Time Travel Theory, Experiments and Consequences
There was a young lady called Bright
Who could travel much faster than light;
She went out one day,
In a relative way,
And came back the previous night.
-- Davies, 1992
There are scientists who think that time travel is possible and that it will be invented in a far away future. Basically the idea is that time machine will project the human body or consciousness into the past or the future.
Time travel and the technology to send messages to the past should be considered part of the same science as both activities are based on the idea that time is not necessarily constant, linear or flowing in one direction only. That could mean that time itself might be manipulated and possibly traveled in both directions. As a result our current accepted theory about what is time and how does it behave, cannot be correct or complete.
If such future technology ever gets invented then the consequences must be already with us. Regardless of how far ahead the technology is going to be developed, we already have seen the effects, and documented some of the communications we must have had with time travelers. However since we have not yet established a theory, it might be difficult to recognize an event as the result of an experiment.
Time travel to the past and sending messages to the past is unique in the sense that the consequences occur before the technology gets developed, before a time travel theory gets adopted and before anybody knows what it will look like or accept that it might be possible.
No matter how far in the future time travel gets invented the effects of experiments must be already with us and documented in our own history.
If we find no reasonable proof of experiments then maybe the technology is never going to be invented, eliminating the need to do any time travel research. A first step in proving the existence of time travel and the consequences of time travel experiments could be to analyze - random - events from the point of view that they are time travel related. Or even better: any recorded mysterious, paranormal or unexplained event could be an excellent potential candidate, even if there seems to be no direct relationship with the future.
Since there are many theories, we could reason how each would leave it's unique marks and tell tales in history which we then could use to indicate the probability that a theory could be viable.