(adsbygoogle = window.adsbygoogle || []).push({}); 1. Question

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

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

so that [tex]\mathcal{L}_{\xi}g_{\alpha\beta} = {L}_{\xi}A_{\alpha} = 0[/tex].

Prove that [tex](u_{\alpha}+eA_{\alpha})\xi^{\alpha}[/tex]

is constant on the world line of the particle.

2. Relevant equations

[tex]\mathcal{L}_{\xi}g_{\alpha\beta} = \xi_{\alpha;\beta}+ \xi_{\beta;\alpha}[/tex]

[tex]\mathcal{L}_{\xi}A_{\alpha} = A_{\alpha;\beta}\xi^{\beta}+ \xi^{\beta}_;\alpha}A_{\beta}[/tex]

3. The attempt at a solution

My approach at this problem was to show that [tex](u_{\alpha}+eA_{\alpha})[/tex]

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

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