### What does one really mean with gravitational Planck constant?

There are important questions related to the notion of gravitational Planck constant, to the identification of gravitational constant, and to the general structure of magnetic body. What gravitational Planck constant really is? What the formula for gravitational constant in terms of CP

_{2}length defining Planck length in TGD does really mean, and is it realistic? What space-time surface as covering space does really mean?

**What does one mean with space-time as covering space?**

The central idea is that space-time corresponds to n-fold covering for h_{eff}=n× h_{0}. It is not however quite clear what this statement does mean.

- How the many-sheeted space-time corresponds to the space-time of QFT and GRT? QFT-GRT limit of TGD is defined by identifying the gauge potentials as sums of induced gauge potentials over the space-time sheets. Magnetic field is sum over its values for different space-time sheets. For single sheet the field would be extremely small in the present case as will be found.

- A central notion associated with the hierarchy of effective Planck constants h
_{eff}/h_{0}=n giving as a special case ℏ_{gr}= GMm/v_{0}assigned to the flux tubes mediating gravitational interactions. The most general view is that the space-time itself can be regarded as n-sheeted covering space. A more restricted view is that space-time surface can be regarded as n-sheeted covering of M^{4}. But why not n-sheeted covering of CP_{2}? And why not having n=n_{1}× n_{2}such that one has n_{1}-sheeted covering of CP_{2}and n_{2}-sheeted covering of M^{4}as I indeed proposed for more than decade ago but gave up this notion later and consider only coverings of M^{4}? There is indeed nothing preventing the more general coverings.

- n=n
_{1}× n_{2}covering can be illustrated for an electric engineer by considering a coil in very thin 3 dimensional slab having thickness L. The small vertical direction would serve and as analog of CP_{2}. The remaining 2 large dimensions would serve as analog for M^{4}. One could try to construct a coil with n loops in the vertical direction direction but for very large n one would encounter problems since loops would overlap because the thickness of the wire would be larger than available room L/n. There would be some maximum value of n, call it n_{max}.

One could overcome this limit by using the decomposition n=n

_{1}× n_{2}existing if n is prime. In this case one could decompose the coil into n_{1}parallel coils in plane having n_{2}≥ n_{max}loops in the vertical direction. This provided n_{2}is small enough to avoid problems due to finite thickness of the coil. For n prime this does not work but one can of also select n_{2}to be maximal and allow the last coil to have less than n_{2}loops.

An interesting possibility is that that preferred extremal property implies the decomposition n

_{gr}=n_{1}× n_{2}with nearly maximal value of n_{2}, which can vary in some limits. Of course, one of the n_{2}-coverings of M^{4}could be in-complete in the case that n_{gr}is prime or not divisible by nearly maximal value of n_{2}. We do not live in ideal Universe, and one can even imagine that the copies of M^{4}covering are not exact copies but that n_{2}can vary.

- In the case of M
^{4}× CP_{2}space-time sheet would replace single loop of the coil, and the procedure would be very similar. A highly interesting question is whether preferred extremal property favours the option in which one has as analog of n_{1}coils n_{1}full copies of n_{2}-fold coverings of M^{4}at different positions in M^{4}and thus defining an n_{1}covering of CP_{2}in M^{4}direction. These positions of copies need not be close to each other but one could still have quantum coherence and this would be essential in TGD inspired quantum biology.

Number theoretic vision suggests that the sheets could be related by discrete isometries of CP

_{2}possibly representing the action of Galois group of the extension of rationals defining the adele and since the group is finite sub-group of CP_{2}, the number of sheets would be finite.

The finite sub-groups of SU(3) are analogous to the finite sub-groups of SU(2) and if they action is genuinely 3-D they correspond to the symmetries of Platonic solids (tetrahedron,cube,octahedron, icosahedron, dodecahedron). Otherwise one obtains symmetries of polygons and the order of group can be arbitrary large. Similar phenomenon is expected now. In fact the values of n

_{2}could be quantized in terms of dimensions of discrete coset spaces associated with discrete sub-groups of SU(3). This would give rise to a large variation of n_{2}and could perhaps explain the large variation of G identified as G= R^{2}(CP_{2})/n_{2}suggested by the fountain effect of superfluidity.

- There are indeed two kinds of values of n: the small values n=h
_{em}/h_{0}=n_{em}assigned with flux tubes mediating em interaction and appearing already in condensed matter physics and large values n=h_{gr}/h_{0}=n_{gr}associated with gravitational flux tubes. The small values of n would be naturally associated with coverings of CP_{2}. The large values n_{gr}=n_{1}× n_{2}would correspond n_{1}-fold coverings of CP_{2}consisting of complete n_{2}-fold coverings of M^{4}. Note that in this picture one can formally define constants ℏ(M^{4})= n_{1}ℏ_{0}and ℏ(CP_{2})= n_{2}ℏ_{0}as proposed for more than decade ago.

**Planck length as CP**

_{2}radius and identification of gravitational constant GThere is also a puzzle related to the identification of gravitational Planck constant. In TGD framework the only theoretically reasonable identification of Planck length is as CP_{2} length R(CP_{2}), which is roughly 10^{3.5} times longer than Planck length. Otherwise one must introduce the usual Planck length as separate fundamental length. The proposal was that gravitational constant would be defined as G =R^{2}(CP_{2})/ℏ_{gr}, ℏ_{gr}≈ 10^{7}ℏ. The G indeed varies in un-expectedly wide limits and the fountain effect of superfluidity suggests that the variation can be surprisingly large.

There are however problems.

- Arbitrary small values of G=R
^{2}(CP_{2})/ℏ_{gr}are possible for the values of ℏ_{gr}appearing in the applications: the values of order n_{gr}∼ 10^{13}are encountered in the biological applications. The value range of G is however experimentally rather limited. Something clearly goes wrong with the proposed formula.

- Schwartschild radius r
_{S}= 2GM = 2R^{2}(CP_{2})M/ℏ_{gr}would decrease with ℏ_{gr}. One would expect just the opposite since fundamental quantal length scales should scale like ℏ_{gr}.

- What about Nottale formula ℏ
_{gr}= GMm/v_{0}? Should one require self-consistency and substitute G= R^{2}(CP_{2})/ℏ_{gr}to it to obtain ℏ_{gr}=(R^{2}(CP_{2})Mm/v_{0})^{1/2}. This formula leads to physically un-acceptable predictions, and I have used in all applications G=G_{N}corresponding to n_{gr}∼ 10^{7}as the ratio of squares of CP_{2}length and ordinary Planck length.

_{2})= n

_{2}ℏ

_{0}, n

_{2}≈ 10

^{7}and nearly maximal except possibly in some special situations? For n

_{gr}=n

_{1}× n

_{2}the covering corresponding to ℏ

_{gr}would be n

_{1}-fold covering of CP

_{2}formed from n

_{1}n

_{2}-fold coverings of M

^{4}. For n

_{gr}=n

_{1}× n

_{2}the covering would decompose to n

_{1}disjoint M

^{4}coverings and this would also guarantee that the definition of r

_{S}remains the standard one since only the number of M

^{4}coverings increases.

If n_{2} corresponds to the order of finite subgroup G of SU(3) or number of elements in a coset space G/H of G (itself sub-group for normal sub-group H), one would have very limited number of values of n_{2}, and it might be possible to understand the fountain effect of superfluidity from the symmetries of CP_{2}, which would take a role similar to the symmetries associated with Platonic solids. In fact, the smaller value of G in fountain effect would suggest that n_{2} in this case is larger than for G_{N} so that n_{2} for G_{N} would not be maximal.

See the article TGD View about Quasars
or the chapter About the Nottale's formula for h_{gr} and the possibility that Planck length l_{P} and CP_{2} length R are identical giving G= R^{2}/ℏ_{eff}.

For a summary of earlier postings see Latest progress in TGD.

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