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How so? Why should faster-than-light travel imply a reversal of the arrow of time? Can you explain the reasoning?

This is an intuitive way of thinking about this without diving into the math. Here are the postulates: speed of light (c) is same for everyone. And there is no special reference frame; your physics is as good as the physics of your friend moving away from you in a space-ship in a constant velocity v. Now consider that your friend is actually moving in a spaceship away from you. He has an experimental setup with him using which he measures speed of light. You are sitting at home postulating about his experiment. Consider he is moving at velocity v, so is his experiment equipment. How can he arrive the same result (3*10^8 m/s) for speed of light? You conclude that only way is that whatever clock he is measuring the experiment with is running slower than yours (velocity=distance/time, he is measuring less distance, so time must be less too). Now imagine he is going at the speed of light. (The actual equations breakdown at this point, but you can imagine he is infinitesimally close to speed of light). How is he arriving at c? Only way is time is not passing for him (according to you). Now imagine he is going faster than speed of light. At this point, the equipment he is sending photons with is traveling faster than photons itself! How does he arrive at c? His clock must be running backwards! And keep in mind, your friend in all these cases is happily measuring speed of light as c without any of these conundrums you have. And now you can publish a paper claiming that your friend travelled back in time! Did he really do that in a "Back to the future" kind of way? Perhaps not.

I did used to ponder over this sort of mystery, but I realized the actual answer to this lied at the particle level as explained in QED by Feynman: A positron can be treated as an electron traveling backwards in time. When it collides with an electron, the two create a photon. Should you ever come close to the speed of light, one way to "exceed" it would require a collision with your anti-matter self, resulting in (self-obliteration and...) photons from everyone's else's point of view. Of course, assuming my understanding is correct, it's also possible that you transform into your anti-matter self... bizarre. Going forward again would require some similar interaction, but I seem to recall formulating a thought experiment where this violates causality, making either the whole idea impossible even from a Relativistic standpoint or forcing some denial of causality, which would make other dependencies on causality (such as science itself) unreliable.

This wiki page [1] explains why whether two events happen at the same time depends on your reference frame. If you can send signals faster than light then you can send a signal to an even far away and cancel that event. In another reference frame the sending of the signal would have happened after the far-way event has already happened. Hence, due to the "relativity of simultaneity" FTL communication is equivalent to sending messages to the past.

[1]: https://en.wikipedia.org/wiki/Relativity_of_simultaneity

By sending a message in two opposite directions, we can achieve 2c speed of information transfer. How that is possible?

No, we can not, but the reason might seem subtle. S sends the same information in opposite directions to A and B. So A learns what B learns, but A can not use that to tell something to B, that S did not already know. A can not influence B, rather it is S that influenced both A and B.

No, sorry, I can't explain the theory of relativity in one paragraph. It takes a book. Spacetime Physics, by Taylor and Wheeler, explains it very well with only high school level maths.

Thanks for the reference.

The idea behind relativity is that the speed of light is constant in all reference frames. [0]. In order for this to work, we "warp" spacetime depending on the velocity of the observer.

More concretely, imagine Alice and Bob are moving away from each other at 50% the speed of light. They both observe event some event, X, occur and note the location in space-time. If Alice observes that X occurs in the location (t,x,y,z), then we can compute precisely where and when Bob observed the event occurring (t',x',y',z'). This conversion is known as the Lorentz Transformation [1], and can be derived mathematically from the assumption that the speed of light is constant regardless of reference frame.

Where this gets weird is the case where Alice and Bob observe 2 events: X and Y. In this case, it is possible that they will disagree about which event happened first. Once you accept this, it should become clear why causality requires some speed limit. You can do the math based on the Lorentz Transform and confirm that this limit is the speed of light. Intuitively, this is a direct consequence of the fact that we defined the Lorentz Transform to make the speed of light constant.

In case the need for a speed limit is not obvious, let as pretend that it does not exist. Suppose that, from Alice's perspective, X happened before Y. Further, suppose that Carol happened to be in a spaceship that passed by X at the instant it occurred, moving at a velocity that would take her by Y at the instant it occurred. [2] From the perspective of Bob, Carol would have traveled backwards, going from Y to X.

We can make this situation even worse by considering Carol's perspective. Recall that we defined Carol as starting at X and traveling to Y. In the same way, we can define Dave as starting at Y and traveling to X (recall that, if we were Bob, we would be convinced that Y happened first, so with a fast enough ship, Dave and make it in time). In this situation, both Carol and Dave exist at Y, so Carol can give Dave a copy of her diary of the trip. However, after Dave makes the trip to X, he will meet Carol again, so he can give her a copy of her diary of the trip she is about to take.

[0] This means that if I am on Earth, and you are in a space-ship moving at 99% the speed of light (from my perspective), and we both measure the speed of light, we will arrive at the same answer.

[1] https://en.wikipedia.org/wiki/Lorentz_transformation

[2] This is possible only because there is no limit to how fast Carol's Spaceship can travel.

> From the perspective of Bob, Carol would have traveled backwards, going from Y to X.

So what? It's Bobs problem.

FTL electrons, in medium where speed of light is much less than c, are traveling exactly as you described.

Suppose Carol has a clock. At X it reads 0, and at Y it reads 100 (100 units in the future). From Bob's perspective, Carol's clock would also read 0 at X and 100 at Y despite the fact that she was at Y first.

In other words, as Carol travels 'forward' in Bob's time, her clock runs backwards. Or, from Carol's perspective, Bob's clock would be running backwards.

This isn't actually a problem in relativity, but is the definition of time travel.

No, it's not a time travel. Bob will see that Carol travels backward and Carol will see that Bob travel backward, like we see EM emission from FTL electron in reverse order. FTL electron is not moving back in time, nor Bob or Carol.

The notion of "backwards" in time is a bit fuzzy once you get into relativity, so the three person example I gave may not clearly show it.

However, in the four person example, Dave is able to hand Carol her diary of the trip that she is about to make. If that is not time travel, I do not know what is.

It's very unlikely. If Dave has Carol diary, then Carol made the trip already, so Carol will notice empty fuel tanks and big holes in the shield. She will may have problems with memory, but she is not stupid.

No she didn't. X is the point in spacetime where Carol begins her trip. From her perspective, Dave came from the future.

The only reason that this cannot happen is that X and Y are so far away in space, and close in time, that it is impossible to travel between them.

Sorry, I cannot follow you here. Maybe it's well known thought experiment, but I'm unable to find it.

Because events that have a spacelike separation -- that is, they are close enough together in time and far enough apart in space that even travelling at the speed of light you couldn't be present at both -- have no absolute order. Depending on your frame of reference -- how fast you're travelling and in what direction -- you might see two such events (let's call them A and B) as simultaneous, as A happening before B, or as B happening before A.

This is commonly illustrated with the following scenario. Imagine Alice is sitting by the train tracks. Her friend Bob comes past riding on top of a train, sitting exactly in the middle of the car's roof. As he passes Alice, they high-five (presumably Alice is on a raised platform of some sort). A split second later, Alice sees lightning strike each end of Bob's carriage at exactly the same moment. Thanks to some very precise measuring equipment, Alice is able to determine that the two lightning strikes occurred at the very moment that she high-fived Bob. At that point, Alice was exactly halfway between the two points that were struck, so it makes sense that the light from those strikes reaches her at exactly the same time.

Alice also knows that Bob would have seen lightning strike the front of the car before it struck the back of it: the light from the front strike would have passed Bob on its way to Alice, and the light from the rear strike would have passed Alice on its way to Bob.

Now let's look at it from Bob's view. As Alice concluded, he sees the lightning strike the front of the car first. But (a) he's exactly the same distance between the two strike-points, and (b) the speed of light is always the same for any observer. So if he sees the light from the strike at the front first, that means the front was struck first. Bob has a different order of events from Alice.

Which order is the "right" one? Answer: both. Or, if you prefer, neither. There are no grounds to prefer Alice's view over Bob's, or vice versa. You cannot say that one lightning strike "really" happened first, or that they "really" happened simultaneously.

To complete the picture, let's imagine Charlie riding another train car on the set of tracks the other side of Bob's, travelling in the opposite direction to Bob. At the exact moment Alice is high-fiving Bob, Charlie is also exactly lined up with them and smacks Bob on the back of the head. For reasons similar to but opposite to Bob, Charlie will first see lightning strike the rear of Bob's car and then the front, which means that in his (equally valid) frame of reference, the rear strike happened first.

Now imagine someone or something used FTL travel to go from the front of Bob's car to the rear, leaving at the moment the front was struck by lightning and arriving at the moment the rear was struck. In Bob's frame, this would be unremarkable, except for the exceptional speed (ignoring for the moment any adverse environmental effects -- Google "what-if xkcd relativistic baseball" for a flavour of what those might be). But in Charlie's frame, this would be travel backwards in time, as the arrival at the rear would occur before the departure from the front.

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