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Raising and Lowering Vectors/Tensors

  1. Apr 16, 2012 #1
    I've been going back over my general relativity problem sheets and have realised I am not raising and lowering vectors/tensors properly. I've written out my calculations in detail below. I've tried to follow exact logical steps, but I must be misunderstanding, because my answers seem wrong (I often end up with just numbers)..

    1. The problem statement, all variables and given/known data

    Imagine we have a tensor Xαβ and a vector Vα with components

    Xαβ = [itex]\begin{pmatrix} 2 & 0 & 1 & -1 \\-1 & 0 & 3 & 2 \\-1 & 1 & 0 & 0 \\-2 & 1 & 1 & -2 \end{pmatrix}[/itex]

    and Vα = [itex]\begin{pmatrix} -1 \\2 \\0 \\-2 \end{pmatrix}[/itex]

    Find Xαβ, Xαβ, X(αβ), X[αβ], Xλλ, VαVα, VαXαβ


    2. Relevant equations

    I have been trying to use the raising/lowering metric g = [itex]\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/itex]

    3. The attempt at a solution


    Xαβ = gααXαβ = [itex]\sumαgαα[/itex]Xαβ = g00X + g11X + g22X + g33X = X - X - X - X = [itex]\begin{pmatrix} 2 & 0 & 1 & -1 \end{pmatrix}[/itex] - [itex]\begin{pmatrix} -1 & 0 & 3 & 2 \end{pmatrix}[/itex] - [itex]\begin{pmatrix} -1 & 1 & 0 & 0 \end{pmatrix}[/itex] - [itex]\begin{pmatrix} -2 & 1 & 1 & -2 \end{pmatrix}[/itex] = [itex]\begin{pmatrix} 6 & -2 & -4 & -1 \end{pmatrix}[/itex]


    Vα = gααVα = g00V0 + g11V1 + g22V2 + g33V3 = V0 - V1 - V2 - V3 = -1 -2 -0 -(-2) = -1

    → VαVα = -Vα


    Xλλ = gλαgλβXαβ

    gλβXαβ = gλ0Xα0 + gλ1Xα1 + gλ2Xα2 + gλ3Xα3 = [itex]\begin{pmatrix} 1 \\ 0 \\ 0 \\0 \end{pmatrix}[/itex]*[itex]\begin{pmatrix} 2 \\ -1 \\ -1 \\-2 \end{pmatrix}[/itex] + [itex]\begin{pmatrix} 0 \\ -1 \\ 0 \\0 \end{pmatrix}[/itex]*[itex]\begin{pmatrix} 0 \\ 0 \\ 1 \\1 \end{pmatrix}[/itex] + [itex]\begin{pmatrix} 0 \\ 0 \\ -1 \\0 \end{pmatrix}[/itex]*[itex]\begin{pmatrix} 1 \\ 3 \\ 0 \\1 \end{pmatrix}[/itex] + [itex]\begin{pmatrix} 0 \\ 0 \\ 0 \\-1 \end{pmatrix}[/itex]*[itex]\begin{pmatrix} -1 \\ 2 \\ 0 \\-2 \end{pmatrix}[/itex] = 2 + 0 + 0 + 2 = 4

    I haven't written my attempts to every equation, because I'm pretty sure my method is wrong and it's taking forever to write out the matrices!

    If anyone can spot where I'm going wrong that would be really helpful. Seeing as this is the most basic part of my general relativity course, if I can't get this bit right I'll be in trouble in the exam in june!

    Thank you.
     
  2. jcsd
  3. Apr 16, 2012 #2
    Your notation is a little dangerous: you should probably use different dummy indices for summation. So for example

    [tex] V_{a} = g_{ab} V^b \equiv \sum_{b} g_{ab} V^b[/tex]

    However, this is not your problem.

    What you should do is to try to calculate the components explicitly. So for example you know that [itex]V_a [/itex]is a (dual) vector, so it has 4 components. Do them one by one: [itex] V_0 = \sum_a g_{0 a} V^a = -1 V^0 [/itex]etc. It seems to me that you keep losing track of what you're calculating, so you're for adding together different vector components, etc.
     
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