0
$\begingroup$

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take

$\Psi_{2}=C_{1342}=C_{pqrs}l^{p}m^{q}\bar{m}^{r}n^{s}$

where $C_{1342}$ is the component of the Weyl tensor and $\lbrace l,n,m,\bar{m}\rbrace$ is the Newman Penrose tetrad.

Now, I'm comfortable with calculating the Weyl tensor, but I don't understand what the product $l^{p}m^{q}\bar{m}^{r}n^{s}$ means. I know if we have a scalar product of two vectors with respect to a metric $g$ then $g(v,w)=g_{ij}v^{i}w^{j}=v^{T}(g)w$. But what about the four vectors?

$\endgroup$

1 Answer 1

0
$\begingroup$

You can think of the metric $g$ (with components $g_{ij}$) as being a (0,2) tensor (0 upper indices, 2 lower indices) that eats two vectors $v$ and $w$ (components $v^{i}$ and $w^{i}$, each vector being a (1,0) tensor; the contraction of the $i$ and $j$ indices is the 'eating') and spits out a number. Similarly, the Weyl tensor eats 4 vectors to spit out a number. Note that the indices on the vectors are contracted with the indices on the Weyl tensor, so it's not exactly a product of 4 vectors.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.