(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the metric for a 3-sphere embedded in 4-space is:

[tex]ds^2=dr^2+R^2 sin^2(\frac{r}{R})(d\theta^2+sin^2\theta d\phi^2)[/tex]

r is the distance from some "pole" and R is the radius of curvature of the 3-sphere.

My question:

I showed this by using the transformations (as suggested by the professor):

[tex]w=Rcos\chi[/tex]

[tex]z=Rsin\chi cos\theta[/tex]

[tex]x=Rsin\chi sin\theta cos\phi[/tex]

[tex]y=Rsin\chi sin\theta sin\phi[/tex]

[tex]r=R\chi[/tex]

So, all I did was differentiate w, z, x, and y implicitly using all the product rules so that:

[tex]dw=-Rsin\chi d\chi[/tex]

[tex]dz=R(cos\chi cos\theta d\chi - sin\chi sin\theta d\theta)[/tex]

[tex]dx=R(cos\chi sin\theta cos\phi d\chi + sin\chi cos\theta cos\phi d\theta - sin\chi sin\theta sin\phi d\phi)[/tex]

[tex]dy=R(cos\chi sin\theta sin\phi d\chi + sin\chi cos\theta sin\phi d\theta + sin\chi sin\theta cos\phi d\phi)

[/tex]

Ok. So, I squared each and set:

[tex]ds^2=dw^2+dx^2+dy^2+dz^2[/tex]

After a lot of algebra, Lo, and behold I got exactly what my professor asked for!

I was overjoyed.

And then I got to thinking. Nowhere in my solution have I even invoked any condition that I am working on a 3-sphere. In fact, my metric before the coordinate transform is for flat space!

This leads me to think that all I have done is a coordinate transformation and NOT finding the metric of a 3-sphere. I tried the same method in just 2 dimensions going from Cartesian coordinates to Polar coordinates and I in fact got back the flat-space metric in polar coordinates.

So...my questions is...how the heck did I arrive at the metric for a 3-sphere embedded in 4-space WITHOUT even invoking the condition of a 3-sphere? How did I get to this metric by just doing a coordinate transformation!?

I refuse to believe that I just made a mistake somewhere and MIRACULOUSLY I got the right answer...There must be something I'm missing here.

Please help! Thanks.

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# Homework Help: Derivation of a metric

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