Very simple index notation question

Question

Trying to understand index notation in the context of spacetime. If I have $x^{\mu}$ and then set $\mu=\phi$ (for example), is it acceptable to then write $x^{\phi}$ or should I just write $\phi$? For example, can I write $$\frac{\partial g_{\mu\nu}}{\partial x^{\mu}}$$ as $$\frac{\partial g_{\phi\nu}}{\partial x^{\phi}},$$ or should I stick with$$\frac{\partial g_{\phi\nu}}{\partial\phi}?$$

Answer 1

I think that there is some problem with your notation.

Disclaimer: what follows are conventions, so they could be not universally respected, although I think this is what is mostly followed.

Reading $$ \frac{\partial g_{\mu\nu}}{\partial x^\mu} = \partial_\mu g_{\mu \nu} $$ in its literal sense, it is not a relativistic expression, since you are contracting two lower indices. So in this case no sum is implied: $\mu$ is fixed. Now you can do two different things.

1. If you set $\mu =\phi$ you are just relabelling the index and so you can write $$ \frac{\partial g_{\mu\nu}}{\partial x^\mu} = \frac{\partial g_{\phi\nu}}{\partial x^\phi} = \partial_\phi g_{\phi \nu} .$$

2. Not being a relativistic expression you can specialise in some frame of reference where you can choose particular coordinates, like spherical coordinates, in which you call $x^\mu = \phi$ for some fixed index $\mu$, for instance $\mu =2$. In this case $\phi$ is not an index but is is a coordinate. Then you'd have $$ \frac{\partial g_{\phi\nu}}{\partial \phi} = \partial_\phi g_{\phi \nu} = \frac{\partial g_{2\nu}}{\partial x^2} = \partial_2 g_{2 \nu} .$$

On the other hand, if by your writing you mean $$ \frac{\partial g_{\mu\nu}}{\partial x_\mu} = \partial^\mu g_{\mu\nu}=\sum_{\mu=0}^3 \frac{\partial g_{\mu\nu}}{\partial x_\mu}= \sum_\mu \partial^\mu g_{\mu\nu} $$ (note that the position of indices, different from yours!) this is a relativistic expression and there is a sum implied unless differently stated. Then you can rename the index $\mu$ calling it $\phi$ because it is a dummy index, and you get $$ \frac{\partial g_{\mu\nu}}{\partial x_\mu} = \frac{\partial g_{\phi\nu}}{\partial x_\phi} =\sum_{\phi=0}^3 \frac{\partial g_{\phi\nu}}{\partial x_\phi} $$ In this context an expression like $$ \frac{\partial g_{\mu\nu}}{\partial \mu} = \frac{\partial g_{\phi\nu}}{\partial \phi} =\sum_{\phi=0}^3 \frac{\partial g_{\phi\nu}}{\partial \phi} $$ is not conventionally defined.

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Other clarifications:

In using spherical coordinates $x^\mu = (x^0,x^1,x^2,x^3)=(t,r,\theta,\phi)$, $\phi$ and $\theta$ are coordinates, not indices. In (consistent) index notation, where $x^3=\phi$ and $x^2=\theta$, you want to compute $$\Gamma_{33}^2 = 1/2 g^{2\mu}\left(\frac{\partial}{\partial x^3} g_{\mu 3}+\frac{\partial}{\partial x^3} g_{3 \mu} - \frac{\partial}{\partial x^\mu} g_{33}\right);$$ here $3$ and $\mu$ are indices; moreover on the previous expression there is a sum over $\mu$ implied.

Since $x^3 = \phi$, then $$\frac{\partial}{\partial x^3}=\frac{\partial}{\partial\phi}$$ and so on. They are just different names. So: $$\Gamma_{33}^2 = 1/2 g^{2\mu}\left(\frac{\partial}{\partial \phi} g_{\mu 3}+\frac{\partial}{\partial \phi} g_{3 \mu} - \frac{\partial}{\partial x^\mu} g_{33}\right).$$

Now it comes the slight abuse of notation. Since it is easier to think and remember the 'real' coordinates rather than confusing indices, we write $\Gamma_{\phi\phi}^\theta$ rather than $\Gamma_{33}^2$, $g_{\mu \theta}$ rather than $g_{\mu 2}$ and so on; I would then write $$\Gamma_{\phi\phi}^\theta= \Gamma_{33}^2 = 1/2 g^{\theta\mu}\left(\frac{\partial}{\partial \phi} g_{\mu \phi}+\frac{\partial}{\partial \phi} g_{\phi \mu} - \frac{\partial}{\partial x^\mu} g_{\phi\phi}\right).$$ Remember that you have still an implied sum over $\mu$. Probably in the particular metric you are studying there are some symmetries that allow you to restrict to just one term, the one with $\mu = 2$ (but this could also not happen in other cases), then $$\Gamma_{\phi\phi}^\theta=1/2 g^{\theta\theta}\left(\frac{\partial}{\partial \phi} g_{\theta \phi}+\frac{\partial}{\partial \phi} g_{\phi \theta} - \frac{\partial}{\partial \theta} g_{\phi\phi}\right),$$ with no sum implied anymore.

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So now that you have clarified the context I can complete the answer by saying that in your question there is a mistake:

You did not put $\mu = \phi$ for $\mu=3$, but rather $x^\mu = \phi$ for $\mu =3$, or $x^3=\phi$.

Answer 2

Yes of course that is completely acceptable. You can say $x^\mu=x^\nu=(ct,r,\theta,\phi)$ The $\mu, \nu$ or whatever else are at your discretion as long as you are consistent. With respect to your derivative the first two make sense. The third derivative is a legitimate expression but it means that you are taking the derivative of $g_{\phi \nu}$ with respect to the coordinate $\phi$. Normally you would write $x^\mu$ where $mu=(0,1,...)$. So with what I've written above $x^\mu=x^\nu=(ct,r,\theta,\phi)$ where $\mu,\nu=(0,1,2,3)$ and then $\frac{\partial g_{\sigma\nu}}{\partial x^2}=\frac{\partial g_{\sigma \nu}}{\partial \theta}$. Note: that is not an $x^2$ like x-squared. I should also say that if you meant that $x^\phi=\phi$ then all three derivatives in your question are good to go.