Subject: Relativity and FTL Travel--PART I   (suggested reading)
Date: 3 Apr 1996 18:20:12 GMT
Summary: Special Relativity

Posting-Frequency: bimonthly for r.a.s.tech, monthly for news.answers


=============================================================================
                          Relativity and FTL Travel

               by Jason W. Hinson (hinson@physics.purdue.edu)
-----------------------------------------------------------------------------

                         PART I: Special Relativity

=============================================================================
Edition: 4.1b 
Last Modified: February 29, 1996 
URL: http://bohr.physics.purdue.edu/~hinson/ftl/FTL_StartingPoint.html
FTP (text version): ftp://ftp.cc.umanitoba.ca/startrek/relativity/



     This is PART I of the "Relativity and FTL Travel" FAQ. It contains
basic information about the theory of special relativity. In the FTL
discussion ( of this FAQ), it is assumed that the reader understands the
concepts discussed below, while it is not assumed that the reader has read
parts II and III of this FAQ as they are "optional reading". Therefore, if
the reader is unfamiliar with special relativity in general (and especially
if the reader is unfamiliar with space-time diagrams) then he or she should
read this part of the FAQ to understand the FTL discussion in PART IV.
     For more information about this FAQ (including copyright information
and a table of contents for all parts of the FAQ), see the 'Relativity and
FTL Travel--Introduction to the FAQ' portion which should be distributed
with this document.


Contents of PART I:

1. An Introduction to Special Relativity
     1.1 Reasoning for its Existence
     1.2 Time Dilation and Length Contraction Effects
     1.3 Introducing Gamma
     1.4 Energy and Momentum Considerations
     1.5 Experimental Support for the Theory
2. Space-Time Diagrams
     2.1 What are Space-Time Diagrams?
     2.2 Basic Information About the Diagrams we will Construct
     2.3 Constructing One for a "Stationary" Observer
     2.4 Constructing One for a "Moving" Observer
     2.5 Interchanging "Stationary" and "Moving"
     2.6 "Future", "Past", and the Light Cone



1. An Introduction to Special Relativity

     The main goal of this introduction is to make relativity and its
consequences feasible to those who have not seen them before. It should also
reinforce such ideas for those who are already somewhat familiar with them.
This introduction will not completely follow the traditional way in which
relativity came about. It will begin with a pre-Einstein view of relativity.
It will then give some reasoning for why Einstein's view is plausible. This
will lead to a discussion of some of the consequences this theory has, odd
as they may seem. Finally, I want to mention some experimental evidence that
supports the theory.


1.1 Reasoning for its Existence

     The idea of relativity was around in Newton's day, but it was
incomplete. It involved figuring out what an observation would seem like to
one observer once you knew what it looked like to another observer who was
moving with respect to the first. This is called transforming from one frame
of reference to another. The transformation as it was done before Einstein
was not completely correct, but it seemed so in the realm of small speeds.
     Before we go on, I wanted to make a note about what we mean when we
talk about occurrences in some frame of reference. By "frame of reference"
we sort of mean the "point of view" of a particular observer. However, some
newcomers to relativity get mislead by this idea, and so I need to mention a
couple of points here. As we talk about when something occurs in some frame
of reference, we don't necessarily mean what the observer in that frame
would actually see. This is because the observer (for example) may see an
event today, but if the event occurred on some star ten light-years away in
this observer's frame, then we must say that the event actually occurred ten
years ago in this observer's frame of reference. I mention this because it
is sometimes tempting for newcomers of relativity to conclude that its odd
effects (like time dilation--which we will discuss later in this chapter)
are only illusions created by the fact that light from an event may reach
one observer before it reaches another. However, here I am clearly stating
that when we talk about when an event occurs in a frame of reference, we are
talking about when it _actually_occurred_ in that frame after all light
signal delays are taken into account.
     Similarly, if I say that event A and event B occur simultaneously in
some frame of reference, I do not mean that an observer in that frame would
necessarily see them occur at the same time, but rather that they actually
happened at the same time. For example, if two explosions really happened at
the same time in our frame of reference, and one occurred on the moon while
the other occurred on the sun, then we would see the one from on the moon
first (because it is closer). However, we must take into account the time it
takes the light to get to us. Then we would conclude that the explosions
actually happened at the same time in our frame. It will be important to
remember that this is the what we mean as we talk about when and where
things occur in different frames of reference (especially in Chapter 2).
     Now, here is an example of the Newtonian idea of transforming from one
frame of reference to another. Consider two observers, you and me, for
example. Let's say I am on a train (in some enclosed, see-through car--if
you want to visualize the situation) that passes you at 30 miles per hour. I
throw a ball in the direction the train is moving such that the ball moves
at 10 mph in MY point of view. Now consider a mark on the train tracks which
starts out ahead of the train. As I am holding the ball (before I throw it),
you will see it moving along at the same speed I am moving (the speed of the
train). When I throw the ball, you will see that the ball is able to reach
the mark on the track before I do. So to you, the ball is moving even faster
than I (and the train). Obviously, it seems as if the speed of the ball with
respect to you is just the speed of the ball with respect to me plus the
speed of me with respect to you. So, the speed of the ball with respect to
you = 10 mph + 30 mph = 40 mph. This was the first, simple idea for
transforming velocities from one frame of reference to another. It tries to
explain a bit about observations of one observer relative to another
observer's observations. In other words, this was part of the first concept
of relativity, but it is incomplete.

     Now I introduce you to an important postulate that leads to the concept
of relativity that we have today. I believe it will seem quite reasonable. I
state it as it appears in a physics book by Serway: "the laws of physics are
the same in every inertial frame of reference." (Note that by "inertial
frame of reference" we basically mean a frame of reference which is not
accelerating.) What the postulate means is that if two observers are moving
at two different, constant speeds, and one observes any physical laws for a
given situation in their frame of reference, then the other observer must
also agree that those physical laws apply to that situation.
     As an example, consider the conservation of momentum (which I will
briefly explain here). Say that there are two balls coming straight at one
another. They collide and go off in opposite directions. Conservation of
momentum says that if you add up the total momentum (which for small
velocities is given by the mass of the ball times its velocity) of both the
balls before the collision and after the collision, then the two should be
identical. Now, let this experiment be performed on a train where one ball
is moving in the same direction as the train, and the other is moving in the
opposite direction. An outside observer would say that the initial and final
velocities of the balls are one thing, while an observer on the train would
say they were something different. However, BOTH observers must agree that
the total momentum is conserved. One will say that momentum was conserved
because the momentums before _AND_ after the collision were both some
number, A; while the other will say that momentum was conserved because the
momentums before _AND_ after were both some other number, B. They will
disagree on what the actual numbers are, but they will agree that the law
holds. We should be able to apply this postulate to any physical law. If
not, (i.e., if physical laws were different for different frames of
reference) then we could change the laws of physics just by traveling in a
particular reference frame.
     A very interesting result occurs when you apply this postulate to the
laws of electrodynamics (the area of physics which deals with electricity
and magnetism). What one finds is that in order for the laws of
electrodynamics to be the same in all inertial reference frames, it must be
true that the speed of electromagnetic waves (such as light) is the same for
all inertial observers. Simply stating that may not make you think that
there is anything that interesting about it, but it has amazing
consequences. Consider letting a beam of light take the place of the ball in
our earlier example (the one where I was on a train throwing a ball, and you
were outside the train). If the train is moving at half the velocity of
light (c), and I say that the light beam is traveling at the speed c with
respect to me, wouldn't you expect the light beam to look as if it were
traveling one and a half that speed with respect to you? Well, because of
the postulate above, this is not the case, and the old ideas of relativity
in Newton's day fail to explain the situation. All observers must agree that
the speed of any light beam is c, regardless of their frame of reference.
Thus, even though I measure the speed of the light beam to be c with respect
to me, and you see me traveling past you and one half that speed, still, you
must also agree that the light is traveling at the speed c with respect to
you. This obviously seems odd at first glance, but time dilation and length
contraction are what account for the peculiarity.


1.2 Time Dilation and Length Contraction Effects

     Now, I give an example of how time dilation can help explain a
peculiarity that arises from the above concept. Again we consider a case
where I am on a train and you are outside the train, but let's give the
train a speed of 0.6 c with respect to you. (Note that c is generally used
to denote the speed of light which is 300,000,000 meters per second. We can
also write this as 3E8 m/s where "3E8" means 3 times 10 to the eighth). Now
I (on the train) shine a pulse of light (a small burst of light) so that (to
me) the light goes straight up, hits a mirror at the top of the train, and
bounces back to the floor of the train where some instrument detects it.
Now, in your point of view (outside the train), that pulse of light does not
travel straight up and straight down, but makes an up-side-down "V" shape
because of the motion of the train. Here is a diagram of what occurs in your
frame of reference::

  Diagram 1-1

                             /|\
                            / | \
                           /  |  \
    light pulse going up->/   |   \<-light pulse on return trip
                         /    |    \
                        /     |     \
                       /      |      \
                      /       |       \
                     ---------|---------->train's motion (v = 0.6 c)


     Let's say that the trip up takes 10 seconds in your point of view. The
distance the train travels during that time is given by its velocity (0.6 *
c) multiplied by that time of 10 seconds:

  (Eq 1:1)
   (0.6 * 3E8 m/s) * 10 s = 18E8 m

The distance that the light pulse travels on the way up (the slanted line to
the left) must be given by its speed with respect to you (which MUST be c
given our previous discussion) multiplied by the time of 10 seconds:

  (Eq 1:2)
   3E8 m/s * 10s = 30E8 m

Since the left side of the above figure is a right triangle, and we know the
length of its hypotenuse (the path of the light pulse) and one of its sides
(the distance the train traveled), we can now solve for the height of the
train using the Pythagorean theorem. That theorem states that for a right
triangle the length of the hypotenuse squared is equal to the length of one
of the sides squared plus the length of the other side squared. We can thus
write the following (note "^2" means "raised to the power of two"):

  (Eq 1:3)
    Height^2 + (18E8 m)^2 =  (30E8 m)^2
   so
    Height = [(30E8 m)^2 - (18E8 m)^2]^0.5 = 24E8 m

(It is a tall train because we said that it took the light 10 seconds to
reach the top, but this IS just a thought experiment.) Now we consider my
frame of reference (on the train). The light MUST travel at 3E8 m/s for him
also, and the height of the train doesn't change because relativity doesn't
affect lengths perpendicular to the direction of motion. Therefore, we can
calculate how long it takes for the light to reach the top of the train in
my frame of reference. That is given by the distance (the height of the
train) divided by the speed of the light pulse (c):

  (Eq 1:4)
   24E8 m / 3E8 (m/s) = 8 seconds,

and there you have it. To you the event takes 10 seconds, while according to
me it must take only 8 seconds. We measure time in different ways.
     You see, to you the distance the light travels is longer than the
height of the train (see the diagram). So, the only way I (on the train)
could say that the light traveled the height of the train while you say that
the _SAME_ light travels a longer distance is if we either (1) have
different ideas for the speed of the light because we are in different
frames of reference, or (2) we have different ideas for the time it takes
the light to travel because we are in different frames of reference. Now, in
Newton's days, they would believe that the former were true. The light would
be no different from, say, a ball, and observers in different frames of
reference can observer different speeds for a ball (remember our first
"train" example in this introduction). However, with the principles of
Einstein's relativity, we find that the speed of light is unlike other
speeds in that it must always be the same regardless of your frame of
reference. Thus, the second explanation must be the case, and in your frame
of reference, my clock (on the fast moving train) is going slower than
yours.
     As I mentioned in the last part of the previous section, length
contraction is another consequence of relativity. Consider the same two
travelers in our previous example, and let each of them hold a meter stick
horizontally (so that the length of the stick is oriented in the direction
of motion of the train). To the outside observer (you), the meter stick of
the traveler on the train (me) will look as if it is shorter than a meter.
One can actually derive this given the time dilation effect (which we have
already derived), but I wont go through that explanation for the sake of
time.
     Now, DON'T BE FOOLED! One of the first concepts which can get into the
mind of a newcomer to relativity involves a statement like, "if you are
moving, your clock slows down." However, the question of which clock is
REALLY running slowly (yours or mine) has NO absolute answer! It is
important to remember that all motion is relative. That is, there is no such
thing as absolute motion. You cannot say that it is the train that is
absolutely moving and that you are the one who is actually sitting still.
     Have you ever had the experience of sitting in a car, noticing that you
seemed to be moving backwards, and then realizing that it was the car beside
which was "actually" moving forward. Well, the only reason you say that
"actually" the other car was moving forward is because you are considering
the ground to be stationary, and it was the other car who was moving with
respect to the ground rather than the other car. Before you looked at the
ground (or surrounding scenery) you had no way of knowing which of you was
"really" moving. Now, if you did this in space (with space ships instead of
cars), and there were no other objects around to reference to, then what
would be the difference in saying that your space ship was the one that was
moving or saying that it was the other space ship that was moving? As long
as neither of you is undergoing an acceleration (which would mean you were
not in an inertial frame of reference) there is no absolute answer to the
question of which one of you is moving and which of you is sitting still.
You are moving with respect to him, but then again, he is moving with
respect to you. All motion is relative, and all inertial frames are
equivalent.
     So what does that mean for us in this "train" example. Well, from my
point of view on the train, I am the one who is sitting still, while I zip
past him at 0.6 c. Since I can apply the concepts of relativity just as you
can (that's the postulate of relativity--all physical laws are the same for
all inertial observers), and in my frame of reference you am the one who is
in motion, that means that I will think that it is YOUR clock that is
running slowly and that YOUR meter sticks are length contracted.
     So, there is NO absolute answer to the question of which of our clocks
is REALLY running slower than the other and which of our meter sticks is
REALLY length contracted smaller than the other. The only way to answer this
question is relative to whose frame of reference you are considering. In my
frame of reference your clock is running slower than mine, but in your frame
of reference my clock is running slower than yours. This lends itself over
to what seem to be paradoxes such as "the twin paradox" (doesn't it seem
like a paradox that we each believe that the other persons clock is running
slower than our own?). Understanding these paradoxes can be a key to really
grasping some major concepts of special relativity. The explanation of these
paradoxes will be given for the interested reader in PART II of this FAQ.


1.3 Introducing Gamma

     Now, the closer one gets to the speed of light with respect to an
observer, the slower ones clock ticks and the shorter ones meter stick will
be in the frame of reference of that observer. The factor which determines
the amount of length contraction and time dilation is called gamma.
     Gamma for an object moving with speed v in your frame of reference is
defined as

  (Eq 1:5)
   gamma = 1 / (1 - v^2/c^2)^0.5

For our train (for which v = 0.6 c in your frame of reference), gamma is
1.25 in your frame. Lengths will be contracted and time dilated (as seen by
you--the outside observer) by a factor of 1/gamma = 0.8. That is what we
demonstrated in or example by showing that the difference in measured times
was 10 seconds for you (off the train) and 8 seconds for me (on the train)
in your point of view. Gamma is obviously an important number in relativity,
and it will appear as we discuss other consequences of the theory (including
the effects of special relativity on energy and momentum considerations).


1.4 Energy and Momentum Considerations

     Another consequence of relativity is a relationship between mass,
energy, and momentum. Note that velocity involves the question of how far
you go and how long it takes. Obviously, if relativity affects the way
observers view lengths and times relative to one another, one could expect
that any Newtonian concepts involving velocity might need to be re-thought.
For example, because of relativity we can no longer simply add velocities to
transform from one frame to another as we did with the ball and the train
earlier. (However, for small velocities like we see every day, the
differences which comes in because of relativity are much to small for us to
notice).
     Further, consider momentum (which in Newtonian mechanics is defined as
mass times velocity). With relativity, this value is no longer conserved in
different reference frames when an interaction takes place. The quantity
that is conserved is relativistic momentum which is defined as

  (Eq 1:6)
   p  =  gamma * m * v

where gamma is defined in the previous section.
     By further considering conservation of momentum and energy as viewed
from two frames of reference, one can find that the following equation must
be true for the total energy of an unbound particle:

  (Eq 1:7)
   E^2  =  p^2 * c^2  +  m^2 * c^4

where E is energy, m is mass, and p is the relativistic momentum as defined
above.
     Now, by manipulating the above equations, one can find another way to
express the total energy as

  (Eq 1:8)
   E  =  gamma * m * c^2

Notice that even when an object is at rest (gamma = 1) it still has an
energy of

  (Eq 1:9)
   E  =  m * c^2

Many of you have seen something like this stated in context with the theory
of relativity ("E equals m c squared"). It says that mass itself contains
energy.
     It is important to note that the mass in the above equations has a
special definition which we will now discuss. As a traveler approaches the
speed of light with respect to an observer, the observer sees the "mass" of
the traveler increase. (By "mass", we mean the property that indicates (1)
how much force is needed to create a certain acceleration and (2) how much
gravitational pull you will feel from that object in Newtonian gravitation).
However, the mass in the above equations is defined as the mass measured in
the rest frame of the object. That mass is always the same. We sometimes
define the properties of mass as they are seen by an outside observer (I
will call these properties "observed mass") as being gamma * m. Thus, since
in Equation 1:8 we wrote E = gamma * m * c^2, we could also write the total
energy as

  (Eq 1:10)
   E  =  (observed mass) * c^2

That observed mass approaches infinity as the object approaches the speed of
light with respect to the observer (because gamma approaches infinity in
that way).
     Further, note the case where the rest mass of an object is zero (such
is the case for a photon). Given the equation for the energy in the form of
Equation 1:8 (E = gamma*m*c^2), one might at first glance think that the
energy was zero when m = 0. However, note that massless particles like the
photon travel at the speed of light. Since gamma goes to infinity as the
velocity of an object goes to c, the equation E = gamma*m*c^2 involves one
part which goes to zero (m) and one part which goes to infinity (gamma).
Thus, it is not obvious what the energy would be. However, if we use the
energy equation in the form of Equation 1:7 (E^2 = p^2*c^2 + m^2*c^4), then
we can see that when m = 0 then the energy is given by E = p*c. Now, a
photon has a momentum (it can "slam" into particles and change their motion,
for example) which is determined by its wavelength. It thus has an energy
given by p*c, even though it has no rest mass.


1.5 Experimental Support for the Theory

     These amazing consequences of relativity do have experimental
foundations. One of these involves the creation of particles called muons by
cosmic rays (from the sun) in the upper atmosphere. In the rest frame of a
muon, its life time is only about 2.2E-6 seconds. Even if the muon could
travel at the speed of light, it could still go only about 660 meters during
its life time. Because of that, they should not be able to reach the surface
of the Earth. However, it has been observed that large numbers of them do
reach the Earth. From our point of view, time in the muon's frame of
reference is running slowly, since the muons are traveling very fast with
respect to us. So the 2.2E-6 seconds are slowed down, and the muon has
enough time to reach the earth.
     We must also be able to explain the result from the muon's frame of
reference. In its point of view, it does have only 2.2E-6 seconds to live.
However, the muon would say that it is the Earth which is speeding toward
the muon. Therefore, the distance from the top of the atmosphere to the
Earth's surface is length contracted. Thus, from the muon's point of view,
it lives a very small amount of time, but it doesn't have that far to go.
This is an interesting point of Relativity--the physical results (e.g. the
muon reaches the Earth's surface) must be true for all observers; however,
the explanation as to how it came about can be different from different
frames of reference.
     Another verification of special relativity is found all the time in
particle physics. The results of having a particle strike a target can be
understood only if one takes the total energy of the particle to be E =
Gamma * m * c^2, which was predicted by relativity.
     These are only a few examples that give credibility to the theory of
relativity. Its predictions have turned out to be true in many cases, and to
date, no evidence exists that would tend to undermine the theory in the
areas where it applies.

     In the above discussion of relativity's effects on space and time we
have specifically mentioned length contraction and time dilation. However,
there is a little more to it than that, and the next section attempts to
explain this to some extent.





2. Space-Time Diagrams

     In this section we examine certain constructions known as space-time
diagrams. After a short look at why we need to discuss these diagrams, I
will explain what they are and what purpose they serve. Next we will
construct a space-time diagram for a particular observer. Then, using the
same techniques, we will construct a second diagram to represent the
coordinate system for a second observer who is moving with respect to the
first observer. This second diagram will show the second observer's frame of
reference with respect to the first observer; however, we will also switch
around the diagram to show what the first observer's frame of reference
looks like with respect to the second observer. Finally, we will compare the
concepts these two observers have of future and past, which will make it
necessary to first discuss a diagram known as a light cone.


2.1 What are Space-Time Diagrams?

     In the previous section we talked about the major consequences of
special relativity, but now I want to concentrate more specifically on how
relativity causes a transformation of space and time. Relativity causes a
little more than can be understood by simple notions of length contraction
and time dilation. It actually results in two different observers having two
different space-time coordinate systems. The coordinates transform from one
frame to the other through what is known as a Lorentz Transformation.
Without getting deep into the math, much can be understood about such
transforms by considering space-time diagrams.
     A space-time diagram gives us a means of representing events which
occur at different locations and at different times. Every event is
portrayed as a point somewhere on the space-time diagram. Because of
relativity, different observers which are moving relative to one another
will have different coordinates for any given event. However, with
space-time diagrams, we can picture these different coordinate systems on
the same diagram, and this allows us to understand how they are related to
one another.


2.2 Basic Information About the Diagrams we will Construct

     In the diagrams we will be using, only one direction in space will be
considered--the x direction. So, the space-time diagram consists of a
coordinate system with one axis to represent space (the x direction) and
another to represent time. Where these two principal axes meet is the
origin. This is simply a point in space that we have defined as x = 0 and a
moment in time that we have defined as t = 0. In Diagram 2-1(below) I have
drawn these two axes and marked the origin with an o.
     For certain reasons we want to define the units that we will use for
distances and times in a very specific way. Let's define the unit for time
to be the second. This means that moving one unit up the time axis will
represent waiting one second of time. We then want to define the unit for
distance to be a light second (the distance light travels in one second). So
if you move one unit to the right on the x axis, you will be looking at a
point in space that is one light second away from your previous location. In
Diagram 2-1, I have marked the locations of the different space and time
units (Note: In my diagrams, I am using four spaces to represent one unit
along the x axis and two character heights to represent one unit on the time
axis).
     With these units, it is interesting to note how a beam of light is
represented in our diagram. Consider a beam of light leaving the origin and
traveling to the right. One second later, it will have traveled one light
second away. Two seconds after it leaves it will have traveled two light
seconds away, and so on. So a beam of light will always make a line at an
angle of 45 degrees to the x and t axes. I have drawn such a light beam in
Diagram 2-2.

  Diagram 2-1                      Diagram 2-2

              t                               t
              ^                               ^
              |                               |       light
              +                               +       /
              |                               |     /
              +                               +   /
              |                               | /
     -+---+---o---+---+---> x        -+---+---o---+---+-> x
              |                               |
              +                               +
              |                               |
              +                               +
              |                               |




2.3 Constructing One for a "Stationary" Observer

     At this point, we want to decide exactly how to represent events on
this coordinate system for a particular observer. First note that it is
convenient to think of any particular space-time diagram as being
specifically drawn for one particular observer. For Diagram 2-1, that
particular observer (let's call him the O observer) is the one whose
coordinate system has the vertical time axis and horizontal space axis shown
in that diagram. Now, other frames of reference (which don't follow those
axes) can also be represented on this same diagram (as we will see).
However, because we are used to seeing coordinate systems with horizontal
and vertical axes, it is natural to think of this space-time diagram as
being drawn specifically with the O observer in mind. In fact, we could say
that in this space-time diagram, the O observer is considered to be "at
rest".
     So if the O observer starts at the origin, then one second later he is
still at x = 0 (because he isn't moving in this coordinate system). Two
seconds later he is still at x = 0, etc. If we look at the diagram, we see
that this means he is always on the time axis in our representation.
Similarly, any lines drawn parallel to the t axis (in this case, vertical
lines) will represent lines of constant position. If a second observer is
not moving with respect to the first, and this second observer starts at a
position two light seconds away to the right of the first, then as time
progresses he will stay on the vertical line that runs through x = 2.
     Next we want to figure out how to represent lines of constant time. To
do this, we should first find a point on our diagram that represents an
event which occurs at the same time as the origin (t = 0). To do this we
will use a method that Einstein used. First we choose a point on the t axis
which occurred prior to t = 0. Let's use an example where this point occurs
at t = -3 seconds. At that time we send out a beam of light in the positive
x direction. If the beam bounces off of a distant mirror at t = 0 and heads
back toward the t axis, then it will come back to the us at t = 3 seconds
(because we know that the light must travel at the same speed going as it
does coming back). So, if we send out a beam at t = -3 seconds and it
returns at t = 3 seconds, then the event of it bouncing off the mirror
occurred simultaneously with the time t = 0 at the origin.
     To use this in our diagram, we first pick two points on the t axis that
mark t = -3 and t = 3 (let's call these points A and B respectively). We
then draw one light beam leaving from A in the positive x direction. Next we
draw a light beam coming to B in the negative x direction. Where these two
beams meet (let's call this point C) marks the point where the original beam
bounces off the mirror. Thus the event marked by C is simultaneous with t =
0 (the origin). A line drawn through C and o will thus be a line of constant
time. All lines parallel to this line will also be lines of constant time.
So any two events that lie along one of these lines occur at the same time
in this frame of reference. I have drawn this procedure in Diagram 2-3, and
you can see that the x axis is the line through both o and C which is a line
of simultaneity (as one might have expected).
     Now, by constructing a set of simultaneous time lines and constant
position lines we will have a grid on our space-time diagram. Any event has
a specific location on the grid which tells where and when it occurs. In
Diagram 2-4 I have drawn one of these grids and marked an event (@) that
occurred 3 light seconds away to the left of the origin (x = -3) and 1
second before the origin (t = -1).

  Diagram 2-3                   Diagram 2-4

              t                                  t
              |                      |   |   |   |   |   |
              B                   ---+---+---+---+---+---+---
              | \                    |   |   |   |   |   |
              +   \               ---+---+---+---+---+---+---
              |     \                |   |   |   |   |   |
              +       \           ---+---+---+---o---+---+--- x
              |         \            |   |   |   |   |   |
     -+---+---o---+---+---C- x    ---@---+---+---+---+---+---
              |         /            |   |   |   |   |   |
              +       /           ---+---+---+---+---+---+---
              |     /                |   |   |   |   |   |
              +   /
              | /
              A
              |




2.4 Constructing One for a "Moving" Observer

     Now comes an important addition to our discussion of space-time
diagrams. The coordinate system we have drawn will work fine for any
observer who is not moving with respect to the O observer. Now we want to
construct a coordinate system for an observer who IS traveling with respect
to the O observer. The trajectories of two such observers have been drawn in
Diagram 2-5 and Diagram 2-6. Notice that in our discussion we will usually
consider moving observers who pass by the O observer at the time t = 0 and
at the position x = 0. Thus, the origin will mark the event "the two
observers pass by one another".
     Now, the traveler in Diagram 2-5 is moving slower than the one in
Diagram 2-6. You can see this because in a given amount of time, the Diagram
2-6 traveler has moved further away from the time axis than the Diagram 2-5
traveler. So the faster a traveler moves, the more slanted this line
becomes.

  Diagram 2-5                      Diagram 2-6

              t                               t
              |  /                            |    /
              +                               +   /
              | /                             |  /
              +                               + /
              |`                              |/
     -+---+---o---+---+--- x         -+---+---o---+---+- x
             ,|                              /|
              +                             / +
            / |                            /  |
              +                           /   +
           /  |                          /    |


     What does this line actually represent? Well, remember that the line
marks the position of our observer at different times on our diagram. But,
also, consider an object sitting right next to our moving observer. If a few
seconds later the object is still sitting right next to him (practically on
that line), then, in his point of view, the object has not moved. So, the
line is a line of constant position for the moving observer. Nothing on that
line is moving with respect to him. But that means that this line represents
the same thing for the moving observer as the t axis represented for the O
observer; and in fact, this line becomes the moving observer's new time
axis. We will mark this new time axis as t' (t-prime). All lines parallel to
this slanted line will also be lines of constant position for our moving
observer.
     Now, just as we did for the O observer, we want to construct lines of
constant time for our traveling observer. To do this, we will use the same
method that we did for the O observer. The moving observer will send out a
light beam at some time t' = -T, and the beam will bounce off some mirror so
that it returns to him at time t' = +T. Now remember, light travels at the
same speed in any direction for ALL observers, so our traveling observer
must conclude that the light beam took the same amount of time traveling out
as it did coming back in his frame of reference. If in his frame the light
left at t' = -T and returned at t' = +T, then the point at which the beam
bounces off the mirror must have occurred simultaneously with the origin,
where t' = t = 0, in the frame of reference of our moving observer.
     There is a very important point to note here. What if instead of light,
we wanted to throw a ball at 0.5 c, have it bounce off some wall, and then
return at the same speed (0.5 c). The problem with this is that to find a
line of constant time for the moving observer, the ball must travel at 0.5 c
_both_ways_ in the reference frame of the MOVING observer. But we have not
yet defined the coordinate system for the moving observer, so we do not know
what a ball moving at 0.5 c with respect to him will look like on our
diagram. However, because of relativity, we know that the speed of light
itself CANNOT change from one observer to the next. In that case, a beam of
light traveling at c in the frame of the moving observer will also be
traveling at c for the O observer. So, a line which makes a 45 degree angle
with respect to the x and t axes will ALWAYS represent a beam of light
traveling at speed c for ANY observer in ANY frame of reference.
     In Diagram 2-7, I have labeled a point A' on the t' axes which occurs
some amount of time before t' = 0 and a point B' which occurs the same
amount of time after t' = 0. I then drew the two light rays (remember, these
are "45 degree angle" lines) as before--one leaving from A and going to the
right, and one moving to the left and coming in to B. I then found the point
where they would meet (C') which marks the point where the ray from A' would
have had to bounce in order to get back to the moving observer at B'. Thus,
C' and o occur at the same time in the frame of the moving observer. Notice
that for the O observer, C' is above his line of simultaneity at o (the x
axis). So while the moving O' observer says that C' occurs when the two
observers pass (at the origin), the O observer says that C' occurs after the
two observers have passed by one another. We will further discuss this
difference in the concepts of future and past in Section 2.6.
     In Diagram 2-8, I have drawn a line passing through C' and o. This line
represents the same thing for our moving observer as the x axis did for the
O observer. So we label this line x'.

  Diagram 2-7                     Diagram 2-8

                 t                               t     t'
                 |    /                          |    /
                 +   B'                          +   /
                 |  /  \                         |  /       __--x'
                 + /     C'                      + /   __C'-
                 |/    /                         |/__--
    -+---+---+---o---/---+---+- x   -+---+---+-__o---+---+---+- x
                /| /                    *  __-- /|
               / /                     __--    / +
              // |                   --       /  |
             A'  +                           /   +
            /    |                          /    |


     From the geometry involved in finding this x' axis, we can state a
general rule for finding the x' axis for any moving observer. First recall
that the t' axis is the line that represents the moving observer's position
on the space-time diagram. The faster O' is moving with respect to O, the
greater the angle between the t axis and the t' axis. So the t' axis is
rotated away from the t axis at some angle (either clockwise or
counterclockwise, depending on the direction O' is going--right or left).
The x' axis is then a line rotated at the same angle away from the x axis,
but in the _opposite_ direction (counterclockwise or clockwise).
     Now, x' is a line of constant time for O', and any line drawn parallel
to x' is also a line of constant time. Such lines, along with the lines of
constant position, form a grid of the space-time coordinates for the O'
observer. I have tried my best to draw such a grid in Diagram 2-9. If you
squint your eyes while looking at that diagram, you can see the skewed
squares of the coordinate grid. You can see that if you pick a point on the
space-time diagram, the two observers with their two different coordinate
systems will disagree on when and where the event occurs.

  Diagram 2-9

                       t'
    +-----------------/-------+
    | /  /_-/""/  /__/-"/  / _|
    |/-"/  / _/--/" /  /_-/""/|
    |  /_-/""/  /__/-"/  / _/-->x'
    |"/  / _/--/" /  /_-/""/  |
    |/_-/""/  /__o-"/  / _/--/|
    |  / _/--/" /  /_-/""/  /_|
    |-/""/  /__/-"/  / _/--/" |
    |/ _/--/" /  /_-/""/  /__/|
    |""/  /__/-"/  / _/--/" / |
    +-------------------------+

     As a final note about this procedure, think back to what really made
these two coordinate systems look differently. Well, the only thing we
assumed in creating these systems is that the speed of light is the same for
all observers. In fact, this is the only reason that the two coordinate
systems look the way they do.


2.5 Interchanging "Stationary" and "Moving"

     In our understanding of space-time diagrams, I also want to incorporate
the idea that all reference frames that move with a constant velocity are
considered equivalent and that all motion is relative. By this I mean that O
was considered as the stationary observer only because we defined him as
such. Remember? We said that this it is natural to think of the diagram
being drawn specifically for the observer whose coordinate system is drawn
with vertical and horizontal axes. We then said that we can think of that
observer (O) to be considered "at rest" in this diagram. Then, when I called
O' the moving observer, I meant that he was moving with respect to O.
     However, we should just as easily be able to define O' as the
stationary observer. Then, to him, O is moving away from him to the left.
Then, we should be able to draw the t' and x' axes as the vertical and
horizontal lines, while the t and x axes become the rotated lines. I have
done this in Diagram 2-10. By examining this diagram, you can confirm that
it makes sense to you in light of our discussion thus far. (For example,
picture grabbing the x' and t' axes in Diagram 2-8 and rotating them around
the origin until they are horizontal and vertical lines. If x and t follow
your rotation, then you can see how they would end up as they are drawn in
2-10.)

  Diagram 2-10

         t     t'
          \    |
           \   +
            \  |
             \ +
              \|
    ---+---+---o-__+---+--- x'
               |   ""--__
               +         ""--x
               |
               +
               |




2.6 "Future", "Past", and the Light Cone

     For the later FTL discussions, it will be important to understand the
way different observers have different notions concerning the future and the
past. This difference comes about because of the way the different
coordinate systems of the two observers compare to one another.
     First, let me note that with what we have discussed we cannot make a
complete comparison of the two observers' coordinate systems. You see, we
have not seen how the lengths which represents one unit of space and time in
the reference frame of O compare with the lengths representing the same
units in O'. This will be covered in the 'More About Relativity' Part (which
is "optional" for those of you just interested in the faster than light
discussions). We can, however, compare the observers' notions of future and
past.
     Back on Diagram 2-8, in addition to the O and O' space and time axes, I
also marked a particular event with a star, "*". Recall that for O, any
event on the x axis occurs at the same time as the origin (the place and
time that the two observers pass each other). Since the marked event appears
under the x axis, then O must believe that the event occurs before the
observers pass each other. Also recall that for O', those events on the x'
axis are the ones that occur at the same time the observers are passing.
Since the marked event appears above the x' axis, O' must believe that the
event occurs after the observers pass each other. So, when and where events
occur with respect to other events is completely dependent on ones frame of
reference. Note that this is not a question of when the events are seen to
happen in different frames of reference, but it is a question of when they
_really_do_ happen in the different frames (recall our discussion of
reference frames in Section 1.1 ). So, how can this make sense? How can one
event be both in the future for one observer and in the past for another
observer. To better understand why this situation doesn't contradict itself,
we need to look at one other construction typically shown on a space-time
diagram.
     In Diagram 2-11 I have drawn two light rays, one which travels in the
+x direction and another which travels in the -x direction. At some negative
time, the two rays were headed towards x = 0. At t = 0, the two rays finally
get to x = 0 and cross paths (at the origin). As time progresses, the two
then speed away from x = 0. This construction is known as a light cone.

  Diagram 2-11

                t
                ^
                |         light
        \       +       /
          \   inside  /
            \   +   /
     outside  \ | /  outside
     ---+---+---o---+---+---> x
              / | \
            /   +   \
          /   inside  \
        /       +       \
                |


     A light cone divides a space-time diagram into two major sections: the
area inside the cone and the area outside the cone (as shown in 2-11 ). (Let
me mention here that I will specifically call the cone I have drawn "a light
cone centered at the origin", because that is where the two beams meet.)
Now, consider an observer who has been sitting at x = 0 (like our O
observer) and is receiving and sending signals at the moment marked by t =
0. Obviously, if he sends out a signal, it proceeds away from x = 0 into the
future, and the event marked by someone receiving the signal would be above
the x axis (in his future). Also, if he is receiving signals at t = 0, then
the event marked by someone sending the signal would have to be under the x
axis (in his past). Now, if it is impossible for anything to travel faster
than light, then the only events occurring before t = 0 that the observer
can know about at the moment are those that are inside the light cone. Also,
the only future events (those occurring after t = 0) that he can influence
are, again, those inside the light cone.
     Now, one of the most important things to note about a light cone is
that its position is the same for all observers (because the speed of light
is the same for all observers). For example, picture taking the skewed
coordinate system of the moving observer and superimposing it on the light
cone I have drawn (note: a diagram which does--in part--show this view will
be given in the 'More About Relativity' Part ). If you were to move one unit
"down" the x' axis (a distance that represents one light second for our
moving observer), and you move one unit "up" the t' axes (one second for our
moving observer), then the point you end up at should lie somewhere on the
light cone. In effect, a light cone will always look the same on our diagram
regardless of which observer is drawing the cone.
     This fact has great importance. Consider different observers who are
all passing by one another at some point in space and time. In general, they
will disagree with each other on when and where different events have and
will occur. However, if you draw a light cone centered at the point where
they are passing each other, then they will ALL agree as to which events are
inside the light cone and which events are outside the light cone. So,
regardless of the coordinate system for any of these observers, the
following facts remain: The only events that any of these observers can ever
hope to influence are those which lie inside the upper half of the light
cone. Similarly, the only events that any of these observers can know about
as they pass by one another are those which lie inside the lower half of the
cone. Since the light cone is the same for all the observers, then they all
agree as to which events can be known about as they are passing and which
can be influenced at some point after they pass.
     Now let's apply this to the observers and event in Diagram 2-8. As you
can see, the marked event is indeed outside the light cone. Because of this,
even though the event is in one observer's past at the time in question (t =
t'= 0), he cannot know about the event at the time. Also, even though the
event is in the other observer's future at the time, he can never have an
effect on the event after. In essence, the event (when it happens, where it
happens, how it happens, etc.) is of absolutely no consequence for these two
observers at the time in question. As it turns out, anytime you find two
observers who are passing by one another and an event which one observer's
coordinate system places in the past and the other observer's coordinate
system places in the future, then the event will always be outside of the
light cone centered at the point where the observers pass.
     But doesn't this relativistic picture of the universe still present an
ambiguity in the concepts of past and future? Perhaps philosophically it
does, but not physically. You see, the only time you can see these
ambiguities is when you are looking at the whole space-time picture at once.
If you were one of the observers who is actually viewing space and time,
then as the other observer passes by you, your whole picture of space and
time can only be constructed from events that are inside the lower half of
the light cone. If you wait for a while, then eventually you can get all of
the information from all of the events that were happening around the time
you were passing the other observer. From this information, you can draw the
whole space-time diagram, and then you can see the ambiguity. But by that
time, the ambiguity that you are considering no longer exists. So the
ambiguity can never actually play a part in any physical situation. Finally,
remember that this is only true if nothing can travel faster than the speed
of light.

     Well, that concludes our introduction to special relativity and
space-time diagrams. The next section deals with these concepts with more
detail; however, if the reader wishes to skip to the FTL discussion, the
information provided in the above sections should be enough to follow that
discussion.
