The Shape of Space
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From the conservation laws we will now devise a basic description of the dynamic geometry of space, which will lead us more or less directly to Albert Einstein's twin postulates of Relativity. This derivation originated in the question Why Relativity? Contrary to common expectation, that question actually has a rational, straightforward answer.
For a century of years now people have accepted Relativity as a given, one whose reason for being true to Reality lies far beyond the realm of human comprehension. But it turns out that, for all of its infamously bizarre, counterintuitive aspects, Relativity takes a fundamental part in Reality for reasons that we regard as simple and easy to understand.
We have before us not one, but two questions: Why is Reality so structured that the laws of physics have the same mathematical expression for me as they have for you, regardless of any relative motion that may come between us? And why do space and time have such a relation with each other that a ray of light that passes you at 299,792.458 kilometers per second passes me at precisely the same speed, regardless of how fast I may move toward or away from you? We must answer both of those questions if we want to understand the why of Relativity. What we want, the answers that satisfy the requirements, consists of deducing Einstein's postulates of Relativity from more primitive propositions.
Ideally that deduction would proceed from a small set of self-evident axioms. But Relativity lies too far up the axiomatic-deductive chain of the Map of Physics to touch any self-evident axioms directly. We must thus deduce Einstein's postulates from propositions that we can use as if they were true axioms. Four theorems will serve us in that regard: the conservation laws pertaining to linear and angular momenta and to energy and the finite-value theorem. We have already deduced the conservation laws, so now I will show you how to deduce the finite-value theorem, a rather nice application of Georg Cantor's transfinite arithmetic.
The Finite-Value Theorem
In a conservation law we have a statement about a closed system, a collection of bodies kept isolated from all influences coming from bodies not in the collection. In stating a conservation law pertaining to some property possessed by the bodies in the closed system, we assert that the total amount of the property that the bodies possess after they have suffered some interaction among themselves equals the total amount of the property that they possessed prior to the interaction. If I let Q represent the amount of the conserved property that the system possesses and let the subscripts i (for initial state) and f (for final state) indicate the values of Q before and after the interaction, respectively, then I must express the conservation law as
Qf - Qi = 0.
Physicists commonly (and correctly) understand that absolute zero to mean that no phenomenon can create or destroy the conserved quantity, but only transform it or transfer it from one body to another. Less obviously, that absolute zero also necessitates that any proper description of the conserved quantity conform to the finite-value theorem, which theorem asserts that no conserved property can exist in infinite quantity.
Georg Cantor identified many different infinities and showed how mathematicians could use set theory to carry out transfinite arithmetic, not for actual calculations but to prove and verify theorems pertaining to those infinities. Of those infinities we want to consider only the one he called Aleph-Null, the infinity that tells us the cardinality of the natural numbers, the magnitude of the set comprising all, absolutely all, of the positive integers. By applying Cantor's techniques with Aleph-Null we will prove and verify as true to mathematics the finite-value theorem.
Imagine that we have a closed system in which one body, labeled A, initially possesses an infinite amount of some conserved property; which means, QAi =א. In concept the conservation law allows us to divide the conserved property into standard units and identify each of them by marking it with an index number. Because nothing can create or destroy the conserved property, we can think of that property as if it were an indestructible body that we can cut into parts upon which we can paint serial numbers. In this case the serial numbers comprise all of the positive integers. Other bodies in the system carry their own amounts of the conserved property, also subdivided and index-numbered.
For the sake of the reduction ad absurdum I want to consider two possible final states that the system can enter consequent to some interaction among the bodies comprising the system. Both final states will involve variations on the Hilbert's Hotel paradox. If I do everything correctly, then Cantor subtraction of the initial state from the final state will, in both cases, yield the empty set, the set-theoretic equivalent of zero, as required by the conservation law.
To create Final State #1 something splits Body-A in two. One part takes those elements of the set QA that bear the odd index numbers and the other part takes those elements that bear the even index numbers. Because I have assigned the index numbers entirely arbitrarily, those numbers have no physical meaning and I can change them as I see fit. In this case I replace each of the even index numbers by its half. Final State #1 of the conserved property possessed by Body-A thus comprises a subset of elements labeled with all of the odd positive integers and a subset of elements labeled with all of the natural numbers. Other bodies in the system remain unaffected and we can ignore them.
I carry out Cantor subtraction of the initial state of Body-A from the final state of Body-A by comparing the respective sets representing the conserved property Q and discarding the elements bearing the same index numbers. In this case that process leaves the set comprising the elements that bear all of the positive integers, not the empty set required by the conservation law.
To create Final State #2 something takes some amount of the conserved property from Body-A and transfers it to some other body that initially carries none of the conserved property. Out of the elements of the set QA I specify that the other body receives only those elements bearing the index numbers 1 through N. That transfer thus leaves the Body-A with the elements of QA that bear the index numbers N+1 through Aleph-Null. Again, because the index numbers have no physical meaning, I can alter them as I see fit and I see fit in this case to replace each index number M of an element remaining in the Body-A with the number M-N. Final State #2 of the system thus comprises a subset of elements bearing the index numbers 1 through N and a subset of elements bearing the index numbers 1 through Aleph-Null. Cantor subtraction of the initial state from that final state leaves as remainder the set of elements indexed 1 through N. Again, the process does not give us the empty set and, thus, represents a violation of the conservation law.
Therefore, anyone can violate a conservation law if they have available to them a body carrying an infinite amount of the conserved property. Indeed, they would have considerable difficulty avoiding a violation of the conservation law if that body participates in any interaction with other bodies. But we have already accepted the premise that Reality is so structured, at the most fundamental level, that it always upholds the conservation laws, so in order to dissolve the contradictions that I obtained above I must dismiss as false to Reality at least one of the premises supporting my reasoning. In this instance I have only one premise to discard - the assumption that any body can carry an infinite amount of a conserved property. In discarding that premise I proclaim the finite-value theorem: no body or collection of bodies can ever hold an infinite amount of a conserved property.
The Shape of Space
We know that Reality conserves angular momentum. Oversimplifying considerably, I can describe the mathematical representation of an angular momentum as the product of a mass, a velocity, and a distance. In accordance with the finite-value theorem, we know that we can never have an instance in which that product yields an infinity, so we know that none of its factors can be infinite. Thus, the Universe cannot contain an infinite mass; no body can move with an infinite speed; and the fullest extent of space must be finite.
Consider the chief implication of that last statement. If you were to begin at some point that you could identify as the center of space and draw a straight line in any direction, after extending that line a finite distance you would come to a place where space simply ends. At that place you have reached a boundary that has space on one side and absolute nothingness on the other. Absolute nothingness necessitates nonexistence and applies to any conceivable object or property of Reality, in particular, in this instance, the geometric properties of space. Area does not exist in absolute nothingness, so the side of the boundary facing absolute nothingness can have no area. In space we have an either/or entity (which means, Reality admits of no degrees of spatialness), so the end of space comes with perfect abruptness (which means, the boundary has zero thickness), so the boundary must have the same area on both sides. We thus infer that the outer boundary of space has the nature of a single, simple mathematical point.
That fact means that all of the straight lines passing through the center of space converge upon the boundary of space and upon each other. Each such line meets its own opposite extension through the boundary. Thus, every such line, fully extended, constitutes a diameter of space and has the closed character of a circle. That circular character notwithstanding, for the purpose of obeying Newton's first law of motion, each of those lines has the character of a straight line.
Now we confront a question about the fundamental geometry of space and assemble a team of imaginary technicians to help us answer it. We ask our Surveyor to draw a straight line through the center of space and designate it the baseline. We then ask the Surveyor to draw a straight line parallel to the baseline and designate it the auxiliary line. Can the Logician tell us whether those lines relate to each other as do lines of latitude or as do lines of longitude? To make clear that we have only those two options, we ask Does the auxiliary line meet the boundary of space or does it not?
Let's first take the latitudinal model as true to Reality; which means, let's assume that the auxiliary line and the baseline relate to each other as do a line of latitude and the Equator on Earth. In this model the auxiliary line does not meet the boundary of space. If someone could make the auxiliary line visible with suitable markers, an observer near the center of space would see the line extending along the baseline, then suddenly veering away from it, crossing the sky to the opposite side of space, curving sharply to come alongside the baseline again, and then extending along the baseline to meet itself again near the center of space. In spite of the apparent veering and curving, the auxiliary line is still a straight line, so any body moving along it will continue to move along it until compelled to do otherwise by an interaction with another body. In that statement, the Logician tells us, lie grounds for a falsification.
We ask the Surveyor to extend a straight line from the center of space to the boundary in such a way that it makes a right angle with the baseline and crosses the auxiliary line. We call that the right line and let it define north while the baseline defines east-west. We ask the Engineer to construct a bar on the right line in such a way that the ends of the bar lie on the two points where the auxiliary line crosses the right line. Included in the bar's design, suitable mechanical devices on the bar's ends enable us to carry out the following imaginary experiment.
At the end of the bar nearer the center of space the mechanism ejects a small body eastward along the auxiliary line. That body carries linear momentum in the amount +P, so the bar, in accordance with Newton's third law of motion, recoils slowly westward, carrying linear momentum in the amount -P. As the small body approaches the boundary of space it veers north, apparently moving away from the baseline, and crosses the sky until it strikes the end of the bar farther from the center of space. With perfect elasticity, it rebounds from the bar with the same speed at which it approached the bar. That collision, as viewed from the center of space, also gives the bar a westward motion, so it gains from the body an additional -2P units of linear momentum. Retracing its path, the body returns to its starting point, where the launch mechanism catches it and absorbs it, giving the bar yet another -P units of linear momentum.
At the end of the experiment the bar and the small body together carry -4P units of linear momentum, but they carried zero units of linear momentum when the experiment began. This imaginary experiment displays a blatant violation of an absolute conservation law. Further, we could have conceived a similar experiment involving two small bodies moving in opposite directions along two auxiliary lines to contrive a violation of the law of conservation of angular momentum. In either case we must take at least one of the premises upon which we have based our experiments and remove it from the chain of logic that we are forging.
Of the many premises that underlie this experiment, we brought in two that are unique to it: 1) that parallel straight lines in space relate to each other as do lines of latitude and 2) that we may move bodies arbitrarily close to the boundary of space. We must declare at least one of these premises false to Reality, but this experiment offers no clue as to which one we must choose for dismissal from the Map of Physics. To gain the perspective we need to make the choice, someone must perform an experiment in the alternate geometry.
Take the longitudinal model of space as true to Reality; which means, let's assume that the baseline and the auxiliary line relate to each other as do two meridian lines on Earth's surface. In this model the auxiliary line does meet the boundary of space. One question that comes readily to mind in this case asks whether two straight lines initially drawn parallel to each other and extended can remain straight and parallel if they eventually meet each other at some point, as the auxiliary line and the baseline do at the boundary of space.
Certainly the two lines do not remain parallel to each other: the definitive sine qua non of parallelism tells us that parallel straight lines initially drawn some distance apart can never pass through a common point. We assume in the longitudinal model of space that the two lines remain straight, in the sense that a free body moves along one or the other of them until compelled to do otherwise by an interaction with some other body.
We oblige our Surveyor to define a baseline through the center of space and then draw two auxiliary lines on opposite sides of it, parallel to it, and equal distances from it. After the Surveyor extends those auxiliary lines to the boundary of space, the Engineer constructs a stiff metal ring around the boundary, making it small enough that the auxiliary lines meet its outer surface at a right angle. At the center of space the Engineer creates a barbell that has mass ejectors built into the weights at its ends. For this experiment the Engineer ensures that the center of the barbell lies on the center of space and that the weights lie on the auxiliary lines.
At some given instant the mass ejectors propel two bodies, each identical to the other, in opposite directions along their respective auxiliary lines. In accordance with the rotary analogue of Newton's third law of motion, the barbell rotates about its center in the sense, say counterclockwise, opposite the sense that we would assign to the two bodies taken as a system. Some time later, at the boundary of space, the bodies strike opposite sides of the ring and rebound with perfect elasticity, returning back the ways whence they came. Exquisite timing brings the bodies back to the barbell at the very instant that the mass ejectors come into position to receive and recapture them. Absorbing the bodies' linear momenta, the barbell spins even faster, doubling the angular momentum that it acquired when it initially ejected the bodies. Thus, we have made our imaginary Universe gain net angular momentum in blatant violation of the conservation law.
Again we have based an experiment upon a number of premises, at least one of which we must dismiss as false to Reality. In particular we have used two premises especially vulnerable to falsification: 1) the assumption that the geometry of space reflects the longitudinal model and 2) the assumption that we may move bodies arbitrarily close to the boundary of space.
Having conducted two imaginary experiments that each yield a violation of a conservation law, we ask the Logician to examine the results. We based both experiments upon premises describing a geometry of space and a common premise that we can move bodies as close to the boundary of space as we wish. We have chosen to make our geometric premises exhaustive and mutually exclusive so that one of them must be true to Reality. Thus the Logician must declare the common premise false to Reality. Equivalently, they declare the premises converse true to Reality: to wit, nothing can move a body arbitrarily close to the boundary of space.
But the Logician feels compelled to go a step further, simply because we cannot specify what we mean by "arbitrarily close to the boundary of space". If at some time a body lies closer to the boundary of space than it did at some previous time, then over a suitable elapse of time someone could move that body to a place that satisfies the statement "arbitrarily close to the boundary of space". That fact necessitates that Reality be so structured that at no time can any body lie closer to the boundary of space than it did at some previous time. (Note to the Archer: we believe that we may have found your lost arrow of time.)
The Dynamic of Space
Bodies move. Space, in its most fundamental nature, exists to enable bodies to move by giving them places to which they may move. Any given body can move closer to any fixed point, so, if we must have the proposition deduced above true to Reality, then the boundary of space cannot be a fixed point. We must assert that the boundary moves: specifically; it must move away from all bodies; it must move in all directions away from the center of space; and it must move at a speed that no body can achieve.
Infinite speed offers us an obvious choice to assert as the speed of the boundary. We have already proven and verified the proposition that no body can move at infinite speed, so this choice would well satisfy the can't-get-closer criterion. However, if the boundary of space were to move away from the center of space at infinite speed, then space would have infinite extent and would thus possess the potential for a violation of the conservation law pertaining to angular momentum. We have already eliminated that possibility, of infinite space, from the chain of logic emanating from the finite-value theorem. We must, therefore, represent the speed at which the boundary of space moves with a finite number.
If we want to ensure that no body exceeds that speed, how should we construct space? In accordance with the Neo-Aristotelian assumption that I have tacitly made thus far (that space has a definite, unique center), the Logician answers that question by asserting that space also has a center of velocity, an inertial frame that defines the state of absolute rest: in that frame and only in that frame the boundary of space moves at the same speed in all directions. Clearly we find the center of space at rest in that frame.
To test this new feature of space the Engineer creates a rocketship at rest at the center of space and launches it. Some time later the Pilot reports finding that, based on measurements made via Doppler didar (DIvine Detection And Ranging), the relative speed between the rocketship and the boundary has diminished and that the rocketship's acceleration has also diminished, though the Pilot has not changed the control settings. If that pattern persists as the rocketship goes ever faster, then the rocketship would need an infinite amount of time to match velocities with the boundary; which means, the rocketship will never come closer to the boundary than it was at any previous time. This gives the Logician and the Engineer what they want in order to test the Logician's assertion about the nature of space.
Sitting in the rocketship's cockpit with nothing to do, the Pilot idly tosses a tennis ball from one hand to the other and back again. It seems strange, but the Pilot notices that the small, hollow rubber ball seems to have the heft of a bowling ball. That fact raises a question in the Pilot's mind: in the moving rocketship has the ball become more massive than it was when the rocketship was at rest or have the forces that the Pilot exerts upon it diminished?
To answer that question the Logician directs our attention to an apparatus that the Engineer built into the rocketship. In its essentials the apparatus comprises two identical, counter-rotating wheels whose axes lie parallel to the rocketship's thrust axis, a long, stiff spring oriented parallel to the rocketship's thrust axis, and a mechanism that can either despin the wheels and use the energy thus gained to stretch the spring or draw energy from the spring to spin up the wheels. No friction interferes with the operation of the apparatus and the mechanism transfers energy with 100% efficiency.
Prior to the rocketship's launch the Engineer sets the wheels to spinning. Because they spin independently of each other, the Engineer must take great care to ensure that each wheel carries as much energy as does the other. At the end of the rocketship's acceleration, when the rocketship moves at its maximum speed relative to the center of space, the apparatus despins the wheels and stores their energy in stretching the spring. With the apparatus locked into that state (wheels not spinning and spring stretched), the Pilot decelerates the rocketship back into the presumed absolute rest frame. Finally the apparatus uses the energy stored in the spring to spin the wheels back up again. Now the Logician has two possible outcomes of this experiment to consider.
If the rocketship and its contents have become more massive, as measured by the Pilot, (ultimately in the proportion R) as acceleration takes the ship farther out of the frame of absolute rest, then the wheels must spin more slowly than they did at launch. Neither wheel's angular momentum changes as the rocketship accelerates, so the product of the wheel's moment of inertia and angular velocity (L=Iω) must also remain unchanged, in obedience to the conservation law. Directly proportional to the wheel's mass, the moment of inertia must increase in the proportion I'=RI, so the angular velocity of the wheel must decrease in the same proportion; which means that ω'=ω/R. If we calculate the kinetic energy manifested in the spinning of both of the wheels (T=Iω2), we find that it diminishes to T'=T/R. We transfer that energy to the spring and have T'=½ kx2, in which equation k represents the stiffness of the spring and x represents the distance we have stretched the spring to store the energy T'. When the rocketship decelerates the spring's stiffness and length do not change, so we have an amount of energy T' to use in respinning the wheels. In this case an amount of energy equal to T(1-1/R) has simply ceased to exist, in violation of the conservation law.
If forces exerted aboard the rocketship diminish in the proportion R when the rocketship accelerates to some speed relative to the absolute rest frame, the wheels will spin unaffected by the acceleration, their combined energy remaining equal to their original energy (T=Iω2). When the energy transfer mechanism despins the wheels and stretches the spring, it puts that energy into stretching the softened spring (k'=k/R) by the distance x': T=½ k'x'2. Deceleration of the rocketship rehardens the spring, but does not change its length, so the energy transfer mechanism has an amount of energy equal to T'=½ kx'2=½ Rk'x'2=RT to put into respinning the wheels. In this case energy in the amount equal to T(R-1) has come into existence ex nihilo, again in violation of the conservation law.
Examination of those results convinces the Logician that masses and forces in the rocketship cannot change when the rocketship accelerates. But without an increase of mass or a weakening of forces, the rocketship has no phenomenon that restricts its acceleration. Nothing prevents the Pilot from piling on the thrust, matching the rocketship's speed to that of the boundary of space, and then exceeding that speed.
In order to preserve the conservation laws, the Logician has inferred that the boundary of space must move, but that it cannot move at infinite speed. It must move at a finite speed that no body can achieve, but the nature of space puts no physical constraints on the acceleration of bodies. Here I want to restate the basic idea of the reductio ad absurdum by paraphrasing Sherlock Holmes's exquisite statement of it: when you have eliminated the impossible, whatever you have left, however improbable and incredible it may seem, must be true to Reality. So what does the Logician have left? Only a finite speed made unreachable by geometric constraints, by acceleration so reshaping space that it puts the boundary forever out of reach. Just such a reshaping lurks implicit in the statement that space and time have such a relation that the boundary of space moves away from all bodies with a finite speed that has the arithmetic character of an infinity. Just as subtracting an finite number, however big, from Aleph-Null leaves Aleph-Null unchanged, so applying any delta-vee, however great, to the rocketship leaves the Pilot measuring the same speed for the boundary of space, at all times and in all directions.
For convenience the Logician wants to restate that result as a simple rule that the Pilot, the Engineer, the Surveyor, and any other interested party might use. Wanting to provide a readily available touchstone to potential users who cannot measure the speed of the boundary of space directly, the Logician assumes the existence of some phenomenon that flies through space at the same speed at which the boundary moves (and here I will drop my feigned ignorance to say that such a phenomenon exists and that we commonly call it light), in essence assuming that the nature of light has a relationship with space and time that gives the speed of light the character of an infinity. With that information in mind, the Logician states as follows:
Any ray of light passes all observers at precisely the same speed (299,792.458 kilometers per second or 186,234.709 miles per second), regardless of how those observers move relative to each other.
This rule constitutes a large part of the logical foundation upon which the Logician will build the transformation equations that will enable observers to translate their spatial and temporal measurements of events into the equivalent measurements made by other observers. But the Logician does not yet know that such equations exist, that the joint nature of space and time necessitates their use. That knowledge will come quickly though.
In the previous deductions the Logician has assumed that space has a Neo-Aristotelian nature; which means, that space has a unique center (the point equidistant from the boundary of space in all directions) and a unique center of velocity (a state of absolute rest from which an observer would measure the boundary moving away in all directions at the same speed). Einstein's second postulate (now theorem) of Relativity eliminates the second part of that description from any description of space that we mean to have true to Reality: every observer appears to occupy the center of velocity, regardless of how they move relative to any other observer, who also appears to occupy the frame at the center of velocity. No acceleration can change that fact, so the Logician wonders whether every observer also occupies the center of space, regardless of how far they lie from any other observer, who also appears to occupy the center of space.
But now the Pilot and the Engineer appeal to the Logician to tell them how they can determine which of them truly moves and which lies truly at rest. Apparently they have not properly understood the rule given above and still believe in absolute velocities referred to an absolute state of rest. But space could still have absolute velocities even if measurements of the recession velocity of the boundary does not reveal them. Can the Pilot and the Engineer perform some experiment that will reveal absolute motion and answer their appeal to the Logician?
Hoping to find at least some small remnant of Neo-Aristotelianism in the nature of space, the Logician reasons thus;
1) Observers can measure only positions and times occupied and marked by bodies participating in events,
2) From any set of measurements taken from a given body observers may derive a mathematical description of the body's motions and changes of motion,
3) From the changes of motion of a suitably large set of bodies observers may deduce general rules, laws of physics, that, at the most fundamental level, the observers must express mathematically,
4) Because the laws of physics describe changes of motion, manifestations of applied forces, the observers must express them in terms of energy or quantities (such as momentum) related to energy,
5) If the laws of physics manifested in one inertial frame differ from the laws of physics manifested in another inertial frame that moves relative to the first, then someone may do the following:
a) that someone may establish a box containing a collection of bodies that move and exert forces upon each other, establishing it as the State-A with the amounts of the conserved quantities carried by the bodies comprising the set KA. Whoever establishes the box must do so in conformity with the proviso that the bodies inside the box have such structures and locations in the box that they cannot gain energy from acceleration of the box in a certain direction.
b) someone accelerates the box from Inertial Frame A to Inertial Frame B. In accordance with the forces and torques applied by the acceleration and the work done in accelerating the bodies, the set KA becomes the set KB. We assume the transformation operation TAB, which makes TABKA = KB by matrix multiplication, changing elements of the set KA into elements of the set KB.
c) someone then rearranges the bodies in such a way that the conserved quantities carried by the bodies change by the set of values ΔB, changing KB into KB' = KB-ΔB.
d) someone accelerates the box from Inertial Frame B back to Inertial Frame A and the inverse transformation TBA (TBATAB=I, the identity operator) makes TBAKB'=KA'.
e) someone rearranges the bodies in the box in a way that restores the bodies' original configuration, thereby changing the conserved quantities held by those bodies by the set of values ΔA, changing KA' into KA'' = KA'+ΔA.
Now in order to ensure that in going around that cycle the procedure upholds the conservation laws, the Logician must determine the conditions that make KA'' = KA. In preparation for that determination the Logician expresses KA'' in its most primitive terms; which means,
If we require that KA'' = KA, then we must also require that TBAΔB=ΔA.
In the Hamiltonian version of dynamics a specification of the distribution of energy over the bodies comprising a closed system provides sufficient information to enable anyone to determine the motions of the bodies. If the Logician takes the Hamiltonian view, then each of the sets K and Δ comprises elements that represent the amount of each kind of energy (e.g. linear-kinetic, rotary-kinetic, spring-potential, etc.) carried by each and every body in the system and the amount by which those energies change. We can draw each such set as a linear array of its elements, in the manner in which mathematicians commonly represent vectors, so the transformation operators TAB and TBA become matrices, each having as many rows and columns as the sets K and Δ have elements. Someone would then carry out the transformation TBAΔB=ΔA, for example, by applying the rules of matrix multiplication. Now the Logician wants to work out the explicit form of the transformation operator TBA.
First, the Logician deduces that each and every element of TBA that lies off the main upper-left to lower-right diagonal necessarily equals zero. Each such non-diagonal element of the matrix represents the transformation of the energy in one mode in one body into energy in another mode in the same body or in a different body consequent to the acceleration of the box from Inertial Frame B to Inertial Frame A. To give the appropriately abstract example, we have the transformation TBAijΔBj=ΔAi, which tells us of the possibility that every single mode in the set ΔB contributes to any one given mode in the set ΔA. We cannot dismiss that possibility outright, but we must insist that it conform to the conservation of energy. Let's assume that the Engineer has distributed energy over the set ΔB in such a way that the sum of all the elements of the set equals the sum of all the elements of the set ΔA produced by the transformation above (ΣjΔBj =ΣiΔAi). That distribution obeys conservation of energy. But now the Logician rearranges the energies in the set ΔB and demands that the new arrangement also obey the conservation law under the transformation. Mathematics gives us only one way to satisfy that demand: every off-diagonal element of the transformation matrix must equal every other off-diagonal element and every diagonal element must equal every other diagonal element.
Second, the Logician infers that the elements of the transformation matrix comprise only numbers, simple multipliers rather than more complex operators, such as the process of differentiation. We might derive those numbers from algebraic formulae, perhaps some functions of the velocities of the Inertial Frames A and B relative to the presumed frame of absolute rest, but we must nonetheless get numbers into the transformation matrix before we can apply it. If that inference were not true to Reality, then we might use operators in the transformation matrix. Because the representation of the energy in each mode comes from an algebraic formula and because those formulae differ for different modes, the requirement that the elements of the transformation matrix change all elements in the same proportion necessitates that the elements of the transformation matrix comprise simple multipliers or operators equivalent to simple multipliers.
Having thus established the fundamental mathematical form of the transformation matrix, the Logician contemplates one off-diagonal element TBAij = N and its conjugate TBAji = N. Those elements represent in the Logician's mind the rotary-kinetic energy gained by one spinning wheel due to the rotary-kinetic energy held by another wheel in the box. If the energies held by the wheels do not come from equal and oppositely directed angular momenta, then the transformation increases the wheels' rotary-kinetic energies in a way that causes the overall angular momentum contained in the box to change gratuitously, in violation of the conservation law. We must thus have N = 0 in order to uphold the conservation law.
That fact, in turn, means that all of the off-diagonal elements of the transformation matrix must equal zero. We know that we must have that inference true to Reality because we could put a pair of spinning wheels into any box we establish in order to carry out this experiment. In anticipation of that possibility, every transformation matrix must have, at least implicitly, those particular elements in it, so all of the off-diagonal elements, explicit or implicit, must equal zero. We should not feel astonished at this deduction, because those off-diagonal elements all represented the purest form of what Einstein called spooky actions at a distance.
But now we know that the diagonal elements of the transformation matrix must each equal one, lest we violate conservation of energy directly. We know that the transformation matrix that we have assumed differs not at all from the identity matrix, the vector analogue of multiplying a number by one. Seen in the light cast by the fundamental fact of Hamiltonian mechanics, that fact necessitates that the invariance thus inferred also apply to the laws of physics in general. As before, the Logician sums up this knowledge in an easy-to-apply rule:
If two observers occupying two different inertial frames of reference perform identical experiments in their respective frames, then the laws of physics that those observers infer from their measurements have the same mathematical expression, regardless of how the observers and their frames move relative to each other.
And that puts the final end to Neo-Aristotelian space. From those two rules that the Logician deduced, we may proceed in a straightforward way to the understanding that if I measure distance and duration between any two events, then you may calculate the distance and duration that come between those same two events in your frame by applying the Lorentz Transformation to my measurements. From the basic conservation laws of classical physics we have deduced the fundamental constraints upon the relationship between space and time and demonstrated that Reality is absolutely relativistic.
And the Logician, the Engineer, the Pilot, the Surveyor, and our other observers can only stare in awe at what they have found. Here we have a truly wondrous spectacle. We see that real magic consists in the faculty of reason confounding the expectations of intuition, of logic revealing the Reality behind Appearance.
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