5.2 Entropy of 3D Banados–Teitelboim–Zanelli (BTZ) black hole
5.2.1 BTZ black-hole geometry
The black-hole solution in three-dimensional gravity with negative cosmological constant was first obtained
in [6] (see also [5] for global analysis of the solution). We start with the black-hole metric written in a form
that makes it similar to the four-dimensional Kerr metric. Since we are interested in its thermodynamic
behavior, we write the metric in the Euclidean form:
where the metric functions
and
read
and we use the notation
Obviously one has that
. The coordinate
in Eq. (224) is assumed to be periodic with
period
.
In order to transform the metric (224) to a Lorentzian signature we need to make the analytic
transformation
,
so that
where
and
are the values in the Lorentzian spacetime. These are the respective radii of the outer
and inner horizons of the Lorentzian black hole in
dimensions. Therefore, we must always apply
the transformation (227) after carrying out all calculations in the Euclidean geometry in order to obtain the
result for the Lorentzian black hole. The Lorentzian version of the metric (224) describes a black hole with
mass
and angular momentum
. The outer horizon is located at
; the respective
inverse Hawking temperature is
In the
sector of the metric (224) there is no conical singularity at the horizon if the Euclidean time
is periodic with period
. The horizon
is a one-dimensional space with metric
,
where
is a natural coordinate on the horizon.
The BTZ space is obtained from the three-dimensional maximally-symmetric hyperbolic space
(sometimes called the global Euclidean anti-de Sitter space) by making certain identifications. In order to
see this one may use the coordinate transformation
In new coordinates
the BTZ metric takes the form
which is the metric on the hyperbolic space
. In this metric the BTZ geometry is defined by
identifications
The outer horizon
in the coordinate system
is located at
and
is the angular
coordinate on the horizon. Notice that the geodesic distance
between two points with coordinates
and
is
5.2.2 Heat kernel on regular BTZ geometry
Consider a scalar field with the operator
. The maximally-symmetric constant-curvature
space is a nice example of a curved space in which the heat equation
has a simple,
exact, solution. The heat kernel in this case is a function of the geodesic distance
between two points
and
. On the global space
one finds
where
. The regular BTZ geometry is defined by identifications
and
defined above. As is
seen from Eq. (231) the geodesic distance and the heat kernel (232), expressed in coordinates
,
are automatically invariant under identification
. Thus, it remains to maintain identification
. This is
done by summing over images
Using the path integral representation of the heat kernel we would say that the
term in Eq. (233)
is due to the direct way of connecting points
and
in the path integral. On the other
hand, the
terms are due to uncontractible winding paths that go
times around the
circle.
5.2.3 Heat kernel on conical BTZ geometry
The conical BTZ geometry, which is relevant to the entanglement entropy calculation, is obtained from
global hyperbolic space
by replacing identification
as follows
and
not changing identification
For
this Euclidean space has a conical singularity at the horizon
(
). The heat kernel on the conical BTZ geometry is constructed via the heat kernel (233) on the
regular BTZ space by means of the Sommerfeld formula (22)
where
is the heat kernel (233). The contour
is defined in Eq. (22).
For the trace of the heat kernel (234) one finds [171
] after computing by residues the contour integral
where
,
and
(
and
). Notice that we
have already made the analytical continuation to the values of
and
in the Lorentzian
geometry.
5.2.4 The entropy
When the trace of the heat kernel on the conical geometry is known one may compute the entanglement
entropy by using the replica trick. Then, the entropy is the sum of UV divergent and UV finite parts [171
]
where the UV divergent part is
This divergence is renormalized by the standard renormalization of Newton’s constant
in the three-dimensional gravitational action.
The UV finite part in the entropy is
where
.
After the renormalization of Newton’s constant, the complete entropy of the BTZ black hole,
, is a rather complicated function of the area of inner and outer horizons.
Approximating in Eq. (239) the infinite sum by an integral one finds [171
]
where
. The second term on the right-hand side of Eq. (240) can be considered to be the
one-loop quantum (UV-finite) correction to the classical entropy of a black hole.
For large enough
the integral in Eq. (240) goes to zero exponentially and we have the
classical Bekenstein–Hawking formula for entropy. On the other hand, for small
, the integral in
Eq. (240) behaves logarithmically so that one has [171
]
This logarithmic behavior for small values of
(provided the ratio
is fixed) is universal
and independent of the constant
(or
) in the field operator and the area of the inner horizon (
)
of the black hole. Hence, the rotation parameter
enters Eq. (241) only via the area
of the outer
horizon.
The other interesting feature of the entropy (240) is that it always develops a minimum, which is a
solution to the equation
This black hole of minimal entropy may be interesting in the context of the final stage of the Hawking
evaporation in three dimensions. As follows from the analysis of Mann and Solodukhin [171], the minimum
of the entropy occurs for a black hole whose horizon area is of the Planck length,
(in threee
dimensions
).