I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear coordinates namely $$ \nabla\times V=\frac{1}{h_1h_2h_3}\sum\limits_{i=1}^3 e_i\sum\limits_{j,k}\epsilon_{ijk}h_i\frac{\partial h_kV_k}{\partial x^j} \tag 1 $$ where $$ h_1=\sqrt{g_{\rho\rho}}, \quad h_2=\sqrt{g_{\phi\phi}}, \quad h_3=\sqrt{g_{zz}}.$$ But what irritates me is that $g_{\rho,z},g_{z,\rho}\neq0$ where $(\rho=r\sin\theta, z=r\cos\theta)$, which is zero in the flat space. Doesn't that mean that the cylindrical coordinate basis vectors aren't mutually orthogonal in curved space? If so that means I can't use $(1)$ for my goal. Why doesn't it happen with the spherical polar coordinate basis? Because this metric is the solution of spherically symmetric solution of Einstein field equation? Should I use the more general expression for the curl operator? (I wanted to get rid of calculations of Christoffel symbols)

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    $\begingroup$ Could you write down what you think the Schwarzschild metric is in cylindrical coordinates? $\endgroup$ – gj255 Dec 24 '17 at 13:34
  • $\begingroup$ @gj255 $\ ds^2=-(1-\frac{r_s}{\sqrt {\rho^2+z^2}})dt^2+\frac{1}{\rho^2+z^2}(z^2+\frac{\rho^2}{(1-\frac{r_s}{\sqrt{\rho^2+z^2}})})d\rho^2+\frac{2\rho z}{\rho^2+z^2}(\frac{1}{(1-\frac{r_s}{\sqrt{\rho^2+z^2}})}-1)d\rho dz+\frac{1}{\rho^2+z^2}(\rho^2+\frac{z^2}{(1-\frac{r_s}{\sqrt{\rho^2+z^2}})})dz^2+\rho^2d\phi^2 $ $\endgroup$ – Iraklolo Dec 24 '17 at 20:37
  • $\begingroup$ @Irakolo The presence of cross-terms in your metric (namely, a $\mathrm{d} \rho \, \mathrm{d} z$ term) means your coordinate system is indeed not orthogonal. And indeed, this means $(1)$ is not valid. $\endgroup$ – gj255 Dec 24 '17 at 20:44

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