The CFT-interpolating Black Hole in Three Dimensions

Abstract

We present a new exact black hole solution in three dimensional Einstein gravity coupled to a single scalar field. This is one of the extended solutions of the BTZ black hole and has in fact ${\text{AdS}}_3$ geometries both at the spatial infinity and at the event horizon. An explicit derivation of Virasoro algebras for ${\text{CFT}}_2$ at the two boundaries is shown to be possible à la Brown and Henneaux's calculation. If we regard the scalar field as a running coupling in the dual two dimensional field theory, and its flow in the bulk as the "holographic" renormalization group flow, our black hole should interpolate the two ${\text{CFT}}_2$ living at the infinity and at the horizon. Following the Hamilton-Jacobi analysis by de Boer, Verlinde and Verlinde, we calculate the central charges $c_{\text{UV}}$ and $c_{\text{IR}}$ for the ${\text{CFT}}_2$ on the infinity and the horizon, respectively. We also confirm that the inequality $c_{\text{IR}}<c_{\text{UV}}$ is satisfied, which is consistent with the Zamolodchikov's c-theorem.

20 pages, 2 figures, references added

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