How does "warp drive" not violate Special Relativity causality constraints?

Question

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.)

Answer 1

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.

Answer 2

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

Answer 3

Dolphus333 mentioned in his answer that Alcubierre demonstrated in his paper that there is no violation of causality in the spacetime he constructed, but I believe this is just for the case of a spacetime containing a single warp bubble--if you have multiple warp bubbles moving in different directions you can indeed violate causality in a manner similar to the tachyonic antitelephone. The wikipedia article links to a paper by Allen Everett demonstrating this, "Warp Drive and Causality", which was published in Physical Review D. It also links to this series of lecture slides by Alcubierre, in which he mentions in the "Conclusions" section at the end that "if [FTL] is possible, we must confront the problem of time travel to the past and the causality problems is causes". However, in the previous slide he brings up the possibility that Hawking's chronology protection conjecture might be true in quantum gravity, so that solutions in general relativity involving closed timelike curves could be destroyed by quantum effects, without ruling out the possibility of constructing FTL solutions that don't involve CTCs.

Also, dolphus333's comment "when the ship reaches Alpha Centauri, 817 years will have passed here on Earth" seems to create the misleading impression that the warp bubble doesn't travel faster than light from the perspective of external observers (i.e. observers on Alpha Centauri wouldn't see the bubble reach them before a light ray sent from Earth at the same time, and which traveled outside the bubble), and thus that causality issues associated with FTL can be avoided, but in fact the Alcubierre bubble definitely does travel faster than light as seen by external observers. This is stated specifically in the book Time Travel and Warp Drives by Allen Everett (the same physicist I mentioned above) and another physicist, Thomas Roman. in the section on the Alcubierre solution on p. 117, the book says:

The bubble and its contents could travel through spacetime at a speed faster than light, as seen by observers outside the bubble.

And from pages 118-119:

So essentially what we've done here is "speed up" light inside the bubble relative to observers outside the bubble. Observers inside the bubble can thus travel at faster than light speeds relative to observers outside, but still slower than the local light speed inside the bubble.

I suspect that when dolphus333 said in his "end edit" that a ship would take 817 years to reach Alpha Centauri for observers on Earth, this comment was falsely assuming that the relationships between time and distance for the Alcubierre drive would be identical to those of a relativistic rocket undergoing uniform proper acceleration in SR. A couple paragraphs before the "end edit" dolphus333 had brought up the case of 'uniform acceleration and no exotic matter whatsoever', i.e. a relativistic rocket rather than an Alcubierre drive (since the Alcubierre drive does require exotic matter, and from what I understand it isn't actually accelerating relative to quasi-inertial observers who are far away in the asymptotically flat spacetime of Alcubierre's solution). Furthermore, the equation for "distance" as a function of acceleration and proper time that dolphus333 gives in the "end edit" is identical to the equation for "d" given on the relativistic rocket page here, and the equation for "Earth time elapsed" that dolphus333 gives is almost identical to the equation for t on that page, except that dolphus333 writes cosh(at/c) where the corresponding relativistic rocket equation has a factor of sinh(at/c)--probably that dolphus333 either misremembered the equation, or made a transcription error when copying it from some other source (if you use the correct equations, then if a ship travels with a uniform acceleration of 188G to Alpha Centauri 4.3 light years away in the Earth's frame, the proper time on the ship would be 14 days just as dolphus333 calculated, but the time in the Earth frame would be 4.32 years, not 817 years).

Apart from this error, dolphus333 seems to just assume without argument that these equations can be generalized to the case of an Alcubierre bubble, since the "end edit" first calculates the time for a relativistic rocket with an acceleration of 188G, and then goes on to say "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." However, the idea that the relativistic rocket equations apply to the Alcubierre bubble is completely unjustified, since the relativistic rocket equations calculate the effects of uniform acceleration in the flat spacetime of special relativity, whereas the Alcubierre bubble is a spacetime that is highly curved in the vicinity of the bubble, requiring general relativity to deal with it. And clearly the relativistic rocket equations do not apply, since in the inertial frame where the rocket initially began to accelerate (the rest frame of the Earth, in the SR simplification where we treat Earth and other planets and stars as moving inertially in flat spacetime), the relativistic rocket can never get to a distant location in a time lower than it would take light to travel the same distance, yet as the quote from Time Warps and Warp Drives makes clear, an Alcubierre warp bubble would be able to do this.

Answer 4

Hypnosifl's answer is good but I want to add some context.

In any discussion of Alcubierre's warp drive, it's important to understand that there are no rules. You can take any smooth spacetime manifold you like, plug it into the GR field equation, and get a corresponding stress-energy tensor. You can then dress up the calculation with language about "metric engineering" and you'll have a paper as publishable as Alcubierre's. When people talk about the extraordinary difficulty of finding exact solutions in general relativity, they're talking about solutions where the stress-energy tensor resembles something that might arise in the real world. If you put no constraints on the stress-energy tensor, then anything is a solution.

The question says

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.

A signal arriving "before it's sent" in coordinate time is actually not a problem, because coordinates are arbitrary and meaningless. What does cause problems is causal loops, that is, timelike or lightlike worldlines that eventually return to their starting points.

Alcubierre's original geometry has no causal loops. You can easily create an Alcubierre-like geometry that does have causal loops by arranging two warp tunnels in a "tachyonic antitelephone" configuration, as Hypnosifl said. However, since we're making up the rules as we go along, there's no reason not to also make up some reason why that won't work. For example, we could appeal to the chronology protection conjecture, which, though very speculative, is better motivated and more likely to be correct than anything in Alcubierre's paper. Or we could invoke the sort of determinism found in many time travel stories, where "everything just works out" and all your efforts to create a paradox fail because they're destined to fail. I doubt the universe works that way, but it's more plausible than the behavior of the exotic matter in Alcubierre's geometry.

Answer 5

1. The warp drive in the form proposed by Alcubierre will violate causality globally, for the reason given in the clear and succinct answer here by Jerry Schirmer.

2. It is always possible in physics to introduce something unphysical, such as negative mass, and then deduce something equally unphysical, such as an infinite energy source or an impossible propulsion mechanism.

3. The mechanism described by Alcubierre both invokes an unphysical stress tensor and also violates causality (see point 1). Together, this is enough to rule it out with close to 100% certainty as a serious possibility.

4. However, general relativity does allow certain types of warp drive. The cosmic expansion is one; gravitational waves are another. In principle one could arrange for two places in the universe which were previously far-separated to be brought near to one another. A large enough gravitational wave, for example, could change the proper separation between two stars from 1 light-year to 1 light-minute, and this need not take very long. While the distortion was present one could then make the journey between those stars quickly. However, this does not mean one can zoom around the universe at the press of a button. The catch is that in order to set up the disturbance in spacetime between any two given locations, one would first have to communicate between those locations without the distortion. The result is that from the global perspective there is no violation of causality: it is a planned trip and the preparations take time. Nevertheless, this does open up some interesting possibilities for futuristic space travel.

Answer 6

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 $c$.

Answer 7

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?