Scientific Research

An Academic Publisher

A Rindler-KAM Spacetime Geometry and Scaling the Planck Scale Solves Quantum Relativity and Explains Dark Energy ()

We introduce an ultra high
energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles
Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive
an effective quantum gravity energy-mass relation and
compute a dark energy density in complete agreement with all cosmological measurements,
specifically WMAP and type 1a supernova. In particular we find that ordinary
measurable energy density is given by *E*_{1}= *mc*^{2} /22 while the dark
energy density of the vacuum is given by *E*_{2} = *mc*^{2} (21/22). The sum of both energies is equal to Einstein’s
energy * E* =

*mc*^{2}. We conclude that

*=*

*E*

*mc*^{2}makes no distinction between ordinary energy and dark energy. More generally we conclude that the geometry and topology of quantum entanglement create our classical spacetime and glue it together and conversely quantum entanglement is the logical consequence of KAM theorem and zero measure topology of quantum spacetime. Furthermore we show via our version of a Rindler hyperbolic spacetime that Hawking negative vacuum energy, Unruh temperature and dark energy are different sides of the same medal.

Share and Cite:

*International Journal of Astronomy and Astrophysics*, Vol. 3 No. 4, 2013, pp. 483-493. doi: 10.4236/ijaa.2013.34056.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | D. J. Gross, “Can We Scale the Planck Scale?” Physics Today, Vol. 42, No. 6, 1989, pp. 9-11. http://dx.doi.org/10.1063/1.2811040 |

[2] | M. S. El Naschie and A. Helal, “Dark Energy Explained via the Hawking-Hartle Quantum Wave and the Topology of Cosmic Crystallography,” Int. J. Astron. & Astrophys, Vol. 3, No. 3, 2013, p. 318. |

[3] | M. S. El Naschie, “Quantum Entanglement as a Consequence of a Cantorian Micro Spacetime Geometry,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 50-53. http://dx.doi.org/ 10.4236/jqis.2011.12007 |

[4] | M. S. El Naschie, “A Resolution of Cosmic Dark Energy via a Quantum Entanglement Relativity Theory,” Journal of Quantum Information Science, Vol. 3, No. 1, 2013, pp. 23-26. http://dx.doi.org/ 10.4236/jqis.2013.31006 |

[5] | 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. http://dx.doi.org/10.1016/ S0960-0779(03)00278-9 |

[6] |
A. Elokaby, “Knot Wormholes and the Dimensional Invariant of Exceptional Lie Groups and Stein Space Hierarchies,” Chaos, Solitons & Fractals, Vol. 41, No. 4, 2009, pp. 1616-1618.
http://dx.doi.org/10.1016/j.chaos.2008.07.003 |

[7] | P. S. Addison, “Fractals and Chaos. An Illustrated Course,” IOP, Bristol, 1997. http://dx.doi.org/10. 1887/0750304006 |

[8] | F. Diacu and P. Holmes, “Celestial Encounters—The Origins of Chaos and Stability,” Princeton University Press, Princeton, 1996. |

[9] | M. S. El Naschie, “A Fractal Menger Sponge Space-Time Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy,” International Journal of Modern Nonlinear Theory and Application, Vol. 2, No. 2, 2013, pp. 107-121. http://dx.doi.org/10.4236/ijmnta.2013.22014 |

[10] | M. S. El Naschie, “Quantum Entanglement: Where Dark Energy and Negative Gravity plus Accelerated Expansion of the Universe Comes from,” Journal of Quantum Information Science, Vol. 3, No. 2, 2013, pp. 57-77. http://dx.doi.org/10.4236/jqis.2013.32011 |

[11] | A. Connes, “Noncommutative Geometry,” Academic Press, New York, 1994. |

[12] | J. Bellissard, “Chapter 12: Gap Labeling Theorems for Schrodinger Operators,” In: M. Waldschmidt, P. Monsa, J. Luck and C. Itzykon, Eds., From Number Theory to Physics, Springer, Berlin, 1992. |

[13] | G. Landi, “An Introduction to Noncommutative Space and Their Geometries,” Springer, Berlin, 1997. |

[14] | L. Marek-Crnjac, M. S. El Naschie and J.-H. He, “Chaotic Fractals at the Root of Relativistic Quantum Physics and Cosmology,” International Journal of Modern Nonlinear Theory and Application, Vol. 2, No. 1A, 2013, pp. 78-88. http://dx.doi.org/10.4236/ijmnta.2013.21A010 |

[15] | M. S. El Naschie and J.-H. He, “Fractal Hilbert Space as the Geometry of Quantum Mechanical Entanglement,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 1, 2012, p. 41. |

[16] | M. S. El Naschie, “Towards a General Transfinite Set Theory for Quantum Mechanics,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 2, 2012, pp. 135-142. |

[17] | M. S. El Naschie, J.-H. He, S. Nada, L. Marek-Crnjac and M. Atef Helal, “Golden Mean Computer for High Energy Physics,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 2, 2012, pp. 80-92. |

[18] | J.-H. He and M. S. El Naschie, “On the Monadic Nature of Quantum Gravity as a Highly Structured Golden Ring Spaces and Spectra,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 2, 2012, p. 94. |

[19] | M. S. El Naschie and J.-H. He, “The Fractal Geometry of Quantum Mechanics Revealed,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 1, No. 1, 2011, p. 3. |

[20] | M. S. El Naschie and S. Olsen, “When Zero Is Equal to One: A Set Theoretical Resolution of Quantum Paradoxes,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 1, No. 1, 2011, p. 11. |

[21] | M. S. El Naschie and O. E. Rossler, “The Zero Measure Fractal Ghost inside Quantum Mechanics,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 2, No. 1, 2012, p. 56. |

[22] | J.-H. He, T. Zhong, L. Xu, L. Marek-Crnjac, S. Nada and M. Atef Helal, “The Importance of the Empty Set and Noncommutative Geometry in Underpinning the Foundations of Quantum Physics,” Nonlinear Science Letters B, Vol. 1, No. 1, 2011, p. 15. |

[23] | M. S. El Naschie, “Quantum Collapse of Wave Interference Pattern in the Two-slit Experiment: A Set Theoretical Resolution,” Nonlinear Science Letters B, Vol. 2, No. 1, 2011, p. 1. |

[24] | M. S. El Naschie, L. Marek-Crnjac, J.-H. He and M. Atef Helal, “Computing the Missing Dark Energy of a Clopen Universe which Is Its Own Multiverse in Addition to Being Both Flat and Curved,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 3, No. 1, 2013, p. 3. |

[25] | T. Zhong and J.-H. He, “El Naschie’s Resolution of the Mystery of Missing Dark Energy of the Cosmos via Quantum Field Theory in Curved Spacetime,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 3, No. 1, 2013, p. 46. |

[26] | M. S. El Naschie, “Dark Energy from Kaluza-Klein Spacetime and Noether’s Theorem via Lagrangian Multiplier Method,” Journal of Modern Physics, Vol. 4, No. 6, 2013, pp. 757-760. http://dx.doi.org/ 10.4236/jmp.2013.46103 |

[27] | M. S. El Naschie, “Topological-Geometrical and Physical Interpretation of the Dark Energy of the Cosmos as a “Halo” Energy of the Schrodinger Quantum Wave,” Journal of Modern Physics, Vol. 4, No. 5, 2013, pp. 591-596. http://dx.doi.org/10.4236/jmp.2013.45084 |

[28] | M. S. El Naschie, “The Quantum Gravity Immirzi Parameter—A General Physical and Topological Interpretation,” Gravitation and Cosmology, Vol. 19, No. 3, 2013, pp. 151-155. http://dx.doi.org/ 10.1134/S0202289313030031 |

[29] |
S. Krantz and H. Parks, “Geometric Integration Theory,” Birkhauser, Boston, 2008.
http://dx.doi.org/10.1007/978-0-8176-4679-0 |

[30] | R. Penrose, “The Road to Reality,” Johnathan Cape, London, 2004. |

[31] | L. Hardy, “Nonlocality for Two Particles without Inequalities for Almost All Entangled States,” Physical Review Letters, Vol. 71, No. 11, 1993, p. 1665-1668. http://dx.doi.org/10.1103/PhysRevLett.71.1665 |

[32] | P. Kwiat and L. Hardy, “The Mystery of the Quantum Cakes,” American Journal of Physics, Vol. 68, No. 1, 2000, p. 33. http://dx.doi.org/10.1119/1.19369 |

[33] | N. D. Mermin, “Quantum Mysteries Refined,” American Journal of Physics, Vol. 62, No. 10, 1994, p. 880. http://dx.doi.org/10.1119/1.17733 |

[34] | E. J. Copeland, M. Sami and S. Tsujikawa, “Dynamics of Dark Energy,” arXiv: hep-th/ 0603057V3, 2006. http://arxiv.org/pdf/hep-th/0603057 |

[35] | S. Perlmutter, et al., “Supernova Cosmology Project Collaboration,” The Astrophysical Journal, Vol. 517, No. 2, 1999, p. 565. http://dx.doi.org/10.1086/307221 |

[36] |
M. S. El Naschie, “A Remark on the Cosmic Microwave Background Radiation and the Hausdorff Dimension of Spacetime,” Chaos, Solitons & Fractals, Vol. 10, No. 11, 1999, pp. 1807-1811.
http://dx.doi.org/10.1016/S0960-0779(99)00008-9 |

[37] | M. S. El Naschie, “Derivation of the Threshold and Absolute Temperature Tc = 273.16 K from the Topology of Quantum Space-Time,” Chaos, Solitons & Fractals, Vol. 14, No. 7, 2002, pp. 1117-1120. http://dx.doi.org/10.1016/S0960-0779(02)00053-X |

[38] | M. S. El Naschie, “Cobe Satellite Measurement, Hyperspheres, Superstrings and the Dimension of Spacetime,” Chaos, Solitons & Fractals, Vol. 9, No. 8, 1998, pp. 1445-1471. http://dx.doi.org/10. 1016/S0960-0779(98)00120-9 |

[39] | M. S. El Naschie, “Fuzzy Multi-Instanton Knots in the Fabric of Space-Time and Dirac’s Vacuum Fluctuation,” Chaos, Solitons & Fractals, Vol. 38, No. 5, 2008, pp. 1260-1268. http://dx.doi.org/10. 1016/j.chaos.2008.07.010 |

[40] | M. S. El Naschie, “Fractal Black Holes and Information,” Chaos, Solitons & Fractals, Vol. 29, No. 1, 2006, pp. 23-35. http://dx.doi.org/10.1016/j.chaos.2005.11.079 |

[41] | M. S. El Naschie, “Knots and Exceptional Lie Groups as Building Blocks of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 41, No. 4, 2009, pp. 1799-1803. http://dx.doi.org/10.1016/ j.chaos.2008.07.025 |

[42] | M. S. El Naschie, “Arguments for the Compactness and Multiple Connectivity of Our Cosmic Spacetime,” Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp. 2787-2789. http://dx.doi.org/10. 1016/j.chaos.2008.10.011 |

[43] | M. S. El Naschie, “Transfinite Neoimpressionistic Reality of Quantum Spacetime,” New Advances in Physics, Vol. 1, No. 2, 2007, p. 111. |

[44] | J. Mageuijo and L. Smolin, “Lorentz Invariance with an Invariant Energy Scale,” arXiv: hep-th/0112090V2, 2001. http://arxiv.org/pdf/hep-th/0112090 |

[45] |
T. Zhong, “From the Numerics of Dynamics to the Dynamics of Numerics and Visa Versa in High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 42, No. 3, 2009, pp. 1780-1783.
http://dx.doi.org/10.1016/j.chaos.2009.03.079 |

[46] | M. S. El Naschie, “Determining the Temperature of the Microwave Background Radiation from the Topology and Geometry of Spacetime,” Chaos, Solitons & Fractals, Vol. 14, No. 7, 2002, pp. 1121-1126. http://dx.doi.org/10.1016/S0960-0779(02)00172-8 |

[47] | M. S. El Naschie, “On the Vital Role Played by the Electron-Volt Units System in High Energy Physics and Mach’s Principle of ‘Denkokonomie’,” Chaos, Solitons & Fractals, Vol. 28, No. 5, 2006, pp. 1366-1371. http://dx.doi.org/10.1016/j.chaos.2005.11.001 |

[48] | M. S. El Naschie, “Holographic Dimensional Reduction: Center Manifold Theorem and E-Infinity,” Chaos, Solitons & Fractals, Vol. 29, No. 4, 2006, pp. 816-822. http://dx.doi.org/10.1016/j.chaos. 2006.01.013 |

[49] | M. S. El Naschie, “Modular Groups in Cantorian E(∞) High-Energy Physics,” Chaos, Solitons & Fractals, Vol. 16, No. 2, 2003, pp. 353-366. http://dx.doi.org/10.1016/ S0960-0779(02)00440-X |

[50] | M. S. El Naschie, “Quantum Loops, Wild Topology and Fat Cantor Sets in Transfinite High-Energy Physics,” Chaos, Solitons & Fractals, Vol. 13, No. 5, 2002, pp. 1167-1174. http://dx.doi.org/10. 1016/S0960-0779(01)00210-7 |

[51] | M. S. El Naschie, “Wild Topology, Hyperbolic Geometry and Fusion Algebra of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 13 No. 9, 2002, pp. 1935-1945. http://dx.doi.org/10.1016/ S0960-0779(01)00242-9 |

[52] | A. Pais, “Subtle Is the Lord: The Science and Life of Albert Einstein,” Oxford University Press, Oxford, 1982. |

[53] | J. H. He and M. S. El Naschie, “On the Shoulders of Giants,” Fractal Space-Time and Non-Commutative Geometry in High Energy Physics, Vol. 3, No. 1, 2013, p. 59. |

[54] | A. Gefter, “Mind-Bending Mathematics: Why Infinity Has to Go,” New Scientist, Vol. 219, No. 2930, 2013, pp. 32-35. http://dx.doi.org/10.1016/S0262-4079(13)62043-6 |

[55] | 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. http://dx.doi.org/10. 1016/j.chaos.2008.09.059 |

[56] | R. Elwes, “Ultimate Logic,” New Scientist, Vol. 211, No. 2823, 2011, pp. 30-33. http://dx.doi.org/ 10.1016/S0262-4079(11)61838-1 |

[57] | J. H. He and L. Marek-Crnjac, “Mohamed El Naschie’s Revision of Albert Einstein’s E = m0c2: A Definite Resolution of the Mystery of the Missing Dark Energy of the Cosmos,” International Journal of Modern Nonlinear Theory and Application, Vol. 2, No. 1, 2013, pp. 55-59. http://dx.doi.org/10. 4236/ijmnta.2013.21006 |

[58] | M. S. El Naschie, “Superstrings, Knots, and Noncommutative Geometry in E(∞) Space,” International Journal of Theoretical Physics, Vol. 37, No. 12, 1998, pp. 2935-2951. http://dx.doi.org/10.1023/ A:1026679628582 |

[59] | S. Chattopadhyay and A. Pasqua, “Various Aspects of Interacting Modified Holographic Ricci Dark Energy,” Indian Journal of Physics, Vol. 87, No. 10, 2013, pp. 1053-1057. |

[60] | S. Kalita, H. L. Duorah and K. Duorah, “Late Time Cosmic Acceleration of a Flat Matter Dominated Universe with Constant Vacuum Energy,” Indian Journal of Physics, Vol. 84, No. 6, 2013, p. 629. |

[61] | M. S. El Naschie, “The Missing Dark Energy of the Cosmos from Light Cone Topological Velocity and Scaling of the Planck Scale,” Open Journal of Microphysics, Vol. 3, No. 3, 2013, pp. 64-70. http://dx.doi.org/10.4236/ojm.2013.33012 |

[62] | E. Goldfain, “On a Possible Evidence for Cantorian Space-Time in Cosmic Ray Astrophysics,” Chaos, Solitons & Fractals, Vol. 20, No. 3, 2004, pp. 427-435. http://dx.doi.org/10.1016/j.chaos.2003.10.012 |

[63] | G. N. Ord, “Quantum Mechanics in a Two-Dimensional Spacetime: What Is a Wave Function?” Annals of Physics, Vol. 324, 209, 2004, p. 1211. |

[64] |
M. S. El Naschie, “‘t Hooft Ultimate Building Blocks and Space-Time as an Infinite Dimensional Set of Transfinite Discrete Points,” Chaos, Solitons & Fractals, Vol. 25, No. 3, 2005, pp. 521-524.
http://dx.doi.org/10.1016/j.chaos.2005.01.022 |

[65] |
A. Elokaby, “On the Deep Connection between Instantons and String States Encoder in Klein’s Modular Space,” Chaos, Solitons & Fractals, Vol. 42, No. 1, 2009, pp. 303-305. http://dx.doi.org/10.1016/j.chaos.2008.12.001 |

[66] | G. Iovane and S. I. Nada, “Strange Non-Dissipative and Non-Chaotic Attractors and Palmer’s Deterministic Quantum Mechanics,” Chaos, Solitons & Fractals, Vol. 42, No. 1, 2009, pp. 641-642. http://dx.doi.org/10.1016/j.chaos.2008.11.024 |

[67] | J. H. He, E. Goldfain, L. D. Sigalotti and A. Mejias, “Beyond the 2006 Physics Nobel Prize for COBE,” China Culture & Scientific Publishing, Shanghai, 2006. |

[68] | M. S. El Naschie, “The Hyperbolic Extension of Sigalotti-Hendi-Sharifzadeh’s Golden Triangle of Special Theory of Relativity and the Nature of Dark Energy,” Journal of Modern Physics, Vol. 4, No. 3, 2013, pp. 354-356. http://dx.doi.org/10.4236/jmp.2013.43049 |

[69] | D. R. Finkelstein, “Quantum Relativity,” Springer, Berlin, 1996. http://dx.doi.org/10.1007/978-3-642-60936-7 |

[70] | C. Rovelli, “Quantum Gravity,” Cambridge Press, Cambridge, 2004. http://dx.doi.org/10.1017/ CBO9780511755804 |

[71] | Y. Gnedin, A. Grib and V. Matepanenko, “Quantum Gravity,” Proceedings of the Third Alexander Friedmann Int. Seminar on Gravitation and Cosmology, Friedmann Lab. Pub., St. Petersberg, 1995. |

[72] | A. Vileukin and E. Shellard, “Cosmic Strings and Other Topological Defects,” Cambridge University Press, Cambridge, 2001. |

[73] | M. S. El Naschie, “On the Uncertainty of Cantorian Geometry and the Two-Slit Experiment,” Chaos, Solitons & Fractals, Vol. 9, No. 3, 1998, pp. 517-529. http://dx.doi.org/10.1016/S0960-0779(97) 00150-1 |

[74] | M. A. Helal, L. Marek-Crnjac and J.-H. He, “The Three Page Guide to the Most Important Results of M. S. El Naschie’s Research in E-Infinity Quantum Physics and Cosmology,” Open Journal of Microphysics, Vol. 3, No. 4, 2013. |

[75] | M. S. El Naschie, “A Note on Quantum Gravity and Cantorian Spacetime,” Chaos, Solitons & Fractals, Vol. 8, No. 1, 1997, pp. 131-133. http://dx.doi.org/10.1016/S0960-0779(96)00128-2 |

[76] | V. Adrian Parsegian, “Van der Waals Forces,” Cambridge University Press, Cambridge, 2006. |

[77] | L. Graham and J. Kantor, “Naming Infinity,” Harvard University Press, Cambridge, 2009. |

[78] | M. Persinger and C. Lavellee, “Theoretical and Experimental Evidence of Macroscopic Entanglement between Human Brain Activity and Photon Emission,” Journal of Consciousness Exploration & Research, Vol. 1, No. 7, 2010, pp. 785-807. |

[79] | S. Vrobel, “Fractal Time,” World Scientific, Singapore, 2011. |

[80] | G. Casati, I. Guarneri and U. Smilansky, “Caos Quantico,” Proceedings of the International School of Physics “Enrico Fermi”, Amsterdam, 1993. |

[81] | L. Nottale, “Scale Relativity and Fractal Spacetime,” Imperial College Press, London, 2011. |

[82] | L. Susskind and J. Lidesay, “Black Holes, Information and the String Theory Revolution,” World Scientific, Singapore, 2010. |

[83] | M. S. El Naschie, “Nash Embedding of Witten’s M-Theory and the Hawking-Hartle Quantum Wave of Dark Energy,” Journal of Modern Physics, Vol. 4, No. 10, 2013, pp. 1417-1428. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.