How Do You Derive the Line Element on a 3-Sphere?

[Biest]

Hello,

I have been trying to figure out the line element on a 3-sphere for my GR class. Problem is that I am trying to go the traditional way of finding dx, dy, dz, and dw, then regroup and find the respective unit vectors and go from there. We are given that the answer as:

$$ds^2 = a^2(d\chi^2 + (\sin^2 \chi)(d\theta^2 + (\sin^2 \theta d\phi^2))$$

and are supposed to find our own transition from $$(x,y,z,w)$$ to $$(r, \chi , \theta , \phi )$$

By now I have defined my variables as:

$$x^2 + y^2 + z^2 + w^2 = a^2 \newline$$

$$x = a \cos \chi \newline$$
$$y = a \cos \phi \sin \theta sin \chi \newline$$
$$z = a \sin \phi \sin \theta sin \chi \newline$$
$$w = a \cos \theta \sin \chi \newline$$

I haven't written out my dx, dy, dz, and dw because I hope we can agree we just have to differentiate with respect to the three variables $$\chi , \theta$$ and $$\phi$$ and then just multiple the respective term after the Leibniz rule with the differential.

Is there are a better method then taking an educated guess as to what the unit vectors in 4-D are and moving on from there?

Thank you very much in advance.

Regards,

Biest

 

[Dick]

Oh, I think that's better than an educated guess. At an angle chi the x coordinate is a*cos(chi) and the remaining coordinates are those of a 2-sphere of radius a*sin(chi). It's basically just induction.

 

[Biest]

I think I just got it. Thank you very much. I simply use the line element of a sphere and find dr in terms of $$d\chi$$