How does faster than light travel violate causality?

(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.

For the tachyon case, you implicitly assume that the tachyons ultimately travel forward in time, just going faster than light. But there exist Lorentz transformations (that is, other inertial frames) in which such a particle would travel backward in time as it traverses space.

You may have trouble believing this, so consider a 1+1 spacetime. This spacetime has four distinct regions: future timelike, +x spacelike, -x spacelike, and past timelike. These regions are cut by two diagonal, lightlike lines, which divide spacelike from timelike and represent the asymptotes of hyperbolas.

Most massive objects have four-velocities in the future timelike region, and a Lorentz transformation will keep them in that region no matter what. They are, however, quite free to move around in that region, provided that they maintain an overall magnitude of $c$.

A tachyon is the same, except it occupies either the +x-spacelike or -x-spacelike regions. This means that, even if you think your tachyon travels forward in time, there exists some reference frame in which it travels backward in time. You may not see causality violated, but someone else will.

The Alcubierre drive gets around this problem by changing the geometry of spacetime itself so the above notions get much more complicated. The basic idea is this: inside the bubble, you can fire off a photon and it will, assuredly, go away from you along a well-defined trajectory, one that is "faster" than yours. Causality is not violated because all observes will agree that you merely took a timelike trajectory in a very unusual spacetime--events before and events after your trip are still well-defined.

The danger here in thinking about the Alcubierre drive is that we often take the perspective of a distant observer and naively think that our coordinates (our measures of time and space) will not be affected by the drive, but they are. The geometry of the drive itself will warp and distort coordinate lines around it, resolving any seeming causality violations.

What everyone else said, but note that this STILL violates causality if you use general relativity to create one of these "warp drive" scenarios--the "warp drive" can always be restricted to an arbitrarily small region of spacetime, and then special relativity will be true over the rest of spacetime, and the problems will still arise.