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Metric Tensor questions

  • Thread starter narfarnst
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  • #1
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Hi. This is example 7.2 from Hartle's "Gravity" if you happen to have it lying around.

Metric of a sphere at the north pole
The line element of a sphere (with radius a) is [tex]dS^{2}=a^{2}(d\theta^{2}+sin^{2}\theta d\phi ^{2})[/tex]
(In [tex](\theta , \phi )[/tex] coordinates).
At the north pole [tex]\theta = 0[/tex] and at the N. pole, the metric doesn't look like a flat space, dS2=dx2+dy2.
Using the coordinate transformations: [tex]x= a(\theta cos \phi , y= a \theta sin \phi [/tex] show that the metric [tex]g_{\alpha\beta} = (1-2y^{2}/(3a^{2}) | 2xy/(3a^{2}, 2xy/(3a^{2}) | 1-2x^{2}/(3a^{2}) ) [/tex]

Where [tex]g_{\alpha\beta}[/tex] is suppose to be a 2x2 matrix.

The book tells you that you rewrite [tex]\theta=1/a \sqrt{x^{2}+y^{2}} , \phi = tan^{-1}(\frac{y}{x})[/tex]
And then use taylor series to expand and keep the first few terms. But it doesn't show the work.

I get the idea, but I'm not sure how the math works out.
So what I'd want to do is, from [tex]\theta[/tex] and [tex]\phi[/tex], find [tex]d\theta[/tex] and [tex]d\phi[/tex], and then plug them into the line element.
But I have two questions.
1. How do I find [tex]d\theta[/tex] and [tex]d\phi[/tex].
2. How do I go from that line element to a 2x2 matrix?


Thanks.
 

Answers and Replies

  • #2
fzero
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For 1., you want to use

[tex]
\theta=1/a \sqrt{x^{2}+y^{2}} , \phi = tan^{-1}(\frac{y}{x})
[/tex]

to write

[tex]d\theta = \frac{\partial \theta}{\partial x} dx + \frac{\partial \theta}{\partial y} dy,[/tex]

with an analogous expression for [tex]d\phi[/tex]. This will give you a 2x2 system of equations that you can solve for [tex]dx,dy[/tex] to rewrite the line element.

For 2, note that

[tex]
dS^{2}=a^{2}(d\theta^{2}+sin^{2}\theta d\phi ^{2}) = \begin{pmatrix} d\theta & d\phi \end{pmatrix} \begin{pmatrix} a^2 & 0 \\ 0 & a^2 \sin^2\theta \end{pmatrix} \begin{pmatrix} d\theta \\ d\phi \end{pmatrix} [/tex]

If you go through the trouble to express the results of part 1 in vector form, it might make computing the metric a bit faster.
 

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