The twin paradox: Is the symmetry of time dilation paradoxical?

The twin paradox uses the symmetry of time dilation to produce a situation that seems paradoxical. In the introductory film clip, we saw that time was dilated when observed from frames of reference with a constant relative velocity v. There is an animation and analysis below, but let's introduce it with a cartoon.

The really strange thing about time dilation is that it is symmetrical: if you and I have relative motion, then I see your clock to be running slow (with respect to my frame), and you see mine to be running slow. (Revise time dilation.) This is just one example of the weird logic of Einstein's theory of Special Relativity. The theory is counter-intuitive, because most of us are unfamiliar with measurements made at speeds approaching c, the speed of light. Because of this, it is fun to attempt to prove that it is wrong. Surely it's possible to make a paradox out of the symmetry of time dilation? Let's see. Jane and Joe are twins. Jane travels in a straight line at a relativistic speed v to some distant location. She then decelerates and returns. Her twin brother Joe stays at home on Earth. The situation is shown in the diagram, which is not to scale. Joe observes that Jane's on-board clocks (including her biological one), which run at Jane's proper time, run slowly on both outbound and return leg. He therefore concludes that she will be younger than he will be when she returns. On the outward leg, Jane observes Joe's clock to run slowly, and she observes that it ticks slowly on the return run. So will Jane conclude that Joe will have aged less? And if she does, who is correct? According to the proponents of the paradox, there is a symmetry between the two observers, so, just plugging in the equations of relativity, each will predict that the other is younger. This cannot be simultaneously true for both so, if the argument is correct, relativity is wrong.

Analysis

Now let's draw graphs of position vs time for the voyages of the two twins, and also for their anniversary messages. These are called space time diagrams. Note that there are three space time diagrams, one for each of the three different inertial reference frames that are involved in this problem. At left is Joe's diagram. At right are Jane's two diagrams, with the diagram for the return journey drawn above that for the outward journey. These diagrams are drawn to scale for v = 0.66 c. The distance is 2.67 light years in each direction, as measured in Joe's frame. Thus Joe concludes the voyage will take (2.67 light years)/0.66 c = 4 years in each direction. In each of Jane's frames, the journey length is shorter by the factor 1/γ = (1 − v²/c²)^1/2 = 0.75, so for her the distance is only 0.75*2.67 = two light years, so at 0.66 c it takes her only 3 years in each direction. According to Joe, he will have aged 8 years and Jane will have aged 6 years when they are reunited. Jane's version of events will depend upon how careful she is about applying Special Relativity.

(The original animation died with Flash. A new animated version will come soon.)

The naive interpretation—the reason why the situation is called a paradox—is to assume that the situation is competely symmetrical. If that were the case, Jane's diagram would simply be a mirror image of Joe's. But the results that we derived in Special Relativity were restricted to the relations between inertial frames of reference. In this regard, the situations of the twins are definitely not symmetrical. Joe is in one inertial frame throughout, but not Jane. (We discuss the partial symmetry below.)

Jane certainly knows that she has not been in the same inertial frame for the whole trip: in order to stop the outward journey and to commence the return, she had to turn the ship around and fire the engines hard and long. During that time she knew that she was not in an inertial frame (the importance of this is discussed below) because suddenly the objects, including her, were all squashed towards the end of the spaceship with the engines. (She had to give up lots of the best yoga positions and hyperdance routines, too.) Then, when the ship was travelling at v towards Earth, she cut the engines, she was back in another, different, inertial frame. In the second of Jane's inertial frames (the homeward trip), she receives a lot of anniversary messages from Joe. If she pretends that she has been in this same frame of reference all along (the dashed line extrapolation of her returning world line), i.e. if she assumes that she has been travelling towards Earth at constant v for six of her years, she would conclude that Joe had been sending them for eight of his years (follow the dashed lines). 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 naïve, 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. 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.) It has shrunk by the factor 1/γ = (1 − v2/c2)1/2 = 0.75, so she now only has to travel for three of her years to get there. 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.

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. 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 (and an acceleration in between). 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? Let's picture what happens. While the engines were on at mid voyage, objects in the spacecraft are no longer in free fall (they are no longer 'weightless'): the objects in Jane's ship collect on the 'floor' (this is the name we might give to the wall in a space ship in the direction of the engine exhaust). During this phase, and with reference to the frame of the ship, any free objects seem to accelerate towards the 'floor'. No force is causing this 'acceleration', so this is not an inertial frame. (For the importance of inertial frames, see this link.) Now if Jane treats this as an acceleration, she will deduce from it that she will no longer be flying away from Joe's messages, but flying towards them, so she will, as we saw above, expect them to arrive at higher frequency, starting immediately. Applying Special Relativity, she will conclude that she will arrive having aged less than Joe.

But what if there are no windows on Jane's ship? Is there an alternative, local explanation for the asymmetry in the clocks and messages? There is, and it involves Einstein's General Theory of Relativity.

If Jane cannot look out of the ship, her sensations and measurements during the deceleration will be just the same is if her ship were at rest on the surface of a planet and that gravity made things fall towards the floor. The local equivalence of a gravitational field and an accelerating frame is a starting point for Einstein's General Theory of Relativity.

One of the consequences of the general theory is that clocks at high gravitational potential run more quickly than those at low potential. (So, for example, very accurate laboratory clocks on Earth run are experimentally observed to run faster when their altitude is increased.) In terms of Jane's local frame during the turn around, Joe is a long way overhead and so, according to her, his clocks run fast during that time, and he ages quickly. Further, Joe's 'height' above her depends on how far she has travelled, so his clocks run more quickly during the turn around in a long voyage. This is quite important, because proponents of the twin paradox sometimes argue that, whatever the effect of the turn around, it can be made negligible by making the journey far enough. Not so. The longer the journey, the greater the effect due to GR. (Similarly, in terms of the SR argument above, the longer the journey, the longer it takes for Jane's change of frames to be observed by Joe, and so the bigger effect.)

Thus, if Jane applies General Relativity as well as Special Relativity, she concludes that Joe will be older and thus resolves the paradox. It is important to point out, however, that appealing to General Relativity is not necessary to resolve the paradox, as demonstrated above. In order to create the twin paradox, one must assume that Jane has been in a single inertial frame throughout her out-and-back trip. As this assumption is false, there is no paradox.