Kundt spacetimes of massive gravity in three dimensions

TL;DR

The paper advances the classification of Kundt spacetimes in three-dimensional massive gravity by deriving a tractable set of equations for TMG and NMG and identifying both CSI and non-CSI solutions. It provides the first explicit non-CSI TMG solutions at Λ = -μ^2 and a comprehensive CSI analysis for NMG, including a pair of CSI families (Type A and Type B) and a special Type C at λ = m^2, with scalar-curvature patterns mapped across parameter spaces. It also clarifies when Kundt solutions solve both full nonlinear and linearized theories, and when TMG and NMG share common solutions, including particular background choices that render the linearized equations exact. The results illuminate how special values of Λ and λ organize the solution landscape, revealing smooth interpolations between AdS, CSI, and non-Einstein conformal cases, and contributing explicit spacetimes relevant to holography and the broader study of 3D massive gravity.

Abstract

We study Kundt solutions of topologically massive gravity (TMG) and the new theory of massive gravity (NMG), proposed recently in arXiv:0901.1766. For topologically massive gravity, only the CSI Kundt solutions (i.e., solutions with constant scalar polynomial curvature invariants) were found very recently in arXiv:0906.3559. We find non-CSI explicit solutions of TMG, when $Λ=-μ^2$, and these are the first such solutions. For the new theory of massive gravity, after reducing the field equations to a manageable system of differential equations, the CSI solutions are discussed in detail, with a focus on a subfamily whose solutions are particulary easy to describe. A number of properties of Kundt solutions of TMG and NMG, such as an identification of solutions which lie at the intersection of the full non-linear and linearized theories, are also derived.

Figures (7)

• Figure 1: The square of the Ricci tensor (in units of $\mu^4$) of the CSI Kundt solutions of TMG versus $\Lambda$ (in units of $\mu^2$). The solid curves correspond to metrics of type A and the dashed curves to metrics of type B.
• Figure 2: Nature of the roots of equation (\ref{['cEuu']}) for the CSI Kundt solutions of TMG of type B. The presence of non-real roots is indicated by the thick dashed curve. Multiple roots occur at $\Lambda=-4\mu^2/27$ and $\Lambda=5\mu^2/27$. $R^{\alpha\beta}R_{\alpha\beta}$ and $\Lambda$ are expressed in units of $\mu^4$ and $\mu^2$ respectively.
• Figure 3: The square of the Ricci tensor (in units of $\mu^4$) of (a) the new Kundt solutions of TMG satisfying (\ref{['new1']}) and (b) the new Kundt solutions satisfying (\ref{['new2']}).
• Figure 4: The scalar curvature of the CSI Kundt solutions of NMG versus (a) $\lambda$ and (b) $\Lambda$, for positive $m^2$. The solid curves correspond to metrics of type A and the dashed curves to metrics of type B. In (a), the two parabolas meet tangentially at the point $(-35/289,-12/17)$. In (b), $\Lambda_\pm=2\pm4\sqrt{3/7}$, and the lines meet the ellipse tangentially at the points $(-2/17,-12/17)$ and $(70/17,-12/17)$. The three quantities $R$, $\Lambda$, and $\lambda$ are all expressed in units of $m^2$.
• Figure 5: The scalar curvature of the CSI Kundt solutions of NMG versus $\lambda$, for negative $m^2$. The solid parabola represents metrics of type A, and the dashed curve (which is part of a parabola) represents metrics of type B. $R$ and $\lambda$ are expressed in units of $|m^2|$.