About the interpretation of the parameter β0 and alternative view of the reduction of the oscillator frequency in Allais effect

There are several longstanding questions related to the parameter β₀ appearing in the formula ℏ_gr= GMm/β₀ introduced originally by Nottale.

1. Is the interpretation of β₀ as a velocity parameter necessary? The gravitational Compton length Λ_gr =r_s/2β₀ has no dependence on the small mass m, which conforms with the Equivalence Principle. Also the cyclotron frequencies at the monopole flux tubes of the gravitational field body are independent of m.
2. There are two preferred values for β₀: β₀∼ 1 assigned with the Earth's gravitational field body an and β₀∼ 2^-11 assigned with the field body of the Sun.
3. The velocity of the solar system with respect to the galaxy is of the same magnitude as β₀, which supports the interpretation as velocity. The interpretation of β₀=v₀/c∼ 1 as a velocity of a massive object does not however look sensible.

There might be a very simple solution to these interpretational problems, which I have failed to notice.

1. In the standard quantum theory two quantum lengths characterize a massive particle. The Compton length Λ_c= h/m and the de-Broglie wavelength Λ_de-B= h/mβ₀, where β₀=v₀/c is the velocity of particle using light velocity as a unit.
2. Could the gravitational Planck constant ℏ_gr(S) assigned to Sun and also planets in the Bohr model for planetary orbits corresponds to de-Broglie wave length and could β₀ correspond to a velocity 220-230 km/s giving β ∈ [(.73, .77) × 10^-3] of the solar system with respect to galactic center. The error is about 20 per cent. The gravitational Planck constant assigned with the Earth would correspond to the gravitational Compton length and the problem with β₀=1 would disappear.

There is however an objection against this proposal.

1. M⁸-H duality for the gravitational Planck constant leads to a fractal generalization of Hubble's law suggesting that Hubble tension might relate to two slightly different values of β₀∼1 in short and long length scales differing by 5-6 percent (see this). This interpretation is not consistent with the interpretation of Λ_gr for β₀=1 as gravitational Compton length.
2. The problem disappears if one can interpret v₀≤ c as light velocity with c_#= g_tt^1/2c≤ c along the space-time surface in the formula for the gravitational Compton length.
3. This interpretation has non-trivial consequences. In the case of the Sun, the disappearance of the 1/β₀(S)∼ 2¹¹ from the formula h_gr reduces the gravitational Compton length and gives Λ_gr(S)= 3× 10⁵Λ_gr(E) rather than Λ_gr(S)∼ 2¹¹× 3× 10⁵× Λ_gr(E). The energy E= h_gr(S)f for a given frequency would be also reduced by β₀(S)∼ 2^-11.

How could the proposal ℏ_gr= GMm/c_# implying the formulas for the gravitational Compton length and time and de-Broglie wavelength be tested?

1. For the dark cyclotron the transitions at the magnetic body, the dependendence of cyclotron energy on m disappears. For other frequencies this is not the case and one would have E=h_grf= (GMm/2πc_#)× f. A possible test is to look whether the energies for slightly different masses m differ.
2. Examples would be proton and hydrogen atom with a relative mass difference of order 2^-11 and proton and neutron with mass difference of .14 per cent. One can imagine an entire spectroscopy allowing to test the notion of gravitational Planck constant by using the effects caused by the transformation of gravitationally dark photons to ordinary ones. Biophotons could be products of this transformation.

There might be a connection with the TGD proposed model explaining the Allais effect. There is a surprisingly large reduction of the value of the oscillation frequency having upper bound Δ f/f≤ 2^-11. This brings in mind β₀(S) and the proposal was that the quantum critical transitions involves fluctuations reducing the oscillator frequency satisfying the formula E= h_gr(E)f: now the mass of the pendulum would be in the role of the small mass.

Could the values of E be invariant under fluctuations and so that the small fluctuations of h_gr induce small fluctuations of f.

1. The fluctuations Δ c_#/c_# required to produce the effect would be quite too large as compared to the reduction of the value of c from its maximal value by GM_S/AU =r_s(S)/2AU∼ 10^-9 and GM_E/R_E= r_s(E)/2R_E∼ 10^-9.
2. The model for the Allais effect proposed that diffraction-like effect for the gravitational flux tubes meaning a deviation of the monopole flux tubes, analogous to the deviation of flow lines of a hydrodynamic flow past solid object, could produce reduction of the effective gravitational flux. This would reduce the effective gravitational mass M_S experienced by the pendulum.
3. But why should this reduction be Δ f/f≤ 2^-11? Could the change of the mass of the pendulum could affect the value of h_gr forcing the change of f if E is invariant? The reduction Δ m/m≤ 2^-11 for the mass of the pendulum is highly implausible.
4. What about the mass of the field body? Δ f/f≤ 2^-11 is not far from the electron-proton mass ratio m_e/m_p∼ 1/1880: the deviation is 9 per cent. If the field body contains hydrogen atoms, their ionization to protons and electrons transforming to ordinary electrons would reduce h_gr by the required amount.

   The hydrogen atoms should be Rydberg atoms with a very small binding energy and therefore with very large size: this is indeed possible at the field body. The dropped electrons should have smaller energy compensating for the energy needed for the energy needed for ionization. The transition could take place by tunnelling and therefore involve a pair of "big" state function reductions (BSFRs).

   This kind of phase transition should occur at quantum criticality assigned with the beginning of the solar eclipse? Why the turning of the monopole flux tubes meeting the Moon should induce a phase transition leading to the transformation of dark electrons to ordinary electrons? Are the electrons so near to ionization state the turning ionizes them?

See the article Allais effect again or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.