Hi, I'm trying to verify that the covariant derivative of the metric tensor is D(g) = 0.(adsbygoogle = window.adsbygoogle || []).push({});

But I have a few questions:

1) This is a scalar 0 or a tensorial 0? Because it is suposed that the covariant derivative of a (m,n) tensor is a (m,n+1) tensor, and g is a (0,2) tensor so I think this 0 should be a (0,3) tensor. Am I right?

2) Following the MTW book in the example of page 341 we have:

[tex]\Gamma^{\theta}_{\phi\phi} = -sin(\theta)cos(\theta)[/tex]

[tex]\Gamma^{\phi}_{\phi\theta} = cos(\theta)/sin(\theta)[/tex]

and all the other [tex]\Gamma = 0[/tex]

But when I try to verify the covariant derivative of the metric tensor for the component [tex]g_{\phi\phi;\theta}[/tex] it doesn't give me 0, but instead:

[tex]g_{\phi\phi;\theta} = g_{\phi\phi,\theta} - \Gamma^{k}_{\theta\phi} g_{k\phi} - \Gamma^{k}_{\theta\phi} g_{\phi k} = 2a^2sin(\theta)cos(\theta) - (0+0) - (0+0) = 2a^2sin(\theta)cos(\theta) \neq 0 [/tex]

I checked it a lot of times and am not sure if this is a conceptual error or a procedure error.

Can someone clarify this to me?

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# Covariant derivative of metric tensor

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