General Relativity/Coordinate systems and the comma derivative

<General Relativity

In General Relativity we write our (4-dimensional) coordinates as (x⁰,x¹,x²,x³). The flat Minkowski spacetime coordinates ("Local Lorentz frame") are x⁰ = ct, x¹ = x, x² = y, and x³ = 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. $T^\alpha_{\ \beta , \gamma} = \frac {\partial T^\alpha_{\ \beta}} {\partial x^\gamma}$

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

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

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

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 $x^\mu_{\ , \nu}$. The transformation rule is $x^{\bar \mu} = x^\mu x^{\bar \mu}_{\ , \mu}$.

Finally, we present the following important theorem:

Theorem: $x^\alpha_{\ , \mu} x^\mu_{\ , \beta} = \delta^\alpha_\beta$

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