Twin Paradox - Viewpoint of The Traveling Twin

Viewpoint of The Traveling Twin

During the turnaround, the traveling twin is in an accelerated reference frame. According to the equivalence principle, the traveling twin may analyze the turnaround phase as if the stay-at-home twin were freely falling in a gravitational field and as if the traveling twin were stationary. A 1918 paper by Einstein presents a conceptual sketch of the idea. From the viewpoint of the traveler, a calculation for each separate leg, ignoring the turnaround, leads to a result in which the Earth clocks age less than the traveler. For example, if the Earth clocks age 1 day less on each leg, the amount that the Earth clocks will lag behind amounts to 2 days. The physical description of what happens at turnaround has to produce a contrary effect of double that amount: 4 days' advancing of the Earth clocks. Then the traveler's clock will end up with a net 2-day delay on the Earth clocks, in agreement with calculations done in the frame of the stay-at-home twin.

The mechanism for the advancing of the stay-at-home twin's clock is gravitational time dilation. When an observer finds that inertially moving objects are being accelerated with respect to themselves, those objects are in a gravitational field insofar as relativity is concerned. For the traveling twin at turnaround, this gravitational field fills the universe. In a weak field approximation, clocks tick at a rate of t' = t (1 + Φ / c2) where Φ is the difference in gravitational potential. In this case, Φ = gh where g is the acceleration of the traveling observer during turnaround and h is the distance to the stay-at-home twin. The rocket is firing towards the stay-at-home twin, thereby placing that twin at a higher gravitational potential. Due to the large distance between the twins, the stay-at-home twin's clocks will appear to be sped up enough to account for the difference in proper times experienced by the twins. It is no accident that this speed-up is enough to account for the simultaneity shift described above. The general relativity solution for a static homogeneous gravitational field and the special relativity solution for finite acceleration produce identical results.

Other calculations have been done for the traveling twin (or for any observer who sometimes accelerates), which do not involve the equivalence principle, and which do not involve any gravitational fields. Such calculations are based only on the special theory, not the general theory, of relativity. One approach calculates surfaces of simultaneity by considering light pulses, in accordance with Hermann Bondi's idea of the k-calculus. A second approach calculates a straightforward but technically complicated integral to determine how the traveling twin measures the elapsed time on the stay-at-home clock. An outline of this second approach is given in a separate section below.