Part IV: Faster Than Light TravelConcepts and Their "Problems"
This is Part IV of the "Relativity and FTL Travel" FAQ. It discusses the various problems involved with FTL travel and how they apply to particular FTL concepts. This part of the FAQ is written under the assumption that the reader understands the concepts discussed in Part I of this FAQ which should be distributed with this document.
In the event that you need more information about this FAQ (including copyright information and a table of contents for all parts of the FAQ), see the Introduction to the FAQ portion.
The following discussion completes the purpose of this FAQ by considering faster than light travel with relativity in mind. After this brief introduction, I will discuss the general problems associated with FTL travel. These problems will apply differently to different FTL concepts, but I need to go over the general idea behind the problems first. After this general discussion of the problems, we will consider their applications to specific FTL concepts. We will also consider possible, conceptual "solutions" to the particular problem that seems to plague all FTL concepts. Finally, because this FAQ is written for the rec.arts.startrek.tech newsgroup, I will go over some notes and arguments for why "warp" drive should be explained in a particular way in order to get around the FTL problems and give us what is seen on the show.
Before we begin the discussion, I wanted to go over the basic idea of what we mean by FTL travel. To do so, we should start by noting that most of spacetime through which we would want to travel is fairly flat. For those who have not read Part III of this FAQ, that means that special relativity describes the spacetime fairly well without having resorting to general relativity (which applies when a gravitational field is present). Sources of gravity are few and far between, and even if you travel "close" to one, it would have to be a significant source of gravity in order to destroy our flat spacetime approximation. Now, some FTL travel concepts we consider will involve using certain areas of spacetime which are not flat (and I will go over them when we get there); however, the important thing for us is that all around these nonflat areas, the spacetime can be approximated fairly well as being flat.
Thus, for our purposes, we can use the following to describe FTL travel. Consider some observer traveling from point A to point B. At the same time this observer leaves A, a light beam is sent out towards the destination, B. This light travels in the area of fairly flat spacetime outside of any effects that might be caused by the method our observer uses to travel from A to B. If the observer ends up at B in time to see the light beam arrive, then the observer is said to have traveled "faster than light".
Notice that with this definition we don't care where the observer is when he or she does the traveling. Also, if some spacetime distortion is used to drive the ship, then even if the ship itself doesn't move faster than light within that distortion, the ship still travels faster than the light which is going through the normal, flat spacetime that is not effected by the ship's FTL drive. Thus, this ship still fits our definition of FTL travel.
So, with this basic definition in mind, let's take a look at the problems involved with FTL Travel.
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 discussed in Part I , Chapter 1 of this FAQ, light speed seems to be a giant, unreachable wall standing in our way. I note that various concepts for FTL travel may deal with this problem, but here we simply want to talk about the problem in general.
To begin, 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 a highly relativistic speed as he (B) heads towards some distant star. If B's speed is 80% that of light with respect to A, then for him (as defined in Section 1.4 ) is 1.6666666... = 1/0.6. So from A's frame of reference, 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 energy as a factor of larger than his restmass energy (Note, I use an equation for energy here defined in Section 1.5, Equation 1:8 ):
(Eq 7:1)
where m(B) is the mass of observer B. 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 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 now 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. Remember that to A, B is traveling past the earth and toward some nearby star. However, in B's frame of reference, 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. B notices nothing odd about his own clock, but in his frame the distance he travels is quite small.) If (in B's frame) that distance shrinks to zero as his speed with respect to A goes to the speed of light, and he is thus able to get there instantaneously, then from B's point of view, c is the fastest possible speed.
From either point of view, it seems that the speed of light cannot be reached, much less exceeded. This, then, is the "light speed barrier", but most concepts people have in mind for producing FTL travel explicitly deal with this problem (as we will see). However, the next problem isn't generally as easy to get away with, and it probably isn't as well known among the average science fiction fan.
In this section we will explore a problem with FTL travel that doesn't always seem to get consideration. The problem involves ones ability to violate causality in certain frames of reference with the use of FTL travel. While this in itself doesn't necessarily make FTL travel impossible, the ability to go further and produce an unsolvable paradox would make the FTL travel prospect logically self contradictory. So, I will start by discussing the meaning of causality and the problems of an unsolvable paradox. I will then try to show how any form of FTL travel will produce violation of the causality principle. Finally, I will explain how, without special provisions being in place, FTL travel can go further to produce an unsolvable paradox.
The principle of causality is fairly straight forward. According to causality, if there is some effect which is produced by some cause, then the cause must precede the effect. So, if for some observer (in some frame of reference) an effect truly happens before its cause occurs, then causality is violated for that observer. Now, recall our discussion in Section 1.1 concerning when occurrences happen in a frame of reference. There I took a moment to explain that when I talk about the order of events in some frame of reference, I mean their actual order, and not necessarily the order in which they are seen. One can imagine a situation whereby I could first receive light from the effect and later receive light from the cause. However, This might be because the effect is simply much closer to me than the cause (so that light takes less time to travel from the effect I observer, and I see it first). After I take into account the time it took the light to travel from each event, then I will find the order in which the events truly occurred, and this will determine whether or not there is a true violation of causality in my frame. This true violation of causality is what I will be talking about, not some trick concerning when observers see events, but a concept concerning the actual order of the events in some frame of reference.
Now, one can argue that the idea of causality violation doesn't necessarily destroy logic. The idea seems oddto have an effect come first, and then have the cause occurbut it doesn't have to produce a selfcontradictory situation. An unsolvable paradox, however, is a selfcontradictory situation. It is a situation which logically forbids itself from being. Thus, when one shows that a particular set of circumstances allows for an unsolvable paradox, then one can argue that those circumstances must logically be impossible.
I refer you back to Diagram 29 (reproduced below as Diagram 81 ) so that I can demonstrate the causality problem involved with FTL travel. There you see two observers passing by one another.
Diagram 81

The origin marks the place and time where the two observers are right next to one another. The x' and t' axes are said to represent the frame of reference of O' (I'll use Opfor Oprimeso that I can easily indicate the possessive form of O as O's and the possessive form of O' as Op's). The x and t axes are then the reference frame of the O observer. We consider the O system to be our rest system, while the Op observer passes by O at a relativistic speed. As you can see from the two coordinate systems, the two observers measure space and time in different ways. Now, consider again the event marked "*". Cover up the x and t axis and look only at the Op system. In this system, the event is above the x' axis. If the Op 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 Op system and look only at the O system. In this system, the event is below the x axis. So to O, the event has already occurred by the time the two observers are passing one another.
Normally, this fact gives us no trouble. If you draw a light cone (as discussed in Section 2.8 ) 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.
However, consider the prospect of FTL travel with this diagram in mind. As O and Op pass by one another, the event "*" has not happened yet in Op's frame of reference. Thus, if he can send an FTL signal fast enough, then he should be able to send a signal (from the origin) which could effect "*". However, in O's frame, "*" has already occurred by the time O and Op pass by one another. This means that the event "Op sends out the signal which effects *" occurs after the event which it effects, "*", in O's frame. For O, The effect precedes the cause. Thus, the signal which travels FTL in Op's frame violates causality for O's frame. Similarly, since "*" has already occurred in O's frame when O and Op pass one another, then in his frame an FTL signal could be sent out from "*" which could reach O and tell him about the event as the two observer's past. However, for Op, the event "O learns about * as O and Op pass one another" comes before * itself. Thus, the signal which is FTL in O's frame violates causality in Op's frame.
In short, for any signal sent FTL in one frame of reference, another frame of reference can be found in which that signal actually traveled backwards in time, thus violating causality in that frame.
Notice that in this example I never mentioned anything about how the signal gets between the origin and *. I didn't even require that the signal be "in our universe" when it was "traveling" ( remember our definition of FTL travel). The only things I required were that (1) the signal's "sending" and "receiving" were events in our universe and (2) the spacetime between the origin and "*" is flat (i.e. it is correctly described by special relativity diagrams). Some FTL ideas may invalidate the second assumption, but we will consider them a bit later. We will find, however, that violation of causality still follows from all the FTL travel concepts.
As I mentioned before, violations of causality (as strange as they may be) do not have to truly, logically contradict themselves. However, it isn't too difficult to show (starting with the above arguments) that FTL travel can be used to produce an unsolvable paradox (a situation which contradicts its own existence). As a note, in the past I have called such situations "gross" violations of causality.
I'll illustrate the point with an example (again referring to Diagram 81 ) Remember we said that as O and Op pass, Op can send an FTL message out (from his frame of reference) which effects "*". However, rather than having him send a message out, let's say that Op sends out a bullet that travels faster than the speed of light. This bullet can go out and kill someone lightyears away in only a few hours (for example) in Op's frame of reference. So, say he fires this bullet just as he passes by O. Then the death of the victim can be the event (*). Now, in O's frame of reference, the victim is already dead ("*" has occurred) when Op passes by. This means that another observer (stationary in O's frame) who was at the position of the victim when the victim was shot could have sent an FTL signal just after the victim's death, and that signal could reach O before Op passed by him. So O can know that Op will shoot his gun as they pass each other.
To intensify the point I will make, we can let the signal which was sent to O be a picture of the victim, or even an ongoing video signal of the victim's body. Thus, O has evidence of the victim's death before Op has fired the weapon (a plain ol' violation of causality). However, at this point O can decide to stop Op from firing the gun. But if the bullet doesn't go out, and the victim never dies, then why (and how) would a video signal/picture of the victim's dead body ever be sent to O? And yet, O has that video/picture.
In the end, it is the death of the victim which causes O to prevent the victim's death, and that is a self contradicting situation. Thus, if there are no special provisions (which we will discuss later ) FTL travel will not only allow violation of causality, but it can also produce unsolvable paradoxes.
At this point, I want to clearly list the various events which must happen to produce an unsolvable paradox in our "FTL bullet" example. Through the rest of our FTL discussion, this will be helpful as a reference listing.
It is important to note that the real crux of this problem does not come from the form of the FTL travel used, but from the relationship between the two, ordinary frames of reference for observers (O and Op) who never themselves travel FTL. This ordinary relationship (determined by relativity) can be demonstrated through experiments today, and as long as the exact same experiments can be performed in the future to yield the same results, then this argument must still hold. This is the power of this problem, and we will see that the special provisions we will discuss later must concern themselves with the ability of the observers to use the relationship between themselves in order to produce unsolvable paradoxes. Thus, the provisions will not be specifically concerned with the form of FTL travel used or the future theories which might suggest FTL travel, because the problem we have discussed here will be present regardless of either of these considerations.
And so, we have discussed the two problems which arise with FTL travel. Our next job is to consider various, specific FTL concepts in light of these problems. If your not interested in the discussion of the various forms of FTL travel, and you want to take my word for it that they will all suffer from the problem discussed above, then you may want to skip to the " Special Provisions " section.. I'll leave that to the reader.
Next, we want to ask about how one might try to get around these problems. Many of you have heard of ideas which get around the light speed barrier problem. For example, if we can do our traveling in some other, parallel "space", then we won't be bothered by the light speed barrier in our own space. However, these ideas have a much harder time getting around the second problem. In fact, to get around the second problem, we will see that special provisions will have to be made.
Therefore, the format of this discussion will involve the following. First, we will look at the various concepts which exist for possibly allowing FTL travel. I will show how each of them allows one to get around the light speed barrier problem, and I will explain how (without special provisions) none of them can bypass the second problemproducing unsolvable paradoxes. Finally, I will introduce some special provisions (beyond the basic assumptions made for the FTL concepts) and show how one can imagine using these provisions in conjunction with some of the FTL concepts to get around the second problem.
Tachyons are hypothetical/theoretical particles which would travel FTL. The concept of the tachyon attempts to get around the infinite energy requirements which the light speed barrier problem poses on a particle as it approaches the speed of light. This was accomplished by demanding that the particle have certain characteristics which we will discuss here.
First, consider the energy and momentum. Recall that we can write the energy (E) and the momentum (p) of a particle of mass m as expressed in Equation 1:8 and Equation 1:6 which are duplicated here:
(Eq 9:1Copy of Eq 1:8)
(Eq 9:2Copy of Eq 1:6)
Where is defined in Equation 1:5 as . From this we find that , which is greater than 1 if v is greater than c. We can thus write
(Eq 9:3)
But since we can also express the energy squared as defined in Equation 1:7 :
(Eq 9:4Copy of Eq 1:7)
we find that the only way to get is if the mass squared is negative (because then reduces the sum in Equation 9:4 ). The mass would then be the square root of a negative number, and such an obviously unreal number is called an imaginary number (imaginary numbers may seem odd, but they have important uses in mathematics). In general we express such imaginary numbers as a product of a real number multiplied by something that symbolizes the imaginary squareroot of negative one: . So, the mass of a tachyon is imaginary. Further, from the equation for , we find that it too is imaginary if v is greater than c, but it is also negative because we have the i in the denominator of , and 1/i = i. (We can show this as follows: start with and multiply and divide the righthand side by (which doesn't change the value): . The top of that equation is just i, and the bottom is . Thus 1/i = i/(1) = i.) That would mean that from Equation 9:1 , the energy would still be a real, positive number (because to get E we multiply the i in the imaginary m by the i in to get ). The same would be true for the momentum, .
I would like to note that I have read elsewhere that the energy would be negative for a tachyon, but this doesn't seem to be the case.
The final interesting property of tachyons I will mention comes
from noting that as their velocity increases, the value of their
will become a smaller, negative, imaginary number
(because when v/c > 1, is a
negative, imaginary number that decreases as v gets larger). That
means that the value of a tachyons energy will decrease as the speed
of the tachyon increasesor in other words, as the tachyon loses
energy, it gains speed. One result of this is that if a charged
tachyon were to exist, then because it would travel faster than light,
it would give off a radiation known as Cherenkov radiation. This would
take energy away from the tachyon and cause it to go faster and
faster, continually giving off more and more energy. Neutral tachyons,
however, wouldn't do this.
In any case, we can consider the possibility that tachyons exist and always travel faster than light. They then never have to cross the light speed barrier, and they do not have infinite energy (but their mass is imaginary and their energy decreases as their velocity increases). However, they still cause trouble because of the second problemif you can use them for FTL communication, they can be used to create unsolvable paradoxes using the same arguments as we used in our "FTL bullet" example.
To explore the question of using tachyons for FTL communication, one can apply quantum mechanics to the energy equation of the tachyon. What one finds is that either (1) the tachyons cannot be localized, or (2) the actual effects of a tachyon cannot themselves move faster than light. In either of these cases, the tachyon cannot be used to produce an FTL signal.
A third idea would also allow the tachyon to exist without the possibility of using the tachyon to send FTL signals. The basic idea is that there would be no way to distinguish between the situation through which you could receive a tachyon and the situation though which you could transmit a tachyon. To show what I mean, consider Diagram 81 yet again. From the O frame of reference, a tachyon could be sent "from" * and "to" the origin. However, as long as you cannot distinguish between the transmitter and the receiver, then the Op observer could reinterpret this as a tachyon being sent "from" the origin "to" *. Neither, then, will believe that the tachyon went backwards in time. Obviously, there is no way for a message to be sent (because then you could identify the sender and decide which way the tachyon "really" went), and it wouldn't be quite right to call this FTL travel. However, it would allow tachyons to exist (though uselessly) without causing any problems.
And so, we find that with tachyons, one of the following must be true:
This next concept is often found in FTL travel methods of science fiction. The basic idea is that a ship (for example) can use a special field or travel in another space/dimension in order to "leave" the physics of our universe and thus not be limited by the speed of light.
Again, we see that this concept is basically designed to get around the light speed barrier problem; however, it doesn't deal very well with the problem of producing unsolvable paradoxes.
Though the FTL observer or signal which travels using this concept would leave the realm of our physics, the relationship between two observers (like O and Op) who stayed behind (within the realm of our physics) would not be effected. This means (if you recall the points made earlier about the "second problem") that the arguments for producing an unsolvable paradox must still hold (unless there are special provisions), because those arguments were based on the relationship between the two observers who themselves never traveled FTL (and thus never left the realm of our physics).
Thus, we very quickly see that with any such methods (as long as no special provisions apply) one can produce an unsolvable paradox.
Another concept which pops into the minds of science fiction lovers when considering FTL travel is that of "folding" space. Basically, the idea is to bring two points in space closer together in some way so that you can travel between them quickly without having to "actually" travel faster than light. Of course, by our definition of FTL travel in Section 6.1 (where the light you are "racing" against goes through normal space between the starting and ending points) this would still be considered FTL travel.
A frequently used approach for picturing this idea is to think of two dimensions of space represented by a flat sheet of paper. Then consider yourself at some point on the paper (call this point "o"). If you want to travel to some distant point ("D"), you simply fold/bend/crumple/etc the paper and place "o" and "D" close to one another. Then its just a matter of traveling the now short distance between the points.
Again, we see an FTL concept which is built in order to get around the problem of the light speed barrier. However, we will see, once again, that the second problem of FTL travel is not so easily fixed.
We begin to understand this when we consider again the sheet of paper discussed above. Every object in that two dimensional space has a place on the paper. However, because objects may be moving, their position depends on the time at which you are considering them. Basically, if you are sitting at "o", you imagine every point on that sheet of paper as representing space as it is "right now" according to your frame of reference. However, as we have discussed, what is going on "right now" at a distant location truly depends on your frame of reference. Two observers at "o" in two different frames of reference will have two different ideas of what events should be represented on the paper as going on "right now". This difference in simultaneity between different frames of reference is what allowed for the "unsolvable paradox" problem to exist in the first place. Thus, even though you "fold" the paper so that you don't "actually" travel faster than light, you don't change the fact that you are connecting two events at distant points (your departure and your arrival) which in another frame of reference occur in the opposite order. (In the other frame of reference, you aren't just bending space, you're bending spacetime such that you travel backwards in time.) It is that fact which allowed the unsolvable paradoxes to be produced.
In the end, unless special provisions are present, one can use this form of FTL travel in our FTL bullet example (I refer you back to the listing of events in Section 8.3 ). Op will fold space in his frame of reference to connect the passing event with the event "*", while the third observer will fold space from his frame of reference to connect the event "he sees the victim die" with an event "O learns of the victims death before the FTL bullet is sent". Thus, you can used this method to produce an unsolvable paradox as we discussed earlier.
The final concept we will discuss before looking at special provisions is what I call spacetime manipulation. The idea is to change the relationship between space and time in a particular region so that the limitation of light speed no longer applies. This is basically confined to the realm of general relativity (though the more simplified concept of "changing the speed of light" can also be handled by the arguments in this section). We won't worry too much about the particulars of how GR can be used to produce the necessary spacetime, because the arguments that will be made will apply regardless of how you manipulate spacetime in the region of interest.
There are two general types of spacetime manipulation to consider. The first I will call "localized", because the spacetime that is effected is that surrounding your ship (or whatever it is that is traveling FTL). A basic example of this is the idea for FTL travel is presented in a paper by Miguel Alcubierre of the University of Wales ( the paper is available via the world wide web ). In the paper, Alcubierre describes a way of using "exotic matter" (matter with certain properties which may or may not exist) to change the space time around a ship via general relativity. This altered spacetime around the ship not only keeps the ship's clock ticking just as it would have if the ship remained "stationary" (in its original frame of reference), but it also "drives" the ship to an arbitrarily fast speed (with respect to the original frame of reference of the ship before it activated the FTL drive).
The second type is thus "nonlocalized", and it involves the manipulation of spacetime which at least effects the departure and arrival points in spacetime (and perhaps effects all the spacetime between). A basic example of this is the idea of a wormhole. A wormhole is another general relativity concept. Again, exotic matter is used, but here spacetime is effected so that two distant locations in space are causally connected. You can enter one "mouth" of the wormhole and exit from the other very distant "mouth" so as to travel FTL (by our definition in Section 6.1 ).
Both of these concepts get around the light speed barrier problem, but again we will argue the case for the problems with unsolvable paradoxes. To do this, we will first carefully describe the situation in which a couple of FTL trips will occur. Let's call the starting point of the first trip "A". B will then be the destination point of that trip. Also, consider a point (C) which is some distance to the "right" of B ("right" being defined by an observer traveling from A to B), and finally consider a corresponding point (D) which is to the right of A. Diagram 91 uses two dimensions of space (no time is shown in this diagram) to depict the situation (at least from some particular frame of reference).

(x and y are spatial dimensions) 
Now, let's go back to the FTL bullet example through which we first explained the unsolvable paradox problem. In this case, the FTL bullet travels from A to B through spacetime manipulation. (The event "the bullet leaves A" is event (1) in our list from Section 8.3 ). This means that all the spacetime along the bullet's path between A and B might be affected by the spacetime manipulation. Thus, we can no longer assume (after the bullet's trip) that a spacetime diagram such as those we have drawn (which only apply to special relativity, not GR) will still apply. However, the space between D and C does not have to be effected by the FTL drive. Because of that we can make our argument by considering the following events:
The above events show that even though the spacetime may be changed between A and B during the bullet's trip, the O observer can still know about and use the fact that the victim was killed in order to prevent the victims death. We use the same arguments we did in the section concerning the "second problem" ( Section 9.1 ), except that the two FTL portions (the bullet and the signal from the third observer) are sent from two different locations so that neither is affected by the other's effects on spacetime. Thus, as long as there are no special provisions, this form of FTL travel will still allow for unsolvable paradoxes.
Thus far, we have seen that the second problem is not easily gotten around using any FTL concept. However, we have also insisted during our arguments that none of these FTL concepts include "special provisions". The specific provisions we were referring to will be discussed here. Basically, these are ideas which allow one to bypass the second problem in some way, and the ideas are generally not specific to any one form of FTL travel. They don't require that you bend spacetime in some way or that you travel in some other universe or that you be made of some specific form of matter when you do your FTL traveling. What they do require is for the universe itself to have some particular property(ies) which, in conjunction with whatever form of FTL travel you use, will prevent unsolvable paradoxes.
There are four basic types of provisions, but we can express the general idea behind them all before we look at each one specifically. Recall that in producing the unsolvable paradox in our "FTL bullet" example, there was a series of events listed , each of which had to occur to produced the paradox. The provisions simply require that at least one of these events be prevented from occurring. With the first and second provisions we will discuss, no restrictions necessarily have to be placed on the actual FTL travel, and any of the events (even those not directly dealing with the FTL travel) can be the "disallowed" event. The other two provisions place restrictions on the actual FTL travel in certain cases in order to prevent the unsolvable paradox.
In the first provision, one of the events in our list is not so much prevented as it is "transferred" to or from another (parallel) universe or reality. For example, say O has just received the information about the victim who dies at the "*" event, and O is waiting to stop Op from firing the FTL bullet. However, before he stops Op, he could find himself transferred to a parallel universe. In this universe he is able to stop Op from firing the bullet. The unsolvable paradox is resolved because the information about the death at "*" was not from the universe in which O stopped Op. Instead, O brought the information from a very similar parallel universe when he came over.
As another example, the bullet which killed the victim could have appeared from a parallel universe rather than being sent from Op in "our" universe. In this case, it is the "other universe bullet" which kills the victim. This bullet could seem to come from Op in our universe, though it actually came from an Op in the parallel universe. So, O is lead to believe that the bullet came from his own Op, and O stops Op from firing the FTL bullet. However, he doesn't prevent the death of the victim because the bullet which did the killing came from the "other universe Op". Again, the paradox is resolved.
Now, in that second case, the FTL bullet wasn't just performing FTL travel, but was involved with interdimensional travel. However, the second FTL signal in which the information is sent from the third observer to O (event number 4 in our list ) was allowed. Thus, though this provision can effect the FTL trips, it doesn't have to forbid either of them.
In the end, as long as one of the events is forced to transfer to or from a parallel universe, there will be no unsolvable paradox (although why or how the interuniverse transfer would occur is left unanswered). Also, we should note that this provision could be applied with any of the FTL concepts we have discussed in order to allow them to exist without being selfinconsistent.
The second provision is what I am calling "consistency protection". The idea is that the universe contains some sort of builtin mechanism whereby some event in our list of events would not be allowed to occur.
An example of such a mechanism can be found when we look at the situation through quantum mechanics. (A theory of Steven Hawking called the "chronology protection conjecture" (CPC) attempts to do just thatthe jury is still out on this theory, by the way, and will probably be out for a long time.) In quantum mechanics (QM), we do not think in certain terms of whether or not an event will occur in the future given everything we can possibly know about the present. Instead we consider the probability of an event (or string of events) occurring. One form of consistency protection would insist that QM prevents the unsolvable paradoxes because the probability of all the events occurring so as to produce an unsolvable paradox is identically zero.
Under this explanation using QM, our bullet example would be resolved through arguments similar to this: It may be that the Op observer is unable to produce the FTL bullet (perhaps his FTL gun fails), thus averting the paradox. If he is able to get the FTL bullet on its way, then perhaps the bullet will end up missing its mark. If it does hit the victim, then perhaps the victim's friend will be unable to send an FTL signal back to the O observer (perhaps his FTL message sender fails). If the signal to O gets sent, it still might not be received by O. If O receives it, he may be unable to stop Op from firing the bullet. In any case, this particular QM explanation would insist that one of these events must not occur, because the quantum mechanics involved forces the probability of all of the events occurring to be zero.
To sum up, this provision requires that some mechanism exists in the universe that would prevent at least one of the events from occurring so that the unsolvable paradox does not come about. This mechanism does not have to specifically target any of the FTL trips/messages which one might want to make/send, but it could disallow any of the events which must be present for the unsolvable paradox to occur. We should also note that this provision (just like the last) can be apply regardless of the FTL concept used.
This provision is sort of an extension on the previous one, but its mechanism specifically targets the FTL travel so as to restrict one of the FTL trips or messages one must use to produce an unsolvable paradox. Remember that in the list of events for our FTL bullet example, there were two different FTL portions (the FTL bullet and the FTL message from the third observer to O). This provision would cause the sending or receiving of one of these "messages" to strictly prohibit the sending or receiving of the other. I will try to illustrate the basic way in which such restrictions could work to always prevent unsolvable paradoxes. I will then give an example where this provision is implemented with a particular FTL concept.
For the illustration, we need to consider each of two possibilities within our FTL bullet example. In the first possibility, the Op observer is allowed to send his FTL bullet which strikes the victim, but that FTL trip must then restrict the third observer's ability to send the FTL message to O. In the second example, the third observer happens to decide to send some FTL signals to O at some point before the event "*" (which is the event in our example that usually marked the victim's death). Now, we let the third observer continue to send those FTL signals until some point after "*". Then, if the victim dies at "*" because of the FTL bullet, then since the third observer is sending FTL signals to O at that point, he would be able to tell O about the victim's death, and the paradox would still be possible. Thus, in this second case, the FTL bullet must not be allowed to strike the victim (the FTL travel of the bullet is restricted because the third observer sends FTL signals to O).
So, how would these restrictions work in these two possible cases? Well, as it turns out, if all unsolvable paradoxes are going to be averted while only placing restrictions on particular FTL trips, then there must be a very specific provision in place. To explain this, we will look at both possible situations, and consider diagrams which explain each one. (Note that these diagrams are drawn a little differently from Diagram 81 so as to better show the point I am trying to make here.)

(Case 1The FTL bullet is allowed to strike at the event "*") 
In this diagram we mean to illustrate case one in which the FTL bullet leaves the "passing event" (i.e. the origin, "o") and is "received" by the victim who immediately dies at event "*". Now, I have also drawn parts of two light cones (marked in yellow). One part is the "upper half light cone of the event '*'," and the other is the "lower half light cone of the passing event, 'o'". The upper half light cone of "*" contains all events which an observer at "*" (like the third observer in our bullet example) can influence without having to travel FTL. All observers agree that all events in this area occur some time after "*" (as discussed in Section 2.8 ). Also, the lower half light cone of "o" contains all the events which could effect "o" (which, remember, is the event at which the FTL bullet is sent) through nonFTL means. Thus, as long as no FTL signal/traveler can leave as an event in the upper half light cone of "*" and be received as an event in the lower half light cone of "o", then all unsolvable paradoxes will be averted. There would be no way for the third observer to witness the death of the victim and afterwards get a signal to O before the bullet is fired.
Now, that seems to be straight forward. We just need to make this provision: When an FTL signal is transmitted as event T, and it is received as event R, then it must be impossible for any information to be sent as an event in R's upper ("future") light cone and end up being received as an event in T's lower ("past") light cone. If the universe restricted FTL travel in this way, it would be impossible to produce unsolvable paradoxes.
However, we can see that the matter can get a little complicated when we consider things from O's frame of reference (which is also the frame of the third observer). In this frame, after the third observer witnesses the victim's death at "*", the event "the bullet leaves" hasn't occurred yet. He might then argue that no FTL signal has yet been sent which would keep him from sending a FTL message to O. The problem with his argument is that he has already witnessed the result of the FTL bullet being sent (even if it hasn't occurred in his frame yet). Thus, any FTL signal he tries to send to O (in the lower half light cone of the origin/passing event/bulletbeingfired event) must be prevented from being received by O.
Ah, but what if he (the third observer) just happened to decide to start sending FTL signals to O (just to chat) before the bullet strikes the victim? That leads to our second case. Here, then, is a diagram we will use to describe this second case.

(Case 2The FTL bullet may not be allowed to strike at the event "*") 
Now, there are a few extra events here. The point "s" marks the point where the third observer starts sending FTL signals to O while "T" marks the point where he finishes sending those FTL signals. The point "R" marks the point where O receives the last message which was sent at "T". Now, here we have drawn the upper and lower half light cones of interest, and according to our discussion above, it would be impossible for Op to send his bullet at the origin, "o" (which is in the upper half light cone of R) and have it "received" by the victim at "*" (which is in the lower half light cone of T). So, according to that argument, the bullet doesn't strike while the third observer is sending FTL signals to O, and so the third observer never tells O about the victim's death.
However, this doesn't have to be what happens, and we might just end up back at the first case. You see, either (1) the signals sent by the third observer are all successful, and the FTL bullet is restricted from striking the victim at "*" (that's the second case); or (2) the FTL bullet does strike the victim at "*" and any FTL signals that the third observer sends after "*" are restricted from reaching the O observer before the bullet is fired (this is the first case, even though the third observer was sending signals to O just before the bullet hit). The obvious question, then, is "which one of these two cases actually occurs?" The answer happens to be, "it really doesn't matter." You see, as long as one or the other does occur, the situation remains self consistent and no self inconsistent paradoxes are produced. Roll some dice and pick one, if you like, or let some unknown force decide which happens. It really doesn't matter for our argument. Is that a bit odd? Yes. Is it selfinconsistent so as to produce unsolvable paradoxes? No.
Finally, as example to show this provision in action with a particular FTL concept, let's consider a case where spacetime manipulation is used via a wormhole. Recall that in our discussion of this FTL concept in
Section 9.4
, we showed that one can still produce unsolvable paradoxes. Notice, that there still must be two FTL parts (we discussed one FTL "trip"the bulletfrom A to B and anotheran FTL messagefrom C to D). Now, to prevent the paradox, the existence of the wormhole that allows the bullet to travel from A to B could forbid the existence of the wormhole that allows the FTL message to go from C to D. This is a situation where case 1 applies, and here the way the provision is satisfied comes from the conceptual ability of one wormhole's existence to forbid the existence of another wormhole.
And so, we have a provision which simply restricts (in a very particular way) certain FTL trips because of other FTL trips. We have found that there doesn't have to be a discernible answer to the question of whether trip A disallows trip B or trip B disallows trip A, but as long as it is one case or the other, this provision will keep all situations self consistent and thus avoid unsolvable paradoxes.
The fourth and final provision is (again) something of an extension to the previous one. This provision also forbids certain FTL signals, but it does so in a very specific and interesting way (there will be no question as to which trips are allowed and which are not). To explain this provision, I will start by describing a situation through which the provision could be applied. I will then explain how the provision works, given that particular situation.
Now, as I describe the situation, I will use the idea of a "special field" to implement the "special frame of reference". However, it isn't necessary to have such a special field to imagine having a special frame of reference. I am simply using this to produce a clear illustration.
So, join me now on a journey of the imagination. Picture, if you will, a particular area of space (a rather large areasay, a few cubic lightyears if you like) which is permeated with some sort of field. Let this field have some very particular frame of reference. Now, in our imaginary future, say we discover this field, and 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 the x and t axes in a spacetime diagram for the observer will be the same as the ones for the special field, regardless 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.
For example, consider two observers, A and B, who both start out stationary in the frame of reference of the field. Under normal circumstances, if A (who starts out next to B) began to travel with respect to B, then later turned around and returned to B, A would have aged less because of time dilation (this is fully explained in Section 4.1 of Part II if you are interested). However, if A uses the special property of this field we have introduced, his frame of reference will be the same as B's even while he is moving. Thus, there will be no time dilation effects, and A's clock will read the same as B's.
Now, for the provision we are discussing to work using this special field, we must require that all FTL travel be done while using this field's special property. How will that prevent unsolvable paradoxes? Well, to demonstrate how, let's go back to our FTL bullet example and consider one of two cases. In case 1, we will let Op's frame of reference be the same as the frame of reference of our special field. With this in mind, let's go through the events listed in Section 8.3 once again; only this time, we will require any FTL travel to use the special property of the field we have discussed.
So, here is the new list of events given that the special frame of reference of the field is the same as Op's frame. Remember, our new provision requires that any FTL trip will have to use the property of our special field, thus the object/person/message traveling FTL will be forced to take on the frame of reference of our special field (Op's frame in this example). (It may be good for you to review the original list before reading this one):
But that is where the "agains" stop. You see, in the original argument event (4) was possible in which the third observer sends this information about the future to O via an FTL signal. In the frame of reference of O (and the third observer), that FTL signal could be sent after the victim's death and arrive at O before the passing event (when the bullet was fired). But now, as the FTL signal is sent, it must take on the frame of reference of the special field. That frame of reference is the frame of Op, and in that frame the victim dies after the bullet is fired. So, in the new reference frame of the message (forced on it by the provision we are making) the bullet has already been sent, and thus the FTL message cannot be received by O before the bullet is sent.
From the frame of reference of the third observer, he simply cannot get the FTL signal to go fast enough (in his frame) to get to O before the bullet is sent. From Op's frame of reference (that of the special field) any FTL signal (even an instantaneous one) can theoretically be sent using our provision. However, from O's frame (and that of the third observer) some FTL signals simply can't be sent (specifically, signals that would send information back in time in Op's frame of referencelook again at Diagram 81 to make this clear). This prevents the unsolvable paradox.
We can also consider case 2 in which the special frame of reference of the field is the same as O's frame of reference. In this case, any FTL traveler/signal/etc must take on O's frame of reference as it begins its FTL trip. Thus, as Op passes O and tries to send the FTL bullet from his frame of reference, the bullet will have to take on O's frame as it begins is FTL trip. But in O's frame of reference, the event "*" has already occurred by the time O and Op pass one another. Therefore, from the FTL bullet's new frame of reference (forced on it by the provision we are making), it cannot kill the victim at the event "*" since that event has already occurred in this frame. Thus, the paradox is obviously averted in this second case as well because of our provision.
So, in the end, if all FTL travelers/etc are required to take on a specific frame of reference when they begin their FTL trip, then there will be no way an unsolvable paradox can be produced. This is because it takes two different FTL trips from two different frames of reference to produce the paradox. Under this provision, if you are sending tachyons, the tachyons must only travel FTL in the special frame of reference. If you are folding space, the folding must be done in the special frame of reference. If you are using the special field itself to allow FTL travel, then you must take on the field's frame of reference. Etc. If these are the cases, then there will be no way to produce an unsolvable paradox using any of the FTL concepts.
As a final note about this provision, we should realize that it does seem to directly contradict the idea of relativity because one particular frame of reference is given a special place in the universe. However, we are talking about FTL travel, and many FTL concepts "get around" relativity just to allow the FTL travel in the first place. Further, the special frame doesn't necessarily have to apply to any physics we know about today. All the physics we have today could still be completely relativistic. In our example, it is a special field that actually has a special place in the physics of FTL travel, and that field just happens to have some particular frame of reference. Thus, the special frame does not have to be "embedded" in the makeup of the universe, but it can be connected to something else which just happens to make that frame "special" for the specific purpose of FTL travel.
And so, we have seen the four provisions which would allow for the possibility of FTL travel without producing unsolvable paradoxes. For the case of the real world, there is no knowing which (if any) of the provisions are truly the case. For the purposes of science fiction, one may favor one of the provisions over the others, depending on the story one wishes to tell.
Since this document is meant for the rec.arts.startrek.tech newsgroup, it seems appropriate to take all we have discussed and apply it to what we see in Star Trek. Of course, it would be foolish to assume (unfortunately) that the writers for the show take the time to learn as much about these concepts as we now know, and I am certainly not implying that a conscious effort was made to incorporate what we know to be true in a consistent way on the show (after all, this is Star Trek :'). However, interestingly enough, if we apply the concepts correctly, we can explain most of what Star Trek has shown us. That is what I will try to do here.
First, we might want to consider the four provisions and try to decide which one would best fit Trek so that everyday warp travel couldn't be used to produce unsolvable paradoxes.
So, let's consider both the first and second provisions. In these cases, neither of the two FTL trips in our FTL bullet example will necessarily be forbidden. So, if we consider that example yet again, we can make the following argument: Let Op be the Enterprise. Then, rather than sending a bullet, the Enterprise could itself travel from the origin to "*". It could then (through ordinary acceleration) change its frame of reference to match O's. Then it could travel from "*" (or just after "*"we have to give them a little time to do their acceleration) back to the O observer, and it could get to O before it ever left for its first FTL trip (i.e. we put the Enterprise in place of the FTL signal sent by the third observer). Thus, since neither the first or second provision has to forbid any of these actions, the Enterprise could use everyday warp travel via this method to easily travel back in time without having to do something as dangerous as zipping around the sun (as they have had to do on the show).
In addition, if the first provision governed normal warp travel, then making different trips from different frames of reference would introduce the possibility that you would find yourself being transferred to another parallel universe to prevent unsolvable paradoxes. Also, if the second provision governed normal warp travel, it would require Star Trek ships to be careful as to which frames of reference they were in when they decided to enter warp. After all, they may not want to accidentally meet themselves from a previous trip (in which case the universe may destroy them to protect self consistency). So, there seems to be some daunting arguments against using either the first or second provision to keep ordinary warp travel from producing unsolvable paradoxes in Trek.
Okay, what about the third provision? With that provision it would be impossible to use ordinary warp travel as a "time machine". However, this provision does cause certain noticeable restrictions on some FTL trips (remember, it allows certain FTL trips to prevent other FTL trips). There could be cases where the Enterprise would be prevented from completing its warp trip on time because of an FTL signal sent by someone else. We certainly don't see that on the show (not surprisingly). So, considering this provision, I can't easily point out any arguments to support using it to keep warp travel from being self inconsistent.
This leaves us with the fourth provision, and I think you will see that it the provision of choice for the purposes of Trek. Of course, this fourth provision must involve some special frame of reference; therefore, we might first ask about where this special frame might come from. Thus, I will make a proposal for answering such a question in the next section, and then I will present what I believe are strong arguments for using the fourth provision to keep normal warp travel from being self inconsistent in Trek.
When we discussed the fourth, "special frame of reference" provision, I introduced the idea of a field which had a particular frame of reference. For Star Trek, we can imagine subspace to be this field, and we can let it pervade all of known space. Then, subspace (or at least some property of subspace) would define a particular frame of reference at every point in space. When you entered warp, you would take on the frame of reference of subspace and keep it, regardless of your velocity with respect to subspace. This would ensure that normal, everyday warp travel would not produce unsolvable paradoxes (as we discussed in Section 9.5.4 ).
So, what does this provision give us that the third provision didn't? Well, by assuming that subspace defines a special frame of reference, we can explain some interesting points on the technical side of Trek. For example, in the "Star Trek the Next Generation Technical Manual" (and in other sources) we see that the different warp numbers correspond (in some way) to different FTL speeds. But when they say that Warp 3 is 39 times the speed of light, we must ask what frame of reference this speed is measured in. With subspace as a special frame of reference, it would be understood to mean "39 times the speed of light in the frame of reference of subspace."
The same idea can be applied to references made to impulsedriveonly speeds. In the Technical Manual, they mention efficiency ratings for "velocities limited to 0.5c." They also mention the need for added power for "velocities above 0.75c." But these velocities are all relative, and so we must ask why these normal, slower than light velocity of the Enterprise should matter when considering efficiencies, etc. After all, the Enterprise is always traveling above 0.5 c in some frame of reference and above 0.75c in some other frame of reference. However, since impulse is supposed to use a subspace field to "lower the mass of the ship" (so that it is easier to propel), we could argue that the speed of the ship with respect to subspace (assuming subspace defines a special frame of reference) would effect efficiencies, etc.
Further, there is a much more documented example which refers to warp 10. As many of you know, warp 10 is supposed to be infinite speed in the Next Generation shows. That means that the event "you leave your departure point" would be simultaneous with the event "you arrive at your destination". But, as we have discussed, the question of whether two events are simultaneous or not truly depends on the frame of reference you are in. So, we ask, in what frame of reference is warp 10 actually infinite speed. Again, we can use the frame of reference of subspace to resolve this issue. Warp 10 would be understood to be infinite speed in the frame of reference of subspace.
Finally, using this provision, there would be a standard, understood definition for measuring times, lengths, etc. Times would be measured just as it would tick on a clock in the frame of reference of subspace, and distances would be measured just as they would be by a ruler at rest in the subspace frame of reference. Basically, the feeling we have for the way things work in every day, nonrelativistic life would be applicable to Trek by using the subspace frame of reference as a standard, understood reference frame.
And so, I believe that the fourth provision gives us the best explanation for how normal, everyday warp travel in Trek could be self consistent.
Given the previous discussion, we see that the fourth provision seems to fit Star Trek like a glove. Thus, it may be best for us to view 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 regardless of your speed. Not only does this mean that normal warp travel cannot be used to produce unsolvable paradoxes, but since in warp your frame of reference would no longer 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.
(As a note, this is similar to Alcubierre's idea for "warp" travel (mentioned earlier), but in his idea the traveler did not take on a "special" frame. Instead, he took on the frame he had before entering warp, but that allows two trips from two different frames of reference to produce an unsolvable paradox. If we add subspace as a special frame of reference to Alcubierre's idea, we could get a self consistent situation which would be very similar to what we see in Trek.)
For more information on how this might conceptually work in the science fiction world of Trek (at least one way I imagine it) you may want to read my other regular post, "Subspace Physics". Here, however, we can at least use this "picture" of warp to consider how the outside universe might appear to someone traveling at warp speed. Remember, at any point the warp traveler's frame of reference it is as if he is sitting still in subspace's reference frame. We could illustrate the way such an observer would picture a particular event by using the following idea: Picture a string of cameras, each a distance (d) away from the one before it. Let these cameras all be stationary in the frame of reference of subspace, and let them all be pointed at the event of interest. Further, let each camera have a clock on it, and let all the clocks be synchronized in the subspace frame. Then, we can set each camera to go off with the time between one camera flash and the next being d/v (where v is the FTL velocity of the observer we want to illustrate). Then, each picture is taken in the subspace frame of reference, but the string of pictures (one from each camera) would form a movie in which each frame was taken from a different place in space from the previous frame. Thus, we can use this to produce a film of how an event would look to a warp traveler.
Of course, in Trek they have subspace sensors which do all their seeing for them (faster than light, of course). However, the above does illustrate one's ability to use this view of warp travel to answer various technical questions.
Now, there are cases in Trek where FTL travel exists without necessarily using subspace (and thus the subspace frame of reference would not apply and would not prevent unsolvable paradoxes). For example, if the wormhole in Deep Space Nine is assumed to be the same as a wormhole we theorize about today, then it wouldn't need to deal with subspace to allow FTL travel. (Now, what they call a wormhole doesn't necessarily have to be what we call a wormhole, but for this illustration, let's assume it is). So, if the wormholes in Trek aren't bounded by the subspace frame of reference, we could imagine a situation whereby they could be used to cause unsolvable paradoxes. This is true for any form of FTL travel in Trek which might not use subspace. However, I propose that in cases where subspace isn't used (so that its special frame of reference could not prevent unsolvable paradoxes) then the first or second provision, "parallel universes" or "consistency protection", would apply. In that way, we can allow for nonwarp/nonsubspaceusing FTL travel in Trek while still preventing unsolvable paradoxes.
Further, consider time travel in Trek. Actual time travel couldn't be accomplished by using subspace alone (the subspace frame along with the fourth provision would prevent it). However, I propose again that such travels in time should not be able to produce unsolvable paradoxes because the "parallel universes" or "consistency protection" provisions would apply (since subspace alone couldn't be in use to produce the time travel).
For example, consider the Star Trek: The Next Generation episode, "Time's Arrow" (in which Data's severed head is found on 24th century Earth, and Data eventually travels back in time to (unintentionally) leave his head behind to be found). Now, after the head was found, one of the crew (let's say Riker, just to use an example) could decide to try to produce an unsolvable paradox. Riker may decide to do everything in his power so as to keep Data from going back in time. He may even try to destroy Data and his head to accomplish this task. Of course, Riker isn't the type of person to do this, but what if he was? Well, in that case, he would be trying to produce an unsolvable paradox, and the first or second provision would prevent it. For the first provision, the head found in the 24th century might have actually come from a parallel universe. For the second provision, we could imagine various ways in which Riker might fail in his task of trying to keep data from going back in time. Further, we could consider the case in which he would succeed in producing an unsolvable paradox and we could insist that such situations would destroy themselves or prevent themselves from ever happening.
Such a situation is seen in a particular Voyager episode. In this episode, members of the crew are caught in a "subspace fissure", and they travel back in time. By the end of the episode their trip back in time has produced a selfinconsistent situation. That series of events then becomes impossible and ceases to exist by the closing credits. This could be seen as a result of having the "consistency protection provision" apply to a case where the subspace frame of reference is bypassed via "subspace fissures".
So, even though we can be relatively sure that this was not the intention of the writers, the situations shown do seem to comply with the concepts we have developed. If you are a scifi writer, understanding these concepts may help you in the future.
To sum up, we have found that by introducing a special frame of reference which would be "attached" to subspace, and by further insisting that any type of FTL/time travel done without using subspace be governed by the "parallel universe" or "consistency protection" provisions, we will not only have a self consistent universe for our Star Trek stories, but we can also (coincidentally) explain many of the "but how come...?" questions which some Star Trek episodes produce.
In Part I of this FAQ, I presented some of major concepts of special relativity, and here in Part IV , we have discussed the considerable havoc they play with the possibility of faster than light travel. I have argued that the possibility of producing unsolvable paradox is a very powerful deterrent to all FTL concepts. Further, we have introduced four basic provisions, at least one of which must be in place so that FTL trips/signals (sent using any of the FTL concepts) cannot be used to produce unsolvable paradoxes. Finally, we looked at the science fiction of Star Trek while considering all that we had discussed. We concluded that warp travel could be governed by the fourth provision (via subspace defining a special frame of reference) while all other FTL travel (or time travel) could be governed by the first or second provisions. This, I believe, best explains what we see on Star Trek.
If you have not read Part II or Part III of this FAQ, and you are interested in learning more about relativity (special and general), then you may want to give them a look.
As the end result of producing this FAQ, I hope that I have at least informed you to some extent (or perhaps just helped to clarified your own knowledge) concerning relativity and the problems it poses for FTL travel.
Jason Hinson