Kaluza-Klein Lorentz Force

[elfmotat]

This came up in another forum, and as of yet we weren't able to come up with an answer. Someone posted the following pdf about KK theory: http://www.weylmann.com/kaluza.pdf . In it, the Lorentz force law is derived from the 5D geodesic equation. What was puzzling us was the appearance of an additional term:

 

[math]\frac{d^2x^\lambda }{ds^2}+\Gamma^\lambda_{\mu \nu }\frac{dx^\mu}{ds} \frac{dx^\nu}{ds}=-kF^\lambda_{~\mu}\frac{dx^\mu}{ds} \frac{dx^4}{ds}-\frac{1}{2}kA_\nu F^\lambda_{~\mu} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}[/math]

 

This is troubling for two reasons: (1) there's no experimental evidence to suggest deviations from the traditional Lorentz force law, and (2) it is directly dependent on the four-potential which suggests it is not gauge-invariant and that the potential is directly observable.

 

Any insight would be appreciated.

[elfmotat]

I was able to resolve this with an another (anonymous) individual. (1) The factor of 1/2 in front of the second term is incorrect. (2) it turns out that the second term on the RHS of the equation isn't gauge-invariant because the first term isn't either! A gauge transformation is equivalent to a change in the 5th coordinate, so the first term on the RHS becomes:

 

[math]k F^{\lambda}_{~\mu} \frac{dx^{\mu}}{ds} \frac{dx^4}{ds} \rightarrow kF^\lambda_{~\mu} \frac{dx^\mu}{ds} \left(\frac{dx^4}{ds}+\frac{d \zeta}{ds}\right)[/math]

 

The second term becomes:

 

[math]k A_{\nu}F^{\lambda}_{~\mu}\frac{dx^{\mu}}{ds} \frac{dx^{\nu}}{ds} \rightarrow k (A_{\nu}-\partial_{\nu} \zeta)F^{\lambda}_{~\mu} \frac{dx^{\mu}}{ds} \frac{dx^{\nu}}{ds} = kA_{\nu}F^{\lambda}_{~\mu} \frac{dx^{\mu}}{ds} \frac{dx^{\nu}}{ds}-kF^{\lambda}_{~\mu} \frac{dx^{\mu}}{ds} \frac{d \zeta}{ds}[/math]

 

So the entire RHS is actually gauge-invariant.

 

(3) The Noether conserved momentum about the "curled up" 5^th dimension turns out to be:

 

[math]k\left (\frac{dx^4}{ds}+A_\mu \frac{dx^\mu}{ds} \right )[/math]

 

So if we identify this conserved quantity with electric charge, we obtain the standard Lorentz force law with no additional term.

Edited February 5, 2013 by elfmotat