Does general relativity allow for faster than light travel?

[Jocko Homo]

I understand special relativity enough to know that for anything traveling faster than light, there will exist a reference frame for which it is traveling back in time...

However, I've heard claims that general relativity allows for the possibility of traveling faster than light by sufficiently warping space-time.

Is this true? If so then how can this be reconciled with causality? If not then what are these physicists talking about?

Thank you!

 

[bcrowell]

The article you linked to is about the Alcubierre metric, but for the issues you're asking about, there are other solutions to the equations of GR whose interpretations are better understood, e.g., the Godel metric http://en.wikipedia.org/wiki/Gödel_metric , which dates back to 1949. The Godel metric is an exact solution of the Einstein field equations, and it has closed timelike curves (CTCs), meaning that time wraps around on itself cyclically. This violates causality. The argument you gave in your post is the standard one that says that if you have FTL (faster than light) in special relativity, it violates causality. Typically people use this to argue that since we believe causality is necessary for a sensible physical theory, FTL must be impossible. But the argument is actually pretty weak, since we know that general relativity allows solutions that violate causality. For people who feel that causality is indispensable, and who also want to keep GR as a foundational theory, there is the chronology protection conjecture, which, if true, would tell us that CTCs can't occur in our own universe. Some good popular-level treatments of this kind of thing are ch. 14 of Kip Thorne's Black Holes and Time Warps, and Gott's Time Travel in Einstein's Universe.

 

[humanino]

Some good popular-level treatments of this kind of thing are ch. 14 of Kip Thorne's Black Holes and Time Warps, and Gott's Time Travel in Einstein's Universe.

in which Kip concludes that CTCs are unstable under QM fluctuations.

 

[bcrowell]

in which Kip concludes that CTCs are unstable under QM fluctuations.

Hmm...rereading the material in the book, I don't think that's right. He tells the story of a paper his group wrote, and their series of flip-flops on what the paper's final conclusion should actually be. He says on p. 521, referring to this paper, "The bottome line, however, was equivocal." Then he goes on to discuss the chronology protection conjecture in general, concluding "...we cannot know for sure until physicists have fathomed in depth the laws of quantum gravity."

 

[Hurkyl]

However, I've heard claims that general relativity allows for the possibility of traveling faster than light by sufficiently warping space-time.

Is this true? If so then how can this be reconciled with causality?

If there is a long way and a short way from point A to point B, it may be possible to send a light pulse to travel the long way, and beat it there by taking the short way.

It's always near impossible to tell anything from articles like that, but it sounds like they're speculating on a way to manufacture a short way.

Of course, no matter what, you still won't outrun any light pulses that happened to take the short way as well.

 

[bcrowell]

If there is a long way and a short way from point A to point B, it may be possible to send a light pulse to travel the long way, and beat it there by taking the short way.

This does happen in our universe when there's gravitational lensing. But that doesn't really tell us anything about whether CTCs or violations of causality exist in our universe. When we talk about FTL in the context of SR, we mean local FTL. In GR, the useful way to talk about it is in terms of CTCs (which are generally equivalent to FTL). Just being able to win a race against a beam of light isn't a useful definition in either SR or GR.

 

[Hurkyl]

Just being able to win a race against a beam of light isn't a useful definition in either SR or GR.

But it's good enough for science entertainment, and what it sounds like the article is proposing.

 

[George Jones]

But it's good enough for science entertainment, and what it sounds like the article is proposing.

At any event on the ship's worldline, the warp-drive ship's worldline always lies within the event's lightcone. so, in this sense, the ship's speed is always less than the speed of light.

 

[humanino]

Hmm...rereading the material in the book, I don't think that's right. He tells the story of a paper his group wrote, and their series of flip-flops on what the paper's final conclusion should actually be. He says on p. 521, referring to this paper, "The bottome line, however, was equivocal." Then he goes on to discuss the chronology protection conjecture in general, concluding "...we cannot know for sure until physicists have fathomed in depth the laws of quantum gravity."

Then I will quote exactly from the epilogue

Wormholes and time machines today are regarded as outrageous by most physicists, even though Einstein's general relativistic laws permit them to exist. Skeptical physicists can take comfort, however, in our newfound knowledge that the existence of wormholes and time machines is controlled not by Einstein's rather permissive laws, but rather by the more restrictive laws of quantum fields in curved spacetime, and quantum gravity. When we understand those laws better, perhaps they will teach us unequivocally that physical laws always protect the Universe against wormholes and time machines—or at least time machines. Perhaps.

In the spirit of Hurkyl's answer, it may be good enough for sales and science entertainment to concentrate on the putative "perhaps" while willingly ignore the "outrageous". There is no known way out Hawking's argument as presented in Thorne's book. A CTC would explode in less than a Planck time, included for whoever attempts to travel along it.

 

[bcrowell]

But it's good enough for science entertainment, and what it sounds like the article is proposing.

Well, the Alcubierre metric is real physics published by a real physicist in a peer-reviewed journal, and later widely discussed in the field. It's not just science entertainment.

But you raise a good point about the scientific issues surrounding FTL in science fiction. The way I've often phrased it is that based on the fundamental structure of relativity, any scheme for FTL is also a method for time travel, and therefore any SF universe in which there is routine FTL but no routine time travel (e.g., Star Trek and Star Wars) is scientifically wrong. (Not that Trek and Star Wars fans really care about the fact that this stuff isn't hard SF. After all, they have starships flying by and going whoosh.) Your example shows that my way of phrasing it is kind of loose, and allows for counterexamples depending on how you interpret it. I would be interested in getting a better understanding of how to deal with this more rigorously.

Here are two examples of arguments that FTL implies causality violation.

Example #1: Suppose you're in a region of flat spacetime, with the standard topology, and you're not going to modify that spacetime at all. Then any technology for FTL allows A to precede B in one frame, while B precedes A in another.

Example #2 (given in ch. 14 of Thorne's Black Holes and Time Warps): Suppose you have a wormhole that allows you to travel from one of its mouths to the other, emerging at a time that is simultaneous as judged by an observer in the two mouths' common rest frame. Since the wormhole is gravitational in nature, the mouths can be acted on by gravitational fields. That means that you can take one mouth and tow it far away at relativistic speeds, then tow it back so it's in its original position and again at rest relative to the other mouth. As in the twin paradox, this implies that the two mouths are now temporally out of sync, so you've constructed a time machine.

I would like to have a better understanding of how an example like 2 differs from examples like these:

Example #3: Spacetime is flat everywhere, but has the spatial topology of a cylinder.

Example #4: In gravitational lensing, you can win a race against a beam of light.

Thorne's towing-and-twin-paradox argument certainly fails in examples 3 and 4. One distinction to make is the distinction between universes with CTCs and those without -- but this only seems to succeed in distinguishing 2 from 1, 3, and 4, whereas we really want something to distinguish 1 and 2 from 3 and 4.

I would also like to have a good, clear understanding of why nothing like #4 is likely to permit SF-style FTL. My initial impression is that if you wanted to set up something like #4 to allow you to get to Alpha Centauri in a week, the expenditure of mass-energy to create the gravitational fields involved would be many orders of magnitude greater than the expenditure of energy needed to simply move Alpha Centauri closer to Earth.

 

[bcrowell]

Then I will quote exactly from the epilogueIn the spirit of Hurkyl's answer, it may be good enough for sales and science entertainment to concentrate on the putative "perhaps" while willingly ignore the "outrageous". There is no known way out Hawking's argument as presented in Thorne's book. A CTC would explode in less than a Planck time, included for whoever attempts to travel along it.

You seem to be asserting that the chronology protection conjecture has been proved. It hasn't. Nobody even knows the best way to state the conjecture rigorously. Both my quote from ch. 14 of the book and your quote from the epilogue clearly present chronology protection as something that still just has the status of a conjecture. As Thorne notes, if it was going to be rigorously proved, we'd have to start from a theory of quantum gravity, which we don't have.

 

[humanino]

You seem to be asserting that the chronology protection conjecture has been proved. It hasn't. Nobody even knows the best way to state the conjecture rigorously. Both my quote from ch. 14 of the book and your quote from the epilogue clearly present chronology protection as something that still just has the status of a conjecture. As Thorne notes, if it was going to be rigorously proved, we'd have to start from a theory of quantum gravity, which we don't have.

I understand that CTC are still a possibility. All I am saying, is that 99% of the people who are going to read this thread have a 99% likelihood of benefit by trying to understand Hawking's argument rather than simply being told "we do not know for sure, it is a possibility". I can keep quoting from Thorne's book by the way :

Hawking suspects that the growing beam of vacuum fluctuations is nature's way of enforcing chronology protection: Whenever one tries to make a time machine, and no matter what kind of device one uses in one's attempt (a wormhole, a spinning cylinder, a "cosmic string", or whatever), just before one's device becomes a time machine, a beam of vacuum fluctuations will circulate through the device and destroy it Hawking seems ready to bet heavily on this outcome.

I am not willing to take the other side in such a bet. I do enjoy making bets with Hawking, but only bets that I have a reasonable chance of winning. My strong gut feeling is that I would lose this one. My own calculations with Kim, and unpublished calculations that Eanna Flanagan (a student of mine) has done more recently, suggest to me that Hawking is likely to be right. However, we cannot know for sure until physicists have fathomed in depth the laws of quantum gravity.

The reasonable expectation and consensus is that CTC are protected out of existence. The reasonable attitude and conclusion is that we have to study quantum gravity to make sure we are not wrong in this expectation.

 

[Hurkyl]

Example #1: Suppose you're in a region of flat spacetime, with the standard topology, and you're not going to modify that spacetime at all. Then any technology for FTL allows A to precede B in one frame, while B precedes A in another.

(nitpick -- you assume FTL technology obeys Poincaré symmetry)

My initial impression is that if you wanted to set up something like #4 to allow you to get to Alpha Centauri in a week, the expenditure of mass-energy to create the gravitational fields involved would be many orders of magnitude greater than the expenditure of energy needed to simply move Alpha Centauri closer to Earth.

Part of the point is that the problem has moved from "theoretically impossible" to "may simply be an incredibly difficult engineering problem". Nobody has ever claimed practicality.

 

[Jocko Homo]

Thank you all for some illuminating and interesting answers! As is common, I'm sure, I have some follow up questions. Answers beget more questions and all that...

Can anyone describe the role of various "metrics" in the context of general relativity? Is it a metric tensor? I'm surprised that the theory itself doesn't specify its own metric. How does one physically interpret the use of different metrics?

If I understand CTC correctly, if space is warped sufficiently, I can just sit here and eventually be back in an "earlier" point in time. Really, being that time would be a loop, the term "earlier" wouldn't really make sense in that context... right?

How does this relate to special relativity and causality? Would it still be true that if you moved faster than light then you would travel in time as well? Would the faster than light travel described in the original post (or at all) be facilitated by the time travel?

Thank you!

 

[Fredrik]

Can anyone describe the role of various "metrics" in the context of general relativity? Is it a metric tensor? I'm surprised that the theory itself doesn't specify its own metric. How does one physically interpret the use of different metrics?

In this context, "metric" means "metric tensor". Actually, "metric tensor field" would be even more accurate, but no one likes to say that every time. Special relativity says that the metric is specifically the one we call the Minkowski metric. General relativity tells us that the metric is a solution to Einstein's equation.

Not sure what sort of description you want of "the various metrics". Roughly speaking, the metric tells us how matter must move. Einstein's equation allows several interesting spacetimes. The Schwarzschild spacetime describes a universe that's completely empty except for a single spherical non-rotating massive object. It's useful as an approximation of the metric near a planet, star or a black hole. The FLRW spacetimes describe universes in which matter is distributed homogeneously and isotropically. They are useful as approximations of the behavior of matter on very large scales (where the distribution of matter can be described as approximately homogeneous and isotropic). In particular, they describe approximately, the expansion of the universe and the associated redshift of distant galaxies.

If I understand CTC correctly, if space is warped sufficiently, I can just sit here and eventually be back in an "earlier" point in time. Really, being that time would be a loop, the term "earlier" wouldn't really make sense in that context... right?

It would be an earlier time in some coordinate systems, but it would be a later time according to the clock you carry with you.

How does this relate to special relativity and causality?

If you could create stable wormholes, I think you could recreate all the problems described here, using two wormholes. There are also paradoxes in even simpler scenarios that involve only one wormhole, but they don't correspond to any FTL paradoxes in SR.

Would it still be true that if you moved faster than light then you would travel in time as well?

I can't answer that better than A.T. did here.

 

[Passionflower]

It would be an earlier time in some coordinate systems, but it would be a later time according to the clock you carry with you.

Assuming you and our clock are part of a worldtube of CTC's you or your clock could not possibly carry any experiental knowledge you are in a CTC.