Solutions with pure radiation and gyratons in 3D massive gravity theories

TL;DR

The work constructs and analyzes exact solutions of 3D massive gravity theories (TMG and NMG) in the presence of a cosmological constant, focusing on pure radiation and gyratons within Robinson-Trautman (GR-like expansion $\Theta=\frac{1}{r}$) and Kundt ($\Theta=0$) spacetimes. By integrating the field equations under these ansätze, the authors obtain GR-like Robinson-Trautman gyraton solutions for both theories (with a notable GR limit) and derive Kundt gyraton solutions, including Type III pp-waves, while highlighting additional constraints that arise in NMG. A key finding is that, in 3D, the metric function $c_1$ remains arbitrary, allowing a larger solution space than in higher dimensions, and several explicit non-GR solutions are presented, including explicit vacuum examples. The results broaden the catalog of exact 3D solutions in TMG/NMG, illuminate the roles of pure radiation and gyratons in these theories, and lay groundwork for extending similar analyses to other 3D massive gravities.

Abstract

We find exact solutions of topologically massive gravity (TMG) and new massive gravity (NMG) in ${2+1}$ dimensions (3D) with an arbitrary cosmological constant, pure radiation, and gyratons, i.e., with possibly non-zero $T_{uu}$ and $T_{ux}$ in canonical coordinates. Since any `reasonable' geometry in 3D (i.e., admitting a null geodesic congruence) is either expanding Robinson-Trautman ($Θ\neq 0$) or Kundt (${Θ=0}$), we focus on these two classes. Assuming expansions ${Θ=1/r}$ (`GR-like' Robinson-Trautman) or ${Θ=0}$ (general Kundt), we systematically integrate the field equations of TMG and NMG and identify new classes of exact solutions. The case of NMG contains an additional assumption of $g_{ux}$ being quadratic in $r$, which is automatically enforced in TMG as well as in 3D GR. In each case, we reduce the field equations as much as possible and identify new classes of solutions. We also discuss various special subclasses and study some explicit solutions.