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What is a principal null direction

  1. Jan 18, 2012 #1
    I am starting my honours project on colliding plane gravitational waves and I am learning about the Petrov-Penrose classification of the Weyl tensor. I can't find any good explanation on what a principal null direction is.

    Thanks

    Chris
     
  2. jcsd
  3. Jan 19, 2012 #2

    Bill_K

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    A principal null vector is an eigenvector of the Riemann tensor. Consider first the Maxwell tensor Fμν. A principal null vector of the Maxwell tensor is a null vector kν such that kαFαν ∝ kν. An equivalent way of writing this is kαFα[νkσ] = 0. Similarly a principal null vector of the Riemann tensor is a null vector such that

    kαkRν]αβ[σkτ]kβ = 0
     
  4. Jan 21, 2012 #3
    Thanks for that!
     
  5. Jan 21, 2012 #4
    I am actually having a little trouble with the anti-symmetric part of your answer. I understand that [tex]T_{[ab]} = \frac{1}{2}(T_{ab}-T_{ba})[/tex]. But how do you expand when the square brackets go over more than one tensor? I.e. in [tex]K^aF_{a[v}K_{\sigma]} = 0[/tex]?
    Thanks again
     
  6. Jan 21, 2012 #5

    George Jones

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    [tex]K^aF_{a[v}K_{\sigma]} = \frac{1}{2} \left( K^aF_{a v}K_\sigma - K^aF_{a \sigma}K_v \right)[/tex]
     
  7. Jan 21, 2012 #6
    Cool, thought that would be the case.
     
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