Building in part on my talk at the time conference, Scott Aaronson has a blog post about entropy and complexity that you should go read right now. It’s similar to one I’ve been contemplating myself, but more clever and original.

Back yet? Scott did foolishly at the end of the post mention the faster-than-light neutrino business. Which of course led to questions, in response to one of which he commented thusly:

Closed timelike curves seem to me to be a different order of strangeness from anything thus far discovered in physics—like maybe 1000 times stranger than relativity, QM, virtual particles, and black holes put together. And I don’t understand how one could have tachyonic neutrinos without getting CTCs as well—would anyone who accepts that possibility be kind enough to explain it to me?

The problem Scott is alluding to is that, in relativity, it’s the speed-of-light barrier that prevents particles (or anything) from zipping around and meeting themselves in the past — a closed loop in spacetime. On a diagram in which time stretches vertically and space horizontally, the possible paths of light from any event define light cones, and physical particles have to stay inside these light cones. “Spacelike” trajectories that leave the light cones simply aren’t allowed in the conventional way of doing things.

What you don’t see in this spacetime diagram is a slice representing “the universe at one fixed time,” because that kind of thing is completely observer-dependent in relativity. In particular, if you could move on a spacelike trajectory, there would be observers who would insist that you are traveling backwards in time. Once you can go faster than light, in other words, you can go back in time and meet yourself in the past. This is Scott’s reason for skepticism about the faster-than-light neutrinos: if you open that door even just a crack, all hell breaks loose.

But rest easy! It doesn’t necessarily follow. Theorists are more than ingenious enough to come up with ways to allow particles to move faster than light without letting them travel along closed curves through spacetime. One minor technical note: if some particle moves faster than light, it’s not “closed timelike curves” that we should be worried about, it’s “closed spacelike curves on which physical particles move.”

But we shouldn’t necessarily even worry about that. The usual argument that faster than light implies the ability to travel on a closed loop assumes Lorentz invariance; but if we discover a true FTL particle, your first guess should be that Lorentz invariance is broken. (Not your only possible guess, but a reasonable one.) Consider, for example, the existence of a heretofore unobserved fluid pervading the universe with a well-defined rest frame, that neutrinos interact with but photons do not. Or a vector field with similar properties. There are various ways we could imagine some background that actually picks out a preferred frame of reference, violating Lorentz invariance spontaneously.

If that’s true, the argument that FTL implies closed loops through spacetime no longer works. Even if neutrinos are able to sneak outside light cones, there may nevertheless be “neutrino cones” to which they are still confined. These neutrino cones could be a little bit broader than ordinary light cones, but they could still define a fixed notion of “going forward in time” that even neutrinos couldn’t violate.

There’s a nice (although technical) discussion of this in a short paper by Robert Geroch. Read Section 2 for the math, Section 3 for the words. From the discussion:

In short, the causal cones of special relativity, from this perspective, have no special place over and above the cones of any other system. This is democracy of causal cones with a vengeance. This, of course, is not the traditional view. That view — that the special relativity causal cones have a preferred role in physics — arises, I suspect, from the fact that a number of other systems — electromagnetism, the spin-s ?elds, etc — employ precisely those same cones as their own. And, indeed, it may be the case that the physical world is organized around such a commonality of cones. On the other hand, it is entirely possible that there exist any number of other systems — not yet observed (or maybe they have been!) — that employ quite di?erent sets of causal cones. And the cones of these “other systems” could very well lie outside the null cones of special relativity, i.e., these systems could very well manifest superluminal signals. None of this would contradict our fundamental ideas about how physics is structured: An initial-value formulation, causal cones governing signals, etc.

The odds are still long against the OPERA result being right at face value. But even if it’s right, it doesn’t immediately imply that neutrinos are time-travelers.

A Cassini view of Enceladus. Click for larger (full res below). Image Credit: NASA/JPL/Space Science Institute

Don’t know what happened to Marian, nary a peep from her.  I assume she has lost internet.

So, being out of town most of the day so far, no time to put a riddle together.  How about a nice Cassini image of Enceladus?

This one was taken on september 13th from a distance of 26,709 miles or 42,823 km.

Want to see a full res version?  Click here (links to NASA).

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