THE TWIN PARADOX
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There's something about Lorentz Rule 2 that should bother you. The Principle of Relativity tells us that there is no experiment that two observers with some relative motion between them can perform that will establish one of them absolutely moving and the other absolutely at rest. What that means, necessarily, is that while we see the pulses in a laser clock mounted on a train tracing a sawtooth pattern in the air, the passengers on the train must see our clock tracing the same patterns. That, in turn, means that we see the train's clocks running slower than our clock does and the passengers on the train see our clock running slower than their clocks do. That statement looks very much like a logical contradiction and it is the basis on which Paul Ehrenfest conceived the best-known paradox in Relativity.
According to Richard Feynman (and to get the full effect of one of the Twentieth Century's greatest physicists you must imagine the following being spoken with a noticeable Brooklyn accent: Professor Feynman was born and raised in Far Rockaway), "A paradox is a situation which gives one answer when analyzed one way, and a different answer when analyzed another way, so that we are left in somewhat of a quandary as to actually what would happen. Of course, in physics there are never any real paradoxes because there is one correct answer." He then adds, "A paradox is only a confusion in our understanding."
So what confusion do we have in our understanding of time dilation? Our derivation of the description of the effect from the sawtooth pattern that a laser clock traces seems clear enough to be free of any logical flaws. And the Principle of Relativity is perfectly clear in its application here. So what have we missed? To answer that question we need to perform some more imaginary experiments.
From Fresno to Modesto the Southern Pacific tracks run more or less straight for about 87 miles. That's a good distance because a train that can go from Fresno to Modesto in one hour in our fantasy world will do so by occupying an inertial frame that has, relative to the frame occupied by the Fresno and Modesto train stations, a Lorentz factor of 2. We can thus figure easily that for the passengers on that train one half hour will elapse between departure from Fresno and arrival in Modesto. But that calculation should bring to mind another thought to distract you momentarily from contemplation of temporal paradoxes.
The Principle of Relativity tells us that if I see you move past me at 87 miles per hour, then you must see me pass you at 87 miles per hour; that is, the ratio of distance crossed to time elapsed must be the same for both of us. Thus I calculate the ratio of 87 miles crossed by the train to one hour elapsed on the station clocks as 87 miles per hour. If you're riding the train, you may make a slightly different calculation: you know that the landscape is moving past you at 87 miles per hour and you measure one half hour on your watch from departure to arrival, so you calculate that the distance from Fresno to Modesto in your frame must have been 43-1/2 miles. What you have detected through your calculation is the Lorentz-Fitzgerald contraction, which we can describe in
LORENTZ RULE 2B: An object moving relative to some observer appears to that observer to be shorter in the direction of relative motion than it does to an observer at rest in its inertial frame, the moving length being equal to the stationary length divided by the Lorentz factor between the two inertial frames.
If you look out the window of your carriage when the train has gained its full speed, you will see a landscape remarkably deformed by the world's adherence to that rule. Objects appear to have their normal heights but are shortened in the direction parallel to that in which they appear to be moving. Telephone poles, trees, and houses all appear to be narrower, thinner than they ought to be. And cars and trucks on the highway appear to be cartoonishly foreshortened. Of course, as you have come to expect, to the people in those cars and trucks it's your train that appears foreshortened. This is all quite amusing, but it isn't helping us resolve the problem with time dilation (though the picture that you have drawn in your mind contains a clue to that resolution), so let's go back to our contemplation of our clocks and look at the twin paradox.
Twins John and Jane have come to the train station in Fresno. John has an errand he needs to run in Modesto, so while he's gone Jane will wait for him in the station's coffee shop. She figures that she will have to wait a little over two hours, but she also figures that for John a little over one hour will elapse; one half hour each way on the train plus the few minutes he needs to run his errand once he gets to Modesto. John also knows about time dilation; thus, when he gets to Modesto and sees that half an hour has elapsed on his watch, he figures that one quarter hour has elapsed for Jane and expects that she will have to wait in the coffee shop a little over half an hour.
Do you see what Feynman meant by a paradox? Analyzing the situation through Jane's perspective, we figure that Jane will wait two hours. Analyzing the situation through John's perspective, we figure that Jane will wait one half hour. We are stuck in a quandary. Which is the one correct answer that we are told that physics has for us?
Two hours would be the unanimous choice of all physicists to whom that question might be put. When asked why they chose that answer, they would all give a reply that involves pointing out that Jane occupied only one inertial frame while John occupied three (the frame marked by the Fresno and Modesto train stations, the one moving north at 87 miles per hour, and the one moving south at 87 miles per hour). But, we might object, doesn't Jane occupy three inertial frames as seen through John's perspective? Isn't that what Relativity tells us? The answer to both of those questions is no: there is no doubt that John has occupied three inertial frames and Jane has occupied only one, regardless of whose viewpoint we use, because there is no doubt that John has undergone accelerations and Jane hasn't. True, through John's perspective Jane appears to have accelerated, but that's only an appearance because acceleration is an absolute state of motion, which means that there are experiments that Jane and John can perform that will determine which of them is actually accelerating and which of them is not.
And how does the
fact of acceleration resolve the twin paradox? It brings into play a distortion
of time even stranger than time dilation. That distortion is often neglected in
presentations of Relativity, even though Einstein regarded it as one of the most
fundamental features of his theory, the one that answers the question, "What
time is it?" Einstein called it the relativity of simultaneity, noting that
events that are simultaneous for one observer won't necessarily be simultaneous
for other observers.
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