To show how the title of this short letter is true, we start by using stringent mathematics to demonstrate how the time dimension is actually a fractal effect that may be called a real and persistent illusion without being inconsistent or esoteric [1] - [15] . We know from many previous publications that the average topological dimension of spacetime is given by the formula [10] - [15]

〈 n 〉 = 1 + d c ( O ) 1 − d c ( O ) (1)

where d c ( O ) is the Hausdorff dimension of the backbone fractal-Cantor unit set forming the space of the physical-mathematical E-infinity theory [12] - [19] . Setting d c ( O ) equal to the rational value d c ( O ) = 3 / 5 = 0.6 [10] [11] [12] , one finds

〈 n 〉 = 1 + ( 3 / 5 ) 1 − ( 3 / 5 ) = ( 8 / 5 ) ( 2 / 5 ) = 4 (2)

The result means that for d c ( O ) = 3 / 5 = 0.6 which is the first rational approximation found from continued fraction expansion of the irrational golden mean ϕ = ( 5 − 1 ) / 2 = 0.618033989 one finds the dimensionality of Einstein’s spacetime [20] [21] . On the other hand, we know that the average d c ( O ) is given not by d c ( O ) = 0.6 but by 〈 d c ( O ) 〉 = 1 / 2 which is equal to 〈 d c ( O ) 〉 = ϕ − k / 10 = 0.618033 − 0.018033989 = 1 / 2 as well known from the E-infinity interpretation of Sir. R. Penrose twistor theory [22] . Inserting 〈 d c ( O ) 〉 = 1 / 2 in 〈 n 〉 one finds [3] [8] [10] [13]

〈 n 〉 = 1 + ( 1 / 2 ) 1 − ( 1 / 2 ) = 3 (3)

which is the topological dimensionality of classical Newtonian space [3, 8, 13]. Remembering that d c ( O ) = 0.6 is the first approximation of d c ( O ) = ϕ and that f is nothing but ( d c ( O ) = 0.6 ) plus ‘tHooft’s renormalon divided by superstring dimensionality D = 10 [11] [12] [13] [20] [21] , we can see the difference between 〈 n 〉 = 4 and 〈 n 〉 = 3 is the first effect of the irrational fractality implicated in d c ( O ) = 0.6 [3] [10] . The preceding result could be reproduced using another manoeuvre by taking 〈 n 〉 = 3 to be n and find the corresponding fractal dimension of spacetime where d c ( O ) = 1 / 2 using the well known bijection formula of E-infinity [8] [10] [11] [12]

d c ( n ) = ( d c ( O ) ) n − 1 (4)

Consequently for d c ( O ) = ( d c ( O ) ) = 1 / 2 and n = 3 one finds [3] [8] [10] [12]

d c ( 3 ) = [ 1 / ( 1 / 2 ) ] 3 − 1 = ( 2 ) 2 = 4 (5)

That means that for n = 3 we find d_c = 4 which means that 4 - 3 = 1 is the difference between the fractal Hausdorff dimension of spacetime, i.e. d_c = 4 and the corresponding topological dimension n = 3 so that we may interpret d = 4 not as the union of n = 3 plus a time dimension n (time) = 1 leading to 3 + 1 = 4 spacetime dimension but rather the difference on average between n = 3 and d c ( 3 ) = 4 , i.e. the effect of hidden on average fractality [3] [8] [10] [12] . To be absolutely convinced of the soundness of our fractality interpretation, we can demonstrate yet a third derivation, this time using the true average of the Hausdorff dimensionality for d c ( O ) = 1 / 2 which is given in E-infinity theory by [10] [12]

〈 d c 〉 = 1 d c ( O ) ( 1 − d c ( O ) ) (6)

and find

〈 d c 〉 = 1 ( 1 / 2 ) ( 1 − ( 1 / 2 ) ) = ( 1 / 4 ) − 1 = 4 (7)

where we saw from Equation (3) that 〈 n 〉 = 3 [3] [8] [10] [12] . Thus we can say without any reservation that time is the extra dimension arising from the fact that the spacetime manifold has an average Hausdorff dimension 〈 d c 〉 = 4 larger than its embedding average topological dimension 〈 n 〉 = 3 [3] [8] [10] [12] .

It is highly instructive and insightful that while we do not see but simply “feel” the time dimension, there is another effect that we can “see” and feel due to a fractality effect, namely the ordinary energy density of the cosmos [21] . Without going into the detail of why the total Hausdorff dimensionality of the universe is not given by D = 4, nor in fact by D = 4 + ϕ 3 of the E-infinity core but by the fractal version of Kaluza-Klein spacetime theory [20] [21] D = ( 4 + ϕ 3 ) + 1 = 5 + ϕ 3 , we can easily demonstrate that the ϕ 3 is the Hausdorff mass of the cosmos while the 5 is the topological mass of the universe [23] [24] so that we can easily reason that the ordinary energy density of the cosmos is given by [23] [24]

ϕ 3 / ( 5 + ϕ 3 ) = ϕ 5 / 2 (8)

in full agreement with all actual cosmic measurements and observations [23] [24] [25] . We note on passing that ϕ 5 is Hardy’s probability of quantum entanglement for two quantum particles. This makes a great deal of sense because entanglement is a very strong form of correlation so that we can measure this correlated energy density with relative ease. By contrast the dark energy sector is anti-correlated, i.e. 1 − ( ϕ 5 / 2 ) and therefore it could not be measured in a direct way leading to the wrong supposition that dark energy does not exist [20] - [25] .

From the preceding analysis and conclusion, we can draw the profound conclusion that one of the most important components of our universe and existence is its chaotic fractal dynamics, geometry and topology [8] - [16] [20] - [25] . In fact, this conclusion is the deep rationale behind our Casimir-dark energy nano reactor proposal [15] [26] [27] as well as our explanation and endorsement of NASA’s electromagnetic drive which opened the possibility of fuelless interstellar spacecraft travel [28] [29] . Last but no means least, we should point out the central and important role played by understanding vital concepts or spacetime and its fractal origin for avoiding or resolving certain class of paradoxes in quantum information science such as the Hardy paradox which we have addressed in a recent paper [30] .

El Naschie, M.S. (2018) Time Dimension and Ordinary Cosmic Energy Density Are Fractal Effects. Journal of Quantum Information Science, 8, 47-51. https://doi.org/10.4236/jqis.2018.82004