There and Back Again

In a research paper with seemingly tragic consequences for science fiction fans, researchers in Hong Kong have confirmed that individual photons can’t move faster than light.

(Hopes had been raised a few years back, when a pair of German physicists claimed to have broken the light speed barrier by propagating waves that effectively arrived before they departed. I said at the time that I could do just as well without all the fancy lab equipment — all I need is a yardstick and an axe. If I chop off the last 12 inches, then the center of the yardstick moves from the 12″ mark to the 18″ mark — a six inch advance in exactly zero time. Of course a sane person might argue that this six inch “advance” involves no actual forward motion, but if I correctly understood the German paper, the same sort of objection would apply there as well.)

It’s being reported that the results from Hong Kong doom all hope for time travel. That’s true or false depending on how restrictively you interpret the phrase “time travel”. Back in 1938, Kurt Goedel (yes, the great logician Kurt Godel — though this particular work has nothing to do with his work in mathematical logic) constructed an example of a universe — that is, a structure that obeys all the laws of physics as laid down by Einstein — that is completely filled with closed timelike curves. If you lived in that universe, you’d be able to travel only forward in time, but still eventually come back to your starting point in both time and space — just as a bug can go in just one direction around a circle and still come back to its starting point. Just like Groundhog Day.

(Well, not just like Groundhog Day, because Bill Murray never returns to the exact same place in space and time — he always returns to a slightly altered version of that place, altered, for example, by the memories he brings along.)

To be more precise: If you lived in Godel’s Universe, you’d always have the option of traveling back into your own past (while always traveling forward in time), but you’d never choose to exercise that option (because while these forward-into-the-past paths exist, no matter ever travels along them). Conceivably there is some variant of Godel’s Universe in which that option does get exercised. Maybe some reader knows more about this.

We (almost surely) don’t live in anything like Godel’s Universe. (Thought question: How would we know?) But the existence of that Universe tells us that the laws of physics can’t rule out Godel-style time travel — and more generally that the laws of physics sometimes allow far more than we expect them to. There are a lot of universes out there, and some of them are pretty weird.

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21 Responses to “There and Back Again”


  1. 1 1 Mike H

    You have previously argued that, since Godel’s universe is a mathematical construct, it must really exist. You can’t logically consistently say that Godel’s universe (almost surely) doesn’t exist, but still believe the Natural numbers (almost surely) do for the reasons you gave… ;-)

  2. 2 2 Xorph

    Mike H : I think Steve meant that given the infinite number of possible universes, the probability that we live in one with Godel’s Universe properties is very low, however all of them exist.

  3. 3 3 Steve Landsburg

    Mike H: Of course Godel’s Universe exists. I think, however, that it’s unlikely we live in it.

    Ah! I see Xorph beat me to this!

  4. 4 4 Harold

    Until we know everything, I don’t think we can conclusively say it dooms all hope for time travel. Maybe one sort of time travel.

  5. 5 5 John Hall

    I think the media was selling the story a bit strong. Perhaps could be limited to “one way of time travel not possible.” I prefer the method of time travel in Michael Crichton’s Timeline. Namely that you’re traveling between parallel universes in the multiverse that are N years before or after the present universe and exactly like this universe except there’s a rounding error in the creation of the earth.

    Nevertheless, my preferred reason to believe time travel isn’t possible is that if our civilization survives sufficiently long into the future we would develop the technology and our future selves would come visit us in the present. The fact that we don’t is either evidence that it is impossible or that humans will be wiped out before we discover how.

  6. 6 6 Ken B

    @Mike H: Steve is notably sloppy with the word “exists”. Functionally he means “we can reason about it using the rules of logic.” And that is ALL he means (though he might deny this). So the smallest positive real number does NOT exist, but Godel’s universe does. You can see that this is the real content of “exists” as follows: The ONLY evidence he cites for the existence of Godel’s universe is logical consistency, and inconsistency is the only reason he cites for the non-existence of the smallest real number greater than 0.

  7. 7 7 Steve Landsburg

    Ken B: We could argue about whether standard usage is sloppy (I suppose that at some level *all* language is sloppy), but this *is* standard usage. Every mathematician I know of would be 100% comfortable characterizing Godel’s paper as a *proof of existence* for a universe with certain properties.

  8. 8 8 Ken B

    @Steve: Yes, I know standard mathematical usage, and existence proofs. And its usage corresponds to my characterization. It is not a heavily freighted metaphysical claim. I am perfectly happy saying that a well ordering of the reals exists (accepting AC for the moment). But as I say it’s metaphysical content is small; I can rephrase in terms of consistency and models etc. You want to do more. That is where you get sloppy, by equivocating on the meaning of “exists”. I know you take an extreme Platonist view (as did Godel) and that for you the number 4 is so real you could spread it on toast. But you will perhaps agree that yours IS an extreme position, by no means endorsed by most thinkers.

    And Mike H is clearly not familiar with this way of speaking.

  9. 9 9 Paul G

    Isn’t the highest law of physics and the foundation of all science is the principle of cause and effect? Science is a study of how events are caused by other, earlier events. With time travel, you lose cause and effect, so it is misleading to say that the laws of physics cannot rule it out. Perhaps we could continue doing something like physics in a world with time travel, but it would have to be radically changed and probably quite limited in its domain.

  10. 10 10 Paul G

    On the other hand, reading over my post, the extra “is” is sure making me wish I could go back in time!

  11. 11 11 Roger Schlafly

    Goedel’s paper is not a proof of existence of a universe that obeys the laws of physics. It is just a model of the relativity field equations. Models are often unphysical for some reason.

    As Paul G notes, causality is also a law of physics. It is not correct to say that the Goedel model obeys the laws of physics, or that his universe exists as anything but a mathematical model of some differential equations.

  12. 12 12 Ken B

    Roger S makes the key point. For Steve the Higgs Boson exists. It is a prediction of a consistent theory. We might never find it, we might learn that the theory makes false predictions; we might discover instead a Hoggs Bison, a very odd particle which fills the gap. We might find any theory with a Higgs boson is unphysical. No matter. For Steve the Higgs boson EXISTS — just perhaps not in the universe we occupy.

    Similarly the god Thor and Beethoven’s 12th symphony “exist”. Just not perhaps in the universe we occupy. That’s why I say Steve’s use of “exists” is sloppy. In the universe we occupy that is; I won’t speak for the others.

  13. 13 13 Roger Schlafly

    Steve is using the word “exists” in the mathematical sense, as in “there exists a real number whose square is 2″. In that sense, Goedel found a solution to a differential equation.

  14. 14 14 Ken B

    @Roger: I think he is using it in two senses. He uses it mathematically and then turns around and uses it metaphysically. This is pretty clear in his book The Big Questions. I think it is clear in his answer to Mike H. And I think it’s implicit in this:
    “But the existence of that Universe tells us that the laws of physics can’t rule out Godel-style time travel.”

  15. 15 15 nobody.really

    That’s why I say Steve’s use of “exists” is sloppy. In the universe we occupy that is; I won’t speak for the others.

    Sure you will. Perhaps not in this universe, but ….

  16. 16 16 Mike H

    Ok, I know practically zilch about general relativity, but here’s a thought experiment…

    We take a universe that exists for an infinite amount of time. Then we find some translational symmetry along the time axis. Then we take the quotient of our eternal universe by that translation, and we get a universe where time is cyclic.

    Was this basically Godel’s trick?

    Second thought – although the quotient universe is distinct, mathematically, from the original eternal one, I can’t see how any of its inhabitants could tell the difference.

  17. 17 17 Harold

    I have only just remembered that the title of this post is that of The Hobbit.

  18. 18 18 Ken B

    @Mike H: As I recall Godel’s universe is discussed in Gravitation by Misner et al. Certainly in some of the more mathematically rigourous treatments of GR. Not that I can recall even those bits of it I understood at the time :<

  19. 19 19 nobody.really

    We take a universe that exists for an infinite amount of time. Then we find some translational symmetry along the time axis. Then we take the quotient of our eternal universe by that translation, and we get a universe where time is cyclic….

    Second thought – although the quotient universe is distinct, mathematically, from the original eternal one, I can’t see how any of its inhabitants could tell the difference.

    “There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.

    There is another theory which states that this has already happened.”

    — Douglas Adams, The Restaurant at the End of the Universe

  20. 20 20 Seth

    nobody.really – …and may be happening all the time.

  21. 21 21 dave

    douglas adams is (was) (always will be) god. i find the answer to the tunnel through the earth problem (42) as further proof.

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