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Vector fields, metrics and two forms on a spacetime.

  1. Apr 27, 2008 #1
    1. The problem statement, all variables and given/known data
    Let (M,g) be a spacetime.
    (a) Let A and A' be vector fields on M such that g(A,B)=g(A',B) for any future-pointing timelike vector field Y. Show that X=X'.
    (b) Let w and w' be two two-forms on M. Suppose that i¬A w = i¬A w' for any future -pointing timelike vector field A on M, where i¬x denotes the contraction with A. Show that w=w'

    2. Relevant equations

    3. The attempt at a solution
    I think that part (a) is based on the transformation law between two metrics. So if we are going from g(A,B) -> g(A',B') we use the tensorial transformation law:
    g(A,B) = d(A',A)d(B',B) g(A',B')
    where d(A',A) mean the partial derivative of A' with respect to A, etc.

    But g(A',B')=g(A,B) (given)
    so g(A,B) = d(A',A)d(B',B) g(A,B)

    Thus d(A',A)d(B',B) = 1

    But B'=B , so d(B',B) = Kroneckerdelta (B',B) = 1

    So d(A',A)=1, thus A'=A.

    I'm not sure if this is even the right way of tackling this question for part (a). As for Part (b), I'm completely lost :S
  2. jcsd
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