Is Newton's 3rd law valid in non-inertial frames?
If so, then are there other cases for which Newton's 3rd law is not applicable?
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fictitious forces when you define a non-inertial frame. For more information, see these Newtonian Dynamics notes by Richard Fitzpatrick. There are two core parts of the answer:
One corollary of Newton's third law is that an object cannot exert a force on itself. Another corollary is that all forces in the Universe have corresponding reactions. The only exceptions to this rule are the fictitious forces which arise in non-inertial reference frames (e.g., the centrifugal and Coriolis forces which appear in rotating reference frames). Fictitious forces do not possess reactions.
It should be noted that Newton's third law implies action at a distance. In other words, if the force that object exerts on object suddenly changes then Newton's third law demands that there must be an immediate change in the force that object exerts on object . Moreover, this must be true irrespective of the distance between the two objects. However, we now know that Einstein's theory of relativity forbids information from traveling through the Universe faster than the velocity of light in vacuum. Hence, action at a distance is also forbidden.
In other words, if the force that object exerts on object suddenly changes then there must be a time delay, which is at least as long as it takes a light ray to propagate between the two objects, before the force that object exerts on object can respond. Of course, this means that Newton's third law is not, strictly speaking, correct. However, as long as we restrict our investigations to the motions of dynamical systems on time-scales that are long compared to the time required for light-rays to traverse these systems, Newton's third law can be regarded as being approximately correct.
The textbook case is the electromagnetic force between two relativistic charged particles. Because the electromagnetic field travels at finite speed, the interaction is delayed and usually the force two particles experience at the same time does not add up to zero. From another perspective, electromagnetic radiation carries away momentum, so the total momentum of just the charged particles is not conserved.
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