# 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. :)

• – twistor59 Jan 26 '13 at 14:31

(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.

• The alcubierre drive doesn't actually move faster than the speed of light due to bending excess space around it, if I recall, it just gets there faster than light... it's reference frame stays neigh motionless, and is more akin to if you had a wormhole connecting the two points, so velocity calculation formulas don't apply in the normal sense, because at any given instance, its velocity is zero? Or am I missing something? – liljoshu Mar 2 '17 at 22:46
• @liljoshu We used the velocity formula to convert between Earth and Tralfamadore's frames of reference. We didn't even mention the traveller's point of view. Maybe I "warped", maybe I "hyperspaced". Whatever that means! It doesn't matter what the rules are in "hyperspace", as long the normal rules of physics apply to the clocks in "normal space" (eg Earth & Tralfamadore). From Earth's frame of reference, I arrived on Tralfamadore 10000 years after I left Earth, at a time when it was 100000 light years away, my (overall) speed was 10 times $c$. – Retarded Potential Mar 11 '18 at 21:30
• @liljoshu Quantum foam is hypothetical too, but why not? It doesn't change the result. If you can create two foamy channels between moving endpoints that allows someone to make the round trip as described, then you have a time machine. The velocities are only being measured with respect to the end points, velocity is just distance between the endpoints, divided by arrival time minus departure time. And "time" still follows the normal rules of relativity at the endpoints, because they're outside the channel. – Retarded Potential Mar 13 '18 at 21:21
• can you explain where the 2796 came from? – frogeyedpeas May 27 '18 at 2:54
• going at 10 Light years per year, it takes you 10000 years to arrive at the Tralfamadore. So 3k to 13k years. You leave the Tralfamadore going 10x the speed of light relative to its speed, which is .2 speed of light. You are approaching earth at 9.8x the speed of light relative to earth. To me this means you will arrive back on earth 10204 years after you leave the Tralfamadore @ year 23204, not 2796. From earths perspective you still wont arrive or leave the Tralfamadore until year 103000. They may even observe you going backwards. Can you explain why the return trip is going back in time? – Erudaki Jul 5 '18 at 17:56

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.

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.

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.

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?

• Spherical spacetime would indeed give us some interesting new possibilities (though not necessarily practical). But we're pretty sure spacetime is almost perfectly flat, with the only significant distortions being near large concentrations of energy. But whatever spacetime topology you imagine, you need to do the math. Any topology that isn't nearly perfectly flat gives you at least a theoretical possibility of shortcuts, but our observable universe doesn't seem to be that way - all we get are roundabouts like swinging close to a black hole :) – Luaan Dec 11 '18 at 9:36
• As I briefly mentioned, the spherical topology isn't essential; its just the most symmetric and easiest to mentally visualise; like a circular wave travelling over a water-planet; just with an extra dimension thrown in. Maybe the universe is spherical and just really huge? Or toroidal and perfectly flat? It's really the topology that would allow triangulation of a special frame, not the curvature. – Eelco Hoogendoorn Dec 11 '18 at 10:38

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