Number Six said:
Suppose EarthTwin and SpaceTwin can observe each other at all times. And suppose each observes the other moving slowly, as if time is moving more slowly for the other. So each thinks time is moving normally for himself and time is moving slowly for the other.
First off, we need to be clear about terminology. In relativity, the term "observe" has a very specific meaning, but it's not completely intuitive. What you "observe" and what you "see" are NOT the same thing. In fact, understanding that distinction is one of the biggest stumbling blocks for beginners in relativity. Here's an example from every-day life to illuminate the difference:
Let's say you're standing outside, watching a jet fly overhead. You hear the roar of the engines, but it sounds like it's coming from behind the jet, because the jet is flying at a significant fraction of the speed of sound. You "hear" sound coming from behind the jet. But if you know how far away the jet is, how fast it's moving, and how fast sound moves, you can deduce that the sound is not actually coming from behind the jet, but from the jet itself. Thus, you "observe" that the jet is emiting the sound, but you "hear" the sound coming from a point behind the jet. Similarly in relativity, what you "see" and what you "observe" are not the same thing.
So, with that out of the way, let's get to your question:
And yet when SpaceTwin lands back on earth, EarthTwin has aged 20 years compared to just 10 for SpaceTwin. If SpaceTwin can observe EarthTwin constantly as SpaceTwin flies through space then it doesn't make sense that he observes time passing slowly for EarthTwin the entire time and then when he lands back on earth suddenly 10 more years have passed for EarthTwin than for SpaceTwin. I must be missing something.
Yes, you are missing something. While traveling, both twins "observe" the other twin's clock slowed down, regardless of direction. But what they "see" does depend on which direction they're going. If the other twin is going away from you, you see that twin's clock slowed down even more from doppler shift, but if it's heading towards you, you "see" it moving faster.
During the acceleration, the traveling twin "sees" the earthbound twin's clock speed up continuously. Nothing particularly weird here. But something very weird does happen to what the traveling twin "observes". When you shift to a reference frame that's moving with respect to your original reference frame, your time axis (the line that marks out a single location at all times) gets tilted. This is easy to understand, and that's what you get in classical physics too. The tricky part is, your space axis (which marks out all points in space at the same moment of time) also tilts. So during the acceleration, the traveling twin's space axis is tilting, so where it intersects the time-line trajectory of the earth-bound twin changes a LOT during the acceleration process. What the traveling twin "observes", then, is a very large passage of time for the earth-bound twin during that acceleration period. Note again, though, that this isn't what the traveling twin "sees", which isn't really very remarkable at all. This huge observed passage of time is a result of the fact that during the acceleration, the traveling twin's reference frame is NOT an inertial reference frame - while physics makes no distinction between different inertial reference frames, there IS a difference between inertial reference frames and non-inertial reference frames.
Now if we go back and think about what each twin sees, not what they observe, we can still figure out that the traveling twin will have experienced less time. For the traveling twin, the outbound and return journeys take the same amount of time, so for half that time, the earthbound twin's clock looks like it's running slow, and half that time it looks like it's running fast. But the earthbound twin doesn't see the outbound twin arrive at the turnaround point until sometime after he actually got there (since the light signal takes time to return), so the earthbound twin "sees" the traveling twin's clock running slow for MORE than half the trip, and running fast for LESS than half the trip. The asymmetry is again due to the fact that it's the traveling twin that accelerates.
