The Elegant Universe (48 page)

By contrast with its tiny, sub-millionth of a degree temperature, when one calculates the entropy of, say, a three-solar-mass black hole, the result is an absolutely enormous number: a one followed by about 78 zeros! And the more massive the hole, the greater the entropy. The success of Hawking’s calculations unequivocally established that this truly reflects the enormous amount of disorder embodied by a black hole.

But disorder of what? As we have seen, black holes appear to be terribly simple objects, so what is the source of this overwhelming disorder? On this question, Hawking’s calculations were completely silent. His partial merger of general relativity and quantum mechanics could be used to find the numerical value of a black hole’s entropy, but offered no insight into its microscopic meaning. For nearly a quarter of a century, some of the greatest physicists tried to understand what possible microscopic properties of black holes could account for their entropy. But without a fully trustworthy amalgam of quantum mechanics and general relativity, glimpses of an answer may have been uncovered, but the mystery remained unsolved.

Enter String Theory

Or, it did until January 1996, when Strominger and Vafa—building on earlier insights of Susskind and Sen—released a paper to the electronic physics archive entitled “Microscopic Origin of the Beckenstein-Hawking Entropy.” In this work, Strominger and Vafa were able to use string theory to identify the microscopic constituents of a certain class of black holes and to calculate precisely their associated entropy. Their work relied on the newfound ability to partially circumvent the perturbative approximations in use during the 1980s and early 1990s, and the result they found agreed exactly with that predicted by Bekenstein and Hawking, finally completing a picture partially painted more than two decades previously.

Strominger and Vafa focused on the class of so-called extremal black holes. These are black holes that are imbued with charge—you can think of it as electric charge—and that, moreover, have the minimal possible mass consistent with the charge they carry. As can be seen from this definition, they are closely related to the BPS states discussed in Chapter 12. In fact, Strominger and Vafa exploited this similarity to the hilt. They showed that they could build—theoretically, of course—certain extremal black holes by starting with a particular collection of BPS branes (of certain specified dimensions) and binding them together according to a precise mathematical blueprint. In much the same way that an atom can be built—theoretically, again—by starting with a bunch of quarks and electrons and then precisely arranging them into protons and neutrons, surrounded by orbiting electrons, Strominger and Vafa showed how some of the newfound ingredients in string theory could similarly be molded together to yield particular black holes.

In actuality, black holes are one possible end product of stellar evolution. After a star has burned all its nuclear fuel through billions of years of atomic fusion, it no longer has the strength—the outward-directed pressure—to withstand the enormous inward force of gravity. Under a broad spectrum of conditions, this results in a cataclysmic implosion of the star’s enormous mass; it violently collapses under its own tremendous weight, forming a black hole. Contrary to this realistic means of formation, Strominger and Vafa advocated “designer” black holes. They turned the tables on black hole formation by showing how they could be systematically constructed—in a theorist’s imagination—by carefully, slowly, and meticulously weaving together a precise combination of the branes that had emerged from the second superstring revolution.

The power of this approach became immediately clear. By maintaining full theoretical control over the microscopic construction of their black holes, Strominger and Vafa could easily and directly count the number of rearrangements of the black hole’s microscopic constituents that would leave its overall observable properties, its mass and force charges, unchanged. They could then compare this number with the area of the black hole’s horizon—the entropy predicted by Bekenstein and Hawking. When Strominger and Vafa did so, they found perfect agreement. At least for the class of extremal black holes, they had succeeded in using string theory to account for the microscopic constituents and the associated entropy precisely. A quarter-century-old puzzle had been solved.6

Many string theorists view this success as an important and convincing piece of evidence in support of the theory. Our understanding of string theory is still too coarse to be able to make direct and precise contact with experimental observations of, say, the mass of a quark or an electron. But we now see that string theory has provided the first fundamental explanation of a long-established property of black holes that has stumped physicists using more conventional theories for many years. And this property of black holes is intimately tied up with Hawking’s prediction that they should radiate, a prediction that, in principle, should be experimentally measurable. Of course, this requires that we definitively find a black hole in the heavens and then construct equipment sensitive enough to detect the radiation that it emits. If the black hole were light enough, the latter step is well within the reach of current technology. Even though this experimental program has not as yet met with success, it does re-emphasize that the chasm between string theory and definitive physical statements about the natural world can be bridged. Even Sheldon Glashow—the archrival of string theory through the 1980s—has said recently, “when string theorists talk about black holes they are almost talking about observable phenomena—and that is impressive.”7

The Remaining Mysteries of Black Holes

Even with these impressive developments, there are still two central mysteries surrounding black holes. The first surrounds the impact black holes have on the concept of determinism. In the beginning of the nineteenth century the French mathematician Pierre-Simon de Laplace enunciated the strictest and most far-reaching consequence of the clockwork universe that followed from Newton’s laws of motion:

An intelligence that, at a given instant, could comprehend all the forces by which nature is animated and the respective situation of the beings that make it up, if moreover it were vast enough to submit these data to analysis, would encompass in the same formula the movements of the greatest bodies of the universe and those of the lightest atoms. For such an intelligence nothing would be uncertain, and the future, like the past, would be open to its eyes.8

In other words, if at some instant you know the positions and velocities of every particle in the universe, you can use Newton’s laws of motion to determine—at least in principle—their positions and velocities at any other prior or future time. From this perspective, any and all occurrences, from the formation of the sun to the crucifixion of Christ, to the motion of your eyes across this word, strictly follow from the precise positions and velocities of the particulate ingredients of the universe a moment after the big bang. This rigid lock-step view of the unfolding of the universe raises all sorts of perplexing philosophical dilemmas surrounding the question of free will, but its import was substantially diminished by the discovery of quantum mechanics. We have seen that Heisenberg’s uncertainty principle undercuts Laplacian determinism because we fundamentally cannot know the precise positions and velocities of the constituents of the universe. Instead, these classical properties are replaced by quantum wave functions, which tell us only the probability that any given particle is here or there, or that it has this or that velocity.

The downfall of Laplace’s vision, however, does not leave the concept of determinism in total ruins. Wave functions—the probability waves of quantum mechanics—evolve in time according to precise mathematical rules, such as the Schrödinger equation (or its more precise relativistic counterparts, such as the Dirac equation and the Klein-Gordon equation). This informs us that quantum determinism replaces Laplace’s classical determinism: Knowledge of the wave functions of all of the fundamental ingredients of the universe at some moment in time allows a “vast enough” intelligence to determine the wave functions at any prior or future time. Quantum determinism tells us that the probability that any particular event will occur at some chosen time in the future is fully determined by knowledge of the wave functions at any prior time. The probabilistic aspect of quantum mechanics significantly softens Laplacian determinism by shifting inevitability from outcomes to outcome-likelihoods, but the latter are fully determined within the conventional framework of quantum theory.

In 1976, Hawking declared that even this softer form of determinism is violated by the presence of black holes. Once again, the calculations behind this declaration are formidable, but the essential idea is fairly straightforward. When anything falls into a black hole, its wave function gets sucked in as well. But this means that in the quest to work out wave functions at all future times, our “vast enough” intelligence will be irreparably shortchanged. To predict the future fully we need to know all wave functions fully today. But, if some have escaped down the abyss of black holes, the information they contain is lost.

At first sight, this complication arising from black holes may not seem worth worrying about. Since everything behind the event horizon of a black hole is cut off from the rest of the universe, can’t we just completely ignore anything that is unfortunate enough to have fallen in? Philosophically, moreover, can’t we tell ourselves that the universe has not really lost the information carried by the stuff that has fallen into the black hole; it is simply locked within a region of space that we rational beings choose to avoid at all costs? Prior to Hawking’s realization that black holes are not completely black, the answer to these questions was yes. But once Hawking informed the world that black holes radiate, the story changed. Radiation carries energy and so, as a black hole radiates, its mass slowly decreases—it slowly evaporates. As it does so, the distance from the center of the hole to the event horizon slowly shrinks, and as this shroud recedes, regions of space that were previously cut off re-enter the cosmic arena. Now our philosophical musings must face the music: Does the information contained in the things swallowed by the black hole—the data we imagined existing within the black hole’s interior—re-emerge as the black hole evaporates? This is the information required for quantum determinism to hold, and so this question goes to the heart of whether black holes imbue the evolution of our universe with an even deeper element of happenstance.

As of this writing, there is no consensus among physicists regarding the answer to this question. For many years, Hawking has strongly claimed that the information does not re-emerge—that black holes destroy information thereby “introducing a new level of uncertainty into physics, over and above the usual uncertainty associated with quantum theory”9 In fact, Hawking, together with Kip Thorne of the California Institute of Technology, has a bet with John Preskill, also of the California Institute of Technology, regarding what happens to the information captured by a black hole: Hawking and Thorne bet that the information is forever lost, while Preskill has taken the opposite position and bet that the information re-emerges as the black hole radiates and shrinks. The wager? Information itself: “The loser(s) will reward the winner(s) with an encyclopedia of the winner’s choice.”

The bet remains unsettled, but Hawking has recently acknowledged that the newfound understanding of black holes from string theory, as discussed above, shows that there might be a way for the information to re-emerge.10 The new idea is that for the kind of black holes studied by Strominger and Vafa, and by many other physicists since their initial paper, information can be stored and recovered from the constituent branes. This insight, Strominger recently said, “has led some string theorists to want to claim victory—to claim that the information is recovered as black holes evaporate. In my opinion this conclusion is premature; there is still much work to be done in order to see if this is true.”11 Vafa concurs, saying that he “is agnostic on this question—it could still turn out either way.”12 Answering this question is a central goal of current research. As Hawking has put it,

Most physicists want to believe that information is not lost, as this would make the world safe and predictable. But I believe that if one takes Einstein’s general relativity seriously, one must allow for the possibility that spacetime ties itself in knots and that information gets lost in the folds. Determining whether or not information actually does get lost is one of the major questions in theoretical physics today.13

The second unresolved black hole mystery concerns the nature of spacetime at the central point of the hole.14 A straightforward application of general relativity, going all the way back to Schwarzschild in 1916, shows that the enormous mass and energy crushed together at the black hole’s center causes the fabric of spacetime to suffer a devastating rift, to be radically warped into a state of infinite curvature—to be punctured by a spacetime singularity. One conclusion that physicists drew from this is that since all of the matter that has crossed the event horizon is inexorably drawn to the center of the black hole, and since once there the matter has no future, time itself comes to an end at the heart of a black hole. Other physicists, who over the years have explored the properties of the black hole’s core using Einstein’s equations, revealed the wild possibility that it might be a gateway to another universe that tenuously attaches to ours only at a black hole’s center. Roughly speaking, where time in our universe comes to an end, time in the attached universe just begins.

We will take up some of the implications of this mind-boggling possibility in the next chapter, but for now we want to stress one important point. We must recall the central lesson: Extremes of huge mass and small size leading to unimaginably large density invalidate the sole use of Einstein’s classical theory and require that quantum mechanics be brought to bear as well. This leads us to ask, What does string theory have to say about the spacetime singularity at the center of a black hole? This is a topic of intense current research, but as with the question of information loss, it has not yet been settled. String theory deftly deals with a variety of other singularities—the rips and tears in space discussed in Chapter 11 and in the first part of this chapter.15 But if you have seen one singularity you have not seen them all. The fabric of our universe can be ripped, punctured, and torn in many different ways. String theory has given us profound insights into some of these singularities, but others, the black hole singularity among them, have so far eluded the string theorists’ reach. The essential reason for this, once again, is the reliance on perturbative tools in string theory whose approximations, in this case, cloud our ability to analyze reliably and fully what happens at the deep interior point of a black hole.