Simple question on Kaluza-Klein

[kdv]

In the Kaluza-Klein approach, 4-D electromagnetism is explained in terms of the curvature of a 5D space.

But I am bothered by an obvious question: what are the sources of the em field?
In GR, matter and energy curve spacetime. Now, if we assume a 5 dimensional version of of GR, it seems as if any matter/energy distribution would curve 5D spacetime.
So how could the theory accounts for the fact that charged matter "curves 5D spacetime" whereas neutral matter curves only the 4D submanifold (so that there is no electromagnetic force observed).

Whenever I have seen derivations of KK, it has always been with the metric only, not the metric coupled to sources. And one indeed ends up with 4D GR plus electromagnetism. But it seems to me that the question of sources is an obvious question.

Anybody can shed some light on this?

 

[JustinLevy]

I was searching for insight on exactly the same question.

Can someone please share a bit of insight here?
If not, has anyone worked with KK before and can suggest some good reading material?

 

[cosmik debris]

The sources of EM are the same as in Maxwell's theory, charge and EM radiation.

 

[granpa]

good question

 

[JustinLevy]

The sources of EM are the same as in Maxwell's theory, charge and EM radiation.

You miss the point of the question.
How do the sources couple to the metric. ... in a "unified" theory why does "charge" cause curvature to the metric in a way that mass can't so that gravitation and electromagnetism effectively decouple?

Consider a particle at rest, verse a charged particle at rest. It seems that mass cannot be a simple scalar in this theory, as it can have multiple testable properties.

I'm sure I'm misunderstanding something fundemental.
I was glad to see someone else was having similar confusion and questions, and was hoping discussion would open up.

Hopefully someone can help.

 

[arivero]

It is even worse: a torus, for instance, has no curvature. So neither the charge nor the mass cause curvature in the extra dimensions.

Modern Kaluza Klein theories (which, btw, were deprecated in 1983, in favour of string gauge groups) get the gauge bosons as the generators of the group of isometries of the compact space. On other hand, matter is got by looking the 4 dimensional part of the fermions that live in the total space. Here is where different fermions are produced, but I am not sure if you can produce different charges or only different generations.

 

[arivero]

This is probably the place to admit the supreme irony of the Kaluza-Klein
unification story. It is that the heterotic superstring [60], which is currently
the favorite way to unify gravity with the other forces, while making use of
(and indeed demanding) extra spacetime dimensions a la Kaluza-Klein, nevertheless
eschews the route of getting the Yang-Mills gauge group from the
general coordinate group. Instead, the Yang-Mills fields are already present
in the D = 10 dimensional formulation! Moreover, the favorite way subsequently
to compactify the theory to D = 4 invokes a Calabi-Yau manifold
[61]; an M6 with no isometries at all! Does this mean that the idea of
non-abelian gauge fields from extra dimensions is dead?

from http://arxiv.org/abs/hep-th/9410046

 

[arivero]

http://backreaction.blogspot.com/2007/06/early-extra-dimensions.html

ans she refers to

http://arxiv.org/pdf/gr-qc/9805018v1

 

[BenTheMan]

You miss the point of the question.
How do the sources couple to the metric. ... in a "unified" theory why does "charge" cause curvature to the metric in a way that mass can't so that gravitation and electromagnetism effectively decouple?

Consider a particle at rest, verse a charged particle at rest. It seems that mass cannot be a simple scalar in this theory, as it can have multiple testable properties.

I'm sure I'm misunderstanding something fundemental.
I was glad to see someone else was having similar confusion and questions, and was hoping discussion would open up.

Hopefully someone can help.

I THINK you're asking why does the 5-d Lorentz Invariance SO(4,1) break to SO(3,1) x U(1). I THINK this is what people mean by "compactification", although I still (after working in the field for some time) don't understand this fully. There is a similar conceptual issue in string theory---specifically, WHY do six dimensions become small? What allows 3 dimensions to be large? Insofar as I understand it, these questions are not (in general) answered, although Vafa and Brandenburger constructed a beautiful statistical mechanics argument in like 1988.

So, I too would like help with this question. How or why does the SO(4,1) Lorentz symmetry break? What one would have to find is a state which is charged under the U(1) which is contained in the SO(4,1), then give it a VEV. This would give you a spectrum of particles at the compactification scale, which correspond to polarizations of the 5-d gravitons, or something. Maybe someone more well versed in symmetries can answer this.

 

[granpa]

You miss the point of the question.
How do the sources couple to the metric. ... in a "unified" theory why does "charge" cause curvature to the metric in a way that mass can't so that gravitation and electromagnetism effectively decouple?
.

I don't know either but it looks to me like it just follows from the nature of tensor multiplication.

 

[cosmik debris]

You miss the point of the question.
How do the sources couple to the metric. ... in a "unified" theory why does "charge" cause curvature to the metric in a way that mass can't so that gravitation and electromagnetism effectively decouple?

Consider a particle at rest, verse a charged particle at rest. It seems that mass cannot be a simple scalar in this theory, as it can have multiple testable properties.

I'm sure I'm misunderstanding something fundemental.
I was glad to see someone else was having similar confusion and questions, and was hoping discussion would open up.

Hopefully someone can help.

Sorry, yes I misunderstood your question. The fifth dimension is curled up into a circle and as such represents a U(1) gauge group just like it does in QED, with all its emergent properties. Because this circle is small it has little effect on the four dimesional spacetime of GR. Have a look at papers on fibre bundles and you will see the different "connections" which are derived from the manifold. There will be one that represents the Christoffel connection of GR and one that represents the "force" of electromagnetism.

 

[granpa]

You miss the point of the question.
How do the sources couple to the metric. ... in a "unified" theory why does "charge" cause curvature to the metric in a way that mass can't so that gravitation and electromagnetism effectively decouple?
.

actually its worse than that. all 3 fields must decouple. the electric, magnetic, and gravitational.

 

[LukeD]

So I might be completely off base here since I've never studied Kaluza-Klein or GR, but it seems to me that if you have a 5D space time, that your fields would all need to be 20 dimensional. That seems a little... odd to me, but whatever. Maybe we'd experience that as 5 separate parts of a force similar to how to experience electricity and magnetism as 2 parts of a force. It would also need to have a 5 dimensional potential and a 5 dimensional source.

So the 5 dimensional source seems to make sense to me. However, I don't at all understand how having a 20 dimensional force matches much with reality. Do we have any 20 dimensional forces?

I don't know much about action principles, but am I just making some weird crazy and incorrect assumptions about how they give rise to forces?

 

[granpa]

it seems to me that if you have a 5D space time, that your fields would all need to be 20 dimensional.

why?

 

[Mentz114]

I don't know much about action principles, but am I just making some weird crazy and incorrect assumptions about how they give rise to forces?

When the 5-D equations of motion are projected into 4-D, there is a term arising from U⁵ ( which may be taken as a constant ) and the {i,j,5} affine connection. This term looks just like the EM force F^abU_a. It's very neat and well documented.