What is a principal null direction

[purakanui]

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

 

[Bill_K]

A principal null vector is an eigenvector of the Riemann tensor. Consider first the Maxwell tensor F_μν. A principal null vector of the Maxwell tensor is a null vector k^ν such that k^αF_αν ∝ k_ν. An equivalent way of writing this is k^αF_α[νk_σ] = 0. Similarly a principal null vector of the Riemann tensor is a null vector such that

k^αk_[μR_ν]αβ[σk_τ]k^β = 0

 

[purakanui]

Thanks for that!

 

[purakanui]

I am actually having a little trouble with the anti-symmetric part of your answer. I understand that $$T_{[ab]} = \frac{1}{2}(T_{ab}-T_{ba})$$. But how do you expand when the square brackets go over more than one tensor? I.e. in $$K^aF_{a[v}K_{\sigma]} = 0$$?
Thanks again

 

[George Jones]

I am actually having a little trouble with the anti-symmetric part of your answer. I understand that $$T_{[ab]} = \frac{1}{2}(T_{ab}-T_{ba})$$. But how do you expand when the square brackets go over more than one tensor? I.e. in $$K^aF_{a[v}K_{\sigma]} = 0$$?
Thanks again

$$K^aF_{a[v}K_{\sigma]} = \frac{1}{2} \left( K^aF_{a v}K_\sigma - K^aF_{a \sigma}K_v \right)$$

 

[purakanui]

Cool, thought that would be the case.