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Wei_Dai comments on Open Thread June 2010, Part 3 - Less Wrong
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Here's another possible experiment. Send a robot to the other planet, cut it in half, and then build a beam to push the two halves apart. If that planet is rotating, then due to conservation of angular momentum, this should cause its rotation to slow down, and you'd see that. If the two planets are just revolving around each other, then you won't observe such a slowdown in the apparent rotation of the other planet.
ETA: I'm pretty curious what the math actually says. Do we have any GR experts here?
Also, if you've asked the right question, would the stresses that would push the halves apart also show up as geological stresses?