Relativistic Doppler effect

# Relativistic Doppler effect

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Encyclopedia
The relativistic Doppler effect is the change in frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

(and wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

) of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

, caused by the relative motion of the source and the observer (as in the classical Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

), when taking into account effects described by the special theory of relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

.

The relativistic Doppler effect is different from the non-relativistic Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

as the equations include the time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

effect of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

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.

## Visualization

In Diagram 2, the blue point represents the observer, and the arrow represents the observer's velocity vector. 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
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

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
Redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...

) and lines appear closer together. Note, the grid itself has not changed, but its appearance for the observer has. The main point in the Doppler Effect is to individualize each wave out sourcing the central point and expanding the wave given due to circulatory vibrations at a certain focal point.

## Analogy

Understanding relativistic Doppler effect requires understanding the Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

, time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

, and the aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

. As a simple analogy, 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 one meter away. 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.

Diagram 1 shows an emitter traveling to the right, whereas Diagram 2 shows the observer traveling right. While the color shift appears similar, the aberration of light is opposite. To understand this effect, again imagine two people playing catch. If the pitcher is moving to the right and the catcher is standing still, then the pitcher must aim behind the catcher. Otherwise the ball will pass the catcher on the right. Also, the catcher must turn in front of the pitcher, or the ball will hit on the catcher's left. Conversely, if the pitcher is stationary and the catcher is moving to the right, then the pitcher must aim in front of the catcher. Otherwise, the ball will pass the catcher on the left. Also, the catcher must turn to the back of the pitcher, or the ball will hit on the catcher's right. The degree to which the pitcher and catcher must turn to the right or left depends on two things: 1) the instantaneous angle between the pitcher-catcher line and the runner's 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 ( is negative if the observers are moving toward each other). Let us consider the problem in the reference frame
Reference frame
Reference frame may refer to:*Frame of reference, in physics*Reference frame , frames of a compressed video that are used to define future frames...

of the source.

Suppose one wavefront
Wavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...

arrives at the observer. The next wavefront is then at a distance away from him (where is the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

, is the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

of the wave the source emitted, and is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

). Since the wavefront moves with velocity and the observer escapes with velocity , the time (as measured in the reference frame of the source) between crest arrivals at the observer is

where is the velocity of the observer in terms of the speed of light (see beta (velocity)
Beta (velocity)
β in special relativity is the speed of an object relative to the speed of light: β = v/c.β is dimensionless and equal to the velocity in natural units. Any expression which involves v, like the Lorentz factor, can be rewritten using β instead....

).

Due to the relativistic time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

, the observer will measure this time to be

where

is the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...

. The corresponding observed frequency is

The ratio

is called the Doppler factor of the source relative to the observer. (This terminology is particularly prevalent in the subject of astrophysics
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...

: see relativistic beaming
Relativistic beaming
Relativistic beaming is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of light...

.)

The corresponding wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

s are related by

and the resulting redshift
Redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...

can be written as

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

corresponding to the classical Doppler effect.

## Transverse Doppler effect

The transverse Doppler effect is the nominal redshift
Redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...

or blueshift predicted by special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

that occurs when the emitter and receiver are at the point of closest approach. Light emitted at this instant will be redshifted. Light received at this instant will be blueshifted.

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
Light received when the objects are closest together was emitted some time earlier, at reception the amount of blueshift is

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.

In the frame of the receiver, θ0 represents 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

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 – in addition to the classical Doppler effect – 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 her/him 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
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....

.

### Experimental verification

In practice, experimental verification of the transverse effect usually involves looking at the longitudinal
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

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
SS 433
SS 433 is one of the most exotic star systems observed. It is an eclipsing X-ray binary system, with the primary most likely a black hole, or possibly a neutron star., pp. 23–24. The spectrum of the secondary companion star suggests that it is a late A-type star...

.

#### Longitudinal tests

The first of these experiments was carried out by Ives and Stilwell in (1938) and although the accuracy of this experiment has since been questioned, many other longitudinal tests have been performed since with much higher precision http://www.mpi-hd.mpg.de/ato/rel/,http://www.mpi-hd.mpg.de/ato/rel/doppler-symposium.tgif.pdf. These usually claim greater certainty than Ives-Stilwell, but also tend to be more complicated.
• Herbert E. Ives and G.R. Stilwell, “An experimental study of the rate of a moving clock”
J. Opt. Soc. Am 28 215–226 (1938) and part II. J. Opt. Soc. Am. 31, 369–374 (1941)

#### Transverse tests

To date, only one inertial experiment seems to have verified the redshift effect for a detector actually aimed at 90 degrees to the object.
• D. Hasselkamp, E. Mondry, and A. Scharmann, "Direct Observation of the Transversal Doppler-Shift"
Z. Physik A 289, 151–155 (1979).

## Motion in an arbitrary direction

If, in the reference frame of the observer, the source is moving away with velocity at an angle relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as
(1)

In the particular case when and one obtains the transverse Doppler effect:

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

Therefore, Eq. (1) can be rewritten as
(2)

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

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

## 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 emitter motion must again be taken into account.

The Doppler shift when observed from an arbitrary inertial frame:

where: is the velocity of the source at the time of emission is the velocity of the receiver at the time of reception is the light velocity vector is angle between the source velocity and the light velocity at the time of emission is angle between the receiver velocity and the light velocity at the time of reception

If is parallel to , then , which causes the frequency measured by the receiver to increase relative to the frequency emitted at the source . Similarly, if is anti-parallel to , , which causes the frequency measured by the receiver to decrease relative to the frequency emitted at the source .

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.

• Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

• Doppler beaming
• Redshift
Redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...

• Blueshift
• Time Dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

• Gravitational Time Dilation
Gravitational time dilation
Gravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the lower the gravitational potential, the more slowly time passes...

• Special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...