From Eternity to Here (74 page)

71
It might seem crazy that tension, which pulls things together, is responsible for the acceleration of the universe, which pushes things apart. The point is that the tension from dark energy is equal at every point throughout space, and precisely cancels, so there is no direct pulling. Instead, we are left with the indirect effect of the dark energy on the curvature of spacetime. That effect is to impart a perpetual push to the universe, because the dark energy density does not dilute away.

72
Here is another way of thinking about it. The fact that energy is conserved in Newtonian mechanics is a reflection of an underlying symmetry of the theory: time-translation invariance. The background spacetime in which particles move is fixed once and for all. But in general relativity that’s no longer true; the background is dynamical, pushing things around, changing their energies.

73
See Michell (1784); Laplace’s essay is reprinted as an appendix in Hawking and Ellis (1974). It is occasionally pointed out, with great raising of eyebrows and meaningful murmurs, that the radius of a “black star” as calculated according to Newtonian gravity is precisely the same size as the predicted Schwarzschild radius of a black hole in general relativity (2
GM
/
c
²
, where
G
is Newton’s constant of gravitation,
M
is the mass of the object, and
c
is the speed of light). This coincidence is completely accidental, due primarily to the fact that there aren’t many ways you can create a quantity with units of length out of
G
,
M
, and
c
.

74
For purposes of this chapter, we are assuming the validity of classical general relativity, even though we know that it must be replaced by a better theory when it comes to singularities. For more on these issues, see Hawking (1988) or Thorne (1994).

75
Feel free to construct your own moral lessons.

6. LOOPING THROUGH TIME

76
Referring, of course, to the time machines in George Pal’s 1960 movie version of H. G. Wells’s
The Time Machine
; Robert Zemeckis’s 1985 film
Back to the Future
; and the long-running BBC serial
Doctor Who
, respectively.

77
In the interest in getting on with our story, we’re not being completely fair to the subject of tachyons. Allowing objects that travel faster than light opens the door to paradoxes, but that doesn’t necessarily force us to walk through the door. We might imagine models that allowed for tachyons, but only in self-consistent ways. For some discussion see Feinberg (1967) or Nahin (1999). To make things more confusing, in quantum field theory the word “tachyon” often simply refers to a momentarily unstable configuration of a field, where nothing is actually traveling faster than light.

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 curves 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).