### Allais effect and TGD

Allais effect is a fascinating gravitational anomaly associated with solar eclipses. It was discovered originally by M. Allais, a Nobelist in the field of economy, and has been reproduced in several experiments but not as a rule. The experimental arrangement uses so called paraconical pendulum, which differs from the Foucault pendulum in that the oscillation plane of the pendulum can rotate in certain limits so that the motion occurs effectively at the surface of sphere.

The articles * Should the Laws of Gravitation Be Reconsidered: Part I,II,III?* of Allais here and here and the summary article The Allais effect and my experiments with the paraconical pendulum 1954-1960 of Allais give a detailed summary of the experiments performed by Allais.

** A. Experimental findings of Allais**

Consider first a brief summary of the findings of Allais.

- In the ideal situation (that is in the absence of any other forces than gravitation of Earth) paraconic pendulum should behave like a Foucault pendulum. The oscillation plane of the paraconic pendulum however begins to rotate.
- Allais concludes from his experimental studies that the orbital plane approach always asymptotically to a limiting plane and the effect is only particularly spectacular during the eclipse. During solar eclipse the limiting plane contains the line connecting Earth, Moon, and Sun. Allais explains this in terms of what he calls the anisotropy of space.
- Some experiments carried out during eclipse have reproduced the findings of Allais, some experiments not. In the experiment carried out by Jeveran and collaborators in Romania it was found that the period of oscillation of the pendulum changes by Δ f/f≈ 5× 10
^{-4}, which happens to correspond to the constant v_{0}=2^{-11}appearing in the formula of the gravitational Planck constant. - There is also quite 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).

** B. TGD inspired model for Allais effect**

The basic idea of the TGD based model is that Moon absorbs some fraction of the gravitational momentum flow of Sun and in this manner partially screens the gravitational force of Sun in a disk like region having the size of Moon's cross section. Screening is expected to be strongest in the center of the disk. The predicted upper bound for the change of the oscillation frequency is slightly larger than the observed change which is highly encouraging.

* 1. Constant external force as the cause of the effect*

The conclusions of Allais motivate the assumption that quite generally there can be additional constant forces affecting the motion of the paraconical pendulum besides Earth's gravitation. This means the replacement **g**→ **g**+Δ**g** of the acceleration g due to Earth's gravitation. Δ**g** can depend on time.

The system obeys still the same simple equations of motion as in the initial situation, the only change being that the direction and magnitude of effective Earth's acceleration have changed so that the definition of vertical is modified. If Δ **g** is not parallel to the oscillation plane in the original situation, a torque is induced and the oscillation plane begins to rotate.This picture requires that the friction in the rotational degree of freedom is considerably stronger than in oscillatory degree of freedom: unfortunately I do not know what the situation is.

The behavior of the system in absence of friction can be deduced from the conservation laws of energy and angular momentum in the direction of **g**+Δ **g**.

* 2. What causes the effect in normal situations?*

The gravitational accelerations caused by Sun and Moon come first in mind as causes of the effect. Equivalence Principle implies that only relative accelerations causing analogs of tidal forces can be in question. In GRT picture these accelerations correspond to a geodesic deviation between the surface of Earth and its center. The general form of the tidal acceleration would thus the difference of gravitational accelerations at these points:

Δ**g**= -2GM[(Δ **r**/r^{3}) - 3(**r**•Δ **r****r**/r^{5})].

Here **r** denotes the relative position of the pendulum with respect to Sun or Moon. Δ**r** denotes the position vector of the pendulum measured with respect to the center of Earth defining the geodesic deviation. The contribution in the direction of Δ **r** does not affect the direction of the Earth's acceleration and therefore does not contribute to the torque. Second contribution corresponds to an acceleration in the direction of **r** connecting the pendulum to Moon or Sun. The direction of this vector changes slowly.

This would suggest that in the normal situation the tidal effect of Moon causesgradually changing force mΔ**g** creating a torque, which induces a rotation of the oscillation plane. Together with dissipation this leads to a situation in which the orbital plane contains the vector Δ**g** so that no torque is experienced. The limiting oscillation plane should rotate with same period as Moon around Earth. Of course, if effect is due to some other force than gravitational forces of Sun and Earth, paraconic oscillator would provide a manner to make this force visible and quantify its effects.

* 3. What happens during solar eclipse?*

During the solar eclipse something exceptional must happen in order to account for the size of effect. The finding of Allais that the limiting oscillation plane contains the line connecting Earth, Moon, and Sun implies that the anomalous acceleration Δ |**g**| should be parallel to this line during the solar eclipse.

The simplest hypothesis is based on TGD based view about gravitational force as a flow of gravitational momentum in the radial direction.

- For stationary states the field equations of TGD for vacuum extremals state that the gravitational momentum flow of this momentum. Newton's equations suggest that planets and moon absorb a fraction of gravitational momentum flow meeting them. The view that gravitation is mediated by gravitons which correspond to enormous values of gravitational Planck constant in turn supports Feynman diagrammatic view in which description as momentum exchange makes sense and is consistent with the idea about absorption. If Moon absorbs part of this momentum, the region of Earth screened by Moon receives reduced amount of gravitational momentum and the gravitational force of Sun on pendulum is reduced in the shadow.
- Unless the Moon as a coherent whole acts as the absorber of gravitational four momentum, one expects that the screening depends on the distance travelled by the gravitational flux inside Moon. Hence the effect should be strongest in the center of the shadow and weaken as one approaches its boundaries.
- The opening angle for the shadow cone is given in a good approximation by Δ Θ= R
_{M}/R_{E}. Since the distances of Moon and Earth from Sun differ so little, the size of the screened region has same size as Moon. This corresponds roughly to a disk with radius .27× R_{E}.The corresponding area is 7.3 per cent of total transverse area of Earth. If total absorption occurs in the entire area the total radial gravitational momentum received by Earth is in good approximation 93.7 per cent of normal during the eclipse and the natural question is whether this effective repulsive radial force increases the orbital radius of Earth during the eclipse.

More precisely, the deviation of the total amount of gravitational momentum absorbed during solar eclipse from its standard value is an integral of the flux of momentum over time:

Δ P

^{k}_{gr}= ∫ Δ(P^{k}_{gr}/dt) (S(t))dt,(ΔP

^{k}_{gr}/dt)(S(t))= ∫_{S(t)}J^{k}_{gr}(t)dS.This prediction could kill the model in classical form at least. If one takes seriously the quantum model for astrophysical systems predicting that planetary orbits correspond to Bohr orbits with gravitational Planck constant equal to GMm/v

_{0}, v_{0}=2^{-11}, there should be not effect on the orbital radius. The anomalous radial gravitational four-momentum could go to some other degrees of freedom at the surface of Earth. - The rotation of the oscillation plane is largest if the plane of oscillation in the initial situation is as orthogonal as possible to the line connecting Moon, Earth and Sun. The effect vanishes when this line is in the the initial plane of oscillation. This testable prediction might explain why some experiments have failed to reproduce the effect.
- The change of |
**g**| to |**g**+Δ**g**| induces a change of oscillation frequency given byΔf/f=

**g**• Δ**g**/g^{2}= (Δ g/g) cos(Θ).If the gravitational force of the Sun is screened, one has |

**g**+Δ**g**| >g and the oscillation frequency should increase. The upper bound for the effect is obtained from the gravitational acceleration of Sun at the surface of Earth given by v^{2}_{E}/r_{E}≈ 6.0× 10^{-4}g. One has|Δ f|/f≤ Δ g/g = v

^{2}_{E}/r_{E}≈ 6.0× 10^{-4}.The fact that the increase(!) of the frequency observed by Jeveran and collaborators is Δf/f≈ 5× 10

^{-4}supports the screening model. Unfortunately, I do not have access to the paper of Jeveran*et al*to find out whether the reported change of frequency, which corresponds to a 10 degree deviation from vertical is consistent with the value of cos(Θ) in the experimental arrangement. - 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 Δ
**g**and thus of the limiting oscillation plane. This quantization should reflect the quantization of the gravitational momentum flux receiving Earth. The flux would be reduced in a stepwise manner during the solar eclipse as the distance traversed by the flux through Moon increases and reduced in a similar manner after the maximum of the eclipse.

** C. What kind of tidal effects are predicted?**

If the model applies also in the case of Earth itself, new kind of tidal effects (for normal tidal effects see this) are predicted due to the screening of the gravitational effects of Sun and Moon inside Earth. At the night-side the paraconical pendulum should experience the gravitation of Sun as screened. Same would apply to the "night-side" of Earth with respect to Moon.

Consider first the differences of accelerations in the direction of the line connecting Earth to Sun/Moon: these effects are not essential for tidal effects proper. The estimate for the ratio for the orders of magnitudes of the these accelerations is given by

|Δ**g**_{p}(Sun)|/|Δ**g**_{p}(Moon)|= (M_{S}/M_{M}) (r_{M}/r_{E})^{3}≈ 2.17.

The order or magnitude follows from r(Moon)=.0026 AU and M_{M}/M_{S}=3.7× 10^{-8}. The effects caused by Sun are two times stronger. These effects are of same order of magnitude and can be compensated by a variation of the pressure gradients of atmosphere and sea water.

The tangential accelerations are essential for tidal effects. The above estimate for the ratio of the contributions of Sun and Moon holds true also now and the tidal effects caused by Sun are stronger by a factor of two.

Consider now the new tidal effects caused by the screening.

- Tangential effects on day-side of Earth are not affected (night-time and night-side are of course different notions in the case of Moon and Sun). At the night-side screening is predicted to reduce tidal effects with a maximum reduction at the equator.
- Second class of new effects relate to the change of the normal component of the forces and these effects would be compensated by pressure changes corresponding to the change of the effective gravitational acceleration. The night-day variation of the atmospheric and sea pressures would be considerably larger than in Newtonian model.

The intuitive expectation is that the screening is maximum when the gravitational momentum flux travels longest path in the Earth's interior. The maximal difference of radial accelerations associated with opposite sides of Earth along the line of sight to Moon/Sun provides a convenient manner to distinguish between Newtonian and TGD based models:

|Δ **g**_{p,N}|=4GM ×(R_{E}/r)^{3} ,

|Δ **g**_{p,TGD}|= 4GM ×(1/r^{2}).

The ratio of the effects predicted by TGD and Newtonian models would be

|Δ **g**_{p,TGD}|/|Δ **g**_{p,N}|= r/R_{E} ,

r_{M}/R_{E} =60.2 , r_{S}/R_{E}= 2.34× 10^{4}.

The amplitude for the oscillatory variation of the pressure gradient caused by Sun would be

Δ|**grad**p_{S}|=v^{2}_{E}/r_{E}≈ 6.1× 10^{-4}g

and the pressure gradient would be reduced during night-time. The corresponding amplitude in the case of Moon is given by

Δ |**grad**p_{S}|/Δ|**grad**p_{M}|= (M_{S}/M_{M})× (r_{M}/r_{S})^{3}≈ 2.17.

Δ |**grad**p_{M}| is in a good approximation smaller by a factor of 1/2 and given by

Δ|**grad**p_{M}|=2.8× 10^{-4}g.

Thus the contributions are of same order of magnitude.

One can imagine two simple qualitative killer predictions.

- Solar eclipse should induce anomalous tidal effects induced by the screening in the shadow of the Moon.
- The comparison of solar and moon eclipses might kill the scenario. The screening would imply that inside the shadow the tidal effects are of same order of magnitude at both sides of Earth for Sun-Earth-Moon configuration but weaker at night-side for Sun-Moon-Earth situation.

** D. An interesting co-incidence**

The measured value of Δ f/f=5× 10^{-4} is exactly equal to v_{0}=2^{-11}, which appears in the formula hbar_{gr}= GMm/v_{0} for the favored values of the gravitational Planck constant. The predictions are Δ f/f≤ Δ p/p≈ 6× 10^{-4}. Powers of 1/v_{0} appear also as favored scalings of Planck constant in the TGD inspired quantum model of bio-systems based on dark matter (see this). This co-incidence would suggest the quantization formula

g_{E}/g_{S}= (M_{S}/M_{E}) × (R_{E}/r_{E})^{2}= v_{0}

for the ratio of the gravitational accelerations caused by Earth and Sun on an object at the surface of Earth.

** E. Summary of the predicted new effects**

Let us sum up the basic predictions of the model.

- The first prediction is the gradual increase of the oscillation frequency of the conical pendulum by Δ f/f≤ 6× 10
^{-4}to maximum and back during night-time. Also a periodic variation of the frequency and a periodic rotation of the oscillation plane with period co-inciding with Moon's rotation period is predicted. - A paraconical pendulum with initial position, which corresponds to the resting position in the normal situation should begin to oscillate during solar eclipse. This effect is testable by fixing the pendulum to the resting position and releasing it during the eclipse. The amplitude of the oscillation corresponds to the angle between
**g**and**g**+Δ**g**given in a good approximation bysin[Θ(

**g**,**g**+Δ**g**)]= (Δ g/g)sin[Θ(**g**,Δ**g**)].An upper bound for the amplitude would be Θ≤ 6× 10

^{-4}, which corresponds to .03 degrees.

- Gravitational screening should cause a reduction of tidal effects at the "night-side" of Moon/Sun. The reduction should be maximum at "midnight". This reduction together with the fact that the tidal effects of Moon and Sun at the day side are of same order of magnitude could explain some anomalies know to be associated with the tidal effects. A further prediction is the day-night variation of the atmospheric and sea pressure gradients with amplitude which is for Sun 6× 10
^{-4}g and for Moon 1.3× 10^{-3}g.

To sum up, the predicted anomalous tidal effects and failure of the explanation of the limiting oscillation plane in terms of stronger dissipation in rotational degree of freedom could kill the model.

For details see the chapter The Relationship Between TGD and GRT of "Classical Physics in Many-Sheeted Space-Time".

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