Warped AdS₃black holes

Abstract

Three dimensional topologically massive gravity (TMG) with a negative cosmological constant −ℓ −2 and positive Newton constant G admits an AdS 3 vacuum solution for any value of the graviton mass µ. These are all known to be perturbatively unstable except at the recently explored chiral point µℓ = 1. However we show herein that for every value of µℓ = 3 there are two other (potentially stable) vacuum solutions given by SL(2, R) × U (1)-invariant warped AdS 3 geometries, with a timelike or spacelike U (1) isometry. Critical behavior occurs at µℓ = 3, where the warping transitions from a stretching to a squashing, and there are a pair of warped solutions with a null U (1) isometry. For µℓ > 3, there are known warped black hole solutions which are asymptotic to warped AdS 3 . We show that these black holes are discrete quotients of warped AdS 3 just as BTZ black holes are discrete quotients of ordinary AdS 3 . Moreover new solutions of this type, relevant to any theory with warped AdS 3 solutions, are exhibited. Finally we note that the black hole thermodynamics is consistent with the hypothesis that, for µℓ > 3, the warped AdS 3 ground state of TMG is holographically dual to a 2D boundary CFT with central charges c R = 15(µℓ) 2 +81 Gµ((µℓ) 2 +27) and c L = 12µℓ 2 G((µℓ) 2 +27) .

What do non-relativistic CFTs tell us about Lifshitz spacetimes?

“…where a dot indicates a derivative with respect to the affine parameter λ. Geodesics then obey 12) with κ = 1 for timelike and κ = 0 for null geodesics, then we find…”

“…There are also other spacetimes in the literature, including the warped AdS spacetimes [12][13][14][15][16], which exhibit temporo-spatial anisometry. In this paper, we will concentrate on spacetimes which have Lifshitz symmetry at least in some region, but many of our conclusions apply to more general spacetimes with scaling differences between space and time.…”

The Brown-York mass of black holes in Warped Anti-de Sitter space

“…Motivated by the similarities between asymptotically WAdS 3 and asymptotically AdS 3 spaces, the authors of [1] proposed the idea of extending AdS/CFT to the former case. The conjecture is that quantum gravity in asymptotically WAdS 3 space would be dual to a twodimensional theory which exhibits partial conformal symmetry, it being symmetric under right -but not leftdilations.…”

“…[1]. In the case of TMG, the large r limit of Q[ξ] is regularized by introducing a boundary cosmological term with coefficient…”

“…(54) Contributions to charges M = 2πρu i T * ij ξ j and J = 2πρu i T * iϕ coming from the large r expansions M ≃ M (1) r + M (0) + ... and J ≃ J (2) r 2 + J (1) where ρ (n) refer to the components of the large expansion of metric function (12) in powers of r; analogously for the large r expansion of u t = u t (0) + u t (−1) r −1 + u t (−2) r −2 , u ϕ ≃ u ϕ (−1) r −1 + u ϕ (−2) r −2 , and the Killing vectors ξ t = ξ t (0) = 1, ξ ϕ ≃ ξ ϕ (−1) r −1 + ξ ϕ (−2) r −2 . For the angular momentum, the analogous expression is J (2) 2π = ρ (1) Then, one finds that the mass of the black hole is given by M (0) while its angular momentum is given by J (0) .…”