- #1
T-chef
- 12
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Homework Statement
Using the Leibniz rule and:
[tex] \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b [/tex]
[tex] \nabla_{a}\Phi=\partial\Phi [/tex]
Show that [itex] \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} [/itex].
The question is from Ray's Introducing Einsteins relativity,
My attempt:
[tex] \nabla_c(X^aX_a)=\nabla_c(X^a)X_a+X^a\nabla_c(X_a) [/tex]
[tex] = (\partial_{c}X^a+\Gamma_{bc}^{a}X^b)X_a+X^a\nabla_c(X_a) [/tex]
From here I'm not sure how to introduce the scaler field phi, or how doing so would help. Cheers for any help!