Journal of Modern Physics, 2013, 4, 1027-1035
http://dx.doi.org/10.4236/jmp.2013.47138 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Gravitational Model of the Three Elements Theory:
University of Nice Sophia Antipolis, Nice, France
Received January 19, 2013; revised February 21, 2013; accepted March 19, 2013
Copyright © 2013 Frederic Lassiaille. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim is to parse the mathematical details related to the gravitational model of the three elements theory . This
model is proven to be coherent and really compatible with relativity. The Riemannian representation of space-time
which is used in this model is proven to be legal. It allows to understand relativity in a more human sensitive manner
than Minkowskian usual representation.
Keywords: Relativity; Gravitation; Newton’s Law
The aim of this article is to depict the mathematical basis
which supports the gravitational model of the three ele-
ments theory . The first analysis addresses the link
between the two space-time representations. Those rep-
resentations are Minkowskian usual representation and a
Riemannian representation. The latter must be detailed,
and usual relativity mechanisms like general relativity
principles and Lorentz transformation are interpreted in
this representation in a particular geometrical manner.
Lorentz transformation is explained by postulate 1 of 
which is interpreted geometrically in a wired manner.
This must be explained here. Also, application of postu-
late 3 of  uses a strange projection rule, which must be
explained. The last analysis is checking that the “follow-
ing geodesic principle” is still valid in this Riemannian
representation, in a specific case. This is mandatory for
the calculations which are done in .
2. Comparison of the Two Metrics
the Minkowskian pseudo-Riemannian
metric coefficients, and h
the corresponding Rieman-
nian metric ones.
As usual we have:
ct, and 123
xxx the three space co-
ordinates. In this equation above
variables are sup-
posed to be space and time coordinates in some given
0 reference frame. Let’s choose a 0 base as part of
this 0 reference frame. In other words, 0 is the
reference frame at some given instant in time.
In each space-time point, a mathematical theorem
states that, for the bilinear form associated with this ds2
quadratic form, it is always possible to find an orthogo-
nal and normalizedbase with respect to some fixed
0 base, for which the metric local matrix is diagonal.
In these orthogonal bases, the metric coefficients be-
comes then such that
. That’s just a
reminder. But let’s notice that time and space units in
these bases are 0 base time and space units. In
those bases, now the Minkowskian metric can be written:
are coordinates in the base. By defi-
nition, the Riemannian coefficients are constructed the
Those equations are a consequence of the differences
between Minkowskian representation and the Rieman-
nian representation which is used in this model. The first
one is of course the metric signature which is + + + + for
the Riemannian one, in place of the + − − − Min-
kowskian one. The second difference is that local time is
inverted. In other words, the diagonal matrix coefficients
opyright © 2013 SciRes. JMP
are inverted when passing from one metric to another.
This allows to apply the “following geodesics” principle
in this Riemannian representation, as this will be shown
in this document. The physics understanding of this sec-
ond difference is the same as the general relativity ex-
planation of the twin paradox. Let’s remind that this ex-
planation is based upon the generation of a global vac-
uum by the universe in the representation of the reference
frame attached to the travelling twin. On the contrary the
twin staying on earth is watching the vacuum generated
by the travelling twin only locally and that’s the explana-
tion of the asymmetrical roles of the twins. This is not a
physical vacuum but a vacuum in the representation of
space-time. Now in the Riemannian representation, with
respect to 0 reference frame, this vacuum is supposed
to be generated globally by the free falling particle (as if
the free falling particle was not less than the universe
itself!). On the contrary, in the Minkowskian representa-
tion in 0 this vacuum is supposed to be generated only
locally to the particle along its trajectory. Of course, only
the latter version is the realistic one. But one may notice
that the mathematical model do not prefer any physical
An illustration of this relationship rule between those
two metrics is given by the case of the Schwarzschild
Of course this is the Minkowskian metric usual ver-
M is supposed to be the Schwarzschild ray of the at-
tracting object as usual. x is as usual in this metric the
spatial physical distance from the attracting object (more
usually written r). , and bb
tively, the time and space coordinates in the system of
bases. ds is the infinitely small space-time length,
calculated in some given space-time point. The 0 ref-
erence frame which generated this system of bases
gets on its origin its time axis parallel to local time axis
in two cases. The first one is when the origin of 0 is
located in the middle of the attracting object of this
Schwarzschild metric. The second one is when the origin
of 0 is located infinitely far from the attracting object.
In the latter case, 0 time axis is tangent to the trajec-
tory of a null mass free falling particle. Because of the
equivalence principle, this free falling particle trajectory
will coincide with a time coordinate curve in the system
of bases. Because of the “following geodesics” prin-
ciple, it will also be a geodesic in the Minkowskian met-
ric. As usual this free falling particle gets a null mass,
because we don’t want it to generate any space-time de-
formations around it. Otherwise, this should modify the
studied metric. And it gets a null speed when located
infinitely far, because this means that its trajectory will
always be perpendicular to space lines with respect to
. In other words, this trajectory is a time curve in this
0 representation. In this document, the expression
“free falling particle” will always means in fact this par-
ticular case of free falling particle, and the expression
“time line” will always means that specific kind of tra-
jectory. The term “space line” will mean perpendicular
curves to such time lines.
The corresponding Riemannian version of this Sch-
warzschild metric is the following.
is also a function of space and time lengths
with respect to the base. As it will be seen,
allows to draw space and time lines in the space-time
Riemannian representation, with respect to 0. It is this
drawing which will allow to measure space-time lengths
in this representation.
Using Equations (3), (4), and (5) there is:
spherical coordinates are not written here.
Their coefficients will get opposite sign when passing
from one metric to the other, because their Euclidean
associated local coordinates have their Minkowskian
coefficients equal to −1. Therefore, they becomes equal
to +1 during the metric transformation when applying
Equations (3) and (4). Hence, the
also their sign modified but that’s all.
Distance no tations
Affine distance generated by Euclidean canonical
Distance given by the Riemannian metric and Equa-
On Figure 1 are drawn qualitatively those space lines
associated with this Schwarzschild Riemannian metric,
with respect to 0. It is only after this drawing, that
space-time lengths can be measured by evaluating the
distances in 0between space lines (for measuring time)
and between time lines (for measuring space).
On Figure 2 are firstly added two space-time points, A,
and B. A and B are supposed to be infinitely close one to
each other, and located on the same time line. It can be
checked on Figure 2 that the space-time length between
A and B correspond roughly to the number of space lines
passed through when going from A to B. Therefore when
space lines get close from each other, Riemannian local
time is elapsing faster with respect to 0. This is the
case as we get x weaker and weaker, that is, as we get
closer to the attracting object. Of course, it goes reversely
for space distances. That’s for a qualitative understand-
Copyright © 2013 SciRes. JMP
F. LASSIAILLE 1029
Figure 1. Space lines drawn with respect to R0 reference
Figure 2. Two couples of points, each of them aligned on
their local time line, B and sharing the same space line,
and such that
In a precise manner, the Riemannian metric distance
between A and B is a
value calculated by Equation
(7). Let’s calculate this space-time length between A and
because here , and then:
Now let’s redo this calculation for
points. Like A and B, they share the same time line, but
they are located far from the attracting object. It is sup-
In other words, those segment distances are equal with
respect to the Euclidean canonical metric given by 0.
is the distance of the
point from the attracting
object, we get:
Because that’s Equation (9) applied to
is supposed to be far greater than M, then the
following approximation can be done.
That’s because here the space-time deformations gen-
erated by the attracting object are vanishing. It can be
written by other means:
Indeed, here the base is the 0 base except for
their origins. Back to B point, Equation (9) can be written
the following way.
This equation gives the time ratio between
local Riemannian time in B point, and its corresponding
time in 0 reference frame. For yielding this equa-
tion, it has been used Equation (9),
, time distance in 0, has been renamed
because this distance can be also evaluated in B point.
Now let’s calculate the local and physical time dilatation
between x and
distances from the attracting object.
The same reasoning with Minkowskian metric yields
relativity time dilatation:
Another way to understand the difference between
Equation (14) and Equation (15) is the following. By
construction, the physical local time ratio is growing in-
versely as compared with the Riemannian metric local
time ratio. In other words, ,,dABDAB ratio must
be inverted in order to get the ddctc
because local times are inverted when passing from Min-
kowskian to Riemannian representation. In geometric
The Riemannian time distance is growing proportion-
ally with the number of crossed space lines;
Local time physical length is growing proportionally
with Euclidean distance between two given space
Those values are inversely proportional to each other
and Equation (15) could be retrieved this way starting
from Equation (14). In other words, the relativistic coef-
Copyright © 2013 SciRes. JMP
is equal to ddt
, as seen projected along
local tangent of space lines. It can be said that it is also
projected along space lines. This rule is illustrated by
represents, everywhere, the local physical dis-
tance between a given couple of space lines. This dis-
tance is measured with 0 Euclidean metric. When
those lines get close to each other, this value decrease
because it is measured with .
Inversely, the Riemannian distance,
those space lines remains the same everywhere. This dis-
tance is calculated with Equation (7) in a covariant man-
ner along space lines.
3. Covariance and Bases
This covariance will be used for understanding this geo-
metrical Riemannian mechanism. For this let’s go back
to Figure 2.
Let’s write m the base which is constructing, in B
point, theinertial reference frame of the free falling
By construction, if
are base vectors, and if
B are base vectors, there is m
other words, this m base is orthogonal with its vectors
normalized in the Minkowskian metric. This is not a new
concept. It comes from general relativity in Minkowskian
Now the same way, let’s write r the following base,
in B point. If r
u are the base vectors, there is
. In other words, this base is orthogo-
nal with its vectors normalized in the Riemannian metric.
distance between two space lines is the Riemannian
length of r vector. By construction this length stays
constant along a given space line.
Inversely, cdt represents the distance between space
lines calculated along 0 time axis (not along local
time axis). The cd
vector is parallel transported along a
Figure 3. Local time, and global time. “cd
” vector is paral-
lel transported along space lines.
given space line. The cdt vector is the projected vector of
vector. This projection is done along the local
space line tangents. But evaluation of this cdt vector pro-
jection can be done everywhere along the corresponding
space lines, in a parallel transport manner. This is also
illustrated by Figure 3.
What about the space length ratio? The rule for the
answer is the famous “constancy of metric determinant”,
which is a classical rule of general relativity. Using it, of
course the consequence is merely an inverted evolution
of the space length ratio as compared to the time ratio.
As an intermediate conclusion, the Riemannian dis-
tance allows drawing space lines. When drawing those
lines, it is supposed that this Riemannian metric distance,
, between two given space lines is always the same
when measured everywhere along those lines. Those
space and time lines are drawn in the 0
frame. This representation is understandable with human
senses. In , it allows to understand the relativistic en-
ergy equation in the simple geometric manner of the Py-
thagore theorem. Let’s remind that it is possible to de-
scribe this Pythagore equation, using surfaces, as a func-
tion of luminous points space-time deformation heights.
This gives the determination of space-time shape in .
4. How to Apply the Rules of the Postulate 3
When applying the third postulate , the projection
used for writing
along time axis, not along time lines.
On Figure 4 this postulate 3 application mode is
That’s because the determinant of the metric is sup-
posed to be constant, as it was discussed above. Indeed, a
geometrical property of this determinant is that it is the
surface of the ABCD rectangle of Figure 5 which is de-
scribed by the base (of course this surface is meas-
ured in ).
But this surface is also the surface of the BDE F paral-
lelogram of Figure 6 which vertical sides are given by
the c dt vector. The other vector side of this parallelo-
Figure 4. Application of Postulate 3 of .
opyright © 2013 SciRes. JMP
F. LASSIAILLE 1031
Figure 5. Geometric interpretation of the metric determi-
nant: surface of the ABCD rectangle drawn in the
Figure 6. BDEF parallelogram sharing the same surface as
the ABCD rectangle.
gram is d
which direction is along local space tangent.
The projection of this vector along 0 time axis is some
dx fixed vector. This dx vector is parallel to 0 space
axis and its length stays constant through space and time.
This is represented on Figures 5-7. Let’s describe this
On Figure 5, the surface of the ABCD rectangle is
equal to , the metric determi-
,dB ctB and lengths are con-
structed from r base vectors, and therefore from
coefficients. There is: 0
DAB u, and
DDB u, with being (therefore )
associated norm. On Figure 6, the surface of the ABCD
rectangle is also equal to the surface of the BDEF paral-
On Figure 7, cdt and dx remain constant when pass-
ing from BDEF (parallelogram on the left) to
Figure 7. Interpreting constancy of metric determinant.
the BDEF parallelogram, as well as the surface of the
parallelogram. In , this paral-
lelogram is equal to the
parallelogram (on the right). Inversely, cd
modified. The metric determinant stay constant, it is
equal to everywhere. It is the surface of
dd ddctx ctx
rectangle is not drawn).
The conclusion of this mechanism is the following.
R time lengths are projected along space lines, and
Physical space lengths are projected along 0
In one word, this strange projection rule is explained
by the constancy of metric determinant. Finally, if v is
the physical speed of the free falling particle with respect
to , and if
is the angle between local time axis
and time axis, there is:
ct ct ct c
, and not tan v
another legal used notation for d
, the infinitely
small physical space distance along its space line with
respect to 0 reference frame. Therefore, the relativistic
coefficient, yielded by the relativistic operator in , is
equal as expected to
, and there-
fore there is .
5. Comparison of Lorentz Transformation
Formulation in the Two Metrics
Let’s go back to the geometrical interpretation of postu-
late 1 which is done in  in the context of the Rieman-
Locally, the situation is exactly the same as above be-
cause of the postulate 1 of . Indeed, in the context of
Lorentz transformation, space-time deformation is lo-
cally exactly the same as studied above. Figure 8 shows
qualitatively those space-time deformations. It could
Copyright © 2013 SciRes. JMP
Figure 8. Space-time deformations in the case of Lorentz
transformation, with respect to R frame.
even be possible to write also metric equations for this
case. Space line has rocked with an
angle such as
,, ,,, ,etc
. Therefore, the same construction of vec-
tors, bases and frames
The usual and
reference frames will be used as usual when writing
For the geometrical understanding of this transforma-
tion in , a projection of space-time lengths along the
time and space axis of referential frame is used. But
the normal usage of a basis should dictate to project
those lengths along the time and space axis of
And in the Riemannian representation which has been
seen above, those projections were even more different.
They were done, for time lengths along space lines
(therefore along space axis), and for physical space
lengths alongtime axis. Indeed, corresponds to
in the description above, and corresponds to
(which is just a reduced version of ).
Let’s write Lorentz transformation in the context of
the Riemannian metric. Using the strange projection rule
of this representation, the space-time deformation of Fig-
ure 8, which is the postulate 1 of  space-time defor-
mation, yields the following equations.
Equations (17) are the result when applying the
strange projection rule seen above, to the Figure 1 of 
(or Figure 8). Let’s remind that the coordinates of the
. Therefore the first
equation is the projection of local space lengths along
0 time axis (and along Lorentzframe time axis). The
second equation is the projection of time lengths along
space lines and along Lorentz frame space axis.
Let’s remind that the final bb
variables of (17) are
the coordinates of space-time events in thebase. Here,
space lines where parallel to 0 space axis, before the
deformation, which is only local to the point. There-
fore the speed vector of the point is parallel to 0
space axis. This remark allows to write Equations (17).
Now let’s write
t B the r base coordinates. This
transformation becomes the following.
These equations are obtained using rb
Equation (17). This is a Galilean transformation, which
corresponds to the motion of the point which is the
local point of the
reference frame. But this motion is
interpreted now with covariant space and time units. In-
coordinates are those of the r
base which is part of a space-time map. By construction,
in this map the resulting metric is the Euclidean trivial
tensor, constant everywhere in space-time. It is therefore
coherent to find here a Galilean transformation. This last
formulation of Lorentz transformation is the correspond-
ing one in this Riemannian representation of space-time.
Now (16) is derived from (18). First of all, (18) is
similar to the affine transformation which is the identity
linear application composed with the O to
tion. But it is noticed that the O point is moving and
its coordinates are function of x and t. Therefore, (18) is
in fact a linear transformation with respect to the x and t
variables. As such, its determinant is no longer 1, which
was the identity linear transformation determinant. Now
its determinant is equal to 22
1vc. Therefore, compo-
sition of this transformation (18) with 11 which
multiplies the Identity transformation yields (16), which
is the same as (18) but with a determinant equal to 1.
And of course this final (16) result is Lorentz transfor-
This correspondence between those different formula-
tions of Lorentz transformation gives the explanation of
the strange projection rule: the geometrical Euclidean
interpretation is driven by Equation (18). But this equa-
tion is another formulation of Lorentz transform.
6. Comparison of the Geodesics between the
Now let’s compare the “following geodesics” principle in
those two metrics. This comparison is mandatory be-
opyright © 2013 SciRes. JMP
F. LASSIAILLE 1033
cause this principle is used with the Riemannian metric
when writing Equation (15) of .
The simplest way to express the “following geodesics”
principle is the following. The free falling particle tra-
jectory is a geodesic in the Riemannian metric. That’s the
most natural and simple way to express the “following
Of course, this is not the official one. The official one
is the following. The free falling particle trajectory is an
extremal trajectory in the Minkowskian metric. Now, the
usual following reasoning must be done. It is always
possible to construct a null trajectory close to any other
trajectory. Since the free falling particle trajectory is a
strictly positive one, and because of the signature + − − −,
(which is not − + + + otherwise the trajectory would be-
come a minimal one), therefore its extremal value cannot
be a minimal one. Therefore this is a maximal one. This
was a reminder. This complicated and mathematical rea-
soning will be compared further with the one given by
the gravitational model of the three elements theory.
Now it must be checked that those two geodesic defi-
nitions coincide. That is to say:
1) maximal trajectory in the Minkowskian metric,
2) minimal trajectory in the Riemannian metric, are
exactly the same.
This is false in the general case. But it is true for a
time line and in the case of the weak space-time defor-
For proving this let’s compare the geodesic trajectories
in the two metrics. Let’s remind usual equations of ex-
tremal trajectories in some given metric:
being 0 for ct, or i for
as usual, and
the Christoffel symbols.
is the exponential map metric
parameter. In the Minkowskian one, it is equal to the
physical local time. In the Riemannian one, along a time
local time is different, related to the first one
. Let’s remind that the difference be-
tween those local times is driven by the importance of
the space-time vacuum generated along time lines. The
extremal (maximal) trajectory in the Minkowskian metric
is the following.
It has been supposed 0011 for i =
2 and i = 3. Therefore, the corresponding equations for i
= 2 and i = 3 become trivial and are not written here. It
has been supposed also that
the trajectory, meaning that this trajectory is a time line.
Under those considerations, (20) equations are coming
from (19) equations after calculations. The details of
these calculations are available in . Now the extremal
(minimal) trajectory in the Riemannian metric is the fol-
lowing (expressed with the Minkowskian coefficients
and Minkowskian local time for comparison).
It has been supposed also that the trajectory is a time
line in the Riemannian metric. The difference between
(20) and (21) trajectory equations is only occurring for
their second equations and is summarized this way:
is the value of
in the Minkowskian met-
ric trajectory, and r
its value in the Riemannian one.
Therefore, it is impossible to detect this difference in
the weak deformations case. For example, in the Schwarz-
schild metric, there is
physical distances. (Of course, x is the distance from the
attracting object, and M is the Schwarzschild ray). In a
more convincing manner, the (20) second equation be-
comes the following.
Which is Newton’s acceleration. Whereas the (21) sec-
ond equation becomes the following.
Which is an approximation of Newton’s acceleration.
Therefore, those equations are approximately equal.
Finally, the second equation of (20) can be written us-
ing the angle of the space curve tangent in this space-
time Riemannian representation. The result is that Equa-
tion (15) of  is also a good approximation of the last
equation of (20), in the weak space-time deformations
Nevertheless, Equations (20) and (21) do not yield ex-
actly the same trajectories. Of course, the question of
which system equation is correct is easy to answer. The
Minkowskian metric trajectory is the correct one. This
choice can be argued with the help of the gravitational
model of the three elements theory. Indeed, in this model
the classical space-time distance
depends upon 00
Copyright © 2013 SciRes. JMP
Copyright © 2013 SciRes. JMP
related to the total energy of the particle, and 11
the height of the asymmetrical space-time vacuum gen-
erated by the motion of this free falling particle. In this
model, this vacuum has always a key importance.
Therefore it is very coherent to have it minimized in
the trajectories. This straightforward physical reasoning
must be compared with the mathematical and compli-
cated official one, which was reminded above.
As a consequence the Minkowskian metric is still of
extreme importance and can’t be replaced by the Rie-
The aim given in the introduction has been achieved in a
coherent manner. This proves that the gravitational model
of the three elements theory is coherent and therefore
really compatible with relativity. The Riemannian repre-
sentation of space-time which is used in this model is legal.
It allows to understand relativity in a more human sensi-
tive manner than Minkowskian usual representation.
Postulate 1 of  geometrical interpretation and postulate
3 of  application rule has been explained. Equation (15)
of  has been explained. It uses the “following ge-
odesics” principle in the context of the Riemannian metric.
And this has been proven to be a correct approximation
for involved physical distances and for time line trajecto-
Moreover, this geometrical sensitive description of re-
ality allows the construction of the three elements theory,
a unifying theory . This theory is fully understandable
in this geometrical and deterministic manner. It gives a
complete traceability of the mathematic models calcula-
tions along their physical explanations. It indicates that a
more intimate link might exist between classical physics
theories and reality.
 F. Lassiaille, Journal of Modern Physics, Vol. 3, 2012, pp.
 S. Carroll, “Space Time and Geometry, an Introduction to
General Relativity,” Addison-Wesley, San Francisco, 2004.
 F. Lassiaille, “Gravitational Model of the Three Elements
Theory: Mathematical Detailed Calculations,” 2013.
 F. Lassiaille, “Three Elements Theory,” 1999.
F. LASSIAILLE 1035
c: Speed of light.
: Space-time coordinates.
: Space coordinates.
t: Time variable with respect to reference frame.
x: Space physical distance between a space-time point
and the center of the attracting object in the Schwarz-
schild metric, with respect to 0. It is calculated with
the help of an integral along a given space line.
0: Inertial reference frame “attached to” the universe.
For example, in the case of the Schwarzschild metric,
is also attached to the attracting object.
0: Base located along0trajectory. The set of those
bases is thereference frame.
: Inertial reference frame of a “free falling particle”
(see definition). This particle is supposed as usual as get-
ting a null mass for avoiding modification of the space-
time structure. It is located at rest with respect to
when located infinitely far.
: Orthogonal and normalized base with respect to
0 Euclidean metric, in which the Minkowskian metric
m: Normalized base with respect to the Minkows-
kian metric. The set of those bases along a given “time
line” (see definition) will construct the reference
frame. This base is constructed from base. Its vec-
tors are parallel tovectors. Their lengths are inverted
from vectors lengths, with respect to the Minkowskian
metric. In other words, we get
r: Normalized base with respect to the Riemannian
metric. In some way it explains time and space values
in reference frame before space-time vacuum genera-
tion by the free falling particle. This base is constructed
from base. Its vectors are parallel to vectors.
Their lengths are inverted from vectors lengths, with
respect to the Riemannian metric. In other words, we get
: The vectors of this base are equal to
which multiplies the vectors of .
Free falling particle: in this document, this always
means a special kind of free falling particle. Its trajectory
coincides with a time coordinate curve in the system of
bases. Because of the “following geodesics” princi-
ple, it is a geodesic in the Minkowskian metric. As usual
this free falling particle gets a null mass, because we
don’t want it to generate any space-time deformations
around it. Otherwise, this should modify the studied met-
ric. And it gets a null speed when located infinitely far,
because this means that its trajectory will always be per-
pendicular to space lines with respect to 0. In other
words, this trajectory is a time curve in this repre-
Space line: a space curve, which is the space three di-
mension manifold, represented by a curve after projec-
tion on the two dimension figures of this document. But
in this document, this curve is always perpendicular to
any local time axis in the representation.
Time line: the trajectory of a free falling particle (refer
to the above definition of “free falling particle”).
: base vectors.
: base vectors.
uB: base vectors.
: Coefficients of the Minkowskian metric with re-
spect to the base.
: Coefficients of the Riemannian metric with re-
spect to base.
Copyright © 2013 SciRes. JMP