How to embed the universe in R^4

  • Context: Undergrad 
  • Thread starter Thread starter Kontilera
  • Start date Start date
  • Tags Tags
    Universe
Click For Summary
SUMMARY

The discussion centers on embedding hyperbolic geometry within R^4, contrasting it with spherical geometry. The equation for the hyperbolic model is established as x_1^2 - x_2^2 - x_3^2 - x_4^2 = 1. This equation defines the hyperbolic space, which exists independently of larger spaces. Additionally, the distance calculation in hyperbolic geometry requires a unique approach, as referenced in the Hyperboloid model on Wikipedia.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with hyperbolic geometry concepts
  • Knowledge of mathematical embedding techniques
  • Basic proficiency in mathematical notation and equations
NEXT STEPS
  • Study the properties of hyperbolic geometry in R^4
  • Explore the Hyperboloid model for distance calculations
  • Investigate applications of hyperbolic geometry in theoretical physics
  • Learn about Riemann surfaces and their embeddings
USEFUL FOR

Mathematicians, physicists, and students interested in advanced geometry, particularly those focusing on hyperbolic structures and their implications in higher-dimensional spaces.

Kontilera
Messages
176
Reaction score
24
Hello!
I got a quick question.

The universe can have different geometries, as I understood it.
And we don't need to embed these structures in any larger space, they "exist on their own" (in a mathematical sense). However my question is the following:

How would one embed the hyperbolic alternative for our spatial universe in R^4?

The spherical alternative seems obvious, x_1^2+x_2^2+x_3^2+x_4^2 = 1.

Whats the corresponding equation for the hyperbolic geometry?Thanks!
 
Space news on Phys.org

Similar threads

Undergrad Is the Universe a 3-sphere or a 4-sphere?
  • · Replies 42 ·
2
Replies
42
Views
7K
Undergrad 3 questions about a paper on density fluctuations in the Universe
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Why does an Empty Universe have to obey Negative Curvature?
  • · Replies 10 ·
Replies
10
Views
3K
High School General explicit solution to polynomial interpolation
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad If a 4th dimension existed would universe expand anisotropically?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
2K
Undergrad An idea for calculating the probability that the Universe is flat
  • · Replies 20 ·
Replies
20
Views
2K
Graduate How Do You Apply Noether Normalization to a Polynomial Ring Ideal?
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Shape of de Sitter Universe: Is Hyperboloid Misleading?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Argument that a maximum density Universe expands linearly
  • · Replies 9 ·
Replies
9
Views
3K