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 c^{2}t^{2}
- (x^{2} + y^{2}
+ z^{2}). (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
x^{2} + y^{2}
= hypoteneuse^{2}, for three dimensions, x^{2}
+ y^{2} + z^{2}
= hypoteneuse^{2},and for four dimensions,.c^{2}t^{2}
- (x^{2} + y^{2}
+ z^{2}) = hypoteneuse^{2}.
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
c^{2}t^{2} -
(x^{2} + y^{2}
+ z^{2}), or, or to say it in a form that
sets the stage for general relativity, c^{2}dt^{2}
- (dx^{2} + dy^{2}
+ dz^{2}).= ds^{2}.
*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, r_{S}
is the Schwartzchild radius at which, for a given mass, a star becomes
a black hole, and t_{l}. 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?"

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.

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.

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.

Suppose that you and
I are walking down a road together when we come to a fork in the road.
The left branch veers off at a 60º angle to the main road, while the
main road goes straight ahead. I decide to explore the left-hand fork while
you continue on the main road. The minute I turn left 60º and start
down the left-hand road, it will look to you as though my ruler has shrunk
to half (= cos 60º) its length (Figure 3b). Every foot that I travel
will only advance me 6 inches along the main road. It will look to you
as though I'm only going half as fast as you are. Of course, from my perspective,
it will look as though *your* ruler has shrunk to half its size, and
that you're going half as fast as I am (Figure 3c). Every foot that you
travel will advance you only 6 inches along my road. Now suppose that after
traveling 1,000 feet down

my fork of the road, I come upon another
fork that branches back toward your fork, and I decide to come back and
tell you what I've found (Figure 3b). I'm going to have to travel another
1,000 feet down two sides of an equilateral triangle to rejoin you on your
road. If we're both walking at the same speed, it will take me twice as
long as it does you to cover the same 1,000-foot distance down *your*
road. On the other hand, if you decided to come over and join me on my
road, then it would be you who had to take the long way around. In a way,
it's like selling stocks. As long you don't sell your equities when the
stock market goes up or down, your gains or losses are "unrealized". But
if you sell when your stock prices have dropped, your losses become all
too real. Similarly, as long as you and I are going our separate ways,
neither direction is better than the other. But if I decide that I want
to come back to your road, then I've "locked in my losses". Your direction
and your road become the preferred direction and road.

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

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**

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 x_{1}, x_{2},
x_{3}, x_{4}, x_{5}, x_{6} 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 x_{1},
x_{2}, x_{3}, x_{4}, x_{5}, x_{6}
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**

To say that two things
happen *simultaneously* is to say that they happen at the same time–at
the same value of "t", or as we're going to keep saying, "ct". The spatial
equivalent of "simultaneous" is " parallel". If something is parallel with
us, its "x" is staying the same as our "x", or in other words, it's keeping
up with us. If we're walking along and our partner is walking beside us,
he or she is remaining parallel or "simultaneous" with us. The same thing
can be said of someone who's twenty feet away but who is directly to our
left or to our right and is keeping up with us. But now look what happens
if we turn and face 30º toward the right . What was directly to our left is now
angled 30º back to our left, and what was directly to our right is now angled
30º ahead and to the right of us. In other words, these no longer
parallel or "simultaneous" with us. A good way to see this is to look at
the situation where you and I come to a fork in the road. I take the left
branch while you go straight ahead. The minute I turn left 60° and
start down the left branch, you're no longer going to be "simultaneous"
or parallel with me. In fact, it's going to look to me as though you're
falling behind me (Figure 3c). When I get over to the turning point, it
will look as though you've only gone half as far down *your *road
as I have. Now suppose I turn back until I'm facing the same direction
you're facing. What a shock! Now instead of appearing to be twice as far
down your road as you are. I'm only half as far down your road (Figure
3d)! That's a factor of four! And then when I turn and start back toward
the main road, it looks as shown in Figure 3e below. Finally, when I get
back to, and turn down the main road, everything looks the same as it did
when we were walking down the road together, except that now I'm now trailing
behind you. Since we parted company, I've made only half as much progress
down the main road as you have.

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; A_{x}, A_{y},
and A_{z} 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 t_{c},
remembering all the while that when we say " t_{c}",
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 = it_{c} (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,
x_{4 }is t, and x_{1},
x_{2}, and x_{3}
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.