Asymptotically flat black holes and gravitational waves in three-dimensional massive gravity

Abstract

Different classes of exact solutions for the BHT massive gravity theory are constructed and analyzed. We focus in the special case of the purely quadratic Lagrangian, whose field equations are irreducibly of fourth order and are known to admit asymptotically locally flat black holes endowed with gravitational hair. The first class corresponds to a Kerr-Schild deformation of Minkowski spacetime along a covariantly constant null vector. As in the case of General Relativity, the field equations linearize so that the solution can be easily shown to be described by four arbitrary functions of a single null coordinate. These solutions can be regarded as a new sort of pp-waves. The second class is obtained from a deformation of the static asymptotically locally flat black hole, that goes along the spacelike (angular) Killing vector. Remarkably, although the deformation is not of Kerr-Schild type, the field equations also linearize, and hence the generic solution can be readily integrated. It is neither static nor spherically symmetric, being described by two integration constants and two arbitrary functions of the angular coordinate. In the static case it describes "black flowers" whose event horizons break the spherical symmetry. The generic time-dependent solution appears to describe a graviton that moves away from a black flower. Despite the asymptotic behaviour of these solutions at null infinity is relaxed with respect to the one for General Relativity, the asymptotic symmetries coincide. However, the algebra of the conserved charges corresponds to BMS$_{3}$, but devoid of central extensions. The "dynamical black flowers" are shown to possess a finite energy. The surface integrals that define the global charges also turn out to be useful in the description of the thermodynamics of solutions with event horizons.