M. S. EL NASCHIE, L. MAREK-CRNJAC

Copyright © 2012 SciRes. IJMNTA

124

c

smooth, continuous and without holes. However actual

space-time at quantum scales and surprisingly again at

intergalactic scale displays a wild Cantorian fractal ge-

ometry and topology. It is a T-duality which we saw in

the unification program at the Plank length, yet this time

the surprising quantum effect of entanglement is showing

its power at the Hubble length scale. The main equation

obtained in the present work which is

2

0QR S

EEm

(36)

is gauge invariant in the widest sense possible, meaning

that it is almost invariant to the use of any mathemati-

cally and physically reasonably meaningful theory. It is a

very robust result not affected by minor details of theo-

retical modeling. Thus we may show here in the conclu-

sion what on reflection should have been presented in the

introduction at the very beginning:

Special relativity implies three strange effects [11]:

1) length contraction;

2) time delineation;

3) mass increase.

All these classically feeble effects become noticeable

only as the speed approaches the speed of light c [11].

We handle this semi-classically, i.e. using common sense

by introducing a boost

1

and anti boost

1

.

Thus we have 1)

1

X

t

, where X is space co-

ordinate, 2)

1t

0

m

, where t is ordinary time and 3)

, where m0 is non-relativistic mass [11].

0

1m

Inserting in Newton’s kinetic energy

2

0

1

2

Emv

we find

2

2

0

11

1.

21

QR

Em

c (37)

Setting 2

51

we find our previous result

2

0

1.

22

Emc Thus in one stroke we reconciled and

fused together classical mechanics with relativity and

quantum mechanics via the non-classical geometry of

fractals [3-8]. This is magically beautiful.

REFERENCES

[1] L. Amendola and S. Tsujikawa, “Dark Energy: Theory

and Observations,” Cambridge University Press, Cam-

bridge, 2010.

[2] Y. Baryshev and P. Teerikorpi, “Discovery of Cosmic

Fractals,” World Scientific, Singapore, 2002.

[3] M. S. El Naschie, L. Nottale, S. Al Athel and G. Ord,

“Fractal Space-Time and Cantorian Geometry in Quan-

tum Mechanics,” Chaos, Solitons & Fractals, Vol. 7, No.

6, 1996, pp. 877-938.

[4] M. S. El Naschie, “The Theory of Cantorian Space-Time

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

[5] M. S. El Naschie, “A Review of E-Infinity and the Mass

Spectrum of High Energy Particle Physics,” Chaos, Soli-

tons & Fractals, Vol. 19, No. 1, 2004, pp. 209-236.

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

[6] J.-H. He, L. Marek-Crnjac, M. A. Helal, S. I. Nada and O.

E. Rössler, “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. Nottale, “Scale Relativity and Fractal Space-Time,”

Imperial College Press, London, 2011.

[8] L. Nottale, “Fractal Space-Time and Micro Physics,”

World Scientific, Singapore, 1993.

[9] J. Polchinski, “String Theory,” Cambridge University Press,

Cambridge, 1998.

[10] C. Rovelli, “Quantum Gravity,” Cambridge University

Press, Cambridge, 2004.

doi:10.1017/CBO9780511755804

[11] R. Penrose, “The Road to Reality—A Complete Guide to

the Laws of the Universe,” Jonathan Cape, London, 2004.

[12] J. Mageuijo and L. Smolin, “Lorentz Invariance with an

Invariant Energy Scale,” 2001.

http://arxiv.org/abs/hep-th/0112090

[13] J. Mageuijo, “Faster than the Speed of Light,” William

Heinemann, London, 2003.

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

quence of a Cantorian Micro Space-Time Geometry,”

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

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

[15] M. S. El Naschie, “On an Eleven Dimensional E-Infinity

Fractal Space-Time Theory,” International Journal of Non -

linear Sciences and Numerical Simulation, Vol. 7, No. 4,

2006, pp. 407-409.

[16] M .S. El Naschie, “The Discrete Charm of Certain Eleven

Dimensional Space-Time Theory,” International Journal

of Nonlinear Sciences and Numerical Simulation, Vol. 7,

No. 4, 2006, pp. 477-481.

[17] A. Connes, “Noncommutative Geometry,” Academic Press,

San Diego, 1994.

[18] M. S. El Naschie, “Transfinite Harmonization by Taking

the Dissonance out of the Quantum Field Symphony,”

Chaos, Solitons & Fractals, Vol. 36, No. 4, 2008, pp. 781-

786. doi:10.1016/j.chaos.2007.09.018

[19] S. T. Yau and S. Nadis, “The Shape of Inner Space,” Basic

Book, Persens Group, New York, 2010.

[20] J. Ambjorn, J. Jurkiewicz and R. Loll, “Lattice Quantum

Gravity—An Update,” 2011.

http://arxiv.org/abs/1105.5582