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Weak field approx raising index using minkowski metric

  1. Mar 26, 2015 #1
    I have the expression ##g_{ab}=\eta_{ab}+\epsilon h_{ab}##,

    The indices on ##h^{ab}## are raised with ##\eta^{ab}## to give ##g^{ab}=\eta^{ab}-\epsilon h^{ab}##

    I am not seeing where the minus sign comes from.

    So I know ##\eta^{ab}\eta_{bc}=\delta^{a}_{c}## and ##h^{ab}=\eta^{ac}\eta^{bd}h_{cd}## ,
    And that ##\eta_{ab}## has signature ##(-,+,+,+)##,

    Any help greatly appreciated.
     
  2. jcsd
  3. Mar 26, 2015 #2

    Mentz114

    User Avatar
    Gold Member

    If ##\epsilon## is an infinitessimal then the sign changes between the covariant and contravariant delta so that the contraction is unchanged to order ##\epsilon##
     
  4. Mar 27, 2015 #3
    Thanks .
     
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