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RobinZ comments on The Graviton as Aether - Less Wrong
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A quick Internet search reveals that there is research being conducted to test this testable conclusion. (Full disclosure: I never understood special relativity very well, much less general relativity; I do not feel qualified to comment on the results described in the link, attractive though they are.)
Interesting. However, I think it's reasonably plausible that this is only a "test" in the same sense that the Brans-Dicke theory is "testable": namely, if the test doesn't work out (as indeed it didn't), you can just cross out "X >= 10^13" and write in "X >= 10^15" (or whatever). I don't know LQG well enough to know if this is the case, though.
If nothing else, such alterations are likely to make the theory weak as a meme - much like Arthur Eddington's theories about the fine-structure constant. (By these arguments, it must be exactly 1/136! What, it's closer to 1/137? Well, by these arguments it must be exactly 1/137!)