The Geometry of Subspace:
A New Approach to a Unified Field Theory
From The Theory of Ubiquity
Abstract: There has been much speculation about the nature of
matter over the recent decades. Presented here is a model that is entirely
different than the quark model or the string model. It is a model based
upon singularities, which are subspaces that have an infinite space on the
inside instead of the outside. From a simple geometric construction it is
possible to model an elementary particle with all of the properties of gravity,
electromagnetism, and nuclear strong forces. This model also provides additional
insight into Quantum Mechanics and General Relativity that may make these
theories understandable to the public at large.
Key Words: Electron Model, General Relativity, Unified Field
theory, singularities, gravity, neutrino, charge, nuclear Strong Force,
electromagnetic.
Introduction
This seems to be a strange starting point, examining how space can contain
space, but in the past we have thought of space as an absolutely uniform and empty substance that can contain matter and various kinds of force fields.
It has been a stage or backdrop to the objects of importance but of itself it's of very little consequence.
However, over the last one hundred years of scientific thinking space has proven to be more than just a backdrop. For example, objects cannot move faster than the speed of light. Why the limitation?
If it's only empty space, how can it force a limitation on objects? Likewise, light cannot move at any other speed than the speed of
light. Also, if space is indeed empty then the boundaries to the objects in this space should be very distinct. An object is either occupying space or it isn't. But the truth is that the boundaries are not very distinct. There's an uncertainty to the measurement of an object's energy and momentum across space.
"Space" appears to be more than just space.*
There is an alternate theory to the concept of matter occupying space. This is the "Theory of Ubiquity" where each particle of matter consists of concentrated space.
And it's this concentrated space alone that accounts for all of the properties of matter. This is the approach taken by this paper.
Removing all concepts of masses and force fields, the simple geometry of subspace is sufficient to explain the interactions of physical matter.
Subspace: The Concept of Concentrated Space
How can Space occupy space? If we can accept the concept that matter can
occupy space then why can't we accept the concept of Space occupying space? One problem that we have
is that we consider space to be infinite in size, without bounds. How can
Space that's infinite in size occupy space that is also infinite in size without consuming all of it?
There would be no uniqueness to this Space. Everywhere would appear the same.
Instead, let's define a Space called a subspace that is somehow trapped within a region of space. The region of space may be fixed, but the subspace contained within it is infinite.
Picture a ball of infinite space. We could stick our whole arm into this ball and it would appear as if we lost our arm but we didn't. We could pull our arm out unharmed. Strangely enough, we could safely place this subspace on a table without losing the table. The table is larger than the outside of the subspace. The subspace is only infinite within itself.
If we could look within the structure of the subspace, we would see an amazing sight. If the width of the sphere was 10 centimeters,
for example, and we could measure its depth, we would see that the first centimeter measured from the outside edge towards the center is actually 10 centimeters long. The second centimeter is 100 meters long, with the third being 100,000 kilometers long, and the next 10 light years long.
The last centimeter is infinite in size with the innermost center a point of infinite size.
These numbers are arbitrary but they help us to see the concept that such a
situation could be an entire universe.
How do we contain this subspace? The sphere, if it were not contained, would grow at some unknown rate consuming the table and everything else that's in the universe. In fact, the only way to contain this subspace is with another subspace, both being infinitely large they couldn't fit into each other. But before we determine how a subspace interacts with another subspace, we need to redefine the concept of space.
The Detection of Space
What is space? Space can be viewed as a set of points that occupy space. Each point is actually spaceless; a point has no dimensions. Rather, it is the distance and direction between the points that defines space. But what connects the dots? What produces this relationship between one point in space and another?
The easiest place to start is with a line. A straight line through space is an instantaneous connection of points. Since I am not talking about energy or information, I am not limited by the speed of light. Now imagine that this line is through a subspace. If it connects the edge of the subspace to its center point it would be infinitely long. Would the connection of dots still be instantaneous? Substitute the words "infinite speed" for instantaneous. The meaning is identical, except we have now provided direction to our dot connecting process. Mathematically we can not derive this dotconnecting period, but as sometimes happens during physical model formulations, we are going to make a leap of logic here.
What we are actually viewing is the definition of the speed of light. It takes time to connect an infinite set of dots even at an infinite speed. And this time will always appear to be the time required for light to move across the corresponding distance of regular space. For the sphere of ten centimeters, this process would be of the order of 3 X 10^{10} seconds. Note also that there is direction to this process; there are two types of subspaces, ones that appears to be sinking light and ones that appear to be a source of light. Light is the viewing of the infinite in the finite.
This is the actual appearance of the subspace in normal space. Its space appears to us as light. This is what I call virtual light. Its wavelength is related to distance that it's traveling. The smaller the subspace the greater the energy within. The larger the subspace the lesser the energy. And since this light is a virtual light, the subspaces are not creating or destroying energy. However, they are holding energy so the concept of stored energy must be related to subspace, in other words, the mass of a particle.
At this point, it is possible to describe an elementary particle made from two subspaces. First, however, we need to provide the seven rules regarding subspace behavior.
Seven Rules of Subspace Behavior
Rule #1 
Subspaces exist within space. 
Rule #2 
Subspaces are always infinite in size and are in expansion. 
Rule #3 
Subspaces consist of virtual light between their center points and their boundaries. 
Rule #4 
There are two types of subspaces, one sourcing virtual light and one sinking virtual light. 
Rule #5 
Two subspaces of the same type can not overlap. 
Rule #6 
Virtual light can be summed together between subspaces.
This light can be viewed as a perpendicular and tangential component to the subspace boundaries. 
Rule #7 
The perpendicular aspect of this light must equal zero or the boundary between the subspaces will move. 
Model of an Elementary Particle
An elementary particle is the basic building block used to construct all other particles. As such it must contain all of the properties of matter; it must have mass, it must have a gravity field, and it must explain the electromagnetic attraction and repulsion between charged particles. The following model meets these criteria. First however we must resolve the puzzle of how to contain the subspaces within this particle.
There are a number of assumptions that we can start with. First we only need to use two subspaces, this is an attempt to keep the model as simple as possible. Secondly the subspaces must be of the same type, rule #5. Only subspaces of the same type, either sourcing or sinking virtual light can contain each other. Thirdly the size of the particle may not be finite. The influence of a particle extends throughout space. It may be that in some multiple dimensional way the particle may be extended through space as well.
If we take the simplest possible model of two spheres of subspace side by side we immediately note that neither subspace is contained. If we surround one sphere with a donut shaped subspace, one who's infinite centers are on a ring, then we see that the sphere is only captured on the plane of the donut but would spill out above and below. We will have to change the donut subspace into one whose infinite centers are on a sphere surrounding the first subspace to keep the first subspace in check. But what would keep this second subspace in check? The sphere of infinite centers would have a tendency to shrink to a point, except that the first subspace is in the way. However, their dimensions are perfect matched to allow this second subspace to collapse around the first without overlapping it. The second subspace would not be contained.
Surprisingly enough a model does exists, one that is derived from Einstein's General Theory of Relativity. This theory reduces the force of gravity to a curvature of space and time around a particle of matter. It is this curvature that causes it to attract other particles. It even attracts light. And it is this light that is the most fascinating part of this attraction.
A gravitational field described by the following equation bends all paths of light.
[1 (2GM)/RC^{2}] d(CT)^{2}  [1 + (2GM)/RC^{2}] dR^{2} = dS^{2} = 0
Many papers describe the effect of gravity upon light as it moves past an object's gravity field. The gravity causes the path of light to bend around the object. What is the effect on light as it moves directly into an object? The gravity actually generates a shell of light that is the model of the elementary particle.
The following diagram shows the math behind the elementary particle model. Taking the special case of light entering the particle, a model is made by adding one additional real dimension that I have labeled as W. The other effects of gravity are not shown in this model. For now it is important to note that gravity exists for the model but it is more like an unseen presence.
This diagram shows a two dimensional slice of the model with the W axis and one of the three dimensions represented by R the radial distance from the center of the object. This shows a plot that is almost parabolic in appearance. Instead of being equal distance between a line (the W axis) and a point (on the sphere) this is equal distance between a line and a circle around a point. This is the basic description of an elementary particle.
This model is comprised of two subspaces that are locked together. The innermost subspace is not a sphere but is cylindrical in shape along the fourth dimensional axis, identified as the W axis. Instead of having just one center of infinite space, there is an infinite set along the W axis. The space is concentrated inward toward the W axis. The virtual light would appear to be moving from this W axis.
The second subspace is wrapped around the first subspace at the W = 0 point. Its infinite centers exist around a sphere, a fixed distance away from the center point of the particle. Its virtual light is moving from this sphere. The easiest way to picture this subspace is in a threedimensional slice as follows:
Note that the slice looks like a ring shown above in green. That is because the infinite centers are not on the sphere but are around it as shown by the circle in the two dimensional drawing. In fact, the only way to comprehend this model is to look at it in threedimensional slices. It is important to remain mindful of the fact that we are looking at slices of the image. It is hard to comprehend the overall fourdimensional diagram.
The overall particle model consists of these two subspaces in conflict with each other. Now picture the line expanding in width and the ring trying to shrink. There would be a point where they would stop each other. This is the model that is shown by the attraction of light by gravity.
In this model the dimensions of the two subspaces are not perfect matched which prevents the second subspace from collapsing around the first. The first subspace can not expand pass the second subspace at the w = 0 coordinates. This produces an hourglass intersection where both subspaces are pressing against each other. I call this the Area of Compression. This intersection is the space described by Einstein's General Theory of Relativity for the case of light being drawn in by the gravity.
Gravity
Before we continue with the particle model description we need to go back to the subspace definition and explore the definition of gravity. In it's simplest terms it is like the concept of a merrygoround. If an object is rotated around a fixed point in space that's is located outside of the object's center, it experiences a centrifugal force. This force will throw the object to the outside towards infinity.
If an object is being rotated in a subspace around a singularity then it experiences a centripetal force. This force will throw the object to the inside towards infinity.
But the real question is what's spinning? It's the dot connecting process, the definition of virtual light. The frequency of the virtual light is directly related to the spinning effect of singularity space. And this spinning is only detected at the edge of the subspace boundary. This is where we see the actual light moving around the subspace.
The Static Particle Model
This area of compression is composed of light. To understand this intersection we need to review how the virtual light from the two subspaces react. For this example the type of subspace that sources light will be used. Both subspaces contain light moving out from the infinite center points out toward the edge of each subspace. Because the two subspaces are different in shape, the patterns of the virtual light are different. The first subspace has light moving perpendicularly from the w axis toward the hourglass intersection. The second subspace, viewed as a circle in the two dimensional crosssection, has light moving perpendicularly from the circle toward the hourglass. As these two virtual lights touch, the vector addition of these two lights appears as a light along the intersection surface. The perpendicular aspect of this light equals zero. Because of this fact, both subspaces are fixed.
If both subspaces are sourcing virtual light, then the parallel addition of this virtual light on the hourglass would appear as light leaving the particle. If both subspaces are sinking virtual light, then this parallel addition appears to be entering the particle. This light is the overall energy of the particle and for an elementary particle equals hc when integrated across the waxis. The energy at the neck of this area, at W = 0, is the mass of the particle.
It is important to note that the phase of this virtual light can come in different frequencies. The wavelength of this light is a multiple set of the distance that the light travels. The wavelength can equal the distance (n=1), half of the distance (n=2), one third of the distance (n=3), etc. Also the phase changes as you move around the particle. The light sheet is continuous without sudden discontinuities or voids.
Elementary particles can only exist in discrete sizes. This is because the gravity field is directly related to the size of the mass that produces it but the energy of the mass is inversely related to the size of the gravity. A larger gravity curvature would result in a smaller mass at the neck of the hourglass; the light around the neck being reduced in frequency. A larger subspace would have a greater centripetal force but the energy around it would be less. And since it would start to attract more light it would not remain in balance and would have to reduce in size giving off energy.
These balanced sizes equal n^{1/2} x 1.58 x 10^{8} KG, where n equals the number of wavelengths of light around the hourglass, matching the number of wavelengths of the virtual light. Note that the electron is only 9 x 10^{31} KG and the proton is 1.7 x 10^{27} KG. Thus, our known particles must be a composition of both matter and antimatter elementary particles. But how can matter and antimatter interact?
Particle Interaction
The basic definition of a matter / antimatter particle depends upon the subspaces that make it up. By convention a sourcing subspace produces a matter particle and a sinking subspace produces an antimatter particle. I may be wrong on this but we will have to leave this for the future.
To understand the interaction between matter and antimatter, we must first understand the effects of the virtual light on the boundary of the subspace. The virtual light can be viewed as a force whose strength is related to n.
As shown in the next set of equations the virtual light on the boundary between the two subspaces cancels out, except for the light that moves along the surface of the intersection surface. The perpendicular component of the light is equal and opposite from both of the subspaces. If there were a perpendicular component of this light, the surface would move accordingly. The influence on another particle's subspace could cause this effect.
This force is the force between two charged particles. Since the virtual light is related to the n (the number of wavelength) times hc, and the finestructure constant shows a relationship between q^{2} and hc, then we can represent this force as:
F @ (nhc)/2pr^{2} @ (nq^{2})/2pr^{2}
The attractive and repulsive nature of the force is dependent upon whether it is matter or antimatter, it's made up from either sourcing or sinking subspaces. The interaction between particles is the force of electromagnetism
Two like particles, mattermatter or antimatterantimatter will repeal each other due to the combination of forces from their inner foci. Likewise, a matterantimatter combination will attract each other up to the 2GM/C^{2} limit. This 2GM/C^{2} limit is related to the Nuclear Strong force that counteracts the force of two charge particles across a short distance. It appears that two like particles can stay within this limit without experiencing a repulsive force.
One last point of interest is this circle around the sphere of the second subspace. Looking at a three dimensional slice there may be a light contained within this space as pictured here as a blue line. This light is made from the virtual light of the 2GM/C^{2} wavelength. This light would be standing alone from any other light and the strength would be related to the mass of the particle.
Composition Particles
It is possible to develop particle models where composite particles consist of both matter and antimatter particles. In these models the summation of masses must add up to zero, since the elementary particles are much heavier than any known particles. The relativistic movement of the masses provides the delta changes that give the particles their masses. Likewise, the forces are added together with the residual forces accounting for the charge of the particles. Since the mass is related to n^{1/2} and the electrical force is proportional to n, the generation of the model is quite simple.
Component 
Neutrino 
Electron 
PiMeson 
PiMeson 
KMeson 
KMeson 
1 
1, 1 
+2 
1 
3 

2 
4 

1 
+2 
+3 
1 

9 


1 
1 
+2 
+2 
16 




1 
1 
Excess n 
0 
2 
2 
0 
2 
0 
The list of particles could go on forever, but a simple list is presented with a possible matching of known particles. Looking at the simplest particle construction shows a particle of no mass. This particle also has no charge. It is hard to determine its stability, but it must move at a speed close to the speed of light, since the two masses cancel each other out. There is no relativistic mass addition to give it any weight.
The next model shows two matter particles moving around an antimatter particle. Each of the outer particles has a mass of 1, and the innermost particle has a mass of 2. The two outer particles rotate around the inner particle billions of times a second at a distance comparable to the Compton wavelength. This produces the relativistic mass change that equals the weight of the electron. Of course, this raises the question about the resultant leftover forces. The inner particle has a spin of 4, and the outermost particles have a spin of 1 each. Is radiation generated from the spinning particles? If we could look at the light, we could see that the gravity actually bends the light so that the effects of the spinning particles are localized and cannot be detected outside of the composite particle. The effect is localized just the charge and the gravity can be detected from afar.
This model does shows various characteristics of an electron. It has a charge and a magnetic moment. The charge is distributed in a ring around the elementary particles that make up the electron.^{ }Its energy is distributed across space except where it connects with a positively charged particle like a proton. At that point the overall energy of the particles are joined together. This is part of the bonding process that holds atoms together.
One last thought about particles that are not on this diagram, those that are identified by the term dark matter. These are particles of matter that have energy and gravity but they have no light shell to interact with light. I feel that there are other models for particles perhaps consisting of more than two subspaces that would react entirely different than the particles listed above.
Quantum Mechanics Model
This model can provide some insight on Dirac’s Equation of Relativistic Quantum Mechanics. These equations describes the quantum effects of physical matter by examining the energy and momentum generated by four waves identified as Y1, Y2, Y3, and Y4. The equations are as follows:
(P_{O}+MC)Y_{1} = P_{1}YY_{2}  P_{2}Y_{4}
+ (2)^{1/2 }P_{3} (Y_{2}+Y_{4})
(P_{O}MC)Y_{2} = P_{1}YY_{1}  P_{2}Y_{3}
+ (2)^{1/2 }P_{3} (Y_{1}+Y_{3})
(P_{O}+MC)Y_{3} = P_{1}YY_{4}  P_{2}Y_{2}
+ (2)^{1/2 }P_{3} (Y_{2}Y_{4})
(P_{O}MC)Y_{4} = P_{1}YY_{3}  P_{2}Y_{1}
+ (2)^{1/2 }P_{3} (Y_{1}Y_{3})
where P_{O} = d/dt and P_{1}=d/dx, P_{2}=d/dy, P_{3}=d/dz
This model of a physical matter can also be described by four waves. Two of these waves relate to matter and two relate to antimatter. When we view a physical particle at rest we have been viewing it in threedimensional slicing. I believe that there are only two slices of light in an elementary particle, not an infinite set.
The gravity of the particle is across all three dimensions. But the light can only be pulled into two perpendicular slices or spins. This is because light can not overlap. In any one piece of space there is only one light sheet. We may think that light is moving in all directions without interference but that is because of the complexity of the gravity. The light that we see around us is the effect of acceleration of charged particles where the gravity has actually been bent or distorted and this distortion is moving out at the speed of light. An energy shell change where an electron jumps from one shell to another is still this same effect. Light actually rides on the gravity.
The light moving around the particle can only occur on two great circles. Anything else is interference and would shake the particle apart. Everything is in balance.
Relativity Model
The Special Theory of Relativity has its origins in the fact that light travels at a constant speed aptly named the speed of light. All of our measurements of time and space are based upon the fact that the information obtained about time or distance came to our viewpoint through light. In other words it took time to get the information.
The simplest way to look at the theory of Relativity is using this diagram. The sin of the angle equals (1V^{2}/C^{2}) ^{1/2 }where V is the velocity of the object and C is the speed of light. Its distance is increasing with time the two long lines of the polygram where X=VT. The major argument against the Theory of Relativity concerns those two small triangles on the right side of the figure. They show a reduction in the size of the object and a slowing of the time of the object.
The relationship between a particle's mass, its momentum, and its overall energy is shown in the following diagram. Note that the C is a conversion factor. Momentum only occurs when an object is moving. At rest the energy of the particle equals MC^{2}. When an object is moving it's overall energy is increasing, at slow speeds this equals 1/2 MV^{2}. Overall energy equals M/(1V^{2}/C^{2})^{1/2} and momentum equals MV/(1V^{2}/C^{2})^{1/2} .
The diagram shows how an elementary particle moves. The critical part is simply the doppler. If we move towards a light that is coming at us, its frequency shifts into the violet region (higher frequency). If we move away a light that is coming at us, its frequency shifts into the red region (lower frequency). The amount of doppler shift depends upon the speed. We are ignoring relativity for the moment.
The model at rest is perfect synchronization with the subspace virtual light
in frequency and in phase matching the light moving out of the particle and around the particle. During movement a mismatch seems to occur, the light moving forward
(toward us) is stronger than the light moving back (away from us). This doesn't effect the light model since the model is at rest to itself. The synchronization remains.
Notice also that there is a distortion in the overall particle model. It is no longer balanced on all sides but it has a stronger gravity on one side and a weaker gravity on the other.
The most noticeable effect is the relativity effect. The constant G has changed with velocity. G = G_{0 }(1V^{2}/C^{2})^{1/2}. This causes the time and space reduction. The subspace actually contracts in all dimensions not just the one it traveling in. Since the time axis has contracted (along the area of compression) the frequency has increased, energy equals M/(1V^{2}/C^{2})^{1/2} .
Conclusion
The theory of Ubiquity as presented here will hopefully add to the discussion
of subspace and its geometry. The concepts presented here are based upon
singularities, which are subspaces that have an infinite space on the
inside instead of the outside. From a simple geometric construction it
appears to be possible to model an elementary particle that has all of the properties of gravity,
electromagnetism, and nuclear strong forces. This model can also bring additional
insight into the areas of Quantum Mechanics and General Relativity that may make these
theories more understandable to the public at large.
*The old concept of "space" as being the infinite void, will be
spelled with a small "s" while the newer concept of
"Space" which has characteristics and geometry will be spelled with
a capital "S."
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