Anisotropic scale invariant spacetimes and black holes in Zwei-Dreibein Gravity

Abstract

We show that Zwei-Dreibein Gravity (ZDG), a bigravity theory recently proposed by Bergshoeff, de Haan, Hohm, Merbis, and Townsend in ref.[1], admits exact solutions with anisotropic scale invariance. These type of geometries are the three-dimensional analogues of the spacetimes which were proposed as gravity duals for condensed matter systems. In particular, we find Schrödinger invariant spaces as well as Lifshitz spaces with arbitrary dynamical exponent z. We also find black holes that are asymptotically Lifshitz with z = 3, showing that these (non-constant curvature) solutions of New Massive Gravity (NMG) are persistent after the introduction of the infinite tower of higher-curvature terms of ZDG, provided a renormalization of the parameters. Black holes in asymptotically warped Anti-de Sitter spaces are also found. Interestingly, in almost all the geometries studied in this work, the metric associated with the second dreibein turns out to be equivalent, up to a constant global factor, to the first one. This phenomenon has been previously observed in other bigravity theories in asymptotically flat and asymptotically Anti-de Sitter backgrounds. However, for the particular case of the z = 3 Lifshitz black hole, here we found that the second metric corresponds to a different black hole that coincides with the former only in the asymptotic region. In fact, we find a new family of z = 3 black holes that corresponds to a one-parameter deformation of the NMG solution.

Interacting spin-2 fields in three dimensions

“…For classical solutions in the 3D bigravity see[32][33][34][35] 4. The constraint e 1 · e 2 = 0, sometimes called the symmetry condition, is of great importance in connecting the frame formulation to the metric form of[6].…”

The mass of a Lifshitz black hole

“…which in particular yields the surface gravity κ = r 3 + / 4 . Now, we are ready to evaluate (15) for the Killing vector ξ = ∂ v and realize the functional variation by varying the parameter δr + .…”

“…It was shown in [2] that, if one considers the parityeven massive 3D gravity proposed in [3], a static Lifshitz black hole solution with dynamical exponent z = 3 can be analytically constructed. While other models admitting Lifshitz black holes are known in 3 dimensions, these either include additional fields [4,5] or exotic gravity field equations [6]. This makes the simple instances of Lifshitz black hole scarce.…”

Conserved charges in timelike warpedAdS3spaces

“…A feature that makes WAdS 3 spaces of particular interest is that these geometries appear as exact solutions of a large variety of models, including string theory [15,16], topologically massive gauge theories [17,18,19,20], higher-derivative theories [21], bi-gravity theories [22], and Einstein gravity non-minimally coupled to matter fields [23]. A minimal setup in which WAdS 3 spaces appear is three-dimensional gravity with no matter fields.…”

“…where the first contribution comes from the pure GR part of the equations of motion, while Q µν K accounts for the contribution of the K µν tensor of NMG, whose explicit expression can be found in equations (22), (28) and (29) in [35], respectively.…”