Searching for AdS₃ waves and Asymptotically Lifshitz black holes in R³-NMG

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In this paper we consider the structure of the AdS₃ vacua in R³ expansion of the New Massive Gravity (R³-NMG). We obtain the degeneracies of the AdS₃ vacua at several points of the parametric space. Additionally, following a specific analysis we show that AdS₃ wave solutions are present. Using these wave solutions, we single out two special points of the parametric space for which logarithmic terms appear in the solutions. The first one is a point at which the effective mass of the wave profile which is interpreted as a scalar mode, completely saturates the Breitenlohner-Freedman bound of the AdS₃ space in which the wave is propagating. The second special point is a point at which the central charge of the theory vanishes. Furthermore, we investigate the possibility of asymptotically Lifshitz black solutions to be present in the three-dimensional R³-NMG. We derive analytically the Lifshitz vacua considering specific relations between the mass parameters of R³-NMG. A certain polynomial equation arises at the first special point where solutions with logarithmic fall-off in the AdS₃ space appear. Solving this polynomial equation, we obtain the values of the dynamical exponent z which correspond to possible asymptotically Lifshitz black hole solutions. However, it is shown that asymptotically Lifshitz black solutions do not exist in the three-dimensional R³-NMG for a specific ansatz of the black hole metric.