Gravitons with negative mass? I have been reading several papers on massive gravity.  All of them have equations that involve the square of the graviton mass, rather than graviton mass itself.  See for example, equations 43 and 44 in 
https://arxiv.org/abs/1505.00743 
or de Rham's review of massive gravity in https://arxiv.org/abs/1401.4173  .
This makes me wonder if a graviton has a negative mass.  Such negative mass gravitons will still obey these equations (because the square of a negative number is a positive number, just like the square of a positive number).
Can anyone in this forum give any reason why gravitons cannot have a negative mass?  Of course, general relativity (GR) is compatible only with gravitons having a zero mass.  But massive gravity theories are a modification of general relativity.  So the question of massive gravitons contradicting GR does not arise.
 A: The universe could be full of negative mass particles that cause dark matter and dark energy, see http://arxiv.org/abs/1712.07962
The graviton mass could then be negative.
A: You are partially correct. Gravitons are considered to be massless. It is because gravitational force propagates with the speed of light and you need to be massless to reach such speeds.
But here is the interesting thing:
Some dude tried to put some negative mass in the Unvierse and tested the hypothesis that resulted in some paradoxes. Read up on "runaway phenomenon".
There is another hypothesis that antimatter may have negative (gravitational) mass. This would indeed violate the principle of equivalence since chunks of antimatter (eg, antiparticles produced in an accelerator) still have positive energy.
Since General Theory of Relativity is not the theory of matter, there is no reason why we could not introduce, in principle, matter with negative mass. People did try that but the problem with such a theory was not due to the equivalence principle, but due to the fact that such negative mass states would have lower energy than empty Minkowski space, so the vacuum itself would be unstable.
In massive gravity theory, gravitational waves obey equations where they travel at a speed lower than the speed of light. Food for thought: if we happen to observe a gravitational wave at the exact same time as we visibly see it, then we would know that they travel at the same speed but if we observe something different, then we have something to work with. I am sure some Noble Prize aspirants are probably working on that theory.
