FTL travel without time travel (again)

Question

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 ?

Answer 1

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.

Answer 2

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.

Answer 3

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.