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Gravitational Stability for a Vacuum Cosmic Space Crystalline Model

Using Heisenberg's uncertainty principle it is shown that the gravitational stability condition for a crystalline vacuum cosmic space implies to obtain an equation formally equivalent to the relation first used by Gamow to predict the present temperature of the microwave background from the matter density. The compatibility condition between the quantum and the relativistic approaches has been obtained without infinities arising from the quantum analysis or singularities arising from the relativistic theory. The action which leads to our theory is the least action possible in a quantum scheme. The energy fluctuation involved in the gravitational stabilization of vacuum space is 10 to the power of (-40) times the energy of the crystalline structure of vacuum space inside the present Universe volume.


INTRODUCTION
The modern concept of a physical quantity in science follows that of Maxwell, where every expression of a quantity consists of two factors or components. One is the chosen standard quantity technically called the unit, and the other is the number of units required to make up the physical quantity [1]. From this conceptual scheme two different lines have been developed. One, the most important from a practical point of view, attends to the necessity to provide the basic units for measurements used in science, technology and everyday life [2]. The other line is devoted to the philosophical search for a deeper foundation of physical constant; see for instance works due to Maxwell [1], Planck [3] and Heisenberg [4].
Heisenberg's proposal of a natural system of units of measurement based on universal constants is a very interesting one, and is justified as follows [4]: "The universal constants determine the scale of nature, the characteristic quantities that cannot be reducing to other quantities. One needs at least three fundamental units for a complete set of units. A unit of length, one of time and one of mass is sufficient to form a complete set. The theory of relativity is connected with a universal constant in nature, the velocity of light, c. The quantum theory is connected with another universal constant of nature, Planck's quantum of action, h. There must exist a third universal constant in nature, this is obvious for purely dimensional reasons. An only a theory which contains this third unit (constant) can possibly determine the masses and other properties of the elementary particles. Judging from our present knowledge of these particles the most appropriate way to introducing the third universal constant would be the assumption of a universal length the value of which should be roughly 10 -13 cm, that is some what smaller than the radii of light nuclei. When from such three units one forms an expression which in its dimensions corresponds to a mass, its value has the order of magnitude of the masses of the elementary particles". Following Heisenberg's proposal, it is possible to consider that some new physical result obtained in a previous paper [5] requires further analysis. In particular the possibility that vacuum cosmic space could have a crystalline structure, with a lattice parameter r N = R OU /10 40 (r N similar to neutron radius) where R OU is the present Universe radius deserves a careful exploration. From our point of view such physical analysis requires the study of compatibility conditions between the General Theory of Relativity and Quantum Theory as applied to the crystalline vacuum cosmic space model. This because a theory which physically describes the metric of cosmic space and its evolution and a quantum theory which allows to determine the masses of elementary particles (neutrons) existing in such metric are simultaneously required to analyze such problem.
The main purpose of this paper is to study the immediate implications about the gravitational stability of a model which considers that the vacuum cosmic space has a crystalline structure with a lattice parameter of the order of the neutron radius r N ~ 10 -13 cm.

FORMALISM
Our model for the vacuum cosmic space is an infinite crystalline structure characterized by a lattice parameter roughly the size of the neutron radius which is the distance between the physical entities that form the crystalline structure. Here vacuum means, by definition, the state of lowest or minimum energy per unit volume. The state > 0 is the state of crystalline structure without deformation. According to Einstein's gravitational theory, in this crystalline structure there is a gravitational attraction between such physical entities, and also there appears an interaction between them due to crystalline lattice deformation. These phenomena lead to a gravitational instability of such vacuum cosmic space.
In nature we have a lot of systems which, in principle, are unstable against some kind of force but thanks to quantum forces which arise from Heisenberg's uncertainty principle the system gets stability. Here we mention two examples: the atom and the nucleus.
As we know, in the hydrogen atom, which classically is an unstable system, we can obtain stability by using Heisenberg's uncertainty principle, which introduces something like a compensatory quantum force, which stabilise, the system.
The main point here is that, in contrast to classical mechanics, the energy is bounded from below because of the uncertainty principle.
Similarly, for the case of nuclear forces, in the deuteron the stability can be explained by using again Heisenberg's uncertainty principle, or in nuclei with several nucleons, the stability can be achieved appealing to this principle as Yukawa did in 1935, explaining the nuclear force by the particles exchange, through the relation Following the same arguments as above, we can extend these ideas to a system which interacts by gravitational forces. In the same way that Heisenberg's uncertainty principle is appealed for stabilizing a system like the hydrogen atom or a light nucleus, in this work this principle is used to prevent the collapse of a crystalline structure, which is our model for the vacuum cosmic space, due to the action of gravitational stresses. In our crystalline model of vacuum cosmic space with lattice parameter of the order of the neutron radius r N , the number of physical entities which exist for such crystal inside a volume CVS OU OU is 10 120 , provided that R OU is the radius associated to the most usual value for the Universe age: 15000 millions years. Einstein's gravitational theory states that such physical arrangement is unstable under the action of long-range gravitational stresses. We can restore the equilibrium or stability of this system around average gravitational stresses with zero value by using Heisenberg's uncertainty principle.
Each of the N CVS entities inside V OU behaves as a linear harmonic oscillator. In general we have 3N CVS degrees of freedom in this system for vibration modes [6], but due to the radial symmetry of the gravitation interaction we have only N CVS degrees of freedom, which correspond to 3N CVS linear one dimensional harmonic oscillators. Each of the N CVS physical entities of the crystalline vacuum cosmic space inside the volume V OU , contributes with a stabilisation energy OU ε Δ against gravitational forces, given by Eq. (5) describes the fundamental quantum of gravitational waves which, in principle are responsible for the gravitational stability of the vacuum cosmic crystalline structure.
It is important to note that in the three cases that we have considered, the stabilization of fundamental physical systems against instabilities arising from electromagnetic forces, nuclear forces and gravitational forces which has led to Eqs. (1), (2) and (4), the stability radii are inversely proportional to the rest energy (self-energy) of the "particle" which is orbiting. For the three cases the De-Broglie matter theory [7] states that each "particle" orbiting around a radius r , satisfies the De-Broglie relation p is the momentum of the orbiting physical entity, circling in a stationary wave.
p For low frequencies, the relation between absolute temperature, T, and the photoenergy [8,9] is given by: , lead to an adiabatic compression process due to the gravitational attraction effect between them.
According to Peebles [11] during an adiabatic expansion of gravitational waves, the fractional change in the frequency ν ν Δ and the fractional change in the radius r r Δ of the volume enclosing the gravitational waves, are related through The same expression applies to the adiabatic gravitational compression process derived by their own gravitational attraction.
For an isentropic process of expansion by electromagnetic radiation [12], we have , Or equivalently, Eq. (14) is a generalization of Eq. Or, equivalently which combined with Eq. (18) gives Eq. (20) resembles an equation previously obtained by Gamow. According to Penzias [13]: "once pair production has ceased ρ , the matter density, varies simply as

DISCUSSION AND CONCLUSIONS
3.1. By using Heisenberg's uncertainty principle it has been shown that the vacuum cosmic space could be gravitationally stable. The model for such vacuum space is crystalline with a lattice parameter of the order of the neutron radius and the volume used to obtain gravitation stability from the collective quantum fluctuations is about the present Universe volume. Then Heisenberg's uncertainty principle allows to stabilise not only microscopical systems against electromagnetic or nuclear forces but also to stabilise macroscopically system against gravitational forces. Stephen Hawking consideration (among few other) about that there is no objective picture at all, and which considers that: actually there is nothing "out there" at the quantum level. Some how, reality emerges only in relation to the results of "measurements" made by human beings. Quantum theory, according to this view, provides merely a calculation procedure and does not attempt to describe the world as it actually "is". Following Penrose's [15] and Barrow's analysis [16] it is clear that both Bohr's and Hawking's analysis mislead the point by confusing between "measurement" made by human observers and interaction. Interactions which occur between any physical entities all the time in all the places of the Universe governed by Heisenberg's uncertainty principle as has been shown here and in many other applications do no require the existence of human beings to take place. In other words, according to our model the physical reality at quantum level as applied to the so called Universe does not require the presence of a conscious being to exist.
3.9. There are many problems which remain to be solved in our theoretical scheme for instance: the thermodynamical aspects related to the formation of quantum matter packages, the evaluation of the cosmological constant implied by our model, the very low entropy value at the starting of the big bang, the quantum aspects of the formation process of the quantum matter packages, the relation between the matter and antimatter production, etc. All these problems will be addressed in further contributions.