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I was reading my GR textbook when I came across the definition of the covariant derivative in coordinate form. Now, to get some experience with it, I decided to play around with it and see if I could construct some definitions from vector calculus with it. Now, for the Laplacian, this works fine, giving $$\nabla^{\mu}\partial_{\mu}\phi=g^{\mu\nu}(\partial_{\nu}\partial_{\mu}\phi-\Gamma^{\lambda}_{\nu\mu}\partial_{\lambda}\phi)$$

This expression works perfectly well for the three-dimensional metrics for spherical, cylindrical, etc.. Note that $\Gamma^{\mu}_{\nu\sigma}$ are the Christoffel coefficients describing the Levi-Civita connection. Now, when I try to find, say, a generalized expression for the divergence the same way, $$\nabla_{\mu}V^{\mu}=\partial_{\mu}V^{\mu}+\Gamma^{\mu}_{\mu\lambda}V^{\lambda}$$ this fails every time, save for on a Cartesian metric. Why does one work and not the other?

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  • $\begingroup$ What is it that goes wrong? $\endgroup$
    – Arthur
    Commented Sep 24, 2019 at 16:05
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    $\begingroup$ @Arthur probably the fact that if you were to grind it out with the Christoffel symbols, the expression does not line up with the accepted defintions of divergence in standard curvilinear coordinates, like what you can find in a vector calculus table. This is because the basis vectors have different defintions than from vector calculus. When you define the basis vectors to be partial derivatives in tensor calculus, they don't line up with the unit vectors used in vector calculus. $\endgroup$ Commented Sep 24, 2019 at 16:11
  • $\begingroup$ It simply gives an incorrect answer for metrics other than Cartesian. I’m not precisely sure what’s going wrong, to be honest. $\endgroup$ Commented Sep 24, 2019 at 16:14
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    $\begingroup$ @Tesseract I have an answer in my reply to Arthur $\endgroup$ Commented Sep 24, 2019 at 16:16

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The reason you get a different (but not wrong) answer from what you might find on the wikipedia page for Del in Cylindrical and Spherical Coordinates, is because the defintions for the basis vectors of the vector fields have changed. In vector calculus we used unit vectors. But on a manifold, unit vectors are not the natural choice, we use partial derivatives.

Take 3D spherical coordinates and consider the basis vector $\partial_\theta$ that you might find in a GR book. If the definitions for vector calculus stuff were to line up with their tensor calculus counterparts then $\partial_\theta$ would have to be a unit vector. But using the defintion of the metric in spherical coordinates,

$$\partial_\theta \partial^\theta = g^{\theta\theta} = \frac{1}{r^2}$$

So in fact the component of the vector field in that direction is actually not the same between the two conventions, they're distorted by a scale factor.

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    $\begingroup$ I found a sentence in Carrol's book: "It is standard in the study of vector analysis in three-dimensional Euclidean space to choose orthonormal bases rather than coordinate bases; you should therefore be careful when applying formulae from GR texts to the study of non-Cartesian coordinates in flat space." Looks like this fits this situation to a T. Thank you. $\endgroup$ Commented Sep 24, 2019 at 16:45
  • $\begingroup$ I find it odd that the normalization of the basis vectors would affect the value of the scaler as presumably it coordinate and then basis independent as it is a tensor after all? Further, more why would such discrepancies not arise with the covariant definition of the laplacian? $\endgroup$
    – Aidan R.S.
    Commented Oct 20, 2023 at 22:16
  • $\begingroup$ @AidanR.S. on what basis (heh) are you proposing that the scalar trace that results from the tensor calculation is in any way a priori related to the divergence operator defined through a dot product when a manifold base structure does not have an inner product space? That they must be equal simply because they are both scalar quantities? And why are you assuming a discrepancy does not arise with the covariant definition of the Laplacian, the discrepancy still exists. $\endgroup$ Commented Oct 20, 2023 at 23:23

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