QUESTION: Relativistic Doppler Effect & Special Relativity

[Shayne T]

Hey all,

I was just wondering something which I couldn't get my head around tonight. If special relativity states that the speed of light is constant for all observers, regardless of the reference frame of any observer, then how is it possible for light to Doppler shift? Wouldn't the frequency be unaffected by the rules special relativity laid out? I am probably just missing something here, so any knowledge or theories on this would be much appreciated!

Cheers

 

[PeterDonis]

Hi, Shayne T, welcome to PF!

If special relativity states that the speed of light is constant for all observers, regardless of the reference frame of any observer, then how is it possible for light to Doppler shift?

Frequency is not the same as speed. That can be seen from the fact that we can find various light sources that emit light of different frequencies, but all of that light travels at the same speed (assume that I set things up so all of the sources are at rest relative to me, so there are no other effects such as Doppler involved).

The Wikipedia page on the relativistic Doppler effect explains the basics reasonably well and has some decent visualizations:

http://en.wikipedia.org/wiki/Relativistic_Doppler_effect

 

[Shayne T]

Thanks Peter,

I figured that the answer was related to frequency not being the same as speed. But what I still don't fully understand, is how the light shifts frequencies (blue shift or red shift) based on the relative motion of the source and observer. If light is the same speed for all observers, then wouldn't that mean that the lightwaves are still spaced out by the same amount, regardless if its moving towards or away from you? Or are there just more or less lightwaves, bunched up within the same beam of light traveling at the constant c? Thus affecting frequency.

Doppler effect of sound is just keeping me in this mindset. Because I understand that for sound, when the source physically moves, it bunches up the sound waves in front of it by "catching up" to its previously emitted. I just don't understand what causes this frequency change in light due to relative motion. Because if light is the same speed for all, then how can one do any sort of "catching up" in order to cause the observed frequency change?

 

[PeterDonis]

I understand that for sound, when the source physically moves, it bunches up the sound waves in front of it by "catching up" to its previously emitted. I just don't understand what causes this frequency change in light due to relative motion. Because if light is the same speed for all, then how can one do any sort of "catching up" in order to cause the observed frequency change?

The basic Doppler effect for light works the same as for sound. The sound waves all move at the same speed, just as the light waves do: the light source "catches up" between wave fronts if it is moving towards you, just as the sound source does.

(The speed of the sound waves will change if the speed of the *air*--the medium in which the sound propagates--changes, relative to you--for example, if there's a wind blowing. But the speed of the *source* relative to you doesn't affect the speed of sound, if the air is at rest relative to you. The difference between light and sound is that there is no "medium" for light corresponding to the air for sound, so there's no way for "motion relative to the medium" to change the speed of light.)

The *relativistic* Doppler effect for light does have some extra complications, compared to the Doppler effect for sound, but you don't need to understand those to understand the basic mechanism.

 

[Shayne T]

Thanks again!

I think I understand how this frequency change is possible now, even with special relativity at play. I just figured one would not be able to catch up to something that is emitted at a constant speed, for whatever reason, call it a brainfart. But now I can clearly understand and visualize a moving object catching up to its light waves causing the shift. I guess special relativity would restrict light from causing a "sonic boom" like sound does, because one wouldn't be able to totally surpass their emitted light waves.

One last question though! At what speeds is a red or blue shift noticeable? Assuming an the observed object is large enough to see clearly through a telescope. Would 0.5C cause a visible shift? Or are speeds closer to C required to determine this?

 

[WannabeNewton]

Keep in mind that you can't catch up with a light wave; you can't travel at the speed of light. All that's happening is there are more surfaces of constant phase hitting you per unit time.

 

[PeterDonis]

I can clearly understand and visualize a moving object catching up to its light waves causing the shift.

Ok, good--but bear in mind that this analysis is being done in your (the observer's) reference frame, *not* the reference frame of the source. A key difference between light and sound is that, for sound, if you look at things in the reference frame of the source, the speed of the sound waves *changes* (because the source is moving relative to the air); but for light, if you look at things in the reference frame of the source, the speed of the light waves is the same. That's really what "the speed of light is constant for all observers" means.

I guess special relativity would restrict light from causing a "sonic boom" like sound does, because one wouldn't be able to totally surpass their emitted light waves.

Not in vacuum, no. But in a material medium, it is possible for a light source to travel faster than the light it emits. The resulting analogue to the "sonic boom" for light is called Cherenkov radiation; you can read about it at these links:

http://en.wikipedia.org/wiki/Cherenkov_radiation

http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/cherenkov.html

At what speeds is a red or blue shift noticeable? Assuming an the observed object is large enough to see clearly through a telescope. Would 0.5C cause a visible shift? Or are speeds closer to C required to determine this?

It depends on what instruments you are using. We can detect very small Doppler shifts with the right instruments, corresponding to relative velocities on the order of meters per second. (For example, the radar speed detectors that police use measure the small Doppler shift associated with your car's speed on the highway.)

 

[Shayne T]

but for light, if you look at things in the reference frame of the source, the speed of the light waves is the same. That's really what "the speed of light is constant for all observers" means.

Not sure if i quoted correctly. So that means, for instance, if your in a ship going 0.999C, and you turn on your headlights, the light beam would still emit in front of you at the full magnitude of C? And if you were an observer watching the ship from a stationary position, on the ground let's assume (and also assume you always have clear visual contact of the ship that's going 0.999C) and he turned on the headlights, the ground observer would only see the light beam travel at 0.001C in front of the ship?

 

[PeterDonis]

if your in a ship going 0.999C, and you turn on your headlights, the light beam would still emit in front of you at the full magnitude of C?

Relative to the ship, yes.

And if you were an observer watching the ship from a stationary position, on the ground let's assume (and also assume you always have clear visual contact of the ship that's going 0.999C) and he turned on the headlights, the ground observer would only see the light beam travel at 0.001C in front of the ship?

Relative to the ground, yes. But these are two different observations; the second observation is of how much the light beam gains on the ship, as seen from the ground. In relativity, that's not the same as the velocity of the beam relative to the ship, because velocities don't add linearly in relativity.

 

[Nugatory]

So that means, for instance, if your in a ship going 0.999C relative to the ground observer, and you turn on your headlights, the light beam would still emit in front of you at the full magnitude of C relative to you? And if you were an observer watching the ship from a stationary position, on the ground let's assume (and also assume you always have clear visual contact of the ship that's going 0.999C) and he turned on the headlights, the ground observer would only see the light beam travel at 0.001C in front of the ship but still C relative to the ground observer?

Yes, with the clarifications that I added in bold to avoid any ambiguity - remember, speeds are only meaningful relative to something else.

There's a general formula here. If you're moving at speed $u$ relative to me, and you fire a missile at speed $v$, I will measure its speed as $\frac{u+v}{1+uv}$ (measuring speed in units of light-seconds per second so that $c=1$). If $u$ and $v$ are both small compared with the speed of light this formula reduces to the $(u+v)$ that you'd expect from your experience in day-to-day life; if either is equal to $c$ that's what the result is no matter the value of the other.