Spoonfedrelativity.com was intended to take what I had managed to glean from reading about Relativity, and make it clearer. Unfortunately I ran into a problem. There seems to be major confusion and controversy on the subject even among the experts, and I found myself in disagreement with the consensus. I've been forced into a Quixotic role. Watch my videos and join me in my tenacious pursuit attacking windmills.
Jonathan Doolin, Generator of Hypotheses (and College Physics and Astronomy Professor)
What always really bugs me about these explanations of the twin paradox is that they usually seem to say "there's something different about the accelerating twin." But they are always very vague about what that difference is.
But the differences need not be vague. Everything can easily be derived from the Lorentz Transformation equations, and there is no mystery to it.
If you have an object receding from you at near the speed of light, it will appear to be moving, at a maximum, of 50% of the speed of light.
Why? (Answer this question for yourself, and if you agree, Upvote, if you don't agree, downvote, and post a comment.)
On the other hand, if you have an object approaching you near the speed of light, there is NO LIMIT to it's maximum apparent speed.
Why? (Answer this question for yourself. If you agree, upvote. If you don't agree, downvote, and post a comment.)
Now, when you look at a fast approaching, or fast receding object are you looking at where the object is now? No. You're looking at where the object WAS when it emitted or reflected the light. That emission or reflection of light is an "event" which happened at a place and time in your perceptions. It has physical coordinates of (t,x,y,z) Space and time.
What happens when you accelerate toward a past event in Special Relativity? It moves away from the observer, and back in time. Again, do the math yourself. If you agree, Upvote. If you don't agree, Downvote and post a comment.)
But yes, if you accelerate toward an event in the past, Lorentz Transformation equations say it moves away and back in time. That's good, because it makes everything consistent with what I said earlier:
As the moving twin is moving away from the sun, he's going to see the sun moving away at less that half the speed of light. When he turns around, he's going to see the image of the sun jump away from him--lurching away from him spatially. And it will also (from his perspective) lurch backward in time... So the emission/reflection event happened much further away and longer ago. So at the "instant of acceleration" is when the earth has suddenly aged in his point-of-view.
(Video explanation added, November 7)
The video explains what is meant by "if you have an object receding from you at near the speed of light, it will appear to be moving at a maximum of 50% of the speed of light."
It also explains why the distance traveled by the earth in the second leg of the inbound frame is much greater than the distance traveled by the earth in the first leg of the outbound frame
Video added November 27, 2015:
In the comments below, I did work through some of the math that I recommended above... I decided to re-post it within the answer, because it's hard to find, looking through the comments.
Alright... I have a lot more Lorentz Transforms for you to do!
I'm trying to give relevance, here, to is the events on the observer's past light cone, and deemphasize the simultaneous events... (See Part 5 below for more discussion of the simultaneous events)
Setup: Let's imagine a scenario where a deep space-station lies exactly one light-year from earth. A twin goes to visit the space-station. He travels out at 0.866c, stays there for an hour, and then travels back at 0.866 c. These coordinates are in years, and light years, so the hour long visit is just going to be a rounding error somewhere past the 3rd significant digit on this scale. (And we'll pretend that the acceleration can be done quickly without killing the pilot, even though accelerating to .866c in less than a day would be quite a few g's, I'm sure.)
Part I: Calculate the spacetime coordinates of these events in the earth/space-station frame. (1) the take-off (2) The space-station visit (3) The return home.
Part 3. There are two other events that I think are quite relevant. Let event X be the event that happens on Earth that can be seen on camera at the space-station during the visit. Let event Y be the event where observers on Earth actually see the traveling twin land at the space-station.
Answers: X: (t,x)=(-1,-1) Y: (t,x)=(1,-1)
Part 4: The really interesting event to look at here, from the traveling twin's perspective is Event X. Because he can be watching that event (1) immediately before the visit. (2) During the visit. And (3) immediately after the visit to the the space station. Finally, here, we're doing a Lorentz Transformation on the event at (-1,-1)... Perform a Lorentz Boost of v=+0.866c on event X to find it's location immediately before the visit, and a Lorentz boost of v=-0.866c on event X to find it's location immediately after the visit.
Answers (1) Before visit: (t,x) = (-0.268 light-years, -0.268 years) (2) During visit: (t,x) = (-1 light-years,-1 years) (3) After visit: (t,x) = (-3.73 light-years, -3.73 years)
Finally, let's try to put that into words... What's happened here? You're looking at the coordinates of the same event before, during, and after the visit. What has happened is that the earth's image jumps back, from 0.268 light-years away, to 3.73 light-years away. In fact everything in that direction suddenly becomes 13.9 times as far away at the beginning of the return journey than it was at the end of the outbound journey. This works out to be a factor of 1+v/c1−v/c.
Now, how do you explain this phenomena? Do you explain it as "real" or do you explain it as an "optical illusion?" Yes it is optical. But is it illusion? Is the event really one-light-year away, but it only "LOOKS LIKE" it is .268 light years away, and then 3.73 light years away? To claim that one point-of-view is more valid than the others is to go against the Principle of Relativity.
To the contrary, they are all valid coordinates: The earth is 0.268 light years away for the outbound observer, 1 light year away for the space-station observer, and 3.73 light years away for the returning observer... When you add together these distances 0.268+3.73 you find that the traveling twin literally sees the earth travel about twice as far (more generally, the traveling observes the earth travel γ−βγtimes as far on the way out, and sees the earth travel γ+βgamma times as far on the way in.
Part 5: Now, is it less "confusing" perhaps to just refer to the simultaneous distance?
For this purpose, let's find the the event which is regarded as simultaneous with the visit during (1) the outbound trip (2) during the visit and (3) on the inbound trip.
(1) A. (t,x)= (0,-0.5) according to the outbound twin (2) B. (t,x) =(0,-1) according to the space-station twin (3) C. (t,x) = (0,-0.5) according to the inbound twin
Now, where do those events occur in the space-station frame?
Answers: (1) A. (t,x) = (-.866,-1) (2) B. (t,x) = (0,-1) (3) C. (t,x) = (0.866,-1)
Event A is simultaneous with E on the outbound trip. Event B is simultaneous with E according to the twin on the space-station. Event C is simultaneous with E on the return trip.
This seems to wrap up the question neatly and tie it up with a bow: "The total distance of the trip... according to these simultaneous distances is LESS than the "real distance" by a factor of gamma." So it makes "perfect sense" that the time experienced by the traveling twin should also be less by a factor of gamma.
The problems with that explanations of the Twin Paradox that chalk-it-up to length contraction are (1) It gives the impression that the traveling twin actually "experiences" this length contracted distance through which he is traveling. (2) it does not not really explain the 1.73 year discontinuity between events A and C, (3) It sometimes gives the impression that that the discontinuity cannot be explained and remains a paradox (especially when the traveling twin is put in an enclosed capsule so they cannot look outside.) (4) They gives the impression that simultaneity is an observable. That somehow events A, B and/or C might be marked in some phenomenological way by the traveling twin that he might say "Oh... That was what was happening on earth when I turned around."