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

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.

The conical BTZ geometry, which is relevant to the entanglement entropy calculation, is obtained from global hyperbolic space by replacing identification as follows

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.

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 ).
Living Rev. Relativity 14, (2011), 8
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