Re: Einstein's Doppler equation wrong?

From: John Kennaugh (JKNG_at_kennaugh2435hex.freeserve.co.uk)
Date: 10/12/04 

Date: Tue, 12 Oct 2004 14:59:03 +0100

sal writes
>(I tried to send this earlier but my new server apparently ate it. So
>I took advantage of the delay to rework it a bit, and I'm posting it
>again -- maybe it'll actually go this time.)
>
>On Mon, 11 Oct 2004 19:47:32 +0100, John Kennaugh wrote:
>
>> Maybe this is something everyone knows except me (and Paul Anderson)
>> but here goes anyway.
>>
>> In ON THE ELECTRODYNAMICS OF MOVING BODIES Sect 7 Einstein says the
>> frequency of the light perceived by the observer is given by the
>> equation
>>
>> f' = f (1 - Cos(Fi)v/c)/Sqr(1-v^2/c^2) ---------------- [1]
>>
>> If the source is travelling directly to or directly from an observer
>> then Cos(Fi) = 1. Einstein states that this gives equation [2] Which
>> it does.
>>
>> f' = f (1 - v/c)/Sqr(1-v^2/c^2) ----------[1a]
>> f' = f Sqr[(1-v/c)^2 /(1-v^2/c^2)]
>> = f Sqr[(1-v/c)(1-v/c)/(1-v/c)(1+v/c)]
>> = f Sqr[(1-v/c)/(1+v/c) ---------------- [2]

>> I have shown the working because the equation which interests me is [1a]
>
>That equation is correct. Einstein's got at least one mistake in the
>1905 paper but it's not here; it's in section 10, which nobody cares
>about anymore anyway. But his derivation of the Doppler formula is a
>little confusing, I think.
>
>Below, I point out what I think the point of disagreement between you and
>E. is. But first here's a quickie "third derivation" of the formula for
>longitudinal doppler.
>
>The easiest way I know to see that E's formula is correct for
>longitudinal motion is to move into the frame of the emitter, moving
>in the +x direction at velocity v.

Relativity is based on a postulate which in affect says "the speed of
light is constant w.r.t the observer observing it." an observer in the
FoR of the emitter is therefore bound to see a different reality to that
of an observer moving w.r.t the source.

As far as I am concerned No 1 rule in relativity, at least for me, is to
make sure you know which FoR you are describing things from and stick to
it. I tend to distrust this kind of approach. It is the bit where having
derived something in one frame you say something similar to "..therefore
in the other FoR.....". I have seen too many people come unstuck that
way. I'm not saying you are wrong just that in taking this approach I am
not going to be sure you are right.

> Then the emitter sees the observer
>moving in the -x direction at v. Let's assume the observer is to the
>"left" of the emitter.

              O S

>
>In the emitter's frame, the wavecrests travel with frequency f and
>velocity c. In this frame, the receiver is running away from them,
>and the crests actually catch up to the receiver at a rate of
>f*(c-v)/c.
>
>In addition, the receiver's clock is running slow (in this frame!) by
>a ratio of 1/gamma.

Here is where my distrust surfaces because the light speed relative to
the observer is c-v in this frame, c in his, therefore I can't be sure
that anything derived in this frame will apply in the observers FoR.

> So, the frequency the receiver sees will be
>"boosted" by a factor of 1/(1/gamma), which is gamma. So overall, if
>we set c=1, the shift will be
>
> f' = f * gamma * (1-v) = f * (1-v)/sqrt(1-v^2)
>
>which is exactly what's in E's paper for this case.

>Now, on to the nit-picking mission!
>
>
>> We know that in acoustics you get Doppler if either the source is
>> moving or the observer is moving or both relative to the air, the
>> propagating medium. We are all familiar with the diagrams/equations
>> in the text book.
>>
>> In Relativity we have two complications, one is the assumed absence
>> of a propagating medium and the other is time dilation. Time
>> dilation is not strictly Doppler shift but never the less represents
>> a change in frequency due to speed so that if you are going to
>> produce an equation which tells you what the frequency is when the
>> source is moving it is necessary to include it. Time dilation makes
>> time intervals increase which is the same as making frequencies
>> reduce - frequency being the reciprocal of time.
>>
>> If we ignore the Doppler component then the frequency observed by an
>> inertial observer due to time dilation would be.
>>
>> f' = f Sqr(1-v^2/c^2) -------------- [3]
>>
>> where v is the velocity of the source relative to the observers FoR
>> independent of the direction of v.
>
>You seem to be saying the source is moving, and we're viewing this
>from the observer's frame.

Again the second postulate says the speed of light is constant w.r.t the
observer observing it so it is sensible, (for me at least) to look at
any problem in relativity from the observers PoV assuming the observer
is stationary. Speed is relative so I can legitimately do that.

> In that case, the source clock is slowed
>down by time dilation, and the "wavecrests" will come out less
>rapidly; the frequency would be f/gamma, which is what you've shown.
>So far so good.
>
>
>> If we try and derive from first principles the Doppler component
>> then the lack of a propagating medium is not the problem it might
>> seem. The second postulate says that the speed of light will be c
>> everywhere in the observers FoR. With or without a propagating
>> medium this is mathematically identical to what you would get if an
>> observer were stationary w.r.t a propagating medium so we know that
>> mathematically it is the same as Acoustic Doppler with the source
>> and not the observer moving and so is.
>>
>> f' = f (c/(c-v)) = f/(1-v/c) -------- [4]
>>
>> for a source moving towards the observer.
>
>Just for laughs, let's note that for a source moving _away_ at v,
>you'd get (setting c to 1 to save typing):
>
> f'' = f/(1 + v)
>
>> One can therefore combine these factors to get an equation for the
>> frequency observed when both affects are included so from [3] and
>> [4]:
>>
>> f' = f Sqr(1-v^2/c^2)/(1-v/c) ---------- [5]
>
>And if we look at a source moving away, rather than toward, the
>observer, we see (again with C set to 1):
>
> f'' = f * sqrt(1 - v^2)/(1+v)
>
>Multiply top and bottom by (1-v), and we get
>
> f'' = f * (sqrt(1-v^2) * (1-v)) / (1-v^2)
>
>divide top and bottom by sqrt(1-v^2) and we get
>
> f'' = f * (1-v)/sqrt(1-v^2)
>
>which is Einstein's formula.

Let me try that myself.
           f' = f Sqr(1-v^2/c^2)/(1-v/c) ---------- [5]

make v = -v
           f' = f Sqr(1-v^2/c^2)/(1+v/c) ---------- [5]
Take it all inside the Sqr

           f' = f Sqr[(1-v^2/c^2)/(1+v/c)^2]
              = f Sqr[(1-v/c)(1+v/c)/(1+v/c)^2]
              = f Sqr[(1-v/c)/(1+v/c)]
which is equation [2] which I know is the same as [1a] OK I agree.

Interesting. Thank you for that. I have always thought of the equation
as being made of two distinct components. Those components are obviously
related mathematically but then I recall something about Lorentz's
starting point (or one of them) being the Doppler equation so there is
perhaps an underlying relationship Doppler - LTs - Time dilation.

>
>> But if we now compare it with Einstein's then clearly something is
>> wrong:
>>
>> f' = f (1-v/c)/Sqr(1-v^2/c^2) ----------[1a]
>
>Yup. Einstein's got the source moving _away_ at v. You've got it
>moving _toward_ the observer at v.
>
>Set gamma to 1, and you'll see that Einstein's formula is for
>redshift, yours is for blueshift. The direction of motion is clearly
>different. It's a swapped sign, nothing more.
>
>Glancing again at section 7 it appears that E. rather annoyingly
>failed to say which direction his observer was moving in.

I noticed that. I would dock him a mark for that if I were you :o)

>
>
>> Clearly one is the inverse of the other. One would naturally assume
>> that Einstein was right and me wrong except for a couple of other
>> things which don't fit the first is Einstein's own statement
>> following [2].
>>
>> "We see that, in contrast with the customary view, when v = -c
>> f'=infinity."
>>
>> Time dilation when v = +/- c means that time has stopped, time intervals
>> are infinity and f' = 0
>>
>> The second comes if we return to Einstein's original equation.
>>
>> f' = f (1 - Cos(Fi)v/c)/Sqr(1-v^2/c^2) ---------------- [1]
>>
>> and now make Fi = 90deg Cos(fi) = 0. This gives:
>>
>> f' = f /Sqr(1-v^2/c^2) ---------------- [6]
>>
>> This is the condition for what is described as transverse
>> Doppler. But we know that that is just time dilation. Again it is
>> the wrong way up and we know it should be as per equation [3]:
>>
>> f' = f Sqr(1-v^2/c^2) ---------------- [3]
>>
>> Have I have missed something or did Einstein have it wrong?

>
>I'm not going to dig around in the transverse Doppler formula tonight,
>but I will point out that it's _not_ just time dilation. Been there,
>done that; chased my tail quite a while before I finally "got it". The
>key is in the "infinitely distant" clause in his description of the setup.
>If the photon is received when the line between source and observer is at
>90 degrees to each, then it must have been emitted much earlier -- when
>the angle between them was quite different!

If it is an infinite distance then the angle between them when it was
emitted is indeterminate surely.

>
>Keep that in mind, work through it again, and I think you'll find it
>comes out as he said.

He defines his co-ordinates as follows:

"the connecting line ``source-observer'' makes the angle Fi with the
velocity of the observer referred to a system of co-ordinates which is
at rest relatively to the source of light"

The infinite distance is surely to avoid complications with parallax,
light coming from a fixed direction.

"a system of co-ordinates which is at rest relatively to the source" to
avoid aberration complications. To make Fi the real angle between the
direction of the light and the direction of the observer.

I agree it is not simple. I for one would not like to argue about any
angle of Fi other than 0 and 90 because surely one of the affects of
length dilation is that an angle in one FoR changes when viewed in
another and I am not sure in which FoR Einstein has defined the angle.
Apparently in the source's yet he says it is what the observer would
measure.

  Let us just say that there is an awful lot of room for confusion.

Thank you for taking the time. Very interesting. We have two equations
which look entirely different and you have shown that in reality they
only differ in which direction you call positive which is arbitrary
anyway. I am never the less gratified that my derivation from 'first
principles' did give a valid solution and I did define the direction of
v.

My head hurts :o)

If you have time I would be interested in why transverse Doppler is
"_not_ just time dilation". 

-- 
John Kennaugh
to email convert the number from hex to decimal

Relevant Pages

  • Re: "The Impossibility of Measuring the Velocity of Light"
    ... it but the speed of the observer would. ... with relativity it supports relativity while in fact the simplest ... "3.3 Tests of Light Speed from Moving Sources". ... to say "they travel in all possible directions at all possible speeds, ...
    (sci.physics.relativity)
  • Re: The reasons why SR is an aether theory
    ... B moving at v towards ... at two different speeds to go to A and B respectively through the same ... Relativity says that in B's ... Some text books are confusing in that they describe light leaving a source from the PoV of the source's FoR and then describing it arriving at the observer from the PoV of the observers FoR. ...
    (sci.physics.relativity)
  • Re: Relativity and Thermodynamics
    ... In a moving reference frame, a container of gas will appear ... velocity and hence a lower temperature. ... relativity, the two systems are moving at a constant velocity. ... the velocity of the observer. ...
    (sci.physics)
  • Re: Read this one, the other is a mess! Re: Is the speed of light really constant ?
    ... the questions I was asking concerning the doppler shifts. ... If an ultrasonic wave is reflected off a moving obstacle, ... Relativity. ... the observer), then the stable properties required for the ...
    (sci.physics.relativity)
  • Re: SR fundamental contradiction
    ... and the principle of relativity. ... Your transformations cannot be derived from any consistent ... What my transformation predicts is that a moving stick is contracted ... Each observer must consistently use his own ...
    (sci.physics.relativity)