what's the deal with rotating frames of reference

[69dodge]

Another question: suppose we were to that since everything is relative, we can consider the Earth to be at rest and the stars to be orbiting around us. Then a star 4 light years away is moving about 25 light years each day, or more than 9000 times the speed of light. Does this contradict the principle that nothing can travel faster than light?

I'm pretty sure it doesn't, because the basic idea of general relativity is that any reference frame is as good as any other, even reference frames that would usually be called "accelerated". But I don't know enough about the details of GR to say exactly why it doesn't. Probably the reason has something to do with the fact that you're here and the star is there, and, as you discussed with Yllanes, one can't compare a vector here with a vector there. (The vectors in this case being velocities.) My understanding is, the relative velocity of two objects close to each other can't exceed the speed of light, but the relative velocity of two objects far from each other isn't well-defined.

 

[Yllanes]

I'm pretty sure it doesn't, because the basic idea of general relativity is that any reference frame is as good as any other, even reference frames that would usually be called "accelerated". But I don't know enough about the details of GR to say exactly why it doesn't. Probably the reason has something to do with the fact that you're here and the star is there, and, as you discussed with Yllanes, one can't compare a vector here with a vector there. (The vectors in this case being velocities.) My understanding is, the relative velocity of two objects close to each other can't exceed the speed of light, but the relative velocity of two objects far from each other isn't well-defined.

I didn't see this originally. Of course, it is not in contradiction with relativity. The fact is that we can define lots of things that are sort of like a velocity, but can be greater than c. However, information is never propagated faster than c.

Let us consider a simpler example. Assume you have a laser pointer in your hand and aim it to a distant wall. You move your hand to the left and to the right and look at the point on the wall. It's obvious that the point is moving faster than your hand. In fact, if the laser point reached the Moon, the dot on the Moon would move faster than c (check this, it is easy). Now, does this mean that we have broken the 'light barrier'? No, because the dot cannot carry any information (or energy) faster than the speed of light. Imagine two observers on the Moon, A and B. The dot from the laser pointer sweeps the distance between them faster than a light signal would. Can we use this to communicate them at superluminal speeds? Think it yourself: We give them the following instructions:

A: When you see the dot, fire a bullet.
B: When you see the dot, take cover, as A has shot at you.

Has the warning message travelled from A to B faster than light? Once you work out this example, you will understand the one about the stars. But I think it is important that I do not give all the details, because the way to 'grok' these things is by doing them.

Another excercise (from Taylor & Wheeler): The maker of an oscilloscope claims a writing speed (the speed with which the spot moves across the screen) in excess of the speed of light. Is this possible, or a claim for the Million Dollar Challenge?

Edited to fix typos, my keyboard is dying...

 

[69dodge]

Actually, GR says that the centrifugal force will be present. It makes a definite prediction. The scenario is a glass of water in an otherwise empty (asymptotically Minkowskian) spacetime. As I said, it rotates with respect to the metric. If you don't believe me, notice that the metric is a field ijust like the electromagnetic field. Only that now, instead of a vector, we have a rank (0,2) tensor at each point. Gravitational waves (provided they exist) are a good indication that the geometry is as good a field as any.

I think we're kind of talking past each other here.

GR predicts that centrifugal force will be present, if the glass of water is rotating. If it's not rotating, GR predicts that centrifugal force won't be present. Correct?

So now the question is, how can we find out whether the glass is rotating, so that we can decide what GR predicts? The metric is considered real in GR, but it's not something that we can see, like we can see stars. Is there some way to distinguish a glass that's rotating with respect to the metric from a glass that's not rotating with respect to the metric, except by the presence or absence of centrifugal force? In other words, is there any way that we could decide in advance whether centrifugal force will be present, rather than just checking to see whether it is or it isn't? I don't see how we could. That's why I said, "GR says that centrifugal force might be present."

If we check and we find that centrifugal force is present, then we can say, "oh, I guess the glass must be rotating with respect to the metric." And if we find that centrifugal force is not present, then we can say, "oh, I guess the glass is not rotating with respect to the metric." But there's no way, beforehand, to look at the metric "directly" and see whether or not the glass is rotating with respect to it.

Do you agree?

Yes, there are observational bounds on the rotation of the universe.

Imagine only SR at play. A rotation of the Universe could be detected by measuring redshift of light from distant sources, looking in a direction perpendicular to the axis of rotation and comparing it with sources on the axis. The redshift would not be the usual (since there is no radial velocity) but a cuadrupolar redshift, due to time dilation at the sources. Roughly the same happens in GR.

In short: a rotating universe is theoretically possible in GR, well defined, and different from a nonrotating one. However, there are strong observational bounds (isotropy) on the rotation velocity.

Yes, I agree with all this.

Lots of different possible universes are consistent with GR. We, of course, live in just one universe. This one universe is not rotating, as far as we can tell (correct?), which is just what Mach's principle says should be the case. So why should we hold against Mach's principle the fact that GR says other universes are possible which contradict it, when the one universe we know about doesn't contradict it?

 

[Art Vandelay]

Has the warning message travelled from A to B faster than light?

No, it hasn't traveled from A to B at all. Are you saying that if the stars are rotating around us, they are not going from one point to another?

Just use it in the normal manner to weigh something.

What is the normal manner?

In the case of your pail, it is only necessary to determine a frame of reference in which light beams trace a straight path (determined say by a stretched string) and fix your pail with respect to that frame of reference to know that the water in your pail will remain flat.

You're confusing space and spacetime. Light travels in a straight line in spacetime, not space. The string is measuring distance in space. How would rotation affect the string?

 

[Yllanes]

Do you agree?

I don't seem to understand what you mean, I'm sorry. I think we are talking about different things. On the one hand, you can mathematically define the universe with the rotating glass and nothing else and get the definite prediction that there will be centrifugal forces. On the other hand (and this may be what you are saying), we cannot, experimentally, decide whether the universe with the glass is rotating except for the existence or absence of centrifugal forces.

So, GR says that there will be forces. We don't need to see anything to predict this because, once Einstein's equations are established, it is just a mathematical result, as the concept of a rotating glass is well defined in a mathematical sense.

Knowing this, if we are in the universe, we can look at the centrifugal forces. If there aren't any we can say the glass is not rotating. If there are, we can say it is. You seem to think this way of determining whether something is rotating is 'weaker' than just looking at the stars. But I disagree. Example: Hubble measured redshift and deduced that the galaxies are moving away from us. He didn't see them moving, just measured a consequence of their movement. This is the same thing, which is why I say that there will be forces if the glass is rotating.

So why should we hold against Mach's principle the fact that GR says other universes are possible which contradict it, when the one universe we know about doesn't contradict it?

I wasn't explicitly holding it against it. If and when we measure gravitational waves, however, I will hold that against Mach. My real issue with the principle is that it is too vague to be useful for predictions, i.e., science. If you formulate it on an unambiguous way, inconsistencies may arise. It is also dangerous for the layman for several reasons, not the least of them being that it may give rise to thoughts that GR is not a very definite and rigourous theory. And also it may make you think that the geometry is a result of the mass configuration alone, when in fact is a dynamical thing.

However, to be fair, the case may be made also for it. Misner, Thorne and Wheeler have several pages about it in their book and the book I quoted on an earlier post by Ciufolini and Wheeler has an extended discussion of it, including one precise formulation via gravitomagnetic effects. Einstein loved it, and wrote to Mach to congratulate him on it. Stephen Hawking said:

The observed isotropy of the microwave background indicates that the universe is rotating very little if at all [...] This could possibly be regarded as an experimental verification of Mach's Principle.

And now for something completely different:

No, it hasn't traveled from A to B at all. Are you saying that if the stars are rotating around us, they are not going from one point to another?

The light beam has passed through both points. This is a very close analogy to the stars. I don't know what you are saying here. Are you saying that relativity is wrong because stars move faster than c? I assume you are not, but believe me, the light beam and the galaxies rotating are really the same example.

 

[Art Vandelay]

The light beam has passed through both points.

One set of photons went through point A. Another set of photons went through point B. No photon went from one to the other.

I don't know what you are saying here. Are you saying that relativity is wrong because stars move faster than c?

I am not saying anything. I am asking questions.

I assume you are not, but believe me, the light beam and the galaxies rotating are really the same example.

How so? And how is it not the same as going from the Earth to Mars in a minute?

 

[davefoc]

Originally Posted by davefoc :
In the case of your pail, it is only necessary to determine a frame of reference in which light beams trace a straight path (determined say by a stretched string) and fix your pail with respect to that frame of reference to know that the water in your pail will remain flat.

You're confusing space and spacetime. Light travels in a straight line in spacetime, not space. The string is measuring distance in space. How would rotation affect the string?

I may be confusing space and spacetime but I also don't think I made my suggestion quite clear.

I propose three ways of determining if something is straight while floating in space.
1. Stretch a string between two points the string will be straight.
2. propel an object. The line formed by the center of gravity of the object will be straight if you are not measuring it relative to a rotating frame.
3. Fire a laser beam. The line formed by the beam will be straight if you are not measuring it relative to a rotating frame.

So my thought was to stretch a string and look to see if the propelled object or the laser beam moved parallel to it. If they are, at least one axis of the frame you are in is non-rotating. Do the same experiment in a direction that neither intersects the original line nor is parallel to it and you can determine if your frame is non-rotating in all three axies.

You could skip the string if you just have a good measurement system so you could set up way points in the frame of reference you are in that are on a line. If you can propel your object so that they go past each of the way points in your frame of reference then your frame of reference is not rotating in at least one axis.

Of course, it is much simpler to just instrument some sort of merry go round that is fixed to your reference frame and look for centrifugal forces. If there aren't any you are in a non-rotating frame.

If I was going to restate the question that I originally started this thread with it would be, "What is the property of space that allows us to detect whether we are in a non-rotating or rotating frame of reference?"

The problem with putting the question quite like that was that when this thread was started I didn't know whether it was a property of space or something else that allowed us to distinguish between rotating and non-rotating.

It now seems like it probably is a property of space that allows us to determine rotating from non-rotating. And that property seems to be the path that space imposes on an object that is moving through it without external forces.

ETA: Another roughtly equivalent statement that applies to the merry go round is that when something is moving in our frame of reference in a curved path a force will be required on the object to cause it to move in that curved path. Hence when the merry go round rotates in our frame of reference and we detect centrifigal forces we know that the merry go round is rotating relative to a non-rotating reference frame of the space we are in.