About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant
Nottale's formula for the gravitational Planck constant hbar_{gr}= GMm/v_{0} involves parameter v_{0} with dimensions of velocity. I have worked with the quantum interpretation of the formula but the physical origin of v_{0} - or equivalently the dimensionless parameter β_{0}=v_{0}/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. A generalization of the Hubble formula β=L/L_{H} for the cosmic recession velocity, where L_{H}= c/H is Hubble length and L is radial distance to the object, is suggestive. This interpretation would suggest that some kind of expansion is present. The fact however is that stars, planetary systems, and planets do not seem to participate cosmic expansion. In TGD framework this is interpreted in terms of quantal jerk-wise expansion taking place as relative rapid expansions analogous to atomic transitions or quantum phase transitions. The TGD based variant of Expanding Earth model assumes that during Cambrian explosion the radius of Earth expanded by factor 2. There are two measures for the size of the system. The M^{4} size L_{M4} is identifiable as the maximum of the radial M^{4} 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 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_{M4}. The identification β_{0}= L_{M4}/L_{H}<1 does not allow the identification 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 β_{0} by approximating the space-time surface near the light-cone boundary as Robertson-Walker cosmology, and expressing the mass density ρ defined as ρ=M/V_{M4}, where V_{M4}=(4π/3) L_{M4}^{3} is the M^{4} volume of the system. ρ can be expressed as a fraction ε^{2} of the critical mass density ρ_{cr}= 3H^{2}/8π G. This leads to the formula β_{0}= [r_{S}/L_{M4}]^{1/2} × (1/ε), where r_{S} is Schwartschild radius. This 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. Also the consistency of the Bohr quantization for dark and ordinary matter is discussed and leads to a number theoretical condition on the ratio of the ordinary and dark masses. See 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}About the Nottale's formula for h_{gr} and the relation between Planck length and CP_{2} length R or the article About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant. |