# What is a principal null direction

1. Jan 18, 2012

### 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

2. Jan 19, 2012

### Bill_K

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

3. Jan 21, 2012

### purakanui

Thanks for that!

4. Jan 21, 2012

### 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

5. Jan 21, 2012

### George Jones

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

6. Jan 21, 2012

### purakanui

Cool, thought that would be the case.