Relativistic Doppler effect

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 the change in frequency, wavelength and amplitude^[1] 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^[2]), when taking into account effects described by the special theory of relativity.

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

In the following table, it is assumed that for ${displaystyle eta =v/c>0}$ the receiver ${displaystyle r}$ and the source ${displaystyle s}$ are moving away from each other, ${displaystyle v}$ being the relative velocity and ${displaystyle c}$ the speed of light, and ${ extstyle gamma =1/{sqrt {1-eta ^{2}}}}$ .

Scenario	Formula	Notes
Relativistic longitudinal Doppler effect	${displaystyle {frac {lambda _{r}}{lambda _{s}}}={frac {f_{s}}{f_{r}}}={sqrt {frac {1+eta }{1-eta }}}}$
Transverse Doppler effect, geometric closest approach	${displaystyle f_{r}=gamma f_{s}}$	Blueshift
Transverse Doppler effect, visual closest approach	${displaystyle f_{r}={frac {f_{s}}{gamma }}}$	Redshift
TDE, receiver in circular motion around source	${displaystyle f_{r}=gamma f_{s}}$	Blueshift
TDE, source in circular motion around receiver	${displaystyle f_{r}={frac {f_{s}}{gamma }}}$	Redshift
TDE, source and receiver in circular motion around common center	${displaystyle {frac {f'}{f}}=left({frac {c^{2}-R^{2}omega ^{2}}{c^{2}-R'^{2}omega ^{2}}} ight)^{1/2}}$	No Doppler shift when ${displaystyle R=R'}$
Motion in arbitrary direction measured in receiver frame	${displaystyle f_{r}={frac {f_{s}}{gamma left(1+eta cos heta _{r} ight)}}}$
Motion in arbitrary direction measured in source frame	${displaystyle f_{r}=gamma left(1-eta cos heta _{s} ight)f_{s}}$

Derivation

Relativistic longitudinal Doppler effect

Relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, is often derived as if it were the classical phenomenon, but modified by the addition of a time dilation term.^[3]^[4] This is the approach employed in first-year physics or mechanics textbooks such as those by Feynman^[5] or Morin.^[6]

Following this approach towards deriving the relativistic longitudinal Doppler effect, assume the receiver and the source are moving away from each other with a relative speed ${displaystyle v,}$ as measured by an observer on the receiver or the source (The sign convention adopted here is that ${displaystyle v,}$ is negative if the receiver and the source are moving towards each other).

Consider the problem in the reference frame of the source.

Suppose one wavefront arrives at the receiver. The next wavefront is then at a distance ${displaystyle lambda _{s}=c/f_{s},}$ away from the receiver (where ${displaystyle lambda _{s},}$ is the wavelength, ${displaystyle f_{s},}$ is the frequency of the waves that the source emits, and ${displaystyle c,}$ is the speed of light).

The wavefront moves with speed ${displaystyle c,}$ , but at the same time the receiver moves away with speed ${displaystyle v}$ during a time ${displaystyle t_{r,s}}$ , which is the period of light waves impinging on the receiver, as observed in the frame of the source. So, ${displaystyle lambda _{s}+vt_{r,s}=ct_{r,s}Longleftrightarrow lambda _{s}=ct_{r,s}(1-v/c)Longleftrightarrow t_{r,s}={frac {1}{f_{s}(1-eta )}},}$ where ${displaystyle eta =v/c,}$ is the speed of the receiver in terms of the speed of light. The corresponding ${displaystyle f_{r,s}}$ , the frequency of at which wavefronts impinge on the receiver in the source's frame, is: ${displaystyle f_{r,s}=1/t_{r,s}=f_{s}(1-eta ).}$

Thus far, the equations have been identical to those of the classical Doppler effect with a stationary source and a moving receiver.

However, due to relativistic effects, clocks on the receiver are time dilated relative to clocks at the source: ${displaystyle t_{r}=t_{r,s}/gamma }$ , where ${ extstyle gamma =1/{sqrt {1-eta ^{2}}}}$ is the Lorentz factor. In order to know which time is dilated, we recall that ${displaystyle t_{r,s}}$ is the time in the frame in which the source is at rest. The receiver will measure the received frequency to be

Eq. 1:

{displaystyle f_{r}=f_{r,s}gamma }

{displaystyle ={frac {1-eta }{sqrt {1-eta ^{2}}}}f_{s}}

{displaystyle ={sqrt {frac {1-eta }{1+eta }}},f_{s}.}

The ratio

{displaystyle {frac {f_{s}}{f_{r}}}={sqrt {frac {1+eta }{1-eta }}}}

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

The corresponding wavelengths are related by

Eq. 2:

{displaystyle {frac {lambda _{r}}{lambda _{s}}}={frac {f_{s}}{f_{r}}}={sqrt {frac {1+eta }{1-eta }}},}

Identical expressions for relativistic Doppler shift are obtained when performing the analysis in the reference frame of the receiver with a moving source. This matches up with the expectations of the principle of relativity, which dictates that the result can not depend on which object is considered to be the one at rest. In contrast, the classic nonrelativistic Doppler effect is dependent on whether it is the source or the receiver that is stationary with respect to the medium.^[5]^[6]