Line Element on a 3-Sphere

In summary, the conversation is about finding the line element on a 3-sphere for a GR class. The goal is to find a transition from (x,y,z,w) to (r, \chi , \theta , \phi) and the variables x, y, z, and w are defined in terms of a and the angles \chi, \theta, and \phi. The speaker is wondering if there is a better method than taking an educated guess for finding the unit vectors in 4-D. The other person suggests using the line element of a sphere and finding dr in terms of d\chi, d\theta, and d\phi.
  • #1
Biest
67
0
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:

[tex] ds^2 = a^2(d\chi^2 + (\sin^2 \chi)(d\theta^2 + (\sin^2 \theta d\phi^2))[/tex]

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

By now I have defined my variables as:

[tex] x^2 + y^2 + z^2 + w^2 = a^2 \newline [/tex]

[tex]
x = a \cos \chi \newline [/tex]
[tex]y = a \cos \phi \sin \theta sin \chi \newline[/tex]
[tex]z = a \sin \phi \sin \theta sin \chi \newline[/tex]
[tex]w = a \cos \theta \sin \chi \newline[/tex]

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 [tex] \chi , \theta [/tex] and [tex] \phi [/tex] 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
 
Last edited:
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  • #2
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.
 
  • #3
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 [tex] d\chi[/tex]
 

1. What is a 3-sphere?

A 3-sphere is a three-dimensional geometric shape that is the surface of a four-dimensional ball. It is a three-dimensional analog of the two-dimensional sphere, or the surface of a three-dimensional ball.

2. How is the line element defined on a 3-sphere?

The line element on a 3-sphere is defined using the Pythagorean theorem in four dimensions. It takes into account the three spatial dimensions and the time dimension, and is used to calculate the distance between two points on the surface of the 3-sphere.

3. What is the curvature of a 3-sphere?

The curvature of a 3-sphere is positive, meaning that it is a positively curved space. This is in contrast to a flat space, where the curvature is zero, or a negatively curved space, where the curvature is negative.

4. How is the line element on a 3-sphere different from the line element on a 2-sphere?

The line element on a 3-sphere is different from the line element on a 2-sphere because the 3-sphere has an extra dimension, the time dimension. This means that the line element on a 3-sphere has four terms, while the line element on a 2-sphere only has three terms.

5. What are some applications of studying the line element on a 3-sphere?

The line element on a 3-sphere has applications in various fields such as cosmology, differential geometry, and general relativity. It is used to understand the curvature of space and the behavior of light in curved spaces. It also has implications in the study of higher-dimensional spaces and their properties.

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