The proportionality correlation that is determined between the inertial mass referred to the event and the time curvature that it generates on spacetime allows us to describe the spacetime variations by operating via quantum field theory (QFT).The invariance to the reference system as defined by the Lorentz transformations is preserved by the event’s frame of reference (Figure 3).

Calculations of time curvature (∆τ) and precursive time ( e 0 − ) as referred to a generic inertial mass.

The theses and concepts from which the main equations were obtained were described in the previous HDTSS publication, [17]; in this study we make some improvements in the calculation and further analyze and evaluate the procedure. We will examine a fundamental correlation that occurs between: < nuclear mass - weak nuclear force-orbital levels >.

The time-curvature (∆qt) is calculated in Planck time.

The Equation (1) allows us to determine time-space curvatures (∆τ) of a generic inertial mass:

M ( P k ) = h t p c 2 → t ( P k ) = h M ( P k ) c 2 → 1 h M ( P k ) c 2 ( s ) − 1 Δ τ ( joules / sec ) (1)

– t p = ℏ G c 5 Planck time

– M(P_k) = Planck mass event

– c² = Speed of Light Squared

From the Equation (1) we get the value of the precursive time ( e 0 − ) (-seconds) referred to the instant time inertial frame of the reference event.

τ v = 1 h m p c 2 joules / sec

As Figure 4 shows, Equation (1) determines the amount of deflected time induced by the time curvature of the inertial mass of the event; we can distinguish:

· (∆τ); time curvature: (∆τ) indicates the deflection effect of the real time coordinate induced by the time curvature.

· ( τ v ); precursive time: the precursive time is specifically the amount of time deflected by the time curvature. We remember that the time deflection effect is dictated by the equation:

< minor space ⇔ minor time > so < less space → less time >. From a relativistic point of view, the dynamics (torsion ® contraction) of spacetime simultaneously determines a deflation effect of space (less space) together with a deflection of time (less time):

|time deflection ® space deflation| Þ spacetime variation

Consequently, the time curvature becomes the primary cause of the spacetimecurvature according general theory of relativity (Figure 5).

It is important to note that ( τ v ) expressed in Planck time (joule/sec) coincides with the amount of energy of the inertial mass:

| energy ( τ v ) ≡ inertial   mass | ⇒ ( Δ τ ) ≡ M i (2)

The equation states that the inertial mass is induced by the time curvature; more specifically we can state that the inertial mass is exactly the value the time curvature.

Two other fundamental parameters are acquired from Equation (1):

1) (−s) time deflection in second (Planck time in precursive second).

2) (σ) space deflation. It coincides with the spacetime curvature of the GR (Figure 5).

Equation (1) shows that spacetime variations can be interpreted as classically described by GR through quantum physics.

Equation (1) also confirms that:

<For each event with [mass > 0] there is an imaginary time component proportional to the value of the inertial mass>.

Equation (2) extends the concept of complex variable mass in which the inertial frame of reference is a function of the time-space curvature (∆τ), the equality (mi º ∆τ) indicates that:

Theinertial mass in the absence of interacting gravitational forces is a physical quantity determined by the time curvature.

Note:

Conceptually, the hypothesis of time curvature deals with two important inexplicable issues that cannot be demonstrated through the classic depiction of space-curved GR:

1) It unifies general relativity and quantum field theory.

2) It unifies inertial and gravitational mass phenomena.

The representation in TimeSpace quadrant allows us to get a photograph of the event in instant time, as a “snapshot” of the imaginary timespace curvature ( σ ϖ ). The coordinates ( x i , y j , z k ) can be resolved as “hyperdimensional” since the complementary reference base ( e 0 → e 0 − ) is placed on an abscissa dimensionally shifted with respect to the event (instantaneous time) (Figure 6).

· Coordinates ( x i , y j , z k ) define the “curvature” of the deflated space.

· Sigma point ( σ ϖ ) which indicates the point of deflated edge space (is the concrete physical space even if calculated with imaginary coordinates).

· ( e 0 − ) is a deflection Imaginary time coordinate assuming the local observables reference frame, induced by the inertial mass (P_k) on time space (time curvature) ( e 0 − ) is positioned in advance (precursive) with respect to ( t 0 ), because < less space → less time >

From Timespace manifold:

∀ | P k | ∈ ℝ ( x 0 , y 0 , z 0 ) ( t 0 ) ⊆ ℂ ( x i , y j , z k , e 0 − )

M 4 ℝ ( x 0 , y 0 , z 0 , t 0 ) ; 3 s ℂ ( x i , y j , z k , e 0 − ) ;

P k ( t 0 ) ( x 0 , y 0 , z 0 ) ; P ′ k ( e 0 − ) ( x i , y j , z k ) ;

Summing up, each event is characterized by “seven” coordinates:

P k ( t 0 ) ( x 0 , y 0 , z 0 ) ; P ′ k ( e 0 − ) ( σ ϖ )

real coordinates referring to the chronotropic position of the M4 event:

P k : ℝ ( x 0 , y 0 , z 0 ) ( t 0 )

and four 3S hyperdimensional coordinates:

σ ϖ : ℂ ( x i , y j , z k ) ( e 0 − )

where, (t₀) is the event time of inertial frame of reference.

From the analyses carried out in this study, we can assume that matter and spacetime are in fact a single entity consisting of two joint parts, real partand spacetime the complex hyperdimensional joint part, the interaction < inertial mass ⇔ spacetime > assumes the concept of a bivalent set. Ref. timespace manifold (Figure 2).

From Equation (1), we can infer correlation (3) that introduces a concept of mass expressed in Planck parameters, directly related to the spacetime interaction properties.

It is a new expression of mass calculation as a function of the precursive time, that is, time-space curvatures, a new concept of mass that we distinguish with the name “mass-matter” ( ψ ).

Equation (3): calculation of inertial mass according to the precursive time equation:

τ v = 1 h m p c 2 → ψ = m p τ v Γ p K π F (3)

– ψ : mass-matter (inertial mass as function oftime curvature)

– m_p: chronotope mass (P_k) in Planckmass

– τ v : precursive time

– Γ p : quantum specific Heat capacity:

– K: Boltzmann Constant

– π F : quantum compensation spacetime, (F) Fibonacci coefficient [18] [19] [20] [21].

The equation of the calculation of the inertial mass as a function of the timespace curvature implies a direct correlation to Quantum Heat capacity ( Γ p ):

Q. Heat capacity:

Γ p = τ v K C 2 (4)

The equation expressed with ( Γ p ) is calculated as Planck temperature referred to ( τ v ).

Equation (3) highlights that: ψ f ( Γ p → τ υ )

this leads to asserting that ( Γ p )is a state property of matter.

From the calculation of Q.Heat capacity it is pointed out that is a characteristic of the mass as a function of the time curvature; in fact the correlation provides us with a reading key to the interpretation of the rebus of absolute zero: ( ψ → Γ p ) > 0   K

From the correlations expressed, we can say that: a generic mass | m i > 0 | has a heat state of Q. Heat capacity determined by its time curvature. This assumption implies that it cannot degrade below the Absolute Zero (0 K).

(F) Fibonacci coefficient

Let’s highlight the remarkable coefficient (π^F) obtained from theoretical processing of spacetime twist dynamics: quantum compensation spacetime (q.c.s.). This coefficient allows us to obtain aprecise measure of the value of the inertial mass as a function of the precursive time. The representation of spacetime calculated as a function of the Fibonacci exponential faithfully describes the variation of spacetime in twist (torsion) dynamics with an accuracy of the time curvatures of (10⁻¹⁵), as shown below in Figure 8 and Figure 9 with regard to nuclear masses. The constant of “quantum compensation spacetime”, describes the development layout of the twist-contraction dynamics of spacetime (Figure 7).

Given the coordinates: ( y i ( e 0 ) ; x i ( e 0 ) ) the graphic elaboration model draws a centripetal cycloid where the space-time (eigenvectors) contracts towards the centre of gravity of the chronotope. The four segments (F) represent equipotential space-time areas with a different frame of reference.

It is important to underline that the different areas or equipotential regions (F₁, F₂, F₃, F₄) are also iso-temporal, this being in accordance with Equations (1) and (2). The summation of Fibonacci’s hyperdimensional regions: complex time + complex space, is equivalent to the value of the inertial mass as referred to the chronotope: ( F n ): τ v n 1 + τ v n 2 + τ v n 3 + τ v n 4 = ψ

The hypothesis under study seems to draw an important meaning as a physical consequence, that is: mass and spacetime are not two different interacting entities as we would interpret classically, but we can state that:

< inertial mass and spacetime are constituents of a single entity in which precursive time is the mediator >.

The energy equation | ψ | function of time curvature, becomes a concept of mass peculiarly related to spacetime interaction. As a result of this, the equation: ( E = m c 2 ), can be reexamined in:

E = ψ c 3 (5)

Equation (5) which confirms the thesis, in fact it can take over from Einstein’s equation.

E = m c 2 ≡ ψ c 3

From the exposed correlations [matter ® precursive time ® energy], we can support the principle stating that: [matter Û spacetime] uniquely phenomena.

Figure 8 shows an example of application of Equations (1) (3)-(5) to the nuclear masses of the following atomic elements.

Figure 8 confirms that the correlations are consistent, in fact the value of the energy calculated in GeV is correct.

The value of the energy GeV confirms: [ E = m c 2 ≡ ψ c 3 ]

– Column (∆τ) J/sec indicates the precursive time (time curvature) calculated from Equation (1).

– Column (Ψ) indicates “mass-matter” calculated from Equation (3).

– The energy calculation (GeV) is obtained from Equation (5).

Figure 8 confirms the thesis that:

1) We can specify spacetime in quantum physics.

2) Spacetime can also be calculated for atomic nuclei.

In Figure 9, it is possible to quantify the spacetime variation even for nuclear masses. We apply Equations (1) (3)-(5) to the nuclear masses of the following atomic elements:

The column (picometers) is calculated:

· ( Δ q t ) : τ v ⇒ P t c 2 = orbitalradius

· nuclear mass ψ c 3 = GeV

Figure 9 correlates time curvatures ( Δ τ )and the distance ( σ ϖ ) in picometers.

We can note that time curvatures generated by the nucleus’ mass determines the distance of the electronic orbitals’ levels ( σ ϖ ).

In the following diagram we have highlighted the correlation ratio determined between the orbitals i.e. Na -sodium (2, 8, 1) and the respective distance from the nucleus. The energy contained in orbitals (2, 8.1) is directly related to the distance between the nucleus and the orbital radius.

– Values in red refer to the energies ((∆τ) J/sec))of the different orbital levels.

– Values in blue refer to the positions (Pm) in picometers attributed to orbitals.

The orbital takes on an energy value as a function of the distance determined from the nucleus.

From the Equation (1) ( τ v ) it is possible to calculate the levels of the orbitals and the number of electrons between orbitals according to the distance from the nucleus.

Figure 10 shows the position of the three sodium levels and the overall energy equal to the number of electrons on that level:

The movement of electrons between orbital levels in emission or absorption form is indeed a variation of the orbital radius, the correlation is:

| τ v | → ( σ ϖ ) → ψ c 3

Each level is characterized by a space position coordinate ( σ μ n ) (Figure 11) and a corresponding amount of energy as a function of precursive time ( τ v ). The complexity of the system is due to the double correlation that specifies the coordinate set ( ℝ + ℂ ) for each Orbital level (K, L, M, N):

- Σ K : ℂ ( x k i , y k j , z k k ) ( τ v k ) ⇒ ψ k c 3

- Σ L : ℂ ( x l i , y l j , z l k ) ( τ v l ) ⇒ ψ l c 3

- Σ M : ℂ ( x m i , y m j , z m k ) ( τ v m ) ⇒ ψ m c 3

- Σ N : ℂ ( x n i , y n j , z n k ) ( τ v n ) ⇒ ψ n c 3

Nucleus:

– chronotope nucleus position (nucleus coordinate): P k : ℝ ( x 0 , y 0 , z 0 ) ( t 0 )

– Nucleus energy: ( Δ q τ ) c 3

– Atom cross sections: σ ϖ : ℂ ( x i , y j , z k ) ( e 0 − )

– orbital (N) stretch dynamic: ( 1 n e 2 = τ N 1 − τ N 2 )

➢ m e 4 8 ε 0 2 h 2 τ N 2 → Δ τ ( − Γ p ) ® orbital contraction (Figure 12).

➢ m e 4 8 ε 0 2 h 2 τ N 1 → Δ τ ( + Γ p ) ® orbital stretch (Figure 12).

In summary, each orbital consists of a space position ( σ μ n ) (hyperspace positioning) and a time quantum level ( τ v ) which determines its energy (Figure 11). The correlation is extremely significant, as it places the nucleus mass as the cause of the energies distributed over the orbital levels (Figure 12 general diagram).

As shown on the chart in Figure 11, the electrons are emitted from quantum time precursive orbital, with respect to the nucleus chronotypal position, (the frame of reference of the nucleus is delayed compared electronicorbitals), it follows that the electron does not have a specific position “x” and momentum “p”, but is in a weak interactions’ regime in the latency energy form of.

The deductive conclusions suggest an understanding of the structure of a four-dimensional atom much more complex than the one presented by the current “standard model”.

The concepts presented so far are coherent with the standard model, Bohr model representation models hitherto analyzed, however the electroweak interaction dynamics defined by the new variable suggest a mathematical structure which goes under the name the Hopf fibration [23] [24] [25].

Hopf’s fibrational dynamics configuration allows the standard model to be extended to hyperspace correlations between nucleus and electronic orbitals. Fibrational mechanics allow a consistent representation of the theory and allow to calculate the energies of the electronic orbitals (Figure 13 referred to Figure 10)

The fibration dynamics explain the reason why there is no conflict between the negative charge of the electron and the positive charge of the nucleus, the system to continue in balance because it is the nucleus itself the one generating the spacetime variations that give rise to the electron charge of the electro-weak interaction.

– fibration positioning ( s 1 → s 3 → s 2 ).

– energy levels ( ψ 1 → ψ 3 → ψ 2 ).

The proposed atom fibration depiction dynamics does not modify the energy levels ascertained by the standard model, it varies the rationality of the origin causality of the Electroweak interactions.

As regards Quantum Chromodynamics (QCD) theory of the strong interaction between quarks and gluons, a review the interpretations in this new key of reading necessarily arises.

The hypothesis studying a “hyperdimensional atom” explains the reason for the incomprehensible conflict “The Wave-particle duality relation” and confirms Louis de Broglie’s Research on Quantum Theory hypothesized that all matter can be represented as a de Broglie wave in the manner of light.

The uncertainty principle is basically evincible through the electron levels in fibrational dynamic, since it is generally not possible to predict the value of a quantity with absolute certainty. In fact, any measurement test induces the parameters (X position and P momentum) change in ( σ ω ¯ ) of the fibers, just as demonstrated by Heisenberg’s uncertainty principle.

Ø Δ χ → f Δ S x n ( τ υ )

Ø Δ P x → f Δ S p n ( σ ω )

We have explained the concept of time curvarture as an effect of “temporal discrepancy” originating between the chronotropic mass referred to the event and the spacetime indicated in Planck time defined as precursive time.

The Equations (1) and (2) indicate that the mass and time curvature are proportional. This leads us to infer that time curvature is primarily responsible for gravitational mass.

In fact, Equation (6) confirms the thesis:

g = G m 1 r 2 ⇒ g = G ψ r Δ τ (6)

– g: gravity.

– r: radius.

– G: gravitational constant.

– Δ τ : time-space curvature.

The reason why the inertial mass and gravitational mass are related is because both forces originate from the same causal event.

The direct correlation exists between inertial and gravitational mass, so the gravitational mass value acquires an objective value defined Gravitation as a 0)>.

A 0 ≡ Δ q τ ≡ g

Time curvature defines the value of the specific attraction capacity degree of a body (A₀).

The two masses attract each other g = ( F A 01 − F A 02 ) so the result is the difference between the two sections of time curvature (Interferential of the Fibonacci areas).

Brief note about it.

In gravitational interactions, areas (F) are consistently added into quaternion matrix as described according to Hurwitz-Lipschitz, since the components are unitary quantities.

However, in this study we limit the exposure to the gravitational mass equation or gravitational attraction force, understood as a specific attraction capacity of a body as a function of the time curvature (space deflated effect). About the interactions between bodies, that is, between gravitational masses, the theory extends the concept to the principle of interference, resulting from the mutual action of the precursive areas ( e q − ) (Figure 14).