Exploring AdS Waves Via Nonminimal Coupling

TL;DR

This work constructs exact AdS-wave solutions in three dimensions sourced by nonminimally coupled scalars, showing that pure radiation constraints uniquely fix both the scalar configuration and its self-interaction potential $U_\xi(\\Phi)$, governed by a single coupling $\\lambda$. In the free case $\\lambda=0$ the scalar realizes a curvature-dependent mass $m_\\xi^2$, producing massive, tachyonic, or massless regimes, with a remarkable $\\xi=1/5$ case where the effective mass vanishes in AdS. The self-interacting case ($\\lambda\neq0$) yields a double Kerr-Schild structure and, at a critical coupling $\\lambda=(\\kappa/(8l))^2$, a nonperturbative strong–weak duality $\\Phi\leftrightarrow 1/\\Phi$, while a conformal-type correspondence with pp waves relates the AdS-wave sources to pp-wave sources via a conformal weight $s=-2\\xi/(1-4\\xi)$. The paper also develops an off-shell and on-shell AdS–pp correspondence, and discusses extensions to topologically massive gravity and conformal gravity, highlighting a unified framework for gravitational waves sourced by nonminimally coupled scalars. The results have implications for exact wave solutions in curved backgrounds and for the broader interplay between AdS, pp, and conformal gravitational theories.

Abstract

We consider nonminimally coupled scalar fields to explore the Siklos spacetimes in three dimensions. Their interpretation as exact gravitational waves propagating on AdS restrict the source to behave as a pure radiation field. We show that the related pure radiation constraints single out a unique self-interaction potential depending on one coupling constant. For a vanishing coupling constant, this potential reduces to a mass term with a mass fixed in terms of the nonminimal coupling parameter. This mass dependence allows the existence of several free cases including massless and tachyonic sources. There even exists a particular value of the nonminimal coupling parameter for which the corresponding mass exactly compensates the contribution generated by the negative scalar curvature, producing a genuinely massless field in this curved background. The self-interacting case is studied in detail for the conformal coupling. The resulting gravitational wave is formed by the superposition of the free and the self-interaction contributions, except for a critical value of the coupling constant where a non-perturbative effect relating the strong and weak regimes of the source appears. We establish a correspondence between the scalar source supporting an AdS wave and a pp wave by showing that their respective pure radiation constraints are conformally related, while their involved backgrounds are not. Finally, we consider the AdS waves for topologically massive gravity and its limit to conformal gravity.

Figures (1)

• Figure 1: The solid graph shows the dependence of the square of the scalar field mass ${m_\xi}^2$ on the nonminimal coupling parameter $\xi$, as fixed by the pure radiation constraints. The dotted graph corresponds to the dependence of the square of the effective mass ${m_{\mathrm{eff}}}^2$, obtained when the contribution of the curvature is taken into account, see Eq. (\ref{['eq:me^2']}). The graphs are valid for $\xi\ne1/4$. The dependence for $\xi>1/4$ is not shown in the graph but it is positive definite in both cases.