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Exploring the Nature of Time
(and Other Occasional Dimensions)

by Thomas L. Atwood
Tom@SugarTreeRidge.com
February 21, 2007

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Introduction

The Wikipedia article on classical mechanics at http://en.wikipedia.org/wiki/Classical_mechanics shows how the concept of time enters into the derivation of kinematic variables such as velocity, acceleration, etc. If you want to know more about the mathematical description of the universe, that would be a good place to start.

This article takes a look at the nature of time by turning those definitions on their heads, so to speak, and by going back and taking a look at what it really means to be a clock. The following sections demonstrate how time can be thought of as a derived quantity, rather than something more fundamental, and how the motion of objects can be considered to exist in a three-dimensional space alone.

The concept of time has essential mathematical utility when we want to describe the physical universe around us. But this article will show how this extra, fourth dimension can come about, and that it may be purely mathematical in nature, and only an indirect manifestation of an underlying reality. Rather, the underlying reality, which is directly perceivable, is the concept of the relative displacement of objects within the three-dimensional space.

This article is merely an exploration. It is not science. It presents a different way of looking at the concept of time. It is purely speculative. You should not ascribe any extra-dimensional physical reality to the ideas mentioned in these words.

Reference Frames

Before getting into this time thing, let's do a little housekeeping. All the universe is considered to be embedded in a three-dimensional space. To describe a position in this space mathematically, we introduce the idea of a reference frame. A reference frame is an abstract measurement grid, with an associated coordinate system. The simplest form of a three-dimensional coordinate system would consist of three mutually orthogonal (at 90-degree angles from each other in all three directions) axes, usually labeled x, y and z. Each axis is considered to be a graduated ruler, with measurements marked off in conventional distance units, such as meters or feet (or centimeters, or inches, or miles or furlongs, etc.). Each axis is considered to extend to infinity in both the positive and the negative directions. The point where the x, y and z intersect is called the Origin of coordinates. Typically the value of each coordinate at the origin is taken to be zero.

Once you have set up a reference frame, you can describe the location of any object anywhere by giving the values of the x, y and z coordinates that describe its position in space, relative to the origin of your reference frame, wherever you may have decided to put it. Every distinct object will have a unique set of coordinate values, to the extent that "no two objects can occupy the same space at the same time".

This article will deal with the movements of objects through space. To keep it simple mathematically, I will rip from your hands the reference frame you have so carefully crafted. Whenever I want to describe the motion of an object, I will arbitrarily rotate your reference frame so that the x-axis is parallel to the direction in which the object is moving at that particular moment. Then I will move the origin of coordinates so that the object is moving right down the x-axis. Then its y and z coordinates will always be zero, and I can ignore them. This means that I can describe the motion using only one dimensional displacement, which eliminates the need for you to know anything about vectors, and which keeps the discussion from getting cluttered up with x, y and z values all over the place.

(However, although I can arbitrarily align the reference frame, I am not allowed to turn it or accelerate it while I'm talking about the motion of the object in question. This is just a minor legal point. Otherwise, I would end up introducing what are sometimes called fictitious forces into my description of the object's motion.)

Another short cut I will take here is to ignore complications that arise concerning the finiteness of the speed of sound or the speed of light. In what follows, you should assume that the velocities being measured are sufficiently small that the the sound or light signals travel extremely fast in comparison. In this way the errors that result from assuming an infinite speed of sound or an infinite speed of light will be negligible. After we have a fundamental understanding of the nature of time for very slow speeds, our understanding can be generalized by taking into account these other effects.

The Conventional Definition of Velocity

There are a lot of different ways to think about the changes that occur around us. Ultimately, at least at the molecular level, any change can be traced to the physical displacement of one or more objects or constituents of objects. For this reason, I look at the concept of the velocity of an arbitrary object as a way to parameterize the idea of change. If an object has a nonzero velocity, something is changing in the universe.

To define velocity as a mathematical construct, consider the position of an object at time t1 and again later at time t2 . Since I have aligned the x-axis of the reference frame along the direction of the object's motion, its y and z coordinate values will remain constant. Thus the motion will involve only changes in the value of its x coordinate.

Let us express the value of the object's location at time t1 as x1 and again later at time t2 . as x2 . Then during the time interval that elapses (Can you feel the time flowing?) between time t1 and the time t2 our object will have traveled a distance of

           distance traveled = x2 - x1

The average velocity with which it travels is defined by dividing this distance by the elapsed time duration:

           average velocity = ( x2 - x1 ) / ( t2 - t1 )

Thus, the velocity is defined as the rate of change of position with respect to time. An automobile speed of 60 miles per hour could exemplify a typical value for a velocity measurement. You can see that the measurement indicates that during a time interval of one hour, the automobile would travel a distance of 60 miles.

In general, to talk about the instantaneous velocity instead of the average velocity, one has to introduce a definition that involves differential calculus. But this is no big deal. In the following definition,

           Instantaneous velocity = dx / dt

the symbol d merely indicates to take a small difference between two nearby values of x or of t. Thus,

           dx = x2 - x1
where positions x1 and x2 are taken to be very close together.

Whether we're talking about average velocity or instantaneous velocity, the concept is essentially the same. Velocity is defined as the rate of change of an object's position with respect to the time.

The Measurement of Time

In a nutshell, we measure time using clocks. Let's take a close look at clocks. A clock is a device that tabulates equal-duration intervals of the regular, cyclic motion of some object, for example, a pendulum. We don't see pendulums used in clocks much these days. Most clocks and watches now use vibrating crystals. Here the regular, cyclic motion that is being tabulated is the dislocation of the surface planes of a crystal, which behave in a very rhythmic way when they are disturbed.

With any clock there is usually some energizing source that causes the periodic motion to be initiated and sustained by overcoming frictional or other energy losses, but this is not essential. The motion of the Earth around the Sun makes a pretty good clock if you want to measure really long intervals of time. Likewise, the rotation of the Earth is a regular, cyclic motion that we use to tabulate the time interval we know as the "day". We don't have to supply any energy to keep the Earth moving.

What is important about a clock is that it always requires a moving system of some kind in order to establish the time intervals that it tabulates. For purposes of illustration, suppose I manufacture a really bizarre kind of clock. Suppose I put a locomotive on a section of railroad track. The locomotive is pulling a flatcar. On the car stands a man with a sledge hammer. Along the track I put up posts at regular intervals, say 100 feet. On each post I hang a gong, positioned so that it can easily be struck by the hammer as the train rolls along. Do you get the picture?

I direct the engineer to set the throttle on the locomotive to a particular speed. He has to have a head start so the speed of the train has become stable as the train approaches my line of posts. The man with the hammer is instructed to strike each gong as he passes it. Anyone within hearing distance can use my clock, but it does have some potential drawbacks. For one, the unit of time, the interval between the striking of two successive gongs, depends upon the speed of the train. If the engineer sets the throttle to a different position on a subsequent pass, the interval between gong strikes will change. This is why we want the motion of a clock to be regular, in addition to being merely cyclic. So for rendering Gong Standard Time, let's require that the engineer always sets the throttle to the same speed. (I'm not going to say anything about what the specific throttle setting ought to be.)

Using Gong Time To Measure Velocity

Now that I have set up my gong timepiece, suppose there is a second train on an adjacent track and that I want to measure its velocity. This is a straightforward application of the definition of velocity given above. When the engineer on the second train hears a gong, I have instructed him to drop a marker on the ground beside the railroad track.

After the two trains have passed, I can walk over to the second track and measure the distance between successive markers. I can measure this distance in feet, for example. Then I use the definition of average velocity and compute the velocity of the second train as the distance in feet between two successive markers, divided by one standard gong interval of time. The units of measurement of this velocity will be feet per gong.

Using Only Markers To Measure Velocity

Suppose I come out to the tracks one day in my timekeeping capacity with the task of measuring the speed of the second train again. But on this day, something terrible has happened! Someone has stolen all my gongs! What's a timekeeper to do?

I have a brilliant idea. I instruct the man with the hammer on the flatcar of the first train (the "clock" train) to drop a marker whenever he passes one of the posts. Thank God those weren't stolen as well! I instruct the engineer of the second train to keep an eye on the man with the hammer. When he sees the hammer man drop a marker, he is to drop one from his train at precisely the same instant. (This is actually overkill, since I could have simply had the hammer man wave his hand as he passed each post. The marker he drops will always correspond with the location of a post. But if you bear with me, it will all turn out nicely in the end.)

After the trains have passed I can conduct my measurement of the second train's velocity the same way as before, since I know that each of his markers was dropped at the end of a time interval of one gong. Nothing has really changed. Even though the gong didn't sound, the dropping of the second train's markers was still synchronized with the clock train. I can still measure its velocity in feet per gong.

The Significance of Simultaneous Events.

In these examples, the dropping of a marker by either of the trains amounts to marking its position at that instant on a ruler running along the tracks. The key to my being able to compute the second train's velocity is that whenever a marker is dropped from the "clock" train, a marker must also be dropped from the second train simultaneously.

We can call the dropping of a marker from the "clock" train an event. The dropping of a marker from the second train would also be an event. This idea of simultaneous events is fundamental to our use of clocks. For example, suppose a master clock at the bureau of standards reads twelve noon. That is an event. If at that very moment we all set our clocks to the reading of the one at the bureau of standards, all these acts of clock setting are simultaneous events.

By using the simultaneous events to synchronize our clocks to the standard clock, we can establish a standard for time measurement for those of us who are not within hearing distance of my gong clock. We may recognize that across a 3000 mile wide country this system isn't ideal. We may choose the event that the Sun is directly overhead to synchronize our clocks. This system would have the advantage that the Sun rises and sets at the same time for everybody in the country. Our actual system of time zones is a compromise between these two approaches that seems to work pretty well.

Measuring Relative Velocity

Now, forget about the posts and the gongs. All we will have is a lot of regularly spaced markers lying around. The hammer man is given no external cues about when he needs to drop a marker. He is simply instructed to find some regular, cyclic motion on his train and to synchronize his marker drops to the cycle of his choice. Since I, the timekeeper, no longer know the time interval, I can no longer compute the velocity of the second train, the absolute velocity, that is, relative to the "stationary" location where I mounted the gongs for my original clock.

Nevertheless, I can still compute the velocity of the second train relative to the velocity of the first train in terms of a ratio. I simply divide the marker distance interval of the second train by the marker distance interval of the first train. If the marker interval of the second train is twice as great as the marker interval of the "clock" train, then the second train is moving twice as fast as the "clock" train. I may not be able to express the speed of the second train in feet per gong, but I can still give someone who didn't see it pass a precise measurement of its speed compared to that of the "clock" train.

The Significance of the Velocity Ratio

When we measure the velocity of an object, it is always measured relative to some observer. A different observer, moving relative to the first observer will get a different measurement for the object's velocity. Each observer measures the velocity relative to himself. The second observer can compensate for his motion relative to the first observer to compute the velocity that he thinks the first observer would measure. But the object's velocities seen by each observer are different.

Let's go back to the two trains. Instead of using Gong Time and dropped markers to compute velocities, for now we will use electronic radar guns. An observer standing on the ground may measure a particular velocity for the clock train. A second observer, chasing the clock train on a handcar would measure a smaller value of the clock train velocity relative to himself. A similar analysis holds for their measurements of the velocity of the second train. The physical quantity that is the same for the two observers is the difference in the velocities of the two trains. (Likewise, the two observers would agree on the value of their motion relative to each other, except for direction.)

Although the two observers would agree on the difference between the velocities of the two trains, they would still express this difference in miles per hour, according to the values displayed on their radar guns. This still requires an absolute time scale. However, as pointed out in the previous section, the ratio of the two train velocities does not require an absolute time scale.

However, the two observers will differ on the values they compute for the ratio of the two train velocities. So the ratio of the train velocities, which does not require an absolute time scale to compute, is not invariant among different observers. Concerning this invariance, the ratio of velocities behaves somewhat like a simple measured velocity in not being observer invariant. But it does not behave the same way under observer changes as a simple velocity, because of the absence of the time scale.

We could create a definition for this velocity ratio such as the following:

           Comparative velocity = dx / dx'

Here dx' is defined to be a small distance traveled by the "clock" train, while dx is a small distance traveled by the train whose velocity is to be computed. These distances correspond to the distance traveled by each train between the occurrences of two separate events (such as the striking of a gong or the dropping of a marker) observable from both trains. Each of these distance measurements has to begin in conjunction with a single event witnessed by both trains or in conjunction with two simultaneous events. The end of the interval must be marked in the same way for this velocity definition to have any meaning in this context. The same would be true for the previous velocity definitions.

This definition is precisely what we used to calculate the ratio of the velocities of the two trains by taking the ratio of the marker distance intervals. It also has a similar mathematical form to the conventional time-based instantaneous velocity given earlier.

This new formulation involves more than just a substitution of "time" variables, namely x' for t. The ratio of distances traveled by two physical systems, which is identical to the ratio of their velocities, is something that is physically different from the velocity of a physical system relative to an observer. Only when one of the systems is designated as the "clock" system do the formulations become mathematically equivalent.

Furthermore, the clock system must be carefully chosen. Ideally, its velocity should be constant, or the "time scale" will vary. Its velocity should also be significantly greater than zero, or any "comparative velocity" calculation based on it will be indeterminate. Thus, the two trains are not fully interchangeable. The clock train is a little special.

The Origin of the Fourth Dimension

For these train examples, we have only been dealing with one-dimensional motion along the x-axis of our coordinate system. For this business, we wouldn't need a four-dimensional space-time. A two-dimensional space-time would suffice instead, one spatial x dimension and one temporal t dimension. Thus, these simple examples could be handled mathematically by a two-dimensional space-time continuum.

Mathematical Dimensions

Mathematically, you can describe physics just as we currently do in terms of a four-dimensional space-"time" using this approach. The "clock" train comparative velocity approach is mathematically identical to the conventional time-based approach. Whether you use x' or t as a "time" index merely amounts to a choice of scale factor.

The distinction of the "clock" train example is that both trains operate in the same 3-dimensional space, the same one we live in. In this example, the additional mathematical dimension required for the "clock" train is just mathematical. It isn't a real, separate spatial dimension. Both trains operate in the same geometric space.

It is common these days to ascribe extra mathematical dimensionality to physical situations. This allows physicists to apply powerful mathematical theorems to physics problems that might otherwise be intractable. That doesn't mean there are mysterious extra "dimensions" in our universe. It just means that there are extra degrees of freedom that appear when describing a physical system that have the mathematical properties equivalent to a spatial dimension in the context of that particular problem. People shouldn't get carried away with this multi-dimensional stuff. It's just a mathematical thing.

As an example, consider that the "clock" train's path could be nonlinear. It could follow a track that goes around curves and up and down hills. Even so, we can use it to measure a scalar (one-dimensional) "time" by measuring its progress in terms of the total distance it travels along the track, regardless of whether the track is straight or not. However, if we chose to parametrize the motion of a physical system as a function of the motion of the clock system in terms of its full three-dimensional position, this would require three spatial coordinates for the "clock" train, and therefore three "time" dimensions. Then you would have to deal with a six-dimensional space-time continuum. Yet both systems would still be moving in the same "real" space of only three dimensions.

Einstein's Theory of Special Relativity

Einstein's theory of special relativity says something important about the behavior of lengths and times measured in different reference frames, so I want to elaborate a little bit on its underlying assumptions.

By the time the mid-1880s rolled around, physicists had developed an essentially full understanding of the physical nature of electricity and magnetism, as formulated in the four "Maxwell's equations". From these equations it is simple to derive an equation that describes the propagation of light waves, so that light could clearly be considered to be an electromagnetic phenomenon. The Maxwell equations even provided a way to calculate what the speed of light ought to be, in terms of the electrical permittivity and the magnetic permeability of free space.

This provided a model for the universe in which light waves were considered to propagate through some kind of a spatial medium, just as sound waves propagate through the air. This medium, called the "ether" would have to permeate all of space, even the vacuum of outer space. In a way the "ether" is like the "dark matter" scientists are currently searching for. You are pretty sure it's there, but you can't see it. In such a medium the speed of light relative to the ether would be independent of the motion of the source from which the light was emitted.

An important property of the laws of physics is that we want them to be the same in all inertial reference frames. An inertial reference frame is one that is not accelerating. One inertial reference frame can be moving at a constant velocity relative to another inertial reference frame, and we want the laws of physics to be the same, remaining in their simplest form as viewed by observers in each reference frame.

Before Einstein came onto the scene there was a concerted effort by physicists to establish the conditions under which Maxwell's equations of electromagnetism would be the same for all inertial reference frames. It was known that in a reference frame moving through the presumably stationary ether, length would have to be contracted and time would have to be slowed down compared to a reference frame that was stationary relative to the ether. If this were compensated for, then Maxwell's equations could be transformed from the coordinates of one reference frame to the coordinates of another without changing their form.

However, this still left the reference frame that was stationary relative to the ether as a special frame of reference. The idea of the stationary ether was in direct conflict with the idea that the laws of physics ought to be invariant for all inertial reference frames. Enter Einstein. He took these two seemingly conflicting principles and deduced from them the behavior of space and time transformations among reference frames moving with respect to one another. He did this in such a way that the results agreed with what was already known about electrodynamics transformations, but that applied to all physical phenomena.

All Einstein used to derive his theory were two simple assumptions:

    1. The laws of physics exhibit no properties that would correspond to the idea of absolute rest.

    2. Light propagates with a definite velocity which is independent of the motion of its source.

The first assumption pretty well eliminates the idea that there is a "stationary" medium of light propagation permeating all of space. The second assumption pretty well suggests that there is such a medium. Einstein essentially said that these two assumptions are actually compatible, and that their apparent conflict is telling us something fundamental about the nature of the motion of objects in the universe.

A remarkable result of Einstein's insight is that the speed of light is the same for all observers, as if each observer were at rest, even though different observers are moving relative to one another. If a pulse of light is approaching you and you increase your speed toward it (through the action of a force, which accelerates your velocity), it will still be traveling toward you at the same speed as before! If you decelerate and reverse your direction so that you are moving away from the approaching pulse of light, it will still be approaching you at the same speed as before!

It is as if the moving pulse of light automatically adjusts for any changes in your speed, even though there is no apparent physical mechanism through which it could do so. How can this be? Perhaps there is something about the application of the force which is needed to change your velocity that distorts the space through which you are traveling. Remember that in the theory of special relativity, the laws of physics are only the same for different systems traveling at constant velocity relative to one another. The assumptions of special relativity don't tell us enough to say for sure what happens if a reference frame is accelerating. (General relativity tries to deal with that topic.)

Einstein would resolve this quandary by saying that distance and time are distorted by the motion of one observer relative to another. It is not that the light pulse is compensating for your changes of speed. Rather, your changes in speed result in compensating changes in the lengths of rulers that you may use to measure distances. Being always at rest relative to your ruler, you don't see these changes. But when you put your ruler in motion and use it to measure the distance traveled by another inertial reference frame, a distortion seems to occur. For a traveling pulse of light, this distortion is precisely sufficient to compress the apparent distance and extend the apparent time it has to cover to reach you (as measured by you) in proportion to your change in velocity.

The Twin Paradox

Special relativity says that a clock that is part of a reference frame that is moving relative to an observer appears to run slower when viewed by that observer, in comparison to an identical clock that is not in motion relative to him. The twin paradox suggests that we take two twins at rest relative to us, and set one of them in motion. For the moving (relative to us) twin, time will appear to us and to the other twin standing there with us, to be moving more slowly. We can see this by watching through a sufficiently powerful telescope the face of a clock that he carries with him. Even after we compensate for the time it takes the light image of the clock to arrive at my location, his clock will still run slower than our clock. If we cause the moving twin to travel in a circle, then as he moves past his "stationary" twin he will appear not to have aged as rapidly.

On the other hand, if we look at the world from the point of view of the moving twin, he will see himself at rest, while his twin will appear to be moving in a circle relative to him. So, when the twins come back together, he will see the twin that has been standing there with us as being the younger one. How can each twin be younger than the other one?

The answer is that for a system to move in a circle, it has to be constantly accelerated to prevent it from moving off in a straight line. The speed may not change, but the direction of movement must be constantly changing in order for it to move in a circle. This requires acceleration, which requires the application of a force. Therefore, the two twins are not equivalent. The one moving in a circle can feel the force being applied to him. The twin standing next to us remained in an inertial reference frame. The moving twin did not. We may presume that the application of the force introduced a compensating distortion in the time, through which this paradox could be resolved.

Einstein's general theory of relativity was designed to uncover the mechanism responsible for this compensating distortion that results from the presence of a force, and he succeeded in the special case of the gravitational force. More than a century after Einstein first broached this problem, a general solution that applies to all types of forces has still not been found.

If, instead of moving him in a circle, we kept the moving twin in an inertial reference frame by letting him move off in a straight line, we could never get the two twins back together again to compare them side-by-side. All we could say is that each twin, looking at the other through a telescope, would see his twin appear to be aging more slowly than himself as he receded into the distance, even after the time for the propagation of the image is taken into account. This case is not a paradox, but is consistent with apparent time distortions among different inertial reference frames predicted by Einstein's special theory, which have been experimentally verified.

The Derivation of Time

Now it is time to come full circle. Suppose we lived in a strange, make-believe world where the science of physics was based on the motion of objects relative to one another, rather than relative to an observer. Then physics models would be confined to a three-dimensional spatial continuum. In addition to a description of the location of objects in that continuum, there would be a dual description of their momenta (mass times velocity), mapped onto the same three-dimensional space. All physics would flow from this fundamental duality.

In such a world, even though the physical concept of time was totally unreal and irrelevant to the mathematical formalism of the laws of physics, how would we go about deriving it? We could start out by attacking the problem of describing the motion of an object relative to a passive, abstract observer.

Without comparing the motion of the object to some other object, the physics in this strange world doesn't provide a way to do this. So we introduce a reference object. We choose for this reference object some object that exhibits a regular, cyclic motion. We develop a consensus that this object shall be treated as a standard object when describing motion. This reference object and any other objects that exhibit similar motions shall be designated as "clock" objects.

When describing motion relative to a passive observer, the velocity ratios will always be given relative to a clock object. We will refer to the standard distance intervals traversed by a "clock" object as time. Mathematically, we will follow the convention of using the symbol t for this coordinate. Instead of using something like a "clock" train for this purpose, we will use a small mechanical device with a rotating pointer that sweeps around a circular face with markings inscribed at regular distances. We will designate the distance traveled by the sweeping pointer between two successive marks as one "second".

An Alternate Derivation of Time

Pretend we are still in that make-believe world where physics is described without a time dimension. In this world the state of any closed (not acted upon from outside the system and not acting upon anything outside the system) system can be completely described by giving the position and momentum (mass times velocity) of each of its components.

The laws of physics could be used to describe how the motions unfold that ensue from some specified initial state. For example, picture a solar system near the edge of the Milky Way galaxy, far enough out from the galactic center to be effectively isolated. If we ignore the very minor effects that result from the solar radiation and small asteroids that enter into or escape from this system, we can treat it as if it were a closed system. To describe an arbitrary initial state for this system, we set up a reference frame, and then locate each planet. We also measure the mass and the velocity of each planet and the central star relative to our coordinate system. As a simplified model of this system we will ignore the effects of planetary rotation and of all the minor bodies and the interplanetary gas and radiation. To this approximation, we have therefore completely described the state of our system.

But all the planets are in motion. They quickly move out of the initial state and through a succession of different states, as their positions and momenta change. Thus, the state of our system evolves in accordance with the laws of physics. In principle, we can use the laws of physics to predict this evolutionary process, how each planet changes its location relative to all the others, under the action of their mutual gravitational attractions.

In reality, we cannot stop the motion within a solar system. We cannot affect it in any significant way. But in our minds we can visualize the concept of taking a snapshot, a picture or image of the system, which has the effect of freezing the motion. The taking of such a snapshot would be an event. We can use the snapshot to measure the position of each of the planets. A planet arriving at the position we see in our snapshot would be another event, simultaneous with our taking of the snapshot and also simultaneous with the arrival events of all the other planets at their positions shown in the snapshot.

From a single snapshot we can't deduce anything about the velocity (the momentum) of any of the planets. Our image is just a "still" picture. But we can take a second snapshot and use it to measure the change in position of each of the planets that occurred between the snapshots. This tells us their average velocities between the two snapshot events.

We can, if we like, model the evolution of this system in terms of a series of regularly spaced snapshots. If we take the snapshots based on the regular, cyclic motion of some moving reference system (our clock system), we can define a time index for use in labeling the snapshots.

Suppose our snapshots are actually holograms that exhibit the location of each of the planets in a full, three-dimensional representation. If we line up the holograms, we can see a representation of the changing state of our planetary system. As we walk along this line of holograms viewing the state changes, we are traversing a distance along a path that is outside any of the holograms and possibly outside the planetary system. Because this walking path is separate from the three-dimensional space inside each snapshot, and our motion along it is independent of any of the states depicted in the snapshots, we could treat it as a fourth dimension mathematically, even though in reality the "time path" is still in the same physical three-dimensional space as the holograms, the planetary system and the rest of the galaxy.

There is still a lot that we don't know about the laws of physics. We've been hampered for almost a century by the constraints imposed upon our measuring processes by the laws of quantum mechanics. We can't say for sure what time really is. Is it an artificial construct which we use to parametrize motion, or do we instead live in a real four-dimensional world?

As a significant aside, let me point out that the idea about the state of a physical system is not from a make-believe world at all, but rather it is an accurate reflection of how we actually practice physics currently, except that our current formulation involves a time parameter explicitly. For a physical system of N interacting bodies, the position of each body is given by 3 coordinates. The total number of coordinates for all N bodies is 3 times N, or 3N. This multiplicity of coordinates can be mapped into a 3N-dimensional mathematical space, called configuration space. If we add another 3N coordinates corresponding to the momenta (mass times velocity) of the objects, we can map both locations and velocities into a 6N-dimensional space called the phase space for the system. Specifying a single point in this phase space completely describes the state of the system of N distinct moving bodies at a given time. As the motions within the system evolve, its state changes trace out a path in this phase space. The distance intervals along this path are what we conventionally regard as "time". The two fields of Lagrangian mechanics and Hamiltonian mechanics use these concepts to describe physical systems.

This concept of the state of a physical system gives physicists the ability to predict future (and past) events with great precision, to the extent that they are working with a mathematical model that accurately reflects the behavior of the real world. (This ability to predict future events gives modern physicists a power that many shamans, priests, witches and sorcerers of the past would have sold their souls to acquire.) However, since the laws of physics act as a constraint to predetermine the state of the system at any past or future time, the state doesn't really depend upon the time but rather upon the laws of physics. In this sense, the time coordinate doesn't really represent a separate degree of freedom in the evolution of the system. Time, taken as a real dimension, is only essential as a replacement to provide a full empirical description of the full life of a system if we don't have knowledge of the laws of physics.

Conclusion

The fact that objects move in space clearly demands a mathematical fourth dimension for its description. The idea of a time dimension is a simple and elegant way to handle this mathematical requirement. In this article I have tried to show that ascribing physical reality to this extra dimension could possibly be a fool's errand.

Empirically at least, the time variable used to parametrize the motion of an observed system is inextricably tied to the motion of some other system, from which the apparent extra dimensionality derives. Rather than considering a mysterious concept of an extra-dimensional "time" as being something fundamental to natural law, I personally am more inclined to consider the corresponding "motion" as being fundamental. Motion I can see!

By approaching in this way the problem of understanding the concept of time, I have not removed the mystery. I have merely substituted another mystery. If you really want to understand how the universe ticks, you need to come up with an answer to this question: What causes objects to change position? The answer is not "Force". We can eliminate that by elaborating the question: By what mechanism does an object change position relative to another object, even when neither object is acted upon by any net force? That is a fundamental mystery. It requires a much deeper understanding of the microscopic behavior of physical objects than we currently have.