I understand that the twin paradox isn't truly a paradox, and that the traveling twin ages since the universe can be "aware" of his traveling. However, my question is how does the time dilation formula break down? Why can't you integrate dT' = dT * Gamma for each velocity as the traveler accelerates, and then actually reach the so called paradox? As both integrals for the traveler and earth stayer would be the same. Does the time dilation formula break down during acceleration?