Bending AdS Waves with New Massive Gravity

TL;DR

This work analyzes exact AdS-wave solutions in three-dimensional New Massive Gravity (NMG) and in its coupling to Topologically Massive Gravity (TMG). By deriving the AdS-wave profile equation, it identifies multiple branches with distinct asymptotics, including logarithmic fall-off at critical masses and Schrödinger-invariant sectors, and shows the wave profile behaves as a massive scalar with $m_{\text{eff}}^2 = m^2 - \frac{1}{2l^2}$. At the critical point $m^2 = -\frac{1}{2l^2}$ the BF bound is saturated, while at $m^2 = +\frac{1}{2l^2}$ logarithmic waves appear as a superposition of two scalar modes. Introducing the Chern–Simons term splits the degeneracy of the NMG modes, producing a mass-splitting structure and enlarging the logarithmic sector, with limits connecting smoothly to pure NMG and to TMG. The results illuminate the interplay between parity-preserving and parity-violating massive gravities and have implications for AdS$_3$/CFT$_2$ and non-relativistic holography.

Abstract

We study AdS-waves in the three-dimensional new theory of massive gravity recently proposed by Bergshoeff, Hohm, and Townsend. The general configuration of this type is derived and shown to exhibit different branches, with different asymptotic behaviors. In particular, for the special fine tuning $m^2=\pm1/(2l^2)$, solutions with logarithmic fall-off arise, while in the range $m^2>-1/(2l^2)$, spacetimes with Schrodinger isometry group are admitted as solutions. Solutions that are asymptotically AdS$_3$, both for Brown-Henneaux and for the weakened boundary conditions, are also identified. The metric function that characterizes the profile of the AdS-wave behaves as a massive excitation on the spacetime, with an effective mass given by $m_{eff}^2=m^2-1/(2l^2)$. For the critical value $m^2=-1/(2l^2)$, the value of the effective mass precisely saturates the Breitenlohner-Freedman bound for the AdS$_3$ space where the wave is propagating on. The analogies with the AdS-wave solutions of topologically massive gravity are also discussed. Besides, we consider the coupling of both massive deformations to Einstein gravity and find the exact configurations for the complete theory, discussing all the different branches exhaustively. One of the effects of introducing the Chern-Simons gravitational term is that of breaking the degeneracy in the effective mass of the generic modes of pure New Massive Gravity, producing a fine structure due to parity violation. Another effect is that the zoo of exact logarithmic specimens becomes considerably enlarged.