(There's a couple of these questions kicking around, but I didn't see anyone give the "two boosted copies" answer. Generically, I'd say that's the right answer, since it gives an actual causality violation.)
In your scenario, the two planets remain a hundred thousand light years apart. The fact is, you won't get any actual causality violations with FTL that way. The trouble comes if the two planets are moving away from each other. So, let's say that your warp drive travels at ten times the speed of light. Except if the two endpoints of the trip are moving, then what does that mean? Ten times the speed of light relative to which end?
Let's say Tralfamadore is moving at a steady 20% of $c$ (the speed of light), away from Earth. (So, Earth is moving at a steady 20% of $c$ away from Tralfamadore.)
If I leave Tralfamadore (in the direction of Earth) and I am travelling at anything less than 20% of $c$ relative to Tralfamadore, then I am still moving away from Earth. I'll never get home.
Let's say instead I am travelling at 60% of $c$ relative to Tralfamadore. I will catch up to Earth. Relative to Earth, how fast am I approaching? You might guess the answer is 40% of $c$, but it's 45.45%.
Generally, the velocity subtraction formula of relativity is: $$w = (u-v)/(1-uv/c^2)$$
Let's say instead I am travelling at 100% of $c$ relative to Tralfamadore. Plug $u=c, v=0.2c$ into the formula and get $w=c$. Relative to Earth, I am approaching at 100% of $c$! The speed of light is the same for everyone.
So finally, let's say instead I am using your warp drive to travel at 1000% of $c$ relative to Tralfamadore. Relative to Earth, I am approaching at -980% of $c$. In Earth's reference frame, I will arrive on Earth before I leave Tralfamadore. Now you may say this in itself isn't a causality violation, because we've applied Earth's calendar to Tralfamadore. And that's true, but I'll make a round trip:
- In the futuristic Earth year of 3000, Tralfamadore is 98,000 light years away, and receding at 20% of $c$. I leave Earth at 1000% of $c$, relative to Earth.
- In Earth year 13000 Tralfamadore is 100,000 light years away, and I catch up to it. I turn around and leave Tralfamadore at 1000% of $c$, relative to Tralfamadore.
- In Earth year 2796, I arrive home.
Earth's calendar certainly applies to Earth, and I arrived home two centuries before I left. No two ways about it, I'm a time traveller!
There is nothing special about ten times the speed of light. Given a warp drive that moves a certain amount faster than light, you can make the above time machine using two endpoints that are moving apart a certain amount slower than light, provided that the warp drive can move faster than light relative to either end. This time machine works for any form of FTL: tachyons, warp drives, wormholes, what have you.
Yes, if a particle would be travelling faster than light, it would always travel faster than light. This is what's called a tachyon, and they have in some sense imaginary mass.
The three regimes, time-like, light-like and space-like (i.e. subluminal, luminal and superluminal space-time distances) are invariant under Lorentz transformation. Therefore anything on a super-luminal 'mass-shell' would always stay there and could not be decelerated to light/ or sub-light speed.
The problem is not that it would violate relativity, but rather causality, since with faster than light information propagation one could 'travel back in time', therefore leading to paradoxes.
For an introduction check out Wikipedia
Best Answer
In order for an idea to lead to a logical contradiction, there needs to be something that it contradicts. Since the idea of faster than light travel is logically conceivable, it has no self-contradiction, and therefore there is no a priori contradiction. Thus, we have the following answer.
The only other way for us to reach a contradiction is for faster than light travel/communication to contradict something else. The obvious candidate is special relativity. So now we reduce our question to, does faster than light travel/communication violate special relativity? As I explain below, faster than light travel/communication does not contradict special relativity. Thus, we have the following answer.
Here I explain the conclusion of the above answer.
Two events $(t_{1}, x_{1}, y_{1}, z_{1})$ and $(t_{2}, x_{2}, y_{2}, z_{2})$ are said to be time-like separated if $$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} < 0, $$ they are said to be light-like separated if $$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} = 0, $$ and they are said to be space-like separated if $$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} > 0. $$
Two events that are time-like or light-like separated are those for which you could go from one to the other in less than or equal to the speed of light. Space-like separated events are those that are too distant compared to the time you would need to traverse to get from one to the other.
The main premise of special relativity is that the laws of physics are invariant under Lorentz transformations. In the passive sense, Lorentz transformations are simply a change of coordinates. What this means is that if $K$ is one inertial coordinate system, and $L(K)$ is another coordinate system obtained by a Lorentz transformation $L$, the fundamental equations of motion should be of the same form regardless whether they are written in terms of coordinates $K$ or coordinates $L(K)$.
To give a related example to make this more understandable, consider taking an inertial coordinate system $K$ and consider rotating the coordinates any angle $\theta$ about the $z$-axis to get new coordinates $R(K)$. This is also a change of coordinates, and because the universe doesn't have any "intrinsically preferred direction," it doesn't matter whether we've written the laws of physics with respect to $K$ or $R(K)$. Hence the laws of physics are rotationally invariant.
To understand Lorentz transformations, it might be better to first think about Galilean transformations. A Galilean transformation is simply a change in the velocity of the coordinates (so if you have an inertial coordinate system and another inertial coordinate system passes by at constant velocity, the two coordinate systems differ by a Galilean transformation), and the classical notion that physics is invariant under Galilean transformations is the idea that there is no meaning to absolute speed.
A Lorentz transformation is similar to a Galilean transformation, except it mixes time and space coordinates. Einstein found out that, upon closer examination, physics is not Galilean invariant but Lorentz invariant. The invariance under partial mixing of time and space coordinates leads to time dilation, Lorentz contraction, and the invariance of the speed of light, but most of all, it leads to the idea that different inertial coordinate systems will disagree on the ordering of space-like separated events.
If events $A$ and $B$ are space-like separated, one coordinate system might say $t_{A} > t_{B}$, but another coordinate system might say $t_{A} < t_{B}$. This is not a problem, however, because no communication can occur between space-like separated events. Also, time-like and light-like separated events never change order. Thus, causality is always preserved.
Now the problem with faster than light communication (of any kind) is that it involves sending signals between space-like separated events. Let's say $A$ and $B$ are space-like separated, and I send a signal from $A$ to $B$. In one inertial coordinate system, $t_{A} < t_{B}$ and I sent a signal from location $A$ to location $B$ in a certain time period. But in another inertial coordinate system, $t_{A} > t_{B}$, and I sent a signal from location $A$ to location $B$ to the past.
But if sending signals to the past is possible in one inertial coordinate system, then by Lorentz invariance it must be possible to do this in any inertial coordinate system. Thus, by performing the same thing I did at event $A$, a person at event $B$ can send that signal back to my past self, and essentially a form of time travel is achieved, leading to the grandfather paradox.
Now the grandfather paradox is not a contradiction by itself, because it is possible that we are deterministically fated to avoid creating various paradoxes in our timeline, but the whole idea of sending signals to the past is too much to entertain.
If signaling at faster than the speed of light is possible, there is another possibility, which is that Lorentz invariance is broken, and that there really is an absolute coordinate system after all. If you go back through my reasoning, you would find that Lorentz invariance was the key component to $(1)$ changing the ordering of events $A$ and $B$, and $(2)$ generalizing "sending a signal back in time according to one coordinate system" to "sending a signal back in time in any coordinate system." If laws of physics ultimately depend on an absolute coordinate system, then it is possible to send signals faster than light according to that one coordinate system without any backwards time traveling.
So to summarize, if faster than light communication existed, then either special relativity is incorrect and there is an absolute reference frame, or signaling into the past is possible. Since we have no reason to think that special relativity is incorrect and no reason to think that signaling into the past is possible, we are left with the conclusion that faster than light communication is probably impossible.
Ultimately, it depends on what empirical evidence will show us in the future. Any one of the possibilities I mentioned is possible (with no logical contradictions), but until then, not much more can be said.