# What is the four-current $J_{\mu}$ of a single charged particle in curved spacetime?

I read this paper Phys.Rev.D 88.084059 which gives the formula (Eq.51) of four-potential using Kirchhoff representation with Green function and parallel propagator in curved spacetime.

Then I want to find out the four-potential of a single charged particle but I don’t know how to write down the four-current $J_{\mu}$ of it in curved spacetime. So could anyone help me?

Is it correct if I write down:

$J_{\mu}(x) = e g_{\mu\nu}(x) \int d\tau g^{\nu}_{\ \nu’}(x,X(\tau))U^{\nu’}(\tau)\delta_{4}(x,X(\tau))$

where the first $g_{\mu\nu}$ is the spacetime metric at point $x$ and the second $g^{\nu}_{\nu’}$ is the parallel propagator from $X(\tau)$ to $x$ (Eq.25 in the paper) . $X(\tau),U(\tau)$ are the trajectory and four-velocity of the charged particle and $\delta_{4}(x,x’)$ is the invariant Dirac distribution.

This review

claimed that in page 113 the propagator term should be added, so, should I?

Thank you.

First, parameterize the trajectory of the particle by proper time: $\mathbf{X}(s)\equiv {{X}^{\alpha }}(s)$ and write ${{u}^{\alpha }}=d{{X}^{\alpha }}/ds$. The 4-current is ${{J}^{\alpha }}(\mathbf{x})=\int{ds}\ {{\delta }^{4}}(\mathbf{x}-\mathbf{X}(s))\ {{u}^{\alpha }}$ as a function of $\mathbf{x}=(t,x,y,z)$.