Our 4-D Universe
Bob Seitz
February 29, 2000

There was a time when ordinary arithmetic was a profound mystery known only to an elite priesthood—a mystery that took years to learn. There was a time when algebra and trigonometry were mysteries known only to a select cognoscenti. Today, every high school student learns these subjects as a matter of course. Today, though, we have face a similar situation with the geometry of space-time. Some day, every high school student will intuitively understand it, but today, it is beyond the reach of most university professors. That’s a shame. So let's see if we can find a way to rectify that presumed deficiency. If you read this exposition and try to follow it, it's my hope that you will be able to see at a glance what the smartest, best-educated men in the world didn’t see a hundred years ago.
This treatment of the special theory of relativity is an outgrowth of my efforts to understand it. Nothing I am going to say is intended to be at odds with Special or General Relativity.. What I’m going to try to provide is what, at least for me, has been a more intuitive way of looking at Special and General Relativity. I have had to work out for myself the simplifications presented here. For me, there have been a lot of "aha" insights as this picture came together. I had noticed early on that some of the paradoxical effects of relativity—your clock is running slower than my clock but you think that my clock is running slower than your clock—were similar to the distortions that occur when we rotate something in three dimensions. But I’m embarrassed to admit how long it took me to realize that if time is a dimension, then the time axis and a spatial axis can be rotated about one of the other spatial axes, and that when we do this, the slope the rotated time axis makes with the unrotated time axis is seen by us as a velocity. Bingo!
I need to warn you that this presentation has not had the benefit of a review by my peers, and although it seems correct to me, could be flawed. I need to also warn you that you're the first guinea pigs to whom this presentation is being made. I will welcome your suggestions for corrections and improvements.
For some reason, there has been a certain amount of popular resistance to the special theory of relativity since its inception. Denial of the validity of relativity often forms a frontispiece of amateur theories of physics. Although the theory is attributed to Einstein, its foundations had already been laid by H. A. Lorentz and Henri Poincaré. In 1905, Einstein arrived at his conclusions using thought-experiments involving clocks, measuring rods,  light-sources and mirrors. However, in 1908, his former math professor, Dr. Rudolph Minkowski, grasped the true meaning of Einstein's results and formulas—that they are really consequences of the fact that we live in a truly four-dimensional universe instead of the "Newtonian universe" of three-dimensions plus time—a four-dimensional universe whose measure for distance (space-time metric) is given by c2t2 - (x2 + y2 + z2). (The term "space-time metric" is just a fancy name for the Pythagorean Theorem that you learned in grade school that says that the square of the base plus the square of the altitude is equal to the square of the hypoteneuse. For two dimensions, it reads x2 + y2  = hypoteneuse2, for three dimensions, x2 + y2 + z2 = hypoteneuse2,and for four dimensions,.c2t2 - (x2 + y2 + z2) = hypoteneuse2. But don't you think it sounds a lot more impressive to say that you're "pondering the space-time metric" than to say that you're "pondering the Pythagorean Theorem"?) Seeing it this way, suddenly, everything snaps into place. The behavior of clocks and light reflections can now be understood on the basis of this four-dimensional geometrical model.
In my opinion, the work performed by Lorentz, Poincaré, and Einstein was spadework leading up to the real breakthrough by Minkowski in 1908—the realization that time is a fourth dimension. The clock, measuring rod, and light signal effects are now seen to be consequences that the proper measure for distances in our universe (neglecting gravitational distortions) is  c2t2 - (x2 + y2 + z2), or, or to say it in a form that sets the stage for general relativity, c2dt2 - (dx2 + dy2 + dz2).= ds2. So the theory of relativity is really about a change in our mathematical modeling of the geometry of the universe rather than about clocks, measuring rods, and light signals. Unfortunately, Dr. Minkowski diedof a cerebral hemorrhage in 1909, the year after he published his landmark paper. One wonders what would have happened had he lived a longer life.
There is a further discussion of this sometimes-contended topic of clocks, simultaneity, and light signals toward the end of this paper in a section entitled, "Einstein Is Wrong!"
In his first 1905 paper, Dr. Einstein observed that, somehow, the Maxwell Equations were already designed with special relativity (four-dimensional space-time) built into them. However, Newton's Laws of Motion were not relativistically compatible, and later that year, Dr. Einstein took the remarkable step of altering Newton's laws of mechanics so that they would also incorporate our four-dimensional space-time. Experiments were then conducted that showed that under the unusual (for that time) conditions under which Einstein's equations would differ from Newton's equations, it was Einstein's equations rather than Newton's equations that predicted the obsreved results. (Einstein's equations approximate Newton's Three Laws of Motion within the limits of experimental error under everyday circumstances.)
Next, Einstein found a way to show that gravity could be explained geometrically by assuming that matter distorts space-time in its vicinity, causing its geometry to become Riemannian or non-Euclidean. This involved an extension of the four-dimensional metric from one that is the same everywhere to one that could change from point to point. For example, for a radially symmetric, time-independent gravitational geometry—viz., a star, the metric becomes,

In this equation, rS is the Schwartzchild radius at which, for a given mass, a star becomes a black hole, and tl. is the time, integrated along a path (since time intervals change at different points in the star's gravitational field).  (We'll chew on this this in a future article.)

Are you ready? Here we go.

Time doesn't flow past Us. It's we who are moving down the time axis.
Sir Isaac Newton spoke of "time flowing like a river", and that's the way we perceive it. It seems to us as though time flows past us. But in reality, it's the other way around: it's we (or perhaps, our "conscious awarenesses") who are "moving". Time is a dimension or direction just like the other three spatial dimensions, and everything up and down the time axis is static or frozen. It's our "motion" down the time axis from the past into the future that animates our world. It's like a 3-D Omnimax movie. The Omnimax film consists of several film reels of two-dimensional images that, when flashed in front of us on a screen, give us the illusion of a 3-D world in motion. If we stop the film(s), what we'll see are static 3-D images, with each successive image differing slightly from the preceding image. It's only when we run the films through the projectors that we get the illusion of motion.
A 3-D virtual reality simulation might be an even better example of our four-dimensional universe. In, maybe, 10 more years or certainly in 20, we should be able to put together some really good computer simulations, with 3-D imagery perhaps fed to eye-mounted displays or a to wide-screen high-definition display, with stereo sound, tactile feedback, a "motion seat", and maybe even the release of various odors ("smell-a-vision?). Much of this is probably available right now at Wright-Patterson Air Force Base, but in 10 or 20 years, it should be greatly improved, and maybe even cheap enough for us. And who knows what will be available in 200 or 300 years? The bottom line is that such simulations consist of successive frames of 3-D imagery flashed fast enough to give the illusion of continuous motion. There have been some virtual-reality simulations conducted in which the subject wears display goggles, earphones, and a tactile-feedback suit. The subject walks around in a large open area like a gymnasium, experiencing a virtual world. As time goes by, these simulations should become better and better and cheaper and cheaper, and most science fiction writers expect to see them become very popular.
The point of all this is that the reality in which we actually live our lives is very much like the steadily improving virtual reality that we're gradually inventing. We are moving down the time-axis of a four-dimensional universe that consists of a continuous series of three-dimensional "frames" or cross-sections that are ourselves and the objects around us at a succession of instants. Just like 70-mm. movie frames in an Omnimax 3-D theater film, each cross-section (three-dimensional tableau) is slightly different from the cross-sections before it and the cross-sections after it. It is these changes from frame to frame, when we zip through them, that give us the illusion of continuous motion. However, whereas a movie film is two-dimensional, and an Omnimax film uses several two-dimensional film strips simultaneously to create magnified 3-D imagery, the universe in which we live consists of a 4-D objects that we perceive as a continuous succession of full-size 3-D objects. It's our mental motion along this continuous succession of gradually changing 3-D cross-sections of 4-D objects that creates the illusion of motion. And the time axis is just like a spatial axis. What makes it unique is that,
1. For some reason, we can't see the past or the future—only the present. Being able to see only the present amounts to our being to able to see only a razor-thin window revealing what lies directly perpendicular to the time axis, but not being able to actually see anything that lies behind us or in front of us in the time direction.
2. For some reason, we're moving down the time axis rather than one of the other three axes.
3. Our speed down the time axis is fixed. We can't stop, speed up, or slow down. ("Stop the world! I want to get off!")
One of the possibilities is that you're really experiencing a super-realistic computer simulation. Maybe in some laboratory beyond our present awareness, you're hooked up neurally to "God's" computer, and the life you're experiencing, including reading this presentation, is really only a hyper-sophisticated computer simulation.

"Oh, that this too, too solid earth should melt,
And leave not stick, nor bone,
Nor curl of vapor."

"Our revels now are ended. These, our actors,
As I foretold you, were all spirits, and
Are melted in air, into thin air:
And like the baseless fabric of this vision,
The cloud-capped towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And, like this insubstantial pageant faded,
Leave not a rack behind.

Four Dimensional Buck Fever
Please don’t be intimidated by the idea that space-time is four-dimensional! It turns out that we only need to use two dimensions to understand all the significant ideas regarding four-dimensional space-time. The other two dimensions are irrelevant. (Actually, it turns out that visualizing in four dimensions isn’t hard to do and we’ll do it so that you’ll know how. But we’ll do it more for esthetic and conceptual purposes than for any practical reasons.)
So how do things look in four dimensions?
One way to "see" objects in four dimensions is with the aid of the kind of "smear photography" that's used to photograph a pitched ball or a bullet in flight. Years ago, Life Magazine ran some high-speed photographs showing ghostly images of a spinning baseball as it whizzed along its path from the pitcher to the catcher, like a long semi-transparent cylinder. At each time instant, the baseball is a three-dimensional sphere, and the outline of its flight path generates a semi-transparent cylinder. (Actually, it generates a kind of droopy cylinder, like an Oscar Mayer wiener, because the ball drops a little as it goes from the pitcher's mound to the catcher's mitt.) The cylindrical "object" generated by all those baseball images at different times and places along the baseball's path could be called a "cylindrical sphere" or a "spherical cylinder". At any given point along the path, the cylindrical sphere has a three-dimensional cross-section (the baseball). The way that I visualize four-dimensional objects is to imagine them being moved from left to right. Then the cross-section at any given instant is the three-dimensional object.
Another example of a four-dimensional object is a balloon or the expanding universe. The universe starts as a point (at the moment of the Big Bang) and expands spherically to generate a "conical sphere" (Figure 1). At any given instant in time, the expanding universe has a three-dimensional cross-section, which is the universe "frozen in time" at that instant.
Generally, any three-dimensional object that doesn't change over time, like a rock, if we could see it extending down the time axis, generates a kind of four-dimensional non-circular cylinder or extrusion whose cross-section at any given instant is a snapshot of the three-dimensional rock.
So how do you look in four dimensions? Seen in four dimensions, you would be appear as a newborn baby over on the left, growing into an adult in the middle of our imaginary field of view, and finally into your elderly self at the right-hand side of our fanciful field of view. At any given instant, or as we're imagining it, at any given point along your four-dimensional self would be your "frozen in time" three-dimensional self at that instant in time.
If you could see in four dimensions, you probably wouldn't see that much. The three-dimensional image closest to you would block the earlier three-dimensional images behind it.

Question: "Are you saying that when I look out and see clouds drifting by and trees swaying in the breeze, they're not actually moving? It's my motion down the time axis that makes them look as though they're moving?"
Question: "Am I moving down the time axis?"
Answer  Not physically. Physically, you're a static four-dimensional object just like everything else in the universe, stretching from conception to old age. Everything that has ever happened to or will ever happen to you lies in between. It's your awareness or your consciousness that is moving down the time axis rather than anything material."

To summarize it all in three basic statements:
1. Time is a Fourth Dimension
We've known since the early years of the twentieth century that time is a dimension, and is almost exactly like the other three (spatial) dimensions.
2. Not "t" But "ct"
Actually, technically speaking, it isn’t time itself that is the fourth dimension, but rather, time multiplied by c, the speed of light. Time is measured in seconds, but time-multiplied-by-the-speed-of-light is a unit of length (e.g., light-seconds, light-years) like x, y, and z. In order for time to be eligible for treatment as a dimension, it must have units of length just like the three spatial dimensions. As it turns out, it always appears in the relativistically correct equations of physics in the form of "ct" rather than as "c" alone. (Later in this discussion, when we look at the space-time metric and the Maxwell Equations, you’ll see why the speed of light is the correct conversion factor and not some other number.)
This little matter of multiplying time by a constant (the speed of light) may seem trivial, but from a conceptual standpoint, it's a momentous change. It probably wasn't possible to come up with a 4-dimensional model of the universe and of the laws of physics without recognizing "c" as the proper multiplier to convert time into distance.
All our speeds should probably be denominated in, for example, "lights" and nano-lights—something other than miles- or kilometers-per-hour. ("But officer, I was only doing 92 nanolights."–about 62 miles per hour.) After all, kilometers are based upon the circumference of the earth, as well as it was known in 1800, and seconds are based upon the Babylonian sexagesimal divisions of the minute and the hour. Measuring time in seconds and speeds in meters per second masks some of the underlying simplifications in physics.

3. Circular versus "Hyperbolic" Rotations
Finally, although the speed-of-light multiplied by time (ct) is similar to the three spatial dimensions, it isn’t interchangeable with the three spatial dimensions. The difference is this. When we rotate something around one of the x, y or z spatial axes, the rotation is circular. The curves of constant radius are circles (Figure 1). However, when we rotate the time axis around one of the spatial axes, the rotation is "hyperbolic", meaning that the curves that correspond to curves of "constant radius" for a circular rotation are hyperbolas for a "hyperbolic rotation" (Figure 2).
The "space-time metric" or Pythagorean theorem for Euclidean space-time is
Equation (1)      distance = .

As you can see from looking at this equation, the time term has a plus sign in front of it and the three spatial signs have minus signs in front of them. I'll talk more about this later, but it's this difference in sign that prevents time from being interchangeable with the three spatial axes, and that causes rotations of the time axis about a spatial axis to be "hyperbolic rotations" rather the circular rotations we get when we rotate a spatial axis around another spatial axis.
We'll be comparing hyperbolic rotations with circular rotations later, in the discussion of Figure 4, but this is just to point out that time isn't quite interchangeable with space.

The Punch Lines
And now for the punch line: The bizarre effects that arise in special relativity—I say that your clock is running slower than my clock, but you say that my clock is running slower than your clock—and we’re both right—are nothing more than the perspective-effect distortions that arise when the time-axis is rotated around one of the spatial axes. I mentioned near the beginning of this opus that I had noted for years that the effects of special relativity seemed analogous to the optical perspective effects that occur when something is rotated toward or away from you. All of a sudden one day, I realized that, from a four dimensional perspective, a moving frame of reference is a frame of reference whose time axis is tilted away from us. Speed is the slope of the space traversed divided by the time required to traverse it—x/t! No wonder the effects of special relativity look like rotational perspective effects! They are rotational perspective effects!
And now for the second punch line: we can use ordinary circular rotations to show us what will happen when we carry out Lorentz transformations. Lorentz transformations require "hyperbolic rotations", but the perspective effects that occur as a result of a Lorentz transformation are the reciprocal of (in other words, "over-over") the effects of ordinary circular rotations (see below). Consequently, we can use circular rotations to understand what's going on with hyperbolic rotations (Lorentz transformations). We can even carry out quantitative calculations using circular rotations and then take the reciprocal of the circular results to convert the circular results to relativistic results. Some examples are given in the discussion below.

Perspective Effects Caused by Rotations
Put both hands out in front of your face, with your palms facing you. Now bend your left hand away from your face. It looks shorter than your right hand, doesn't it? Is it? Yes and no. In a sense, it is shorter, in terms of letting light pass that would otherwise be blocked by your unrotated hand. On the other hand, we know that it’s not really shorter. If you swing it back, it will return to its normal length. Or if you view it from above, you can see that both hands are the same length. One is simply rotated with respect to the other.
Or imagine two yardsticks. One is perpendicular to us and one has been rotated 60°away from (or toward) us. The one that's rotated 60° looks only half as long to us. But to an observer who's at a 60 angle to us, the rotated yardstick will look full-length and our yardstick will look half as long. In other words, how long the yardstick looks depends on where you’re standing and which direction you're looking.. (In a way, you can say that each yardstick looks shorter than the other.)
If you swing the yardstick 90º away from you, its length will appear to go to zero.
All this is second nature to us. These are simply perspective effects that depend upon where we're standing. . Our brains automatically interpret these distortions as arising from three-dimensional rotations.
. We don't normally get into philosophical discussions about whether or not a rotated hand is shorter than an unrotated hand because everything about this common situation is so obvious. But when we get to the theory of relativity and are asked to visualize perspective effects of a rotation of a fourth dimension that we aren't biologically equipped to see, our intuition needs some help.
To give an example of the way we can use circular rotations to help understand relativistic "rotations", let's look at the "twins paradox". We're going to begin by considering its circular-rotation analog.

In this example, time is passing at the same rate for both of us, independently of how rapidly or slowly we're traveling. But if our movements down our roads represented motion through time, then I would have to travel two feet (nanoseconds) down my time-path to advance one foot (nanosecond) down your time-path, and I would age twice as fast as you do in getting to the point where my path rejoined yours because I would have to travel twice as far through time as you do to reach the same point. To say it another way, if our clocks ticked every nanosecond, my clock would have to tick twice to advance me one foot or nanosecond down your time path. As seen by you, my clock intervals would be only half as long as your clock intervals, and my clock would be running twice as fast as your clock.
Before carrying through this analogy to encompass the Relativistic-Twins Paradox, it may be time to look at circular and hyperbolic rotations. However, we'll note in passing that when it's actually the time axis that's being rotated, the rotation is hyperbolic, and the perspective effects are "inverted" (1÷) from those that occur with a circular rotation. In the example of the circular-rotation "twins paradox" above, where I'm pretending that rotations of the time axis are ordinary circular rotations, my clock would run twice as fast than yours. But in the real-world situation in which my time axis is hyperbolically-rotated to the left, and then is hyperbolically-rotated back to the right to rejoin your time path, my clock would run only half as fast as your clock, and I, the rotated or moving observer, would only age half as fast as you, the unrotated or stationary observer. (You remember I said that the perspective effects in hyperbolic rotations are generally the inverse of the perspective effects in a circular rotation? "Twice as fast" for a purely spatial detour becomes "one-halfas fast" for a time axis detour.)
Figure 4 below, shows a 60º circular rotation (in blue) and the corresponding hyperbolic rotation (in magenta). If the radius of a circle is rotated 60º, its projection upon the horizontal axis will be ½ (= cos 60º) its horizontal radius. In other words, if you rotate your left hand so that it angles away from you at 60, it will only look half as long as your unrotated right hand. The corresponding hyperbolic rotation is one in which the

Figure 4 - 60º Circular Rotation    Figure 5 - The Same Rotation shown in Figure 4,
(Blue) and Its Corresponding        Only Now Seen from the Perspective
(~40.89º) Hyperbolic Rotation       (Magenta) of the Rotated Observer
Looking at the Unrotated Observer.

horizontal projection of the hyperbola's radius is  times 1, or 2  its radius (= cosh 40.89º). (You remember I mentioned that the effects of carrying out a hyperbolic rotation are the reciprocal of the effects that we see when we carry out a circular rotation?) For the hyperbolic rotation, the angle between its radius and the horizontal cannot exceed the 45º angle that the hyperbola's asymptote makes with the horizontal, and is 40.89º. Notice how its "radius" has had to stretch to remain on the hyperbola.
The blue circle in this figure is defined by the equation

or .

"ct" in these figures could be considered to be the z-axis, but as you can imagine, I've got a sneaky reason for labeling it "ct", instead of "cz", with "c" =1.

"Apparent length" I’m going to define "apparent length" to mean how long your hand appears to be when you swing it away from you. In other words, the "apparent length" of a rotated ruler is how long the ruler appears to be to someone who hasn’t rotated with it (Figure 4). To say it as it might be said in a textbook on relativity, it is the projection (cos if it's a spatial axis being rotated, or cosh if it's the ct axis being rotated) of the rotated observer’s ct-axis (what we're labeling the ct’-axis or rotated axis in the drawing) upon the unrotated observer’s ct-axis.

Figure 6 - The Area Subtended by the Arc of a Hyperbola (the Stippled Area)

1. The Range of Apparent Lengths for Circular and for Hyperbolic Rotations
Circular rotation: The apparent length of a rotated ruler appears to go from 1 to 0 as we rotate it from 0 to 90º—(where it becomes perpendicular to the unrotated ruler).
Hyperbolic rotation:
The apparent length of a rotated ruler appears to go from 1 to ¥ as we rotate it from 0º to 45º (where it coincides with the upper asymptote of the hyperbola).

2. The Behavior of the Radius for Circular and for Hyperbolic Rotations
For a circular rotation: When we carry out a circular rotation, the tip of the radius follows the arc of a circle, and the length of the radius remains constant. But..
For a hyperbolic rotation:
When we carry out a hyperbolic rotation, the tip of the radius has to stretch to follow the arc of a hyperbola (Figure 4). In other words, not only does the apparent length of a hyperbolically rotated ruler appear to stretch, but so does the apparent length of the radius itself. It’s as though when you swung your hand away from you, it looked longer instead of shorter. (There is nothing in our everyday experience to simulate this but it might be done with an anamorphic lens or a virtual reality program.) Of course, if you’ve rotated with the rotated ruler, it looks to you as though your ruler has its normal length, while "my" (unrotated) ruler looks as though it has stretched (Figure 4). This is just like the rotational-perspective effects with a circular rotation, except that, instead of each ruler looking shorter than the other, each ruler looks longer than the other.

3. The Behavior of the Axes for Circular and for Hyperbolic Rotations
When we carry out a circular rotation, the axes rotate rigidly and remain perpendicular to each other. But when we carry out a hyperbolic rotation, the axes swing toward each other and the angle between them becomes less than 90º (Figure 5).

Figure 5 - 30º Circular, and Corresponding Hyperbolic Rotations, with x & ct Axes

You can see in Figure 5 how the blue axes remain perpendicular to each other as we carry out a circular rotation, and how steeply they tilt toward each other as we perform a hyperbolic rotation.

4. One Direction is as Good as Another
We've drawn this circle and this hyperbola as though the horizontal direction were a preferred direction. However, in intergalactic space, there isn't any preferred direction. Furthermore, it's characteristic of a circle that you can go around it a thousand times and not be any farther ahead than someone who's turned around it less than once. And it probably won't come as a surprise to anyone to say that physical laws will be the same in intergalactic space in all frames of reference, whether rotated or unrotated. It's in the same league with saying that the laws of nature are the same no matter where you are in the universe. A similar situation exists with hyperbolic rotations. No matter how close (as seen by us) the hyperbolic radius vector seems to get to an asymptote, someone who's hyperbolically-rotated with (moving with) the hyperbolic radius vector always sees themselves at the center of the hyperbola, and sees us hyperbolically rotated in the opposite direction almost to the other asymptote. It's like chasing a rainbow or trying to reach the horizon. Or it's like looking into a mirrored garden globe. From whatever angle you look, your nose is always the biggest thing in your reflection, with everything else falling away on either side. To say it in the language of special relativity, no matter how close to the speed of light we think someone is traveling, as far as they're concerned, they're at rest and it's we who are traveling close to the speed of light in the opposite direction. No matter how close to the speed of light we think they're traveling, the speed of light is always as far away from them as ever

Now we're going to pull some rabbits out of hats. The cosine can be written:

Equation (1):

Now in Figures 4 and 5, .

Supposing we use the symbol "v" to represent the slope . Then can be written .

Consequently, we can rewrite Equation (1) in the form:

Equation (2)

(Both  and are equivalent ways of saying the same thing!)

Look familiar? Equation (2) would be the well-known Lorentz contraction formula except that it has a "+" sign where the Lorentz contraction formula has a "-" sign. That's understandable, since circular trigonometric functions apply to circular rotations, while hyperbolic functions apply to hyperbolic rotations.
Before going further, we note that is the speed at which a rotated coordinate system seems to be moving away from our coordinate system. A system whose time axis is tilted away from ours seems to be moving away from us. As we move through time down our time axis able to see only through a very narrow slit at right angles to our time axis, things in a system whose time axis is tilted away from ours will seem to be moving away from us (see Figure 6).

Figure 6 - Points x1, x2, x3, x4, x5, x6 along a tilted time axis ct' appear to be moving away from us.

As we move down the time axis ct, since we can only see what is directly perpendicular to us, the points x1, x2, x3, x4, x5, x6 appear to be moving away from us. We can't see that they're points along an line that's tilted away from us. Instead, they would appear to us to be a dot moving farther and farther away from us as we move down the ct axis.
The best analogy might be a situation in which you’re moving on a ship in crystal-clear, unruffled water in pitch darkness. There is a bright, narrow-beam lamp that illuminates a very narrow window at right angles to the boat in the water under the boat. Let's suppose that underneath the boat, there is a straight pipe or a sunken road that angles from right to left as we move along in the boat. We don't know it's a pipe. All we can see is the sliver of a cross-section that's illuminated by the ship's underwater wedge-light. It looks to us as though we're seeing an object that's moving from right to left, when in reality, it's a stationary "filament" that only appears to be moving because we're seeing different points along its length as we move with the boat.
Exactly the same situation exists with respect to your motion down the time axis. Your narrow-beam spotlight is the present. The world is constructed of filaments stretching down the time axis, but because of your motion down the time axis, you see their three-dimensional cross-sections moving away from or toward you when these filaments (time lines) are angled away from or toward you. And this generates what we perceive as motion. The only difference is that the cross-sections that we see as we move down the time axis are three-dimensional rather than two-dimensional.
Now let's carry out the analogous transformation for the cosh function, bearing in mind that the cosh function gives the projection of the hyperbolically-rotated time axis upon the unrotated time axis.

Equation (3) .

But tanh u = , and Equation (3) becomes:

Equation (4) cosh u = , often written , where b = v/c.

So cosh u is the familiar Lorentz-Fitzgerald contraction factor: .
We perceive something that is hyperbolically tilted toward or away from us as moving toward or away from us. In other words, motion and hyperbolic rotation are the same thing. The slope of the tilt of something else's time axis relative to ours is given by . But that's precisely the definition of its speed. So we're we say that something is moving toward us or away from us, we're really saying that its time axis is tilted toward or away from us

Simultaneity
It's probably time to say something about the difference between reality, and our models of reality. Since the introduction of Euclidean geometry, mankind has developed ever-more-accurate mathematical models of the real world. At the same time, it's important to distinguish between our mathematical models and the real world—between theory and experiment. For example, Euclidean plane and solid geometry isn't currently designed to model a preferred direction. On the earth, we have what we might consider the absolute directions: north, east, south, west, up, and down. However, since the days of Copernicus, we realize that these absolute directions apply only when we're on the surface of the Earth. When we get into space, it's a whole new ballgame. Here, there's no dramatic evidence of a preferred direction. There's our galactic lens, there are clusters of galaxies, and there are super-clusters of galaxies. The general theory of relativity derives its Schwartzchild solutions by assuming that the universe is homogeneous and isotropic. That seemed reasonable in 1916 when Einstein published his first paper on general relativity, and it's not unreasonable today. However, we now know that we live in a lumpy universe that isn't entirely homogeneous, and we know that the background microwave radiation isn't completely isotropic. The point of all this is simply that one has to be careful not to confuse theory with reality.
In a similar way, the fact that the special theory of relativity doesn't provide for an absolute velocity.doesn't mean that an absolute velocity can't exist. Experimentally, as far as we know, absolute velocities, along with absolute coordinates and absolute orientations, don't exist, in keeping with our mathematical model. However, that determination is ultimately the prerogative of experimental physics and not of the theoretician. Just as special relativity emended the laws of mechanics and our ideas of real-world geometry, and just as general relativity showed the applicability of a non-Euclidean geometry to our universe, so future updates to our mathematical models may be needed to accommodate future discoveries that we'll make. (Or at least, I hope so.) I'm emphasizing this because, in everyone's zeal to

Our Filamentary Universe
If you could see in four dimensions, people would be about 5 to 6 feet tall, 1½-feet wide, ½ to 2/3rds of a foot thick, and, if they were 30 years old, 30 light-years or 180,000,000,000,000 miles, or about 1,000,000,000,000,000,000 (one sextillion) feet long down the time axis. At one end (birth), you would see a newborn baby and at the other end, you would see the person at age 30. Seen in four dimensions, you appear to be this incredibly long-enduring filament (this sculpture) whose cross-section at any given instant is your three-dimension self at that instant.
How can we know that our bodies extend like this down the time axis? Because length along the time axis is (intuitively) a measure of how long something exists. A magnetic pulse in a semiconductor chip that lasted for one picosecond would extend about 30 microns down the time axis. We last for decades, so we extend decades (tens of light years) down the time axis. How do we know that we stretch out so far along the time axis? Because the conversion factor that converts time into a distance in the–let's call it the "Einstein-Minkowski metric for flat space-time" (Equation 1)–is the speed of light, "c".
In the language of relativity, these filamentary time lines are called "world lines". I like "time lines" better but since "world line" is the accepted nomenclature, we'll use it.
Of course, your world line doesn't end with your death. Your skeleton and eventually, the elements of which you are made will stretch—or, in a God's-eye view, are already stretching—into the indefinitely far future. And your spatial dimensions are incredibly small compared to your temporal dimensions. The ratio of your temporal extent to your spatial extent—of the order of a sextillion to one—is beyond anything in our experience. And the same thing is true of everything around us.
From a four-dimensional viewpoint, even our Milky Way galaxy is an extremely slender filament . Our galaxy is about 100,000 light years across and several billion years old, so that the ratio of its temporal extent to its spatial extent is of the order of 100,000:1. That's a lot less than a sextillion to one—more like a long spider thread—but it's still filamentary.
Is there anything that isn't a filament? One thing that isn't is the universe itself. The universe is as broad, spatially, as it is long down its time axis. In general, anything that travels at the speed of light, such as a spherically expanding light wave, will have that as-broad-as-it-is-long characteristic.

The Relativistic Twins Paradox

The Twins' Paradox refers to a situation in which one twin–the "traveling twin"–heads for a neighboring star at almost the speed of light, while the other twin–the stationary twin–stays home. ("This little piggie goes to market, while that little piggie stays home.") Relativity predicts that the traveling twin's clock will run slower than that of the stationary twin, so the traveling twin will age less than the stationary twin. The paradox arises when it's reasonably argued that since uniform motion is relative, it's just as correct to say that the traveling twin can be regarded as stationary, while that "stationary" twin can be regarded as the one who's traveling. So how can we conclude that either twin will age less than the other?
This subject triggered endless debate in the 50's and 60's when someone by the name of "Dingle"
In the 60's, when a couple of us puzzled over this, we speculated that the resolution of this paradox might reside in the fact that the traveling twin has to be accelerated and decelerated, and that this situation would be subject to general, rather than special relativity. However, that's not the reason. In the circular-rotation analog to the twins' paradox, as shown in Figures 3 above, the traveling twin travels a greater distance (by a factor of 2) than the stationary twin. So if rotations of the time axis were circular rotations like the rotations of the other three space-time axes, the twins' paradox would still arise, only the traveling twin would be older than the stationary twin. However, because rotations of the time axis about the spatial axes are "hyperbolic rotations", the temporal distance that the traveling twin ("I") has to travel is shorter than the temporal distance the stationary twin ("you") has to travel. Consequently, the traveling twin will have aged less than the stationary twin because the path the traveling twin followed in space-time was shorter than the path the stationary twin followed in space-time. (This is so contrary to everyday experience and so counter-intuitive that it's hard to imagine and difficult to depict.)
Another way to envision the Twins Paradox is this. When the traveling twin accelerates to about 7/8ths of the speed of light, the distance to her target star will appear to be halved. Proxima Centauri, instead of being 4.26 light-years distant will now appear to be only 2.13 light-years away. Proxima's color will be shifted into the violet, while Sol's light will have shifted into the red. Furthermore, all the distances along her line of flight will appear to be halved. She'll know that distances in the universe didn't suddenly become only half as great as they were just because she has just accelerated to a near-light speed. She'll realize that this is an artifact of her motion relative to the rest of the visible universe. After traveling about than 2.13 years (neglecting acceleration and deceleration times), she'll arrive in the neighborhood of Proxima Centauri, where she'll rapidly "decelerate" to a state of rest relative to Proxima. Now distances will look normal to her again. Sol will appear to be 4.26 light-years away, while only 2.13 years will have passed for her. Turning around, she'll "put the pedal to the metal" and  re-accelerate to ~7/8ths light-speed back toward the Solar System, which will once again appear to be 2.13 light-years away. After another 2.13 years, she'll reach the Solar System. For her, 4.26 years will have elapsed for the round trip (plus a little acceleration time). But when she reaches Earth, she'll find that 9.52 years have elapsed (plus acceleration, deceleration times, and dwell time at Centauri).
The third and conventional way to resolve this paradox is by counting the pulses sent out by the stationary twin to the moving twin, and sent by the moving twin to the stationary twin. Let's suppose that, according to each twin's clock, a pulse is emitted once every second. And let's suppose that the total number of pulses emitted by the stationary observer during the trip out to Proxima (153,640,430) is taken as unity. In that case, the total number of pulses emitted by the stationary twin during the round trip is twice that, or 2.
The first pulse from the stationary twin is emitted one second after the traveling twin departs. By this time, the traveling twin is ~7/8ths light-seconds away. The light pulse continues chasing the taveling twin, catching up with it 8 seconds after the traveling twin departs, or 7 seconds after the pulse is emitted (since the first pulse is emitted 1 second after the traveler departs.). By now, the traveling twin, after traveling 8 seconds at 7/8ths c, is 7 light-seconds away. Meanwhile, the second light-pulse, emitted after 2 seconds, is chasing the traveler, catching up at the rate of 1/8th light-second per second. At the end of 16 seconds, the traveler will have traveled 14 light-seconds, while the 2-second light pulse will have traveled 16 light-seconds, and will have overtaken the traveler. And in this way, as seen by the stationary twin, the traveling twin will receive a light signal from the stationary twin every 8 seconds. Consequently, by the time the traveling twin reaches Proxima, she will have received only 1/8th of the clock pulses emittted by the stationary twin back in the Solar System. (Of course, because her clock is running only half as first as the stationary twin's lock, it will appear to her as though she's receiving a pulse every 4 seconds for the 2.13 years that, according to her clock, she's "on the road". In other words, the traveling twin will see radiation from the sun red-shifted to 1/4th the frequency seen by the stationary twin.) 7/8ths of the pulses are still on their way from Sol to Proxima when the traveling twin arrives at Proxima in 4.87 years.. Now the traveling twin turns around, puts the pedal to the metal, and heads back at 7/8ths c. Now her speed relative to the oncoming wave fronts (as seen by a "stationary" observer) is 1 7/8ths c. Consequently, on her way home she will encounter the 7/8ths of the pulses that were chasing her to Proxima plus the 4.87 years worth of pulses that will be emitted during the 4.87 years that she spends on her return voyage. Of course, since her clock is running at only half the speed of the stationary twin's clock, it will seem to her as though the frequency of the pulses she's receiving are blue-shifted to 3.75 times what they are to a "stationary" observer.

Now let's look at the pulses that the traveling twin sends to the stationary twin. The traveling twin will emit her first pulse when 1 second has passed on her clock, and 2 seconds have elapsed on the sttionary twin's clock. At that moment, she will appear to the stationary observer to be 1 3/4ths light-seconds from her starting point, while appearing to herself to have traveled 7/8ths of a light-second.. It will take 3 3/4ths seconds for that pulse to reach the stationary twin. After that, another pulse will arrive every 3 3/4ths seconds. By the time the traveling twin reaches Proxima, she will have emitted 1/2 or 15/30ths as many pulses as the stationary observer. 1/3.75th or 8/30ths of them will have reached the stationary twin, leaving 15/30ths - 8/30ths, or 7/30ths of them still in transit, to be received over a period of 4.26 years (since the nearest will be arriving every 3.5 seconds, and the last of them will have been emitted 4.26 light-years away. Now when our traveling twin turns around and puts the pedal to the metal, her emissions will only be moving 1/8th c faster than she will. So for the next 4.26 years, the signals that she emitted on her way to Centauri will continue to arrive at Sol, until all the pulses that she transmitted on her way to Proxima have been received, totaling 1/2 as many pulses as the stationary observer sent her on her outbound trip. Then in 4.87 - 4.26 years = 0.61 years or 1/7th of the 4.26 years, the other 1/2 of the pulses emitted by the traveling observer will arrive at the stationary observer's viewing station. The traveling twin is 7/8ths of the way home when the pulses she has emitted on her way home finally begin to reach the stationary twin. So half of the total number of pulses that she has broadcast on her entire round trip will arrive in the last 1/8th of the time .

The Maxwell Equations
Now it's time to show you some equations. Don't panic! I'm not going to include them in your final exam. What? You say you didn't know there was going to be a final exam? Darn! Did I forget to mention that again?
All we're going to be concerned about is the format of these equations. You don't need to understand them  or what the little ¶ symbols mean. What's important are the similarities (and the differences) between the time term and the spatial (x, y, and z) terms in these equations. What's startling about these equations is that the Special Theory of Relativity and the idea that time is the fourth dimension are implied by these equations. But these equations were published years before 1905, when Einstein published his famous relativity paper: "On the Electrodynamics of Moving Bodies". Apparently, nobody interpreted these equations before Einstein. But you and I, with 20/20 hindsight, are going to able to easily see what even the greatest minds on Earth couldn’t see before 1905.
What I'm going to show you now are the famous Maxwell Equations (in what's called their "Lorentz form"). The Maxwell Equations describe all electric and electronic phenomena in the universe in which we live. Well.. Maybe I'm stretching it a little. Maybe you wouldn't want to try to wire your house using the Maxwell Equations. But they are the defining equations for one of the four known forces in the universe: the electromagnetic force.
These landmark Maxwell Equations are:

(1)

You don't need to know what these equations mean, but those who want to know a little more about them may look in Appendix A. In these equations, the little ¶ squiggle is the partial derivative sign, t is the time, in seconds; c is the speed of light (= 300,000 kilometers meters per second = 186,000 miles a second); j is the electric potential (the voltage), in volts; x, y and z are the x, y and z coordinates, in meters, of any point in space where we are located; Ax, Ay, and Az are the x, y, and z components of the magnetic vector potential, r is the charge density in Coulombs per cubic meter; and the j‘s are the x, y and z components of the current density, in amperes per square meter,
As I’ve mentioned previously, to convert time into distance along the time axis, you have to multiply time intervals by the speed of light. This has some startling implications. Something lasting a billionth of a second would stretch a little less than a foot along its time axis. Something like a soap bubble that lasts a second will extend about 300,000,000 meters or nearly a billion feet along its time axis. That would be a pretty long soap bubble.
Now it's time to get back to our equations. Let's change ct to tc, remembering all the while that when we say " tc", we really mean "ct". Since it's ct that we should be using in these equations, anyway, that probably makes good sense. (The only reason the equations are written the way they are is because when they were first derived, people didn't understand what they were all about.)

(2)

Now look at how similar these equations are to each other and how similar the time terms (the terms) in them are to the spatial terms (the  terms). The only remaining differences are the presence of the minus signs in front of the time terms. and the fact that, in writing down the velocity components on the right hand sides of the equations, the time is the variable on the bottom of all four of them. These differences, though small, are very important. Before we go on, we'll make one more "change of variables". Einstein pointed out that this one hasn't proved very useful but conceptually, it might lead to something. We can write the minus sign in the above equations as . If we do that, then our equations become

(3)

and since i is a constant, now we'll do what we did with c and bring it under the ¶ sign:

(4)
.

Now what we can do if we want to make the equations look the same in all four variables is to define some new variable t = itc (t is typically used for this purpose), so that our equations become:

(5)
,

and lo and behold, the left-hand side of our equations looks exactly the same for space as it does for time. In fact, if we really wanted to emphasize the interchangeability of t with space, we could write our equations in the form

Of course, in this formulation, x4 is t, and x1, x2, and x3 are x, y and z, respectively.

You remember I said at the beginning of this disquisition that time is a dimension like, but not interchangeable with the three spatial dimensions? If we use ict for our time variable, then time is interchangeable with the three spatial dimensions. There are some problems with using ict as our measure of time, which I'll discuss later, but conceptually, it puts all four dimensions on an equal footing.
I want to emphasize that the fact that time is a 4th dimension like the three spatial dimensions is immanent in these equations! If the Maxwell Equations are valid for electromagnetic phenomena, no Michelson-Morley experiment or tests of relativity are necessary to prove the validity of relativistic effects. All the strange effects of relativity—the Lorentz-Fitzgerald contraction, time dilatation and the twins paradox—follow as almost-trivial results!

"Einstein Is Wrong!"
One popular pastime among amateur relativists is seek a combination of clocks, mirrors and light signals that reveals an error in Einstein's reasoning, and especially, his reasoning about simultaneity. As we've seen, Einstein's relativistic reasoning led to a higher level of insight about our universe: the fact that it is four-dimensional. Einstein's thought experiments with clocks, measuring rods, and light signals then become results that may be derived and calculated from the underlying concepts. As I see it, these underlying concepts are built into the very structure of the Maxwell Equations, Newton's relativistically-revised Three Laws of Motion, and finally, into the general-relativistic model of gravitation. Anyone who proposes to change the concept of simultaneity or any other aspect of relativity must either show compatibility with the Maxwell Equations and with the mathematical formulations of relativistic mechanics, or must alter these equations in such a way that they predict undiscovered phenomena under extreme conditions (since we know that these equations apply within the limits of experimental error under less-extreme conditions). Unfortunately, it can be a time-consuming and error-prone task to find the flaws in some of these thought experiments, a little like demonstrating that a Rube Goldberg contraption isn't a perpetual motion machine.
Even if Einstein made a mistake, the resulting clockwork works. at least within the limits of experimental error. There may well be emendations to special and general relativity, but if so, one would expect them  to take the form of extensions to the present model.
One reservation about general relativity concerns gravitational propagation speeds. It is apparently common knowledge in celestial mechanics that gravitational influences must be assumed to propagate instantaneously. If they did not, the planets would have fallen into the sun shortly after they were formed. This is in direct violation of everything I know about relativity. This is a subject for future exploration.

Miscellaneous Earlier Notes

How Time Flies!
Does it seem as though time rushes by breathtakingly fast? Perhaps it’s because your consciousness is rushing down the time axis at the speed of light. Yes, I know: material objects cannot attain the speed of light. But this is the "speed" at which our individual awarenesses must be travelling down the time axis in order to experience our "filamentary universe" the way we do.
This realization raises some interesting metaphysical questions. Why are all our awarenesses translating down the time axis at the speed of light? Could our "reality" be just a "virtual reality" on some cosmic computer? Could it be that I am (or you are) the only real person in existence, with everyone and everything else just computer simulations (like the Holodeck on the Starship Enterprise). Or are all of us role players in some cosmic game? For answers to these and other stimulating questions, tune in again around 3000 A.D..
Seen from this perspective, the universe is a frozen four-dimensional sculpture whose future unfolds as you "move" down the time axis inside the sculpture. Nothing moves in the universe except our conscious awareness. It's like a computer program that's static until it's animated by the computer. And your conscious awareness is "moving" down the time axis at the speed of light! You've heard that nothing material can go as fast as light? That's true insofar as motion through space is concerned. But this "motion" down the time axis is different. We cannot move in spatial directions at the speed of light, but our "awarenesses" move down the time axis at—and only at—the speed of light. We are well aware of our motion through time We've always known that time flows by but until the twentieth century, we didn't know how fast time flows by, or how to convert time intervals to distances. Why are we moving down the time axis? Why are we all moving down the time axis at the speed of light? Why is the speed of light what it is and not something else? I don't know. To the best of my knowledge, no one knows. There's nothing in Special Relativity or classical physics that says this has to happen. Maybe you can figure it out.
Another comparable situation might be one in which you're flying at night and all of a sudden, a cloud jumps in front of you. Of course, you're savvy enough to know that it's really you who are moving into the cloud. The cloud is virtually standing still. But if you didn't know that you were moving, you might think the cloud had jumped in front of you (the way trees and telephone poles can sometimes jump in front of cars in spite of our best efforts to avoid them).
To me, the theological implications of this "revelation" are awesome. For example, from the standpoint of relativity, the future and the past are as "realized" and as unchangeable as the present.
All our speeds should probably be denominated in, for example, "lights" and nano-lights—something other than miles-or-kilometers per hour. ("But officer, I was only doing 92 nanolights.") After all, kilometers are based upon the circumference of the earth, as well as it was known in 1800, and seconds are based upon the Babylonian sexagesimal division of the length of the day into 86,400 seconds..

Visualizing in Four Dimensions
It's really not that difficult to visualize in four dimensions. After all, our eyes take in two-dimensional images. Our brain then combines them, based upon experience, into three-dimensional imagery. And we can't actually see three dimensions. What we see are projections of three dimensions. You have already visualized yourself from birth to death in 4-space (or space-time). One way to "see" objects in four dimensions is with the aid of the kind of "smear photography" that's used to photograph a pitched ball or a bullet in flight. A pitched ball shows up as one of those old Life Magazine high-speed photographs with ghostly images of the spinning baseball along its length, like a long semi-transparent cylinder from the pitcher to the catcher. At any point along the way, the cross section is that of the three-dimensional baseball. (By analogy with a right circular cylinder, we could probably call this a right spherical cylinder.) Of course, the ball drops somewhat as it goes from the pitcher to the catcher, so we could probably call it a droopy right spherical cylinder—like an Oscar Meyer wiener. Generally, any three-dimensional object that doesn't change over time, like a rock, will form a kind of cylinder, if we could see it extending down the time axis, whose cross-section at any given instant is a snapshot of the three-dimensional rock.
The universe is an example of something that is expanding with time. So would an expanding balloon. They would show up as a "spherical cone" or "conical sphere". Of course, these visualization methods involve imagining one of the spatial axes as the time axis.
If you could see in four dimensions, you probably wouldn't see that much. The three-dimensional image closest to you would block the earlier three-dimensional images behind it.