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Topological Geometrodynamics: an Overview

Note: Newest contributions are at the top!

Year 2011


Sean Carroll writes about breakdown of classical gravity in Cosmic variance. Recall that the galactic dark matter problem arose with the observation that the velocity spectrum of distance star is constant rather than behaving as 1/r as Newton's law assuming that most mass is in the galactic center predicts.

The MOND theory and its variants predict that there is a critical acceleration below which Newtonian gravity fails. This would mean that Newtonian gravitation is modified at large distances. String models and also TGD predict just the opposite since in this regime General Relativity should be a good approximation.

  1. The 1/r2 force would transform to 1/r force at some critical acceleration of about a=10-10 m/s2: this is a fraction of 10-11 about the gravitational acceleration at the Earth's surface.

  2. What Sean Carroll wrote about was the empirical study giving support for this kind of transition in the dynamics of stars at large distances and therefore breakdown of Newtonian gravity in MOND like theories.

In TGD framework critical acceleration is predicted but the recent experiment does not force to modify Newton's laws. Since Big Science is like market economy in the sense that funding is more important than truth, the attempts to communicate TGD based view about dark matter have turned out to be hopeless. Serious Scientist does not read anything not written on silk paper.

  1. One manner to produce this spectrum is to assume density of dark matter such that the mass inside sphere of radius R is proportional to R at last distances. Decay products of and ideal cosmic strings would predict this. The value of the string tension predicted correctly by TGD using the constraint that p-adic mass calculations give electron mass correctly.

  2. One could also assume that galaxies are distributed along cosmic string like pearls in necklace. The mass of the cosmic string would predict correct value for the velocity of distant stars. In the ideal case there would be no dark matter outside these cosmic strings.

    1. The difference with respect to the first mechanism is that this case gravitational acceleration would vanish along the direction of string and motion would be free motion. The prediction is that this kind of motions take place along observed linear structures formed by galaxies and also along larger structures.

    2. An attractive assumption is that dark matter corresponds to phases with large value of Planck constant is concentrated on magnetic flux tubes. Holography would suggest that the density of the magnetic energy is just the density of the matter condensed at wormhole throats associated with the topologically condensed cosmic string.
    3. Cosmic evolution modifies the ideal cosmic strings and their Minkowski space projection gets gradually thicker and thicker and their energy density - magnetic energy - characterized by string tension could be affected
TGD option differs from MOND in some respects and it is possible to test empirically which option is nearer to the truth.

  1. The transition at same critical acceleration is predicted universally by this option for all systems-now stars- with given mass scale if they are distributed along cosmic strings like like pearls in necklace. The gravitational acceleration due the necklace simply wins the gravitational acceleration due to the pearl. Fractality encourages to think like this.

  2. The critical acceleration predicted by TGDr depends on the mass scale as a ∝ GT2/M, where M is the mass of the object- now star. Since the recent study considers only stars with solar mass it does not allow to choose between MOND and TGD and Newton can continue to rest in peace in TGD Universe. Only a study using stars with different masses would allow to compare the predictions of MOND and TGD and kill either option or both. Second test distinguishing between MOND and TGD is the prediction of large scale free motions by TGD option.

TGD option explains also other strange findings of cosmology.

  1. The basic prediction is the large scale motions of dark matter along cosmic strings. The characteristic length and time scale of dynamics is scaled up by the scaling factor of hbar. This could explain the observed large scale motion of galaxy clusters -dark flow- assigned with dark matter in conflict with the expectations of standard cosmology.

  2. Cosmic strings could also relate to the strange relativistic jet like structures meaning correlations between very distant objects. Universe would be a spaghetti of cosmic strings around which matter is concentrated.

  3. The TGD based model for the final state of star actually predicts the presence of string like object defining preferred rotation axis. The beams of light emerging from supernovae would be preferentially directed along this lines- actually magnetic flux tubes. Same would apply to the gamma ray bursts from quasars, which would not be distributed evenly in all directions but would be like laser beams along cosmic strings.

For more about TGD based vision about cosmology and astrophysics see the chapter Cosmology and Astrophysics in Many-Sheeted Space-Time.

Is the effective metric defined by modified gamma matrices effectively one- or two-dimensional?

The following argument suggests that the effective metric defined by the anti-commutators of the modified gamma matrices is effectively one- or two-dimensional. Effective one-dimensionality would conform with the observation that the solutions of the modified Dirac equations can be localized to one-dimensional world lines in accordance with the vision that finite measurement resolution implies discretization reducing partonic many-particle states to quantum superpositions of braids. This localization to 1-D curves occurs always at the 3-D orbits of the partonic 2-surfaces.

The argument is based on the following assumptions.

  1. The modified gamma matrices for Kähler action are contractions of the canonical momentum densities Tαk with the gamma matrices of H.

  2. The strongest assumption is that the isometry currents

    J =Tα kjAk

    for the preferred extremals of Kähler action are of form

    JA α= ΨA (∇Φ)α

    with a common function Φ guaranteeing that the flow lines of the currents integrate to coordinate lines of single global coordinate variables (Beltrami property). Index raising is carried out by using the ordinary induced metric.

  3. A weaker assumption is that one has two functions Φ1 and Φ2 assignable to the isometry currents of M4 and CP2 respectively.:

    JA α1 = Ψ1A (∇Φ1)α ,

    JA α2 = Ψ2A (∇Φ2)α .

    The two functions Φ1 and Φ2 could define dual light-like curves spanning string world sheet. In this case one would have effective 2-dimensionality and decomposition to string world sheets (for the concrete realization see this). Isometry invariance does not allow more that two independent scalar functions Φi.

Consider now the argument.

  1. One can multiply both sides of this equation with jAk and sum over the index A labeling isometry currents for translations of M4 and SU(3) currents for CP2. The tensor quantity ∑A jAkjAl is invariant under isometries and must therefore satisfy

    A ηABjAkjAl= hkl ,

    where ηAB denotes the flat tangent space metric of H. In M4 degrees of freedom this statement becomes obvious by using linear Minkowski coordinates.

    In the case of CP2 one can first consider the simpler case S2=CP1= SU(2)/U(1). The coset space property implies in standard complex coordinate transforming linearly under U(1) that only the the isometry currents belonging to the complement of U(1) in the sum contribute at the origin and the identity holds true at the origin and by the symmetric space property everywhere. Identity can be verified also directly in standard spherical coordinates. The argument generalizes to the case of CP2=SU(3)/U(2) in an obvious manner.

  2. In the most general case one obtains

    Tα k1 =∑AΨ1A jAk × (∇Φ1)α == fk1 (∇Φ1)α ,

    Tα k2 =∑AΨ1A jAk × (∇Φ2)α ≡ fk2 (∇Φ2)α .

    Here i=1 refers to M4 part of energy momentum tensor and i=2 to its CP2 part.

  3. The effective metric given by the anti-commutator of the modified gamma matrices is in turn is given by

    Gα β = mklfk1fl1 (∇Φ1)α(∇Φ1)β +skl fk2 fl2 (∇Φ2)α(∇Φ2)β .

    The covariant form of the effective metric is effectively 1-dimensional for Φ12 in the sense that the only non-vanishing component of the covariant metric Gα β is diagonal component along the coordinate line defined by Φ≡ Φ12. Also the contravariant metric is effectively 1-dimensional since the index raising does not affect the rank of the tensor but depends on the other space-time coordinates. This would correspond to an effective reduction to a dynamics of point-like particles for given selection of braid points. For Φ1≠ Φ2 the metric is effectively 2-dimensional and would correspond to stringy dynamics.

For background see the chapter Overall View about TGD from Particle Physics Perspective.

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