Water in bucket and GR

exponent137

I am reading about rotating water in bucket - about Mach's principle. But, how general relativity explains this bucket.
1. Let us say, that bucket with water is the only object in universe. What GR says.
2. Let us say that one star in distance is in this universe. How rotating water "know" that it rotates and so curves it surface.
3. etc.

PAllen

1) Same as Newton. Rotation is absolute.
2) Same as Newton.

GR does not incorporate a naive Mach principle.

However, there is a sense in which is Machian. In Newtonian theory, the question of what frames are inertial is somehow God given. In GR, the totality of matter influences what frames are inertial, and in a closed universe, completely determines what frames are inertial.

It is presumably beyond you, but the following discusses different formulations of Mach's Principle and discusses how at least one form of it is encompassed in GR:

http://arxiv.org/abs/1007.3368

pervect

1) Gr says that the surface of the water curves if it rotates, and doesn't curve if it doesn't. It doesn't particularly matter whether the bucket is alone in the universe or not. Assuming an asymptotically flat space-time (you need to specify this as well as the matter distribution), you can write down solutions where the bucket's surface curves, and ascribe rotation to these solutions and you can write down solutions where the bucket's surface doesn't curve, and ascribe no rotation to those solutions, regardless of the rest of the universe,

2) This may or may not be compatible with a specific interpretation of Mach's principle, depending exactly on what you interpret Mach's principle to mean. It seems to be incompatible with the average person's interpretation of the principle from my observations but not necessarily incompatible with all possible intepretations of the principle.

Passionflower

I have trouble with 'frames being inertial' in curved spacetime. Perhaps you should explain what you mean by that.

PAllen

There are more modern definitions I am not expert in, but the older definition is simply local Fermi-Normal coordinates along a timelike world line (I guess, possibly including rotating the frame). Juilian Barbour's idea is that in a closed universe, which frames, so defined, are inertial is completely determined by the matter distribution (in an open universe, there are boundary conditions as well, so matter distribution is not fully determinative).

Chronos

If you could rotate a bucket of water in an 'empty' [Minkowski] universe, the results might be surprising.

exponent137

I also think so. There is no reference non-rotating system...

It is also a question why it is easier to find relativity in linear direction than in circular direction!!!

exponent137

Thanks for a link. I hope that It will clarify a lot.

What was Einstein's opinion about this?

It seems to me, that if we remove all from universe, there no space-time remains - according to difeomorphism and general covariance.

There are claims that gravitational waves defines empty space - but they need one rest inertial sistem, but if we remove all matter, there is not reference rest inertial system...

pervect

There are some fundamental difficulties with carrying out such an experiment, so it's not surprising that we haven't performed such an experiment, and its more problematic in that we probably will never be able to actually perform such an experiment.

While we can't actual perform the experiment, we CAN easily analyze the situation to see what special and/or general relativity predicts. The short version is that SR and GR predict "nothing surprising".

So the way I see it is that we are currently left with no theoretical and no experimental support for any such surprises.

It's possible that some other theory will someday give theoretical support for "suprises", but that theory won't be SR or GR.

If you call the idea that there are "suprising" results in this circumstance "Mach's principle" (this isnt universally accepted, but it seems common, though I might add it seems more common on PF than it does in the literature), then we can say that SR and GR are "not Machian".

Now we haven't disproved this particular interpretation of Mach's principle, the problem with the principle as stated is that it can't be disproved in any experiment we can actually perform, putting it into the realm of philosophy. But what we can say is that SR and GR are not "Machian" in this particular sense.

exponent137

I do not understand, how SR or GR determines the preffered situation this means when bucket does not rotate?

Maybe it is a cause because relativistically the bucket cannot rotate as a rigid body, thus we can always decipher zero position?