# Spherical polar coordinates in a tetrad frame

I am looking at a paper which writes the spatial components of a vector $$S_i$$ in terms of spherical polar coordinates w.r.t the local tetrad frame as (Eq 33 in the linked paper),

$$S_1 = s \sin \theta \cos \phi, S_2 = -s \cos \theta , S_3 = s \sin \theta \sin \phi$$

Now, intuitively, I think of $$S_{1,2,3}$$ as corresponding to the $$x,y,z$$ components of the vector.

If this is correct I would then expect $$S_2 = s \sin \theta \sin \phi$$ and $$S_3 = s \cos \theta$$, in accordance with the usual definition of spherical coordinates.

Why are the $$S_2, S_3$$ components 'switched' in this way?

Thanks

• There's more than one convention for how to write spherical components, as noted correctly at the Wikipedia entry that you linked. Is it explained by that alone? (I haven't dug into the paper that you referenced to try to trace it through.) – Brick Aug 5 '19 at 16:35
• Unfortunately, no – user1887919 Aug 6 '19 at 17:17

But, the coordinates they use make sense. This is a right-handed coordinate system. It looks like the following one where $$\vec{S}=(S_1, S_2, S_3)$$ with $$s=|\vec{S}|$$: 