From General Relativity to A Simple-Harmonically Oscillating Universe, and Vice-Versa: A Review

In this paper two different lines of reasoning are followed in order to discuss a Universe that belongs to the so-called oscillatory class. In the first section, we start from the general writing of the first Friedmann – Lemaître equation. Taking into account mass – energy equivalence, the so-called fluid equation is immediately deduced, with the usual hypotheses of homogeneity and isotropy, once identified the evolution of the Universe with an isentropic process. Considering equal to zero the curvature parameter, and carrying out an opportune position concerning the so-called cosmological constant, we obtain an oscillating class, to which a simple-harmonically oscillating Universe evidently belongs. In the second section, we start from a simple-harmonically oscillating Universe, hypothesized globally flat and characterized by at least a further spatial dimension. Once defined the density, taking into account a global symmetry elsewhere postulated, we carry out a simple but noteworthy position concerning the gravitational constant. Then, once established the dependence between pressure and density, we deduce, by means of simple mathematical passages, the equations of Friedmann – Lemaître, without using Einstein’s Relativity.

For a uniform Universe, with the usual hypotheses of homogeneity and isotropy, we can write the first Friedmann -Lemaître equation (Friedmann, 1922) as follows: (1) R represents the scale factor, G the gravitational constant, ρ the density, λ the so-called cosmological constant, k the curvature parameter, whose value depends on the hypothesized geometry, and c the speed of light.
As well known, if we denote with E the energy, with T the thermodynamic temperature, with S the entropy, with p the pressure, and with V the volume, we can write the following: = − (2) If we identify the evolution of the Universe with an isentropic process, from the previous relation we obtain: According to mass-energy equivalence (Einstein, 1916), we have: Obviously, we can write: Taking into account the foregoing relation and mass-energy equivalence, from (3) we immediately obtain: From the previous relation, we may immediately deduce the so-called fluid equation: According to Zeldovich (Zeldovich, 1961), the relation between pressure and density can be expressed as follows: (10) The value of ν, hypothesized as being constant, depends on the fluid, evidently postulated as being ideal, with which we identify the Universe.
From the previous equation, taking into account (8), we can immediately write: As a consequence, if we denote with C the constant of integration, we can immediately write: 1) can be written as follows: We can now define the following new constant (Harrison, 1967): Finally, by substituting the previous identity into (16), we obtain: If we denote with ω the pulsation of the Universe we want to describe, we can carry out the following position: If we set the curvature parameter equal to zero, by substituting (19) into (18), we finally obtain: = − (20) From the previous equation, we can deduce as follows: If we impose that the radius of curvature assumes a null value when t=0, from the prior equation we can deduce: From ( 17) and ( 27), we easily obtain: By taking into account (19) we can write the foregoing equation as follows: If we set ν equal to 1/3, from ( 27) we obtain In other terms, we have found a simple-harmonically oscillating universe, characterized by a variable density whose value, taking into account (29), is provided by the following relation: Obviously, if we denote with A the amplitude of the motion, we can write: Finally, denoting with R m the mean radius, we obtain: = 2 → = ( ) = − 4 (36)

An Alternative Deduction of the Friedmann -Lemaître Equations
We now want to follow an inverse line of reasoning.In other terms, we start from postulating a Universe, homogeneous and isotropic, whose evolution is described by (35).We can evidently write the following: By virtue of ( 37) and ( 40), we can write the so-called Hubble parameter (Hubble, 1929), denoted by H, as follows: Obviously, when t=0 we have a singularity.This and other problems related to the Universes belonging to the oscillatory class (Turok & Steinhardt, 2005) may be solved by assigning a different meaning to the metric variation of distances.Although the topic will not be herein considered, we may state that the discussion would have started by postulating a Universe actually static, characterized, net of symmetry, by at least five dimensions.
We hypothesize a closed Universe, globally flat, characterized by at least a further spatial dimension (Cataldo, 2016a).Such a Universe, identifiable with a four-dimensional ball, is described by the following inequality: However, the Universe we are allowed to perceive, when we are at rest, can be assimilated to the hyper-surface described by the underlying equation: Consequently, we can write the hyper-volume of the Universe in its entirety as follows: As for the volume of the curved space we believe to live in, we can evidently write: Obviously, we are not allowed to perceive such a Universe in its entirety.Net of the symmetry, what we perceive as a material point at rest may be nothing but one of the intersections between a material segment, that crosses the center of the 4-ball defined by (43), and the hyper-surface defined by (44) (Cataldo, 2016b).If we denote, as usual, with a(t) the scale factor, and with χ the angular distance, as perceived by an ideal observer placed right at the center of the 4-ball defined by (43), between two points, one of which is taken as origin, belonging to the hyper-surface defined by (44), the curved space we perceive can be effectively described, with obvious meaning of symbols and signs, by means of the well-known Robertson-Walker metric: Our analysis will be carried out by considering one amongst the scenarios that arise from (43) and (44), once having set equal to zero one of the coordinates.In other terms, the Universe in its entirety is identified with a ball, and the curved space we are allowed to perceive is assimilated to a spherical surface.
Taking into account the symmetry (Cataldo, 2016c), denoting with M the mass of the whole Universe, we can now define the density as follows: