Section 17–4 More about four-vectors

(Energy is mass / Composite velocity / Transformation of energy and momentum)

In this section, Feynman discusses four-momentum from the perspectives of mass-energy equivalence, composite velocity, and the transformation of energy and momentum. This section could be renamed as “four-momentum” instead of “more about four-vectors.”

1. Energy is mass:

“Energy and mass, for example, differ only by a factor c² which is merely a question of units, so we can say energy is the mass (Feynman et al., 1963, section 17–4 More about four-vectors).”

There are many examples of four-vectors such as four-velocity and four-force. Another example of four-vectors is four-momentum that has three spatial components (linear momentum) and a temporal component (energy). Because it is inconvenient to write c’s everywhere in the equations, Feynman introduces E = m by using the same trick concerning units of the energy. He explains that energy and mass differ only by a factor c² which is merely a question of units, and says that energy is the mass. On the contrary, Okun (1989) argues that mass is not equivalent to energy and emphasize that E = m₀c² is the correct equation instead of E = mc². Some physicists have also argued that the term relativistic mass is obsolete and suggested that it is no longer fashionable to teach this concept.

In The Evolution of Physics, Einstein and Infeld (1938) write that “according to the theory of relativity, there is no essential distinction between mass and energy. Energy has mass and mass represents energy. Instead of two conservation laws we have only one, that of mass-energy” (pp. 197-198).” Currently, some physicists explain that the definition of relativistic mass is redundant because mass and energy would refer to the same thing. In a sense, the crux of the problem is a matter of definition and one may argue whether the definition is useful or not. However, Galison (1997) observes that there are three sub-groups of physicists within the particle physics community (experimentalists, instrument developers, and theorists) that have a different specialized language (or definitions) for their internal communication.

2. Composite velocity:

“What is v′, the velocity as seen from the space ship? It is the composite velocity, the “difference” between v and u (Feynman et al., 1963, section 17–4 More about four-vectors).”

To find out the momentum and energy in another inertial reference frame, we have to know how the velocity transforms. If an object has a velocity v and an observer is in a space ship that is moving with a velocity u (with respect to the Earth), we can use v′ to designate the observed velocity of the object as seen from the space ship. Feynman adopts the concept of the composite velocity, the “difference” between v and u, and states it as v′ = (v−u)/(1−uv). Alternatively, we can explain that the four-momentum, P ≡ mV = (E/c, p) is obtained by multiplying the four-velocity by the rest (invariant) mass. More important, we need the Lorentz factor g in the expression of momentum and energy.

Feynman’s method to obtain the 4-momentum may seem unnatural by using a mathematical trick, v′² = (v²−2uv+u²)/(1−2uv+u²v²). As a suggestion, we should clarify that Feynman needs to derive the Lorentz factor that can be expressed as 1/√(1−v′²) = (1−uv)/(√1−v²)(√1−u²). For example, we can directly substitute the composite velocity, v′ = (v−u)/(1−uv) into the Lorentz factor, g = 1/√(1−v′²). Thus, we can get g = 1/√(1−[(v−u)/(1−uv)]²) = (1−uv)/√([1−uv]²–[v−u]²). The expression g_v′ could be quickly simplified as (1−uv)/√(1−u²)√(1–v²) and it is equal to g_ug_v(1−uv). In essence, we need the Lorentz factor of the composite velocity v′ (or observed velocity) that is from the perspective of a moving space ship (an inertial frame).

3. Transformation of energy and momentum:

“… transformations for the new energy and momentum in terms of the old energy and momentum are exactly the same as the transformations for t′ in terms of t and x, and x′ in terms of x and t… (Feynman et al., 1963, section 17–4 More about four-vectors).”

Feynman explains that the transformations for the new energy and momentum in terms of the old energy and momentum are exactly the same as the Lorentz transformations of space and time. That is, we simply replace t by E and replace x by p_x. We may elaborate that for a massive particle, the four-momentum (E, p_x, p_y, p_z) is derived using the particle’s invariant mass m multiplied by the particle’s four-velocity (cdt/dt, dx/dt, dy/dt, dz/dt). The four-velocity of a particle can be defined as the rate of change of its four-position (ct, x, y, z) with respect to the proper time, t. In short, the four-velocity is represented as (cdt/dt, dr/dt) and the four-momentum is simply (E, p), but it should be (E/c, p) if we remember to check the units.

Feynman concludes that we have discovered the four-vector momentum which can transform like x, y, z, and t. Furthermore, the “arrow” of the four-momentum has a temporal component equal to the energy and its spatial components are the three-vector momentum; but it may not be very clear how this arrow is more “real” than either the energy or the momentum. In a sense, the meaning of real is related to the concept of Lorentz invariance that does not depend on how we look at the space-time diagram. Specifically, the invariance of E² – (cp)² is really the rest energy m₀c² that remains unchanged under the Lorentz (energy-momentum) transformations. This invariance is also associated with the concept of rest mass (or invariance mass) that is the same in all inertial frames of reference.

Questions for discussion:

1. Would you explain that energy is mass in the transformation of four-momentum?

2. Would you adopt the concept of composite velocity (instead of observed velocity or Lorentz factor) in the derivation of four-momentum?

3. Why is the arrow of four-momentum more “real” than the energy or momentum?

The moral of the lesson: the “arrow” of the four-momentum has a time component equal to the energy (scalar) and three spatial components are equal to the three-vector momentum (vector); this arrow is more “real” than the energy or momentum.

References:

1. Einstein, A., & Infeld, L. (1938). The Evolution of Physics. New York: Simon and Schuster.

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. Galison, P. (1997). Image and Logic: A Material Culture of Microphysics. Chicago: University of Chicago Press.

4. Okun, L. B. (1989). The concept of mass. Physics Today, 42(6), 31-36.