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Index raising and lowering: GR

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider a 2-dimensional line element: [tex]ds^2=x^2dx+2dxdy-dy^2[/tex]

    Raise and lower the indices of given vectors by finding the raised index metric, etc, blah blah


    2. Relevant equations
    [tex]V_a = (1,-1)[/tex] and [tex]W^a = (0,1)[/tex]


    3. The attempt at a solution
    Solution is given and I understand how to raise and lower indices, but I'm just wondering if my book has a massive exemplary typo. They raise V by multiplying it by the raised metric, which is fine. I get that. I don't get why they use the raised metric for lowering the W vector. Shouldn't you just use the lower metric given by the line element?
     
  2. jcsd
  3. Mar 21, 2012 #2
    That's right. The whole point of the index notation is you can immediately see what you have to do.
    [itex] V^a = g^{ab} V_b, W_a = g_{ab}W^b [/itex]

    Which book is that? You can usually find an errata online.
     
  4. Mar 21, 2012 #3
    It's Relativity Demystified by McMahon
     
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