One of the mysteries in modern physics is why black holes have an entropy. This entropy, known as the
Bekenstein–Hawking entropy, was first introduced by Bekenstein [18, 19, 20] as a rather useful analogy.
Soon after that, this idea was put on a firm ground by Hawking [128] who showed that black holes
thermally radiate and calculated the black-hole temperature. The main feature of the Bekenstein–Hawking
entropy is its proportionality to the area of the black-hole horizon. This property makes it rather different
from the usual entropy, for example the entropy of a thermal gas in a box, which is proportional to the
volume. In 1986 Bombelli, Koul, Lee and Sorkin [23] published a paper in which they considered the
reduced density matrix, obtained by tracing over the degrees of freedom of a quantum field that are inside
the horizon. This procedure appears to be very natural for black holes, since the black hole
horizon plays the role of a causal boundary, which does not allow anyone outside the black
hole to have access to the events, which take place inside the horizon. Another attempt to
understand the entropy of black holes was made by ’t Hooft in 1985 [214]. His idea was to
calculate the entropy of the thermal gas of Hawking particles, which propagate just outside the
horizon. This calculation has uncovered two remarkable features: the entropy does turn out to
be proportional to the horizon area, however, in order to regularize the density of states very
close to the horizon, it was necessary to introduce the brick wall, a boundary, which is placed
at a small distance from the actual horizon. This small distance plays the role of a regulator
in the ’t Hooft’s calculation. Thus, the first indications that entropy may grow as area were
found.
An important step in the development of these ideas was made in 1993 when a paper of Srednicki [208]
appeared. In this very inspiring paper Srednicki calculated the reduced density and the corresponding
entropy directly in flat spacetime by tracing over the degrees of freedom residing inside an
imaginary surface. The entropy defined in this calculation has became known as the entanglement
entropy. Sometimes the term geometric entropy is used as well. The entanglement entropy, as was
shown by Srednicki, is proportional to the area of the entangling surface. This fact is naturally
explained by observing that the entanglement entropy is non-vanishing due to the short-distance
correlations present in the system. Thus, only modes, which are located in a small region close to
the surface, contribute to the entropy. By virtue of this fact, one finds that the size of this
region plays the role of the UV regulator so that the entanglement entropy is a UV sensitive
quantity. A surprising feature of Srednicki’s calculation is that no black hole is actually needed: the
entanglement entropy of a quantum field in flat spacetime already establishes the area law. In an
independent paper, Frolov and Novikov [99] applied a similar approach directly to a black hole. These
results have sparked interest in the entanglement entropy. In particular, it was realized that the
brick-wall model of ’t Hooft studies a similar entropy and that the two entropies are in fact related.
On the technical side of the problem, a very efficient method was developed to calculate the
entanglement entropy. This method, first considered by Susskind [211], is based on a simple
replica trick, in which one first introduces a small conical singularity at the entangling surface,
evaluates the effective action of a quantum field on the background of the metric with a conical
singularity and then differentiates the action with respect to the deficit angle. By means of this
method one has developed a systematic calculation of the UV divergent terms in the geometric
entropy of black holes, revealing the covariant structure of the divergences [33
, 197
, 111
]. In
particular, the logarithmic UV divergent terms in the entropy were found [196
]. The other aspect,
which was widely discussed in the literature, is whether the UV divergence in the entanglement
entropy could be properly renormalized. It was suggested by Susskind and Uglum [213
] that the
standard renormalization of Newton’s constant makes the entropy finite, provided one considers the
entanglement entropy as a quantum contribution to the Bekenstein–Hawking entropy. However, this
proposal did not answer the question of whether the Bekenstein–Hawking entropy itself can be
considered as an entropy of entanglement. It was proposed by Jacobson [141
] that, in models in
which Newton’s constant is induced in the spirit of Sakharov’s ideas, the Bekenstein–Hawking
entropy would also be properly induced. A concrete model to test this idea was considered
in [97
].
Unfortunately, in the 1990s, the study of entanglement entropy could not compete with the booming
success of the string theory (based on D-branes) calculations of black-hole entropy [209]. The second wave
of interest in entanglement entropy started in 2003 with work studying the entropy in condensed matter
systems and in lattice models. These studies revealed the universality of the approach based on the replica
trick and the efficiency of the conformal symmetry to compute the entropy in two dimensions. Black holes
again came into the focus of study in 2006 after work of Ryu and Takayanagi [189] where a holographic
interpretation of the entanglement entropy was proposed. In this proposal, in the frame of the AdS/CFT
correspondence, the entanglement entropy, defined on a boundary of anti-de Sitter, is related to the
area of a certain minimal surface in the bulk of the anti-de Sitter spacetime. This proposal
opened interesting possibilities for computing, in a purely geometrical way, the entropy and for
addressing in a new setting the question of the statistical interpretation of the Bekenstein–Hawking
entropy.
The progress made in recent years and the intensity of the on-going research indicate that entanglement
entropy is a very promising direction, which, in the coming years, may lead to a breakthrough in our
understanding of black holes and quantum gravity. A number of very nice reviews appeared in recent years
that address the role of entanglement entropy for black holes [21, 90, 146, 54]; review the calculation of
entanglement entropy in quantum field theory in flat spacetime [81, 37] and the role of the conformal
symmetry [31]; and focus on the holographic aspects of the entanglement entropy [185
, 11]. In the present
review I build on these works and focus on the study of entanglement entropy as applied to black
holes. The goal of this review is to collect a complete variety of results and present them in a
systematic and self-consistent way without neglecting either technical or principal aspects of the
problem.
http://www.livingreviews.org/lrr-2011-8 |
Living Rev. Relativity 14, (2011), 8
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