Resolution to the Tachyonic Antitelephone paradox

[industry7]

Wikipedia has this interesting article about a physics "paradox" that Einstein came up with:
https://en.wikipedia.org/wiki/Tachyonic_antitelephone
It shows that if you were able to send a signal faster than the speed of light, then you could send a message back in time leading to a causality paradox. Einstein said this wasn't a problem since you can't send FTL signals to begin with.

I don't understand why the result comes out the way it does though. The one way communication example doesn't really have an explanation, other than if you pick the right variables to plug into this equation then you can get a negative result. But I can't figure out how to translate that to a hypothetical experiment that someone could carry out. The provided illustration didn't really help me either.

Then there's the two way communication example. I found that one to be even more confusing, because you have to keep track of two different "timelines", and the example keeps switching back and forth between them. But what really confused me about the two way communication example, is that it seems to me that you can play out all the events from just one person's perspective, and if you do you end up with a consistent (paradox free) accounting of the events. But obviously that must not be possible for some reason, and I don't know why.

 

[mfb]

Something that is faster than light (FTL) for some observer is going backwards in time for other observers. Do you understand that? If not, see the one-way example and learn how to read those 2D diagrams.

The two-way telephone combines two FTL transmissions as seen by different reference frames. For the person sending the first signal (A), the first signal is FTL: It goes a bit forward in time but mainly away in space. For the other person (B) it goes backwards in time. The answer sent by B is FTL for B, and going backwards in time for A.
If the FTL transmissions are fast enough, and the relative speed between A and B is right, the answer arrives before the first signal is sent.

 

[PeterDonis]

it seems to me that you can play out all the events from just one person's perspective, and if you do you end up with a consistent (paradox free) accounting of the events. But obviously that must not be possible for some reason, and I don't know why.

The reason is the principle of relativity. Basically, if we allow FTL signals, we are allowing signals that can travel on spacelike paths through spacetime. But the time ordering of events on spacelike paths is not invariant under Lorentz transformations. So we have to come up with some rule that determines the time ordering of the events of FTL signal "emission" and "reception".

If we accept the principle of relativity, then this rule has to be relativistically invariant. For example, we could say that the FTL signal propagates instantaneously according to the emitter--the time of emission is the same as the time of reception in the emitter's frame. But then we get the paradox: if A and B are in relative motion, and A sends B an FTL signal, and then B sends an FTL signal back to A, the second signal can arrive at A before the first was sent, because of relativity of simultaneity--the second signal is instantaneous according to B, which if B is moving in the right way relative to A, can mean that signal goes backwards in time according to A. And since the first signal was instantaneous according to A, the overall effect is a round-trip signal that goes backwards in time period--it arrives on A's worldline before it was sent, and that will be an invariant, true in all frames.

If, OTOH, we want to prevent the paradox, we have to adopt a rule that fixes some particular inertial frame and says that FTL signals are instantaneous only in that frame. For example, suppose it's A's frame. Then A can send a signal to B, and B can send one right back to A, and the second signal will arrive a tiny bit after the first (however long it takes B's signal processor to receive the first signal and emit the second). To B, if B is moving relative to A, it will look like one signal went forward in time and one signal went backward in time (which is which will depend on the direction of relative motion), but there will be no paradox because no signal can ever arrive before it was sent. However, this rule violates the principle of relativity because it picks out one particular inertial frame (in this case A's frame) and puts it into a fundamental law of physics, the law that governs FTL signals.

 

[industry7]

Something that is faster than light (FTL) for some observer is going backwards in time for other observers. Do you understand that? If not, see the one-way example and learn how to read those 2D diagrams.

I will work on that first, thanks :-)

 

[industry7]

I've been playing around with the Minkowski diagrams in order to understand them better. But there is something that I'm finding fundamentally really confusing. Let's say that you set up two transceivers 4 light seconds apart. And maybe they use normal luminal signals, or maybe they use superluminal signals. You know they are 4 light seconds apart, and as you go zipping by at some significant percentage of c, you see some weird effects.

In the case of a luminal signal (call this Case 1), what you see appears to be a superluminal signal. Ie, you know the transceivers are 4 light seconds apart, but from your view the signal appears to propagate between them in less than 4 seconds. Therefore it would appear that the signal traveled faster than the speed of light. In the case of a superluminal signal (call this Case 2), what you see appears to be acausal. Ie, you see the signal arrive before it leaves. Therefore it would appear that the signal violated causality.

So in Case 1, everyone agrees that no faster than light travel "actually" occurred. So I would expect that in Case 2, no violations of causality would "actually" occur. However, the wiki article asserts that actual backward time travel is predicted in Case 2, and that actual violations of causality are predicted too. Or in other words, even though it appears to Bob that Alice received his warning before he sent it, I would expect that Bob could always perform the calculations backwards in order to determine that's not "really" what happened.

Also, I keep putting "really" and "actually" in quotes because we all already accept relativity as true, which means we all already accept that the time ordering of events is observer dependent due to Relativity of Simultaneity. But what I mean by "what actually happened", is that (without bringing superluminal signals into the picture) despite the fact that several observers may all see a different ordering of events, if you hooked up a bunch of bombs to explode based on the ordering of events, you'll never end up with a situation where a bomb goes off from one person's point of view but not from another. Reality always agrees. So I'm still having a hard time seeing why superluminal signals lead to the prediction that reality doesn't agree. It seems to me that the only thing being predicted is the appearance of disagreement, which is not a problem.

 

[Nugatory]

n the case of a luminal signal (call this Case 1), what you see appears to be a superluminal signal. Ie, you know the transceivers are 4 light seconds apart, but from your view the signal appears to propagate between them in less than 4 seconds.

What you see does not appear to be a superluminal signal. The time you measure is less than four seconds but the distance the light covers is less than four light-seconds; because of length contraction the transceivers are less than four light-seconds apart.

 

[mfb]

In the case of a luminal signal (call this Case 1), what you see appears to be a superluminal signal.

No. Light-like signals appear light-like in every reference frame. The signal takes a different time because (a) the two transceivers are now moving and (b) their distance decreased.

In the case of a superluminal signal (call this Case 2), what you see appears to be acausal. Ie, you see the signal arrive before it leaves.

It depends on the signal and your velocity.
If the signal arrive before it leaves, this is one part of the violation of causality.

There is no reference frame that is "more real" than others.

if you hooked up a bunch of bombs to explode based on the ordering of events

You cannot do this. There is no unique time-ordering of events. No bomb can figure out "did X happen before or after Y?" in a coordinate-independent way if X and Y were spacelike separated.

 

[industry7]

What you see does not appear to be a superluminal signal.

No. Light-like signals appear light-like in every reference frame.

Apparent superluminal motion of astronomical bodies is a well known phenomenon, and people have been talking about it for over a hundred years. My example was based on the work of Martin Rees. The crux of the problem in both my example and in Rees', is a mistaken assumption about one of the distances involved because you didn't actually measure it.

Anyway, it looks like this thread has pretty much died, as nobody else ever joined in the discussion. So I guess we don't really need to try to continue it. But, both of you took the time to respond to my post, and I really do appreciate it. So thank you both for trying to help me out.

 

[mfb]

Apparent superluminal motion of astronomical bodies is a well known phenomenon

That is a different case of "apparent". It only appears superluminal if you don't keep track of its distance to us and the (changing) time light needs to reach us. All the previous post assume that you take this time into account.

 

[industry7]

That is a different case of "apparent".

Yes, I'm aware that it is not the exact same scenario. The exact scenario Rees described would not have worked for the point that I was trying to make. In Rees' example, you have to actually measure the distance more than once. The point of my example was that you could use relativity to calculate the distance that's needed to get the right answer (you don't have to measure it while you're flying past).

No bomb can figure out "did X happen before or after Y?" in a coordinate-independent way if X and Y were spacelike separated.

Of course not. If that were possible, then my example wouldn't have worked for the point I was trying to make. The point being that reality always agrees. So to be crystal clear, the bomb on the train determines whether or not the doors closed at the same time using ordinary means, like a person would, perhaps using a camera system to watch for the doors, or an electronic switch that sends an electrical signal down a wire. So for the sake of this example, we'll say that the bomb (on the train) explodes if it sees the doors drop at the same time. So the woman on the train breathes a sigh of relief because she (and the bomb) see the doors drop at different times.

However, the man on the train platform sees the doors drop at the same time. So he expects the bomb to blow up (because, and this was the fundamental similarity between my two examples, he's not taking into account a specific aspect of relativity). But it doesn't blow up. In fact, no matter how you setup the scenario, either the bomb goes off in both frames, or it doesn't go off in both frames. It will never happen that the bomb explodes from one person's point of view but not the other. Furthermore, this fact is formalized under relativity.

 

[rkolter]

You said you found something fundamentally confusing, but then went on to describe a scenario that has a fundamental error in it. Because of this, you are not listening to what you are being told. The fundamental error in your original post was:

Let's say that you set up two transceivers 4 light seconds apart. And maybe they use normal luminal signals, or maybe they use superluminal signals. You know they are 4 light seconds apart...

You do not know they are 4 light seconds apart, and the fact that in a frame of reference you happen to measure them as 4 light seconds apart is utterly irrelevant to anything that happens after that. There is no "In Reality" distance. Length is not an absolute. You have to wrap your mind completely around this to see what is fundamentally confusing you.

The reality is that you could well be in a frame of reference where those two transmitters are absolutely 4 light seconds apart. But from my frame of reference they are not 4 light seconds apart. They are 3.9999999999999999999999999999998 light seconds apart. Because I am walking 2 miles an hour parallel to the transmitters as I approach you. Length is not an absolute. There is no value you can offer that is more or less correct than any other value.

You and I would agree they are 4 light seconds apart only because the actual difference between our frames of reference is too small for us to comprehend. But it is there. So long as you continue to say that you "know" something is some distance from something else, you are granting one frame of reference a privileged place that it has not earned.

So, when you are traveling at relativistic velocities and you measure the distance between the transmitters at 1.7 light-seconds, that is literally their distance. That is "reality". Reality is not 4 light-seconds for you. It is 1.7 light seconds for you. And the sole reason there is an "apparent" superluminal transmission is because you have made the mistake of assigning a different frame of reference to be "reality" thus making your frame of reference "not reality".

An observer with you, who didn't know you had set up the transmitters, would be confused as to why you keep saying "in reality" they are 4 light seconds apart. They might try to teach you how to do what they see as rudimentary measuring. Only when you said "Oh, well I meant that in a different frame of reference they are 4 light seconds apart, and I've decided that frame of reference is reality and not this frame of reference" would they understand... that you have gone nuts.

Hope this helps. Or if not, hope you read the post quickly - since that'd reduce its length. ;)