Deeper discussion of Schrödinger invariant and logarithmic sectors of higher-curvature gravity

Abstract

The aim of this paper is to explore D-dimensional theories of pure gravity whose space of solutions contains a certain class of anti-de Sitter waves including, in particular, Schrödinger invariant spacetimes. This leads to consider higher-order theories, and the natural case to start with is to analyze generic square-curvature corrections to the Einstein-Hilbert action. In this case, the Schrödinger invariant sector in the space of solutions arises for a special relation between the coupling constants appearing in the action. On the other hand, besides the Schrödinger invariant configurations, logarithmic branches similar to those of the so-called log-gravity are also shown to emerge for another special choice of the coupling constants. Interestingly enough, these log solutions can be interpreted as the superposition of the massless mode of general relativity and two scalar modes that saturate the Breitenlohner-Freedman (BF) bound of the AdS space on which they propagate. These solutions are higher-dimensional analogues of those appearing in three-dimensional massive gravities with relaxed AdS3 asymptotic, which are candidates to be gravity duals for logarithmic conformal field theories (CFTs). Other sectors of the space of solutions of higher-curvature theories correspond to oscillatory configurations, which happen to be below the BF bound. Also, there is a fully degenerated sector, for which any wave profile is admitted. We comment on the relation between this degeneracy and the nonrenormalizability of the dynamical exponent of the Schrödinger spaces. Our analysis also includes more general gravitational actions with nonpolynomial corrections consisting of arbitrary functions of the square-curvature invariants. By establishing a correspondence with the quadratic gravity model, the same sectors of solutions are shown to exist for this more general family of theories. We finally consider the parity-violating Chern-Simons modified gravity in four dimensions, for which we derive both the Schrödinger invariant as well as the logarithmic sectors. © 2011 American Physical Society.