Section 15–8 Relativistic dynamics

(Relativistic momentum / Relativistic mass / An approximate formula)

In this section, Feynman discusses relativistic momentum, relativistic mass, and an approximate formula.

1. Relativistic momentum:

“Momentum is still given by mv, but when we use the new m this becomes p = mv = m₀v/√1−v²/c²… (Feynman et al., 1963, section 15–8 Relativistic dynamics).”

According to Feynman, the relativistic momentum of an object is given by the same mathematical expression mv. In addition, under Einstein’s modification of Newton’s laws, the modified mass (m) of the relativistic momentum can be expressed as m₀/√(1−v²/c²) such that conservation of momentum holds. This is not quite correct because Einstein’s definitions of transverse mass and longitudinal mass are not equal to m₀/√(1−v²/c²). Historically, Lewis and Tolman (1909) first derived the relativistic momentum using conservation of momentum in a paper titled The Principle of Relativity and non-Newtonian mechanics. Feynman held the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology, but he could have acknowledged Tolman’s contribution to relativistic momentum.

Textbook authors may define relativistic momentum in terms of p = gmv in which g is equal to 1/√(1−v²/c²). For example, Halliday et al. (2014) state that p = mDx/Dt₀ = gmv and explain that “Dt₀ is the time required to travel that distance, measured not by the observer watching the moving particle but by an observer moving with the particle (p. 1138).” This should not be viewed as a rigorous derivation or complete explanation of relativistic momentum. Although observers in all inertial frames of reference observe different values of dx/dt of a particle, they agree with the term dx/dt₀ where the proper time t₀ is measured by the moving observer. In a sense, the gamma factor may mix with rest mass to become relativistic mass or proper length (instead of only proper time), however, there could be only a modification of mass, time, or length in p = gmv.

2. Relativistic mass:

“Of course, whenever a force produces very little change in the velocity of a body, we say that the body has a great deal of inertia, and that is exactly what our formula for relativistic mass says… (Feynman et al., 1963, section 15–8 Relativistic dynamics).”

Feynman uses the relativistic mass formula m₀/√(1−v²/c²) to explain that the inertia is comparatively higher when the velocity of an object (v) is nearly as fast as c. He relates the effect of relativistic mass to the deflection of high-speed electrons in the synchrotron of Caltech. For instance, the relativistic mass of an electron in the synchrotron is explained to be about 2000 times of the rest mass that is comparable to the mass of a proton. Thus, there is a need for a much stronger magnetic field to deflect the high-speed electrons. Some physics teachers may prefer to use the concept of relativistic energy instead of relativistic mass that may cause a distraction in understanding the relativistic effect.

Lewis and Tolman (1909) derive a mathematical expression of relativistic momentum that needs the concept of relativistic mass using a thought experiment involving an elastic collision. We can also derive the relativistic momentum formula p = gmv without the concept of relativistic mass (Adkins, 2008). This can be achieved using a Lewis-Tolman inelastic collision of an object (m) with another identical object (initially at rest) that coalesce and form a particle of mass M. The relativistic momentum is conserved in this collision. This is in contrast to the Newtonian momentum that is not conserved. Some physicists may argue that Feynman could have avoided using the obsolete concept of relativistic mass to discuss relativistic momentum.

3. An approximate formula:

“An approximate formula to express the increase of mass, for the case when the velocity is small, can be found by expanding m₀/√(1−v²/c²)−in a power series, using the binomial theorem (Feynman et al., 1963, section 15–8 Relativistic dynamics).”

Feynman discusses the motion of molecules in a small tank of gas using the concept of relativistic mass. When we heat the gas in the small tank, it increases the average speed of the molecules and therefore, the total mass of the gas. Using the binomial theorem, Feynman suggests expanding m₀/√(1−v²/c²) in a power series to obtain an approximate formula for the increase of mass. Because the average speed of the molecules is relatively slow in comparison to the speed of light, the terms after the first two are negligible and we can get m = m₀+ ½m₀(v²/c²). In other words, the increase in the mass of the gas is approximately equal to the total increase in kinetic energy of molecules divided by c², or Δm = Δ(K.E.)/c².

We may not be able to measure the increase in mass because the gain in kinetic energy is relatively small compared to the term c². Proponents of relativistic mass would conceptualize every gas molecule has a mass that can be expressed in terms of m₀/√(1−v²/c²) and explain that the total mass of the gas may be expressed as M = Sm_i. To support their position, they may cite the following words of Einstein (1922), “Mass and energy are therefore essentially alike; they are only different expressions for the same thing (p. 47).” However, proponents of invariant mass may argue that the definition of mass in terms of energy (m = E/c²) is a redundant concept. In a sense, physicists that disagree with the concept of relativistic mass prefer the equation E² = (pc)² + (m₀c²)² instead of E = mc².

Questions for discussion:

1. Would you agree with Feynman in using the mathematical expression of relativistic momentum in terms of p = mv instead of p = gmv?

2. Would you explain the relativistic mass of an electron in a synchrotron to be about 2000 times of its rest mass that is comparable to the mass of a proton?

3. How would you explain that the increase in the mass of a gas is proportional to the increase in temperature?

The moral of the lesson: when a gas is heated, the average speed of molecules is increased and the approximate increase in mass of the gas can be estimated using the formula Δm = Δ(K.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. Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Hoboken, NJ: Wiley.

4. 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.

5. Einstein, A. (1922). The meaning of relativity. Princeton: Princeton University Press.