## Tree like structure of the extended imbedding space
The quantization of hbar in multiples of integer n
characterizing the quantum phase q=exp(iπ/n) in
M
Two imbedding spaces with different scalings
factors of metrics are glued directly together only
if either M
- In the phase transition between different
hbar(M
^{4}):s the projection of the 3-surface to M^{4}becomes single point so that a cross section of CP_{2}type extremal representing elementary particle is in question. Elementary particles could thus leak between different M^{4}:s easily and this could occur in large hbar(M^{4}) phases in living matter and perhaps even in quantum Hall effect. Wormhole contacts which have point-like M^{4}projection would allow topological condensation of space-time sheets with given hbar(M^{4}) at those with different hbar(M^{4}) in accordance with the heuristic picture. - In the phase transition different between
CP
_{2}:s the CP_{2}projection of 3-surface becomes point so that the transition can occur in regions of space-time sheet with 1-D CP_{2}projection. The regions of a connected space-time surface corresponding to different values of hbar (CP_{2}) can be glued together. For instance, the gluing could take place along surface X^{3}=S^{2}× T (T corresponds time axis) analogous to black hole horizon. CP_{2}projection would be single point at the surface. The contribution from the radial dependence of CP_{2}coordinates to the induced metric giving ds^{2}= ds^{2}(X^{3})+g_{rr}dr^{2}at X^{3}implies a radial gravitational acceleration and one can say that a gravitational flux is transferred between different imbedding spaces.Planetary Bohr orbitology predicting that only 6 per cent of matter in solar system is visible suggests that star and planetary interiors are regions with large value of CP _{2}Planck constant and that only a small fraction of the gravitational flux flows along space-time sheets carrying visible matter. In the approximation that visible matter corresponds to layer of thickness Δ R at the outer surface of constant density star or planet of radius R, one obtains the estimate Δ R=.12R for the thickness of this layer: convective zone corresponds to Δ R=.3R. For Earth one would have Δ R≈ 70 km which corresponds to the maximal thickness of the crust. Also flux tubes connecting ordinary matter carrying gravitational flux leaving space-time sheet with a given hbar (CP_{2}) at three-dimensional regions and returning back at the second end are possible. These flux tubes could mediate dark gravitational force also between objects consisting of ordinary matter.
Concerning the mathematical description of this
process, the selection of origin of M For details see the chapter Does TGD Predict the Spectrum of Planck Constants. |