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:

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

 

[sardar]

As a fellow scientist, I can understand your difficulty in understanding the notation used in this stationary metric. It is important to note that notation can vary between different fields and even within the same field, so it is not uncommon to come across unfamiliar symbols or expressions.

In this case, the notation used for the spin connections, $$\omega^{0i}$$ and $$\omega^{ij}$$, may seem confusing at first. However, these symbols represent the components of the spin connection matrix, which is used to describe the rotational properties of a curved space. The subscripts, i and j, represent the spatial dimensions, while the superscripts, 0 and i, represent the time and space components, respectively.

The notation $$\partial_{[_i\omega _j]}$$ may seem unfamiliar, but it simply means the partial derivative of the spin connection with respect to the spatial coordinates. Similarly, $$\partial_{[_ie^{-2U}\delta_j]k}$$ represents the partial derivative of the exponential function with respect to the spatial coordinates, multiplied by the Kronecker delta function.

The book mentions that $$\partial_{[_i\omega _j]}= - \frac{1}{2} \epsilon _{ijk}\partial_kb$$, where b is a constant. This is a common way to express the spin connection in terms of the Levi-Civita symbol, which is used to represent the cross product in three-dimensional space. So, the author is essentially saying that the spin connection can be expressed in terms of the cross product of the spatial coordinates.

I would suggest consulting with your colleagues or referring to other sources to gain a better understanding of this notation. It is important to familiarize yourself with the different symbols and expressions used in your field of study, as it will greatly aid in your understanding and interpretation of scientific literature.