How does a rotatiing reference frame relate to the large scale distribution of matter? It sounds like typical physic-y type stuff until it talks about my arms moving when I spin being related to the stars.
Aside from being really cool sounding, does this have any explanation? Is it more than just philosophical in nature? Why did Einstein find this important?
This comes up now and then in discussions with Biblical Astronomers – the obscure little group who believes that the Earth doesn’t rotate, but that the cosmos whirls around us. The fascinating thing is that the relativistic equations permit this! The Earth could be still, with the cosmos rotating about us.
Yes, parts of the cosmos would be moving faster than light…but this doesn’t matter, as they aren’t moving ftl with respect to the cosmos itself. Relativity permits it. (The early expansionary phase of the Big Bang may have exceeded the speed of light: that’s okay too.)
What about geostationary satellites? If the earth isn’t rotating, then they aren’t actually orbiting, just “hovering” over some spot on the earth’s surface. Whee! The vast rotating cosmos produces an outward attraction which exactly balances this. Frame dragging and so forth.
The Bad Astronomer has dealt with it. The same equations would also permit the Earth to be rotating with a period of only 12 hours, and the cosmos also rotating to make it up to an apparent 24 hours. Or Mars could be stationary, and the cosmos (including us!) all spinning about Mars.
Since this would focus on one point in the cosmos as “preferred,” and would violate various cosmological principles, the idea is not taken seriously by anyone worth listening to. But it is permitted by the equations.
So, yeah, you could say that the spinning stars out there lifted your arms up while you were perfectly stationary. No laws are violated in this interpretation (except for the titanic accelerations when you started and stopped spinning.) It’s nonsense, but it’s valid nonsense.
Imagine that you lived your life in a closed laboratory floating in space somewhere with some undetermined motion. Even though you have no external reference points, you can still tell the difference between non-rotating and rotating reference frames – if you’re in a rotating reference frame, you feel centrifugal force, but if you’re in a non-rotating frame you don’t. Maybe your lab is rotating, maybe your lab is accelerating, but you can still, using experiments solely inside your lab, figure out what a non-rotating inertial reference frame is – it’s the frame where the laws of mechanics are really simple without these funny “fictitious force” terms.
Now suppose you’ve done all these physics experiments in your lab, so you know what a non-rotating frame is, and all of the sudden a window opens in your lab and you can see the distant galaxies. You’d notice right away that the distant galaxies define a non-rotating reference frame – that is these reference frames that were interesting to you as making it particularly easy to do mechanics experiments in your lab, are also “cosmically interesting” in that the distribution of distant matter in the universe seems to define one.
Mach’s principle is essentially that this is not a coincidence, and that which frames are inertial are somehow determined by the motion of the distant galaxies and that if we could somehow make the distant galaxies rotate around us, then the results of local experiments we do to tell the difference between a rotating and non-rotating frame would change, too.
I find this very interesting and unsettling. What are other good instances/examples?..and then I’m off to try to find out why it’s unsettling…
I wish I could offer more… I first saw this in an old “Mathematical Games” column by Martin Gardner, and it came up a lot on The Bad Astronomy web site. The Bad Astronomer himself acknowledged it, but also pointed out that it could apply to any point in the cosmos, about which the rest of the cosmos would rotate, and at any given rate of rotation. It’s meaningless, for being applicable anywhere, in any fashion.
FWIW, Frame Dragging seems to have been confirmed observationally, by satellite experiments.
(If modern science isn’t disturbing, it isn’t being done right!)
The reason that the principle is important is that you don’t need spacetime for relativity to work. Relativity will work as long as you have curved space with the right topology. Adding time as a dimension, as best as I can understand it, was a sort of shortcut Einstein took so as to avoid using a completely dimensionless or Machian dynamic model - which would have been substantially more complex.
Here’s a good article on the subject and the man who is trying to kill the concept of spacetime.
I once asked here (but got an answer to a different question, at least I think it was to a different question):
“Does centrifugal force result from the mass of the rest of the Universe, or does it arise from the mere presence of other matter in the Universe providing a remote frame of reference independent of its mass?”
Are you saying that it is, indeed, the latter? That remote matter in the Universe provides a frame of reference that allows for centrifugal force to be manifest?
Thanks Karl, this is just the question I was trying to formulate here. It makes sense up until this point, and maybe it does at this point, just not the way I’ve ever thought of things, but this may begin to explain why centrifugal force is called a* fictitious force*.
Mach’s principle unto itself doesn’t say how inertial frames arise from distant matter, so it doesn’t really say whether the effect it is mass-dependent or not – just that there has to be some interaction between bodies that gives rise to the distinction between inertial and non-inertial frames.
Of course, if I had to guess I’d say that frame dragging in GR is probably a good part of the explanation of Mach’s principle. I don’t know enough about GR to say whether it is thought that frame dragging could be sufficient to explain it, though.
Can you go into more detail here? Suppose the stars were rotating in sync with the earth such that they seemed frozen in the sky, would there be no centrifugal effect on the planet making it more spherical instead of somewhat squashed at the poles?
Unfortunately I’m not the right person to be going into a lot of detail here – I’m sure one of the working physicists that the board is lousy with will be around eventually with a better answer, but by my understanding:
Yes (well, eventually – rock doesn’t just change shape instantaneously if you remove the force 
). The classic example of a “local rotating frame detector” is a bucket of water. If it is not rotating, the surface of the water will be more or less flat, but if it is rotating the surface will be made concave by centrifugal force. So by making observations solely of the bucket of water, you can tell whether you’re in a rotating frame or not.
So suppose you have a bucket of water, and you can tell by the concave surface that it is rotating. You can then perform a completely independent verification of the “bucket is rotating” observation, you can attach a telescope to the bucket and observe distant galaxies seeming to rotate from the point of view of the bucket. So you now have two seemingly independent experiments that show that the bucket is rotating.
Now suppose Q decides to fuck with you, snaps his fingers, and sets every galaxy more that half a gigaparsec from your bucket rotating in sync with the bucket. So now your telescope observations say that the bucket is not rotating.
If you don’t believe in Mach’s principle, you’d say that the telescope observations are off only because Q was fucking with you and that if you view the surface of the water you’d still see a concave surface because the bucket is really still spinning.
If you do believe in Mach’s principle, you’d say that by rotating the distant galaxies, Q actually changed the universe such that the frame the bucket is in is no longer a rotating frame, so if you looked at the surface of the water it would be flat.
The idea Mach’s principle is kind of analogous to the equivalence between gravitational and inertial mass. There are two ways to measure something (mass or rotation) that always seem to come out the same way there’s no real reason to expect them to. A theory that explains why this observed fact is true is more parsimonious than a theory that doesn’t. GR does a good job of saying why inertial mass and gravitational mass are the same, and maybe provides some hints as to why Mach’s principle seems to be true.
I don’t believe Mach’s principal ever went beyond being a qualitative statement. As you implied, it doesn’t tell you how to calculate the non-rotating frame from a given mass distribution in the universe. It was a counter to the idea of there being absolute space, and therefore absolute rotation with respect to that space.
I absolutely agree. Mach was talking about a good property that a theory should have, not a working theory by itself.
This was debated! As I was taught in cosmology class (a damn long time ago) Einstein and Mach disagreed. (But I don’t recall who held which opinion.)
The thought-experiment is: a spinning planet in an otherwise empty cosmos. Can it be said to spin at all? If so, does it develop a tidal bulge? If an empty cosmos has an intrinsic space-time “metric” – if it is meaningful to speak of an X-Axis and a Y-Axis in spatial geometry when there aren’t any distant stars upon which to base your reference direction – then “distance” or “direction” have meaning. But if not, then what?
Some said that empty space itself has a metric. Others said no.
Has it been resolved? The satellite confirmation of frame dragging is good evidence…but in which direction?
Mach’s principle is that inertia (the resistance of a body to a change in it’s state in motion) is due to the large scale arrangement of matter.
Einstein was influenced by Mach’s ideas, but general relativity is clearly not Machian.
But it should be. Were it so, every prediction of relativity would be preserved and all of the unnecessary baggage that is forced on us by spacetime would be gone.
A dimensionless version of relativity might be more amenable to some form of unification as opposed to what appears to the present intractable situation.