A CED: Solving the Full-blown Classical Electromagnetism Problem

[dextercioby]

Is it me, or the overall problem in CED is not addressed anywhere?
Statement of the problem: Let there be two massive electrically charged particles of invariant masses m₁ and m₂ and electric charges q₁ and q₂, respectively. They are free to move arbitrarily in Minkowski spacetime endowed with (+---) metric tensor. One is further given the full vacuum set of Maxwell's equations written in terms of a generic electric field E and magnetic field B. Please, write down (in covariant form, i.e. using spacetime (pseudo)tensors) the full set of dynamical equations for the motion of particles and the 2 fields that they generate. How are all these equations written in the presence of a gravitational field described by GR?

What do you think? Is the problem written somewhere, together with the solution, or can we (you) write the down the solution here?

EDIT after 3 days: NO TAKE? Ok, I will spend more time reading/working on this.

 

[Markus Hanke]

I don't know about the full treatment for the "two moving charges" problem in relativistic electrodynamics ( I leave that to the experts here ), but I can give you at least the basic equations in the presence of gravity. Maxwell's electric equation is

\displaystyle{F{^{\alpha \beta }}_{||\beta }=4\pi J^\alpha }

The magnetic equation is

\displaystyle{F_{\alpha \beta ||\gamma }+F_{\beta \gamma ||\alpha }+F_{\gamma \alpha ||\beta }=0}

And the equation of motion then is

\displaystyle{ma^\alpha =qF^{\alpha \beta }u_\beta }

wherein F denotes the usual electromagnetic field tensor, and the "||" means covariant differentiation, which is what accounts for the metric. You could alternatively write the Maxwell equations with differential forms, which would probably be more intuitive.