- #1
mtak0114
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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.
[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]
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
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.
Homework 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]
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