_{1}

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The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.

Post modernistic research in theoretical physics [

In the present work which is anticipating a sweeping new revolution in the way, we will be doing physics in 10 to 20 years, [

To keep the length of the present paper to a minimum we start from what we called in a recent paper [^{2} where k is essentially related to the familiar Lorentzian factor

where E is the energy, m is the mass and c is the speed of light while ^{2} [^{2} where v is the classical velocity of a particle as well as Einstein’s leap to a fully fledged generalization of similar earlier discoveries of E = mc^{2} by Poincare and others will not be discussed here in the depth it requires [

We are invoking here nothing connected directly to the theory of relativity or quantum mechanics and yet we will arrive to a result which can be understood deeply only via these two pillars of modern physics [^{(n)} one could take the view with considerable justification, that R^{(5)}, i.e. five dimensions is about the limit of accessible physics in the experimental possibilities of a 3 + 1 dimensional conscious and well equipped observer. At the same time it is an educated guess that M-theory is real and R^{(11)} is probably one of the best ways to describe not only theoretical high energy physics but the entire cosmos. To put this to a pragmatic test we calculate the vital independent components of the most important driving force in Einstein’s relativity, namely the Riemann tensor. For n dimensions this number is given by [

Setting n = 5 and n = 11 one finds [

and

respectively. Assuming that all these components have almost the same statistical weight, then the difference between 50 and 2110 measures clearly the sparseness of the associated space and consequently the totality of the average curvature. Similarly the ratio between 50 and 2110 is a measure of the density of the energy which is likened in the theory of relativity mainly to the curvature as it is in the case a simple elastic wire in the theory of engineering elasticity. Viewing the complex problem in this quite simplistic way leads us directly to estimating

Consequently the rest of the energy filling not only D = 5 but also D = 11 must be given by the so called dark energy density which is given logically by [

For two important reasons the preceding rough result is truly striking. First it is quite close, in fact very close to highly accurate cosmic measurements and observations connected to the famous COBE, WMAP and Type 1a Supernova [

That means E(O) is not 4% but rather 4.5% while E(D) is not 96% but 95.5% to a very high degree of accuracy and in astounding agreement with measurements [

Our first estimate of ^{(5)} = 50 and analogously D = 11 to R^{(11)} = 1210. This is an obvious and trivial embedding problem because R^{(5)} and R^{(11)} are treated as quasi-dimensions estimating the size of our spacetime manifold. Consequently a more accuret

This is almost the familiar exact rational (1/22) value as document in my previous papers using different methods. Clearly

almost exactly as expected.

Although not trivial, it is not difficult to obtain the truly exact formulas of energy density ^{(5)} = 5 to R^{(5)} but the full space which embeds the SL(2, 7) Lie symmetry group of the holographic boundary of our universe, i.e. D = 7 and similarly the vacuum of Witten’s D = 11 which is the corresponding pure gravity with

In other words we have [

which is this time the truly exact integer value. For

We may remark on passing that 57 is actually the intrinsic dimension of the fundamental E(8) exceptional group [

Readers familiar with E-infinity Cantor spacetime theory [

where

From the above we see that the pre-quantum particle is identified by the bi-dimension [

while the pre-quantum wave is given by

From all these previous results, it is easily reasoned that the “topological” volume of the pre-particle in 5D is given by the obvious multiplication formula [

The corresponding additive volume of the pre-quantum wave on the other hand is given by

The total volume is thus equivalent to world sheet of string theory [

Inserting mean volume = 2/2 = 1 in Einstein’s formula E = mc ^{2} one finds [

That means [

This is the exact expression which leads to the integer solution by disregarding k = 0.18033989 compared to 21 and 22. We note that this k was interpreted physically as ‘tHooft’s renormalon hypothetical particle which is equal to

Although we have not invoked in all the preceding analysis any Lorentzian transformations or Einsteinian conception relating to the meaning of simultaneousity [^{2}. In the following solution all these things will be dispensed of so that the reader may see clearly the main message of the present paper that even a low dimensional fractal is essentially infinite quasi dimensional because of the involved infinite iteration and self similarity and that even a harmless conventional fractal curve in 3 dimensions like the familiar Menger sponge [

This fractal, although it looks like a cubic sponge in 3D is essentially a curve, not a real 3D and possesses in our case the disadvantage of being continuous and could therefore be expected to deliver a good approximation only because continuity violates one of our main E-infinity theory principles, namely being the “pointless” point-set theory as emphasized in the pioneering work of von Neumann’s continuous geometry where continuity is not referring to the geometry [_{H} = 2.726833028 and D_{T} = 3 will give us the density of the involved empty set which represents dark energy. Consequently we may write

Consequently the ordinary energy density must be

Now, and this is a crucial point, we do not insert

For a basically almost entirely classical analysis this result reinforces our conjectured equation:

and together with the rest of the paper shows beyond reasonable doubt that there is far more than meets the eye to dimensionality of spacetime and fractal Cantorian geometry.

There are many shortcut derivations and radically different solutions all converging directly or indirectly towards the main thesis of the present work, namely that of measure concentration of volume in a sufficiently high dimensional manifold with fractal-Cantorian features. Thus, we have all three fractal spacetime theories of ‘tHooft-Veltman-Wilson dimensional regularization spacetime D = 4 − k as well as Kaluza-Klein fractal space

where

The analysis is in complete conformity with the result based on pure mathematical theorems such as Dvoretzky’s theorem as well as accurate measurements and observations such as COBE, WMAP and Type 1a supernova which is awarded the 2011 Nobel Prize [

From all the above we conclude that higher dimensionality and fractality of spacetime are a reality of the small and large scale structure of spacetime and that our conceptual equation constituting the title of the present paper is far from being esoteric or mathematical abstraction with no tangible content. Hardy’s quantum entanglement [

Mohamed S. El Naschie, (2016) On a Quantum Gravity Fractal Spacetime Equation: QRG ≃ HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion. Journal of Modern Physics,07,729-736. doi: 10.4236/jmp.2016.78069