Covariant Derivative derivation.

  • Thread starter T-chef
  • Start date
  • #1
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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!
 

Answers and Replies

  • #2
938
9
You already introduced it!
 
  • #3
12
0
Of course! The left hand side is exactly the scaler field I need. All comes out nicely after that! Thank you sir,
 

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