Section 16–2 The twin paradox

(Describing twin paradox / Resolving twin paradox / Comparing muons)

In this section, the three main points may be classified as descriptions of the twin paradox, resolution of twin paradox, and experimental verifications using mu-mesons (muons).

1. Describing twin paradox:

“This is called a ‘paradox’ only by the people who believe that the principle of relativity means that all motion is relative … (Feynman et al., 1963, section 16–2 The twin paradox).”

Feynman describes a so-called “paradox” of Peter and Paul: when they are old enough to drive a space ship, Paul flies away at a very high speed and then comes back later. Peter is left on the ground and sees Paul moving so fast that Paul’s clock appears to tick slower. According to Feynman, if you believe that “the principle of relativity means that all motion is relative,” you may argue from Paul’s view that Peter was moving and Peter should also appear to age more slowly. To be precise, Paul must accelerate with respect to the Earth during parts of his trip in order to leave the Earth initially, turn around, and return to the Earth finally. On the other hand, one may add that Peter is on the surface of the Earth and he continuously experiences an acceleration that is equivalent to the Earth’s gravitational field.

It was Einstein (1905) who first presented a “clock problem,” whereas Langevin (1911) extended Einstein’s problem to human observers and the aging effect. The paradox can be expressed as follows: “if the effects of absolute motion are unobservable and only relative motion can be detected, one might just as well say that the earth with B on it went away from the spaceship and came back so that A would be younger. Thus the argument seems to require A on her return to be both older and younger than B (Park, 1988, p. 297).” Essentially, if you adopt relativism, it becomes an apparent “paradox” because each twin deduces the other twin to be younger when they meet together. By applying an idea of symmetry, you may argue that the age of the twins should be the same after the relative motion. It is a paradox because you have made an incorrect assumption.

2. Resolving twin paradox:

“… the rule is to say that the man who has felt the accelerations, who has seen things fall against the walls, and so on, is the one who would be the younger (Feynman et al., 1963, section 16–2 The twin paradox).”

Feynman explains that the motions of Peter and Paul are not really symmetrical because Paul has felt the accelerations during the motion while Peter felt nothing at all. Importantly, the rule is the man who has felt the accelerations is the one who would be younger. That is, there is a difference in motion between Peter and Paul in an “absolute” sense. Alternatively, some may prefer Feynman to explain that Peter remains in an idealized inertial frame of reference, whereas Paul’s reference frame must be changed from inertial to non-inertial, and vice versa, during his motion. Furthermore, Feynman could have clarified whether there is a need for the general theory of relativity to resolve this paradox.

In resolving the paradox, Langevin (1911) emphasizes the idea of acceleration that caused the distinction. On the other hand, Max von Laue (1913) suggests that the idea of reference frame alone (and “quasi-stationary acceleration”) is enough to explain the paradox. Ideally, we should provide quantitative calculations using the general theory of relativity that gives a complete picture in understanding the paradox. However, some physicists prefer to emphasize that the paradox can be qualitatively resolved using special relativity. For example, in his seminal paper, Einstein (1905) writes that “… we conclude that a balance-wheel clock that is located at the Earth’s equator must be very slightly slower than an absolutely identical clock, subjected to otherwise identical conditions, that is located at one of the Earth’s poles.”

3. Comparing muons:

“Although no one has arranged an experiment explicitly so that we can get rid of the paradox, one could compare a mu-meson which is left standing with one that had gone around a complete circle… (Feynman et al., 1963, section 16–2 The twin paradox).”

Feynman elaborates that it is not necessary to carry out an experiment to resolve the twin paradox because everything fits together all right. This position seems to contradict his position that emphasizes the importance of empirical evidence (Feynman, 1965). More importantly, he suggests that we can create mu-mesons (it is now known as muons) in a laboratory and use a magnet to accelerate them to move in a curve. Currently, it is not true to say that no one has arranged an experiment in order to get rid of the twin paradox. Physicists have already compared muons (microscopic clocks) that are stationary with respect to the Earth’s frame to those muons that are moving circularly.

In October 1971, four cesium atomic clocks were flown on two commercial jet flights around the world twice (one eastward and one westward) to test Einstein’s theory of relativity (Hafele & Keating, 1972). Using the actual flight paths of each trip, the theory predicted that the flying atomic clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds (ns) after the eastward flight and should have gained 275+/-21 ns after the westward flight. In the experiment, the flying clocks lost 59+/-10 ns after the eastward flight and gained 273+/-7 ns during the westward flight, relative to the stationary clocks on the Earth. The results indicate an empirical resolution of the clock paradox. However, the general theory of relativity was used to predict the difference in time between the clocks.

Questions for discussion:

1. How would you describe the twin paradox?

2. How would you resolve the twin paradox?

3. Do we need an experiment to resolve the problem of twin paradox?

The moral of the lesson: the twin paradox can be theoretically resolved by explaining the asymmetry in the motions and empirically verified by using atomic clocks.

References:

1. Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der Physik, 17, 891-921.

2. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Hafele, J. C., & Keating, R. E. (1972). Around-the-world atomic clocks: predicted relativistic time gains. Science, 177(4044), 166-168.

5. Langevin, P. (1911). L’evolution de l’espace et du temps. Scientia, 10, 31-54

6. Park, D. (1988). The How and the Why. Princeton: Princeton University Press.

7. von Laue, M. (1913). Das Relativitätsprinzip, Jahrbücher der Philosophie, 1, 99-128.