**1. Simplest candidate for the metric of a rotating star**

The simplest situation for the metric of rotating start is obtained by assuming that vacuum extremal imbeddable to M^{4} × S^{2}, where S^{2} is the geodesic sphere of CP_{2} with vanishing homological charge and induce Kähler form. Use coordinates Θ,Φ for S^{2} and spherical coordinates (t,r,θ,φ) in space-time identifiable as M^{4} spherical coordinates apart from scaling and r-dependent shift in the time coordinate.

- For Schartschild metric one has
Φ= ωt
and

u= sin(Θ)= f(r),

f is fixed highly uniquely by the imbedding of Schwartschild metric and asymptotically one must have

u =u

_{0}+ C/rin order to obtain g

_{tt}= 1-2GM/r (=1+Φ_{gr}) behavior for the induced metric. - The small deformation giving rise to the gravitomagnetic field and metric of rotating star is given by
Φ = ωt+nφ

There is obvious analogy with the phase of Schödinger amplitude for angular momentum eigenstate with L

_{z}=n which conforms with the quantum classical correspondence. - The non-vanishing component of A
^{g}is proportional to gravitational potential Φ_{gr}A

^{g}_{φ}= g_{tφ}= (n/ω)Φ_{gr}. - A little calculation gives for the magnitude of B
_{g}^{θ}from the curl of A^{g}the expressionB

_{g}^{θ}= (n/ω)× (1/sin(θ)× 2GM/r^{3}.In the plane θ=π/2 one has dipole field and the value of n is fixed by the value of angular momentum of star.

- Quantization of angular momentum is obtained for a given value of ω. This becomes clear by comparing the field with dipole field in θ= π/2 plane. Note that GJ, where J is angular momentum, takes the role of magnetic moment in B
_{g}(see this). ω appears as a free parameter analogous to energy in the imbedding and means that the unit of angular momentum varies. In TGD framework this could be interpreted in terms of dynamical Planck constant having in the most general case any rational value but having a spectrum of number theoretically preferred values. Dark matter is interpreted as phases with large value of Planck constant which means possibility of macroscopic quantum coherence even in astrophysical length scales. Dark matter would induce quantum like effects on visible matter. For instance, the periodicity of small n states might be visible as patterns of visible matter with discrete rotational symmetry (could this relate to strange goings on in Saturn? See also the Red Square!).

**2. Comparison with the dipole field**

The simplest candidate for the gravitomagnetic field differs in many respects from a dipole field.

- Gravitomagnetic field has 1/r
^{3}dependence so that the distance dependence is same as in GRT. - Gravitomagnetic flux flows along z-axis in opposite directions at different sides of z=0 plane and emanates from z-axis radially and flows along spherical surface. Hence the radial component of B
_{g}would vanish whereas for the dipole field it would be proportional to cos(θ). - The dependence on the angle θ of spherical coordinates is 1/sin(θ) (this conforms with the radial flux from z-axis whereas for the dipole field the magnitude of B
^{θ}_{g}has the dependence sin(θ). At z=0 plane the magnitude and direction coincide with those of the dipole field so that satellites moving at the gravitomagnetic equator would not distinguish between GRT and TGD since also the radial component of B_{g}vanishes here. - For other orbits effects would be non-trivial and in the vicinity of the flux tube formally arbitrarily large effects are predicted because of 1/sin(θ) behavior whereas GRT predicts sin(θ) behavior. Therefore TGD could be tested using satellites near gravito-magnetic North pole.
- The strong gravimagnetic field near poles causes gravi-magnetic Lorentz force and could be responsible for the formation of jets emanating from black hole like structures and for galactic jets. This additional force might have also played some role in the formation of planetary systems and the plane in which planets move might correspond to the plane θ=π/2, where gravimagnetic force has no component orthogonal to the plane. Same applies in the case of galaxies.

**3. Consistency with the model for the asymptotic state of star**

In TGD framework natural candidates for the asymptotic states of the star are solutions of field equations for which gravitational four-momentum is locally conserved. Vacuum extremals must therefore satisfy the field equations resulting from the variation of Einstein's action (possibly with cosmological constant) with respect to the induced metric. Quite remarkably, the solution representing asymptotic state of the star is necessarily rotating (see this).

The proposed picture is consistent with the model of the asymptotic state of star. Also the magnetic parts of ordinary gauge fields have essentially similar behavior. This is actually obvious since CP_{2} coordinates are fundamental dynamical variables and the field line topologies of induced gauge fields and induced metric are therefore very closely related.

**Addition**: Lubos Motl's blog tells that the error bars are still twice the size of the predicted frame dragging effect. Already this information would have killed TGD inspired (strongly so) model unless the satellite had been at equator! The sad conclusion is that unless my blog page inspires new many year project with satellite nearer to either pole, which does not seem very plausible, we lose the possibility to kill GRT or TGD for years to come.

For the background see the chapter TGD and GRT of "Classical Physics in Many-Sheeted Space-Time".

## 5 comments:

Great stuff, Matti. I wonder how much time there is to think about this before they do come up with a figure for (2).

Dear Kea,

I found no material from web about

the talk of Everitt. Seems that not much interesting has been revealed.

Matti

Dear Kea,

as the little addition to the posting makes clear, the results of experiment could have killed TGD (inspired model at least) unless the satellite would have been on equator! Pity that the experiment will not be repeated at Northern latitudes.

If the experiment had been carried out at higher latitudes I would of course asked how to modify the model;-) rather than concluding that TGD is dead.

The basic alternative is inspired by the topological quantization of (gravi-)magnetic fluxes. The key question is in what scale this quantization becomes visible. I would guess that this scale is rather large as compared to Earth and these effects are not important.

The close relationship between ordinary magnetic field and gravitomagnetic forces however to ask at least half-rhetorically whether gravimagnetic field of Sun could resemble in nearby region that of solar magnetic field. Could same flux tubes carry both of these fields? Also in the case of Earth one can could ask in the same spirit whether gravimagnetic field rotates within Earth inside gravi-magnetosphere so that the simple description would apply only outside it. If gravimagnetic flux quanta correspond to magnetic ones then it might happen that solar wind affects also gravimagnetic flux quanta. I however think that this is not the case.

Matti

I won't pretend I understand the physics in your post, but could you summarize in laymans terms the way that the angular momentum of the planets, particularly Jupiter, could affect the sun. There is an interesting correlation between Jovian orbital periodicity and the cycles of sunspot maxima and minima, and tidal forces aren't strong enough to provide a mechanism which would explain it.

The work of Charvakova elucidates the correlation, but doesn't explain the underlying cause.

Thanks

Dear tallbloke,

you posed an interesting question which I cannot answer. The correlation you mentioned is unknown to me. Could you give some link?

Matti

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