Re: Why Einstein is the founder of special relativity

Continuing discussion of :
http://www.soso.ch/wissen/hist/SRT/P-1908.pdf

"Martin Ouwehand" <see.URL@xxxxxxxxxxxxxx> wrote in message
news:4337fbc2$1@xxxxxxxxxxxxxxxxxxx
> In article <43328457$1@xxxxxxxxxxxxxxxx>, I wrote:
>
> ] I understand from his arguments around pages 565-566
> ] that he uses the following transformation:
> ]
> ] x' = (x - v * t) / sqrt(1 - v^2/c^2) (Lorentz
contraction)
> ] t' = t - (v * x) / (c^2 - v^2) ("local time")
>
> I made a mistake (it should be x-vt in the equation for t'), the full
> transformation between ether frame (unprimed [x, y, t]) and moving frame
> (primed [x', y', t']) is:
>
> x' = (x - vt) / sqrt(1 - v^2/c^2)
> y' = y
> t' = t - ((v * (x - vt)) / (c^2 - v^2))
> = (t - vx/c^2) / (1 - v^2/c^2)

Dear Martin, Poincare certainly didn't abandon the Lorentz transformations.

And pasting here what you concluded with:

> Note that Einstein uses the same synchronisation procedure as
> Poincaré (who must indeed be credited for this nice idea) but
> comes to different conclusions because he postulates that the
> speed of light is the same in all inertial frames.

Note also that Einstein postulated what Poincare had concluded, so that they
could not disagree about that.

According to Poincare, the full transformation between the measured values
in any moving inertial frame and in the ether frame as well as between those
in any set of inertial frames is:

x' = (x + vt) / sqrt(1 - v^2/V^2)
y' = y
z' = z
t' = (t + vx/V^2) / sqrt(1 - v^2/V^2)

The sign depends on the choice of which coordinates one chooses as the
primed ones.
Thus, according to your interpretation, there is an error in the time
calculation of his M-M example. Logically, any interpretation of
explanations by Poincare in the context of Lorentz' theory that disagrees
with the LT must be either due to an interpretation error or an error in the
text.

In order to not contradict his Lorentz transformation, tau must be equal to
t*sqrt(1-v^2/V^2), so that he had to end with the last equation just as it
appeared in print (BTW, I misread that the first time):

AB = V*t*sqrt(1-v^2/V^2)

>>From that he concluded that Michelson's experiment will yield an *apparent
transmission duration* that is proportional to the apparent distance
(independent of the angle).

I'll now follow you and also have a closer look at it, to try to spot where
the error is. See below.

> Let me show how this is consistent with what Poincaré says on pages
565-566
> of the article http://www.soso.ch/wissen/hist/SRT/P-1908.pdf. He considers
> two events: A (emission of a light wave) and B (detection of the light
> on an arbitrary point of the wave front at some later time).

I read: the light goes "de B en A", which I think can only mean from B to
A.

Also, I had some difficulty with:

"We choose the contraction law, such that the point S is at the base of the
meridian section of the ellipse."

It's in cursive, thus important.
I think that he meant that we shift the local time such that the contraction
ellipse is apparently centered around the source.
Then, the detector A is located at the source S, and we look at the light
that returns from the apparent position of B back to A.

Note that this apparent position is due to length contraction and time
offset, while still using absolute time.
Apparently, he wanted to illustrate local time measurement by at first not
accounting for it - IMO a bit cumbersome, but typical for him. As he stated:

"How will we then proceed, to evaluate the time that the light takes to go
from B to A?"

Thus IMO he does not, at that point, use his Lorentz transformations, but
instead he illustrates the consequences for local time measurement, taking
the well-known assumption of length contraction for granted.

IOW, fig.2 sketches only half of the Lorentz transformation, as he uses
local (apparent) coordinates A and B with the true duration t (watch out:
this "t" is *not* a time coordinate; it corresponds to duration delta_t of
the Lorentz transformations!).

That is confirmed by his second equation on p.566:

AB + v/V *AB' = V*t*sqrt(1-v^2/c^2)

> Let
> the space-time coordinates of A and B in the ether frame be:
>
> x_A = 0
> y_A = 0
> t_A = 0
> and
> x_B = ct cos(a)
> y_B = ct sin(a)
> t_B = t
>
> (the angle a is arbitrary and serves as parameter to denote any
> point on the wave front.)
>
> With the help of the above formulas, I find that the space-time
> coordinates in the moving frame are:
>
> x_A' = 0
> y_A' = 0
> t_A' = 0
> and
> x_B' = (ct cos(a) - vt) / sqrt(1 - v^2/c^2)
> y_B' = ct sin(a)
> t_B' = (t - (vct cos(a) /c^2)) / (1 - v^2/c^2)
>
> Then I can check that the circular (in the ether frame) wave
> front goes into an ellipse
>
> (x_B' * sqrt(1 - v^2/c^2) + vt)^2 + y_B'^2 = (ct)^2
>
> with ellipcity e = v/c and center [-vt * sqrt(1 - v^2/c^2), 0],
> corresponding to the picture on page 566.
>
> Next I compute Poincaré's AB and AB':
>
> AB = sqrt(x_B'^2 + y_B'^2) = t * (c - v * cos(a)) / sqrt(1 -
v^2/c^2)
> AB' = x_B' = t * (c cos(a) - v)/ sqrt(1 - v^2/c^2)
>
> and check that indeed:
>
> AB + v/c * AB' = ct * sqrt(1 - v^2/c^2)
>
> which is the second equation on page 566.
>
> Then I can find the
> next-to-last equation for the local time by rewriting:
>
> tau = t_B'
> = (t - (vct cos(a)/c^2)) / (1 - v^2/c^2)
> = (t - t* v^2/c^2 + t * v^2/c^2 - (vct cos(a)/c^2))/(1 -
v^2/c^2)
> = t - ((ct cos(a) - vt)*v/c^2)/(1 - v^2/c^2)
> = t - ((AB' v/c) / c sqrt(1 - v^2/c^2))
>
> Finally, it's easy to check that:
>
> AB = c tau sqrt(1 - v^2/c^2)
>
> which is the last equation (with t corrected to tau -- that this is
> a misprint can be seen by comparing the next-to-last and the
> second equation).

You overlooked that that would not only contradict everything he said before
but even his conclusion in the next paragraph.

Now I'll do a first attempt to finish following Poincare's reasoning,
instead of yours:

The direction of motion is along PP'.
His first equation is trigonometry with v/V=e, the excentricity.

Next he states that, as there is no effect on lateral dimensions,

OQ = Vt

I first didn't spot his mistake, but here it happened!
Ironically, it's the same mistake that Michelson made in 1881 and that
Lorentz corrected before Michelson repeated it with Morley.
He should have stated:

OQ = Vt *sqrt(1-v^2/V^2)

Overlooking that, follows his second equation,

AB + e *AB' = V*t*sqrt(1-v^2/V^2)

with AB' = x'_B - x'_A
(x' = moving frame coordinates)

It should have been:

AB + e *AB' = V*t*(1-v^2/V^2)

He states in words (no symbols, so I generate them here but it's
inconvenient as the symbol t is already taken, leading to confusing
notation - which is no doubt why he skipped it):

Supposing that the difference between true and local time at a point in time
and space depends on the location of the clock along PP' as well as on a
constant that is a function of the speed, as follows:

tau_p - t_p = x'_p * C

With tau_p and t_p indicating instances,
and C = e/(V*sqrt(1-v^2/V^2))

(Note: here I corrected what looks like sloppy phrasing, as from his words I
first guessed that he meant "t_p - tau_p", but then the sign doesn't come
out right. Thus that would be a little glitch on his part.)

>>From that the apparent duration of the transmission from B to A is as
follows:

tau_A - t_A = x'_A * C
tau_B - t_B = x'_B * C
-------------------------- -
tau_A - tau_B - t_A + t_B = (x'_A - x'_B) * C

tau_A - tau_B = t_A - t_B + (x'_A - x'_B ) * C

So that, going back to his notation,

tau = t - AB' * C

He next claims that from that follows that :

AB = V*t*sqrt(1-v^2/V^2)

However, this is in direct contradiction with his second equation, due to
his mistake in that second equation.
Perhaps he was so sure that it would work out (as he knew that it should),
that he didn't bother to verify it.

Concistency check (I won't bother to derive it):

Taking the correct equations and writing g for 1/sqrt(1-v^2/c^2),

AB + e *AB' = V*t/g^2 }
t = tau + e* AB' /(V/g) }

AB + e *AB' = V*t/g^2 }
t *(V/g) = tau*(V/g) + e*AB' }
----------------------------------- -
AB - t *(V/g) = V*t/g^2 - tau * (V/g)

With tau = t/g we get:

AB - t *(V/g) = V*t/g^2 - t/g * (V/g)
AB - t *(V/g) = 0

So that indeed:

AB = V* t/g = V*tau

Which is what he attempted to show.

> As AB is the distance travelled by the ligth wave and tau the time of
> travel, *as measured by an observer in the moving frame*, I
> deduce that the speed of light for that observer is:
>
> c' = AB / tau = c sqrt(1 - v^2/c^2)
>
> While it is true that "you don't find such a nonsense" (as Homo
> Lykos says) in the article, it's just a division away :-/
>
> Now I'd like to follow-up to some remarks by Harry and Homo Lykos.
>
> In article <43328457$1@xxxxxxxxxxxxxxxx>,
> Harry <harald.vanlintel@xxxxxxx> writes:
>
> ] > you seem to be saying that after discovering Special Relativity in
1905,
> ] > he changed his mind... It is easy to see that what is missing in his
1908
> ] > article is time dilatation:
> ]
> ] I am flabbergasted - "time dilatation" is represented by the tau on page
> ] 566.
>
> there is "time delay" but no time dilatation in the equation linking t and
> tau: for an observer at rest in the moving frame (hence: constant AB')
> a phenomenon won't start at the same time as for an observer in the ether
> frame (because of the second term -- that's time delay), but its
*duration*
> will be the same (t and tau come with the same unit factor in the
equation.)

Many people, incl. Poincare, sometimes use t for duration and sometimes for
point in time. IMO that's not good practice, as some may overlook it, as now
happend to you:

"t [étant] la durée de transmission".

Thus you were right to state that tau is the "time of travel" in the moving
frame - that means, duration.

As he stated here in two equations, the apparent local duration in the
moving frame is less than the duration as determined in rest. That happens
to be called "time dilation".

> Hence, in Poincaré's theory time durations have an absolute
> meaning. Not so in Einstein's theory.

SNIP

It's a pity that he messed up, as one rarely sees an example that neatly
shows the invariance of "light speed" for any angle.

Best regards,
Harald


.

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