Why Does FTL Travel Lead to Time Paradoxes?

[Ryan_m_b]

I'm not a physics student but have always been interested. That said my knowledge is very slim. I've always wondered why traveling faster than light would lead to time travel, I've poured over websites and other forums but they tend to descend into physics that is beyond me. Here is an outline of my current understanding,

Travelling faster than light would cause events to occur in a non-causal manner to some observers. This can lead to paradoxes.

What I don't understand is why observing something happening out of sequence is a problem and secondly how that can lead to time travel/paradoxes?

I'm best at understanding these things as real world examples so here's one I've thought up;

Alice, Bob, Claire and Dan are all located exactly 1 light minute away from each other in a straight line. I'm chatting to Bob telling him that I've invented an instantaneous teleporter. He dares me to try it so I activate it and instantly teleport to Claire. I understand that this would happen:

*Alice would see me hanging out with Bob, then she would see me disappear. Precisely 1 minute later she would see me appear by Claire.

*Bob would see me disappear, looking at Claire he would see me appear in 1 minute the same as Alice .

*Claire would see me with Bob before all of a sudden I appear in front of her. We can both watch for another minute at me and Bob before I disappear.

*Dan would see me with Bob when a second me would appear by Claire. Dan could see me and Claire looking at Bob until after a minute the me by Bob disappears.

The two things I struggle to understand is why this is a bad thing and how exactly it can be used to effect the past. In my mind if I teleport to Claire and teleport back to Bob before I see myself leave I won't bump into my past, ill arrive just after I left.

Any help would be most appreciated!

 

[JesseM]

Travelling faster than light would cause events to occur in a non-causal manner to some observers. This can lead to paradoxes.

What I don't understand is why observing something happening out of sequence is a problem and secondly how that can lead to time travel/paradoxes?

It's because of the requirement that the laws of physics work the same way in all inertial frames (the first postulate of relativity), so if you can send a signal which moves backwards in time in one frame (i.e. the time the signal is received is earlier than the time it was sent), then this must be possible in all frames. So suppose Alice and Bob are moving apart at relativistic (but slower-than-light) speed, and Alice sends a message to Bob which moves FTL (but forwards in time) in her her frame and backwards in time in Bob's frame. Then Bob can send a reply back--which might just be a copy of the original message--which moves FTL in his frame but backwards in time in Alice's frame, with the net result being that Alice receives Bob's reply at an earlier time than she sent the original message, as measured in all frames (i.e. there is a timelike separation between the event of her receiving the reply and the event of her sending the message, so all frames agree on the order of these events). If you're familiar with how spacetime diagrams work there are some illustrating this here, and I could give you a numerical example if you know how to use the Lorentz transformation.

Alice, Bob, Claire and Dan are all located exactly 1 light minute away from each other in a straight line. I'm chatting to Bob telling him that I've invented an instantaneous teleporter. He dares me to try it so I activate it and instantly teleport to Claire. I understand that this would happen:

*Alice would see me hanging out with Bob, then she would see me disappear. Precisely 1 minute later she would see me appear by Claire.

*Bob would see me disappear, looking at Claire he would see me appear in 1 minute the same as Alice .

*Claire would see me with Bob before all of a sudden I appear in front of her. We can both watch for another minute at me and Bob before I disappear.

*Dan would see me with Bob when a second me would appear by Claire. Dan could see me and Claire looking at Bob until after a minute the me by Bob disappears.

When you say "sees", do you mean when each person actually receives the light from these events, or do you mean the time-coordinates they assign to these events in their own rest frames? The relativity of simultaneity is not about visual appearances, it's about the time coordinates retroactively assigned to events in different frames (for example, if in 2010 according to my clock I receive light from an event 10 light-years away according to my measurements, then I retroactively say this event occurred in 2000, and is therefore simultaneous with an event whose light I received in 2005 coming from 5 light-years away). It's not clear if you understand this, since you don't specify anything about these four characters having different velocities (if they are all at rest relative to each other, they all share the same rest frame and therefore all agree about simultaneity).

 

[Ryan_m_b]

Thank you for your reply. My knowledge of physics is limited so whilst I have seen some spacetime diagrams before i haven't fully understood them, and I don't even know what a Lorentz transformation is lol.

Ok, with my example Alice, Bob, Claire and Dan are completely at rest. When I say "see" I do mean observe the light (obviously they could all realize the distance and figure out what time the event actually occurred)

With your example of Alice and Bob moving apart I'm still confused. I've looked at the link you sent me which shows how the lightcone on the graph shifts but I'm still having trouble getting my head around how that would work in real life.

Alice and Bob start back to back and then move away from each other at relativistic speeds. If they both have a clock with them they could decide before they set out that Alice will send Bob an instantaneous message after ten minutes of travel. If they are traveling at the same speed then they will undergo the same time dilation, if that is a factor of 2 then Alice will send the instant message at 10mins on her clock, 10mins on Bobs clock and 20mins on Claires clock (who stayed still at the start line).

What i can't get my head around is why Bob receiving the message from outside his lightcone and sending it back means absolute time travel

 

[Ryan_m_b]

Also having read your comment over more and looked at other websites i wonder if my confusion is rooted in the idea of FTL. In my example the traveller wasn't meant to have a faster than light speed, merely he could travel faster than light. If one could travel faster than light (without having superluminal velocity) would that still cause you to travel back in time?

 

[JesseM]

Thank you for your reply. My knowledge of physics is limited so whilst I have seen some spacetime diagrams before i haven't fully understood them, and I don't even know what a Lorentz transformation is lol.

Well, I think it'll be difficult to really answer your questions if you don't understand the basics of SR, maybe you could take a look at some of the introductory material listed on this thread.

Ok, with my example Alice, Bob, Claire and Dan are completely at rest.

Do you understand there is no notion of absolute speed in relativity, that speed can only be defined in a relative manner? You can say they are at rest relative to one another, but in relativity there is no objective answer to whether a given object is "really" at rest or moving, for any slower-than-light object there will always be one inertial frame where it's at rest (perhaps only instantaneously, if it's accelerating) while in other inertial frames its velocity is nonzero, and all these different inertial frames are considered equally valid.

When I say "see" I do mean observe the light (obviously they could all realize the distance and figure out what time the event actually occurred)

OK, but the question of when they observe the light is actually completely irrelevant to understanding why the possibility of FTL signals would also imply backwards-in-time signals according to relativity, so talking about visual appearances is a blind alley here. You need to learn about the "relativity of simultaneity" (discussed here and here for example) which says that different inertial frames can disagree about whether a given pair of events happened at "the same time" or at "different times", and also on the order of the events, as long as there is a spacelike separation between them, meaning the distance and time between them are such that it would be impossible for a signal traveling at the speed of light or slower to get from one event to the other (on the other hand, if there is a timelike or lightlike separation between a pair of events, then all inertial frames agree on their order and no inertial frame thinks they happened simultaneously). Here the times of the events in each inertial frame already correct for delays an observer at rest in that frame might experience in seeing light from the event, due to the finite speed of light; think of the example I mentioned earlier where in 2005 according to my clock I see light from an event 5 light-years way according to a ruler at rest relative to me, and in 2010 I see light from an event 10 light-years away according to my ruler, then if I correct for the delays I will say that in my frame both these events happened simultaneously in 2000 in my frame, even though I didn't see them simultaneously. But if an observer in motion relative to me uses the same type of procedure to correct for light delays, he will conclude that the events did not happen simultaneously in his frame. The links I gave above about the "relativity of simultaneity" discuss this, as do the various links in the thread about introductions to SR.

Alice and Bob start back to back and then move away from each other at relativistic speeds. If they both have a clock with them they could decide before they set out that Alice will send Bob an instantaneous message after ten minutes of travel. If they are traveling at the same speed then they will undergo the same time dilation, if that is a factor of 2 then Alice will send the instant message at 10mins on her clock, 10mins on Bobs clock and 20mins on Claires clock (who stayed still at the start line).

But because of the relativity of simultaneity, "instantaneous message" does not have any frame-independent meaning--if a message travels instantaneously according to Alice's definition of simultaneity, being received by Bob at the "same time" it was sent by Alice in this frame, then in Bob's frame that same message does not travel instantaneously, in his frame it was received by Bob before it was sent by Alice (that's assuming they are moving apart--if Bob was moving towards Alice, then it would be received by Bob after it was sent by Alice in his frame, again assuming that in Alice's frame both events were simultaneous). Spacetime diagrams typically show the lines of simultaneity in two different frames, for example this page gives a diagram drawn from the perspective of the frame of "John", with time in John's frame as the vertical axis and space as the horizontal axis (so two events at the same vertical height, both lying along a single horizontal line, happened simultaneously in John's frame); then the slanted line labeled "Bill's space coordinate" show a set of events that are simultaneous in the frame of different observer, "Bill", in motion relative to John:

So if you draw two dots along this slanted line, both occur simultaneously in Bill's frame but at different times in John's frame.

What i can't get my head around is why Bob receiving the message from outside his lightcone and sending it back means absolute time travel

As I said the idea is that the laws of physics must work the same way in all inertial frames, so if it's possible in Bob's frame for Alice to send a signal which reaches him at an earlier time then she first sent the signal, it must likewise be possible in Alice's frame for Bob to send a reply which reaches her at an earlier time than he first sent the reply, and so if Bob sends a reply as soon as he first receives Alice's message then Alice can receive Bob's reply before she sent her original message. This is absolute time travel because the event of Alice receiving Bob's reply and the event of Alice sending the original message both occur on Alice's worldline, so one is in the past light cone of the other, and they have a "timelike separation" (meaning a slower-than-light object, such as Alice, can travel from one event to the other) which means all inertial frames agree on the order of the events.

Also having read your comment over more and looked at other websites i wonder if my confusion is rooted in the idea of FTL. In my example the traveller wasn't meant to have a faster than light speed, merely he could travel faster than light. If one could travel faster than light (without having superluminal velocity) would that still cause you to travel back in time?

I don't understand, how can you travel faster than light without having a superluminal velocity? Speed in a given frame is just distance/time, and in every frame light has a distance/time of c, so for example if you travel from Earth to a star 4 light-years away in a time of less than 4 years, then you have traveled faster than light and by definition you also have a superluminal speed.

 

[Ryan_m_b]

Thank you very much for your explanation. Ill use it and the links and try to educate myself more on this, I think I'm starting to get it now (though it's still weird). My interest stemmed from wondering if I would actually want to live in a universe where FTL was possible, not sure if I would if it meant traveling in time as well.

As for non-superluminal speed I was changing the thought experiment from an instantaneous message that travels between point A and B instantly that one that cuts through space without having to be in between either point. Its speed could be STL but it's traveled less of a distance.

 

[FtlIsAwesome]

As for non-superluminal speed I was changing the thought experiment from an instantaneous message that travels between point A and B instantly that one that cuts through space without having to be in between either point. Its speed could be STL but it's traveled less of a distance.

Like a http://en.wikipedia.org/wiki/Wormhole" ? Both bend space so that an object will be STL relative to local space, but FTL relative to distant space.

 

[icester]

You can only travel a distance. Time does not exist.
The concept of time is simply ratio of motions. We compare observed motion with planetary periodic motion and use it to synchronize motion events.

Mathew Orman