Relativistic Doppler effect

Visualization

In Diagram 2, the blue point represents the observer, and the arrow represents the observer's velocity vector relative to its surroundings. When the observer is stationary, the x,y-grid appears yellow and the y-axis appears as a black vertical line. Increasing the observer's velocity to the right shifts the colors and the aberration of light distorts the grid. When the observer looks forward (right on the grid), points appear green, blue, and violet (blueshift) and grid lines appear farther apart. If the observer looks backward (left on the grid), then points appear red (redshift) and lines appear closer together. The grid has not changed, but its appearance for the observer has.

Diagram 3 illustrates that the grid distortion is a relativistic optical effect, separate from the underlying Lorentz contraction which is the same for an object moving toward an observer or away.

Analogy

Understanding relativistic Doppler effect requires understanding the Doppler effect, time dilation, and the aberration of light. As a simple analogy of the Doppler effect, consider two people playing catch. Imagine that a stationary pitcher tosses one ball each second (1 Hz) at one meter per second to a catcher who is standing still. The stationary catcher will receive one ball per second (1 Hz). Then the catcher walks away from the pitcher at 0.5 meters per second and catches a ball every 2 seconds (0.5 Hz). Finally, the catcher walks towards the pitcher at 0.5 meters per second and catches three balls every two seconds (1.5 Hz). The same would be true if the pitcher moved toward or away from the catcher. By analogy, the relativistic Doppler effect shifts the frequency of light as the emitter or observer moves toward or away from the other.

To understand the aberration effect, again imagine two people playing catch on two parallel conveyor belts (moving sidewalks) moving in opposite direction. The pitcher must aim differently depending on the speed and the spacing of the belts, and where the catcher is. The catcher will see the balls coming at a different angle than the pitcher chose to throw them. These angle changes depend on: 1) the instantaneous angle between the pitcher-catcher line and the relative velocity vector, and 2) the pitcher-catcher velocity relative to the speed of the ball. By analogy, the aberration of light depends on: 1) the instantaneous angle between the emitter-observer line and the relative velocity vector, and 2) the emitter-observer velocity relative to the speed of light.

Motion along the line of sight

Assume the observer and the source are moving away from each other with a relative velocity v ( v is negative if the observer and the source are moving towards each other). Considering the problem in the reference frame of the source, suppose one wavefront arrives at the observer. The next wavefront is then at a distance λ = c / f s away from the observer (where λ is the wavelength, f s is the frequency of the wave the source emitted, and c is the speed of light).

The wavefront moves with velocity c , but at the same time the observer moves away with velocity v , so λ + v t = c t . This gives us

t = λ c − v = c ( c − v ) f s = 1 ( 1 − β ) f s ,

where β = v / c is the velocity of the observer in terms of the speed of light.

Due to the relativistic time dilation, the observer will measure this time to be

t o = t γ ,

where

γ = 1 1 − β 2

is the Lorentz factor. The corresponding observed frequency is

f o = 1 t o = γ ( 1 − β ) f s = 1 − β 1 + β f s .

The ratio

f s f o = 1 + β 1 − β

is called the Doppler factor of the source relative to the observer. (This terminology is particularly prevalent in the subject of astrophysics: see relativistic beaming.)

The corresponding wavelengths are related by

λ o λ s = f s f o = 1 + β 1 − β ,

and the resulting redshift

z = λ o − λ s λ s = f s − f o f o

can be written as

z = 1 + β 1 − β − 1.

In the non-relativistic limit (when v ≪ c ) this redshift can be approximated by

z ≃ β = v c ,

corresponding to the classical Doppler effect.

Systematic derivation for inertial observers

Let us repeat the derivation more systematically in order to show how the Lorentz equations can be used explicitly to derive a relativistic Doppler shift equation for waves that themselves are not relativistic.

Let there be two inertial frames of reference, S and S ′ , constructed so that the axes of S and S ′ coincide at t = t ′ = 0 , where t is the time as measured in S and t ′ is the time as measured in S ′ . Let S ′ be in motion relative to S with constant velocity v ; without loss of generality, we will take this motion to be directed only along the x-axis. Thus, the Lorentz transformation equations take the form

x = γ ( x ′ + β c t ′ ) y = y ′ z = z ′ c t = γ ( c t ′ + β x ′ ) d x d t = v + d x ′ d t ′ 1 + v c 2 d x ′ d t ′ .

See velocity-addition formula, where β = v / c and γ = ( 1 − β 2 ) − 1 2 , and c is the speed of light in a vacuum.

The derivation begins with what the observer in S ′ trivially sees. We imagine a signal source is positioned stationary at the origin, O ′ , of the S ′ system. We will take this signal source to produce its first pulse at time t 1 ′ = 0 (this is event 1) and its second pulse at time t 2 ′ = 1 / f ′ (this is event 2), where f ′ is the frequency of the signal source as the observer in S ′ reckons it. We then simply use the Lorentz transformation equations to see when and where the observer in S sees these two events as occurring:

The period between the pulses as measured by the S observer is not, however, t 2 − t 1 because event 2 occurs at a different point in space to event 1 as observed by the S observer (that is, x 2 ≠ x 1 ) — we must factor in the time taken for the pulse to travel from x 2 to x 1 . Note that this complication is not relativistic in nature: this is the ultimate cause of the Doppler effect and is also present in the classical treatment. This transit time is equal to the difference x 2 − x 1 divided by the speed of the pulse as the S observer sees it. If the pulse moves at speed − u ′ in S ′ (negative because it moves in the negative x-direction, towards the S observer at O ), then the speed of the pulse moving towards the observer at O , as S sees it, is:

− u = − u ′ + v 1 + ( − u ′ ) v c 2 ,

using the Lorentz equation for the velocities, above. Thus, the period between the pulses that the observer in S measures is:

τ = t 2 − t 1 + γ v f ′ ( u ′ − v 1 − v u ′ c 2 ) − 1 = γ f ′ + γ f ′ v u ′ − v ( 1 − v u ′ c 2 ) .

Replacing τ with 1 / f and simplifying, we get the required result that gives the relativistic Doppler shift of any moving wave in terms of the stationary frequency, f ′ :

f = γ ( 1 − v u ′ ) f ′ .

Ignoring the relativistic effects by taking v ≪ c or c → ∞ (equivalent to γ → 1 ) gives the classical Doppler formula:

f = ( 1 − v u ′ ) f ′ .

For electromagnetic radiation where u ′ = c the formula becomes

f = γ ( 1 − v c ) f ′ = γ ( 1 − β ) f ′ = f ′ 1 − β 1 + β

or in terms of wavelength:

λ = λ ′ 1 + β 1 − β ,

where λ ′ is the wavelength of the source at the origin O ′ as the observer in S ′ sees it. In these equations v (and thus β) is assumed positive when the source is receding from the observer, and negative when approaching.

For electromagnetic radiation, the limit to classical mechanics, c → ∞ , is instructive. The Doppler effect formula simply becomes f = f ′ . This is the "correct" result for classical mechanics, although it is clearly in disagreement with experiment. It is "correct" since classical mechanics regards the maximum speed of interaction — for electrodynamics, the speed of light — to be infinite. The Doppler effect, classical or relativistic, occurs because the wave source has time to move by the time that previous waves encounter the observer. This means that the subsequent waves are emitted further away (or closer) to the observer than they otherwise would be if the source were not in motion. The effect of this is to stretch (or compress) the wavelength of the wave as the observer encounters them. If however the waves travel instantaneously, the fact that the source is further away (or closer) makes no difference because the waves arrive at the observer no later or earlier than they would anyway since they arrive instantaneously. Thus, classical mechanics predicts that there should be no Doppler effect for light waves, whereas the relativistic theory gives the correct answer, as confirmed by experiment.

Transverse Doppler effect

The transverse Doppler effect is the nominal redshift or blueshift predicted by special relativity that occurs when the emitter and receiver are at the point of closest approach. Light emitted at closest approach in the source frame will be blueshifted at the receiver. Light received at closest approach in the receiver frame will be redshifted relative to its source frequency.

Assuming the objects are not accelerated, light emitted when the objects are closest together will be received some time later. At reception, the amount of redshift will be

1 γ = 1 − v 2 c 2 .

If the light is received when the objects are closest together, then it was emitted some time earlier. At reception, the amount of blueshift will be

γ = 1 1 − v 2 c 2 .

Classical theory does not make a specific prediction for either of these two cases, as the shift depends on the motions relative to the medium.

The transverse Doppler effect is a consequence of the relativistic Doppler effect.

f o = f s γ ( 1 + v c cos ⁡ θ o )

In the frame of the receiver, θ₀ represents the angle between the direction of the emitter at emission, and the observed direction of the light at reception. In the case when θ₀ = π/2, the light was emitted at the moment of closest approach, and one obtains the transverse redshift

f o = f s γ

The transverse Doppler effect is one of the main novel predictions of the special theory. As Einstein put it in 1907: according to special relativity the moving object's emitted frequency is reduced by the Lorentz factor, so that the received frequency is reduced by the same factor.

Reciprocity

Sometimes the question arises as to how the transverse Doppler effect can lead to a redshift as seen by the "observer" whilst another observer moving with the emitter would also see a redshift of light sent (perhaps accidentally) from the receiver.

It is essential to understand that the concept "transverse" is not reciprocal. Each participant understands that when the light reaches them transversely as measured in terms of that person's rest frame, the other had emitted the light afterward as measured in the other person's rest frame. In addition, each participant measures the other's frequency as reduced ("time dilation"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity.

Experimental verification

In practice, experimental verification of the transverse effect usually involves looking at the longitudinal changes in frequency or wavelength due to motion for approach and recession: by comparing these two ratios together we can rule out the relationships of "classical theory" and prove that the real relationships are "redder" than those predictions. The transverse Doppler shift is central to the interpretation of the peculiar astrophysical object SS 433.

The first longitudinal experiments were carried out by Herbert E. Ives and Stilwell in (1938), and many other longitudinal tests have been performed since with much higher precision. Also a direct transverse experiment has verified the redshift effect for a detector actually aimed at 90 degrees to the object.

Motion in an arbitrary direction

If, in the reference frame of the observer, the source is moving away with velocity v at an angle θ o relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as

f o = f s γ ( 1 + v c cos ⁡ θ o ) .             (1)

In the particular case when θ o = 90 ∘ and cos ⁡ θ o = 0 one obtains the transverse Doppler effect:

f o = f s γ .

Due to the finite speed of light, the light ray (or photon, if you like) perceived by the observer as coming at angle θ o , was, in the reference frame of the source, emitted at a different angle θ s . cos ⁡ θ o and cos ⁡ θ s are tied to each other via the relativistic aberration formula:

cos ⁡ θ o = cos ⁡ θ s − v c 1 − v c cos ⁡ θ s .

Therefore, Eq. (1) can be rewritten as

f o = γ ( 1 − v cos ⁡ θ s c ) f s .             (2)

For example, a photon emitted at the right angle in the reference frame of the emitter ( cos ⁡ θ s = 0 ) would be seen blue-shifted by the observer:

f o = γ f s .

In the non-relativistic limit, both formulæ (1) and (2) give

Δ f f ≃ − v cos ⁡ θ c .

Doppler effect on intensity

The Doppler effect (with arbitrary direction) also modifies the perceived source intensity: this can be expressed concisely by the fact that source strength divided by the cube of the frequency is a Lorentz invariant (here, "source strength" refers to spectral intensity in frequency, i.e., power per unit solid angle and per unit frequency, expressed in watts per steradian per hertz; for spectral intensity in wavelength, the cube should be replaced by a fifth power). This implies that the total radiant intensity (summing over all frequencies) is multiplied by the fourth power of the Doppler factor for frequency.

As a consequence, since Planck's law describes the black body radiation as having a spectral intensity in frequency proportional to ν 3 / ( e h ν / k T − 1 ) (where T is the source temperature and ν the frequency), we can draw the conclusion that a black body spectrum seen through a Doppler shift (with arbitrary direction) is still a black body spectrum with a temperature multiplied by the same Doppler factor as frequency.

Accelerated motion

For general accelerated motion, or when the motions of the source and receiver are analyzed in an arbitrary inertial frame, the distinction between source and receiver motion must again be taken into account.

The Doppler shift when observed from an arbitrary inertial frame:

f o f s = 1 − ∥ v o → ∥ ∥ c → ∥ c o s ( θ c o ) 1 − ∥ v s → ∥ ∥ c → ∥ c o s ( θ c s ) 1 − ( v s c ) 2 1 − ( v o c ) 2

where:

v s → is the velocity of the source at the time of emission v o → is the velocity of the receiver at the time of reception c → is the light velocity vector θ c s is the angle between the source velocity and the light velocity at the time of emission θ c o is the angle between the receiver velocity and the light velocity at the time of reception

If c → is parallel to v s → , then θ c s = 0 ∘ , which causes the frequency measured by the receiver f o to increase relative to the frequency emitted at the source f s . Similarly, if c → is anti-parallel to v s → , θ c s = 180 ∘ , which causes the frequency measured by the receiver f o to decrease relative to the frequency emitted at the source f s .

This is the classical Doppler effect multiplied by the ratio of the receiver and source Lorentz factors.

Due to the possibility of refraction, the light's direction at emission is generally not the same as its direction at reception. In refractive media, the light's path generally deviates from the straight distance between the points of emission and reception. The Doppler effect depends on the component of the emitter's velocity parallel to the light's direction at emission, and the component of the receiver's velocity parallel to the light's direction at absorption. This does not contradict Special Relativity.

The transverse Doppler effect can be analyzed from a reference frame where the source and receiver have equal and opposite velocities. In such a frame the ratio of the Lorentz factors is always 1, and all Doppler shifts appear to be classical in origin. In general, the observed frequency shift is an invariant, but the relative contributions of time dilation and the Doppler effect are frame dependent.