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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