From Eternity to Here: The Quest for the Ultimate Theory of Time

by Sean M. Carroll

  78 Gödel (1949). In doing research for their massive textbook Gravitation (1973), Charles Misner, Kip Thorne, and John Wheeler visited Gödel to talk about general relativity. What Gödel wanted to ask them, however, was whether contemporary astronomical observations had provided any hints for an overall rotation in the universe. He remained interested in the possible relevance of his solution to the real world.

  79 Kerr (1963). The Kerr solution is discussed at a technical level in any modern textbook on general relativity, and at a popular level in Thorne (1994). Thorne relates the story of how Kerr presented his solution at the first Texas Symposium on Relativistic Astrophysics, only to be completely (and somewhat rudely) ignored by the assembled astrophysicists, who were busily arguing about quasars. To be fair, at the time Kerr found his solution he didn’t appreciate that it represented a black hole, although he knew it was a spinning solution to Einstein’s equation. Later on, astrophysicists would come to understand that quasars are powered by spinning black holes, described by Kerr’s spacetime.

  80 Tipler (1974). The solution for the curvature of spacetime around an infinite cylinder was actually found by Willem Jacob van Stockum, Dutch physicist (and bomber pilot), in 1937, but Van Stockum didn’t notice that his solution contained closed timelike curves. An excellent overview of both research into time machines in general relativity, and the appearance of time travel in fiction, can be found in Nahin (1999).

  81 Erwin Schrödinger, one of the pioneers of quantum mechanics, proposed a famous thought experiment to illustrate the bizarre nature of quantum superposition. He imagined placing the cat in a sealed box containing a radioactive source that, in some fixed time interval, had a 50 percent chance of decaying and activating a source that would release poison gas into the box. According to the conventional view of quantum mechanics, the resulting system is in an equal superposition of “alive cat” and “dead cat,” at least until someone observes the cat; see Chapter Eleven for discussion.

  82 Kip Thorne has pointed out the “grandfather paradox” seems a bit squeamish, with the introduction of the extra generation and all, not to mention that it’s somewhat patriarchal. He suggests we should be contemplating the “matricide” paradox.

  83 This rule is sometimes raised to the status of a principle; see discussions in Novikov (1983) or Horwich (1987). Philosophers such as Hans Reichenbach (1958) and Hilary Putnam (1962) have also emphasized that closed timelike curves do not necessitate the introduction of paradoxes, so long as the events in spacetime are internally consistent. Really, it’s just common sense. It’s perfectly obvious that there are no paradoxes in the real world; the interesting question is how Nature manages to avoid them.

  84 In Chapter Eleven we’ll backtrack from this statement just a bit, when we discuss quantum mechanics. In quantum mechanics, the real world may include more than one classical history. David Deutsch (1997) has suggested that we might take advantage of multiple histories to include one in which you were in the Ice Age, and one in which you were not. (And an infinite number of others.)

  85 Back to the Future was perhaps the least plausible time-travel movie ever. Marty McFly travels from the 1980s back to the 1950s, and commences to change the past right and left. What is worse, whenever he interferes with events that supposedly already happened, ramifications of those changes propagate “instantaneously” into the future, and even into a family photograph that Marty has carried with him. It is hard to imagine how that notion of “instantaneous” could be sensibly defined. Although perhaps not impossible—one would have to posit the existence of an additional dimension with many of the properties of ordinary time, through which Marty’s individual consciousness was transported by the effects of his actions. There is probably a good Ph.D. thesis in there somewhere: “Toward a Consistent Ontology of Time and Memory in Back to the Future, et seq.” I’m not sure what department it would belong to, however.

  86 More or less the final word in consistent histories in the presence of closed timelike curves was explored in Robert A. Heinlein’s story “All You Zombies—” (1959). Through a series of time jumps and one sex-change operation, the protagonist manages to be his/her own father, mother, and recruiter into the Temporal Corps. Note that the life story is not, however, a self-contained closed loop; the character ages into the future.

  87 For a discussion of this point see Friedman et al. (1990).

  88 Actually, we are committed determinists. Human beings are made of particles and fields that rigidly obey the laws of physics, and in principle (although certainly not in practice) we could forget that we are human and treat ourselves as complicated collections of elementary particles. But that doesn’t mean we should shrink from facing up to how bizarre the problem of free will in the presence of closed timelike curves really is.

  89 This is a bit more definitive-sounding than what physicists are able to actually prove. Indeed, in some extremely simplified cases we can show that the future can be predicted from the past, even in the presence of closed timelike curves; see Friedman and Higuchi (2006). It seems very likely (to me, anyway), that in more realistically complicated models this will no longer be the case; but a definitive set of answers has not yet been obtained.

  90 We might be able to slice spacetime into moments of constant time, even in the presence of closed timelike curves—for example, we can do that in the simple circular-time universe. But that’s a very special case, and in a more typical spacetime with closed timelike curves it will be impossible to find any slicing that consistently covers the entire universe.

  91 The exception, obviously, is the rotating black hole. We can certainly imagine creating such a hole by the collapse of a rotating star, but there is a different problem: The closed timelike curves are hidden behind an event horizon, so we can’t actually get there without leaving the external world behind once and for all. We’ll discuss later in the chapter whether that should count as an escape hatch. Perhaps more important, the solution found by Kerr that describes a rotating black hole is valid only in the ideal case where there is absolutely no matter in spacetime; it is a black hole all by itself, not one that is created by the collapse of a star. Most experts in general relativity believe that a real-world collapsing star would never give rise to closed timelike curves, even behind an event horizon.

  92 Abbott (1899); see also Randall (2005).

  93 The original paper was Gott (1991); he also wrote a popular-level book on the subject (2001). Almost every account you will read of this work will not talk about “massive bodies moving in Flatland,” but rather “perfectly straight, parallel cosmic strings moving in four-dimensional spacetime.” That’s because the two situations are precisely equivalent. A cosmic string is a hypothetical relic from the early universe that can be microscopically thin but stretch for cosmological distances; an idealized version would be perfectly straight and stretch forever, but in the real world cosmic strings would wiggle and curve in complicated ways. But if such a string were perfectly straight, nothing at all would depend on the direction of spacetime along that string; in technical terms, the entire spacetime would be invariant with respect to both translations and boosts along the string. Which means, in effect, that the direction along the string is completely irrelevant, and we are free to ignore it. If we simply forget that dimension, an infinitely long string in three-dimensional space becomes equivalent to a point particle in two-dimensional space. The same goes for a collection of several strings, as long as they are all perfectly straight and remain absolutely parallel to one another. Of course, the idea of pushing around infinitely long and perfectly straight strings is almost as bizarre as imagining that we live in a three-dimensional spacetime. That’s okay; we’re just making unrealistic assumptions because we want to push our theories to the edge of what is conceivable, to distinguish what is impossible in principle from what is merely a daunting technical challenge.

  94 Soon after Gott’s paper appeared, Curt Cutler (1992) showed that the closed timelike c
urves extended to infinity, another signal that this solution didn’t really count as building a time machine (as we think of “building” as something that can be accomplished in a local region). Deser, Jackiw, and ’t Hooft (1992) examined Gott’s solution and found that the total momentum corresponded to that of a tachyon. I worked with Farhi, Guth, and Olum (1992, 1994) to show that an open Flatland universe could never contain enough energy to create a Gott time machine starting from scratch. ’t Hooft (1992) showed that a closed Flatland universe would collapse to a singularity before a closed timelike curve would have a chance to form.

  95 Farhi, Guth, and Guven (1990).

  96 Think of a plane, seen from the perspective of some particular point, as stretching around for 360 degrees of angle. What happens in Flatland is that every bit of energy decreases the total angle around you; we say that every mass is associated with a “deficit angle,” which is removed from the space by its presence. The more mass, the more angle is removed. The resulting geometry looks like a cone at large distances, rather than like a flat piece of paper. But there are only 360 degrees available to be removed, so there is an upper limit on the total amount of energy we can have in an open universe.

  97 “Something like” because we are speaking of the topology of space, not its geometry. That is, we’re not saying that the curvature of space is everywhere perfectly spherical, just that you could smoothly deform it into a sphere. A spherical topology accommodates a deficit angle of exactly 720 degrees, twice the upper limit available in an open universe. Think of a cube (which is topologically equivalent to a sphere). It has eight vertices, each of which corresponds to a deficit angle of 90 degrees, for a total of 720.

  98 Sagan (1985). The story of how Sagan’s questions inspired Kip Thorne’s work on wormholes and time travel is related in Thorne (1994).

  99 As should be obvious from the dates, the work on wormhole time machines actually predates the Flatland explorations. But it involves a bit more exotic physics than Gott’s idea, so it’s logical to discuss the proposals in this order. The original wormhole-as-time-machine paper was Morris, Thorne, and Yurtsever (1988). A detailed investigation into the possible consistency of time travel in wormhole space- times was Friedman et al. (1990), and the story is related at a popular level in Thorne (1994).

  100 I once introduced Bob Geroch for a talk he was giving. It’s useful in these situations to find an interesting anecdote to relate about the speaker, so I Googled around and stumbled on something perfect: a Star Trek site featuring a map of our galaxy, prominently displaying something called the “Geroch Wormhole.” (Apparently it connects the Beta Quadrant to the Delta Quadrant, and was the source of a nasty spat with the Romulans.) So I printed a copy of the map on a transparency and showed it during my introduction, to great amusement all around. Later Bob told me he assumed I had made it up myself, and was pleased to hear that his work on wormholes had produced a beneficial practical effect on the outside world. The paper that showed you would have to make a closed timelike curve in order to build a wormhole is Geroch (1967).

  101 Hawking (1991). In his conclusion, Hawking also claimed that there was observational evidence that travel backward in time was impossible, based on the fact that we had not been invaded by historians from the future. He was joking (I’m pretty sure). Even if it were possible to construct closed timelike curves from scratch, they could never be used to travel backward to a time before the closed curves had been constructed. So there is no observational evidence against the possibility of building a time machine, just evidence that no one has built one yet.

  7. RUNNING TIME BACKWARD

  102 See O’Connor and Robertson (1999), Rouse Ball (1908). You’ll remember Laplace as one of the people who were speculating about black holes long before general relativity came along.

  103 Apparently Napoleon found this quite amusing. He related Laplace’s quip to Joseph Lagrange, another distinguished physicist and mathematician of the time. Lagrange responded with, “Ah, but it is a fine hypothesis; it explains so many things.” Rouse Ball (1908), 427.

  104 Laplace (2007).

  105 There is no worry that Laplace’s Demon exists out there in the universe, smugly predicting our every move. For one thing, it would have to be as big as the universe, and have a computational power equal to that of the universe itself.

  106 Stoppard (1999), 103-4. Valentine, one presumes, is referring to the idea that the phenomenon of chaos undermines the idea of determinism. Chaotic dynamics, which is very real, happens when small changes in initial conditions lead to large differences in later evolution. As a practical matter, this makes the future extremely difficult to predict for systems that are chaotic (not everything is)—there will always be some tiny error in our understanding of the present state of a system. I’m not sure that this argument carries much force with respect to Laplace’s Demon. As a practical matter, there was no danger that we were ever going to know the entire state of the universe, much less use it to predict the future; this conception was always a matter of principle. And the prospect of chaos doesn’t change that at all.

  107 Granted, physicists couldn’t actually live on any of our checkerboards, for essentially anthropic reasons: The setups are too simplistic to allow for the formation and evolution of complex structures that we might identify with intelligent observers. This stifling simplicity can be traced to an absence of interesting “interactions” between the different elements. In the checkerboard worlds we will look at, the entire description consists of just a single kind of thing (such as a vertical or diagonal line) stretching on without alteration. An interesting world is one in which things can persist more or less for an extended period of time, but gradually change via the influence of interactions with other things in the world.

  108 This “one moment at a time” business isn’t perfectly precise, as the real world is not (as far as we know) divided up into discrete steps of time. Time is continuous, flowing smoothly from one time to another while going through every possible moment in between. But that’s okay; calculus provides exactly the right set of mathematical tools to make sense of “chugging forward one moment at a time” when time itself is continuous.

  109 Note that translations in space and spatial inversions (reflections between left and right) are also perfectly good symmetries. That doesn’t seem as obvious, just from looking at the picture, but that’s only because the states themselves (the patterns of 0’s and 1’s) are not invariant under spatial shifts or reflections.

  Lest you think these statements are completely vacuous, there are some symmetries that might have existed, but don’t. We cannot, for example, exchange the roles of time and space. As a general rule, the more symmetries you have, the simpler things become.

  110 This whole checkerboard-worlds idea sometimes goes by the name of cellular automata. A cellular automaton is just some discrete grid that follows a rule for determining the next row from the state of the previous row. They were first investigated in the 1960s, by John von Neumann, who is also the guy who figured out how entropy works in quantum mechanics. Cellular automata are fascinating for many reasons having little to do with the arrow of time; they can exhibit great complexity and can function as universal computers. See Poundstone (1984) or Shalizi (2009).

  Not only are we disrespecting cellular automata by pulling them out only to illustrate a few simple features of time reversal and information conservation, but we are also not speaking the usual language of cellular-automaton cognoscenti. For one thing, computer scientists typically imagine that time runs from top to bottom. That’s crazy; everyone knows that time runs from bottom to top on a diagram. More notably, even though we are speaking as if each square is either in the state “white” or the state “gray,” we just admitted that you have to keep track of more information than that to reliably evolve into the future in what we are calling example B. That’s no problem; it just means that we’re dealing with an automaton where the “cells” can take on more than t
wo different states. One could imagine going beyond white and gray to allow squares to have any of four different colors. But for our current purposes that’s a level of complexity we needn’t explicitly introduce.

  111 If the laws of physics are not completely deterministic—if they involve some random, stochastic element—then the “specification” of the future evolution will involve probabilities, rather than certainties. The point is that the state includes all of the information that is required to do as well as we can possibly do, given the laws of physics that we are working with.

  112 Sometimes people count relativity as a distinct theory, distinguishing between “classical mechanics” and “relativistic mechanics.” But more often they don’t. It makes sense, for most purposes, to think of relativity as introducing a particular kind of classical mechanics, rather than a completely new way of thinking. The way we specify the state of a system, for example, is pretty much the same in relativity as it would be in Newtonian mechanics. Quantum mechanics, on the other hand, really is quite different. So when we deploy the adjective classical, it will usually denote a contrast with quantum, unless otherwise specified.

  113 It is not known, at least to me, whether Newton himself actually played billiards, although the game certainly existed in Britain at the time. Immanuel Kant, on the other hand, is known to have made pocket money as a student playing billiards (as well as cards).

  114 So the momentum is not just a number; it’s a vector, typically denoted by a little arrow. A vector can be defined as a magnitude (length) and a direction, or as a combination of sub-vectors (components) pointing along each direction of space. You will hear people speak, for example, of “the momentum along the x-direction.”

  115 This is a really good question, one that bugged me for years. At various points when one studies classical mechanics, there are times when one hears one’s teachers talk blithely about momenta that are completely inconsistent with the actual trajectory of the system. What is going on?