Zeno’s Relativity

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"We are caught here for a moment like an imprint of a hand

on an old and frosted window or a footprint in the sand."

-Koppången-

The above is sung by Sissel Kirkjebø on her album Northern Lights.

***

    Before we can understand and resolve the paradoxes of motion presented by Zeno of Elea (ca. 490 BC - ca. 430 BC) in the Fifth Century BC, we must understand the context in which he conceived them. We need to know how the Ancient Greeks conceived the fundamental structure of Reality and how that concept evolved into the foundation of classical physics, then how Isaac Newton’s vision evolved into our modern concept of the substructure of Reality.

    Begin with Xenophanes (ca. 570 BC - 478 BC), the founder of the Eleatic school of philosophy. He founded his doctrine on the concept that "the all is one". That statement implies that distance and duration do not truly exist, so the differences that we see must exist only as illusions in our minds. Indeed, any appearance of multiplicity, change, or motion exist as illusions only.

    Parmenides (ca. 515 BC - 450 BC) extended Xenophanes’ doctrine of the absolute unity and constancy of Reality by inventing the concept of logical proof. As a pupil of Xenophanes, Parmenides had absorbed his teacher’s idea that "the all is one". In that doctrine the Universe exists as a singular, eternal, and unchanging entity. Because of the apparent absurdity of the doctrine, Parmenides augmented his statements of belief in the permanent unalterable oneness of Reality with logical statements supporting that belief. The Eleatic belief in the absolute unity and constancy of Reality strikes us, as it did the Greeks, as absurd, so Parmenides had to reason that plurality and change exist in our minds only as illusions. Thus Parmenides originated the distinction that we make between the thing-as-perceived (the what-we-see) and the thing-in-itself (the actual what-really-exists). We thus have the Aristotelian-Platonic distinction between function and essence; what it does versus what it is.

    Zeno, one of Parmenides’ students, presented his paradoxes as logical proof that change/motion and plurality cannot possibly exist and must, therefore, come to us as illusions. Zeno thus devised the technique of reductio ad absurdum, the reasoning that if we can prove and verify the impossibility of a proposition, then we thereby necessitate the truth of its negation.

    Zeno’s main contribution to the Eleatic doctrine consisted of several paradoxes of motion, which paradoxes he presented as proof that motion cannot actually occur and, thus, must exist only as an illusion. Of the absurdities that Zeno presented, three in particular address the paradoxes of motion in the clearest way. We have those paradoxes as:

    I. The Dichotomy; Aristotle (in his Physics VI:9, 239b10) said, "... that which is in locomotion must arrive at the half-way stage before it arrives at the goal."

    Imagine a tableau in which a hunter has seen a deer, put an arrow into his bow, drawn the bow, and has just released the bowstring. Zeno claims that the arrow cannot move: it cannot leave the bow. He reasoned that, in order to strike the deer, the arrow must first cross half the distance to the deer; but before that the arrow must cross one quarter of the distance; but before that the arrow must cross one eighth of the distance; and so on in an infinite regress. In order to leave the bow, the arrow must take a first step, but in the infinite regress there is no first step; therefore, the arrow cannot begin to move.

    II. The Arrow; Aristotle (in Physics VI:9, 239b5) said, "If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless."

    Somehow the arrow has left the hunter’s bow and flies toward the hapless deer. But at any given instant, Zeno argues, the arrow occupies an extent of space equal precisely to its own extent. Because we define an instant as an elapse of time having zero duration, the arrow does not have time to move, so at every instant the arrow is at rest. This fact is true for every instant, so the arrow is always at rest and, therefore, does not move.

    Note that, in a way, this reasoning anticipates the phenomenon behind the motion picture. The motion picture film bears a sequence of still photographs, in which the images stand perfectly motionless. But when those photographs are projected one after another at an appropriate rate, then the projected images appear to move.

    III. The Pursuit; Aristotle (in Physics VI:9, 239b15) said, "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."

    This paradox is usually explained through the little story of Achilles racing a tortoise, but to maintain consistency let’s return to our story of the hunter. As the hunter releases the arrow the deer panics and runs straight away from the hunter. If the arrow overtakes the deer, the hunter’s family will have venison for dinner. But Zeno says that the arrow cannot overtake the deer, for in order to do so it must first cross the original distance between the deer and the bow. While the arrow crosses that distance the deer moves a second distance. As the arrow crosses the second distance the deer moves over a third increment of distance, and so on in an infinite regress. In crossing the distance to the deer the arrow can only strike the deer if it takes a final step, but in the infinite regress there is no final step. We must thus infer that the arrow did not reach the deer, which means (in addition to the hunter’s family going hungry) that the arrow didn’t really move.

    Eleatic physics: the Universe has a state of absolute rest, which all things occupy. All things are illusory manifestations of the one and only simple entity that exists.

    Because Zeno conceived those paradoxes as criticisms of what we have called the Aristotelian-Newtonian picture of Reality, we can reasonably expect that resolving those paradoxes will lead us to a more correct picture of Reality.

    In fact, those paradoxes reveal flaws in our concepts of space and time and of their relationship to each other. Zeno failed to understand that motion is a relationship between distance and duration, which failure should have obliged him to analyze both space and time together. He also lacked the advantage of Descartes’ analytical geometry with its coordinated grid on which to project algebraic functions.

    Philosophers continued for centuries to struggle with the Eleatic doctrine. In those struggles they devised a description of the substructure of Reality, the foundation underlying classical physics. They did so by examining the fundamental dichotomy, that of matter existing in space and time.

    For example, Leucippos and his student Democritos sought to reconcile Eleatic stasis and observable flux by asserting that matter consists of changing arrays of unchanging particles called atoms. Likewise, Plato devised a similar mixed solution of the Eleatic problem, reconciling changing and unchanging, plural and singular, in his assertion that matter consists of changeable substance spattered upon unchangeable Forms. In Plato’s metaphor of the cave matter appears to us as a blurry shadow of the Ideal, cast on the wall by a flickering fire.

    The Greeks also had a theory of space, which we call geometry. Consisting of assumptions about the nature of space, relations among distances and angles, and logical proofs of those relations, geometry provides a means of surveying the Universe. In the Euclidean conception space comprises an infinite set of elements (points, indexed uniquely by trios of real numbers) organized by a characteristic function (distance; D2=Ax2+By2+Cz2). Technically the spatial point is an infinitesimal of extent. The mathematical point is the limit as infinitesimal goes to zero. Ironically, Euclid’s definition of a point as a thing that has zero extent necessitates that space, conceived as an infinite set of points, also has no extent, thereby conforming to a monad of Xenophanes. Distance seems to exist only as an illusion. If the points are ordered, then we certainly have the illusion of distance.

    How far does that illusion extend? Archytas, a Pythagorean friend of Plato, imagined going to the edge of the Universe and stretching out his hand. He would feel empty space beyond the edge or some barrier would stop him; either way the edge wouldn’t be much of an edge since there was always a beyond. He concluded that the Universe must be spatially infinite.

    Aristotle objected that an actual infinity was an impossible irrationality and made a peerless claim about it. He had no quarrel with any kind of infinity existing as long as it remained only potentially infinite. A sum could grow larger, a universe older, a space could expand without end, provided that the infinity was never there all at once. And yet....

    Aristotle resolved the paradoxes of The Dichotomy and The Pursuit by noting that the elapsed time occurs in an infinite series that matches the infinite series of spatial steps taken in the story. Just as the infinite series of spatial steps adds up to a finite distance, so too the infinite series of temporal leaps adds up to a finite duration. The deer is doomed.

    Aristotle also conceived space as having a dynamic aspect. In Aristotelian physics space has a unique, well-defined center, which marks and occupies an absolute state of rest, toward which all bodies tend. Because the Ancient Greeks did not have a consistent means to measure time with the precision needed for basic physics, natural philosophers had to wait two millenia to incorporate time into their physics. Only when mechanical clocks had developed to the necessary precision was Isaac Newton able to conceive the incorporation of time into his dynamic geometry, declaring that time flows "evenly and equably" for all observers. Again we note the irony that if we define an instant of time as having no duration, in analogy with the mathematical point of space, then an eternity of instants also has no duration, in accordance with the Eleatic doctrine.

    Aristotelian physics: Matter consists of arrays of particles whose motions change in accordance with applied forces. Material bodies act out events in an infinite space that has a unique center that defines an absolute state of rest and do so in a universal time. Aristotelian physics is based on a state of absolute rest referred to the center of the Universe.

    Resolving the Paradox of the Arrow: at a given instant an arrow occupies a set of points with ends at x1 and x2 (a single instant requires that t2-t1=0). Per Zeno, the arrow cannot move. But consider two arrows side by side at an instant, one arrow motionless and the other launched from a bow. They are identical, so how can one move and the other not?

    To prepare the solution imagine tilting one arrow away from the other in the x-y plane. In the first arrow (lying on the x-axis) we have y1-y0=0 for any two points x1 and x0 on the arrow and on the second arrow we have y1'-y0' 0. For any point on the second arrow we can write y’=y0'+αx’, in which x’ represents the x-ward distance measured along the arrow from the point x0' that corresponds to y0' and alpha represents a measure of the degree to which the arrow tilts away from the first arrow (or the x-axis). That last equation, if we let x’ extend beyond the arrow, describes the straight line on which that arrow lies. If we keep the arrows on the lines thus defined, then when we increase their x-coordinates, the y-ward distance between the arrows will change.

    We note that the Greeks of Zeno’s time did not use the Cartesian coordinate grid, but they had available to them something just as good – latitude, longitude, and altitude. Lines representing the directions north-south, east-west, and up-down cross each other at right angles, thereby producing the equivalent of the more abstract Cartesian grid. If Zeno or someone close to him had conceived the idea of past-future as a direction analogous to those three spatial directions, they would have rendered time static (in accordance with Xenophanes’ doctrine) and incorporated it into geometry. That incorporation would have enabled that person to resolve the arrow paradox.

    The Greeks also did not have available to them the symbolic algebra that we use to construct and solve equations. Instead the used rhetorical algebra, a purely verbal means of solving equations by way of invoking Euclid’s five common notions. Occasionally they would use geometric diagrams as aids to the imagination, but fundamentally their version of algebra was worked as word problems. That fact made mathematics, like that displayed in the equations below, more difficult than it is for us, but not impossible. That difficulty was increased by the fact that when they had to carry out arithmetic calculations the Greeks did not use a place-notation system based on something like our Hindu-Arabic numerals; rather, they used the Greek equivalent of Roman numerals, which make even basic arithmetic much more difficult than what we are used to doing.

    With those caveats in mind, we can imagine telling our Ancient Greek philosopher (let’s call him Monopetros (and, yes, that’s a very bad pun. It means "one stone", which is rendered in German as ein stein)) to lay two arrows on the ground with one pointing due east (the positive x-direction on our Cartesian grid) and the other pointing slight north of due east (partly into the y-direction). If we move the arrows in the directions in which they point, the north-south distance between them will increase.

N    ow imagine that Monopetros lays two arrows on the ground, both pointing due east, and then tilts the bowshot arrow into the temporal (past-future) direction. Any given point on the stationary arrow, such as the tip of the arrowhead, traces a straight line parallel to the past-future line (the temporal axis or t-axis), so as the arrow goes from past to future it does not move to the east. But the tip of the bowshot arrow’s head traces a straight line that tilts away from the temporal axis, so as the arrow goes from past to future it also moves to the east. If we designate a point on the east-west line as x0 and specify that the tip of the bowshot arrow occupies that point at some instant (which we call t=0, though the Greeks, without the concept of zero as a number, would more likely have used something like t=1), then we can describe the location of the arrow’s tip at any other instant by writing

(Eq’n 1)

In that equation c represents a factor that converts units of time into units of distance and phi represents a number that measures the degree to which the straight line traced by the arrow’s tip tilts away from the temporal axis.

    We measure motion with a number v that we call speed (or velocity if it has a specific direction associated with it), which equals the ratio of distance crossed to time elapsed measured over a segment of the moving object’s path. We know that if an object moves at a uniform speed, we can calculate the distance that it moves in a given elapse of time by multiplying the speed by the elapsed time. Thus we can write Equation 1 as

(Eq’n 2)

which leads us to infer that φ=v/c.

    We said that we have tilted the bowshot arrow into the temporal direction. That statement means that on the arrow itself clocks placed at different points will show different times. Note that the Ancient Greeks did not have mechanical clocks, so this imaginary experiment would have been difficult for them to conceive. However, Monopetros might have conceived the idea of using a rapidly repeating phenomenon, such as the beating of a gnat’s wings, to divide time finely enough for him to count time in this experiment. All he would need is something to keep count of the gnatbeats.

    Sitting by the stationary arrow, he would then see the clock on the bowshot arrow’s tail showing a time later than the time shown on the clock occupying the arrow’s tip; in other words, the rear clock appears to lie in the future relative to the fore clock. That difference between the clocks maintains the right angle between the arrow and the straight line traced in time by any point on the arrow, provided that the arrow tilts to the same degree. We then know that if we designate one clock as showing t0, then any other clock on the arrow shows

(Eq’n 3)

in which x represents the distance (x-x0) between the t0-clock and that other clock.

    If Monopetros wants to use the arrow’s clocks to determine the time at which two simultaneous events separated by a distance x occur, then he must compensate the temporal offset between the clocks. He must thus calculate the time on one clock in terms of the time shown on the other clock through the equation

(Eq’n 4)

That equation, derived from Equation 3, tells Monopetros the time that he would see on the second clock if the arrow were not tilted in time, a time related to the time T displayed on Monopetros’ own clocks.

    Now Monopetros understands that his concept of geometry must change. He has been accustomed to using a geometry based on points separated by distances. Now he must devise a version of geometry based on instants (entities with zero extent in time) separated by durations. Instants seem as static as points, so the new geometry appears to conform to the Eleatic doctrine. And just as we use dots or other small things to mark points, so we can use brief events, such as the beat of a gnat’s wings, to mark instants. And just as we use rulers to measure distances, so we use clocks to measure durations.

    Equations 2 and 4 look like the operative equations of the Lorentz Transformation, the equation of motion and the equation expressing the relativity of simultaneity. But it’s not the Lorentz Transformation. First, it lacks the Lorentz factor, so it also lacks time dilation and the Lorentz-Fitzgerald contraction. Second, we derived it through a circular transformation of coordinates, while the authentic Lorentz Transformation reflects an hyperbolic transformation of coordinates. Yes, it legitimately resolves the arrow paradox, but to the extent that it gives us any kind of relativity theory, that theory is Lorentzian; that is, the velocity in the equations refers to a state of absolute rest, the state presumably occupied and marked by the lumeniferous aether. But like any good theory of relativity it would have let the Greeks know that Reality is weirder than they imagined.

    To obtain a proper theory of relativity, we may notice in passing, Monopetros would need to know and understand the fundamental nature of light, possessing something the Ancient Greeks did not have. He would have needed to know, first of all, that the basic form of light mimics the basic form of waves on the ocean. He would have also needed to know that, unlike waves on the ocean, light is its own medium; in particular, he would have needed to know that light consists of a vibrating electric field, which generates a vibrating magnetic field, which generates a vibrating electric field, which generates.... Lacking that knowledge, Monopetros would not have known that light must fly at a certain set speed in order for the two fields to support each other mutually.

    Having resolved Zeno’s version of the arrow paradox, Monopetros would soon see that he had just created another such paradox. He need only imagine some ants riding on the bowshot arrow. Surely in the ants’ experience the bowshot arrow does not move but the stationary arrow does move. In the ants’ view, then, the clocks on the bowshot arrow should appear in synchrony with each other and the clocks on the stationary arrow should appear out of synchrony by the appropriate temporal offset. Is it possible for the bowshot arrow to appear tilted in time to Monopetros and appear not tilted in time to the ants?

    If Monopetros were to abstract his latitude-longitude-altitude frame into a generalized rectangular coordinate frame and if he understood how two such frames might have some angle between them, that they could lie tilted relative to each other, then he would have to answer that question in the affirmative. A geometer using one of those grids would see the lines defining the other grid tilted away from the lines that define his grid but he also knows that a geometer using that second grid would see the lines of his grid tilted away from the second grid’s lines. The two geometers would discern a symmetry in the difference that they see between their respective reference grids.

    Monopetros wants to deepen his understanding of that symmetry as it applies to grids tilted in time. To that end he considers how the clocks on the bowshot arrow came to be out of synchrony. The ants want their clocks to be synchronized and they have an easy way to achieve that end: one ant sets the two clocks on the head and tail of the arrow to show the same time, positions one of her sisters by each of the clocks, and has the two clocks started at the same instant after the arrow leaves the bow.

    With the arrow in flight, the first ant, standing on the midpoint of the arrow, gives the signal to start the clocks. Light carrying the image of the signal propagates both forward and backward along the arrow and when it reaches the fore and aft ants, those ants immediately start their clocks. In order for those clocks to be out of synchrony, the aft clock must start some time before the fore clock does and that fact entails, as Monopetros discerns to his astonishment, that light travels at a finite speed, which we will represent as s for convenience. Given that the arrow moves at a speed v, Monopetros infers that the signal goes to the aft ant at a net speed of s+v and to the fore ant at a net speed of s-v. Given the arrow’s length as x, he can calculate the time difference between the clocks, ta-tf, by first calculating ta=½x/(s+v) and tf=½x/(s-v) and then obtaining

(Eq’n 5)

That looks like it has the same algebraic form as does the second term on the right side of Equation 4, but Monopetros does not want to assert an identity just yet, simply because the denominator doesn’t look quite right.

    Again as Monopetros calculates it, light carrying an image of the rear clock propagates to the middle ant at a net speed of s-v and light carrying an image of the fore clock propagates to the middle ant at a net speed of s+v. Because the velocities of the light are interchanged between the front and rear halves of the arrow, the time difference between the clocks is reversed and the clocks look to the middle ant as though they are telling the same time; they appear to be in synchrony, even though they are out of synchrony for Monopetros. The arrow appears tilted in time (moving) for Monopetros and not tilted in time (stationary) for the ants. Of course Monopetros has assumed that the speed of light is referred to the state of absolute rest: that assumption is about to create a dilemma for him.

    A trio of small spiders stands on the stationary arrow and they have their own miniature clocks on the arrow’s head and tail. As the ants have done, so the spiders synchronize their clocks: the spider standing on the midpoint of the arrow gives a signal, the light carrying the signal propagates to the spiders at the arrow’s head and tail, and those spiders immediately start their pre-set clocks when they see the signal. Monopetros, sitting on a line drawn perpendicular to the arrow from its midpoint, confirms that the clocks are counting time in perfect synchrony (the light paths are of equal length, so there’s no time lag to take into account).

    The ants have observed the synchronization of the spiders’ clocks and come to a remarkable conclusion – that the clocks are synchronized in their view as well. If light moves at the speed s relative to the state of absolute rest, then the ants would measure it traveling toward the tail of the stationary arrow at the speed s+v and toward the head of the stationary arrow at the speed s-v. But in the ants’ view the stationary arrow’s tail should be moving away from the middle spider’s signal at the speed v, so the light carrying the image of the signal should travel to the tail with a net speed of s; likewise, the arrow’s head should be moving toward the middle spider’s signal at the speed v, so the light carrying the image of the signal should travel to the head with a net speed of s; thus, in the ants’ view the spiders’ clocks start simultaneously, which puts them into perfect synchrony, which means that the stationary arrow is not tilted in time as seen by the ants, which means that the arrow does not move as the ants move past it. To enlarge the absurdity, Monopetros discerns that the ants, in their derivation, had assumed that the stationary arrow was moving relative to their point of view when they deduced that it is not moving relative to their point of view.

    Monopetros easily sees that the dilemma goes away if light travels at the same speed in all directions for both the ants and the spiders (thereby expressing the gist of Einstein’s second postulate of Relativity). That proposition conforms to the symmetry that Monopetros saw in the diagram he drew to show the bowshot arrow tilted in time, thereby conforming to the fundamental notion that the Platonic Form of Reality is geometric. Yes, it would seem strange to him, but light is an immaterial thing and, therefore, cannot be expected to behave as matter does. But he would be disturbed by the necessary implication that the world has no universal state of absolute rest.

    If the world has no state of absolute rest, one part of Aristotle’s physics must go away. No longer can Monopetros believe that matter has a propensity to come to rest. So what propensity does matter have with regard to motion?

    Reviewing his imaginary experiment, Monopetros notices a certain feature of the bowshot arrow. As the bowstring pushed against it, it gained motion. But once it left the bow, the arrow gained no more motion. From that realization and other examples, Monopetros infers that matter has a propensity to not change its motion (which is the gist of Newton’s first law of motion).

    In Aristotle’s physics bodies have propensities; in particular, propensities to seek their proper place in the system of the world and to seek the state of absolute rest. In Newton’s physics bodies still have propensities, but the basic propensity is to remain in a given state of motion until compelled to change that state of motion by the application of a force. Instead of seeking a state of absolute rest, of non-motion, the body seeks the state of absolute non-acceleration, of non-change-of-motion.

    Newtonian physics: The Universe has no state of absolute rest and is filled with matter that has a propensity to remain in a given state of motion unless forced to change it. Light travels through space at the same speed for all observers.

    Monopetros looks again at his diagram and notices that, although he has been told that both arrows have the same length, the projection of the bowshot arrow onto the spaceline occupied by the stationary arrow is shorter than the stationary arrow. As the bowshot arrow flies past the stationary arrow it appears shrunk (Monopetros has just discovered the Lorentz-Fitzgerald contraction). After a little thought he sees that he has a ready explanation for how that shrinkage comes about.

    The clock on the arrow’s tail lies in the future relative to the clock on the arrow’s head, so because the arrow is moving that rear clock has partly overtaken the fore clock. That effect is prorated along the length of the arrow, so the arrow is uniformly shrunk. And as strange as it seems, Monopetros can understand the effect in terms of perspective.

    He imagines two hoplites standing on a north-south line facing each other. Then he imagines each of them turning slightly to the east. Each soldier would then see the other soldier’s shield foreshortened. The soldiers understand that the widths of their shields haven’t actually shrunk; they’re merely seeing each other’s shield from an oblique perspective. In the same way Monopetros sees the arrow tilted in time from a different perspective.

    So how long does the arrow appear to Monopetros? If he looks at his diagram, he sees that he can use the Pythagorean theorem: the length of the arrow measured by the ants, the proper length x, corresponds to the hypotenuse of a right triangle and the length X of the arrow as seen by Monopetros and the spiders corresponds to one of the triangle’s sides. The length of the triangle’s other side equals xsinè, the product of the arrow’s proper length and the sine of the angle between the spacelines occupied (or the timelines traced) by the observers. Thus Monopetros can calculate

(Eq’n 6)

or

(Eq’n 7)

Monopetros understands sines, of course, so now he wants to calculate the sine in the above equations. He doesn’t yet have the means to make that calculation, so he sets it aside for now and goes to the next phenomenon that for which he needs an explanation.

    As the head of the bowshot arrow flies past the tail of the stationary arrow a minuscule twinkle of sunlight glints off the arrowhead and goes straight into one of Monopetros’ eyes. As seen by the ants, that tiny pulse of light flies due south while Monopetros moves into its path and intercepts it. In Monopetros’ view the twinkle passes the clock on the tail of the stationary arrow and comes to his eye on a path that slants a little east of due south. Thus, for Monopetros the twinkle travels a little farther than it does for the ants. That fact doesn’t bother our philosopher; after all, if someone riding by Monopetros on a chariot were to toss him a drachma, he would understand that the drachma would travel different distances for him and for the charioteer. But the twinkle travels at the same speed for him and for the ants and that fact has an unsettling consequence.

    A pulse of light traveling different distances at the same speed must necessarily cross those distances in different elapses of time. But if those different distances measure the same path, the same traverse between two given points, then the pulse’s motion must have been measured on clocks that count time at different rates (Monopetros has just discovered time dilation, though he doesn’t know it just yet). To Monopetros, then, the ants’ clocks appear to count time more slowly than his clocks (or the spiders’ clocks) do. But shouldn’t time elapse at the same rate for everyone, as Newton asserted in the Principia?

    Even more problematic for Monopetros is the consequence of applying the principle of symmetry (which we call the principle of Relativity) that he has assumed as an axiom of Reality. Thus, given that he sees the ants’ clocks counting time more slowly than his clocks do, he must infer that the ants see his clocks counting time more slowly than their clocks do. He appears to have an unresolvable paradox, but he thinks he sees a way to avoid it.

    Having deduced the proposition that moving objects shrink in the direction of motion, Monopetros thinks that, perhaps, moving objects also change their dimensions in directions perpendicular to the direction of relative motion. He even has a simple model for such distortions. He imagines holding a ball of clay between his hands and pressing his hands together. The ball shrinks in one dimension and expands in the other two, becoming an oblate spheroid. Perhaps motion imposes a similar double distortion upon moving objects?

    But, no, the principle of symmetry strongly supports an argument against such a thing. If motion caused lateral distances to expand, then the fletching on the bowshot arrow would be wider for the spiders than it is for the ants. But by symmetry a ring set up by the spiders would be wider for the ants than it is for the spiders. If the bowshot arrow were to pass through the ring and if the spiders and the ants agreed that the fletching and the ring had the same widths by their respective measurements, then the ants would see their arrow pass through the ring without touching it, but the spiders would see the fletching brush the ring as the arrow passed through. Monopetros has no way to resolve that contradiction, so he must conclude that relative motion does not affect lateral distances (a proposition expressed algebraically by y=Y and z=Z).

S    o now Monopetros wants to calculate the distance that the twinkle traveled from the arrowhead to his eye and correlate it with the time elapsed in that traverse. For the ants the distance is simply y=st, with y being measured purely in the north-south direction. To calculate the distance in his view Monopetros must use the Pythagorean theorem,

(Eq’n 8)

In that equation vT represents the distance that the arrow and, thus, the twinkle moves in the east-west direction in the time T that the twinkle takes in going from the arrowhead to Monopetros’ eye. A little algebraic rearranging of that equation gives our philosopher

(Eq’n 9)

the formula for time dilation.

    Substituting from Equation 4 in Equation 9 yields

(Eq’n 10)

Comparing the second term on the right side of that equation with Equation 5 leads Monopetros to make the tentative equation c=s, asserting that the time-to-space conversion factor is identical to the speed of light. In the verbal equivalent of Equation 10 he has a general formula for transforming measurements made on the moving arrow into the equivalent measurements made by his own timekeeper.

    Equation 5 still has the extra factor of

(Eq’n 11)

but it goes into the correct form if Monopetros writes

(Eq’n 12)

Substituting from Equation 2 transforms that equation into

(Eq’n 13)

Given the symmetry between space and time, Monopetros accepts this result as making good sense. Thus Monopetros obtains four equations that transform measurements of events made by the ants on the moving arrow into the equivalent measurements of the same events that would be made by the spiders on the stationary arrow. And, of course, he figures out the reverse transformation:

(Eq’ns 14)

    Two extremely small, extremely brief events mark point-instants that can serve as the endpoints of a straight line that can, in concept, be drawn through space and time. Monopetros then recognizes Equations 14 as a description of the sides of a four-dimensional equivalent of a right triangle, for which the aforementioned straight line serves as an hypotenuse. After multiplying the fourth equation by the speed of light and then squaring all four of the equations, Monopetros discovers that he can combine them in a simple sum. He struggles with his perplexity for a time and then discovers that he must do this:

(Eq’n 15)

That equation gives him the four-dimensional equivalent of the Pythagorean theorem.

    Finally Monopetros compares Equation 12 to Equation 7 and discovers that he has calculated a cosine with a value greater than one, a mathematical impossibility. But if he instead combines Equation 13 with Equation 10, he obtains a different value for the length of the moving arrow, one that gives him a cosine less than one. In that way he has calculated the proper form of the Lorentz-Fitzgerald contraction.

    Thus Monopetros has devised the theory of Special Relativity and all he had to do in order to obtain it was to resolve one of Zeno’s paradoxes.

    Einsteinian physics: The Universe has a state of absolute motion (the speed of light) referred to the boundary enclosing the Universe. Space consists of an infinite set of inertial frames of reference, all moving relative to each other and none having an absolute center. Matter has the propensity to remain at rest in some inertial frame until compelled by an applied force to occupy some other inertial frame. Contrast that with the Aristotelian view of a Universe that has a state of absolute rest referred to the center of the Universe.

    Astoundingly enough, the Ancient Greeks had the possibility of discovering the theory of Special Relativity. Of course, they didn’t actually make that discovery and there’s only a minuscule chance that they would have done so in any alternate history that we might conceive. But suppose that they had made the discovery and suppose further that they had also discovered the quantum theory. What vision of Reality would they have abstracted from those two theories?

    In the Allegory of the Cave Plato illustrated his Doctrine of the Forms by describing people who are chained to a wall in a cave and compelled to watch shadows projected onto an opposite wall by the light from a fire passing over objects behind them. In Plato’s view we and everything we observe are merely shadows projected onto space by an aetherial light washing over the Eternal Forms.

    In the quantum theory, in contrast to Plato’s cave, the patterns of light and shadow are the ultimate Reality and the matter riding those patterns is the illusion. Whereas in Plato’s cave matter shapes light into shadows, in the quantum world waves shape matter by way of interference patterns (the wave-theory analogues of shadows). If we combine Relativity with the quantum theory, we see that all is waves moving at the speed of light in all directions and that we only experience the interference patterns.

    Now look at the world through the lenses of Relativity and the quantum theory.

    Relativity tells us that nothing real can go faster than light. Quantum theory tells us that nothing real can go slower than light. What we perceive as Reality is, as Einstein said, merely an illusion, albeit a rather persistent one. As part of Reality we are also illusions, fleeting imprints impressed upon aetherial waves of possibility. And as such we watch as the endless Now ravels up the world-lines of potentiality and weaves them into the vast pattern of historical actuality.

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