General Relativity/Covariant Differentiation

Now that we have established some of the basic of tensor algebra and curved space, lets try to do something within that curved space. We start by taking a derivative. In Einstein notation, taking a derivative looks like

$\frac{\partial}{\partial x^j} f (x^i)$

In the spirit of seeing how things work when we transform coordinates, we convert the coordinates from $x^i$ to $x^{\prime i}$ by a tensor transformation $x^{\prime i} = A^i_k x^k$

So let's figure out what our derivatives look like in our new coordinate system.

$\frac{\partial}{\partial x^{\prime j}} f(x^{\prime i}) =$

$\frac{\partial}{\partial x^{k}} f(x^{\prime i}) \frac{\partial x^{k}}{\partial x^{\prime j}} =$

$\frac{\partial}{\partial x^{k}} f(x^{l}) \frac{\partial x^{l}}{\partial x^{\prime i}} \frac{\partial x^{k}}{\partial x^{\prime j}}$

So if the transform is a constant we get a very nice result.....

The result is much less nice if the transform changes with location, that is to say instead of transform $A^{i}_{k}$ we use the transform $A^{i}_{k} (x^{j})$

What we have here is a nice part, and a not so nice part. If only there was a way to get rid of the not so nice part. At this point we do an interesting trick and that is to redefine the notion of derivative. Instead of defining derivative as simply the way we do in Euclidean space, we create a new type of derivative called the covariant derivative. The covariant derivative is like the normal derivative, except that we add a "fudge factor" to get rid of the not nice parts of the equation so that the result transforms nicely.