# A Full-blown classical electromagnetism (in vacuum)

1. Mar 15, 2016

### 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.

Last edited: Mar 18, 2016
2. Mar 16, 2016

### 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.