Special relativity and Newton's 3rd law

When we think about Newton's 3rd law we assume that a change in action leads to a simultaneous change in the reaction. If there are two objects at a distance and this change happens at the same time at an inertial frame of reference , at one other this change does not happen at the same time. How can we explain that? Do we consider that Newton's third law still works but the change just does not happen at the same time? I do not know if this thought is right. I would appreciate if you could explain to me the relation between special relativity and the 3rd law.

If there are two objects at a distance and this changes happens at the same time at an inertial frame of reference at one other this change does not happen at the same time . How can we explain that ?

You have identified the key problem with action at a distance. This is explained simply by saying that forces acting at a distance is not permitted in the laws of physics. This is usually called “locality”.

The main reason that Newtonian gravity is incompatible with relativity is this notion of locality. Newtonian gravity is non-local so it needed to be replaced by a local theory of gravity, like general relativity.

As @Dale points out correctly, this is certainly a problem for forces like this whose effects are felt instantaneously. Events that are simultaneous for one observer are necessarily not simultaneous for any observer moving relative to them, and so if two events happen "instantaneously" in one frame, they won't happen instantaneously in any other frame!

This has an important consequence to the notion of rigid bodies in Special Relativity, for example. Our naive idea of a rigid body says that when we push one end of it the other end moves simultaneously, but as should be clear from the above argument, the object would not appear rigid in any other frame!

More importantly, the third law (as usually understood) is not true in general in Special Relativity. The best example (well, the only one I know) for this is given in my favourite chapter of the Feynman Lectures Chapter 26, where he shows that the forces on two charges moving perpendicularly to each other are not equal and opposite. (See Fig. 26-6 of the link posted.)

The reason for this is that in classical mechanics I believe the third law is just a restatement of the principle of conservation of momentum. It turns out (as often happens in physics) that while the third law does not hold here, momentum is still conserved: electromagnetic fields have energy and momentum, and the total momentum of the field and the particles is conserved, it's just that some of the momentum of the particles has been transferred to the field.

• If something's wrong with this answer, do let me know. I don't see anything incorrect in it, but I could be wrong. Commented Aug 29, 2020 at 15:57
• The answer is basically right, and good enough that I don't feel prompted to give another +1. In relativity it is much easier to use conservation of momentum, whereas care is needed to give a correct definition of force. We don't usually talk of force in relativity, because conservation of momentum covers both the second and third laws. Commented Aug 29, 2020 at 18:15
• @CharlesFrancis I agree, thanks for the comment. The answer initially had only a negative score, so I was concerned that there was something wrong in it that needed to be corrected. :) Indeed the four-force seems to be much a more more annoying four-vector to work with than the four-momentum, and with much less physical intuition associated with it. However I'm not sure what you mean when you say that the conservation of momentum "covers both the second and third law". Wouldn't one still need the second law to quantitatively associate a force with a phenomenon (say the Lorentz Force)? Commented Aug 29, 2020 at 18:24
• Yes, if one wants to associate a force, for example to show the Lorentz force, then one uses the second law. The second law can be seen as a definition of force, but as a definition it doesn't add anything fundamental to conservation of momentum which is the underlying physical principle. For example, conservation of momentum holds in qm, and can actually be proven in relativistic qm, but concepts like velocity and acceleration do not even make sense in qm. Commented Aug 29, 2020 at 18:47
• But we don't AFAIK have any examples of forces whose effects are felt instantaneously. Even without considering special relativity, when we push on one end of that rigid bar, the force isn't transmitted instantaneously to the other end. Commented Aug 30, 2020 at 3:10