There's a great series of papers by Franklin Felber which predicts a repulsive gravity effect measurable in the beam tunnel, but not in a way which interferes with the normal operation of the LHC.
The consequences of verifying this prediction, which is based on GR, cannot be overstated, because it is quite likely that some sort of space drive could be engineered from it.
I wonder if anyone is aware of plans to conduct this experiment?
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Interesting idea. I am skeptical although I didn't go through the relativistic physics in detail. However, I can say that he makes a couple of pretty basic errors with regards to the LHC, which leads me to believe that he hasn't discussed the idea with anyone at CERN.