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Need help!!! MTW's Gravitation exercise 16.1!!!
I am working on MTW's Gravitation and I came across a problem. In the attachment View attachment 50292, I show my calculation and, finally, I show the contradiction that is,
you can get a vanishing [itex]\Gamma^k_{\phantom{k}kk}[/itex] with Cartan's equation
but, at the same time,
you can also get a nonvanishing [itex]\Gamma^k_{\phantom{k}kk}[/itex] by simply inserting metric into the definition of connection coefficients.
What's wrong with my calculation? I need your help. Thank you!
I am working on MTW's Gravitation and I came across a problem. In the attachment View attachment 50292, I show my calculation and, finally, I show the contradiction that is,
you can get a vanishing [itex]\Gamma^k_{\phantom{k}kk}[/itex] with Cartan's equation
but, at the same time,
you can also get a nonvanishing [itex]\Gamma^k_{\phantom{k}kk}[/itex] by simply inserting metric into the definition of connection coefficients.
What's wrong with my calculation? I need your help. Thank you!
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