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Uku Jun26-10 04:44 AM

Special relativity: 2d metric components
 
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]?

gabbagabbahey Jun26-10 04:53 AM

Re: Special relativity: 2d metric components
 
Quote:

Quote by Uku (Post 2776398)
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)

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

Uku Jun26-10 04:56 AM

Re: Special relativity: 2d metric components
 
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!

gabbagabbahey Jun26-10 05:08 AM

Re: Special relativity: 2d metric components
 
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].

Uku Jun26-10 05:59 AM

Re: Special relativity: 2d metric components
 
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?

Uku Jun26-10 06:28 AM

Re: Special relativity: 2d metric components
 
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.


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