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Metric tensor division

  1. Dec 2, 2012 #1
    If you know that
    [tex]{{x}^{a}}{{g}_{ab}}={{x}_{b}}[/tex]

    is it proper to say that you also know
    [tex]{{g}_{ab}}=\frac{{{x}_{b}}}{{{x}^{a}}}[/tex]
     
  2. jcsd
  3. Dec 2, 2012 #2

    PeterDonis

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    2016 Award

    Staff: Mentor

    No, because the expression [itex]{{x}^{a}}{{g}_{ab}}={{x}_{b}}[/itex] is not a single term, it's a sum of terms. Repeated indexes are summed over, so what your first equation really means is

    [tex]\Sigma_a {{x}^{a}}{{g}_{ab}} = x^0 g_{0b} + x^1 g_{1b} + x^2 g_{2b} + x^3 g_{3b} = {{x}_{b}}[/tex]

    (I've assumed that we're working in a 4-dimensional manifold.) There's no way to transform that into your second equation.
     
  4. Dec 3, 2012 #3
    Ah, yes, of course. Thank you.
     
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