Chaos, Solitons & Fractals Vol. 5, No. I, PP. 1391-1392, 1995 Elsevier Science Ltd Printed in Great Britain 0960-0?79/95$9.50 + .oo On the Initial Singularity and the Banach-Tarski Theorem US. EL NASCHIJI DAMPT-Cambridge, U.K. In what follows, we would like to summarize in an informal way the main ideas and concepts underlying a forthcoming paper [l] in which we argue that the vacuum, and consequently matter, was created at the initial singularity via what we term the Banacl+Tarski mechanism. The vacuum, which we regard as an ‘insubstantial something’ in contradistinction to nothingness, which is ‘insubstantial nothing’, was discovered relatively recently to be an extremely rich and complex entity for a starting point of theoretical physics. Pair creation and annihilation together with a plethora of virtual and extremely short-lived elementary particles belong nowadays to the daily routine of theoretical and experimental physicists alike. We learned to accept all that as given facts yet some nagging questions always remained as to how it all began, and more fundamentally, if there was a beginning, how could it ever happen that ‘being’ could be created from ‘non-being’ or something out of nothing. A partial but an important answer to these questions was the initial singularity scenario of Gamow which was developed and refined by many scientists, in particular, Penrose and Hawking [2]. Nevertheless, it is still not clear at all within this theory or, in fact, any other theory, how the violation of the basic and most fundamental of all physical laws, namely the conservation of mass and energy, could be overcome to explain the inflation of the universe out of the initial singularity. The answer which we are proposing here is based upon an incredible theorem in transfinite set theory, which was proven in general form by two legendary Polish mathematicians, Banach and Tarski [3]. The theorem, which they proved around 1924, is a relatively speaking simple extension of an earlier theorem by Hausdorff. Loosely speaking, the Banach-Tarski theorem asserts that we could take one sphere, dissect it into several parts (a minium of 5) and then reassemble these parts into two new spheres with a total volume exceeding the original one by an arbitrary amount. Stated plainly, we are getting something out of nothing. There are no trivial tricks involved in this theorem. We are not creating holes or leaving gaps inside the spheres-it is a genuine mathematical theorem. Now with a theorem like this at the initial singularity, we can handle something like the inflation theory of the universe. However, at this point, one may rightly inquire why only at a singularity could we have the Banach-Tarski theorem at our disposal. The answer is that there are, of course, very sophisticated tricks associated with those incredible volume doubling processes which may be available at first only where many of the laws of geometry cease to exist in the form known to us. For the Banach-Tarski theorem to work, the geometrical environment must be so complex to the extent 1391 1392 MS. EL NASCHIE that simple notions such as volume cease to have any well-defined meaning. This process we call complexification. It is not difficult to imagine the incredible possibilities which the Banach-Tarski theorem can open on the level of sub-elementary particles and mysteries surrounding the creation of space-time itself and how it acquired dimensionality. The possible collapse into a black hole or the end of the universe in the ‘big crunch’ may also be interpreted as a Banach-Tarski mechanism in reverse [l]. In recent times many scientists have contemplated the meaning of vacuum, its structure and how it came into being. Of very special relevance in this respect are the early and more recent investigations of Finkelstein and it may be of interest to draw some parallels between our concept of Cantorian space-time [4] and his theory of quantum network dynamics [5]. He suggested, for instance, that what he called ‘balls of vacuum II’ are produced in experiments today as well as in the creation of the universe [5]. Noting that his theory is based on quantum sets, the connection to our transfinite Cantor space-time and Banach-Tarski mechanism becomes plausible. Furthermore, he distinguished between the ordered phase, vacuum I, and the disordered phase, vacuum II. In other words, vacuum II must be related to our fourdimensional random Cantorian space-time when interpreted using the Mauldin-Williams theorem of random sets 141. We could go on drawing further parallels between both theories, for instance, our fivedimensional Cantor sets and the five-dimensional Kaluztt-Klein theory of which the quantum network theory is a discrete version in some respect. Also, the connection between our DNAlike informational ether and the way information is carried in quantum network dynamics is another important parallel. Having emphasized the similarity, there are of course many aspects where the two theories differ, also on a formal ground. Our set theory is based on intersection of sets and not the disjoint union of them to mention but one point [4,5]. Nevertheless, the most important and novel aspect of what we are trying to do in [l] is not the developing of a consistent or a new quantum set theory, but is in showing that the completely esoteric theorem of Banach and Tarski could well have a real physical application with far-reaching consequences to the way we imagine the universe to have begun and how chaos entered into it for the first time. REZ’ERENCES 1. MS. El Naschie, Banach-Tamki theman and Cantorian space-time, Cti, SoIitons & Fmctak (in press, 1995). 2. S. Hawking, On the Big Bang atui Blackffoles. World Scientific, Singapore (1993). 3, S. Banach and A. Tarski, Fwrdanenta Mathmatica 6,244 (1924). 4. M.S. El Naschie, 0. R&sler and I. Prigogine, Quantum Mechanicr. Dt@sion and Chaotic FmctaLr. Elsevier, Oxford (1995). 5. D. Finkelstein, Theq of vacuum, in Zke Philosophy of Vbmun, edited by S. Saunders and H. Brown. Clamdon Press, Oxford (1991).
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