Help Needed: Rewriting Covariant Derivative to Killing Equations

[trv]

A little stuck while working through a derivation. Hope someone can help.

Homework Statement

Starting from

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

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

$\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=0$

Homework Equations

The Attempt at a Solution

Working backwards...

Rewriting the covariant derivative in terms of the partial derivative gives

$\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=0$

Lowering the vector in the partial derivatives gives...

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

I don't however know how to go from

$-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c$

to

$-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad}$)

Can someone help?

 

[xaos]

A little stuck while working through a derivation. Hope someone can help.

Homework Statement

Starting from

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

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

$\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=0$

Homework Equations

The Attempt at a Solution

Working backwards...

Rewriting the covariant derivative in terms of the partial derivative gives

$\bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=0$

Lowering the vector in the partial derivatives gives...

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

I don't however know how to go from

$-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c$

to

$-\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad}$)

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.

 

[trv]

Thanks, it does make sense.

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