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If I say an electron has "a speed of 0.999c", I think I mean that the
electron will move a distance of 299,792,458*0.999 meters (by my
meterstick) in one second (by my watch).
I don't see why I need to redefine v.
It seems I am still using v==dx/dt. Sure, someone in a
different frame will get a different answer, but that is true even in
Galilean transformations.
Certainly, F <> ma in the Newtonian sense.
I would tend to look to F = dp/dt.
Then I would say that applying a force (say from a uniform E) to an
electron that becomes highly relativistic means:
Force is constant
speed approaches c
acceleration approaches 0 (since the speed mostly stops changing)
a = F/gamma*(rest mass)
momentum continues to increase linearly: dp = qE dt
Is this not a reasonable way to view things?
I would keep the gamma explicit, rather than bury it
in either m or v or t.