General Relativity/Coordinate systems and the comma derivative

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<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^\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