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The Large Hadron Collider and Gravitational Waves
When you slam protons together at very high speed as a Large Hadron Collider, if there are these extra dimensions. The debris would have taken energy away from the place where our detectors can detect it. So that's one real concrete way that you could find evidence for extra dimensions. But yeah, since extra dimensions are of space, gravitational waves in principle can provide a kind of CAT scan of the extra dimensions if you had sufficient control over those processes. We don't yet, or perhaps one day we will.