(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.
In any curved spacetime we can still talk about local reference frames that are small enough scale we can ignore the curvature. We also can ask if there are closed timelike curves (CTC) which basically is asking whether we can time-travel to our past selves. CTCs are strongly thought to be impossible in reality.
The universe is thought to be spatially flat, but the spacetime as a whole is curved. CTC's are impossible: at each point in spacetime you have an "age of the universe". To be precise, this is maximum path-length (proper time) a geodesic could have between the big-bang singularity and said point. Any time-like or light-like curve is moving in the direction of increasing age of the universe; this is just as strong a concept of "future" and "past" as in flat spacetime.
With a single warp-drive you don't have CTC's. But you can still get CTC's with multiple warp-drives. Suppose you build a warp-drive on Earth and send it out into space. You start with an (almost) flat initial-condition and then generate a strongly curved spacetime (your warp bubble). Starting from a flat spacetime (or for very large scales from the spacetime of the universe), is much more physically realistic than starting from any other spacetime. You have to make your weird and wonderful curvature from an "empty canvas" !
With a warp-bubble, the highly curved spacetime is on a small scale. This allows us to glue two bubble spacetimes together so long as the ships don't get very close to each-other. If we consider two Earths, moving relative to each-other, that each make a warp-drive, we can set up the system to generate CTCs. This is one reason we suspect this to be impossible.
There is another reason to suspect making warp drives is impossible: Geodesics would have to diverge in some region, which is an anti-gravity effect. Neither matter nor light can make anti-gravity (antimatter has positive mass just like matter). The "attractive gravity only" rule is more precisely defined as an energy condition and at least one of these is violated by warp drives. Violating certain energy conditions would make the speed of sound faster than light which also allows for time-travel paradoxes.
In general, no known solution with CTC's is physically realistic. They either involve infinitely large systems that cannot be setup from an "empty canvas" or violations of energy conditions. For example, the Kerr metric concentrates it's energy condition violation in it's singularity. Real black holes are thought to lack this feature and be much deadlier instead.
Best Answer
Edit regarding 3+1 spacetimes and causality
I'll keep adding to the answer as I get more information, and hopefully everything will just evolve along. At the very least, I'll have a set of notes to work from in the future :) This is also the first, broadest, cut at an actual answer regarding causality.
Alcubierre sets out to find his warp drive metric using a 3+1 formulation of spacetime. In the 3+1 formulation, spacetime is described as a set of constant coordinate time spacelike hypersurfaces, (foliations, for the fancy). In doing this, you wind up with a line element that looks like (see erudite comments from @Jerry Schirmer below, I'm playing catchup):
$ds^2 = -d\tau^2 = \gamma_{ij}dx^idx^j + 2\beta_i dx^i dt - \left(\alpha^2 - \beta_i\beta^i\right)dt^2$,
where $\alpha$ is the lapse function, and is positive, and $\beta$ is the shift vector between spatial foliations. $\alpha$ describes how quickly time evolves, while $\beta$ describes how the spatial coordinates evolve in time. In other words $\alpha$ and $\beta$ describe how your ship moves through space and time per incremental step.
What's important here is that $ds^2$ is positive and for real space, $\gamma_{ij}$ is as well. Remember, hyperbolas look like $\dfrac{x^2}{a^2} - \dfrac{t^2}{b^2} = 1$. So, the line element equation above describes a globally hyperbolic system in space time. What's that mean? It means you can't close a curve in spacetime, so you can't violate causality. Note that $\beta^i$ squares up where it's important to maintain sign to maintain a hyperbola. I'd think there should be another requirement that $\alpha^2 > \beta_i\beta^i$, but Alcubierre doesn't mention this, so I'm guessing we don't actually need it.
Alcubierre isn't done yet, he's still got to find a metric that will fit in a 3+1 spacetime and do what he wants, (provide faster than light propulsion), but if he does, the above property of 3+1 spacetimes will guarantee causality.
Edit I Stand Corrected Regarding the Alcubierre Drive
@Superbest pointed out, that the claims for the drive were that it could go faster than the speed of light with regard to the laboratory frame, and hence with laboratory velocity. I found the original paper by Alcubierre on arxiv[2], and...
he's absolutely right!
The paper is amazingly well written and folks that have had a grad level general relativity class should be able to easily traipse through it. Alcubierre even shows that causality won't be violated. I haven't had time to digest the material enough to say why causality isn't violated except with the very unsatisfying statement, "Well, the math works out." Alcubierre was also quick to point out that he felt that with a bit of effort he could come up with an example that would violate causality:
"As a final comment, I will just mention the fact that even though the spacetime described by the metric (8) is globally hyperbolic, and hence contains no closed causal curves, it is probably not very difficult to construct a spacetime that does contain such curves using a similar idea to the one presented here."
OK, so to summarize. The math explanation and associated formulas I wrote below are correct. With uniform acceleration and no exotic matter whatsoever, you can travel more than x light years in x proper time years. In the case of the Alcubierre drive, however, that's not the trick they're playing. I hope to have more details soon, but in the meantime I'll leave you with this quote from Schild regarding the twin paradox and general relativity.
"A good many physicists believe that this paradox can only be resolved by the general theory of relativity. They find great comfort in this, because they don't know any general relativity and feel that they don't have to worry about the problem until they decide to learn general relativity."
End Edit
The explanation given in the Washington post article triggers a pretty common misconception:
"If an object reaches a distance x light years away in under x years, then it must be travelling faster than the speed of light."
What the article failed to mention is that the 14 days quoted is in the reference frame of the ship. The equation for the distance travelled with respect to time in the frame of the ship, (known as proper time), is
$$\mathrm{distance} = \dfrac{c^2}{a}\cosh\left(\dfrac{at}{c}\right)-\dfrac{c^2}{a},$$
where $a$ is the acceleration of the ship and $c$ is the speed of light.
Using this formula, it can be shown that at an acceleration of 188g, (188 times the acceleration due to gravity), the ship could reach Alpha Centauri in 14 days of ship time. You might point out that 188 g's would surely smush everyone against the back wall of the ship, but the beauty of the theoretical drive described is that you carry your own gravity well along with you and therefore, you're always in freefall and don't feel the acceleration.
Here's the problem though. The time that will have elapsed here on Earth will be much, much greater than the 14 days that elapsed on the ship. The expression for the time elapsed on Earth is
$$\mathrm{Earth\ time\ elapsed}= \dfrac{c}{a}\cosh\left(\dfrac{at}{c}\right),$$
which can be used to show that when the ship reaches Alpha Centauri, 817 years will have passed here on Earth.
The calculations shown here are nothing new, by the way. Rindler applied them to the problem of relativistic space travel for the first time in 1960 in a Physical Review article titled "Hyperbolic Motion in Curved Space Time" [1].
References
Rindler, W., "Hyperbolic Motion in Curved Space Time", Phys. Rev. 119 2082-2089 (1960).
Alcubierre's original warp drive paper http://arxiv.org/abs/gr-qc/0009013v1