5.3 Entropy of d-dimensional extreme black holes
The extremal black holes play a special role in gravitational theory. These black holes are
characterized by vanishing Hawking temperature
, which means that in the metric (165) the
near-horizon expansion in the metric function
starts with the quadratic term
.
Topologically, the true extremal geometry is different from the non-extremal one. Near the
horizon the non-extremal static geometry looks like a product of a two-dimensional disk (in the
plane
) and a
-dimensional sphere. Then, the horizon is the center in the polar
coordinate system on the disk. Contrary to this, an extremal geometry in the near-horizon
limit is a product of a two-dimensional cylinder and a
-dimensional sphere. Thus, the
horizon in the extremal case is just another boundary rather than a regular inner point, as in the
non-extremal geometry. However, one may consider a certain limiting procedure in which one
approaches the extremal case staying all the time in the class of non-extremal geometries. This
limiting procedure is what we shall call the “extremal limit”. A concrete procedure of this type
was suggested by Zaslavsky [226]. One considers a sequence of non-extreme black holes in a
cavity at
and finds that there exists a set of data
such that the limit
is well defined. Even if one may have started with a rather general non-extremal
metric, the limiting geometry is characterized by very few parameters. In this sense, one may
talk about “universality” of the extremal limit. In fact, in the most interesting (and tractable)
case the limiting geometry is the product of two-dimensional hyperbolic space
with the
-dimensional sphere. Since the limiting geometry belongs to the non-extreme class, its classical
entropy is proportional to the horizon area in accord with the Bekenstein–Hawking formula. Then,
the entanglement entropy of the limiting geometry is a one-loop quantum correction to the
classical result. The universality we have just mentioned suggests that this correction possesses a
universal behavior in the extreme limit and, since the limiting geometry is rather simple, the
limiting entropy can be found explicitly. The latter was indeed shown by Mann and Solodukhin
in [172
].
5.3.1 Universal extremal limit
Consider a static spherically-symmetric metric in the following form
where
is the metric on the
-dimensional unit sphere, describing a non-extreme hole with
an outer horizon located at
. However, the analysis can be made for a more general metric, in
which
, the limiting geometry is the simplest in the case we consider in Eq. (243). The function
in Eq. (243) can be expanded as follows
It is convenient to consider the geodesic distance
as a radial coordinate. Retaining the first
two terms in Eq. (244), we find, for
, that
In order to avoid the appearance of a conical singularity at
, the Euclidean time
in Eq. (243)
must be compactified with period
, which goes to infinity in the extreme limit
. However,
rescaling
yields a new variable
having period
. Then, taking into account
Eq. (245), one finds for the metric (243)
where we have introduced the variable
. To obtain the extremal limit one just takes
.
The limiting geometry
is that of the direct product of a 2-dimensional space and a
-sphere and is characterized
by a pair of dimensional parameters
and
. The parameter
sets the radius
of the
-dimensional sphere, while the parameter
is the curvature radius for
the
2-space. Clearly, this two-dimensional space is the negative constant curvature
space
. This is the universality we mentioned above: although the non-extreme geometry
is in general described by an infinite number of parameters associated with the determining
function
, the geometry in the extreme limit depends only on two parameters
and
. Note that the coordinate
is inadequate for describing the extremal limit (247) since
the coordinate transformation (245) is singular when
. The limiting metric (247) is
characterized by a finite temperature, determined by the
periodicity in angular coordinate
.
The limiting geometry (247) is that of a direct product
of 2d hyperbolic space
with
radius
and a 2D sphere
with radius
. It is worth noting that the limiting
geometry (247) precisely merges near the horizon with the geometry of the original metric (243) in the
sense that all the curvature tensors for both metrics coincide. This is in contrast with, say, the situation in
which the Rindler metric is considered to approximate the geometry of a non-extreme black hole: the
curvatures of both spaces do not merge in general.
For a special type of extremal black holes
, the limiting geometry is characterized by just one
dimensionful parameter. This is the case for the Reissner–Nordström black hole in four dimensions. The
limiting extreme geometry in this case is the well-known Bertotti–Robinson space characterized by just one
parameter
. This space has remarkable properties in the context of supergravity theory that are not the
subject of the present review.
5.3.2 Entanglement entropy in the extremal limit
Consider now a scalar field propagating on the background of the limiting geometry (247) and described by
the operator
where
is the Ricci scalar. For the metric (247) characterized by two dimensionful parameters
and
, one has that
. For a
-dimensional conformally-coupled scalar
field we have
. In this case
The calculation of the respective entanglement entropy goes along the same lines as before. First, one
allows the coordinate
, which plays the role of the Euclidean time, to have period
. For
the metric (247) then describes the space
, where
is the hyperbolic space
coinciding with
everywhere except the point
, where it has a conical singularity with an
angular deficit
. The heat kernel of the Laplace operator
on
is given by the
product
where
and
are the heat kernels, of the Laplace operator on
and
respectively.
The effective action reads
where
is a UV cut-off. On spaces with constant curvature the heat kernel function is known
explicitly [34]. In particular, on a 2D space
of negative constant curvature, the heat kernel has the
following integral representation:
where
. In Eq. (251)
is the geodesic distance between the points on
. Between two
points
and
the geodesic distance is given by
. The heat
kernel on the conical hyperbolic space
can be obtained from (251) by applying the Sommerfeld
formula (22). Skipping the technical details, available in [172
], let us just quote the result for the trace
where
.
Let us denote
the trace of the heat kernel of the Laplace operator
on a
-dimensional sphere of unit radius. The entanglement entropy in the extremal limit then takes the
form [172
, 206
]
The function
has the following small-
expansion
The trace of the heat kernel on a sphere is known in some implicit form. However, for our purposes a
representation in a form of an expansion is more useful,
where
is the area of a unit radius sphere
. The first few coefficients in this
expansion can be calculated using the results collected in [219
],
We shall consider some particular cases.
d=4.
The entanglement entropy in the extreme limit is
where
is the UV finite part of the entropy. For minimal coupling
this result was obtained
in [172]. The first term in Eq. (257) is proportional to the horizon area
, while the second term
is a logarithmic correction to the area law. For conformal coupling
, the logarithmic term is
d=5.
The entropy is
To simplify the expressions in higher dimensions we consider only the case of the conformal coupling
.
The entropy takes the form:
d=6.
d=7.
d=8.
Two examples of the extreme geometry are of particular interest.
Entanglement entropy of the round sphere in Minkowski spacetime.
Consider a sphere of radius
in flat
Minkowski spacetime. One can choose a spherical coordinate system
so that the surface
is
defined as
and
, and variables
are the angular coordinates on
. The
-metric reads
where
is a metric on
sphere of unit radius. Metric (264) is conformal to the metric
which describes the product of two-dimensional hyperbolic space
with coordinates
and the
sphere
. Note that both spaces,
and
, have the same radius
. Metric (265) describes
the spacetime, which appears in the extremal limit of a
-dimensional static black hole. In the hyperbolic
space
we can choose a polar coordinate system
with its center at point
,
(for small
one has that
as in the polar system in flat spacetime) so
that the metric takes the form
In this coordinate system the surface
is defined by the condition
. In the entanglement entropy
of a conformally-coupled scalar field the logarithmic term
is conformally invariant. Therefore, it is the
same [206] for the entropy of a round sphere of radius
in Minkowski spacetime and in the extreme
limiting geometry (267). In various dimensions
can be obtained from the results (258) – (262) by
setting
. One finds
in
,
in
and
in
.
For
the logarithmic term in the entropy of a round sphere has been calculated by Casini and
Huerta [38
] directly in Minkowski spacetime. They have obtained
in all even dimensions up to
. Subsequently, Dowker [72
] has extended this result to
and
. In arbitrary
dimension
the logarithmic term
can be expressed in terms of Bernulli numbers as is shown in [38]
and [72]
Entanglement entropy of the extreme Reissner–Nordström black hole in d dimensions.
As was shown by
Myers and Perry [180
] a generalization of the Reissner–Nordström solution to higher dimension
is
given by Eq. (243) with
In the extreme limit
. Expanding Eq. (268) near the horizon one finds, in this limit, that
. Thus, this extreme geometry is characterized by values of radii
and
. In dimension
we have
as in the case considered above. In dimension
the
two radii are different,
. For the conformal coupling the values of logarithmic term
in various
dimensions are presented in Table 1.
Table 1: Coefficients of the logarithmic term in the entanglement entropy of an extreme
Reissner–Nordström black hole.