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negru
negru is offline
#3
Mar7-10, 06:46 PM
P: 308
Well I assume
[tex]
\Delta^c\xi_c=g^{ac}\Delta_a\xi_c
[/tex]
I mean that's the only thing I can imagine..otherwise I don't know what P^{ab} would be.

I'm not sure how to get to your equation. If I write everything out I get:
[tex]
P_{ab}P^{ab}=(\Delta_a\xi_b+\Delta_b\xi_a)(\Delta^a\xi^b+\Delta^b\xi^a) +g_{ab}g^{ab}\Delta_c\xi^c\Delta_c\xi^c+ \text{etc}
[/tex]
and
[tex]
(\Delta_a\xi_b+\Delta_b\xi_a)(\Delta^a\xi^b+\Delta^b\xi^a)=2\Delta_a\xi _b(\Delta^a\xi^b+\Delta^b\xi^a)
[/tex]

[tex]
g_{ab}g^{ab}\Delta_c\xi^c\Delta_c\xi^c= 2 \Delta_c\xi^c\Delta_c\xi^c
[/tex]

[tex]
etc=-2\Delta_c\xi^cg^{ab}(\Delta_b\xi_a+\Delta_a\xi_b)
[/tex]
It doesn't seem to add up to what you have :-?
For example, where would
[tex]
4\xi_{b;a}g^{ab}
[/tex]
come from?