# How does faster than light travel violate causality?

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. :)

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– twistor59 Jan 26 '13 at 14:31
That was a helpful link. Mark Eichenlaub's answer, in particular, was interesting. Could you explain how a frame moving from me to my friend would perceive these events from happening? And how would these spacelike separation of events violate causality? – markovchain Jan 26 '13 at 14:40
Also, why the downvote? I believe the question was specific and clear enough, and generally I've only read that clocks would revese if it went beyond c, but without exploring why exactly. Also, obviously, it assumes tachyons and Alcubierre drives exist -- is there a problem with that assumption, and why? Thank you. – markovchain Jan 26 '13 at 14:49
Not sure why it was downvoted - possibly because it could benefit from splitting up into distinct scenarios: (1)I do .... with tachyons (2) I do... with an Alcubierre drive? – twistor59 Jan 26 '13 at 14:59
You mention thousand-year differences in your post, but then the numbers you use are different by only one year. 99,999 is not one-thousand years before 100,000, it is one year before. – TylerH Feb 18 '15 at 15:28

(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.

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Thank you. This completely answered my question, and any follow up questions that could have emerged. :) – markovchain Feb 19 '13 at 14:32
Using v of 0.2c it seems fine to travel at any speed below 5c to Earth. u = c, w = c; u = 2c, w = 3c; u = 3c, w = 7c; u = 4c, w = 19c; u = 5c, w = (division by zero)c [infinite speed]; u = 6c, w = -29c; u = 7c, w = -17c; u = 8c, w = -13c; u = 9c, w = -11c; u = 10c, w = -9.8c. It seems that if your start speed is 0.2c away from Earth then only travelling faster than 5c towards Earth would violate causality. As your speed approaches infinity, your speed relative to Earth approaches -5c. – CJ Dennis Dec 12 '14 at 12:01
In your example you allow the speed to be a % of c, which I understand can allow you to then arrive back before you left. I wonder if that is an invalid way measuring the limits of speed? ie: if you travelled at 500% of c, would you not violate causality? If so, what is that limit? – Andrew Bickerton Jul 18 '15 at 9:22
I'd like to see the math you used to arrive at this answer. Plugging in anything greater than c should blow up the equations so how are you getting at things like -980%c relative to earth? Perhaps moving at such speeds (Which is physically impossible but lets just use pretend land energy and mass for now) just functions in ways we cannot understand yet. And if I'm correctly remembering, Einstein said you can never accelerate to C, but says nothing about instantaneous jumps. If we assumed that was possible. (wormholes / warpdrives etc) – David Oct 8 '15 at 15:00
You point is invalid, please read users.physics.ox.ac.uk/~smithb/website/coursenotes/rel_A.pdf page 34 quote "That equation also serves as a general proof that the velocity addition formulae never result in a speed w > c when u, v ≤ c. For, if u ≤ c and v ≤ c then the right hand side of (3.13) is real and non-negative, and therefore γ(w) is real, hence w ≤ c." using values above 1c produces non real values. aka -9.8c... you can only travel through time if you do your maths wrong, sorry – Arthur Feb 3 at 14:27

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.

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The Alcubierre drive violates causality globally even if not locally. It is just as much a work of science fiction as tachyons for this reason. – Chris White Jan 26 '13 at 23:26
@ChrisWhite how do you define "global" causality? To me, the question is simple. Either the effect lies within the light cone or it does not. If you can create an Alcubierre bubble, you can impose an ordering of events because photons can go into the bubble too and define a light cone that surrounds the effect. There are plenty of reasons to consider the Alcubierre drive to be unphysical (exotic matter, for instance), but I don't know if I can say the causality considerations are among them. – Muphrid Jan 27 '13 at 6:24
That answer makes sense. But could you give me an example where an outsider will see causality violated because of faster than light travel, while I don't see the same? @ChrisWhite, I share the same question as Murphid to you. If you could elaborate a bit? :) – markovchain Jan 28 '13 at 4:01
@Muphrid: the superluminal light cone will be tilted relative to the minkowski background. Draw a spacetime diagram, and it'll be easy to see how you can have backward causality with two Alcubierre travellers communicating while travelling in opposite spatial directions. – Jerry Schirmer Aug 1 '13 at 14:25
@Muphrid: also, all of the existence and uniqueness and causlality theorems require assumptions about the matter distribution that ultimately reduce to the abscence of exotic matter. Throw that out, and none of the causality theorems hold. – Jerry Schirmer Aug 1 '13 at 14:26

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.

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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.

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## protected by Qmechanic♦Feb 4 '13 at 16:54

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