Since Kaluza-Klein theory is made out of general relativity - a gravitational theory in 4 dimensions which is only attractive, then how does it takes into account the attraction/repulsion of opposite/same charges in 5 dimensions? What does Kaluza-Klein theory say about the reason behind the attraction/repulsion of opposite/same charges?

  • $\begingroup$ +1. I also wonder how the Lorentz force law manifests itself in KK theory. (My SWAG says it'll be $\nabla_a G^{ab}=0$ for charged dust. But how is "charged" defined?) $\endgroup$ – Ryan Unger Dec 4 '15 at 13:29

In the context of Kaluza-Klein theory, the charge of a particle is its momentum in the compactified direction. Particles traveling in one direction around the circle are positively charged. Particles traveling in the other are negatively charged.

In Kaluza-Klein theory, the five dimensional metric is re-expressed in terms of a four dimensional metric, a four dimensional gauge field, and a four dimensional dilaton. A massive particle in five dimensions will couple to all three four dimensional fields from the point of view of a four dimensional observer. From a five dimensional point of view, two massive particles of course attract one another because of five dimensional graviton exchange. From a four dimensional point of view, the net force between the particles must continue to be attractive. Both graviton and dilaton exchange will make the particles attract. However, the gauge field contribution can be positive or negative.

I suspect a good way to think about the gauge field contribution from a five dimensional point of view is in terms of frame dragging. The orbiting of one particle in the compact direction produces frame dragging. The effective mass of the other particle then depends on which way it orbits the compact direction compared to the first particle. On the linked Wikipedia page, one finds the statement, "Another interesting consequence [of frame dragging] is that, for an object constrained in an equatorial orbit, but not in freefall, it weighs more if orbiting anti-spinward, and less if orbiting spin ward."

I like to think about the gauge field effects using the identification $g_{\mu\theta} \approx A_\mu$ where $g_{ab}$ is the five dimensional metric, $\theta$ the compact direction, and $A_ \mu$ the four dimensional vector potential. Depending on your background, this identification either makes the previous discussion more precise or less intuitive. But there is a straightforward way to work out the Lorentz force law from this perspective: Simply write the five dimensional geodesic equation using $A_\mu$.


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