- #1
maxverywell
- 197
- 2
With the Levi-Civita connection, the Killing's equation is:
[itex]\xi_{a;b}+\xi_{b;a}=\xi_{a,b}+\xi_{b,a}-2{\Gamma^{c}}_{ab}\xi_{c}=0[/itex]
I can't understand why there is a minus sign in front of the Christoffel symbols.
We have that:
[itex]\xi_{a;b}=\xi_{a,b}+{\Gamma^{c}}_{ab}\xi_{c}[/itex]
[itex]\xi_{b;a}=\xi_{b,a}+{\Gamma^{c}}_{ba}\xi_{c}[/itex]
and because of [itex]{\Gamma^{c}}_{ab}={\Gamma^{c}}_{ba}[/itex], it should be
[itex]\xi_{a;b}+\xi_{b;a}=\xi_{a,b}+\xi_{b,a}+2{\Gamma^{c}}_{ab}\xi_{c}=0[/itex]
and not with a minus sign.
[itex]\xi_{a;b}+\xi_{b;a}=\xi_{a,b}+\xi_{b,a}-2{\Gamma^{c}}_{ab}\xi_{c}=0[/itex]
I can't understand why there is a minus sign in front of the Christoffel symbols.
We have that:
[itex]\xi_{a;b}=\xi_{a,b}+{\Gamma^{c}}_{ab}\xi_{c}[/itex]
[itex]\xi_{b;a}=\xi_{b,a}+{\Gamma^{c}}_{ba}\xi_{c}[/itex]
and because of [itex]{\Gamma^{c}}_{ab}={\Gamma^{c}}_{ba}[/itex], it should be
[itex]\xi_{a;b}+\xi_{b;a}=\xi_{a,b}+\xi_{b,a}+2{\Gamma^{c}}_{ab}\xi_{c}=0[/itex]
and not with a minus sign.