Section 16–5 Relativistic energy

(Energy has mass / Mass has energy / Rest energy)

In this section, Feynman discusses the concept of “energy has mass” and “mass has energy,” as well as the rest energy that may not be known in a physical process.

1. Energy has mass:

“… the excess mass of the composite object is equal to the kinetic energy brought in. This means, of course, that energy has inertia (Feynman et al., 1963, section 16–5 Relativistic energy).”

According to Feynman, if a proton and a neutron are “stuck together” (and can still be “seen”), the total mass M should be to 2m_w instead of 2m₀. This is because the excess mass of the composite object is due to the kinetic energy and it means that energy has inertia. In chapter 7, Feynman explains that “anything which has energy has mass—mass in the sense that it is attracted gravitationally (Feynman et al., 1963).” In the last chapter, Feynman has also shown that moving gas molecules are heavier because of the kinetic energy. Similarly, in volume II, Feynman 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).” It is essentially related to the principle of equivalence of energy and mass.

According to Wilczek (2005), “[s]tated as m = E/c², Einstein’s law suggests the possibility of explaining mass in terms of energy. That is a good thing to do because in modern physics energy is a more basic concept than mass (p. 863).” Feynman elaborates that if two particles join together and produce potential energy or any other form of energy, then the mass of the composite object is equivalent to the total energy that has been put in. In other words, the conservation of mass is equivalent to the conservation of energy and there is no strictly inelastic collision in the special theory of relativity. An inelastic collision is a collision in which the total kinetic energy of the two colliding particles is not the same after the collision as it was before; the so-called loss of kinetic energy may appear as part of the mass of the composite object.

2. Mass has energy:

“How much energy will they have given to the material when they have stopped? Each will give an amount (m_w−m₀)c²… (Feynman et al., 1963, section 16–5 Relativistic energy).”

The concept of “mass has energy” can be shown by the famous equation E = Dmc². Feynman explains that the total energy released can be calculated using the equation (m_w−m₀)c² and some energy is left in the material as thermal energy, potential energy, or other forms of energy. Furthermore, Einstein was thinking more about the origin of inertia or mass instead of making bombs. One may clarify this using Wilczek’s (2005) words: “[t]he usual way of writing the equation, E = mc², suggests the possibility of obtaining large amounts of energy by converting small amounts of mass. It brings to mind the possibilities of nuclear reactors or bombs… Actually, Einstein’s original paper does not contain the equation E = mc², but rather m = E/c² (p. 863).”

Feynman used the equation E = mc² to estimate the energy liberated under fission in an atomic bomb. Historically, the energy that should be liberated when an atom of uranium undergoes fission was estimated before the first direct test of atomic bomb. Interestingly, Feynman adds that if Einstein’s formula had not worked, they would have measured it anyway. One may be surprised that Feynman used the word “they.” In his autobiography, Feynman (1997) says that “we decided that the big problem -- which was to figure out exactly what happened during the bomb’s implosion, so you can figure out exactly how much energy was released and so on -- required much more calculating than we were capable of. A clever fellow by the name of Stanley Frankel realized that it could possibly be done on IBM machines… (p. 125).”

3. Rest energy:

“…we do not have to know what things are made of inside; we cannot and need not identify, inside a particle, which of the energy is rest energy of the parts into which it is going to disintegrate (Feynman et al., 1963, section 16–5 Relativistic energy).”

Feynman discusses the question of whether we could always add the rest energy m₀c² to the kinetic energy to determine the total energy of an object is mc². In a sense, this is possible if we are sure of the component pieces of rest mass m₀ inside a composite object that has a mass of M. Generally speaking, we may not be always sure of (or cannot “see”) the parts inside, for example, a K-meson may disintegrate into two pions or three pions. The K-mesons are also known as kaons that helped to understand the problem of parity violation (and CP violation). Feynman cites this example possibly because he and Gell-Mann (1958) developed a theory of weak interactions to explain the parity violation.

Although Feynman promotes the concept of relativistic mass, he ends the chapter by providing two equations that only include rest mass: E²−p²c² = m₀²c⁴ and Pc = Ev/c. The two equations are useful because we can use them to find the velocity v, momentum P, or the total energy E of an object. One may argue that the two equations are also useful because they do not require the concept of relativistic mass and are applicable to photons. However, Feynman did not seem to be aware of particle physicists that prefer the concept of invariant mass. Physicists that oppose the use of relativistic mass may cite the last two sentences of this chapter and explain that relativistic mass is rarely used (as mentioned by Feynman).

Questions for discussion:

1. How would you explain that “energy has mass” or “energy has inertia”?

2. How would you explain that “mass has energy” or “how energy is released”?

3. Would you cite the last two sentences of the chapter to explain that the relativistic mass is useless?

The moral of the lesson: “energy has mass” and “mass has energy,” but it is debatable whether the concept of relativistic mass is useful.  

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. Feynman, R. P., & Gell-Mann, M. (1958). Theory of the Fermi interaction. Physical Review, 109(1), 193-198.

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. Wilczek, F. (2005). Nobel Lecture: Asymptotic freedom: From paradox to paradigm. Reviews of Modern Physics, 77(3), 857-870.