We have this stationary metric, $$ds^2 = e^{2U}(dt+\omega_idx^i)^2 -e^{-2U}dx^2$$(adsbygoogle = window.adsbygoogle || []).push({});

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).}

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Difficulty in understanding the notation

Tags:

**Physics Forums | Science Articles, Homework Help, Discussion**