Is Mach's Principle truly a paradox in rotational frames?

[Leo Liu]

I have been reading Kleppner's An Intro to Mech recently and have found an interesting discussion on the nature of rotational motion in the book.

The authors wrote:
Newton described this puzzling question in terms of the following experiment: if a bucket contains water at rest, the surface of the water is flat. If the bucket is set spinning at a steady rate, the water at first lags behind, but gradually, as the water’s rotational speed increases, the surface takes on the form of the parabola of revolution discussed in Example 9.6. If the bucket is suddenly stopped, the concavity of the water’s surface persists for some time. It is evidently not motion relative to the bucket that is important in determining the shape of the liquid surface. So long as the water rotates, the surface is depressed. Newton concluded that rotational motion is absolute, since by observing the water’s surface it is possible to detect rotation without reference to outside objects.
And:
Nevertheless, there is an enigma. Both the rotating bucket and the Foucault pendulum maintain their motion relative to the fixed stars. How can the fixed stars determine an inertial system? What prevents the plane of the pendulum from rotating with respect to the fixed stars? Why is the surface of the water in the rotating bucket flat only when the bucket is at rest with respect to the fixed stars?

I am intrigued by this paradox (or property?), which is named Mach's principle, because I think it is bizarre that we can't know whether a frame is inertial in such cases. Would you mind sharing your insight into it?

 

[anuttarasammyak]

Fixed stars determine non rotational frame. If the bucket is still and all the stars rotate around the bucket, the center water surface may be depressed but we have no way to check it, Mach's principle, in experiment.
And more in detail stars move from their "fixed" position. It is called as proper motion.

 

[A.T.]

I am intrigued by this paradox (or property?), which is named Mach's principle, because I think it is bizarre that we can't know whether a frame is inertial in such cases. Would you mind sharing your insight into it?

There are many threads about Mach's principle here. Try a search.

Regarding this bit:

How can the fixed stars determine an inertial system?

It doesn't have to be a causal influence by the current distal masses. But note that before the expansion all the mass in the universe in was closer together.PS: Did you mean "mystery" in the thread title? Makes sense both ways though.