The Boundary of Space
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Does space actually exist and, if so, does it have a boundary: that is, does it have only finite extent? If we can answer that question, what consequences can we infer from the answer?
As a first step we refine the question. What does it mean to ask whether space actually exists? It means that I want to know whether space exists as a thing with an existence independent of my existence. But now I need to ask whether anything has an existence independent of mine.
We conceive ourselves as existing in a realm that we call Reality, which consists of objects separated from each other and from us in space. We base that conception upon the perceptions, which we receive through our senses, of objects and of their relationships. I see a chair and a clock, I"see" an invisible something that exists between them and keeps them apart, and I call that invisible something "space". Do the percepts that lead to those concepts emanate from something outside me and independent of me? Or do they merely give me an illusion of a Reality separate from me?
I know that I actually exist, because I can invoke Rene Descartes' "cogito; ergo, sum" (I think; therefore, I exist) to prove and to verify the proposition that I do indeed exist; after all, I must exist in order to think (or to do anything else for that matter), so the observation that I am thinking suffices to verify my existence. My thinking consists of the manipulation of percepts from all of my senses and of the concepts that organize those percepts into a worldview. That fact, in itself, cannot tell me whether the percepts emanate from something separate from me or originate within me.
But I also have desires and when percepts satisfy those desires I feel pleasure (and a desire to prolong the percept) and when percepts dissatisfy my desires I feel discomfort, even pain (and a desire to rid myself of the percept). My observation of the existence of percepts that go against my desires (such as the sensations of being sick) tells me of the existence of something that I cannot control. What I cannot control is not me. (OK, that gets us into the question of Free Will, but that's a different subject). The observation that something exists outside my direct control proves and verifies the proposition that something exists outside me. I identify that something with Reality.
Now I ask: When I perceive some feature of Reality, does the feature-as-perceived differ significantly from the feature-in-itself? I say that my perceptions of a thing must mimic the properties and relations of the thing-in-itself insofar as the thing-in-itself has a relationship with the other features of Reality (including me). Here it helps to review a statement that Albert Einstein (1879 - 1955), Boris Podolsky (1896 - 1966), and Nathan Rosen (1909 - 1995) included in their 1935 paper on the quantum paradox that we now call entanglement:
"If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."
Superficially that statement resembles Bishop George Berkeley's doctrine of immaterialism, which asserts that only what is perceived exists. Because we can only know percepts, Berkeley maintained that matter, the foundation of the thing-in-itself, does not exist, that the thing-in-itself does not exist. But we may well ask whence come the percepts that tell us about a thing if the thing-in-itself does not exist as a source of those percepts. Einstein, Podolsky, and Rosen answered that question with their own version of ex nihilo, nihil fit (from nothing, nothing is made): if we can predict with certainty what we will perceive in some given circumstance, then the percepts we receive must emanate from some thing that exists in itself in the realm of Reality.
Of course, we can never know the thing-in-itself. We cannot bring it into our minds as we do the percepts that emanate from it. We can only know a thing through its percepts. That fact brings us back to the question: Does the thing-as-perceived differ significantly from the thing-in-itself? How well do the percepts represent the thing from which they emanate? And then we can ask the meta-question: Does that question actually ask for any meaningful information?
What would constitute meaningful information in this case? What does it mean to ask how well a percept represents the thing from which it emanates? At its simplest a percept gives me information that I can use to guide my interaction with the underlying object. For example, I receive a percept from my pen and one from my hand and those percepts enable me to reach my hand to the pen, to pick up the pen (using tactile percepts to confirm what the visual percepts tell me), and to write on my notepad with the pen. I now have a new percept -- this note on my notepad.
Using percepts to guide my actions and successfully altering a small piece of Reality to match what I intended implies that those percepts accurately represented the underlying things-in-themselves. Thus I answer the above question in the only way possible. If I use the information from certain percepts to contrive an event and if I subsequently perceive the outcome of the event matching my expectation, then I can say that the percepts accurately represent the Reality behind them (though I leave unexamined the question of how I took information from the percepts, used it to contrive an event, and then extracted a "Realistic" expectation). In that statement I have established a criterion similar to that offered by Einstein, Podolsky, and Rosen for proving and verifying the existence of physical Reality.
All of my percepts share one feature: all of the objects in them display a property that we call extent. That property also goes beyond the objects and separates the objects from each other. It does not itself give me a percept; I infer it from the relationships I see within and among the other objects of my perception. It gives me information that I can use successfully to contrive events and anticipate their outcomes, so, in accordance with the criterion above, I infer the existence of a real entity that corresponds to it and call that entity space. (I also, in like manner, infer the existence of an entity that separates events from one another and call that entity time). What more can I say about space?
Space gives us an example of a physical field (not to be confused with a mathematical field, an entity that exists only in the realm of abstract algebra). The field consists of indexed elements and a field equation (or equations) that assign to each element the field's characteristic property in accordance with some function of the elements' indices. In the case of space the characteristic property is extent or distance from an arbitrarily chosen element and the field equation is what we call a metric equation, something that looks like it's related to the equation that expresses the Pythagorean theorem (it is, indeed, and in Euclidean space the metric equation is identical to the Pythagorean equation). But the field equation, though necessary, is not sufficient to give each element a unique value of the characteristic property. To get a truly unique field we must constrain the field equation with boundary conditions (or initial conditions, which are just temporal boundary conditions).
Consider a simple example. Imagine that we have inscribed a set of circles on a sphere, orienting them at random. Some circles will cross others and we can say that the distance between the points where two circles cross each other have multiple values, depending upon which circle we follow to get from one point to the other. If we have drawn enough circles, then we can find a multitude of distances between any two points. But if we impose the boundary condition that the centers of the circles must all lie on the same straight line passing through the center of the sphere, then the circles will all lie in parallel planes, like the lines of latitude on Earth's surface, and between any two points we will find only one distance.
We have boundary conditions when each element of a subset of the field elements is given a unique value of the field property by something that is not part of the field. For example, the form of an electrostatic field in a volume enclosed within an electrically conducting surface is constrained by the fact that adjacent to the boundary (the conducting surface) the field can have no components parallel to the boundary and that the voltage of the field has the same value over the entire boundary. Constrained by the boundary conditions, the field equation then assigns a unique value of the field property to every element of the field.
Now we can turn that reasoning around (though we still need a strong proof and verification of the assumed proposition that no alternatives exist) and note that every point in space has a unique value of the field property, distance from some arbitrarily chosen point. From that fact we infer that the value assigned by the metric equation is constrained by a boundary condition. From that fact, in its turn, we infer necessarily that space has a boundary. Now we must determine what condition that boundary imposes upon the field of space.
We know that the boundary must be a closed manifold extending into one less dimension than space does. It therefore has an inside and an outside. We know that space lies on one side, so we infer that not-space lies on the other side. That inference tells us that one side of the boundary has no area, because area denotes a spatial relationship and thus it cannot exist in not-space. Because the boundary has no thickness, in must have the same area on the other side (zero area in this case, which corresponds to the no area on the far side). So now we know that the boundary of space is a single perfect zero-dimensional mathematical point. That fact gives us a seeming paradox, because we know that the boundary must enclose space completely: we think that it must have a vast area in order to cover the entire sky at whatever distance it lies from us. But the paradox will dissolve when we discover that space has a non-Euclidean nature and only displays a Euclidean "flatness" locally.
We have tacitly assumed that space exists on the inside of the boundary (like air in a balloon) and that not-space exists on the outside. But if space exists outside the boundary, then it can certainly extend endlessly away from the boundary. We then conceive the boundary as the beginning of space rather than its end. Given that the boundary constitutes a single point, we may feel a strong temptation to accept as true to Reality the proposition that space lies outside the boundary, but consider what we have if not-space thus exists inside the boundary.
In that case the boundary, enclosing not-space, exists as a point in infinite space. That point can move in space, so it can give not-space the property of motion. But not-space cannot possess properties of any kind, certainly not dynamic properties like motion. Therefore, we must either assert that no motion exists in space or that not-space does not lie inside the boundary of space. We receive percepts that we interpret as emanating from bodies moving in space, so we must dismiss the first of those possibilities and assert that not-space lies outside the boundary of space.
Now we infer that space has finite extent. Of course, infinite means without boundary, but that linguistic fact does not, in itself, necessitate the finitude of space. But we use infinite to denote the mathematical concept of endlessness and that fact does tell us that bounded space has only finite extent, however big it may be or become. Put simply: infinite space never ends; the boundary is a place where space ends (or begins, though we have eliminated that possibility from consideration); therefore, space with a boundary cannot have infinite extent.
Motion does not exist with respect to not-space (I keep wanting to say"motion does not exist in not-space", but "in" does not go with "not-space", so I must use an alternative way of expressing an association between motion and not-space in order to negate it). Thus, having nowhither to go, space and its contents, the Universe, can have no motion. We know that the motion of the whole equals the sum of the motions of its parts (the parts of a potter's wheel move as the wheel turns, but the wheel itself does not move from place to place), so we infer that the no motion of the Universe necessitates that the motions of all of its parts always add up to a net zero. We do not concern ourselves with how that situation came about and we do not intend to add up the motions of all the bodies in the Universe, but we need only know that the laws that govern Reality and its workings have such a form that they automatically maintain the non-motion of the Universe. From that fact we infer the further fact that if a body's motion changes, then at the same instant in time another body's motion must change by the same amount in the opposite direction: thus we obtain Isaac Newton's first and third laws of motion combined.
Imagine throwing something at the boundary of space and ask what happens when it reaches the boundary. We conceive three, and only three, possibilities:1) the object bounces off the boundary and returns to the thrower, 2) the object penetrates the boundary, touches not-space, and ceases to exist, and 3) the object traverses the boundary and emerges on the opposite side of the Universe. Which of these possibilities holds true to Reality?
We dismiss the first possibility because it clearly violates the laws of motion. Because the boundary of space is not a moveable body, it cannot provide the equal and opposite reaction to the thrown body's rebound. Also, because only moveable bodies can exert forces, the boundary of space cannot act as an intermediary and transfer the reaction to another body. Therefore, no body can ever strike the boundary of space and rebound from it.
We dismiss the second possibility for essentially the same reason. If the body penetrates the boundary of space, it comes into contact with what lies on the opposite side of the boundary. Because nothing can exist "in" not-space, the body and all of its properties (particularly motion) would cease to exist the instant the body penetrated the boundary of space. But that would leave the net motion of the Universe unbalanced and thus violate the laws of motion. Therefore, no body can penetrate the boundary of space.
That analysis would seem to leave the third possibility true to Reality by default (which is the purpose of the reductio ad absurdum). But now we must attend to a subtlety in the laws of motion.
The fact that all of the motions within the Universe must always add up to a net zero necessitates that the weighted positions (a body's position multiplied by the body's mass) of all the bodies in the Universe add up to a constant, a number that never changes. It does so because we calculate the body's motion by taking the derivative with respect to time of the body's weighted position. That fact and the fact that differentiation commutes with integration gives us a calculation that resembles the calculation that physicists and engineers make when they locate the center of mass of a composite body, so now we can say that the Universe has a center of mass and we can say further that it can never move. That latter statement encodes what we already know; that the Universe has no location with respect to not-space, so any location that we can identify uniquely with the Universe as a whole (such as its center of mass) cannot change, ever.
If a body were to traverse the boundary of space, it would disappear from one part of space and immediately reappear on the opposite side of the Universe. Such a jump would alter the center-of-mass calculation in a way that tells us that the jump made the location of the Universe's center of mass move, albeit by a minuscule distance. Nonetheless, our previous inferences allow no movement, of any magnitude, of the Universe's center of mass, so now we infer that the Universe has such a structure that no body can ever traverse the boundary of space.
Now we know that we have an impossible situation if a body reaches the boundary of space. The body cannot rebound from the boundary; it cannot pass through the boundary; nor can it traverse the boundary. We must infer, then, that the Universe has such a structure that no body can ever reach the boundary of space.
We can conceive only two ways in which to uphold that rule: either the boundary repels bodies that approach it or the boundary moves away from bodies that approach it. We already know that neither the boundary nor space itself can exert a force upon a body to keep it away from the boundary, so we infer that the boundary upholds the motionlessness of the Universe by moving away from any bodies that may approach it. But the boundary cannot simply react to each approach; it must pre-empt all possible approaches. If that statement did not stand true to Reality, then the boundary would require the capability of detecting approaching bodies, of reaching across space to touch bodies and determine their motions. But that, along with the fact that the boundary cannot simply exert and react to a force upon an approaching body, would make the boundary at least semi-animate and we require that it be perfectly inanimate, a thing perfectly incapable of reacting to anything. Thus we deduce that the boundary moves continuously away from all bodies at a speed that no body can ever achieve.
Space must have finite extent, so the boundary of space cannot move at infinite speed. Thus we infer the existence of a finite speed that no body can ever reach. That fact necessitates that the speed so described represent an absolute state of motion, one that cannot be altered by anything under our control. If some phenomenon were to move at that absolute speed, then that phenomenon can never slow down, not by any amount, not by any means, either direct (the exertion of a force upon the phenomenon) or indirect (the acceleration of the observer relative to the phenomenon). Thus we must infer that, no matter how much we accelerate ourselves, the boundary will nonetheless recede from us in all directions at the same absolute speed.
That fact means that any phenomenon (such as light) that travels at the speed of the boundary must pass any two observers at the same absolute speed, regardless of any motion that may exist between those observers. That is, both observes calculate that the phenomenon moves at precisely the same number of kilometers per second for each of them based on measurements they make as the phenomenon crosses the grid that they have created with rulers and clocks to make distance and duration countable. That proposition corresponds to Einstein's second postulate of Relativity.
We must thus conceive the recession speed of the boundary of space as an unreachable absolute, equal to a kind of modern updating of Plato's Forms. Every observer, then, must have the same unchanging relationship with the boundary, but can only do so through the mediation of space and time. Thus, any phenomenon that measures space and time must have the same form for all observers. If that proposition did not stand true to Reality, if the phenomena did not display perfect symmetry between any two observers, then differences in the manifestations of the phenomena would indicate differences in the respective observers' relationships with the boundary of space, an impossibility with an absolute entity.
Now we know that if observers A and B conduct identical experiments and if A sees B's experiment differing from his in a certain way, the B must see A's experiment differing from hers in exactly the same way. From the data that their experiments yield observers extract mathematical descriptions of the things that we call the laws of physics. Those extractions must have the same forms in order that their manifestations in phenomena display the same differences from one observer to another, so now we know that the laws of physics have an absolute nature and, thus, constitute Platonic Forms encoded into the structure of Reality. The extractions take us from the contingent to the necessary. That proposition corresponds to Einstein's first postulate of Relativity.
That fact, the absolute nature of the motion of the boundary of space, enables us to explore how the perception of space and time differs between different observers. Consider the case of observers A and B holding their rulers parallel to each other and perpendicular to the direction of the relative motion between A and B. When the rulers pass each other A and B can compare their lengths. Assume that observer A sees the upper end of B's ruler pass the upper end of his ruler within some arbitrarily small distance and sees the lower end of B's ruler pass the lower end of his ruler within some arbitrarily small distance. Observer B must see the same close encounters, but in her view A's ruler moves past her ruler. That fact necessitates the inference that distances perpendicular to a relative motion remain unaffected by that motion, neither dilating nor contracting. If that inference did not stand true to Reality, then A would see B's ruler altered in the same way that B would see A's ruler altered B a perfect impossibility for any alteration other than none at all.
Imagine that a number of observers move relative to each other in the direction parallel to their common x-axis and imagine that one of them has set up a long straight rod with clocks at both ends perpendicular to that x-axis. A spark emits a flash of light next to one of the clocks and part of the light flies up along the rod and illuminates the second clock, giving us two events with both spatial and temporal intervals between them measured. For any observer who sees the rod moving, the path that the light follows from the spark to the second clock constitutes the hypotenuse of a right triangle whose sides coincide with the length of the rod (y) and the distance that the rod moves between the two events (x=vt). Applying Pythagoras' theorem gives us
But y must have the same value for all of the observers in this imaginary experiment, so we have in general
If we multiply that equation by minus one to reverse the algebraic signs, we get the mathematical expression of Minkowski's theorem, the metric equation describing the geometry of "flat" spacetime,
That equation gives us a local approximation to the field equation for space itself and it enables us to calculate a kind of distance between any pair of events, the same distance for all observers.
Suppose that observer B has a bar of length x' laid along the x-axis and both she and observer A measure the time that elapses between the events defined by the front of the bar passing A's clock and the rear of the bar passing A's clock. The distance between those events in A's frame is x=0, so Minkowski's theorem gives us
If we solve that equation for t or t', we get a description of time dilation.
Observer A sees observer B moving past him at the speed v and observer B sees observer A moving past her in the opposite direction at the speed v'. But no measurement or calculation from measurement can imply that one observer has a relationship with the boundary of space different from the relationship any other observer has with the boundary of space, so we must have as true to Reality v=v'. Our observers calculate their respective relative velocities by dividing the length of B's bar by the time elapsed between the events marked by its ends passing A's clock,
But time dilation necessitates that
so we also have as true to Reality
which describes the Lorentz-Fitzgerald contraction.
Observer A wants to confirm that calculation by direct measurement, so he projects a brief pulse of light in the form of a plane wave across the x-axis and onto a sheet of photographic film. He can then measure the length of the bar's shadow on the developed film. But now he must understand that the act of measurement consists of two events, however contrived, that the shadow encoded in the light striking the ends of the bar. That interruption of the light is perfectly equivalent to swinging down a calipers in such a way that its points just pass the ends of the bar. Those events occur simultaneously for observer A (t=0), so Minkowski's theorem describes the measurement of the bar as
in whichΔt' represents the time that elapses between the measurements for observer B. But Equation 7 converts that into
which necessitates that
That represents a temporal offset between the measurement events for B and illustrates the relativity of simultaneity, which Einstein regarded as the beating heart of Relativity. If B had put synchronized clocks on the ends of her bar, it gives us the difference in their readings that A would see in his photograph if he made one from the light reflected off the bar.
Now from those pieces we can assemble the equations of the Lorentz Transformation. We have
in which I made the substitution from Equation 10 in the second term on the right side of the equation. The product of velocity and the temporal offset represents the distance that A's clock moves relative to B between the two events that the observers are measuring.
Of course we have y=y' and z=z' for the directions perpendicular to the direction of relative motion between the observers.
And we also have
Though we devised that equation for the special events described above, it must apply to measurements that A and B make between any pair of events.
And now we know one more thing about the boundary of space: time does not elapse on it. Equation 12 tells us that if a clock were to move at the speed v=c, then an infinite time would elapse on all other clocks before any time at all elapsed on that moving clock. We might have expected this result, because we can guess that not-space, which the boundary of space touches, also correlates with not-time. On the other side of the boundary of space we have total non-existence. Nothing that exists in this Universe exists on the other side of the boundary.
Compare all of this to Aristotle's physics. Aristotle derived his physics from the belief that the Universe and its phenomena are organized around a state of absolute rest. This new Rationalist physics that I am working out derives from the belief that the Universe and its phenomena are organized relative to a state of absolute motion. Not surprisingly, both Aristotelian physics and the Rationalist physics of these essays are both Platonic.
From Descartes'"Cogito" and the interaction between thought and feeling I infer the existence of a Reality of which I am merely a part. The percepts that I receive from that Reality accurately mimic the things from which they emanate, things-in-themselves very much like Plato's Forms underlying the Reality that we perceive. From the relationships we conceive from those percepts we infer the existence of space and time, things that offer no percepts of their own, but which we measure through the percepts that we obtain from rulers and clocks. From the observation that space is a physical field with unique values of its characteristic property, extent, between pairs of its elements we infer that space has a boundary and, thus, finite extent. We infer that the absolute motionlessness of the Universe necessitates that the boundary move away from all objects at an absolute, but finite, speed in all directions. And then we imagine how two observers would measure the distance and duration between a pair of contrived events and deduce a set of equations that will convert one observer's measurements between any pair of events into the other observer's measurements between the same pair of events. Thus we establish the foundation upon which we stand to deduce the rest of modern physics.
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