Can Galileons support Lorentzian wormholes?

We discuss the possibility of constructing stable, static, spherically symmetric, asymptotically flat Lorentzian wormhole solutions in general relativity coupled to a generalized Galileon field $\pi$. Assuming that the Minkowski space–time is obtained at $\partial\pi=0$, we find that there is tension between the properties of the energy–momentum tensor required to support a wormhole (violation of the average null energy conditions) and stability of the Galileon perturbations about the putative solution (absence of ghosts and gradient instabilities). In three-dimensional space–time, this tension is strong enough to rule out wormholes with the above properties. In higher dimensions, including the most physically interesting case of four-dimensional space–time, wormholes, if any, must have fairly contrived shapes.