_{1}

The present paper is basically a synthesis resulting from incorporating Kerr spinning black hole geometry into E-infinity topology, then letting the result bares on the vacuum zero point Casimir effect as well as the cosmic dark energy and dark matter density. In E-infinity theory a quantum particle is represented by a Hausdorff dimension
*Φ* where
*Φ* =2/(
√5+1) . The quantum wave on the other hand is represented by
*Φ*
^{2} . To be wave and a particle simultaneously intersection theory leads us to (
*Φ*) (
*Φ*)
^{2}=
*Φ*
^{3} which will be shown here to be twice the value of the famous Casimir force of the vacuum for a massless scalar field. Thus in the present work a basically topological interpretation of the Casimir effect is given as a natural intrinsic property of the geometrical topological structure of the quantum-Cantorian micro spacetime. This new interpretation compliments the earlier conventional interpretation as vacuum fluctuation or as a Schwinger source and links the Casimir energy to the so called missing dark energy density of the cosmos. From the view point of the present work Casimir pressure is a local effect acting on the Casimir plates constituting the local boundary condition while dark energy is nothing but the global combined effect of infinitely many quantum waves acting on the Möbius-like boundary of the holographic boundary of the entire universe. Since this higher dimensional Möbius-like boundary is one sided, there is no outside to balance the internal collective Casimir pressure which then manifests itself as the force behind cosmic expansion, that is to say, dark energy. Thus analogous to the exact irrational value of ordinary energy density of spacetime
*E*(
*O*)=(
*Φ*
^{5}/2)
*mc*
^{2} we now have P (Casimir) = (
*Φ*
^{3}/2)(
*ch*/
*d*
^{2}) where
*c* is the speed of light,
*m* is the mass,
*h* is the Planck constant and
*d*
* * is the plate separation. In addition the new emerging geometry combined with the topology of E-infinity theory leads directly to identifying dark matter with the quasi matter of the ergosphere. As a direct consequence of this new insight
*E*=
*m*c
^{2} which can be written as
*E* =
*E* (
*O*) +
* E* (
*D*) where the exact rational approximation is
*E *(
*O*)=
*m*c
^{2}/22 is the ordinary energy density of the cosmos and the exact rational approximation
*E *(
D)=
*m*c
^{2}/(21/22) is the corresponding dark energy which could be subdivided once more albeit truly approximately into E(
D)=
*m*c
^{2}/(5/22) +
*m*c
^{2}/(16/22) where 5 is the Kaluza Klein spacetime dimension, 16 are the bosonic extra dimensions of Heterotic superstrings and 5/22 □ 22% is approximately the density of the dark matter-like energy of the ergosphere of the Kerr geometry. As for the actual design of our nano reactor, this is closely related to branching clusters of polymer, frequently called lattice animals. In other words we will have Casimir spheres instead of Casimir plates and these spheres will be basically nano particles modelling lattice animals. Here
*D*= 4 will be regarded as spacetime dimensionality while
*D*=6 of percolations are the compactified super string dimensions and
*D*=8 is the dimension of a corresponding super space.

The present work has three different aims to achieve and three different messages to communicate. The scientific part is centred around the quantum vacuum and its singularities as a source of energy [

Unlike in previous publications, in the present work we rely heavily on Kerr spacetime geometry of rotating black holes (see

In the present analysis however, and as far as the Casimir effect is concerned, we opted for a rather different way of viewing the Casimir effect as a natural topological necessity of a Cantorian spacetime fabric that was

woven from an infinite number of zero Cantor sets and empty Cantor sets [

difference between the Hausdorff dimension of the particle zero set

entanglement

We can improve our model considerably by drawing into the preceding picture the possibilities offered by the Kerr spacetime geometry in

We give here a general outline of the theoretical principle and basic design concepts of a proposed Casimir dark energy nano reactor [

wave

It is an almost trivial matter to visualize and understand intuitively the physics behind the Casimir effect if we

replace in our mind quantum waves and quantum spacetime by the mental picture of real sea waves. Imagining two large walls brought just a meter apart and placed in relatively shallow water, then clearly the two walls are more likely to be pushed together by the outside waves because the water between the walls is obviously much calmer than in the open se water outside. In other words, the Casimir effect is not a mystery. The real mystery is in the quantum wave and quantum spacetime is the solution. In

fluctuation, i.e.

particle. Rather than cluttering

value of the Hausdorff dimension is

to the approximate value found by A. Zee for the same arrangement assuming a massless scalar field rather than an electromagnetic field, namely

Based on his E-infinity Cantorian spacetime theory [

a real physical entity [

ferent bounded regions of space compared to the unbounded outside of the same space is behind the Casimir forces which in the limit can be show to be equal to the difference between the quantum zero set

wave empty set

the edge of the universe there is only internal Casimir quantum wave pressure not balanced by outside pressure which is the dark energy concentration of 96 percent as per the consequences of Dvoretzky’s theorem and the present author’s dissection of Einstein’s

The sources of the ideas contained in the present work go back to many years ago when we attempted to improve on the traditional fast and slow fission reactors using the modern mathematics of fractals and nonlinear dynamics [^{th} century is also recommended [

To keep the present paper short and yet to cover the large amount of the needed prerequisites we decided in favour of a condensed presentation coupled to a large number of references as mentioned earlier on in an apologetic way [

The holographic boundary theory goes back to the pioneering work of ‘tHooft and Susskind [

thingness” fixed by the well known E-infinity formula

preceding elementary reasoning it is clear that Casimir-effect and dark energy have the same cause, namely the topology of a Banach-spacetime like manifold and, as we repeatedly mentioned earlier on, the only difference is the difference of local exophysics and global endophysics and the respective associated boundary conditions [

There has been no want of imaginative experimental set ups for measuring, testing and visualizing the Casimir effect since it was proposed by Dutch physicist, H. Casimir [

We can start for instance with a highly complex sub-structuring of space using nano tubes and nano particles and create that way fractal-like nanosphere packing modelling the moonshine conjecture which relates superstrings to other fields of theoretical physics. We stress in this connection that we have a clear model for our nano reactor based on two important facts. The first fact is the equivalence between branching polymer clusters and Cantorian-fractal spacetime [

In noncommutative geometry as well as E-infinity theory, the Penrose universe plays a significant role as a generic concrete model for both theories [

It may come as a pleasant mild surprize that exact limits could easily be established for Casimir-dark energy using nothing more than the topology of our E-infinity Cantorian spacetime [

where

[

the condition

The dimensional expression is of course

Disturbing the quantum vacuum with the objective of causing a shift in its energy density was an ingenious idea that came to Dutch physicist H. Casimir when he was originally following a completely down to earth applied physics problem in the Philips Research and Development Laboratory where he was employed in 1948. Following again

By varying the distance d we must be keeping an eye on the variation of the energy density outside the two Casimir plates. However we can avoid this complex situation following A Zee by introducing the clever trio of an intermediate third plate. By holding the two outer plates 1 and 3 fixed and moving only the inner plate number 2, we can take L as large as we wish and come around the problem that way. Now two more simplifications: First we ignore polarization by regarding only a massless scaled field instead of an electromagnetic one. Second we reduce dimensionality of space to only two so that the integration over

where we have noted that the wave modes are given by

It is the merit of E-infinity theory with its monadic random Cantor sets structure and the in built transfinite Turing computer and golden mean renormalization binary number system that these terms do not appear in the first place. However in 1948 Casimir had to deal with it in a different way mingling mathematics with physics as in the old pre ‘tHooft-Veltman and Wilson dimensional regularization groups days. So while E-infinity proceeds in a purely mathematical way utilizing the result of pure mathematics regarding the continuum hypothesis and ultimate L [

This cut off term could of course be justified on the physical grounds that the assumption of perfectly conducting Casimir plates breaks down at smaller wave lengths than atom dimensions. However the better argument is that spacetime is a Cantorian fractal. In fact ‘tHooft’s dimensional regularization is only possible in the author’s opinion because ‘tHooft is tacitly using a fractal spacetime in the indirect form of a Borel summation [

The Casimir force is then given by the negative value of the first derivation of E with respect to d where d is much smaller than L so that we find approximately the classical result for a massless scalar field

This extremely small force was measured with high accuracy experimentally and is real [

The preceding derivation must strike mathematical purists as a kind of voodoo and it may really be to a certain extent. In fact W. Pauli, with his superior intellect, sharp tongue and slightly over the top sense of humour, used the joke about this procedure by calling it theoretical physics circumcision. We cannot resolve this issue within the present paper nor a dozen more papers, however as a whole, we think that quantum gravity and theoretical physics is going in the general right direction of the ideas of Witten’s topological quantum field theory as well as Grothendieck’s conception of a monadic covering spacetime which harmonizes with E-infinity theory and is invading all of mathematical and theoretical physics in a steady pace. In the spirit of the above, we would like to show how some of the ideas and formalisms of E-infinity could be drawn in, albeit in a heuristic form to solve the problem at hand. Contemplating again

mated by the ratio of

where

where our “classical” approximate result reported in the majority of the literature on the subject and given earlier on in this section is for a massless scalar field [

Another heuristic E-infinity way to look at the same problem is to take the ratio of the generators of

Again not a bad approximation to

The following section is firmer than scientific speculation and a little less than a presentation of a definite model connecting a Kerr rotating black hole and its ergosphere to the quantum wave and quantum particle energy model of E-infinity theory [

In the case of Kerr singularity (see

However our new equation for dark matter

(see

The present work makes quite a leap from E-infinity dissection of Einstein’s

E_{3} is the dark matter density related to the ergosphere [

The author, who was born and raised in the Middle East with its unrivalled rich history and unparalleled chequered present day turbulent and logic defying politics feels morally obliged to call all the governments of the region to participate in a new dawn of science and dignified life for all living beings on this planet.

In conclusion of the conclusion we must stress the beauty of the theoretical grand picture reflected in Einstein’s iconic equation

In concluding this conclusion it must be said that without the inspiration of Nobel Laureate G. Binnig via his remarkable book about fractals, nano technology and creativity [