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Tensor Algebra

  1. May 14, 2014 #1
    1. The problem statement, all variables and given/known data

    Show that the tensor
    θ[itex]_{ik}[/itex] = [itex] g_{ik} - U_{i}U_{k}[/itex]
    projects any vector, [itex]V^{k}[/itex], into a 3-surface orthogonal to the unit time-like
    vector [itex] U_{i} [/itex] (By a projection, the vector [itex]θ_{ik}V_{k}[/itex], is implied).



    2. Relevant equations



    3. The attempt at a solution

    The projection should be,
    [itex] θ_{ik} V^k = g_{ik} V^k - U_i U_k V^k
    \Rightarrow θ_{ik} V^k U_i = g_{ik} V^k U_i - U_i U_k V^k U_i [/itex]. This should equal zero, for the projection to be orthogonal. But, i'm not being able to proceed.
    By timelike, the problem means [itex] U_i U^i \ge 0 [/itex], right? But, i dont see how that helps me.
     
    Last edited: May 14, 2014
  2. jcsd
  3. May 14, 2014 #2

    strangerep

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    Science Advisor

    The phrase "unit time-like vector ##U_i##" is important. What do you think "unit" means here?

    Yes, but (again), what does "unit" mean here?

    Whenever you're contracting indices in such contexts, it should be between an upstairs and a downstairs pair of indices. So your last part should be
    $$\theta_{ik} V^k U^i ~=~ g_{ik} V^k U^i - U_i U_k V^k U^i $$Now, can you simplify these subexpressions:
    $$g_{ik} U^i ~~~\text{and}~~~ U_i U^i ~~~?$$Then consider further my remarks about the meaning of "unit"...
     
  4. May 16, 2014 #3
    Ohho, sorry! I missed the 'unit' bit. [itex] U_i U_k [/itex] can be written as [itex] g_{ik} U_i U^i[/itex]. Now, [itex] U_i [/itex] is a unit timelike vector. By definition, then, [itex] U_i U^i = [/itex] 1. Then, the terms cancel and the result follows, right! Thanks for pointing that out! :)
     
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