International Journal of Modern Nonlinear Theory and Application, 2012, 1, 118-124
http://dx.doi.org/10.4236/ijmnta.2012.14018 Published Online December 2012 (http://www.SciRP.org/journal/ijmnta)
Deriving the Exact Percentage of Dark Energy
Using a Transfinite Version of Nottale’s
Scale Relativity
M. S. El Naschie1, L. Marek-Crnjac2*
1Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt
2Technical School Center of Maribor, Maribor, Slovenia
Received September 19, 2012; revised November 10, 2012; accepted November 18, 2012
ABSTRACT
In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions
of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Ein-
stein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual
measurements.
Keywords: Dark Energy; Lorentz Factor; Scale Relativity; Cantor Set; Hausdorff Dimension; Hardy’s Quantum
Entanglement
1. Introduction
The mysterious major fundamental problem of cosmo-
logy and theoretical physics [1-20], i.e. the missing hy-
pothetical dark energy is tackled and solved in the pre-
sent paper.
Nottale’s theory of scale relativity is a powerful Weyl-
like general gauge theory with applications in high en-
ergy particle physics as well as cosmology [3,7,8]. In that
respect it is quite similar to the mathematical and physic-
cal K and E-infinity theory [17-19]. The main difference
comes only from the systematic use of logarithmic scal-
ing in Nottale’s scale relativity [3,7,8] where as E-infi-
nity is exclusively based on a transfinite Weyl scaling [4,
5]. In particular Nottale’s theory gave up differentiability
but not continuity [3,7,8]. By contrast E-infinity gave up
both differentiability as well as continuity but preserved
the cardinality of the continuum [4,5] using the geometry
of random elementary Cantor sets [3-5,18]. Since the
Hausdorff dimension of such random elementary Cantor
sets is the golden mean and its powers, the scaling expo-
nent of the theory are combinatorics of these infinitely
many golden mean random Cantor sets [3-6,14-18]. In
this regard we stress that it is generally wrong to think
that discontinuity of space-time introduces something un-
physical or a-physical to a theory because an empty set is
physical and present in nature as it is present in the fun-
damental axioms of set theory upon which our entire
mathematical methods are based.
In the present work we use Nottale’s theory to give
first an accurate approximate solution to the problem of
the missing dark energy in the cosmos [1,2]. Subsequent-
ly we minimally deform Nottale’s scale relativity [3,7,8]
making it transfinitely almost exact. The so obtained re-
sults are in superb agreement with the cosmological mea-
surements [1,2].
2. Scale Relativity—Preliminary Remarks
Scale relativity is a profound general theory which was
developed toward the end of the eighties last century [3,
7,8]. The theory builds heavily on the tradition of Ein-
stein-Minkowski geometrization of physics and simulta-
neously makes extensive use of what at the time was the
new science of nonlinear dynamics and the great pio-
neering spirit of deterministic chaos and fractal geometry.
That way scale relativity combined the great ideas of
Einstein with those of H. Weyl’s original gauge theory
[4], R. Feynman and Garnet Ord’s proposal for a fractal
space-time [3]. Scale relativity and fractal space-time
sparked quite a revolution in the way we think about
foundational problems and cutting edge research in theo-
retical physics and although it is not as visible as super-
strings [9] or loop quantum mechanics [10], it is in no
way less original or insightful. In fact, it may be more in
a complimentary way as we will attempt to show in the
present paper.
*Corresponding author. There are many parallels and equally differences between
C
opyright © 2012 SciRes. IJMNTA

M. S. EL NASCHIE, L. MAREK-CRNJAC 119
scale relativity and E-infinity Cantorian space-time the-
ory [3-8]. Let us concentrate on the most important com-
mon aspect of the two theories. No doubt it is fractality
and scaling. These in reverse order are the quintessence
of the two theories from a conceptual view point. How-
ever when we look at the quantitative analytical treat-
ment, then the two theories differ in slightly less than a
minor way. This is because scale relativity employs loga-
rithmic scaling more or less similar to the logarithmic
scaling of the standard model of elementary particles and
quantum field theory. One only needs to remember the
logarithmic running of the coupling constant as a func-
tion of the energy scale used in the electroweak and
strong interaction [9-13] to get the idea. Such logarithmic
scaling is a powerful approximation to the unattainable
non-perturbative exact solution but none the less, it is an
approximation. By contrast E-infinity theory employs a
golden mean based exact renormalization semi-group
and exact golden mean scaling exponents [4-6,14]. None
the less, without Weyl and Nottale’s insight into the in-
terrelation between gauge theory and fractals, we could
not have developed E-infinity theory in its present form
which depends crucially on the many excellent results of
not only non-commutative geometry [17], superstrings
[9], M-theory [15,16] and loop quantum gravity [10] but
also in a fundamental way on Nottale’s scale relativity
and of course Ord’s fractal space-time [3].
A second important aspect of scale relativity is that it
gives the Planck length the same status which only the
velocity of light enjoys in Einstein’s theory of relativity.
Such a proposal seems at first sight to be controversial
because unlike the speed of light, the Planck length can-
not be measured experimentally in any realistic set up [7,
12,13]. Never the less, it seems to us that the theory of
varying speeds of light [12,13] which was clearly influ-
enced by Nottale’s scale relativity and gives a convincing
mathematical argument for Planck energy invariance
while preserving the Lorentzian symmetry group invari-
ance although velocity and energy could be made arbi-
trarily much larger than the light velocity and the Planck
energy without violating both [12,13].
In the present work we will adopt scale relativity the-
ory in substantially its original form to determine the dark
energy content of the universe [1,2] which boils down to
revising Einstein’s energy mass relation in order to ex-
tend its applicability to the realm of quantum gravity [7,
8,10]. Subsequently we will show how Nottale’s scale re-
lativity could gradually be made transfinitely exact and
obtain the same result obtained using E-infinity theory
and the fractal 11 dimensional M-theory [15,16]. We start
in the next section with a few explicit examples to illus-
trate the main ideas of scale relativity in the light of the
competing theories such as E-infinity theory and Het-
erotic superstrings [9].
3. Calculus in Non-Commutative Geometry,
Scale Relativity and Cantorian Fractal
Space-Time
3.1. Background
To be able to use our calculus as developed by Newton
and Leibniz, smoothness and of course continuity are
absolutely indispensible. However fractals are not smooth
and Cantor sets are totally disjoint. In such a case and
where continuity is assumed as in Nottale’s scale relati-
vity and partially in Ord’s anti-Bernoulli stochastic frac-
tal space-time, one could resort to the non-standard ana-
lysis developed by Robinson as done initially by Nottale
[8] or use a form of quantum calculus as done by Ord
[3,7,8,18]. These methods break down completely in the
case of non-continuity and Cantor sets. From a classical
pure quantum view point, only a generalization of the
Heisenberg truly non-continuous view point rather than
Schrödinger’s pseudo continuous theory is to resort to
non-classical measure theoretical methods, K-theory and
categories [17]. In non-commutative geometry A. Con-
nes replaced differentiation of real or complex variables
by a Poisson bracket of the form [17]


,Df Ff FffF (1)
while the opposite or reciprocal operation, i.e. integration
is replaced by the Dixmier trace [17]

.
w
DixTTr T (2)
Here we follow the standard notation used also in [17].
3.2. Scaling in E-Infinity Theory
In E-infinity Cantorian fractal space-time on the other
hand we discovered quite early that the H. Weyl failed
original gauge theory is valid in space with no scale at all
such as the infinite dimensional but hierarchical Cantor
set modelling E-infinity space-time [4,5]. Consequently
scaling down is analogous to differentiation while scaling
up is analogous to the opposite operation namely integra-
tion. It is as simple a duality as that between adding and
subtracting or multiplying and dividing. Maybe a simple
example makes the idea clearer. Let us take the low en-
ergy electromagnetic coupling constant


55
00
137137 1137.082039325,k   
where 5

is Hardy’s generic probability of quantum en-
tanglement and


512   . Now we scale 0

up
twice using

2
12.



 This is the opposite number
so to speak of “integrating” 0

twice. That way one
finds [4]

M. S. EL NASCHIE, L. MAREK-CRNJAC
120
2
0
1358.8854382.




 (3)
We note that pure gravity


3
2
d
p
dd
G  for d = 8
dimensions and the Riemann tensor



22
1
12
nnn
R


for n = 4 are equal [4]
 
48
20.
p
RG
(4)
The strong interaction is given by the compactified Lie
symmetry group SL(2,7) [4,5]
 
2, 72, 716
336 2.88543824339
c
SSLk
  (5)
and we could infer that scaling up “ergo pseudo inte-
grate” of 0

results in obtaining all the 339 gluons as
well as gravity. In fact we could be more accurate and
write that [4,5]
  


2
0
3
2, 72,72
1
336 324
c
SSLSUk



k
(6a)
where

55
01 0.082039325k
  (6b)
and

33 5
12 0.1803398k    (6c)
are Hardy type transfinite quantum entanglement correc-
tions [6,14].
In addition we have [4]


 
2
2,717 48336SLnn n (7)
for the quarks-like state of the strong force [4,5] and fi-
nally [4]

2
21SU nn3 (8)
for the electroweak force.
To sum up the insight of this section we say that while
in scale relativity calculus is replaced by non-standard ana-
lysis and logarithmic scaling, in our approach we need
only the golden mean scaling operation down scaling re-
places differentiation and up scaling replacing integra-
tion. Before the end of this part however we give a very
important and instructive down scaling which starts from
the number of the first level of massless particles like
quantum states in a transfinite Heterotic string theory.
We can determine the Ambjorn-Loll [20] extremely im-
portant spectral dimension of quantum gravity Ds =
4.019999.
We know that the classical value of No in Heterotic
strings is found from (504)(16) = 8064. This is actually
the multiplication of the holographic boundary


2,7 336SL  with the instanton number n = 24.
However, in the exact transfinite theory we have 336 
336 + 16k and 24  26 + k. Therefore the exact is [4,5]




026 3368872.135962.Nkk (9)
This is nothing but up scaling of the modular space
M(80) with dim M(80) = 80 using 5
5
111


 of a
fractal 11 dimensional M-theory space [4,16,17]






5
01080 118872.135962.N
  (10)
Differentiating, i.e. scaling using

sixteen times,
one finds [4,20]


16
04.01999 S
DN   (11)
exactly as the value found by Ambjorn and Loll using an
efficient computer [20]. It is really curious to see that di-
gital computers are far more accurate than calculus when
it comes to high energy physics. However, golden mean
computers are even far more accurate and efficient than
digital computers [4]. Next we look at a simple example
of how Nottale’s theory deals with scaling and non-stan-
dard analysis.
3.3. Some Examples from Scale Relativity
Calculus
Let us start with the fundamental optimized coupling
constant of scale relativity, namely unification coupling
[3,7,8]
2
4π39.4784176.
g
  (12)
To bring this value in line with numerical experiments
as well as E-Infinity’s exact prediction of the non-super
symmetric grand unification of all fundamental forces
except for gravity, we must realize the following and
change things accordingly:
1) The factor 4 stands in reality for the four topologi-
cal dimensions of space-time. It must therefore be chang-
ed to the fractal-Hausdorff dimension of the core of space-
time, i.e. to that of a Hilbert 4D cube 4 + 3

=
4.2360679.
2) Second π2 must be changed to one of its transfinite
opposite numbers. In this case π2 = 9.869604401 must be
changed:
2
π10
From the above we find




3
transfinitely exact
41042 242.36067
g
k

   (13)

M. S. EL NASCHIE, L. MAREK-CRNJAC 121
where 51
2


, .

335
12 0.1803398k   
There is a very simple and elementary way to show that
this is the exact value as well as how to obtain the super
symmetric
g
s

which we know to be 26
gs k


.
The value of inverse electromagnetic coupling at low
energy 00
137 137.0820393k

 should be divided
equally among the number of fundamental equations at a
certain energy scale. When gravity is out and the electri-
cal force and the magnetic force are counted as one force,
then we have only 3 fundamental forces with a fractal
weight due to the fractality of space-time equal to
. The common coupling or unifica-
tion grand unification coupling is thus
3
3 3.23606799  
0
33
137 42 2
33 k

 
 (14)
exactly as anticipated. Now if we admit gravity and
count electrical force and magnetic force as two forces,
then the number is 5 and the fractal weight is 3 5


.
Consequently the total unification inverse coupling of all
fundamental forces becomes
0
326 26.18033989.
5k

 
 (15)
Finally in the case of only 4 fundamental forces the
unification coupling is given by
0
332 232.18033989.
4k

 
 (16)
This 32.18033989.

 is what we include approxi-
mately in our renormalization equation of unification us-
ing the logarithmic scaling as in Nottale’s theory of scale
relativity. We see this clearly from [18]
34 ln u u
x
M
M


 


(17)
where 39,

 41,
QG

  Mu = 1016 GeV, Mx = 91
GeV, 1n

, i.e. 1

 for non-super symmetric in-
teraction, 12

 for super symmetric interaction [18]
and

1
01 2 3
0
1
2
137 137.082039325k

4










(18)
for 160

 and 230

. Now inserting in the loga-
rithmic term one finds [18]
16
0
3
10 GeV
ln 32.33050198
91 GeV4




 
 (19)
exactly as anticipated. Here 1016 GeV is the mass of the
GUT monopole and 91 GeV is the mass of the electro-
weak unification. In fact Nottale’s scale relativity has ge-
neralized this logarithmic scaling and used Levy-Gillmann
operators skillfully to achieve his result which although
not exact, paved the way for our work and for the excep-
tionally beautiful work of Magueijo and Smolin [12,13]
on varying speed of light theory (VSL). In the next sec-
tion we will show how E-infinity as well as scale relati-
vity can resolve the mysterious dark energy problem
[1,2].
4. Resolution of the Missing Hypothetical
Dark Energy Using Scale Relativity and
E-Infinity
Scale relativity puts the running value of 0

at 1016
GeV of scale relativity [3,7,8] for 105
GUT

. Clearly
at GUT

we have everything except gravity. Scaling 105
logarithmically and squaring it gives us now a measure
for the error in Einstein’s special relativity energy mass
resolution when applied at ultra high energy and dis-
tances. That way we find the scaling exponent needed for
, namely
2
0
Emc


2
11
4.65396036
ln
10.04616944.
21.65934694
GUT
 

(20)
Einstein’s energy-mass equation now reads as follows:
2
0
Emc   (21)
where
1
21.65934694
 .
The corresponding dark energy is therefore


dark
1
1100 95.383%.
21.65934694
E  

 (22)
Before giving an exact interpretation for this approxi-
mate result let us first revise the numerics. The value
which should have been used for GUT

is (10)


11
F
D
which means [4,5]
 
5
1
1010 11.09016995
110.9016995.
GUT






(23)
Logarithmic scaling and squaring then leads to

22
1ln110.9016995 4.70864419
22.17133038 22.




(24)
The result is almost the exact one, namely (22 + k)
where k = 0.18033989 as we can show using exact methods

M. S. EL NASCHIE, L. MAREK-CRNJAC
122
In other words 1
22
  is the reciprocal value of the
non-visible “dark” dimension of our Bosonic section of
the transfinite version of Heterotic string theory. That
means for “dark” dimensions we have [9]


dark The total number of the dimensions
space-time dimensions
264 2222.18033939.
D
kk



(25)
E-infinity scaling reaches the exact result without lo-
garithmic scaling. Let us first recall that the entire He-
terotic superstrings dimensional hierarchy is readily found
for 0

for a Cooper pair as follows, starting from [4,
5,16-18]
 
068.54101966
2
n

n


 

 (26)
and setting one finds [4] 1,2, 3,n
42 242.36067977k
26 26.18033939k
16 16.18033939k
10
6 6.18033939k
4 3.819660122.k
Setting X ±
k X one finds the classical Heterotic
string dimensional hierarchy 26, 16, 10, 6 and 4. This

was a down scaling of 0
2

. Now the up scaling leads to
the following 01
2
n




 


. For n = 1 one finds [4]


5
011110 110.9016995
2GUT



  




 . (27)
Dividing through all the five interactions using the DT
= 5 one finds [4]
22 22.18033989.
5
GUT k

 (28)
This is of course the exact result and shows the high
quality of accuracy in the Nottale method. Should we
have used the fractal weight 3
5

 rather than 5 we
would have found [4,5]
321 21.18033989.
5
GUT k



 (29)
In the first case we look at an Einstein 4 dimensional
space-time with 22 + k “dark” dimensions while in the
second case we have a 5 dimensional Klein-Kaluza
space-time with only 21 + k “dark” dimensions. Based on
this analysis our tangible space is exactly four dimen-
sional topologically and Haus-
dorffly. However it is the larger
3
4 4.23606799  
5
11


3
4
core of our
space which encapsules the

 smaller core which
decides on Hardy’s quantum entanglement [6,14] being
exactly 5
5
1
11



and also decides on the reduction
factor or the scaling exponent 51
222 k  
 of Ein-
stein’s equation The scaled new quantum
relativity or effective quantum gravity equation
2
0.Emc

2
20
0
5
11
22
11
mc
Emc
k





 


 2
(30)
predicts that we have a missing dark energy of exactly
E(dark) = 95.49150281%, almost the same as in the appro-
ximate scale relativity analysis following Nottale’s the-
ory. This reduction could be interpreted in a variety of
intuitive ways which will be discussed in the conclusion
of the paper.
It is instructive for a deep understanding of the present
work to ponder the implication of a comparison between
Nottale’s theory of scale relativity and El Naschie’s E-
infinity theory which is summarized in Table 1. In Table
2, we give another instructive comparison between work-
ing in the bulk and working with the holographic bound-
ary to derive the scaling which elevates Einstein’s special
relativity equation to an effective quantum gravity equa-
tion.
5. Discussion
Following the picture adopted by Heterotic string theory
compactified on a Calabi-Yau manifold, every point in
our spacetime is joined to a Calabi-Yau 6 dimensional
real manifold containing internal symmetry and compac-
tified dimensions [19]. On this account we would have
all in all (4)(6) = 24 dimensions and adding the string
Table 1. Comparison in calculating the Lorentz factor using
scale relativity and E-infinity.
Scale Relativity (Nottale) E-infinity (El Naschie)
Grand
unification
coupling
 
2
0
ln542.6411  
where
0137.08203932
 

 
0
3
5 42.3606
10
41




 0
where
0137.08203932
 


Lorentz
factor

 
2
2
ln
ln 111
22.1796
GUT




0
110
22
26 4
22.18033
k
k








M. S. EL NASCHIE, L. MAREK-CRNJAC 123
Table 2. Comparison in predicting the Lorentz factor for
dark energy step by ste p using the bulk and using the holo-
graphic boundary [4,5].
88
Bulk 496EE Holographic boundary ≡ SL(2,C)
 
88
1
321 EE SUSUU

 
44
4
2,74
RD
SLRD


1
496 12

20 4
33620 4


1
484 16 336 16

2
1
22
1
352 16
1
22 1 22
world sheet to it arrives at the 24 + 2 = 26 Bosonic di-
mension. These dimensions move in the opposite direc-
tion of another 16 Fermionic dimensions from which one
finds 26 – 16 = 10 super symmetric dimensions. How-
ever in our transfinite version of Heterotic strings we do
not need the 2 dimensional world sheet to arrive at 26.
This is because the Hausdorff dimension of our core
space is not 4 but and the 6 di-
mensions of the Calabi-Yau manifold [19] are not 6 but 6
+ k = 6.18033898. Consequently the total dimension is
given by
3
4 4.23606799  



3
Heterotic 4626
26.18033989.
s
D   

kk
(31)
Now Einstein’s energy-mass equation was based on a
mere 4 dimensional flat non-fractal, non-fuzzy Euclidean
manifold. Subtracting these 4 dimensions from DS = 26 +
k we are left with 26 + k 4 = 22 + k hidden dimen-
sions.

This is a wonderfully simple and intuitive picture and
is numerically identical with our analysis which was
based on superficially completely different theories such
as Nottale’s scale relativity [3,7,8] or E-infinity theory
[3-6]. It is now clear that must be scaled us-
ing
2
0
Emc
10.0450849718
22 k
  
 (32)
which fully agrees with the measurement of WMAP and
supernova analysis by predicting that exactly 95.4915028%
of the energy of the cosmos must be dark energy [1,2].
To gain a deeper insight into the roots of scale relati-
vity we should apply the original energy mass relation of
scale relativity directly to the problem of dark energy.
Even a fleeting glance at these equations reveal that they
are in almost one to one correspondence with Einstein’s
equation and are also the inspiration to Magueijo-Smolin’s
beautiful energy-mass Planck length invariant equation.
Following Nottale’s notation we have [7,8]




2
0
22
200
0
1ln1 lnln.
2
N
Emc
mc




  

  

  
0

(33)
Here

is the Plank length . From Si-
galotti’s analysis of the classical relativistic transition we
know that when we set

33
10 cm





2
0
mc 1, then one finds that
 

61211
0
65
2
0
11
2
1.
22
1
Em
mc









 

2
c
(34)
Exactly as in the previous analysis which means a re-
duction of 95.49150281% in energy which matches al-
most exactly the missing dark energy measurements [1,
2].
There is even an outrageously simple way of arriving
at 1
22
  from semi classical considerations as sug-
gested by El Naschie. The argument goes as follows.
Special relativity is a one degree of freedom theory
where the photon is the only elementary particle involved.
The standard model however has 12 elementary pho-
ton-like particles. Thus we have here a factor of 12 − 1=
11 involved. Inserting in Newton’s kinetic energy we
find our previous result
22
00
11 1.
211 22
Emcmc 
 

 
 
2
0
mc (35)
6. Conclusions
Scale relativity gives yet another very constructive men-
tal picture to understand what the missing dark energy
means apart of answering the quantum question quantita-
tively with remarkable accuracy. Scale relativity is com-
pletely embedded in the scale invariance of fractal ge-
ometry [3-8]. We do not need to go from general relati-
vity via quantum mechanics to arrive at quantum gravity.
We could do the same by starting with special relativity
however after freeing it from traditional prejudice and
putting it in the right space-time setting, namely fractal
geometry. The Lorentz factor does the rest and one finds
that Einstein’s celebrated equation maintains its form and
the change is a mere down scaling by a minimal Lorentz
factor equal to half of the value of Hardy’s quantum en-
tanglement [6,14]. In turn this causes a reduction of al-
most 95.5% of the classically predicted energy. This is
what we call missing dark energy. It is the energy which
would have been there if the space-time fabric were

M. S. EL NASCHIE, L. MAREK-CRNJAC
124
c

smooth, continuous and without holes. However actual
space-time at quantum scales and surprisingly again at
intergalactic scale displays a wild Cantorian fractal ge-
ometry and topology. It is a T-duality which we saw in
the unification program at the Plank length, yet this time
the surprising quantum effect of entanglement is showing
its power at the Hubble length scale. The main equation
obtained in the present work which is
2
0QR S
EEm   (36)
is gauge invariant in the widest sense possible, meaning
that it is almost invariant to the use of any mathemati-
cally and physically reasonably meaningful theory. It is a
very robust result not affected by minor details of theo-
retical modeling. Thus we may show here in the conclu-
sion what on reflection should have been presented in the
introduction at the very beginning:
Special relativity implies three strange effects [11]:
1) length contraction;
2) time delineation;
3) mass increase.
All these classically feeble effects become noticeable
only as the speed approaches the speed of light c [11].
We handle this semi-classically, i.e. using common sense
by introducing a boost

1

 and anti boost


1

.
Thus we have 1)

1

X
X  

t
, where X is space co-
ordinate, 2)

1t


0
m
, where t is ordinary time and 3)
, where m0 is non-relativistic mass [11].

0
1m


Inserting in Newton’s kinetic energy

2
0
1
2
Emv 


we find


2
2
0
11
1.
21
QR
Em


 



 
 

c (37)
Setting 2
51
  we find our previous result
2
0
1.
22
Emc Thus in one stroke we reconciled and
fused together classical mechanics with relativity and
quantum mechanics via the non-classical geometry of
fractals [3-8]. This is magically beautiful.
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