M. S. EL NASCHIE

168

dark energy of the cosmos [16]. In short, it turns out that

the famous relativity formula relating mass (m)

to energy (E) via the speed of light (c) does not distin-

guish between measurable real ordinary energy E(O) and

missing dark energy of the cosmos E(D) which cannot be

detected or measured directly using any of present day

technology [17,18]. The simple explanation for this un-

paralleled challenge to the foundations of modern theo-

retical physics and cosmology is again intimately con-

nected to Hardy’s quantum entanglement and conse-

quently to random Cantor sets and their golden mean

Hausdorff dimensions. Since at the quantum resolution

level, spacetime is made up of totally disjoint random

Cantor sets with an infinite number of graded dimensions,

the probability of quantum entanglement is governed by

the zero measure of this Cantorian space and is given

generally by

2

Emc

3n

3n

P

where n is the num-

ber of particles [5,7]. For two particles, one finds 5

P

which is the theoretically and experimentally confirmed

Hardy quantum probability [5-7]. Intersecting the three

fundamental equations of physics, namely Newton’s ki-

netic energy 2

1

2

N

Emv where v is the velocity, Eins-

tein’s equation and

2

Emc5

P

one finds a

quantum relativity energy-mass formula

255 22

122

2

QR

Em vcmcmc

2

.

This is exactly the energy density found via accurate

cosmic measurements COBE and WMAP which amount

to only 1224.5% of what E(Einstein) pred icted [10],

[16-19]. The rest namely

11 2295.5%is the

presumed missing dark energy of the Universe. It then

turned out that ordinary energy i.e. the measurable 4.5%

is simply the energy of the quantum particle face of the

particle wave duality of quantum mechanics [7,10] while

the missing 95.5% dark energy E(D) is the energy of the

other face of the duality, namely the quantum wave

[10,14,18]. Since the quantum particle is modeled by a

five dimensional zero set and the wave is modeled by a

five dimensional empty set we have here a zero set-

empty set duality [19].

Together

222Ec mc of the particle and

222

52 2122E Dmcmc

of the wave add

up to a total exactly equa l to Einstein’s energy [16-19]:

522

total25 2E

2

mcmc. (2)

Two major conclusions follow after the above. First,

Einstein’s is correct but completely blind to

the distinction between dark energy and ordinary energy.

Second, since measurement collapses the quantum wave,

it is natural that our present day technology cannot detect

dark energy [10,16-18]. From the above we can see fu-

ture technological research going towards developing

quantum nondemolition measurement instruments and

nuclear dark energy reactors. None the less it is vital to

understand that we may have a wavy spacetime at the

quantum resolution simulating a quantum wave of a knot

in the fabric of spacetime. Consequently harvesting the

quantum wave may be harvesting the real vacuum mod-

eled by the empty set, similar to what KAM theorem

implies [8].

2

Emc

Finally we note that E(0) and E(D) could be readily

found from the ground energy state of the hydrogen atom

22

1

2

Emc

where 1/137

[4,20] by running

as a function of energy as in quantum field theory and

replacing 2

by 23 5

so that we find

25 2

122.

2

Emcmc

REFERENCES

[1] V. M. Petruševski, “The H-Atom and the Golden Ratio:

A Possible Link,” Journal of Chemical Education, Vol.

83, No. 1, 2006, p. 40.

[2] V. M. Petruševski, “The First Excited State of the Hy-

drogen Atom and the Golden Ratio: A Link or a Mere

Coincidence?” Bulletin of the Chemists and Technologists

of Macedonia, Vol. 25, No. 1, 2006, pp. 61-63.

[3] C. L. Devito and W. A. Little, “Fractal Sets Associated

with Function: The Spectral Lines of Hydrogen,” Physi-

cal Review A, Vol. 38, No. 12, 1988, pp. 6362-6364.

doi:10.1103/PhysRevA.38.6362

[4] A. C. Phillips, “Introduction to Quantum Mechanics,”

John Wiley & Sons Ltd., Chichester, 2003.

[5] M. S. El Naschie, “Quantum Entanglement as a Conse-

quence of a Cantorian Micro Spacetime Geometry,”

Journal of Quantum Information Science, Vol. 1, No. 2,

2011, pp. 50-53. doi:10.4236/jqis.2011.12007

[6] J.-H. He, et al., “Quantum Golden Mean Entanglement

Test as the Signature of the Fractality of Micro Space-

time,” Nonlinear Science Letters B, Vol. 1, No. 2, 2011,

pp. 45-50.

[7] L. Hardy, “Nonlocality of Two Particles without Ine-

qualities for Almost All Entangled States,” Physical Re-

view Letters, Vol. 71, No. 11, 1993, pp. 1665-1668.

doi:10.1103/PhysRevLett.71.1665

[8] M. S. El Naschie, “A Review of E-Infinity Theory and

the Mass Spectrum of High Energy Particle Physics,”

Chaos, Solitons & Fractals, Vol. 19, No. 1, 2004, pp.

209-236. doi:10.1016/S0960-0779(03)00278-9

[9] M. S. El Naschie, “The Theory of Cantorian Spacetime

and High Energy Particle Physics (An Informal Review),”

Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp.

2635-2646. doi:10.1016/j.chaos.2008.09.059

[10] R. Penrose, “The Road to Reality,” Jonathan Cape, Lon-

don, 2004.

[11] R. Mauldin and S. Williams, “Random Recursive Con-

Copyright © 2013 SciRes. IJMNTA