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]?
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!
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].
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?
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.