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 that consists of many parts has been developed that answers these questions quantitatively. These parts have been collected into a self-consistent set, which will be referred to as Model 1. 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 characteristics of the barrier shell needed to be detailed. There are some questions that have not been answered about them:

In this paper, these questions will be answered in terms of Model 1.

 

 

The Problem

The barrier shell has been described thus far as a potential energy barrier that contains the super particle within the barrier. In the Standard Model, nuclear and atomic structures are described as fermion particles held together in orbiting structures with forces mediated by boson exchange particles. The forces are the electromagnetic, the weak and the strong forces. The shell structure is different in kind, and does not appear to be an orbiting particle structure. The barrier shell is all about gravity, mass and the Higgs field. It is formed in the strong gravitational force of a black hole. Thus its structure is curved space as described by General Relativity. Mass or a scalar field influences the curvature. The Higgs field influences the mass. There are five important questions to answer about the barrier shell.

• What is the barrier shell made of and what are its characteristics?
• How is it related to the Higgs field?
• How is it formed?
• Can the barrier shell exist without the super particle?
• What are the relative lifetimes of the super particle and its barrier shell?

In the next section, we will answer each question in terms of  the structures of Model 1.

 

The Solution

To explain the construction and operation of the barrier shell, and then answer the five questions posed above, we must first explore the pressure and density equations from general relativity, and then connect them with the equations that describe the Higgs field.

The Effect of a Real Scalar Field on Space in General Relativity

Consider a real scalar field f threading through space. This field controls the expansion and contraction of space as shown by the following density (ρ_ø) and pressure (p) equations from General Relativity (see Peebles, 396):

            ρ_ø = ø’ ²/2+ V (ø) = density

            p = ø’ ²/2 – V (ø) = pressure               (1)

            Where:

           V = a potential energy density

            ø = a new real scalar field

            ø‘= the time rate of change of the field

            ø‘ ²/2 = a field kinetic energy term

The Effect of Mass on Space in General Relativity

The cosmological equation for the time evolution of the space expansion parameter a(t) is determined by the mass density (ρ _m) and the pressure (p) as shown in the equation (see Peebles, 75):

            ä/a = -4/3πG (ρ_m+ 3p)               (2)

            Where:

            ä = the acceleration of the cosmological expansion parameter

            ρ _m = mass density = Σ m_η

                  m_η = particle mass

            p = pressure

We note that the pressure here is a pressure due to mass. On the other hand, the pressure in the prior section is a pressure due to the new real scalar field. The universe, however, cannot differentiate between a pressure from mass or from a field. Both are pressures, so it responds to either pressure or the sum of both. We will not differentiate hereafter.

Note also from the field pressure equation, that if the potential energy exceeds the field kinetic energy, the field pressure is negative. Also from the expansion parameter equation, if the field pressure is large enough, it can dominate the mass density. Then if the negative field pressure term is large enough to exceed the mass density term, the acceleration of the cosmological expansion ä/a becomes positive, and space expands. If the potential energy V is small compared to the field kinetic energy term, mass density controls, the acceleration is negative, and space contracts.

The Effect of the Higgs Field on Mass

The relation between the potential energy density, the rest mass of particles and the Higgs field f_h is as follows. Recall that in reference 2, AP4.7G, a connection was made between the above-mentioned new real scalar field of general relativity f and the Higgs field f_h, so we will only use one field f. We start with the Lagrangian (Kane, 98):

            Lagrangian = T – V = ½∂_μø∂ ^μø– (½μ²ø²+ ¼ λø⁴)                (3)

            Where:

            T = kinetic energy

            V = ½μ²ø²+ ¼ λø⁴ = potential energy               (4)

By expanding around the minimum of V, and using perturbation theory, the equation is found to describe a particle with mass m_η (Kane, 100) where:

            m_η² = -2 μ²  

If the kinetic energy of the particle is zero, m_η is the rest mass. This is only one example of several that describe the complete Higgs field operating in the standard model. It is generally typical.

This rest mass can be modeled by a screening effect with the Higgs field (see Shumm, 293-299) that explains mass formation. To explain this effect, we compare it with the screening effect of electrons on photons. If a photon passes through a medium filled with free electrons, its oscillating electric field oscillates the electron charges, and they generate opposing photons that tend to cancel the original photon. This tendency is called screening, and it gives a finite range to a photon in conducting media. We can model this effect by saying that the screening electric field generates an “effective mass” for the photon even though the photon has zero mass and infinite range (see Kane, 29). In a similar manner, we can model the mass of a particle with a Higgs field that “drags” on the particle as it accelerates. If an accelerating particle passes through a medium filled with a uniform Higgs field in such a way as to reduce its acceleration, it gives its accelerated motion a finite range. We can model this effect by saying that the screening Higgs field generates an “effective mass” for the accelerating particle even if the particle has zero intrinsic mass (see Kane, 29). If the particle has zero kinetic energy, we will call this effective mass the rest mass of the particle.

To complete this description, we must describe where the Higgs field comes from. The electric field has a source-the electric charge. Schumm points out (Schumm, 10) that mass appears to be the charge for the gravitational field. Does the Higgs field then have as its source the mass charge? If so, the field diverges from its particle mass source and becomes less with distance.

            ø= ø_o m_η /r²               (5)

We might therefore expect the Higgs field to be lumpy. But to get the mass generation effects we observe, the Higgs field should be relatively uniform. To explain why this conflict does not exist, we must look at three cases.

• The Higgs field in the universe comes from the total distribution of mass in the universe. That distribution is smooth on a large scale. Thus the Higgs field from mass at a long distance is smooth, and there is a significant amount of it because of the huge mass in the universe (see equation 2 with m_η dominant). The Higgs field from a medium distance mass is only slightly lumpy, and it is important because the source is nearer, and the interaction frequent. The Higgs field from very close mass could be important because it is close, but it is not. A particle only glimpses the Higgs source from close masses for a very short time because of the high relative velocity of the particle and the mass, and such a glimpse is rare, so the particle does not experience a screen that gives an effective mass. Thus the field comes from a distance, it is smooth, and the screen is effective. Note that our expanding universe is finite as shown by the Microwave Background Radiation (ref 5, AP4.7F), or the Higgs field we experience would be infinite, because the mass of the universe would be infinite.
• The Higgs field from the super particle mass dominates the Higgs field in the bubble inside its barrier shell. The mass of the super particle is large and the distance is small, so the field and its potential energy is huge. It is smooth because the super particle moves rapidly within the shell (v~c), and the shell diameter is small (r~10¹¹ cm). However, the super particle cannot move outside the bubble, so the time averaged mass density and its field is high, but only within the shell (see equation 2 with both m_η and f large). Thus the potential energy V_vs generated by the field is large in the bubble (10³⁴ GeV/cc). Also, the super particle moves slower and reverses its direction in the shell, so the time averaged mass density and its field and potential energy V_bs is even higher in the shell (10⁷¹ GeV/cc) than in the bubble. This potential energy forms the barrier that reflects the super particle and holds it in the vacuum space bubble.
• The Higgs field in a black hole is dominated by the mass of the black hole. Consider a test particle falling toward its center. The mass surrounding the particle increases as it falls. The Higgs field increases in proportion to the black hole mass surrounding the particle. The particle mass increases with the Higgs field. The potential energy of the particle increases with the Higgs field and mass until it reaches the super particle potential (10¹⁷ GeV), where it can convert to a super particle. The extra potential energy of the particle keeps it from collapsing in the gravitational field of the black hole. As a particle moves toward the black hole center, it gains more kinetic energy and potential energy (Misner, 911) until it enters the diffusion zone. There the particles are close enough so that the Higgs field is smooth. There also, an equilibrium creation-destruction cycle for super particles starts (see ref 7, AP4.7C for details). The high-energy vacuum bubbles with a vacuum potential energy density of ~10⁴⁹GeV/cc begin to form, but they cannot maintain themselves because there is no means to define the boundary. The particles and bubbles exist in a Gaussian distribution of kinetic and potential energy. Thus a small fraction of the bubbles will see a potential energy near the Planck energy. Then the high potential energy bubbles will form a shell of higher potential energy, which seals the boundary, forming a stable vacuum space bubble. At the same time, the particles with kinetic energy near the Planck energy have enough energy (activation energy) to penetrate the shell, and form a creation-destruction cycle within the bubble. Thus a stable shielded super particle is born ready to move out of the black hole diffusion zone.  

Now we can answer the questions we posed in the Problem section.

What is the barrier shell made of and what are its characteristics?

According to Model 1, the barrier shell is filled with a real scalar field f that functions as a barrier to super particle transmission. Thus it limits the movement of the super particle to high velocity travel within the bubble bounded by the barrier shell. The shielded super particle also has an average motion as a movement of the center of gravity of the super particle and its barrier.

How is the barrier shell related to the Higgs field?

The motion of the super particle within its barrier shell generates a large, time averaged mass within the bubble. Because it is reflected within the shell, the super particle also generates an even larger time averaged mass within the shell. These time averaged masses generate Higgs fields in both the bubble and the shell (equation 5). The Higgs fields and the super particle mass generate high potential energy zones in the bubble and the shell.  We recall that in reference 2, AP4.7G, a connection was made between the above-mentioned new real scalar field of general relativity f and the Higgs field f_h. Thus the potential energy of the barrier shell is generated by the super particle mass through the Higgs field.    

How is the barrier shell formed?

The conditions that allow for the formation of a super particle generation zone around the black hole center are as follows:

1). The particles gain potential energy as they fall from the event horizon toward the center of the black hole (Misner, 911) until the radius is small enough so that the potential energy is large enough (>10¹⁷ GeV) to generate super particles (R < R_sp). Also the mass-energy density r_m increases as R decreases until R reaches R_sp. Here, the collision rate is high, and this becomes a diffusion zone. Then, the growth of r_m flattens due to the net loss of shielded super particles through diffusion out of that zone. The loss of shielded super particles through diffusion occurs because the collision cross section for shielded super particles is much smaller and the speed of the shielded super particles is much larger than that of incoming particles. Growth of the black hole depends on the ratio of the diffusion rate of shielded super particles out of the black hole to the entry rate of particles in through the event horizon. Growth of the black hole would show as an increase in 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 formation zone where shielded super particles are formed. When formed, the shielded super particles diffuse out, and eventually pass through the event horizon and travel out to become dark matter at the edge of the galaxy and in intergalactic space. More details on this diffusion-super particle generation zone including the diffusion equations are given in ref 5, AP4.7F, Appendix 5. Here we will describe the basic operation characteristics of the super particle and the barrier shell formation zones.

2). The Higgs field (ø) is proportional to the mass of a particle (ø= ø_o m_η /r²) as shown in equation 5 above, but here in the central zones of the black hole, it is the sum of the mass of the particles that determines ρ_m and thus ø, so:

            ρ_m = Σ m_η/2πRΔR, R< R_sp

            ρ_m = Σ m_η/2πR_plΔR, R_pl<R< R_sp  (diffusion and super particle formation zone)

                  ρ_m = Σ m_η/2π R_pl², R< R_pl  (barrier shell formation zone)

            Where:

            R = radius from the center of the black hole

            ΔR = thickness of the spherical shell zone of interest

            Σ m_η  = total number of particles in the zone

Thus the mass density increase from the event horizon to R_sp, then it will flatten and be highest in the diffusion zone, and remain flat in the barrier shell formation zone.  

3). Here we must determine the dependence of V₀ on ø. The potential energy of the particle or super particle is a complex function of f (Kane, 98).

            V₀   = (½ μ²ø ² + ¼ λø ⁴);

            Where:

            m_η² = -2 μ²  

Kane shows that V₀ forms a Mexican hat shaped potential energy function of ø (Kane, 99). As a particle moves toward the black hole center, it gains kinetic energy and potential energy (Misner, 911). Thus V₀(r) where r is the distance from the particle to the test position varies as follows:

1. For very small R (barrier shell formation zone), ρ_m is high, ø is high, and V₀ is high, so barrier shell mass formation is high. 
2. For small R (diffusion and super particle formation zone), ρ_m is medium, ø is medium, and V₀  is medium, so super particle mass formation is medium. 
3. For medium R (gap near the diffusion zone), ρ_m is high, ø is low, and V₀is low, so mass formation is low.
4. For large R, (Super particle expulsion zone), ρ_m is very high, ø is low, and V₀is low, so mass formation is low.

Thus the super particle mass formation zone is confined to the diffusion zone, and the barrier shell formation zone is confined to the central zone around the black hole center.

4). Now we must determine what keeps gravity from crushing the particles into a singularity. This has to do with the factors that make space expand or contract (equations 1 and 2). In the explanation for equations 1 and 2, we noted that if the potential energy exceeds the field kinetic energy, the field pressure is negative. Also from the expansion parameter equation, if the field pressure is large enough, it can overcome the mass density. Then when the negative field pressure term is large enough to exceed the mass density term, the acceleration of the cosmological expansion ä/a becomes positive, and space expands. On the other hand, if the potential energy V is small compared to the field kinetic energy term, mass density controls, the acceleration is negative, and space contracts. Thus two zones are formed, a contraction zone, and an expansion zone inside it. We noted in 3 above, that V is inversely proportional to R in both zones. In this same pair of zones, ø’ due to mass change is small because the rate of change of black hole mass is small. On the other hand, ø‘ due to the motion of particles across zones varies according to the zone:

1. Contraction zone—For small R, øis medium and decreasing with R because mass is decreasing with R, so a particle with high average velocity will pass through a zone where ø’ is high, making ø‘ ²/2 higher. Thus ø‘ ²/2 exceeds V, and this is a contraction zone.
2. Expansion zone—For very small R, ø‘ is high and flat with R because mass is high and flat, so a particle with high average velocity will pass through a zone where ø‘ is low. Thus ø‘ ²/2 is smaller than V, and it is an expansion zone.

Now in the zone where V₀ nears the Planck energy (R > R_pl), ø‘ becomes flat with R, so Case 2 obtains, and ø‘ is small as particles travel inward.. V₀ is 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 ø is high, so Case 1 obtains. Here particles collect and form super particles (see ref 7, AP4.7C for details), and this is the diffusion zone.  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.

Thus there will be a shell zone where space will contract, and just inside it there will be a zone where space will expand. So a particle will descend through the contraction zone toward the black hole center, and then be pushed back by the expansion zone and then cycle again. This cycling characterizes the diffusion zone. Another way of saying this is the curvature of space is large in the outer contraction zone, but becomes negative in the expansion zone and pushes super particles back toward the event horizon, so particles stop 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.

5). As a particle moves toward the black hole center, it gains kinetic energy and potential energy (Misner, 911) until it enters the diffusion zone. There, the potential energy reaches ~10¹⁷ GeV, and there is enough to form super particles, and so an equilibrium creation-destruction cycle for super particles starts (see ref 7, AP4.7C for details). The particle moves on to the Planck zone, and the high-energy vacuum bubbles form with a vacuum potential energy density of ~10⁴⁹GeV/cc inside, but they cannot maintain themselves because there is no means to define the boundary.

6). The particles and bubbles exist in a Gaussian distribution of kinetic and potential energy. Thus a small fraction of the bubbles will have a potential energy near the Planck energy. Then the high potential energy bubbles will form a shell of higher potential energy, which seals the boundary, forming a stable vacuum space bubble. At the same time, the particles with kinetic energy near the Planck energy have enough energy (activation energy) to penetrate the shell, and form a creation-destruction cycle within the bubble. Thus a stable shielded super particle is born ready to move out of the black hole diffusion zone. 

7). The shielded super particles then diffuse out to the event horizon with energy near the Planck energy, and velocity greater than c (light speed at moderate energy) (see ref 8, AP4.7M). This velocity is greater than the escape velocity for the black hole at the event horizon, so the particle escapes.

Can the barrier shell exist in particle space without the super particle?

When the super particle exits the barrier shell in particle space, it is unstable, and it disrupts into an ultra high energy proton. When the super particle leaves the bubble, there is little mass left in it, and thus no means to generate the Higgs field. Without the Higgs field, there is no means to generate the potential energy in the barrier, so the barrier collapses, and its residual potential energy is added to the potential energy of particle space.

What are the relative lifetimes of the super particle and its barrier shell?

When the super particle exits the barrier shell, each resulting component must last long enough to balance the momentum of the other before it disintegrates. According to the uncertainty principle:

            ΔEΔt ~ ħ

Since the super particle and the barrier shell separately have masses (energies) less than the combined mass (energy) of the shielded super particle, their lifetimes are longer. Thus the super particle and the barrier shell will last longer than the process of super particle exit. So there is enough time to accomplish the process of balancing the momentum before the super particle and the barrier shell break down.

 

Summary and Conclusion

In this paper, we have answered five important questions about the barrier shell of Model 1, namely:

• What is the barrier shell made of and what are its characteristics?
• How is it related to the Higgs field?
• How is it formed?
• Can the barrier shell exist without the super particle?
• What are the relative lifetimes of the super particle and its barrier shell?

This paper then completes the description of this important segment of Model 1.

 

References

1. P. J. E. Peebles, Principles of Physical Cosmology, Princeton, New Jersey, Princeton University Press, 1993.
2. L. H. Wald, “AP4.7G ORIGIN OF THE NEW SCALAR FIELD” www.Aquater2050.com/2015/12/
3. G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing, 1993.
4. B. A. Schumm, Deep Down Things, Baltimore, Johns Hopkins University Press, 2004.
5. L. H. Wald, “AP4.7F GATHERING DARK MATTER FOR THE BIG BANG AND ITS IMPACT ON MBR” www.Aquater2050.com/2015/12/
6. Misner, Thorne and Wheeler, Gravitation, New York, Freeman and Co., 1973.
7. L. H. Wald, “AP4.7C DARK MATTER RATE EQUATIONS” www.Aquater2050.com/2015/11/
8. L. H. Wald, “AP4.7M VARIABLE LIGHT SPEED IN MODEL 1” www.Aquater2050.com/2015/12/