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.