Section 34–7 The ω, k four-vector

(Wave four-vector / Wave vector / Phase)

 

In this section, Feynman discusses wave four-vector, wave vector, and phase (dot product of wave four-vector and position four-vector). This section could be titled as “wave four-vector” (instead of “ω, k four-vector”) that is useful for analyzing relativistic effects related to waves.

 

1. Wave four-vector:

“They constitute what we call a four-vector; when a quantity has four components transforming like time and space, it is a four-vector…. What is the formula for such a wave? The answer is clearly cos (ωt−ks), where k = 2π/λ and s is the distance along the direction of motion of the wave—the component of the spatial position in the direction of motion. Let us put it this way: if r is the vector position of a point in space, then s is r⋅e_k, where e_k is a unit vector in the direction of motion (Feynman et al., 1963, p. 34–9).”

 

The wave four-vector is a mathematical construct used to describe wave phenomena in the context of special relativity. Perhaps Feynman could have distinguished four key features of the wave four-vector as follows: (1) Four-Dimensional: The wave four-vector is a four-dimensional mathematical object, typically denoted by the symbol k_μ, where the Greek index μ ranges from 0 to 3. (2) Temporal & Spatial Components: The temporal component (k_₀ = w) represents the temporal frequency of the wave and the spatial components (k_₁, k_₂, k_₃) represent the spatial wave vector. (3) Lorentz Invariance: The wave four-vector is Lorentz invariant, meaning its components vary in accordance with Lorentz transformations, but the phase is an invariant quantity. (4) Energy-Momentum Relation: The wave four-vector is related to the energy and momentum of the wave through its components.  

 

“We have seen that ω and k are like time and space in one space direction, but not in all directions, and so we must next study the problem of the propagation of light in three space dimensions, not just in one direction, as we have been doing up until now (Feynman et al., 1963, p. 34–9).”

 

It should be worth mentioning that the wave four-vector is useful for analyzing relativistic effects related to light waves, such as the Doppler effect and the stellar aberration. By utilizing the four-vector components of the source and observer, one can determine the frequency shift of the light waves accurately, accounting for relativistic Doppler effect. This is particularly relevant in astrophysics when studying objects moving at relativistic speeds, such as quasars and gamma-ray bursts. To describe stellar aberration in the context of special relativity, the wave four-vector can be used to characterize the incident light wave from a star that is dependent on the velocity of an observer on Earth. The apparent direction of the star as seen from Earth can be determined using relativistic transformations of the wave four-vector.

 

2. Wave vector:

“Now it turns out to be very convenient to define a vector k, which is called the wave vector, which has a magnitude equal to the wave number, 2π/λ, and is pointed in the direction of propagation of the waves… Therefore the rate of change of phase, which is proportional to the reciprocal of λ_x, is smaller by the factor cos α; that is just how k_x would vary—it would be the magnitude of k, times the cosine of the angle between k and the x-axis! (Feynman et al., 1963, p. 34–9).”

 

According to Feynman, it is very convenient to define a wave vector, which has a magnitude equal to the wave number, 2π/λ, and is in the direction of wave propagation. We can represent the wave vector as (k_x, k_y, k_z) = (2p/l)(cos a, cos b, cos g) where cos a, cos b, and cos g are the direction cosines. The direction cosines of the wave vector are the cosines of the angles between the wave vector and the x-axis, y-axis, and z-axis, where cos² a + cos² b + cos² g = 1. Some argue that it is meaningless to define the concept of wavelength vector because the wavelength in the direction of x-axis does not equal to lcos a, i.e.,  l_x = l/cos a instead (Koks, 2006). The apparent wavelength is elongated due to the “projection” of wave vector in the x-axis direction (see fig. below). We can rotate the axis of observation by an angle a such that the apparent wavelength would be reduced (by “inverse projection”) to l_x cos a = l.

Source: Koks, 2006

“Using this vector, our wave can be written as cos(ωt−k⋅r), or as cos(ωt−k_xx−k_yy−k_zz). What is the significance of a component of k, say k_x? Clearly, k_x is the rate of change of phase with respect to x (Feynman et al., 1963, p. 34–9).”

 

The physical significance of a component of wave vector, say k_x, could be viewed from three perspectives. (1) Wave number: By the definition of wave number, k_x determines the rate of change of phase (of the wave) with respect to distance along the x-axis. (2) Reciprocal of wavelength (k_x = 2p/l_x): The wave vector in the x-direction is directly related to the reciprocal of the wavelength, i.e., the number of wavelengths within a unit distance in the x-direction; it is inversely proportional to the wavelength. (3) Linear momentum (p_x = ħk_x): In the context of quantum mechanics, ħk_x is the linear momentum of waves in the x-direction. For example, by applying the equation p = ħk, we can relate the wave properties of light (described by the wave vector) to its particle-like properties (described by the linear momentum).

 

3. Phase:

“This dot product is an invariant, independent of the coordinate system; what is it equal to? By the definition of this dot product in four dimensions, it is ∑′k_μx_μ = ωt−k_xx−k_yy−k_zz (34.21) (Feynman et al., 1963, p. 34–9).”

“But this quantity is precisely what appears inside the cosine for a plane wave, and it ought to be invariant under a Lorentz transformation (Feynman et al., 1963, p. 34–10).”

 

The phase of a wave under a Lorentz transformation, which includes spatial rotations (in three dimensions) and spacetime rotations (in four dimensions), remains invariant. In a 3D-space, we may write k_x¢Dx¢+ k_y¢Dy¢ + k_z¢Dz¢ = k_xDx, which means that the x¢-axis is rotated to x-axis whereby k_xDx is the phase difference between two points on a wave. Similarly, ∑′k_μDx_μ = ωDt−k_xDx−k_yDy−k_zDz refers to the phase difference between two events, which is an invariant quantity under a spacetime rotation. (It is analogous to the spacetime interval Ds.) In other words, the x-axis and ct-axis can be rotated inwardly with respect to the 45^o light ray line, which preserves the phase difference between two points in the spacetime diagram. However, the crest or trough of a wave is a uniquely identifiable point on the wave, but it is unnecessary for Feynman to explain invariance using nodes and maxima (in the previous section) because they are associated with two waves that do not have a distinct phase.

 

Note: The phase (or absolute phase) of a wave refers to the specific phase angle of the wave at a particular point in time, often referenced to a starting point or a certain time. The relative phase is the phase difference between two waves or points in space, measured at the same time. In short, absolute phase is a subset of relative phase.

 

We know from our study of vectors that ∑′k_μx_μ is invariant under the Lorentz transformation, since k_μ is a four-vector. But this quantity is precisely what appears inside the cosine for a plane wave, and it ought to be invariant under a Lorentz transformation. We cannot have a formula with something that changes inside the cosine, since we know that the phase of the wave cannot change when we change the coordinate system (Feynman et al., 1963, p. 34–10).”

 

Perhaps Feynman could have elaborated on why the phase of the wave is invariant when we change the coordinate system. In a sense, a coordinate system in 3D-space is an arbitrary map that helps to locate a crest (or trough) that is independent of the axis of observation (or spatial rotation), but we need a different map depending on the speed of the observer. If we observe the wave from different reference frames, the k and ω will vary in accordance with Lorentz transformation, but the invariance of the spacetime interval ensures that kΔx − ωΔt remains the same. In essence, the phase of a wave cannot change under the Lorentz transformation is a consequence of the spacetime symmetry of special relativity. However, the relativistic invariance* of the phase of a wave is closely related to the operational definition of space and time that is based on the Poincaré-Einstein synchronization convention.

 

*There are different definitions of relativistic invariance, e.g., “The property that physical laws maintain their form under Lorentz transformations, which describe the transition from one inertial reference frame to another. This property of physical laws is known as Lorentz invariance. Where it is essential to emphasize that relativistic invariance includes invariance under translation in time and space, one speaks about Poincaré invariance. Lorentz invariance expresses equivalence of all inertial systems and uniformity of space-time… (Hazewinkel, 2012, p. 61).”

 

Review Questions:

1. How would you define the concept of wave four-vector?

2. How would you explain the physical significance of a component of wave vector?

3. How would you explain the dot product ∑′k_μx_μ is an invariant that is independent of the coordinate system?

 

The moral of the lesson: The crest and trough of a wave would remain the same as well as any phase that lies somewhere in between, no matter how you rotate the axis of observation or change the relative velocity between the source and observer.

 

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. Hazewinkel, M. (Ed.). (2012). Encyclopaedia of Mathematics: Reaction-Diffusion Equation-Stirling Interpolation Formula (Vol. 8). Springer Science & Business Media.

3. Koks, D. (2006). Explorations in mathematical physics: the concepts behind an elegant language. New York: Springer.