I Del with Superscript in Carroll's Equation

[Pencilvester]

In Carroll’s “Spacetime and Geometry” his equation (1.116) for $\partial_{\mu} T^{\mu \nu}$ for a perfect fluid ends with the term $... + ~\partial^{\nu} p$. First of all, in order for this equation to really be general, it would need to use the covariant derivative instead of the simple partials, right? I assume he’s assuming a flat Minkowski spacetime where dels work just fine because it’s at the beginning of the book. But here’s the main question: is $\partial^{\nu} p$ just the same as $\eta^{\nu \sigma} \partial_{\sigma} p$ here? I’ve just never seen a del with a superscript before, and he offers no explanation for it.

 

[PeterDonis]

in order for this equation to really be general, it would need to use the covariant derivative instead of the simple partials, right?

I think he's assuming flat spacetime in Cartesian coordinates at this point in the book, in which case the two are the same. For a general curved spacetime or general curvilinear coordinates, yes, you would have to use the covariant derivative.

is $\partial^{\nu} p$ just the same as $\eta^{\nu \sigma} \partial_{\sigma} p$ here?

Yes. You can use the metric to raise an index on anything that has a lower index, including derivative operators. (The same applies to covariant derivatives when you are in a general curved spacetime or general curvilinear coordinates.)

 

[Pencilvester]

Yes.

Thanks!