Deeper discussion of Schr\"odinger invariant and Logarithmic sectors of higher-curvature gravity

TL;DR

This paper investigates higher-curvature gravity theories in D dimensions, revealing special solution sectors including Schrödinger invariant and logarithmic solutions, and explores their properties, degeneracies, and extensions to more general theories and Chern-Simons gravity.

Contribution

It identifies and analyzes specific solution sectors in higher-curvature gravity theories, including Schrödinger invariant and logarithmic branches, and extends the analysis to non-polynomial and Chern-Simons modified gravities.

Findings

Schrödinger invariant solutions arise at specific coupling relations.

Logarithmic solutions are interpreted as superpositions of scalar modes.

Degenerate sectors admit arbitrary wave profiles, linked to non-renormalization properties.

Abstract

The aim of this paper is to explore D-dimensional theories of pure gravity whose space of solutions contains certain class of AdS-waves, including in particular Schrodinger invariant spacetimes. This amounts to consider higher order theories, and the natural case to start with is to analyze generic square-curvature corrections to Einstein-Hilbert action. In this case, the Schrodinger 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 Schrodinger 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. 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 bound (BF) of the AdS space on which they propagate. These solutions are higher-dimensional analogues of those appearing in three-dimensional massive gravities with relaxed AdS_3 asymptotic. 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 non-renormalization of the dynamical exponent of the Schrodinger spaces. Our analysis also includes more general gravitational actions with non-polynomial corrections consisting of arbitrary functions of the square-curvature invariants. The same sectors of solutions are shown to exist for this more general family of theories. We finally consider the Chern-Simons modified gravity in four dimensions, for which we derive both the Schrodinger invariant as well as the logarithmic sectors.