Special relativity: 2d metric components

  1. 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
  2. jcsd
  3. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    Why no [itex]g_{01}[/itex] and [itex]g_{10}[/itex] terms?
     
  4. 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. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    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. 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. 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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