Jupiter and Mars in their orbits – Source: NASA

Isaac Newton believed that space is the background where motions are taking place in it. Therefore, he took space as an actual body and absolute benchmark for the universe. On the contrary, Gottfried Leibnitz believed that heavenly bodies are moving relative to each other, but there is no background. Thus, he denied space as being an actual entity.   Ernst Mach had a similar idea, but he introduced the acceleration as a contributing factor for the motion of stars.  He believed the relative motion of heavenly bodies is affected by mass distribution throughout the universe.
Albert Einstein however, introduced the combination of space and time as the cosmos background. He declared space and time as actual entities. In his belief, although space and time individually are relative and malleable, the space-time combination builds a solid background for the universe. In his General Relativity Theory, space-time is vibrant and active in the evolving world. The gravity of stars changes the shape of space and curves it. The curved space bends the trajectory of other stars. Time can dilate or shrink depending on objects relative speed. Hence space-time is not just a rigid and passive background but a dynamic entity.  
Einstein’s model has passed the test of time and many experiments and observations confirm its precise predictions. Underneath, I will adopt Einstein’s space-time fabric and build my model around it. 

Singularity and Space-Time 
My conjecture is that space-time universe is enclosed inside the proposed singularity. In this scenario, singularity has to have an internal dimension, which is against our original assumption. Maybe we can assume that our universe dissolves in singularity. But 'dissolve' is not the right word either. Even overlapping does not define the actual meaning. How then can I suggest that space and time is -in a way- embedded in singularity, so does matter and everything else, which exists in our universe?
The singularity, as defined raises new questions and concerns that we have to offer reasonable answers for.
The first question which may come to mind is:
If singularity is a mathematical point, how can our enormous space-time universe grow inside it?
For the sake of argument one can look at the image of a three dimensional object in a zero dimension (0).

Singularity

As the diagram shows, the image of a three-dimensional object in a two dimensional world (book page, monitor screen) would be reduced. That same image in a one dimension (X-axis) is just a line and in no dimensions (point zero) the image coincides with zero. No matter how big the object is, its image in 0 has no size. In other words, no-dimension (0) can accommodate the image of any size object. Formerly, We have decided that matter can not exit in singularity but its image does.
Physical reality of the above can be seen in holography. In holography, a two-dimensional picture can contain all of the information of a three- dimensional object. “Even a tiny fragment of the 2-D holographic plate contains enough information to reconstruct the whole 3-D image of the object.” ⁶
The algebraic can be written as:

Any number times Zero = Zero

X * 0 = 0

As long as we deal with finite numbers, the above equation holds. In space-time we are dealing with finite numbers.
From the above arguments, I want to conclude that if we deliver the universe to singularity, the dimension of it in singularity domain is reduced to zero. At this time let our imagination fly and help us to visualize the relation between space-time and singularity. Just suppose you are looking at an object in a mirror. Can you claim any interface between image and mirror itself? In this scenario, when you look at the image, you will also see the mirror everywhere. However, the three-dimentiality of the object and two-dimentiality of image make them substantially different entities although they are very closely related. Similarly, we can limit the image to a zero-dimension arena. Are the above very simple substantiations? Roger Penrose believes that as we develop deeper and deeper theories, the mathematics become simpler. Reality in its basic level has to be simple and beautiful.

Be yeki naghsh bar in khako, bar an nagsh degar
Dar behesht abadio shekarestan mano to
 

In one mold we are in earth, but in another mold
You and I are in an infinite sweet paradise
Rumi

Shrinking World 
Einstein's Special Relativity tells us that when an object is moving relative to a stationary framework, its size shrinks for someone residing in that framework. The amount of shrinkage depends on speed of the object. If we take this principle to extreme, for a framework which is not moving at all, the moving object downsizes to zero.
If the notion of distance is not present in singularity, logically there is no movement for such an entity. Therefore, we can take it fixed and motionless. For such an entity, the size of moving space-time will be reduced to zero.
I have adapted this concept from The Universal Theory of Relativity by Roland Michel Tremblay and Big Shrink by Richard Quist. It is interesting to note that the above authors reflected and expressed the non-locality and shrunk world that they sense in their minds’ domain to a new model for the universe.

Super-space
The need to look beyond the ordinary four-dimensional Minkovsky space-time to explain many physical phenomena has been noticed for some time.  Many phenomena, for example, electromagnetic field cannot be explained in the context of a four dimensional universe alone.
To explain these mysteries, main stream physicists chose to theorize another space-like manifold attached to ordinary space-time. This manifold is called super-space. In basic terms the super-space idea presumes that the points in space-time are actually cross sections of bundles which are extended in this proposed super-space. 

Every point in space-time is a cross section of an extended thing.

Extra dimensions in super space are called internal dimensions. Therefore, supposedly, every point in space-time has internal dimensions that are out of site.  Interestingly these bundles (fibers) are frequently represented by a complex vector one-dimensional space, which include an imaginary portion.  
The introduction of super-space opens up a can of worms and exposes theories to the doldrums of extra dimensions.  The Super String Theory is proposing up to 7 extra dimensions. Did super-space solve the mysteries? Not quite. On the contrary, it created a lot of chaos and dragged the theoretical physics to places that are far from objectivity.  Maybe it is time to forget about extra dimensions. Maybe it is time to think about a non-space like entity adjacent to our universe. Here, I am proposing the singularity as the non-spatial essence adjacent to the space-time.

If the above assumption is true , where are we going to find the singularity? Where are the boundaries of space-time? One can further speculate, if space-time universe is expanding, does it crystallize and push the singularity away? Underneath I will try to use simple mathematics to offer a solution for above questions.

Real and Natural Numbers
Our world is built based on mathematical fundamentals. Therefore, we frequently look at mathematics to build physical models.
We have two different mathematical choices to adopt in order to build a model for space-time.
If we take real numbers to represent space, we may take the world as a continuum. Real numbers with infinite amount of decimals (2.567854332234…) stands for continuity in the field. Then space is continuous as well.
In contrast, natural numbers (1, 2, 3, 4, 5……) with their discrete nature symbolize discontinuity of the field. If we adopt natural number system as the model then our space-time at fundamental level has to have units and therefore it is discrete.
But as it was mentioned earlier the correct mathematics at fundamental level is the mathematics of complex numbers. In fundamental level, we need to include imaginary numbers in the equations. So perhaps we have to build our model base on the complex system.

Imaginary Numbers
In mathematics, the square root means the sides of a square with area a. Imaginary numbers (square root of minus numbers) are frequently being used as an integral part of quantum mechanical mathematics.

By definition, square root of a (√-a) is the side of a square with area equal to -a. What kind of a square would have negative sides? It cannot have a meaningful real (tangible) area. It cannot have a meaningful dimension either. Dimensions of such an square are not similar to real world four dimensions. They are imaginary dimensions.

In theoretical physics these numbers are representing the hyper-space (a space beyond 4 familiar dimensions). Assuming a type of hyper-space is essential in building theories about nature of the universe. So, It seems that in contrary to string theory assumptions, hyper-space cannot have a real dimension or area. I call the Hyper-space in this model the singularity which does not posses any real dimension. However, it can accommodates image of real dimensions.

Complex Number Mathematics
Now let us go back to complex number system (a two dimensional system which consists of a real and an imaginary coordinates). It has been explained in detail in the first chapter.
Assertion C#3 indicates that discontinuity of real numbers always happen around point zero. One of the strange characteristics of imaginary numbers is the fact that the real value of any real number coupled with (multiplied by) them will be reduced to zero.

 

As shown in diagram above when we multiply any real quantity by i its real value (Its real number coordinate value) is reduced to zero. Its algebraic can be written as:

(X+0i) i = Xi + 0(ii) = Xi

Therefore, the real value disappears and imaginary value fully demonstrates itself . In trigonometry we can show the fact as;
X = r Cos a,   Since we took a = 90 and, Cos a = 0 then, X=0

The complex number equation Z = R [cos a + i sin a] indicates that these numbers also have a periodic nature. So they loose their real number value and hit zero twice in each period which indicates discontinuity in real number field. Therefore, any space-time element (space, time, matter) shown by X coordinate, as it couples (multiplies) by imaginary number, looses and regains its real value periodically. For example, if x indicates dimension and distance, because of the function of the complex system, the space has to disappear and reappear during each period. This is the basis for our assumption that space and time are discrete and not continuous.
Erwin Schrodinger was one of the first physicists who suggested a discrete space. Einstein in his last published paper, having quantum theory in mind, also proposed that a discretely based theory might be the way forward for the future physics.
As it was mentioned before, in this model we take the zero point on the complex number plane to represent singularity. The imaginary number (i) represents the singularity effect on different space-time phenomenon.
Assertion C2 indicates that continuity of real number breaks down intermittently. What are we going to get out of these argument?
If we take the space-time as being discrete and if in every period the real value hits 0, then we may conclude that space-time is spread over singularity like a web. But then we assumed that the singularity is zero size. How are we going to imagine a web spreading over zero?  We may find the answer by noting that the X coordinate denotes the value for dimension in space. Dimension is not defined in singularity. Assertion # S1 denotes that singularity is a separate entity. We are talking about two separate domains. If we cannot expect and assume dimension to singularity, then we have to depart from thinking objectivity as the only tool. Next we let our mind’s eye lead us and imagine a 4-dimensional universe. Then we come to realization that the universe intermingles and amalgamates with our mind (a 0-dimensional entity). This is a very suitable analogy.

One can also argue that, singularity is a point, because we are not comparing it with anything of its kind, in its own domain. Trying to assess the singularity with space-time parameters is wrong. However, we need to elaborate more about the elation of these two separate entities. When talking about incomputable parameters, Douglas Hofstader concludes, 

Undecidable propositions run through mathematics like threads of gristle that criss-cross a steak in such a dense way that they cannot be cut out without the entire steak being destroyed ⁷

What does this actually mean in real world?
Assertion C4 indicates that any point in the plane can be considered point zero. This supports are assumption that point zero is present in proximity of any miniscule of the space-time meshwork. Or we can further assume that zero is present next to each atom of pace and time. It also interweaved with them.
Considering every point of the domain as zero is called blowing up the origin.This is done with a mind-state that localizes the origin somewhere in space-time. Then spreading zero all over the domain is considered explosion of the origin. But in this model zero point is a separate entity which is accessible in every point of space-time. So the origin remains intact. In addition, Assertion C6 supports the notion that the singularity and our universe are two separate domains.

To make this concept more tangible and objective, imagine that you are looking at a piece of sponge with a magnifier.  First, we see the roughness of the surface.  By increasing the magnification, we can also see empty holes in between.  At this point we can claim that the sponge is perforated by many empty holes. Nevertheless this is not an ideal analogy for the correlation of singularity and space-time because empty space requires dimensions whereas the proposed singularity does not possess any dimension.  Perhaps the best way to analogize this concept is to imagine the sponge immersed in an infinite entity which cannot be measured or understood with our existing limitations. On the basis of the above arguments, we may conclude that in the gaps between space-time webs, we are faced with the 0-dimensional singularity.

Ultraviolet Catastrophe
At the turn of the twentieth century, classical physics faced a major paradox.  Ludwig  Boltzmann showed that entropy S of a physical system is directly related to its volume T.
S = k Ln T
Where k is Boltzmann constant and Ln is the logarithm.  This implies that if the volume turns to zero entropy also turn to zero. Actually, Walter Nernst restated the Third Principle of Thermodynamics as,
The entropy of any physical system vanishes when the absolute temperature vanishes.
Classically, if an atom is at rest, its momentum and its energy is zero. Therefore the subspace corresponding to the temperature of it shrinks to zero. Here comes the problem,
Ln 0 = Infinity (1)
This is impossible. classical physics predicts that when the temperature vanishes, entropy must diverge. Emilio Del Giudice from National Institute of nuclear physics, Italy calls it ” complete nonsense. It implies universal death in a hot furnace created by ultra cold”⁷⁵.
So a way out of it was needed. Max Planck came out with a solution. He suggested that space does not include zero points. It is made of tiny regions as unit of space.  This unit of space is controlled by a fundamental constant of universe. This constant is named after him and is shown by symbol h. The proposal solved the problem because now,

Ln 1 = 0 (2) So there is no infinite temperature catastrophe.
However it opened a can of worm. The above suggestion changed classical physics drastically and turned to act as a pillar for modern physics and quantum mechanics. The Planck discovery confirms that point zero cannot be a part of space. Then what does zero representing? What is this zero that according to equation (1) has infinite temperature?

Fabric of the Universe

Many schools and physical theories are considering the space as a continuum. But others like Loop Quantum Gravity picture time and space as discrete entities. Mathematical discontinuity represents holes in the fields. Dividing a real number by zero creates discontinuity in the math domain. This same problem renders itself in the General Theory of Relativity. This gives many theoretical physicist the sense that there should be holes in space-time.  In the present model, I also take space and time as discrete elements.
The Planck discovery mentioned above resulted findings of other fundamental elements. Planck constant is the origin of a system of natural units known as Planck Units. The Planck length is the ‘atom of length', the smallest possible size of length. It roughly equals to 1.6 x 10^-35 m or about 10^-20 times the size of a proton. This is the scale at which classical ideas about gravity and space-time cease to be valid.
Inside of the Planck length, the notion of space is not valid anymore. Can we suppose that at the boundaries of each Planck distance our matter-space-time universe ends?
Planck time is the time it would take a photon, traveling at the speed of light, to cross a distance equal to Planck length. This is the ‘atom of time', the smallest unit of time, and is equal to 10 ^-43 seconds. No smaller division of time has any meaning.

In other words, Planck time is the building block of time and, as such, it is the absolute unit of time. With the same token, Planck length is the building block for space and as such is the absolute unit of space. Out of this boundaries, the meaning of space and time breaks down.
If no distance less than Planck length and no time less than Planck time has any meaning, can we assume that Inside Planck unit limits is out of our space-time universe, because it does not contain meaningful space or time?

The above arguments substantiate and favor a space-time with discrete fabric over a continuous one.

Exposure to Singularity
We have assumed that universe is contained inside the singularity. Therefore, based on our assumption we can claim that, our exposure to singularity in each centimeter of space is 1.6 * 10 ³⁵ absolute units of length. In other words, the exposure is in every minuscule of space, or everywhere.
One also can claim that, in every second we are exposed to naked singularity 10 ⁴³ absolute unit of time. That means all the time.
Formerly, I have assumed that singularity contained merely energy and information and it is benign entity.  How is it possible to be exposed to infinite energy and not experience catastrophe?  The laws of physics tell us that, if blasts of energy were confined to short enough intervals, these blasts would have no effect in our space-time. furthermore, it would not even be detectable.
As mentioned, we are exposed to singularity in each interval of Planck Time. This explains why the singularity is benign for us.  The energy exposure is very brief.

 

 

 

 

 

 

 

 

If we confine our definition of physics to the limits of space-time, then we will need to renormalize the undesirable findings, frequently. Re- normalization means using mechanisms to omit zero and infinities and other problematic findings which is not acceptable with our space-time knowledge and logic. However, the Big Bang theory, Dark Energy, and the non-zero Cosmological Constant are directing us to the beyond. There is more and more evidences now to force us to come out of our ken.
Pertaining to the sponge analogy, the Planck distance corresponds to the holes in the sponge. This implies that our universe is not continuous; rather, it has a discrete atomic structure. In the 1970’s Jacob Bekenstein proved it by his discovery, the so-called “Bekenstein’s Bound”. ²⁷ This law states that the amount of information within any horizon, forming a boundary, is finite and proportionate to the area of the horizon. If this is the case, the space inside the horizon cannot be continuous. It has to be finite and as a result space has to be discrete.

Loop Quantum Gravity
Loop Quantum Gravity Theory is the next candidate in line for the Theory of Everything. The first being the Super String theory. Lee Smolin, one of the main advocates of the Loop Quantum Gravity, claims that black holes thermodynamics, loop quantum gravity and string theory all agree that, on the Planck scale, space appears to be composed of fundamental discrete units. He adopts Roger Penrose's spin network as a model for fabric of the universe.

Spin Networks28
In this picture, each node represents a Planck space, which is separated from each other by the edges.

The loop Quantum Gravity calculations also suggest that space is quantized.  This theory postulates that space is made of building blocks that are Planck distance cubic or (10 ^-33)³ or 10 ^-99 cubic centimeter. Thus it predicts that there are

10 ⁹⁹ quanta of space in each cubic centimeter. Fortunately, this assumption may likely be testable in the near future.

Examining the radiation from distant cosmic explosions called gamma ray bursts might provide a test whether fabric of space is discrete. If it is, radiation with different energy will arrive at earth at different times. The difference is minimal but for faraway bursts (3-4 billion light years away), it can be significant enough to be measured.
There is a modified version of Einstein’s theory, by Lee Smolin and researchers from the Imperial College of London that accommodates high-energy photons traveling at different speeds. Our technology today is not precise enough to measure such a minute difference, but fortunately, the Glast Satellite, which is launched June 11, 2008 by NASA has the required sensitivity for this measurement.
So, if the Glast Satellite shows that space is discrete, we can suppose each Planck volume (10 ^-99 cubic centimeters) defines internal boundaries of space-time.

The Riddle of Infinities
In the introduction to this book I claimed that re-normalizing within the theoretical physics mathematics is actually ignoring the road signs to explore reality. I further claimed that by normalization we create blockades for ourselves along the road to discovery. Steven Weinberg explains renrmalization ,

This term goes back to the 1940's when physicists where learning how to use the first quantum field theories to calculate small shifts of atomic energy levels. They discovered that calculations using quantum field theory kept producing infinite quantities. ⁴⁹

The emergence of infinities in calculations were interpreted as flaws.  Their appearance is considered that the process was pushed beyond its limits of validity. This is done because the finite universe cannot include infinities. Since infinity is not physically possible, the answer was interpreted as impossible and flaw. 
Physicists consider infinities, which occur in ultra-short scales, a major problem. We can view the gravity field equation as an example:

F = G m₁ m₂/d ²

In the above equatin, F is gravity force, m₁ and m₂ are masses of the two objects, G is Gravitational constant and d is distance between them. (As distances between the sun, earth, moon, etc...).⁴⁰

As d approaches to zero, gravity force increases and in zero distance, if we could reach it, the force of gravity would equal infinity.

F = G m₁ m₂/d ², if d=0, then g = G m₁ m₂/0 = ∞

Here gravity force is used as an example. In usual situations, because of electromagnetic deflection between atoms, reaching to zero distance is not attainable. However, at the center of black holes atoms are destroyed and matter are to compact and reach to zero size. The above example is also true for electromagnetic force that is inversely related to distance square.

Infinity in this model is defined. It points to energy and information in the proposed singularity. The act of ignoring and bypassing zero and infinity in calculations is called re-normalization. Theoretical physicists use the Bekenstein Law (mentioned above) to get rid of infinities in the calculations.
If we confine ourselves to space-time physics, we have to frequently renormalize our abnormal!! findings. However, the Big Bang model, the proposed dark energy, and non-zero cosmological constant make us look beyond space-time.  More and more evidence is there to force us to come out of our ken and believe that actual physics should extend beyond the familiar space-time universe. If this is accepted, there would be no further need to renormalize.
Let us use common sense mathematics. Imagine that we have a bunch of pebbles. Now, we remove them from the scene one by one. At the end, there will not be any pebbles left.  The reality is that there are no pebbles in the scene, but zero is still there. The scene still exists.
In conclusion, zero and infinity should also be considered a part of reality. They cannot be ignored and avoided.  Zeros and Infinities have profound effects on our world.  We just need to give them an identity and include them in our theories, not dismiss them as nothing or meaningless entities.  The fifteenth century mathematicians had to expand the mathematical arena to include negative numbers. It took us until the nineteenth century to relate a physical meaning to them, namely the negative charges in electromagnetic. Just remember that there are also imaginary numbers, which we have not found an exact physical meaning for them yet. The actual physics extends to territories, which are vast, active and effective.
The presence of constants in our calculations indicates the unknown factors, which are affecting our universe. It is not humanlike to accept that the meanings of these factors are out of our reach. We are hunters in the dark; and, so far, we have been finders.

Final Theory
Mark McCutcheon, in his book The Final Theory,⁶⁰ raised some very interesting questions. If gravitational force of say our planet has been at work curving space in the way and therefore pulling objects for billions of years   (like the gravitational force between earth and moon which lasted at least four billion years), its energy has had to diminish as time goes by. If the force comes from the mass of the earth, we should have to see or foresee the end of it. Apparently, this force has been in effect for more than four billion years without any change. Where is the source of  this unending energy? This is obviously a breech of the conservation of energy law. McCutcheon answers this question by introducing the expanding world model. He rejects the notion of gravity and reasons that the attraction is actually because of the expanding planet, which swells out and reaches the free flowing objects. It seems that this model has serious flaws. A more logical answer is the aforementioned zero point energy. Precisely the energy from singularity that is infinite in this model.  The mass of particles somehow introduces this energy to space-time, curving it according to the General Relativity theory.

Dissipative system
With this definition of singularity, infinities in physics are not problematic. They are defined and explained. By assuming, the benign naked singularity as a medium that accommodates the core fabric of the universe, many paradoxes in physics obtain a deterministic and comprehensible explanation. In this view our material world and ourselves are connected to singularity at all times and places. In this model, singularity is not 15 billion years away (birthdate of the universe). Singularity is here.

Singularity is here
 

Summary
Space in this model is a complex Minkowsky manifold. Which means it has three spatial dimensions and one time dimension? But each one of these dimensions are represented with a complex number (containing an imaginary element i). As such it mimics twister theory, which is a Theory of Everything supported by Roger Penrose the famous British mathematician.  But in my model the imaginary element is common between all four dimensions.
In this view the outer boundary of space-time and Planck distance and time are the interface where our universe meets singularity. I have also attributed the notions of zero and infinity to this entity. The concept of imaginary numbers is also accredited to physical activities related to proposed singularity.

The arguments presented are open for debate. The reader is encouraged to email his/her inputs to correct, modify or develop the contents. Please send your emails to; zpfields@yahoo.ca

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