Thread: Conformal killing vector View Single Post
 P: 308 Well I assume $$\Delta^c\xi_c=g^{ac}\Delta_a\xi_c$$ 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: $$P_{ab}P^{ab}=(\Delta_a\xi_b+\Delta_b\xi_a)(\Delta^a\xi^b+\Delta^b\xi^a) +g_{ab}g^{ab}\Delta_c\xi^c\Delta_c\xi^c+ \text{etc}$$ and $$(\Delta_a\xi_b+\Delta_b\xi_a)(\Delta^a\xi^b+\Delta^b\xi^a)=2\Delta_a\xi _b(\Delta^a\xi^b+\Delta^b\xi^a)$$ $$g_{ab}g^{ab}\Delta_c\xi^c\Delta_c\xi^c= 2 \Delta_c\xi^c\Delta_c\xi^c$$ $$etc=-2\Delta_c\xi^cg^{ab}(\Delta_b\xi_a+\Delta_a\xi_b)$$ It doesn't seem to add up to what you have :-? For example, where would $$4\xi_{b;a}g^{ab}$$ come from?