Difficulty in understanding the notation

[PhyAmateur]

We have this stationary metric, $$ds^2 = e^{2U}(dt+\omega_idx^i)^2 -e^{-2U}dx^2$$

The book wrote down the spin connections of this:

$$ \omega^{0i}=\partial_ie^{U}e^0 +e^{3U}\partial_{[_i\omega _k]}e^k $$
and $$ \omega^{ij}= e^{3U}(\partial_{[_i\omega _j]}e^0-\partial_{[_ie^{-2U}\delta_j]k} )$$
it is this $$ \partial_{[_i\omega _j]}$$ that I didn't understand along with the $$\partial_{[_ie^{-2U}\delta_j]k}$$ . If we unwrapped these, what do we get? I am only having problem with the notation.

Note please that the book mentioned that $$\partial_{[_i\omega _j]}= - \frac{1}{2} \epsilon _{ijk}\partial_kb$$ where I have no idea what he meant by b. The first time I saw this b was in this note.

{I can attach the page of the book if needed (if my writings here are not clear as upper indices or lower ones).}

 

[Matterwave]

I think...probably attach the page of the book.

 

[PhyAmateur]

Yes of course. Here it is. @Matterwave

 

[Matterwave]

Oh...I see what is happening lol. Your latex made the omega's subscripts and made the notation much more confusing! The bracket notation means the "anti-symmetric part". So, say I have a tensor $T_{ij}$ then $T_{[ij]}=\frac{1}{2}(T_{ij}-T_{ji})$ (some references might have the 1/2 there as a normalization factor, while others might not, check which convention is being used by your book). With more indices, you just have to be alternate signs, a + sign will appear for terms which are even permutations of the indices, and a - sign will appear for odd permutations. That's basically all there is to it. If I have a rank two co-variant tensor which is the outer product of two one-forms, then the notation just looks a little weird, but it means the same thing:

$$A_{[i}\omega_{j]}=\frac{1}{2}(A_i\omega_j-A_j\omega_i)$$