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RE: Tidal bulge, Bohr atom & other myths



Item Subject: RE: Tidal bulge, Bohr atom $ other myths
Dave Dockstader asked what it is about most explanations of rainbows given in
many introductory textbooks that is erroneous.
OK Dave, I'll bite: What's wrong with rainbows!

First of all, most descriptions imply, either explicitly or through a diagram,
that sunlight rays inside the raindrops suffer total internal reflection on
the back side of the drop. This is not correct. Most of this internal light
goes right through the drop and only a fraction of it is reflected from the
back internal surface. This is one reason why rainbows are as faint as they
are. At best, the description ignores this transmission and the diagram shows
no transmitted ray leading the student to think that the internal reflection
is total. (In a similar vein, usually the reflected ray off the outside of
the drop is also ignored for the externally incident ray.) It is
geometrically impossible for a ray to enter a drop (assuming it is
sufficiently accurately modeled by a sphere) by refraction and then after it
cuts a cord across the drop for it to suffer total internal reflection. This
is because each angle a cord across a circle makes with the tangent to the
circle at each of the two intersection points on the circle is congruent with
the other one.

Secondly, most descriptions say (or imply) that each wavelength of light is
backscattered (after refracting upon entering, reflecting off the back and
then refracting upon exiting) only into a particular direction relative to the
incident ray. This is not true. The backscattering angle depends *strongly*
on the impact parameter the ray makes on the side of the drop. The back-
scattering angle (the outgoing angle with respect to total 180 degree
scattering back towards the sun) is so impact parameter-dependent that the
backscattering angle is non-monatonic with respect to the impact parameter and
actually has a maximum backscattering angle. What this maximum backscattering
angle is is dependent on the (dispersive) index of refraction of the water for
the wavelength of the ray. Thus, when summing (integrating) over all possible
impact parameters on all of the drop positions in the sky we see that the
backscattered light is sent into a cone (interior and surface) whose axis
points back to the sun. Because of dispersion violet light is sent into a
narrower cone than red light. This means that when looking at a rainbow one
sees backscattered light of all visible wavelengths in the interior of the
violet ring, of all wavelengths except violet in the interior of the blue
ring, all wavelengths except violet and blue in the interior of the green
ring, ..., and only red light throughout the inside of the arc up to and
including the red ring. The intensity of the backscattered light of a
particular wavelength is strongest near the maximum angle of its cone and
weaker in the interior of the cone. This is because at the maximum angle, a
first order change in impact parameter results in a second order change in the
backscattering angle--effectively concentrating the backscattered rays near
the maximum angle. Therefore when we see a rainbow we see a weak colorless
(grayish) glow on the interior of the rainbow from the overlapping region of
all the wavelengths' backscattering cones. The reason that this glow is not
noticeable is that the intensity in these interior directions is significantly
weaker than near the respective cone maxima at the main rainbow rings. On top
of this is a weak colorless glow at nearly all angles from the rays reflected
at initial incidence off of the outer surface of the drops.

Jack U. wrote:
Hi all and David Bowman-
Hmmm, I dunno, David. Suppose that the photon did have a small
mass while still being the lightest particle in nature. What would special
relativity look like then? What kind of argument would lead us to the
Lorentz transformation? We would, of course, discover the constant "c"
by balancing energy-momentum and mass differences for unstable nuclei.

My point was that if the photon did have a small mass then that is not
sufficient to invalidate special relativity. Nature would still have a "speed
limit of causation" c and no informative influence could propagate faster than
this speed. The main effect (on causality) would be that if you wanted to
have high speed EM communications (near the speed limit) we would need to use
gamma rays, and if we insisted on using radio waves we would have to wait
longer for our communications. Of course there would be other collateral side
effects such as the 1/r Coulomb potential would have imposed on top of it an
exponential decay envelope whose decay length is given by the Compton wave
length of the photon. This would mean that electric/electronic circuits would
not work normally, esp. if they were built over a length scale bigger than the
photon Compton wave length. Since normal matter (atoms and molecules) is held
together with EM forces there would be changes in chemistry and in atomic
structure if the photon Compton wave length was of atomic sizes. But since
this corresponds to x-ray sizes we see that unless the photon rest energy was
greater than the order of keV there would be not much of an effect. I suppose
that when a plasma cooled, the time it would take for the atoms to form would
be longer since the opposite charges would have a harder time finding each
other. We would not describe E&M with Maxwell's equations but with the Proca
equations. In this case we would be stuck using the EM 4-potentials (A,phi)
rather than E and B in our equations because the local U(1) gauge invariance
of E&M would have broken down. Suppose we had a technology based on neutrino
physics. The validity of special relativity is not in the balance whether or
not neutrinos have a mass. I suppose that if both neutrinos and photons had
nonzero mass we could still (in principle) use gravity waves to propagate
informative influences AT the speed limit c.

As far as what kind of argument that would lead us to the Lorentz
transformation goes, the argument is exactly the same whether or not the
photon has a mass. The need to use Lorentz transformations when changing
between different inertial reference frames is a consequence of 1. the laws of
nature being form-invariant in all inertial reference frames, and 2. there is
no instantaneous-action-at-a-distance (and thus there exists a speed limit to
causation) in nature. If one insisted that postulate #2 was that the speed of
*light* (i.e. EM waves) was the same in all inertial frames and the photon had
a nonzero mas, we would have a contradiction of special relativity since then,
among other things, photons would travel at a speed less than c while light
travels (i.e. its group velocity) at c which is impossible. Since Jack U.
seems to be a fan of classic texts, I suggest that he and any other interested
souls consult the derivation of special relativity in Landau & Lifshitz' book
_The_Classical_Theory_of_Fields. Here is one of the few places where it is
right.

Also, the consequences for cosmology, with all of that mass hanging around,
would be profound. But your insistence on a particular postulate as
"fundamental" intrigues me. So let's change one of the facts and see
what happens.
You are right here about cosmology. All that extra mass would change the
SOLUTION of Einstein's equations of GENERAL relativity so that the universe
would, not only be closed, but it would go through its expansion/shrinkage
cycle very rapidly, and there probably wouldn't be enough time for us to be
here (Anthropic argument). This would still have no effect on the LAWS of
SPECIAL relativity however.

David Bowman
dbowman@gtc.georgetown.ky.us