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I Schnutz Special Relativity Tensors Question

  1. Apr 17, 2016 #1
    There's a question in Schnutz - A first course in special relativity
    Consider a Velocity Four Vector U , and the tensor P whose components are given by
    Pμν = ημν + UμUν .
    (a) Show that P is a projection operator that projects an arbitrary vector V into one orthogonal to U . That is, show that the vector V⊥ whose components are
    Vα ⊥ = Pα βVβ = (ηα β + UαUβ)Vβ is
    (i) orthogonal to U

    Now I've attempted the solution and it is the following

    PβαVα = Vβ+UβUαVα

    So now if I calculate

    Vα ⊥ ⋅ U = VαUα+UαUαUαVα

    which is orthogonal if c=1 ... as |U|^2= -c^2

    but.. this is just in the metric -+++ , if I change metrics to +--- then it won't be orthogonal? Also it's not orthogonal if c=/=1 .. which doesn't seem right to me either
    how can that be?

    Thank for you help!

    Adam
     
  2. jcsd
  3. Apr 17, 2016 #2

    robphy

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    In general, you should include a factor of ##U\cdot U## in the denominator.
    That is to say, you should do the normalization explicitly, in your signature.
     
  4. Apr 17, 2016 #3
    Hmmm, how do you mean?
    do you mean I need to normalise ##U\cdot U## with itself?
     
  5. Apr 17, 2016 #4

    robphy

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    [tex]P_{\mu\nu}=\eta_{\mu\nu}- \frac{U_{\mu}U_{\nu}}{\eta_{\alpha\beta}U^{\alpha}U^{\beta}}[/tex]
     
  6. Apr 17, 2016 #5
    Thanks for that, but I don't understand why you do this?
     
  7. Apr 17, 2016 #6

    robphy

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    Does this operator do what you want it to do, independent of signature convention?
    Use it and see.
     
  8. Apr 17, 2016 #7
    I think it should, but I can't get it to work out.
    The operator is given in the question in the text book.
     
  9. Apr 18, 2016 #8

    Orodruin

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    Yes, and the textbook uses all of the conventions which makes the operator a projection operator. If it did not, it would have had to write out the form given in post #4.
     
  10. Apr 18, 2016 #9
    Ahhh right I see, that makes sense!

    thanks for the help guys
     
  11. Apr 18, 2016 #10
    Do you mean Schutz, A First Course in General Relativity?
     
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