On the Physical Interpretation of the Velocity Parameter in the Formula for Gravitational Planck Constant

Abstract

Nottale's formula for the gravitational Planck constant ħ_gr = GMm/v₀  involves parameter v₀ with dimensions of velocity. I have worked with the quantum interpretation of the formula but the physical origin of  v₀ - or equivalently the dimensionless parameter β₀ = v₀ /c (to be used in the sequel) appearing in the formula  has remained open hitherto. In the following, a possible interpretation based on many-sheeted space-time concept, many-sheeted cosmology, and zero energy ontology (ZEO)  is discussed. In ZEO, the non-changing parts of zero energy states are assigned to the passive boundary of CD and β₀ should be assigned to it. There are two measures for the size of the  system. The  M⁴ size L_M4 is identifiable as the maximum of the  radial M⁴ distance  from the tip of CD  associated with the  center of mass of the system along the light-like  geodesic at the boundary of CD. System has also size L_ind defined in terms of the induced metric of  the space-time surface, which is space-like at the boundary of CD.  One has L_ind <L_H.    The identification β₀ = L_M4 / L_H does not allow the identification of L_H = L_M4. L_H would however naturally corresponds to the size of the magnetic body of the system in turn identifiable as the size of CD. One can deduce an estimate for β₀ by approximating the space-time surface as Robertson-Walker cosmology expected to be a good approximation near the passive light-like boundary of CD. The resulting formula is  tested for planetary system and Earth.   The dark matter assignable to Earth can be identified as the innermost part of inner core with volume,  which is .01 per cent of the volume of Earth.