Comments on (super)luminality

TL;DR

The paper challenges the claim that a superluminal fluctuation on a specific background rules out massive gravity, showing that the background is physically pathological due to a rapidly growing mode that can overwhelm the background with negligible energy cost. It provides an alternative exact, Minkowski-like background in a boosted frame where fluctuations are subluminal and the BD ghost is absent, illustrating that superluminality here is background-dependent. It also demonstrates that treating the background as a perfect fluid yields complex stress-tensor eigenvalues and no rest frame, making such a configuration unphysical and not a generic feature of the theory. Together these results argue for a careful, background-dependent assessment of superluminality in massive gravity and highlight the viability of ghost-free backgrounds in constraining the theory's phenomenology.

Abstract

Recently, in an interesting work arXiv:1106.3972 a solution of the equations of motion of massive gravity was discussed, and it was shown that one of the fluctuations on that solution is superluminal. It was also stated that this rules out massive gravity. Here we find that the solution itself is rather unphysical. For this we show that there is another mode on the same background which grows and overcomes the background in an arbitrarily short period of time, that can be excited by a negligible cost in energy. This solution is triggered by the parameter governing the superluminality. Furthermore, we also show that the solution, if viewed as a perfect fluid, has no rest frame, or that the Lorentz transformation that is needed to boost to the rest frame is superluminal itself. The stress-tensor of this fluid has complex eigenvalues, and could not be obtained from any physically sensible matter. Moreover, for the same setup we find another background solution, fluctuations of which are all stable and subluminal. Based on these results, we conclude that the superluminality found in arXiv:1106.3972 is an artifact of using an inappropriate background, nevertheless, this solution represents an instructive example for understanding massive gravity. For instance, on this background the Boulware-Deser ghost is absent, even though this may naively appear not to be the case.