_{1}

^{*}

We hypothesize a closed Universe belonging to the oscillatory class. More precisely, we postulate a Universe that evolves following a simple harmonic motion whose pulsation is equal to the ratio between the speed of light and the mean radius of curvature. The existence of at least a further spatial dimension is contemplated. Although the space we are allowed to perceive is curved, since it is identifiable with a hypersphere whose radius depends on our state of motion, the Universe in its entirety, herein assimilated to a four-dimensional ball, is to be considered as being flat. All the points are replaced by straight line segments: In other terms, what we perceive as being a point is actually a straight line segment crossing the center of the above mentioned four-dimensional ball. In the light of these hypotheses, we can easily obtain the identity that represents the so called relativistic energy. In this paper we discuss, more thoroughly than elsewhere, the deduction of the so called mass-energy equivalence. Moreover, by carrying out a simple comparison with the way in which we perceive a bi-dimensional surface, the noteworthy concept of dimensional thickness is introduced.

Our universe is hypothesized as belonging to the so called oscillatory class [

In

If we denote with R the radius of curvature, with

The beginning of a new cycle (

By taking into account Equations (1) and (4), we can immediately write the so-called Hubble parameter [

As a consequence, it is quite evident how the Hubble parameter may have assumed in the past, and could possibly still assume in the future, negative values.

Let’s consider a material point whose motion is defined by Equation (1) (in other terms, a simple harmonic oscillator consisting of a mass and an ideal spring).

If we denote with m the mass of the above-mentioned point, the elastic constant, denoted by k, can be written as follows:

Consequently, the total mechanical energy acquires the following form:

Now, by solely modifying the amplitude of the motion, denoted by z_{m}, and by keeping the values of mass and pulsation constant, we obtain:

Once fixed the value of z_{m}, from Equations (1) and (9) we obtain:

At any given time, the value of R is obviously univocally determined by means of Equation (1), being R_{m} a constant. On the contrary, the value of z, provided by Equation (9), depends on the amplitude of the motion_{m}.

The total mechanical energy of a material point, whose motion is defined by Equation (9), acquires the following form:

The material point can be replaced by a material segment (in other terms, it is as if we consider a spring, no longer ideal, whose length at rest is equal to R_{m}). The length (R) of the segment evolves in accordance to Equation (3). If we denote with M the mass of the segment, the linear density can be defined as follows:

Denoting with M_{z} the mass of a portion of the segment, characterized, at any given time, by a length equal to z, we can write the following:

Taking into account Equations (11) and (13), the energy related to an infinitesimal material segment can be written as follows:

Therefore, the final expression for the energy of a material segment, whose length, at any given time, is equal to z, acquires the underlying form:

Let’s consider a 4-ball, centered at the origin, whose radius, denoted by R, evolves in accordance to Equation (1). The corresponding boundary, that represents the space we are allowed to perceive (when we are at rest), is a three dimensional surface (a hyper sphere) characterized by the following identity:

The Universe in its entirety is described by the following inequality:

Let’s consider the point

and its antipode (the one diametrically opposite)

Now, let’s consider the straight line segment bordered by the above mentioned points.

If we set

For example, we can set

Now, let’s consider the straight line segment bordered by the center of the ball and the point defined by Equation (22). If the segment in question, whose length evolves in accordance with Equation (1), is provided with a mass equal to M, its energy can be immediately deduced by Equation (15) by setting z = R. Consequently, underlining how the same procedure can be banally carried out by considering the point defined by Equation (23), we can write, with obvious meaning of notation, as follows:

Generalizing the outcome just obtained, we can write:

Therefore, for the material segment in its entirety, bordered by the points defined by Equations (22) and (23), characterized by a length equal to 2R and a mass equal to 2M, the energy is provided by the underlying relation:

Finally, by superposition, we can easily write the total amount of energy related to the material segment bordered by the points defined by Equations (18) and (19) as follows:

The points defined by Equations (18) and (19) are nothing but the interceptions between the material segment, whose energy is provided by Equation (27), and the hyper surface described by Equation (16). The latter represents the Universe we are allowed to perceive when we are at rest. As far as our perception of reality is concerned, each point and its antipode are to be actually considered as being the same thing, since they both belong to the same straight line segment [

In other terms, we may state that the one provided by Equation (27) represents the total energy that can be ascribed to what we perceive as being a material point at rest.

To carry out an opportune rewriting of the conservation of energy principle, we have to suppose that the total amount of energy of a material segment must remain the same.

On this purpose, let’s generalize Equation (27). Firstly, we have to replace, in Equations (18) and (19), R with z. Now, if we take into account Equations (13) and (15), by following the procedure previously discussed, we can deduce, for one among the scenarios that arise from Equation (20), the energy that should be ascribed to a material segment characterized by a length equal to 2z and a mass equal to 2M_{z}. Finally, we can write, with obvious meaning of notation, the following:

Let’s suppose that the above-mentioned material segment starts rotating around an orthogonal axis. If we denote with v the tangential speed of the endpoints, and with I the moment of inertia, we can write the correspondent kinetic energy as follows:

The moment of inertia is banally provided by the following relation:

From Equations (29) and (30) we immediately obtain:

Taking into account Equations (26), (28) and (31), we may express the conservation of energy, for the considered scenario, as follows:

By multiplying by three all the members of Equation (32), we immediately obtain the underlying noteworthy relation:

As far as the last member of Equation (33) is concerned, we may state that the first term represents the kinetic energy, the second term, that could be in appropriately defined as the potential energy, represents an energetic amount evidently related to the curvature, while the third term represents the energy actually needed to produce the motion.

From Equation (33) we immediately obtain:

From Equation (34) we can deduce the following fundamental relation:

We can interpret Equation (35) as follows: if a point, or better what we perceive as being a point, starts moving with a constant speed equal to v, it is as if it were instantly dragged towards an inner surface characterized by a radius of curvature equal to z. Actually, as qualitatively shown in

From Equation (35), if we introduce the so-called Lorentz factor [

Fundamentally, summarizing the results up to now obtained, we may state that our perception of reality depends on the state of motion. Very simply, if we start moving with a constant speed equal to v, the Universe we are allowed to perceive is no longer represented by Equation (16), but rather by the following identity:

Now, from Equations (13), (34) and (36), we immediately obtain:

If we multiply the first and second members of Equation (38) by the Lorentz factor, taking into account Equation (27), we can write the following:

From Equation (39), taking into account Equation (13) and (36), we obtain:

Once fixed the mass at rest of the material segment (M), the energy provided by Equation (40) represents, obviously, a constant. On the contrary, the reduced mass (M_{z}) does not represent an invariant, since it evidently depends on the state of motion. Bearing in mind that, due to motion, the material segment undergoes a radial contraction, we can divide the energy by z, that is to be always considered as greater than zero. This way, we banally obtain the energetic density (in other terms, the amount of energy per length unit). The result can be multiplied by a length, denoted by l, that could be identified with the dimensional thickness (the thickness that can be ascribed to our three-dimensional curved space), so obtaining the following relation:

Very roughly, we should admit that, from our point of view, a plane represents a pure abstraction. As much as one can reduce, for example, the thickness of a paper sheet, it will be in any case different from zero. In other terms, a bi-dimensional surface will actually be always characterized by an extension along the third dimension. Following the same line of reasoning, we could state that our universe can never be actually represented by Equation (37), since any three-dimensional surface is to be considered as being characterized by an extension along the fourth dimension. As a consequence, the mass of a material point, denoted by m, may be nothing but the product between the linear mass density and the dimensional thickness, whose value can be greater than or equal to the so called Planck Length (1,616,252 × 10^{−35} meters). From Equations (13) and (41), taking into account the definition of punctual mass we have just provided, we obtain the relation for the so called relativistic energy [

The one obtained in Equation (42) represents the energy that can be ascribed to a point characterized by a mass, that now can be considered as being a relativistic invariant, equal to m, that moves with a speed equal to v.

By following the same line of reasoning, bearing in mind that, net of the symmetry, the radial extension of the material segment at rest is to be considered as being equal to R, we can immediately deduce the expression for the so called energy at rest:

From Equations (42) and (43) we can deduce the following well known relation:

The one provided by Equation (44), usually identified with kinetic energy, actually represents the energy needed to produce the motion.

In this paper, starting from the well-known concept of oscillating Universe [_{1} type in Harrison’s classification), have not been addressed in this paper. On this subject, we specify how the metric variation of cosmological distances is not considered as a real phenomenon: in other terms, the amount of space between whatever couple of points remains the same with the passing of time, and the Universe can be considered as being actually static. However, to thoroughly discuss the metric variation of cosmological distances (and, as a consequence, the interpretation of cosmological red shift), it would have been necessary to speculate about the real nature of radiation, as well as about the well-known problem related to dark energy: We can herein just declare that the total number of dimensions we would have had to hypothesize, net of the symmetry, should have been actually greater than four. Nonetheless, such a speculation would have led the discussion far beyond the aim of this paper. It is quite evident how all the equations we have obtained are formally identical to the ones that characterize Einstein’s Special Relativity: Nonetheless, it should be rather clear how their meaning is deeply different. The hypothesized absoluteness of time should immediately suggest how the aim of this paper does not consist in a mere deduction of equations, routinely classified as relativistic, surely well-known in literature. The equations arisen from Special and General relativity have proven to be able to effectively describe the phenomenological reality, but it does not necessarily mean that the above-mentioned theories are absolutely valid in their entirety, at least as far as their meaning is concerned. Relativistic time dilation, for example, is herein considered as being nothing but an apparent phenomenon: it may be simply related to the radial contractions of material segments, perceived as material points, induced by motion. For example, the alleged increase of the lifetime of muons, although coherent with Special Relativity, could be easily explained avoiding time dilations. Muons evidently succeed in covering a distance clearly not compatible with their mean lifetime: this is unconfutable. On the one hand, we could admit that time, for muons, starts slowing down due to the high value of their speed. On the other hand, and for the same reason, we could imagine that, for muons, the distances undergo a contraction. In the latter case, the speed perceived by an observer at rest is greater that the real one, but time does not undergo any dilation whatsoever. Two different explanations, one of which based upon the absoluteness of time, both fully compatible with the Lorentz Equations, that consequently, though, acquire a completely different meaning in the two cases [

Cataldo, C. (2017) From the Oscillating Universe to Relativistic Energy: A Review. Journal of High Energy Physics, Gravitation and Cosmology, 3, 68-77. http://dx.doi.org/10.4236/jhepgc.2017.31010