I am trying to understand if it is possible to have a consistent solution to Maxwell's equations and Lorentz force equation simultaneously with no other forces present. The Lorentz force density is $f^a = {F^a}_b j^b$, where ${F_a}_b$ is the electromagnetic tensor.
a) How to write the equation of motion? I want to write something like $v^{a}\nabla_a (mv^b) = f^b$, where $m$ is electromagnetic mass density and $v^{a} = j^{a}/ \rho$ is 4-velocity of the charge distribution, $m$ and $\rho$ are proper. Does this equation make sense?
b) How to calculate electromagnetic charge density?
c) I pose the following initial conditions problem. Fix a reference frame so that $t=0$ is defined, define $j^a$ for $t=0$. I want to obtain ${F_a}_b$, $j^a$ for the whole of $R^4$ satisfying Maxwell's equations and the Lorentz force equation. Is there solutions? I understand that solutions (if there are any) will not be very interesting because without other forces concentrated particles will disperse, but maybe someone can help me satisfy my curiosity. Thank you.
Edit: I partially answered the question myself. There is no need to add electromagnetic mass in the equation of motion because it's been already accounted for by the action of the field on the charge distribution which in turn produces the field.
Therefore, there is no need to calculate electromagnetic mass to pose the question (c), but we need some rest mass for equation $v^{a}\nabla_a (mv^b) = f^b$ to make sense. I guess to get solutions for massless charge distribution, we can solve it for some $m$ and then take the limit as $m \to 0$, hoping that solutions will converge. I think they should because the standard examples in textbooks show that (apparent) electromagnetic mass adds to rest mass in equations of motion.