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

  1. May 22, 2016 #1
    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?
     
  2. jcsd
  3. May 22, 2016 #2

    Orodruin

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    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.
     
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