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Physics in Many-Sheeted Space-Time

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Year 2006



About the interpretation of the parameter v0

The formula for the gravitational Planck constant contains the parameter v0/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 v0/c=2.68× 10-5/3: also other units of quantization have been proposed but they are multiples of v0 (see this).

The strange behavior of graphene includes high conductivity with conduction electrons behaving like massless p"../articles/ with light velocity replaced with v0/c=1/300. The TGD inspired model explains the high conductivity as being due to the Planck constant h(M4)= 6h0 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 M4× CP2 with constant CP2 coordinates can become periodically warped in time direction because of the bending in CP2 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 M4→ S1, S1 a geodesic circle of CP2. Let the angle coordinate of S1 depend linearly on time: Φ= ω t. gtt} component of metric becomes 1-R2ω2 so that the light velocity is reduced to v0/c=(1-R2ω2)1/2. No gravitational field is present.

The fact that M4 Planck constant nah0 defines the scaling factor na2 of CP2 metric could explain why dark matter resides around strongly warped imbeddings of M4. The quantization of the scaling factor of CP2 by R2→ na2R2 implies that the initial small warping in the time direction given by gtt=1-ε, ε=R2ω2, will be amplified to gtt= 1-na2ε if ω is not affected in the transition to dark matter phase. na=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/v0 quantized in terms of ruler and compass rationals?

The known cases suggests that c/v0 is always a rational number expressible as a ratio of integers associated with n-polygons constructible using only ruler and compass.

  1. c/v0=300 would explain graphene. The nearest rational satisfying the ruler and compass constraint would be q= 5× 210/17≈ 301.18.

  2. If dark matter space-time sheets are warped with c0/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/v0= 5× 211.

  3. If Tifft's red-shifts relate to the warping of dark matter space-time sheets, warping would correspond to v0/c=2.68× 10-5/3. c/v0= 25× 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= nη0 of the hyperbolic angle η0≈ v0/c. This would give v/c= sinh(nη0)≈ nv0/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 p"../articles/ 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 η0 should be quantized as ruler and compass rationals? In the case of Planck constants the quantum phases q=exp(imπ/nF) are number theoretically simple for nF 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 η0= n/nF would give rise to number theoretically simple phases. Note that this quantization is more general than η0= nF,1/nF,2.

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



Orbital radii of exoplanets and Bohr quantization of planetary orbits

Orbital radii of exoplanets save as a test for the Bohr quantization of planetary orbits. Hundreds of them are already known and in tables (Masses and Orbital Characteristics of Extrasolar Planets using stellar masses derived from Hipparcos, metalicity, and stellar evolution) basic data for for 136 exoplanets are listed. The tables also provide references and links to sources giving data about the stars, in particular star mass M using solar mass MS as a unit. Hence one can test the formula for the orbital radii given by the expression

r/rE= (n2/52) ×(M/MS)× X ,

X= (n1/n2)2, ni=2ki× ∏siFsi ,

Fsi in the set {3,5,17,257, 216+1} . Here a given Fermat prime Fsi can appear only once.

It turns out that the simplest option assuming X=1 fails badly for some planets: the resulting deviations of of order 20 per cent typically but in the worst cases the predicted radius is by factor of ≈ .5 too small. The values of X used in the fit correspond to X having values in {(2/3)2, (3/4)2, (4/5)2, (5/6)2, (15/17)2, (15/16)2, (16/17)2} ≈ {.44, .56,.64,.69,.78, .88,.89} and their inverses. The tables summarizing the resulting fit using both X=1 and X giving optimal fit are here. The deviations are typically few per cent and one must also take into account the fact that the masses of stars are deduced theoretically using the spectral data from star models. I am not able to form an opinion about the real error bars related to the masses.

The Appendix of the chapter TGD and Astrophysics contains more details.



Dark matter based model for Pioneer and flyby anomalies

This has been very enjoyable period for dark matter afficionado. During last month I have had an opportunity to apply TGD based vision about dark matter to about five existing or completely new anomalies. Just yesterday I learned about the new findings related to Pioneer and flyby anomalies which challenge the standard theory of gravitation.

I have proposed earlier a model for Pioneer anomaly resulting as a by-product of an explanation of another anomaly which can be understood if cosmic expansion is compensated by a radial contraction of solar system in local Robertson-Walker coordinates. The recent findings reported here allow to sharpen the model suggesting a universal primordial mass density associated with the solar system. The facts about flyby anomaly lead to a model in which anomalous energy gain of the space-craft in Earth-craft rest system results when it passes through a spherical layer of dark matter containing Earth's orbit (this is of course too stringent model). These structures are predicted by the model explaining the Bohr quantization of planetary orbits to served as templates for the condensation of visible matter around them.

1. Explanation of Pioneer anomaly in terms of dark matter

I have proposed an explanation of Pioneer anomaly as a prediction of a model explaining the claimed radial acceleration of planets which is such that it compensates the cosmological expansion of planetary system. The correct prediction is that the anomalous acceleration is given by Hubble constant. The precise mechanism allowing to achieve this was not proposed.

A possible mechanism is based the presence of dark matter increasing the effective solar mass. Since acceleration anomaly is constant, a dark matter density behaving like ρd = (3/4π)(H/Gr), where H is Hubble constant giving M(r) propto r2, is required. For instance, at the radius RJ of Jupiter the dark mass would be about (δa/a) MS≈ 1.3× 10-4MS and would become comparable to MS at about 100RJ=520 AU. Note that the standard theory for the formation of planetary system assumes a solar nebula of radius of order 100AU having 2-3 solar masses. For Pluto at distance of 38 AU the dark mass would be about one per cent of solar mass. This model would suggest that planetary systems are formed around dark matter system with a universal mass density. For this option dark matter could perhaps be seen as taking care of the contraction compensating for the cosmic expansion by using a suitable dark matter distribution.

Here the possibility that the acceleration anomaly for Pioneer 10 (11) emerged only after the encounter with Jupiter (Saturn) is raised. The model explaining Hubble constant as being due to a radial contraction compensating cosmic expansion would predict that the anomalous acceleration should be observed everywhere, not only outside Saturn. The model in which universal dark matter density produces the same effect would allow the required dark matter density ρd= (3/4π)(H/Gr) be present only as a primordial density able to compensate the cosmic expansion. The formation of dark matter structures could have modified this primordial density and visible matter would have condensed around these structures so that only the region outside Jupiter would contain this density. There are also diurnal and annual variations in the acceleration anomaly (see the same article). These variations should be due to the physics of Earth-Sun system. I do not know whether they can be understood in terms of a temporal variation of the Doppler shift due to the spinning and orbital motion of Earth with respect to Sun.

2. Explanation of Flyby anomaly in terms of dark matter

The so called flyby anomaly might relate to the Pioneer anomaly. Fly-by mechanism used to accelerate space-crafts is a genuine three body effect involving Sun, planet, and the space-craft. Planets are rotating around sun in an anticlockwise manner and when the space-craft arrives from the right hand side, it is attracted by a planet and is deflected in an anticlockwise manner and planet gains energy as measured with respect to solar center of mass system. The energy originates from the rotational motion of the planet. If the space-craft arrives from the left, it loses energy. What happens is analyzed the above linked article using an approximately conserved quantity known as Jacobi's integral

J= e- ω ez · r× v.

Here e is total energy per mass for the space-craft, ω is the angular velocity of the planet, ez is a unit vector normal to the planet's rotational plane, and various quantities are with respect to solar cm system.

This as such is not anomalous and flyby effect is used to accelerate space-crafts. For instance, Pioneer 11 was accelerated in the gravitational field of Jupiter to a more energetic elliptic orbit directed to Saturn ad the encounter with Saturn led to a hyperbolic orbit leading out from solar system.

Consider now the anomaly. The energy of the space-craft in planet-space-craft cm system is predicted to be conserved in the encounter. Intuitively this seems obvious since the time and length scales of the collision are so short as compared to those associated with the interaction with Sun that the gravitational field of Sun does not vary appreciably in the collision region. Surprisingly, it turned out that this conservation law does not hold true in Earth flybys. Furthermore, irrespective of whether the total energy with respect to solar cm system increases or decreases, the energy in cm system increases during flyby in the cases considered.

Five Earth flybys have been studied: Galileo-I, NEAR, Rosetta, Cassina, and Messenger and the article of Anderson and collaborators gives a nice quantitative summary of the findings and of the basic theoretical notions. Among other things the tables of the article give the deviation δeg,S of the energy gain per mass in the solar cm system from the predicted gain. The anomalous energy gain in rest Earth cm system is δeEv·δv and allows to deduce the change in velocity. The general order of magnitude is δv/v≈ 10-6 for Galileo-I, NEAR and Rosetta but consistent with zero for Cassini and Messenger. For instance, for Galileo I one has vinf,S= 8.949 km/s and δvinf,S= 3.92+/- .08 mm/s in solar cm system.

Many explanations for the effect can be imagined but dark matter is the most obvious candidate in TGD framework. The model for the Bohr quantization of planetary orbits assumes that planets are concentrations of the visible matter around dark matter structures. These structures could be tubular structures around the orbit or a nearly spherical shell containing the orbit. The contribution of the dark matter to the gravitational potential increases the effective solar mass Meff,S. This of course cannot explain the acceleration anomaly which has constant value.

For instance, if the space-craft traverses shell structure, its kinetic energy per mass in Earth cm system changes by a constant amount not depending on the mass of the space-craft:

δE/m ≈ vinf,E×δv= δVgr = GδMeff,S/R.

Here R is the outer radius of the shell and vinf,E is the magnitude of asymptotic velocity in Earth cm system. This very simple prediction should be testable. If the space-craft arrives from the direction of Sun the energy increases. If the space-craft returns back to the sunny side, the net anomalous energy gain vanishes. This has been observed in the case of Pioneer 11 encounter with Jupiter (see this).

The mechanism would make it possible to deduce the total dark mass of, say, spherical shell of dark matter. One has

δM/MS ≈δv/vinf,E ×(2K/V) , K= v2inf,E/2 , V=GMS/R .

For the case considered δM/MS≥ 2× 10-6 is obtained. Note that the amount of dark mass within sphere of 1 AU implied by the explanation of Pioneer anomaly would be about 6.2× 10-6MS from Pioneer anomaly. Since the orders of magnitude are same one might consider the possibility that the primordial dark matter has concentrated in spherical shells in the case of inner planets as indeed suggested by the model for quantization of radii of planetary orbits.

In the solar cm system the energy gain is not constant. Denote by vi,E and vf,E the initial and final velocities of the space-craft in Earth cm. Let δv be the anomalous change of velocity in the encounter and denote by θ the angle between the asymptotic final velocity vf,S of planet in solar cm. One obtains for the corrected eg,S the expression

eg,S= (1/2)[(vf,E+vPv)2-(vi,E+vP)2] .

This gives for the change δeg,S

δeg,S≈(vf,E+vP)· δv≈ vf,S δv× cos(θS)= vinf,Sδv × cos(θS).

Here vinf,S is the asymptotic velocity in solar cm system and in excellent approximation predicted by the theory. Using spherical shell as a model for dark matter one can write this as

δeg,S= vinf,S/vinf,E × G δM/R × cos(θS) .

The proportionality of δeg,S to cos(θS) should explain the variation of the anomalous energy gain.

For a spherical shell δv is in the first approximation orthogonal to vP since it is produced by a radial acceleration so that one has in good approximation

δeg,Svf,S· δvvf,E· δv≈ vf,Sδv × cos(θS)= vinf,E δv× cos(θE).

For Cassini and Messenger cos(θS) should be rather near to zero so that vinf,E and vinf,S should be nearly orthogonal to the radial vector from Sun in these cases. This provides a clear cut qualitative test for the spherical shell model.

For TGD based view about astrophyscs see the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-Time".



New Results in Planetary Bohr Orbitology

The understanding of how the quantum octonionic local version of infinite-dimensional Clifford algebra of 8-dimensional space (the only possible local variant of this algebra) implies entire quantum and classical TGD led also to the understanding of the quantization of Planck constant. In the model for planetary orbits based on gigantic gravitational Planck constant this means powerful constraints on the number theoretic anatomy of gravitational Planck constants and therefore of planetary mass ratios. These very stringent predictions are immediately testable.

New results in planetary Bohr orbitology

1. Preferred values of Planck constants and ruler and compass polygons

The starting point is that the scaling factor of M4 Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant.

One expects that quantum phases q=exp(iπ/n) which are expressible using only square roots of rationals are number theoretically very special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature.

These polygons are obtained by ruler and compass construction and Gauss showed that these polygons, which could be called Fermat polygons, have

nF= 2ks Fns

sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes Fn=22n+1 correspond to n=0,1,2,3,4 with F0=3, F1=5, F2=17, F3=257, F4=65537. It is not known whether there are higher Fermat primes. n=3,5,15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter.

2. Application to planetary Bohr orbitology

The understanding of the quantization of Planck constants in M4 and CP2 degrees of freedom led to a considerable progress in the understanding of the Bohr orbit model of planetary orbits proposed by Nottale, whose TGD version initiated "the dark matter as macroscopic quantum phase with large Planck constant" program.

Gravitational Planck constant is given by

hbargr/hbar0= GMm/v0

where an estimate for the value of v0 can be deduced from known masses of Sun and planets. This gives v0≈ 4.6× 10-4.

Combining this expression with the above derived expression one obtains

GMm/v0= nF= 2kns Fns

In practice only the Fermat primes 3,5,17 appearing in this formula can be distinguished from a power of 2 so that the resulting formula is extremely predictive. Consider now tests for this prediction.

  1. The first step is to look whether planetary mass ratios can be reproduced as ratios of Fermat primes of this kind. This turns out to be the case if Nottale's proposal for quantization in which outer planets correspond to v0/5: TGD provides a mechanism explaining this modification of v0. The accuracy is better than 10 per cent.

  2. Second step is to look whether GMm/v0 for say Earth allows the expression above. It turns out that there is discrepancy: allowing second power of 17 in the formula one obtains an excellent fit. Only first power is allowed. Something goes wrong! 16 is the nearest power of two available and gives for v0 the value 2-11 deduced from biological applications and consistent with p-adic length scale hypothesis. Amusingly, v0(exp)= 4.6 × 10-4 equals with 1/(27× F2)= 4.5956× 10-4 within the experimental accuracy.

    A possible solution of the discrepancy is that the empirical estimate for the factor GMm/v0 is too large since m contains also the the visible mass not actually contributing to the gravitational force between dark matter objects. M is known correctly from the knowledge of gravitational field of Sun. The assumption that the dark mass is a fraction 1/(1+ε) of the total mass for Earth gives 1+ε= 17/16 in an excellent approximation. This gives for the fraction of the visible matter the estimate ε=1/16≈ 6 per cent. The estimate for the fraction of visible matter in cosmos is about 4 per cent so that estimate is reasonable and would mean that most of planetary and solar mass would be also dark as TGD indeed predicts and for which there are already now several experimental evidence (consider only the evidence that photosphere has solid surface discussed earlier in this blog ).

To sum up, it seems that everything is now ready for the great revolution. I would be happy to share this flood of discoveries with colleagues but all depends on what establishment decides. To my humble opinion twenty one years in a theoretical desert should be enough for even the most arrogant theorist. There is now a book of 800 A4 pages about TGD at Amazon: Topological Geometrodynamics so that it is much easier to learn what TGD is about.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of the book "TGD: an Overview" and the chapter TGD and Astrophysics of this book.



Higgs particle as wormhole contact and weakened form of Equivalence Principle

Quantum classical correspondence has turned out to be the magic tool leading to the a surprisingly detailed understanding of the physics predicted by TGD. I am continuing the updating of TGD forced by the dark matter revolution and newest result relate to a more detailed understanding of particle massivation, the manner how Equivalence Principle is weakened in TGD framework, color confinement, and the variant of Higgs mechanism involved with super-conductivity. The really big surprise was that Higgs particle is nothing but a wormhole contact carrying left handed weak isospin.

1. Higgs as wormhole contact, electro-weak symmetry breaking, the weakening of Equivalence Principle, and color confinement

The proper understanding of the concepts of gauge charges and fluxes and their gravitational counterparts in TGD space-time has taken a lot of efforts. At the fundamental level gauge charges assignable to light-like 3-D elementary particle horizons surrounding a topologically condensed CP2 type extremals can be identified as the quantum numbers assignable to fermionic oscillator operators generating the state associated with horizon identifiable as a parton. Quantum classical correspondence requires that commuting classical gauge charges are quantized and this is expected to be true by the generalized Bohr orbit property of the space-time surface.

There are however non-trivial questions. Do vacuum charge densities give rise to renormalization effects or imply non-conservation so that weak charges would be screened above intermediate boson length scale? Could one assign the non-conservation of gauge fluxes to the wormhole (#) contacts, which are identifiable as pieces of CP2 extremals and for which electro-weak gauge currents are not conserved so that weak gauge fluxes would be non-vanishing but more or less random so that long range correlations would be lost?

It indeed turns that one can understand the non-conservation of weak gauge fluxes in terms of wormhole contacts carrying pairs of right/left handed fermion and left/right handed antifermion having interpretation as Higgs bosons. The average non-conserved light-like gravitational four-momentum of wormhole contact representing Higgs boson can be identified as the inertial four-momentum apart from the sign factor so that one can also understand particle massivation at fundamental level and a connection with p-adic thermodynamics based description of Higgs mechanism emerges. Also a detailed understanding about how Equivalence Principle is weakened in TGD framework emerges.

Also color confinement can be understood using only quantum classical correspondence and general properties of classical color gauge field. Spin glass degeneracy allows to understand the generation of macro-temporal quantum coherence and the same mechanism allows also to understand more quantitatively color confinement by applying unitarity conditions.

2. Dark matter hierarchy and fractal copies of standard model physics

The most dramatic prediction obvious from the beginning but mis-interpreted for about 26 years is the presence of long ranged classical electro-weak and color gauge fields in the length scale of the space-time sheet. The only interpretation consistent with quantum classical correspondence is in terms of a hierarchy of scaled up copies of standard model physics corresponding to p-adic length scale hierarchy and dark matter hierarchy labelled by arbitrarily large values of dynamical quantized Planck constant. Chirality selection in the bio-systems provides direct experimental evidence for this fractal hierarchy of standard model physics.

3. Wormhole contacts, super-conductivity, and biology

Wormhole contacts, feeding gauge fluxes from a given sheet of the 3-space to a larger one, which are a necessary concomitant of the many-sheeted space-time concept. # contacts can be regarded as p"../articles/ carrying classical charges defined by the gauge fluxes but behaving as extremely tiny dipoles quantum mechanically in the case that gauge charge is conserved. # contacts must be light, which suggests that they can form Bose-Einstein condensates and coherent states. The real surprise (after 27 years of TGD) was that the formation of these rather exotic macroscopic quantum phases could be identified as formation of vacuum expectation value of Higgs field for various scaled up copies of standard model physics. This kind of macroscopic quantum phases could be in a central role in the TGD inspired model for a bio-system as a macroscopic quantum system. Electromagnetically charged # contacts are also possible and would explain the massivation of photons in super-conductors implying that long ranged exotic W boson exchanges play a key role in super-conductivity.

For more details see the chapter General Ideas about Topological Condensation and Evaporation.



Empirical support for the p-adic evolution of cosmological constant?

The evolution of the cosmological constant Λ is different at each space-time sheet, and the value of Λ is determined by the p-adic length scale size of the space-time sheet according to the formula Λ (k)= Λ (2)× (L(2)/L(k))2, where L(k)=2kL0, k integer, is the p-adic length scale associated with prime p≈ 2k. L0 is apart from a numerical constant CP2 geodesic length. Prime values of k are especially interesting.

The formula is derived in the chapter TGD and Cosmology of TGD from the requirement that gravitational energy identified as the difference of inertial energies and matter and antimatter (or vice versa) is non-negative. The result means discrete evolution of cosmological constant with jumps in which cosmological constant is reduced by a power of 2.

In standard physics context piecewise constant cosmological constant would be naturally replaced by a cosmological constant behaving like 1/a2 as a function of cosmic time. p-Adic prediction is consistent with the study of Wang and Tegmark according to which cosmological constant has not changed during the last 8 billion years: the conclusion comes from the reshifts of supernovae of type Ia. If p-adic length scales L(k)= p ≈ 2k, k any positive integer, are allowed, the finding gives the lower bound TM > 21/2/( 21/2-1))× 8= 27.3 billion years for the recent age of the universe.

Now Brad Shaefer from Lousiana University has studied the red shifts of gamma ray bursters up to a red shift z=6.3, which corresponds to a distance of 13 billion light years, and claims that the fit to the data is not consistent with the time independence of the cosmological constant. In TGD framework this would mean that a phase transition scaling down the value of the cosmological constant by a power of 2 can be located in cosmological past at a temporal distance in the range 8--13 billion years.

For more details see the chapter Cosmic Strings.



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