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From: hinson@bohr.physics.purdue.edu (Jason W. Hinson)
Newsgroups: rec.arts.startrek.tech
Subject: Relativity and FTL Travel
Summary: A detailed look at the problem
Message-ID: <8974@dirac.physics.purdue.edu>
Date: 18 Dec 92 21:35:54 GMT
Organization: Purdue University Physics Department
Lines: 785
Finally, here it is. It is a length discussion, 794 lines by my count,
but it is fairly complete for what I intended to do. Also, if you are
only interested in a particular part, you can just skip the rest.
What is it about, and who should read it:
This is a detailed explanation about how relativity and that
wonderful science fictional invention of faster than light travel do not
seem to get along with each other. It begins with a simple introduction
to the ideas of relativity. This section includes some important
information on space-time diagrams, so if you are not familiar with
them, I suggest you read it. Then I get into the problems that
relativity poses for faster than light travel. If you think that there
are many science fictional ways that we can get around these problems,
then you probably do not understand the second problem which I discuss
in the third section, and I strongly recommend that you read it to
educate yourself. Finally, I introduce my idea (the only one I know of)
that, if nothing else, gets around this second problem in an interesting
way.
The best way to read the article may be to make a hard copy. I
refer back a few times to a Diagram in the first section, and to have it
readily available would be nice.
I hope you can learn a little something from reading this, or at
least strengthen your understanding of that which you already know.
Your comments and criticisms are welcome, especially if they indicate
improvements that can be made for future posts.
And now, without further delay, here it is.
Relativity and FTL Travel
Outline:
I. An Introduction to Special Relativity
A. Reasoning for its existence
B. Time dilation effects
C. Other effects on observers
E. Space-Time Diagrams
D. Experimental support for the theory
II. The First Problem: The Light Speed Barrier
A. Effects as one approaches the speed of light
B. Conceptual ideas around this problem
III. The Second Problem: FTL Implies The Violation of Causality
A. What is meant here by causality, and its importance
B. Why FTL travel of any kind implies violation of causality
C. A scenario as "proof"
IV. A Way Around the Second Problem
A. Warped space as a special frame of reference
B. How this solves the causality problem
C. The relativity problem this produces
D. One way around that relativity problem
V. Conclusion.
I. 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. For future
reference, it will also introduce the reader to the basics of space-time
diagrams. Finally, I want to mention some experimental evidence that
supports the theory.
The idea of relativity was around in Newton's day, but it was
incomplete. It involved transforming from one frame of reference to
another frame which is moving with respect to the first. The
transformation was not completely correct, but it seemed so in the realm
of small speeds. I give here an example of this to make it clear.
Consider two observers, you and me, for example. Lets say I am
on a train which passes you at 30 miles per hour. I through a ball in
the direction the train is moving, and the ball moves at 10 mph in MY
point of view. Now consider a mark on the train tracks. You see the
ball initially moving along at the same speed I am moving (the speed of
the train). Then I through the ball, and before I can reach the mark on
the track, the ball is able to reach it. 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. In other words, this was part of the first concept
of relativity.
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."
What it means is that if you observer any physical laws for a given
situation in your frame of reference, then an observer in a reference
frame moving with a constant velocity with respect to you should also
agree that those physical laws apply to that situation.
As an example, consider the conservation of momentum. 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 (mass times velocity) before the collision and
after the collision, that the two should be identical. Now, let this
experiment be preformed on a train where the balls are moving along the
line of the train's motion. 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 the same before and
after the collision. We should be able to apply this 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. 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 the first example given in this introduction. If
the train is moving at half the velocity of light, wouldn't you expect
the light beam (which is traveling at the speed of light with respect to
the train) to look as if it is traveling one and a half that speed with
respect to an outside observer? Well this is not the case. The old
ideas of relativity in Newton's day do not apply here. What accounts
for this peculiarity is time dilation and length contraction.
Here I give an example of how time dilation can help explain a
peculiarity that arises from the above concept. Again we consider a
train, but let's give it a speed of 0.6 c (where c = the speed of light
which is 3E8 m/s). An occupant of this train shines a beam of light so
that (to him) the beam goes straight up, hits a mirror at the top of the
train, and bounces back to the floor of the train where it is detected.
Now, in my point of view (outside of the train), that beam of light does
not travel straight up and straight down, but makes an up-side-down "V"
shape since the train is also moving. Here is a diagram of what I see:
/|\
/ | \
/ | \
light beam going up->/ | \<-light beam on return trip
/ | \
/ | \
/ | \
/ | \
---------|---------->trains motion (v = 0.6 c)
Lets say that the trip up takes 10 seconds in my point of view. The
distance the train travels during that time is:
(0.6 * 3E8 m/s) * 10 s = 18E8 m.
The distance that the beam travels on the way up (the slanted line to
the left) must be
3E8 m/s * 10s = 30E8 m.
Since the left side of the above figure is a right triangle, and we know
the length of two of the sides, we can now solve for the height of the
train:
Height = [(30E8 m)^2 - (18E8 m)^2]^0.5 = 24E8 m
(It is a tall train, but this IS just a thought experiment). Now we
consider the frame of reference of the traveler. The light MUST travel
at 3E8 m/s for him also, and the height of the train doesn't change
because only lengths in the direction of motion are contracted.
Therefore, in his frame the light will reach the top of the train in 24E8 m / 3E8 (m/s) = 8 seconds,
and there you have it. To me the event takes 10 seconds, while
according to him it must take only 8 seconds. We each measure time in
different ways.
To intensify this oddity, consider the fact that all inertial
frames are equivalent. That is, from the traveler's point of view he is
the one who is sitting still, while I zip past him at 0.6 c. So he will
think that it is MY clock that is running slowly. This lends itself
over to what seem to be paradoxes which I will not get into here. If
you have any questions on such things (such as theJ"twin paradox" --
which can be understood with special relativity, by the way) feel free
to ask me about them, and I will do the best I can to answer you.
As I mentioned above, 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, the meter stick of the traveler on the train will
look as if it is shorter than a meter. Similarly, the observer on the
train will think that the meter stick of the outside observer is the one
that is contracted. The closer one gets to the speed of light with
respect to an observer, the shorter the stick will look to that
observer. The factor which determines the amount of length contraction
and time dilation is called gamma.
Gamma is defined as (1 - v^2/c^2)^(-1/2). For our train (for
which v = 0.6 c), gamma is 1.25. Lengths will be contracted and time
dilated (as seen by the outside observer) by a factor of 1/gamma = 0.8,
which is what we demonstrated with the difference in measured time (8
seconds compared to 10 seconds). Gamma is obviously an important number
in relativity, and it will appear as we discuss other consequences of
the theory.
Another consequence of relativity is a relationship between
mass, energy, and momentum. By considering conservation of momentum and
energy as viewed from two frames of reference, one can find that the
following relationship must be true for an unbound particle:
E^2 = p^2 * c^2 + m^2 * c^4
Where E is energy, m is mass, and p is relativistic momentum which is
defined as
p = gamma * m * v (gamma is defined above)
By manipulating the above equations, one can find another way to express
the total energy as
E = gamma * m * c^2
Even when an object is at rest (gamma = 1) it still has an energy of
E = m * c^2
Many of you have seen something like this stated in context with the
theory of relativity.
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).
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. The
mass seen by the observer (which I will call the observed mass) is given
by gamma * m. Thus, we could also write the total energy as
E = (observed mass) * c^2
That observed mass approaches infinity as the object approaches the
speed of light with respect to the observer.
So far 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 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 are known as Lorentz
Transformation. Without getting deep into the math, much can be
understood about such transforms by considering space-time diagrams.
A space-time diagram consists of a coordinate system with one
axis to represent space and another to represent time. Where these two
principle axes meet is the origin (see Diagram 1 below), and for the
most part, we consider ourselves to be at that point. Anything above
the principle space axis is in our future, while anything below that
axis is in our past. Any event can be described as a point in this axis
system. For example, consider an event that took place 3 seconds ago
and was 2 light seconds (the distance light travels in 2 seconds) away
from you to the left (x = -2 light seconds). This event is marked in
Diagram 1 as a "*".
Now consider a traveler going away from the origin to the right.
As time progresses forward, the traveler gets further and further from
the time axis. The faster he goes, the more slanted the line he makes
will be as he is able to get far down the x axis in a short amount of
time. One important traveler to consider here is light. If we define
the x axis in light seconds and the time axis in seconds, then light
will speed away from the origin creating a line at a 45 degree angle to
the two axes. On diagram 2, I have drawn two lines which represent a
pulse of light going away from the origin in the plus and minus x
directions. The two pulses are extended back into the past, as if they
started from far off, came to the origin, and sped away in the future.
This figure is known as a light cone.
A light cone divides a space-time diagram into two major
sections: the area inside the cone and the area outside the cone. If it
is impossible for anything to travel faster than light, then the only
events in the past that you can know about at this moment are those that
are inside the light cone. Also, the only events that you can influence
in the future are, again, those inside the light cone.
Let us now consider (again) an arbitrary traveler who is going
slower than the speed of light. As a consequence of the Lorentz
transforms that I have mentioned, the line he makes on the space-time
diagram becomes his new time line (t'). Because of relativity, his
space axis will also be transformed. As can be seen in Diagram 3, his
time axis has been rotated by some angle clockwise, while his space axis
(x') has been rotated by the same angle counterclockwise. The faster
the speed, the greater this angle, and as you approach the speed of
light, the two axis come closer and closer to being the same line (a
line on the light cone which is at 45 degrees). This gives him a skewed
set of space-time coordinates that I have tried my best to show on
Diagram 4 (squint your eyes, and you can see the skewed squares of the
new coordinate system). It is important to note that in this
transformation, the position of the light cone does not change. If you
move one unit down the space axis, and one unit up the time axis, that
point will still lie somewhere on the light cone. This shows that the
speed of light has not changed for the moving observer (it still travels
one light second per second).
Now let us compare the different ways that each observer views
space and time. Look at the event marked "*" on Diagram 3. For the
observer in the x',t' system, the event is in his future (above his
principle x' axis). For the observer in the x,t system, the event is in
his past. So how does this make since? Recall two things: (1) you can
only know about and influence events that are inside the light cone, and
(2) the light cone does not change for the moving observer. So even if
an event is in one observers past and in another observers future, it
will be outside the light cone, and neither observer will be able to
know about it or influence it. It is the fact that nothing travels
faster than light that causes this to be true.
Diagram 1 Diagram 2
t t
| | light
future \ inside /
| \ cone /
| \ | /
| outside \ | / outside
| cone \ | / cone
-------------+------------- x -------------+------------- x
| / | \
| / | \
event * | / | \
| / inside \
past / cone \
|
Diagram 3
t t'
| /
| /
| /
| / ___---> x'
|/___---'''
-------------+------------- x
* ___ ---'''|
''' / | note: * = event
/ |
/ |
/ |
Diagram 4 principle t' axis
/
+---------------------/-----------+
|__---/"" / / / / __/--|
| / / / /__--/""" / |
| / /___-/-"""/ / / |
|__/---"/" / / / /__--/|
| / / / / ___/--""/ / |
|/ / _/_---/"" / / / | ___--->principle x' axis
|___-/-"""/ / / / __/---"""
| / / / /__--/""" / |
| / / ___O--""/ / / /|
|_/_---/"" / / / /___-/-| O = Origin
|/ / / / __/---"/" / |
| / /__--/""" / / / |
|___/--""/ / / / _/_---|
| / / / /___-/-"""/ |
+---------------------------------+
These amazing consequences of relativity do have experimental
foundations. One of these involves the creation of muons by cosmic rays
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 traveling at the
speed of light, it could still only go 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 muons frame of
reference is running slow, 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 muons frame
of reference. So in its point of view, it does only have 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 its
point of view, it lives a very small amount of time, but it doesn't have
that far to go.
Another verification is found all the time in particle physics.
The results of having a particle strike a target can only be understood
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 exits that would tend to undermine
the theory.
Well, that was a fairly lengthy look at relativity, but how does
it all apply to faster than light travel? This is what we will look at
next.
II. The First Problem: The Light Speed Barrier
In this section we discuss the first thing (and in some cases
the only thing) that comes to mind for most people who consider the
problem of faster than light travel. I call it the light speed barrier.
As we will see by considering ideas from the previous section, light
speed seems to be a giant, unreachable wall standing in our way. I also
introduce a couple of fictional ways to get around this barrier;
however, part of my reason for introducing these solutions is to show
that they do not solve the problem discussed in the next section.
Consider two observers, A and B. Let A be here on Earth and be
considered at rest for now. B will be speeding past the A at highly
relativistic speeds. If B's speed is 80% that of light with respect to
A, then gamma for him (as defined in the previous section) is
1.6666666... = 1/0.6
So from A's point of view B's clock is running slow and B's lengths in
the direction of motion are shorter by a factor of 0.6. If B were
traveling at 0.9 c, then this factor becomes about 0.436; and at 0.99 c,
it is about 0.14. As the speed gets closer and closer to the speed of
light, A will see B's clock slow down infinitesimally slow, and A will
see B's lengths in the direction of motion becoming infinitesimally
small.
In addition, If B's speed is 0.8 c with respect to A, then A
will see B's observed mass as being larger by a factor of gamma (which
is 1.666...). At 0.9 c and 0.99 c this factor is about 2.3 and 7.1
respectively. As the speed gets closer and closer to me speed of light,
A will see B's observed mass (and thus his energy) get infinitely large.
Obviously, from A's point of view, B will not be able to reach
the speed of light without stopping his own time, shrinking to
nothingness in the direction of motion, and taking on an infinite amount
of energy.
Now lets look at the situation from B's point of view, so we
will consider him be at rest. First, notice that the sun, the other
planets, the nearby stars, etc. are not moving very relativistically
with respect to the Earth; so we will consider all of these to be in the
same frame of reference. Let B be traveling past the earth and toward
some near by star. In his point of view, the earth, the sun, the other
star, etc. are the ones traveling at highly relativistic velocities with
respect to him. So to him the clock on Earth are running slow, the
energy of all those objects becomes greater, and the distances between
the objects in the direction of motion become smaller.
Lets consider the distance between the Earth and the star to
which B is traveling. From B's point of view, as the speed gets closer
and closer to that of light, this distance becomes infinitesimally
small. So from his point of view, he can get to the star in practically
no time. (This explains how A seems to think that B's clock is
practically stopped during the whole trip when the velocity is almost
c.) If B thinks that at the speed of light that distance shrinks to
zero and that he is able to get there instantaneously, then from his
point of view, c is the fastest possible speed.
So from either point of view, it seems that the speed of light
cannot be reached, much less exceeded. However, through some inventive
imagination, it is possible to come up with fictional ways around this
problem. Some of these solutions involve getting from point A to point
B without traveling through the intermittent space. For example,
consider a forth dimension that we can use to bend two points in our
universe closer together (sort of like connecting two points of a "two
dimensional" piece of paper by bending it through a third dimension and
touching the two points directly). Then a ship could travel between two
points without moving through the space in between, thus bypassing the
light speed barrier.
Another idea involves bending the space between the points to
make the distance between them smaller. In a way, this is what highly
relativistic traveling looks like from the point of view of the
traveler; however, we don't want the associated time transformation. So
by fictionally bending the space to cause the space distortion without
the time distortion, one can imagine getting away from the problem.
Again I remind you that these solutions only take care of the
"light speed barrier" problem. They do not solve the problem discussed
in the next section, as we shall soon see.
III. The Second Problem: FTL Implies The Violation of Causality
In this section we explore the violation of causality involved
with faster than light travel. First I will explain what we mean here
by causality and why it is important that we do not simply throw it
aside without a second thought. I will then try to explain why
traveling faster than light by any means (except the one introduced in
the next section) will produce a violation of causality. Finally,
attempting to remove any doubt, we will preform a thought experiment to
show that FTL travel does imply the violation of causality.
When I speak of causality, I have the following particular idea
in mind. Consider an event A which has an effect on another event B.
Causality would require that event B cannot in turn have an effect on
event A. For example, let's say that event A is a murderer making a
decision to shoot and kill his victim. Let's then say that event B is
the victim being shot and killed by the murderer. Causality says that
the death of the victim cannot then have any effect on the murderer's
decision. If the murderer could see his dead victim, go back in time,
and then decide not to kill him after all, then causality would be
violated. In time travel "theories," such problems are reasoned with
the use of multiple time lines and the likes; however, since we do not
want every excursion to a nearby star to create a new time line, we
would hope that FTL travel could be done without such causality
violations. As I shall now show, this is not a simple problem to get
around.
I refer you back to the diagrams in the first section so that I
can demonstrate the causality problem involved with FTL travel. In
Diagram 3, two observers are passing by one another. At the moment
represented by the principle axes shown, the two observers are right
next to one another an the origin. The x' and t' axes are said to
represent the K-prime frame of reference (I will call this Kp for
short). The x and t axes are then the K frame of reference. We define
the K system to be our rest system, while the Kp observer passes by K at
a relativistic speed. As you can see, the two observers measure space
and time in different ways. For example, consider again the event
marked "*". Cover up the x and t axis and look only at the Kp system.
In this system, the event is above the x' axis. If the Kp observer at
the origin could look left and right and see all the way down his space
axis instantaneously, then he would have to wait a while for the event
to occur. Now cover up the Kp system and look only at the K system. In
this system, the event is below the x axis. So to the observer in the K
system, the event has already occurred.
Normally, this fact gives us no trouble. If you draw a light
cone (as discussed in the first section) through the origin, then the
event will be outside of the light cone. As long as no signal can
travel faster than the speed of light, then it will be impossible for
either observer to know about or influence the event. So even though it
is in one observers past, he cannot know about it, and even though it is
in the other observers future, he cannot have an effect on it. This is
how relativity saves its own self from violating causality.
Now consider what would happen if a signal could be sent
arbitrarily fast. From K's frame of reference, the event has already
occurred. For example, say the event occurred a year ago and 5 light
years away. As long as a signal can be sent at 5 times the speed of
light, then obviously K can receive a signal from the event. However,
from Kp's frame of reference, the event is in the future. So as long as
he can send a signal sufficiently faster than light, he can get a signal
out to the place where the event will occur before it occurs. So, in
the point of view of one observer, the event can be know about. This
observer can then tell the other observer as they pass by each other.
Then the second observer can send a signal out that could change that
event. This is a violation of causality. Basically, when K receives a
signal from the event, Kp sees the signal as coming from the future.
Also, when Kp sends a signal to the event, K sees it as a signal being
sent into the past.
As a short example of this, consider the following. Instead of
sending a message out, let's say that Kp sends out a bullet that travels
faster than the speed of light. This bullet can go out and kill someone
light-years away in only a few hours (for example) in Kp's frame of
reference. Now, say he fires this bullet just as he passes by K. Then
we can call the death of the victim the event (*). Now, in K's frame of
reference, the victim is already dead when Kp passes by. This means
that the victim could have sent a signal just after he was shot that
would reach K before Kp passed by. So K can know that Kp will shoot his
gun as he passes, and K can stop Kp. But then the victim is never hit,
and he never sends a message to K. So K doesn't know to stop Kp and Kp
does shoot the bullet. Obviously, causality is not very happy about
this logical loop that develops.
If this argument hasn't convinced you, then let me try one more
thought experiment to convince you of the problem. Here, to make
calculations easy, we assume that a signal can be sent infinitely fast.
Person A is on earth, and person B speeds away from earth at a
velocity v. To make things easy, lets say that v is such that for an
observer on Earth, person B's clock runs slow by a factor of 2. now,
person A waits one hour after person B has passed earth. At that time
person A sends a message to person B which says "I just found a bomb
under my chair that will take 10 minutes to defuse, but goes off in 10
seconds ... HELP" He sends it instantaneously from his point of view...
well, from his point of view, B's clock has only moved half an hour. So
B receives the message half an hour after passing earth in his frame of
reference.
Now we must switch to B's point of view. From his point of
view, A has been speeding away from him at a velocity v. So, to B, it
is A's clock that has been running slow. Therefore, when he gets the
message half an hour after passing earth, then in his frame of
reference, A's clock has only moved 1/4 an hour. So, B sends a message
to A that says: "There's a bomb under your chair." It gets to A
instantaneously, but this time it is sent from B's frame of reference,
so instantaneously means that A gets the message only 1/4 of an hour
after B passed Earth. You see that A as received an answer to his
message before he even sent it. Obviously, there is a causality
problem, no matter how you get the message there.
OK, what about speeds grater than c but NOT instantaneous?
Whether or not you can use the above argument to find a causality
problem will depend on how fast you have B traveling. If you have a
communication travel faster than c, then you can always find a velocity
for B (v < c) such that a causality problem will occur. However, if you
send the communication at a speed that is less than c, then you cannot
create a causality problem for any velocity of B (as long as B's
velocity is also less that c).
So, it seems that if you go around traveling faster than the
speed of light, causality violations are sure to follow you around.
This causes some very real problems with logic, and I for one would like
to find a way around such problems. This next section intends to do just
that.
IV. A Way Around the Second Problem
Now we can discuss my idea for getting around the causality
problem produced by FTL travel. I will move through the development of
the idea step by step so that it is clear to the reader. I will then
explain how the idea I pose completely gets rid of causality violations.
Finally, I will discuss the one "bad" side effect of my solution which
involves the fundamentals of relativity, and I will mention how this
might not be so bad after all.
Join me now on a science fictional journey of the imagination.
Picture, if you will, a particular area of space about one square light-
year in size. Filling this area of space is a special field which is
sitting relatively stationary with respect to the earth, the sun, etc.
(By stationary, I mean relativistically speaking. That means it could
still be moving at a few hundreds of thousands of meters per second with
respect to the earth. Even at that speed, someone could travel for a
few thousand years and their clock would only be off by a day or two
from earth's clocks.) So, the field has a frame of reference that is
basically the same as ours on earth. In our science fictional future, a
way is found to manipulate the very makeup (fabric, if you will) of this
field. When this "warping" is done, it is found that the field has a
very special property. An observer inside the warped area can travel at
any speed he wishes with respect to the field, and his frame of
reference will always be the same as that of the field. In our
discussion of relativity, we saw that in normal space a traveler's frame
of reference depends on his speed with respect to the things he is
observing. However, for a traveler in this warped space, this is no
longer the case.
To help you understand this, lets look at a simple example.
Consider two ships, A and B, which start out sitting still with respect
to the special field. They are in regular space, but in the area of
space where the field exists. At some time, Ship A warps the field
around him to produce a warped space. He then travels to the edge of
the warped space at a velocity of 0.999 c with respect to ship B. That
means that if they started at one end of the field, and A traveled to
the other end of the field and dropped back into normal space, then B
says the trip took 1.001001... years. (That's 1 light-year divided by
0.999 light-years per year.) Now, if A had traveled in normal space,
then his clock would have been moving slow by a factor of 22.4 with
respect to B's clock. To observer A, the trip would have only taken
16.3 days. However, by using the special field, observer A kept the
field's frame of reference during the whole trip. So he also thinks it
took 1.001001... years to get there.
Now, let's change one thing about this field. Let the field
exist everywhere in space that we have been able to look. We are able
to detect its motion with respect to us, and have found that it still
doesn't have a very relativistic speed with respect to our galaxy and
its stars. With this, warping the field now becomes a means of travel
within all known space.
The most important reason for considering this as a means of
travel in a science fiction story is that it does preserve causality, as
I will now attempt to show. Again, I will be referring to Diagram 3 in
the first section. In order to demonstrate my point, I will be doing
two things. First, I will assume that the frame of reference of the
field (let's call it the S frame) is the same as that of the x and t
system (the K system) shown in Diagram 3. Assuming that, I will show
that the causality violation discussed in the previous section will not
occur using the new method of travel. Second, I will show that we can
instead assume that the S frame is the same as that of the x' and t'
system (the K-prime--or Kp for short--system), and again causality will
be preserved.
Before I do this, let me remind you of how the causality
violation occurred. The event (*) in the diagram will again be focussed
on to explore causality. This event is in the past of the K system, but
it is in the future of the Kp system. Since it is in the past according
to the K observer, a FTL signal could be sent from the event to the
origin where K would receive the signal. As the Kp observer passed by,
K could tell him, "Hay, here is an event that will occur x number of
light years away and t years in your future." Now we can switch over to
Kp's frame of reference. He sees a universe in which he now knows that
at some distant point an event will occur some time in the future. He
can then send a FTL signal that would get to that distant point before
the event happens. So he can influence the event, a future that he
knows must exist. That is a violation of causality. But now we have a
specific frame of reference in which any FTL travel must be done, and
this will save causality.
First, we consider what would happen if the frame of the special
field was the same as that of the K system. That means that the K
observer is sitting relatively still with respect to the field. So, in
the frame of reference of the field, the event "*" IS in the past. That
means that someone at event "*" can send a message by warping the field,
and the message will be able to get to origin. Again, the K observer
has received a signal from the event. So, again he can tell the Kp
observer about the event as the Kp observer passes by. Again, we switch
to Kp's frame of reference, and again he is in a universe in which he
now knows that at some distant point an event will occur some time in
the future. But here is where the "again's" stop. Before it was
possible for Kp to then send a signal out that would get to that distant
point before the event occurs. But NOW, to send a signal faster than
light, you must do so by warping the field, and the signal will be sent
in the field's frame of reference. But we have assumed that the field's
frame of reference is the same as K's frame, and in that frame, the
event has already occurred. So, as soon as the signal enters the warped
space, it is in a frame of reference in which the event is over with,
and it cannot get to the location of the event before it happens. What
Kp basically sees is that no matter how fast he tries to send the
signal, he can never get it to go fast enough to reach the event. In
K's frame, it is theoretically possible to send a signal
instantaneously; but in Kp's frame, that same signal would have a non-
infinite speed. So we see that under this first consideration,
causality is preserved.
To further convince you of my point, I will now consider what
would happen if the frame of the special field was the same as that of
the Kp system instead of the K system. Again, consider an observer at
the event "*" who wishes to send a signal to K before Kp passes by K.
The event of K and Kp passing one another has the position of the origin
in our diagram (as I hope you understand). In order to send this
signal, the observer at "*" must warp the field and thus enter the
system of the Kp observer. But in the frame of reference of Kp, when he
passes by K, the event "*" is in the future. Another way of saying this
is that in the Kp frame of reference, when the event "*" occurs, Kp will
have already passed K and gone off on his merry way. So when the signal
at "*" enters the warped space, it's frame of reference switches to one
in which K and Kp have already passed by one another. That means that
it is impossible for "*" to send a signal that would get to K before Kp
passes by. The possibility of creating a causality violation thus ends
here.
Let me summarize the two above scenarios. In the first
situation, K could know about the event before Kp passes. So Kp can
know about the event after he passes K, but Kp could not send a signal
that would then influence the event. In the second situation, Kp can
send a signal that would influence the event after he passed by K.
However, K could not know about the event before Kp passed, so Kp cannot
have previous knowledge of the event before he sends a signal to the
event. In either case, causality is safe. Also notice that only one
case can be true. If both cases existed at the same time, then
causality would be no safer than before. Therefore, only one special
field can exist, and using it must be the only way that FTL travel can
be done.
Many scenarios like the one above can be conceived using
different events and observers, and (under normal situations) FTL
travel/communication can be shown to violate causality. However, in all
such cases, the same types of arguments are used that I have used here,
and the causality problem is still eliminated by using the special
field.
So, is the the perfect solution where FTL travel exists without
any side effects that make it logically impossible? Does this mean that
FTL travel in Star Trek lives, and all we have to do is accept the idea
that subspace/warped space involves a special frame of reference? Well,
not quite.
You see, there is one problem with all of this which involves
the basic ideas which helped form relativity. We said that an observer
using our special mode of transportation will always have the frame of
reference of the field. This means that his frame of reference does not
change with respect to his speed, and that travel within the warped
field does not obey Einstein's Relativity. At first glance, this
doesn't seem too bad, it just sounds like good science fiction. But
what happens when you observer the outside world while in warp? To
explore this, let's first look back at why it is necessary for the frame
of reference to change with respect to speed. We had assumed that the
laws of physics don't simply change for every different inertial
observer. It had been found that if the laws of electrodynamics look
the same to all inertial observers, then the speed of an electromagnetic
wave such as light must be the same for all observers. This in turn
made it necessary for different observers to have different frames of
reference. Now, lets go backwards through this argument. If different
observers using our special mode of transportation do not have different
frames of reference, then the speed of light will not look the same to
all observers. This in turn means that if you are observing an
electromagnetic occurrence from within the warped space, the laws
governing that occurrence will look different to you that they would to
an observer in normal space.
Perhaps this is not that big of a problem. One could assume
that what you see from within warped space is not actually occurring in
real space, but is caused by the interaction between the warped space
and the real universe. The computer could then compensate for these
effects and show you on screen what is really happening. I do not,
however pretend that this is a sound explanation. This is the one part
of the discussion that I have not delved into very deeply. Perhaps I
will look further into this in the future, but it seems like science
fiction could take care of this problem.
V. Conclusion.
I have presented to you some major concepts of relativity and
the havoc they play with faster than light travel. I have show you that
the violation of causality alone is a very powerful deterrent to faster
than light travel of almost any kind. So powerful are its effects, in
fact, that I have found only one way to get around them. I hope I have
convinced you that (1) causality is indeed very hard to get around, and
(2) my idea for a special field with a particular frame of reference
does get around it. For the moment, I for one see this as the only way
I want to consider the possibility of faster than light travel. Though
I do not expect you to be so adamant about the idea, I do hope that you
see it as a definite possibility with some desirable outcomes. If
nothing else, I hope that I have at least educated you to some extent on
the problems involved when considering the effects of relativity on
faster than light travel.
Jason Hinson
-Jay