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.


1 Answer 1


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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.