- #1
Damidami
- 94
- 0
Hi, I'm trying to verify that the covariant derivative of the metric tensor is D(g) = 0.
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?
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?