The standard story
Faster-than-light (FTL) communication is thought by some to allow signalling back in time, thereby violating causality.(It's worth noting that no FTL communication is actually known to be possible in the real world, however it's interesting from a metaphysical point of view, and also for science-fiction authors!)
It works like this: imagine you have two space stations floating in space, stationary relative to each other (s.s. a and s.s. b on illustration below). Then imagine you have a rocket that flies past the space stations at some velocity v.
Then imagine that you have some device that can transmit information faster than light; let's suppose that it can transmit with an infinite velocity. So space station a sends a FTL pulse to space station b. This is the horizontal green line from O to A in the illustration.
As space station b receives the FTL pulse, we've arranged it so that the rocket is at that exact moment flying right past space station b. So space station b communicates with the rocket. (Just with normal light is fine since they are basically at the same position)
Here's the tricky bit: Special relativity says that the rocket has a different coordinate frame than the space stations. In particular the x axis, which corresponds to all the points in space at the same rocket-time, is not the same as the x-axis for the space stations.
So the standard story is that the infinite velocity FTL commuication pulse, sent back from the rocket to the left (lower green arrow from A to B), will travel along the skewed x axis of the rocket-time.
Space station a then receives the pulse at B. Crucially, from the point of view of space station a, it has received the return communication before it sent it!
This is where the paradoxes come in. For example, what if the message said something like 'do not send the message', or 'kill this person, who is the grandfather of the message sender'. Then a contradiction arises. Much ink has been spilled (or keys mashed) trying to resolve this seeming contradiction.
v:
Interactive illustration - try changing the v value. A depiction of how FTL signalling violates causality in the standard special relativity story. Green lines = FTL communication. Blues lines = trajectory of space stations. Bold red line = trajectory of rocket.
With absolute space and time
With an absolute space and time, sending messages back in time is prohibited. Let us suppose that the space stations are at rest with respect to absolute space. Then the message from the rocket back to space station a does not point down in the space-time plot, but is instead just horizontal. Therefore no contradiction can arise.
v:
Interactive illustration - try changing the v value. A depiction of how causality violation with FTL signalling is avoided with absolute time. Green lines = FTL communication. Blues lines = trajectory of space stations. Bold red line = trajectory of rocket.
What happens instead of a contradiction, is that with FTL communication, you can determine your velocity relative to the absolute reference frame (absolute space).
Imagine a setup where you have two meter rulers, one pointing in the positive x direction, one in the negative x direction. You simultaneously send a FTL pulse to both ends of the rulers. There is a device at each end of the rulers that when it receives the FTL pulse, sends a light pulse back to you.
If you are at rest relative to the absolute reference frame, the light pulses will return to you at exactly the same time. However if you are at motion relative to the absolute reference frame, the light pulses will return at different times. In particular, you will receive the light pulse from a ruler pointing in the direction of travel earlier than the pulse from a ruler pointing away from the direction of travel. See the illustration below. An observer can calculate their speed relative to the absolute reference frame by measuring the difference in times between the return of the light pulses.
Physicists might refer to this as 'symmetry breaking' - the apparent equality of each reference frame is broken, and you can detect the absolute frame from the comfort of your laboratory.
v:
Interactive illustration - try changing the v value. A depiction of an observer measuring their speed relative to absolute space with FTL signalling. Green lines = FTL communication. Blues lines = light pulses. Red line = world line of observer moving relative to absolute reference frame.
Special relativity and absolute time and space
Special relativity does not eliminate the possibility of an absolute reference frame - rather it is an axiom of special relativity that it is impossible to detect the velocity of the observer relative to any possible absolute reference frame.It is an additional metaphysical step to discard the undetectable absolute reference frame as non-existing.
To quote Matt Visser:
Warning: the "Einstein relativity principle" does not imply that there's no such thing as "absolute rest"; it does however imply that the hypothetical state of "absolute rest" is not experimentally detectable by any means. In other language, the "Einstein relativity principle" does not imply that the "aether" does not exist, it implies that the "aether" is undetectable. It is then a judgement, based on Occam's razor, to dispense with the aether as being physically irrelevant.
Some further food for thought regarding a potential absolute reference frame that not everyone is aware of - did you know that what is thought to be the afterglow of the Big Bang, called the Cosmic Microwave Background (CMB) is visible in all directions from Earth, and that scientists have measured the speed of the Earth relative to the CMB, which is possible due to the red-shift of the CMD spectrum in the direction of Earth's travel? Earth is travelling at about 0.002c (0.2% the speed of light) through the Universe relative to the CMB. Personally I regard the CMB as some kind of evidence for absolute space and time.
General relativity (the theory of gravity which explains gravity as curvature of space-time) also has profound things to say about space and time, but I have restricted myself to special relativity in this post.
Edit: Added some text in the 'With absolute space and time' section clarifying that the space station is stationary with respect to absolute space. Added a space-time diagram in the 'With absolute space and time' section to illustrate what I propose would happen with the FTL signals.