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Minkowski Inner Product and General Tensor/Matrix Question

  1. Jul 26, 2012 #1
    Hello all.

    I have a fairly rudimentary knowledge of matrices and broader linear algebra. This gets me in a lot of trouble when I'm following along the math of something fine and then I run into some matrix stuff and get stumped, like this. I'm a little bit confused on taking the inner product from the Minkowski Tensor to the actual number. I understand why (in the context of spacetime intervals) it makes sense to define the metric as diag(-1,1,1,1) = [itex]\eta[/itex][itex]\alpha\beta[/itex]

    What I don't get is that if you define the inner product of two vectors [itex]A[/itex] and [itex]B[/itex] as [itex]\eta[/itex][itex]\alpha\beta[/itex]A[itex]\alpha[/itex]B[itex]\beta[/itex] (I hope I got the summation convention right), how do you get from the matrix form to the number -t[itex]\alpha[/itex]t[itex]\beta[/itex]+x[itex]\alpha[/itex]x[itex]\beta[/itex] ....(and so on)?
    It is just a rule of matrices I don't know? Or is it a specific thing in this context.

    Thank you
    Also, unrelated: my textbook didn't explicitly say that the spacetime interval (squared) is equal to the inner product, is that true?
     
  2. jcsd
  3. Jul 26, 2012 #2

    HallsofIvy

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    It is, in fact, the standard matrix multiplication that you do know, but the order of multiplication is different- the only order in which those matrices can be multiplied. This would be interpreted as
    [tex]\begin{pmatrix}t & x & y & z\end{pmatrix}\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}t \\ x\\ y \\ z\end{pmatrix}[/tex]
    [tex]= \begin{pmatrix}t & x & y & z\end{pmatrix}\begin{pmatrix}-t \\ x \\ y \\ z\end{pmatrix}= -t^2+ x^2+ y^2+ z^2[/tex]
     
  4. Jul 26, 2012 #3
    Ah, you're totally right. I wasn't thinking of the vectors as matrices.

    Thank you
     
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