On a Quantum Gravity Fractal Spacetime Equation: QRG ≃ HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion


The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.

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El Naschie, M. (2016) On a Quantum Gravity Fractal Spacetime Equation: QRG ≃ HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion. Journal of Modern Physics, 7, 729-736. doi: 10.4236/jmp.2016.78069.

Received 18 April 2016; accepted 25 April 2016; published 28 April 2016

1. Introduction

Post modernistic research in theoretical physics [1] - [20] notably that connected to superstrings [21] - [25] , loop quantum gravity [26] [27] , fractal-Cantorian spacetime [28] - [30] , M-theory [24] [30] [31] and a host of other theories [1] [2] [17] - [19] is most if not all pointing towards a rather firm fact that higher dimensionality and fractal geometry can be used to simulate relativity as well as quantum mechanics and possibly replace them, at least partially and at a minimum in basic situations where relativity and quantum mechanics are both relevant in equal measure [1] - [4] [32] [33] . Looking a little back in history, this is actually the achievement of visionaries and a few towering figures of science which are not credited sufficiently with pointing to what in our opinion is the superior direction of geometrizing and “topologizing” physics and cosmology in a most general way [1] [2] [15] such as F. Gauss [3] , J. Bolya, N. Lobachevsky, H. Minkowski [3] , J. von Neumann and A. Connes [34] . All apart from these pioneers, the recent contributions to the theory of fractal spacetime and the E-infinity theory of G. Ord, L. Nottale and the present author are almost exclusively going in the same direction constituting the subject of the paper at hand [4] [6] [8] .

In the present work which is anticipating a sweeping new revolution in the way, we will be doing physics in 10 to 20 years, [1] - [54] , we will illustrate the above by giving in a relatively short and concentrated form various, and on its face value, classical derivations of the fundamental major problem of the supposedly “missing” dark energy density of the cosmos amounting to 95.5 to 96 percent of the total theoretically expected value [33] - [36] .

To keep the length of the present paper to a minimum we start from what we called in a recent paper [37] “The real Einstein” beauty E = kmc2 where k is essentially related to the familiar Lorentzian factor with a twist. Thus in the present paper we introduce as being the topological energy density factor of a generalization of Einstein’s famous equation [1] [38] [39]


2. A More than Noteworthy Hidden Connection between Riemann’s Powerful Curvature Tensor [42] - [44] and I. Dvoretzky’s Magnificent Theorem Regarding Measure Concentration [45] - [48]

We are invoking here nothing connected directly to the theory of relativity or quantum mechanics and yet we will arrive to a result which can be understood deeply only via these two pillars of modern physics [1] [3] [4] . The main idea behind the following analysis of dark energy is the intuitive picture that extra dimensions are where things can be there but not directly seen nor in fact measured by us 3-dimensional beings and with time being only a parameter not given to us in a tangible, physical way let alone the fifth dimension of Kaluza and Klein nor Witten’s eleven dimensional spacetime of his M-theory which is way above our non-mathematical intuitive grasp. Confining ourselves to n dimensional Riemannian tensor R(n) one could take the view with considerable justification, that R(5), i.e. five dimensions is about the limit of accessible physics in the experimental possibilities of a 3 + 1 dimensional conscious and well equipped observer. At the same time it is an educated guess that M-theory is real and R(11) is probably one of the best ways to describe not only theoretical high energy physics but the entire cosmos. To put this to a pragmatic test we calculate the vital independent components of the most important driving force in Einstein’s relativity, namely the Riemann tensor. For n dimensions this number is given by [42] - [44] [54]


Setting n = 5 and n = 11 one finds [54]




respectively. Assuming that all these components have almost the same statistical weight, then the difference between 50 and 2110 measures clearly the sparseness of the associated space and consequently the totality of the average curvature. Similarly the ratio between 50 and 2110 is a measure of the density of the energy which is likened in the theory of relativity mainly to the curvature as it is in the case a simple elastic wire in the theory of engineering elasticity. Viewing the complex problem in this quite simplistic way leads us directly to estimating, i.e. the density of the probably accessible energy which we can normally measure directly and call it aptly and logically, ordinary energy as follows [42] - [44] [54] :


Consequently the rest of the energy filling not only D = 5 but also D = 11 must be given by the so called dark energy density which is given logically by [35] - [39]


For two important reasons the preceding rough result is truly striking. First it is quite close, in fact very close to highly accurate cosmic measurements and observations connected to the famous COBE, WMAP and Type 1a Supernova [33] [39] which was awarded the Nobel Prize in Physics or Cosmology in 2011. These measurements establish the existence of 4.5% ordinary energy density while the expected but missing 95.5% energy density was dubbed dark energy and concluded that it is behind the accelerated rather than previously believed deceleration of cosmic expansion [33] [39] . The second reason for the profundity of our simple estimation is connected to the magnificent theorem of the late, great Ukrainian born legendary mathematician and past President of the Wiseman Institute, I. Dvoretzky [45] - [48] . This theorem states that in a high dimensional manifold almost 96 percent of the volume is at the surface leaving only 4 percent in the bulk. The analogy and connection is obvious. Now we have solved this problem in numerous previous publications and came to a definite answer, namely that E of Einstein may be dissected into two quantum components E(O) of the quantum particle and E(D) of the quantum wave. Within an exact integer solution one finds that [35] - [39]


That means E(O) is not 4% but rather 4.5% while E(D) is not 96% but 95.5% to a very high degree of accuracy and in astounding agreement with measurements [33] [30] . This leads us to the next section where we will attempt to improve our first estimation presented at the beginning of this section.

3. An Almost Exact Integer Solution of the Ordinary and Dark Energy Density Problem Based on the Number of Independent Riemannian Curvature

Our first estimate of clearly missed the fact that we needed to add D = 5 to R(5) = 50 and analogously D = 11 to R(11) = 1210. This is an obvious and trivial embedding problem because R(5) and R(11) are treated as quasi-dimensions estimating the size of our spacetime manifold. Consequently a more accuret must be [35] - [39] [45] - [48]


This is almost the familiar exact rational (1/22) value as document in my previous papers using different methods. Clearly is given by the self explanatory values


almost exactly as expected.

4. The Exact Integer Value of Ordinary and Dark Energy Density

Although not trivial, it is not difficult to obtain the truly exact formulas of energy density and using the Riemannian tensor independent components method. The trick is to realize that we have to add not D(5) = 5 to R(5) but the full space which embeds the SL(2, 7) Lie symmetry group of the holographic boundary of our universe, i.e. D = 7 and similarly the vacuum of Witten’s D = 11 which is the corresponding pure gravity with. For n = 8 we have the famous D = 20 case but in the case of eleven dimensions we set n = 11 and find a 44 degrees of freedom vacuum. Consequently our exact must be given by [35] - [39] [45] - [48] .


In other words we have [35] - [38] [52]


which is this time the truly exact integer value. For we have naturally also the exact integer value


We may remark on passing that 57 is actually the intrinsic dimension of the fundamental E(8) exceptional group [52] with its famous 248 isometries which gives E8E8 of D. Gross et al. heterotic superstring symmetry group, the famous |E8E8| = (2) (248) = 496 dimensions [50] - [52] .

5. A Short Introduction to the Von Neumann-Connes Theory and Comparison with the Exact “Transfinite” Solution of Ordinary Dark Energy Density

Readers familiar with E-infinity Cantor spacetime theory [4] [8] know that the key to all exact results of this theory, including the ordinary and the dark energy density section of the cosmos, is the deceptively simple dimensional function due to the work of J. von Neumann in his unsung papers and book published posthumously “Continuous Geometry” as well as the work of the creator of noncommutative geometry, the great French pure and applied mathematician A. Connes [34] . This function as is well known is given by [34]


where is the Hausdorff dimension, and. In E-infinity we made our first move by identifying the zero set as being necessarily be the empty set which we equate physically and mathematically to the pre-quantum wave. Therefore the Hausdorff dimension of the empty set is given by and can be found together with all other sets recursively in a Fibonacci-like way as shown below where we start with the two trivial seeds, namely the zero set and the unity set which are indicated in Equation (14) by two small arrows. Proceeding in this way we find [34] [39]


From the above we see that the pre-quantum particle is identified by the bi-dimension [47]


while the pre-quantum wave is given by


From all these previous results, it is easily reasoned that the “topological” volume of the pre-particle in 5D is given by the obvious multiplication formula [39]


The corresponding additive volume of the pre-quantum wave on the other hand is given by


The total volume is thus equivalent to world sheet of string theory [21] - [25]


Inserting mean volume = 2/2 = 1 in Einstein’s formula E = mc 2 one finds [35] - [39]


That means [35] - [39] [48] - [52]


This is the exact expression which leads to the integer solution by disregarding k = 0.18033989 compared to 21 and 22. We note that this k was interpreted physically as ‘tHooft’s renormalon hypothetical particle which is equal to where is hardy’s famous quantum probability of two entangled quantum particles which is the exact solution of the corresponding Dirac’s equations of the problem. In addition we must stress that was verified experimentally and found to a very high degree of precision. Thus our theory is well founded theoretically and experimentally on all fundamental levels. The more interesting it must be that one can find the exact integer solution without direct reference to the theory of quantum relativity. In the next section we will look at the same problem using nothing more than our ordinary three dimensional space coupled to a rather familiar fractal curve [53] .

6. Solving Dark Energy in a Classical Newtonian Three Space Dimensions

Although we have not invoked in all the preceding analysis any Lorentzian transformations or Einsteinian conception relating to the meaning of simultaneousity [40] nor of course a Schrödinger equations [1] - [9] , we did make extensive use of higher dimensionality of spacetime as well as the basic final results of Einstein’s special relativity marvel, namely E = mc2. In the following solution all these things will be dispensed of so that the reader may see clearly the main message of the present paper that even a low dimensional fractal is essentially infinite quasi dimensional because of the involved infinite iteration and self similarity and that even a harmless conventional fractal curve in 3 dimensions like the familiar Menger sponge [53] could simulate quantum effects involved in the physics geometry and topology of ordinary and dark energy density of the cosmos [47] - [57] . Thus we draw in our following analysis on the classical three dimensional sponge named after the great Austria- American mathematician Karl Menger [53] who together with the outstanding young Russian mathematician P. Urysohn discovered the inductive dimensional theory which is one of our main tools in erecting E-infinity theory [28] [50] . The Hausdorff dimension in this case is [53]


This fractal, although it looks like a cubic sponge in 3D is essentially a curve, not a real 3D and possesses in our case the disadvantage of being continuous and could therefore be expected to deliver a good approximation only because continuity violates one of our main E-infinity theory principles, namely being the “pointless” point-set theory as emphasized in the pioneering work of von Neumann’s continuous geometry where continuity is not referring to the geometry [34] [39] [47] but to the spectrum of the most important topological invariant of a manifold, namely the dimension. Thus one should not be misled by the word “continuity” with which von Neumann means fractal dimensionality spectrum but at the time, the word fractal was not invented yet by Mandelbrot nor were fractals part of the mathematical science culture [3] [46] . Since the Hausdorff dimension refers here not to the dark non-differentiable lines remaining from the Menger sponge iteration but to the space encased between these 3 dimensional lines, we see that the ratio between DH = 2.726833028 and DT = 3 will give us the density of the involved empty set which represents dark energy. Consequently we may write


Consequently the ordinary energy density must be


Now, and this is a crucial point, we do not insert in Einstein’s formula which we used all along because Einstein’s formula belongs to D = 4 while the previous analysis is in D = 3. Here we remain truly classical and insert in of Newton and take the limit to find


For a basically almost entirely classical analysis this result reinforces our conjectured equation:


and together with the rest of the paper shows beyond reasonable doubt that there is far more than meets the eye to dimensionality of spacetime and fractal Cantorian geometry.

7. Discussion and Conclusion

There are many shortcut derivations and radically different solutions all converging directly or indirectly towards the main thesis of the present work, namely that of measure concentration of volume in a sufficiently high dimensional manifold with fractal-Cantorian features. Thus, we have all three fractal spacetime theories of ‘tHooft-Veltman-Wilson dimensional regularization spacetime D = 4 − k as well as Kaluza-Klein fractal space and Witten’s fractal M-theory [55] with the remarkable dimensionality all leading to exactly the same result, namely [35] - [39]


where. The “integer” exact value is


The analysis is in complete conformity with the result based on pure mathematical theorems such as Dvoretzky’s theorem as well as accurate measurements and observations such as COBE, WMAP and Type 1a supernova which is awarded the 2011 Nobel Prize [35] - [39] .

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. Tegmark: Our Mathematical Universe. Allen Lane, Penguin Books, London, 2014.
[2] J. Changeux and A. Connes: Conversation on Mind, Matter and Mathematics. Princeton University Press, Princeton, New Jersey, USA, 1989.
[3] R. Penrose: The Road to Reality. J. Cape, London, UK, 2004.
[4] J.-H. He, et al.: Transfinite Physics: A Collection of Publications on E-Infinity Cantorian Spacetime Theory. China Science & Culture Publishing, 2005.
[5] J. Mageuijo: Faster than the Speed of Light. Heinemann, London, UK, 2003.
[6] L. Nottale: Fractal Space-Time and Micro Physics. World Scientific, Singapore, 1993.
[7] S. Vrobel: Fractal Time. World Scientific, Singapore, 2011.
[8] P. Weibel, G. Ord and O. Rössler (Editors): Spacetime Physics and Fractality. Springer, Vienna-New York, 2005.
[9] L. Smolin: The Trouble with Physics. Allen Lane-Penguin, London, 2006.
[10] HRH The Prince of Wales, Tony Juniper and Ian Skelly: Harmony: A New Way of Looking at Our World. Harper Collins Publications, London, UK, 2010. (see in particular page 113).
[11] P. Woit: Not Even Wrong. Jonathan Cape, London, 2006.
[12] L.M. Wapner: The Pea and the Sun. A.K. Peters Ltd., Wellesley, MA, USA, 2005.
[13] H. Diebner, T. Druckrey and P. Weibel (Editors): Science of the Interface. GenistaVerlag, Tübingen, Germany, 2001.
[14] A. Connes, M. Heller, S. Majid, R. Penrose, J. Polkinghorne and A. Taylor: On Space and Time. Cambridge University Press, Cambridge, 2009.
[15] A. Connes, A. Lichnerowicz and M.P. Schutzeuberger: Triangle of Thoughts. American Mathematical Society, Rhode Island, USA, 2001.
[16] P. Cartier, B. Julia, P. Moussa and P. Vanhove (Editors): Frontiers in Number Theory, Physics and Geometry Vol. I. Springer, Berlin, 2006.
[17] L. Smolin: Three Roads to Quantum Gravity. Weindenfeld & Nicolson, London, UK, 2000.
[18] A.B. Balachandran, S. Kürkcüoglu and S. Vaidya: Lectures on Fuzzy and Fuzzy Susy Physics. World Scientific, Singapore, 2007.
[19] M.S. El Naschie, O.E. Rössler and I. Prigogine (Editors): Quantum Mechanics, Diffusion and Chaotic Fractals. Pergamon Press/Elsevier, Oxford, 1995, (see in particular pp. 191-205).
[20] S. Vongehr: Post Modern Physics Description of Relativity. 24 September 2012.
[21] M. Green, J. Schwarz and E. Witten: Superstring Theory. Vol. II. Cambridge University Press, Cambridge, 1987.
[22] J. Polchinski: String Theory, Vol. II. Cambridge University Press, Cambridge, 1998.
[23] K. Becker, M. Becker and J.H. Schwarz: String Theory and M-Theory. Cambridge University Press, Cambridge, 2007.
[24] M. Kaku: Introduction to Super-strings and M-Theory. Springer, New York, USA, 1999.
[25] P. West: Introduction to Strings and Branes. Cambridge University Press, Cambridge, UK, 2012.
[26] R. Gambini and J. Pulling: Loop Quantum Gravity. Oxford University Press, Oxford, 2011.
[27] C. Rovelli: Quantum Gravity. Cambridge Press, Cambridge, 2004.
[28] M.S. El Naschie: A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics. Chaos, Solitons & Fractals, 19(1), 2004, pp. 209-236.
[29] M.S. El Naschie: On the Uncertainty of Cantorian Geometry and the Two-Slit Experiment. Chaos, Solitons & Fractals, 19(3), 1998, pp. 517-529.
[30] M.S. El Naschie: On the Unification of Heterotic Strings, M Theory and E (∞) Theory. Chaos, Solitons & Fractals, 11(14), 2000, pp. 2397-2408.
[31] M. Duff (Editor): The World in Eleven Dimensions. IOP Publishing, Bristol, 1999.
[32] G.W. Gibbons and S.W. Hawking: Euclidean Quantum Gravity. World Scientific, Singapore, 1993.
[33] L. Amendola and S. Tsujikawa: Dark Energy: Theory and Observation. Cambridge University Press, Cambridge, UK, 2010.
[34] A. Connes: Noncommutative Geometry. Academic Press, San Diego, USA, 1994.
[35] M.S. El Naschie: From E = mc2 to E = mc2/22—A Short Account of the Most Famous Equation in Physics and Its Hidden Quantum Entangled Origin. Journal of Quantum Information Science, 4, 2014, pp. 284-291.
[36] A. Connes and M. Marcolli: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, Rhode Island, USA, 2008.
[37] A.J. Babchin and M.S. El Naschie: On the Real Einstein Beauty E = kmc2. World Journal of Condensed Matter Physics, 6(1), 2016, pp. 1-6.
[38] M.S. El Naschie: Einstein-Rosen Bridge (ER), Einstein-Podolski-Rosen Experiment (EPR) and Zero Measure Rindler KAM Cantorian Spacetime Geometry (ZMG) Are Conceptually Equivalent. Journal of Quantum Information Science, 6, 2016, pp. 1-9.
[39] L. Marek-Crnjac: On El Naschie’s Fractal-Cantorian Space-Time and Dark Energy—A Tutorial Review. Natural Science, 7(13), 2015, pp. 581-598.
[40] W. Rindler: Relativity (Special, General and Cosmological). Oxford University Press, Oxford, 2006.
[41] L.B. Okun: Energy and Mass in Relativity Theory. World Scientific, Singapore, 2009.
[42] G.S. Hall: Symmetries and Curvature Structure in General Relativity. World Scientific, Singapore, 2004.
[43] J.L. Martin: General Relativity. Prentice Hall, London, 1995.
[44] S. Sternberg: Curvature in Mathematics and Physics. Dover Publications, New York, USA, 2012.
[45] M.S. El Naschie: Application of Dvoretzky’s Theorem of Measure Concentration in Physics and Cosmology. Open Journal of Microphysics, 5, 2015, pp. 11-15.
[46] M.S. El Naschie: The Counterintuitive Increase of Information Due to Extra Spacetime Dimensions of a Black Hole and Dvoretzky’s Theorem. Natural Science, 7(10), 2015, pp. 483-487.
[47] M.S. El Naschie: Mathematical Models and Methods in Dark Energy Theory: Dvoretzky’s Theorem, Casimir Effect, Mobius Geometry. Problems of Nonlinear Analysis in Engineering Systems, 2(44), 2015, pp. 1-16. University of Kazan Press, Russia. (Published in Russian and English language versions).
[48] M.S. El Naschie: Banach Spacetime-Like Dvoretzky Volume Concentration as Cosmic Holographic Dark Energy. International Journal of High Energy Physics, 2(1), 2015, pp. 13-21.
[49] M.S. El Naschie: If Quantum Wave of the Universe then Quantum Particle of the Universe: A Resolution of the Dark Energy Question and the Black Hole Information Paradox. International Journal of Astronomy and Astrophysics, 5(4), 2015, pp. 249-260.
[50] M.S. El Naschie: The theory of Cantorian Spacetime and High Energy Particle Physics (An Informal Review). Chaos, Solitons & Fractals, 41(5), 2009, pp. 2635-2646.
[51] M.S. El Naschie: The Exceptional Lie Symmetry Groups Hierarchy and the Expected Number of Higgs Bosons. Chaos, Solitons & Fractals, 35(2), 2008, pp. 268-273.
[52] M.S. El Naschie: Exceptional Lie Groups Hierarchy and the Structure of the Micro Universe. International Journal of Nonlinear Sciences and Numerical Simulation, 8(3), 2007, pp. 445-450.
[53] M.S. El Naschie. A Fractal Menger Sponge Spacetime Proposal to Reconcile Measurements and Theoretical Predictions of Cosmic Dark Energy. International Journal of Modern Nonlinear Theory & Application, 2, 2013, pp. 107-121.
[54] M.S. El Naschie: Cosmic Dark Energy Density from Classical Mechanics and Seemingly Redundant Riemannian Finitely Many Tensor Components of Einstein’s General Relativity. World Journal of Mechanics, 4(6), 2014, pp. 153-156.
[55] M.S. El Naschie: On a fractal version of Witten’s M-Theory. Journal of Astronomy & Astrophysics, 6(2), 2016, pp. 135-144.
[56] M.S. El Naschie: From Witten’s 462 Supercharges of 5-D Branes in Eleven Dimensions to the 95.5 Percent Cosmic Dark Energy Density behind the Accelerated Expansion of the Universe. Journal of Quantum Information Science, 6(2), 2016, pp. 57-61.
[57] M.S. El Naschie: Negative Norms in Quantized Strings as Dark Energy Den-sity of the Cosmos. World Journal of Condensed Matter Physics, 6, 2016, pp. 63-67.

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