Abstract

There are currently eight connected major unanswered questions in astrophysics. The most important of these are:

How can dark matter be explained and described?
How can dark energy be explained and described?
Where do the extremely high-energy cosmic rays that occur in the energy range beyond the GZK cutoff come from?

A self-consistent theory called Model 1 has been developed that appears to answer these questions quantitatively. The unique features of this model are:

There are two spaces in the universe, particle space and quantum vacuum space. A potential barrier separates them. One space contains visible matter, and the other space contains dark matter.

There is a cycling of mass-energy between these spaces through the black holes that connect them. Particles pass from our space through black holes where they are converted into super particles that operate with unified force. They then pass into the high-energy vacuum space where they become dark matter operating behind the potential barrier.

Dark matter particles interact with each other and form a slowly building and moving bubble centered on a galaxy. Corridors of dark matter forming a cosmic web, which guide the development of new galaxies connect the bubbles of dark matter to each other.

There, behind a potential barrier, the dark matter particles gain energy, build up in number and eventually exceed the ability of the barrier to contain them. They then explode back into particle space as a big bang. This process repeats to make a long series of new universes.

After the big bang exhausts itself, super particles continue to tunnel through the barrier into particle space. The super particles are unstable and break down into particles with extreme kinetic energy. In doing so, they give up potential energy into particle space. The potential energy gradually builds up to become the dark energy that we observe as the cause of our accelerating, expanding universe. The extreme energy protons are observed as cosmic rays with energy between the energy of force unification and the Planck energy, which is beyond the GZK cutoff.

In working out this model, it became clear that the universe may recycle, and mass-energy may be the key to understanding this recycling. In this paper, a new concept of a mass-energy cycle will be described, analyzed, and connected to Model 1.

The Problem

There are several pieces to the explanation of mass as it rotates through the spaces in the universe that have been described in Model 1. The pieces have not yet been connected properly, however, to make a comprehensive description. In this paper we will make these connections. In addition, we will describe a local rotation that connects to the comprehensive one. This local rotation appears to be complete, reversible and re-entrant. We recall that a complete, reversible and re-entrant description of electromagnetism in atomic structure has been developed. Here, we will show that it is possible to construct a model for mass that is almost as complete, reversible and re-entrant.

The Solution

We will divide this study of mass-energy in to two parts, the comprehensive rotation of mass-energy through its states in the universe, and the local rotation of mass-energy through its states in a black hole.

A Comprehensive Rotation of Mass-energy Through its States in the Universe

Here we will provide the proper connections for a complete and comprehensive cycle of mass-energy through its states in the universe according to Model 1. In doing this, we will use the theories of general relativity, quantum mechanics and classical (Newtonian) mechanics in each step where it appears to apply. There will be steps where quantum mechanics and general relativity are both used in the same physical situation, so properly; a complete theory of quantum gravity should be applied, but currently, none exists. The justification for these applications is given in Appendix 1.

The connections for the rotation of mass-energy are given here. To do this task, we must start with an unjustified set of initial conditions, but we will end with a set of conditions that justify the initial conditions.

We start with a huge, concentrated, totally symmetric blob of energy from disrupted super particles set in a granulated space etched with a shadow cosmic web of dark matter from a prior universe. The energy blob is seeded with the charges for the particles and forces we observe in particle space. General relativity demands appropriate curvature in space from the mass-energy. Granulated space is included in the description providing curvature of both space and mass-energy of disrupted super particles combined (see Appendix 2). In this description, the mass-energy tells space how to curve, and the curvature of space then tells the mass-energy how to behave. The curvature of space at this initial time is enormous.
There are conditions when the net pressure in space is negative. When these conditions apply, the Robertson-Walker line elements and thus the spatial distances diverge. This divergent condition applies when the potential energy V dominates over the kinetic energy f’ ²/2 according to the equation (Peebles, 395):

p = f’ ²/2 – V

Where:

f = a new real scalar field that controls this expansion.

Particle space starts high in potential energy compared to the kinetic energy from the super particles with their high V that pass through the barrier, collide and disrupt. So the potential energy in particle space dominates during the initial stage of the big bang, and highly curved space expands (see Appendix 2).

The energy blob cools as it expands. Recall that the blob is seeded with the charges for the forces and particles. The symmetries are broken, and all the particles and forces of the Standard Model freeze out as the energy mixture cools, according to the laws stored in those charges. This freeze-out process is described in more detail by Kane (Kane, 97).
The high potential energy uses the Higgs field f to generate particles with rest mass m at this initial time. The particles are excitations created in the field f (see Appendix 3). The minimum of the field is the ground or vacuum state in particle space. The Higgs field f appears to be the same as the expansion field (ref 6, AP4.7G). Thus new particles of matter and anti matter are created in a thermodynamic mix with rest mass m. Charge is conserved, but potential energy is used to make particles, and so the high initial level of V is steadily reduced.
The matter and anti matter particles annihilate each other and form photons as they freeze out of the thermodynamic mix. The speed of light, which starts high in the early, high-energy phase of the expansion, maintains thermodynamic equilibrium throughout the mix. However, the light speed reduces as the kinetic energy drops (see Appendix 4), and thermodynamic equilibrium is lost.
At this point, an excess of matter is generated. The details of the process that generates matter particles in particle space are important. In the formation of matter and anti matter, our current particle data show that there is roughly one matter baryon excess for 10¹⁰ photons in particle space. This result can happen only if (Kane, 290):

1. Conservation of baryon number is violated.

2. Charge-parity (CP) is violated.

Particle space is not in thermodynamic equilibrium while the above conditions are satisfied.

We see then that the conservation of baryon number was violated as soon as the super particles flowed over the potential barrier and were converted to energy. The potential energy from super particles was reformed as matter and anti matter particles while still in thermodynamic equilibrium due to the high light speed at the high energy existing then. Since there is thermodynamic equilibrium at that time, a roughly equal number of particles and anti particles are produced. The matter and anti matter particles annihilate each other producing photons that eventually become our microwave background radiation. As this process continues, light speed drops as energy drops, and then thermodynamic equilibrium is lost. Charge-parity is violated, and without thermodynamic equilibrium to overcome it an excess of matter particles is produced. Particle space is then left with a large number of photons, and a small excess of matter particles, from which galaxies are formed.

At this stage, the potential energy from the big bang is going down, but tunneling super particles from the residual cosmic web are adding potential energy. Thus, the pressure goes through a minimum according to the equation:

p =f’ ²/2 – V

So the expansion of space goes through a minimum (see Appendix 2).

The shadow web of residual dark matter from a prior universe provides the nucleation material needed to form galaxies out of the thin soup of matter left from the particle formation step (ref 2, AP4.7B). Some of this matter then moves toward the high curvature of the center of mass of the galaxy, and gains circular speed ν_c around the center until the mass reaches a speed of 2.99 x 10¹⁰cm/sec according to the equation (Peebles, 47),

M(r) = mass within radius r = ν_c²r /G

At this speed, the energy increases, but the speed ν_c cannot because it has reached the speed of light limit for particle space, so the increased energy goes into mass. The mass keeps falling toward the center of mass of the galaxy, since it cannot maintain that radius with centrifugal force. This is the event horizon for a black hole. The matter that passes through the event horizon is torn apart into particles by tidal forces, and a black hole is born. Note that this is not the only way a central, super massive black hole is formed. For example, a very massive star near the center of a galaxy may run out of fuel and collapse and blow off its outer layers, leaving a massive black hole. In all cases, the black hole near the center of a galaxy will feed on dust, stars and gas near the center of the galaxy and become a growing, super massive black hole.

As a particle moves toward the black hole center, its potential energy increases inversely with the distance from that center (Misner, 911). Thus, the potential energy turns dominant near the black hole center; then the spatial pressure turns negative, and space expands. The particles continue to fall toward the black hole center, and a flat zone in the curvature of space forms near the center of the black hole. Here captured particles will temporarily collect in a diffusion zone or shell around the black hole center where they can be converted to super particles. See Appendix 5 for more details.

The high potential energy in the diffusion shell allows for the formation of super particles and barrier shells in a rate process, but the super particles are in a dynamic equilibrium both forming and disrupting (ref 3, AP4.7C). Note that the barrier shells constitute the formation of a new, high-energy space. Super particles will form and disrupt constantly as long as the potential energy is high enough to provide for formation, but they will disrupt permanently if they move away from the black hole and the potential energy falls too low. See Appendix 6 for more details.
As more particles are captured by the black hole, they are added to the diffusion shell, which increases the density and average energy (temperature). The energy increases until it nears the Planck energy, then there is enough excitation energy to force each super particle into a barrier shell in a second rate process (ref 3, AP4.7C). Note that this action is equivalent to particles sinking into black holes, bouncing at the Plank energy and expanding into a new space (inside the barrier shell) as described by Loop Quantum Gravity (see Appendix 1). Inside the barrier shell, the potential energy is high enough to provide for the dynamic equilibrium of super particles, so the super particles are stable. They can move anywhere, taking their barrier shells and the potential energy along to maintain their dynamic equilibrium (see Appendix 7).
The super particles inside their barrier shells in the highest energy range (near Planck energy) diffuse out of the black hole diffusion shell with circular speeds higher than 2.99 x 10¹⁰cm/sec (see Appendix 3). Thus the super particle has a circular speed that is beyond the particle space event horizon speed and is free from the black hole, speeding toward the galaxy edge. Some kinetic energy is lost by the super particles in the electromagnetic fields immediately surrounding the event horizon, and some is lost in the shock zones around solar systems in the galaxy, and in the shock zone just beyond the edge of the galaxy at the interface with intergalactic space. In a band near the galaxy’s edge, the super particle’s mass becomes dominant over the visible matter, and becomes the dark matter halo around the galaxy (ref 2, AP4.7B).
Many of the super particles that are gathered by black holes in the low to mid-energy range of the energy distribution remain within the black hole’s event horizon as super particles. Thus the super particles that result from conversion of particles escape more slowly than they are formed, because the black hole gravity limits escape to those in the high-energy range. Therefore the super particles that remain within the event horizon and in the diffusion shell gain energy and density
The super particles that escape from all the galaxies finally end in intergalactic space (and especially in the cosmic web), as high energy shielded super particles. Some of these super particles will tunnel through their shells into particle space providing ultra high-energy protons (UHECRs or cosmic rays) and potential energy, but most add to the energy of the super particles in particle space. The energy of the super particles does not degrade because the collision probability is low. By addition of new high-energy super particles the average energy of the super particles in intergalactic space (especially the cosmic web) eventually heats up to near the barrier potential energy (see Appendix 8).
The diffusion zones in the black holes are also being heated by the addition of particles through the event horizon. The diffusion zone in one of these black holes in a dense group of galaxies finally gains the average energy needed (near Planck energy) to start a large-scale flow of super particles over the barrier shell into particle space. If the average energy in the cosmic web is also near the Planck energy, a sustainable big bang has started. This is the first, principal black hole.
The flow of super particles in the first principal black hole reverses the super particle diffusion direction from flow toward the galaxy edge to flow toward the black hole center. This draws the super particles in from the event horizon and eventually from the edge of the galaxy. Then the super particles come from other galaxies along the cosmic web corridors. All of this movement happens at a speed much greater than 2.99 x 10¹⁰ cm/sec because the average super particle energy in the web has become close to the Planck energy (Appendix 3). Note that this rapid accumulation of mass-energy in a small zone causes space to curl tightly around the first principal black hole, which reduces the distances between galaxies dramatically, and facilitates the flow. Now we have the flow from the web needed to sustain a true big bang. Note also that the flow from the first principal black hole cannot become a true big bang until the average super particle energy and density in intergalactic space is high enough (near the Planck energy) to sustain it. Until this energy is achieved, flows from many black holes will start, send high-energy particles into intergalactic particle space and then sputter and die, thus contributing to the space potential energy and kinetic energy of intergalactic space and the cosmic web. We see that the progress toward a big bang is inevitable (see Appendix 9).
Once the super particle kinetic energy has actually exceeded the barrier shell potential energy, the super particles have enough kinetic energy (at the Planck energy) after flowing into particle space, that collisions can cause complete disruption of the super particles into energy. The charges that define the particles and forces remain in the resulting energy blob as seeds, and are conserved. At this energy, however, they cannot manifest their properties, so total symmetry is maintained. Note finally, that gathering super particles from intergalactic space cannot go to completion because some of the super particles will be at too low energy to flow rapidly to the principal black hole, so a residual cosmic web of super particles with high entropy remain in particle space after the big bang (see Appendix 10).
The cosmic microwave background photons that came originally from super particles have become high-energy photons as the universe curls around the gathering mass of the super particles. Those that do not reach close enough to Planck energy are not carried into the new cycle, and remain in particle space.
Thus we have a blob of energy seeded with charges in granulated space with a residual cosmic web ready to start the cycle again, and the space is etched with the dark matter of the cosmic web from the prior universe. At this stage, the universe has reproduced the conditions of the starting big bang, so we start a new cycle. Note that the charges in the energy blob are the way nature carries the physical laws from cycle to cycle, so the laws do not change with each cycle.
Note that with each cycle, more particles remain at low energy (high entropy) in the shadow web. Eventually, the universe will not have enough high-energy particles to start a new big bang and the cycles will end with the universe in a high entropy state (ref 12, AP4.7J). There the universe must remain until a new shot of high temperature energy is found and given to it. When the new shot of high temperature energy is provided, the charges must also be provided to determine the physical laws for this new universe. Only at this initial stage can the physical laws of the universe be changed.

We note that the processes described above for matter (both photons and particles with rest mass) is:

Complete in that potential energy is converted into rest mass and kinetic energy, and back again.
Mostly reversible. The process goes from particles to super particles, and super particles to particles, but the process of conversion is not complete. A residuum of super particles remains as a shadow web (see ref 12, AP4.7J for more details).
Re-entrant in that the process happens in nature automatically, creating and destroying photons and particles in a reentering process.

A Local Rotation of Mass-energy Through its States

We have moved through a complete cycle of mass-energy with all the proper connections. To illustrate its completeness, let us compare it with an example that most would consider to be a properly connected model—electromagnetism in an atom. For electromagnetism in an atom:

Electrical charge determines the electric field, and the electrical field determines potential energy in an atom.
The position of the electron changes so the electric field changes and the potential energy is decreased. The change in the electric field also forms the magnetic field, and the magnetic field reenters to reform the electric field, and so a photon is born.
The photon propagates at the speed of light and carries kinetic energy away from the atom equal to the potential energy change.
An atom absorbs a photon so the potential energy is increased and that changes the electric field. The change in the electric field changes the position of the electron. Note that the time required for this process is controlled by the uncertainty principle for energy and time because photons are not conserved
The increase in the potential energy is equal to the energy of the photon absorbed.

Now, we will show that mass-energy rotates through its various forms in a similar fashion by considering super particle formation in a diffusion shell near a black hole. We will use Newtonian notation here because it is easier to follow, but we will include the notation of general relativity in parentheses.

The mass (gravitational charge) determines the gravitational field (the curvature of space), which determines the potential energy in the black hole. Here the potential energy is directly proportional to the mass and inversely proportional to the distance to the center of mass (the potential energy is proportional to mass and the curvature of space) in a black hole (see Misner, 911).

The position of a massive particle changes, so the gravitational field (the curvature of space) changes, and the potential energy is decreased. The change in the potential energy forms the excitations in the Higgs field (see Appendix 1), which generates the rest mass needed to form a super particle from a particle. The potential energy change is partitioned equally between the rest mass and the kinetic energy of the super particle (following the equipartition of energy), and a massive super particle is born.
The super particle propagates at the speed determined by the kinetic energy, and carries away a mass and kinetic energy equal to the change in potential energy.
A massive super particle is absorbed in the diffusion zone near the black hole center, and that changes the gravitational field (the curvature of space), which changes the potential energy near the black hole center. The change in the potential energy changes the Higgs field, which absorbs the rest mass of the super particle and its kinetic energy (see Appendix 1). Note that this is a rate process dependent on the concentration of super particles and particles because the number of baryons must be conserved (see ref 3, AP4.7C).
The increase in the potential energy and thus the gravitational field (curvature of space) is equal to the mass and kinetic energy of the particle absorbed.

The creation and destruction operators for particles describe the formation and absorption of particles and super particles quantitatively as shown in Kane (Kane, 21).

We note that the processes described above for particles with mass energy is:

Complete in that potential energy is converted completely into rest mass and kinetic energy, and back again.
Reversible in that potential energy is converted into rest mass and kinetic energy and back by use of the same process both forward and backward.
Reentrant in that the process happens in nature automatically in a black hole creating and destroying particles and super particles in a reentering process.

Working toward Experimental Proof of Mass Rotation

There is reason to take Model 1 (which includes mass rotation) seriously. It quantitatively explains:

The origin, characteristics and operation of dark matter.
The origin, characteristics and operation of dark energy.

The origin, characteristics and operation of the Ultra High Energy Cosmic Rays (UHECR) that have been observed in the energy range beyond the GZK cutoff.

The huge disparity in the different estimates of the vacuum potential energy.
The large-scale cutoff and asymmetry in the Microwave Background Energy.
The origin, characteristics and values of light speed.

For more details, see ref 5, AP4.7D.

There may even be a way to provide definitive proof of Model 1 by extracting some dark matter (a super particle) from behind its barrier (see ref 14, AP4.7L).

Model 1 still needs more work however. Most of the areas in the description of the mass-energy cycle such as General Relativity and Quantum Mechanics are firm and supported by considerable theoretical effort and experimental evidence. One area is newly discovered and is less well developed. It needs more work. The melding of physics on the scale of Planck granules with that of physics on the scale of particles as well as physics on the scale of galaxies needs work (see Appendix 2).

Summary and Conclusions

In this paper, we have taken the results of prior papers describing Model 1, and provided the proper connections for a complete and comprehensive cycle of mass-energy through its states in the universe. In addition, we have taken the local rotation of mass-energy through its states in a black hole. These results were accomplished to gain understanding and to demonstrate the completeness, reversibility and re-entrant character of nature.

Appendix 1

Description of the Mathematical Concepts Used in This Paper

In doing this work, it was necessary to answer certain basic questions about physics in black holes, and the early universe. This requires the use of General Relativity, Schrödinger’s equation of quantum physics, the results of the Standard Model of particle physics, and the kinetic theory of gasses in a zone close to a black hole, and in the early stages of the big bang. Loop Quantum Gravity theory provides some justification for this use. Some of these results (Smolin, 250) are already available from Loop Quantum Gravity, and they appear to be compatible with this concept of mass. For example, loop Quantum Gravity is finite. It is background independent. It fits into the notation used for the Standard Model, and for General Relativity as well. It predicts gravitons at low energy. Especially, it predicts a Newtonian type gravitational force. It can also be used to predict some important states in black holes and the early universe. For example, it shows particles sinking into black holes, bouncing at the Plank energy and expanding into a new space. However, this theory is not clearly connected to Model 1 as described in these papers. It hints at a basis for Model 1, but does not clearly provide one.

In addition to these results, Peebles (Peebles 368) describes a partial, but incomplete transition of small-scale (Planck quantum physics level) particle physics into the large-scale description of the universe of galaxy formation (General Relativity level). He shows that there are several versions of this theory, but none appear to be complete and widely accepted. All, however, are background independent and all satisfy the requirements of general relativity. Unfortunately, none appear to describe a complete theory of granulated space on the Planck level that moves smoothly into large-scale space as required by Model 1, although some hint at it.

These are fundamental steps in the Model 1 papers, and so more work is needed here. However, the existing body of work, with its hints of connections to the results needed to support Model 1 appears to justify assuming the existence of these results to see if Model 1 satisfies the existing body of experimental data. Model 1 has indeed been shown to satisfy the existing data (see ref 13, AP4.7D). Thus there is justification for taking Model 1 seriously even though not all of the underlying theoretical structure is in place.

Appendix 2

Conditions at the Beginning of the Big Bang

In 1955, John Wheeler proposed that a concept called Quantum Foam be used to describe the foundation of the fabric of the universe (ref 7, AP4.7M Appendix 3, also Peebles, 368). At scales of the order of a Planck length, the Heisenberg uncertainty principle allows energy to decay into virtual particles and anti particles and then annihilate back into energy without violating physical conservation laws. At the Planck size, the energy density involved is extremely high, and so according to General Relativity, it would curve space-time tightly and cause a significant departure from the smooth space-time observed at larger scales. Note that the potential energy of the tightly curled grain would be stored in the curvature of space. Thus at these tiny scales, space-time would have a “foamy” or “grainy” character. The resulting grains would pop in and out of existence with Planck energy and Planck size for a Planck time. According to this proposal, the Planck granules would have size, time and energy characteristics as follows (Peebles, 368):

l_pl= 1.62 x 10^-33 cm

t_pl= 5.38 x 10^-44 sec

E_pl = 1.22 x 10¹⁹ GeV

These are the limiting characteristics of both particle and vacuum space. The mass-energy of the universe is embedded in this granulated space (see Appendix 1 for limitations to this concept).

The local energy conservation equation of general relativity (Peebles, 395) is,

r’ = -3 (r + p) a’/a

Where:

p = pressure

r = energy density

r’ = rate of change of energy density

a = space expansion factor

a’ = rate of change of space expansion factor

There are conditions when the net pressure is negative, i,e.

p < –r/3

Then the Robertson-Walker line element and thus the spatial distances diverge. Space expands.

In order to achieve this condition, particles must exist in particle space controlled by a single new real scalar field, f. The energy density ρ and pressure p equations that result are as follows (Peebles, 396):

r = f’ ²/2+ V

p = f’ ²/2 – V

And so:

f’ ² = p + r

Where:

V = a potential energy density

f = a new real scalar field

f’ ²/2 = a field kinetic energy term

These equations allow us to draw certain conclusions.

1). General relativity teaches us that there is space with potential energy V and kinetic energy f’ ²/2 controlled by a field f associated in a relatively smooth space on a large scale. Quantum mechanics teaches us that the space is granulated on a small scale (Peebles, 368). Peebles describes the melding of the physics at these scales and ends with a description of the current state of this theory (Peebles, 391). Although incomplete, it appears possible to describe matter and granulated space in one complete theory. We start with such a description of matter in granulated space.

2). The high kinetic energy of super particles after passage through the barrier in the big bang is far in excess of the binding energy of its components, so collisions convert the super particles into energy, and baryon conservation is lost.

3). In converting to energy, the disrupting super particles dump a large amount of potential energy V into particle space.

4). The speed of light is extremely high at the extreme super particle energy of the early stage of the big bang (see Appendix 3). Thus all of the super particles and particles in the big bang remain in thermodynamic contact.

5). The rate of change of the field (f’) is small compared to V in particle space in the early stage of the big bang because the density increases while the pressure decreases when V is increased, thus reducing the net change in f’. As a result, when V gets large enough, it exceeds f’ ²/2, so p turns negative and space expands

6). As time goes on, and potential energy is converted into photons and matter, the potential energy level goes down, and the speed of light goes down to its standard particle space value. Thus the particles from the big bang lose contact, and thermodynamic contact is lost.

7). The net effect of these processes is that the pressure in space starts negative and accelerated expansion of space in the early stage of the big bang results, but thermodynamic equilibrium is maintained. The pressure gradually increases as the potential energy is used up forming particles. Then potential energy is added by tunneling super particles so negative pressure goes through a minimum.

Appendix 3

Construction of Particles

Here, we describe the formation of the rest mass of particles by the Higgs field for two symmetry cases, reflective symmetry, and global symmetry.

Reflective symmetry

We start with the Lagrangian (see Kane, 122):

½∂_μf ∂ ^μf – (½μ²f ²+ ¼ λf ⁴)

Note that the potential energy V controls the Higgs field as follows

V = ½μ²f ²+ ¼ λf ⁴

Note the reflection symmetry; f = –f. To find the ground state and the excitations, it is necessary to expand the field around the minimum and establish the perturbation terms. In field theory, it is conventional to call the minimum the ground or vacuum state, and the perturbation terms are excitations, which are the particles. The form of the Lagrangian determines the mass.

To get the minimum of the potential, we take the derivative, set it equal to zero, and get,

f ² = – μ²/ λ = ν²

The value ν is the vacuum expectation value, and f is called the Higgs field. We have to work with one value, so we choose the positive root. To expand the function, we set:

f(x) = ν + η(x)

We then get for the Lagrangian:

½∂_μ η ∂ ^μ η – (λ ν² η ²+ λ ν η ³ + ¼ λ η ⁴) + constant

This Lagrangian represents the description of a particle with rest mass

m _η² = 2 λ ν² = -2 μ²

The reflection symmetry is gone here. The symmetry was broken when a specific vacuum state was chosen, so the vacuum does not have the symmetry of the original Lagrangian, and therefore the solutions do not. When this occurs, it is called “spontaneous symmetry breaking”.

Global symmetry

Now we take f to be a complex scalar.

f = (f₁ + if₂)/2^1/2

Then the Lagrangian is invariant under a global gauge transformation where:

f becomes e ^iχ f

Again we expand around the minimum. To get the minimum of the potential, we take the derivative, and set it equal to zero, and get,

f₁² + f₂² = – μ²/ λ = ν ²

Then we expand the Lagrangian around the minimum as above by setting:

f =[(ν + η(x) + i ρ(x)]/2^1/

We find that the expansion represents the description of a μ field particle with mass

m _η² = -2| μ²|

Note that the term with ρ² has gone, so the ρ field particle has zero mass. The continuous global symmetry (U(1) invariance under rotation) was spontaneously broken when we chose a particular vacuum or ground state. As a result, the spectrum will contain a massless spin-zero boson.

The Albelian and Standard Model Higgs Mechanism

The Albelian and Standard Model Higgs mechanisms are described by Kane (Kane, 104), but are not directly applicable to Model 1 as described here, so they will not be detailed further. Eventually, they will be important in detailing the internal energy states of the super particle. These states may be important in trying to extract a super particle from its barrier shell for observation and study (ref 14, AP4.7L).

The Higgs vacuum potential

Kane (Kane, 112) estimated the contribution of spontaneous symmetry breaking to the vacuum energy of the universe (particle space for Model 1). He found it to be roughly 2 x 10⁴⁹ GeV/cc, with a possible error of perhaps an order of magnitude based on uncertainty in the value of the Higgs self-coupling.

Appendix 4

The Speed of Light at High Energy

When super particles gain enough kinetic energy to operate freely in barrier space, they are operating in the zone 10¹⁹< E_ske < 1.22 x 10¹⁹ GeV. At the same time, they are passing over the barrier into particle space in a big bang. Here we see that particles have about the same kinetic energy as the potential energy in a Planck granule, so they operate freely in Planck space as well. They do not notice the potential energy edge of a Planck granule until it meets a disruption in space such as that at an edge of a solar system, a galaxy, a boundary of a corridor in the cosmic web or at the edge of the observable universe. Thus since the length of travel for a particle is n Planck lengths long, and N is the number of disruptions along the travel length, then the distance traveled is nl_p,and time used for this travel is just Nt_p,for each disruption, then:

v = nl_p/ Nt_p ½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_so)]

= 2.99 x 10¹⁰ n/N ½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_so)]

Now, since:

(E_smo+E_ske) > V_so

Then:

½[(1 + (E_smo+ E_ske)/(E_smo+E_ske– V_so)] ~ 1

Also,

n/N >> 1

So:

v >> 2.99 x 10¹⁰ cm/sec during the early stage of the big bang, and the late stage of the gathering of super particles for the big bang.

Note that the same argument holds for super photon velocity. See ref 7, AP4.7M for more details.

Thus we conclude that according to Model 1, the super particle (and super photon) velocity is large at high energy, and drops off to 2.99 x 10¹⁰cm/sec or less when the energy falls off.

Appendix 5

Formation of the Super Particle Generation Zone in a Black Hole

The conditions that allow for the formation of a super particle generation shell are as follows:

1). The mass density m in a shell Δr thick around a black hole is:

m r²Δr = n_p’m_pΔt / r² + (-R_spgn_pm_p)Δt + (R_spg n_spm_sp)Δt + (-R_spd n_spm_sp)Δt+ (-R_pd n_pm_p)Δt

Where:

n_p’m_p/ r² = mass entry rate into the shell through the black hole event horizon

(-R_pgm_p) = mass loss rate due to super particle generation from particles

(R_spg n_s)= mass appearance rate due to super particle generation from particles

(-R_spd n_spm_sp) = mass loss rate due to super particle diffusion out of the shell

(-R_pd n_pm_p)Δt = mass loss rate due to particle diffusion out of the shell

Δt = average time needed for a particle to diffuse across the shell

Note that particles gain gravitational energy as they fall toward the center of the black hole. Then the last four terms in the equation are zero until the radius is small enough that the potential energy is large enough to generate super particles (r < r_sp). Thus m increases as r² decreases until r reaches r_sp.Then, m decreases due to the loss of particles and super particles through diffusion out of the shell. The loss of super particles through diffusion is much higher than the loss of particles because the collision cross section for super particles is much smaller and the speed of the super particles is much larger. Growth of the black hole depends on the ratio of the diffusion rate of super particles out of the black hole to the entry rate of mass through the event horizon. Growth of the black hole would show as an increase in r_spwith the new particles inside r_sp.Inside r_sp, the particles would form super particles and gain energy until they near the Planck energy at a radius of r_pl. There, they pass inside the barrier shell (see 4 below). Then the shielded super particles would diffuse out, and eventually pass through the event horizon and travel out to become dark matter at the edge of the galaxy. Thus in this high-energy zone, m flattens as a function of r. More details on this diffusion-particle generation zone including the diffusion equations are given in ref 5, AP4.7F, Appendix 5.

2). The vacuum expectation value (ν) of the Higgs field (f ) is proportional to the mass, so it, too is inversely proportional to r² except for a central zone within r_sp where f falls off with reducing r.

f ² = m²/ 2λ = ν²

3). The potential energy of the particle or super particle is a complex function of f (Kane, 122; and Appendix 2).

V₀ = (½ μ²f ² + ¼ λ f ⁴)

Kane shows that V₀ forms a Mexican hat shaped potential energy function versus f. Thus V₀forms the following function versus r.

V₀rises to a peak in a spherical shell around the black hole center as r is reduced.
It then falls off to a minimum as r is further reduced.
It then rises further, and flattens out at the Planck energy because it can rise no further. The Planck energy is the highest possible (ref 7, AP4.7M-App 3).
As the total mass of the black hole gets larger, r_spgets larger.

4). Recall from ref 6, AP4.7G that the Higgs field of quantum physics appears to be the same as the expansion field of general relativity. Then from Appendix 2 we have:

p = f’ ²/2 – V₀

In order to determine whether space is expanding or not, we must establish the relative values of f’ ²/2 and V₀. Two cases are important for f’ ²/2.

Case 1. A test particle has high average velocity in a zone where the gradient in f is high, making f’ ²/2 high.

Case 2. A test particle has low average velocity in a zone where the temporal change in f is low, making f’ ²/2 low.

Now in the zone where V₀nears the Planck energy (r > r_pl), f becomes flat with r, so Case 2 obtains, and f’ is small as particles travel inward. V₀is large (near the Planck energy) in this zone, however, so p is negative, and space expands. In the zone between r_pl and r_sp, particles are moving toward the black hole center and the gradient in f is high, so Case 1 obtains, space stabilizes and diffusion dominates. Here particles collect and form super particles (see ref 3, AP4.7C for details of this formation). Inside r_pl, the particle density thins out because space is expanding, and super particles are gaining their barrier shells and diffusing out toward the event horizon. Another way of saying this is the curvature of space becomes small because of the expansion of space and the diffusion of super particles toward the event horizon, so particles reduce their motion toward the black hole center. Note that there is no mass singularity at the center of the black hole. The expansion zone excludes it.

Appendix 6

Formation of Super Particles in the Diffusion Zone

Here we study how the potential energy is partitioned in the formation of super particles in black holes. First, we note from Appendix 1, taking the reflective symmetry case as an example, that the potential energy V₀ is related to the Higgs field f as follows:

V₀ = (½ μ²f ² + ¼ λ f ⁴)

The minimum of the potential is:

f ² = – μ²/ λ = ν²

If we expand around the minimum, we find the particle rest mass as the following excitations:

m _η² = 2 λ ν² = -2 μ²

The argument is the same for the other symmetry. We note from these equations that the particle rest mass and the field are intimately related.

The mass of a boson in particle space is made up of the mass-energy of internal motion m_ke and the rest mass-energy from the Higgs field m_ho.

m_o = m_ke+ m_ho

Where:

m_ho= m _η

The mass of a super particle in vacuum space is made up of the above masses plus the rest mass of the barrier shell m_bsh

m_so= m_ske+ m_sho+ m_bsh

We see that mass-energy rotates through its various forms in the following way. (Here, we will use Newtonian notation because it is easier to follow, and include the notation of general relativity in parentheses).

Mass (gravitational charge) determines the gravitational field (the curvature of space), which determines the potential energy in a black hole (see Misner, 911). The potential energy is directly proportional to the mass and inversely proportional to the distance to the center of mass (the curvature of space) of a black hole.
The positions of the particles move toward the black hole center, and that changes the potential energy, which changes the Higgs field, which forms the excitation that generates the rest mass needed to form a super particle from a particle. The potential energy change is partitioned equally between the rest mass and the kinetic energy of the super particle (following the equipartition of energy), and a massive super particle is born.
The super particle propagates at the speed determined by the kinetic energy, and carries away a mass and kinetic energy equal to the change in potential energy.
A massive super particle interacts near a black hole, and that changes the gravitational field (the curvature of space), which changes the potential energy, which changes the Higgs field near the black hole, which absorbs the rest mass of the super particle and leaves a particle. Note that this is a rate process dependent on the concentration of particles and super particles because of the rest mass change involved (see ref 3, AP4.7C).
The increase in the potential energy and thus the gravitational field (curvature of space) is equal to the mass and kinetic energy of the particle absorbed.
These two processes feed each other in a dynamic equilibrium as long as the potential energy is high enough to form super particles.

In passing, we see that this mass-energy cycle is similar to that of the photon

Appendix 7

Formation of Super Particles Inside Barrier Shells

We notice from Appendix 6 that as the particle moves toward the black hole center, the kinetic energy increases with the potential energy according to the principal of equipartition of energy, and eventually reaches the value necessary to pass through the super particle barrier into the interior, forming a shielded super particle in a high potential energy zone. These shielded super particles are stable even in low potential energy zones outside the diffusion zone of the black hole. They also have high kinetic energy and a low interaction cross-section, so they rapidly pass out of the diffusion zone. The particle speed at this energy is also higher than the event horizon speed, so the shielded super particles escape from the black hole, and are observed as dark matter.

Appendix 8

The Heating of Vacuum Space

The shielded super particles inside the event horizon of a black hole are in a broad energy distribution. Those particles in the high end of this distribution have a high enough speed to pass through the event horizon, but those in the medium to low end do not. They must gain energy to pass out into the galaxy. Thus the super particles that reach intergalactic space are high energy. Some tunnel through the barrier into particle space and decay into UHECR’s and the potential energy that causes the accelerated expansion of particle space. To explain further, we must use the expansion equation.

p = f’ ²/2 – V₀

During the later part of the expansion phase of the big bang, there are still old super particles in the shadow web of residual dark matter that the new particles from the big bang meet as they expand. They continue to tunnel into particle space and provide high-energy particles and potential energy to particle space. The potential energy V₀is dwindling, since it is being used to form particles, but it is increasing also from the tunneling super particles. The field f is increasing too as follows.

1). Tunneling super particles bring rest mass and kinetic energy (mass energy m) into particle space as well as potential energy. In fact, more mass energy (10¹⁷ GeV <m< 10¹⁹GeV) is brought in than potential energy (V₀= 10¹⁷ GeV).

2). The vacuum expectation value (ν) or minimum of the Higgs field (f_m) is proportional to the mass (m), which is building up.

ν²= f_m ² = m²/ 2λ

Since f_m ² is increasing, f ²must increase. Thus more f ² is added to particle space than potential energy, but it is being added slowly in the early stage of the universe development because of the extremely low density of residual super particles from the prior universe.

3). New particles form into stars and galaxies, and eventually form into new black holes, and start to form new super particles. The density of super particles in intergalactic particle space builds up (see ref 2 AP4.7B).

4). The new potential energy from tunneling super particles forces the total potential energy of particle space to go through a minimum and build up to the level we observe now, but the field f is also building. This is the stage we are in now. As the tunneling continues, the space kinetic energy term (f’ ²/2) will eventually exceed the potential energy term (V₀), and the pressure p will turn positive. Then the acceleration will stop, the expansion will slow, and if there is sufficient mass, the universe will start to shrink.

5). The average super particle energy and density will build up by addition of new high-energy super particles from the new black holes.

6). Eventually, the average super particle energy in intergalactic space and especially the web will near the barrier potential energy (~10¹⁹ GeV), and the stage will be set for a new big bang. The flow of particles toward the principal black hole will concentrate the mass of the universe, and start the shrinkage of the universe toward it. Note that the flow from a principle black hole cannot become a true big bang until the average super particle energy and density in intergalactic space is high enough to sustain the flow.

Appendix 9

The Collection of Particles into the Flow of the Big bang

Note that the increase in super particle density and super particle mean kinetic energy in intergalactic space and the cosmic web toward the Planck level is inevitable. Black holes must preferentially send out high-energy super particles to overcome the black hole gravitational attraction. Thus the super particles in intergalactic space must increase in average energy until halted by the Planck energy. Furthermore, as black holes reach the critical energy (near the Planck energy) and begin a large-scale flow into particle space, they add more high-energy super particles unto intergalactic space. If the average energy of the super particles is not high enough to sustain a big bang, then the added super particles from the flow increases the particle density and mean kinetic energy in intergalactic space. This must continue until the particle density and mean kinetic energy is high enough to sustain a big bang, then the majority of the mass-energy of vacuum space is cleared out into particle space, and the new cycle begins. Remember, however, that there will be a residual number of low energy super particles that remain in a shadow web in particle space because they don’t have enough energy to obtain the speed needed to reach the principal black hole and flow out into particle space before the big bang shuts down. Thus after each cycle there are fewer high-energy super particles and photons available for recycling

The cosmic microwave background that originally came from super particles becomes a background of energetic photons as the universe shrinks with the gathering of the mass for the big bang.

Appendix 10

Conversion of Big Bang Particles into Energy

Particles that have passed over the potential barrier in a big bang have energy very near the Planck energy. These particles occur at very high densities as well. Thus collisions are common. The energy of collision drives the energy of the particles beyond the Planck energy, and the structure of the particles cannot contain the energy, so the super particle is converted to energy. It cannot maintain this form, so it must decay into a particle in a time determined by the Uncertainty Principal. It cannot decay into super particles because they require a particle to build on to form a barrier shell. Super particles must be the end result of a complete cycle of mass rotation through a black hole.

References

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