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.
10 comments:
You have probably seen this before, but if not you might find it an interesting read:
http://dx.doi.org/10.1119/1.1937666
Anon. - Thanks; I hadn't seen that. The comment (http://dx.doi.org/10.1119/1.1942027) is also interesting and rather persuasive, at least to me.
I'm convinced that the graviton flux from distant bodies defines the local Minkowski reference frame. So without the distant bodies, there would be no special relativity (and also no clock to define time). Then the big bang corresponds to the time before clocks began running. See Unzicker's paper: A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke for some hints along this line.
Mechanics is a wonderful thing when it hits you on the head. Knowing how inertia works or knowing our current understanding of it does open your mind to just how the universe can work. If someone hadn't told us that inertia properties are due to the distribution of matter within our visible sphere across the universe then we might not have ever realised. It's a tremendous concept.
But i feel that, given we don't obviously know everything yet, what will be the most shocking and exciting stories in physics will be our understanding of these most fundamental concepts. I welcome any further progress :).
"What if there were no other matter in the universe than the bucket and liquid?"
Well, then the liquid would most likely form a little sphere like you see in movies from zero-g experiments.
Your focus on the fixed stars seems to forget that theres a roughly 6*10^24 kg big rotating solid sphere underneath the bucket in the reference frame where it isn't rotating.
Vincent - If you like, consider the earth to be there, too. Does it matter if the earth is co-rotating? One could do the same bucket argument without the earth, but with the bucket accelerating at 1g. Indeed, let's leave out rotation and just have the accelerating bucket and fluid. If there is no other matter in the universe, how can you tell if the bucket is accelerating?
Doesn't space exist even in the absence of matter? I'm no cosmologist, but I remember learning that space itself is expanding and that expansion occurs concurrently with matter motion. Namely, objects move farther apart AND space expands,too. To me this implies that even if there weren't far away stars, there still would be an inertial reference frame defined by that space.
If there is no other matter in the universe how can you tell if the bucket is accelerating?
Well, lets assume that with respect to my reference frame the bucket is accelerating with 1g, giving a nice smooth water surface. Taking the simplest picture we have F=m*a, leading to the conclusion that there is an earth sized planet near the bucket, or a much larger planet further away. To remove all mass from the universe we could assume that the force comes from an infinitely large planet positioned far away, with the mass constantly adjusting to give a constant force on the bucket, however then we would not be justified in considering the region containing the reference frame and the bucket as the entire universe.
Simpler still we could say, that considering the universe a closed system, the bucket cannot constantly accelerate (gain energy) unless we have some other energy source constantly decreasing, this would in the above case be potential energy due to the presence of another mass.
Hi Doug,
The question of: is the rest frame of the Universe a special frame of reference? Has not yet been answered.
Tests of Lorentz invariance (for example at UWashington:
http://link.aps.org/doi/10.1103/PhysRevD.78.092006) are one way to determine if that frame is special or not. And when I say special I mean that the laws of physics are somehow different in that reference frame, for example maybe there is a (v - v_CMB) term in the Hamiltonian so that this term vanishes in the CMB reference frame. Otherwise the rest frame of the Universe is just like any other frame of reference, it is not special (it just so happens that the CMB is stationary in that frame).
I'm not sure if that frame is special or not. I want to say that it isn't.
Nevertheless, Mach's Principle is still very compelling because if you were in an empty universe, then you wouldn't know if you were rotating or not. I'm of the opinion that you need the distant stars in order to define your reference frame, and therefore a problem with locality exists in the macroscopic world just as it does in the microscopic world.
@Vincent: If there is vacuum energy in the Universe then I believe you can accelerate the bucket at the expense of creating a gradient in the zero-point energy. You don't need to invoke a planet (which would invalidate the original presumption of an otherwise empty universe -- can an empty universe still have vacuum energy?).
How about an empty Universe and a rotating ball of liquid? In the case of a Universe filled with matter the ball should not be a uniform sphere, it will have a density gradient. So in the empty Universe (a truly empty one without vacuum energy) will the rotating liquid have a density gradient or be uniform?
It's a tremendous concept.But i feel that, given we don't obviously know everything yet, what will be the most shocking and exciting stories in physics will be our understanding of these most fundamental concepts.
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