the states in question are invariant with respect to $\{\lambda_{a}\}$
and the generator of the corresponding strongly continuous unitary
representation of $\{\lambda_{a}\}$ in the GNS-representation space must have
nonnegative spectrum.}
and which is also a KMS-state with respect to $\{\alpha_{t}\}$ on $\Ns_R$
{\it must} have the (inverse) Hawking temperature $\beta = 2\pi / \kappa$.
\\\\
\noindent {\bf Theorem 3}: Let $\omega$ be a state on a net $\net$ admitting a
symmetry-improving restriction $\nnet$ to ${\bf h}_A$ (or ${\bf h}_B$)
with the additional properties: \par
(i) $\Gs_1 \neq \Gs_2$ entails $\Ns(\Gs_1) \neq \Ns(\Gs_2)$; \par
(ii) the restriction of $\omega$ to $\Ns$ is a ground state with respect
to the affine translations $\lambda_a$; \par
(iii) the restriction of $\omega$ to $\Ns_R$ (resp. $\Ns_L$) is a KMS-state
at inverse temperature $\beta \neq 0$ with respect to the Killing flow
$\alpha_t$. \par
Then $\beta = 2\pi/\kappa$ (resp. $\beta = -2\pi/\kappa$). \\
\\
{\it Proof:} 1.
Note that the collection $\Ks_R$ is invariant
under the induced action of the Killing flow and under nonpositive affine
translations. This is readily seen by noting that one has the decomposition
${\bf h}_A \cong {\bf R} \times \Sigma$ and
${\bf h}_A^R \cong {\bf R}^+ \times \Sigma$.
Let ${\rm pr}_1$ denote the corresponding projection onto the first component,
${\rm pr}_1(U,p) = U$, for $(U,p) \in {\bf h}_A$. In order to show that
$\tau_t({\bf h}_A^R \cap \Gs) \in \Ks_R$ for $\Gs \in \Ks$, it suffices to
verify that ${\rm pr}_1(\tau_t ({\bf h}_A^R \cap \Gs)) \subset \bf{R}^+$,
which is evident from Lemma 1. Similarly, it is evident that
${\rm pr}_1(\ell_a ({\bf h}_A^R \cap \Gs)) \subset {\bf R}^+$ whenever
$a\leq 0$. It may be seen in a like manner that $\Ks_L$ is invariant under the
induced action of the Killing flow and under nonnegative affine translations.
The $C^*$-algebras $\cal N(G)$ will be notationally identified with their
images under $\pi_{\omega}$, $\pi_{\omega}(\cal N(G))$, where
$\rep$ is the GNS-representation of $\cal N$ with respect to
the state $\omega$. $\sf{U}(\bf{R})$ will denote the continuous unitary group
implementing the affine translations in the GNS-representation
of $\omega$, whose existence is assured by assumption (ii). The
previous paragraph entails that the von Neumann algebra $\Ns_R ''$ is invariant
under the action of ${\rm Ad}{\sf U}(a)$,
$a\leq 0$. Moreover, $\Ns_R ''$ is
also invariant under the action of the Killing flow, which by assumption (iii)
is implemented on $\Ns_R ''$ by ${\rm Ad}\Delta^{it/\beta}$, for all
$t \in \bf{R}$, where $\Delta$ is the modular operator associated with
$(\Ns_R '',\Omega)$.\footnote{Note that (iii) implicitly assumes that the
vector $\Omega$ is cyclic and separating for $\Ns_R ''$.} Since assumption
(ii) implies that the generator of the group $\sf{U}(\bf{R})$ satisfies the
spectrum condition, one may appeal to Theorem 2 to conclude that
$$ \Delta^{it}{\sf U}(a)\Delta^{-it} \: = \: {\sf U}({e}^{2\pi t}a) \, ,$$
for all $t, a \in {\bf R}$. Similarly, when considering $\Ns_L ''$ these
arguments lead to the equality
$ \Delta^{it}{\sf U}(a)\Delta^{-it} = {\sf U}({e}^{-2\pi t}a)$. \par
2. For an arbitrary $\Gs \in \Ks_R$ one therefore has
\begin{eqnarray*}
\Ns((\tau_t \circ \ell_a) \Gs) &=& (\alpha_t \circ \lambda_a) \Ns(\Gs) \\
\quad &=& \Delta^{it/\beta}{\sf U}(a) \Ns(\Gs){\sf U}(a)^* \Delta^{-it/\beta}\\
\quad &=&{\sf U}(e^{2\pi t/\beta} a)\Delta^{it/\beta} \Ns(\Gs)\Delta^{-it/\beta}
{\sf U}(e^{2\pi t/\beta} a)^* \\
\quad &=& (\lambda_{e^{2\pi t/\beta} a} \circ \alpha_t) \Ns(\Gs) \\
\quad &=& \Ns((\ell_{e^{2\pi t/\beta} a} \circ \tau_t) \Gs) \quad .
\end{eqnarray*}
In light of assumption (i), this entails
$$
(\tau_t \circ \ell_a) \Gs = (\ell_{e^{2\pi t/\beta} a} \circ \tau_t) \Gs,
$$
for all $\Gs \in \Ks_R$. Applying the projection $\rm{pr}_1$ to both sides
of this equation and using the fact that both $\tau_t$ and $\ell_a$ act only
on the first component, one finds
$$
e^{\kappa t}({\rm pr}_1 \Gs - a)
= e^{\kappa t}({\rm pr}_1 \Gs) - e^{2\pi t/\beta}
a,
$$
for all $t \in \bf{R}$, $a \leq 0$, and all $\Gs \in \Ks_R$. One concludes
that $\beta = 2\pi/\kappa$. If one had considered $\Ns_L$, one would have
concluded $\beta = -2\pi/\kappa$. $\Box$ \\
\noindent {\bf Remark}:
Suppose that we assume in (ii) of Theorem 3 only that
$\omega$ induces a $\lambda_a$-invariant state on
$\cal N$. Then the automorphisms
\begin{equation}
\alpha_t \circ \lambda_a \circ \alpha_t^{-1}
\circ \lambda_{e^{\kappa t}a}^{-1}
\end{equation}
are, in the GNS-representation of $\omega | \cal N$,
implemented by unitaries $V_{t,a}$ which leave the
GNS-vector $\Omega$ invariant, and which are inner
automorphisms of the net, $V_{t,a}{\cal N(G)}
V_{t,a}^{-1} = \cal N(G)$ for all ${\cal G} \in \cal K$,
$t,a \in {\bf R}$. If these automorphisms are trivial
and if $\omega |{\cal N}_R$ is already at inverse
temperature $\beta =2\pi / \kappa$ with respect to
the Killing flow, then using Wiesbrock's converse
(extended in [5]) to Theorem 2, one concludes that
$\omega | \cal N$ is actually a ground state for the
affine translations. We point out that in the case that the net $\nnet$ is
generated by a quantum field $\Phi$ over {\bf h}, more precisely by an
injective, operator valued distribution
${\cal T} \owns f \mapsto \Phi(f)$, with a sufficiently
large class $\cal T$ of testfunctions on {\bf h}, the automorphisms (4) are, in
fact, trivial. \\
One therefore sees that the Hawking temperature emerges quite naturally
in a model-independent manner
\footnote{In particular, one sees that the
statement and proof of the theorem do not rely upon using the special
structure of free quantum fields, as do most of the rigorous discussions of the
Hawking temperature in the literature.}
on space-times with a bifurcate Killing horizon -- if the states on the
horizon are ``hot'', then they are at the Hawking temperature. However, what
is necessarily a model-dependent question is how large the region of dependence
of the horizon is, i.e.\
how large the algebra $\Ns_R$ is in $\As$. This
question is, in principle, of physical interest, since one wishes to know where
one may observe the temperature of the horizon. \par
The first paper [15] to connect the Hawking temperature with modular
objects can be seen in hindsight to have posited the existence of something
which looks very much like a symmetry-improving restriction (though in the
setting of Wightman fields) to a horizon. However, we feel that Theorem 3
more clearly isolates the structures required. \par
Note that the situation adressed in Theorem 3 is analogous to the
Rindler-Fulling-scenario (see, e.g., [12]),
where the Minkowski vacuum, which
is a ground state with respect to lightlike affine translations, restricts to
a thermal equilibrium state on the algebra of local observables localized in
the right Rindler wedge with respect to the Lorentz boosts leaving this
region invariant. In fact, this scenario is simply an {\it example} of a
space-time with a bifurcate Killing horizon where the assumptions of our
theorem hold. \par
In their fundamental work [14],
Kay and Wald have studied thermal
and uniqueness properties
of Killing-flow invariant,
Hadamard states of the linear Klein-Gordon field
on manifolds with a bifurcate Killing horizon.
In the next section we illustrate that their setting provides
an example of a net of local algebras over a spacetime
with a bifurcate Killing horizon admitting symmetry-improving restrictions
to ${\bf h}_A$ (or ${\bf h}_B$) to which our Theorem 3 applies.
It is, in particular, worth
noting that assumptions (ii) and (iii) have been shown in
[14] to follow from the
Hadamard condition and Killing-flow
invariance for quasifree states of the Klein-Gordon field on a spacetime
with a bifurcate Killing horizon; see also Section 4 below. \par
A further result of relevance here is that of Haag, Narnhofer and Stein
[10], who proved that if a (linear) quantum field theory on a
Lorentzian manifold
with timelike Killing vector field and horizons satisfying certain conditions
admits a KMS-state with respect to the Killing flow and the state satisfies
their conditions of local definiteness and stability on the horizon, then
the temperature of the said state must equal the Hawking temperature. In
[9] the collapse of a star into a black hole was treated dynamically, and
it was shown that for a linear quantum field there is radiation at large times
at the Hawking temperature.
\\[24pt]
%%
{\bf 4. Example:
The Weyl Algebra of the Linear Hermitean Scalar Field in a
Space-time with a Bifurcate Killing Horizon} \\[18pt]
%%
We next provide an example which illustrates the notions and satisfies
the assumptions of the previous section by considering a net of local algebras
corresponding to the linear Hermitean scalar field, following Dimock [7] and
Kay and Wald [14], and showing that it does admit a symmetry-improving
restriction. The relevant field equation on $(M,g)$ is the Klein-Gordon
equation:
\begin{equation}
(\Box_g + m^{2})\varphi \: = \: 0 \, ,
\end{equation}
for $m \geq 0$.
Here, $\Box_g$ is the D'Alembertian with respect to the Lorentzian
metric $g$ (which equals $\nabla^{\mu}\nabla_{\mu}$ in
index notation with $\nabla$ = covariant derivative of $g$).
As explained in the ``Note added in proof" in [14], it is
necessary to consider special spaces of solutions of (5), since we wish to view
certain Weyl algebras associated with symplectic spaces formed by
characteristic data of (5) on the bifurcate Killing horizon as subalgebras of
the Weyl algebra over the symplectic space of solutions of (5) whose
restrictions to Cauchy surfaces have compact support.
Let $\cal C$ be a Cauchy surface for $(M,g)$ and let, for $C^{1}$-functions
$\psi$ on $M$, $\rho_{0}\psi \equiv \psi | \cal C$ and
$\rho_{1}\psi \equiv n\cdot\psi| \cal C$, with $n$ denoting the
future-directed unit-normal field of $\cal C$. We define $S$ as the space
of all {\it real-valued} $C^{2}$-solutions $\varphi$ of (5) such that
their Cauchy-data
$\rho_{0}\varphi $ and
$\rho_{1}\varphi $ are of compact support on any Cauchy surface
$\cal C$. The set $S$ will be endowed
with the symplectic form
$$
\sigma(\varphi,\psi) \: \equiv \: \int_{\cal C}(\varphi\,(n\cdot\psi) -
\psi\,(n\cdot\varphi))\,d\eta_{\cal C} \, ,
$$
where $d\eta_{\cal C}$ denotes the induced measure on $\cal C$. That
$\sigma$ is indeed a symplectic form and independent of $\cal C$ follows
from standard theorems on existence and uniqueness of initial-value
solutions of (5) in globally hyperbolic space-times (cf.\ [7]) and from Green's
formula.
We next introduce some symplectic subspaces of $(S,\sigma)$ (following the
``Note added in proof'' of [14]).
Let $S_{A}$ consist of all solutions $\varphi$ in $S$ such that there is
a function $f \in C_{0}^{\infty}({\bf h}_{A})$ so that the characteristic
data $\varphi | {\bf h}_{A}$ of $\varphi$ on ${\bf h}_{A}$ have the form
\begin{equation}
(\varphi|{\bf h}_{A})(U,p) \: = \: U^{5}\frac{\partial^{5}}{\partial U^{5}}
f(U,p) \, ,
\end{equation}
for all $U \in {\bf R}$ and all $p \in \Sigma$ (by the results of the
``Note added in proof" in
[14], one obtains that equation (6) indeed implies
$\varphi \in S$). Moreover, we shall say that $\varphi$ is in the set
$S_{A}^{R}$ if the function $f$ in (6) lies in
$C_{0}^{\infty}({\bf h}_{A}^{R})$.
The subspaces $S_{B}$ and $S_{X}^{Y}$ for $X = A,B$ and $Y = R,L$ are defined
analogously. The symplectic form $\sigma(\varphi,\psi)$ for elements
$\varphi,\psi \in S_{A}$ takes the form
$$
\sigma(\varphi,\psi) \: = \: \int_{{\bf h}_{A}} \left (
\tilde{\varphi}(U,p)\,\frac{\partial}{\partial U}\tilde{\psi}(U,p) -
\tilde{\psi}(U,p)\,\frac{\partial}{\partial U} \tilde{\varphi}(U,p)
\right ) \,dU\,d\eta_{\Sigma}(p) \, ,
$$
where $\tilde{\varphi},\tilde{\psi}$ denote the restrictions of $\varphi,
\psi$ to ${\bf h}_{A}$ in the coordinatization of ${\bf h}_{A}$ which we
have chosen; note that we shall henceforth maintain this notation. The
expression for $\sigma(\varphi,\psi)$ when $\varphi,\psi \in S_{B}$ is
analogous. Therefore one can show that $S_{A},S_{B}$ and $S_{X}^{Y}$
$(X = A,B$; $Y = R,L)$ are symplectic subspaces of $(S,\sigma)$ (see
[14] for further details). If we write
$$
T_{t}\varphi \: \equiv \: \varphi \circ \tau_{-t} \, ,
$$
then $\{T_{t}\}$ is a symplectomorphism group on $(S,\sigma)$. It is also
clear from Lemma 1 that the action of $\{T_{t}\}$ on $S_{X}^{Y}$
$(X = A,B$; $Y=R,L)$ and on $S_{A}$ and $S_{B}$ leaves these symplectic
subspaces of $(S,\sigma)$ invariant.
%Also, one can define on $S_{A}$ (and
%likewise on $S_{B}$) an antisymplectic involution $I$ by
%$$
%\widetilde{(I\varphi)}(U,p) \: \equiv \: \tilde{\varphi}(-U,p)\,;
%$$
%note that on $S_{A}$, and also on $S_{B}$, $I$ and $\{T_{t}\}$ commute,
%i.e. $T_{t} \circ I \: = \: I \circ T_{t}$.
%Notice furthermore that $I$ maps $S_{X}^{R}$ onto $S_{X}^{L}$, for $X = A,B$.
Another group of symplectomorphisms on $S_{A}$, and on $S_{B}$, is given by the
{\it affine translations},
$$
\widetilde{(\Lambda_{a}\varphi)}(U,p) \: \equiv \: \tilde{\varphi}(U+a,p) \, ,
$$
for $a \in {\bf R}$. Observe that $S_{X}^{Y}$ $(X = A,B$; $Y= R,L)$ are not
left invariant under the action of $\{\Lambda_{a}\}$, but are left
{\it half-sided invariant}:
\begin{equation}
\Lambda_{a}\varphi \in S_{A}^{R}\ \ \ {\rm for}\ \ \ \varphi \in S_{A}^{R}\,,
a \in {\bf R}^{-}
\end{equation}
$$ \Lambda_{a}\varphi \in S_{A}^{L}\ \ \ {\rm for}\ \ \ \varphi \in
S_{A}^{L}\,,\ a \in {\bf R}^{+}\,,\ {\rm etc.}$$
By $\cal A$, ${\cal A}_{A}$, ${\cal A}_{B}$, ${\cal A}_{X}^{Y}$ we shall
denote the Weyl algebras corresponding to the symplectic spaces
$(S,\sigma)$, $(S_{A},\sigma|{S_{A}})$, $(S_{B},\sigma|{S_{B}})$,
$(S_{X}^{Y},\sigma|{S_{X}^{Y}})$, and by $\alpha_{t}$, % $\cal I$,
$\lambda_{a}$ the induced actions of $\tau_{t}$, % $I$,
$\Lambda_{a}$ on the appropriate Weyl algebras.
\par
It is known (cf.\ [7] and references quoted there)
that there are unique advanced \linebreak (+)/retarded(--)
distributional fundamental solutions of the Klein-Gordon
equation (5), that is, continuous linear operators
$$
(E')^{\pm} : {\cal E}'({M}) \longrightarrow
{\cal D}'( M) $$
with the property that $(\Box_g + m^2)(E')^{\pm}u =
u = (E')^{\pm}(\Box_g + m^2)u$ for all
$u \in {\cal E}'(M)$ (to be understood in the sense
of distributions), and
${\rm supp}((E')^{\pm}u) \subset J^{\mp}({\rm supp}\,u)$%
\footnote{This is in contrast to
${\rm supp}(E^{\pm}f) \subset J^{\pm}({\rm supp}f)$ for
smooth functions $f$, where $E^{\pm}$ are the fundamental
solutions defined on smooth test functions, which is due
to the way the smooth test functions are embedded in the
distributions.}. Their difference gives the distributional
propagator $E' := (E')^+ - (E')^-$ of the Klein-Gordon
equation. Generalizing the the proofs presented in
[7] for smooth functions, one can show that, whenever
$\varphi$ is in $S$, with data $\rho_0\varphi,\rho_1\varphi$
on some Cauchy-surface $\cal C$, there exists for
every open neighborhood $\cal O$ of
\newline ${\rm supp}\,\rho_0\varphi \cup {\rm supp}\,\rho_1\varphi$
a continuous function $u$ on $M$ with ${\rm supp}\,u \subset \cal O$
and $E'u = \varphi$. We define for each open, relatively
compact subset $\cal O$ of $M$ the set $S(\cal O)$
as consisting of all $\varphi \in S$ such that there is
$u \in C^0(M)$ with $E'u = \varphi$ and ${\rm supp}\,u \subset \cal O$.
Then, defining $\cal A(O)$ as the $C^*$-subalgebra of
$\cal A$ generated by Weyl-operators $W(\varphi)$ with
$\varphi \in S(\cal O)$, one proves by an appropriate
generalization of the methods given in [7] that the family
of $C^*$-algebras $\Os \mapsto \As(\Os)$, as $\Os$ ranges through the set
$\Rs$ of open, relatively compact
subsets of $M$, is a net of local algebras. (The local net of
the Klein-Gordon field.) \par
We shall need the following lemma. \\
\noindent {\bf Lemma 4:} Let $\varphi \in S_A$, let $E'$
denote the distributional propagator
of the Klein-Gordon equation (5), and let $\overline{\Gs}$ be the support
of the restriction of $\varphi$ to ${\bf h}_A$.
Given any neighborhood $\Os$ of $\overline{\Gs}$ in $M$, there exists a
continuous function $u$ with support in $\Os$ such that $\varphi = E'u$. The
same result is true if the subscripts $A$ are everywhere replaced by $B$. \\
\noindent {\it Proof:} For any $p,q \in M$ with $p \in I^+(q)$, let
$D_{p,q}$ denote the interior of the set $J^-(p) \cap J^+(q)$. Since the
space-time $M$ is assumed to be globally hyperbolic, the set of all
such ``double cones'' $D_{p,q}$ forms a basis for the topology of $M$.
Let $\Os$ be a neighborhood of $\overline{\Gs}$ in $M$. Since $\overline{\Gs}$
is compact, it possesses a finite cover $\{D^{(k)}\}_{k=1,\ldots,n}$
of double cones $D^{(k)} = D_{p_k,q_k}$ satisfying
$D^{(k)} \subset \Os$, $k=1,\ldots,n$. The sets
$L^{(k)} \equiv D^{(k)} \cap {\bf h}_A$ form a finite open cover of
$\overline{\Gs}$ in the submanifold ${\bf h}_A$. For this open cover, there
exists a smooth decomposition of $1$; denote this by the collection
$\{ \chi^{(k)} \}_{k=1,\ldots,n}$. Then one has
$\varphi = \sum_{k=1}^n \varphi_k$, with
$\varphi_k \in S_A$,
$$
(\varphi_k | {\bf h}_A)(U,p) \: = \:
U^{5}\frac{\partial^{5}}{\partial U^{5}} (\chi_k f)(U,p) \, ,
$$
whenever
$$
(\varphi | {\bf h}_A)(U,p) \: = \:
U^{5}\frac{\partial^{5}}{\partial U^{5}} f(U,p) \, .
$$
Of course, the restriction $\varphi_k | {\bf h}_A$ has compact support
contained in $L^{(k)}$. It remains to be shown that for each $\varphi_k$
there exists a continuous $u_k$ with support in $D^{(k)}$ such that
$\varphi_k = E'u_k$. \par
For a given $k = 1,\ldots,n$, choose a Cauchy surface $C_k$
such that $\overline{L^{(k)}} \subset I^+(C_k)$.
The submanifold $M_k$, defined to be the interior
of the set $J^-(p_k) \cup J^-(C_k)$, is also a globally hyperbolic space-time
with respect to the restricted metric, so there exists a Cauchy surface
$C$ such that the (compact) closure of $L^{(k)}$ is contained in the
interior (with respect to $M_k$) of $J^-(C)$ (which coincides with the
interior with respect to $M$ of $J^-(C)$). $C$ is also a
Cauchy surface for $M$. Since
the intersection of the support of $\varphi_k$ with $C$ is compact and
contained in $D^{(k)}$, there exists a continuous $u_k$ with support in
$D^{(k)}$ satisfying $\varphi_k = E'u_k$. $\Box$ \\
This entails that if $\Ks$ denotes the set of open, relatively compact
subsets of ${\bf h}_A$ (or ${\bf h}_B$), and
if for each ${\cal G} \in \cal K$, $\cal N(G)$
is defined as the $C^*$-subalgebra of ${\cal A}_A$ (or
${\cal A}_B$) generated by all Weyl-operators $W(\varphi)$
with $\varphi \in S_A$ (or $S_B$) and
${\rm supp}(\varphi | {\bf h}_A) \subset \cal G$
(resp., ${\rm supp}(\varphi | {\bf h}_B) \subset \cal G$),
then the net $\nnet$ is a restriction
of the local net $\net$ to ${\bf h}_A$ (or ${\bf h}_B$).
Hence states on the net
$\net$ may be restricted to the net $\nnet$. Note that in
this case one has $\Ns = \As_A$ (or $\Ns = \As_B$), and $\Ns_X = \As_A ^X$ (or
$\Ns_X = \As_B ^X$), for $X = R,L$. Note also that the collection $\Ks$ is
$\tau_t$- and $\ell_a$-invariant. In addition, it is evident from the
construction that $\Ns(\Gs_1) \neq \Ns(\Gs_2)$ whenever $\Gs_1 \neq \Gs_2$.
We therefore have the immediate corollary: \\
\noindent {\bf Corollary 5:} The above-constructed net of $C^*$-algebras
of the Klein-Gordon field
admits a symmetry-improving restriction to the horizon ${\bf h}_A$ (and
to ${\bf h}_B$) which satisfies the condition that
$\Gs_1 \neq \Gs_2$ implies $\Ns(\Gs_1) \neq \Ns(\Gs_2)$. \\
We now come to the question whether states on this symmetry-improving
restriction which satisfy the assumptions of Theorem 3 exist at all. In fact,
we may take
\begin{equation}
w_{\omega}(\varphi,\psi) \: =
\: - \frac{1}{\pi} \lim_{\epsilon \to 0+}
\int \frac{\widetilde{\varphi}(U_{1},p)\,\widetilde{\psi}(U_{2},p)}
{(U_{1} - U_{2} - i\epsilon)^{2}}\,
d\eta_{\Sigma}(p) \, dU_{1} \, dU_{2}
\end{equation}
as the two-point function of a quasifree state $\omega$ on ${\cal A}_{A}$
(or $\As_B$), and this state has all the desired properties (cf.\ [14])
on the net $\nnet$.
Note that Kay and
Wald [14] proved that every quasifree Hadamard state which is invariant under
the action of $\{\alpha_{t}\}$ on $\cal A$ must restrict to a quasifree state
on ${\cal A}_{A}$ with the two-point function given in (8). (In fact, with
the new argument of Kay [13], the assumption that the state be quasifree may
be dropped). Hence, any extension of this state to the net $\net$ (which
always exists, since $\Ns$ is a subalgebra of $\As$) yields an example
satisfying the assumptions of Theorem 3. \\
\nind {\bf Corollary 6:} On every globally hyperbolic space-time with
bifurcate Killing horizon there exist a state $\omega$ and a net $\net$
satisfying the hypotheses of Theorem 3. \\
A different question, however, is if states on the larger algebra $\cal A$
exist such that their restrictions to more than one horizon algebra
satisfy the assumptions of Theorem 3. It may well be that this is not possible
if the space-time contains more than one bifurcate Killing horizon which have
different surface gravities. And, in fact, as in Section 6.3 in [14], if one
considers as an example the Schwarzschild-deSitter space-time, where there is
a pair of neighboring bifurcate Killing horizons with unequal surface
gravities, then it is clear from Theorem 3 that there cannot exist a state on
any net, whose causal support contains both horizons and has
symmetry-improving restrictions to both bifurcate Killing horizons, and which
is simultaneously a ground state for the affine
translations and a KMS-state for the Killing flow on these restrictions.
There are other such examples, and it is obvious how
a theorem analogous to Theorem 6.5 in [14] can be
formulated in our setting. \\
\noindent {\bf Acknowledgements}:
{\footnotesize SJS wishes to thank the
Sonderforschungsbereich `Differential Geometry and Quantum Physics' at the
three Berlin universities for invitations in the Summer of 1992 and 1993 as
well as the Second Institute for Theoretical Physics at the University of
Hamburg for invitations in the Summer of 1993 and 1994. These
invitations and their financial support made this collaboration possible.
In addition, part of this work was completed while SJS was the Gauss Professor
at the University of G\"ottingen in 1994. For that opportunity SJS wishes to
thank Prof. H.-J. Borchers and the Akademie der Wissenschaften zu G\"ottingen.
RV gratefully acknowledges financial support by the DFG.}\\
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\end{document}