_{0}/c=2

^{-11}. This velocity defines the rotation velocities of distant stars around galaxies. The presence of a parameter with dimensions of velocity should carry some important information about the geometry of dark matter space-time sheets.

Velocity like parameters appear also in other contexts. There is evidence for the Tifft's quantization of cosmic red-shifts in multiples of v_{0}/c=2.68× 10^{-5}/3: also other units of quantization have been proposed but they are multiples of v_{0} (see this).

The strange behavior of graphene includes high conductivity with conduction electrons behaving like massless particles with light velocity replaced with v_{0}/c=1/300. The TGD inspired model explains the high conductivity as being due to the Planck constant h(M^{4})= 6h_{0} increasing the delocalization length scale of electron pairs associated with hexagonal rings of mono-atomic graphene layer by a factor 6 and thus making possible overlap of electron orbitals. This explains also the anomalous conductivity of DNA containing 5- and 6-cycles (same reference).

** 1. Is dark matter warped?**

The reduced light velocity could be due to the warping of the space-time sheet associated with dark electrons. TGD predicts besides gravitational red-shift a non-gravitational red-shift due to the warping of space-time sheets possible because space-time is 4-surface rather than abstract 4-manifold. A simple example of everyday life is the warping of a paper sheet: it bends but is not stretched, which means that the induced metric remains flat although one of its components scales (distance becomes longer around direction of bending). For instance, empty Minkowski space represented canonically as a surface of M^{4}× CP_{2} with constant CP_{2} coordinates can become periodically warped in time direction because of the bending in CP_{2} direction. As a consequence, the distance in time direction shortens and effective light-velocity decreases when determined from the comparison of the time taken for signal to propagate from A to B along warped space-time sheet with propagation time along a non-warped space-time sheet.

The simplest warped imbedding defined by the map M^{4}→ S^{1}, S^{1} a geodesic circle of CP_{2}. Let the angle coordinate of S^{1} depend linearly on time: Φ= ω t. g_{tt}} component of metric becomes 1-R^{2}ω^{2} so that the light velocity is reduced to v_{0}/c=(1-R^{2}ω^{2})^{1/2}. No gravitational field is present.

The fact that M^{4} Planck constant n_{a}h_{0} defines the scaling factor n_{a}^{2} of CP_{2} metric could explain why dark matter resides around strongly warped imbeddings of M^{4}. The quantization of the scaling factor of CP_{2} by R^{2}→ n_{a}^{2}R^{2} implies that the initial small warping in the time direction given by g_{tt}=1-ε, ε=R^{2}ω^{2}, will be amplified to g_{tt}= 1-n_{a}^{2}ε if ω is not affected in the transition to dark matter phase. n_{a}=6 in the case of graphene would give 1-x≈ 1- 1/36 so that only a one per cent reduction of light velocity is enough to explain the strong reduction of light velocity for dark matter.

** 2. Is c/v _{0} quantized in terms of ruler and compass rationals?**

The known cases suggests that c/v_{0} is always a rational number expressible as a ratio of integers associated with n-polygons constructible using only ruler and compass.

- c/v
_{0}=300 would explain graphene. The nearest rational satisfying the ruler and compass constraint would be q= 5× 2^{10}/17≈ 301.18. - If dark matter space-time sheets are warped with c
_{0}/v=^{11}one can understand Nottale's quantization for the radii inner planets. For dark matter space-time sheets associated with outer planets one would have c/v_{0}= 5× 2^{11}. - If Tifft's red-shifts relate to the warping of dark matter space-time sheets, warping would correspond to v
_{0}/c=2.68× 10^{-5}/3. c/v_{0}= 2^{5}× 17× 257/5 holds true with an error smaller than .1 per cent.

** 3. Tifft's quantization and cosmic quantum coherence**

An explanation for Tifft's quantization in terms of Jones inclusions could be that the subgroup G of Lorentz group defining the inclusion consists of boosts defined by multiples η= nη_{0} of the hyperbolic angle η_{0}≈ v_{0}/c. This would give v/c= sinh(nη_{0})≈ nv_{0}/c. Thus the dark matter systems around which visible matter is condensed would be exact copies of each other in cosmic length scales since G would be an exact symmetry. The property of being an exact copy applies of course only in single level in the dark matter hierarchy. This would mean a delocalization of elementary particles in cosmological length scales made possible by the huge values of Planck constant. A precise cosmic analog for the delocalization of electron pairs in benzene ring would be in question.

Why then η_{0} should be quantized as ruler and compass rationals? In the case of Planck constants the quantum phases q=exp(imπ/n_{F}) are number theoretically simple for n_{F} a ruler and compass integer. If the boost exp(η) is represented as a unitary phase exp(imη) at the level of discretely delocalized dark matter wave functions, the quantization η_{0}= n/n_{F} would give rise to number theoretically simple phases. Note that this quantization is more general than η_{0}= n_{F,1}/n_{F,2}.

For more details see the chapter TGD and Astro-Physics.

## 4 comments:

11 06 06

Matti:

I was just thinking about you as I wrote my latest post. It is an extension of the finger counting games, but this time I invoke linguistics. You have likely come to this conclusion already, but I believe that the most basic spoken language has a 3-adic basis, and that our perception of mathematics and nature shapes our language use and formation. I will develop this idea further, but let's just say that we now see applications for p-adic formalism in not only the sciences, but also linguistic analysis AND accounting... :) Just a few thoughts, but they need to be further developed:) Have a nice day.

BTW: See Kea's latest blog post for a discussion on dark energy that she started on Cosmic Variance. I am quite curious as to your take:)

I visited in Kea's blog but did not find anything about dark matter. I wrote a little comment to your blog about Zeta and p-adics in TGD framework.

I have not thought about 3-adics and language. I have however thought about possibility of mapping 64 genetic codons (consisting of 3 base pairs!) to phonemes as something analogous to representation of DNA in terms of aminoacids.

The fact is that trialities fill archaic thinking: think only fairy tales, holo trinity, etc... This might relate to 3-adicity as next level in cognitive hierarchy after the most primitive 2-adic level which I tend to assign to our engineering achievements;-).

11 06 06

Oh Matti:

Thanks for the responses. I just updated my post with a note about 5-adic physics and DNA strands. It turns out that there is a relationship between 2adic distances between codons... Here is the title and link to the article, as you may really enjoy it!

"A p-Adic Model of DNA Sequence and Genetic Code", by Dragovich and Dragovich.OK, I must now go to bed, so have a great week:)

I looked at D-D model (I think I have years ago had some communication with D:s) for genetic code. I have also proposed a model in which base 4 plays a key role. This is based on observation that first nucleotide represents the most weighty 4-digit, second less weighty and third least important one. This is consistent with almost symmetries of genetic code with respect to 3 nucleotide. I did not formulate this notion in terms of 5-adicity since it looks me somewhat un-natural because the digit has only 4 values (most naturally 0,1,2,3) rather than 5 values (0,1,2,3,4).

One of the three models for genetic codes that I have proposed is based on the observation that the number of primes smaller than 64 is 18. Adding to this 0 and 1 gives 20, the number of aminoacids.

This observation led to a number theoretical model for genetic code maximizing number theoretic information measure defined by number theoretic variant of Shannon entropy. The model assigns to number characterizing DNA codon prime p <64 or number 0,1 characterizing aminoacid.

Information measure is Shannon entropy with log(probability) replaced with log(p-adic norm of probability): this gives a well-defined and additive information measure assuming that probabilities are rationals or algebraic numbers. The king idea is that now entropy can be also negative and thus means information!

In the case of entanglement entropy the interpretation as information is natural: entanglement carries indeed information as is obvious from the fact that people are busily constructing quantum computers!

Model is able to reproduce genetic code and also its small variations.

Matti

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