Metric Tensor Questions: Understanding Hartle's "Gravity" Example 7.2

[narfarnst]

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

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

The book tells you that you rewrite \theta=1/a \sqrt{x^{2}+y^{2}} , \phi = tan^{-1}(\frac{y}{x})
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 \theta and \phi, find d\theta and d\phi, and then plug them into the line element.
But I have two questions.
1. How do I find d\theta and d\phi.
2. How do I go from that line element to a 2x2 matrix? Thanks.

 

[fzero]

For 1., you want to use

<br /> \theta=1/a \sqrt{x^{2}+y^{2}} , \phi = tan^{-1}(\frac{y}{x})<br />

to write

d\theta = \frac{\partial \theta}{\partial x} dx + \frac{\partial \theta}{\partial y} dy,

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

For 2, note that

<br /> dS^{2}=a^{2}(d\theta^{2}+sin^{2}\theta d\phi ^{2}) = \begin{pmatrix} d\theta &amp; d\phi \end{pmatrix} \begin{pmatrix} a^2 &amp; 0 \\ 0 &amp; a^2 \sin^2\theta \end{pmatrix} \begin{pmatrix} d\theta \\ d\phi \end{pmatrix}

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.