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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.

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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.

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    $\begingroup$ The theory has long been considered. The first experimental result is from the recent kilonova detection which saw gravitational waves and EM waves only 1.2 sec later, over a distance of more than a billion light years. That is the same speed to within 1 part in about $10^{15}$, indeed a nice limit on the graviton mass. $\endgroup$ – Bob Bee Nov 9 '17 at 1:08
  • $\begingroup$ @BobBee I think you are talking about the recent neutron stars collision. Don't think your numbers are correct. The collision happened 130 million light-years away. The gravitational wave signal lasted about 100 secs. And our Fermi gamma-ray telescope identified gamma rays 1.7 sec later. Now, if gravitational waves travel slower, then why did we observe the light after the waves? I think we need precise calculation on how long does it take for light to emerge after the gravitational wave is created and then go from there. $\endgroup$ – LostCause Nov 9 '17 at 15:37
  • $\begingroup$ We need to be very precise because 1.7 light seconds is a bigger distance than the distance between earth and moon. Nevertheless, these types of detection help us correct our understanding of science. $\endgroup$ – LostCause Nov 9 '17 at 15:41
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    $\begingroup$ I can't remember the exact number, I posted on it a few days after and it is one part in about $10^{15}$. You can also see the same numbers in a LIGO papaer on it. The 1.7 seconds is from threshold to threshold, but the calculations added more time because of the uncertainty of the model of when the gamma rays were produced. But yes, they did use some model results also. So, not exact but the number is still pretty amazing. Even 100 seconds would still be the same speed with the exponent being 13. The graviton mass limits were also calculated but some model dependency also. $\endgroup$ – Bob Bee Nov 10 '17 at 4:38
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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.

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