Hi. If the reason for the time dilation measured by an observer (twin A) at rest on Earth of an observer (twin B) on a spaceship travelling near to the speed of light is that twin B's frame of reference isn't always inertial (there is a large amount of acceleration at points of the journey), then what calculation should be made to measure the age difference for the two twins when twin B comes back to Earth if the dilation is not a result of the average speed of the space-ship, but its periods of acceleration?
Definition of twin paradox: if two events occur at the same place in an inertial frame S then an observer in S measures the 'proper time' interval between them, t0, and an observer in an inertial frame S' moving relative to S seeing the events at different locations measures the time interval t = γt' (where γ is the Lorentz factor). This may be switched round if the events occur at the same place in S' and at different locations in S, i.e. an observer in S' measures t0. However, in the case of twins aging both see a start age and an end age (start and end events) at different locations relative to each other and in their own frames aging occurs at the same location. Thus this raises the question of why the twin moving relative to the other at a speed approaching that of light has aged less than the one who's effectively stationary when (s)he returns to Earth, and the textbook reason given is that the moving frame is not always inertial, as I said above.