Section 16–4 Relativistic mass

(Reasonable assumptions / Elastic collision / Inelastic collision)

In this section, Feynman discusses reasonable assumptions of relativistic mass, and the concept of relativistic mass from the perspective of an elastic collision and inelastic collision.

1. Reasonable assumptions:

“However, we can show that, as a consequence of relativity plus a few other reasonable assumptions, the mass must vary in this way (Feynman et al., 1963, section 16–4 Relativistic mass).”

According to Feynman, the concept of relativistic mass can be shown by making a few reasonable assumptions: Firstly, we shall assume the conservation of momentum and energy by analyzing a thought experiment involving a collision. Secondly, we shall assume that the momentum of a moving particle is a vector and it is always directed in the direction of the velocity. Thirdly, we shall assume the momentum is a function of velocity instead of a constant times the velocity. In the previous chapter, Feynman discusses the assumption of the energy of a body always equals mc². In short, one may also mention that the two main assumptions in deriving the relativistic mass expression are conservation of momentum and Lorentz transformation of velocities.

Kaufmann–Bucherer experiments were performed in the early 1900s to test different mathematical models of relativistic mass. Specifically, experimental physicists used deflections of electrons by magnetic fields to determine the expression of variation of mass with speed. These are indirect experiments that have limitations in determining the mass of an object. Similarly, we do not have a direct experiment that can prove the mass of a photon is strictly zero. For example, physicists are unable to place a photon on a weighing scale to measure its weight. In his Lectures on Gravitation (for postgraduates), Feynman suggests how the mass of a photon is infinitesimally small and “discuss the possibility that the mass is not of a certain definite size (Feynman et al., 1995, p. 22).”

2. Elastic collision:

“Two views of an elastic collision between equal objects moving at the same speed in opposite directions (Feynman et al., 1963, section 16–4 Relativistic mass).”

Feynman deduces the relativistic mass formula using an elastic collision between two identical objects moving at the same speed in opposite directions. Instead of quickly asking what is the vertical velocity u tan α, he could state the velocity of particle 2 in Fig 16–3(a) is v, the horizontal component is u and the vertical component is x. Therefore, tan α = x/u (vertical component of v divided by horizontal component of v) and it becomes trivial to deduce x = u tan α. If we let w be the vertical velocity of object 2 in the second frame of reference, then the expected velocity of object 2 in the first frame of reference would be wÖ(1–u²/c²). This expected velocity of object 2 can be calculated using the Lorentz transformation of velocity or explained by the time dilation effect.

Based on the same elastic collision thought experiment, Lewis and Tolman (1909) first derived the relativistic mass using conservation of momentum. Feynman could have cited Tolman’s paper in teaching the concept of relativistic mass because his pay was above the $20,000 mark in 1960 after he had been appointed Richard Chace Tolman Professor of Theoretical Physics (Gribbin & Gribbin, 1997). However, the formula of relativistic mass may also be expressed as m = p/v, m = BqR/v, m = E/c², or m = hf/c² instead of m₀/√(1−v²/c²). Theoretically, the expression of relativistic variation of mass with speed can be derived using the principles of electrodynamics or the Lagrangian approach. Empirically, the dependence of the mass of an object on its speed was also verified using the Kaufmann–Bucherer experiments.

3. Inelastic collision:

“Let us consider what is commonly called an inelastic collision. For simplicity, we shall suppose that two objects of the same kind, moving oppositely with equal speeds w… (Feynman et al., 1963, section 16–4 Relativistic mass).”

Feynman discusses an inelastic collision in which two identical objects moving in opposite directions with equal speeds w, hit each other and stick together, to become a new object. Essentially, he demonstrates the additive property of relativistic mass in which the total mass M of the new object is equal to the sum of two relativistic mass of two particles: M = m₀/√(1−u²/c²) + m₀/√(1−u²/c²). One may elaborate the combined mass using the words of Wilczek (1999), “How is it possible that massive protons and neutrons can be built up out of strictly massless quarks and gluons? The key is m = E/c². There is energy stored in the motion of the quarks, and energy in the color gluon fields that connect them. This bundling of energy makes the proton’s mass (p. 11).” In other words, the quarks are not stationary with respect to the combined object, proton.

Feynman says that the mass of the object which is formed when two identical objects collide must be twice the mass of the objects which come together. Then, he adds that these masses have been enhanced such that the total mass “must” be greater than the rest masses of the two objects. Alternatively, it is possible to discuss the inelastic collision without using the concept of relativistic mass. For example, Adkins (2008) suggests a variant of the inelastic collision of an object (m) with another identical object (initially at rest) that coalesce and form another new object of mass M. Using this thought experiment, he derives the expression of relativistic momentum p = gmv and total energy E = gmc².

Questions for discussion:

1. What are the reasonable assumptions needed for the concept of relativistic mass?

2. How would you derive the expression of relativistic mass using an elastic collision?

3. How would you explain the concept of relativistic mass using an inelastic collision?

The moral of the lesson: Feynman assumes the mass of an object which is formed when two equal objects collide must be twice the mass of the objects which come together and in accordance with the equation, m = E/c².

References:

1. Adkins, G. S. (2008). Energy and momentum in special relativity. American Journal of Physics, 76(11), 1045-1047.

2. 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.

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

4. Gribbin J., & Gribbin, B. (1997). Richard Feynman: A Life in Science. New York: Dutton.

5. Lewis, G. N., & Tolman, R. C. (1909). The Principle of Relativity, and non-Newtonian mechanics. Proceedings of the American Academy of Arts and Sciences. 44(25), 711-724.

6. Wilczek, F. (1999). Mass without mass. I: Most of matter. Physics Today, 52(11), 11-13.