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Physics in Many-Sheeted Space-Time

Note: Newest contributions are at the top!

Year 2008

Anyonic view about black holes

A new element to the model of black hole comes from the vision that black hole horizon as a light-like 3-surface corresponds to a light-like orbit of light-like partonic 2-surface. This allows two kinds of black holes. Fermion like black hole would correspond to a deformed CP2 type extremal which Euclidian signature of metric and topologically condensed at a space-time sheet with a Minkowskian signature. Boson like black hole would correspond to a wormhole contact connecting two space-time sheets with Minkowskian signature. Wormhole contact would be a piece deformed CP2 type extremal possessing two light-like throats defining two black hole horizons very near to each other. It does not seem absolutely necessary to assume that the interior metric of the black-hole is realized in another space-time sheet with Minkowskian signature.

Second new element relates to the value of Planck constant. For (h/2p)gr = 4GM2 the Planck length LP((h/2p)) = {(h/2p) G} equals to Schwartschild radius and Planck mass equals to MP((h/2p)) = {(h/2p)/G} = 2M. If the mass of the system is below the ordinary Planck mass: M mP((h/2p)0)/2 = {(h/2p)0/4G}, gravitational Planck constant is smaller than the ordinary Planck constant.

Black hole surface contains ultra dense matter so that perturbation theory is not expected to converge for the standard value of Planck constant but do so for gravitational Planck constant. If the phase transition increasing Planck constant is a friendly gesture of Nature making perturbation theory convergent, one expects that only the black holes for which Planck constant is such that GM2/4p(h/2p) < 1 holds true are formed. Black hole entropy -being proportional to 1/(h/2p)- is of order unity so that TGD black holes are not very entropic.

If the partonic 2-surface surrounds the tip of causal diamond CD, the matter at its surface is in anyonic state with fractional charges. Anyonic black hole can be seen as single gigantic elementary particle stabilized by fractional quantum numbers of the constituents preventing them from escaping from the system and transforming to ordinary visible matter. A huge number of different black holes are possible for given value of (h/2p) since there is infinite variety of pairs (na,nb) of integers giving rise to same value of (h/2p).

One can imagine that the partonic surface is not exact sphere except for ideal black holes but contains large number of magnetic flux tubes giving rise to handles. Also a pair of spheres with different radii can be considered with surfaces of spheres connected by braided flux tubes. The braiding of these handles can represent information and one can even consider the possibility that black hole can act as a topological quantum computer. There would be no sharp difference between the dark parts of black holes and those of ordinary stars. Only the volume containing the complex flux tube structures associated with the orbits of planets and various objects around star would become very small for black hole so that the black hole might code for the topological information of the matter collapsed into it.

For details and background see the updated chapter Quantum Astrophysics.

Revised vision about quantum astrophysics

I had deduced earlier a formula for the quantized Planck constant from the requirement that it represents algebraic homomorphism. Two options for which Planck constants were inverses of each other were possible. As usual, I chose the wrong one! The development of a detailed model for fractional quantum Hall effect fixed the choice on basis of physical arguments. The next task is to go through all applications and make the needed modifications. I started from Quantum Astrophysics. A glue below the abstract.

The vision that the quantum dynamics for dark matter is behind the formation of the visible structures suggests that the formation of the astrophysical structures could be understood as a consequence of gravitational Bohr rules. The origin of these rules has remained a little bit mysterious until the discovery that the hierarchy of Planck constants relates very closely to anyons and fractionization of quantum numbers.

  1. Key element is the notion of partonic 2-surface, which for large values of Planck constant can have astrophysical size. This surface contains dark matter in anyonic many particle state if it surrounds the tip of so called causal diamond (the intersection of future and past directed light-cones). Also flux tubes surrounding the orbits of planets and other astrophysical objects containing dark matter would be connected by radial flux tubes to central anyonic 2-surface so that the entire system would be anyonic and quantum coherent in astrophysical scale. Visible matter is condensed around these dark matter structures.

  2. Since space-times are 4-surfaces in H=M4×CP2 (or rather, its generalization to a book like structure), gravitational Bohr rules can be formulated in a manner which is general coordinate invariant and Lorentz invariant.

  3. The value of the parameter v0 appearing in gravitational Planck constant varies and this leads to a weakened form of Equivalence Principle stating that v0 is same for given connected anyonic 2-surface, which can have very complex topology. In the case of solar system inner planets would be connected to an anyonic surface assignable to Sun and outer planets with different value of v0 to an anyonic surface assignable to Sun and inner planets as a whole. If one accepts ruler-and-compass hypothesis for allowed values of Planck constant very powerful predictions follow.

This general conceptual framework is applied to build simple models in some concrete examples.

  1. Concerning Bohr orbitology in astrophysical length scales, the basic observation is that in the case of a straight cosmic string creating a gravitational potential of form v12/r Bohr quantization does not pose any conditions on the radii of the circular orbits so that a continuous mass distribution is possible. This situation is obviously exceptional. If one however accepts the TGD based vision that the very early cosmology was cosmic string dominated and that elementary p"../articles/ were generated in the decay of cosmic strings, this situation might have prevailed at very early times. If so, the differentiation of a continuous density of ordinary matter to form the observed astrophysical structures would correspond to an approach to a stationary situation governed by Bohr rules for dark matter and in the first approximation one could neglect the intermediate stages.

  2. This general picture is applied by considering some simple models for astrophysical systems involving planar structures. There are several universal predictions. Velocity spectrum is universal and only the Bohr radii depend on the choice of mass distribution. The inclusion of cosmic string implies that the system associated with the central mass is finite. Quite generally dark parts of astrophysical objects have shell like structure like atoms as do also ring like structures.

  3. p-Adic length scale hypothesis provides a manner to obtain a realistic model for the central objects meaning a structure consisting of shells coming as half octaves of the basic radius: this obviously relates to Titius-Bode law. Also a simple model for planetary rings is obtained. Bohr orbits do not follow cosmic expansion which is obtained only in the average sense if phase transitions reducing the value of basic parameter v0 occur at preferred values of cosmic time. This explains why v0 has different values and also the decomposition of planetary system to outer and inner planets with different values of v0.

TGD Universe is quantum critical and quantum criticality corresponds very naturally to what has been identified as the transition region to quantum chaos.

  1. The basic formulation of quantum TGD is consistent with what has been learned from the properties of quantum chaotic systems and quantum chaotic scattering. Wave functions are concentrated around Bohr orbits in the limit of quantum chaos, which is just what dark matter picture assumes.

  2. The model for the emission and detection of dark gravitons allows to conclude that the transition to chaos via generation of sub-harmonics of fundamental frequency spoiling the original exact periodicity corresponds to a sequence of phase transitions in which Planck constant transforms from integer to a rational number whose denominator increases as chaos is approached. This gives a precise characterization for the phase transitions leading to quantum chaos in general.

  3. In this framework the chaotic motion of astrophysical object becomes the counterpart of quantum chaotic scattering and the description in terms of classical chaos is predicted to fail. By Equivalence Principle the value of the mass of the object does not matter at all so that the motion of sufficiently light objects in solar system might be understandable only as quantum chaotic scattering. The motion of gravitationally unbound comets and rings of Saturn and Jupiter and the collisions of galactic structures known to exhibit the presence of cart-wheel like structures define possible applications.

The description of gravitational radiation provides a stringent test for the idea about dark matter hierarchy with arbitrary large values of Planck constants. In accordance with quantum classical correspondence, one can take the consistency with classical formulas as a constraint allowing to deduce information about how dark gravitons interact with ordinary matter. The standard facts about gravitational radiation are discussed first and then TGD based view about the situation is sketched.

For details and background see the updated chapter Quantum Astrophysics.

Flyby anomaly as relativistic transverse Doppler effect?

For half year ago I discussed a model for the flyby anomaly based on the hypothesis that dark matter ring around the orbit of Earth causes the effect. The model reproduced the formula deduced for the change of the velocity of the space-craft at qualitative level, and contained single free parameter: essentially the linear density of the dark matter at the flux tube.

From Lubos I learned about a new twist in the story of flyby anomaly. September twelfth 2007 Jean-Paul Mbelek proposed an explanation of the flyby anomaly as a relativistic transverse Doppler effect. The model predicts also the functional dependence of the magnitude of the effect on the kinematic parameters and the prediction is consistent with the empirical findings in the example considered. Therefore the story of flyby anomaly might be finished and dark matter at the orbit of Earth could bring in only an additional effect. It is probably too much to hope for this kind of effect to be large enough if present.

For background see the chapter TGD and Astrophysics.

Could symplectic QFT allow to understand the fluctuations of CMB?

Depending on one's attitudes, the anomalies of the fluctuation spectrum of the cosmic microwave background (CMB) can be seen as a challenge for people analyzing the experiments or that of the inflationary scenario. I do not pretend to be deeply involved with CMB but as I read about one these anomalies in Sean Carrol's blog and next day in Lubos's blog, I felt that I could spend some days by clarifying myself what is involved.

What interests me whether the replacement of inflation with quantum criticality and hbar changing phase transitions could provide fresh insights about fluctuations and anomalies of CMB. In the following I try first to explain to myself what the anomalies are and after that I will consider some TGD inspired crazy (as always) ideas. My motivations to communicate are indeed strong: the consideration of the anomalies led to a generalization of the notion of conformal QFT to what might be called symplectic QFT having very natural place also in quantum TGD proper.

A brief summary about my views is as follows.

  1. There are several types of anomalies (with respect to the expectations motivated by inflation theory). Fluctuation spectrum shows hot and cold spots; there is so called hemispherical asymmetry in the spectrum; rotationally averaged two-point correlation function C(q) is vanishing for angle separations above 60 degrees; the so called cosmic axis of evil means that 3 of the multipole area vectors assignable to the l=2 and l=3 spherical harmonics in the expansion of C(q) are aligned and in the galactic plane (very near to ecliptic) and in roughly the same direction as the dipole corresponding to the motion of the Milky Way with respect to cosmic frame of reference; there is also evidence for non-Gaussianity meaning non-vanishing 3-point functions. Especially strange finding is that the features of the local geometry seems to reflect themselves in CMB at the surface of last scattering. If these findings are not artifacts of the analysis or pure accidents, the consequences for our understanding of the Cosmos would be dramatic.

  2. In TGD framework quantum criticality replaces inflation. This means that the fluctuations of CMB do not correspond to primordial fluctuations of inflaton field evolved into large scale fluctuations during rapid expansion but to long range fluctuations involved with a phase transition increasing Planck constant and occurring at the time of decoupling. The p-adic length scale involved with the sphere of last scattering and the amplitude of the fluctuations provide two dimensionless couplings and allow to make estimates for the scaling of Planck constant in this transition.

  3. I have suggested earlier a conformal field theory defined at the sphere of last scattering as a TGD based model for the anomalies. The analysis of the situation however demonstrates that a more natural approach is based on symplectic variant of conformal QFT at the sphere of last scattering. By combining generalized fusion rules with the knowledge about symplectic invariants associated with 3 or 4 points of sphere, one can deduce surprisingly detailed information about the n-point functions of symplectic QFTs. There are two variants of the theory: the first one is rotation- and reflection symmetric. The fact that the generalization of the notion of imbedding space allows to identify preferred quantization axis, allows to formulate also a variant theory breaking these symmetries. Fusion rules determine n-point functions highly uniquely in terms of 3-point functions expressible as functions of simple symplectic invariants. Symplectic QFT makes special predicions distinguishing it from inflationary models: a sizable non-Gaussianity is predicted and correlation functions vanish when any two arguments are very near to each other. It is certainly possible to reproduce the vanishing of C(q) for large values of q but it is not clear whether fusion rules allow this.

  4. Quite generally, symplectic QFT provides a long sought-for manner to describe the vacuum degeneracy of TGD in terms of n-point functions. What is of special importance is that the n-point functions have no singularities at the limit when some arguments co-incide. This means a profound distinction from quantum field theories and something like this is required by general arguments demonstrating that quantum TGD is free of the standard divergences due to the non-locality of Kähler function as a functional of 3-surface. The classification of symplectic QFTs should be a fascinating challenge for a mathematician. The basic challenge is to determine how uniquely fusion rules determine the 3-point functions generating all other n-point functions.

  5. The possibility of having gigantic values of gravitational Planck constant and zero energy ontology suggests that quantum measurements in cosmological scales are possible. This would mean that time-like entanglement between positive and negative energy parts of zero energy state could correlate galactic geometry with the geometry of fluctuation spectrum so that the hemispherical asymmetry with respect to galactic plane could be produced by this kind of quantum measurement. This would mean a dramatic proof of the notion of participatory Universe introduced by Wheeler.

For details see the chapter Quantum Astrophysics or the article Could symplectic quantum field theory allow to model the fluctuations cosmic microwave background?.

From Equivalence Principle to Zero Energy Ontology

The tension between analytic and constructive approaches is present even in Einstein's own theory and was the basic stimulus leading to TGD. Equivalence Principle states that gravitational and inertial masses are identical. This statement is however rather problematic as Einstein himself was first to admit. Einstein's equations express the identity for gravitational and inertial energy momentum densities as a consequence of a variational principle. There is however no global version of this statement because one cannot define the notions of inertial and gravitational four-momenta without adherence to perturbative approach.

The hypothesis that space-times are 4-D surfaces of a higher-dimensional space-time of form H=M4×S resolves the problem: since Poincare symmetries become symmetries of H rather than space-time itself. Inertial four-momentum can be defined as a conserved Noether charge and also gravitational four-momentum can be regarded as a Noether charge albeit non-conserved. Equivalence Principle can hold true only under some additional conditions. For instance, for the imbeddings of Robertson-Walker cosmologies inertial four-momentum density vanishes unlike gravitational four-momentum density, which for a long time remained quite a mystery. The real understanding of the situation became possible only after the introduction of what I call zero energy ontology.

In zero energy ontology one replaces positive energy states with zero energy states with positive and negative energy parts of the state at the boundaries of future and past direct light-cones forming a causal diamond. All conserved quantum numbers of the positive and negative energy states are of opposite sign so that these states can be created from vacuum. Äny physical state is creatable from vacuum" becomes thus a basic principle of quantum TGD and together with the notion of quantum jump resolves several philosophical problems (What was the initial state of universe?, What are the values of conserved quantities for Universe, Is theory building completely useless if only single solution of field equations is realized?).

At the level of elementary particle physics positive and negative energy parts of zero energy state are interpreted as initial and final states of a particle reaction so that quantum states become physical events. Equivalence Principle would hold true in the sense that the classical gravitational four-momentum of the vacuum extremal whose small deformations appear as the argument of configuration space spinor field is equal to the positive energy of the positive energy part of the zero energy quantum state. Equivalence Principle is expected to hold true for elementary p"../articles/ and their composites but not for the quantum states defined around non-vacuum extremals.

More precisely, the inertial four-momentum assignable to the 3-D Chern-Simons action is non-vanishing only if one adds to the CP2 Kähler form a pure gauge part Aa=constant, where a denotes light cone proper time . A breaking of Poincare invariance is implied which is however compensated by the fact that configuration space corresponds to the union of configuration spaces associated with future and past directed light-cones. If the vacuum extremal is also an extremal of the curvature scalar, gravitational four-momentum is conserved.

In the case of CP2 type vacuum extremal gravitational stationarity transforms the M4 projection of the extremal from a random light-like curve to a light-like geodesic allowing an interpretation as incoming or outgoing on mass shell particle. General vacuum extremal corresponds to a virtual particle. At the classical level Equivalence Principle requires that the light-like gravitational four-momentum of CP2 vacuum extremal co-incides with the light-like inertial four-momentum associated with Chern-Simons action in this situation. This condition relates the value of Aa to gravitational constant G and CP2 radius R. G would thus appear as a fundamental constant and quantum criticality should dictate the ratio G/R2. The topologically condensed CP2 type vacuum extremal necessarily creates a non-vacuum region around it and the resulting inertial four-momentum corresponds to the gravitational four-momentum.

The strong form of Equivalence Principle would require that the classical 4-momentum associated with Kähler action of allowed small deformations co-incides with the conserved gravitational four-momentum of the vacuum extremal extremizing curvature scalar: this might have a natural interpretation in terms of Bohr orbitology but is not be consistent with zero energy ontology inspired picture unless one has double sheeted structure with sheets possessing opposite energies such that double sheeted structure is approximated by single sheet with Robertson-Walker cosmology in GRT framework. The identification of gauge bosons as wormhole contacts and gravitons as pairs of wormhole contacts supports double sheeted structure with sheets possessing opposite arrows of geometric time. Near the vicinity of wormhole contacts (pieces of CP2 type vacuum extremals) the sheets which are originally vacuum extremals are deformed to non-vacuum extremals and carry inertial four-momentum which should be equal to the gravitational four momentum of the vacuum extremal.

For background see the chapter TGD and GRT . See also the article "Topological Geometrodynamics: an Overall View".

The mystery of mini galaxies and the hierarchy of Planck constants

New Scientist informs that a team led by Pieter van Dokkum at Yale University measured the light of distant galaxies from around 3 billion years after the big bang. They had the same mass as the Milky Way, but were 10 times smaller (The Astrophysical Journal, vol 677, p L5). Peering at younger regions of the sky shows that galaxies this size are no longer around, but it's not clear what happened to them. "This is a very puzzling result," says Simon White of the Max Planck Institute for Astrophysics in Garching, Germany. "Galaxies cannot disappear." Team member Marijn Franx of Leiden Observatory, the Netherlands, suspects they have since merged with extremely massive galaxies.

The evidence of Bohr quantization of planetary orbits with a gigantic value of Planck constant led originally to the idea that Planck constant in TGD Universe has a spectrum of discrete values. The hierarchy of Planck constants leads to generalization of the imbedding space H=M4×CP2 so that it becomes a book like structure with pages characterized partially by the value of Planck constant (see this). This generalization is not anymore ad hoc but is essential for the realization of quantum criticality and the construction of quantum TGD proper: for instance, Higgs mechanism and the manner how space-time sheet is assigned with light-like 3-surfaces defining the fundamental dynamical objects of quantum TGD can be understood in this framework (see this ).

The matters at different pages would be dark relative to each other since the p"../articles/ from different pages cannot appear in the same interaction vertex (see for instance this). The p"../articles/ at different pages can however interact via classical gauge- and gravitational fields leaking between the pages via the back of the book, and by the exchange of p"../articles/ tunneling in the same manner and thus experiencing a phase transition changing Planck constant. The model for living matter and gravitational interactions suggests that the space-time sheets containing dark matter are magnetic and gravimagnetic flux tubes (perhaps the attribute "wormhole magnetic" is more accurate characterization) (see this).

These assumptions guarantee consistency with what is known about dark matter (long range classical gauge fields do not - or at least are not believed to- exist) so that only the classical gravitational interaction and graviton exchanges necessarily involving hbar changing phase transition remain in astrophysical length scales. The interactions with p"../articles/ of visible matter via graviton exchanges are predicted to differ from what GTR predicts (see this): ordinary gravitons could appear as bursts corresponding to the decays of gigantic gravitons with large hbar to ordinary gravitons. These kind of bursts would be interpreted as external perturbations and would not be counted as gravitons.

This framework also explains various really weird anomalies in neuroscience and biology, in particular those associated with living cell, if considerable part of biologically important ions are in phase with large Planck constant phase at what I call wormhole magnetic flux tubes (see this, this, this, and this).

Consider now the mysterious disappearance of mini galaxies. The smooth cosmic expansion of standard cosmology is in TGD replaced by quantum leaps analogous to quantum transitions of atoms in which the Planck constant associated with the space-time sheet increases by a factor which is most naturally power of 2. The expansion of dark matter forces the visible matter condensed around it to follow. There are several applications: in particular, the piecewise expansion of Earth explains those forgotten facts about the evolution of continents which Wegener's theory failed to explain.

The disappearance of the mini galaxies would be due to this mechanism. From the assumption that this mechanism gives rise to the same outcome as smooth expansion within factor of two at given moment, one could estimate their recent size. If the galaxies are assumed to have roughly the size of Milky Way now, an upwards scaling would be roughly by a factor 8. This would mean that recent age of Universe would be about 24 billion years.

For background see the chapter TGD and Astrophysics .

Dark matter based model for Flyby anomaly

The so called flyby anomaly provides a test for any theory of gravitation, quantal or not. I have already earlier discussed a model of this anomaly based on dark matter located either at spherical shell or tube around Earth's orbit. The recent data (see this and this ) allowed to fix the model to a tube around the orbit of Earth deformed by gravimagnetic force of Earth to the direction of equatorial plane of Earth.

1. Flyby 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 δv inf,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. One can also consider dark matter rings associated with planets and perhaps even Moon's orbit is an obvious candidate now. It turns out that the tube associated with Earth's orbit and deformed by Earth's presence to equatorial plane of Earth explains qualitatively the known facts.

2. Dark matter at the orbit of Earth?

The almost working model is based on dark matter on the orbit of Earth. One can estimate the change of the kinetic energy in the following manner.

  1. Assume that the the orbit is not modified at all in the lowest order approximation and estimate the kinetic energy gained as the work done by the force caused by the dark matter on the space-craft.

    ΔE/m= -Gdρdark/dl × ∫γEdl EγS drSrSE /rSE3 ,

    rSE== rS-rE .

    Here γS denotes the portion of the orbit of space-craft during which the effect is noticeable and γE denotes the orbit of Earth.

    This expression can be simplified by performing the integration with respect to rS so that one obtains the difference of gravitational potential created by the dark matter tube at the initial and final points of the portion of γS: ΔE/m= V(rS,f)-V(rS,i),

    V(rS)=-G×(dρdark/dl)×∫γEdl E /rSE

  2. Use the standard approximation (briefly described in (see this)) in which the orbit of the spacecraft consists of three parts joined continuously together: the initial Kepler orbit around Sun, the piece of orbit which can be approximate with a hyperbolic orbit around Earth, and the final Kepler orbit around Sun. The piece of the hyperbolic orbit can be chosen to belong inside the so called sphere of influence, whose radius r is given in terms of the distance R of planet from Sun by the Roche limit r/R= (3m/MSun)2/5. γS could be in the first approximation taken to correspond to this portion of the orbit of spacecraft.

  3. The explicit expression for the hyperbolic orbit can be obtained by using the conservation of energy and angular momentum and reads as

    u=rs/r= 2GM/r= (u02/2v02)×(1+X1/2],

    X=1+4u 02×v2v02/sin2(φ),

    u0== rs/a , |v×r|== vr ×sin(φ) .

    The unit c=1 is used to simplify the formulas. rs denotes Schwartschild radius and v the asymptotic velocity. v0 and a are the velocity and distance at closest approach and the conserved angular momentum is given by L/m= v0 a. In the situation considered value of rS is around 1 cm, the value of a around 107 m and the value of v of order 10 km/s so that the approximation

    u ≈ u0× (v/v0)×sin(φ)

    is good even at the distance of closest approach. Recall that the parameters characterizing the orbit are the distance a of the closest approach, impact parameter b, and the angle 2θ characterizing the angle between the two straight lines forming the asymptotes of the hyperbolic orbit in the orbital plane PE.

Consider first some conclusions that one can make from this model.

  1. Simple geometric considerations demonstrate that the acceleration in the region between Earth's orbit and the part of orbit of spacecraft for which the distance from Sun is larger than that of Earth is towards Sun. Hence the distance of the spacecraft from Earth tends to decrease and the kinetic energy increases. In fact, one could also choose the portion of γS to be this portion of the spacecraft's orbit.

  2. ΔE depends on the relative orientation of the normal nS of the the orbital plane PE of spacecraft with respect to normal nO the orbital plane PO of Earth. The orientation can be characterized by two angles. The first angle could be the direction angle Θ of the position vector of the nearest point of spacecraft's orbit with respect to cm system. Second angle, call it Φ, could characterize the rotation of the orbital plane of space-craft from the standard orientation in which orbital plane and space-craft's plane are orthogonal. Besides this ΔE depends on the dynamical parameters of the hyperbolic orbit of space-craft given by the conserved energy Etot =E and angular momentum or equivalently by the asymptotic velocity v and impact parameter b.

  3. Since the potential associated with the closed loop defined by Earth's orbit is expected to resemble locally that of a straight string one expects that the potential varies slowly as a function of rS and that ΔE depends weakly on the parameters of the orbit.

The most recent report (see this ) provides additional information about the situation.

  1. ΔE is reported to be proportional to the total orbital energy E/m of the space-craft. Naively one would expect (E/m)1/2 behavior coming from the proportionality ΔE to 1/r. Actually a slower logarithmic behavior is expected since a potential of a linear structure is in question.

  2. ΔE depends on the initial and final angles θi and θ f between the velocity v of the space-craft with respect to the normal nE of the equatorial plane PE or Earth and the authors are able to give an empirical formula for the energy increment. The angle between PE and P O is 23.4 degrees. One might hope that the formula could be written also in terms of the angle between v and the normal nO of the orbital plane. For θi ≈ θf the effect is known to be very small. A particular example corresponds to a situation in which one has θi=32 degrees and θf =31 degrees. Obviously the PO≈ PE approximation cannot hold true. Needless to say, also the model based on spherical shell of dark matter fails.

3. Is the tube containing the dark matter deformed locally into the equatorial plane?

The previous model works qualitatively if the interaction of Earth and flux tube around Earth's orbit containing the dark matter modifies the shape of the tube locally so that the portion of the tube contributing to the anomaly lies in a good approximation in PE rather than P O. In this case the minimum value of the distance rES between γ E and γS is maximal for the symmetric situation with θi f and the effect is minimal. In an asymmetric situation the minimum value of rES decreases and the size of the effect increases. Hence the model works at least qualitatively of the motion of Earth induces a moving deformation of the dark matter tube to PE. With this assumption one can write ΔE in a physically rather transparent form showing that it is consistent with the basic empirical findings.

  1. By using linear superposition one can write the potential as sum of a potential associated with a tube associated with Earths orbit plus the potential associated with the deformed part minus the potential associated with corresponding non-deformed portion of Earth's orbit:

    ΔE/m= V(rS,f)-V(rS,i) ,

    V(rS)=-G×(dρdark/dl)Z(rS) ,

    Z(rS)= X(γorb;rS)+ X(γd;rS) -X(γnd;rS) ,

    X(γi;rS) = ∫γidl/rSi.

    Here the subscripts "orb", "d" and "nd" refer to the entire orbit of Earth, to its deformed part, and corresponding non-deformed part. The entire orbit is analogous to a potential of straight string and is expected to give a slowly varying term which is however non-vanishing in the asymmetric situation. The difference of deformed and non-deformed parts gives at large distances dipole type potential behaving like 1/r2 and thus being proportional to v 2 by the above expression for the u=rs/r. The fact that ΔE is proportional to v2 suggests that dipole approximation is good.

  2. One can therefore parameterize ΔE as

    ΔE/m= V(rS,f)-V(rS,i) ,

    V(rS)=-G×(dρ dark/dl)×Z ,

    Z(rS)= X(γorb;rS)+ d×cos(Θ)/rS 2,

    where Θ is the angle between r and the dipole d, which now has dimension of length. The direction of the dipole is in the first approximation in the equatorial plane and and directed orthogonal to the Earth's orbit.

Consider now the properties of ΔE.

  1. In a situation symmetric with respect to the equator Ed vanishes but End is non-vanishing which gives as a result potential difference associated with entire Earth's orbit minus the part of orbit contributing to the effect so that the result is by the definition of the approximation very small.

  2. As already noticed, dipole field like behavior that the large contribution to the potential is proportional to the conserved total energy v02/2 at the limit of large kinetic energy.

  3. From the fact that potential difference is in question it follows that the expression for the energy gain is the difference of parameters characterizing the initial and final situations. This conforms qualitatively with the observation that this kind of difference indeed appears in the empirical fit. 1/r2-factor is also proportional to sin2(φ) which by the symmetry of the situation is expected to be same for initial and final situation. Furthermore, ΔE is proportional to the difference of the parameter cos(Θf)-cos(Θi) and this should correspond to the reported behavior. Note that the result vanishes for the symmetric situation in accordance with the empirical findings.

To sum up, it seems that the qualitative properties of ΔE are indeed consistent with the empirical findings. The detailed fit of the formula of the recent paper should allow to fix the shape of the deformed part of the orbit.

4. What induces the deformation?

Authors suggest that the Earth's rotation is somehow involved with the effect. The first thing to notice is that the gravimagnetic field of Earth, call it BE, predicted by General Relativity is quite too weak to explain the effect as a gravimagnetic force on spacecraft and fails also to explain the fact that energy increases always. Gravito-Lorentz force does not do any work so that the total energy is conserved and ΔE=-ΔV=-grad V.•Δ r holds true, where Δr is the deflection caused by the gravimagnetic field on the orbit during flyby. Since Δr is linear in v, ΔE changes sign as the velocity of space-craft changes sign so that this option fails in several manners.

Gravimagnetic force of Earth could be however involved but in a different manner.

  1. The gravimagnetic force between Earth and flux tube containing the dark matter could explain this deformation as a kind of frame drag effect: dark matter would tend to follow the spinning of Earth. If the dark matter inside the tube is at rest in the rest frame of Sun (this is not a necessary assumption), it moves with respect to Earth with a velocity v=-vE , where vE is the orbital velocity of Earth. If the tube is thin, the gravito-Lorentz force experienced by dark matter equals in the first approximation to F=-vE × BE with BE evaluated at the axis of the tube. TGD based model for BE (see this) does not allow BE to be a dipole field. BE has only the component Bθ and the magnitude of this component relates by a factor 1/sin(θ) to the corresponding component of the dipole field and becomes therefore very strong as one approaches poles. The consistency with the existing experimental data requires that BE at equator is very nearly equal to the strength of the dipole field. The magnitude of BE and thus of F is minimal when the deformation of the tube is in PE, and the deformation occurs very naturally into P E since the non-gravitational forces associated with the dark matter tube must compensate a minimal gravitational force in dynamical equilibrium.

  2. BθE at equator is in the direction of the spin velocity ω of the Earth. The direction of vE varies. It is convenient to consider the situation in the rest system of Sun using Cartesian coordinates for which the orbital plane of Earth corresponds to (x,y) plane with x- and y-axis in the direction of semi-minor and semi-major axes of the Earth's orbit. The corresponding spherical coordinates are defined in an obvious manner. vE is parallel to the tangent vector eφ(t)=-sin (Ωt) ex+ cos(Ωt)ey of the Earth's orbit. The direction of B E at equator is parallel to ω and can be parameterized as eω= cos(θ) ez+ sin(θ)(cos(α) ex+sin(α)ey). F is parallel to the vector -cos(θ) eρ(t) + sin(θ)cos(Ωt-α)ez, where eρ(t) is the unit vector directed from Sun to Earth. The dominant component is directed to Sun.

Happy note added: The price of "../articles/ of PRL are really dirty, 25 dollars per article. Hence I decided to take the risk and put the prediction on blog without knowing whether it is correct. Today I got the article in email. The prediction for the increment of kinetic energy was correct! Champagne for that! Flyby will be for the TGD based view about dark matter what the shift of perihelion of Mercury was for General Relativity! Let us now cross our fingers and hope that it would not take too many decades for this message to diffuse through the cognitive immune system of the super string hegemony.

For TGD based view about astrophysics see the chapter TGD and Astrophysics .

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