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Vector notation / manipulation question

  1. Sep 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Xuv is a 4x4 tensor and Vu is a vector.

    Vu = (-1, 2, 0, -2) (i.e. it is a 1x4 vector).

    Find the quantities Vu Vu and Vu Xuv



    2. Relevant equations
    Given above

    3. The attempt at a solution
    I'm having trouble finding Vu. Initially I thought that it should be the transpose of Vu (so a 4x1 vector) but then you can't multiply a 4x1 matrix with a 4x4 matrix, which the tensor is.

    Also, my book says that you can turn a vector into a dual vector by doing
    Vu = ηuv Vv. However, I don't kow Vv; I only know Vu

    Can somebody point me in the right direction? I've taken linear algebra before so I should understand this but my teacher used a different notation than the general relativity class I'm taking right now uses.
     
  2. jcsd
  3. Sep 19, 2012 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    [itex]\nu[/itex] is just an index, it will range from 0 to 3 (or 1 to 4) just like [itex]\mu[/itex]. If you know [itex]V^{\mu}[/itex], then you know [itex]V^{\nu}[/itex]. The components of the vector do not change just because one uses a different index to refer to them.

    What is the metric tensor in this case (i.e. what type of space-time are you using)? What is your tensor [itex]X^{\mu\nu}[/itex]?
     
  4. Sep 19, 2012 #3
    Superscripts and subscripts range from 0-3 (or 1-4, whatever your preference). Calling a superscript a different letter has no significance. It's only when an index is repeated that you should interpret this to mean that the index should be the same for both.
     
  5. Sep 19, 2012 #4
    I'm in Minkowski flat space-time so the metric tensor is \begin{array}{ccc}
    -1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 \\
    0 & 0 & 0 & 0 \end{array}

    and the tensor is
    \begin{array}{ccc}
    2 & 0 & 1 & -1 \\
    -1 & 0 & 3 & 2 \\
    -1 & 1 & 0 & 0 \\
    -2 & 1 & 1 & -2 \end{array}

    I'm afraid I'm still confused. So
    Vu = ηuvVu? This still makes no sense to me because you can't multiply a 4x4 by a 1x4 matrix.
     
  6. Sep 19, 2012 #5
    Sure you can. You just multiply the 1x4 on the left of the matrix. (Whether this is what your instructor wants you to do is a different story. It's a matter of convention whether up-index vectors represent row matrices multiplied on the left of the transformation matrix or column matrices multplied on the right.)


    Really, though, you're trying to jump into representing things with matrices when you don't have a grasp of the underlying math yet. Index notation frees you from dealing with actual matrix multiplication at all.

    Expand the implied summation to see what you need to do.

    [tex]v_\mu = \eta_{\mu \nu} v^\nu = \eta_{\mu 1} v^1 + \eta_{\mu 2} v^2 + \eta_{\mu 3} v^3 + \eta_{\mu 4} v^4[/tex]
     
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