Journal of Modern Physics, 2012, 3, 388-397 Published Online May 2012 (
Gravitational Model of the Three Elements Theory
Frederic Lassiaille
University of Nice Sophia-Antipolis, Valbonne, France
Received March 8, 2012; revised March 25, 2012; accepted April 15, 2012
The gravitational model of the three elements theory is an alternative theory to dark matter. It uses a modification of
Newton’s law in order to explain gravitational mysteries. The results of this model are ex planations for the dark matter
mysteries, and the Pioneer anomaly. The disparity of the gravitational constant measurements might also be explained.
Concerning the Earth flyby anomalies, the theoretical order of magnitude is the same as the experimental one. A very
small change of the perihelion advance of the planet orbits is calculated by this model. Meanwhile, this gravitational
model is perfectly compatible with restricted relativity and general relativity, and is part of the three element theory, a
unifying theory.
Keywords: Relativity; Gravitation; Newton
1. Introduction
This article addresses an issue yielded by general relati-
vity, which is a determination of a global space’s shape.
This absolute space-time is the general relativity space-
time, and this space deformation in space-time must be in
conformity with Newton’s law at least for long distances.
The adopted point of view is a Euclidean relativity. A
Euclidean mathematical context is used, with 4 dimen-
sions (three of space, x, y, z, and one of time: ct). This is
used for restricted relativity. For general relativity, of
course we apply the same and we extend it with a tensor,
except that here locally it is a Euclidean metric used to
represent space-time. In one word, a Riemannian tensor
is used, in place of the usual pseudo-Riemannian Min-
kowskian tensor.
The physics principles within this mathematical frame-
work will be exactly the principles of relativity. There
exists, however, a difference between relativity and the
approach of this document. Indeed, constancy of space-
time distance, in a global representation of space-time.
Minkowski’s representation is not used, and the Lor-
entz’s invariant length is left off.
At first, the method is postulating th at Lorentz’s equa-
tions are simply a consequence of space-time deforma-
tions by energy. In other words we try to express the
general relativity “deformation” principle, in the context
of restricted relativity. Once this is done, physics incon-
sistencies are revealed. Of course a solution is searched.
This will finally lead to postulate the existence of indi-
visible particles, from which matter is made of. Thus, the
space-time determination is calculated, by means of rela-
tivity energy equation. By construction, this determina-
tion is coherent with Lorentz’ s equations. This is the fi-
nal determination of the shape of global space inside
space-time. After this theoretical construction, this arti-
cle will describe some results of this gravitational
Galaxy speed profiles mystery,
Galaxy velocit y my st ery ,
Pioneer anomaly,
Saturn flyby by Pioneer 11,
Earth flyby anom alies,
Perihelion advance or precession of Mercury and
Disparity of gravitational constant measurements.
The detailed about this article can be found in [1-3].
Reference [1] was the first article about the gravitational
model of the three elements theory. It describes the theo-
retical basis of the model. It has been published in the
17th NPA proceedings, vixra database, and on the site of
the three elements theory, since 2010. It has also been
published on Amazon site. Reference [2] was only pub-
lished on the site of the three elements theory. But fig-
ures where extracted from it for publishing in the pro-
ceedings below. Reference [3] was published on the site
of the three elements theo ry, vixra database, and leads to
a short version of it, which is in progress in the fo llowing
proceedings: 2011 Dark matter symposium (poster), 18th
NPA proceedings, PIRT 2011, ECLA 2011, FFP12, and
may be SMFNS2011, STARS2011, and the 8th Interna-
tional Conference on Progress in Theoretical Physics.
These “proceeding versions” are also reminding briefly
the content of [1], and using some figures of [1].
opyright © 2012 SciRes. JMP
2. Retrieving Lorentz’s Equations
Lorentz’s physical context is used. Let us remind it.
There are two inertial frames, R (O, x, y, z, ct) and R' (O',
x', y', z', ct'), in uniform rectilinear motions at the v speed
one compared to another, along Ox axis. The O' point is
moving along Ox axis in the direction of x increasing.
Here, only x dimension, ct, and x', ct', are important. At t
= x = 0 there is also t' = x' = 0.
In order to find Lorentz’s equations within this ph ysics
framework, and since our representations are Euclidean,
we are bound to suppose that Ox' axis rocked with an
angle compared to Ox axis, such as sin(
) = v/c. See
Figure 1. In the same way it is necessary to have O' co-
ordinates equal to: (x = vt, and ct = v²t/c). Conversely
under these conditions the reader will be able to calcu late
that Lorentz’s equations are found.
On the basis of this observation, we are tempted to
suppose a coherent physics postulate. This postulate is
the following.
Postulate 1. Any particle with a non null mass m,
moving with v speed along Ox axis, x increasing, com-
pared to an inertial frame R (O x y z ct), deforms space-
time around it with a rotation of the Ox- Oct plan around
the Oy Oz axis, with an
angle between Ox and Ox', such
as sin(
) = v/c. During the displacement of this particle
from O to A(x=vt, ct), a vacuum appeared inside space-
time. The location of this vacuum is the (O, O', H) triangle,
such as: O' coordinates are O'(vt, v²t/c), H coordinates
are H(vt, O).
In the borderline case of a photon, with v = c, the
swing becomes maximum:
= /2, and the vacuum is
the (O, A, H) triangle.
Figure 2 represents the effect of postulate 1. A P par-
ticle is moving at v speed in the inertial reference frame
R, parallel to Ox axis, and in the direction of x increasing.
At the t instant, the particle is located coordinates x and
ct in R (A point). Hence R' is also moving uniformly
along Ox axis. It is the same case as the one of Figure 1,
except that here exists the P particle on A point. On Fig-
ure 2, the space line rocked with the
angle, locally in A
and O'.
On the other h a nd, far from A point this line of space is
Figure 1. Lorentz transformation euclidean representation.
Figure 2. Postulate 1.
parallel to Ox axis. This is indeed the only realistic pos-
sibility! It is difficult to imagine the movement of a par-
ticle deforming the entire universe this way along Ox
With this postulate, Lorentz transformation now ex-
presses a local deformation of space-time, caused by the
energy of the moving particle (postulate 1 above). This
respect of Lorentz’s equations is only local to the parti-
cle. It can be noticed that those restrictions explain the
Sagnac effect.
3. Luminous Points
However at this stage a problem of coherence arises
since a particle is constituted of smaller particles. Indeed,
how to ensure that space-time deformation generated by
the movement of a big particle, composed by a heap of
smaller particles, can rise from the deformations of th ese
smaller particles? To ensure this coherence a solution
consists in supposing that matter is made up of a re-
stricted group of very small “indivisible” particles. This
is postulate 2 below.
Postulate 2. Each particle consists of a certain number
of smaller particles, called the luminous points. These
luminous points are moving constantly at the c speed,
inside the first particle, and with respect to any inertial
frame of reference.
These small particles are conceived in such a way that
they can explain space-time deformations generated by
any other composed particle. For this explanation a sim-
ple operation must calculate the final deformation gener-
ated by the large particle, from the small particles it is
made of. Thus defined, this operation must allow, by
construction, calculation of the shape of absolute curves
of space, starting from the positions and energies of these
“small indivisible particles”. At the same time, this op-
eration must be, of course, compatible with postulate 1
and Lorentz transformation.
From this postulate 2 it is possible to determine in a
single way any deformation generated by any particle.
For that, we apply postulate 1 to these “luminous point”
particles. For these luminous points the
angle is equal
Copyright © 2012 SciRes. JMP
to its limit value /2. The shape of space is thus at any
moment the result of successive combinations of these
small deformations caused by all these “luminou s poin ts”.
What remains to be specified is the way of combining
those various deformations. This will be specified by th e
postulate 3 which follows. After that, it will possible to
check that the
angle calculated from postulates 2 and 3
is well given by the formula of postulate 1.
For that let us return to Lorentz transformation. A first
mathematical observation is essential. The energy con-
servation equation of restricted relativity is found by
quantifying the luminous point trajectory lengths, inside
a given P parti cle. It is what we will see.
It will be supposed that this P particle is modelled as
consisting of only one luminous point. Consequently the
obtained model is the one described by Figure 3. It can
be che cked that the r easo nin g r ema in s v al id in the ge ne-
ral case of a particle made up of several luminous
When the P particle moves from O point (on Figure 3)
to A point, along OA segment, the contained luminous
point follows a trajectory having a V shape, that is: 1)
First stage: displacement at the speed +c along Ox,
(milked in fat on Figure 3); 2) Second stage: displace-
ment at the speed –c along Ox (milked in fat on Figure
3). For the first stage, L1 is the displacement length, and
L2, (positive), is the displacement length of the second
stage. If x is the position of the A point, we can write: x =
vt = L1 L2. This x position is also the space coordinate
of P in R at this time t.
12 1
,ct LLvt LL  2
12 1212
 
 (2)
This last equation is the Pythag ore equation using sur-
faces. It is also nothing more than the relativistic equation
of energy:
EEE 2
with 222
1Emc vc, 22
Emvc vc, and
. To obtain the equivalent equation for the en-
ergy densities, each term of th is equation is divided by t he
122LL which is the valu e of the to tal energ y
of the particle:
12 12
LL oper LL
 
12 12
oper LL LL
  (5)
The introduced o perator is the relationship between the
algebraic average and the arithmetic mean. It is equal to
Figure 3. Luminous point trajectory in a moving particle.
the relativistic coefficient 22
Indeed, from Equati on (1): 1
21Lct vc ,
 , and
22 12 12
14vcLLL L . Let
us remind that Equation (4) is also written: 1 = sin²(
) +
) where
is the angle of the space-time swing of
postulate 1.
Finally, this last study of Lorentz transformatio n led us
to define an o perator. It will be used to postulate, finally,
the mode of determination of general relativity absolute
space-time. By construction, this determination will be
compatible with restricted relativity.
4. Relativistic Operator
This is done by the following postulate. It generalizes the
previous observation done about relativity energy equa-
Postulate 3. Space shape in space-time is given at any
point by the ratio of the infinitesimal space lengths, ds
along space line, and dx its length projected on Ox axis.
This ratio is equal at any point to the relativistic operator
applied to the two following values:
1: sum of the heights of vacuum of space-time de-
formations propagated in Ox direction, x increasing
2: sum of the heights of vacuum of space-time de-
formations propagated in Ox direction, x decreasing.
That is to say:
121 2
dd 2
 where
and are the 2 above-ment i oned sums. 1
This operator is neither linear n or associative. But this
doesn’t matter since it is calculated once, at any space-
time point. Its value remains the same, when calculated
in different inertial frames. (This can be either deduced
or directly calculated).
It is written above that and are the sums of
Copyright © 2012 SciRes. JMP
the heights of vacuum of the “propagated deformations”.
It is necessary to describe how these “space-time defor-
mations” are propagated. The mechanism is very similar
to waves propagated by the movement of a boat over a
water surface. Let us consider Figure 1. The initial de-
formation relates to the Ox-Oct plan. The propagations of
this deformations in space-time are carried out on re-
maining space dimensions, i.e. Oy and Oz, more gener-
ally on any Or direction, half-line based on O and con-
tained in the Oy-Oz plan. The form of these propagated
deformations is each time exactly the same as the one of
the initial deformation. The initial deformation was done
on Ox-Oct plan (space-time swing represented Figure 1).
Now the propagated deformation is the same but it re-
lates to the (Ox+Or)-Oct plan in place of Ox-Oct plan.
The height of this propagated deformation attenuates
progressively as r increases. The attenuation law, g, will
be given further. At each moment, the “luminous point”
thus emits this deformation. Therefore, like in the case of
the boat, the finally overall propagated deformation is th e
envelope of all these propagated deformations. (In the
case of a boat this envelope has a V shape which is the
final shape of these waves over the water surface). Here
this envelope is a cone whose axis is Ox axis.
Let us study the overall result of all the propagated
deformations which are received in the same M point at
the same moment. The postulate 3 above expresses the
length ratio along Ox axis only. But at one M point there
are numerous such directions coming to. Hence it is ne-
cessary to calculate the relativistic operator of postulate 3
for each space direction. The final deformation is then
obtained. The only qu estion is: “What is the combination
rule for the deformations of all these directions in order
to obtain the result?” This result will be the resulting
space-time deformation at M point. It will be thus neces-
sary to generalize the above operator with a second more
generic operator, which will take in to account each sp ace
directions. The result given by this second generalized
operator must be the famous final space-time deforma-
tion at M point. It will be probably useful to use a mathe-
matical base like the quaternions for that. In this article
this complexity will not be seen because fortunately not
We thus found relativity starting from postulates 1, 2,
and 3. The
angle of the postulate 1 rotation is calcu-
lated, by applying postulates 2 and 3. Overall, we ob-
tained a way for calculating space shape inside space-
time. We can now study Newton’s law.
5. Modification of Newton’s Law
The studied case is a particle of mass M isolated in a
space filled uniformly with a constant energy density.
The particle coordinates are x = y = z = 0, which are
those of the O point in our usual inertial frame R of re-
ference. The studied case being invariant by any rotation
of center O, only the Ox axis with x > 0, and the axis of
times Oct, are important.
How does evolve the local slope tan(
of space, along
Ox axis? The postulate 3 above is applied. Let us con-
sider a space-time P point to which comes at least one
deformation from a luminous point pertaining to the M
mass. We suppose P x-coordinate positive strict that is x
> 0. The M mass particle propagates on P point the fol-
lowing defo rmations, with pr opagation directions give n:
Lgx x increasing (6)
x decreasing (7)
For Equation (6), g is the attenuation function given
further. There is no deformation propagated in the direc-
tion of x decreasing, coming from M, because x > 0. The
surrounding universe with constant energy density propa-
gates on P point the following deformations, with pro-
pagation directions given:
x increasing (8)
x decreasing (9)
Therefore, the and sums are the following.
11 1umu
 (10)
22 2um
 (11)
xoperLg xL
 (12)
The last equation is the application of postulate 3. Af-
ter calculations:
cos( )1
  (13)
In addition let’s ap ply the formula of the expr ession of
a force, to an m mass moving. The traditional relativistic
equation is the following one:
This is a very classical relativity result. Now let us
take the case of a particle with a negligible mass at rest.
It is with this particular case that is applied the principle
of general relativity: the trajectory of this particle will
follow a space-time geodesic. Moreover, if the particle is
located at rest infinitively far at t = 0, then we have, for
any x, v = ctan (
), where
is the slope angle of the re-
quired curve ct = h(x). This curve is the searched space
curve. From where:
dtan( )tan( )
Fmc x
Copyright © 2012 SciRes. JMP
For x large, this must be equal to 2
mMG x ,
which is Newton’s equation. Hence, after calculation, we
have, for x large:
() u
gx L
 (16)
With 2
RMGc . Now it will be postulated that this
Equation (16) is correct not only for long distances but
for any values of x. It will be supposed also that space is
no longer filled uniformly with a constant energy density.
Let us write f as being the normalized space-time height
of the propagated deformations which are coming from
the closed surrounding matter. For example, in the case
of a galaxy, this f contribution will comes from the gal-
axy’s stars. It will be us ed also s = 1 + f, where 1 stands
here for the normalized contribution coming from the
universe. With this notation, the and sums be-
comes the following. 1
LL f
 
 (18)
After calculations, Equation (15) becomes the fol-
sss s
 
 
In this equation, G' stands for the value of G valid for
long distances and in extra-galactic locations, and R' is
equal to MG'/c2. It must be noticed that, if f = 0, it still
remains a correction of Newton’s law expressed by
Equation (19). But this modification is only noticed for x
close to the Schwarzschild radius (2R').
In this document, if f = 0 then this Newton’s law
modification will be called the “first modification of
Newton’s law”. If f
0, it will be called the “second
modification of Newton’s law”.
6. The Pioneer Anomaly
This modification of Newton’s law has been using a
fitting of Newton’s law for long distances. But for
studying the Pioneer anomaly, this fitting must not be
done for long distances. It must be done for a distance
from the sun where the heliocentric gravitational constant
is known to be perfect. This value is around the sun to
Saturn distance. This can be seen on the measured curve
of the Pioneer anomaly, in [4]. With this supposition the
calculations yields a theoretical anomaly value equal to
10 2
7.2510 ms
 , in place of the measured value
10 2
8.7410 ms
 . But the shape of the theoretical
curve is not perfect. Figure 4 shows this theoretical
curve. It has been plotted using Equation (19) and the
“first mo dification of Newton ’s law”.
There is also a negative predicted anomaly for the probe’s
trajectory between the sun and Saturn (x-coordinate
lesser than 10 AU). This work is in progress, and notice-
ably exact comparison with ephemerides must be done.
In order to retrieve the perfect curve, it must be
supposed that f is not null nor constant but varies (“sec-
ond modification of Newton’s law”). The Kuiper belt
will be taken into account. The Kuiper belt is a belt of
asteroids located beyond the location of Saturn , along the
ecliptic plane. Now the result, on Figure 5, is very
encouraging. On this theoretical curve, the maximum
value is exactly 10 2
 , the measured
value. But this theoretical curve has been obtained with
fitted values for the Kuiper belt space-time deformations
contributions. On the contrary, the curve of Figure 4 was
calculated without any fitting.
Figure 4. Theoretical curve of the Pioneer anomaly, using
“first modification of Newton’s law”. X-coordinate is the
distance from the sun, in AU, y-coordinate is the added ac-
celeration toward the sun, in 10–10 m/s2. This theoretical
curve must be compared with the experimental one located
in [4].
Figure 5. Theoretical curve of the Pioneer anomaly, using
the Kuiper belt, and the “second modification of Newton’s
law” for calculation. X-coordinate is the distance from the
sun, in AU, y-coordinate is the added acceleration toward
the sun, in 10–10 m/s2.
Copyright © 2012 SciRes. JMP
7. The Pioneer 11 Flyby of Saturn
When applying this gravitational model to the case of
Pioneer 11 trajectory, an added anomaly is found. It is an
acceleration anomaly during the flyby of Saturn. The
model is calculating an anomalous decrease before the
Saturn encounter, and an anomalous increase after the
encounter. This is shown on Figure 6. An important pa-
rameter has been fitted. It is 0
, the distance at which the
Saturn gravitational constant,
GM , is perfect (
stands for Saturn mass). Indeed, in this gravitational
model, this distance is always of great importance: the
distance between the attracted masses, where Newton’s
law is perfect.
The values of Figure 6 are very strong, as compared to
the Pioneer an omaly values, but the distanc es range of this
anomaly is very short. This range is roughly [9.550 AU,
9.555 AU]. It has been checked that the global Pioneer
anomaly curve is still correct for this Pioneer 11 travel,
after taking into account this Saturn flyby anomaly. In fact,
this global curve shape depends strongly on the 0
8. Speed Profile Mystery
In order to obtain the explanation of the stars speed in a
galaxy it is necessary to take into account these star
masses. For this, a P point is supposed located in the
middle of these stars i.e. inside the studied galaxy. It will
be supposed that the deformation propagated by these
stars and received in P is only coming from the sur-
rounding stars closed to P. This supposition is conf irmed
later on, outside of this article.
It will be supposed th at this matter density in a galaxy
evolves following a 1/x2 law. (x is the distance from the
galactic center). As a consequence, the f contribution of
Equation (19) is equal to r/x, where r is the ray from
which the gravitational effect of the surrounding stars is
noticed. With those suppositions and notations, Equation
(19) becomes:
Now, of course M stands for the mass of the galactic
center. It is noticed that the gravitational force can be
null, and even negative for x < r. As usual the tangential
speed of the stars is calculated using the classical equa-
tion vFxm .
This Equation (20) yields the curve located on the
bottom of Figure 7. It has been used: r = 1 kpc. This
value has been adjusted in order to obtain the best possi-
ble curve.
The variation of the speed of the stars in a galaxy is
theoretically explained. Hence we retrieve the global
Figure 6. Theoretical curve of the Pioneer 11 anomaly, in
the vicinity of Saturn. X-coordinate is the distance from the
sun, in AU, and y-coordinate is the anomalous acceleration
toward the sun, in m/s2. The represented anomaly is the
sum of the Pioneer anomaly and the Saturn flyby anomaly.
Figure 7. Theoretical speed profile for the stars in a galaxy,
calculated using a 1/x2 matter density curve. X-coordinate
represents, in kpc, the distance between the star and the
galactic center. The unit of the ordinate, y, is km/s. The
curve located on the bottom represents the speed resulting
from the new model. The curve located on the top repre-
sents the speed resulting from traditional Newton’s law.
shape of a typical galaxy speed profile.
More practically, a program calculating the NGC 7541
speed profile yields the curve of Figure 8. This curve has
been computed using the matter density profile available
in [5]. The corresponding measured speed profile of
NGC 7541 is also found in [5]. The shape of this mea-
sured curve is the same as the theoretical curve of Figure
8. But there is an issue of sign for the theoretical speed
profiles of those real galaxies. Each time, locally, the
Copyright © 2012 SciRes. JMP
Figure 8. Theoretical speed profile of the NGC 7541 galaxy.
X-coordinate is in kpc, and y ordinate is a relative speed.
The sign of the gravitational force has been changed, and
the speed values has been raise d with a fitted constant value,
before normalization.
shape of the curve is not retrieved, but another one,
which is exactly the same as the measured one, except
that it is reversed along the y-coordinate. Hence, using an
opposite sign for the gravitational force equation, the
measured curve shape is retrieved. This is true for each
of the five real galaxies which has been calculated: NGC
3310, NGC 106 8, NGC 157, NG C 7541, and N GC 7331.
This is calculated in [2]. A plausible explanation of this
error is an occultation mechanism during the propagation
of the space-time deformations coming from the galaxy
luminous points. In this mechanism, the galaxy dust is
acting like a fog, occulting the space-time deformation
propagations coming from the matter located beyond it.
This mechanism has been confirmed by calculations, but
still remains to be explained fully.
For some galaxies, it is possible to calculate the exact
values of the theoretical speed, for the maximums of the
speed profiles. For NGC 3310, the measured value in [6]
for the first maximum is around 175 km/s. The corre-
sponding theo retical value is in the rang e [20, 700] km/s.
For NGC 1068, the measured value in [6] is 210 km/s,
the theoretical value is 260 km/s. The precision of those
theoretical calculations has not been evaluated. But those
calculations are not fitted.
9. Galaxy Speed Mystery
Let us study the mystery of the velocities of the galaxies.
From the above equations we have, after calculations:
This sum is done for each luminous point, p, along a
half-cone. This half-cone is centered on the location in
which we want to calculate G.
e is the energy of each
luminous point.
is the distance of each luminous
point from the location in which we calculate G. Let us
inside outside
 
is for the p luminous points inside the Milky Way.
o is for luminous points outside the Milky Way. There-
 
It has been written G as the gravitational constant valid
inside the Milky Way, and G' when located outside any
galaxy. Therefore this modeling of relativity explains
qualitatively the “dark matter” mystery for the velocities
of the galaxies. Also, this exp lanation is the same for the
mystery of light beam deviation in the vicinity of a gal-
10. PPN Parameters
The PPN parameters (Parameterized Post-Newtonian for-
malism) are exactly the same as for relativity. Indeed, the
only differences between relativity and the gravitational
model of the three elements theory are the following.
Lorentz equations are true only between inertial ref-
erence frames which get “energy attached locally”.
“Ener gy attached locally” to a reference fra me ( O, ct, x,
y, z) means that there is a particle or a group of parti-
cles whom inertial poi nt i s const antly equal to O.
Newton’s law is not used, like in relativity, but re-
trieved. There is a slight difference between Newton’s
law and this corrected Newton’s law.
There exists nonlinearity in the superposition law for
gravity. This work is in progress.
Each relativistic equation remains also the same, ex-
cept Einstein’s equation because it is calculated using
Newton’s law. In the gravitational model of the three
elements theory, Einstein’s equation is only an approxi-
11. Earth Flyby Anomalies
The issue of the flyby anomalies is explained in [7]. When
applying the “first modification of Newton’s law”, the
order of magnitude is well retrieved for these anomalies.
Table 1 shows the theoretical added ve locities, which are
roughly of the same order of magnitude as the measured
However, those calculations are much too simpler. The
Copyright © 2012 SciRes. JMP
Table 1. Comparison between the real and the theoretical
Earth flyby anomalies for 6 probes. The values represents
added perigee velocitie s. Units in mm/s.
Probe Measured anomaly Theoretical Anomaly
Galilleo I 2.56 2.53
Galilleo II –2 –11
NEAR 7.21 –6.6
Cassini –1.7 4.5
Rosetta I 0.67 22
Messenger 0.008 29
calculations must be done at least in the Schwarzsch ild’s
metric of the sun, using the motion of the Earth, the exact
trajectory of the probes, and the contributions of the sur-
rounding matter (asteroids, planets, etc.). Moreover, the
“second modification of Newton’s law must be used also,
as the Pioneer anomaly analysis shows. Noticeably, the
location of the ecliptic plane as compared to the exact
probes trajectories, has an important impact on the final
result. Indeed, the Kuiper belt is located on this plane, and
the Kuiper belt influence has been proven in the Pioneer
anomaly analysis above. This remark might explain the
experim ental verification of the importance of the location
of the equator plane as c ompared to the probes trajectories.
Indeed, the equator and the ecliptic planes are not far away
from each other.
12. Perihelion Advance
When applying the “first modification of Newton’s law”
to the perihelion advance of Mercury and Saturn, the
results of Table 2 are retrieved. The theoretical values are
very close from the general relativity values. They do not
explain the anomaly of the precession of the Saturn peri-
helion explained in [8], which is of –0.006 arc-second by
But, since the calculated values are very close to t he GR
values, it results that the three elements theory model
seems to be compatible here with gravitation experimental
measurements. Moreover, as well as for Earth flyby ano-
malies, the “second modification of Newton’s law” must
be conducted, i n order to check if t he Saturn anomaly of [8]
is theoretically explained.
13. Measurements of G
The issue is well defined. For nearly three centuries, G has
been measured, without getting roughly a better precision
than 0.7%. The reason of this poor precision is the fact that
many measurements values are contradicting each others,
taking into account their con fid ent interval. The details of
this issue is, for example, well described in [9]. The “first
modification of Newton’s law” is not able to exp lain this
issue. Indeed, the order of magnitude of the theoretical
Table 2. Theoretical modification of the perihelion advance
or precession of Mercury and Saturn. On the left are the
general relativity results calculated with a computer pro-
gram. On the right are the added values from these general
relativity values. Units are arc-second by century.
Planet GR value 3elt value
Mercury 42.7848 GR value – 0.0014
Saturn 1.66291 GR value + 0.00056
error is far below the experimental one.
But the “second modification of Newton’s law” re-
trieves the same order of magnitude as the measured one.
Let’s compare first with data coming from [10]. A pre-
diction of the model of this document is that the G value is
depending of the surrounding matter distribution. This is
not a ne w i dea , it has bee n s u gge ste d i n [ 1 1]. M ore ov e r, i t
has been proven by experimental data in [10], that the
value of G depends of the orientation. This behavior is
also a consequence of this theoretical idea. In [10], the
amplitude of the variation of the G value is more than
0.054% as compared to its absolute value. The prediction
of the gravitational model of the three elements theory for
this value is be tween 1% and 0.01 3% depending o f galaxy
matter distribution. Therefore, the order of magnitude of
this model prediction is compatible with experimental
Now let’s try another estimation, not tak ing in account
the presence of the stars, like in [10], but the presence of
mountains around the apparatus during the measurement
of G. The ratio below is the relative difference between
two values of G, 1m, and 2m. 1m is the measured
value of G in [12], and is the measured value in
6.5 10
 (24)
Those two official measurements of G have been cho-
sen in order to get the greatest possible ratio above.
Let us assume that those two experiments have been
done in completely different places. And the important
difference between them is the distribution of matter in the
surrounding neighbourhood.
Experiment {1}, for example, is done at the very top
of a hill on the floor of a desert, and this floor is com-
pletely plane outside of the hill on which we are lo-
Experiment {2}, is done in the middle of a valley,
which is surrounded by mountains.
The interesting thing is that the measured value of G
will be completely different between those two cases even
if exactly the same experiment apparatus is used and the
same measurement procedure is applied. Indeed, the pre-
sence of the surrounding mountains in the second mea-
surement has an important effect on the final measured
Copyright © 2012 SciRes. JMP
value. After calculations, the corresponding theoretical
ratio of G is estimated by the following formula.
r stands for the maximum distance between the sur-
rounding mountains, and the location of experiment {2}.
It will be used .
4 km
r stands for the greatest distance between a galactic
object and the location of the experiments. It will be used
the same value as for the Pioneer anomaly calculation:
. This value is based on the occultation me-
chanism solving the sign issue which has been noticed
above for the speed profiles. In other words, galactic ob-
jects located beyond this distance will not be noticed.
Their space-time deformation will vanish, during propa-
gation, before arrival, because of the occultation mecha-
nism. This
140 pc
r value has been fitted in a coherent man-
ner by the model when comparing its predictions to ex-
perimental data.
is the mean matter density of the surrounding
mountains of experime nt {2}. It will be used ρs = 1.7 g/cm 3,
which is weaker than granite density .
granite 2.7 g/cm
is the matter density of the galaxy. We wil l use the
value , which is the matter den-
sity in the galaxy, near the solar system.
20 3
0.70910 kg/m
With those numerical values, the final result is the fol-
This theoretical value is not far from the meas ure d o ne,
of Equation (24). This proves that the order of magnitude
of the measured difference can be explained by our cor-
rection of Newton’s Law.
As an intermediate conclusion, the gravitational model
of this study might explain the disparity between the
measurements of G. This theoretical value of G is de-
pending on the distribution of matter in the surrounding
neighborhood (buildings, hills, mountains, and sea) of the
place where the measurement of G is done.
A linearity violation of gravitational forces might be
predicted by the m odel. If predicted, it could explain, also,
this historical disparity in the measurements of G. This
work is in progress.
14. Conclusions
As a conclusion, this new modeling of space-time retri-
eves general and restricted relativity. Nevertheless, it is
more than a simple Euclidean representation of relativity,
as shown by postulate 1 and 2.
A direct explanation of the Sagnac effect is given by
this modeling. Moreover, it is enough to add a third pos-
tulate in order to explain those “dark matter mysteries”: 1)
galaxy speed profiles, 2) speed of the galaxies them-
selves, inside their group, and deformation of light tra-
jectories in the vicinity of a galaxy, 3) Pionee r anomaly.
In more details, this third postulate conducts a modify-
cation of Newton’s law. This modification is conceived
in order to find exactly Newton’s law in the specific case
of pin pointed masses inside an homogeneous universe,
and long distances. Using the model’s equation, immedi-
ately a correction of Newton’s law is noticed in the case
of short distances. This first correction occurs in fact for
relativistic speed. Now, adjusting this correction in order
to fit Newton’s law with this new law for some particular
distance, yields immediately a theoretical explanation of
Pioneer anomaly. This model is also predicting an anom-
aly for Saturn flyby b y Pioneer 11.
Another result of this correction of Newton’s law oc-
curs when the galaxy’ s stars are introduced in the model.
After this, a strong difference appears between the cal-
culated force and Newton’s law. The predicted galaxy
speed profiles are close to measured speed profiles.
For the mystery of galaxy velocities, the explanation is
more direct. This mystery is explained by a different
value of G, between our case inside the Milky Way, and
the case outside any galaxy. Here again, the “third spea-
ker”, which decreases G constant, is the stars of our gal-
This model m ight ex plain als o, a fter s ome calculations,
the following anomalies:
Earth flyby anomalies,
perihelion precession of Saturn,
disparity of the gravitational constant measurements.
Moreover, and from a theoretical point of view, this
study finds a way for space-time determination. At any
space-time point, this determination is based upon the
matter density distribution throughout the whole un iverse.
This model is compatible by construction with restricted
and general relativity.The PPN parameters are exactly
the same as for relativity.It seems to be compatible also
with gravitation experim e ntal m easurements.
As a conclusion, the gravitational model of the three
elements theory seems to be validated. As such, this is a
validation of the three elements theory itself. This theory
is described in [14]. A next step will be to solve the sign
issue for the dark matter mystery (speed profiles). The
exact comparison with ephemerides must be done also.
Another interesting work will be to calculate the linearity
violation of gravitational forces predicted by the model,
and to compare it to experimental data. It remains also to
check if the model described in this article is coherent
with other actual physics theories (electromagnetism,
quantum mechanics, etc). As an answer to this last ques-
tion one will notice that this m odel is in conformity with a
Copyright © 2012 SciRes. JMP
Copyright © 2012 SciRes. JMP
unifying theory called three elements theory.
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