Over at Cosmic Variance, I learned that FQXi (the organization that paid for me to go to Iceland) sponsored an essay contest on “The Nature of Time”, and the submission deadline was last week. Because of deep and fundamental properties of time (at least as perceived by human observers), this means that I will not be able to enter the contest. However, by exploiting the timeless nature of the blogosphere, I can now tell you what I would have written about if I had entered.(Warning: I can’t write this post without actually explaining some standard CS and physics in a semi-coherent fashion. I promise to return soon to your regularly-scheduled programming of inside jokes and unexplained references.)

I’ve often heard it said—including by physicists who presumably know better—that “time is just a fourth dimension,” that it’s no different from the usual three dimensions of space, and indeed that this is a central fact that Einstein proved (or exploited? or clarified?) with relativity. Usually, this assertion comes packaged with the distinct but related assertion that the “passage of time” has been revealed as a psychological illusion: for if it makes no sense to talk about the “flow” of x, y, or z, why talk about the flow of t? Why not just look down (if that’s the right word) on the entire universe as a fixed 4-dimensional crystalline structure?

In this post, I’ll try to tell you why not. My starting point is that, even if we leave out all the woolly metaphysics about our subjective experience of time, and look strictly at the formalism of special and general relativity, we still find that time behaves extremely differently from space. In special relativity, the invariant distance between two points p and q—meaning the real physical distance, the distance measure that doesn’t depend on which coordinate system we happen to be using—is called the interval. If the point p has coordinates (x,y,z,t) (in any observer’s coordinate system), and the point q has coordinates (x’,y’,z’,t’), then the interval between p and q equals

(x-x’)2+(y-y’)2+(z-z’)2-(t-t’)2

where as usual, 1 second of time equals 3×108 meters of space. (Indeed, it’s possible to derive special relativity by starting with this fact as an axiom.)

Now, notice the minus sign in front of (t-t’)2? That minus sign is physics’ way of telling us that time is different from space—or in Sesame Street terms, “one of these four dimensions is not like the others.” It’s true that special relativity lets you mix together the x,y,z,t coordinates in a way not possible in Newtonian physics, and that this mixing allows for the famous time dilation effect, whereby someone traveling close to the speed of light relative to you is perceived by you as almost frozen in time. But no matter how you choose the t coordinate, there’s still going to be a t coordinate, which will stubbornly behave differently from the other three spacetime coordinates. It’s similar to how my “up” points in nearly the opposite direction from an Australian’s “up”, and yet we both have an “up” that we’d never confuse with the two spatial directions perpendicular to it.

(By contrast, the two directions perpendicular to “up” can and do get confused with each other, and indeed it’s not even obvious which directions we’re talking about: north and west? forward and right? If you were floating in interstellar space, you’d have three perpendicular directions to choose arbitrarily, and only the choice of the fourth time direction would be an “obvious” one for you.)

In general relativity, spacetime is a curved manifold, and thus the interval gets replaced by an integral over a worldline. But the local neighborhood around each point still looks like the (3+1)-dimensional spacetime of special relativity, and therefore has a time dimension which behaves differently from the three space dimensions. Mathematically, this corresponds to the fact that the metric at each point has (-1,+1,+1,+1) signature—in other words, it’s a 4×4 matrix with 3 positive eigenvalues and 1 negative eigenvalue. If space and time were interchangeable, then all four eigenvalues would have the same sign.

But how does that minus sign actually do the work of making time behave differently from space? Well, because of the minus sign, the interval between two points can be either positive or negative (unlike Euclidean distance, which is always nonnegative). If the interval between two points p and q is positive, then p and q are spacelike separated, meaning that there’s no way for a signal emitted at p to reach q or vice versa. If the interval is negative, then p and q are timelike separated, meaning that either a signal from p can reach q, or a signal from q can reach p. If the interval is zero, then p and q are lightlike separated, meaning a signal can get from one point to the other, but only by traveling at the speed of light.

In other words, that minus sign is what ensures spacetime has a causal structure: two events can stand to each other in the relations “before,” “after,” or “neither before nor after” (what in pre-relativistic terms would be called “simultaneous”). We know from general relativity that the causal structure is a complicated dynamical object, itself subject to the laws of physics: it can bend and sag in the presence of matter, and even contract to a point at black hole singularities. But the causal structure still exists—and because of it, one dimension simply cannot be treated on the same footing as the other three.

Put another way, the minus sign in front of the t coordinate reflects what a sufficiently-articulate child might tell you is the main difference between space and time: you can go backward in space, but you can’t go backward in time. Or: you can revisit the city of your birth, but you can’t (literally) revisit the decade of your birth. Or: the Parthenon could be used later to store gunpowder, and the Tower of London can be used today as a tourist attraction, but the years 1700-1750 can’t be similarly repurposed for a new application: they’re over.

Notice that we’re now treating space and time pragmatically, as resources—asking what they’re good for, and whether a given amount of one is more useful than a given amount of the other. In other words, we’re now talking about time and space like theoretical computer scientists. If the difference between time and space shows up in physics through the (-1,+1,+1,+1) signature, the difference shows up in computer science through the famous

P ≠ PSPACE

conjecture. Here P is the class of problems that are solvable by a conventional computer using a “reasonable” amount of time, meaning, a number of steps that increases at most polynomially with the problem size. PSPACE is the class of problems solvable by a conventional computer using a “reasonable” amount of space, meaning a number of memory bits that increases at most polynomially with the problem size. It’s evident that P ⊆ PSPACE—in other words, any problem solvable in polynomial time is also solvable in polynomial space. For it takes at least one time step to access a given memory location—so in polynomial time, you can’t exploit more than polynomial space anyway. It’s also clear that PSPACE ⊆ EXP—that is, any problem solvable in polynomial space is also solvable in exponential time. The reason is that a computer with K bits of memory can only be 2K different configurations before the same configuration recurs, in which case the machine will loop forever. But computer scientists conjecture that PSPACE ⊄ P—that is, polynomial space is more powerful than polynomial time—and have been trying to prove it for about 40 years.

(You might wonder how P vs. PSPACE relates to the even better-known P vs. NP problem. NP, which consists of all problems for which a solution can be verified in polynomial time, sits somewhere between P and PSPACE. So if P≠NP, then certainly P≠PSPACE as well. The converse is not known—but a proof of P≠PSPACE would certainly be seen as a giant step toward proving P≠NP.)

So from my perspective, it’s not surprising that time and space are treated differently in relativity. Whatever else the laws of physics do, presumably they have to differentiate time from space somehow—since otherwise, how could polynomial time be weaker than polynomial space?

But you might wonder: is reusability really the key property of space that isn’t shared by time—or is it merely one of several differences, or a byproduct of some other, more fundamental difference? Can we adduce evidence for the computer scientist’s view of the space/time distinction—the view that sees reusability as central? What could such evidence even consist of? Isn’t it all just a question of definition at best, or metaphysics at worst?

On the contrary, I’ll argue that the computer scientist’s view of the space/time distinction actually leads to something like a prediction, and that this prediction can be checked, not by experiment but mathematically. If reusability really is the key difference, then if we change the laws of physics so as to make time reusable—keeping everything else the same insofar as we can—polynomial time ought to collapse with polynomial space. In other words, the set of computational problems that are efficiently solvable ought to become PSPACE. By contrast, if reusability is not the key difference, then changing the laws of physics in this way might well give some complexity class other than PSPACE.

But what do we even mean by changing the laws of physics so as to “make time reusable”? The first answer that suggests itself is simply to define a “time-traveling Turing machine,” which can move not only left and right on its work tape, but also backwards and forwards in time. If we do this, then we’ve made time into another space dimension by definition, so it’s not at all surprising if we end up being able to solve exactly the PSPACE problems.

But wait: if time is reusable, then “when” does it get reused? Should we think of some “secondary” time parameter that inexorably marches forward, even as the Turing machine scuttles back and forth in the “original” time? But if so, then why can’t the Turing machine also go backwards in the secondary time? Then we could introduce a tertiary time parameter to count out the Turing machine’s movements in the secondary time, and so on forever.

But this is stupid. What the endless proliferation of times is telling us is that we haven’t really made time reusable. Instead, we’ve simply redefined the time dimension to be yet another space dimension, and then snuck in a new time dimension that behaves in the same boring, conventional way as the old time dimension. We then perform the sleight-of-hand of letting an exponential amount of the secondary time elapse, even as we restrict the “original” time to be polynomially bounded. The trivial, uninformative result is then that we can solve PSPACE problems in “polynomial time.”

So is there a better way to treat time as a reusable resource? I believe that there is. We can have a parameter that behaves like time in that it “never changes direction”, but behaves unlike time in that it loops around in a cycle. In other words, we can have a closed timelike curve, or CTC. CTCs give us a dimension that (1) is reusable, but (2) is also recognizably “time” rather than “space.”

Of course, no sooner do we define CTCs than we confront the well-known problem of dead grandfathers. How can we ensure that the events around the CTC are causally consistent, that they don’t result in contradictions? For my money, the best answer to this question was provided by David Deutsch, in his paper “Quantum Mechanics near Closed Time-like Lines” (unfortunately not online). Deutsch observed that, if we allow the state of the universe to be probabilistic or quantum, then we can always tell a consistent story about the events inside a CTC. So for example, the resolution of the grandfather paradox is simply that you’re born with 1/2 probability, and if you’re born you go back in time and kill your grandfather, therefore you’re born with 1/2 probability, etc. Everything’s consistent; there’s no paradox!

More generally, any stochastic matrix S has at least one stationary distribution—that is, a distribution D such that S(D)=D. Likewise, any quantum-mechanical operation Q has at least one stationary state—that is, a mixed state ρ such that Q(ρ)=ρ. So we can consider a model of closed timelike curve computation where we (the users) specify a polynomial-time operation, and then Nature has to find some probabilistic or quantum state ρ which is left invariant by that operation. (There might be more than one such ρ—in which case, being pessimists, we can stipulate that Nature chooses among them adversarially.) We then get to observe ρ, and output an answer based on it.

So what can be done in this computational model? Long story short: in a recent paper with Watrous, we proved that

P CTC = BQP CTC = PSPACE.

Or in English, the set of problems solvable by a polynomial-time CTC computer is exactly PSPACE—and this holds whether the CTC computer is classical or quantum. In other words, CTCs make polynomial time equal to polynomial space as a computational resource. Unlike in the case of “secondary time,” this is not obvious from the definitions, but has to be proved. (Note that to prove PSPACE ⊆ P CTC ⊆ BQP CTC ⊆ EXP is relatively straightforward; the harder part is to show BQP CTC ⊆ PSPACE.)

The bottom line is that, at least in the computational world, making time reusable (even while preserving its “directionality”) really does make it behave like space. To me, that lends some support to the contention that, in our world, the fact that space is reusable and time is not is at the core of what makes them different from each other.

I don’t think I’ve done enough to whip up controversy yet, so let me try harder in the last few paragraphs. A prominent school of thought in quantum gravity regards time as an “emergent phenomenon”: something that should not appear in the fundamental equations of the universe, just like hot and cold, purple and orange, maple and oak don’t appear in the fundamental equations, but only at higher levels of organization. Personally, I’ve long had trouble making sense of this view. One way to explain my difficulty is using computational complexity. If time is “merely” an emergent phenomenon, then is the presumed intractability of PSPACE-complete problems also an emergent phenomenon? Could a quantum theory of gravity—a theory that excluded time as “not fundamental enough”—therefore be exploited to solve PSPACE-complete problems efficiently (whatever “efficiently” would even mean in such a theory)? Or maybe computation is also just an emergent phenomenon, so the question doesn’t even make sense? Then what isn’t an emergent phenomenon?

I don’t have a knockdown argument, but the distinction between space and time has the feel to me of something that needs to be built into the laws of physics at the machine-code level. I’ll even venture a falsifiable prediction: that if and when we find a quantum theory of gravity, that theory will include a fundamental (not emergent) distinction between space and time. In other words, no matter what spacetime turns out to look like at the Planck scale, the notion of causal ordering and the relationships “before” and “after” will be there at the lowest level. And it will be this causal ordering, built into the laws of physics, that finally lets us understand why closed timelike curves don’t exist and PSPACE-complete problems are intractable.

I’ll end with a quote from a June 2008 Scientific American article by Jerzy Jurkiewicz, Renate Loll and Jan Ambjorn, about the “causal dynamical triangulations approach” to quantum gravity.

What could the trouble be? In our search for loopholes and loose ends in the Euclidean approach [to quantum gravity], we finally hit on the crucial idea, the one ingredient absolutely necessary to make the stir fry come out right: the universe must encode what physicists call causality. Causality means that empty spacetime has a structure that allows us to distinguish unambiguously between cause and effect. It is an integral part of the classical theories of special and general relativity. Euclidean quantum gravity does not build in a notion of causality. The term “Euclidean” indicates that space and time are treated equally. The universes that enter the Euclidean superposition have four spatial directions instead of the usual one of time and three of space. Because Euclidean universes have no distinct notion of time, they have no structure to put events into a specific order; people living in these universes would not have the words “cause” or “effect” in their vocabulary. Hawking and others taking this approach have said that “time is imaginary,” in both a mathematical sense and a colloquial one. Their hope was that causality would emerge as a large-scale property from microscopic quantum fluctuations that individually carry no imprint of a causal structure. But the computer simulations dashed that hope. Instead of disregarding causality when assembling individual universes and hoping for it to reappear through the collective wisdom of the superposition, we decided to incorporate the causal structure at a much earlier stage. The technical term for our method is causal dynamical triangulations. In it, we first assign each simplex an arrow of time pointing from the past to the future. Then we enforce causal gluing rules: two simplices must be glued together to keep their arrows pointing in the same direction. The simplices must share a notion of time, which unfolds steadily in the direction of these arrows and never stands still or runs backward.

By building in a time dimension that behaves differently from the space dimensions, the authors claim to have solved a problem that’s notoriously plagued computer simulations of quantum gravity models: namely, that of recovering a spacetime that “behave[s] on large distances like a four-dimensional, extended object and not like a crumpled ball or polymer”. Are their results another indication that time might not be an illusion after all? Time (hopefully a polynomial amount of it) will tell.