What is the difference between:

$\nabla _{\sigma} $ and $ \nabla^{\sigma}$?

I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and am still a bit unsure of the notation.


$\nabla_\sigma$ is the covariant derivative. $\nabla^\sigma$ means $g^{\sigma\rho}\nabla_\rho$. It's pretty much the same as raising any other index. The covariant derivative when acting on any tensor adds a down index, and you can raise it as with any other index. Since the covariant derivative of the metric is 0, you can work with either $\nabla_\sigma$ or $\nabla^\sigma$ without worrying about derivatives of the metric showin up.

  • $\begingroup$ sorry, to be more precise, why does: $\nabla^{\sigma}[\nabla_{\sigma},\nabla_{\nu}]f = 0$ Where f is a scalar? $\endgroup$ – sarahusher Apr 26 '14 at 19:02
  • $\begingroup$ The high-level answer is that $[\nabla_\mu, \nabla_nu]$ represents the commutator of infinitesimal parallel transport, curvature. But since a scalar is unaffected by parallel transport, the effect of curvature on it is zero. For a more pedestrian approach you can calculate it in coordinates where the Christoff symbols vanish (such coordinates always exist). $\endgroup$ – Robin Ekman Apr 26 '14 at 19:17
  • $\begingroup$ @sarahusher $\nabla_a f = \partial_a f$ for $f$ scalar. $\endgroup$ – auxsvr Apr 26 '14 at 19:24

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