- #1

- 12

- 0

## 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!