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General Relativity/Coordinate systems and the comma derivative
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<General Relativity
In General Relativity we write our (4-dimensional) coordinates as . The flat Minkowski spacetime coordinates ("Local Lorentz frame") are , , , and , where is the speed of light, is time, and , , and 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.
2.
3.
4.
If several indices appear after the comma, they are all taken to be part of the differentiation. Here are some examples:
1.
2.
Now, we change coordinate systems via the Jacobian . The transformation rule is .
Finally, we present the following important theorem:
Theorem:
Proof: , which by the chain rule is , which is of course .