As we know, bi-directional faster than light (FTL) travel can cause paradoxical situations. One possible way to resolve this would be a multiverse theory: Whenever a paradox could arise, the universe splits into two (or maybe more) more or less internally consistent universes, and whenever a paradox actually would arise this split becomes observable.
Let's take the Alice and Bob example from the wikipedia page on the tachyon antitelephone:
Alice and Bob are travelling at 0.8c away from each other from a common starting point. After 300 days subjective time, She eats bad shrimp and sends a message to that effect "I ate some bad shrimp" to Bob. The message travels to Bob at 2.4c. Bob receives the message at his subejctive time of 270 days (due to time dilation as seen from Alice's standpoint, math behind the link). He immediatly sends back a warning "Don't eat the shrimp" at 2.4c relative to his frame of reference which reaches her at 243 days "Alice-time". Provided Alice doesnt forget the warning in the following two months, she won't eat the shrimp, won't message Bob, won't receive a warning, will eat shrimp, will message Bob ...
(Edit to add: The preceding paragraph summarizes a thought experiment that was first thought up by Einstein in 1910 and worked on by other physicists. If your answer boils down to "Bidirectional FTL travel cannot create paradoxes, Einstein was wrong on relativity" your answer is bad and you should feel bad)
How could this play out in a Multiverse as sketched above? From Bob's perspective, he will receive a message from a futre that will never happen. Weird but not non-sensical. The same holds true for the Alice who receives the warning.
The possible paradox happens from the Alice-primes perspective: She sends a message to Bob but has not received the return message. When Bob and Alice communicate via non-FTL means she will learn he sent a warning that she did not receive (it simply vanished). The only way I see to resolve this is that return messages or trips that are quick (and close in time) enough to cause a paradox vanish from their senders perspective, and consequently from some receivers. While this requires a serious rethinking of causality (things happen for which the cause is outside this universe), within one timeline there are no causality violations.
Most times this will only happen when someone wants it to: Say ESA launches a space telescope at .99 c to observe the future history of the universe, or commanders on opposing sites of a space battle use relativistic spysats to adopt their battle plans by looking at one outcome (that is rendered impossible immediately).
The biggest downside from a storytelling perspective I see is this: Anyone capable of accelerating many tons of matter to relativistic speed and have it travel back with an FTL drive can duplicate matter: Instead of eating a shrimp Alice sends it to Bob, who sends it back. Now secondary Alice has two shrimp. Unless we limit FTL to information.
My question is: Do the preceding paragraphs make sense or is there a serious flaw in my thinking?
For storytelling purposes this works best with FTL devices where the travelling ship vanishes into a sort of warp bubble and cannot communicate with the rest of the universe until it pops back into sublight space. Maybe it could work with wormhole-like travel.
p.s. Let me explain the spirit of this question: FTL is impossible in two distinct ways. There's (IMO) no mechanism in sight that could feasibly allow any signal to travel FTL - that's one impossibility. The other is that unless relativity is grossly wrong (and we'd know by now?) FTL could lead to paradoxes as described above. The latter is IMO the "deeper" problem: you could add a new observation that allows an FTL signal or maybe the exotic matter required for an Alcubierre-style warp drive is suddenly observed. That would still require relativity, specifically the experimentally well established time dilation aspects, to be grossly wrong.
So the idea of this question is to ignore the first impossibility for now and look soleyly at the second one.
There's a reason I ask this here, not on Physics: An SF Author can apply an infinity of handwaves to tell their story. This question hopefully helps those who for whatever reason don't want to handwave relativity away.
Or to put it yet another way: If one breaks relativity as we know it in an SF story, the story ceases to be about an unlikely future and starts to be about an alternate universe. There's nothing wrong with that, it's just not what I'm going for.
p.p.s. This, by Demigan, ist too good to reamin hidden in the comments:
I never understood why this is a paradox. It is so easy to describe what happens that you do it right here! Bad shrimp->message->return message->doesnt eat shrimp->doesnt send message (somehow creates a problem with her own timeline), recreates the old timeline where she eats bad shrimp. Cause and effect, clear and simple (except for the twist where not sending the message recreates the old timeline. Why would it?).