What are some scenarios where FTL information transfer would violate causality? I've always heard people saying, "Faster than light information transfer can't happen because it would violate causality!  Effects can precede their causes!"
However, I'm trying to think of a situation where this would happen.  I'm sure it has something to do with relativity or something like that.  I sort of understand that people going faster perceive time slower.
Can someone help illuminate this for me by showing a scenario where causality is clearly violated due to FTL information transfer?
 A: Basically, this is because simultaneity is not an invariant notion in SR.
You probably know the classical example of two lightnings striking at the same time but different spots with respect to an observer who is at rest. But for an observer moving towards one of the lightnings, the lightning he is moving towards will have struck first. To an observer moving in the opposite direction, it will be the lightning he's moving towards. So each observer arranges the events in a different temporal sequence. This is true for space-like separated events, to use the technical jargon.
This is not problematic because space-like separated events should not be causally related... unless there is FTL communication/travel/whatever... That's when the funny things happen, when Jack responds to the phone call before Judy ever made it.
I've always wondered if the show writers of Star Trek, or any other SF series using FTL com, thought about it. For all the time travel stories these shows have, I can't remember a story that exploited this particular effect.
EDIT: Since some people have a hard time understanding special relativity, I thought it'd be nice to add a simple diagram to illustrate the relativity of the concept of simultaneity:

In this drawing, I represent two lightning strikes happening at a distance 1 (doesn't matter what units you choose, but the drawing is such that 1 unit on the position axis (horizontal) is equal to 1 unit on the time axis (vertical) such that the speed of light is 1 in this picture), on the left and right of an observer sitting at position zero. He perceives both lightning strikes at time 1, when the light from both strikes has traveled to him (blue lines crossing at coordinates (0,1)). 
But, another person is traveling at half the speed of light towards the lightning strike on the right (red line). He therefore sees that strike first, and only after that sees the second strike (the green intersections of the red line with both blue lines).
A: Let's assume you are stationary and are observing a spaceship flying by at a velocity $v$. 


*

*Case $v \lt c$: If the spaceship flies at relativistic speeds, you will notice their clocks going slower, but everything would happen in the same order as if they were stationary with respect to you. The causal relationships between events is preserved, so is the second law of thermodynamics.

*Case $v = c$: If the spaceship flies at the speed of light, you will notice their clocks stopped. Nothing is moving, so, in a limit sense, the causal relationships between events is preserved, so is the second law of thermodynamics.

*Case $v \gt c$: If the spaceship flies faster than light, you will notice their clocks going backwards. All the causal relationships are reversed and the second law of thermodynamics is violated.
A: Oh boy it's my favorite topic, causality violations!
Depending on the definition you give to causality violations and tachyons, it is fairly easy to give a causality violation on a spacetime. Here are some nice examples. Spacetime here is assumed flat (although topology may change), no math included as they are mostly diagrams that speak for themselves (just your basic Minkowski spacetime diagrams), and no coordinate change involved : this is just causality violation in a coordinate-invariant way.
Tachyon trajectories are in red, observers in blue, coordinate axis in black. 
Here's a simple example involving two observers : Observer $A$ emits a tachyon (very slanted) in the direction of observer $B$, which emits back a tachyon to observer $A$. 

You can check that the observers are all timelike while the tachyons are all spacelike trajectories. Given enough distances between $A$ and $B$, you can send tachyons arbitrarily far back in the past of the emission of $T_1$, and of course send arbitrarily many tachyons to compose whatever message you want. 
One may object that the fact that $T_1$ points to the "past" is cheating, but this is entirely a coordinate artefact : a boosted observer $A$ will see $T_1$ as future-pointing, with respect to its own spacelike hypersurface.
Slightly fancier example : take the spacetime to be the Minkowski cylinder $\mathbb{R} \times S^1$, with a single observer. 

A single observer can communicates with itself. This is not possible to do in $1+1$ dimensions in Minkowski space (it can be shown somewhat easily by the fact that in $1+1$ dimensions, timelike and spacelike dimensions are interchangeable and there are no closed timelike curves, so there are also no closed (smooth) spacelike curves). 
If we allow more dimensions, things become easier. Consider the $2+1$ dimensional example with (non-free) tachyons.

It is possible to have some helicoidal shape in $2+1$ dimension that is entirely spacelike, but goes back in the past of its own lightcone, which is a fairly bad thing. 
Once you have those various scenarios, it's not hard to construct one of the classic horrible causality paradox to show the various Cauchy development problems involved.
A: Suppose you and I have a conversation from a long distance away.  We're at rest with respect to each other and communicate much faster than light.  I say "How are you", and you wait a short time and say, "I'm fine thanks."
From our point of view, you were responding to my question.  However, from a reference frame moving from me to you at relativistic speed, your clock is significantly ahead of mine (a relativistic effect).  This means that although you thought you received the message shortly after I sent it, in this frame you didn't.  You actually received the message at an earlier time (before I sent it), but you thought it was later because your clock is ahead.
From your and my point of view, the order of events is


*

*I say "How are you?"

*You hear me say "How are you?"

*You pause a short time.

*You say, "I'm fine thanks."

*I hear you say, "I'm fine thanks."


From the frame moving from me to you, the order of events is


*

*You hear me say "How are you?"

*You pause a short time.

*You say "I'm fine thanks."

*I say, "How are you?"

*I hear you say, "I'm fine thanks."


The fact that the order of events changes between reference frames is simply part of relativity, with or without faster-than-light communication.  However, it seems strange in this scenario because you are responding to me.  Presumably, if I had said, "Where are my car keys?", you would have chosen a different response than "I'm fine thanks."  How then is it possible that you responded to my greeting before I uttered it, at least in some frame?
I'm not sure if this "violates causality", but it's unintuitive.
