Classic 4D Model Of the Discrete Universe (C3:B). I:Quantum

Este post es la continuación del capítulo 3 del libro 

Classic 4D Model Of the Discrete Universe. I:Quantum

3.4 Variation of the gravitational and magnetic forces

Suppose we have a circular coil, traversed by a electric current I. Suppose further that we reduce the loop to the size of Planck (r»10^-33 m), and that the current flowing is equal to 1A. Under these conditions, the coulombs, which is an arbitrary unit of electric charge (it was un agreement), coincide with the time that the particle is late in giving one spin

Quantum fluctuations produce a four-dimensional Planck atom that can rotate in three-dimensional space, and also in the fourth dimension. The Planck energy is

3.6 Conclusion

The hypothesis of a universe formed by four-dimensional Planck atoms impedes the formation of singularities inside a black hole, because a force is needed to compress the atoms of space and time. The aforementioned force is also deduced from the Schwarzschild radius.

Moreover, the gravitational potential energy with respect to the origin verifies the Heisenberg Uncertainty Principle. This gives rise to the fact that the relationship between the total mass of the universe, and its radius, fulfills the relation of Schwarzschild, which clearly indicates that we live in inside a black hole.

Finally, the four-dimensional Planck atom evolves so that its mass decreases until reaching the mass of the electron. At the same time, the time it takes to make a turn in the fourth dimension increases, giving rise to the charge of the electron, so that mass and charge are joined as are the gravitational and electric forces. While the gravitational force due to the rotation decreases, the electric force increases. The expansion of the universe separates Planck's four-dimensional atoms, producing a decrease in both forces.

In discreet space-time, neither the renormalization nor singularity are necessary, since the infinites disappear.

As the electric charge turns out to be the period in the fourth dimension, electric charge is imaginary, and therefore, while the masses of the same sign attract, the electric charges of the same sign are repelled.

Classic 4D Model Of the Discrete Universe (C3:A). I:Quantum

Este post está dedicado al capítulo 3 del libro 

Classic 4D Model Of the Discrete Universe. I:Quantum

   En dicho capítulo se demuestra que las fuerzas de largo alcance, gravitatoria y magnética son la misma fuerza. La fuerza gravitatoria es producida por una velocidad de rotación del electrón del orden de 10^-12  m/s, mientras que la fuerza magnética es producida por una velocidad de traslación del orden de 10⁺⁶ m/s. Esa diferencia de velocidades junto con la diferente orientación del campo es lo que origina que observemos los campos gravitatorio y magnético como diferentes. 

Como el capítulo 3 es bastante largo lo he dividido en dos entradas.

3. The origin of long-range forces         

The current theories are based on the assumption that the Big Bang had a unique strength, which, along with the expansion of the universe, gave rise to four forces, two of which are of infinite range. In the model developed here, there is only a single force, which is the magnetic force. Gravity and the electromagnetic force are connected at all times in the electron, both being different manifestations of the same phenomenon.

3.1 Introduction

In nature there are four interactions that are responsible for all phenomena in the universe: force or gravitational interaction, weak nuclear force, electromagnetic force, and the strong nuclear force. According to the Standard Model of Particle Physics (SMPP), the four interactions manifest themselves through an exchange of particles called bosons.

The electromagnetic and gravitational interactions have infinite extent, due to the fact that the interacting particles, photons and gravitons respectively, have zero mass. The weak and strong interactions have finite range, because the interacting particles have non-zero mass. They are the vector bosons and gluons. In the long-range forces, the time variable does not appear, suggesting that the action is performed instantaneously, i.e. at infinite speed. In turn, this implies that there is no material in between interacting bodies. That infinite speed requires the existence of the vacuum.

This action at a distance was a problem even for Newton himself: “That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.” ^[20]

Current physics has solved this problem by including particles that can not be detected, such as virtual photons and gravitons. It should be noted however, that the graviton has not yet been discovered, and the photons involved in the electromagnetic force are virtual photons, which have not been discovered either. However, according to the current physics, it is accepted that virtual photons must exist because the electromagnetic force can be measured. According to the SMPP, when energy is high enough, the electromagnetic interaction and the weak interaction are combined into a single interaction called the electroweak interaction. The grand unified theories (GTU) combine the electroweak interaction with the strong nuclear force, but leave aside gravity.

       Initially, according to the SMPP, four forces were unified into a single force. Figure 3.1 shows roughly the times when the various forces separated.

Figure 3.1 Evolution of forces.

In this monograph we will see that the long-range forces such as the gravita-tional and electro-magnetic forces still remain connected, since the electric charge and mass are two different manifes-tations of the rotation of the Planck particle ^{[1, 2]}.

3.2 Black holes

In recent years a group of physicists have developed the theory that within each black hole there is a universe, and that our universe could be the result of a black hole. From a modification of the equations of Einstein's general relativity, and the analysis of the motion of particles entering a black hole, Poplawski concludes that our entire universe may have originated inside a black hole existing in a bigger universe ^[21]. He says that “the interior of every black hole becomes a new universe”. The new universe is a natural consequence of a simple new assumption about the nature of space-time ^[22].

Several other authors have also proposed cosmological scenarios in which our universe emerges from a black hole ^[23-26]. Poplawski suggests that the universe formed inside of each black hole may be due to the coupling between rotation and torsion ^[27]. Space-time with torsion prevents the formation of singularities, where the universe expands from a low, but finite, radius, which corresponds to the dynamics of matter inside a black hole ^{[28, 29]}. Pourhasan, Afshordi, and Manna suggest that our universe could be a spherical 3-brane- formed by the explosion of a four-dimensional black hole ^[30].

When a star consumes all of its fuel gravity shrinks the star. If the mass of a star is large enough gravitation can overcome the neutron repulsion. In that case, in the current model, there is nothing that can oppose the collapse. According to Susskind, “The material squeezes into an unimaginably small region called a singularity, the density inside of which is essentially infinite” ^[31].

Suppose that the universe is formed by discrete atoms, of four spatial dimensions, and that the particles also have four spatial dimensions with a diameter equal to the Planck radius r_p=(Gℏ/c³)^1/2. Of the four dimensions, three are observed as space (x, y, z), and the fourth dimension (u = ct) is observed as time. These are the atoms of space and time from Smolin [3].

Therefore, in every atom we have the potential due to Planck's gravitational field, or any other field, to consider:

Here, m_p is the Planck mass, r_p the Planck radius , G the gravitational constant, and c is the speed of light. To compress the atom of space-time (Figure 3.2) we need to apply the force of Planck (Equation (2.3),F_p=c⁴/G ).

Figure 3.2 Atom of space and time (u=ct).

If we suppose that a star of mass M has surpassed the neutronica repulsion its radius is reduced until it reaches the Schwarzschild radius R, which we can write as:

Dividing Equation (3.2) by the radius squared (R²), and multiplying by the Planck mass (m_p), gives us

In Equation (3.3) the first term represents the gravitational force exerted by the mass M of the star, in gravitational collapse, on the Planck mass located on the surface of star of radius R.

It is possible to write the second term of Equation (3.3) as

We obtain an equation corresponding to the Heisenberg Uncertainty Principle

UGP is the Planck potential that exists in the past (t = R / c), at the distance R in the forth-dimension being R the radius of mass M observed in three-dimensional space.

Therefore, from the Heisenberg Uncertainty Principle, the same relationship to the radius of a black hole (or Schwarzschild radius) is obtained.

3.3 Quantum universe

Suppose a quantum fluctuation in energy occurs, that verifies the Heisenberg Uncertainty Principle, so that

where E_i, is the initial energy, t_p is Planck time, and ħ is the reduced Planck constant.

If, in addition to the quantum fluctuation, an expansion of space formed by discrete space-time atoms, of four dimensions, with diameter equal to the Planck wavelength, is produced (Figure 3.4).

where U_po is the Planck potential at the origin, R = ct is the radius of the universe at time t and U_p is the Planck potential that exists in every atom of space-time.

At the initial time, there is a quantum fluctuation that produces mass m (E_i). The expansion of space (Figure 3.4) causes that gravitational potential energy of the mass m, from the origin (O point), to decrease. The gravitational potential energy of the mass m, with respect to Planck´s potential at the origin, will be

Here m_p is the Planck mass, and c is the speed of light.

Considering the gravitational potential energy of the mass m, with respect to the Planck potential Up, local energy is obtained

The principle of conservation of energy implies that, at any time, the total energy, E(t), of the universe must be equal to the initial energy, E_i, therefore

where M is the total mass of the universe at time t. From this:

This equation corresponds to the Schwarzschild radius of a black hole.

As the radius of the universe increases to the speed of light, the gravitational potential energy with respect to the origin decreases. This generates an increase in mass to compensate for the decrease the energy. 

At all times, the total gravitational energy of the universe, with respect to the origin, verifies the Heisenberg Uncertainty Principle.

If we consider the gravitational potential of the mass m, with respect to the Planck potential that existed in the past, then the potential energy is less than or equal to the energy given by the Heisenberg Uncertainty Principle.

Moreover, considering the discrete four-dimensional space formed by atoms whose diameter is the Planck radius, the radius of the universe will always be an integer multiple of the Planck radius

Substituting Equation (3.18) in Equation (3.17), the mass of the universe can be expressed as:

Therefore, the mass of the universe can only take discrete values, equal to half integer multiples of the Planck mass. That means that, for each time increment of Planck time, an increase, equal to half a Planck mass, in the mass of the universe, occurs. It can be considered that at every Planck time, a quantum fluctuation occurs, of amount half of a Planck mass, and that instead of disappearing, it persists in time due to the expansion of space. This causes a decrease in the total gravitational potential energy. The Planck’s fluctuation, mass  length  time and energy, creates an expanding universe that at all times is a cosmological quantum black hole. This converts  Planck’s fluctuation into the cosmological Planck’s fluctuation (the Universe today) ^[32] of length  and  time.

Classic 4D Model Of the Discrete Universe (C2). I:Quantum

En esta entrada os dejo el capitulo 2 del libro

Classic 4D Model of the Universe Discrete. I:Quantum

2. Discrete space-time

The current theories, based on a space-time continuum, sometimes give rise to the appearance of infinites masked with renormalization, and the denomination of singularity, as in the case of black holes. Instead, in the discrete 4D model, the infinities disappear because at no time can the space-time be zero.

2.1 Introduction

General relativity implies that space-time is a continuum. However, there is no experimental evidence for this. Are space and time a continuum or are they composed of indivisible discrete units? We're probably convinced of continuity as a result of education. In recent years however, both physicists and mathematicians have asked if it is possible that space and time are discrete? Smolin states that space is formed from “irreducible pieces of volume that cannot be broken into anything smaller” that he calls “Atoms of Space and Time” ^[3].

Minimum volume, length or area are measured in units of Planck ^[3]. Planck's constant, h, which represents the elementary quantum of action, has an important role in quantum mechanics. There are several theories that predict the existence of a minimum length ^[4,5]. These theories are related to quantum gravity, such as string theory and double special relativity, as well as black hole physics ^[6–8].

Quantification of space-time maintains relativistic invariance ^[9] and causation, and allows us to distinguish elementary particles from each other in a simple and natural way ^[10].

There is evidence of discrete structures on the largest scales, for example superclusters and the redshift ^[11]. Cowan already said in 1969 that the redshift can only occur with discrete values ^[12]. This was subsequently confirmed by Karlsson ^[13].

Heisenberg said that physics must have a fundamental length scale, and with Planck's constant, h, and the speed of light, c, allow derivation of the masses of the particles ^{[14, 15]}. Planck’s length can be considered as the shortest distance having any physical meaning. To Sprenger, “a fundamental (minimal) length scale naturally emerges in any quantum theory in the presence of gravitational effects that accounts for a limited resolution of space-time.” The Planck scale appears to combine gravity (G), quantum mechanics (h), and special relativity (c) ^[16]. Padmanabhan shows that the Planck length provides a lower limit of length in any suitable physical space-time ^{[17, 18]}. Also, Messen starts from a minimum length he calls a, and a four-dimensional space, which allows him to characterize the different types of particles by quantum numbers. Then, different states of the particles correspond to different excitations of space-time ^[19].

From Planck units Planck force is derived, which is associated with the gravitational potential energy and electromagnetic energy. Planck force can be expressed as

where G is the gravitational constant, mp the Planck mass, c the speed of light, ħ the reduced Planck constant (h/2π), and rp the radius of Planck, which can be expressed in terms of G, ħ and c as

Substituting all of these into Equation (2.1), results in

2.2 Definition of time

What relates time to speed? The answer is obvious, space. Here space is not the distance between two points, but the space that fills the universe. We can thus define time as: the variation of the space-universe with speed. Planck’s particle expands in four spatial dimensions at the speed of light, giving rise to the appearance of time as a result of the variation of space with the speed of expansion, c. 

Let us now consider the gravitational force between the sun and the Earth. To simplify matters, let us suppose that the earth describes a flat circular orbit. To calculate this path we use the coordinate x-y axes. From the solar perspective, the only important thing for the sun to exert a force on the Earth is the distance r between them. It does not matter whether the Earth is on the right, left, behind or in front, as this force does not vary with position. The same happens with the expansion of the universe as only one of the three spatial dimensions is important: the direction between the two points or objects of reference considered. This direction is at all times perpendicular to the fourth dimension (radial direction). The increase of space is two dimensional and for an observer at rest time, will be given by:

If the surface increase, ΔS, is positive, time will be real and positive. This is proper time, pertaining to relativity and to the time we can observe and measure. However, if the surface increase is negative, the time is imaginary. This is the time that must be considered, for example, when we measure particles with energy whose wavelength is less than that of the particle, or when we apply Schrödinger’s equation.

2.3 Real and imaginary time

When we measure electron mass precisely, we need a lot of time, or conversely, little energy, which is equivalent to being at a distance greater than its wavelength. However, when we want to measure its position accurately, we use a lot of energy or short wavelength, which is equivalent to being at distances smaller than the electron’s wavelength. According to the given definition of time, the spherical surface will be formed by the fourth dimension (radial direction), and the direction of the observation (figure 2.1), 

so that

will be real and positive for distances greater than the wavelength.

When we get closer to the particle (a distance shorter than its wavelength) it turns out that the surface variation in the interior of the surface of radius l=cto  is negative, so that:

Here, t_i, is the time in the interior of the surface of radius l and t_e is the time on the exterior of the surface. t₀ is the time on the surface, or time at rest, or ct₀, which is the wavelength of the quantum mechanics.

2.4 Conclusion

Time is a two-dimensional physical quantity. The fourth dimension we measure is time due to the expansion of the universe. There is also another time, which is the distance between two points in space. The expansion of the universe generates real and positive time. In addition, when we measure particles with long wavelengths, we have a real and positive time. But when we measure particles with short wavelengths, smaller than the wavelength of the particle, then the time is imaginary.

References

[1] Garrigues-Baixauli, J., Derivation of the Schrodinger Equation from Classical Physics, Int. J. Sci. Res., 5(5), pp.77-80 , (2016).

[2] Garrigues-Baixauli, J., Relation Between the Gravitational and Magnetic Fields, J. Phys. Math., 7(2), 169, (2016).

[3] Smolin, L., Atoms of Space and Time, Sci. Am., 290(1), pp. 66-75, (2004).

[4] Maggiore, M., Quantum Groups, Gravity, and the Generalized Uncertainty Principle, Phys. Rev. D, 49(10), pp. 5182-5187, (1994).

[5] Maggiore, M., The Algebraic Structure of the Generalized Uncertainty Principle, Phys. Lett. B, 319, pp. 83-86, (1993).

[6] Ali., A. F., Minimal Length in Quantum Gravity, Equivalence Principle and Holographic Entropy Bound, Class. Quantum Grav., 28(6), (2011).

[7] Jalalzadeh, S. and Vakili, B., Quantization of the Interior Schwarzschild Black Hole. Int.J. Theor. Phys. 51 (2012) 263-275, (2012)

 [8] Kempf, A., Mangano, G., and Mann, R. B., Hilbert  Space  Representation  of  the Minimal Length Uncertainty Relation,  Phys. Rev. D, 52(2), pp. 1108-1118, (1995).

[9] Meessen, A., Space-Time Quantization, Elementary Particles and Dark Matter, (2011). arXiv:1108.4883

[10] Meessen, A., Spacetime Quantization, Elementary Particles and Cosmology, Found. Phys., 29(2), pp. 281-316, (1999).

[11] Roberts, M. D., Quantum Perturbative Approach to Discrete Redshift, Astrophys. Space Sci., 279(4), pp. 305-342, (2002).

[12] Cowan, C. L., Periodic Clustering of Red-shifts in the Spectra of Quasi-stellar and other Unusual Objects, Nature, 224, pp. 665-656, (1969).

[13] Karlsson, K. G., Possible Discretization of Quasar Redshifts, Astron. Astrophys., 13, pp. 333-335, (1971).

 [14] Heisenberg, W., Die Leobachtbaren Grossen in der Theorie der Elemntarteilchen, Z. Phys., 120, pp. 513-538, (1943).

[15] Heisenberg, W., Quantum Theory of Fields and Elementary Particles, Rev. Mod. Phys., 29, pp. 269-278 (1957).

[16] Sprenger, M., Nicolini, P., and Bleicher, M., Physics on the Smallest Scales– An Introduction to Minimal Length Phenomenology, Eur. J. Phys., 33(4), pp. 853-862, (2012).

[17] Padmanabhan, T., Planck Length As The Lower Bound To All Physical Length Scales, Gen. Rel. Grav., 17(3), pp. 215-221, (1985).

[18] Padmanabhan, T., Physical Significance of Planck Length, Ann. Phys., 165(1), pp. 38-58, (1985).

[19] Meessen, A., Space-Time Quantization, Elementary Particles and Dark Matter, (2011). arXiv:1108.4883