The Fundament of Spacetime
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Whence does it all come? We talk about space and time, but why do they exist? And why do they take the form that we infer from our observations? Astoundingly, we can deduce answers to those questions.
Imagine something that separates existence from non-existence, a kind of boundary. In contact with non-existence, that boundary can have no properties. In contact with existence, it must have the existent equivalent of no properties; that is, its properties must all equal zero. Thus we infer that the boundary consists of a single mathematical point, a thing with no properties at all except that it exists.
The existence of existence necessitates that something exist. Given that we have a point, we can infer that the minimum something that can exist is a set of additional points. We have no basis for the existence of a certain specific number of points, so we must infer that the number of points must be indeterminate; that is, because there is only one indeterminate number, we must assert that the points constitute an infinite set.
In order to be different points, as they certainly must be, the elements of that set must differ from one another in some way. That way must constitute an infinite set of mutually different possibilities. Whatever that way may be, we can make an analogy between it and another infinite set Ė the set of the real numbers. If we assign a unique real number to each and every point, we transform the inchoate set of points into an ordered set, which set we call space. We achieve that ordering in particular by assigning numbers with adjacent values to points that lie adjacent to each other. In that ordering we make the boundary between existence and non-existence the point of absolute zero, the origin of space.
Imagine that we put something, call it a particle, on one of the points, which point we label A. We can describe the location of the particle by stating the number associated with that point, the index number of Point A. Point B has a different index number. If we want to change the location of the particle from Point A to Point B, the particle must touch each and every point between Points A and B and do so in their proper sequence, thereby tracing out a line segment. We can describe that line segment through some function of the index numbers of Points A and B and we call that description extent, either as the length of the line segment or as the distance the particle moved.
Although an infinite subset of points lies between Point A and Point B, the extent that we calculate for the line they constitute must come out as a finite number. Point A and Point B are definite points with definite index numbers, so the difference between them, the distance, must also be definite; that is, finite. We can see this fact reflected on the number line, where we have an infinite subset of decimal fractions between one and two and yet the difference between one and two is a finite number. If we imagine an object whose ends occupy Point A and Point B, which object we call a ruler, then we can imagine using that object to measure distance, such as the distance between Point C and Point D.
Notice that each and every point in space has two other points lying adjacent to it, one with a lower index number and one with a higher index number. The sole exception is the boundary of space, the absolute zero point, which can only have a point with a higher index number adjacent to it. But then what can we say about the point with the highest possible index number, a point that we call the farpoint?
If that farpoint has only one point lying adjacent to it, then it is a second boundary with non-existence. But that proposition necessitates that the second boundary differ from the first boundary in some way in both existence and non-existence. Differences cannot exist in non-existence, so a second boundary cannot exist. We must assert, instead, that a second point lies adjacent to the farpoint and that it has an index number smaller than that of the farpoint, whose index number, whatever it may be, is the maximum of all of the index numbers. But then the points extending beyond that point must also have progressively smaller index numbers until they come to zero, which must lie on the boundary.
We must note that the farpoint itself cannot touch the boundary. If it did, it would mark a definite place. But the farpoint, representing infinity, is necessarily indefinite. Thus, it must lie adjacent to another indefinite point, the farpoint of another line.
We now have two lines extending from the farpoint to the boundary. To keep them distinct from one another, as we must, we assign to each and every point a second index number. Those second index numbers come from an infinite set, so we infer the existence of an infinite set of lines extending between the farpoint and the boundary. We now have a two-dimensional space.
An infinite set of endpoints lying adjacent to the boundary of space constitute a line of no length. That line can have a definite beginning but it must have an indefinite end while remaining adjacent to the boundary. From that fact we infer the existence of angular measure. The set of endpoints forms a line of no length that goes around the boundary endlessly.
That line must, in some sense, go around the farpoint. As a consequence, an observer at the farpoint would see the boundary encircling the farpoint. That circle would display the illusion of a large, nonzero, but finite length. That circle would have to be absolutely perfect, with the farpoint in the precise center. The reasoning behind that statement originates in the fact that, as a pure mathematical point, the boundary can have no parts and no features that would distinguish parts. In order for the lines extending from the boundary to the farpoint to have different lengths, there would have to be distinguishing parts on the boundary, so we necessarily infer that those lines all have the same length and, thus, that the farpoint exists at the center of what appears to be a perfect circle. Further, the circle has no actual extent, no circumference, so a point lying some distance from one apparent part of the circle must necessarily lie the same distance from all of the apparent parts of the circle.
Pick two arbitrary points on that circle. There is no actual distance between those faux points, but there appears to be a non-zero distance between them. That appearance reflects the fact that an infinite subset of points spans that illusory distance. Because it emerges from nothing, that illusory distance must be indeterminate; that is, it must have multiple values available to it, each value being manifested by a different subset of points. Because of their different values, those subsets must be distinguishable, so we assign to all of our points a third index number. Our space now has three dimensions and the boundary appears to someone at the farpoint as a spherical shell with an infinite set of lines connecting our two chosen illusory points.
Now we turn our attention back to the lines connecting the farpoint to the boundary and note that they must also have indefinite lengths. That fact necessitates that we assign a fourth index number to each and every point in space in order to distinguish the different possible lengths from each other. For convenience we can assign the numbers in such a way that an increase in the number corresponds to a lengthening of the distance and a decrease in the number corresponds to a shortening of the distance. We then refer to the increase in the fourth index number as the elapse of time and note that as time elapses, the boundary appears to move farther away from an observer at the farpoint. We also note that the fourth index number must have a value of absolute zero on the boundary, which means that time does not elapse on the boundary.
The four index numbers define loci in what we now call spacetime. Each locus provides the place and time of a potential or hypothetical event. If we have an object that produces identical events occurring uniform intervals apart, then that object constitutes a clock, which enables us to count time. With rulers and clocks we can measure distances and durations between pairs of events.
Consider a particle that occupies two different loci. We infer that in the temporal interval between those loci the particle has crossed the spatial distance between them point by point. To describe that crossing we impute to the particle a velocity, the measure of motion, which consists of the spatial distance crossed and the direction of the crossing, taken together, divided by the time elapsed. And we know that an observer remaining with that particle would see us moving in the opposite direction at some speed.
We have inferred that our farpoint occupies and marks the center of the Universe by virtue of the fact that the distance between that point and the boundary is the same in all directions. That equality of distances is necessitated by the nature of the boundary as a featureless mathematical point. That fact tells us that any point we can occupy must necessarily appear to float at the center of the Universe.
That statement also applies to the observer accompanying the moving particle described above. That person must see themself as occupying the point at the center of an infinite set of points that do not move relative to that central point. We see ourselves in the same way regarding our farpoint and the infinite set of points surrounding it. We thus have two infinite sets of points moving relative to each other. Each of those sets constitutes an inertial frame of reference with its own coordinates (index numbers). The velocity between the sets is arbitrary, so we infer that spacetime consists of an infinite set of inertial frames moving in all available directions at all possible speeds.
Each and every inertial frame originates on the boundary of space. In order for the boundary to be completely and absolutely featureless, the inertial frames must all be identical, each to another. Nothing can distinguish one inertial frame from another except the velocity between them and only the velocity between them. That statement is equivalent to Einsteinís first postulate of Relativity: the laws of physics must be the same and have the same mathematical expression in all inertial frames.
Each and every observer occupies and marks a point that appears to float at the center of the Universe: the distance to the boundary is the same in all directions. As time elapses, the distance to the boundary increases in all inertial frames. The speed with which the boundary appears to recede from any observer must have the same value for all observers, regardless of any velocity between the inertial frames that they occupy. If we identify that speed of recession with the speed of light, then we see that statement as a form of Einsteinís second postulate of Relativity.
If two observers occupying different inertial frames measure distance and duration between the same two events, they must inter-relate their measurements through a set of transformation equations. There are four such equations and together they are known as the Lorentz Transformation, which we will explore in the next chapters.
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