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If the twins embark with equal and opposite velocities and return with equal and opposite velocities, which one is older when they reunite? (My contention is that they age by the same amount, but both will be younger than an observer who remains at their centre of mass. More generally, the reference frame with the maximal lapse of proper time is that of the centre of momentum.)

user140847
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1 Answers1

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There is no frame-independent center of mass.

In the diagram below, I've taken the earth to be massless. If you don't like that assumption, feel free to redraw the diagram giving the earth some positive mass; it won't change the conclusion.

enter image description here

The black frame is the frame of the earthbound twin, the dark blue line is the worldline of the outbound twin, and the gold line is, according to the earthbound twin, the worldline of the center of mass. (That is, along any fixed line $t=t_0$, it is equidistant from the worldlines of the two twins.)

Now let $t'$ be the time coordinate of the outbound twin. Along any line $t'=t'_0$ (such as the dashed line in the drawing), it's clear that the distances to the gold line and the black vertical axis are not equal. Therefore the outbound traveler will not agree that the gold line represents the center of mass.

WillO
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