I know that for particles carrying no electric charge, given a Killing vector $K_{\mu}$, we have a conserved quantity $K_{\mu}p^{\mu}$ along geodesics, where $p^{\mu}$ is a tangeant vector. However, if $e \neq 0$, the equation of motion isn't simply $p^{\nu}\nabla_{\nu}p^{\mu} = 0$ anymore and so the above doesn't hold. I thought that the quantity $K_{\mu}(p^{\mu} - e A^\mu$), with $A^\mu$ the vector potential, would be conserved instead, but I'm having trouble showing this. Is this quantity really conserved and, if not, what is the correct quantity?

  • $\begingroup$ A general procedure would be to start with lagrangian of the system :(free particle+electromagnetic field+interaction $J^{\mu}A_{\mu}$), then evaluate $\delta S=0$ by considering variation of coordinates $\delta x^{\mu}=\alpha K^{\mu}$ and then solve for E-L equations of motion $\endgroup$
    – KP99
    Jul 12, 2021 at 16:48

1 Answer 1



  1. Write down the action $S[x]$ for a relativistic point charge in a background metric $g=g_{\mu\nu}\mathrm{d}x^{\mu}\odot\mathrm{d}x^{\nu}$ and a background gauge field $A=A_{\mu}\mathrm{d}x^{\mu}$.

  2. Assume that the vertical infinitesimal transformation $\delta x^{\mu}=\epsilon K^{\mu}$ is a symmetry, i.e. that the Lie derivatives $$ {\cal L}_Kg~=~0\quad\text{and}\quad{\cal L}_KA~=~0 $$ vanish.

  3. Deduce that the conserved Noether charge is $$ Q ~=~K^{\mu}p_{\mu}, $$ where $$ p_{\mu}~:=~\frac{\partial L}{\partial \dot{x}^{\mu}}~=~ p^{\rm kin}_{\mu} \pm q A_{\mu} $$ is the canonical/conjugate 4-momentum. Here we are using the signature convention $(\mp,\pm,\pm,\pm)$, respectively.

  • $\begingroup$ Right, so you need the Killing vector to also generate a symmetry of the vector potential for this to hold. Thanks a lot! $\endgroup$
    – johnny
    Jul 14, 2021 at 18:29
  • $\begingroup$ Right. That's the main point. Btw $\odot$ stands for the symmetric tensor product. $\endgroup$
    – Qmechanic
    Jul 14, 2021 at 18:35
  • $\begingroup$ Interesting, never came across that notation before! $\endgroup$
    – johnny
    Jul 14, 2021 at 18:42

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