pp Waves of Conformal Gravity with Self-Interacting Source

TL;DR

The paper extends 3D conformal gravity by including a conformally invariant self-interacting scalar (λψ^6) and shows that pp-wave solutions still exist, with the scalar source becoming inhomogeneous and independent of retarded time due to the self-interaction. It derives the pp-wave solutions, revealing two branches depending on λ: a power-law dependence on (2√λ y+f(u)) and, at the critical λ=(κ/16)^2, a logarithmic form, alongside explicit scalar field profiles. A conformal-frame mapping to topologically massive gravity with a cosmological constant Λ = -4λ/(π^2 G^2) and μ = -κ/(8π G) is established, enabling explicit TM gravity solutions for Λ<0, though some mapped spacetimes are no longer pp waves. The results illuminate a direct link between conformal 3D gravity with self-interaction and TM gravity, providing concrete solutions and suggesting further exploration of such configurations.

Abstract

Recently, Deser, Jackiw and Pi have shown that three-dimensional conformal gravity with a source given by a conformally coupled scalar field admits pp wave solutions. In this letter, we consider this model with a self-interacting potential preserving the conformal structure. A pp wave geometry is also supported by this system and, we show that this model is equivalent to topologically massive gravity with a cosmological constant whose value is given in terms of the potential strength.