Graduate How can curl of 4-vector or 6-vector be writen?

Click For Summary
The discussion focuses on how to express the curl of a 4-vector and 6-vector, specifically examining the 4-vector A4=(a1,a2,a3,a4). It highlights that the traditional curl operation is limited to three dimensions, but in four-dimensional space-time, the exterior derivative can be used to derive a two-form that corresponds to the three-dimensional curl. This approach is relevant in the context of the electromagnetic field tensor, which is derived from the 4-potential and contains six independent components. The conversation emphasizes the relationship between these mathematical constructs and their applications in physics. Understanding these concepts is crucial for advanced studies in fields like electromagnetism and differential geometry.
The Count
Messages
27
Reaction score
0
How can a curl of 4-vector or 6-vector be writen? Let's say that we have a 4-vector A4=(a1,a2,a3,a4)
how can we write in details the ∇×A4

Can we follow the same procedure for 6-vector?
 
Physics news on Phys.org
The normal curl is an operation which is restricted to three-dimensions. If you look at the 4-vector as a one-form in space-time you can find the exterior derivative, which will be a two-form. Essentially, this will be the thing corresponding to the three-dimensional curl and the set of two-forms is 6-dimensional in a 4-dimensional space-time.

This construction appears, eg, when considering the electromagnetic field tensor as the exterior derivative of the 4-potential ##F = dA##. As you may be aware, the field tensor ##F## has six independent components, three for the electric field and three for the magnetic.
 

Similar threads

  • · Replies 9 ·
Replies
9
Views
3K
  • · Replies 2 ·
Replies
2
Views
4K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 2 ·
Replies
2
Views
4K
  • · Replies 7 ·
Replies
7
Views
2K
Replies
54
Views
4K
Replies
33
Views
4K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 9 ·
Replies
9
Views
1K
Replies
3
Views
2K