Metric in Manifold Homework: 3-sphere in 4D Euclidean Space

In summary, the standard method for finding the line element in higher dimensions is to use spherical coordinates and integrate the line element to find the radial distance between two points on the curved surface.
  • #1
kau
53
0

Homework Statement



I am confused if there is any standard way to check what should be the line element $$ds^{2}$$ when the dimensions are more than three ( since we don't have the option to draw things as we usually do in case of 3d or less dimensional cases. I am following Hobson's book. I am giving you an example. consider a 3sphere embedded in 4d euclidean space. show that line element can be written as following

Homework Equations


##ds^{2} = a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]##
now here can you introduce the concept of radial distance?? what it would be??
In that case if I have a metric ##ds^2=\frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]##
tell me what should be the radial distance between a sphere at r=2##\mu## and r=3##\mu##??
these are ques from hobson's book basically.
if any of you take the pain to explain me little bit in detail to visualise these stuff i would be grateful.
thanks

The Attempt at a Solution

 
Physics news on Phys.org
  • #2
For the first case involving a 3-sphere embedded in 4D Euclidean space, the line element can be written as follows:##ds^2 = a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]##Where a is the radius of the 3-sphere and ##\chi, \theta, \phi## are the usual spherical coordinates. The radial distance between two points on the 3-sphere can then be found by integrating the line element:##r = \int_0^a ds = \int_0^a a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]##For the second case involving the metric ##ds^2=\frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]##, the radial distance between a sphere at radius ##r_1=2\mu## and ##r_2=3\mu## can be found by integrating the line element from ##r_1## to ##r_2##:##r_{12} = \int_{2\mu}^{3\mu} ds = \int_{2\mu}^{3\mu} \frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]##
 

FAQ: Metric in Manifold Homework: 3-sphere in 4D Euclidean Space

1. What is a 3-sphere in 4D Euclidean space?

A 3-sphere in 4D Euclidean space is a mathematical concept that represents a three-dimensional hypersphere in a four-dimensional space. It is essentially a spherical object that exists in a four-dimensional world.

2. How is the metric of a 3-sphere in 4D Euclidean space calculated?

The metric of a 3-sphere in 4D Euclidean space is calculated using the Pythagorean theorem, just like a regular sphere in three-dimensional space. However, in 4D space, there is an extra dimension, so the metric equation becomes: ds^2 = dx^2 + dy^2 + dz^2 + dw^2.

3. What is the significance of studying the 3-sphere in 4D Euclidean space?

Studying the 3-sphere in 4D Euclidean space has applications in various fields such as physics, mathematics, and computer science. It helps us understand higher dimensions and their properties, which can be useful in fields like quantum mechanics and string theory.

4. Can the 3-sphere in 4D Euclidean space be visualized?

No, the 3-sphere in 4D Euclidean space cannot be visualized in the same way as we can visualize a regular sphere in three-dimensional space. This is because our brains are limited to comprehending only three dimensions, and the fourth dimension is beyond our perception.

5. Are there any real-world examples of the 3-sphere in 4D Euclidean space?

While we cannot directly observe the 3-sphere in 4D Euclidean space, it has been used in physics to describe the curvature of space-time in Einstein's theory of general relativity. It has also been used in computer graphics and animation to create 3D models and simulations.

Similar threads

Replies
10
Views
2K
Replies
3
Views
1K
Replies
11
Views
3K
Replies
42
Views
5K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
7
Views
2K
Replies
20
Views
2K
Replies
1
Views
2K
Back
Top