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Planck Unit Packing

On another page, I introduced the Plank length in my musings about how many Wheeler-Everett "Many World" universes could have evolved since the beginning of time. The Planck length is 1.616 X 10-35 meters long. Thatís about 1 ten-octillionth the diameter of an atom.

The Plank length was conceived by Max Planck in 1899, in his attempt to establish "natural units" of mass, length and time. (He didnít name it after himself, of course; others did.) The Planck length is actually derived from the Planck mass, which -- forgive me for the technical divergence here Ė calculates the mass at which the Schwarzschild radius (the size of a bodyís Black Hole event horizon) is the same size as its quantum-mechanical Compton wavelength (the spread of a particleís wave-function). Well, hmmmm. It all ends up saying that the Planck mass Ė which is roughly equivalent to the mass of a largish biological cell Ė can become a Black Hole if itís compressed to the size of a Plank length.

The Planck length has become a very important player in modern physics, because it describes the size at which quantum mechanical and relativistic effects become equal partners in determining how things behave. This is the domain where quantum mechanics happen, where the universe changes from a smooth analogic continuum to a grainy region, where all events occur in sudden jumps with no intervening transitions. There is no physical reality to any length smaller than the Planck length. (My gosh, who would need a smaller space than 1 ten-octillionth the diameter of an atom to define the basic dimensional building block of the universe?)

In my calculation of possible Many Worlds universes, I figured the number of Planck-length volumes that would fit inside the universe, considered (with present knowledge) as a sphere of diameter 1026 km. I assumed the Planck volumes to be little cubes, one Plank length on a side. That was OK for the purposes of what I was doing at the time, but itís definitely unsatisfactory when you think about it more. Surely, a real Planck volume must be a sphere, not a cube. You canít accept a "quantum passage" through a Planck region thatís larger than a Planck length Ė certainly not a diagonal trip through from one cube vertex to another, which would be significantly longer -- 1.732 times longer, in fact -- than along the side of a cube one Planck length long.

Now this is a real dilemma. You can close-pack cubes, but you canít close-pack spheres. There are only 2 ways to pack spheres tightly: a cubic-centered packing (ccp), or a hexagonal-centered packing arrangement (hcp). In either case, the packing density, using unit-diameter spheres, is pi/181/2 = 0.7408Ö Thereís a significant amount of space left over. And in the 3-dimensional geometry of our universe, you canít just pretend that 26% of the unfilled space is really there and usable for quantum events to occur inside of.

Now, I suppose you could say that geometry may change at the Planck length level; that somehow, at that magnification, the diagonal through a cube is equal to the side length, or that the spheres of Planck-length diameter can close-pack. That doesnít seem satisfactory to me. It makes better sense to say that the concave-planar octahedronal and cuboctahedronal solids that correspond to the empty spaces between the close-packed Planck unit spheres are not to be considered part of the universal volume, and in fact donít exist at all except as small houses for wee ghosts:

Quantum mechanics obey the laws of general relativity, and they also obey "localistic" physical laws. The flow of information derived from any quantum mechanical transaction is limited by the speed of light in a vacuum. (Another way of saying this is that it must remain within the ever-expanding Minkowski light-cone that marks out the boundaries of our universe.) We normally consider the path of a photon as being a straight line Ė except when influenced by local gravity or by the curvature of space itself. But consider that a photon has to be recapitulated -- reborn -- within every single Planck-volume it "passes through". (I believe the quantum odds say that almost surely a photon going in one side will still be a photon coming out the other.) In a cubic-centered packing, each Planck sphere is surrounded and contacted, at one point on its surface, by 12 others. The contact points in a large set of close-packed spheres donít line up in a straight line. Therefore, it seems unavoidable that the photonís trip must consist of a series of doglegs from the original angle of flight, dictated by wherever the contact points between adjacent Planck-spheres occur which are closest to the original photon incidence angle. Itís not permissible to talk about "increments of travel" less than one Planck length long, so you have to presume that the travel length of a photon "through" each Planck volume must be exactly 1 Planck length long -- that is, a virtual journey from the outer contact point to the center of the sphere, plus the virtual journey to another surface contact point that is the closest one to the photonís original incidence angle vector Ė the angle as if the photon were to go "straight through" the sphere. Here is a 2-d representation of what I am talking about:

Or, as an alternate, here is another representation that allows the photon to "jump across" the concave-planar octahedronal and cuboctahedronal gaps between the Plank volume-spheres (since we earlier established that these were not considered part of the universe's volume, remember?):

In either case, the photon trip length is the same, 5 Planck lengths long.  (I won't speculate here on how, in either example, the photon knows exactly where on the circumference its original bearing line enters or exits!)

Of course, in these 2-d representations, there are incidence angles which produce a "straight-through shot", where the scalar length is equal to the number of Planck lengths traversed. I think thatís not the case in a 3d packing arrangement, however.

You can see there are a number of alternate paths from one end to the other which yield the same total photon Planck travel length. Thatís true in the 3d scenario also.

The number of Planck-lengths between two points cannot simply be their scalar straight-line distance divided by the Planck length.  That's because Planck volumes, which have to be considered as an arrangement of close-packed spheres, donít line up nicely along any straight line. You canít have a fractional Planck length, so you have to calculate the length as the total integer number of close-packed Planck spheres laying between the two points. That will always be a greater number than the straight scalar distance, since packed spheres must jigger themselves in all 3 spatial dimensions to nest properly.  I would hazard a guess that the actual "Plank scalar length" would be some function of the inverse of the ccp packing density number, perhaps a cube root of that, and rounded up to the next whole integer, e.g., int(scalar length * 1/(.7408))^1/3 + 1) or something like that.

In other words, as the crow flies, the photon tarries. The further away the origin of the photon, the tardier it will be in reaching us.  The Planck time unit is the time it takes for light to travel the distance of one Planck length -- 10-43 seconds, to be precise.  As with the Planck length, there is no physical meaning to time durations less than the Plank time.  The photon has to navigate through each Planck unit volume it encounters -- that is, it has to take a full Planck time unit to pass it, regardless of how short the imaginary chord through the Planck sphere is, that describes the photon's original bearing.  (Remember, you can't talk about any lengths shorter than a Planck length!)  But our imaginary specular crow can "cut the corners", so to speak.  Were he also to be traveling at the speed of light, he would get to his destination much, much quicker than the poor plodding photon.

Iíve heard criticisms of physicists' explanations of the spectrographic frequency "red shift" (Doppler Effect) phenomenon that cosmologists use to base the size of the universe upon. Some alternate (mostly crackpot) explanations talk about "tired light" causing the apparent shift. Maybe those distant photons are indeed tired, after having to zig-zag their way through so many Planck-volume spheres to reach us. (And if the size of the universe and the speed of light is flummoxed up, maybe the universe really was created by God in October 4004 BC, as Bishop Usher so craftily calculated long ago. And maybe itís not so big after all.)

One problem (among many) with this thought experiment is how a photon can "remember" its original bearing as it passes through the complex network of intervening Planck volumes. As it enters any individual Planck volume, its exit point could be any one of 12 adjacent sphere contact points (including its original entry point) -- with no energy penalty at all associated with any of the 12 possible exit choices. If you consider that a photon has so many degrees of freedom at every Planck volume sphere it passes through, itís hard to imagine how any photons at all get from the thing I am looking at to my eye. Conversely, many photons emanating from other, completely different objects would end up winding their way into the Planck volume-network path to my retina, interfering with the thing Iím supposed to be looking at.

Plus, what exactly does "photon incidence angle" or "original bearing" mean? The photon itself originated in a close-packed Planck sphere, and if thatís the case, then it would be constrained to have only one of 12 total exit incidence angles!

Maybe those wee ghosts that live in the ("shall-not-be named, but-I-did-it-anyway") concave-planar octahedronal and cuboctahedronal gaps between the Plank volume-spheres contribute an essential service by shepherding photons through their neighboring Planck sphere-fields.  Afterwards, work done, perhaps they meander down to the neighborhood cuboctahedronal pub to imbibe a pint or two.  Stranger things have been postulated in the annals of Physics...

These problems arise if one considers the Planck volume packing arrangement to be static. But nothing appears to be static in the universe; all the physical events that we can observe are characterized as a teeming, writhing, seething, vibrating, foaming condition of constant change. If the Planck universal volume packing also jiggles, then perhaps the "lowest energy solution" of a photonic journey through it is more selective and directive. In quantum mechanical terms, perhaps the probability wave function in that scenario would favor a photon's exit point from a Planck volume sphere that is more correlated with its original entry incidence angle. However, I canít think of a definite reason why that would be the case, so Iím not completely convinced that jiggling would solve anything.  After all, I myself jiggle a lot more than I used to, and it hasn't done me any good at all...

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The sphere packing images are from The Geometrical Foundation of Natural Science, Robert Williams (Dover, 1979). I canít even begin to count the times Iíve gone back to this book as reference material for all sorts of different things.

I have no real problem with the physical theory behind the Doppler (red-shift) effect, except for the common example always given to illustrate it.  That is the one about the train whistle, that seems to drop in pitch as the train rushes past and away from you.  It's much more likely that the perceived pitch drop is due to the decline in sound magnitude (amplitude), not by a drop in frequency.  Anybody who listens to music using headphones knows that if you move the headphones further away from your ears, the song pitch appears to drop.  This is strictly an auditory-perceptual phenomenon, not a Doppler Effect.  When the amplitude of an auditory sound signal drops, the pitch -- as perceived and interpreted by the human brain -- also drops.

The following information on the Planck realm is excerpted from The Vacuum, Light Speed, and the Redshift, by Australian Astronomer Barry Settlefield (http://www.ldolphin.org/setterfield/vacuum.html).  I should point out that Mr. Settlefield's ideas are generally discredited within the scientific community as being "bad science", and Settlefield himself is associated with the conservative Christian Creationist movement.  But the background information in the early part of this paper is a quite well-written review for the layman of accepted modern physical theory relating to this subject -- as far as it goes:

"ÖWhen dealing with the vacuum, size considerations are all-important. On a large scale the physical vacuum has properties that are uniform throughout the cosmos, and seemingly smooth and featureless. However, on an atomic scale, the vacuum has been described as a "seething sea of activity", or "the seething vacuum". It is in this realm of the very small that our understanding of the vacuum has increased. The size of the atom is about 10-8 centimeters. The size of an atomic particle, such as an electron, is about 10-13 centimeters. As the scale becomes smaller, there is a major change at the Planck length (1.616 x 10-33 centimeters), which we will designate as L. In 1983, F. M. Pipkin and R. C. Ritter pointed out in Science (vol. 219, p.4587), that "the Planck length is a length at which the smoothness of space breaks down, and space assumes a granular structure."

This "granular structure" of space, to use Pipkin and Ritter's phrase, is considered to be made up of Planck particles whose diameter is equal to L, and whose mass is equal to a fundamental unit called the Planck mass, M, (2.177 x 10-5 grams). These Planck particles form the basis for various cosmological theories such as strings, super strings, 10-dimensional space, and so on. During the last hundred years, physicists have discovered that atomic particles such as electrons or protons, have a wave-form associated with them. This is termed the wave/particle duality of matter. These waves are called de Broglie waves and vary inversely with mass. That is to say, the heavier the particle, the shorter its wavelength. This means that because a proton is more massive, its wavelength is shorter than an electron's. What is interesting and important is that Planck particles have a diameter L that is equal to their de Broglie wavelength.

The physical vacuum of space therefore appears to be made up of an all-pervasive sea of Planck particles whose density is an unbelievable 3.6 x 1093 grams per cubic centimeter. It might be wondered how anything can move through such a medium. It is because de Broglie wavelengths of elementary particles are so long compared with the Planck length, L, that the vacuum is 'transparent' to these elementary particles. It is for the same reason that long wavelength infrared light can travel through a dense cloud in space and reveal what is within instead of being absorbed, and why light can pass through dense glass. Therefore, motion of elementary particles through the vacuum will be effortless, as long as these particles do not have energies of the magnitude of what is referred to as Planck energy, or M* c2 ('c' is the velocity of light). Atomic particles of that energy would simply be absorbed by the structure of the vacuum. From the figures for the density given above, the energy associated with this Planck particle sea making up the physical vacuum can be calculated to be of the order of 10114 ergs per cubic centimeter. This is the maximum value for the "Zero Point Energy" (ZPE) inherent in the vacuum..."

"...In order to appreciate the magnitude of the ZPE in each cubic centimeter of space, consider a conservative estimate of 1052 ergs/cc. Most people are familiar with the light bulbs with which we illuminate our houses. The one in my office is labeled as 150 watts. (A watt is defined as 107 ergs per second.) By comparison, our sun radiates energy at the rate of 3.8 x 1020 watts. In our galaxy there are in excess of 100 billion stars. If we assume they all radiate at about the same intensity as our sun, then the amount of energy expended by our entire galaxy of stars shining for one million years is roughly equivalent to the energy locked up in one cubic centimeter of space."

I don't know why or if each Planck-length unit sphere can truly be considered to be a "Planck particle of Planck Mass".  Since a particle of such characteristics and size turns out to be a Black Hole, that would mean that the Universe is packed tightly with Black Holes!  But the concept of ZPE itself seems to be valid in the general scientific community.  See http://www.calphysics.org/zpe.html.




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