Fig. 1: Bertha sends a message back in time
Here's the essential setup.
1. We have two pairs of distantly separated lovers: a) Bill and Bertha and b) Jim and Jane. Woes is them.
2. If you haven't figured it out from the graph yet, Bill and Bertha are colocal (not moving with respect to one another), as are Jim and Jane.
3. Jim and Jane are also moving at some velocity with respect to Bill and Bertha (that's why the x' and t' axes for Jim and Jane are tweezed inward with respect to the x and t axes for Bill and Bertha). This is very important, and is the second most essential geometrical intuition you must develop to understand SR.
4. Missing from the graph are lines indicating the "world (or light) cone". Simply imagine four lines intersection points A at opposing 45 degree angles from the x and t axes. That represents the motion of light in each of our four traveller's frame of reference. You'll note that the light cone lines are the same for Jim and Jane as they are for Bill and Bertha despite Jim and Jane's odd set of axes (x' and t'). This is also very important, and is the most essential geometric intuition behind special relativity.
So for the past week or so a few diarists have pointed out that FTL motion violates causality in some frame of reference. Let's see why that is.
Let's say after some interval B, Bertha hands a message to Jane (who is flying right past her window). Jane then sends an instantaneous signal (that is, the message propagates entirely parallel to Jim and Jane's x' axis) such that Jim receives it at point A on the t axis. Jim then hands off the message to Bill as he flies past Bill's window. Bill then immediately sends an instantaneous response directly to Bertha (this time, the signal runs along the x axis).
Congratulations, you've just sent a message into Bertha's past. That is, Bertha received Bill's reply to a message she would not send until some time B in the future.
Remember those light cone lines you imagined earlier? Well, here's just one line (red, marked C) that should help get the point across:
Fig. 2: Bill and Bertha just relaxing and shit. Without some sort of ansible, that red line represents the minimum interval in time and space Bertha must endure to receive a message from her long absent friend Bill.
As it turns out, the only way to guarantee causality is maintained in all frames of references is to require all signals to travel along paths that do not fall between the a light cone line and the spatial axis (or axes if you generalize to higher dimensions). This notion is represented by something called the spacetime interval. That is, the interval between two events on our graph. Take my word for it right now, we will represent that interval as:
1) ds^2 = dx^2 - (c*dt)^2
In our 3 spatial dimension + 1 time dimension experience, we'd write (in Cartesian coordinates):
2) ds^2 = dx^2 + dy^2 + dz^2 - (c*dt)^2
Look familiar? Looks like the Pythagorean theom, except for that weird -1 * (c*dt)^2 term at the end. That term, and the observation that the spacetime interval between two events should be invariant from any frame of reference, is the essence of special relativity. That tweezing we see for Jim and Jane's x' and t' axes? It is a consequence of ensuring that their measurement of some interval is the same as Bill and Bertha's.
A few things come to mind. One, because the we have a negative term for the time differential, the interval (ds) can be positive, negative, or zero. Under this convention, we label timelike intervals for those where ds < 0, lightlike intervals for those where ds = 0, and spacelike ones when ds > 0.
Motion and/or communication across spacelike intervals--or FTL--is where we run into time travel like the thought experiment we geometrically explored above. In at least one frame of reference, the order of two events separated by a spacelike interval will flip. If you presume one event caused the other, then in at least one observer's frame of reference they will perceive the effect preceding the cause. This is a very basic definition of time travel into the past.
If you're interested in learning more, you can work through this article on Wikibooks (from which the image in the introduction is provided). I'll do a piece on other time traveling oddities (well, probably just one other) in relativistic physics whenever I fucking feel like it.
Go read something else now.