Weyl scalar calculation

Question

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?

Answer 1

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.