NumberTenOx

Assuming that the universe is finite and connected, imagine the view of a high velocity observer. The direction of motion would undergo the Lorentz contraction, in other words, the size of the universe along the direction of motion would appear smaller than the other two dimensions (wouldn't it?).

Ultimately, at some speed, the direction-of-motion dimension would shrink altogether, somewhat resembling what is theorized as the curled up dimensions for String Theory.

Is there a discussion somewhere of this similarity? Is it at all possible that the extra dimensions appear curled up because all real particles are traveling through those dimensions at a high velocity? Or am I being too simplistic?

Anglican

Well the theories that use these 'curled up' dimensions are defintely Lorentz covariant if that's what you're asking, but dimensions do not disappear as a result of Lorentz boost, rather the fact they are curled up is to do with the structure of spacetime.

NumberTenOx

Let me back up a step.

So, given my assumptions, would the observer not measure the dimension of motion as smaller than the other dimensions?

Answerer

No, they are curled up because

1) Superstring theorists calculated the solutions to their equations to have eleven dimensions

2) Only four large dimension are observable which mean other seven dimensions must be too small to be seen

3) And in order to visualize the other seven dimensions that are small and mixing together, physicist find a Calubi-shaped manifolds that solve their problems

4) Manifolds are said to be "curved" most of the time

Jesse

If the universe is closed, so that if you travel far enough in one direction you'll return to where you started, then it might be true that this distance would appear smaller as you travel faster (But faster relative to what? Maybe the average rest frame of all the matter in the universe, like the rest frame of the CMBR...), although I'm not sure about this. But if the universe is flat or open, it should be infinite in all 3 non-curled spatial dimensions no matter how fast you go.

NumberTenOx

Thanks. Yes, if the universe is infinite, it doesn't matter how fast you travel.

However, if it's closed, couldn't you in theory make the size of the universe in the direction of travel arbitrarily small, just by increasing your velocity?

This would, I suppose, establish a universal frame of reference, i.e. the velocity that makes the universe look the largest in all directions. However this is already established by the CMBR, the velocity that makes the CMBR look the most uniform.

Archie Goodwin

Not really. In fact Lorentz symmetry is only a symmetry in an infinite flat geometry.

When we say a direction is curled up, we mean that if you travel a fixed spatial distance, we return to the same point. For a circle of circumference C, if you go a distance C, you're back where you started. Now for a moving observer, this is not the case. Since a Lorentz transformation mixes up time and space, what this observer will see is that if you travel a spatial distance C', and travel through time a distance t, then you return to where you started. Hence for a moving observer, the curled up dimensions look quite different from what an observer at rest sees. The frames are not equivalent.

So when for example string theorists talk about curled up dimensions, they are referring to the situation where you travel a spatial distance and no time distance, and still return to the original point. This is unambiguous.

Jesse

In a spacetime that's possible to foliate, can't you foliate it in different ways with respect to each observers' definition of time and space? If so, couldn't you define the size of the universe relative to an observer in terms of "the situation where you travel a spatial distance and no time distance, and still return to the original point", making use of the way time and space are defined in that observer's preferred foliation?

easychair

I thought time was the stuff that spatial distance slides on. How is it possible to move from a point without creating time distance? Never mind....