electromagnetism - Geodesic for Electromagnetic forces

Considering the fact that electrons tend to take the maximum conductance path to flow from A to B. This is justified by saying that $\vec{E}$ is larger in conductors. But once similarly it was thought for gravitation, that if in a region the gravity was stronger, the mass more likely took that path, then later it was found it is actually a geodesic in space time as gravity curves space time. So is there some underlying geodesic for motion caused by electromagnetic force?

Answer

One way to formulate the equations of motion of a charged particle as a geodesic equation is through the Kaluza–Klein theory. In it we add additional dimension (just one, if we are only interested in the electromagnetism) and write the 5D metric

$$dS^2 = ds^2 + \epsilon \Phi^{2}(dx^{4} + A_{\mu}dx^{\mu})^2,$$ where $ds^{2} = g_{\mu \nu} dx^{\mu} dx^{\nu}$ is the 4D (curved) metric, $\epsilon=+1$ or $-1$ is a sign choice for either space-like or time-like dimension, $A_\mu$ is identified with the 4-potential of electromagnetic field and $\Phi$ is an additional scalar field . The geodesic equation written in this 5D metric is: $$\begin{multline} \frac{d^2 x^{\mu}}{d{\cal S}^2}+ {\Gamma}^{\mu}_{\alpha \beta}\frac{dx^{\alpha}}{d{\cal S}}\frac{dx^{\beta}}{d{\cal S}}= n F^{\mu}_{\;\;\nu}\frac{dx^{\nu}}{d{\cal S}}+ \epsilon n^2 \frac{\Phi^{;\mu}}{\Phi^{3}} - A^{\mu}\frac{dn}{d{\cal S}}-\\- g^{\mu\lambda}\frac{dx^4}{d{\cal S}}\left(n \frac{\partial{A_{\lambda}}}{\partial{x^4}}+\frac{\partial{g_{\lambda\nu}}}{\partial{x^4}}\frac{dx^{\nu}}{d{\cal S}}\right), \end{multline}$$ and the same rewritten so that particle motion is parametrized through 4D proper interval $s$, rather than 5D $S$: $$\begin{multline} \frac{d^2 x^{\mu}}{ds^2}+{\Gamma}^{\mu}_{\alpha \beta}\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}=\\= \frac{n}{(1-\epsilon{n^2}/{\Phi^2})^{1/2}}\left[ F^{\mu}_{\;\;\nu}\frac{dx^{\nu}}{ds} - \frac{A^{\mu}}{n}\frac{dn}{ds}- g^{\mu\lambda}\frac{\partial{A_{\lambda}}}{\partial{x^4}}\frac{dx^4}{ds} \right]+ \\ + \frac{\epsilon n^2}{(1-\epsilon n^2/\Phi^2)\Phi^3}\left[\Phi^{;\mu} + \left(\frac{\Phi}{n}\frac{dn}{ds}- \frac{d\Phi}{ds}\right)\frac{dx^{\mu}}{ds}\right]-\\-g^{\mu\lambda}\frac{\partial{g_{\lambda\nu}}}{\partial{x^4}}\frac{dx^{\nu}}{ds}\frac{dx^4}{ds}. \end{multline}$$ Here the $F_{\mu\nu}$ tensor is the usual EM strength 4-tensor: $$F_{\mu\nu} = A_{\nu,\mu}-A_{\mu,\nu},$$ and $n$ is the (covariant) 4-speed component along the additional dimension: $$n =u_4 = \epsilon {\Phi}^2\left(\frac{dx^4}{d{\cal S}} + A_{\mu}\frac{dx^{\mu}}{d{\cal S}}\right).$$ These equations are taken from the paper:

Ponce de Leon, J. (2002). Equations of Motion in Kaluza-Klein Gravity Reexamined. Gravitation and Cosmology, 8, 272-284. arXiv:gr-qc/0104008.

which in turn refers to the book:

Wesson, P. S. (2007). Space-time-matter: modern higher-dimensional cosmology (Vol. 3). World Scientific google books.

We see in these equations many new terms absent in the equations of motion of a charge in a 4D curved spacetime. To eliminate these terms we impose constraints on the 5D metric by requiring independece of all metric component of the $x^4$ coordinate, and assuming the scalar $\Phi$ is simply constant. Then the quantity $n$ is an integral of motion and the geodesic equation now looks like this:

$$\frac{d^2 x^{\mu}}{ds^2}+{\Gamma}^{\mu}_{\alpha \beta}\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}= \frac{n}{\left(1-\epsilon{n^2}/{\Phi^2}\right)^{1/2}}\left[F^{\mu}_{\;\;\nu}\frac{dx^{\nu}}{ds} \right],$$ which is exactly the equation of motion for the charge in curved space-time in the presence of EM field. With the (now) constant factor $n(1-\epsilon{n^2}/{\Phi^2})^{-1/2}$ having the role of a charge to mass ratio $e/m$.

I have left out numerous questions arising from this simple treatment, for them you should look into the relevant books and papers, but for the purpose of casting equations of motion of a test charge as a geodesic equations the answers to them are not needed.