Self-similar and oscillating solutions of Einstein's equation and other relevant consequences of a stochastic self-similar and fractal Universe via El Naschie's ε(∞) Cantorian space–time

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Abstract

In this paper we make some suggestion regarding the unification of the fundamental forces and the age of the Universe in the context of the a stochastic self-similar and fractal Universe using El Naschie's ε(∞) Cantorian space–time. We also show how Einstein's equation can admit for the scale factor a(t) a self-similar solution in agreement with our stochastic self-similar, fractal Universe and El Naschie's ε(∞) Cantorian space–time. In addition, this solution is found to be oscillating one. Thanks to the first quantization it is possible to recast the equations in a Schrödinger-like form. Consequently, the presently observed large scale structure reflects the phenomenology of the Early Universe or of the microscopic world. Again it appears clear that the Universe and the structures inside must have a memory of its quantum origin as conjectured sometime ago.

Introduction

Observation shows that the Universe has a structure with scaling rules, where the clustering properties of cosmological objects reveals a form of hierarchy. In the previous paper, the author considers the compatibility of a stochastic self-similar, fractal Universe with the observation and the consequences of this model. In particular, it was demonstrated that the observed segregated Universe is the result of a fundamental self-similar law, which generalizes the Compton wavelength relation, R(N)=(h/Mc)Nφ, where R is the radius of the astrophysical structures, h is the Planck constant, M is the total Mass of the self-gravitating system, c the light speed, N the number of nucleons within the structures, and φ=5−12 is the Golden mean value [1]. This expression agree with the Golden mean and with the gross law of Fibonacci and Lucas [2], [3].

In [4] Iovane showed the relevant consequences of a stochastic self-similar and fractal Universe, such as the time dependence of the gravitational constant G, and how a G=G(t) in El Naschie's ε(∞) Cantorian space–time could imply an accelerated Universe. Starting from an universal scaling law, the author showed its agreement with the well–known Random Walk equation or Brownian motion relation that was used by Eddington [5], [6], [7]. Consequently, he arrived at a self-similar Universe. It appears that the Universe has a memory of its quantum origin as suggested by Penrose with respect to quasi-crystal [8]. Particularly, our model is related to Penrose tiling and thus to ε(∞) theory (Cantorian space–time theory) as proposed by El Naschie [9], [10] as well as with Connes Noncommutative Geometry [11]. In [1] the authors presented a descriptive model of segregated universe, then considered a dynamical model to explain the results and to give the evolution of the structures [12]. In [13] a waveguiding and mirroring effects are considered with respect to the large scale structure of the Universe.

In the present paper we find a general expression that links the energy at the macroscopic scale with the microscopic one. Moreover as consequence of this model of Universe we see that there exist a kind of resonant frequency. This frequency is peculiar with respect to the scale for each object in the Universe, and it is linked to its mass and dimension. Of course the point can be seen in a reversed way; in the sense that the mass or the dimension of an object is linked to its quantum nature in agreement with the Planck law. We also analyze to explain the oscillating behaviour of several observational quantities, using the above generalization of the Compton wavelength relation. In particular, the scale factor a(t) in the Friedman–Robertson–Walker metric can be understood in the context of a self-similar Universe and presents a fractal expression. In addition, we see how the age of the Universe can be connected to some fundamental quantum quantities and its density. In conclusion, we clearly see the unitary of the Universe and the universality of its law at all scales.

The paper is organized as follows: we discuss first the unification of all fundamental interactions in ε(∞) Cantorian Universe in Section 2; Section 3 presents the Fractality of the scale factor a(t) and its consequences; Section 4 is devoted to the oscillating scale factor a(t) and its consequences; finally conclusions are drawn in Section 5.

Section snippets

The unification of the fundamental interactions in ε(∞) Cantorian Universe

Following El Nashie [5], let us start from Planck's fundamental energy quantization relationEP=hν=hcλC,where E is the energy, h the Planck constant, ν the frequencies, c the speed of light and λC is the Compton wavelength. On the other hand, the Einstein mass–energy equation of special relativity isEE=mc2,where m is the mass. Consequently, we obtainλC=hmc,this is the well-known Compton wavelength relation [14].

Iovane in [1] has demonstrated that (3) is a special case of the self-similar lawR(N)=

Fractal scale factor a(t) and its consequences

Below the Planck scale, we find the quantum fluctuations of the space–time geometry. For this reason, the scale of the order 10−40 cm represents the quantum memory that we re-find in the present Universe. In fact, it is well-known that at the Planck scale, or equivalently after 10−43 s after the Big Bang, the Universe starts to emerge from the fluctuating quantum geometry in the way that we know and then producing the “building blocks” of nature, like quarks, leptons, fermions and bosons [17],

Oscillating scale factor a(t) and its consequences

In the usual approach to quantum gravity the Universe is assumed to be a quantum system [20], [21], [22], [23] or as a classical background with primordial quantum processes, like as in the context of quantum field theory on curved spacetime [24]. Here we have considered the Universe as a fractal spacetime background, where primordial quantum processes have given rise to the present segregated macroscopic structures.

Eq. (26) corresponding to the (0,0)-Einstein field equation can be formally

Conclusions

In this paper we have studied the effect of a stochastic self-similar and fractal Universe on some physical quantities and relations. Following the El Naschie's suggestions we have analyzed the unification of the fundamental interactions in ε(∞) Cantorian Universe. With respect to this aspect two interesting result are found.

  • The mass and the extension of a body are connected with its quantum properties, via the relation EE,N(N)=EPN1+φ, that links Planck's energy with that of Einstein. The

Acknowledgements

The author wish to thank E. Laserra for comments and discussions.

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