Extra Dimensions: Lorentz Covariance, String Theory, Symmetries & More

[Khashishi]

We know that the 3+1 dimensions all fit together in a nicely with Lorentz covariance. We can rotate and apply Lorentz boosts without breaking the laws of physics. How do string theory's extra dimensions fit in with this? Does it make sense to rotate between a normal space dimension and one of these "extra" dimensions, or between two extra dimensions? In relativity, a 4-vector is something special. Do we have to go to a 10-vector in 10D theory? Or are all these extra dimensions not part of the same "space-time"? Are there additional symmetries and conserved quantities associated with these dimensions?

Sorry--that's a lot of questions at once.

 

[Nugatory]

There's a short but interesting discussion of this in (of all places!) Hartle's GR textbook for undergrads.

Think of an "extra" dimension as an extra row and column in the metric tensor, so that instead of being 4x4 it's 5x5 with one "extra dimension" and our indices range over 0..4 instead of 0..3. If the metric coefficients are such that even a large displacement along this extra dimension doesn't change the value of $ds^2$ much (that's the "tightly rolled up" bit), we might never never notice... but all of the formalism will continue to work.

 

[dslowik]

Nugatory, Isn't there also something special about the topology of the 'extra' dimensions? i.e., if the are 'curled up' so that , e.g., if you move 1 cm along the extra dimension you return to your starting point (in both the extra dimension and in 3d space)?