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The paper presents a detailed analysis of ordinary and dark energy
density of the cosmos based on two different but complimentary theories. First,
and starting from the concept of the speed of light being an average over
multi-fractals, we use Magueijo-Smolin’s ingenious revision of Einstein’s
special relativity famous formula *E* = *mc*^{2} to a doubly special
formula which includes the Planck energy as invariant to derive the ordinary
energy density *E*(*O*) = *mc*^{2}/22
and the dark energy density *E*(*D*) = *mc*^{2}(21/22)
wheremis the mass andcis the speed of light. Second we use
the topological theory of pure gravity to reach the same result thus confirming
the correctness of the theory of varying speed of light as well as the COBE,
WMAP and Type 1a supernova cosmological measurements.

The familiar speed of light is the well-known constant value only in the vacuum. The speed of light is otherwise dependent upon the density of the medium which the light passes through. The situation with a fractal space- time is not dissimilar. Imagining a fractal space-time to be infinitely many layers, each with a different number of dimensions depending upon the resolution of observation, we could regard the speed of light as variable but with a constant expectation value taken as the statistical averaging over infinitely many speeds of slight ranging from infinitely small to infinitely large. Looking at things in this way we will apply the idea of varying speed of light to obtain an accurate estimation for the missing dark energy and the surprising acceleration rather than deceleration of cosmic expansion.

In the present paper we use Magueijo-Smolin’s varying speed of light (VSL) theoretical modification of Einstein’s energy mass relation to derive an exact value for the missing dark energy which is found to be in astonishing agreement with the latest result of the WMAP measurement and the independent super nova analysis. Thus while Einstein’s formula predicts 95.5% more energy than found in highly precise astrophysical measurement, our VSL based calculation indicates an exact theoretical value of only 4.508497% real energy. Consequently the exact conjectured missing dark energy must be 95.491502%. This is a clear cut confirmation for both the cosmological measurement and the VSL theory. However, this is far from being the final resolution of the riddle because the missing dark energy of the cosmos has a profound real effect reflected in the observed increased rather than decreased rate of the cosmic expansion of the universe mentioned earlier on. It then turned out that the dark energy is the 95.5% energy of the quantum wave which is a Legendre transformation in dual complementarity to the ordinary 4.5% energy of the quantum particle. On the other hand measurement leads to quantum wave collapse preventing the detection of its propagation energy. Thus we are left with a single rational conclusion, namely that quantum non-demolition measurement instruments must be developed first if we are to have a chance in finding the missing 95.5% dark energy. We stress that the ability of the present theory to make such accurate predictions is uniquely tied up with incorporating into it, the measure zero Cantor set behind Hardy’s quantum entanglement and the associated empty set of the vacuum or pure gravity of the general theory of relativity.

The issue of missing dark matter and worse still missing dark energy in the cosmos is since some time now a colossal problem threatening the very foundations of theoretical physics and cosmology [

In the present work we give an exact analysis based on a topological theory of varying speed of light pioneered by J. Magueijo and L. Smolin showing that the maximal energy density value predicted by Einstein’s theory of relativity E = mc^{2} where E is the energy, m is the rest mass and c is the velocity of light gives the wrong answer when it comes to calculating the intergalactic energy scales [^{2} which should be E(Ordinary) @ mc^{2}/22. However, where is E(Dark) @ mc^{2}/(21/22) which together with E(Ordinary) gives E(Einstein)? This is still a more than valid and meaningful question because this dark energy has a real profound effect reflected in the measured increase rather than decrease of the rate of expansion of the cosmos [

The present analysis is structured logically as follows:

1) After the introduction and the background information we give the quintessence of the VSL theory in Section 3.

2) In Section 4 we show the consistency of the Magueijo-Smolin formula with Newtonian mechanics and Ein- stein’s relativity, then derive the ordinary energy density and the dark energy density.

3) In Section 5 the importance of quantum entanglement interpretation of the obtained results is explained in the light of Hardy’s celebrated formula for quantum entanglement of two quantum particles.

4) Finally the set theoretical interpretation of ordinary energy and dark energy is explained in Section 6 while Section 7 wraps it all in terms of the topological theory of pure gravity.

The completely new idea behind the derivation of the energy-mass relationship using the VSL theory basically comes from H. Weyl inspired theory of scale relativity [

where l_{p} is the Planck length as invariant. Consequently J. Magueijo and L. Smolin arrive at their replacement of Einstein’s single equation E = mc^{2} by three remarkable and thoroughly ingenious equations [

with the usual m transformation

where

and

The preceding equations give us the possibility of admitting a ratio m_{0}c^{2}/E_{p} far larger than unity. In other words through the postulates of VSL theory m_{0}c^{2} can become many times larger than E_{p} without violating Lorentzian invariance, nor the 1/l_{p} invariance of the theory [

(A) Ordinary Energy, Dark Energy and Einstein’s Energy from the view point of set theory and quantum wave collapse as well as pure gravity in the absence of matter field associated with Einstein’s field equation degrees of freedom D = d(d − 3)/2 where d is the dimension. The contra part of the zero set quantum particle in this case is given by d = 3 and D = 0 while the contra part of the quantum wave empty set is given by d = 2 and D = −1; (B) Quantum Measurement and dark energy of the quantum wave

. Explicit correspondence transforming Joao Magueijo-Lee Smolin, El Naschie and Einstein formulas into each other

Joao Magueijo-Lee Smolin | El Naschie | Einstein |
---|---|---|

There is an entire web of relations in the above constituting a complete topological “physical” dimensionless theory behind the conventional facade. For instance the dimensionless topological Plank energy

From the first equation of Magueijo-Smolin [_{0}c^{2}/E_{p} could take three vital values:

i.e. we have Einstein’s classical equation of special relativity.

i.e. we have the Newtonian expression for energy when setting c = v and find E = (1/2)mv^{2}.

a)

This is a crucial situation for us as we will explain shortly. The energy in this case could be written as

Let us consider the above from the viewpoint of space-time dimensions [

In case (a) above we could say that space and time fuse into 3 + 1 = 4 dimensions. There are zero hidden dimensions in this theory. The total energy E is not reduced or diluted by extra dimensions.

In case (b) above we could say, somewhat surprisingly, that we have a hidden dimension, namely time appearing only as a parameter. The energy mc^{2} → mv^{2} is divided therefore by 2.

On the other hand in the most important case concerning the present work i.e. case (c), we could interpret the factor 21 as the number of dark dimensions of 26 bosonic string theory [

Table 2. Bosonic strings interpretation of and varying speed of light theory [9] [10] .

Starting from Einstein’s 4D space-time and the 26D of Bosonic strings we have | Starting from Kaluza-Klein 5-D space-time and the 26D of Bosonic strings we have |
---|---|

Note that: E_{p} is the Planck energy E_{p} = (hc^{5}/G)^{1/2}. Setting

Therefore we can write [

This result is tantamount to saying that (1 − 0.04545) (100) = 95.454% of the energy in the universe is dark energy. The result is very close indeed to the experimental measurement of 95.5% [

The classical Heterotic string theory [

where _{max} is consequently [

The remarkable thing is that

In addition, the same previous result can be found directly from a delicate light cone quantized Lorentzian transformation [

Set theoretical reformulation of orthodox quantum mechanics leads to some truly insightful realization concerning wave collapse, measurement and the fractal-Cantorian non-classical nature of the geometry of the fabric of space-time [

set

mension of the same particle. The quantum wave on the other hand is described by the empty set

five dimensional zero set

(see

Naїve geometrical interpretation of Vol(D_{0}) = f^{5} of the ordinary energy of the quantum particle E(Ordinary) = (f^{5}/2)mc^{2}

Naїve geometrical interpretation of Vol(D_{‒1}) =5f^{2} which decides upon the magnitude of dark energy of the quantum wave [24] -[27] E(Dark) = (5f^{2}/2)mc^{2}

The first result is that Einstein’s formula E = mc^{2 }is correct but totally blind to any distinction between the ordinary energy

Added together one finds a new confirmation of Einstein’s equation without its limitation, namely [

Here E(O) plays the role of potential energy while E(D) plays the role of kinetic energy.

The second result is that the only hope to measure dark energy is to develop new non-demolition measurement instruments [

When the present Authors obtained the ordinary energy density E = mc^{2}/22 they were quite mindful of the fact that this unexpected fundamental result is coming primarily from the special theory of relativity rather than Einstein’s general theory of gravity. In retrospect it is now clear why. Special relativity does not depend in any fundamental way or include a matter field, only 4D space-time is all what we start and end with. By contrast we tend generally to think of Einstein’s field equation in the absence of any matter field as trivial. However, many deep thinkers of theoretical physics were intrigued by the fact that pure gravity is governed by various differential equations arising from general relativity with remarkable consequences. It is one of the most important findings of the present work that we can give a physical interpretation of the zero set of the quantum particle and the empty set of the quantum wave via the degrees of freedom of pure gravity D in dimension d = 3 and d = 2. The fundamental equation governing this behavior is the same equation for the Vierbein as well as that of the massless graviton, namely

for d = 3 corresponding to a classical particle D = 0 converting it to a de facto quantum particle being deprived of local degrees of freedom. The corresponding Hausdorff dimension is thus

In the present case we find first D = 4(4 ‒ 3)/2 = 2 for d = 4, which represents classical space-time, then we find D for d = 11 of Witten’s M-theory which comes to D = 11(11 ‒ 3)/2 = 44. Our ^{2}/22. The Legendre transformation

We give a topological interpretation and a hidden dark dimension interpretation to the Magueijo-Smolin energy mass equation of VSL theory [^{2}. However this is a total maximal energy representing the 100% theoretical density. The VSL equation on the other hand predicts the measurable ordinary energy which is approximately 1/22 of this value. Accurate calculation shows that 22 = 26 − 4 hidden dark dimensions of bosonic string theory should be 22.18033989 when taking the “fine structures” of the strings into account and consequently the exact reduction is not 1/22 but

where

actly by

If we were asked to summarize the importance and deep meaning of the present work in a very few lines, we would say the following:

Remembering J. A. Wheeler’s slogan, the boundary of a boundary is empty [