From Eternity to Here (80 page)

255
See Guth (1997), also Overbye (1991).

256
Space can be curved even if spacetime is flat. A space with negative curvature, expanding with a size proportional to time, corresponds to a spacetime that is completely flat. Likewise, space can be flat even if spacetime is curved; if a spatially flat universe is expanding (or contracting) in time, the spacetime will certainly be curved. (The point is that the expansion contributes to the total curvature of spacetime, and the curvature of space also contributes. That’s why an expanding negatively curved space can correspond to a spacetime with zero curvature; the contribution from spatial curvature is negative and can precisely cancel the positive contribution from the expansion.) When cosmologists refer to “a flat universe” they mean a
spatially
flat universe, and likewise for positive or negative curvature.

257
They add up to less than 180 degrees.

258
One way of measuring the curvature of the universe is indirectly, using Einstein’s equation. General relativity implies a relationship between the curvature, the expansion rate, and the amount of energy in the universe. For a long time, astronomers measured the expansion rate and the amount of matter in the universe (which they assumed was the most important part of the energy), and kept finding that the universe was pretty close to flat, but it should have a tiny amount of negative curvature. The discovery of dark energy changed all that; it provided exactly the right amount of energy to make the universe flat. Subsequently, astronomers have been able to measure the curvature directly, by using the pattern of temperature fluctuations in the cosmic microwave background as a kind of giant triangle (Miller et al
.,
1999; de Bernardis et al
.,
2000; Spergel et al
.,
2003). This method indicates strongly that the universe really is spatially flat, which is a nice consistency check with the indirect reasoning.

259
Nobody else calls it that. Because this form of dark energy serves the purpose of driving inflation, it is usually postulated to arise from a hypothetical field dubbed the “inflaton.” It would be nice if the inflaton field served some other purpose, or fit snugly into some more complete theory of particle physics, but as yet we don’t know enough to say.

260
You might think that, because the Big Bang itself is a point, the past light cones of any event in the universe must necessarily meet at the Big Bang. But that’s misleading. For one thing, the Big Bang is not a point in space—it’s a moment in time. More important, the Big Bang in classical general relativity is a singularity and shouldn’t even be included in the spacetime; we should talk only about what happens after the Big Bang. And even if we included moments immediately after the Big Bang, the past light cones would not overlap.

261
The original papers are by Andrei Linde (1981) and Andreas Albrecht and Paul Steinhardt (1982). See Guth (1997) for an accessible discussion.

262
See, for example, Spergel et al. (2003).

263
See Vilenkin (1983), Linde (1986), Guth (2007).

264
This scenario was invented under the slightly misleading name of “open inflation” (Bucher, Goldhaber, and Turok, 1995). At the time, before the discovery of dark energy, cosmologists had begun to get a bit nervous—inflation seemed to robustly predict that the universe should be spatially flat, but observations of the density of matter kept implying that there wasn’t enough energy to make it work out. Some people panicked, and tried to invent models of inflation that didn’t necessarily predict a flat universe. That turned out not to be necessary—the dark energy has exactly the right amount of energy density to make the universe flat, and observations of the cosmic microwave background strongly indicate that it really is flat (Spergel et al., 2003). But that’s okay, because out of the panic came a clever idea—how to make a realistic universe inside a bubble embedded in a false-vacuum background.

265
In fact, the early papers on eternal inflation were set in the context of new inflation, not old-inflation-with-new-inflation-inside-the-bubbles. In new inflation it is actually more surprising that inflation is eternal, as you would think the field would just roll down the hill defined by its potential energy. But we should remember that the rolling field has quantum fluctuations; if conditions are right, those fluctuations can be quite large. In fact, they can be large enough that in some regions of space the field actually moves
up
the hill, even though on average it is rolling down. Regions where it rolls up are rare, but they expand faster because the energy density is larger. We end up with a picture similar to the old-inflation story; lots of the universe sees an inflaton roll down and convert to matter and radiation, but an increasing volume stays stuck in the inflating stage, and inflation never ends.

266
See Susskind (2006), or Vilenkin (2006). An earlier, related version of a landscape of different vacuum states was explored by Smolin (1993).

267
In the original papers about inflation, it was implicitly assumed that the particles in the early universe were close to thermal equilibrium. The scenario described here, which seems a bit more robust, goes under the name of “chaotic inflation,” and was originally elucidated by Andrei Linde (1983, 1986).

268
See for example Penrose (2005), Hollands and Wald (2002).

269
This is not to imply that choosing a configuration of the universe randomly from among all possible allowed states is something we are ordered to do, or that there is some reason to believe that it’s actually what happens. Rather, that if the state of the universe is clearly
not
chosen randomly, then there must be something that determines how it is chosen; that’s a clue we would like to use to help understand how the universe works.

270
You may object that there is another candidate for a “high-entropy state”—the chaotic mess into which our universe would evolve if we let it collapse. (Or equivalently, if we started with a typical microstate consistent with the current macrostate of the universe, and ran the clock backward.) It’s true that such a state is much lumpier than the current universe, as singularities and black holes would form in the process of collapse. But that’s exactly the point; even among states that pack the entire current universe into a very small region, an incredibly small fraction take the form of a smooth patch dominated by dark super-energy, as required by inflation. Most such states, on the contrary, are in a regime where quantum field theory doesn’t apply, because quantum gravity is absolutely necessary to describe them. But “we don’t know how to describe such states” is a very different statement than “such states don’t exist” or even “we can ignore such states when we enumerate the possible initial states of the universe.” If the dynamics are reversible, we have no choice but to take those states very seriously.

271
For example, Guth (1997).

15. THE PAST THROUGH TOMORROW

272
Pascal (1995), 66.

273
What would be even better is if some young person read this book, became convinced that this was a serious problem worthy of our attention, and went on to solve it. Or an older person, it doesn’t really matter. In either case, if you end up finding an explanation for the arrow of time that becomes widely accepted within the physics community, please let me know if this book had anything to do with it.

274
Perhaps the closest to something along these lines is the “Holographic Cosmology” scenario advocated by Tom Banks and Willy Fischler (2005; also Banks, 2007). They suggest that the effective dynamical laws of quantum gravity could be very different in different spacetimes. In other words, the laws of physics themselves could be time-dependent. This is a speculative scenario, but worth paying attention to.

275
A related strategy is to posit a particular form for the wave function of the universe, as advocated by James Hartle and Stephen Hawking (1983). They rely on a technique known as “Euclidean quantum gravity”; attempting to do justice to the pros and cons of this approach would take us too far afield from our present concerns. It has been suggested that the Hartle-Hawking wave function implies that the universe must be smooth near the Big Bang, which would help explain the arrow of time (Halliwell and Hawking, 1985), but the domain of validity of the approximations used to derive this result is a bit unclear. My own suspicion is that the Hartle-Hawking wave function predicts that we should live in empty de Sitter space, just as a straightforward contemplation of entropy would lead us to expect.

276
Penrose (1979). When you dig deeply into the mathematics of spacetime curvature, you find that it comes in two different forms: “Ricci curvature,” named after Italian mathematician Gregorio Ricci-Curbastro, and “Weyl curvature,” named after German mathematician Hermann Weyl. Ricci curvature is tied directly to the matter and energy in spacetime—where there’s stuff, the Ricci curvature is nonzero, and where there’s not, the Ricci curvature vanishes. Weyl curvature, on the other hand, can exist all by itself; a gravitational wave, for example, propagates freely through space, and leads to Weyl curvature but no Ricci curvature. The Weyl curvature hypothesis states that singularities in one direction of time always have vanishing Weyl curvature, while those at the other are unconstrained. We would assign the descriptive adjectives
initial
and
final
after the fact, since the low-Weyl-curvature direction would have a lower entropy.

277
Another problem is the apparent danger of Boltzmann brains if the universe enters an eternal de Sitter phase in the future. Also, the concept of a “singularity” from classical general relativity is unlikely to survive intact in a theory of quantum gravity. A more realistic version of the Weyl curvature hypothesis would have to be phrased in quantum-gravity language.

278
Gold (1962).

279
For a brief while, Stephen Hawking believed that his approach to quantum cosmology predicted that the arrow of time would actually reverse if the universe re-collapsed (Hawking, 1985). Don Page convinced him that this was not the case—the right interpretation was that the wave function had two branches, oriented oppositely in time (Page, 1985). Hawking later called this his “greatest blunder,” in a reference to Einstein’s great blunder of suggesting the cosmological constant rather than predicting the expansion of the universe (Hawking, 1988).

280
Price (1996).

281
See, for example, Davies and Twamley (1993), Gell-Mann and Hartle (1996). A different form of future boundary condition, which does not lead to a reversal of the arrow of time, has been investigated in particle physics; see Lee and Wick (1970), Grinstein, O’Connell, and Wise (2009).

282
Once again, the English language lacks the vocabulary for nonstandard arrows of time. We will choose the convention that the “direction of time” is defined by us, here in the “ordinary” post-Big-Bang phase of the universe; with respect to this choice, entropy decreases “toward the future” in the collapsing phase. Of course, organisms that actually live in that phase will naturally define things in the opposite sense; but it’s our book, and the choice is simply a matter of convention, so we can make the rules.

283
Greg Egan worked through the dramatic possibilities of this scenario, in his short story “The Hundred Light-Year Diary” (reprinted in Egan, 1997).

284
Cf. Callender’s Fabergé eggs, discussed in Chapter Nine.

285
See also Carroll (2008).

286
One of the first bouncing scenarios was simply called the “Pre-Big-Bang scenario.” It makes use of a new field called the “dilaton” from string theory, which affects the strength of gravity as it changes (Gasperini and Veneziano, 1993). A related example is the “ekpyrotic universe” scenario, which was later adapted into the “cyclic universe.” In this picture, the energy that powers what we see as the “Bang” comes when a hidden, compact dimension squeezes down to zero size. The cyclic universe idea is discussed in depth in a popular book by Paul Steinhardt and Neil Turok (2007); its predecessor, the ekpyrotic universe, was proposed by Khoury et al. (2001). There are also bouncing cosmologies that don’t rely on strings or extra dimensions, but on the quantum properties of spacetime itself, under the rubric of “loop quantum cosmology” (Bojowald, 2006).

287
Hopefully, after the appearance of this book, that will all change.

288
The same argument holds for Steinhardt and Turok ’s cyclic universe. Despite the label, their model is not recurrent in the way that the Boltzmann-Lucretius model would be. In an eternal universe with a finite-sized state space, allowed sequences of events happen both forward and backward in time, equally often. But in the Steinhardt-Turok model, the arrow of time always points in the same direction; entropy grows forever, requiring an infinite amount of fine-tuning at any one moment. Interestingly, Richard Tolman (1931) long ago discussed problems of entropy in a cyclic universe, although he talked about only the entropy of matter, not including gravity. See also Bojowald and Tavakol (2008).