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Basis vectors and abstract index notation

  1. May 1, 2012 #1
    First of all, I'd like to say hi to all the peole here on the forum!

    Now to my question:

    When reading some general relativity articles, I came upon this strange notation:

    T[itex]^{a}[/itex][itex]_{b}[/itex] = C(dt)[itex]^{a}[/itex](∂[itex]_{t}[/itex])[itex]_{b}[/itex] + D(∂[itex]_{t}[/itex])[itex]^{a}[/itex](dt)[itex]_{b}[/itex]. Can someone please explain to me what this means? Clearly the author is trying to use the abstract index notation but I'm used to think of dx[itex]^{i}[/itex] as the covector basis and ∂[itex]_{i}[/itex] as the vector basis thus you're not allowed to change the co- or contravariance of these in an expression.

    Thank you,
  2. jcsd
  3. May 1, 2012 #2


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    Science Advisor
    Gold Member

    Follow the usual rules for raising and lowering indices, e.g.[tex]
    (dt)_a = g_{ab}\,(dt)^b
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