# V>c violates causality

1. Jul 11, 2012

### GarageDweller

It is well known that traveling at superluminal speeds violates causality, this can be shown by a space time diagram, where two frames would disagree on which event came after the other.
My question is, if such particle were to exist (called tachyons), what would they see in their frame? Since tachyons have v =/= c, their proper time is non-zero, hence such must exist.

2. Jul 11, 2012

### Demystifier

Hypothetic observers made of tachyons might see the rest of Universe looking as a movie playing backwards. It looks strange when you see it for the first time, but later you get used to it and there is nothing paradoxical about it.

3. Jul 11, 2012

### DrGreg

It's not possible to find a coordinate system in which a tachyon's 3 space coordinates are all zero and its time coordinate increases. It is possible a coordinate system in which 2 of the tachyon's 3 space coordinates are zero and its time coordinate is zero and its 3rd space coordinate increases. Proper time is not defined for a tachyon, but proper distance travelled is. Given all the above, there is no inertial frame in which a tachyon is at rest. There is a frame in which a tachyon exists momentarily as a line in space. I don't think the original question has any meaningful answer.

4. Jul 11, 2012

### Boston_Guy

If I remember correctly, observers who move less than the speed of light also disagree on the order in which events occur. They just can't alter cause and effect whereas FTL observers can.

5. Jul 11, 2012

By proper time do you mean $d\tau{}^2$?

6. Jul 11, 2012

### codeman_nz

I know what it says in relativity but this whole thing about light cones and causality confuses me.

Say a star 10 light years away explodes so we will not know about it for 10 years. If you could jump instantaneously from Earth to the star you would find that it is no longer there. Just because the end of the stars light has not reached Earth does not change the fact that the star is gone.

7. Jul 11, 2012

### Staff: Mentor

The reverse causality arguments don't directly involve "travel", whether faster than light or slower than light.

Consider two points in space that I observe to be some distance, say ten light-years apart. Now suppose something happens at one of those points, and less than ten years later (by my clock) something else happens at the other point. In the lingo we'd say these two events were "space-like separated"., meaning that a beam of light emitted when the first event happened at the first point wouldn't get to the second point in space until after the second event had happened.

Different observers moving at different speeds will come up with different values for the distance and time between the two events, but all will agree that they are space-like separated: for example someone who measures the distance at five light-years will see a time separation of less than five years. It would be pretty weird if we didn't all agree about this... for example, the second event could be "I defuse a bomb that's been sitting there for the past thousand years"... but the bomb could be wired to explode if it is exposed to a flash of light coming from the direction of the first event... When we all get together later and compare notes, we'd better all agree about whether the bomb exploded or was safely defused.
(There's also a notion of "time-like separation". If the second event ten light-years away happens more than ten years after the first, then although the various observers will still see different distances and times, all will agree that first event happened before the second).

However, if the two events are space-like separated, different observers will see them happening at different times. Some observers will see them both happening at the same time; others will see the event at the first point happening before the event at the second point, and yet others will see the second event happening first. - even though there's no faster-than-light involved and all of these observers are moving at perfectly ordinary speeds less than the speed of light. We just have two events happening at different places, and no meaningful way of saying which of them happened first.

But if A did not necessarily happen before B (indeed, some of us can prove that B happened before A), it seems reasonable to believe that A cannot have caused B, or equivalently that B could happen even if A never did. ("Reasonable"? Well, the alternative leads to various bizarre logical contradictions - and remember that we're talking about ordinary observers moving at ordinary less-than-light speeds, so if any of this bizarreness really happens, we might have noticed by now).

So what does all of this have to do with faster-than-light travel? Consider the two events "I start on my journey" and "I arrive at my destination". On any faster-than-light journey, whether realized by enormous powerful rockets, warp drives, falling through a wormhole, teleportation, or outright magic.... These events have to be space-like separated, or it's not a faster-than-light journey.

8. Jul 12, 2012

### Demystifier

There IS a coordinate system in which 3 coordinates of tachyon are constant and only one coordinate changes. The "problem" is that this one changing coordinate is "spacelike", not "timelike". But the word "timelike" only tells something about the sign of the corresponding component of the metric tensor. That word tells nothing about the property of "flowing", which one usually assigns to an intuitive concept of time.

Therefore, even though the one changing coordinate of the tachyon is not "timelike", it is fully justified to think of that coordinate as TIME experienced by the tachyon. In other words, a tachyon does make a physical sense.

9. Jul 12, 2012

### Boston_Guy

If one had a wormholle then it would provide another way to calculate the distance between the events. Couldn't it also be used to calculate the spacetime seperation between the two events and come up with a timelike spacetime seperation?

10. Jul 12, 2012

### Staff: Mentor

That's a rather big "if"... But even if it were to come true, then all you'd have is another path between the two events. It wouldn't change the space-like separation already seen by those of us who haven't noticed the wormhole and aren't going through it. We're still stuck with the fact that some of us can prove that B happened before A, therefore that A could not have been the cause of B and indeed that B can happen even if A never does.

That's generally just fine and creates no particular problems for anyone if the two events are not causally related. But it is a problem if event B is "traveller arrives at destination" and A is "traveller departs at start of journey"; and it's a problem no matter what mechanism we assume the traveler used. The problem is with how various disinterested observers view the endpoints of the journey, not the journey itself.

But with all of that said.... There could be a way around this argument. If you could produce an internally consistent and contradiction-free model of the world that works for all observers and in which space-like separated events could be causally related, then this particular argument against FTL travel would fail. However, that is quite a challenge - the exploding/defused bomb of my earlier post is just the tip of the iceberg.

11. Jul 12, 2012

### DrGreg

I can't accept that it's fully justified to describe a spacelike quantity as "time".

12. Jul 12, 2012

### Staff: Mentor

I would go further than DrGreg and say that this is unjustified unless you can come up with a physical account of how proper length along the tachyon's worldline is interpreted as "time experienced by the tachyon". The standard justification for that interpretation for a timelike worldline is that we can construct "clocks" that physically represent proper time along the worldline--for example, a light clock. Can you construct a light clock that ticks off "time" along a tachyon's worldline?

13. Jul 12, 2012

### jbriggs444

Well, here is one that does...

Suppose we have a hypothetical faster-than-light craft.

We are sitting here on Earth and we want to get to Alpha Centauri, four light years away. We want to get there yesterday (in coordinate time in a frame of reference in which both Earth and Alpha Centauri are at rest).

No problem. A trajectory which is faster than light in one frame will be backwards in [coordinate] time as viewed from another frame. All we have to do is pick an appropriate inertial frame of reference so that FTL in that frame is back one day in the Earth rest frame and launch our craft from a platform that is at rest in that selected frame. [By the premises of special relativity, there is no preferred frame. If our craft performs in a certain way as referenced against one frame, it can perform in that same way as referenced against any other]

So we climb out of our craft at Alpha Centauri, one day earlier [in coordinate time] from when we left Earth. There is nothing stopping us from taking another faster-than-light hop back to Earth, arriving there the day before yesterday [in coordinate time]

But this puts us back on Earth unambiguously before we left. If we wait two days, a closed timelike loop will be completed.

14. Jul 12, 2012

### Staff: Mentor

I agree - that's a delightful example of the sort of contradictions that the hypothetical faster-than-light craft leads to.

But I also think that this example supports the point I've been trying to make, that the causality problems with FTL travel have to do with the endpoints of the journey, not the mechanics of the movement itself. This matters (at least to me) because we hear arguments along the lines of "yes, I know Einstein proved that nothing can go faster than light... But suppose that I could warp space/take a shortcut through some other dimension/fall through a wormhole/popsci buzzword.... So that I could get from point A to point B without traveling through the light-years that separate them in normal space...."

And your example works just as well on those arguments. No matter what "what if" magic is invoked, if the arrival at alpha centauri happens less than four years after the departure from earth, your scenario has to be accepted.

15. Jul 13, 2012

### Demystifier

It's not a problem to construct such a clock. For example, let the "spacelike" coordinate which is changing along the tachyon trajectory be x. Then the corresponding clock is a physical process periodic in x. It's easy to find solutions of tachyonic field equations periodic in x.

In fact, I don't see any physical reason why t would behave more as a "time" for non-tachyonic matter than x for tachyonic matter.

16. Jul 13, 2012

### Demystifier

By fully, I mean not less then it is justified to interpret t as time for non-tachyons. And by justified, I mean by theoretical arguments. In the case of non-tachyons we also have a direct empirical evidence that t behaves as "time", but empirical evidence is not a theoretical argument.

17. Jul 13, 2012

### Staff: Mentor

Sure, and it's also easy to find solutions of the *non*-tachyonic field equations periodic in x. But those solutions, in both cases, will also be periodic in t, correct? And yet in the non-tachyonic case, we use the t periodicity to measure time and the x periodicity to measure *distance*, not time. Are you saying that this is reversed in the tachyonic case? In other words, that in that case we use the x periodicity to measure "time", and the t periodicity to measure "distance"? How can this work when spacetime itself is not completely symmetric between "timelike" and "spacelike" dimensions--there are three of the latter but only one of the former?