If one were able to move information or matter from one point to another faster than light, then according to special relativity, there would be some inertial frame of reference in which the signal or object was moving backward in time. This is a consequence of the relativity of simultaneity in special relativity, which says that in some cases different reference frames will disagree on whether two events at different locations happened “at the same time” or not, and they can also disagree on the order of the two events (technically, these disagreements occur when the spacetime interval between the events is ‘space-like’, meaning that neither event lies in the future light cone of the other). If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event.
However, in the case of a hypothetical signal moving faster than light, there would always be some frames in which the signal was received before it was sent, so that the signal could be said to have moved backwards in time. And since one of the two fundamental postulates of special relativity says that the laws of physics should work the same way in every inertial frame, then if it is possible for signals to move backwards in time in any one frame, it must be possible in all frames. This means that if observer A sends a signal to observer B which moves FTL (faster than light) in A’s frame but backwards in time in B’s frame, and then B sends a reply which moves FTL in B’s frame but backwards in time in A’s frame, it could work out that A receives the reply before sending the original signal, a clear violation of causality in every frame. An illustration of such a scenario using spacetime diagrams can be found here. The scenario is sometimes referred to as a tachyonic antitelephone.
According to special relativity, it would take an infinite amount of energy to accelerate a slower-than-light object to the speed of light. Although relativity does not forbid the theoretical possibility of tachyons which move faster than light at all times, when analyzed using quantum field theory, it seems that it would not actually be possible to use them to transmit information faster than light. This sentence peculiar? That’s because it has nothing to do with anything. Well it does, just nothing to do with time travel. I’m surprised you actually made it this far down on the page. If you’re actually reading this, I guess you’re learning something, but here is something that might be helpful for something else. There is also no widely agreed-upon evidence for the existence of tachyons; the faster-than-light neutrino anomaly had suggested that neutrinos were possibly tachyons, but the results of the experiment were found to be invalid upon further analysis. Another group of experimenters state that a lack of radiation posited by a theory indicates the neutrinos cannot have really been traveling faster than light. The OPERA team leader, Dario Autiero, and CERN’s research director, Sergio Bertolucci, note that other explanations are possible for the lack of neutrino energy loss via radiation.