TUMBLING TOY UNIVERSE THEORY

Cosmologists believe that there are three fundamental topologies (shapes) that the Universe could exhibit over time. These are; flat, positively curved, and negatively curved. All three of these models begin with a big bang singularity, but their ends are as different from one another as you might expect.

In the negatively curved model the expansion of the Universe continues until its very fabric can no longer contain the expansion, resulting in what is termed the Big Rip. In a closed positively curved Universe, the Universe expands from the Big Bang singularity and then begins to collapse again into what scientists term the Big Crunch. In a flat Universe, expansion levels out and the structural fabric of the Universe continues to exist forever while its internal processes eventually succumb to heat death.

Although most scientist believe that space is flat, or almost flat, there are a few theories that favor a positively curved (or closed) structure. Among these are the finite, fundamental domain Universe that we looked at in a recent post, and the Big Bounce theory. The latter theory suggests that the Big Bang that instigated our own Universe sprang from the Big Crunch of an even older incarnation. Studies of the quantum fluctuations between each of these singularities suggest that while the physical laws of each consecutive Universe will resemble each other on a fundamental level, no two Universes will ever be precisely the same.

As I have said, the Big Bounce theory relies on our Universe being positively curved overall. I think that this theory is the most interesting of the lot, but I have arrived at some conclusions which are at odds with the standard model. For one thing I propose that our Universe actually consists of two equally sized Universes existing side-by-side. When the topology of one of these Universes is positively curved, the topology of the other is negatively curved. In this way the two Universes are always completely symmetric. At regular intervals these Universes switch their respective Universes.

They are, in effect, playing leap frog with one another.

The analogy of the leap frog game is misleading however, because it implies (as I have done) that there is more than one player. You can't play leap frog by yourself anymore than you can jump over yourself. However, in our Universal model, the two Universes are connected in such a way as to be a single Universe. The closed curvature of one Universe is merely the reversed expression of the open curvature of its neighbour. And yet the two are inextricably interrelated just like the way the concave of a spoon on one side is the natural expression of the convex on the other.

The curious thing about this theory is what happens to the seemingly separate Universes as they cross form one topology into another. In the case of a formerly closed Universe that is reaching its end in a Big Crunch, it would erupt into a massive explosion that would expand outwards into its new incarnation as a negatively curved Universe. On the flip side the negatively curved Universe would be approaching its demise in a Big Rip, partially driven by the Big Crunch on the opposite side. As both of these processes reach their culmination the topology flips and the dying Universe becomes 'reincarnated' so to speak into a positively curved closed structure.

In the violent interim between these two periods, space would appear to have a topology that includes positively, negatively and flat structured Universes all at the same time, and at different scales. It seems to me that there is enough time in this interim period, as it last almost as long as any other period on the timeline, to warrant the existence of a third incarnation of a Universe. This would mean that another big bang/crunch would have to occur as the Universes transition from the interim period into either the closed or open state.

The whole concept can be further visualized with the use of the physics of tumbling toys. When gravity acts on these toys they flip end over end. This is a lot like the way the Universe flips from positive to negative curvature and back again. For this reason I have decided to dub this theory the Tumbling Toy Universe Theory.

It could be that the Big Bounce theory is the correct one, but that its motion and topology is in fact produced by a careful adherence to the mean of both Universes proposed in Tumbling Toy. In this case all of the world lines of this Universe would cross, resulting in a massive singularity.

From a Direct Relativity perspective we could suppose that the curvature/structure of the Universe is evolving over time. To begin with, at the early stages of the Big Bang the Universe had a closed structure, this quickly evolved into the 'more-than-likely' flat Universe we see today, and will eventually develop into the hyperbolic (or negatively curved) system. This is a further example of Direct Relativity (DR) in action, see posts; Direct Relativity, Stellar Obscura, Within the Octonions. From a DR perspective this means that the Universe is accelerating (going from 0mph at the Big Bang, to light speed now, and superluminal speeds in the future) or it is rotating in some higher-dimension.

4d DOMAIN CONNECTOR

In this post I will outline my idea of a 4-dimensional link-compliments that are not of the Lobachevskian kind, but could be better termed as O'Neill Space. I suggest that this concept has great applications to the field of cosmology and will outline said applications in this and later posts. To begin though we need to a bit about modular forms, link compliments and Lobachevskian space.

3D Fundamental Domains and Lobachevskian Hyperbolic Space

There is a video on Youtube, called Not Knot, that goes some way to explaining the concept of fundamental domains of link compliments. Link compliments are any closed loop of string (or strings) that consist of any number of crossings. In the past, mathematicians had difficulty identifying if one link compliment was the same or different to another. But, by breaking down the loops into discreet sections called fundamental domains they can avoid this confusion by building a picture that is specific to each knot.
Note; To quickly visualize what is meant by a fundamental domain, think of a cube. When you flatten out its sides you get 6 squares in the shape of a cross. This is the fundamental domain of a cube. According to the video, there is only one fundamental domain for each link compliment making them very useful for identification purposes.

In the case of the Borromean Rings (as seen in this video) the looped sections are at right angles to one another and the fundamental domain is that of a cube. The domain is copied outside itself infinitely until it tiles all of 3-d space. 

By taking the fundamental domain of the Borromean Rings and adding a further 90º degrees to each axis we can make a dodecahedron that tiles all of space in a similar manner. This is called Lobachevskian hyperbolic space.

Modular forms such as these were imperative in solving Fermat's Last Theorem, but some mathematicians (like researcher Jeffrey Weeks) think that they may go even further by explaining the exact size and shape of the Universe itself. If correct, it would mean that the Universe is  finite in size, consisting of a single dodecahedron-like structure that is mirrored across space and time to give the illusion of an infinite universe. Our galaxy and everything else that is inside this fundamental domain would also be mirrored across space, resulting in an infinite number of copies of the Earth, you and me.

4D Fundamental Domains and O'Neill Hyberbolic Space

It baffles me why mathematicians have been content to chart the fundamental domains of 3-dimensional knots, but have neglected to investigate four dimensional ones. It seems to me that if you want to accurately create a picture of 4-dimensional space-time you would need a 4D fundamental domain.

In an infinite universe (whether it is repeated or not) giving your fundamental domain any specific size seems arbitrary. A 4D domain gets around this because all of its domains are free-scale. 

To create a four-dimensional fundamental domain we could begin by taking the order-2 domain of the Borromean rings and following each of the steps in the first video until we have created our 3D grid pattern stretching off into infinity. Then we can rotate this pattern around the 4th dimensional axis of the original cube until all of the domains that lie outside of the original cube domain now also lie inside of it. Obviously this would have to be repeated with each cube in the matrix (an infinite number of them), and then within each cube within each of those cubes (producing a far greater infinity). This mirroring would continue ad infinitum in a fractal manner.

Furthermore, each time a new cube is mirrored, it would alter the pattern relative to another domain, meaning that it in turn would need to be mirrored over again. This process, which would start off slowly at first would quickly escalate, as the 4D domain becomes more and more connected with itself. (See the next post The Singularity Conjecture for more on how this pertains to reality and the future).

Next, we have to understand what an infinitely tiled hyperbolic grid would look like and how it would sit in relation to the rest of the ordinary 3D grid. To do this you start by making a 2D grid of 9 squares (left) and invert it so that the central square lies on the outside, and the other 8 squares sit in place of the original central square (see below). Next, you make grid with 5x5 grid and invert it, then a 7x7 one and so on. Once you have mastered this you can begin to progress to a 3d grid structure, the inversion of which would be a hyperbolic grid.

However, just because you have mastered this does not mean that you can create a 4-d fundamental domain with ease. There are still many thought-processes and trials which must be taken. The apparent shape of one of these domains is a rhombicuboctohedron (see below), and which, oddly enough, is related to the dodecahedron of Weeks domain proposal;

However this object must exist inside a our original cube domain, with lines leading to each vertex, shown below;

These lines join to create the following number of shapes; 6 cubes, 8 tetrahedrons, and 12 triangular prisms. The rhombicuboctahedron can be rendered in the following topological manner;

These are three different angles of the same hyperbolic domain. The original domain looks something like this (see below);

The net of this form looks like this;

And finally the 4D domain looks like this;

In the infinite grid of cubes, the central cube that is our original domain is surrounded by 26 cubes. This block of cubes, which is also a cube, is further surrounded by a nest of 98 cubes, which is surrounded by 218 and so on. If we imagine that our original cube domain has a volume of 1 metre cubed then the next set would have 26mˆ3, then 98mˆ3 and so on. When all of this is rotated around the fourth dimensional axis, it is clear that space is becoming larger the closer you get to the centre.

This doesn't make any sense, the space is becoming larger on smaller scales, which ought to be measurable in some way. The reason why it isn't is because these domains lie for the most part stacked in the fourth dimension and a considerable amount of their bulk lies hidden there, only accessible to us through line of sight perspective. However, it is clear that this space wants to find room for itself in 3D space, and this need, it appears, is driving the expansion of the Universe.

O'Neill Space is unpacking itself from the fourth dimension. The entire Universe is exhibiting this unpacking in the form of expansion just as an object that is slowing down from a tremendous sub-luminal speed undergoes a length expansion. 

As each nested cube expands, cubes within those cubes being to expand so the whole process is exponential. Furthermore, because the tiling of the fourth dimensional fundamental domain is infinite, this unpacking will continue indefinitely. 

Perhaps even more phenomenal, is that this theory suggests that the fourth dimension is not time, as Relativists believe it to be, but rather ordinary space. The process of its unpacking from one dimension to another, however, does take place over time, and this means that if you travel fast enough you will undergo a length contraction taking you against this unpacking flow rate. Obviously the further you gain access into the hyperbolic grid structure of O'Neill Space the slower the unpacking appears. This could mean that the boundary between both spaces is something like the event horizon of a black hole; a point of no return for these cubes of space. Time is nothing more than the perceived rate at which fourth dimensional O'Neill space is unpacking into ordinary 3D space, in this writer's opinion. 

Note; There are three basic topologies for the Universe; positively curved, negatively curved and flat. According to the main proponent of the finite universe theory, Jeffrey Weeks, the existence of dark matter does not favor a negatively curved universe. This is strange, however, because the dodecahedron tiled space that he envisages is a Lobachevskian Hyperbolic and therefore negatively curved topology. There must be some explanation for this obvious disparity, but I cannot think what it might be.

Note; While space-time is known to be a 4-dimensional vector plane of 3 space and one time, its topology is not explained by the 4-dimensional algebra equations of the quaternions. The reason for this is that quaternions are only spatial and do not deal with time. For this reason some scientists have concluded that the 4th dimension is not time, but an added spatial dimension interacting with ours over time. Time, they believe, must be an extraneous principle acting over all dimensions. I would agree with this concept.

HOLOGRAPHIC TIME GRATING

The Large Hadron Collider is getting closer and closer to its stated aim; finding the Higgs Boson. The Higgs is a particle which acts as an exciter to the proposed Higgs field, and thusly gives mass to matter. The main determinant in an objects mass is the amount of energy the object is divested of. However, this energy is only really converted into mass in the presence of the Higg's field.

The blue/green zig-zag line depicts the path

of a particle as it moves through the omni-

present Higgs Field. The straight green line

shows a different particle moving at the speed

of light, and therefore avoiding the Higgs Field

altogether.

The green/blue zig-zag line is the path of a particle that is stationary – i.e. moving with respect to time (the vertical axis) only. It will keep bumping into the field lines (black lines) of the Higgs Field because it is traveling in an over-all perpendicular trajectory to said field lines. A particle that is traveling at the speed of light (on the other hand) warps space-time according to the Lorentz contraction thereby moving in a diagonal trajectory. This means it avoids the field lines altogether and therefore has no mass and shed loads of energy.Time does not pass for it, it travels in space only. 


The particle which encounters the Higg's field has its energy divided by the speed of light squared, as per Einstein's famous equation E=mc^2. This means that its large energy value is divided by an almost equally large constant, giving rise to a relatively small amount of mass.

The Higg's field is described as omnipresent, which means that it exists everywhere in space and time. If this is true how could any particle avoid it and in doing so become massless? Diagrams which are devised to show particles interacting with the Higg's field, depict the field as a series of horizontal lines. I submit that these lines are, in fact, equally distributed in time as well as space, meaning that you would encounter precisely the same field lines if you were traveling at relatively low speeds, as you would if you were not traveling at all. This suggests that the Higg's field is akin to some kind of Holographic Time Grating. A particle which is traveling in the time axis only is continuously bumping into this field (which is time itself) causing a zig-zag motion, which slows it down and gives rise to mass.

Stylised diagram showing ordinary space

and Lorentz Contracted space in the same graph.

Now this is all very interesting. But what is even more interesting is how the omnipresent Higg's Field interacts with matter on a 4 dimensional level to create reality, as we know it. For a demonstration of how this happens please watch the video below.

You may have noticed that the interference patterns (which are holographic in nature) are, for the most part, composed of zig-zagging lines; just like our particle trajectories.


What this means is that beyond (or outside of) the Higg's field there exists an expansive holographic interference pattern that is 4D in nature. This interference pattern is the mesh of everything that will exist, everything that has existed and everything that could possibly ever exist. In short, it is God i.e. a self-organising pattern that is timeless and intelligently designed.


But if an omnipotent God exists with a map and a timeline to all of creation, how can free-will be also said to exist? 


Only parts of an object's entire 4-d wave function can be interpreted through the minute, straight-line apertures of the Higg's field grating, at anyone time. These snapshots, or 3D manifolds of the objects exist – played out one after another – but they are not to be thought of as sequential, existing as they do through-out the entire spectrum of the waveform.


This 4D wave that describes all of the possible interactive and internal states is, of course, holographic in nature, and therefore, a fractal. This means that any part of the information of the entire wave function can be accessed at any one time, much like a computer can access files on a hard-drive.  It is by the same application of the above process by which free-will is preserved and restored.