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\begin{center}{\Large {\bf Allais Effect and TGD}}\end{center}
\begin{center} \vspace{.7cm} M. Pitk\"anen$^1$,
August
1, 2007\\ %\end{center}
\vspace{.7cm} \indent {Email:
matpitka@rock.helsinki.fi,\\ URL:
http://www.physics.helsinki.fi/$\sim$matpitka/.\\
\indent Recent address: Puutarhurinkatu 10,10960,
Hanko, Finland.}
\end{center}
\tableofcontents
\section{Introduction}
Allais effect \cite{Allais,Allaisnasa} 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.
\subsection{Experimental findings}
Consider first a brief summary of the findings of
Allais and others \cite{Allaisnasa}.
a) In the ideal situation (that is in the absence
of any other forces than gravitation of Earth)
paraconical pendulum should behave like a Foucault
pendulum. The oscillation plane of the
paraconical pendulum however begins to rotate.
b) 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.
c) Some experiments carried out during eclipse
have reproduced the findings of Allais, some
experiments not. In the experiment carried out by
Jeverdan and collaborators in Romania it was found
that the period of oscillation of the pendulum
decreases by $\Delta f/f\simeq 5\times 10^{-4}$
\cite{Allais,Jeverdan} which happens to correspond
to the constant $v_0=2^{-11}$ appearing in the
formula of the gravitational Planck constant. It
must be however emphasized that the overall
magnitude of $\Delta f/f$ varies by five orders
of magnitude. Even the sign of $\Delta f/f$ varies
from experiment to experiment.
d) There is also quite recent finding by Popescu
and Olenici, which they interpret as a quantization
of the plane of oscillation of paraconical
oscillator during solar eclipse \cite{Olenici}.
\subsection{TGD based models for Allais effect}
I have already earlier proposed an explanation of
the effect in terms of classical $Z^0$ force
\cite{Zanom}. If the $Z^0$ charge to mass ratio of
pendulum varies and if Earth and Moon are $Z^0$
conductors, the resulting model is quite flexible
and one might hope it could explain the high
variation of the experimental results.
The rapid variation of the effect during the
eclipse is however a problem for this approach and
suggests that gravitational screening or some more
general interference effect might be present.
Gravitational screening alone cannot however
explain Allais effect. Also the combination of
gravitational screening and $Z^0$ force assuming
$Z^0$ conducting structures causing screening fails
to explain the discontinuous behavior when massive
objects are collinear.
A model based on the idea that gravitational
interaction is mediated by topological light rays
(MEs) and that gravitons correspond to a gigantic
value of the gravitational Planck constant however
explains the Allais effect as an interference
effect made possible by macroscopic quantum
coherence in astrophysical length scales.
Equivalence Principle fixes the model to a high
degree and one ends up with an explicit formula
for the anomalous gravitational acceleration and
the general order of magnitude and the large
variation of the frequency change as being due to
the variation of the distance ratio
$r_{S,P}/r_{M,P}$ ($S, M$,and $P$ refer to Sun,
Moon, and pendulum respectively). One can say that
the pendulum acts as an interferometer.
\section{Could gravitational screening explain Allais effect}
The basic idea of the screening 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. The
screening is expected to be strongest in the center
of the disk. Screening model happens to explain the
findings of Jevardan but fails in the general case.
Despite this screening model serves as a useful
exercise.
\subsection{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 $\overline{g}\rightarrow
\overline{g}+\Delta\overline{g}$ of the
acceleration $g$ due to Earth's gravitation.
$\Delta\overline{g}$ 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 $\Delta
\overline{g}$ 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
$\overline{g}+\Delta \overline{g}$. The explicit
formulas are given by
\begin{eqnarray}
E&=& \frac{ml^2}{2}(\frac{d\Theta}{dt})^2 +
sin^2(\Theta)(\frac{d\Phi}{dt})^2+mglcos(\Theta)\per
, \nonumber\\
L_z&=& ml^2sin^2(\Theta)\frac{d\Phi}{dt}\per .
\end{eqnarray}
\noindent and allow to integrate $\Theta$ and
$\Phi$ from given initial values.
\subsection{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:
\begin{eqnarray}
\Delta\overline{g}&=& -2GM[\frac{\Delta
\overline{r}}{r^3} - 3\frac{
\overline{r}\cdot\Delta
\overline{r}\overline{r}}{r^5}] \per .
\end{eqnarray}
\noindent Here $\overline{r}$ denotes the relative
position of the pendulum with respect to Sun or
Moon. $\Delta \overline{r}$ 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 $\Delta
\overline{r}$ 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 $\overline{r}$ 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 causes gradually changing
force $m\Delta\overline{g}$ 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
$\Delta \overline{g}$ 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, paraconical
oscillator would provide a manner to make this
force visible and quantify its effects.
\subsection{What would happen during the 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 $\Delta
\vert{g}$ 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.
a) 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.
b) 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.
c) The opening angle for the shadow cone is given
in a good approximation by $\Delta \Theta=
R_M/R_E$. 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 $.27\times R_E$.
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 92.7 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.
%Check the numbers!
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:
\begin{eqnarray}
\Delta P^k_{gr} &=& \int \frac{\Delta P^k_{gr}}{dt}
(S(t))dt\per ,\nonumber\\
\frac{\Delta P^k_{gr}}{dt} (S(t))&=& \int_{S(t)}
J^k_{gr}(t)dS\per .
\end{eqnarray}
\noindent 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.
d) 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 initial plane of
oscillation. This testable prediction might
explain why some experiments have failed to
reproduce the effect.
e) The change of $\vert \overline{g}\vert$ to
$\vert \overline{g}+\Delta \overline{g}\vert$
induces a change of oscillation frequency given by
\begin{eqnarray}
\frac{\Delta f}{f}&=& \frac{\overline{g}\cdot
\Delta \overline{g}}{g^2}= \frac{\Delta g}{g}
cos(\theta)\per .
\end{eqnarray}
\indent If the gravitational force of the Sun is
screened, one has $\vert \overline{g}+\Delta
\overline{g}\vert>g$ and the oscillation frequency
should increase. The upper bound for the effect
corresponds to vertical direction is obtained from
the gravitational acceleration of Sun at the
surface of Earth:
\begin{eqnarray}
\frac{\vert \Delta f\vert}{f}&\leq& \frac{\Delta
g}{g} = \frac{v^2_E}{r_E}\simeq 6.0\times
10^{-4}\per .
\end{eqnarray}
f) One should explain also the recent finding by
Popescu and Olenici, which they interpret as a
quantization of the plane of oscillation of
paraconical oscillator during solar eclipse
\cite{Olenici}. A possible TGD based explanation
would be in terms of quantization of
$\Delta\overline{g}$ 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.
\subsection{What kind of tidal effects are predicted?}
If the model applies also in the case of Earth
itself, new kind of tidal effects 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. The estimate for the ratio for the orders
of magnitudes of the these accelerations is given
by
\begin{eqnarray}
\frac{|\Delta\overline{g}_{\perp}(Moon)|}{|\Delta\overline{g}_{\perp}(Sun)|}&=&
\frac{M_S}{M_M} (\frac{r_M}{r_E})^3\simeq 2.17\per
.
\end{eqnarray}
\noindent The order or magnitude follows from
$r(Moon)=.0026$ AU and $M_M/M_S=3.7\times 10^{-8}$.
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 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. They decompose as
$$\frac{1}{r^3}\left[\Delta \overline{r}- 3|\Delta
\overline{r}| cos(\Theta)
\frac{\overline{r}}{r}\right]\per .$$
\noindent $\pi/4\leq \Theta\leq \pi/2$ is the angle
between $\Delta \overline{r}$ and $\overline{r}$.
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.
a) 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.
b) 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:
\begin{eqnarray}
|\Delta \overline{g}_{\perp,N}|&=&4GM
\times\frac{R_E}{r^3}\per ,\nonumber\\
|\Delta \overline{g}_{\perp,TGD}|&=& 4GM
\times\frac{1}{r^2}\per .
\end{eqnarray}
\noindent The ratio of the effects predicted by
TGD and Newtonian models would be
\begin{eqnarray}
\frac{|\Delta \overline{g}_{\perp,TGD}|}{|\Delta
\overline{g}_{\perp,N}|}&=& \frac{r}{R_E}\per
,\nonumber\\
\frac{r_M}{R_E} &=&60.2\per , \per \frac{r_S}{R_E}=
2.34\times 10^4\per .
\end{eqnarray}
\noindent The amplitude for the oscillatory
variation of the pressure gradient caused by Sun
would be
$$\Delta\vert\nabla p_S\vert=\frac{v^2_E}{r_E}\simeq 6.1\times 10^{-4}g$$
\noindent and the pressure gradient would be
reduced during night-time. The corresponding
amplitude in the case of Moon is given by
$$\frac{\Delta \vert\nabla p_s\vert}{\Delta \vert\nabla p_M\vert}= \frac{M_S}{M_M}\times
(\frac{r_M}{r_S})^3\simeq 2.17\per .$$
\noindent $\Delta \vert\nabla p_M$ is in a good
approximation smaller by a factor of 1/2 and given
by $\Delta\vert\nabla p_M\vert=2.8\times 10^{-4}g
$. Thus the contributions are of same order of
magnitude.
\vl
\begin{tabular}{||l|l|l|l|l||}\hline\hline
$M_M/M_S$ &$M_E/M_S$& $R_M/R_E$ &
$d_{E-S}/AU$&$d_{E-M}/AU$
\\ \hline
$3.0\times 10^{-6}$&$3.69\times 10^{-8}$&.273 &1
&.00257 \\\hline
%
$R_E/d_{E-S}$& $R_E/d_{E-M}$& $g_S/g$ &$g_M/g$&
\\ \hline
%
$4.27\times 10^{-5}$&$01.7\times 10^{-7}$&
$6.1\times 10^{-4}$ &$2.8\times 10^{-4}$&
\\\hline\hline
\end{tabular}
\vl
Table 1. The table gives basic data relevant for
tidal effects. The subscript $E,S,M$ refers to
Earth, Sun, Moon; $R$ refers to radius; $d_{X-Y}$
refers to the distance between $X$ and $Y$ $g_S$
and $g_M$ refer to accelerations induced by Sun and
Moon at Earth surface. $g=9.8$ m/s$^2$ refers to
the acceleration of gravity at surface of Earth.
One has also $M_S=1.99\times 10^{30}$ kg and $AU=
1.49\times 10^{11}$ m, $R_E= 6.34\times 10^6$ m.
One can imagine two simple qualitative killer
predictions assuming maximal gravitational
screening.
a) Solar eclipse should induce anomalous tidal
effects induced by the screening in the shadow of
the Moon.
b) 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.
\subsection{An interesting co-incidence}
The value of $\Delta f/f=5\times 10^{-4}$ in
experiment of Jeverdan 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
$\Delta f/f\leq \Delta p/p\simeq 3\times 10^{-4}$.
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
\cite{eegdark}. This co-incidence would suggest
the quantization formula
\begin{eqnarray}
\frac{g_E}{g_S}&=& \frac{M_S}{M_E} \times
\frac{R_E^2}{r_E^2}= v_0 \per
\end{eqnarray}
\noindent for the ratio of the gravitational
accelerations caused by Earth and Sun on an object
at the surface of Earth.
It must be however admitted that the larger
variation in the magnitude and even sign of the
effect does not favor this kind of interpretation.
\subsection{Summary of the predicted new effects}
Let us sum up the basic predictions of the model
assuming maximal gravitational screening.
a) The first prediction is the gradual increase of
the oscillation frequency of the conical pendulum
by $\Delta f/f\leq 3\times 10^{-4}$ to maximum and
back during night-time in case that the pendulum
has vanishing $Z^0$ charge. 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. Already
Allais observed both 24 hour cycle and cycle which
is slightly longer and due to the fact that Moon
rates around Earth.
b) 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 $
\overline{g}$ and $ \overline{g}+\Delta
\overline{g}$ given in a good approximation by
\begin{eqnarray}
sin[\Theta(\overline{g},\overline{g}+\Delta
\overline{g})]&=& \frac{\Delta g}{g}sin[\Theta(
\overline{g},\Delta \overline{g})]\per .
\end{eqnarray}
\noindent An upper bound for the amplitude would be
$\Theta\leq 3\times 10^{-4}$, which corresponds to
.015 degrees. $Z^0$ charge of the pendulum would
modify this simple picture.
c) 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 \cite{tides}. A further prediction is the
day-night variation of the atmospheric and sea
pressure gradients with amplitude which is for Sun
$3\times 10^{-4}g$ and for Moon $1.3\times
10^{-3}g$.
To sum up, the predicted anomalous tidal effects
and the explanation of the limiting oscillation
plane in terms of stronger dissipation in
rotational degree of freedom could kill the model
assuming only gravitational screening.
\subsection{Comparison with experimental results}
The experimental results look mutually
contradictory in the context provided by the model
assuming only screening. Some experiments find no
anomaly at all as one learns from \cite{Allais}.
There are also measurements supporting the
existence of an effect but with varying sign and
quite different orders of magnitude. Either the
experimental determinations cannot be trusted or
the model is too simple.
a) The {\it increase} (!) of the frequency
observed by Jeverdan and collaborators reported in
Wikipedia article \cite{Allais} for Foucault
pendulum is $\Delta f/f\simeq 5\times 10^{-4}$
would support the model even quantitatively since
this value is only by a factor $5/3$ higher than
the maximal effect allowed by the screening model.
Unfortunately, I do not have an access to the paper
of Jeverdan {\it et al} to find out the value of
$cos(\Theta)$ in the experimental arrangement and
whether there is indeed a decrease of the period as
claimed in Wikipedia article. In \cite{Mihaila} two
experiments supporting an effect $\Delta g/g=
x\times 10^{-4}$, $x=1.5$ or $2.6$ but the sign of
the effect is different in these experiments.
b) Allais reported an anomaly $\Delta g/g\sim
5\times 10^{-6}$ during 1954 eclipse \cite{Nasa}.
According to measurements by authors of
\cite{Mihaila} the period of oscillation increases
and one has $\Delta g/g\sim 5\times 10^{-6}$.
Popescu and Olenici report a decrease of the
oscillation period by $(\Delta g/g)
cos(\Theta)\simeq 1.4\times 10^{-5} $.
c) In \cite{Wang} a {\it reduction} of vertical
gravitational acceleration $\Delta g/g= (7.0\pm
2.7)\times 10^{-9}$ is reported: this is by a
factor $10^{-5}$ smaller than the result of
Jeverdan.
d) Small pressure waves with $\Delta p/p= 2\times
10^{-5}$ are registered by some micro-barometers
\cite{Nasa} and might relate to the effect since
pressure gradient and gravitational acceleration
should compensate each other. $\Delta
gcos(\Theta)/g$ would be about 7 per cent of its
maximum value for Earth-Sun system in this case.
The knowledge of the sign of pressure variation
would tell whether effective gravitational force
is screened or amplified by Moon.
\section{Allais effect as evidence for large
values of gravitational Planck constant?}
One can represent rather general counter arguments
against the models based on $Z^0$ conductivity and
gravitational screening if one takes seriously the
puzzling experimental findings concerning
frequency change.
a) 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.
b) 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.
c) 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.
\subsection{What one an conclude about general
pattern for $\Delta f/f$?}
The above findings allow to make some important
conclusions about the nature of Allais effect.
a) 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.
b) If stellar objects and also ring like dark
matter structures possibly assignable to their
orbits are $Z^0$ conductors, one obtains screening
effect by polarization and for the ring like
structure the resulting effectively 2-D dipole
field behaves as $1/\rho^2$ so that there are hopes
of obtaining large screening effects and if the
$Z^0$ 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.
c) 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 $\Delta g/g$ 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 $Z^0$ force, which can have
arbitrarily long range in TGD Universe.
d) 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.
e) Interference suggests macroscopic quantum effect
in astrophysical length scales and thus
gravitational Planck constants given by
$\hbar_{gr}= GMm/v_0$, where $v_0=2^{-11}$ is the
favored value, should appear in the model. Since
$\hbar_{gr}= GMm/v_0$ 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.
\subsection{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.
a) The picture based on topological light rays
suggests that the gravitational force between two
objects $M$ and $m$ has the following expression
\begin{eqnarray}
F_{M,m}&=& \frac{GMm}{r^2}= \int \vert
S(\lambda,r)\vert^2
p(\lambda)d\lambda \nonumber\\
p(\lambda)&=&\frac{h_{gr}(M,m)2\pi}{\lambda}\per ,
\per \hbar_{gr}= \frac{GMm}{v_0(M,m)}\per .
\end{eqnarray}
\noindent $p(\lambda)$ denotes the momentum of the
gravitational wave propagating along ME. $v_0$ can
depend on $(M,m)$ pair. The interpretation is that
$\vert S(\lambda,r)\vert^2$ gives the rate for the
emission of gravitational waves propagating along
ME connecting the masses, having wave length
$\lambda$, and being absorbed by $m$ at distance
$r$.
b) Assume that $S(\lambda,r)$ has the decomposition
\begin{eqnarray}
S(\lambda,r)&=&
R(\lambda)exp\left[i\Phi(\lambda)\right]\frac{exp\left[ik(\lambda)r\right]}{r}\per
, \nonumber\\
exp\left[ik(\lambda)r\right]&=&exp\left[ip(\lambda)r/\hbar_{gr}(M,m)\right]\per
, \nonumber\\
R(\lambda)&=& \vert S(\lambda,r)\vert\per .
\end{eqnarray}
\noindent The phases $exp(i\Phi(\lambda))$ might
be interpreted in terms of scattering matrix. The
simplest assumption is $\Phi(\lambda)=0$ turns out
to be consistent with the experimental findings.
The substitution of this expression to the above
formula gives the condition
\begin{eqnarray}
\int \vert R(\lambda)\vert^2
\frac{d\lambda}{\lambda} &=&v_0 \per .
\label{condition}\end{eqnarray}
Consider now a model for the Allais effect based on
this picture.
a) 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.
b) In the collinear case the interference takes
place. If interference occurs for identical
momenta, the interfering wavelengths are related
by the condition
\begin{eqnarray}p(\lambda_{S,P})&=&p(\lambda_{M,P}\per .
\end{eqnarray}
\noindent This gives
\begin{eqnarray}\frac{\lambda_{M,P}}{\lambda_{S,P}}&=&
\frac{\hbar_{M,P}}{\hbar_{S,P}}\per
=\frac{M_M}{M_S}\frac{v_0(S,P)}{v_0(M,P)}\per .
\end{eqnarray}
c) The net gravitational force is given by
\begin{eqnarray}
F_{gr}&=& \int \vert Z(\lambda,r_{S,P})+
Z(\lambda/x,r_{M,P})\vert^2
p(\lambda) d\lambda\nonumber\\
&=&F_{gr}(S,P)+ F_{gr}(M,P) + \Delta F_{gr}\per , \nonumber\\
\Delta F_{gr}&=& 2\int
Re\left[S(\lambda,r_{S,P})\overline{S}(\lambda/x,r_{M,P}))\right]
\frac{\hbar_{gr}(S,P)2\pi}{\lambda}d\lambda\per
,\nonumber\\ x&=&\frac{\hbar_{S,P}}{\hbar_{M,P}}=
\frac{M_S}{M_M} \frac{v_0(M,P)}{v_0(S,P)}\per
.\end{eqnarray}
\noindent Here $r_{M,P}$ is the distance between
Moon and pendulum. The anomalous term $\Delta
F_{gr}$ would be responsible for the Allais effect
and change of the frequency of the oscillator.
d) The anomalous gravitational acceleration can be
written explicitly as
\begin{eqnarray}
\Delta a_{gr}&=&
2\frac{GM_S}{r_Sr_M}\frac{1}{v_0(S,P)}\times
I\per , \nonumber\\
I&=& \int R(\lambda)R(\lambda/x)
cos\left[\Phi(\lambda)-\Phi(\lambda/x)+2\pi\frac{(y_Sr_S-xy_Mr_M)}{\lambda}\right]
\frac{d\lambda}{\lambda} \per ,\nonumber\\
y_M&=& \frac{r_{M,P}}{r_M} \per ,\per
y_S=\frac{r_{S,P}}{r_S} \per .\end{eqnarray}
\noindent 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
$2\pi(y_Sr_S-xy_Mr_M)/\lambda$. 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.
e) 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 $\Delta f/f$. One
can also consider the possibility that MEs satisfy
periodic boundary conditions so that only wave
lengths $\lambda_n= 2r_S/n$ are allowed: this
implies $sin(2\pi y_Sr_S/\lambda)=0$. Assuming
this, one can write the magnitude of the anomalous
gravitational acceleration as
\begin{eqnarray}
\Delta a_{gr}&=&
2\frac{GM_S}{r_{S,P}r_{M,P}}\times\frac{1}{v_0(S,P)}
\times I\per , \nonumber\\
I&=&\sum_{n=1}^{\infty}
R(\frac{2r_{S,P}}{n})R(\frac{2r_{S,P}}{nx}) (-1)^n
cos\left[\Phi(n)-\Phi(xn)+n\pi\frac{xy_M}{y_S}\frac{r_M}{r_S}\right]
\per
.\nonumber\\
\end{eqnarray}
\noindent If $R(\lambda)$ decreases as
$\lambda^k$, $k>0$, at short wavelengths, the
dominating contribution corresponds to the lowest
harmonics. In all terms except cosine terms one can
approximate $r_{S,P}$ {\it resp.} $r_{M,P}$ with
$r_S$ {\it resp.} $r_M$.
f) 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:
$$\frac{xy_M}{y_S}\frac{r_M}{r_S}= \frac{y_M}{y_S}
\frac{M_S}{M_M}\frac{r_M}{r_S}\frac{v_0(M,P)}{v_0(S,P)}
\simeq 6.95671837\times 10^4\times
\frac{y_M}{y_S}\times \frac{v_0(M,P)}{v_0(S,P)}
\per .$$
\noindent 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 values of $y_M/y_S$
and $\frac{v_0(M,P)}{v_0(S,P)}$.
The value of $y_M/y_S$ varies from experiment to
experiment and this alone could explain the high
variability of $\Delta f/f$. The experimental
arrangement would act like interferometer measuring
the distance ratio $r_{M,P}/r_{S,P}$. Hence it
seems that the condition
\begin{eqnarray}
\frac{v_0(S,P)}{v_0(M,P)}&\neq& const.
\end{eqnarray}
\noindent implying breaking of Equivalence
Principle is not necessary to explain the variation
of the sign of $\Delta f/f$ and one can assume
$v_0(S,P)=v_0(M,P)\equiv v_0$. One can also assume
$\Phi(n)=0$.
\subsection{Scaling law}
The assumption of the scaling law
\begin{eqnarray}
R(\lambda)&=&R_0 (\frac{\lambda}{\lambda_0})^k
\end{eqnarray}
\noindent is very natural in light of conformal
invariance and masslessness of gravitons and allows
to make the model more explicit. With the choice
$\lambda_0=r_S$ the anomaly term can be expressed
in the form
\begin{eqnarray}
\Delta a_{gr}&\simeq& \frac{GM_S}{r_Sr_M}
\frac{2^{2k+1}}{v_0}(\frac{M_M}{M_S})^k
R_0(S,P)R_0(M,P)\times \sum_{n=1}^{\infty}
\frac{(-1)^n}{n^{2k}}cos\left[\Phi(n)-\Phi(xn)+n\pi K\right]\per , \nonumber\\
K&=& x\times \frac{r_M}{r_S}\times \frac{y_M}{y_S}
\per
.\nonumber\\
\end{eqnarray}
\noindent The normalization condition of Eq.
\ref{condition} reads in this case as
\begin{eqnarray}
R_0^2&=&v_0\times \frac{1}{2\pi\sum_n
(\frac{1}{n})^{2k+1}}
=\frac{v_0}{\pi\zeta(2k+1)}\per .
\end{eqnarray}
\noindent Note the shorthand $v_0(S/M,P)= v_0$. The
anomalous gravitational acceleration is given by
\begin{eqnarray}
\Delta
a_{gr}&=&\sqrt{\frac{v_0(M,P)}{v_0(S,P)}}\frac{GM_S}{r_S^2}
\times X Y\times \sum_{n=1}^{\infty}
\frac{(-1)^n}{n^{2k}}cos\left[\Phi(n)-\Phi(xn)+n\pi
K\right]\per , \nonumber\\
X&=& 2^{2k} \times \frac{r_S}{r_M}\times
(\frac{M_M}{M_S})^k \per , \per\nonumber\\ Y&=&
\frac{1}{\pi \sum_n (\frac{1}{n})^{2k+1}
}=\frac{1}{\pi\zeta(2k+1)}\per .
\end{eqnarray}
\noindent 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 $v_0(M,P)= v_0(S,P)$ and
$\Phi(n)=0$ and gives
\begin{eqnarray}
\frac{\Delta
a_{gr}}{gcos(\Theta)}&=&\frac{GM_S}{r_S^2g} \times
X Y\times
\sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2k}}cos(n\pi K)\per , \nonumber\\
X&=& 2^{2k} \times \frac{r_S}{r_M}\times
(\frac{M_M}{M_S})^k \per , \per\nonumber\\ Y&=&
\frac{1}{\pi \sum_n (\frac{1}{n})^{2k+1}
}=\frac{1}{\pi\zeta(2k+1)}\per ,\nonumber\\
K&=&x\times \frac{r_M}{r_S}\times
\frac{y_M}{y_S}\per , \per x=\frac{M_S}{M_M}\per .
\end{eqnarray}
\subsection{Numerical estimates}
To get a numerical grasp to the situation one can
use $M_S/M_M\simeq 2.71\times 10^7$, $r_S/r_M\simeq
389.1$, and $(M_Sr_M/M_Mr_S)\simeq 1.74\times
10^4$. The overall order of magnitude of the effect
would be
\begin{eqnarray}
\frac{\Delta g}{g}\sim XY\times
\frac{GM_S}{R_S^2g}cos(\Theta)\per ,\nonumber\\
\frac{GM_S}{R_S^2g} &\simeq&6\times 10^{-4}\per .
\end{eqnarray}
The overall magnitude of the effect is determined
by the factor $XY$.
a) For $k=0$ the normalization factor is
proportional to $1/\zeta(1)$ and diverges and it
seems that this option cannot work.
b) The table below gives the predicted overall
magnitudes of the effect for $k=1,2/2$ and $1/4$.
\vl
\begin{tabular}{||l|l|l|l|||}\hline\hline
k &1& 1/2 & 1/4 \\ \hline
%
$\frac{\Delta g}{g cos(\Theta)}$&$1.1\times
10^{-9}$&$4.3\times 10^{-6}$ &$1.97\times 10^{-4}$
\\\hline\hline
\end{tabular}
\vl
For $k=1$ the effect is too small to explain even
the findings of \cite{Wang} since there is also a
kinematic reduction factor coming from
$cos(\Theta)$. Therefore $k<1$ suggesting fractal
behavior is required. For $k=1/2$ the effect is of
same order of magnitude as observed by Allais. The
alternating sum equals in a good approximation to
-.693 for $y_S/y_M=1$ so that it is not possible to
explain the finding $\Delta f/f\simeq 5\times
10^{-4}$ of Jeverdan.
c) For $k=1/4$ the expression for $\Delta a_{gr}$
reads as
\begin{eqnarray}
\frac{\Delta
a_{gr}}{gcos(\Theta)}&\simeq&1.97\times
10^{-4}\sum_{n=1}^{\infty}
\frac{(-1)^n}{n^{1/2}}cos(n\pi K)\per
\per ,\nonumber\\
K&=& \frac{y_M}{y_S}u\per , \per
u=\frac{M_S}{M_M}\frac{r_M}{r_S}\simeq
6.95671837\times 10^4\per .
\end{eqnarray}
\noindent The sensitivity of cosine terms to the
precise value of $y_M/y_S$ gives good hopes of
explaining the strong variation of $\Delta f/f$
and also the findings of Jeverdan. Numerical
experimentation indeed shows that the sign of the
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 {\it
resp.} Earth are $e_M=.0549$ {\it resp.}
$e_E=.017$. Denoting semimajor and semiminor axes
by $a$ and $b$ one has
$\Delta=(a-b)/a=1-\sqrt{1-e^2}$. $\Delta_M=15\times
10^{-4}$ {\it resp.} $\Delta_E=1.4\times 10^{-4}$
characterizes the variation of $y_M$ {\it resp.}
$y_M$ due to the non-circularity of the orbits of
Moon {\it resp.} Earth. The ratio $R_E/r_M= .0166$
characterizes the range of the variation $\Delta
y_M =\Delta r_{M,P}/r_M\leq R_E/r_M$ 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.
The duration of full eclipse is of order 8 minutes
which corresponds to angle $\phi=\pi/90$ and at
equator roughly to a $\Delta
y_N=(\sqrt{r^2_M+R^2_Esin^2(\pi/90)}-r_M)/r_M\simeq
(\pi/90)^2R^2_E/2r_M^2
\simeq 1.7\times 10^{-7}$. Thus the change
of argument of $n=1$ cosine term during full eclipse
is of order $\Delta \Phi=.012\pi$ at equator. The duration
of the eclipse itself is of order two 2 hours
giving $\Delta y_M\simeq 3.4\times 10^{-5}$ and
the change $\Delta \Phi=2.4\pi$ of the argument of
$n=1$ cosine term.
\section{Could $Z^0$ force be present?}
One can understand the experimental results without
a breaking of Equivalence Principle if the pendulum
acts as a quantum gravitational interferometer. One
cannot exclude the possibility that there is also
a dependence on pendulum. In this case one would
have a breaking of Equivalence Principle, which
could be tested using several penduli in the same
experimental arrangement. The presence of $Z^0$
force could induce an apparent breaking of
Equivalence Principle. The most plausible option is
$Z^0$ MEs with large Planck constant. One can
consider also an alternative purely classical
option, which does not involve large values of
Planck constant.
\subsection{Could purely classical $Z^0$ force
allow to understand the variation of $\Delta f/f$?}
In the earlier model of the Allais effect (see the
Appendix of \cite{Zanom}) I proposed that the
classical $Z^0$ force could be responsible for the
effect. TGD indeed predicts that any object with
gravitational mass must have non-vanishing em and
$Z^0$ charges but leaves their magnitude and sign
open.
a) If both Sun, Earth, and pendulum have $Z^0$
charges, one might even hope of understanding why
the sign of the outcome of the experiment varies
since he ratio of $Z^0$ charge to gravitational
mass and even the sign of $Z^0$ charge of the
pendulum might vary. Constant charge-to-mass ratio
is of course the simplest hypothesis so that only
an effective scaling of gravitational constant
would be in question. A possible test is to use
several penduli in the same experiment and find
whether they give rise to same effect or not.
b) If Moon and Earth are $Z^0$ conductors, a $Z^0$
surface charge cancelling the tangential component
of $Z^0$ force at the surface of Earth is generated
and affects the vertical component of the force
experienced by the pendulum. The vertical component
of $Z^0$ force is $2F_Zcos(theta)$ and thus
proportional to $cos(\Theta)$ as also the effective
screening force below the shadow of Moon during
solar eclipse. When Sun is in a vertical direction,
the induced dipole contribution doubles the radial
$Z^0$ force near surface and the effect due to the
gravitational screening would be maximal. For Sun
in horizon there would be no $Z^0$ force and
gravitational tidal effect of Sun would vanish in
the first order so that over all anomalous effect
would be smallest possible: for a full screening
$\Delta f/f\simeq \Delta g^2/4g^2 \simeq 4.5 \times
10^{-8}$ would be predicted. One might hope that
the opposite sign of gravitational and $Z^0$
contributions could be enough to explain the
varying sign of the overall effect.
c) It seems necessary to have a screening effect
associated with gravitational force in order to
understand the rapid variation of the effect during
the eclipse. The fact that the maximum effect
corresponds to a maximum gravitational screening
suggests that it is present and determines the
general scale of variation for the effect. If the
maximal $Z^0$ charge of the pendulum is such that
$Z^0$ force is of the same order of magnitude as
the maximal screening of the gravitational force
and of opposite sign (that is attractive), one
could perhaps understand the varying sign of the
effect but the effect would develop continuously
and begin before the main eclipse. If the sign of
$Z^0$ charge of pendulum can vary, there is no
difficulty in explaining the varying sign of the
effect. An interesting possibility is that Moon,
Sun and Earth have dark matter halos so that also
gravitational screening could begin before the
eclipse. The real test for the effect would come
from tidal effects unless one can guarantee that
the pendulum is $Z^0$ neutral or its $Z^0$
charge/mass ratio is always the same.
d) As noticed also by Allais, Newtonian theory does
not give a satisfactory account of the tidal forces
and there is possibility that tides give a
quantitative grasp on situation. If Earth is $Z^0$
conductor tidal effects should be determined mainly
by the gravitational force and modified by its
screening whereas $Z^0$ force would contribute
mainly to the pressure waves accompanying the
shadows of Moon and Sun. The sign and magnitude of
pressure waves below Sun and Moon could give a
quantitative grasp of $Z^0$ forces of Sun and Moon.
$Z^0$ surface charge would have opposite signs at
the opposite sides of Earth along the line
connecting Earth to Moon {\it resp.} Sun and
depending on sign of $Z^0$ force the screening and
$Z^0$ force would tend to amplify or cancel the net
anomalous effect on pressure.
e) A strong counter argument against the model
based on $Z^0$ force is that collinear
configurations are reached in continuous manner
from non-collinear ones in the case of $Z^0$ force
and the fact that gravitational screening does not
conform with the varying sign of the discontinuous
effect occurring during the eclipse. It would seem
that the effect in question is more general than
screening and perhaps more like quantum mechanical
interference effect in astrophysical length scale.
\subsection{Could $Z^0$ MEs with large Planck constant
be present?}
The previous line of arguments for gravitational
MEs generalizes in a straightforward manner to the
case of $Z^0$ force. Generalizing the expression
for the gravitational Planck constant one has
$\hbar_{Z^0}= g_Z^2Q_Z(M)Q_Z(m)/v_0$. Assuming
proportionality of $Z^0$ charge to gravitational
mass one obtains formally similar expression for
the $Z^0$ force as in previous case. If $Q_Z/M$
ratio is constant, Equivalence Principle holds true
for the effective gravitational interaction if the
sign of $Z^0$ charge is fixed. The breaking of
Equivalence Principle would come naturally from the
non-constancy of the $v_0(S,P)/v_0(M,P)$ ratio also
in the recent case. The variation of the sign of
$\Delta f/f$ would be explained in a trivial manner
by the variation of the sign of $Z^0$ charge of
pendulum but this explanation is not favored by
Occam's razor.
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\end{document}