Paradoxes in the theory of relativity,
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Now this is a strange assumption, because she would also conclude that she and Joe had never been together - that Jane and Joe never occupied the same position before! Making this strange assumption, she would calculate that, when she started the trip the bottom line on her diagram Joe was four light years away from her, had already been travelling towards her for two years, and had already sent a couple of anniversary messages!
See the dashed lines in the diagram. Another very asymmetrical observation is that Jane notices a sudden change in the rate of arrival of the messages, just after she turns.
We discuss this further below. Let's now assume that Jane is not naive, that she knows about relativity, that she remembers the acceleration, that she remembers being with Joe at the beginning of the trip and that she uses this knowledge in analysing her version of the space time diagrams.
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First, once she has left the Earth, accelerated and is travelling without acceleration towards her destination, she can apply Special Relativity. She observes that the distance between the Earth and her destination has shrunk. See Relativistic time dilation, simultaneity and length contraction for an explanation. Similarly, in her return trip another inertial frame so she can use Special Relativity again the distance is also shorter, so she only has to travel for three years to get there.
So Jane's space time diagrams are those shown at right. We repeat the diagram. Are the space-time diagrams symmetrical? Parts of them are. The first three years of the diagrams for Joe's frame and Jane's departing frame are symmetrical: each twin sends three greetings but only receives one. The last year and a half of Joe's frame and Jane's returning frame are also symmetrical: each sends two greetings and receives four.
But the diagrams are not symmetrical in between.
Why not? Look at Jane's diagram.
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From Jane's point of view, immediately after she has fired her engines, she begins receiving Joe's greetings more frequently. This does not surprise her: she has gone from travelling away from the sender of the greetings and is now travelling towards him.
Jane observes this change as soon as she turns around, which is for her the midpoint of her voyage. She now receives blue shifted messages instead of red shifted ones. One could apply the same relativistic Doppler factor to the frequency of arrival of the messages. Joe, on the other hand, doesn't start to receive messages at a higher frequency blue shifted messages until considerably after the midpoint between Jane's departure and arrival, simply because the effect of Jane's acceleration and changed reference frame takes a while to get to him : he doesn't see the high frequency arrival of messages until the arrival of the first message that Jane sends after she turns around.
This is a clear example of where the asymmetry of the twins appears. The causes of this asymmetry are the fact that Jane reverses direction and Joe does not, and the finite time that light takes to transmit this information to Joe means that Joe doesn't get the news immediately. Jane leaves one inertial frame and joins another, and she has the effect of that change immediately.
Joe, on the other hand, doesn't notice the effects of Jane being in a different inertial frame until much later because she is a long way away from him when it happens. The asymmetry is as simple as that. In these diagrams, we have resolved the paradox by pointing out that the problem is not symmetrical: Jane actually has two different inertial frames of reference, the outgoing voyage and the return.go to link
Some Paradoxes of Relativity
Two different clock synchronisation events are required, and the easist examples of these are at their separation for the outward journey and their reunion for the return. To understand the importance of synchronising clocks in Special Relativity, see Relativistic time dilation. Why is the accleration in mid voyage so important?
As we saw above, it marks the point at which Jane goes from one inertial frame to another. Does this have a direct, physical effect on her?
The first point to understand is that acceleration in SR is a little tricky it's actually handled better in Einstein's Theory of General Relativity - GR. I don't mean to say that SR can't handle acceleration, because it can.
In SR, you can describe the acceleration in terms of locally "co-moving" inertial frames. This allows SR to view all motion to be uniform, meaning constant velocity non-accelerating. The second point is that SR is a "special" theory. By this, I mean that it is applicable in situations where there is no gravity, hence where space-time is flat. In GR, Einstein unifies acceleration and gravity so actually my previous statement is redundant.
Anyway, the lack of gravity in SR is why it is called "Special Relativity". Now, back to the paradox… While both did view the other as shrinking and slowing down, the person that actually underwent the acceleration to reach the high speed is the one that aged less. If you dig deeper into the world of SR, you will realize that it's not really the acceleration that is important; it's the change of frame. Until John and Hunter returned to a frame of reference where their relative motion was zero where they are standing beside each other they would always disagree with what the other said he saw.
As strange as this seems, there really isn't a conflict - both did observe that the other was experiencing the relativistic effects. One technique that is used to show the dynamics of the Twin Paradox is a concept is called the Relativistic Doppler Effect. What's the Fascination With Number 23? Switching Shampoos: Necessary or Nah? Myth Everyday Myths. How Special Relativity Works.