General Relativity/Comma derivative

< General Relativity

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

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 $x^{\overline{\mu }}=x^{\mu }{x}_{,\mu }^{\overline{\mu }}$.

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 ${\delta }_{\beta }^{\alpha }$. $◻$

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