Why would an FTL-drive imply time travel? (other answers and questions unsatisfactory)

Question

I've been spending quite some time trying to understand why an FTL-drive would also imply time-travel, but every answer I can find seems to mainly be about semantics and perception. I will break it down to a very simple question:

I have a watch. Jean-Luc Picard leaves on the Enterprise from my position at my current time (10:05, 2021-09-13), his mission is to go to [place] and then return to my position. No matter how fast or slow his FTL-drive is, I am going to claim that there is no way he will ever achieve time-travel and thus violate causality according to my time at my current position, i.e he will never arrive to me before he left in a way where it's not just the perception of light-lag to my retina, and thus he will never be able to interact with himself in any way (meaning: no time travel). Can anyone refute this?

On John Donnes request I will clarify a few things. What spurred me to write my own (according to me) simpler question was reading through the answer given here: How does faster than light travel violate causality?.

This answer is basically "plug x numbers into equation y and then you have time travel", but it allows zero intuitive understanding, and I can still not see how it would allow Jean-Luc to arrive at my position before he left. Erudaki exemplifies my confusion in his comment to that answer, where he calculates that the ship in that question would return to earth at year 23204, not year 2796 as the answer claims.

Here's the specific part of that answer which irks me:

• In the futuristic Earth year of 3000, Tralfamadore is 98,000 light years away, and receding at 20% of $c$. I leave Earth at 1000% of $c$, relative to Earth.
• In Earth year 13000 Tralfamadore is 100,000 light years away, and I catch up to it. I turn around and leave Tralfamadore at 1000% of $c$, relative to Tralfamadore.
• In Earth year 2796, I arrive home.

I do not understand why in this example the traveller would arrive at 2796 (earth calendar), rather than 13000 + x (where x is travel time from Tralfamadore).

Answer 1

An important caveat: FTL implies time travel only if the FTL mechanism obeys the principle of relativity (that is if there is no "absolute" speed, only relative speed). All known physical laws obey this principle, and it dates back to Galileo, so this is a reasonable assumption.

For simplicity let's consider an arbitrary FTL communication mechanism, and again for simplicity let's assume it is so fast it is "instantaneous" (we'll relax this constraint later). Suppose Bob and Alice are in deep space, moving apart at 0.87c relative to one another, and each is equipped with identical FTL message devices. At time t=0 they passed one another and synchronized clocks.

Suppose Alice sends a signal to Bob when her clock reads 10 hours. What will Bob's clock read when he receives the signal? The principle of relativity says we can assume Alice is at rest, and Bob is moving at 0.87c, so he experiences time dilation relative to Alice with a factor of 2x. So according to Alice, Bob's clock is showing 5 when Alice's shows 10. Thus Alice knows that when her clock reads 10 Bob's will read 5. Since Alice's FTL message is instantaneous, Bob will receive the FTL signal at 5 o'clock (his time).

But the situation is completely symmetric: we could equally well assume Bob to be at rest and Alice to be moving, so in Bob's coordinates Alice's clock is slow and when his clock reads 10 Alice's will show 5. That both observers see the other's clock as running slowly is the root of the so called twin paradox, which we won't go into here (there are many, many questions and answers about it) but it is an established fact of relativity.

Thus: Alice uses her FTL machine to send a message from her 10 o'clock to Bob's 5 o'clock. Bob can then wait 5 hours and send a message from his 10 o'clock to Alice's 5 o'clock. Thus, Alice can use Bob to send a message back in time to herself!

OK, suppose the FTL isn't instantaneous, just very very fast. This doesn't help much: the message gets from Alice to Bob a bit slower, so instead of Bob's clock showing 5 maybe it'll show 6 or 7, and vice-versa. Backwards in time messages are still possible.

If the message is slow enough though, it'll get to Bob after his 10 o'clock, and the paradox is resolved. In this particular case if the FTL message is slower than 2c then no paradox is possible. But if Bob and Alice are moving apart faster, the time dilation effect is stronger and the FTL has to go slower still to avoid paradoxes. In the limit as Alice and Bob are moving apart close to c, the FTL message has to slow down to c to avoid paradox.