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I'm talking about this nonsense: http://www.washingtonpost.com/news/post-nation/wp/2014/06/11/this-is-the-amazing-design-for-nasas-star-trek-style-space-ship-the-ixs-enterprise/

Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here: Doesn't Warp theory violate causality?

But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality?

SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?)

So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR?

Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.)

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Just my 2 cents here. Special relativity deals with flat Miskowskian spacetime hence why we use inertial reference frames. In the warp drive case we are not dealing with flat space anymore but curved space hence losing the benefit of inertial frames. Hence we need to use GR instead. –  PhotonicBoom Jun 16 at 18:50
    
@PhotonicBoom : Yes, but what of an observer in an inertial frame that is far enough away from the spacecraft that the observer is in essentially flat spacetime. Will they not see the events as being non-causal? Is that not a fairly fatal problem? What aspect of GR explicitly allows violations of causality? (I'm not a GR expert.) –  Larry Gritz Jun 16 at 20:08
    
I'm not an expert either but I believe the two frames will not be able to communicate since the warp drive will be outside of the inertial frame's light cone. –  PhotonicBoom Jun 16 at 23:26
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It unquestionably violates causality, and that's why Harold White is a charlatan. Every cent of NASA money given to his research has been nothing but a way of defrauding US taxpayers. –  Chris White Jun 17 at 1:23
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So is the conclusion of all this that "exotic matter with negative mass* cannot exist, because if it did, we could use it to build an Alcubierre Drive and violate causality" or "we don't know if causality is an inviolable law, it depends on whether we find any exotic matter"? (*: and whatever other properties are needed) –  Superbest Jun 17 at 6:38

4 Answers 4

Edit regarding 3+1 spacetimes and causality

I'll keep adding to the answer as I get more information, and hopefully everything will just evolve along. At the very least, I'll have a set of notes to work from in the future :) This is also the first, broadest, cut at an actual answer regarding causality.

Alcubierre sets out to find his warp drive metric using a 3+1 formulation of spacetime. In the 3+1 formulation, spacetime is described as a set of constant coordinate time spacelike hypersurfaces, (foliations, for the fancy). In doing this, you wind up with a line element that looks like (see erudite comments from @Jerry Schirmer below, I'm playing catchup):

$ds^2 = -d\tau^2 = \gamma_{ij}dx^idx^j + 2\beta_i dx^i dt - \left(\alpha^2 - \beta_i\beta^i\right)dt^2$,

where $\alpha$ is the lapse function, and is positive, and $\beta$ is the shift vector between spatial foliations. $\alpha$ describes how quickly time evolves, while $\beta$ describes how the spatial coordinates evolve in time. In other words $\alpha$ and $\beta$ describe how your ship moves through space and time per incremental step.

What's important here is that $ds^2$ is positive and for real space, $\gamma_{ij}$ is as well. Remember, hyperbolas look like $\dfrac{x^2}{a^2} - \dfrac{t^2}{b^2} = 1$. So, the line element equation above describes a globally hyperbolic system in space time. What's that mean? It means you can't close a curve in spacetime, so you can't violate causality. Note that $\beta^i$ squares up where it's important to maintain sign to maintain a hyperbola. I'd think there should be another requirement that $\alpha^2 > \beta_i\beta^i$, but Alcubierre doesn't mention this, so I'm guessing we don't actually need it.

Alcubierre isn't done yet, he's still got to find a metric that will fit in a 3+1 spacetime and do what he wants, (provide faster than light propulsion), but if he does, the above property of 3+1 spacetimes will guarantee causality.

Edit I Stand Corrected Regarding the Alcubierre Drive

@Superbest pointed out, that the claims for the drive were that it could go faster than the speed of light with regard to the laboratory frame, and hence with laboratory velocity. I found the original paper by Alcubierre on arxiv[2], and...

he's absolutely right!

The paper is amazingly well written and folks that have had a grad level general relativity class should be able to easily traipse through it. Alcubierre even shows that causality won't be violated. I haven't had time to digest the material enough to say why causality isn't violated except with the very unsatisfying statement, "Well, the math works out." Alcubierre was also quick to point out that he felt that with a bit of effort he could come up with an example that would violate causality:

"As a final comment, I will just mention the fact that even though the spacetime described by the metric (8) is globally hyperbolic, and hence contains no closed causal curves, it is probably not very difficult to construct a spacetime that does contain such curves using a similar idea to the one presented here."

OK, so to summarize. The math explanation and associated formulas I wrote below are correct. With uniform acceleration and no exotic matter whatsoever, you can travel more than x light years in x proper time years. In the case of the Alcubierre drive, however, that's not the trick they're playing. I hope to have more details soon, but in the meantime I'll leave you with this quote from Schild regarding the twin paradox and general relativity.

"A good many physicists believe that this paradox can only be resolved by the general theory of relativity. They find great comfort in this, because they don't know any general relativity and feel that they don't have to worry about the problem until they decide to learn general relativity."

End Edit

The explanation given in the Washington post article triggers a pretty common misconception:

"If an object reaches a distance x light years away in under x years, then it must be travelling faster than the speed of light."

What the article failed to mention is that the 14 days quoted is in the reference frame of the ship. The equation for the distance travelled with respect to time in the frame of the ship, (known as proper time), is

$$\mathrm{distance} = \dfrac{c^2}{a}\cosh\left(\dfrac{at}{c}\right)-\dfrac{c^2}{a},$$

where $a$ is the acceleration of the ship and $c$ is the speed of light.

Using this formula, it can be shown that at an acceleration of 188g, (188 times the acceleration due to gravity), the ship could reach Alpha Centauri in 14 days of ship time. You might point out that 188 g's would surely smush everyone against the back wall of the ship, but the beauty of the theoretical drive described is that you carry your own gravity well along with you and therefore, you're always in freefall and don't feel the acceleration.

Here's the problem though. The time that will have elapsed here on Earth will be much, much greater than the 14 days that elapsed on the ship. The expression for the time elapsed on Earth is

$$\mathrm{Earth\ time\ elapsed}= \dfrac{c}{a}\cosh\left(\dfrac{at}{c}\right),$$

which can be used to show that when the ship reaches Alpha Centauri, 817 years will have passed here on Earth.

The calculations shown here are nothing new, by the way. Rindler applied them to the problem of relativistic space travel for the first time in 1960 in a Physical Review article titled "Hyperbolic Motion in Curved Space Time" [1].

References

  1. Rindler, W., "Hyperbolic Motion in Curved Space Time", Phys. Rev. 119 2082-2089 (1960).

  2. Alcubierre's original warp drive paper http://arxiv.org/abs/gr-qc/0009013v1

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See also the song " '39 " by Brian May of Queen (who has a PhD in Astrophysics, I believe) –  Jim Garrison Jun 17 at 4:44
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I thought the whole point of the Alcubierre Drive was that it could go faster than light, meaning that you would get to Alpha Centauri in two weeks of Earth time. Otherwise, the question is a very trivial one which can be restated "How does an object traveling slower than light not violate causality". –  Superbest Jun 17 at 6:34
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Interesting fact: if I did the math correctly. It would take 5.5 days to Mars at 1G, while on Earth it would pass 8.5 days. That's pretty reasonable I guess. –  jnovacho Jun 17 at 9:03
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The article refers to an Alcubierre drive, so the ship would move at superluminal speeds in the inertial frame of an external observer. It would cause violations of causality and indeed would allow closed timelike curves. –  John Rennie Jun 17 at 9:39
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Also, you don't get causality for free from the 3+1 formulation, because, in general, 3+1 spacetimes can only evolve over a finite subdomain. In principle, it could break down in such a way, or have coordinate singularities in places that block out "bad" regions. For example, the $a > M$ solution to the Kerr metric is known to have CTCs, but it also has a global timelike Killing vector, and thus can be written using the 3+1 formulation perfectly cleanly. –  Jerry Schirmer Jun 17 at 20:40

It will violate causality globally. There is no way around that. This is the best argument about why building these things is impossible. To see this, all you have to do is zoom out to some scale where the warp drive field becomes a point. Then, the warp ship is just a point moving superluminally against a SR background, and all of the causlity problems derived from there arise again.

As PhotonicBoom says, local causality is preserved, fwiw.

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What exactly is the constraint on spacetime curvature from this argument? What spacetimes are forbidden? what energy-momentum tensors are forbidden? –  innisfree Jun 17 at 13:45
    
@innisfree: Spacetimes with closed causal curves are forbidden. That's really it. –  Jerry Schirmer Jun 17 at 15:08
    
Well, the Alcubierre drive does not include closed causal curves. –  innisfree Jun 17 at 15:35
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@innisfree: his spacetime doesnt, because it includes only one of these that doesn't turn around. My argument still stands. All you need is to have two of these warp drives going in opposite directions relative to a static observer with intersecting paths. Outside of the warp drive region, the background geometry still applies. If that's special relativity, you still have strong constraints on this. –  Jerry Schirmer Jun 17 at 16:10
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causality might kick in in chronology-protection a la Hawking to destabilise the vacuum fields before the CTC is formed. So strictly speaking, causality can still hold over a specific foliation. But the point is moot because there is no exotic matter that can enable warps or Lorentzian wormoles.. that we know of –  lurscher Jun 17 at 20:44

I looked into this a little bit more and this is what I have gathered. We need General Relativity to describe this warp drive machine (bends spacetime after all). But locally the drive is travelling slower than $c$ because spacetime directly "below" it is flat. This I believe would preserve causality even though the drive does appear to be travelling at speeds higher than $c$.

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Let's say that Earth sends a radio signal (in all directions) when the ship leaves, and Alpha Centauri sends a signal when the ship arrives there. If the ship is FTL in proper time, are there not inertial frames (far from the ship) that would see the arrival as occurring before the ship's departure, clearly violating global causality? –  Larry Gritz Jun 17 at 21:29
    
Global causality is violated. Local causality though is preserved. Thats why I said locally the drive is travelling with $v < c$ which implies that something like causality is preserved locally as well. –  PhotonicBoom Jun 17 at 21:51
    
It definitely breaks causality, provided you have more than one drive, or the ability for the thing to turn around. –  Jerry Schirmer Aug 15 at 15:29

This is a question that is asked the world over and the answer seems obvious to me, yet everyone deems it to be from perception which for me seems to be the issue apply some 1st grade logic to the situation and the answers reveal themselves quite simply.

For instance, travelling faster than light from a single point of perception this is absolutely possible and for those nay-sayers consider this:

Person A heads in one direction at 0.75*c and person B travels in the opposite directs at the same speed, neither is breaking the speed of light but taken from a relative perception of person A, person B has to be travelling at a resultant speed of 1.5*c. That much everyone should agree on.

Now consider this in terms of causality because again this is all about perception.

If Person A and Person B both start in a mutual place and go one light year away from the starting point at the same speed and acceleration in opposite directions they are 2 light years away from each other.

Now if person A and B look at their watches it would show the same time and date as they have both experienced the relative distortion through acceleration from the central point, hopefully, you're still with me.

Now if person A sets off at 2*c he will get to person B in 1 year and a year later he will be able to see his own launch. Causality hasn't been evaded as he did not arrive earlier than he set off he travelled for a year and it would show on person B's watch, in this system.

I do think that the mind's limitation is due the thought that time moves either away from a point or towards one, rather I perceive existence as a plane and time to pass simultaneously across the plane, yes time may be perceived to be distorted by gravity by a body near such gravitational forces, however when said body moves away from the gravity causing this effect a 'Chronotic Snap' would happen where their time distortion would catch up to the plane's own time. This ties quite well to the thought of riding the event horizon of a black hole to simulate time travel, or rather sit in a distorted time stream while the rest of the universe moves past.

An interesting musing is the possible side effect as experienced in sound where a plane travels at Mach1 we get sonic booms cuased by compression of sound waves at the same speed of thier creation, would a photonic boom also happen in light?

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