Exotic massive gravity: Causality and a Birkhoff-like theorem

We study the local causality issue via the Shapiro time-delay computations in the on-shell consistent exotic massive gravity in three dimensions. The theory shows time delay as opposed to time advance despite having a ghost at the linearized level both for asymptotically flat and anti–de Sitter spacetimes. We also prove a Birkhoff-like theorem: any solution with a hypersurface orthogonal non-null Killing vector field is conformally flat; and we find some exact solutions.

Figure 1

Spacetime diagram depicting the shockwave geometry created by a massless particle with momentum $p$ moving in the $+x$ direction. Another massless test particle with momentum $q$ (blue line) propagating through this geometry experiences a time delay when it crosses the constant $u$ line. The time delay is denoted as a discontinuous jump $\mathrm{\Delta }v$ in the null $v$ direction. Time advance would violate causality.