Vol.3, No.3, 165-185 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.33023
Copyright © 2011 SciRes. OPEN ACCESS
The foundation of the theory of the universe dark energy
and its nature
Murad Shibli
Mechanical Engineering Department, College of Engineering, United Arab Emirates University, Al-Ain, UAE; malshibli@uaeu.ac.ae
Received 1 December 2010; revised 5 January 2011; accepted 8 January 2011.
ABSTRACT
Surprisingly recent astronomical observations
have provided strong evidence that our uni-
verse is not only expanding, but also is ex-
panding at an accelerating rate. This paper pre-
sents a basis of the theory of universe space-
time dark energy, a solution of Einstein’s cos-
mological constant problem, physical interpre-
tation of universe dark energy and Einstein’s
cosmological constant Lambda and its value ( =
0.29447 × 1052 m2), values of universe dark en-
ergy density 1.2622 × 1026 kg/m3 = 6.8023 GeV,
universe critical density 1.8069 × 1026 kg/m3 =
9.7378 GeV, universe matter density 0.54207 ×
1026 kg/m3 = 2.9213 GeV, and universe radiation
density 2.7103 × 1031 kg/m3 = 1.455 MeV. The
interpretation in this paper is based on geomet-
ric modeling of space-time as a perfect four-
dimensional continuum cosmic fluid and the
momentum generated by the time. In this mod-
eling time is considered as a mechanical vari-
able along with other variables and treated on
an equal footing. In such a modeling, time is
considered to have a mechanical nature so that
the momentum associated with it is equal to the
negative of the universe total energy. Since the
momentum associated with the time as a me-
chanical variable is equal to the negative sys-
tem total energy, the coupling in the time and its
momentum leads to maximum increase in the
space-time field with 70.7% of the total energy.
Moreover, a null paraboloid is obtained and in-
terpreted as a function of the momentum gen-
erated by time. This paper presents also an in-
terpretation of space-time tri-dipoles, gravity
field waves, and gravity carriers (the gravitons).
This model suggests that the space-time has a
polarity and is composed of dipoles which are
responsible for forming the orbits and storing
the space-time energy-momentum. The tri-di-
poles can be unified into a solo space-time di-
pole with an angle of 45 degrees. Such a result
shows that the space-time is not void, on the
contrary, it is full of conserved and dynamic
energy-momentum structure. Furthermore, the
gravity field waves is modeled and assumed to
be carried by the gravitons which move in the
speed of light. The equivalent mass of the gra-
viton (rest mass) is found to be equal to 0.707 of
the equivalent mass of the light photons. Such a
result indicates that the lightest particle (up to
the author’s knowledge) in the nature is the
graviton and has an equivalent mass equals to
2.5119 × 1052 kg. Based on the fluidic nature of
dark energy, a fourth law of thermodynamics is
proposed and a new physical interpretation of
Kepler’s Laws are presented. Additionally, based
on the fact that what we are observing is just the
history of our universe, on the Big Bang Theory,
Einstein’s General Relativity, Hubble Parameter,
cosmic inflation theory and on NASA’s obser-
vation of supernova 1a, then a second-order
(parabolic) parametric model is obtained in this
proposed paper to describe the accelerated ex-
pansion of the universe. This model shows that
the universe is approaching the universe cos-
mic horizon line and will pass through a critical
point that will influence significantly its fate.
Considering the breaking symmetry model and
the variational principle of mechanics, then the
universe will witness an infinitesimally station-
ary state and a symmetry breaking. As result of
that, our universe will experience in the near
future, a very massive impulse force in the order
1083 N. Subsequently, the universe will collapse.
Finally, simulation results are demonstrated to
verify the analytical results.
Keywords: Dark Energy; Nature of Dark Energy;
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
166
Expansion of the Universe; Einstein’s
Cosmological Constant; Universe Mass/Energy
Densities; Space-Time Dipoles; Gravitons;
Fourth Law of Thermodynamics;
Fate of the Universe; Kelper’s Laws
1. INTRODUCTION
Recent astronomical observations by the Supernova
Cosmology Project, the High-z Supernova Search Team
and cosmic microwave background (CMB) have pro-
vided strong evidence that our universe is not only ex-
panding, but also expanding at an accelerating rate [1-8].
It was only in 1998 when dark energy proposed for the
first time, after two groups of astronomers made a survey
of exploding stars, or supernovas Ia, in a number of dis-
tant galaxies [1,3]. These researchers found that the su-
pernovas were dimmer than they should have been, and
that meant they were farther away than they should have
been. The only way for that to happen, the astronomers
realized, was if the expansion of the universe had sped up
at some time in the past, as well as accounting for a sig-
nificant portion of a missing component in the universe
[9]. The only explanation is that there is a kind of force
that has a strong negative pressure and acting outward in
opposition to gravitational force at large scales which was
proposed for the first time by Einstein in his General
Relativity and given the name the cosmological constant
Lambda [10]. This force is given the name Dark Energy,
since it is transparent and can not be observed or detected
directly. The fourth law of thermodynamics is proposed
by the author to account for the dark energy [11].
Moreover, in Nov. 2006 Scientists using NASA’s Hu-
bble Telescope have discovered that the universe has
been expanding as long as nine billion years ago. Mem-
bers of High-z Supernova Team and Supernova Cos-
mology Project used Hubble to detect the acceleration of
the expansion of the space from observations of distant
supernovae aged between three to ten billion years. The
objective was to uncover two of dark energy’s most fun-
damentals properties: its strength and its permanence.
This paper will be published by NASA’s group in March
2007. Before that, it is thought that the expansion was
decelerating, due to the attractive influence of dark matter
and baryons. The density of dark matter in an expanding
universe disappears more quickly than dark energy, and
eventually the dark energy dominates. If the acceleration
continues indefinitely, the ultimate result will be that
galaxies outside the local supercluster will move beyond
the cosmic horizon: they will no longer be visible, be-
cause their line-of-sight velocity becomes greater than the
speed of light.
These cosmological observations strongly suggest that
the universe is dominated by a smoothly homogenous
distributed dark energy component [9,12-21]. The quan-
tity and composition of matter and energy in the universe
is a fundamental and important issue in cosmology and
physics. Based on the Lambda-Cold Dark Matter Model
(Lambda-CDM 2006), dark energy contributes about
70% of the critical density and has a negative pressure.
The cold dark matter contributes 25%, Hydrogen, Helium
and stars contributes 5% and, finally the radiation con-
tributes 5 × 10–5. The measurements of the Wilkinson
Microwave Anisotropy Probe (WMAP) satellite indicate
the universe geometry is very close to flat [22,23].
The dark energy that is causing the accelerated expan-
sion of the universe is still poorly understood. What is
known so far is that it contributes about 70% of the crit-
ical density and has a negative pressure [9,19,20]. Ac-
cording to the theory of General Relativity, the effect of
such a negative pressure is qualitatively similar to a force
acting in opposition to gravity at large scales. Such as-
tronomical observations have raised fundamental issues:
1) what is the nature of smoothly-distributed energy
which apparently dominates the universe (70%), 2) why
the vacuum energy is much smaller than the theory can
predict? 3) why dark energy density is approximately
equal to the matter density today? A Physical insight of
dark energy and how to detect it using micro-nano space
robotic system was proposed in [24-26].
Modern theoretical scientific exploration of the ulti-
mate fate of the universe became possible with Albert
Einstein’s 1915 theory of general relativity. Alexander
Friedmann proposed one such solution in 1921. This
solution implies that the universe has been expanding
from an initial singularity; this is, essentially, the Big
Bang.
According to the Big Bang Model, inflation suggests
that there was a period of exponential expansion in the
very early universe probably lasted roughly 10–33 seconds
[27]. During inflation, the universe undergoes exponen-
tial expansion so that large regions of space are pushed
beyond our observable horizon. After inflation, the uni-
verse expands according to Hubble’s law, and regions
that were out of casual contacts come back into the ho-
rizon. During inflation, the universe is flattened and the
universe enters a homogeneous and isentropic rapidly
expanding phase is which seeds of structure formation are
laid down in form of a primordial spectrum of nearly-
scale-invariant fluctuations. This explains the observed
isotropy of CMB.
The cosmic inflation may occur during the grand uni-
fication symmetry is broken. Symmetry breaking in
physics describes a phenomenon where infinitesimally
fluctuations acting on the universe crossing a critical
point decide its fate by determining which branch of a
bifurcation is taken. This process is called symmetry
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
167167
breaking because such transitions usually bring the uni-
verse from a disorderedly state into one that is more order
or vice versa.
Based on the measurements of the expansion of the
universe using type 1a supernovae, measurements of the
cosmic microwave background, the measurements of the
correlation function of galaxies, the universe has a cal-
culated age of 13.7 +/– 0.2 billion years. Such a result is
presented in the lambda-Cold Dark Matter model (Lamb-
da-CDM) which is a mathematical model of the Big Bang
with six free parameters.
The discovery of cosmic microwave background ra-
diation (CMB) by Arno Penzias and Robert Wilson in
1965 was a straightforward prediction of the Big Bang
theory. The Big Bang immediately became the most
widely held view of the origin of the universe. Based on
the Big bang theory, the Cosmic Microwave Background
(CMB) receiving the photons just 300,000 years after the
Big Bang, Lambda-CDM [22,23] which estimates that
the age of the universe by 13.7 billion years, the Ein-
stein’s General theory that both space and time are one
entity and are not separated, and on Friedman proposed
solution of Einstein’s General theory which implies that
the universe has been expanding from an initial singular-
ity; this is, essentially, the Big Bang, based on all of that it
can be concluded that the universe which is structured
and composed of homogeneous conserved energy-mass
space-time structure. Then, before the Big Bang it was
null and then it was created and has been expanding since
then. Accordingly, the space and time has a beginning
signed by the Big Bang.
There are some very speculative ideas about the future
of the universe. One suggests that phantom energy causes
divergent expansion, which would imply that the effec-
tive force of dark energy continues growing until it do-
minates all other forces in the universe. Under this sce-
nario, dark energy would ultimately tear apart all gravi-
tationally bound structures, including galaxies and solar
systems, and eventually overcome the electrical and nu-
clear forces to tear apart atoms themselves, ending the
universe in a Big Rip [27]. On the other hand, dark energy
might dissipate with time, or even become attractive.
Such uncertainties leave open the possibility that gravity
might yet rule one day and lead to a universe that con-
tracts in on itself in a “Big Crunch”. Some scenarios, such
as the cyclic model suggest this could be the case.
An important parameter in fate of the universe theory is
the density parameter, Omega (), defined as the average
matter density of the universe divided by a critical value
of that density [24-26]. This creates three possible ulti-
mate fates of the universe, depending on whether is
equal to, less than, or greater than 1. These are called,
respectively, the Flat, Open and Closed universes. These
three adjectives refer to the overall geometry of the uni-
verse, and not to the local curving of spacetime caused by
smaller clumps of mass (for example, galaxies and stars).
Astronomers using the Hubble Space Telescope in 2004
unveiled the deepest look into the universe yet, a portrait
of what could be the most distant galaxies ever seen just
300-800 million years after the big bang from. This image
is called the Hubble Ultra Deep Field (HUDF). Fur-
thermore, rec universe’s first trillionth of a second.ent
results from the WMAP probe have allowed scientists to
see what is thought.
This paper presents a basis of the theory of universe
space-time dark energy, a solution of Einstein’s cosmo-
logical constant problem, physical interpretation of uni-
verse dark energy and Einstein’s cosmological constant
Lambda and its value (= 0.29447 × 1052 m2), values of
universe dark energy density (= 1.2622 × 1026 kg/m3 =
6.8023 GeV), critical density (1.8069 × 1026 kg/m3 =
9.7378 GeV), matter density (= 0.54207 × 1026 kg/m3 =
2.9213 GeV), and universe radiation density (= 2.7103 ×
10–31 kg/m3 = 1.4558 MeV). The interpretation in this
paper is based on geometric modeling of space-time as a
perfect four-dimensional continuum cosmic fluid and the
momentum generated by the time. In this modeling time
is considered as a mechanical variable along with other
variables and treated on an equal footing. In such a
modeling, time is considered to have a mechanical nature
so that the momentum associated with it is equal to the
negative of the universe total energy. It is found that dark
energy is a property of the space-time itself.
Since the momentum associated with the time as a
mechanical variable is equal to the negative system total
energy, the coupling in the time and its momentum leads
to maximum increase in the space-time field. The amount
of energy which contributes to this increase is found to be
70.7% of the total energy. Moreover, a null paraboloid is
obtained and interpreted as a function of the momentum
generated by time. This paper presents also an interpre-
tation of space-time tri-dipoles, gravity field waves, and
gravity carriers (the gravitons). This model suggests that
the space-time has a polarity and is composed of dipoles
which are responsible for forming the orbits and storing
the space-time energy-momentum. The tri-dipoles can be
unified into a solo space-time dipole with an angle of 45
degrees. Such a result shows that the space-time is not
void, on the contrary, it is full of conserved and dynamic
energy-momentum structure. Furthermore, the gravity
field waves is modeled and assumed to be carried by the
gravitons which move in the speed of light. The equiva-
lent mass of the graviton (rest mass) is found to be equal
to 0.707 of the equivalent mass of the light carrier (the
photon). Such a result indicates that the lightest particle
(up to the author’s knowledge) in the nature is the gra-
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
168
viton and has an equivalent mass equals to 2.5119 × 10–52
kg. Moreover, based on the fluidic nature of dark energy,
the fourth law of thermodynamics is proposed, a new
formulation and physical interpretation of Kepler’s Three
Laws are presented.
Additionally, based on the fact that what we are ob-
serving is just the history of our universe, on the Big Bang
Theory, Einstein’s General Relativity, Hubble Parameter,
the estimated age of the universe, cosmic inflation theory
and on NASA’s observation of supernova 1a, then a
second-order (parabolic) parametric model is obtained in
this proposed paper to describe the accelerated expansion
of the universe. This model shows that the universe is
approaching the universe cosmic horizon line and will
pass through a critical point that will influence signifi-
cantly its fate. Considering the breaking symmetry model
and the variational principle of mechanics, then the uni-
verse will witness an infinitesimally stationary state and a
symmetry breaking. Considering the breaking symmetry
model and the variational principle of mechanics, then the
universe will witness an infinitesimally stationary state
and a symmetry breaking. As result of that, our universe
will experience in the near future (relative to the age of
the universe) a very massive impulse force in the order
1083 N. Finally, simulation results are demonstrated to
verify the analytical results.
This proposed paper is organized as follows. In Section
2 modeling of the space-time universe is presented, Sec-
tion 3 describes the modeling of gravity waves and gra-
vitons. Section 4 proposes space-time dipoles. Seven
independent proofs of the nature of dark energy and its
nature, the fourth law of thermodynamics are presented in
Section 5. Section 6 explains why the universe is ex-
panding and calculates Einstein’s cosmological constant
and the value of dark energy density and other mass/
energy densities. Section 7 presents a new formulation of
Kepler’s laws. The parametric model of the ultimate fate
of universe is suggested in Section 8. Simulation re- sults
of the analytical models are demonstrated in 9. Finally,
the paper is concluded by Section 10.
2. MODELING OF THE SPACE-TIME
UNIVERSE
Great advantage occurs sometimes through letting the
time t become one of the mechanical variables. Instead
of considering the generalized position coordinates i
as
a function of the time t, we consider the position coor-
dinates i
q and the time t as mechanical variables,
giving them as a function of some unspecified parame-
ter
. In relativistic mechanics this procedure is an ab-
solute necessity since space and time are united into
four-dimensional world of Einstein and Minkovski. Hence
the Lagarngian configuration space has 1N dimen-
sions, we now add the time t to the generalized coor-
dinates i
q as
1N
tq
(1)
Then, the corresponding phase space must comprise of
22N
corresponding to the 1N pairs of canonical
variables
112
1
12
,,, ,
NN
NN
qq
qq
pp
pp
 
  

  
(2)
The system now has 1N degrees of freedom. De-
noting derivative with respect to
by a prime “'”, the
system action integral appears in the form
2
1
'12
12
,,,; ,,,
N
N
q
qq
WLqq qtd
tt t



 

 (3)
For this purpose let the momentum associated with the
time t denoted as 1tN
pp
be formed as
2
1
N
i
t
ii
Lt q
L
pL t
tq
t





(4)
1
N
tii
i
pL pq

(5)
1
N
tii
i
ppqL

 


(6)
The last Eq.6 is exactly the negative of the total energy
of the system. This leads to the conclusion that the mo-
mentum associated with the time is the negative of the
system total energy [6]. That is, every system is con-
served when the time is considered a mechanical variable.
In other words
tTOTAL
pE
(7)
Now the Hamiltonian function can be extended by
adding the time as a mechanical variable as
1211 21
,,,;,,,
EXTN N
K
qqqp ppconst

  (8)
In the absence of external forces that the conservation
of momentum of a mechanical system can be extended to
1
12 1
1
0
N
iN
i
Pppp p

(9)
or
12 1
1
N
iNNTOT
i
ppppp E
 
(10)
The last Eq.10 indicates that total momentum of the
system is conserved but with a drift. This drift is due to
considering the time as a mechanical variable. As men-
tioned before in this section the objective to describe the
gravitational field generated by the rotating of the mother
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
169169
spacecraft with high relative mass comparing to the micro
spacecraft.
The line-element in this field can be described in terms
of four variables: three for the space 123
,,qqqand one for
the time 4
q. The surface of the gravitational field can be
given as


222 2
12341 2 34
,,, 0
t
fqqqqqqqf qR  (11)
where

4t
qis a function of the variable time and will
be determined later, and R is the radius of the sphere.
The motion of the micro spacecraft robot is restricted to
this surface.
Now let us introduce the line element

12341234
,,,; ,,,ds Fqqqqdqdqdqdq (12)
where the function
F
is an arbitrary function of eight
variables, except for the restriction that to is assumed to
be homogeneous differential form of the first order in
term of the displacements i
dq . This means that


12341234
12341234
,,,; ,,,
,,,; ,,,
dsFqq qq adqadq adqadq
Fqqqqdqdqdqdq
(13)
An arbitrary curve of this manifold can be given in a
parametric form as follows
ii
qf
(14)
where
is an arbitrary parameter. The line-element ds
can be modified to

12341234
,,,;,,,dsFqqqqqqqq d

(15)
Hence the problem of minimizing the length of a curve
between two points 1
and 2
, leads to the minimizing
the volitional integral

2
1
12341234
,,,;,,,
I
Fqqqqqqqq d

(16)
Let the momenta i
p be defined as
i
i
F
pq
(17)
Taking into consideration that the function
F
is ho-
mogeneous function of first order in terms of the vari-
ables i
q, then the Hamiltonian function
H
can be con-
structed as
4
1
0
i
ii
F
HqF
q

(18)
along with the identity

12341 2 34
,,,;, ,,0Kqqqqpppp (19)
Now going back to the surface manifold to wish the
micro spacecraft robot is restricted. Let us assume that
two points come arbitrary near to each other such that
iii
qqdq
(20)
It is interesting to find the shortest distance of the point
i
q from an arbitrary point on the surface. Then the mi-
nimization of a function

,
ii
Wqq subjected to the
constraint (11) where

123411223344
,
,,,, ,,,
ii
Wqq
qq qq qdqqdq qdqqdq 
(21)
Now the minimization problem can be given by
0
ii
Wf
qq

 (22)
Making a benefit of the properties of Hamilton’s prin-
ciple function then the momenta is
i
i
f
pq
(23)
This equation represents the direction cosines of the
normal in terms of the gradient of
f
.
It is possible to rewrite (18) in the form
ii
pdq ds
(24)
Substituting (23) into (24) yields
i
i
f
dsdq df
q

(25)
From (25),
1grad
df
f
ds
 (26)
where 1/
is the maximum rate of change of the func-
tion
f
at that given point along the normal (because it is
known from vector calculus that the directional derivative
characterizes the maximum increase in function
f
).
Considering (26) then (23) can be written as
1
|grad |
i
i
f
p
f
q
(27)
According to the theorem that the momentum associ-
ated with the time is the negative of the system total en-
ergy, then (27) becomes
4
grad
TOTAL
fEf
q

(28)
Note that if the gradient value is other than zero it
characterizes the maximum increase, then grad
f
> 0.
For example for a spherical field, the maximum rate of
change (directional derivative) is 1 because the gradient is
the normal to the sphere surface and taken along the ra-
dius, in this case (28) becomes
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
170
4
4
TOTAL
f
pE
q

(29)
Using (29) the function

4t
qcan now be determined
as

44tTOTAL
f
qEq (30)
By this the surface of the sphere can be represented as
222 2
1234 0
TOTAL
qqqE qR  (31)
As a general case, the gravitational surface should be
modified to
222 2
123 4
grad 0
TOTAL
qqqfEqR 
(32)
Eq.32 has a physical interpretation, specifically the
term 4
grad |TOTAL
f
Eq, it shows that time as a mechanical
variable along with its associated momentum which is
equal to the negative of the system total energy play a
significant role in determining the size of the gravitational
field under consideration and characterized by the
maximum increase |grad |
f
.
Recalling the kinetic energy of a unit mass which can
be given in the form
2
1
2
ds
Tdt



(33)
where ds is the line element of the configuration space.
Since now the time t is no longer an independent vari-
able but
, the kinetic energy (33) should be modified to
2
2
1
2
ds
Tt
d


 (34)
or
2
2
2
ds Tt
d



 (35)
From (35) it can be seen that
2ds (36)
Using (36) and (26) it yields
1
grad 0.707
2
f (37)
Making a benefit of (37) and substituting this value in
(28) and (32), respectively, one gets
4
0.707 TOTAL
fE
q

(38)
222 2
123 4
0.707 0
TOTAL
qqqEqR  (39)
The physical significance of (38) and (39) is that the
amount of energy that contributes to the maximum in-
crease is proportional to the total energy by 0.707. This
energy is proposed as the universe dark energy which is
behind the expansion. The value 0.707 (= 70.7% of total
energy) agrees with the measured values reported re-
cently in [1-4,10]. It is found for a flat universe the dark
energy = 0.71 in [15]. Eq.39 can be modified to
222 2
123 4
0.707 TOTAL
qqqE qR
  (40)
For a given conserved mechanical system, the right
hand side of (40) shows that the surface that describes the
space-time field is expanding. This expansion is due to
the coupling in time and its momentum and characterized
by the maximum increase gard 0.707f. The more the
system has energy the more it is expanding. That explains
why galaxies and clusters are departing much faster that
our solar system. For a conserved mechanical system, the
line-element of this manifold now can be given in the
form
2 222
123 4
0.707 TOTAL
dsdqdq dqEdq (41)
To compare this manifold with Robertson-Walker and
Minkowski’s four-dimensional world are defined as fol-
lows. For a flat universe represented in Robertson-Walker
cosmology defined as
222222
dsatdxdydzcdt (42)
where c is the speed of light and

at is the expansion
rate. Meanwhile, for a Minkowski’s four-dimen- sional
world it is defined as
222222
123 4
dsdqdqdqc dq (43)
The difference occurs in the coefficient of the time
displacement related terms so that in (42) and (43) it is
equal to the (22
4
cdq), meanwhile in the proposed model
(41) it is equal to (4
0.707 TOTAL
Edq
). The previous anal-
ysis can be summarized in the following theorem.
Theorem 1: Considering the time as a dependent me-
chanical variable along with other generalized variables,
the gravitational field described in (39) is expanding due
to the coupling in time and its momentum (34) and
characterized by the maximum increase gard 0.707f.
Three cases can be also concluded from (43)
Case 1: if 20ds
, then

222
4123
0.707 TOTAL
E dqdqdqdq (44)
and the curve is time-like in that interval.
Case 2: if 20ds, then

222
4123
0.707 TOTAL
Edqdqdqdq (45)
and the curve is space-like in that interval.
Case 3: if 20ds
, then
222
4123
0.707 TOTAL
E dqdqdqdq (46)
and the curve is null in that interval. The reduced line-
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
171171
element (46) represent a null paraboloid which depends
on the system total energy TOTAL
E and the maximum
increase. In this case if a particle is moving with the speed
of light then the null rays will lie on the surface of the
paraboloid.
3. MODELING OF GRAVITY WAVES AND
GRAVITONS
In the following analysis we assume that we are deal-
ing with the motion of a particle in the gravitational field
of a total energy E and a potential energy V. Based
on the previous analysis and the line-element, we consider
now a gravitational field function

1234
,,,SSqqqq,
a function of 3 space coordinates 123
,,qqq and a time
coordinate 4
q, treated on an equal footing and satisfies
the following wave equation follows

2
22 2
1234
2
SSSS
mE V
qq qq

 



 

 
 (47)
where m is the equivalent mass of the particle that
carries the gravitational waves and will be calculated in
the following analysis. It is known now from the previ-
ous analysis that the gradient of the gradient of a surface
function S has the maximum increase and has the di-
rection of the normal to the surface

1234
,, ,const.SSqqqq (48)
As derived in the previous section, the directional de-
rivative (the absolute value of S) can be defined as
gard S
Sq
(49)
and taken in the direction of the normal. Referring to
Figure 1, it possible to write (49) in the form
gardS
(50)
The gradient (47) can be rewritten as follows

2mE V

(51)
The gradient (49) represents the linear momentum
such that
gard S
Smv
q

(52)
Based on (41) the maximum increase is found to be
0.707, then (51) and (52) yield to:
0.707
Smv
q

(53)
Since the Einstein’s Relativity assumes that the
maximum speed in the nature is the speed of light, and
CS
CS
1
p
2
p
Figure 1. Gravitational wave construction.
assuming that the gravitational waves moves in the speed
of light
speed of lightvc
(54)
and assuming that the speed of light is considered as a
unity and from (53) we conclude that the mass of the
gravitons which carries the gravity waves is defined as
0.707
g
raviton photon
mm
(55)
This indicates that the lightest particle in the nature is
the graviton and has an equivalent mass equals to 2.5119
× 10–52 kg (the equivalent mass of the photon is 3.9 ×
10–22 of the mass of the electron). That is, the electron is
heavier than the graviton by 3.6267 × 1021 times.
Theorem 2: (Gravity Waves and Gravitons): The
gravity waves described by (47) is carried by the gravi-
tons which move in the speed of light and the graviton has
an equivalent mass equals to 2.5119 × 10–52 kg.
4. MODELING OF SPACE-TIME DIPOLES
The Euler-Lagrangian equation takes the following
form in case of a 4-Dimnesional space-time world, which
depends on the coordinates: 1
x
q, 2
yq, 3
zq
,
4
tq
,
3
12 4
1234
div 0
Total
p
pp p
qqqq
 


P (56)
When dealing with a particle, only 4
tq is consid-
ered, and (56) is reduced to
4
4
0
p
q
(57)
Eq.57 implies that the time momentum is constant
which by the virtue of Theorem (1) is interpreted as the
system total energy, that is
4.
Total
pE const
 (58)
Eq.57 suggests that 4
p is not a function of the time
coordinate 4
tq
.
Now by the virtue of Neother’s Invariant Theorem [6,8]
and Divergence Transformation Theorem of Gauss, (56)
can be interpreted as follows:
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
172
div
VS
dV ndA
 
PP
(59)
where P is the momentum vector of the 3-domensional
space, n is the normal vector to the surface S and dV is the
segment of volume V. Now by considering that the mo-
mentum of time (as a mechanical variable) is constant and
equals to the system total energy as in (58), then (59) can
be modified as
4
4
SV
d
dAp dV
dq

 
Pn (60)
The physical interpretation of (59) and (60) as follows:
the space-time total energy-momentum contained in a
certain volume is equal to the negative of the total
space-time energy-momentum that flows through the
boundary surface of that certain volume. In sense of the
previous analysis an equivalent equation can be drawn
from (56) as:
4
4
div 0
p
q

P (61)
The importance of (61) is the condition for continuity
equation of the space-time.
The world line-element proposed in (41) and Minko-
visky world shows that there is a polarity in the space-
time description (+, +, +, –): a plus (+) is a space signa-
ture, meanwhile a minus (–) is a time signature. This
polarity can also be suggested by applying the system
conservation of momentum with zero initial conditions
1234
0pp pp (62)
or by virtue of (21)
123 0
Total
pp pE (63)
This suggests that space-time exists in a dipole form
originated at the time 4
q and ended at the space resultant
vector

123
,,qq qq. The space-time dipole moment
vector is denoted by e. Assuming also that the dipole is
placed in a uniform gravity field and assuming that
the dipole makes an angle
with the field. This pro-
duces a net torque on the dipole expressed as the cross
product of the space-time dipole moment and the field:
sinee

 (64)
The potential energy can be expressed as a function of
the orientation of the space-time dipole with the gravity
field:
cosUe e
  (65)
Eqs.64 and 65 are another representation of (62,63).
Since the time momentum is equal to the negative of
system total energy, then from (64) and (65) the orienta-
tion angle should be 45 degrees (that is sin
cos 0.707
). The figure 0.707 has a special signifi-
cance in the space-time. The space-time is made up of
dipoles. The dipoles are oriented with 45 degrees. When a
gravity field is applied, a torque on the dipoles is pro-
duced. The space-time dipoles align themselves with the
gravity field. The degree of alignment of the space-time
dipoles with the field depends on strength of the gravity
field and the mass and angular velocity. This insight can
be added to Theorem 1 where only 0.707 of the total
energy contributes to the expansion of the universe and
this energy is given the name Dark Energy.
Theorem 3: (Space-Time Dipoles): the space-time
described by (60), (62) and (63) is composed of space-
time dipoles and the dipole moment is oriented by 45
degrees originated from the time and ended at the space
and these dipoles are behind the formation of orbits
structure.
Theorem 3 is derived based on Noether’s Invariant
Variational Theorem, Euler-Lagarangian equations and
the Divergence Theorem of Gauss. This theorem can also
be proved by physical tracking of planetary, solar and
galactic orbits as shown in Figures 2 and 3.
5. NATURE OF DARK ENERGY
The purpose of this section to reveal the nature of
themysterious dark energy and its governing law that is
causing the universe to expand at an accelerating rate.
Seven independent proofs are provided.
Proof 1 (Classical Thermodynamics): Since the uni-
verse is expanding, then its volume is increasing and its
Figure 2. Saturn and its rings, uranus and its rings.
Figure 3. Spiral galaxy and the milky way galaxy.
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
173173
boundaries are moving. A closed system composed of
homogenous isotropic cosmic fluid (space-time universe),
with moving boundaries is then considered in this proof.
In classical thermodynamics the expansion work is often
called moving boundary work [28].
The objective is now to determine the moving bound-
ary work of expanding universe assuming a quasi-equi-
librium expansion process. Consider that the system is
experiencing a negative pressure , and the pressure
force acting on a differential area dA is dA acts
outward normal to the surface. Under the influence of the
negative pressure the system will move outward a dif-
ferential distance ds and the system is undergoing a
change in its volume equal to dV dAds, where V is the
total volume as shown in Figure 4. Noting that work is
force times distance, then the boundary work done on the
system is expressed by
WFds dAds dV
 (66)
Note that both
and d are used to indicate differ-
ential quantities, but
is typically used for quantities
that are path functions (such as work) and have inexact
differentials, while d is used for quantities that are point
functions and have exact differential. Taking the integra-
tion of both sides of Eq.66 yields to
VW  (67)
where V is the net final total volume. Defining the
amount of energy exists per unit volume as the energy
density
. Then, the amount of dark energy contained
inside the universe system undergoing an increase in its
volume is
EV
(68)
By the virtue of first law of thermodynamics of energy
conservation and by comparing (67) with (68) yields
 (69)
Figure 4. The differential surface of a
space-time universe system with arbitrary
shape under the influence of expansion by
the negative pressure of dark energy.
The physical interpretation of (69) is that the negative
pressure that is causing the universe to expand is equal to
the negative of energy volumetric density. This force is
behind the expansion of the universe.
Proof 2 (The Thermodynamics Equation of State of
an Ideal Gas): this equation relates the pressure P, tem-
perature T and the volume V of a substance behaves as an
ideal gas [28], that is
VmRTE
 (70)
Note that both sides of the equation have the units of
energy (work done by pressure P). Assume now that dark
energy behaves like an ideal gas with a negative pressure
that causes the universe to expand with a total vol-
umeV, then by dividing both side of the equation of state
(71) by V, it becomes
mRT E
VV
 (71)
It is worth to mention that NASA’s Cosmic Microwave
Background Explorer (CMB) in 1992 estimated that the
sky has a temperature close to 2.7251 Kelvin. Moreover,
the Wilkinson Microwave Anisotropy Probe (WMAP) in
2003 has made a map of the temperature fluctuations of
the CMB with more accuracy [8].
Proof 3 (The Cosmological Equation of State of a
Perfect Fluid): the equation of state of a perfect fluid in
cosmology is characterized by a dimensionless number
w [9], equal to the ratio of the pressure of the fluid to its
energy density as follows
w
(72)
Recent observations of Lambda-Cold Dark Matter
model (L-CDM) estimate the value of wof a flat uni-
verse close to 1, that is
 (73)
Proof 4 (Invariant Variational Principle of Me-
chanics): for the importance of this proof some details
here will be repeated so that all proofs together can be
treated at the same time. Hence the Lagarngian configu-
ration space has 1N
dimensions, we now add the time
t to the generalized coordinates i
as
1N
tq
(74)
Then, the corresponding phase space must comprise of
22N
corresponding to the 1N pairs of canonical
variables:
112
1
12
,,, ,
NN
NN
qq
qq
pp
pp
 
  

  
(75)
The system now has 1N
degrees of freedom (DOF).
Denoting derivative with respect to
by a prime “'”,
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
174
the system action integral appears in the form
2
1
1
12
12 1
,,,; ,,,
N
N
q
qq
WLqq qtd
tt t



 

 (76)
For this purpose let the momentum associated with the
time t denoted as 1tN
pp
be formed as

4
2
1
i
t
ii
Lt q
L
pL t
tq
t





(77)
4
1
tii
i
pL pq

(78)
4
1
tii
i
ppqL




(79)
The last Eq.79 is exactly the total energy of the system.
This leads to the conclusion that the momentum associ-
ated with the time is the negative of the system total en-
ergy [29]. That is, every system is conserved when the
time is considered a mechanical variable. In other words
tTOTAL
pE (80)
Noting that the momentum associated with the time as
a mechanical variable has the units of work, considering
that the system is expanding such that it has a total net
volume V and by dividing both side of (15) by V,
one obtains
tTOTAL
pE
VV
 (81)
Recalling that the work of the pressure done on the
system as its volume changes is defined as /WV ,
then (81) can be expressed as
 (82)
The last equation is equivalent to Eq.69. The impor-
tance of Eq.82 is that it has the following physical inter-
pretation: the momentum associated with time per unit
volume (the pressure associated with time) is equal to the
negative of energy density that is the negative force be-
hind the expansion of the universe.
Proof 5 (Euler-Lagrangian Equation): in addition to
the previous analysis that has proved that the momentum
associated with time is the dark energy, the objective now
is to show that the flow of dark energy of the expanding
universe is the momentum associated withe time. The
Euler-Lagrangian equation takes the following form in
case of a 4-D space-time world, which depends on the
coordinates: 1
x
q, 2
yq, 3
zq, 4
tq,
3
12 4
1234
div 0
Total
p
pp p
qqqq
 


P (83)
where i
p is the momentum associated with coordinate
i
q. When dealing with a particle, only 4
tq is con-
sidered, and (83) is reduced to
4
4
0
p
q
(84)
Eq.84 implies that the time momentum is constant
which by the virtue of (80) is interpreted as the system
total energy, that is
4.
Total
pE const
 (85)
Eq.84 suggests that 4
p is not a function of the time
coordinate 4
tq
.
Now by the virtue of Neother’s Invariant Theorem and
Divergence Transformation Theorem of Gauss [29-31],
(84) can be interpreted as follows:
div dd
VS
VA
 
PPn
(86)
where P is the momentum vector of the 3-domensional
space, n is the normal vector to the surface S and dV is
the segment of volume V. Now by considering that the
momentum of time (as a mechanical variable) is constant
and equals to the negative of the system total energy as in
(85), then (86) can be modified as
4
4
0
SV
d
dAp dV
dq

 
Pn (87)
or
4
4
d
dd
d
SV
A
pV
q

 
Pn (88)
The physical interpretation of (88) is as follows: the
flux of energy (work flow) flowing through the boundary
surface of the expanding space-time universe is equal to
the negative of the universe total energy (the momentum
associated with time):
 . In sense of the previous
analysis an equivalent equation can be drawn from (83)
as:
4
4
div p
q

P (89)
Proof 6 (Einstein’s Equation of General Relativity):
The main goal of this proof is to show that Einstein’s
Cosmological constant Lambda is equivalent to the dark
energy by applying Einstein’s Equation of General Rela-
tivity. One of the candidates for dark energy is vacuum
energy, or the Einstein cosmological constant, charac-
terized by a pressure equal to the negative of the energy
density [9,10]:
18
2
RRgg GT

 (90)
where the left-hand side of Einstein’s equation charac-
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
175175
terizes the geometry of the space-time and the right-hand
side represents the energy sources,
g
is the space-
time metric, R
is the Recci tensor, R is the space-
time curvature scalar, is a free parameter called Ein-
stein’s cosmological constant (which equivalent to the
dark energy), G is the Newtonian Gravitational Uni-
versal Constant and T
is the energy-momentum tensor.
If the vacuum (space-time) is Lorentz-invariant, then the
energy momentum will take the form
vac
vac
Tg
 (91)
where vac
is the vacuum density (dark energy). Lor-
entz-invariant energy momentum is associated with an
negative isotropic pressure such that
vac vac
 (92)
where vac
is the vacuum pressure. It can be proved that
Einstein’s cosmological constant is proportional to
dark energy so that
2
3
8
vac
c
G
 (93)
The physical interpretation of (93) is that the dark en-
ergy density constant Lambda is equivalent to the cos-
mological constant Lambda by the constants 2
38cG
.
Proof 7 (Einstein’s Mass-Energy Equivalence Prin-
ciple): consider that the universe has a total energy Total
E
with an equivalent mass
E
q
M
, then by the virtue of Ein-
stein’s Mass-Energy Equivalence Principle, the relation
is given as
2
Total Eq
EMc (94)
where c is the speed of light. By dividing the last equa-
tion by the universe volume, one gets
2
Eq
Total M
Ec
VV
(95)
or
2
,
Eqmass c

According to Eqs.69, 71, 73, 88 and 92:
, then
2
,
Eqmass c
 (96)
where ,
E
qmass
is the equivalent dark energy mass den-
sity. Eq.96 shows that the pressure of the dark energy is
equal to the negative of the equivalent dark energy mass
density times the square of the light speed.
According to the previous proofs, the fourth law of
thermodynamics is proposed and stated as follows:
Proposed Fourth Law of Thermodynamics: “Consid-
ering time as mechanical variable, for a closed system
with moving boundaries composed of homogenous iso-
tropic cosmic fluid, the system will have a negative
pressure equal to the energy density that causes the sys-
tem to expand at an accelerated rate. Moreover, the mo-
mentum associated with time is equal to the negative of
the system total energy” [22].
6. COSMOLOGICAL CONSTANT AND
UNIVERSE MASS-ENERGY
DENSITIES
Based on the observation that the universe is flat
(WMAP), and previous proofs that the dark energy is a
property of the space-time itself and it is homogenous
and uniformly distributed over the universe and behaves
like an ideal perfect cosmic fluid. Considering time as a
mechanical variable, and based on the definition of the
total kinetic energy of universe
2
1
2eq
ds
KM
dt



(97)
where eq
M
is the equivalent mass of the universe and
ds is the line-element of the configuration space, and
t stands for time. Since the kinetic energy of the uni-
verse is defined astotal
K
EV
where V is the po-
tential energy, and defining
dt
dt d
d


 (98)
where
is a cosmic space-time parameter that repre-
sents the coupling in the space-time and has the units of
.secm, taking into the account the Hubble expansion
parameter, then the rate of change of time as a mechani-
cal variable with respect to the coupled space-time cos-
mic parameter using (97) can be given as

2
eq
total
M
dt ds
dd
EV



(99)
Now assuming for a flat universe (as in WMAP) a
zero reference potential energy 0V, and considering
the well-known equation of Einstein 2
totaleq
EMc,
where c is the speed of light. Then, Eq.34 can be re-
duced to
11 1
0.7071
2
dt dsds
dcd cd

 

 
 
(100)
Since the universe is expanding and Hubble parameter
is a measure of this expansion, then ds d
is equiva-
lent to the Hubble parameter
H
and have the units of
msm
or 1sec. The value of the present Hubble
parameter 0
H
considered is

70kmsMpc which is
equivalent to
18
2.310m sm
based on the WMAP
2006. Then Eq.10 0 yields to
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
176


26
0.70710.7673 10secsec
dt m
d
  (101)
Since 26 1
0.5426510dt dm
 has the units of
1
m and since Einstein’s Cosmological constant Lambda
has the units of 2
m in (90) and (93), then

2
dt d
is
equivalent to Lambada, that is 52 2
0.29447 10m
 
which complies with the astronomical observation that
estimated its value 52 2
~10 m

. Now it is possible to
modify Einstein’s Cosmological Constant to have the
form

2
2
8
3
G
dt dc

 (102)
then,

2
2
3
8
cdtd
G
(103)
where
is the density of the cosmological constant
and G is the Universal Gravitation Constant and has
the value 1122
6.6710N mkg
 . According to (103),
the universe dark energy density has the value
26 3
1.262210kgm6.8023 GeV
 . It can be con-
cluded that rate of change of time with respect to the
cosmological coupled parameter
is behind the ex-
pansion of the universe and, accordingly, represents a
physical meaning of the cosmological constant. By this,
the second issue raised above has been solved. This
model solves the 120-orders-of-magnitude discrepancy
between the theoretical and observed values of the cos-
mological constant.
As it can be seen from the previous analysis, universe
dark energy density is 26 3
1.262210kg m
 as cal-
culated theoretically by the proposed model which totally
agrees with the astronomical observations that estimate
the dark energy density in the range of 26 3
~10kgm
.
Furthermore, the energy density of each component i
of the universe is characterized by its density parameter
i
and the critical density c
as
i
i
c
 (104)
Recent observation of Supernovae by WMAP shows
that 0.70
 , 0.30
matter
 ,5
510
radiation
. Now
benefiting from (103) and (104), the average critical
density of the universe is 26 3
1.806910kg m
c
 =
9.7378 GeV which is very close to 26 3
0.7410kg m
estimated by cosmological observations. According to the
proposed model in this paper the average density of the
matter in the universe is 26 3
0.5420710kgm
m
 =
2.9213 GeV. By this, the third issue raised above has been
solved as well. Finally, the average density of the radia-
tion in the universe is 31 3
2.710310kg m
r
 =
1.4558 MeV.
7. NEW FORMULATION OF KEPLER’S
LAWS: DARK ENERGY AND OUR
SOLAR SYSTEM
This section presents a new formulation of Kepler’s
Laws in terms of Dark Energy. Kepler’s complete analy-
sis of planetary motion is summarized in three state-
ments known as Kepler’s laws as follows: Kepl er’s First
Law: All plants move in elliptical orbits with the Sun at
one Focus. Keple r s Second Law: The radius vector
drawn from the Sun to a planet sweeps out equal areas
in equal time intervals. Keplers Third Law: The square
of the orbital period of any planet is proportional to the
cube of the semi-major axis of elliptical orbit.
7.1. New Interpretation of Kepler’s Second
Law
The New Newtonian Kepler’s Second Law:For a
planet orbiting the Sun, the rate of change of the area
swept by the planet with respect to the cosmic coupled
space-time parameter is proportional to the square root
of the Einstein’s cosmological constant”.
Proof: Kepler’s Second law can be derived as a se-
quence of the principle of conservation of angular mo-
mentum. Consider a planet of mass
P
M
moving about
the Sun in an elliptical orbit. Let the planet to be consid-
ered as a system and the Sun is so heavy so that it does
not move relatively to the planet. The gravitational force
acting on the planet is a central force, always along the
radius vector and directed toward the Sun. Taking the
torque on the planet due to this central force which is
always zero because the central force
F
is parallel to
R. That is
0RF
 (105)
Recalling that the time rate of change of the angular
momentum of the system equals the external net torque
on the system; that is, /dLdt
, where L is the
angular momentum. Therefore, and because torque
0
, the angular momentum L of the system (the
planet) is a constant
.
P
LRPMRvconst
  (106)
By the geometric of the motion, (106) can be modified
as follows. In a time interval dt, the radius R sweeps
out the area dA , which equals half the area RdR of
the parallelogram formed by the vectors Rand dR.
Because the displacement of the planet is the time inter-
val dt is given by dR vdt
, one gets
11
22 2A
P
L
dARdRRvdtdtKdt
M
 (107)
By dividing both sides of Eq.107 by the coupling pa-
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
177177
rameter variation d
yields
2A
P
dALdtdt
K
dMd d




 (108)
By the virtue of (101) the value of dA d
is propor-
tional to square root of Einstein’s Constant, i.e.,
26
0.54265 10
. Note that dAd
is a vector and has the
units of velocity and directed upward, perpendicular to
the Sun-planet plane and along the angular momentum
vector. From vectors view point and as a result of the rate
change of time with respect to the coupled space-time
parameter
and the angular motion of the plant around
the Sun, the planet tends to move in a vertical direction
perpendicular to the elliptic plane. Consequently, the
planet tends to leave the plenary motion, but the existence
of the dark energy with a very small value (
26 3
1.262210kg m
) damps such an upward motion
and keeps it so small comparing to the planetary rota-
tional velocity. That is why a stable circular path is al-
ways obtained and planets always move in a stable
co-plenary motion.
Calculations results of such a model shows that in the
case of the Erath-Sun system, the vertical motion of the
Erath is approximately 1.5 × 10-11 m/sec which is very
negligible comparing to its rotational velocity. Moreover,
in case of the Mercury-Sun system, the Mercury vertical
velocity is in the range of 2.5 × 10-12 m/sec. Meanwhile,
such a vertical motion of the Sun orbiting our Milky Way
Galaxy is approximately close to 2.5 × 10-31 m/sec. Ac-
cording to the previous analysis, then, it can be seen that
Einstein’s Constant (and dark energy) is behind the sta-
bility of elliptic orbits. For given planet orbiting the sun,
the momentum is constant, its mass is constant, then the
swept area is maintained fixed due to Einstein’s Constant
Lambda and constant dark energy density at given orbital
location.
7.2. New Interpretation of Kepler’s Third
Law
The New Newtonian Kelper’s Third Law:The
square of the orbital period of any planet orbiting the
Sun is inversely proportional to the dark energy density
at that orbit”.
Proof: Consider a planet of mass
P
M
that is as-
sumed to be moving about the Sun with mass S
M
in a
circular orbit. Because of the gravitational force provides
the centripetal acceleration of planets as it moves in a
circle. Using Newton’s second law for an object in uni-
form circular motion then,
2
2
SP P
GM M
M
v
R
R (109)
The orbital speed of the planet is 2/RT
, where T is
the period; then (109) becomes

2
2
2
SRT
GM
R
R
(110)
2
233
4
S
S
TRKR
GM



 (111)
where
2
192 3
42.97 10sec
S
S
K
m
GM



 for our so-
lar system. Eq.111 is still valid if we replace the radius
R with the length of the semi-major axis for elliptical
orbits. Note that the constant is inversely proportional to
the mass of the central body (the Sun in our case) and
independent of the mass of the planet.
By the virtue of Eq.111, the square of the orbital period
of any planet is proportional to the cube of the semi-major
axis of elliptical orbit; that is 23
TR
. In other words, it is
a volumetric measure. Now since Dark Energy is very
homogeneous and uniformly distributed all over the
universe, then it is very reasonable to calculate the dark
energy density consisted in sphere centred at the Sun and
have radiusR. To do so, multiply both sides of Eq.11 1
by
43
to obtain
23
44
33
S
TKR

(112)
Now assume that the equivalent mass of the dark en-
ergy contained in that sphere is eq
M
, then by dividing
both sides of (112) by this equivalent mass eq
M
yields
 
23
11
43 43
S
eq eq
TK R
MM
(113)
The expression

3
14/3
eq
R
M
represents the recipro-
cal of the dark energy density
E
in 3
kg m. Accord-
ing to this analysis (113) can be rewritten as

211
43
Seq
E
E
E
KM
TK


(114)
where

192 3
3
43
0.70910kg sec
Seq
Eeq
S
eq
KM
KM
GM
M
m




 
for our solar system.
The heavier the central body is, the more dark energy
is dragged to it. It is then very possible to estimate the
value of dark energy density by knowing the orbiting
period. Calculation results show that dark energy density
at locations close to the earth is 0.25 × 1025 kg/m3.
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
178
Meanwhile, it is in the range of 18 × 10–25 kg/m3 for
Mercury which agrees with the observations. The more
the planets are close to Sun, the faster the orbiting speed
will be. This is due to the dark energy density which
behaves as a perfect fluid. Hence, more of dark energy
fluid exits near to the sun.
According to the previous that shows a fluidic nature
of the dark energy, then new physical interpretation of
Kepler’s First Law is as follows: The New Newtonian
Kepler’s First Law: Revolving of the Sun around itself
causes elliptic streamlines in the fabric of the cosmic
space-time fluid on which planets orbit.
8. THE ULTIMATE FATE OF THE
UNIVERSE: THE BIG IMPULSE
This part presents a parametric modeling of the ulti-
mate fate of the universe. Such a model is based on: the
Big Bang theory, Einstein’s General Relativity, Fried-
mann-Robertson-Walker model, Lambda-Cold Dark Mat-
ter model (CDM), Hubble expansion parameter, Cosmic
Microwave Background radiation (CMB), Hubble Ultra
Deep Field (HUDF) observation of the deep space, Wil-
kinson Microwave Anisotropy Probe (WMAP) [22,23],
the exponential inflation model, cosmic horizon problem,
breaking symmetry model [27], the estimation of the age
of the universe by 13.7 billion years, the NASA’s obser-
vation of dark energy that caused the expansion of the
universe 9 billion years ago and before that time the ex-
pansion was decelerating.
Based on all of that and on the fact that what we are
observing is just the history of our universe, a second-
order (parabolic) parametric model is obtained in this
proposed paper. Such a parabolic model describes the rate
of change of time with the space-time cosmic coupled
parameter, that is, the expansion of the universe. This
model shows that the universe is approaching the uni-
verse cosmic horizon line and will pass through a critical
point that will influence significantly its fate. Considering
the breaking symmetry model and considering a very
infinitesimally period of time similar to that of the infla-
tion 33
10 sec
happened early in the universe, then the
universe will witness an infinitesimally stationary state
by the virtue the variational principle of mechanics and as
result of that the a very massive impulse will occur and
correspondingly the universe will collapse. The physical
meaning of such a model is that as the universe time
(clock) stops, the universe will collapse.
Proof: Considering time as a mechanical variable, and
based on the definition of the total kinetic energy of uni-
verse as

11 1
0.7071
2
dtdsds f
dcd cd
 
 
 
 
  (115)
Based on the fact that our observation of the universe is
a record of its history, based on the cosmic exponential
inflation model that universe brought back to our ob-
servable horizon within 33
10 sec
after the Big Bang,
based on the Hubble observation that after the cosmic
inflation the universe expansion decelerated for 4.7 bil-
lion years then it has started to accelerate up until these
days for 9 billion years, taking into account of the age of
the universe is 13.7 billion years by CDM model, based
on Hubble Ultra Deep Field (HUDF) observation of the
deep space, based on the astronomical observations us-
ing Cosmic Microwave Background radiation (CMB) of
the photons signals had sent to us 300,000 years after the
Big Bang that shows that the universe is observable
within the cosmic horizon, it can be concluded that the
universe is still within the observable horizon line, then,
a second order parabolic parametric model is proposed
to describe the expansion behavior of the universe.
This curve is open up and bounded by the cosmic ho-
rizon line, and the left portion of the curve accounts for
the deceleration of the expansion of the universe for the
first 4.7 billion years, meanwhile, the right portion of this
parabolic curve accounts for the acceleration of the uni-
verse expansion for 9 billion years. Moreover, this model
intersects with the cosmic horizon line at the first critical
breaking symmetry point within the cosmic inflation, that
is, the Big Bang. The minimum point (turning point be-
tween deceleration and acceleration) is characterized in
terms of Hubble parameter when the universe starts ac-
celerating at 18 1
~10 sec
rate. Furthermore, the second
intersection point of this parametric parabolic curve with
the universe cosmic horizon line represents the second
breaking symmetry critical point at which our proposed
Big Impulse occurs and would be estimated as follows.

2
dt abc
d

 (116)
The constant a, b, and c are constants to be deter-
mined according to the initial and given condition. It is
clear that the constant c is zero since the /0dt d
at
0
. Now the model (116) is reduced to

22
dt b
aba
da


 


(117)
It is very useful now to write (117) in completing-the-
square form because it can represent the minimum point
and if the curve is open-up in a simple way
22
11
24
dtb b
a
daa

 

 (118)
Note that the vertex point is

min
,
cH
, where mi n
H
represents the Hubble parameter when the universe starts
to accelerate 9 billion years go 18 1
10 sec

, and c
is
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
179179
its corresponding space-time parameter value. Note also
that 3
t
, then at 4.7 billion years, 1.6751
. Since
11.6751
2
c
b
a
 , then
3.3502
b
a (119)
From the vertex-form (completing the square),
2
min
11
44
bb
H
b
aa



 (120)
or

min
4
bH
ba
(121)
It can be seen that the ratio 3.3502ba is not a
function of mi n
H
contrary to the constant b which a
function of min
H
as given in (121). Solving for a and b
in (119), (120) and (121), one obtains b = 1.194 mi n
H
and a = 0.3564 min
H
. Since the constant a is positive
then the second order parabolic curve is open up.
Substituting for a and b in the expansion parabolic
parametric Eq.117 yields

2
min 0.3564 1.194
dt H
d

(122)
Simulation of the parametric expansion (122) with
respect to
in Figure 6 shows the Big Bang singular-
ity, the left portion of the decelerated expansion, then the
universe started to accelerate at 1.6751
and heading
up towards the cosmic horizon and the breaking symme-
try critical point.
Since the Hubble parameter is a slowing down pa-
rameter and since it is characterized by 18 1
10 sec
when the universe starts to accelerate, it should be men-
tioned that Hubble parameter is not a constant but it is
time changing and slowly decreasing with respect the
universe age. Eq.120 shows that the constant b is Hubble
Parameter dependent and because Hubble parameter
recently is slower than of that one 9 billion years ago,
then it should be modified as
*
min
1.194bHk (123)
where k is s correction factor to account for the de-
crease in the value of Hubble parameter over ages. The
most interesting value of it is when 0.7071 12k
as in (120) which leads to the correspondent parabolic
parametric expansion equation

2
min 0.3564 0.8443
dt H
d

(124)
Analyzing Eq.124 shows that it has one root at zero
singularity (Big Bang) and another root at 2.369
which is corresponding to 13.3295t billion years
since 3
t
which is very close to the estimations of
the age of the universe. Remember that the LCDM model
estimates the age of the universe by 13.7 +/– 0.2 billion
years. Other astronomical calculations estimate the age
of the universe between 13-14 billion years. Simulation
of the parametric expansion (124) with respect to
in
Figure 7 shows the Big Bang singularity, the left portion
of the decelerated expansion, then the universe started
accelerating at 1.6751
heading up towards the cos-
mic horizon and will cross it at the second critical point
2.369
as a breaking symmetry critical point.
Assuming now that the universe started its acceleration
8.85 billion years ago, that is at the age 4.85 billion years,
then the constants in the parametric equation are as fol-
lows. At 4.85 billion years ~1.6927
, and
min
11.6927
2
b
a
 , 3.3854
b
a,

min min
41.1815
/
bH H
ba

, 0.349
3.3854
b
a,
and considering 0.7071k
, then the parametric equa-
tion will take the following form

2
min 0.349 0.8354
dt H
d
 (125)
Eq.125 will have two roots at the zero singularity (the
Big Bang) and another root at 2.3937
which is
corresponding to 13.7154t
billion years since 3
t
.
Such value is very close to the current estimated value of
the age of the universe using Hubble parameter and
LCDM model of 13.7 +/– 0.2 billion years. Simulation of
the parametric expansion (125) with respect to
in
Figure 8 shows the Big Bang singularity, the left portion
of the decelerated era, then the universe started to accel-
erate at 1.6927
c
and heading up towards the uni-
verse cosmic horizon line where the universe is very close
to the cosmic horizon line and will have a breaking
symmetry critical point at 2.3937
. By zooming out
the upper right portion of the parametric expansion curve
it can be seen how close the universe is to the Big Impulse
as shown in Figure 9 and correspondingly it will dra-
matically influence its fate and cause it collapse.
It should be emphasized that this parametric model
cannot predict the exact time of the second critical point,
that is, the Big Impulse. That is because the parametric
model is based on several assumptions: 1) This model is a
parametric model and is not explicitly a function of the
time, 2) It is assumed that the order of the expansion
parametric model is two, 3) The parametric transforma-
tion from the time scale to the parametric scale is of third
order, 4) This parametric model is Hubble parameter
dependent and a correction factor is assumed to account
for the slowly decreasing Hubble parameter, 5) The ac-
curacy in the astronomical observations of the age of the
universe is in the range of +/– 0.2 billion years, 6) The
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
180
astronomical observation that the expansion of the uni-
verse as long as 9 billion years ago has also some meas-
urement error. But in general, the model is significantly
correct in terms of describing the behavior and the trend
of the expansion of the universe and its closeness to the
cosmic horizon and the symmetry breaking point.
Moreover, taking into account the very massive
equivalent mass-energy of the universe, the very infini-
tesimal stationary time and principle of impulse force-
momentum principle, then the universe will experience a
very massive impulse given the name the Big Impulse
just to distinguish it from the Big Bang. This can be
verified by the virtue of the variational principle of me-
chanics as follows. For the 4-diemnsional space-time, the
stationary action integral (the universe expansion) would
be (considering that 0tdtd

)
2
1
0ds Ltd


(126)
According the principle of impulse-momentum, a very
massive impulse would occur as follows

,21 ,1impulseuinverse Equinverse Eq
F
tMv vMv  (127)

,1impulseuinverse Eq
F
Mvt (128)
The negative sign shows that the line of action of the
Impulse force is in the opposite of the outward expan-
sions, that is, it would act towards the universe system.
Such a Big Impulse will determine the fate of the universe
and cause to break the symmetry of the universe and push
it beyond the cosmic horizon. It is estimated that the mass
of the universe is 3 × 1050 kg as by Jeanne Hopkins in his
article “Universe” Glossary of Astronomy and Astrophys-
ics, Chicago: the University of Chicago, 183, 1980. A
very infinitesimally period of time similar to that of the
inflation 1033 sec happened early in the universe. Cal-
culations of (128) show that, our universe will experience
an impulse force in the order 1083 N. Because the impulse
force will act in a direction opposite to gravitational force
line of action, then, the gravitational bounding force will
collapse, and so the universe will.
9. SIMULATION RESULTS
This section is dedicated to simulate some of the pro-
posed models presented earlier in this paper as follows.
9.1. Can Dark Energy Be Generated?
Macro Scale: astronomical observations have brought
the evidence that dark energy is homogenous, isotropic
and uniformly fills the space with a density approximated
10-26 kg/m3. It is known that it does not interact with
forces other than gravity. For example, for a space-time
sphere with 1 m radius, the total amount of dark energy
contained in it is 4.188 × 10-26 kg which is equivalent to
3.7699 × 10-9 Joules (= 23.562 GeV). One Joule of dark
energy is equivalent to a space-time sphere of 642.5258 m
radius. One kilo Joules of dark energy is equivalent to a
space-time sphere of 6425.3 m radius. Meanwhile, One
Mega Joules of dark energy is equivalent to a space-time
sphere of 64252 m radius. Furthermore, One Giga Joules
of dark energy is equivalent to a space-time sphere of
642.520 km radius. According to this analysis it seems that
massive objects are needed to drag reasonable amounts of
dark energy as galaxies, stars or planets as shown in
Figure 2.
Micro-Nano Scale: previous analysis is based on the
macro scale level. To test now the existence of dark en-
ergy at the micro-nano level, the following experiment is
proposed. This system is composed of a closed cubic box
of edge length equal to 1 m and a micro piston-cylinder is
then to be placed inside that box. Such a micro pis-
ton-cylinder might be fabricated using Micro-Electro-
Mechanical Systems technology (MEMS). The total
amount of dark energy contained inside the cubic box is
in the range of 10-26 kg which is equivalent to 0.90 × 10-9
Joules (or 5.625 GeV). If the volume of the micro pis-
ton-cylinder is chosen such that it has a 1 cm3, it contains
an amount of energy equivalent 10-32 kg or 9 × 10-16
Joules or 5625 eV. Now if the micro piston-cylinder is
placed inside the cubic box, then the piston will move to
the left as shown in Figure 5. That is because the dark
energy inside the cubic box is much higher that that inside
Figure 5. Micro piston-cylinder system to test the pressure
work of dark energy: the piston will move by 0.18 nanometer.
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
181181
Figure 6. Dark energy associated with time as a mechanical
variable of a 6 dof micro space robot (10-5 j).
Figure 7. Drifted momentum (dark energy effect) by consid-
ering time as a mechanical variable of a 6 dof micro space robot
(10-4 j).
Figure 8. Micro-space antenna detector.
Figure 9. The micro space antenna (top), The angular position
of the micro space antenna (10-33) rad (down).
the micro piston-cylinder. Such a displacement is pro-
portional to the dark energy density. The work of the dark
energy is W FdsAds
 as in (1). According to
the principle of conservation of energy, this work should
be equal to the kinetic energy of the piston
K
E

2
0.5 p
mds. Hence,

2
0.5 p
K
EAdsm ds , then,
2
p
dsA m
 , where
p
m is the piston mass, if the
piston area is chosen as A=104 m2 , its mass is 103 kg,
and estimating the dark energy pressure as 0.90 × 109
J/m3 (5.625 GeV), then the piston will move 0.18 nano-
meter. By this, it is possible that dark energy can be util-
ized to operate mico or nano systems. Such systems can
have several applications in space, biomedical or elec-
tronics applications.
9.2. A Micro Free Flying Space Robot
The purpose of simulation is to show the dark energy
associated with a mechanical system by considering time
as a mechanical variable. The dark energy is equal to
the-momentum associated with time. Such energy (nega-
tive pressure) drifts the system a way. The system is
composed of 6-DOF micro space robot arm and mounted
on a micro base satellite and used to demonstrate the
analytical results. The mass of the base servicing satellite
is chosen as 10 g, the masses of the 6-robot arm as [1 1 1 1
1 1] g. All initial conditions are assumed to be zero. The
desired values for the robot angular position are chosen
as

0.3 0.2 0.1 0.6 0.5 0.4
des
q
. A PD controller is
used to control the micro space-satellite robot with 0.01
value selected for both the proportional and derivative
parameters. The simulation in Figure 6 shows the mo-
mentum associated with the time by considering it as a
mechanical variable (dark energy), meanwhile, Figure 7
shows a comparison between the classical momentum
and the drift in momentum by considering time as a me-
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
182
chanical variable. This comparison indicates that there is
indeed a drift (expansion) equal to the negative of the
system total energy (dark energy). This drift is a com-
mon phenomena for free-flying space robot.
9.3. Micro-Space Antenna Detector
A space robotic mission can be proposed to probe the
gravity waves and gravitons. The micro space antenna is
located at a proper distance from the mother spacecraft.
The mass of the mother spacecraft is relatively very huge
comparing to the mass of the antenna. According to the
previous analysis and the General Relativity, the mother
spacecraft will generate a gravity field and carried out by
the gravitons. The micro antenna will be deflected as a
result of the graviton torque acting on it. Now, consider a
space antenna shown in Figure 8, where l denotes the
length of the antenna and m denotes to its mass. Let
denote the angle subtended by the antenna and the verti-
cal axis through the pivot point. Using Newton’s second
law of motion, we can write the equation of motion in the
tangentional direction as T
mlbl l


 where T is
the torque acted by the gravitons on the antenna. This
equation will serve the simulation later. A micro space
antenna model is used to verify the analysis results of
gravity waves and gravitons. The mass of the antenna is
chosen as 0.000150 kg, its length chosen as 0.0004 m. All
initial conditions are assumed to be zero. It is assumed
that the mother spacecraft generated a gravity field due to
its huge mass and velocity comparing to the antenna. The
model (29) is used in the simulation based on the as-
sumption that the graviton is 0.707 of the mass of the
photon (that 2.5119 × 10–52 kg). The position response of
the micro space inverted antenna is shown in Figure 9. It
shows that the angular position is increasing in the scale
of 10-35 rad. Such value cannot be measured using nowa-
days technology. Future developments might make this
measurement possible.
9.4. Space-Time parboliod
Simulation results in Figures 10-14 show four cases
when 1
TOTAL
E, 1
TOTAL
E, 1
TOTAL
E,2
TOTAL
Ec
for the proposed Null Paraboliod (46). Finally in Figure
15 the Minkovisky Null cone (43), respectively.
These figures compare paraboloid manifold with
Robertson-Walker and Minkowski’s four-dimensional
world are defined as follows for a flat universe. The re-
duced line-element (46) represent a null paraboloid
which depends on the system total energy TOTAL
E and
the maxi-mum increase. In this case if a particle is mov-
ing with the speed of light then the null rays will lie on
the surface of the paraboloid.
Figure 10. Null Paraboloid for E = 1.
Figure 11. Null paraboloid for E = 1e20.
Figure 12. Null paraboloid for E = 0.001.
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
183183
Figure 13. Null light paraboloid 2
Ec.
Figure 14. Minkovisky null cone 2
Ec.
9.5. The Ultimate Fate of the Universe
Simulation of the ultimate fate of the universe in terms
of the parametric expansion model is demonstrated in
Figures 15-17 for Eqs.124 and 125, respectively. It can
be seen that the universe is approaching to a breaking
symmetry state and a Big Impulse that will cause its
collapse. Such a Big Impulse will determine the fate of
the universe and cause to break the symmetry of the
universe and push it beyond the cosmic horizon. It is
estimated that the mass of the universe is 3 × 1050 kg as
by Jeanne Hopkins in his article “Universe” Glossary of
Astronomy and Astrophysics, Chicago: the University of
Chicago, 183, 1980. Considering a very infinitesimally
period of time similar to that of the inflation 10–33 sec
happened early in the universe. Calculations of (128)
Figure 15. The expansion parametric model of the Eq.124.
Figure 16. The expansion parametric model of Eq.125.
Figure 17. Zooming the upper right portion of Eq.125.
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
184
show that, our universe will experience an impulse force
in the order 1083 N. Because the impulse force will act in a
direction opposite to gravitational force line of action,
then, the gravitational bounding force will collapse, and
so the universe will.
10. CONCLUSIONS
This paper has presented a foundation of the theory of
universe dark energy, a solution of Einstein’s cosmo-
logical constant problem, physical interpretation of uni-
verse dark energy and Einstein’s Cosmological Constant
and its value, values of universe dark energy density,
universe matter density, and universe radiation density.
Einstein’s Cosmological constant Lambda Lambada, is
calculated as 26 3
0.7410kg m
which complies with
the astronomical observation that estimated its value
52 2
10m

. The universe dark energy density has been
calculated to have the value 26 3
1.262210kg m
 =
6.8023 GeV.The average critical density of the universe
is 26 3
1.806910kg m
c
 = 9.7378 GeV which is
very close to 26 3
0.7410kg m
estimated by cosmo-
logical observations. According to the proposed model in
this paper the average density of the matter in the universe
is 26 3
0.5420710kg m
m
 = 2.9213 GeV. By this,
the third issue raised above has been solved as well. Finally,
the average density of the radiation in the universe is
31 3
2.710310kgm
r
 = 1.4558 MeV.
It is found that the momentum associated with time is
behind the expansion of the universe since it is equal to
the negative of the universe total energy. The negative
pressure of the dark energy is equal to the energy density.
It has been proved also that dark energy is a property of
the space-time itself. Based on the fluidic nature of dark
energy, the fourth law of thermodynamics is presented; a
new formulation of the three laws of Kepler is proposed
as well. A second order parametric model of the expan-
sion of the universe is presented and estimated that the
universe is acceleration to its fate and will soon experi-
ence a Big Impulse that will cause its collapse.
Moreover, in this paper a modeling approach of a
symmetrical gravitational field is proposed. In the prop-
soed model time is treated as a mechanical dependent
variable. In the proposed field tensor, the coupling in
time and its associated momentum, which equal to the
negative of system total energy leads to maximum in-
crease in the gravitational field. The amount of energy
that contributes to the maximum increase is 70.7% of the
total energy. A Null Paraboliod is proposed and inter-
preted in terms on the space and momentum of time.
Modeling of what is so-called gravity field waves, grav-
ity carriers (the gravitons) is presented in this proposed
paper. This model suggests that the space-time has a
polarity and is composed of dipoles which are response-
ble for forming the orbits and storing the space-time
energy-momentum. The tri-dipoles can be unified into a
solo space-time dipole with an angle of 45 degrees. Such
a result shows that the space-time is not void, on the
contrary, it is full of conserved and dynamic energy-
momentum structure. In this model the gravity field
waves is assumed to be carried by the graviton and
moves in the speed of light. The equivalent mass of the
graviton (the lightest particle in nature up to the author’s
knowledge) is found to be equal to 0.707 of the equivalent
mass of the light carrier (the photon) (2.5119 × 52
10
kg).
Considering a very infinitesimally period of time
similar to that of the inflation 10–33 sec happened early in
the universe. Calculations show that, our universe will
experience an impulse force in the order 1083 N. Because
the impulse force will act in a direction opposite to
gravitational force line of action, then, the gravitational
bounding force will collapse, and so the universe will.
REFERENCES
[1] Einstein, A. (1997) The foundation of the general Theory
of Relativity. In: English translation edited by A. J. Kox,
M. J. Klien, and R. Schulmann, The Collected Papers of
Einstein, 6, Princeton University Press, New Jersey, pp.
146-200.
[2] Knop, R.A., Aldering, G., Amanullah, R., et al. (2003)
New constraints on
M
,
, and
from an inde-
pendent set of eleven high-redshift supernovae ob-
served with HST1. The Astrophysical Journal.
http://sonic.net/~rknop/php/astronomy/papers/knop2003/
knopetal2003.pdf
[3] Permutter, S., et al. (1999) Measurements of omega and
lambda from 42 high redshift supernovae. Astrophysical
Journal, 516, 565-586.
[4] Riess, A.G., et al. (1998) Observational evidence from
supernovae for an accelerating universe and cosmologi-
cal constant. Astronomical Journal, 116, 1009-1038.
doi:10.1086/300499
Maartens, R. and Majerotto, E. (2006) Observa- tional
constraints on self-accelerating cosmology. Journal of
Astrophysics. arXiv:astro-ph/0603353v4
[5] Perlmutter, S., Turner, M.S. and White, M. (1999) Con-
straining dark energy with SNe Ia and large-scale struc-
ture. Journal of Astrophysics.
arXiv:astro-ph/9901052v2
[6] Freedman, W.L. et al. (2000) Final results from the Hub-
ble Space Telescope Key Project to measure the Hubble
Constant. Journal of Astrophysics, 553, 47-72.
doi:10.1086/320638
Tonry, J.L., et al. (2003) Cosmological results from
High-z Supernovae. Journal of Astrophysics, 594, 1-24.
doi:10.1086/376865
[7] Carroll, S.M. Sawicki, I. Silvestri, A. and Trodden, M.
(2006) Modified-source gravity and cosmological struc-
ture formation. Journal of Astrophysics, 8.
[8] Cengel, Y.A. and Boles, M.A. (2006) Thermodynamics:
M. Shibli et al. / Natural Science 3 (2011) 165-185
Copyright © 2011 SciRes. OPEN ACCESS
185185
An engineering approach. 5th Edition, McGraw Hill, Co-
lumbus.
[9] Noether, E. (1918) Inavariante variationsprobeleme.
Nachrichten von der Gesellschaft der Wissenschaften zu
Goettingen, 235-257.
[10] Spergel, D.N., et al. (2006) Willkinson Microwave Ani-
sotropy Probe (WMAP) three years results: implications
for cosmology. NASA publications.
[11] Turner, M.S. (1999) Dark matter and dark energy in the
universe. The Third Astronomy Symposium: The Galactic
Halo ASP Conference Series, 666.
[12] Huterer, D. and Turner, M.S. (1999) Prospects for prob-
ing the dark energy via supernova distance measurements.
Physical Review D, 60.
[13] Carroll, S.M., Hoffman, M. and Trodden, M. (2003) Can
the dark energy equation-of-state parameter w be less
than 1? Physical Review D, 68.
[14] Huterer, D. and Turner, M.S. (2001) Probing dark energy:
Methods and strategies. Physical Review D, 64.
[15] Peebles, P.J.E. and Ratra, B. (2003) The cosmological
constant and dark energy. Reviews of Modern Physics, 75,
559-606.
doi:10.1103/RevModPhys.75.559
Copeland, E.J., Sami, M. and Tsujikawa, S. (2006) Dy-
namics of Dark Energy. International Journal of Modern
Physics D, 15, 1753-1935.
doi:10.1142/S021827180600942X
[16] Volovik, G.E. (2006) Vcauum energy: myths and reality.
International Journal of Modern Physics A, 15, 1987-
2010.
[17] Carroll, S.M. (2003) Why is the universe accelerating?
Journal of Astrophysics.
[18] Carroll, S.M. (2006) The cosmological constant. Living
Reviews of Relativity, 4.
[19] Shibli, M. (2007) The foundation of the Fourth Law of
Thermodynamics: Universe Dark Energy and its nature
can Dark Energy be generated? To be presented at In-
ternational Conference on Renewable Energies and
Power Quality (ICREPQ’07), Spain.
[20] Jarosik, N., et al. (2007) Three-year Wilkinson Micro-
wave Anisotropy Probe (WMAP) observations. Astro-
physics Journal, 170, in Press.
[21] Dvali, G. and Turner, M. (2003) Dark energy as a modi-
fications of the Friedmann Equation. Journal of Astro-
physics.
[22] Shibli, M. (2006) Canonical modeling approach of a
micro/nano free-flying space robot: a proposal towards
detecting the nature of space-time. 1st IEEE ISSCAA,
Co-sponsored by AIAA, Harbin, China.
[23] Shibli, M. (2006) Physical insight of universe Dark En-
ergy: the space mission. 1st IEEE International Sympo-
sium on Systems and Control in Aerospace and Astro-
nautics (ISSCAA 2006), Co-sponsored by AIAA, Harbin.
[24] Spergel, D.N., et al. (March 2006) Wilkinson Microwave
Anisotropy Probe (WMAP) three year results: implications
for cosmology. WMAP collaboration.
[25] Zhurkin, V.B. (1983) Specific alignment of nucleosomes
on DNA correlates with periodic distribution of purine-
pyrimidine and pyrimidine-purine dimers. FEBS Letters,
158, 293-297. doi:10.1016/0014-5793(83)80598-5
[26] Goldstien, H. (1980) Classical mechanics, 2nd Ed., Addi-
son-Wesley, Boston.
[27] Shibli, M. (2006) Physical insight of space-time and
modeling of space-time dipoles, gravity waves and
gravitons: A micro space antenna to detect the nature of
gravity waves. 1st IEEE International Conference on
Advances in Space Technologies (ICAST2006), Pakistan.
[28] Lanczos, C. (1966) The variational principle of mechan-
ics, University of Toronto Press, Toronto.