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Anisotropic Scale Invariant Spacetimes and Black Holes in Zwei-Dreibein Gravity

We show that Zwei-Dreibein Gravity (ZDG), a bigravity theory recently proposed by Bergshoeff, de Haan, Hohm, Merbis, and Townsend in Phys.Rev.Lett. 111 (2013) 111102, 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.