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How Special Relativity Works

Twin Paradox

Now let's look at the twins again, but this time Hunter will send a signal every hour by his clock. What will John see? When Hunter sees the outgoing leg of John's trip end, his clock reads 15 hours and he has sent 15 signals. John, however, will say that he received 6 signals separated by 2-hours (relativistic Doppler shift) for a total of 12 hours. What happened to the other 9 signals? They are still in transit to John. Therefore, when John changes to his return leg, he will now encounter the missing 9 signals plus the 15 signals Hunter sent for the 15 hours his clock recorded for the return leg. So John receives 24 signals that are 30 minutes apart for a total of 12 hours. Like the previous example, these 24 signals have all been doppler shifted to a higher frequency because John is now approaching them. Now if we total the whole trip, Hunter sent one signal every hour for thirty hours, but John received 6 signals that were 2 hours apart and 24 signals that were 30 minutes apart. Hunter sent 30 signals in 30 hours; John received 30 signals in 24 hours. The result is the same as before, but the twins do not agree on when the first leg ended and the last leg began. So from this we can conclude that the change of frame for John (from outgoing to return) is what distinguishes him from Hunter. For Hunter, nothing changes at all. Anyway you look at it; he waits 30 hours without a change. John, however, does change. He changes from a frame in which he is moving away to a frame in which he is moving back. It is this change that breaks the symmetry between John and Hunter, thus removing the paradox as well.

Before going on to the next concept, I want to make sure that a couple things about SR and the speed of light are properly understood. First, SR predicts doom for anything with mass approaching the speed of light from a slower speed due to length contraction and time dilation, but it does allow for speeds greater than the speed of light. Consider the speed of light as a barrier. SR allows for existence on both sides of the barrier, but neither side can cross over to the other. As of yet, nothing has been discovered on the faster-than-light side, and all that we have are theories on particles (tachyons) that may have the ability to exist there. Maybe one day someone will discover their existence.

Secondly, velocities from a different frame of reference can not be summed. For example, if I run 5 miles/hour and at the same time, throw a rock 5 miles/hour, the only reason you (standing still) can say the rock is travelling 10 miles/hour is because the speed is so small with respect to the speed of light. We use the Lorentz Transformations to transform from one frame to another using the relative velocity of the frames. These transformations tell us mathematically that while at slow speeds the error in straight addition is much too small for us to detect, at very fast speeds, the error would become quite large. So classical mechanics, which teaches us to sum these velocities, is actually incorrect. We can do it, but it's a case of getting the right answer for the wrong reason.

We'll look at the twin paradox and simultaneity in the next section.