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Why there is a minus sign in the Killing's equation

  1. May 19, 2013 #1
    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.
     
  2. jcsd
  3. May 19, 2013 #2
    Hi

    For tesor in variant components

    [itex]\xi_{a;b}=\xi_{a,b}-{\Gamma^{c}}_{ab}\xi_{c}[/itex]
     
  4. May 19, 2013 #3
    Oops, right, [tex]\xi_a[/itex] has lower index i.e. it's a component of covector and its covariant derivative has a minus sign in front of the connection coefficients. Thnx!
     
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