Fluid Turbulence Batchelor's Law Implies Spacetime Unification Of

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International Journal of Applied Science and Mathematics Volume 7, Issue 3, ISSN (Online): 2394-2894

Fluid Turbulence Batchelor’s Law Implies Spacetime Unification of Classical and Quantum Physics*

Mohamed S. El Naschie Distinguished professor, Dept. of Physics, Faculty of Science, University of Alexandria, Egypt.

Date of publication (dd/mm/yyyy): 27/06/2020 Abstract – Classical mechanics and quantum mechanics are looked upon within an identical framework via a new physical-mathematical interpretation of a fluid turbulence model based upon a golden spiral extension of Batchelor’s law and the associated E-infinity-like picture of self-similar eddies. The work amounts to a sweeping unification of mechanics in the spirit of E-infinity Cantorian spacetime and the golden mean number system. In particular it is concluded that quantum wave collapse may be looked upon as a phenomenon related to the Prandtl boundary layer of fluid mechanics.

Keywords – Batchelor’s Law, Self-Similar Eddies, Leonardo Da Vinci’s turbulence, Golden Logarithmic Spirals, E- Infinity, Random Cantor Sets, Unification of Classical and Quantum Mechanics, F. Hoyle, Golden Mean Number System, DAMTP Cambridge, Steady State Theory, Big Bang, Von Karman Vortex Street, Tacoma Narrow Bridge Disaster, Prandtl Boundary Layer, Homoclinic Soliton, Spatial Chaos, Buckling.

I. INTRODUCTION

Evidently one could not talk about mechanics and the motion of anything, let alone devise any mathematical description constituting a law for this motion [1-73] unless we possess a clear picture of the space in which this motion is taking place [1-30] as well as the meaning and measurement of the time duration which elapses to move from a point A to a point B in space or more generally spacetime [11-34]. Equally evident is the fact that we cannot speak about the geometry of space or spacetime unless we can define what we mean exactly when we speak of a point, a line, an area and a volume and the corresponding dimensions for hyper spaces and hyper volumes and so on particularly since we are moving from a point to another point [31-73]. It is well known that many deep thinkers regard a satisfactory definition of a point or the lack of such, as the real hurdle in the face of constructing a consistent spacetime theory as noted for instance by D. Bohm and B. Hiley in their renowned classic “The undivided universe” [33], [72] where they wrote: “The principle current difficulty in the attempt to make theoretical physics coherent is the concept of a mathematical point in spacetime without extension or duration”. In fact a point must be a mental abstraction due to the limitation on our physical biological inability to have a natural optical resolution of infinite intensity. In more mundane words, take any picture of say the entire house which the writer is sitting in right now writing his paper and let us reduce the size of the picture again and again using highly efficient scanning. It may take a short or a long time but sooner or later the picture of the house will appear as a little point before it almost disappears. Consequently at a quasi-infinite size reduction all shapes reduce to quasi points. However because everything is inside this point on a perfect „blow up” as in the classical film of Michelangelo autonioni [35], everything is back including the Author in the house. These kinds of thoughts may have been in the back of the mind of another great mind, namely John von Neumann when he invented “or discovered” the pointless geometry of his fractal-like continuous geometry [33], [36] which is the predecessor of our E-infinity Cantorian spacetime geometry [29-33]. The plausible question is thus the following: could a fractal random Cantor set [37-41] with its logical self *) this paper is dedicated to

Copyright © 2020 IJASM, All right reserved 85 International Journal of Applied Science and Mathematics Volume 7, Issue 3, ISSN (Online): 2394-2894 three great scientists who are unfortunately no longer with us, Prof. G. Batchelor, Prof. F. Hoyle and my friend and mentor Prof. Sir Herman Bondi. Referential nature [33] be our ideal point definition which eluded physics for a long time until is was discovered accidentally our purposefully by J. von Neumann and A. Connes [27] in connection with Sir R. Penrose fractal tiling universe [42-45]? The answer is of course yes but with a footmark, namely that it was discovered in a vague but profound way much earlier in the magnificent drawings of the greatest universal genius Leonardo da Vinci [46-47]. In the following sections we will attempt to make the preceding ideas more specific.

II. BACKGROUND INFORMATION

In the present work we will be de facto returning to Leonardo‟s drawing [46] and use it to “build” the main features of our spacetime where of course all “fractalized” shapes are equal in the previous sense and purpose. However, and paraphrasing the words of G. Orwell in his Animal Farm [48], we stress that although all shapes are equal, some shapes are more equal than others. Without much more ado let us announce that the shape we “rediscovered” must be ideally the famous logarithmic spiral [49-53] which should have been obvious from the very beginning because as we will show clearly here in the main body of this paper “assay” it contains by construction the main ingredients we need to apply our E-infinity golden mean number system [54-57], namely the golden mean scaling of its very geometry and self-similar, self-referential nature [33]. This fact is basic to fluid turbulence and not only transparent from Leonardo‟s drawing and Leonardo‟s familiarity with the golden mean as part of the artists tools of composition [46] but also and most importantly more than transparent from the remarkable George Batchelor‟s famous law [58].

III. G. BATCHELOR, F. HOYLE AND DAMTP, CAMBRIDGE

With the permission of the reader the author takes the liberty of mentioning some personal points which are none the less of general relevance to appreciating the importance of Batchelor‟s law of turbulence for the present work and otherwise [58-61]. The point is that this law was very recently given stringent mathematical proof that involved randomness [58] exactly as in the E-infinity Mauldin-Williams random Cantor sets [58]. Second, it was a pleasure and an honour for the author to have come in touch with G. Batchelor at Cambridge, DAMTP [62-64] of which he was the first head of department while at odds with another awesome scientific personality, namely F. Hoyle [63-64], one of the three pillars of the steady state theory of the eternal universe and who incidentally coined the name big bang theory with which he meant to be restrained facetious [64]. Scientifically speaking the two great men could not be more apart from their research backgrounds or was it? Batchelor worked in technical almost engineering problems connected to fluid mechanics and particularly turbulence [62] while Hoyle was obsessed with fundamental seemingly esoteric theories of cosmology and whether the universe had a beginning [64]. It therefore provides the Author an immense pleasure to show here that the research of both of these two great men of DAMTP needed each other and provided mutual validation without anyone realizing or could ever realized. Incidentally the first problem Sommerfeld gave his student Werner Heisenberg was to solve turbulence [26]. The problem has technical applications in ships, submarines, aircraft and rocket design that were of great importance at the time and particularly during the war [70-71], [73]. Heisenberg did as good a job as he could but failed to solve turbulence completely and went on to solve quantum mechanics with flying colours [12], [26]. The lesson that can be drawn here is that freedom to research is extremely important that each serious research effort should be respected and never played down or ridiculed j

-ust because it does not agree with ones own research, inclination or prejudice [12], [26].

IV. NOTE ON THE REFERENCES

A note on the large list of references of the present paper is probably in order at this point. The intention was to make the work self-contained despite the condensed amount of information. Therefore we opted for a middle of the way solution by keeping the length of the paper reasonably short and in turn added a large, but not very large number of references [1-75] to fill in the inevitable gaps.

V. THE GOLDEN SPIRAL AS THE FUNDAMENTAL BUILDING BLOCKS OF A POINTLESS, SELF-

SIMILAR AND SELF-REFERENTIAL CANTORIAN FRACTAL SPACETIME

In this section we proceed to meticulously construct the elementary fundamental building blocks of spacetime [11]. Let us construct a nested golden mean rectangular [53]. That means we consider a golden mean rectangular that contains another golden mean rectangular and so on like a Russian doll as shown in Fig. 9 of Ref. [53]. The way to do that is well known and will not be repeated here but what is not well known are two facts: first this is precisely the way to construct a logarithmic spiral as shown also in Fig. 9 of Ref. [53] and second, that this spiral gets wider by a golden mean factor [67-69].

1/휙 = (1 + 5)/2 (1)

= 휙 + 1

= 1.618033988

On the other hand even much less known than the preceding two facts is that the golden mean growth scaling factor is a consequence of the latent random Cantor set implicit in the very construction procedure generating the said logarithmic spiral [53]. As we mentioned earlier, randomness was crucial in the proof of Batchelor‟s law [58] but it is in our case basically related to a theorem by the two American mathematicians Mauldin and Williams stating that the Hausdorff dimension of a randomized, i.e. perturbed invariant set of a dynamical system with a certain probability distribution is with a probability equal one, the solution of the following equation [53]:

푎 푎 Ex (푡1 +…+푡푛 ) = 1 (2)

Note here that the left hand side is the expected value of the sum of the scaling ratios to the power of a. This rationale leads to the following unitary integral [53].

1 1 1 푥훼 + 1 − 푦 훼 푑푥 = 1 (3) 1−푥 푥 표

Where 훼 = 휙 is equal 5 – ½, i.e. the golden mean as explained in great detail in the original paper of Mauldin and Williams [53].

VI. DISCUSSION OF THE CENTRAL IDEA OF THE WORK

E-infinity Cantorian spacetime theory was so far associated with two fundamental pictures: a) the first picture is almost purely topological-geometrical [29-34] and b) the second is almost mechanical-physical [29-34]. However in both cases a) and b) it follows computationally the golden mean number system [54]. More

Copyright © 2020 IJASM, All right reserved 87 International Journal of Applied Science and Mathematics Volume 7, Issue 3, ISSN (Online): 2394-2894 accurately we recall that in the first mental picture a) we start by building a Cantorian spacetime that is made of infinite numbers of union and intersections of elementary random Cantor sets of Mauldin-Williams type and that way we arrive at an Einstein-like four dimensional spacetime that is in contrast to the original Einstein spacetime D = 4 is scale invariant and has D = 4 + 휙3 with the continued fractional representation [36-45].

1 D = 4 + 1 (4) 4+ 4+⋯

By contrast in the second mental picture b) the universe is envisaged as an infinite number of elementary golden mean oscillators joined in parallel and in succession using the Southwell and Dunkerley theorems on Eigenvalues [36-45], [54] and that way spanning the same spacetime given by D = 4 + 휙3.

The central idea and objective of the present work is to outline a third mental picture with a profound implication, namely the unification of classical and quantum mechanics via the most difficult problem, which is not really classical nor quantum but we may still call mechanics, namely fluid turbulence [55-62], [68-71]. It turned out that the golden eddies model of Prof. G. Batchelor is not only an exact solution to turbulence but in fact a unification scheme for the entire fundamental physics and constitutes a third mental picture c) of E- infinity Cantorian spacetime as a turbulence of a five dimensional empty set [65] representing the Aether which some wrongly thought to be disqualified and abandoned by Einstein‟s relativity [66]. This point will be made even clearer in section 7.

VII. SPACETIME AS A MULTI-FRACTAL EMPTY SET FLUID IN WHICH THE PRE-QUANTUM

ZERO SET PARTICLE IS SUBMERGED SURROUNDED BY THE EMPTY SET PRE-QUANTUM

WAVE AS A QUASI PRANDTL BOUNDARY LAYER [71], [73], [75]

In this section we will look very briefly to an aspect of our theory that we either underestimated in the past or overlooked for some reason that is, in all honesty, not clear to the Author at all. This aspect is the relevance of some very profound work on the engineering side of hydrodynamics that has never the less quite a substantial relevance to the understanding of fundamental physics as addressed and presented in our work in general and the present paper in particular [59-61]. We mean here the theoretical and experimental work of engineering scientists like Prandtl and his students von Karman and Schlechting to mention only a few outstanding names from many others [66-71]. In particular von Karman‟s vortex street [70], [73] is in a manner of speech a highway that can take us from the flutter dynamical collapse of the Tacoma bridge in USA [65], [76] to spacetime physics in a fractal Cantorian setting via golden mean-like spiralling air and water vortices [68-71], [73]. Building on the boundary layer theory of Prandtl and the vortices street of von Karman and relating all that to E-infinity Cantorian spacetime theory we could logically develop a new picture of our platonic quantum set theory. In this new picture the empty set, which was modelling the pre-quantum wave as the cobordism, i.e. the surface of the zero set pre-quantum particle will now be considered a Prandtl-like boundary layer for the pre- quantum particle. The rest of spacetime may be likened to mille-feuille of empty sets with increasing degrees of emptiness (in Russian it is called sloika cack). Thus except for the boundary layer, we have a multi-fractal spacetime “fluid” in which the particle moves surrounded by not really a wave but in fact a boundary layer. That way wave collapse and state vector reduction are misconceptions and we think that the time has come to put them, with all due respect, to rest as being out dated.

VIII. CONCLUSION

In the present relatively short paper we addressed a host of different important subjects of a general and philosophical, mathematical and specific technological nature. Science is essentially the absolute freedom to think freely. Thus in the limit only this scientific freedom bring humanity forward and all artificial division into theoretical, experimental, applied and fundamental science would and should disappear. In this sense and on a purely personal note, Prof. G. Batchelor and Prof. F. Hoyle are two sides of the same coin of integrity and excellent scientific enquiry. In the same spirit there is no classical nor quantum mechanics as it was shown here that both are special cases of fluid turbulence and essentially resonates Leonardo da Vinci‟s magnificent drawing which in turn unifies art and science in the most tangible and delightful way. Finally we may mention that invoking fluid mechanics in quantum particles physics may give those who are inclined to think in engineering moulds a both charming and instructive picture, namely that of a zero set pre-quantum particle [40- 45] moving like a material body in the fluid of a multi-fractal empty set while the pre-quantum wave [40-45] empty set is practically the Prandtl boundary layer [71], [73] of a zero set pre-quantum particle. In other words the pre-quantum wave empty set may be likened to a boundary layer empty set of the multi-fractal fluid, which we call Cantorian spacetime [68-71]. The implication of this conclusion opens new vistas not only in our philosophical understanding of the true meaning of quantum wave collapse but also in the technology and possibilities on non-demolition quantum measurement using a variant of boundary layer suction [74-75].

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AUTHOR’S PROFILE

Professor M.S. El Naschie was born in Cairo, Egypt on 10th October 1943. He received his elementary education in Egypt. He then moved to Germany where he received his college education and then his undergraduate education at the Technical University of Hannover where he earned his (Dipl-Ing) diploma, equivalent to a Master‟s degree in Engineering plus being a professional chartered engineer. After that he moved to the UK where he enlisted as a post graduate student in the stability research group of the late Lord Henry Chilver and obtained his Ph.D. degree in structural mechanics under the supervision of Professor J.M.T. Thompson, FRS. After his promotions up to the rank of full professor, he held various positions in the UK, Saudi Arabia and USA and was a visiting professor, senior scholar or adju- -nct professor in Surrey University, UK, Cornell, USA, Cambridge University, UK and Cairo University, Egypt. In 2012 he ran for the Presidency of Egypt but withdrew at the final stage and returned to academia and his beloved scientific research. He is presently a Distinguished professor at the Dept. of Physics, Faculty of Science of the University of Alexandria, Egypt. Professor El Naschie is well known for his research in structural stability in engineering as well as for his work on high energy physics and more recently for his work is cosmology and elucidation of the secret of dark energy and dark matter as well as for proposing a dark energy Casimir nanoreactor. He is the creator of E-infinity theory, which is a physical theory based on random Cantor sets and can be applied to micro, macro and mesoscopic systems. Professor El Naschie is the single or joint author of about one thousand publications in engineering, physics, mathematics, cosmology and political science. His current h-index is 81 and his i-10 index is 796 and total citations are 36206 according to Google Scholar Citation.