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If we are given a line element;

$$ds^2=dr^2+r^2d\theta^2+r^2sin^2\theta d\varphi^2$$

We can easily then see that the metric and the inverse metric are;

$$g=\begin{pmatrix}1&0&0\\0&r^2&0\\0&0&r^2sin^2\theta\\\end{pmatrix}$$

$$g^{-1}=\begin{pmatrix}1&0&0\\0&r^{-2}&0\\0&0&r^{-2}(\sin\theta)^{-2}\\\end{pmatrix} $$

I have used the formula;

$$\Gamma^m_{ij}=\frac12 g^{ml}(\partial_jg_{il}+\partial_ig_{lj}-\partial_lg_{ji} )$$

Where upper indices indictate the inverse matrix to calculate my Christoffel symbols. My results are as follows:

$$\Gamma^1=\begin{pmatrix}0&0&0\\0&-r&0\\0&0&-rsin^2\theta\\\end{pmatrix}$$ $$\Gamma^2=\begin{pmatrix}0&\frac1r&0\\\frac1r&0&0\\0&0&-sin\theta cos\theta\\\end{pmatrix}$$ $$\Gamma^3=\begin{pmatrix}0&0&\frac1r\\0&0&cot\theta\\\frac1r&cot\theta&0\\\end{pmatrix}$$

However the results quoted by Wolfram Mathworld (http://mathworld.wolfram.com/SphericalCoordinates.html) are different, but only in that some entries have been swapped or moved about - all the values in both are the same.

I notice that the article uses the labelling convention $\{r,\varphi,\theta\}$, whereas I have used $\{r,\theta,\varphi\}$ (which is annoying, since the article uses my preferred notation which I have made a conscious effort to move away from due to none of my professors using it), and was wondering whether or not I made a fundamental mistake, or if the different labeling systems could account for the discrepancy.

Thanks in advance!

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    $\begingroup$ So you want us to go through all steps to see if the labeling system is the cause for the conflicting answers, when you could just as well do it yourself? $\endgroup$ Oct 26, 2016 at 13:14
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    $\begingroup$ No, I wondered if there was anybody who knew off hand whether this would result in the problem. I understand it's a long laborious problem, and that's exactly why I didn't expect someone to redo it. If somebody said "yes, that's the problem, because of x property of the Christoffel symbol." that would be great, and equally if somebody said "it's impossible to know without doing it again." then I'd do it again. Thank you for your contribution though. $\endgroup$
    – arcturus7
    Oct 26, 2016 at 13:28

1 Answer 1

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The Christoffel symbols you dervied are indeed the correct ones for a spherical coordinate system $(r, \theta, \varphi)$.

If you do the same procedure for a system $(r, \varphi, \theta)$ (in the metric tensor, the entries $(22)$ and $(33)$ are now swapped) you will get the Christoffel symbols as stated on Wolfram Mathworld.

It is simply due to the order of $\theta$ and $\varphi$.

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