SPACE UNBOUNDED

At its greatest extent space is finite. It cannot be otherwise, lest the conservation law pertaining to angular momentum be invalidated. But if space is finite, then it must end somewhere. And where it ends there must be a boundary that separates space from nonspace. That boundary encloses space completely, so we are tempted to imagine the boundary as being like the skin of a balloon, with the air inside the balloon representing space and the air outside the balloon representing nonexistence. But that analogy is a bad one and we know that it's a bad one because of two things we know about the boundary.

First, we know that the boundary must have absolutely zero thickness. If it had a nonzero thickness, then some of the points that comprise it would have other points all around them: those points would be in space rather than bounding it. Only those points that have space on one side and nonspace on the other side can be part of the boundary. Second, we know that any surface that has zero thickness has exactly the same amount of area on one of its sides as it has on the other. But one side of the boundary faces absolute nothingness and we know that space does not exist in absolute nothingness, so none of the properties of space (such as area) can exist there either. No area on one side of the boundary corresponds to zero area on the other side. Thus, the boundary of space has zero thickness and zero area, which means that the boundary of space is a single point.

Is that possible? Can the boundary actually be smaller than what it encloses? Common sense rebels against the idea. After all, I can look to the east and know that I am looking toward the boundary some billions of lightyears distant and then I can turn around and look to the west and know that I am also looking toward the boundary some billions of lightyears distant. Surely I must be looking toward different parts of an extended object. But reason says no, that's not true. In whatever direction you gaze, you are looking toward the same single point. And what reason tells us reveals a strange geometry underlying what we see in space.

Our tacit assumption in our commonsense view is that our sightlines are perfectly straight. That assumption is founded upon the fact that in our experience light travels in straight lines. It remains a good assumption because, even though we know that gravity bends light, the perturbations that will be imposed upon light traveling great distances through intergalactic space will be small enough to ignore. But now we have before us the proposition that two straight lines, extended away from a common point, come together again at some other point. How can that proposition be true to Reality?

To paraphrase Sherlock Holmes, once we have eliminated what cannot be true to Reality, then whatever we have left, however incredible, must be at least a reflection of what is true to Reality. What we have left in this case, to use the simplest example, is a straight line that is extended in both directions and eventually meets itself on the single point that comprises the boundary of space. In that property of closing on itself the line displays the character of a circle. But we know that the line is straight, so we must infer that it curves in a direction that we cannot sense; that is, there must be a fourth dimension to space, in which dimension straight lines display curvature. The existence of that fourth dimension is necessary for space to exist with a boundary that's a single point. Beyond that proposition there is nothing more that I can say here about the fourth dimension except to point out that this fourth dimension is not to be confused with time, which is also called the fourth dimension in some presentations of the theory of Relativity.

There is something more to say about that straight line that has the character of a circle. Imagine that you have drawn a set of such lines from a point and extended them all the way to the boundary of space. If space were small enough (and we can imagine it to be so, confident that the rules that we infer for such a microspace will apply with equal validity to our own much deeper cosmos), then you would reasonably believe that you could follow one to the boundary and look back at the point from which those lines radiate. Tacitly assuming that the point that you have chosen is the center of the Universe, I picture what you would see as this: the boundary would appear to you as a true point with your lines radiating away from it in all directions and your beginning point would appear to be spread across the sky, just the reverse of what I see if I remain at the center of the Universe. That situation implies that at some point on your travel to the boundary you would have seen a transition, in which the boundary of space collapses down to a true point and the point at the center of the Universe blossoms across the sky. In so collapsing and blossoming the two points carry the endpoints of the lines that you have drawn with them and that fact creates a problem.

Let's assume that the line that you are following defines East and that two lines that you drew perpendicular to it point North and South. During the transition, you will see those North and South lines appear to reorient themselves. Whether they remain straight while they reorient or whether they bend and then straighten out is irrelevant; when you reach some point on your East line some part of the North and South lines will appear to you to be oriented parallel to the East line. But that automatically violates our theorem that all of the motions in the Universe must always add up to a net zero.

Imagine that I have taken two dynamically equivalent bodies and so pushed them apart that they move away from me in opposite directions, one along the North line and the other along the South line, at the same speed. The linear momenta of the bodies are of equal magnitude but are oriented in opposite directions, so the net momentum of the two bodies is equal to zero, as we should expect. But you would be obliged to disagree with that description. As seen from your position, at some time the two bodies will traverse the parts of the North and South lines that run parallel to each other; thus, the bodies will be traveling in the same direction and their linear momenta will add up to something that's not equal to zero. That's bad enough, given that there's no velocity between you and I in this imaginary experiment, but, worse, the Eastward action acquired by the bodies is not accompanied by a equal Westward reaction upon me. I see the situation as being in perfect accord with the conservation of linear momentum, but you see in it a blatant violation of the conservation law.

Again we have a contradiction that we must resolve by deciding which of the premises underlying it is false to Reality. A quick review of the experiment tells us that if we are to keep all of the propositions that we have deduced up to now, then the only premise available for dismissal is the tacit assumption that you could move to a place where the curved nature of space becomes overtly manifest, that you could move away from the center of the Universe. We are thus obliged to assert as true to Reality the proposition that no thing can move away from the center of the Universe.

Has logic failed us? The path of deduction seems clear enough and our axioms are truly self-evident, but we seem to have deduced a description of the Universe that is simply not true to Reality. How can we go any further? Is there anything more that we can reason out about space that will get us out of this jam? Indeed there is and it will astound and delight you.

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