The velocity redshift and
the Bohr radius
Even though
photons are emitted from electrons, Bohr was able to tie the frequency of
emitted photons in a stationary atom to electron shell radii. Electrons in
moving atoms still must orbit the nucleus, so here we considering the impact
of velocity upon the wavelengths of a moving atom in terms of the Bohr radius(2,3)
, which
will be used as a proxy for the behavior of all atoms. The reason is that we
are not concerned here with electron jumps between energy levels, but the
effect of velocity on all of the quantum orbits.
One finds that the complete story cannot be told if everything but mass is
treated as constant. Wavelength is associated in a positive sense with the Bohr
radius, with an increased Bohr radius producing an increased wavelength,
while a decreased Bohr radius is associated with a shorter wavelength(1).
Thus, if the mass of the electron in the denominator of the above formula is
treated as the only non-constant and increases with velocity, as special
relativity asserts, the result is a decreased wavelength in contradiction
with experiment.
There are two reasons for this discrepancy. First, the mass in the
Bohr radius calculations originated from consideration of the inertial mass
of the electron(2,3). Since all bodies fall at the same rate
when dropped from the same position in a gravitational field, the effect in
the increased inertial mass of the electron is canceled by its increased
gravitational mass. Second, because gravitational forces are so many orders
of magnitude smaller then electric forces, gravitational forces were ignored
in the calculation of the Bohr radius. It is not appropriate to consider them
now, without factoring them into the original calculations. The only thing to
do is to treat mass as a constant here.
A likely candidate for contributing to the correct effect would appear to
be the adjustable parameter the permittivity of the vacuum є(7),
which fortuitously is located in the numerator. Thus, its increase would be
associated with an increased Bohr radius.
To those who would object to having a constant like permittivity change,
the author notes that the “foot has already be stuck in the door” at least
twice
The first foot through the door is that permittivity has
been shown to be dependent on spacetime geometry(4), which happily
is consistent with this model’s assertion that the velocity of light varies
with the state of expansion of the universe.
The second foot through the door is the fact that the
value of the fine structure constant has been shown to be dependent on the
energy of the probe that senses the charge. The value is 1/128(8)
in high energy experiments as apposed to the conventional 1/137. The fine
structure constant is so called because it is a composition of other
constants. So, it is obvious that one or more constants change. This issue
will be examined more thoroughly shortly.
At any rate,
the change in the effective permittivity on the Bohr radius may be reflective
of an effective weakening of the propagation of forces related to timeliness
factors discussed below.
Permittivity has been shown to be
dependent on spacetime geometry(18), which as noted above,
supports this model’s assertion that the velocity of light varies with the
state of expansion of the universe.
Permittivity Є has also been shown to be related to the energy
and the Friedmann radius in the following
manner
.
The permittivity є is a form of resistance. Insight can be gained by
considering the analogous situation in electricity where the current I = E/R.
The same relationship exists between columns E and D in table A. The
propagability of light in column E is the result of dividing column D into
unity (or C0). Thus column D can be thought of in part as a
measure of external response to the condensation process and in part as a
measure of the internal resistance to photon propagation. Thus, column D is a
measure of the permittivity of free space, expressed here in terms of the
same velocity relationship Einstein used, namely, 1 + v/c. Thus, the formula
for the Bohr radius is modified to 
, which shall be justified further below.
The frequency of light emitted by the hydrogen atom in quantum transition
between n1 and n2 is
This can also be written: 
, where f is the orbital frequency of the electron. But we
are not concerned here with the dynamics of orbital energy jumps; but what
the impact of velocity is on the emitted frequency after these dynamics have
been taken into account for any given orbital jump. Thus the quantities in
which n is involved should be treated as constant and the above formula
should be written ve=πfk. So the emitted frequency is
directly proportional to the orbital frequency which in turn is directly
proportional to orbital velocity vo.
Timeliness
But how does increased velocity bring about this same relationship? One
way to understand this phenomenon is to do a thought experiment similar to
those Einstein did.
It is apparent that the change in frequency can be explained in terms of
timeliness and relative motions. Increased forward velocity of an atom
affects the relative orbital velocity of an electron around the nucleus,
since the nucleus has only to increase its forward velocity while the
electron "maintains" an orbit in addition to increasing its forward
velocity. Consider a fictional hydrogen atom traveling at the speed of light.
It is obvious that since matter cannot exceed the speed of light, the electron
could not orbit any longer. The best it could do would be to travel a
parallel path with the proton. Odds are that it would have headed off into
the sunset long before reaching the speed of light. In short as the velocity
of the atom is increased, some of the orbital velocity of the electron is
converted to forward velocity and the relative time for orbiting the nucleus
is decreased.
Another way of viewing the same phenomenon is to think of the orbital
velocity remaining constant while the forward velocity increases. If the
values in column D are used as representative forward velocities, then the
values in column E become representative of the relative orbital
velocities.
This model posits that the velocity redshift and the gravitational
redshift are tied to the same dynamics, namely the relative flow between
atoms and the spatial medium, by whatever name it is called. In the case of
the gravitational redshift and a stationary atom, the flow is that of the
gravitational field. It seems reasonable that the parameters є
and a should be reflective of this dynamic. This relationship also
applies to rotational processes in the nucleus.
Further support for the relationships derived above comes from an analysis
of the Plank constant which can be written 
. The directly inverse relationship between permittivity and
orbital velocity vo is that of this model.
Once it is recognized that gravitational and electromagnetic forces
operate at the speed of light, then the issue of the timeliness(6)
and the delay of communication of and responses to forces with increasing
velocity becomes important. One thing that occurs is a weakening of the gravitational
field which is accounted for by the increase in the Bohr radius. The
inability of adjustment's to a particle's behavior to occur in a timely
manner as its velocity approaches that of light is the cause of much
non-intuitive phenomena, including the slowing of the aging process and of
atomic clocks with increased velocity.
The Fine Structure Constant
The formula for the constant is:
Current conventional wisdom holds it is the charge that
varies in high energy experiments as the result of the energetic electron
being able to shed hangers on virtual particles that mask its true charge.
Whether this is the case or not, it helps with renormalization. It also seems
likely that changes in the value of the fine structure constant in high energy
experiments and that, if any, due to cosmic expansion result from different
causes. Regardless, it is still a second foot through the door of variably
measured constants.
It is interesting that permittivity is found in the
denominator of the formula. Thus, an increase in permittivity due to cosmic
expansion would result in a decrease in the fine structure constant which in
turn would lengthen emission wavelengths.
But the speed of light c and Plank’s constant h are also
found in the denominator, as potential third and forth feet through the door.
This model posits that the internal speed of light through the spatial medium
varies with the density of the spatial medium, so although the units of
measurement are significantly different, the affects of changes in
permittivity and the speed of light tend to offset each other in the fine
structure formula.
There is no compelling reason from the Plank formula
referenced above to conclude that h is influenced by permittivity as the
orbital velocity is present in just the right place to offset changes in
permittivity. However there is also no compelling reason, except convenience,
why any of the so called constants should not vary as space expands. That h
should not vary as space expands seems a stretch. The author suspects that the
third and forth feet are also stuck in the door.
What makes all of this relevant is that cosmological
tests of the fine structure constant are being made to determine if constants,
including the speed of light, have varied with the expansion of the universe.
The fine structure constant may well be the poorest entity to measure such
changes as any change in one component would likely be masked by cumulative
changes in the others resulting in little or no overall net change. Obviously
high energy experiments are a different matter, as may be the case for cosmic
measurements if they could reach back close to the big bang.
Time and rod lengths
One can analyze the above in other ways, including analysis of energy and
mass considerations, but timeliness of expression of forces that propagate
with finite velocity is insightful. Both gravitational and velocity dependent
redshifts are posited to be a direct result of this timeliness factor. Timeliness
causing the slowing of clocks and lengthening of rods (contrary to
conventional wisdom) is posited to be the basis for the constant speed of
light which special relativity requires, as the next section clarifies
mathematically.
In a moving frame clocks run slow. Let To equal the number of
ticks given by a standard clock i.e. a master clock that does not change
under any circumstances. Let Tm equal the number of ticks in a
moving clock. Let Lo equal the length of a standard rod that does
not change under any circumstances and Lm equal the
length of a moving rod.
Let Einstein's relationship referenced above be 
The Lorentz Fitzgerald formula is not used as the underlying fundamentals
depend upon the relative orbital time of an electron, not the relative travel
time of photons oriented perpendicular to each other. Einstein had already
derived his own version of the Lorentz Fitzgerald formula, when he derived
the above relationship. Moreover, the use of a Doppler shift to measure the
gravitational redshift by Pound and Rebka(19) in their famous
gravitational redshift experiment supports Einstein’s rationale. The use of
the Einstein relationship fortunately permits the use of values similar to
those in Table A.
Let Tm = To/Q. Then Tm has fewer
ticks and the associated clock runs slower per special relativity than To.
One way to keep c constant is to let L = Lo/Q, where L is shorter
than Lo, then c = L/Tm equals a constant ratio.
Supposing that Lo = 100 units, To = 10 ticks, and c
= 10 units. These numbers are contrived
and chosen purely for reasons of simple math. Let’s say that Tm
slows to 9.0909..ticks and L shortens to 90.909.., then as Einstein requires
90.909../9.0909.. = 10.
This is the case when the observer in the moving frame is looking at the
standard rod in the stationary frame, which the moving observer thinks is
shorter because the use of a constant speed of light in special relativity
forces this conclusion. The L in the previous two paragraphs is Lo the
standard rod of the rest frame. Although the rod has not changed in fact, the
observer in the moving frame measures Lo as shorter.
But this cannot be the case when the observer looks at a rod in his or her
own frame as unacceptable results occur. If no change in L occurs and Tm
slows to 9.0909...ticks. Now c = 100/9.0909... = 11 units, an unsatisfactory
result in both special relativity and this theory.
But this model and redshift theory suggest Lm = LoQ
that is Lm becomes longer in the moving frame. If no change in
length had occurred a moving clock would click only Tm times
during the measurement of Lo. But it will click TmQ
times or To times during the measurement of the longer Lm
and the velocity of light remains constant.
To illustrate let Lm= 110 units and Tm slow to
9.0909...ticks. Now it will take 9.0909...ticks to cover the first 100 units
of Lm. To cover the whole distance the total number of ticks Tt
will be 9.090909..x/(110/100) or 10 ticks, so c remains constant in the
observer’s own frame as special relativity requires, because although
the rod has lengthened so has the ruler and the measurer thinks that he or
she is measuring 100 units.
For
the total measured number of ticks Tt the following relationship
holds: 
Therefore, 
Thus,
which
is what special relativity and this model require within the moving frame. It
is hard to see how Einstein, given his velocity redshift, can do other than
have rods in the accelerated frame lengthen.
Once upon a Time - a light
hearted tale treating the above serious thoughts
King Henry recognizing the need for a system of
measurement has modestly declared that the master unit of distance shall be
one tenth of that run by his own royal self in ten seconds using one of his
two favorite clocks which tick at the same rate. The king has also asked
Prince Henry Junior to do likewise in the Land of Oz using the second clock.
Both distances are to be run and marked off on royal carpets which are to be
maintained as official standards of measurement.
The following events are testified to by Merlin, King
Henry’s royal counselor, and the Wizard of Oz, the Prince’s advisor. They use
magic whenever necessary to communicate and ascertain the Real Truth. But they
decided to play a trick on the king and prince by making clocks in Oz tick
only five times per each ten clicks of the clock in the Home Kingdom and
stretching the carpet in Oz to double the length of that in the home kingdom.
The magicians made sure that both the king and the prince
started their races at the same universal time and verified that their
velocities were identical. The king quit at the end of ten clicks and Merlin
marked off the Home Kingdom’s carpet into 10 equal master units.
But the prince’s clock had only clicked five times so he
kept running for five more clicks, after which the Wizard marked off 10 master
units on the carpet which became the official standard for the Land of Oz.
The king believing that by definition he was running 1
unit per tick was certain that he had run ten units. Viewing the prince’s
effort, he felt that Junior had run twice as far as he was instructed to do.
The Prince likewise regarded his own achievement as
accurate and he could not understand why the king had quit when his job was
only half done. He must be getting old.
Merlin and the Wizard, who were beside themselves with
mischievous joy, asked the prince to measure the standard reference rod in the
home kingdom using the reference rod from Oz, which they froze in length. The
prince found the rod in the home kingdom to have apparently contracted, which
is what Merlin and the Wizard knew that special relativity predicted.
They told the king to measure the reference rod in the
Land of Oz using the standard reference rod of the home kingdom, which they
also froze in length. The result confirmed that the reference rod in the Land
of OZ had expanded. The King called in the magicians to explain what was
happening. Using their magic they were able to show king and prince that the
clock in Oz was ticking at half the rate of the Home clock and that
calculations based on said clock used with the longer rod produced the same
velocity as would the king’s Home clock and shorter rod.
The king asked the magicians why he should not lop off
their heads. Their defense was that they wished to provide the king with an
important insight regarding the way the universe functioned and a concept
called timeliness. They explained that their trick was accomplished by having
the Land of Oz fly through the universe with a much greater velocity than the
Home Kingdom possessed.
DISCUSSION
These considerations are why Paper I predicts only that a difference in
compared measured speeds of light will occur with different elevations in a
gravitational field and does not predict which elevation will be measured as
faster. I also predict that radial versus tangential measurements of the speed
of light will also differ. But the interplay of factors in the two theories
could cause measured results to mask actual results.
But Paper I predicts:
1. The velocity of the spaceship will cause an on
board clock to slow.
2. Decreased gravity on the spaceship will cause on board a
clock to speed up.
The theory presented here suggests that:
3. Rod lengths will lengthen due to increased velocity in orbit.
4. The decreased gravity will cause rod lengths to shorten.
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BIBLIOGRAPHY
(1) Paul Marmet, http://www.newtonphysics.on.ca/
(2)F. J. Blatt, 1983 Principles of Physics
(3) H.E. White, 1962 Modern College Physics
(4) William. Q. Sumner, 1994 The Astrophysical Journal, Vol. 429,
No 2, Part 1 On the Variation of Vacuum Permittivity in Friedman
Universes
(5) Clifford M. Will, 1986 Was Einstein Right Basic Books, inc.
(6) James DeMeo, http://www.orgonelab.org/miller.htm
(7) Tom Van Flandern, 1993, Dark Matter, Missing planets and New
Comets, North Atlantic Books
(8) Particle Physics Booklet,
extracted from the Review of Particle Physics S. Eidelman et Al,
Physical Letters B592, 1 (2004)
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