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.