Relativistic Doppler effect

The relativistic Doppler effect is the change in frequency, wavelength and amplitude of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect, first proposed by Christian Doppler in 1842), when taking into account effects described by the special theory of relativity.

Figure 1. A source of light waves moving to the right, relative to observers, with velocity 0.7c. The frequency is higher for observers on the right, and lower for observers on the left.

The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.

Astronomers know of three sources of redshift/blueshift: Doppler shifts; gravitational redshifts (due to light exiting a gravitational field); and cosmological expansion (where space itself stretches). This article concerns itself only with Doppler shifts.

Summary of major results

The relativistic Doppler effect describes how the frequency and color of light change when there is motion between the source of light and the observer, considering the effects of special relativity. This effect is different from the regular Doppler effect because it includes changes due to time dilation and does not need a medium like air or water to work.

In simple terms, if a light source and an observer are moving away from each other, the light appears redder, and if they are moving toward each other, the light appears bluer. This happens because of their relative speeds and the properties of space and time described by Einstein's theory of relativity.

Scenario	Formula
Relativistic longitudinal Doppler effect	λ r λ s = f s f r = 1 + β 1 − β {\displaystyle {\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}}}
Transverse Doppler effect, geometric closest approach	f r = γ f s {\displaystyle f_{r}=\gamma f_{s}}
Transverse Doppler effect, visual closest approach	f r = f s γ {\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}
TDE, receiver in circular motion around source	f r = γ f s {\displaystyle f_{r}=\gamma f_{s}}
TDE, source in circular motion around receiver	f r = f s γ {\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}
TDE, source and receiver in circular motion around common center	f ′ f = ( c 2 − R 2 ω 2 c 2 − R ′ 2 ω 2 ) 1 / 2 {\displaystyle {\frac {f'}{f}}=\left({\frac {c^{2}-R^{2}\omega ^{2}}{c^{2}-R'^{2}\omega ^{2}}}\right)^{1/2}}
Motion in arbitrary direction measured in receiver frame	f r = f s γ ( 1 + β cos ⁡ θ r ) {\displaystyle f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}}
Motion in arbitrary direction measured in source frame	f r = γ ( 1 − β cos ⁡ θ s ) f s {\displaystyle f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}}

Derivation

The relativistic Doppler effect describes how the frequency and wavelength of light change when there is motion between a light source and an observer, taking into account Einstein's theory of relativity. Unlike the classical Doppler effect, this version includes time dilation, a key part of relativity.

When the source and observer move directly toward or away from each other, the effect is similar to the classical version but includes an extra factor for time dilation. This means that clocks moving relative to an observer tick slower, affecting how we perceive the frequency of the light.

In cases where the source and observer move at angles to each other, the effect becomes more complex. Special relativity predicts that even when the source and observer are at the closest points in their paths, there can be a shift in the observed frequency. This shift can be either towards blue (shorter wavelength) or red (longer wavelength), depending on the exact motion and viewpoint.

f r = f r , s γ = 1 − β 1 − β 2 f s = 1 − β 1 + β f s . {\displaystyle f_{r}=f_{r,s}\gamma ={\frac {1-\beta }{\sqrt {1-\beta ^{2}}}}f_{s}={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}

Eq. 1

λ r λ s = f s f r = 1 + β 1 − β , {\displaystyle {\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}},}

Eq. 2

f r = γ f s {\displaystyle f_{r}=\gamma f_{s}}

Eq. 3

f r = f s γ {\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}

Eq. 4

f ′ f = ( c 2 − R 2 ω 2 c 2 − R ′ 2 ω 2 ) 1 / 2 {\displaystyle {\frac {f'}{f}}=\left({\frac {c^{2}-R^{2}\omega ^{2}}{c^{2}-R'^{2}\omega ^{2}}}\right)^{1/2}}

Eq. 5

f r = f s γ ( 1 + β cos ⁡ θ r ) . {\displaystyle f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}.}

Eq. 6

f r = γ ( 1 − β cos ⁡ θ s ) f s . {\displaystyle f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}.}

Eq. 7

cos ⁡ θ r = cos ⁡ θ s − β 1 − β cos ⁡ θ s {\displaystyle \cos \theta _{r}={\frac {\cos \theta _{s}-\beta }{1-\beta \cos \theta _{s}}}}

Eq. 8

Visualization

Figure 8. Comparison of the relativistic Doppler effect (top) with the non-relativistic effect (bottom).

Figure 8 shows how the relativistic Doppler effect and relativistic aberration differ from the non-relativistic Doppler effect and non-relativistic aberration of light. Imagine you are moving through space surrounded by stars that shine with a steady yellow light. When moving very fast, the light from stars ahead of you appears bluish, while the light from stars behind you looks reddish.

In both relativistic and non-relativistic cases, stars ahead and behind you shift to wavelengths we can't see. However, in the relativistic case, these shifts are much more extreme, and the stars appear more crowded in front of you compared to behind. Real stars emit a mix of colors, so the exact color changes depend on how our eyes see light and the types of stars we observe.

Doppler effect on intensity

Further information: Black-body radiation § Doppler effect

When the Doppler effect happens, it not only changes the frequency of light but also affects how bright the source appears. This happens because the strength of the source, divided by the cube of its frequency, stays the same for all observers, no matter how they are moving. This means that the total amount of light coming from the source increases by the fourth power of the Doppler factor.

Because of this, a black body — an object that radiates heat in a special way — still looks like a black body after a Doppler shift. Its temperature appears higher by the same factor that changes its frequency. This idea helps support the Big Bang theory as an explanation for the cosmological redshift, which is the stretching of light wavelengths from distant objects.

Experimental verification

Main article: Ives–Stilwell-, Mössbauer rotor-, and Spectroscopy tests of time dilation

Scientists have done many experiments to check if the ideas of special relativity, including the transverse Doppler effect, are true. One important early experiment was done by Ives and Stilwell in 1938. They used a special setup with mirrors to look at light from moving particles and found results that matched what special relativity predicts.

With newer technology, scientists have been able to test these ideas even better. They have used very fast particle beams and special methods to look directly at light from particles moving at right angles. These experiments have shown that the predictions of special relativity, including the transverse Doppler effect, are correct.

Relativistic Doppler effect for sound and light

Many physics books explain the Doppler effect—the change in sound or light frequency as a source moves toward or away from an observer—using simple, everyday examples for sound but a more complex theory for light. This can make it seem like sound and light need completely different explanations. However, the full story shows that both can be understood using similar ideas, even when we consider the rules of Einstein’s theory of relativity.

The Doppler effect for light changes not just because of motion but also because of time dilation, a key part of relativity. This means that even if a source of light or sound moves very fast, the way we measure its frequency depends on both its speed and how time itself changes for moving objects. This gives us a more complete picture of how waves behave, whether they are sound waves traveling through air or light waves moving through space.

Main article: Relativistic Doppler effect

f r f s = | O A | | O B | = 1 − v r / c s 1 + v s / c s 1 − ( v s / c ) 2 1 − ( v r / c ) 2 {\displaystyle {\frac {f_{r}}{f_{s}}}={\frac {|OA|}{|OB|}}={\frac {1-v_{r}/c_{s}}{1+v_{s}/c_{s}}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}}}

Eq. 9

f r f s = 1 − | v r | | C | cos ⁡ ( θ C , v r ) 1 − | v s | | C | cos ⁡ ( θ C , v s ) 1 − ( v s / c ) 2 1 − ( v r / c ) 2 {\displaystyle {\frac {f_{r}}{f_{s}}}={\frac {1-{\frac {|\mathbf {v_{r}} |}{\mathbf {|C|} }}\cos(\theta _{\mathbf {C,v_{r}} })}{1-{\frac {|\mathbf {v_{s}} |}{\mathbf {|C|} }}\cos(\theta _{\mathbf {C,v_{s}} })}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}}}

Eq. 10