Comments on (super)luminality

TL;DR

This paper critiques a superluminal solution in massive gravity, showing it is unphysical due to instabilities and non-physical stress-tensors, and presents a stable, subluminal alternative background.

Contribution

It demonstrates that the superluminal solution is unphysical and introduces a stable, subluminal background in massive gravity.

Findings

The superluminal solution is unstable and unphysical.

A stable, subluminal background exists in the same setup.

The superluminal solution has a non-physical stress-tensor with complex eigenvalues.

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.