Geometric Finiteness, Holography and Quasinormal Modes for the Warped

 AdS_3 Black Hole

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Abstract

We show that there exists a precise kinematical notion of holography for the Euclidean warped \(AdS_3\) black hole. This follows from the fact that the Euclidean warped \(AdS_3\) black hole spacetime is a geometrically finite hyperbolic manifold. For such manifolds a theorem of Sullivan provides a one-to-one correspondence between the hyperbolic structure in the bulk and the conformal structure of its boundary. Using this theorem we obtain the holographic quasinormal modes for the warped \(AdS_3\) black hole.

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Geometry of the 2+1 Black Hole

Jorge Zanelli, Claudio Teitelboim, Marc Henneaux … (1993)

The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti-de Sitter space by a discrete subgroup of \(SO(2,2)\). The generic black hole is a smooth manifold in the metric sense. The surface \(r=0\) is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at \(r=0\) to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti-de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum . A thorough classification of the elements of the Lie algebra of \(SO(2,2)\) is given in an Appendix.

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Conformal Field Theory Interpretation of Black Hole Quasi-normal Modes

Sergey N. Solodukhin, Danny Birmingham, Ivo Sachs (2001)

We obtain exact expressions for the quasi-normal modes of various spin for the BTZ black hole. These modes determine the relaxation time of black hole perturbations. Exact agreement is found between the quasi-normal frequencies and the location of the poles of the retarded correlation function of the corresponding perturbations in the dual conformal field theory. This then provides a new quantitative test of the AdS/CFT correspondence.

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Scalar Wave Falloff in Asymptotically Anti-de Sitter Backgrounds

, (2009)

Conformally invariant scalar waves in black hole spacetimes which are asymptotically anti-de Sitter are investigated. We consider both the \((2+1)\)-dimensional black hole and \((3+1)\)-dimensional Schwarzschild-anti-de Sitter spacetime as backgrounds. Analytical and numerical methods show that the waves decay exponentially in the \((2+1)\) dimensional black hole background. However the falloff pattern of the conformal scalar waves in the Schwarzschild-anti-de Sitter background is generally neither exponential nor an inverse power rate, although the approximate falloff of the maximal peak is weakly exponential. We discuss the implications of these results for mass inflation.