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Tensors help

  • Thread starter trv
  • Start date
  • #1
trv
73
0
A little stuck while working through a derivation. Hope someone can help.

Homework Statement



Starting from

[itex]
-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad})+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0
[/itex]

I need to obtain the Killing equations, i.e.

[itex]
\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=0
[/itex]

Homework Equations



The Attempt at a Solution



Working backwards...

Rewriting the covariant derivative in terms of the partial derivative gives

[itex]
\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=0
[/itex]

Lowering the vector in the partial derivatives gives...

[itex]
\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0
[/itex]

I don't however know how to go from

[itex]
-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c[/itex]

to

[itex]
-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad}[/itex])

Can someone help?
 

Answers and Replies

  • #2
165
4
A little stuck while working through a derivation. Hope someone can help.

Homework Statement



Starting from

[itex]
-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad})+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0
[/itex]

I need to obtain the Killing equations, i.e.

[itex]
\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=0
[/itex]

Homework Equations



The Attempt at a Solution



Working backwards...

Rewriting the covariant derivative in terms of the partial derivative gives

[itex]
\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=0
[/itex]

Lowering the vector in the partial derivatives gives...

[itex]
\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0
[/itex]

I don't however know how to go from

[itex]
-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c[/itex]

to

[itex]
-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad}[/itex])

Can someone help?
its a little difficult to show. first you should replace Xi with Xi*metric, then use this metric to lower the index on Gamma, then replace this Gamma with Gamma*metric, which is what we want. hopefully that makes some sense.
 
  • #3
trv
73
0
Thanks, it does make sense.

[itex]
\xi^c\Gamma^d_{ca}g_{bd}=\xi_eg^{ce}\Gamma^d_{ca}g_{bd}=\xi_eg^{ce}\Gamma_{bca}=\xi_e\Gamma^e_{ba}=\xi_c\Gamma^c_{ba}
[/itex]
 

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