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From: hinson@bohr.physics.purdue.edu (Jason W. Hinson)
Subject: Relativity and FTL Travel
Message-ID: <D3CKFI.17J@physics.purdue.edu>
Sender: usenet@physics.purdue.edu (News Administration)
Organization: Physics Department, Purdue University
Date: Thu, 2 Feb 1995 00:38:06 GMT
Lines: 1114
NOTE: I HOPE this is the last time I'll be posting edition 3.1 of this
post, however I make no promises. My spare time is scarce these days,
and I am still working on getting edition 4.0 completed. It will
include the following: a new sectioning method; an entirly new section
(not manditory reading if you just want to understand the FTL
arguments) which gives more information on special relativity,
paradoxes, and even takes a look at general relativity; and a
re-worked FTL discussion which will talk about the various ideas for
how one could imagine FTL travel (like wormholes, "changing the speed
of light", etc.). This re-worked FTL section will also show why I
think the best method for explaining FTL travel as it is depicted on
Trek is still, by far, the one given in the edition below.
So, be looking for edition 4.0--if not in March then in April. Thanks
for the patience, and if you'd like to be placed on a mailing list to
receive 4.0 via e-mail when it is finished, just let me know.
This is edition 3.1 of this post. Only slight corrections have been
made since version 3.0. As planned, it has become a regular monthly
post on the rec.arts.startrek.tech newsgroup. Again, let me know if
you think that any changes should be made.
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. The next 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 ways for science fiction to get around these
problems, then you may not understand the problem that I discuss in
the forth section, and I strongly recommend that you read it to
increase your understanding of the FTL problem. Finally, I introduce
my idea (the only one I know of) that, if nothing else, gets around
the second problem I discuss in an interesting way.
The best way to read the article may be to make a hard copy. I
refer a few times to a diagram in the second section, and to have it
readily available would be helpful.
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 I can make 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
D. Experimental support for the theory
II. Space-Time Diagrams
A. What are Space-Time Diagrams?
B. Constructing one for a "stationary" observer
C. Constructing one for a "moving" observer
D. Interchanging "stationary" and "moving"
E. Introducing the light cone
F. Comparing the way two observers view space and time
III. The First Problem: The Light Speed Barrier
A. Effects as one approaches the speed of light
B. Conceptual ideas around this problem
IV. 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"
V. 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
VI. 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. 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.
Here is an example of this to make it clear. Consider two
observers, you and me, for example. Let's say I am on a train that
passes you at 30 miles per hour. I throw 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
throw the ball, and 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. 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 observe 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 performed 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 conserved. They
will disagree on what the actual numbers are, but they will agree that
the law holds. 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.
Now, 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--3E8 means 3 times 10 to the eighth). 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 some instrument detects it. Now, in my
point of view (outside the train), that beam 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 I see:
/|\
/ | \
/ | \
light beam going up->/ | \<-light beam on return trip
/ | \
/ | \
/ | \
/ | \
---------|---------->trains motion (v = 0.6 c)
Let's 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 relativity contracts only lengths in the direction of motion.
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 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 the "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.
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 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 muons 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 muons 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 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 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 above discussion of relativity's effects on space and time
we have looked at only 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.
II. 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 way these two observers view
space and time, which will make it necessary to first discuss a diagram
known as a light cone.
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 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 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. For the
space part of the diagram, we will be looking in only one direction, 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 principle 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 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 1, I have marked the locations of
the different space and time units (Note: In my diagrams, I am using
four spaces to be one unit along the x axis and two character heights
to be 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.
Diagram 1 Diagram 2
t t
^ ^
| | light
+ + /
| | /
+ + /
| | /
-+---+---o---+---+---> x -+---+---o---+---+-> x
| |
+ +
| |
+ +
| |
At this point, we want to decide exactly how to represent events
on this coordinate system. First, when we say that we are using this
diagram to represent the reference frame of a particular observer, we
mean that in this diagram the observer is not moving. We will call this
observer the O observer. So if the O observer starts at the origin,
then one second later he is still at x = 0. Two seconds later he is
still at x = 0, etc. So, 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 is 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. 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.
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 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
simultaneous position lines we will have a grid on our space-time
diagram. Any event has a specific location on the grid which tells when
and where it occurs. In Diagram 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 light seconds before the origin (t = -1).
Diagram 3 Diagram 4
t t
| | | | | | |
B ---+---+---+---+---+---+---
| \ | | | | | |
+ \ ---+---+---+---+---+---+---
| \ | | | | | |
+ \ ---+---+---+---o---+---+--- x
| \ | | | | | |
-+---+---o---+---+---C- x ---@---+---+---+---+---+---
| / | | | | | |
+ / ---+---+---+---+---+---+---
| / | | | | | |
+ /
| /
A
|
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 Diagrams 5 and 6. Notice that in our discussion we will
always consider moving observers who pass by the O observer at the time
t = 0 and at the position x = 0. Now, the traveler in Diagram 5 is
moving slower than the one in Diagram 6. You can see this because in a
given amount of time, the Diagram 6 traveler has moved further away from
the time axis than the Diagram 5 traveler. So the faster a traveler
moves, the more slanted this line becomes.
Diagram 5 Diagram 6
t t
| / | /
+ + /
| / | /
+ + /
|` |/
-+---+---o---+---+--- x -+---+---o---+---+- x
,| /|
+ / +
/ | / |
+ / +
/ | / |
What does this line actually represent? Well, consider an object
sitting on this line, right next to our moving observer. If a few
seconds later the object is still sitting on that line (right next to
him), then in his point of view, the object has not moved. The line is
a line of constant position for the moving observer. 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 at time t' = +T. Then the point
at which the beam bounces off the mirror will be simultaneous with the
origin, where t' = t = 0.
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, then 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 no matter what observer we are representing on our
diagram, a beam of light will ALWAYS make a 45 degree angle with respect
to the x and t axes.
In Diagram 7, I have labeled a point A' 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 as before and found
the point where they would meet (C'). Thus, C' and o occur at the same
time in the eyes of the moving observer. Notice that for the O
observer, C' is above his line of simultaneity (the x axis). So while
the moving observer says that C' occurs when the two observers pass (at
the origin), the stationary observer says that C' occurs after the two
observers have passed by one another. In Diagram 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'.
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 at some angle (either clockwise or
counterclockwise, depending on the direction O' is going--left or right)
away from the t axis. The x' axis is a line rotated at the same angle,
but in the _opposite_ direction (counterclockwise or clockwise) away
from the x axis.
Diagram 7 Diagram 8
t t t'
| / | /
+ B' + /
| / \ | / __--x'
+ / C' + / __C'-
|/ / |/__--
-+---+---+---o---/---+---+- x -+---+---+-__o---+---+---+- x
/| / * __-- /|
/ / __-- / +
// | -- / |
A' + / +
/ | / |
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 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.
As a final note about this, 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.
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. By this I mean that O was
considered as the stationary observer only because we defined him as
such. 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 10. By examining this
Diagram, you can confirm that it makes sense to you in light of our
discussion thus far.
Diagram 9 Diagram 10
t' t t'
+-----------------/-------+ \ |
| / /_-/""/ /__/-"/ / _| \ +
|/-"/ / _/--/" / /_-/""/| \ |
| /_-/""/ /__/-"/ / _/-->x' \ +
|"/ / _/--/" / /_-/""/ | \|
|/_-/""/ /__o-"/ / _/--/| ---+---+---o-__+---+--- x'
| / _/--/" / /_-/""/ /_| | ""--__
|-/""/ /__/-"/ / _/--/" | + ""--x
|/ _/--/" / /_-/""/ /__/| |
|""/ /__/-"/ / _/--/" / | +
+-------------------------+ |
The last thing I want to do in this discussion is to compare the
way our two observers view a particular event. 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 compares with the lengths representing the same units in O'.
I will tell you that the lengths are in fact different; however, I will
not take up any more of your time by going into exactly how they
compare. Also, to do this comparison one would use the fact that for
the observers we have defined, if an event occurs at a point (x,t) for O
and (x',t') for O', then x^2 - t^2 = x'^2 - t'^2. The best way to show
this on the diagram is to draw hyperbolas represented by these
equations, and I don't even want to consider how to do this with my
limited experience with ASCII graphics.
There is, however, one comparison that we can make, and it will
be of importance in later discussions. In Diagram 8, in addition to the
O and O' space and time axes, I have 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 who is observing
the events. Now, how can this make sense? How can one event be both in
the future for one observer and in the past to 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 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. As time progresses, the
two then speed away from x = 0. This construction 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 (as
shown in Diagram 11). Let me mention here that specifically I will 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 it's 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. 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 reguardless 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, reguardless 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.
Now let's apply this to the observers and event in Diagram 8. As
you can see, the event in question is indeed outside the light cone.
Because of this, even though the event is in one observers past, he
cannot know about the event at this time. Also, even though the event
is in the other observer's future, he can never have an effect on the
event. In essence, the event (when it happens, where it happens, how it
happens, etc.) is of absolutely no consequence for these two observers at
this time. As it turns out, any time 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 for the observers.
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.
Diagram 11
t
^
| light
\ + /
\ inside /
\ + /
outside \ | / outside
---+---+---o---+---+---> x
/ | \
/ + \
/ inside \
/ + \
|
Well, that concludes our look at relativity and space-time
diagrams. Now, we can use these concepts to discuss the problems
presented by FTL travel.
III. 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 first 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 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 first 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 the speed of
light, A will see B's observed mass (and thus his energy) become
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 let's look at the situation from B's point of view, so we
will consider him to 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 nearby 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 clocks 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.
Let's 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.
IV. 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 any
faster than light method that allows you to travel faster than light in
any frame you wish will also allow you to violate 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 second section so that I
can demonstrate the causality problem involved with FTL travel. In
Diagram 8, 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 second 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 observer's past, he cannot know about it, and even though it is
in the other observer's 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 known 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. In one frame of
reference the signal is moving faster than light, while in the other
frame it is going backwards in time. Also notice that in this example I
never mentioned anything about how the signal gets between two points.
I didn't even require that the signal be "in our universe" when it is
traveling. The only thing I required is that the signal starts and ends
as events in our universe. As long as this is true, and as long as
either observer (K or Kp) can send any faster than light signal in their
own frame of reference, then the causality problem can be produced.
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 him. But then the victim is never hit,
so 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, let's 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 moved only 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 moved only 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.
V. 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 be off by only 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. This means that
x and t axes in a space time diagram will be the same as the ones for
the special field, reguardless of the observer's motion. 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, let's 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 8 in
the second 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 8. 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 focused
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, an 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 "agains" 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 any signal, even an
instantaneous one in any direction; but in Kp's frame, some signals
which would appear to him to be FTL cannot be sent (specifically,
signals which would go back in time in the K frame). 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. 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. In general, this is because no observer can ever send a signal
which goes backward in time in the frame of the special field.
I thus see warp travel in Star Trek like this: Subspace is a
field which defines a particular frame of reference at all points in
known space. When you enter warp, you are using subspace such that you
keep its frame of reference reguardless of your speed. Not only does
this mean that normal warp travel cannot be used to grossly violate
causality, but since your frame of reference does not depend on your
speed as it does in relativity, relativistic effects in general do not
apply to travelers using warp. Since relativistic effects don't apply,
you also have a general explanation as to why you can exceed the speed
of light in the first place.
So, is this 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 observe 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, let's 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 event occurring in normal space while you are within the
warped space, the laws governing that occurrence will look different to
you than 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 as if science fiction
could take care of this problem.
VI. Conclusion.
I have presented to you some major concepts of relativity and the
havoc they play with faster than light travel. I have shown 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 if we wish to have
faster than light travel readily available. 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 that I would
ever 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