5.1 Entropy of a 2D black hole
In two dimensions the conformal symmetry plays a special role. This has many manifestations. In
particular, the conformal symmetry can be used in order to completely reproduce, for a conformal field
theory (CFT), the UV finite part of the corresponding gravitational effective action. This is done by
integration of the conformal anomaly. For regular two-dimensional spacetimes, the result is the well-known
non-local Polyakov action. In the presence of a conical singularity the derivation is essentially the same,
although one has to take into account the contribution of the singularity. Consider a two-dimensional CFT
characterized by a central charge
. For a regular two-dimensional manifold, the Polyakov action can be
written in the form
where the field equation for the field
is
. On a manifold
with a conical singularity
with angle deficit
the Polyakov action is modified by the contribution from the
singularity at the horizon
(which is just a point in two dimensions) so that [196
, 112
]
where
is the value of the field
on the horizon. Applying the replica method to the Polyakov
action (214) one obtains that the corresponding contribution to the entanglement entropy from the UV
finite term in the effective action is
This result agrees with a derivation of Myers [179] who used the Noether charge method of Wald [221] in
order to calculate the entropy. The easiest way to compute the function
is to use the conformal gauge
in which
. Together with the UV divergent part, the complete entanglement
entropy in two dimensions is
where
is an IR cut-off.
Let the black-hole geometry be described by a 2D metric
where the metric function
has a simple zero at
. Assume that this black hole is placed
inside a box of finite size
so that
. In order to get a regular space, one closes the
Euclidean time
with period
,
. It is easy to see that Eq. (217) is conformal to the flat
disk of radius
(
):
where
(
),
. So that the entanglement entropy of the 2D black hole takes
the form [199
, 98]
where we omit the irrelevant term that is a function of
but not of the parameters of the black hole
and have retained dependence on the UV regulator
.
As was shown in [199], the entanglement entropy (219) is identical to the entropy of the
thermal atmosphere of quantum excitations outside the horizon in the “brick-wall” approach of
’t Hooft [214].
The black hole resides inside a finite-sized box and
is the coordinate of the boundary of
the box. The coordinate invariant size of the subsystem complimentary to the black hole is
. Two limiting cases are of interest. In the first, the size of the system
is taken to infinity. Then, assuming that the black-hole spacetime is asymptotically flat,
we obtain that the entanglement entropy (219) approaches the entropy of the thermal gas,
This calculation illustrates an important feature of the entanglement entropy of a black hole placed in a box
of volume
. Namely, the entanglement entropy contains a contribution of the thermal gas that, in the
limit of large volume in dimension
, takes the form (7). This is consistent with the thermal nature of the
reduced density matrix obtained from the Hartle–Hawking state by tracing over modes inside the
horizon.
The other interesting case is when
is small. In this case, we find the universal behavior
The universality of this formula lies in the fact that it does not depend on any characteristics
of the black hole (mass, temperature) other than the value of the curvature
at the
horizon.
Consider two particular examples.
2D de Sitter spacetime is characterized by the metric function
and the Hawking
temperature
. In this spacetime the size of the box is bounded from above,
. The
corresponding entanglement entropy
is a periodic function of
.
The string inspired black hole [222, 169] is described by the metric function
. It
described an asymptotically-flat spacetime. The Hawking temperature is
. The entanglement
entropy in this case is
The entropy in these two examples resembles the entanglement entropy in flat spacetime at zero
temperature (11) and at a finite temperature (12) respectively.