As I recall, Mach would have said that a planet alone in the cosmos could not spin (and possibly not even accelerate) since there was nothing to spin (or accelerate) relative to. Einstein started out with a Machian view, but moved away from it later in life (again as I recall).
Of course, Gödel then went on to point out that if you have a planet alone in the cosmos, and it is spinning, you’d get all sorts of weird effects (to put it in context, time travel would be one of the less weird effects).
The juxtaposition of these two ideas opens up some fascinating rabbit trails. Do you have any approachable readings for us undergrads?
Would the spin be a quantity of energy, even if there was nothing else in the universe? Would there be effects of relativity from the center to the surface? The surface is moving faster inner parts.
If suddenly something else came into existence in the universe and collided with the spinning object. There would be a different exchange of energy depending if the collision was with or against the spin. So the energy of the spin was there, even if nothing else was. The existence or nonexistence of another object would not change that.
Yes…for the reason you note.
You can’t have it both ways. If something “came into existence” then that changes the universe, so you cannot use that new situation to argue about what the situation is in the case of the “nonexistence” of another object.
(The magical disappearance of an object is a commonplace thought-experiment in relativity, especially relating to the “speed of gravity.” If the sun suddenly disappeared, how long would it take earth’s orbit to straighten out? Nearly everyone says gravity would still “travel” through space at the speed of light, but there are a tiny handful of dissidents who say it propagates instantaneously. Unfortunately, physics makes no provision for masses “suddenly” disappearing, so the thought-experiment is null and moot.)
The Godel solution is probably most accurately described as a swirling homogeneous dust with a cosmological constant, but it is general regarded as unphysical, not least because it is it allows closed timelike curves.
A lone spinning planet would be expected to be described by a Kerr solution in the exterior matched to a suitable interior solution (though nobody has yet found such an interior solution).
If you have a planet alone in the universe, then what is time traveling? It seems unfair to say a part of the planet is time traveling as you’re then treating the single planet as at least two things. And if it’s not a part of the planet doing the time traveling, there’s nothing else there at all to do it.
One can imagine a piece of the planet (say, a spaceship) separating from the planet, flying along an appropriate path, and then returning.
As I said the Godel solution isn’t really anything like a planet alone in the Universe - it is full of matter, whereas an isolated planet is surrounded by a vacuum. Also it is homogeneous (the same everywhere), whereas you’d expect the field of an isolated planet to change with proximity to the planet. It is more like a cosmological solution where the particles swirl around each other.
It would be worrying if such an eminently physically reasonable situation like an isolated planet resulted in time travel as we’d probably have to accept that time travel was a generic physical prediction of GR or that the properties of a region of spacetime depended so heavily on what was going on outside of it that the whole theory had little predictive value.
However there are solutions which describe isolated objects which lead to time travel. A Tipler cylinder is an infinitely long rotating rod and hence unphysical, but in theory there is a region surrounding the cylinder that a hypothetical observer could go into and emerge from and meet a past version of themselves.
Don’t the time-travel equations work for a truncated Tipler cylinder? It doesn’t have to be infinitely long, just “sufficiently” long. And massive. And spinning insanely fast.
Still never gonna happen in the real world, but at least physically possible, which an infinitely long rod isn’t.
Direction is relative. In space which way is North?
You can define a direction as North in many different ways, such as the direction of the vector from Sol to alpha Ursa Minoris.
If you know the axis of rotation of some object, there are not that many ways to define north, just two.
If you are navigating in deep space, it is wise to pay attention to details like whether your coordinates are with respect to galactic north or something more local, which galaxy you are actually in, etc.
You’d have to create North to get a direction. When we say Polaris is North then that way is North. 
It’s not known if a long, but finite length, cylinder could reproduce the CTCs seen in a Tipler cylinder and it is at least plausible that the CTCs are a result of the unphysical aspects of the Tipler cylinder.
On the other hand, the gravitational field near a large but finite cylinder is very close to the field for an infinite cylinder (and become even closer to the infinite cylinder result as you get closer to the finite cylinder) as long as you avoid the ends of the cylinder.
A Tipler cylinder has very weird asymptotic properties so it is dangerous to assume too much about it’s similarity to a finite length cylinder. In addition the theorem Hawking used to argue for his chronology protection hypothesis suggests a link between the infinite length/age of a Tipler cylinder and the closed timelike curves in the vacuum surrounding it.
That doesn’t completely kill the idea that general relativity might predict that it is possible to construct a time machine from a rotating cylinder, but there are good reasons to suspect that time travel does not occur in the physically reasonable solutions of general relativity.
Thanks.
I remember a PBS program where a theoretical theorist claimed as much. Paraphrasing: “Space isn’t the absence of everything, but the “medium” in which matter can exist.” Thus objects spun relative to the space “medium”.
I was watching an episode last night of “How the Universe Works” about exploding stars, supernovas, gamma ray bursts, etc. and also wondered the same. Some the more violent explosions seem to be the results of stars spinning very rapidly. Presumably, these would happen regardless of the existence of any other bodies in the universe.