{"id":368,"date":"2008-12-12T16:54:37","date_gmt":"2008-12-12T20:54:37","guid":{"rendered":"https:\/\/scottaaronson.blog\/?p=368"},"modified":"2008-12-12T16:54:37","modified_gmt":"2008-12-12T20:54:37","slug":"time-different-from-space","status":"publish","type":"post","link":"https:\/\/scottaaronson.blog\/?p=368","title":{"rendered":"Time: Different from space"},"content":{"rendered":"

Over at Cosmic Variance<\/a>, I learned that FQXi (the organization that paid for me to go to Iceland<\/a>) sponsored an essay contest on “The Nature of Time”<\/a>, and the submission deadline was last week.\u00a0 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.\u00a0 However, by exploiting the timeless nature of the blogosphere, I can now tell you what I would have written about if I had<\/em> entered.(Warning:<\/em> I can’t write this post without actually explaining some standard CS and physics in a semi-coherent fashion.\u00a0 I promise to return soon to your regularly-scheduled programming of inside jokes and unexplained references.)<\/p>\n

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.\u00a0 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?\u00a0 Why not just look down (if that’s the right word) on the entire universe as a fixed 4-dimensional crystalline structure?<\/p>\n

In this post, I’ll try to tell you why not.\u00a0 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.\u00a0 In special relativity, the invariant distance<\/em> 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<\/a>.\u00a0 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<\/p>\n

(x-x’)2<\/sup>+(y-y’)2<\/sup>+(z-z’)2<\/sup>-(t-t’)2<\/sup><\/p>\n

where as usual, 1 second of time equals 3×108<\/sup> meters of space.\u00a0 (Indeed, it’s possible to derive special relativity by starting with this fact as an axiom.)<\/p>\n

Now, notice the minus sign in front of (t-t’)2<\/sup>?\u00a0 That minus sign is physics’ way of telling us that <\/em>time is different from space—or in Sesame Street terms, “one of these four dimensions is not like the others.”\u00a0 It’s true that special relativity lets you mix together<\/em> 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.\u00a0 But no matter how you choose the t coordinate, there’s still going to be<\/em> a t coordinate, which will stubbornly behave differently from the other three spacetime coordinates.\u00a0 It’s similar to how my “up” points in nearly the opposite direction from an Australian’s “up”, and yet we both have<\/em> an “up” that we’d never confuse with the two spatial directions perpendicular to it.<\/p>\n

(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?\u00a0 If you were floating in interstellar space, you’d have three<\/em> perpendicular directions to choose arbitrarily, and only the choice of the fourth time direction would be an “obvious” one for you.)<\/p>\n

In general relativity, spacetime is a curved manifold, and thus the interval gets replaced by an integral over a worldline.\u00a0 But the local<\/em> 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.\u00a0 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.\u00a0 If space and time were interchangeable, then all four eigenvalues would have the same sign.<\/p>\n

But how does that minus sign actually do the work of making time behave differently from space?\u00a0 Well, because of the minus sign, the interval between two points can be either positive or negative (unlike Euclidean distance, which is always nonnegative).\u00a0 If the interval between two points p and q is positive, then p and q are spacelike separated<\/em>, meaning that there’s no way for a signal emitted at p to reach q or vice versa.\u00a0 If the interval is negative, then p and q are timelike separated<\/em>, meaning that either a signal from p can reach q, or a signal from q can reach p.\u00a0 If the interval is zero, then p and q are lightlike separated<\/em>, meaning a signal can get from one point to the other, but only by traveling at the speed of light.<\/p>\n

In other words, that minus sign is what ensures spacetime has a causal structure<\/em>: 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”).\u00a0 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.\u00a0 But the causal structure still exists—and because of it, one dimension simply cannot be treated on the same footing as the other three.<\/p>\n

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.\u00a0 Or: you can revisit the city of your birth, but you can’t (literally) revisit the decade of your birth.\u00a0 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.<\/p>\n

Notice that we’re now treating space and time pragmatically, as resources<\/em>—asking what they’re good for, and whether a given amount of one is more useful than a given amount of the other.\u00a0 In other words, we’re now talking about time and space like theoretical computer scientists.\u00a0 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>\n

P<\/strong> \u2260 PSPACE<\/strong><\/p>\n

conjecture.\u00a0 Here P<\/strong> 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.\u00a0 PSPACE<\/strong> 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.\u00a0 It’s evident that P<\/strong> \u2286 PSPACE<\/strong>—in other words, any problem solvable in polynomial time is also solvable in polynomial space.\u00a0 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<\/strong> \u2286 EXP<\/strong>—that is, any problem solvable in polynomial space is also solvable in exponential time.\u00a0 The reason is that a computer with K bits of memory can only be 2K<\/sup> different configurations before the same configuration recurs, in which case the machine will loop forever.\u00a0 But computer scientists conjecture that PSPACE<\/strong> \u2284 P<\/strong>—that is, polynomial space is more powerful than polynomial time—and have been trying to prove it for about 40 years.<\/p>\n

(You might wonder how P<\/strong> vs. PSPACE<\/strong> relates to the even better-known P<\/strong> vs. NP<\/strong> problem.\u00a0 NP<\/strong>, which consists of all problems for which a solution can be verified<\/em> in polynomial time, sits somewhere between P<\/strong> and PSPACE<\/strong>.\u00a0 So if P<\/strong>\u2260NP<\/strong>, then certainly P<\/strong>\u2260PSPACE<\/strong> as well.\u00a0 The converse is not known—but a proof of P<\/strong>\u2260PSPACE<\/strong> would certainly be seen as a giant step toward proving P<\/strong>\u2260NP<\/strong>.)<\/p>\n

So from my perspective, it’s not surprising that time and space are treated differently in relativity.\u00a0 Whatever else the laws of physics do, presumably they have<\/em> to differentiate time from space somehow—since otherwise, how could polynomial time be weaker than polynomial space?<\/p>\n

But you might wonder: is reusability really<\/em> 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?\u00a0 Can we adduce evidence<\/em> for the computer scientist’s view of the space\/time distinction—the view that sees reusability as central?\u00a0 What could such evidence even consist of?\u00a0 Isn’t it all just a question of definition at best, or metaphysics at worst?<\/p>\n

On the contrary, I’ll argue that the computer scientist’s view of the space\/time distinction actually leads to something like a prediction<\/em>, and that this prediction can be checked, not by experiment but mathematically.\u00a0 If reusability really is the key difference, then if we change the laws of physics so as to make time reusable<\/em>—keeping everything else the same insofar as we can—polynomial time ought to collapse with polynomial space.\u00a0 In other words, the set of computational problems that are efficiently solvable ought to become PSPACE<\/strong>.\u00a0 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<\/strong>.<\/p>\n

But what do we even mean<\/em> by changing the laws of physics so as to “make time reusable”?\u00a0 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.\u00a0 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<\/strong> problems.<\/p>\n

But wait: if time is reusable, then “when” does it get reused?\u00a0 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?\u00a0 But if so, then why can’t the Turing machine also go backwards in the secondary time?\u00a0 Then we could introduce a tertiary<\/em> time parameter to count out the Turing machine’s movements in the secondary time, and so on forever.<\/p>\n

But this is stupid.\u00a0 What the endless proliferation of times is telling us is that we haven’t really<\/em> made time reusable.\u00a0 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.\u00a0 We then perform the sleight-of-hand of letting an exponential amount of the secondary<\/em> time elapse, even as we restrict the “original” time to be polynomially bounded.\u00a0 The trivial, uninformative result is then that we can solve PSPACE<\/strong> problems in “polynomial time.”<\/p>\n

So is there a better way to treat time as a reusable resource?\u00a0 I believe that there is.\u00a0 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.\u00a0 In other words, we can have a closed timelike curve<\/em>, or CTC.\u00a0 CTCs give us a dimension that (1) is reusable, but (2) is also recognizably “time” rather than “space.”<\/p>\n

Of course, no sooner do we define CTCs than we confront the well-known problem of dead grandfathers<\/a>.\u00a0 How can we ensure that the events around the CTC are causally consistent<\/em>, that they don’t result in contradictions?\u00a0 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).\u00a0 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.\u00a0 So for example, the resolution of the grandfather paradox is simply that you’re born with 1\/2 probability, and if<\/em> you’re born you go back in time and kill your grandfather, therefore you’re born with 1\/2 probability, etc.\u00a0 Everything’s consistent; there’s no paradox!<\/p>\n

More generally, any stochastic matrix S has at least one stationary distribution<\/em>—that is, a distribution D such that S(D)=D.\u00a0 Likewise, any quantum-mechanical operation Q has at least one stationary state<\/em>—that is, a mixed state \u03c1 such that Q(\u03c1)=\u03c1.\u00a0 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<\/em> probabilistic or quantum state \u03c1 which is left invariant by that operation.\u00a0 (There might be more than one such \u03c1—in which case, being pessimists, we can stipulate that Nature chooses among them adversarially.)\u00a0 We then get to observe \u03c1, and output an answer based on it.<\/p>\n

So what can be done in this computational model?\u00a0 Long story short: in a recent paper with Watrous<\/a>, we proved that<\/p>\n

PCTC<\/sub><\/strong> = BQPCTC<\/sub><\/strong> = PSPACE<\/strong>.<\/p>\n

Or in English, the set of problems solvable by a polynomial-time CTC computer is exactly PSPACE<\/strong>—and this holds whether the CTC computer is classical or quantum.\u00a0 In other words, CTCs make polynomial time equal to polynomial space as a computational resource<\/em>.\u00a0 Unlike in the case of “secondary time,” this is not obvious from the definitions, but has to be proved.\u00a0 (Note that to prove PSPACE<\/strong> \u2286 PCTC<\/sub><\/strong> \u2286 BQPCTC<\/sub><\/strong> \u2286 EXP<\/strong> is relatively straightforward; the harder part is to show BQPCTC<\/sub><\/strong> \u2286 PSPACE<\/strong>.)<\/p>\n

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.\u00a0 To me, that lends some support to the contention that, in our<\/em> world, the fact that space is reusable and time is not is at the core of what makes them different from each other.<\/p>\n

I don’t think I’ve done enough to whip up controversy yet, so let me try harder in the last few paragraphs.\u00a0 A prominent school of thought in quantum gravity regards time as an “emergent phenomenon”: something that should not<\/em> 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.\u00a0 Personally, I’ve long had trouble making sense of this view.\u00a0 One way to explain my difficulty is using computational complexity.\u00a0 If time is “merely” an emergent phenomenon, then is the presumed intractability of PSPACE<\/strong>-complete problems also an emergent phenomenon?\u00a0 Could a quantum theory of gravity—a theory that excluded time as “not fundamental enough”—therefore be exploited to solve PSPACE<\/strong>-complete problems efficiently (whatever “efficiently” would even mean in such a theory)?\u00a0 Or maybe computation is also<\/em> just an emergent phenomenon, so the question doesn’t even make sense?\u00a0 Then what isn’t<\/em> an emergent phenomenon?<\/p>\n

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.\u00a0 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<\/em> emergent) distinction between space and time.\u00a0 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.\u00a0 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<\/strong>-complete problems are intractable.<\/p>\n

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

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.<\/p>\n

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.<\/p>\n

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.<\/p><\/blockquote>\n

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”.\u00a0 Are their results another indication that time might not<\/em> be an illusion after all?\u00a0 Time (hopefully a polynomial amount of it) will tell.<\/p>\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

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.\u00a0 Because of deep and fundamental properties of time (at least as perceived by human observers), this means that I will not […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false},"categories":[30,4],"tags":[],"class_list":["post-368","post","type-post","status-publish","format-standard","hentry","category-mirrored-on-csail-blog","category-quantum"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts\/368","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=368"}],"version-history":[{"count":0,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=\/wp\/v2\/posts\/368\/revisions"}],"wp:attachment":[{"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=368"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=368"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/scottaaronson.blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=368"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}