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:
If several indices appear after the comma, they are all taken to be part of the differentiation. Here are some examples:
Now, we change coordinate systems via the Jacobian . The transformation rule is .
Finally, we present the following important theorem:
Proof: , which by the chain rule is , which is of course .