Proving Conservation of (u_{α}+eA_{α})ξα on Particle World Line in Relativistic Mechanics

[mtak0114]

1. Question

A particle with an electric charge $$e$$ moves in a spacetime with the metric $$g_{\alpha\beta}$$ in the presence of a vector potential $$A_{\alpha}$$.

The equations of motion are $$u_{\alpha;\beta}u^{\beta} = eF_{\alpha\beta}u^{\beta}$$, where $$u^{\alpha}$$ is the four-velocity and $$F_{\alpha\beta} = A_{\beta;\alpha}-A_{\alpha;\beta}$$. It is assumed that the spacetime possesses a killing vector $$\xi^{\alpha}$$,

so that $$\mathcal{L}_{\xi}g_{\alpha\beta} = {L}_{\xi}A_{\alpha} = 0$$.

Prove that $$(u_{\alpha}+eA_{\alpha})\xi^{\alpha}$$

is constant on the world line of the particle.

Homework Equations

$$\mathcal{L}_{\xi}g_{\alpha\beta} = \xi_{\alpha;\beta}+ \xi_{\beta;\alpha}$$

$$\mathcal{L}_{\xi}A_{\alpha} = A_{\alpha;\beta}\xi^{\beta}+ \xi^{\beta}_;\alpha}A_{\beta}$$

The Attempt at a Solution

My approach at this problem was to show that $$(u_{\alpha}+eA_{\alpha})$$
satisfies the geodesic equation

and hence the inner product with the killing vector would be constant but doing so did not lead to any useful results.

Any suggestions or comments to this approach would be greatly appreciated

thanks

mtak

Note: this problem was from: A Relativists toolkit: The mathematics of black-hole mechanics
by Eric Poisson

 

[AEM]

Just a quick question: How did you express the geodesic equation in the presence of the electromagnetic field as well as the gravitational field?

 

[AEM]

Here is an amplification on my quick question in the previous post. The equation of motion for your charged particle is not the geodesic equation of the form of Poisson's equation (1.14). Rather, if you have access to Misner, Thorne, and Wheeler's "Gravitation", look at page 898. In the middle of the page is the equation of motion of your charged particle in the elementary coordinate representation. The lefthand side of that equation is the standard geodesic equation form, but the righthand side is a "source term" --the Lorentz force. But note the sentence after this term: "The Hamiltonian formalism enables one to discover immediately two constants of motion; the elementary Lorentz-force equation does not." I interpret this to mean that perhaps rather than a calculational proof that $$(u_{\alpha} + eA_{\alpha}) \xi^{\alpha}$$ is a constant on the world line, you need a logical argument.

I hope these comment are of use to you

 

[AEM]

One more though: You could read the whole of section 33.5 in MTW and obtain the results you want using the Hamiltonian formulation, but I'm sure that's NOT what Poisson had in mind when he posed that problem!