D DAVID PUBLISHING. Bits, Mass and Acceleration Issues in BSC. 1. Introduction. Giuseppe Fazio, Mauro Giaconi, Angela D Arienzo and Davide Quatrini

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1 Journal of Physical Science and Application 4 (8) (2014) doi: / / D DAVID PUBLISHING Giuseppe Fazio, Mauro Giaconi, Angela D Arienzo and Davide Quatrini Electronic Engineering Department, University of Rome, Tor Vergata Received: June 25, 2014 / Accepted: August 05, 2014 / Published: August 15, Abstract: The goal of the present paper is to expand already published works in the frame of Banded speed cosmology (BSC). In particular this paper gives validated values for physical quantities not so far investigated in previous publications, i.e., the number of individual physical entity in the universe, as well as the maximum value for acceleration. Validates values mean identical quantities from a numerical point of view obtained with different theoretical procedures, additionally compared with data based on NASA observations with Planck probe. Key words: Banded speed cosmology, minimum and maximum value for the mass, number of mass unities, maximum value for acceleration, Bekenstein. 1. Introduction The main goal of this paper is to estimate the number of the smallest physical units in our universe, without having, at the moment, any particular definition for these units, deferring a possible physical interpretation for them. The proposed method is based on the following assumptions: Nowadays we consider classical physical theory as a particular case of quantum theory. The two theories are connected by the following principle quantum theory must agree with classic physical one in the same way that the classical theory must agree with experiments [1]; Planck expresses radiation law conjecturing atoms behaviour similar to oscillators, each of them with a characteristic oscillation frequency. In many physical fields it is very useful to apply mathematical relations which describe the behaviour of known phenomena (electrical, mechanical, thermal ) to unknown ones. Starting from the above assumptions, we suppose that matter behave like harmonic oscillators, that all cinematic and dynamic quantities be quantized, limited Corresponding author: Giuseppe Fazio, professor, research fields: electrical measurements, railway signaling and theoretical physics. fazio@ing.uniroma2.it. and measurable and all said above be in tune with both mechanical model here presented and the theory of Banded speed cosmology, whose hypotheses, already explained in previous papers [2-5], are consistent with Bekenstein studies; in this respect, in the paper the maximum number of visible units in the universe is estimated; to this aim, the value of minimal mass in the universe is derived, in tune with the same result already estimated and juridically protected [6]. We point out that, though by means of two different models, with proper hypotheses, it is possible to obtain the same result and that this result is compatible with the data of NASA Planck probe. Moreover, starting from mechanical model, it is possible to estimate the maximum value for the acceleration in the universe, in tune with Capozziello result [7]. More specifically, in Subsection 1.1 the reference harmonic oscillator is presented, that is the basis for subsequent discussions; in Subsection 2.3 an estimate on minimum and maximum value for the mass in the universe is evaluated; in Subsection 2.1 the greatest number of mass units in the universe is estimated, a value which is confirmed applying Bekenstein theory, which is discussed in Subsection 2.2; in Subsection 2.3

2 532 the value of maximum acceleration is evaluated, using both oscillator model and a more easy computation. 1.1 Harmonic Motion Hypothesis We can then consider at a first analysis a harmonic motion of a mass with time equations on any couple of Cartesian axes equal in amplitude and with 90 phase difference, so we can assume a simple circular motion for the mass. We have then the following relationship u = ωr, K = Iω 2 /2, a = ω 2 R with u, ω, R, K, I, a respectively tangential speed, angular frequency, radius, kinetic energy, inertia momentum and centripetal acceleration. Considering the previous reference system, let us consider a material point moving along a circumference with radius R, assuming that the values of time and space be multiple of the related e Planck Physical quantities (t p, l p ) and the greatest possible speed be light speed c = l p /t p. Let we put R = Ll p /2, t = Tt p and u = Vc being L, T, 1/V positive integers (this choice will be justified hereinafter). Then: ω = u/r = 2Vc/Ll p = 2Vl p /Ll p t p = 2V/Lt p It is possible to obtain an angular frequency ω = 2V/Lt p for which in one Planck time t 0 = t p the material point travels the length S along the circumference equal to S = R θ = Rω t = utp = Vl p. Then the maximum move in one Planck time is obtained when V is maximum, and in this situation, owing to our hypotheses, V must be one, then the maximum move in one Planck time must be l p and the related speed is c = l p /t p. Moreover, considering a mass M times Planck mass m p = ħ/cl p then I = Mm p R 2 equal to the momentum of inertia of a mass turning around an axis placed at the rotation centre. 1.2 Quantization Hypothesis Considering the wave particle duality, the obtained results make no sense because for uncertainty principle a wave packet cannot occupy a definite space position unless the energy of that packet is infinite. Then considering a fundamental wavelength λ 0 = l p and a wave packet with length x = L x 0 = 2R the minimum uncertainty about the position of a particle must be equal to the elementary Planck quantum x 0 = l p With the same reasoning about time we can obtain t 0 = t p. We can easily infer that also angular frequency, momentum of inertia and energy are differential quantities. With this reasoning we can justify the choice of considering space, time and speed as functions with constant coefficients lp, tp, c and discrete variables L,T, (1/V), positive integers. 1.3 Hypothesis on Minimum and Maximum Value for the Mass in the Universe From the previous relationships the change in kinetic energy is: K = Iω 2 /2 = Mm p R 2 ω 2 /2 = Mm p u 2 /2 = Mm p V 2 c 2 /2 = MV 2 ħc/2l p = MV 2 ħ/2t p On the basis of Heisenberg uncertainty principle and putting t = Tt p K t = MV 2 Tħ/2 ħ/2 if and only if MV 2 T 1 From which V 2 1/MT and because V > 0 the solution is V 1/ MT We suppose that V is a bounded set with upper and lower limit V min and V max respectively, we have found V max = 1 and considering V min = 10-61, a value obtained in previous publication [2] equal to Planck length divided by universe s age. We have obtained: 1. then, 1/, MT 1/V 2 min MT 1/V 2 max, therefore, MT (10 61 ) 2 MT 1 Supposing the maximum uncertainty on the time, we can set the universe s age as the extreme value for T [2]. So we can obtain: Ua = t p from which T max = or M Then it is possible get the maximum and minimum values of the mass of the universe:

3 533 m Max = m p ; m min =10-61 m p What is the mass we have just found? The measures performed by Planck probe [8] show that only 4.9% of the universe is made by atoms and the remaining 95.1% is made by dark energy and dark matter, 26.8% and 68.3%, respectively. If we suppose that the found value m Max can be assimilated to the total amount of matter/energy of the universe, then 4.9% of this one could be the visible matter. = m p = ; m p since m p = kg then kg which is a value already found in a previous paper [5] applying the Heisenberg uncertainty principle and verified by NASA results. In regard to the value of minimum value of the mass m min = m p, since mp = kg this entity could have, as the universe, only a small percentage of visible mass, then probably this entity is a mass/energy type that is a entity made by visible mass and other unknown forms of energy. Referring to the results published by Journal of Physical Science and Application the smallest visible mass is m vmin = kg [6] which is equal to the half of the value here found. From the calculation of associated energy K = Iω 2 /2 in the case of V = 1 and M = 10-61, we obtain: having found the well known Einstein equation in the case of a mass placed in its balance point with uncertainty equal to l p /2 and with zero travel speed, for which it is possible the connection with a Lorentz factor equal to 1, this can confirm the hypothesis of the existence of a smallest inseparable mass/energy, hypothesis already adopted by Einstein in its study of rest mass [9]. 2 Results and Discussion 2.1 Counting the Number of Mass Unities Assuming = m p /2 as smallest visible mass unit, the definition of the mass as a quantity constrained by two values multiple of Planck allows us to estimate the greatest number of mass units in the universe I / 10, a value which is easily confirmed applying Bekenstein theory evaluating the greatest number of information in the universe. Considering the results of our model for the evaluation for the number of single units of mass/energy we can obtain I = m Max /m min = m p /10-61 m p = and comparing the results wit previous one it is possible to argue that 1/10 of the total information is given by visible matter. Those Units can be identified as the number of bits required for the description of a space. time with physical properties that can be assimilated to the ones of observable universe. Such assimilation gets further strength from the considerations made in the following paragraph. 2.2 Evaluation of the Number of Units, Bekenstein Method The Bekenstein bound can represent the missing link between Banded Speed Cosmology and the world of mass which was out of the numerical results obtained in the last years. This is true because of the following reason: the quantity of information of a sphere which diameter is the Hubble diameter and the mass is that obtained through Sir Hoyle method (i.e., through the application of Friedmann equations) has the same value both if calculated by the means of the Bekenstein bound and by the Banded Speed Cosmology approach. In fact: I = (2RE)/(ħ c ln2) = mu ru where, mu is the said total mass-energy of the whole Universe (about kg) and ru is half Hubble diameter (about m). This number (dimensionless) is If we try to calculate the Information of a sphere large and heavy as our Universe is through Banded Speed Cosmology we can say that: The mass of such a sphere is about if calculated by the means of Heisenberg s Uncertainty Principle

4 534 (HUP), and the number is the same of the one obtained by Hoyle-Friedmann, as we shown by our research group during the 2011 Fundamentals of Physics symposium in Udine. Besides the minimum acceptable mass in our model, calculated using HUP as usual, is 10 (-69) kg as shown in the papers published by Journal of Physical Science and Application [5]. The ratio between the two quantities or, in other words, the number of minimal masses that can be found in the Universe mass is (dimensionless). It is easy to see that the number of bits in our Universe is the same, both if we try to calculate it using Bekenstein bound and if we instead calculate it simply counting the number of minimal masses available in our Universe. In other words the physical meaning of the minimal mass we found in our previous researches is the following: the smallest information entity in our Universe has a mass obtainable applying HUP, and that mass is about 10 (-69) kg. Add a bit (basic unit of information) of information builds up the area of the event horizon of a black hole of Planck unit of area that is a Planck length squared. Imagine building the black hole a bit at a time. Every time we add a bit of information the area of the event horizon increases by one unit of Planck. When the black hole is finished, the area of its horizon will be equal to the total number of information bits hidden in the black hole. So the physicist Jacob Bekenstein arrived at the following result: the entropy (a measure of information hidden) of a black hole, measured in bits, is proportional to the area of its event horizon measured in Planck units. 2.3 On the Value of Maximum Acceleration Continuing the research for meaningful values of the greatest and smallest physical quantities in the frame work of Banded speed cosmology, we focus on acceleration. Hereinafter, following the harmonic motion model previously utilized, we obtain the greatest value for acceleration. The validation is given by quantities already found by an independent research team. Centripetal acceleration, with the previous hypotheses, is given by: a = ω 2 R = (2V/Lt p ) 2 Ll p /2 = 2cV 2 /Lt p With regard to angular frequency: ω = 2V/Lt p If: V min V V max Then: 2V min /Lt p ω 2V max /Lt p And then: 2 / /2 2 / /2 2cV 2 min /Lt p ω 2 R 2cV 2 Max /Lt p From a physical point of view the greatest value is related to light speed c and length l p. The found value is a = 2c/tp m/s 2. This greatest value of acceleration is comparable with the one published by S. Capozziello, G. Lambiaseay e G. Scarpetta a = 2m p c 3 / ħ [7]. There is a different, easier, way to find the last result; let us consider three Planck Time units; in the first one we can measure the speed of a particle, and let c its value; in the second Planck time the particle can change its speed; in the third Planck Time we can measure the speed of the particle. Obviously, the maximum speed variation is from c to -c, so the maximum value of acceleration is 2c/tp. 3. Conclusions Several reasons have prompted us, engineers not physicians, to publish various papers in the area of theoretical physics. First of all we think that the characterization of physical phenomena starting from different backgrounds can help us in enlightening pros & cons of the theories which attempt the explanation of the nature. Moreover even if with pure mathematical description of physical phenomena we can get complex equation describing with great accuracy the behaviour

5 535 of the matter, if mathematics is not supported by plausible physical models we cannot progress in the nature s knowledge. What would happen if Bohr and its group had elaborated only mathematically Blaumer s series in order to explain atomic world without making physical models? Furthermore, making parallelisms among at first sight quite different phenomena, for instance mechanical electro dynamical or electronic ones, greatly helps us in gain understanding of physical phenomena. Borrowing a sentence of a colleague often nature when finds a good trick for a problem, tends to replay it in other fields with last changes. In this sense, when laws can explain physical phenomena, frequently they can be proved in various ways and we think that in this case in addition to show the effective truth of the laws we are helped in finding physical models for phenomena. Nowadays, it often happens to read disconcerting assertions that take us back in time when it was thought that all the physical phenomena where clarified and at most some physical constant where not quite exact. On the contrary, we think we must continue to look for new possible theories and explanations, without rejecting new hypotheses only because in conflict with traditional theories. In this respect we would not wish to read that a formula can prove or deny the existence of God. Nowadays only 5% of matter can be explained, partially, with known theories. But unfortunately any proposals not perfectly matched with standard Theories are labelled as heretical. Let us consider, for instance, the dark energy, at a first glance, non from a relativistic point of view, the analysis of the distribution of speeds and accelerations of the galaxies could be useful to find a possible unique rotating axis of the universe which could explain the noticed accelerations. Like the famous Newton s buckle, avoiding the need for strange energies. We hope that the present and previous our papers in this area could represent effective starting point in finding new ways for setter understanding the marvellous reality in which we live. References [1] D. Halliday, R. Resnick, K.S. Krane, Physics, Vol. 2, 4th ed., chapter 49, p [2] G. Fazio, M. Giaconi, D. Quatrini, An elegant universe is a necessity, not just a case, Journal of Physical Science and Application 4 (2012) [3] G. Fazio, M. Giaconi, D. Quatrini, Band structure: A solid-state physics model with cosmological implications, Journal of Physical Science and Application 7 (2012) [4] G. Fazio, M. Giaconi, D. Quatrini, Dark matter and other phenomena in the framework of banded speed cosmology, Journal of Physical Science and Application 2 (10) (2012). [5] G. Fazio, M. Giaconi, D. Quatrini, New strength to Planck s length choice, Frontiers of Fundamental Physics and Physics Education Research, Springer Proceedings in Physics 145 (2014) [6] G. Fazio, M. Giaconi, D. Quatrini, S. Martellucci, Bands of kinetic energy in our Universe: A working hypothesis, SIAE registered text number , date [7] S. Capozziello, G. Lambiase, G. Scarpetta, The Generalized Uncertainty Principle from Quantum Geometry, [8] Planck collaboration, Planck 2013 results, XVI. Cosmological parameters, Astronomy & Astrophysics March 21, 2014, [9] A. Einstein, The Principle of Relativity, Methuen and Company, Ltd., London,1923. [10] A.K. Pati, A Note on Maximal Acceleration, Europhysics Letters, 1992.