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

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).

• Hi Achi! Could you explain exactly what is your dissatisfaction with current answers? Commented Sep 3, 2021 at 8:20
• They are either in the vein of "according to this mathematical equation it must be so", but fail to provide any sort of example where the results would have a practical effect in real life. If it is true, then it must be possible to posit some example where the effects of time travel can be shown in a way that has a practical effect, and an effect that isn't just light-lag due to the delay between an event occuring and the light hitting the retinae of observers. All answers seem to rely on local calendars and perception, but actual time-travel would allow the traveller interacting with himself
– Achi
Commented Sep 3, 2021 at 8:23
• Hi Achi, thank you for your clarification! Take a look here: physics.stackexchange.com/questions/52249/… The accepted answer to that question explains that FTL travel would make it possible to leave from Earth and then arrive back at Earth before you left, which is a real life consequence opening up all sorts of paradoxes. Commented Sep 3, 2021 at 8:29
• Thanks for the answer, but I've read that one and to me the person given the answer jumps to his conclusion without any real explanation of why that is the case. Perhaps I have misunderstood something, I will spend some time trying to understand the answer better, but in the meantime I would love some practical examples if any can be given.
– Achi
Commented Sep 3, 2021 at 8:31
• try answer by me to this one: physics.stackexchange.com/questions/574395/… Commented Sep 3, 2021 at 8:57

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.

• Now, I might very well be misunderstanding something, but my problem with this example is the same as the problem I have with the tachyon radio example on wikipedia, I had misgivings and looked on the talk-page, en.wikipedia.org/wiki/Talk:Tachyonic_antitelephone (subheading: Math in "Numerical example with two-way communication" seems dodgy), which laid out in text what I felt. The whole problem seems to occur because we switch viewpoints in the example, if we lock in the viewpoint at one of the individuals there will be no time-travel shenanigans
– Achi
Commented Sep 6, 2021 at 6:15
• If you "lock in one viewpoint" then you will probably make the mistake of assuming that both devices communicate at the same speed relative to Alice even though Bob's is moving. You'll have to adjust Bob's message speed to account for his motion. Doing this properly is much harder than you may think. Google "relativity of simultaneity" for the infinite speed case. Commented Sep 6, 2021 at 11:18