How does faster than light travel violate causality?

Question

Let's say I have two planets that are one hundred thousand lightyears away from each other. I and my immortal friend on the other planet want to communicate, with a strong laser and a tachyon communication device.

I record a message on the tachyon communication device and release the message at exactly the same time as I activate the laser, both of which are directed to the other planet which is one hundred thousand lightyears away. Say it is the year 0 for both of us at the time I did this.

If tachyons existed, then the message would arrive to my friend before the photons in the laser. It would arrive, say, a thousand years earlier. From my vantage point, that message will arrive to her at year 99,999; the same would be true for my friend's vantage point. However, she will only see the laser at year 100,000.

So since she got the message at year 99,999, she immediately sends me a reply back going through the same procedure as I did. She records a message and releases it at the same time as the laser. The tachyons will arrive 1,000 years earlier than the laser, so for me, I will receive the message at year 199,998. I will receive the laser, however, at year 199,999.

It seems to me that communication this way does not violate causality. I will still have received the message after I had sent it.

If tachyons truly violated causality, though, I realize it should arrive at year -1 for her, and so she can reply to me at year -2, which would mess me up by year 0 as I will ask her how she knew I was planning on sending her a message before I sent it. I could send her a different message, which she would end up receiving at year -1, and will end up confusing her as she would have received one message asking her out, and the other asking her how did she know I was asking her out. She then decides I am crazy and sends me a message at year -2 that she does not want to date me, and so she will have both turned me down and entertained me before I have even asked her out.

On the other hand, let's go back to year 0 and add a third device to our list: an Alcubierre drive. After I send out the message and the laser, I get impatient and do not feel like waiting 99,999 years, so I get on my Alcubierre drive spaceship and arrive on her planet at the same year 0. My friend is not in her office, so I leave a note to her also immortal secretary saying I dropped by and that she should expect a message for her in year 99,999.

I then get back on my Alcubierre drive and land back on my planet, still on year 0. Meanwhile, the tachyons and photons I sent out are still racing to arrive to her. By year 99,999, she receives the message just as I Alcubierre drive back to her, and I pick her up for dinner.

But the point of my question is, it seems to me that just going faster than light, if that alone was what you had, would not violate causality. It must be something else. I understand time dilation and that things with mass cannot travel at the speed of light, but using the Alcubierre drive, hypothetically speaking, I was still able to outpace the photons while also having mass. It still did not produce causality problems. Alcubierre drives are also valid solutions to GR.

It seems circular to me to say that what makes traveling faster than light violate causality is because it violates causality (if faster than light communication was divorced from causality problems, then the causality problem would cause itself -- thereby violating causality and, hence, we would scrap it and conclude that there is no causality problem after all).

What is it that I am missing? If someone could help me out, that would be excellent. I've been itching to ask my friend out for a few millenia now. :)

Answer 1

(There's a couple of these questions kicking around, but I didn't see anyone give the "two boosted copies" answer. Generically, I'd say that's the right answer, since it gives an actual causality violation.)

In your scenario, the two planets remain a hundred thousand light years apart. The fact is, you won't get any actual causality violations with FTL that way. The trouble comes if the two planets are moving away from each other. So, let's say that your warp drive travels at ten times the speed of light. Except if the two endpoints of the trip are moving, then what does that mean? Ten times the speed of light relative to which end?

Let's say Tralfamadore is moving at a steady 20% of $c$ (the speed of light), away from Earth. (So, Earth is moving at a steady 20% of $c$ away from Tralfamadore.)

If I leave Tralfamadore (in the direction of Earth) and I am travelling at anything less than 20% of $c$ relative to Tralfamadore, then I am still moving away from Earth. I'll never get home.

Let's say instead I am travelling at 60% of $c$ relative to Tralfamadore. I will catch up to Earth. Relative to Earth, how fast am I approaching? You might guess the answer is 40% of $c$, but it's 45.45%.

Generally, the velocity subtraction formula of relativity is: $$w = (u-v)/(1-uv/c^2)$$

Let's say instead I am travelling at 100% of $c$ relative to Tralfamadore. Plug $u=c, v=0.2c$ into the formula and get $w=c$. Relative to Earth, I am approaching at 100% of $c$! The speed of light is the same for everyone.

So finally, let's say instead I am using your warp drive to travel at 1000% of $c$ relative to Tralfamadore. Relative to Earth, I am approaching at -980% of $c$. In Earth's reference frame, I will arrive on Earth before I leave Tralfamadore. Now you may say this in itself isn't a causality violation, because we've applied Earth's calendar to Tralfamadore. And that's true, but I'll make a round trip:

• In the futuristic Earth year of 3000, Tralfamadore is 98,000 light years away, and receding at 20% of $c$. I leave Earth at 1000% of $c$, relative to Earth.
• In Earth year 13000 Tralfamadore is 100,000 light years away, and I catch up to it. I turn around and leave Tralfamadore at 1000% of $c$, relative to Tralfamadore.
• In Earth year 2796, I arrive home.

Earth's calendar certainly applies to Earth, and I arrived home two centuries before I left. No two ways about it, I'm a time traveller!

There is nothing special about ten times the speed of light. Given a warp drive that moves a certain amount faster than light, you can make the above time machine using two endpoints that are moving apart a certain amount slower than light, provided that the warp drive can move faster than light relative to either end. This time machine works for any form of FTL: tachyons, warp drives, wormholes, what have you.

Answer 2

For the tachyon case, you implicitly assume that the tachyons ultimately travel forward in time, just going faster than light. But there exist Lorentz transformations (that is, other inertial frames) in which such a particle would travel backward in time as it traverses space.

You may have trouble believing this, so consider a 1+1 spacetime. This spacetime has four distinct regions: future timelike, +x spacelike, -x spacelike, and past timelike. These regions are cut by two diagonal, lightlike lines, which divide spacelike from timelike and represent the asymptotes of hyperbolas.

Most massive objects have four-velocities in the future timelike region, and a Lorentz transformation will keep them in that region no matter what. They are, however, quite free to move around in that region, provided that they maintain an overall magnitude of $c$.

A tachyon is the same, except it occupies either the +x-spacelike or -x-spacelike regions. This means that, even if you think your tachyon travels forward in time, there exists some reference frame in which it travels backward in time. You may not see causality violated, but someone else will.

The Alcubierre drive gets around this problem by changing the geometry of spacetime itself so the above notions get much more complicated. The basic idea is this: inside the bubble, you can fire off a photon and it will, assuredly, go away from you along a well-defined trajectory, one that is "faster" than yours. Causality is not violated because all observes will agree that you merely took a timelike trajectory in a very unusual spacetime--events before and events after your trip are still well-defined.

The danger here in thinking about the Alcubierre drive is that we often take the perspective of a distant observer and naively think that our coordinates (our measures of time and space) will not be affected by the drive, but they are. The geometry of the drive itself will warp and distort coordinate lines around it, resolving any seeming causality violations.

Answer 3

There is a simple answer; faster than light travel does not violate causality.

What faster than light travel does, is contradicts the usual axiomization of relativity; and hence allows you to derive all kinds of paradoxial 'conclusions'. But shifting the blame on causality is more fashionable convention than anything else.

What would happen 'in practice' in the case of faster than light travel, is that itd become possible to triangulate a 'special' reference frame. The inability to triangulate such a frame really need not be taken as an axiom; it can infact be derived from the apparent empirical fact that all fundamental forces travel at the unfortunately named 'speed of light'. If they wouldnt, triangulating a special 'rest' frame really wouldnt be that hard.

However, to see that our inability to triangulate a 'rest' frame probably doesn't make for all that good an axiom, it is easier to just consider a spherical universe. If I flash a spherical light in a topologically euclidian universe, no set of contraptions or mirrors will allow me to triangulate a rest frame.

But in a spherical universe, different observers that flash a spherical light might see that light come back to them in different ways. Some, in a 'special', 'preferred', or 'rest' frame, if you will, will see the flash come back at them from all directions at the same time. If you want to call it a 'rest' frame; whatever. But it sure is a 'unique' or 'special' frame. Any change in momentum from that state will make the roundtrip time of your lightflash a function of direction.

Note that this argument does not require the universe to be spherical; it can even be flat, like a toroidal topology. The same conclusions follow for all topological scenarios that allow multiple independent geodesics between two points in spacetime, like wormholes. They either force you to believe in timetravel (kindof a friendly way of saying blatant-self-contradiction); or alternatively youd be forced to come to the conclusion you can indeed triangulate a 'special' frame.

And so what? You still cant do it with local measurements using fundamental forces all traveling at the same speed though, and thats all thats needed to be consistent with all our current experimental evidence.

Not that anyone has observed a wormhole or light going around the universe; but it seems wrong to axiomatically throw the latter under the bus at least until we have experimentally excluded it. And should we find any such topological feature, id neither get out my disproving-relativity-contraption or my time-traveling-gear, but rather dust off my rest-frame-triangulating-apparatus. But your opinion may vary.

Disclaimer: I would not be surprised if someone has argued this case better before, but this specific line of argumentation is original to me. So I am aware it is not orthodoxy, and I am aware my argument leaves some things as an exercise to the reader; but feel free to give your prediction of what a lightflash going around a spherical universe would look like. Or maybe it is better explained in terms of 'time travel' after all?

Answer 4

What everyone else said, but note that this STILL violates causality if you use general relativity to create one of these "warp drive" scenarios--the "warp drive" can always be restricted to an arbitrarily small region of spacetime, and then special relativity will be true over the rest of spacetime, and the problems will still arise.

Answer 5

There are a few misconceptions in your scenario that cause the misunderstanding. First of all, by definition, causality means that if the time interval between two events is positive in one reference frame, then it is positive in any other reference frame of your choice and viceversa, provided the velocity the events propagate to be smaller than $c$. If, on the other hand, you allow events to propagate faster than light, then there might be reference frames wherein the orders of the events is switched. This does not imply, as you assume, that travelling faster than light makes things arrive back to you earlier than when you sent them. Violation of causality means that this might happen in at least one other reference frame; in yours things will stay as they are.

Most important: notice that for events to go "back and forth" you need to change sign to the velocity ($a \neq 0$), which implies that the notion of time itself depends on the space-time path that you follow and on the point in the space-time you are in. The time that you hence measure has in principle nothing to do with the notion of "biological" time that you have in mind, but it is a mere parameter in the metric. In the case at hand you would have to solve the Alcubierre metric and integrate between the two points you are considering along the path you want to follow. This in general will give you something that is totally unrelated to the notion of biological time, to which all your concepts apply.

Last but not the least, besides the violation of causality, travelling at $v>c$ will produce diverging physical observables (energy, momentum and so on) violating all the other conservation laws that must nevertheless hold true.

Answer 6

Since I haven't seen this example, I'll add to the pile. As has been said, the problem occurs once you have relative motion between two objects. I had a hard time pinning down special relativity until I went back to the original premise: The result of any experiment has to be the same in all non-accelerating reference frames and looked at this example experiment.

Let's say that 2 ships, ship A and ship B, leave the surface of a non-accelerating planet going opposite directions at 0.8c. Each ship is equipped with a device that will emit a unique signal once it receives an initializing signal from the planet. The planet will detect whether the signals from the ships are received at the same time or not.

In the reference frame of the planet: After one year the planet emits the initializing signal. 4 years later each ship receives that signal at the same time. 5 years after that, the people on the planet receive signal A and signal B at the same time.

In the reference frame of ship A: The planet will be moving away at a speed of 0.8c while ship B is moving away even faster than that (let's not worry about the exact speed of ship B in this frame.) After some amount of time, t, in reference frame A, the planet emits the initializing signal. This signal has to travel a distance of 0.8ct to reach ship A. However, the signal will have to catch up to ship B, and since it's moving close to the speed of light (definitely over 0.8c), that will take quite a while. The numbers here get complicated, but conceptually I would say it's pretty easy to see that the initializing signal will reach ship B after it reaches ship A. However, since the planet is also moving, this won't be a problem, because now signal A has to catch up to the planet while signal B is going to head back towards the planet, which will take less time. It's still true in this reference frame that the planet receives both signals at the same time.

Now, imagine the people on ship A had a way to communicate at faster than the speed of light. In the reference frame of A, after they received the signal, they could contact ship B and tell them to turn their machine off. Meaning that the planet would only receive a signal A, and there's our violation of causality.

The issue is that things happening simultaneously in one reference frame don't necessarily happen simultaneously in another frame, and, as we see in this example, if you could affect things faster than the speed of light, you could change the result between reference frames.

Answer 7

Not all Faster Than Light (FTL) drives violate causality. Alcubierre drive does not since this spacetime is Lorentzian and does not allow for closed timelike curves (CTC). Allen E. Everett and Thomas A. Roman 1 demonstrated that with two non-overlapping Krasnikov tubes 2, it is possible to construct a time machine.

The Krasnikov tube is a FTL drive proposed by S.V. Krasnikov with metric $$ds^2 = - dt^2 + (1 - k) dx dt + k dx^2 \,,$$ where k = k(t,x) can be determined in terms of step functions, describing a junction of three regions in spacetime resembling a tube. The outside region is flat, with two simple killing fields determined by time and space translations defining lightcones. The transition region is the only one that is curved. The interior region is also flat, with killing fields that are linear combinations of space and time translations. Still, one of these fields carries a proportionality factor that can be taken arbitrarily small, giving the lightcones a 'wider' opening. This interior lightcone does not possess an event horizon like the Alcubierre warp drive metric.

Everett and Roman demonstrated that one Krasnikov tube does not possess issues with causality, but two Krasnikov tubes can create a time machine with causality violation, causing CTC. This time machine can only allow going backward in time, though. This CTC can be avoided with boundary conditions on the Krasnikov metric. imposing limitations on the direction of movement for the observers inside the tube. Sort of like a 'one-way tube'; once the observers start moving, they can never return to the departure point.

[1] A Superluminal Subway: The Krasnikov Tube
[2] Hyperfast Interstellar Travel in General Relativity