TIME IS OF THE ESSENCE IN SPECIAL RELATIVITY
PART 2, THE TWIN PARADOX
by Dr. S. Peter Rosen
Office of Science
U.S. Department of Energy
No other aspect of Einstein’s Special Relativity conflicts so much with our intuition as does the Twin Paradox. It boggles the mind that one twin can end up younger than the Earth-bound member of the pair simply by taking a voyage to a nearby star at speeds approaching that of light. Even though inanimate versions of this paradox take place around us every second of every day, we still find it hard to believe that mere travel can change the rate at which we humans age. This feeling is further compounded by the fact that we will never be able to perform the experiment with real twins and settle the question once and for all. The twin paradox will remain forever a “gedanken,” or “thought” experiment.
The subject started out neither as a paradox, nor about twins. In his original paper on Special Relativity, Einstein noted what he called a theorem about synchronized clocks: if two such clocks, A and B, are initially at the same point, and B travels round a closed path back to A, the clock B will run slow relative to A. Behind this theorem is the time dilation property of Special Relativity: if clock B is moving with constant speed v relative to clock A, a time interval measured by clock B will be amplified by the factor Γ = 1/√{1-(v/c)2 } when measured by clock A, where c is the speed of light (c=300,000 kilometers per second). Suppose, for example, that v is 3/5 times c and that clock B emits light signals every 4 seconds. The factor Γ will be 5/4, and clock A will see the light signals occurring every 4Γ seconds, that is, every 5 seconds. The time on clock B is thus stretched out, or dilated, relative to clock A whenever clock B is moving with constant velocity relative to clock A.
Einstein’s theorem was known as the “clock paradox” until 1911, when Paul Langevin and others re-expressed it in terms of twins. One twin, Alice, stays at home on Earth, while the other, Bob, takes a journey to a nearby star at a significant fraction of the speed of light, for example (5/13) c, turns around and returns to Earth. According to the travelling twin Bob’s clock, the round trip has taken, say, 12 years, while the stay-at-home twin Alice’s clock registers 13 years. Bob has thus become 1 year younger than his stay-at-home twin sister Alice.
But what does Bob have to say about Alice’s clock? According to him, Alice is traveling in the opposite direction at (5/13) c and so he will see her clock as running slower than his. Will he conclude therefore that Alice is one year younger than he is?
Now, the first step in assessing the issue of the Twin Paradox is to ask is whether time dilation takes place in the real world. The answer is a resounding “Yes!” During the past 60 years, we have discovered families of elementary particles with short lifetimes in the range of a few microseconds (millionths of a second) all the way down to several picoseconds (millionths of a microsecond). Among these particles is the muon, a particle similar in many ways to the electron, but about 200 times heavier. At rest, or moving very slowly compared with the speed of light, the muon decays into an electron and two other particles, known as neutrinos, typically in 2.2 microseconds.
Muons are created in the upper atmosphere, roughly 10 kilometers above the surface of the Earth, when cosmic rays collide with the atoms in the atmosphere. Muons can subsequently be detected at the surface of the Earth, or even as much as 1 kilometer below the surface. How can they travel such long distances in 2.2 microseconds, when muons cannot exceed that of light, and so the farthest they can travel is less than 2/3 of a kilometer (2.2 microseconds multiplied by 300,000 km/second = 0.66 km)? The answer has to be that, to an observer on the surface of the Earth, the muon is travelling very close to the speed of light, and time dilation enables it to survive far longer than its 2.2 microseconds lifetime.
The next step in assessing the nature of the paradox is to argue that Alice and Bob do not have identical experiences, and therefore they are not equivalent to one another. There is a real difference between them because it is only the star-going twin, Bob, who goes through a reversal of direction and therefore through a deceleration, followed by re-acceleration in the opposite direction. In other words, he has to fire the thrusters of his space ship to come to rest and then to come up to the same speed but in the opposite direction (with all maneuvers performed in a short time compared with the total journey). Alternatively, Bob might do a turning maneuver around the star. Either way, his travel history is distinctly different from, and not equivalent to that of his stay-at-home sister Alice.
Arguments based on General Relativity indicate that acceleration will tend to speed up Bob’s clock compared to Alice’s during the turn-around period, and this can be estimated using the equivalence principle. The increase is roughly proportional to (v/c)2 and so, as long as the turn around time is short compared with the main journey, it is a negligible effect. Hence, the reciprocal argument is not valid, and it is the travelling twin who is younger after the trip is over.
We can actually realize the situation envisaged by Einstein in his original 1905 paper using muons. Muons can be produced in the laboratory through the same types of collision as take place in the upper atmosphere. Because they are charged, we can steer them into circular storage rings, and keep them going round and round at speeds exceedingly close to the speed of light, say 0.995 c. In experiments at laboratories in both Europe and the United States, such muons have been found to survive typically for 30 times the 2.2 microsecond lifetime before decaying, in very good numerical agreement with the predictions of Special Relativity. The muons going around the accelerator are like clock B in Einstein’s theorem, and a stationary muon in the laboratory would be the equivalent of clock A.
We may therefore conclude from such experimental observations that, insofar as the inanimate muons are concerned, the Twin Paradox is valid, and that to an observer at rest on Earth, travelling muons remain truly younger than their stationary brethren. So what does this say about human twins?
Our unease with this result for humans stems from our intuitive notion of time, and the way in which we measure it. Psychologically, we tend to think of time as an endless river, and we humans as passengers on a boat flowing down the river at a constant speed. Relativity forces us to take a different view of time, a view that does not correspond to our everyday experience.
In the case of space, we measure distances essentially by comparing the length in question with a standard meter or yardstick, when the object is at rest with respect to the
yardstick . We measure time by observing motion: some object that is presumed to repeat its path, or its cycle, with exact regularity. The second hand makes one full cycle around the face of the clock in one minute, or the Earth makes one rotation about its axis in one day, and one full journey on its orbit around the sun in one year. Nowadays, our most precise measurement of time comes from the collective vibrations of atoms held at constant temperatures inside a cavity. Time is thus an abstraction drawn from these mechanistic models.
Since we humans are made of atoms, we can ask whether we can actually use our bodies as clocks. Do our hearts beat with sufficient regularity throughout our lives, and are the heart beats of exactly the same duration for each and every one of us? If so, then we could use heart beats as the basic measure of time: meet me one thousand heart beats from now, or let’s go to the five-thousand heart-beat show tomorrow.
In the case of twins, especially identical twins, it seems reasonable to assume that their heart-beats are identical and therefore play the role of the synchronized clocks in Einstein’s original theorem. The beats of Bob’s heart will indeed slow down relative to Alice’s heart as he travels to a star and back. Thus, even though we cannot perform the experiment, the outcome of the Twin Paradox appears to be inescapable: the star-going twin will indeed be younger than his stay-at-home sister.
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