How are curvature and field strength exactly the same?

In summary, in this lecture series by Fredric Schuller, he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold. He also shows the relationship between general relativity and electromagnetism in the classical theory, emphasizing that the curvature of space-time is exactly the same as the non-abelian Yang-Mills field strength. However, it is not clear in what sense they are "exactly" the same, especially since the physical electromagnetism field strength tensor is a real valued and abelian object. It is also noted that the Lie group U(1) for electromagnetism is different from the Lie group GL for general relativity.
  • #1
victorvmotti
155
5
I am watching these lecture series by Fredric Schuller.

[Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller][1] @minute 34:00

In this part he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold.

He shows the relationship between general relativity and electromagnetism in the classical theory.

He emphasize that for instance that the curvature of space-time is exactly the same as the non-abelian Yang-Mills field strength.

It is not clear to me in what sense they are "exactly" the same.

Isn't the physical electromagnetism filed strength tensor a real valued object and abelian?

So is it right to say that in the case of electromagnetism the Lie group U(1) is different from the case of the general relativity Lie group which is GL? Otherwise, curvature and field strength are exactly the same?

[1]:
 
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  • #4
So, the field strength in this (GR) case is the field strength of local Lorentz transformations. The general coordinate transformations show up differently, as the insight above emphasizes. The field strength of the local translations is just the torsion, which in GR vanishes.
 

1. What is the relationship between curvature and field strength?

Curvature and field strength are directly related, meaning that they are essentially the same concept. This is because curvature is a measure of how much a field (such as an electromagnetic field) curves or bends in space, and field strength is a measure of how strong that field is at a given point. So, the stronger the field, the more it will curve, and vice versa.

2. How does the concept of curvature apply to different fields?

The concept of curvature can apply to various fields, such as gravity, electromagnetism, and even the curvature of space-time in general relativity. In each case, it refers to the way the field curves or bends, and how this curvature affects the behavior of particles or objects within the field.

3. Can you give an example of how curvature and field strength are the same?

One example of how curvature and field strength are equivalent is in the case of a strong magnetic field. The curvature of this field can be seen in the way it bends the path of charged particles, such as electrons, passing through it. And the strength of the field directly determines the amount of curvature observed.

4. Are there any practical applications of understanding the relationship between curvature and field strength?

Yes, there are many practical applications of understanding the relationship between curvature and field strength. For example, this understanding is crucial in the design and operation of technologies such as MRI machines, particle accelerators, and electric motors.

5. Is the relationship between curvature and field strength always the same?

The relationship between curvature and field strength can vary depending on the specific field being considered and the context in which it is being studied. However, the general concept of curvature and field strength being equivalent remains consistent across different fields and situations.

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