Relativistic Doppler effect

The relativistic Doppler effect is the change in frequency, wavelength and amplitude of light. This happens because of the motion of the source and the observer. It is like the Doppler effect, first talked about by Christian Doppler in 1842. But it also uses ideas from 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 regular Doppler effect. Its equations include the time dilation effect from special relativity. They also do not need a medium to work as a reference point. These equations show the total difference in observed frequencies and have the needed Lorentz symmetry.

Astronomers know of three reasons for redshift/blueshift: Doppler shifts; gravitational redshifts (when light leaves a gravitational field); and cosmological expansion (when space itself stretches). This article only talks about Doppler shifts.

Summary of major results

The relativistic Doppler effect explains how the color of light changes when there is motion between the light source and the person seeing it. This idea comes from Einstein’s theory of relativity.

If the light source and the observer are moving away from each other, the light looks redder. If they are moving toward each other, the light looks bluer. This change happens because of how fast they are moving and the way space and time work.

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 explains how the frequency and wavelength of light change when there is movement between a light source and someone watching it. This idea comes from Einstein's theory of relativity. It is different from the regular Doppler effect because it includes time dilation.

When the source and observer move straight towards or away from each other, the effect looks similar to the regular version but has an extra part for time dilation. This means that moving clocks seem to tick slower, which changes how we see the light's frequency.

If the source and observer move at angles to each other, the effect becomes more complicated. Special relativity says that even when they are closest to each other, there can still be a change in the seen frequency. This change can make the light look blue (shorter wavelength) or red (longer wavelength), depending on how they move and from where we look.

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 with stars that shine with steady yellow light. When moving very fast, the light from stars ahead of you looks bluish, while the light from stars behind you looks reddish.

In both relativistic and non-relativistic cases, stars ahead and behind you shift to colors we can't see. In the relativistic case, these shifts are more extreme, and the stars look more crowded in front of you compared to behind. Real stars have many colors, so the exact color changes depend on what we see and the types of stars.

Doppler effect on intensity

Further information: Black-body radiation § Doppler effect

When the Doppler effect happens, it changes the frequency of light and also how bright the light looks. This is because the strength of the light source, divided by the cube of its frequency, stays the same for everyone, no matter how they are moving. Because of this, the total amount of light 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 looks higher by the same amount 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 tests to see if the ideas of special relativity are true. One early test was done by Ives and Stilwell in 1938. They used mirrors to study light from moving particles and found results that matched what special relativity predicts.

With better tools, scientists have tested these ideas more carefully. They have used very fast particle beams and special methods to study light from particles moving in straight lines. These tests have shown that the ideas of special relativity, including the transverse Doppler effect, are correct.

Relativistic Doppler effect for sound and light

Many physics books talk about the Doppler effect. This is when the pitch of sound or the color of light changes as something moves toward or away from us. They often use easy examples for sound but a harder theory for light. This can make it seem like sound and light need totally different ideas to explain them. But the real story shows that both can be understood using similar thoughts, even when we think about Einstein’s theory of relativity.

The Doppler effect for light changes because of motion and also because of time dilation. Time dilation is a big idea in relativity. It means that when something moves very fast, time for that thing seems to go slower to us. So, even if a light or sound source moves quickly, how we see its frequency depends on both its speed and how time changes for moving things. This helps us understand waves better, whether they are sound waves in air or light waves in 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