The unavoidable spatial imperfections of figure 3 aside, what is important for understanding the relationship between the twins' clocks is that their relative time-frames are accurately represented, consistent with the temporal expression of the Lorentz transformations. The twins' clocks are not simultaneous except at the beginning and end of their relative motion, but they are correlative at every moment in between, in alignment along Twin A's space axis (x), as for example when the distance from A's x-axis to Twin B's position in spacetime is at the 3 yr mark, A's clock is likewise 3 yr away from its original point in time on its space-axis. Therefore, regardless of the periods and velocities in uniform relative motion during the twins' separation, their clocks for such periods will necessarily remain correlative, and will be synchronous at their reunion, when they again share the original space axis.
Given the inherent spatial distortion in figure 3, it may be helpful to consider a rotate-able, although physically impossible perspective on the twins' relative motion, one that can transcend the limitation of a realistic single reference frame as with figure 3.
Figure 4 takes an unnatural but more comprehensible perspective as of a demiurge, or if it is preferred, a God's-Eye view of the spatial relationship between the twins, illustrating that except when Twin B is accelerating, one twin isn't at rest while the other is in motion; both are at once relatively at-rest and relatively in-motion.
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figure 4
A composite and transcendent perspective on the twins' adventureprovides a comprehensible but unnatural representation of the
un-accelerated portions of their mutual separation and
reconvergence. The lengths of their world-lines are equal, as are
the durations of their clocks.
The illustration of uniform motion in figure 4 balances the to-and-from segments for the sake of clarity, but the periods of uniform motion in each direction needn't be equal for the final reckoning of clocks to be in agreement. Given the correlative relationship between clocks discussed above, the convergent vectors can have a different length than the divergent vectors. And there needn't be any uniform motion at all in one direction for concurrency to be maintained; such a situation can be envisioned as involving only one period of uniform motion, using one pair of the vectors in figure 4 -- vectors A1 and B1 or vectors A2 and B2. In each case, no matter how long or how relatively fast their uniform motion in either direction, there is always a correlation between clocks.
Conclusion
The
"Twin Effect" has been shown to be entirely explicable in terms of Relativity Theory,
and supported by the principle of correlative clocks entailed by relative
motion. The confirmed effects of acceleration are neither complicated nor in
any way affected by intermittent periods of uniform motion in space due to the
correlation of uniform motion in time.
References
Arnold J; "Time as the Dynamic Aspect of the Continuum", ESJ, Vol 11, No 9, 2015.
Botermann B, et al; "Test of Time Dilation Using Stored Li+ Ions as Clocks at Relativistic Speed", Phys. Rev. Lett. 113, 120405, 2014
Einstein A; Zur Elektrodynamik bewegter KÃ ¶rper ("On the electrodynamics of moving bodies"), Annalen der Physik 17, 1, 1905, pp. 904-905
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