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

In summary: Yes, that's what I was trying to get at when I said "tightly rolled up" but you're right, I was being pretty vague and your summary is a much better and more succinct way of putting it.In summary, the 3+1 dimensions fit together nicely with Lorentz covariance, allowing for rotations and Lorentz boosts without breaking the laws of physics. In string theory, the extra dimensions can be thought of as extra rows and columns in the metric tensor, with indices ranging over 0..4 instead of 0..3. If these extra dimensions are tightly rolled up, we might not notice them in our everyday experience, but they still affect the formalism and could have their own symmetries and cons
  • #1
Khashishi
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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.
 
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  • #2
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.
 
  • #3
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)?
 

1. What are extra dimensions?

Extra dimensions refer to theoretical spatial dimensions beyond the three dimensions of length, width, and height that we experience in our everyday lives. These dimensions are postulated by some scientific theories to explain phenomena that cannot be explained by the currently accepted laws of physics.

2. How do extra dimensions relate to Lorentz covariance?

Lorentz covariance is a principle in physics that states that the laws of physics should be the same for all observers, regardless of their relative motion. Extra dimensions are often incorporated into theories that aim to be Lorentz invariant, as these dimensions can help explain the behavior of particles at high energies and in different frames of reference.

3. What role does string theory play in understanding extra dimensions?

String theory is a theoretical framework that attempts to unify all the fundamental forces of nature, including gravity, into one coherent theory. In string theory, the fundamental building blocks of the universe are not particles, but rather tiny vibrating strings. This theory predicts the existence of extra dimensions to explain certain physical phenomena.

4. How do symmetries play a role in the study of extra dimensions?

Symmetries are mathematical patterns that describe the relationships between different physical quantities. The study of extra dimensions often involves exploring the symmetries within different theories and how they can help explain the behavior of particles and forces in these dimensions.

5. Are there any experiments or observations that support the existence of extra dimensions?

While there is currently no direct evidence for the existence of extra dimensions, some theories, such as string theory, make predictions that could potentially be tested through experiments. Some experiments, such as those conducted at the Large Hadron Collider, have attempted to detect the effects of extra dimensions by looking for deviations from the expected behavior of particles at high energies.

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