Deep thoughts....

Pondering introductory mechanics has made me think again about some foundational issues that I've wondered about in the past.  Mach's Principle is the idea, put forward by Ernst Mach, that the inertial properties of matter depend somehow on the distribution of matter at far away points in the universe.  The classic thought experiment toted out to highlight this idea is "Newton's bucket".  Imagine a bucket filled with water.  Start rotating the bucket (relative to the "fixed stars") about it's central axis of symmetry.  After transients damp away due to viscosity of the water, the water's surface will have assumed a parabolic shape.  In a (non-inertial) frame of reference that co-rotates with the bucket, an observer would say that the surface of the liquid is always locally normal to the vector sum of the gravitational force (which wants to pull the liquid down relative to the bucket) and the (fictitious, and present because we're working in a rotating frame) centrifugal force (which is directed radially outward from the rotation axis).  [In an inertial frame of reference, the water has arranged itself so that the gradient in hydrostatic forces provides the centripetal force needed to keep the water rotating about the axis at a constant radius.]  This rotating bucket business, by the way, is a great way to make parabolic mirrors for telescopes.

Mach was worried about what rotation really means here.  What if there were no "fixed stars"?  What if there were no other matter in the universe than the bucket and liquid?  Moreover, what if the bucket were "still", and we rotated the whole rest of the universe about the bucket?  Would that somehow pull the liquid into the parabolic shape?  This kind of thinking has been difficult to discuss mathematically, but was on Einstein's mind when he was coming up with general relativity.  What does acceleration mean in an otherwise empty universe?  There seems to be reason to think that what we see as inertial effects (e.g., the appearance of fictitious forces in rotating reference frames) has some deep connection with the distribution of matter in the far away universe.  This is very weird, because a central tenet of modern physics that physics is local (except in certain very well defined quantum mechanical problems).    

The thing that's been knawing away at the back of my mind when thinking about this is the following.  There is a big overall dipole moment in the cosmic microwave background.  That means, roughly speaking, that we are moving relative to the center-of-mass frame of reference of the matter of the universe.  We could imagine boosting our velocity just so as to null out the dipole contribution to the CMB; then we'd be in an inertial frame co-moving with the overall mass distribution of the universe.  If inertial properties are tied somehow to the overall mass distribution in the universe, then shouldn't the center-of-mass frame of reference of the universe somehow be special?  Some high energy theorist may tell me this is all trivial, but I'd like to have that conversation.   Ahh well.  It's fun that basic undergrad physics can still raise profound (at least to me) issues.