### More precise interpretation of gravitational Planck constant

The notion of gravitational Planck constant hgr=GMm/v0 was introduced originally by Nottale. In TGD it was interpreted in terms of astrophysical quantum coherence. The interpretation was that hgr characterizes a gravitational flux tube connecting masses M and m and v0 is a velocity parameter - some characteristic velocity assignable to the system.

It has become clear that a more precise formulation of the rather loose ideas about how gravitational interaction is mediated by flux tubes is needed.

1. The assumption treats the two masses asymmetrically.
2. A huge number of flux tubes is needed since every particle pair M-m would involve a flux tube. It would be also difficult to understand the fact that one can think the total gravitational interaction in Newtonian framework as sum over interactions with the composite particles of M. In principle M can be decomposed into parts in many manners - elementary particles and their composites and larger structures formed from them: there must be some subtle difference between these different compositions - all need not be possible - not seen in Newtonian and GRT space-time but maybe having representation in many-sheeted space-time and involving hgr.
3. Flux tube picture in the original form seems to lead to problems with the basic properties of the gravitational interaction: namely superposition of gravitational fields and absence or at least smallness of screening by masses between M and m. One should assume that the ends of the flux tubes associated with the pair pair M-m move as m moves with respect to M. This looks too complex.

Linear superposition and absence of screening can be understood in the picture in which particles form topological sum contacts with the flux tubes mediating gravitational interaction. This picture is used to deduce QFT-GRT limit of TGD. Note that also other space-time sheets can mediate the interaction and pairs of MEs and flux tubes emanating from M but not ending to m are one possible option. In the following I however talk about flux tubes.

These problems find a solution if hgr characterizes the magnetic body (MB) of a particle with mass m topologically condensed to a flux tube carrying total flux M. m can also correspond to a mass larger than elementary particle mass. This makes the situation completely symmetric with respect to M and m. The essential point is that the interaction takes place via touching of MB of m with flux tubes from M.
1. In accordance with the fractality of the many-sheeted space-time, the elementary particle fluxes from a larger mass M can combine to a sum of fluxes corresponding to masses Mi<M with ∑ Mi=M at larger flux tubes with hbargr=GMMi/v0,i> hbar. This can take place in many manners, and in many-sheeted space-time gives rise to different physical situations.

Due to the large value of hgr it is possible to have macroscopic quantum phases at these sheets with a universal gravitational Compton length Lgr= GMim/v0. Here m can be also a mass larger than elementary particle mass. In fact, the convergence of perturbation theory indeed makes the macroscopic quantum phases possible. This picture holds true also for the other interactions. Clearly, many-sheeted space-time brings in something new, and there are excellent reasons to believe that this new relates to the emergence of complexity - say via many-sheeted tensor networks (see this).

2. Quantum criticality would occur near the boundaries of the regions from which flux runs through wormhole contacts from smaller to larger flux sheets and would be thus associated with boundaries defined by the throats of wormhole contacts at which the induced metric changes from Minkowskian to Euclidian.
3. This picture implies that fountain effect - one of the applications of large hgr phase is a kind of antigravity effect for dark matter - maybe even for non-microscopic masses m - since the larger size of MB implies larger average distance from the source of the gravitational flux and the experienced gravitational field is weaker. This might have technological applications some day.
This picture is a considerable improvement but there are still problems to ponder. In particular, one should understand why the integer n= heff/h= hgr/h interpreted as a number of sheets of the singular covering space of MB of m emerges topologically. The large value of hgr implies a huge number of sheets.

Could the flux sheet covering associated with Mi code the value of Mi using as unit Planck mass as the number of sheets of this covering? One would have N=M/MPl sheeted structure with each sheet carrying Planckian flux. The fluxes experienced by the MB of m in turn would consist of sheets carrying fusion nm= MPlv0/m Planckian fluxes so that the total number of sheets would be reduced to n= N/nm= GMm/v0 sheets.

Why this kind of fusion of Planck fluxes to larger fluxes should happen? Could quantum information theory provide clues here? And why v0 is involved?

See the chapter Criticality and dark matter.