The symbol W' has been removed in this draft. It has been replaced
with the symbol s to eliminate the redundancy found in the previous
versions.
____________
Fizeau's results are correctly accounted for as such:
Attachment 1: Fizeau: Table of Comparisons
In reference to Fizeau's experimental results of the speed of light in
moving media:
Following is a table of results obtained from three various equations
using a detector at rest wrt the Earth, speed of water measured wrt
Earth, and
sodium light.
n = 1.332 c = 3e8 w = c/n = 225225225.225
Light and media moving in the same direction wrt detector.
Fizeau W = w + v(1 - 1/n^2)
Galilean W = v + w(c^2 - vc)/(vw + c^2 - vc)
= v + c(c - v)/(v + n(c - v))
Relativistic W = (v + w)/(1 + vw/c^2)
= (v + c/n)/(1 + v/nc)
All speeds below are in m/s
_____________________________________________________________
Water W-Fizeau W-Galilean W-Relativistic
1m/s 225225225.6616m/s 225225225.6616m/s 225225225.6616m/s
2 225225226.098 225225226.098 225225226.098
3 225225226.5343 225225226.5343 225225226.5343
4 225225226.9707 225225226.9707 225225226.9707
5 225225227.4071 225225227.4071 225225227.4071
6 225225227.8435 225225227.8435 225225227.8435
7 225225228.2798 225225228.2798 225225228.2798
8 225225228.7162 225225228.7162 225225228.7162
9 225225229.1526 225225229.1526 225225229.1526
10 225225229.589 225225229.589 225225229.589
Interesting outcome, to say the least.
The three equations diverge at very high values of v, but the Galilean
and Relativistic equations both give W = c when v = c. OTOH wrt these
two, ironically, only the Galilean derivation holds the speed of light
to maintain its speed of c wrt the 'medium' space.
______________________________________________
Attachment 2: Fizeau: The Galilean Derivation
A necessary introduction:
It should be kept in mind that the semantic interpretation of the
following mathematical arguments is that light propagates through the
local medium at c wrt the medium, and that the Earth is relatively at
rest wrt that medium, or at least enough so that any variations in light
speed have no measurable effect on the derived outcomes in the table
above in the preceding attachment. The physical media (in this case
water) is not the medium, it's particles simply interfere with the light
in its translation through the actual medium, which is the vacuum itself
(it should also be clarified that the vacuum is just the extension of
all surrounding matter, so in effect the surrounding matter is
collectively the medium). The observed refractive index is simply
another form of stating the observed speed of light through the media,
and it is therefore a redundant inclusion into the theory, and in fact
harmful in that it lends to the illusion that n is constant irrespective
of motion of the media wrt the medium. It is not. Although if
arbitrarily defined as constant, then it follows deductively that an
adjustment factor is required to compensate for this incorrect
definition. That factor is provided in the final conclusion. What we
will find is that w in the equations below is not the speed of light wrt
the moving media, but rather it is the speed only wrt the stationary
media, hence if we substitute the variable s for the defined-as-constant
w, we get
W = v + s as a correct statement.
's' is the speed of light wrt the 'moving' media, and is not constant
through changes in v, in contrast to Einstein's assumption that it is
constant.
The light, in its propagation from source to detector, is periodically
absorbed by the material media, and subsequently re-emitted. During the
intervals when the light energy is contained as PE within the media, it
necessarily propagates at the same speed as the media over that
interval, thus we obtain the reduced 'effective' speed of light through
the media. IOW for a portion of the total time interval the light is not
propagating at all, in fact for that portion of the total time interval
there is no light. We can form an analogy to a jet that periodically
lands on moving ships for refueling; it's effective speed is less than
its propagational speed, and moreover, the propagational speed wrt the
ship will
depend entirely upon the speed of the ship wrt the medium, in this case
air.
This addendum (this entire introduction) was added to clarify the
argument; it consists of definitions, and thus it is inarguable, so
please reserve your arguments for the math that follows]
The following is the correct Galilean derivation of Fizeau's empirical
results, which were obtained from his measurements of the 'speed of
light in moving media' (water).
x = length of column of water between the source and detector
x'= segment of column of water traversed by the source photons
t_o = total duration of the event
(emission of photon by source/detection at the detector)
dt'= time delay introduced by a stationary volume of water
introduced into the medium of propagation, i.e. into
the space through which the photon propagates as derived from w.
dt'= time delay introduced by a moving volume of water introduced into
the medium of propagation, i.e. into the space through which
the photon propagates.
w = net speed of propagation through 'stationary' column of water
as derived from n
W = net speed of propagation through moving column of water wrt Earth
s = net speed of propagation wrt the 'moving' column of water
n = c/w = index of refraction of media (water)
c = speed of light in vacuum at rest wrt Earth
v = speed of moving water column wrt vacuum
We begin with a source of light and a detector located at each end
respectively of a pipe through which water flows. The direction of water
flow will be from source to detector. The object is to derive an
equation based upon the following premises, that will accurately provide
for
Fizeau's empirical data, and this will be accomplished without reference
to any experimental data, or
to any other formulations pertaining to this effect that have been thus
far derived, with the
exception of the obvious data, e.g. index of refraction of the media,
the speed of light in a vacuum wrt Earth, and premise 1 below.
Premises:
1) Light moving through a homogenous material medium, which is in
uniform
motion at a constant velocity, has a speed wrt that medium that is
independent of the length of the path through
that medium. From this follows that any time delay associated with the
speed of propagation through a given volume of space, that is altered by
the
presence of a homogenous material media within that volume, is
proportional to
the path length through that media. Viz.
dt/x = dt'/x'
2) The medium of propagation is vacuum itself, and is considered to be
'at rest' wrt Earth when located
within an Earth environment. I.e. An observer in motion wrt Earth is
moving equally wrt the local 'space' medium.
3) The Galilean transformation is valid for the description
of all natural phenomena.
A photon is emitted from the source and propagates through the space
between the source and the detector, (both of which are at rest wrt the
vacuum) and the photon is absorbed at the detector. Its time of flight
is recorded. The tube length is immaterial to the following argument,
per
Premise 1 above.
The path length traversed by the photon, through a stream of water
moving
from source to detector at a speed v, is given by the Galilean
transform as:
x - vt_o = x'
The time delay experienced by the photon during its flight per above is:
dt x'/x = dt'
dt, over the length x, is in turn given by
x/w - x/c = dt = x(1/w - 1/c)
Thus dt/x = (1/w - 1/c) = dt'/x'
That is, dt is the difference in the actual time of flight through water
that is at rest wrt the vacuum, source, and detector, as compared to the
time of flight when the space between the source and detector is
evacuated.
From the frame of reference of the water, the water itself is
motionless, i.e. it is at rest. However, from this frame of reference
the medium of propagation is in motion, i.e. the vacuum, source, and
detector are moving away from the observer at v, and any light
propagating through the vacuum is, according to the Galilean transform,
moving relative to our observer at
c-v.
From the frame of reference of the water we have:
t_o = x'/(c-v) + dt'
t_o = (x'/(c-v)) + dtx'/x
t_o = x'((1/(c-v)) + dt/x)
t_o = x'((1/(c-v)) + (1/w - 1/c))
s = x'/t_o
Hence
s = x'/ x'((1/(c-v)) + (1/w - 1/c))
s = 1/((1/(c-v)) + (1/w - 1/c))
s = 1/(1/(c-v) + 1/w - 1/c))
s = 1/(1/(c-v) - 1/c + 1/w))
s = 1/(v/(c^2-vc) + 1/w))
s = 1/(vw +(c^2-vc)/(wc^2-wvc))
s = w(c^2 - vc)/(vw + c^2 - vc)
The Galilean transform gives:
s = W - v
Hence
W = v + w(c^2 - vc)/(vw + c^2 - vc)
Or just
W = v + s (see above definition of s in the intro)
And thus the results are perfectly consistent with normal velocity
addition and with the Galilean transform.
Moreover the corrections of SR to this solution will provide a doubling
of the predicted shift in W, and thus SR has been proved by the
experiment to be false, in contrast to the heretofore unsubstantiated
claims quite to the contrary.
As already shown, the above equation agrees immaculately with the
results of
Fizeau's experiments, to several decimal places when v is small, as was
the case in all of those experiments.
(_x_)
The correct relativistic treatment of electromagnetic forces is derived
here:
http://www.cswnet.com/~rper
There is simply no need for a change in relativistic framework in order
to clarify what are ambiguities only wrt those who
failed to reason the problems through without assuming an egocentric
view.
--
Richard Perry
http://www.cswnet.com/~rper
.
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