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

by Sean M. Carroll

  12. BLACK HOLES: THE ENDS OF TIME

  2 06 Bekenstein (1973).

  207 Hawking (1988), 104. Or, as Dennis Overbye (1991, 107) puts it: “In Cambridge Bekenstein’s breakthrough was greeted with derision. Hawking was outraged. He knew this was nonsense.”

  208 For discussion of observations of stellar-mass black holes, see Casares (2007); for supermassive black holes in other galaxies, see Kormendy and Richstone (1995). The black hole at the center of our galaxy is associated with a radio source known as “Sagittarius A*”; see Reid (2008).

  209 Okay, for some people the looking is even more fun.

  210 Way more than that, actually. As of January 2009, Hawking’s original paper (1975) had been cited by more than 3,000 other scientific papers.

  211 As of this moment, we have never detected gravitational waves directly, although indirect evidence for their existence (as inferred from the energy lost by a system of two neutron stars known as the “binary pulsar”) was enough to win the Nobel Prize for Joseph Taylor and Russell Hulse in 1993. Right now, several gravitational-wave observatories are working to discover such waves directly, perhaps from the coalescence of two black holes.

  212 The area of the event horizon is proportional to the square of the mass of the black hole; in fact, if the area is A and the mass is M, we have A = 16πG2M2/c4, where G is Newton’s constant of gravitation and c is the speed of light.

  213 The analogy between black hole mechanics and thermodynamics was spelled out in Bardeen, Carter, and Hawking (1973).

  214 One way to think about why the surface gravity is not infinite is to take seriously the caveat “as measured by an observer very far away.” The force right near the black hole is large, but when you measure it from infinity it undergoes a gravitational redshift, just as an escaping photon would. The force is infinitely strong, but there is an infinite redshift from the point of view of a distant observer, and the effects combine to give a finite answer for the surface gravity.

  215 More carefully, Bekenstein suggested that the entropy was proportional to the area of the event horizon. Hawking eventually worked out the constant of proportionality.

  216 Hawking (1988), 104-5.

  217 You may wonder why it seems natural to think of the electromagnetic and gravitational fields, but not the electron field or the quark field. That’s because of the difference between fermions and bosons. Fermions, like electrons and quarks, are matter particles, distinguished by the fact that they can’t pile on top of one another; bosons, like photons and gravitons, are force particles that pile on with abandon. When we observe a macroscopic, classical-looking field, that’s a combination of a huge number of boson particles. Fermions like electrons and quarks simply can’t pile up that way, so their field vibrations only ever show up as individual particles.

  218 Overbye (1991), 109.

  219 For reference purposes, the Planck length is equal to (Għ/c3)½, where G is Newton’s constant of gravitation, ħ is Planck’s constant from quantum mechanics, and c is the speed of light. (We’ve set Boltzmann’s constant equal to 1.) So the entropy can be expressed as S = (c¾ħG)A. The area of the event horizon is related to the mass M of the black hole by A = 8πG2M2. Putting it all together, the entropy is related to the mass by as S = (4πGc3/ħ) M2.

  220 Particles and antiparticles are all “particles,” if that makes sense. Sometimes the word particle is used specifically to contrast with antiparticle, but more often it just refers to any pointlike elementary object. Nobody would object to the sentence “the positron is a particle, and the electron is its antiparticle.”

  221 “Known” is an important caveat. Cosmologists have contemplated the possibility that some unknown process, perhaps in the very early universe, might have created copious amounts of very small black holes, perhaps even related to the dark matter. If these black holes were small enough, they wouldn’t be all that dark; they’d be emitting increasing amounts of Hawking radiation, and the final explosions might even be detectable.

  222 One speculative but intriguing idea is that we could make a black hole in a particle accelerator, and then observe it decaying through Hawking radiation. Under ordinary circumstances, that’s hopelessly unrealistic; gravity is such an incredibly weak force that we’ll never be able to build a particle accelerator powerful enough to make even a microscopic black hole. But some modern scenarios, featuring hidden dimensions of spacetime, suggest that gravity becomes much stronger than usual at short distances (see Randall, 2005). In that case, the prospect of making and observing small black holes gets upgraded from “crazy” to “speculative, but not completely crazy.” I’m sure Hawking is rooting for it to happen.

  Unfortunately, the prospect of microscopic black holes has been seized on by a group of fearmongers to spin scenarios under which the Large Hadron Collider, a new particle accelerator at the CERN laboratory in Geneva, is going to destroy the world. Even if the chances are small, destroying the world is pretty bad, so we should be careful, right? But careful reviews of the possibilities (Ellis et al., 2008) have concluded that there’s nothing the LHC will do that hasn’t occurred many times already elsewhere in the universe; if something disastrous were going to happen, we should have seen signs of it in other astrophysical objects. Of course, it’s always possible that everyone involved in these reviews is making some sort of unfortunate math mistake. But lots of things are possible. The next time you open a jar of tomato sauce, it’s possible that you will unleash a mutated pathogen that will wipe out all life on Earth. It’s possible that we are being watched and judged by a race of super-intelligent aliens, who will think badly of us and destroy the Earth if we allow ourselves to be cowed by frivolous lawsuits and don’t turn on the LHC. When possibilities become as remote as what we’re speaking about here, it’s time to take the risks and get on with our lives.

  223 You might be tempted to pursue ideas along exactly those lines—perhaps the information is copied, and is contained simultaneously in the book falling into the singularity and in the radiation leaving the black hole. A result in quantum mechanics—the “No-Cloning Theorem”—says that can’t happen. Not only can information not be destroyed, but it can’t be duplicated.

  224 Preskill’s take on the black hole bets can be found at his Web page: http://www.theory.caltech.edu/people/preskill/bets.html. For an in-depth explanation of the black hole information loss paradox, see Susskind (2008).

  225 You might think we could sidestep this conclusion by appealing to photons once again, because photons are particles that have zero mass. But they do have energy; the energy of a photon is larger when its wavelength is smaller. Because we’re dealing with a box of a certain fixed size, each photon inside will have a minimum allowed energy; otherwise, it simply wouldn’t fit. And the energy of all those photons, through the miracle of E = mc2, contributes to the mass of the box. (Each photon is massless, but a box of photons has a mass, given by the sum of the photon energies divided by the speed of light squared.)

  226 The area of a sphere is equal to 4π times its radius squared. The area of a black hole event horizon, logically enough, is 4π times the Schwarzschild radius squared. This is actually the definition of the Schwarzschild radius, since the highly curved spacetime inside the black hole makes it difficult to sensibly define the distance from the singularity to the horizon. (Remember—that distance is timelike!) So the area of the event horizon is proportional to the square of the mass of the black hole. This is all for black holes with zero rotation and no net electric charge; if the hole is spinning or charged, the formulas are slightly more complicated.

  227 The holographic principle is discussed in Susskind (2008); for technical details, see Bousso (2002).

  228 Maldacena (1998). The title of Maldacena’s paper, “The Large N Limit of Superconformal Field Theories and Supergravity,” doesn’t immediately convey the excitement of his result. When Juan came to Santa Barbara in 1997 to give a seminar, I stayed in my office to work, having not been especi
ally intrigued by his title. Had the talk been advertised as “An Equivalence Between a Five-Dimensional Theory with Gravity and a Four-Dimensional Theory Without Gravity,” I probably would have attended the seminar. Afterward, it was easy to tell from the conversations going on in the hallway—excited, almost frantic, scribbling on blackboards to work out implications of these new ideas—that I had missed something big.

  229 The good thing about string theory is that it seems to be a unique theory; the bad thing is that this theory seems to have many different phases, which look more or less like completely different theories. Just like water can take the form of ice, liquid, or water vapor, depending on the circumstances, in string theory spacetime itself can come in many different phases, with different kinds of particles and even different numbers of observable dimensions of space. And when we say “many,” we’re not kidding—people throw around numbers like 10500 different phases, and it could very well be an infinite number. So the theoretical uniqueness of string theory seems to be of little practical help in understanding the particles and interactions of our particular world. See Greene (2000) or Musser (2008) for overviews of string theory, and Susskind (2006) for a discussion (an optimistic one) of the problem of many different phases.

  230 Strominger and Vafa (1996). For a popular-level account, see Susskind (2008).

  231 While the Strominger-Vafa work implies that the space of states for a black hole in string theory has the right size to account for the entropy, it doesn’t quite tell us what those states look like when gravity is turned on. Samir Mathur and collaborators have suggested that they are “fuzzballs”—configurations of oscillating strings that fill up the volume of the black hole inside the event horizon (Mathur, 2005).

  13. THE LIFE OF THE UNIVERSE

  232 In the eighteenth century, Gottfried Wilhelm Leibniz posed the Primordial Existential Question: “Why is there something rather than nothing?” (One might answer, “Why not?”) Subsequently, some philosophers have tried to argue that the very existence of the universe should be surprising to us, on the grounds that “nothing” is simpler than “something” (e.g., Swinburne, 2004). But that presupposes a somewhat dubious definition of “simplicity,” as well as the idea that this particular brand of simplicity is something a universe ought to have—neither of which is warranted by either experience or logic. See Grünbaum (2004) for a discussion.

  233 Some would argue that God plays the role of the Universal Chicken, creating the universe in a certain state that accounts for the low-entropy beginning. This doesn’t seem like a very parsimonious explanatory framework, as it’s unclear why the entropy would be quite so low, and why (for one thing among many) there should be a hundred billion galaxies in the universe. More important, as scientists we want to explain the most with the least, so if we can come up with naturalistic theories that account for the low entropy of our observed universe without recourse to anything other than the laws of physics, that would be a triumph. Historically, this has been a very successful strategy; pointing at “gaps” in naturalistic explanations of the world and insisting that only God can fill them has, by contrast, had a dismal track record.

  234 This isn’t exactly true, although it’s a pretty good approximation. If a certain kind of particle couples very weakly to the rest of the matter and radiation in the universe, it can essentially stop interacting, and drop out of contact with the surrounding equilibrium configuration. This is a process known as “freeze-out,” and it is crucially important to cosmologists—for example, when they would like to calculate the abundance of dark matter particles, which plausibly froze out at a very early time. In fact, the matter and radiation in the late universe (today) has frozen out long ago, and we are no longer in equilibrium even when you ignore gravity. (The temperature of the cosmic microwave background is about 3 Kelvin, so if we were in equilibrium, everything around you would be at a temperature of 3 Kelvin.)

  235 The speed of light divided by the Hubble constant defines the “Hubble length,” which works out to about 14 billion light-years in the current universe. For not-too-crazy cosmologies, this quantity is almost the same as the age of the universe times the speed of light, so they can be used interchangeably. Because the universe expands at different rates at different times, the current size of our comoving patch can actually be somewhat larger than the Hubble length.

  236 See, for example, Kofman, Linde, and Mukhanov (2002). That paper was written in response to a paper by Hollands and Wald (2002) that raised some similar issues to those we’re exploring in this chapter, in the specific context of inflationary cosmology. For a popular-level discussion that takes a similar view, see Chaisson (2001).

  237 Indeed, Eric Schneider and Dorion Sagan (2005) have argued that the “purpose of life” is to accelerate the rate of entropy production by smoothing out gradients in the universe. It’s hard to make a proposal like that rigorous, for various reasons; one is that, while the Second Law says that entropy tends to increase, there’s no law of nature that says entropy tends to increase as fast as it can.

  238 Also in contrast with the gravitational effects of sources of energy density other than “particles.” This loophole is relevant to the real world because of dark energy. The dark energy isn’t a collection of particles; it’s a smooth field that pervades the universe, and its gravitational impact is to push things apart. Nobody ever said things would be simple.

  239 Other details are also important. In the early universe, ordinary matter is ionized—electrons are moving freely, rather than being attached to atomic nuclei. The pressure in an ionized plasma is generally larger than in a collection of atoms.

  240 Penrose (2005), 706. An earlier version of this argument can be found in Penrose (1979).

  241 Most of the matter in the universe—between 80 percent and 90 percent by mass—is in the form of dark matter, not the ordinary matter of atoms and molecules. We don’t know what the dark matter is, and it’s conceivable that it takes the form of small black holes. But there are problems with that idea, including the difficulty of making so many black holes in the first place. So most cosmologists tend to believe that the dark matter is very likely to be some sort of new elementary particle (or particles) that hasn’t yet been discovered.

  242 Black-hole entropy increases rapidly as the black hole gains mass—it’s proportional to the mass squared. (Entropy goes like area, which goes like radius squared, and the Schwarzschild radius is proportional to the mass.) So a black hole of 10 million solar masses would have 100 times the entropy of one coming in at 1 million solar masses.

  243 Penrose (2005), 707.

  244 The argument here closely follows a paper I wrote in collaboration with Jennifer Chen (Carroll and Chen, 2004).

  245 See, for example, Zurek (1982).

  246 It’s also very far from being accepted wisdom among physicists. Not that there is any accepted answer to the question “What do the highest-entropy states look like when gravity is taken into account?” other than “We don’t know.” But hopefully you’ll become convinced that “empty space” is the best answer we have at the moment.

  247 This is peeking ahead a bit, but note that we could also play this game backward in time. That is: start from some configuration of matter in the universe, a slice of spacetime at one moment in time. In some places we’ll see expansion and dilution, in others contraction and collapse and ultimately evaporation. But we can also ask what would happen if we evolved that “initial” state backward in time, using the same reversible laws of physics. The answer, of course, is that we would find the same kind of behavior. The regions that are expanding toward the future are contracting toward the past, and vice versa. But ultimately space would empty out as the “expanding” regions took over. The very far past looks just like the very far future: empty space.

  248 Here in our own neighborhood, NASA frequently uses a similar effect—the “gravitational slingshot”—to help accelerate probes to the far reaches of the Solar System. If a s
pacecraft passes by a massive planet in just the right way, it can pick up some of the planet’s energy of motion. The planet is so heavy that it hardly notices, but the spacecraft gets flung away at a much higher velocity.

  249 Wald (1983).

  250 In particular, we can define a “horizon” around every observable patch of de Sitter space, just as we can with black holes. Then the entropy formula for that patch is precisely the same formula as the entropy of a black hole—it’s the area of that horizon, measured in Planck units, divided by four.

  251 If H is the Hubble parameter in de Sitter space, the temperature is T = (ħ/2πk)H, where ħ is Planck’s constant and k is Boltzmann’s constant. This was first worked out by Gary Gibbons and Stephen Hawking (1977).

  252 You might think this prediction is a bit too bold, relying on uncertain extrapolations into regimes of physics that we don’t really understand. It’s undeniably true that we don’t have direct experimental access to an eternal de Sitter universe, but the scenario we have sketched out relies only on a few fairly robust principles: the existence of thermal radiation in de Sitter space, and the relative frequency of different kinds of random fluctuations. In particular, it’s tempting to wonder whether there is some special kind of fluctuation that makes a Big Bang, and that kind of fluctuation is more likely than a fluctuation that makes a Boltzmann brain. That might be what actually happens, according to the ultimately correct laws of physics—indeed, we’ll propose something much like that later in the book—but it’s absolutely not what happens under the assumptions we are making here. The nice thing about thermal fluctuations in eternal de Sitter space is that we understand thermal fluctuations very well, and we can calculate with confidence how frequently different fluctuations occur. Specifically, fluctuations involving large changes in entropy are enormously less likely than fluctuations involving small changes in entropy. It will always be easier to fluctuate into a brain than into a universe, unless we depart from this scenario in some profound way.