question : $\nabla_a \nabla_b \sqrt{g} \phi =\partial_a \sqrt{g} \partial_b \phi$ is true ?

because $\nabla_a \sqrt{g}=0$ so we can write $\sqrt{g} \nabla_a \nabla_b \phi$ , but because metric determinant is not transfers like scalar, we can not write partial derivative instead of covariant derivative. what do think guys ?

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    $\begingroup$ $\sqrt{g}$ is not a tensor and there we cannot define a covariant derivative action on it, so I don't know what you mean by $\nabla_a \sqrt{g}$. Can you clarify that? $\endgroup$ – Prahar Apr 21 '15 at 18:44
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    $\begingroup$ @Prahar It is possible to define a covariant derivative on a scalar density $\rho(x)$ as $\nabla_i\rho=(\partial_i-\Gamma^l{}_{il})\rho$, cf. N. Straumann General Relativity (2013), pp. 663. It may be easily verified that this satisfies $\nabla_i\sqrt{g}=0$. $\endgroup$ – Ryan Unger Apr 22 '15 at 2:11
  • $\begingroup$ @0celo7 - cool. Did not know that. $\endgroup$ – Prahar Sep 19 '15 at 0:51

$$ g=\frac{1}{4!}\varepsilon^{abcd}\varepsilon^{efgh}g_{ae}g_{bf}g_{dg}g_{dh}\\ \therefore \quad \nabla_m g = \frac{1}{3!} \varepsilon^{abcd}\varepsilon^{efgh}g_{ae}g_{bf}g_{dg}\nabla_m g_{dh}\\=0\;. $$ Note that $\varepsilon^{abcd}$ is Levi-Civita symbol, it is constant. It is tensor density of weight 0.5 and make $g$ be a tensor density of weight 1.


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