The entropy of a quantum-statistical system which is classically approximated
by a general stationary eternal black hole is studied by means of a
microcanonical functional integral. This approach opens the possibility of
including explicitly the internal degrees of freedom of a physical black hole
in path integral descriptions of its thermodynamical properties. If the
functional integral is interpreted as the density of states of the system, the
corresponding entropy equals ${cal S} = A_H/4 - A_H/4 =0$ in the semiclassical
approximation, where $A_H$ is the area of the black hole horizon. The
functional integral reflects the properties of a pure state.Comment: To appear in the proceedings of the Sixth Canadian Conference on
  General Relativiy and Relativistic Astrophysics, 7 pages, Late

Martinez, Erik A.

English

arXiv

The entropy of a quantum-statistical system which is classically approximated by a general stationary eternal black hole is studied by means of a microcanonical functional integral. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals {cal S} = A_H/4 - A_H/4 =0 in the semiclassical approximation, where A_H is the area of the black hole horizon. The functional integral reflects the properties of a pure state

Martínez, E A

CERN Document Server

gr-qc/9508057
Alberta-Thy-18-95
Entropy of eternal black holes
1
Erik A. Martinez
2
Theoretical Physics Institute, University of Alberta
Edmonton, Alberta, Canada T6G 2J1
martinez@phys.ualberta.ca
Abstract
The entropy of a quantum-statistical system which is classically
approximated by a general stationary eternal black hole is studied by
means of a microcanonical functional integral. This approach opens the
possibility of including explicitly the internal degrees of freedom of a
physical black hole in path integral descriptions of its thermodynamical
properties. If the functional integral is interpreted as the density of
states of the system, the corresponding entropy equals calS = A
H
=4 
A
H
=4 = 0 in the semiclassical approximation, where A
H
is the area of
the black hole horizon. The functional integral reects the properties
of a pure state.
1 Introduction
The relationship between the entropy of a physical black hole and its in-
ternal degrees of freedom remains a subject of active research. A natural
question to ask in this regard is: can these degrees of freedom be eectively
included in a functional integral description of black hole entropy? In an
attempt to give an armative answer to this question, we investigate in
this paper a microcanonical functional integral when applied to a quantum
self-gravitating statistical system that includes spacetimes whose topology
and boundary conditions coincide with the ones of (either distorted or Kerr-
Newman) eternal black holes.
1
To appear in the proceedings of the Sixth Canadian Conference on General Relativity
and Relativistic Astrophysics, Fredericton, Canada, May 25-27, 1995
2
Address after October 1st, 1995: Center for Gravitational Physics and Geometry,
Department of Physics, The Pennsylvania State University, University Park, PA 16802-
6300, USA
1
A proposal for the density of states of a gravitational system dened in
terms of a microcanonical functional integral has been suggested recently
in Ref. [1]. This integral is dened as a formal sum over Lorentzian ge-
ometries. The black hole density of states is obtained from this functional
integral when the latter is approximated semiclassically by using a complex
metric whose boundary data at its single boundary surface coincide with
the boundary data of a Lorentzian, stationary black hole. The density of
states dened accordingly equals the exponential of one fourth of the area
of the black hole horizon. This proposal opens the possibility of determin-
ing the thermodynamical properties of black hole systems starting from a
sum over real Lorentzian geometries. However, several problems remain in
this approach. First, a spacelike hypersurface  that describes the initial
data of a Lorentzian black hole has to cross necessarily the event horizon
and eventually intersect the interior singularity. Second, a Lorentzian, sta-
tionary black hole is not a extremum of the microcanonical action for a
spacetime region with a single timelike boundary surface. This implies that
the black hole density of states whose boundary data correspond to the
ones of a Lorentzian, stationary black hole cannot be approximated semi-
classically by using the same Lorentzian metric that motivates its boundary
conditions. It is therefore necessary to use complex metrics to evaluate the
Lorentzian functional integral in a steepest descents approximation. This
procedure yields the correct result for the entropy but conceals its origin:
the interior of the Lorentzian black hole literally disappears by virtue of
this procedure, leaving eectively only a periodically identied Euclidean
version of the \right" wedge region of a Kruskal diagram. The properties of
the black hole interior become encoded in a set of conditions at the so-called
\bolt" of the complex geometry. In this approach the statistical origin of
entropy and its relationship to the internal degrees of freedom of a black
hole remain obscure.
The problems mentioned above and the role of internal degrees of free-
dom in functional integral descriptions of black hole thermodynamics can
be addressed by explicitly considering the eternal version of a black hole [2].
The excitations of the physical black hole can be associated with the defor-
mations of an initial global Cauchy surface  of the eternal black hole. In
general, the spatial slices  that foliate an eternal black hole are (deformed)
Einstein-Rosen bridges with wormhole topology R
1
 S
2
. The spacetime is
composed of two wedges M
+
and M
 
located in the right and left sectors
of a Kruskal diagram. Internal and external degrees of freedom of the black
hole can be easily identied in this approach since the hypersurfaces  are
2
naturally divided in two parts 
+
and 
 
by a bifurcation two-surface S
0
where the lapse function N vanishes. While the \external" degrees of free-
dom of the original black hole are naturally given by the initial data at 
+
,
its \internal" degrees of freedom can be identied with initial data dened
at 
 
[3].
2 Microcanonical action and functional integral
Consider a spacetime region whose three-dimensional timelike boundary sur-
face B consists of two disconnected parts B
+
and B
 
. The microcanonical
action S
m
is the action appropriate to a variational principle in which the
xed boundary conditions at the timelike boundaries B

are not the space-
time three-geometry but the surface energy density ", surface momentum
density j
a
, and boundary two-metric 
ab
[1, 4]. The covariant form of the mi-
crocanonical action for a general spacetime whose timelike surfaces B
+
and
B
 
are located in the regions M
+
and M
 
respectively has been presented
in Ref. [2]. Its Hamiltonian form is easily obtained under a 3+ 1 spacetime
split by recognizing that there exists a direction of time at the boundaries
B

[2]. By introducing the momentum P
ij
conjugate to the three-metric h
ij
of  for the so-called \tilted" foliation [3] and integrating the kinetic part of
the volume integral, the action becomes S
m
=
R
M
d
4
x[P
ij
_
h
ij
 NH V
i
H
i
],
where the dot denotes dierentiation with respect to time, V
i
denotes the
shift vector, and the gravitational contributions to the Hamiltonian and mo-
mentum constraints are given by the usual expressions. Observe that the
action vanishes identically for stationary solutions of the vacuum Einstein
equations describing stationary eternal black holes. In this case
_
h
ij
= 0,
the constraint equations are satised, and no boundary terms remain in the
action.
Consider now the microcanonical functional integral for a gravitational
system whose timelike boundary surfaces B

are located in M

. The path
integral
["
+
; j
+
; 
+
; "
 
; j
 
; 
 
] =
X
M
Z
DH exp(iS
m
) (1)
is a functional of the energy density "

, momentum density j

, and two-
metric 

at the boundaries B
+
and B
 
. The sum over M refers to a
sum over manifolds of dierent topologies whose boundaries have topologies
B
+
 S
+
S
1
= S
2
S
1
and B
 
 S
 
S
1
= S
2
S
1
. The element S
1
is due
to the periodic identication in the global time direction at the boundaries
3
when the initial and nal hypersurfaces are identied. The integral is a sum
over periodic Lorentzian metrics that satisfy the boundary conditions at B
+
and B
 
.
The eternal black hole functional integral 

is obtained when the bound-
ary data ("
+
; j
+
; 
+
) and ("
 
; j
 
; 
 
) of the geometries summed over cor-
respond to the boundary data of a general Lorentzian, stationary eternal
black hole. The boundary data of this solution can be determined at S
+
and S
 
for each slice . Observe that by virtue of the gravitational con-
straint equations, these data determine uniquely the size of the black hole
horizon [5] and are such that the two-metric is continuous at this horizon.
We evaluate now the functional integral in the semiclassical approxima-
tion. This requires nding a four-metric that extremizes the action S
m
and
satises the boundary conditions ("
+
; j
+
; 
+
) at S
+
and ("
 
; j
 
; 
 
) at S
 
.
Since the classical Lorentzian eternal black hole metric can be periodically
identied and placed on a manifold whose two spatial boundaries have the
desired topologies, the resulting metric can be used to approximate the path
integral. The periodic identication alters neither the constraint equations
nor the boundary data. In the semiclassical approximation the functional
integral 

becomes [2]


["
+
; j
+
; 
+
; "
 
; j
 
; 
 
]  exp (iS
m
[
~
N;
~
V ;
~
h])  exp (0) ; (2)
since the microcanonical action S
m
[
~
N;
~
V ;
~
h] evaluated at the periodically
identied geometry vanishes identically if the stationarity condition and the
constraints are satised.
It is illustrative to consider now a complex four-metric which also ex-
tremizes the microcanonical action for eternal black hole boundary condi-
tions and which can be used to reevaluate the path integral (1) in a steepest
descent approximation. This alternative approximation of the quantity 

is useful in understanding the relationship of the result (2) with the density
of states for an ordinary (that is, non-eternal) black hole [1]. The complex
metric can be obtained from the Lorentzian eternal black hole metric by a
complexication map 	 dened by 	(N) =  iN , 	(V
i
) =  iV
i
. This map
preserves the reection symmetry and the canonical variables h
ij
and P
ij
of the Lorentzian solution. The complex geometry consists of two complex
sectors

M
+
and

M
 
which join at the locus of points at which the lapse
vanishes. This geometry is also a solution of Einstein equations if one re-
quires that conical singularities do not exist at that locus for every . To do
this, it is necessary to puncture each complex sector and to close smoothly
4
the geometry at the inner boundaries
3
H

=
2
H

 S
1
of

M

, where
2
H

denotes the intersection of the slices 

with the black hole horizon for the
Lorentzian metric [2]. After imposing these regularity conditions, the topol-
ogy of each sector

M

becomes R
2
 S
2
. However, each element
3
H
+
and
3
H
 
does contribute a term to the microcanonical action for the complex
geometry.
The regular complex metric is not included in the sum over Lorentzian
geometries 

in (1) but can be used to approximate it by distorting the con-
tours of integration for both lapse and shift into the complex plane [1]. In
this approximation the path integral becomes 

 exp(iS
m
[ i
~
N; i
~
V ;
~
h]),
where S
m
[ i
~
N; i
~
V ;
~
h] is the action of the complex metic when the smooth-
ness of the geometries at
3
H
+
and
3
H
 
is inforced. This action turns out
to be S
m
[ i
~
N; i
~
V ;
~
h] =  iA
+
=4 + iA
 
=4, where A
+
and A
 
denote the
surface area of the elements
2
H
+
and
2
H
 
[2]. Since the periodic identi-
cation and the complexication 	 do not alter the boundary data nor the
gravitational constraint equations, the area A
+
of
2
H
+
coincides with the
area A
 
of
2
H
 
: A
+
("
+
; j
+
; 
+
) = A
 
("
 
; j
 
; 
 
)  A
H
. This implies
that, in agreement with (2), the eternal black hole functional integral is


["
+
; j
+
; 
+
; "
 
; j
 
; 
 
]  exp (A
H
=4 A
H
=4) = exp(0) in the \zero-loop"
approximation.
If the microcanonical functional integral (1) is interpreted as the density
of states of the statistical system, it is possible to express 

approximately
as 

["
+
; j
+
; 
+
; "
 
; j
 
; 
 
]  exp(S["
+
; j
+
; 
+
; "
 
; j
 
; 
 
]), where S repre-
sents the total entropy of the system. The above result implies that the
entropy for the system in the semiclassical approximation is
S 
1
4
A
H
 
1
4
A
H
= 0 ; (3)
where A
H
is the area of the horizon of the physical eternal black hole solution
that classically approximates the system [2]. The total entropy is given
formally by the subtraction S = S
+
["
+
; j
+
; 
+
]   S
 
["
 
; j
 
; 
 
], where S
+
and S
 
can be interpreted as the semiclassical entropies associated with the
external (M
+
) and internal (M
 
) regions respectively of the eternal black
hole system.
3 Conclusions
The functional integral (2) refers to a quantum-statistical system which is
classically approximated by a general stationary, eternal black hole solution
5
of Einstein equations within a region bounded by two timelike surfaces B
+
and B
 
. Its semiclassical value is a consequence of the choice of boundary
data, the gravitational constraint equations, and the vanishing of the micro-
canonical action for the four-geometries that satisfy the boundary conditions
and approximate the path integral. The calculation presented above applies
to any distorted black hole in the strong gravity regime. It indicates that a
pure state (of zero entropy) can be dened not only for matter elds pertur-
bations propagating in the spacetime of an eternal black hole but also for
the gravitational eld itself. This is physically appealing: the initial data for
the eternal black hole specied at the spacelike hypersurface  contain all
the information required for the evolution of both the exterior and interior
parts of a physical black hole. The entropy associated with  must therefore
equal zero. Since in a microcanonical description it seems natural to relate
the external and internal degrees of freedom of a black hole with the bound-
ary data at the surfaces B
+
and B
 
respectively [3], we believe that the
microcanonical functional integral for eternal black hole systems opens the
possibility of extending path integral formulation of black hole thermody-
namics to situations when internal degrees of freedom are present and allows
the study of gravitational statistical properties in terms of a single pure state
[2]. These conclusions are in complete agreement with thermoeld dynamics
descriptions of quantum processes and, in particular, with the application of
this approach to black hole thermodynamics developed originally by Israel
[6] for small perturbations. They strongly suggest that the thermoeld dy-
namics description of quantum eld processes in a curved background can be
extended beyond perturbations to the gravitational eld itself of distorted
eternal black holes.
I am grateful to Valeri Frolov and Werner Israel for helpful comments.
This work was supported by the Natural Sciences and Engineering Research
Council of Canada.
References
[1] J. D. Brown and J. W. York, Jr., Phys. Rev. D 47, 1420 (1993).
[2] E. A. Martinez, Phys. Rev. D 51, 5732 (1995).
[3] V. Frolov and E. A. Martinez, \Action and Hamiltonian for eternal black
holes", preprint, Alberta-Thy-32-94, gr-qc/9411001, (1994).
6
[4] J. D. Brown, E. A. Martinez, and J. W. York, Jr., Phys. Rev. Lett., 66,
2281 (1991).
[5] J. D. Brown, G. L. Comer, E. A. Martinez, J. Melmed, B. F. Whiting,
and J. W. York, Class. Quantum Grav. 7, 1433 (1990).
[6] W. Israel, Phys. Lett. 57A, 107 (1976).
7


Entropy of eternal black holes

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