Tensor Calculus and Differentiation

Question

I've been reading various texts on tensor algebra and calculus in preparation for applications of it, and I find myself continuously having issues with the index calculus. I've been seeing:

$$ \partial^{\mu}T^{\nu} = g^{\rho\mu} \frac{\partial T^{\nu}}{\partial x^{\rho}} $$

ie the metric appears explicitly with a dummy index to match $ \partial x^{\rho} $. Everything appears to be contravariant but the appearance of the metric with a dummy index matching the denominator leads me to believe that indices in the denominator of differentiation are of opposite character than what their index states. My question arises because I don't see the metric when taking a derivative with respect to a covariant index and the denominator still appears as contravariant. Am I correct in my assumption?

Answer 1

$\partial^\mu$ is often a shortcut for $\partial/\partial x_\mu$ and $\partial_\mu$ for $\partial/\partial x^\mu$ because of this. See here for example.

$$ g^{\rho\mu} \frac{\partial}{\partial x^\rho} = g^{\rho\mu} \partial_\rho = \partial^\mu $$

Answer 2

The simple answer is that the naive differentiation operator $\frac{\partial}{\partial x^\alpha}$ when acting on a tensor with $n$ covariant indices results, if the metric is that of flat space, in a tensor with $n+1$ covariant indices. Thus that differential operator is often written as $\partial_\alpha$.

For example, if $\phi$ is a scalar-valued field, $\frac{\partial \phi}{\partial{x^\mu}} \equiv \partial_\mu \phi$ is a covariant tensor, and in fact this is the prototypical example of a one-index covariant tensor.

And so in your equation, $$ \frac{\partial T^\nu}{\partial x^\rho} $$ is, at least in flat space, a mixed tensor with one covariant and one contravariant index, and you can then use $g^{\rho\mu}$ to raise the covariant index.

You need to be careful, however, because (except for trivial metrics) the operator $\frac{\partial}{\partial x^\alpha}$ is not the covariant derivative, so you don't actually get a covariant tensor incurved space (the case of $\partial_\mu \phi$, which is a covariant tensor, is an important special case). The covariant derivative operator is often written as $D_\mu$, is the sum of $\partial_\mu$ plus correction terms depending on the tensor indices, involving the Christofel symbols. For example: $$ D_\rho T^\nu = \partial_\rho T^\nu + \Gamma^{\nu}_{\rho\mu}T^\mu $$ $ \Gamma^{\nu}_{\rho\mu}$ is a simple combination of (ordinary) derivatives of the metric tensor.