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?
Best Answer
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.