Section 17–5 Four-vector algebra

(Relativistic invariance / Scalar product / Momentum of a photon)

In this section, Feynman discusses relativistic invariance, scalar product of four-vectors, and momentum of a photon (an exception of four-vector).

1. Relativistic invariance:

“...complete the law of conservation of momentum by extending it to include the time component. This is absolutely necessary to go with the other three, or there cannot be relativistic invariance (Feynman et al., 1963, section 17–5 Four-vector algebra).”

According to Feynman, we must complete the law of conservation of momentum by extending it to include the conservation of energy (time component) to obtain a valid four-vector relationship in the space-time geometry. That is, it is absolutely necessary to have four components of momentum such that there is relativistic invariance. We can define relativistic invariance (or Lorentz invariance) in terms of a four-vector a_μ under a Lorentz transformation whereby a_t′²−a_x′²−a_y′²−a_z′² = a_t²−a_x²−a_y²−a_z². In chapter 16, Feynman explains that “the conservation of mass which we have deduced above is equivalent to the conservation of energy” in the context of collisions within the special theory of relativity. In a sense, this implies that we have a single principle involving the conservation of momentum, mass, and energy.

In Volume II, Feynman adds that “the quantity which is analogous to r² for three dimensions, in four dimensions is t²−x²−y²−z². It is an invariant under what is called the “complete Lorentz group”—which means for transformation of both translations at constant velocity and rotations (Feynman et al., 1964, section 25–2 The scalar product).” He also discusses the four-velocity vector with components v_x = dx/dt, v_y = dy/dt, v_z = dz/dt, and mentions that an incorrect guess of the time component is v_t = dt/dt = 1. It turns out that the four “velocity” components that we have written down also have an invariant quantity (the speed of light) if we include the proper time t in all denominators (cdt/dt, dx/dt, dy/dt, dz/dt). The invariant quantity of the four-velocity can be calculated easily using (cdt/dt)² – (dv/dt)² = g²(c² – v²) = c².

2. Scalar product:

“if a_μ is one four-vector and b_μ is another four-vector, then the scalar product is S′a_μb_μ = a_tb_t−a_xb_x−a_yb_y−a_zb_z. It is the same in all coordinate systems (Feynman et al., 1963, section 17–5 Four-vector algebra).”

Feynman states the notations that are used for a scalar product (or inner product) in terms of S′_μA_μA_μ = A_t²−A_x²−A_y²−A_z². The prime on S′ means that the first term, the “time” term, is positive, but the other three terms have minus signs. This invariant quantity is the same in any coordinate system (or inertial frame), and we may call it the square of the length (or norm) of a four-vector. In addition, Feynman (1964) mentions that “the only real complication is the notation.” However, some authors complicate the situation by changing the sign of all the terms and state the square of the length of the four-vector as +a_x²+a_y²+a_z²−a_t². Alternatively, one may state Sa_μ² = +a_x²+a_y²+a_z²+a_t² in which all have the same signs, and define a_t in terms of ict instead of ct (i is an imaginary number).

Feynman elaborates that the square of the length of a four-vector momentum of a single particle is equal to p_t²−p_x²−p_y²−p_z², or in short, E²−p². He says that the invariant quantity must be the same in every coordinate system (or inertial frame) and it is purely its energy, which is the same as its rest mass. In other words, the norm of the four-momentum vector is equal to m₀²c⁴ (i.e., E²−p²c² =m₀²c⁴) and the invariant quantity can be stated as rest energy instead of rest mass. In essence, we may save time in problem-solving if we use the four-momentum vector with a scalar product (relativistic invariant) that is the same in all inertial frames of reference. Simply put, the knowledge of an invariant quantity allows us to choose an inertial frame in which a problem can be solved more easily.

3. Momentum of a photon:

“Such a photon also carries a momentum, and the momentum of a photon (or of any other particle, in fact) is h divided by the wavelength: p = h/λ (Feynman et al., 1963, section 17–5 Four-vector algebra).”

Feynman ends the chapter by discussing the momentum of a photon. This is an interesting example because the rest mass of a photon is zero. Thus, the equation E²−p²c² = m₀²c⁴ can be simplified as E²−p²c² = 0 or E = pc. However, Feynman could mention that the momentum of any (massless) particle is equal to its total energy times its velocity (p = vE/c²) can be reduced to E = p if c = 1. More importantly, physics teachers should explain that the equations E = gm₀c² and p = gm₀v are not really useful for massless particles because m₀ = 0 and g approaches infinity for objects moving at the speed of light (i.e., E and p are not equal to zero). The energy of a photon would change and this is related to the relativistic Doppler effect that is elaborated later in the section 34–6 The Doppler effect (Feynman et al., 1963).

Feynman mentions that the rest mass of a photon is zero and ask whether the photon’s energy is zero using the formula m₀/√(1−v²) instead of E = gm₀c². He adds that the photon really can have energy even though it has no rest mass, but it possesses energy by always moving at the speed of light. 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).” On the other hand, from an interpretation of Meissner effect, “[t]he photon mass inside a conventional superconductor is 10^–11 proton masses, or less (Wilczek, 2008, p. 213).” It is good to end the chapter by discussing a special case (or exception) of four-vector using the momentum of a photon.

Questions for discussion:

1. How would you define the concept of relativistic invariance?

2. Would you define the scalar product of a four-vector that has the same sign or opposite signs?

3. How would you explain the mass and momentum of a photon?

The moral of the lesson: the relativistic invariance of a four-vector a_μ is related to the principle of the conservation of momentum, mass, and energy.

References:

1. 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.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

3. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.