What is the covariant derivative of a metric tensor, this particular one to be specific $\nabla_{\mu} g^{\mu\nu}$? Notice we've got repetitive indices here. Is it zero and has it got to do anything with $$\nabla_{\alpha} g^{\mu\nu}=0~?$$

Here $\nabla_{\mu}$ is the covariant derivative and the connection is given by the Christoffel symbol $\Gamma^{\mu}_{\alpha \beta}$.


Yes, it is zero. $\nabla_{\alpha} g^{\mu\nu}$ is a three-index tensor. If any tensor is zero, all of its contractions are zero.

  • $\begingroup$ Could you elaborate that mathematically? $\endgroup$ – Geeth Chandra May 27 at 17:00
  • 3
    $\begingroup$ A contraction is a particular sum of components in a basis. But if all components are 0, then this particular sum is 0. $\endgroup$ – DanielC May 27 at 17:01

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