Re: Einstein's Doppler equation wrong?

From: V ertner Vergon (vergon_enterprises_at_highstream.net)
Date: 11/04/04 

Date: 3 Nov 2004 20:39:27 -0800

John Kennaugh <JKNG@kennaugh2435hex.freeserve.co.uk> wrote in message news:<DdMEioHEVtaBFwf4@kennaugh2435hex.freeserve.co.uk>...
> 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.

Vergon:

Here's your mistake. Directly to is 180 deg, the cos is -1
                     Directly from is 0 deg, the cos is 1

So that affects your equation [2] below

> 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]

Your [2] is for directly from, i.,e, directly receding.
For direct approach you need to reverse the signs:
sqrt[(1+v/c)/(1-v/c)]f

 
> I have shown the working because the equation which interests me is [1a]
>
> 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.

Vergon:

Sorry, but the change in frequency is NOT due to time dilation but due
to Doppler. Following is an excerpt from a monograph currenly on this
NG.

It shows that time dilation does not exist -- that what is observed in
the night sky by astronomers is Doppler time.

You will notice that the transverse Doppler time is exactly the same
as that given for time dilation. It's just a coincidence. One of
Mother Nature's dirty tricks. :-)
******************

"Next, we examine a counter argument which states that the
relativistic Doppler rate is the result of the following: If one takes
the non-relativistic Doppler rate and modifies it by the time dilation
they, come up with the relativistic Doppler rate. This, supposedly,
confirms time dilation. Upon further examination, we perceive that
even if we were to accept that explanation, what we have is the
situation whereby an approaching clock is observed to run fast -- not
as fast as non-relativistic mechanics would have -- but fast. This is
contrary to time dilation which requires an approaching clock to run
slowly. That a reversal in direction results in a reversal of time
rate is to be expected since time variation is the result of velocity
vectors – and vectors are directional.

If a theory conflicts with empiricism, it has to be discarded.

Next we point out that we need not necessarily accept the above
explanation. We note that above, the non-relativistic Doppler is
modified by applying time dilation. We also note that time dilation is
written t' = t sqrt(1 – v2/c2), where sqrt(1 – v2/c2) is the
Lorentz transformation.

Now non-relativistic Doppler in direct approach is written (where V =
v/c and nu is frequency).
nu'/nu = 1/1- V

Next, we apply time dilation:
nu'/nu = (1/ 1 – V) x t sqrt1 – V2).

It is immediately apparent how clumsy this equation is. Frequency
times time is n/t x t = n (where n is a number) To avoid this problem
we may write:
nu'/nu = (1/1 – V) sqrt(1 – V2)

Now the argument presents itself that the equation immediately above
does not represent the application of time dilation but does instead
simply apply the Lorentz transformation to the frequency as it does to
other parameters in transposing from one coordinate system to another.
Thus relativistic Doppler is simply that – and not modified by time."
*************************************************

> 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.
>
> 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.
>
> 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]
>
> 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]
>
> 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?

Vergon:

Einstein has it wrong. I wrote a paper on that 17 years ago. The
reason he had it wrong was because he was confused as to the
directions from which the angle measurements were made. It's a long
story -- but briefly the 90 deg observation of the incoming ray (s')
has to be between the velocity vector and the direction of the
incoming ray s',O.

Since the s',O (s' being the forward observed position of the
source)axis is slanted forward that means the 90 deg velocity vector
has to be slanted downward. And that means it is receding from S. In
other words the velocity vector is no longer orthogonal to the s,O
axis.

The corrected measure gives f' = f sqrt(1-v^2/c^2)for 90 deg.

For all velocities:

                        f' f sqrt(1-v^2/c^2)/ 1+cos fi * v/c

If you will look in the books you will find they have quietly patched
up Al's boo boo. In part he calculated 0 deg as direcly behind the
vector instead of directly away from the source, S.

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