Weyl Tensor Notation in Dimension 4

[jfy4]

Hi,

I'm getting used to the anti-symmetric bracket notation used with indices and I can't seem to find the Weyl Tensor written fully out. So I want to make sure I get it. Here is my attempt in dimension 4.

$$W_{abcd}=R_{abcd}-\frac{1}{2}\left[g_{ac}R_{db}-g_{ad}R_{cb}-g_{bc}R_{da}+g_{bd}R_{ca}\right]+\frac{1}{6}R\left(g_{ac}g_{db}-g_{ad}g_{cb}\right)$$

It is written on wiki as

$$W_{abcd}=R_{abcd}-\frac{2}{n-2}\left(g_{a[c}R_{d]b}-g_{b[c}R_{d]a}\right)+\frac{2}{(n-1)(n-2)}R\, g_{a[c}g_{d]b}$$

Did I do it correct for n=4?

Thanks,

 

[cristo]

Looks good to me.

 

[Bill_K]

Looks right. Try contracting, g^acW_abcd, and see if you get 0.

 

[jfy4]

Looks good to me.

Looks right. Try contracting, g^acW_abcd, and see if you get 0.

Thanks, I got 0 when I contracted.