Abstract

In a previous paper (ref 1, AP4.7), a self-consistent, cyclical theory for the universe called Model 1 was developed, that is complete, self-constructing and regenerating. 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 moving halo centered on a galaxy. The halos of dark matter are connected to each other by corridors of dark matter which form a cosmic web. The corridors guide the development of new galaxies.

• 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 an endless 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 (protons) 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 (UHECRs) with energy between the energy of force unification and the Planck energy, which is beyond the GZK cutoff.  

Several observables were noted in the paper that support the model. To augment these observables, an experiment was proposed (ref 1, AP4.7L) to extract a super particle from vacuum space and study its properties. In working out this experiment, it became obvious that our understanding of the result of adding energy to particles up to the Planck energy is not well understood. The purpose of this paper is to gain a better understanding of this process.

 

The Problem

Model 1 describes a separate vacuum space filled with high vacuum energy as well as high-energy particles (super particles). Furthermore, this space can export its energy into particle space under some circumstances (by tunneling, if the particle energy is low, and spilling over the barrier if the super particle kinetic energy exceeds the shell potential energy). If the super particle enters particle space, it would appear as an Ultra High Energy Cosmic Ray (UHECR) (see ref 2 AP4.7I). The problem is what will happen to matter that is accelerated to Planck energy and how can this acceleration be accomplished. This paper will address this problem by following the acceleration of:

1. A particle
2. A small group of particles (a small spacecraft)
3. A large group of particles (a starship with astronauts)

 

The Solution

First it is necessary to define Planck mass, Planck energy, Planck length, and Planck time.

            l_p = (ћG/c³)^½ = 1.62 x 10^-33 cm

            t_p = (ћG/c⁵)^½ = 5.38 x 10^-44 sec

            E_p = (ћc⁵/G)^½ = 1.22 x 10¹⁹ GeV

            m_p = (ћc/G)^½ = 2.18 x 10^-5 gm

We note that these physical constants are related to quantum mechanics (ћ ) and gravity (G). We note also, that because of the Einstein foreshortening relation l_p(1- v²/c²)^½, either c or l_p is not a constant. In reference 1,AP4.7L, we describe a solution to this problem in which c is found to vary as energy approaches the Planck energy. We note finally that these constants are self-consistent, and represent the limiting value of each variable for spacetime. In a sense, they are the basic quanta of a quantized spacetime. They are the smallest (l_p, t_p) and the largest (E_p, m_p) values possible in spacetime.

In 1955, John Wheeler proposed that a concept called Quantum Foam be used to describe the foundation of the fabric of the universe (ref 9). 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. Any particle with energy less than E_p (and thus all particles) can accomplish this trick, but the decay-annihilation cycle must be done one particle at a time. 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 size is the smallest possible size, and the Planck energy is the largest possible energy, and the separation of these grains must be at least a Planck distance. Note that the time of existence as determined by the uncertainty principle is extremely small, so the flickering of the energy to virtual particles and back within the grains would not be noticeable to the real particles or photons passing through them, but they would notice the grainy structure due to the high potential energy. Real particles and photons operate on a much longer time scale unless the energy is near the Planck energy. Thus we have defined a vacuum composed of granules with high potential energy that flicker in and out of existence so fast that a particle or photon passing through will only see the high potential energy grains. 

If energy is provided in the form of high kinetic energy particles, and if a collision occurs to generate unorganized energy from the kinetic energy, the Planck granules will provide the potential energy, and virtual particles generated according to the appropriate symmetry will convert to real particles using the kinetic and potential energies provided. The real particles cannot have energy greater than the kinetic energy provided, and must have the proper symmetry. Since the potential energy is limited to the Planck energy, no particle can be produced with energy higher than the Planck energy.       

1). Accelerating a Particle.

If we accelerate a particle to the Planck energy, and no collisions happen, the symmetry will be conserved, and we will get the same particle at high energy. As an example, we start with a proton in particle space. We can accelerate the proton up to the Planck energy without changing anything but its kinetic energy and its mass. The proton with its symmetry is stable. If no collisions happen that disrupt the proton and produce unorganized energy, the proton will remain a proton. Similarly, an electron with its symmetry will remain an electron when accelerated. 

As the proton is accelerated, velocity and kinetic energy is increased. As more acceleration is provided, the velocity increase flattens near c (3.99 x 10¹⁰ cm/sec), and mass increases until the mass nears the Planck mass. Then the velocity rapidly increases again, and gains values greater than c (see Appendix A).

If collisions occur at high energy, the proton is disrupted, and sub particles, unorganized energy and potential energy are produced. The uncertainty principle allows for the production of virtual particles and anti particles for a short time, and the available energy along with the virtual particle produces a real particle. Thus a new real particle can form if the energy produced is large enough to form it as with the formation of the Higgs particle (Kane, 248) and the super particle (ref 3, AP4.7C). 

2). Accelerating a small group of particles (a small spacecraft)

When we accelerate to Planck energy, we must view the task as accelerating particles as a group rather than a connected starship mass. When we explored the relationship between the speed of light and the Planck granules of space (ref 6, AP4.7M) it was particles moving through granules that we analyzed. It was particles gaining Planck energy that changed the speed of light. The energy tying particles together is small compared to the kinetic energy carried by the particles near Planck energy. In order to determine the energy needed to accelerate a spacecraft to Planck energy, we must determine the number of particles and which particles they are. Then we can accelerate one particle to Planck energy and sum over all particles of the spacecraft. 

The starship in particle space consists primarily of protons and neutrons. These particles have nearly the same mass, so a small spacecraft in particle space can be represented by a cluster of protons. The proton is quite stable. In fact, the decay of a proton has not been observed. Thus the proton will remain a proton during the acceleration as long as a high-energy collision does not occur. So if the cluster is accelerated, and no collisions occur, the spacecraft will remain a spacecraft. Even if a high-energy collision does occur (by collision with a hydrogen atom in space), the damage will be localized to one proton (or neutron), and will not involve the whole starship structure. Therefore, we will think of accelerating a starship as accelerating a proton from ~1 GeV to ~10¹⁹ GeV summed over all the protons (and neutrons) of the starship.

As an example, we take a 10 KG starship, powered by a super particle thruster (ref 1, AP4.7L). Now we need to see what happens at the top speed of the starship. The momentum of the starship must equal the momentum of the exhaust products, so:

            M_s U_s / (1- U_s²/v_o²)^½ = mn_oћk_oΔt

            Where:

            M_s = mass of the starship = 10⁴ gm

            U_s = velocity of the starship = 10¹⁰ cm/sec

            v_o = velocity of light = 10¹⁰ cm/sec (energy < Planck energy)

            c = velocity of light at low energy = 10¹⁰ cm/sec

            Δt = thrust time for the starship drive in sec

            n_oћk_o = thrust of the unit super particle thruster = 10⁷ dynes = 10⁴ gmf

            m = number of unit thrusters needed to obtain a 1 G thruster = 1 

Now the 10⁴ gm starship can be seen as a collection of neutrons and protons each of which has a mass-energy of ~1 GeV. In order for the starship to attain Planck energy, each of the neutrons and protons must obtain an energy of 10¹⁹ GeV.  The number of neutrons and protons the starship has, then, is:

            N = 10⁴ 10²³ / 1 = 10² 

So the energy needed is:

            E = 10²⁷ 10¹⁹ = 10⁴⁶ GeV = 10⁴³ erg   

Using a super particle thruster that gives 1 G acceleration on the 10⁴ gm starship to attain Planck energy, the acceleration time is 

            M_s U_s = 10⁴³/10¹⁰ = 10⁷Δ 

            Δt = 10²⁷ sec = 10²⁰ yr

Even an increase of the thrust to 10³ G would not reduce the time very much. In its travel to a star, the starships velocity will remain just under 3.99 x 10¹⁰ cm/sec. Clearly, the particles of a starship will not attain Planck energy in its travel to a star.

3). Accelerating a large group of particles (a starship with astronauts)

A large starship will have an even greater problem than a small starship has in that it will take an even longer number of years to accelerate the starship to Planck energy. In addition, the starship mass and the speed achieved ensures that a large number of collisions with hydrogen atoms will occur in interstellar space while the starship travels to a star. The radiation from the collisions would probably be fatal to the astronauts inside.

 

Summary and Conclusions

We have explored the maximum speed that can be achieved by a starship, and found that it is limited only by the time that can be used for acceleration rather than the speed of light at low energy (2.99 x 10¹⁰ cm/sec). However, this time is so huge (~10²⁰ yr or greater), that achieving such superluminal speeds appears impractical.

 

Appendix A

For the proton to go beyond the low energy speed of light, we see from ref 6, AP4.7M that,

            v_o = nl_p/ Nt_p

            Where:

            n = the number of Planck granules crossed = number of Planck lengths traveled

            N = the number of disruptions along the travel length

            l_p = Planck length

            t_p = Planck time

When the spaceship energy is less than the Planck energy, N is equal to the number of Planck granules of space crossed, so:

            N = n, so        

            v_o = nl_p/ Nt_p = c

Now recall that the potential energy of a Planck granule is the Planck energy. Particles with low kinetic energy are not reflected at a Planck granule because the granule is too thin. As the super particle energy moves higher still toward the Planck energy, the super particle can pass right through the potential energy barrier of the Planck granule just like a super particle passes through the super particle potential barrier shell when the kinetic energy equals the barrier potential energy. Thus the particle cannot detect the potential energy edge of a Planck granule. Then the number of disruptions along the travel length (N) is reduced from n to the large edges like disruptions in space such as the shock at an edge of a solar system, a galaxy, a boundary of a corridor in the cosmic web or an edge of the observable universe. This is a much smaller number than n, but is never less than 1, so:

            n/N >> 1, so

            M_s U_s / (1- U_s²/v_o²)^½ = M_s U_s / (1- U_s²/v_o²)^½

                                                           = M_s U_s / (1- U_s²N²/c²n²)^½     

Now, U_s² increases slower than c²n² / N² because the large mass of the starship slows the increase of U_s², so the term U_s²N²/c²n² will eventually go to zero as the energy of the particles of the starship approaches the Planck energy, and the term M_s U_s will continue to increase. Thus the starship can achieve speeds beyond the speed of light at low energy (c). However, the acceleration time needed to move beyond light speed c is large.

 

 

References

1. L. H. Wald, “AP4.7L EXTRACTING SUPER PARTICLES FROM THE BARRIER SHELL” www.Aquater2050.com/2016/01/
2. L. H. Wald, “AP4.7I THE SUPER PARTICLE AS A COSMIC RAY” www.Aquater2050.com/2015/12/
3. L. H. Wald, “AP4.7C DARK MATTER RATE EQUATIONS” www.Aquater2050.com/2015/11/
4. L. H. Wald, “AP4.7 DARK MATTER AND ENERGY-FUNDAMENTAL PROBLEMS IN ASTROPHYSICS” www.Aquater2050.com/2015/11/
5. G. Kane, Modern Elementary Particle Physics, Ann Arbor, Michigan, Perseus Publishing
6. L. H. Wald, “AP4.7M VARIABLE LIGHT SPEED AND MODEL 1” www.Aquater2050.com/2015/12/