_{1}

^{*}

The 95.5 percent
of discrepancy between theoretical prediction based on Einstein’s theory of relativity and the accurate cosmological measurement of WMAP and various supernova analyses is resolved classically using Newtonian mechanics in conjunction with a fractal Menger sponge space proposal. The new energy equation is thus based on the familiar kinetic energy of Newtonian mechanics scaled classically by a ratio relating our familiar three dimensional space homology to that of a Menger sponge. The remarkable final result is an energy equation identical to that of Einstein’s
E=mc
^{2}
but divided by 22 so that our new equation reads as
. Consequently the energy Lorentz-like reduction factor of
percent is in astonishing agreement with cosmological measurements which put the hypothetical dark energy including dark matter at
percent of the total theoretical value. In other words our analysis confirms the cosmological data putting the total value of measured ordinary matter and ordinary energy of the entire universe at 4.5 percent. Thus ordinary positive energy which can be measured using conventional methods is the energy of the quantum particle modeled by the Zero set in five dimensions. Dark energy on the other hand is the absolute value of the negative energy of the quantum Schrodinger wave modeled by the empty set also in five dimensions.

The discrepancy between theoretical prediction and cosmological measurements of the entire energy content of our universe [1-3] is resolved in the present work. This is achieved by combining classical Newtonian mechanics with a novel fractal interpretation of our familiar classical space. We start by assuming that space itself is a Cantorian set-like fractal akin to a Menger sponge [4,5]. This immediately leads us to qualitative and equally important, if not more important, quantitative results [

thus mathematics and consequently physics [7,8].

The present analysis starts by showing that Einstein’s [9,10] must be revised to _{ }and conclude that

in complete agreement with the WMAP and supernova measurements [1-3]. This means that only 4.5% of the expected energy exists while the rest of 95.5% must be assumed to be missing and is therefore referred to as “dark” or missing energy [1-3,6]. Subsequently the analysis is refined and extended to find the exact which turned out to be 1/22.18033989 being the ratio of

and where

. Consequently the exact is given by

It should be noted that is the well known Hardy’s probability of quantum entanglement [

A Menger sponge is basically a three dimensional fractal [4,5,7] constructed by drilling infinitely many cubic holes into it iteratively, the result of which is shown in

merely matter to be a Menger sponge fractal, then the Hausdorff dimension of this space could be set equal to the Menger sponge (see

Let us now ponder carefully what D_{H }(M) really measures and refers to. Since the original cube was obviously 3 dimensions and we have at least in theory removed almost the entire substance, i.e. space which makes it upthen it follows that the large dimension of refers basically the quasi-Hausdorff value to the space removed rather than the sparse Cantor point set left. Said in a different way the volume of the Menger sponge space is now zero and nothing is left except a zero measure infinitely long and infinitely thin fractal line in three dimensional classical spaces. What could be said to have remained from this 3D space is a zero volume Menger fractal of a Hausdorff dimension equal to that of the complement space of the Menger sponge and given by (see

It is important to realize that the relative ratio of what is left of real space to the original 3D cube is obviously the difference between 3D “solid” and “smooth” Euclidean space and a cotton candy-like (see

It is thus imperative to understand that this must be included in the classical kinetic energy expression of Newton which presupposes a “smooth” “solid” nonfractal space. Consequently

must be logically extended to

and therefore

That means

This is only 4.5% from what the relativistic nonquantum equation of Einstein predicts. However it is clear from the full agreement of the energy predicted by E_{QR}_{ }with the accurate experimental measurement of WMAP and others [1-3] that does not apply to extreme situations like when considering the cosmos as a whole.

In the next section we will give some deeper and mathematically more sophisticated reasons why E_{QR} is the correct equation for calculating the energy of the cosmos and that could be seen as resulting from accounting for a fundamental quantum mechanical effect, namely quantum entanglement [8,11].

It is well known that Hardy’s quantum probability [8,11] is generic and is given by

where [8,11]. At least in theory the two particles which were tested to very high accuracy experimentally lead to the conclusion that for a single particle we would have

.

Now Einstein’s equation is a one particle equation

.

Intersecting this relativistic formula with the quantum formula, a quantum relativistic energy formula is easily found to be (see Figures 5 and 6)

This is almost the same result obtained earlier on using classical mechanics and the Menger sponge space in the previous Section 2.1.

The Kähler manifolds are used for compactification in superstrings and related theories [

space and time are fused together and modeled by such a Kähler manifold. The Betti number b_{2} for K3 is given by [12,13]

This number could be thought of as counting the number of 3D holes in K3. Thus compared with Einstein’s 4D smooth manifold for which b_{2} = 1, our K3 has 22 times more 3D holes in it [12,13]. Thus we could write the ratio as [12,13]

This is obviously a very useful scaling exponent and we see that and consequently multiplied with ^{ }of Einstein we find again our E_{QR} energy formula

Thinking deeply about this result one may be yet again surprised to realize that in retrospect, it should have been expected for the following obvious reason. The difference between Newton’s kinetic energy formula

and Einstein’s maximal energy is formally a factor half and setting v = c. Subsequently we showed that by assuming a different Menger fractal geometry instead of the smooth geometry of Newton’s space. Here again E_{QR} kept the same form of Newton and Einstein and everything else was taken care of by a simple factor 1/22. Then in our second derivation the same result was found after fusing quantum entanglement with special relativity. Again if we remember that gauge theory started with the idea of Weyl scaling and that Nottale’s high energy particle physics and cosmology theory is based on scale relativity principle, then we realize that this was also to be expected in our case. For these reasons the ratio of the homology of a classical geometry such as b_{2} = 1 of Einstein’s space and the b_{2} = 22 of a complex manifold like our K3 used here [12,13] harbors more than meets the eyes in the harmless appearance of a simple scaling factor.

To connect all the preceding three different derivations with the original theory of Lorentz and Einstein, it is instructive to see that a similar derivation in the spirit of Lorentz-Einstein transformation holds and leads to the same result of quantum relativity. Accepting the three fundamental phenomenological effects of special relativity, the following transformations are evidently consistent, i.e. [

where is a boost which does not need to be defined by anything related directly to v/c where c is the phenomenologically and experimentally accepted constant speed of light. Inserting in Newton’s kinetic energy we find

On the other hand we could use the conventional Lorentz transformation in the unconventional form of light cone velocity used in superstrings quantization [14,15] and extend it to encompass a light cone mass as follows:

Inserting again the Newton kinetic energy we find

Setting v = c, c = m = 1 and equating E_{1 }and_{ }E_{2 }one finds

This leads to a quadratic equation in with the only positive root [_{1} one finds immediately that

which confirms without any doubt the correctness of all the previous three derivations of Sections 2.1, 2.2 and 2.3 of the present work. In Chart Nos. 4 and 5 we give an overview comparing different Lorentz-like transformations leading to the same robust result.

When an elastic surface is acted upon with a load, it curves [17,18]. The theory of such elastic surfaces is highly developed in a remarkably successful theory called theory of elasticity [17-20]. This theory and its sister, theory of plasticity, is the basis of all structural engineering science which gave us shell structures [17-19] covering large sports and airport halls without supporting columns and thin fuselages which carry passengers across the Atlantic in a few hours. When such an elastic or elasto-plastic surface is sufficiently thick, long and narrow then an interesting curvature phenomena takes place called anticlastic curvature [

Cosserat [

The thread connecting the different themes of all the preceding sections is the profound impact of non-classical and hyperbolic geometry on physics. In this section we stress this point by referring to the explicit impact of non-classical geometry and its Lie symmetry groups as presented in overview Chart 7 on physics [12-16].

Assuming that space-time is akin to a Menger sponge fractal we were able to show that a purely classical energy expression changes to

The result of this Newtonian non-relativistic and nonquantum derivation is confirmed using a variety of sophisticated mathematical methods including a Lorentzlike transformation as well as an intersection between Hardy’s quantum entanglement

and Einstein’s maximal energy. Thus

may be regarded as a quantum relativity formula and therefore may be viewed in various ways as:

1) A Weyl-Nottale scaling expression for quantum relativity [

2) A measure for the hypothetical dark energy of the cosmos

in full agreement with measurements [1-3,6].

2) The magnitude of quantum entanglement involved in quantum relativity at the Hubble radius scale of the universe [

3) A measure for the negative gravity or anticlastic curvature effect responsible for the increasing rate of expansion of the universe.

4) The ratio of two Betti numbers characterizing the homology of Einstein’s space and a K3 Kähler space namely [12,13].

It is important to note that recent investigation by the present author has revealed that is the energy of the quantum particle while is the dark energy of the quantum wave. The sum is Einstein’s energy. Thus Einstein’s formula is blind to any distinction between ordinary energy and dark energy. (See also Overview Charts 1-3 and Figures 6 and 7).