It is the last part that I am having trouble understanding. It says that if(adsbygoogle = window.adsbygoogle || []).push({}); uandware tangent vectors then,

[tex]\nabla_{\bold{u}}\bold{w} - \nabla_{\bold{w}}\bold{u} = [\bold{u},\bold{w}][/tex].

Now,

[tex][\bold{u},\bold{w}] = \partial_{\bold{u}}\partial_{\bold{w}} - \partial_{\bold{w}}\partial_{\bold{u}} = (u^\beta\,v^{\alpha}_{,\beta} - v^\beta\,u^{\alpha}_{,\beta})\bold{e}_\alpha[/tex].

But,

[tex]\nabla_{\bold{u}}\bold{w} - \nabla_{\bold{w}}\bold{u} = (u^\beta\,w^{\alpha}_{,\beta} - w^\beta\,u^{\alpha}_{,\beta})\bold{e}_\alpha + (u^\beta\,w^{\alpha} - w^\beta\,u^{\alpha})\bold{e}_{\alpha}_{,\beta}[/tex].

So how do I reason that the second term disappears? Is the derivative of the vector basis zero b/c this is in a local Lorentz frame? In other words, there are no correction terms??

Thanks in advance for any help you may give me.

-LD

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# MTW: Exer. 8.11, pg. 215

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