Relativistic Doppler Effect

Claimed - Dev Sharma (Fall 2024)
The relativistic Doppler effect describes the wavelength and frequency of light due to relative motion between the source and an observer.

The Main Idea

The classical Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave. A common example is the change in pitch of the sound of a passing vehicle -- the noise has a higher pitch when the vehicle is approaching and a lower pitch when it is receding. This occurs due to the compression/stretching of the sound wave relative to the observer as it move through its medium (READ MORE HERE).

In the case with light in a vacuum however, there is no medium of propagation. Instead the effect occurs due to relativistic effects, primarily time dilation, which is described by Einstein's theory of Special Relativity.

While we experience the classical Doppler effect as a change in a sound's pitch, the eelativistic Doppler effect causes a change in the frequency and wavelength of the light wave. If the light source is moving towards the observer, the light's frequency will increase and wavelength will decrease, causing it to "blueshift". If the light source is moving away from the observer, the light's frequency will decrease and wavelength will increase, causing it to "redshift".

It is important to note that there are other cosmological effects that can cause redshifting and blueshifting of light.

• Gravitational redshift occurs due to the influence of strong gravitational fields on light (READ MORE HERE).
• Cosmological redshift is caused by the expansion of the universe "stretching" light over large distances (READ MORE HERE).

These effects operate through very different principles than the relativistic Doppler effect, and so are not covered in any greater detail in this article.

Derivation of General Relativistic Doppler Formula

Consider a given light source moving at relativistic speed [math]\displaystyle{ v }[/math], and an observer that is observing the light at angle [math]\displaystyle{ \theta }[/math] relative to the source's motion. Assume that the distance between the source and the observer is large enough that [math]\displaystyle{ \theta }[/math] does not change any meaningful amount as the source moves.

The source emits some light at [math]\displaystyle{ t_1 }[/math], which is then received by the observer at time [math]\displaystyle{ t_1^r }[/math]. After emitting the light, the source keeps moving at velocity [math]\displaystyle{ v }[/math] for time [math]\displaystyle{ \Delta t }[/math] before emitting more light at time [math]\displaystyle{ t_2 }[/math], which is then received by the observer at time [math]\displaystyle{ t_2^r }[/math]; at this time the source and observer are distance [math]\displaystyle{ L }[/math] away. All of these measurements are measured in the observer reference frame.

It is trivial to see that [math]\displaystyle{ t_2^r = t_2 + \frac{L}{c} }[/math] and that [math]\displaystyle{ t_1^r = t_1 + \frac{L + v \Delta t \cos(\theta)}{c} }[/math]

We can combine these equations to find the change time for the observer between receiving the two signals. [math]\displaystyle{ t_2^r - t_1^r = (t_2 + \frac{L}{c}) - (t_1 + \frac{L + v \Delta t \cos(\theta)}{c}) = (t_2 - t_1) + (\frac{L}{c} - \frac{L + v \Delta t \cos(\theta)}{c}) = (t_2 - t_1) + (\frac{L - L - v \Delta t \cos(\theta)}{c}) = (t_2 - t_1) - \frac{v \Delta t \cos(\theta)}{c} }[/math]

We can now rewrite this as [math]\displaystyle{ \Delta t^r = \Delta t (1 - \frac{v \cos(\theta)}{c}) }[/math]

We previously stated that [math]\displaystyle{ t }[/math] is from the observer's reference frame. Due to time dilation effects from the light source moving at relativistic speeds, we can use a Lorentz Transformation to find the relationship between the time in the observer's reference frame [math]\displaystyle{ t }[/math] and the time in the light source's reference frame [math]\displaystyle{ \tau }[/math], such that [math]\displaystyle{ t = \frac{\tau}{\sqrt{1 - \frac{v^2}{c^2}}} }[/math]

From this, we have that [math]\displaystyle{ \Delta t^r = \frac{\Delta \tau}{\sqrt{1 - \frac{v^2}{c^2}}} (1 - \frac{v \cos(\theta)}{c}) }[/math]

The frequency of the light wave is the inverse of the period. Thus, if we let [math]\displaystyle{ f_o = \frac{1}{\Delta t} }[/math] represent the frequency of the light from the observer's frame of reference and [math]\displaystyle{ f_s = \frac{1}{\Delta \tau} }[/math] represent the frequency of the light from the source's reference frame, we have [math]\displaystyle{ \frac{1}{f_o} = \frac{1}{f_s} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} (1 - \frac{v \cos(\theta)}{c}) }[/math]

We can rewrite this to find the general formula for the relativistic Doppler effect: [math]\displaystyle{ f_o = f_s \frac{\sqrt{1 - \frac{v^2}{c^2}}}{1 - \frac{v \cos(\theta)}{c}} }[/math]

It is clear to see that [math]\displaystyle{ f_o \neq f_s }[/math] and that their relationship depends on both the velocity of the moving light source and the angle at which the source is moving with respect to the line of sight of the observer.

A Mathematical Model

We have the general formula for the relativistic Doppler effect: [math]\displaystyle{ f_o = f_s \frac{\sqrt{1 - \frac{v^2}{c^2}}}{1 - \frac{v \cos(\theta)}{c}} }[/math]

We can simplify this formula to consider some common cases of the relativistic Doppler effect: the transverse Doppler effect, and the longitudinal Doppler effect.

The transverse Doppler effect is when the light source is traveling perpendicular to the observer (i.e. [math]\displaystyle{ \theta }[/math] = 90). In this case, the general formula simplifies to [math]\displaystyle{ f_o = f_s \sqrt{1 - \frac{v^2}{c^2}} }[/math]

The longitudinal Doppler effect is when the light source is moving directly towards/away from the observer (i.e. [math]\displaystyle{ \theta }[/math] = 0). In the case that the source is traveling directly towards the observer, the general formula simplifies to [math]\displaystyle{ f_o = f_s \frac{\sqrt{1 + \frac{v}{c}}}{\sqrt{1 - \frac{v}{c}}} }[/math]. In the case that the traveling directly away from the observer, the general formula simplifies to [math]\displaystyle{ f_o = f_s \frac{\sqrt{1 - \frac{v}{c}}}{\sqrt{1 + \frac{v}{c}}} }[/math]

A Computational Model

This Desmos model represents the Relativistic Doppler Effect: https://www.desmos.com/calculator/9laabrg6qr.

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