Allais effect as evidence for large values of gravitational Planck constant?

I have considered two models for Allais effect. The first model was constructed for several years ago and was based on classical Z⁰ force. For a couple of weeks ago I considered a model based on gravitational screening. It however turned that this model does not work. The next step was the realization that the effect might be a genuine quantum effect made possible by the gigantic value of the gravitational Planck constant: the pendulum would act as a highly sensitive gravitational interferometer.

One can represent rather general counter arguments against the models based on Z⁰ conductivity and gravitational screening if one takes seriously the puzzling experimental findings concerning frequency change.

1. Allais effect identified as a rotation of oscillation plane seems to be established and seems to be present always and can be understood in terms of torque implying limiting oscillation plane.

2. During solar eclipses Allais effect however becomes much stronger. According to Olenici's experimental work the effect appears always when massive objects form collinear structures.

3. The behavior of the change of oscillation frequency seems puzzling. The sign of the frequency increment varies from experiment to experiment and its magnitude varies within five orders of magnitude. There is also evidence that the effect is present also before and after the full eclipse. The time scale is 1 hour.

The above findings allow to make some important conclusions about the nature of Allais effect.

1. Some genuinely new dynamical effect should take place when the objects are collinear. If gravitational screening would cause the effect the frequency would always grow but this is not the case.

2. If stellar objects and also ring like dark matter structures possibly assignable to their orbits are Z⁰ conductors, one obtains screening effect by polarization and for the ring like structure the resulting effectively 2-D dipole field behaves as 1/\rho² so that there are hopes of obtaining large screening effects and if the Z⁰ charge of pendulum is allow to have both signs, one might hope of being to able to explain the effect. It is however difficult to understand why this effect should become so strong in the collinear case.

3. The apparent randomness of the frequency change suggests that interference effect made possible by the gigantic value of gravitational Planck constant is in question. On the other hand, the dependence of Δg/g on pendulum suggests a breaking of Equivalence Principle. It however turns out that the variation of the distances of the pendulum to Sun and Moon can explain the experimental findings since the pendulum turns out to act as a sensitive gravitational interferometer. An apparent breaking of Equivalence Principle could result if the effect is partially caused by genuine gauge forces, say dark classical Z⁰ force, which can have arbitrarily long range in TGD Universe.

4. If topological light rays (MEs) provide a microscopic description for gravitation and other gauge interactions one can envision these interactions in terms of MEs extending from Sun/Moon radially to pendulum system. What comes in mind that in a collinear configuration the signals along S-P MEs and M-P MEs superpose linearly so that amplitudes are summed and interference terms give rise to an anomalous effect with a very sensitive dependence on the difference of S-P and M-P distances and possible other parameters of the problem. One can imagine several detailed variants of the mechanism. It is possible that signal from Sun combines with a signal from Earth and propagates along Moon-Earth ME or that the interferences of these signals occurs at Earth and pendulum.

5. Interference suggests macroscopic quantum effect in astrophysical length scales and thus gravitational Planck constants given by hbar_gr= GMm/v₀, where v₀=2^-11 is the favored value, should appear in the model. Since hbar_gr= GMm/v₀ depends on both masses this could give also a sensitive dependence on mass of the pendulum. One expects that the anomalous force is proportional to hbar_gr and is therefore gigantic as compared to the effect predicted for the ordinary value of Planck constant.

2. Model for interaction via gravitational MEs with large Planck constant

Restricting the consideration for simplicity only gravitational MEs, a concrete model for the situation would be as follows.

1. The picture based on topological light rays suggests that the gravitational force between two objects M and m has the following expression

   F_M,m=GMm/r²= ∫|S(λ,r)|² p(λ)dλ

   p(λ)=h_gr(M,m)2π/λ , hbar_gr= GMm/v₀(M,m) .

   p(λ) denotes the momentum of the gravitational wave propagating along ME. v₀ can depend on (M,m) pair. The interpretation is that |S(λ,r)|² gives the rate for the emission of gravitational waves propagating along ME connecting the masses, having wave length λ, and being absorbed by m at distance r.

2. Assume that S(λ,r) has the decomposition

   S(λ,r)= R(λ)exp[iΦ(λ)]exp[ik(λ)r]/r,

   exp[ik(λ)r]=exp[ip(λ)r/hbar_gr(M,m)],

   R(λ)= |S(λ,r)|.

   To simply the treatment the phases exp(iΦ(λ)) are assumed to be equal to unity in the sequel. This assumption turns out to be consistent with the experimental findings. Also the assumption v₀(M,P)/v₀(S,P) will be made for simplicity: these conditions guarantee Equivalence Principle. The substitution of this expression to the above formula gives the condition

   ∫ |R(λ)|²dλ/λ =v₀ .

1. In the non-collinear case one obtains just the standard Newtonian prediction for the net forces caused by Sun and Moon on the pendulum since Z_S,P and Z_M,P correspond to non-parallel MEs and there is no interference.

2. In the collinear case the interference takes place. If interference occurs for identical momenta, the interfering wavelengths are related by the condition

   p(λ_S,P)=p(λ_M,P) .

   This gives

   λ_M,P/λ_S,P= hbar_M,P/hbar_S,P =M_M/M_S .

3. The net gravitational force is given by

   F_gr= ∫ |Z(λ,r_S,P)+ Z(λ/x,r_M)|² p(λ) dλ

   =F_gr(S,P)+ F_gr(M,P) + ΔF_gr ,

   ΔF_gr= 2∫ Re[S(λ,r_S,P)S^*(λ/x,r_M,P))] (hbar_gr(S,P)2π/λ)dλ,

   x=hbar_S,P/hbar_M,P= M_S/M_M.

   Here r_M,P is the distance between Moon and pendulum. The anomalous term Δ F_gr would be responsible for the Allais effect and change of the frequency of the oscillator.

4. The anomalous gravitational acceleration can be written explicitly as

   Δa_gr= (2GM_S/r_Sr_M)×(1/v₀(S,P))× I ,

   I= ∫ R(λ)×R(λ/x)× cos[2π(y_Sr_S-xy_Mr_M)/λ] dλ/λ ,

   y_M= r_M,P/r_M , y_S=r_S,P/r_S.

   Here the parameter y_M (y_S) is used express the distance r_M,P (r_S,P) between pendulum and Moon (Sun) in terms of the semi-major axis r_M (r_S)) of Moon's (Earth's) orbit. The interference term is sensitive to the ratio 2π(y_Sr_S-xy_Mr_M)/λ. For short wave lengths the integral is expected to not give a considerable contribution so that the main contribution should come from long wave lengths. The gigantic value of gravitational Planck constant and its dependence on the masses implies that the anomalous force has correct form and can also be large enough.

5. If one poses no boundary conditions on MEs the full continuum of wavelengths is allowed. For very long wave lengths the sign of the cosine terms oscillates so that the value of the integral is very sensitive to the values of various parameters appearing in it. This could explain random looking outcome of experiments measuring Δf/f. One can also consider the possibility that MEs satisfy periodic boundary conditions so that only wave lengths λ_n= 2r_S/n are allowed: this implies sin(2π y_Sr_S/λ)=0. Assuming this, one can write the magnitude of the anomalous gravitational acceleration as

   Δa_gr= (2GM_S/r_S,Pr_M,P)×(1/v₀(S,P)) × I ,

   I=∑_n=1^∞ R(2r_S,P/n)×R(2r_S,P/nx)× (-1)ⁿ × cos[nπx×(y_M/y_S)×(r_M/r_S)].

   If R(λ) decreases as λ^k, k>0, at short wavelengths, the dominating contribution corresponds to the lowest harmonics. In all terms except cosine terms one can approximate r_S,P resp. r_M,P with r_S resp. r_M.

6. The presence of the alternating sum gives hopes for explaining the strong dependence of the anomaly term on the experimental arrangement. The reason is that the value of xyr_M/r_S appearing in the argument of cosine is rather large:

   x(y_M/y_S))r_M/r_S)= (y_M/y_S) (M_S/M_M)(r_M/r_S)(v₀(M,P)/v₀(S,P)) ≈ 6.95671837× 10⁴× (y_M/y_S).

   The values of cosine terms are very sensitive to the exact value of the factor M_Sr_M/M_Mr_S and the above expression is probably not quite accurate value. As a consequence, the values and signs of the cosine terms are very sensitive to the value of y_M/y_S.

   The value of y_M/y_S varies from experiment to experiment and this alone could explain the high variability of Δf/f. The experimental arrangement would act like interferometer measuring the distance ratio r_M,P/r_S,P.

The assumption of the scaling law

R(λ)=R₀ (λ/λ₀)^k

is very natural in light of conformal invariance and masslessness of gravitons and allows to make the model more explicit. With the choice λ₀=r_S the anomaly term can be expressed in the form

Δ a_gr≈ (GM_S/r_Sr_M) × (2^2k+1/v₀)×(M_M/M_S)^k × R₀(S,P)× R₀(M,P)× ∑_n=1^∞ ((-1)ⁿ/n^2k)× cos[nπK] ,

K= x× (r_M/r_S)× (y_M/y_S).

The normalization condition reads in this case as

R₀²=v₀/[2π∑_n (1/n)^2k+1]=v₀/πζ(2k+1) .

Note the shorthand v₀(S/M,P)= v₀. The anomalous gravitational acceleration is given by

Δa_gr=(GM_S/r_S²) × X Y× ∑_n=1^∞ [(-1)ⁿ/n^2k]×cos[nπK] ,

X= 2^2k × (r_S/r_M)× (M_M/M_S)^k ,

Y=1/π∑_n (1/n)^2k+1=1/πζ(2k+1).

It is clear that a reasonable order of magnitude for the effect can be obtained if k is small enough and that this is essentially due to the gigantic value of gravitational Planck constant.

The simplest model consistent with experimental findings assumes v₀(M,P)= v₀(S,P) and Φ(n)=0 and gives

Δa_gr/gcos(Θ)=(GM_S/r_S²g)× X Y× ∑_n=1^∞ [(-1)ⁿ/n^2k]×cos(nπ K) ,

X= 2^2k × (r_S/r_M)× (M_M/M_S)^k,

Y=1/π ∑_n (1/n)^2k+1 =1/πζ(2k+1) ,

K=x× (r_M/r_S)× (y_M/y_S) , x=M_S/M_M .

Θ denotes in the formula above the angle between the direction of Sun and horizontal plane.

4. Numerical estimates

To get a numerical grasp to the situation one can use M_S/M_M≈ 2.71× 10⁷, r_S/r_M≈ 389.1, and (M_Sr_M/M_Mr_S)≈ 1.74× 10⁴. The overall order of magnitude of the effect would be

Δ g/g≈ XY× GM_S/R_S²gcos(Θ) ,

(GM_S/R_S²g) ≈6× 10^-4 .

The overall magnitude of the effect is determined by the factor XY.

For k=1 and 1/2 the effect is too small. For k=1/4 the expression for Δ a_gr reads as

(Δa_gr/gcos(Θ))≈1.97× 10^-4∑_n=1^∞ ((-1)ⁿ/n^1/2)×cos(nπK),

K= (y_M/y_S)u , u=(M_S/M_M)(r_M/r_S)≈ 6.95671837× 10⁴ .

The sensitivity of cosine terms to the precise value of y_M/y_S gives good hopes of explaining the strong variation of Δf/f and also the findings of Jeverdan. Numerical experimentation indeed shows that the sign of cosine sum alternates and its value increases as y_M/y_S increases in the range [1,2].

The eccentricities of the orbits of Moon resp. Earth are e_M=.0549 resp. e_E=.017. Denoting semimajor and semiminor axes by a and b one has Δ=(a-b)/a=1-(1-e²)^1/2. Δ_M=15× 10^-4 resp. Δ_E=1.4× 10^-4 characterizes the variation of y_M resp. y_M due to the non-circularity of the orbits of Moon resp. Earth. The ratio R_E/r_M= .0166 characterizes the range of Δy_M =Δr_M,P/r_M< R_E/r_M due to the variation of the position of the laboratory. All these numbers are large enough to imply large variation of the argument of cosine term even for n=1 and the variation due to the position at the surface of Earth is especially large.

5. Other effects

1. One should explain also the recent finding by Popescu and Olenici, which they interpret as a quantization of the plane of oscillation of paraconic oscillator during solar eclipse (see this). A possible TGD based explanation would be in terms of quantization of Δg and thus of the limiting oscillation plane. This quantization could reflect the quantization of the angular momentum of the dark gravitons decaying into bunches of ordinary gravitons and providing to the pendulum the angular momentum inducing the change of the oscillation plane. The knowledge of friction coefficients associated with the rotation of the oscillation plane would allow to deduce the value of the gravitational Planck constant if one assumes that each dark graviton corresponds to its own approach to asymptotic oscillation plane.

2. There is also evidence for the effect before and after the main eclipse. The time scale is 1 hour. A possible explanation is in terms of a dark matter ring analogous to rings of Jupiter surrounding Moon. From the average orbital velocity v = 1.022 km/s of the Moon one obtains that the distance traversed by moon during 1 hour is R₁ = 3679 km. The mean radius of moon is R=1737.10 km so that one has R₁=2R with 5 per cent accuracy (2×R = 3474 km). The Bohr quantization of the orbits of inner planets discussed in with the value (^h/_2p)_gr = GMm/v₀ of the gravitational Planck constant predicts r_n � n²GM/v₀² and gives the orbital radius of Mercury correctly for the principal quantum number n=3 and v₀/c = 4.6×10^-4 @ 2^-11. From the proportionality r_n � n²GM/v₀² one can deduce by scaling that in the case of Moon with M(moon)/M(Sun) = 3.4×10^-8 the prediction for the radius of n=1 Bohr orbit would be r₁ = (M(Moon)/M(Sun))×R_M/9 @ .0238 km for the same value of v₀. This is too small by a factor 6.45×10^-6. r₁=3679 km would require n ~ 382 or n=n(Earth)=5 and v₀(Moon)/v₀(Sun) @ 2^-4.