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.

Section 16–3 Transformation of velocities

(1-D Velocity addition / Constant speed of light / 2-D Velocity addition)

In this section, Feynman discusses velocities addition in one dimension and two dimensions, as well as verifies the constant speed of light using the velocities addition formula.

1. 1-D Velocity addition:

“Next we discuss the interesting problem of the addition of velocities in relativity (Feynman et al., 1963, section 16–3 Transformation of velocities).”

Feynman presents a general problem as follows: “an object is moving with velocity v inside a space ship and the space ship itself is moving with velocity u with respect to the ground. What is the apparent velocity v_x of this object from the point of view of a man on the ground?” Feynman derives the one-dimensional velocity addition formula (or longitudinal velocity addition) of v_x using the ratio of x to t instead of the formula v_x = Dx/Dt. He clarifies that the velocity of a moving object as seen by the outside observer is equal to the observer’s distance divided by the observer’s time and it should not be the moving man’s time. In general, we can substitute Dx = g(Dx¢+uDt¢) and Dt = g(Dt¢+uDx¢/c²) into v_x = Dx/Dt = (Dx¢+uDt¢)/(Dt¢+uDx¢/c²) and simplify it into (u+v_x¢)/(1+uv_x¢/c²).

The symbols used in velocities addition may be revised as follows: let u be the velocity of an object measured by an observer in S frame and u¢ be the velocity measured by another observer in S¢ frame. We may also use the symbol v to represent the relative velocity between the S frame and S¢ frame. Alternatively, we can use u_A and u_B to represent the velocities that are measured by A and B respectively. Thus, the formula of relative velocity of an object (or observer) A with respect to another object B can be revised from u_A – u_B to (u_A – u_B)/(1 – u_Au_B/c²) that has a minus sign. If the two objects are moving in an opposite direction, the formula can be expressed as (|u_A| + |u_B|)/(1+|u_A||u_B|/c²) that has a plus sign.

2. Constant speed of light:

“… suppose that inside the space ship the man was observing light itself. In other words, v = c, and yet the space ship is moving. How will it look to the man on the ground? (Feynman et al., 1963, section 16–3 Transformation of velocities).”

Feynman’s question may be rephrased as follows: “suppose that a light beam is in a space ship that is moving at a constant velocity u with respect to the ground. What is the apparent speed of light observed by a man on the ground?” By using the longitudinal velocity addition formula, we obtain v = (u+c)/(1+uc/c²) = c. Feynman explains that this result is good because Einstein’s theory of relativity was designed to do this in the first place. In a sense, it is unnecessary to verify the velocity addition formula because Lorentz’s equations are derived from the light postulate. Essentially, the transformation of velocities is based on the “invariant” speed of light in vacuum that is independent of the motion of an observer in any inertial frame of reference.

One may relate Einstein’s beam of light thought experiment to the velocities addition formula. In his Autobiographical Notes, Einstein (1949) writes that “if I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such thing, however, neither on the basis of experience nor according to Maxwell’s equations... (pp. 49-51)” One may also introduce this problem: “if an observer moves at the speed of light, what will be the speed of light that is observed in the mirror and the speed of light that is moving in the same direction as observed by the observer?” Furthermore, physics teachers could let students verify the velocities addition formula using two different velocities of light: v = (u±c)/(1±uc/c²) = c.

3. 2-D Velocity addition:

“… since the speed c_y is less than the speed of light, the speed v_y of the particle must be slower than the corresponding speed by the same square-root ratio (Feynman et al., 1963, section 16–3 Transformation of velocities).”

To illustrate the addition of velocities in two dimensions (or transverse velocity addition), Feynman gives an example of an object inside a ship which is just moving “upward” with the velocity v_y′ with respect to a horizontally moving ship. He shows that using the relevant Lorentz transformations and the results, y = y¢ = v_y¢t′, we can deduce that if v_x′ = 0, the vertical velocity of the object as measured by an observer on the ground, v_y = v_y′Ö(1−u²/c²). However, Feynman did not explain how the factor Ö(1−u²/c²) is related to time dilation. To be more comprehensive, we can explain that: if the object’s velocity has a horizontal component, then Dx¢ is not zero and Dt = g(Dt¢ + uDx¢/c²). Thus, the velocity of the object moving in two dimensions measured by an observer on the ground is v_y = Dy/Dt =Dy¢/g(Dt¢+uDx¢/c²) = v_y′/g(1+uv_x¢/c²).

According to Feynman, each click of the “particle clock” will coincide with each n-th “click” of the light clock because the physical phenomenon of coincidence will be a coincidence in any frame. He concludes that the speed c_y is less than the speed of light c and the speed v_y of the particle must be slower than the corresponding speed by the same square-root ratio. Importantly, the square root ratio may be explained by Pythagoras theorem: a² + b² = c². The apparent vertical speed of light (v_y¢) in the moving frame, the horizontal speed of the moving system (u) and the speed of the light (c) are related by the equation v_y¢² + u² = c². We may deduce that v_y¢² = c² - u² and thus, the ratio of the two speeds: v_y¢/c = Ö(1 – u²/c²). This is based on the light postulate that is applicable to a light beam that is moving horizontally or any direction.

Questions for discussion:

1. How would you derive the relative velocity of two objects in the context of special theory of relativity?

2. Do we need to verify that the speed of light remains constant using the velocities addition formula?

3. How would you explain that the speed v_y of the particle in a moving space ship must be slower than the corresponding speed v_y′ by the same square-root ratio?

The moral of the lesson: the velocity of a moving object as observed by the outside observer is equal to the observer’s distance divided by the observer’s time and it should not be the moving man’s time.

References:

1. Einstein, A. (1949/1979). Autographical notes (Translated by Schilpp). La Salle, Illinois: Open court.

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.