# Does SR actually forbid FTL travel?

Does SR actually imply that FTL travel would allow for violations of causality? Here is some reasoning to suggest that it does not:

(I would be surprised if there were no holes in the reasoning. My desire to find these holes is what motivated me to post this here.)

Consider the following thought experiment (which demonstrates why FTL travel would violate causality):

1) Observers A and B are in a shared frame of reference at the same location. Let us call this location "the origin".
2) B teleports some distance away from the origin, but does so without accelerating (thus remaining in a shared frame of reference with A).
3) B then accelerates in a direction away from the origin (and A). By doing so, B effectively moves into A's past (to a time when B was still at the origin).
4) Finally, B teleports back to the origin. B has now returned to the origin before it left (and could stop itself from leaving if it so chose).

On the surface, this violation seems to support the idea that FTL travel is forbidden by SR. However, it has a problem. Consider what happens when we add a third observer to the thought experiment:

1) Observer C joins A and B at the origin in their common reference frame.
2) B teleports away as before.
3) B accelerates away from the origin as before. At the same time, C accelerates in the same direction by the same amount, thus remaining in a shared reference frame with B. As before, B believes that it has traveled to A's past, but it remains in C's present. From C's perspective, however, both A (co-located) and B (same reference frame) share its same time.
4) B teleports back to the origin. Is B now in A's past, C's present, or both?

When step 3) occurs, and B accelerates, a paradox arises. From B's perspective, A's time (At) is less than B's time (Bt), but C's time (Ct) is equal to B's time. Here are the perceived time relationships for each observer after step 3:

A) At = Ct, At > Bt
B) Bt = Ct, Bt > At
C) At = Bt = Ct

The solution to this paradox is that all of the above are true. We can go further, however, and say that from B's perspective, it simultaneously exists at every point in time at the origin, depending on an observer at the origin's frame of reference. This explanation works fine if you do not allow FTL travel.

However, if B is able to teleport back to the origin (via entanglement or some undiscovered physics), SR does not actually predict what A's time will be when B arrives. While we could excuse the notion that B was simultaneously in A's past and C's present while B was far away from them, we can no longer do so when it is co-located. According to SR, all co-located observers must agree on the order of events (and time) at their shared location.

I would suggest that this agreed upon time amongst all origin-located observers is, in fact, the "proper" time at the origin. If B were able to teleport back to the origin, this would be B's time as well (from B's step 3 perspective, this would be C's present and not A's past).

If true, that would mean that SR does not forbid FTL travel. (Yes, the speed of light is still the limit to how fast matter can move, but there are other potential methods that get around this, such as entanglement and wormholes).

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fzero
Homework Helper
Gold Member
Your logic is faulty. If you assume that teleportation is possible, then that is already FTL travel. You can't prove what you've already assumed.

I am not assuming that teleportation is possible. I am suggesting that SR does not forbid it.

Jonathan Scott
Gold Member
Does SR actually imply that FTL travel would allow for violations of causality? ...
If it is possible to communicate faster than light by at least a finite amount in every frame (so the laws of physics are the same in every frame, as required by SR) then violations of causality are certainly possible as this makes it possible to send a signal backwards in time.

If there is only some "preferred frame" in which some influence can travel at unlimited speed (which is sometimes suggested as an explanation for quantum entanglement, and is probably the basis for the "subspace" idea in science fiction) this does not create a causality problem, but it violates SR anyway by having a preferred frame.

I am not assuming that teleportation is possible. I am suggesting that SR does not forbid it.
But you are, your thought experiment for proving the FTL is not forbidden requires a method of FTL (teleportation). You can't prove something is possible by using a situation where that things existence is given.

mfb
Mentor
3) B accelerates away from the origin as before. At the same time, C accelerates in the same direction by the same amount, thus remaining in a shared reference frame with B.
At the same time in which reference frame?
Think about it, this is the problem in your argument. If they begin to accelerate at the same time (viewed by A, for example), they will have a different time-scale in their inertial frames after the acceleration. Just by accelerating, B gets into C's past.

With FTL travel (and your teleportation is similar to that, with "infinite" speed), you can violate causality. This is not a fundamental problem. But it has a lot of strange consequences.

If it is possible to communicate faster than light by at least a finite amount in every frame (so the laws of physics are the same in every frame, as required by SR) then violations of causality are certainly possible as this makes it possible to send a signal backwards in time.
The point I was attempting to illustrate with my thought experiment is that if you could travel faster than light, you would not be traveling an absolute amount into the past. Instead, you would be traveling different amounts into the past depending on the frame of reference of the observer into whose past you were traveling. However, because all observers at a single location all agree on the time regardless of their frame of reference, the idea of simultaneously traveling to multiple points in time at a single location does not make any sense.

But you are, your thought experiment for proving the FTL is not forbidden requires a method of FTL (teleportation). You can't prove something is possible by using a situation where that things existence is given.
There is a subtle distinction between proving that something is possible (which I am not doing) and proving that a disproof of something is flawed (which is what I am trying to do). When we say that SR forbids FTL information transfer, we do so because we can set up thought experiments that show it to violate causality if we assume FTL information transfer is possible. I am merely stating that those same thought experiments do not necessarily violate causality and thus cannot be used to prove that SR unequivocally forbids FTL information transfer.

I hope that makes sense.

At the same time in which reference frame?
Think about it, this is the problem in your argument. If they begin to accelerate at the same time (viewed by A, for example), they will have a different time-scale in their inertial frames after the acceleration. Just by accelerating, B gets into C's past.

With FTL travel (and your teleportation is similar to that, with "infinite" speed), you can violate causality. This is not a fundamental problem. But it has a lot of strange consequences.
1) Viewed by A, B and C will appear to begin accelerating at the same time (because A, B, and C will all start in the same reference frame and so will agree on simultaneity). However, as B and C continue to accelerate, A will think that C is accelerating faster than B (and that B and C are not traveling at the same speed).

2) From both B's and C's perspective, however, they will be accelerating at the same time and at the same rate (ie. the distance between them will not change). Or another way to put it is that a light signal sent from B to C and bounced back to B will have a constant round trip time throughout the acceleration and afterwards (from both B and C's perspective, but not A's).

The fundamental problem that I am trying to show is that thought experiments that show a violation of causality are nonsensical because they involve simultaneously traveling to an infinite number of points in time (one corresponding to every possible frame of reference), not just a single point in some observer's past.

Now to counter my own argument, you could say that SR forbids FTL information transfer BECAUSE it involves simultaneously traveling to an infinite number of points in time. That is, however, not the same as saying that it would violate causality.

DaveC426913
Gold Member
DrSnarl's logic is flawless.

If we are able to travel faster than the speed of light to do our experiments then we can do any number of experiments in which we will be able demonstrate that faster-than-light travel is possible.

PAllen
2019 Award
You can't strictly say SR prohibits FTL. You can say that the following lead to causality violation:

1) FTL travel
2) equivalence of all inertial reference frames, and Lorentz transform being valid between reference frames with relative velocity < c.

All you need to do is:

- pick a reference frame; in the frame suppose there is a rocket going near c.
- travel FTL to the rocket, from some start position in the starting reference frame.
- In the rocket's reference frame, travel FTL back to the start position in the starting reference frame.
- you will arrive there before you left, in the starting reference frame.

DrSnarl's logic is flawless.

If we are able to travel faster than the speed of light to do our experiments then we can do any number of experiments in which we will be able demonstrate that faster-than-light travel is possible.
Nice. :)

I am not claiming that FTL transfer of information is possible. It probably is not. I do not how to state that any more strongly. My only point is that you cannot use thought experiments (at least not like the one I presented) to support the claim that SR forbids it.

I should also reiterate that I am not trying to push some "new" theory. I am merely trying to refine my understanding of SR.

In my experience, most people who are educated enough to know something about relativity can tell you that SR forbids FTL information transfer. Those same people will say that a scheme that uses quantum entanglement and interference patterns (or lack thereof) to instantaneously transfer information cannot possibly work BECAUSE it violates SR. My question is, why? What is the basis for such a dismissal? Is it because of thought experiments such as the one I described? My understanding is that it is.

PAllen
2019 Award
Nice. :)

I am not claiming that FTL transfer of information is possible. It probably is not. I do not how to state that any more strongly. My only point is that you cannot use thought experiments (at least not like the one I presented) to support the claim that SR forbids it.

I should also reiterate that I am not trying to push some "new" theory. I am merely trying to refine my understanding of SR.

In my experience, most people who are educated enough to know something about relativity can tell you that SR forbids FTL information transfer. Those same people will say that a scheme that uses quantum entanglement and interference patterns (or lack thereof) to instantaneously transfer information cannot possibly work BECAUSE it violates SR. My question is, why? What is the basis for such a dismissal? Is it because of thought experiments such as the one I described? My understanding is that it is.
On the last paragraph, nonsense. No one knowledgeable says entanglement can't transfer information FTL because it would violate SR. Instead they say entanglement can't transfer information because the actual mathematics of entanglement says it can't. Period. Note, especially, that you can't even verify that you have successfully produced entanglement without transferring information from one place to another by some other means. Then, (long) after the fact, you can verify you successfully entangled interactions.

Nabeshin
Nice. :)

I am not claiming that FTL transfer of information is possible. It probably is not. I do not how to state that any more strongly. My only point is that you cannot use thought experiments (at least not like the one I presented) to support the claim that SR forbids it.

I should also reiterate that I am not trying to push some "new" theory. I am merely trying to refine my understanding of SR.

In my experience, most people who are educated enough to know something about relativity can tell you that SR forbids FTL information transfer. Those same people will say that a scheme that uses quantum entanglement and interference patterns (or lack thereof) to instantaneously transfer information cannot possibly work BECAUSE it violates SR. My question is, why? What is the basis for such a dismissal? Is it because of thought experiments such as the one I described? My understanding is that it is.
Hasn't this already been outlined by PAllen? FTL leads to causality violation, which is not something we're willing to give up. Quantum entanglement simply does not transfer information, and anyone who claims that it does is mistaken. It's not like QM predicts FTL information travel but we discard those results because of special relativity. Rather, QM tells us entanglement schemes do not transmit information FTL, in accord with our intuition from SR.

Edit: Ugh! PAllen has bested me!

You can't strictly say SR prohibits FTL. You can say that the following lead to causality violation:

1) FTL travel
2) equivalence of all inertial reference frames, and Lorentz transform being valid between reference frames with relative velocity < c.

All you need to do is:

- pick a reference frame; in the frame suppose there is a rocket going near c.
- travel FTL to the rocket, from some start position in the starting reference frame.
- In the rocket's reference frame, travel FTL back to the start position in the starting reference frame.
- you will arrive there before you left, in the starting reference frame.
Thank you. I understand exactly what you are saying, and your thought experiment is very similar to my initial one (with only A and B). I completely agree that you will arrive before you left in the starting reference frame. However, here is the rub: you will arrive even earlier in some reference frames, and you will arrive after you left in yet others. That violates SR in and of itself because all observers must agree on the order of events if the observers and the events are all at the same location.

PAllen
2019 Award
Thank you. I understand exactly what you are saying, and your thought experiment is very similar to my initial one (with only A and B). I completely agree that you will arrive before you left in the starting reference frame. However, here is the rub: you will arrive even earlier in some reference frames, and you will arrive after you left in yet others. That violates SR in and of itself because all observers must agree on the order of events if the observers and the events are all at the same location.
Well actually, this last argument is not compelling. There is no frame dependence of the order of events on along a timelike world line. There is frame dependence of the order of a sequence events on a spacelike path (the FTL away and back). Frame dependence of the order of events on spacelike paths is a given in SR, not an anomaly. It is only a causality problem if you propose the matter or information can follow these spacelike paths.

Hasn't this already been outlined by PAllen? FTL leads to causality violation, which is not something we're willing to give up. Quantum entanglement simply does not transfer information, and anyone who claims that it does is mistaken. It's not like QM predicts FTL information travel but we discard those results because of special relativity. Rather, QM tells us entanglement schemes do not transmit information FTL, in accord with our intuition from SR.

Edit: Ugh! PAllen has bested me!
Thank you (and PAllen as well) for your time in thinking about this. I would like to discuss quantum entanglement further (including experiments that people have done), but as that is really a detour from this discussion, I will refrain. Just to head that off for now, I will accept that you are correct in your assertion about QM and the FTL transmission of information.

However, you said "FTL leads to causality violation, which is not something we're willing to give up." Many people much smarter than I have been studying this since before I was born, and they agree with you, so there must be a clear explanation as to why we are not willing to give it up. I am looking for that explanation.

Again, I will accept that the reason SR forbids FTL transmission of information is that it leads to the nonsensical traveling through time to an infinite number of points in time at a given location. That, however, is not a "violation of causality". Rather, it is merely because you logically cannot be at two different points in time within your own frame of reference (ie. your clock cannot simultaneously have more than one value when you observe it.)

Well actually, this last argument is not compelling. There is no frame dependence of the order of events on along a timelike world line. There is frame dependence of the order of a sequence events on a spacelike path (the FTL away and back). Frame dependence of the order of events on spacelike paths is a given in SR, not an anomaly. It is only a causality problem if you propose the matter or information can follow these spacelike paths.
The fact that there is "no frame dependence of the order of events along a timelike world line" is central to my entire point. That is the reason why all observers must agree on the sequence of events if the observers and the events are all at the same location - the sequence of events does not depend on the observers' frames if the observers are all co-located with the events. Or did I misunderstand what you are saying?

PAllen
2019 Award
However, you said "FTL leads to causality violation, which is not something we're willing to give up." Many people much smarter than I have been studying this since before I was born, and they agree with you, so there must be a clear explanation as to why we are not willing to give it up. I am looking for that explanation.
The only reason fundamental reason to reject causality violation is paradoxes. Some physicists have explored how far we can go to admitting FTL travel in limited ways (e.g. by tachyons; alcubierre drive; wormholes). One technique for limiting the impact of paradoxes is to propose, e.g. the Novikov Consistency conjecture (see, for example http://en.wikipedia.org/wiki/Novikov_self-consistency_principle ). However, even this principle fails to prevent the following type of paradox:

I transmit (via tachyon round trip) the content of a current edition of a Shakespere play to Shakespere before he wrote it. He looks at it, likes it, and publishes it as his own. None of this violates Novikov. So who wrote it? If you are fine with the answer that nobody wrote it, it just 'is', then you can happily pursue FTL theory.
Again, I will accept that the reason SR forbids FTL transmission of information is that it leads to the nonsensical traveling through time to an infinite number of points in time at a given location. That, however, is not a "violation of causality". Rather, it is merely because you logically cannot be at two different points in time within your own frame of reference (ie. your clock cannot simultaneously have more than one value when you observe it.)
Again, I disagree with this summary. SR per se, only says FTL information transfer or travel produces causality violations, it does not say the things you claim above. SR also says various other things: even infinite energy would allow ordinary matter to exceed the speed of light. But you can develop mathematically consistent theories of tachyons.

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PAllen
2019 Award
The fact that there is "no frame dependence of the order of events along a timelike world line" is central to my entire point. That is the reason why all observers must agree on the sequence of events if the observers and the events are all at the same location - the sequence of events does not depend on the observers' frames if the observers are all co-located with the events. Or did I misunderstand what you are saying?
I think you misunderstand that the existence of a spacelike round trip path between two events on a timelike world line has no effect whatsoever on the order of those events for any observer. Actually, perhaps there is something I failed to question earlier in your various analyses. For a round trip FTL path that connects two points on a given timelike world line, there is no frame dependence at all on the order of the events it connect (say e1 and e2). Let us call the turnaround event et. There is only disagreement on the relative ordering of e1, and et, and on e2 and et. Despite this disagreement between frames, all agree on the ordering of e1 and e2 irrespective of the existence or nature of the proposed FTL path between them.

[edit: For example, frame 1 thinks e2->et is forward in time, and et->e1 is back in time more, and e1->e2 is forward in time. Meanwhile, frame 2 thinks e2->et is back in time more, et->e1 is forward in time less, and e1->e2 is forward in time. ]

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Again, I disagree with this summary. SR per se, only says FTL information transfer or travel produces causality violations, it does not say the things you claim above.
Now I think we have arrived at the heart of the matter. I think that SR does say the following (and if one of these is wrong, then that will expose my premise):

1) When an observer A is separated by observer B in space, any acceleration by B relative to A will cause B to travel through time relative to A.

2) The amount and direction of time travel by B relative to A depends on a) the distance between A and B, and b) the difference in velocity between A and B.

3) Consider an infinite continuum of observers co-located with A (let's call this the point of interest, or POI, and define it as A's location) across every frame of reference (with each observer having a different velocity relative to B). Because of 1) and 2), when B accelerates, it will be traveling a different amount of time relative to each of those observers.

So far, there is no problem. Sure, after B accelerates, it is simultaneously (when considered from its own reference frame) at different points in time for each of the observers at the POI, but because we cannot transmit information faster than light, we cannot turn this into a logical problem. If B then travels towards the POI at any velocity, it will perceive each observer on the continuum having a different degree of time dilation, so by the time B arrives at the POI, B will think that all clocks are in agreement. By agreement, I mean that B will be able to correctly predict to value of any observer's clock and vice versa.

NOTE: I realize that all but one of the observers at the POI will no longer be there by the time B arrives - if this is a problem for you, then assume that each "observer" is actually a stream of observers with a shared frame of reference (ie. velocity) moving through the POI.

4) SR does state that if B could travel fast enough to get out of the light cone (while preserving a reference frame with a velocity < c relative to A, such as via a wormhole), then the time dilation experienced by the observers at the POI relative to B WOULD NOT be sufficient to bring the clocks into agreement by the time B arrived. Hence time travel.

5) The amount of time travel experienced by B relative to an observer at the POI would depend on that observer's frame of reference.

6) All observers at the POI should agree on the value of each other's clocks at the exact time when they are all at the POI.

5) and 6) conflict with each other, yet both are supported by SR. That is my premise. What gives? Is 5) wrong or is 6) wrong?

Thanks again for taking the time to have this discussion. I very much appreciate your insights, and I know it takes effort to try to figure out what someone else is saying, particularly if there is a faulty assumption underlying their argument somewhere.

I think you misunderstand that the existence of a spacelike round trip path between two events on a timelike world line has no effect whatsoever on the order of those events for any observer. Actually, perhaps there is something I failed to question earlier in your various analyses. For a round trip FTL path that connects two points on a given timelike world line, there is no frame dependence at all on the order of the events it connect (say e1 and e2). Let us call the turnaround event et. There is only disagreement on the relative ordering of e1, and et, and on e2 and et. Despite this disagreement between frames, all agree on the ordering of e1 and e2 irrespective of the existence or nature of the proposed FTL path between them.

[edit: For example, frame 1 thinks e2->et is forward in time, and et->e1 is back in time more, and e1->e2 is forward in time. Meanwhile, frame 2 thinks e2->et is back in time more, et->e1 is forward in time less, and e1->e2 is forward in time. ]
What you are saying seems like it has to be true, but that also would imply that the existence of an FTL spacelike round trip path between ANY two events on a timelike world line cannot be used to violate causality. This is precisely the point I set out to make, though you did it in a much more concise fashion.

So that said, how can FTL travel violate causality?

PAllen
2019 Award
What you are saying seems like it has to be true, but that also would imply that the existence of an FTL spacelike round trip path between ANY two events on a timelike world line cannot be used to violate causality. This is precisely the point I set out to make, though you did it in a much more concise fashion.

So that said, how can FTL travel violate causality?
No, my example shows the round trip violates causality by a frame independent amount. All observers agree that the path (e2,et,e1) violates causality by connecting e2 on given timelike world line to earlier e1 on the timelike worldline. They all agree on the proper time interval between e1 and e2, and that e1 occurs before e2. The only thing they disagree on is which part(s) of the path (e2,et,e1) go backwards in time, and by how much - but the total violation along the (e1,e2) world line is invariant.

[edit: what makes it impossible to prevent this type of thing is the assumptions I gave a number of posts ago. In frame of e1,e2 being stationary, e2->et is a forward in time FTL path. Then in rapidly moving frame (which should have all the same capabilities), et->e1 is forward in time FTL. So two actions that must be allowed if you have both FTL and the principle of relativity produce causality violation. The key is the in frame of (e1,e2), the path et->e1 is the one that shoots back in time; while in the rocket frame, it is the path e2-et that shoots back in time.]

[edit2: If you add to SR the axiom that no possible FTL path can appear to go back in time to any inertial observer, then you conclude that no FTL path is possible at all. But this would be an additional assumption to the normal axioms of SR.]

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mfb
Mentor
2) From both B's and C's perspective, however, they will be accelerating at the same time and at the same rate (ie. the distance between them will not change).
This is not possible, unless B accelerates really slowly, as "at the same time" shifts with increasing speed. But in that case, he wastes the time he wanted to go into the past. Of course, if you wait long enough, you won't come into the past.

Demystifier