Can 5D Kaluza-Klein Theory Be Adapted for 2D Space-Time Models?

Click For Summary

Discussion Overview

The discussion centers on the possibility of adapting 5D Kaluza-Klein theory to work within a 2D space-time framework. Participants explore the implications of reducing dimensions and how this might relate to classical electrodynamics and general relativity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question the feasibility of applying 5D Kaluza-Klein theory results directly to a 2D model, suggesting that the original theory's intent is to incorporate more than four dimensions.
  • Others propose that it might be possible to compactify three spatial dimensions instead of one, although this raises questions about the definition of 2D general relativity.
  • A participant suggests that a model with one large space dimension, one compact space dimension, and one time dimension could help understand the geometrical aspects of electrodynamics in lower dimensions.
  • There is a discussion about the motion of charged particles in this framework, with some proposing that the motion could be represented as a helix around a cylinder defined by the dimensions.
  • Concerns are raised regarding the application of general relativity in 2D, noting that the Einstein equations become identities and suggesting alternative approaches like dilatonic gravity.
  • Some participants express interest in explicitly carrying out the compactification process to explore its implications for electrodynamics.
  • One participant mentions that while a model can be constructed with the specified dimensions, it may not accurately reflect real-world physics, such as reconstructing Coulomb's law.

Areas of Agreement / Disagreement

Participants express a range of views on the adaptation of Kaluza-Klein theory to 2D models, with no consensus reached on the feasibility or implications of such adaptations. Multiple competing perspectives remain regarding the definitions and outcomes of such theories.

Contextual Notes

Limitations include the unclear definitions of what it means for the theory to "work" in 2D, the unresolved mathematical steps in compactification, and the dependence on specific interpretations of dimensions and their implications for physical laws.

Who May Find This Useful

This discussion may be of interest to those exploring theoretical physics, particularly in the context of dimensional analysis, Kaluza-Klein theory, and the geometrical aspects of electrodynamics.

Spinnor
Gold Member
Messages
2,231
Reaction score
419
Should one be able to take the results of 5D Kaluza-Klein theory and turn it into a theory that works with two less space dimensions?

Thanks!
 
Physics news on Phys.org
It is unclear what you are asking. The entire point of KK theory is to have more than four space-time dimensions.
 
Orodruin said:
It is unclear what you are asking. The entire point of KK theory is to have more than four space-time dimensions.

Could you take the results from the 5D theory and simplify it to work in a made up space of one time, one space, and one compact dimension.

Thanks!
 
You will have to be more precise. FWhat do you mean by "work"?
 
You mean to compactify three spatial dimensions instead of one? I think that should be possible, e.g. compactifying 5D GR to 2D by taking your compactification space being spheres. Normally, 2D GR is not defined in terms of a metric because the Einstein eqns are identically satisfied, so I have to think about that or do the compactification explicitly :P
 
I guess GR is maybe boring with one space and one time dimension? Here I'm thinking we have one large space dimension and one compact space dimension in addition to time. I was hoping that KK theory could be simplified so that I might better understand the geometrical aspect of electrodynamics in lower dimensions.

Thanks!
 
Orodruin said:
You will have to be more precise. FWhat do you mean by "work"?

It should reproduce classical electrodynamics of one space and one time dimension by proper interpretation of an additional compact dimension.
 
Last edited:
So for a charged particle with no external forces the particles motion is a helix of a fixed pitch around a cylinder defined by the one large space dimension and the one small compact dimension? The greater the pitch of the helix the greater the momentum in the large dimension, assume momentum is constant in the compact dimension? If there is an external field which causes a force the pitch of the helix is no longer constant in time? The path the particle takes is the shortest by some measure?

Thanks!
 
Spinnor said:
I guess GR is maybe boring with one space and one time dimension? Here I'm thinking we have one large space dimension and one compact space dimension in addition to time. I was hoping that KK theory could be simplified so that I might better understand the geometrical aspect of electrodynamics in lower dimensions.
Thanks!
Well, it's not that GR is 'boring' in 2D, it is simply not defined because the Einstein eqns are identities. So what people try to do (at least, what I know of), is to describe gravity by a scalar field (dilatonic gravity). The reason is that if you fix your 2 diffeomorphisms in two dimensions, a metric has 1/2*2*(2+1) - 2= 1 degree of freedom. But if I were you, I would carry out the compactification explicitly and see what you get :) There are several interestic geometric identities in 2 and 3 dimensions, and I'm not sure how these mix up with compactification.
 
  • Like
Likes   Reactions: Spinnor
  • #10
Spinnor said:
It should reproduce classical electrodynamics of one space and one time dimension by proper interpretation of an additional compact dimension.
Actually, that's an interesting question, because GR in D=3 doesn't have gravitational waves with the Einstein eqn's; the Riemann tensor is completely fixed by the Ricci tensor and the metric. So the question is how that translates itself into the electrodynamics.
 
  • Like
Likes   Reactions: Spinnor
  • #11
So to answer your question in the OP which I now understand: I think you have to be very careful in just copy-pasting the 5D results to the 3D results. You have to do the compactification explicitly to find out because of the reasons stated earlier.
 
  • Like
Likes   Reactions: Spinnor
  • #12
Thanks for the replies! Been a bit under the weather. I guess to try and make things a bit more simple I wonder if one can start with 2D Minkowski space-time and then just add a single compact dimension as Kaluza did and formulate classical electrodynamics in this new space? I am really at this point only interested in the geometrical aspects of electrodynamics in this space. I was told in a thread that QED in 2D space-time is simple but not trivial.

6th post in thread below,

https://www.physicsforums.com/threads/qed-in-3d-qed-in-1d-what-changes.403293/#post-2721506

Thanks again for the help!
 
  • #13
You can construct a model with as many dimensions as you wish. It may be 1 spatial, 1 temporal and 1 compactified if you wish.

As whether this theory would be applicable to the real world: no. You will not reconstruct the Coulomb's law from it, for instance.

It might be useful however for such setups as quantum dots.
 

Similar threads

High School Energy Stored in Charged Capacitor: Kaluza-Klein Theory
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Kaluza-Klein Theory and 5D Einstein Equations
  • · Replies 5 ·
Replies
5
Views
5K
Graduate How Did Clifford's Theories Influence Modern Physics and String Theory?
  • · Replies 5 ·
Replies
5
Views
2K
High School Is Einstein's Space-Time Theory Overfitted or Simply a Visualization Tool?
  • · Replies 29 ·
Replies
29
Views
3K
Undergrad Kaluza–Klein metric, space between charged capacitor?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate How Does Einstein's Corner Term Challenge 5D General Relativity?
  • · Replies 2 ·
Replies
2
Views
2K
High School Can Kaluza-Klein theory accommodate magnetic charge?
  • · Replies 9 ·
Replies
9
Views
3K
High School Visualize 2D Intrinsic Curvature of Spacetime (1s+1t) in 3D
  • · Replies 14 ·
Replies
14
Views
5K
Graduate Does String Theory incorporate Kaluza Klein? How?
  • · Replies 3 ·
Replies
3
Views
3K
High School What Is the Fourth Dimension and How Does It Work?
  • · Replies 3 ·
Replies
3
Views
3K