In spherical polar coordinates, I am looking at the curvature tensor wrt the diagonal metric tensor whose elements are:

$$g_{tt}={-u^2}, g_{rr}=\frac{1}{u^2}, g_{\theta\theta}=\frac{1}{r^2}, g_{\phi\phi}=\frac{1}{r^2sin^2\theta} $$

Where $u=\sqrt{1-2m/r}$, with $m=M/c^2$ ($M$ is the mass generating the gravitational field and $c$ is the speed of light)

I am interested in calculating elements of the curvature tensor namely $R^t_{rtr}$ and $R^t_{\theta t \theta}$.

But when I use the definition of $R_{ijkl}$ it appears to me that $$R_{t\theta t \theta}=\frac{1}{2}(d_\theta d_tg_{t\theta}-d_t d_tg_{\theta\theta}-d_\theta d_\theta g_{tt}+d_t d_\theta g_{\theta t})=0?$$

But I know that this element cannot be zero. What am I missing?

  • 1
    $\begingroup$ It seems to me like your $g_{\theta\theta}$ and $g_{\phi\phi}$ should be upside down. But anyway, where did you get that formula from? $\endgroup$ – Javier Mar 26 '18 at 13:54
  • $\begingroup$ It is a formula that can be derived from the usual definition which invovles Christoffel symbols of 2nd Kind. $\endgroup$ – dahaka5 Mar 26 '18 at 13:56
  • $\begingroup$ Can you give me a source? That formula is linear in the metric tensor, which the Riemann tensor is certainly not. It's just too simple. $\endgroup$ – Javier Mar 26 '18 at 13:57
  • $\begingroup$ Unfortunately, I cannot supply a source. Regardless, the context of this question is that it's in a Schwarzschild metric. I know that I have to use the definition of the fully covariant curvature tensor, then raise indices to get the required element $\endgroup$ – dahaka5 Mar 26 '18 at 14:07
  • 2
    $\begingroup$ Well, since it seems to me like the formula is wrong, the answer is "that's not how you calculate a Riemann tensor". Go straight from the definition, or look up the Cartan method with differential forms for some extra efficiency. $\endgroup$ – Javier Mar 26 '18 at 14:08

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