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Covariant characterization of causality in continuum

  1. Jan 23, 2016 #1
    Hi!

    Let ##T^{ik}## be the stress-energy-tensor, and ##v_k## some future-pointing, time-like four vector.

    How can I see that the object ##T^{ik}v_k## is future-pointing and not space-like?

    Thank you for your help!
     
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  3. Jan 23, 2016 #2

    robphy

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    What is the dot product of two future pointing timelike vectors in your signature convention?
     
  4. Jan 23, 2016 #3
    ##v_k v^k##
     
  5. Jan 23, 2016 #4

    PeterDonis

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    Compute its squared length. What do you get?
     
  6. Jan 23, 2016 #5
    The thing is, I am not really sure how to do that :)

    What I'd do is the following:

    ##T^{ik}v_k T_{il}v^l=T^{ik}v_k \eta_{im}\eta_{ln}T^{mn} \eta^{lj}v_j## but I'm not sure if that leads anywhere....
     
  7. Jan 23, 2016 #6

    PeterDonis

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    This is correct as an expression for the squared length of the vector, yes. If that vector is timelike, then the sign of its squared length must be the same as the sign of ##v_k v^k##. How would you go about comparing the signs of the two?

    (One hint: you should find that, in order for the signs of the two to be the same, you have to impose conditions on the components of ##T^{ij}##; i.e., the vector you're looking at is not always timelike, it only is if ##T## satisfies certain conditions.)
     
  8. Jan 24, 2016 #7
    Thanks for your help.

    According to my calculations [tex]T^{ik}v_k T_{ij}v^j=(T^{00}v_0+T^{01}v_1+T^{02}v_2+T^{03}v_3)^2-(T^{10}v_0+T^{11}v_1+T^{12}v_2+T^{13}v_3)^2-(T^{20}v_0+T^{21}v_1+T^{22}v_2+T^{23}v_3)^2-(T^{30}v_0+T^{31}v_1+T^{32}v_2+T^{33}v_3)^2.[/tex]

    However, I am struggling to find conditions for ##T^{ik}## from this...
     
  9. Jan 24, 2016 #8

    PeterDonis

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    These don't look right. The expression ##T^{ik} v_k T_{ij} v^j## should expand to terms that look like ##( T^{00} v_0 + T^{01} v_1 + T^{02} v_2 + T^{03} v_3 ) ( T_{00} v^0 + T_{01} v^1 + T_{02} v^2 + T_{03} v^3 )##, with the index positions on ##T## and ##v## switching from one factor to the other.
     
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