Introduction

    The newtonian model of universe is already well-known [1] and it requires the introduction of Einstein´s cosmologic constant in order to avoid whatever expansion and the same thing occurs with the relativistic model of Einstein.

    The cosmological constant is a force proportional to the distance extending between particles. Up to now there has not been found some simple model avoiding the introduction of these forces that are imparting an hypothetical existence and that are extremely minute and thus cannot be measured directly.

   The novel model discloses herein is purporting to eliminate this void thanks to the absence of this defect. An earlier paper of ours [2] showed that from within a cell of Planck-Poincaré space of phases a certain amount of energy E = h/t , where h is Planck constant and t is time, can originate spontaneously owing to the principle of uncertainty. When t = t₀ Planck time then cell energy is null in a certain case. Mass energy E₀ = m₀c², where m₀ is Planck mass and c is ligth velocity, is compensated with negative gravitational energy f(m₀²/l₀), where f is the constant of universal gravitation and l₀ is Planck length. Hence if the system is exhibiting an overall null energy then it may not only originate from Nothingness, but in addition it may evolve as will be seen hereinafter.
 

The (restricted) relativism model exhibiting an overall null energy

    Let us assume that the observer is located at coordinates origin O and at a distance r a matter density r (r,v) at a velocity v = dr/dt. The whole discussion supposes the existence of a spherical symmetry. Hence volume-unit energy amounts to c²r (r,v) 4p r²dr while mass amount to r (r,v) 4p r²dr. It should be remembered that for the observer located at the origin r (r,v) = r (r,0)/Ö (1 – (v/c)²).Gravitational energy amounts to f r (r,v) 4p r²drm/r where m is the mass of the sphere of radii r and M is the whole mass referred to the observer located at the origin. Extending the foregoing to the whole space of density r¹ 0, in 0£ r £ R, yields
 
 

(1)

     This equation (1) means that overall energy of the system is null, and thus it follows that
 

(2)

     Inasmuch as r (r,v) r² ¹ 0 for r ¹ 0, we have
 

(3)

wherein the value of r is whatsoever when r (r,v) ¹ 0 and hence putting r (r,0) = constant = r_u , where r_u is the density of universe, yields
 

(4)

     Equation (4) is transformed into the following one:
 

(5)

     Equation (5) allows to easely determine the maximum radius of the universe taking v = dr/dt = 0 and thus
 

(6)

     This formula of R_max supplied by equation (6) had been obtained by Einstein before[3] but for the actual size of universe. The values of the constants are:

(7)

and all the cosmologist are in agreement with this value. The minimum time T_min required to reach this size is:
 

(8)

     This minimum time of 1.05x10⁴x10⁶ years given by equation (8) is considered to be correct between 10.000 to 15.000 millions years. However, replacing v by dR/dt in equation (5) solving the equation allows to compute the exact time T/2 required to reach the maximum radius
 

(9)

     The change of variable x = R Ö (4pr_u f/c²) = R/R_max gives:
 

(10)

where G is Euler gamma function. Hence:
 

(11)

     Hubble constant H = v/R has now the value H_o = 2.5x10^-18 s^-1 accepted by cosmologists. Expressing the velocity v in equation (5) in terms of Hubble constant allow to get the present radius R of universe. Hence
 

(12)

     Replacing in equation (12) the values of H_o , c and R_max yields the present radius of universe:
 

(13)

     Then it is possible to determine the time elapsed since universe beginning up to the present moment that exhibits the radius given by equation (13). That time t present is obtained in a similar fashion as equation (9):
 

(14)

The implications of the model

     The model disclosed herein is affording the advantage that v = c when R = 0. Hence the universe started with light in a Planck-Poincaré cell. As overall energy is null, its configuration is as indicated herein above. In this case
 

(15)

where r_o is the density of planck-Poincaré cell and h = 6.62 x 10^-27 cm²g/s. The mass M that the universe can exhibit is:
 

(16)

     According to matter density shown in equation (1) and with equation (5),(6), it follows that
 

(17)

     As density r_u = r (r,0) can be taken as constant in all r, it follows that r (r,v) = r (r,0)Ö (1 – (v/c)²) and r (r,0) = constant for 0£ r £ R. It can be seen that universe density is a function of R.

     Equations (15), (16), (17) lead to the following notable relationships:
 

(18)

     According to its principal characteristics, the present novel model supplies values that are consistent with those recognized at the present time. Universe radius increases from time 0 up to T/2 were it reaches its maximum value R_max and expansion velocity v is annulled, then universe radius decreases from time T/2 up to T and a new expansion started, embodying thus successively a periodical universe. In addition the foregoing solves the problem of universe creation from nothingness. The hypothesis that overall energy of the universe is null, supporting this relativist model of universe creation and development, was formulated by Nernst [4] and here through equation (1). As up to now no new universe seems to have appeared, the probability of some null-energy configuration in a Planck-Poincaré cell is equal to t₀/t_a =1.33x10^-43 s/0.29x10¹⁸ s » 4.6x10^-61, where t_a is universe time elapsed from zero to its present radius.

     Finally, equations (18) establishes a relationship between r_u and r₀ which allows to say that universe amount M/R has been remaining constant since its birth. Constant M/R indicates that universe mass determines its natures. The foregoing was conceived first by Mach and confirmed thereafter by Einstein, and it is appearing here again. It must be pointed out that the observer located at the origin can be present in whatever place of the universe and that radius R refers to the most distant observations available. The foregoing makes thinking, as general relativity [3], that 3-dimensions tangible universe is submerged into a 4-dimensional one. This is confirmed by the fact that Hubble law applies exactly on spherical surface. The length of some geodesic line s = Rf where f is the angle at the center of a maximum circle, is exhibiting the velocity  where = ds/dt and = dR/dt and therefore s/= R/= constant.
 

A sphere and it's metris in the 4-dimensinal space

     In order to be more precise let us consider the case of some sphere in the 4-dimensional space [5]. Consider a system of orthogonal coordinates x,y,z,u and a coordinates transformation where sphere equation is parametrically expressed by coordinates R, f , q , j as follows
 

(19)

     In this 4-dimensional space the line element is obtained through the differentiation of variables change expressed in equation (19), obtaining thus
 

(20)

     Coordinates system R, f , q , j is orthogonal and if R is fixed we get an orthogonal system f , q , j , that is to say a 3-dimensional space, which for R large, can be assimilated to our space. The radius of universe curvature is R and the coordinates Rf , Rq , Rj with the origin at whatever point of the 4-dimensional sphere are forming the 3-dimensional space or a surface submerged within a 4-dimensional space. On the 3-dimensional surface, a geodesic passing through f₀, q₀, j₀ will correspond to equation (21) as follows
 

(21)

where g is a function of coordinates f , q , j and of geodesic position, these coordinates and geodesic position not being a function of time. Hence
 

(22)

     Dividing the equations (21) and (22), and fixing the time t then Hubble law is satisfied
 

(23)

     The values f = p /2, q ® 0 are considered in order to measure universe radius. This can be made always because the center of coordinates system is located at any point of the space, and any orientation can be adopted for this point. The measurement of the area S_M of a circle whose radius is r = R sinq yields
 

(24)

     The computation of this same area using euclidian geometry yields the value S_C
 

(25)

     Expanding in Taylor series the trigonometric functions appearing in equations (24) and (25), the following approximations for S_M and S_C are obtained at the limit
 
 

(26)

     The difference D S between measured and computed values amounts to
 

(27)

     Besides the relative value of surfaces difference D S with respect to measured surface S_M is approximately equal to
 

(28)

     Hence, according to the conditions imposed here, namely f = p /2, q ® 0, and with equation (20) the length of the geodesic in this space amounts to
 

(29)

     Therefore, equations (27) to (29) yield
 

(30)

     Where s is the approximated radius of solar system, namely s = 6x10¹⁴ cm while R is the present radius of universe curvature amounting to R = 10²⁸ cm. Thus it can be seen that the possibility of some geometrical measurement of curvature radius is extremely difficult, if not impossible. Finally if is pointed out that the geodesic lines are the only movement permitted in this universe though R measurement will be almost impossible, the measuring of some smaller R will not be possible.
 

Extention of the relativistic model to 4-dimensinal space

     In a previous title the case of a sphere in a space of 4-dimensional was treated. Now let us extend the relativistic model of the creation and development of a periodical universe in a 4-dimensional space. In this manner will be possible to precise more the ideas developed until here.

     Remember the expressions for surface and volume of a sphere in a n-dimensional space
 

(31)

     Where S_N is the sphere surface and V_N is his volume. In our case N = 4 and then
 

(32)

     Thinking in an sphere of 4-dimensional, where his surface of 3-dimensional is our real space, that have the following energy mass
 

(33)

     In equation (33) it is necessary to taking account that r (r,v) is a "superficial" density of matter. The same sphere have a potential energy gives by
 

(34)

     Equating the mass energy with potential energy, given by equations (33) and (34), respectively, and rewriting yields
 

(35)

     Equation (35) differs of equation (4) only in the coefficient 2p² in place of 4p . From a numerical point of view there is a little difference. Thus, we have news values to R_max, T_min, T/2, R y t_a obtained in analogous manner as equations (6), (8), (11), (12) and (14), respectively
 

(36)

     Now, the maximum mass, M_max, that can be the universe and his actual mass M, in 4-dimensional space, are, respectively
 

(37)

     In accord with the new expression for R_max given by equation (36), and considering equation (15) yields
 

(38)

then, considering the expression of r (R, v)
 

(39)

     From equation (37) and (39) the following relationship is obtained
 

(40)

     The equation (40) is really notable, because the ratio m₀/l₀ leave constant during all evolution of the universe and, moreover can be predicted his fundamental aspects. Due to that in M/R relation not appear the Planck´s constant h we can assure that quantum mechanics only acts in the creation and death of the universe and not in his evolution. But fine structure (galaxies, stars, etc.) must came from quantum field fluctuations.
 

The acceleration that stop the growing of the universe

     Then the universe begin as light in a Planck-Poincare´s cell in the fases space and explode in the 4-dimensional space as a shell of 3-dimensional. On such shell, as we seen, the Hubble law is valid in all directions. Moreover exist an acceleration directed to the center of sphere. Rewriting equation (5) that acceleration can be determined
 

(41)

     Where v = dv/dt. The value of acceleration is negative and cannot be directly measured. But as the length of a geodesic change, leaving the point fixed to the surface of 3-dimensional, yields
 

(42)

where = d²s/dt² and = d²R/dt². If s/r is very little, the acceleration value is the same order. Evaluating for s » planetary system diameter and for s » R the following values are obtained
 

(43)

     This negative acceleration is allows that the system back to the origin after to rise at maximum size, and is applied at the space with matter.
 

The number of particles and the temperature of the universe

      It is possible to compute the number of particles in the universe, considering the number of cells in the Planck-Poincare´s phases space [2]. This number is, as easily can see
 

(44)

where V is the surface sphere of 4-dimensional, r₀ the Planck´s density, R is the universe radius at the beginning, that is to say l₀, v is the velocity at the beginning and h is the Planck´s constant. Then
 

(45)

     We assume tat this number N₀, that we named Planck number of the universe, leave constant and is
 

(46)

     With this idea the temperature of the universe can be computed using the theorem of the equal partition of the energy
 

(47)

     This is the temperature of the universe at the present time.

     In relation to actual theories of creation and evolution of the universe not only we can consult [1] but Hawking and Penrose too (6). Recently Glanz [7] pay attention about of experimental evidence of a repulsive force, that correspond to negative acceleration that shows qualitative and quantitatively in this work. This repulsive force correspond to Einstein cosmological constant. As a result of this negative acceleration the withdrawal velocity, of the galaxies, is increasing when increase the time elapsed between lightly signal was emitted by them and when was detected and allows to known the velocities of the galaxies. That time is measured by the distance of the galaxy divided by the light velocity. At the beginning the withdrawal velocity was light velocity and the time the actual age of the universe.
 

References

1.  Reeves, H.: Dernieres nouvelles du cosmos ( Vers la premiere seconde). Ultimas noticias del cosmos ( A eso del primer segundo de tiempo) Editions du Seuil, 1995.
2. Kittl, P.: Some observations on quantum mechanics history, on Planck`s elemental cell, on the universe beginning and ending, on mimi black holes, and on massive binary atom. Anales de la Sociedad Científica Argentina, 228 (1998) 89.
3. Einstein, A.: The meaning of relativity. Princeton, 1950.
4. Westphal, W. H.: Physik ein lehrbuch. Springer Verlag, 1950.
5. Loedel, E.: Física relativista. Editorial Kapeluz, 1955.
6. Hawking, S and Penrose, R.: The nature of the space and time. Princeton University Press, 1996.
7. Glanz, J.: Exploding stars point to a universal repulsive force. Science, 279 (1998) 651.Note: From the actual work elemental versions had been published (8,9).
8.  Kittl, P. Y Dìaz, G. : Sobre un modelo elemental del universo periòdico con una duraciòn de 22.400 millones de años. Ciencia Abierta, N^o 5, 1999. (http://tamarugo.cec.uchile.cl/~ cabierta/revista5/universo1.htm)
9. Kittl, P.: Las dimensiones de Planck y el nacimiento y muerte del universo. Ciencia Abierta, N^o 5, 1999. (http://tamarugo.cec.uchile.cl/~ cabierta/revista5/Planck.htm)