Numinations — March, 1998

The Twin Paradox

© 1998, by Gary D. Campbell

The Twin Paradox takes its name from two hypothetical twins that live in our distant future. These twins, born on April Fools Day, have just reached their 21st birthday. One of the twins has graduated from the space academy and is now being sent out into the final frontier. The other will remain on earth.

Twenty years go by (on earth, at least), and the space traveling twin has just returned. During the time she was away, this twin spent much of her time traveling at extremely high rates of speed. As they get back together, the twins realize they are no longer the same age! The earthbound twin is about to celebrate his 41st birthday, but the space traveling twin is about to celebrate her 39th birthday! Is this an example of a woman trying to hide her true age, or is this a genuine paradox?

It’s been called the Twin Paradox because speed is relative. From earth, the spaceship appears to move away at a high rate of speed, but from the spaceship’s point of view the earth appears to move away. From an observer’s point of view, an object moving at a speed close to that of light undergoes a “time slowdown.” If the observer’s point of view were symmetrical with the object’s point of view, the total time elapsed for each would be the same when they came back together, but it isn’t. Experiments have been done along these lines, and the object that has traveled the greater distance is always the one that ages the least.

One twin could actually become younger than the other. But, for this to happen, speed can’t be relative after all. Let’s see if we can find the asymmetry and “solve” this paradox. Some have claimed that the traveler must undergo huge accelerations that the “stay-at-home” does not experience. True, there are two factors that can throw relative aging out of synch. One is a difference in acceleration, explained by general relativity. The other has to do with a distance-time tradeoff, explained by special relativity. We could keep the effects of general relativity to a minimum by accelerating at exactly 1 G in our spaceships. There would be no difference between this and staying behind on earth at 1 G. Special relativity explains the paradox of the twins, but it is not an easy thing to see. Let’s give it a try, anyway.

To do this, we’ll take an imaginary journey. We will travel to Proxima Centauri and back. This star is 4.3 light-years away. Light can travel there and back in 8.6 years. When objects (such as our space ship and the earth) move away from one another, any waves of electromagnetic energy passing between them are spread out. The color of light from one object would be shifted to a lower frequency when observed by the other object. This is known as the Doppler effect. It’s used to clock the speed of moving objects (such as your car, by the police, for example).

The color of light is related to its frequency. Red light has a lower frequency than blue light. Light is simply one part of the entire electromagnetic spectrum. The lowest frequencies are radio waves. TV waves have somewhat higher frequencies. Higher yet are microwaves and then infra-red. Then come the familiar colors of light: red, orange, yellow, green, blue, and violet. Beyond visible light is the ultra-violet region. Then come X-rays and cosmic rays. We have different names for various parts of the spectrum, but each of these are simply higher and higher frequencies of electromagnetic energy.

As the relative speed that two objects separate increases to a significant fraction of the speed of light, any electromagnetic energy (light) passing between them is red shifted. Objects moving toward each other will see a blue shift due to the Doppler effect.

As we board ship, we activate two separate TV systems. One will broadcast back to earth a live picture of life aboard the ship; the other TV signal will be beamed from earth to the ship. It will show what’s happening in real time somewhere on earth. Now let’s blast off. Our huge (and completely fanciful) engines slingshot us to nearly the speed of light (using a huge, momentary, gravitational warp).

Once moving, we notice a major change in our TV signal. It is red-shifted almost infinitely. What were 100 megahertz signals passing between us and earth are now shifted down, let’s say, to only 100 hertz. Since it takes on the order of a hundred million waves or cycles of a TV signal to encode a one-second picture, when a TV signal showing one second’s worth of picture is red shifted to 100 Hz, it then takes a million seconds to collect it. For a live broadcast, this means in effect that time has slowed down by a factor of a million between the source of the broadcast and where it’s being received. This shift is symmetric. The relative speed of departure determines the red shift, and it’s the same in both directions. So, let’s take it one direction at a time.

First, let’s think about the people on earth watching us on their TVs, and where our broadcast is coming from. After two years our signal is emitted from a point about two light-years away; it will take another two years to get back to earth. After 4.3 years earth gets a signal showing us only halfway to our destination star. After 8.6 years, the signal shows us arriving at Proxima Centauri and turning around to come home. This signal originated 4.3 years earlier, and took an additional 4.3 years to get back to earth. How much time is represented by 8.6 years of signal slowed to a millionth of normal speed? Only 4.5 minutes! That’s how much time must have elapsed for us, because that’s exactly how much picture they receive. Now, after 8.6 years, all of a sudden they get a TV signal blue shifted by a factor of a million. This signal, which lasts only 270 microseconds, contains the same amount of information as the 8.6 year red-shifted signal. Moments after receiving this signal, before they have even had time to slow it down and view it at normal speed, our ship brakes to a stop in the space near the earth. And, what have we been watching on our TV screen? During the trip away, we also got a signal red shifted by a factor of a million. This signal, lasting 4.5 minutes (by the inference we made just above), contains only 270 microseconds of “real time” when played at normal speed (not even a single frame of a TV picture). On the whole way back, however, we got a million-fold blue shifted signal. This signal showed virtually all of the 8.6 years that elapsed on earth during our entire trip. Again, some simple math shows that a live broadcast 8.6 years long, blue shifted by a factor of a million, will be a broadcast so tightly compressed that it will take only 4.5 minutes to receive it.

Thus, we have the asymmetry. We see 270 microseconds of time go by on earth in the 4.5 minutes it takes us to get to Proxima Centauri, traveling at nearly the speed of light. Then we see 8.6 years of earth time in the 4.5 minutes it takes us to return. Those on earth see 4.5 minutes go by on our space ship during the 8.6 years it takes for them to receive that signal. Then, they see another 4.5 minutes on board our ship during the 270 microseconds just before we return.

That’s right, when our trip is over, we’re less than 10 minutes older! Those on earth have aged more than eight years!

This “thought experiment” demonstrates at least three important things. It shows the asymmetry that “explains” the paradox. It shows that less time elapses in the frame we call the moving frame of reference (and how distance and time are related by velocity in the special theory of relativity). And, it shows that there is no limit to how quickly we can get somewhere (if we have nearly infinite amounts of energy to spend, and we don’t care how old the people we left behind are when we return). With distance measured from the point of view of those on earth, and time measured from the point of view of those aboard ship, we traveled over eight light-years in less than 10 minutes. This is “warp” speed in anybody’s book!

Our knowledge of relativity gives us some perspective about this imaginary journey, but the new lesson is about information transfer. Time in some distant place or frame of reference only passes as far and as fast as signals actually communicate. The amount of time is equal to the number of clock ticks, or the number of vibrations of the signal received. The rate that time is passing in the other frame can only be observed as a red or blue shifted signal coming from it. There is no better handle we can grasp. The rate of exchange of information is the defining phenomenon.

The asymmetry is explained by the distance in space over which the signals propagate and by the fact that the moving ship sweeps its signals up, while the stationary earth has to wait for its signals to arrive. Observers in each frame see time slowed down in the other frame as the frames separate, and they see it speeded up as they approach, but the amount of time observed is a function of the red or blue shifted signals they receive. The amount of subjective time each spends receiving the other’s signal, and the amount of “time” shown by that signal, can be very different.

So, there is a difference between remaining stationary in space and moving through it. Relativity isn’t so relative after all! The speed of light may be a universal constant in all frames of reference, but time and distance are not. The real handle we have on time has to do with the transfer of information (a live TV broadcast being a perfect example), and the shifting of frequencies up and down the spectrum. The bottom line is the rate of information transferred in each direction. This is given by the total amount of red or blue shift in the signals exchanged. This, in turn, is the sum of the effects of special (and sometimes general) relativity, and the Doppler effect.

If two systems maintain contact with each other by exchanging a known signal, such as a TV broadcast, they can observe the slowdown of time in the other system and they can infer when events happen in the other system with respect to their own by simple considerations involving how long it takes light to get there.

And now, a definition to pique your “Numination” process: Let’s define absolute rest as a frame of reference in which matter ages at the maximum possible rate with respect to matter in any other frame.

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