_{1}

This tutorial review is dedicated to the work of the outstanding Egyptian theoretical physicist and engineering scientist Prof. Mohamed El Naschie. Every physics student knows the well-known Einstein’s mass-energy equation,
*E=mc*
^{2}, but unfortunately for physics, few know El Naschie’s modification,
*E(O)=mc*
^{2}/22, and El Naschie’s dark energy equation
*E(D)=*
*mc*
^{2}(21/22) although this new insight has truly far reaching implications. This paper gives a short tutorial review of El Naschie’s fractal-Cantorian space-time as well as dark energy. Emphasis is put on the fundamental concept of Cantor set, fractal dimensions, zero set, empty set, and Casimir effect.

Modern theoretical physics has a truly fascinating and marvellous story to tell and teach everyone, particularly physics students, regarding its logical structure and development [

Einstein’s well-known iconic equation of relativity relating total energy to mass and the speed of light is given by

It was derived for a photon. It does not model the energy of our real world. Einstein’s theory was established on the assumption of absolute smooth 3 + 1 dimensional space-time. However, space and time are not at all smooth. To demonstrate the discontinuity of space-time [

Now as we mentioned earlier on, Einstein’s mass-energy equation is derived under the assumption of absolute smooth space-time. In reality space-time is intrinsically discontinuous when it tends to a quantum scale [

Hilbert cube can excellently model the actual fractal space-time [

When we construct a Cantor set, whether deterministic or random, we end up with two Cantor sets, the zero set and the empty set, the former consists of infinite points, and the latter is the left of the unit interval [

We consider an extremely large plan with discontinuous boundary at an extremely small scale, see

The average Hausdorff dimension of the plan given in

where 0.2618 is the fluctuation of plan dimensions. The fluctuation dimensions for our space-time at quantum scale is

The average Hausdorff dimension of our space-time is [

The general El Naschie’s dimensional form is [

For our real n = 4 space-time, Equation (4) gives Equation (3).

The fractal explanation of Equation (3) is given in

It is instructive to relate in a visually impressive way how the dimensionality of space progressed from Newton to El Naschie via Einstein following a proposal by Prof. Ji-Juan He:

Similarly the five dimensional Kaluza-Klein space-time [

Further more El Naschie was able to show that Einstein’s iconic E = mc^{2} is in fact the sum of two basically quantum parts [

and that of the quantum wave energy

so that Einstein, without realizing, did indeed hit the nail on its head, quantumly speaking [

The quantum particles are not occupied within a 1 × 1 × 1 × 1 × 1 hyper volume but a

This is El Naschie’s meantime equation [^{th} international symposium on nonlinear dynamics was held in Shanghai, China on October 30, 2012, for celebrating El Naschie’s greatest finding, see

We recall first Einstein’s relativistic mass equation, which is given by

In the above equation, the velocity v relates to that of the photon. Now we consider a photon moving in x-di- rection, while a particle moves on a 2 dimensional plane with same projection velocities in x- and y-directions, that is the velocity in x-direction is

and its energy can be approximately written in the form

Using Mitten’s 11 dimensional M-theory [

Using fractal M-theory [

which is exactly the same as El Naschie’s equation, Equation (10). We believe that El Naschie’s theory gives a bridge joining our visible world with the invisible quantum world, as illustrated in

It is fair to say that only a few would place the field of deterministic chaos and fractals as the next mile stone or revolution after quantum mechanics. However, the work of G. Cantor and his transfinite theory are by far the most fundamental mathematics which quantum physics requires and this fact at long last becomes known via the work of the pioneer of nonlinear dynamics, chaos and fractals, notably Lorenz, Ruelle, Feigenbaum, Mandelbrot, Takens, York and El Naschie to mention only a few [

namely the Heisenberg uncertainty principle [

The main two elements or building blocks of E-infinity diagrams are the zero set and the empty set [

a) The zero set

b) The empty set

where D(H) is the Hausdorff dimensions and

From the diagrams of

1) When the distance between the two plates of ^{(−∞)} = zero. In the surrounding space we have a non-empty set with the average latent pressure everywhere equal

less than the Casimir latent energy of space-time and at this scale it is not related in any way to the Riemann curvature of space-time but to the chaotic fractality of space-time in full agreement with our picture which we adopted based on Feynman’s conjecture that gravity is similar to van der Waals fluctuation of micro space-time frequently termed Feynman-El Naschie van der Waals quantum gravity conjecture [

2) Now, we look at the other extreme where the distance between the Casimir plates is equal to the diameter of our universe as shown schematically in

We start from our exact picture of quantum space-time [

The first is the zero set of the quantum pre-particle (0; f). The boundary of the points in the zero set has a negative dimension, −1 [

Around this pre-particle we have a topological surface representing the empty set pre-quantum wave (?1;

Finally, we have the third layer which happens to be an expectation value

Now, f,

Consequently we have

where

Similar to the quasi probability of Wigner’s quantum mechanics in phase space [

The internal logic of our approach rivals that of Weyl-Wigner quantum mechanics in phase space and its logical underpinning by Groenewold and Moyal [

To obtain E(O) from E(D) and vice versa we have the following transformation [

The same applies to a fractal de Sitter universe [

That means ordinary energy is

and dark energy is

Let us start with Heisenberg’s uncertainty [

It would be a mistake to think this result is trivial. However, written in symbolic form

other words E-infinity transfinite Cantorian quantum mechanics has at its disposal and in a natural way, a transfinite golden computer [

or

when compared to its totally chaotic decimal expansion [

and find on the top of that a well-known fact, namely that

At the risk of appearing facetious, we would like to seriously propose that in no minor measure a draw back of E-infinity Cantorian space-time theory is that it is excessively simple. A theory should be simple but not excessively so. Being excessively simple puts a theory at risk of being called trivial as an easy shot by members of the “voluntary opposition”. In his early days working in applied mechanics, the author was in awe vis-à-vis the work of a great scholar Prof. Cliff Truesdell who coined the word rational mechanics [

In the following, we show a simple deep connection between a few fundamental aspects of E-infinity and Weyl- Wigner theory [

a) If we take

b) Seen that way, then

c) Since the VAK is a Hamiltonian “strange” attractors conjectured by Rene Thom to represent the equilibrium states of quantum mechanics [

In very recent work El Naschie used the entire mathematical machinery of E-infinity theory [

In the present work we show that due to the well-known theorem on measure concentration the above conclusion is fallacious [

The moral which we can learn, in fact relearn from this theorem is a well-known wisdom from many counterintuitive results of geometry in higher dimensions, namely that we should in general never generalize an obvious conclusion from a low dimensional space to a higher one. For instance on a flat two dimensional space any two lines will intersect in a point unless they are parallel. However, the spectacular failure of this simple obvious result in three dimensional space is embarrassingly clear. Now let us start with a Euclidean ball [

Working in the usual way to find the volume of this n dimensional ball we arrive via gamma function and Stirling formula to [

That means for V = 1 the radius is a very large one equal approximately to

Now, we proceed to the distribution of the mass, i.e. how the “volume” of this ball is distributed. To do that, we estimate first the (n − 1) volume of a slice through the centre of the unit ball. Since the radius of the ball is

then the volume of the slice (n − 1) dimensional ball is given by [

Using the Stirling formula again we find that the slice has the volume

Since r is approximately

one finds

That means we obtain “mass” distribution that is almost Gaussian, with variance which surprisingly does not depend upon n:

That way we conclude the following remarkable result, namely that almost all the “volume” stays within a flab of fixed width and our result announced in the introduction of the present paper follows that about 96% of the “mass”, i.e. the volume lies in the slab [

That means 96% is concentrated near the subspace (n − 1) which may be regarded as the hyper surface of an n dimensional black hole. This is a clear failure of our low dimensional intuition to anticipate what happens in high dimensional cases.

In E-infinity theory, the pre-quantum particle as well as the pre-quantum wave follows from the fundamental equation fixing the invariants of the noncommutative E-infinity space-time [

where

which models the quantum wave. Transferring this result to Kaluza-Klein “quantum” space-time we note that the “inner” volume must be correlated, i.e. intersectional which is appropriate for a volume and leads to [

where D(Kaluza-Klein) = 5. The outer surface, i.e. the quantum wave on the other hand is additive and non- correlated so that the union operation is what leads to the volume

A typical volume representative for both would be clearly the arithmetic mean

In turn looking at the above as energy density we see that [

for m = c = 1 while

In other words E(O) is our familiar ordinary measureable energy density of the quantum particle

while E(D) is our dark energy density of the quantum wave which we cannot measure [

Adding both together we obtain the celebrated result [

Now remembering that energy and information are directly related via entropy, the preceding result is confirmation of what we obtained earlier on using Dvoretzky’s theorem, namely that 96% of the information is drawn to the surface higher dimensionality rather than “diluted” by it. Needless to say, the preceding results remain valid for a rotating Kerr black hole [

We presented in this relatively short trivial review a general theory for quantum space-time and zero point energy fluctuation based on E-infinity and related mathematical concepts. The main results and conclusions may be summarized in the following rather important points:

1) El Naschie’s E-infinity theory is a pointless self-referential geometrical-topological space-time theory related to von Neumann-Connes’ noncommutative geometry of Penrose fractal tiling universe.

2) Casimir energy and ordinary energy density of the cosmos are not only identical conceptually but identical numerically.

3) Casimir energy and cosmic dark energy are complimentary in the most strict mathematical and physical meaning.

4) The difference between Casimir energy and dark energy is a difference of boundary condition where the boundary of the holographic boundary of the universe is a one-sided Möbius-like manifold (see

5) Using a heap of space filling fractal nano spheres, we can build in principle a mini universe and use it as an energy reactor (see

6) The main conclusion is a natural consequence of mirror symmetry and Witten’s T-duality (see Figures 2-4 of Ref. [

7) The famous Einstein equation E = mc^{2} is in fact the sum of two quantum relativity parts E(O) = mc^{2}/22 of the quantum particle and E(D) = mc^{2} (21/22) of the quantum wave. In particular, E(O) is 4.5% of the total E and is the measured ordinary energy density of the cosmos while E(D) is 95.5% of the total E and represents the dark energy and dark matter density of the cosmos [

8) The formal logic proposed by M.S. El Naschie is the only ingredient missing from Hawking’s theory and that of ‘tHooft and Susskind [

The author is truly grateful to Prof. El Naschie for providing generous guidance and supplying important diagrams and figures to illustrate the present paper. I also thank Prof. Ji-Huan He for many helpful contributions to this paper.

LeilaMarek-Crnjac, (2015) On El Naschie’s Fractal-Cantorian Space-Time and Dark Energy—A Tutorial Review. Natural Science,07,581-598. doi: 10.4236/ns.2015.713058