Part I described the history of relativity and showed support for two
educated guesses. One is that the principle of relativity is true, that
there really is no way to determine or measure speed except relative to something
else. From this we get the idea that the physics of motion works the
same way for everyone no matter how they are moving (so long as they
aren't accelerating). Of course, we also need some idea of what the physics
of motion are, but despite the rocky history of the study of motion, most of this is intuitive, and we can show it in due course.
The other
educated guess is that the speed of light depends neither on the speed
of the light source nor on the speed of the observer.
These two premises don't seem so strange. Many philosophers argue that
the relativity of motion is selfevident, which certainly seems consistent
with experience. Sitting in an airplane in smooth skies feels like not moving. The independence of the
speed of light from the speeds of both the source and observer is the
only premise that Newton and Galileo didn't have. At least it doesn't seem to contradict any experience.
Following Einstein's theory of relativity, using these
premises and reason, results in conclusions that seem very strange
indeed. Here are a few, stated in informal form as they tend
to be filtered through popular media:
 When objects go fast, time slows down.
 Nothing can go faster than light.
 When objects go fast, they become more massive.
 When objects go fast, they contract in the direction of travel.
 Even though it's the Theory of Relavitiy, the speed of light seems absolute.
 Time is the fourth dimension.
 E = mc^{2}
Now, these do seem strange! Many people, seeing them, find them so bizarre
and counterintuitive that they conclude that Special Relativity must be
wrong. Part of the problem is how the ideas are popularly
presented, and so we may find that they are subtly or even profoundly
different from what Special Relativity actually says.
Remember also that if we assert that the implications of a
scientific theory are wrong, then logically it must mean that either the
reasoning in the theory is wrong or that one or more of the premises
is wrong.
The rest of this article shows the reasoning behind Einstein's
Special Theory of Relativity and how the conclusions follow from
the premises. Many of the examples are adaptations of the
thought experiments that Einstein himself used, but they are presented in an unusual way. The goal is to give an intuitive
feel for the reasoning behind Special Relativity. It relies on insight, vision, informal logic, common sense, and pattern matching as
much as possible, only using equations when there seems to be
no other choice. When used, they are not derived, although links to the derivations are provided. It is a nontraditional approach, an attempt at a shortcut to understanding. Those who prefer traditional presentations or wish to supplement their intuition can use those links.
First Things First
Special relativity is "special" because it only deals with the special case of constant motion in
a straight line. We'll imagine everything happening out in deep space in the middle of nowhere, far away from any gravity
that might change the results. It doesn't matter exactly where we do it; anywhere is fine as long as it is far enough away from stars and planets that their influence does not matter. A lightyear or two in any direction away from Earth there are lots of places that are more than good enough.
Einstein thought in terms of railroad cars, but we can use rocket ships. We'll use a simple square rocket ship, moving to the right. We'll concentrate on the interior of the ship, so we don't have to worry about the thickness of the hull. The engines are simple and always produce the same force. We won't have it carry any of its fuel so we don't have to worry about the change in mass as it is used; maybe there's another rocket ship going alongside with a big hose to give it fuel.
++
 
 
 
  >
 
 
K 
++
Kirk (K) is sitting in the rocket ship and will do experiments while we watch.
Of course, we're also sitting in a rocket ship watching Kirk's ship. We'll make it the same size and mass as Kirk's ship. We'll do most of the thought experiments with Kirk's ship coasting at constant speed in a straight line to
the right relative to us. Because of relativity, it would be as
accurate to say that we are coasting to the left relative to Kirk or that we're
both coasting at different speeds; only the relative motion is important.
This ship looks a bit like a coordinate system with Kirk sitting near the origin, where +X is toward the right and +Y is toward the top of the page. This coordinate system provides a frame of reference. It's called an "inertial" frame of reference because it is coasting from inertia.
Next, we need something to measure space. We'll use a stick, something like a meter stick, as long as the ship.
We'll also need a clock to measure time. To make sure that everything
that affects the stick also affects the clock, we'll make the clock from the stick. Since we're presuming that relative motion does not affect
the speed of light, we'll use light. At one end of the stick we'll mount a light emitter, a light detector,
and some electronics. At the other end of the stick we'll mount a mirror. The
electronics will send out a pulse, which will travel up to the mirror. That's a tick. Then the light pulse will bounce back from the mirror and be detected.
bounce back, and be detected. That's a tock. The round trip is a ticktock. Immediately, the electronics
will send out another pulse of light toward the mirror. Every time the electronics send a light pulse toward the mirror, it will also send out another signal (maybe another flash of light), so that we can see it at a distance.
We'll make the electronics fast
enough so that we can ignore its latency and think of a single pulse of
light bouncing back and forth. We could also make it like a laser so that the next pulse is stimulated automatically. Now we have a lightstick that can measure space
and time. One ticktock is the time it takes light to go twice
the length of the stick, which is also the length or breadth of the ship.
It also provides a natural way of measuring speed, so we don't have
to use meters per second or furlongs per fortnight. Light speed is
two sticks per ticktock.
Kirk can hold the stick vertically, transverse to the direction of travel:
++
^v 
^v 
^v 
^v  >
^v 
^v 
^vK 
++
or horizontally, along the axis of travel:
++
 
 
 
  >
K 
 > > > > > > >
< < < < < < < 
++
The symbols in the diagram suggest the direction the light pulse travels.
Now we have everything we need.
1. When objects go fast, time slows down.
We start our first experiment and tell Kirk to set up his lightstick vertically, across the breadth of the ship, and remain stationary relative to us. We ignore the rest of the ship and concentrate on the lightstick:
^v
^v
^v
^v
^v
^v
^v
So far, so good. A light pulse goes up; a light pulse goes down. Up and down, over and over again. We compare it with our own lightstick, and they're perfectly synchronized.
Then we tell Kirk to fly off to the left, speed up with his rockets, and then
coast by us with his lightstick going. We're not so sure about speeds, yet, but using trial and error we figure out how long he has to run the engines so that
we see the signals from his lightstick one stick apart. We'll call running the engines for this amount of time one impulse, and we'll put one button in the cockpit to deliver one impulse to the ship. We'll have his path always be at about the same distance from us, so that we don't have to worry about how long it takes for his signals to get to it. In our frame of reference, the pulse of Kirk's light stick traces this path:
^v ^v
^ v ^ v
^ v ^ v
^ v ^ v ^ etc.
^ v ^ v ^
^ v ^ v ^
^ v^ v^
The light pulse goes in a zigzag pattern. It still goes up and down, but also goes to the right. The path the light has to take is
longer than when Kirk was stationary relative to us, because it has to move
along a diagonal. We have presumed that
the speed of light is independent of the speed of the source, so it can't
go any faster simply because Kirk and the lightstick are moving fast past us. The distance that Kirk's light pulse travels and the distance that
our light pulse travels must be the same.
Our clock makes one ticktock, like this:
^v
^v
^v
^v
^v
^v
^v
The light in Kirk's clock has to travel the same distance, about this much:
^v
^ v
^ v
^
^
^
^
Our lightstick has already made one ticktock. His light stick, with the
path of light traveling a greater distance at the same speed, hasn't finished yet. We see Kirk's lightstick as going more slowly than ours.
Of course, Kirk doesn't see anything strange about his clock. In fact,
when he looks at our clock, he sees it as going more slowly than his, in exactly the same way.
All of his clocks have to work at the same rate in his frame of
reference. (Of course, they could be broken or not very good, but this can't depend on his speed relative to us.) Why? Well, if he had a pocketwatch that read a different time when he was moving, then he would be able to tell that he was moving by looking at
the difference between his pocketwatch and the lightstick. If the premise of relativity
is correct, that would be impossible. All of his clocks, provided they are all at the same location, also have to work the same when
viewed from our frame of reference. If his lightstick sent out a red flash as a signal
when it read 6:30, then Kirk could also send out a blue flash when his
pocketwatch read 6:30. The flashes would come from the same place at the
same time, and they both have to travel at the same speed of light, so we
would see one purple flash.
Around now, someone suggests another idea. Maybe Kirk's ship got skinnier, contracted up and down transverse to the direction of travel. Things are already strange, so why not? Maybe as our lightstick made one ticktock, so did his, but more like this:
^v
^ v
^ v
^ v
That is, although the path is still stretched out horizontally, it is contracted vertically enough to make the total length the same.
This cannot be. To test it, we attach a hoop to our ship, just bigger than Kirk's ship and ask him to fly through it. He's a pretty good pilot, so he manages the trick with ease. We put the same hoop on his ship and have him fly the hoop around us.
If he contracted transverse to his direction of travel, the hoop would contract as well, and it would break
on our hull. Yet assuming that Kirk is as skillful this time, it would
have to come out the same, because according to relativity, it is not possible
to say which one of us is the one who is moving.
We're left with the conclusion that Kirk's clocks slow down relative
to us when viewed from our frame of reference. Because all of his clocks
slow down, in some sense time must slow down in his frame relative
to ours. Of course, he sees time in our frame of reference slow down
relative to his as well. Otherwise, it wouldn't be relative.
This is called "time dilation."
Those who like formulas and equations may notice that the path of Kirk's light pulses can be seen as forming two
right triangles:
/\
/\
/  \
/  \
/  \
/  \
/++\
This observation plus a modified Pythagorean theorem and highschool algebra
are enough to derive the Lorentz Transformations, the bulk of the mathematics of Special Relativity.
2. Nothing can go faster than light.
Kirk can't go faster than light in his frame of reference. If he sends out
a light pulse forward, he will perceive it as going ahead of him at the speed of light. Since the speed of light does not depend on the speed of the observer, no matter how fast he tries to go, the light pulse will still be going away from him at the speed of light.
Can he go faster than the speed of light in our frame of reference? This wouldn't work, because then we'd see something different about the speed of light, which we have presumed is impossible. There's another way of looking at this. We have already seen what happens to Kirk's clock when he gives his engines one impulse:
^v
^ v
^ v
^ v
^ v
^ v
^ v
We tell him to give his engines two impulses:
^v
^ v
^ v
^ v
^ v
^ v
^ v
Three impulses:
^v
^ v
^ v
^ v
^ v
^ v
^ v
Four impulses:
^v
^ v
^ v
^ v
^ v
^ v
^ v
We might be tempted to say that Kirk is going double, triple, and
quadruple his speed in the subsequent diagrams, but we don't really know that. All we can see are the light flashes from his clock in space, and
we draw the triangles to trace out the path his light pulse makes in our frame of reference.
As Kirk tries to go faster and faster, the triangle becomes more and more stretched out. It can never stretch so far that it breaks. However fast Kirk tries to go, in our frame of reference, light still has to get there, and it must get there at the same speed. The shortest distance is a straight horizontal line. The zigzag pattern of his lightstick will always be longer than that. He can never catch up to the speed of light.
This property of the speed of light as a maximum speed follows
from the premise that the speed of light is constant. It isn't a
barrier, like the speed of sound. It isn't the maximum speed because light
happens to go at that speed. Rather, light goes at that speed because, as
will be discussed later, the maximum speed is the only one at which light
can go.
3. When objects go fast, they become more massive.
In our normal experience, masses resist being pushed. A car is more massive than a bicycle, and it's easier to push a bicycle than to push a car. Another way to look at this is to say that if we push a bicycle and a car with the same force, the bicycle will accelerate faster than the car.
When we push a massive object and accelerate it, it builds up momentum. Like velocity, momentum has an amount and a direction. Each impulse of Kirk's ship adds the same amount of momentum, as long as it is in the same direction.
Newton's three laws of motion describe how momentum works. The first law says that momentum remains the same without acceleration. The second law says that an applied force changes momentum proportional to the force and the time the force is applied, inversely proportional to the mass. The third law states that
momentum is conserved. The conservation of momentum is also supported by the work of Emmy Noether, who showed that every symmetry in physics required a conservation law, and
vice versa. We've already decided where exactly we
do our experiments doesn't matter. This goes by the name of "translational symmetry." According to Noether's Theorem, it means that momentum is conserved as well.
In the previous section, we saw what happened when Kirk gave two, three, and four impulses. We saw that it was appealing to think that he went two, three, and four times as fast. After all, every impulse that Kirk puts out, he sees himself increasing in speed by the same amount. He can't tell any difference between impulse number 1 and impulse number 4. He also sees the blast from his rocket going away from him in the opposite direction, precisely fast enough
and in the right direction to conserve momentum. So in his frame of reference, Kirk sees himself as speeding up by the same amount every impulse.
We see a different picture, but we also have to see momentum conserved. As has been shown, we can't see him going faster than light, so his impulses can't keep adding up speed. In our frame of
reference, the faster he goes, the less difference an impulse makes on
his speed. His inertia, his resistance to acceleration, is given by his
mass. Since his resistance to acceleration increases with speed, we can say in some sense that as he goes faster relative to us, his mass increases in our
frame of reference.
This is sometimes called the relativistic mass, as distinguished
from the rest mass, the mass of Kirk's ship when it isn't moving relative
to us.
Relativistic mass is related to rest mass by a factor called
γ or "gamma." This factor appears in many equations in relativity and
can be derived from the modified Pythagorean Theorem mentioned earlier. For our current
purposes, suffice it to say that it is 1 for a relative speed of zero, climbs
up slowly as the speed increases, and approaches infinity (actually, 1/0) as a limit
as the speed aproaches the speed of light. It starts to climb so slowly
that something has to be going more than 14% of the speed of light for
the relativistic numbers to vary as much as 1% from the Galilean numbers.
This is why Galilean/Newtonian motion works so well at low speed.
This shows another reason why an object cannot ever accelerate to the
speed of light in any frame of reference. If it ever did, its mass would become infinite.
It's also possible to avoid the notion of relativistic mass, as long
as we make sure that the factor of γ is in the right equations. Many physicists prefer to reserve the word "mass" only for the rest mass. There are many ways of formulating the equations, and which one to choose is a matter of convenience and preference.
4. When objects go fast, they contract in the direction of travel.
We give Kirk a second lightstick and ask him to hold it along the length
or the ship and go past us at his original speed. The lightsticks are synchronized, so they flash at the same time.
Since the speed of light is the same, the lengths of the paths must be the same,
like this:
^v
^ v
^ v
^ v
^ v
^ v
^ v
> > > > > > > > > >
< < <
1 3 2
Three events are marked on the diagram:
 A pulse of light is emitted from the lightstick at the stern of the ship.
 The pulse hits the mirror at the prow of the ship and bounces back.
 The pulse is detected at the stern of the ship.
Bringing back the pictures of the ship, we can see these three events as follows. The path of the light from the horizontal lightstick is still there
as a guide, and the number of the illustrated event is underlined. Only the part of the ship
we know about is drawn.
+







+
> > > > > > > > > >
< < <
1 3 2
+







+
> > > > > > > > > >
< < <
1 3 2
+







+
> > > > > > > > > >
< < <
1 3 2
At event 2, where is the stern of Kirk's ship in our frame of reference? If we assume that the ship does not contract in the direction of travel, we might guess this:
++
 
 
 
 
 
 
 
++
> > > > > > > > > >
< < <
1 3 2
This can't be right. The stern of the ship is as far from 3 as 3 is from 2. Since light has to come back from 2 to 3, and the stern cannot be going as fast as light, there will not be enough time for the stern to catch up. Instead, the picture must be more like this:
++
 
 
 
 
 
 
 
++
> > > > > > > > > >
< < <
1 3 2
Kirk's ship appears contracted in our frame of reference when it moves
past us. This might be expected from the description of Lorentz contraction in Part I.
Some find ASCII diagrams unconvincing and are encouraged to try this at home or the office. Get a piece of string or yarn already stretched to its limit. Loop the ends with a slipknot, and using three pushpins, make a triangle like the path of the light on Kirk's vertical clock. The string is the path of the light. The wider and shorter the triangle, the more obvious the effect will be. Moving only the center pushpin, like drawing an ellipse, pull the string out to the right. This will be the path of the light on Kirk's horizontal clock.
We also notice something else from the horizontal path of the light on Kirk's lightstick: in our frame of reference, his ticks take longer than his tocks! If Kirk sat between the ends of the light stick, he would see the ticks and tocks as taking the same time. If he turned another synchronized lightstick the other way, he would see the signals as simultaneous. We would not see them as simultaneous.
The problems with simultaneity suggest that our common sense notions of "now" don't quite work with relativity. Barnard's Star is a bit more than four light years away, so in one sense, we see it as it "was" a bit more than four years ago. I wonder what it's like now. However, "now" doesn't really work with relativity. An observer the same distance between us and Barnard's Star might get some relative notion of now, but any other observer would disagree.
5. Even though it's the Theory of Relavitiy, the speed of light seems absolute.
We are used to seeing time and space as related through velocity.
A moving object in our normal experience has a velocity that describes
how it moves through space relative to time.
Velocity is a slightly different idea from speed. 40 mph is a speed. Velocity combines speed with direction.
40 mph due North is a velocity. The speed component relates distance (one measurement of space) and time, while the direction component adds more information related to space. In an important sense,
the history of relativity described in Part I is the history of the
increasing respect for the importance of speed.
Velocity, space, and time are all interrelated in the case of an
object moving in a straight line at a constant speed. Given any two,
it is easy to calculate the third in common experience. The most
natural idea is that two of them are somehow basic or fundamental, and the third is a consequence of the two. Which two?
Space is pretty obvious; we can see it. On the other hand, we can ride a horse and watch
the trees pass. Still, maps are useful after a couple of years, so perhaps we can count on space. Time we can't really see, but it seems basic in
another way: it passes no matter what we do. It always seemed natural to define velocity and speed
in terms of space and time.
Aristotle didn't consider velocity very important. He thought the
cosmos was organized into concentric spheres, with the Earth at the
center. Objects had their natural places, rock in the earth, and smoke
in the air. An object moved away tried to return according to its
natural motion. Based on Aristotle's ideas, medieval thinkers thought
that invisible angels pushed the planets around the earth.
By the time of Galileo and then Newton, velocity was seen as more
important. Galileo showed that the acceleration, understood as the
change of velocity over time, of falling objects did not depend on
the mass of the objects.
Newton, with his first law of motion, showed that in the absence of
acceleration objects kept the same velocity. Velocity seemed much
more basic.
As Richard Feynman pointed out, the medieval theory of invisible angels
had to be modified; the angels only needed to push inward. The velocity around can be taken for granted; all that is needed is a change of the velocity toward another mass, such as the Sun.
During the 19^{th} century, one particular speed, the
speed of light, started to seem very important indeed. It is a speed
that does not depend on the velocity of either the source or the observer. The speed of light seems, in a sense, absolute.
One way of looking at relativity is, when in doubt, respect the
importance of speed, whether the constant speed of light or the relative
speeds of ordinary objects. There are many different ways of looking
at motion, but since speed seems so important, we try to use speed as a basic concept.
Just speed isn't enough; we need to understand where objects go as well as how fast. One number from speed cannot by itself explain
the complexity of motion. Fortunately, velocity also has
the idea of a direction in space. It also has a direction in time,
which we usually ignore. Yet 1 meter per second North is the
same as 2 meters per 2 seconds North. There are
many ways to represent the same velocity as long as the change in space
(called "translation") and the change in time is kept consistent.
Using these ideas, plus the idea of an observer to give a
reference in space and time, we can define
everything we can observe of motion around us, including relative space
and time. It seems different from using space and time as the absolute
bedrock of reality, but it works even better. It's the only way to look at motion that preserves the idea that the speed of light is constant.
Since speed is so important, we might intuitively expect it to work a bit differently
from the direction component of the velocity. We have already seen that it does.
It affects measurements of time, as in time dilation, and space, as in
Lorentz contraction. We had Kirk going to the right, but it would have
worked as well if he had moved to the left or top of the page
or at an angle. We had him move in the plane of the page so that
we didn't have to worry about the differences in distance from us.
If he had gone toward or away from us, we would have had to take
the differences in distance from us into account, which makes the
math much harder, but the effects are the same.
The speed of light is presumed constant, but speed is
not quite the same as velocity. We might expect that, although the speed
cannot change, the other component of velocity, the direction, can change. This is true.
The velocity of light can change, so long as it only changes direction and
not speed. Other attributes of the light, such as the color (related to the energy and momentum), can also
change. This is the source of the famous "red shift" of galaxies that
are moving away from us. (Or we're moving away from them. We never
can tell which; it's the basis of modern cosmology.)
Relativity also ties up one more loose end. Speed
affects both time and space, but we normally use
seemingly unconnected units for time and space, like seconds and meters
or hours and miles. It would be nice to have a natural way to measure
time and space in relation to each other. Fortunately, the speed of
light is always constant no matter how we move, so we can use
that speed to relate space and time. We have already used units where
the speed of light is 2 sticks per ticktock. Physicists prefer to use units where the speed of light is 1. This is very close to measuring
space in feet and time in nanoseconds.
6 . Time is the fourth dimension.
After so many seemingly bizarre implications, it's a relief to come across one that doesn't seem so odd. Since time and space are both affected by speed, it might be elegant to try to look as time and
space as different aspects of the same thing.
We are used to seeing time used as a dimension with ordinary graphs. Consider sliding a ring randomly back and forth over a meter stick for two seconds. The path of the ring can be represented in a single graph like this:
1 m ^
 ***
 ** * *
 * * *
* *
 * **
 * *
 **
0 m +>
0 s 2 s
In the previous diagram, space is represented vertically, and time is represented horizontally. When talking about relativity,
it's conventional to show time vertically and space horizontally.
Based on this, we can make a graph of space and time:
^


Our  Future




To the Left <+> To the Right




Our  Past


v
The top half is our future and holds everything that can happen
as a result of what we do. The bottom half is our past and holds
everything in the past that can affect us. The left and the right show
one direction in space. Of course, our space has three dimensions.
We could make a model with sticks balls for two dimensions plus time,
but we have to imagine a model with three dimensions plus
time. In any event, additional dimensions in space work the same as
the one drawn here.
We imagine ourselves sitting at the center, on the +. We always
draw the diagram like this, with us at the origin, showing time up and down and space left and right. Other
objects have paths in our diagram, but whatever we do, we use this diagram to look at the world. We always know that the + at the origin is where we are now. The other objects can construct
their own diagrams.
This is a spacetime diagram of Galilean and Newtonian physics.
It represents a world where speeds can be as fast as we like. Kirk, as well as any other object, traces a "world line" through this diagram. Objects
that do not move relative us trace vertical world lines.
Instantaneous light traces a horizontal world line. Objects moving between these speeds have world lines at an angle. As long as the object remains inertial and doesn't accelerate, the line is straight.
A diagram like this is fine, but is it real? All talk of dimensions is a human abstraction, but is it totally arbitrary, like
plotting sales of ice cream in Boston against the price of tea in China, or is there some deeper physical connection? We'll try to look at this as an analogue of an ordinary map. In a map, we can find a point and call that a location on the map. In this diagram, we'll extend the idea of location to include time and call it an event.
In the Galilean/Newtonian spacetime diagram, our future consists of every event
above the horizontal line. Our past consists of every event below the line.
The discovery that light had a finite speed poses a problem for this
diagram. An infinite speed is horizontal, but we shouldn't expect to be
able to draw a horizontal line for the speed of a real object. We need to modify the diagram by drawing two world lines for a pulse of light from where and when we are:
^
*  *
*  *
* Our  Future *
*  *
*  *
*  *
*  *
To the Left <+> To the Right
*  *
*  *
*  *
*  *
* Our  Past *
*  *
*  *
v
In one dimension of space, we can only send a pulse of light in one of two directions. In two dimensions of space, we can send a flash of light in any direction on a 360 degree circle. The possible world lines of light form a cone, so this is sometimes called a light cone. In three dimensions of space, it becomes hard to draw.
Our future is now the area in the top, and our past is the area to the bottom. What of the areas to the left and the right? They are neither
part of our past nor part of our future. Certainly there are events there, but we cannot see or affect them from here and now.
Using this diagram, we once again tell Kirk to pass us from left
to right. We make a diagram of his world line, a snapshot relative
the event where he is closest to us. It comes out like this:
^
*  / *
*  / *
*  / *
*  / *
*  / *
*  / *
* /*
<+>
*/ *
* /  *
* /  *
* /  *
* /  *
* /  *
* /  *
v
We might think, from looking at this diagram, that Kirk doesn't see
light as receding from him in all directions at the same speed. After all,
that / looks a lot closer to the * on right than it does to the * on the
left in our future. Remember, though, that he sees us relative to him
as we see him relative to us. He, too, can draw a spacetime diagram for
what he sees of us moving relative to him. Assuming that we agree on the
directions we call "left" and "right," his diagram looks like this:
^
* \  *
* \  *
* \  *
* \  *
* \  *
* \  *
*\ *
<+>
* \*
*  \ *
*  \ *
*  \ *
*  \ *
*  \ *
*  \ *
v
It seems almost as if we saw Kirk's vertical line rotated relative
to ours based on his speed relative to us, and he saw ours rotated in the opposite
direction for the same reason. It can't be a simple rotation, because we
can't rotate past the light cone: that would mean exceeding the speed
of light. Still, the idea of rotation is appealing.
Ordinary maps have some interesting properties. Take a map small enough that the curvature of the Earth does not matter much, and we
can pretend that it's flat. Say it's a map of Central Florida, and we
want to fly from Tampa to Orlando. We draw an arrow from Tampa to Orlando on the map. A friend comes by and looks over our shoulders. Even
though the point of view is different from ours, rotated and translated
a bit, our friend still sees the same arrow going from Tampa to Orlando. It may be at a different angle, but the distance between Tampa and Orlando is the same as we rotate the map. If many people viewed the map around a round
table, they would each see the arrow at a different angle, but they would also all agree on the distance.
It would be nice to have something like the distance that was the same in relativity for all observers, not only no matter what their position, but also no matter what their velocity. It turns out there is, and it's called the interval.
We're used to finding the distance between two points based on the Pythagorean Theorem. Consider this right triangle:
b
/
/ 
D /  D_{y}
/ 
/ 
a+
D_{x}
The distance D between events a and b is related to D_{x} and D_{y} by the Pythagorean formula D^{2} = D_{x}^{2} + D_{y}^{2}. Of course, it works in three dimensions as well. We want to put a difference of time, a D_{t}, in there somehow. Time already seems different from space. As Kirk's ship
moves through space, his ship seems to get shorter, while his times seem to get longer, so we might expect it to be treated differently in the formula. It turns out that the interval is related to the distance in space and time as I^{2} = D_{x}^{2}  D_{t}^{2} in one dimension
of space. The minus sign makes all the difference. This also works as well in three dimensions.
Another way of looking at this is that the minus sign comes from squaring D_{t}. If you square something and get a negative number, that means the original number was imaginary, some real number multiplied by i, the square root of 1.
Hermann Minkowski showed that the interval would not change using a coordinate system where the time axis was imaginary, scaled according to the speed of light. Furthermore, he showed that the Lorentz contraction and time dilation could be viewed as simple rotations in such a coordinate system. It turns out our intuition about rotation was correct.
Time can be considered the fourth dimension, in physically
meaningful way that preserves the properties of maps, but only if it time is
considered an imaginary number. This property also helps us understand why the speed of light is a constant. This spacetime diagram shows
a few events:
^
*  *
*  d
*  b *
*  *
*  *
*  * c
*  *
<ae>
*  *
*  *
*  *
*  *
*  *
*  *
*  *
v
If this were a map of space, the distances would all be positive
or zero, and there wouldn't be anything to distinguish those angles at
45 degrees. With the definition of an interval, things look more sensible. Going from a to b, there is a bigger difference in time than space, so the square of the interval is negative. We call this a timelike interval. Similarly, from a to c, there is a bigger difference in space than time, so square of the interval is positive. This is a spacelike interval. The interval from a to d is 0, and we call that a lightlike interval. The interval of 0 distinguishes those 45 degree angles.
Note that if there is no difference in time between events, as with events a and e, the interval is the same as the distance in space.
The problems with simultaneity describe earlier only apply to spacelike
intervals. For every spacelike interval between two events e_{1} and e_{2}, there are
some observers who will see the events at the same time, some who will see e_{1} before e_{2}, and some who will see e_{1} after e_{2}. Fortunately, for timelike intervals, all observers will agree on the same order.
Many physicists prefer not to consider time imaginary and use different operators that handle the minus signs. Furthermore, it works as well to consider space imaginary and time real. Even better, there is a kind of number called a quaternion that has a single real number and three imaginary numbers and handles the righthand rule by definition, but
quaternions never really caught on. Mathematics has many different ways to express the
same idea; which one to use is a matter of convention and preference.
7. E = mc^{2}
This is an equation that most people know, so we need to use equations to understand it.
Remember the use of γ or "gamma" in the description of momentum and relativistic mass. Gamma depends on relative speed. Considering only massive objects, the momentum works out to be p = γm_{0}v, where m_{0} is the rest mass of the object, and v is the velocity. This can also be written p = m_{r}v, where m_{r} is the relativistic mass, the same as γm_{0}.
Moving objects have something other than momentum: kinetic energy. This is the amount of energy needed to get an object moving at a certain speed.
For example, the amount the brakes of a car have to heat up to stop it is given by the kinetic energy of the car. There's another way to think about it. A force over a period of time imparts a change in momentum, while a force over a distance imparts a change in kinetic energy.
It's also possible to derive kinetic energy in relativity. It turns out to be E_{k} = (γ  1)m_{0}c^{2}
That " 1" seems out of place. This led Einstein to suggest that an object had some energy, not because it is moving, but simply because it is there and has mass. That energy would be the rest energy, given by the 1, and the kinetic energy would be the difference between that and the total energy when moving. This whole
mess of equations can be summarized by one simple one: E = m_{r}c^{2}, or E = mc^{2} if we don't bother to write the "_{r}". (Physicists seldom use this formulation, preferring to reserve m for the rest mass, m_{0}.)
If we have the right kind of insight, we might notice
a few things:
 Energy uses one (scalar) number. Momentum uses three numbers.
 Space is measured using three numbers. Time is measured using one number, even it it's imaginary.
 Energy is changed by force over distance (space). Momentum is changed by force over time.
 Space and time are related to each other, through Minkowski space
or thinking of time as imaginary.
 Maybe we can think of energy and momentum as related to
each other in a similar way.
This is true. We can put time and energy into a 4vector or quaternion, rotate it the same way according to relative motion, and get behavior similar to what we get from rotating intervals.
The only thing yet to understand is light. Light has energy.
If it did not, it wouldn't cost anything to run a light bulb. Light also has
momentum. At the speed of light, γ approaches 1/0.
We notice that γ is also multiplied by the rest mass for objects that have rest mass.
If the rest mass is 0, then multiplying it by γ at the limit of the speed of light gives 0/0. 0/0 is undefined. It's the same as asking what number, when multiplied by 0, gives 0. Any number works, so 0/0 doesn't help us pick the right one. We have to conclude that light or anything going at the speed of light must have a
rest mass of 0, and vice versa.
Fortunately, quantum thinking and experiment give good values for the momentum and energy of light, which depends on its color and the number of photons.
Summary
Einstein's Special Theory of Relativity, which results in some seemingly weird implications, results from some rather simple and straightforward reasoning based on two premises given in Part I. Although a lot of the mathematics and equations have been left out of this description, it is still possible to get a pretty good quantitative idea of all the important aspects of the Special Theory.