The Mass of Bodies as a Physical Dimension

The gap between the quantum approach and the general relativistic approach interpreted as a consequence of the mass of bodies

Matter is contained in bodies and it is made up of particles, which, in their entirety, quantitatively define matter in the body. The amount of matter therefore becomes the mass of a body.

A body undergoes physical effects due to its mass. These effects have been studied and are consolidated through Newton's Second Law, Newton's Law of Universal Gravitation and Principle of Equivalence.

However, mass as an element present in the physical phenomena of movement and energy is not an absolute value, but variable depending on the conditions of movement, or the relativity of movement.

Mathematically and experimentally it has been proven that every body has, even at rest, an energy value called rest energy associated with its amount of matter, in such a way that there is an algebraically described mass-energy relationship.

What we propose here is not to contradict the above principles. But to derive a new understanding of mass or matter in the sense of establishing a more elementary principle by geometric means immune to dependence on physical time or movement, that is, more comprehensive.

Cavendish's experiment proved the existence of the gravitational force between two bodies. There is no physical experiment like that of Cavendish whose purpose is to make one body orbit around another, even because, perhaps, of the technical difficulties of doing so. However, on a larger scale it is perfectly possible, since this is already a fact present in the routine of navigation of communication satellites.

Let us consider a body constituted with a certain mass. And another body in its vicinity constituted with a mass three hundred times less than the first. If we minimally approach such bodies they will experience the forces of gravitational origin. We could describe the less massive body as being in free fall relative to the larger body.

Under idealized conditions, physics describes that there are gravitational effects directed towards the geometric center of the planet. Let us consider, from this geometric center, that an observer A mentally establishes a simultaneous view of the 4π steradians of 3D space. Let's call it the omnipolar view or the ϕ view. As it is not a physical dimension, we will call it a pseudo-dimension. So, describing the situation of the soccer ball through this pseudo-dimension, our observer A would not distinguish the ball, because it is heading towards the ground and at any moment of its trajectory it is part of the view ϕ. If this ball happens to receive enough momentum to enter a gravitational orbit, despite being part of the ϕ view, it will be distinct from the free-fall situation experiencing its ϕ₂ view. The most basic peudogeometric figure for writing the relationship between the two views would be a line, which we will call here a pseudoline or σ-line. Consequently the ball in its supposed orbit would be in a pseudopoint or λ-point. These two pseudogeometric elements lead us to determine the figure of the pseudoplane or π-plane. These three elements constitute an infrastructure for a geometric theory. We will call it omnipolar geometry.

In order to return time t as part of the gravitational mechanism, we will do so through ordered pairs (x,t) and a graphical representation paradigm:

• the x coordinate corresponds to a component of reality, where there is no distinction between the present and the past, and where the simultaneity of things in general would prevail;

Figure 1. The triangle inscribed in the circle corresponds to element B1.

• the interposed, or perhaps intrinsic, t (time-like) coordinate determines a specific effect.

We then chose a paradigm for the analytical representations of ordered pairs (x,t), and although the Cartesian representation is a universal method, it will be modified here.

1. The x coordinate will be associated with a polygon inscribed in a circle.
2. the t coordinate will be associated with a geometric segment.

Therefore, the physical time represented by the t coordinate will have the character of a geometric entity such as the x coordinate. The x coordinate will have as values ​​polygonal geometric figures inscribed on the circumference. The circumference is therefore representing the Universe.

Figure 2 denotes a symbolic result of the Cartesian product (x,t).

Figura 2. Cartesian product between sets in analytic geometry.

According to the arranged data shown there, we have to suggest the following considerations:

• Elements F1 and G1 belong to the visible (observable) Universe, that is, the Universe with three-dimensionality;
• Elements B1, C1, and D1 belong to the unobservable, pseudo-plane, or two-dimensional pseudo Universe.
• Geometric elements inscribed in the circle are simultaneous.
• Non-simultaneous geometric elements must be semi-inscribed in the circle.

Figure 3. The “t” type coordinate is drawn in its position (generic example) determining a vertex at P.

In figure 3 the Cartesian product (x₁, t₁) generates an inscribed polygon that passes through M, which, by paradigm, also generates a coordinate, for example xᴊ≡(x₁, t₁) on the circumference that represents the pseudodimensional Universe where the property prevails of the simultaneity of everything.

As we have established only geometric and analytical representations, including physical time as such, this leads us to establish the physical mass as a connection information between the ϕ view and the 3D view due to the phenomenon of gravitation. Figure 4 shows a configuration of two bodies and their masses in a 3D view of and the corresponding diagram of their representation in the ϕ view. They realize that in the ϕ view the mass represents the possibility of the object being represented by one λ-point in relation to the other.

Figure 4. Schematic comparison of the mass of bodies as an intrinsic factor to determine a λ-point in the φ-view

Figure 5. All things or events fall within the present time of the Universe in the pseudo two-dimensional view. From this conjecture then the beginning of the ball’s fall is omnipresent upon its arrival which suggests that the ball’s mass is indistinguishable from the mass of the Earth (which in fact is true).

Thus, in figure 5, the soccer ball does not have enough information brought by its mass in order to establish itself as a λ-point in the free fall condition. This could be analyzed qualitatively based on the expression of Newton's law of universal gravitation:

By doing some manipulation, we arrive at the expression

interpreting the meaning of the term in parentheses as omnipolar distance, and adopting a notation for this

transforming now, in terms of “omnipolar vision”, that expression, in the form:

so that we can extract from there a rational relation for the mass “M”, whose denominator of the fraction has a distance meaning (omnipolar)

Such an expression can be interpreted as a linear density of “information”, since, in the omnipolar view, there is no meaning of volume and area.

Thus, the situation of free fall of the ball suggests that there is a relativity of the mass of the body in free fall (the soccer ball) to the point that the mass of the ball or even the body is indistinguishable from the mass of the Earth in informational terms.

On the other hand, we can also say that a set of information derived from Cartesian products between elements of the x coordinate domain and elements of the t coordinate (time) are not enough to distinguish the mass of the ball in relation to the mass of the Earth.

Figure 6. Hypothetical situation where the planet Earth, an artificial satellite orbiting it and the Moon also in its natural orbit around the planet are schematized.

In figure 6, we have a schematic drawing of the Earth-Moon system, also counting on the presence of an artificial satellite orbiting the Earth. This configuration is physically described through the Law of Universal Gravitation, which postulates that matter attracts matter in direct proportion to its masses and in inverse proportion to the distance that separates them. That being clear, we now insert the interpretation into the pseudogeometric view.

In the satellite situation, unlike the soccer ball, the set of information of Cartesian products between the coordinates of type x and those of type t (time) are sufficient to distinguish the satellite in relation to the Earth's mass. . In this way we say that the satellite is positioned in a pseudo point, or in a λ-point. The same reasoning applies to the Moon's mass situation.

It should be noted that, in the case of the satellite, its visibility at the λ-point level is due to so-called activated geometries and its λ-point is said to be a pseudo point obtained by "homothety". In the case of λ-point visibility for the Moon, this is due to intrinsic reasons of the Universe, that is, the "natural" cases come from a legacy of the Universe.

Also on account of this previous conjecture, we suggest that the masses of bodies, distinguishable at the level of λ-points, and which, consequently, will reach a possible gravitational orbital configuration around another (massive) body, are so informatively combined with each other, via Cartesian product (x, t) that such combinations may establish a permissive limit for the distinctiveness condition, corroborating what we said earlier about the dimensional relativity character of the mass of bodies. In other words, mass, in pseudo-dimensional geometry, has the character of "distance".

Once the mass has a character of distance, in cases of bodies distinguishable as λ-points, then a supposed action at a distance of the gravitational force can be substituted geometrically as a (static) alignment by means of λ-point on a pseudo line or σ-line.

The σ-line is not infinite, since its meaning ends until it finds a λ-point. In pseudo-dimensional geometry there is no meaning for the term infinity in dimensional terms.

As for the case of a particle configuration such as one atom in relation to another, it probably suggests that they do not have enough mass-level information to lead them to establish λ-points. In the same way, a particle in relation to a macroscopic body would not have enough information to establish a λ-point, becoming indistinguishable in relation to the macroscopic body.