Why is there a preferred frame in string theory with curled dimensions?

[CuriousKid]

I saw a show about string theory on TV, and I don't understand why this hasn't already been ruled out. My thought is this:
If there is a dimension that is curled up, wouldn't this violate relativity because moving would cause it to length contract? This seems to violate relativity since there is a preferred frame in which the curled dimensions have a maximum length. But we know relativity is correct! And if you don't have curled dimensions, then string theory can be ruled out since we don't see 10 dimensions. Either way, there seems to be a problem.

How can string theory with curled dimensions match the amazingly precise relativity experiments?


EDIT: I see now that there was a thread on 'preferred frames in a closed universe'
https://www.physicsforums.com/showthread.php?t=375432
and the consensus was YES there is a preferred frame in curled dimensions. In that example an experiment would have to go all the way around the universe to be able to detect it, but in String Theory's case ... "all the way around" is just a Planck length or something. Strings themselves can go all the way around a curled dimension. So reality built from these curled dimensions would obviously have a preferred frame for experiments at lengths greater than the distance around such a dimension. So why isn't string theory ruled out already?

 

[bcrowell]

This seems like an interesting question to me. My general impression is that string theory is seen as having SR built in, but there is some controversy over whether string theory is background-independent as required by GR.

I doubt that the length of the curled-up dimensions in string theory is really a distance that you can measure, in the sense that you can measure cosmological dimensions with macroscopic measuring techniques.

My guess is that if this was moved to the Beyond the Standard Model forum, you'd get more helpful answers.

 

[CuriousKid]

My general impression is that string theory is seen as having SR built in

Well yes, but so did the flat spacetime of the cylinder universe discussed in the other thread. The point is that despite having SR built in, a preferred frame is still seen if experiments can be done around the entire closed direction. Since string theory demands the curled dimensions to be small (to even have a chance of matching experiment), this requirement that signals must travels around the entire dimension before the preferred frame is apparent is not restrictive at all. In fact, the strings themselves can loop around the dimensions several times.

My guess is that if this was moved to the Beyond the Standard Model forum, you'd get more helpful answers.

I was hoping to discuss this in a purely classical sense.

If the answer is: if it wasn't for quantum mechanics, curled dimensions couldn't fit with precise measurements showing there is no preferred frame ... then fine, I'd be interested in seeing how quantum mechanics saves this. However, I have a feeling adding QM into the bunch is not useful here. I expect that the punch line can be discussed completely classically since the symmetries of relativity are classical as well as the "background metric + topology" of string theory describing the curled dimensions (at least appears to be classical).

 

[CuriousKid]

No one?
If not, can someone move this to the "Beyond the Standard Model" forum as bcrowell suggested?

Again, since this was approaching it from a classical / non-quantum question viewpoint, I was hoping the relativity forum would be more suited. Maybe someone here knew a "loophole" that this global preferred frame can be ignored. The only loophole I know of is: if experiments don't involve information going around the full dimension, it can be ignored. That loophole can't apply here. So I really don't see how String Theory can get around this.

 

[Fredrik]

If there is a dimension that is curled up, wouldn't this violate relativity because moving would cause it to length contract? This seems to violate relativity since there is a preferred frame in which the curled dimensions have a maximum length. But we know relativity is correct!

To get length contraction in the extra dimensions, you would actually have to be moving in those directions. Get in a rocket and fly as fast as you can in any direction (in the usual 3 dimensions), and you're still moving perpendicularly to the extra dimensions. So your argument clearly doesn't imply that there's a preferred frame on Minkowski spacetime.

You say that "we know relativity is correct". I would object to any claim that a theory is "correct". There's no such thing as a correct theory. All theories that have been found so far, and probably all that will ever be found, are wrong. The ones that we consider "good" are just less wrong than others. But your claim has two problems that are far more serious than that. (The second one is what really kills your argument).

1. What we know is just that SR (which is a theory about 3+1 spacetime dimensions) makes very accurate predictions about results of experiments. This doesn't in any way imply that a 9+1-dimensional version of the theory would be accurate.

2. If I was asked to try to write down a version of SR for 9+1 dimensions with six of the spatial dimensions "curled up", I would choose the appropriate underlying manifold and see if it makes logical sense to try to define a 9+1-dimensional version of the Minkowski metric on it. I don't expect such a theory to be consistent with a 9+1-dimensional version of Einstein's "postulates", but you clearly do. Why would it be? Einstein's "postulates" are just guesses about what a good theory of space, time and motion should look like, which are based on our experiences in 3+1 dimensions, and on the earlier theories that describe those experiences pretty well. (I'm talking about the role of inertial frames in Newtonian mechanics/Galilean spacetime, and the invariance properties of Maxwell's equations). There's certainly no reason to demand that a theory of 9+1 dimensions of space and time, with six of them curled up, would be consistent with that.

 

[CuriousKid]

To get length contraction in the extra dimensions, you would actually have to be moving in those directions.

If string theory is correct, we ARE moving in those directions. We are built of strings, which constantly are moving around those dimensions, and can even be wound around them multiple times. Every time we send a light signal, it will go through these dimensions as well. String theory demands that things move and vibrate in all the dimensions.

Relativity shows there will be a preferred frame for spacetimes with curled dimensions.

2. If I was asked to try to write down a version of SR for 9+1 dimensions with six of the spatial dimensions "curled up", I would choose the appropriate underlying manifold and see if it makes logical sense to try to define a 9+1-dimensional version of the Minkowski metric on it. I don't expect such a theory to be consistent with a 9+1-dimensional version of Einstein's "postulates", but you clearly do. Why would it be? Einstein's "postulates" are just guesses about what a good theory of space, time and motion should look like, which are based on our experiences in 3+1 dimensions, and on the earlier theories that describe those experiences pretty well. (I'm talking about the role of inertial frames in Newtonian mechanics/Galilean spacetime, and the invariance properties of Maxwell's equations). There's certainly no reason to demand that a theory of 9+1 dimensions of space and time, with six of them curled up, would be consistent with that.

Let me get this straight, your argument against the reasoning that string theory and relativity do not fit together, is that, we can't expect string theory and relativity to fit together?

WHAT!? How is that not agreeing with the point?

 

[Fredrik]

You seem to have the wrong idea about what "special relativity" means. It's defined by Minkowski spacetime, not by Einstein's "postulates". If you're going to generalize SR to 9+1 dimensions with 6 spatial dimensions curled up, you need to forget about the postulates. The appropriate generalization is a 10-dimensional manifold with a Lorentzian metric that induces the Minkowski metric on some of its 4-dimensional submanifolds.

This doesn't mean that string theory is inconsistent with special relativity. It means that special relativity with curled up dimensions is inconsistent with Einstein's postulates.

 

[bcrowell]

I agree with CuriousKid that this is a nontrivial issue, and I haven't seen anything so far in this thread that would resolve it.

 

[Fredrik]

What do you think needs to be resolved?

 

[dx]

My thought is this:
If there is a dimension that is curled up, wouldn't this violate relativity because moving would cause it to length contract? This seems to violate relativity since there is a preferred frame in which the curled dimensions have a maximum length.

Length contraction applies to bodies moving in space, not to the size or other properties characterizing the spacetime itself. A given body can appear to have different sizes depending on the reference frame from which it is observed, but the properties of spacetime are invariant. This is exactly analogous to the radius of curvature of a sphere being independent of which coordinate system you use to describe it. The size of the sphere is not a property of the coordinate system. In the same way, the size of a curled up dimension is not a property of the reference frame. For example, a cylindrical spacetime with metric dt² - R²dθ² can be said to have a size R, but this property is not tied to any reference frame; it is a property of the underlying Riemannian manifold.

 

[Fredrik]

Length contraction applies to bodies moving in space

Yes, but you can imagine a "body" that extends all the way around the universe along one of the curled up dimensions, and the question is then what lengths it would be measured to have by observers with different velocities.

I don't think there's an issue here that needs to be resolved. A theory of 9+1 spatial dimensions with six of them curled up that in any way resembles special relativity will say that different observers will measure different lengths if they move with different speeds along one of the extra dimensions. This doesn't contradict anything that's at all relevant.