We already have an answer why physically traveling faster than light would violate causality (the clock on board our hypothetical FTL spaceship would tick backwards to some outside observers).
However, would a wormhole-based FTL violate causality? In this case, never does an object actually move faster than light.
If this violated causality, then how could I send back information from, let's say, 2014 Earth to 2004 Earth?
Ok, so let's say you had a wormhole. How long would it take to get from point A to point B using it? Let's say it's instantaneous. A traveller would arrive the same moment they left, spend some time at point B (it's really a nice place; the B-ian people are friendly and the food is great), then use the wormhole to go back to point A. No problem right? No violation of causality? Perhaps, but you need to ask yourself "when is right now at point B?" Consider this diagram:
This is a Minkowski diagram. The red axes represent the reference frame we're in and the green axes represent a reference frame at some high velocity relative to ours. So now ask yourself that question, when is right now? By the red frame, right now is the x-axis. The wormhole could take you to any point along it. But what if I enter the wormhole travelling fast enough to be in the green frame? In that frame, instantaneous travel is anything along the x'-axis. Notice that accordingly, that would put me into the red frame's future (we're just looking at the first quadrant). So you say "well that's simple, the wormhole isn't moving in my frame so it would make use of my definition of instantaneous". Here's the bigger problem. Now if I'm in the green frame and enter the wormhole, I travel along the x-axis and the point I end up is actually in my past (trace a line from somewhere on the x-axis back to the ct'-axis that is parallel to the x'-axis, it leads to the past).
How does this explain how I can send a message back to 2004? Say I have the wormhole, I enter it in the red frame (let's assume that's the Earth frame). Then I get to point B, accelerate to be in the green frame and go back through the wormhole to our point A at x=0. So let's run through this. I start at x=0, I used the wormhole to travel to some point on the x-axis, I speed up (so shift the green frame so that the green origin is on the x-axis at our chosen point), then I return through the wormhole to x=0 except remember I'm travelling along the x'-axis now. Voila, I'm in 2004.
But hold on, you say. Didn't I already establish that the wormhole uses the definition of instantaneous from its own frame; the red one? True, I did say that. But what if at point B I find another wormhole travelling in the green frame that links back to x=0? Or what if I found a way of speeding up the other end of the wormhole? Then I could certainly travel back to 2004.
The only way to prevent me from using a wormhole to travel to the past is to make all wormholes exist in the same reference frame and transport objects using that frame's definition of instantaneous. But that would mean there is a preferred reference frame in the universe. And that would be a matter for another question on this site.
Some preliminaries:
(1) Wormholes are a type of curved spacetime. Therefore there can't be any complete analysis of this problem in special relativity. You need general relativity.
(2) The basic logic is not that causality violation implies FTL, it's that FTL implies causality violation. There are other reasons besides FTL why you can get causality violation. One of them is if your spacetime has closed timelike curves (CTCs).
(3) Because this is GR, not SR, there is no unambiguous way to define the velocity of an object relative to some other, distant object. The question assumes that going through a wormhole does not involve FTL velocities, but in fact there is simply no way to define what velocity it would be.
There are straightforward arguments to the effect that if wormholes are possible, then CTCs are possible as well. The basic idea is that either mouth of the wormhole can be acted on by gravitational forces (I guess in the sense that it has a certain Komar or ADM mass), and therefore you can manipulate it and move it around (at least in theory, if you had the ability to manipulate huge amounts of matter). So you take one mouth, accelerate it away from the other, and then bring it back. This is very much like the twin paradox; the two mouths are no longer synchronized temporally. This kind of thing is discussed in [Friedmann 1990] and [Echeverria 1991]. See Echeverria's figure 1.
The Echeverria paper discusses some toy models involving billiard balls, and shows that, surprisingly, one does not always get causality violation despite the presence of CTCs. Echeverria was a student of Thorne at CalTech, and Thorne also gives a popular-level discussion of this idea in ch. 14 of his book Black Holes and Time Warps. IIRC Echeverria's paper proposed a research program to investigate this kind of thing in more detail, but as far as I know it was never followed up. This may have been because the research program didn't work out, or because Echeverria didn't get a permanent job doing relativity.
Friedman, Cauchy problem in spacetimes with closed timelike curves, http://authors.library.caltech.edu/3737/
Echeverria, Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory, http://authors.library.caltech.edu/6469/
I had this same question for a long time and I had some new idea how to get hold of it in laymen terms.
I think the ingredient that was missing for me - and I might be totally wrong here - is, that creating a wormhole between two points A and B will take time and can not happen instantaneous. So connecting two points will take at least as much time as traveling there with light speed.
Once the connection is there, it is just normal curved space where some points are connected twice via a long and short route. But beside the heavily curved spacetime, I think there is really nothing special anymore. It is just hard to grasp as we are used to 2D diagram abstractions of flat 3D-Space. And here we suddenly work with curved 3D space and it seems like this could violate our light cone diagrams. But most likely it just doesn't
Sending information through the wormhole is not "fast" in any way. And violating causality is not happening here.
It would be IMHO a totally different matter, if wormholes could be opened spontaneously over arbitrary distances.
If the ends of the wormholes would also move relative to one another this might be a different story though.
Now please go ahead and downvote me without explanation ;-)
