### p-Adic length scale hypothesis and the weakness of the gravitational constant

In the posting Reduction of long length scale real physics to short length scale p-adic physics and vice versa I discussed the idea that long range p-adically fractal physics can be reduced to local p-adic physics with p-adic continuity and smoothness alone implying the universal characteristics of long scale physics. Real field equations would determine local real physics and their p-adic counterparts the global real physics.

I mentioned also how multi-p-fractality and the pairing p≈2^{k}, k prime, in p-adic length scale hypothesis can be understood in this scenario.
There is however a problem involved with the understanding of the origin of the p-adic length scale hypothesis if the correspondence via common rationals is assumed.

**1. How p-adic length scale hypothesis can be understood?**

Here the problem and its resolution is discussed (I have discussed this point already earlier but from different view point).

- The mass calculations based on p-adic thermodynamics for Virasoro generator L
_{0}predict that mass squared is proportional to 1/p and Uncertainty Principle implies that L_{p}is proportional to p^{1/2}rather than p, which looks more natural if common rationals define the correspondence between real and p-adic physics. - It would seem that length d
_{p}≈ pR, R or order CP_{2}length, in the induced space-time metric must correspond to a length L_{p}≈ p^{1/2}R in M^{4}. This could be understood if space-like geodesic lines at real space-time sheet obeying effective p-adic topology are like orbits of a particle performing Brownian motion so that the space-like geodesic connecting points with M^{4}distance r_{M}has a length r_{X}propto r_{M}^{2}. Geodesic random walk with randomness associated with the motion in CP_{2}degrees of freedom could be in question. The effective p-adic topology indeed induces a strong local wiggling in CP_{2}degrees of freedom so that r_{X}increases and can depend non-linearly on r_{M}. - If the size of the space-time sheet associated with the particle has size d
_{p}≈ pR in the induced metric, the corresponding M^{4}size would be about L_{p}propto p^{1/2}R and p-adic length scale hypothesis results. - The strongly non-perturbative and chaotic behavior r
_{X}propto r_{M}^{2}is assumed to continue only up to L_{p}. At longer length scales the space-time distance d_{p}associated with L_{p}becomes the unit of space-time distance and geodesic distance r_{X}is in a good approximation given byr

_{X}= (r_{M}/L_{p})d_{p}propto p^{1/2}× r_{M},and is thus linear in M

^{4}distance r_{M}.

**2. How to understand the smallness of gravitational constant?**

The proposed explanation of the p-adic length scale hypothesis allows also to understand the weakness of the gravitational constant as being due to the fact that the space-time distance r_{X} appearing in gravitational force as given by Newtonian approximation is by a factor p^{1/2} times longer than r_{M} so that the strong gravitational force proportional to L_{p}^{2}/r_{X}^{2} scales down by a factor p as r_{X} is expressed in terms of r_{M}. M^{4} distance r_{M} is indeed the natural variable since distances are measured using space-time sheets as units and their sizes are always measured in M^{4} metric or almost flat X^{4} metric.

A more precise argument goes as follows.

- Assume first that the space-time sheet is characterized by single prime p: also multi-p-fractality is possible. The strong gravitational constant G
_{s}characterizes the interactions involving exchanges of string like objects of size scale measured naturally using L_{p}as a unit. In this case the force is mediated in M^{4}as an exchange of a particle. By dimensional estimate G_{s}is proportional to L_{p}^{2}and string model picture gives a precise estimate for the numerical factor n in G_{s}=nL_{p}^{2}. - The classical gravitational force is mediated via induced metric inside the space-time sheets and in long length scales is proportional to G
_{s}/r_{X}^{2}propto L_{p}^{2}/r_{X}^{2}propto R^{2}/r_{M}^{2}, where R≈ 10^{4}G^{1/2}is CP_{2}length. Hence the effective gravitational constant is reduced by a factor 1/p and is same for all values of p. - The value of the gravitational constant is still by a factor of order R
^{2}/G≈ 10^{8}too high. A correct value is obtained if multi-p-fractality prevails in such a manner that p^{1/2}is replaced by n^{1/2}with n=2× 3× 5...× 23× p. One can visualize the situation as hierarchy of wavelets: to p-adic wavelets very small q=23-adic wawelets are superposed to which in turn q=19-adic ... The earlier estimates for the gravitational constant are consistent with this result and fix the numerical details. - This approach predicts a p-adic hierarchy of strong gravitons and unstable spin 2 hadrons are excellent candidates for them. It is however not clear whether Newtonian graviton is predicted at all: in other words could the gravitation inside space-time sheets be a purely classical phenomenon? One can certainly imagine the exchange of topologically condensed Newtonian gravitons moving along light-like geodesics along space-time sheets and the lengths of spatial projections of these geodesics would be indeed scaled up by p
^{1/2}factor.

Matti Pitkanen

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