Angular Momentum of the Universe Our planet revolves around its axis with a period of roughly 24 hours. But with respect to what? Is there an inertial frame that we can attach at the earth's position with respect to which we can measure the angular velocity unambiguously? And I think we can answer that by looking at the sun and neglecting for a moment the Earth's rotation around the sun. Every time we see the Sun rising we conclude that the Earth has completed a full revolution around it's axis.
Now suppose we ask this question about the Sun's revolution. Is there an inertial system with respect to which we can measure the Sun's total angular momentum? And moving on to larger length scales, what about the Milky way center that the sun rotates around? Does this structure continue and for how many levels? 
And when we finally ask the question about the angular momentum of the entire universe (trying to leave out gravity and space-time curvature for a moment, just the usual flat space-time) must we not conclude that the total angular momentum of the universe is plain zero because space is rotating itself with the universe? I mean there is no structure greater than the universe to give us a handle, so what would such a frame depend upon?
Any impressions, thoughts or ideas appreciated!
 A: The linear velocity of a reference frame is a relative quantity. It is always measured relative to another reference frame. In contrast, the angular velocity of a reference frame is an absolute quantity, which can be measured unambiguously.
The reason is that only non-rotating reference frames are inertial (they satisfy Newton's first law and fictitious forces are absent). In a rotating reference frame, we can measure fictitious forces (centrifugal and Coriolis), and conclude that the frame is non-inertial. The strength of the fictitious forces is directly proportional to the absolute angular velocity of the frame.
A concrete implementation of this measurement is Foucault's pendulum, which measures the rotation of the Earth's reference frame. If the Earth was rotating with period 24 hours in an otherwise empty universe (with no sun, stars or other objects), we would still use Foucault's pendulum to detect this rotation.
A: But with respect to what? Is there an inertial frame that we can attach at the earth's position wrt which we can measure the angular velocity without ambiguity?
No, there is not. Its all relative. But rather than the rising sun, if you use a distant star then your accuracy level is greatly increased.
must we not conclude that the total angular momentum of the universe is plain zero because space is rotating itself with the universe? I mean there is no structure greater than that.
Not at all, you could equally well conclude it is infinite, or 42. If you have no frame of reference to measure something that requires a frame of reference you cannot measure it. If you cannot measure it, you still cannot conclude it is zero. And you would have to define your 'Universe' in this case because as an advocate of the multiverse, I would certainly argue that the classic 'universe' is by no means the largest structure or construct.
However, let's consider as a thought experiment, a universe with only two planets of equal masses revolving around their center of mass. Where is the inertial frame?
You are asking "If we take out anything we can use for a reference, what can we use as a reference?". It is not a well formed question. I suspect that beings in this system would find it very hard to deduce they were orbiting with PlanetB, and yes, their impression of gravity would be very strange indeed. Although I suspect they would get there eventually, especially when they started trying to calculate their own satellite' trajectories.
An interesting question. I don't think this is an 'answer' of the quality I normally see on this site but you did ask for thoughts or impressions.... 
Great question though. :-)
