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.

Section 34–6 The Doppler effect

(Moving source / Moving observer / Relativistic invariance)

 

In this section, Feynman discusses the Doppler effect from the perspective of moving source, moving observer, and relativistic invariance. This section could be titled as “Derivations of relativistic Doppler effect” because there are three different derivations of relativistic Doppler effect involving a source or an observer that is moving at a relativistic speed. The term Doppler-Fizeau effect is sometimes used because Fizeau discovered independently the same phenomenon on electromagnetic waves in 1848.

 

1. Moving source:

“The shift in frequency observed in the above situation is called the Doppler effect: if something moves toward us the light it emits appears more violet, and if it moves away it appears more red (Feynman et al., 1963, p. 34–7).”

 

Feynman mentions that the shift in frequency observed for a moving source is called the Doppler effect. Perhaps he could have adopted the terms, e.g., classical Doppler effect, relativistic Doppler effect, and longitudinal Doppler effect. Firstly, one may describe the shift in frequency for a source that is moving toward or away from an observer as longitudinal Doppler effect. This is different from the transverse Doppler effect whereby the source is moving perpendicularly to the line joining the source and observer. Furthermore, one may explain that the relativistic Doppler effect formula is contributed by the classical Doppler effect and time dilation. However, the Doppler effect for light waves and sound waves are not quite the same because light waves can travel in vacuum, but sound waves would need a medium.

 

“In a given amount of time τ, when the oscillator would have gone a distance vτ, on the x′ vs. ct diagram it goes a distance (c−v)τ. So all the oscillations of frequency ω₁ in the time Δτ are now found in the interval Δt = (1−v/c)Δτ; they are squashed together, and as this curve comes by us at speed c, we will see light of a higher frequency, higher by just the compression factor (1−v/c) (Feynman et al., 1963, p. 34–7).”

 

Feynman suggests an interesting way to understand the relativistic Doppler effect using the x′ vs. ct diagram. He explains the equation Δt = (1−v/c)Δτ in which the compression factor accounts for the classical Doppler effect due to the relative motion between the source and observer. When the source is moving toward or away from the observer, this factor may lead to an apparent compression or stretching of the waves. On the other hand, the Lorentz factor (γ) in the relativistic Doppler effect formula accounts for the time dilation due to the source moving at a relativistic velocity. When the source moves at a velocity close to the speed of light, time dilation becomes significant, affecting the apparent rate at which time flows for the moving object as compared to the stationary observer. Thus, the compression factor increases the observed frequency, but the Lorentz factor reduces the rate of flow of time, and thus the observed frequency.

 

2. Moving observer:

“Is the frequency that we would observe if we move toward a source different than the frequency that we would see if the source moved toward us? Of course not! The theory of relativity says that these two must be exactly equal (Feynman et al., 1963, p. 34–8).”

 

According to special relativity, the observed frequency must be exactly equal whether an observer is moving toward a source, or vice versa, at the same speed. Specifically, this is based on the first postulate of special relativity which states that the laws of physics are the same in all inertial reference frames. It implies the equivalence of inertial frames, which means that observers within a box moving at a constant speed cannot determine their absolute velocity by any experiment. However, the Doppler effect for sound waves is not the same for a moving source or moving observer at the same speed with respect to the air medium or wind velocity. The presence of the air medium or the influence of wind creates an asymmetry in the measured frequency, resulting in different Doppler shifts for the moving source and moving observer in various scenarios.

 

“If we were expert enough mathematicians we would probably recognize that these two mathematical expressions are exactly equal! In fact, the necessary equality of the two expressions is one of the ways by which some people like to demonstrate that relativity requires a time dilation, because if we did not put those square-root factors in, they would no longer be equal (Feynman et al., 1963, p. 34–8).”

 

It seems to be a humor that it would require mathematicians to recognize that the two mathematical expressions are exactly equal. To simplify the mathematical expressions, we may let b = v/c. Thus, a simple trick involved in the derivation is to apply the product of the two square roots Ö(1 - b)Ö(1 + b) = Ö1 - b² = Ö(1 – v²/c²). With this in mind, we may express (1 + b)/Ö(1 - b²) as (1 + b)/Ö(1 - b)Ö(1 + b) and simplify it to Ö(1 + b)/Ö(1 - b) or Ö(1 - b²)/(1 - b).

In short, Ö(1 + b)/Ö(1 - b) = (1 + b)/Ö(1 - b²) by multiplying both sides by Ö(1+b). Alternatively, Ö(1 + b)/Ö(1 - b) = Ö(1 - b²)/(1 - b) by multiplying both sides by Ö(1-b).

This mathematical exercise is unnecessary if Feynman simply shows that the expression Ö(1 - b²)/(1 - b) in 34.12 is also equal to Ö(c + v)/(c - v).

  

3. Relativistic invariance:

“But what would a man in motion, observing the same physical wave, see? Where the field is zero, the positions of all the nodes are the same (when the field is zero, everyone measures the field as zero); that is a relativistic invariant (Feynman et al., 1963, p. 34–8).”

 

Perhaps explaining the relativistic invariance of the phase of a wave using nodes is inappropriate because it can lead to misunderstanding. The concept of nodes is used to describe stationary points in wave patterns, such as standing waves, where the amplitude is always zero (See figure below). Although the nodes are located at the same positions for an inertial observer, they are formed by the superposition of two oppositely traveling waves that are always anti-phase with each other. Feynman’s statement about nodes being relativistic invariant is also misleading because the positions of nodes vary in accordance with Lorentz contraction. The crux of the matter is about the phase of a wave instead of nodes that are due to two waves. (Feynman gives a better explanation on the relativistic invariance of the phase of a wave in the next section.)

Modified from 9.5 Superposition and Interference – review – Douglas College Physics 1207 (bccampus.ca)

 

In Jackson’s (1999) words, “The phase of a wave is an invariant quantity because the phase can be identified with the mere counting of wave crests in a wave train, an operation that must be the same in all inertial frames (p. 529).” However, the phase of a wave may refer to a crest that is the same for all inertial observers, but it is not necessarily determined by the counting operation. Instead of counting, there could be a reference when making an observation on the absolute phase or relative phase of a wave. The wave’s frequency and wavelength may vary due to relativistic effects, but the phase difference between any two points remains the same for all inertial observers. In essence, the relativistic invariance of the phase is the invariance of the scalar product of two four-vectors k_μx_μ that is a scalar. The constantly varying combination of k and ω as well as space (x) and time (t) in the phase (kx - wt) ensures that it remains invariant under the Lorentz transformation.

 

Review Questions:

1. Would you use the term Doppler effect, classical Doppler effect, or relativistic Doppler effect? Should Feynman mention "blue shift" instead of "violet shift"?

2. How would you explain the observed frequency must be exactly equal if an observer is moving toward a source, or vice versa, at the same speed?

3. Would you explain relativistic invariance using the concept of nodes that are formed by the superposition of two waves (instead of the phase of a wave)?

 

The moral of the lesson: The relativistic Doppler effect is an extension of the classical Doppler effect, taking into account the effects of both the compression factor (classical Doppler effect) and the Lorentz factor (time dilation) from special relativity.

 

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. Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). John Wiley & Sons, New York.