bcrowell said:
The third law applies to mechanics, not electrodynamics. This is not an "unresolved question." In any case, discussion of this kind of thing is off topic for this thread. I just wanted to correct Gadhav's incorrect statement about normal forces, since they seemed to be contributing to his/her confusion about inertial frames.
The third law is very tricky in the relativistic context, and I think it shouldn't be used, because it's superfluous in relativistic physics (and also in Newtonian for that way as soon as one formulates everything in terms of Hamilton's principle and argues with space-time symmetries).
The reason is simple to understand: in Newtonian physics the third law says that if there's an interaction between two bodies, the force excerted by body 1 on body 2, ##\vec{F}_{21}##, is opposite to the force excerted by body two on body 1, ##\vec{F}_{12}##, i.e., at any instant of time
$$\vec{F}_{21}=-\vec{F}_{12}.$$
Now, if you try to naively use that principle in relativistic mechanics, you run into trouble with causality: E.g., suppose the bodies are charged and moving with respect to each other, this would mean you had an action at a distance, because the Force acting on body 1 due ot body 2 (the Coulomb interaction) would be immideatly changing also the force acting on body 2 due to body 1, when I use an external force to push body 1 apart from body 2 (let alone the point that such a system couldn't be stable at rest if the bodies are not hold by other forces due to the Coulomb interaction).
That's the reason, why traditionally one gives up the action-at-a-distance models in relativistic mechanics but invokes the field picture. To understand this idea better, we switch to arguments based on fundamental space-time symmetries. In SRT, I'm discussing here, that's Poincare symmetry (homogeneity of space and time, isotropy of space for any inertial observer, form-invariance of natural laws under Lorentz boosts). Now the 3rd law in non-relativistic physics can be substituted easily by assuming momentum conservation for interacting closed systems of bodies, and this conservation law is nothing else than the result of spatial homogeneity according to Noether's theorem on the relation between continuous (Lie) symmetries and conservation laws. Now that holds true also in special relativity.
However, it's also not so easy to accommodate for interacting particles that are some distance apart. Of course to argue like this we have to keep the system closed. Now assume we have the two charged particles leting them move freely. The mutial Coulomb interaction will lead to an accelerated motion of both particles. So particle 1 changes its speed and thus its momentum as does particle 2, and this must go instantaneously such that the total momentum is conserved. Again this would imply an instantaneous exchange of information, which is forbidden in relativity.
The traditional way out is to introduce the electromagnetic field as a dynamical quantity. So the interaction is mediated by this dynamical field, and body 1 "feels a force" due to the presence of body 2 because body 2 contributes to the total electromagnetic field at body 1's location and vice versa. So I don't need the assumption of "action at a distance" anymore and can still save total-momentum conservation, but now for a closed system consisting of the bodies and the electromagnetic field! Indeed, due to the acceleration of the particles they radiate off electromagnetic radiation fields, which are described by the Lienard-Wiechert retarded potentials. Thus at every time energy and momentum is conserved, because this radiation field carries momentum too and not only the moving bodies. Also there's no action at a distance anymore, but the force on body 1 is entirely due to the presence of the electromagnetic field at its location, i.e., it is a local interaction between the field and the body (and this local interaction is vice versa, i.e., the motion of the body also creates a field as the field makes the body moving accelerated!).
