General Relativity/Comma derivative

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In General Relativity we write our (4-dimensional) coordinates as $(x^{0},x^{1},x^{2},x^{3})$. The flat Minkowski spacetime coordinates ("Local Lorentz frame") are $x^{0}=ct$, $x^{1}=x$, $x^{2}=y$, and $x^{3}=z$, where $c$ is the speed of light, $t$ is time, and $x$, $y$, and $z$ are the usual 3-dimensional Cartesian space coordinates.

A comma derivative is just a convenient notation for a partial derivative with respect to one of the coordinates. Here are some examples:

1. $T_{\ \beta ,\gamma }^{\alpha }={\frac {\partial T_{\ \beta }^{\alpha }}{\partial x^{\gamma }}}$

2. $f_{,\mu }={\frac {\partial f}{\partial x^{\mu }}}$

3. $w_{\ ,\nu }^{\mu }={\frac {\partial w^{\mu }}{\partial x^{\nu }}}$

4. $\Gamma _{\ \beta \gamma ,\mu }^{\alpha }={\frac {\partial \Gamma _{\ \beta \gamma }^{\alpha }}{\partial x^{\mu }}}$

If several indices appear after the comma, they are all taken to be part of the differentiation. Here are some examples:

1. $S_{\alpha \ ,\mu \nu }^{\ \beta }=\left(S_{\alpha \ ,\mu }^{\ \beta }\right)_{,\nu }={\frac {\partial }{\partial x^{\nu }}}\left({\frac {\partial S_{\alpha }^{\ \beta }}{\partial x^{\mu }}}\right)={\frac {\partial ^{2}S_{\alpha }^{\ \beta }}{\partial x^{\nu }\partial x^{\mu }}}$

2. $f_{,\alpha \beta \beta }=\left[\left(f_{,\alpha }\right)_{,\beta }\right]_{,\beta }={\frac {\partial ^{3}f}{\partial ^{2}x^{\beta }\partial x^{\alpha }}}$

Now, we change coordinate systems via the Jacobian $x_{\ ,\nu }^{\mu }$. The transformation rule is $x^{\bar {\mu }}=x^{\mu }x_{\ ,\mu }^{\bar {\mu }}$.

Finally, we present the following important theorem:

Theorem: $x_{\ ,\mu }^{\alpha }x_{\ ,\beta }^{\mu }=\delta _{\beta }^{\alpha }$

Proof: $x_{\ ,\mu }^{\alpha }x_{\ ,\beta }^{\mu }=\sum _{\mu =0}^{3}{\frac {\partial x^{\alpha }}{\partial x^{\mu }}}{\frac {\partial x^{\mu }}{\partial x^{\beta }}}$, which by the chain rule is ${\frac {\partial x^{\alpha }}{\partial x^{\beta }}}$, which is of course $\delta _{\beta }^{\alpha }$. $\square$

So, as a matrix, the matrix inverse of $\left(x_{\ ,\mu }^{\alpha }\right)$ is $\left(x_{\ ,\beta }^{\mu }\right)$.