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As far as I can see most of the questions about rotation of a black hole refer to the appearance of a hole to an outside observer.

What about the region within the Schwarzschild radius?

According to Mach’s principle, the frame of rotation can only be defined in relation to distant masses. In the general theory of relativity, would distant masses outside the hole have any influence inside the Schwarzschild radius or how would rotation otherwise be defined?

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This probably ought to be a comment rather than an answer, but I have insufficient reputation yet to comment.

Standard GR is not Machian in the sense you describe. Rotation can be locally defined, because it is a form of acceleration, and this is unambiguous (acceleration means deviation from geodesic motion).

Also, most standard GR texts discuss the interior region of the Kerr geometry. There is a ring-shaped singularity surrounded by closed timelike curves.

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  • $\begingroup$ Note: the interior Kerr solution is also known to not be stable to some perturbations, and the closed timelike curves makes the space acausal. Also, as @Ben Niehoff also said, GR is not strictly Machian although some effects are Machian like such as frame dragging outside near a black hole. Finally, I think there are some solutions to the EFEs for a black hole (can't remember if spherical or axially symmetric/rotating), with a rotation at infinity which then has some Machian like effect on the horizon. But it's all one solution at a time some effects, nothing exactly like a Mach principle. $\endgroup$ – Bob Bee Mar 2 '17 at 4:56
  • $\begingroup$ Not sure about black hole solutions, but the Gödel universe is a globally rotating spacetime. It is also full of closed timelike curves, but the point is: if one can unambiguously say the entire universe is rotating, then the theory is not Machian! en.wikipedia.org/wiki/G%C3%B6del_metric $\endgroup$ – Ben Niehoff Mar 2 '17 at 10:21
  • $\begingroup$ You can already comment, since you've got 50 rep. You have now 81. $\endgroup$ – peterh Mar 3 '17 at 15:26
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    $\begingroup$ At the time of writing, I had 1 rep. :) $\endgroup$ – Ben Niehoff Mar 3 '17 at 15:57

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