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.

Section 16–1 Relativity and the philosophers

(Relativism / Empiricism / Mach’s Positivism)

In this section, Feynman discusses relativism, empiricism, and Mach’s positivism.

1. Relativism:

“That all is relative is a consequence of Einstein, and it has profound influences on our ideas (Feynman et al., 1963, section 16–1 Relativity and the philosophers).”

According to Feynman, cocktail-party philosophers simply interpret Einstein’s theory as “all is relative.” These philosophers would explain that phenomena depend on one’s frame of reference has been demonstrated in physics. That is, the idea of “things depend upon your frame of reference” has a profound effect on modern thought. The philosophical view that “all is relative” is sometimes described as relativism. In a sense, this misconception of the special theory of relativity can be attributed to the term relativity. On the contrary, there are absolute truths and invariant quantities in special relativity.

One may expect Feynman to elaborate on the importance of invariants that is in contrast to relativism. For example, in Jammer’s (1999) words, “mathematician Felix Klein and physicist Arnold Sommerfeld suggested that the name ‘theory of relativity’ should be replaced by ‘theory of invariants’ because the theory is merely a theory of the invariants of the Lorentz transformation (p. 33).” On the other hand, Minkowski (1907) has developed the concept of invariant (absolute) space-time interval and suggested the name “the postulate of the absolute world (p. 117).” Importantly, the light postulate states that the speed of light is invariant (or absolute) in all inertial frames of reference. Furthermore, the equations (e.g., four-momentum) in the special theory of relativity are invariant.

2. Empiricism:

“Our inability to detect absolute motion is a result of experiment and not a result of plain thought, as we can easily illustrate (Feynman et al., 1963, section 16–1 Relativity and the philosophers).”

Feynman explains that a consequence of relativity was the development of empiricism which said, “You can only define what you can measure.... The physicists should have realized that they can talk only about what they can measure.” In other words, the whole problem of whether one can define absolute velocity is the same as the problem of whether one can measure absolute velocity in an experiment without looking outside. However, another consequence is the development of operationalism: the idea of an operational definition in which “the concept is synonymous with the corresponding set of operations (Bridgman, 1927, p. 5).” Bridgman further suggests that a dozen operating procedures in measuring a physical quantity may lead to a dozen of different concepts.

Empiricism is a philosophy of science that emphasizes the importance of experimental evidence. In a lecture delivered in Cornell University, Feynman (1965) says that “[i]t does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is - if it disagrees with experiment it is wrong (p. 156).” Thus, some may argue that Feynman is an empiricist because he advocates the role of empirical evidence in the development of physical laws. However, in “Surely, You’re Joking, Mr. Feynman!,” Feynman questioned empirical results of Telegdi’s experiment on parity violation. Interestingly, Telegdi (1989) praises Feynman and writes that “[h]e understood experiments deeply and could suggest sources of error that had escaped the experimenters themselves (p. 85).”

3. Mach’s Positivism:

“Now that the motion is no longer absolute, but is a motion relative to the nebulae, it becomes a mysterious question, and a question that can be answered only by experiment (Feynman et al., 1963, section 16–1 Relativity and the philosophers).”

Feynman states Mach’s philosophy as one cannot detect any motion except by looking outside. (Mach distrusts concepts that cannot be verified by observable evidence.) Moreover, Feynman argues that Mach’s philosophy is not true because the Earth’s rotation on its axis can be determined using Foucault pendulum without looking at the stars. However, Bondi and Samuel (1997) applies Mach’s principle to explain a small precession (or Lense-Thirring precession) of the Foucault plane. More important, Einstein recognized Mach’s influence in questioning the notion of absolute space and time that helped to develop the special theory of relativity. Mach’s philosophy of science is more appropriately known as Mach’s positivism instead of simply positivism or phenomenalism.

Mach’s positivism can be described by Einstein’s words: “Science is nothing else but the comparing and ordering of our observations according to the methods and angles which we learn particularly by trial and error... As results of this ordering abstract concepts and the rules of their connection appear... Concepts have meaning only if we can point to objects to which they refer and to the rules by which they are assigned to these objects (Frank, 1952, p. 271).” Despite his criticisms of Mach’s positivism, Feynman explores how to incorporate Mach’s principle in quantum mechanics and gravitation theory (Feynman, Morinigo, & Wagner, 1995). Furthermore, Feynman discussed Mach’s principle with Wheeler before performing a sprinkler’s experiment that resulted in Feynman banished from the laboratory (Wheeler, 1989).

Note: The philosophical underpinning of Poincaré’s principle of relativity is essentially geometric conventionalism. For example, Poincaré (1902) writes that ‘[e]xperiment guides us in this choice, which it does not impose on us. It tells us not what is the truest, but what is the most convenient geometry (pp. 71-72).” Poincare’s conventionalism can be described as a “philosophy asserted that fundamental scientific principles are not reflections of the ‘real’ nature of the universe but are convenient ways of describing the natural world insofar as they are not contradicted by observation or experiment (Nye, 1979, p. 107). Thus, Feynman could have clarified whether the method of synchronizing clocks in special relativity is also a matter of convention.

Questions for discussion:

1. Why is relativism (“all is relative”) a misconception of Einstein’s special theory of relativity?

2. How would you justify empiricism which is related to “you can only define what you can measure?”

3. How do you explain that “one cannot detect any motion except by looking outside” (using a smartphone that has a built-in GPS)?

The moral of the lesson: currently, we have a much more humble point of view of our physical laws—everything can be wrong! (Mach’s philosophy may seem silly to physicists, but it helps to question the notion of absolute space and time.)

References:

1. Bondi, H., & Samuel, J. (1997). The Lense-Thirring effect and Mach’s principle. Physics Letters A, 228(3), 121-126.

2. Bridgman, P. W. (1927). The Logic of Modern Physics. New York: Macmillan.

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

4. 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.

5. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

6. Frank, P. (1951). Einstein, Mach, and logical positivism. In Paul Schilpp (Ed.), Albert Einstein: Philosopher-Scientist. La Salle, Illinois: Open Court Press. pp. 270-286.

7. Jammer, M. (1999). Einstein and Religion: Physics and Theology. Princeton: Princeton University Press.

8. Minkowski, H. (1907). Space and Time. In Petkov, V., (Ed.), Minkowski’s Papers on Relativity. Moscu: Minkowski Institute Press.

9. Nye, M. J. (1979). The Boutroux circle and Poincaré's conventionalism. Journal of the History of Ideas, 40(1), 107-120.

10. Poincaré, H. (1902/1952). Science and hypothesis. Mineola, NY: Dover.

11. Telegdi, V. (1989). A Lowbrow's View of Feynman. Physics Today, 42(2), 85.

12. Wheeler, J. A. (1989). The Young Feynman. Physics Today, 42(2), 24-28.

Section 15–9 Equivalence of mass and energy

(Mass-energy equivalence / “Derive” m₀/√(1−v²/c²) / Experimental verification)

In this section, Feynman discusses mass-energy equivalence, a “derivation” of the formula m₀/√(1−v²/c²), and experimental verifications of mass-energy equivalence.

1. Mass-energy equivalence:

“This theory of equivalence of mass and energy has been beautifully verified by experiments in which matter is annihilated—converted totally to energy… (Feynman et al., 1963, section 15–9 Equivalence of mass and energy).”

According to Feynman, Einstein interpreted the term m₀c² to be a part of the total energy of a body that is known as the “rest energy.” Feynman elaborates that by assuming the energy of a body always equals to mc², we can derive the formula m₀/√(1−v²/c²) for the variation of mass with speed. However, some physicists do not agree with Feynman in using the phrase “equivalence of mass and energy.” To be precise, one may use Einstein’s words, “equivalence between mass at rest and energy at rest (Einstein, 1922, p. 45).” Some physicists (e.g., Lederman & Hill, 2004) prefer to explain that “energy and mass are not equivalent”. (In the context of particle physics, a photon has energy, but it has no mass.)

Einstein compares the concept of inertial mass and gravitational mass using seven different terms: proportionality (Proportionalität), identity (Identität), physical identity (physikalische Wesengleichheit), equivalence (Äquivalenz), equality (Gleichheit), and equality proportionality (Gleichheit Proportionalität) (Baierlein, 2007). Einstein’s inconsistent use of different terms may cause confusion in understanding the principle of mass-energy equivalence. When a physicist uses the word equivalence to describe the mass-energy relation, it does not necessarily exclude incomplete equivalence. Some textbook authors also use the phrase mass-energy equivalence and explain that “energy and mass are the same thing” or “energy and mass are not the same thing.”

2. “Derive” m₀/√(1−v²/c²):

“As an interesting result, we shall find the formula (15.1) for the variation of mass with speed, which we have merely assumed up to now (Feynman et al., 1963, section 15–9 Equivalence of mass and energy).”

Feynman “finds” the mathematical expression of relativistic mass using the equation dE/dt = F.v that means the rate of change of energy with time is equal to the force times the velocity. This “derivation” requires a substitution of E = mc² and simple mathematical tricks. Besides, we need to choose a special case where v = 0 and state that the mass is m₀. Importantly, Feynman did not explain that this is not a rigorous derivation of the relativistic mass formula. In volume II, he states that “the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c², and is ‘attracted’ by the earth (Feynman et al., 1964, section 42–6 The speed of clocks in a gravitational field).” Although the formula of relativistic mass is shown to be m₀/√(1−v²/c²), it can be expressed as m = BqR/v, m = p/v, and m = E/c² (or m = hf/c²).

The expression of relativistic variation of mass with speed can be theoretically derived using the principles of electrodynamics, a collision between two identical particles in which their total momentum and energy are conserved, and the Lagrangian approach. However, it was not conclusive what should be the formula for the relativistic mass of electrons. For example, Einstein incorrectly defines longitudinal mass as m₀/(1−v²/c²)^3/2 and transverse mass as m₀/(1−v²/c²) using F = ma. Historically, Kaufmann–Bucherer–Neumann experiments were performed between 1901 and 1915 to test different theoretical models of relativistic mass. Specifically, physicists used deflections of electrons by magnetic fields to empirically determine the expression of relativistic variation of mass with speed.

3. Experimental verification:

“This experiment furnishes a direct determination of the energy associated with the existence of the rest mass of a particle (Feynman et al., 1963, section 15–9 Equivalence of mass and energy).”

Feynman explains that the energy changes represent extremely slight changes in mass because most of the time we cannot generate much energy from a given amount of material. He adds that in an atomic bomb of explosive energy equivalent to 20 kilotons of TNT, it can be shown that the released energy had a mass of 1 gram, according to the relationship ΔE = Δ(mc²). In short, the mass-energy relation means that “mass has energy” and “energy has mass.” In other words, we may explain that there is a numerical proportionality between the mass and energy of a system. Furthermore, one may emphasize that there is no new energy generated because it is mainly the conversion of rest energy (or mass-energy) to other forms of energy (e.g., kinetic energy).

Feynman gives an example of how the theory of mass-energy equivalence has been beautifully verified by experiments in which annihilation of matter can result in energy released. That is, when an electron and a positron come together at rest (each has a rest mass m₀), they disintegrate into two gamma rays each with the measured energy of m₀c². However, when a collision between a high-energy electron and a high-energy positron occurs, it is possible that many particles emerge from the event. In the words of Wilczek (2003), “[t]he total mass of these particles can be thousands of times the mass of the original electron and positron. Thus mass has been created, physically, from energy (p. 29).” Alternatively, Feynman suggests conceptualizing a moving positron as an electron traveling backward in time using a Feynman diagram.

Questions for discussion:

1. What does the principle of the equivalence of mass and energy mean?

2. Do you agree with Feynman’s derivation of the formula of relativistic mass that is in terms of m₀/√(1−v²/c²)?

3. How do experiments verify the equivalence of mass and energy?

The moral of the lesson: the principle of the equivalence of mass and energy can be verified experimentally by the annihilation of an electron and a positron that releases pure energy.

References:

1. Baierlein, R. (2007). Does nature convert mass into energy?. American Journal of Physics, 75(4), 320-325.

2. Einstein, A. (1922/2013). The meaning of relativity. London: Chapman & Hall.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

5. Lederman, L. M. & Hill, C. T. (2004). Symmetry and the Beautiful Universe. Amherst, NY: Prometheus.

6. Wilczek, F. (2003). The origin of mass. Annual Physics @ MIT, 24-35.