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Special relativity: 2d metric components

  1. Jun 26, 2010 #1

    Uku

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    An SR question again, exam on monday.

    1. The problem statement, all variables and given/known data
    I'm given a 2D metric as:

    [tex]ds^{2}=x^{2}dx^{2}+2dxdy-dy^{2}[/tex]

    I have to first find the contravariant and covariant components of the metric, or [tex]g_{ab}[/tex] and [tex]g^{ab}[/tex]

    2. Relevant equations
    General expression of a metric tensor

    [tex]ds^{2}=g_{\mu\nu}dx^{\nu}dx^{\mu}[/tex]

    3. The attempt at a solution
    Since the metric is 2D, I can write the above as (with significance to me)

    [tex]ds^{2}=g_{00}dx^{0}dx^{0}+g_{11}dx^{1}dx^{1}[/tex] 1)

    Now this is assuming that the metric is Euclidean, with the components not on the main diagonal being zero.

    Now using "common sense" I know that in Euclidean space [tex]ds^{2}=dx^{2}+dy^{2}[/tex]
    Comparing the two I can assume that [tex]g_{00}=1[/tex] and [tex]g_{11}=1[/tex], which seems to make sense, because then the Phythagoras theroem emerges from 1)

    But! The lecturer has written down the metric formally as:

    [tex]g_{ab}=\left[ \begin{array}{cc} g_{xx} & g_{xy} \\ g_{yx} & g_{yy} \end{array} \right][/tex]

    And now, out of the blue for me, he has written [tex]g_{xx}=x^{2}[/tex] and [tex]g_{yy}=-1[/tex] Why so?

    Further, he has written that the metric is non-diagonal, meaning that

    [tex]\left[ \begin{array}{cc} g_{xx} & g_{xy} \\ g_{yx} & g_{yy} \end{array} \right]=\left[ \begin{array}{cc} x^{2} & 1 \\ 1 & -1 \end{array} \right][/tex]

    the elements aside the main diagonal are not zero. I'm puzzled at this point. The non-diagonal metric means that the summation 1) is a false assumption by me, because the components are not zero. How do I approach this 2D metric to find the [tex]g_{ab}[/tex] and [tex]g^{ab}[/tex]?
     
    Last edited: Jun 26, 2010
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  3. Jun 26, 2010 #2

    gabbagabbahey

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    Why no [itex]g_{01}[/itex] and [itex]g_{10}[/itex] terms?
     
  4. Jun 26, 2010 #3

    Uku

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    Because I assumed the metric to be Euclidean, where the components not on the main diagonal are zero, meaning that the [tex]g_{01}[/tex] and [tex]g_{10}[/tex] are zero, meaning I do not have to consider them in the summation. But now that you put my attention to it, ill look into it.

    EDIT: I see some light!
     
  5. Jun 26, 2010 #4

    gabbagabbahey

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    Why assume the metric is Euclidean (although you should assume it is symmetric)? The metric is defined by the equation [itex]ds^{2}=x^{2}dx^{2}+2dxdy-dy^{2}[/itex].
     
  6. Jun 26, 2010 #5

    Uku

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    Solved it! I assumed because I wanted to start solving the assignment from somewhere.

    Why should I assume symmetry? Because I can't prefer any direction over others?
     
  7. Jun 26, 2010 #6

    Uku

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    I have a second question about symmetry. My my course material, I have a following statement about symmetric tensors:

    [tex]S^{\mu}_{\;\nu}=S^{\;\mu}_{\nu}\equiv S^{\mu}_{\nu}[/tex]

    What does the spacing in the indexes mean?

    EDIT: I see it means that the tensor has mixed components, but what does that mean when I start the summation? I'm seeing that I can't sum, because the indexes are both cotravariant or covariant.
     
    Last edited: Jun 26, 2010
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