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In discussions about FTL communication and time travel, there is a simple thought experiment: the relativistic duel which is used to demonstrate that FTL travel implies time travel (I know it has a proper name, but can't find it back.)

Two observers move in opposite directions at relativistic speed. After 10 seconds in its reference frame, the first one (o1) shoots an FTL bullet at the other one (o2), and it is supposed that the bullet reaches him instantly from the viewpoint of o1, which means that o2 receives the bullet before the 10 seconds count. He then feels entitled to retaliate before the 10 second mark and o1 receives o2's retaliation bullet before it has even fired his own (and causality goes down the drain).

In my opinion, this is not an example of FTL displacement but an example of time travel in proper.

It seems to me we can change the thought experiment so that it is real FTL travel without time travel by slowing the FTL speed to less fast but still FTL speed:

  • o2 is hit by o1’s FTL bullet slightly after his 10 seconds mark and so slightly after he has fired his own FTL bullet on o1 (which does not prevent his bullet to be properly shot)

  • conversely o1 is hit by o2’s FTL bullet slightly after his 10 seconds mark and so slightly after he has fired his own FTL bullet on o2 (which does not prevent his bullet to being properly shot)

No causality issues.

If it gives the feeling that I think there might be an absolute time which allows to synchronize both duelists, I know it is not the case (there is no absolute reference frame), I am merely using a kind of FTL travel that does not break the time travel barrier. Note also that I did not specify what were the bullets speed when they reached or missed their targets in the target's reference frame (it allows for any kind of FTL travel)

It seems to me there is FTL travel without time travel for the bullets because the duel referee who stayed at the dueslist's start point will see o2’s bullet being fired and o1 dying from the blow almost at the same time (the reverse is also true) regardless of the distance.

Now the questions:

  • Am I even right in thinking bullets have travelled FTL?

  • Is there some fundamental issue with this thought experiment?

  • In the theory, what is the difference between both setups and how could we constrain FTL speeds to allow a general use of this kind of FTL displacement, but not time travel regardless of the practicality of said means of FTL travel? What is special about the kind of FTL travel (FTL speed limit) I drafted ?

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  • $\begingroup$ Where did the magic FTL bullet come from ? $\endgroup$
    – StephenG - Help Ukraine
    May 5, 2022 at 17:03
  • $\begingroup$ What do you mean ? Who is the designer/builder :) ? If the question is who shot it/them : I updated the description : o1 is hit by o2's FTL bullet, o1 FTL bullet misses o2 because he did not maintain his straight trajectory and o1 was not able to shoot in the proper direction, so o1's FTL bullet closely misses o2's ear while o2 is firing his own FTL bullet. If you think that the fun takes the clarity out, sorry for that. $\endgroup$
    – Carm
    May 5, 2022 at 17:14
  • $\begingroup$ I don't understand why dodging out of the way is relevant to whether time travel is involved; and if it's not, I don't understand the difference between your version of events and the original. In your first version, you say "o2 receives the bullet before the 10 seconds count"; but in the second, "o2 ... sees o1 FTL bullet ... at his 10 seconds mark". That appears to be the only difference in the timeline, but you don't explain why it's different, making the whole thing a circular argument - "if you could travel FTL without time travel, you could travel FTL without time travel". $\endgroup$
    – IMSoP
    May 5, 2022 at 18:10
  • $\begingroup$ Dodging was for fun, I could have removed it, I did not think it would raise eyebrows, but I did not mean to write « o2 sees o1 FTL bullet at his 10s mark », but « o2 is reached (and closely missed) by o1 FTL bullet at his 10s mark ». Give me 2 minutes to remove the fun and remove the ambiguity. $\endgroup$
    – Carm
    May 5, 2022 at 20:37
  • $\begingroup$ Have you drawn a spacetime diagram for this? $\endgroup$
    – m4r35n357
    May 6, 2022 at 9:14

3 Answers 3

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It seems to me we can change the thought experiment so that it is real FTL travel without time travel by slowing the FTL speed to less fast but still FTL speed

I think this answer does a good job of explaining why that doesn't help, with some numerical examples. This is just me re-hashing that with your example:

  • We start with two bullets travelling at speed $B_1$, and duellists travelling apart at speed $D_1$. The bullets are exchanged so fast that the retaliation comes before the first shot, and causality is violated. (A side note about frames of reference: if somebody perceives their death as happening at time $t$, and them pulling the trigger at time $t+1$, it doesn't matter what anyone else perceives, they're too dead to pull the trigger.)
  • So, we pick a slower speed, $B_2$. The bullets don't catch up in time, the shots happen in the right order in all reference frames, and everything's fine.
  • But then we can pick a new speed for the duellists, $D_2$, which is faster, so has more extreme time dilation. Now, even bullets travelling at $B_2$ violate causality.
  • It turns out that for every speed $B$ above the speed of light, there is some speed $D$ below the speed of light, such that the bullets travelling at $B$ kills the duellists travelling at $D$ in the wrong order.

So it becomes a bit like "man goes back and tries to kill his grandfather but misses" - the particular incident with bullets that weren't quite fast enough didn't violate causality, but the same bullets might violate causality next time. The Novikov self-consistency principle can be roughly translated as "the universe just ensures that you always do miss", but that's probably more useful for science fiction writers than physicists.

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You say "both bullets reached/missed their target at the same instant each duellist fired". You need to carefully clarify which reference frame you are using here. In special relativity, events that are simultaneous in one reference frame need not be simultaneous in another reference frame. So if A thinks their bullet hit B at the same instant it was fired in A's reference frame then in B's reference frame the bullet may have arrived before it was fired - which would imply time travel from B's point of view.

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  • $\begingroup$ Not sure it’s necessary : if two events (o2’s bullet reaching o1 and o1 firing) occur at the same place, they are either simulteanous in all reference frames or in none. Regardless, if o2’s bullet reaching o1 at the same time as o1 firing was perceived by o2 as o2’s bullet reaching o1 before o1 firing, it wouldn’t imply causality violation and time travel, just weird perception. $\endgroup$
    – Carm
    May 5, 2022 at 17:50
  • $\begingroup$ @Carm The events that we're comparing are not in the same place: a bullet is fired by 01 in one place, and reaches 02 in another place; that bullet is perceived by 02 as arriving before it was fired. So 02 knows 01 is going to fire even though they haven't yet; when they retaliate, we have a second bullet, which is perceived by 01 to reach 01 before either bullet is fired. So 01 "perceives" that they are dead, and therefore cannot "perceive" themselves firing the original bullet. $\endgroup$
    – IMSoP
    May 5, 2022 at 18:24
  • $\begingroup$ I was comparing o2’s bullet reaching o1 and o1 firing, not o1 firing and o1’s bullet teaching o2. I made no assumption of simultaneity of o1 firing and o1’s bullet reaching o2. $\endgroup$
    – Carm
    May 5, 2022 at 18:26
  • $\begingroup$ o2 will rightly perceive o1’s bullet before perceiving o1 firing but it does not mean anything to say that it is before o1 actually firing, and in my setup, o1’s bullet will not reach o2 before o2 fires but at the exact same time o2 fires. That means that is no causal relatin between o2 receiving o1’s bullet and o2 firing $\endgroup$
    – Carm
    May 5, 2022 at 18:32
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    $\begingroup$ I removed my initial answer to your comment because in fact this linked answer solves the issue exactly as I tried to do it : if you slow sufficiently, while remaining FTL, there is no time travel. What I had missed is that the closer the relative velocity is to c the closer you have to slow FTL speed back to c .. it seems to both solve the causality issue and remove the possibility of FTL speeds because it is probably always possible to find for any object in space an other one moving away from it at a speed close to c. $\endgroup$
    – Carm
    May 5, 2022 at 21:50
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Is there some fundamental issue with this thought experiment?

the first one (o1) shoots an FTL bullet at the other one (o2)

This is a magic bullet from a physics point of view.

If you start with an FTL bullet you have no basis to apply relativity to the problem as there is no way a bullet can reach FTL in relativity.

Why ? Because it cannot even reach light speed because that would imply it has infinite kinetic energy. Kinetic energy in special relativity is given by :

$$E_K=m_0c^2\left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} -1\right)$$

That tells you firstly that $v<c$ always because otherwise that square root become an imaginary number (implying an unphysical result) or you end up dividing one by zero and secondly that while $E_K$ can get as large as you like, you still cannot get FTL.

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  • $\begingroup$ I did not say that the bullet would reach light velocity and accelerate beyond it, I merely said it would travel faster than light. AFAIK, FTL travel is not forbidden by special relativity, altough travelling at light speed is and accelerating continuously from STL to FTL therefore also is impossible. There are numerous exemples of theoretical means of traveling FTL, even if there is none realistically proven as feasible. $\endgroup$
    – Carm
    May 5, 2022 at 18:37
  • $\begingroup$ Said otherwise, moving FTL (id est going from A to B faster than light would) is not possible by standard means but would for instance be possible with an alcubiere warp drive. The possibility of building such a warp drive is highly improbable but not theoretically impossible, is it ? And it seems to me that it would even be possible to build the warp tunnel in the setup I laid out (I mean with the time constraints), but I do not master the topic that much to be definitive $\endgroup$
    – Carm
    May 5, 2022 at 18:49
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    $\begingroup$ @Carm An Alcubierre drive does not move at FTL speeds, the practical effect of one might be equivalent to moving at those speeds (using a classical understanding of speeds and not a relativistic one), but the distinction is important. $\endgroup$
    – agaminon
    May 5, 2022 at 18:54
  • $\begingroup$ @Carm All the models of so-called warp drives I have seen require unphysical conditions to operate. They are interesting theoretical studies but not as far as I know more than that. $\endgroup$
    – StephenG - Help Ukraine
    May 5, 2022 at 19:34
  • $\begingroup$ @StephenG: I am not able to analyze these publication but Harold White in 2012 mentions small amounts of exotic matter and Erik Lentz in 2021 even goes as far as saying that it would be possible with limited amounts of purely positive energy. See en.m.wikipedia.org/wiki/Alcubierre_drive for references. Even if it’s bullshit, other means unknown to us ATM could provide a way to travel FTL and resolving so called causality paradoxes seems interesting in itself $\endgroup$
    – Carm
    May 5, 2022 at 20:24

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