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For the sake of the argument lets use a black hole so the time dilation effects are greater and the planet in question is orbiting close enough to have significant TD effects but far enough to not fall in. So, in this scenario one part of the planet would be closer to the BH and therefore would experience a slower passage of time relative to the part of the planet on the other side. Would this cause the planet to "roll on time" due to the time dilation effects?

Assume it is not tidally locked.

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    $\begingroup$ Small point, but in reality, I think the tidal effects would be many thousand times stronger, if not millions of times stronger, so it would be hard to detect. That and the Roche limit for your average planet around a black hole would be well outside any significant time dilation. In theory, I don't see why it wouldn't happen and even be measurable. $\endgroup$ – userLTK Mar 8 '15 at 18:08
  • $\begingroup$ This would be interesting math. How do you figure it could be calculated? $\endgroup$ – Joe Mar 10 '15 at 23:33
  • $\begingroup$ And does that mean the mass that is creating the time dilation is losing energy? $\endgroup$ – Joe Mar 11 '15 at 0:09
  • $\begingroup$ It would depend on a few factors. The size of the planet and how close it was orbiting. I can give running the numbers a try a bit later. As to the 2nd question, Er, hmm. I don't think that's how it works. The mass doesn't spend any energy creating the time dilation - that wouldn't make sense. Similar to how the magnet doesn't spend any energy creating the magnetic field. What changes is the potential and/or kinetic energy of the object in orbit. But, I'm not a professional, so, you might want to verify that with someone who is. $\endgroup$ – userLTK Mar 11 '15 at 12:35
  • $\begingroup$ Any luck? I tried running the numbers and quickly got stuck. $\endgroup$ – Joe Mar 17 '15 at 23:36
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So we are to assume that the planet is not gravitationally locked for some period of time. This could possibly happen for a neutron star because it's gravitational bulging would be minimal owing to its structural integrity. We also have to assume that $t_{rot}\ll t_{orbit}$, because of $g^{tt}$ varies appreciably from one side of the planet to the other, then so does $g^{rr}$ and I'm not sure what kind of orbit that would be (presumably different than the particle orbit one finds in a GR exercise).

That being said I think different stories will be told by different observers. So let's focus on someone looking at the planet from a far away distance, he will see the far side of the planet rotating faster than the one close to the horizon. If this side was really close to the horizon, then the observer will basically see a very small percentage of the planet's mass stretched out on the far side rotating at normal speed, and most of the planets mass saturated on the other side almost motionless. So even though the speed at which land is moving on the far side is relatively fast, it would take a single point on the surface a very long time to complete one orbit. This asymmetric mass distribution I think will cause gravitational locking which is peculiar in that there was no gravitational bulging involved in the traditional sense of the word. So in other words this would imply that time dilation will be responsible somehow for slowing down the overall rotation.

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