Special relativity: 2d metric components

[Uku]

An SR question again, exam on monday.

Homework Statement

I'm given a 2D metric as:

$$ds^{2}=x^{2}dx^{2}+2dxdy-dy^{2}$$

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

Homework Equations

General expression of a metric tensor

$$ds^{2}=g_{\mu\nu}dx^{\nu}dx^{\mu}$$

The Attempt at a Solution

Since the metric is 2D, I can write the above as (with significance to me)

$$ds^{2}=g_{00}dx^{0}dx^{0}+g_{11}dx^{1}dx^{1}$$ 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 $$ds^{2}=dx^{2}+dy^{2}$$
Comparing the two I can assume that $$g_{00}=1$$ and $$g_{11}=1$$, which seems to make sense, because then the Phythagoras theroem emerges from 1)

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

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

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

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

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

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 $$g_{ab}$$ and $$g^{ab}$$?

 

[gabbagabbahey]

An SR question again, exam on monday.

Homework Statement

I'm given a 2D metric as:

$$ds^{2}=x^{2}dx^{2}+2dxdy-dy^{2}$$

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

Homework Equations

General expression of a metric tensor

$$ds^{2}=g_{\mu\nu}dx^{\nu}dx^{\mu}$$

The Attempt at a Solution

Since the metric is 2D, I can write the above as (with significance to me)

$$ds^{2}=g_{00}dx^{0}dx^{0}+g_{11}dx^{1}dx^{1}$$ 1)

Why no $g_{01}$ and $g_{10}$ terms?

 

[Uku]

Because I assumed the metric to be Euclidean, where the components not on the main diagonal are zero, meaning that the $$g_{01}$$ and $$g_{10}$$ 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]

Why assume the metric is Euclidean (although you should assume it is symmetric)? The metric is defined by the equation $ds^{2}=x^{2}dx^{2}+2dxdy-dy^{2}$.

 

[Uku]

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]

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

$$S^{\mu}_{\;\nu}=S^{\;\mu}_{\nu}\equiv S^{\mu}_{\nu}$$

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.