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One more try :)
If the two electrons are initially travelling at constant speeds inopposite
directions (which they are in the CM frame) and the net forces on themare
zero at all times, then the center of mass is inertial. Why do youdisagree
1.) You agree that the lab frame is an inertial frame.
2.) You agree that the CM of the electron system is accelerating, as
viewed from this inertial (lab) frame
(and therefore from all other inertial frames).
3.) Therefore, an observer travelling with this CM is viewing the world
from a non-inertial frame.
He will see objects accelerate with no forces
acting on them;
he will of course see objects at rest relative to him NOT
accelerate even though there is a net force on them
(as there must be on him).
He cannot apply Newton's laws to his observations without adding
corrections which take into account his own acceleration.
4.) I think that by your phrase "then the center of mass is inertial"
you are asserting something other than the (false)statement that an
observer travelling with the CM is at rest in an inertial frame. What is
it that you really mean?