Q:

I had a discussion with my highschool physics teacher and he was trying to explain to me the difference between 'extrinsic' curvature and 'intrinsic' curvature in spacetime. One example that really helped me was a "closed"/cylinder universe. We picture it as 'curved' because to picture it we intend to embed it in a higher dimensional spacetime. Yet in-reality, the closed spacetime can be everywhere flat.
Because it is everywhere flat, we can use special relativity. My teacher and I worked it out (and he found some papers as well), that even though there is no local preferred frame since special relativity works in flat spacetimes, the topology of the spacetime (I don't know what that means besides closed+finite in a direction vs open+infinite in a direction) gives it a preferred _global_ frame. That is, two clocks can move inertially and meet up at two different events, and not agree on the time spent between meeting. This preferred "global" frame is (as we were able to work out, and as the paper stated), the one in which a surface of constant time goes around the closed direction of the universe and meets back up (in other frames, it will instead "helix" around in spacetime).
I'm sorry, for all the background, but I'm really enjoying this, and wanted to make it clear the background behind this question which would probably sound very odd otherwise.
The question is:
Since spacetime in string theory has many curled up dimensions, while there is no preferred local frame (locally SR remains exact -- local poincare invariance is exact), there will be a global preferred frame in the same sense as above. Since strings can wind all the way around these dimensions, wouldn't their dynamics be affected by this global preferred frame. Furthermore, since we zoom out to see 4-dimensional spacetime in ordinary life, wouldn't these curled dimensions (since we kind of 'average' over all of them, their global stucture matters when we zoom out) induce a preferred frame in the remaining 4-dimensions? So does string theory predict what appears to be 'effectively' modification to SR when we try to describe the world as only 4-dimensional?
My teacher said we'd have to go to experts for this one, so we are both very interested in your response. Thank you!

- Samantha Hersh (age 18)

Harperville, Mississippi, US

- Samantha Hersh (age 18)

Harperville, Mississippi, US

A:

Samantha- Your background introduction is beautifully clear and spot-on. Please consider coming to study physics at UIUC. Your teacher sounds amazing as well.

As for your question, it's too hard for me, so I'll forward it to a string theorist. I'm sure the answer is that SR emerges as correct on spacetime scales large compared to the Planck scale in regions which are intrinsically flat, but I'm not sure what the limits are on that conclusion. Certainly if there are some other extended dimensions you can get effects which would never show up in plain SR. There may be other symptoms as well, which I hope we'll both soon learn about.

Mike W.

I guess what we're talking about here is an example where spacetime is S^1xR. There is a frame in which space is S^1 and time is R (the reals). The best way to think of this is as a 'compactification' of flat spacetime, RxR. We regard the circle S^1 as R/Z (reals mod integers), which means if we take a coordinate x along R, we regard a point x and a point x+2\pi m R as equivalent (for m an integer and R the radius of the circle).

This compactification breaks the Lorentz invariance of the 'covering space' RxR. To see this, start in the frame that I was discussing above, but impose the identification (x,t) ~ (x+2\pi R, t). If we boost to another frame, this identification becomes

(x',t') ~ (x',t') + 2\pi R (\gamma,v\gamma)

where v is the velocity of the frame (in units c=1). Note that (\gamma, v\gamma) is a unit spacelike Lorentz vector (which in the origin frame was just (1,0)). Thus, to specify the compactification, we have to select a fixed unit spacelike Lorentz vector, and this is where the notion of a 'preferred frame' comes from.

In string theory, what often happens is that we soup up this example to include extra large flat dimensions (that are not compact). So spacetime might be of the form S^1xR^3xR (compact direction, space, time). The 5-dimensional covering space has 4+1 Lorentz symmetry, but the compactified space has only 3+1 Lorentz symmetry (one can still boost along the large flat directions) -- one spatial direction is distinguishable from the other 3).

R

As for your question, it's too hard for me, so I'll forward it to a string theorist. I'm sure the answer is that SR emerges as correct on spacetime scales large compared to the Planck scale in regions which are intrinsically flat, but I'm not sure what the limits are on that conclusion. Certainly if there are some other extended dimensions you can get effects which would never show up in plain SR. There may be other symptoms as well, which I hope we'll both soon learn about.

Mike W.

I guess what we're talking about here is an example where spacetime is S^1xR. There is a frame in which space is S^1 and time is R (the reals). The best way to think of this is as a 'compactification' of flat spacetime, RxR. We regard the circle S^1 as R/Z (reals mod integers), which means if we take a coordinate x along R, we regard a point x and a point x+2\pi m R as equivalent (for m an integer and R the radius of the circle).

This compactification breaks the Lorentz invariance of the 'covering space' RxR. To see this, start in the frame that I was discussing above, but impose the identification (x,t) ~ (x+2\pi R, t). If we boost to another frame, this identification becomes

(x',t') ~ (x',t') + 2\pi R (\gamma,v\gamma)

where v is the velocity of the frame (in units c=1). Note that (\gamma, v\gamma) is a unit spacelike Lorentz vector (which in the origin frame was just (1,0)). Thus, to specify the compactification, we have to select a fixed unit spacelike Lorentz vector, and this is where the notion of a 'preferred frame' comes from.

In string theory, what often happens is that we soup up this example to include extra large flat dimensions (that are not compact). So spacetime might be of the form S^1xR^3xR (compact direction, space, time). The 5-dimensional covering space has 4+1 Lorentz symmetry, but the compactified space has only 3+1 Lorentz symmetry (one can still boost along the large flat directions) -- one spatial direction is distinguishable from the other 3).

R

*(published on 01/25/2010)*