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?

• This has the makings of a good question I think, but what does "but once similarly it was though for gravitation" mean? – joshphysics Jan 18 '14 at 0:01
• To have the geodesic, you need to firstly define a metric. (because geodesic is defined as the critical path on a manifold with metrics.) But the metric is defined by the spacetime. The variation of the spacetime metric is a 2-form symmetric tensor, is the graviton. So how can one talk about only the EM metric or EM geodesic without considering the spacetime metric? – wonderich Jan 18 '14 at 4:10
• Is there a notion of EM metric? – wonderich Jan 18 '14 at 4:12
• since user37569 who offered a link has departed and the answer deleted by Community, I am copying it as a comment: " Estakhr's Material-Geodesic Equation, meetings.aps.org/Meeting/DFD13/Session/R8.4 , which is Unification between Lorentz Force and Einstein Geodesic Equation. – anna v Jan 20 '14 at 6:04

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.

I believe so, that is, I believe there is a mathematical formulation for electromagnetic geodesics. But to me, it takes a jump all the way back to the Equivalence Principle (UP). In the gravitational UP, F = m*a equals Newton’s F = G*m1*m2/r^2 (inertial forces equal gravitational forces). Bohr used an electronic extension of this principle in his simple nonrelativistic hydrogen model, where he set F = m*a equal to Coulomb’s “electronic universal law" F = (1/4*pi*eps)*e1*e2/r^2. Note how these two universal laws, electronic and gravitational, are of essentially the same form. Many scientists have noted this, but I think it has significance. The ultimate generalization of Newton’s gravity is Einstein’s GR, with its defined geodesics, based on the UP. The limit in this generalized gravitational theory is gravitational Newtonian mechanics. It is therefore possible to construct a type of non-Euclidean field theory which has Coulomb’s “electronic universal law” as its limit, once the UP has been extended to incorporate electronic forces.

Consider (perhaps) the “ultimate” differential geometry based mass/charge field theory in GR today, that is, consider the charged Kerr-Newman field theory. The physical characteristics of the central body are mass, charge and spin. Consider a small “test body” coasting along a geodesic in this charged Kerr-Newman field. The proton in hydrogen has a mass, a charge, and a spin (the three physical parameters needed to completely define a charged Kerr-Newman field). The electron in hydrogen, then, according to GR, coasts in a charge Kerr-Newman field (or “jitters about,” according to QM, that’s ok, GR still says the proton “generates” a charged Kerr-Newman field). For the particular values of the proton’s rest mass, charge and spin, its generated charged Kerr-Newman field is pathetically weak in curvature. But here is an important fact, even though weak, it is not zero. According to charged Kerr-Newman GR, the spacetime within which the electron “moves” in hydrogen is not theoretically Minkowski, as assumed in all of QM. It is at least partially “moving” along an electromagnetic charged Kerr-Newman geodesic.

• The equations user23660 presented can be made specific for a central body containing mass, charge and angular momentum (spin). An important set of EM geodesics of interest (to me, and I hope to others) are the geodesics around central bodies possessing mass, charge and spin. The metric structure obtained by imposing static spherical symmetry and asymptotic flatness to the central body's exterior gravitoelectronic field (no magnetism) was first worked out by Reissner and Nordstrom by the 1920s (see p. 158 of Wald’s “General Relativity”). They follow from user23660’s equations for central bodies – user37024 Jan 19 '14 at 5:29