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 M4 × S2, where S2 is the geodesic sphere of CP2 with vanishing homological charge and induce Kähler form. Use coordinates Θ,Φ for S2 and spherical coordinates (t,r,θ,φ) in space-time identifiable as M4 spherical coordinates apart from scaling and r-dependent shift in the time coordinate.
- For Schartschild metric one has
u= sin(Θ)= f(r),
f is fixed highly uniquely by the imbedding of Schwartschild metric and asymptotically one must have
u =u0 + C/r
in order to obtain gtt= 1-2GM/r (=1+Φgr) 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 Lz=n which conforms with the quantum classical correspondence.
- The non-vanishing component of Ag is proportional to gravitational potential Φgr
Agφ= gtφ = (n/ω)Φgr.
- A little calculation gives for the magnitude of Bgθ from the curl of Ag the expression
Bgθ= (n/ω)× (1/sin(θ)× 2GM/r3.
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 Bg (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/r3 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 Bg 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 Bθg 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 Bg 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 CP2 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".