Starting with this definition of the Reimann tensor(adsbygoogle = window.adsbygoogle || []).push({});

[tex]R^a_{mbn}=\Gamma ^{a}_{mn,b}-\Gamma ^{a}_{mb,n}+\Gamma ^{a}_{rb}\Gamma ^{r}_{mn}-\Gamma ^{a}_{rn}\Gamma ^{r}_{mb}

[/tex]

Can I contract on indices a,b and r to get [tex]R_{mn}[/tex] ?

It bothers me that the expression on the right is not symmetric in m,n. I worked out

[tex]R_{12}=\Gamma ^{2}_{12}\Gamma ^{3}_{23}-\Gamma ^{3}_{13,2}-\Gamma ^{3}_{13,2}-\Gamma ^{3}_{13,2}-\Gamma ^{3}_{13,2}-\Gamma ^{3}_{13}\Gamma ^{3}_{32}[/tex]

and

[tex]R_{21}=\Gamma ^{2}_{21}\Gamma ^{3}_{23}-\Gamma ^{3}_{23,1}-\Gamma ^{3}_{23,1}-\Gamma ^{3}_{23,1}-\Gamma ^{3}_{23,1}-\Gamma ^{3}_{23}\Gamma ^{3}_{31}[/tex].

I thought they should be equal. I'm Using the Schwarzschild metric, with signature (-1, 1, 1, 1), x0 = t, x1=r, x2=theta, x3=phi.

This is no doubt due to some misunderstanding on my part - please help me out.

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# Ricci tensor problem

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