As we know, the twin paradox states that an astronaut that accelerates away from the Earth, stops, turns around, and then accelerates back to the Earth ages less than his twin that remains behind (on Earth). How about in the case where the acceleration isn't linear? Let's say our astronaut accelerates away from the Earth and immediately begins a long arc in a path that returns him to the earth in a circular trajectory. Does he age the same as if he went linearly to the farthest point of the arc? How do you reconcile the fact that he is both, always accelerating away from the Earth and accelerating toward it at the same time? (he never has to stop and turn around) P.S. I know that on the face of it this seems similar to the closed universe version, but consider that space-time is flat in this universe. Are the effects the same?