Isaac Newton believed that space is the background in which motion takes place. Therefore, he took space as an actual body and as the absolute benchmark for the universe. On the contrary, Gottfried Leibnitz believed that heavenly bodies move relative to each other and that there is no background. Thus, he denied that space is an actual entity. Ernst Mach had a similar idea, but he introduced acceleration as a contributing factor for the motion of stars. He also believed that the relative motion of heavenly bodies is affected by the distribution of mass throughout the universe.

Albert Einstein, however, introduced the idea that the cosmic background is a combination of space and time. He declared that space and time are actual entities. Although individually, space and time are relative and malleable, the space-time combination forms a solid background for the universe. In his relativity theories, space-time is vibrant and active in the evolving world. Time and space can dilate or shrink depending on object’s relative speed. In other words, objects are behaving differently based on their space and time frame of reference. The gravity of stars curves the shape of space. The curved space bends the trajectory of other stars. 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. Below, I propose a model for universe that adopts Einstein’s space-time fabric but includes it into a bigger scheme.

Super-Space

For some time we have realized the need to look beyond ordinary four-dimensional Minkovsky space-time to explain many physical phenomena. Many mechanisms—for example, electromagnetic fields—cannot be explained in the context of a four-dimensional universe alone. To explain these mysteries, mainstream physicists chose to theorize another space-like manifold in addition to ordinary space-time. This manifold is called super-space. In basic terms, the idea of super-space presumes that the points in space-time are actually cross-sections of bundles which are extended into this proposed super-space.

Extra dimensions in super-space are called internal dimensions. Therefore, every point in space-time should have internal dimensions that are out of site. Interestingly these bundles (fibres) are frequently represented by a complex vector, which include an imaginary portion.

The introduction of super-space opens up a can of worms and subjects our theories to the doldrums of extra dimensions. Super-string theory proposes up to seven extra dimensions. Did super-space solve existing mysteries in theoretical physics? On the contrary, it created a lot of chaos and dragged 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 space-time.

Mathematics has been very helpful in evaluating and understanding the world around us. Therefore, we frequently look at mathematics to build physical models. We have two different mathematical choices we can adopt to build a model for space-time.

Real numbers with infinite amount of decimals (2.567854332234…) stand for continuity in the field. If we take real numbers to represent space, we may see the world as a continuum. The problem is as far as we know the universe is finite. therefore, real number system can not represent the universe. In contrast, natural numbers (1, 2, 3, 4, 5…), with their discrete nature, symbolize discontinuity of the field. If we adopt the natural number system as the model, then our space-time at a fundamental level has to be grainy and discrete.Then the question arises, what is the nature of the void in between the units?

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

By definition, the 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? Such a square cannot have a meaningful, real (tangible) area. It cannot have a meaningful dimension either. Dimensions of such a square are not similar to the four dimensions of the real world. They are therefore imaginary dimensions.

In theoretical physics, imaginary numbers represent components of hyper-space (a space beyond our four familiar dimensions). The assumption that hyper-space exists is necessary to build theories about nature of the universe. So, it seems that, contrary to the assumptions of string theory, hyper-space cannot have a real dimension, or area. In this model, I call hyper-space the singularity, which does not possess any real dimension. However, it can accommodate the abstract quantities.

Let us return now to the complex number system (a two-dimensional system that consists of real and imaginary coordinates). The system is explained in detail in the article “Complex Numbers”

Assertion C3 postulates that the discontinuity of real numbers always occurs around zero point. One of the strange characteristics of imaginary numbers is the fact that in unit circle, the real value of any parameter when coupled with (multiplied by) any imaginary number will be reduced to zero.

The algebraic formula can be written as follows:

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

X in the unit circle is one. Therefore,

X*i* =* i*

As we go counter-clock wise around the circle the real value gradually disappears and the imaginary value fully expresses itself at 90^{0} degrees.This is analogous to our attention span. The more we focus on quality of an object the more we lose focus on its quantity or measurable features.

In trigonometry we can show this fact as follows:

X (the real Value) = 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. Therefore, they lose their real number value and hit zero twice in each period, which indicates discontinuity in the real number field.

Therefore, any space-time element (space, time, matter) shown by the X coordinate, as it couples (multiplies) by imaginary number, loses and regains its real value periodically. For example, if X indicates distance in space, 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 Schroedinger was one of the first physicists to suggest a discrete space. Einstein, in his last published paper, having quantum theory in mind, also proposed that a theory based on discrete space might be the way forward for physics.

As mentioned, 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.

Paul Davis writes, “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" ^{[1]}

What can we make out of the above statement? The undecidable propositions to me represent physical findings which are not directly related to the pre-determined or expected outcome of calculations. They represent outcomes that are ambivalent and therefore cannot point to a single definite answer related to space-time elements.

Assertion C4 indicates that any point in the plane can be considered zero point. Therefore zero point is interwoven with any minuscule portion of the space-time fabric. Or we can further assume that zero is present in between each atom of space-time.

In mathematics considering every point of a domain as zero is called “blowing up the origin.” This is done with a mind-state that localizes the origin somewhere inside space-time. Spreading zero all over the domain, on the other hand, is considered “exploding the origin.” However, in this model zero point is a separate entity, which is accessible at every point of space-time. Therefore, the origin remains intact. Assumption C_{6} (in the section "Complex Numbers") speculates that the singularity and our universe are two separate domains.

"There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition; and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call "the Twilight Zone"^{[2]}

Using common sense mathematics, imagine that we have a bunch of pebbles and 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.

My conjecture is that the space-time universe is enclosed within the proposed singularity. If we see space-time as 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. Then again, we have assumed that the singularity is zero size. How are we going to imagine a web spreading over zero? We may find the answer if we take X coordinate to denote space dimension. At zero point value of space is zero. space is not defined in singularity. Assertion S1 (in the section “Singularity”) denotes that singularity is a separate entity. We are talking about two separate domains.

In the enclosed universe scenario, singularity has to have an internal dimension, which runs contrary to our original assumption. When explaining this idea, choosing a metaphor to explain what happens is challenging. We might be tempted to say that our universe dissolves in singularity. However, “dissolve” is not the right word. Nor does the word “overlap” accurately describe what is happening. How then can I suggest that space and time is, in a way, embedded in singularity, as is all matter and everything else that exists in our universe?

If we cannot expect singularity to have dimension, then we have to depart from objective thinking as our only tool. Next we must let our mind’s eye see an extra arena adjunct to the universe, one that appears in our imagination. Can this analogy guide us to the realization that the universe can intermingle with a 0-dimensional entity (mind itself). 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.

The definition of singularity presented in these articles raises a new series of questions and concerns: 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).

The above diagram shows that the image of a three-dimensional object in a two-dimensional world (book page, computer monitor) would be reduced to a series of lines that provides the illusion of a 3-D cube. That same image in a one-dimensional world (X-axis) is just a line, and in a zero-dimensional world (zero point), the image would coincide 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. Here, we can deduce that tangible qualities of things cannot exist in singularity, but their image does.

The physical reality of the above can be seen in holography. In holography, a two-dimensional picture can contain all 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.

Leonard Susskind and Gerard ‘t Hooft have stretched the holographic principle even further by exploring the possibility that the objective universe is the projected image of the data which exist in the two dimensional boundaries of the universe.

The algebraic can be written as “any number times zero equals zero,” or,

X × 0 = 0

If we couple any number with zero, the result is still zero. This simple equation can lead us to a deeper concept.

"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 ^{[3]}

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?

To make this concept more tangible and objective, imagine that you are looking at a piece of sponge at different magnifications. 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 that cannot be measured or understood with existing axioms. 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.

In the Introduction section, I questioned if re-normalizing within the mathematics of theoretical physics is actually blocking path to exploring reality. Steven Weinberg explains the origin of re-normalization:

“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.^{[4]}”

The infinities that emerged in these calculations were considered flaws. Given that the finite universe cannot include infinities, their appearance seemed to indicate that the process had been pushed beyond the limits of its validity. in calculus we insert limits to avoid zeros and infinities.

Physicists consider infinities, which occur at the scale of the ultra-short, a major problem. Infinity in this model is defined. It refers to energy and information in the proposed singularity. As mentioned before, re-normalization is the act of ignoring and bypassing zero and infinity in calculations. Theoretical physicists use the Bekenstein bound (mentioned above) to get rid of the infinities in the calculations.

**Dirac's Delta Function**

The delta function is a generalized function, on the real number line where it is zero everywhere else except at the zero point. At the origin it can spike to infinity. One can consider delta function as a model for the singularity containing infinite energy or information.

In Algebra we write,

X/0 = infinity

In mathematics, the above equation is called undefined. In this model infinity is defined, therefore the equation is valid.

If we confine ourselves to space-time physics, we have to frequently re-normalize our abnormal(!) findings. More and more evidence compels us to leave our ken and believe that actual physics extends beyond the familiar space-time universe. Once we can accept the possibility, maybe there will be no further need to re-normalize.

In conclusion, zero and infinity cannot be ignored and avoided. They should also be considered a part of reality. Zero and infinity have profound effects on our world. We just need to speculate on an identity for them and include them in our theories, not dismiss them as meaningless entities. Fifteenth-century mathematicians had to expand the mathematical arena to include negative numbers. It took us until the nineteenth century to describe a physical meaning to them, for example the negative charges in electromagnetism. We should keep in mind that there are also imaginary numbers for which we have not yet found an exact physical meaning. The actual physics extends to unkown territories, which are vast, active and effective.

The presence of constants in our calculations points to unknown factors that affect our universe. It is not humanlike to accept that the meanings of these factors are beyond our reach. We are hunters in the dark, and so far we have been finders.

3.) Craig Hogan, “Is Space Digital,” Scientific American (February 2012): 32–37

4.) Steven Weinberg, “A Unified Physics by 2050” Scientific American 13, no. 1.

As mentioned before, holographic principle asserts that the information of any region exists at the boundaries of that region. We don't need to look at the far away boundaries. According to the holographic universe theory,the region can be as small as we choose, even as small as a Planck cube (about 10^{-99} cm^{3}). This is the only way that boundaries of any region can contain the information of that region.Therefore the entity at the boundaries of Planck cubes are supposed to accommodate the data. Supposedly, within the Planck cubes space is not defined and we are facing with absolute void. This void accommodates the information. In this view this entity is the space-free singularity.

Many physical theories consider 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 arises in the general theory of relativity. Because of this, many theoretical physicists believe there should be holes in space-time.

The Planck discovery mentioned above led to the discovery of other fundamental constants. The Planck constant is the origin of a system of natural units known as Planck units. The Planck length is the “atom of length,” or the smallest length possible. It equals roughly 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 the Planck length, the notion of space is no longer valid.

____________________

2.) Gordon G. Globus, Brain and Being (City: John Benjamins Publishing Company, 2004).

Planck time is the time it would take a photon travelling at the speed of light to cross a distance equal to the Planck length. This is the “atom of time,” the smallest unit of time, equal to 10 ^{-43} seconds. No smaller division of time has any meaning. We can therefore think of the Planck time as the building block of time. Outside these boundaries, the meaning of space and time break down.Can we then suppose that at the boundaries of each Planck distance, our matter-space-time universe ends?

If a distance less than Planck length and a time less than Planck time have no meaning, can we assume that whatever lies inside Planck unit limits is out of our space-time universe, because it does not contain meaningful space or time?

We have assumed that the universe is contained inside the singularity. Based on this assumption, our exposure to singularity in each centimetre of space is 1.6 × 10^{35} times. In other words, we are exposed to singularity in every minuscule of space, or everywhere. We may also claim that in every second we are exposed to naked singularity 10^{43} times. That means all the time.

Formerly, I posited that singularity contained merely energy and information, and it is a 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 are confined to short enough intervals, these blasts would have no tangible effect in our space-time. Furthermore, it would not even be detectable. As mentioned, we are exposed to the singularity in each interval of Planck Time. This explains why the singularity is benign for us. The energy exposure is very brief.

Going back to the sponge analogy, the Planck distance corresponds to the size of holes in the sponge. This implies that our universe is not continuous; rather, it has a discrete atomic structure. In the 1970s Jacob Bekenstein proved it by his discovery, known as the Bekenstein bound. 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 any horizon cannot be continuous. It has to be finite, and as a result space has to be discrete.

The 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 loop quantum gravity, mentions that black hole thermodynamics, loop quantum gravity,and string theory all agree that, on the Planck scale, space appears to be composed of fundamental discrete units. He suggests Roger Penrose’s spin network as one model for fabric of the universe. Loop quantum gravity calculations also suggest that space is quantized, that it is made of building blocks that are Planck distance—(10^{-33})3 or 10^{-99}—cubic centimetres. Thus it predicts that there are 10^{99} quanta of space in each cubic centimetre. It was suggested that this assumption may be testable.

The assumption goes that by examining the radiation from distant cosmic explosions, called gamma ray bursts, it is possible to test whether fabric of space is discrete. Lee Smolin and researchers from the Imperial College of London have modified Einstein’s theory to accommodate high-energy photons travelling at different speeds. Base on the modification, radiation with different energy should arrive at earth at different times. The difference is supposedly minimal, but for faraway bursts (3–4 billion light years away), it may be significant enough to measure. It was suggested that, the GLAST satellite, launched June 11, 2008, by NASA, has the required sensitivity for this measurement.

Renamed the Fermi Gamma-ray Space Telescope, its findings so far indicates that the fabric of space-time is smooth. This is what Fermi Telescope found on 2009.

"On May 10, 2009, Fermi and other satellites detected a so-called short gamma ray burst, designated GRB 090510. Astronomers think this type of explosion happens when neutron stars collide. Ground-based studies show the event took place in a galaxy 7.3 billion light-years away. Of the many gamma ray photons Fermi's LAT detected from the 2.1-second burst, two possessed energies differing by a million times. Yet after travelling some seven billion years, the pair arrived just nine-tenths of a second apart.

"This measurement eliminates any approach to a new theory of gravity that predicts a strong energy dependent change in the speed of light," Michelson said. "To one part in 100 million billion, these two photons travelled at the same speed. Einstein still rules."^{[5]}

Apparently other bursts have provided a similar result. one may ask if the radiation travels just along the space-time grains while bypassing the intervals why should there be a difference. In a discrete fabric model, there is one special case where the above kind of result can be obtained. In a case where there is no space in between the grains. In such a case the radiation travels on top of the space-time grains while bypassing the intervals. The Einstein's speed limit for electromagnetic waves pertains to speed in space. This is another indication to consider a non space-time entity as a medium.

It makes a lot of sense if space-time is digital. Craig Hogan, the director of the Fermi lab Particle Astrophysics centre is also currently working in an experiment to see if space is made of chunks, blocks and bits. Hogan and his colleague Aaron Chou are using a holometer to find out if space is discrete at the Planck scale. They also want to know if information comes as discrete, quantized bits at that scale 3.

By looking at the benign, naked singularity as a medium that accommodates the core fabric of the universe, we can obtain a deterministic and comprehensible explanation for many paradoxes in physics. In this view, our material world and ourselves are connected to the singularity at all times and at all places. In this model, Singularity is not 13.7 billion years away (the birth date of the universe) or inside black holes). Singularity is here.

Space in this model is a complex Minkowski manifold, which means it has three spatial dimensions and one temporal dimension. However, each of these dimensions is represented with a complex number (containing an imaginary element i). As such it mimics twister theory, a theory suggested by Roger Penrose. But in my model, the imaginary element is shared in common between all four dimensions.

In this view, the outer boundary of space-time and Planck distance/Planck time is 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 visit The Feedback Page, discuss and share your views.

1.

Paul Davies, The Mind of God ^{↑}

2.

From Twilight Zone Show by Rod Serling ^{↑}

3.

4.

Steven Weinberg, “A Unified Physics by 2050?” Scientific American 13, no. 1. ^{↑}

5.

http://www.nasa.gov/mission_pages/GLAST/news/first_year.html ^{↑}