Is Relativity the Key to Understanding Doppler Shift on a Moving Train?

[MisterBig]

I’m on a train sitting next to an open window - it’s a bit stuffy - traveling towards a station. A siren is sounding at the station. As I travel toward the siren its pitch is higher that it would be if I were stood on the platform. Once I have passed the siren and I’m moving away, its pitch becomes lower.

The reason for this change in pitch, as I understand it, is that as I travel toward the source of the sound it could be viewed that I am stationary but the speed of the siren’s sound has increased by my speed toward the siren. More cycles per second enter my ear (although the wavelength remains the same) and so the pitch of the sound increases.

Now if there was a red flashing light on this siren and the train was fast enough (and I was willing to stick my head out of the window of a train traveling at a significant potion of the speed of light) I would also see a frequency increase on the siren’s flashing light. However relativity tells me that I would also measure the speed of that light to be the same as if the train were stationery.

Why is this?

 

[Jonathan]

Time and space warp so that the math ends up working out.

 

[HallsofIvy]

"Time and space warp so that the math ends up working out."

LOL. In other words, "that's just the way it is"!

 

[Jonathan]

no, lol...

...that's not just the way it is, my anwser is the basic concept of how it is!

 

[quartodeciman]

"The wavelength {of sound} remains the same" is granted because it is assumed that the speed of the train is too small compared to the speed of light to make a significant difference.

Let's look at the red light.

Assign the following literals:

l := wavelength from the station platform
l' := wavelength from the train
n := wavelength from the station platform
n' := wavelength from the train

The two wavelengths are corresponding space displacements along an x axis. To make things simpler, use waveperiod instead of frequency.
Waveperiod is time per cycle instead of cycles per time. So, they are mutual reciprocals.

T := waveperiod from the station platform
T' := waveperiod from the train

and

T = 1/n
T' = 1/n'.

Since the wavelengths are space displacements along an x-axis and the waveperiods are time differences, lorentz transformation equations from relativity can be used for them.

Set g := 1/(1 - v²/c²)^.5, with v the speed of the train and c the speed of light on the train platform.

Use x = l, x' = l, t = T, t' = T' in the lorentz transformation equations and produce

l' = g(l - vT)
T' = g(T - vl/c²)

Now, l is a wavelength of light, so the speed ought to be c. So the relationship between wavelength and waveperiod is simple:

l = cT.

Substitute and simplify.

l' = g(cT - vT)
T' = g(T - vcT/c²)

l' = g(c - v)T
T' = g(1 - v/c)T

Now for the finale. Multiply the second of these equations by c.

cT' = gc(1 - v/c)T
cT' = g(c - v)T

Substitute from the first of the equations.

cT' = l'
l'/T' = c

That shows the speed of light from the train, and it is the same as from the station platform. That ought to be no surprise, since the lorentz transformation equations were made to render the speed of light invariant to transformations. It is just nice to see at the end.

Just for closure, let's make a transformation equation for the frequencies. Since they are reciprocals of the waveperiods of the light, just substitute.

1/T' = 1/(g(1 - v/c)T)
n' = n/(g((1 - v/c))

Use a tad more algebra.

n' = n(1 + v/c)^.5/(1 - v/c)^.5

*****

 

[zoobyshoe]

There may be some confusion here.

The rise in the pitch of the siren
is not caused by a virtual increa-
se in the speed of sound. If you
were to be able to measure the
speed of sound via Lorentz trans-
formations you would find it had-
nt changed.

What has changed is the frequency.
It is, in fact, accelerating as
you approach the station. The
frequency of the sound and the speed of sound aren't the same.

For the same reason the speed of
light doesn't change but if the
train were going fast enough the
frequency of the light would
change-it would shift toward the
blue frequencies.

Zoob

 

[Loren Booda]

"c": more a standard than an absolute

The speed of light is the same in all reference frames not so much because we measure it thus, but because with all measurements we utilize light (E-M waves) to define observable quantities, uncertainty and spacetime. This is why the most accurate and stable standards of physical parameters are the Planck units, based soley upon radiative constants like c.

If we experienced our physical system according to pervasive tachyons traveling at speed v>c, then we would perceive their velocity as a constant. (One could also consider a "speed of cognition" or "speed of socialization," although variable and small, setting an upper limit on the systems of the brain or interrelationships and establish a relativity of intra- or interpersonality.)

 

[jcsd]

Zooby Shoes, Lorentz transformations are nothing to do with sound.

Quite frankly for the doppler effect you needen't consider relativity at all because your only concerned with how the stationery observer views things. Just as in Newtonian physics the staionery observer would not measure a change in the speed of light, but a change in the wavelength, in relativity from his reference framespeed of light wouldn';t cahnge (as it is invariant in all refernce frames) but the wavelemgth would.

 

[zoobyshoe]

jcsd,

I understand Lorentz transformat-
ions have nothing to do with
sound. I was suggesting a fiction-
al use for them. Any fiction that
gives MisterBig a way to measure
the speed of the sound he is
hearing from the moving train
could have been plugged in there.

MisterBig said "...the speed of the siren's sound has increased by
my speed toward the siren..."

I percieve him to be confused
about this. The speed of the
sound has not increased. The
frequency has increased.

-Zoob

 

[Jonathan]

Exactly, the frequency was artificially increased, which is the equivalent of a virtual decrease in the speed of sound.

 

[chroot]

Originally posted by Loren Booda
The speed of light is the same in all reference frames not so much because we measure it thus, but because with all measurements we utilize light (E-M waves)... If we experienced our physical system according to pervasive tachyons traveling at speed v>c, then we would perceive their velocity as a constant.

It is disconcerting to me that you post frequently with a tone of authority, yet do not know your ass from a hole in the ground.

- Warren

 

[russ_watters]

Originally posted by Loren Booda
The speed of light is the same in all reference frames not so much because we measure it thus, but because with all measurements we utilize light (E-M waves) to define observable quantities, uncertainty and spacetime.

Wow. I won't be so blunt as Warren, but that represents a severe misunderstanding of Relativity.

 

[Loren Booda]

Forgive my speculation, for that is all it is. What one needs to disprove my statements is physical argument, not personal commentary.

 

[arcnets]

Originally posted by MisterBig
I would also see a frequency increase on the siren’s flashing light.

Correct, I think.

However relativity tells me that I would also measure the speed of that light to be the same as if the train were stationery.

Why is this?

Let me try and give a simple answer to this.

1) Special relativity (which is, basically, the consequence of c being constant) says that the observed frequency is lower than the original one, because of time dilatation.
2) Doppler effect says that the observed frequency is higher than the original one, because you are approaching the source.

If you do the math (see quartodeciman's post), you get the so-called relativistic Doppler effect. It turns out that 'Doppler always wins over time-dilatation' (Please don't bash me anyone because I put it like that. I think it's a good way to memorize this.) Meaning, you always get blue-shift when approaching.

Some more aspects:
- In nonrelativistic Doppler effect, it makes a difference whether the source or the observer is at rest relative to the medium. In relativistic D.e., it doesn't. So, from observing a r.D.e., you can tell nothing about the medium. Well - SR states that there is no medium (ether) at all...won't go into this, now...
- Since you have also lenght-contraction, the observed wavelength is also smaller than the original one. The factor is the same, so that c remains constant.

OK?

 

[Loren Booda]

That the speed of light is constant in vacuum in all reference frames is a generally accepted postulate (which I too believe is a correct statement) of relativity that has been regurgitated by most every physicist since Einstein. I attempted to support it by explaining possible physics behind the assumption.

This is a case of convention (a postulate) vs creativity (a possible proof). Do all of you believe the postulates of relativity to be absolute and unprovable? Can you not differentiate here between accepted doctrine and potentially constructive ingenuity?

Maybe I should try in the future to identify all whimsical statements as such, lest Physics Forums become a specter of imagination.

 

[MisterBig]

Thanks for all the responses.

Originally posted by Jonathan
Time and space warp so that the math ends up working out.

At first I was disapointed with this answer but now I see it was for the best

If I understand this correctly the basic answer to my question seems to be that wavelength is decreased due to length contraction and since c is always constant this means that f had to increase.

This doesn't sit that well with me. I would expect to observe the station platform as contracted but would have imagined light it has emitted to be immune from the effect (what with it already traveling at c).

 

[quartodeciman]

I would expect to observe the station platform as contracted but would have imagined light it has emitted to be immune from the effect (what with it already traveling at c).

Actually, the contraction of a body is computed based on observation, but the actual appearance of the body is subject to distortions, including rotation. James Terrell published this over 40 years ago. Here is a link to a writeup:

http://www.math.ubc.ca/people/faculty/cass/courses/m309-01a/cook/index.html

---
[AN EXTRA BIT]

Ms. Cook writes in a note: "It would be nice if there were a more pictorial way to derive the Lorentz transformation, instead of relying on the algebra."
Then she follows this with even more obscure-sounding mathematical suggestions than just plain algebra. :)

In fact, there are more pictorial ways to derive Lorentz transformations than the straight algebra method. But first, the value of using straight algebra is that you can derive LT easily as a generalization of the Galilean transformation (GT) by the old time-tested method of introducing another variable, (g), or several variables, then forcing the resultant equations to preserve relativity of motion and the invariance of lightspeed (c), then solving for the new unknown(s).

But some alternative methods allow more graphing. Unfortunately, I am inept at using drawing tools. These methods are:

(1) Langevin/Tolman light clock method: this is really just an idealization of the transverse piece of the Michelson-Morley apparatus. Since c is believed to be invariant, it comprises an ideal perfect clock (as long as its length doesn't change). A lot of modern physics books use this method to derive time dilation and length contraction. Tolman wrote a book back around 1912 that took it all the way through the Lorentz transformation equations. You do have to believe that the device oriented at 90º has the same length for the two observers. It is best used not as a clock, but as an demonstration apparatus supplied with its own ordinary stationary clock.

(2)Milne/Bondi Radar method: This uses lightsignals bouncing back and forth between observers (or observer platforms). It is especially good at time dilation, relativistic doppler shifts and the twin problem. The idea is exploited in Hermann Bondi's book, "Relativity and Common Sense: A New Approach to Einstein".

(3)Electrodynamic method: I have seen bits of this here and there, but not a complete presentation. It uses some of the classical components that went into the Maxwell-Lorentz formulation of electromagnetic laws. For instance, two close parallel currents are described, using Oersted's law and the Lorentz force law. The contraction factor (g) becomes a necessity in order to avoid inconsistent results in computing the net effect of the given electrostatic forces and the induced Lorentz forces.

None of these will satisfy a really math-sharp physics student, but they might help the rest of us.

---

 

[russ_watters]

Originally posted by Loren Booda
That the speed of light is constant in vacuum in all reference frames is a generally accepted postulate (which I too believe is a correct statement) of relativity that has been regurgitated by most every physicist since Einstein. I attempted to support it by explaining possible physics behind the assumption.

Its not a postulate (assumption). It is experimental data first recorded in the Michelson/Morley experiment.

 

[arcnets]

Originally posted by MisterBig
I would expect to observe the station platform as contracted but would have imagined light it has emitted to be immune from the effect

Imagine you have 2 mirrors on the platform, and standing waves between them. You must, I think, admit that the wave is contracted as the platform is. (This is, basically, Michelson-Morley).

 

[Loren Booda]

R. W., do a google on "postulates of relativity". I'll try not to be blunt.