An SR question again, exam on monday.(adsbygoogle = window.adsbygoogle || []).push({});

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 isassumingthat 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]?

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# Special relativity: 2d metric components

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