### 7.3 The holographic entanglement entropy of 2D black holes

In order to check the proposal (304) one needs a solution to the bulk Einstein equations that describes a
black hole on the boundary of AdS. In three dimensions a solution of this type is known explicitly [194],
where the AdS radius is set to unity. At asymptotic infinity () of the metric (305) one has the 2D
black hole metric
where has a simple zero in . The cavity is placed at so that .
The regularity of the metric (306) at the horizon requires that , .
Note that Eq. (305) is a vacuum solution of the Einstein equations for any function . The regularity
of the 3D metric (305) requires that the constant should be related to the Hawking temperature of the
two-dimensional horizon by . The geodesic lies in the hypersurface of constant
time . The induced metric on the hypersurface has a constant curvature equal to
for any function and is, thus, related by a coordinate transformation to the metric
The exact relation between the two coordinate systems is
where . The equation for the geodesic in the metric (307) is . The
geodesic length between two points lying on the geodesic with radial coordinates and
is
The saddle point of the geodesic is at . The constant is determined from the
condition that the geodesic joins points and lying on the
regularized (with regularization parameter ) boundary. In the limit of small one finds
that
The
length of the geodesic joining point corresponding to and thus the saddle point
is .
Now, one has to take into account that, in the AdS/CFT correspondence, the value of Newton’s
constant in the bulk is related to the number of quantum fields living in the boundary . In the
AdS_{3}/CFT_{2} case, one has that , where is the central charge of boundary CFT. Thus, one
finds that the holographic entropy (304)

where , indeed coincides with expression (219) for the holographic entanglement entropy of the
2D black hole in conformal field theory. In particular, for large values of the holographic formula for the
entropy correctly reproduces the entropy of thermal gas (we remember that
). This is a consequence of the choice of the minimal surface made in the
proposal (304).