The next question is how to choose the minimal surface . Emparan’s proposal [83
] consists of the
following. Suppose the metric on the boundary of asymptotically-AdS spacetime describes a black hole with
a Killing horizon at surface
. Presumably, the horizon on the boundary
is extended to
the bulk. The bulk horizon is characterized by vanishing extrinsic curvature and is a minimal
-dimensional surface. Thus, one can choose the bulk horizon to be that minimal surface
, the area of which should appear in the holographic formula (303
). In this construction
the Killing horizon
is the only boundary of the minimal surface
. This prescription is
perfectly eligible if one computes the entanglement entropy of a black hole in infinite spacetime.
In [83
] it was applied to the entropy of a black hole residing on the 2-brane in the 4d solution
of [84
]. In [139
] a similar prescription is used to compute entanglement entropy of the de Sitter
horizon.
On the other hand, in certain situations it is interesting to consider a black hole residing inside a cavity,
the “black hole in a box”. Then, as we have learned in the two-dimensional case, the entanglement entropy
will depend on the size of the box so that, in the limit of large
, the entropy will have a thermal
contribution proportional to the volume of the box. This contribution is additional to the pure
entanglement part, which is due to the presence of the horizon
. In order to reproduce this dependence
on the size of the box, one should use a different proposal. A relevant proposal was suggested by Solodukhin
in [202
].
Let a -dimensional spherically-symmetric static black hole with horizon
lie on the regularized
boundary (with regularization parameter
) of asymptotically anti-de Sitter spacetime AdSd+1 inside a
spherical cavity
of radius
. Consider a minimal d-surface
, whose boundary is the union of
and
.
has saddle points, where the radial AdS coordinate has an extremum. By spherical
symmetry the saddle points form a
-surface
with the geometry of a sphere. Consider the
subset
of
whose boundary is the union of
and
. According to the prescription of [202
], the
quantity
It should be noted that as far as the UV divergent part of the entanglement entropy is concerned, the
two proposals [83, 139] and [202
] give the same result. This is due to the fact that the UV
divergences come from that part of the minimal surface, which approaches the boundary
of the AdS spacetime. In both proposals this part of the surfaces
and
is the same.
Thus, the difference is in the finite terms, which are due to global properties of the minimal
surface.
From the geometrical point of view, the holographic calculation of the logarithmic term in the entanglement entropy is related to the surface anomalies studied by Graham and Witten [123] (this point is discussed in [191]).
http://www.livingreviews.org/lrr-2011-8 |
Living Rev. Relativity 14, (2011), 8
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