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Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity:

$$\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt{|g|}} \partial_{\mu} \left( \sqrt{|g|} A^{\mu}\right),$$

where $g = \det(g_{\alpha\beta})$. I was wondering if this formula holds if $A^{\mu}$ is replaced with a general rank $(n,m)$ tensor

$$T^{\mu \mu_1\mu_2 \cdots \mu_{n-1}}_{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\nu_1\cdots \nu_m}?$$

If not, could you point me to any references that have divergence formulas for higher rank tensors?

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    $\begingroup$ It does not. It holds when $A^\mu$ is replaced with a general $p$-form (which is a totally antisymmetric $(p,0)$ tensor. $\endgroup$ – Prahar Oct 1 '18 at 14:40
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No, this does not hold in general for higher-rank tensors. The general equation for the divergence of a completely contravariant tensor in terms of a coordinate derivative operator $\partial_\mu$ is $$ \nabla_\mu T^{\mu \nu_1 \dots \nu_n} = \partial_\mu T^{\mu \nu_1 \dots \nu_n} + \Gamma^\mu {}_{\mu \rho} T^{\rho \nu_1 \dots \nu_n} + \sum_{i = 1}^n \Gamma^{\nu_i} {}_{\mu \rho} T^{\mu \nu_1 \dots \rho \dots \nu_n}. $$ We also have the fact that $$ \Gamma^\mu {}_{\mu \rho} = \frac{1}{\sqrt{|g|}} \partial_\mu \sqrt{|g|}. $$ Thus, $$ \nabla_\mu T^{\mu \nu_1 \dots \nu_n} = \frac{1}{\sqrt{|g|}} \partial_\mu \left( \sqrt{|g|} T^{\mu \nu_1 \dots \nu_n} \right) + \sum_{i = 1}^n \Gamma^{\nu_i} {}_{\mu \rho} T^{\mu \nu_1 \dots \rho \dots \nu_n}. $$ This last sum will not vanish for a general tensor. However, some or all of the terms may vanish for tensors with a particular symmetry structure. In particular, if $T^{\mu \nu_1 \dots \nu_n}$ is antisymmetric in all of its indices, then any contraction of two of its indices with the symmetric indices of the Christoffel symbols automatically vanishes; and thus the entire sum goes away.

For references as to how to take the covariant derivative of a general tensor, see Chapter 5 of Schutz's A First Course in General Relativity for a coordinate-based approach, or Chapter 3 of Wald's General Relativity for a more general approach.

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  • $\begingroup$ I believe that for general Christoffel symbols and non-zero $T$, total antisymmetry is the only case where the extra sum vanishes. $\endgroup$ – Void Oct 1 '18 at 15:06

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