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George Keeling
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- TL;DR Summary
- Peculiar Coordinate conversion in Orodruin's Insight "Explore Coordinate Dependent Statements in an Expanding Universe"
I am studying @Orodruin's Insight "Explore Coordinate Dependent Statements in an Expanding Universe". It looks pretty interesting. About three pages in it reads "expanding ##x^a## to second order in ##\xi^\mu## generally leads to$$
x^a=e_\mu^a\xi^\mu+c_{\mu\nu}^a\xi^\mu\xi^\nu+\mathcal{O}_3
$$where we have introduced the notation ##\mathcal{O}_n## for terms that are of order three [##n##?] or higher in the coordinates." I don't know why one would expand ##x^a## like that.
##x^a## are general curved coordinates, ##\xi^\mu## are local Minkowski coordinates at some point ##p##. ##e_\mu## are orthonormal vectors at ##p## in the ##\xi^\mu## system. They might be basis vectors, I am not sure. ##e_\mu^a## are the coordinates of ##e_\mu## in the ##x^a## system. ##c_{\mu\nu}^a## is a mysterious thing to be discovered. It turns out to be not a tensor but more like a Christoffel symbol. I think. We are also told earlier that $$
e_\mu^a=\frac{\partial x^a}{\partial\xi^\mu}
$$so the first part of the first equation is $$
x^a=\frac{\partial x^a}{\partial\xi^\mu}\xi^\mu
$$which is just the tensor transformation law.
Where does the first equation come from and why don't we use the ordinary tensor transformation equation?
x^a=e_\mu^a\xi^\mu+c_{\mu\nu}^a\xi^\mu\xi^\nu+\mathcal{O}_3
$$where we have introduced the notation ##\mathcal{O}_n## for terms that are of order three [##n##?] or higher in the coordinates." I don't know why one would expand ##x^a## like that.
##x^a## are general curved coordinates, ##\xi^\mu## are local Minkowski coordinates at some point ##p##. ##e_\mu## are orthonormal vectors at ##p## in the ##\xi^\mu## system. They might be basis vectors, I am not sure. ##e_\mu^a## are the coordinates of ##e_\mu## in the ##x^a## system. ##c_{\mu\nu}^a## is a mysterious thing to be discovered. It turns out to be not a tensor but more like a Christoffel symbol. I think. We are also told earlier that $$
e_\mu^a=\frac{\partial x^a}{\partial\xi^\mu}
$$so the first part of the first equation is $$
x^a=\frac{\partial x^a}{\partial\xi^\mu}\xi^\mu
$$which is just the tensor transformation law.
Where does the first equation come from and why don't we use the ordinary tensor transformation equation?