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

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

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

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

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.