The Perfect Theory

by Pedro G. Ferreira

  Ashtekar was a committed relativist working at Syracuse University. He came up with an ingenious approach to untangling Einstein’s field equations, rewriting them so that most of the fiendish nonlinearities disappeared and general relativity looked much, much simpler. Ashtekar’s trick unlocked Einstein’s equations in an unexpected way and opened the door for three young relativists to tease out their quantum nature.

  Just like Bryce DeWitt, Lee Smolin fell in love with quantum gravity the moment he arrived at Harvard for graduate school in the 1970s. His adviser, Sidney Coleman, let Smolin get his hands dirty in quantum gravity by working with Stanley Deser at Brandeis. As a student, Smolin failed miserably to quantize gravity, but he remained passionate about solving the problem. It was only when he headed to Yale as an assistant professor that he realized how Ashtekar’s trick made his job much easier. At Yale, Smolin teamed up with Theodore Jacobson, an ex-student of Cécile DeWitt-Morette from the Texas relativity group. Smolin and Jacobson found that instead of talking about the quantum properties of geometry at isolated points in space as they evolved over time, it was much easier to work with the geometry of a collection of points, effectively focusing on chunks of space at any given moment. In their case, the natural building blocks for the quantum theory were loops, like ribbons, in space that could be used to build solutions to the Wheeler-DeWitt equation. Things just seemed to fall into place, and a whole new way of thinking about quantum geometry emerged. The loops could link up and intertwine themselves like chain mail or an intricate fabric. As with a piece of fabric, from a distance the weaves and links disappeared and the smooth, curved spacetime of Einstein’s theory would emerge. Smolin and Jacobson’s approach became known as loop quantum gravity.

  Smolin was joined in his quest by an iconoclastic young Italian physicist named Carlo Rovelli who had also cut his teeth working on the impossible algebra of quantum gravity. Rovelli enjoyed being a rebel. He had set up an alternative radio station during his student days in Rome, had been pursued by the Italian authorities for his political views, and had risked imprisonment for refusing conscription. Alternative views suited him. Smolin and Rovelli took the loop picture even further and looked at how the loops could be linked, braided, and knotted together. In doing so, they wandered from their starting point, the geometry of space, toward an even more broken-up and shattered view of geometry. In the mid-1990s, they stumbled upon an old idea Roger Penrose had for describing a quantum system in terms of a simple mathematical scaffolding, what Penrose called a spin network. Just like a crazy climbing frame in a children’s park, the structure would be a network of links and vertices, each of which carried with it some special quantum properties. Rovelli and Smolin showed that these networks were even better solutions to the Wheeler-DeWitt equation. Yet these selfsame networks had no resemblance to the intuitive picture of space and time that any self-regarding relativist would work with.

  Rovelli and Smolin’s spin networks were a completely new way of looking at quantum gravity. In their model, space didn’t exist at a quantum level—it was atomized or molecularized like water. Water, which looks smooth and continuous at a macroscopic level, is actually made up of molecules, little clusters of protons, electrons, and neutrons that float in empty space, loosely bound to each other through electric force. In the same way, according to Rovelli and Smolin, while space may seem smooth, it shouldn’t exist if you peer at it with an extremely powerful microscope. In Rovelli and Smolin’s theory, if you were able to look at distances of a trillionth of a trillionth of a centimeter, there would be no space, just the frame or network.

  Loop quantum gravity was the plucky competitor to string theory in its attempts to quantize gravity. Loop quantum gravity and its progeny offered a canonical alternative to string theory’s covariant approach. The devotees of loop quantum gravity made no attempt at unifying all the forces, but in taking geometry as their starting point, they tried to preserve some of the beauty of Einstein’s original idea in general relativity. Ironically, in the process, they abandoned the idea of spacetime as something fundamental.

  In a lecture Bryce DeWitt gave in 2004, shortly before his death, he marveled at how far quantum gravity had come along: “In viewing string theory one is struck by how completely the tables have been turned in fifty years. Gravity was once viewed as a kind of innocuous background, certainly irrelevant to quantum field theory. Today gravity plays a central role. Its existence justifies string theory! There is a saying in English: ‘You can’t make a silk purse out of a sow’s ear.’ In the early seventies string theory was a sow’s ear. Nobody took it seriously as a fundamental theory. . . . In the early eighties, the picture was turned upside down. String theory suddenly needed gravity, as well as a host of other things that may or may not be there. Seen from this point of view string theory is a silk purse.”

  DeWitt had never worked on string theory, but it was clear where his allegiance lay. About the canonical approach he was much less enthusiastic. Despite having created it, DeWitt hated the Wheeler-DeWitt equation. He thought it “should be confined to the dustbin of history” for, among other things, “it violates the very spirit of relativity.” In fact, according to DeWitt, “the Wheeler-DeWitt equation is wrong. . . . It is wrong to use it as a definition of quantum gravity or as a basis for refined and detailed analysis.” He acknowledged Abhay Ashtekar’s work on the equation as “elegant,” but, he said, “apart from some apparently important results on so-called ‘spin foams’ I tend to regard the work as misplaced.” DeWitt’s antipathy reflected the popular view in the world of theoretical physics: string theory was winning.

  The string theorists revel in what they perceive as their success. Mike Duff, now back in London, declares, “We have made tremendous progress with string and M-theory. . . . And it is the only attempt at unification.” Many string theorists are convinced supersymmetry and extra dimensions will soon be discovered and that string theory is the only acceptable approach. Stephen Hawking himself has said that “M-theory is the only candidate for a complete theory of the universe.” When asked about the rival canonical approach, seen by many as the rightful heir of Wheeler’s philosophy of quantizing geometry, Duff accuses them of claiming that “quantum gravity” is synonymous with “loop quantum gravity.” Duff is not alone. “They can’t even calculate what a graviton does. How are they ever going to know that they are right?” argues Philip Candelas, who is firmly entrenched in the string theory camp.

  In the mid-2000s, the deep-rooted antagonism between the different camps in the quest for quantum gravity came out into the open. For years, the odd op-ed articles by a few outspoken pundits had been cropping up in blogs and popular physics magazines questioning the hegemony of string theory in theoretical physics. Around 2006, two books came out claiming that string theory was, in fact, destroying the future of physics. The authors, Lee Smolin, one of the champions of loop quantum gravity, and Peter Woit, a mathematical physicist at Columbia, claimed that impressionable young physicists were being lured into working in a field that, after almost thirty years, had yet to deliver tangible hard results that would unify the forces and explain quantum gravity. According to them, academia was dominated by string theorists who hired more string theorists and kept out bright young people who didn’t toe the party line. As Smolin put it in 2005, “A lot of people are frustrated that this community that styles itself as dominant—and is dominant in many places in the U.S.—is uninterested in other good work. Look, when we have quantum gravity meetings, we try to invite a representative from each of the major opposing theories, including string theory. It’s not that we’re so very moral; it’s just what you do. But at the annual international string theory meeting, they’ve never done this.” The blogosphere blazed with the debate while the pro–string theory camp, flustered by the attacks, took it upon themselves to set the record straight. Statements posted on physics websites were followed by hundreds of comments, a messy mélange of technical details, punditry, and pure ignoranc
e. Everyone had an opinion.

  The hostility toward string theory was palpable in 2011 when Michael Green, who had replaced Stephen Hawking as the Lucasian Professor in Cambridge, came to give a public lecture on string theory at Oxford. Green had, with John Schwartz, kick-started string theory’s growth in 1984, and I had seen him give a colloquium in London in the early 1990s to enormous acclaim. String theorists were riding high then. This time, at Oxford, the atmosphere was much cooler. While most of the questions were about the specifics of his talk, a few were needling jibes. No public string theory talk can now get by without the inevitable question: “Is this theory testable?” The question always comes from someone sympathetic to the anti-string camp.

  It is too early to tell how the antagonism between the different tribes working on quantum gravity will play out. For a while those working on non-string formulations of quantum gravity found it difficult to thrive, but it now seems that string theorists working on quantum gravity are being hounded, too.

  A remarkable result of the debate has been that many more people are familiar with the idea of quantum gravity than before. The war between the canonical and covariant approaches has even made network TV. On the popular show Big Bang Theory, two characters broke off their relationship because they couldn’t agree on which approach to teach their children. As Leslie Winkle says to Leonard Hofstadter as she storms out of the room, “It’s a deal breaker.”

  Thirty years after Stephen Hawking predicted the end of physics and then unleashed his black hole information paradox on an unsuspecting world, there isn’t an agreed-upon theory of quantum gravity, let alone a complete unified theory of all the fundamental forces. Yet, despite the acrimony in the quest for quantum gravity, there is common ground. A radically new and almost shared view of the nature of spacetime is emerging. From string theory to loop quantum gravity to all the other niche attempts at quantizing general relativity, almost all approaches give up on spacetime as something truly fundamental. This insight can be directly related to Hawking’s discovery of black hole radiation and may help resolve the problem of information loss in black holes and the end of predictability in physics. One of the key steps in resolving Hawking’s paradox is to understand how black holes actually store the information that they gobble up and how they might release it to the outside world. This requires a more complicated black hole than general relativity’s naive picture of a horizon and nothing else. Somewhat surprisingly, both loop quantum gravity and string theory, as well as other more esoteric and more marginalized proposals for quantum gravity, seem to shed light on this problem.

  In loop quantum gravity, spacetime is atomized and there is a minimum size below which it makes no sense to even talk about the concepts of area and volume. Lee Smolin, Carlo Rovelli, and Kirill Krasnov from Nottingham University have each shown how this theory makes it possible to subdivide the area of the black hole into microscopic pieces, each of which stores a bit of information like a screen of digitized information. According to the champions of loop quantum gravity, it all adds up exactly to give the right entropy of the black hole.

  The string theorists see things slightly differently. Andrew Strominger and Cumrun Vafa from Harvard have shown that with M-theory, the current incarnation of string theory, it is also possible to derive an exact relationship between the entropy, information, and the area of a black hole. For a particular type of black hole they were able to show how assembling particular types of branes together allows the black hole to store just the right amount of information. The branes gave black holes exactly the right microstructure to solve Hawking’s paradox. More generally, they believe that a black hole can be seen as a seething mess of strings and branes, like a tangled ball, with the ends and edges flailing about on the horizon. These bits of branes and string that bounce around on the horizon can be used to reconstruct all the information contained in the black hole. And, again, the numbers add up to give the right entropy.

  While radically different, both loop quantum gravity and string theory seem to be on the right track to solve the information paradox. For, if the information actually lives on the horizon, it can feed the Hawking radiation that the black hole gradually emits, releasing information to the outside world as the black hole slowly glows. And so, by the time the black hole finally evaporates, it will have released all the information that it originally sucked in and no information will have been lost.

  The string theorists are even bolder and more adventurous and claim that what they have found about Hawking radiation is an even more profound property of physical theories. Black holes seem odd because the amount of information that a black hole can store, while related to its entropy, is actually a function of its area, not its volume as one might naively expect—indeed, Bekenstein and Hawking had already argued that was so in the mid-1970s. But this means that, more generally, the maximum amount of information that can be stored in any volume of space will always be bounded. To find what that maximum amount of information is, just take a hypothetical black hole that contains exactly that volume of space and work out how much information can be stored on its surface. And so, instead of having to describe physics in a chunk of space, it should be enough to determine what happens on a surface that encompasses it, much as a two-dimensional hologram can encode all the information of a three-dimensional scene. But if this is true for a piece of space, it should be true everywhere, for the whole of the universe. In such a holographic universe, the details of what spacetime is doing at each point in the universe become irrelevant. This property is so striking that it has led Edward Witten and some of his string theory colleagues to argue that spacetime is an “approximate, emergent, classical concept” that doesn’t have meaning at the quantum level. It seems that for any of the approaches to quantum gravity, at the most fundamental level spacetime might not actually exist.

  When, in the 1950s, John Wheeler and his students started thinking about spacetime and the quantum, he speculated that if one were able to look really closely at space, with an inconceivably ultra-powerful microscope, one might see that “geometry in the small would seem to have to be considered as having a foam-like character.” He was remarkably prescient, but from what we are beginning to understand, even Wheeler of all people might have been too conservative. Not even a foam begins to capture the complexity of where spacetime comes from.

  It looks as if one of the main ideas that underpins Einstein’s great theory, the geometry of spacetime itself, needs to be revisited. The quantum seems to push general relativity beyond what it is capable of describing, and a completely new way of thinking may need to be developed. But there are other hints that we may be reaching the limits of what Einstein’s theory can tell us about space, time, and even the universe on the whole. It is, as Wheeler pointed out, when a theory is pushed to its extremes that we learn something new and surprising. In those regimes we may get a glimpse of something bigger and better that may in the end supersede Einstein’s great discovery.

  13

  A Spectacular Extrapolation

  I HAD JUST GIVEN my lecture and now stood with the audience in the atrium of the Institute of Astronomy at the University of Cambridge drinking cheap wine out of plastic cups. We gathered in small clusters, shuffling our feet, trying to fan conversation into life. The talk I had been invited to deliver that day had been about modifying gravity, describing a class of theories that proposed to dethrone general relativity as an explanation for some cosmological conundrums. The lecture itself had been uneventful. Early on, I had stumbled in refuting a comment about dark matter but had thankfully recovered. No one had told me I was wrong, nor had the questions dragged, and I was now ready to head home to Oxford.

  The institute’s director, George Efstathiou, strode up, eyes gleaming, brandishing his white plastic cup like a weapon. “Thank you for coming,” he said. “That was an interesting talk. In fact I would say it was a good lecture about a really crap subject.” I smiled politely as he slapped me on the back. It was
n’t the first time I had faced this reaction and I wasn’t surprised. Efstathiou had been instrumental in working out the details of how dark matter might have evolved in the formation of large-scale structure. He had also been one of the first to claim that there was evidence for a cosmological constant in the distribution of galaxies. Having risen fast in his career, Efstathiou was successful and confident. “When I took over the institute, I tried to declare it a zone free of modified gravity. And on the whole, I think I have been pretty successful.” He beamed as the small group of people around us looked down at the ground. “Why on earth do you work on it?” he asked me, not really expecting an answer.

  A few months earlier, I had attended a small workshop at the Royal Observatory in Edinburgh entirely devoted to discussing alternative theories of gravity. The crowd that day had included a strange mix of astronomers, mathematicians, and physicists. This meeting was different. Whenever a speaker finished a presentation, there was a round of warm applause of a kind common to self-help groups. There was also a buzz in the air, as if all the talks that day were groundbreaking revelations of some divine law of physics. Everyone was a prophet. Everyone was Einstein. The camaraderie reminded me of my brief flirtation with a Trotskyite organization in my youth, when I had experienced a heady sense of community as my fellow agitators and I agreed implicitly with one another on the innate corruption of the world.

  The evangelical zeal of the workshop made me deeply uncomfortable, part of a deluded cult. After my own talk, I felt almost sickened by the applause and had to leave the room. I was being unfair; the people in that room had been working on alternative theories of gravity for years, fighting against a mainstream that believed piously in Einstein. These were scientists who would regularly have their papers rejected simply because they were about a deeply unfashionable topic. They were used to facing hostile audiences. At this meeting, their zeal fell on sympathetic ears, and they could freely discuss their goal: to overthrow Einstein’s general relativity.