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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.

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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 $\mathbf{x}=(t,x,y,z)$.

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  • $\begingroup$ I know this result and it is true in a flat background. But what I am doubt is whether it remains this form if the background is curved. I mean should we modif the definition of delta function or take the parallel translation that links two spacetime point into acount?Thank you. $\endgroup$ Mar 6, 2018 at 23:28
  • $\begingroup$ Valid point. In non-flat, or even non-Cartesian, coordinates, we might need to insert a Jacobian, sqrt(det(g)), which I might insert upside down because I'm an algebraic klutz. Don't worry about parallel transport; it only affects the given trajectory. $\endgroup$ Mar 7, 2018 at 14:47

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