THE UNREACHABLE BOUNDARY

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This is absurd! From the observation that we exist we have deduced the proposition that every body that exists must exist at the center of space for all time. But the very concepts of space and time necessarily entail the possibility of motion from one place to another, a possibility that we have exploited in our reasoning so far. And the concept of multiple bodies in space necessarily entails the possibility of spatial separation. If two or more bodies are separated each from another by some distance, how can they satisfy the requirement that they all be at the center of space?

We can start to approach an answer to that question by asking how we know that a given point is the center of space. What is it about that point that tells us that it's the center of the Universe? You already know the answer to that question: the point's distance from the boundary of space must be the same in all directions. Implicit in that statement is a proposition that I now make explicit; that is, that space must have a shape that reflects the geometric character of a sphere. At first that proposition may seem to claim too much. In order for our chosen point to be the well-defined center of space it is sufficient only that the distances to the boundary in opposite directions be equal to each other: if we draw a line to the boundary in one direction, then the line drawn to the boundary in the opposite direction must have the same length. But that criterion, strictly interpreted, would apply equally to a sphere, to an ellipsoid, or, for that matter, to a cube. If space is a figure other than a sphere, then, the boundary will display features characteristic of that figure; that is, it will have the edges and vertices of a cube or the varying curvatures of an ellipsoid or the features of some other figure. We have already discovered, however, that the boundary of space must have the same geometric properties on both sides. The side facing Absolute Nothingness can have no features whatsoever, so our side can also have absolutely no features that would in any way or to any degree point a special direction in space. The only figure that satisfies that further criterion is a sphere, so we must conclude that space is absolutely, perfectly spherical.

So every object that exists sits forever at the center of a perfectly spherical space. How is that possible? If space had infinite extent, then objects separated by any finite distance would nonetheless each still be an infinite distance from the boundary in all directions and the criterion above would be satisfied. But an infinitely extended space is incompatible with conservation of angular momentum, as we deduced earlier, so space cannot have infinite extent. The image that I just evoked offers a hint of a way past that barrier by way of an analogy. Imagine that you and I are standing on a featureless plain that extends for many miles in all directions and that I walk away from you. When we are together we use appropriate surveying techniques to determine that the distance to the horizon is the same in all directions. Thereafter, when I have walked some distance away from you, I measure the distance to the horizon again and discover that it has not changed, regardless of how far I have walked. I seem to have remained at the center of the world, but I understand that the chief fact underlying that conception is that I have been walking on a curved surface. The analogy is spoiled by the fact that the horizon is different for both of us: the boundary of space must be the same for both of us, so we can't hypothesize that space is curved in the way that Earth's surface is curved.

That spoiled analogy is still useful in that it tells us what we need to discover about space that will solve our problem. It tells us that we must hypothesize that space is so structured that when I travel in any direction some part of the space ahead of me will expand and some part of the space behind me will contract and that they will do so in a way that maintains an unchanging finite distance between me and the boundary in both directions. In such a space you would see me taking progressively shorter steps (though I have stated that all of my steps are of the same length) as I go progressively farther from you: though the distance that you measure to the boundary is finite, I must take an infinite number of steps and thus cross an infinite distance to reach it. As a simple example that doesn't necessarily reflect the actual distortion of space, I offer the following: I take a step and then make each subsequent step half as long as the one preceding it. If you recall Zeno's paradox of Achilles and the Tortoise, you will see that even if I take an infinite number of steps, I will not reach a point twice as far from my starting point as the length of my first step.

But if I can walk away from you and appear to approach the boundary, doesn't that bring us back to our original problem of straight lines bending in ways that violate the conservation laws? I answer no, because the situation is different from our original situation. In this distorted space I will never reach a point at which the boundary ceases to cover the entire celestial sphere and begins to collapse to a true point. In such a circumstance space can be so distorted that straight lines remain straight, regardless of how far they appear to be removed from what appears to us to be the center of space, and thus preserve the validity of the conservation laws. Without actually needing to describe it, we simply assert that space is so deformed from the simplest spherical geometry.

That distortion of space solves our problem by putting the regions of space in which curvature of straight lines would permit violations of the conservation laws at a distance from us that, though finite, is effectively infinite. That is, we have asserted that space has a finite extent that has the character of an infinity. But we haven't yet exhausted our supply of problems with the geometry of space. Suppose that after I have walked a certain distance you throw a ball to me. If I fail to catch the ball, what's to prevent it from reaching and crossing the boundary?

If space contracts as it gets closer to the boundary, as we have inferred, then we would expect that the ball would slow down as it passes me and travels onward. That would solve our problem but for the fact that moving objects cannot change their velocities spontaneously, even if their masses change in a way that maintains their original linear momenta. As I showed in the essay on linear momentum, there is no way to make that process work in a way that makes the momentum change by zero for all observers. That proposition remains true even in our deformed space. Thus, once the ball leaves your hand and until it collides with another object, it will fly at the same speed. Because an object moving with a finite speed crosses a finite distance in a finite elapse of time, the ball will reach the boundary unless some other phenomenon prevents it from doing so.

One obvious candidate for that other phenomenon is that of the boundary moving away from us and doing so at a speed that the ball cannot achieve. The expansion of space necessary for such a phenomenon is certainly allowable: in my discussion of the finite-value theorem I noted that space itself is not subject to a conservation law and, thus, may expand or contract as other laws may require. However, the finite-value theorem also tells us that the boundary cannot move at infinite speed. The boundary must move away from us at a finite speed that no object can ever surpass, though some objects may equal it. What we need, then, is a finite speed that has the dynamical character of an infinite speed.

If we can declare that space is so shaped to permit an infinite length to fit into a finite distance, then we can certainly claim that space and time taken together are so shaped that the boundary of space moves away from us in all directions at a finite speed that has the arithmetic character of an infinity. What that means is that if, instead of walking away from you, I accelerate away from you until I reach some speed, I will nonetheless measure the boundary moving away from me at the same speed in all directions, just as I measure the distance to the boundary to be the same in all directions. As we expect of something with the character of an infinity, in particular of Aleph-Null, we see that adding a finite speed to the boundary's speed (in the direction opposite my acceleration) or subtracting a finite speed from the boundary's speed (in the direction of my acceleration) does not change the boundary's speed, not one little bit.

So now we have a Universe in which space and time are so shaped that they uphold the conservation laws. In obtaining that description we have put ourselves, in imagination, into a strange situation. You see yourself at rest at the center of the Universe and observe me moving away from you, but from my perspective you're the one who's off-center and moving. If we were to argue over which of us is right, our description of space and time necessitates that the laws of Nature not help either of us in resolving the argument. If every observer must necessarily see themself as being at the center of the Universe, then two theorems must be true to Reality:

COSMIC THEOREM 1: The results of any given experiment or observation and the laws of physics derived from them are the same for all experimenters or observers, regardless of how those experimenters or observers are positioned, oriented, or moving relative to each other.

COSMIC THEOREM 2: Any phenomenon that moves at the same speed at which the boundary of space moves passes all observers at that same speed, regardless of how those observers are positioned, oriented, or moving relative to each other.

If we recognize the speed of the boundary as the speed of light, then we must also recognize those theorems as being slightly broader versions of Einstein's two postulates of Relativity.

From the bare fact that we exist I have shown you how to deduce the proposition that the Universe must be so structured that the theory of Relativity based on Einstein's postulates correctly reflects the shape of space and time. Beginning with those postulates, I will now show you how Einstein deduced the features of the Lorentz Transformation, which describes the relationship between space and time. Before I do that, though, I want to digress a bit to tell you what we will be transforming and why.

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