FLATLAND UNIVERSE

I know I am being preposterous in writing this small "essay" of a blog post. Somehow, an idea got stuck in my mind that I simply had to put it out in order for someone to explain to me where I got it wrong, in logic or observation.

The idea came suddenly to me as I watched Sagan's Cosmos episode ten, “The Edge of Forever.” The basic premise of the idea is to consider the Flatland of Sagan’s presentation as having the same natural laws and the same observations as our own three dimensional universe, minus one space dimension as I imagine the flatlanders would observe, and to speculate how that would work.

I’m not a scientist. I’m a layman with only a very rudimentary understanding of science. I don’t know much about neutrinos, quasars, or photons. To be honest, everything I learned was from only a select few of readings – Wikipedia articles, A Short History of Nearly Everything, and the Cosmos series by Carl Sagan. Therefore, all the observational evidence and physics that I understand are limited to how they are explained to the layman. 

Of course, I naturally am expecting that most of the ideas I have to be completely in error. I'm quite embarrassed to have written this at all. It is just that the idea is latched on to my mind without any understanding on how to test the speculations that my embarrassment is not enough to stop me in writing out the ideas.

In case you have figured out where I am in total error, I would very much appreciate it if you could point it out to me. I wonder, even maybe wish, that there’s a tiny smidgen of merit in any of the idea, but I’m not wishing much.

FLATLAND UNIVERSE 

How a Two Dimensional Representation of Our Universe Should Look Like 

Gravity, as Einstein explained in his theory of Relativity, is the result of the bending of space time caused by mass. Space time is explained as a three-dimensional space that we occupy, plus a dimension called Time. Of course, we don’t have much problem with conceptualizing a three dimensional space, but not so for the additional time dimension. We’re usually asked to imagine a matter on top of a mattress with a significant mass to bend the mattress. Things moving in straight line would “fall” towards the object with mass due to the bending of this mattress.

Carl Sagan in Cosmos episode ten, The Edge of Forever, explained about the fourth dimension by asking us to envision how a two dimensional creature would experience the additional third dimension. He asked us to imagine a ‘flatland,’ a universe of two dimensions (The idea of flatland itself was derived from a novel by Edwin Abbott). He then asked us to imagine a third dimensional creature descending on the flatland to meet with a two dimensional creature, appearing before the flatlanders as if magically inside an enclosed space. The three dimensional being then whisked the flatlanders to the third dimension where it could see the innards of everything, and returning to the flat universe where it was unable to point to the third dimension when prodded by its friends. From there, he explained how we as three dimensional beings are trapped in our own three dimensionality, just like the flatlander is trapped in its two dimensionality.

However, there are two flaws in the presentation. First of all, the two dimensional universe, which I will call the “Flatland universe” from now for brevity and clarity’s sake, wouldn’t look like how Sagan presents it. The flatlanders in Sagan’s thought experiment is enclosed on a wall and is assumed to be standing on something. That something, the floor, is an extra dimension of height that should not have existed in the true two-dimensional universe.

Assuming the same laws of physics applies in this Flatland universe, with the only difference being that space is defined in two dimensions rather than three, mass in the flatland universe would have gravitational pull with each other. Matter would clump together to form planetary and stellar objects and they would more or less be circles. Any being living on such worlds would have the concept of up and down, down being the direction where gravity pulls it.

Any wall in this universe, which is a construct of height that is relative to the center of gravity, would become a formidable barrier to any beings to move in the forward and backwards direction – they would not have the concept of left and right, although we can change the terms used between left right and back forward and it would make no difference to the flatlanders. Or, in terms familiar to gamers, the universe would look more like a “side-scroller” like Super Mario Bros rather than a top-down game like Zelda, only that there is no background sky and cloud in such a world (I apologize for this analogy, it just seems like a perfect analogy to me).

The Third Dimension in Flatland?

Another aspect in Sagan’s presentation of Flatland that I consider to be flawed is that the three dimensional being that looks like an apple is moving freely within the third dimension as if it is another spatial dimension. If the fourth dimension of space time is a time dimension, then the third dimension in the flatland universe should also be a time dimension. I do not think that the time dimension should be a dimension where matter can freely move back and forth. Matter should only travel through the dimension of time in one direction only. It should have come from the past in the dimension and is travelling towards the future.

Let us expand the exercise of imagining flatland further, while keeping in mind that the third dimension is the dimension of time. Now, the citizen of flatland can point forward and backwards, and up and down (down being the direction of where gravity pulls relatively). But the flatlanders has no concept of left and right and they would be unable to point to the direction of the third dimension in space, which in this exercise is the time dimension.

Fortunately for us, as a three-dimensional being, we could point to that other direction in Flatland’s time dimension. Now if we consider time as travelling in one direction only, then one direction in the time dimension would be the past, and the other direction of the time dimension would be the future. To better visualize this third time dimension, we can label the past in this third dimension as “down” and the future as “up.” In other words, time is moving “up” while the direction where the present came from is “down.” It would be different from the flatlanders’ relativistic concept of up and down. Our up and down would be absolute in a three dimensional sense. At any rate, we could label the direction as something else, and it would not change the following assumptions.

If we, as a three dimensional creature, can see the two dimensional universe with the additional of time dimension, we could see that the movement of a flatland object as a column. If the flatland object splits in half, we would see parts of the being splitting from the being into two separate columns. The edge of the column, the “top,” would constitute the present, which continually grows. However, this assumes that we are beings who can see through time, and that there is something to be seen in the spacetime of the past, which may not be the case as we speculate further.

The Shape of Flatland Universe

Now I will continue the assumption that the laws of physics and observed state of nature apply equally to the Flatland universe as it is in our three dimensional universe. One observed phenomena of our universe is that of an expanding space, with galaxies moving away from us and faster the further away they are. The conclusion of such observation is that there was once a point in time where all space and matter that ever exist were once a singularity, and the point where it ceased from being a singularity has been known as the moment of Big Bang.

Now suppose the Flatland universe also has a moment where everything was a singularity. That moment lies in the past – we will say 13.7 billion years in the past just like our universe. The point has been made that as a three dimensional creature, we say that the past in the time dimension of the flatland is down. Now following that everything that ever was, and all space that ever exists, were once a singularity in the beginning, it would mean that all down direction of all space where time is 0 is a single point in the down direction. An area where all the downwards (or inwards) direction is a singular point would describe an area of a sphere, or something close to a sphere depending on the distance between a point in the area to the central point downwards.

This goes to show that our flatland universe would have its space curved into something like a sphere. How apt then, that Sagan in the Cosmos episode The Edge of Forever, shows a visualization of a sphere to explain the expansion of space and the apparent movement of galaxies accelerating away from any one point. If time could serves as measurement, then the radius of such sphere would be 13.7 billion years as observed by the acceleration rate of galaxies.

As we know it in our universe, the basis of the big bang theory is the observation that the universe is expanding, and that galaxies are racing away from us, and faster the further they are from us. However, there is not enough force to explain what drives the expansion. As I understand it, it has been theorized that the force that drives this expansion is dark energy.

Now imagine again the visualization that Sagan gave us through the expanding sphere in a flatland universe. I am struck by how similar it looks to an expanding beach ball, or any kind of ball that uses air pressure to maintain its form for that matter. What drives the expansion of the beach ball, of course, is the air that we are pumping into it. But there is no cosmic pump from outside of the sphere plugged into the fabric of cosmos to drive the expansion. It should have come somewhere from the inside of the sphere.  As we have established, we are calling the inside of the sphere as the “down” direction of time dimension, or in other words, the past.

So there is something from the past, some force that is driving the expansion of space itself. This hypothetical force, aside from driving the expansion of time, is also driving the two dimensional space towards the “up” direction, or towards the future. Without this force, there would be no expansion of space and there would be no sense of time progressing. The citizens of flatland, being trapped in two dimensions, would not be able to point to the direction of this force.

So this force would be coming from past and drives both the expansion of space and the progress of time itself. For brevity’s sake, I will call it “Time force” or Ft in my illustrations. So what Dark Energy is, is actually the Time force itself that is pushing the fabric of space time, which is space as it is in a given moment, outwards. This time force would be pushing space everywhere, in all of space, presumably in equal force.

Mass and Black Holes in This Understanding of Flatland

Mass is understood to have created the force of gravity, but the explanation of what mass actually is, is still being debated (or has it been clarified by the findings in Quantum Theory?). Regretfully, my understanding of quantum theory and the study of the seat of mass is woefully inadequate. But from what little I know, that gravity is the result of the bending of space time by mass, I would like to make a leap in speculation that is based solely on my own thought and not based on any observations.

I speculate that mass is the manifestation of interaction between the Time force and some subatomic substance that tugs the fabric of space time at towards the direction of the past. Dark Energy is simply Time force that does not interact with such subatomic substance. What we perceive as mass then is truly a construct of the bending of space; a warp in the “fabric” of space time at the present. This means that somehow, in Flatland universe, gravity works not only on the traditional spatial dimensions, but also on the time dimension where there’s an actual “down” direction.

Let’s take all these speculations further to consider the phenomenon of black holes, where the force of gravity is so great that even light cannot escape. Considering that in flatland we use the direction of “down” as the past, assuming that black holes form wormholes that connect to another opening of black holes in some exotic dimension or another universe or another point of space and time would be incorrect. Instead, in a black hole, the fabric of space time at present bends towards the “down” direction or the past.

It is said of black holes that they occupy a singularity of space. Also, there are many possibilities of the fate of a black hole theorized. As the gravitational pull of a black hole is too strong even for light to escape, then can matter and information be truly destroyed from the universe? Could the amount of radiation released by a black hole account for all the matters and energy that is consumed by a black hole? And then, could we one day detect evidence of black hole evaporation events when enough energy escapes the black hole until the mass of a black hole is reduced until light can finally escape?

Back in the flatland universe, if we accept that the black hole occupies a singularity of space, this would mean that the flatland space time has a curvature that is infinite, and the curvature goes infinitely “down,” or rather, it would bend only to the point of the original singularity of t=0.

Considering this, I speculate that the matter and information trapped by the black hole go “down” through the curved fabric of space time of present to reach t=0, or at least t=certain point in the past, where in a process beyond my understanding and is admittedly just pure speculation, is processed to become the energy that fuel the Time force that expands space and time. In a sense, physical information is not lost, but processed to become the Time force (since mass and energy is equivalent – E = mc²).

I imagine that the Flatland universe would probably work like a star, with the core of stars having nuclear fusion process to create the energy and photons that stars eject as heat and light. Matter is disintegrated at the core of the Flatland universe, and the energy is released back as the energy that pushes space and time outwards. I wonder how this guess works can really apply to our actual universe.

How the End and Beginning of Flatland Might Look Like

I will now speculate on the ultimate fate of the flatland universe as space expands further away into the future. Space would one day expand so greatly that matters began to disintegrate. The universe would be empty of matter, with only black holes scattered throughout space. At that point, there would be no more matter that black holes can devour. If Time force is dependent on mass to become energy, then one day there would be no source of energy to expand space time. Perhaps this would be the start of the evaporation event of black holes.

I imagine that such event would happen more or less at the same moment. There would be no more bending of the fabric of space time. There would be no interaction between Time force and the subatomic substance that created the illusion of mass and gravity.

Let me wildly speculate further on the fate of flatland universe after there’s no more Time force and no more bending of the fabric of space time. Just like a star that has exhausted its Hydrogen fuel and starts to consume Helium instead, the singularity would begin to consume back Time force that it has produced. At this moment, the fabric of space time would lose its meaning, both as a concept of space and a concept of time. There would be no more time force to expand space and move time forward (“outward” in the three dimensional map of flatland), and there would be no more matter also. It’s not difficult to imagine the absence of expansion of space – we are well aware on the limitations of space around us. However, without matter and forward movement of time, then truly there would be nothing left to define an existence of anything.

So in the end, even all the energy of Time force is consumed back to the singularity of t=0, the exact same condition as the big bang. This would mean that the end of our universe would be exactly the beginning of the next reincarnation of the same universe, which would mean that the flatland universe is an oscillating universe. However, the oscillating universe doesn’t have the fabric of space time crunching back through the past to become a singularity. It would simply create a new fabric of space time, completely separate from the previous fabric of space and time which, to repeat, would have lost its meaning.

If black holes do not evaporate, then at the end of the Flatland universe’s life, everything would just be a huge collection of black holes. I am guessing that if this is the case, that the fabric of space time itself would be bent back and consumed towards singularity. This doesn’t mean that events will start to travel backwards through time – no matter would reappear in existence in a fabric of space time that moves towards the past. I personally find this guess to be more difficult to envision than the previous one where I assumed that black holes evaporate. I have no idea on which scenario is more plausible, or if none is. Again, I have to admit the limitations of my understanding about the nature of black holes and end of universe and leave the speculations as they are now.

Explaining the Distribution of Galaxies Using Assumptions and Speculations of the Flatland Universe

It has been noted that the distribution ofgalaxy is not evenly scattered throughout space and time, but clumped together and is distributed almost “like on the surface of bubbles” as Sagan noted.  There are amazingly large areas of complete emptiness between the strings of galaxies, which was noted as being similar to “the inside of the bubbles.”

How would the Flatland universe look like if the distribution of galaxies is anything like the observation made in our universe? If we visualize the sphere of flat land universe, we would see it more like a standard soccer ball, with lumps on the ball and galaxies distributed on the crevasses of the ball (this analogy doesn’t work on some soccer ball designs like the Jabulani). Or, to make a simpler analogy, we can imagine it like bubbles on a flat surface, while keeping in mind that only where the surface of the bubbles touch the flat surface do the galaxies are distributed.

Considering mass as a manifestation of the warping of space and time, in the flat land universe, this would push mass towards the bent space. Meanwhile, Time force keeps on acting to expand the fabric of space time outwards.  As a result, bulges form on spaces where there is no mass that warp the fabric of space time. These bulges in Flatland universe would become the empty space where no galaxies with enough mass could venture into. These bulges would, in our three dimensional universe, be the inside of bubbles where there is no concentration of galaxies.

If this is true, then in the flatland universe, it would appear that there are some imaginary central lanes forming between the bulges. Then galaxies, following the bending of space time, would be pulled towards it and then move past the central lane, only to stop and move back. The apparent movement of the galaxies in the Flatland universe as observed by three dimensional being would be like a pendulum. Perhaps then, like the pendulum, we can see that a galaxy that moves towards the central lane would be accelerating relative to the central lane and a galaxy that moves away from the central lane would be decelerating relative to the central lane. 

We can make a similar test in our three dimensional world. We can plot out an imaginary tube, which would be more or less cylindrical, based on the distribution of galaxies, with the outer most galaxy of a certain region forming the outer wall of the imaginary tube, and the imaginary central lane on the axis of the imaginary tube. The area outside the imaginary tube would be the empty space. For this speculation to have any merit, it has to be observed that a galaxy that moves towards the central lane should accelerate relative to the imaginary central lane and a galaxy that moves away from the central lane should decelerate relative to the central lane. The galaxy on the outer region of the tube should appear still relative to the position of the central lane.

The Apparent Flatness of Space in Our Universe

According to observations from WilkinsonMicrowave Anisotropy Probe, it has been observed that “the observable universe is flat with only a margin of error of 0.5%.” Unfortunately, again, my understanding of the nature of observation is woefully inadequate to fully comprehend it. I could only infer that my assumption about our universe being like the Flatland universe I have been conjecturing is flat out wrong.

The only way this observation could be reconciled is to take the margin of error and surmise that the universe as a whole is amazingly large that the apparent flatness is such that the observable universe follows a local geometry.

So let’s say that a flatland observer observed that that the observable universe is flat with only a margin of error of 0.5%. I know that the margin of error of 0.5% does not mean that it is curved to a 0.5% of a full circle, but for simplicity’s sake and absent further understanding I will assume that 0.5% is the curve of the flatland universe. This means that far as the observable flatland universe is concerned, which is 27.4 billion light year in radius (or 13.7? Should we account for the space that has moved away from us since 13.7 billion years ago?), the local geometry of the observable universe is 99.5% flat and 0.5% curved.

Considering that, could we then surmise that 0.5% of the sphere of the flatland universe (figure 4) is 54.8 billion light years long? We take a point of the event horizon in the flatland universe (edge of the observable universe) and take the opposite point, then we make imaginary lines from those points towards t=0 in the third dimension of time. Then I just wonder if with the assumptions made, the two intersecting lines would make an angle of 1.8 degrees, which is 0.5% of 360 degrees of a full circle.

If, and it’s a very big if, all these assumptions are near correct, then we can roughly calculate the size of the flatland universe! The full circumference of the sphere would be 54.8 billion light years times 360 degrees divided by 1.8 degrees, which would be 10,960 billion light years.  This means that the radius of the sphere in terms of distance (not time) is 1.745 trillion light years. The area of the sphere then would be 3.82 x 10²⁵ Light years².

I wonder then if we could use the radius of the sphere to infer the volume of space in our three dimensional universe. What is a four dimensional sphere? There is a concept of spheres in higher dimensions, called the n-Sphere where the n denotes the number of dimensions of the sphere. For example, in flatland universe, the area of the universe is a two dimensional area on a three dimensional sphere (the third dimension being the time dimension). This is understood as the sphere being 3-Sphere, and the area being in n-1 dimension.

In our universe then, with the fourth dimension being the direction of time and accepting the assumption that the shape is spherical, then it would be 4-Sphere where the volume of the universe would be the three dimensional “area” of the 4 sphere. The formula for the three dimensional space of the 4-Sphere is 2Π²R³ (since the area of a 3-Sphere is 4ΠR², then the additional dimensional space is the 2D area of 3-Sphere times (2Π²R³ / 4ΠR² = ½ ΠR). If these were all true, then the total volume of the universe would be approximately 1.046 x 10³⁸ Light years³.

Of course, admittedly these are very presumptuous conjectures. I am only trying to show the possible implications of inferring the size of the universe if we were to accept that the universe is shaped into a sphere.

There Wouldn’t Be Time Travel in Flatland Universe

Many have speculated whether it is physically possible to time travel in reality. In the hypothetical Flatland universe with properties as I have presumed, I do not think that is the case. The only space that exists at any given time would be the space at one certain position at t=present. Some of the fabric of space time may be curved more towards the past due to gravity, but they still represent the same point of time t=present.  This would be true even on the extremes of space time bending like the black holes. In another sense, everything is travelling through the time dimension to the future, but not the past.

The only thing that exists in the past, or t=past is only the Time force that is pushing space and the matter it contains outwards. There would be neither space nor matter in the dimension of time whether in the past or the future.  In the oscillating universe of flatland, matter and even Time force do not survive to the next reincarnation of the universe.

So Many Questions Unanswered

Due to severe limitations of my understanding, I haven’t discussed many aspects of physics in this flatland universe model. Electromagnetism, nuclear force, quantum theory, Quasars, et cetera are beyond what I can plausibly presume. Even the speculations I’ve made are stretching what’s plausible. I couldn’t comprehend enough to offer any method on how to test all these ideas (with a possible exception on the motion of galaxies). I hope that someone who has a more thorough understanding of these subjects to point out to me where the logical or observational fallacies lie so I can properly correct, or even discard, some if not all of these ideas.

Appendix

Additional Random Considerations and Disjointed Thoughts of the Flatland Universe

On 4^th of July 2012, experiments at the Large Hadron Collider have confirmed on the existence a subatomic particle of Higgs Boson. I wonder if this could be considered the subatomic particle that interacts with Time force to create mass, and that Higgs field could be considered the fabric of space time itself?

I wonder about how to explain quantum theory and string theory with this model of Flatland universe: how would subatomic particles and quarks behave absent one spatial dimension? I wonder if it’s realistic to presume that the behavior eludes observers because they move about in that extra dimension of time. I'm suspicious that the way scientists are trying to describe the characteristic of elementary particles in the standard model as having flavors is due to the elementary particles moving about the fourth dimension of time. Imagine a flatlander trying to describe the elementary particles that move about the time dimension up and down, while only being able to observe the two dimension of space. The flatlanders would be hard-pressed to explain energy that strikes the two space dimension from differing angles within the time dimension. I wonder if the flatlanders would then describe the elementary particles as having flavors. 

Any living creature on the flatland would not be able to have a digestion tract which has the food intake at one side and the excretion at the other side. That would split the creature in half, with no extra dimension to put them together. So any predatory creature that eats of “plants” or “animals” would either have a digestion system where the intake and excretion at the same point, or it would have a completely exotic system yet to be imagined. Perhaps evolutionary forces would produce many “flying” and “jumping” creatures since relative horizontal movement would be difficult on planets with bumps and valleys.