Proof of impossibility of superluminal signals?

[kochanskij]

Does the following argument prove that any method of faster than light communication is impossible?

1. A transmitter sends a signal at 12:00 from point A to B faster than light by some hypothetical method.
2. A transmitter at point B is moving at a high speed (but less than speed of light) relative to A. When it receives the signal, it sends a reply signal faster than light back to point A.
3. Receiver at A detects the reply at 11:00 - before original signal was transmitted. According to special relativity, this backward in time effect can always be set up using a pair of superluminal signals.
4. Electronic mechanism is set up to turn off the transmitter as soon as receiver detects a signal.
5. If a signal is received at 11:00, then no signal can be sent at 12:00. If no signal is received at 11:00, then a signal is sent at 12:00. A paradox is created. If no original signal is sent, then how can you get a reply? If no reply was received, then what happened to the original
signal? This is a version of the "grandfather paradox".
6. If a theory has a logical contradiction in it, then it is not self-consistent. So it must be rejected on logic alone. No experiments are needed.

Therefore, the theory that some form of faster than light signaling exists must be false.
This argument would prove not only that all current ideas for superluminal signaling are impossible (wormholes, warp drives, quantum mechanisms) but also that all new clever ideas that will be thought of in the future must be impossible too.

Did I make any mistake in logic? Did I make any unproven assumptions? What is your opinion??

 

[ghwellsjr]

Does the following argument prove that any method of faster than light communication is impossible?

1. A transmitter sends a signal at 12:00 from point A to B faster than light by some hypothetical method.
2. A transmitter at point B is moving at a high speed (but less than speed of light) relative to A. When it receives the signal, it sends a reply signal faster than light back to point A.
3. Receiver at A detects the reply at 11:00 - before original signal was transmitted. According to special relativity, this backward in time effect can always be set up using a pair of superluminal signals.
4. Electronic mechanism is set up to turn off the transmitter as soon as receiver detects a signal.
5. If a signal is received at 11:00, then no signal can be sent at 12:00. If no signal is received at 11:00, then a signal is sent at 12:00. A paradox is created. If no original signal is sent, then how can you get a reply? If no reply was received, then what happened to the original
signal? This is a version of the "grandfather paradox".
6. If a theory has a logical contradiction in it, then it is not self-consistent. So it must be rejected on logic alone. No experiments are needed.

Therefore, the theory that some form of faster than light signaling exists must be false.
This argument would prove not only that all current ideas for superluminal signaling are impossible (wormholes, warp drives, quantum mechanisms) but also that all new clever ideas that will be thought of in the future must be impossible too.

Did I make any mistake in logic? Did I make any unproven assumptions? What is your opinion??

My opinion is that you made a big mistake.

If you send a signal at the speed of light from A to B and then back from B to A at the speed of light, there will be a time delay from sending the first signal at A to receiving the second signal at A. If you could make the signal go faster than the speed of light, it would just reduce the time from sending to receiving but even at an infinite speed, it only get back to A at the same time it was sent, not before.

And I don't know why you think the speed of B is significant.

 

[mfb]

@ghwellsjr: Faster than light in one system is backwards in time in another system. The described setup is well-known and would lead to a violation of causality, if special relativity is correct and superluminal transmissions are possible.

There are multiple ways to avoid this paradox but save most of the physics, however.
- special relativity could be incorrect in some (but not all) ways. Something like "A can send superluminal for him, but B cannot".
- such a device switching in the described way together with the transmissions just cannot be set up. This is basically the self-consistency principle. And the principle could be realized - see the thread Simulating Closed Timelike Curves through Quantum Optics.

 

[Nugatory]

6. If a theory has a logical contradiction in it, then it is not self-consistent. So it must be rejected on logic alone. No experiments are needed.

Therefore, the theory that some form of faster than light signaling exists must be false.
This argument would prove not only that all current ideas for superluminal signaling are impossible (wormholes, warp drives, quantum mechanisms) but also that all new clever ideas that will be thought of in the future must be impossible too.

Did I make any mistake in logic? Did I make any unproven assumptions? What is your opinion??

Your conclusion is valid (any superluminal signal will imply a contradiction) but I'm not sure you've correctly derived the contradiction... Try googling for "tachyonic antitelephone", compare that with your thought experiment.

(I'll look more carefully at your derivation later today unless someone else gets to it first)

 

[Fredrik]

And I don't know why you think the speed of B is significant.

If B is moving away from A, then B's simultaneity line through the event where he receives the original message, may intersect A's world line at an event earlier than the one where the original message was sent. This makes it conceivable that the reply message (if it's near instantaneous in B's rest frame) reaches A before the original message was sent.

 

[Fredrik]

Does the following argument prove that any method of faster than light communication is impossible?

1. A transmitter sends a signal at 12:00 from point A to B faster than light by some hypothetical method.
2. A transmitter at point B is moving at a high speed (but less than speed of light) relative to A. When it receives the signal, it sends a reply signal faster than light back to point A.
3. Receiver at A detects the reply at 11:00 - before original signal was transmitted. According to special relativity, this backward in time effect can always be set up using a pair of superluminal signals.
4. Electronic mechanism is set up to turn off the transmitter as soon as receiver detects a signal.
5. If a signal is received at 11:00, then no signal can be sent at 12:00. If no signal is received at 11:00, then a signal is sent at 12:00. A paradox is created. If no original signal is sent, then how can you get a reply? If no reply was received, then what happened to the original
signal? This is a version of the "grandfather paradox".
6. If a theory has a logical contradiction in it, then it is not self-consistent. So it must be rejected on logic alone. No experiments are needed.

Therefore, the theory that some form of faster than light signaling exists must be false.
This argument would prove not only that all current ideas for superluminal signaling are impossible (wormholes, warp drives, quantum mechanisms) but also that all new clever ideas that will be thought of in the future must be impossible too.

Did I make any mistake in logic? Did I make any unproven assumptions? What is your opinion??

You have to be more precise about the precise location of A and B, B's velocity relative to A, the velocities of the messages (and in what rest frames those velocities are given). But it looks like you have essentially reproduced the standard argument against FTL messages. (I didn't examine all the details in your argument, since the details I just mentioned were missing at the start). There are some loopholes though. For example:

1. What if the time it takes to receive or transmit a tachyonic message increases rapidly with the distance the message has to travel?

2. What if there's a theory of matter in Minkowski spacetime that doesn't forbid FTL messages, or any specific event you described, but still doesn't have any solutions (to the equations of motion) that describe all the events in your scenario. (Maybe there's a solution that describes a universe where B gets hit by a meteor just before he tries to send the reply back in time).

 

[ghwellsjr]

@ghwellsjr: Faster than light in one system is backwards in time in another system. The described setup is well-known and would lead to a violation of causality, if special relativity is correct and superluminal transmissions are possible.

And I don't know why you think the speed of B is significant.

If B is moving away from A, then B's simultaneity line through the event where he receives the original message, may intersect A's world line at an event earlier than the one where the original message was sent. This makes it conceivable that the reply message (if it's near instantaneous in B's rest frame) reaches A before the original message was sent.

Well-known or not, I wasn't aware that the OP was describing a scenario in two different frames, so thanks for filling in that detail.

But since the scenario is impossible in and of itself, I don't know why it should be used as a proof of the impossibility of faster than light signals.

 

[PeterDonis]

Does the following argument prove that any method of faster than light communication is impossible?

I don't think so; I think it only proves that faster than light communication does not let you create a scenario that is not self-consistent. Your argument does not rule out the possibility of FTL communication being possible, but only in the context of a scenario that is self-consistent when FTL (and therefore backwards in time) communication is included. It is perfectly possible to construct such scenarios; for example, see here:

http://en.wikipedia.org/wiki/Novikov_self-consistency_principle

 

[Fredrik]

But since the scenario is impossible in and of itself, I don't know why it should be used as a proof of the impossibility of faster than light signals.

The idea is to prove that no special relativistic theory allows FTL messages, by deriving an absurdity from an arbitrary theory of that sort.

 

[kochanskij]

Fredrik and Peter ask some good questions and make some good points.

I could put in specific numbers into my argument and use relativity to calculate the time the reply signal is received at A. But my point is that if A and B are far enough apart, B is moving away from A at a large relativistic speed, and both signals travel much faster than light, then A will receive the reply BEFORE he transmits the original signal. (Note: If the signal goes at infinite speed relative to A and the reply goes at infinite speed relative to B, then A will always get the reply before it transmits.)

Both Fredrik and Peter suggest that spacetime has a global consistency principle - all the events in my argument are permitted by laws of nature but they can not all occur together. Something would always happen to prevent my paradox from happening. It is hard to comprehend how this conspiracy would work. (How does Fredrik's meteor know when and where to strike point B if point A and the paradox is light years away?)

I agree that a global consistency principle is possible but it would be fun to see exactly what happens to prevent the paradox in a variety of different situations. Is there any possibility of doing this actual experiment in the lab? Perhaps we could use quantum particle tunneling through a barrier as the superluminal signal??

 

[verty]

@ghwellsjr: Faster than light in one system is backwards in time in another system. The described setup is well-known and would lead to a violation of causality, if special relativity is correct and superluminal transmissions are possible.

Regarding this, suppose Saturn's rings are sufficiently reflective for us to use them as a mirror. We send some photons toward them at the speed of light, they return and we detect them. There will be a time delay, ~170 minutes (sorry, this is not the correct delay, it'll be somewhat shorter). So there is a window of maybe 2 hours for faster communications to return. Does SR imply that no communication of any hypothetical speed can arrive in this time window? It seems odd to say that faster communications could in fact arrive in negative time (violating causality) but not in this window.

Similarly, does SR imply that light-speed communications take no time at all?

 

[ghwellsjr]

Fredrik and Peter ask some good questions and make some good points.

I could put in specific numbers into my argument and use relativity to calculate the time the reply signal is received at A. But my point is that if A and B are far enough apart, B is moving away from A at a large relativistic speed, and both signals travel much faster than light, then A will receive the reply BEFORE he transmits the original signal.

You did it again, even after being admonished by Fredrik to be more precise in stating what frame the velocities are being specified in. You can't assume that someone reading your scenario will understand it the way you understand it unless you state what you mean. Don't you think it is reasonable to assume unless otherwise stated, that a scenario is described according to a single frame?

(Note: If the signal goes at infinite speed relative to A and the reply goes at infinite speed relative to B, then A will always get the reply before it transmits.)

Finally, for this scenario, you state what the speeds are relative to.

 

[ghwellsjr]

@ghwellsjr: Faster than light in one system is backwards in time in another system. The described setup is well-known and would lead to a violation of causality, if special relativity is correct and superluminal transmissions are possible.

Regarding this, suppose Saturn's rings are sufficiently reflective for us to use them as a mirror. We send some photons toward them at the speed of light, they return and we detect them. There will be a time delay, ~170 minutes (sorry, this is not the correct delay, it'll be somewhat shorter). So there is a window of maybe 2 hours for faster communications to return. Does SR imply that no communication of any hypothetical speed can arrive in this time window?

Yes, it does. At least that's what I understand. But mfb's quote states the opposite. He thinks that special relativity can be correct and superluminal transmissions can be possible.

It seems odd to say that faster communications could in fact arrive in negative time (violating causality) but not in this window.

I don't think either one is possible.

Similarly, does SR imply that light-speed communications take no time at all?

No, SR defines light-speed communications to take place at light speed. Not faster, not slower, not instantly.

 

[Fredrik]

You did it again, even after being admonished by Fredrik to be more precise in stating what frame the velocities are being specified in. You can't assume that someone reading your scenario will understand it the way you understand it unless you state what you mean. Don't you think it is reasonable to assume unless otherwise stated, that a scenario is described according to a single frame?

If you want to see the scenario described in more detail, see posts 133 and 134 here: https://www.physicsforums.com/showthread.php?p=2588832. Post 133 is a description of a scenario with infinite-speed messages (infinite speed in the sender's rest frame). I'm afraid I was too lazy to draw a diagram. Post 134 has a link to a spacetime diagram for a scenario with finite-speed FTL messages.

 

[Fredrik]

Regarding this, suppose Saturn's rings are sufficiently reflective for us to use them as a mirror. We send some photons toward them at the speed of light, they return and we detect them. There will be a time delay, ~170 minutes (sorry, this is not the correct delay, it'll be somewhat shorter). So there is a window of maybe 2 hours for faster communications to return. Does SR imply that no communication of any hypothetical speed can arrive in this time window? It seems odd to say that faster communications could in fact arrive in negative time (violating causality) but not in this window.

An FTL message that's just reflected and returns with the same speed (in Earth's rest frame) just arrives less than 170 minutes (or whatever) after the message was sent.

To see anything funny in a thought experiment, the recipient of the original message has to be moving away from the sender at close to the speed of light, and it's essential that the reply he's sending is a very fast FTL message in his rest frame. (You need to look at a spacetime diagram to see why this leads to weirdness).

Similarly, does SR imply that light-speed communications take no time at all?

A communication that take no time at all would have to move as described by a horizontal line in a spacetime diagram (so that both endpoints will have the same time coordinate). Light moves as described by a line drawn at a 45° angle.

 

[mfb]

Regarding this, suppose Saturn's rings are sufficiently reflective for us to use them as a mirror. We send some photons toward them at the speed of light, they return and we detect them. There will be a time delay, ~170 minutes (sorry, this is not the correct delay, it'll be somewhat shorter). So there is a window of maybe 2 hours for faster communications to return. Does SR imply that no communication of any hypothetical speed can arrive in this time window? It seems odd to say that faster communications could in fact arrive in negative time (violating causality) but not in this window.

With hypothetical faster-than-light signals, a return signal could arrive anywhere in this timeframe, that would be no problem. But that is the boring part of FTL - the more interesting part is the possibility to get the answer before we even send the message out.

With SR as we know it, you can have either causality or FTL signals, but not both.

Similarly, does SR imply that light-speed communications take no time at all?

No. Why do you ask?

Yes, it does. At least that's what I understand. But mfb's quote states the opposite. He thinks that special relativity can be correct and superluminal transmissions can be possible.

I don't see causality as part of SR, so yes.

To see anything funny in a thought experiment, the recipient of the original message has to be moving away from the sender at close to the speed of light, and it's essential that the reply he's sending is a very fast FTL message in his rest frame. (You need to look at a spacetime diagram to see why this leads to weirdness).

That depends on the (completely unknown) way to send those signals. Sure, all the scenarios consider faster-than-light signals that still move forward in time in the frame of the emitter - but why should this be the limit? We know for sure that you do not need a large spacecraft moving away: a small emitter inside (maybe even a single atom), moving away for a short period of time, could be sufficient in this hypothetical scenario. Maybe this emitter can even do the same without moving - directly sending messages that travel backwards in time in the frame of the emitter. Sure, it violates causality, but that's what all those scenarios are doing.

 

[phyti]

Does the following argument prove that any method of faster than light communication is impossible?

1. A transmitter sends a signal at 12:00 from point A to B faster than light by some hypothetical method.
2. A transmitter at point B is moving at a high speed (but less than speed of light) relative to A. When it receives the signal, it sends a reply signal faster than light back to point A.
3. Receiver at A detects the reply at 11:00 - before original signal was transmitted. According to special relativity, this backward in time effect can always be set up using a pair of superluminal signals.
4. Electronic mechanism is set up to turn off the transmitter as soon as receiver detects a signal.
5. If a signal is received at 11:00, then no signal can be sent at 12:00. If no signal is received at 11:00, then a signal is sent at 12:00. A paradox is created. If no original signal is sent, then how can you get a reply? If no reply was received, then what happened to the original
signal? This is a version of the "grandfather paradox".
6. If a theory has a logical contradiction in it, then it is not self-consistent. So it must be rejected on logic alone. No experiments are needed.

Therefore, the theory that some form of faster than light signaling exists must be false.
This argument would prove not only that all current ideas for superluminal signaling are impossible (wormholes, warp drives, quantum mechanisms) but also that all new clever ideas that will be thought of in the future must be impossible too.

Did I make any mistake in logic? Did I make any unproven assumptions? What is your opinion??

A signal faster than light requires less time than one at light speed, but it is not instantaneous or zero. A finite amount of time ocurrs between emission and reception. Your assumption (red) is incorrect.
The simultaneity axis for B is a calculation convention, in effect for the local observer B.
B’s motion does not affect the clock rate of A. The A clock still runs forward, accumulating ‘time’.

 

[stevendaryl]

A signal faster than light requires less time than one at light speed, but it is not instantaneous or zero. A finite amount of time ocurrs between emission and reception. Your assumption (red) is incorrect.
The simultaneity axis for B is a calculation convention, in effect for the local observer B.
B’s motion does not affect the clock rate of A. The A clock still runs forward, accumulating ‘time’.

If a signal is faster than light in one frame, then it can be instantaneous or back-in-time in another frame. This follows from the Lorentz transformations:

$\delta t' = \gamma (\delta t - \frac{v}{c^2} \delta x)$

If $v = \dfrac{c^2 \delta t}{\delta x}$, then $\delta t' = 0$

 

[Fredrik]

A signal faster than light requires less time than one at light speed, but it is not instantaneous or zero. A finite amount of time ocurrs between emission and reception. Your assumption (red) is incorrect.

It's not wrong, just poorly explained. He just failed to mention that the original FTL message is near instantaneous in A's rest frame, that the reply is near instantaneous in B's rest frame, that B is moving away from A at close to the speed of light, and that B is already far from A.

 

[ghwellsjr]

You did it again, even after being admonished by Fredrik to be more precise in stating what frame the velocities are being specified in. You can't assume that someone reading your scenario will understand it the way you understand it unless you state what you mean. Don't you think it is reasonable to assume unless otherwise stated, that a scenario is described according to a single frame?

If you want to see the scenario described in more detail, see posts 133 and 134 here: https://www.physicsforums.com/showthread.php?p=2588832. Post 133 is a description of a scenario with infinite-speed messages (infinite speed in the sender's rest frame). I'm afraid I was too lazy to draw a diagram. Post 134 has a link to a spacetime diagram for a scenario with finite-speed FTL messages.

My problem had nothing to do with the lack of a diagram, it had only to do with an inadequate explanation, as I have repeatedly stated and as you stated. Once it was pointed out to me that the scenario was specified in terms of two frames, I understood at once what was going on.

I made my own diagrams to illustrate a similar scenario. First we start with the rest frame of A who sends an instantaneous signal to B at the Coordinate Time of zero:

Then we switch to the rest frame of B who sends an instantaneous signal back to A at the Coordinate Time of -6.67 nanoseconds:

A receives this signal at his Proper Time of -4 nanoseconds as indicated by the dots or as indicated in his rest frame by the Coordinate Time:

But as we can also see in the above diagram, the return signal is traveling backwards in time in A's rest frame so it's no wonder it arrives before A sent his signal. And in B's rest frame, the first signal goes backwards in time so it's no wonder the return signal arrives before the first signal was sent:

And as long as we're going to permit signals changing their speeds in different frames, why do we need B to be traveling at a relativistic speed? Why can't he just be stationary in A's rest frame and merely reflect the signal back to A like this?

 

[verty]

Similarly, does SR imply that light-speed communications take no time at all?

No. Why do you ask?

If FTL or causality but not both are possible, then FTL must occur, it would seem, at the boundary of causality. The boundary of causality (as I'm describing it) is when a communication is received at the instant it was sent. But there are two ways to look at this.

In the sender's frame, the causality boundary is when the message is received at exactly the time it was sent. In the message's frame, the causality boundary is with a light-speed communication, time is dilated to infinity (I believe) meaning that the message never leaves the Earth according to the message. And if an FTL communication is involved, the message sees a discontinuity at the instant the message is received (it would seem).

So I wondered if it was a matter of definition that light-speed communications (LSC's) take no time to travel, but a reflected LSC takes time because it reaches a different region of the manifold which is somehow at the same time. That would make the FTL versus causality dilemma trivial.

And in some sense, this could be the case because the LSC message experiences no passage of time. But I realize now that this kind of absolute time is almost certainly not defined in a theory based around the tenet that time is relative.

 

[Fredrik]

And as long as we're going to permit signals changing their speeds in different frames, why do we need B to be traveling at a relativistic speed? Why can't he just be stationary in A's rest frame and merely reflect the signal back to A like this?

If tachyons exist, then we can probably build a tachyon transmitter. If we can build one, then by the principle of relativity, we can build two, that are identical except for their velocities. If one of them emits tachons at speed kc (with k>1) in its own rest frame, then the other one will emit tachyons at speed kc (with the same k) in its own rest frame.

A signal reflected off a mirror that's stationary in A's rest frame should have the same speed as the incoming signal, due to conservation of momentum. At least that's how it works for massive particles. I haven't tried to think it through for tachyons, but my first guess is that it's the same.

 

[mfb]

If FTL or causality but not both are possible, then FTL must occur, it would seem, at the boundary of causality. The boundary of causality (as I'm describing it) is when a communication is received at the instant it was sent. But there are two ways to look at this.

If your signal goes to some other place, this "signal gets back instantly" has no special (boundary) meaning, as then one way has to be backwards in time for some observers. The boundary is "the signal gets back as fast as light can" - the light cone, the universal boundary of SR.

In the sender's frame, the causality boundary is when the message is received at exactly the time it was sent.

This is backwards in time for some observers, not at a boundary.

So I wondered if it was a matter of definition that light-speed communications (LSC's) take no time to travel, but a reflected LSC takes time because it reaches a different region of the manifold which is somehow at the same time.

The proper time for a light-like signal is zero, but in the frame of the emitter, it still needs time.

 

[ghwellsjr]

The proper time for a light-like signal is zero, but in the frame of the emitter, it still needs time.

I thought proper time did not apply to a light-like signal, but in this thread, who knows?

 

[Fredrik]

I thought proper time did not apply to a light-like signal, but in this thread, who knows?

We're discussing the possibility that theories of matter that moves FTL can be stated in the framework of Minkowski spacetime, so all the usual concepts like proper time still apply. You're right that proper time is only defined for timelike curves.

 

[ghwellsjr]

We're discussing the possibility that theories of matter that moves FTL can be stated in the framework of Minkowski spacetime, so all the usual concepts like proper time still apply. You're right that proper time is only defined for timelike curves.

Does that mean we're not discussing Einstein's theory of Special Relativity?

 

[Fredrik]

Does that mean we're not discussing Einstein's theory of Special Relativity?

Depends on what exactly you mean by "Einstein's theory of Special Relativity". I prefer to think of SR as the framework in which special relativistic theories are defined. This framework is defined by mathematical definitions of terms like "Minkowski spacetime" and "proper time", and (if we prefer) also by the correspondence rules that all theories in this framework have in common (e.g. the rule that tells us how to interpret statements about proper time as predictions about measurements with a clock).

Classical theories of matter in this framework can be defined by specifying the matter content of spacetime, and the complete set of correspondence rules. I would say that we can also define quantum theories of matter in this framework. It's mathematically more complicated, but there is a way to make sense of the concept "special relativistic quantum theory".

So to me, this discussion is about SR. You may prefer to define SR as a smaller class of theories. If your definition of "theory" is such that what you consider a theory is what I'd consider a class of theories, then you can also think of SR as one specific classical theory.

 

[mfb]

I thought proper time did not apply to a light-like signal, but in this thread, who knows?

Take the limit for v->c from v<c, if you like.
That's just a naming issue.

 

[stevendaryl]

Take the limit for v->c from v<c, if you like.
That's just a naming issue.

That gives you zero for every lightlike path. You can say that zero is a perfectly fine answer to the question of what is the proper time to get from point A to point B, but it's not very informative, since it's the same answer for any pair of points.

 

[mfb]

since it's the same answer for any pair of points.

The proper time experienced in unaccelerated motion (which is also the maximal proper time you can have) is positive if A and B are timelike separated. And the limit for light-like separation is zero.

 

[stevendaryl]

The proper time experienced in unaccelerated motion (which is also the maximal proper time you can have) is positive if A and B are timelike separated. And the limit for light-like separation is zero.

I meant for any two points along a light-like path.

 

[phyti]

kochanskij

1. A transmitter sends a signal at 12:00 from point A
3. Receiver at A detects the reply at 11:00

Why would the A clock run in reverse?

(Note: If the signal goes at infinite speed relative to A and the reply goes at infinite speed relative to B, then A will always get the reply before it transmits.)

Instantaneous is the speed limit.
The time interval would equal zero for the A clock.
A fundamental principle: you cannot detect a signal before it is emitted.

 

[Fredrik]

Why would the A clock run in reverse?

It's not running in reverse. The world line of the reply message is (or is close to) B's simultaneity line, which (assuming that B is moving away from A) intersects A's world line before A sent the original message. If you want to see a version of this scenario with the details written out, click the link in post #14.

 

[stevendaryl]

kochanskij
Why would the A clock run in reverse?
Instantaneous is the speed limit.
The time interval would equal zero for the A clock.
A fundamental principle: you cannot detect a signal before it is emitted.

I think there might be some miscommunication going on. The point is not to demonstrate that back-in-time communication is possible, but to demonstrate that FTL is impossible. It's a principle of logic that if $A \wedge B \Rightarrow C$, and $C$ is impossible, then either $A$ is false, or $B$ is false.

In this case:
$A$ = the principle of relativity
$B$ = the assumption that FTL communication is possible
$C$ = the conclusion that back-in-time communication is possible.

If you have relativity, and you also have faster-than-light communication, then you have back-in-time communication (signals arriving before they are sent).

 

[phyti]

In drawing ftl2 the ftl signal moves at 2c. B moves at .5c, with c=1
Using the SR synch convention (radar or pinging method), the emission event A(x,t)=(0,1) is assigned to B(x,t)=(-.58,1.15),
and the detection event A(x,t)=(0,1.67) is assigned to B(x,t)=(-.96,1.92).
B calculates the incoming signal as instantaneous,
and the outgoing signal at .96/(1.92-1.15)=1.25.

In drawing ftl3 the ftl signal is instantaneous. B moves at .5c, with c=1
Since the time of travel is zero, and equivalent to zero distance, emission, reflection, and detection events are simultaneous and coincident for A. Thus must be coincident for all frames.
There could be any number of cycles relative to A in zero time. No information can be obtained in zero time. The same radar method does not work for this case because it requires a finite amount of time.
This is the view of universal time, before SR, i.e. observers see events in real time(as it happens).