[Certhas]

If you take a non-relativistic model of your matter content, then the theory becomes non-relativistic. That's trivial. So take a relativistic model for your matter and you have no problem [1]. EM gives you a theory of EM Fields and their interaction with matter. It shouldn't be surprising that EM doesn't give you a theory of matter.

I maintain there is no conceptual problem with EM, the problem is with your electron model which is unphysical. It might seem reasonable to you, but that's because of your intuition to build up matter from point particles, which is only justified by QFT considerations that came almost a century after EM.

[1] A simple matter model often used in GR is dust. I'm sure this would work for EM even better.

[jules]

I don't think making a relativistic theory of charged matter that approximates anything in the real world is as easy as you think. Charged dust will behave in very complicated ways, so I'd have to see a differential equation that models it.

I'm not saying that the point particle model is reasonable. I'm saying that it seems reasonable given what is said in a standard EM course.

Let me phrase it in a different way. In classical mechanics you have lots of problems of the form "the state of the system at time 0 is X, what is the state at time t?".

The problem with EM is that it doesn't have a relativistically invariant answer to such questions when point charges are involved. And, as far as I am aware, there also isn't a standard relativistically invariant answer involving a charge distribution, or at the very least it's not commonly taught.

Maybe you think that I shouldn't find this surprising, but given how EM is taught, I'd say that my surprise is fully justified.

[Certhas]

The teaching will vary greatly depending on teacher. But I remember that I was told that rigid bodies, and hence centre of mass thinking did not work in relativistic mechanics. It's very possible that as a student I never put that together with the inadmissibility of point charges in EM.

I must insist though that EM has no problem with initial value formulations. It simply doesn't provide you with a theory of matter. It turns out that that theory of matter really requires QM, hence in EM we never bother with non-QM models of relativistic matter. That's why you have to look in the GR literature.

As a pedagogical point I can agree that the limits of the conceptual foundations of our theories are never really explored enough. EM turns out to be fine, the field tensors are completely measurable, but that's a really cool paper that isn't taught either:

https://link.springer.com/chapter/10.1007/978-94-009-9349-5_...

As for charged dust, it works even if you switch on GR as well:

http://www.numdam.org/article/AIHPA_1973__18_2_137_0.pdf