B Implications of FTL signals in SR

As I understand if it were possible to send a single faster than light then one can send a signal back into the past. What is the simplest way to explain how this would happen?
 

.Scott

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The issue is how things look from different inertial reference frames.

Something that is traveling at the speed of light will appear to be traveling at that speed in all inertial reference frames.
Something traveling slower than the speed of light will appear to be traveling at some speed less than the speed of light in all inertial reference frames - but not necessarily the same speed.
But something traveling faster than light will appear to be traveling either faster than light or backwards through time depending on which inertial reference frame it is viewed from.

So it's not just that FTL means you can send the signal back in time, it means you have already sent it back in time - depending on the reference frame.

So let's say that you have a station on Pluto. Then you have a spacecraft capable of sending FTL signals shoot past Earth as it transmits such a signal. The velocity would have to be such that it would appear to be an FTL signal from the spacecrafts reference frame, but a signal that goes back through time as viewed from Earth. Then you would have another FTL transmitter on a spacecraft swoopong past Pluto just as the signal arrives there. It woul then relay the signal back towards Earth. Once again, this would be a signal that would appear to be FTL to the spacecraft but one that trvelled back through time as seen from Earth.

The final result would be a signal that appears to start on Earth goes out to Pluto and back, arriving on Earth well before it was transmitted.
 

Nugatory

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As I understand if it were possible to send a single faster than light then one can send a signal back into the past. What is the simplest way to explain how this would happen?
Google for "tachyonic antitelephone" - the Wikipedia article is pretty good,
 
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The issue is how things look from different inertial reference frames.

Something that is traveling at the speed of light will appear to be traveling at that speed in all inertial reference frames.
Something traveling slower than the speed of light will appear to be traveling at some speed less than the speed of light in all inertial reference frames - but not necessarily the same speed.
But something traveling faster than light will appear to be traveling either faster than light or backwards through time depending on which inertial reference frame it is viewed from.
Do you have any calculation for this ? As I viewed, Lorentz transform for faster than light induces an imaginary spacetime, from ##\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}##. If ##v\gt c## then ##\gamma\in i\mathbb{R}##.

How do we get backward in time from this ?
 

.Scott

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Do you have any calculation for this ? As I viewed, Lorentz transform for faster than light induces an imaginary spacetime, from ##\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}##. If ##v\gt c## then ##\gamma\in i\mathbb{R}##.

How do we get backward in time from this ?
I'm not sure if you can get it from the Lorentz transform, but if a clock is approaching you at greater than the speed of light, the first you see of it will be as it passes you. Then you will see it at earlier and earlier times as the light reaches you from those more distant locations. So you would see the hands on the clock move backwards.

If you can send messages faster than the speed of light, there is a procedure for relaying the message among three reference frames that allows the message to be drilled back through time.
 

Nugatory

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How do we get backward in time from this ?
You don't. You get nonsense, because the Lorentz transformations are derived from assumptions that are equivalent to assuming that the speed is less than ##c## so using a larger value is internally inconsistent. However....
If you can send messages faster than the speed of light, there is a procedure for relaying the message among three reference frames that allows the message to be drilled back through time.
The "tachyonic antitelephone" mentioned above, for example.
 

Ibix

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You can't use the Lorentz transforms to define frames with relative velocities exceeding light. However, you can easily use them to show that something exceeding the speed of light is moving backwards in time in some frame. Choose the object to move through the origin of coordinates and arrive at coordinates (x,t). Lorentz transform them to get ##t'=\gamma(t-vx/c^2)##. If ##x>ct## (i.e. the thing travelled faster than light), it's easy enough to show that you can pick a ##v<c## such that ##t'<0##. And that you cannot if ##x\leq ct##.
 
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For the purposed of a novel I'm writing, does the 'FTL = time travel' situation change if the FTL is via a wormhole to effect instantaneous travel and the end-points are a significant distance apart? Such as light days or even light years.

In this case, the ship is not actually moving faster than c the conventional sense. It does not accelerate to faster than c, or even anywhere near c. And there is no tachyonic antitelephone, so Alice and Bob have no way to communicate in the way that page describes.

Surely the ship is only ever in the 'now', it is just that its 'now' shifts from A to B instantaneously. So right now it is in orbit around Earth, then an instant later it is in orbit around Proxima Centauri b. If the ship broadcast its departure and arrival as a radio burst, those would still take 4 years to arrive at each end.
 

Nugatory

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For the purposed of a novel I'm writing, does the 'FTL = time travel' situation change if the FTL is via a wormhole
No. The problem with FTL is the relationship between the start of the journey and the arrival at the destination, no matter how the journey happens. If something leaves point A and arrives at point B before a flash of light travelling in the ordinary manner could, then someone in the universe will find that the arrival happens before the departure - and will be able to stop the departure (perhaps by blowing up the FTL ship on the launchpad) after the ship has already arrived at the destination.

If you haven't already about "tachyonic antitelephone" mentioned twice above already, give it a try.
 

Ibix

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For the purposed of a novel I'm writing, does the 'FTL = time travel' situation change if the FTL is via a wormhole to effect instantaneous travel and the end-points are a significant distance apart? Such as light days or even light years.
It depends on how it works. The problem with FTL travel in relativity is that there's no global definition of what "at the same time" means for things that aren't in the same place. The tachyonic anti-telephone exploits the fact that "ship arrives at the same time as it leaves" can mean different things in order to build a (potentially paradoxical) time loop.

You can easily avoid this in an SF setting by insisting that your FTL drive respect a chosen global simultaneity rule (which Einsteinian physics says in nonsense, but follows obviously from the theoretical work done by John Smith in the mid-2030s, or whatever name and time frame is suitable for your story). Some frames will still regard the ship as arriving before it leaves, but it doesn't matter because it can neither return before it leaves nor send a message that arrives home before it leaves. So there's never a kill-my-own-grandfather paradox of the kind Nugatory mentions.

There's no evidence of such a global simultaneity rule in reality, so any kind of FTL travel remains problematic.
 

Mister T

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As I understand if it were possible to send a single faster than light then one can send a signal back into the past. What is the simplest way to explain how this would happen?
Who is the explanation targeted at?
 
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does the 'FTL = time travel' situation change if the FTL is via a wormhole to effect instantaneous travel and the end-points are a significant distance apart? Such as light days or even light years.

In this case, the ship is not actually moving faster than c the conventional sense. It does not accelerate to faster than c, or even anywhere near c.
This is general relativity rather than special relativity so things are a little bit more complicated. For a bit of background, if an object/particle/signal travels slower than light locally it is said to have a timelike curve or worldline. If it is like a tachyon that goes faster than light locally it is said to have a spacelike curve. Massive objects must travel on timelike curves.

As you mention, an object going through a wormhole travels on a timelike path. At each point it is going locally slower than light, although there are non-local paths where it arrives to the destination faster than light going “the long way”.

In GR, you can in principle use a wormhole for time travel. Such time travel is called a closed timelike curve. If you take a wormhole and move one entrance at relativistic speeds then when it gets back near the other closed timelike curves become possible.

If you wanted to prevent time travel you could just say that accelerating the wormhole mouth makes it unstable.
 

pervect

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As I understand if it were possible to send a single faster than light then one can send a signal back into the past. What is the simplest way to explain how this would happen?
I think one thing that is confusing is a good notion of "past" and "future" in relativity. There is a phenomenon known as "the relativity of simultaneity", which makes this concept a bit difficult to pin down. There's a lot of discussion of the relativity of simultaneity, see for instance any of the numerous threads on "Einstein's train" here or elsewhere, but it's difficult topic. I'll restrict myself to pointing out it's existence here, rather than trying to explain it in any more detail than giving it's name.

While "the past" and "the future" are a bit problematic, what is well defined in special relativity is is the "past light cone" of an event and "the future light cone" of an event, this is different from 'the past" and "the future" because one needs to specify a pariticular event in order to specify the past and future light cones.

Past light cones, and future light cones, form the basis for causality in special relativity, rather than "the past" and "the future". If ight cannot propage from one event to another, then causal influences cannot exist between the two events.
 
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and will be able to stop the departure (perhaps by blowing up the FTL ship on the launchpad) after the ship has already arrived at the destination.
Even working through the tachyonic anti-telephone concept, this is the part that I'm struggling to get my head around. There is only one ship, so if it's on the launch pad, it is not also orbiting Proxima Centauri b. So someone can't slip in before it launches and blow it up...or if they do, it doesn't then end up orbiting Proxima Centauri b. It is clearly non-intuitive to have a situation where the ship launches, goes into orbit elsewhere, then is blown up before it launches, all because of FTL travel o_O

Anyway, I appreciate the attempts to explain what seems to be a deeply subtle concept, and much as I want to keep my sci-fi hard, I'm going to have to ignore time-travel effects and just get on with telling the story.
 
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There is only one ship
If the ship's path through spacetime has a "loop" in it, so it returns to the same spacetime region, then in that region, there could be more than one of it--an earlier version and a later version. This is one of many counterintuitive things that makes it very difficult to think about such scenarios.

It is clearly non-intuitive to have a situation where the ship launches, goes into orbit elsewhere, then is blown up before it launches, all because of FTL travel
This is not just "non-intuitive", it's not possible, at least not if there is one spacetime reality. If there is one spacetime reality, it has to be self-consistent; FTL travel would not change that. This idea has even been formalized as the Novikov self-consistency principle:

 
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This is not just "non-intuitive", it's not possible, at least not if there is one spacetime reality.
Aha, that's good to know. And it highlights the subtly inherent in time travel, I guess. Nugatory possibly assumed mutiverse type conditions, but irrespective, the whole situation with FTL = time travel is too complex for me to write up, so I'll stick to the basics :smile:
 

Ibix

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Aha, that's good to know. And it highlights the subtly inherent in time travel, I guess. Nugatory possibly assumed mutiverse type conditions
No - the point is that if you allow faster than light travel in relativity, without restrictions of the type I proposed, you do get paradoxes. That's one of the reasons we believe it's impossible, because we have no evidence of anything that would stop paradoxes if FTL were possible. And we'd like to think the universe makes some kind of sense.
the whole situation with FTL = time travel is too complex for me to write up, so I'll stick to the basics
If you want to acknowledge it, simply put in a line about physicists' worries about FTL leading to paradoxes turning out to be unfounded as a result of (imaginary) future theories. And otherwise ignore it as you suggest.
 

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